Datasets:

reichenbach

/

arxiv_ppr_embeds

like 0

evaluation of the transport coefficients for coupled plasmas is greatly complicated by the many - body physics of particle collisions . the recently proposed effective potential theory ( ept ) addresses this issue by stipulating that , as far as transport is concerned , collisions can be considered effectively binary even at finite coupling , with the many - body physics manifesting itself solely through modifying the interaction potential between the two colliding particles @xcite . in turn , the effective potential enters expressions for the transport coefficients through the so - called generalized coulomb logarithms " , which are closely related to the standard gas - kinetic cross sections . the resulting transport predictions for a one component plasma ( ocp ) prove in a remarkable agreement with molecular dynamics ( md ) simulations , encouraging extension of the ept concept to the case of a plasma with multiple ion species . while kinetic calculations for multi - component systems are more complex , the problem is well explored in the literature on diluted gas mixtures . in this note we summarize the existing transport results in the form convenient for practical use . local transport formalisms for systems with binary collisions assume that the distribution function @xmath0 of a given species @xmath1 weakly deviates from equilibrium , @xmath2 , due to the knudsen number @xmath3 being small , where @xmath4 and @xmath5 are the characteristic mean free path and background scale , respectively . the linearized boltzmann equation is solved for @xmath6 whose moments give transport coefficients of interest . in the commonly used chapman - enskog approach , the solution for @xmath6 is obtained by expanding it over a set of orthogonal polynomials of the particle velocity @xmath7 . accordingly , precision of the resulting transport coefficients is governed by the number @xmath8 of the so - called sonyne polynomials kept in the expansion over the radial component of the velocity . following earlier works we will denote the approximation level , in which transport quantity @xmath9 is evaluated , by @xmath10_{\xi}$ ] . transport calculations based on grad s method use different precision nomenclature , but it is straightforward to observe that the orthogonal polynomials employed there are the same and therefore local transport results are identical to those obtained with the chapman - enskog approach . in particular , grad s 21n results by zhdanov @xcite , in chapman - enskog s nomenclature correspond to @xmath11 for the heat and diffusive fluxes and @xmath12 for viscosity . one difference between the neutral gas mixtures and unmagnetized plasmas with multiple ion species is presence of electrons . due to their small mass the energy exchange between them and ions is much slower than equilibration within them or any of the ion species . consequently , electron temperature should generally be distinguished in fluid plasma models . also , in a vast range of scenarios plasmas are quasi - neutral and so , if @xmath13 were the total number of plasma species , there would be only @xmath14 independent species concentrations as opposed to @xmath15 in an @xmath13-component gas mixture . however , these issues can be easily circumvented by considering separately the two subsystems : all the ion species ( ionic mixture ) on the one hand and electrons on the other hand , which interact through collisions and fields . evaluation of the ion transport then reduces to the classical problem of a mixture under external forces , making it possible to use the well - established prescriptions from the conventional kinetic theory of diluted gases . accordingly , in what follows we let @xmath13 denote the number of the ion species and exploit results from various sources obtained with either chapman - enskog @xcite or grad @xcite methods . the resulting compact representation for the transport coefficients is summarized in sec . [ app : formulary ] . these formulas involve matrix elements , whose expressions in terms of the generalized coulomb logarithms @xmath16 are given in sec . [ app : matrix - elem ] . once the effective potential , and therefore @xmath16 , are known equations of sections [ app : formulary ] and [ app : matrix - elem ] provide explicit transport results . in particular , in the weakly coupled limit considered in sections [ app : matrix - elem - weak ] and [ app : diff - weak ] , @xmath16 can be calculated analytically , thereby giving fully analytical expressions for all the transport coefficients . finally , section [ sec : routines ] describes the numerical routines , which implement the formalisms for weakly and arbitrarily coupled plasmas . in what follows , @xmath17 and @xmath18 denote the number and mass fractions of the ion species @xmath1 , respectively , where @xmath19 and @xmath20 are the number and mass densities of the ion species @xmath1 , respectively , and @xmath21 and @xmath22 are the total number and mass densities of the ionic mixture , respectively . partial pressure of the ion species @xmath1 is denoted by @xmath23 and the total pressure of the ionic mixture is denoted by @xmath24 . finally , @xmath25 and @xmath26 are the particle mass and charge number of the ion species @xmath1 , respectively . we also define the collision frequency between plasma species @xmath1 and @xmath27 by @xmath28 in eq . ( [ eq : nu ] ) , @xmath29 is the reduced mass and @xmath30 with @xmath31 and @xmath32 need to be set for ion species with comparable masses . finally , @xmath33 is the lowest order generalized coulomb logarithm , which was introduced in ref . equation ( [ eq : nu ] ) reduces to the familiar expression in the weakly coupled limit , in which @xmath34 becomes the conventional coulomb logarithm @xmath35 @xcite . diffusive velocity of the ion species @xmath1 is given by @xmath36 where @xmath37 includes all the thermodynamic forces other than @xmath38 : @xmath39 in eq . ( [ eq : diff - force ] ) , @xmath40 are forces that are external with respect to the ionic mixture . in the absence of the magnetic field and forces that are external with respect to the plasma as a whole , @xmath40 only includes the thermal force @xmath41 exerted by electrons on the ion species @xmath27 and the electric field @xmath42 : @xmath43 since the effect of the electron - ion dynamic friction on the ion transport can be neglected for realistic electron currents @xcite . to evaluate the diffusive flux from the background gradients one thus needs to know the ordinary and thermal diffusion coefficients , @xmath44 and @xmath45 . a number of equivalent representations for these can be found in literature @xcite . here we utilize the formalism by ferziger and kaper @xcite and use the kramers rule to write the @xmath8th chapman - enskog approximation to the ordinary and thermo - diffusion coefficients in the form of ref . @xcite @xmath46_{\xi } & = - \frac{4}{25 n_i |\tensor{m}| } \times \\ \label{eq : ordin - diff } & \begin{vmatrix } \tensor{m}^{(0,0 ) } & \tensor{m}^{(0,1 ) } & \ldots & \tensor{m}^{(0,\xi-1 ) } & \vec{\delta}_{k\beta } - \vec{c } _ { k } \\ \tensor{m}^{(1,0 ) } & \tensor{m}^{(1,1 ) } & \ldots & \tensor{m}^{(1,\xi-1 ) } & \vec{0 } \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ \tensor{m}^{(\xi-1,0 ) } & \tensor{m}^{(\xi-1,1 ) } & \ldots & \tensor{m}^{(\xi-1,\xi-1 ) } & \vec{0}\\ \vec { \delta } _ { k\alpha } & \vec{0 } & \ldots & \vec{0 } & 0 \end{vmatrix}\end{aligned}\ ] ] and @xmath47_{\xi}= - \frac{2}{5 n_i |\tensor{m}| } \times \\ \begin{vmatrix } \tensor{m}^{(0,0 ) } & \tensor{m}^{(0,1 ) } & \ldots & \tensor{m}^{(0,\xi-1 ) } & \vec{0 } \\ \tensor{m}^{(1,0 ) } & \tensor{m}^{(1,1 ) } & \ldots & \tensor{m}^{(1,\xi-1 ) } & \vec{x}_{k } \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ \tensor{m}^{(\xi-1,0 ) } & \tensor{m}^{(\xi-1,1 ) } & \ldots & \tensor{m}^{(\xi-1,\xi-1 ) } & \vec{0}\\ \vec { \delta } _ { k\alpha } & \vec{0 } & \ldots & \vec{0 } & 0 \end{vmatrix},\end{gathered}\ ] ] where blocks @xmath48 are @xmath49 matrices , whose elements are provided in the next subsection . in eqs . ( [ eq : ordin - diff ] ) and ( [ eq : thermo - diff ] ) , @xmath50 denotes the determinant of the @xmath51 matrix @xmath52 composed of @xmath48 . the determinants in the numerator are obtained by appending @xmath52 with a row and a column that are , in turn , composed of @xmath13-element vectors indicated by the arrow sign and the last element , scalar 0 . the @xmath53-th element in such a vector is given by the corresponding expressions , in which @xmath54 is the kronecker delta and @xmath55 appearing in the upper right corner in the numerator on the right side of eq . ( [ eq : ordin - diff ] ) is equal to @xmath56 for @xmath57 and @xmath58 for @xmath59 . in the employed formalism the ordinary diffusion coefficients are symmetric , @xmath60 , and also satisfy the constraints @xmath61 for @xmath62 , so there are only @xmath63 independent coefficients . thermo - diffusion coefficients satisfy the constraint @xmath64 , so there are @xmath15 independent coefficients . it can be observed straight from eq . ( [ eq : thermo - diff ] ) that thermo - diffusion vanishes in the lowest order approximation . finally , we notice that the above expressions provide the kinetic part of the diffusive flux . at finite coupling there should generally be a thermodynamic prefactor . in the presented formalism it is included through the partial ionic pressure in the first term on the right side of eq . ( [ eq : diff - force ] ) and can be retrieved from the equation of state of coupled multi - component plasma . the viscous stress tensor of the ion species @xmath1 is written as @xmath71 , where the rate - of - strain tensor @xmath72 is defined by @xmath73 with @xmath74 being the plasma center - of - mass velocity . the partial viscosity coefficient is given by @xmath75_{\xi}= - \frac{1}{|\tensor{l}| } \times \\ \begin{vmatrix } \tensor{l}^{(0,0 ) } & \tensor{l}^{(0,1 ) } & \ldots & \tensor{l}^{(0,\xi-1 ) } & \vec{x_k \delta_{k\alpha } } \\ \tensor{l}^{(1,0 ) } & \tensor{l}^{(1,1 ) } & \ldots & \tensor{l}^{(1,\xi-1 ) } & \vec{0 } \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ \tensor{l}^{(\xi-1,0 ) } & \tensor{l}^{(\xi-1,1 ) } & \ldots & \tensor{l}^{(\xi-1,\xi-1 ) } & \vec{0 } \\ \vec{x}_k & \vec{0 } & \ldots & \vec{0 } & 0 \end{vmatrix},\end{gathered}\ ] ] and the total viscosity of the ionic mixture is given by @xmath76_{\xi}= - \frac{1}{|\tensor{l}| } \times \\ \begin{vmatrix } \tensor{l}^{(0,0 ) } & \tensor{l}^{(0,1 ) } & \ldots & \tensor{l}^{(0,\xi-1 ) } & \vec{x}_k \\ \tensor{l}^{(1,0 ) } & \tensor{l}^{(1,1 ) } & \ldots & \tensor{l}^{(1,\xi-1 ) } & \vec{0 } \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ \tensor{l}^{(\xi-1,0 ) } & \tensor{l}^{(\xi-1,1 ) } & \ldots & \tensor{l}^{(\xi-1,\xi-1 ) } & \vec{0 } \\ \vec{x}_k & \vec{0 } & \ldots & \vec{0 } & 0 \end{vmatrix}.\end{gathered}\ ] ] it is convenient to introduce @xmath77 . then , elements of matrix @xmath52 can be written as follows : for the first row of the uppermost leftmost block ( @xmath78 ) @xmath79 for the first rows of the remaining uppermost blocks ( @xmath80 ) @xmath81 and for all other elements @xmath82 where @xmath83 and @xmath84 are related to standard bracket integrals @xcite so one can find @xmath85 and @xmath86 with @xmath87 defined by eq . ( [ eq : nu ] ) and @xmath88\\ \bar{a}_{\alpha\beta}^{(0,2 ) } & = 4 \mu_{\beta}^3 \bigl ( \frac{35}{4 } - 7 \bar{\xi}_{\alpha \beta}^{(1,2 ) } + \bar{\xi}_{\alpha \beta}^{(1,3 ) } \bigr)\\ \bar{a}_{\alpha\beta}^{(1,2 ) } & = 8 \mu_{\beta}^2 \bigl[\frac{35}{16 } ( 12 \mu_{\alpha}^2 + 5 \mu_{\beta}^2 ) - \frac{21}{8 } ( 4 \mu_{\alpha}^2 + 5 \mu_{\beta}^2 ) \bar{\xi}_{\alpha \beta}^{(1,2 ) } + \frac{19}{4 } \mu_{\beta}^2 \bar{\xi}_{\alpha \beta}^{(1,3 ) } - \frac{1}{2 } \mu_{\beta}^2 \bar{\xi}_{\alpha \beta}^{(1,4 ) } + 7 \mu_{\alpha } \mu_{\beta } \bar{\xi}_{\alpha \beta}^{(2,2 ) } - 2 \mu_{\alpha } \mu_{\beta } \bar{\xi}_{\alpha \beta}^{(2,3 ) } \bigr]\\ \nonumber \bar{a}_{\alpha\beta}^{(2,2 ) } & = 8 \mu_{\beta } \bigl[\frac{35}{64 } ( 40 \mu_{\alpha}^4 + 168 \mu_{\alpha}^2 \mu_{\beta}^2 + 35 \mu_{\beta}^4 ) - \frac{7}{8 } \mu_{\beta}^2 ( 84 \mu_{\alpha}^2 + 35 \mu_{\beta}^2 ) \bar{\xi}_{\alpha \beta}^{(1,2 ) } + \frac{1}{8 } \mu_{\beta}^2 ( 108 \mu_{\alpha}^2 + 133 \mu_{\beta}^2 ) \bar{\xi}_{\alpha \beta}^{(1,3 ) } \\ & - \frac{7}{2 } \mu_{\beta}^4 \bar{\xi}_{\alpha \beta}^{(1,4)}+ \frac{1}{4 } \mu_{\beta}^4 \bar{\xi}_{\alpha \beta}^{(1,5 ) } + \frac{7}{2 } \mu_{\alpha } \mu_{\beta } ( 4 \mu_{\alpha}^2 + 7 \mu_{\beta}^2 ) \bar{\xi}_{\alpha \beta}^{(2,2 ) } - 14 \mu_{\alpha } \mu_{\beta}^3 \bar{\xi}_{\alpha \beta}^{(2,3 ) } + 2 \mu_{\alpha } \mu_{\beta}^3 \bar{\xi}_{\alpha \beta}^{(2,4 ) } + 2 \mu_{\alpha}^2 \mu_{\beta}^2 \bar{\xi}_{\alpha \beta}^{(3,3 ) } \bigr]\\ \bar{b}_{\alpha\beta}^{(0,0 ) } & = - 8 \mu_{\alpha}^{1/2 } \mu_{\beta}^{1/2 } \\ \bar{b}_{\alpha\beta}^{(0,1 ) } & = - 8 \mu_{\alpha}^{3/2 } \mu_{\beta}^{1/2 } \bigl ( \frac{5}{2 } - \bar{\xi}_{\alpha \beta}^{(1,2 ) } \bigr ) \\ \bar{b}_{\alpha\beta}^{(1,1 ) } & = - 8 \mu_{\alpha}^{3/2 } \mu_{\beta}^{3/2 } \bigl ( \frac{55}{4 } - 5 \bar{\xi}_{\alpha \beta}^{(1,2 ) } + \bar{\xi}_{\alpha \beta}^{(1,3 ) } - 2 \bar{\xi}_{\alpha \beta}^{(2,2 ) } \bigr ) \\ \bar{b}_{\alpha\beta}^{(0,2 ) } & = - 4 \mu_{\alpha}^{5/2 } \mu_{\beta}^{1/2 } \bigl ( \frac{35}{4 } - 7 \bar{\xi}_{\alpha \beta}^{(1,2 ) } + \bar{\xi}_{\alpha \beta}^{(1,3 ) } \bigr ) \\ \bar{b}_{\alpha\beta}^{(1,2 ) } & = - 8 \mu_{\alpha}^{5/2 } \mu_{\beta}^{3/2 } \bigl ( \frac{595}{16 } - \frac{189}{8 } \bar{\xi}_{\alpha \beta}^{(1,2 ) } + \frac{19}{4 } \bar{\xi}_{\alpha \beta}^{(1,3 ) } - \frac{1}{2 } \bar{\xi}_{\alpha \beta}^{(1,4 ) } - 7 \bar{\xi}_{\alpha \beta}^{(2,2 ) } + 2 \bar{\xi}_{\alpha \beta}^{(2,3 ) } \bigr ) \\ \nonumber \bar{b}_{\alpha\beta}^{(2,2 ) } & = - 8 \mu_{\alpha}^{5/2 } \mu_{\beta}^{5/2 } \bigl ( \frac{8505}{64 } - \frac{833}{8 } \bar{\xi}_{\alpha \beta}^{(1,2 ) } + \frac{241}{8 } \bar{\xi}_{\alpha \beta}^{(1,3 ) } - \frac{7}{2 } \bar{\xi}_{\alpha \beta}^{(1,4 ) } + \frac{1}{4 } \bar{\xi}_{\alpha \beta}^{(1,5 ) } \label{eq : bracket-1-end } - \frac{77}{2 } \bar{\xi}_{\alpha \beta}^{(2,2 ) } + 14 \bar{\xi}_{\alpha \beta}^{(2,3 ) } \\ & - 2 \bar{\xi}_{\alpha \beta}^{(2,4 ) } + 2 \bar{\xi}_{\alpha \beta}^{(3,3 ) } \bigr).\end{aligned}\ ] ] elements of matrix @xmath89 are given by @xmath90 where @xmath91 and @xmath92 and @xmath93\\ \bar{k}_{\alpha\beta}^{(0,0 ) } & = -\frac{16}{3 } \mu_{\alpha } \mu_{\beta } \bigl ( 5 - \frac{3}{2 } \bar{\xi}_{\alpha \beta}^{(2,2 ) } \bigr)\\ \bar{k}_{\alpha\beta}^{(0,1 ) } & = \frac{16}{3 } \mu_{\alpha}^2 \mu_{\beta } \bigl ( -\frac{35}{2 } + 7 \bar{\xi}_{\alpha \beta}^{(1,2 ) } + \frac{21}{4 } \bar{\xi}_{\alpha \beta}^{(2,2 ) } -\frac{3}{2 } \bar{\xi}_{\alpha \beta}^{(2,3 ) } \bigr)\\ \label{eq : bracket-2-end } \bar{k}_{\alpha\beta}^{(1,1 ) } & = -\frac{16}{3 } \mu_{\alpha}^2 \mu_{\beta}^2 \bigl ( \frac{385}{4 } - 49 \bar{\xi}_{\alpha \beta}^{(1,2 ) } + 8 \bar{\xi}_{\alpha \beta}^{(1,3 ) } - \frac{301}{8 } \bar{\xi}_{\alpha \beta}^{(2,2 ) } + \frac{21}{2 } \bar{\xi}_{\alpha \beta}^{(2,3 ) } - \frac{3}{2 } \bar{\xi}_{\alpha \beta}^{(2,4 ) } + 3 \bar{\xi}_{\alpha \beta}^{(3,3 ) } \bigr ) . \ ] ] in eqs . ( [ eq : bracket-1])-([eq : bracket-1-end ] ) and ( [ eq : bracket-2])-([eq : bracket-2-end ] ) , @xmath94 and @xmath95 and due to symmetry properties of the bracket integrals @xmath96 , @xmath97 , @xmath98 and @xmath99 @xcite . the explicit expressions for the matrix elements provided in this section can thus be used for evaluating the first to third order chapman - enskog approximations to the diffusive and heat fluxes and the first to second order chapman - enskog approximations to the viscosities . if higher accuracy is desired , one can retrieve the bracket integrals for larger @xmath100 and @xmath101 from prescription of ref . however , as known from earlier works @xcite , going to higher order does not result in significant changes in the transport coefficients for weakly coupled plasmas and , as the more recent study @xcite revealed , the role of higher order corrections can only diminish with coupling within the ept framework . in the weakly coupled limit @xcite @xmath102 greatly simplifying eqs . ( [ eq : bracket-1])-([eq : bracket-1-end ] ) and ( [ eq : bracket-2])-([eq : bracket-2-end ] ) and making representation for the transport coefficients particularly compact : @xmath103 the matrix elements @xmath104 and @xmath105 are obtained by using the above expressions in eqs . ( [ eq : m - elem-2 ] ) and ( [ eq : l - elem ] ) , respectively . here we illustrate application of the presented formulary by considering transport in weakly coupled plasmas . we begin by noticing that @xmath106 and so the first term on the right side of eq . ( [ eq : diff - force ] ) becomes @xmath107 next , we evaluate the second term on the right side of eq . ( [ eq : diff - force ] ) with the help of eq . ( [ eq : ext - force ] ) . to do so we first notice that @xmath108 , where @xmath109 is the total thermal force exerted on electrons by all the ion species . we also use that contribution to this total force from an ion species @xmath27 is proportional to @xmath110 , making @xmath111 where @xmath112 is the effective ion charge number and the electron - ion thermal force coefficient can be found in literature @xcite : @xmath113 as a proof of principle one can also obtain electron - ion thermal force from formulas of subsection [ app : diff ] by taking @xmath114 and interpreting component 1 " as the electrons and component 2 " as the single ion species with the charge number @xmath115 . then , by setting @xmath116 to enforce quasi - neutrality and evaluating @xmath117 with @xmath115 equal to @xmath118 and 4 one can reproduce braginskii s results for the thermal force coefficient @xcite . by utilizing @xmath119 one can also simplify the determinant on the right side of eq . ( [ eq : thermo - diff ] ) to recover the analytical expression ( [ eq : el - thermal ] ) , which was originally obtained by zhdanov by separating the electron and ion moment equations @xcite . of course , eq . ( [ eq : el - thermal ] ) evaluated at @xmath120 and 4 gives braginskii s results as well . using eq . ( [ eq : press - grad ] ) and eq . ( [ eq : ext - force ] ) along with eq . ( [ eq : ion - el - thermal - force ] ) , we find the diffusion driving force from eq . ( [ eq : diff - force ] ) @xmath121 eq . ( [ eq : ext - force - contrib ] ) and formulas of section [ app : formulary ] with the matrix elements of section [ app : matrix - elem - weak ] give fully analytical expressions for all the transport coefficients in a weakly coupled plasma with @xmath13 ion species . we now consider the case @xmath114 and write the diffusive _ mass _ flux @xmath122 of the lighter ion species in the landau - lifshitz form @xcite @xmath123 where @xmath124 and @xmath125 with subscripts @xmath126 " and @xmath127 " denoting the light and heavy ion species , respectively . using that @xmath128 and comparing eq . ( [ eq : ll - flux ] ) and eqs . ( [ eq : diff - flux - gen])-([eq : ext - force ] ) it is straightforward to see that the ordinary diffusion coefficient given by eq . ( [ eq : ordin - diff ] ) is related to the classical diffusion coefficient @xmath129 of landau and lifzhitz through @xmath130 and is also equal to the binary diffusion coefficient " @xmath131 of ferziger and kaper . in turn , the thermo - diffusion ratio with the ion temperature gradient can be recovered from @xmath132 one can then utilize eqs . ( [ eq : ordin - diff ] ) and ( [ eq : thermo - diff ] ) to reproduce the results for the thermo - diffusion ratio with the ion temperature gradient from fig . 2 of our earlier work @xcite . the dynamic friction results , shown in fig . 1 of the same publication , can be reproduced by computing @xmath133_1/[d_{lh}]_3 $ ] from eq . ( [ eq : ordin - diff ] ) . finally , our earlier result for the baro - diffusion ratio @xmath134 can be reproduced by evaluating the right side of eq . ( [ eq : press - grad ] ) with the help of eq . ( [ eq : grad - x ] ) , and the results for the electro - diffusion ratio @xmath135 and thermo - diffusion ratio with the electron temperature gradient @xmath136 by evaluating the right side of eq . ( [ eq : ext - force - contrib ] ) with @xmath114 . the presented expressions for the transport coefficients have been implemented in matlab routines , which can be downloaded from the corresponding links in the ancillary files " field on this article webpage . these routines do not involve symbolic math operations and can be readily adopted in any other standard programming environment . for coupled plasmas one has to use formulas of section [ app : formulary ] along with matrix elements of section [ app : matrix - elem ] . these are implemented in * heatdiffusion.m * " and * viscosity.m * " , which return the corresponding transport coefficients along with the reduced masses @xmath137 and effective binary collision frequencies @xmath138 . the arguments are the temperature , @xmath13-element vectors of the species masses , charge numbers and number densities , and an array of @xmath139 . the generalized coulomb logarithms @xmath139 are to be calculated separately as described in ref . the matlab data file * xilkocp.mat * " with the generalized coulomb logarithms for the one - component plasma ( ocp ) of hydrogen is also included . since the effective potential introduced in ref . @xcite is not sensitive to relative concentrations of species with the same charge number , one can use the same data for evaluating transport in a mixture of hydrogen isotopes as well as in ocp . to illustrate application of the transport formalism for coupled plasmas , we include routine * examplecoupled.m * " , which submits the data from * xilkocp.mat * to * heatdiffusion.m * and * viscosity.m * to evaluate the transport in ocp and in the binary dt mixture . the output is then plotted to reproduce the ocp results shown in fig . 7 of ref . @xcite and the dt results shown in fig . 1 of the same publication . to calculate the transport coefficients for weakly coupled plasmas one can use the same routines by submitting the right side of eq . ( [ eq : coul - log - weak ] ) for @xmath140 . however , it is more computationally efficient to directly code the much simplified expressions for the bracket integrals from section [ app : matrix - elem - weak ] of this note . the simplified routines * heatdiffusionweak.m * " and * viscosityweak.m * " are also included as ancillary files . the matlab script * exampleweak.m * " illustrates their application by using * heatdiffusionweak.m * to calculate the dynamic friction coefficient @xmath141 and thermo - diffusion ratio @xmath142 as functions of @xmath143 for a binary ionic mixture . by appropriately setting the ion masses and charge numbers one can then reproduce the results of figs . 1 and 2 of ref . @xcite . the authors would like to thank j. daligault of lanl and a.a . stepanenko and v.m . zhdanov of mephi for many useful discussions . this work was partially supported by the asc thermonuclear burn initiative under the auspices of the u.s . dept . of energy by the los alamos national security , llc , los alamos national laboratory under contract no . de - ac52 - 06na25396 .

the recently proposed effective potential theory [ phys . rev . lett . 110 , 235001 ( 2013 ) ] allows evaluating transport in coupled plasmas with the well - developed formalisms for systems with binary collisions . to facilitate practical implementation of this concept in fluid models of multi - component plasmas , compact expressions for the transport coefficients in terms the generalized coulomb logarithms are summarized from existing prescriptions . for weakly coupled plasmas , characterized by debye - shielded coulomb interaction potential , expressions become fully analytical . in coupled plasmas the generalized coulomb logarithms need to be evaluated numerically . routines implementing the described formalisms are included as supplemental material .

the process of crumpling is everywhere in nature and human activities , including the formation of mountains and valleys in tectonics@xcite , packing of dna strands in viruses@xcite , car wreckage after an accident@xcite , or the noisy food wraps that drive us nut in the theater , etc .. in spite of its ubiquity , the mechanism behind many of its properties has remained unclear@xcite . for instance , how does the labyrinthian internal structure evolve such that it can withstand extraordinary pressure while more than 80% of its interior remains vacant ? also , why is it that there exists a power law between the external force and the sphere radius with an exponent that varies with material@xcite but , otherwise , is insensitive to the thickness and size of the thin sheet ? scientists have studied crumpled wires ( cw ) both theoretically@xcite and experimentally@xcite . as is shown in fig.[fig : photo ] , the wire is smooth and rid of the complicated ridges and vertices , which is different from a crumpled sheet@xcite . in the last few years , more and more interesting properties of cw have emerged . for instance , donato _ et al . _ found a scaling law in the size - mass relation@xcite , while stoop _ et al . _ offered the associated morphological phase diagram and reported a power law for the number of loops with different exponents in each morphology@xcite . however , there are still few quantitative studies of cw either on the energy aspect or macroscopic properties . furthermore , the former authors all worked on stuffing the wire into a fixed two - dimensional cavity . it is not clear whether this shares the same properties as by increasing the strength of the confining potential while fixing the wire length . aside from this , another motivation for us is to check how sensitive is the cw on the specific form and relative amplitude of stretching and bending energies . in this report , simulations are carried out to study the mechanical properties of cw under the semiflexible polymer model@xcite and the minimal energy model@xcite . to prevent the entropy from dominating the statistical behavior@xcite , the wire is chosen to be of medium length . in our simulations , the optimal shape of a wire is obtained by minimizing the total energy of cw . we begin with a wire in a weak external field , and gradually increase the field strength . at each stage with a certain field strength , we recursively minimize the energy and move to the next stage with a larger external field after an equilibrium state has been achieved . this process is carried out throughout the whole simulation until the wire structure no longer changes with the increasing external field . the wire consists of _ n _ number of monomers and is of length @xmath4 . the total energy of a cw includes bending energy @xmath5 , stretching energy @xmath6 , and external potential @xmath7 . in addition , a hard - core potential @xmath8 is incorporated to keep the wire from self - crossing . explicitly , the energy function can be written as : @xmath9+e_{hc } \label{eq : model}\ ] ] where @xmath10 is the bending angle between two successive segments and @xmath11 is the bending rigidity which is proportional to the young s modulus @xmath12 and second moment of inertia @xmath13 . the stretching modulus @xmath14 depends on the wire cross section @xmath15 , and @xmath16 is the deviation of the @xmath17-th segment from its equilibrium length @xmath18 . the distance @xmath19 of the _ i_-th monomer is measured from the the center of the external parabolic potential . since most research interests fall into cases with insignificant extension , we mainly focused on conditions that @xmath20 . typical parameters for simulations are listed in tab . [ tb : parameter ] . the steady state of cw is determined by minimizing its total energy with powell s algorithm . to make sure that the local minimum we found was not a special case , we adopted a random set of searching directions in the powell s algorithm and repeated the simulations for ten times to get the averaged results . most of the time , cw are randomly folded rather than forming a perfect spiral - the apparent global minimum@xcite . this indicates that those folded configurations are metastable states . the modified powell s minimization is equivalent to the monte carlo simulation in this aspect . .the value of parameters used in simulations [ cols="^,^",options="header " , ] the loops of cw in figs.[fig : photo ] and [ fig : demon ] resemble the boundary of a water - drop - like structure . compare with the simulations for a semi - flexible polymer@xcite which explicitly put in finite temperatures and consequently resulting in many small fluctuations , the configuration of cw is characteristically smooth , and so we exclude the temperature effect in this simulation . we believe the smoothness is credited to the stretching force which causes the relaxation of roughness , analogous to the surface tension which disfavors a kinky surface . because the stretching energy is found to be insignificant when @xmath21 , we can neglect it when determining the mathematical form of the configuration . for simplicity and without the loss of generality , we consider the case of @xmath22 , i.e. , the continuous limit . eq.([eq : model ] ) then can be re - written as : @xmath23 ds \label{eq : lagrangian}\end{aligned}\ ] ] the search for shape function @xmath24 with minimal energy is done by introducing the variational method . fix both @xmath25 and @xmath26 and this total energy is identical to the classical action once we identify the arc length parameter @xmath2 as the time . the euler - lagrange equation of motion immediately gives out the shape function as @xmath27 , where the @xmath28 term is not hard to expect by the dimensional analysis in retrospect . this matches the pattern in the initial stage of the crumpling observed experimentally@xcite as well as in our simulations . however , when the loops start to touch each other as the crumpling proceeds further , the excluded volume or steric interaction@xcite is expected to squeeze and distort the shape of loops . the radius of the circle is rescaled by @xmath29 . apparently , all mass will already be included in the circle when @xmath30 . the rescaling process only shifts the data on a log - log plot and does not affect the value of exponent @xmath0 . since we set the origin of the circle at the center of mass of the cw , there is a bias in the data at small r. this bias is avoided by fitting only from @xmath31=0.5 to @xmath31=1 as shown by the solid straight line@xcite.,scaledwidth=35.0% ] the mass function @xmath32 is a common property@xcite to characterize the configuration of cw . it is defined as the total mass encompassed within a circular area of radius @xmath31 centering at the cw s center of mass : @xmath33 where @xmath34 denotes the step function , @xmath35 stands for the kronecker delta - function which equals 1 when @xmath36 and zero otherwise , and @xmath37 is the portion of length between @xmath38 and @xmath39 within the circle . after performing a series of simulations , the mass - size relation is computed and shown in fig.[fig : rmdemon ] for six distinct combinations of @xmath40 and @xmath41 . we can see that the mass of cw grows in a power - law fashion when the radius increases and , disregarding@xcite the fluctuations at small external field , the growing exponent saturates at @xmath42 for @xmath43 , see fig.[fig : rmdemon ] . the saturating behavior is in accordance with donato _ s conjecture@xcite that in a loose - packing situation the corresponding mass - size exponent has insignificant dependence on how the wires are injected . and the value of @xmath0 is also consistent with donato _ s result . we characterize the allocation of elastic energies in cw by the ratio @xmath44=@xmath45 . data in fig . [ fig : energy ] establish a scaling relation between @xmath44 and the three parameters in the hamiltonian as : @xmath46 where @xmath47 , @xmath48 , and @xmath49 . the numerical results can be deduced analytically . these exponents can be obtained by doing a dimensional analysis on the bending energy and the external potential in eq.([eq : lagrangian ] ) , @xmath50 , which gives the characteristic size for each loop as @xmath51 . multiplying the bending energy stored in each loop , @xmath52 , by the number of loops @xmath53 enable us to estimate the bending energy as @xmath54 . in a similar fashion , the stretching energy can be written as @xmath55 . minimizing the sum of these two energies with respect to @xmath56 gives @xmath57}\ ] ] the value of @xmath58 matches the simulation result . although the exact magnitudes are not the same , the signs of @xmath59 and @xmath60 and their relative magnitude are still captured . one possible explanation for the discrepancy is our failure to include the hard - core potential in the analytic arguments , which has been known to be difficult@xcite but crucial for a realistic polymer . is plotted against several parameters . for ( a ) and ( b ) , @xmath44 grows with the strength of external field in a power - law fashion with exponent @xmath47 . in ( a ) the stretching modulus is set at @xmath61 , and in ( b ) the bending rigidity is @xmath62 . panels ( c ) and ( d ) depict @xmath44 versus @xmath63 and @xmath64 respectively , and they share the same denotation of data . notice that the power - law can only be found for strong external field ( @xmath65 ) . the exponents of the power - law are determined to be @xmath48 and @xmath49,scaledwidth=45.0% ] versus the strength of confining potential for cw with different rigidity and modulus , @xmath63 and @xmath64 . all data collapse to the parameters @xmath66 with the scaling @xmath67 . the solid line is the fitting result @xmath68.,scaledwidth=35.0% ] how to estimate the resistance of a crumpled structure is a well - known difficult problem . for instance , what is the main reason why such a loose crumpled structure can be so hard ? as a starting point , people has studied the power law between the structure size and the applied force both experimentally and theoretically@xcite . our simulation result in fig.[fig : size ] showed that the radius of gyration @xmath69 of cw decreases with @xmath70 in a power - law form . to prevent the structure from becoming too complicated , we limit the field strength @xmath70 at 2000 to optimize the resolution and accuracy . the data are found to collape into the scaling relation @xmath71 , which suggests that the stretching energy not only correlates those monomers , but also contributes to the stiffness . intuitively , the dependence of @xmath41 is expected to drop out instead of diverging as it approaches infinity . the reason why we obtained such a scaling relation is a consequence of the discrete model since the length of monomers is bound to be compressed as they all scramble to be near the center of the external potential . in fig.[fig : size ] , @xmath72 is determined to be independent of @xmath63 and @xmath64 and , consequently , the thickness and young s modulus of the wire . this result is the same as the predictions made for an elastic thin sheet@xcite . it can be derived analytically by minimizing with respect to @xmath31 the sum of the external potential , @xmath73 , and the total repulsive interaction , @xmath74 where @xmath75 is a measure of the hard - core potential , @xmath76 is the monomer density , and the other @xmath77 factor comes from integrating over the cavity . since stoop _ _ has demonstrated the importance of plasticity for cw , we also include it in our simulation . the bending rigidity @xmath63 is revised with a linear approximation , @xmath78 in the plastic regime , so that the bending energy becomes @xmath79 when @xmath80 is larger than the yield threshold angle @xmath3@xcite . we analyzed the @xmath70-@xmath81 relation of cw for @xmath82 and @xmath83 , and it behaves very different from that of the elastic one . in fig.[fig : plasticity ] , the value of @xmath84 decays with the bending rigidity in the plastic regime which is consistent to the experimental findings in @xcite . it also decays exponentially with the yield threshold @xmath3 . when large deformations cost less energy due to the plasticity , they appear in abundance and cause many vertex - like structures which makes the cw stiffer . -@xmath81 relation of the cws(@xmath85 ) with plasticity for different @xmath2 which are denoted in the figure . the value of the exponents @xmath84 are -0.21 , -0.24 , and -0.29 for s=0.4 , 0.2 , and 0.05 respectively . the inset illustrates that @xmath84 decays with @xmath3 exponentially and eventually saturates at value of @xmath86 for @xmath87.,scaledwidth=35.0% ] in conclusion , we introduced a minimal one - dimensional model to simulate the crumpled wires . the simulation results agree excellently with the experimental observation , which include the pattern of loops , the universal value of mass - size exponent @xmath0@xcite . we checked the exponent @xmath84 that characterizes the power - law relation between the crumping force and the cw radius to be independent of the bending rigidity and the stretching modulus which is consist to @xcite . plasticity is found to suppress @xmath84 which is in the right trend as the recent experiment@xcite on crumpled sheets . we benifit from correspondence and discussions with professors p. c. chen , h. j. herrmann , p. y. hsiao , y. kantor , h. h. lin , and e. terentjev . support by the national science council in taiwan under grant 95 - 2120-m007 - 008 is acknowledged .

an energy - minimal simulation is proposed to study the patterns and mechanical properties of elastically crumpled wires in two dimensions . we varied the bending rigidity and stretching modulus to measure the energy allocation , size - mass exponent , and the stiffness exponent . the mass exponent is shown to be universal at value @xmath0 . we also found that the stiffness exponent @xmath1 is universal , but varies with the plasticity parameters @xmath2 and @xmath3 . these numerical findings agree excellently with the experimental results .

it has long been known that the wave function of an electron moving in a random potential becomes spatially localized . this effect was first predicted by anderson @xcite and is termed _ anderson localization_. in one and two dimensions _ all _ quantum states are localized in the presence of _ any _ amount of disorder while in three dimensions localization occurs only above some critical disorder . while this phenomenon is now fairly well understood in a single - particle picture , the inclusion of interactions in the disordered many - body system is a non - trivial problem . insight can be gained by studying the simpler case of just two interacting particles in a random potential . in this context , it has recently been claimed that the interaction can actually lead to a _ delocalization _ effect , in the sense that the spatial extent of the two - body wave function is larger than the single - particle localization length @xcite . this delocalization was found for both attractive and repulsive interactions , at least for some of the eigenstates in the continuous spectrum of energy eigenvalues . some authors objected to this finding @xcite and this point is still being debated . furthermore , the relevance of these results for the many - body problem has not yet been clarified . in this paper our main concern is the case of an _ attractive _ interaction . the concept of a two - particle bound state ( the cooper pair ) plays an important role in the theory of superconductivity . there is theoretical and experimental evidence for the existence of superconductor - insulator transitions , where localized states combine coherently into a superconducting condensate with a finite superfluid density . this finding motivates the question addressed in this paper : will the bound state of two attracting particles be extended over distances larger than the single - particle localization length ? we find that a result analogous to the anderson theorem for localized superconductors is valid for this problem in the limit of large single - particle localization lengths . by increasing the disorder we find a transition from a regime in which the interaction _ increases _ the localization length to a regime in which the interaction _ reduces _ the localization length . the hamiltonian for a disordered conventional ( @xmath0-wave ) superconductor is @xmath1 where @xmath2 represents the single - electron part including a spatially random external potential and the interaction parameter @xmath3 is taken as positive . the attractive interaction could be due to exchange of phonons or purely electronic mechanisms but its origin does not concern us here . depending on the disorder , the eigenstates @xmath4 of @xmath2 can be extended or localized ( with localization length @xmath5 at the fermi level ) . at t=0 the system described by ( [ bcs ] ) is a superconductor with spatially constant order parameter @xmath6 when the condition @xmath7 holds @xcite . @xmath8 is then given by the same expression as that for a clean superconductor . this is the anderson theorem . the important message of equation ( [ valanderson ] ) is that if the disorder is strong enough to localize the single - particle states , the superconducting order parameter @xmath8 and critical temperature @xmath9 will remain unaffected as long as the number of single - particle states in the energy range @xmath8 contained in the localization volume @xmath10 is still large . ( the superfluid density , on the other hand , is greatly reduced . ) if the amount of disorder is further increased so that ( [ valanderson ] ) is no longer valid the superconducting order parameter will fluctuate strongly in space and the critical temperature will be lowered . note that because @xmath11 is the wave function of the condensate , the anderson theorem tells us that the attractive interaction is delocalizing the cooper pairs . keep in mind , however , that in this derivation the cooper pairs are strongly interacting ( and actually overlapping ) with each other , forming a correlated liquid . in bcs theory the concept of cooper pair only has the formal significance that strong pair correlations exist between the particles in phase space . in the regime of validity of the anderson theorem @xmath8 is given by @xmath12 the denominator of this expression is the quasi - particle excitation energy . the binding energy of an isolated cooper pair is given by an expression of the same form as this one ( see eq.([cond ] ) below ) but with a different denominator . this is because equation ( [ delta ] ) takes into account the interactions between cooper pairs . experimentally , it has been found that superconductivity persists up to the anderson metal - insulator transition @xcite . these systems exhibit activated conductivity above @xmath9 . the coexistence of superconductivity and localization has also been observed in underdoped high-@xmath9 superconductors @xcite . in what follows the problem of two interacting electrons in a random potential will be addressed for the specific case of an attractive interaction . the aim is to see how the interplay of disorder and interaction affects the coherent propagation of the electrons in the ground state . the attractive interaction will be assumed to be short - ranged . we can describe this system by an anderson - hubbard hamiltonian @xmath13 with @xmath14 representing the site energies randomly distributed over a width @xmath15 , and @xmath16 . the single - particle eigenfunctions @xmath17 are assumed to be localized by the disorder with a localization length @xmath18 . we search for the two - particle ground state @xmath19 where @xmath20 and @xmath21 are the coordinates of the electrons . because the bound state is a spin singlet , @xmath22 is a symmetric function of its arguments and has the general form @xmath23 where @xmath17 are normalized single - particle eigenfunctions of ( [ ham ] ) with energy eigenvalues @xmath24 . in order to simulate the presence of a fermi sea , it is assumed that the two electrons can only be paired in states @xmath17 which lie above the fermi surface . according to the values of the parameters in ( [ ham ] ) we recognize several different regimes . if @xmath25 ( strong attraction ) then the electrons are tightly bound and move together like a heavy boson with hopping amplitude @xmath26 in an environment with disorder w. this boson would then become easily localized . in the remainder of the paper we concentrate on the regime in which the interaction is not strong ( @xmath27 ) . in this case @xmath28 is essentially the result of electron pairing in time - reversed single - particle eigenstates of ( [ ham ] ) for not too strong disorder . this can be seen as follows . the schrdinger equation for the cooper pair wave function is @xmath29 it admits the solution @xmath30 if the condition @xmath31 holds . ( here @xmath32 denotes the local density of states at the point @xmath33 ) . thus a result analogous to the anderson theorem is valid for the function @xmath34 . in such a case the solution to ( [ sch ] ) is @xmath35 and @xmath28 has no overlap with the state @xmath36 if @xmath37 . the condition ( [ cond ] ) can be satisfied if the integral of the local density of states @xmath38 over an energy interval of order @xmath39 does not depend on @xmath33 . this is possible even if the disorder is strong enough to localize the single - particle states as long as the condition @xmath40 holds . the binding energy @xmath39 will then be the same as that for a clean system . so we reach the conclusion that the attractive interaction can delocalize the pair or , at least , increase its localization length . the delocalization by interactions due to the attractive interaction can be understood within the block - scaling picture of localization introduced by thouless @xcite . in a recent paper @xcite , imry has used the block - scaling picture in order to argue that interactions ( attractive or repulsive ) should , in some cases , delocalize some of the eigenstates of the continuous spectrum of the pair of electrons . in what follows we extend the argument to the case of a cooper pair . suppose the system is divided into blocks of linear size @xmath18 ( measured in units of the lattice spacing ) so that the mean level spacing in a block is @xmath41 with @xmath42 equal to the band width . we can then solve the cooper problem , as above , for each block by pairing the two electrons in time - reversed states localized inside the block . if the fluctuation of the binding energy is smaller than the effective scattering amplitude of the pair between blocks then the pair will be extended over many blocks . we denote the cooper pair in the @xmath43-th block by @xmath44 . next , we estimate the scattering amplitude @xmath45 , _ due to the interaction _ , of the cooper pair between two neighbouring blocks as @xmath46 we now note that @xmath47 depends smoothly on @xmath48 and has no nodes because it is the wave function of a ground state , namely @xmath49 the normalization of @xmath44 implies @xmath50 and the number of terms in the sum in ( [ hopp ] ) is large , of the order of @xmath51 . all those terms interfere constructively yielding @xmath52 the fluctuation of the binding energy @xmath39 due to the randomness in @xmath24 is of the order of @xmath53 ( see the appendix ) . thus the condition for delocalization is obtained as @xmath54 the ratio @xmath55 is the bcs product @xmath56 . the following points should be noted : @xmath57 there are no phase correlations to consider in the sum ( [ hopp ] ) because all the terms are real and negative ; @xmath58 the large effective hopping amplitude @xmath45 resulted from @xmath47 being a smooth function of @xmath48 with no nodes . in other words , _ because we have been considering the ground state_. @xmath59 if the sign of the interaction @xmath60 is reversed , to make it repulsive , we do not expect any delocalization to occur in the ground state . the reason for this is that @xmath44 for each block ( using time - reversed pairing ) would now essentially involve only the single - particle eigenstate @xmath61 with the lowest energy : @xmath62 . then @xmath63 if @xmath64 . if we remove the constraint of time - reversed pairing then imry s argument would predict delocalization only if the pair has a certain excitation energy above the fermi level @xcite . so it would not be in the ground state . we have also performed numerical calculations of the cooper pair wave function for 1d systems of up to @xmath65 sites using the lanczos algorithm . in order to impose the constraint of a filled fermi sea the single - particle eigenstates for a given realization of the disorder potential are calculated and the eigenenergies of the states below the fermi energy are shifted by a large amount . thus only the one - particle states above the fermi surface are accessible for the cooper pair . before doing the lanczos diagonalization of the two - particle problem the hamiltonian is transformed back to real space representation where the number of nonzero matrix elements is much smaller than in the basis of the eigenstates of the non - interacting disordered system . in order to determine the spatial extent of the cooper pair wave function with respect to both the relative and the center - of - mass coordinates we have calculated two different quantities . the first one is the participation ratio @xmath66 where @xmath34 is normalized to unity . since only the diagonal part @xmath67 of the wave function is involved in the calculation the participation ratio can also be interpreted as the localization length of the cooper pair . the second quantity , which is related to the relative coordinate of the two electrons , is the average size @xmath68 of the cooper pair defined as @xmath69 our results are obtained for a system of 100 sites and each data point represents the average over 200 realizations of the disorder potential . since the fluctuations of the participation ratio @xmath70 are very large we found it more convenient to average @xmath71 instead of @xmath70 itself . 1 shows the behavior of @xmath71 as a function of @xmath72 for different values of the disorder strength @xmath73 . while in the case of strong disorder the interaction leads to a decrease of the participation ratio , for smaller values of @xmath73 an enhancement of @xmath71 is observed , at least for not too large @xmath72 . this is in qualitative agreement with our analytical arguments . one should however keep in mind that the calculations are done for a rather small system and that finite size effects can be of importance , especially in the weak disorder region where the localization length becomes large . in fig . 2 we see that the size of the cooper pair decreases rapidly with increasing interaction strength . the values for @xmath74 which should be of the order of the one - particle localization length are considerably reduced due to finite size effects . the interplay of disorder and interaction is a complex problem and therefore it is useful to start with the two - body problem in a disordered medium . the effect of a repulsive interaction seems to be very different from that of an attractive interaction since the latter can induce propagation of the bound state of two particles ( their ground state ) . the binding energy and spatial extent of the bound state is insensitive to disorder beyond the point where single - particle states become localized . this result for the cooper pair corresponds to the familiar anderson theorem for the many - body problem of dirty superconductors and is valid in a regime where the attraction and the disorder are not too strong so that electrons are paired in time - reversed single - particle eigenstates . these localized states can then be combined coherently into an extended pair wave function ( eq.([extended ] ) ) . in analogy with the case of dirty superconductors discussed in @xcite where the superfluid density is greatly reduced by the disorder , we also expect the energy of the wave function ( [ extended ] ) to be much less sensitive to the boundary conditions than that in a clean system . in other words , disorder has a stronger effect on the sensitivity to boundary conditions than on the localization length . it is possible to go away from this anderson regime in two ways . if the interaction is increased then both the binding energy and the effective mass of the pair increase , so the pair becomes more localized . on the other hand , if the interaction is kept not too strong but the disorder is increased , the binding energy and the localization length are reduced . we want to prove that @xmath75 . the binding energy e is obtained from the equation @xmath76 the density of states is @xmath77 where @xmath78 is the average density of states and @xmath79 the noise due to disorder . then @xmath80 where @xmath81 is the binding energy derived from @xmath78 . expanding the right - hand side of the above equation for small @xmath82 and taking into account the definition of @xmath81 we arrive at @xmath83 we make the following assumptions about the moments ( averages ) of @xmath79 : @xmath84 and @xmath85 . this latter condition is only intended to express the fact that the correlation persists over an energy of the order of the single - particle level spacing . taking the square of ( [ de ] ) and averaging we obtain r. a. rmer and m. schreiber , phys . lett * 78 * , 515 ( 1997 ) and phys . lett . * 78 * , 4890 ( 1997 ) ; k. frahm et al . lett . * 78 * , 4889 ( 1997 ) ; t. vojta , r. a. rmer and m. schreiber , cond - mat/9702241

we discuss the effect of disorder on the coherent propagation of the bound state of two attracting particles . it is shown that a result analogous to the anderson theorem for dirty superconductors is also valid for the cooper problem , namely , that the pair wave function is extended beyond the single - particle localization length if the latter is large . a physical justification is given in terms of the thouless block - scaling picture of localization . these arguments are supplemented by numerical simulations . with increasing disorder we find a transition from a regime in which the interaction delocalizes the pair to a regime in which the interaction enhances localization .

investigations of the @xmath0-nucleus interaction are motivated by various reasons . some of them , such as the possibility of forming quasi - bound states or resonances @xcite in the @xmath0-nucleus system , are purely of nuclear nature . the others are related to the study of the properties and structure of the @xmath4 resonance which is strongly coupled to the @xmath5 channel . for example , it is interesting to investigate the behavior of the @xmath0-meson in nuclear media where , after colliding with the nucleons , it readily forms the @xmath6 resonance . the interaction of this resonance with the surrounding nucleons can be described in different ways @xcite , depending on whether the structure of this resonance is defined in terms of some quark configurations or by the coupling of meson - baryon channels , as suggested in ref . the estimation by tiwari _ et al . _ @xcite shows , that in case of pseudoscalar @xmath7 coupling there is an essential density dependent reduction of the @xmath0-meson mass and of the @xmath8 mixing angle . the importance of the influence of the nuclear medium on the mesons passing through it , was recently emphasized by drechsel _ if this influence is described in terms of self - energies and effective masses , then the passing of @xmath9-mesons through the nucleus provides `` saturation '' of the isobar propagator ( or self - energy ) . this phenomenon manifests itself even in light nuclei @xcite . similar ideas were discussed also in ref . @xcite . in other words , the propagation of @xmath0-mesons inside the nucleus is a new challenge for theorists . another interesting issue related to the @xmath0-nucleus interaction is the study of charge symmetry breaking , which may partly be attributed to the @xmath10 mixing ( see , for example , refs . @xcite ) . in principle , one can extract the value of the mixing angle from experiments involving @xmath0-nucleus interaction and compare the results with the predictions of quark models . however , to do such an extraction , one has to make an extrapolation of the @xmath0-nucleus scattering amplitude into the area of unphysical energies below the @xmath0-nucleus threshold . this is a highly model dependent procedure requiring a reliable treatment of the @xmath0-nucleus dynamics . in this respect , few - body systems such as @xmath11 , @xmath0@xmath12he , and @xmath0@xmath13he , have obvious advantages since they can be treated using rigorous faddeev - type equations . to the best of our knowledge , the exact ags theory @xcite has been used in the few calculations ( see refs . @xcite ) for the @xmath11 and in one recent calculation @xcite for the @xmath14h and @xmath14he systems . a solution of the few - body equations presupposes the knowledge of the corresponding two - body @xmath3-matrices @xmath15 and @xmath16 off the energy shell . due to the fact that at low energies the @xmath0 meson interacts with a nucleon mainly via the formation of the @xmath6-resonance , the inclusion of the higher partial waves ( @xmath17 ) is unnecessary . furthermore , since the @xmath2 interaction is poorly known , the effect of the fine tuned details of the `` realistic '' @xmath18 potentials would be far beyond the level of the overall accuracy of the @xmath19 theory . in contrast to the well - established @xmath18 forces , the @xmath2 interaction is constructed using very limited information available , namely , the @xmath2 scattering length and the parameters of the @xmath6-resonance . furthermore , only the resonance parameters are known more or less accurately while the scattering length ( which is complex ) is determined with large uncertainties . moreover , practically nothing is known about the off - shell behavior of the @xmath2 amplitude . it is simply assumed that the off - shell behavior of this amplitude could be approximated ( like in the case of @xmath9 mesons ) by appropriate yamaguchi form - factors ( see , for example , refs . however , if the available data are used to construct a potential via , for example , fiedeldey s inverse scattering procedure @xcite , the resulting form factor of the separable potential is not that simple . the problem becomes even more complicated due to the multichannel character of the @xmath2 interaction with the additional off - shell uncertainties stemming from the @xmath9-meson channel . in such a situation , it is desirable to narrow as much as possible the uncertainty intervals for the parameters of @xmath2 interaction . this could be done by demanding consistency of theoretical predictions based on these parameters , with existing experimental data for two- , three- , and four - body @xmath0-nucleus processes . this is one of the objectives of the present work . to do this , we calculate the cross sections of coherent @xmath0-photoproduction on @xmath1he and @xmath1h and study their sensitivity to the parameters of the @xmath2 amplitude . we start by assuming that the compton scattering on a nucleon , @xmath20 as well as the processes of multiple re - appearing of the photon in the intermediate states , @xmath21 give a negligible contribution to the coherent @xmath0-photoproduction on a nucleus @xmath22 . in our model , we also neglect virtual excitations and breakup of the nucleus immediately after its interaction with the photon . with these assumptions , the process @xmath23 can be formally described in two steps : in the first one , the photon produces the @xmath0 meson on one of the nucleons , @xmath24 in the second step ( final state interaction ) the @xmath0 meson is elastically scattered off the nucleus , @xmath25 an adequate treatment of the scattering step is , of course , the most difficult and crucial part of the theory . the first microscopic calculations concerning the low - energy scattering of the @xmath0-meson from @xmath12h , @xmath12he , and @xmath13he were done in refs . @xcite where the few - body dynamics of these systems was treated by employing the finite - rank approximation ( fra ) @xcite of the nuclear hamiltonian . this approximation consists in neglecting the continuous spectrum in the spectral expansion @xmath26 of the hamiltonian @xmath27 describing the nucleus . since the three- and four - body nuclei have only one bound state , fra reduces to @xmath28 physically , this means that we exclude the virtual excitations of the nucleus during its interaction with the @xmath0 meson . it is clear that the stronger the nucleus is bound , the smaller is the contribution from such processes to the elastic @xmath19 scattering . by comparing with the results of the exact ags calculations , it was shown@xcite that even for @xmath11 scattering , having the weakest nuclear binding , the fra method works reasonably well , which implies that we obtain sufficiently accurate results by applying this method to the @xmath0@xmath12h , @xmath0@xmath12he , and even more so to the @xmath0@xmath13he scattering . in essence , the fra method can be described as follows ( for details see ref.@xcite ) . let @xmath29 be the total @xmath19 hamiltonian , where @xmath30 describes the free @xmath0-nucleus motion and @xmath31 the sum of the two - body @xmath0-nucleon potentials . the lippmann - schwinger equation @xmath32 for the @xmath0-nucleus @xmath3-matrix can be rewritten as @xmath33 where @xmath34 and the auxiliary operator @xmath35 is split into @xmath22 components of faddeev - type , @xmath36 satisfying the following system of equations @xmath37 with @xmath38 being the two - body @xmath3-matrix describing the interaction of the @xmath0-meson with the @xmath39-th nucleon , _ i.e. _ , @xmath40 it should be emphasized that up to this point no approximation has been made and , therefore , the set of equations ( [ teq]-[tetani ] ) is equivalent to the initial equation ( [ lsinitial ] ) . however , to solve eq . ( [ teq ] ) , we have to resort to the approximation ( [ fraapprox ] ) which simplifies its kernel ( [ kernel ] ) to @xmath41 with this approximation , the sandwiching of eq . ( [ teq ] ) between @xmath42 and @xmath43 and the partial wave decomposition give a one - dimensional integral equation for the amplitude of the process ( [ etaaaeta ] ) . although this one - dimensional equation may look similar to the integral equation of the first - order optical - potential theory , the fra approach is quite different . firstly , in contrast to the optical potential of the first order , the operator @xmath35 includes all orders of rescattering via solution of eq . ( [ t0i ] ) . secondly , the @xmath2 amplitudes @xmath44 entering eq . ( [ t0i ] ) , are taken as operators in the many - body space and off the energy shell ( note that @xmath45 in eq . ( [ tetani ] ) is four - body propagator with @xmath19 reduced mass and @xmath46 is total four - body energy ) , i.e. the fra method does not involve the so - called `` impulse approximation '' ( using free two - body amplitudes for @xmath38 ) which is an indispensable part of the optical theory . the question then arises how a photon can be included in this formalism in order to describe the photoproduction process ( [ gaaeta ] ) . this can be achieved by following the same procedure as in ref.@xcite where the reaction ( [ gaaeta ] ) with @xmath47 was treated within the framework of the exact ags equations , and the photon was introduced by considering the @xmath2 and @xmath48 states as two different channels of the same system . this implies that the operators @xmath38 should be replaced by @xmath49 matrices . it is clear that such replacements of the kernels of the integral equations ( [ t0i ] ) and subsequently of the integral equation ( [ teq ] ) lead to solutions having a similar matrix form @xmath50 here @xmath51 describes the compton scattering , @xmath52 the photoproduction process , and @xmath53 the elastic @xmath0 scattering on the @xmath39-th nucleon . what is finally needed is the cross section @xmath54 of the reaction ( [ gaaeta ] ) averaged over orientations @xmath55 of the initial nuclear spin and photon polarization @xmath56 and summed over spin orientations @xmath57 in the final state . here @xmath58 and @xmath59 are the spatial and spin - isospin parts of @xmath60 ( with the third components of the nuclear spin and isospin being @xmath55 and @xmath61 respectively ) , @xmath62 and @xmath63 are the momenta of the photon and @xmath0 meson , @xmath64 is the energy of the photon , @xmath65 the mass of the nucleus , and @xmath66 the reduced mass of the meson and the nucleus . however , it is technically more convenient to consider radiative @xmath0-absorption , _ i.e. _ , the inverse reaction . then the photoproduction cross section can be obtained by applying the principle of detailed balance . the reason for this is that all the processes in which the photon appears more than once , _ i.e. _ , the terms of the integral equations of type @xmath67 or @xmath68 involving more than one electromagnetic vertex , can be neglected . omission of these terms in ( [ teq ] ) results in decoupling the elastic scattering equation @xmath69 from the equation for the radiative absorption @xmath70 once @xmath71 is calculated , the radiative absorption @xmath3-matrix ( [ absorption ] ) can be obtained by integration . therefore , the procedure of calculating the photoproduction cross section ( [ sechenie ] ) consists of the following steps : * solving the system of equations @xmath72 for the auxiliary elastic - scattering operators @xmath73 . * calculating ( by integration ) the auxiliary matrices @xmath74 from @xmath75 * solving the integral equation @xmath76 for the elastic - scattering @xmath3-matrix . * calculating ( by integration ) the radiative absorption @xmath3-matrix @xmath77 * substituting this @xmath3-matrix into eq . ( [ sechenie ] ) to obtain the differential cross section for the photoproduction . this is possible because the absolute values of the photoproduction and radiative absorption @xmath3-matrices coincide . to implement the steps described in the previous section , we need the two - body @xmath3-matrices @xmath78 and @xmath79 for the elastic @xmath2 scattering and the radiative absorption @xmath80 on a single nucleon , respectively . furthermore , all equations ( [ proc1]-[proc4 ] ) have to be sandwiched between @xmath42 and @xmath43 ( ground state of the nucleus ) . since at low energies both the elastic scattering and photoproduction of the @xmath0 meson on a nucleon proceed mainly via formation of the @xmath6 resonance , we may retain only the @xmath81-waves in the partial wave expansions of the corresponding two - body @xmath3-matrices . the problem of constructing an @xmath2 potential or directly the corresponding @xmath3-matrix @xmath78 has no unique solution since the only experimental information available consists of the @xmath6-resonance pole position @xmath82 and the @xmath2 scattering length @xmath83 . in the present work we use three different versions of @xmath78 . without any scattering data it is practically impossible to construct a reliable @xmath2 potential . in the low - energy region , however , the elastic scattering can be viewed as the process of formation and subsequent decay of @xmath6 resonance , _ i.e. _ , @xmath84 this implies that the corresponding breit - wigner formula could be a good approximation for the @xmath2 cross section . therefore , we may adopt the following ansatz @xmath85 where the propagator @xmath86 , describing the intermediate state of the process ( [ diagram ] ) , is assumed to have a simple breit - wigner form @xmath87 which guaranties that the @xmath3-matrix ( [ tetan ] ) has a pole at the proper place . the vertex function @xmath88 for the processes @xmath5@xmath89 is chosen to be @xmath90 which in configuration space is of yukawa - type . the range parameter @xmath91@xmath92 was determined in ref . @xcite while the parameters of the @xmath6-resonance @xmath93 are taken from ref . the strength parameter @xmath94 is chosen to reproduce the @xmath0-nucleon scattering length @xmath83 , @xmath95 the imaginary part of which accounts for the flux losses into the @xmath96 channel . here @xmath97 is the @xmath2 reduced mass . the two - body scattering length @xmath83 is not accurately known . different analyses @xcite provided values for @xmath83 in the range @xmath98 in most recent publications the value used for i m @xmath83 is around @xmath99fm . however , for re@xmath83 the estimates are still very different ( compare , for example , refs . @xcite and @xcite ) . in the present work we assume that @xmath100 the @xmath3-matrix @xmath78 constructed in this way reproduces the scattering length ( [ etanlength ] ) and the @xmath6 pole , but apparently violates the two - body unitarity since it does not obey the two - body lippmann - schwinger equation . an alternative way of constructing the two - body @xmath3-matrix @xmath78 is to solve the corresponding lippmann - schwinger equation with an appropriate separable potential having the same form - factors ( [ ffactor ] ) . however , a one - term separable @xmath3-matrix obtained in this way , does not have a pole at @xmath101 . to recover the resonance behavior in this case , we use the trick suggested in ref . @xcite , namely , we use an energy - dependent strength of the potential @xmath102 \ , g(k')\ ] ] where @xmath103 is complex while @xmath104 and @xmath105 are real constants . with this ansatz for the potential , the lippmann - schwinger equation gives the @xmath3-matrix in the form ( [ tetan ] ) with @xmath106 / ( 1 - i\sqrt{2 z \mu_{\eta n } } /\alpha)^2}\ . \label{tdeloff}\ ] ] the constants @xmath103 , @xmath104 , and @xmath105 can be chosen in such a way that the corresponding scattering amplitude reproduces the scattering length @xmath83 and has a pole at @xmath101 . this version of @xmath78 also reproduces the scattering length ( [ etanlength ] ) and the @xmath6 pole . moreover , it is consistent with the condition of two - body unitarity . we can also construct @xmath78 in the form ( [ tetan ] ) , with the same @xmath86 as in ( [ tau1 ] ) , but obeying the unitarity condition @xmath107 of course , with the simple form ( [ tetan ] ) we can not satisfy the condition ( [ unitarity ] ) at all energies . to simplify the derivations , we impose this condition on @xmath78 at @xmath108 . ( [ unitarity ] ) is real , it can fix only one parameter and we need one more condition to fix both the real and imaginary parts of the complex @xmath94 . as the second equation , we used the real or imaginary part of eq . ( [ t000 ] ) ( version iii - a or iii - b respectively ) with @xmath83 given by ( [ etanlength ] ) . this procedure guaranties two - body unitarity and gives the correct position of the resonance pole , but the resulting @xmath78 provides a value of @xmath83 which , of course , slightly differs from ( [ etanlength ] ) , namely , @xmath109 in what follows we use these three versions of the matrix @xmath78 . all of them have the same separable form ( [ tetan ] ) but different @xmath86 . comparison of the results obtained with these three @xmath3-matrices can give an indication of the importance of two - body unitarity in the photoproduction processes . in constructing the radiative absorption @xmath3-matrix @xmath79 , the @xmath6 dominance in the near - threshold region also plays an important role . it was experimentally shown @xcite that , at low energies , the reaction ( [ gnneta ] ) proceeds mainly via formation of the @xmath6-resonance , @xmath110 by the @xmath81-wave @xmath111 multipole . this means that in the standard cgln decomposition of the @xmath112 amplitude ( see , for example , ref.@xcite ) only the term proportional to the dot - product @xmath113 of the nucleon spin and photon polarization can be retained , i.e. @xmath114 the dominance of the process ( [ diagramgamma ] ) implies that @xmath115 in this energy region can be written in a separable form similar to ( [ tetan ] ) . to construct such a separable @xmath3-matrix , we use the results of ref . @xcite where @xmath116 was considered as an element of a multi - channel @xmath3-matrix which simultaneously describes experimental data for the processes @xmath117 on the energy shell in the @xmath6-channel ( the @xmath2 scattering length obtained in ref . @xcite is the same as we use for constructing versions i and ii of @xmath118 ) . in the present work , we take the @xmath3-matrix @xmath119 from ref . @xcite and extend it off the energy shell via @xmath120 where @xmath121 is a parameter . the yamaguchi form - factors used in this ansatz go to unity on the energy shell . since @xmath121 is not known , this parameter is varied in our calculations within an interval @xmath122 which is a typical range for meson - nucleon forces . it is known that @xmath116 is different for neutron and proton . in this work we assume that they have the same functional form ( [ toff ] ) but differ by a constant factor , @xmath123 multipole analysis @xcite gives for this factor the estimate @xmath124 . therefore , if we direct the @xmath46-axis along the photon momentum @xmath125 , the radiative absorption @xmath3-matrix entering our equations , can be written as @xmath126 where @xmath127 and @xmath128 are transverse components of the photon polarization vector while @xmath129 and @xmath130 are the operators projecting onto the proton and neutron isotopic states respectively . since the @xmath3-matrices @xmath78 and @xmath79 are poorly known and their uncertainties significantly limit the overall accuracy of the theory , it is not necessary to use any sophisticated ( `` realistic '' ) potential to describe the @xmath18 interaction . therefore we may safely assume that the nucleons interact with each other only in the @xmath81-wave state . to obtain the necessary nuclear wave function @xmath60 , we solve the few - body equations of the integro - differential equation approach ( idea ) @xcite with the malfliet - tjon potential @xcite . this approach is based on the hyperspherical harmonic expansion method applied to faddeev - type equations . in fact , in the case of @xmath81-wave potentials , the idea is fully equivalent to the exact faddeev equations . therefore , the bound states used in our calculations are derived , to all practical purposes , via an exact formalism . the wave function @xmath131 of the @xmath1h/@xmath1he system obtained by solving the idea equations with the malfliet - tjon potential , has only the symmetric @xmath81-wave spatial component @xmath58 multiplied by the antisymmetric spin - isospin part @xmath132 where @xmath133 , @xmath134 and @xmath135 , @xmath136 are the mixed symmetry states in the spin and isospin sub - spaces . the matrix element of @xmath137 in eq . ( [ sechenie ] ) involves the average not only over the spatial part of @xmath60 but over @xmath138 as well . the average @xmath139 can be done before we start solving equations ( [ proc1]-[proc4 ] ) . since @xmath78 , @xmath45 , and @xmath140 do not involve spin - isospin operators , the averaging of eqs . ( [ proc1 ] ) and ( [ proc3 ] ) over @xmath138 is trivial : it does not produce any additional coefficients . ( [ proc2 ] ) , however , changes . indeed , for each nucleon ( @xmath141 ) , it involves the operator ( [ general ] ) which causes nucleon spin to flip over . formal averaging of @xmath142 ( for @xmath143 ) over the states @xmath59 having definite values of the @xmath46-components of total spin ( @xmath55 ) and isospin ( @xmath61 ) , gives the same results @xmath144 \displaystyle \delta_{-s'_z , s_z}f_{\rm off}^{\gamma \eta } \cdot\frac13(\epsilon_x+i\epsilon_y ) & , & \quad \mbox{for } & t_z=-1/2\ & ( ^3{\rm h } ) \end{array } \right.\ ] ] for all three nucleons . this means that all three matrix elements @xmath145 ( @xmath143 ) acquire the same coefficient , namely , @xmath146 or @xmath147 depending on @xmath61 . via eq . ( [ proc4 ] ) , the same coefficient goes to the matrix element @xmath148 . since @xmath149 , this gives the factor @xmath150 ( for the case of @xmath1he ) or @xmath151 ( for the case of @xmath1h ) in the final eq . ( [ sechenie ] ) for the photoproduction cross section . thus , the cross section for the @xmath1he target quadratically depends on @xmath22 while for the case of @xmath1h it is independent of @xmath22 . figures [ fig1.fig ] and [ fig2.fig ] show the results of our calculations for the total cross section of the coherent process ( [ gaaeta ] ) . the calculations were done for two nuclear targets , @xmath1h and @xmath1he , using the three versions of @xmath78 described in the section [ twobodyint ] . the curves corresponding to these three @xmath3-matrices are denoted by ( i ) , ( ii ) , and ( iii - a , iii - b ) , respectively . we found that the coherent @xmath0-photoproduction on these targets is strongly enhanced in the near - threshold region as compared to higher photon energies ( @xmath152mev ) . this can be attributed to strong final state interaction caused , for example , by a pole of the scattering @xmath81-matrix , situated in the complex - energy plane not far from the threshold energy , or in other words , to formation of @xmath0-nucleus resonance . in order to emphasize this finding and to remove the insignificant but distracting differences among different curves , we present the results in a normalized form . each curve shows the ratio @xmath153 with @xmath154 being the corresponding cross section at @xmath155mev , i.e. at the energy where the near - threshold enhancement dies out . at this energy , all the curves become flat and are not far from each other as well as from the value of 59.812nb obtained in ref . the normalization values @xmath154 are given in table [ tab1.tab ] . as can be seen in fig . [ fig1.fig ] , the two versions of @xmath78 , ( i ) and ( ii ) , give significantly different results despite the fact that both of them reproduce the same @xmath83 and the @xmath6-resonance . this indicates that the scattering of the @xmath0 meson on the nucleons ( final state interaction ) is very important in the description of the near - threshold photoproduction process . this conclusion is further substantiated when comparing our curves with the corresponding points ( circles ) calculated for the @xmath1he target in ref . @xcite . there the final state interaction was treated using an optical potential of the first order . it is well - known that the first - order optical theory is not adequate at energies near resonances . this is the reason why the calculations of ref . @xcite underestimate @xmath156 near the threshold where , with @xmath157 , the systems @xmath0@xmath12h and @xmath0@xmath12he show a resonance behavior @xcite . significant differences between the corresponding curves ( i ) and ( ii ) in fig . [ fig1.fig ] imply that two - body unitarity is important as well . actually , due to the resonant character of the final state interaction , all the details of @xmath78 have strong influence on the photoproduction cross section in the near - threshold region . [ fig2.fig ] where we compare the results corresponding to the three choices of @xmath86 in ( [ tetan ] ) , serve as another illustration of this statement . since nothing is known about the parameter @xmath121 , we assume @xmath158 as its basic value . this can be motivated by the fact that both the elastic scattering and radiative absorption ( photoproduction ) of the @xmath0 meson on the nucleon go via formation of the same @xmath6 resonance . this means that at least one vertex , namely , @xmath159 should be the same for both the elastic scattering and radiative absorption . to find out how crucial the choice of @xmath121 is , we did two additional calculations with @xmath160 and @xmath161 . we found that even with this wide variation , the corresponding @xmath162 curves show practically identical enhancement of the crosss section ( less than 1% difference ) . the cross section only slightly increases when the range of the interaction becomes smaller ( when @xmath121 grows ) . therefore , the dependence on @xmath121 is not very strong and the choice @xmath158 gives a reasonable estimate for the photoproduction cross section . as far as the dependence of @xmath156 on the choice of the parameter @xmath163 is concerned , we found ( see sec.[spinisospina ] ) that for @xmath0 photoproduction on @xmath1h the cross section in our model does not depend on @xmath22 , while for the @xmath1he target the @xmath22-dependence is quadratic . this means that among these two nuclei , the helium is preferable candidate for experimental determination of the ratio @xmath22 . the sign or any phase factor of @xmath22 , however , has no influence on the cross section if the electromagnetic vertex is taken into account only in the first order as it was done in our calculation . q. haider and l. c. liu , phys . * b172 * , 257 ( 1986 ) . l. frankfurt _ et al . _ , phys . rev . * c60 * , 055202 ( 1999 ) . n. kaiser , p. b. siegel , w. weise , phys . b362 * , 23 ( 1995 ) . j. nieves and e. r. arriola , phys . rev . * d64 * , 116008 ( 2001 ) . v. k. tiwari and a. kundu , e - print lanl , _ nucl - th/9811064_. d. drechsel , l. tiator , s. s. kamalov , shin nan yang , nucl * a660 * , 423 ( 1999 ) . a. fix and h. arenh@xmath164vel , nucl a697 * , 277 ( 2002 ) . s. a. coon and m. d. scadron , phys . rev . * c26 * , 562 ( 1982 ) . c. wilkin , phys b331 * , 276 ( 1994 ) . a. magiera , h. machner , nucl . phys . * a674 * , 515 ( 2000 ) . _ , j.phys . * g25 * , l1 ( 1999 ) . t. ueda , phys . lett . * 66 * , 297 ( 1991 ) . n. v. shevchenko , v. b. belyaev , s. a. rakityansky , w.sandhas , and s. a. sofianos , eur . j. * a9 * , 143 ( 2000 ) . a. fix and h. arenh@xmath164vel , eur . j. * a9 * , 119 ( 2000 ) . a. fix and h. arenh@xmath164vel , phys * b492 * , 32 ( 2000 ) . e. o. alt , p. grassberger and w. sandhas , nucl . * b2 * , 167 ( 1967 ) ; e. o. alt , p. grassberger , and w. sandhas , phys . rev . * c1 * , 85 ( 1970 ) . a. fix and h. arenh@xmath164vel , e - print lanl , _ nucl - th/0206038_. h. garcilazo and m. t. pena , phys . rev . * c61 * , 064010 ( 2000 ) . a. deloff , phys . rev . * c61 * , 024004 ( 2000 ) . h. fiedeldey , nucl . phys . * a135 * , 353 ( 1969 ) . s. a. rakityansky , s. a. sofianos , w. sandhas , v. b. belyaev , phys . lett . , * b359 * , 33 ( 1995 ) . v. b. belyaev , s. a. rakityansky , s. a. sofianos , m. braun , w. sandhas , few body systems suppl . , * 8 * , 309 ( 1995 ) . s. a. rakityansky , s. a. sofianos , v. b. belyaev , w. sandhas , few - body systems suppl . , * 9 * , 227 ( 1995 ) . s. a. rakityansky , s. a. sofianos , m. braun , v. b. belyaev , w. sandhas , phys.rev . , * c53 * , r2043 ( 1996 ) . s. a. rakityansky , s. a. sofianos , m. braun , v. b. belyaev , w. sandhas , chinese j. phys . , * 34 * , 998 ( 1996 ) . s. a. sofianos , s. a. rakityansky , proceedings of the european conference on advances in nuclear physics and related areas , thessaloniki - greece 8 - 12 july 1997 , pp . 570 - 581 , giahoudi - giapouli publishing , thessalonoki , 1999 . s. a. sofianos , s. a. rakityansky , m. braun , in : exciting physics with new accelerator facilities , world scientific , singapore , pp . 111 - 116 ( 1998 ) . v. b. belyaev , _ lectures on the theory of few - body systems _ , springer verlag , heidelberg , 1990 . n. v. shevchenko , s. a. rakityansky , s. a. sofianos , v. b. belyaev , and w. sandhas , phys . rev . * c58 * , r3055 ( 1998 ) . n. v. shevchenko , v. b. belyaev , s. a. rakityansky , w. sandhas , and s. a. sofianos , nucl . phys . * a689 * , 383 ( 2001 ) . c. bennhold and h. tanabe , nucl . phys.*a530 * , 625 ( 1991 ) . particledatagroup , phys . rev . * d50 * , 1173 ( 1994 ) . m. batinic , i. slaus , and a. svarc , phys . rev . * c52 * , 2188 ( 1995 ) . a. m. green , s. wycech , phys . rev . * c55 * , r2167 ( 1997 ) . grishina , l. a. kondratyuk , m. buescher , c. hanhart , j. haidenbauer , and j. speth , phys . lett . * b475 * , 9 ( 2000 ) . b. krusche _ et al . _ , lett . * 74 * , 3736 ( 1995 ) . a. m. green and s. wycech , phys . * c60 * , 035208 ( 1999 ) . n. c. mukhopadhyay , j. f. zhang , m. benmerouche , phys . b364 * , 1 ( 1995 ) . m. fabre de la ripelle , h. fiedeldey , and s. a. sofianos , phys . rev . * c38 * , 449 ( 1988 ) . w. oehm , h. fiedeldey , s. a. sofianos , and m. fabre de la ripelle , phys . rev . * c44 * , 81 ( 1991 ) . r. a. malfliet and j. a. tjon , nucl . phys . * a127 * , 161 ( 1969 ) ; ann . ( n.y . ) * 61 * , 425 ( 1970 ) . l. tiator , c. bennhold , and s. s. kamalov , nucl . phys . * a580 * , 455 ( 1994 ) ; and _ private communication_. . values of the total cross section of the coherent process ( [ gaaeta ] ) at @xmath155mev calculated with the four versions of @xmath78 which are denoted as i , ii , iii - a , and iii - b . these values are used to normalize the curves shown in figures [ fig1.fig ] and [ fig2.fig ] . [ cols="^,^,^,^,^,^",options="header " , ]

a microscopic few - body description of near - threshold coherent photoproduction of the @xmath0 meson on tritium and @xmath1he targets is given . the photoproduction cross - section is calculated using the finite rank approximation ( fra ) of the nuclear hamiltonian . the results indicate a strong final state interaction of the @xmath0 meson with the residual nucleus . sensitivity of the results to the choice of the @xmath2 @xmath3-matrix is investigated . pacs numbers : 25.80.-e , 21.45.+v , 25.10.+s

studying the dynamics of nearly one - dimensional structures has various scientific and industrial applications , for example in biophysics ( cf . @xcite and the references therein ) and visual computing ( cf . @xcite ) as well as in civil and mechanical engineering ( cf . @xcite ) , microelectronics and robotics ( cf . @xcite ) . in this regard , an appropriate description of the dynamical behavior of flexible one - dimensional structures is provided by the so - called special cosserat theory of elastic rods ( cf . @xcite , ch . 8 , and the original work @xcite ) . this is a general and geometrically exact dynamical model that takes bending , extension , shear , and torsion into account as well as rod deformations under external forces and torques . in this context , the dynamics of a rod is described by a governing system of twelve first - order nonlinear partial differential equations ( pdes ) with a pair of independent variables @xmath0 where @xmath1 is the arc - length and @xmath2 the time parameter . in this pde system , the two kinematic vector equations ( ( 9a)(9b ) in @xcite , ch . 8) are parameter free and represent the compatibility conditions for four vector functions @xmath3 in @xmath0 . whereas the first vector equation only contains two vector functions @xmath4 , the second one contains all four vector functions @xmath3 . the remaining two vector equations in the governing system are dynamical equations of motion and include two more dependent vector variables @xmath5 and @xmath6 . moreover , these dynamical equations contain parameters ( or parametric functions of @xmath1 ) to characterize the rod and to include the external forces and torques . because of its inherent stiffness caused by the different deformation modes of a cosserat rod , a pure numerical treatment of the full cosserat pde system requires the application of specific solvers ; see e.g. @xcite . in order to reduce the computational overhead caused by the stiffness , we analyzed the lie symmetries of the first kinematic vector equation ( ( 9a ) in @xcite , ch . 8) and constructed its general and ( locally ) analytical solution in @xcite which depends on three arbitrary functions in @xmath0 and three arbitrary functions in @xmath2 . in this contribution we perform a computer algebra - based lie symmetry analysis to integrate the full kinematic part of the governing cosserat system based on our previous work in @xcite . this allows for the construction of a general analytical solution of this part which depends on six arbitrary functions in @xmath0 . we prove its generality and apply the obtained analytical solution in order to solve the dynamical part of the governing system . finally , we prove its practicability by simulating the dynamics of a flagellated microswimmer . to allow for an efficient solution process of the determining equations for the infinitesimal lie symmetry generators , we make use of the maple package sade ( cf . @xcite ) in addition to desolv ( cf . @xcite ) . this paper is organized as follows . section [ sec:2 ] describes the governing pde system in the special cosserat theory of rods . in section [ sec:3 ] , we show that the functional arbitrariness in the analytical solution to the first kinematic vector equation that we constructed in @xcite can be narrowed down to three arbitrary bivariate functions . our main theoretical result is presented in section [ sec:4 ] , in which we construct a general analytical solution to the kinematic part of the governing equations by integrating the lie equations for a one - parameter subgroup of the lie symmetry group . section [ sec:5 ] illustrates the practicability of this approach by realizing a semi - analytical simulation of a flagellated microswimmer . this is based on a combination of the analytical solution of the kinematic part of the cosserat pde and a numerical solution of its dynamical part . some concluding remarks are given in section [ sec:6 ] and limitations are discussed in section [ sec:7 ] . in the context of the special cosserat theory of rods ( cf . @xcite ) , the motion of a rod is defined by a vector - valued function @xmath7\times { \mathbb{r}}\ni ( s , t ) \mapsto \left({{\boldsymbol{r}}}(s , t),\,{{\boldsymbol{d}}}_1(s , t),\,{{\boldsymbol{d}}}_2(s , t)\right)\in { \mathbb{e}}^3\ , . \label{rd1d2}\ ] ] here , @xmath2 denotes the time and @xmath1 is the arc - length parameter identifying a _ material cross - section _ of the rod which consists of all material points whose reference positions are on the plane perpendicular to the rod at @xmath1 . moreover , @xmath8 and @xmath9 are orthonormal vectors , and @xmath10 denotes the position of the material point on the centerline with arc - length parameter @xmath1 at time @xmath2 . the euclidean 3-space is denoted with @xmath11 . the vectors @xmath12 , and @xmath13 are called _ directors _ and form a right - handed orthonormal moving frame . the use of the triple @xmath14 is natural for the intrinsic description of the rod deformation whereas @xmath15 describes the motion of the rod relative to the fixed frame @xmath16 . this is illustrated in figure [ fig1 ] . from the orthonormality of the directors follows the existence of so - called _ darboux _ and _ twist _ vector functions @xmath17 and @xmath18 determined by the kinematic relations @xmath19 the _ linear strain _ of the rod and the _ velocity of the material cross - section _ are given by vector functions @xmath20 and @xmath21 . ( d1 ) at ( 609.500 mm , 393.654 mm ) ; ( d2 ) at ( 451.588 mm , 30.955 mm ) ; ( d3 ) at ( 839.662 mm , 92.118 mm ) ; ( o1 ) at ( 631.054 mm , 174.816 mm ) ; ( ax ) at ( 887.632 mm , 543.492 mm ) ; ( ay ) at ( 1369.466 mm , 541.844 mm ) ; ( az ) at ( 1126.066 mm , 941.984 mm ) ; ( o2 ) at ( 1118.542 mm , 639.549 mm ) ; ( r ) at ( 1495.859 mm , 221.154 mm ) ; ( s0 ) at ( 608.083 mm , 533.215 mm ) ; ( sl ) at ( 1832.694 mm , 552.899 mm ) ; ( 0.3 , 0 ) rectangle ( 0.12 * 0.0352777778 * 1920- 0.3 , 0.12 * 0.0352777778 * 1080 ) ; at ( 0 , 0 ) forms a right - handed orthonormal basis . the directors @xmath22 and @xmath23 span the local material cross - section , whereas @xmath24 is perpendicular to the cross - section . note that in the presence of shear deformations @xmath24 is unequal to the tangent @xmath25 of the centerline of the rod.,title="fig : " ] ; ( o2 ) ( r ) node[black , pos=0.65 , right , inner sep=5pt , xshift=0pt , yshift=0pt ] ; ( o2 ) circle ( 0.1 ) ; ( o1 ) ( d1 ) node[black , anchor = west , inner sep=2pt , xshift=0pt , yshift=0pt ] ; ( o1 ) ( d2 ) node[black , anchor = south east , inner sep=0pt , xshift=0pt , yshift=0pt ] ; ( o1 ) ( d3 ) node[black , anchor = west , inner sep=0pt , xshift=0pt , yshift=0pt ] ; at ( ax ) ; at ( ay ) ; at ( az ) ; at ( s0 ) ; at ( sl ) ; the components of the _ strain variables _ @xmath26 and @xmath27 describe the deformation of the rod : the flexure with respect to the two major axes of the cross - section @xmath28 , torsion @xmath29 , shear @xmath30 , and extension @xmath31 . the triples @xmath32 are functions in @xmath0 , that satisfy the _ compatibility conditions _ @xmath33 the substitution of into the left equation in leads to @xmath34 on the other hand one obtains , @xmath35 and @xmath36 with @xmath37 and @xmath38 , and therefore @xmath39 similarly , the second compatibility condition in is equivalent to @xmath40 with @xmath41 and @xmath42 . the first - order pde system ( [ ke1])([ke2 ] ) with independent variables @xmath0 and dependent variables ( [ kv ] ) forms the kinematic part of the governing cosserat equations ( ( 9a)(9b ) in @xcite , ch . 8) . the construction of its general solution is the main theoretical result of this paper . the remaining part of the governing equations in the special cosserat theory consists of two vector equations resulting from newton s laws of motion . for a rod density @xmath43 and cross - section @xmath44 , these equations are given by @xmath45 \partial_t{{\boldsymbol{h}}}(s , t)=\partial_s{{\boldsymbol{m}}}(s , t)+{{\boldsymbol{\nu}}}(s , t)\times { { \boldsymbol{n}}}(s , t)+{{\boldsymbol{l}}}(s , t)\ , , \end{array } \label{nl}\end{aligned}\ ] ] where @xmath46 are the _ contact torques _ , @xmath47 are the _ contact forces _ , @xmath48 are the _ angular momenta _ , and @xmath49 and @xmath50 are the _ external forces _ and _ torque densities_. the contact torques @xmath51 and contact forces @xmath52 corresponding to the _ internal stresses _ , are related to the extension and shear strains @xmath53 as well as to the flexure and torsion strains @xmath54 by the _ constitutive relations _ @xmath55 under certain reasonable assumptions ( cf . @xcite ) on the structure of the right - hand sides of , together with the kinematic relations and , it yields to the governing equations ( cf . @xcite , ch . 8 , ( 9.5a)(9.5d ) ) @xmath56 { { \boldsymbol{\nu}}}_t={{\boldsymbol{v}}}_s+{{\boldsymbol{\kappa}}}\times{{\boldsymbol{v}}}-{{\boldsymbol{\omega}}}\times { { \boldsymbol{\nu}}}\,,\\[0.15 cm ] \rho{{\boldsymbol{j}}}\cdot { { \boldsymbol{\omega}}}_t=\hat{{{\boldsymbol{m}}}}_s+{{\boldsymbol{\kappa}}}\times\hat{{{\boldsymbol{m}}}}+{{\boldsymbol{\nu}}}\times \hat{{{\boldsymbol{n}}}}-{{\boldsymbol{\omega}}}\times(\rho{{\boldsymbol{j}}}\cdot{{\boldsymbol{\omega}}})+{{\boldsymbol{l}}}\,,\\[0.15 cm ] \rho a{{\boldsymbol{v}}}_t={{\boldsymbol{n}}}_s+{{\boldsymbol{\kappa}}}\times \hat{{{\boldsymbol{n}}}}-{{\boldsymbol{\omega}}}\times ( \rho a{{\boldsymbol{v}}})+{{\boldsymbol{f}}}\ , , \end{array } \label{sct}\end{aligned}\ ] ] in which @xmath57 is the inertia tensor of the cross - section per unit length . the dynamical part of contains parameters characterizing the rod under consideration of @xmath58 and the external force and torque densities @xmath59 and @xmath60 , whereas the kinematic part is parameter free . in @xcite , we constructed a general solution to that is the first equation in the pde system . in so doing , we proved that the constructed solution is ( locally ) analytical and provides the structure of the twist vector function @xmath61 and the darboux vector function @xmath26 : @xmath62 where @xmath63 and @xmath64 are arbitrary vector - valued analytical functions , and @xmath65 . it turns out that the functional arbitrariness of @xmath66 and @xmath67 is superfluous , and that with @xmath68 is still a general solution to . this fact is formulated in the following proposition . [ pro:1 ] the vector functions @xmath69 and @xmath70 expressed by @xmath71 with an arbitrary analytical vector function @xmath72 , are a general analytical solution to . let @xmath73 be a fixed point . the right - hand sides of and satisfy for an arbitrary vector function @xmath72 analytical in @xmath73 . it is an obvious consequence of the fact that is a solution to for arbitrary @xmath74 analytical in @xmath75 . also , the equalities and can be transformed into each other with @xmath76 reflecting the invariance of under . the equalities and are linear with respect to the partial derivatives @xmath77 and @xmath78 , and their corresponding jacobians . the determinants of the jacobian matrices @xmath79 and @xmath80 coincide because of the symmetry and read @xmath81 let @xmath82 and @xmath54 be two arbitrary vector functions analytical in @xmath73 . we have to show that there is a vector function @xmath72 analytical in @xmath73 satisfying and . for that , chose real constants @xmath83 such that @xmath84 and set @xmath85 . then and are solvable with respect to the partial derivatives of @xmath77 and @xmath78 in a vicinity of @xmath73 , and we obtain the first - order pde system of the form @xmath86 where the vector function @xmath87 is linear in its first argument and analytical in @xmath67 at @xmath88 . also , the system inherits the symmetry under the swap and is _ passive _ and _ orthonomic _ in the sense of the riquer - janet theory ( cf . @xcite and the references therein ) , since its vector - valued _ passivity ( integrability ) condition _ @xmath89 holds due to symmetry . therefore , by riquier s existence theorems @xcite that generalize the cauchy - kovalevskaya theorem , there is a _ unique _ solution @xmath72 of analytical in @xmath73 and satisfying @xmath90 . in this section , we determine a general analytical form of the vector functions @xmath53 and @xmath92 in describing the linear strain of a cosserat rod and its velocity . these functions satisfy the second kinematic equation of the governing pde system under the condition that the darboux and the twist functions , @xmath54 and @xmath82 , occurring in the last equation , are given by and which contain the arbitrary analytical vector function @xmath72 . similarly , as we carried it out in @xcite for the integration of , we analyze lie symmetries ( cf . @xcite and the references therein ) and consider the _ infinitesimal generator _ @xmath93 of a lie group of point symmetry transformations for . the coefficients @xmath94 with @xmath95 in are functions of the independent and dependent variables . the _ infinitesimal criterion of invariance _ of reads @xmath96 where @xmath97 in addition to those in , the _ prolonged _ infinitesimal symmetry generator @xmath98 contains extra terms caused by the presence of the first - order partial derivatives in and . the invariance conditions lead to an overdetermined system of linear pdes in the coefficients of the infinitesimal generator . this _ determining _ system can be easily computed by any modern computer algebra software ( cf . we make use of the maple package desolv ( cf . @xcite ) which computes the determining system and outputs 138 pdes . since the completion of the determining systems to involution is the most universal algorithmic tool of their analysis ( cf . @xcite ) , we apply the maple package janet ( cf . @xcite ) first and compute a janet involutive basis ( cf . @xcite ) of 263 elements for the determining system , which took about 80 minutes of computation time on standard hardware . then we detected the functional arbitrariness in the general solution of the determining system by means of the differential hilbert polynomial @xmath99 computable by the corresponding routine of the maple package differentialthomas ( cf . it shows that the general solution depends on eight arbitrary functions of @xmath0 . however , in contrast to the determining system for which is quickly and effectively solvable ( cf . @xcite ) by the routine _ pdesolv _ built in the package desolv , the solution found by this routine to the involutive determining system for needs around one hour of computation time and has a form which is unsatisfactory for our purposes , since the solution contains nonlocal ( integral ) dependencies on arbitrary functions . on the other hand , the use of sade ( cf . @xcite ) leads to a satisfying result . unlike desolv , sade uses some heuristics to solve simpler equations first in order to simplify the remaining system . in so doing , sade extends the determining systems with certain integrability conditions for a partial completion to involution . in our case the routine _ liesymmetries _ of sade receives components of the vectors in and outputs the set of nine distinct solutions in just a few seconds . the output solution set includes eight arbitrary functions in @xmath0 which is in agreement with . each solution represents an infinitesimal symmetry generator . among the generators , there are three that include an arbitrary vector function , which we denoted by @xmath100 , with vanishing coefficients @xmath101 , @xmath102 . the sum of these generators is given by @xmath103 it generates a one - parameter lie symmetry group of point transformations ( depending on the arbitrary vector function @xmath104 ) of the vector functions @xmath53 and @xmath92 preserving the equality for fixed @xmath54 and @xmath82 . in accordance to lie s first fundamental theorem ( cf . @xcite ) , the symmetry transformations @xmath105 generated by , are solutions to the following differential ( lie ) equations whose vector form reads @xmath106 the equations can easily be integrated , and without a loss of generality the group parameter can be absorbed into the arbitrary function @xmath107 . this gives the following solution if one takes and into account . ] to : @xmath108 the vector functions @xmath82 , @xmath54 , @xmath53 , and @xmath92 expressed by and with two arbitrary analytical functions @xmath72 and @xmath104 form a general analytical solution to . the fact that and form a general analytical solution to was verified in proposition [ pro:1 ] . we have to show that , given analytical vector functions @xmath53 and @xmath92 satisfying with analytical @xmath82 and @xmath54 satisfying , there exists an analytical vector function @xmath104 satisfying . consider the last equalities as a system of first - order pdes with independent variables @xmath0 and a dependent vector variable @xmath109 . according to the argumentation in the proof of proposition [ pro:1 ] , this leads to the fact , that the equations in are invariant under the transformations @xmath110 this symmetry implies the satisfiability of the integrability condition @xmath111 without any further constraints . therefore , the system is passive ( involutive ) , and by riquier s existence theorem , there is a solution @xmath109 to analytical in a point of analyticity of @xmath112 . to demonstrate the practical use of the analytical solution to the kinematic cosserat equations , we combine it with the numerical solution of its dynamical part . the resulting analytical solutions and for the kinematic part of ( [ sct ] ) contain two parameterization functions @xmath72 and @xmath104 , which can be determined by the numerical integration of the dynamical part of ( [ sct ] ) . the substitution of the resulting analytical solutions and into the latter two ( dynamical ) equations of ( [ sct ] ) , the replacement of the spatial derivatives with central differences , and the replacement of the temporal derivatives according to the numerical scheme of a forward euler integrator , leads to an explicit expression . iterating over this recurrence equation allows for the simulation of the dynamics of a rod . in order to embed this into a scenario close to reality , we consider a flagellated microswimmer . in particular , we simulate the dynamics of a swimming sperm cell , which is of interest in the context of simulations in biology and biophysics . since such a highly viscous fluid scenario takes place in the low reynolds number domain , the advection and pressure parts of the navier - stokes equations ( cf . @xcite ) can be ignored , such that the resulting so - called _ steady stokes equations _ become linear and can be solved analytically . therefore , numerical errors do not significantly influence the fluid simulation part for which reason this scenario is appropriate for evaluating the practicability of the analytical solution to the kinematic cosserat equations . the _ steady stokes equations _ are given by @xmath113 in which @xmath114 denotes the fluid viscosity , @xmath115 the pressure , @xmath116 the velocity , and @xmath117 the force . similar to the fundamental work in @xcite we use a regularization in order to develop a suitable integration of ( [ eq : stokes1])([eq : stokes2 ] ) . for that , we assume@xmath118 in which @xmath119 is a smooth and radially symmetric function with @xmath120 , is spread over a small ball centered at the point @xmath121 . let @xmath122 be the corresponding green s function , i.e. , the solution of @xmath123 and let @xmath124 be the solution of @xmath125 , both in the infinite space bounded for small @xmath126 . smooth approximations of @xmath122 and @xmath124 are given by @xmath127 for @xmath128 and @xmath129 , the solution of the biharmonic equation @xmath130 . the pressure @xmath115 satisfies @xmath131 , which can be shown by applying the divergence operator on ( [ eq : stokes1])([eq : stokes2 ] ) , and is therefore given by @xmath132 . using this , we can rewrite ( [ eq : stokes1 ] ) as @xmath133 with its solution @xmath134 the so - called _ regularized stokeslet_. for multiple forces @xmath135 centered at points @xmath136 , the pressure @xmath115 and the velocity @xmath116 can be computed by superposition . because @xmath122 and @xmath124 are radially symmetric , we can additionally use @xmath137 and obtain , @xmath122 , and @xmath124 only depend on the norm of their arguments , we change the notation according to this . ] @xmath138\,.\nonumber\end{aligned}\ ] ] the flow given by ( [ eq : stokessolution2 ] ) satisfies the incompressibility constraint ( [ eq : stokes2 ] ) . because of @xmath139 the integration of @xmath140 leads to @xmath122 similarly , @xmath141 leads to the expression @xmath142 to determine @xmath124 . we make use of the specific function @xmath143 which is smooth and radially symmetric . up to now , this regularized stokeslet ( [ eq : stokessolution1])([eq : stokessolution2 ] ) allows for the computation of the velocities for given forces . similarly , we can tread the application of a torque by deriving an analogous _ regularized rodlet _ ; see e.g. @xcite . in the inverse case , the velocity expressions can be rewritten in the form of the equations @xmath144 for @xmath145 which can be transformed into an equation system @xmath146 with a @xmath147-matrix @xmath148 . since in general @xmath149 is not regular , an iterative solver have to be applied . a flagellated microswimmer can be set up by a rod representing the centerline of the flagellum ; see @xcite . additionally , a constant torque perpendicular to the flagellum s base is applied to emulate the rotation of the motor . from forces and torque the velocity field is determined . repeating this procedure to update the system state iteratively introduces a temporal domain and allows for the dynamical simulation of flagellated microswimmers ; see figures [ fig : bacteria ] and [ fig : spermcell ] . compared to a purely numerical handling of the two - way coupled fluid - rod system , the step size can be increased by four to five orders of magnitude , which leads to an acceleration of four orders of magnitude . this allows for real - time simulations of flagellated microswimmers on a standard desktop computer . and [ fig : spermcell ] can be carried out in real - time on a machine with an intel(r ) xeon e5 with 3.5 ghz and 32 gb ddr - ram . ] + , the flagellum of a sperm cell does not have its motor at its base as simulated here . instead several motors are distributed along the flagellum ( cf . @xcite ) , for which reason this simulation is not fully biologically accurate , but still illustrates the capabilities of the presented approach.,scaledwidth=100.0% ] we constructed a closed form solution to the kinematic equations of the governing cosserat pde system and proved its generality . the kinematic equations are parameter free whereas the dynamical cosserat pdes contain a number of parameters and parametric functions characterizing the rod under consideration of external forces and torques . the solution we found depends on two arbitrary analytical vector functions and is analytical everywhere except at the values of the independent variables @xmath0 for which the right - hand side of vanishes . therefore , the hardness of the numerical integration of the cosserat system , in particular its stiffness , is substantially reduced by using the exact solution to the kinematic equations . the application of the analytical solution prevents from numerical instabilities and allows for highly accurate and efficient simulations . this was demonstrated for the two - way coupled fluid - rod scenario of flagellated microswimmers , which could efficiently be simulated with an acceleration of four orders of magnitude compared to a purely numerical handling . this clearly shows the usefulness of the constructed analytical solution of the kinematic equations . because of the presence of parameters in the dynamical part of the cosserat pdes , the construction of a general closed form solution to this part is hopeless . even if one specifies all parameters and considers the parametric functions as numerical constants , the exact integration of the dynamical equations is hardly possible . we analyzed lie symmetries of the kinematic equations extended with one of the dynamical vector equations including all specifications of all parameters and without parametric functions . while the determining equations can be generated in a reasonable time , their completion to involution seems to be practically impossible . this work has been partially supported by the max planck center for visual computing and communication funded by the federal ministry of education and research of the federal republic of germany ( fkz-01imc01/fkz-01im10001 ) , the russian foundation for basic research ( 16 - 01 - 00080 ) , and a biox stanford interdisciplinary graduate fellowship . the reviewers valuable comments that improved the manuscript are gratefully acknowledged . y. blinkov , c. cid , v. gerdt , w. plesken , d. robertz : the maple package janet : ii . linear partial differential equations . computer algebra in scientific computing , casc 2003 , v. ganzha , e. mayr , e. vorozhtsov ( eds . ) , 4154 , springer , ( 2003 ) . j. butcher , j. carminati , k.t . vu . : a comparative study of some computer algebra packages which determine the lie point symmetries of differential equations . 155 , 92114 ( 2003 ) . w. hereman : review of symbolic software for lie symmetry analysis . crs handbook of lie group analysis of differential equations , vol . 3 : new trends in theoretical developments and computational methods , chap . 13 , n. h. ibragimov ( ed . ) , 367413 , boca raton , fl , crs press ( 1996 ) . d. michels , d. lyakhov , v. gerdt , g. sobottka , a. weber : lie symmetry analysis for cosserat rods . computer algebra in scientific computing , casc 2014 , v. p. gerdt , w .koepf , w .m . seiler , e. v. vorozhtsov ( eds . ) , 324334 , springer , ( 2014 ) . d. michels , d. lyakhov , v. gerdt , g. sobottka , a. weber : on the partial analytical solution to the kirchhoff equation . computer algebra in scientific computing , casc 2015 , v. p. gerdt , w. koepf , w. m. seiler , e. v. vorozhtsov ( eds . ) , 320331 , springer , ( 2015 ) .

based on a lie symmetry analysis , we construct a closed form solution to the kinematic part of the ( partial differential ) cosserat equations describing the mechanical behavior of elastic rods . the solution depends on two arbitrary analytical vector functions and is analytical everywhere except a certain domain of the independent variables in which one of the arbitrary vector functions satisfies a simple explicitly given algebraic relation . as our main theoretical result , in addition to the construction of the solution , we proof its generality . based on this observation , a hybrid semi - analytical solver for highly viscous two - way coupled fluid - rod problems is developed which allows for the interactive high - fidelity simulations of flagellated microswimmers as a result of a substantial reduction of the numerical stiffness . rods , differential thomas decomposition , flagellated microswimmers , general analytical solution , kinematic equations , lie symmetry analysis , stokes flow , symbolic computation .

a periodic two - dimensional lorentz gas ( sinai billiard ) is a billiard system on the two - dimensional torus with one or more circular regions ( scatterers ) removed . this model in classical mechanics was introduced by lorentz @xcite in 1905 to describe the dynamics of electrons in metals . the associated dynamical system is simple enough to allow a comprehensive study , yet complex enough to exhibit chaos . according to gutzwiller @xcite : `` the original billiard of sinai was designed to imitate , in the most simple - minded manner , a gas of hard spherical balls which bounce around inside a finite enclosure . the formidable technical difficulties of this fundamental problem were boiled down to the shape of a square for the enclosure , and the collisions between the balls were reduced to a single point particle hitting a circular hard wall at the center of the enclosure . '' the model was intensively studied from the point of view of dynamical systems @xcite . our primary goal here is to estimate the _ free - path length _ ( _ first return time _ ) in this periodic two - dimensional model in the small - scatterer limit . we solve the following three open problems : * the existence and computation of the distribution of the free path length , previously considered in @xcite . * the existence and computation of the distribution of the geometric free path length , previously shown , but not fully proved , in @xcite . * the existence and computation of the second ( constant ) term in the asymptotic formula of the ks entropy @xmath2 of the billiard map in this model , previously studied in @xcite . for each @xmath3 let @xmath4 denote by @xmath5 the boundary @xmath6 of @xmath7 , and define the _ free path length _ ( also called _ first exit time _ ) as the borel map given by @xmath8 if @xmath9 is irrational , then @xmath10 for every @xmath11 . we consider the probability space @xmath12 , with @xmath13 and @xmath14 the normalized lebesgue measure on @xmath15 . let @xmath16 denote the characteristic function of @xmath17 . for every @xmath18 the probability that @xmath19 is given by @xmath20 lower and upper bounds for @xmath21 of correct order of magnitude were established by bourgain , golse and wennberg @xcite , using the rational channels introduced by bleher @xcite . more recently , caglioti and golse @xcite have proved the existence of the cesaro @xmath22 and @xmath23 means , proving for large @xmath24 that @xmath25 in sections 2 - 7 below we prove the existence of the limit @xmath26 of @xmath27 as @xmath28 and explicitly compute it . [ t1.1 ] for every @xmath18 and @xmath29 @xmath30 with @xmath31 after a direct computation the above formula for @xmath26 yields @xmath32 and thus for large @xmath24 we find @xmath33 which agrees with . the related homogeneous " problem when the trajectory starts at the origin @xmath34 and the phase space is a subinterval of the velocity range @xmath35 was studied by gologan and the authors . the limit distribution @xmath36 where @xmath37 denotes the lebesgue measure , was shown to exist and explicitly computed in @xcite . unlike @xmath38 , the function @xmath39 is compactly supported on the interval @xmath40 $ ] . interestingly , in the particular situation where the scatterers are vertical segments , this case is related to some old problems in diophantine approximation investigated by erd " os , sz " usz and tur ' an @xcite , friedman and niven @xcite , and by kesten @xcite . the main tools used to prove theorem [ t1.1 ] are a certain three - strip partition of @xmath41 and the weil - sali ' e estimate for kloosterman sums @xcite . the latter is used in infinitesimal form with respect to the parameter @xmath42 to count the number of solutions of equations of form @xmath43 in various regions in @xmath44 . this approach , somehow reminiscent of the circle method , produces good estimates , allowing us to keep under control the error terms . it was developed and used recently in many situations to study problems related to the spacing statistics of farey fractions and lattice points in @xmath44 @xcite . a possible source for getting better estimates for the error terms might come from further cancellations in certain sums of kloosterman sums , of the form @xcite @xmath45 the three - strip partition of @xmath46 is related to the continued fraction decomposition of the slope of the trajectory . following work of blank and krikorian @xcite on the longest orbit of the billiard , caglioti and golse explicitly introduced this partition and used it in conjunction with ergodic properties of the gauss map @xcite to prove . we will use it in section 3 in a suitable setting for our computations . one can also consider the phase space @xmath47 with @xmath48 the inward unit normal at @xmath49 and the probability measure @xmath50 on @xmath51 obtained by normalizing the liouville measure @xmath52 to mass one . consider also the distribution @xmath53 of the _ geometric free path length _ @xmath54 . the first moment ( _ geometric mean free path length _ ) of @xmath55 with respect to @xmath50 can be expressed as @xmath56 equality is a consequence of a more general formula of santal ' o @xcite who extended earlier work of p ' olya on the mean visible distance in a forrest @xcite . the formulation from appears in @xcite . knowledge of the mean free path does not give however any information on other moments or on the limiting distribution of the free path in the small - scatterer limit . our number theoretical analysis leads to the following solution of this limiting distribution problem , proved in sections 8 - 11 below . [ t1.2 ] for every @xmath18 and @xmath29 @xmath57 with @xmath58 , @xmath59 , and respectively @xmath60,title="fig : " ] , @xmath59 , and respectively @xmath60,title="fig : " ] , @xmath59 , and respectively @xmath60,title="fig : " ] we note the equalities @xmath61 and @xmath62 the latter also yields @xmath63 remarkably , formulas and were found by dahlqvist @xcite . that approach however does not provide a rigorous proof for the existence of the limit distribution , because it fails to control in a quantitative way the uniform distribution of his variable @xmath64 ( see the comments after formulas ( 75 ) and ( 86 ) in @xcite ) . in the final section we use some standard analysis arguments and properties of the dilogarithm and trilogarithm to estimate @xmath65 it was conjectured by friedman , kubo and oono @xcite that @xmath66 is convergent as @xmath28 . its hypothetical limit @xmath67 was estimated to be @xmath68 in @xcite and @xmath69 in @xcite . this conjecture was known to imply @xcite the asymptotic formula @xmath70 for the ks entropy of the associated billiard map . in @xcite chernov proved that @xmath66 remains bounded when @xmath28 , without giving however any estimate for the bounds . the constant @xmath67 was identified by dahlqvist ( * ? ? ? * formula ( 73 ) ) as being @xmath71 . the conjecture of friedman , kubo and oono , in the more precise form provided by dahlqvist , follows now from theorem [ t1.2 ] . [ t1.3 ] in the small scatterer limit @xmath28 the following holds : * @xmath72 * @xmath73 these methods work for any convex scatterer due to the good error control they give when integrating over the velocity in very short intervals . to keep the presentation of the paper neat we have chosen to only consider circular scatterers . in dimension @xmath74 the problem of the existence of the limiting distribution of the free - path length in the small - scatterer limit remains open and is manifestly difficult . partial results in this direction have appeared in @xcite . in this section we collect some basic properties of farey fractions and outline the summation method that will allow us to estimate the limit distribution of the free path length when the size of scatterers tends to zero . for each positive integer @xmath75 , let @xmath76 denote the set of farey fractions of order @xmath75 . these are the rational numbers @xmath77 with coprime integers @xmath78 such that @xmath79 . for each interval @xmath80 $ ] the number of elements in the set @xmath81 can be expressed , using elementary arguments on m " obius and euler - maclaurin summation , as @xmath82 if @xmath83 are two consecutive elements in @xmath76 , then @xmath84 this shows on the one hand that the denominators of consecutive farey fractions of order @xmath75 are exactly the primitive integer points in the set @xmath85 and on the other hand that denominators uniquely determine consecutive farey fractions . for instance , @xmath86 is the unique integer in @xmath87 $ ] for which @xmath88 . in many instances in this paper we will seek to estimate sums of type @xmath89 where @xmath80 $ ] is an interval , @xmath90 a region , and @xmath91 a @xmath92 function . these kinds of sums can be roughly approximated by some integrals , with control on error terms given by the following two results which will be systematically used in this process . the first one is a standard fact and is a plain consequence of the m " obius summation ( for a proof see ( * ? ? ? * lemma 2.3 ) ) . [ lemma1 ] let @xmath93 and @xmath91 be a @xmath92 function on @xmath94 $ ] . then @xmath95 where @xmath96 denotes euler s totient function . the second one is a consequence of weil s type bounds for kloosterman sums ( cf . * lemma 2.2 ) ) . [ lemma2 ] let @xmath97 be an integer , @xmath98 and @xmath99 intervals with @xmath100 , @xmath91 a @xmath92 function on @xmath101 , and @xmath102 an integer . then for all @xmath29 @xmath103 with @xmath104 where we denote @xmath105 and @xmath106 . when @xmath107 is a subset of @xmath108 , the above mentioned properties of farey fractions lead to @xmath109 where we denote @xmath110 \subseteq ( 1-x,1],\qquad x\in ( \alpha,\beta].\ ] ] the inner sum above is mastered by lemma [ lemma2 ] , being approximated by @xmath111 thus @xmath112 where we take @xmath113 when the error term is small enough , this sum is mastered by lemma [ lemma1 ] , giving @xmath114 in this section we give an account on the three - strip partition mentioned in the introduction . this approach , slightly different from that in @xcite , is suitable for computations involving farey fraction partitions of the unit interval . in the first part of this section we shall consider a fixed ( small ) @xmath115 and let @xmath116 $ ] be the integer part of @xmath117 for each @xmath118 , consider the points @xmath119 and @xmath120 . consider also the points @xmath121 and @xmath122 , and denote by @xmath123 the strip determined by the lines @xmath124 and @xmath125 . a segment does not interfere with an open strip when their intersection is empty . throughout this section @xmath126 will be two consecutive fractions in @xmath76 , so that is fulfilled . in particular this gives @xmath127 the slope of a segment @xmath128 is denoted by @xmath129 . set @xmath130 . 0.4 mm ( 200,70)(15,0 ) ( 0,0)(0,8)(165,65)(165,57)(0,0)(220,27)(220,35)(0,8 ) ( 0,-5)(0,0)@xmath122 ( -5,15)(0,0)@xmath121 ( 185,70)(0,0)@xmath131 ( 185,52)(0,0)@xmath132 ( 235,40)(0,0)@xmath133 ( 235,22)(0,0)@xmath134 [ lemma3 ] the segment @xmath135 does not interfere with the strip @xmath136 , and the segment @xmath137 does not interfere with the strip @xmath123 . first , we show that @xmath138 lies above the line @xmath139 of equation @xmath140 which amounts to @xmath141 the latter is equivalent to @xmath142 , which is true by . furthermore , @xmath143 lies below the line @xmath144 of equation @xmath145 as a result of @xmath146 being equivalent to @xmath147 for each @xmath148 set @xmath149 the following three relations hold for every @xmath150 @xmath151 @xmath152 @xmath153 as a result of and , it is seen that @xmath154 and that @xmath155 so putting @xmath156,\quad i_{\gamma , k}=(t_k , t_{k-1 } ] , \quad i_{\gamma ,- k}= ( u_{k-1},u_k],\qquad k\in { { \mathbb{n}}},\ ] ] we end up with a partition @xmath157 of the interval @xmath158 . next we consider the points @xmath159 and @xmath160 , proving [ lemma4 ] the following inequalities hold for every @xmath161 _ : _ _ ( i ) _ @xmath162 . _ ( ii ) _ @xmath163 . the inequalities in ( i ) are equivalent to @xmath164 which follow from , , , and from . the inequalities in ( ii ) are equivalent to @xmath165 which follow from , , , and @xmath166 . consider the half - infinite strip @xmath167 of direction @xmath42 , top line passing through @xmath168 , and bottom line passing through @xmath169 . assume that @xmath170 . for each @xmath171 $ ] we wish to find the first vertical segment of form @xmath172 $ ] , @xmath173 , that intersects the line of slope @xmath9 passing through @xmath174 . in other words , we wish to calculate @xmath175 where we denote @xmath176 , @xmath177 . we shall assume that @xmath9 is irrational and split the discussion according to the three cases where the slope of @xmath42 belongs to one of the intervals @xmath178 $ ] , @xmath179 $ ] or @xmath180 $ ] . [ prop1 ] let @xmath126 be consecutive fractions in @xmath76 . suppose @xmath181 $ ] is irrational and @xmath182 $ ] for some @xmath183 . set @xmath184=({\varepsilon}-w_{b_k},{\varepsilon}],\\ & i_{c_k } : = [ -{\varepsilon}+w_{a_+},-{\varepsilon}+w_{a_+}+w_{c_k } ] , \end{split}\ ] ] and @xmath185 then for any @xmath186 we have @xmath187 and @xmath188 furthermore , if @xmath189 denotes the parallelogram of height @xmath190 , angle @xmath42 between its side and the horizontal direction , and side length @xmath191 , then @xmath192 moreover , @xmath193 mod @xmath194 provides a partition of the unit square @xmath41 _ ( _ we allow the boundaries of these three sets to intersect_)_. 0.35 mm ( 200,115)(50,0 ) ( 0,0)(85,30)(85,35)(300,110.882)(300,113.382)(215,83.382)(215,85.8823)(0,10)(0,0 ) ( 0,0)(280,0 ) ( 0,0)(85,25)(85,35)(0,10)(0,0 ) ( 0,0)(85,30 ) ( 85,35)(0,10 ) ( 215,83.3823)(215,93.3823)(300,118.382)(300,108.382)(215,83.3823 ) ( 215,85.882)(300,115.8823 ) ( 130,57.8823)(130,67.8823 ) ( 0,7.5)(300,113.3823 ) ( 0,5)(300,110.88236 ) ( -5,-6)(0,0)@xmath122 ( -8,16)(0,0)@xmath121 ( -5,1)(0,0)@xmath195 ( -5,5)(0,0)@xmath196 ( -5,9.5)(0,0)@xmath197 ( 0,0)(0,0)@xmath198 ( 0,10)(0,0)@xmath198 ( 85,25)(0,0)@xmath198 ( 85,35)(0,0)@xmath198 ( 300,108.382)(0,0)@xmath198 ( 300,118.382)(0,0)@xmath198 ( 215,83.3823)(0,0)@xmath198 ( 215,93.3823)(0,0)@xmath198 ( 130,58.05)(0,0)@xmath198 ( 130,67.3823)(0,0)@xmath198 ( 117,58.05)(0,0)@xmath199 ( 117,67.3823)(0,0)@xmath200 ( 108,32)(0,0)@xmath201 ( 108,22)(0,0)@xmath202 ( 302,101)(0,0)@xmath203 ( 302,124)(0,0)@xmath204 ( 215,100)(0,0)@xmath205 ( 220,76)(0,0)@xmath206 ( 25,3.5)(0,0)@xmath42 ( 0,0 ) taking stock on lemma [ lemma4 ] , we notice that the line of slope @xmath9 through @xmath169 intersects the vertical line @xmath135 at a point between @xmath143 and @xmath207 ( see figure [ figure3 ] ) . also , because @xmath208 , the line of slope @xmath209 through @xmath168 ( respectively through @xmath143 ) intersects the line @xmath210 ( respectively @xmath211 ) between @xmath205 and @xmath206 ( respectively between @xmath204 and @xmath203 ) . the segment @xmath212 is placed above these two parallel lines because @xmath213 . next , we find that the intersections with the vertical axis of the lines @xmath214 and @xmath215 which have slope @xmath9 and pass through @xmath143 and respectively @xmath206 , are @xmath216 and respectively @xmath217 , whence the required values of @xmath218 , @xmath219 and @xmath220 follow . notice that @xmath221 besides one clearly has @xmath222 and it is easy to check by a direct calculation that @xmath223 it remains to check that the interiors of the subsets @xmath224 , @xmath225 , are disjoint . if not , there exist two points @xmath226 inside @xmath227 such that @xmath228 . the latter is preserved by translating the segment @xmath229 to a parallel segment . owing to the shape of @xmath227 we may thus assume that , say , @xmath230 lies on the @xmath231-axis ; hence @xmath232 and @xmath233 for some @xmath171 $ ] , @xmath234 . the line of slope @xmath9 which passes through @xmath235 intersects the @xmath231-axis at @xmath236 . hence @xmath237 , which shows that @xmath238 . by the first part of the proposition this gives @xmath239 , thus @xmath235 must belong to the boundary , which is a contradiction . [ prop2 ] let @xmath126 be consecutive fractions in @xmath76 . suppose @xmath240 is irrational . _ ( i ) _ if @xmath241 $ ] , then the analog of proposition [ prop1 ] holds true , with or @xmath242 we get @xmath243 . ] @xmath244=({\varepsilon}-w_{b_0},{\varepsilon}],\\ & i_{c_0}:= [ -{\varepsilon}+w_{a_0},-{\varepsilon}+w_{a_0}+w_{c_0}],\\ & l(\omega , y_0)=\begin{cases } l_{a_0 } ( \omega):=q & \mbox{if $ y_0 \in i_{a_0};$ } \\ l_{c_0}(\omega):=q^\prime + q & \mbox{if $ y_0 \in i_{c_0};$ } \\ l_{b_0}(\omega):=q^\prime & \mbox{if $ y_0 \in i_{b_0}$. } \end{cases } \end{split}\ ] ] _ ( ii ) _ if @xmath183 and @xmath245 $ ] , then the analog of proposition [ prop1 ] holds true , with @xmath246,\\ & i_{b_- } = ( -{\varepsilon}+w_{a_{-k}}+w_{c_{-k}},{\varepsilon}]=({\varepsilon}-w_{b_-},{\varepsilon}],\\ & l(\omega , y_0)= \begin{cases } l_{a_{-k } } ( \omega):=q_k^\prime & \mbox{\sl if $ y_0 \in i_{a_{-k}};$ } \\ l_{c_{-k } } ( \omega):=q^\prime_{k+1 } & \mbox{\sl if $ y_0 \in i_{c_{-k}};$ } \\ l_{b_- } ( \omega):=q^\prime & \mbox{\sl if $ y_0 \in i_{b_- } .$ } \end{cases } \end{split}\ ] ] \(i ) follows as in the proof of proposition [ prop1 ] , using @xmath247 0.4 mm ( 200,60)(10,0 ) ( 0,0)(85,22.5)(85,25)(225,62.05882)(225,67.058823)(110,36.61764)(110,39.11764)(0,10)(0,0 ) ( 0,10)(0,0)(200,0 ) ( 0,2.5)(225,62.058823 ) ( 0,7.5)(225,67.058823 ) ( 85,15)(85,25 ) ( 225,59.5588)(225,69.5588 ) ( 110,36.6176)(110,46.6176 ) ( -5,-6)(0,0)@xmath122 ( -8,16)(0,0)@xmath121 ( -5,1)(0,0)@xmath195 ( -5,5)(0,0)@xmath196 ( -5,9.5)(0,0)@xmath197 ( 0,0)(0,0)@xmath198 ( 0,10)(0,0)@xmath198 ( 85,25)(0,0)@xmath198 ( 85,15)(0,0)@xmath198 ( 225,59.5588)(0,0)@xmath198 ( 225,69.5588)(0,0)@xmath198 ( 110,36.6176)(0,0)@xmath198 ( 110,46.6176)(0,0)@xmath198 ( 91,22)(0,0)@xmath143 ( 91,12)(0,0)@xmath207 ( 103,48)(0,0)@xmath248 ( 116,42)(0,0)@xmath249 ( 225,74)(0,0)@xmath250 ( 225,54)(0,0)@xmath251 ( 0,0 ) ( 25,3.5)(0,0)@xmath42 0.4 mm ( 200,60)(10,0 ) ( 0,0)(170,45)(170,47.5)(225,62.05882)(225,67.058823)(110,36.61764)(110,39.11764)(0,10)(0,0 ) ( 0,10)(0,0)(200,0 ) ( 0,2.5)(225,62.058823 ) ( 0,7.5)(225,67.058823 ) ( 170,47.5)(170,37.5 ) ( 225,59.5588)(225,69.5588 ) ( 110,36.6176)(110,46.6176 ) ( 140,34.058823)(140,24.058823 ) ( -5,-6)(0,0)@xmath122 ( -8,16)(0,0)@xmath121 ( -5,1)(0,0)@xmath195 ( -5,5)(0,0)@xmath196 ( -5,9.5)(0,0)@xmath197 ( 0,0)(0,0)@xmath198 ( 0,10)(0,0)@xmath198 ( 170,47.5)(0,0)@xmath198 ( 170,37.5)(0,0)@xmath198 ( 225,59.5588)(0,0)@xmath198 ( 225,69.5588)(0,0)@xmath198 ( 110,36.6176)(0,0)@xmath198 ( 110,46.6176)(0,0)@xmath198 ( 140,34.058823)(0,0)@xmath198 ( 140,24.058823)(0,0)@xmath198 ( 152,32)(0,0)@xmath252 ( 152,22)(0,0)@xmath253 ( 178,43)(0,0)@xmath254 ( 178,33)(0,0)@xmath255 ( 103,48)(0,0)@xmath248 ( 116,42)(0,0)@xmath249 ( 232,72)(0,0)@xmath256 ( 232,52)(0,0)@xmath257 ( 0,0 ) ( 25,3.5)(0,0)@xmath42 \(ii ) follows as in the proof of proposition [ prop1 ] using @xmath258 we now start investigating the case where the scatterers are vertical slits . propositions [ prop1 ] and [ prop2 ] will only be applied for @xmath259 , corresponding to the case of vertical slits of height @xmath260 . the lebesgue measure of a borel set @xmath261 in @xmath262 , @xmath263 , will be denoted by @xmath264 . throughout the paper @xmath265 will denote the free path length in the periodic two - dimensional lorentz gas with vertical slits of height @xmath266 as scatterers centered at all integer lattice points . given @xmath267 , @xmath268\subseteq [ 0,1]$ ] with @xmath269 , and @xmath270 integer , we denote @xmath271 although the cases @xmath272 , @xmath273 , @xmath274 , will be considered separately , applying propositions [ prop1 ] and [ prop2 ] to @xmath275 , can write for all @xmath276 @xmath277 [ lemma5 ] for any interval @xmath268 \subseteq [ 0,1]$ ] such that @xmath278 with fixed @xmath279 and small @xmath115 , and any _ ( _ large _ ) _ integer @xmath280 , the estimate @xmath281 holds uniformly in @xmath24 on compact subsets of @xmath282 . here @xmath283 is obtained by substituting @xmath284 in place of @xmath285 in , that is @xmath286 with @xmath287 using the inequality @xmath288 , which is a consequence of @xmath289 and of the similar relations for @xmath290 and @xmath291 , the estimate ( see also ) @xmath292 and the inequalities @xmath293 and @xmath294 it follows that we can replace @xmath295 by @xmath284 in at a cost which is @xmath296 equality will be at the center of most of the forthcoming computations because it shows how the estimation of distribution of the free path length reduces to estimates on sums involving farey fractions . there is an alternative approach to estimating @xmath297 , by using a monotonicity argument instead of the continuity argument which based on . such an argument will be used in the proof of theorem [ t1.2 ] . in the remainder of the paper given @xmath298\subseteq [ 0,1]$ ] we denote @xmath299 the aim of this section is to prove the following result [ prop3 ] suppose @xmath301 is a subinterval of @xmath40 $ ] of size @xmath302 for some @xmath279 . then for every @xmath303 with @xmath304 and @xmath305 @xmath306 with @xmath307 the estimate is uniform in @xmath308 $ ] . before starting to estimate @xmath309 , the following remark is in order . _ if @xmath80 $ ] is an interval with @xmath310 , then as a consequence of @xmath311 we have @xmath312 as a result , replacing the condition @xmath313 by @xmath314 only produces an error of order @xmath315 , which has no impact in any of the forthcoming estimates . thus in propositions [ prop3 ] , [ prop4 ] , [ prop5 ] , [ p8.2 ] , [ p9.1 ] , [ p10.1 ] the assumption @xmath316 can be replaced by the weaker assumption @xmath317 and @xmath318 . _ then we notice that since @xmath319 for all @xmath161 , we can write , according to and , @xmath320 with @xmath321 here the formulas for the width of the strips are as in , and we take @xmath322 taking into account the equalities @xmath323 @xmath324 and @xmath325 we can write @xmath326 with @xmath327 rewriting the terms in a convenient way we arrive at @xmath328 where @xmath329 remark first that @xmath330 , as a result of @xmath331 the next elementary statement will be repeatedly used . [ lemma6 ] for any @xmath332 we have , uniformly in @xmath333 $ ] as @xmath334 , @xmath335 @xmath336 applying to our situation taylor s formula @xmath337 together with @xmath338 we get @xmath339 whence follows for @xmath340 . the case @xmath341 is a direct consequence of , while is derived from by changing @xmath342 into @xmath343 . this result will only be applied in cases where @xmath344 . we shall also use the following weaker form of : @xmath345 it remains to estimate @xmath346 , @xmath347 and @xmath348 . by it is immediate that @xmath349 this shows in conjunction with @xmath350 and with the subsequent lemma [ l4.4 ] that @xmath351 [ l4.4 ] let @xmath352 such that @xmath304 . then for any interval @xmath80 $ ] with @xmath353 and @xmath354 @xmath355 we decompose the sum above as @xmath356 , according to whether @xmath357 or @xmath358 . thus we can write @xmath359 \\ a\in { { \mathcal{j}}}:=qi \\ -aq^\prime = 1 \hspace{-6pt } \pmod{q } } } \hspace{-20 pt } f_q ( q^\prime , a),\ ] ] where we put @xmath360.\ ] ] the inclusion @xmath361 $ ] gives @xmath362 applying lemma [ lemma2 ] with @xmath363 $ ] , the inner sum in can be expressed as @xmath364 where @xmath365.\ ] ] the function @xmath366;$}\\ 1 & \mbox{\rm if $ x=0 $ , } \end{cases}\ ] ] is bounded and has finite total variation on @xmath40 $ ] , hence @xmath367 } \vert w(x)\vert+\int_0 ^ 1 \vert w^\prime ( x)\vert \ , dx\ = o(1).\ ] ] since @xmath368 , lemma [ lemma1 ] yields @xmath369 } \vert v(q)\vert + \int_0^q \vert v^\prime ( q)\vert\ , dq\bigg ) \right ) \\ & = \frac{1}{\zeta(2)}\int_0 ^ 1 w(x)\ , dx+o(q^{-1}\ln q ) . \end{split}\ ] ] hence @xmath370 using a familiar identity of euler ( cf . formula ( 1.8 ) in @xcite ) we find that @xmath371 which we combine with to get @xmath372 finally we employ @xmath373 and to write @xmath374 using and we see that @xmath375 \\ a^\prime \in q^\prime i \\ a^\prime q=1\hspace{-6pt}\pmod{q^\prime } } } \frac{1}{qq^\prime ( 1+a^{\prime 2}/q^{\prime 2 } ) } .\ ] ] changing @xmath376 to @xmath377 , reversing the roles of @xmath378 and @xmath379 , and using @xmath380 it follows that @xmath381 is given by the same expression as in . next we estimate @xmath347 and find , taking @xmath382 , @xmath383 , @xmath384 , @xmath385 in , that @xmath386 since @xmath387 we infer from and the definition of @xmath347 that @xmath388 applying lemma [ lemma2 ] to @xmath389 $ ] , @xmath390 , @xmath391 for which @xmath392 and @xmath393 , and taking @xmath363 $ ] , the inner sum above becomes @xmath394 which inserted back into gives that @xmath395 may be written as @xmath396 applying now lemma [ lemma1 ] to the main term above with @xmath397 , @xmath398 $ ] , we find that @xmath399 in a similar way we find @xmath400 and therefore @xmath401 which proves proposition [ prop3 ] . in this section we shall evaluate the contribution of the integrals on @xmath402 $ ] in when @xmath403 and @xmath274 . in this situation there is a unique nonnegative integer , given by @xmath404 \geq 0,\ ] ] for which @xmath405 when @xmath406 it follows that @xmath407 , and we prove [ prop4 ] suppose @xmath301 is a subinterval of @xmath40 $ ] of size @xmath302 for some @xmath408 . then for every @xmath303 with @xmath304 and @xmath29 @xmath409 with @xmath410 as in theorem _ [ t1.1 ] _ and @xmath411 as in proposition _ [ prop3]_. the estimate is uniform in @xmath24 on compacts of @xmath412 . next , @xmath413 and @xmath414 will be as in and the widths @xmath415 as in . since @xmath274 , then @xmath416 and the second sum in is zero . in the beginning we fix @xmath417 and estimate @xmath418 using and , taking @xmath419 , @xmath420 , @xmath421 , @xmath422 in , and also owing to @xmath423 we infer that @xmath424 on the other hand , taking @xmath425 , @xmath426 , @xmath427 , @xmath428 in , and also using @xmath429 we estimate @xmath430 using also @xmath431 , this gives whenever @xmath274 ( so @xmath407 ) @xmath432 since @xmath274 , the sum of integrals on @xmath402 $ ] in becomes @xmath433 making use of @xmath434 and of @xmath435 we find @xmath436 next for each integer @xmath161 consider the sets @xmath437 = k\right\ } \quad \mbox{\rm and } \quad i_k=\bigg [ \frac{t-1}{k},\frac{t-1}{k-1}\bigg ) \cap [ 0,1),\ ] ] and for @xmath438 and @xmath161 , respectively @xmath439 , the intervals ( see figure [ figure6 ] ) @xmath440=\left\ { \frac{q^\prime}{q } \ , ; \ , \bigg ( \frac{q}{q},\frac{q^\prime}{q } \bigg ) \in \omega_{k-1 } \cap { { \mathcal{t}}}\right\ } \subseteq \bigg ( 1-\frac{q}{q},1\bigg ] , \\ & j_{k , q}^{(1 ) } = \bigg ( 1-\frac{q}{q},t-\frac{kq}{q } \bigg ] = \left\ { \frac{q^\prime}{q}\ , ; \ , \bigg ( \frac{q}{q},\frac{q^\prime}{q } \bigg ) \in \omega_k \cap { { \mathcal{t}}}\right\ } \subseteq \bigg ( 1-\frac{q}{q},1\bigg ] . \end{split}\ ] ] note that @xmath441 , that @xmath442 - 1\geq 1 $ ] , and that @xmath443 unless @xmath444\geq 2 $ ] . 0.5 mm ( 100,170)(0,0 ) ( 22.2222,100)(33.3333,100)(33.3333,66.6666)(22.2222,100 ) ( 33.3333,100)(33.3333,66.6666)(66.6666,33.3333)(33.3333,100 ) ( 0,100)(100,100)(100,0)(0,100 ) ( 22.2222,100)(33.3333,66.6666 ) ( 33.3333,100)(66.6666,33.3333 ) ( 45,55)(45,100 ) ( 66.6666,100)(66.6666,33.3333 ) ( 45,55)(45,0 ) ( 0,100)(0,166.6666)(22.2222,100 ) ( 0,166.6666)(33.3333,100 ) ( 45,77)(100,77 ) ( 45,55)(100,55 ) ( 0,166.6666)(0,0)@xmath198 ( 0,100)(0,0)@xmath198 ( 22.2222,100)(0,0)@xmath198 ( 33.3333,100)(0,0)@xmath198 ( 66.6666,100)(0,0)@xmath198 ( 100,100)(0,0)@xmath198 ( 33.3333,66.6666)(0,0)@xmath198 ( 66.6666,33.3333)(0,0)@xmath198 ( 100,0)(0,0)@xmath198 ( 45,0)(0,0)@xmath198 ( -9,100)(0,0)@xmath445 ( 9,166.66)(0,0)@xmath446 ( 109,100)(0,0)@xmath447 ( 109,0)(0,0)@xmath448 ( 55,0)(0,0)@xmath449 ( 18,60)(0,0)@xmath450 ( 66,29)(0,0)@xmath451 ( 28,145)(0,0)@xmath452 ( 18,120)(0,0)@xmath453 ( 66,106)(0,0)@xmath454 ( 38,106)(0,0)@xmath455 ( 16,106)(0,0)@xmath456 ( 109,87)(0,0)@xmath457 ( 109,65)(0,0)@xmath458 we also consider the function @xmath459 \ni ( q , q^\prime , a ) \mapsto f_k(q , q^\prime , a ) & = \frac{(1-q / q)^2 ( 2q_k - tq)}{2qq_k q_{k-1}(1+\gamma^2 ) } \\ & = \frac{(1-q / q)^2 ( 2q^\prime+2kq - tq)}{2q(q^\prime + kq ) ( q^\prime+(k-1)q ) ( 1+a^2/q^2 ) } . \end{split}\ ] ] using the one - to - one correspondence between the primitive integer points in @xmath460 and the set of consecutive farey fractions @xmath77 and @xmath461 in @xmath76 with @xmath462=k$ ] , we derive using the summation method described in section 2 that @xmath463 with @xmath464 we aim to estimate @xmath465 and @xmath466 applying lemma [ lemma2 ] to the intervals @xmath467 , @xmath390 and the function @xmath468 , and respectively to @xmath469 , @xmath390 and @xmath470 . for @xmath471 we have @xmath472 , or equivalently @xmath473 as a result , we see that ( here @xmath439 ) @xmath474 } \frac{1}{q(q+q^\prime ) } < \frac{1}{qq } , \\ \| f_{k-1}(q,\cdot,\cdot)\|_{\infty } & \leq \sup\limits_{q^\prime \in qj_{k , q}^{(0 ) } } \frac{q_{k-1}}{qq_{k-1}q_{k-2 } } \leq \sup\limits_{q^\prime > ( t-2)q } \frac{1}{qq^\prime } \leq \frac{1}{(t-2)qq } \ll_t \frac{1}{qq } . \end{split}\ ] ] the last estimate holds without the factor @xmath475 whenever @xmath476 $ ] . in the remainder of this section we will simply write @xmath477 with the understanding that this holds uniformly in @xmath24 on compacts of @xmath412 . we also need to estimate the @xmath478-norm of @xmath479 . it is easily seen that @xmath480 } \frac{1}{q(q+q^\prime)^2 } < \frac{1}{qq^2 } \leq \frac{1}{q^2 q } , \end{split}\ ] ] and similarly @xmath481 } \frac{1}{q^{\prime 2 } } \leq \frac{1}{(t-2)^2 qq^2 } \ll_t \frac{1}{q q^2 } \leq \frac{1}{q^2 q } .\ ] ] applying now lemma [ lemma2 ] to this situation with @xmath482 $ ] , where @xmath483 is to be determined later , we approximate @xmath484 within error @xmath485 by @xmath486 where @xmath487 is as in and @xmath488 by a direct computation we find that @xmath489 is independent of @xmath490 . since the error terms sum up to @xmath491 we arrive at @xmath492 with @xmath493.\ ] ] for @xmath274 consider the function @xmath494.\ ] ] using the taylor series of the logarithm we obtain for small @xmath495 @xmath496 which shows that @xmath91 extends to a @xmath92 function on @xmath40 $ ] , and so @xmath497 uniformly for @xmath24 in compacts of @xmath412 . the equality @xmath498 , @xmath499 $ ] , implies now that both @xmath500 and the total variation of @xmath501 on @xmath502 $ ] are @xmath503 . thus we may apply lemma [ lemma1 ] to and conclude , also using @xmath304 , that @xmath504 one can see in a similar way that the contribution of integrals on the intervals @xmath505 $ ] in for @xmath161 and @xmath274 is @xmath506 proposition [ prop4 ] now follows from and . in this section we prove [ prop5 ] suppose @xmath301 is a subinterval of @xmath40 $ ] of size @xmath302 for some @xmath279 . then for any @xmath508 with @xmath304 and @xmath29 @xmath509 with @xmath510 and @xmath410 as in theorem _ [ t1.1]_. the estimate holds uniformly in @xmath24 on compacts of @xmath511 . in this case gives @xmath512 we break the main term above according as to whether @xmath513 or @xmath514 . thus we first estimate @xmath515 using and we may also write @xmath516 with @xmath517 0.5 mm ( 100,100)(0,0 ) ( 66.6666,100)(100,66.6666)(100,0)(66.6666,33.3333)(66.6666,100 ) ( 0,100)(66.6666,33.3333)(66.6666,100)(0,100 ) ( 0,100)(100,100)(100,0)(0,100 ) ( 22.2222,100)(33.333,66.6666 ) ( 33.333,100)(66.666,33.3333 ) ( 66.666,100)(100,66.6666 ) ( 33.3333,100)(33.3333,66.6666 ) ( 0,100)(0,0)@xmath198 ( 22.2222,100)(0,0)@xmath198 ( 33.3333,100)(0,0)@xmath198 ( 66.6666,100)(0,0)@xmath198 ( 100,100)(0,0)@xmath198 ( 33.3333,66.6666)(0,0)@xmath198 ( 66.6666,33.3333)(0,0)@xmath198 ( 66.6666,100)(0,0)@xmath198 ( 100,0)(0,0)@xmath198 ( 100,66.6666)(0,0)@xmath198 ( -9,100)(0,0)@xmath445 ( 109,100)(0,0)@xmath447 ( 109,0)(0,0)@xmath448 ( 113,66.6666)(0,0)@xmath518 ( 66,106)(0,0)@xmath519 ( 53,29)(0,0)@xmath520 ( 22,61)(0,0)@xmath521 ( 40,106)(0,0)@xmath522 ( 16,106)(0,0)@xmath523 we proceed to estimate @xmath524 , @xmath525 , @xmath526 and @xmath527 by noticing that yields @xmath528 next @xmath525 is estimated in a similar way as @xmath347 was in section 4 , only with the difference that the summation over @xmath529 is being done under the additional requirement @xmath530 . this is not going to produce any change in the error , and will only affect the main terms . as in and we obtain @xmath531 then , as in the proof of lemma [ l4.4 ] , we find that @xmath532\\ a\in { { \mathcal{j}}}:=qi \\ -aq^\prime = 1\hspace{-6pt}\pmod{q } } } \hspace{-15pt } f_q(q^\prime , a ) = c_i(1-t)\hspace{-13pt } \sum\limits_{(t-1)q < q\leq q}\hspace{-6pt } \frac{\varphi(q)}{q } v(q)+o_\delta ( e_{c , c_1,\delta}(q)),\ ] ] where this time we take @xmath533.\ ] ] but @xmath534 and the function @xmath535,\ ] ] is @xmath92 on @xmath536 $ ] . hence both the @xmath537-norm and the total variation of @xmath501 on the interval @xmath538 $ ] are @xmath539 , uniformly in @xmath24 on compacts of @xmath511 . lemma [ lemma1 ] applies now and yields @xmath540 proceeding as in section 4 ( see ) we find @xmath541 this immediately gives @xmath542 in a similar way we find @xmath543 from we now collect @xmath544 it remains to estimate the contribution of farey fractions of order @xmath75 with @xmath514 to @xmath283 , which is @xmath545 where @xmath546 denotes @xmath547 and @xmath548 denotes @xmath549 in this case one also has @xmath550 \geq 1,\ ] ] and as in section 5 we find @xmath551 and thus @xmath552 in a similar way we find that @xmath548 can too be expressed as in , and thus @xmath553 proposition [ prop5 ] follows now from and . we may assume without loss of generality that @xmath554 $ ] , thus estimate for small @xmath115 the quantity @xmath555 we partition the interval @xmath40 $ ] as a union of @xmath556 intervals @xmath557 $ ] of equal size , with @xmath558 $ ] , thus with @xmath559 , where @xmath279 is to be chosen later . for each @xmath560 we set @xmath561 , \quad q_j^+=\left [ \frac{\cos \omega_j}{2{\varepsilon}-2{\varepsilon}^{c+1}}\right]+1.\ ] ] since @xmath562 $ ] , we have @xmath563 , and thus @xmath564 . moreover , for @xmath565 $ ] we have @xmath566 from the definition of @xmath567 and from @xmath568,\ ] ] we infer that @xmath569 and @xmath570 _ if @xmath571 $ ] and @xmath572 are such that @xmath573 , then for all @xmath574 we have @xmath575 this shows in turn that if for each interval @xmath576 \subseteq [ 0,1]$ ] we denote @xmath577 then for any integers @xmath578 such that @xmath579 we have @xmath580 _ by the previous remark we infer @xmath581 for small @xmath115 we also have @xmath582 uniformly in @xmath24 on compacts of @xmath282 . thus , , and lemma [ lemma5 ] yield @xmath583 and @xmath584 by the definition of @xmath38 we see that for any compact interval @xmath585 , there exists a constant @xmath586 such that @xmath587 now by propositions [ prop3 ] , [ prop4 ] , [ prop5 ] we know that for any @xmath588 we have for small @xmath115 @xmath589 uniformly in @xmath24 on compacts of @xmath590 . here @xmath26 is defined as in theorem 1.1 . summing over @xmath560 the inequalities and , and using also @xmath591 @xmath592 , and , we gather @xmath593 for obvious symmetry reasons we can only consider @xmath594 $ ] . thus , after normalizing the lebesgue measure @xmath14 on @xmath15 by dividing by @xmath595 , we get @xmath596 the proof of theorem [ t1.1 ] is completed by taking @xmath597 . identifying @xmath51 with @xmath671 \right\ } , \ ] ] the ( non - normalized ) liouville measure on the phase space @xmath51 is expressed as @xmath672 0.5 mm ( 0,130)(20,-60 ) ( 0,0 ) ( -60,0)(80,0 ) ( 0,-60)(0,60 ) ( 0,0)(65,37.52777 ) ( -3.3495,12.5005)(43.30127,25 ) ( 15,-55.9807)(-20,74.64102 ) ( 90,-27.64838)(-10,-54.4433 ) ( 90,75.8792)(-40,41.04584 ) ( 65,-49,39814)(30,81.22364 ) ( 0,51.7638)(0,0)@xmath198 ( 0,-51.7638)(0,0)@xmath198 ( 43.30127,25)(0,0)@xmath198 ( 48.2963,12.94095)(0,0)@xmath198 ( 12.94095226,-48.2963)(0,0)@xmath198 ( -12.94095226,48.2963)(0,0)@xmath198 ( 0,13.3975)(0,0)@xmath198 ( -3.34936,12.5005)(0,0)@xmath198 ( 0,0)(0,0)@xmath198 ( 37,27)(0,0)@xmath230 ( 4,11)(0,0)@xmath673 ( -4,-4)(0,0)@xmath34 ( -2.5,24)(0,0)@xmath42 ( 27,18)(0,0)@xmath674 ( 20,8.5)(0,0)@xmath674 ( 24,3)(0,0)@xmath42 ( 2,-56)(0,0)@xmath675 ( 2,60)(0,0)@xmath676 ( 0,0 ) ( 0,0 ) ( 43.30127,25 ) ( 0,0 ) ( 0,0 ) ( 0,0 ) ( 0,-51.7638)(0,51.7638 ) ( 0,0)(85,22.7757 ) next we shall consider a fixed interval @xmath298\subseteq [ 0,1]$ ] , define @xmath677 and estimate @xmath678 to each point @xmath679 we associate ( see figure [ figure8 ] ) the point @xmath680 , where @xmath681 $ ] . note that @xmath682 since @xmath229 has slope @xmath9 , we have the obvious inequality @xmath683 and as a consequence we can write @xmath684 when @xmath621 , obvious monotonicity properties yield @xmath685 with @xmath686 as defined in . using similar arguments we infer @xmath687 take now @xmath115 small , and suppose that @xmath688 and @xmath689 are two integers such that @xmath690 such integers can be chosen for instance as at the beginning of section 7 with @xmath691 . fix also a compact @xmath692 . applying successively , remark 3 , propositions [ p8.2 ] , [ p9.1 ] , [ p10.1 ] , and inequality , we infer that @xmath693 in a similar way we infer from and the previous arguments that @xmath694 consider now a partition of @xmath40 $ ] with intervals @xmath695 , where @xmath696 $ ] and @xmath697 . summing over @xmath560 we find as a result of , and that @xmath698 } ( t)=\sum\limits_{j=1}^n { \mathbb{g}}_{{\varepsilon},i_j}(t ) = \frac{\pi}{2}\ , { \mathbb{g}}(t)+ o_\delta ( { \varepsilon}^{1/8-\delta}),\ ] ] and thus @xmath699}(t)}{\lambda_{\varepsilon } ( \sigma^+_{{\varepsilon},[0,1 ] } ) } & = \frac{\lambda_{\varepsilon } ( \ { ( x,\omega)\in \sigma^+_{{\varepsilon},[0,1 ] } \ , ; \ , 2{\varepsilon}\tau_{\varepsilon}(x,\omega)>t \})}{\lambda_{\varepsilon } ( \sigma_{{\varepsilon},[0,1]}^+ ) } \\ & = \frac{\pi}{2}\cdot \frac{{\varepsilon}{\mathbb{g}}(t)}{2{\varepsilon}c_{[0,1]}}+o_\delta ( { \varepsilon}^{1/8-\delta } ) = { \mathbb{g}}(t)+o_\delta ( { \varepsilon}^{1/8-\delta } ) . \end{split}\ ] ] for obvious symmetry reasons we can only consider @xmath700 $ ] , therefore @xmath701 which ends the proof of theorem [ t1.2 ] . in this and the next two sections we shall take @xmath594 $ ] , and analyze the geometric free path length in the case of vertical scatterers of height @xmath266 centered at integer lattice points . in this setup we will consider the phase space @xmath598 , where @xmath29 , @xmath268\subseteq [ 0,1]$ ] is an interval , @xmath599 \times [ \omega_0,\omega_1]$ ] and @xmath600 is the ( non - normalized ) lebesgue measure on @xmath601 . the trajectory will therefore start at a point @xmath602 , @xmath603 $ ] , under angle @xmath42 , with @xmath604 . recall that the free path length is denoted by @xmath605 in this case . given @xmath267 , consider @xmath606 actually it will suffice to take @xmath607 for properly chosen integers @xmath75 . the first goal will be to estimate the distribution of the free path length @xmath608 when we average over @xmath609 , under the assumptions that @xmath268 \subseteq [ 0,1 ] $ ] is a short interval of length @xmath610 for small @xmath611 , and that @xmath75 is a ( large ) integer such that @xmath612 . concretely , we will be interested in the quantity @xmath613 in the remainder of the paper we take @xmath597 . we set @xmath614 a direct application of propositions [ prop1 ] and [ prop2 ] , with widths @xmath415 given by and @xmath413 , @xmath414 by , yields the following formula , derived from by replacing @xmath615 with @xmath616 , and valid for any @xmath617 : @xmath618 this quantity will be compared with the one obtained by substituting @xmath284 in place of @xmath285 in , as in section 3 . for this purpose we shall consider @xmath619 with @xmath620 _ if @xmath268 \subseteq [ 0,1]$ ] and @xmath621 , then owing to , and to the fact that @xmath622 is monotonically decreasing we have @xmath623 _ the argument , based on inequality used to compare @xmath624 with @xmath283 in lemma [ lemma5 ] , is not going to apply here because @xmath625 is not a lipschitz function . nevertheless , we can overcome this problem by appealing again to a soft monotonicity argument , based on remark 3 and on the fact ( which can be seen directly from the definition of the function @xmath59 ) that for any compact @xmath585 , there exists a constant @xmath626 such that @xmath627 in this and the the next two sections we will analyze the asymptotic of the quantity @xmath628 for large integers @xmath75 and short intervals @xmath301 such that @xmath629 . we note at this point that the relation is hinted by formula and by @xmath630 for the sake of space , the error estimates which are similar to the ones already derived in the first part of the paper are going to be more sketchy . [ p8.2 ] for every interval @xmath80 $ ] of size @xmath631 and every @xmath29 @xmath632 the estimate holds uniformly in @xmath308 $ ] . since @xmath300 , we have @xmath633 for all @xmath161 . thus we infer from , as in formula , that @xmath634 with @xmath635 @xmath636 from we gather @xmath637 on the other hand , gives @xmath638 we can show in a similar way that @xmath639 from the formulas for @xmath640 , @xmath641 , @xmath642 and from we infer @xmath643 finally we show as at the end of section 4 that @xmath644 a similar formula holds for the second sum in , and therefore we get @xmath645 it is clear that these estimate hold uniformly in @xmath646 $ ] . in this section we prove in the setting of section 8 the following result [ p9.1 ] for every interval @xmath80 $ ] of size @xmath647 and every @xmath29 @xmath648 the estimate holds uniformly in @xmath24 on compacts of @xmath412 . we proceed as in section 5 , estimating first @xmath649 and then @xmath650 in this case we find from , and , as in section 5 , that @xmath651 with @xmath652 employing the same technique as in section 5 and the fact that one gets a similar result while integrating between @xmath653 and @xmath414 , we find that @xmath628 can be expressed , up to an error term of order @xmath654 , as @xmath655 this is further equal to @xmath656 which is the desired conclusion . in this section we prove in the setting of section 8 the following result [ p10.1 ] for every interval @xmath80 $ ] of size @xmath647 and @xmath29 @xmath657 the estimate holds uniformly in @xmath24 on compacts of @xmath511 . since @xmath273 , we have @xmath658 and we infer from that @xmath659 where @xmath660 , respectively @xmath661 , contains the contribution of farey fractions in @xmath662 with @xmath530 , respectively with @xmath514 . when @xmath530 we have @xmath663 , @xmath161 , and therefore @xmath664 standard considerations as in sections 6 and 8 show that , uniformly in @xmath24 on compacts of @xmath511 and up to an error term of order @xmath665 , @xmath666 can be expressed as @xmath667 next we estimate @xmath668 and find as in sections 5 , 6 and 9 that @xmath661 can be expressed , up to an error term of order @xmath669 , as @xmath670 we conclude the proof by adding the formulas for @xmath661 and @xmath660 . in this section we prove theorem [ t1.3 ] ( i ) . part ( ii ) then follows from ( i ) and from relation ( 2.8 ) in @xcite . we consider the probability measures @xmath703 and @xmath704 on @xmath705 defined by @xmath706 as a result of theorem [ t1.2 ] @xmath707 which implies @xmath708 meaning that @xmath709 vaguely as @xmath710 . since @xmath711 and the map @xmath712 belongs to @xmath713 because @xmath714 , @xmath715 , the lebesgue dominated convergence theorem yields @xmath716 using fubini s theorem , these double integrals can also be expressed as @xmath717 } ( x)\ , d\widetilde{\nu}_{\varepsilon}(u)\ dx & = \int_1^\infty \int_1^\infty \frac{1}{x } e_{[1,u ] } ( x ) \ , dx\ d\widetilde{\nu}_{\varepsilon}(u ) \\ & = \int_1^\infty \int_1^u \frac{dx}{x } \ d\widetilde{\nu}_{\varepsilon}(u ) = \int_1^\infty \ln u\ , d\widetilde{\nu}_{\varepsilon}(u ) , \end{split}\ ] ] and respectively as @xmath718}(x)\ , d\nu_0(u)\ dx = \int_1^\infty \ln u\ , d\nu_0 ( u).\ ] ] it follows that for any ( small ) @xmath115 @xmath719 and also that @xmath720 we show in a similar way that @xmath721 by using fubini s theorem which gives in turn @xmath722 } ( x)\ , dx\ d\widetilde{\nu}_{\varepsilon}(u ) \\ & = -\int_0 ^ 1 \int_0 ^ 1 \frac{1}{x } e_{[u,1 ] } ( x)\ , d\widetilde{\nu}_{\varepsilon}(u)\ dx = -\int_0 ^ 1 \frac{1}{x } \int_0^x d\widetilde{\nu}_{\varepsilon}(u ) \ dx . \end{split}\ ] ] by and we get @xmath723 since yields @xmath724 we collect @xmath725 finally we outline the proof of the identity @xmath726 first we note that @xmath727 so we may write @xmath728 where @xmath729 the substitution @xmath730 leads to @xmath731 by a direct computation , the integral above is equal to @xmath732 , thus @xmath733 next , a direct computation shows that @xmath734 the relations provide @xmath735 but @xmath736 thus we get @xmath737 with @xmath738 where @xmath739 by a direct computation we find @xmath740 as a result we gather @xmath741 and so @xmath742 by a careful computation we find @xmath743 where @xmath744 denotes the trilogarithm function @xmath745 using the equality ( cf . * formula ( 6.12 ) ) ) @xmath746 we infer @xmath747 inserting this back into we finally find @xmath748 we are grateful to professor giovanni gallavotti for bringing to our attention reference @xcite in 2002 . bunimovich , l. : billiards and other hyperbolic systems . in : _ dynamical systems , ergodic theory and applications _ , ya.g . sinai ( ed . ) , encyclopaedia math . sci . vol . * 100*. berlin : springer - verlag , 2000 , pp . 192233 gallavotti , g. : lectures on the billiard . in : _ dynamical systems , theory and applications ( rencontres , battelle res . seattle , wash . , 1974 ) _ , j. moser ( ed . ) , lecture notes in phys . vol . * 38 * , springer - verlag , 1975 , pp . 236295 . golse , f. : on the statistics of free - path lengths for the periodic lorentz gas . in : _ xiv international congress on mathematical physics ( lisbon , 2003 ) _ zambrini ( ed . ) , world sci . publ . , 2006 , pp . 439446 . gutzwiller , m. : physics and arithmetic chaos in the fourier transform . in : _ the mathematical beauty of physics ( saclay , 1996)_. j.m . drouffe , j.b . zuber ( eds . ) , adv . series in math . . vol . * 24*. river edge , nj : world sci . publ . , 1997 , pp .

we study the free path length and the geometric free path length in the model of the periodic two - dimensional lorentz gas ( sinai billiard ) . we give a complete and rigorous proof for the existence of their distributions in the small - scatterer limit and explicitly compute them . as a corollary one gets a complete proof for the existence of the constant term @xmath0 in the asymptotic formula @xmath1 of the ks entropy of the billiard map in this model , as conjectured by p. dahlqvist .

magnetic fields contribute to the dynamical behavior of ionized astrophysical fluids such as those in the upper solar and stellar atmospheres , the interstellar medium and star - forming regions . their influence is carried out by hydromagnetic waves which efficiently propagate perturbations , ensure a turbulent pressure or may even cause the development of instabilities ( @xcite ) . however , kulsrud & pearce ( @xcite ) showed that in the magnetized and weakly ionized interstellar medium hydromagnetic waves are heavily damped in a frequency range ( and thus scale ) associated with ambipolar diffusion . at low frequency the neutrals are well coupled to the ions ( which are tied to the magnetic field lines ) and hydromagnetic waves propagate at the alfvn speed defined by the total inertia ( given by ions+neutrals ) . at high frequency neutrals and ions are totally decoupled , and alfvn waves involve only the ions , which define a larger alfvn velocity . in the intermediate range ( the ` ambipolar range ' , between the ion - neutral and neutral - ion collision frequencies @xmath0 and @xmath1 ) the neutrals are imperfectly coupled to the ions ; this results in a drag which strongly damps the waves . the non - linear evolution of this process can cause an _ ambipolar filamentation _ of the magnetic field when a magnetized and weakly ionized plasma is stirred by hydromagnetic turbulence in the ambipolar range ( @xcite ) . if such a plasma presents small variations in the ionization fraction ( @xmath2 ) , the turbulent velocity of the neutrals is higher in the most ionized regions , since they are better coupled to the ions . this gives rise to a force ( given by the average of the @xmath3 term ) driving the neutrals out of the most ionized regions . by reaction the ions and the magnetic flux are compressed in these regions , so that the initial ionization inhomogeneities are strongly amplified . as a consequence a concentration of the flux tubes is expected to occur , producing a filamentary structure , so that turbulent energy would be converted into magnetic energy associated with the concentration of the magnetic field . ( 1995 ) provided only order of magnitude estimates of the expected amplification of the ionization fraction . in this work we present a fully consistent 2-d non - linear numerical simulation of the mechanism in order to test its efficiency . the non - linear analysis is a fundamental tool to study the physics in certain astrophysical environments , such as molecular clouds , where the observed amplitudes of the turbulent velocities are comparable with the mean field velocities . the ambipolar filamentation mechanism might help to explain some well known problems arising in magnetized , partially ionized astrophysical plasmas . one of them is related with the observations of turbulence in molecular clouds . observations show a filamentary structure , and strong supersonic motions resulting in turbulent and magnetic energies in approximate equipartition , i.e. , much larger than the thermal energy ( @xcite ) . the ambipolar filamentation mechanism would concentrate the magnetic field in intense flux ropes surrounded by essentially neutral clouds . another possible application relates to the fibrilled structure observed in the magnetic field emerging from the solar photosphere , organized in very narrow flux tubes . the ambipolar filamentation mechanism might provide an explanation for the spicules emerging from the photosphere : let us consider magnetic field lines raising from the photosphere . then an alfvn wave of a given frequency , produced in the photosphere and initially below the local ambipolar frequency range , will propagate upward along the field lines and reach at high altitudes a plasma of much lower density , i.e. , lower collision frequencies . it will thus be damped by ambipolar effects and can expel the neutrals from the most ionized flux tubes , concentrating the magnetic flux in narrow tubes where strong vertical motions can be expected . this would occur together with the mechanism discussed by de pontieu & haerendel ( @xcite ) . these prospects will be discussed in more detail in the last section of this work . we have carried out numerical simulations in which a weakly ionized and magnetized gas inside a cartesian box is submitted to a high amplitude oscillation emitted from one of its sides . the perturbation propagates inside the box as an alfvn wave with a frequency chosen to be in the ambipolar range , so that it will be strongly damped . in section 2 we describe the dynamical equations that govern the evolution of a two fluid gas , together with the numerical code and the boundary conditions used to solve them . we also discuss the numerical constraints present in our simulations . the results from the numerical experiments are presented in section 3 and discussed in the context of the problems cited above in section 4 . the magnetohydrodynamics ( mhd ) equations describing a two fluid ( ions and neutrals ) system are ( @xcite ) : @xmath4 @xmath5 @xmath6 @xmath7 @xmath8 for simplicity we assume an isothermal equation of state : @xmath9 @xmath10 ( 8,10.5 ) where @xmath11 , @xmath12 and @xmath13 are , respectively , the density , velocity and partial pressure of the ions ( with subscript i ) and neutrals ( with subscript n ) , @xmath14 is the gravity , @xmath15 is a constant such that @xmath16 and @xmath17 are the ion - neutral and neutral - ion collision frequencies , and @xmath18 is the sound velocity ( assumed the same for ions and neutrals ) . we assume that ionization and recombination occur on a longer time scale than the one we consider . this should of course be checked for applications to specific astrophysical situations . we have also checked that in these conditions the characteristics of the problems in which we are interested , namely the high electron densities in the case of solar spicules and the large spatial dimensions of molecular clouds , allow us to use these simplified two fluid mhd equations instead of the full set of the three fluids ( electrons , ions and neutrals ) equations which describe the dynamics of a weakly ionized gas . in order to allow for the long time scales considered ( hundreds of alfvn times ) , associated with numerical constraints giving a short time step , we simplify the problem by making it two - dimensional : all quantities are @xmath19-invariant , and only the perturbed current has a component along @xmath19 . therefore all quantities depend only on @xmath20 ( the vertical direction ) and @xmath21 ( the horizontal one ) on a cartesian grid . in this geometry the distinction between shear - alfvn and magnetosonic waves disappears , but waves propagating along @xmath20 retain the properties of the usual alfvn waves , that is , they twist the field lines and to lowest order they are not compressional . the numerical code used in our simulations is based on the same general methods as the zeus-2d code ( @xcite ) . it solves the mhd equations on a staggered mesh using the method of finite differences with a time explicit , operator split scheme . densities and pressures are zone centered quantities while the velocity components are centered at their corresponding zone faces . as in zeus-2d the solution procedure is arranged in two main steps , first taking into account the source terms in the right - hand side of the equations , and then solving for the advection terms in the left - hand side . the main difference with zeus-2d is the treatment of the gravitational and magnetic terms . an initial state of hydrostatic equilibrium is assumed , by fixing the vertical density and pressure profiles , from which we derive a value for the gravity ; this results in an ad - hoc gravity profile leaving us some freedom to optimize the density profiles , limiting the numerical difficulties associated with the emission of the wave ( see below ) . the 2-d geometry allows us to describe the magnetic field in eqs . ( [ eq3 ] ) and ( [ eq5 ] ) , in terms of the flux function @xmath22 as follows : @xmath23 so that the induction equation , eq . ( [ eq5 ] ) becomes : @xmath24 the momentum and induction equations are thus first solved without the advection terms , using time explicit operator split schemes . the moc - ct algorithms included in zeus-2d ( @xcite ) are not used to evolve the magnetic terms , since they are not required by this simple problem . tests were carried out to verify the correct propagation of the waves . the second step in the numerical code , still following zeus-2d , solves finite difference versions of the integral equations coming from the advection terms in eqs . ( [ eq1 ] ) , ( [ eq2 ] ) , ( [ eq3 ] ) and ( [ eq4 ] ) : @xmath25 @xmath26 @xmath27 @xmath28 which allows us to obtain a conservative scheme by computing the fluxes of the advected quantity ( @xmath29 , @xmath30 , @xmath31 , @xmath32 respectively in the equations above ) at every interface on the grid and using the same flux to update adjacent zones . to solve the problem of calculating values of the advected quantities on the grid interfaces maintaining numerical stability ( @xcite ) , we used the second order van leer interpolation method ( @xcite ) . this method is fast and accurate enough for the requirements of our simulations . as in zeus-2d the two - dimensional advection problem is simplified by using directional splitting ( @xcite ) , that means using two one - dimensional advection steps to construct the full solution . the boundary conditions used to solve the mhd equations are determined by the physics of the phenomenon we are studying . we use periodic boundary conditions in @xmath21 ( note that with the definition of @xmath22 in equation ( [ eqpsi ] ) , the periodicity of @xmath22 ensures conservation of the total vertical magnetic flux through the simulation zone ) . at @xmath33 we launch an alfvn wave by giving the whole fluid ( neutrals , ions and magnetic field lines ) a motion in @xmath21 which , in this first numerical test of the mechanism , will be limited to a single periodic oscillation : @xmath34 this will thus propagate upward as an alfvn wave . at the opposite boundary ( @xmath35 ) we impose reflective boundary conditions ; they have no real effect since the total length in @xmath20 is chosen such that the waves are heavily damped before they reach that point . at the moment , in order to isolate the effect we want to study , we impose that there is no flux of matter from @xmath33 . in this manner we ensure that the variations in the ion or neutral densities do not result from inflow of matter at the lower boundary . on the other hand this results in severe constraints on the code because the wave pressure pushes matter upward , resulting in very low densities at the first grid point . this turns out to be the most stringent limit on the parameters we can use ( _ e.g. _ the perturbed velocity @xmath36 ) . the simulations have other numerical constraints . to ensure that our calculations are fully consistent the dimensions of our spatial grid must be large enough to allow the ion - neutral interaction , that is , larger than the ion - neutral ( neutral - ion ) mean free path , @xmath37 , where @xmath38 is the collision frequency . for the characteristic values used in the simulations the grid is more than 10 times the maximum mean free path for the neutrals , which is enough for the necessary ion - neutral interaction . ( 8,7 ) we also wish to launch the wave in a manner such that it is damped not right in its emission zone ( near @xmath33 , where we create it ) but away from it , so that we can clearly identify the processes obtained . we do this by imposing a strong vertical density gradient . thus the density at @xmath33 can be taken such that the wave initially propagates well ( @xmath39 ) , but later reaches an altitude where @xmath40 so that it is strongly damped and ambipolar processes can act . at higher altitude we maintain a small but constant density . finally , in order to initiate the filamentation process we need a transverse density profile : the neutral density is initially independent of @xmath21 , but the ion density has a horizontal profile , so that the ionization fraction is higher along the central field line ( @xmath41 ) . the magnetic field is chosen to ensure mhd equilibrium in the horizontal direction , that is : @xmath42 thus we expect that ions in the central flux tube will be compressed , and the neutrals expelled , by the ambipolar filamentation process . on the other hand , in order to save computation time we must take the largest time step that guarantees the numerical stability for the set of difference equations . our time explicit code has to satisfy a courant - friedrichs - lewy ( cfl ) stability condition , @xmath43 derived applying a von neumann stability analysis ( @xcite ) . physically , this condition sets the higher limit of the distance that information can travel in one time step ( waves or fluid motion ) to be the minimum size of the discrete elements or zones in the spatial grid . for multidimensional systems , a suitable time step limit is the smallest of all one dimensional cfl conditions in each coordinate direction ( @xcite ) . this gives strong constraints since ( a ) we need to consider the highest alfvn velocity , obtained where the density is lowest ( b ) the cfl condition involves waves with the shortest wavelengths , _ i.e. _ high frequencies . at high frequency , as explained in the introduction , the relevant alfvn velocity involves only the ion density . therefore compared with the alfven velocity at @xmath33 the one used for the cfl criterion is higher by the density ratio @xmath44 along the vertical density profile , and by the inverse of the ionization ratio @xmath45 . these two quantities are large and force us to use very short time steps , typically less than @xmath46 of the alfvn time through a grid cell at @xmath33 . let us now consider the time scales associated with the mechanism of ambipolar filamentation . three time - scales appear . the first one is the period of the wave . the second one involves the response of the fluid to the wave pressure ; in the basic mechanism of ambipolar filamentation , the neutrals feel a turbulent pressure : @xmath47 ( where the angular brackets mean a time average ) , which has a gradient along @xmath21 associated with the gradient of @xmath48 ; they respond by a pressure perturbation to re - establish equilibrium . this is established after a time @xmath49 , where @xmath50 is the scale of the horizontal ionization inhomogeneity . ( 8,9 ) however , at this stage the situation keeps evolving since the new neutral and ions density profiles are more peaked , resulting in a further peaking of @xmath51 . thus , on a much longer time scale , the ionization contrast will keep growing ( @xcite ) ; in our simulations we find that typically @xmath52 alfvn times are needed to reach an equilibrium . such long time scales mean that any small numerical resistivity will result in a diffusion of the magnetic flux , so that the filamentation of the magnetic field associated with that of the ions can not be observed . the simulations were carried out using the numerical model described in the previous section . they follow the evolution of a 2-d ( @xmath21-horizontal and @xmath20-vertical directions ) , two fluid system ( ions and neutrals ) with low ionization fraction . the gas is initially threaded by a vertical constant magnetic field and perturbed by horizontal waves excited at the footpoints of the magnetic field lines ( @xmath33 ) . since no approximations were made except the invariance in @xmath19 , the following parameters have to be imposed on each simulation : @xmath53 , the initial magnetic field , @xmath54 , the sound velocity , @xmath15 , the ion - neutral collisional coefficient , @xmath36 and @xmath55 , the amplitude and frequency of the wave and the profiles of @xmath29 and @xmath30 in the @xmath21 and @xmath20 directions . in order to clarify the results presented in all figures , we must briefly explain the scaling and units used by our numerical code . we have normalized the parameters to the characteristic scales of the problem we are solving . therefore we have taken as units the initial alfvn velocity at @xmath33 , @xmath56 , and the alfvn time @xmath57 , defined as the time that takes such an alfvn wave in crossing one wavelength @xmath58 . in the simulation presented in figs . 1 - 6 we used a 61@xmath59200 numerical grid . it is longer in the vertical direction in order to allow the complete damping of the wave . as it was established in the previous section , we assume initial equilibrium in a fluid vertically stratified and supported by gravity , with ions and neutrals densities decreasing sharply ( by a factor @xmath60 10 ) with @xmath20 ( shown in fig . initially the neutral density is independent of @xmath21 , but the ion density shows a small enhancement in the @xmath21 direction , along which lie the perturbed velocities associated with the wave . the magnetic field is adjusted so as to ensure mhd equilibrium in the @xmath21 direction , given by ( [ peq ] ) . the resulting initial ionization fraction @xmath61 is constant over @xmath20 but shows a maximum at @xmath41 , as shown in fig . 2 ( left ) . ( 8,5.6 ) a high amplitude alfvn wave is launched from @xmath33 , by making the whole fluid ( neutrals , ions and magnetic field lines ) oscillate in @xmath21 at a single frequency @xmath55 ; this perturbation propagates upwards as an alfvn wave . its frequency is chosen so that the wave propagates without damping at the lower part of the simulation grid ( @xmath62 ) , but is strongly damped at intermediate altitudes , where @xmath48 ( and thus @xmath1 ) was taken to decrease sharply . therefore the filamentation will occur only at the intermediate altitude where the wave is damped ( @xcite ) but still retains a strong perturbed velocity . in the simulation presented in figs . 2 - 4 and 6 , the perturbed velocity of the wave at @xmath33 is of the order of the sound velocity ( @xmath63 ) , which is taken to be a half of the alfvn velocity . ( 8,5.6 ) fig . 2 ( right ) shows the spatial profile of the ionization fraction at the end of the calculation . the initial contrast in the horizontal direction has grown and shrunk at a certain height . at higher altitude small traces of the wave still appear , together with a reversed profile along @xmath21 . an explanation to this behavior will be given later in this section . in the vertical profile of the ionization fraction ( fig . 3b ) we note that the strong peak in fig . 2 ( right ) is located at @xmath64 , actually where the wave begins to be damped ( see fig . 4 ) but still keeps enough amplitude to make the neutral expulsion mechanism effective . therefore we find a strong amplification ( by a factor @xmath65 ) of the contrast in @xmath66 , at the altitude where the wave is damped ( fig . 3a ) . at the same altitude the ion density contrast along @xmath21 , @xmath67 , grows and the more ionized region shrinks ( fig . 3c ) , as expected . moreover the ion density decreases on the sides of the peak , suggesting ion motion towards the most ionized regions . in fig . 5 we can clearly quantify the final increase in ion density . we also see in the simulation how the neutrals are expelled from the most ionized regions ( fig . 3e ) , generating in the density profile a central minimum of the same order as the increase achieved by the ions . in all horizontal profiles of fig . 3 , the enhanced minimum / maximum are not centered at @xmath41 because of the lateral motion due to the wave . on fig . 4 we compare the ion and neutral velocity . a phase lag ( corresponding to the damping of the wave by ion / neutral friction ) appears around @xmath68 . there is an additional effect acting to expel the neutrals from the most ionized regions . since along those field lines the wave is less damped , its perturbed velocity at a given height is larger . however this contributes with a @xmath3 term on the ions as well as on the neutrals , so that it should not contribute to the growth of the ionization contrast . besides the strong peak in the spatial distribution of the ionization fraction , there is another feature which is a direct output of the enhancement of the ion density in the central magnetic field lines . the condensation of the ions must be accompanied by a compression of the magnetic field lines at the same point . then magnetic tension causes the compression of the field lines upwards . but as noted before , numerical resistivity has allowed the field to diffuse almost totally . a small residual intensification at the central field lines is left but it is invisible at low altitudes , where the wave dominates the field dynamics . however , at the highest grid zones , where the wave is almost completely damped , the increase in @xmath69 can be barely detected . that residual enhancement in @xmath69 causes the rise of the magnetic pressure , which results in an expulsion of ions from the central field lines at high altitudes . this causes a reversed ion density profile , as observed in fig . 2 . in our simulations the filamentation process seems only limited by two facts . firstly , @xmath48 can reach zero ( so that the code crashes ) at large @xmath21 . in this case the process is so efficient that all the ions at large @xmath21 are evacuated towards the most ionized regions , at the altitude where the filamentation occurs . @xmath48 can reach zero in the lowest grid zones because the matter is pushed upwards by the wave pressure . both processes impose severe restrictions to the model parameters , in particular to the highest value of @xmath36 we can use . the second limitation could be easily overcome by changing the boundary conditions to the more realistic case of allowing flux of matter at @xmath33 . however , as discussed above , we prefer not to do it in these first simulations since it would make it difficult to distinguish the enhancements of ion density due to the filamentation process from the ones due to this vertical flow . this will be necessary , on the other hand , in future realistic simulations of spicules . 6 shows the evolution of the maximum value of the ionization fraction for several initial wave amplitudes . for the lowest values of the perturbed velocity a state of equilibrium is achieved early in the calculation . however , for the highest velocities the filamentation is more efficient and causes the simulation to crash . in the simulation shown in fig . 3 ( @xmath70 ) , this is due to the complete depletion of ions at the first grid zones , although very small values of @xmath48 are also achieved at high @xmath21 , at the altitude where z is maximum . in fact the curves corresponding to the highest velocities show an oscillating behavior over @xmath71 , which can be a consequence of the extremely low values of the ion density in those regions . 6 shows that the process acts faster for higher velocities . the asymptotic value of @xmath72 varies quadratically with @xmath73 , as expected from the theory ( @xcite ) . however we find that this value ( and accordingly the horizontal pressure gradients obtained for @xmath29 and @xmath30 ) depends on the initial profiles , _ i.e. _ , more peaked initial profiles result in more peaked final ones . theory would lead us to expect that the final equilibrium , balancing the ponderomotive force of the wave with the pressure gradient of the neutrals , should be independent of the initial state . we believe that this may be due to the vertical transport of ions and neutrals from the lower and higher regions of the simulation grid . in these regions the filamentation process does not act , so that they retain a memory of the initial conditions and can feed the region of wave absorption with additional ions and neutrals . however we have been unable to prove it with the present limitations , due to the boundary condition at @xmath33 . we thus defer the treatment of this question to future work where the detailed physics will be considered in more realistic conditions . the aim of the simplified 2-d simulations presented in this paper was to provide a first numerical test of the ambipolar filamentation mechanism . we have tried to limit the physics involved to the minimal ingredients needed , in order to clearly separate the expected physical effect ( the filamentation ) from other sources of variation of ion and/or neutral density . in particular we have limited ourselves to the excitation of a monochromatic wave and forbidden any inflow of matter from the lowest grid boundary , although this turned out to cause the most severe numerical limitation , by creating a vanishing ion density on the first grid points . we have found that the mechanism is strong and efficient , even with these constraints and simplifying assumptions . in the most efficient of the simulations , the perturbed velocity was close to the sound velocity , taken to be a half the alfvn velocity ( so the conditions are similar to that of the interstellar medium ) . however such velocities are still lower than the observed supersonic motions which would make the mechanism even more efficient , since we have found that its efficiency goes with the square of the perturbed ion velocities . we have obtained similar results in cases closer to the conditions of the solar spicules , where the sound velocity is a lower fraction of the alfvn velocity than in the results presented here ( @xmath74 ) . future work will go in two directions : we will excite a more complete spectrum of waves from the lowest boundary . this should cause the filamentation to be less localized , since waves of different frequencies will be damped ( and cause filamentation ) at different heights . a more important evolution will be to relax the condition of no vertical flow from @xmath33 . this will make the simulation more efficient since we found that this boundary condition , chosen to clarify the physics , caused the most severe numerical limitation . it will also allow us to make more realistic simulations of solar spicules . in that case , we expect our mechanism to act together with the vertical flow of matter discussed in haerendel ( @xcite ) , tagger _ et al _ ( 1995 ) and de pontieu & haerendel ( @xcite ) . upward traveling alfvn waves generated in the photosphere are expected to cause both ambipolar filamentation of the field lines and vertical acceleration of the gas , so that a boundary condition allowing matter to flow vertically from @xmath33 is necessary . first tests actually show an intense growth of the ionization fraction in the most ionized flux tubes . we also expect that , with these boundary conditions , we will be able to use higher perturbed velocities making the mechanism much more efficient . the effects of a more realistic equation of state and , of course , the inclusion of ionization and recombination , are also obvious extensions to this work . this should make us able to address more realistically the role of our mechanism in the filamentary structure of the interstellar medium and the star formation process . models of star formation ( since the early work of @xcite ) invoke molecular clouds supported essentially by turbulent magnetic pressure and an ambipolar flow of the neutrals toward dense cores ( @xcite ; @xcite ; @xcite ; @xcite ; and references therein ) . in the molecular clouds conditions , where the turbulence is supersonic , we expect our mechanism to result in an additional pressure from the most ionized to the less ionized regions that efficiently separates the ions from the neutrals , favoring the gravitational collapse of the latter . the turbulent magnetic fields can be generated as a consequence of the jeans - parker instability during the molecular cloud formation process , or ( as suggested by recent models ) by the outflow lobes associated with young stellar objects ; in the former case , large scale waves could feed the turbulent cascade in the cloud ( @xcite ) . in the latter case , the outflows act as the sources of turbulence and the energy released heats the cloud by the ambipolar drift ( @xcite ) . in those contexts , there would be localized sources for the turbulence , that would propagate in a stratified medium , so that the physics studied here could be readily applied . on the other hand for a detailed study of the turbulent cascade and its effects on the interstellar medium , a pseudo - spectral technique would be better adapted to allow a random , non - localized excitation of the waves . the relevance of the role of the ambipolar filamentation in those conditions will be the object of future studies . the authors wish to thank f. masset and a. hetem , who have been involved in the early stages of the numerical developments used in this work , and n. hulamo for the constructive discussions . mf has been supported by a predoctoral research fellowship of the ministerio de educacin y cultura ( mec ) and by a c.i.e.s . ( centre international des etudiants et stagiaires ) grant . this work has been partially supported by the mec through the projects pb93 - 491 and pb97 - 269 .

we present the results of a 2-d , two fluid ( ions and neutrals ) simulation of the ambipolar filamentation process , in which a magnetized , weakly ionized plasma is stirred by turbulence in the ambipolar frequency range . the higher turbulent velocity of the neutrals in the most ionized regions gives rise to a non - linear force driving them out of these regions , so that the initial ionization inhomogeneities are strongly amplified . this effect , the ambipolar filamentation , causes the ions and the magnetic flux to condense and separate from the neutrals , resulting in a filamentary structure . # 1 # 1 0.5 cm

spectroscopic studies of hyperfine manifolds in alkalies , such as measurements of energy separations , have benefitted by the high precision of the experimental techniques available to interrogate atoms @xcite . their hydrogen - like structure makes interpretation of experimental results straightforward in terms of electromagnetic fields generated by the valence electron and nuclear moments . precise measurements in higher excited states accessible through two - step transitions@xcite have appeared in recent years . this has renewed interest in improving calculations in other states where theoretical methods such as many - body perturbation theory ( mbpt ) ( see for example the recent book of w. r. johnson @xcite ) are yet to be tested against experimental results . precise measurements in excited states , beyond the first one , have several experimental complications . standard spectroscopic techniques rely on the high population of atoms in the ground state to guarantee a good signal to noise ratio of the fluorescence or absorption of the atomic sample . in two - step transitions this is no longer the case . the amount of population transferred to the intermediate level , for reasonable powers of the lasers , tends to be small , and detectors at the desired frequency might no be readily available . we present in this paper two - color modulation transfer spectroscopy as a tool for studies of atomic properties of higher excited states . the method consist of two lasers ( pump and probe ) counter - propagating through a thermal vapour . before being directed to the interaction region , one of the lasers is modulated . the first step of the transition _ i.e. _ the pump , connects the ground state to a resonant intermediate state while the probe scans over the desired energy manifold . we monitor the absorption of the pump laser as a function of probe laser detuning . the non - linear interaction of the lasers burns a hole " in the atomic ground state population . the generated spectra presents sub - doppler peaks ( sometimes called lamb - bennett dips ) corresponding to the atomic resonances with the trademark sidebands at their side . this technique overcomes the two main inconveniences of direct absorption of the probing laser _ i.e. _ low signal to noise ratio and non - availability of detectors at the desired wavelength . we present two ladder systems in @xmath0rb to illustrate the main features of the technique and two different applications of the modulation . we select the @xmath1 and the @xmath2 ladder transitions to illustrate their different uses . the amplitude of the probe laser is modulated for the first system while the second system has its pump frequency modulated . the frequency modulation of the pump laser and good signal to noise ratio allows us to lock the probe laser to the @xmath3 excited atomic resonance . in this case the probe laser remains modulation free . this is highly desired since the electronic modulation of the laser itself can carry unwanted effects such as sidebands at higher or lower frequencies as well as bandwidth problems . the method we are presenting is , of course , not limited to these two cases and can be extended to other atomic levels . the organization of the paper is as follows : section ii contains the theoretical model , section iii explains the experimental setup and results , section iv has a summary of the precise measurements using this method , and section v presents the conclusions . we start with a three level model that can show some of the qualitative features of the experimental spectra . we use a density matrix formalism to describe a three level atom in ladder configuration interacting with two lasers , one of which has sidebands . we model our system as doppler - free ignoring zeeman sublevels to keep it tractable . the experimental situation is more complex and for quantitative analysis it is necessary to take into account those same effects that we are ignoring . figure [ figure energy levels theory ] shows our theoretical model . we treat two cases . fig [ figure energy levels theory ] ( a ) is a ladder type system with an amplitude modulated probe ( amp ) . fig ( b ) presents the same system except it has a frequency modulated pump ( fmp ) . the intermediate and last levels are coupled by a single laser with three frequencies : a carrier and two sidebands separated form the carrier by @xmath4 ( in mhz ) . we represent the amplitude of the carrier by a rabi frequency @xmath5 and the sidebands by a modulation depth @xmath6 . the ground and intermediate states are coupled by @xmath7 . the detuning of the carrier between levels @xmath8 and @xmath9 is zero in the model as it is for our experiment and we let the detuning between levels @xmath9 and @xmath10 vary as @xmath11 . the total population is normalized to unity . [ figure energy levels theory ] ( b ) follows the same nomenclature except that the sidebands arise from frequency modulation and they appear in the pump laser @xmath7 . for the fmp systems the sidebands have the appropriate sign difference . we have a set of nine linear equations for the slowly varying elements of the density matrix @xmath12 after using the rotating wave approximation with the sidebands rotating - one clockwise , one counter clockwise - at a frequency @xmath4 . the equations are : @xmath13\sigma_{nm}~+}\\ & & \frac{i}{2}\sum_{k}(\alpha_{nk}\sigma_{km}-\sigma_{nk}\alpha_{km})=\dot{\sigma}_{nm}~for~n\neq m,\nonumber\end{aligned}\ ] ] where @xmath14 is the transition frequency , and @xmath15 is the laser frequency connecting the levels . the damping rate is given by : @xmath16 and @xmath17 for the fmp system and @xmath18 for the amp system . the time dependence of the rabi frequency makes the standard approach of obtaining the steady state solution of the system not feasible . instead , we use a floquet basis expansion of the density matrix @xcite to solve our system of equations . we replace each of the slowly rotating elements of the density matrix by : @xmath19 where @xmath20 is the fourier amplitude of the component oscillating at @xmath21 . the system is now a series of @xmath22 coupled equations for some large @xmath23 that have to be solved recursively . it is necessary to set @xmath24 for some @xmath23 to cut off the infinite number of coupled equations . by solving the equations in terms of their predecessors we can extract @xmath25 . . the parameters are ( in units of @xmath26 ) : @xmath27 , @xmath28 , @xmath29 , @xmath30 , and @xmath31,width=283 ] . the parameters are ( in units of @xmath26 ) : @xmath32 , @xmath33 , @xmath34 , @xmath35 , and @xmath36,width=283 ] for our experiment we are interested in the terms @xmath37 , @xmath38 , and @xmath39 which are proportional to the absorption of the first laser carrier and sidebands , respectively . we plot the absolute value of the imaginary part as a function of @xmath11 to recover the absorption . this is necessary to take into account the square - law nature of the photodiode . our three level model reproduces the resonance features of the absorption observed as the second excitation goes into resonance for both amp and fmp systems ( see fig . [ figure absorption am ] and fig . [ figure absorption fm ] , respectively ) . the demodulation of the fmp signal yields the expected error - like feature shown in fig . [ figure demodulated absorption ] . . the parameters are ( in units of @xmath26 ) : @xmath32 , @xmath33 , @xmath40 , @xmath41 , and @xmath36,width=283 ] the size of the sidebands in our model depends on the modulation index ( separation from resonance and strength ) , as well as the specific decay rates of the levels which set up the rabi frequencies @xmath42 in the amp and fmp systems . figure [ figure experimental setup pump ] and fig . [ figure experimental setup probe ] present block diagrams of the fmp and amp systems , respectively . a coherent 899 - 01 ti : sapphire laser with a linewidth of less than 100 khz is the pump laser in both cases . a small amount of laser power from the pump laser is frequency modulated by a small bandwidth electro - optical modulator at @xmath4315 mhz and sent to a glass cell filled with rubidium at room temperature to lock the laser frequency to the @xmath44 crossover line of the @xmath45 and @xmath46 hyperfine levels for the fmp system and to the on resonance @xmath47 transition of the @xmath48 level for the amp system at 795 nm with a pound - drever - hall lock . level @xmath8 in the amp system corresponds to the lower hyperfine state of the 5s@xmath49 level ( @xmath45 ) while @xmath9 is the highest hyperfine state of the 5p@xmath49 level ( @xmath50 ) of @xmath0rb . the decay rate between the two levels is @xmath51 5.7 mhz @xcite . we simplify the hyperfine states of the @xmath52 level to just one level with decay rate @xmath53 3.5 mhz @xcite . for the fmp system , the probe laser is an sdl diode laser with a linewidth of 5 mhz at 776 nm . the lasers overlap inside an independent rubidium glass cell at room temperature wrapped in @xmath54-metal in lin - perp - lin polarization configuration . their @xmath55 power diameter of the laser beams is 1 mm . we scan the probe laser over the @xmath56 level hyperfine manifold and observe the absorption of the pump laser as a function of the probe laser detuning using a fast photodetector . we send the signal to a bias - t and record the dc and demodulated ac components with a wavesurfer digital oscilloscope with an 8-bit resolution from lecroy . we keep the power of the pump laser and the modulation depth fixed to a value of 100 @xmath54w and @xmath57 , respectively . we change the power of the probe beam and observe its influence on the spectra . it is possible to observe the resonant features of the @xmath56 hyperfine manifold with little as 100 @xmath54w of probe power . higher probe power increases the signal size and the width of the features . playing with the polarization and powers we also observe eit features @xcite . we restrict ourselves to a space parameter where these very narrow features are absent . figure [ figure absorption with sidebands ] ( fmp ) and [ figure absorption ] ( fmp ) show typical experimental traces of the absorption of the 780 nm laser . the spectrum has been offset to zero transmission for convenience . the first of these , fig . [ figure absorption with sidebands ] , has the dc component of the absorption with the sidebands appearing on both sides of the main resonances . no doppler background is observed for any of the experimental conditions explored , showing that this is a doppler free spectrum . [ figure absorption ] ( a ) shows the lower hyperfine states of the @xmath56 level manifold with no sidebands for clarity . fig . [ figure absorption](b ) has the demodulated ac component of the absorption . the dashed lines identify the error - like features with their corresponding hyperfine levels . we use this spectrum to stabilize the frequency of the probe laser . level in @xmath0rb . it presents the main resonances as well as the indicated sidebands ( sb ) . the power of the probe and pump beam are 4.3 mw and 100 @xmath54w , respectively . , width=283 ] resonances in @xmath0rb of ( a ) absorption without sidebands and ( b ) demodulated absorption of 780 nm laser as a function of detuning of the 776 nm laser . the power of the probe and pump beam are 4.3 mw and 100 @xmath54w , respectively.,width=283 ] we monitor the laser frequency of the probe beam using a coherent confocal fabry - perot cavity with a free spectral range of 1.5 ghz to test the performance of the laser lock . [ figure error signal ] shows the fringe - side transmission of the probe laser through the cavity . we monitor the behavior of the laser before and after it has been locked . the reduction of the frequency excursions is quite evident as the laser is locked to the atomic resonance . under normal experimental conditions we have observed locking times of 30 minutes , and a significant reduction of the rms noise of more than a factor of seven . excited atomic transition using fmp.,width=283 ] a thick glass plate splits into two the main beam at 795 nm in the amp system before entering an independent rubidium vapor glass cell inside a three layered magnetic shield . a grating narrowed diode laser at 1.324 @xmath54 m ( from here on referred to as 1.3 @xmath54 m laser ) with a linewidth better than 500 khz excites the second transition . we scan the frequency of the 1.3 @xmath54 m laser over the hyperfine manifold of the @xmath52 level . a fiber - coupled semiconductor amplifier increases the power of the 1.3 @xmath54 m laser before it goes to a large bandwidth ( @xmath4310 ghz ) electro - optic modulator ( eom ) that generates the sidebands . the power of each 795 nm beam is approximately 10 @xmath54w with a diameter of 1 mm . we operate in the low intensity regime to avoid power broadening , differential ac stark shifts and line splitting effects such as the autler - townes splitting . both beams are circularly polarized by a @xmath58 waveplate . the counter propagating 1.3 @xmath54 m laser beam with a power of 4 mw and approximately equal diameter overlaps one of the 795 nm beams . after the glass cell an independent photodiode detects each 795 nm beam . the outputs of the detectors go to a differential amplifier to reduce common noise . a digital oscilloscope records the output signal for different values of modulation . [ figure whole scan ] shows an absorption spectrum of the 795 nm laser as a function of the detuning of the 1.3 @xmath54 m laser that shows the signature sidebands of the technique . [ figure linear regression ] shows a plot of the distance between the sidebands as a function of the modulation of the 1.3 @xmath54 m laser . the sidebands that appear on the absorption spectra provide _ in situ _ calibration for the energy spacing of the hyperfine splittings . this effectively translates a measurement of energy spacings from the optical region to a much easier measurement in the radio frequency range . we observe a rich atomic behavior such as eit and reversal of the peaks that depends on the power of the lasers , relative polarization and magnetic field intensity ( see for example fig [ figure bad absorption ] ) . this points towards a stringent control on experimental parameters for precision studies of energy separations . , f=1 and f=2 hyperfine states of @xmath0rb with sidebands . the big sideband belongs to the f=1 peak . the small feature on the side of the f=2 peak corresponds to the second sideband of the f=1 peak . the glass cell is in a magnetic field of 0.37 g.,width=283 ] m laser for @xmath0rb . the arrow point to the value of the modulation that corresponds to the overlap of the sidebands and half the hyperfine splitting of the @xmath52 level hyperfine splitting.,width=283 ] table [ table theory and experiment ] shows the values of the magnetic dipole constants using relativistic mbpt @xcite with single double ( sd ) and single double partial triple ( sdpt ) wave functions and values extracted from measurements of the hyperfine splitting in other electronic states currently in the literature for @xmath59=1/2 @xcite . we have not been able to find in the literature values for higher levels with adequate precision to include them in the figure . the agreement of the theory with the experiment , for @xmath59=1/2 levels , is well within the 1% level . the sdpt relativistic wave functions do indeed improve the accuracy of the calculations of the single double wave functions . .single double ( sd ) and partial triple ( sdpt ) excitation calculated from _ ab intio _ mbpt in ref . @xcite and experiment magnetic dipole constants for the first @xmath59=1/2 levels in @xmath60rb . ( adapted from ref . @xcite ) . [ cols="<,^,^,^",options="header " , ] the accuracy of the 6@xmath61 measurement is high enough to extract a hyperfine anomaly @xcite in an excited state , which shows that the effect is independent of the n state of the level , as originally predicted by bohr and weiskopf @xcite . we have presented two - color modulation transfer spectroscopy as reliable and simple method for studies of atomic properties in excited states . the characteristic sidebands appearing the spectra have the two - fold utility of working as an _ in situ _ ruler for measurements of energy separations or to lock the frequency of a laser to an excited transition . the good quality of the data presented is due to monitoring of the absorption of the pump beam instead of direct absorption of the probe beam . the absorption of the pump beam ( or lack thereof ) , is always guaranteed since a vast amount of atoms are always in the ground state and even small changes _ i.e. _ excitation to the last step of the transition , will be noticeable even for small powers of the pump beam . in addition , the spectra does not show a doppler background due to the lack of an equilibrium thermal population in the intermediate state . it is the hope that the method will stimulate studies of atomic properties of excited states and further push the experimental precision and theoretical work in excited atomic states . work supported by nsf . a.p.g . would like to thank e. gomez for discussions on the subject of this article and p. barberis for help on the theory of three level atoms .

we present two - color modulation transfer spectroscopy as a tool for precision studies of atomic properties of excited states . the bi - colored technique addresses a narrow set of velocity groups of a thermal atomic vapour using a two - step transition to burn a hole " in the velocity distribution . the resulting spectrum presents sub - doppler linewidths , good signal to noise ratio and the trademark sidebands that work as an _ in situ _ ruler for the energy spacing between atomic resonances . the spectra obtained can be used for different applications such as measurements of energy splittings or stabilization of laser frequencies to excited atomic transitions . = 10000

all stars later than f5 possess convective zones that drive hot corona heated to 1 - 10 mk . for this standpoint , the sun has a moderately heated corona ( 1 - 3 mk ) extending from the transition zone to a few solar radii . the solar coronal heating is observed in the soft x - ray ( sxr ) and euv bands and plays a critical role in controlling the thermodynamics and chemistry of the earth s upper atmosphere ( meier 1991 ) . the corona s variable radiative output is associated with flares and coronal mass ejections that affect space weather , and eventually , life on earth . variations in the radiation affect radio signal propagation and satellite drag thereby impacting communication , navigation , surveillance , and space debris collision avoidance . predicting the spectral irradiance from the global sun is therefore a major goal of the national space weather program . having this capability requires an understanding of the puzzling physical mechanism that heats the outermost part of the solar atmosphere , the solar corona , to multi - million degree temperatures . stellar sxr observations have revealed that the coronal heating processes are not unique to the sun , but are common in magnetically active stars . therefore , understanding the origin of high - temperature plasma in the solar / stellar coronal environments is one of the fundamental problems of solar physics and stellar astrophysics . while stellar observations show a large variety of coronal environments characterized by up to four orders of magnitude larger heating rates ( for example on rs cvn stars or coronal giants ) , higher spatial and spectral resolution euv / sxr observations of the solar corona provide the critical data for resolving this puzzle . specifically , first sxr yohkoh and later soho observations of the global sun have revealed that the solar coronas represent a highly inhomogeneous environment filled with plasma frozen to magnetic structures of two basic configurations : open and closed . magnetically open structures extend from the solar photospheres into the heliosphere , while closed structures are signified as loop - like structures filled with relatively dense ( 10@xmath0 @xmath1 ) and hot ( few mk ) plasma emitting in euv lines of highly ionized metals . while the quite - sun regions are associated with weak magnetic fields ( a few gauss ) , euv / sxr emitting plasma in active regions ( ar ) is formed in magnetic structures that can be traced back to strong ( over 1 kg ) surface magnetic fields . the strongest magnetic field in ars is usually associated with hotter ( @xmath2 5 mk ) and denser plasma which is observed as higher contrast in aia and sxr images , while regions with weaker fields show signatures of cooler plasma . this association clearly relates the problem of coronal heating to the energy stored and released in the solar coronal magnetic field . energy into the magnetic field is likely supplied from the mechanical energy of photospheric convective motions . the coronal loops observed in the ar core are usually shorter , denser with higher temperature and associated with stronger magnetic fields . the footpoints of core loops are observed in euv structures called `` moss '' ( fletcher & de pontieu 1999 ; de pontieu et al . studies of the temperature evolution of ar coronal loops in time suggested that their emission in euv results from impulsive heating events occurring at sub - resolution scale ( or strands ) and ignited a new heating scenario of coronal loops through `` nanoflare storms '' ( klimchuk 2006 ) . the recent evidence in favor of impulsive heating in coronal loops comes from observations of time - lag of peaks of emission observed in high - temperature lines compared to cooler lines suggesting that these loops can be explained by so - called long nanoflare storms occurring in many strands within a coronal loop ( klimchuk 2009 ; viall & klimchuk 2012 ) . recent high spatial resolution sdo and the latest high - resolution coronal imager ( hi - c ) observations of one active region imply that a magnetic loop is not a monolithic structure , but consists of many ( possibly hundreds ) of unresolved `` strands , '' with the fundamental flux tubes thinner than 15 km ( peter et al . 2013 ; brooks et al . 2013 ) . moreover , a nanoflare scenario was further specified from analysis of cool , dense and dynamic loops observed by hi - c observations in lower parts of coronal loops ( winebarger et al . 2013 ) . two leading theories provide an explanation for how `` nanoflares '' release magnetic energy in the corona . magnetic energy dissipated in coronal loops is supplied by the photospheric convection either in the form of upward propagated mhd waves ( asgari - targhi & van ballegooijen 2012 ) or formation of current sheets driven by twisting and braiding of coronal field lines forming a nanoflare storm ( parker 1988 ) . in either of these proposed scenarios , energy can dissipated at small scales on a single `` strand '' ( a flux tube ) in a series of transient heating events . two important questions are : what is the time scale between two successive `` nanoflares '' ( or frequency of nanoflares ) within an ar coronal loop ? to what extent are waves or current sheets responsible for nanoflare heating ? these two theories predict distinctive scaling laws of the heating rates with magnetic field and characteristic spatial scales of coronal loops ( mandrini et al . 2000 ) . all coronal loop models presented to date can be divided into three categories . early models of equilibrium loops by rosner et al . ( 1978 ) and craig et al . ( 1978 ) suggested that loops are symmetric , semi - circular monolithic loops with uniform cross section in static equilibrium . these and later studies of individual loops were successful in explaining many signatures of sxr and euv loops ( porter & klimchuk 1995 ; cargil & priest 1980 ; aschwanden & schriver 2002 ; winebarger et al . 2003 ; reep et al . 2013 ) . this approach is useful in studying detailed response of individual loops to different heating scenarios ; however , it is difficult to compare them directly to observations of active regions with collections of loops `` contaminated '' by selection and line of sight ( los ) effects . another approach is to construct three dimensional mhd models of an active region that will accommodate the above mentioned effects ( lionello et al . 2005 ; gudiksen & nordlund 2005 ; bourdin et al . these models are extremely useful in understanding a general geometry and dynamics of magnetic structures and can be directly compared to observations . however , they are computationally expensive especially when it comes to resolving physically important scales as well as in treating thermal conduction at small scales in individual loops . the third class of emerging models incorporates the advantages of individual loop models with geometry and los effects . this class includes forward models of active regions ( lundquist , fisher & mctiernan 2008a ; 2008b , patsourakos & klimchuk 2008 ; airapetian & klimchuk 2009 ) . airapetian & klimchuk ( 2009 ) have developed a new class of impulsive coronal heating models that are based on introducing magnetic field extrapolation of active regions using hmi / sdo magnetograms . they make use of the `` 0d '' hd code , ebtel , which provides a computationally fast way to derive loop averaged temperature and density and construct 2d synthetic images of an active region driven by nanoflare storms . however , that model assumed a uniform cross section of modeled loops as well as uniform heating along each loop . in the current paper , we have significantly expanded on the capabilities of forward models of active regions to construct realistic synthetic images of individual ars and the global sun by applying our state - of - the - art fully non - linear 1d hydrodynamic code . first , we developed a fundamentally new class of active region models based on parametrically specified impulsive heating of individual strands ( flux tubes ) comprising coronal loops . we begin with constructing a `` magnetic skeleton '' of an active region using the most sophisticated methods to extrapolate non - linear force free coronal magnetic fields ( nlfff ) from high resolution vector hmi / sdo and solis observations ( tadesse et al 2013 ) . we then study how the entire active region ( with los projection effects ) responds to the heating function ( volumetric heating rate ) scaled with magnetic field and spatial scale parameters and find the best match between synthetic and actual ( reconstructed ) dems obtained by sdo . in this paper we construct synthetic em images of specific ars in euv and sxr bands , we need first to construct a 3d equilibrium magnetic loop model of an entire ar or the `` magnetic skeleton '' of an active region . the magnetic skeleton in the solar corona can be realistically constructed by using sdo / hmi vector magnetograms and extrapolating them into the inner solar corona . reliable magnetic field measurements are still restricted to the level of the photosphere , where the inverse zeeman effect in fraunhofer lines is observable . as an alternative to measurements in these super - photospheric layers , we must rely on numerical computations ( known as extrapolation ) ( amari et al . , 2006 ) of the field that use the observed photospheric magnetic field vector as a boundary condition . these numerical computations can be carried out using potential field , force - free field or magneto - hydrodynamics ( mhd ) models . force - free models do include electric current , and so they can include free - magnetic energy . force - free models make the simplifying assumption that these currents are field aligned . a force - free model gives static representations of the state of the solar corona at a given instant . this is a good approximation in the low- corona because the vanishing lorentz - force does not allow currents perpendicular to the magnetic field . by applying a force - free model to a time sequence of magnetograms we can study the changes in magnetic configuration that results from a flare or eruption . in nonlinear force - free field ( nlfff ) models , there are no forces in the plasma which can effectively balance the lorentz force , @xmath3 , ( where @xmath4 and @xmath5 have the standard definitions of current density and magnetic field , respectively ) . nlfff extrapolation is a realistic way to model the non - potential coronal fields in active regions . we use an optimization procedure to calculate 3-d magnetic field solutions into the corona from photospheric boundary . we implement cartesian or spherical geometry depending on the size of area of region of interest . we have developed idl tools which help us trace the magnetic field . to describe the equilibrium structure of the static coronal magnetic field , the force - free assumption is appropriate : @xmath6 @xmath7 subject to the boundary condition @xmath8 on photosphere where @xmath9 is the magnetic field and @xmath10 is measured vector field on the photosphere . using the three components of b as a boundary condition requires consistent magnetograms , as outlined in aly ( 1989 ) . the photospheric vector magnetograms , obtained by the synoptic optical long - term investigations of the sun survey ( solis)/vector spectromagnetograph ( vsm ) or hmi / sdo are used as the boundary conditions . meanwhile , those measured data are inconsistent with the above force - free assumption . therefore , one has to apply some transformations to these data before nonlinear force - free extrapolation codes can be applied . this procedure is known as preprocessing . this preprocessing scheme modifies the boundary data so that they are consistent with necessary conditions for a force - free field , namely so that integrals representing the net force and torque on the coronal volume above the photosphere are closer to zero ( wiegelmann et al . 2006 ; tadesse et al . we solve the force - free equations using an optimization principle ( wheatland et al . 2000 ; wiegelmann 2004 ) in spherical geometry ( wiegelmann 2007 ; tadesse et al . 2009 , 2012 ; 2013 ) . for our test calculations , we have selected ar 11117 observed by sdo on oct 26 , 2010 at 04:00 ut . the image in 171 is presented in figure 2 . using the described technique , we have constructed a `` magnetic skeleton '' of the active region containing over 12,000 strands . we then imposed a background heating rate in each strand and evolve them using time - dependent hydrodynamics until they have reached equilibrium . coronal loops are be treated as bundles of magnetic field lines ( or elementary flux tubes ) that expand into the corona but are rooted in the solar photosphere . their lengths are much greater than their widths and their orientation is along the direction of the magnetic field . the expansion factor ( cross - section ) of each individual flux tube is controlled by the condition of the magnetic flux conservation along the tube , specified at the photosphere from the local magnetic field derived from a magnetogram and the minimum size of the magnetic element resolved by hmi observations , 350 km . once the magnetic skeleton of the active regions is constructed , we populate each strand of the active region with an initial atmospheric state . to do this we apply uniform background heating , @xmath11 , that provides the temperature of 0.5 mk . this allows density and temperature to reach a steady - state equilibrium . to simulate the thermodynamics in each coronal loop driven by a storm of impulsive `` nanoflare '' events , we use a time - dependent heating rate applied to each strand . the heating rate we use has a general form allowing us to model energy release due to a number of different physical mechanisms . the time - dependence of the impulsive heating from each nanoflare were modeled as triangular pulse with a maximum value given by , @xmath12 after heating has been applied for a specified duration we continue to simulate the strands as they cool . the duration of each pulse as well the number of heating pulses applied , and the cooling time are also free parameters . varying these allows to study the frequency with which nanoflare heating occurs . therefore , the physical size of the cell varies for each loop with its length with the grid resolution of a few tens of km at the loop base . the heating function , @xmath13 , is scaled with the local value of the magnetic field within each _ nth _ cell as @xmath14 and the physical extent of the cell as @xmath15 . therefore , in each cell the local heating function is defined as @xmath16 in the low solar corona , the magnetic forces dominate over gas pressure . in this regime , plasma is constrained to flow along magnetic field lines and the magnetic field remains static over the time scales which we simulated . thus , the full 3d mhd equations can be well - approximated by one - dimensional hydrodynamics with that dimension being the axis of magnetic field lines . to model solar coronal loop dynamics , we solve the 1d hydrodynamic equations using a modified form of the arc7 code ( allred & macneice 2012 ) . arc7 was created to solve the equations of mhd in 2.5d geometry . it solves the equations of mhd explicitly using a 2nd order accurate in time and space flux - corrected transport algorithm . a radiative loss term is included in the energy conservation equation this term is proportional to @xmath17 , where @xmath18 is the electron number density and @xmath19 is the radiative loss function and is obtained from the chianti package ( dere et al . 2009 ) . field - aligned thermal conduction is included in the energy conservation equation and is assumed to have the classical spitzer formulation . however , during our impulsive heating simulations temperature gradients can occasionally become large enough that the spitzer formula predicts fluxes which would exceed the free electron streaming rate . this is unphysically large and we cap the heat flux at the free streaming rate . in order to capture the effect of the expansion of the magnetic field from the footpoints into the corona , we scale the cross sectional area of arc7 s grid cells so that magnetic flux is conserved . at the loops boundaries ( i.e. , footpoints ) we have implemented a non - reflecting boundary condition so that waves can pass through . the boundary of our loops is held at a temperature of 20,000 k and start with sufficient mass density so that material can be evaporated into the corona in response to impulsive heating without significantly changing the boundary density . we perform an impulsive heating simulation using the following algorithm . an initial background heating rate is specified to obtain an equilibrium temperature 0.5mk . we use the rtv scaling laws ( rosner et al . 1978 ) to setup a starting atmospheric state within loops depending on the background heating rate and loop length . we allow arc7 to evolve the loop until it reaches equilibrium . we then turn on the impulsive heating term which linearly ramps the heat function up until it reaches a maximum value and then linearly ramps it down over a time @xmath20 t . the maximum value heating function is assumed to have the form @xmath21 , where @xmath22 is a coefficient , @xmath9 is the magnetic field strength and @xmath23s is the length of the element along a flux tube . we also specify n , @xmath24 , and @xmath25 , where n=4 is the number of heating pulses we applied during the simulation , @xmath24 is the time interval between heating pulses and @xmath26 is the time we allow the loop to cool after the impulsive heating has been applied . we have chosen to use arc7 because of its high - speed performance . as noted , arc7 is a 2.5d mhd code . our proposed method requires hydrodynamics in only one spatial dimension because plasma is frozen - in the magnetic field in a low-@xmath27 low corona we have simplified arc7 to take advantage of these assumptions which results in a vast improvement in performance . using a standard single processor computer , we can model the evolution of a single loop in response to impulsive heating in a few seconds . our model active regions have on the order of 104 individual strands . we performed these strand simulations in parallel using 100 processors simultaneously on nasa s pleiades supercomputer and completed the simulations over an entire active region in about an hour . to reproduce the magnetic structure of the active region , we have used 12,800 individual strands and ran individual trains of nanoflares ( low frequency events ) on each of them . we then ran nanoflare trains on each of the strands . we selected the duration of a nanoflare as @xmath23 t = 200 s with the time interval between two successive events , @xmath28 = 200 s. for this example we have used @xmath29 = 2 and @xmath27 = 2 . in the selected extrapolation model of the active region , the loop lengths vary between 5 mm and 200 mm . we ran simulations with 5 consecutive pulses then another 5000 s of cooling time . figure 3 shows the temperature and density at the flare peak in a single strand compared with the background temperature and density . the temporal evolution of the peak of that strand is shown in figure 4 . we combined the results of all of our hd simulations to form a 2d picture of the dem for that active region . we calculated the dem in each grid cell of each strand using our 1d simulations . we then averaged these dem in time over the duration of the simulations . time - averaging captures the assumption that these impulsive heating events occur at random intervals and are independent of each other . these time - averaged dems were projected along the line - of - sight back onto hmi pixels forming a 2d representation of the temperature and density structure of that ar . once the dem is known , we calculated the optically - thin radiation spectrum , i(@xmath30 ) , using the most recent version of chianti atomic database package ( dere et al . 2009 ) . the right panel of figure 5 shows an example of the dem constructed from our simulations . our model results can be compared with observations in two ways . first , we can convolve our dem with aia filter passbands to produce synthetic images which can be compared directly with aia images . we can also construct a dem from aia images and compare that directly with our simulated dem . developing methods for constructing dems from aia images is a very active topic of research . we have used the tool developed by hannah & kontar ( 2012 ) . this tool uses a regularized inversion method and has the advantage that it provides uncertainties in both the dem and temperature ( i.e. , it provides both horizontal and vertical error estimates ) . the aia images were obtained and processed using solarsoftware ( ssw ) idl packages . we downloaded level 1 aia images for all passbands for the time interval over which our hmi magnetogram was observed using the ssw routines vso@xmath31search and vso@xmath31get . next , we converted them to level 1.5 and co - aligned them with the hmi magnetogram using the aia@xmath31prep function . finally , we ran the dem construction program data2dem@xmath31reg provided by hannah & kontar ( 2012 ) on all pixels which modeled in our simulations . the left panel of figure 5 shows this dem reconstruction at a temperature of log t = 6.5 . this simulated dem distribution will be compared with the observationally derived dem for the active region in the near future . we have constructed the first realistic synthetic em images of the entire coronal active region , ar 11117 driven by a storm of nanoflares . each nanoflare event was modeled by using our 1d fully non - linear time - dependent single - fluid hydrodynamic code . we simulated the response of the entire active region to a storm of nanoflares specified by impulsive ( time - dependent ) heating function occurring on over 12,000 strands within the active region . the heating function is scaled with the magnetic field and spatial scale parameters with @xmath29=2 , @xmath27=2 power indices . the reconstructed dem for this ar will be compared with the observationally derived dem for the active region in the near future . we will also construct dems for a range of @xmath29 and @xmath27 values to determine the sensitivity of its shape to the specified shape of the heating function .

recent progress in obtaining high spatial resolution images of the solar corona in the extreme - ultraviolet ( euv ) with hinode , trace , sdo and recent hi - c missions and soft x - ray ( sxr ) bands opened a new avenue in understanding the solar coronal heating , the major goal of solar physics . the data from euv / sxr missions suggest that solar corona is a non - uniform environment structured into active regions ( ar ) represented by bundles magnetic loops heated to temperatures exceeding 5 mk . any viable coronal heating model should be capable of reproducing euv and sxr emission from coronal active regions well as dynamic activity . measurements of emission measures ( em ) for ars provide clues to time dependence of the heating mechanism : static versus impulsive . while static equilibrium coronal loop models are successful in reproducing sxr emission within an ar , they can not adequately predict the bright euv loops . meantime , impulsive heating is capable in reproducing both euv and sxr loop emission . the major goal of this paper is to construct realistic synthetic em images of specific solar corona active region , ar 11117 by using our 1d fully non - linear time - dependent single - fluid hydrodynamic code . we first construct a magnetic skeleton for the entire active region using the hmi / sdo magnetogram for ar 11117 and populate magnetic field lines with plasma . we then parametrically specify impulsive heating of individual strands ( flux tubes ) comprising coronal loops . next , we simulated the response of the entire active region ( with los projection effects ) to the heating function ( volumetric heating rate ) scaled with magnetic field and spatial scale parameters and find the best match between synthetic and actual ( reconstructed ) dems obtained by sdo .

the active binary capella ( @xmath3 aurigae , hd 34029 , hr 1708 ) was observed with the high energy transmission grating spectrometer ( hetgs ) on the chandra x - ray observatory ( cxo ) . we present a first analysis of the spectra with the goals of demonstrating the hetgs performance , and of applying plasma diagnostics to infer physical parameters of the capella corona . a complementary analysis of the corona of capella based on high resolution spectra obtained using the cxo low energy transmission grating spectrometer ( letgs ) has been presented by @xcite . further analysis of diagnostic emission lines from these and other chandra grating data of capella are underway with the goal of obtaining refined temperature - dependent emission measures , abundances , and densities , leading to a self - consistent determination of the coronal structure . [ [ the - chandra - hetgs ] ] the chandra hetgs : + + + + + + + + + + + + + + + + + + the high energy transmission grating assembly @xcite consists of an array of periodic gold microstructures that can be interposed in the converging x - ray beam just behind the chandra high resolution mirror assembly . when in place , the gratings disperse the x - rays according to wavelength , creating spectra that are recorded at the focal plane by the linear array of ccds designated acis - s . there are two different grating types , designated meg and heg , optimized for medium and high energies ( partially overlapping in spectral coverage ) . the hetgs provides spectral resolving power of @xmath4 - 1000 for point sources ( corresponding to a line fwhm of about 0.02 for meg , and 0.01 for heg ) and effective areas of 1 - 180 @xmath5 over the wavelength range 1.2 - 30 ( 0.4 - 10 kev ) . multiple overlapping orders are separated using the moderate energy resolution of the acis detector . the hetgs complements the letgs , which is optimized for lower energy x - rays . ( for detailed descriptions of the instruments see http://chandra.harvard.edu ) . preliminary analysis of in - flight calibration data including those presented here indicates that the hetgs is performing as predicted prior to the chandra launch . the spectral resolution is as expected and effective areas are within 10% of the expected values except from 612 where there are systematic uncertainties of up to 20% . ongoing calibration efforts will reduce these uncertainties . [ [ the - coronal - structure - of - capella ] ] the coronal structure of capella : + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + capella is an active binary system comprised of g1 and g8 giants in a 104 d orbit at a distance of 12.9 pc . the g1 star rotates with an @xmath6 d period @xcite . capella has been studied by many previous x - ray telescopes , including einstein @xcite , exosat @xcite ; rosat @xcite , beppo - sax @xcite , and asca @xcite . the fundamental parameters of capella , some activity indicators , and primary references may be found in @xcite . the corona of capella appears intermediate in temperature , being cooler than those of rs cvn stars such as hr 1099 or ii peg , but significantly hotter than a less active star like procyon . x - ray observations obtained at low to moderate spectral resolution are generally consistent with emission from an optically thin , collisionally dominated plasma with two temperature components @xcite . spectra obtained by the extreme ultraviolet explorer ( euve ) have provided more discriminating temperature diagnostics , showing plasma over a continuous range of temperatures , with the peak emission measure near @xmath7 @xcite . simultaneous measurements using euve and asca spectra did not require emission from plasma hotter than @xmath8 @xcite . euve observations show variability by factors of 3 to 4 in lines formed above @xmath9 @xcite . @xcite have estimated plasma electron densities in the range from @xmath10 to @xmath11 from lines of fe xxi formed near @xmath12 , implying that the scale of the emitting volume is @xmath13 , although @xcite question the reliability of this diagnostic . @xcite use euv lines of fe xviii to constrain the optical depth in the strong x - ray emission line , fe xvii @xmath1415.014 , to @xmath15 . from high - resolution uv spectra from the hubble space telescope , @xcite concluded that both stars have comparable coronal emission , based on measurements of the fe xvii ( 1354 ) coronal forbidden line , and that the plasma is magnetically confined . thus the `` corona '' of capella is actually a composite of two `` coronae . '' we combined data from three hetgs observations ( from 1999 august 28 , september 24 & 25 ) for a total exposure of 89 ks . data were processed with the standard chandra x - ray center software ( versions from july 29 ( r4cu3upd2 ) and december 13 ( ciao 1.1 ) ) . the image of the dispersed spectrum is shown in figure [ fig : image ] . each photon is assigned a dispersion angle , @xmath16 , relative to the undiffracted zero - order image . the angle is related to the order , @xmath17 , and wavelength , @xmath14 , through the grating mean period , @xmath18 , by the grating equation , @xmath19 . the spectral order is determined using the acis - s ccd pulse height for each photon event ( with wide latitude to avoid sensitivity to variations in ccd gain or pulse height resolution ) . the positive and negative first orders were summed separately for heg and meg for all observations and divided by the effective areas to provide flux - calibrated spectra ( figure [ fig : spectrum ] ) . listed in table [ tab : linelist ] . the fe xvii @xmath20 line strength is , within the uncertainties , identical to that observed in 1979 with the einstein crystal spectrometer by @xcite , while the o viii @xmath21 line is roughly half the previous value . [ [ emission - measure - distribution ] ] emission measure distribution : + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + some properties of the coronal temperature structure can be deduced from a preliminary analysis of the spectrum . the data warrant a full analysis of the volume emission measure distribution with temperature , @xmath22 , ( @xmath23 in which @xmath24 is the electron density of plasma at temperature @xmath25 which occupies the volume , @xmath26 ) , which will be the subject of a future paper . as table [ tab : linelist ] illustrates , the spectrum contains lines from different elements in a range of ionization states , demonstrating that the emitting plasma has a broad range of temperature . further evidence of multi - temperature emission comes from two line ratios . first , ratios of h - like to he - like resonance lines , o viii / vii , mg xii / xi , and si xiv / xiii indicate ionization ratios corresponding to @xmath27 = 6.55 - 6.60 , 6.75 - 6.85 , and 6.95 - 7.00 , respectively . second , the he - like ions provide temperature - sensitive ratios involving the resonance ( @xmath28 ) , forbidden ( @xmath29 ) and intersystem ( @xmath30 ) lines @xcite . for the observed o vii , mg xi , and si xiii lines , the ratio @xmath31 corresponds to temperatures @xmath32 , @xmath33 , and @xmath34 , respectively , using the theoretical models of smith et al . ( 1998 , in the low density limit ) . in both cases , the ratios indicate that the corona has a broad range of temperature . an approximate upper envelope to the true @xmath35 distribution is given by the family of curves formed by plotting the ratio of line strength to corresponding emissivity for a collection of lines . for a given element , its abundance affects only the overall normalization of the envelope of all lines from that element . for this initial analysis , we assumed solar abundances @xcite , which is consistent with previous analyses except possibly for ne @xcite . the vem envelope of figure [ fig : vemt ] , indicatates that plasma must be present over nearly a decade in temperature . the absence of lines from he - like and h - like ions of fe provides an upper limit to the @xmath35 above @xmath36 . although the envelope does not trace closely the peaked distribution derived from euv lines , such a distribution is not excluded , [ [ density - diagnostics ] ] density diagnostics : + + + + + + + + + + + + + + + + + + + + the he - like @xmath37 ratio is primarily sensitive to density using the theoretical line ratios of smith et al . ( 1998 ) , our measured o vii ratio of @xmath38 implies an electron density within the range @xmath39@xmath40 . similarly , the mg xi and si xiii ratios of @xmath41 and @xmath42 give upper limits near @xmath43 and @xmath44 , respectively . we note that our ratio @xmath37 for o vii is somewhat lower than that obtained by @xcite from letgs spectra . the hetgs and letgs observations were not simultaneous ; however , based on evidence from prior euve observations @xcite , we would be surprised if this difference represented actual changes in the mean coronal plasma density . instead , we suggest that this results from different treatements of the continuum plus background , which particularly affects the strength of the intercombination line . these x - ray data confirm that capella s corona contains plasma at multiple temperatures in the accessible range from @xmath45 to @xmath46 , and set stringent constraints on the amount of plasma hotter than @xmath47 at the time of this observation . these properties are generally consistent with the results found with euve and asca @xcite and the line strengths are close those seen 20 years earlier by @xcite . the preliminary results presented here have implications for the structure of capella s corona : they suggest that the characteristic dimensions of the coronal loops at @xmath48 are small compared to the stellar radius , @xmath49 . for simple semi - circular loops of constant circular cross - section of radius @xmath28 , we use the measured density and @xmath35 for oxygen to estimate loop heights @xmath50 , where @xmath51 is the ratio of @xmath28 to loop length in units of 0.1 , and @xmath52 is @xmath53 the number of loops . detailed loop modeling of @xcite also required compact structures , though variable cross - section loops were needed to increase the proportion of hot to cool plasma . work at mit was supported by nasa through the hetg contract nas8 - 38249 and through smithsonian astrophysical observatory ( sao ) contract svi61010 for the chandra x - ray center ( cxc ) . jjd and nsb were supported by nasa nas8 - 39083 to sao for the cxc . we thank all our colleagues who helped develop the hetgs and all members of the chandra team . anders , e. , & grevesse , n. , 1989 , geochimica et cosmochimica acta 53 , 197 . brickhouse , n. s. 1996 , in proc . iau colloq . 152 , _ astrophysics in the extreme ultraviolet _ , ed . s. bowyer & r. f. malina , ( dordrecht : kluwer ) , 105 . brickhouse , n. s. , dupree , a. k. , edgar , r. j. , liedahl , d. a. , drake , s. a. , white , n. e. , & singh , k. p. 2000 , , 387 . brinkman , a.c . , et al . , 2000 , , ( submitted ) brown , g.v . , beiersdorfer , p. , liedahl , d.a . , et al . , 1998 , , 502 , 1015 . canizares , c.r . 2000 in preparation dempsey , r.c . , linsky , j.l . , schmitt , j.h.m.m . , and fleming , t.a.,1993 , , 413,333 dupree , a.k . , brickhouse , n.s . , doschek , g.a . , green , j.c . , and raymond , j.c . , 1993 , , 418 , l41 dupree , a.k . , brickhouse , n.s . , and sanz - forcada , j. , 2000 , in preparation . favata , f. , mewe , r. , brickhouse , n. s. , pallavicini , r. , micela , g. , & dupree , a. k. 1997 , , 324 , l37 gabriel , a. h. 1972 , , 160 , 99 gabriel , a. h. , & jordan , c. 1969 , , 145 , 241 griffiths , n.w . , jordan , c. , 1998 , 497 , 883 holt , s. s. , white , n. e. , becker , r. h. , boldt , e. a. , mushotzky , r. f. , serlemitsos , p. j. , & smith , b. w. 1979 , , 234 , l65 hummel , c. a. , armstrong , j. t. , quirrenbach , a. , buscher , d. f. , mozurkewich , d. , & elias ii , n. m. 1994 , , 107 , 1859 lemen , j. r. , mewe , r. , schrijver , c. j , & fludra , a. 1989 , , 341 , 474 linsky , j.l . , wood , b.e . , brown , a. , & osten , r.a . , 1998 , , 492 , 767 markert , t.h . , canizares , c.r . , dewey , d. , mcguirk , m. , pak , c.s . , & schattenburg , m.l . , 1994 , proc . spie , 2280 , 168 mewe , r. et al . 1982 , , 260 , 233 pradhan , a. k. and shull , j.m 1981 , , 249 , 821 saba , j.l.r . , schmelz , j.t . , bhatia , a.k . , and strong , k.t . , , 510 , 1064 . schmitt , j.h.m.m . , collura , a. , sciortino , s. , vaiana , g.s . , harnden , f.r . , jr . , and rosner , r. , 1990 , , 365 , 307 . schrijver , c.j . , mewe , r. , van den oord , g.h.j . , and kaastra , j.s . 1995 , , 302 , 438 . smith , r. k. , brickhouse , n. s. , raymond , j. c. , & liedahl , d. a. 1998 , in proceedings of the first xmm workshop on `` science with xmm '' , ed . m. dahlem ( noordwijk , the netherlands ) strassmeier , k.g . , hall , d.s . , fekel , f.c . , and scheck , m. , 1993 , , 100 , 173 swank et al . , 1981 , , 246 , 214 van den oord , g.h.j . , schrijver , c.j . , camphens , m. , mewe , r. , & kaastra , j.s . , , 326 , 1090 vedder , p.w . , canizares , c.r . , 1983 , , 270 , 666 rrrrr fe xxv & 1.85 & @xmath54 & @xmath55 & 7.8 + s xv & 5.040 & 33 & 110 & 7.2 + s xv & 5.060 & 16 & 55 & 7.2 + s xv & 5.100 & 26 & 85 & 7.2 + si xiii & 5.680 & 25 & 93 & 7.0 + si xiv & 6.180 & 48 & 472 & 7.2 + si xiii & 6.650 & 182 & 1228 & 7.0 + si xiii & 6.690 & 47 & 374 & 7.0 + si xiii & 6.740 & 121 & 834 & 7.0 + al xii & 7.750 & 16 & 204 & 6.9 + mg xii & 8.419 & 152 & 1947 & 7.0 + mg xi & 9.170 & 348 & 2818 & 6.8 + mg xi & 9.230 & 63 & 618 & 6.8 + mg xi & 9.310 & 190 & 1425 & 6.8 + ne x & 10.240 & 92 & 740 & 6.8 + ni xxii & 10.791 & 62 & 426 & 7.0 + ne x & 12.132 & 929 & 4171 & 6.8 + fe xvii&12.134 & & & 6.8 + fe xix & 13.515 & 530 & 1587 & 6.9 + fe xix & 13.524 & & & 6.9 + fe xvii & 15.013 & 3043 & 7476 & 6.7 + fe xvii & 15.272 & 1119 & 2919 & 6.7 + fe xviii & 15.641 & 410 & 938 & 6.8 + o viii & 16.003 & 898 & 1885 & 6.5 + fe xvii & 16.796 & 2004 & 3669 & 6.7 + fe xvii & 17.071 & 2641 & 4554 & 6.7 + fe xvii & 17.119 & 2443 & 4191 & 6.7 + o viii & 18.967 & 2634 & 2810 & 6.5 + o vii & 21.600 & 967 & 396 & 6.3 + o vii & 21.800 & 255 & 102 & 6.3 + o vii & 22.100 & 736 & 249 & 6.3 + n vii & 24.779 & 549 & 327 & 6.3 +

high resolution spectra of the active binary capella ( g8 iii + g1 iii ) covering the energy range 0.4 - 8.0 kev ( 1.5 - 30 ) show a large number of emission lines , demonstrating the performance of the hetgs . a preliminary application of plasma diagnostics provides information on coronal temperatures and densities . lines arising from different elements in a range of ionization states indicate that capella has plasma with a broad range of temperatures , from @xmath0 , generally consistent with recent results from observations with the extreme ultraviolet explorer ( euve ) and the advanced satellite for cosmology and astrophysics ( asca ) . the electron density is determined from he - like o vii lines , giving the value @xmath1 at @xmath2 ; he - like lines formed at higher temperatures give only upper limits to the electron density . the density and emission measure from o vii lines together indicate that the coronal loops are significantly smaller than the stellar radius .

embedded control systems are ubiquitous and can be found in several applications including aircraft , automobiles , process control , and buildings . an embedded control system is one in which the computer system is designed to perform dedicated functions with real - time computational constraints @xcite . typical features of such embedded control systems are the control of multiple applications , the use of shared networks used by different components of the systems to communicate with each other for control , a large number of sensors as well as actuators , and their distributed presence in the overall system . the most common feature of such distributed embedded control systems ( des ) is shared resources . constrained by space , speed , and cost , often information has to be transmitted using a shared communication network . in order to manage the flow of information in the network , protocols that are time - triggered @xcite and event - triggered @xcite have been suggested over the years . associated with each of these communication protocols are different set of advantages and disadvantages . the assignment of time - triggered ( tt ) slots to all control - related signals has the advantage of high quality of control ( qoc ) due to the possibility of reduced or zero delays , but leads to poor utilization of the communication bandwidth , high cost , overall inflexibility , and infeasibility as the number of control applications increase . on the other hand , event - triggered ( et ) schedules often result in poor control performance due to the unpredictable temporal behavior of control messages and the related large delays which occurs due to the lack of availability of the bus . these imply that a hybrid protocol that suitably switches between these two schedules offers the possibility of exploiting their combined advantages of high qoc , efficient resource utilization , and low cost @xcite . such a hybrid protocol is the focus of this paper . to combine the advantage of tt and et policies , hybrid protocols are increasingly being studied in recent years . examples of such protocols are flexray and ttcan @xcite , used extensively in automotive systems . while several papers have considered control using tt protocols ( see for example , @xcite ) and et protocols ( see for example , @xcite ) , control using hybrid protocols has not been studied in the literature until recently . the co - design problem has begun to be addressed of late as well ( see for example , @xcite ) . in @xcite , the design of scheduling policies that ensure a good quality of control ( qoc ) is addressed . in @xcite , the schedulability analysis of real - time tasks with respect to the stability of control functions is discussed . in @xcite , modeling the real - time scheduling process as a dynamic system , an adaptive self - tuning regulator is proposed to adjust the bandwidth of each single task in order to achieve an efficient cps utilization . the focus of most of the papers above are either on a simple platform or on a single processor . a good survey paper on co - design can be found in @xcite . our focus in this paper is on the co - design of adaptive switching controllers and hybrid protocols so as to ensure good tracking in the presence of parametric uncertainties in the plant being controlled while utilizing minimal resources in the des . the hybrid protocol that is addressed in this paper switches between a tt and a et scheme . the tt scheme , which results in a negligible delay in the processing of the control messages , is employed when a control action is imperative and the et scheme , which typically results in a non - zero delay , is employed when the controlled system is well - behaved , with minimal tracking error . the latter is in contrast to papers such as @xcite and @xcite where the underlying _ event _ is associated with a system error exceeding a certain threshold , while here an _ event _ corresponds to the case when the system error is small . the controller is to be designed for multiple control applications , each of which is subjected to a parametric uncertainty . an adaptive switching methodology is introduced to accommodate these uncertainties and the hybrid nature of the protocol . switched control systems and related areas of hybrid systems and supervisory control have received increased attention in the last decade ( see e.g. , @xcite ) and used in several applications ( see e.g. @xcite ) . adaptive switched and tuned systems have been studied as well ( see @xcite ) . the combined presence of uncertainties and switching delays makes a direct application of these existing results to the current problem inadequate . the solution to the problem of co - design of an adaptive swtiched controller and switches in a hybrid protocol was partially considered in @xcite , where the control goal was one of stabilization . in this paper , we consider tracking , which is a non - trivial extension of @xcite . the main reason for this lies in the trigger for the switch , which corresponds to a system error becoming small . in order to ensure that this error continues to remain small even in the presence of a non - zero reference signal , we needed to utilize fundamental properties of the adaptive system with persistent excitation , and derive additional properties in the presence of reference signals with an invariant persistent excitation property . these properties in turn are suitably exploited and linked with the switching instants , and constitute the main contribution of this paper . in section [ sec : problem ] the problem is formulated , and preliminaries related to adaptive control and persistent excitation are presented . in section [ sec : switchingadaptivecontroller ] , the switching adaptive controller is described and the main result of global boundedness is proved . concluding remarks are presented in section [ sec : conclusion ] . the problem that we address in this paper is the simultaneous control of @xmath0 plants , @xmath1 , @xmath2 , in the presence of impulse disturbances that occur sporadically , using a hybrid communication protocol . we assume that each of these @xmath0 applications have the following problem statement . the plant to be controlled is assumed to have a discrete time model described by @xmath3 + b_0u(k - d)+\sum_{l=1}^{m_2}b_lu(k - l - d)+d(k - d)\end{gathered}\ ] ] where @xmath4 and @xmath5 are the input and output of the @xmath6-th control application , respectively , at the time - instant @xmath7 and @xmath8 is a time - delay . the disturbance @xmath9 are assumed to be impulses that can occur occasionally with their inter - arrival time lower - bounded by a finite constant . the parameters of the @xmath6-th plant are given by @xmath10 , @xmath11 , @xmath12,@xmath13 and are assumed to be unknown . it is further assumed that the sampling time of the controller is a constant @xmath14 , so that @xmath15 . the goal is to choose the control input @xmath16 such that @xmath5 tracks a desired signal @xmath17 , with all signals remaining bounded . the model in ( [ eq : model ] ) can be expressed as @xmath18 where @xmath19 is the backward shift operator and the polynomials @xmath20 and @xmath21 are given by @xmath22 the following assumptions are made regarding the plant poles and zeros : \1 ) an upper bound for the orders of the polynomials in ( [ eq : polyab ] ) is known and 2 ) all zeros of @xmath23 lie strictly inside the closed unit disk . [ ass : fixeddelay ] for any delay @xmath24 , eq . ( [ eq : model ] ) can be expressed in a _ predictor form _ as follows @xcite : @xmath25 with @xmath26 where @xmath27 and @xmath28 are the unique polynomials that satisfy the equation @xmath29 equation ( [ eq : predictorform ] ) can be expressed as @xmath30 where @xmath31 , @xmath32 , @xmath33 , and @xmath34 are defined as @xmath35 @xmath36 with @xmath37 , @xmath38 , @xmath39 , @xmath40 , and @xmath41 and @xmath42 the coefficients of the polynomials in ( [ eq : alphabeta ] ) with respect to the delay @xmath24 and finite initial conditions @xmath43 from eqs . ( [ eq : regressor])-([eq : phiandtheta ] ) , we observe that a feedback controller of the form @xmath44 realizes the objective of stability and follows the desired bounded trajectory @xmath17 in the absence of disturbances . designing a stabilizing controller @xmath4 essentially boils down to a problem of implementing ( [ eq : feedbackcontroller ] ) with the controller gain @xmath32 . two things should be noted : ( i ) controller ( [ eq : feedbackcontroller ] ) is not realizable as @xmath32 and @xmath45 are not known , and ( ii ) the dimension of @xmath31 , @xmath32 as well as the entries of @xmath32 depend on the delay @xmath24 . since @xmath32 and @xmath45 are unknown , we replace them with their parameter estimates and derive the following adaptive control input @xmath46 where @xmath47 denotes the @xmath48-th element of the parameter estimation @xmath49 and is the estimate of @xmath45 . @xmath49 is adjusted according to the adaptive update law @xcite : @xmath50 with @xmath51^t$ ] . equation ( [ eq : avoidzero ] ) is necessary to avoid division by zero in the control law ( [ eq : adaptivecontroller ] ) . theorem [ thm : fixeddelay ] addresses the stability of the adaptive system given by ( [ eq : predictorform ] ) , ( [ eq : adaptivecontroller ] ) , and ( [ eq : updatelaw])-([eq : updateepsilon ] ) . the reader is referred to theorem 6.3.1 in @xcite or theorem 5.1 in @xcite for the proof of theorem [ thm : fixeddelay ] . [ thm : fixeddelay ] let @xmath52 . subject to assumption [ ass : fixeddelay ] and given a fixed delay @xmath24 , the adaptive controller ( [ eq : adaptivecontroller ] ) with the update law ( [ eq : updatelaw ] ) guarantees that the plant given by ( [ eq : predictorform ] ) follows the reference @xmath53 , i.e. , @xmath54 , and that the sequences @xmath55 , @xmath56 and @xmath57 are bounded for all @xmath58 . the following definitions related to persistent excitation are needed to introduce our switching controller . we define the terms _ persistently exciting _ and _ sufficiently rich _ in the following way : [ def : pe ] a sequence @xmath59 is said to be _ persistently exciting ( pe ) ( in @xmath60 steps ) _ , if there exists @xmath61 such that @xmath62 uniformly in @xmath63 . [ def : sr ] a sequence @xmath59 is said to be _ sufficiently rich ( sr ) of order @xmath64 ( in @xmath60 steps ) _ , if there exists @xmath61 such that @xmath65 with @xmath66 uniformly for all @xmath63 . the following lemma is useful to prove theorem [ thm : persistentexcitation ] . suppose that @xmath67 and @xmath68 are two bounded sequences taking values in @xmath69 satisfying @xmath70 . then @xmath67 is sr of order @xmath71 if and only if @xmath68 is sr of order @xmath71 . the reader is referred to @xcite for the proof of theorem [ thm : persistentexcitation ] . [ lem : reachability ] consider the discrete time system @xmath72 with @xmath73 , @xmath74 , @xmath75 , and @xmath76 . assume that ( [ eq : reachable ] ) is completely reachable and that the input @xmath77 is sr of order @xmath78 . then , @xmath79 for all @xmath80 . we first rewrite ( [ eq : reachable ] ) as @xmath81 and define the characteristic polynomial of @xmath20 to be @xmath82 and let @xmath83 then , from the cayley - hamilton theorem it follows that @xmath84 where @xmath85 . let @xmath86 then , @xmath87 where `` @xmath88 '' denotes the direct sum , @xmath89 is the @xmath90 identity matrix , and @xmath91 is the minimal singular value of @xmath92 . @xmath93 can be also stated in terms of @xmath94 in the following way @xmath95 where @xmath96 is given by @xmath97 then , @xmath98 and hence using ( [ eq : kleinergamma ] ) , @xmath99 that is , @xmath100 from theorem 1 in @xcite , it follows that an input signal which is sr of order @xmath71 implies a persistent excitation of at most @xmath71 directions in a @xmath0-dimensional space . thus , @xmath101 _ remark : _ much of the existing results pertaining to persistent excitation pertain to the case when the external input @xmath77 is sr of order @xmath0 . lemma [ lem : reachability ] above as well as corollary [ cor : inomega ] stated below address the case when @xmath77 is sr of order @xmath71 , where @xmath102 , which to our knowledge has not been examined in the literature . as our goal is tracking of an arbitrary signal and not identification , we do not need the sr - order to be @xmath0 , but arbitrary and fixed at some @xmath71 . [ cor : inomega ] consider the discrete time system @xmath103 with @xmath73 , @xmath74 , @xmath75 , and @xmath76 . assume that ( [ eq : reachable2 ] ) is completely reachable and that the input @xmath77 is sr of a fixed order @xmath78 . then , there exists a subspace @xmath104 such that @xmath105 that is , the columns of @xmath94 span the subspace @xmath106 . this follows directly from lemma [ lem : reachability ] and the fact that for any complex matrix @xmath107 the following is true @xmath108 where @xmath109 denotes the image of the linear transformation @xmath107 . we make the following assumption which refers to an invariant property of persistent excitation . [ ass : pe ] @xmath17 is sufficiently rich of constant order @xmath110 for all @xmath58 . theorem [ thm : persistentexcitation ] connects the sufficient richness of @xmath53with the tracking error and the parameter convergence in an adaptive system . [ thm : persistentexcitation ] let @xmath52 . suppose the adaptive controller ( [ eq : adaptivecontroller])-([eq : updateepsilon ] ) is used to control the plant in ( [ eq : thetaregressor ] ) and let assumptions [ ass : fixeddelay ] and [ ass : pe ] hold . then 1 . @xmath111 , and 2 . @xmath112 as @xmath113 3 . @xmath114 converges to @xmath115 where @xmath115 is defined as @xmath116 where @xmath33 is given in ( [ eq : phiandtheta ] ) . item ( i ) follows directly from theorem [ thm : fixeddelay ] as it is independent of any persistent excitation of the reference signal @xmath53 . item ( ii ) follows by noting that the adaptive system in ( [ eq : model ] ) and ( [ eq : adaptivecontroller])-([eq : updateepsilon ] ) becomes asymptotically linear , and this linear system in turn has a state that satisfies ( [ eq : dimmx ] ) due to assumption [ ass : pe ] . item ( iii ) follows from ( i ) and the fact that @xmath117 . hybrid communication protocols such as flexray @xcite provide time - triggered and event - triggered bus schedules . time - triggered communication offers highly predictable temporal behavior , and event - triggered communication provides efficient bandwidth usage . to exploit their combined advantages , we propose the use of a hybrid communication protocol in this paper . to illustrate our proposed scheme , we use flexray as it has been established as the de - facto standard for future automotive in - vehicle networks . the flexray protocol is organized in a sequence of communication cycles of fixed length . further , every such cycle is subdivided into a _ static _ segment ( st ) and a _ dynamic _ segment ( dyn ) . the static segment is partitioned into time windows of fixed and equal length which are referred as _ slots_. each processing unit is assigned one or more slots indexed by a slot number @xmath118 that indicates available time windows for bus access in the static segment . due to the predictable temporal behavior we use the static segment schedules for communication in the time - triggered mode and dynamic segment schedules in the event - triggered mode . the dynamic segment is partitioned into _ minislots _ of much smaller duration than the static slots . similar to the static segment , the minislots are indexed by a slot number to indicate allowable message transmissions . however , dynamic slots are of varying size depending on the size of the message which is transmitted in a certain slot @xmath119 , where @xmath120 is the set of available slot numbers in the dynamic segment . if no message is ready for transmission in a particular slot only one small minislot is consumed and the slot number is incremented with the next minislot . however , if a message is transmitted in a slot @xmath119 then the slot number increments with the next minislot after which the message transmission has been completed . hence , bus resources are only utilized if messages are actually transmitted on the bus ; otherwise only one minislot is consumed . dynamic segment schedules are used for communication in the event - triggered mode . the focus of this problem is the simultaneous control of several applications for stabilization . that is , the goal is to choose @xmath16 , the input of the @xmath6th control application such that @xmath5 , its output , converges to @xmath17 which is zero . in the context of the problem under consideration , all control applications are partitioned into a sensor task @xmath121 , a controller task @xmath122 , and an actuator task @xmath123 ( figure [ fig : bus ] ) . we consider a communication protocol where each communication cycle is divided into time - triggered and event - triggered segments . using _ time - triggered _ communication schedules , denoted as @xmath124 , applications are allowed to send messages only at their assigned slots and the tasks are triggered synchronously with the bus , i.e. , we assume that the communication delay due to the finite speed of the bus is negligible and hence the delay @xmath24 in ( [ eq : predictorform ] ) is equal to @xmath125 . on the other hand , in an _ event - triggered _ schedule , denoted as @xmath126 , the tasks are assigned priorities in order to arbitrate for access to the bus . note that in our setup , multiple control applications share the same bus and hence multiple control messages have to be sent using a common bus and thus the messages might experience a communication delay @xmath127 when the higher priority tasks access the event - triggered segment . we choose the event - triggered communication schedules such that the sensor - to - actuator delay @xmath127 is within @xmath128 sample intervals , i.e. , @xmath129 for the control - related messages and hence the delay @xmath24 is at most equal to @xmath130 with @xmath131 . in summary , the delay @xmath132 if @xmath133 and @xmath134 if @xmath135 where @xmath136denotes the protocol used at time @xmath58 . the properties of the varying delay of the tt and et protocol are directly exploited in the control design in the following way . whenever the error between the plant output and its desired value is above some threshold @xmath137 , we send the control messages over the tt protocol , as this guarantees an aggressive control action with minimal communication delay . otherwise , the control messages are sent using the et protocol . that is , @xmath138 that is , the protocol switches depending on the state of the control application , as in ( [ eq : protocol ] ) . [ cc][cc]@xmath139 [ cc][cc]@xmath140 [ cc][cc]@xmath121 [ cc][cc]@xmath122 [ cc][cc]@xmath123 [ cc][][0.8]@xmath5 [ tc][cc][0.7]sensor [ cc][cc][0.7]controller [ cc][cc][0.7]actuator [ cc][cc]shared communication network commensurate with the switching protocol in ( [ eq : protocol ] ) , we propose a switch in the adaptive controller as well , and is defined below : @xmath141where @xmath142 is given in eq . ( [ eq : phiandthetaklein ] ) , @xmath143 is given in eq . ( [ eq : phiandtheta ] ) , @xmath144^t$ ] is the estimation of the controller gains @xmath145 ( eq . [ eq : phiandtheta ] ) , and @xmath146 . if @xmath147 , the adaptive controller is given by @xmath148where @xmath149 is given in eq . ( [ eq : phiandthetaklein ] ) , @xmath150 is given in eq . ( [ eq : phiandtheta ] ) , @xmath151^t$ ] is the estimation of the controller gains @xmath152 ( eq . [ eq : phiandtheta ] ) , and @xmath153 . the following definitions are useful for the rest of the paper . we denote the instants of time when the switch from tt to et occurs with @xmath154 , @xmath155 , and the instants of time when the switch from et to tt occurs with @xmath154 , @xmath156 . that is , the tt protocol is applied for @xmath157,p\in{\ensuremath{\mathbb{n}}}_0 $ ] and the et protocol is applied for @xmath158,p\in{\ensuremath{\mathbb{n}}}_0 $ ] with @xmath159 and switches occurring between @xmath160,p\in{\ensuremath{\mathbb{n}}}$ ] ( see figure [ fig : error ] ) . [ ass : dimpulse ] the disturbance @xmath9 in ( [ eq : predictorform ] ) is an impulse train , with the distance between any two consecutive impulses greater than a constant @xmath161 . this is the main result of the paper : [ thm : betaknown ] let the plant and disturbance @xmath162 in ( [ eq : predictorform ] ) satisfy assumptions [ ass : fixeddelay ] , [ ass : pe ] , and [ ass : dimpulse ] . consider the switching adaptive controller in ( [ eq : ttcontroller ] ) and ( [ eq : etcontroller ] ) with the hybrid protocol in ( [ eq : protocol ] ) and the following parameter estimate selections at the switching instants @xmath163 then there exists a positive constant @xmath164 such that for all @xmath165 , the closed loop system has globally bounded solutions . a qualitative proof of theorem [ thm : betaknown ] is as follows : + first , theorem [ thm : fixeddelay ] shows that if either of the individual control strategies ( [ eq : ttcontroller ] ) or ( [ eq : etcontroller ] ) is deployed , then boundedness is guaranteed . that is , for a sufficiently large dwell time @xmath161over which the controller stays in the tt protocol , with the controller in ( [ eq : ttcontroller ] ) , boundedness can be shown . after a finite number of switches , when the system switches to an et protocol , it is shown that the regressor vector remains in the same subspace as in the earlier switch to et and hence , the corresponding tracking error remains small even after the switch to et . hence the stay in et is ensured for a finite time , guaranteeing boundedness with the overall switching controller . _ proof of theorem [ thm : betaknown ] : _ we define an equivalent reference signal @xmath166that combines the effect of both @xmath53and the disturbance @xmath162 as @xmath167 where @xmath168 is given by @xmath169 and @xmath170 is the transfer function of the plant ( [ eq : model ] ) . also , we define a reference model signal @xmath171 given by @xmath172 where the transfer functions @xmath173 is given by @xmath174 and the optimal feedback gain @xmath175 is given by ( [ eq : phiandthetaklein ] ) . the overall ideal closed - loop system is given by the block diagram shown in figure [ fig : blockdiagram01 ] . we note that when there is no disturbance , the output @xmath171 corresponds to the desired regressor vector , and its first element of the vector corresponds to @xmath53 . ] when the algorithm is in mode @xmath124 , the underlying error equation is given by @xmath176 with @xmath177 . when the system is in mode @xmath126 , the error equation is given by @xmath178 with @xmath179 . define @xmath180 as @xmath181 choose lyapunov function @xmath182 where @xmath183 . let @xmath184 . the proof consists of the following four stages : * stage :* let there exist a sequence of finite switching times @xmath185 with the properties described above . then the errors @xmath186 and @xmath187 are bounded for all @xmath58 . + the proof of stage 1 is established using the following three steps : * step - * there exists a @xmath188 such that @xmath1890;{\ensuremath{e_{\text{th}}}\xspace}]:\;|e_1(k_1)|<\varepsilon\leqslant{\ensuremath{e_{\text{th}}}\xspace}$ ] where @xmath190 . during @xmath124(@xmath126 ) , the error @xmath186 ( @xmath187 ) is bounded . there exists a constant @xmath191 with @xmath192 , for @xmath156 . the length of the interval @xmath193 $ ] is greater than 2 , i.e. , @xmath194 + stage 2 is established using the following steps : * step - * if @xmath195 then @xmath196 if @xmath197 , then @xmath198 @xmath199 for @xmath200 @xmath201 for @xmath200 @xmath202 is bounded for all @xmath203 . + the following steps will be used to establish stage 3 : * step - * @xmath204 and @xmath205 , for all @xmath206 . @xmath207 during @xmath124 and during @xmath126 the control input is bounded for all @xmath58 and hence all signals are bounded . + the following two steps will be used to prove stage 4 : * step - * @xmath208 all signals are bounded . we note that the proofs of stages 1 , 3 , and 4 are identical to that in @xcite and are therefore omitted here . since stage 2 differs significantly from its counterpart in @xcite due to @xmath209 , we provide its proof in detail below . [ ] [ ] [ 0.8]time are assumed to occur at @xmath210.,title="fig : " ] @xmath211 in this step we show that if the tracking error @xmath212 is small the state signal error @xmath213 is also small . the signal @xmath214 is the output produced by the following transfer function @xmath215 with @xmath212 as the input : @xmath216 where @xmath217 is the inverse of the plant transfer function @xmath218 with the input signal @xmath219 and @xmath220 given in ( [ eq : phistern ] ) . from assumption [ ass : fixeddelay ] , it follows that @xmath217 is a stable transfer function . hence , as @xmath212 tends to zero , @xmath214 also tends to zero . if @xmath197 , then @xmath198 we first show that @xmath221 for @xmath222 and @xmath223 . we note that the reference model given in ( [ eq : phistern ] ) is a linear system and hence there exists a state space representation @xmath224 with @xmath225 being completely reachable . then it follows directly from lemma [ cor : inomega ] that @xmath226 for @xmath222 and @xmath223 . together with step 2 - 1 it follows that if @xmath197 , then @xmath198 . @xmath199 for @xmath200 first , we show that the error of the signal generated by the reference model signal @xmath227 together with the last parameter estimation value @xmath228 at the end of the previous et phase is small and therefore the output error @xmath229 is below the threshold @xmath137 . from step 2 - 2 we know that @xmath227 is in the same subspace @xmath230 as @xmath231 . from step 2 - 1 we know that @xmath231 is close to @xmath232 which in turn generates together with @xmath233 and @xmath234 an error which is @xmath235 according to theorem [ thm : fixeddelay ] . hence , @xmath236 from step 2 - 1 we know that @xmath237 is close to @xmath227 hence , according to step 2 - 4 we have @xmath238 . @xmath239 and @xmath240 this step shows that the error at the beginning of the et mode is below the threshold for at least @xmath241 steps . from step 2 - 3 we know that @xmath242 . according to the parameter choice in ( [ eq : thetachoice2 ] ) , the controller uses a constant initial value for the first @xmath243 steps . thus , the error @xmath244 because steps 2 - 1 to 2 - 5 can be applied . theorem [ thm : betaknown ] implies that the plant in ( [ eq : predictorform ] ) can be guaranteed to have bounded solutions with the proposed adaptive switching controller in ( [ eq : ttcontroller ] ) and ( [ eq : etcontroller ] ) and the hybrid protocol in ( [ eq : protocol ] ) , in the presence of disturbances . the latter is assumed to consist of impulse - trains , with their inter - arrival lower bounded . we note that if no disturbances occur , then the choice of the algorithm in ( [ eq : protocol ] ) implies that these switches cease to exist , and the event - triggered protocol continues to be applied . and switching continues to occur with the onset of disturbances , with theorem [ thm : betaknown ] guaranteeing that all signals remain bounded with the tracking errors @xmath245 converging to @xmath246 before the next disturbance occurs . the nature of the proof is similar to that of all switching systems , in some respects . a common lyapunov function @xmath202 was used to show the boundedness of parameter estimates , which are a part of the state of the overall system ( in stage 3 ) . the additional states were shown to be bounded using the boundedness of the tracking errors @xmath247 and @xmath248 ( in stage 1 ) and the control input using the method of induction ( in stage 4 ) . since the switching instants themselves were functions of the states of the closed - loop system , we needed to show that indeed these switching sequences exist , which was demonstrated in stage 2 . to this end , the sufficient richness properties of the reference signal were utilized to show that the signal vectors of a reference model and the system converge to the same subspace . next , it was shown that the error generated by the reference model is small and thus concluded that the tracking error at the switch from tt to et stays below the threshold @xmath137 . it is the latter that distinguishes the adaptive controller proposed in this paper , as well as the methodology used for the proof , from existing adaptive switching controllers and their proofs in the literature . in this work we considered the control of multiple control applications using a hybrid communication protocol for sending control - related messages . these protocols switch between time - triggered and event - triggered methods , with the switches dependent on the closed - loop performance , leading to a co - design of the controller and the communication architecture . in particular , this co - design consisted of switching between a tt and et protocol depending on the amplitude of the tracking error , and correspondingly between two different adaptive controllers that are predicated on the resident delay associated with each of these protocols . these delays were assumed to be fixed and equal to @xmath125 for the tt protocol and greater than @xmath249 for the et protocol . it was shown that for any reference input whose order of sufficient richness stays constant , the signal vector and the parameter error vector converge to subspaces which are orthogonal to each other . the overall adaptive switching system was shown to track such reference signals , with all solutions remaining globally bounded , in the presence of an impulse - train of disturbances with the inter - arrival time between any two impulses greater than a finite constant .

the focus of this paper is on the co - design of control and communication protocol for the control of multiple applications with unknown parameters using a distributed embedded system . the co - design consists of an adaptive switching controller and a hybrid communication architecture that switches between a time - triggered and event - triggered protocol . it is shown that the overall co - design leads to an overall switching adaptive system that has bounded solutions and ensures tracking in the presence of a class of disturbances .

the energy gap formation is an ubiquitous phenomena in condensed matter systems . when the band structure appears in the one - particle hamiltonian with a periodic potential , the band gap is the region in the spectrum where there is no density of states . on the other hand , the repulsion interaction generates the energy gap in the fractional quantum hall systems . generally speaking , systems with an energy gap are more stable against perturbations . the systems with the energy gap are , however , not good nurturing cradles for the superconductivity , which arises in the systems with fermi surface ( gapless ) . because of the instability of the interaction with the phonons , the electrons pair up and condense to the superconducting state . however , there are some classes of superconductors which were obtained by doping the antiferromagnetic insulators with mobile carriers , for example high transition temperature superconductors in the cooper - based transition metal oxides ( cuprates ) @xcite . by the chemical doping , the systems enter the phase where the energy gap structure is anisotropic in the momentum space , before becoming the superconductors @xcite . the enigmatic gap phase has agonised condensed matter community for three decades . recently , one of us ( chern ) developed a weak - coupling theory based on the hubbard model for the gap formation in cuprates @xcite . introducing the spin berry s phase as the gauge interaction @xcite , the hubbard model in two dimensions can be formulated in the renormalizable theory in the continuous limit . considering the antiferromagnetic fluctuation additionally , the gauge field acquires the mass via the stckelberg mechanism . the 2 + 1 dimensional lagrangian density is given by @xmath2[(\frac{\vec { \nabla}}{i}-g\vec a){\psi}_{\sigma}(x ) ] \nonumber \\ -\frac{1}{4}f_{\mu\nu}f^{\mu\nu}+m_{0}(d_{0}{\phi}(x))^{\dagger}(d_{0}{\phi}(x))-m_{1}(\vec d{\phi}(x))^{\dagger}{\cdot}(\vec d{\phi}(x ) ) , \label{u1}\end{aligned}\ ] ] where @xmath3 are the electrons , @xmath4 are the gauge fields , @xmath5 is the gauge coupling , @xmath6 is the antiferomagnetic fluctuation , @xmath7 are the covariant derivatives , and @xmath8 and @xmath9 are the mass parameters . the antiferromagnetic fluctuation is parameterised by a complex phase field @xmath10 , where @xmath11 is the coupling between the gauge field and the antiferromagnetic fluctuation . in two dimensions , the @xmath6 field takes place an infinite order phase transition at the finite temperature , so called the berezinski - kosterlitz - thouless transition @xcite . combining with the gauge fields , the @xmath6 field becomes the longitudinal mode of the gauge fields . as the transition of the mass acquisition takes place , the electronic energy structure opens a gap without breaking the translational and the time reversal symmetry . the gap formation is not the patent for cuprates but has found in many other strongly - correlated electron systems , for example the iron pnictides and the heavy fermion systems @xcite . unlike the cuprates , the iron pnictides and the heavy fermion materials are the multi - band systems . it inspires us to generalise the current u(1 ) scheme to the su(@xmath12 ) cases , where the multiple @xmath12-flavours of electrons can be considered . furthermore , while the stckelberg mechanism works in the u(1 ) case , we generalise the mass acquisition scheme to the higgs mechanism . restricting ourself to the simplest fundamental representation for both electrons and the higgs , we found that there is always one flavour of the electrons which is not degenerate to the other for @xmath1 . this robust behaviour can be understood by the group theory . in this paper , the sections are organised as the following . in the second section , the su(2 ) case will be discussed . in the third section , the results of the su(@xmath12 ) cases are provided . the last section is the discuss and the conclusion . for a system with multi - flavours of electrons that are degenerate to each other , we can possibly consider the su(2 ) gauge symmetry . for simplicity , we consider the electrons to be in the su(2 ) fundamental representation . the u(1 ) lagrangian in eq . ( [ u1 ] ) can be generalised to the su(2 ) form , @xmath13^{\dagger}[(\frac{\vec { \nabla}}{i}-g\vec a){\psi}(x ) ] \nonumber \\ -\frac{1}{4}f_{\mu\nu}f^{\mu\nu}+m_{0}^{2}(d_{0}{\phi}(x))^{\dagger}(d_{0}{\phi}(x))-m_{1}^{2}(\vec d{\phi}(x))^{\dagger}{\cdot}(\vec d{\phi}(x ) ) , \label{lagrangian}\end{aligned}\ ] ] where @xmath14 , @xmath15 , @xmath16 , @xmath5 and @xmath17 are the gauge couplings for the electrons and the higgs boson respectively , and @xmath18 , @xmath4 , and @xmath19 are matrix - valued , @xmath20 where @xmath21 are the pauli spin matrices . the higgs field can be stabilised by the following terms @xmath22 where @xmath23 is the higgs mass and @xmath24 is self - interaction parameter . the total lagrangian density is given by @xmath25 . the mass generation of the su(2 ) gauge bosons via the higgs mechanism is a textbook story . for example , the mass acquisition of the gauge boson is related to the group representation of the higgs field . in the fundamental representation , three gauge bosons acquire the equal mass , and in the adjoint representation , only two gauge bosons obtain the mass . on the other hand , different from the high - energy physics , the condensed matter community cares more about the length scale . the gauge bosons of zero mass produce a long - ranged interaction , and the ones of finite mass produce a short - ranged interaction . in the condensed matter systems , the long - ranged interaction is often screened and becomes short - ranged . in the systems with the gauge symmetry , it corresponds to the gauge bosons of finite mass @xcite . as the gauge bosons acquire the mass , the short - ranged interaction modifies the electronic specturm , opening a gap - like structure in the non - relativistic band structure @xcite . in the condensed matter language , the notion of the energy gap is different from the mass , which is determined by the curvature of the dispersion relation . the nature of the phase transition to the gap phase is , however , different from the u(1 ) case . in the higgs mechanism given by eq . ( [ higgs ] ) , it favors a second - order phase transition . in the real materials , it may take place at the finite temperature , if the two dimensionality of the space is only an approximation . similar to the u(1 ) case , we compute the energy gap using the single - particle green s function . the leading diagrams contributing to the self - energy term @xmath26 are given in the fig . ( [ diagram ] ) . in the fundamental representation of the higgs mechanism , the electronic gap , the energy at the bottom of the band , is @xmath27 for both flavors of the electrons . although the diagram in fig . ( [ diagram]b ) modifies the dispersion relation , it does not contribute to the gap generation . on the other hand , in the adjoint representation of the higgs mechanism , it becomes @xmath28 for all flavors of the electrons . the su(2 ) theory may be realized in the condensed matter system with the non - abelian holonomy @xcite and the magnetism . the non - abelian holonomy plays the role of the su(2 ) gauge fields . on the other hand , the ferromagnetic or the antiferromagnetic fluctuations may serve as the higgs field . if the non - abelian holonomy is in the particle - hole channel of the degrees of freedom , for example the spin berry s phase , it may be able to couple to the ( anti)-ferromagnetic fluctuation and manifests the effect of the electronic gap generation . the mechanism of the non - relativistic gap generation can be generalized to the su(@xmath12 ) case . the formalism of the su(@xmath12 ) lagrangian is the same as the ones in eq . ( [ lagrangian ] ) and eq . ( [ higgs ] ) . in addition , the electrons are considered in the su(@xmath12 ) fundamental representation , namely @xmath29 . if the higgs field is also considered in the fundamental representation , the mass spectrum of the @xmath30 gauge bosons can be given as the following . @xmath31 where @xmath32 gauge bosons remain massless , and the rest of them become massive . among the massive gauge bosons , there is always one boson acquiring different mass . the self energy of the electrons is also computed using the diagram in fig . ( [ diagram ] ) . we obtain @xmath33 for @xmath34 , we reproduce the results of the su(2 ) case . different from the su(2 ) case , however , there is always one flavor of the electron that is not degenerate to the rest of the @xmath0 electrons . this robust structure may be considered as the signature of the su(@xmath12 ) gauge symmetry for @xmath35 . the current results can be understood by the group theory . before the symmetry breaking of the higgs field , the theory is su(@xmath12 ) symmetric . in the fundamental representation , there are @xmath36 degrees of freedom in the @xmath12 multiplet of the higgs field . after the spontaneous symmetry breaking , there are @xmath37 goldstone modes which combine with the gauge bosons and become the longitudinal modes of the massive bosons . consequently , in the @xmath38 gauge bosons , there are @xmath32 boson remaining massless as shown in eq . ( [ higgsn ] ) . interestingly , the remaining @xmath32 bosons preserve the su(@xmath0 ) symmetry . after the symmetry breaking , the remnant symmetry becomes su(@xmath0 ) . therefore , spectrum of the @xmath12 electrons splits into @xmath39 , reflecting the su(@xmath0 ) symmetry . the nonrelativistic gap formation is generalized from the u(1 ) gauge symmetry with the stckelberg mechanism to the su(@xmath12 ) gauge symmetry with the higgs mechanism . in the u(1 ) case , the phase transition is the berezinskii - kosterlitz - thouless - like transition at the finite temperature in the 2 + 1 dimensional spacetime . namely , there is no significant signature of the phase transition . on the other hand , in the su(@xmath12 ) case , the gap spectrum of the @xmath12-plet of the electrons splits into @xmath39 , as the consequence of the remnant su(@xmath0 ) symmetry . the su(@xmath12 ) theory may be applicable to the system with non - abelian holonomy . we are grateful for the stimulated discussions with chong - der hu and pei - ming ho . this work is supported by ministry of science and technology of taiwan under the grant : most 103 - 2112-m-002 - 014-my3 and by na- tional taiwan university under the grant : 103r7831 and 104r7831 .

we demonstrate that the non - relativistic fermions open the energy gap when the su(n ) gauge bosons , mediating the interaction between fermions , acquire the mass . surprisingly , even though there is the su(n ) gauge symmetry , there is always one fermionic energy gap which is not degenerate to the rest of the @xmath0 fermions for @xmath1 in the fundamental representation . energy gap , non - abelian gauge systems , strongly - correlated electrons

binary models like ising - type simulation have a long history . they have been applied by schelling to describe the ghetto formation in the inner cities of the usa , i.e. , to study phase separation between black and white @xcite . in the sociophysics context , recently , many social phenomena such as election , propagation of information , predicting features of traffic , migration , opinion dynamics and formation in a social group have been successful modelled based on ising spin systems using models and tools of statistical physics . with this respect , particularly successful models have been developed by sznajd @xcite , deffuant et al.@xcite and hegselmann and krause @xcite . among those three models , the one developed by sznajd is the most appropriate for simulation in networks and lattices , since it consider just the interactions between the nearest neighbors . indeed , the sznajd model has been successfully applied to model sociophysical and economic systems @xcite . on the other hand , several modifications of the sznajd model have been studied using different rules or topologies starting from different initial opinion densities @xcite . all these models are static ( i.e. not dynamic ) and they allow for consensus ( one final opinion ) , polarization ( two final opinion ) , and fragmentation ( more than two final opinions ) , depending on how tolerant people are to different opinions . more recently the striking sociophysical model has been suggested by aydiner @xcite in order to explain the time evolution of resistance probability of a closed community in a one - dimensional sznajd like model based on ising spin system . it has been shown that resistance probability in this model decay as a stretched exponential with time . in that model spins does not move on the lattice sites during the simulation , so this model was so - called static . however , in a realistic case , spins i.e. , people move in the community i.e. , in the space . social or opinion formation formed depend upon dynamics of the system . because , there must be a direct connection between opinion dynamics and formation in a social system since the social formation is determined by the dynamics . meyer - ortmanns @xcite studied recent work in which the condition for ghetto formation in a population with natives and immigrants by using kawasaki - exchange dynamics in a two dimensional ising model . she showed that ghetto formation can be avoided with a temperature increasing with time . similarly , schulze have also generalized meyer - ortmanns work to up to seven different ethnic groups to explain ghetto formation in a multi - cultural societies in a potts - like model @xcite . in this study , we have developed a dynamic version of the aydiner @xcite model by combining the aydiner and meyer - ortmanns @xcite models based on one - dimensional ising model . in one - dimensional static model @xcite , each site carriers a spin which is either spin up ( + 1 ) or spin down ( -1 ) randomly . spin up ( + 1 ) represent the host people and spin down ( -1 ) represent the soldier . the host people always against occupation , and , on the other hand , soldier always willing to continue occupation , who always have the opinion opposite of that of the host people . furthermore , the community member i.e. , spins does nt also move on the lattice during the process . in this model , initially , it was assumed that there was a over all consensus among member of the community against occupation even if some exceptions exist . one expects that host people obey to this consensus at least initially . in this sense , community behaves as polarized at zero social temperature @xcite against occupation just like ising ferromagnet at zero temperature . it was conjectured that host people are influenced by soldiers even though they against occupation owing to they are exposed to intensive biased information or propagation . soldiers affect the host people and force to change their opinion about occupation . effected people may change their own opinions depending on resistance probability of the nearest neighbors about occupation . moreover , effected host people affect neighbors . such a mechanism depolarize the polarization ( resistance probability ) of all host people . hence social polarization destroy . however , soldiers , unlike host people , have not been influenced by the host people . their opinion about justifying the occupation does not change during the occupation process , since they may be stubborn , stable or professional etc . , who behaves like persistent spins in ising spin system . it is means that the probability of the against occupation of a soldier is always zero . if we summarize , we can say that none spins does flip fully in the system . spin up always remains spin up , and spin down always remains spin down . in this respect , the probability of against occupation of host people can be interpreted as a survival probability of opinion of host people about occupation under above considerations . in this sense , the survival probability @xmath0 of opinion of host people indicate equal to @xmath1 at least initially and , on the other hand , the probability of against occupation of soldier equal to zero , which means that soldier behaves as a trap point lattice which depolarize the survival probability of opinion of host people . of course , one may suggest that there are many different number of opinions in society , however , it is possible to find that a society being formed two - state opinion in a real case . therefore this model is a good example for two - state opinion model as well galam contrarian model @xcite even though it seems that it is very simple . furthermore , in real social systems , people move on the space , i.e. , lattice . therefore , in this study , we assumed that people i.e. , spins randomly move on the lattice through the kawasaki - exchange dynamics contrary to previous model . the survival probability @xmath2 for a people at site @xmath3 at the next time @xmath4 is determined with the survival probability of nearest - neighbors with previous time @xmath5 as @xmath6.\label{eq1}\ ] ] we note that the survival probability for all site are calculated as synchronously . randomly motion of the spins i.e. , people on the lattice through the kawasaki - exchange dynamics . firstly , a spin pair is chosen randomly and then it is decided whether spin pair exchange with each other or not . in this approach , the nearest - neighbor spins are exchanged under heat - bath dynamics , i.e. , with probability @xmath7 , where @xmath8 is the energy change under the spin exchange , @xmath9 is the boltzmann constant , and @xmath10 is the temperature i.e. , social temperature or tolerance . hence , to obtain probability @xmath11 we need to calculate @xmath12 and @xmath13 which correspond to energy of the spin pair at first position and after exchange with position of spins , respectively . energy @xmath12 and @xmath13 can be calculated in terms of the survival probability instead of spin value as @xmath14 @xmath15 where @xmath16\ ] ] and @xmath17\ ] ] energy difference is written as @xmath18 from eq . ( [ eq2a ] ) and ( [ eq2b ] ) . in addition , the total survival probability of opinion of host people at the any time @xmath5 can be obtained over each person for any @xmath19 configuration as @xmath20 where @xmath21 is the initial number of host people . on the other hand , the averaged survival probability at the any time @xmath5 can be obtained from eq . ( [ eq3 ] ) over the independent configuration as @xmath22 where @xmath23 is the number of different configurations . we have adopted the monte carlo simulation technique to the one - dimensional sociophysical model using the lattice size @xmath24 with periodic boundary condition , and independent configuration @xmath25 for the averaged results . the simple algorithm for the simulation is as follows : i ) at the @xmath26 , eq . ( [ eq4 ] ) is initially calculated , ii ) for @xmath27 a spin pair is randomly chosen , and then it is decided whether the spin pair exchange or not with the probability @xmath28 , this step is repeated @xmath29 times , iii ) after ii - steps are completed , eq . ( [ eq4 ] ) is recalculated again , and to continue this procedure goes to step ii . the simulation results are as follow : we have firstly plotted simulation data versus time in fig . [ fig1 ] in a several manner . it is explicitly seen from figs . [ fig1](a)-(c ) that there are no power , exponential and logarithmic law dependence in our simulation data , respectively . however , as seen fig . [ fig1](d ) , data well fit to the stretched exponential function as @xmath30 where @xmath31 is the relaxation constant , and @xmath32 is the decay exponent of the survival probability . this result indicate that the time evaluation of survival probability of the opinion of the host people in a closed community has stretched exponential character i.e. , kohlraush - william - watts ( kww ) decay law @xcite . it should be tested whether fig . [ fig1](d ) satisfies to stretched exponential or not @xcite . because , as noted by stauffer , the fig . [ fig1](d ) would work as stretched exponential , if pre - factor of eq . ( [ eq5 ] ) is equal to 1 . however , if pre - factor is less than @xmath1 , it may give the impression of stretched exponential form , even for @xmath33 . therefore , it can be plotted @xmath34 versus suitable powers of @xmath5 , like @xmath35 , @xmath36 , etc . , and find out the best straight line among the powers of @xmath5 for long times . hence , @xmath34 was plotted versus powers of @xmath5 for @xmath37 then the best straight fitting line for long times was obtained for @xmath38 for @xmath39 , @xmath40 for @xmath41 , and @xmath42 for @xmath43 as seen in fig . [ fig3](a)-(c ) respectively . these results confirm to this method used to find out stretched exponential exponents in fig . [ fig1](d ) , and also all figures in fig . [ fig2 ] as mentioned . also , this test indicates that prefactor in eq . ( [ eq5 ] ) does not effect results presented in this paper . it is concluded that results for high temperatures also consistent with static model @xcite . but , unlike the static model , time crossover has been observed in dynamic model at low temperatures . in order to investigate the transition we have plotted survival probability versus time for different social temperature @xmath10 in fig . it is clearly seen that the time crossover occurs depend on social temperature . when social temperature decreases , the crossover become more clear . such a behavior was not observed in a static model . we can bridge the short time regime and the long time regime by a scaling function @xmath44 @xmath45 where @xmath46 indicates the time crossover . for our simulation data , the scaling relation ( 6 ) can be written for very long and very short time intervals as @xmath47{c}e^{-\left ( t/\tau\right ) ^{\beta_{1}}}\\ e^{-\left ( t/\tau\right ) ^{\beta_{2}}}\end{array } \begin{array } [ c]{l}if \hspace{0.3 cm } t<<t_{c}\\ if \hspace{0.3 cm } t>>t_{c}.\end{array}\ ] ] on the other hand , in order see how the decay exponent @xmath32 depend on soldier density @xmath48 , and social temperature @xmath10 , we have plotted @xmath32 versus soldier density @xmath48 in fig . [ fig4](a ) for @xmath49 and @xmath50 in account to taken different social temperatures , and social temperature @xmath10 in fig . [ fig4](b ) for a fixed value of density @xmath48 , respectively . as seen from fig . [ fig4](a ) that @xmath51 and @xmath52 are linearly depend on soldier density both of two regimes at low social temperature . on the other hand , the decay exponent has two different character for @xmath49 and @xmath50 depend on social temperature @xmath10 in fig . [ fig4](b ) , the decay exponent @xmath51 decreases with increasing temperature @xmath10 for @xmath49 , whereas @xmath52 increases with increasing temperature @xmath10 for @xmath50 at low temperatures . however , for relatively high temperatures we roughly say that @xmath51 approach to @xmath52 for both two regimes obey to eq . ( [ eq7 ] ) . finally , to understand the social temperature and soldier density dependence of the time crossover @xmath53 , we have plotted @xmath53 versus social temperature @xmath10 in fig . [ fig5](a ) for a fixed soldier density @xmath48 , and versus soldier density in fig . [ fig5](b ) for fixed social temperature @xmath10 , respectively . it seems from fig . [ fig5](a ) that the crossover transition @xmath53 quite rapidly decrease with increasing @xmath10 , on the other hand , it seems from fig . [ fig5](b ) that it slowly decrease with increasing soldier density @xmath48 . we note that as seen inserted figure in fig . [ fig5](b ) the crossover transition @xmath53 depends on soldier density with power law for fixed social temperature . we suggest that the stretched exponential behavior of decay must be originated from model system . the persistent spins i.e. , the soldiers does nt flip during simulation , therefore they behave as a trap in the system . hence they play a role diminishing the survival probability of the neighbor spins in the system . consequently , decay characteristic of the system can be explain due to the trapping states . another say , this characteristic behavior does nt depend on either diffusion dynamics of spins or interaction rules between spins . another unexpected behavior is the time crossover in @xmath32 contrast to previous model @xcite . we supposed that this amazing result originated from opinion dynamics depend on social temperature . model allows to the opinion formation with time . indeed , there is a direct connection between opinion dynamics and formation in a social system since the social formation is determined by the dynamics as depend on the social temperature . for example , in a real spin system , decreasing temperature phase separation may occur in the system . in the sociophysical sense , it means that people who have different opinion are separated each other with decreasing social tolerance , and therefore the ghetto formation or polarization may occur in the system . it is expected that interactions between soldier and host people is maximum when soldiers are randomly distributed in the community . as social temperature , i.e. , tolerance is decreased , however , phase separation occur with time , so this leads to decreasing of the interactions . in our opinion , the ghetto formation in the system does nt leads crossover transition in time because of the ghetto formation is randomly distributed relatively . on the other hand , the time average of survival probability over different configuration effect of ghetto formation may probably destroy . so we do nt hope that ghetto formation is not responsible crossover transition . however , polarization must be occurred at low temperature leads to meaningful phase separation in the system . such a polarization may leads to crossover transition in time . stretched exponential behavior indicates mathematically that decay for the relatively short times is fast , but for relatively long times it is slower . one can observe that this mathematical behavior corresponds to occupation processes in the real world . in generally , a military occupation is realized after a hot war . the community does not react to occupation since it occurs as a result of defeat . people are affected easily by propaganda or other similar ways . therefore , it is not surprised that resistance probability decrease rapidly at relatively short times . on the other hand , spontaneous reaction may begin against occupation in the community after the shock . hence , community begins by regaining consciousness and more organized resistance may display difficulties for occupants . for long times , the resistance probability decreases more slowly . this means that resistance against occupation extends to long times in practice . at this point , the number of soldiers is also important , because the density of soldiers determines the speed of decaying . the different regimes have been observed in the decay of the survival probability . these regimes clearly appear particularly at low temperatures . in the case of the social temperature is very low , @xmath51 is bigger than @xmath52 which indicates the decay of the survival probability for relatively short time is slower than for relatively long time . this can be interpreted that the resistance of host people against occupation may be broken spontaneously if soldier can wait enough time . of course , the mechanism considered in this work can be regarded as simple , but , it would be useful to understand the time evolution of the resistance probability of the community against to occupation in the one - dimensional model under some considerations . we remember that simple social rules lead to complicated social results , hence we believe that the obtained results and model can be applied the real social phenomena in the societies to understand the basis of them . authors are grateful dietrich stauffer for the suggestions in the preparation of this paper . 000 g. deffuant , d. neau , f. amblard , and g. weisbuch , adv . compl . syst . * 3 * , 87 ( 2000 ) ; g. deffuant , f. amblard , g. weisbuch , and t. faure , artificial societies and social simulation * 5 * ( 4 ) , 1 ( 2002 ) ( jass.soc.surrey.ac.uk ) .

the time dependence of the survival probability of an opinion in a closed community has been investigated in accordance with social temperature by using the kawasaki - exchange dynamics based on previous study in ref . [ 1 ] . it is shown that the survival probability of opinion decays with stretched exponential law consistent with previous static model . however , the crossover regime in the decay of the survival probability has been observed in this dynamic model unlike previous model . the decay characteristics of both two regimes obey to stretched exponential . * keywords : * ising model ; politics ; random walk ; sociophysics ; sznajd model .

going hand - in - glove with analytic models of accretion disks , discussed in chapter 2.1 , are direct numerical simulations . although analytic theories have been extremely successful at explaining many general observational properties of black hole accretion disks , numerical simulations have become an indispensable tool in advancing this field . they allow one to explore the full , non - linear evolution of accretion disks from a first - principles perspective . because numerical simulations can be tuned to a variety of parameters , they serve as a sort of `` laboratory '' for astrophysics . the last decade has been an exciting time for black hole accretion disk simulations , as the fidelity has become sufficient to make genuine comparisons between them and real observations . the prospects are also good that within the next decade , we will be able to include the full physics ( gravity + hydrodynamics + magnetic fields + radiation ) within these simulations , which will yield complete and accurate numerical representations of the accretion process . in the rest of this chapter i will review some of the most recent highlights from this field . one of the most exciting recent trends has been a concerted effort by various collaborations to make direct connections between very sophisticated simulations and observations . of course , observers have been clamoring for this sort of comparison for years ! perhaps the first serious attempt at this was presented in @xcite . schnittman produced a simulation similar to those in @xcite and coupled it with a ray - tracing and radiative transfer code to produce `` images '' of what the simulated disk would look like to a distant observer . by creating images from many time dumps in the simulation , schnittman was able to create light curves , which were then analyzed for variability properties much the way real light curves are . following that same prescription , a number of groups have now presented results coupling general relativistic mhd ( grmhd ) simulations with radiative modeling and ray - tracing codes ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? more recent models have even included polarization measurements @xcite . this approach is most applicable to very low - luminosity systems , such as sgr a * and m87 . a sample light curve for sgr a * covering a 16-hour window is shown in figure [ fig : lightcurve ] . in the case of m87 , modeling has focused on accounting for the prominent jet in that system @xcite . along with modeling light curves and variability , this approach can also be used to create synthetic broadband spectra from simulations ( e.g. * ? ? ? * ; * ? ? ? * ) , which can be compared with modern multi - wavelength observing campaigns ( see chapter 3.1 ) . this is very useful for connecting different components of the spectra to different regions of the simulation domain . for example , figure [ fig : spectrum ] shows that the sub - mm bump in sgr a * is well represented by emission from relatively cool , high - density gas orbiting close to the black hole , while the x - ray emission seems to come from comptonization by very hot electrons in the highly magnetized regions of the black hole magnetosphere or base of the jet . as important as the radiative modeling of simulations described in section [ sec : matching ] has been , its application is very limited . this is because , in most cases , the radiative modeling has been done after the fact ; it was not included in the simulations themselves . therefore , the gas in the accretion disk was not allowed to respond thermodynamically to the cooling . this calls into question how much the structure obtained from the simulation reflects the true structure of the disk . fortunately , various groups are beginning to work on treating the thermodynamics of accretion disks within the numerical simulations with greater fidelity . thus far , two approaches have principally been explored : 1 ) _ ad hoc _ cooling prescriptions used to artificially create _ optically thick , geometrically thin _ disks and 2 ) fully self - consistent treatments of radiative cooling for _ optically thin , geometrically thick _ disks . we review each of these in the next 2 sections . for the _ ad hoc _ cooling prescription , cooling is assumed to equal heating ( approximately ) everywhere locally in the disk . since this is the same assumption as is made in the shakura - sunyaev @xcite and novikov - thorne @xcite disk models , this approach has proven quite useful in testing the key assumptions inherent in these models ( e.g * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? in particular , these simulations have been useful for testing the assumption that the stress within the disk goes to zero at the innermost stable circular orbit ( isco ) . a corollary to this is that the specific angular momentum of the gas must remain constant at its isco value inside this radius . both of these effects have been confirmed in simulations of sufficiently thin disks @xcite , as shown in figure [ fig : isco ] . another approach to treating the thermodynamics of accretion disks has been to include _ physical _ radiative cooling processes directly within the simulations . so far there has been very limited work done on this for optically thick disks , but an optically - thin treatment was introduced in @xcite . similar to the after - the - fact radiative modeling described in section [ sec : matching ] , the optically - thin requirement restricts the applicability of this approach to relatively low luminosity systems , such as the quiescent and low / hard states of black hole x - ray binaries . recently this approach has been applied to sgr a * @xcite , which turns out to be right on the boundary between where after - the - fact radiative modeling breaks down and a self - consistent treatment becomes necessary @xcite . figure [ fig : sgra ] illustrates that this transition occurs right around an accretion rate of @xmath0 for sgr a*. ( _ black _ ) , @xmath1 ( _ blue _ ) , and @xmath2 ( _ red _ ) . for each accretion rate , two simulations are shown , one that includes cooling self - consistently ( model names ending in `` c '' ) and one that does not . the spectra begin to diverge noticeably at @xmath0 . figure from @xcite.,scaledwidth=70.0% ] another area where a lot of interesting new results have come out is in the study of how magnetic field topology and strength affect black hole accretion . although there is now convincing evidence that the blandford - znajek mechanism @xcite works as predicted in powering jets ( e.g. * ? ? ? * ; * ? ? ? * ) , one lingering question is still how the accretion process supplies the required poloidal flux onto the black hole . simulations have demonstrated that such field can , in many cases , be generated self - consistently within mri - unstable disks ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . however , this is strongly dependent on the initial magnetic field topology , as shown in @xcite . figure [ fig : beckwith ] nicely illustrates that when there is no net poloidal magnetic flux threading the inner disk , the magnetically - driven jet can be 2 orders of magnitude less energetic than when there is . at this time it is unclear what the `` natural '' field topology would be , or even if there is one . ( _ top left _ ) , magnetic field strength @xmath3 ( _ top right _ ) , electromagnetic energy flux @xmath4 ( _ bottom left _ ) , and angular momentum flux @xmath5 ( _ bottom right _ ) . dashed lines show @xmath6 standard deviation from the average . ] although strong poloidal magnetic fields are useful for driving powerful jets , they can also create interesting feedback affects on an accretion disk . in the case where a black hole is able to accumulate field with a consistent net flux for an extended period of time , it is possible for the amassed field to eventually `` arrest '' the accretion process @xcite . an example of an arrested state is shown in figure [ fig : arrest ] . in a two - dimensional simulation where plasma with a constant net flux is fed in from the outer boundary , a limit - cycle behavior can set in , where the mass accretion rate varies by many orders of magnitude between the arrested and non - arrested states . figure [ fig : arrested_mdot ] provides an example of the resulting mass accretion history . it is straightforward to show that the interval , @xmath7 , between each non - arrested phase in this scenario grows with time according to @xmath8 where @xmath9 is the radial infall velocity of the gas , @xmath10 is the strength of the magnetic field , and @xmath11 is the density of the gas . . starting from @xmath12 , a pattern of cyclic accretion develops ( seen as a sequence of spikes ) . figure from @xcite.,scaledwidth=70.0% ] in three - dimensions , the magnetic fields are no longer able to perfectly arrest the in - falling gas because of a `` magnetic '' rayleigh - taylor effect . basically , as low density , highly magnetized gas tries to support higher density gas in a gravitational potential , it becomes unstable to an interchange of the low- and high - density materials . indeed , such a magnetic rayleigh - taylor effect has been seen in recent simulations by @xcite . ) and meridional ( @xmath13 ) snapshots of the gas density of a magnetically arrested flow in 3d . black lines show field lines in the image plane . ( panel e ) : time evolution of the mass accretion rate . ( panel f ) : time evolution of the large - scale magnetic flux threading the bh horizon . ( panel g ) : time evolution of the energy outflow efficiency @xmath14 . figure from @xcite . ] the results of @xcite are important for another reason . these were the first simulations to demonstrate a jet efficiency @xmath15 greater than unity . since the efficiency measures the amount of energy extracted by the jet , normalized by the amount of energy made available via accretion , a value @xmath16 indicates more energy is being extracted than is being supplied by accretion . this is only possible if some other source of energy is being tapped in this case the rotational energy of the black hole . this was the first demonstration that a blandford - znajek @xcite process _ must _ be at work in driving these simulated jets . there is observational evidence that several black - hole x - ray binaries ( bhbs ) , e.g. gro j1655 - 40 @xcite , v4641 sgr @xcite and gx 339 - 4 @xcite , and active galactic nuclei ( agn ) , e.g. ngc 3079 @xcite , ngc 1068 @xcite , and ngc 4258 @xcite , may have accretion disks that are tilted with respect to the symmetry plane of their central black hole spacetimes . there are also compelling theoretical arguments that many black hole accretion disks should be tilted @xcite . this applies to both stellar mass black holes , which can become tilted through asymmetric supernovae kicks @xcite or binary captures and will remain tilted throughout their accretion histories , and to supermassive black holes in galactic centers , which will likely be tilted for some period of time after every major merger event @xcite . close to the black hole , tilted disks may align with the symmetry plane of the black hole , either through the bardeen - petterson effect @xcite in geometrically thin disks or through the magneto - spin alignment effect @xcite in the case of geometrically thick , magnetically - choked accretion @xcite . however , for weakly magnetized , moderately thick disks ( @xmath17 ) , no alignment is observed @xcite . in such cases , there are many observational consequences to consider . chapter 4.3 of this book discusses the two primary methods for estimating the spins of black holes : continuum - fitting and reflection - line modeling . both rely on an assumed monotonic relation between the inner edge of the accretion disk ( assumed to coincide with the radius of the isco ) and black hole spin , @xmath18 . this is because what both methods actually measure is the effective inner radius of the accretion disk , @xmath19 . one problem with this is that it has been shown @xcite that tilted disks do not follow such a monotonic behavior , at least not for disks that are not exceptionally geometrically thin . figure [ fig : fragile09 ] shows an example of the difference between how @xmath19 depends on @xmath18 for untilted and tilted simulated disks . similar behavior has been confirmed using both dynamical @xcite and radiative @xcite measures of @xmath19 . the implication is that spin can only be reliably inferred in cases where the inclination of the inner accretion disk can be independently determined , such as by modeling jet kinematics @xcite . of simulated untilted ( _ circles _ ) and tilted ( _ diamonds _ ) accretion disks as a function of black - hole spin using a surface density measure @xmath20 . the _ solid _ line is the isco radius . figure from @xcite.,scaledwidth=70.0% ] for geometrically thin , shakura - sunyaev type accretion disks , the bardeen - petterson effect @xcite may allow the inner region of the accretion disk to align with the symmetry plane of the black hole , perhaps alleviating concerns about measuring @xmath18 , at least for systems in the proper state ( `` soft '' or `` thermally dominant '' ) and luminosity range @xmath21 , where @xmath22 is the eddington luminosity . extremely low luminosity systems , though , such as sgr a * , do not experience bardeen - petterson alignment . further , for a system like sgr a * that is presumed to be fed by winds from massive stars orbiting in the galactic center , there is no reason to expect the accretion flow to be aligned with the black hole spin axis . therefore , a tilted configuration should be expected . in light of this , @xcite presented an initial comparison of the effect of tilt on spectral fitting of sgr a*. figure [ fig : dexter12 ] gives one illustration of how important this effect is ; it shows how the probability density distribution of four observables change if one simply accounts for the two extra degrees of freedom introduced by even a modestly tilted disk . the take away point should be clear ignoring tilt artificially constrains these fit parameters ! one remarkable outcome of considering tilt in fitting the spectral data for sgr a * is that tilted simulations seem able to naturally resolve a problem that had plagued earlier studies . spectra produced from untilted simulations of sgr a * have always yielded a deficit of flux in the near - infrared compared to what is observed . for untilted simulations , this can only be rectified by invoking additional spectral components beyond those that naturally arise from the simulations . tilted simulations , though , produce a sufficient population of hot electrons , _ without any additional assumptions _ , to produce the observed near - infrared flux ( see comparison in figure [ fig : tilted_spectrum ] ) . they are able to do this because of another unique feature of tilted disks : the presence of standing shocks near the line - of - nodes at small radii @xcite . these shocks are a result of epicyclic driving due to unbalanced pressure gradients in tilted disks leading to a crowding of orbits near their respective apocenters . figure [ fig : henisey12 ] shows the orientation of these shocks in relation to the rest of the inner accretion flow . orders of magnitude in comparable untilted simulations . figure from @xcite.,scaledwidth=80.0% ] a worthwhile future direction to pursue in this area would be a robust comparison of tilted disk simulations using both grmhd and smoothed - particle hydrodynamics ( sph ) numerical methods . the grmhd simulations ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) enjoy the advantage of being `` first principles '' calculations , since they include all of the relevant physics , whereas the sph simulations ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) enjoy the advantage of being more computationally efficient , though they make certain assumptions about the form of the `` viscosity '' in the disk . thus far , the grmhd and sph communities have proceeded separately in their studies of tilted accretion disks , and it has yet to be demonstrated that the two methods yield equivalent results . this would seem to be a relatively straightforward and important thing to check . a few years ago , it might have been very ambitious to claim that researchers would soon be able to perform global radiation mhd simulations of black hole accretion disks , yet a lot has happened over that time , so that now it is no longer a prediction but a reality . in the realm of newtonian simulations , a marvelous study was published by @xcite , showing global ( though two - dimensional ) radiation mhd simulations of accretion onto a black hole in three different accretion regimes : @xmath23 , @xmath24 , and @xmath25 . the remarkably different behavior of the disk in each of the simulations ( illustrated in figure [ fig : ohsuga11 ] ) is testament to how rich this field promises to be as more groups join this line of research . the specifics of this work are discussed more in chapter 5.3 . the other big thing to happen ( mostly ) within the past year is that a number of groups have now tackled , for the first time , the challenge of developing codes for _ relativistic _ radiation mhd in black hole environments @xcite . so far none of these groups have gotten to the point of simulating accretion disks in the way @xcite did ( they are still mostly at the stage of code tests and simple one- and two - dimensional problems ) , but with so many groups joining the chase , one can surely expect rapid progress in the near future . one result of some astrophysical interest is the study of bondi - hoyle ( wind ) accretion onto a black hole , including the effects of radiation @xcite ( see figure [ fig : zanotti11 ] ) . work presented in this chapter was supported in part by a high - performance computing grant from oak ridge associated universities / oak ridge national laboratory and by the national science foundation under grant no .

as the title suggests , the purpose of this chapter is to review the current status of numerical simulations of black hole accretion disks . this chapter focuses exclusively on _ global _ simulations of the accretion process within a few tens of gravitational radii of the black hole . most of the simulations discussed are performed using general relativistic magnetohydrodynamic ( mhd ) schemes , although some mention is made of newtonian radiation mhd simulations and smoothed particle hydrodynamics . the goal is to convey some of the exciting work that has been going on in the past few years and provide some speculation on future directions . = 1

nanoscopic physics has been a subject of increasing experimental and theoretical interest for its potential applications in nanoelectromechanical systems ( nems)@xcite . the physical properties of these devices are of crucial importance in improving our understanding of the fundamental science in this area including many - body phenomena@xcite . one of the most striking paradigms exhibiting many body effects in mesoscopic science is quantum transport through single electronic levels in quantum dots and single molecules@xcite coupled to external leads . realizations of these systems have been obtained using semiconductor beams coupled to single electron transistors ( set s ) and superconducting single electron transistors ( ssets)@xcite , carbon nanotubes@xcite and , most recently , suspended graphene sheets@xcite . such systems can be used as a direct measure of small displacements , forces and mass in the quantum regime . the quantum transport properties of these systems require extremely sensitive measurement that can be achieved by using set s , or a resonant tunnel junction , and sset s . in this context , nems are not only interesting devices studied for ultrasensitive transducers but also because they are expected to exhibit several exclusive features of transport phenomena such as avalanche - like transport and shuttling instability@xcite . the nanomechanical properties of a resonant tunnel junction coupled to an oscillator@xcite or a set@xcite coupled to an oscillator are currently playing a vital role in enhancing the understanding of nems . the nanomechanical oscillator coupled to a resonant tunnel junction or set is a close analogue of a molecule being used as a sensor whose sensitivity has reached the quantum limit@xcite . the signature of quantum states has been predicted for the nanomechanical oscillator coupled to the sets@xcite and ssets@xcite . in these experiments , it has been confirmed that the nanomechanical oscillator is strongly affected by the electron transport in the circumstances where we are also trying to explore the quantum regime of nems . in this system , electrons tunnel from one of the leads to the isolated conductor and then to the other lead . phonon assisted tunneling of non resonant systems has mostly been shown by experiments on inelastic tunneling spectroscopy ( its ) . with the advancement of modern technology , as compared to its , scanning tunneling spectroscopy ( sts ) and scanning tunneling microscopy ( stm ) have proved more valuable tools for the investigation and characterization of molecular systems@xcite in the conduction regime . in sts experiments , significant signatures of the strong electron - phonon interaction have been observed@xcite beyond the established perturbation theory . hence , a theory beyond master equation approach or linear response is necessary . most of the theoretical work on transport in nems has been done within the scattering theory approach ( landauer ) but it disregards the contacts and their effects on the scattering channel as well as effect of electrons and phonons on each other@xcite . very recently , the non equilibrium green s function ( negf ) approach@xcite has been growing in importance in the quantum transport of nanomechanical systems@xcite . an advantage of this method is that it treats the infinitely extended reservoirs in an exact way@xcite , which may lead to a better understanding of the essential features of nems . negf has been applied in the study of shot noise in chain models@xcite and disordered junctions@xcite while noise in coulomb blockade josephson junctions has been discussed within a phase correlation theory approach@xcite . in the case of an inelastic resonant tunneling structure , in which strong electron - phonon coupling is often considered , a very strong source - drain voltage is expected for which coherent electron transport in molecular devices has been considered by some workers@xcite within the scattering theory approach . inelastic effects on the transport properties have been studied in connection with nems and substantial work on this issue has been done , again within the scattering theory approach@xcite . recently , phonon assisted resonant tunneling conductance has been discussed within the negf technique at zero temperature@xcite . to the best of our knowledge , in all these studies , time - dependent quantum transport properties of a resonant tunnel junction coupled to a nanomechanical oscillator have not been discussed so far . the development of time - dependent quantum transport for the treatment of nonequilibrium system with phononic as well as fermionic degree of freedom has remained a challenge since the 1980s@xcite . generally , time - dependent transport properties of mesoscopic systems without nanomechanical oscillator have been reported@xcite and , in particular , sudden joining of the leads with quantum dot molecule have been investigated@xcite for the case of a noninteracting quantum dot and for a weakly coulomb interacting molecular system . strongly interacting systems in the kondo regime have been investigated@xcite . more recently@xcite , the transient effects occurring in a molecular quantum dot described by an anderson - holstein hamiltonian has been discussed . to this end , we present the following study . in the present work , we shall investigate the time evolution of a quantum dot coupled to a single vibrational mode as a reaction to a sudden joining to the leads . we employ the non - equilibrium green s function method in order to discuss the transient and steady state dynamics of nems . this is a fully quantum mechanical formulation whose basic approximations are very transparent , as the technique has already been used to study transport properties in a wide range of systems . in our calculation inclusion of the oscillator is not perturbative as the sts experiments@xcite are beyond the perturbation theory . so a non - perturbative approach is required beyond the quantum master equation@xcite or linear response . hence , our work provides an exact analytical solution to the current voltage characteristics , including coupling of leads with the system , very small chemical potential difference and both the right and left fermi level response regimes . for simplicity , we use the wide - band approximation@xcite , where the density of states in the leads and hence the coupling between the leads and the dot is taken to be independent of energy . although the method we are using does not rely on this approximation . this provides a way to perform transient transport calculations from first principles while retaining the essential physics of the electronic structure of the dot and the leads . another advantage of this method is that it treats the infinitely extended reservoirs in an exact way in the present system , which may give a better understanding of the essential features of nems in a more appropriate quantum mechanical picture . we consider a single quantum dot connected to two identical metallic leads . a single oscillator is coupled to the electrons on the dot and the applied gate voltage is used to tune the single level of the dot . in the present system , we neglect the spin degree of freedom and electron - electron interaction effects and consider the simplest possible model system . we also neglect the effects of finite electron temperature of the lead reservoirs and damping of the oscillator . our model consists of the individual entities such as the single quantum dot and the left and right leads in their ground states at zero temperature . the hamiltonian of our simple system@xcite is@xmath0 c_{0}^{\dag}c_{0}+\hbar\omega(b^{\dagger}b+{{\mathchoice{{\textstyle{\frac12}}}{{\textstyle{\frac12}}}{{\scriptstyle{1/2}}}{{\scriptscriptstyle{1/2}}}}})\ , , \label{1}\ ] ] where @xmath1 is the single energy level of electrons on the dot with @xmath2 the corresponding creation and annihilation operators , the coupling strength , @xmath3 with @xmath4 is the electrostatic field between electrons on the dot and an oscillator , seen by the electrons due to the charge on the oscillator , @xmath5 is the zero point amplitude of the oscillator , @xmath6 is the frequency of the oscillator and @xmath7 are the raising and lowering operator of the phonons . the remaining elements of the hamiltonian are @xmath8 where we include time - dependent hopping @xmath9 to enable us to connect the leads @xmath10 to the dot at a finite time . for the time - dependent dynamics , we shall focus on sudden joining of the leads to the dot at @xmath11 , which means @xmath12 , where @xmath13 is the heaviside unit step function . @xmath14 is the total number of states in the lead , and @xmath15 represents the channels in one of the leads . for the second lead the hamiltonian can be written in the same way . the total hamiltonian of the system is thus @xmath16 . we write the eigenfunctions of @xmath17 as@xmath18\operatorname{h}_{m}(lk)\exp[-\mathrm{i } kx_{0 } ] \label{4}\\\psi_{n}(k , x_{0}=0)&=&a_{n}\exp[-{\textstyle\frac{l^{2}k^{2}}{2}}]\operatorname{h}_{n}(lk)\ , , \label{5}\ ] ] for the occupied , @xmath19 and unoccupied , @xmath20 , dot respectively , where @xmath21 is the shift of the oscillator due to the coupling to the electrons on the dot , where @xmath22 , @xmath23 , and @xmath24 are the usual hermite polynomials . here we have used the fact that the harmonic oscillator eigenfunctions have the same form in both real and fourier space ( @xmath25 ) . in order to transform between the representations for the occupied and unoccupied dot we require the matrix with elements @xmath26 which may be simplified@xcite as@xmath27 for @xmath28 , where @xmath29 and @xmath30 are the associated laguerre polynomials . note that the integrand is symmetric in @xmath31 and @xmath32 but the integral is only valid for @xmath28 . clearly the result for @xmath33 is obtained by exchanging @xmath31 and @xmath32 in equation ( [ 7 ] ) to obtain @xmath34!}{\max[n , m]!}}\exp\left ( -\textstyle\frac{1}{4}x^{2}\right ) \left ( { \mathchoice{{\textstyle{\frac12}}}{{\textstyle{\frac12}}}{{\scriptstyle{1/2}}}{{\scriptscriptstyle{1/2}}}}\mathrm{i}x\right ) ^{|m - n|}\operatorname{l}_{\min[n , m]}^{|m - n|}\left ( \textstyle\frac{1}{2}{x^{2}}\right ) \ , . \label{8}\ ] ] in order to calculate the analytical solutions and to discuss the numerical results of the transient and steady state dynamics of the nanomechanical systems , our focus in this section is to derive an analytical relation for the time dependent effective self - energy and the green s functions . in obtaining these results we use the wide band approximation only for simplicity , although the method we are using does not rely on this approximation , where the retarded self energy of the dot due to each lead is given by@xcite @xmath35 where @xmath10 represent the left and right leads and the green s function in the leads for the uncoupled system is @xmath36,\ ] ] with the fact that @xmath37 where @xmath15 stands for every channel in each lead and @xmath38 is the constant number density of the leads . now using the uncoupled green s function into equation ( [ 9 ] ) , the retarded self energy may be written as @xmath39v_{\alpha}(t_{2}),\label{10}\\ & = & -\mathrm{i}n_{\alpha}v_{\alpha}^{\ast}(t_{1})v_{\alpha}(t_{2})\theta(t_{1}-t_{2})\overset{+\infty}{\underset{-\infty}{{\displaystyle\int } } } \mathrm{d}\varepsilon_{\alpha}\exp[-\mathrm{i}\varepsilon_{\alpha}(t_{1}-t_{2})],\nonumber\\ & = & -\mathrm{i}n_{\alpha}v_{\alpha}^{\ast}(t_{1})v_{\alpha}(t_{2})\theta(t_{1}-t_{2})\times2\pi\delta(t_{1}-t_{2 } ) , \label{11}\ ] ] now we use the fact that @xmath40 , @xmath41 . then the above expression can be written as @xmath42 where @xmath43 is the damping factor ( @xmath44 ) . similarly @xmath45 ^{\ast } = + { \mathchoice{{\textstyle{\frac12}}}{{\textstyle{\frac12}}}{{\scriptstyle{1/2}}}{{\scriptscriptstyle{1/2}}}}\mathrm{i}\gamma_{\alpha}$ ] @xmath46 we solve dyson s equation using @xmath47 , as a perturbation . in the presence of the oscillator , the retarded and advanced green s functions on the dot , with the phonon states in the representation of the unoccupied dot , may be written as @xmath48 where @xmath49 is the retarded ( advanced ) green s function on the occupied dot coupled to the leads may be written as,@xmath50,\text { } t_{1}>0 \label{14}\\ \operatorname{g}_{k}^{a}(t_{2},t^{\prime } ) & = & + \mathrm{i}\theta(t_{2}-t^{\prime})\times\exp[-\mathrm{i}(\varepsilon _ { k}+\mathrm{i}\gamma)(t_{2}-t^{\prime})],\text { \ \ } t_{2}>0 \label{15}\ ] ] with @xmath51 , @xmath52 , and @xmath53 . the above eqs . ( [ 12 ] , [ 13 ] , [ 14 ] , [ 15 ] ) will be the starting point of our examination of the time - dependent response of the coupled system . these functions are the essential ingredients for theoretical considerations of such diverse problems as low and high voltage , coupling of electron and phonons , transient and steady state phenomena . the density matrix is related to the dot population through @xmath55 , where the density matrix @xmath56 , for @xmath57 and @xmath58 . @xmath59 is the lesser green s function@xcite on the dot including all the contributions from the leads . the lesser green s function for the dot in the presence of the nanomechanical oscillator is given by@xmath60 whereas , for @xmath61 and @xmath62 , the @xmath59 is equal to zero , and @xmath63 includes all the information of the nanomechanical oscillator and electronic leads of the system , and @xmath64 are the oscillator indices . the lesser self - energy , @xmath65 , contains electronic and oscillator contributions . the electronic contributions are non - zero only when @xmath66 and @xmath67 . as the oscillator is initially in its ground state , only the @xmath68 term gives a non - zero contribution to the lesser self - energy . the lesser self energy for the dot may be written as @xmath69 with @xmath70,\ ] ] where @xmath71 is the fermi distribution functions of the left and right leads , which have different chemical potentials under a voltage bias . for the present case of zero temperature the lesser self energy may be recast in terms of the heaviside step function @xmath72 as @xmath73\ , , \label{17}\ ] ] where @xmath74 are all non - zero only when both times ( @xmath75 ) are positive @xmath76 and @xmath77 is the fermi energy on each of leads . the density matrix @xmath78 can be calculated by using eqs . ( [ 12 ] , [ 13 ] , [ 14 ] , [ 15 ] , [ 17 ] ) in eq . ( [ 16 ] ) at @xmath57 and @xmath58 as @xmath79\\ & & \times\{\mathrm{i}\gamma{\displaystyle\int_{-\infty}^{\epsilon_{\mathrm{f}\alpha } } } \frac{\mathrm{d}\varepsilon_{\alpha}}{2\pi}\exp[-\mathrm{i}\varepsilon_{\alpha}(t_{1}-t_{2})\,\}\phi_{0,k}\phi_{n , k}^{\ast}\exp[-\mathrm{i}(\varepsilon_{k}+\mathrm{i}\gamma ) ( t_{2}-t)],\end{aligned}\ ] ] although @xmath80 is non - zero for @xmath81 , it is never required due to the way it combines with @xmath74 . by carrying out the time integrations , the resulting expression is written as@xmath82-\exp[-\mathrm{i}(\varepsilon_{\alpha}-\varepsilon_{k}-\mathrm{i}\gamma)t]-\exp [ \mathrm{i}(\varepsilon_{\alpha}-\varepsilon_{m}+\mathrm{i}\gamma)t]\right\}\end{aligned}\ ] ] the integral over the energy in the above equation is carried out@xcite . the final result for the density matrix is written as@xmath83 , \label{18}\ ] ] where we have added the contribution from the right and the left leads , which can be written in terms of@xmath84 as @xmath85\right\ } \left\ { \ln(\epsilon_{\mathrm{f}\alpha}-\varepsilon_{k}-\mathrm{i}\gamma ) -\ln(\epsilon_{\mathrm{f}\alpha}-\varepsilon_{m}+\mathrm{i}\gamma ) \right\}\\ & = & \left\{1+\exp[\mathrm{i}(\varepsilon_{k}-\varepsilon_{m}+2\mathrm{i}\gamma)t]\right\}\\ & & \times\left\{{\mathchoice{{\textstyle{\frac12}}}{{\textstyle{\frac12}}}{{\scriptstyle{1/2}}}{{\scriptscriptstyle{1/2}}}}\frac{\ln[(\epsilon_{\mathrm{f}\alpha } -\varepsilon_{k})^{2}+\gamma^{2}]}{\ln[(\epsilon_{\mathrm{f}\alpha } -\varepsilon_{m})^{2}+\gamma^{2}]}+\mathrm{i}\left[\tan^{-1}\left(\frac{\varepsilon_{f\alpha}-\varepsilon_{k}}{\gamma}\right)+\tan^{-1}\left(\frac{\varepsilon_{f\alpha}-\varepsilon_{m}}{\gamma}\right)+\pi\right]\right\},\\ z^{\alpha}_{mk } & = & \exp[\mathrm{i}(\varepsilon_{k}-\varepsilon_{m}+2\mathrm{i}\gamma ) t]\left\{-\operatorname{ei}[\mathrm{i}(\epsilon_{\mathrm{f}\alpha}-\varepsilon_{k}-\mathrm{i}\gamma)t]+\operatorname{ei}[-\mathrm{i}(\epsilon_{\mathrm{f}\alpha}-\varepsilon _ { m}+\mathrm{i}\gamma)t]\right\}\\ & & + \left\{\operatorname{ei}[\mathrm{i}(\epsilon_{\mathrm{f}\alpha}-\varepsilon_{m}+\mathrm{i}\gamma)t]-\operatorname{ei}[-\mathrm{i}(\epsilon_{\mathrm{f}\alpha}-\varepsilon _ { k}-\mathrm{i}\gamma)t]\right\},\end{aligned}\ ] ] with @xmath86 being the right and the left fermi levels and @xmath87 the exponential integral function . special care is required in evaluating the @xmath87 to choose the correct riemann sheets in order to make sure that these functions are consistent with the initial conditions @xmath88 and are continuous functions of time and chemical potential . the same applies to complex logarithms in the first , apparently simpler , form for @xmath89 . now using equation ( [ 18 ] ) , the dot population may be written as@xmath90.\ ] ] the particle current @xmath92 into the interacting region from the lead is related to the expectation value of the time derivative of the number operator @xmath93 as@xcite @xmath94\right > \label{19}\ ] ] and the final result for the current through each of the leads is written as ( see appendix [ app.a ] ) @xmath95 where @xmath96+\frac{\pi}{2}\right)\\ & & -\mathrm{i}\left\ { \operatorname{ei}[+\mathrm{i}(\epsilon_{\mathrm{f}\alpha}-\varepsilon_{m}+\mathrm{i}\gamma)t ] -\operatorname{ei}[-\mathrm{i}(\epsilon_{\mathrm{f}\alpha}-\varepsilon_{m}-\mathrm{i}\gamma)t]\right\ } , \\ i^{2\alpha}_m & = & -\left ( 1+\exp[-2\gamma t]\right ) \left(\tan^{-1}\left[\frac{\varepsilon_{f\alpha}-\varepsilon_{m}}{\gamma}\right]+\frac{\pi}{2}\right)\\ & & -{\mathchoice{{\textstyle{\frac12}}}{{\textstyle{\frac12}}}{{\scriptstyle{1/2}}}{{\scriptscriptstyle{1/2}}}}\mathrm{i}\exp[-2\gamma t]\left\ { \operatorname{ei}[+\mathrm{i}(\epsilon_{\mathrm{f}\alpha}-\varepsilon_{m}-\mathrm{i}\gamma)t ] -\operatorname{ei}[-\mathrm{i}(\epsilon_{\mathrm{f}\alpha}-\varepsilon_{m}+\mathrm{i}\gamma)t ] \right\}\\ & & + { \mathchoice{{\textstyle{\frac12}}}{{\textstyle{\frac12}}}{{\scriptstyle{1/2}}}{{\scriptscriptstyle{1/2}}}}\mathrm{i}\left\ { \operatorname{ei}[+\mathrm{i}(\epsilon_{\mathrm{f}\alpha}-\varepsilon_{m}+\mathrm{i}\gamma)t ] -\operatorname{ei}[-\mathrm{i}(\epsilon_{\mathrm{f}\alpha}-\varepsilon_{m}-\mathrm{i}\gamma)t ] \right\},\end{aligned}\ ] ] where in calculating the left current we need @xmath97 and both the contributions @xmath98 and @xmath99 and for the right current @xmath97 is replaced by @xmath100 . as before , special care is required in evaluating the @xmath87 to choose the correct riemann sheets in order to make sure that these functions are consistent with the initial conditions @xmath101 and are continuous functions of time and chemical potential . to calculate the energy transferred from the electrons to the nanomechanical oscillator , we return to the density matrix @xmath78 given in eq . ( [ 18 ] ) . we may therefore use the lesser green s function or density matrix to calculate the energy transferred to the oscillator as@xmath102 note that the normalisation in equation ( [ 21 ] ) is required as the bare density matrix contians both electronic and oscillator contributions . the trace eliminates the oscillator part , leving the electronic part . in order to further characterize the state of the nanomechanical oscillator we investigate the fano factor for the change of average occupation number , @xmath103 as a function of time . the corresponding relation for the fano factor is given by@xcite@xmath104 where @xmath105 and @xmath106 , with the average evaluated using the diagonal element of the density matrix on the quantum dot . the dot population , net current through the system , total current into the system , average energy and fano factor of a resonant tunnel junction coupled to a nanomechanical oscillator are shown graphically as a function of time for different values of coupling strength , tunneling rate , and voltage bias . the following parameters@xcite were employed : the single energy level of the dot @xmath107 , and the characteristic frequency of the oscillator @xmath108 . these parameters will remain fixed for all further discussions and have same dimension as of @xmath109 . we are interested in small and large values of tunneling from the leads , different values of the coupling strength between the electrons and the nanomechanical oscillator , and of the left chemical potential @xmath110 . time - dependent dot population @xmath54 against time for different pairs of the right and the left fermi energies ( 0,0 ) , ( 0,1 ) , ( 1,1 ) . the dotted line correspond to empty , dashed line correspond to half full and solid line corresponds to almost full state of the dot . parameters : @xmath111 . units : all the parameters have same dimension as of @xmath109.,scaledwidth=50.0% ] total current ( @xmath112 ) flowing onto the dot as a function of time for fixed values of @xmath113this current ( solid line ) is equivalent to the rate of change of dot population @xmath114 ( dashed line ) as a function of time for same parameters as of current . in this figure , solid and dashed lines have same values at all points . units : all the parameters have same dimension as of @xmath109.,scaledwidth=50.0% ] the nanomechanical oscillator induced resonance effects are clearly visible in the numerical results . it must be noted that we have obtained these results in the regime of both strong and zero or weak coupling between the nanomechanical oscillator and the electrons on the dot . the tunneling of electrons between the leads and the dot is considered to be symmetric ( @xmath115 ) and we assume that the leads have constant density of states . the dot population is shown in fig . [ fig.1 ] , as a function of time in order to see the transient and steady state dynamics of the system . we consider here empty , half full and occupied states of the system for fixed values of @xmath116 @xmath117 by choosing the right and the left fermi levels pairs ( 0 , 0 ) , ( 0 , 1 ) and ( 1 , 1 ) respectively . firstly , when both the fermi levels are below the dot energy then the dot population rises initially for a short time and for long times settles at a small but finite value . this is not quite empty because the finite @xmath118 allows some tunneling onto the dot . secondly , when the left fermi level is above the dot energy then the dot population settles in a partially full ( half full ) state . thirdly , when both the fermi levels are above the dot energy , it is completely full for a short time but for long time is not quite full , again due to the dot coupling with the leads . these results are consistent with the particle - hole symmetry of the system as the empty state of the system is not empty and the occupied state is not completely full , while the partially full is roughly half full . in fig . [ fig.2 ] , we have shown the total current flowing onto the dot as a function of time for fixed values of @xmath119 and of the left fermi level 1 . this current ( solid line ) is equivalent to the rate of change of the dot population ( dashed line ) for the same parameters . in this figure , we can not distinguish the solid and the dashed line . this confirms that our analytical results are consistent with the equation of continuity , @xmath120 , and hence , with the conservation laws for all parameters . net current ( @xmath121 ) flowing through the system as a function of both time and of the left fermi level for two different values of coupling strength : @xmath122 ( fig . [ fig.3](a))@xmath123 and @xmath124 ( fig . [ fig.3](b ) ) . parameters : @xmath125 . units : all the parameters have same dimension as of @xmath109.,title="fig:",scaledwidth=70.0% ] net current ( @xmath121 ) flowing through the system as a function of both time and of the left fermi level for two different values of coupling strength : @xmath122 ( fig . [ fig.3](a))@xmath123 and @xmath124 ( fig . [ fig.3](b ) ) . parameters : @xmath125 . units : all the parameters have same dimension as of @xmath109.,title="fig:",scaledwidth=70.0% ] net current ( @xmath121 ) flowing through the system as a function of time for two different values of coupling strength : @xmath122 ( dotted line)@xmath123 and @xmath126 ( solid line ) . parameters : units : all the parameters have same dimension as of @xmath109.,scaledwidth=50.0% ] net current ( @xmath121 ) flowing through the system as a function of time for two different values of coupling strength : @xmath122 ( dotted line)@xmath123 and @xmath126 ( solid line ) , and @xmath128 . all the parameters are same as in fig . [ fig.4 ] and have same dimension as of @xmath109.,scaledwidth=50.0% ] in fig . [ fig.3 ] we have shown the net current ( @xmath121 ) flowing through the system as a function of both time and of the left fermi level for two different values of coupling strength : @xmath122 to @xmath129 and for small and large values of @xmath118 . we observe simple oscillations in the net current flowing through the system for weak coupling strength and weak tunneling . with increasing coupling strength the structure of the oscillations becomes more complicated as shown in fig . [ fig.3](b ) . in order to interpret this complicated structure , we have a two step discussion : firstly , we have plotted the net current as a function of time in fig . [ fig.4 ] with fixed values of the fermi level , @xmath130 , @xmath131 tunneling energy , @xmath132 and for different values of coupling strength : @xmath122 and @xmath133 in this figure , in the limit of weak coupling the oscillations are again simple while for the strong coupling limit , there is a beating pattern in the oscillations . we note that the frequency of the simple oscillations is ( @xmath134 ) and these oscillations are present even in the limit of weak coupling . we conclude that this is a purely electronic process ( plasmon oscillations ) . it is clear from the figure that in the strong coupling case , it contains two beating frequencies , therefore we interpret this as due to a mixture of electronic and mechanical frequencies . secondly , in fig . [ fig.5 ] , we have plotted the net current for fixed values of @xmath130 , @xmath135 tunneling energy , @xmath136 and for different values of coupling strength : @xmath122 and @xmath133 we have found that in the regime ( @xmath137 ) , the effects of the oscillator are not apparent and the period of the nanomechanical oscillator can not be resolved . why can the period of the oscillator not be resolved by the electrons in this limit ? in this regime , electrons spend less time on the dot than the period of the oscillator . therefore , electrons do not resolve the period of the nanomechanical oscillator . now we will focus only in the regime of small tunneling @xmath138 , for further discussion in order to analyze the dynamics of the nanomechanical oscillator and the effects of coupling between the electrons and the nanomechanical oscillator . average energy transferred to the oscillator as a function of time and left fermi level for fixed values of @xmath139 and for different values of coupling strength : @xmath122 ( fig . [ fig.6](a))@xmath123 and @xmath126 ( fig . [ fig.6](b ) ) . units : all the parameters have same dimension as of @xmath109.,title="fig:",scaledwidth=70.0% ] average energy transferred to the oscillator as a function of time and left fermi level for fixed values of @xmath139 and for different values of coupling strength : @xmath122 ( fig . [ fig.6](a))@xmath123 and @xmath126 ( fig . [ fig.6](b ) ) . units : all the parameters have same dimension as of @xmath109.,title="fig:",scaledwidth=70.0% ] next we have shown the average energy of the nanomechanical oscillator as a function of time and of the left fermi energy in fig . [ fig.6 ] for fixed values of tunneling @xmath132 , @xmath131 and for different values of coupling strength @xmath140 @xmath129 . we found damped oscillations for short times and constant energy for long times . this constant average energy increases with increasing fermi level . why have we found this particular type of structure ? we know that the nanomechanical oscillator potential seen by the electrons on the dot is independent of time when the oscillator is in any of its pure eigenstates . otherwise , when the oscillator is not in a pure state , the potential seen by the electrons is time dependent . in the former case , the electrons are scattered elastically by the time independent potential and in the latter case the scattering process is inelastic because the time dependent potential allow the transfer of energy between the two . we observe that the constant average energy also has steps as a function of the left fermi level which become more pronounced with increasing coupling strength . hence , the oscillatory part of the behavior of the mechanical oscillator is damped by coupling with the electrons on the dot but the constant part is not . the damping mechanism in the transient dynamics is due to transfer of energy from the nanomechanical oscillator to the electrons on the dot while when the oscillator is in any of the pure eigenstate then there is no mechanism for the transfer of energy between the two . this same physical phenomenon also applies to the net current flowing through the dot as well . this appear to be a specifically new quantum phenomena in the study of nanomechanical systems . average energy transferred to the oscillator as a function of @xmath141 and for fixed values of @xmath142 and @xmath122 . units : all the parameters have same dimension as of @xmath109.,scaledwidth=50.0% ] can we compare this quantum phenomena with the classical mechanical oscillator ? yes , the nanomechanical oscillator has to enter the classical regime in the limit of small @xmath143 . for this , we study the dynamics of the quantum oscillator in the classical limit , in which @xmath143 in the mechanical oscillator part of the hamiltonian given in eq . ( [ 1 ] ) goes to zero , where @xmath144 . to see this , we have plotted the average energy as a function of @xmath145 in the nanomechanical part of the system in fig . [ fig.7 ] for fixed values of tunneling @xmath146 , @xmath147 and coupling strength @xmath148 . we found that the average energy of the quantum nanomechanical oscillator scales as @xmath149 . we set the average energy in the limit @xmath150 to see what happen to the system for long time . it implies that in this limit , the energy transferred to the nanomechanical oscillator is zero for long time . hence , we conclude that the long time dynamics of the classical mechanical oscillator is always zero . fano factor as a function of time for two different values of coupling strength : @xmath122 ( dotted line)@xmath123 and @xmath126 ( solid line ) . parameters : @xmath151 . units : all the parameters have same dimension as of @xmath109.,scaledwidth=50.0% ] finally , in fig . [ fig.8 ] , we have shown the fano factor as a function of time for two different values of @xmath152 and for fixed values of @xmath153 . in the limit of weak coupling , the nanomechanical oscillator shows thermal like behavior and poissonian statistics while in the limit of strong coupling its dynamics is non - thermal which leads to super - poissonian statistics . in this figure , the short time behavior is always thermal , but this is trivial as the nanomechanical oscillator is initially in its ground state . in conclusion , we have found mixed and pure states in our results which confirm the quantum dynamics of our model with the following justifications : in a classical mechanical oscillator model@xcite all states give rise to a time dependent potential . hence , all states of the classical mechanical oscillator are damped . thus , we confirm the new quantum dynamics of the nanomechanical oscillator that will be helpful for further experiments beyond the classical limit to develop better understanding of nems devices . in this work , we analyzed the time - dependent quantum transport of a resonant tunnel junction coupled to a nanomechanical oscillator by using the nonequilibrium green s function approach without treating the electron phonon coupling as a perturbation . we have derived an expression for the full density matrix or the dot population and discuss it in detail for different values of the coupling strength and the tunneling rate . we derive an expression for the current to see the effects of the coupling of the electrons to the oscillator on the dot and the tunneling rate of electrons to resolve the dynamics of the nanomechanical oscillator . this confirms that electrons resolve the dynamics of nanomechanical oscillator in the regime @xmath154 while they do not in the opposite case @xmath155 furthermore , we discuss the average energy transferred to oscillator as a function of time . we also discuss the fano factor as a function of time , which shows thermal behavior and poissonian to non - thermal and super - poissonian behavior . we have found new dynamics of the nanomechanical oscillator : pure and mixed states , which are never present in a classical oscillator . these results suggest further experiments for nems to go beyond the classical dynamics . the particle current @xmath92 into the interacting region from the lead is related to the expectation value of the time derivative of the number operator @xmath156 , as@xcite@xmath157\rangle \label{23}\\ i_{\alpha}(t ) & = & \frac{e}{\hbar}\{\operatorname{g}_{0,\alpha}^{<}(t , t)v_{\alpha,0}(t)-v_{0,\alpha}^{\ast}(t)\operatorname{g}_{\alpha,0}^{<}(t , t)\ } , \label{24}\ ] ] where we have the following relations@xmath158 where @xmath159 refers to the unperturbed states of the leads and given as@xmath160,\ ] ] with the fact that @xmath161 with @xmath38 being the constant number density of the leads and other uncoupled green s function in the leads are@xmath162 , \\ \operatorname{g}_{\alpha,\alpha}^{<}(t , t^{\prime } ) & = & \frac{1}{n}\underset{j}{{\displaystyle\sum } } \operatorname{f}_{\alpha}(\varepsilon_{\alpha})g_{\alpha , j}^{<}(t , t^{\prime})=\overset { + \infty}{\underset{-\infty}{{\displaystyle\int } } } \mathrm{d}\varepsilon_{\alpha}\operatorname{f}_{\alpha}(\varepsilon_{\alpha})\mathrm{i}n_{\alpha}\exp[-\mathrm{i}\varepsilon_{\alpha}(t - t^{\prime})],\end{aligned}\ ] ] now using equations ( [ 25 ] & [ 26 ] ) in the equation ( [ 24 ] ) of current through lead @xmath91 as@xmath163 using the fact that @xmath164 , we can simplify the above equation as@xmath165\right\ } , \label{28}\ ] ] where @xmath166 are non - zero only when both the times ( @xmath167 ) are positive @xmath168 . although @xmath169 is non - zero for @xmath81 , it is never required due to the way it combines with @xmath166 . here we note that we require @xmath169 from eq . ( [ 14 ] & [ 15 ] ) for positive times only ( @xmath170 . the first integral on right hand side of eq . ( [ 28 ] ) may be solved by using eq . ( [ 13 ] , [ 14 ] & [ 17 ] ) as@xmath171\exp[-\mathrm{i}\varepsilon_{\alpha}(t^{\prime}-t)]\nonumber\\ & = & \frac{\mathrm{i}\gamma}{2\pi}\sum_{m}\phi_{0,m}\phi_{0,m}^{\ast}{\displaystyle\int\limits_{-\infty}^{\epsilon_{\mathrm{f}\alpha } } } \mathrm{d}\varepsilon_{\alpha}\left\{\frac{1- \exp[\mathrm{i}(\varepsilon_{\alpha}-\varepsilon_{m}+\mathrm{i}\gamma)t]}{\varepsilon_{\alpha}-\varepsilon_{m}+\mathrm{i}\gamma}\right\}\nonumber\\ & = & \frac{\mathrm{i}\gamma}{2\pi}\sum_{m}\phi_{0,m}\phi_{0,m}^{\ast}\left\{\ln(\epsilon_{\mathrm{f}\alpha}-\varepsilon_{m}+\mathrm{i}\gamma ) -\operatorname{ei}[\mathrm{i}(\epsilon_{\mathrm{f}\alpha}-\varepsilon_{m}+\mathrm{i}\gamma)t]\right\ } , \label{29}\ ] ] where the final result is obtained using standard integrals@xcite . we note once again that special care is required in evaluating the @xmath172 and @xmath87 to choose the correct riemann sheets in order to make sure that these functions are consistent with the initial conditions and are continuous functions of time and chemical potential . this statement will also apply to all further discussions . the second & third integral on right hand side of eq . ( [ 28 ] ) are written as@xmath173 this integral can be solved in the same way as for the dot population . the final result is written as@xcite@xmath174\right)\left(\tan^{-1}\left[\frac{\epsilon_{\mathrm{f}\alpha}-\varepsilon_{m}}{\gamma}\right]+\frac{\pi}{2}\right)\nonumber\\ & & + { \mathchoice{{\textstyle{\frac12}}}{{\textstyle{\frac12}}}{{\scriptstyle{1/2}}}{{\scriptscriptstyle{1/2}}}}\mathrm{i}\exp[-2\gamma t]\left(-\operatorname{ei}[+\mathrm{i}(\epsilon_{\mathrm{f}\alpha}-\varepsilon_{m}-\mathrm{i}\gamma)t ] + \operatorname{ei}[-\mathrm{i}(\epsilon_{\mathrm{f}\alpha}-\varepsilon_{m}+\mathrm{i}\gamma)t]\right)\nonumber\\ & & + { \mathchoice{{\textstyle{\frac12}}}{{\textstyle{\frac12}}}{{\scriptstyle{1/2}}}{{\scriptscriptstyle{1/2}}}}\mathrm{i}\left ( \operatorname{ei}[+\mathrm{i}(\epsilon_{\mathrm{f}\alpha}-\varepsilon_{m}+\mathrm{i}\gamma)t ] -\operatorname{ei}[-\mathrm{i}(\epsilon_{\mathrm{f}\alpha}-\varepsilon_{m}-\mathrm{i}\gamma)t]\right)\biggr\ } , \label{30}\ ] ] and the fourth integral on right hand side of equation ( [ 28 ] ) can be solved by using eq . ( [ 13 ] , [ 15 ] , & [ 17 ] ) as@xmath175\exp [ -\mathrm{i}\varepsilon_{\alpha}(t - t^{\prime})]\nonumber\\ & = & \frac{-\mathrm{i}\gamma}{2\pi}\sum_{m}\phi_{0,m}\phi_{0,m}^{\ast}{\displaystyle\int\limits_{-\infty}^{\epsilon_{\mathrm{f}\alpha } } } \mathrm{d}\varepsilon_{\alpha}\left\{\frac{1-\exp[-\mathrm{i}(\varepsilon_{\alpha}-\varepsilon _ { m}-\mathrm{i}\gamma)t}{\varepsilon_{\alpha}-\varepsilon_{m}-\mathrm{i}\gamma}\right\}\nonumber\\ & = & \frac{-\mathrm{i}\gamma}{2\pi}\sum_{m}\phi_{0,m}\phi_{0,m}^{\ast}\left\ { \ln(\epsilon_{\mathrm{f}\alpha}-\varepsilon_{m}-\mathrm{i}\gamma ) -\operatorname{ei}[-\mathrm{i}(\epsilon_{\mathrm{f}\alpha}-\varepsilon_{m}-\mathrm{i}\gamma)t ] \right\ } \label{31}\ ] ] using equations ( [ 29 ] , [ 30 ] & [ 31 ] ) in eq . ( [ 28 ] ) , the final expression for the current is written as@xmath176 where components of current are written as@xmath96+\frac{\pi}{2}\right)\\ & & -\mathrm{i}\left\ { \operatorname{ei}[+\mathrm{i}(\epsilon_{\mathrm{f}\alpha}-\varepsilon_{m}+\mathrm{i}\gamma)t ] -\operatorname{ei}[-\mathrm{i}(\epsilon_{\mathrm{f}\alpha}-\varepsilon_{m}-\mathrm{i}\gamma)t ] \right\}],\\ i^{2\alpha}_m & = & -\left(1+\exp[-2\gamma t]\right ) \left\{\tan^{-1}\left[\frac{\epsilon_{\mathrm{f}\alpha}-\varepsilon_{m}}{\gamma}\right]+\frac{\pi}{2}\right\}\\ & & -{\mathchoice{{\textstyle{\frac12}}}{{\textstyle{\frac12}}}{{\scriptstyle{1/2}}}{{\scriptscriptstyle{1/2}}}}\mathrm{i}\exp[-2\gamma t]\left\ { \operatorname{ei}[+\mathrm{i}(\epsilon_{\mathrm{f}\alpha}-\varepsilon_{m}-\mathrm{i}\gamma)t ] -\operatorname{ei}[-\mathrm{i}(\epsilon_{\mathrm{f}\alpha}-\varepsilon_{m}+\mathrm{i}\gamma)t ] \right\}\\ & & + { \mathchoice{{\textstyle{\frac12}}}{{\textstyle{\frac12}}}{{\scriptstyle{1/2}}}{{\scriptscriptstyle{1/2}}}}\mathrm{i}\left\ { \operatorname{ei}[+\mathrm{i}(\epsilon_{\mathrm{f}\alpha}-\varepsilon_{m}+\mathrm{i}\gamma)t ] -\operatorname{ei}[-\mathrm{i}(\epsilon_{\mathrm{f}\alpha}-\varepsilon_{m}-\mathrm{i}\gamma)t ] \right\},\end{aligned}\ ] ] where in calculating the left current we need @xmath97 together with both @xmath98 and @xmath99 whereas for the right current @xmath97 is replaced by @xmath100 . a. schliesser , et . al . , nature physics 5 , 509 ( 2009 ) ; , k. l. ekinci and m. l. roukes , review of scientific instruments 76 , 061101 ( 2005 ) . ; k. l. ekinci , small 2005,1 , no . 8 - 9 , 786 - 797 . ; m. l. roukes , technical digest of the 2000 solid state sensor and actuator workshop ; nanoelectromechanical systems . ; h. g. craighead , science 290 , 1532 ( 2000 ) . ; p. kim and c. m. lieber , science 126 , 2148 ( 1999 ) . s. d. bennett , and a.a . clerk , phys . b 78 , 165328 ( 2008 ) . ; s. akita , y. nakayama , s. mizooka , y. takano , t. okawa , y. miyatake , s. yamanaka , m. tsuji , and t. nosaka , appl . 79 , 1691 ( 2001 ) . ; a. m. fennimore , t. d. yuzvlnsky , w. q. han , m. s. fuhrer , j. cummings , and a. zettl , nature 424 , 408 ( 2003 ) . j. kinaret , t. nord , and s. viefers , appl . 82 , 1287 ( 2003 ) . ke and h. d. espinosa , appl . 85 , 681 ( 2004 ) . ; v. sazonova , y. yaish , h. stnel , d. roundy , t. arias , and p. mceuen , nature 431 , 284 ( 2004 ) . h. park et al . , nature ( london ) 407 , 57 ( 2000).j . koch and f. von oppen , phys . 94 , 206804 ( 2005 ) . ; j. koch , m. e. raikh , and f. von oppen , ibid . 95 , 056801 ( 2005 ) . ; j. koch , f. von oppen , and a. v. andreev , phys . rev . b 74 , 205438 ( 2006 ) . m. a. reed , c. zhou , c. j. muller , t. p. burgin , and j. m. tour , science 278 , 252 ( 1997 ) . ; r. h. m. smit , y. noat , c. untiedt , n. d. lang , m. c. van hemert , and j. m. van ruitenbeek , ibid . 419 , 906 ( 2002 ) . l. h. yu , z. k. keane , j. w. ciszek , l. cheng , m. p. stewart , j. m. tour , and d. natelson , phys . rev . 93 , 266802 ( 2004 ) . ; l. h. yu and d. natelson , nano lett . 4 , 79 ( 2004 ) . ; m. elbing , r. ochs , m. koentopp , m. fischer , c. von hnisch , f. weigend , f. evers , h. b. weber , and m. mayor , proc . 102 , 8815 ( 2005 ) . ; m. poot , e. osorio , k. oneill , j. m. thijssen , d. vanmaekelbergh , c. a. van walree , l. w. jenneskens , and h. s. j. van der zant , nano lett . 6 , 1031 ( 2006 ) . e. a. osorio , k. oneill , n. stuhr - hansen , o. f. nielsen , t. bjrnholm , and h. s. j. van der zant , adv . ( weinheim , ger . ) 19 , 281 ( 2007 ) . ; e. lrtscher , h. b. weber , and h. riel , phys . 98 , 176807 ( 2007 ) . j. repp , g. meyer , s. m. stojkovi , a. gourdon , and c. joachim , phys . 94 , 026803 ( 2005 ) . ; j. repp , g. meyer , s. paavilainen , f. e. olsson , and m. persson , phys . 95 , 225503 ( 2005 ) . a. shimizu and m. ueda , phys . 69 , 1403 ( 1992 ) ; o. l. bo and yu . galperin , phys . b 55 , 1696 ( 1997 ) ; b. dong , h. l. cui , x. l. lei , and n. j. m. horing , phys . rev . b 71 , 045331 ( 2005 ) ; y .- c . chen and m. di ventra , phys . lett . 95,166802 ( 2005 ) . n. nishiguchi , phys . 89 , 066802 ( 2002 ) ; a. a. clerk and s. m. girvin , phys . rev . b 70 , 121303(r ) ( 2004 ) ; t. novotn , a. donarini , c. flindt , and a .- p.jauho , phys . 92 , 248302 ( 2004 ) . a. d. armour and a. mackinnon , phys . b 66 , 035333 ( 2002 ) ; c. flindt , t. novotny , and a .- p . jauho , phys . rev . b 70 , 205334 ( 2004 ) ; j. wabnig , d. v. khomitsky , j. rammer , and a. l. shelankov , phys . rev . b 72,165347 ( 2005 ) . s. dallakyan and s. mazumdar , appl . 82 , 2488 ( 2003 ) ; k. walczak , phys . status solidi b 241 , 2555 ( 2004 ) ; y .- c . chen and m. di ventra , phys . rev . b 67 , 153304 ( 2003 ) ; j. lagerqvist , y .- c . chen , and m. di ventra , nanotechnology 15 , s459 ( 2004 ) . v. aji , j. e. moore , and c. m. varma , arxiv : cond - mat/0302222 ( unpublished ) . ; dmitry a. ryndyk and gianaurelio cuniberti , phys . b 76 , 155430 ( 2007 ) . ; j .- x . zhu and a. v. balatsky , phys . rev . b 67 , 165326 ( 2003 ) . ; m. tahir and a. mackinnon , phys . rev . b 77 , 224305 ( 2008 ) . v. moldoveanu , v. gudmundsson and a. manolescu , phys . b 76 , 085330 ( 2007 ) . ; j. maciejko , j. wang and h. guo , phys . rev . b 74 , 085324 ( 2006 ) . ; y. wei and j. wang , phys . rev . b 79 , 195315 ( 2009 ) . ; p. myhnen , a. stan , g. stefanucci and r. van leeuwen , phys . b 80 , 115107 ( 2009 ) . ; a. r. hernndez , f. a. pinheiro , c. h. lewenkopf , and e.r . mucciolo , phys . b 80 , 115311 ( 2009 ) . h. hbener and t. brandes , phys . b 80 , 155437 ( 2009 ) . ; s. ramakrishnan , y. gulak and h. benaroya , phys . rev . b 78 , 174304 ( 2008 ) . ; g. kiesslich , e. schll , t. brandes , f. hohls , and r. j. haug , phys . rev . lett . 99 , 206602 ( 2007 ) . ; h. hbener and t. brandes , phys . 99 , 247206 ( 2007 ) .

we present a theoretical study of time - dependent quantum transport in a resonant tunnel junction coupled to a nanomechanical oscillator within the non - equilibrium green s function technique . an arbitrary voltage is applied to the tunnel junction and electrons in the leads are considered to be at zero temperature . the transient and the steady state behavior of the system is considered here in order to explore the quantum dynamics of the oscillator as a function of time . the properties of the phonon distribution of the nanomechnical oscillator strongly coupled to the electrons on the dot are investigated using a non - perturbative approach . we consider both the energy transferred from the electrons to the oscillator and the fano factor as a function of time . we discuss the quantum dynamics of the nanomechanical oscillator in terms of pure and mixed states . we have found a significant difference between a quantum and a classical oscillator . in particular , the energy of a classical oscillator will always be dissipated by the electrons whereas the quantum oscillator remains in an excited state . this will provide useful insight for the design of experiments aimed at studying the quantum behavior of an oscillator .

first principles calculations of the free induction decay ( fid ) measured by nuclear magnetic resonance ( nmr ) in solids is a long - standing theoretical problem@xcite still lacking a controllable solution@xcite . the most challenging aspect of this problem is the prediction of the long - time behavior of the fids . recently some progress in this direction was made on the basis of the notion of microscopic chaos@xcite . namely , it was predicted that the generic long - time behavior of fids in single crystals has the character of exponential decay with or without sinusoidal oscillations . in the most common case of magnetic dipolar interaction between nuclear spins , the oscillatory regime is realized , and hence , the long - time fid behavior can be parameterized as @xmath3 where @xmath4 , @xmath5 , @xmath6 and @xmath7 are some constants whose values were not predicted . it was only estimated@xcite that , generically , the values of @xmath5 and @xmath6 fall on the timescale of the spin - spin interactions often referred to as @xmath8 . it was also estimated that the long - time behavior ( [ ltform ] ) becomes dominant after a time on the order of several times @xmath8 from the beginning of the fid . the above predictions agree with the experimental@xcite and numerical@xcite results for quantum and classical spin systems . the situation becomes somewhat more involved theoretically for polycrystalline samples or crystal powders . different orientations of single crystallites in polycrystals / powders with respect to an external magnetic field imply different microscopic hamiltonians , and hence different values of @xmath5 and @xmath6 , which in turn leads to the additional averaging over the oscillation frequencies . at sufficiently long times , the crystallites exhibiting the smallest value of @xmath5 should start dominating the overall response , and , therefore , the well - defined frequency of these crystallites should also control the overall decay . we call the latter regime the asymptotic long - time behavior . it is to be distinguished from the intermediate behavior , which we define as the regime , when the individual crystallites have reached their respective long - time regimes but the asymptotic polycrystalline long - time behavior is not yet reached . the challenge here is to understand how long the above transition to the asymptotic behavior takes , and what the intermediate behavior looks like . it is , in particular , possible that the intermediate behavior exhibits a tentative `` washing out '' of the fid beats . on the experimental side , the available facts about the long - time fid behavior in polycrystals / powders do not reveal a consistent picture . on the one hand , no well - defined long - time beats of form have been observed in the caf@xmath2 powder ( within the range limited by the experimental signal - to - noise ratio)@xcite . on the other hand , in hyperpolarized solid xenon , which is supposedly polycrystalline , the experiments reveal well - defined beats of form ( [ ltform ] ) appearing rather quickly@xcite . in the latter case , the situation is complicated by the fact that hyperpolarized solid xenon is prepared in convection cells@xcite by first optically polarizing xenon gas@xcite and then rapidly cooling it into a liquid phase and subsequently quenching the liquid into the solid phase . as a result , an uncertainty remains about the proper thermalization of the resulting solid . in addition , the formation of crystal structure in solid xenon is controlled by the relatively weak van der waals interaction , which is known to allow significant residual atomic motion@xcite that further complicates the theoretical analysis . a related unclear issue is the strength of the exchange coupling between xenon nuclei . in this paper , we assume that hyperpolarized solid xenon samples investigated in refs.@xcite can be described as polycrystalline fcc lattices of immobile nuclear spins coupled by magnetic dipole interaction . we perform the first principles calculations of @xmath0xe fid on the basis of the approximation procedure introduced in refs.@xcite . we also perform the first principles @xmath1f fid calculation for the powder of caf@xmath2 , where @xmath1f nuclei form a simple cubic lattice . our goal is to verify whether the above calculations are sufficient to explain why the well - defined beats of form ( [ ltform ] ) were seen in polycrystalline solid xenon@xcite but not in caf@xmath2 powder@xcite . we will use the approximation scheme for fid calculations that was introduced in ref.@xcite with small modifications added in ref.@xcite . this scheme is quite similar to the one introduced earlier in ref.@xcite . alternative attempts to calculate powder fids were made in refs.@xcite . the approximation technique of ref . @xcite results in a very accurate description of the extended initial behavior of single crystal fids in caf@xmath2 . it also leads to the long - time behavior of form , but with constants noticeably different from those observed experimentally ( see below ) . as explained in ref.@xcite , an accurate prediction of the parameters in eq.([ltform ] ) is not expected here due to the oversimplified nature of the approximation . we are , however , mainly interested in the qualitative question of the difference the solid xenon and the caf@xmath2 powders posed at the end of the preceding section . answering this question presumably depends on the qualitative differences in the distributions of @xmath5 and @xmath6 for different orientations of single crystallites in the external magnetic field . the approximations used should , therefore , be adequate for detecting such differences , if they exist . in caf@xmath2 , @xmath1f nuclei are characterized by spin 1/2 , gyromagnetic ratio @xmath9 and abundance @xmath10 . these nuclei form simple cubic lattice with period @xmath11 ( at 293 k ) . for solid xenon , we perform the calculation for the fcc lattice with the nearest neighbor distance @xmath12 and abundance @xmath13 of @xmath0xe nuclei . this abundance is representative of the sample most studied in refs.@xcite . other nuclear isotopes present in this xenon sample are assumed to be non - magnetic . ( here , in particular , we neglect the contribution of the magnetic isotope @xmath14xe , which has spin 3/2 with a smaller gyromagnetic ratio . its abundance is 2 per cent in the sample analyzed . ) the @xmath0xe nuclei have spin 1/2 with gyromagnetic ratio @xmath15 . we obtain the powder fid as the average over large number of single crystallite fids . the orientation of each crystallite in the external magnetic field is selected randomly . for each crystallite , we calculate the fid as the infinite temperature correlation function@xcite @xmath16 for the microscopic hamiltonian of the truncated magnetic dipole interaction in the larmor rotating reference frame : @xmath17 , \label{h}\ ] ] where @xmath18 and @xmath19 are the lattice site indices , @xmath20 is the operator of the @xmath21th ( @xmath22 , @xmath23 , or @xmath24 ) component of the @xmath18th nuclear spin 1/2 with the @xmath24-axis chosen along the direction of the external static magnetic field , and @xmath25 are the coupling constants given by @xmath26 here , @xmath27 is the position vector of the @xmath18th nucleus , and @xmath28 is the angle between vector @xmath29 and the @xmath24-axis . extending the approximation scheme of refs.@xcite to the case of isotopic abundance @xmath30 , we obtain the fid function @xmath31 as the numerical solution of the following integral equation : @xmath32 where @xmath33 \prod_n \left [ 1 - \nu + \nu \hbox{cos}\left ( { 3 \over 4 \hbar } j_{mn } \eta t \right ) \right ] , \label{g}\ ] ] @xmath34 @xmath35 @xmath36 @xmath37 @xmath38 ( see also @xcite ) , @xmath39 @xmath40 . \label{m4g}\ ] ] the initial condition for eq.([fint ] ) is @xmath41 . parameter @xmath42 given by eq . does not depend on @xmath43 . therefore , one can first set @xmath44 , then calculate @xmath42 and finally use eq . to calculate the actual value of @xmath43 . in order to illustrate the performance of the above approximation scheme , we show in fig . [ fig - caf2-cryst ] the results of the calculations of the initial and the long - time behavior of single crystal fids in caf@xmath2 for three directions of the external magnetic field . while the linear plots ( insets ) in each panel of fig . [ fig - caf2-cryst ] illustrate that the overall agreement of the theoretical and the experimental curves is very good , the semilog plots ( main panels ) amplify the discrepancy in the long - time tails . the comparison of the theoretical and the experimental values of @xmath5 and @xmath6 for the long - time fits of form ( [ ltform ] ) is presented in table [ tab - cryst ] . it indicates a typical discrepancy of about 20 percent . ( 100 , 204 ) ( 0 , 136 ) = 3.3 in ( 0 , 67 ) = 3.3 in ( 0 , -2 ) = 3.3 in ( 49 , 32 ) = 1.3 in ( 49 , 101 ) = 1.3 in ( 49 , 170 ) = 1.3 in ( 0,61 ) ( 0,128 ) ( 0,195 ) .table summarizing the experimental and theoretical values of parameters @xmath5 and @xmath6 for the single crystal caf@xmath2 . the parameters are obtained by fitting the long - time tails of the fids presented in fig . [ fig - caf2-cryst ] by eq.([ltform ] ) . [ cols="^,^,^,^,^,^,^ " , ] let us first examine the discrepancies between the theoretical calculations and the experimental curves . for the caf@xmath2 powder , the discrepancies appear starting from the intermediate section of the fid . we believe that these discrepancies are due to the limitations of the theoretical approximation scheme based on eq.([fint ] ) . on the other hand , the discrepancy for the solid xenon appears from the very beginning of the fid . it is related to the fact that the theoretical and the experimental values of the second moment @xmath45 are different from each other ( @xmath46 and @xmath47 , respectively ) . since the theoretical value of @xmath45 is the input rather than the output parameter for the theoretical approximation scheme , the above discrepancy indicates the inadequacy of our initial assumptions about either the form or the parameters of the hamiltonian ( [ h ] ) . it may be related to the insufficient thermalisation and/or atomic motions in the quenched solid xenon samples@xcite . leaving this discrepancy to be investigated in a later experimental work , below we focus on the outcome of the theoretical calculation and examine the differences between the long - time fid behavior for caf@xmath2 powder and polycrystalline solid xenon . figures [ fig - caf2-powder](b ) and [ fig - xenon](b ) include fits of the true theoretical long - time behavior to the asymptotic formula , while figs . [ fig - caf2-powder](c ) and [ fig - xenon](c ) attempt to fit the middle section of the theoretical fids with eq.([ltform ] ) . in caf@xmath2 , the intermediate fid behavior is not well described by eq .. at the same time , the asymptotic long - time behavior becomes pronounced relatively quickly after about three beats . on the contrary , the intermediate behavior of the solid xenon fid is well described by eq . , which covers about 6 beats and 5 orders of magnitude , while the asymptotic long - time behavior emerges only at relatively late times and small values of fid . it is expected that the behavior of the middle section of the fids is controlled by the typical single crystallite values of @xmath5 , while the true long - time behavior is controlled by the crystallites with the smallest value of @xmath5 . in order to clarify this issue further , we present in figs . [ fig - gmom - caf2](a ) and [ fig - gmom - xenon](a ) the theoretical values of the long - time parameters @xmath5 and @xmath6 for the single crystallites included in the powder average , while figs . [ fig - gmom - caf2](b ) and [ fig - gmom - xenon](b ) show the histogram of the resulting points . ( 100 , 180 ) ( 0 , 93 ) = 3.3 in ( 10 , 3 ) = 2.9 in ( 0,175 ) ( 0,80 ) ( 100 , 180 ) ( 0 , 93 ) = 3.3 in ( 10 , 3 ) = 2.9 in ( 0,175 ) ( 0,80 ) one can now appreciate the qualitative difference between the powder of simple cubic crystallites and the powder of fcc crystallites . the long - time parameters @xmath5 and @xmath6 for the simple cubic lattice are much broader and increase or decrease roughly proportionally to each other . therefore , the typical values of @xmath5 and @xmath6 are sufficiently different from those representing the asymptotic decay . the poor performance of the middle section fit is in large part due to the larger difference of frequency @xmath6 between a typical value and the asymptotic long - time value . on the contrary , all possible values of @xmath5 and @xmath6 are more clustered for the fcc polycrystal and do not exhibit much of a systematic dependence on each other . as a result , the typical value of @xmath5 is rather close to the true long - time value . this explains why the fit to the intermediate fid behavior works so well over an extended time interval . the more clustered behavior of parameters @xmath5 and @xmath6 for the fcc powder was , in fact , expected . the differences in @xmath5 and @xmath6 originate from the differences in the truncated hamiltonians for different orientations of the magnetic field with respect to single crystallites . the orientation - dependent differences are expected to be smaller for the fcc lattice , because the fcc lattice is in a sense more isotropic : each lattice site has 12 nearest neighbors as opposed to 6 nearest neighbors in the case of simple cubic lattice . the 12-neighbor environment is obviously more isotropic than 6-neighbor environment . the higher sensitivity of the simple cubic lattice to different orientations of the magnetic field can be illustrated by the example of the magnetic field oriented along [ 111 ] crystal direction , in which case the coupling constants to all six nearest neighbors become equal to zero the so called `` magic angle '' condition . in principle , the polycrystal / powder average also depends on the distribution of parameters @xmath4 and @xmath7 in eq.([ltform ] ) , but we found that the parameter @xmath4 has comparable values for all orientations and that its distribution does not add any new qualitative insight to the above discussion . likewise , we were not able to find any particularly important aspect associated with the distribution of @xmath7 , apart from the observation that it makes the frequency @xmath6 of the intermediate section fit for caf@xmath2 powder smaller than the minimal value of @xmath6 for individual single crystallites . we have presented first principles fid calculations for the powder of caf@xmath2 and for polycrystalline solid xenon . the long - time fid decay for powders / polycrystals is the superposition of the long time decays for individual single crystallites . the typical single crystallite values of the long - time parameter @xmath5 control the middle section of the resulting fids , whereas the true long - time behavior is controlled by single crystallites with the smallest value of @xmath5 . we have found that the single crystallite parameters @xmath5 and @xmath6 are rather broadly distributed caf@xmath2 , and as a result , the intermediate section beats become washed out and relatively quickly evolve to the asymptotic long - time behavior . such a behavior might be observable in the future caf@xmath2 powder experiments with improved signal - to - noise ratio . on the contrary , in the case of solid xenon , the single crystallite values are more clustered , and as a result the middle section is characterized by well defined beat frequency and exponential decay constant over several orders of magnitude , while the true long - time behavior appears only at relatively later times . we explain the above clustering of parameters @xmath5 and @xmath6 by the more isotropic character of the fcc lattice in comparison with the simple cubic lattice . our findings suggests that the experiments conducted so far in solid xenon have been able to access only the intermediate section of the powder / polycrystalline fids , and hence observed the well - defined behavior ( [ ltform ] ) . it is clear that , although observing well - defined behavior ( [ ltform ] ) in the intermediate fid section requires suitable crystal structures , such a behavior would be extremely unlikely , if the long - time behavior of single crystallites were different from ( [ ltform ] ) . therefore , the experiments accessing the intermediate section of fids in polycrystalline fcc solids are appropriate to test the theoretical long - time predictions@xcite originally made mostly for single crystals . as discussed in ref.@xcite , the same conclusion is likely true for solids with disordered arrangements of magnetic nuclear sites , but further experimental and theoretical investigation of this situation is necessary .

free induction decay ( fid ) measured by nuclear magnetic resonance ( nmr ) in a polycrystalline solid is the isotropic average of the fids for individual single crystallites . it has been recently proposed theoretically and verified experimentally that the long - time behavior of single - crystal fids has the universal form of exponentially decaying sinusoidal oscillations . polycrystalline averaging complicates the situation theoretically , while the available experimental evidence is also ambiguous . exponentially decaying sinusoidal oscillations have been observed for @xmath0xe in polycrystalline solid xenon but not for @xmath1f in the powder of caf@xmath2 . in this paper , we present the first principles fid calculations for the powders of both caf@xmath2 and solid xenon . in both cases , the asymptotic long - time behavior has the expected form of exponentially decaying sinusoidal oscillations , which is determined by the single crystallite fid with the slowest exponential decay . however , this behavior appears only at rather small values of the signal that have not yet been measured in experiments . at intermediate times accessible experimentally , a polycrystalline fid depends on the distribution of the exponential decay constants and oscillation frequencies for single crystallite fids . in caf@xmath2 , these parameters are relatively broadly distributed , and as a result , the sinusoidal long - time oscillations become somewhat washed out . in contrast , the single crystallite parameters are more clustered in solid xenon , and , as a result , the experimentally observable range is characterized by well - defined oscillation frequency and exponential decay constant even though both of these parameters do not represent the true long - time behavior . the above difference of the intermediate fid behavior originates from the difference of the crystal structures of solid xenon and caf@xmath2 .

superwinds are galaxy scale outflows , caused by supernovae in nuclear starburst regions or active galactic nuclei ( agns ) . they are so powerful that interstellar matter within the galaxies is blown out . some of the material may escape to the intergalactic or group medium , while some of the material may be recycled throughout the galactic halo @xcite . superwinds are expected to quench star - formation activity ( feedback ) and to enrich the external medium with new metals . generally , galactic winds are diffuse and difficult to observe . m82 , one of the nearest starburst galaxies ( 3.63 mpc , * ? ? ? * ) , is one of the most well known examples of the superwind phenomenon . its large inclination angle and proximity allow us to see many details of the wind phenomenon far from the galactic plane . the source has been observed in hot gas ( @xmath7 k ; e.g. , @xcite ) , ionized gas ( @xmath8 k ; e.g. , @xcite ) , and molecular gas ( @xmath9 k ; e.g. , @xcite ) . the kinematics and ionization of the wind material over the inner few kiloparsecs have been investigated in detail . @xcite and @xcite modeled the outflow structure using position - velocity diagrams in optical emission lines . the emission line ratios of the inner region indicate that photoionization by the nuclear starburst plays a significant role in the excitation @xcite . in recent years , new observational methods such as integral field spectroscopy ( e.g. , @xcite ) and spectropolarimetry ( e.g. , @xcite ) have revealed its more intricate structure . our goal is to shed light on processes behind large - scale galactic winds . very little is known about their total extent , energetics and importance in the context of galaxy evolution . by studying the most spatially extended emission , we can obtain a better understanding of the total kinetic energy of the wind . there are many questions that remain unanswered for m82 s outflow . how old is the wind and how far does it extend ? is it powered by radiation pressure or wind pressure , or a combination of both ? is the source of energy impulsive or sustained over many dynamical times ? is most of the outflowing material swept up or entrained from the disk ? does the wind material escape the galaxy or fall back to the disk ? to have any chance of answering these questions , we need a better understanding of the most basic properties of the large - scale wind . the most distant gas cloud in m82 is the ` cap ' originally discovered in h@xmath0 and x - ray emission at a radius of 11.6 kpc along the minor axis of m82 @xcite . strong uv emission provides evidence for reflecting dust in the cloudlets that make up the cap @xcite . the metal abundances of o , ne , mg , si , and fe of x - ray emitting gas in the cap suggest that most of the metals arise from a circumnuclear starburst dominated by type ii supernovae @xcite . we now show that the dominant ionization source in the cap provides an important clue to the wind s origin and history . @xcite suggested the cap is either photoionized by uv photons from the nuclear starburst region or by a shock being driven by the hot wind into a dense halo cloud , or a combination of both . the x - ray observations already support the idea that the wind reaches the distance of the cap , but are the optical emission line diagnostics consistent with a wind - driven shock ? therefore , in order to obtain emission line intensity map and line ratio maps at high spatial resolution , we carried out fabry - perot observations of m82 s cap with the subaru telescope . this combination enables us to detect weak emission with a larger field of view than that of integral field spectroscopy . through a comparison of the observed line ratios and those calculated by photoionization and shock models , we discuss the ionization source of the m82 cap and a likely evolution history for the large - scale galactic wind . on 2011 november 22 , we observed the central part of the m82 cap , @xmath1010 n of the nucleus of m82 , with the kyoto3dii fabry - perot mode @xcite mounted on the cassegrain focus of the subaru telescope . figure [ fig : image - m82-whole ] displays the position of the cap relative to the m82 center , and indicates the region where we observed in this observation . this mode uses an et-50 etalon manufactured by queensgate instruments . the field of view is @xmath101.9 and the pixel scale is 0.112 pixel@xmath4 after 2 @xmath11 2 on - chip binning . the spectral resolution @xmath12 corresponds to 19 at 6598.95 . we obtained 14 object frames for h@xmath0 + [ ] @xmath26548,6583 , five for [ ] @xmath26716,6731 , and two for the off bands . the observed wavelengths at the field centres are summarized in table [ tb : obs - wavelength ] . the exposure time for each frame was 300 seconds . we also observed a standard star eggr247 for flux calibration @xcite . bias subtraction and flat fielding were performed for the target and standard star frames . because the center wavelength in fabry - perot observations depends on the distance from the center of the field of view , simple sky subtraction results in some residuals due to sky emission lines . we measured sky emission fluxes in blank regions of the object frames , and subtracted it from the regions at the same distance from the center . flux calibration and distortion correction were carried out for the target frames . we used a spectrum catalog of @xcite for flux calibration for each wavelength setting . the positional offsets among the object frames were detected , because the cassegrain auto guider was unavailable due to repairs and we did not use it in this observation run . we corrected the offsets by using the stars in the target frames . we matched the spatial resolution of the target frames to the worst one , 0.9 , and carried out 4 @xmath11 4 binning , resulting in the pixel scale of 0.45 pixel@xmath4 . cc h@xmath0 + [ ] @xmath26548,6583 & 6546 , 6554 , 6562 , 6570 , 6578 , 6586 , 6594 , 6602 + [ ] @xmath26716,6731 & 6714 , 6722 , 6730 , 6738 , 6746 + continuum & 6656 because of the relatively low spectral resolution , h@xmath0 and [ ] @xmath26548,6583 were blended in h@xmath0 + [ ] band . we fitted these lines pixel by pixel with the transmission curve of the fabry - perot interferometer ( airy function : * ? ? ? * ) and decomposed them ( see @xcite ) . better wavelength sampling than that of the previous observation enables us to find the best velocity center . we fitted the emission line fluxes at each velocity , from 100 km s@xmath4 to 700 km s@xmath4 , and selected the velocity for which the fitting residual is the smallest . for line decomposition , we assumed that the [ ] @xmath16548 flux is one - third of [ ] @xmath16583 flux , and that the velocity centers of h@xmath0 and [ ] @xmath26548,6583 are same . the velocity dispersion is fixed to 0 km s@xmath4 because it is much smaller than the spectral resolution of the instrument . this assumption is reasonable , since the observed velocity dispersion of h@xmath0 at the cap is small ( @xmath10100 km s@xmath4 ; @xcite ) . the same fitting was performed for the [ ] band data . however , the [ ] @xmath16716/[]@xmath16731 can not be determined well , because the wavelength difference between these two lines is smaller than the spectral resolution in this observation . therefore we only use the total flux of [ ] @xmath26716,6731 in this study . our 1@xmath13 detection limit in h@xmath0 surface brightness is estimated to be 6.5 @xmath11 10@xmath14 erg @xmath15 s@xmath4 arcsec@xmath16 or an emission measure of roughly 1 rayleigh ( 3.3 @xmath17 pc ) at a temperature of 10@xmath18 k. we adopt a distance of 3.63 mpc to m82 @xcite . figure [ fig : image-3dii ] ( a ) displays the continuum surface brightness map of the central part of the m82 cap . only stars in the galaxy and distant galaxies are detected . the number counts are consistent with the freely available galaxycount program which provides source statistics for any window function down to 28 ab mag @xcite . the relative positions of objects in our image coincide with those of @xcite and sdss dr7 @xcite . in contrast , we can not detect continuum emission from the cap . the upper limit in surface brightness at 6656 is 23.7 mag arcsec@xmath16 ( ab , 5@xmath13 ) , which corresponds to stellar mass of approximately 3 @xmath11 10@xmath19 @xmath20 using the cap size as 0.5 kpc@xmath21 and mass - to - luminosity ratio at solar metallicity , star - formation history of ssp , and an age of 1 gyr estimated from the @xcite model . this fact indicates that the cap is not a dwarf galaxy @xcite . in the h@xmath0 intensity map ( figure [ fig : image-3dii ] ( b ) ) , clumpy and filamentary structures in the cap are clearly detected . our h@xmath0 map is roughly consistent with that of @xcite , but bright knots c and d identified by them are not confirmed in our h@xmath0 image ; instead these are detected in our continuum image . the h@xmath0 image of @xcite is clearly contaminated by continuum emission . knots c and d are not related to the cap , and appear to be more distant disk galaxies . our high resolution (= 0.9 ) h@xmath0 map enables us to resolve some bright h@xmath0 knots . the typical size of h@xmath0 knots is 510 , which corresponds to 90180 pc at the distance of m82 . we renamed the h@xmath0 knots after @xcite , as shown in the h@xmath0 image ( figure [ fig : image-3dii ] ( b ) ) . the h@xmath0 flux and luminosity of the brightest knot h - c in 5 @xmath11 5 aperture are 1.7 @xmath11 10@xmath22 erg @xmath15 s@xmath4 and 2.8 @xmath11 10@xmath23 erg s@xmath4 , respectively . the electron density and mass of the brightest knot h - c are estimated as 1.0 @xmath24 @xmath25 and 9 @xmath11 10@xmath26 , respectively , where @xmath27 indicates the filling factor of the knot . they are estimated from its h@xmath0 luminosity , size ( 5 = 90 pc ) , and the h@xmath0 recombination rate @xmath28 = 8.7 @xmath11 10@xmath29 @xmath30 s@xmath4 @xcite , with the assumption of spherical symmetry and a completely ionized gas . the observed h@xmath0 flux at each knot is displayed in table [ tb : data - knot ] . the total h@xmath0 flux of the cap region in our field of view is 7.3 @xmath11 10@xmath29 erg @xmath15 s@xmath4 . this flux is about half of that estimated by @xcite and @xcite ( @xmath101.3 @xmath11 10@xmath31 erg @xmath15 s@xmath4 ) consistent with the fact that about half of the cap region falls within our field of view . cccc e - c & 4.3 @xmath32 0.1 & 0.34 @xmath32 0.03 & 0.63 @xmath32 0.06 + e - n & 4.0 @xmath32 0.1 & 0.22 @xmath32 0.03 & 0.60 @xmath32 0.06 + e - e1 & 3.3 @xmath32 0.1 & 0.32 @xmath32 0.04 & 0.63 @xmath32 0.08 + e - e2 & 2.8 @xmath32 0.1 & 0.27 @xmath32 0.04 & 0.45 @xmath32 0.08 + f - c & 3.2 @xmath32 0.1 & 0.18 @xmath32 0.03 & 0.55 @xmath32 0.07 + f - n & 2.6 @xmath32 0.1 & 0.20 @xmath32 0.04 & 0.60 @xmath32 0.09 + f - s & 2.5 @xmath32 0.1 & 0.20 @xmath32 0.05 & 0.55 @xmath32 0.08 + g - c & 2.3 @xmath32 0.1 & 0.21 @xmath32 0.05 & 0.56 @xmath32 0.10 + g - se1 & 3.1 @xmath32 0.1 & @xmath33 0.11 & 0.63 @xmath32 0.08 + g - se2 & 3.1 @xmath32 0.1 & 0.12 @xmath32 0.04 & 0.62 @xmath32 0.08 + g - sw & 2.9 @xmath32 0.1 & 0.18 @xmath32 0.04 & 0.52 @xmath32 0.08 + h - c & 5.7 @xmath32 0.1 & @xmath33 0.06 & 0.55 @xmath32 0.04 + h - w & 3.3 @xmath32 0.1 & @xmath33 0.10 & 0.66 @xmath32 0.07 + j - c & 2.8 @xmath32 0.1 & 0.24 @xmath32 0.04 & 0.55 @xmath32 0.09 + j - e & 2.9 @xmath32 0.1 & @xmath33 0.12 & 0.46 @xmath32 0.07 + j - s & 2.6 @xmath32 0.1 & 0.17 @xmath32 0.04 & 0.50 @xmath32 0.08 figures [ fig : image-3dii ] ( c ) , ( d ) , ( e ) , and ( f ) displays [ ] @xmath16583 intensity , [ ] @xmath26716,6731 intensity , [ ] /h@xmath0 ratio , and [ ] /h@xmath0 ratio maps of a part of the m82 cap , respectively . the [ ] flux map is generally similar to the h@xmath0 map . all counterparts of h@xmath0 knots are also found in the [ ] map . [ ] /h@xmath0 flux ratios are almost constant among these knots , 0.450.66 . the [ ] flux map is quite different from the h@xmath0 flux map . knots e - c and e - e are clearly detected in [ ] , but the counterparts of the other knots , even the h@xmath0 brightest knot h - c , are barely detected . the observed [ ] /h@xmath0 flux ratios of knots e - c and e - e1 are the largest and peak at @xmath100.33 . [ ] /h@xmath0 and [ ] /h@xmath0 line ratios at these knots are summarized in table [ tb : data - knot ] and plotted in figure [ fig : n2ha - s2 ha ] . small [ ] /h@xmath0 and [ ] /h@xmath0 ratios are consistent with the previous result of non - detection of forbidden lines at the cap @xcite . we do not find correlations between the h@xmath0 flux , [ ] /h@xmath0 and [ ] /h@xmath0 ratios . we compare the [ ] /h@xmath0 and [ ] /h@xmath0 ratios of the cap with those at other regions in m82 . the emission line ratios at a radius of 1 kpc from m82 s center @xcite , and in the circumnuclear regions @xcite , are also plotted in figure [ fig : n2ha - s2 ha ] . we find an interesting trend of the line ratios with galactic radius . the [ ] /h@xmath0 ratio tends to decrease , while the [ ] /h@xmath0 ratio tends to increase with distance from the m82 nucleus . this fact suggests that some parameters , such as metallicity and shock velocity , gradually change , if the ionization source is the same in these regions . in figure [ fig : n2ha - s2 ha ] , the line ratios of the m82 starburst regions and the cap are significantly different . therefore , dust reflection of the m82 starburst regions , suggested by strong uv emission @xcite , is not the dominant emission mechanism in optical wavelength . figure [ fig : n2ha - s2ha - coma ] compares the observed [ ] /h@xmath0 and [ ] /h@xmath0 ratios of the m82 cap with those of regions @xcite , blue compact galaxies @xcite , liners @xcite , and very extended ionized gas ( eig ) in the coma cluster @xcite . we find that the line ratios of the m82 cap are similar to those of some eig knots which have larger [ ] /h@xmath0 ratios than the main sequence of star - forming galaxies . this fact suggests that the emission line ratios of the m82 cap are not peculiar , and that the m82 cap and eigs in the coma cluster are ionized by the same mechanism . first we discuss the dominant ionization source of the m82 cap . the observed [ ] /h@xmath0 and [ ] /h@xmath0 line ratios fall within the range of star - forming galaxies ( e.g. , @xcite ) and diffuse ionized gas ( e.g. , @xcite ) . the ionization source of star - forming galaxies is uv photons from massive stars ( e.g. , @xcite ) , whereas there are some possible sources for diffuse ionized gas and eig , such as photoionization , shock ( e.g. , @xcite ) , and turbulent mixing layers ( e.g. , @xcite ) . in the case of turbulent mixing layers , the h@xmath0 emitting ionized gas should exist at the boundary of the hot gas , but they appear to be spatially coincident ; thus , we rule out this model . in order to reveal the ionization source of the cap , we compare the observed line ratios with theoretical values for photoionization and shock models . photoionization of the cap by the m82 nuclear starburst region was suggested by @xcite . they calculated the number of ionizing photons at the cap region , and found that there are enough ionizing photons relative to that estimated from h@xmath0 luminosity . this model assumes that ionizing photons are not strongly absorbed or scattered by interstellar matter between the m82 starburst regions and the cap region , presumably because the wind phenomenon has cleared the sight line of obscuring matter . the observed [ ] /h@xmath0 and [ ] /h@xmath0 ratios ( table [ tb : data - knot ] ) are similar to those of galaxies ( e.g. @xcite ) , and the observed ratios do not exceed the maximum starburst line defined by @xcite , unless the [ ] @xmath15007/h@xmath34 ratio is larger than @xmath10 3 . but the observed [ ] /h@xmath0 ratio is roughly equal to the largest value of @xmath10 40,000 sdss star - forming galaxies @xcite making uv photoionization less probable . in order to clarify whether photoionization can produce the observed larger [ ] /h@xmath0 ratios , we calculated emission line ratios with a _ cloudy _ photoionization model @xcite . we used two electron densities for the calculation : 1 @xmath25 for the cap , and 2000 @xmath25 for the m82 center @xcite . due to its secondary nature , we assume that the nitrogen abundance scales with metallicity ( @xmath35 ) as @xmath36 and that the stellar metallicity ( @xmath37 ) is the same as @xmath35 ( see @xcite ) . the sed of the stars is taken from @xcite . figure [ fig : n2ha - s2 ha ] ( a ) indicates that most of the observed line ratios at the m82 cap are reproduced by the photoionization model of ionization parameter @xmath38 = 10@xmath3910@xmath40 and metallicity @xmath41 0.4 @xmath42 . @xmath38 represents the dimensionless ratio of the ionizing photon density to the electron density . to constrain the ionization parameter @xmath38 of the m82 cap , we follow the @xcite condition : the dimension of 3.7 @xmath11 3.7 kpc@xmath21 , electron density of 4.3 @xmath11 10@xmath43 @xmath25 , where @xmath44 indicates the filling factor of ionized gas averaged over the whole m82 cap , and the number of ionizing photons at the cap of 2 @xmath11 10@xmath45 photon s@xmath4 . the calculated ionization parameter @xmath46 is consistent with that estimated from emission line ratios , if we assume @xmath47 . the larger distance from the ionization source than for typical regions leads to smaller ionization parameter and larger [ ] /h@xmath0 ratios . the number of the ionizing photons which each knot receives is almost the same , and the ionization parameter depends on the knot electron density only . the calculated line ratios whose electron densities are smaller than 10 @xmath25 are almost the same . for these reasons , we can consider that the difference in line ratios among the knots in the cap is explained by the difference in the electron density and/or metallicity , as seen in figure [ fig : n2ha - s2 ha ] ( a ) . although the lower density model can not reproduce the ratios at the central regions , log @xmath38 = @xmath48 and @xmath49 = 2.0 of the higher density model fits the observed ratios ( figure [ fig : n2ha - s2 ha ] ( b ) ) . however , taking account of the escape fraction of ionizing photons , we find that this picture can not be correct . @xcite uses h@xmath0 surface brightness averaged over the projected length of the m82 cap for the calculation of the escape fraction of ionizing photons . since the required escape fraction is only @xmath103 % , they considered that the m82 starburst regions can provide enough ionizing photons for the m82 cap . meanwhile , our high resolution image reveals that m82 cap is patchy and its filling factor @xmath44 is much smaller than 1 ( figure [ fig : image-3dii ] ( b ) ) . this means that the h@xmath0 surface brightness of the brightest knot h - c must be explained by the ionizing photon flux from the m82 starburst region . knot h - c is smaller and denser than the whole m82 cap , and therefore much larger escape fraction is required . we estimate the required escape fraction of ionizing photons from the m82 starburst regions . we assume that the knots are spherical symmetric ( figure [ fig : image-3dii ] ) . then we can use the same calculation method as @xcite . the h@xmath0 surface brightness of knot h - c is 1 @xmath11 10@xmath50 erg @xmath15 s@xmath4 arcsec@xmath16 in 2.25 @xmath11 2.25 aperture ( table [ tb : data - knot ] ) . since the surface brightness of 6 @xmath11 10@xmath14 erg @xmath15 s@xmath4 arcsec@xmath16 corresponds to 1 rayleigh at the h@xmath0 wavelength , the h@xmath0 emission measure of the knot h - c is 16.5 rayleigh . the required number of ionizing photons is 3.7 @xmath11 10@xmath19 @xmath15 s@xmath4 at the distance of m82 , using the h@xmath0 recombination rate @xmath28 = 8.7 @xmath11 10@xmath29 @xmath30 s@xmath4 @xcite . whereas the ionizing photon flux from the nuclear starburst region is 10@xmath51 photons s@xmath4 @xcite , and 6 @xmath11 10@xmath19 @xmath15 s@xmath4 at the distance of knot h - c from the m82 nuclear starburst region . thus the required escape fraction for knot h - c is 60 % . this escape fraction is an order of magnitude larger than what has been measured in the galaxy ( @xmath106 % : @xcite ) and dwarfs to date ( @xmath103 % : @xcite ) . in passing , we note that 3d simulations of uv radiative transfer in superwinds indicate that higher values are possible in extreme cases @xcite . thus , it is quite unlikely that the m82 cap clouds are ionized by photons from the m82 starburst regions . we consider shock ionization to be a more likely explanation , as we discuss in the next section . in order to explain the h@xmath0 emission in the cap , @xcite first suggested that the m82 superwind can drive a shock into the underlying gas cloud . the gas metallicity , which is presently unknown , is likely to fall in the range 0.11 @xmath52 depending on whether the gas is infalling or entrained with the wind flow . the threshold metallicity for infalling gas at the present epoch appears to be close to 0.1@xmath52 in all observations of the local universe to date @xcite . we regard this value as a lower limit because the outer envelope in the m81 group appears to be material stripped from the outer disk of one or more galaxies @xcite . the cap is unlikely to be entrained gas from the disk because dense material is broken up very quickly by rayleigh - taylor instabilities @xcite , as observed in the wind filaments close to the disk . we compare the observed line ratios with those calculated from a fast shock model ( @xmath53 200 km s@xmath4 ) of @xcite . the [ ] /h@xmath0 ratios in the shock model are larger than 0.3 , while the observed values are mostly less than 0.3 ( figure [ fig : n2ha - s2 ha ] ( b ) ) . but this may reflect the lower expected metallicity in the cap . fast shocks in a low metallicity gas may be able to explain the observed [ ] /h@xmath0 ratios . the computed [ ] /h@xmath0 ratios fall in the range 0.1 to 0.3 , which is similar to the observed values . but if fast shock excitation is dominant , the observed [ ] /h@xmath0 ratios should correlate with [ ] /h@xmath0 , because both ratios increase in lock step with an increase in shock velocity ( figure [ fig : n2ha - s2 ha ] ( c ) ) ; however , no such correlation between these ratios is found . a more compelling argument against fast shocks is the kinematically ` cold ' line emission observed across the ` cap ' region , an issue we return to below . thus fast shocks are unlikely to be the dominant ionization source of the cap today . next we compare the observed ratios with those calculated from slow to intermediate shock models , i.e. , 40 km s@xmath54 130 km s@xmath4 , given by @xcite ( figure [ fig : n2ha - s2 ha ] ( a ) ) . in this shock velocity range , [ ] /h@xmath0 ratios increase as shock velocity increases , while [ ] /h@xmath0 ratios neither increase nor decrease monotonically . in figure [ fig : n2ha - s2 ha ] ( a ) , the observed points at the cap knots are distributed along the line of slow shock model . the calculated [ ] /h@xmath0 and [ ] /h@xmath0 ratios are 0.10 and 0.58 at @xmath55 = 60 km s@xmath4 , and 0.27 and 0.44 at @xmath55 = 80 km s@xmath4 , respectively . therefore , shock velocities of 60 km s@xmath4 and 80 km s@xmath4 can also explain the observed [ ] /h@xmath0 and [ ] /h@xmath0 ratios . model comparisons for shock velocities higher than 80 km s@xmath4 are ruled out . the superwind is powered by kinetic energy from some combination of stellar winds , radiation pressure and supernovae in the m82 starburst region . here we examine whether the momentum is enough to produce the observed shock . we compare the inferred range of shock velocities ( 4080 km s@xmath4 ) with the superwind velocity of a spherically - symmetric model for m82 by @xcite . their model fits well with the observed thermal pressure profile of m82 within @xmath101 kpc , and the pressure profiles at the larger radii than 0.5 kpc are consistent with the @xmath56 dependence expected for ram pressure @xcite . using their model , the gas density , wind velocity , and ram pressure at the distance of m82 cap are estimated as 4 @xmath11 10@xmath57 @xmath25 , 5600 km s@xmath4 , and 2.1 @xmath11 10@xmath58 dyne @xmath15 , respectively . since the observed electron density is 1.0 @xmath25 at the knot h - c , the thermal pressure of ionized gas is calculated as @xmath59 dyne @xmath15 with the assumption of @xmath60 k. this indicates that the model ram pressure and the observed thermal pressure are well balanced , and the observed pressure follows @xmath56 law even at the distance of the m82 cap , 11.6 kpc ( figure [ fig : pressure - profile ] ) . furthermore , given that some of the shock motion is expected to generate local turbulent motions of the same order @xcite , the measured narrow emission line widths ( @xmath3 km s@xmath4 fwhm , @xcite ) are consistent with a slow shock . therefore , it is quite likely that the cap clouds are ionized by a slow shock produced by the m82 superwind . future deep observations of the m82 cap , e.g. , [ ] @xmath16300 , will enable us to directly confirm its dominant ionization source . in the case of a slow shock , the [ ] /h@xmath0 ratio larger than 0.1 is to be expected @xcite , while a smaller ratio is evidence for photoionization by massive stars ( e.g. , @xcite ) . diffuse x - ray emission was detected at the m82 cap @xcite . the electron density , temperature , and thermal pressure of hot gas are @xmath61 @xmath25 , 9 @xmath62 k , and 1.3 @xmath63 dyne @xmath15 , respectively , where @xmath64 is the filling factor of hot gas . however , a slow shock is not the main heating mechanism , because these can only heat to @xmath65 k. additionally , thermal pressures of ionized and hot gas are inconsistent at the m82 cap , though they are consistent at the m82 center @xcite . hence , another heating source for hot gas is required . a reasonable explanation for the hot gas is found when considering a 2-component shock model . an cloud in isolation develops a core - halo structure where the core is dense and the halo is relatively diffuse @xcite . in this interpretation , the x - ray emission is produced by the superwind triggering a shock in the diffuse gas surrounding the knots in the cap ( see @xcite ) . to produce gas with @xmath66 k requires a fast shock whose shock velocity @xmath67 is 820 km s@xmath4 . but the 2-component shock by the same superwind can not explain the discrepancy between the thermal pressures of these gas phases . a 2-component shock by different superwinds from the m82 starburst region may be the case for the m82 cap . in this model , the slow shock by the present superwind ionizes the cap clouds , while hot gas observed today was produced by a fast shock driven by a past superwind outburst . what is important is that the cooling timescales of these gas phases are drastically different : @xmath68 yr and @xmath69 yr for ionized and hot gas , respectively @xcite . the history of this model is as follows : @xmath70 yrs ago , a fast shock driven by a past superwind produced hot ionized gas ; after @xmath68 yrs , the ionized gas cooled while the hot gas continued to emit x - rays , and now the slow shock ionizes the cap clouds again to produce the h@xmath0 and x - ray emission from the cap . it is undoubtedly true that the physics of superwinds is complicated . however , we are able to deduce some basic properties of the wind . as discussed by @xcite , there is a well determined series of events that lead to a superwind taking hold . we believe that the new observations support the developing paradigm . when a critical surface density of dense molecular clouds is reached in the galaxy , massive stars are born . these stars evolve rapidly and their strong uv radiation fields produce a warm gaseous medium that encircles the remaining molecular clouds . after a few million years , the cores of the most massive stars collapse leading to multiple supernova explosions . this huge impulse of mechanical energy heats the diffuse medium to extremely high temperatures ( @xmath71k ) where it expands to form a powerful superwind . ( if the mechanical energy or uv radiative energy were entirely absorbed by the dense gas , most of the energy would simply be re - radiated as ir emission . ) the hot flowing gas entrains cooler gas from the disk with the flow ; we know this because of well defined rotation of the entrained filaments about the wind axis @xcite . the entrained gas is very clumpy because it is mostly entrained from the surviving dense clouds in the disk @xcite . m82 is engulfed by a large cloud complex and some of this material appears to be accreting onto the dwarf close to the minor axis @xcite . the cap material is almost certainly supplied by infalling gas . the cap ionization almost certainly arises from the expanding superwind interacting with infalling gas . the spatial coincidence of the h@xmath0 and x - ray emission may be explained with the model of a 2-component shock by different m82 superwinds . further constraints on the cap may come from uv absorption line spectroscopy using distant quasars as a background light source . the expected warm - hot medium should be visible in low to intermediate ionization states of c , n , o , ne and fe , even for gas in a non - equilibrium state @xcite . our new highly resolved observations show that the cap is very clumpy presumably as a consequence of the wind - cloud interaction . our expectation is that the cap will be disrupted in a shock crossing time of about @xmath72 myr . the fact that the outer cap is presently ionized by local shocks , and not by nuclear uv radiation , supports the findings of @xcite . unlike agn - driven wind filaments which are all found to be ionized by nuclear uv radiation , the starburst wind filaments are ionized by local shocks far from the nucleus . this is easily understood in terms of a ` starburst ' because the hot young stars must evolve to supernovae before the wind gets going , and therefore few remain to ionize gas clouds in the direction of the flow . we fully anticipate that future deep optical , uv and x - ray imaging and spectroscopy will reveal further details about the cap region , and in turn about the nature of the superwind in m82 . this work is based on data collected at subaru telescope , which is operated by the national astronomical observatory of japan . we thank m. yoshida for useful discussions . this work was financially supported in part by the japan society for the promotion of science ( nos . 17253001 , 19340046 , 23244031 , and 23654068 ) . abazajian , k. n. , adelman - mccarthy , j. k. , ageros , m. a. , et al . 2009 , , 182 , 543 allen , m. g. , groves , b. a. , dopita , m. a. , sutherland , r. s. , & kewley , l. j. 2008 , , 178 , 20 barger , a , haffner , m & bland - hawthorn , j. 2012 , , in press bland , j. , & tully , r. b. 1988 , , 334 , 43 bland , j. , & tully , r. b. 1989 , , 98 , 723 bland - hawthorn , j. 1995 , iau colloq . 149 : tridimensional optical spectroscopic methods in astrophysics , 71 , 72 bland - hawthorn , j. , & maloney , p. r. 1999 , , 510 , l33 bland - hawthorn , j. , & maloney , p. r. 2002 , extragalactic gas at low redshift , 254 , 267 bland - hawthorn , j. , sutherland , r. , agertz , o. , & moore , b. 2007 , , 670 , l109 bruzual , g. , & charlot , s. 2003 , , 344 , 1000 cappi , m. , persic , m. , bassani , l. , et al . 1999 , , 350 , 777 chevalier , r. a. , & clegg , a. w. 1985 , , 317 , 44 chynoweth , k. m. , & langston , g. 2007 , bulletin of the american astronomical society , 39 , 905 cooper , j. l. , bicknell , g. v. , sutherland , r. s. , & bland - hawthorn , j. 2008 , , 674 , 157 cooper , j. l. , bicknell , g. v. , sutherland , r. s. , & bland - hawthorn , j. 2009 , , 703 , 330 devine , d. , & bally , j. 1999 , , 510 , 197 ellis , s. c. , & bland - hawthorn , j. 2007 , , 377 , 815 ferland , g. j. , korista , k. t. , verner , d. a. , et al . 1998 , , 110 , 761 field , g. b. 1965 , , 142 , 531 freedman , w. l. , hughes , s. m. , madore , b. f. , et al . 1994 , , 427 , 628 gnat , o. , & sternberg , a. 2004 , , 608 , 229 greve , a. 2004 , , 416 , 67 ho , l. c. , filippenko , a. v. , sargent , w. l. w. , & peng , c. y. 1997 , , 112 , 391 heckman , t. m. , armus , l. , & miley , g. k. 1990 , , 74 , 833 hoopes , c. g. , heckman , t. m. , strickland , d. k. , et al . 2005 , , 619 , l99 jansen , r. a. , fabricant , d. , franx , m. , & caldwell , n. 2000 , , 126 , 331 kewley , l. j. , dopita , m. a. , sutherland , r. s. , heisler , c. a. , & trevena , j. 2001 , , 556 , 121 kong , x. , cheng , f. z. , weiss , a. , & charlot , s. 2002 , , 396 , 503 lehnert , m. d. , heckman , t. m. , & weaver , k. a. 1999 , , 523 , 575 leitherer , c. , schaerer , d. , goldader , j. d. , et al . 1999 , , 123 , 3 liang , y. c. , yin , s. y. , hammer , f. , et al . 2006 , , 652 , 257 lynds , c. r. , & sandage , a. r. 1963 , , 137 , 1005 mccarthy , p. j. , van breugel , w. , & heckman , t. 1987 , , 93 , 264 mckeith , c. d. , greve , a. , downes , d. , & prada , f. 1995 , , 293 , 703 mcleod , k. k. , rieke , g. h. , rieke , m. j. , & kelly , d. m. 1993 , , 412 , 111 matsubayashi , k. , sugai , h. , hattori , t. , et al . 2009 , , 701 , 1636 moustakas , j. , & kennicutt , r. c. , jr . 2006 , , 651 , 155 nagao , t. , maiolino , r. , marconi , a. , & matsuhara , h. 2011 , , 526 , a149 nakai , n. , hayashi , m. , handa , t. , et al . 1987 , , 39 , 685 oconnell , r. w. , & mangano , j. j. 1978 , , 221 , 62 osterbrock , d. e. , & ferland , g. j. 2006 , astrophysics of gaseous nebulae and active galactic nuclei , 2nd . ed . by d.e . osterbrock and g.j . ferland . sausalito , ca : university science books , 2006 oke , j. b. 1990 , , 99 , 1621 ryan - weber , e. v. , pettini , m. , madau , p. , & zych , b. j. 2009 , , 395 , 1476 sharp , r. g. , & bland - hawthorn , j. 2010 , , 711 , 818 shopbell , p. l. , & bland - hawthorn , j. 1998 , , 493 , 129 shull , j. m. , & mckee , c. f. 1979 , , 227 , 131 slavin , j. d. , shull , j. m. , & begelman , m. c. 1993 , , 407 , 83 smith , l. j. , westmoquette , m. s. , gallagher , j. s. , et al . 2006 , , 370 , 513 sternberg , a. , mckee , c. f. , & wolfire , m. g. 2002 , , 143 , 419 strickland , d. k. , & heckman , t. m. 2007 , , 658 , 258 sugai , h. , hattori , t. , kawai , a. , et al . 2010 , , 122 , 103 tsuru , t. g. , ozawa , m. , hyodo , y. , et al . 2007 , , 59 , 269 walter , f. , weiss , a. , & scoville , n. 2002 , , 580 , l21 westmoquette , m. s. , smith , l. j. , gallagher , j. s. , iii , et al . 2009 , , 696 , 192 westmoquette , m. s. , gallagher , j. s. , smith , l. j. , et al . 2009 , , 706 , 1571 yajima , h. , umemura , m. , mori , m. , & nakamoto , t. 2009 , , 398 , 715 yoshida , m. , kawabata , koji s. , & ohyama , y. 2011 , , 63 , 493 yoshida , m. , yagi , m. , komiyama , y. , et al . 2012 , , 749 , 43 zastrow , j. , oey , m. s. , veilleux , s. , mcdonald , m. , & martin , c. l. 2011 , , 741 , l17

the m82 ` cap ' is a gas cloud at a projected radius of 11.6 kpc along the minor axis of this well known superwind source . the cap has been detected in optical line emission and x - ray emission and therefore provides an important probe of the wind energetics . in order to investigate the ionization source of the cap , we observed it with the kyoto3dii fabry - perot instrument mounted on the subaru telescope . deep continuum , h@xmath0 , [ ] @xmath16583/h@xmath0 , and [ ] @xmath26716,6731/h@xmath0 maps were obtained with sub - arcsecond resolution . the superior spatial resolution compared to earlier studies reveals a number of bright h@xmath0 emitting clouds within the cap . the emission line widths ( @xmath3 km s@xmath4 fwhm ) and line ratios in the newly identified knots are most reasonably explained by slow to moderate shocks velocities ( @xmath5 = 4080 km s@xmath4 ) driven by a fast wind into dense clouds . the momentum input from the m82 nuclear starburst region is enough to produce the observed shock . consequently , earlier claims of photoionization by the central starburst are ruled out because they can not explain the observed fluxes of the densest knots unless the uv escape fraction is very high ( @xmath6 60% ) , i.e. , an order of magnitude higher than observed in dwarf galaxies to date . using these results , we discuss the evolutionary history of the m82 superwind . future uv / x - ray surveys are expected to confirm that the temperature of the gas is consistent with our moderate shock model .

as the most commonly observed objects , stars remain at the forefront of astrophysical research . advances in optical detector technology , computer processing power , and data storage capability have enabled new sky surveys ( e.g. , the sloan digital sky survey ; * ? ? ? * ) ; triggered many new optical transient surveys , such as the palomar transient factory @xcite and pan - starrs1 @xcite ; and allowed for space missions ( e.g. , _ kepler _ ; * ? ? ? * ) that continuously monitor more than 100,000 stars . the stellar discoveries from these surveys include revelations about rare stars , unusual explosive outcomes , and remarkably complex binaries . the immediate future holds tremendous promise , as both the space - based survey _ gaia _ @xcite and the ground based large synoptic survey telescope ( lsst ; * ? ? ? * ) come to fruition . these developments have created a new demand for a reliable and publicly available research and education tool in computational stellar astrophysics . we introduced the open source community tool ` mesa ` ( * ? ? ? * hereafter paper i ) to meet these new demands . this first `` instrument '' paper described the design , implementation , and realm of validity of ` mesa ` modules for numerics , microphysics , and macrophysics , and introduced the stellar evolution module , ` mesa ` ` star ` . we presented a multitude of tests and code comparisons that served as our initial verification and demonstrated ` mesa ` ` star ` s initial capabilities . since paper i , ` mesa ` has attracted over 500 registered users , witnessed over 5,000 downloads from http://mesa.sourceforge.net/ , started an annual summer school program , and provided a portal ( http://mesastar.org ) for the community to openly share knowledge ( e.g. , the specific settings for a published ` mesa ` ` star`run ) , codes , and publications . this paper describes the major new ` mesa ` capabilities for modeling giant planets , asteroseismology , and the treatment of rotation and evolution of massive stars . we also describe numerous advances since paper i. these include the incorporation of composition gradients in the determination of convective mixing and additional verification for evolution of intermediate mass stars and the white dwarfs they create . our improvements to ` mesa ` ` star ` for gas giant planets were motivated by the dramatic growth in this field . over 800 exoplanets have been confirmed , and their study has prompted enormous progress in our understanding of the formation and migration of giant planets , and of the importance of factors such as stellar mass @xcite , composition @xcite , and binarity @xcite . puzzles remain , though , both in our solar system and in the studies of the plethora of these newly discovered exoplanets , including the characteristics of the planet - hosting stars and the interiors , atmospheres , surface gravities , temperatures , and compositions of the planets ( e.g. , * ? ? ? * ; * ? ? ? many of these variations can now be numerically explored , as can the incorporation of an inert core in an otherwise regular gas giant and the impact of irradiation . the ability to infer stellar properties ( e.g. , mass , radius , internal state , and rotation ) from measurements of the radial and non - radial oscillation modes has been dramatically improved by two space - based optical telescopes ( convection rotation and planetary transits , _ corot _ ; @xcite and _ kepler _ ; @xcite ) . the high cadences and precision ( often better than ten parts per million ) reveal and accurately measure multitudes of oscillation frequencies for over 10,000 stars , substantially raising the need for accurate and efficient computations of stellar mode frequencies and the resulting eigenfunctions . the intrinsic flexibility of ` mesa ` ` star ` allows for the exploration of model - space required to precisely infer stellar properties from the observed frequencies . an important new addition to ` mesa ` is the incorporation of stellar rotation and magnetic fields in radiative regions . as stars are not solid bodies , they undergo radial differential rotation @xcite and also rotate at different angular velocities at different latitudes @xcite . these rotational shears have a significant impact on the evolution of the stellar magnetic field . despite the resulting 3d nature of magnetism and rotation , the stellar evolution community has come a long way in understanding stars with 1d simulations @xcite , thus motivating our need to fully incorporate rotation within ` mesa ` . the new flexibility in angular momentum transport mechanisms allows for numerical exploration of alternate rotational outcomes should the observations ( e.g. , asteroseismology ) require it . the paper is outlined as follows . section [ s.planets ] describes the new capability of ` mesa ` to evolve models of giant planets , while [ s.astroseismology ] discusses the new asteroseismology capabilities . the ` mesa ` implementation of composition gradients in stellar interiors and their impact on convective mixing is described in [ s.mixing ] . the status of the evolution of intermediate mass stars and the ` mesa ` ` star ` construction and evolution of white dwarfs is described in [ s.agb-wd ] . the new capabilities for evolving rotating stars is described in [ s.rotation ] . the onset of near eddington luminosities and radiation pressure dominance in the envelopes of evolving massive stars has been a challenge for many stellar evolution codes ever since the realization of the iron opacity bump at @xmath1 @xcite . we discuss in [ s.massive ] the resulting improvements for evolving massive stars . this allows for the uninterrupted evolution of rotating massive stars to the onset of core collapse . we conclude in [ s.conclusions ] by highlighting where additional improvements to ` mesa ` are likely to occur in the near future . appendix [ s.input-physics ] describes the many improvements to the physics modules since paper i ; appendix [ s.nuts-and-bolts ] presents `` nuts and bolts '' information on the primary components of evolution calculations ; and appendix [ s.mesasdk ] presents the ` mesa ` software development kit ( ` sdk ` ) . all of our symbols are defined in table [ t.list-of-symbols ] . we denote components of ` mesa ` , such as modules and routines , in courier font , e.g. , ` evolve_star ` . cll @xmath2 & atomic mass number & [ s.chem ] + @xmath3 & mass excess of the @xmath4th isotope & [ s.chem ] + @xmath5 & wind mass loss coefficient & [ s.completing-evolution ] + @xmath6 & day - side flux incident on an irradiated planet & [ sec : irradiation ] + @xmath7 & coulomb coupling parameter & [ s.completing-evolution ] + @xmath4 & specific moment of inertia & [ s.nuts-rotation ] + @xmath8 & opacity & [ s.construction-models ] + @xmath9 & stellar luminosity & [ s.astero ] + @xmath10 & lagrangian mass coordinate & [ s.construction-models ] + @xmath11 & stellar mass & [ s.construction-models ] + @xmath12 & brunt - visl frequency & [ s.ledoux ] + @xmath13 & number density of the @xmath4th isotope & [ s.chem ] + @xmath14 & turbulent viscosity & [ s.nuts-rotation ] + @xmath15 & radial coordinate & [ sec : irradiation ] + @xmath16 & total stellar radius & [ s.construction-models ] + @xmath17 & baryon mass density & [ s.chem ] + @xmath18 & specific entropy & [ s.construction-models ] + @xmath19 & mass column & [ sec : irradiation ] + @xmath20 & depth for heating from irradiation & [ sec : irradiation ] + @xmath21 & optical depth & [ s.completing-evolution ] + @xmath22 & magnitude of changes during a timestep & [ s.timestep-controls ] + @xmath23 & target value for @xmath22 & [ s.timestep-controls ] + @xmath24 & atomic weight & [ s.chem ] + @xmath25 & h mass fraction & [ s.astero ] + @xmath26 & baryon mass fraction of the @xmath4th isotope & [ s.ledoux ] + @xmath27 & he mass fraction & [ s.planets ] + @xmath28 & electrons per baryon ( @xmath29/@xmath30 ) & [ s.chem ] + @xmath31 & abundance of the @xmath4th isotope & [ s.chem ] + @xmath32 & metallicity & [ s.planets ] + @xmath32 & atomic number & [ s.chem ] + @xmath33 & mixing length parameter & [ s.planets-no-core ] + @xmath34 & semiconvection efficiency parameter & [ s.semiconvection ] + @xmath35 & thermohaline efficiency parameter & [ s.thermohaline ] + @xmath36 & smoothing parameter for mlt++ & [ s.superadiabatic ] + @xmath37 & mlt++ parameter used in construction of @xmath36 & [ s.superadiabatic ] + @xmath38 & @xmath39 & [ s.superadiabatic ] + @xmath40 & @xmath41 & [ s.ledoux ] + @xmath42 & @xmath43 & [ s.ledoux ] + @xmath44 & specific heat at constant pressure & [ s.semiconvection ] + @xmath45 & adiabatic sound speed & [ s.astroseismology ] + @xmath46 & large frequency separation of pulsation modes & [ s.astero ] + @xmath47 & overshoot diffusion coefficient & [ s.ledoux ] + @xmath48 & thermohaline diffusion coefficient & [ s.thermohaline ] + @xmath49 & fermi energy at center & [ s.planets-no-core ] + @xmath50 & gravitational heating rate & [ s.completing-evolution ] + @xmath51 & nuclear heating rate & [ s.reactions ] + @xmath52 & neutrino loss rate & [ s.rotation-modifications ] + @xmath53 & convective flux & [ s.massive-evol ] + @xmath54 & convective overshoot parameter & [ s.astero ] + @xmath55 & radiative flux & [ s.massive-evol ] + @xmath56 & reduction factor for @xmath57 & [ s.superadiabatic ] + @xmath58 & @xmath59 & [ s.astroseismology ] + @xmath60 & opacity for thermal radiation orig . in planet & [ sec : irradiation ] + @xmath61 & opacity for irradiation from star & [ sec : irradiation ] + @xmath62 & boltzmann constant & [ s.planets-no-core ] + @xmath63 & accretion luminosity & [ s.compressional ] + @xmath64 & @xmath65 & [ s.superadiabatic ] + @xmath66 & core luminosity & [ s.planets-cores ] + @xmath67 & eddington luminosity & [ s.rotation-mass-loss ] + @xmath68 & log surface gravity & [ s.atmospheres ] + @xmath69 & luminosity at which the onset of convection occurs & [ s.massive-evol ] + @xmath70 & radiative luminosity & [ s.massive-evol ] + @xmath71 & luminosity at which a density inversion occurs & [ s.massive-evol ] + @xmath72 & atomic mass unit & [ s.chem ] + @xmath73 & core mass & [ s.planets-cores ] + @xmath74 & mass - loss rate & [ s.compressional ] + @xmath75 & modeled mass & [ s.mesh-controls ] + @xmath76 & avogadro number & [ s.planets-no-core ] + @xmath77 & adiabatic temperature gradient & [ s.ledoux ] + @xmath78 & ledoux criterion & [ s.semiconvection ] + @xmath79 & radiative temperature gradient & [ s.ledoux ] + @xmath80 & actual temperature gradient & [ s.ledoux ] + @xmath81 & baryon density & [ s.chem ] + @xmath82 & frequency of maximum power & [ s.astero ] + @xmath83 & surface angular velocity & [ s.rotation-mass-loss ] + @xmath84 & angular velocity & [ s.rotation-modifications ] + @xmath85 & surface critical angular velocity & [ s.rotation-mass-loss ] + @xmath86 & central pressure & [ s.construction-models ] + @xmath87 & gas pressure & [ s.compressional ] + @xmath88 & radiation pressure & [ s.compressional ] + @xmath89 & core radius & [ s.planets-cores ] + @xmath90 & central density & [ s.construction-models ] + @xmath91 & pressure scale height & [ s.ledoux ] + @xmath92 & stefan - boltzmann constant & [ s.construction-models ] + @xmath93 & lamb frequency & [ s.astroseismology ] + @xmath94 & superadiabaticity , @xmath95 & [ s.superadiabatic ] + @xmath96 & controls when mlt++ is applied & [ s.superadiabatic ] + @xmath97 & central temperature & [ s.construction-models ] + @xmath98 & effective temperature & [ s.construction-models ] + @xmath99 & numerical timestep & [ s.compressional ] + @xmath100 & thermal ( kelvin - helmholtz ) timescale & [ s.rotation-mass-loss ] + @xmath101 & equatorial velocity & [ s.rotation ] + evolutionary models of giant planets and low - mass stars differ from their higher - mass stellar counterparts in both the microphysics needed to describe the interior and the role of stellar irradiation in the outer boundary condition . for masses @xmath102 , hydrogen burning is insufficient to prevent cooling and contraction . deuterium burning can briefly slow the cooling for @xmath103 , where @xmath104 is jupiter s mass , but has a negligible influence on the cooling for smaller masses . hence nuclear burning can be ignored in the planetary mass regime . for hydrogen - helium rich objects with @xmath105 , an ideal gas equation of state ( eos ) , with arbitrary degeneracy , is a good approximation while for @xmath106 particle interactions play an important role . specifically , pressure ionization of hydrogen at @xmath107 and @xmath108 causes a sudden change from a h@xmath109-dominated phase to an ionized phase . ` mesa ` employs the @xcite equation of state ( scvh eos ) , smoothly interpolated from the low to high pressure phase , for this complicated region of parameter space where thermal , fermi , and electrostatic energies may all be comparable . the scvh eos includes pressure ionization of hydrogen , but not helium . the temperature range covered by the tables is @xmath110 , and the pressure ranges from @xmath111 to a maximum value @xmath112 dependent on the temperature . smooth interpolation to other eos occurs near the scvh boundaries ( for more details see paper i ) . at the low temperatures in planetary atmospheres , abundant species such as cno atoms will be in molecular form , and may condense into clouds . ` mesa ` does not follow the transition from atomic to molecular form for these species in the eos they are currently included by increasing the helium abundance from @xmath27 to @xmath113 when calling the scvh eos . ` mesa ` does , however , include the effect of molecules in the rosseland opacities . currently , the @xcite and @xcite tables , which include the opacity from molecules , but ignore condensates , are available . lastly , for planets in close - in orbits about their parent star , the external irradiation flux may be orders of magnitude larger than the cooling flux from the planet s interior . this may dramatically increase the surface temperature and affect the outer boundary condition . ` mesa ` now implements several options for this surface heating , including the flexibility to include user - supplied prescriptions . in the following subsections , we discuss a new ` mesa ` module that creates initial models in the planetary mass range @xmath114 , and present a suite of evolutionary calculations . we discuss how surface irradiation may be included , as well as an inert core at the center of the planet . we also show what ` mesa ` ` star ` yields for the mass - radius relation for sub - solar mass stars in [ s.mass-radius ] . for stellar mass objects , the ` pre_ms_model ` routine constructs pre - main - sequence ( pms ) models assuming @xmath115 , where @xmath116 is the luminosity at radius @xmath15 , by iterating on the starting conditions at the center to find a model with a given @xmath11 and central temperature @xmath117 . this pms routine works well for @xmath118 , but lower masses may not converge when the guess for central density @xmath119 and luminosity are not close enough to the ( unknown ) true values . as a result , it is difficult and time consuming to create models with @xmath120 using the same routine for giant planets as for stars . a new routine called ` create_initial_model ` builds a model of given @xmath11 and radius @xmath16 using an adiabatic temperature profile . given the central pressure @xmath86 and specific entropy @xmath18 , the equation of hydrostatic balance is integrated outward , and the temperature at each step determined from the equation of state using @xmath121 . the values of @xmath86 and @xmath18 are iterated to attain the desired @xmath11 and @xmath16 . the luminosity profile is then derived treating @xmath18 as constant in space for the fully convective planet ( e.g. , @xcite ) , so @xmath122 the luminosity at the surface , @xmath123 , is estimated using the radius @xmath16 and temperature @xmath124 at the @xmath125 point as @xmath126 . given @xmath123 , the luminosity at interior points is found by @xmath127 this procedure works well for @xmath11 down to @xmath128 and over a range of initial radii , allowing the user to choose either a @xmath129 radius appropriate for a cold planet , to radii @xmath130 appropriate for young or inflated planets ( e.g. , @xcite ) . here @xmath131 is the equatorial radius of jupiter . versus @xmath90 during the evolution . each line is labeled on the left by the mass in units of @xmath132 . the dotted red lines show constant values of @xmath133)$ ] , labeled at the base of each line . the blue dashed lines show fixed values of @xmath134 , labeled at the top of each line . the large black dots show the position of maximum @xmath97 along the evolutionary track . ] versus @xmath135 at fixed values of @xmath133)$ ] , labeled on the left of each curve . the dashed blue curves are for fixed entropy , with each curve labeled by @xmath134 on the right . the dotted black curves are for fixed luminosity , with each curve labeled by @xmath136)$ ] above @xmath137 . the green curve at the bottom is the @xmath138 @xmath139 relation from @xcite for a solar mixture of h and he . ] figures [ fig : tc_vs_rhoc_isoage_isoent ] and [ fig : r_vs_m_iso_tsl ] show evolutionary calculations for models with masses @xmath140 . all models were evolved for @xmath141 gyr . the initial models from ` create_initial_model ` had a large radius @xmath142 . the other parameters used are @xmath143 , @xmath144 and @xmath145 . the opacity and eos tables used are ` eos_file_prefix ` = ` mesa ` , ` kappa_file_prefix ` = ` gs98 ` and ` kappa_lowt_prefix ` = ` lowt_freedman11 ` . the atmosphere model is ` which_atm_option ` = ` simple_photosphere ` . figure [ fig : tc_vs_rhoc_isoage_isoent ] is a low mass extension of figure 16 from paper i , showing evolution in the @xmath90-@xmath97 plane . each track ( solid black curve ) is labeled on the left by the planet s mass , and evolution goes from left to right . initially the planet is non - degenerate and contraction increases both @xmath146 and @xmath147 . a maximum @xmath97 is reached when @xmath148 , where @xmath49 is the electron fermi energy at the center , beyond which @xmath90 approaches a constant as @xmath97 decreases further . ignoring coulomb interactions in the eos , @xmath18 is a function of the electron degeneracy parameter @xmath149 , where @xmath150 is the electron chemical potential and all models should have maximum @xmath151 at the same @xmath18 . the line labeled @xmath152 indeed coincides with maximum @xmath97 down to @xmath153 , but at smaller masses where non - ideal effects are more important , maximum @xmath97 occurs when @xmath154 . also shown in figure [ fig : tc_vs_rhoc_isoage_isoent ] are lines of constant age , shown as dotted red lines , and labeled on the bottom of the plot . the same evolutionary calculations are used in figure [ fig : r_vs_m_iso_tsl ] to show radius versus mass at fixed values of age , entropy or luminosity . at late times , or low entropy and luminosity , the radius approaches the zero - temperature value ( green curve ; @xcite ) for which thermal support is insignificant . the maximum radius occurs where gravitational and coulomb energies , per ion , are comparable . the solid red lines , labeled by age on the left , show that contraction down to @xmath155 is rapid , taking less than 10 myr for @xmath156 . this initial rapid cooling phase occurs because the initial luminosity is orders of magnitude higher than the luminosity around one gyr . this can been seen in the black dotted contours of constant @xmath157 , where @xmath9 is larger by a factor of 100 for @xmath158 and @xmath159 for @xmath160 , as compared to @xmath161 . the blue dashed lines show contours of constant entropy , labeled on the right by @xmath162 . in the core accretion model of planet formation ( e.g. , @xcite ) , a rock / ice core is first assembled . once this core grows to @xmath163 , where @xmath164 denotes an earth mass , it can initiate rapid accretion of nebular gas , which could then dominate the mass of the planet . for studies of planetary radii , a central core composed of high mean molecular weight material can decrease the radius of the planet by a significant amount ( @xmath165 ) . the ` mesa ` ` star ` inert core feature allows one to add a core of specified mass @xmath73 and radius @xmath89 , or more conveniently , density @xmath90 . a luminosity @xmath66 may also be specified , although the high mean molecular weight of the core , as compared to the overlying h / he envelope , implies that even large cores will tend to have small heat content @xcite . this inert core is not presently evolved in any way , and changes in @xmath86during evolution are neglected as @xmath16 changes . while cores of mass @xmath166 are commonly used for modeling solar system giants ( e.g. @xcite ) , the large masses and small radii of some exoplanets may imply far larger core masses ( e.g. hd 149026 ; @xcite ) . in addition , neptune - like planets with smaller ratios of envelope to core masses may be modeled with ` mesa ` @xcite . -@xmath20 surface heat source . the data point with error bars is the observed value of the radius for hd 209458b quoted in . the two sets of curves are deep heating ( upper three curves ) and shallow heating ( lower three curves ) . ] surface heating by stellar irradiation changes the boundary condition for the planet s cooling and contraction . this modifies the planetary radius versus age for exoplanets at orbital separation @xmath167 . ` mesa ` provides several ways to implement surface heating with varying degrees of fidelity to the true solution . these presently include : \a ) an energy generation rate @xmath168 applied in the outer mass column @xmath169 . here @xmath6 is the day - side flux from the star , and @xmath170 is the mass column . in steady - state , this generates an outward flux @xmath171 , which is meant to simulate the angle - averaged flux over the planetary surface . this model implicitly assumes that day - night heat transport is efficient , and at the depths of interest the temperature is uniform over the surface . the parameters @xmath6 and @xmath20 are specified through the user - specified variables ` irradiation_flux ` and ` column_depth_for_irradiation ` , making this the simplest method to use . this heating mechanism represents absorption of stellar optical radiation well below the photosphere of the planet s thermal radiation and gives rise to greenhouse heating of the atmosphere where @xmath172 . \b ) ` mesa ` s ` grey_irradiated ` atmosphere model ( see also [ s.atmospheres ] ) implements the angle - averaged temperature profile of . this approximate solution to the transfer equation assumes two frequency bands : optical radiation from the star ( with user - specified opacity @xmath173 ) and thermal radiation originating in the planet ( with user - specified opacity @xmath174 ) . the temperature profile is derived using the eddington approximation , assuming an external flux from the star as well as a flux from the planetary interior . while the model implemented in ` mesa ` uses a single temperature as a function of depth , it is derived allowing for local temperature variations over the surface which are then averaged over angle . this temperature profile is shown to be valid in the presence of horizontal heat transport by fluid motions . this is the only ` mesa ` atmosphere model that uses pressure instead of optical depth to determine the surface boundary condition . as this pressure may be relatively deep in the atmosphere , a correction to the radius may be required to give either the vertical thermal photosphere , or the optical photosphere in transit along a chord . lastly , the ` relax_irradiation ` routine improves initial convergence by providing a starting model closer to the irradiated one . \c ) finally , ` mesa ` allows user - specified heating functions ( e.g. , @xmath6-@xmath20 surface heating ) or atmosphere models ( e.g. , ` grey_irradiated ` ) . user - supplied routines may be easily implemented by using the ` other_energy ` module . figure [ fig : rph_vs_age_compare_to_cepam_guillot2010_fig9 ] shows radius versus age for the planet hd 209458b . the two groupings of lines are for different heating depths , and within each grouping of lines , there are three calculations : ` mesa ` using ` grey_irradiated ` surface boundary condition ( solid red line ) , ` mesa ` using the @xmath6-@xmath20 surface heating profile ( dashed blue line ) , and cepam using the same grey irradiated boundary condition ( dotted black line ; kindly provided by tristan guillot ) . the lower curves , corresponding to shallow heating , use fiducial values @xmath175 and give a model radius significantly smaller than the observed radius . the upper curves , corresponding to deep heating , use @xmath176 , yielding significantly hotter temperatures deep in the surface radiative zone , which slow the cooling enough to agree with the observed radius . the choice @xmath177 gives agreement between the grey irradiated and @xmath6-@xmath20 methods , where the factor of 2 accounts for the fact that the grey irradiated boundary condition has some heating below @xmath178 . the radii are at the @xmath179 photosphere for a vertical path into the atmosphere . the agreement between all three methods is excellent , at the 12% level after 100@xmath180@xmath181 . the remaining discrepancy between the ` mesa ` and cepam grey irradiated results are likely due to different opacity tables , with the ` mesa ` result using an update of @xcite ( freedman 2011 , priv . comm . ) while the cepam run uses the @xcite cond table . the differences at ages @xmath182 are due to different starting conditions . the cepam calculation started with initial radius @xmath183 , whereas the mesa calculations started with @xmath184 . the ` mesa ` grey irradiated and @xmath6-@xmath20 calculations differ at @xmath185 , likely because the former has a fixed thermal opacity while the latter allows the opacity to change . most of @xmath186 s capability to evolve low - mass ( @xmath187 ) stars was demonstrated in section 7.1 of paper i. ` mesa ` has seen use in the asteroseismology of helium core flashing stars @xcite and the discovery of a new instability from the onset of @xmath188 burning @xcite . we expect the future use of ` mesa ` ` star ` for asteroseismic investigations of these stars to be substantial ( see [ s.astroseismology ] ) . from ` mesa ` ` star ` ( solid lines ) and ( * ? ? ? * dashed lines ) in the mass - radius plane . the data points plotted are the same as shown by @xcite . ] the derivation of accurate planetary radii based on transits requires accurate radii of the host stars ; this motivates ` mesa ` ` star ` investigations of low - mass stars @xcite . figure [ fig : stellar_m_r ] shows 1 and 5 gyr isochrones at solar composition ( @xmath189 ) from ` mesa ` ` star ` ( solid lines ) and ( * ? ? ? * dashed lines ) in the mass - radius diagram . data points shown in figure [ fig : stellar_m_r ] are taken from , @xcite , @xcite , and @xcite . this figure is a reproduction of the upper panel of figure 11 from @xcite . figure [ fig : stellar_m_r ] indicates that ` mesa ` ` star ` is capable of producing mass - radius relations for main sequence stars that are consistent with other widely - used models as well as observational data . the ` mesa ` ` star ` models were computed using , as much as possible , the same physical assumptions as the models used by @xcite . the main difference is the equation of state , for which @xcite used freeeos and ` mesa ` ` star ` uses a combination of the opal @xcite and scvh eos for thermodynamic parameters relevant to this diagram . with its highly configurable output options , and its ability to calculate asteroseismic variables , ` mesa ` ` star ` can readily produce models suitable for use with a range of oscillation codes . in addition to its own text output files , ` mesa ` can produce outputs in formats widely used by stellar oscillation codes , such as ` fgong ` and ` osc ` @xcite . in figure [ astero:1m_pre_ms_to_wd ] we show the evolution of a @xmath190 model in the hertzsprung - russell diagram ( hrd ) and in @xmath97-@xmath90 space . these were evolved following the test case found in ` 1m_pre_ms_to_wd ` , which was modified to include diffusion . this runs without user intervention from pre - main sequence to white dwarf . to demonstrate the changing stellar structure as the model evolves from the main sequence to post helium - core burning on the asymptotic giant branch ( agb ) , we show in figure [ astero : profiles ] some of the fundamental quantities extracted from the corresponding ` profile.data ` files for the models marked in figure [ astero:1m_pre_ms_to_wd ] . these include the lamb and brunt - visl frequencies defined respectively by @xmath191 , \label{n1}\end{aligned}\ ] ] where @xmath45 is the adiabatic sound speed and @xmath192 is the spherical harmonic degree . the seismic properties of the sun provide a test of stellar evolution models , and an opportunity to calibrate @xmath193 for any particular set of input physics and other assumptions . the ` mesa ` ` star ` test case ` solar_calibration ` produces a calibrated standard solar model . figure [ astero : solar ] shows the difference between the helioseismically - inferred solar sound speed profile and this model . we also show `` model s '' from @xcite . both models employ comparable input physics and assume solar abundances from @xcite and @xcite . one clear improvement since paper i is a smoother sound speed profile at small @xmath194 , which is primarily due to improvements in the ` diffusion ` module . this is particularly important for asteroseismology , where sharp features in the sound speed profile can influence the stellar oscillation frequencies . the results are based on the solar calibration test case compiled with the gnu fortran compiler version 4.7.2 on mac os x 10.7.5 ; appendix [ s.compiler_os ] provides information about how the solar calibration results may depend on different operating systems and compilers . the `` astero '' extension to ` mesa ` ` star ` implements an integrated approach that passes results automatically between ` mesa ` ` star ` and the new ` mesa ` module based on the adiabatic code adipls ( * ? ? ? * release ) . the ` mesa ` module ` adipls ` also supports independent use for post - processing , including the calculation of pulsation frequencies . this astero extension enables calculation of selected pulsation frequencies by ` mesa ` ` star ` during the evolution of the model . this allows fitting to the observations that can include spectroscopic constraints ( e.g. , [ fe / h ] and @xmath98 ) , asteroseismic constraints , such as the large frequency separation ( @xmath46 ) and the frequency of maximum power ( @xmath82 ) , and even individual frequencies . a variety of approaches for finding a best - fitting model are available , including grid searches and automatic @xmath195 minimization by the hooke - jeeves algorithm @xcite or by the `` bound optimization by quadratic approximation '' ( bobyqa ; * ? ? ? * ) technique . these searches are user controlled through a number of parameter bounds and step sizes . users also have full control over the relative weight assigned to the seismic and spectroscopic parts of the @xmath196 statistic . for the automated @xmath195 minimization , astero will evolve a pre - main sequence model from a user defined starting point , and find the best match along that single evolutionary track . the code then recalculates the track , again initiated at the pre - main sequence , with different initial parameters such as mass , composition , mixing length parameter and overshoot , and repeats until the lowest @xmath195 has been found . calculating specific mode frequencies is computationally intensive . hence , a number of options exist to improve the efficiency of the minimization when individual frequencies are included . bounds can be established on stellar parameters ( e.g. , @xmath98 , central h mass fraction , @xmath46 ) , so that ` adipls ` is invoked only when the model falls within these bounds . this enables certain evolutionary stages to be skipped when other observational diagnostics rule them out if a star is known to be a red giant , for instance , there is no sense in invoking ` adipls ` when models are on the main sequence . the large frequency separation , @xmath46 , of the model is calculated as the inverse of the sound travel time through the star , @xmath46 @xmath197^{-1}$ ] @xcite . there is also the option to derive @xmath46 using simple solar scaling : @xmath46 @xmath198 @xcite . to obtain @xmath82 , ` mesa ` scales the solar value with the acoustic cut - off frequency : @xmath82 @xmath199 @xcite . moreover , hierarchical approaches to the frequency fitting can be selected , saving large amounts of computational time . in one case the radial modes are first calculated , and only when they match reasonably well are the non - radial mode frequencies derived and included in the @xmath196 . this is particularly beneficial for red giants where the calculation of the non - radial frequencies is extremely time consuming . another example is when the time steps in the stellar evolution calculations are too large to find an accurate minimum of @xmath196 . hence , as a further option to increase efficiency while attaining accuracy , the time steps can be set to automatically reduce when the model comes close to the `` target box '' of the observational constraints . as for other modules used in ` mesa ` ` star ` , astero offers a range of graphical outputs including an chelle diagram where the fitting process can be followed in real time . there is also an option for including corrections to the model frequencies on - the - fly to compensate for the inadequate modelling of the near surface layers of the star . the effect , known as the `` surface term , '' is seen as a frequency dependent offset between the modelled and observed acoustic frequencies of the sun ( e.g. * ? ? ? the offset increases towards higher frequencies and is well described by a power law @xcite . ` mesa ` ` star ` follows the approach described by @xcite for correcting the surface term . to illustrate the performance of astero , we show here a fit to the star hd49385 . the input frequencies and the spectroscopic constraints are from @xcite . we first ran a wide - range grid search over @xmath11 , @xmath193 , [ fe / h ] , and @xmath27 , including only [ fe / h ] , @xmath98 , and @xmath46 as observational constraints . the results of this initial search guided our starting parameters and ranges for the next automatic @xmath195 minimization . we first compare our grid results with those of the radius grid search routine @xcite , which is based on a grid of astec models @xcite and find agreement within uncertainties . we then include the individual oscillation frequencies and use the hooke - jeeves algorithm for the @xmath195 minimization . model frequencies were corrected for the surface term , and the part of the @xmath195 coming from the frequencies was given 2/3 of the weight in the final @xmath195 , similar to that used by @xcite . to ensure we adequately sample the parameter space , we initiate the search at several initial values within a broad range . by starting the search from multiple initial values , we aim to reduce the chance of ending up in a local minimum , which could potentially provide unphysical results , such as the spuriously low helium abundances reported by @xcite . current developments in astero further seeks to overcome such problems and improve the robustness of the results by including frequency ratios @xcite in the @xmath195 minimization . each `` hooke '' search generates several stellar evolution tracks , each with a best @xmath195 value . we then combine the data from about 1400 tracks to estimate the 1-@xmath200 uncertainties in the varied parameters following the approach by @xcite . the lowest ( reduced ) @xmath195 value we obtained was 2.4 with a few tens of models in the 2.44.0 range , which all fit the frequencies similarly well . among these models there are two families of results , one of which has slightly lower [ fe / h ] and @xmath27 , and a slightly increased value for the spectroscopic part of the @xmath195 . the comparison of the observed and modeled frequencies for the realization with the lowest @xmath195 is shown in the chelle diagram format in figure [ astero : echelle ] . a plot of the internal structure including the brunt - visl and lamb frequencies is shown in figure [ astero : hd49385profile ] , and the parameters of the model are listed in table [ t.astero:hd49385 ] . we set @xmath201 and use the gn98 solar abundances . our results can be best compared to those listed as `` low @xmath202 '' and `` gn93 '' in table 4 of @xcite and agree within the uncertainties . rr @xmath203 & @xmath204 + @xmath205 & @xmath206 + @xmath207 & @xmath208 + @xmath68 & @xmath209 + @xmath124/k & @xmath210 + age / gyr & @xmath211 + @xmath33 & @xmath212 + @xmath213_i$ ] & @xmath214 + @xmath213_s$ ] & @xmath215 + @xmath216 & @xmath217 + @xmath218 & @xmath219 + @xmath195 & @xmath220 + including the effect of composition gradients in the calculation of the brunt - visl frequency is important for two reasons . first , it is necessary for implementing the ledoux criterion for convection , which is used to determine the chemical mixing and convective heat transport in a region ( see [ s.semiconvection ] ) . second , a smooth and accurate method for calculating @xmath221 is crucial for studies of g - mode pulsation in stars . in a highly degenerate environment , the pressure is nearly independent of temperature , and @xmath222 , so from eq . ( [ n1 ] ) we see that @xmath223 depends on the difference of two large and nearly equal quantities . this can lead to a loss of precision and a noisy @xmath223 . to eliminate this problem , @xmath221 is re - written into a form that depends on the difference of the adiabatic and true temperature gradients and on the composition gradient : @xmath224 the term @xmath225 explicitly takes into account the effect of composition gradients and is commonly called the ledoux term ( e.g. , * ? ? ? * ; * ? ? ? for the general case of an @xmath12-component plasma with mass fractions @xmath226 , the standard formula for @xmath225 is ( e.g. , * ? ? ? * ) @xmath227 since @xmath228 , one of the mass fractions can be eliminated , so that the sum in eq . ( [ bsum ] ) runs from 1 to @xmath229 . we note that the partial derivatives in eq . ( [ bsum ] ) hold all the \{@xmath230 } constant except for @xmath26 and @xmath231 , where @xmath231 is varied so as to maintain @xmath228 . although eq . ( [ bsum ] ) is correct as written , we have developed a new , formally - equivalent prescription that is both numerically robust and simpler to implement . we define a new ledoux term by taking a directional derivative along the radial composition gradient in the stellar model , @xmath232 the implementation of the above derivative typically involves the use of quantities on neighboring mesh points . using the subscript @xmath233 to denote the value of a given quantity on the @xmath233th mesh point , we therefore have @xmath234 this is the form of the ledoux term that is implemented in ` mesa ` and we term it the `` new ledoux '' formulation . since ` mesa ` ensures that @xmath228 at each mesh point , this condition does not have to be separately enforced . this formulation requires just one numerical difference along @xmath235 that is consistent with the stellar model and equation of state . because ` mesa ` s eos does not directly supply the partial derivatives required for the formulation in eq . [ bsum ] , an implementation of that method would suffer in both accuracy and efficiency from having to do a large number of numerical differences . @xcite dealt with a similar problem by using a restricted form of eq . [ bsum ] that included only the helium composition gradient . they showed that for cases where their restricted form applied , it gave significantly better numerical results than an implementation of eq . [ n1 ] based on finite differences . figure [ bvdiff ] shows that our new ledoux prescription ( grey heavy curve ) retains their good results compared to eq . [ n1 ] ( thin black curve ) while extending the applicability to cases that can not be dealt with using only helium gradients . we described the implementation of mixing - length theory ( mlt ) in paper i , including the allowance for overshoot beyond the boundaries of the convective zones as determined by the standard schwarzschild condition , @xmath236 . overshooting is implemented via an exponential decay of the convective diffusion coefficient beyond the boundary of full convection , following : @xmath237 where @xmath238 is the diffusion coefficient at the convective border , @xmath239 is the distance from the start of overshoot , and @xmath240 is the local pressure scale height . the user - adjusted dimensionless parameter @xmath241 then determines the extent of the overshooting region . ` mesa ` also allows for the adoption of a step - function overshooting model , where the mixing region extends a distance @xmath242 beyond the convective boundary with a constant specified diffusion coefficient . in paper i we did not implement the influence of composition gradients on mixing and the resulting diffusion coefficients when instabilities are operative . the description of how ` mesa ` ` star ` calculates the ledoux criterion is in [ s.ledoux ] . in this section , we describe the implementation of mixing due to composition gradients in stellar interiors . we refer to [ s.nuts-free-parameters ] for a discussion of the free parameters involved in the implementation of these mixing mechanisms . semiconvection refers to mixing in regions unstable to schwarzschild but stable to ledoux , that is @xmath243 where @xmath244 is the sum of the adiabatic gradient and the brunt composition gradient term ( see eqs . [ [ bled ] ] and [ [ new ] ] ) , @xmath245 once @xmath244 is calculated , regions satisfying equation ( [ eq : ledoux_stable ] ) undergo mixing via a time - dependent diffusive process with a diffusion coefficient calculated by the ` mlt ` module following , @xmath246 where @xmath247 is the radiative conductivity , @xmath44 is the specific heat at constant pressure , and @xmath34 a dimensionless efficiency parameter . see [ s.nuts-free-parameters ] for a discussion of the range of values for @xmath248 . we stress that semiconvection and overshooting have distinct implementations in ` mesa ` . both are time - dependent diffusive processes . as an example , in figure [ 3m_grad_dmix ] we display profiles of thermodynamic gradients and their resulting diffusion coefficients during core helium burning in a semiconvective model with @xmath249 and in an exponentially overshooting model with @xmath250 . sample profiles of semiconvective ( left ) and exponentially overshooting ( right ) @xmath251 models undergoing core helium burning . top panels show the radiative , adiabatic , temperature , and ledoux gradients that determine mixing boundaries and diffusion coefficients . bottom panels show the resulting diffusion coefficients for energy and chemical transport . in either case , a thin dotted line spanning a single intermediate cell joins the convective and semiconvective / overshoot curves . this is intended merely as a guide for the eye , as diffusion coefficients are defined only at the two boundaries of a cell . in particular , diffusion for this intermediate cell is governed by convection at its interior boundary and semiconvection / overshoot at the exterior . the semiconvective model shown here was run with @xmath249 ; the exponentially overshooting model with @xmath250 . the profiles are taken at the points marked in figure [ 3m_he_burn_mixing ] . ] thermohaline mixing arises in the presence of an inversion of the mean molecular weight in regions that are formally stable against convection according to the ledoux criterion , @xmath252 in ` mesa ` thermohaline mixing is treated in a diffusion approximation , with a diffusion coefficient motivated by the linear stability analysis of @xcite and @xcite @xmath253 the quantity @xmath35 is a dimensionless efficiency parameter . in the linear analysis it depends on the aspect ratio of the blobs / fingers arising from the instability . in the case of salt fingers such a value is calibrated using laboratory experiments in water ( e.g. * ? ? ? * ) , where the fingers have an aspect ratio of @xmath254 . in the stellar case the value of this parameter is vexatious ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , with recent 2d and 3d hydrodynamical calculations pointing toward a much reduced value of @xmath35 relative to the salt fingers case @xcite . figure [ f.thermohaline ] shows a calculation including the effects of thermohaline mixing during the rgb phase of a @xmath255 star after the luminosity bump ( e.g. * ? ? ? * ; * ? ? ? * ) . for this calculation a value @xmath256 has been adopted , but see [ s.nuts-free-parameters ] for a discussion of the range of options . the duration of the hydrogen and helium core burning depends on the extent of the convective core , so we focus here on exhibiting the ` mesa ` capabilities during these phases . as we noted above , there are many physical effects that change the size of the convective core , such as semiconvection , overshooting , and rotation - induced mixing . for example , the schwarzschild criterion implies larger cores than the ledoux criterion , but when using ledoux alone , the region above the convective boundary is overstable and so semiconvection occurs ( see [ s.semiconvection ] ) . we evolved a non - rotating @xmath257 star with @xmath258 through central hydrogen burning using ledoux , ledoux plus semiconvection , schwarzschild , and schwarzschild plus overshoot . as is evident in figure [ 1p5m_main_sequence_mcc ] , this set of physical processes leads to a large range of convective core masses and thereby main sequence lifetimes . for the parameters explored we found that overshooting increases the lifetime by a factor @xmath2591.2 for schwarzschild and @xmath2592.5 for ledoux . figure [ 1p5m_main_sequence_hrd ] shows an hr diagram for each of the @xmath257 models undergoing core hydrogen burning , showing the impact of convective core extent on main - sequence turnoff morphology . history of convective core extent during the main sequence for a non - rotating @xmath257 star with various mixing options . the plot shows the boundary of convection not including the extent of semiconvection or overshooting . ] the hr diagram for the non - rotating @xmath257 star with various mixing options . tracks are displayed from zams until depletion of core hydrogen to @xmath260 . ] we also evolved a non - rotating @xmath261 star with @xmath262 through central helium burning . overshooting extends the burning lifetime by a factor @xmath2631.6 for schwarzschild and @xmath2632.8 for ledoux . although this lengthening of the core burning phase is always true of convective overshoot , we find that the extension of the overshoot and convective regions is sensitive to the temporal resolution adopted . with sufficiently large values of @xmath54 the upper boundary develops oscillatory behavior which can also affect the lifetime . this behavior also occurs with the step - function implementation of overshoot . this instability is not seen in overshoot during hydrogen burning and has yet to be studied in detail . history of convective core extent during the core helium burning phase for a non - rotating @xmath261 star with various mixing options , as in figure [ 1p5m_main_sequence_mcc ] . time is measured relative to the onset of the convective core burning . efficient semiconvection ( @xmath249 ) and inefficient overshooting ( @xmath250 ) coincide with the pure schwarzschild model . the filled ( open ) circle indicates the time for which we display a profile detailing semiconvection ( overshooting ) in figure [ 3m_grad_dmix ] . ] extending the verification of paper i , we now compare to other available codes for intermediate - mass stars , @xmath0 . we describe the techniques used by ` mesa ` ` star ` to evolve stars through the agb phase to the white dwarf cooling sequence . we also demonstrate how ` mesa ` ` star ` incorporates compressional heating from accretion . we start by comparing the results of ` mesa ` ` star ` to those from the dartmouth stellar evolution program ( dsep ; @xcite ) for stars with @xmath264 . in both cases , the models were evolved from the pre - main sequence to the depletion of helium in their cores . for completeness , the ` mesa ` ` star ` models were further evolved to the occurrence of the first helium thermal pulse . all models have an initial composition @xmath265 , @xmath144 , and no mass loss or rotation was included . the boundaries of mixing zones are determined by the schwarzschild criterion with @xmath145 . in order to compare the codes , we do not allow overshooting or semiconvection . we adopt the @xcite rate for @xmath266@xmath267@xmath268 and the @xcite rate for @xmath269@xmath270@xmath271 ; for all other rates we use the nacre compilation @xcite . we use the opal type 2 opacity tables ( @xcite ) to account for the carbon- and oxygen - enhanced opacities during helium burning . the resulting tracks in the hr diagram of figure [ fig : mesa_vs_dsep_hrd ] and the evolution in the @xmath97-@xmath90 plane of figure [ fig : mesa_vs_dsep_tc - rhoc ] show excellent agreement between the codes . figures [ fig : mesa_vs_dsep_4 m ] and [ fig : mesa_vs_dsep_6 m ] show the hydrogen - burning luminosity , the helium - burning luminosity , and the extent of the convective core during convective helium core burning for a @xmath272 model ( fig . [ fig : mesa_vs_dsep_4 m ] ) and a @xmath273 model ( fig . [ fig : mesa_vs_dsep_6 m ] ) . table [ mesa_dsep_table ] gives a summary of the core hydrogen burning lifetime , the core helium burning lifetime , the final extent of the convective core during central helium burning , and the final carbon mass fraction @xmath274 in the core for each model . for the ` mesa ` models , we also show the maximum extent of the convective core during central hydrogen burning , the mass of the helium core before helium ignition , and the mass of the c / o core at the time of the first helium thermal pulse . hertzsprung - russell diagram for evolution of @xmath275 stars from the pre - main sequence through core helium depletion . models are from ` mesa ` ( thick grey lines ) and dsep ( dashed black lines ) . each curve is labeled with its corresponding initial mass in solar units . ] same as fig . [ fig : mesa_vs_dsep_hrd ] , but in the @xmath97-@xmath90 plane . the ` mesa ` models ( thick grey lines ) are evolved until the occurrence of the first thermal pulse . ] history of hydrogen burning luminosity ( top ) , helium - burning luminosity ( center ) , and convective core extent ( bottom ) during the core helium burning phase for the @xmath272 models . time is measured relative to the onset of the convective core . ] history of hydrogen burning luminosity ( top ) , helium - burning luminosity ( center ) , and convective core extent ( bottom ) during the core helium burning phase for the @xmath273 models . time is measured relative to the onset of the convective core . ] rrrrrrrrrrrrr 3.0 & 320.6 & 0.69 & 0.36 & 83.59 & 0.097 & 0.426 & 0.466 & & 312.0 & 80.81 & 0.098 & 0.456 + 4.0 & 152.7 & 1.01 & 0.47 & 29.78 & 0.149 & 0.490 & 0.667 & & 147.3 & 28.91 & 0.153 & 0.516 + 5.0 & 85.61 & 1.34 & 0.59 & 15.52 & 0.214 & 0.511 & 0.827 & & 84.75 & 15.19 & 0.210 & 0.507 + 6.0 & 55.98 & 1.68 & 0.72 & 9.62 & 0.288 & 0.514 & 0.870 & & 55.41 & 9.61 & 0.289 & 0.505 + 7.0 & 39.91 & 2.03 & 0.86 & 6.51 & 0.375 & 0.511 & 0.915 & & 39.69 & 6.79 & 0.401 & 0.454 + 8.0 & 30.42 & 2.40 & 1.02 & 4.67 & 0.480 & 0.504 & 0.966 & & 30.26 & 4.71 & 0.482 & 0.515 + we close with an additional comparison of the helium core burning phase of a @xmath276 , @xmath144 model computed by ` mesa ` to that of @xcite . both models were evolved using the @xcite rate for @xmath266@xmath267@xmath268 . the results for ` mesa ` ` star ` are a helium core burning lifetime of 83.6 myr and final c / o mass fractions of @xmath277 , @xmath278 ; @xcite find a lifetime of 88 myr and @xmath279 , @xmath280 . in the previous section , we discussed the evolution of @xmath275 stars up to the occurrence of the first he thermal pulse . in paper i we showed detailed comparisons of the evolution of a @xmath281 star to the evol code @xcite , exhibiting the ability of ` mesa ` ` star ` to calculate multiple helium shell pulses . we now illustrate the final evolution of intermediate - mass stars , and how to construct white dwarfs ( wds ) by using winds . we evolve @xmath261 , @xmath282 , and @xmath283 stars from the zams using the test suite case ` make_co_wd ` . this makes use of rgb mass loss following @xcite with an efficiency parameter @xmath284 and agb mass loss following using @xmath285 until the occurrence of the first helium shell flash . at that time , an increased bloecker @xmath286 is adopted to allow only a small number of thermal pulses before the wind mass loss eliminates the envelope . such intervention allows ` mesa ` ` star ` to make a high - mass wd . to avoid shortening of timesteps due to radiation - dominated envelopes , these cases also use the mlt++ capability described in [ s.superadiabatic ] . figure [ fig:357_zams_to_wd_hr ] shows the resulting tracks on the hr diagram . the @xmath261 star underwent eight thermal pulses after the enhancement of bloecker winds , while the @xmath282 and @xmath283 stars lost their envelopes so quickly that thermal pulses were immediately halted . the @xmath282 star ended up as an @xmath287 c / o wd with a helium shell of thickness @xmath288 and a hydrogen envelope of @xmath289 . note that the c / o wd mass is only slightly larger than the c / o mass at the first thermal pulse ( @xmath290 ) reported in table [ mesa_dsep_table ] . evolution of @xmath291 , @xmath292 and @xmath283 models from zero - age main sequence to cooling white dwarfs . a bloecker mass loss strips the stars of their envelopes on the thermally pulsing agb to make the three c / o white dwarfs . the single @xmath293 he white dwarf was made with mass loss after the hydrogen main sequence for the @xmath261 model was completed . ] profiles in @xmath294-@xmath295 space of the cooling @xmath296 c / o white dwarf evolved from a @xmath282 progenitor . each model is labeled to the right by @xmath98 . the outermost point of the model is at @xmath297 . dotted curves denote convective regions . going toward the interior , open circles designate the transition into the helium - rich shell , and filled circles designate the transition into the c / o core . ] surface luminosity as a function of central temperature for the cooling @xmath298 , @xmath299 , @xmath300 , and @xmath301 wds evolved from @xmath291 , @xmath292 , and @xmath283 progenitors . ] after removal of the envelope , the evolution of the white dwarf is continued through its cooling phase past solidification . we include gravitational settling and chemical diffusion of the outermost layers . figure [ fig : co_wd_from_5m_trho ] shows @xmath302@xmath17 profiles taken at various effective temperatures during the cooling of the @xmath287 c / o wd made from the @xmath282 star . the growing depth of the convection zone is shown by the dashed line , and the open circles designate the h / he transition , while the filled circles denote the he / co transition . figure [ fig:357_cool_l_tc ] illustrates the resulting @xmath9-@xmath97 relation as these models cool . the test suite case ` wd_diffusion ` uses the implementation of diffusion described in paper i to evolve a wd of mass @xmath303 until the @xmath304 hydrogen layer and the @xmath305 helium layer approach diffusive equilibrium . at this point , the wd has an effective temperature of @xmath306 . we show the resulting abundance profiles in figure [ wddiffuse ] , and , for comparison , the abundance profiles derived from the analytic form for diffusive equilibrium ( eq . ( 22 ) of * ? ? ? this formula is obtained by integrating equation ( a.5 ) of @xcite and assuming an ideal gas equation of state and complete ionization of both species . a comparison of time - dependent diffusion calculations for a @xmath307 wd with @xmath308 and @xmath309 with ` mesa ` ` star ` ( solid lines ) to those assuming diffusive equilibrium and an ideal gas equation of state ( dashed lines ) . ] the specific treatment of convection can also impact wd evolution . in paper i , ` mesa ` used the @xcite prescription for convection as its default convective mlt , with the optional extension of @xcite . since paper i , we have added support for the formulations of @xcite , @xcite , and @xcite . in particular , the bhm & cassinelli prescription , often referred to as `` ml2 , '' is frequently employed in wd studies ( e.g. , * ? ? ? * ) . in figure [ wdconv ] we show a comparison of the brunt - visl frequency calculated with ` mesa ` to that using the warsaw envelope code @xcite , assuming the ml2 prescription . this is the same wd as in figure [ wddiffuse ] , but now at a lower @xmath310 . to more accurately integrate these opaque but thin layers , we reduce @xmath21 at the boundary of the model by a factor of 1000 from its photospheric value of @xmath311 . this calculation is a sensitive test of the envelope integrations because @xmath221 is a derivative of the envelope structure . the two codes give indistinguishable results for this case and all other cases that we have calculated . ` mesa ` now includes atmospheric tables based on the non - grey model atmospheres for hydrogen - atmosphere wds @xcite , spanning the following range of parameters : @xmath312 and @xmath313 . such an approach is necessary at @xmath314 , where wds develop deeper convection zones . when the convection zone comes in contact with the degenerate , nearly isothermal core , energy is able to flow out of the core much more efficiently . the use of non - grey atmosphere models results in shallower convection zones , so this convective coupling of the core and envelope is delayed . for reliable cooling ages , we therefore recommend using non - grey atmospheres when @xmath314 . figure [ age_diff ] demonstrates the impact of non - grey atmospheres with the @xmath303 wd , which was cooled with and without the non - grey atmosphere . ` mesa ` currently treats crystallization by employing the @xcite eos ( pc eos ) . the pc eos is callable for arbitrary mixtures of chemical species and for densities with @xmath315 ; it is applicable in the domains of non - degenerate and degenerate , non - relativistic and relativistic electrons , weakly and strongly coupled coulomb liquids , and classical and quantum coulomb crystals . the phase transition is first - order , so the pc eos exhibits a latent heat between the solid and liquid phases , i.e. , the entropy and internal energy both experience finite jumps . this energy is included in ` mesa ` ` star ` models of cooling white dwarfs through the gravitational source term in the energy equation , @xmath316 this form for @xmath50 replaces the default one ( see eq . ( [ eq : eps_grav_lnd ] ) below ) in cells where @xmath317 ( @xmath7 is the coulomb coupling parameter ) , and is smoothly interpolated with the default form in cells where @xmath318 . the pc eos uses the criterion @xmath319 to determine crystallization , but it is straightforward to include explicit crystallization curves for c / o and other mixtures ( e.g. , * ? ? ? * ; * ? ? ? for example , using the parameters of the model in figure [ wddiffuse ] , the age difference at late times ( @xmath320 ) between a model with and without the latent heat of crystallization is @xmath321 ; a slightly larger value would be obtained using the phase diagram of @xcite . ` mesa ` does not currently treat phase separation of different chemical species upon crystallization . low mass wds ( @xmath322 ) with helium cores and hydrogen envelopes may be produced in binary systems when the envelope is stripped by the companion as the primary evolves up the giant branch ( * ? ? ? * and references therein ) . he - core wds of mass @xmath323 may also be produced through strong rgb winds ( dcruz et al . 1996 ) , although we do not discuss this possibility further here . here we discuss the prescription for stripping the envelope used in the test case ` make_he_wd ` . the first step is to evolve a star , @xmath324 in this example , from the pms until a he core of the correct size has been made . the remnant total mass is determined by the mass interior to where the h abundance has dropped below a preset value , for example , @xmath325 , moving in from the surface . next , the routine ` relax_mass ` is used to remove mass from the model until it has the desired remnant mass . after the initial remnant has been constructed , diffusion can then be turned on to allow an outer h layer to form . after this stage , normal evolution of the wd occurs , as shown in figures [ fig:357_zams_to_wd_hr ] and [ fig:357_cool_l_tc ] . accretion onto stars occurs in many contexts and requires special treatment for the outermost layers added in each timestep . in particular , a special evaluation of the @xmath326 term is required for fluid parcels that were not present in the previous timestep . prior to addressing that subtlety , we restate ( as discussed in 6.2 of paper i ) that ` mesa ` ` star ` calculates @xmath50 of eq . ( [ epsent ] ) in terms of the local thermodynamic variables ( @xmath302 and @xmath17 ) used by ` mesa ` , @xmath327.\ ] ] ` mesa ` ` star ` takes the quantities in this equation as provided by ` eos ` , and computes the lagrangian time derivatives to find @xmath50 . ` mesa ` ` star ` can alternatively work under the assumption that @xmath328 , in which case ` mesa ` ` star ` treats @xmath87 rather than @xmath17 as its basic variable ( see [ s.solve-burn-and-mix ] for a discussion ) . in that case , @xmath329.\ ] ] either formulation can be used deep within the star , as long as the location is safely removed from any phase transition . paper i described the validation of these formulations . we now turn to the complication which arises when @xmath50 needs to be evaluated in material that was not present in the previous timestep . defining the envelope mass coordinate @xmath330 , we need to resolve the entropy for @xmath331 , as the explicit lagrangian time derivatives of eqs . ( [ eq : eps_grav_lnd ] ) and ( [ eq : eps_grav_prho ] ) can not be numerically evaluated . since there can be important physics that needs to be resolved for these mass shells for @xmath332 , an approximation must be derived that allows for accurate modeling of the star s outermost layers without having to result to a dramatic shortening of @xmath99 . the luminosity @xmath333 from the accretion shock ( or boundary layer ) goes outwards and does not determine the entropy of the material as it becomes part of the hydrostatically adjusting star . rather , the entropy of the material at @xmath332 is determined by the the transport of @xmath9 ( @xcite ; @xcite ; @xcite ) . consider such an outermost layer , where there are two relevant timescales , the thermal time , @xmath334 , and the local accretion time , @xmath335 . in nearly all relevant cases , the ratio @xmath336 ; this implies that the fluid element adjusts its temperature to that needed to transport the stellar luminosity from deep within . this simplifies @xmath337 in that part of the star ( following @xcite ) to @xmath338 enabling accurate modeling within ` mesa ` ` star ` of nearly all fluid elements that become part of the star during each timestep , many of which have envelope mass coordinates @xmath339 . we give an explicit example of this thin - shell radiative calculation of @xmath50 in a c / o white dwarf accreting hydrogen - rich material and undergoing classical nova ( cn ) cycles . we present two models accreting at rates of @xmath340 and @xmath341 . both cases were evolved from a @xmath342 starting model with @xmath343 which had undergone a few flashes while accreting at @xmath340 . the accreted material has solar - like composition @xmath344 , @xmath345 , and @xmath346 where the metal mass fractions are taken from @xcite . profiles of the envelope during the mass accumulation phase between cn outbursts for the two accretion rates are displayed in figures [ co_wd_accrete_1d-11 ] and [ co_wd_accrete_1d-10 ] . each line represents a different time in the accumulation cycle up to the unstable ignition , when the hydrogen mass reaches @xmath347 . all material at pressures smaller than that shown by the open circle is new to the model in that timestep ( e.g. , it has @xmath348 ) and employs the modified @xmath50 of eq . ( [ eq : eps_grav_thin_radiative ] ) . this highlights the significance of this approximation as it allows ` mesa ` ` star ` to calculate material properties at @xmath349 . the solid points show where @xmath50 switches to the explicit form employing the lagrangian time derivatives , such as eq . ( [ eq : eps_grav_lnd ] ) . the middle panel shows @xmath350 , which reflects the contribution of @xmath50 to the outward luminosity . the discontinuity of @xmath50 at the solid point reflects the error associated with the abrupt transition in the calculational approach . the substantially larger luminosity of the early ( @xmath351 ) stages is due to the ongoing transfer of heat from the previous outburst . the near - discontinuous drop in @xmath50 occurs at the base of the hydrogen - rich envelope , and reflects the jump in composition from the accreted material to the nearly pure @xmath352 layer . the expected amplitude of the jump in @xmath50 depends on both the composition jump and the local degree of electron degeneracy ( see appendix b of @xcite for a discussion ) . envelope profiles as a function of pressure of the accreting white dwarf for three instants during the mass accumulation phase ; @xmath340 model . the top panel shows temperature , the central panel shows the gravitational energy release rate , and the bottom shows the luminosity . material to the right of the open circle is newly accreted . the code treats material to the right of the filled circle using the thin - shell radiative calculation of @xmath50 . ] same as fig . [ co_wd_accrete_1d-11 ] , but for model accreting @xmath341 . ] a star s rotational energy is usually a small fraction of the gravitational energy : for the sun it is @xmath353 and for a @xmath354 star rotating with a typical equatorial velocity @xmath355 on the main sequence it is @xmath356 . therefore the effects on the stellar hydrostatic equilibrium are marginal , with the exception of stars close to critical rotation ( see [ s.rotation-mass-loss ] ) . even in the case of a small perturbation to hydrostatic equilibrium , rotation induces a modification to the star s thermal equilibrium @xcite . together with the emergence of rotationally - induced dynamical and secular instabilities , this can significantly affect the evolution of stars @xcite . due to the destabilizing effect of increasing radiation pressure , rotation is particularly important in massive stars ( see , e.g. , * ? ? ? * ; * ? ? ? * ) . moreover , the final fate of a massive star depends chiefly on the relative importance of rotation during its evolution ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . here we describe the implementation of rotation in ` mesa ` ` star ` . we briefly discuss the modification to the stellar structure equations and the inclusion of rotationally- and magnetically - induced mixing . magnetic fields generated by differential rotation in radiative regions have been implemented following the work of @xcite and in the same fashion as in @xcite and @xcite . rotationally enhanced mass loss is also discussed . we compare rotating massive - star models calculated with ` mesa ` ` star ` to previous calculations performed with kepler @xcite . we also directly compare runs from ` mesa ` ` star ` and stern @xcite . the purpose of these tests is to verify our implementation of rotation , which is derived from stern . we do not compare to codes that have a different implementation of rotation ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? although beyond the scope of this paper , such comparisons are critical when coupled to observations of the effects of rotation in stars ( e.g. , * ? ? ? * ; * ? ? ? * ) including asteroseismology @xcite . stellar structure deviates from spherical symmetry in the presence of rotation . while the structure is inherently three - dimensional , it suffices to solve the stellar structure equations in one dimension if the angular velocity , @xmath84 , is constant over isobars ( the so - called shellular approximation ; see , e.g. , * ? ? ? this is expected in the presence of strong anisotropic turbulence acting along isobars . in radiative regions such turbulence is a consequence of differential rotation @xcite and efficiently erases gradients along isobars and enforces shellular rotation @xcite . turbulence in the vertical direction ( i.e. , perpendicular to the isobars ) is much weaker due to the stabilizing effect of stratification . in ` mesa ` ` star ` we adopt the shellular approximation @xcite and calculate the modification to the stellar equations due to centrifugal acceleration @xcite . an isobar with volume @xmath357 and surface area @xmath358 deviates from spherical symmetry in the presence of rotation . however one can retain a 1d approximation by re - defining the radius coordinate as the radius of a sphere containing the same volume @xmath359 , allowing an equation of continuity in the usual form @xmath360 with @xmath17 being the density and @xmath361 the mass enclosed by @xmath358 . the energy equation also retains its usual , non - rotating form @xmath362 where @xmath363 is the rate of energy flow through the equipotential surface @xmath358 . then the next step is to define mean values for the quantities varying on isobars , @xmath364 where @xmath365 is an isobaric surface area element . the equation of momentum balance can be written as @xmath366 where @xmath367 is the pressure , @xmath368 is the gravitational constant and @xmath369 the time . the last term in the equation is the inertia term . rotation enters the momentum equation through the quantity @xmath370 @xmath371 where @xmath372 , with @xmath373 the effective gravitational acceleration ( @xmath374 is normal to @xmath358 ) . then the radiative temperature gradient becomes @xmath375}}^{\!-1 } \;,\ ] ] with @xmath376 the radiation constant , @xmath8 is the opacity , @xmath302 the temperature and @xmath363 , the energy flux through @xmath358 . the last factor on the right hand side accounts for inertia , and @xmath377 in rotating models the values of @xmath378 and @xmath370 differ from 1 mostly in the outer stellar layers . limits to the minimum values of @xmath378 and @xmath370 are set in the code ( default values are 0.95 and 0.75 , respectively ) . this prevents numerical instabilities in models approaching critical rotation ( @xmath379 , see [ s.rotation-mass-loss ] ) . in such cases the outer layers greatly deviates from spherical symmetry and the results from 1d calculations should be considered particularly uncertain . transport of angular momentum and chemicals due to rotationally - induced instabilities is implemented in a diffusion approximation ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? this choice has also been adopted by other stellar evolution codes ( e.g. , kepler , @xcite ; stern , @xcite ) . we stress that this is not the only possibility , and other groups have implemented a diffusion - advection approach ( e.g. , geneva , @xcite ; rose , @xcite ) . the rose code can switch between the two different implementations . the two approaches are equivalent for the transport of chemicals . potentially large differences can arise , however , for the transport of angular momentum . a detailed description of the advection - diffusion equation for angular momentum is given in @xcite and @xcite . in ` mesa ` ` star ` the turbulent viscosity @xmath14 is determined as the sum of the diffusion coefficients for convection , semiconvection , and rotationally - induced instabilities . in convective regions , the very large diffusion coefficient implies that the rotation law is not far from solid body . this is a very common assumption in stellar evolution codes ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) ; note however that helioseismology has clearly shown this is not the case for the solar convection zone ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? ` mesa ` ` star ` calculates diffusion coefficients for five different rotationally - induced mixing processes : dynamical shear instability , solberg - hiland instability , secular shear instability , eddington - sweet circulation , and the goldreich - schubert - fricke instability . see @xcite for a detailed description of the physics of the different instabilities and the calculation of the respective diffusion coefficients . these enter the angular momentum and abundance diffusion equations that are solved at each timestep ( see [ s.nuts-rotation ] ) . it has been suggested that differential rotation in the radiative layers of a star can amplify a seed magnetic field . such a dynamo process has been proposed by ( * ? ? ? * spruit - tayler dynamo ) ; a theoretical debate on this is still ongoing @xcite . from the observational point of view , pure hydrodynamic models fail to predict the solar core rotation ( e.g. , * ? ? ? * ) , with the exception of models that include transport of angular momentum by gravity waves @xcite . models that include the spruit - tayler dynamo can reproduce the flat rotation profile of the sun . note however that these have difficulty explaining the core - envelope decoupling observed in low - mass , young cluster stars @xcite . on the other hand , observations of the final spins of both wds and neutron stars @xcite suggest that angular momentum transport with an efficiency similar to the torques provided by the spruit - tayler dynamo operates . models that only include angular momentum transport through rotational instabilities do not produce the core - envelope ratio of angular velocity observed through the splitting of mixed modes in red giant stars @xcite . ` mesa ` ` star ` accounts for transport by magnetic fields of angular momentum and chemicals due to the spruit - tayler dynamo . we refer to @xcite for a description of the physics of the dynamo loop and to @xcite , @xcite and @xcite for a discussion of its inclusion in stellar evolution codes . we implement the spruit - tayler dynamo in ` mesa ` ` star ` following kepler @xcite and stern @xcite . rotating stars that have a significant outer convective zone can produce surface magnetic fields through a dynamo ( see e.g. , * ? ? ? * for a review on astrophysical dynamos ) . this is the case for low - mass main sequence stars below about 1.5 @xmath380 , and observationally the break in the rotation properties around this mass is attributed to the presence of magnetized stellar winds ( e.g. , * ? ? ? * ; * ? ? ? note that dynamo action in a subsurface convective layer is in principle possible also in early - type stars @xcite . surface magnetic fields can also be of fossil origin , as is usually discussed in the context of ap stars @xcite . whatever the origin of surface magnetic fields , these are expected to couple to the wind mass - loss and , if strong enough , produce magnetic braking ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? . such magnetic braking has been directly observed in the case of the main sequence massive star @xmath200-ori e @xcite . here we do not include the physics of magnetic braking , as we only consider the evolution of stars without surface magnetic fields . we include the rotational modification to the wind mass loss rate @xcite . similar to other codes ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , in ` mesa ` ` star ` the stellar mass loss is enhanced as the rotation rate increases according to the prescription @xmath381 where @xmath382 is the value of the surface angular velocity and @xmath383 is the critical angular velocity at the surface . this last quantity is defined as @xmath384 , where @xmath385 is calculated as a mass - weighted average in a user - specified optical depth range ( default value @xmath386 $ ] ) . in ` mesa ` ` star ` the default value for the exponent @xmath387 is 0.43 @xcite . other implementations of rotationally enhanced mass loss can be found in @xcite and @xcite . for stars approaching @xmath379 , the mass loss calculated using equation ( [ mdotomega ] ) diverges . notice that luminous stars can approach this limit without having to rotate very rapidly as @xmath388 when @xmath389 . following @xcite we limit the mass loss timescale to the thermal timescale of the star @xmath390 @xmath391\ ] ] where @xmath392 is an efficiency factor of order unity ( default value is @xmath393 ) . in all the rotating models presented in this paper , rotation is initialized by imposing a solid body rotation law on the zero - age main sequence ( zams , @xmath394 ) . in these massive stars this is motivated by the presence of rotationally - induced angular momentum transport during the pre - main sequence evolution . this alone is able to enforce a state of close - to - rigid rotation by the time the star reaches the zams @xcite . overall initial solid body rotation is a common choice in stellar evolution codes , but other rotational laws are certainly possible . * 15mag includes the effects of rotation and spruit - tayler magnetic fields on both the transport of chemicals and angular momentum . * 15rot includes only the effect of rotation on both the transport of chemicals and angular momentum ; the initial conditions have been calibrated to match as closely as possible the kepler @xmath395 models @xcite . moreover , we directly compare the ` mesa ` ` star ` models with calculations from stern ( see e.g. , * ? ? ? * ; * ? ? ? in particular we adopt a value of @xmath396 for the ratio of the turbulent viscosity to the diffusion coefficient and a value @xmath397 for the sensitivity to @xmath398-gradients ( see * ? ? ? * for a discussion of these calibration parameters ) . the ledoux criterion is used for the treatment of convective boundaries , together with semiconvection ( @xmath399 ) . we use @xmath400 , mass loss as in @xcite with rotational enhancement as described in [ s.rotation-mass-loss ] . in fig . [ fig : hrdfig15 ] we show the evolutionary track and the evolution of surface equatorial rotational velocity for the 15mag model . results of a similar calculation using stern are shown as a dashed curve . the two results are in excellent agreement . small differences in luminosity and lifetimes are not unexpected , as we have only matched the physics of rotation between the two calculations and not other ingredients . values for the diffusion coefficients for rotationally induced mixing and magnetic torques during the main sequence of 15mag are shown in fig . [ fig : msfig15 ] . the comparison reveals a very good agreement . both stars are kept in solid - body rotation during the main sequence by the efficient transport of angular momentum provided by the eddington - sweet circulation and spruit - tayler magnetic fields . the amplitude and location of the azimuthal ( @xmath401 ) and radial ( @xmath402 ) components of the magnetic fields during different phases of the evolution of 15mag are shown in fig . [ fig : bfig15 ] . as expected , these fields are generated only in radiative regions of the star and @xmath403 @xcite . as the star evolves away from the main sequence its structure departs from solid - body rotation with the core rotating faster than the envelope . during this stage the role of magnetic fields is very important in transporting angular momentum from the core to the envelope . the effect can be seen in fig . [ fig : jfig15 ] , which shows the evolution of the internal specific angular momentum in models 15rot and 15mag . the presence of magnetic torques results in a dramatic spin - down of the core of 15mag with respect to 15rot ( see also table [ tab:15evo ] ) . these results are in very good agreement with the ones obtained by stern and kepler . to address this issue we have created the ` mesa ` software development kit ( ` sdk ` ) , which packages everything necessary to establish a unified and maintained build environment . the principal components of the ` sdk ` are summarized in table [ tab : sdk ] ; all of these are distributed under an open - source license ( detailed in the table ) , permitting their redistribution without financial or copyright encumbrances . perhaps the most important component is the gfortran compiler , part of the gnu compiler collection . gfortran implements almost all of the fortran 2003 ( f2003 ) standard , and benefits from a high level of community support . the ` sdk ` is available for intel x86 and x86 - 64 cpu architectures running the linux and mac os x operating systems ( these platforms comprise most of the ` mesa ` user base ) . installation of the kit is straightforward , requiring a tar archive to be unpacked ( linux ) or an application folder to be copied ( os x ) , followed by the initialization of a few environment variables . by default , ` mesa ` is configured to compile `` out - of - the - box '' with the ` sdk ` . ` mesa ` can also be compiled without the ` sdk ` , using any alternate compiler which supports the f2003 standard . in this respect , gfortran should not be viewed as _ the _ ` mesa ` compiler ( nor the full ` sdk ` as _ the _ ` mesa ` build environment ) . ` mesa ` will adhere to fortran standards rather than rely on vendor - specific extensions . uptake of the ` sdk ` has been very rapid : at the time of writing , we estimate over 90% of the ` mesa ` community ( over 500 users ) are using the ` sdk ` . this growth has been matched by a significant decline in the number of installation support requests , and a corresponding reduction in the time taken to resolve these requests . with these maintenance overheads curbed , the ` mesa ` developers are able to devote more of their time to refining and extending the code .

we substantially update the capabilities of the open source software package modules for experiments in stellar astrophysics ( ` mesa ` ) , and its one - dimensional stellar evolution module , ` mesa ` ` star ` . improvements in ` mesa ` ` star ` s ability to model the evolution of giant planets now extends its applicability down to masses as low as one - tenth that of jupiter . the dramatic improvement in asteroseismology enabled by the space - based _ kepler _ and _ corot _ missions motivates our full coupling of the adipls adiabatic pulsation code with ` mesa ` ` star ` . this also motivates a numerical recasting of the ledoux criterion that is more easily implemented when many nuclei are present at non - negligible abundances . this impacts the way in which ` mesa ` ` star ` calculates semi - convective and thermohaline mixing . we exhibit the evolution of @xmath0 stars through the end of core he burning , the onset of he thermal pulses , and arrival on the white dwarf cooling sequence . we implement diffusion of angular momentum and chemical abundances that enable calculations of rotating - star models , which we compare thoroughly with earlier work . we introduce a new treatment of radiation - dominated envelopes that allows the uninterrupted evolution of massive stars to core collapse . this enables the generation of new sets of supernovae , long gamma - ray burst , and pair - instability progenitor models . we substantially modify the way in which ` mesa ` ` star ` solves the fully coupled stellar structure and composition equations , and we show how this has improved the scaling of ` mesa ` s calculational speed on multi - core processors . updates to the modules for equation of state , opacity , nuclear reaction rates , and atmospheric boundary conditions are also provided . we describe the mesa software development kit ( ` sdk ` ) that packages all the required components needed to form a unified , maintained , and well - validated build environment for ` mesa ` . we also highlight a few tools developed by the community for rapid visualization of ` mesa ` ` star ` results .

to reveal the nature of the extremely metal - poor ( emp ) stars in the galactic halo is the key to the understanding of the formation process of the galaxy as well as of the mechanism of star formation in the primordial and very metal - poor gas clouds . because of the very low abundances of iron and other metals , these stars are thought to be survivors from the early days , and hence , are expected to carry the precious information about the early universe when they were born while they reside in our nearby space . for a past decade , a lot of emp stars have been discovered by hk survey @xcite and hamburg / eso ( he s ) survey @xcite , which enables us to use halo emp stars as a probe into the early universe . the number of known emp stars exceeds several hundreds even if we limit the metallicity range below @xmath0}\lesssim -2.5 $ ] @xcite . one of their observed characteristics is very low frequency of stars below the metallicity @xmath0}\simeq -4 $ ] . despite that more than @xmath2 stars have been registered in the metallicity range of @xmath3}\lesssim -3 $ ] by high - dispersion spectroscopy ( e.g. , see saga database ; * ? ? ? * ) , only three stars were found well below this metallicity ; two hyper metal - poor ( hmp ) stars of @xmath0 } < -5 $ ] , he 0107 - 5240 ( @xmath0}= -5.3 $ ] ; * ? ? ? * ) and he 1327 - 2326 ( @xmath0}= -5.4 $ ] ; * ? ? ? * ) , and one ultra metal - poor ( ump ) star of @xmath4 } < -4 $ ] , he 0557 - 4840 ( @xmath0}= -4.8 $ ] ; * ? ? ? has attracted wide interest , in particular , before the discovery of he 0557 - 4840 in - between metallicity of @xmath5 } < -4 $ ] . @xcite points out that such a metallicity cut - off can be interpreted as a result of metal spreading process in the stochastic and inhomogeneous chemical - enrichment model . @xcite then introduce a period of low or delayed star formation due to the negative feedback by the population iii stars , during which metals spread to explain very low iron - abundance of hmp with the carbon yield from rotating stellar models by @xcite . @xcite argues an early infall phase of primordial gas to alleviate the paucity of low - metallicity stars . @xcite adopts a semi - analytic approach for the hierarchical structure formation and presents the model of inhomogeneous galactic chemical evolution in an attempt of reproducing the statistical features of emp stars and the re - ionization of the universe . he addresses the constraints on the imf of population iii stars , arguing high - mass imf of the mean mass at @xmath6 . @xcite also take a similar approach to investigate the chemical evolution of our galaxy with the mass outflow from mini - halos . in these former works , is introduced in rather arbitrary ways , and the proper explanation is yet to be devised about the nature and origin of hmp / ump stars . one of the decisive ingredients in studying the structure formation and chemical evolution of galactic halo is the initial mass function ( imf ) of stars in the early days . most of existent studies have assumed the imf of emp stars more or less similar to that of the metal - rich populations except for hmp and ump stars . from the observations , however , we know that the emp stars have the distinctive feature that than the stars of younger populations @xcite . in addition , it is revealed that the carbon - enhanced extremely metal - poor ( cemp ) stars are divided into two sub - groups , cemp-@xmath7 and cemp - no@xmath7 according to the presence and absence of the enhancement of @xmath7-process elements @xcite . assuming this binary scenario , @xcite argue an imf with the typical mass of @xmath8 for emp stars from the surplus of cemp-@xmath7 stars . previously , @xcite have also asserted an imf peaking in the intermediate - mass range of @xmath9 for population iii stars from the consideration of galactic chemical evolution with the cn enrichment among the emp stars . furthermore , an imf with @xmath10 has been is discussed for the old halo stars from the macho observation in relation to the prospect that the observed micro - lensing may be caused by an alleged population of white dwarfs @xcite . in order to use the carbon - enhancement to constrain the imf , we should properly take into account the evolutionary peculiarity of emp stars . the stars of @xmath0}\lesssim -2.5 $ ] , there are two mechanisms of carbon enhancement , while only one mechanism for the stars of younger populations , pop . i and ii , and also , that a different mode of s - process nucleosynthesis works @xcite . these theoretical understandings , ( * ? ? ? * referred to as paper i in the following ) find that the imf for emp stars has to be high - mass with the typical mass of @xmath11 to explain the observed statistic features of both cemp-@xmath7 and cemp - no@xmath7 stars . that the majority of emp stars , including cemp stars , were born as the low - mass members of binary systems with the primary stars which have shed their envelope by mass loss to be white dwarfs and have exploded as supernovae . the purpose of this paper is twofold , first to demonstrate the robustness of the high - mass imf derived in paper i , and then to discuss the implications to the formation and early evolution of galaxy . in the following , we make a distinction between the total assembly of emp stars that were born in the early galaxy , including massive stars which were already exploded as supernovae , and the low - mass emp stars that are still alive in the nuclear burning stages by calling the former emp population " and the latter emp survivors " . in deriving the constraints on the imf of stars for the emp population , one has to make the assumptions on the binary characteristics , among which the most crucial is the distribution function of mass ratio between the primary and secondary stars in binaries . paper i adopts a flat distribution for simplicity . it seems plausible from the observations of the stellar systems of younger populations @xcite , and yet , it is true that the mass - ratio distribution is yet to be properly established both observationally and theoretically even for the binaries of younger populations . several different mechanisms have been proposed for the binary formation , such as the fragmentation during the collapse and the capture of formed stars , and are thought to give different mass - ratio distributions ( see also e.g. , * ? ? ? * and the references therein ) . the distribution may increase or decrease with the mass - ratio , or the two stars may form in the same imf as suggested for the capture origin . in this paper , we examine the dependence of the resultant imf on the assumed mass - ratio distributions of various functional forms , including the independent coupling of the both stars in the same imf to demonstrate that the high - mass nature of imf of emp population is essentially unaltered . the recent large - scaled surveys of emp stars provide the additional information on the early history of galactic halo . a fairly large number of known metal - poor stars ( 144 and 234 stars of @xmath0 } < -3 $ ] by the hk and he s surveys , respectively ) makes it feasible to discuss the metallicity distribution function @xcite . moreover , the significant coverage of celestial sphere ( 6900 and @xmath12 by the hk and he s surveys , respectively ; * ? ? ? * ; * ? ? ? * ) allows to consider the total number of emp survivors in the galactic halo . we demonstrate that the latter also places an independent constraint on the imf of emp population in combination with the metal yields produced by the emp supernovae if the binary contribution is properly taken into account . we then apply the imf , thud derived , to discuss the chemical evolution in which the stars of emp population take part . the resultant imfs can reproduce the number and slope of observed metallicity distribution functions ( mdf ) for emp stars , and also , to give an explanation to the scarcity and origin of hmp / ump stars with the effects of hierarchical structure formation process included . in this paper , we , and discuss the basic characteristics of hierarchical structure formation by using simple analytic approximations . this paper is organized as follows . in 2 , we discuss the constraints on the imf of emp population from the statistics of cemp stars and from the total number of emp survivors in our galaxies . in 3 , we investigate the metallicity distribution of emp stars in galactic halo with the formation process of the galaxy taken into account . then our conclusions follow with discussion of the origin of observed mdf and also of hmp stars . in appendix , we re - discuss the relationship between the number of emp survivors , estimated from the surveys , and the metal production by the emp supernovae with the binary contribution taken into account , to demonstrate that they entail the same imfs as drawn independently from the statistics of cemp stars . in this section , we revisit the problem of constraining the imf for the stars of emp population from the observations of emp survivors , studied in paper i. the method is based on the analysis of statistics of cemp stars in the framework of binary scenario , and hence , involves the assumptions of emp binary systems . we start with reviewing the method and assumptions used in paper i in deriving the constraints on the imf of emp population stars . we first investigate the dependence of resultant imf on these assumptions , in particular of the mass - ratio distribution of binary members . we then discuss the iron production by emp population stars in relation to the total number of emp survivors , estimated from the hk and he s surveys , to assess the constraints on the imf through the chemical evolution of galactic halo . we give the outline of our method in studying the statistics of cemp stars and chemical evolution of galactic halo with the discussion of the assumptions involved , and a brief summary of the observational facts that our study rely on . [ [ section ] ] consequently , the origins of two sub - groups of cemp stars are identified with these two mechanisms . the cemp-@xmath7 and cemp - no@xmath7 stars stem from the low - mass members of emp binaries with the primaries in the mass ranges of @xmath13 and @xmath14 , respectively . here @xmath15 is the upper limit to initial mass of stars for the formation of white dwarfs . @xmath16 ( * ? ? ? * see also siess 2007 ) , which is also taken to be the lower mass limit to the stars that explode as supernova . this is the fundamental premise of our study . for the formation of cemp stars in the binary systems , the initial separation , @xmath17 , has to be large enough to allow the primary stars to evolve through the agb stage without suffering from the roche lobe overflow , but small enough for the secondary stars to accrete a sufficient mass of the wind to pollute their surface with the envelope matter processed and ejected by the agb companion . the lower bound , @xmath18 , to the initial separation is estimated from the stellar radii of emp stars taken from the evolutionary calculation @xcite , where @xmath19 and @xmath20 are the masses of primary and secondary stars . the agb star is assumed to eject the carbon enhanced matter of @xmath21 with the wind velocity @xmath22 until it becomes a white dwarf , and we define cemp stars as @xmath23 . the upper bound , @xmath24 , is estimated by the amount of accreted matter calculated by applying the bondi - hoyle accretion rate , @xmath25 in the spherically symmetric wind from the companion , and @xmath26 is the relative velocity of the secondary star to the wind . accreted matter is mixed in surface convection of depth @xmath27 and @xmath28 in mass for giants and dwarfs , respectively . e.g. , for the stellar metallicity @xmath29 , the mass of accreted matter has to be larger than @xmath30 and @xmath31 , and hence , the upper bounds are @xmath32 @xmath33 for dwarfs , respectively . if we specify the initial mass function , @xmath34 , and the distributions of binary parameters , therefore , we can evaluate the frequency of cemp-@xmath7 and cemp - no@xmath7 stars , and through the comparison with the observations , we may impose the constraints on the imf and on the binary parameters . the numbers of cemp-@xmath7 and cemp - no@xmath7 stars currently observable in flux - limited samples are given by @xmath35 ) \nonumber\\ \times \int_{0.8 m_\odot}^{3.5 m_\odot } & dm_1 & \xi_b ( m_1 ) \frac{n(q)}{m_1 } \int^{a_m(m_1,m_2)}_{a_{\rm min}(m_1,m_2 ) } f(p ) \frac{dp}{da}da \\ \psi_{{\rm cemp\hbox{-}no}s } & = & f_b \int^{0.8 \ , m_\odot } _ { 0.08 \ , m_\odot}d m_2 n_s ( l[m_2 ] ) \nonumber \\ \times \int_{3.5m_\odot}^{m_{up } } & dm_1 & \xi_b ( m_1 ) \frac{n(q)}{m_1 } \int^{a_m(m_1,m_2)}_{a_{\rm min}(m_1,m_2 ) } f(p ) \frac{dp}{da}da , \end{aligned}\ ] ] where @xmath36 is the binary fraction : @xmath37 is the distribution of the mass - ratio , @xmath38 , and @xmath39 is the distribution of the period of binaries : and @xmath40 is the probability of the stars in the galactic halo with the luminosity @xmath41 in the survey volume of he s survey . similarly the total number of emp survivors is given by @xmath42 ) [ ( 1-f_b ) \xi_s ( m ) + f_b \xi_b ( m ) ] \nonumber \\ & & + f_b \int^{0.8 \ , m_\odot } _ { 0.08 \ , m_\odot}dm_2 n_s ( l[m_2 ] ) \int_{0.8 \ , m_\odot } ^{\infty } dm_1 \xi_b ( m_1 ) \frac{n(q)}{m_1 } , \label{eq : surv}\end{aligned}\ ] ] the rest of the terms give the number of emp survivors formed as binary , @xmath43 . for the form of imf , we may well assume a lognormal function with the medium mass , @xmath44 , and the dispersion , @xmath45 , as parameters @xmath46.\ ] ] in addition , we assume the binary fraction @xmath47 in this paper . our results are little affected by the assumption about @xmath36 since not only the cemp stars but also most of the emp survivors come from the secondary companions of binaries unless @xmath48 , as seen later . as for the binary period , we may adopt the distribution derived for the nearby stars by @xcite , the binary fractions and period distributions of halo stars are observed to be not significantly different from those of nearby disk stars @xcite . additionally , it is shown in paper i that this period distribution is consistent with the observations of cemp stars for the periods of @xmath49 yr confirmed to date ( see fig . 3 in paper i ) . the mass ratio distribution is an essential factor in discussing the evolution of binary systems , and yet , it is not well understood . the mass ratio distribution of metal - poor halo stars is investigated observationally ( e.g. , see * ? ? ? * ; * ? ? ? * ) , and yet , subject to large uncertainties . especially for the binary with intermediate - mass or massive primary stars , it is hard to know the mass ratio distribution from the observations . theoretically , neither the fragmentation of gas cloud nor the accretion process onto proto - binaries are yet well understood even for population i stars ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . in order to test the assumption on the mass - ratio distribution , we investigate the constraints on the imf for different mass - ratio distributions . in paper i , the simplest flat distribution is assumed in paper i among the possible distributions . we may define the coupling mass distribution function , @xmath50 , as the fraction of the binaries with a primary and secondary star in the mass range of @xmath51 $ ] and @xmath52 $ ] @xmath53 and write it in the form ; @xmath54 mass function , @xmath55 , of the primary star is assumed to be the same as imf of single stars : and @xmath37 is the mass ratio distribution , for which we assume both extremities of increase and decrease functional forms in addition to the constant one , adopted in paper i ; @xmath56 & ( \rm{case~b } ) \\ q^{-1 } / \ln ( m_1/0.08 { \ , m_\odot } ) & ( \rm{case~c } ) . \end{array } \right.\ ] ] furthermore , we take up a different type of mass - ratio distribution that the primary and secondary stars independently obey the same imf such as assumed by @xcite . in this case , the coupling mass distribution function is given as a product of the same imf as ; @xmath57 we shall refer to this distribution function as independent " coupling . from the comparison with eq . ( [ eq : masscouple1 ] ) , we may write the mass - ratio function in the form @xmath58 ; it is should be noted , however , that the frequency of binaries with a primary star of mass @xmath19 is not normalized and increases with @xmath19 from zero to 2 , as given by the integral @xmath59 . with these specification and with the assumed mass - ratio distribution function , we may compute the fractions of emp survivors , @xmath60 and of both cemps stars , @xmath61 and @xmath62 , and search the ranges of the imf parameters , medium mass @xmath44 and dispersion @xmath45 , that can reproduce the statistics of cemp stars consistent with observations we can pose another constraint from the total iron yield , @xmath63 , of emp population and the total number , @xmath64 , of the giant emp survivors . for @xmath64 , estimated from the results of existent surveys , the total stellar mass , @xmath65 , of emp population for an assumed imf is given by , @xmath66 where @xmath67 is the fraction of giant emp survivors in all the stellar systems , born as emp population , and @xmath68 is the averaged mass of emp population stars : @xmath69 \delta m_{\rm g } , \label{eq : frac - giants}\ ] ] @xmath70.\ ] ] the first terms of both equations denote the contributions by the stars born as the single stars and as the primary stars in the binaries and the second terms denote the contributions by the stars born as the secondary stars in the binaries . the mass and mass range of emp stars now on the giant branch are taken to be @xmath71 and @xmath72 , based on the stellar evolution calculation of stars with @xmath0}= -3 $ ] , as in paper i. the massive stars of emp population have exploded as supernovae to enrich the interstellar gas with metals . the amount of iron , @xmath73 , ejected by all the supernovae of emp population of the total mass , @xmath65 , is given by @xmath74 where @xmath75 is the fraction of the stars that have exploded as supernovae and given by , @xmath76 : \label{eq : frac - sn}\ ] ] and @xmath77 is the averaged iron yield per supernova , taken to be @xmath78 in the following calculations . with these evaluations and the observed number of emp giants , we can give the total iron yield of stars of emp population as a function of imf parameters . the comparison with the total amount of iron estimated from the chemical evolution of galactic halo may impose constraint on the imf parameters . the first constraint is the number fraction of cemp-@xmath7 stars . the hk and he s observations tell that the cemp stars with @xmath79 account for @xmath80 of emp stars ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? @xcite suggest a slightly lower fraction of @xmath81 with the errors in the abundance analysis taken into account while in this paper , we adopt the observational constraint on the fraction of the cemp-@xmath7 stars at @xmath82 ; @xmath83 the second constraint is the number ratio between cemp - no@xmath7 and cemp-@xmath7 stars . the observed frequency of cemp - no@xmath7 to cemp-@xmath7 stars is @xmath84 or more , ( e.g. , * ? ? ? * ; * ? ? ? @xcite point out that it increases for lower metallicity , reporting the ratio as large as @xmath85 for @xmath0}\le -2.5 $ ] . in addition , emp stars enriched with nitrogen are found in number comparable with , or more than , cemp - no@xmath7 stars ( mixed stars ; * ? ? ? * ) , whose origin can be interpreted in terms of the same mechanism but with more massive primary companions that experience the hot bottom burning ( hbb ) in the envelope of the agb . some other scenarios for cemp stars have been proposed @xcite but we assume all cemp stars are formed in binaries with agb in this paper . we adopt the observational constraint on the relative frequency of cemp - no@xmath7 to cemp-@xmath7 stars at @xmath86 ; @xmath87 we note that the above two constraints are not dependent on the total mass nor on the spatial distribution of the stellar halo because they are concerned with the relative number ratios . the third constraint is the total iron yield from the emp population . the he s survey obtained 234 stars of @xmath0 } < -3 $ ] @xcite as a result of the medium - resolution , follow - up observations of 40% of the candidates , selected by the objective - prism survey of the nominal area @xmath88 @xcite . taking the relative frequency between the giants and dwarfs ( @xmath89 ) and the ratio of the stars of @xmath0}<-3 $ ] and @xmath0}<-2.5 $ ] ( @xmath90 ) from their table 3 , we may estimate the total number of emp giants in the galactic halo in the flux limited sample at : @xmath91 we assume that all giant stars in the survey areas are observed because of the fairly large limiting magnitude of he s survey ( @xmath92 ) , about two magnitude deeper than for the hk survey , and neglect the spatial distribution of emp giant for simplicity since the sufficient information is not yet available ( see 7.3 in paper i for the detail ) . then we have the total number of emp giants @xmath93 in the galaxy . on the other hand , the amount of iron necessary to promote the chemical evolution of the whole gas in galaxy of mass , @xmath94 , up to @xmath0}= -2.5 $ ] is as much as @xmath95 and the supernovae of emp population should have provided this amount of iron unless there were other population(s ) of stars which made iron without producing low - mass stars . @xmath96 for the four mass - ratio distributions , formulated in [ subsec - modelpara ] , we can figure out , as the function of @xmath44 and @xmath45 , the portion of stars that survive to date ( @xmath97 ) , and then , the fractions of stars in these emp survivors that evolved to cemp-@xmath7 and cemp - no@xmath7 stars according to the masses of primary stars and to the orbital separations . figure [ fig : psi ] compares the fractions of cemp-@xmath7 stars in the emp survivors [ @xmath98 and the ratios of cemp - no@xmath7 to cemp-@xmath7 stars [ @xmath99 , predicted with use of these four different mass - ratio distributions , as a function of medium mass @xmath44 of imfs with the dispersion of @xmath100 , taken to be same as the present - day imf of galactic spheroid component @xcite . figures [ fig : s ] and [ fig : nos ] present the contour maps on the @xmath44-@xmath45 diagram for the fractions of cemp-@xmath7 stars and the ratio between the cemp - no@xmath7 and cemp-@xmath7 stars , respectively . cc + left top panels on these figures show the results for the flat mass - ratio distribution of case a , which reproduces the results obtained in paper i. in fig . [ fig : psi ] , the cemp-@xmath7 fraction peaks at @xmath101 , slightly above the upper mass limit of the primary stars for cemp-@xmath7 . note that when the secondary mass is specified , the mass distribution of primary stars peaks at mass smaller than @xmath44 for this mass - ratio function [ @xmath102 , see fig . 12 in paper i ) . two ranges of @xmath44 , @xmath103 and @xmath104 ( light shaded parts ) gives the imfs compatible with the observations , separated by the overproduction of cemp-@xmath7 stars . the relative frequency of cemp - no@xmath7 to cemp-@xmath7 stars is a steep increase function of @xmath44 , and excludes the lower range of @xmath44 compatible with the cemp-@xmath7 fraction . the imfs with @xmath105 ( dark shaded part ) gives compatible ratio with the observations this range of @xmath44 lies in the mass range of primary stars of cemp - no@xmath7 stars or even larger . accordingly , the intersection of the light and dark shaded parts designates the ranges ( @xmath106 ) that can explain the both statistics of cemp stars , and hence , high - mass imfs results for a dispersion @xmath107 . on the @xmath108 diagram of fig.[fig : s ] , the parameter space compatible with the observed cemp-@xmath7 fraction separates into two ranges for the dispersion smaller than @xmath109 , converging to the narrow ranges around @xmath110 and @xmath111 , respectively , as @xmath45 decreases . for larger dispersion , on the other hand , it merges into one part to cover wider range . as for the ratio between the cemp - no@xmath7 and cemp-@xmath7 stars , fig . [ fig : nos ] shows that the medium mass compatible with observed ratio increases with the dispersion to cover wider range , from @xmath112 at @xmath113 through @xmath114 at @xmath115 . accordingly , for the imfs that satisfy the both statistical constraints , the medium mass increases with the dispersion from @xmath116 for @xmath117 and beyond @xmath118 for @xmath119 . for the mass - ratio distribution function increasing with @xmath120 of case b ( right top panel ) , the portion of binaries that have the secondary stars surviving to date decreases with the mass of primary stars in proportion to @xmath121 , more steeply than in proportion to @xmath122 for a flat mass - ratio distribution in case a. since the average mass of the primary stars is smaller for a given emp star , therefore , the fraction of cemp-@xmath7 stars is larger for a given @xmath123 , and the peak shifts to larger @xmath123 , as compared with case a. in fig . [ fig : psi ] , the @xmath123 of imfs compatible with the observed fraction of cemp-@xmath7 stars separates into two ranges , as in case a , but the in - between gap is larger ; the higher mass range shifts upward in mass to greater extent ( @xmath124 ) than the smaller mass range shifts downward ( @xmath125 ) . this also causes a smaller ratio of cemp - no@xmath7 to cemp-@xmath7 stars for a given @xmath44 , and hence , the imfs compatible with the observed ratio shift to a larger mass of @xmath126 , as compared with that for case a. as a result , the imfs consistent with the both statistics of cemp stars turns out to be higher mass by a factor of @xmath127 than for case a ( @xmath128 for @xmath100 ) . in the fig . [ fig : s ] , we see that the range of @xmath44 , compatible with the observed fractions of cemp-@xmath7 star ( shaded area ) , separates into two and the higher range shifts to larger mass for a given @xmath45 . similarly , in the fig . [ fig : nos ] , the observed ratio of the cemp - no@xmath7 to cemp-@xmath7 stars also demands larger @xmath44 , and the range of @xmath44 of imfs compatible with the observation increases rapidly with @xmath45 to exceed @xmath129 for @xmath130 . in order to satisfy the both conditions of cemp star observations , the imfs fall in the range of higher medium mass and in a rather narrow range of dispersion , lying in the parameter space of @xmath131 , larger by a factor of @xmath132 than for case a , and @xmath133 and of @xmath134 for @xmath135 . for a mass ratio function decreasing with @xmath120 of case c , the portion of emp binaries whose low - mass members survive to date depends only weakly on the primary mass ( @xmath136 ) so that the fraction of cemp-@xmath7 stars reduces because of larger contributions from the binaries with more massive primaries . as seen in fig . [ fig : psi ] , the fraction of cemp-@xmath7 stars in the total emp survivors is well below the upper bound of the observations , and hence , the @xmath44 compatible with the observations merges into one narrower range of @xmath137 for @xmath100 . the observed ratio of cemp - no@xmath7 to cemp-@xmath7 stars can be reproduced also by the imfs with a smaller @xmath44 by a factor of @xmath127 than in case a ( @xmath138 ) . accordingly , the @xmath44 for the imfs consistent with the both cemp star statistics are smaller by a factor of @xmath139 than for case a ( the mass range @xmath140 for @xmath141 ) . in the fig . [ fig : s ] , the range of @xmath44 for the imfs , compatible with the observed cemp-@xmath7 fraction varies only little with @xmath45 , and is restricted in the range between @xmath142 , though it separates into two for small @xmath143 . as shown in fig . [ fig : nos ] the dependence of the ratio of cemp - no@xmath7 to cemp-@xmath7 stars on @xmath45 is also weaker than for case a. consequently , the imfs can reproduce the both cemp star statistics with the mass as small as @xmath144 , smaller by a factor of @xmath145 than for case a , but differently from the above two cases , an upper bound is placed at @xmath146 , regardless of the dispersion with a lower bound of @xmath147 . bottom right panels depict the results for the independent " coupling of case d. for this case , the number of emp survivors produced per binary is independent of the mass , @xmath19 , of primary stars , while the binary frequency itself increases with @xmath19 . the former is similarly to case c , and then , the production of emp survivors from the binaries with massive primary poses a severe constraint on the high - mass side of imfs . on the other hand , the latter favors the production of cemp-@xmath7 stars as compared with the low - mass binaries of @xmath148 . the both shift the imfs , compatible with the observed fraction of cemp-@xmath7 stars , to smaller @xmath44 . in addition , the single stars , born in the same number of binaries , contribute to significant fraction of emp survivors , increasing from @xmath149 up to @xmath150 for smaller @xmath44 for @xmath48 since the low - mass binaries are counted as one object . as a result , the maximum fraction of cemp-@xmath7 stars remains below the upper limit of the observed range , which makes the @xmath44 for the imfs that can reproduce the observation lie in a single range within a relatively small upper bound . the observed ratio of cemp - no@xmath7 to cemp-@xmath7 stars demands also lower - mass imfs , as for case c. accordingly , the imfs that can reproduce the both cemp star statistics fall in the narrowest range of @xmath151 with rather small upper mass limit , almost irrespective of the dispersion , on the @xmath152 diagram in fig.[fig : nos ] . the cemp-@xmath7 star faction remains smaller than @xmath153 because of the contribution of the stars born as single . in conclusion , the statistics of cemp stars demand the imfs for the emp population , peaking at the intermediate - mass stars or the massive stars , by far higher mass than those of pop . i and ii stars , irrespectively of the assumed mass - ratio distribution . the presence of cemp - no@xmath7 stars in a significant number of the cemp-@xmath7 stars excludes the imfs of small mass . the derived mass range varies by a factor of @xmath127 , from the highest @xmath131 for the mass - ratio distribution of increase function of the mass ratio ( case b ) to the lowest @xmath154 for the mass - ratio distribution of " independent coupling ( case d ) . this tendency is explained in terms of the difference in the averaged mass of the primary companion of the emp survivors ; if the contributions to the emp survivors decrease rapidly with the mass of primaries , a relatively higher - mass imfs result without an upper mass limit imposed , while if the contribution to the emp survivors are weakly dependent or independent on the primary masses , an upper limit is set with the relatively smaller - mass on the imfs . dditional constraints can be derived the total number of emp survivors and the iron yields from the emp population . figure [ cefig ] shows the contours of the total iron mass , @xmath73 , produced by the massive stars of emp population , the total mass and the amount of iron production of emp population are sensitive both to @xmath44 and @xmath45 , especially for small @xmath45 and large @xmath44 in contrast to with the other cases . for small @xmath45 , therefore , the fraction of low - mass stars varies greatly with @xmath44 , and the both contours of @xmath65 and @xmath155 converge to @xmath156 . as @xmath45 increases , the differences from case a diminish since the imfs tend to extend into the low - mass stars , and in particular , for @xmath157 and @xmath158 , the contours in the both panels resemble each other to run through the similar parameter spaces . from the comparison with the total amount of iron @xmath159 , necessary for the chemical evolution , in this diagram , the parameter space where @xmath160 is excluded by the overproduction of iron or by the underproduction of emp survivors . for the parameter space where @xmath161 , on the other hand , the stars of emp population can leave the number of emp survivors currently observed but are short of iron production , so that the chemical evolution demands other sources of iron production without producing the low - mass stars that survive to date . for a flat mass - ratio distribution , the imfs that can satisfy the condition of iron production coincide the imfs , derived above from the statistics of cemp stars ( shaded area ) in the parameter range of @xmath162 and @xmath163 . for the independent " coupling , the parameter range of imfs that satisfy the condition of iron production also overlap the shaded area of parameter range , derived above from the statistics of cemp stars , but with the mass @xmath164 , slightly smaller than for case a and only for a small dispersion of @xmath165 . for larger @xmath45 , even the highest - mass imfs of @xmath166 result to be slightly short of , or marginally sufficient at the most , iron production . for the two other mass - ratio distributions of @xmath167 ( case b ) and @xmath168 ( case c ) , the iron production , @xmath73 , with a given imf results to be larger or smaller than for case a because of the difference in the number of massive stars exploded as supernova per low - mass survivor ( e.g. , by a factors of 1.38 and 0.47 , respectively , per a star of @xmath169 and the imf of @xmath170 and @xmath171 ) . the iron production then demands smaller - mass ( higher - mass ) imfs for case b ( case c ) as compared with case a , the shift of imfs in an opposite direction , discussed from the statistics of cemp stars . accordingly , for these two extreme cases , the parameter ranges for the imfs derived from the statistics of cemp stars and the iron production are marginally overlapped ( case c ) or are dislocated with a narrow gap ( case b ) , although a definite conclusion waits for future observations , in particular , to improve the estimate of total numbers of emp stars ( see appendix ) . the relative production rate of carbon to iron may also impose additional constraint since the intermediate - mass stars enrich intergalactic matter with carbon through the mass loss on the agb , as discussed by @xcite . in particular , when @xmath45 is small and @xmath44 is in the range of intermediate- and low - masses , the intermediate - mass stars much surpass the massive stars in number and eject more carbon than the latter eject iron . we compute the amount of carbon ejected by agb stars by taking the carbon abundance in the wind ejecta of agb stars at @xmath172 , and the remnant mass at @xmath173 . contours of @xmath174 are plotted in the figure ( dashed lines ) , for which only the carbon from the agb stars are taken into account . the overabundance of carbon excludes the imfs with low dispersion and low medium mass ; it excludes the parameter space in the range of @xmath175 , derived by the cemp star statistics for case d , but has nothing to do with the high - mass imfs derived for case a. we demonstrate that the imfs , derived from the observed properties of cemp stars , have the parameter ranges that can explain the chemical evolution and the production of low - mass stars , consistent with the observations , both for the flat mass - ratio distribution and for the independent " coupling . in appendix we will discuss the converse to demonstrate that the argument based on the total number of emp survivors and the total iron production can potentially provides more stringent constraint on imfs , independent of the argument based on the cemp star statistics . in this paper , therefore , we the different assumptions on the mass - ratio distributions admit the parameter ranges of high - mass imfs that can reproduce the statistics of cemp stars and the chemical evolution , consistent with the existent observations . the predicted mass ranges differ by a factor of 2 or more between @xmath176 . although hardly distinguishable from the observations discussed so far , they surely make the differences in the properties of emp survivors . we discuss the imprints that the mass - ratio distributions have left on the current emp survivors and investigate the possibility of discriminating the mass coupling of binary systems in the emp population , especially for the two distinct distributions of the flat mass - ratio distribution and the `` independent '' coupling . firstly , an obvious difference is the mass distribution function of emp survivors . for a given imf , @xmath177 , the mass distribution , @xmath178 , of emp survivors is given by ; @xmath179 here a low - mass binary , whose components are both less massive than @xmath180 , is counted as one object with the primary star . figure [ fig : survivors ] shows the mass distributions of emp survivors ( @xmath181 for different assumptions of mass - ratio distributions cases a - c . for these mass - ratio functions , the mass distribution of emp survivors is nearly proportional to the mass - ratio distribution @xmath37 because almost all of them come from the secondary stars ; the contribution from the primary components are denoted by thin solid line , and the same contribution comes from the stars born as single . for the independent " coupling , in contrast , the @xmath178 , has the same form as the imf and the number of emp survivors decreases rapidly as the stellar mass decreases . secondly , the fraction of double - lined binary and the contribution of stars born as single among emp survivors may differ according to the mass - ratio distribution . for the flat mass - ratio distribution , this gives a significant fraction of @xmath182 for @xmath183 and @xmath171 , and increases with @xmath45 to @xmath184 for @xmath185 and with decreasing @xmath44 to 18% for @xmath186 , respectively . the number of low - mass binary decreases rapidly for smaller masses while the number of emp survivors , formed as the low - mass members of white dwarf binaries or supernova binaries , remains constant . for the independent " coupling , the fraction of low - mass binaries in the emp survivors reduces to ; @xmath187 , \end{aligned}\ ] ] which gives a much smaller fraction of @xmath188 for @xmath186 and @xmath189 as compared with the flat mass - ratio distribution . the fraction may increase for smaller medium mass , to 5.5% at @xmath190 , and for larger dispersion , to 3.9 % and 9.5% at @xmath191 and 0.7 , respectively , although these may cause underproduction of iron , in particular for smaller @xmath44 , as seen from fig . [ cefig ] ( bottom panel ) . in this case , the proportion of the emp survivors , born as single stars , is fairly large as given by @xmath192.\ ] ] consequently , nearly one third of emp stars were born as single stars , for @xmath193 , which is much larger fraction than in the case of the flat mass - ratio distribution . thirdly , the fraction , @xmath194 , of supernova binaries with the primary stars of mass @xmath195 also differs between the two mass - ratio distributions . for the flat mass - ratio distribution , almost all of the emp survivors belong , or have been belonged , to the binary systems , and the fraction is given by @xmath196 and amounts to @xmath197 . for the independent " coupling , on the other hand , one third of emp survivors are single stars from their birth , and the percentage of supernovae binaries is relatively small , as given by @xmath198,\end{aligned}\ ] ] and turns out to be @xmath153 . the emp survivors from the supernova binaries have experienced a supernova explosion of the erstwhile primary stars at close distances and are thought to suffer from some abundance anomalies , affected by supernova ejecta . accordingly , these stars , in particular from the binaries of sufficiently small separations , may be discriminated by a large enhancement of elements , characteristic to the supernova yields . these differences in the properties of remnant emp survivors may potentially serve as tools to inquire into the nature of emp binaries and to distinguish the mass - ratio distributions . among the emp stars , several double - lined spectroscopic binaries are reported in the literature . if we restricted to the metallicity range of @xmath0}<-3 $ ] , for which the observations with high - resolution spectroscopy may be regarded as unbiased , there are two stars cs22876 - 032 and cs 22873 - 139 with the detailed analyses and one star he 1353 - 2735 ( @xmath0}\simeq -3.2 $ ] , @xmath199 ; * ? ? ? * ) without the binary parameter . so far 39 dwarf stars of @xmath0 } < -3 $ ] are confirmed by the high - resolution spectroscopy ( we define the dwarf as @xmath200 \ge 3.5 $ ] ) , and hence , the fraction of low - mass binaries , composed of two unevolved emp stars , turns out to be @xmath201 . it may be more straightforward to compare our results with the mass distribution function of emp survivors . from the existent observations , however , it is rather hard to determine since the observed dwarfs are mostly concentrated near to the upper end of main sequence . an exception is a carbon dwarf g77 - 61 of @xmath0}= 4.03 $ ] @xcite whose mass is inferred at @xmath202 , but it was found among the proper - motion - parallax stars @xcite , not from the surveys . we have to wait for the larger - scaled surveys in near future to reveal the distribution of emp survivors of low masses . as for the supernovae binaries , they are expected to be related to the large star - to - star variations in the surface elemental abundances , in particular , with those of r - process elements , ranging more than by two orders of magnitude . it is necessary , however , to understand the nature of interactions between the supernova ejecta colliding at very high velocity and the near - by low - mass stars before the meaningful conclusions can be drawn from the observations . we have shown that the high - mass imfs with the binary provide a reasonable explanation of the observed properties of emp stars in the galactic halo , revealed by the recent large - scaled hk and he s surveys . in this section we discuss the consequence of derived imf on the metal enrichment history of galactic halo up to @xmath0}=-2.5 $ ] to study their relevance to the metallicity distribution function ( mdf ) , observed for the emp stars . under the assumption that matter ejected from supernovae spreads homogeneously and is recycled instantaneously , the iron abundance , @xmath203 , of our galaxy of ( baryonic ) mass @xmath204 can simply be related to the cumulative number , @xmath205 , of the stars born before the metallicity reaches @xmath203 as ; @xmath206 where @xmath77 is the averaged iron yield per supernova and @xmath207 is the fraction of emp stars that have exploded as supernovae , defined in eq . ( [ eq : frac - sn ] ) . by differentiating it with respect to @xmath0}= \log ( x_{\rm fe}/ x_{\rm fe , \odot})$ ] , the number distribution of emp survivors is written as a function of metallicity in the form @xmath208 } ) = \frac { d n ( x_{\rm fe } ) } { d{[{\rm fe } / { \rm h } ] } } = \frac{m_h } { \langle y_{\rm fe } \rangle f_{\rm sn } } \ln(10 ) x_{{\rm fe}\odot } 10^{{[{\rm fe } / { \rm h } ] } } .\ ] ] this shows that the number distribution of emp survivors is simply proportional to the iron abundance apart from the variation of @xmath209 through the imf and the latter is small enough to be neglected for @xmath210 ( see fig . [ fig : yield ] in appendix ) . figure [ mdf ] depicts the number distribution of emp survivors and compares it with the observed mdf provided by the he s survey @xcite . we assume stars of mass @xmath211 become type ii supernova and eject @xmath212 of iron . in this figure , the theoretical mdf , @xmath213 , is evaluated under the same flux - limited condition as the observed mdf is derived ; @xmath214 } ) = n ( { [ { \rm fe } / { \rm h } ] } ) f_{\rm g } \times ( 40 \% ) \times ( 8225 \hbox { degree}^2 / 4 \pi \hbox { sr } ) \times 1.93 $ ] . here the fraction of follow - up observation and the sky coverage are taken into account : as for the contribution of to stars , we take the same ratio to the giants as in the observed sample under the assumption that the giant survivors are all reached in the survey area . solid line shows the mdf for the imf with @xmath215 and @xmath171 with the 50% binary fraction , derived above for emp population stars for the flat mass - ratio distribution , and it is similar to the other mass - ratio functions , as discussed in 2.3 . this reasonably reproduces the observed mdf between the metallicity @xmath3}\lesssim -2.5 $ ] , as expected from the discussion in the previous section . in this figure , we also plot the mdf using the low - mass imfs , the salpeter s power - law mass - function as observed among the present - day stellar populations and that derived only from the statistics of cemp-@xmath7 stars by ( * ? ? ? * @xmath216 and @xmath217 ) . they bring about the overproduction of emp survivors by a factor of more than a few hundreds not only from the low - mass members of binaries but also from the primary stars and the single stars ; both the imfs give the similar mdf since our flux - limited samples are dominated by the giants and luminous dwarfs of mass @xmath218 . this means that the emp survivors is by far a small population as compared with the stellar systems of pop . i and ii , and it is only with the high - mass imfs that can make the emp population produce sufficient amount of metals to enrich the early galactic halo without leaving too many low - mass survivors now observable in galactic halo . in addition , we see in this figure that the slope of observed mdf is consistent with the prediction from the simple one - zone approximation at least for @xmath0 } > -4 $ ] . it implies that the imfs have little changed while the galactic halo has evolved through these metallicities . beyond @xmath0}\simeq -2 $ ] , the observed mdf derived from the hk and he s surveys seems to be underestimated since those objects are out of the metallicity range sought after by the survey and subject to imperfect selection . the observed mdf of galactic halo stars has a sudden drop at @xmath219}\lesssim -4 $ ] , and only three stars are found below it}<-4 $ ] ; cd @xmath220 with @xmath0}= -4.19 \pm 0.10 $ ] @xcite and g77 - 61 with @xmath0}= -4.03 \pm 0.1 $ ] @xcite . we will omit these two stars in our discussion since larger abundances of @xmath0}= -4.07 \pm 0.15 $ ] @xcite and @xmath221 @xcite have been reported for the former , and hence , their abundances are closer to the emp stars of @xmath0}\gtrsim -4 $ ] than to the other three hmp / ump stars . ] . we propose the mechanism responsible for this depression of low - metallicity stars from the consideration of the galaxy formation process . in the current cold dark matter ( cmd ) model , galaxies were formed hierarchically . they started from low mass structures and grew in mass through merging and accreting matter , finally to be large - scale structures like our galaxy . in the hierarchical structure formation scenario with @xmath222cdm cosmology , the typical mass of first star forming halos is @xmath223 for the dark matter and @xmath224 for the gas ( e.g. , see * ? ? ? * ; * ? ? ? * ) . in these first collapsed gas clouds , the first stars contain no pristine metals except for lithium . when the first star explodes as supernova , it ejects @xmath225 of iron , which enriches the gas cloud of mass @xmath224 where it was born up to the metallicity of @xmath0}\sim -3.5 $ ] if the ejecta is well mixed in the gas cloud . we call this event the first pollution " . consequently , the 2nd generation stars have the metallicity of @xmath0}\sim -3.5 $ ] . in the course of time , the mini - halos that host the gas clouds merge with each other and accrete the intergalactic gas to form early galactic halo with the baryonic mass of @xmath226 . we may take the metallicity of this early galactic halo to be @xmath0}\simeq -4 $ ] because of the scarcity of stars of metallicity @xmath0}<-4 $ ] . the cumulative number of stars born before the early galactic halo is enriched up to @xmath0}=-4 $ ] is estimated at @xmath227 with taking into account the supernova fraction @xmath207 . if the mini - halos of larger masses stand between the first collapsed halos and the galactic halo , the dilution of iron with unpolluted primordial gas can give birth to the stars of smaller metallicity of @xmath0}\simeq -4 $ ] , and then , the metallicity at the formation of galactic halo can be larger to increase the cumulative number of stars in accordance ( see below ) . we may estimate the fractions of both the first generation stars without metals and the 2nd generation stars of the metallicity @xmath0}\sim -3.5 $ ] , respectively , assuming that stars are born with an equal probability whether in the gas clouds , polluted with metals , or in the primordial gas clouds . accumulated number , @xmath228 , of pop iii stars , born of gas unpolluted by sn ejecta , when the average metallicity reaches @xmath203 , is given by @xmath229 , \label{eq : num - popiii}\ ] ] where @xmath230 is the mass of gas in the first star forming clouds . if we assume the same imf and binary parameters as in the stars of emp population , then , we expect that the number of pop iii stars is @xmath231 and the number of pop iii survivors is @xmath232 \nonumber \\ & & = 1.3 \times 10 ^ 4 , \end{aligned}\ ] ] and similarly we have @xmath233 and @xmath234 of the 2nd generation stars and their survivors , formed before the averaged metallicity of the galaxy reaches @xmath0}=-4 $ ] . the imf of pop . iii stars may differ from emp stars but the existence of the stars with @xmath0}<-5 $ ] suggests that the low - mass stars can be formed before the first pollution . figure [ mdfpop3 ] illustrates an expected mdf with the hierarchical structure formation . after the formation of large galactic halo , the metal enrichment process is thought to follow the argument of the previous subsection . thus , we can explain the cutoff around @xmath0}\sim -4 $ ] naturally . shaded columns indicate the initial distributions of pop . iii stars and of the 2nd generation stars formed in the low - mass clouds . the 2nd stars were mixed and observationally lost their identities among the stars formed in the merged halo . on the other hand , pop . iii stars should form a distinctive class . from the above estimate , we expect @xmath235 pop . iii survivors in the existing flux - limited samples of he s surveys . similarly , the number of second - generation of stars is estimated at @xmath236 in the same flux - limited he s sample , indicative that most of emp stars are formed of mixture of the ejecta from plural supernovae . this has direct relevance to the study of the nucleosynthetic signatures on the emp survivors and the imprints of supernovae of the first and subsequent generations . we have studied the initial mass function ( imf ) and chemical evolution of the galactic halo population on the basis of the characteristics of extremely metal - poor ( emp ) stars , revealed by the recent large - scaled hk and he s surveys ; the observational facts that we make use of are ; ( 1 ) the overabundance of carbon - enhanced emp ( cemp ) stars , ( 2 ) the relative frequencies of cemp stars with and without the enrichment of s - process elements , ( 3 ) the estimate of surface density or total number of emp stars in the galactic halo , and ( 4 ) the metallicity distribution function ( mdf ) . we take into account the contribution of binary stars properly , as expected from the younger populations . in paper i , the high mass imf peaking around @xmath237 is derived for the stars of emp population and it is shown that the binary population plays a major role in producing the low - mass stars that survive to date , but by using the flat mass - ratio distribution between the component stars . in this paper , we examine these properties of the stars of emp population and emp survivors for the different types of mass - ratio distributions and investigate the constraints on the imfs of the stars of emp population and discuss the observational tests of discriminating them . the derived imfs are applied to understand the characteristics of mdf and the nature of emp stars including hmp / ump stars , provided by the surveys . our main conclusions are summarized as follows ; \(1 ) the statistics of cemp stars are explained by the high - mass imfs with the binaries of significant fraction . predicted typical mass is significantly larger than population i or ii stars , ( 2 ) \(3 ) the mass - ratio distribution of binaries in the emp population can be discriminated by the imprints left on the emp survivors such as the mass function , the binary fraction , and the fraction of stars influenced by the supernova explosion of primary stars . \(4 ) the observed mdf of emp survivors is consequent upon the derived imf with the contribution of the binaries . there is no indication of significant change in the imfs between the metallicity of @xmath3}\lesssim -2 $ ] . the depression of stars below @xmath0 } < -4 $ ] is naturally explicable within the current favored framework of the hierarchical structure formation model . then , the pop . iii stars born of primordial gas , and also , the stars in the primordial clouds before they are contaminated by their own supernovae , should form the distinct class other than emp stars , and may have the relevance to hmp and ump stars observed at lower metallicity , as discussed below in this subsection . the feature of our approach is to take into account the stars born in binary systems properly in discussing the low - mass star formation in early universe , based on the finding in paper i. in addition , we make full use of available information from the existent large - scaled surveys and to draw the maximal constraint on the early evolution of our galactic halo . the known emp stars ( @xmath0}\lesssim -2.5 $ ] ) with the detailed stellar parameters amount to @xmath238 in number ( saga database ; * ? ? ? * ) , and allow us to discuss the averaged properties as studied in this paper . discussion in 2.3 through the iron production consistent with the number of emp survivors will be left largely unaffected even quantitatively . in order to improve and sharpen our conclusions , we have to wait for the future larger - scaled surveys such as sdss / segue @xcite and lamost @xcite . also the high dispersion spectroscopy is necessary to understand the characteristics of emp stars . the constraints on the imfs derived in this work may serve as the basis of understanding the formation and early evolution of the galaxy . accordingly , there should be the transition from the high - mass imf to the low - mass one . our result suggests that the transition is postponed until high metallicity even beyond @xmath0}\simeq -2 $ ] is reached . it is likely that the transition may not be simply determined by the metallicity alone , in discussing the primordial stars or hmp / ump stars in the present work , we assume the metallicity at the formation of galactic halo at @xmath0}\simeq -4 $ ] . the detailed chemical evolution with the merger history taken into account is discussed in a subsequent paper ( komiya et al . 2008 , in preparation ) , in the similar ways as done by @xcite and by @xcite , but taking into account the high - mass imf , derived above , and the contribution of binaries . we end by discussing the consequences of the present study on the understanding of the origin of stars found below the cut - off of mdf . in our model , the stars made after the first pollution have the metallicity @xmath0}\simeq -3.5 $ ] and the stars with slightly lower metallicity of @xmath0}\simeq -3.5 - -4 $ ] are made in the merged clouds where metals are diluted with the primordial gas unpolluted by supernova ejecta . after the halos merge , the 2nd generation stars mingle and observationally lose their identities among the stars formed in the merged halo . @xcite argue the effects of surface pollution of pop . iii stars through the accretion of interstellar gas to show that the main - sequence pop . iii stars can be polluted to be @xmath0}\simeq -3 $ ] while the giants to be @xmath0}\simeq -5 $ ] since the pollutant is diluted by the surface convection deepening @xmath239 times in mass on the giant branch . thus , the pop . iii survivors have evolved to giants to be observed as hmp / ump stars . then some of pop . iii stars become carbon - enriched hmp / ump stars with @xmath0}\sim -5 $ ] through binary mass transfer . if the mass of primary star is @xmath240 and @xmath241 , the primary star enhances the surface abundances of carbon and nitrogen though the he - fddm and of carbon and/or nitrogen through tdu and hot bottom burning in the envelope , respectively , which are transferred onto the secondary stars through the wind accretion . it is to be noted that the primary stars of @xmath242 have the accreted pollutants mixed inward into the whole hydrogen - rich envelope at the second dredge - up , and thereafter , evolve like the stars with the pristine metals . at the same time , the accreted matter is diluted in the envelope and the iron abundance is reduced to @xmath0}\sim -5 $ ] in the primary stars . we estimate that @xmath243 of pop . iii stars become carbon - rich hmp / ump stars under the same assumptions on the binary parameters as in paper i. in fig . [ mdfpop3 ] , solid lines denote the expected mdf at the present days with the surface pollution taken into account . the basic form of observed mdf is reproduced , i.e. , the cutoff around @xmath0}\sim -4 $ ] , the scarcity of stars for the metallicity below it and the existence of a few hmp / ump stars . from the above estimates , there should be @xmath235 pop . iii stars in the existent flux - limited samples of he s surveys ; about a half of them may be discovered as giants with the surface metal pollution and one third as carbon stars . the above estimates are made , however , under the assumption that the pop . iii stars are formed in the same imf as emp stars and with the same binary parameters . this may not be warranted and rather we may take that this deficiency may suggest a still higher - mass imf and/or less efficiency of binary formation for pop . iii stars than the emp stars . in the above discussion , we assume the closed box chemistry in the collapsed object before merging . it is shown that the hypernovae , exploded with a large energy of @xmath244 erg , blow off the first collapsed objects of mass @xmath245 @xcite ; if the first stars are sufficiently massive , the metal yields are spread into larger masses , and pollute the ambient gas before they collapse to form mini - haloes , as discussed by @xcite . after that , the first stars in the collapsed clouds are no longer metal - free . nevertheless , those stars which are formed before each collapsed clouds are polluted by their own supernova form a distinct class from those which suffer from the first pollution . further study is necessary to make clear the present appearance of the possible pop iii survivors and to settle the origin of hmp / ump stars , in particular , for tiny amounts of iron - group metals and the overwhelming carbon - enhancement , shared by all these stars known to date . we benefit greatly from discussion with dr . w. aoki . this paper is supported in part by grant - in - aid for scientific research from japan society for the promotion of science ( grant 18104003 and 18072001 ) . one of the important findings of the recent large - scaled surveys is the scarcity of emp stars in the galactic halo . the he s survey gives the total number of emp stars in our galactic halo at @xmath246 ( giants of @xmath247 in eq . ( [ eq : hes - g ] ) plus turn - off stars @xmath248 within the limiting magnitude @xmath249 . similarly , the hk survey gives @xmath250 within the limiting magnitude of @xmath251 ; 114 stars of @xmath0 } < -3 $ ] are found by the medium - resolution , follow - up spectroscopy of 50% of the candidates , selected from the objective prism survey covering the @xmath252 and @xmath253 areas in the northern and southern hemisphere @xcite . because of the significantly large areas covered by these surveys ( @xmath254 of all sky with the follow - up observations ) , we may place reliance on these results , granted that they may not be complete . this also constrains on the imf of stellar population that promoted the chemical evolution , or more specifically , the formation of metals and the low - mass survivors . in the paper , we have discussed the chemical evolution starting with the statistics of cemp stars . in this appendix , we show that the chemical evolution with the total number of emp survivors provides more stringent constraints on the imf of emp population with the aid of the amount of ejecta from supernova models , independently of the statistics of cemp stars . our basic premise is that the same stellar population is responsible both for the production of metals and of low - mass survivors . in discussing the low - mass survivors , it is indispensable to take into account the contribution from the binaries . this is one of the major conclusions in paper i. we assume that the stars are born not only as single stars but also as the members of binaries in an equal number and with the primary stars in the same imf as the single stars . for a given imf , then , the total number , @xmath255 of emp survivors , currently observed in the galactic halo , is related to the cumulative number , @xmath256 , of stars of emp population as ; @xmath257 , \label{eq : frac - surv}\end{aligned}\ ] ] and hence , to the cumulative number of emp supernovae as @xmath258 . these supernovae have to supply the amount of iron , @xmath73 in eq . ( [ eq : emp - ironprod ] ) , in order to enrich the gas in the galaxy of mass @xmath204 with iron to promote the chemical evolution up to the metallicity @xmath0}= -2.5 $ ] . then , we may derive the averaged iron yield , @xmath259 , per supernova of emp population , necessary to explain the chemical evolution of galaxy , by the relation @xmath260 for an assumed imf with the mass - ratio distribution function . we show in figure [ fig : yield ] the averaged yield , @xmath259 , as a function of @xmath44 for @xmath189 : upper panel for the observations of emp stars of different evolutionary stages from the he s survey and of the total emp stars from the hk survey with use of the imfs with the flat mass - ratio function , and lower panel for the different mass - ratio functions with use of the observation of emp giants from the he s survey . in order to compare the stars of different evolutionary stages , we include the effects of the limiting magnitude of the surveys by assuming the de vaucouleurs density distribution , @xmath261 , with the radial distance , @xmath262 , from the galactic center , the same as the stars in the galactic halo and by assigning the luminosity of @xmath263 and @xmath264 to dwarfs and giants , respectively . the amount of iron demanded by the chemical evolution turns out to be a steep decrease function of @xmath44 since in order to leave a fixed number of low - mass survivors , the total number of stars of emp populations , and hence , the supernova fraction increase rapidly with @xmath44 in particular near @xmath265 . the necessary yields computed from the different samples in upper panel show a fairly good agreement with each other . the difference between the giants and dwarfs for the he s samples is indicative of a relatively deficiency of dwarf stars compared with giants by a factor of @xmath266 in number , which may be attributed to rather crude assignment of averaged giant luminosity , and/or to the different efficiency of identifying giants and turn - off stars in the survey plates , and/or to the uncertainties in the spatial density distribution . the results for the hk survey and the he s survey also agree within the difference by a factor of @xmath267 in number despite the difference in the limiting flux by 2 mag , and hence , to the difference in the searched volume by a factor of @xmath268 . the variations with the mass - ratio functions in the lower panel are caused by the difference in the number of supernovae per emp survivor . as compared to the flat mass - ratio function , the mass - ratio function increasing ( decreasing ) with @xmath120 give a larger ( smaller ) number of supernovae to produce one emp survivors ; the difference of which increases for higher - mass imfs . these iron yields necessary to promote the chemical evolution may be compared with the theoretical iron yields predicted from the supernova models . the imf - weighted iron yields , @xmath269 , per supernova is given by using the iron mass , @xmath270 , ejected from a massive star of initial mass @xmath271 as ; @xmath272 } { \int_{m_{up } } dm_1 \xi ( m_1 ) [ 1 + f_b \int_{m_{up}/m_1}^1 n(q ) d q ] } . \label{eq : yieldsn}\ ] ] the imf averaged yield @xmath269 is also shown in this figure , for which the theoretical yields are taken from the metal - deficient supernova models computed by @xcite , and by @xcite and @xcite . it is a slowly increase function of @xmath44 for @xmath210 with the increase in the fraction of more massive stars that ended as supernovae , while beyond it , the gradient grows steeper owing to the contribution of the electron pair - instability supernovae of @xmath273 . the averaged yields , demanded by the chemical evolution , and the theoretical imf - weighted iron yields both meet with each other near @xmath274 and with the iron yield @xmath275 per supernova . as typically seen for the flat mass - ratio distribution , the parameter range coincides with that we have derived for the imfs from the cemp statistics in figs . [ fig : psi]-[fig : nos ] . for higher - mass imfs , the emp stellar population can not produce the sufficient number of low - mass survivors by themselves , while for lower - mass imfs , it results short of iron production . the differences arising from the mass - ratio distributions seem discernible but not large enough to differentiate these mass - ratio distributions in view of the uncertainties of current observations . as compared with the flat distribution , the mass - ratio distribution increasing ( decreasing ) with @xmath120 demands smaller ( larger ) @xmath44 , the opposite tendency derived from the cemp statistics . these distributions prefer smaller ( larger ) number of emp survivors , larger ( smaller ) fraction of cemp-@xmath7 stars and smaller ( larger ) ratio of cemp - no@xmath7 to cemp-@xmath7 stars . in principle , however , we can discriminate the mass - ratio functions in the emp binaries , including those destructed already by the evolution , with use of the survey and observations of emp stars in sufficiently large number and with sufficient accuracy , which waits for future works . in the above discussion , we assume a single log - normal imf with the binary fraction for the stellar population . it is possible to assume the bi - modal imf and to explain the production of iron and the formation of low - mass stars , separately , in terms of the combination of two stellar populations , one with a higher - mass imf responsible for the iron production and the other with a lower - mass imf for the low - mass survivors , respectively . in the case of bi - modal imfs , the constraints , derived here , place an upper mass limit to the imf of lower - mass population and an lower mass limit to the imf of higher - mass population . it is to be noted that the imf with the binary mass function of cases a - c is regarded as a sort of bi - modal imf with the primary plus single stars as the higher - mass population and the secondary stars as the lower - mass population ( see fig . 12 in paper i ) ; the separation of two imfs differs with the mass - ratio function and the relative contributions of two populations vary with the binary fraction . in any case , as for the emp stars in the galactic halo , the statistics of cemp stars , in particular , the ratio of cemp - no@xmath7 and cemp-@xmath7 stars , place the lower mass limit to the imf of the lower - mass population , and hence , endorses the high - mass imf , which narrows , if any , the contribution of the higher - mass population . abia , c. , domnguez , i. , straniero , o. , limongi , m. , chieffi , a. , & isern , j. 2001 , , 557 , 126 abt , h. a. 2008 , , 135 , 722 adams , f. c. , & laughlin , g. 1996 , , 468 , 586 aoki , w. , beers , t.c . , christlieb , n. , norris , j. e. , ryan , s.g . , & tsangarides , s. , 655 , 492 bate , m. r. , & bonnell , i. a. 1997 , , 285 , 33 beers , t. c. , preston , g. w. , & shectman , s. a. 1992 , , 103 , 1987 beers , t. c. 1999 , asp conf . ser . 165 : stromlo workshop on galactic halo , 165 , 202 beers , t.c . , allende prieto , c. , wilhelm , r. , yanny , b. , & newberg , h. 2004 , pasp , 21 , 207 beers , t.c . & christlieb , n. 2005 , ara&a , 43 , 451 beers , t.c . & christlieb , n. 2005 , ara&a , 43 , 451 beers , t. c. , christlieb , n. , norris , j. e. , bessell , m. s. , wilhelm , r. , allende p. a. , yanny , b. , rockosi , c. , newberg , h. j. , rossi , s. , & lee , y. s. 2005 , in from lithium to uranium : elemental tracers of early cosmic evolution , iau symp . 228 , held in paris , france , pp.175 carney , b. w. , latham , d. w. , stefanik , r. p. , laird , j. b. , morse & j. a. 2003 , , 125 , 293 cassisi , s. & castellani , v. 1993 , , 88 , 509 catelan , m. de freitas pacheco , j.a . & horvath , j.e . 1996 , , 461 , 231 cayrel , r. , depagne , e. , spite , m. , hill , v. , spite , f. , franois , p. , plez , b. , beers , t. , primas , f. , andersen , j. , barbuy , b. , bonifacio , p. , molaro , p. , & nordstrm , b. 2004 , , 416 , 1117 chabrier , g. 2003 , , 115 , 763 chabrier , g. , segretain , l. , & mra 1996 , , 468 , l21 christlieb , n. , green , p. j. , wisotzki , l. , & reimers , d. 2001 , , 375 , 366 christlieb , n. , bessell , m. s. , beers , t. c. , gustafsson , b. , korn , a. , barklem , p. s. karlsson , t. , mizuno - wiedner , m. & rossi , s. 2002 , , 419 , 904 christlieb , n. 2003 , rev . 16 , 191 clark , p. , bonnel , i.a . , klessen , r. 2008 , , 383 , 3 cohen , j. g. , shectman , s. , thompson , i. , mcwilliam , a. , christlieb , n. , melendez , j. , zickgraf , f .- j . , ramirez , s. , swenson , a. , 2005 , , 633 , l109 dahn , c.c . , liebert , j. , kron , r. g. , spinrad , h. , & hintzen , p. m. 1977 , , 216 , 757 depagne , e. , hill , v. , christlieb , n. , & primas , f. 2000 , , 364 , l6 duquennoy , a. , & mayor , m. 1991 , , 248 , 485 elmergreen , b.g . 2008 , 681 , 365 franois , p. , depagne , e. , hill , v. , spite , m. , spite , f. , plez , b. , beers , t. c. , barbuy , b. , cayrel , r. , andersen , j. , bonifacio , p. , molaro , p. , nordstrm , b. , & rimas , f. 2003 , 403 , 1105 frebel , a. , et al . 2005 , , 434 , 871 fujimoto , m. y. , iben , i. j. , & hollowell , d. 1990 , , 349 , 580 fujimoto , m. y. , ikeda , y. , & iben , i. jr . 2000 , , 529 , l25 goldberg , d. , mazeh , t. & latham , d. w. 2003 , , 591 , 397 gonzlez hernndez , j.i . , bonifacio , p. , ludwig , h .- g . , caffau , e. , spite , m. , spite , f. , cayrel , r. , molaro , p. , hill , v. , franois , p. , plez , b. , beers , t.c . ; sivarani , t. , andersen , j. , barbuy , b. , depagne , e. , nordstrm , b. , & primas , f. 2008 , , 480 , 233 goodwin , s. p. , kroupa , p. , goodman , a. , & burkert , a. 2007 , in protostars and planets , des . , v , b. reipurth , d. jewitt , and k. keil ( university of arizona press , tucson ) , p.133 heger , a. & woosley , s.e . 2002 , , 567 , 532 heger , a. & woosley , s.e . 2008 , aarxiv0803.3161 hollowell , d. , iben , i. j. , & fujimoto , m. y. 1990 , , 351 , 245 iwamoto , n. , kajino , t. , mathews , g.j . , fujimoto , m.y . , & aoki , w. 2004 , , 602 , 377 karlsson , t. 2005 , , 439 , 93 karlsson , t. 2006 , , 641 , l41 komiya , y. , suda , t. , minaguchi , h. , shigeyama , t. , aoki , w. , & fujimoto , m.y . 2007 , 658 , 367 latham , d. w. , stefanik , r. p. , torres , g. , davis , r. j. , mazeh , t. , carney , b. w. , laird , j. b. & morse , j. a. , 124 , 1144 lucatello , s. , gratton , r. g. , beers , t. c. , & carretta , e. 2005b , , 625 , 833 lucatello , s. , beers , t. c. , christlieb , n. , barklem , p.s . , rossi , s. , marsteller , b. , sivarani , t. , lee , y.s . 2006 , , 652 , l37 machida , m.n . , tomisaka , k. , nakamura , f. & fujimoto , m.y . 2005 , , 622 , 39 machida , m. n. 2008 , , 682 , l1 machida , m.n . , omukai , k. , matsumoto , t. , inutsuka , s. , , 677 , 813 majewski , s. r. 1993 , , 31 , 575 mayor , m. , duquennoy , a , halbwachs , j .- l . , & mermilliod , j .- c . 1992 , asp con . 32 , iau colloquium 135 , p.73 mcwilliam . a. 1997 , ara & a , 35 , 503 meynet , g. & maeder , a. 2002 , , 381 , l25 meynet , g. , ekstrm . s. & maeder , a. 2006 , , 447 , 623 nishimura , t. , aikawa , m. , suda , s. , fujimoto , m.y . 2008 , submitted to norris , j.e . , beers , t.c . & ryan , s.g . 2000 , , 540 , 456 norris , j. e. , ryan , s. g. , & beers , t. c. 2001 , , 561 , 1034 norris , j.e . , christlieb , n. , korn , a.j . , eriksson , k. , bessell , m.s . , beers , t.c . , wisotzki , l. , reimers , d. , 2007 , , 670 , 774 ochi , y. , sugimoto , k. & hanawa , t. 2005 , , 623 , 922 paczynski , b. 1971 , , 9 , 183 plez , b. , & cohen , j. g. 2005 , , 434 , 1117 prantzos , n. 2003 , , 404 , 211 preston , g.w . 1994 , 108 , 2267 preston , g.w . & sneden , c , 2000 , , 120 , 1014 rossi , s. c. f.,parameter range for the imfs that can give rise to the observed beers , t. c. , & sneden , c. 1999 , asp conf . ser . 165 : stromlo workshop on the galactic halo , 264 ryan , s. g. , & norris , j. e. 1991 , , 101 , 1085 ryan , s. g. , aoki , w. , norris , j. e. , & beers , t. c. 2005 , , 635 , 349 salvadori , s. , schneider , r. , & ferrara , a. 2007 , , 381 , 647 scannapieco , e. , & bildsten , l. 2005 , , 629 , 85 siess , l. 2007 , , 476 , 893 spergel , d. n. , bean , r. , dor , o. , nolta , m. r. , bennett , c. l. , dunkley , j. , hinshaw , g. , jarosik , n. , komatsu , e. , page , l. , peiris , h. v. , verde , l. , halpern , m. , hill , r. s. , kogut , a. , limon , m. , meyer , s. s. , odegard , n. , tucker , g. s. , weiland , j. l. , wollack , e. , & wright , e. l. 2007 , , 170 , 377 spite , m. , depagne , e. , nordstrm , b. , hill , v. , cayrel , r. , spite , f. , & beers , t. c. 2008 , , 360 , 1077 spite , m. et al . 2005 , , 430 , 655 stancliffe , r. j. , glebbeek , e. , izzard , r. g. & pols , o. r. 2007 , , 464 , 57 suda , t. , aikawa , m. , machida , m. n. , fujimoto , m. y. , & iben , i. jr . 2004 , , 611 , 476 suda , t. , & fujimoto , m. y. 2006 , , 643 , 897 suda , t. , & fujimoto , m. y. 2007 , aipc , 990 , 276 suda , t. , katsuta , y. , yamada , s. , suwa , t. , ishizuka , i. , komiya , y. , sorai , k. , aikawa , m. , & fujimoto , m.y . 2008 , , in press tegmark , m. , silk , j. , rees , m.j . , blanchard , a. , abel , t. , & palla , f. 1997 , , 474,1 thorburn , j.a . & beers , timothy c. 1993 , , 404 , l13 tominaga , n. , umeda , h. , & nomoto , k. , 660 , 516 tsujimoto , t. & shigeyama , t. 2001 , , 561 , l97 tumlinson , j. 2006 , , 641 , 1 tumlinson , j. 2007 , , 665 , 1361 tumlinson , j. 2007 , , 664 , l63 umeda , h. , & nomoto , k. 2002 , , 565 , 385 umeda , h. , & nomoto , k. 2003 , nature , 422 , 871 umeda , h. , & nomoto , k. , 619 , 427 wanajo , s. , nomoto , k. , iwamoto , n. , ishimaru , y. , & beers , t.c . 2006 , , 636 , 842 weiss , a. , cassisi , s. , schlattl , h. , & salaris , m. 2000 , , 533 , 413 woosley , s. e. , weaver , t. a. 1995 , , 101 , 18 zhao , g. , chen , y .- q . , liang , y .- c . , hou , j .- l . , chen , l. , zhang , h .- w . , & li , a .- g . 2006 , chjaa , 6 , 265

we exploit the recent observations of extremely metal - poor ( emp ) stars in the galactic halo and investigate the constraints on the initial mass function ( imf ) of the stellar population that left these low - mass survivors of @xmath0}\lesssim -2.5 $ ] and the chemical evolution that took part in . a high - mass nature of imf with the typical mass @xmath1 and the overwhelming contribution of low - mass members of binaries to the emp survivors are derived from the statistics of carbon - enriched emp stars with and without the enhancement of s - process elements ( komiya et al . 2007 , ) . that the same constraints are placed on the imf from the surface density of emp stars estimated from the surveys and the chemical evolution consistent with the metal yields of theoretical supernova models . apply the derived high - mass imf with the binary contribution metallicity distribution function ( mdf ) of emp stars not only for the shape but also for the number of emp stars . in particular , the scarcity of stars below @xmath0}\simeq -4 $ ] is naturally explained in terms of the hierarchical structure formation , and there is no indication of significant changes in the imf for the emp population . the present study indicates that 3 hmp / ump stars of @xmath0 } < -4 $ ] are the primordial stars that were born as the low - mass members of binaries before the host clouds were polluted by their own supernovae .

integration of the form @xmath6 , where @xmath1 is either @xmath2 or @xmath7 , is widely encountered in many engineering and scientific applications , such as those involving fourier or laplace transforms . often such integrals are approximated by numerical integrations over a finite domain @xmath4 , resulting in a truncation error @xmath8 , in addition to the discretization error . one example is a discrete fourier transform ( dft ) , where there is a truncation error due to cut - off in the tail , in addition to the discretization error . in theory the cut - off error can always be reduced by extending the finite domain at the expense of computing time . however , in many cases a sufficiently long integration domain covering a very long tail can be computationally expensive , such as when the integrand @xmath9 itself is a semi - infinite integration ( e.g. forward fourier or laplace transform ) , or when the integrand decays to zero very slowly ( e.g. a heavy tailed density or its characteristic function ) . much work has been done to directly compute the tail integration in order to reduce the truncation error . examples include nonlinear transformation and extrapolation ( wynn 1956 , alaylioglu et al 1973 , sidi 1980 , 1982 , 1988 , levin and sidi 1981 ) and application of special or generalized quadratures ( longman 1956 , hurwitz and zweifel 1956 , bakhvalov and vasileva 1968 , piessens 1970 , piessens and haegemans 1973 , patterson 1976 , evans and webster 1997 , evans and chung 2007 ) , among many others . this paper describes a very simple , perhaps the simplest , end - point correction to account for the tail integration over the entire range @xmath10 . the treatment of the tail reduces the usual truncation error significantly to a much smaller discrete error , thus increasing overall accuracy of the integration , while requiring virtually no extra computing effort . for the same accuracy , this simple tail correction allows a much shorter finite integration domain than would be required otherwise , thus saving computer time while avoiding extra programming effort . to our knowledge this result is not known in the literature and we believe it deserves to be published for its elegant simplicity and broad applicability . though it is possible that our formula is a rediscovery of a very old result hidden in the vast literature related to numerical integration . the paper is organized as follows . in section 2 , we derive the tail integration approximation and its analytical error . a few examples are shown to demonstrate the effectiveness of the tail integration approximation in section 3 . concluding remarks are given in section 4 . consider integration @xmath11 . without loss of generality , we assume @xmath12 ( a change of variable @xmath13 results in the desired form ) . for @xmath14 the derivation procedure and the resulting formula are very similar . in the following , we assume that * the integral @xmath11 exists ; * all derivatives @xmath15 exist and @xmath16 as @xmath17 . the truncation error of replacing @xmath18 by @xmath19 is simply the tail integration @xmath20 for higher accuracy , instead of increasing truncation length at the cost of computing time , we propose to compute the tail integration @xmath21 explicitly by a very economical but effective simplification . assume @xmath9 approaches zero as @xmath22 and the truncation point @xmath23 can be arbitrarily chosen in a numerical integration . let @xmath24 , where @xmath25 is some large integer . dividing integration from @xmath26 to @xmath27 into cycles with an equal length of @xmath28 yields @xmath29 now assume that @xmath9 is piecewise linear within each @xmath28-cycle , so that each of the integrals @xmath30 in ( 2 ) can be computed exactly . that is , in the range @xmath31 $ ] , we assume that @xmath9 is approximated by @xmath32 where @xmath33 . substitute ( 3 ) into ( 2 ) , then analytical integration by parts of each @xmath34 in ( 2 ) gives @xmath35 this elegant result given by ( 4 ) means that we only need to evaluate the integrand @xmath9 at one single point @xmath36 ( the truncation point ) for the entire tail integration , replacing the truncation error with a much smaller round - off error . as will be demonstrated later , this one - point formula for the potentially demanding tail integration is remarkably effective in reducing the truncation error caused by ignoring @xmath21 . formula ( 4 ) can be derived more generally through integration by parts , and a recursive deduction gives us higher order correction terms and thus error estimates . integrating ( 1 ) by parts with @xmath37 , we have @xmath38 where @xmath39 . if we assume @xmath9 is linear within each @xmath28-cycle in the tail , then the integration @xmath40 vanishes , because within each @xmath28-cycle @xmath41 is constant from the piecewise linear assumption and @xmath42 for any integer @xmath43 , and @xmath44 as @xmath45 . thus , under the piecewise linear assumption , ( 5 ) and ( 4 ) are identical . continuing with integration by parts in ( 5 ) and noting @xmath46 at infinity , we further obtain @xmath47 where @xmath48 . equation ( 6 ) , as well as ( 5 ) , is exact no approximation is involved . the recursive pattern in ( 6 ) is evident . if we now assume that the second derivative @xmath49 is piecewise linear in each @xmath28-cycle in the tail , then ( 6 ) becomes @xmath50 with the additional correction term , ( 7 ) is more accurate than ( 4 ) . in general , without making any approximation , from the recursive pattern of ( 6 ) we arrive at the following expression for the tail integral @xmath51 where @xmath52 , @xmath53 is the 2@xmath43-th order derivative of @xmath9 at the truncation point . as will be shown later with examples , typically the first few terms from ( 8) are sufficiently accurate . the error in using formula ( 4 ) is readily obtained from ( 8) @xmath54 in deriving ( 8) , we have assumed all derivatives exist and @xmath55 . under certain conditions , the infinite series in ( 8) and ( 9 ) represents the integral asymptotically as @xmath56 , i.e. we have the asymptotic expansion @xmath57 for example , if we assume that , for some @xmath58 , @xmath59 then the integral term on the right - hand side of ( 8) can be bounded by @xmath60 as @xmath61 , for some positive constant @xmath62 , and the series converges to the integral . the derivatives approaching zero as @xmath22 is a consequence of the existence of integral ( 1 ) . otherwise , if @xmath63 , integral ( 1 ) does not exist , which is evident form ( 5 ) . applying this argument recursively , all derivatives @xmath64 , if they exist . obviously if @xmath9 is a power function ( e.g. @xmath65 ) , the ratio @xmath66 is of the order @xmath67 as @xmath22 , so is the ratio @xmath68 . this implies that , for a power - like function , each error term in ( 9 ) decreases by two orders of magnitude from its preceding term as the index number @xmath43 increases by one . * remark*. note that there is no truncation error in ( 4 ) and the error is a discretization error in nature . in theory , the tail integration error can be estimated by ( 9 ) . in practice , however , derivatives of integrand at the truncation point may only be evaluated numerically . the assumption of piecewise linearity , although reasonable for @xmath9 at large @xmath69 , may appear to be rather crude for a high precision computation . however , we recall that we are only trying to reduce the already small truncation error @xmath21 and a reasonable approximation in @xmath70 could lead to significant improvement in the overall accuracy of integration . for example , suppose a relative error of 1% due to ignoring truncation and 10% error in evaluating the tail integration using the very simple formula ( 4 ) . the overall accuracy with this tail integration added is now improved from 1% to 0.1% ( 1% times 10% ) . this improvement by an order of magnitude is achieved by simply evaluating the integrand at the truncation point . the assumption of a piecewise linearity applies to a broad range of functions , thus the special tail integration approximation can have a wide application . note , piecewise linear assumption does not even require monotonicity - @xmath9 can be oscillating , as long as its frequency is relatively small compared with the principal cycles in @xmath71 , as demonstrated in one of the examples below . if the oscillating factor is @xmath72 instead of @xmath71 , we can still derive a one - point formula similar to ( 4 ) by starting the tail integration at @xmath73 instead of @xmath26 . in this case , the tail integration is @xmath74 also , the tail integration approximation can be applied to the left tail ( integrating from @xmath75 to @xmath76 as well , if such integration is required . it is known from the literature that truncation is better at extrema of the oscillatory part than at the zeros ( lyness 1986 , espelid and overholt 1994 and sauter 2000 ) . truncating at @xmath77 , the extrema for @xmath78 , we obtain an expression for the tail integration or the truncation error similar to ( 8) @xmath79 the leading term of the truncation error is now @xmath80 in ( 11 ) , compared with @xmath81 in ( 8) . assuming @xmath82 for some large @xmath83 , e.g. when @xmath9 is a power - like function , then it is obvious truncation at extrema has a smaller truncation error . however , our formula is about the reduction of the truncation error by including an approximation of the tail integration . if truncation is done at @xmath84 instead of @xmath85 , then the first correction term will be @xmath86 , involving the first derivative of @xmath9 . in many important applications the first derivative of @xmath9 can not be evaluated accurately . for example , when inverting a characteristic function of a compound distribution , @xmath9 itself is a semi - infinite integration of an oscillatory function , which could only be obtained numerically . taking finite difference of a numerically evaluated function will in general reduce the accuracy by an order of magnitude . so for general purposes the truncation is chosen at the zeros , i.e. at @xmath85 . of course , if derivative of @xmath9 is in closed form and can be accurately evaluated , truncation and correction at extrema will indeed be more accurate , with a leading error term of @xmath87 . but we could also include the second derivative term for the truncation at zeros , with a leading error term of @xmath88 , and so on . in general when higher order derivatives can be computed precisely , then one can include some higher order terms to reduce truncation error further and it does not matter much whether the truncation is done at extrema or at zeros . the effectiveness of the above tail integration approximation is now demonstrated in a few examples . introduce the following notations @xmath89 in all the following examples the exact semi - infinite integration @xmath90 is known in closed form , and its truncated counterpart @xmath91 is either known in closed form or can be computed accurately . for simplicity in all the examples @xmath25 is taken to be an even number , i.e @xmath92 . the exact tail integration @xmath93 can be computed from @xmath94 . we compare @xmath95 with @xmath91 and compare both of them with the exact semi - infinite integration @xmath90 . the error reduction can be quantified by comparing the `` magic '' point value given by formula ( 4 ) with the exact tail integration @xmath93 . also note that the analytic formula for the error of using ( 4 ) , @xmath96 $ ] , is given by ( 9 ) . * _ example 1 : _ * @xmath97 in this example , the closed form results are @xmath98 figure 1 compares the `` magic '' point value @xmath99 representing simplified tail integration with the exact tail integration @xmath100 as functions of parameter @xmath101 for @xmath102 , i.e. the truncated lengths @xmath103 . the figure shows that a simple formula ( 4 ) matches the exact semi - infinite tail integration surprisingly well for the entire range of parameter @xmath104 . corresponding to figure 1 , the actual errors of using formula ( 4 ) are shown in table 1 , in comparison with the truncation errors without applying the correction term given by ( 4 ) . figure 2 shows the same comparison at an even shorter truncated length of @xmath105 . the error of using ( 4 ) is @xmath106 . if @xmath104 is large , the function @xmath107 is `` short tailed '' and it goes to zero very fast . the absolute error @xmath108 is very small even at @xmath109 . the relative error ( against the already very small tail integration ) , given by @xmath110 , is actually large in this case . but this large relative error in the tail approximation does not affect the high accuracy of the approximation for the whole integration . what is important is the error of the tail integration relative to the whole integration value . indeed , relative to the exact integration , the error of using ( 4 ) is @xmath111 , which is about @xmath112 at @xmath113 . the condition @xmath114 as @xmath17 is not satisfied in this case if @xmath115 . however , as discussed above , the application of formula ( 4 ) does not cause any problem . for a small value of parameter @xmath104 , the truncation error will be large unless the truncated length is very long . for instance , with @xmath116 the truncation error ( if ignore the tail integration ) is more than 70% at @xmath103 ( @xmath102 , as the case in figure 1 ) , and it is more than 88% at @xmath117 ( @xmath113 , as the case in figure 2 ) . on the other hand , if we add the `` magic '' value from formula ( 4 ) to approximate the tail integration , the absolute error of the complete integration @xmath118 due to this approximation is less than 0.01% , and the relative error is @xmath119 at both @xmath103 and @xmath117 . in other words , by including this one - point value , the accuracy of integration has dramatically improved by several orders of magnitude at virtually no extra cost , compared with the truncated integration . for the truncated integration @xmath91 to have similar accuracy as @xmath120 , we need to extend the truncated length from @xmath105 to @xmath121 for this heavy tailed integrand . * _ example 2 : @xmath122 _ * . this example has a heavier tail than the previous one . here , we have closed form for @xmath90 , but not for @xmath123 or @xmath21 , @xmath124 @xmath123 or @xmath21 can be accurately computed by adaptive integration functions available in many numerical packages . here we used _ imsl _ function based on the modified clenshaw - curtis integration method ( clenshaw and curtis 1960 ; piessens , doncker - kapenga , berhuber and kahaner 1983 ) . figure 3 compares the `` exact '' tail integration @xmath125 with the one - point value @xmath99 . again the one - point approximation does an extremely good job . even at the shortest truncation length of just @xmath126 the one - point approximation is very close to the exact semi - infinite tail integration . applying the analytical error formula ( 9 ) to @xmath127 * * , * * we have @xmath128 taking the first three leading terms we get @xmath129 at @xmath130 and @xmath131 at @xmath132 . the relative error @xmath133 is about 1% at @xmath134 and it is about 0.002% at @xmath132 . apparently , if the extra correction term @xmath135 is included as in ( 7 ) , the error @xmath136 reduces further by an order of magnitude at @xmath130 and by several orders of magnitude at @xmath132 . corresponding to figure 3 , the actual errors of using formula ( 4 ) are shown in table 2 , in comparison with the truncation errors without applying the correction term given by ( 4 ) . figure 4 shows the truncated integration @xmath91 and the truncated integration with the tail modification ( 4 ) added , i.e. @xmath137 , along with the correct value of the full integration @xmath138 . the contrast between results with and without the one - point tail approximation is striking . at the shortest truncation length of @xmath126 ( @xmath139 , the relative error due to truncation for the truncated integration @xmath140 is more than 30% , but with the tail approximation added , the relative error @xmath141 reduces to 0.5% . at @xmath142 , the largest truncation length shown in figure 4 , the relative error due to truncation is still more than 4% , but after the `` magic '' point value is added the relative error reduces to less than @xmath143 . another interesting way to look at these comparisons , which is relevant for integrating heavy tailed functions , is to consider the required truncation length for the truncated integration to achieve the same accuracy as the one with the `` magic '' value added . for the truncated integration @xmath144 to achieve the same accuracy of @xmath145 ( integration truncated at one - cycle plus the magic point value ) , we need to extend the integration length to @xmath146 . for @xmath91 to achieve the same accuracy of @xmath147 , the integration length has to be extended to more than @xmath148 ! on the other hand , if we add the tail approximation @xmath149 to @xmath150 , the relative error reduces from 0.5% to less than @xmath151 ! this error reduction requires no extra computing , since @xmath149 is simply a number given by @xmath152 . * _ example 3 : _ * @xmath153**. * * we have remarked that the piecewise linear assumption does not require monotonicity , i.e. @xmath9 can be oscillating , as long as its frequency is relatively small compared with the principal cycles . for example , when the function @xmath9 is the characteristic function of a compound distribution , it oscillates with its frequency approaching zero in the long tail . in the current example with @xmath154 , there is a closed form for @xmath155 , but not for @xmath123 or @xmath21 , @xmath156 figure 5 compares the `` exact '' tail integration @xmath93 with the one - point approximation @xmath99 for the case @xmath157 . again the one - point approximation performs surprisingly well , despite @xmath9 itself is now an oscillating function , along with the principal cycles in @xmath71 . the piecewise linearity assumption is apparently still valid for relatively mild oscillating @xmath9 . corresponding to figure 5 , the actual errors of using formula ( 4 ) are shown in table 3 , in comparison with the truncation errors without applying the correction term given by ( 4 ) . not surprisingly , the errors are larger in comparison with those in examples 1 and 2 , due to the fact that @xmath9 now is itself an oscillating function . still , table 3 shows the truncation error is reduced by an order of magnitude after applying the simple formula ( 4 ) . figure 6 compares the truncated integration @xmath91 against @xmath158 , along with the correct value of the full integration @xmath159 . at truncation length @xmath160 , the shortest truncation length shown in figures 5 and 6 , the relative error @xmath161 is less than 0.06% and it is less than 0.01% at @xmath162 . in comparison , the truncated integration without the end point correction has relative error of 2.7% and 0.2% , respectively for those two truncation lengths . applying the analytical error formula ( 9 ) to @xmath163 and noting @xmath164 and @xmath165 with @xmath157 and @xmath166 , we obtain @xmath167 where only the first two leading terms corresponding to the 2@xmath168 and 4@xmath169 derivatives are included , leading to @xmath170 at @xmath171 that agrees with the actual error . similar to the previous example , if we include the extra correction term @xmath135 , the error reduces further by two orders of magnitude at @xmath162 . the purpose of example 3 is to show that the piecewise linear approximation in the tail could still be valid even if there is a secondary oscillation in @xmath9 , provided its frequency is not as large as the principal oscillator . if the parameter @xmath172 is larger than one , then we can simply perform a change of variable with @xmath173 and integrate @xmath174 in terms of @xmath175 . better still , for any value of @xmath172 , we can make use of the equality @xmath176 to get rid of the secondary oscillation altogether before doing numerical integration . in practice , the secondary oscillation often has a varying frequency with a slowly decaying magnitude , such as in the case of the characteristic function of a compound distribution with a heavy tail . in this case it might be difficult to effectively apply regular numerical quadratures in the tail integration , but the simple one - point formula ( 4 ) might be very effective . all these examples show dramatic reduction in truncation errors if tail integration approximation ( 4 ) is employed , with virtually no extra cost . if the extra correction term @xmath177 is included , i.e. using ( 7 ) instead of ( 4 ) , the error is reduced much further . we have derived perhaps the simplest but efficient tail integration approximation , first intuitively by piecewise linear approximation , then more generally through integration by parts . analytical higher - order correction terms and thus error estimates are also derived . the usual truncation error associated with a finite length of the truncated integration domain can be reduced dramatically by employing the one - point tail integration approximation , at virtually no extra computing cost , so a higher accuracy is achieved with a shorter truncation length . under certain conditions outlined in the present study , the method can be used in many practical applications . for example , the authors have successfully applied the present method in computing heavy tailed compound distributions through inverting their characteristic functions , where the function @xmath9 itself is a semi - infinite numerical integration ( luo , shevchenko and donnelly 2007 ) . of course there are more elaborate methods in the literature which are superior to the present simple formula in terms of better accuracy and broader applicability , such as some of the extrapolation methods proposed by wynn 1956 and by sidi 1980 , 1988 . the merit of the present proposal is its simplicity and effectiveness - a single function evaluation for the integrand at the truncation point is all that is needed to reduce the truncation error , often by orders of magnitude . it can not be simpler than that . also , in some applications the function @xmath9 may not even exist in closed form , for instance when @xmath9 is the characteristic function of some compound distributions as mentioned above , then @xmath9 itself is a semi - infinite integration of a highly oscillatory function , which could only be obtained numerically . in such cases some of the other more sophisticated methods relying on a closed form of @xmath9 may not be readily applicable . we would like to thank david gates , mark westcott and three anonymous refrees for many constructive comments which have led to significant improvements in the manuscript . 0.5 @@ccc@ @xmath69 & @xmath136 & @xmath179 + @xmath184 & @xmath185 & 0.2241 + @xmath186 & @xmath187 & 0.1422 + @xmath188 & @xmath189 & 0.1006 + @xmath190 & @xmath191 & 0.0637 + @xmath192 & @xmath193 & 0.0318 + 0.5 @@ccc@ @xmath69 & @xmath136 & @xmath179 + @xmath188 & @xmath194 & 0.0105 + @xmath195 & @xmath196 & 0.0053 + @xmath197 & @xmath198 & 0.0035 + @xmath199 & @xmath187 & 0.0026 + @xmath192 & @xmath200 & 0.0021 + and the truncated integration plus the one - point approximation of tail integration , @xmath95 , as functions of the truncated length @xmath203 , where @xmath204 . the solid line represents the exact value of the full integration without truncation error , @xmath205 . ] and the truncated integration plus the one - point approximation of tail integration , @xmath95 , as functions of the truncated length @xmath209 , where @xmath210 , @xmath208 . the solid line represents the exact value of the full integration without truncation error , @xmath211 . ]

integration of the form @xmath0 , where @xmath1 is either @xmath2 or @xmath3 , is widely encountered in many engineering and scientific applications , such as those involving fourier or laplace transforms . often such integrals are approximated by a numerical integration over a finite domain @xmath4 , leaving a truncation error equal to the tail integration @xmath5 in addition to the discretization error . this paper describes a very simple , perhaps the simplest , end - point correction to approximate the tail integration , which significantly reduces the truncation error and thus increases the overall accuracy of the numerical integration , with virtually no extra computational effort . higher order correction terms and error estimates for the end - point correction formula are also derived . the effectiveness of this one - point correction formula is demonstrated through several examples . * keywords : * numerical integration , fourier transform , laplace transform , truncation error .

the search for the possibility of unidirectional motion in a periodic system without the application of any obvious bias is of current research interest@xcite . such possibility requires the system to be out of equilibrium in order for the process to be consistent with the second law of thermodynamics . several physical models have been proposed to obtain such motion . in all the models noise plays the central role . one of the most discussed models is the one in which an asymmetric periodic potential system is adiabatically rocked@xcite by applying constant forces @xmath0 and @xmath1 at regular intervals of time . one obtains unidirectional motion because in such a system the current @xmath2 . eventhough the time averaged applied force over a period vanishes the averaged current @xmath3 $ ] becomes finite in the presence of noise ( thermal fluctuations ) . moreover , the average current @xmath4 was found to peak at an intermediate noise strength ( or temperature ) . in this model it has been further shown that by suitably choosing the asymmetric periodic potential one may obtain current reversal@xcite as a function of temperature provided the rocking frequency is high . similar results , however , can be obtained in the presence of a unbiased colored noise instead of the oscillating force . there are several other interesting models to obtain unidirectional motion including models where potential barriers themselves are allowed to fluctuate@xcite or models wherein symmetric potential system is driven by temporally asymmetric forces@xcite , etc . the result that thermal noise helps to obtain unidirectional current in a periodic system was quite important . but later on it was pointed out that obtaining mere current does not necessarily mean that the system does work efficiently@xcite . doing work involves flow of current against load and hence one must , in the spirit of the model , obtain current up against a tilted ( but otherwise periodic ) potential system . analysis shows , however , that the efficiency of an adiabatically rocked system ( ratchet ) monotonically decreases with temperature . therefore though such a ratchet system helps extract a large amount of work at an intermediate temperature ( where the current peaks ) the work is accomplished at a larger expense of input energy ; thermal fluctuation does not facilitate efficient energy conversion in this model ratchet system . in a subsequent work@xcite this deficiency was rectified but in a different model wherein the asymmetric potential oscillates in time , instead of its slope being changed ( rocked ) between @xmath5 and @xmath1 adiabatically . in both these models the friction coefficient was constant and uniform in space . the present work makes a detailed study of the rocked ratchet system with nonuniform friction coefficient which varies periodically in space@xcite . in this work we take the friction coefficient to vary with the same periodicity as the potential but with a phase difference , @xmath6 . the phase difference @xmath6 , the amplitude @xmath7 of variation of friction coefficient , the amplitude @xmath8 of rocking , the load , etc . affect the functioning of the ratchet in an intricate and nontrivial manner . the two of the important results we obtain are : ( 1 ) the efficiency of the adiabatically rocked ratchet shows a peak as a function of temperature , though the peak ( which may or may not exist in case of spatially asymmetric potentials ) position does not coincide with the temperature at which the current peaks , and ( 2 ) the current could be made to reverse its direction as a function of noise strength and the amplitude @xmath8 even at low frequencies of rocking . these attributes are solely related to the medium being inhomogeneous with space dependent friction . it is worth noting that the introduction of space dependent friction , though does not affect the equilibrium properties ( such as the relative stability of the locally stable states ) , changes the dynamics of the system in a nontrivial fashion . recently it has been shown that these systems exhibit noise induced stability , it shows stochastic resonance in washboard potentials without the application of external periodic input signal,@xcite and also unidirectional motion in periodic symmetric potential ( non ratchet - like ) systems.@xcite in the next section we describe our model and obtain an expression for current and efficiency in the quasi - static limit . in sec . iii we present our results . the nature of correct fokker - planck equation in the presence of space - dependent diffusion coefficient ( inhomogeneous medium ) was much debated earlier . later on the correct expression was found from a microscopic treatment of system - bath coupling . the motion of an overdamped particle , in a potential @xmath9 and subject to a space dependent friction coefficient @xmath10 and an external force field @xmath11 at temperature @xmath12 is described by the langevin equation @xcite @xmath13^{2 } } + \sqrt{\frac{k_{b}t}{\gamma ( q ) } } \xi ( t ) , \ ] ] where @xmath14 is a randomly fluctuating gaussian white noise with zero mean and correlation : + @xmath15 . here @xmath16 denotes an ensemble average over the distribution of the fluctuating noise @xmath17 . the primes in eq . ( 1 ) denote the derivative with respect to the space variable @xmath18 . it should be noted that the above equation involves a multiplicative noise with an additional temperature dependent drift term . the additional term turns out to be essential in order for the system to approach the correct thermal equilibrium state . we take @xmath19 , where @xmath20 , @xmath21 being any natural integer . @xmath22 is a constant force ( load ) representing the slope of the washboard potential against which the work is done . also , we take the friction coefficient @xmath10 to be periodic : + @xmath23 , where @xmath6 is the phase difference with respect to @xmath24 . the equation of motion is equivalently given by the fokker - planck equation @xmath25 .\ ] ] this equation can be solved for the probability current @xmath26 when @xmath27 = constant , and is given by @xcite @xmath28 in the presence of space dependent friction and the phase lag @xmath29 and in the absence of load @xmath30 even for a spatially periodic symmetric potential . thus when the system is subjected to an external ac field @xmath11 the unidirectional particle flow ( or rectification of the current ) takes place . the phase lag @xmath6 brings in the intrinsic asymmetry in the dynamics of the system . when an externally applied ac force changes slowly enough ( quasi - static or adiabatic limit ) i.e. , when the time scale of variation of @xmath11 is much larger compared to any other time scales involved in the system we can readily obtain an expression for the unidirectional current . for a field @xmath11 of a square wave amplitude @xmath31 , an average current over the period of oscillation is given by , @xmath32 $ ] . this particle current can even flow against the applied load @xmath22 and thereby store energy in useful form . in the quasi - static limit following the method of stochastic energetics@xcite it can be shown @xcite that the input energy @xmath33 ( per unit time ) and the work @xmath34 ( per unit time ) that the ratchet system extracts from the external noise are given by @xmath35 $ ] and @xmath36 $ ] respectively . thus the efficiency ( @xmath37 ) of the system to transform the external fluctuation to useful work is given by @xmath38}{f_{0}[j(f_{0 } ) - j(-f_{0})]}.\ ] ] henceforth all our variables like @xmath39 are made dimensionless . the amplitude of potential @xmath40 is set to unity as all other energy scales are scaled with respect to @xmath40 . we evaluate @xmath41 , and @xmath37 numerically@xcite using eq . first , we present our results for average ( net ) unidirectional current in a symmetric periodic potential induced by adiabatic rocking and in the absence of load . we emphasize here that to obtain these currents the system must be inhomogeneous . the phase lag @xmath6 ( except for @xmath42 , for which unidirectional current is not possible ) plays an important role in determining the direction and magnitude of @xmath4 . for @xmath43 , in the presence of external quasistatic force @xmath11 and in the absence of load @xmath22 , we have a forward moving ratchet ( current flowing in the positive direction ) and for @xmath44 we have the opposite . for instance , if we examine the effect of friction coefficient close to the minimum of the potential two different situations are encountered depending on the value of @xmath6 . when @xmath45 and @xmath46 the particle experiences lower friction near the barriers in the direction of acquired velocity . the situation is reverse when @xmath47 . from eq . ( 3 ) it follows that in a static force @xmath8 , @xmath48 , hence rectification of current occurs in the presence of external adiabatic drive . moreover , it should be emphasized that the magnitude of current or mobility , in the static field @xmath8 , depends sensitively on the potential and the frictional profile over the entire period . depending on the system parameters the current or mobility can be much larger or smaller than the current or mobility of a particle moving in a homogeneous medium characterised by the space averaged frictional coefficient . in fact , in the intermediate values of temperature and @xmath8 the mobility can be made much larger than their asymptotic limits . this leads to stochastic resonance in a washboard potential in the absence of ac signal @xcite . in fig . 1 the average unidirectional current @xmath4 is presented as a function of @xmath6 and @xmath12 . for this figure the value of amplitude of square - wave ac field @xmath49 , and the amplitude of frictional modulation @xmath50 . it can be seen that @xmath4 changes sign as a function of @xmath6 and current exhibits either a minimum or a maximum as a function of temperature depending on the value of @xmath6 . at the two limits of temperature ( @xmath51 and @xmath52 ) the currents vanish . this is the case for the value of @xmath31 less than the critical value @xmath53 of @xmath31 where the barrier to motion in either direction vanishes . in our case the critical value of @xmath31 is equal to 1 . this stochastic resonance - like phenomenon has been observed in rocked ratchet systems characterised by asymmetric periodic potentials @xcite . from the contours of the plot it is clear that as we move away from phase shift @xmath54 positive ( or negative ) direction the temperature at which maxima ( minima ) occurs shifts to a larger value and the absolute value of the current at the peak decreases . however , the present symmetric potential situation does not lead to multiple current reversals as a function of temperature in the quasi - static limit of an external drive . the current as a function of @xmath12 and @xmath31 is shown in fig . 2 for @xmath55 and @xmath50 . for smaller fields @xmath8 compared to the critical field @xmath53 the current exhibits a maximum as a function of temperature . as we increase @xmath31 the temperature at which the peak occurs decreases . for fields larger than @xmath31 the current , however , decreases monotonically with temperature because the barrier to motion disappears . in this high field region mobility of a particle decreases with increase of temperature @xcite . however , as we increase the field the net current @xmath4 monotonically changes and saturates to a finite value . this is in contrast @xmath4 vanishes for large @xmath31 in the ratchet subjected to a rocking force in the absence of space dependent friction . in fig . 3 we have plotted @xmath4 versus @xmath56 and @xmath6 for fixed values of @xmath57 and @xmath58 . it can again be seen clearly that the current monotonically varies and saturates to a value given by @xmath59 independent of temperature . this result @xcite follows from the analysis of eq.(3 ) . as expected current reversal can be seen as a function of @xmath6 for large @xmath31 . it can also be verified that the current @xmath4 increases monotonically as a function of the amplitude @xmath7 of the friction coefficient and hence we do not present variation of @xmath4 , etc . , with @xmath7 . we now discuss the efficiency of a symmetric periodic potential system in the presence of space dependent friction driven by an adiabatic periodic field . the efficiency of such a system with uniform friction has been studied earlier and it has been shown that temperature does not facilitate the efficiency @xmath60 of energy conversion in the system @xcite . to calculate @xmath37 we make use of eqs . ( 3 ) and ( 4 ) . in our analysis the load @xmath22 is applied against the direction of net current ( in the absence of load ) . in this situation particle current can flow against the applied load @xmath22 less than some critical value @xmath61 thereby storing energy in useful form . for @xmath62 one can not talk meaningfully the concept of efficiency as the current flows in the direction of the load and hence no storage of useful energy takes place . 4 we have plotted efficiency @xmath37 , input energy @xmath63 and work done @xmath34 ( scaled up by a factor 60 for convenience of comparison ) as a function of @xmath12 for the parameter values , @xmath64 , @xmath65 , @xmath58 , and the load @xmath66 . the figure shows that the efficiency exhibits a maximum as a function of temperature indicating that thermal fluctuation facilitates energy conversion . this in contrast to the case of uniform friction coefficient where @xmath37 decreases monotonically with the increase of temperature in the same adiabatic limit @xcite . it is to be mentioned that the temperature corresponding to the maximum efficiency is not the same as the temperature at which the average current @xmath4 becomes maximum in the absence of load . the temperature at which the extracted work maximizes is not the same as the temperature at which the efficiency becomes maximum for the same parameter values . the input energy increases with temperature monotonically and saturates at the high temperature limit . @xmath37 , @xmath34 , and @xmath33 show similar qualitative behaviour for other parameter values . the above important observation of temperature facilitating the energy conversion is applicable for the spatially symmetric potential . in general in adiabatically rocked systems with frictional nonuniformities the increasing thermal noise need not increase the efficiency . the efficiency is sensitive to the qualitative nature of the periodic potential ( and also to the nonuniformity of friction ) . for instance , asymmetric potential exhibits quite complex behaviour of @xmath37 and @xmath4 . to illustrate this we take @xmath67 ( where @xmath68 lies between -1 and 1 , and is the asymmetry parameter ) . with this potential we discuss three separate cases : case a - system in a symmetric potential ( @xmath69 ) in an inhomogeneous medium ( @xmath70 ) , case b - system in an asymmetric potential ( @xmath71 ) in an inhomogeneous medium ( @xmath70 ) , and case c - system in an asymmetric potential ( @xmath71 ) in a homogeneous medium ( @xmath72 ) . in fig.5 , we have presented results of @xmath37 versus @xmath12 for all the three cases described above . for this we have taken @xmath73 , @xmath74 , and @xmath55 . for case a @xmath58 , for case b @xmath75 and @xmath58 , and for case c @xmath75 . as discussed earlier for the case a temperature maximizes the efficiency . case c , where the medium is homogeneous , efficiency monotonically decreases with temperature . these observations have been emphasized in earlier literature . the case b , where potential asymmetry and frictional inhomogeneity are present , the efficiency decreases monotonically in this parameter regime . in general whether the temperature facilitates the energy conversion in case b depends sensitively on the system parameters . in some limited parameter range the peaking behaviour is seen as in fig.6 . the parameter values are indicated in the caption . the presence of asymmetry in the potential may or may not help in enhancing the efficiency of the system . this can be seen from figs . 5 and 6 . in all these cases the work done @xmath34 and the input energy @xmath33 show similar qualitative features as shown in the inset of fig . , we discuss the variation of efficiency as a function of load . on general grounds it is expected that the efficiency too exhibits maximum as a function of load . it is obvious that the efficiency is zero when load is zero . at the critical value @xmath76 ( beyond which current flows in the direction of the load ) the value of current is zero and hence the efficiency vanishes again . in between these two extreme values of load the efficiency exhibits maximum . beyond @xmath77 the current flows down the load and therefore the idea of efficiency becomes invalid . in fig . 7 , we have plotted @xmath37 versus load for all the three cases for chosen values of parameters as mentioned in the figure . in all these cases current monotonically decreases as a function of load . the work done against load @xmath34 exhibits a maximum as a function of load . the load at which @xmath34 shows maximum does not coincide with the load at which @xmath37 becomes maximum . the input energy @xmath33 as a function of load varies non monotonically exhibiting a minimum . however , depending on the case under consideration the value of the load at which the minimum in the input energies observed may be larger than @xmath61 above which efficiency is not defined . in fig . 8 , we have plotted the efficiency versus the amplitude of the adiabatic forcing @xmath31 for all the three cases . it can be seen from the figure that for the system in an inhomogeneous medium , namely for cases a and b , @xmath37 exhibits a maxima and saturate to the same value in the large amplitude limit . in contrast , for the case c after exhibiting maximum @xmath37 goes to zero . this follows from the simple fact that in the large amplitude limit in the absence of frictional inhomogeneities the net unidirectional current tends to zero . the peculiar feature of saturation of efficiency in inhomogeneous media is related to the fact that the average current saturates to a constant value in the high amplitude limit as discussed earlier . this somewhat counter - intuitive result is typical to inhomogeneous media . having discussed efficiency of energy conversion we now study the nature of net current @xmath4 in the presence of spatially asymmetric potential to examine if current reversals take place in the adiabatic limit in the absence of load . it is known from the earlier literature @xcite that in an adiabatically rocked asymmetric potential ratchet system net current does not exhibit reversals as a function of @xmath12 . in these systems current reversals are possible when the frequency of the applied ac field is large . we show here that in the presence of frictional inhomogeneities in addition to asymmetry in the potential one can observe current reversal as a function of thermal noise . in fig . 9 , we have plotted the magnitude of net current @xmath4 versus @xmath12 for all the three cases a , b , and c. the corresponding parameter values are mentioned in the caption of the figure . the cases a and c do not exhibit current reversal . this is a general result independent of parameter values . however , in case b current reversal is observed . to obtain current reversal both asymmetry in potential and nonuniform friction coefficient are essential , that is , current reversals arise due to the combined effect of @xmath6 and @xmath68 . moreover , it should be noted that to observe current reversals the parameter range should be such that the net current in case a is in the opposite direction to that in case c. for the case b for which current reversal is observed , the plot of efficiency separates into two disjoint branches as the load should be reversed keeping the magnitude same when the current reversal takes place . in the presence of @xmath71 and @xmath70 , where the current reversals are observed , the efficiency as a function of temperature is less than the maximum value of efficiency in either of the two cases a and c. that is , @xmath78 $ ] . to further analyze the nature of current reversals , in fig . 10 , we have plotted @xmath4 as a function of @xmath6 and @xmath12 for fixed values of @xmath79 , @xmath58 , and @xmath80 . from the contour plots it is clear that as a function of @xmath6 the current reverses sign twice in the intermediate temperature range . thus the current exhibits reentrant behaviour as a function of phase @xmath6 which is special to the case b. as we decrease the asymmetry in the potential the @xmath81 contour line shifts towards @xmath82 thereby enhancing the domain of current reversal to a lower value of temperature as a function of @xmath6 . as a function of temperature the current reversals occur in a definite range of phase @xmath6 which , in turn , depends on other material parameters . the qualitative behaviour of @xmath4 remains unaltered for different @xmath31 as long as @xmath31 is less than the critical value . in fig . 11 , @xmath4 is plotted as a function of @xmath12 and @xmath8 for @xmath79 , @xmath83 , and @xmath58 . as opposed to the case of symmetric potential ( fig . 2 ) , currents in the small temperature regime do exhibit maxima and then saturate to a constant value as noted earlier . as a function of @xmath12 the current exhibits similar features as in the case a ( fig . figure 12 , shows @xmath4 as a function of @xmath8 and @xmath6 for @xmath84 , @xmath79 , and @xmath58 . as opposed to case a ( see fig . 3 ) @xmath4 shows current reversal as a function of @xmath8 in the range @xmath85 . however , in the asymptotic limit of @xmath8 current saturates to a value @xmath59 independent of the value of the asymmetry parameter @xmath68 . from the contour plot it follows that as a function of phase @xmath6 we observe the reentrant behaviour of current at high values of @xmath8 . as we decrease @xmath68 the @xmath81 contour shifts towards smaller values of @xmath31 thus making it possible to observe the double reversals at even smaller values of @xmath31 . this reentrant behaviour as a function of @xmath6 and the current reversal as a function of @xmath31 is very specific to the case b alone ( compare fig . thus we conclude from our studies that the dynamics of a particle in an inhomogeneous medium is rich and complex . in the presence of adiabatic forcing and asymmetry in the potential current reversals can be observed as a function of @xmath12 and @xmath31 . and depending on the system parameters thermal fluctuations facilitate the energy conversion . the above behaviour can not be seen in the homogeneous medium in the same adiabatic limit . however , it is possible to observe these in homogeneous media in the presence of finite frequency ac drive ( nonadiabatic regime ) . this seems to suggest that @xmath6 may play the characteristic role of frequency in our model in the absence of nonadiabatic ac drive . this has been noted earlier in the context of observation of stochastic resonance phenomena in inhomogeneous media in the presence of static tilt alone @xcite . as a function of phase @xmath6 , we observe reentrant behaviour for the current which arises because of interplay between asymmetry , inhomogeneity , thermal noise , and strength of the adiabatic forcing . some of the phenomena can be understood at best at a qualitative level only . the effect of nonadiabatic forcing may be of further interest . work on this line is under investigation . mcm acknowledges partial financial support and hospitality from the institute of physics , bhubaneswar . mcm and amj acknowledge partial financial support from the board of research in nuclear sciences , dae , india . p. hanggi and r. bartussek , _ nonlinear physics of complex systems - current status and future trends _ , lecture notes in physics , vol . 476 , ed . by j. parisi , s.c . mueller , and w. zimmermann ( springer , berlin , 1996 ) , pp . 294 - 308 . fig . the net current @xmath4 as a function of @xmath6 and @xmath12 , for parameter values @xmath86 , @xmath58 . in the base plane contour of surface plot @xmath87 are given , dotted line indicates @xmath88 contour . efficiency @xmath89 and @xmath34 as a function of @xmath12 for @xmath65 , @xmath86 , @xmath50 , and @xmath90 . @xmath34 has been scaled up by a factor @xmath91 to make it comparable with @xmath37 and @xmath33 . y - axis is in dimensionless units . fig . 9 . current @xmath4 versus temperature @xmath12 for ( i ) case a ( @xmath92 , and @xmath58 ) , ( ii ) case b ( @xmath97 , @xmath50 ) , and ( iii ) case c ( @xmath97 , @xmath94 ) , for fixed @xmath64 , @xmath99 , and @xmath100 . fig . the net current @xmath4 as a function of @xmath6 and @xmath12 , for parameter values @xmath86 , @xmath58 , and @xmath101 . in the base plane contour of surface plot @xmath87 are given , dotted line indicates @xmath88 contour . fig . @xmath4 as a function of the amplitude of the rocking force @xmath31 , and phase @xmath6 , for @xmath57 , and @xmath101 . in the base plane contour of surface plot @xmath87 are given , dotted line indicates @xmath88 contour .

we present a detailed study of the transport and energetics of a brownian particle moving in a periodic potential in the presence of an adiabatic external periodic drive . the particle is considered to move in a medium with periodic space dependent friction with the same periodicity as that of the potential but with a phase lag . we obtain several results , most of them arising due to the medium being inhomogeneous and are sensitive to the phase lag . when the potential is symmetric we show that efficiency of energy transduction can be maximised as a function of noise strength or temperature . however , in the case of asymmtertic potential the temperature may or may not facilitate the energy conversion but current reversals can be obtained as a function of temperature and the amplitude of the periodic drive . the reentrant behaviour of current can also be seen as a function of phase lag . + pacs number(s ) : 05.40+j , 05.60+w , 82.20mj = 0.3 in = 0.6 in

recently there has been much experimental and theoretical interest in the properties of granular media . in such systems the thermal energy available is not sufficient to allow the rearrangement of a single particle and hence the system is effectively at zero temperature in the thermal sense . the fact that the problem is not trivial lies in the fact that such systems have an exponentially large number of such metastable states , which may be also called blocked or jammed configurations . edwards associated an entropy to these configurations @xmath3 where @xmath4 is the total number of metastable states of the system @xcite . it is reasonable to assume that in complex systems such as granular media @xmath5 is extensive meaning that @xmath6 where @xmath7 is the entropy per particle , alternatively one may work with an entropy per unit of volume which is clearly a more natural choice in granular media . because the system has an extensive number of blocked configurations , if it is prepared from a random initial state it will lower its energy via only energy lowering rearrangements until it becomes stuck in a metastable state . normally this first encountered blocked state will not be that of lowest energy ( or most dense packing ) . in order to change the state of the system an external perturbation such as tapping or shearing is required . in between perturbations the system relaxes into new configurations . a natural and practically very important question concerning this sort of dynamics is : what are the properties of the steady state regime obtained via such mechanical perturbation schemes ? recently it has been shown that spin glasses and ferromagnets on random graphs have an extensive entropy of metastable states and one may calculate this entropy at fixed values of the energy @xcite . therefore , though they are quite different physically to granular materials , these systems have an extensive entropy of metastable states as do granular media . the motivation of this paper is to see if one can understand certain steady state properties of mechanically perturbed systems in terms of their organization of metastable states . the possibility of using spin glasses as a paradigm for granular material was first introduced in @xcite . let us recall an example of an experiment on a system of hard spheres reported in @xcite . a system of dry hard soda glass spheres is placed in a glass tube . the system is tapped by using a piston to move the tube vertically through a sine cycle . the tapping parameter @xmath8 is defined to be the ratio of the maximal acceleration due to the piston in the cycle to @xmath9 the acceleration due to gravity . after an initial irreversible curve , obtained by increasing the tapping rate slowly , the system arrives on a reversible curve where the density is a monotonic function of @xmath8 , the highest packing densities being obtained at lowest tapping rate . numerical simulations on granular media @xcite reveal similar behavior ( though the irreversible part of the experimental curve corresponding to a loosely packed _ fluffy _ metastable state was not seen ) . it was also observed that at small tapping the relaxation to the final density is extremely slow and is well fitted by an inverse logarithmic decay of the form @xmath10 where @xmath11 ( the final density ) , @xmath12 , @xmath13 ( the characteristic relaxation time ) and @xmath14 are fitting parameters . it should however be remarked that the behavior of granular systems is strongly dependent on the tapping mechanism and that horizontal shearing @xcite leads to behavior qualitatively different to vertical tapping . in this paper we extend and elaborate a preliminary report of the results of @xcite . the philosophy of the paper is to examine spin glasses as paradigms for granular media . here the quantity corresponding to the density is the energy of the system . we allow the system to evolve under a random sequential zero temperature single spin flip dynamics where only moves which reduce the energy are allowed . when the system is blocked we tap it with strength @xmath15 $ ] , that is to say each spin is flipped with a probability @xmath0 , the updating at this point being parallel . the system is then evolved by the zero temperature dynamics until it becomes once again stuck , the tapping is then repeated . physically this corresponds to assuming that in granular media the relaxation time to a new metastable state is much shorter than the time between taps . a similar , though not identical , tapping dynamics has also been introduced independently in the context of three spin ferromagnetic interactions on thin hypergraphs @xcite , also in the goal of studying the dynamics of granular media . we find that a stationary regime is reached after a sufficiently large number of taps , characterized by a steady state energy @xmath1 ( analogous to the stationary density the same analogy as used in @xcite ) . the initial dynamics from the random initial configuration into the first metastable state is examined analytically for the one dimensional @xmath2 spin glass or ferromagnet ( the two are equivalent by a gauge transformation ) . we call this the initial fall and the average energy of the first metastable state visited @xmath16 is computed . we then develop a mean field theory for the dynamics under falling then tapping , interestingly this theory appears to be exact in the case of the one dimensional system and one may calculate @xmath1 within this scheme , the results being in excellent agreement with the numerical simulations . numerically we examine the tapping of spin glasses and ferromagnets of higher connectivity . for the spin glass we find that @xmath1 is , as in the experiments , a decreasing function of @xmath0 . for small @xmath0 we define the exponent @xmath17 by @xmath18 , with @xmath19 constant . in the one dimensional case we show analytically that @xmath20 , hence @xmath21 , whereas for spin glasses on thin graphs for connectivity superior to two we find that @xmath22 . however for @xmath23 we find that the time to reach the steady state is extremely long and not accessible numerically . in this slow dynamical regime we find a slow relaxation of the time dependent energy , reminiscent of that observed in experiments on granular media @xcite and hence compatible with eq . ( [ eqlog ] ) . in the case of the ferromagnet we find numerically that there exists a critical value @xmath24 of @xmath0 such that for @xmath25 , @xmath26 where @xmath27 is the energy of the ground state and the inequality is strict , and that for @xmath28 @xmath29 . hence in the ferromagnetic system there is a first order phase transition under tapping dynamics ( in contrast to the usual thermodynamic ferromagnetic transition in these systems which is second order @xcite ) . there have of course been many models studied to understand the compaction process in granular media @xcite , which reproduce many of the experimental features . here the spin glass is clearly far from a realistic realization of a granular media , however the fact that it has extensive entropy of blocked states and the obviously natural form of the tapping dynamics implemented makes it a natural testing ground for ideas about dynamics and possible thermodynamics of systems such as granular media . moreover , it has been argued in @xcite that the slow compaction regime is well explained if we assume that particles can rearrange themselves in such a way as to create a particle size void , which is quickly filled by a new grain . this mechanism involves crossing of energy barriers and leads to a logarithmic compaction before the asymptotic steady state regime @xcite . we expect that the local rearrangements which occur during the tapping dynamics on spin glasses random graphs will be lead to behavior analogous to the slow glassy dynamics of systems as granular media . of course one would ultimately like to obtain a theoretical understanding of the asymptotic , steady state regime of lightly tapped granular media . edwards has proposed @xcite that a light tapping dynamics on granular type systems leads to a steady state whose properties are determined by a flat measure over the blocked or metastable states satisfying the macroscopic constraints involved ( _ e.g. _ fixed internal energy and compactivity ) . this idea has recently attracted much interest and has been examined in the context of various models @xcite . in this paper we shall concentrate simply on the asymptotic energy of the final tapped state , the study of the dynamics leading to this final regime is deferred for further investigation @xcite . we shall see that the calculation of the edwards entropy as a function of energy gives us a possible explanation of the first order ferromagnetic transition . the models we shall consider are spin systems on random thin graphs . a random thin graph is a collection of @xmath30 points , each point being linked to exactly @xmath31 of its neighbors , @xmath31 therefore being the connectivity of the graph . the distribution of metastable states in these systems has been recently considered in @xcite . the spin glass / ferromagnet model we shall consider has the hamiltonian @xmath32 where the @xmath33 are ising spins , @xmath34 is equal to one if the sites @xmath35 and @xmath36 are connected and zero otherwise . the fact that the local connectivity is fixed as @xmath31 imposes the local constraints @xmath37 , for all sites @xmath35 . in the spin glass case the @xmath38 are taken from a binary distribution where @xmath39 with probability half and @xmath40 with probability half . in the ferromagnetic case @xmath41 . here we define a metastable state as a spin configuration where any single spin flip does not increase the energy of the system . mathematically the total number of these metastable states is expressed as : @xmath42 it should be pointed out here that the definition of metastable states is of course dependent on the dynamics of the system , in contrast with micro - states in classical statistical mechanics . whether or not , in certain cases , the information about the dynamics encoded in the calculation of the entropy of metastable states is enough to allow one to predict the properties of the steady state regime is an open question . the fact that , in our definition of metastable states , we include the marginal case ( where the energy change is zero ) implies that here @xmath43 , the heaviside step function , is taken such that @xmath44 . in the context of granular media , where friction plays an important role , this is a natural choice as one certainly needs a non zero force in order to make a grain move . with this definition , the total number of metastable states of internal energy @xmath45 per spin is formally given by @xmath46 the corresponding edwards entropy per spin , at fixed energy @xmath45 per spin , is then given by @xmath47 we remark that by a gauge transformation the one dimensional ferromagnet and @xmath2 spin glass are equivalent and place ourselves for transparency in the context of the ferromagnet . let us remark that the zero temperature glauber dynamics of the one dimensional ferromagnet can be explicitly solved @xcite , here diffusion of domain walls occurs and the dynamics does not get blocked . in the glauber case one may close the dynamical equations , however here such a closure scheme does not seem possible . the zero temperature kawasaki dynamics ( conserving the total magnetization ) of the one dimensional ising model , where the system can freeze , has been solved in @xcite . to solve the dynamics of the one dimensional ferromagnet we consider the dynamics from the point of view of the bonds . we define a fault of length @xmath48 to be a sequence of @xmath48 neighboring adjacent domain walls . the zero temperature dynamics takes place within these faults via the flipping of one of the @xmath49 spins contained between the @xmath48 domain walls . we define @xmath50 to be the indicator function that starting from bond @xmath35 there are exactly @xmath48 consecutive domain walls ( there being no domain wall on bond @xmath51 and no domain wall on bond @xmath52 but all the intervening bonds have a domain wall ) . in the initial configuration we take the probability that a given spin is different to its left neighbor ( that is to say the probability of a domain being between two spins ) to be @xmath53 . hence if @xmath54 we have an initially ferromagnetic configuration , if @xmath55 it is an antiferromagnetic configuration , the case @xmath56 corresponds to a completely random configuration of maximal entropy . the total energy of the initial configuration is then given by @xmath57 as the energy is given by the ground state energy @xmath58 plus two times the number of domain walls ( excitations ) . defining @xmath59 to be the probability that one has @xmath48 consecutive domain walls starting from a given site ( hence @xmath60 where @xmath61 indicates the average over the initial conditions ) , with the initial conditions introduced above one finds @xmath62 ( the length of the defaults has a geometric distribution ) . the initial energy per site of a configuration generated in this manner is therefore ( using the translational invariance of the system when @xmath63 ) given by @xmath64 for the distribution of initial configurations considered here we find therefore that @xmath65 we define by @xmath66 the average number of isolated domain walls left by a fault of @xmath48 consecutive domain walls after the zero temperature dynamics described above has finished . the final energy of the system per site @xmath16 is therefore @xmath67 by the definition of the spin dynamics , domain walls disappear by pairs of two neighboring domain walls . it is clear that @xmath68 , @xmath69 and we set @xmath70 . within such a fault the dynamics proceeds by flipping one of the @xmath49 spins between the domain walls . by recurrence , after a random flip we obtain @xmath71 we solve equation ( [ eq : rec ] ) by introducing the generating functional @xmath72 the resulting equation for @xmath73 is @xmath74 solving this with the appropriate boundary conditions one obtains @xmath75 and substituting this into eq . ( [ eqef ] ) yields @xmath76 thus giving the result @xmath77 this yields a value of @xmath16 for the completely random initial configuration where , @xmath78 , of @xmath79 . in fact the value of @xmath16 is maximal for the case @xmath78 . for the totally antiferromagnetic initial condition , where @xmath80 , here we find @xmath81 . clearly when @xmath82 the system is already in its ground state and we find @xmath83 as we should . we note that these values ( and those for all @xmath53 ) have been checked with and are in perfect agreement with our numerical simulations . this calculation with the one dimensional ferromagnet demonstrates two important points : * the final value of the energy @xmath16 depends strongly on the initial configuration . * the system does not fall into a state of energy corresponding to the maximum of @xmath84 . in @xcite it was shown that @xmath85 where @xmath86 is a concave function peaked at @xmath87 . hence even if the total number of metastable states is dominated ( in the statistical sense ) by those of energy @xmath88 , generic initial conditions always seem to lead to an energy lower than this @xcite . in @xcite the value of @xmath16 for a variety of zero temperature dynamics ( sequential , greedy and reluctant ) in the fully connected sherrington kirkpatrick ( sk ) spin glass model @xcite was studied , similar behavior was found . when the one dimensional system is tapped we find results in line with those described later for the spin glasses at higher connectivity . the curve of @xmath1 , the asymptotic stationary value of the energy at a given @xmath0 , is shown in fig.([fig1 ] ) from @xmath89 systems of size @xmath90 spins . the time taken to reach a stationary value for @xmath1 were rapid for larger @xmath0 but for small @xmath0 there is a very slow relaxation to the final asymptotic state which is of the form @xmath91 , where @xmath92 is the number of taps . this is easily understood as at very slight tapping order @xmath0 effects dominate at early times , this means that : ( i ) isolated pairs of domain walls within large domains are immediately destroyed once tapping is stopped . ( ii ) flipping a spin either side of a domain wall creates domain wall diffusion and with this annihilation by coalescence of domain walls . hence the dynamics at small @xmath0 and early times is qualitatively the same as that for the low temperature ising model coarsening @xcite . in order to go beyond our first calculation of @xmath16 and solve the tapping dynamics we consider a mean field theory for the dynamics of a system of connectivity @xmath31 . we shall see that at @xmath93 this theory gives the analytic result ( [ ef ] ) and reproduces to within numerical errors the numerical tapping results . once again we concentrate on the dynamics on the bonds . for a given site define @xmath94 to be the difference between the number of unsatisfied and satisfied bonds . hence @xmath94 is the local field on the spin at this site and @xmath95 . if @xmath96 then the spin can flip bringing about the change @xmath97 . in addition we denote by @xmath98 the probability that the site of interest has local field @xmath94 after a total of @xmath99 attempted random sequential spin flips under the zero temperature falling dynamics ( that is to say the dynamics in between taps ) . we define @xmath100 and @xmath101 the probabilities that a given spin can flip conditional on the fact that the bond with a given neighboring site is not satisfied or satisfied respectively . formally we have @xmath102 we may turn around this conditional probability using bayes theorem to obtain @xmath103 given that a site has local field @xmath94 , it must have @xmath104 unsatisfied bonds and @xmath105 satisfied bonds . therefore we find that @xmath106 and @xmath107 putting these results into eq . ( [ eqfpm ] ) then gives @xmath108 if we are interested in the spin at site @xmath35 the possibilities between time @xmath99 and @xmath109 are * the spin at site @xmath35 is chosen and @xmath110 , then the spin at site @xmath35 will flip and @xmath94 goes to @xmath111 . * the spin at site @xmath35 is chosen and @xmath112 , then the spin at site @xmath35 can not flip and @xmath94 does not change . * a neighbor of site @xmath35 with positive local field is chosen and so flips . in this case , @xmath94 goes to @xmath113 or to @xmath114 depending whether or not the bond with site @xmath35 was satisfied or not satisfied . * a neighbor of site @xmath35 with negative or zero local field is chosen and so does not flip . in this case , @xmath94 does not change * one chooses neither the spin at site @xmath35 nor any of its neighbors , and so @xmath94 stays @xmath94 . assuming that the distribution at every site is given by @xmath98 and assuming independence between the values of @xmath94 from site to site ( the mean field approximation ) we obtain @xmath115 in this equation we have to define @xmath116 , the choice compatible with the conservation of probability is @xmath117 . taking the limit @xmath118 we may introduce the continuous time @xmath119 and obtain @xmath120 the average energy per site at time @xmath13 is then given by @xmath121 and one can show that the above mean field equation ( [ eqn ] ) respects the exact identity for the evolution of the average energy per spin @xmath122 the case where @xmath93 ( the one dimensional case ) is accessible to analytic solution and we proceed by defining @xmath123 one finds that @xmath124 and the full mean field evolution equations become @xmath125 if we look for a stationary solution of ( [ eqn1 ] ) and ( [ eqn2 ] ) , we find @xmath126 , which expresses the fact that when @xmath13 is infinite , the system is in a metastable state . to solve ( [ eqn1 ] ) and ( [ eqn2 ] ) , we introduce @xmath127 so that @xmath128 . then @xmath127 obeys the equation : @xmath129 with the initial condition @xmath130 . then ( [ eqn2 ] ) becomes : @xmath131 and @xmath132 and @xmath133 are given by @xmath134 the probability to have a positive value for the local fields then goes to zero at infinite @xmath13 as expected and the limit of @xmath135 is @xmath136 . if we consider the geometric initial conditions used in the previous exact calculation of @xmath16 , the induced initial conditions are : @xmath137 , @xmath138 and @xmath139 . in this case we obtain @xmath140 reproducing the exact result ( [ ef ] ) . tapping the system with tapping probability @xmath0 , starting from the values @xmath141 , we obtain the new _ tapped _ values @xmath142 . defining @xmath143 , the relations between the old and _ tapped _ probabilities are : @xmath144 then , after another zero temperature evolution of the system , it reaches a new local energy probability distribution with @xmath145 and : @xmath146 at this stage of computation one should remark that this recursive equation contains one of the main features of our numerical simulations , that is _ reversibility_. indeed , the process involved in ( [ eqn4o ] ) will reach an asymptotic value which is independent of the initial conditions and depends on @xmath0 . hence in the steady state regime under tapping , the probability @xmath147 ( the subscript _ s _ indicating steady state ) for sites to have zero local field is solution of the fixed - point equation : @xmath148 this equation can be solved numerically and the result is shown in fig . ( [ fig1 ] ) in comparison with the numerical simulations which we see is excellent . the small @xmath0 behavior of @xmath1 from ( [ eqn4 ] ) is : @xmath149 , indicating that in this case @xmath21 . given the mean field nature of the above calculation we have used we do not expect this approximation to correctly describe the approach towards the steady state , by direct comparison with the numerical simulations we have verified that this is indeed the case . let us remark here that a defect of the mean field approximation scheme is that it can not distinguish between a spin glass and a ferromagnet , this is clearly not a problem for the one dimensional situation where the two are identical . the systems which we study are @xmath2 spin glasses or uniform ferromagnets on random graphs with fixed connectivity @xmath31 . let us first recall some analytical results of @xcite . it has been found that the mean number of metastable states increases exponentially with the number of sites in both cases . in addition in @xcite an annealed approximation to the edwards entropy per spin of metastable states at fixed energy @xmath45 was carried out : @xmath150 which may be exact for the ferromagnet as the calculated entropy is always positive . moreover , there is an energy threshold @xmath88 above which the results are the same for the @xmath2 spin glass and the ferromagnet ; below the ferromagnet has more metastable states and a non - zero magnetization . hence , as far as the energy density of metastable states is concerned , both ferromagnet and spin glass are the same above @xmath88 that is the effect of loop frustration is negligible . in this regime , one also suspects that the zero temperature dynamics are the same . in particular , numerical simulations with @xmath89 samples of @xmath151 sites for connectivities of @xmath152 , @xmath153 and @xmath154 have found the same @xmath16 for the spin glass and ferromagnet with very good accuracy ( the relative error is about @xmath155 ) . the results are show in table ( 1 ) . the result of tapping experiments on the systems with @xmath156 is displayed in fig.([fig2 ] ) . there is some critical tapping rate , @xmath24 , above which the curves of @xmath1 versus @xmath0 are the same for the spin glass and the ferromagnet . moreover , the ferromagnet is subject to a phase transition under tapping dynamics at @xmath24 such that for @xmath157 , the steady state reached is the ground state . finite size effects have been studied and revealed that the transition is first order ( in as far that @xmath158 ) , in contrast to the usual thermodynamic ferromagnetic transition in these systems which is second order @xcite . for the ferromagnet in the region close to @xmath24 one finds an excellent scaling of the energy as a function of @xmath30 , @xmath159 as shown in the inset of fig . ( [ fig2 ] ) . this scaling may be used to optimize the determination of @xmath24 . the first order nature of the transition may be seen explicitly by looking at the histogram over time ( in the steady state regime ) of the average energy per spin at @xmath160 ; one sees in fig . ( [ fig3 ] ) two separated peaks in the distribution and not a single peak which splits into two as one would expect for a second order transition . near @xmath24 , for systems of finite size , there is therefore coexistence of the two phases . the time dependence of the average energy per spin in the simulation leading to the histogram fig . ( [ fig3 ] ) is shown in fig.([fig4 ] ) . one sees that the system tunnels between the two coexisting states . the typical time for this tunneling increases as the system size increases , indicating , in thermodynamic language , a _ free energy _ barrier between the two phases . as the system size is increased the occupation of the intermediate states of energy between the two phase of energy @xmath27 and @xmath161 ( between the two peaks in fig.([fig3 ] ) ) is suppressed . we have also measured @xmath1 by studying single systems of very large size ( @xmath162 ) , the results are shown in fig . ( [ fig5 ] ) . here again above @xmath24 the curve for the spin glass and the ferromagnet are completely indistinguishable and the ferromagnet reaches the ground state below @xmath24 . for such large sizes , one no longer sees a coexistence of two phases around @xmath24 as presumably the tunneling time has become much larger than the simulation time . moreover , for @xmath162 , for @xmath163 the full temporal plots ( and not just the steady state values ) of @xmath164 and @xmath165 ( where @xmath92 is the number of taps ) are indistinguishable . for @xmath166 the two curves are identical up till a time @xmath167 , which depends on the initial configuration and the sequence of spins flipped during the tapping process , and diverge after @xmath167 , when the ferromagnetic system reaches quickly the ground state ( see fig.([fig6 ] ) ) . once the ferromagnetic system has broken the @xmath168 symmetry the easiest way to lower the energy is to flip the spins which are opposed to the global magnetization ( because they are more probable not to be in the direction of their local field ) until all the spins are @xmath169 or @xmath170 . identical behavior was found in the systems with @xmath171 and @xmath154 . the comparison of @xmath172 and @xmath173 for @xmath171 is shown in fig . ( [ fig5 ] ) . remark that if we compare the different values of @xmath24 when increasing @xmath31 , and considering only odd ( or even ) connectivities , we find that @xmath24 grows , and we expect that it goes to @xmath174 when @xmath31 is very large , as the metastable states are more and more magnetized when @xmath31 grows ( in the case of the fully connected ferromagnetic ising model , only the two ground states are metastable , so @xmath175 ) . the behavior of the spin glass systems is similar to that for the system with @xmath176 . the steady state energy @xmath172 is a monotonically decreasing and continuous function of @xmath0 . for small @xmath0 one finds that here @xmath177 giving @xmath22 in contrast with @xmath21 in the one dimensional case . _ a tentative explanation for the ferromagnetic transition _ : in @xcite it was also shown that for the ferromagnet the edwards entropy as a function of @xmath45 is concave for @xmath178 and convex for @xmath179 . the value of @xmath161 obtained from the tapping experiments are very close to those obtained for @xmath88 in @xcite , the energy at which @xmath180 becomes convex . the results are shown in table ( 1 ) . encouraged by this striking observation we will try to make a tentative link with a possible thermodynamics for such systems . if we imagine that the energy of the system is governed by a partition function inspired by the flat edwards measure over metastable states @xcite @xmath181 where @xmath182 is a lagrange multiplier corresponding to the inverse edwards temperature which depends solely on @xmath0 and not on @xmath45 and is a monotonically decreasing function of @xmath0 for @xmath183 $ ] . the monotonicity hypothesis is supported by the simulation results that @xmath1 decreases with decreasing @xmath0 . clearly the energy which dominates in the sum is that obeying @xmath184 however if this saddle gives a true maximum of the action one must have also that @xmath185 and hence the edwards entropy must be concave for the energy considered to be thermodynamically stable . hence for @xmath186 , this suggests that the only stable energy is the ground state . if one would like to push the analogy of the thermodynamics of first order phase transitions to its limits and assume that @xmath182 is a continuous function of @xmath0 ( it is clear that if this is not the case one could trivially obtain a first order transition ) , one would also expect that the _ free energy _ of the two phases is equal at @xmath187 and hence @xmath188 where @xmath27 is the energy of the ground state of the ferromagnet given here by @xmath189 . it is also clear that @xmath190 hence one obtains @xmath191 however one also has that @xmath192 and hence one should obtain @xmath193 however the fact that the calculated annealed approximation for @xmath194 is convex for @xmath179 means that @xmath195 and hence the equality ( [ eq : fot ] ) is not respected . it is possible that the exact quenched calculation of @xmath194 would give a different value from the annealed calculation of @xcite , however one should not expect the result to be too different qualitatively to that of @xcite . in addition we remark that the annealed calculation @xmath196 gives an upper bound for the quenched value @xmath197 ( from jensen s inequality ) . also the value of @xmath88 found with this approximation seems to be greater than that found numerically for @xmath161 by a very small amount ( @xmath198 for @xmath199 ) . if we accept the possibility that @xmath200 , a more probable hypothesis is the collapse of validity of edwards hypothesis in the region of energy where the @xmath168 symmetry is broken on metastable states , as has been found for the three dimensional random field ising model in @xcite . let us mention here that we measured the energy in the simulations over a few hundred time steps after the energy appeared to stop to decay . to be sure that the systems considered here were in a stationary regime ( and that the energy was not decaying extremely slowly as a function of time ) we measured the correlation function at different waiting times @xmath201 ( the number of taps after the initial preparation of the system ) , that is to say @xmath202 in the stationary regime this should be a function only of @xmath92 . in out of equilibrium systems the fact that the system is not in equilibrium shows up strongly as aging in the correlation function _ i.e. _ @xmath203 ( see @xcite and references within ) , even though the energy may be decaying so slowly that it appears to have reached its asymptotic equilibrium value . the time translational invariance of @xmath204 is thus quite a rigorous test of whether the steady state regime has been attained . for example for the case @xmath205 , with the waiting times @xmath206 and @xmath207 is shown in fig . ( [ fig7 ] ) , one sees , clearly that after the appropriate translation of the @xmath92 axis , the two functions collapse perfectly onto one another . one also sees that the decay of @xmath208 ( in the longtime regime we can now eliminate the @xmath201 dependence ) is exponential at large @xmath92 and also that @xmath208 decays to zero , indicating a form of ergodicity in the system . as pointed out in @xcite , this behavior of the correlation function seems a necessary condition for the validity of the scenario of edwards , that under tapping all metastable states satisfying the relevant macroscopic constraints ( fixed energy and compactivity ) are equiprobable in the stationary regime of gently tapped or perturbed system . in order to test the accuracy of the mean field approximation at high energies ( where the system does not distinguish between the ferromagnet and the spin glass ) , we have compared the value of @xmath16 obtained in the numerical simulations with the result @xmath209 obtained by numerical integration of eq . ( [ eqn ] ) . the comparison is in table ( 1 ) and we see that the agreement is quite good . finally we mention that we have also examined the reversibility of the tapping mechanism . if the system is tapped for a sufficiently long time , compatible with the relaxation times discussed above , the system is completely reversible . this reversibility was found in the experiments in @xcite once the system had left the initial fluffy state . granular media are a natural example of systems having an extensive entropy of metastable states . in such systems the role of thermal fluctuations are negligible and in order to evolve one must apply some external tapping mechanism . one would ultimately like to be able to formulate some sort of thermodynamics for such systems . the proposition of edwards @xcite for a thermodynamics of such systems is an important step in this direction and has had some success @xcite but it has been shown not to be generically true @xcite . a more general understanding of the asymptotic states of tapped systems has far reaching implications for computer science as the tapping mechanism studied here is similar to certain algorithms used in optimization problems . we have presented what appears to be an exact calculation of the steady state energy of a tapped one dimensional spin glass or ferromagnet . for this problem we have obtained the fixed point equations for the distribution of local fields under tapping . these equations also explain the reversibility observed in the numerical simulations . in a wide context of models we confirm the observations of @xcite , that if one reduces the _ strength _ of tapping , then the compaction process , corresponding here to the reduction of the energy of the system , becomes more efficient . the existence of a first order type phase transition for tapped ferromagnets on random thin graphs is of great interest , the possible explanation using the calculations of @xcite on the edwards entropy for this system indicates the possibility that one may eventually construct a more general theory for the thermodynamics and even phase transitions in tapped systems . one is tempted to speculate that generically the convexity of the edwards entropy below a certain energy threshold ( denoted here by @xmath88 ) leads to a _ collapse _ to the ground state energy @xmath27 , the metastable states in the intervening energy values being unable to support a stable thermodynamics . in terms of granular media this sort of transition would correspond to a transition between a random close packed state to a crystalline close packed state . it would be interesting to find other systems ( both theoretical and experimental ) showing the same collapse phenomena in order to test this idea . finally let us mention some of the open questions posed by this study we believe to be of interest for future investigation . clearly a general goal would be , in the spirit of edwards , to develop a thermodynamics to describe the stationary regime of tapped systems such as those studied here . the exact results presented on one dimensional systems here provide a completely analytic understanding of the tapping dynamics which one may be able to rederive from static considerations . indeed it has been shown @xcite in this simple context that several steady state observables may be predicted using edwards measure . the phase transition found in the case of the ferromagnetic systems studied here is extremely novel , an analytic understanding of this phenomena would be desirable , perhaps there exists a percolation type argument which would allow one to evaluate @xmath24 . also of interest is the decay of a system towards its final steady state energy . the slow logarithmic decay described by eq . ( [ eqlog ] ) has been used successfully to fit the experimental data of @xcite and the simulation data of @xcite . our preliminary study of finite connectivity spin glasses and the sk model @xcite indicates the presence of a slow dynamical regime for small values of the tapping parameter @xmath0 which is also compatible with a slow logarithmic decay , but the curves can also be well fitted by power law decays ( with the same number of fitting parameters ) . there exist however phenomenological arguments @xcite and exact calculations and simulations on toy models @xcite which support logarithmic decay . tab . 1 . comparison of the numerical values of @xmath88 and @xmath161 for different values of the local connectivity c. the result for @xmath88 when @xmath199 is a truncation of the analytical value @xmath210 . 2 . numerical simulations of tapping experiments for the spin glass ( c ) and the ferromagnet ( ( a ) , ( b ) and ( d ) ) for @xmath199 for @xmath211 ( ( c ) and ( d ) ) , @xmath212 ( b ) and @xmath151 ( a ) . the inset shows the scaling @xmath213 for @xmath214 for @xmath215 , @xmath211 and @xmath212 . single run for a ferromagnet of local connectivity c=3 , for @xmath151 at @xmath216 . one sees that because the size is not too large , the energy switches between two values , one not far from that of the ground state @xmath27 and the other not far from @xmath88 . 5 . numerical simulations of tapping experiments for @xmath199 ( ( c ) : spin glass , ( d ) : ferromagnet ) and @xmath171 ( ( a ) : spin glass , ( b ) : ferromagnet ) . here , we have computed the asymptotic energy for only one sample of very big size @xmath162 . the results are quite the same as for @xmath217 and do not change if we average over several samples , which indicates that at these sizes , we are very near the thermodynamic limit and we can study only single runs to compute the energy . comparison of the energy versus time ( number of taps ) for the @xmath218 spin glass ( a ) and the ferromagnet ( b ) for @xmath162 spins at @xmath219 . we have displayed the magnetization for the ferromagnet ( c ) , whose absolute value increases with time , whereas that of the spin glass remains zero . correlation function for the @xmath2 spin - glass with local connectivity @xmath199 versus number of taps @xmath92 for two values of the waiting time @xmath201 : @xmath220 and @xmath221 . the tapping value is @xmath222 , the system contains @xmath211 spins and we have averaged over @xmath223 samples . with triangles is shown the right part of the curve , which corresponds to @xmath221 , shifted to the left by @xmath224 , it superimposes perfectly over the curve for @xmath220 demonstrating the time translation invariance of the correlation function .

a zero temperature dynamics of ising spin glasses and ferromagnets on random graphs of finite connectivity is considered , like granular media these systems have an extensive entropy of metastable states . we consider the problem of what energy a randomly prepared spin system falls to before becoming stuck in a metastable state . we then introduce a tapping mechanism , analogous to that of real experiments on granular media , this tapping , corresponding to flipping simultaneously any spin with probability @xmath0 , leads to stationary regime with a steady state energy @xmath1 . we explicitly solve this problem for the one dimensional ferromagnet and @xmath2 spin glass and carry out extensive numerical simulations for spin systems of higher connectivity . the link with the density of metastable states at fixed energy and the idea of edwards that one may construct a thermodynamics with a flat measure over metastable states is discussed . in addition our simulations on the ferromagnetic systems reveal a novel first order transition , whereas the usual thermodynamic transition on these graphs is second order . 0.5 cm pacs numbers : 05.20 , 75.10 nr , 81.05 rm . 2

the atmosphere is the most important part of the detector in ground - based gamma - ray astronomy , but it is also the part that has the greatest systematic uncertainty and over which we have the least control . it falls upon us to instead monitor and characterise the atmospheric conditions at the time of observations so that we can either feed this information into monte carlo simulations or reject data when conditions go out of acceptable parameters . after being generated in the upper atmosphere cherenkov light will either reach the ground or be attenuated through the process of rayleigh scattering on the molecular component of the atmosphere , or mie scattering on the aerosol component ( variously dust . silicates , pollens , etc ) . the molecular component tends to change relativiely slowly , through seasonal variations ; whereas the aerosol component can change more rapidly , depending on eg wind conditions . it becomes vitally important to characterise this aerosol component of the atmosphere through regular monitoring . a lidar is generally used to measure the atmospheric transmission ( eg @xcite ) from backscattered laser light . at the h.e.s.s . site a lidar centred at 355 and 532 nm has been running in conjunction with observations since mid-2011 . whilst lidars are excellent instruments for determining the presence of aerosols they are not without complications . firstly a lidar , due to geometric viewing considerations , only becomes effective above a minimum altitude . secondly , in order to obtain a transmission profile relevant to the cherenkov spectrum the laser wavelengths are close to the peak in the emission , this means the lidar is operated only inbetween observing runs to avoid any light contamination to the telescope images . in this paper we look at utilising another piece of the h.e.s.s . atmospheric monitoring equipment to fill in some of this missing information . the atmosphere is split into regions according to its temperature behaviour . the troposphere is the lowest , most dense , part of the atmosphere where most of the weather happens and is characterised by a linear decline in temperature with increasing altitude and vertical mixing . the molecular density profile falls off exponentially , with a scale height of a few km ; the vertical air motion in this region mixes in the larger aerosols which have a smaller scale height of order a km . the molecular component is an inefficient black - body radiator in the 8 - 14@xmath2 m region of the spectrum , water vapour and aerosols are slightly more efficient and clouds are very efficient . this makes an infra - red radiometer an effective cloud monitor , with clouds showing up as a large brightness temperature compared to a relatively cold " sky @xcite . employ heitronics kt19.82 radiometers with 2@xmath3 field of view to monitor for the presence of clouds , with each telescope having a paraxially mounted unit and a further one continuosly scanning the whole sky . the infra - red luminosity of the sky ( @xmath4 ) is a collective sum of the emission of a number of different constituent parts @xmath5 where @xmath6 is the emissivity of the lens ( @xmath7 ) and the water vapour @xmath8 , the aerosols @xmath9 , and the molecular ( @xmath10 ) profiles of the atmosphere , etc and t is the relevant integrated temperature profile in the line of sight . according to @xcite the aerosol component can contribute up to 30wm@xmath0 to the bolometric luminosity , which can mean the difference between a brightness temperature of -56@xmath3c or -70@xmath3c in the presence or absence of aerosols respectively . this leads to the prospect of changing aerosol conditions leading to a noticeable change in the sky brightness temperature ( @xmath11 ) measurements . the august to september period at the h.e.s.s . site often has noticeable aerosol contamination due to biomass burning in neighbouring countries and the resultant smoke being blown downwind . in figure [ fig:20110820 ] we see an `` ideal '' night which has no measurable aerosol contribution ( the large particles having sedimented out of the atmosphere ) ; within the space of a week figure [ fig:20110829 ] shows `` hazy '' conditions , with a prominent aerosol boundary layer that extends up to about @xmath12 km ; a couple of days later figure [ fig:20110901 ] shows the aerosols sedimenting out once more , with the boundary layer close to the lidar effective altitude threshold at @xmath13 km ( characteristic of `` normal '' observing conditions ) . in figure [ fig : rates ] we show the telescope trigger rates as a function of zenith angle for all observing runs for that osberving period that have 4 telescopes participating , stable rates ( ie no clouds or data acquisition issues ) and noted as clear by the observers in the shift logs . the data points are sub - divided according to the aerosol boundary layer conditions and the @xmath11 at zenith for that run , the correlation between warm sky temperature , aerosol presence and lowered telescope trigger rate is clearly apparent . but for the night of 29/08/2011 . there is a prominent aerosol component up to a boundary layer of @xmath14 km and the infra - red lumonisity is substantially increased.,title="fig : " ] but for the night of 29/08/2011 . there is a prominent aerosol component up to a boundary layer of @xmath14 km and the infra - red lumonisity is substantially increased.,title="fig : " ] but for the night of 01/09/2011 . there is a noticeable aerosol component up to a boundary layer of @xmath15 km and the infra - red lumonisity is moderately increased.,title="fig : " ] but for the night of 01/09/2011 . there is a noticeable aerosol component up to a boundary layer of @xmath15 km and the infra - red lumonisity is moderately increased.,title="fig : " ] km , squares when the boundary layer reaches @xmath12 km and crosses for when there are no measurements available . the red points are when @xmath16 at zenith is @xmath17c , blue points when it is lower . ] the atmospheric clarity conditions according to lidar and infra - red radiometer measurements have been presented here . the presence of aerosols in the atmosphere show up clearly in the lidar returns and also as a clear increase in @xmath16 . the data selected here come from a single two week period to ensure no seasonal temperature effects can bias the dataset . the @xmath16 will still change somewhat due to the day - to - day ambient temperature variation , however this would be expected to produce no more than a @xmath18% difference not the observed @xmath19200% . during the most severe periods of aerosol contamination the boundary layer can be seen to extend to relatively high altitudes . as the production height for air shower photons is above these aerosol layers they should act like filters only , but since the light of muon ring images ( commonly used to determine the atmosphere s contribution to the systematic uncertainty ) develop within these layers they will have a distinctly different and more complicated response to different boundary layer altitudes . this will be something to examine in future work . in summary , the lidar is extremely useful in determining the presence of aerosol layers and measuring the transmission profiles , but has limited resolution at altitudes @xmath201 km and limitations as to when it can be operated ; the infra - red radiometer is sensitive to the presence or absence of aerosols , operates all of the time and will be most sensitive to low altitude aerosols . together they have the potential to quantify atmospheric opacity entirely independently of the telescope systematics .

the attenuation of atmospheric cherenkov photons is dominated by two processes : rayleigh scattering from the molecular component and mie scattering from the aerosol component . aerosols are expected to contribute up to 30 wm@xmath0 to the emission profile of the atmosphere , equivalent to a difference of @xmath1c to the clear sky brightness temperature under normal conditions . here we investigate the aerosol contribution of the measured sky brightness temperature at the h.e.s.s . site ; compare it to effective changes in the telescope trigger rates ; and discuss how it can be used to provide an assessment of sky clarity that is unambiguously free of telescope systematics . address = department of physics , university of durham , durham , dh1 3le . u.k . address = lupm , un . montpellier ii cc-072 , place eugenie bataillon , 34095 montpellier , france .

in this paper , we continue our study of the simplest gross - neveu ( gn ) model @xcite , a 1 + 1 dimensional model field theory of @xmath1 species of massless , self - interacting dirac fermions with lagrangian @xmath2 we restrict ourselves from the outset to the t hooft limit @xmath3 const . semiclassical methods have revealed a number of fascinating properties of this model over the years , see the review articles @xcite and references therein . a key quantity in these studies is the scalar mean field @xmath4 . it plays a role similar to witten s master field " in large @xmath1 gauge theories @xcite , namely as saddle point of the functional integral from which all observables can be computed . for fermions in the large @xmath1 limit , it can be identified with the self - consistent hartree - fock ( hf ) potential . most of the results for @xmath4 obtained so far are related to static problems . in the vacuum , the hf potential is homogeneous and can be interpreted as dynamical fermion mass @xcite . localized , spatially varying hf potentials describe individual baryons @xcite . spatially periodic solutions appear in investigations of baryonic matter , both at zero @xcite and finite temperature @xcite . the most difficult problem is to find solutions of the time - dependent hartree - fock approach ( tdhf ) , at least non - trivial solutions which are not simply boosted , static solutions . the only known analytical solutions of this type to date are the breather @xcite and kink - antikink scattering @xcite . since both are related by analytical continuation , there is in fact only one non - trivial time - dependent solution known . this reflects the lack of systematic methods to derive time - dependent , self - consistent mean fields for fermions . recently , it was pointed out that the situation is more favorable for a class of particularly simple tdhf solutions , classified as type i " in @xcite . they are defined as those solutions where the scalar density of each single particle level is proportional to the full self - consistent potential @xmath4 , @xmath5 where @xmath6 may vanish for some states . if property ( [ a3 ] ) is satisfied , the tdhf problem reduces to the classical @xmath7 gn model , for which neveu and papanicolaou have uncovered a relationship with the sinh - gordon equation some time ago @xcite . as a consequence , the self - consistent tdhf potential of the gn model ( [ a1 ] ) can be shown to satisfy the classical sinh - gordon equation @xcite . this is surprising at first sight , as the sinh - gordon equation possesses only singular solitons . owing to a non - linear field transformation however , these singularities are mapped onto zeros of @xmath4 , @xmath8 so that the scalar mean field @xmath4 is perfectly regular . one can easily check that the mean fields for the kink baryon @xcite , kink - antikink scattering @xcite and the kink crystal , the ground state of the gn model at finite density @xcite , are indeed all related to known soliton solutions of the sinh - gordon equation . this raises immediately the question : are there other soliton solutions of the sinh - gordon equation which might yield physically sensible , new tdhf solutions of the gn model ? if one thinks about this problem , one encounters two potential obstacles . the first has to do with the singularities of all sinh - gordon solitons , the second with the fact that the sinh - gordon equation is a necessary condition for type i solutions , but perhaps not sufficient . the first difficulty can be handled as follows . if one inspects the available solutions of the sinh - gordon equation in the literature , one finds in all cases that the argument of the logarithm in eq . ( [ a2 ] ) has either zeros , or poles , or both . this reflects the fact that all solitons are singular . in order to get a bounded @xmath4 , we should only allow for zeros . as already pointed out in @xcite , the most interesting solution of this type is presumably the @xmath0 soliton solution constructed by the inverse scattering method @xcite ( throughout this paper we use @xmath1 to denote the number of flavors and @xmath0 to denote the number of baryons , to avoid confusion ) . in the gn model the @xmath0 soliton solution is expected to describe time dependent scattering of @xmath0 kink- and antikink - baryons . the second difficulty simply means that solutions of the sinh - gordon equation should only be taken as candidates for tdhf potentials in the gn model . given any such solution , one still has to solve the dirac equation for all continuum states in the dirac sea and the valence bound states and verify self - consistency of the mean field . in this paper , we propose to do just this for the @xmath0 soliton solution . our main goal is to find the most general type i solution of the tdhf equations for the gn model . from the particle physics point of view , one is rarely interested in scattering problems involving more than two incident particles . a time - dependent @xmath0 soliton solution on the other hand describes a scattering process involving @xmath0 incident and @xmath0 outgoing objects . as a purely theoretical problem , we find it nevertheless challenging to solve the dynamics of @xmath0 composite , relativistic bound states at the elementary fermion level , in full generality . our motivation is not primarily particle physics phenomenology , but the desire to find new exact , analytical solutions of a relevant model quantum field theory . finally , let us try to relate our work to another important property of the gn model , integrability . as is well known , the gn model ( [ a1 ] ) is an example of an integrable quantum field theory for any value of @xmath1 . the exact @xmath4 matrix , including kinks and antikinks , has been constructed some time ago @xcite . nevertheless we find it worthwhile to attack this problem with entirely different methods in the large @xmath1 limit . first of all , the @xmath4 matrix for the finite @xmath1 gn model is only known in principle . the examples worked out in the original references deal with low values of @xmath1 ( 2 - 8 majorana flavors , corresponding to 1 - 4 dirac flavors ) and few particles only . since the algebraic complexity rapidly increases with increasing number of flavors and participants , it is not easy to infer the large @xmath1 limit of the collision of @xmath0 bound states from the published @xmath4 matrix . secondly , the full dynamical tdhf solution has more information than the @xmath4 matrix which encodes only asymptotic , on - shell scattering information . finally , although integrability certainly helps to find the tdhf solution , it is apparently not a prerequisite . thus for instance , although the massive version of the gn model is not integrable , hf solutions have been found for baryons @xcite and baryonic crystals @xcite in closed analytical form . for all of these reasons we have decided to make a dedicated effort to solve the @xmath0 kink - antikink scattering problem in the large @xmath1 limit of the gn model . the paper is organized as follows . in sec . ii , we give a rather detailed introduction into the single kink baryon in an arbitrary lorentz frame and set up our notation in light - cone coordinates . sec . iii briefly recalls the @xmath0 soliton solution of the sinh - gordon equation . in sec . iv we describe how we get to the tdhf spinors and prove self - consistency . v is needed to put the formal results into a form better suited for practical applications , which then follow in sec . we characterize the general @xmath0 baryon scattering process qualitatively and exhibit a few illustrative examples involving dynamics of up to eight solitons . we end with a concluding section . the kink baryon of the gn model , originally discovered by callen , coleman , gross and zee ( cited in @xcite ) , is at the same time the simplest and the most exotic baryon . its properties are well studied @xcite . we review it here because of its role as main actor in the dynamical @xmath0 baryon problem addressed in this work . an important aspect in which we differ from all previous works except @xcite is the fact that we consider the kink in an arbitrary lorentz frame , not just its rest frame . this is of course a prerequisite for treating scattering processes . the outline of this chapter is the following : we will introduce light - cone coordinates and present first the vacuum and then the boosted kink in the tdhf approach . the scalar hf potential @xmath4 and the self - consistency issue will be addressed . we then compute expectation values of other relevant fermion bilinears , namely the density @xmath9 , the pseudoscalar density @xmath10 and the axial charge density @xmath11 , resolving contributions from the dirac sea and the bound state . next we briefly recall the derivation of the sinh - gordon equation from ref . @xcite for type i tdhf solutions , of which the kink is a paradigm . finally we summarize the essential physics properties of the kink . this section presents no new results , but serves to introduce light - cone coordinates and set up the notation to be used in later chapters for the @xmath0 baryon problem . starting point is the tdhf equation of the gn model , expected to become exact in the large @xmath1 limit , @xmath12 the sum over occupied states runs over the whole dirac sea as well as possible valence states and includes flavor degrees of freedom . a non - vanishing scalar mean field @xmath4 signals breakdown of the z@xmath13 chiral symmetry @xmath14 . we choose a chiral basis for the dirac matrices , @xmath15 , where @xmath16 is diagonal . in conjunction with light cone coordinates @xmath17 this simplifies the dirac equation in ( [ c1 ] ) to @xmath18 in terms of upper , left - handed ( @xmath19 ) and lower , right - handed ( @xmath20 ) spinor components . consider first the vacuum problem where @xmath21 is the dynamical fermion mass in natural units . here , the tdhf equation reduces to the free , massive dirac equation with solutions @xmath22 labeled by a spectral parameter @xmath23 . this parameter contains the information on momentum @xmath24 and energy @xmath25 via @xmath26 a relation which allows us to cast the plane wave factor in ( [ c4 ] ) into the standard form @xmath27 ( if @xmath28 is interpreted as light cone time , then @xmath23 is the light cone energy , but we shall not use this language in the following . ) the gap equation arises from the self - consistency equation for the scalar condensate in the vacuum . the continuum spinor @xmath29 yields the scalar density @xmath30 the ( cutoff regularized ) summation over the dirac sea can be performed conveniently after the following change of integration variables , @xmath31 the resulting gap equation , @xmath32 yields the relation between bare coupling and cutoff characteristic for dimensional transmutation . we now turn to the simplest baryon solution of eqs . ( [ c1 ] ) , the kink or antikink . without loss of generality , we consider the antikink moving with velocity @xmath33 . in ordinary coordinates it is given by @xmath34 interpolating between the vacua @xmath35 at @xmath36 and @xmath37 at @xmath38 ( the results for the kink @xmath39 can simply be generated by a @xmath40 transformation ) . in what follows , it will be advantageous to express @xmath4 through exponentials , @xmath41 switching to lightcone coordinates , the basic building block , @xmath42 , can be seen to be closely related to a plane wave " with imaginary spectral parameter , @xmath43 this structural element will be important later on . the tdhf spinors for the antikink can easily be found . in lightcone notation , the continuum states read @xmath44 where @xmath45 and @xmath42 differ only by a constant , complex phase , @xmath46 as is well known , the potential @xmath4 is reflectionless , a crucial property for everything we shall do in this work . the kink at rest possesses one normalizable zero energy bound state , in agreement with the expectation based on its topological properties . the corresponding boosted bound state can be obtained from the continuum spinor by setting @xmath47 ( i.e. , analytic continuation to imaginary spectral parameter ) and normalizing , @xmath48 the scalar densities for continuum and bound states , @xmath49 show that we are dealing with a type i solution according to the classification of ref @xcite every occupied state yields a contribution to the scalar condensate proportional to the full hf potential @xmath4 . the self - consistency condition simply reduces to the vacuum gap equation ( [ c9 ] ) , @xmath50 consider the expectation value of the fermion density in the kink next . it consists of two contributions , one from the continuum states ( the dirac sea ) and one from the bound state . an individual continuum state @xmath29 gives the following ( vacuum subtracted ) contribution to the density @xmath51 performing the @xmath52-integration and multiplying by the number of flavors ( each state is fully occupied ) , we find the continuum fermion density @xmath53 and hence the following contribution from the dirac sea to the total fermion number , @xmath54 this result can be understood heuristically as follows : the midgap state receives one half of its strength from the negative , the other half from the positive energy continuum . this half missing state in the dirac sea manifests itself in the peculiar value of the induced fermion number ( [ c21 ] ) . this effect has been discussed extensively in the context of fractional fermion number and gives rise to observable consequences in condensed matter systems , such as unusual spin - charge assignments in solitonic excitations of polymers @xcite . next we turn to the contribution to the fermion density from the bound state , assuming that the valence level is filled with @xmath55 fermions . the bound state fermion density is @xmath56 normalized to the number of fermions in the valence state , @xmath57 the continuum and bound state densities ( [ c20 ] ) and ( [ c22 ] ) are proportional to each other , so that the total fermion density becomes @xmath58 the total fermion number @xmath59 of the kink lies between @xmath60 and @xmath61 . in particular , if the bound state is half filled , the density vanishes identically . we are then dealing with a time - dependent excitation of the scalar condensate , a pure domain wall " moving with constant velocity @xmath33 . if the bound state is fully occupied or empty , the kink carries @xmath62 fermions and may be thought of somewhat loosely as half a baryon or antibaryon . for the sake of completeness , let us also evaluate the pseudoscalar condensate along similar lines , once again assuming @xmath63 valence fermions , @xmath64 this quantity is finite and vanishes in the vacuum , so that no subtraction is needed . finally , the last independent bilinear is the axial density ( or vector current ) @xmath65 , where we must once again subtract the vacuum contribution , @xmath66 notice that in all 3 cases ( [ c24],[c25],[c26 ] ) , the sum over continuum states is proportional to the contribution from the bound state , with identical relative weights ( discrete and continuum parts can be identified via the factors @xmath63 and @xmath1 , respectively ) . this fact can be understood with the help of the divergence of vector and axial vector currents @xcite , @xmath67 invoking large @xmath1 factorization and using @xmath68 characteristic for 1 + 1 dimensions , we get @xmath69 showing that the three bilinears @xmath70 are linearly related . as a test of the above calculations , one can verify that the kink results for the bilinears do satisfy eqs . ( [ c29 ] ) . the evaluation of mass , energy and momentum of the kink baryon is delicate due to vacuum subtraction and subtleties in the counting of modes . we refer to ref . @xcite where it was shown in detail that the tdhf approach gives a covariant energy - momentum relation for the baryon in the gn model , @xmath71 ( in natural units ) . the mass of the kink is independent of the number of fermions carried by it , since the bound state has zero energy in the rest frame and vanishing chiral condensate . so far , we have only dealt with the dirac equation involving @xmath72 and @xmath73 . as shown in @xcite , the other two derivatives , @xmath74 and @xmath75 , can also be expressed linearly in @xmath76 with coefficients depending on @xmath4 and its first derivatives . the result , valid for type i solutions if @xmath4 approaches a vacuum value @xmath77 for @xmath78 , is a kind of extended dirac equation " @xmath79 with @xmath80 the integrability condition of the system ( [ c31 ] ) , @xmath81 = 0 , \label{c33}\ ] ] yields the sinh - gordon equation for @xmath82 , @xmath83 or , in normal coordinates , @xmath84 the linearized form of this last equation is the klein - gordon equation for a scalar field with mass 2 which may be identified with the well - known @xmath85 meson of the gn model . hence the kink can be thought of as a large amplitude excitation of the @xmath85 field , thereby extending the skyrme picture to the case of a discrete chiral symmetry . finally , it is easy to check that eqs . ( [ c31],[c32 ] ) hold for continuum states ( real @xmath23 ) as well as for the bound state ( imaginary @xmath23 , @xmath47 ) . summarizing , let us enumerate some properties of the kink which will turn out to be important for the case of @xmath0 interacting kinks as well : 1 . the tdhf solution is reflectionless and of type i. 2 . there is a single bound state with vanishing scalar density , related to the continuum states by analytic continuation in the spectral parameter . the contributions to the fermion density from the continuum states and the bound state have the same functional form . the fermion density vanishes identically for a half filled valence level . 4 . shape , mass and motion of the kink are independent of the number of fermions it carries in this sense , there is no backreaction of the fermions . we should like to point out that in spite of the solvability of the model and the peculiar properties of the kink , we are dealing with a relativistic , composite object with an interesting internal structure reminiscent of hadrons . in ref . @xcite , the structure function , derived analytically from the fermion momentum distribution in the infinite momentum frame , was shown to display non - trivial contributions from valence quarks " , sea quarks " and antiquarks " , with a slight abuse of language . it is therefore a non - trivial question to ask how such composite , relativistic objects interact with each other . as discussed above , the kink of the gn model is akin to the one - soliton solution of the sinh - gordon equation . similarly , the kink - antikink scattering problem can be mapped onto the two - soliton solution @xcite . if the @xmath0 baryon tdhf solution @xmath4 of the gn model is of type i , then @xmath86 must also be a solitonic solution of the sinh - gordon equation . an obvious candidate is the known @xmath0 soliton solution of the sinh - gordon equation , constructed with inverse scattering methods @xcite . here we collect all formulae needed to solve the @xmath0 baryon problem later on . we closely follow the notation of jevicki and jin @xcite . since the focus of our work is not on classical soliton theory itself but rather on the role solitons play in the tdhf approach , we postpone the discussion of the physics to sec . [ sect7 ] . it is inherent in the inverse scattering method that the soliton solution of a nonlinear partial differential equation is accompanied by a linear problem involving 2-component spinors " . these auxiliary spinors depend on a spectral parameter @xmath23 . in the case of the sinh - gordon equation , they are given in light - cone coordinates ( [ c2 ] ) by @xmath87 here , @xmath88 and @xmath89 are @xmath0 component vectors , @xmath90 whereas @xmath91 is the symmetric @xmath92 matrix @xmath93 the spinor @xmath94 satisfies the system of differential equations @xmath95 with @xmath96 @xmath97 is the solution of the sinh - gordon equation @xmath98 as can be shown with the help of the integrability condition @xmath99 = 0 , \label{c42}\ ] ] and is related to @xmath94 via @xmath100 it does not depend on the spectral parameter @xmath23 , as can be seen more easily from the equivalent expression @xmath101 ^ 2 . \label{c44}\ ] ] like all soliton solutions of the sinh - gordon equation , the function @xmath97 of eqs . ( [ c43],[c44 ] ) is singular in fact the @xmath0 soliton solution has @xmath0 singularities . we identify @xmath102 with @xmath103 , the square of the tdhf potential , and will derive the tdhf wave functions from @xmath104 . in this process , singularities of @xmath97 are mapped onto zeros of @xmath4 which is bounded . by comparing @xmath102 with @xmath103 in the one soliton case , we can identify the parameters @xmath105 as follows [ see eq . ( [ c12 ] ) ] , @xmath106 @xmath107 is the ( asymptotic ) velocity of the @xmath24-th soliton , @xmath108 is related to its initial position . hence the solution is general enough to describe the @xmath0 soliton problem with arbitrary initial positions and velocities of the solitons . furthermore , one can verify that @xmath104 satisfy @xmath109 for all @xmath0 . indeed , by differentiation one finds that the left - hand side is independent of @xmath110 , using eqs . ( [ c39],[c40 ] ) . the integration constant can be taken from the asymptotic region . property ( [ c46 ] ) will be crucial for the proof of self - consistency in the following section . the sinh - gordon equation provides us with candidates for the simplest class of tdhf solutions ( type i ) of the large @xmath1 gn model . in each case one still has to verify self - consistency of the result . to this end one has to solve the dirac equation with the scalar potential inferred from soliton theory . furthermore , summation of the scalar condensates of all continuum states in the dirac sea and the partially filled bound states must be performed to check self - consistency . since the @xmath0 soliton solutions are rather complicated , this might seem hopeless . remarkably , as we shall show in this section , soliton theory provides us with exactly the information needed to perform this task in closed analytical form . the tdhf dirac spinor @xmath111 for any type i solution satisfies the extended dirac equation ( [ c31],[c32 ] ) . on the other hand , the auxiliary spinor @xmath94 in the inverse scattering problem of the sinh - gordon equation solves eqs . ( [ c39],[c40 ] ) . as originally exploited in @xcite for the classical @xmath7 gn model and applied to type i solutions of the large @xmath1 gn model in @xcite , this implies that the two problems are related by a non - abelian gauge transformation . the language of gauge transformations is adequate here because the integrability conditions have the mathematical form of a vanishing non - abelian field strength tensor . similar ideas have been used recently to map the sinh - gordon theory onto string theory in anti de sitter space ads@xmath112 @xcite , or the gn model onto string theory @xcite . we introduce a gauge transformation @xmath113 relating @xmath94 and @xmath111 as follows , @xmath114 upon identifying @xmath97 with @xmath86 , we find ( modulo an arbitrary normalization factor ) @xmath115 with the tdhf spinors at hand , we are now in a position to address the issue of self - consistency . let us start with the continuum spinors . using the gauge transformation ( [ c47],[c48 ] ) , we first write @xmath116 with @xmath117 . notice that the linear combinations @xmath118 are actually simpler than @xmath119 . the normalization factor @xmath120 will be chosen such as to recover the free dirac spinor ( [ c4 ] ) at @xmath121 . using @xmath122 this yields @xmath123 the scalar density can now easily be evaluated with the help of eq . ( [ c46 ] ) , @xmath124 owing to the vacuum gap equation , the self - consistency condition is fulfilled by the negative energy continuum states alone , see eqs . ( [ c8],[c9 ] ) . it remains to be shown that the bound states do not destroy this result . if the solitons are far apart , each of them possesses a normalizable bound state . one therefore expects the presence of @xmath0 bound states in the @xmath0 baryon problem . following an observation made in sec . [ sect2 ] in the one soliton case , we try to generate the bound state spinors from the continuum spinors by analytical continuation to imaginary spectral parameters . we find that the bound state originating from the @xmath24-th soliton can indeed be obtained by setting @xmath125 , @xmath126 the fact that the @xmath127-terms in ( [ c50 ] ) have disappeared is instrumental for the normalizability of the bound states . for @xmath128 , @xmath89 vanishes so that the spinor also vanishes . for @xmath129 , @xmath89 increases exponentially but @xmath91 behaves as @xmath130 , so that again the spinor vanishes . according to eqs . ( [ c37 ] ) and ( [ c45 ] ) , @xmath131 and @xmath132 have the phase ( @xmath133 ) , @xmath134 has the phase @xmath135 and @xmath136 is real . this shows already that @xmath137 and @xmath138 are in phase . the components of the dirac - hf spinor for @xmath125 , @xmath139 then differ by a phase @xmath140 so that the scalar density indeed vanishes for the bound states . hence the situation is the same as for the single kink : the valence fermions play no role for the issue of self - consistency . the explicit spinors @xmath141 will be needed nevertheless to evaluate the fermion density . the only missing piece is the normalization constant @xmath142 , to be determined from the integral over the density , @xmath143 it can easily be found by considering times when the solitons are well separated , where it reduces to the one - soliton case , cf . ( [ c16 ] ) , @xmath144 this completes the proof that the @xmath0 soliton solution of the sinh - gordon equation yields a self - consistent solution of the tdhf equation in the gn model . it covers the kink baryon reviewed in sec . [ sect2 ] and the kink - antikink scattering solution of @xcite as special cases . the @xmath0 baryon solution describes the general scattering problem of an alternating succession of @xmath0 kinks and antikinks . each one can carry an arbitrary number of fermions in the allowed range and has arbitrary initial positions and velocities , parametrized by the constants @xmath145 . the fact that this problem can still be solved in closed analytical form , including the polarization of the dirac sea , is remarkable . in the remaining sections we will first cast the results in a form more suitable for applications and then discuss the physics of the @xmath0 baryon collision in more detail . the preceding section contains all the ingredients needed for the full tdhf solution of @xmath0 interacting kinks and antikinks . yet these results are not yet in a form well suited for practical computations with computer algebra . if one tries to evaluate them , for example with maple , one notices that the number of terms increases rapidly with @xmath0 and algebraic manipulations become prohibitive for rather small @xmath0 values already . the aim of the present section is to present an alternative formulation which has proven more convenient for applications . it is adapted from a work of bowtell and stuart on the sine - gordon equation @xcite and makes the structure of the @xmath0-soliton solution more transparent . it also facilitates the computations of time delays in sec . [ sect6 ] and has proven to be a prerequisite for practical calculations of sizeable number of solitons to be discussed in sec . [ sect7 ] . besides developing this approach for both scalar potential and tdhf spinors in general case , we have also included in this section the proof that the total fermion density is proportional to the bound state contribution , generalizing eq . ( [ c24 ] ) to @xmath0 baryons . this will also be of great help for the computations described in sec . [ sect7 ] . we start with the construction of the @xmath0-soliton potential @xmath4 . since @xmath4 and @xmath39 differ only by a @xmath40 transformation , they describe the same physics and we can choose @xmath146 without loss of generality . the single antikink can been written in the form @xmath147 see eq . ( [ c11 ] ) . following the approach of bowtell and stuart in the sine - gordon case @xcite , we first note that @xmath0 non - interacting solitons are described by simply taking the product of @xmath0 one - soliton solutions , @xmath148 with @xmath149 clearly , this 2@xmath0-parameter ansatz will solve the sinh - gordon equation as long as all solitons are far apart . physically it may be thought of as initial or final configuration of an @xmath0 baryon scattering process . @xmath4 exhibits an alternating sequence of @xmath0 kinks and antikinks . its behavior at spatial asymptotics for fixed time is @xmath150 next , we expand the numerator and denominator of @xmath4 . to explain the general construction of the interacting soliton solution , it is sufficient to consider @xmath151 , @xmath152 numerator and denominator are multivariate polynomials of order @xmath0 in the @xmath153 . in order to arrive at the interacting soliton solution , inspect each monomial of numerator and denominator . if it contains @xmath153 and @xmath154 , multiply it by @xmath155 where @xmath156 is the relative velocity of solitons @xmath24 and @xmath157 ( more precisely , the velocity of soliton @xmath24 in the center - of - velocity frame of solitons @xmath24 and @xmath157 ) @xmath158 in our example ( @xmath151 ) , this prescription yields @xmath159 this is already the full 3-soliton solution . in other words , @xmath86 solves the sinh - gordon equation for all values of ( @xmath160 ) . notice that the velocities @xmath161 all have to be chosen differently . if @xmath162 , the result collapses to the @xmath163 soliton solution . more generally , we write the @xmath0 soliton scalar potential as @xmath164 with @xmath165 etc . the relationship between this notation and the one in previous sections is made by the following useful equations @xmath166 one can now check that the functions @xmath167 can equivalently be expressed as determinants , @xmath168 where @xmath169 is the matrix @xmath91 of eq . ( [ c69 ] ) for the @xmath0 soliton case . thus we recover the result ( [ c44 ] ) , confirming that @xmath170 ^ 2 $ ] with @xmath171 from eq . ( [ c67 ] ) is the @xmath0 soliton solution of the sinh - gordon equation . the advantage of the present algorithm is the fact that it is very easy to implement in computer algebra and makes the structure of the potential more transparent . a similar procedure works for the tdhf spinors as well . the continuum spinors for the @xmath0 soliton problem can be written as @xmath172 with @xmath173 to get the bound state which belongs to the @xmath24-th soliton , replace the normalization factor in ( [ c71 ] ) by ( [ c58 ] ) and @xmath23 by @xmath131 , @xmath174 this is significantly simpler than the continuum state , since all monomials in the numerators containing a factor @xmath175 vanish . if one evaluates the fermion densities with maple for small @xmath0 values , one finds that the simple relation between induced and valence fermion density found in the one- and two - soliton cases generalizes to @xmath0 solitons ( @xmath176 ) , @xmath177 hence one can reconstruct the full fermion density from the discrete states alone . ( [ c74 ] ) can be proven for general @xmath0 with the help of cauchy s theorem . since the analytic structure of the fermion density is rather complicated , we demonstrate the corresponding relation for the simpler case of the pseudoscalar condensate . as pointed out in sec . [ sect2 ] , the divergence of the vector and axial currents establishes a close relationship between various fermion bilinears . eliminating the axial density @xmath178 from eqs . ( [ c29 ] ) , we can express the fermion density directly in terms of the pseudoscalar condensate , @xmath179 so that it is sufficient to prove the analogue of eq . ( [ c74 ] ) for @xmath180 . the pseudoscalar density for a single orbit reads @xmath181 for a continuum state [ see eq . ( [ c71 ] ) ] , we get @xmath182 note the useful relations @xmath183 we perform the sum over modes as an integral over the spectral parameter , using the residue theorem in the complex @xmath23 plane . the integrand is an analytic , even function of @xmath23 falling off like @xmath184 at infinity , so that we can extend the @xmath52 integration from @xmath185 to @xmath186 and apply cauchy s theorem . in the lower half - plane there are simple poles at @xmath125 arising from @xmath187 . since @xmath188 is linear in each @xmath187 , we can evaluate the @xmath24-th residue by setting @xmath189 ( the argument @xmath190 is missing in @xmath191 ) . when applying the residue theorem , @xmath192 in this expression has to be evaluated at the pole @xmath125 , @xmath193 so that @xmath194 this amounts to substituting @xmath195_{\zeta=\zeta_k}. \label{c82}\ ] ] inserting the non - singular factors evaluated at the pole and summing over all poles at @xmath125 , the residue theorem gives the following contribution from the continuum states to the pseudoscalar condensate @xmath196 for the bound states on the other hand , a straightforward evaluation of the pseudoscalar condensate using the wave functions ( [ c73 ] ) yields @xmath197 so that the total condensate becomes @xmath198 where @xmath199 denotes the @xmath24-th term in the sum of eq . ( [ c84 ] ) . due to ( [ c75 ] ) this also proves ( [ c74 ] ) . summarizing this section , we note that the most important results are eqs . ( [ c67 ] ) for the scalar mean field , ( [ c71 ] ) for the continuum spinors and ( [ c73 ] ) for the bound state spinors , together with the constructive algorithm illustrated in eqs . ( [ c68 ] ) and the final expression for the fermion density , eq . ( [ c74 ] ) this is the basis for all the concrete applications discussed in sec . [ sect7 ] . moreover , eq . ( [ c67 ] ) is helpful for deriving the asymptotics in sec . [ sect6 ] . the only observable in an elastic scattering process in 1 + 1 dimensions is the time delay . here we shall compute the time delay experienced by each of the @xmath0 baryons . in order to determine the time delay , we need the asymptotics of the scalar potential @xmath4 for @xmath200 . as a byproduct , this will teach us how to translate the parameters @xmath201 into initial positions and velocities of the baryons . we order the solitons according to the velocities @xmath161 , @xmath202 then for @xmath203 , @xmath4 describes @xmath0 incoming ( anti-)solitons with the functional form ( only valid in the vicinity of the corresponding soliton ) @xmath204 ( @xmath205 ) . they are ordered from left to right , starting with an anti - soliton . for @xmath206 , @xmath4 describes @xmath0 outgoing ( anti-)solitons with the functional form ( again only valid in the vicinity of each soliton ) @xmath207 ( @xmath205 ) . they are also ordered from left to right , starting with an anti - soliton . if one follows the baryon density , one finds that it is exchanged in each two - body collision . this is a direct consequence of the fact that the scalar potential is transparent . hence a particular fermion cluster gets transferred from the incoming soliton @xmath24 to the outgoing soliton @xmath208 ; the spatial order is inverted . physically relevant is presumably only the time delay for the fermion clusters , not the ( anti-)kinks . this is equivalent to computing the time delay from the asymptotic form of @xmath4 , comparing kinks with the same @xmath153 at @xmath209 . the result for @xmath210 is @xmath211 for @xmath212 and @xmath213 , one finds @xmath214 in the special case of two solitons in the center - of - velocity frame , we recover the result of @xcite @xmath215 in the soliton literature , one also introduces a phase shift " related to the time delay by @xcite @xmath216 the total phase shift is the sum of the phase shifts induced by independent collisions with all other solitons . the @xmath217 satisfy @xmath218 to specify the initial conditions , it is helpful to note the equation of motion of the @xmath24-th incoming soliton , @xmath219 similarly , the equation of motion of the @xmath24-th outgoing soliton ( numbered in inverse order , i.e. , according to the fermion clusters they carry ) reads @xmath220 denoting the initial time by @xmath221 , the initial positions of the solitons are given by @xmath222 this tells us how to choose the parameters @xmath108 , given the initial soliton velocities and positions , namely as @xmath223 with @xmath224 from eq . ( [ c94 ] ) . what happens if one prepares @xmath0 alternating , well separated kinks and antikinks with different initial velocities , carrying different numbers of fermions or antifermions ? we are now in a position to predict the time evolution of this initial configuration in the gn model . in general , it would be very hard to characterize such a complex collision process . in our case there are several simplifying features which enable us to draw a full picture . the initial and final states of an @xmath0-body collision may be described as in the previous section the solitons are widely spaced and ordered according to their velocities , the fastest one being leftmost in the incoming and rightmost in the outgoing state . we illustrate such a process schematically in fig . [ fig1 ] for the case of @xmath225 solitons , in a frame where all velocities @xmath161 are positive . since a kink and an antikink can not pass through each other , it looks as if the solitons repel and stay in the same order . however , due to the fact that the self - consistent potential is transparent , the fermions carried by each kink or antikink can only move forward . in every two - soliton collision , the fermions get exchanged as discussed in @xcite . inelastic processes are suppressed due to the integrability of the gn model . in fig . [ fig1 ] , the fermions move roughly along the straight lines ( ignoring interaction effects ) . the intersection points of two straight lines signal two - body collisions . obviously , every baryon interacts with every other one exactly once . the complete time evolution of @xmath4 including interaction effects is shown in fig . [ fig2 ] , where one recognizes time delays . [ fig3 ] shows the corresponding time evolution of the fermion density . to simplify the picture , we have assumed that solitons 1 and 4 have maximal fermion number @xmath226 , whereas solitons 2 and 3 carry no fermions at all . we see that the fast , lorentz contracted fermion cluster of soliton 1 passes through the collision zone almost unaffected . the wider peak corresponding to the slower fermions of soliton 4 suffers stronger interaction effects , being also scattered by the empty " solitons 2 and 3 . if we had loaded any of the solitons with antifermions by choosing an occupation of the valence level @xmath227 , we would observe that fermions and antifermions also pass through each other , due to the absence of annihilation processes . note also that the graph shown in fig . [ fig2 ] is independent of the fermion content of the solitons . it would even hold in the case where all solitons have vanishing fermion number , so that neither baryons nor bosons are involved . nevertheless we would be dealing with a valid solution of a quantum field theory . this underlines the non - perturbative character of the whole approach . let us now consider some further illustrative examples . our original motivation for studying the gn model came form strong interaction physics . in real life , natural many - baryon problems would involve nuclei . if the 2-soliton scattering is taken as a toy model for nucleon - nucleon scattering , one would like to address next nucleon - nucleus or nucleus - nucleus collisions at the elementary fermion level . unfortunately , the gn model has no nuclei " , i.e. , bound states of baryons . the baryon - baryon interaction is repulsive . nuclear matter " exists in the form of a soliton crystal , but it is neither self - bound , nor does it saturate . scattering problems with more than two incident particles on the other hand have no obvious analogue in particle physics . therefore , the best we can do to mock up nuclear targets or projectiles in our toy world is to use trains " of solitons with nearly equal velocities . although unstable , such a configuration will stay together for a time long enough to study scattering processes . these trains of solitons may be thought of as chunks of soliton crystals ( nuclear matter " ) . in applications of the present model to other fields like condensed matter physics , the interest would presumably be in a different kind of @xmath0-soliton problem . the formulae given in sec . [ sect5 ] should enable the reader to produce easily any desired result by choosing appropriate parameters . proceeding in this spirit , we show in figs . [ fig4 ] and [ fig5 ] an example of the analogue of a baryon - nucleus collision for 1 + 5 solitons , in the ( approximate ) rest frame of the target nucleus " . the kinks behave much like classical hard spheres , i.e. , the incoming projectile gets stopped when it hits the first target baryon , and the last target baryon leaves , carrying away the momentum . this can be inferred from the scalar potential in fig . [ fig4 ] . to illustrate the fate of the fermions , we fully load the projectile and target baryons with @xmath226 fermions each . as shown in fig . [ fig5 ] , the fast projectile fermions then hop from one soliton to the next one repeatedly during the collision , until they emerge in the emitted , rightmost soliton and move along with it . owing to the relativistic invariance of the formalism we can study these collision processes in any desired lorentz frame . in our last example , we choose the center - of - mass frame of two nuclei " , each one consisting of 4 solitons carrying the maximal number of fermions . this is the closest we can come to simulate a relativistic nucleus - nucleus collision " in the gn model . figs . [ fig6 ] and [ fig7 ] show again that the solitons repel each other , whereas the fermions keep moving forward . a combination of integrability and transparent mean field is behind this simple scenario . by comparing fig . [ fig6 ] for the scalar potential and fig . [ fig7 ] for the density , one may be tempted to conclude that both figures show multi - soliton collisions . indeed , in both cases all the lumps emerge unchanged from the collision process . however this interpretation is only valid for @xmath4 and the underlying sinh - gordon equation . that the density has no solitonic character already follows from the fact that the normalization of each fermion cluster can be arbitrarily chosen . formally , whereas @xmath4 is the solution of a non - linear differential equation , the density can be thought to arise from a linear equation where @xmath4 enters as an external field , similar to the spinors in the tdhf equation . in any case , the fact that our solitons carry fermions is an interesting aspect not shared by standard applications of solitons in physics , but reminiscent of early soliton bag models @xcite in 3 + 1 dimensions . the observation that solitons share some properties with elementary particles is as old as soliton theory . in the gn model , this relationship can now be made very precise . the underlying quantum field theory is purely fermionic . it produces dynamically multi - fermion bound states . in the large @xmath1 limit , the appropriate semiclassical setting is the relativistic hf approach . the scalar hf potential is a classical field with solitonic character , but the bound fermions are also relevant for understanding the structure of hadrons " . this is of course well known since long time already . the new insight which we can add now is the fact that for a certain class of particularly simple solutions ( called type i ) , the whole dynamics can be decoupled from the fermions and cast into the form of a non - linear differential equation for the scalar mean field . this equation turns out to be the sinh - gordon equation . apparently one can bypass the complicated self - consistency issue for these particular solutions and arrive at the self - consistent solution by just solving a single , nonlinear differential equation for the master field " @xmath4 . the fermions then follow the motion of the solitons , but do not react back in any way . since the relevant soliton equation is well known , this enabled us to solve a rather involved problem in closed analytical form , namely the dynamics of @xmath0 kink and antikink baryons with arbitrary fermion number , initial positions and velocities . we have analyzed this type of scattering process and found that it has many unrealistic features from the point of view of particle physics . however , here we have no choice since we are not dealing with a phenomenological model , but the solution of a given quantum field theory , eq . ( [ a1 ] ) , in the large @xmath1 limit . actually , examples in theoretical physics where the dynamics of a number of composite particles can be analyzed exactly at the elementary constituent level are extremely rare , even in non - relativistic many - body physics . in our case , lorentz covariance is strictly maintained and the polarization of the dirac sea fully taken into account . the methods developed here in a concrete example may have some potential for generalizations . one striking observation is the fact that the tdhf spinors are apparently closely related to auxiliary spinors introduced in soliton theory when one looks for solutions via the inverse scattering method . it is very likely that there is a more general principle behind this apparent coincidence . it was certainly important that we restricted ourselves to type i solutions of the tdhf equations . all other analytically known solutions of the massless or massive gn model are actually type ii and therefore related to the @xmath228 classical gn model . it would be interesting to generalize our approach to this more general case , thereby extending the pool of exact solutions , perhaps even to non - integrable field theories like the massive gn model . this work has been supported in part by the dfg under grant th 842/1 - 1 . 99 d. j. gross and a. neveu , phys . d * 10 * , 3235 ( 1974 ) . v. schn and m. thies , _ at the frontier of particle physics : handbook of qcd , boris ioffe festschrift _ , vol . m. shifman ( singapore : world scientific ) , ch . 33 , p. 1945 j. feinberg , ann . ( n.y . ) * 309 * , 166 ( 2004 ) . m. thies , j. phys . a : math . gen . * 39 * , 12707 ( 2006 ) . e. witten , in _ recent developments in gauge theories _ , 1979 cargese lectures , ed . g. t hooft et al . , plenum press , n.y . r. f. dashen , b. hasslacher and a. neveu , phys . d * 12 * , 2443 ( 1975 ) . m. thies , phys . d * 69 * , 067703 ( 2004 ) . o. schnetz , m. thies , k. urlichs , ann . * 314 * , 425 ( 2004 ) . a. klotzek and m. thies , j. phys . a * 43 * , 375401 ( 2010 ) . a. neveu and n. papanicolaou , comm . * 58 * , 31 ( 1978 ) . d. j. gross , in les houches 1975 , proceedings , _ methods in field theory _ , amsterdam ( 1976 ) , p. 141 - 250 m. j. ablowitz , d. j. kaup , a. c. newell , h. segur , stud . appl . math . * 53 * , 249 ( 1974 ) . a. k. pogrebkov , lett . phys . * 5 * , 277 ( 1981 ) . a. jevicki and k. jin , jhep 0906 , 064 ( 2009 ) . a. b. zamolodchikov and al . b. zamolodchikov , phys . b * 72 * , 481 ( 1978 ) . r. shankar and e. witten , nucl . b * 141 * , 349 ( 1978 ) . m. karowski and h. j. thun , nucl . b * 190 * , 61 ( 1981 ) . j. feinberg and a. zee , phys . b * 411 * , 134 ( 1997 ) . m. thies and k. urlichs , phys . d * 71 * , 105008 ( 2005 ) . m. thies and k. urlichs , phys . d * 72 * , 105008 ( 2005 ) . o. schnetz , m. thies , and k. urlichs , ann . ( n.y . ) * 321 * , 2604 ( 2006 ) . a. klein , phys . d * 14 * , 558 ( 1976 ) . r. pausch , m. thies , and v. l. dolman , z. phys . a * 338 * , 441 ( 1991 ) . j. feinberg , phys . d * 51 * , 4503 ( 1995 ) . w. brendel and m. thies , phys . d * 81 * , 085002 ( 2010 ) . r. jackiw and c. rebbi , phys . d * 13 * , 3398 ( 1976 ) . su , j. r. schrieffer , and a. heeger , phys . lett . * 42 * , 1698 ( 1979 ) . r. jackiw and j. r. schrieffer , nucl . b * 190 * , 253 ( 1981 ) . f. karbstein and m. thies , phys . d * 76 * , 085009 ( 2007 ) . g. bowtell and a. e. g. stuart , phys . d * 15 * , 3580 ( 1977 ) . r. hirota , j. phys . japan * 33 * , 1459 ( 1972 ) . w. a. bardeen , m. s. chanowitz , s. d. drell , m. weinstein , and t .- m . yan , phys . d * 11 * , 1094 ( 1975 ) .

recently it was shown that kink baryons and kink - antikink scattering in the massless gross - neveu model are closely related to one- and two - soliton solutions of the sinh - gordon equation . here we generalize these findings to the case of @xmath0 kinks and antikinks . using the known @xmath0 soliton solution of the sinh - gordon equation , we solve the general @xmath0 kink - antikink scattering problem in the large @xmath1 gross - neveu model analytically , mapping the time - dependent hartree - fock approach onto inverse scattering theory .

the mass - radius relationship of neutron stars ( nss ) is of prime importance to understand the high - density low - temperature regime of the hadronic equation of state ( eos ) . depending on this relationship , certain models for the hadronic eos can either be confirmed or ruled out . several attempts have been made on measuring the radii and masses of nss to constrain the uncertainties in the high density behavior of the eos . the observations on double nss @xcite , glitches in radio pulsars @xcite , thermal emission @xcite from accreting nss and from millisecond x - ray pulsars lead to constraints on mass - radius relationship of nss . recently the pressure of neutron star matter at supranuclear density is measured by zel et al . @xcite directly from observations using advanced astrophysical techniques and ns atmosphere modeling . the pressure extracted from ns mass - radius data crucially constrains the extension of the eos to high density low temperature regime for stellar matter ruling out those who fail to comply with the recent mass - radius data . the quark matter can support a star as massive as @xmath32 m@xmath2 only if the quarks are strongly interacting and are therefore not ` free ' quarks . to overcome this situation , dexheimer et al . @xcite have recently employed a hadronic su(3 ) sigma - omega model including delta - resonances and hyperons to describe the properties of neutron star matter by softer equation of state . delta - resonances have a repulsive vector potential which works to counteract gravity in a compact star . they successfully reproduce both the measured mass - radius relationship and the extrapolated eos by slightly lowering the coupling strength of the delta resonances to the vector mesons . in our previous works , the density dependent m3y effective interaction ( ddm3y ) which provides a unified description of the elastic and inelastic scattering @xcite , cluster @xcite , @xmath4 @xcite and proton radioactivities @xcite , the symmetric and asymmetric nuclear matter @xcite , was employed to obtain nucleonic eos of the @xmath0-equilibrated ns matter @xcite . at high densities , the energy density of this @xmath0-equilibrated charge neutral ns matter is higher than that of quark matter signalling the onset of deconfinement transition to quark matter inside the star . in the present work , we solve the einstein s equations for rotating stars using pure nuclear matter without and with quark matter core . a systematic study of the static as well as rotating compact stars with quark matter inside is presented in view of the recent observations of the massive compact stars . we shall find later that the present eos unlike other eos @xcite can explain successfully the recently observed mass - radius data . the effect of the presence of the quark core on the determination of maximum mass of ns will be investigated for both static and rotating stars . with the energies and interaction rates foreseen at fair , the compressed baryonic matter ( cbm ) will create highest baryon densities in nucleus - nucleus collisions to explore the properties of superdense baryonic matter and the in - medium modifications of hadrons . the compact stars provide natural testing laboratory for highly compressed matter . the stiffness of the high - density matter controls the maximum mass of compact stars . the analyses of mass - radius data on nss by zel et al . @xcite favor smaller masses lying within 1.6 - 1.9 m@xmath2 with radii 8 - 10 kilometers . recent mass measurement of the binary millisecond pulsar j1614 - 2230 by p.b . demorest et al . @xcite rules out the eos which fail to predict the masses within 1.97@xmath10.04 m@xmath2 . most of the currently proposed eos @xcite involving exotic matter , such as kaon condensates or hyperons failed to produce such a massive star . the measured mass of psr j1748 - 2021b , a millisecond pulsar in the globular cluster ngc 6440 , is claimed to be as high as 2.74@xmath5m@xmath2 ( 2@xmath6 ) @xcite . moreover , a pulsar rotating faster ( e.g. , psr j17482446ad ) than the limit set by the r - mode instability has already been observed @xcite . further observations and a better r - mode modeling may shed more light on this issue . if rapidly rotating compact stars were nonaxisymmetric , they would emit gravitational waves in a very short time scale and settle down to axisymmetric configurations . therefore , we need to solve for rotating and axisymmetric configurations in the framework of general relativity . for the matter and the spacetime the following assumptions are made . the matter distribution and the spacetime are axisymmetric , the matter and the spacetime are in a stationary state , the matter has no meridional motions , the only motion of the matter is a circular one that is represented by the angular velocity , the angular velocity is constant as seen by a distant observer at rest and the matter can be described as a perfect fluid . the energy - momentum tensor of a perfect fluid @xmath7 is given by @xmath8 where @xmath9 , @xmath10 , @xmath11 and @xmath12 are the energy density , pressure , four velocity and the metric tensor , respectively . to study the rotating stars the following metric is used @xmath13 where the gravitational potentials @xmath14 , @xmath15 , @xmath4 and @xmath16 are functions of polar coordinates @xmath17 and @xmath18 only . the einstein s field equations for the three potentials @xmath14 , @xmath15 and @xmath4 have been solved using the greens - function technique @xcite and the fourth potential @xmath16 has been determined from other potentials . all the physical quantities may then be determined from these potentials . obviously , at the zero frequency limit corresponding to the static solutions of the einstein s field equations for spheres of fluid , the present formalism yields the results for the solution of the tolman - oppenheimer - volkoff ( tov ) equation @xcite . we use the ` rns ' code @xcite for calculating the compact star properties which requires eos in the form of energy density versus pressure along with corresponding enthalpy and baryon number density and since we are using various eos for different regions , these are smoothly joined . the different regions of a compact star are governed by different eos . these can be broadly divided into two regions : a crust that accounts for about 5@xmath19 of mass and about 10@xmath19 of the radius of a star and the core is responsible for the rest of the mass and radius of a star . the outer layers are a solid crust @xmath3 1 km thick , consisting , except in the outer few meters , of a lattice of bare nuclei immersed in a degenerate electron gas . as one goes deeper into the crust , the nuclear species become , because of the rising electron fermi energy , progressively more neutron rich , beginning ( ideally ) as @xmath20fe through @xmath21kr at mass density @xmath224.3@xmath2310@xmath24 g @xmath25 . at this density , the ` neutron drip ' point , the nuclei have become so neutron rich that with increasing density the continuum neutron states begin to be filled , and the lattice of neutron - rich nuclei becomes permeated by a sea of neutrons . the eos that cover the crustal region of a compact star are feynman - metropolis - teller ( fmt ) @xcite , baym - pethick - sutherland ( bps ) @xcite and baym - bethe - pethick ( bbp ) @xcite . the most energetically favorable nucleus at low densities is @xmath20fe , the endpoint of thermonuclear burning . the fmt is based on fermi - thomas model to derive the eos of matter at high pressures and covers the outermost crust which is essentially made up of iron and a fraction of the electrons bound to the nuclei . the major difficulty in deriving the equation of state is the calculation of the electronic energy . at subnuclear densities , from about 10@xmath26 g @xmath25 up to the neutron drip density 4.3@xmath2310@xmath24 g @xmath25 the eos of bps is applicable which includes the effects of the lattice coulomb energy on the equilibrium nuclide . the domain from neutron drip density to about nuclear density 2.5@xmath2310@xmath27 g @xmath25 , is composed of nuclei , electrons and free neutrons where eos of bbp is applicable which is based on a compressible liquid drop model of nuclei with conditions that nuclei must be stable against @xmath0-decay and free neutron gas must be in equilibrium with neutrons in nuclei . the nuclear matter eos is calculated @xcite using the isoscalar and the isovector components of m3y interaction along with density dependence . the density dependence of the effective interaction , ddm3y , is completely determined from nuclear matter calculations . the equilibrium density of the nuclear matter is determined by minimizing the energy per nucleon . the energy variation of the zero range potential is treated accurately by allowing it to vary freely with the kinetic energy part @xmath28 of the energy per nucleon @xmath29 over the entire range of @xmath29 . in a fermi gas model of interacting neutrons and protons , the energy per nucleon for isospin asymmetric nuclear matter @xcite is given by @xmath30 f(x ) + ( \frac{\rho j_v c}{2 } ) ( 1 - \beta\rho^{\frac{2}{3 } } ) \label{seqn3}\ ] ] where @xmath31=@xmath32 which equals fermi momentum in case of symmetric nuclear matter ( snm ) , the isospin asymmetry @xmath33 with @xmath34 , @xmath35 and @xmath36 being the neutron and proton number densities respectively , and the kinetic energy per nucleon @xmath28=@xmath37 f(x)$ ] with @xmath38=@xmath39 $ ] . @xmath40=@xmath41 , @xmath42 and @xmath43 represent the volume integrals of the isoscalar and the isovector parts of the m3y interaction . the isoscalar @xmath44 and the isovector @xmath45 components of m3y interaction potential are , respectively , given by @xmath46 and @xmath47 where @xmath48 mev@xmath49 , @xmath50 mev@xmath49 , @xmath51 mev@xmath52 . the ddm3y effective nn interaction is given by @xmath53 where the density dependence @xmath54 and the constants @xmath55 and @xmath0 of density dependence have been obtained from the saturation condition @xmath56 at @xmath57 and @xmath58 where @xmath59 and @xmath60 are the saturation density and the saturation energy per nucleon respectively @xcite . this eos evaluated at the isospin asymmetry @xmath61 determined from the @xmath0-equilibrium proton fraction @xmath62 [ @xmath63 , obtained by solving @xmath64 , provides eos for the @xmath0-equilibrated ns matter where nuclear symmetry energy @xmath65 is given by @xmath66 the areas enclosed by the continuous and dashed lines in fig.-1 correspond to the pressure regions for neutron matter consistent with the experimental flow data after inclusion of the pressures from asymmetry terms with weak ( soft nm ) and strong ( stiff nm ) density dependences , respectively @xcite . although , the parameters of the density dependence of ddm3y interaction are tuned to reproduce the saturation energy per nucleon @xmath67 and the saturation density @xmath68 of symmetric nuclear matter which are obtained from finite nuclei , the agreement of this eos with the experimental flow data @xcite , where the high density behaviour looks phenomenologically confirmed , justifies its extrapolation to high density @xcite . for cold and dense quark ( qcd ) matter , the perturbative eos @xcite with two massless and one massive quark flavors and a running coupling constant , is used . the constant @xmath69 is treated as a free parameter , which allows to take into account non - perturbative effects not captured by the weak coupling expansion . in fact , using the free quark number density , one recovers the expression for the pressure in the original mit bag model @xcite , with @xmath69 taking the role of the bag constant . due to physics criteria ( e.g. requiring the energy density to be positive ) , the possible values for @xmath69 are , however , typically rather restricted , allowing to make quantitative statements that are not possible in the original mit bag model . the rotating compact star calculations are performed using the crustal eos , fmt + bps + bbp upto number density of 0.0458 @xmath70 and @xmath0-equilibrated ns matter beyond . it is worthwhile to mention here that a star may not rotate as fast as keplerian frequency due to r - mode instability . there have been suggestions that the r - mode instability may limit the time period to 1.5 ms @xcite . however , a pulsar rotating faster ( e.g. , psr j17482446ad ) than this limit has already been observed @xcite . the variation of mass with central density for static and rotating neutron stars at keplerian limit and also maximum frequencies limited by the r - mode instability with pure nuclear matter inside is shown in fig.-2 . in fig.-3 , the mass - radius relationship for static and rotating neutron stars at keplerian limit and also at maximum frequencies limited by the r - mode instability with pure nuclear matter inside fig.-3 depicts that nss with pure nuclear matter inside , the maximum mass for the static case is 1.92 m@xmath2 with radius @xmath39.7 km and for the star rotating with kepler s frequency it is 2.27 m@xmath2 with equatorial radius @xmath313.1 km @xcite . however , for stars rotating with maximum frequency limited by the r - mode instability , the maximum mass turns out to be 1.95 ( 1.94 ) m@xmath2 corresponding to rotational period of 1.5 ( 2.0 ) ms with radius about 9.9 ( 9.8 ) kilometers @xcite . the energy density of the quark matter is lower than that of the present eos for the @xmath0-equilibrated charge neutral ns matter at densities higher than 0.405 @xmath70 for bag constant @xmath71=110 mev @xcite implying presence of quark core . the energy densities of the present eos for the @xmath0-equilibrated charge neutral ns matter and the quark matter eos for bag constant @xmath71=110 mev are shown in fig.-4 as functions of baryonic densities . for lower values of bag constant such as @xmath71=89 mev , energy density for our eos is lower and makes a cross over with the quark matter eos at very high density @xmath31.2 @xmath70 causing too little quark core and therefore we choose @xmath71=110 mev for representative calculations . the common tangent is drawn for the energy density versus density plots where pressure is the negative intercept of the tangent to energy density versus density plot . however , as obvious from fig.-4 , the phase co - existence region is negligibly small which is represented by part of the common tangent between the points of contact on the two plots @xcite implying constant pressure throughout the phase transition . the variation of mass with central density for static and rotating neutron stars at keplerian limit and also maximum frequencies limited by the r - mode instability with nuclear and quark matter inside is shown in fig.-5 . in fig.-6 , the mass - radius relationship for static and rotating neutron stars at keplerian limit and also at maximum frequencies limited by the r - mode instability with nuclear and quark matter inside fig.-6 depicts that when quark core is considered , the maximum mass for the static case is 1.68 m@xmath2 with radius @xmath310.4 km and for the star rotating with kepler s frequency it is 2.02 m@xmath2 with equatorial radius @xmath314.3 km . in a similar study , it was concluded that compact stars with a quark matter core and an hadronic outer layer , can be as massive as 2.0 m@xmath2 but stay below the pure quark stars and pure neutron stars @xcite . however , they have used two different relativistic mean - field parameter sets tm1 and nl3 @xcite to explore the influence of the hadronic part of the eos whose high density behaviour do not satisfy the criteria extracted from the experimental flow data @xcite . for our case stars rotating with maximum frequency limited by the r - mode instability , the maximum mass turns out to be 1.72 ( 1.71 ) m@xmath2 corresponding to rotational period of 1.5 ( 2.0 ) ms with radius about 10.7 ( 10.6 ) kilometers . it is worthwhile to mention here that the consideration of rotation at the kepler frequency is not of much relevance for a comparison with observational data since even the shortest observed rotation period is still far from the keplerian limit and does not result in a sufficient increase in the maximum mass . the recent observation of psr j1614 - 2230 with a mass of 1.97@xmath10.04 m@xmath2 rotating with a period of 3.1 ms shows that either an eos or transition to quark matter should be excluded that does not allow for star configurations with maximum mass reaching such high values . since the hadronic eos used in the present work is not soft ( incompressibility 274.7@xmath17.4 mev ) @xcite at high densities and neutron star masses reach as high as @xmath32 m@xmath2 for such rotational frequencies , implies that theoretically robust quark matter eos @xcite used in this work is not stiff enough at high densities which was obtained from first principle calculations based on perturbation theory taking terms up to @xmath72 with quark chemical potentials and strange quark mass non - zero . the effects of quark pairing were incorporated by adding to the pressure a term accounting for the condensation energy of the cooper pairs in the color - flavor - locked phase . however , a more detailed and realistic inclusion of the effect of pairing may make quark eos stiffer . in summary , the energy density of the present eos for @xmath0-equilibrated charge neutral ns matter using ddm3y effective nn interaction turns out to be higher than that of quark matter at densities above 0.405 @xmath70 implying possibility of quark core . we have applied our nucleonic eos with a thin crust to solve the einstein s field equations to determine the mass - radius relationship of neutron stars with and without quark cores . we have obtained the masses of neutron ( hybrid ) stars rotating with keplerian frequencies , around 2.27 ( 2.02 ) m@xmath2 with equatorial radii around 13 ( 14 ) kilometres . the result for ns without quark core is in excellent agreement with recent astrophysical observations . the neutron star matter can further undergo deconfinement transition to quark matter , thereby reducing compact star masses considerably . although such hybrid compact stars rotating with kepler s frequency have masses up to @xmath32 m@xmath2 , but rotating with maximum frequency limited by the r - mode instability ( or with 3.1 ms as observed for pulsar j1614 - 2230 ) , the maximum mass @xmath31.7 m@xmath2 turns out to be lower than the observed mass of 1.97@xmath10.04 m@xmath2 and thus rules out quark cores for such massive pulsars but not for pulsars with masses @xmath31.7 m@xmath2 or less . obviously , in order not to conflict with the mass measurement @xcite , either there must be some mechanism to prevent nuclear matter to deconfine into quark matter or the quark eos should be made stiffer by several possible realistic improvements @xcite . the nucleon - nucleon effective interaction used in the present work , which is found to provide a unified description of elastic and inelastic scattering , various radioactivities and nuclear matter properties , also provides an excellent description of the @xmath0-equilibrated ns matter which is stiff enough at high densities to reconcile with the recent observations of the massive compact stars @xmath32 m@xmath2 while the corresponding symmetry energy is supersoft @xcite as preferred by the fopi / gsi experimental data .

-0.0 cm we present a systematic study of the properties of pure hadronic and hybrid compact stars . the nuclear equation of state ( eos ) for @xmath0-equilibrated neutron star matter was obtained using density dependent effective nucleon - nucleon interaction which satisfies the constraints from the observed flow data from heavy - ion collisions . the energy density of quark matter is lower than that of this nuclear eos at higher densities implying the possibility of transition to quark matter inside the core . we solve the einstein s equations for rotating stars using pure nuclear matter and quark core . the @xmath0- equilibrated neutron star matter with a thin crust is able to describe highly massive compact stars but find that the nuclear to quark matter deconfinement transition inside neutron stars causes reduction in their masses . recent observations of the binary millisecond pulsar j1614 - 2230 by p. b. demorest et al . @xcite suggest that the masses lie within 1.97@xmath10.04 m@xmath2 where m@xmath2 is the solar mass . in conformity with recent observations , pure nucleonic eos determines that the maximum mass of ns rotating with frequency below r - mode instability is @xmath31.95 m@xmath2 with radius @xmath310 kilometers . although compact stars with quark cores rotating with kepler s frequency have masses up to @xmath32 m@xmath2 , but if the maximum frequency is limited by the r - mode instability , the maximum mass @xmath31.7 m@xmath2 turns out to be lower than the observed mass of 1.97@xmath10.04 m@xmath2 , by far the highest yet measured with such certainty , implying exclusion of quark cores for such massive pulsars . 0.2 cm _ pacs numbers _ : 26.60.-c , 21.65.cd , 21.65.ef , 26.60.kp , 12.38.-t , 12.39.-x , 21.65.qr -0.0 cm and -0.0 cm -0.29 cm -0.0 cm neutron star ; nuclear eos ; quark eos ; hybrid star . -0.0

a complete mapping of the nucleon excitation spectrum is the key to a detailed understanding of the effective degrees of freedom of the nucleon and its associated dynamics . the most comprehensive predictions of this spectrum have come from various implementations of the constituent quark model incorporating broken su(6 ) symmetry @xcite . additional dynamical contributions from gluonic excitations in the wavefunction may also play a central role @xcite and resonances may be dynamically generated through baryon - meson interactions @xcite . quark model calculations of the nucleon spectrum have predicted more states than have been seen experimentally @xcite . this has been termed the `` missing '' resonance problem , and the existence of these states is tied in directly with the underlying degrees of freedom of the nucleon that govern hadronic production at moderate energies @xcite . ideally we should expect that the fundamental theory that governs the strong interaction , quantum chromodynamics ( qcd ) , should provide a reliable prediction of the nucleon excitation spectrum . however , due to the non - perturbative nature of qcd at these energies , this expectation has not yet been fully realized . there has been notable recent progress in calculations of qcd on the lattice that has led to predictions of the nucleon excitation spectrum with dynamical quarks , albeit with unphysical pion masses @xcite . calculations with improved actions , larger volumes , and smaller quark masses continue to progress . in parallel , the development of coupled - channel models , such as those developed by the groups at bonn - gatchina @xcite , giessen @xcite , jlich @xcite , and ebac @xcite , have made significant progress toward deconvoluting the nucleon spectrum . these multi - channel partial wave analyses have employed partial wave fits from said @xcite based on @xmath10 elastic data to determine the properties of most @xmath11 and @xmath12 resonances listed in the particle data group ( pdg ) @xcite . further critical information on the decay modes was obtained by including the inelastic reactions @xmath13 , @xmath14 , @xmath15 , and @xmath16 . recently the data landscape has undergone significant change with the publication of a vast amount of precision data in the photoproduction sector from jlab , spring-8 , mami , bonn , and graal . data sets spanning a broad angular and energy range for @xmath17 , @xmath18 , @xmath19 , @xmath20 , @xmath21 , @xmath22 , @xmath0 , and @xmath1 have provided high precision differential cross sections and polarization observables . furthermore , new observables with polarized beams on both polarized proton and neutron targets have recently been acquired at several facilities and will be published over the next several years . in the @xmath0 and @xmath1 electroproduction sector , dramatic changes to the world s database occurred with the publications from the clas collaboration . these include ( i ) beam - recoil transferred polarization for @xmath0 @xcite and for @xmath0 and @xmath1 @xcite , ( ii ) separated structure functions @xmath23 , @xmath3 , and @xmath4 for @xmath0 and @xmath1 , as well as @xmath24 and @xmath25 @xcite , and ( iii ) polarized structure function @xmath5 for @xmath0 @xcite . this paper now adds to and extends this database with the largest data set ever acquired in these kinematics for polarized electrons on an unpolarized proton target . this work includes measurements of the separated structure functions @xmath2 , @xmath3 , @xmath4 , and @xmath5 for the @xmath0 and @xmath1 final states at a beam energy of 5.499 gev , spanning @xmath9 from threshold to 2.6 gev , @xmath7 from 1.4 to 3.9 gev@xmath8 , and nearly the full center - of - mass angular range of the kaon . the full set of differential cross sections @xmath26 included in this work consists of 480 ( 450 ) bins in @xmath7 , @xmath9 , and @xmath27 for the @xmath0 ( @xmath1 ) final state and 3840 ( 3600 ) data points in @xmath7 , @xmath9 , @xmath27 , and @xmath6 for @xmath0 ( @xmath1 ) . the organization for this paper is as follows . in section [ theory ] , the different theoretical models that are compared against the data are briefly described . in section [ formalism ] , the relevant formalism for the expression of the electroproduction cross sections and separated structure functions is introduced . section [ analysis ] details the experimental setup and describes all analysis cuts and corrections to the data . section [ systematics ] details the sources of systematic uncertainty on the measured cross sections and separated structure functions , which are presented in section [ results ] along with a series of legendre polynomial fits to the structure function data . finally , we present a summary of this work and our conclusions in section [ conclusions ] . to date the pdg lists only four @xmath11 states , @xmath28 , @xmath29 , @xmath30 , and @xmath31 , with known couplings to @xmath14 and no @xmath11 states are listed that couple to @xmath15 @xcite ; only a single @xmath12 state , @xmath32 , is listed with coupling strength to @xmath15 . the branching ratios to @xmath33 provided for these states are typically less than 10% with uncertainties of the size of the measured coupling . while the relevance of this core set of @xmath11 states in the @xmath34 reaction has long been considered a well - established fact , this set of states falls short of reproducing the experimental results below @xmath9=2 gev . furthermore , recent analyses @xcite have called the importance of the @xmath29 state into question . beyond the core set of @xmath11 states , the pdg lists the @xmath31 state as the sole established @xmath11 near 1900 mev . however , with a 500-mev width quoted by some measurements , it is unlikely that this state by itself could explain the @xmath0 cross sections below @xmath9=2 gev , unless its parameters are significantly different than those given by the pdg . recent analyses @xcite have shown this state to be necessary to describe the clas beam - recoil polarization data @xcite . note that the @xmath31 state is predicted by symmetric quark models and its existence is not expected in diquark models . in the recent fits of @xmath35 data , all @xmath11 resonances found to be necessary to fit the @xmath0 data have been included . however , the existing @xmath1 database is smaller than the @xmath0 database , with significantly larger statistical uncertainties . a recent development in understanding the @xmath11 spectrum was provided by the bonn - gatchina coupled - channel partial wave analysis of the hadronic @xmath10 channels and the photoproduced @xmath36 channels @xcite . this work presents an up - to - date listing of pole parameters and branching fractions for all @xmath11 and @xmath12 states up to @xmath372 gev with uncertainties at the level of a few percent . that analysis provided a list of ( i ) six @xmath11 states with coupling to @xmath14 , @xmath28 , @xmath29 , @xmath38 , @xmath39 , @xmath40 , @xmath31 , ( ii ) five @xmath11 states with coupling to @xmath15 , @xmath38 , @xmath39 , @xmath40 , @xmath31 , @xmath41 , and ( iii ) four @xmath12 states with coupling to @xmath15 , @xmath42 , @xmath43 , @xmath32 , @xmath44 . for more on this list of states that couple to @xmath14 and @xmath15 , see ref . @xcite . the findings of ref . @xcite are based on a significant amount of precision experimental data and the sophisticated coupled - channel fitting algorithms . however , in general , the issue of how to extract nucleon resonance content from open strangeness reactions is a long - standing question . various analyses have led to very different conclusions concerning the set of resonances that contribute ( e.g. compare results from refs . @xcite , @xcite , and @xcite , as well as the statements made regarding the resonant set from ref . @xcite ) . furthermore , lack of sufficient experimental information , incomplete kinematic coverage , and underestimated systematics are still responsible for inconsistencies among the different models that fit the data to extract the contributing resonances and their properties @xcite . the indeterminacy for the open strangeness channels is in contrast to the pionic channels , where the contributing resonances can be more reliably identified by means of a partial wave analysis for @xmath45 gev . in open strangeness channels , this technique is less powerful as the non - resonant background contributions are a much larger fraction of the overall response . several groups have stressed that the importance of the background contributions calls for a framework that accounts for both the resonant and non - resonant processes and that provides for a means to constrain both of these classes of reaction mechanisms independently @xcite . while there have been a number of publications of precision cross sections and spin observables for both the photo- and electroproduction reactions , the vast majority of the theoretical effort has focused on fitting just the photoproduction data . although @xmath33 photoproduction is easier to treat theoretically than @xmath33 electroproduction , and is thus more amenable to a detailed quantitative analysis , the electroproduction reaction is potentially a much richer source of information concerning hadronic and electromagnetic interactions . the electroproduction observables have been shown to yield important complementary insights @xcite . some of the most important aspects of electroproduction include : * the data are sensitive to the internal structure of baryon resonances through the @xmath7 dependence of the electromagnetic form factors of the intermediate hadronic resonances associated with the strangeness production mechanism @xcite . * the structure functions are particularly powerful to gain control over the parameterization of the background diagrams @xcite . * studies of finite @xmath7 processes are sensitive to both transverse and longitudinal virtual photon couplings , in contrast to the purely transverse response probed in the photoproduction reactions . * the longitudinal / transverse interference structure functions provide signatures of interfering partial wave strengths that are often dramatic and have been shown to be useful for differentiating between models of the production amplitudes @xcite . * the beam - recoil transferred polarizations in the @xmath0 and @xmath1 reactions , as well as the recoil polarization in the @xmath0 reaction , have been shown to provide important new constraints to models that describe well the photoproduction data @xcite . at the medium energies of this work , perturbative qcd is not yet capable of providing predictions of differential cross sections . to understand the underlying physics , effective models must be employed that represent approximations to qcd . ultimately , it will be most appropriate to compare the electroproduction measurements against the results of a full coupled - channel partial wave analysis that is constrained by fits to the available data . although output from such models is expected in the electroproduction sector in the future @xcite , as of now , these data have not yet been included in the fits . thus comparisons of the electroproduction observables to single - channel models currently represent the best option to gain insight into the electroproduction realm . this analysis highlights three different theoretical model approaches . the first is a traditional hadrodynamic model and the second is based on @xmath46 and @xmath47 regge trajectory exchange . the third model , a hybrid regge plus resonance approach , amounts to a cross between the first two model types . comparison of the different model predictions to the data can be used to provide indirect support for the existence of the different baryonic resonances and their branching ratios into the strange channels , as well as to improve constraints on the phenomenology of the different strangeness production reactions . the following subsections provide a brief description of the models included in this work . hadrodynamic models provide a description of the reaction based on contributions from tree - level born and extended born terms in the @xmath48 , @xmath49 , and @xmath50 reaction channels ( see fig . [ born ] ) . the born diagrams include the exchange of the proton , kaon , and ground - state hyperons , while the extended born diagrams include the exchange of the associated excited states . this description of the interaction , which involves only first - order terms , is sensible as the incident and outgoing electrons interact rather weakly with the hadrons . a complete description of the physics processes requires taking into account all possible channels that could couple to the initial and final states , but the advantages of the tree - level approach are to limit complexity and to identify the dominant trends . the drawback in this class of models is that very different conclusions about the strengths of the contributing diagrams may be reached depending on which set of resonances a given model includes . maxwell _ et al . _ @xcite have developed a tree - level effective lagrangian model ( referred to as mx ) for @xmath51 that incorporates the well - established @xmath48-channel resonances up to 2.2 gev with spins up to 5/2 . the model also includes four @xmath52 @xmath50-channel states , @xmath53 , @xmath54 , @xmath55 , @xmath56 , four @xmath57 @xmath50-channel states , @xmath58 , @xmath59 , @xmath60 , @xmath61 , and the @xmath62 and @xmath63 @xmath49-channel resonances . the model was initially developed and fit to the available @xmath36 photoproduction data up to @xmath9=2.3 gev @xcite . the most recent published version of the model @xcite included fits to the available @xmath0 separated structure function data from clas @xcite . an extension of this model that also includes fits to the available clas @xmath0 @xmath5 data has been made available for this work as well . overall the fits yield reasonable representations of both the photo- and electroproduction data . however , when compared to the results of a fit to the photoproduction data alone , the combined @xmath36 and @xmath64 fit yields significantly different coupling parameters for an equally good overall fit to the data . this indicates that the photoproduction data alone are not adequate to uniquely constrain effective lagrangian models of electromagnetic strangeness production . our @xmath33 electroproduction data are also compared to the regge model from guidal , laget , and vanderhaeghen @xcite ( referred to as glv ) . this calculation includes no baryon resonance terms at all . instead , it is based only on gauge - invariant @xmath49-channel @xmath46 and @xmath47 regge - trajectory exchange . it therefore provides a complementary basis for studying the underlying dynamics of strangeness production . it is important to note that the regge approach has far fewer parameters compared to the hadrodynamic models . these include the @xmath46 and @xmath47 form factors and the coupling constants @xmath65 and @xmath66 . the glv model was fit to higher - energy photoproduction data where there is little doubt of the dominance of kaon exchanges , and extrapolated down to jlab energies . an important feature of this model is the way gauge invariance is achieved for the kaonic @xmath49-channel exchanges by reggeizing the @xmath48-channel nucleon pole contribution in the same manner as the @xmath49-channel diagrams . no counter terms need to be introduced to restore gauge invariance as is done in the hadrodynamic approach . the glv regge model reasonably accounts for the strength in the clas @xmath0 differential cross sections and separated structure functions @xcite . although the reasonable performance of a pure regge description in this channel suggests a @xmath49-channel dominated process , there are obvious discrepancies between the regge predictions and the data , indicative of @xmath48-channel strength . in the @xmath1 channel , the same regge description significantly underpredicts the differential cross sections and separated structure functions @xcite . the fact that the regge model fares poorly when compared to the @xmath1 data is indicative that this process has a much larger @xmath48-channel content compared to @xmath0 production . the final model included in this work was developed by the ghent group @xcite , and is based on a tree - level effective field model for @xmath0 and @xmath1 photoproduction from the proton . it differs from traditional isobar approaches in its description of the non - resonant diagrams , which involve the exchange of @xmath46 and @xmath47 regge trajectories . a selection of @xmath48-channel resonances is then added to this background . this `` regge plus resonance '' model ( referred to as rpr ) has the advantage that the background diagrams contain only a few parameters that are tightly constrained by high - energy data . furthermore , the use of regge propagators eliminates the need to introduce strong form factors in the background terms , thus avoiding the gauge - invariance issues associated with traditional effective lagrangian models . in addition to the kaonic trajectories to model the @xmath49-channel background , the rpr model includes the same @xmath48-channel resonances as for the mx model below 2 gev . the model does include several missing @xmath11 states at 1.9 gev , @xmath67 , @xmath67 , and @xmath68 . the separated structure functions @xcite and beam - recoil transferred polarization data from clas @xcite were compared to model variants with either a @xmath67 or a @xmath68 state at 1.9 gev . only the @xmath67 state assumption could be reconciled with the data , whereas the @xmath68 option could clearly be rejected . in the @xmath1 channel , four @xmath12 states , @xmath69 , @xmath42 , @xmath43 , and @xmath32 , have been included . in a new version of the rpr model ( referred to as rpr-2011 ) @xcite , several changes relative to the previous model version ( referred to as rpr-2007 ) @xcite are noteworthy . the main difference is the implementation of an unbiased model selection methodology based on bayesian inference . this inference is used as a quantitative measure of whether the inclusion of a given set of @xmath11 states is justified by the data . additionally , in this version of the model , the exchange of spin-3/2 resonances is described within a consistent interaction theory and the model has been extended to include the exchange of spin 5/2 resonances . the regge background amplitude of rpr-2007 is constrained by spectra above the resonance region ( @xmath70 gev ) at forward angles ( @xmath71 ) . by extrapolating the resulting amplitude to smaller @xmath9 , one gets a parameter free background for the resonance region . the @xmath48-channel resonances are coherently added to the background amplitude . rpr-2007 describes the data for forward - angle photo- and electroproduction of @xmath0 and @xmath1 . the resonance parameters of the rpr-2007 model are constrained to the @xmath71 data . the rpr-2011 model with the highest evidence has nine well - established @xmath11 states and the `` missing '' states at 1.9 gev with quantum numbers @xmath67 and @xmath68 , and has been fit to photoproduction data over the full @xmath72 center - of - mass ( c.m . ) angular range . neither version of the model has been constrained by fits to any of the electroproduction data . in kaon electroproduction a beam of electrons with four - momentum @xmath73 is incident upon a fixed proton target of mass @xmath74 , and the outgoing scattered electron with momentum @xmath75 and kaon with momentum @xmath76 are measured . the cross section for the exclusive @xmath77 final state is then differential in the scattered electron momentum and kaon direction . under the assumption of single - photon exchange , where the virtual photon has four - momentum @xmath78 , this can be expressed as the product of an equivalent flux of virtual photons and the @xmath64 c.m . virtual photoabsorption cross section as : @xmath79 where the virtual photon flux factor @xmath80 depends upon only the electron scattering process . after integrating over the azimuthal angle of the scattered electron , the absorption cross section can be expressed in terms of the variables @xmath7 , @xmath9 , @xmath81 , and @xmath6 , where @xmath82 is the squared four - momentum of the virtual photon , @xmath83 is the total hadronic energy in the c.m . frame , @xmath81 is the c.m . kaon angle relative to the virtual photon direction , and @xmath6 is the angle between the leptonic and hadronic production planes . a schematic illustration of electron scattering off a proton target , producing a final state electron , @xmath72 , and hyperon @xmath84 is shown in fig . [ fig - kin ] . introducing the appropriate jacobian , the form of the cross section can be rewritten as : @xmath85 where @xmath86 is the flux of virtual photons ( using the definition from ref . @xcite ) , @xmath87 is the polarization parameter of the virtual photon , and @xmath88 is the electron scattering angle in the laboratory frame . for the case of an unpolarized electron beam ( helicity @xmath89=0 ) with no target or recoil polarizations , the virtual photon cross section can be written ( using simplifying notation for the differential cross section ) as : @xmath90 where @xmath91 are the structure functions that measure the response of the hadronic system and @xmath92 , @xmath93 , @xmath94 , and @xmath95 represents the transverse , longitudinal , and interference structure functions . the structure functions are , in general , functions of @xmath7 , @xmath9 , and @xmath81 only . in this work the unseparated structure function is defined as @xmath23 . in contrast to the case of real photons , where there is only the purely transverse response , virtual photons allow longitudinal , transverse - transverse , and longitudinal - transverse interference terms to occur . each of the structure functions is related to the coupling of the hadronic current to different combinations of the transverse and longitudinal polarization of the virtual photon . @xmath24 is the differential cross section contribution for unpolarized transverse virtual photons . in the limit @xmath96 , this term must approach the cross section for unpolarized real photons . @xmath25 is the differential cross section contribution for longitudinally polarized virtual photons . @xmath4 and @xmath3 represent contributions to the cross section due to the interference of transversely polarized virtual photons and from transversely and longitudinally polarized virtual photons , respectively . for the case of a polarized electron beam with helicity @xmath89 , the cross section form of eq.([sigma0 ] ) is modified to include an additional term : @xmath97 the electron beam polarization produces a fifth structure function @xmath5 that is related to the beam helicity asymmetry via : @xmath98 where the @xmath99 superscripts on @xmath100 correspond to the electron helicity states of @xmath101 . the polarized structure function @xmath5 is intrinsically different from the structure functions of the unpolarized cross section . this term is generated by the imaginary part of terms involving the interference between longitudinal and transverse components of the hadronic and leptonic currents , in contrast to @xmath3 , which is generated by the real part of the same interference . @xmath5 is non - vanishing only if the hadronic tensor is antisymmetric , which will occur in the presence of rescattering effects , interferences between multiple resonances , interferences between resonant and non - resonant processes , or even between non - resonant processes alone @xcite . @xmath5 could be non - zero even when @xmath3 is zero . when the reaction proceeds through a channel in which a single amplitude dominates , the longitudinal - transverse response will be real and @xmath5 will vanish . both @xmath3 and @xmath5 are necessary to fully unravel the longitudinal - transverse response of the @xmath77 electroproduction reactions . the measurement was carried out with the cebaf large acceptance spectrometer ( clas ) @xcite located in hall b at jlab . the main magnetic field of clas is provided by six superconducting coils , which produce an approximately toroidal field in the azimuthal direction around the beam axis . the gaps between the cryostats are instrumented with six identical detector packages . each sector consists of drift chambers ( dc ) @xcite for charged particle tracking , cherenkov counters ( cc ) @xcite for electron identification , scintillator counters ( sc ) @xcite for charged particle identification , and electromagnetic calorimeters ( ec ) @xcite for electron identification and detection of neutral particles . a 5-cm - long liquid - hydrogen target was located 25 cm upstream of the nominal center of clas . the main torus was operated at 60% of its maximum field value and had its polarity set such that negatively charged particles were bent toward the electron beam line . a totally absorbing faraday cup located at the end of the beam line was used to determine the integrated beam charge passing through the target . the efficiency of detection and reconstruction for stable charged particles in the fiducial regions of clas is greater than 95% . the solid angle coverage of clas is approximately 3@xmath102 sr . the polar angle coverage for electrons ranges from 8@xmath103 to 45@xmath103 , while for hadrons it is from 8@xmath103 to 140@xmath103 , with an angular resolution of @xmath104 of better than 2 mr . the clas detector was designed to track particles having momenta greater than roughly 200 mev with a resolution @xmath105 of about 1% . the data in this paper were collected as part of the clas e1f running period in 2003 . the incident electron beam energy was 5.499 gev . the live - time corrected integrated luminosity of this data set is 10.6 fb@xmath106 . the data set contains @xmath107 @xmath108 events and @xmath109 @xmath110 events in the analysis bins included in this work . the data were taken at an average electron beam current of 7 na at a luminosity of about @xmath111 @xmath112s@xmath106 . the event readout was triggered by a coincidence between a cc hit and an ec hit in a single sector , generating an event rate of @xmath372 khz . the electron beam was longitudinally polarized with polarization determined by a coincidence mller polarimeter . the average beam polarization was about 75% . this analysis sought to measure the differential cross sections for the electroproduction reactions @xmath113 and @xmath114 in bins of @xmath7 , @xmath9 , @xmath27 , and @xmath6 . exploiting the @xmath6 dependence of the differential cross sections @xmath115 as given by eq.([sigma0 ] ) , a @xmath6 fit in each bin of @xmath7 , @xmath9 , and @xmath27 provides the separated structure functions @xmath2 , @xmath3 , and @xmath4 . finally , a @xmath6 fit to the beam spin asymmetry as given by eq.([eq : sigltp ] ) in each bin of @xmath7 , @xmath9 , and @xmath27 gives access to the polarized structure function @xmath5 . the bin - centered differential cross section for each hyperon final state in each kinematic bin @xmath116 was computed using the form : @xmath117 where @xmath118 is the virtual photon flux factor computed according to eq.([eq : flux ] ) for each bin at the bin - averaged mean of the bin and @xmath119 is the volume of each analysis bin computed using the bin sizes listed in section [ anal - pid ] ( the bin sizes are corrected for kinematic limits in the threshold @xmath9 bins ) . @xmath120 is the radiative correction factor , @xmath121 is the background - subtracted @xmath0 and @xmath1 yield in each bin , @xmath122 is the factor that evolves the measured bin - averaged differential cross section over each bin to a specific kinematic point within the @xmath7 , @xmath9 , @xmath27 , @xmath6 bin , and @xmath123 accounts for the detector geometrical acceptance and efficiency corrections . @xmath124 is the live - time corrected incident electron flux summed over all data runs included in this analysis determined from the faraday cup charge . for this experiment , the data acquisition live time ranged between 80 and 85% . the incident electron flux was measured to be @xmath125 . finally , @xmath126 represents the target number density , where @xmath127 is avogadro s number , @xmath128=0.07151 g/@xmath129 is the target density , @xmath49=5.0 cm is the target length , and @xmath130=1.00794 g / mol is the atomic weight of the target . the statistical uncertainty on the cross section in each bin @xmath116 includes contributions from the statistical uncertainty on the hyperon yield and the acceptance function and is given by : @xmath131^{1/2}.\ ] ] the @xmath132 and @xmath133 reaction channels were identified by detecting a scattered electron in coincidence with a @xmath72 and then using the missing mass technique to identify the hyperons . event reconstruction required the identification of both a final state electron and @xmath72 candidate within the well - understood fiducial regions of the detector . details on the algorithms employed to minimize the particle misidentification at this stage are included in ref . @xcite . before computing the missing mass spectrum , vertex cuts were employed to ensure that the particles originated from the target . in addition , corrections to the electron and kaon momenta were devised to account for reconstruction inaccuracies that arose due to to relative misalignments of the drift chambers in the clas magnetic field , as well as from uncertainties in the magnetic field map employed during charged track reconstructions . these corrections were typically less than 1% . the algorithm used for hadron identification relied on comparing the measured velocity @xmath134 for the track candidate to that expected for an assumed @xmath135 , @xmath72 , and @xmath136 track . the assumption that resulted in the minimum @xmath137 was used to identify the species of the track . [ beta - plot ] shows @xmath138 versus momentum for the @xmath72 track assumption . for the data included here , the kaon momentum range was between 0.35 gev ( software cut ) and @xmath139 4.5 gev ( kinematic limit ) , with a typical flight path of 5.5 m. the measured mass resolution was primarily due to the reconstructed time - of - flight resolution , which was @xmath139100 ps ( @xmath140 ) on average ; it also included contributions from the momentum and path length uncertainties of clas . [ beta - plot ] shows that unambiguous separation of @xmath72 tracks at the 2@xmath140 level is possible up to about 2 gev . for higher momenta , the background due to particle misidentification increases . detailed background subtractions are necessary to determine the final event yields . fig . [ missing - mass ] shows the @xmath141 missing mass ( @xmath142 ) distribution for the final event sample after all cuts have been made . this distribution contains a background continuum beneath the hyperons that arises due to multi - particle final states where the candidate @xmath72 results from a misidentified pion or proton . the data were binned in a four - dimensional space of the kinematic variables @xmath7 , @xmath9 , @xmath27 , and @xmath6 . the bin definitions used in this analysis are listed in table [ tab : bins ] . [ kin ] shows the kinematic extent of the data in terms of @xmath7 versus @xmath9 and @xmath6 versus @xmath27 . these plots are overlaid with a grid indicating the bins in this analysis . the bin widths in @xmath9 and @xmath6 were chosen to be uniform . note that the maximum @xmath9 bin at each @xmath7 was limited to where the hyperon yield fits were not dominated by systematic uncertainties . l|l @xmath7 : [ 1.4,2.2 gev@xmath8 ] & @xmath9 : [ 1.6,2.6 gev ] ( 20 50-mev - wide bins ) + [ 2.2,3.0 gev@xmath8 ] & @xmath9 : [ 1.6,2.4 gev ] ( 16 50-mev - wide bins ) + [ 3.0,3.9 gev@xmath8 ] & @xmath9 : [ 1.6,2.2 gev ] ( 12 50-mev - wide bins ) + + + + the three components of the @xmath142 spectra are the @xmath0 events , the @xmath1 events , and the particle misidentification background ( dominated by pions misidentified as kaons ) . these individual contributions must be separated to extract the @xmath0 and @xmath1 differential cross sections in each analysis bin . the approach to separate the signal from the background events employed a fitting process based on hyperon template shapes and a polynomial to account for the particle misidentification background . the form for the spectrum fits was given by : @xmath144 where @xmath145 and @xmath146 are the simulated hyperon distributions with scaling factors @xmath147 and @xmath148 , respectively , and @xmath149 is a polynomial describing the background . the hyperon templates were derived from a geant - based monte carlo that included radiative processes and was matched to the detector resolution ( see section [ monte ] ) . the background contributions for this fitting were studied with a number of different assumptions ( see discussion in section [ systematics ] ) . ultimately , a linear form for the background was chosen . the template fits to the missing mass spectra were carried out using a maximum log likelihood method appropriate for the statistical samples of our data . [ samples ] shows two sample fits to illustrate the typical fit quality to the data . the final yields in each kinematic bin were determined by taking the number of counts determined from the fits that fell within a mass window around the @xmath52 ( 1.07 to 1.15 gev ) and @xmath150 ( 1.17 to 1.22 gev ) peaks . hyperon events in the tails of the distributions that fell outside of the mass windows were accounted for by the acceptance and radiative corrections . the number of @xmath52 and @xmath150 hyperons in both the @xmath0 and @xmath1 mass windows relative to the total number of counts in the mass windows was found to be independent of @xmath7 and @xmath6 in each bin of @xmath9 and @xmath27 . thus the final yields in each bin were determined by scaling the raw yields in the @xmath0 and @xmath1 mass windows by a background factor determined from fits in each bin of @xmath9 and @xmath27 . monte carlo simulations were carried out for this analysis for four distinct purposes . the first was to determine the detector acceptance in each bin , the second was as a cross check of the radiative correction factors , the third was to generate the hyperon templates for the spectrum fits , and the fourth was to determine the tracking efficiency corrections . for this analysis we employed two different event generators for the exclusive @xmath0 and @xmath1 event samples . the first generator , fsgen @xcite , generates @xmath151 events according to a phase space distribution with a @xmath49-slope scaled by a factor of @xmath152 . this generator did not include radiative effects . the nominal choice of the @xmath49-slope parameter of @xmath153=1.0 gev@xmath154 was chosen to best match the @xmath27 dependence of the data . the generated data were then weighted with ad hoc functions so that they matched well to the kinematic distributions of the data ( see fig . [ mc - comp ] ) . the second generator , genev @xcite , generates events for various meson production channels . it was modified for this analysis to include the @xmath0 and @xmath1 channels , reading in cross section tables for @xmath0 and @xmath1 photoproduction based on the data of refs . @xcite and @xcite , respectively . it extrapolates to finite @xmath7 by introducing a virtual photon flux factor and electromagnetic form factors based on a simple dipole form . radiative effects based on the formalism of mo and tsai @xcite are part of the generator as an option . here too , the input distributions of the model were weighted with ad hoc function so that they matched the data ( see fig . [ mc - comp ] ) . the monte carlo suite is based on a geant-3 package @xcite . the generated events were processed by this code based on the clas detector . the events were then subjected to additional smearing factors for the tracking and timing resolutions to match the average experimental resolutions . the analysis of the monte carlo data used the same code as was used to analyze the experimental data . ultimately more than 1 billion monte carlo events were generated to determine the correction factors and the associated systematic uncertainties , which are discussed in section [ systematics ] . in order to relate the experimental yields to the cross sections , we require the detector acceptance to account for various effects , such as the geometric coverage of the detector , hardware and software inefficiencies , and resolution effects from the track reconstruction . the acceptance is defined separately for the @xmath0 and @xmath1 reaction channels as a function of the kinematic variables as : @xmath155 where @xmath156 is the reconstructed number of events in each bin and @xmath157 is the generated number of events in each bin . the fsgen simulation was used to determine the acceptance function for the final analysis . typical acceptances for clas for the @xmath141 final state vary from @xmath1391% to 30% . [ accep ] shows examples of this computed acceptance for the @xmath0 final state as a function of @xmath6 and @xmath27 for one @xmath7 and @xmath9 bin . for this analysis several standard clas efficiency corrections were applied to the yields on an event - by - event basis . the first correction accounted for the efficiency of the cherenkov counter for registering electron tracks based on the number of detected photoelectrons in each sector in a fine grid of the @xmath158 and @xmath159 angles of the electron at the face of each cc detector . the average cc efficiency within the electron geometric fiducial cuts for this analysis is 96% . the remaining efficiency corrections account for hadron tracking inefficiencies . the first correction accounts for the single track reconstruction efficiency in clas that is not 100% due to inefficient sc paddles and dc tracking regions . this efficiency function was assigned based on the relative ratio of data counts to monte carlo counts as a function of clas sector and sc paddle number . these corrections are at the level of about 10% on average . another efficiency correction related to tracking is necessary for events in which two charged tracks of the same charge and similar momenta lie very close to each other . for such events the tracking algorithm may not successfully identify two separate tracks . for this analysis , a correction was applied to the small fraction of events in which the @xmath72 and @xmath136 from the decay of the @xmath52 were in the same clas sector within 10@xmath160 of each other in polar angle . this efficiency factor is necessary even for the @xmath141 analysis due the presence of the decay protons in the final state . the systematics associated with each of these efficiency corrections are discussed in section [ sys - acc ] . radiative effects must be considered when determining the @xmath161 cross sections . radiative effects result in bin migration such that the measured @xmath7 and @xmath9 are not the true @xmath7 and @xmath9 to which the event should be properly associated . for this analysis , two different approaches to determine these correction factors have been employed . the first uses the stand - alone program exclurad @xcite and the second uses the event generator genev @xcite in combination with the clas monte carlo . the radiative correction factor that multiplies the measured bin - averaged differential cross section in each bin is defined as the ratio of the computed bin - averaged cross section with radiation off to that with radiation on . more details on each program are included below . exclurad represents a covariant technique of cancellation of the infrared divergence that leads to independence of any parameter that splits the soft and hard regions of phase space of the radiated photons . it uses an integration technique that is exact over the bremsstrahlung photon phase space , and thus does not rely on the peaking approximation @xcite . this approach is an exact calculation in that it specifically accounts for the exclusive nature of the reactions as the detection of hadrons in the final state , in addition to the electron , reduces the phase space allowed for the final radiative photons . the program exclurad was based on the measured structure functions from this analysis for @xmath0 and @xmath1 . the structure functions @xmath2 , @xmath3 , @xmath4 , and @xmath5 were read into the program and the cross section ratio for each bin in @xmath7 , @xmath9 , @xmath27 , and @xmath6 was computed with radiation off to that with radiation on , giving the radiative correction factor @xmath120 for that bin . the trends of the correction ( shown in fig . [ rc - plot1 ] ) are such that it has its largest value near threshold and then quickly falls off to a near constant average value with increasing @xmath9 . note that the radiative correction factors including the helicity - dependent structure function @xmath5 for the two helicity states have no impact on the helicity asymmetry computation in eq.([eq : sigltp ] ) and are not included in the analysis . the event generator genev @xcite was introduced in section [ monte ] as it was used to compute the clas acceptance function . this program also allows for radiative correction factors to be determined . it includes radiative effects based on the formalism for inclusive electron scattering from ref . @xcite and employs the peaking approximation @xcite in the computation . as genev is based on an evolution of the photoproduction cross sections , it does not have an explicit @xmath6 dependence and thus the @xmath120 factors in eq.([dcs ] ) were determined in bins of @xmath7 , @xmath9 , and @xmath27 . this model has several shortcomings . the first is that the phase space for the radiated photons is not properly computed as this is modified by the detected hadrons . secondly , the model is based on only the longitudinal and transverse response and does not include the interference structure functions @xmath3 or @xmath4 . finally , the approach relies on an unphysical parameter to split the hard and soft regions of the radiated photon phase space to cancel the infrared divergence . due to the known limitations with this approach , it was used only to provide a qualitative cross check to the exclurad results and to explore the associated systematic uncertainties ( see section [ sys - acc ] ) . [ rc - plot1 ] shows a comparison of the radiative correction factors computed by genev to those computed from exclurad . apart from the region near threshold , the correspondence between the two approaches is within 10% . the goal of this analysis is to measure cross sections and separated structure functions for the @xmath77 final states at specific kinematic points . however , the analysis proceeds from using finite bins in the relevant kinematic quantities @xmath7 , @xmath9 , @xmath27 , and @xmath6 ( see section [ anal - pid ] ) . the virtual photon flux factor @xmath118 defined in section [ formalism ] is computed for each bin using the bin - averaged values of @xmath7 and @xmath9 . if the cross sections were computed at this point using eq.([dcs ] ) with the @xmath122 terms set to unity , we would have completed a measurement of the bin - averaged cross sections that we could quote at the corresponding bin - averaged kinematic points . to quote the cross section at specific kinematic points of our choosing , namely , the geometric centers of the defined bins , we must evolve the cross sections from the bin - averaged kinematic points to the geometric bin centers . these evolution factors are the bin - centering correction factors @xmath122 in eq.([dcs ] ) . the bin - centering corrections are then applied for each bin as : @xmath162 where @xmath122 are the ratios of the bin - centered cross section to the bin - averaged cross section . studies of the bin - averaged kinematic quantities versus the geometric bin - centered values show that there is no need for bin - centering corrections in @xmath9 or @xmath27 . for this work the threshold @xmath9 bin for @xmath0 is quoted at 1.630 gev and for @xmath1 at 1.695 gev . to determine the bin - centering factor @xmath122 for each bin , we have fit the measured structure functions @xmath2 for each @xmath9 and @xmath27 bin versus @xmath7 for both the @xmath0 and @xmath1 final states . to bin center the data at specific @xmath7 points , we have used the following dipole evolution factor : @xmath163 the bin centering factors using this form were in the range from 0.95 to 1.05 across the full kinematic phase space . the differential cross sections computed using eq.([sigma0 ] ) are the mean values within the finite size of the @xmath6 bins and therefore do not reflect the value at the bin center . thus directly fitting these data with eq.([sigma0 ] ) to extract the structure functions @xmath23 , @xmath4 , and @xmath3 would be inappropriate . integrating eq.([sigma0 ] ) over the finite bin size , @xmath164 , where @xmath165 and @xmath166 are the upper and lower limits of the bin , respectively , gives : @xmath167 @xmath168 now represents the value of the measured bin - averaged cross section in a given @xmath6 bin and fitting the data with eq.([eq - csecfit ] ) yields the separated structure functions . the `` @xmath169 '' pre - factors were evaluated at the bin center and divided out . note that prior to the @xmath6 fits , the statistical uncertainty on each cross section point was combined linearly with that portion of the systematic uncertainty arising from the yield extraction procedures ( see section [ yield - sys ] for details ) . in fig . [ phi - fits1 ] we show a sample of the @xmath6-dependent differential cross sections for the @xmath0 final state at @xmath9=1.725 gev for @xmath7=1.8 gev@xmath8 . the different shapes of the differential cross sections versus @xmath6 in each of our bins in @xmath7 , @xmath9 , and @xmath27 reflect differences of the interference terms , @xmath3 and @xmath4 , while the differences in scale reflect the differences in @xmath2 . the extraction of @xmath5 in each bin of @xmath7 , @xmath9 , and @xmath27 requires knowledge of both the asymmetry @xmath170 and the unpolarized cross section @xmath115 , which can be seen by rearranging eq.([eq : sigltp ] ) into a normalized asymmetry @xmath171 as : @xmath172 @xmath170 is determined by forming the asymmetry of the @xmath0 and @xmath1 yields for the positive and negative beam helicity states ( @xmath101 ) as : @xmath173 where @xmath174 is the average longitudinal polarization of the electron beam . as with the cross sections , the measured asymmetries are the average values over the span of the given @xmath6 bins . integrating eq.([eq : sigltp ] ) over the size of the @xmath6 bin results in : @xmath175 to extract @xmath5 , a @xmath176 fit was performed according to eq.([normb ] ) , where the kinematic @xmath169 factor was calculated at the bin - centered values of @xmath7 and @xmath9 for each bin . a sample of these distributions is shown in fig . [ phi - fits2 ] for the @xmath0 final state at @xmath9=1.725 gev for @xmath7=1.8 gev@xmath8 . similar to the case for the unpolarized structure function extraction discussed in section [ sfsep ] , prior to the @xmath6 fits the statistical uncertainty on the helicity - gated yields was combined linearly with that portion of the systematic uncertainty arising from the yield extraction procedure ( see section [ yield - sys ] for details ) . the statistical uncertainty on the data points in each bin @xmath116 are a combination of the contributions from both @xmath170 and @xmath115 and are given by : @xmath177 to obtain a virtual photoabsorption cross section , we extract the yields for the @xmath0 and @xmath1 reactions from the the missing - mass spectra for each of our bins in @xmath7 , @xmath9 , @xmath27 , and @xmath6 . the yields are corrected for the acceptance function of clas including various efficiency factors , radiative effects , and bin - centering factors . finally , we divide by the virtual photon flux factor , the bin volume corrected for kinematic limits , and the beam - target luminosity to yield the cross section . each of these procedures is subject to systematic uncertainty . we typically estimate the size of the systematic uncertainties by repeating a procedure in a slightly different way , e.g. by varying a cut parameter within reasonable limits , by employing an alternative algorithm , or by using a different model to extract a correction , and noting how the results change . @xmath2 & @xmath3 , @xmath4 & @xmath5 + 1 . yield extraction & + signal fitting / binning effects & + fiducial cuts & 0.4 - 2.6% & 0.4 - 2.6% & - & 0.7 - 4.4% + electron identification & 1.1% & 0.1% & 4.0% & 1.4% + 2 . detector acceptance & + mc model dependence & 4.0 - 9.3% & 3.6 - 7.8% & 6.8% & 3.6 - 7.0% + tracking efficiencies & 5.3% & 5.3% & 5.5% & 5.3% + close track efficiencies & 2.8% & 1.6% & 4.7% & 2.6% + cc efficiency function & 1.5% & 1.5% & 1.5% & 1.5% + 3 . radiative corrections & 2.0% & 2.0% & 4.4% & 2.0% + 4 . bin centering & 0.5% & 0.5% & 0.5% & 0.5% + 5 . scale uncertainties & + beam polarization & - & - & - & 2.3% + photon flux factor & 3.0% & 3.0% & - & 3.0% + luminosity & 3.0% & 3.0% & - & 3.0% + total @xmath178 & 12.5% & 11.1% & 11.7% & 11.6% + total @xmath179 & 9.2% & 8.2% & 11.7% & 9.2% + total @xmath180 & 8.9% & 8.5% & 11.7% & 9.0% + in this section we describe our main sources of systematics . the five categories of systematic uncertainty studied in this analysis include yield extraction , detector acceptance , radiative corrections , bin centering corrections , and scale uncertainties . each of these categories is explained in more detail below . in assigning the associated systematic uncertainties , we have compared the differential cross sections and extracted structure functions , @xmath2 , @xmath3 , @xmath4 , and @xmath5 , with the nominal cuts and the altered cuts . the fractional uncertainty for each bin @xmath116 was calculated via : @xmath181 the relative difference in the results @xmath182 is then used as a measure of the systematic uncertainty . in this analysis we have carefully studied the kinematic dependence of the systematics and conclude that there is no evidence within a given @xmath7 bin of systematic variations with @xmath9 , @xmath27 , or @xmath6 . table [ syserror ] lists the categories , specific sources , and the assigned systematic uncertainties on our measurements . overall the scale of the systematic uncertainties is at the level of about 10% . the procedure to determine the @xmath77 yields in each analysis bin employs hyperon templates derived from monte carlo simulations that have been tuned to match the data . the background fit function has been studied using two different approaches . the first uses a polynomial ( either linear or quadratic ) and the second uses the @xmath183 data sample purposefully misidentifying the detected @xmath135 as a @xmath72 . we have concluded that all systematic effects associated with the spectrum fitting get larger in direct proportion to the size of the statistical uncertainty . we estimated that the systematic uncertainty due to the yield extraction is roughly equal to 20% of the size of the statistical uncertainty in any given bin . we added these correlated uncertainties linearly with the statistical uncertainties on our extracted yields before performing the @xmath6 fits . the other sources of systematic uncertainty considered in this category are associated with the defined electron and hadron fiducial cuts and the cuts on the deposited energy in the calorimeter used to identify the candidate electron sample . variations in the definitions of the fiducial cuts and the ec energy cuts over a broad range showed that the observables were stable for each cut type to within 5% . in the category of detector acceptance , the associated systematics include that due to the model dependence of the acceptance function , the stability of the tracking efficiency corrections , and the cc efficiency function . for this analysis both the fsgen and genev physics models were used to generate the monte carlo events . because of the finite bin sizes used in this analysis , it is necessary to study how the derived acceptance function based on the different event generators impacts the extracted observables . for both models we determined the acceptance function and stepped through the full analysis chain to extract the observables . the systematics assigned for the model dependence were in the range from about 4% to 9% . the approach to assign a systematic associated with the clas tracking efficiency corrections was to employ slightly different algorithms and then to step through the full analysis chain . the tracking efficiency gave stable results at the level of 5% . the systematic associated with the close track efficiency was stable in the range from 2 to 5% . to study the systematic uncertainty associated with the cc efficiency function , we compared the measured observables with the nominal cc efficiency corrections to an analysis with the cc efficiency set to 100% for all events . the differences were within 1.5% for all observables . two very different approaches have been used to study the radiative corrections for the @xmath0 and @xmath1 electroproduction reactions . the first was the exclusive approach based on the exclurad program @xcite and the second was based on the inclusive approach based on the genev program @xcite . comparison of the extracted radiative corrections between exclurad and genev were within about 8% of each other . however , due to the shortcomings of the genev model as discussed in section [ genev - sec ] , this comparison was only used as a cross check of the overall scale of the corrections . to assign a systematic uncertainty for the radiative corrections for this analysis , we compared the measured observables using the exclurad approach but varying the energy range of integration of the radiated photon over a broad range . the corrections were stable in the range from 2 to 5% . to assign a systematic uncertainty to the bin centering corrections , the mass term in the dipole form ( see eq.([bc - eq ] ) ) was varied over a broad range . the maximum variation seen in any of the extracted observables was 0.5% . in the category of scale uncertainties , the associated systematics include that due to the beam - charge asymmetry and uncertainties in the beam polarization , the photon flux factor , and the luminosity . the estimated beam - charge asymmetry is at the level of a few times @xmath184 and is thus entirely negligible . the uncertainty in the beam polarization affects only the systematic assigned to @xmath5 . this is given by : @xmath185 where @xmath186=0.03 and @xmath187 = 0.754 is the average beam polarization . thus the assigned systematic for @xmath5 due to the beam polarization uncertainty is 2.3% . the uncertainties in the average virtual photon flux factor across our phase space were estimated by propagating through the flux definition the uncertainties associated with @xmath9 and @xmath7 that arise from the uncertainty in the reconstructed electron momentum and angles . the uncertainty in the flux factor was determined to be 3% . this scale - type uncertainty affects only the differential cross section and the structure functions @xmath2 and @xmath5 . we estimated uncertainties in the beam - target luminosity based on the analysis of clas @xmath188 elastic scattering cross sections from ref . the overall systematic uncertainty of the faraday cup charge measurement has been assigned to be 3.0% . this scale - type uncertainty affects only the differential cross section and the structure functions @xmath2 and @xmath5 . the nominal analysis for the @xmath0 and @xmath1 differential cross sections and separated structure functions required only the detection of the electron and @xmath72 in the final state . in order to check the overall systematic assignment , the observables were also extracted when detecting an additional @xmath136 . the detection of the proton from the @xmath52 decay gives rise to an analysis sensitive to the same systematic uncertainties as the nominal analysis , and thus should yield consistent results . however , requiring the proton reduces the acceptance by roughly a factor of three , therefore this comparison can only be used as a cross check of the nominal analysis . the agreement between the cross sections extracted using the @xmath141 and @xmath189 final states is at the level of @xmath995 - 10% and independent of kinematics to within the statistical uncertainties . the differences are driven by the marginal statistics in some of the analysis bins for the @xmath189 analysis . these comparisons show that the assigned systematic uncertainties are reasonable . in figs . [ lam_q1_ca ] and [ lam_q1_cb ] we show the extracted structure functions @xmath2 , @xmath3 , @xmath4 , and @xmath5 versus @xmath27 for the @xmath0 final state . figs . [ sig_q1_ca ] and [ sig_q1_cb ] show the same plots for the @xmath1 final state . these plots are for our lowest @xmath7 point at 1.80 gev@xmath8 . the general conclusions that can be drawn from studying the angular dependence are similar for the two higher @xmath7 points at 2.60 and 3.45 gev@xmath8 . however , the full set of our data is available in the clas physics database @xcite . the following curves are overlaid on the data : * the hadrodynamic model of maxwell _ et al . _ ( mx ) ( red / dashed curves - thinner line type from refs . @xcite , thicker line type is an extension of that model including fits to @xmath5 data from ref . note that this model is only available for the @xmath0 final state and calculations go to a maximum @xmath9 of 2.275 gev . * the regge model of guidal _ et al . _ ( glv ) @xcite ( green / dotted ) . * the regge plus resonance model of ghent ( rpr ) @xcite ( black / solid curves - rpr-2007 thinner line type , rpr-2011 thicker line type ) . for the @xmath1 comparison , only the rpr-2007 version is presently available . a number of observations can be made independent of the model calculations : 1 . the production dynamics for @xmath0 and @xmath1 are quite different for @xmath190 gev . however , as @xmath9 increases further , the production mechanisms become similar . this is to be expected as @xmath33 production is known to be dominated by @xmath49-channel exchanges at higher energies . the @xmath0 production dynamics are dominated by @xmath49-channel exchange over the full resonance region as indicated by the strong forward peaking of @xmath2 in figs . [ lam_q1_ca ] and [ lam_q1_cb ] . however , given the mid - angle peaking of @xmath2 for @xmath1 below 2 gev , clearly @xmath48-channel contributions play a much more significant role for this final state . the forward peaking of @xmath2 and @xmath3 for @xmath0 compared to @xmath1 can be qualitatively explained by the effect of the longitudinal coupling of the virtual photons . we note that the two channels are of nearly equal strength at @xmath7=0 gev@xmath8 @xcite , while here at @xmath7=1.80 gev@xmath8 , the @xmath0 channel is stronger than the @xmath1 channel at forward angles by a factor of 3 to 4 . for transverse ( real ) photons , the @xmath49-channel mechanism at low @xmath49 is dominated by vector @xmath191 exchange , which relates directly to the magnitudes of the coupling constants @xmath66 relative to @xmath65 . as @xmath7 rises from zero , the photon can acquire a longitudinal polarization and the importance of pseudoscalar @xmath72 exchange increases . given that @xmath192 @xcite , this effect increases the cross section for @xmath0 relative to @xmath1 ( this is consistent with the arguments presented in ref . this argument is consistent with our observation of a sizable @xmath3 for @xmath0 and a @xmath3 consistent with zero for @xmath1 . it should also be the case that since @xmath193 , @xmath47 exchange should dominate the @xmath1 channel . because @xmath47 exchange must vanish at forward angles due to angular momentum conservation , the @xmath1 cross section should also decrease at forward angles @xcite . 4 . for @xmath0 , @xmath4 is consistent with zero up to about @xmath9=1.9 gev then develops a strong forward peaking that abruptly changes sign at about @xmath9=2.2 gev . for @xmath1 , @xmath4 peaks at mid - range angles up to @xmath9=2 gev and then looks very similar to @xmath0 for higher @xmath9 . this higher @xmath9 response is well explained by the interference of the @xmath46 and @xmath47 regge trajectories . 5 . for @xmath0 , @xmath5 is relatively flat over the full angular range up to @xmath9=2 gev and then develops a strong forward peaking for higher @xmath9 very similar to the other interference structure functions . we also note that it is significantly reduced at this @xmath7 compared to the results at @xmath7=0.65 and 1.0 gev@xmath8 shown in ref . @xmath5 for @xmath1 is consistent with zero over the full angular range . comparing the data in figs . [ lam_q1_ca ] to [ sig_q1_cb ] to the different single - channel model calculations , it is apparent that none of the models is successful at fully describing all of the data . a few general remarks are in order : 1 . in general the models agree better with the @xmath0 data than with the @xmath1 data . this likely arises , in part , due to the fact that better quality data for @xmath0 is available than for @xmath1 . however , as the resonance content is stronger in @xmath1 compared to @xmath0 for @xmath45 gev given that the regge predictions for @xmath0 are in much closer agreement with the @xmath2 measurements compared to @xmath1 , the reaction mechanism for @xmath1 is most certainly more complicated compared to @xmath0 , and thus more difficult to model correctly . the models reproduce reasonably well the forward peaking strength in @xmath2 , @xmath3 , and @xmath4 for @xmath0 and @xmath1 for both final states for higher @xmath9 . at @xmath45 gev where the resonance contributions are a larger contribution relative to the non - resonant background , the agreement is noticeably worse . 3 . none of the models reproduces the trends in @xmath5 for either final state across the full @xmath9 spectrum . interestingly , the hadrodynamic model of maxwell _ et al . _ that includes the available @xmath5 data from ref . @xcite has by far the worst agreement with these data , although the available @xmath5 data only go up to @xmath7=1.0 gev@xmath8 . 4 . the glv regge model that includes no @xmath48-channel resonance terms , does as well as any of the other models in describing these data . for the @xmath1 final state for @xmath45 gev , which has strong @xmath48-channel contributions , the glv model significantly underpredicts @xmath2 . however , for @xmath0 , which has a much more significant @xmath49-channel exchange component within the resonance region , the glv model underpredicts @xmath2 for @xmath194 gev . but for @xmath195 gev , the glv model well matches the data for both final states over our full kinematic phase space . 5 . for @xmath0 , the rpr-2011 model fares noticeably worse than for the rpr-2007 model over all angles for @xmath196 gev for all of the structure functions . for higher @xmath9 , where the response is essentially fully @xmath49-channel , the rpr-2007 and rpr-2011 models agree well with the data and with each other . to more directly look for @xmath48-channel resonance evidence , the extracted structure functions are presented as a function of the center - of - mass energy @xmath9 for our ten values of @xmath27 . [ lam_q1_w ] and [ sig_q1_w ] show the results for our @xmath0 and @xmath1 data , respectively , at @xmath7=1.80 gev@xmath8 . a number of observations can be made regarding the data : 1 . for @xmath0 production , @xmath2 shows a broad peak at about 1.7 gev at forward angles , and two peaks separated by a dip at about 1.75 gev for our two backward angle points . this corroborates similar features seen in recent photo- and electroproduction results @xcite . within existing hadrodynamic models , the structure just above the threshold region is typically accounted for by the known @xmath28 , @xmath29 , and @xmath30 nucleon resonances . however , there is no consensus as to the origin of the bump feature at @xmath371.9 gev that was first seen in the @xmath0 photoproduction data from saphir @xcite . it is tempting to speculate that this is evidence for a previously `` missing '' , negative - parity @xmath197 resonance at 1.96 gev predicted in the quark model of capstick and roberts @xcite . this explanation was put forward in the work of bennhold and mart @xcite , in which they postulated the existence of a @xmath198 state at 1.9 gev . however , in ref . @xcite it was shown that a @xmath31 state is required to explain the beam - recoil polarization data for @xmath0 . in ref . @xcite this broad bump in the @xmath0 cross section could be explained by accounting for @xmath50-channel hyperon exchanges . 2 . for @xmath0 , @xmath3 has about 20% of the strength of @xmath2 and is consistently negative . for @xmath1 , @xmath3 is nearly zero everywhere except for @xmath9=1.9 gev at back angles . the @xmath4 structure function is quite similar for @xmath0 and @xmath1 over all kinematics with a strength comparable to @xmath3 . 4 . for @xmath0 , @xmath5 shows significant structure for @xmath9 below 2.2 gev . for higher @xmath9 it is consistent with zero . 5 . in the @xmath1 channel , @xmath2 is peaked at about 1.9 gev , which also matches the photoproduction result @xcite . @xmath4 , while small , shows a broad feature in this same region . these features are consistent with a predominantly @xmath48-channel production mechanism . in this region , beyond the specific @xmath11 resonances believed to contribute to @xmath0 production ( and hence are strong candidates to contribute to @xmath1 production ) , there are a number of known @xmath12 resonances near 1.9 gev @xcite that can contribute to the @xmath1 final state , particularly the @xmath42 and @xmath43 . these @xmath12 states are forbidden to couple to the @xmath0 state due to isospin conservation . the comparisons of the model calculations to the data clearly indicate that significant new constraints on the model parameters will be brought about when these new electroproduction data are included in the fits . we conclude that the @xmath9 dependence of @xmath0 and @xmath1 production provides strong evidence for baryon resonance activity within the reaction mechanism , but that the data in comparison to present models do not allow any simple statement to be made . we further conclude that at the current time the models that are limited to fits of the photoproduction data only , can not adequately describe the electroproduction data . our data set provides a large @xmath7 reach and it is instructive to study the @xmath9 spectra for increasing values of @xmath7 . these data are shown in figs . [ lam - q2 ] and [ sig - q2 ] for the @xmath0 and @xmath1 final states at two representative @xmath9 points , 1.725 and 1.925 gev . included on these plots are the photoproduction differential cross sections for @xmath0 from ref . @xcite and @xmath1 from ref . @xcite at @xmath7=0 for the kinematic points where they are available . also shown are the data from @xmath2 from ref . @xcite from two different data sets , ( i ) . @xmath199=2.567 gev , @xmath7=0.65 , 1.0 gev@xmath8 and ( ii ) . @xmath199=4.056 gev , @xmath7=1.0 , 1.55 , 2.05 , 2.55 gev@xmath8 at kinematic points that are reasonably close to the present data . what is seen by studying the @xmath7 evolution of @xmath2 is a reasonably smooth fall - off from the photon point . as the photoproduction data involve a purely transverse response , this smooth fall - off to finite @xmath7 in these kinematics predominantly indicates a small longitudinal response . this is also indicated by the small strengths of @xmath3 and @xmath5 relative to @xmath2 in figs . [ lam_q1_ca ] to [ sig_q1_w ] for back- and mid - range angles for the @xmath0 final state and for all angles for the @xmath1 final state . however , there is clearly a non - negligible longitudinal response in the @xmath0 data at forward angles and for higher @xmath9 as seen in these data ( and also seen in the data of ref . note that the comparisons shown in figs . [ lam - q2 ] and [ sig - q2 ] are only for qualitative comparisons as the kinematics are not a perfect match in all cases from refs . @xcite to the present data . the smooth fall - off of @xmath2 with increasing @xmath7 is consistent with the findings of the lower @xmath7 analysis of @xmath0 and @xmath1 electroproduction from ref . as was the case in that work , it is seen that the interference structure functions @xmath3 , @xmath4 , and @xmath5 for both final states do not demonstrate any strong @xmath7 dependence . however , detailed comparisons with available models will be important to gain insight into the associated form factors for the @xmath11 resonances found from fits to the photoproduction data . in order to investigate the possible evidence for the presence of @xmath48-channel resonance contributions in the separated structure functions , we have considered two different approaches . the first is with a fit of the individual structure functions @xmath2 , @xmath3 , @xmath4 , and @xmath5 versus @xmath27 for each @xmath7 and @xmath9 point for the @xmath0 and @xmath1 final states using a truncated series of legendre polynomials as : @xmath200 the fit coefficients for @xmath201 are shown for @xmath0 in fig . [ lam - leg - incoherent ] and for @xmath1 in fig . [ sig - leg - incoherent ] for @xmath202 gev@xmath8 . the structures seen in these coefficients versus @xmath9 are likely indicative of @xmath48-channel contributions . note that the appearance of a structure at a given value of @xmath9 in each of the different @xmath203 coefficients most likely suggests the presence of a dynamical effect rather than the signature of an @xmath11 contribution . instead , the appearance of a structure in a single @xmath203 coefficient at the same @xmath9 value and in each of the @xmath7 points is more likely a signal of an @xmath11 contribution . the fits for @xmath0 show structures at @xmath9=1.7 gev in @xmath204 for both @xmath2 and @xmath3 , @xmath9=1.9 gev in @xmath205 and @xmath206 for @xmath2 , and @xmath9=2.2 gev in @xmath206 for @xmath2 . the fits for @xmath1 show structures at @xmath9=1.9 gev in @xmath204 and @xmath205 for @xmath2 and @xmath4 . of course , making statements regarding the possible orbital angular momentum of the associated @xmath48-channel resonances requires care as interference effects among the different partial waves can cause strength for a given orbital angular momentum value to be spread over multiple legendre coefficients . in a second approach , each of the legendre coefficients can be further expanded in terms of products of pairs of multipole amplitudes , but these expansions quickly become unwieldy as the number of participating partial waves increases . however , one simple thing that can be done for additional insight is to fit the structure functions with a coherent legendre series of the form : @xmath207 ^ 2 + c_x^2.\ ] ] here the @xmath208 are the usual legendre polynomials . the coefficients @xmath209 are the amplitudes of the coherent @xmath210 , @xmath211 , and @xmath212-wave contributions , respectively , while @xmath213 takes into account a incoherent `` background '' connected with higher - order terms that are not taken into account in the truncated sum . of course , one must take care against making too much of the fit results using the simplistic form of eq.([coherent ] ) . this approach is not meant to be an attempt at a true amplitude fit . rather the point is to look for structures that appear at a given @xmath9 and for each @xmath7 for a given @xmath203 coefficient as suggestive evidence for possible @xmath11 resonance contributions . [ lam - leg - coherent ] shows the legendre coefficient from this approach for @xmath2 for the @xmath0 reaction for the three @xmath7 points in this analysis . [ sig - leg - coherent ] is the corresponding figure for @xmath1 . the fit coefficients for @xmath2 shown in figs . [ lam - leg - coherent ] and [ sig - leg - coherent ] show reasonable correspondence among all three @xmath7 points . for the @xmath0 fits , strength is seen at : @xmath9=1.7 gev in @xmath204 , @xmath9=1.9 gev in @xmath214 , and @xmath9=2.2 gev in @xmath205 . while it might be tempting to view this as corroboration of the findings of the @xmath0 photoproduction amplitude analysis from ref . @xcite , obviously more detailed work is required . for the @xmath1 fits , strength is seen at @xmath9=1.85 gev in @xmath204 and @xmath9=1.9 gev in @xmath205 . it is interesting that there is no signature of strength in the @xmath211-wave as seen through the coefficient @xmath214 , but again a higher - order analysis will be required to make more definite statements . we have measured @xmath0 and @xmath1 electroproduction off the proton over a wide range of kinematics in the nucleon resonance region . we have presented data for the differential cross sections and separated structure functions @xmath2 , @xmath3 , @xmath4 , and @xmath5 for @xmath7 from 1.4 to 3.9 gev@xmath8 , @xmath9 from threshold to 2.6 gev , and spanning nearly the full center - of - mass angular range for the @xmath72 . in addition to the increased kinematic reach of these data relative to the previously published @xmath77 electroproduction structure functions from clas in ref . @xcite , this new data set is an order of magnitude larger , allowing for finer binning in @xmath9 and @xmath27 . the structure function data for both @xmath0 and @xmath1 indicates that for @xmath9 below 2.2 gev and back angles , there is considerable strength of contributing @xmath48-channel resonances for @xmath0 and @xmath1 . for higher @xmath9 , the @xmath49-channel non - resonant background dominates and the reaction dynamics are well described solely through interference of @xmath46 and @xmath47 regge trajectories . a legendre analysis confirms these qualitative statements . for the @xmath0 final state , the legendre moments of the structure functions indicate possible @xmath48-channel resonant contributions in the @xmath210-wave near 1.7 gev , in the @xmath211-wave near 1.9 gev , and in the @xmath212-wave near 2.2 gev . this is in qualitative agreement with the more detailed amplitude analysis of ref . @xcite . for the @xmath1 final state , strong @xmath210-wave strength is seen at 1.8 gev and strong @xmath212-wave strength is seen above 1.9 gev , precisely where several @xmath12 states are expected to couple . of course more detailed and quantitative statements await including these data into the coupled - channel partial wave fits . such analyses would help to provide important complementary cross checks to the fit results of the recent bonn - gatchina coupled - channels results from ref . @xcite that seem to favor a much richer mix of states to describe the available photoproduction data . finally , detailed comparisons of our data have been made with several existing models . these include the hadrodynamic model of maxwell _ et al . _ @xcite that has been constrained by both the clas photo- and electroproduction data sets ( both cross sections and spin observables ) , the regge model of guidal _ et al . _ @xcite that has only been constrained by high - energy photoproduction data to fix the parameters of the regge trajectories , and the regge plus resonance model from ghent @xcite that has been constrained by the existing high statistics photoproduction data . none of the available models does a satisfactory job of describing the structure functions below @xmath215 gev for either @xmath0 or @xmath1 . in fact , several of the more recent models ( e.g. rpr-2011 and the mx model including the clas @xmath5 data ) actually are in worse agreement with the data below 2 gev than for earlier versions of the models . clearly more work on the modeling and possibly the fitting / convergence algorithms is required to be able to fully understand the contributing @xmath216 and @xmath217 states and to reconcile the results from the single - channels models with the currently available coupled - channel models . we would like to acknowledge the outstanding efforts of the staff of the accelerator and the physics divisions at jlab that made this experiment possible . this work was supported in part by the u.s . department of energy , the national science foundation , the italian istituto nazionale di fisica nucleare , the french centre national de la recherche scientifique , the french commissariat lenergie atomique , the united kingdom s science and technology facilities council , the chilean comisin nacional de investigacin cientfica y tecnolgica ( conicyt ) , and the national research foundation of korea . the southeastern universities research association ( sura ) operated jefferson lab under united states doe contract de - ac05 - 84er40150 during this work .

we report measurements of the exclusive electroproduction of @xmath0 and @xmath1 final states from an unpolarized proton target using the clas detector at the thomas jefferson national accelerator facility . the separated structure functions @xmath2 , @xmath3 , @xmath4 , and @xmath5 were extracted from the @xmath6-dependent differential cross sections acquired with a longitudinally polarized 5.499 gev electron beam . the data span a broad range of momentum transfers @xmath7 from 1.4 to 3.9 gev@xmath8 , invariant energy @xmath9 from threshold to 2.6 gev , and nearly the full center - of - mass angular range of the kaon . the separated structure functions provide an unprecedented data sample , which in conjunction with other meson photo- and electroproduction data , will help to constrain the higher - level analyses being performed to search for missing baryon resonances .

the analysis of nuclear spectra has produced ample evidence for chaotic motion . indeed , near neutron threshold , the spectra of medium weight and heavy nuclei display fluctuations which agree with those of random matrices drawn from the gaussian orthogonal ensemble ( goe ) @xcite . similar agreement has been found for nuclei in the @xmath0shell ( both in experimental data @xcite and in shell model calculations @xcite ) , and in the ground state domain of heavier nuclei @xcite , although here there exists strong evidence , too , for regular motion as predicted by the shell model and the collective models . calculations in ce @xcite have produced similar evidence for chaotic motion in atoms . thus , chaos appears to be an ubiquitous feature of interacting many body systems . what is the origin of this behavior ? in the present paper , we address aspects of this question . we do so using the nuclear shell model , a theory with a mean field and a residual two body effective interaction @xmath1 . ( we do not include three body forces , although there is evidence @xcite that these may be needed to attain quantitative agreement with data . it will be seen that qualitatively , our arguments would not change with the inclusion of such forces . ) in many nuclei , the mean field is ( nearly ) spherically symmetric . thus , single particle motion is largely regular . chaos in nuclei seems a generic property and , hence , must be due to @xmath1 . we focus attention entirely upon the effects of @xmath1 . therefore , we assume that we deal with a single major shell in which the single particle states are completely degenerate and in which there is a fixed number of valence nucleons . ( a lack of complete degeneracy would reduce the mixing of states due to @xmath1 and , thus , drive the system towards regular motion ) . generic results are expected to be independent of the details of @xmath1 . therefore , we assume that the two body matrix elements ( tbme ) of @xmath1 are uncorrelated gaussian distributed random variables with zero mean value and unit variance . our results then apply to almost all two body interactions with the exception of a set of measure zero . ( the integration measure is the volume element in the parameter space of the tbme . ) the resulting random matrix model is commonly referred to as the two body random ensemble ( tbre ) @xcite . we ask : how does @xmath1 produce chaos in the framework of the tbre ? the two body interaction @xmath1 has two characteristic features . ( i ) it connects pairs of nucleons . ( ii ) it possesses symmetries : it conserves spin , isospin , and parity . we wish to elucidate the role of both features in producing chaos in nuclei . the relevance of the first feature is brought out by comparing the tbre with the goe . we recall that in the latter , the matrix elements of the hamiltonian couple every state in hilbert space to every other such state . these matrix elements are assumed to be uncorrelated random variables . in the context of many body theory , such independent couplings between all pairs of states can be realized only in terms of a many body interaction the rank of which equals the number of valence particles . put differently , with @xmath2 the dimension of the hamiltonian matrix , the number of independent random variables in the goe is @xmath3 and , for @xmath4 , grows much faster than @xmath2 . thus , it is intuitively clear that the goe hamiltonian will produce a thorough mixing of the basis states which is tantamount to chaos . in contradistinction , the number of independent two body matrix elements in a single shell with half integer spin @xmath5 is only @xmath6 while the number of many body states with fixed total spin @xmath7 grows with @xmath5 like @xmath8 where @xmath9 is the number of valence particles . ( the simple estimates leading to these statements are given in the appendix . the statements apply for @xmath10 , and @xmath11 ) . thus , in the tbre the number of independent random variables is much smaller than the dimension of typical matrix spaces , and it is a non trivial fact that @xmath1 produces as much mixing of the basis states as the goe hamiltonian . we wish to elucidate the mechanism which is responsible for this mixing . as for the second feature ( the role of symmetries ) , we compare the tbre with another random matrix model which lacks the symmetries of the tbre but likewise assumes a random two body interaction . this is the embedded two body ensemble of gaussian orthogonal random matrices ( egoe(2 ) ) @xcite . ( for a recent review we refer the reader to ref . @xcite ) . in this model , @xmath9 fermions are distributed over @xmath12 degenerate single particle states . hilbert space is spanned by the resulting @xmath13 slater determinants . the two body interaction connects only those slater determinants which differ in the occupation numbers of not more than two single particle states . therefore , the representation of the two body interaction in the hilbert space of slater determinants yields a sparse matrix ( most non diagonal matrix elements vanish ) . this model does not respect the symmetries of the shell model . neither the single particle states nor the two body interaction carry any quantum numbers . obviously , the model is very different from the goe . it is likewise very different from the tbre . in the latter , the single particle states do carry spin , isospin , and parity quantum numbers , and @xmath1 conserves these symmetries . total hilbert space decays into orthogonal subspaces carrying these same quantum numbers . each subspace is spanned by states which are linear combinations of ( many ) slater determinants . as a result , the matrix representation of @xmath1 in any such subspace becomes fairly dense , even though it remains true that @xmath1 connects only slater determinants which differ in the occupation numbers of not more than two single particle states . in comparing the egoe(2 ) and the tbre , one may consider several options . ( i ) one might use the fixed hilbert space of all many body states that exist within a given major shell , with all possible quantum numbers for total spin , parity , and isospin . these states are coupled either via a symmetry preserving random interaction ( this is the tbre ; here the hamiltonian has block diagonal structure ) , or via a symmetry breaking random interaction ( this is the egoe(2 ) ) . one might compare the spectral statistics of the eigenvalues and the mixing of the eigenfunctions in both models . at this point in time , such a comparison is impractical because of the huge dimension of the matrices involved for the egoe(2 ) . ( ii ) one might use a hilbert space of many body states with fixed values for total spin , parity and isospin . these states are coupled again either via a symmetry preserving random interaction ( this is the tbre within the subspace of many body states with fixed quantum numbers ) or , in the case of a symmetry breaking interaction , via effective interactions that take into account the coupling to many body states with different quantum numbers . here the construction of the effective interaction poses severe difficulties and is practically impossible for realistic cases . both options ( i ) and ( ii ) ask questions relating to the physical role of conserved quantum numbers in the shell model . even if these options were open , we would probably not have used them . the aim of this paper is different . we wish to elucidate the mechanisms which are operative in different random matrix models ( goe , egoe(2 ) , tbre ) in generating chaos . this is why we have followed option ( iii ) : the role of symmetries can also be displayed by comparing the structure of the hamiltonian matrices which are typical for the tbre and the egoe(2 ) , in both cases for hilbert spaces of large dimensions . this option does not require the diagonalization of huge matrices or the difficult construction of effective interactions . it dodges the issue of the physical role of symmetry in shell model calculations and focusses instead upon the structural aspects of the matrices involved in the two approaches . in section [ abs ] we elucidate the structural features of the egoe(2 ) . in section [ pres ] we then turn to a corresponding analysis for the tbre in the case of a single @xmath5shell , and of the @xmath0shell . for both models we aim at displaying those structural elements which are absent in the goe and which are essential for producing a thorough mixing of the basis states in hilbert space and , thus , chaos , in spite of a strongly reduced number of independent random variables . the comparison between the egoe(2 ) and the tbre will then display the role of symmetries in the tbre . ( this aspect of the tbre has been coined `` geometric chaoticity '' by zelevinsky _ et al . _ @xcite . to the best of our knowledge , however , the actual role of symmetries in the tbre has never been investigated . ) we are led to the conclusion , that symmetries are a vital element of the tbre and , in this sense , instrumental in producing chaos in nuclei . we are aware of a huge body of literature addressing issues closely related to our theme . some of these are reviewed in refs . as mentioned above , there is ample evidence for chaos in nuclear shell model calculations in the @xmath0shell and beyond . likewise , there is evidence for chaos in the egoe(2 ) , at least in the center of the spectrum . even deviations from complete chaos have been understood since a long time . as early as 1979 , it was , f.i . , shown @xcite that the non degeneracy of the single particle states in the @xmath0shell prevents a complete mixing of all the states of given total spin and causes the partial width amplitudes ( given as projections of the eigenstates of the shell model hamiltonian onto a fixed vector in hilbert space ) to deviate from the porter thomas distribution predicted by the goe . we are not concerned with adding to this impressive body of evidence . we take it for granted that chaos exists in the shell model and in the egoe(2 ) . rather , we wish to understand the structural aspects of the tbre and of the egoe which produce chaos . we believe that our investigation offers novel insights into the origin of chaos in nuclei in the following two respects : ( i ) we demonstrate that chaos is generic and ubiquitous both in a single @xmath5shell and in the @xmath0shell . to the best of our knowledge , previous arguments have always relied upon numerical results based upon a _ specific _ choice of the two body interaction . in contradistinction , we use _ generic _ aspects of the tbre to show that we must always expect strong mixing of the basis states . ( ii ) we exhibit the role played by symmetries in the tbre for the generation of chaos by comparing this ensemble with the egoe(2 ) , and with the goe . we are not aware of any previous work on this topic . our emphasis on the role of symmetry in generating chaos may be surprising . in fact , the existence of a complete set of quantum numbers ( usually connected to symmetries of the hamiltonian ) is equivalent to integrability and , thus , diametrically opposed to chaos . the case of nuclei ( and , for that matter , of atoms ) is different . the symmetries that dominate nuclei ( invariance under rotation , mirror reflection , and proton neutron exchange ) are incomplete ( they do not form a complete set of integrals of the motion ) . taking account of these symmetries , we can write the hamiltonian in block diagonal form . chaotic motion does seem to exist in every such block . therefore , it is meaningful to ask : in which way is the origin of chaos influenced by the presence such an incomplete set of quantum numbers ? it is this question which we answer by comparing the tbre with the egoe(2 ) and the goe . in this section , we analyse the egoe(2 ) and then compare it with the goe . in the absence of symmetries , the appropriate random two body interaction model is the egoe(2 ) . the egoe(2 ) is often used to model stochastic aspects of realistic systems like small metallic grains or quantum dots . this is justified since in those systems the single particle wave functions themselves are chaotic , and the resulting two body matrix elements reflect this property and transport the information of the underlying one body chaos into the many body system . several numerical studies for matrix dimensions up to a few thousand or so have shown that the egoe(2 ) exhibits goe statistics in the center of the spectrum . for infinite matrix dimensions , the situation is less clear . evidence from previous work on the egoe(2 ) @xcite suggests that the level statistics is poissonian . however , no firm conclusion has yet been reached @xcite . here , we supplement the previous analysis by a different approach and focus on the matrix structure . in the egoe(2 ) , the random variables are the @xmath14/2 $ ] independent two body matrix elements @xmath15 . a hamiltonian drawn from the egoe(2 ) has matrix elements [ hamegoe ] h_=_=1^a v_d _ ( ) , where the matrices @xmath16 transport the information contained in the two body matrix elements @xmath17 into the @xmath2dimensional hilbert space spanned by slater determinants labeled @xmath18 or @xmath19 . the matrices @xmath20 play a central role in the understanding of the egoe(2 ) . indeed , these matrices are the structural elements of this ensemble , while the @xmath17 s are just a set of random variables that change from realization to realization . pictorially speaking , the matrices @xmath20 form the scaffolding which supports the random variables @xmath17 . the properties of the egoe(2 ) ( averages , higher moments and correlation functions ) of both the hamiltonian and the green s functions are completely determined by the matrices @xmath20 . the ensemble average of the hamiltonian is obviously zero . the second moment is [ hameg ] = _ ( d _ ( ) ) ^2 . the properties of the matrices @xmath21 are obtained by counting . let @xmath18 and @xmath19 differ in the occupation numbers of @xmath22 single particle states . ( a ) if @xmath23 , then @xmath24 . ( b ) if @xmath25 , then @xmath26 . ( c ) if @xmath27 , then @xmath28 . ( d ) if @xmath29 or @xmath30 , then @xmath31 . for the number of times each of these alternatives is realized , we find ( a ) @xmath32 , ( b ) @xmath33 , ( c ) @xmath34 , and ( d ) @xmath2 . the number of zero matrix elements dominates ( it is close to @xmath35 ) . the number of unit values comes next and is approximately @xmath36 . the number of values @xmath37 is @xmath34 . the number of values @xmath38 is trivially equal to @xmath2 , the dimension of the matrix and the number of diagonal elements . the correlators @xmath39 can also be worked out easily but are not given here . they do not all vanish . therefore , the random variables @xmath40 are not independent . this reflects the fact that the same matrix element of the two body interaction may couple different pairs of slater determinants . the individual matrices @xmath16 have a very simple structure and barely mix many - body states . let the tbme corresponding to index @xmath41 change the occupation of @xmath22 particles . for @xmath42 the matrix @xmath16 is diagonal . for @xmath43 the matrix @xmath16 can be ordered to have @xmath44 block matrices of dimension two and is zero otherwise . this shows that individual matrices @xmath16 can not generate mixing and chaos . we consider the limit of large matrix dimension @xmath2 , attained by taking the limit @xmath45 . this can be done in two ways : ( i ) by keeping the ratio @xmath46 fixed ; ( ii ) by keeping @xmath9 fixed . ( i ) the ratio of the number of non diagonal elements with variance unity to the total number of matrix elements is @xmath47 . this ratio tends to zero as @xmath45 . the same ratio calculated for the non diagonal elements with variance @xmath37 vanishes even faster . the correlators of the diagonal elements are @xmath48 where @xmath49 is the number of single particle states that are occupied in both @xmath18 and @xmath19 . the correlation is maximal for @xmath50 and vanishes for @xmath51 and @xmath52 . for fixed @xmath18 and @xmath49 , the number of states @xmath19 for which the occupation of @xmath49 single particle states is the same as in @xmath18 is @xmath53 . for @xmath54 , this yields @xmath55 and for @xmath56 , we find @xmath57 . the ratio of both expressions to @xmath2 vanishes exponentially fast as @xmath45 . thus , the matrices approach diagonal form with uncorrelated diagonal elements that all have the same variance . this is suggestive of the poissonian distribution . ( ii ) the ratio of the number of non diagonal elements with variance unity to the total number of matrix elements is @xmath58 and vanishes exponentially fast . the same holds true _ a fortiori _ for the elements with variance @xmath59 . again , the correlators of the diagonal elements are given by @xmath60 , and the number of states which correlate to a given one with this correlator is @xmath61 . the ratio of this value to @xmath2 vanishes as a power of @xmath62 . again , this is suggestive of the poissonian distribution . we note , however , that sparseness of a random matrix is no guarantee for poissonian statistics . indeed , fyodorov and mirlin @xcite have shown that sparse random matrices with @xmath63 non vanishing matrix elements that have a frequency @xmath64 and no preference for large diagonal elements may have either poissonian or goe statistics , depending on the value of @xmath65 . while these results are quite suggestive , the question about spectral fluctuations of the egoe(2 ) in the limit of large matrix dimension remains open . we apply these considerations to the half filled @xmath0-shell ( disregarding , of course , all conserved quantum numbers ) . the number of single particle states is @xmath66 ; the number of nucleons is @xmath67 . the number of independent tbme is @xmath68 . the corresponding matrices of the embedded ensemble have dimension @xmath69 . the variance of the diagonal elements is @xmath70 . the fraction of non diagonal elements with variance unity is approximately @xmath71 , that of those with variance @xmath72 is approximately @xmath73 . the hamiltonian matrices of the egoe(2 ) are thus characterized by their sparsity , strong diagonal structure , and a relatively large number of independent tbme . this structure is very different indeed from that of the goe . in analogy to eq . ( [ hamegoe ] ) , the matrix elements of the goe hamiltonian can be written in the form h^goe _ = _ j < l = 1^n v_j l d^goe_(j , l ) . [ hamgoe ] the random variables @xmath74 are defined for @xmath75 where @xmath2 is the dimension of @xmath76 and are uncorrelated gaussian distributed random variables with zero mean and a common second moment . the matrices @xmath77 again determine the structure of the ensemble and are given by d^goe_(j , l ) = _ j _ [ dgoe ] each such matrix is symmetric and has only one non vanishing matrix element above or in the main diagonal . again , all ensemble averages of the goe are determined by the matrices @xmath77 . in view of the extreme simplicity of these matrices , one usually does not write the goe in the form of eq . ( [ hamgoe ] ) . this form is , however , useful for purposes of comparison . we observe that in the goe , the number of independent random variables and , thus , of matrices @xmath77 is as large as is consistent with the basic symmetry ( invariance under time reversal ) of the ensemble . in the egoe(2 ) and for matrices of the same dimension @xmath2 , the number of independent random variables and , thus , of matrices @xmath20 is strongly reduced . this strong reduction is accompanied by a strong increase in the number of non vanishing matrix elements of each of the matrices @xmath20 . although sparse , the matrices @xmath20 are much less so than their goe counterparts . in this section we consider the tbre , both for a single @xmath5shell and for the @xmath0shell . we first investigate the tbre for the case of a single @xmath5shell . this is the simplest case and already exhibits the major difference to the egoe . then we consider the more realistic ( and more complex ) case of the @xmath0shell where we encounter several sub shells and also have to include isospin in our analysis . we consider @xmath9 fermions in a single @xmath5shell . ( later , we will consider the example @xmath78 and @xmath79 ) . there are @xmath80 tbme as two identical particles can have spins @xmath81 . the corresponding spin conserving two body matrix elements are denoted by @xmath17 , with @xmath82 . the matrix elements of the @xmath5-shell hamiltonian in the space of many body states with total spin @xmath7 and projection @xmath83 are [ hamj ] h_^j = _ = 1^a v_c_^j ( ) . again , the matrices @xmath84 transport the information about the tbme @xmath85 into the space of the many body states . with the @xmath17 considered as uncorrelated gaussian distributed random variables with zero mean value and a common second moment , all properties of the tbre ( mean values , higher moments and correlation functions of both hamiltonian and green s functions ) are again completely determined by the matrices @xmath86 , in full analogy to the cases of the goe and of the egoe(2 ) . to see how chaos is generated in the tbre it is , thus , neccessary to understand the properties and structure of these matrices . in contrast to the egoe(2 ) , the matrices @xmath86 are determined by both , the spin symmetry and the fermionic nature of the @xmath9body system . each element is given in terms of sums over products of angular momentum coupling coefficients and coefficients of fractional parentage and is , thus , a rather complex quantity . therefore , the properties of the matrices @xmath84 can not be inferred as easily as those of the matrices @xmath21 in eq . ( [ hameg ] ) . some facts can be established analytically . for what remains , we rely on numerical investigations . for @xmath10 , the number @xmath87 of @xmath9body states of fixed total spin @xmath7 is approximately given by n(j ) ( _ j,0 + 2 j ) \ { - } . [ numb ] the right hand side is the leading term in an asymptotic expansion in inverse powers of @xmath5 and @xmath9 . this expression is derived in the appendix . we observe that for fixed values of @xmath5 and @xmath9 , the number of states increases monotonically with @xmath7 until the gaussian cutoff becomes relevant . the maximum spin has the value @xmath88 . the cutoff sets in much below this value . for @xmath7 fixed and below the cutoff , @xmath89 grows strongly with @xmath5 . thus , for @xmath90 the dimension of the matrices @xmath86 is very much larger than their number @xmath80 . we note that the trend seen in the comparison between the goe and the egoe(2 ) continues unabatedly : in comparison with the matrix dimension , the number of independent random variables is reduced much below the egoe(2 ) value . to achieve complete mixing of the states , this small number must be compensated by an increased density of the matrix elements of the matrices @xmath86 . the @xmath91 matrices @xmath92 can be viewed as matrix representations of @xmath91 operators @xmath93 . as shown in the appendix , the latter are given by ^j ( ) = * p*(j ) x ( ) * p*(j ) [ oper ] where @xmath94 is the orthonormal projector onto the subspace of many body states with fixed total spin @xmath7 . the operators @xmath95 are scalar two body operators normalized in such a way that with @xmath96 the number operator , we have @xmath97 . using this representation , it is easy to see ( appendix ) that the @xmath93 s do not commute : for @xmath98 , the commutator @xmath99 $ ] is a three body operator projected onto the space of states with spin @xmath7 . we also observe that by definition of the operators @xmath100 , we have @xmath101 . we turn to a numerical determination of the matrices @xmath92 for a shell with @xmath79 and with @xmath78 fermions . to this end we have to define the basis . total hilbert space is spanned by slater determinants of single particle states . these have spin projection @xmath102 ( but not well defined total spin @xmath7 ) . basis states with definite angular momentum are constructed numerically by diagonalizing the total angular momentum operator @xmath103 . on the one hand , this basis is not unique since the spectrum of @xmath103 is highly degenerate . on the other hand , there is no preferred basis , and our results can therefore be viewed as rather generic . the second moment [ mom2 ] = _ ( c^j_())^2 exhibits almost constant and dominant diagonal elements , and considerably smaller off diagonal elements . the dominance of the diagonal elements is not surprising in view of the identity @xmath104 , and in this aspect the tbre is similar to the egoe . the off diagonal elements of the second moment ( [ mom2 ] ) are depicted in fig . [ fig0 ] for the total spin @xmath105 . ( this is the largest dimensional sector in hilbert space ) . the left part of fig . [ fig0 ] shows a contour plot of the off diagonal elements , while the right part of fig . [ fig0 ] shows the corresponding histogram . clearly , all off diagonal elements are non zero , and this is in stark contrast to the sparse egoe matrices . we recall that the corresponding histogram for the egoe would have a ( giant ) peak at zero and two smaller peaks . this suggests that already a two body operator corresponding to a single non vanishing tbme will strongly mix the basis states in the @xmath5shell tbre . we recall that the matrices @xmath106 do not commute . thus , the mixing is expected to be strong for almost all hamiltonians of the tbre . moreover , the non commutativity of the @xmath86 s and the sum rule @xmath107 ( which is invariant under a rotation of the basis ) together strongly suggest that the results shown in fig . [ fig0 ] are generic and independent of the basis chosen . left : ensemble averaged second moment of the tbre ( diagonal elements suppressed ) for 6 fermions in a single @xmath108 shell with total spin @xmath105 . right : same data shown in a histogram . , title="fig:",height=264 ] left : ensemble averaged second moment of the tbre ( diagonal elements suppressed ) for 6 fermions in a single @xmath108 shell with total spin @xmath105 . right : same data shown in a histogram . , title="fig:",height=264 ] to lend further substance to these arguments , we have diagonalized the operators @xmath95 in the space of the 1242 slater determinants @xmath109 with @xmath102 ( but with no fixed spin ) where @xmath110 with @xmath111 . the eigenstates carry the labels @xmath41 and @xmath7 as well as a running label @xmath18 where @xmath112 and @xmath113 is the dimension of the subspace with spin @xmath7 . they are given by @xmath114 . the complexity of the eigenstates @xmath115 is measured by the number of principal components ( @xmath116 ) defined as ( npc)^-1 = _ i = 1^d c_i^4(,j , ) . table [ tab2 ] shows the @xmath116s , averaged over all two body spins ( i.e. , over all values of @xmath41 ) and over all @xmath113 states with spin @xmath7 . we recall that the goe expectation for @xmath116 is @xmath117 . we see that the spin conserving two body operators @xmath95 individually yield a strong mixing of the basis states . this mixing is induced entirely by the rotational symmetry and independent of any particular choice of the two body interaction . it may be argued that the complexity of the coefficients @xmath118 reflects just the need to couple the states @xmath119 to total spin @xmath7 , and is not indicative of strong mixing due to @xmath95 . inspection of the individual coefficients @xmath118 shows , however , strong variations with the index @xmath41 which invalidates this argument . moreover , calculation of the @xmath116s in a basis with fixed @xmath7 confirms our picture . [ cols=">,>,>,>,>,>,>,>,>",options="header " , ] although the mixing between partitions is not as strong as that within each partition , @xmath120 will , for @xmath121si and @xmath122 @xmath123 , generically generate chaos . to show this , we have compared our figure [ fig1 ] with the corresponding figure for the wildenthal two body interaction ( which was used in ref . @xcite and shown there to produce chaos ; see , e.g. , figs . 23(a ) and 23(c ) of that reference ) . the two figures are indistinguishable . the aims and results of our paper can be summarized from two points of view , from that of random matrix theory and from that of nuclear structure theory . from the viewpoint of random matrix theory , we have compared three random matrix ensembles that have been used in the past to deal with interacting many body systems and , in particular , with nuclei : the goe , the egoe(2 ) , and the tbre , the latter applied to a single @xmath5shell and to the @xmath0shell . in our comparison , we have emphasized the central role of the structure matrices denoted by @xmath77 , @xmath20 , @xmath124 , and @xmath125 , respectively . these matrices provide the scaffolding of the underlying random matrix ensemble . averages , higher moments and correlation functions of all observables can be expressed in terms of and are completely determined by these matrices . these matrices then must form the central object of study of the said random matrix ensembles . in the framework of a many body problem , use of the goe is tantamount to assuming many body forces the rank of which equals the number of valence particles . this unrealistic aspect of the goe has to be weighed against the great advantage of its structural simplicity . this simplicity is due to the orthogonal invariance of the goe . it allows for a complete analytical calculation of all moments and correlation functions . hence the predictive power of the goe . the orthogonal invariance manifests itself in the large number ( @xmath126 with @xmath2 the matrix dimension ) of independent random variables and in the extreme simplicity of the structural matrices @xmath77 . in the goe , no reference is made to the possible existence of quantum numbers like spin or isospin . the egoe(2 ) is designed to deal with many body systems which are governed by two body forces . in this respect the egoe(2 ) is a much more realistic model than the goe . the model does not possess the orthogonal invariance of the goe . this is why all attempts at calculating spectral fluctuations analytically for this ensemble have failed so far . for the same matrix dimension , the number of uncorrelated random variables is much smaller than in the goe . to achieve complete mixing of the basis states , this reduction must be made up for by a greater complexity of the matrices @xmath20 . we have displayed the structure of these matrices which individually are quite sparse but jointly provide for strong mixing of the basis states . like the goe , the egoe(2 ) does not allow for the possible existence of quantum numbers like spin or isospin . the tbre is more realistic yet than the egoe(2 ) in that it does take account of the two essential properties of the residual interaction of the nuclear shell model : the interaction is a two body interaction , and it preserves spin , parity , and isospin . the existence of symmetries associated with these quantum numbers has an essential influence on the structure of the ensemble and of the matrices @xmath124 and @xmath125 which embody this structure : compared to the egoe(2 ) with the same matrix dimension , the number of independent random variables is much reduced once again , but the complexity of the matrices @xmath124 and @xmath125 is much increased . it appears that the tbre is even harder to deal with than the egoe(2 ) . in any case , we are not aware of any previous attempts to study this ensemble analytically . in the tbre for a single @xmath5shell , the non commuting matrices @xmath124 have strong diagonal elements ( a feature already encountered for the egoe(2 ) and related to sum rules ) . the non diagonal elements are not sparse but , on the contrary , dense . this is how a complete mixing of the basis states is achieved in the @xmath5shell tbre . in the @xmath0shell tbre , the existence of sub shells and the associated block structure of the matrices @xmath125 lends greater complexity yet to the ensemble . mixing is very strong within the diagonal blocks and weaker for the off diagonal ones . in both these ensembles , symmetries play an important role and , together with the exclusion principle , define the structure of the matrices @xmath124 and @xmath125 . it goes without saying that the relevant symmetry operators must not from a complete set of commuting operators as otherwise there would be no room left for a random matrix ensemble . from the point of view of nuclear structure theory , we have uncovered generic features of shell model calculations . these are embodied in the matrices @xmath124 and @xmath125 . every shell model calculation for a single @xmath5shell or for the @xmath0shell amounts to choosing a specific linear combination of these matrices ( the same for every value of spin @xmath7 ) , and diagonalizing the resulting hamiltonian matrix . corresponding statements apply for other shells . inasmuch as there is evicence for chaos in one such calculation using a specific set of two body matrix elements , the properties of the matrices @xmath124 and @xmath125 displayed above guarantee that spectra and eigenfunctions calculated for most other choices of the two body interaction will likewise be chaotic . the two body interactions for which this statement does not apply form a set of measure zero . it is in this sense that we have demonstrated that chaos is a generic property of the nuclear shell we have also shown that symmetries ( which are due to the existence of an incomplete set of commuting operators ) determine the structure of the tbre in an essential way . thus , such symmetries are vital for the occurrence of chaos in the tbre . another aspect of our work ( not emphasized in the present paper but a natural spin off ) is the existence of correlations between many body spectra having different quantum numbers like total spin @xmath7 . it is immediately obvious from eqs . ( [ mom2 ] ) and ( [ hamsm ] ) that hamiltonian matrices pertaining to different spin values in the same nucleus are correlated since they depend on the same set of random variables . this fact has been used to explain the observed preponderance of spin zero ground states in calculations using the tbre @xcite . our considerations are restricted to spherical nuclei and totally degenerate major shells . we have remarked in the introduction that lifting this degeneracy by taking into account the differences of the single particle energies in individual sub shells , will drive the system toward regularity . our considerstions do not apply to deformed nuclei . here the nilsson model provides a single particle basis in which it is meaningless to assume degenerate single particle energies . such nuclei possess well developed collective motion . however , collective motion exists also beyond the regime of well deformed nuclei . it has been notoriously difficult in the past to understand this fact in the framework of the spherical shell model , with the exception of certain types of collectivity like that of the giant dipole resonance . understanding both , collectivity and chaos , within a common framework is , thus , a goal of future work . we observe , however , that collectivity typically involves levels with different quantum numbers ( like those forming a rotational band ) while chaos is a property displayed by levels with identical quantum numbers . thus , in nuclei chaos and collectivity need not be antagonistic . we are grateful to o. bohigas and u. smilansky for helpful discussions . this research was supported in part by the u.s . department of energy under contract nos . de - fg02 - 96er40963 ( university of tennessee ) and de - ac05 - 00or22725 with ut - battelle , llc ( oak ridge national laboratory ) . in the framework of the tbre for a single @xmath5shell , we derive eq . ( [ numb ] ) , and we define the operators @xmath95 . the calculation of @xmath87 , the number of many body states with spin @xmath7 , is rather standard . we first calculate the number @xmath127 of states with @xmath128 and then @xmath87 from the identity @xmath129 . we focus attention on large values of @xmath9 and @xmath5 , @xmath130 and @xmath131 . we have @xmath132 here the delta symbol stands for a kronecker delta and not for the dirac delta function . we write the sum as @xmath133 \bigg ) \delta(m - \sum_{i = 1}^m \mu_i ) \ . \label{a2}\ ] ] we expand the product in powers of the kronecker deltas and consider first the term of zeroth order @xmath134 , i.e. , the term with only one constraint ( @xmath135 ) . it is @xmath136 the kronecker delta is written as @xmath137 { \rm d}\phi$ ] . this yields @xmath138 \ . \label{a4}\ ] ] the summation yields n_0(m ) = _ - ^+ d ( ) ^m . [ a5 ] we replace @xmath139 by @xmath140 and use that @xmath141 . then the function ( ) ^m [ a7 ] has maxima at @xmath142 , @xmath143 , @xmath144 , @xmath145 with absolute values @xmath146 , @xmath147 , @xmath148 , @xmath145 . for @xmath130 , the maximum at @xmath142 gives the dominating contribution . we take account of this maximum only and write @xmath149 } { \sin [ x / ( 2 j ) ] } \biggr)^m & = & \exp \bigg\ { m \ln \sin [ ( 1 + 1/(2j))x ] - m \ln \sin [ x / ( 2j ) ] \bigg\ } \nonumber \\ & & \qquad \approx ( 2j+1)^m \exp \ { - ( m/6 ) x^2 \ } \ . \label{a8}\end{aligned}\ ] ] in the exponent we have omitted terms of higher order than the first in @xmath150 . this is justified because upon using a taylor expansion and performing the gaussian integration , such terms will produce inverse powers of @xmath9 and are , therefore , negligible . thus , our procedure amounts to an asymptotic expansion in inverse powers of @xmath5 and @xmath9 . hence , @xmath151 \exp \ { - ( m/6 ) x^2 \ } \nonumber \\ & = & \frac{(2j+1)^m}{2 \pi j m ! } \sqrt { \frac{6 \pi}{m } } \exp \big\ { - \frac{3m^2}{2mj^2 } \big\ } \ . \label{a9}\end{aligned}\ ] ] we turn to the terms which are linear in the kronecker delta s in eq . ( [ a2 ] ) . their sum is denoted by @xmath152 and can be calculated along quite similar lines . in the same asymptotic limit we find n_1(m ) - . [ a10 ] this shows that @xmath152 is negligible in comparison with @xmath134 if @xmath90 . that same statement holds _ a fortiori _ for the contributions which are of higher order in the kronecker delta s . thus , @xmath153 , and @xmath129 then gives the result ( [ numb ] ) . we come to the definition of the operators @xmath95 . let @xmath154 and @xmath155 be the destruction and creation operators for a fermion in a state with @xmath156component @xmath157 with @xmath158 . when acting upon the vacuum state , the operators ( a^s_m)^ = _ c(j j s ; , m- ) a^_a^_m- [ b1 ] create a pair of fermions coupled to total spin @xmath159 and @xmath160component @xmath161 . here , @xmath162 . the hermitean conjugate operators are a^s_m = - _ c(j j s ; , m- ) a^_a^_m- . [ b2 ] the scalar operators x(s ) = ( 1/2 ) _ m ( a^s_m)^ a^s_m [ b3 ] describe the interaction of two fermions coupled to spin @xmath159 . with @xmath163 , these are the operators @xmath95 introduced in eq . ( [ oper ] ) . from an identity for the clebsch gordan coefficients , it follows trivially that _ s x(s ) = ( 1/2 ) n ( n - 1 ) . [ b4 ] this is the relation used below eq . ( [ oper ] ) . for @xmath164 , the commutator @xmath165 $ ] does not vanish and has the from of a three body interaction term . since the projection operators @xmath166 trivially commute with @xmath167 for all @xmath7 and @xmath159 , the commutator of two operators @xmath168 with @xmath169 is = * p*(j ) [ x ( ) , x ( ) ] * p*(j ) [ b5 ] and it follows that the @xmath168 s likewise do not commute . 99 r. u. haq , a. pandey , and o. bohigas , phys . * 48 * , 1086 ( 1982 ) ; o. bohigas , r. u. haq , and a. pandey , in _ nuclear data for science and technology _ , k. h. bchhoff ( ed . ) , reidel , dordrecht ( 1983 ) . g. e. mitchell , e. g. bilpuch , p. m. endt , and f. j. shriner , jr . lett . * 61 * , 1473 ( 1988 ) ; f. j. shriner , jr . , e. g. bilpuch , p. m. endt , and g. e. mitchell , z. phys . a * 335 * , 393 ( 1990 ) . v. zelevinsky , b. a. brown , n. frazier , and m. horoi , phys . rep . * 276 * , 85 ( 1996 ) . magd , m. simbel , h.l . harney , and h. a. weidenmller , phys . lett . b * 579 * , 278 ( 2004 ) , nucl - th/0212057 . v. v. flambaum , a. a. gribakina , g. f. gribakin , and m. g. kozlov , phys . a * 50 * , 267 ( 1994 ) . s. c. pieper , r. b. wiringa , ann . nucl . part . sci . * 51 * , 53 ( 2001 ) , nucl - th/0103005 . j. b. french and s. s. m. wong , phys . b * 33 * , 449 ( 1970 ) . o. bohigas and j. flores , phys . b * 34 * , 261 ( 1971 ) . k. k. mon and j. b. french , ann . phys . ( ny ) * 95 * , 90 ( 1975 ) . v. k. b. kota , phys . rep . * 347 * , 223 ( 2001 ) . l. benet and h. a. weidenmller , j. phys . a * 36 * , 3569 ( 2003 ) , cond - mat/0207656 . j. j. m. verbaarschot and p. j. brussaard , phys . lett . * 87 b * ( 1979 ) 155 . l. benet , t. rupp , h. a. weidenmller , phys . rev * 87 * , 010601 ( 2001 ) , cond - mat/0010425 ; ann * 292 * , 67 ( 2001 ) , cond - mat/0010426 . m. srednicki , phys . e * 66 * , 046138 ( 2002 ) , cond - mat/0207201 . y. v. fyodorov und a. d. mirlin , j. phys . 2273 ( 1991 ) . b. a. brown , a. etchegoyen , and w. d. m. rae , _ the computer code oxbash _ , msu - nscl report number 524 ( 1988 ) . t. papenbrock and h. a. weidenmller , phys . * 93 * ( 2004 ) 132503 , nucl - th/0404022 .

to elucidate the mechanism by which chaos is generated in the shell model , we compare three random matrix ensembles : the gaussian orthogonal ensemble , french s two body embedded ensemble , and the two body random ensemble ( tbre ) of the shell model . of these , the last two take account of the two body nature of the residual interaction , and only the last , of the existence of conserved quantum numbers like spin , isospin , and parity . while the number of independent random variables decreases drastically as we follow this sequence , the complexity of the ( fixed ) matrices which support the random variables , increases even more . in that sense we can say that in the tbre , chaos is largely due to the existence of ( an incomplete set of ) symmetries . shell model , symmetry , complexity 21.10.-k , 05.45.mt , 24.60.-k

in constructing a model for the self - assembly of addressable structures , we note that the designed interactions should be much stronger than any attractive interactions between subunits that are not adjacent in a correctly assembled structure . the designed interactions that stabilize the target structure can be described by a connectivity graph , @xmath0 , in which each vertex represents a distinct subunit and each edge indicates a correct bond . this graph allows us to describe the connectivity of the structure independently of the geometry and spatial organization of the building blocks . for structures constructed from dna bricks , the edges of @xmath0 indicate the hybridization of dna strands with complementary sequences that are adjacent in the target structure . an example three - dimensional dna - brick structure is shown along with its connectivity graph in figures [ fig : ramp_example]a and [ fig : ramp_example]b . in an ideal solution with exclusively designed interactions , the subunits assemble into clusters in which all allowed bonds are encoded in the connectivity graph of the target structure . in order to compute the free - energy difference between a particular cluster size and the unbonded single - stranded bricks , we must consider all the ways in which a correctly bonded cluster with a given number of monomers can be assembled . these ` fragments ' of the target structure correspond to connected subgraphs of the connectivity graph . in a dilute solution with strong designed interactions , the numbers of edges and vertices are the primary factors determining the stability of a particular fragment . we therefore identify all of the possible on - pathway assembly intermediates by grouping fragments into sets with the same number of edges and vertices and counting the total number of fragments in each set.@xcite this theoretical approach is powerful because it can predict the free - energy landscape as a function of the degree of assembly between the monomers and the target structure . furthermore , the predicted landscape captures the precise topology of the target structure , which is essential for understanding the assembly of addressable , finite - sized structures . in the case of dna - brick structures , we can assign dna hybridization free energies to the edges of the target connectivity graph in order to determine the temperature dependence of the free - energy landscape ; for example , figure [ fig : ramp_example]d shows the free - energy profile of the 86-strand dna - brick structure with random dna sequences at three temperatures . our theoretical approach allows us to calculate the nucleation barrier , @xmath1 , by examining the free energies of clusters corresponding to fragments with exactly @xmath2 vertices . the critical number of strands required for nucleation is @xmath3 : transient clusters with fewer than @xmath3 strands are more likely to dissociate than to continue incorporating additional strands . the presence of a substantial nucleation barrier therefore inhibits the proliferation of large , partially assembled fragments that stick together to form non - target aggregates . over a significant range of temperatures , we find that the free - energy profiles of dna - brick structures exhibit both a nucleation barrier and a thermodynamically stable intermediate structure . the nucleation barrier is associated with the minimum number of subunits that must be assembled in order to complete one or more _ cycles _ , i.e. closed loops of stabilizing bonds in a fragment . for example , the critical number of monomers in the example structure at 319 k , @xmath4 , is one fewer than the nine subunits required to form a bicyclic fragment of the target structure . under the conditions where nucleation is rate controlling , the minimum free - energy structure is _ not _ the complete 86-particle target structure , but rather a structure with only @xmath5 particles . this incomplete structure is favored by entropy , since it can be realized in many more ways than the unique target structure . hence , the temperature where nucleation is rate controlling is higher than the temperature where the target structure is the most stable cluster . the existence of thermodynamically stable intermediates is a typical feature of dna - brick structures and of complex addressable , finite - sized structures in general . this behavior is not compatible with classical nucleation theory ( cnt ) , which predicts that , beyond the nucleation barrier , large clusters are always more stable than smaller clusters . as a consequence , in ` classical ' nucleation scenarios such as crystallization , there is a sharp boundary in temperature and concentration at which the largest - possible ordered structure , rather than the monomeric state , becomes thermodynamically stable . typically , a simple fluid must be supersaturated well beyond this boundary in order to reduce the nucleation barrier , which arises due to the competition between the free - energy penalty of forming a solid liquid interface and the increased stability due to the growth of an ordered structure.@xcite yet in the case of addressable self - assembly , and dna bricks in particular , a nucleation barrier for the formation of a stable partial structure may exist even when the target structure is unstable relative to the free monomers . an experiment to assemble such a structure requires a _ protocol _ : first nucleation at a relatively high temperature , and then further cooling to complete the formation of the target structure . this behavior can be seen in figure [ fig : ramp_example]e , where we identify a narrow temperature window in which there is a significant yet surmountable nucleation barrier . unlike cnt , the nucleation barrier does not diverge as the temperature is increased . instead , there is a well - defined temperature above which all clusters have a higher free energy than the free monomers . as the temperature is lowered further , the nucleation barrier disappears entirely before the equilibrium yield , defined as the fraction of all clusters that are correctly assembled as the complete target structure , increases measurably above zero . the equilibrium yield tends to 100% at low temperatures , since we have thus far assumed that only designed interactions are possible . therefore , because of the presence of stable intermediate structures , it is typically impossible to assemble the target structure completely at any temperature where nucleation is rate controlling . in order to examine the importance of a nucleation barrier for preventing misassembly , we calculate the free - energy difference between all off - pathway intermediates and all on - pathway intermediates , @xmath6 , by estimating the probability of incidental interactions between partially assembled structures.@xcite from the connectivity graph of the example dna - brick structure , we can calculate the total free energy of aggregated clusters by considering all the ways that partially assembled structures can interact via the dangling ends of the single - stranded bricks , as shown in figure [ fig : ramp_example]c . we also estimate this free - energy difference in the case of slow nucleation , @xmath7 , by only allowing one of the interacting clusters in a misassembled intermediate to have @xmath8 . the above analysis supports our claim that a substantial nucleation barrier is essential for accurate self - assembly . our calculations show that even with very weak incidental interactions , incorrect bonding between the multiple dangling ends of large partial structures prevents error - free assembly at equilibrium , since @xmath9 . the presence of a nucleation barrier slows the approach to equilibrium , maintaining the viability of the correctly assembled clusters . these theoretical predictions are confirmed by extensive monte carlo simulations of the structure shown in figure [ fig : ramp_example]a . in these simulations , the dna bricks are modeled as rigid particles that move on a cubic lattice , but otherwise the sequence complementarity and the hybridization free energies of the experimental system are preserved.@xcite using realistic dynamics,@xcite we simulate the assembly of the target structure using a single copy of each monomer . in figure [ fig : ramp_example]f , we compare a representative trajectory from a simulation using a linear temperature ramp with a trajectory from a constant - temperature simulation starting from free monomers in solution . we also report the largest stable cluster size averaged over many such trajectories in figure [ fig : ramp_example]g . nucleation first occurs within the predicted nucleation window where @xmath10 . at 319 k , the size of the largest stable cluster coincides precisely with the predicted average cluster size at the free - energy minimum in figure [ fig : ramp_example]d . intermediate structures assembled via a temperature ramp continue to grow at lower temperatures , while clusters formed directly from a solution of free monomers become arrested in conformations that are incompatible with further growth ( figure [ fig : ramp_example]f,_inset _ ) . in agreement with our theoretical predictions , the simulation results demonstrate that a time - dependent protocol is essential for correctly assembling a complete dna - brick structure . in the modular assemblies reported in ref . , the maximum coordination number of bricks in the interior of the structure is four . however , one can envisage other building blocks , such as functionalized molecular constructs or nano - colloids , that have a different coordination number . to investigate the effect of the coordination number on the nucleation barrier , we compare the free - energy profile of a 48-strand dna - brick structure with those of two higher - coordinated structures ( figure [ fig : coordination_number]a ) : a simple cubic structure with coordination number @xmath11 and a close - packed structure with @xmath12 . ( for a discussion of two - dimensional structures , see sec . [ sec:2d ] . ) in figure [ fig : coordination_number]b , we show the free - energy profiles at 50% yield assuming identical bond energies within each structure . one striking difference between the @xmath13 structure and the higher - coordinated examples is the stability of the target at 50% yield . in the dna - brick structure , the target structure coexists in nearly equal populations with a partial structure that is missing a single cycle . in the structures with higher coordination numbers , however , the target has the same free energy as the free monomers at 50% yield . intermediate structures are therefore globally unstable at all temperatures , as predicted by cnt . a second point of distinction among these structures lies in the relative stability of intermediate cluster sizes . whereas the dna - brick structure assembles by completing individual cycles , the cubic structure grows by adding one face at a time to an expanding cuboid . with @xmath12 , the greater diversity of fragments with the same number of vertices smooths out the free - energy profile near the top of the nucleation barrier . the fitted black line in figure [ fig : coordination_number]b shows that the assembly of this structure does in fact obey cnt ( see sec . [ sec : cnt ] ) . the differences among these free - energy profiles originate from the topologies of the connectivity graphs of the example structures . the most important determinant of the nucleation behavior is simply the number of vertices required to complete each additional cycle in the target connectivity graph , which is controlled by the maximum coordination number of the subunits . ) and thus does not affect the shape of the free - energy profile . ] our findings imply that controlled self - assembly of three - dimensional addressable structures is unlikely to be achieved straightforwardly using subunits with coordination numbers higher than four . in higher - coordinated structures , which are well described by cnt , it would be necessary to go to high supersaturation in order to find a surmountable nucleation barrier ; however , such an approach is likely to fail due to kinetic trapping.@xcite yet in dna - brick structures , the nucleation barrier is surmountable at low supersaturation and is relatively insensitive to the size of the target structure ( figure [ fig : coordination_number]c ) . the reliable self - assembly of large dna - brick structures is thus a direct consequence of the small number of bonds made by each brick . recent publications have argued that equal bond energies should enhance the stability of the designed structure@xcite and reduce errors during growth.@xcite by contrast , we find that the kinetics of dna - brick assembly are actually worse if one selects dna sequences that minimize the variance in the bond energies . our observation is consistent with the successful use of random dna sequences in the original experiments with dna bricks.@xcite here again , the nucleation behavior is responsible for this unexpected result . to demonstrate the difference between random dna sequences and sequences chosen to yield monodisperse bond energies , we consider the relatively simple non - convex dna - brick structure shown in figure [ fig : central_hole]a . this 74-brick structure , constructed by removing the interior strands and two faces from a cuboidal structure , assembles roughly face - by - face when using random dna sequences . the relevant nucleation barrier , as predicted theoretically in figure [ fig : central_hole]b and confirmed with monte carlo simulations in figure [ fig : central_hole]c , is the completion of the third face . with monodisperse bond energies and an equivalent mean interaction strength , a much larger nucleation barrier appears before the first face forms . attempts to reduce this nucleation barrier by increasing the mean bond energy result in kinetic trapping and arrested growth . despite promising fluctuations in the largest cluster size in the simulation trajectory with monodisperse energies , multiple competing nuclei appear , and the largest cluster remains poorly configured for further assembly ( figure [ fig : central_hole]c,_inset _ ) . the use of sequences with a broad distribution of hybridization free energies results in a more suitable nucleation barrier because such a distribution selectively stabilizes small and floppy intermediate structures . this is a statistical effect : since there are far fewer ways of constructing a maximally connected fragment with a given number of monomers , the chance that randomly assigned sequences concentrate the strongest bonds in a compact fragment is vanishingly small in a large structure . as a result , the dominant nucleation pathways no longer need to follow the maximally connected fragments . the use of a broad distribution of bond energies therefore tends to reduce nucleation barriers , since unstable fragments near the top of a barrier contain fewer cycles and are thus affected more significantly by the variance in the bond - energy distribution . the insights provided by our predictive theory allow us to understand the general principles underlying the unexpected success of dna - brick self - assembly . slow , controlled nucleation at low supersaturation is achieved for large structures since each brick can only make a small number of designed connections . because of an appreciable nucleation barrier that appears in a narrow temperature window , monomer depletion does not pose a significant problem for one - pot assembly . surprisingly , complex structures with randomly selected complementary dna sequences experience enhanced nucleation , making larger intermediate structures kinetically accessible at higher temperatures . the use of a temperature ramp plays a more crucial role than previously thought . cooling the dna - brick solution slowly is not just a convenient way of locating good assembly conditions , as in the case of conventional crystals ; rather , it is an essential non - equilibrium protocol for achieving error - free assembly of finite - sized structures . the explanation of slow nucleation and fast growth that was originally proposed in refs . and is therefore incomplete : fast growth allows the dna bricks to assemble into a stable , on - pathway intermediate that must be annealed at lower temperatures to complete the target structure . remaining out of equilibrium throughout the assembly protocol , as is necessary in order to avoid the aggregation of partial structures , relies on the slow diffusion of large intermediates . this is a reasonable assumption , since the rate of diffusion changes approximately inversely with the radius of a fragment in solution.@xcite our approach also suggests how to improve the design of dna - brick nanostructures beyond the random selection of uniformly distributed dna sequences . for a given target structure , it is easy to tune the nucleation barrier by adjusting the statistical distribution of bond energies . complementary dna sequences can then be assigned to the structure in order to achieve the desired distribution of hybridization free energies . furthermore , with an understanding of the origin of the nucleation barrier in a particular structure , it is possible to optimize the annealing protocol rationally in order to increase the yield of the target assembly . our approach also provides a means of systematically investigating how local modifications to the coordination number through the fusing of adjacent strands affect the nucleation behavior of dna - brick structures.@xcite the theoretical method used here greatly simplifies the quantitative prediction of nucleation barriers and intermediate structures with widespread applications for controlling the self - assembly of biomolecular or synthetic building blocks . addressable self - assembly holds great promise for building intricate three - dimensional structures that are likely to require optimization on a case - by - case basis . because our predictive theory is sensitive to the details of a particular target structure , performing these calculations for nanostructures of experimental interest will enable the precise engineering of assembly properties at the design stage . in order for potential users to perform such experimental protocol design , we provide a user - friendly software package online at https://github.com/wmjac/pygtsa . this work was carried out with support from the european research council ( advanced grant 227758 ) and the engineering and physical sciences research council programme grant ep / i001352/1 . w.m.j . acknowledges support from the gates cambridge trust and the national science foundation graduate research fellowship under grant no . we compute the hybridization free energies of complementary 8-nucleotide dna sequences using established empirical formulae@xcite assuming salt concentrations of [ na@xmath14 = 1 moldm@xmath15 and [ mg@xmath16 = 0.08 moldm@xmath15 . for the calculations with monodisperse bond energies , we use the sequences provided in ref . . the strengths of incidental interactions are estimated based on the longest attractive overlap for each pair of non - complementary sequences . in calculations of the equilibrium yield and free - energy profiles , we report the average thermodynamic properties using 1000 randomly chosen complete sets of dna sequences . see sec . [ sec : distributions ] for further details . constant - temperature lattice monte carlo simulations are carried out using the virtual move monte carlo algorithm@xcite in order to produce physical dynamics . rigid particles , each with four distinct patches fixed in a tetrahedral arrangement , are confined to a cubic lattice . a single copy of each required subunit is present in the simulation box with @xmath17 lattice sites . complete details are given in ref . . for comparison with the results of these simulations , the theoretical calculations reported here assume the same dimensionless monomer concentration , @xmath18 , lattice coordination number , @xmath13 , and fixed number of dihedral angles , @xmath19 ( see sec . [ sec : theory ] ) . 29ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop * * , ( ) @noop * * , ( ) in @noop _ _ ( , ) pp . @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop _ _ , vol . we construct the connectivity graph , @xmath0 , from the designed bonds between adjacent subunits in the target structure . an example connectivity graph is shown in figure [ fig : ramp_example]b . from this graph we are able to determine all the relevant thermodynamic properties of the intermediate and target structures in a near - equilibrium assembly protocol . a thorough explanation of this theoretical method is presented in ref . ; here we summarize the key equations . the connected subgraphs ( ` fragments ' ) of the target connectivity graph are grouped into sets , @xmath20 , in which all fragments have precisely @xmath21 edges and @xmath2 vertices . assuming that subunits can only form clusters with designed bonds , the dimensionless grand potential , @xmath22 , can be written in terms of a sum over all sets of fragments , @xmath23 the average fugacity of the fragments in each set , @xmath24 , depends on the topologies of the fragment graphs as well as the geometry of the subunits and the solution conditions . ignoring excluded volume interactions , we can approximate @xmath25 as @xmath26 where @xmath27 is the rotational entropy of a monomer , @xmath28 is the dihedral entropy of an unconstrained dimer and @xmath29 is the dimensionless concentration . the mean dimensionless dihedral entropy of a fragment is @xmath30 , where @xmath31 is the number of bridges@xcite in the fragment @xmath32 . the exponentially weighted mean bond energy within each set , @xmath33 , is @xmath34 where @xmath35 is the hybridization free energy of bond @xmath36 , @xmath37 is the inverse temperature and @xmath38 is the edge set of fragment @xmath32 . the inner average runs over all fragments in the set @xmath20 with _ quenched _ random bond energies @xmath39 . in the case of dna - brick structures , the outer average samples dna sequences so that each complete set of bond energies for the target structure is chosen independently from the same distribution of hybridization free energies . the free energy of a correctly bonded cluster of @xmath2 monomers is thus @xmath40 since we do not distinguish among equally sized clusters with varying compositions . this definition is appropriate for studying nucleation , as any subset of monomers has the potential to serve as a nucleation site . the equilibrium yield , @xmath41 , is defined as the fraction of all clusters in solution that are correctly formed , @xmath42 where @xmath43 is the grand - canonical average number of copies of fragment @xmath32 in solution and @xmath44 is the fugacity of the target structure . in the case of structures with higher coordination numbers , a few edges may be removed from the connectivity graph without allowing any subunit to disassociate or rotate . for these structures , we replace @xmath44 in eq . ( [ eq : equilibrium_yield ] ) with a sum over the fugacities of all fragments that enforce the correct geometry of the target structure . classical nucleation theory predicts a free - energy barrier for the nucleation of a stable , ordered structure from an unstable , fluid phase . the height of the barrier and the size of the critical nucleus vary with the degree of supersaturation of the fluid phase.@xcite assuming spherical nuclei , the classical prediction for the free - energy difference between a nucleus of the ordered phase containing @xmath2 monomers and the bulk fluid phase is @xmath45 where @xmath46 is the bulk free - energy difference per particle between the fluid phase and the ordered phase , @xmath47 is the free - energy cost per unit area of forming an interface between the two phases and @xmath48 is the number density of the ordered phase . in figure [ fig : coordination_number]b , the black line shows the fit to eq . ( [ eq : cnt ] ) with @xmath49 . in the main text , we report all results of the theoretical calculations assuming a dimensionless concentration of @xmath50 for comparison with the lattice monte carlo simulations . changing this concentration shifts both the equilibrium yield and the nucleation barrier linearly with @xmath51 . the concentration dependence of the nucleation barrier of the example 86-strand structure at several temperatures is shown in figure [ fig : concentration_dependence ] . although we have assumed equal concentrations of all monomers , polydispersity in the monomer concentrations can be easily incorporated into the theoretical treatment in much the same way as the distributions of designed interaction energies . , on the dimensionless concentration , @xmath29 , for the example 86-strand structure shown in figure [ fig : ramp_example]a.,width=321 ] we find that the shapes of the free - energy profiles of two - dimensional structures are also strongly affected by their coordination numbers . in figure [ fig : two_dimensional_structures ] , we show the free - energy profiles at 50% yield of two similarly sized two - dimensional structures with coordination numbers @xmath52 and @xmath13 . the behavior of the two - dimensional structure with @xmath52 is similar to that of the four - coordinated three - dimensional structures examined in the main text . the two - dimensional structure with @xmath13 exhibits the same face - by - face assembly as three - dimensional structures with octahedral coordination ; in the two - dimensional case , however , coexistence at 50% yield occurs between the target structure and fragments with the same number of monomers but fewer bonds . two - dimensional dna - tile structures with @xmath13 have been successfully assembled.@xcite in general , the nucleation barriers in two - dimensional structures are much lower than in three - dimensional structures with a similar number of monomers . consequently , a lower supersaturation is required in order for the target structure to become kinetically accessible , and kinetic trapping is therefore less likely to interfere with accurate assembly . nevertheless , these calculations suggest that lower coordinated two - dimensional structures , such as the hexagonal lattice pictured in figure [ fig : two_dimensional_structures]a , might assemble more robustly in experiments . dna hybridization free energies are strongly temperature - dependent.@xcite in figure [ fig : hybridization_free_energies]a , we compare the two hybridization free - energy distributions used in the main text . the mean and the variance of the hybridization free energies of 8-nucleotide sequences are shown for both the case of randomly chosen sequences and the case of sequences selected to yield monodisperse bond energies.@xcite designed interactions occur between complementary sequences , while incidental interactions are calculated based on the most attractive overlapping regions of two non - complementary sequences . in the lattice monte carlo simulations , all monomers on adjacent lattice sites experience a weak 100 k repulsion at all temperatures . in calculations involving designed interactions , this repulsion is subtracted from the mean interaction strength . when estimating incidental interactions , we ignore associations between non - complementary sequences that have a maximum attractive interaction of less than 100 k. the means and variances reported in figure [ fig : hybridization_free_energies]a are thus calculated based on the fraction of pairs of non - complementary sequences that attract more strongly than 100 k ; this fraction is shown in figure [ fig : hybridization_free_energies]b . these incidental interaction distributions are clearly approximate and are defined in order to match the lattice monte carlo simulations . nevertheless , the choices made in defining these distributions have a negligible effect on the calculated values of @xmath53@xcite and are irrelevant to the prediction of nucleation barriers and equilibrium yields . in order to apply this theoretical method to an experimental system , the designed interaction distributions should be recalculated in accordance with the experimental solution conditions .

the field of complex self - assembly is moving toward the design of multi - particle structures consisting of thousands of distinct building blocks . to exploit the potential benefits of structures with such ` addressable complexity , ' we need to understand the factors that optimize the yield and the kinetics of self - assembly . here we use a simple theoretical method to explain the key features responsible for the unexpected success of dna - brick experiments , which are currently the only demonstration of reliable self - assembly with such a large number of components . simulations confirm that our theory accurately predicts the narrow temperature window in which error - free assembly can occur . even more strikingly , our theory predicts that correct assembly of the complete structure may require a time - dependent experimental protocol . furthermore , we predict that low coordination numbers result in non - classical nucleation behavior , which we find to be essential for achieving optimal nucleation kinetics under mild growth conditions . we also show that , rather surprisingly , the use of heterogeneous bond energies improves the nucleation kinetics and in fact appears to be necessary for assembling certain intricate three - dimensional structures . this observation makes it possible to sculpt nucleation pathways by tuning the distribution of interaction strengths . these insights not only suggest how to improve the design of structures based on dna bricks , but also point the way toward the creation of a much wider class of chemical or colloidal structures with addressable complexity . recent experiments with short pieces of single - stranded dna@xcite have shown that it is possible to assemble well - defined molecular superstructures from a single solution with more than merely a handful of distinct building blocks . these experiments use complementary dna sequences to encode an addressable structure@xcite in which each distinct single - stranded ` brick ' belongs in a specific location within the target assembly . a remarkable feature of these experiments is that even without careful control of the subunit stoichiometry or optimization of the dna sequences , a large number of two- and three - dimensional designed structures with thousands of subunits assemble reliably.@xcite the success of this approach is astounding given the many ways in which the assembly of an addressable structure could potentially go wrong.@xcite any attempt to optimize the assembly yield or to create even more complex structures should be based on a better understanding of the mechanism by which dna bricks manage to self - assemble robustly . the existence of a sizable nucleation barrier , as originally proposed in refs . and , would remedy two possible sources of error that were previously thought to limit the successful assembly of multicomponent nanostructures : the depletion of free monomers and the uncontrolled aggregation of partially formed structures . slowing the rate of nucleation would suppress competition among multiple nucleation sites for available monomers and give the complete structure a chance to assemble before encountering other partial structures . recent simulations of a simplified model of a three - dimensional addressable structure have provided evidence of a free - energy barrier for nucleation,@xcite suggesting that the ability to control this barrier should enable the assembly of a wide range of complex nanostructures . we therefore need to be able to predict how such a barrier depends on the design of the target structure and on the choice of dna sequences . until now , however , there have been no reliable techniques to predict the existence , let alone the magnitude , of a nucleation barrier for self - assembly in a mixture of complementary dna bricks . here we show that the assembly of three - dimensional dna - brick nanostructures is indeed a nucleated process , but only in a narrow range of temperatures . the nucleation barrier in these systems is determined entirely by the topology of the designed interactions that stabilize the target structure . controllable nucleation is therefore a general feature of addressable structures that can be tuned through the rational choice of designed interactions . we find that the reliable self - assembly of three - dimensional dna bricks is a direct consequence of their unusual nucleation behavior , which is not accounted for by existing theories that work for classical examples of self - assembly , such as crystal nucleation . we are thus able to provide a rational basis for the rather unconventional protocol used in the recent dna - brick experiments by showing that they exploit a narrow window of opportunity where robust multicomponent self - assembly can take place .

in this appendix , we explain how the line width was extracted from the numerical data . we begin by determining the spectral function , defined by @xmath115 this consists of a set of delta functions . we then define the integrated spectral function @xmath116 . this consists of a set of step functions ( see fig . [ steps](a ) ) . for each step , we identify the energy values corresponding to @xmath117 of the step , @xmath118 of the step , and @xmath119 of the step . the energy spacing between the @xmath117 and @xmath119 points is taken to be the linewidth of this spectral line . we track how this line width scales with @xmath0 . we note that there is in general a wide distribution of line widths for any @xmath0 ( fig . [ steps](b ) ) . as a result , the mean and the median linewidth scale very differently ( see fig.5 of the main text ) . an understanding of the difference between the scaling of the mean and typical line width is an important challenge for future work . ( a ) the procedure for determining the linewidth . the blue curve is an integrated spectral function . the green squares divide each step into half , the red diamonds mark @xmath117 and the light blue circles mark @xmath120 of each step . ( b ) probability distribution of the linewidth @xmath109 for different values of coupling to the bath @xmath0 for a system with @xmath69 and @xmath121 averaged over 10 disorder configurations . lines are a guide to the eye . ]

we use exact diagonalization to study the breakdown of many - body localization in a strongly disordered and interacting system coupled to a thermalizing environment . we show that the many - body level statistics cross over from poisson to goe , and the localized eigenstates thermalize , with the crossover coupling decreasing with the size of the bath in a manner consistent with the hypothesis that an infinitesimally small coupling to a thermodynamic bath should destroy localization of the eigenstates . however , signatures of incomplete localization survive in spectral functions of local operators even when the coupling to the environment is sufficient to thermalize the eigenstates . these include a discrete spectrum and a gap at zero frequency . both features are washed out by line broadening as one increases the coupling to the bath . we also determine how the line broadening scales with coupling to the bath . isolated quantum systems with quenched disorder can enter a ` localized ' regime where they fail to ever reach thermodynamic equilibrium @xcite . while we have an essentially complete understanding of localization in non - interacting systems @xcite , the theory of many - body localization ( mbl ) is still under construction @xcite . numerical investigations using exact diagonalization @xcite _ do _ indicate that all eigenstates of a strongly interacting disordered system can be localized . most of the theoretical research so far has been in the limit of a perfectly isolated system . however , experimental tests of mbl ( @xcite ) will always include some finite coupling to the environment . what then can we expect to see in experiments designed to probe many body localization ? a recent theory of mbl systems weakly coupled to heat baths proposed that while eigenstates are delocalized by an infinitesimally weak coupling to a heat bath , signatures of localization persist in spectral functions of local operators for weak coupling to a bath @xcite . this theory has yet to face stringent numerical tests . moreover , it did not discuss the spectral functions of the physical degrees of freedom , the quantities of direct relevance for experiments , focusing instead on the spectral functions of certain localized integrals of motion that are believed to exist @xcite , but which are related to the physical degrees of freedom by an unknown unitary transformation . this work directly addresses these issues . we use exact numerical diagonalization to establish the behavior of many body localized systems weakly coupled to heat baths . we show that coupling @xmath0 to a bath results in a crossover from poisson to gaussian orthogonal ensemble ( goe ) eigenvalue statistics , which becomes exponentially steeper with increasing bath size . a similar rapid crossover to thermalization is seen in the eigenstates . however , the prospect for seeing mbl in experiments is still realistic because signatures of incomplete localization remain in the spectral functions of local ( in real space ) operators . indeed , we find that the spectral functions of the microscopic degrees of freedom look completely different in the localized and thermal phases ( see fig . 1 ) . the thermal phase has a continuous spectrum whereas the local spectral function in the localized phase is discrete , with a hierarchy of gaps , and a gap at zero frequency that survives even after spatial averaging . increasing @xmath0 causes lines to broaden and fill in these gaps . however , as long as the typical line broadening is less than the largest gaps , gap - like features remain . our work also reveals how the line broadening scales with @xmath0 . _ the model _ : we choose the antiferromagnetic heisenberg spin-@xmath1 chain with random fields along @xmath2 : @xmath3 we set the interaction @xmath4 . the on - site fields @xmath5 are independent random variables , uniformly distributed between @xmath6 and @xmath7 ; @xmath7 measures the disorder strength in the system . this model with periodic boundary conditions has been shown to have a many - body localization transition at @xmath8 in the infinite temperature limit @xcite . the hamiltonian in eq . [ eq : h_pbit ] is written in terms of the physical degrees of freedom @xmath9 ( ` @xmath10-bits , ' in the language of @xcite , where @xmath10=physical ) . in general , its eigenstates are quite complicated and non - trivial . as shown @xcite , one can perform a unitary transformation to rewrite @xmath11 in terms of localized constants of motion @xmath12 . the @xmath13 are dressed versions of the @xmath9 operators , which are localized in real space , with exponential tails , and are referred to in @xcite as ` @xmath14-bits ' ( @xmath14=localized ) . a unitary transformation to this ` @xmath14-bit ' basis can always be performed , if the system is in the regime where all the many body eigenstates are localized . in this @xmath14-bit basis , the hamiltonian becomes @xmath15 the values of the coefficients @xmath16 and @xmath17 will depend upon the parent hamiltonian ( 1 ) , although these coefficients all fall off exponentially with distance . the eigenstates of ( 2 ) are just products of @xmath18 . motivated by the representation ( 2 ) of the hamiltonian ( 1 ) , it is instructive to consider the simpler hamiltonian @xmath19 where the @xmath20 and @xmath21 as independent random variables taken from a log - normal distribution with @xmath22 and @xmath23 , and similarly for @xmath24 . we take @xmath25 and work with open - boundary conditions . this hamiltonian also has the feature that eigenstates are product states of @xmath26 , and is simpler to work with numerically . for the bath , we use a non - integrable hamiltonian that has been recently studied @xcite . it consists of @xmath27 interacting spins with the hamiltonian : @xmath28 while using open boundary conditions , we add a boundary term @xmath29 to @xmath30 . we use @xmath31 , @xmath32 and @xmath33 , values for which @xmath30 has been numerically shown by @xcite to have fast entanglement spreading . ( we use periodic boundary conditions only for @xmath10-bits with @xmath34 . ) the interaction between the system and bath should be local for both @xmath10- and @xmath14-bits . we first study @xmath14-bit eigenstates , choosing the coupling : @xmath35 later we examine @xmath10-bit spectra , using the coupling @xmath36 the total hamiltonian is thus @xmath37 , where @xmath11 and @xmath38 are given by eq . ( 3 ) and ( 5 ) in the first part of this work , and by eq . ( 1 ) and ( 6 ) in the latter part of this work . we will indicate the transition clearly in the text . we use open boundary conditions except where periodic boundaries are explicitly mentioned . we start by analyzing the breakdown of localization when the @xmath14-bit hamiltonian ( 3 ) is coupled to a bath according to ( 5 ) , by examining the many - body eigenvalue statistics as @xmath0 is increased from @xmath39 . we perform exact diagonalization on a system with @xmath40 spins coupled to @xmath41 spins in the bath . the many body level - spacing is @xmath42 , where @xmath43 is the energy of the @xmath44th eigenstate . following @xcite , we define the ratio of adjacent gaps as @xmath45 . we average this over eigenstates and several different realizations of the disorder to get a probability distribution @xmath46 at a particular value of @xmath0 . in fig . [ fig : level_space ] , we show how @xmath46 evolves from poisson to goe like as @xmath0 is increased . in a localized system we expect that @xmath47 , and for a thermalizing system , we expect that @xmath48 . -bit hamiltonian as @xmath0 is increased . results are for a system with @xmath40 spins and bath with @xmath41 spins averaged over @xmath49 eigenstates obtained from several disorder configurations . the dark blue solid line is the poisson distribution expected for localized systems , and the light blue dashed line is the goe distribution expected for thermalizing systems . ] the transition from poisson to goe statistics happens gradually for this finite size system . a simple analytical estimate of the characteristic value of @xmath0 at the crossover point proceeds as follows ( see also @xcite ) : if @xmath50 is the bandwidth of the bath and @xmath51 is the many body level spacing in the bath , then the system couples to @xmath52 states , with a typical matrix element to each state of order @xmath53 . the coupling to the bath will be effective in thermalizing the system when this matrix element becomes of order the level spacing in the bath , i.e. when @xmath54 . this indicates that the crossover coupling @xmath55 . since @xmath56 , the critical value of @xmath0 is expected to scale as @xmath57 . to quantitatively compare this crossover estimate to the data , we define @xmath58 . after averaging over disorder distributions , @xmath59 should be @xmath60 in the goe regime and @xmath61 in the localized regime @xcite . it is convenient to define the normalized quantity @xmath62=(@xmath63 , such that @xmath64 if the level statistics are goe and @xmath65 if they are poisson . fig . 3(a ) shows how @xmath66 varies with @xmath0 for systems of size @xmath67 . fig . [ fig : r_trans](b ) shows that scaling of the form @xmath68 is successful in making the data for different @xmath27 in fig . [ fig : r_trans](a ) collapse onto one curve . data collapse occurs also for @xmath69 and @xmath70 , indicating clearly that it is @xmath27 which controls the finite size scaling . we get the best collapse when the constant in the exponential is @xmath71 which is in good agreement with the analytical estimate @xmath72 . this implies that the crossover to thermalization is at a coupling @xmath73 that is exponentially small in system size , so that level statistics become goe at infinitesimal @xmath0 in the thermodynamic limit . ( defined in the text ) in the @xmath14-bit hamiltonian as @xmath0 is increased for system sizes @xmath74 and @xmath75 . data is averaged over @xmath49 eigenstates obtained from several disorder configurations . ( b ) collapse of data in ( a ) is in good agreement with analytic arguments for the finite size scaling presented in the main text , and depends only on @xmath27 . ] another test of thermalization is checking whether the eigenstates obey the eigenstate thermalization hypothesis ( eth ) @xcite . the eth states that the expectation value of a local operator should be the same in every eigenstate within a small energy window . for a localized system this will not be the case . in fig . [ fig : eth ] , we show how eigenstate thermalization sets in as @xmath0 is increased . we choose an energy window around the center of the band and calculate the standard deviation of the expectation value of @xmath76 for all eigenstates within the window . explicitly , we define @xmath77,\ ] ] where the overline denotes averaging over an energy window of width @xmath78 in the middle of the band and @xmath79 is an eigenstate of the coupled system and bath . we choose @xmath80 . after averaging over disorder distributions , we expect to find @xmath81 for a thermalized system . fig . [ fig : eth](a ) shows how @xmath82 approaches 0 as @xmath0 is increased for different system sizes . fig . [ fig : eth](b ) shows that @xmath82 scales with @xmath0 similar to @xmath66 . the exponent here is @xmath83 , also close to the estimated analytical value . -bit hamiltonian as @xmath0 is increased for system sizes @xmath69 , @xmath84 , @xmath85 , @xmath86 , @xmath87 and @xmath75 . @xmath82 as defined in the text is measured at the site of the central spin . data is averaged over @xmath49 eigenstates obtained from several disorder configurations . ( b ) collapse of data in ( a ) agrees with analytical estimates of finite size scaling for @xmath88 . for a finite size system with @xmath27 spins in the bath , the eigenstates become effectively thermal for @xmath89 , implying that eigenstates in the thermodynamic limit become thermal for infinitesimal @xmath0 . ] we now turn to an analysis of the spectral functions of local operators . henceforth we are working with the physical degrees of freedom , eq . ( 1 ) and ( 6 ) . we examine the spectral function from an exact eigenstate @xmath90 where @xmath91 is the @xmath92 eigenstate of the combined system and bath . we note that since we are working with a finite size system with a discrete spectrum , the spectral function will always consist of a set of delta functions . at @xmath93 , the delta functions should have minimum spacing @xmath94 , equal to the many body level spacing in the system . at non - zero @xmath0 , each ` parent ' delta function will split into exponentially many descendants , with a typical spacing @xmath95 . a fine binning in energy with bin size greater than @xmath95 will then yield a smooth spectral function , with the ` parent ' delta functions of the system having been ` broadened ' by coupling to the bath . to investigate this broadening , it is convenient to take @xmath96 . we therefore take @xmath97 and @xmath98 , and investigate how the ` line broadening ' evolves with @xmath0 for @xmath99 . details of the procedure are outlined in the supplementary material , and the results are illustrated in fig . [ fig : linewidth ] for @xmath100 . the mean and median linewidth at a particular value of @xmath0 are significantly different . this is a result of the long tails in the distribution of the linewidth ( see supplement ) . fig . [ fig : linewidth ] shows that at the larger values of @xmath0 we study , a log - log plot of the median vs @xmath0 appears to fit well to a straight ( dashed ) line . for the system sizes that we are able to access , the straight line fit suggests @xmath101 , where @xmath102 increases as the size of the bath increases , reaching @xmath103 for @xmath104 . we note that while a simple application of the golden rule predicts @xmath105 , a more careful analysis @xcite suggests that the true scaling should be @xmath106 . the solid lines in fig . [ fig : linewidth ] are a fit to this theoretical prediction , and are consistent with the data , except at smallest @xmath0 . the discrepancy at smallest @xmath0 and the difference between median and mean are worthwhile topics for future work . for a system of @xmath10-bits with @xmath69 and @xmath107 averaged over more than 38000 eigenstates obtained from several disorder configurations at @xmath100 . @xmath108 for the sizes shown here . the mean and the median of the probability distribution of the linewidth @xmath109 are extracted from the data as discussed in the appendix . the dotted lines are linear fits to the data . the solid lines are fits to the theoretical prediction . [ fig : linewidth ] ] finally , we analyze the behavior of the spectral function averaged over all sites and eigenstates of the system , for @xmath110 . we note that the hamiltonian ( 1 ) has a delocalization - localization phase transition at @xmath8 . fig . [ fig : pbits](a ) shows @xmath111 on the delocalized side of the transition for a small value of @xmath0 . @xmath111 is smooth everywhere . ( the graininess is a result of the small system size . ) fig . [ fig : pbits](b ) is on the localized side of the transition , with the system almost decoupled from the bath . here , @xmath111 consists of clusters of narrow spectral lines , with a hierarchy of energy gaps , just as was shown to be the case for @xmath14-bit spectral functions in @xcite . @xmath111 vanishes at @xmath112 . thus , local spectral functions can distinguish between extended and localized phases . in fig . [ fig : pbits](c - e ) we examine how the @xmath10-bit spectral functions evolve as @xmath0 increases . we see that the line broadening increases and different lines start to overlap with each other , washing out the weaker spectral features , but larger gaps remain . the zero - frequency gap also fills in with increasing @xmath0 . the spectral functions retain signatures of localization even for @xmath113 when the eigenstates of the combined system and bath are effectively thermal , and get washed out when @xmath0 becomes comparable to the characteristic energy scales in the system ( i.e. @xmath114 ) . in conclusion , we have investigated the signatures of localization in a disordered system weakly coupled to a heat bath using exact diagonalization . the wave functions are found to exhibit a crossover to thermalization as a function of coupling to the bath . the crossover coupling is proportional to the many body level spacing in the bath , and vanishes exponentially fast in the limit of a large bath size . in contrast , the spectral functions of local operators are found to show more robust signatures of proximity to a localized phase . while the spectral functions are smooth and continuous in the delocalized phase ( after coarse graining on the scale of the many body level spacing ) , the spectral functions in the localized phase consist of narrow spectral lines , and contain a hierarchy of gaps , as well as a gap at zero frequency that persists even after spatial averaging . increasing the coupling to the bath increases the line broadening ( in a manner that we calculate ) and washes out these features . however , signatures of localization survive in the spectral functions even at couplings to the bath where the exact eigenstates are effectively thermal ( fig . 1 ) . _ acknowledgments : _ rn would like to thank sarang gopalakrishnan and david huse for a collaboration on related ideas . this work was supported by doe grant de - sc0002140 . rnb . acknowledges the hospitality of the institute for advanced study , princeton while this work was being done . rn was supported by a pcts fellowship . sj was supported by the porter ogden jacobus fellowship of princeton university . 99 p. w. anderson , phys . rev . * 109 * , 1492 ( 1958 ) . b. l. altshuler , y. gefen , a. kamenev and l. s. levitov , phys . rev . lett . * 78 * , 2803 ( 1997 ) . i. v. gornyi , a. d. mirlin and d. g. polyakov , phys . rev . lett . * 95 * , 206603 ( 2005 ) . d. m. basko , i. l. aleiner and b. l. altshuler , annals of physics * 321 * , 1126 ( 2006 ) . v. oganesyan and d. a. huse , phys . rev . b * 75 * , 155111 ( 2007 ) . m. znidaric , t. prosen and p. prelovsek , phys . rev . b * 77 * , 064426 ( 2008 ) a. pal and d. a. huse , phys . rev . b * 82 * , 174411 ( 2010 ) . j.z . imbrie , arxiv : 1403.7837 d. a. huse , r. nandkishore , v. oganesyan , a. pal and s. l. sondhi , phys . rev . b * 88 * , 014206 ( 2013 ) . b. bauer and c. nayak , j. stat . mech . p09005 ( 2013 ) . d. pekker , g. refael , e. altman , e. demler and v. oganesyan , phys . rev . x * 4 * , 011052 ( 2014 ) . r. vosk and e. altman , arxiv:1307.3256 . y. bahri , r. vosk , e. altman and a. vishwanath , arxiv:1307.4192 . r. nandkishore and a.c . potter , arxiv : 1406.0847 s. gopalakrishnan and r. nandkishore , arxiv : 1405.1036 r. vasseur , s.a . parameswaran and j.e . moore , arxiv : 1407.4476 b. bauer and c. nayak , arxiv : 1407.1840 d. a. huse and v. oganesyan , arxiv:1305.4915 ; d.a . huse , r. nandkishore and v. oganesyan , arxiv : 1408.4297 maksym serbyn , z. papic and dmitry a. abanin , phys . rev . lett . 110 , 260601 ( 2013 ) m. serbyn , z. papic and d. a. abanin , phys . rev . lett . * 111 * , 127201 ( 2013 ) . r. nandkishore and d. a. huse , arxiv : 1404.0686 and references contained therein e. altman and r. vosk , annual reviews of condensed matter physics ( to appear ) and references contained therein d. shahar , presentation at princeton workshop on many body localization ( 2014 ) ( unpublished ) b. de marco , presentation at princeton workshop on many body localization ( 2014 ) ( unpublished ) r. nandkishore , s. gopalakrishnan and d.a . huse , arxiv:1402.5971 . hyungwon kim and david a. huse , phys . rev . lett . * 111 * , 127205 j. m. deutsch , phys . rev . a * 43 * , 2046 ( 1991 ) . m. srednicki , phys . rev . e * 50 * , 888 ( 1994 ) . m. rigol , v. dunjko and m. olshanii , nature * 452 * , 854 ( 2008 ) .

simple fluids are made of atomic particles . these are identical , spherically symmetric particles which interact via a well defined interaction potential of which the lennard - jones formula provides a very good approximation . the classic states of matter are a consequence of this nature : interaction decays fast at long distance , hence we have a gas when the density is low ; the potential has an attractive well at short distances , and this causes the liquid when the density is high enough ; and finally , the interaction becomes strongly repulsive at very short distances and this makes the fluid freeze into a crystalline structure when it becomes very dense , due to entropic considerations ( see e.g. @xcite for further details ) . in contrast to the ` simple fluid ' paradigm provided by atomic fluids we have colloids . these are suspensions of big around one micron particles , which are actually aggregates of smaller particles , in a solvent which may also contain other elements ( like ions , polymers , etc . ) . because of this , particles are all different in shape , size , charge , etc . , and the interactions are the result of adding up the separate contribution of each of the elements of the aggregate that we call a colloidal particle , as well as the entropic forces that the solvent and its constituents exert on them @xcite . because of this , colloidal science has become the laboratory of liquid theory : almost anything in a colloid can be tuned so as to fit experimentally theoretical models that would otherwise be considered highly academic ( like hard spheres , to name the most famous one ) . in particular , by varying the composition of the solvent one can modify entropic forces and gauge in this way the effective interactions between the colloidal particles @xcite . one of the most interesting aspects of a colloid is its inherent _ polydispersity _ , i.e.the fact that colloidal particles have different shape , charge , size , etc . the impact of this on the phase behaviour of the colloidal suspension is still a matter of active research . the study of polydispersity is not new : onsager himself paid attention to it in his famous 1949 article on the isotropic - nematic transition of infinitely thin hard cylinders @xcite . however , it is only in the last decade that the issue has regained the attention of liquid theorists , probably due to the fundamental problems of formulating a statistical mechanics of polydisperse systems @xcite . most theoretical studies of polydisperse systems have focused on demixing and transitions between homogeneous phases @xcite . the reason is two - fold : on the one hand , specific techniques have been developed for those very common cases in which the free energy depends on the polydispersity distribution through a finite set of its moments @xcite , or when polydispersity is small @xcite ; on the other hand , experimental data are available for colloidal liquid crystals and their transitions between the isotropic , nematic and non - uniform phases @xcite . however , when it comes to finding theoretical approaches to spatial ordering transitions , results are more scarce due to the inherent difficulty of discerning how the ordering occurs in the continuum of species that form a polydisperse mixture . in spite of this , several of these transitions have been tackled with different techniques . interfaces and wetting have been successfully addressed with density functional theory @xcite , and so has been freezing of polydisperse hard spheres , despite its higher difficulty @xcite . liquid - crystalline spatially ordered phases ( smectic and columnar ) in polydisperse colloidal mixtures have received considerably less attention from the theoretical point of view . monodisperse fluids of hard rods are known to have a nematic - smectic transition @xcite . this is a continuous transition for parallel rods @xcite which becomes first - order when rods are allowed to freely rotate @xcite . in spite of some initial results that seemed to show a window of stability of the columnar phase @xcite , it turned out that it was a finite - size effect , so that the smectic is more stable than the columnar for any aspect ratio of the rods @xcite . it is also known that the addition of a second species of rods can destabilise the smectic phase in favour of a columnar phase @xcite . the same effect has been shown to occur in grand - canonical simulations of freely rotating , length - polydisperse , infinitely long rods @xcite when polydispersity is larger than @xmath0 . there is also recent experimental confirmation of the enhancement of the stability of columnar ordering by polydispersity @xcite . the terminal polydispersity for the smectic phase had been predicted from a density - functional theory @xcite for a system of parallel hard cylinders . despite the orientational constraint , infinitely long rods are expected to be strongly aligned and thus behave very much as perfectly aligned rods although not quite because the order of the nematic - smectic transition changes to first order for freely rotating rods , no matter their infinite length . the density functional used in @xcite was a very simple version of a weighted - density approximation ( see e.g. @xcite for a recent review ) , in which the weighting function is just proportional to the mayer function . simple as it might be , at the time there was no alternative density functional theory for carrying out this kind of analysis . but very recently a new functional for mixtures of parallel hard cylinders based on rosenfeld s fundamental - measure theory @xcite has been proposed @xcite . the functional has been shown to provide an excellent estimate of the phase diagram of the monodisperse system @xcite . it is thus our aim in this paper to perform a bifurcation analysis of the smectic and columnar instabilities in this more accurate functional . suppose that we have an inhomogeneous polydisperse mixture characterised by density profiles @xmath1 , where @xmath2 is a parameter ( or set of parameters ) which characterises the species ( the length in our case ) . then there is a helmholtz free - energy functional @xmath3 $ ] which can be split into an ideal bit , @xmath4=\int dl\int d{\bf r}\,\rho({\bf r},l)\big\ { \ln[\mathcal{v}(l)\rho({\bf r},l)]-1\big\},\ ] ] plus an excess @xmath5=f[\rho]-f_{\rm id}[\rho]$ ] . here @xmath6 , with @xmath7 the temperature and @xmath8 the boltzmann constant , and @xmath9 stands for the thermal volume of species @xmath2 . the equilibrium density for a given chemical potential @xmath10 is obtained from the euler - lagrange equation @xmath11}{\delta\rho({\bf r},l)}.\end{aligned}\ ] ] if we specialise this equation for the uniform phase , of density profile @xmath12 , corresponding to the same chemical potential @xmath10 , then @xmath13 here @xmath14 stands for the normalised probability density of particles of species @xmath2 . from these two equation we obtain @xmath15 which will be the starting point of the bifurcation analysis . let us assume that we have a length - polydisperse mixture of aligned hard cylinders in a nematic phase . a convenient choice for @xmath2 is @xmath16 , where @xmath17 is the length of the cylinders and @xmath18 its average over the whole mixture . let @xmath12 be the density distribution of lengths in the nematic phase . suppose that we reach a value of @xmath19 at which the nematic fluid is no longer stable . then the inhomogeneous profile that emerges at the onset of the instability can be expressed as @xmath20 , where @xmath21 is a small perturbation . using this expression in eq . ( [ uno ] ) we obtain that , near the bifurcation point , @xmath22 where @xmath23 is the direct correlation function of the nematic phase . in fourier space , @xmath24 where as usual @xmath25 . in order to proceed we need to specify @xmath26 or , equivalently , @xmath27 . we will analyse three choices : ( i ) @xmath27 taken from the fundamental - measure density functional of ref . @xcite , ( ii ) a second virial approximation and ( iii ) a third virial approximation . the expressions for the direct correlation function for this case appears as eq . ( 39 ) in ref . @xcite . for the case of a continuous polydisperse mixture , this is given , in fourier space , by @xmath28 where @xmath29 and @xmath30 are , respectively , the lengths of the perpendicular and parallel components of the wave vector in units of radius @xmath31 and mean cylinder length @xmath18 , @xmath32 and @xmath33 , while @xmath34,\label{psi01}\\ \psi_{11}(q)&=&4y^2\left[\frac{j_1(2q)}{q}+2(1 + 2y)j_0(q)\frac{j_1(q)}{q } \right.\nonumber\\&+&\left . ( 1 + 6y+6y^2)\left(\frac{j_1(q)}{q}\right)^2\right].\label{psi11}\end{aligned}\ ] ] here @xmath35 , where @xmath36 is the total packing fraction , and @xmath37 is the @xmath38-th order bessel functions of the first kind . the functional proposed in ref . @xcite is based on tarazona & rosenfeld s functional for the fluid of hard disks @xcite . we can use rosenfeld s proposal for such a fluid instead @xcite . then the direct correlation function will still have the form ( [ cuatro ] ) , but the functions @xmath39 will be defined as @xcite @xmath40 and @xmath41,\label{yasha1}\\ & & \psi_{11}^{(\rm{r})}(q)=2y^2\left[j_0 ^ 2(q)-j_1 ^ 2(q)+ 2(3 + 4y)j_0(q)\frac{j_1(q)}{q } \right.\nonumber\\ & & \left . + 2(1 + 6y+6y^2)\left(\frac{j_1(q)}{q}\right)^2\right].\label{yasha2}\end{aligned}\ ] ] if we now substitute ( [ cuatro ] ) into ( [ cinco ] ) , multiply the result by @xmath42 and integrate over @xmath2 , we obtain @xmath43 where we have introduced the new functions @xmath44 eq . ( [ seis ] ) can be rewritten in matrix form as @xmath45\boldsymbol{\xi}({\bf q})={\bf 0 } , \label{siete}\end{aligned}\ ] ] where @xmath46 is the @xmath47 identity matrix , @xmath48 and @xmath49 are the matrices with elements @xmath50 and @xmath51 respectively and @xmath52 is the vector with coordinates @xmath53 . denoting @xmath54 $ ] , the first non - trivial solution of ( [ siete ] ) for which @xmath55 follows from the couple of equations @xmath56 the first equation yields the value(s ) of @xmath57 for which @xmath58 , while the second one imposes that @xmath59 has a minimum at this value of @xmath57 . these two equations determine the values of @xmath60 and @xmath57 at the bifurcation point . from ( [ siete ] ) , using ( [ psi01 ] ) , ( [ psi11 ] ) and ( [ ocho ] ) , we obtain @xmath61\psi_{01}^2(q_{\perp}).\nonumber\\ \label{det}\end{aligned}\ ] ] finally , the nematic - nematic demixing spinodal can be obtained as the value of @xmath60 at which @xmath62 . let us now consider three possible scenarios for a bifurcation : ( i ) nematic - nematic ( n - n ) demixing , ( ii ) nematic - smectic ( n - sm ) bifurcation and ( iii ) nematic - columnar ( n - c ) bifurcation . both versions , ( [ psi01 ] ) and ( [ psi11 ] ) , and ( [ yasha1 ] ) and ( [ yasha2 ] ) , yield the same value for the functions @xmath63 also , from eqs . ( [ ocho ] ) and ( [ omegas ] ) we have @xmath64 where @xmath65 is the standard deviation which characterises the degree of polydispersity of the system ( remember that from our choice for @xmath2 we have @xmath66 ) . we have introduced the short hand notation @xmath67 for the mean value of a general function @xmath68 with respect to the distribution function @xmath14 . thus we find @xmath69 \delta^2.\nonumber\\\end{aligned}\ ] ] the equation @xmath70 leads to an analytic formula for the n - n demixing spinodal , namely @xmath71 demixing appears for any @xmath72 where @xmath73 is the solution of ( [ delta ] ) for the maximum value of the polydispersity parameter @xmath74 ( for the sake of reference , a schultz distribution see below has @xmath75 ) . a smectic instability is to be found by setting @xmath76 and @xmath77 . then introducing ( [ omegas ] ) in ( [ ocho ] ) we obtain @xmath78 , \label{ocho - nueve } \\ n_{01}(q ) & = & n_{10}(q)=\frac{1}{q}\langle \sin(ql)\rangle,\label{nueve}\\ n_{11}(q)&=&\frac{2}{q^2}\left[1-\langle \cos(ql)\rangle\right ] . \label{diez}\end{aligned}\ ] ] thus we obtain from ( [ det ] ) , ( [ psies0 ] ) and ( [ ocho - nueve])([diez ] ) , @xmath79}{q^2}\nonumber\\ & + & y^2(4 + 5y+2y^2)^2\frac{\left[\langle\sin(ql)\rangle^2 + \langle\cos(ql)\rangle^2 - 1\right]}{q^2}.\nonumber\\\end{aligned}\ ] ] as a length - polydispersity model we make the standard choice of a schultz distribution function @xmath80 , \quad \nu\ge 0 , \label{schultz}\end{aligned}\ ] ] whose mean is set to unity , i.e. @xmath81 . for this choice @xmath82 . it is easy to show that for this particular distribution function we obtain @xmath83 } { \left[1+q^2\delta^4\right]^{1/(2\delta^2)}},\nonumber\\ \langle \cos(ql)\rangle&= & \frac{\cos\left[\delta^{-2}\arctan(q\delta^2)\right ] } { \left[1+q^2\delta^4\right]^{1/(2\delta^2)}}.\end{aligned}\ ] ] a columnar instability is to be found by setting @xmath84 and @xmath85 . then using ( [ nes ] ) in ( [ det ] ) we find @xmath86.\nonumber\\ \label{lala}\end{aligned}\ ] ] interestingly , ( [ lala ] ) implies that the bifurcation to the columnar phase is independent of the particular functional form of @xmath14 , it only depends on @xmath87 , as it happens for the n - n demixing . the third virial approximation of the direct correlation function for the system we are analysing has the expression @xmath88 , \nonumber\\\end{aligned}\ ] ] where @xmath89 with @xmath90 expressed in units of @xmath18 , is the overlap function ( minus the mayer function ) of two cylinders of the same radius @xmath31 and reduced lengths @xmath2 and @xmath91 . @xmath92 is the heaviside step function ( @xmath93 if @xmath94 and @xmath95 if @xmath96 ) . @xmath97 , the overlap volume between two cylinders of radius @xmath98 and lengths @xmath99 and @xmath100 , @xmath101 being the vector joining their centres of mass , is given by @xmath102 where @xmath103 and @xmath104 \theta\left((l+l')/2+l''-|z|\right).\end{aligned}\ ] ] the fourier transform of @xmath105 can be written in the same form ( [ cuatro ] ) , where now @xmath106 , \\ \psi_{11}(q)&=&\frac{32}{\pi}\eta^2u(2q),\label{psinew}\\ u(q)&=&16\int_0^{1/2}dxxj_0(2qx)\left(\arccos x - x\sqrt{1-x^2}\right ) , \nonumber\\ \label{uu}\end{aligned}\ ] ] and the functions @xmath107 are given by eq . ( [ omegas ] ) . thus , eq . ( [ det ] ) becomes @xmath108 we should point out that the second virial approximation can be obtained from ( [ psinew ] ) just replacing @xmath109 by zero ( thus @xmath110 ) . the n - sm ( @xmath76 ) and n - c ( @xmath85 ) bifurcations can be obtained from ( [ psinew])([b3 ] ) taking into account that @xmath111 ) in eq . ( [ uu ] ) can be calculated analytically as @xmath112 . the uniform limit of ( [ b3 ] ) yields @xmath113\delta^2 , \label{b3a}\end{aligned}\ ] ] where @xmath114 . thus the spinodal of the n - n demixing is @xmath115 the second virial approximation is obtained by setting @xmath116 in ( [ b3a ] ) , which transforms the spinodal into @xmath117 the spinodals obtained from the second virial approximation of the direct correlation function are plotted in fig . [ fig1 ] . as already mentioned , both the n - n demixing spinodal and the n - c spinodal are independent of the details of @xmath14 , so they are valid for any polydisperse mixture . on the contrary , the n - sm spinodal does depend on @xmath14 . the curve of fig . [ fig1 ] has been obtained using the schultz distribution ( [ schultz ] ) , but in order to check what the effect of this choice is on this line we have also plotted the n - sm spinodal for the distribution function @xmath118^{\nu+1}}{\gamma[(\nu+1)/2]^{\nu+2 } } l^{\nu}\exp\left\{-\left(\frac{\gamma[(\nu+2)/2 ] } { \gamma[(\nu+1)/2]}l\right)^2\right\},\nonumber\\ \label{gauss}\end{aligned}\ ] ] which decays as a gaussian and not as an exponential for long rods . for this choice @xmath119\gamma[(\nu+3)/2 ] } { \gamma[(\nu+2)/2]^2}-1}\end{aligned}\ ] ] ( and therefore @xmath120 ) . the comparison of the n - sm spinodal obtained with this distribution function with that obtained with the schultz one reveals a weak dependence on @xmath14 . for this reason we have stuck to the schultz for the rest of the paper . the plot shows a crossover polydispersity , @xmath121 , below which the nematic bifurcates into a smectic and above which it does so into a columnar ( from the @xmath14 given by ( [ gauss ] ) we obtain @xmath122 instead ) . on the other hand , the n - n spinodal line reveals that n - n demixing can occur for very polydisperse mixtures ( with @xmath123 ) . this is a defect of this approximation , as n - n demixing has never been observed in polydisperse systems of hard rods with a unimodal length distribution . and not the only one , since an even more obvious drawback is the unphysical , high values of the packing fraction @xmath60 at which the spinodals appear . in striking contrast , the results provided by the fundamental - measure density functional proposed in ref . @xcite ( c.f.eqs . ( [ psi01 ] ) , ( [ psi11 ] ) and ( [ det0 ] ) , ( [ det ] ) ) , depicted in fig . [ fig2](a ) , show a very different scenario . we also find a crossover polydispersity , at a slightly higher value @xmath124 . however , the n - n demixing is always metastable , as is consistent with simulations and experiments , and the values of the packing fraction at which the bifurcations occur are not far from the transition lines found in simulations . the same figure also shows the n - c spinodal resulting from the fundamental - measure density functional based on rosenfeld s approximation for hard disks . it is most remarkable that for this functional no crossover is found . thus this result either leads to the wrong conclusion that the smectic phase is more stable than the columnar phase for any polydisperse mixture , or it seems to suggest that the crossover polydispersity might be shifted to higher values . however , a definitive conclusion can only be achieved trough a coexistence calculation . finally , fig . [ fig2](b ) shows the results obtained from the third virial approximation of the direct correlation function ( c.f . ( [ det0 ] ) and ( [ b3 ] ) ) . we can see a dramatic improvement with respect to the second virial approximation in all details . n - n demixing becomes metastable and the values of the packing fraction at which the bifurcations occur are much more reasonable . in fact , the scenario this approximation shows is rather close to the one obtained from the fundamental - measure density functional , the differences being only quantitative . the phase behaviour of polydisperse mixtures of hard rods had received little theoretical attention mainly because no good density functional theory was available for such a system , not even for the simplest model of aligned hard rods . only very simple approximations , based on the parsons - lee rescaling , had been used . despite the merit of these studies in finding a terminal polydispersity at which the n - sm transition is preempted by a n - c one , this approximation is contingent on the accuracy of the second virial one which we have seen not to be reliable for large polydispersity . recently a functional based on rosenfeld s fundamental measure theory has been put forward for mixtures of parallel hard cylinders . in the present paper we have analysed its reliability in predicting the phase behaviour of polydisperse mixtures . with this functional we also find a terminal polydispersity for the n - sm transition and we confirm that n - n demixing can at best be metastable with respect to spatial ordering . we have also compared with the results obtained with second and third virial approximations . although we find the former to have serious defects like predicting n - n demixing at high polydispersity the latter yields very reasonable results , close to those obtained with the fundamental - measure functional . interestingly , a variant of the fundamental - measure functional constructed on rosenfeld s proposal for the system of hard disk is not even able to predict the terminal polydispersity of the n - sm transition . this calls for some caution in the use of rosenfeld s functional to study the hard disk fluid . as for the validity of a bifurcation analysis , it obviously provides the location of the phase transition if this is continuous , but it can be far from the coexistence line of the disordered phase for first order phase transitions . in the polydisperse system of hard rods , both the n - sm and the n - c transitions are first order @xcite . in the n - sm transition this seems to be caused by the presence of particles aligned parallel to the smectic layers @xcite . for this reason , in systems of perfectly aligned rods this transition becomes continuous , so the n - sm bifurcation line is the location of the transition predicted by the corresponding theories . the n - c is always found to exhibit a wide coexistence region both in simulations and in theory and therefore the terminal polydispersity found through a bifurcation analysis is but an upper bound of the true one . polydispersity widens this coexistence region hence worsening the estimate provided by this bound . locating this n - c coexistence is thus a necessary step to determine the n - sm to n - c crossover . in the present state of the art this is a non - trivial task because the parallel hard cylinders functional contains a two - particle kernel which hinders the inclusion of polydispersity in inhomogeneous phases . how to circumvent this problem is a matter of current research . this work has been supported by the ministerio de educacin y ciencia under project mosaico and by the comunidad autnoma de madrid under project mossnoho . hansen and i. r. mcdonald , _ theory of simple liquids _ ( academic press , london 2006 ) . n. pusey , _ colloidal suspensions _ , in _ liquids , freezing and glass transition _ , edited by j. p. hansen , d. levesque and j. zinn - justin ( north - holland , amsterdam 1989 ) , pp 763942 . c. n. likos , phys . rep . * 348 * , 267 ( 2001 ) . l. onsager , ann . n. y. acad . sci . * 51 * , 627 ( 1949 ) . p. b. warren , phys . lett . * 80 * , 1369 ( 1998 ) . z. y. chen . e * 50 * , 2849 ( 1994 ) . n. clarke , j. a. cuesta , r. sear , p. sollich and a. speranza , j. chem . phys . * 113 * , 5817 ( 2000 ) . a. speranza and p. sollich , phys . e * 67 * , 061702 ( 2003 ) . a. speranza and p. sollich , j. chem . phys . * 118 * , 5213 ( 2003 ) . h. h. wensink and g. j. vroege , j. chem . phys . * 119 * , 6868 ( 2003 ) . p. sollich , j. chem . phys . * 122 * , 214911 ( 2005 ) . y. martnez - ratn and j. a. cuesta , phys . lett . * 89 * , 185701 ( 2002 ) . y. martnez - ratn and j. a. cuesta , j. chem . phys . * 118 * , 10164 ( 2003 ) . j. a. cuesta , eur . * 46 * , 197 ( 1999 ) . p. b. warren , eur . . lett . * 46 * , 295 ( 1999 ) . r. p. sear , phys . lett . * 82 * , 4244 ( 1999 ) . p. sollich and m. e. cates , phys . lett . * 80 * , 1365 ( 1998 ) . p. sollich , p. b. warren and m. e. cates , in _ advances in chemical physics _ , edited by i. prigogine and s. a. rice , vol . 116 ( john wiley & sons , new york 2002 ) , pp 265336 . p. sollich , j. phys . : condens . matter * 14 * , r79 ( 2002 ) . c. rascn and m. e. cates , j. chem . * 118 * , 4312 ( 2003 ) . r. m. l. evans , phys . e * 59 * , 3192 ( 1999 ) . r. m. l. evans , j. chem . phys . * 114 * , 1915 ( 2001 ) . f. m. van der kooij , k. kassapidou and h. n. w. lekkerkerker , nature * 406 * , 868 ( 2000 ) . f. m. van der kooij and h. n. w. lekkerkerker , phys . lett . * 84 * , 781 ( 2000 ) . i. pagonabarraga , m. e. cates and g. j. ackland , phys . lett . * 84 * , 911 ( 2000 ) . o. pizio , a. patrykiejew and s. sokolowski , molec . phys . * 99 * , 57 ( 2001 ) . m. baus , l. bellier - castella and h. xu , j. phys . : condens . matter * 14 * , 9255 ( 2002 ) . n. b. wilding , phys . e * 71 * , 066126 ( 2005 ) . m. buzzacchi , i. pagonabarraga and n. b. wilding , j. chem . phys . * 121 * , 11362 ( 2004 ) . m. buzzacchi , n. b. wilding and p. sollich , phys . lett . * 97 * , 136104 ( 2006 ) . r. p. sear , eur . . lett . * 44 * , 531 ( 1998 ) . p. bartlett and p. warren , phys . lett . * 82 * , 1979 ( 1999 ) . m. fasolo and p. sollich , phys . * 91 * , 068301 ( 2003 ) . h. maeda and y. maeda , phys . lett . * 90 * , 018303 ( 2003 ) . j. a. c. veerman and d. frenkel , phys . a * 43 * , 4334 ( 1991 ) . p. bolhuis and d. frenkel , j. chem . phys . * 106 * , 666 ( 1997 ) . j. a. capitn , y. martnez - ratn and j. a. cuesta , j. chem . phys . * 128 * , 194901 ( 2008 ) . a. stroobants , phys . lett . * 69 * , 2388 ( 1992 ) . s. m. cui and z. y. chen , phys . e * 72 * , 031405 ( 2005 ) . m. a. bates and d. frenkel , j. chem . phys . * 109 * , 6193 ( 1998 ) . g. j. vroege , d. m. e. thies - weesie , a. v. petukhov , b. j. lemaire and p. davidson , adv . mater * 18 * , 2565 ( 2006 ) . a. m. bohle , r. hoyst and t. vilgis , j. chem . phys . * 106 * , 666 ( 1996 ) . p. tarazona , j. a. cuesta and y. martnez - ratn , in _ theory and simulations of hard - sphere fluids and related systems _ , edited by a. mulero , vol . 753 ( springer , berlin , 2008 ) , pp . . y. rosenfeld , phys . lett . * 63 * , 980 ( 1989 ) . y. martnez - ratn , j. a. capitn and j. a. cuesta , phys . e * 77 * , 051205 ( 2008 ) . p. tarazona and y. rosenfeld , phys . e * 55 * , r4873 ( 1997 ) . y. rosenfeld , phys . a * 42 * , 5978 ( 1990 ) . r. van roij , p. bolhuis , b. mulder , and d. frenkel , phys . rev . e * 52 * , r1277 ( 1995 ) .

we apply a recently proposed density functional for mixtures of parallel hard cylinders , based on rosenfeld s fundamental measure theory , to study the effect of length - polydispersity on the relative stability between the smectic and columnar liquid crystal phases . to this purpose we derive from this functional an expression for the direct correlation function and use it to perform a bifurcation analysis . we compare the results with those obtained with a second and a third virial approximation of this function . all three approximations lead to the same conclusion : there is a terminal polydispersity beyond which the smectic phase is less stable than the columnar phase . this result is in agreement with previous monte carlo simulations conducted on a freely rotating length - polydisperse hard spherocylinder fluid , although the theories always overestimate the terminal polydispersity because the nematic - columnar phase transition is first order and exhibits a wide coexistence gap . both , the fundamental - measure functional and the third virial approximation , predict a metastable nematic - nematic demixing . conversely , according to second virial approximation this demixing might be stable at high values of the polydispersity , something that is observed neither in simulations nor in experiments . the results of the fundamental - measure functional are quantitatively superior to those obtained from the other two approximations . thus this functional provides a promising route to map out the full phase diagram of this system .

the pulsar psr b1951 + 32 , located at the center of the morphologically peculiar radio nebula ctb 80 , is a 39.5-msec radio pulsar ( clifton et al . 1987 ; kulkarni et al . 1988 ) with a characteristic age of @xmath5 yr and an inferred surface dipole magnetic field of @xmath6 g. an x - ray point source was observed within the x - ray nebula related to the radio core of ctb 80 ( becker , helfand & szymkowiak 1982 ; seward 1983 ; wang & seward 1984 ) . search for x - ray pulsation from this source with exosat yielded inconclusive evidence for pulsation ( confidence level of 97% , by gelman & buccheri 1987 , and 93% by angelini et al . the pulsed emission was detected by rosat at a 99% confidence level ( safi - harb , gelman & finley 1995 ) , which shows a single peak roughly consistent in phase with the radio emission . the overall spectrum of this point source in 0.1 - 2.4 kev is best fitted by a power law with a photon spectral index @xmath7 , and an estimated pulsed fraction of 0.35 . the egret instrument on cgro observed and detected gamma - ray pulsation from psr b1951 + 32 above 100 mev ( ramanamurthy et al . 1995 ) , making it a member of egret hard gamma - ray pulsar family with a similar age to vela and geminga . the gamma - ray lightcurve shows two peaks at phase 0.16 and 0.60 with phase 0.0 being the radio peak . its spectrum in the egret energy range follows a power law with a photon spectral index of about @xmath8 ( ramanamurthy et al . 1995 ; fierro 1995 ) over about two decades of photon energy . recently , pulsed emission is reported from the comptel instrument in the 0.75 - 10.0 mev band ( kuiper et al . the osse and batse instruments on cgro only reported upper limits of pulsed emission in the lower energy band ( schroeder et al.1995 ; wilson et al . 1992 ) . there have been a number of models proposed to explain the gamma - ray emission with dramatically different emission sites , some at or very near the surface of the neutron star and some very far away . recently , ho & chang ( 1996 ) proposed a geometry - independent argument to constrain the possible site and mechanism of the gamma - ray emission based on the commonality of power - law emission in the egret ( and possibly comptel ) pulsars . in such arguments , it is important to know whether and how the gamma - ray power - law spectra turn over towards low energy . ( see section 4 for more discussions . ) to gain better understanding of the overall spectral behavior , especially between kev and mev , we conducted an observation of psr b1951 + 32 using both pca and hexte on board rxte during cycle 1 . analysis of the 19k - second pca data does not yield conclusive evidence for pulsation from 2.0 to 13.0 kev . the derived 2-@xmath1 upper limits provide support for the hard turn - over for the high - energy gamma - ray emission . it also indicates that the soft x - ray pulsation observed by rosat has a very soft spectrum . we described the observation in section 2 . the analyses and results for the pca data are discussed in section 3 . we discuss the theoretical implications of this observation and future work in section 4 . the pca and hexte on board rxte were pointed at psr b1951 + 32 on march 24 , 1996 ( mjd 50166 ) , for about 10.5 hours including earth occultations . the rxte mission , spacecraft and instrument capabilities are described in swank et al . ( 1995 ) , giles et al.(1995 ) and zhang et al . ( 1993 ) the pca consists of five essentially identical pcus with a total effective area of 6729 @xmath9 , with no imaging capability . the field of view is one degree . after examining the data , two exclusions were applied to the data set . first , data from the pca pulse - height channel 36 - 255 ( 13.0 - 90.0 kev ) are excluded due to high instrumental noise . second , we observed unexplained anomalous increase during two intervals of our exposure . under the advice of rxte guest observer facility experts , data obtained during these two intervals were excluded . in the second half of the observation , two of the five pcus were turned off . the overall usable data used for this analysis contain two segments of @xmath10 and @xmath11 for a total of @xmath12 , or equivalently , a total integration time of 19232 seconds and an average effective area of 5363.3 @xmath9 . around the same epoch of the rxte observation , psr b1951 + 32 was also monitored at jodrell bank radio observatory . the radio ephemeris is summarized in table 1 and used as the input for pulsation search . the data were reduced to the barycenter and analyzed using the jpl de200 ephemeris , the pulsar position listed in table [ ephemeris ] , and standard rxte reduction package ( ftools v.3.5.2 and xanadu / xronos v.4.02 ) . lightcurve folding was performed separately for each of four typical energy bands and various combinations using the radio ephemeris in table [ ephemeris ] . the four typical energy bands are commonly designated as band 1 through 4 , with each covering pca channels 0 - 13 ( 2.0 - 4.8 kev ) , 14 - 17 ( 4.8 - 6.3 kev ) , 18 - 23 ( 6.3 - 8.5 kev ) , and 24 - 35 ( 8.5 - 13.0 kev ) , respectively . none of the folded lightcurves showed significant deviation from a model steady distribution under the pearson s @xmath13-test ( leahy et al . 1983a , b ) . specifically , the @xmath13 values for the folded lightcurves shown in figure [ lightcurve ] are , for 19 degrees of freedom , 27.4 for band 1 , 21.1 for band 2 , and 8.38 for the combined band of 3 and 4 . in addition to instrumental and cosmic x - ray background , the dc component is mostly likely the contribution from the rosat point source and its associated x - ray nebula . to further ascertain the absence of pulsation , we performed the bin - independent parameter - free @xmath0-test ( de jager , swanepoel & raubenheimer 1989 ) . in this analysis , all detected raw photons with the corrected arrival time are used . the @xmath0-test was applied to the data in different energy bands and various combinations . the results of the @xmath0-test all show high probability that the data are consistent with steady source except for band 1 and 2 . the @xmath0-values are 7.286 and 9.334 for bands 1 and 2 ( both at @xmath14=1 ) . this yields a 5.5% and 2.4% probability of being consistent with a steady source . applying the straight @xmath15-test ( the rayleigh test , which is more appropriate if the underlying pulse profile is sinusoidal ) , the probability of the data being consistent with a steady source is 2.9% and 0.9% for bands 1 and 2 . based on these analyses , we do not consider the null probability , although intriguingly small using the @xmath0-test , provides conclusive evidence of pulsation from psr b1951 + 32 in the xte / pca energy band . the upper limit of pulsed flux is estimated following the prescription given by ulmer et al . ( 1991 ) . assuming a duty cycle of 0.5 and combining bands 2 and 3 to yield a comparable total number of counts to those in bands 1 and 4 individually , we obtain the following 2-@xmath1 upper limits , which are also shown in figure [ spectrum ] : @xmath2 for 2.0 - 4.8 kev , @xmath3 for 4.8 - 8.5 kev , and @xmath16 for 8.5 - 13.0 kev . _ gamma - ray pulsar : _ the current xte / pca upper limits provide support to the combined cgro observation that the gamma - ray pulsed emission from psr b1951 + 32 follows a power law with a significant break towards low energy ( egret and comptel detection along with osse and batse upper limits ) , as indicated in figure [ spectrum ] . such spectral behavior is seen in the vela and geminga pulsars . for psr b1951 + 32 , the break energy ( photon energy at which there is a significant break in photon spectral indices to , say , harder than @xmath17 ) is estimated to be between 70 kev and 3 mev . as noted in ho & chang ( 1996 ) , this common trait could play an important role in the theoretical modeling of this family of `` egret pulsars . '' the power law of these pulsars typically covers two orders of magnitude in the egret band with best - fit photon spectral indices in the range of @xmath18 to @xmath8 . these photon spectral indices can not be produced by a mono - energetic relativistic electron distribution under currently proposed radiation mechanisms . the most likely origin of this power - law behavior is the cooling ( energy loss ) through the dominant radiation mechanism responsible for the gamma - rays . for example , the most simplistic cooling model for a steady state electron distribution will yield a photon spectral index of @xmath19 for synchrotron radiation and @xmath20 for curvature radiation . the cooling spectrum will continue towards low energy until the electron distrbution ( in energy space ) is no longer affected by cooling : i.e. the cooling spectrum turns hard at the break energy which corresponds to the location and electron energy where the radiative cooling time scale is comparable to the dynamical time scale of the relativistic motion of the electrons . such an argument allows us to constrain the radiation mechanism and , more importantly , the emission site , which to date remains unsettled with great bifurcation among gamma - ray pulsar models . following this argument and examining various radiation mechanisms , we find that , for psr b1951 + 32 , the gamma - ray pulsations are most likely generated by synchrotron radiation with a typical pitch angle of 0.1 to 0.001 and emission site is about @xmath21 to @xmath22 cm from the star : i.e. in the outer magnetosphere . _ x - ray pulsar : _ safi - harb , gelman & finley ( 1995 ) reported rosat observation of psr b1951 + 32 with an estimated pulsed fraction of 0.35 and an overall spectrum following a power law with a photon spectral index of @xmath7 . to date , there is no published spectrum for the pulsed component . safi - harb et al . ( 1995 ) estimated the soft x - ray pulsation duty cycle to be 0.1 . such a duty cycle will lower the xte upper limit estimated above . it is clear from figure [ spectrum ] that the xte / pca observation necessitates a steep drop - off at around 2 kev for the pulsed component . this is consistent with safi - harb et al.s report of no pulsation near 2 kev . it is almost certain that , for psr b1951 + 32 , the rosat observed pulsation below 1.3 kev is separate from the egret / comptel observed gamma - ray pulsation . more than likely , they are from different origin and emitted at different location . in summary , xte / pca observation over 19k seconds shows no definitive evidence of pulsation . the upper limits can be used to help constrain theoretical models . our analysis does show tantalizing hints . in addition to the small null probability from the @xmath0-test , the peaks for bands 1 and 2 in figure [ lightcurve ] , taken at face value , are separated by 0.45 in phase , reminiscent to that of the geminga pulsar in 0.07 - 1.5 kev ( halpern & ruderman 1993 ) . a long exposure on xte , e.g. 100 ksecs , will provide better statistics and help advance our understanding of this and other similar pulsars . we thank the xte team in xte / gof at gsfc , especially james lochner , for their help in data reduction and analysis . we also thank andrew lyne for providing the radio ephemeris and comments on the manuscript . many useful discussions with ed fenimore and james theiler are gratefully acknowledged . we are appreciative of the referee , f. seward , for his many helpful comments in improving this paper . this work was performed under the auspices of the us department of energy and was supported in part by the rxte guest observer program and cgro guest investigator program . angelini , l. , white , n.e . , parmar , a.n . , smith , a. , & stevens , m.a . 1988 , apj , 330 , l43 becker , r.h . , helfand , d.j . , & szymkowiak , a.e . 1982 , apj , 255 , 557 clifton , t.r . , et al . 1987 , iau circ . 4422 de jager , o.c . , swanepoel , j.w.h . , & raubenheimer , b.c . 1989 , a&a , 221 , 180 fierro , j.m . 1995 , ph.d . thesis , stanford university giles , a.b . , jahoda , k. , swank , j.h . , & zhang , w. 1995 , publ . australia , 12 , 219 halpern , j.p . , & ruderman , m. 1993 , apj , 415 , 286 ho , c. , & chang , h .- k . 1996 , apj , in preparation kuiper , l. , et al . 1996 , a&a , submitted kulkarni , s.r . , et al . 1988 , nature , 331 , 50 leahy , d.a . , et al . 1983a , apj , 266 , 160 leahy , d.a . , elsner , r.f . , & weisskopf , m.c . 1983b , apj , 272 , 256 gelman , h. , buccheri , r. 1987 , a&a , 186 , l17 ramanamurthy , p.v . , et al . 1995 , apj , 447 , l109 safi - harb , s. , gelman , h. , & finley , j.p . 1995 , apj , 439 , 722 schroeder , p.c . , et al . 1995 , apj , 450 , 784 seward , f.d . 1983 , in supernova remnants and their x - ray emission , ed . j. danziger & p. gorenstein ( proc . iau symp . 101)(dordrecht : reidel ) , 405 strickman , m.s . 1996 , apj , 460 , 735 swank , j.h . , jahoda , k. , zhang , w. , & giles , a.b . 1995 , in the lives of the neutron stars , ed . alpar , . kizilolu , & j. van paradijs ( nato asi series c , 450)(boston : kluwer ) , 525 ulmer , m.p . , purcell , w.r . , wheaton , w.a . , & mahoney , w.a . 1991 , apj , 369 , 485 wang , z.r . , & seward , f.d . 1984 , apj , 285 , 607 wilson , r.b . , et al . 1992 , proc . aip conf . 280 , 291 zhang , w. , et al . 1993 , proc . spie , 2006 , 324 llr validity interval & ( mjd ) & 50057 - 50207 epoch , @xmath23 & ( mjd ) & 50132.000000201 @xmath24 & & @xmath25 @xmath26 & & @xmath27 @xmath28 & ( hz ) & 25.2963865292632 @xmath29 & ( hz / s ) & @xmath30 @xmath31 & ( hz / s / s ) & @xmath32

we report results of rxte observations of psr b1951 + 32 using the pca instrument for 19k seconds during 1996 march 24th . we applied the contemporaneous radio ephemeris and various statistical tests to search for evidence of pulsation . these analyses yield intriguing yet inconclusive evidence for the presence of the pulsation in the time series : confidence level for the presence of pulsation is 94.5% in the 2.0 - 4.8 kev band and 97.6% in the 4.8 - 6.3 kev band based on the @xmath0-test . under the premise of non - detection of pulsation , we derive estimated 2-@xmath1 upper limits for the pulsed flux to be @xmath2 in the 2.0 - 4.8 kev band , @xmath3 in the 4.8 - 8.5 kev band , and @xmath4 in the 8.5 - 13.0 kev band . these upper limits are consistent with the trend of spectral turn - over from high - energy gamma - ray emission as suggested by the osse upper limit . such turn - over strongly suggests the outer magnetosphere as the emission site for pulsed gamma - rays . these rxte upper limits for x - ray pulsation are , on the other hand , not consistent with the extrapolation of reported power - law spectra from the point source observed by rosat in the 0.1 - 2.4 kev band , assuming a constant pulse fraction : the pulsed soft x - ray emission detected by rosat must follow a much softer spectrum than that of the overall point source .

the abrikosov vortex lattice melts into an extended vortex - liquid phase in high - temperature superconductors subject to an external magnetic field oriented perpendicular to the conducting copper - oxygen planes that make them up@xcite@xcite . the large size in temperature and magnetic field of the vortex - liquid phase can be attributed to such layer anisotropy@xcite@xcite@xcite . a cross - over from a vortex - line liquid at temperatures just above the melting point of the abrikosov vortex lattice to a decoupled vortex liquid at higher temperature that shows negligible correlations of the superconducting order parameter across layers is predicted if the vortex lattice in isolated layers melts through a continuous or a weakly first - order phase transition@xcite . such dimensional cross - over is observed experimentally in electronic transport studies of the vortex - liquid phase in moderately anisotropic high - temperature superconductors@xcite . the abrikosov vortex lattice is predicted to sublimate directly into a decoupled vortex liquid at large enough layer anisotropy , on the other hand , if the vortex lattice in isolated layers melts through a first - order phase transition@xcite . electronic transport studies of the mixed phase in extremely layered high - temperature superconductors are consistent with the last sublimation scenario@xcite . an anomalous nernst effect is also observed in the vortex - liquid phase of high - temperature superconductors@xcite . in particular , a gradient in temperature along the copper - oxygen planes generates an electric field perpendicular to it along the copper - oxygen planes as well . the low - temperature onset of the anomalous nernst signal coincides with the melting point of the abrikosov vortex lattice , while the high - temperature onset can lie above the critical temperature of the superconducting state at zero field . the authors of ref . @xcite argue that this effect is principally due to vortex excitations in the mixed phase of high - temperature superconductors . it is then tempting to identify the cross - over between three - dimensional ( 3d ) and two - dimensional ( 2d ) vortex - liquid behavior that is predicted for layered superconductors in certain instances@xcite with the peak in the nernst signal . the fact that anomalous nernst signals are also observed in the vortex - liquid phase of extremely layered high - temperature superconductors that do not show the former dimensional cross - over@xcite@xcite rules out that interpretation , however . the anomalous nernst effect observed in the vortex - liquid phase of high - temperature superconductors may instead be principally due to vortex excitations in copper - oxygen planes that are virtually isolated from one another@xcite . in this letter , the theoretical consequences of that proposal are examined through a duality analysis of the uniformly frustrated @xmath0 model for the mixed phase of extremely type - ii superconductors@xcite@xcite . we find first that weak collective pinning of the vortex lattice results in a melting / decoupling temperature that does _ not _ extrapolate to the mean - field transition in zero field . instead , a relatively big region of vortex liquid that is stabilized by random pinning centers is predicted to exist at temperatures below the mean - field transition . second , a high - temperature expansion of the uniformly frustrated @xmath0 model yields linear diamagnetism at temperatures just below the mean - field transition . the temperature dependence of the predicted equilibrium magnetization is found to agree quantitatively with recent experimental reports of a diamagnetic signal extracted from the vortex - liquid phase of high - temperature superconductors@xcite . last , we emphasize that an anomalous nernst effect is generally expected inside of the vortex liquid phase@xcite , where it tracks the temperature dependence shown by the diamagnetism in the vicinity of the mean - field phase transition . the @xmath0 model with uniform frustration is the minimum theoretical description of vortex matter in extremely type - ii superconductors . both fluctuations of the magnetic induction and of the magnitude of the superconducting order parameter are neglected within this approximation . the model hence is valid deep inside the interior of the mixed phase . its thermodynamics is determined by the superfluid kinetic energy @xmath3|_{r } , \label{3dxy}\ ] ] which is a functional of the phase of the superconducting order parameter , @xmath4 , over the cubic lattice , @xmath5 . here , @xmath6 and @xmath7 denote the local phase rigidities over nearest - neighbor links within layers . these are equal and constant , except over links in the vicinity of a pinning center . the josephson coupling across adjacent layers , @xmath8 , shall be assumed to be constant and weak . it can be parameterized by @xmath9 , where @xmath10 is the gaussian stiffness of the @xmath0 model for each layer in isolation , and where @xmath11 is the model anisotropy parameter . the vector potential @xmath12 represents the magnetic induction oriented perpendicular to the layers , @xmath13 . here @xmath14 denotes the square lattice constant , which is of order the coherence length of the cooper pairs , @xmath15 denotes the flux quantum , and @xmath16 denotes the concentration of vortices per site . the thermal / bulk average of the josephson coupling between adjacent layers is given by the expression@xcite@xcite @xmath17 } \label{cos1}\ ] ] in the decoupled vortex liquid to lowest order in the fugacity @xmath18 . here @xmath19 is the gauge - invariant phase difference across adjacent layers @xmath20 and @xmath21 , and @xmath22 is the autocorrelation function of the superconducting order parameter within layer @xmath20 in isolation ( @xmath23 ) . short - range correlations on the scale of @xmath24 following @xmath25 yields the result@xcite @xmath26 ^ 2 \label{cos2}\ ] ] for the inter - layer `` cosine '' ( [ cos1 ] ) . here , @xmath27 is a quenched disorder scale for the vortex lattices _ across _ adjacent pairs of isolated layers that appears through the autocorrelation @xmath28 \cdot { \rm exp } [ { -i\phi_{l , l + 1}^{(0 ) } ( 2 ) } ] } = % e^{i b_{\parallel } x_{m , \bar m } } e^{-r_{1,2}/l_{\phi}}. \label{form3}\ ] ] of the quenched inter - layer phase difference , @xmath29 . it is set by the density of dislocations quenched into the 2d vortex lattices found in each layer at zero temperature in the present case of uncorrelated pinning centers . also , above we have @xmath30 and the josephson penetration depth @xmath31 . in the absence of inter - layer coupling , arbitrarily weak random point pins result in a stack of 2d vortex lattices with dislocations quenched in@xcite . let us assume that each 2d vortex lattice is in a _ hexatic vortex glass _ state@xcite , such that dislocations do _ not _ arrange themselves into grain boundaries . the quenched disorder scale @xmath27 that renormalizes down the interlayer josephson coupling ( [ cos2 ] ) is then set by the density of such dislocations@xcite . recent theoretical calculations find that each isolated layer shows a net superfluid density near zero temperature in the collective pinning regime , where the number of dislocations quenched into each 2d vortex lattice is small in comparison to the number of pinned vortices@xcite . application of collective pinning theory to the 2d vortex lattices found in isolated layers yields a density of quenched - in dislocations identical to the density of larkin domains@xcite @xmath32 , where @xmath33 denotes the density of pinned vortices per layer , where @xmath34 denotes the maximum pinning force , and where @xmath35 denotes the shear modulus of the 2d vortex lattice . here the critical state is assumed to be limited by plastic creep of larkin domains by an elementary burgers vector @xmath36 of the 2d vortex lattice . consider now the limit of weak pinning centers that do not crowd together : @xmath37 and @xmath38 , respectively , where @xmath39 denotes the density of pinning centers per layer , and where @xmath40 denotes the range of each pinning center . simple probabilistic considerations then yield the identity @xmath41 between the fraction of occupied pinning centers and the ratio of the effective area of each pinning center to the area per vortex , @xmath42 . this yields the result @xmath43 for the density of pinned vortices@xcite , where @xmath44 is the density of vortices per layer . finally , substitution of the estimate @xmath45 for the shear modulus@xcite yields the result @xmath46 for the density of larkin domains , which is independent of magnetic field . note , however , that all of the above is valid only in the 2d collective pinning regime that exists at perpendicular magnetic fields above the threshold @xmath47 , in which case many vortices are pinned in each layer within a larkin domain of dimensions @xmath48 @xcite . single - vortex pinning exists at magnetic field below that threshold , on the other hand , in which case each larkin domain contains only a single pinned vortex : @xmath49 . assembling the above suggests the profile for the density of pinned vortices per layer versus the density of vortices that is depicted by fig . it implies that the quenched disorder scale @xmath50 is independent of magnetic fields above the threshold @xmath51 . we are finally in a position to determine the melting / decoupling line of the 3d vortex lattice at temperatures outside of the 2d critical regime , @xmath52 , at big enough perpendicular magnetic fields such that larkin domains can be defined , @xmath53 . the identification of the separation between dislocations quenched into each 2d vortex lattice with the 2d larkin scale@xcite , @xmath50 , necessarily yields the inequality @xmath54 in such case . at temperatures lying inside of the interval @xmath55 $ ] bounded by melting of the 2d vortex - lattice and by the kosterlitz - thouless transition in isolated layers , yet lying outside of the 2d critical regime , a partial duality analysis of the pristine layered @xmath0 model with uniform frustration ( [ 3dxy ] ) finds a first - order melting / decoupling transition of the 3d vortex lattice at interlayer josephson coupling@xcite @xmath56 the first - order nature of this melting / decoupling line and its coincidence with the contour defined above is consistent both with monte carlo simulations of the same model@xcite and with elastic medium descriptions of the vortex lattice in layered superconductors@xcite . observe now that the criterion ( [ xover ] ) for first - order melting / decoupling should remain valid in the present regime of weak pinning such that @xmath57 . substitution of expression ( [ cos2 ] ) for the inter - layer `` cosine '' in the decoupled vortex liquid then yields the melting / decoupling field @xmath58 where @xmath59 is the melting / decoupling field in the pristine limit@xcite@xcite@xcite . these results are summarized by the phase diagram shown in fig . the short sections of dashed and solid lines that emanate perpendicularly from the horizontal axis originate respectively from the decoupling cross - over ( [ xover ] ) and the second - order phase transition shown by the layered @xmath0 model in the absence of uniform frustration ( cf . we conclude this section by observing that the melting / decoupling line does _ not _ extrapolate to the mean - field critical temperature at zero - field [ @xmath60 due to the presence of dislocations quenched into the weakly pinned vortex lattices found in isolated layers . the phase diagram for the mixed phase of layered superconductors shown by fig . [ f.2 ] implies a large region of vortex liquid in the vicinity of the meanfield transition at zero magnetic field because of the effects of random point pins . in particular , the equilibrium diamagnetic susceptibility due to the emergence of cooper pairs is well defined at temperatures inside of the window @xmath61 $ ] . the former quantity can be obtained from the uniformly frustrated @xmath0 model ( [ 3dxy ] ) in the vicinity of the meanfield transition via a high - temperature expansion in powers of the fugacity @xmath62 @xcite . in particular , a duality analysis yields that the corresponding partition function is approximated by @xmath63 as @xmath64 , where @xmath65 , and where @xmath66 . here @xmath67 is the ratio between a first - order and a zero - order modified bessel function , which is approximately @xmath68 for @xmath69 . also , @xmath70 represents nearest - neighbor links within layers , and @xmath71 represents elementary plaquettes within layers . the equilibrium magnetization is given by @xmath72 in the extreme type - ii limit , where @xmath73 is the gibbs free energy , and where @xmath74 is the volume . substitution of the previous high - temperature approximation yields @xmath75 where @xmath76 , and where @xmath77 denotes the spacing between layers@xcite . the magnetization therefore varies linearly with vanishing magnetic field like @xmath78 , with a diamagnetic susceptibility @xmath79 here @xmath80 is the usual ratio of the london penetration depth to the coherence length of the cooper pairs . the former is related to the gaussian phase stiffness of each layer by @xmath81 . non - linear diamagnetism is observed experimentally in the vortex liquid phase of the extremely layered high - temperature superconductor bi@xmath82sr@xmath82cacu@xmath82o@xmath83 ( bscco ) , at temperatures just above the superconducting transition in zero field@xcite . linear diamagnetism is displayed at yet higher temperature in the same samples , on the other hand . by eq . ( [ m ] ) , the uniformly frustrated @xmath0 model ( [ 3dxy ] ) predicts such linear diamagnetism , @xmath84 , at perpendicular magnetic fields that are small compared to the upper - critical scale @xmath85 , at temperatures just below the mean - field phase transition . the corresponding diamagnetic susceptibility predicted by the @xmath0 model is given by eq . ( [ chi1 ] ) . use of the relation quoted previously between the gaussian phase rigidity within planes and the london penetration length in conjunction with physical parameters @xmath86 , @xmath87 , and @xmath88 nm appropriate for bscco@xcite yields the estimate @xmath89 ^ 3 ( j_{4}/j_0)^4 ( a/\xi)^2 \ , { \rm a / tm } \label{chi2}\ ] ] for the diamagnetic susceptibility of that material in the vicinity of the mean - field phase transition . here @xmath90 is the superfluid fraction . the mean - field superfluid density expected from a pristine @xmath77-wave state in 2d is approximately @xmath91 the corresponding @xmath92-wave result in the vicinity of the meanfield transition at zero field@xcite ; i.e. , @xmath93 , where @xmath94 is the reduced temperature . equation ( [ chi2 ] ) then implies that the diamagnetic susceptibility vanishes like @xmath95 with temperature as it approaches the mean - field transition . figure [ f.3 ] displays the cube - root of the diamagnetic signal extracted experimentally in ref . @xcite from an underdoped sample of bscco with @xmath96 k , in perpendicular magnetic field @xmath97 t , as a function of temperature . the solid line is a fit to the linear diamagnetism , @xmath84 , predicted by the high - temperature expansion of the uniformly frustrated @xmath0 model , eq . ( [ chi2 ] ) , with @xmath98 , @xmath99 k , and with @xmath0 model parameter @xmath100 . the success of the fit indicates that the onset of the diamagnetic signal observed in the vortex liquid phase of high - temperature superconductors reflects nothing other than the mean - field phase transition at which cooper pairs emerge . the large suppression of @xmath101 compared to the meanfield transition temperature @xmath2 obtained here can be accounted for by quenched disordering of the superconducting order parameter , which could be generic to under - doped high - temperature superconductors@xcite . a gradient in temperature along the layers in the vortex liquid phase of high - temperature superconductors generates a voltage in the perpendicular direction within the layers@xcite . in particular , the nernst signal defined by the ratio @xmath102 between the electric field that is generated and the gradient in temperature peaks inside of the vortex liquid . standard transport theory yields the identity@xcite @xmath103 between the nernst signal and the product of the flux - flow electrical resistivity @xmath104 with the off - diagonal peltier coefficient @xmath105 . also , application of ginzburg - landau theory for the superconducting order parameter yields the estimate@xcite @xmath106 for the peltier coefficient near the mean - field transition , where @xmath107 is of order @xmath108 . observe now that the flux - flow resistance increases with temperature in the vortex liquid , while the equilibrium magnetization decreases with temperature there [ cf . ( [ chi1 ] ) ] . substitution of the estimate ( [ alpha ] ) into the identity ( [ e_y ] ) then yields ( _ i _ ) that the low - temperature onset of the anomalous nernst signal is given by the melting / decoupling temperature of the vortex lattice . also , the linear diamagnetism ( [ chi1 ] ) extracted from the high - temperature regime of the frustrated @xmath0 model implies ( _ ii _ ) that the anomalous nernst signal vanishes with temperature at the mean - field transition as @xmath1 . where exactly the nernst signal peaks inside of the vortex - liquid phase depends on how pinning affects the flux - flow resistance@xcite , which is beyond the scope of the paper . in conclusion , a high - temperature analysis of the layered @xmath0 model with uniform frustration finds that the simultaneous onset of linear diamagnetism and of an anomalous nernst effect in the normal phase of high - temperature superconductors@xcite can be identified with the mean - field transition for cooper pairing . the low - temperature onset of the anomalous nernst signal at the melting / decoupling line of the vortex lattice was also found to be depressed substantially by the presence of dislocations quenched into the vortex lattice in isolated layers . the author thanks louis taillefer for discussions and p.w . anderson for correspondence . this work was supported in part by the us air force office of scientific research under grant no . fa9550 - 06 - 1 - 0479 . a natural worry is that the cubic dependence of the magnetization with the fugacity @xmath109 in eq . ( [ m ] ) is an artifact of the square - lattice grid . the same cubic dependence is obtained if a triangular grid is used , however . and although a honeycomb ( graphene ) grid yields a @xmath110 dependence for the magnetization in the high - temperature limit , the energy functional of that @xmath0 model ( [ 3dxy ] ) is _ not _ the superfluid kinetic energy @xmath111 at long wavelength , but a massless dirac functional instead .

linear diamagnetism is predicted in the vortex - liquid phase of layered superconductors at temperatures just below the mean - field phase transition on the basis of a high - temperature analysis of the corresponding frustrated @xmath0 model . the diamagnetic susceptibility , and the nernst signal by implication , is found to vanish with temperature as @xmath1 in the vicinity of the meanfield transition at @xmath2 . quantitative agreement with recent experimental observations of a diamagnetic signal in the vortex - liquid phase of high - temperature superconductors is obtained .

the physics potential of forward proton tagging at the lhc has attracted a great deal of attention in recent years @xcite . a main focus of interest is the central exclusive production ( cep ) process , @xmath7 , in which the protons remain intact and the central system @xmath8 is separated from the outgoing protons by a large rapidity gap . to a very good approximation , @xmath8 is constrained to be in a colour singlet , @xmath9 , state . observation of any particle , such as a standard model higgs boson , in the central exclusive channel would therefore provide a direct observation of its quantum numbers . furthermore , by detecting the outgoing protons and measuring their energy loss accurately , it is possible to measure the mass of the centrally produced particle regardless of its decay products @xcite . because of these unique properties , it has been proposed that forward proton detectors should be installed either side of the interaction points of atlas and/or cms . the fp420 collaboration has proposed to install detectors in the region 420 m from the interaction points @xcite . these detectors would allow the detection of central systems in the approximate mass range 70 gev @xmath10 gev . proposals also exist to upgrade the capabilities of atlas and cms to detect protons in the 220 m region @xcite . these detectors , when used in conjunction with 420 m detectors , would extend the accessible mass range well beyond @xmath11 gev . in this paper , we focus on the central exclusive production of the standard model ( sm ) higgs boson and a supersymmetric ( mssm ) higgs boson , with @xmath12 gev . the cep process is shown schematically in figure [ centralexclusive ] . for this mass region , the dominant decay channel of the higgs boson is to @xmath13 , which is very difficult to observe in conventional higgs searches at the lhc because of the large qcd background . this is not the case in central exclusive production due to the @xmath9 selection rule , which suppresses the leading order central exclusive @xmath13 background by a factor of @xmath14 , where @xmath15 is the mass of the @xmath13 di - jet system . as we shall see , this renders the @xmath13 decay channel observable at the lhc in certain scenarios if appropriate proton tagging detectors are installed . the structure of the paper is as follows . firstly , we give a brief overview of the proposed forward detector upgrades at 220 m and 420 m at atlas and cms , including a simulation of the acceptance of the detectors . we then discuss the predicted signal cross sections and survey the background processes . taking a 120 gev standard model higgs boson as the benchmark , we perform a simulated analysis including an estimation of the detector acceptance and smearing effects and level 1 trigger strategies . the analysis is then extended to the mssm for a particular choice of parameters . , there are two properties of the proposed forward detector systems that are critical to this analysis ; the acceptance of the detectors in the mass range of interest and the ability of the forward detectors to correctly associate the detected outgoing protons with a higgs boson candidate event measured in the central detector . this matching is critical at high luminosities , where the large number of proton - proton collisions per bunch crossing ( often referred to as pile - up ) leads to a high probability that forward protons from single diffractive or double pomeron ( dpe ) collisions not associated with a higgs candidate event will enter the forward detectors during the same bunch crossing . the proposed forward detectors aim to associate particular protons with the higgs candidate event by making a measurement of the outgoing proton time - of - flight ( tof ) from the interaction vertex to the detectors . the difference in the arrival times of the protons on opposite sides of the central detector , @xmath16tof , allows a vertex measurement to be made under the assumption that the detected pair of protons originate from a single hard interaction . this vertex can then be matched with the vertex of a candidate higgs event reconstructed using the central detector alone . the current design goal of the forward detectors is to achieve a 10ps accuracy in the tof measurement @xcite , which translates into a vertex measurement accurate to 2.1 mm . the use of fast - timing measurements is discussed in more detail in section [ olap ] . it has been suggested that the central detector could also be used provide a third timing measurement @xcite , which could allow for an improved rejection of pile - up events . we discuss the effect of this possibility in section [ sec : results ] . the acceptance of the forward detectors is governed by the distance of the active edge of the detector from the beam , which determines the smallest measurable energy loss of the outgoing protons , and the aperture of the lhc beam elements between the interaction point and the forward detectors ; protons that lose too much momentum will be absorbed by beam elements , imposing an upper limit on the measurable momentum loss of the protons . the distance of the active edge of the detector from the beam depends primarily on the beam size at each detector location . previous estimates @xcite have assumed that the closest distance , @xmath17 , is given by @xmath18 where @xmath19 is the gaussian beam size at the detector location and 0.5 mm is a constant term that accounts for the distance from the sensitive edge of the detector to the bottom edge of the window . the beam size @xmath20 is approximately 250 @xmath21 m at 420 m and 100 @xmath21 m at 220 m , leading to a distance of closest approach of 3 mm for detectors at 420 m and 1.5 mm for detectors at 220 m . it is likely that the detectors will begin operation at a larger distance from the beam , at least until the detectors and machine background conditions are well understood @xcite . figure [ forwardaccept ] ( a ) shows the acceptance for events in which both outgoing protons are detected at 420 m around ip1 ( atlas ) , as a function of the mass of the central system for different detector distances from the beam . the protons are generated using the exhume monte carlo @xcite and tracked through the lhc beam lattice using the fptrack program with version 6.500 of the lhc optics @xcite . the central system mass , @xmath15 , is calculated from the forward proton momenta , @xmath22 where the @xmath23 are the fractional momentum losses of the protons and @xmath24 is the centre - of - mass energy of the collision . the acceptance for a 120 gev higgs boson is independent of the distance of approach of the detectors from the beam up to approximately 7 mm in the case where both protons are detected at 420 m . this is consistent with the findings of @xcite . the situation is very different for events in which one proton is tagged at 220 m and the other at 420 m , as shown in figure [ forwardaccept ] ( b ) . in this case , the acceptance is increased when either detector is moved closer to the beam . for a 120 gev higgs boson , with 420 m detectors at 5 mm and 220 m detectors at 2 mm , the acceptance is 28% if both protons are tagged at 420 m ( symmetric tag ) with an additional 16% acceptance if one proton is tagged at 220 m and one at 420 m ( asymmetric ) . moving the 420 m detectors inwards to 3 mm and the 220 m detectors to 1.5 mm increases the asymmetric acceptance by up to a factor of three , as shown in figure [ forwardaccept ] ( b ) . figures [ forwardaccept ] ( a ) and ( b ) also demonstrate the increasing importance of 220 m detectors as the mass of the central system increases . at ip5 ( cms ) the symmetric acceptance is identical to that at ip1 . for the asymmetric tags , however , the acceptance is worse by a factor of @xmath25 across the mass range of interest . this is caused by the horizontal ( rather than vertical ) plane of the crossing angle of the beams at ip5 @xcite . in this paper we concentrate on the measurement around ip1 ( atlas ) . central exclusive signal and background events are simulated with parton showering and hadronisation effects using exhume v1.3.4 @xcite . exhume contains a direct implementation of the khoze , martin and ryskin ( kmr ) calculation of the central exclusive production process @xcite . the cross section for the cep of a standard model higgs boson decaying to @xmath27 as a function of the higgs mass is shown in figure [ sigmamh ] . the rapid decrease in cross section at @xmath28 gev occurs because the primary decay channel changes from @xmath13 to @xmath29 . for masses above @xmath30 gev , it is expected that the higgs boson should be observed in the @xmath29 channel with a luminosity of 30 fb@xmath4 using forward proton tagging @xcite . the primary uncertainties in the predicted cross section come from two sources ; the parton distribution functions ( pdf ) and the soft survival ( often termed rapidity gap survival ) factor . the cep cross sections are relatively sensitive to the pdfs because the derivative of the gluon density enters to the fourth power . figure [ sigmamh ] shows the cross section prediction for three different pdf choices , cteq6 m , mrst2002nlo and cteq6l . the cross section for the cep of a standard model higgs boson of mass @xmath31 gev decaying into b - quarks varies from 1.86 fb ( mrst2002nlo ) to 7.38 fb(cteq6l1 ) . the spread of a factor of @xmath32 is consistent with the findings of @xcite . for the purposes of this analysis , we chose cteq6 m as our default pdf as it lies between the two extremes . there is some justification for choosing an nlo pdf because the kmr calculation contains a nlo qcd k - factor ( 1.5 ) for sm higgs boson production . the soft survival factor , @xmath33 , is the probability that there are no additional hard scatters in a single @xmath34 collision and is expected to vary from process to process . for cep processes , we take the exhume default of @xmath35 at lhc energies . until very recently , there was a consensus that @xmath33 should be between @xmath36 and @xmath37 for the cep of a higgs boson at the lhc @xcite . two very recent studies have predicted a lower value @xcite and it remains to be seen whether a new theoretical consensus can be reached . in any case , @xmath33 will be measurable in early lhc data . , for three different proton parton distribution functions.[sigmamh],scaledwidth=50.0% ] the mssm contains three neutral higgs bosons - two scalar and one pseudo - scalar . the @xmath38 , parity - even selection rules strongly suppress cep of the pseudo - scalar . this can be advantageous in areas of mssm parameter space where the pseudo - scalar is almost degenerate in mass to one ( or both ) of the scalar higgs bosons , since cep would provide a clean and complimentary measurement of the mass of the scalar only @xcite , and allow nearly - degenerate higgs bosons to be distinguished @xcite . furthermore , at large values of tan@xmath39 , the cross section for the cep of the scalar higgs bosons can be strongly enhanced relative to the cep of a sm higgs boson of the same mass @xcite . we choose a point in parameter space defined by the @xmath40 scenario @xcite with @xmath41 gev and tan@xmath42 , resulting in the scalar higgs boson having a mass of 119.5 gev and a decay width of 3.3 gev . with this choice of parameters , the lightest scalar higgs boson has an enhanced cep cross section and is almost degenerate in mass with the pseudo - scalar . the cep cross section for the lightest higgs boson decaying to b - quarks in this scenario is predicted to be @xmath43 fb . the uncertainty on this prediction is the same as that for the sm higgs boson . the backgrounds to central exclusive higgs production can be broken down into three categories ; central exclusive di - jet production , double pomeron exchange and overlap . the most difficult backgrounds to deal with are the central exclusive ( cep ) di - jet backgrounds . central exclusive @xmath13 production is suppressed at leading order by the @xmath45 selection rule , but is still present and forms an irreducible continuum background beneath the higgs mass peak . central exclusive glue - glue production has a much larger cross section than @xmath13 production since it is not suppressed . it contributes to the background when the gluon jets are mis - identified as b - jets ( 1.3% probability for each mis - tag at atlas - see section [ detector ] ) . central exclusive @xmath46 events do not contribute to the background for two reasons . firstly , @xmath46 production is suppressed with respect to the @xmath13 process by @xmath47 . secondly , the @xmath46 background is further suppressed by a factor of @xmath48 relative to the @xmath13 cross section assuming that the probability of mis - identifying a c - quark is approximately 0.1 for a b - tag efficiency of 0.6 @xcite . higher order cep di - jet backgrounds such as the 3-jet final state @xmath49 process have been studied in @xcite . it is expected that these backgrounds will be lower than the lo @xmath50 and @xmath27 backgrounds after experimental cuts , but they should be implemented into the exhume monte carlo and a full study performed before definitive conclusions can be drawn . double pomeron exchange ( dpe ) is defined as the process @xmath51 , where @xmath52 is a central system produced by pomeron - pomeron fusion ( @xmath53 ) . the pomeron is assigned a partonic structure and so there are always pomeron remnants accompanying the hard scatter as shown in figure [ dpepomeron ] . the relevant background processes are separated into @xmath13 and @xmath54 , where @xmath55 represents light - quark and gluons jets . the dpe events are simulated using the pomwig v2.0 event generator @xcite , which implements the diffractive parton distribution functions measured by the h1 collaboration @xcite . we use h1 2006 fit b , although we discuss the effect of using different diffractive pdfs . we choose the pomwig option to treat the valence partons in the pomeron as gluons . pomwig is normalised to the h1 data , and therefore does not account for the soft survival factor . for hard double pomeron events , we take @xmath35 . pomwig is also capable of generating scatters involving sub - leading ( non - diffractive ) exchanges ( @xmath56 ) . we do not include this contribution because the cross section for @xmath57 @xmath13 events is @xmath58 0.034 fb for @xmath59 , @xmath60 gev and @xmath61-quark @xmath62 gev . this is negligible compared to the cross section for @xmath63 @xmath13 , which is @xmath5865 fb in the same kinematic region . system via double pomeron exchange ( dpe ) . [ dpepomeron],scaledwidth=50.0% ] overlap events are defined as a coincidence between an event that produces a central system of interest , @xmath52 , and one or more diffractive events ( single diffractive @xmath64 or dpe ) which produce protons in the acceptance range of a forward detector in the same bunch crossing . on average there will be 3.5 interactions per bunch crossing at low instantaneous luminosity ( 10@xmath65 @xmath3 s@xmath4 ) and 35 interactions at high instantaneous luminosity ( 10@xmath66 @xmath3 s@xmath4 ) . we investigate three types of overlap event ; [ p][x][p ] , [ pp][x ] and [ px][p ] . the square brackets specify the interaction to which a part of the overlap event belongs - in this notation both the cep and dpe events would be [ pxp ] . the cross section , @xmath67 , for the overlap background may be estimated by @xmath68 } \ , \left [ \ , \sum_{n=3}^{\infty } \frac{\lambda^{n } e^{-\lambda}}{n ! } \ , p_{2[p]}\left(n-1\right ) \ , + \ , \sum_{n=2}^{\infty } \frac{\lambda^{n } e^{-\lambda}}{n ! } \ , p_{[pp ] } \left(n-1\right ) \right ] \nonumber \\ & & + \ , \ , \sigma_{[px ] } \ , \sum_{n=2}^{\infty } \frac{\lambda^{n } e^{-\lambda}}{n ! } \ , p_{[p ] } \left(n-1\right ) , \end{aligned}\ ] ] where @xmath69}$ ] is the inclusive ( @xmath70 ) di - jet cross section , @xmath71 is the average number of @xmath34 interactions per bunch crossing and @xmath72 is the actual number of interactions in a specific bunch crossing . because the actual number of interactions is not fixed , we sum over all possible numbers and weight each configuration by a poisson distribution . in the first term , @xmath73}\left(n\right)$ ] is the probability that , given @xmath74 interactions , there are at least two events that produce a forward proton ( one on each side of the interaction point ) by any mechanism . this is dominated by soft single diffractive events @xmath64 . @xmath73}\left(n\right)$ ] is given by a trinomial distribution , i.e. @xmath75}\left(n\right ) = \sum_{r+q=2}^{n } \sum_{q=1}^{r+q-1 } \frac{n!}{\left(n-\left[r+q\right]\right ) ! \ , r ! \ , q ! } \ , \left ( f_{[p]}^{+}\right)^{r } \ , \left(f_{[p]}^{-}\right)^{q } \ , \left ( 1 - f_{[p]}^{+ } - f_{[p]}^{-}\right ) ^{n - r - q}\ ] ] where , for example , @xmath76}^{+}$ ] is the fraction of events at the lhc that produce a forward proton in the @xmath77 direction and within the forward detector acceptance . in the second term , @xmath78}(n)$ ] is defined as the probability that there is at least one event that contains an outgoing proton on each side of the interaction point within the acceptance of the forward detectors ( dominated by the soft double pomeron process @xmath79 ) . @xmath78}(n)$ ] is a binomial distribution that utilises the event fraction @xmath80}$ ] . note that double pomeron events can also contribute to the first term in equation [ olapxs ] , in the case where only one of the outgoing protons falls within the detector acceptance . the third term deals with a two - fold coincidence between a single diffractive di - jet event ( @xmath64 ) , which produces a proton within the forward detector acceptance and a hard central diffractive system @xmath52 that mimics the signal , and an overlap event that produces a proton on the opposite side . @xmath81}$ ] is the total single diffractive di - jet cross section ( @xmath64 ) , i.e the outgoing proton can be on either side of the interaction point . @xmath82}(n)$ ] is defined as the probability that there is at least one event with a forward proton in the forward detector acceptance on the opposite side of the ip to the single diffractive proton from the hard event . this is defined in a similar way to @xmath78}(n)$ ] , but using the event fraction @xmath76}$ ] . for small proton momentum losses , the cross section for events at the lhc that contain a forward proton is dominated by the single diffractive cross section , @xmath83 @xcite ; @xmath84 where @xmath85 is the total cross section at the lhc , @xmath86 is the soft - survival factor for soft single diffraction ( 0.087 ) , @xmath87 is the pomeron trajectory , @xmath88 is the squared 4-momentum transfer at the proton vertex , @xmath89 is the pomeron nucleon coupling and @xmath90 is the triple pomeron vertex . the value of @xmath76}$ ] is calculated by integrating equation [ sdxs ] using standard monte carlo techniques , for a specific acceptance range in @xmath91 and @xmath88 . the cross section for the [ p][x][p ] di - jet events , for parton @xmath92gev and @xmath93 , @xmath94 is shown in figure [ olaplumidep ] ( a ) . the cross section increases by two orders of magnitude from low to high luminosity . at larger values of @xmath91 , there will be an additional contribution from non - diffractive ( i.e. reggeon exchange ) events . this contribution is estimated using the pythia @xcite and phojet @xcite event generators . the predictions for the diffractive and non - diffractive contributions to @xmath76}$ ] are compared to equation [ sdxs ] in table [ fractions ] . as expected , the diffractive contribution dominates at small @xmath91 . the non - diffractive contribution becomes increasingly important at higher @xmath91 , but remains smaller than the diffractive contribution for @xmath95 . phojet predicts a higher fraction of non - diffractive events than pythia and also predicts that a large fraction of forward protons are due to dpe events . in this analysis we use the prediction for single diffraction given by equation [ sdxs ] for @xmath96 ( which would give hits only at 420 m ) , and take the non - diffractive contribution to be negligible . for @xmath97 , we add the pythia prediction for the non - diffractive component , since this agrees more closely with theoretical expectations that the non - diffractive fraction of @xmath34 collisions at the lhc in this kinematic range should be between 1.0% and 1.7% @xcite . .the fraction of events at the lhc that produce a forward proton on one side of the interaction point in a specific kinematic range . the pythia and phojet event generators are compared to the single diffractive cross section given in equation [ sdxs ] . sd labels the outgoing proton from single diffractive scatters and nd labels the protons produced from non - diffractive scatters . dpe labels double pomeron exchange events . [ fractions ] [ cols="^,^,^,^,^,^ " , ] we have shown that the central exclusive production of the lightest scalar higgs boson , for the choice of mssm parameter space described in section [ scenarios ] , can be observed with a significance of at least @xmath1 in the @xmath13 decay channel within 3 years of data taking at the lhc , if suitable proton tagging detectors are installed around atlas and cms and the current predictions for the cep cross section are correct . we have evaluated the most important backgrounds and shown that they can be rejected with high efficiency using a set of exclusivity variables . we have also shown that a fraction of b - jet events can be retained by the currently forseen level 1 trigger hardware , if the trigger strategies we outline are adopted . whilst we have only considered a particular choice of parameters in detail , the general conclusions should hold for a wide range of scenarios . as a general rule , if the cep cross section for the production of the higgs boson is greater than @xmath98 fb , and the higgs boson decays predominantly to b - quarks , then the analysis presented here should apply if the decay width is not too large . the analysis will also apply to any new particle that decays predominantly to b - quarks . it is worth speculating what the future experimental strategy might be if a higgs sector such as the @xmath0 scenario of the mssm is discovered at the lhc or tevatron , and the cep channel proves to be observable with the forward detector configurations currently proposed . the largest loss of cep signal events comes from the limited acceptance of the proton detectors ( between @xmath99 and @xmath100 depending on detector configuration ) and the l1 trigger efficiency , which is at best around @xmath101 at high luminosity for the strategies we consider in this paper . if a hardware upgrade of the l1 trigger systems of atlas and cms were to increase the trigger latency such that the 420 m detector signals could be included , then a trigger efficiency of close to @xmath102 could be achieved . figure [ upgrade ] ( a ) shows a typical mass fit , with 300 fb@xmath4 of data taken at high luminosity using 420 m detectors alone ( i.e. symmetric events ) , assuming 100% trigger efficiency . the significance of this signal is approximately @xmath103 . if it is assumed that , in addition , fast timing detector improvements can be made as described in section [ mssm ] , then the significance rises to nearly @xmath104 , as shown in figure [ upgrade ] ( b ) . in this case , a measurement could be made within 100 fb@xmath4 . combining the symmetric and asymmetric analyses increases the significance such that a 5@xmath105 measurement could be achieved within 30 fb@xmath4 . the mass of the lightest scalar higgs can be measured with an accuracy of better than @xmath106 gev . for the @xmath0 scenario considered here , the width of the lightest higgs is not much larger than the mass resolution of the proton detectors , and therefore the extraction of the higgs width from the fits is marginal . in other scenarios with widths in excess of @xmath107 gev , however , a direct measurement of the width would also be possible . we remind the reader that observation of any particle in the cep channel would provide a direct measurement of the quantum numbers of the particle . furthermore , if the pseudo - scalar higgs is close in mass to the lightest scalar higgs , then cep would provide an unambiguous separation of the two states since the pseudo - scalar can not be produced . finally , with such a trigger strategy in addition to improved fast timing to reduce the overlap backgrounds , the sm higgs may be obervable in the b - jet channel . we would like to thank mike albrow , michele arneodo , andrew brandt , albert deroeck , jeff forshaw , valery khoze , henri kowalski , paul newman , will plano , misha ryskin , marek tasevsky and chris tevlin for interesting discussions and suggestions throughout this project . this work was funded in the uk by stfc and the royal society . a. bialas and p. v. landshoff , phys . b * 256 * ( 1991 ) 540 . j. r. cudell and o. f. hernandez , nucl . b * 471 * ( 1996 ) 471 [ arxiv : hep - ph/9511252 ] . v. a. khoze , a. d. martin and m. g. ryskin , eur . j. c * 23 * ( 2002 ) 311 [ arxiv : hep - ph/0111078 ] . b. cox , j. r. forshaw and b. heinemann , phys . b * 540 * ( 2002 ) 263 [ arxiv : hep - ph/0110173 ] . a. de roeck , v. a. khoze , a. d. martin , r. orava and m. g. ryskin , eur . j. c * 25 * ( 2002 ) 391 [ arxiv : hep - ph/0207042 ] . a. b. kaidalov , v. a. khoze , a. d. martin and m. g. ryskin , eur . j. c * 33 * , 261 ( 2004 ) [ arxiv : hep - ph/0311023 ] . b. e. cox , j. r. forshaw , j. s. lee , j. monk and a. pilaftsis , phys . rev . d * 68 * ( 2003 ) 075004 [ arxiv : hep - ph/0303206 ] . b. e. cox , aip conf . * 753 * ( 2005 ) 103 [ arxiv : hep - ph/0409144 ] . j. r. forshaw , arxiv : hep - ph/0508274 . m. boonekamp , c. royon and r. peschanski , nucl . a * 755 * , 599 ( 2005 ) . j. r. ellis , j. s. lee and a. pilaftsis , phys . rev . d * 71 * ( 2005 ) 075007 [ arxiv : hep - ph/0502251 ] . p. j. bussey , t. d. coughlin , j. r. forshaw and a. d. pilkington , jhep * 0611 * , 027 ( 2006 ) [ arxiv : hep - ph/0607264 ] . v. a. khoze , a. d. martin and m. g. ryskin , arxiv:0705.2314 [ hep - ph ] . v. a. khoze , a. d. martin and m. g. ryskin , phys . b * 650 * , 41 ( 2007 ) [ arxiv : hep - ph/0702213 ] . s. n. white , `` on the correlation of subevents in the atlas and cms / totem experiments , '' arxiv:0707.1500 [ hep - ex ] . v. avati and k.osterberg , _ totem forward measurements : leading proton acceptance _ , in hera and the lhc proceedings part b , arxiv : hep - ph/0601013 . r. appleby , r. m. jones and f. roncarolo , the proceedings of pac07 , albuquerque , us , june 25 - 29 2007 . s. alekhin _ et al . _ , `` survival probability of large rapidity gaps '' , p221 , `` hera and the lhc - a workshop on the implications of hera for lhc physics : proceedings part a '' , arxiv : hep - ph/0601012 . e. gotsman , e. levin and u. maor , arxiv:0708.1506 [ hep - ph ] . l. frankfurt , c. e. hyde - wright , m. strikman and c. weiss , phys . d * 75 * ( 2007 ) 054009 [ arxiv : hep - ph/0608271 ] . m. strikman , private communication a. b. kaidalov , v. a. khoze , a. d. martin and m. g. ryskin , eur . j. c * 31 * ( 2003 ) 387 [ arxiv : hep - ph/0307064 ] . s. heinemeyer , v. a. khoze , m. g. ryskin , w. j. stirling , m. tasevsky and g. weiglein , arxiv:0708.3052 [ hep - ph ] . m. s. carena , s. heinemeyer , c. e. m. wagner and g. weiglein , eur . j. c * 26 * ( 2003 ) 601 [ arxiv : hep - ph/0202167 ] . atlas : detector and physics performance technical design report , volume 1 , p317 - 346 , 1999 . cern - lhcc-99 - 14 a. aktas _ et al . _ [ h1 collaboration ] , eur . j. c * 48 * ( 2006 ) 749 [ arxiv : hep - ex/0606003 ] . a. aktas _ et al . _ [ h1 collaboration ] , eur . j. c * 48 * ( 2006 ) 715 [ arxiv : hep - ex/0606004 ] . v. a. khoze , a. d. martin and m. g. ryskin , phys . b * 643 * , 93 ( 2006 ) [ arxiv : hep - ph/0609312 ] . t. sjostrand , l. lonnblad and s. mrenna , arxiv : hep - ph/0108264 . r. engel and j. ranft , phys . d * 54 * ( 1996 ) 4244 [ arxiv : hep - ph/9509373 ] . v. a. khoze and m. g. ryskin , private communication . g. corcella _ et al . _ , arxiv : hep - ph/0210213 . j. m. butterworth , j. r. forshaw and m. h. seymour , z. phys . c * 72 * ( 1996 ) 637 [ arxiv : hep - ph/9601371 ] . m. grothe et al . , `` triggering on forward physics , '' cern - cms - note-2006 - 54 k. terashi [ cdf collaboration ] , arxiv:0705.3804 [ hep - ex ] . b. e. cox and a. pilkington , phys . d * 72 * ( 2005 ) 094024 [ arxiv : hep - ph/0508249 ] . v. a. khoze , a. d. martin and m. g. ryskin , eur . j. c * 48 * , 467 ( 2006 ) [ arxiv : hep - ph/0605113 ] . g. c. blazey _ et al . _ , arxiv : hep - ex/0005012 . j. m. butterworth , j. p. couchman , b. e. cox and b. m. waugh , comput . phys . commun . * 153 * ( 2003 ) 85 [ arxiv : hep - ph/0210022 ] . atlas : detector and physics performance technical design report , volume 1 , p99 - 175 , 1999 . cern - lhcc-99 - 14 t. aaltonen _ et al . _ [ cdf collaboration ] , arxiv:0707.2374 [ hep - ex ] . a. d. martin , m. g. ryskin and g. watt , phys . b * 644 * ( 2007 ) 131 [ arxiv : hep - ph/0609273 ] . c. m. buttar _ _ , the underlying event " , hera and the lhc proceedings part a , 2006 . [ hep - ph/0601013 ]

a detailed study is presented of the search for higgs bosons in the b - decay channel in the central exclusive production process at the lhc . we present results for proton tagging detectors at both 220 m and 420 m around atlas or cms . we consider two benchmark scenarios ; a standard model ( sm ) higgs boson and the @xmath0 scenario of the minimal supersymmetric standard model ( mssm ) . detector acceptance , smearing and event trigger strategies are considered . we find that the sm higgs will be challenging to observe in the b - jet channel without improvements to the currently proposed experimental configuration , but a neutral scalar mssm higgs boson could be observable in the b - jet channel with a significance of @xmath1 or greater within three years of data taking at all luminosities between @xmath2 @xmath3 s@xmath4 and @xmath5 @xmath3 s@xmath4 , and at @xmath6 or greater after three years in certain scenarios .

one of the simplest ways to model the proton is as three very light quarks confined in a spherical well . choosing the radius of the well to be @xmath0 fm leads to moderately good agreement with experiment for its electromagnetic properties , such as the charge radius , magnetic moment , and static electric and magnetic polarizabilities . this model , a simplified version of the mit bag model @xcite , will be referred to in the following as the static - well model . it allows an alternative approach to the calculation of the electromagnetic properties of the proton , generally treated with methods quite different in character , that uses the methods of conventional bound - state qed . the latter theory is characterized by wave functions that satisfy the dirac equation in an external field along with electron propagators defined in terms of the same field . when the external field is that of a point coulomb source , a modification of the interaction representation introduced by furry @xcite allows a systematic feynman diagram treatment of radiative corrections . this approach can also be applied to many - electron systems , and a feynman diagram treatment of electron - electron interactions is also possible . as will be explained below , the present paper is patterned on a calculation of these interactions in heliumlike ions involving two - photon exchange @xcite . the approach we will use in this paper was applied some time ago @xcite to the computation of the electromagnetic self energy of the proton and neutron . in that work , both the effect of exchange of a photon between quarks along with the electromagnetic self energy of the quarks were evaluated and found to sum to @xmath1 mev for the proton and @xmath2 mev for the neutron for the case of nearly zero - mass quarks . the fact that the proton is lighter than the neutron remains explained by the fact that the down quark is heavier than the up quark , but it is of note that the electromagnetic correction to the mass splitting , @xmath3 mev , is the same order - of - magnitude as the neutron - proton mass difference , @xmath0 mev . the proton can be studied with electron - scattering experiments , which have a long history of providing information about its properties , in particular the root - mean - square ( rms ) radius , @xmath4 . the proton size has recently received considerable attention because of unexpected results for the @xmath5 transition energy of muonic hydrogen @xcite . the issue of determining @xmath4 from scattering data can be problematic , as extrapolating the slope of the dirac form factor to @xmath6 involves a number of assumptions @xcite . an alternative approach is to determine the proton size by doing precise measurements of atomic transitions that are sensitive to the effect of the size . the 2010 codata result @xcite in fact uses this procedure with hydrogen and deuterium , where the experiment and theory are so accurate that the proton size can be inferred with an accuracy comparable to that available from scattering experiments as of 2010 . because of its smaller size , muonic hydrogen has long been recognized as a system whose spectrum could be used to determine a much more accurate rms radius of the proton than that obtained from hydrogen and deuterium , but the associated experimental obstacles have only recently been overcome . while indeed much more accurate , the result of ref . @xcite for the proton size , @xmath7 is significantly smaller than the codata result , @xmath8 this discrepancy is referred to as the muonic hydrogen puzzle . one possible explanation of the puzzle involves the electromagnetic structure of the proton , and the largest theoretical uncertainty comes from an effect called proton polarizability . this is generally evaluated by relating the energy shift to forward virtual photon - proton scattering . the amplitude describing this scattering , @xmath9 , can then be related to proton form factors through dispersion relations . a recent paper that covers all contributions to muonic hydrogen with particular attention to proton polarizability is ref . @xcite ; in the conclusion , we compare our results to results quoted in that paper . a number of issues involving convergence of the dispersion theory integrals and the need for experimental data complicate that approach . the purpose of the present paper is to provide an alternative analysis patterned after bound - state field theory calculations in atomic physics . this will be done by using the static - well model of the proton together with standard bound - state qed . as we will show , there is a natural way of setting up a consistent qed calculation for hydrogen and muonic hydrogen , with the proton treated as a bound state of three quarks interacting with an electron or a muon , that requires no scattering information for its predictions ; rather it depends only on the radius of the well . regarding the proton as three relativistic particles confined to a small volume is closely analogous to treating three electrons in highly - charged ions , where the electrons for large nuclear charge @xmath10 are quite relativistic and the ion has a size of @xmath11 bohr radius . this problem has recently been addressed with techniques similar to those used for heliumlike ions mentioned above @xcite , and have been shown to provide an accurate description of these ions @xcite , the spectra of which have been measured with high accuracy @xcite . in these calculations almost all of the important physics is described by feynman diagrams with one or two photons . the same turns out to hold for the present calculation , though in this paper , while we will show all relevant diagrams , we concentrate our attention on two effects dependent on proton structure , the polarizability of the proton and the screening of the proton electromagnetic self energy . our model of the proton is extremely simple , but there are three reasons we have chosen it . the first is that proton structure effects are generally very small , with even the largest , the effect of its finite size , accounting for about 2 percent of the transition energy in muonic hydrogen . thus even a crude determination of a proton structure effect will have a small relative theoretical error . the second is that while the proton polarizability correction has been evaluated with other methods , a contribution we term the proton lamb shift has not , and the results presented here may stimulate more sophisticated calculations . the final reason is that mentioned above , to explore a method of calculating the effect of proton structure on atomic energy levels that does not require the use of dispersion theory . we will in the following consider the effect of proton structure on both electronic and muonic hydrogen . because our formalism does not include recoil , we will present results in terms of the electron mass @xmath12 and the muon mass @xmath13 even though reduced - mass effects on the latter are about 10 percent . when we give a general formula , we refer to a lepton with mass @xmath14 . the state of the lepton , in practice either @xmath15 or @xmath16 , will be denoted @xmath17 , and the index @xmath18 will be used for sums over intermediate leptonic states : for the corresponding case of quarks , we use @xmath19 to denote a ground - state quark and reserve @xmath20 for sums over intermediate quark states . the plan of the paper is as follows . we begin in section ii with a quantum mechanical ( qm ) treatment of the shift in energy levels arising from the perturbation of replacing the potential of a point proton coulomb field with that of a general distribution of charge @xmath21 . this perturbation theory is evaluated through second order . in section iii we turn to a quantum - field - theoretic approach to the problem in the context of the static - well model . we do this by modifying the standard furry representation @xcite through forcing the lowest - order hamiltonian to be same as that used in that representation , but having the quarks in the proton provide the coulomb field instead of assuming a point source . this requires the introduction of a new term in the interaction hamiltonian we call the counter term , the effects of which however are quite simple to evaluate . we also define the static - well model and briefly review the calculation of the proton electromagnetic self energy . in section iv we use bound - state field theory to treat one photon exchange , and show that the results agree with the first order qm energy . we note here that in this paper we use the coulomb gauge and treat only coulomb photons for all exchanged photons . for the qm approach this corresponds to ignoring magnetic effects , and in field theory to leaving out transverse - photon exchange . in section v we then turn to two - photon exchange diagrams , which we break into two classes , one in which only one photon attaches to the lepton , with the other being emitted and reabsorbed in the proton , and a second in which each photon is exchanged between the lepton and a quark . in section va we treat the first class , which has no qm analog , and present a calculation of the contribution , which we call the proton lamb shift . in section vb we treat the second class , but again make a breakup of the diagrams into firstly a part in which the proton is left unchanged , and secondly a part where it is excited . ( in our model this means that in the spectral representation of the quark propagator it is either saturated with the @xmath22 state , or else that state is excluded ) . the first part will be shown to correspond exactly to the second - order qm energy . the second part , the proton polarizability , is then evaluated . the related calculation of the proton s static electric polarizability is carried out in section vi , and it is shown that in the @xmath23 angular momentum channel a complete cancellation between positive- and negative - energy state terms occurs , leaving only contributions from the @xmath24 intermediate states . in the conclusion we compare our results to the results of other calculations and describe directions for future progress . we consider a central potential for hydrogen or muonic hydrogen coming from a finite charge distribution @xmath25 , normalized to unity . the corresponding static potential is @xmath26 while we have @xmath27 , the following discussion can also be applied to the case @xmath28 . we start with a point - coulomb binding field , so this distribution leads to the perturbation @xmath29 the first - order correction , @xmath30 is valid for either a relativistic or nonrelativistic calculation . we first consider the nonrelativistic limit . then in leading order the wave functions may be replaced by their value at the origin and the first - order energy is @xmath31&= & - z\alpha \ , 4\pi\left(\frac{1}{u^2 } - \frac{1}{6}\,\bm x^{\prime 2 } + \dots\right ) \left[\rho(\bm x^\prime ) - \delta(\bm x^\prime)\right ] \nonumber\\[10 pt]&= & \frac{2\pi z\alpha}{3 } \ , \bm x^{2 } \rho(\bm x ) = \frac{2(z\alpha)^4}{3n^3}\,m_l^3 \int\rd \bm x \ , \bm x^{2 } \rho(\bm x ) , \label{rms1}\end{aligned}\ ] ] where in the last step we have assumed @xmath17 to be an @xmath32-state . the integral over @xmath33 has been carried out using a cutoff procedure which we now describe . we introduce a parameter @xmath34 , understood to ultimately be taken to zero , and work with the basic identity @xmath35 which for small @xmath34 has the expansion @xmath36 by differentiating once or twice with respect to @xmath34 , we have @xmath37 and @xmath38 we have used eq . ( [ eq : rint1 ] ) in the derivation of eq . ( [ rms1 ] ) , and use eq . ( [ eq : rint2 ] ) to evaluate the correction coming from the variation of the wave function to leading order . for s states , this arises from @xmath39 which yields an additional contribution of @xmath40&= & 2 ( z\alpha)^2 m_l\ , 4\pi\left(\frac{2}{u^3 } - \frac{1}{12}\,|\bm x^\prime|^3 + \dots\right ) \left[\rho(\bm x^\prime ) - \delta(\bm x^\prime)\right ] \nonumber\\[10 pt]&= & - \frac{2\pi ( z\alpha)^2}{3 } \ , m_l \ , = -\frac{2(z\alpha)^5}{3n^3}\,m_l^4 \int\rd \bm x \ , \bm x^{3 } \rho(\bm x ) . \label{rms2}\end{aligned}\ ] ] this term cancels a corresponding term in second - order perturbation theory , which is given by the standard form @xmath41 the sum over terms involving @xmath18 is @xmath42 times the reduced green function . this expression is again valid for either a relativistic or nonrelativistic calculation . the correction to the potential is only non - zero outside the nucleus , which means that the wave functions and reduced green function are evaluated for small arguments , because the bohr radius for both the muon and electron is large compared to the nuclear size . we again take the nonrelativistic limit . then the wave functions may be evaluated at the origin and the reduced green function may be replaced by the nonrelativistic free green function to give @xmath43 & = & - \frac{(z\alpha)^2m_l}{2\pi}\,|\phi_v(0)|^2 \int\rd\bm x_2^\prime \int\rd\bm x_1^\prime \ , \nonumber\\[10 pt]&&\times \int\rd\bm x_2 \int\rd\bm x_1\ , \frac{\rho(\bm x_2^\prime)-\delta(\bm x_2^\prime ) } { |\bm x_2 - \bm x_2^\prime| } \ , \frac{1}{|\bm x_2 - \bm x_1| } \ , \frac{\rho(\bm x_1^\prime)-\delta(\bm x_1^\prime ) } { |\bm x_1 - \bm x_1^\prime| } \,.\end{aligned}\ ] ] introducing cutoffs allows us to carry out the integrals over the unprimed variables with the formulas given above , resulting in @xmath44&&\times \int\rd\bm x_1\ , \left[\rho(\bm x_2^\prime)-\delta(\bm x_2^\prime)\right ] \frac{\rho(\bm x_1^\prime)-\delta(\bm x_1^\prime ) } { |\bm x_1 - \bm x_1^\prime| } \,,\end{aligned}\ ] ] which yields @xmath45 \qquad \nonumber \\ & = & - \frac{(z\alpha)^5}{3 n^3}\,m_l^4 \bigg [ \int\rd\bm x_2 \int\rd\bm x_1 \ , \rho(\bm x_2 ) \rho(\bm x_1 ) -2 \int\rd\bm x \ , \rho(\bm x ) \bigg ] . \qquad\label{eq : sopt}\end{aligned}\ ] ] as alluded to above , the second term in the square brackets in eq . ( [ eq : sopt ] ) is cancelled by eq . ( [ rms2 ] ) . the first term in the square brackets is the third zemach moment , which we denote @xmath46 . an interesting feature about this term is that it too is cancelled by a term that arises when the nucleus is allowed to undergo low - energy excitations ( proton polarizability ) , though we will not use this fact directly , and instead just evaluate the entire effect . we turn now to a field - theoretic approach based on the static - well model , and begin by introducing a formalism for bound - state field theory . while the formalism we use here is to our knowledge novel , it is a simple extension of the furry representation @xcite , which we now briefly review . we will use this representation both for leptons and quarks , and begin by describing how it is used for the former . the full qed hamiltonian used for describing the scattering of free leptons is @xmath47 , with ( @xmath48 ) , @xmath49 \psi(x)\ ] ] and @xmath50 with @xmath51 . we suppress normal ordering and the self - mass counter term for simplicity . the furry representation is used when @xmath52 is replaced by @xmath53 \psi(x).\ ] ] this builds in a classical coulomb field from an infinite mass proton . carrying out a unitary transformation to eliminate @xmath54 rather than @xmath52 leads to the furry representation in place of the interaction representation . while the interaction hamiltonian @xmath55 of a lepton with photons keeps the same form , the lowest order spectrum now consists of hydrogenic bound and scattering states , and the lepton green function obeys the relation @xmath56 which has the spectral representation @xmath57 in the present case we treat proton structure using the static - well model to specify the wave functions and green functions of the constituent quarks . however , the influence of the proton on a lepton in a bound state can not be treated perturbatively , so we need to build the binding of the lepton into the formalism nonperturbatively . we do this by modifying the breakup of the hamiltonian given above to @xmath58 . if we choose @xmath59 then this breakup is @xmath60 . in the following we refer to @xmath61 as the counter term , though of course it is to be distinguished from the electron mass counter term . we stress that we do _ not _ assume that a classical coulomb field is present . however , because we have added a new term to the interaction hamiltonian , @xmath54 is unchanged from the usual furry representation , and the same wave functions and green functions used in feynman diagram calculations in that representation can be used , although extra feynman diagrams involving @xmath61 need to be included . the use of the furry representation must be extended to quarks in order to account for the proton s coulomb field . in this case @xmath52 and @xmath55 are almost identical to the free case , but in @xmath52 we assume the presence of a static well that confines the quarks , the details of which are given below , so that the up and down quark fields are expanded in terms of solutions to the dirac equation in this well . the proton then consists of the usual two up quarks and one down quark , and the sum of their charges leads to the coulomb field felt by the lepton . since the lowest - order problem describes the basic physics of the atom , there has to be cancellation between diagrams in which a coulomb photon is exchanged between the lepton and the quarks in the proton and diagrams with one counter term . in second order another cancellation between two - photon exchange diagrams and diagrams involving one and two counter terms must take place , and so on . because the proton is now modeled as a finite object , the cancellation will not be complete , and we will identify the parts remaining after the near cancellation as the subject of this paper , proton structure effects . it is of course vital for this procedure to make sense that the cancellation not only takes place , but that the perturbation expansion converges . we will present results for the first and second terms in the expansion below . in determining the order of the expansion we note that the counter term @xmath61 is of the same order as two @xmath55 s . the static - well model has well - known solutions which we show partly to establish notation . we represent the solution to the free dirac equation in a spherically symmetric well , centered at the same origin as used in furry representation , by @xmath62 here @xmath63 is a spherical spinor and the radial wavefunctions obey the equations @xmath64f_2(r ) & = & 0 \\ \left({\partial \over \partial r } + { 1 - \kappa \over r } \right)f_2(r ) + \left[e - m(r)\right]f_1(r ) & = & 0.\end{aligned}\ ] ] confinement is enforced by choosing @xmath65 to be constant inside the well and tending to infinity for @xmath66 , which leads to the mit bag model boundary conditions @xcite . the ground - state solution for the case @xmath67 for @xmath68 , which will be used throughout this paper , is @xmath69,\end{aligned}\ ] ] with @xmath70 and @xmath71 here @xmath72 for the ground state , which gives @xmath73 mev for @xmath74 fm . when this wave function is used for the up and down quarks the second and third moments , which can be calculated analytically , are @xmath75 and @xmath76 while an analytic form can be derived for the third zemach moment , it is lengthy and involves the sine integral , so we give only its numerical value , @xmath77 the wave functions of the up and down quarks are identical in this zero mass case , and we denote them as @xmath78 . this approximation leads to the important simplification that the charge density of the proton , even though it consists of three quarks , can be written in terms of @xmath78 , @xmath79 we will see that in our calculations of one- and two - coulomb photon exchange this density will enter in exactly the same way as it does in sec . ii in parts of the calculation , thereby reproducing the results of that section with the static - well model charge density . extra terms arising from the field theory approach will be identified with polarizability effects . were we to use different masses , the charge densities of the up and down quarks would differ , and in that case we would use @xmath80 the @xmath81 and @xmath16 atomic wavefunctions for the lepton are the standard dirac - coulomb solutions and will be denoted as @xmath82 . as proton structure effects are strongly suppressed for the @xmath16 case , even though we will continue to use @xmath17 in formulas , in practice we will always assume @xmath83 . the radius @xmath84 is the only variable in this calculation , and we will use different values to study the @xmath84 dependence of what is by far the largest proton structure correction , the finite size correction from one - photon exchange . however , because all other proton structure effects are much smaller , the value 1.2 fm is understood to be used for those corrections . for the calculation carried out here , which involves a quark propagating in the well , we use the same kind of spectral decomposition as given in eq . ( [ spectral ] ) , @xmath85 the sum over @xmath18 for the lepton and @xmath20 for the quark green functions can be carried out using the method of finite basis sets @xcite , which have been used extensively for atomic calculations , with only minor modifications of the associated computer code required for application to the quark green function . this is because the atomic calculations were set up in the same kind of confining well as used here , but in that case only for the purpose of discretizing the spectrum , with the well radius chosen to be much larger than the atom or ion being considered . we will need the explicit form for the spin - up and spin - down proton wave function , with the former being @xmath86\left|0 \right>. \label{eq : qwf}\end{aligned}\ ] ] in eq . ( [ eq : qwf ] ) @xmath87 and @xmath88 denote spin up and down states of an up quark and @xmath89 and @xmath90 spin up and down states of a down quark ; @xmath91 is the levi - civita symbol , which makes the proton a color singlet after the implicit sum over colors @xmath92 is carried out . because we are taking the up and down quark masses equal to zero , the associated wave function @xmath93 can be replaced by @xmath94 and @xmath95 by @xmath96 , which simplifies later formulas . when the spin state of the quark is not important , we simply use @xmath78 . energy shifts are calculated with the use of @xmath97-matrix techniques , where we use sucher s generalization of the gell - mann low formula @xcite , @xmath98\right|v \right>.\ ] ] here @xmath99 indicates that a factor @xmath100 is included in the time integral over the hamiltonian density in order to adiabatically turn off the interaction at large positive and negative times . the advantage of this formula is that the @xmath97-matrix can be described with standard feynman diagram techniques , with the adiabatic factors usually trivially leading to a factor @xmath101 that cancels the @xmath102 in the numerator of the above formula , though when we deal with two - photon diagrams , the formalism is needed to cancel disconnected diagrams . details of how this works along with other technical issues can be found in ref . that work described a calculation of two - photon exchange diagrams contributing to energy shifts of excited states of heliumlike ions , but the basic approach is almost identical . the most important difference is that while in that work @xmath103 describes two electrons , here it describes one electron or muon and three quarks , and is given by @xmath104 the diagrams that involve one photon are given in figs . [ hlamb ] , [ qlamb ] , and [ 1xqh ] . [ hlamb ] is the standard one - loop lamb shift , which has been evaluated to spectroscopic accuracy . [ qlamb ] is the electromagnetic self energy of the proton , which was calculated using the feynman gauge in ref . the simplest diagram to evaluate is fig . [ qlamb1 ] , photon exchange between pairs of quarks . it contributes @xmath105 mev to the proton electromagnetic self energy , all of which comes from the vector part of the photon exchange . more difficult to calculate is the self - energy diagram of fig . [ qlamb2 ] . this diagram in general requires the inclusion of a self - mass counter term , though not for the zero mass case . after this subtraction an ultraviolet - divergent vertex term generally remains , but because the quarks move freely within the well the ultraviolet divergent part of this term vanishes , and numerical evaluation yields @xmath106 mev ( the results given in ref . @xcite referred to in the introduction are based on the radius @xmath107 fm , but because they scale as @xmath108 a factor @xmath0 must be inserted for comparison to the present work ) . we note that vacuum polarization terms do not contribute for an isolated proton , but may be of interest for muonic hydrogen , an issue that will be discussed further in the conclusion . finally , figs . 3a and 3b describe exchange of a photon between the lepton and the quarks in the proton and the first - order effect of @xmath61 respectively , and we now turn to their numerical evaluation . we evaluate fig . [ 1xqh ] in the coulomb gauge , and consider only coulomb photon exchange . this leads to the energy shift @xmath109 we have used our approximation of having the up and down quark wave functions being equal , so the sum of the contribution of the three quarks gives the single term @xmath110 . a direct evaluation of this diagram for a state with principal quantum number @xmath20 gives a result very close to @xmath111 , with the difference attributable to relativistic effects and the finite size of the proton built into our model . the associated counter term in fig . [ ct ] contributes @xmath112 for the @xmath81 state it has the value @xmath113 where @xmath114 . comparing the sum of @xmath115 and @xmath116 to @xmath117 [ eq . ( [ firstorderpt ] ) ] , one sees that it is exactly reproduced by the field - theory expression . however , because we now have a specific model for @xmath25 , we do not make the approximations made in the qm treatment and instead numerically evaluate it . we find @xmath118 \nonumber \\ e^{(1 ) } & = & 2.006\,26 \times 10^{-4}~~ { \rm a.u . } ~~[\rm{muonic~hydrogen}],\end{aligned}\ ] ] where all digits are significant . the root - mean - square charge radius @xmath119 for @xmath74 fm is @xmath120 we emphasize that this is not meant to be a prediction of the proton s rms radius , it is simply the static - well result when @xmath74 fm . however , if we use this in the standard nonrelativistic expression for the finite size effect in hydrogen given in eq . ( [ rms1 ] ) we get @xmath121 \nonumber \\ e^{(1)}_0 & = & 2.013\,00 \times 10^{-4}~~ { \rm a.u . } ~~[\rm{muonic~hydrogen}],\end{aligned}\ ] ] which differ from the exact result by 0.07% and 0.34% respectively . thus we see that the calculation reproduces the bulk of the nonrelativistic expression for the effect of finite nuclear size on @xmath81 energy levels . in the following , while the basic parameter of the static - well model is the well radius @xmath84 , we use eq . ( [ rvsrp ] ) to replace it by @xmath122 in all formulas dependent on @xmath84 . as we have a complete model of the charge distribution , these small deviations can be attributed to higher moments and relativistic effects . by carrying out the calculation for a range of @xmath84 around 1.2 fm , we find the fits @xmath123\ , { \rm a.u.}\ ] ] for hydrogen and @xmath124 \ , { \rm a.u.}\ ] ] for muonic hydrogen , where @xmath125 denotes @xmath119 in units of fermis ( femtometers ) . we first note that the coefficients of the first and second terms for hydrogen increase by close to a factor of @xmath126 and @xmath127 respectively for muonic hydrogen , consistent with the dependence on @xmath14 shown in eqns . ( [ rms1 ] ) and ( [ rms2 ] ) . the coefficients agree at a level of the order of one tenth of a percent . we originally attempted the fit with a quadratic term in @xmath125 instead of a term with the exponent @xmath128 , but were forced to use the latter form for hydrogen to get a proper fit . ( the effect is less important for muonic hydrogen ) . in fact , it is known that the actual dependence of leading finite - size correction on @xmath125 is not the nonrelativistic quadratic form , but instead the relativistic form used above , as shown in ref . @xcite . in perturbation theory , the leading effect of the fractional power is a correction given by the taylor expansion of @xmath129 , which leads to a logarithmic term of relative order @xmath130 . as mentioned in the introduction , the proton electromagnetic self energy has contributions from the feynman diagrams of fig . 2 . before discussing how these diagrams are modified when the lepton interacts with the proton , which can be thought of as the lamb shift of the proton , we mention another lamb shift related term . this other effect , while negligible for hydrogen because it is of relative order @xmath131 , makes a small contribution for muonic hydrogen , and is not suppressed at all for positronium , accounting for the self energy of the positron in that system . it was first derived by fulton and martin @xcite , and is given by @xmath132.\ ] ] this recoil effect , which shifts the @xmath81 energy in muonic hydrogen by @xmath133 mev , is not included in our approach . an isolated proton can of course emit and reabsorb a photon , giving rise to the electromagnetic self energy of the proton just mentioned . this contributes to the mass of the proton , but when the proton is in a bound state an additional shift arises , described in lowest order by the feynman diagrams in figs . [ pqls1 ] and [ pqls2 ] . we note in passing that these diagrams do not have an analog in the qm treatment , as they involve internal electromagnetic interactions in the proton . in this paper we restrict our attention to the second of these diagrams , which we refer to as exchange corrections to the em self energy of the proton and label as @xmath134 . this is justified by the behavior of the lowest - order proton electromagnetic self energy , where the size of the exchange term and the quark self energy are of the same magnitude . the photon propagators are both taken to be coulomb propagators , and it is straightforward to show that this set of diagrams gives the energy shift @xmath135 in this equation @xmath136 and @xmath137 are understood to be summed over the two flavors and appropriate sums over color indices are implicit ; @xmath138 is the charge of the up or down quark in units of @xmath139 depending on whether @xmath140 or @xmath141 respectively ; @xmath142 , @xmath143 , and their adjoints are field operators , but @xmath144 and its adjoint are wavefunctions , with the sum over @xmath20 going over all allowed values of angular momentum , @xmath145 and @xmath146 , and the positive- and negative - energy states associated with @xmath145 . for two coulomb photons , one can show that only @xmath147 yields a non - zero contribution . after taking the normal - ordered product of the quark fields between the spin up wave function of the proton and noting the integrations over angles are all elementary , one is left with the sum @xmath148 } \frac{1}{\epsilon_\rg - \epsilon_n } \int_0^r \rd x \,x^2 r_{\rg\rg}(x ) \int_0^r \rd y \,y^2 r_{\rg n}(y ) \nonumber \\ \int_0^r \rd z \,z^2 r_{n\rg}(z ) \int_0^{\infty } \rd w \,w^2 r_{vv}(w ) \,\frac{1}{{\max}(x , y ) } \ , \frac{1}{{\max}(z , w)}\ , , \label{eq : expls}\end{aligned}\ ] ] where @xmath149 $ ] denotes summation over all @xmath150-states except the @xmath22 state , and @xmath151 we find , using finite basis sets to carry out the sum over @xmath149 $ ] , that @xmath152 the smallness of this effect is related to the fact the integral over @xmath153 in eq . ( [ eq : expls ] ) is @xmath154 which vanishes when @xmath155 from the orthogonality of the radial wave functions . this restricts the muon wave function to lie within the nucleus , which gives a large suppression factor that scales as @xmath156 . this same suppression factor should make the diagram of 4b , which is more difficult to evaluate , numerically unimportant . however , an interesting application of our approach would be the calculation of vacuum polarization . this was shown in ref . @xcite to vanish for a free proton , but when bound in an atom the arguments for its vanishing no longer apply , and in fact this should correspond to the effect of hadronic vacuum polarization , which is treated as a small effect , estimated in ref . @xcite to be 0.011 mev . however , we are not aware of any direct calculation of this term with zero - mass quarks , and will discuss how such a calculation could be carried out in the conclusions section . before turning to two - photon effects , we give more details about the use of finite basis sets . as described in ref . @xcite , atomic finite basis set calculations are carried out in a well much larger than the atom , but with the same boundary conditions as used for the quarks . we continue to use an atomic basis set appropriate for hydrogen , but add a second basis set for the quarks , which is obtained by simple modifications of the atomic code , involving changing the fermion mass to zero , eliminating the potential , and changing from atomic units to mev - fm units . atomic grids are created on an exponential grid of the form @xmath157 $ ] . we use the same grid for both quark and lepton wavefunctions . this is done by choosing parameters such that if , for example , a 1000 point grid were used for the atom , the two hundredth point would be at @xmath158 fm , so that the quark wave function would be put on a 200 point grid that matched the atomic grid , though of course the quark wave function vanishes for @xmath159 . several grids were used to test numerical stability . as is also the case for leptons , a complete set of positive and negative energy states result for each possible value of @xmath145 , in this case @xmath160 positive energy @xmath150-states and @xmath160 negative energy @xmath150-states , with a typical value of @xmath160 being 50 . for leptons the effect of the negative - energy states is generally very small , entering at the order of the lamb shift . however , for quarks they play a more important role . we now turn our attention to diagrams shown in fig . 5 . in this section we will show that they in part reproduce the second order perturbation theory expression for @xmath161 , eq . ( [ secondorderpt ] ) in section ii , but have in addition extra terms we identify as proton polarizability . to compare with individual diagrams , it is useful to employ eq . ( [ vparts ] ) to represent @xmath162 in terms of @xmath163 , which yields four terms for eq . ( [ secondorderpt ] ) : @xmath164 @xmath165 @xmath166 and @xmath167 the simplest diagram , fig . 5a , is easily shown to give @xmath168 . the diagrams of fig . 5b and its complex conjugate give @xmath169 and @xmath170 if we identify @xmath171 as in the treatment of one - photon exchange . this leaves only @xmath172 to be accounted for . in general one contribution to it comes from fig . 5c , given by @xmath173 where the definition of @xmath138 is the same as in sec . [ sec : spse ] . the various contributions to this term happen to sum to zero for the proton in our model , but we note they would not were we considering the neutron , or if we were taking the up and down quark wave functions to be different . the final two diagrams , 5d and 5e , are referred to as the ladder ( l ) and crossed ladder ( xl ) respectively . the closed loop in these diagrams is associated with an integration over a virtual energy . because in this paper we consider only two - coulomb photon exchange , the analysis of the loop integral is considerably simpler than the case where the photon propagators have energy dependence . that complication was encountered in the full feynman gauge analysis of ladder and crossed ladder diagrams in excited states of heliumlike ions @xcite , upon which the present calculation is patterned . issues involved in carrying out the full calculation will be discussed in the conclusion . in this simpler case we can carry out the integral over the timelike component of the loop momentum with cauchy s theorem , which requires identifying the poles coming from the propagators . we partition the lepton and quark propagators into positive and negative energy parts , which could lead to four contributions for each diagram , but the position of the poles is such that only two contributions fail to vanish . the surviving terms are @xmath174 and @xmath175 for the ladder diagram , and @xmath176 and @xmath177 for the crossed ladder . the relative minus sign between the two ladder and the two crossed ladder terms comes from closing the contour in different ways , and we note that the coulomb propagators are different for the ladder and crossed ladder diagrams . in the following we will give formulas only for @xmath178 and @xmath179 as the other terms differ from these only by an overall minus sign and a reversed role of positive and negative energy states . at this point we divide the sum over intermediate quark states into a part with @xmath180 , which leaves the proton unchanged and corresponds to elastic scattering , and the remaining part , corresponding to inelastic scattering , which gives the polarizability contribution . we now show that the elastic term contains @xmath172 . as @xmath181 is positive energy , only @xmath178 and @xmath179 contribute . because the quark states are now all @xmath150-states , the angular - momentum dependence is particularly simple . the coulomb propagators can be simplified with the replacements @xmath182 with @xmath183 , @xmath184 . ( for later use we define @xmath185 , @xmath186 . ) integration over angles then yields @xmath187 with @xmath18 being forced to be an @xmath150 state . we have introduced the shorthand @xmath188 because @xmath189 , this reproduces the part of @xmath172 in which the electron propagator involves sums over positive energy states . in order to include the part with negative energy states the crossed ladder must be considered . the minus sign mentioned above is needed for the energy denominators to match , and the fact that @xmath190 is symmetric under interchange of @xmath191 and @xmath192 is also needed . thus part of the ladder and crossed ladder diagrams together with the other diagrams of fig . 5 simply reproduce second - order perturbation theory with a particular form for the charge distribution of the proton . we evaluated these terms numerically as a test of the coding , since the qm treatment yields a nonrelativistic ( nr ) limit for comparison . the result for muonic hydrogen is @xmath193 which is within @xmath194 of the nr formula . we recall that formula has a mixture of the zemach term and a @xmath195 term , where the latter cancels part of the one - loop result . we now turn to the evaluation of the remaining parts of the ladder and crossed ladder . the quark propagator is treated in the same manner as described in sec . [ sec : spse ] . the feature of the basis set mentioned in that section , whereby it automatically breaks into a positive and a negative energy part , makes separation of the various diagrams a simple matter . we begin with the case where @xmath196 , but where we do not allow the intermediate quark to be in the ground state . the result for the ladder is @xmath197 mev , and is almost completely cancelled by the crossed ladder , which contributes @xmath198 mev , making this channel completely negligible . in order to treat higher partial waves we must now specify the spin state of the atom . polarizability is important only for the @xmath81 state , which of course can be a triplet or singlet state once the spin of the proton is considered . for one - coulomb - photon exchange these states have the same energy , as hyperfine splitting comes from transverse photon exchange , which is not treated here . however , with two - photon exchange , even when both are coulomb photons , the triplet and singlet energies differ except for the @xmath199 partial wave discussed above . we therefore present in the following formulas for the energy levels of the @xmath81 singlet and triplet . while each coulomb propagator has its own partial wave expansion , leading in general to a double sum over @xmath200 and @xmath201 , the fact that the quarks are all in @xmath150-states leads to the simplification that these values are equal , @xmath202 . another simplification is that for a given @xmath203 , only @xmath204 and @xmath145 values associated with that value give a non - vanishing contribution . we define angular factors @xmath205 for the ladder and @xmath206 for the crossed ladder diagrams , which are rational fractions resulting from the integrations over angles and sums over magnetic quantum numbers , tabulated in table ii for @xmath207 and @xmath208 . both the triplet and singlet state results are given , though in this paper we are only interested in fine structure . @xmath207 corresponds to the dominant dipole transition . in terms of these coefficients , the formula for the ladder is @xmath209 and for the crossed ladder is @xmath210 where @xmath211 and @xmath212 . ' '' '' ' '' '' ' '' '' ' '' '' these can be evaluated with the techniques described above with only simple changes , as the spline basis set contains all values of @xmath213 needed and the radial integrals over @xmath214 are evaluated with techniques valid for arbitrary @xmath203 . the main numerical problem was ensuring that the typical rapid convergence of splines in atomic physics carried over to this problem . the calculation shows that the effect of the @xmath208 channel is very small , with almost the entire effect of polarizability coming from the @xmath207 dipole channel , which shifts the @xmath81 energy down by 0.026 mev . this leads to our main polarizability result for the splitting , @xmath215 consistent with the results from dispersion relation analyses , as will be discussed in the conclusions . one of the basic electromagnetic properties of the proton is its static polarizability , which has an electric and a magnetic component . we concentrate on the static electric polarizability , which is given by the particle data group as @xmath216 while this value is extracted from compton scattering data together with dispersion theory arguments @xcite , in the static - well model the lowest - order expression comes from the diagram of fig . 6 , where the line with a cross indicates a constant electric field with magnitude @xmath217 . the energy shift is related to the static electric polarizability through @xmath218 we note two similar calculations using the closely related mit bag model @xcite , @xcite have been presented , but detailed comparison with their results is not possible . a factor of @xmath219 has been absorbed into the external field . the basic equation for the energy shift of a proton in the presence of a constant electric field described by the potential @xmath220 is @xmath221 which reduces to @xmath222 for this case the orientation of the quark spin does not matter , so the expression simplifies to @xmath223 where we have now isolated the factor @xmath224 to give the formula in terms of @xmath225 . we emphasize at this point that this is a field theoretic derivation , and that the sum over @xmath20 is complete , including negative - energy states . introducing the notation [ see eq . ( [ eq : sprod ] ) ] @xmath226 after integration over coordinate angles we find @xmath227 we calculated the two terms with basis set techniques . the most striking result found was that the @xmath23 channel vanishes . if one sums over only positive energy states one obtains a nonzero result , but inclusion of the negative energy states leads to an exact cancellation . this situation does not occur for the @xmath24 channel , which in our model is solely responsible for the proton static polarizability . we also calculated @xmath225 using a form for the quark propagator that does not rely on the spectral decomposition given in eq . ( [ quarkprop ] ) . after rewriting eq . ( [ alphapdef ] ) as @xmath228 we use the fact that the quark green function satisfies the differential equation @xmath229 with appropriate boundary conditions . without those conditions this is the same equation that a free fermion green function satisfies , and a simple modification of the well - known partial - wave expansion of the latter propagator can be made to solve for the quark propagator . we illustrate this for the green function of a massless quark with positive @xmath230 and @xmath231 , in which case @xmath232 \chi_{\kappa \mu}(\bm{\hat{x } } ) \\ \ri \left[h_{l-1}(\epsilon_\rg x ) + a_l j_{l-1}(\epsilon_\rg x)\right ] \chi_{-\kappa \mu}(\bm{\hat{x } } ) \end{array } \right ) \nonumber\\[10 pt]&&\qquad\qquad\qquad\times \left(\begin{array}{c@{\qquad}c } j_l(\epsilon_\rg y ) \chi^{\dagger}_{\kappa \mu}(\bm{\hat{y } } ) & -\ri j_{l-1}(\epsilon_\rg y ) \chi^{\dagger}_{-\kappa \mu}(\bm{\hat{y}})\end{array}\right).\end{aligned}\ ] ] the coefficients @xmath233 are determined by the boundary condition . the fact that the propagator ranges over a finite volume allows the admixture of the solution proportional to @xmath233 which is forbidden for a free propagator because of the boundary condition as @xmath234 . when @xmath23 , the integrand in eq . ( [ eq : spol ] ) includes the factor @xmath235 which vanishes . an analogous factor on the left - hand - side when @xmath236 also vanishes . however , for @xmath24 there is no such cancellation , and using @xmath237 we find the numerical result @xmath238 the spectral decomposition discussed previously gives the same result , and one can see the sum is dominated by the first @xmath239 state , with higher-@xmath20 positive - energy and negative - energy states entering at under one tenth of a percent . this result is a factor of two larger than the experimental value in eq . ( [ eq : epol ] ) . however , we note that the relation of the pdg result quoted to static polarizability determined from energy shifts involves some subtleties , discussed in @xcite and more recently in @xcite . in this paper we present an approach to calculating the effect of the electromagnetic structure of the proton on energy levels of muonic hydrogen that uses a simple bound - state model for the proton , with only one free parameter , the radius of the well , which we have chosen to be @xmath74 fm . once the formalism is set up , standard techniques of quantum field theory can be used to evaluate proton structure effects using this parameter , in contrast to the standard approach , which involves the analysis of forward photon proton scattering . we do not claim our approach is better , only that it introduces a different way of looking at the problem . we have considered only coulomb photons in this paper . this is because our primary concern was in setting up the basic formalism , and testing it in the relatively tractable case of coulomb photon exchange . to continue , while one could stay in the coulomb gauge and introduce transverse photons , it is simpler to simply change to the feynman gauge . because the quarks all have the same energy , much of the work here is effectively already in the feynman gauge , with @xmath240 continuing the calculation to this more complete state will allow the treatment of the interesting case of hyperfine splitting . the difficulties of proton structure are well known to be exacerbated in this case , with even ground - state hydrogen hfs uncertain at the sixth digit . our approach should allow a direct calculation of what is often referred to as dynamic proton polarizability , as well as allowing a systematic treatment of other spin dependent effects that sometimes are difficult to disentangle . these calculations are complicated by the fact that the loop energy integral can no longer be evaluated with cauchy s theorem . instead one must carry out a wick rotation to the imaginary axis and carry out the integral numerically . this rotation requires care because of the presence of poles and cuts in the complex plane , which leads to a number of extra terms . we have shown that in our method a term , while extremely small , that can be thought of as the lamb shift of the proton arises . the smallness of the effect had very much to do with the fact that all three quarks are taken to be in the @xmath22 state , which limits the @xmath213 values allowed for the propagator . it is an interesting open question as to how this effect would change if corrections to the proton wave function involving non-@xmath150 states were present , as they presumably are because of gluon exchange , but that is outside the scope of this paper . we have shown that the one - photon diagrams together with the part of the two - photon diagrams where the proton is unchanged give the result @xmath241~\mbox{mev}.\ ] ] this was calculated without the large recoil corrections present in muonic hydrogen : multiplying the appropriate factors of @xmath242 and @xmath243 into the first and second terms , our result becomes @xmath244~\mbox{mev}.\ ] ] the polarizability of the proton was calculated in a novel fashion . the basic result of this paper is that despite the difference in approach , a similarly small result is found . specifically , again restoring recoil corrections by multiplying our result by @xmath243 , we have found the proton polarizability correction @xmath245 while inclusion of transverse photons may lead to quantitative changes , we consider it unlikely that a qualitative change will result . we now compare our results with ref . @xcite , where the formula @xmath246 is given in their eq . ( 32 ) , with the breakdown of @xmath247 mev given in table i along with results from other calculations that differ by under 2 @xmath248ev . the quadratic term compares well when a term @xmath249 arising from radiative corrections to finite size , not treated here , is removed . the breakup of @xmath250 involves three terms , a subtraction term of @xmath251 mev , an elastic term of @xmath252 mev , and an inelastic term of @xmath253 mev . the first has no counterpart in our calculation , but if we identify the elastic term with our zemach contribution of @xmath254mev and the inelastic with our polarization of @xmath255 mev we see fairly good agreement . perhaps the most interesting calculation left undone in this framework is vacuum polarization . this of course dominates the muonic hydrogen @xmath5 splitting when an electron is in the vacuum polarization loop , but in our framework we could also put in the zero - mass quarks confined in the well into the loop . for an isolated proton , symmetry arguments @xcite show that vacuum polarization vanishes , but once in an atom the arguments no longer hold , and a finite effect should be present . the standard approach is to introduce pion loops , in which case the previously mentioned very small `` hadronic vacuum polarization '' value of 0.011 mev results , but this approach is quite different . such calculations are , however , particularly challenging because of the high degree of divergence present , which always presents difficulties for bound - state methods . we are investigating whether techniques that have proved useful in studying vacuum polarization effects in atoms @xcite , where careful grouping of angular momentum contributions allows an accurate treatment of highly - divergent terms , can be extended to this novel case . a. chodos , r.l . jaffe , k. johnson , c.b . thorn , and v.f . weisskopf , phys . d * 9 * , 3471 ( 1974 ) . w. furry , phys . rev . * 81 * , 115 ( 1951 ) . mohr and j. sapirstein , phys . a * 62 * , 052501 ( 2000 ) p.j . mohr and j. sapirstein , phys . lett . * 54 * , 514 ( 1985 ) . r. pohl _ et . nature * 466 * , 213 ( 2010 ) ; a. antognini _ et . _ , _ science _ , 339 ( 6618 ) ( 2013 ) . r.j . hill and g. paz , phys . rev . d*82 * , 113005 ( 2010 ) . r. pohl , r. gilman , g.a . miller , pohl , and k. pachucki , arxiv:1301.0905 , ( 2013 ) . mohr , b.n . taylor , and d.b . newell , rev . phys . * 84 * , 1527 ( 2012 ) . artemyev , v.m . shabaev , and v.a . yerokhin , phys . a * 52 * , 1884 ( 1995 ) . j. sapirstein and k.t . cheng , phys . rev . a*83 * , 012504 ( 2011 ) . p. beiersdorfer , a.l . osterheld , j.h . scofield , j.r . crespo lopez - urrutia , and k. widmann , phys * 80 * , 3022 ( 1998 ) . mohr , at . data nucl . data tables * 29 * , 453 ( 1983 ) . t. fulton and p.c . phys . rev . * 95 * , 811 ( 1954 ) . k. pachucki , phys . a * 52 * , 1079 ( 1995 ) . johnson , s.a . blundell , and j. sapirstein , phys . a * 37 * , 307 ( 1988 ) . j. sucher , phys . rev . * 109 * , 1010 ( 1958 ) ; m. gell - mann and f. low , phys . rev . * 84 * , 350 ( 1951 ) . blundell , k.t . cheng , and j. sapirstein , phys . a * 55 * , 1857 ( 1997 ) . lvov , int . phys . a * 8 * , 5267 ( 1993 ) . a. schafer , b. muller , d. vasak and w. greiner , phys . lett . b*143 * , 323 ( 1984 ) . b. hecking and g.f . bertsch , phys . b*99 * , 237 ( 1981 ) . hill , gabriel lee , gil paz , and mikhail p. solon , arxiv:1212.4508 ( 2012 ) . gerhard soff and peter j. mohr , phys . rev . a*38 * , 5066 ( 1988 ) .

a bound - state field theory approach to muonic hydrogen is set up using a variant of the furry representation in which the lowest - order hamiltonian describes a muon in the presence of a point coulomb field , but the origin of the binding field is taken to be three charged quarks in the proton which are modeled as dirac particles that move freely within a spherical well . bound - state field theory techniques are used to evaluate one- and two - photon effects . particular attention is paid to two - photon exchange diagrams , which include the effect of proton polarizability . in addition the modification of the electromagnetic self energy of the proton by the electric field of the muon is examined . finally , the model is used to carry out a calculation of the static electric polarizability of the proton . # 1#2 # 1#2 # 1

a few years after paczyski s proposal ( paczyski 1986 ) , the collaboration engaged in long term microlensing observations towards the magellanic clouds in order to probe the galactic halo . 1 and experiments set strong limits on the maximum contribution of low mass objects to the halo of the milky way ( alcock et al . 1998 ) . towards the , the optical depth has been estimated by as @xmath0 , from 8 events ( alcock et al . 1997a ) ; the time scales associated with these events indicate high mass lenses ( @xmath1 ) that are not observed visually . based on 2 candidates , 1 gave an upper limit on the halo mass fraction in s ( ansari et al . 1996 ) that is below that required to explain the rotation curve of our galaxy .. ] it has been suggested that the lenses might be in the bar / disk of the itself ( sahu 1994,wu 1994 ) ; but simple dynamical arguments seem to rule out this possibility ( gould 1995 ) . nevertheless , more complicated models allow for a larger optical depth ( @xmath2 ) . microlensing search provides a test of the halo - lens hypothesis ; in this model both the optical depth and the typical durations should be similar towards the and the . to date , two events have been observed ; they are significantly longer than the average for events . however , no definite conclusion can be drawn from this without more events . one candidate , -97 - 1/-97 - 1 was found in this analysis ( alcock 1997b , palanque - delabrouille 1998 ) . the result of a microlensing fit leads to an einstein radius crossing time @xmath3 days . the @xmath4 is 261 for 279 d.o.f . , taking into account the @xmath5 intrinsic variability of the amplified star ( @xmath6 days , see palanque - delabrouille et al . 1998 , udalski et al . this single event allows us to constrain the halo composition , in particular we exclude that more than 50 % of the standard dark halo is made of @xmath7 objects . the collaboration sent a first level alert for this event on may @xmath8 , 1998 , followed by an announcement that a caustic crossing had occurred on june @xmath9 . a second caustic crossing was predicted around june @xmath10 . after a planned technical maintenance , we could only observe in great detail the end of the second caustic crossing . using this data alone , we could extract a limit on the caustic crossing time , which together with public data from enabled us to determine ( at a 90% likelihood ) that the deflector is in the ( afonso et al . this result has been confirmed and improved by other groups , leading to a common publication ( afonso et al . 1999 , and references therein ) . since august 1996 , we have been monitoring 66 one - square - degree fields towards the . of these , data prior to may 1998 from 25 square - degrees spread over 43 fields are being analyzed . this represents 450 gbytes of raw data , and about 100 days of to produce the light curves . .results of microlensing fits to the candidate 2 - -1 and 2 - -2 . @xmath11 is the einstein radius crossing time , @xmath12 is the impact parameter , and @xmath13 are the blending coefficients in both colors . [ cols="<,<,^,^,^,^,^",options="header " , ] about 90 images of each field were taken , with exposure times from 3 min in the center to 12 min in the outermost regions ; the sampling is one point every 5 days on average . we report a preliminary analysis of the light curves of 17.5 million stars using a new set of selection criteria to isolate microlensing candidates . starting from the images we built a star catalog using the photometry package , and then removed the 90% most stable stars . among stars with the most significant variations , we used the quality of the microlensing fit to select the candidates . in order to maximize the number of surveyed stars and to study the background of microlensing searches , we did not remove any star based solely on its position in the color - magnitude diagram . with this strategy , we characterized the blue bumper stars that mimic a microlensing signal . in this way we removed the stars , located in the upper left of the color - magnitude diagram , that pass all cuts , but have the following features : @xmath14 , and @xmath15 , where @xmath16 are the red(blue ) observed amplifications . among the 17.5 million light curves , two events passed all the cuts ( see table [ cand1 ] ) . event 2 - -1 is a main sequence star blended in red ( 76% of the visible flux was magnified ) . event 2 - -2 is located just under the red giant clump , and necessitates a deeper photometry study to confirm its validity ; it is consistent nevertheless with being achromatic . to set conservative limits on the halo mass fraction @xmath17 comprised of compact objects of mass @xmath18 , we can assume that the observed events are in the dark halo . we only consider the standard spherical halo model described in palanque - delabrouille et al . the most probable mass associated with both candidates is determined by finding the mass for which the ( near gaussian ) distribution of @xmath19 peaks at the geometric mean @xmath20 the resulting mass is found to be @xmath21 . we can also define a 68% confidence interval as follows : the upper ( lower ) bound is determined as the mass for which 16% of detected events would have durations greater ( less ) than @xmath22 . this mass interval is found to be : @xmath23 \ , { \rm m}_\odot$ ] . let @xmath24 be the total expected number of events for the standard halo model ( considering our detection efficiency ) . to be conservative , we simply consider our two candidates without taking their mass into account . in this way , the 95% cl poisson limit for a given mass is obtained by computing the expected number of events @xmath25 compatible with the observations : @xmath26 , where @xmath27 is the poisson probability of observing @xmath28 events where @xmath29 are expected . the fraction @xmath17 for each mass @xmath18 is given by @xmath30 . this allows us to put a preliminary constraint excluding ( at 95 % cl ) that 60 % of the dark halo is composed of objects in the range @xmath31 \ ; \rm{m}_\odot$ ] ( see fig . [ exclusion ] ) . there is growing evidence , from three different data sets ( 1- , 2- , and 2- ) that the standard spherical halo model fully comprised of @xmath32 \ ; m_{\odot}$ ] s is inadequate . the only way to evade this limit is to suppose that the masses of the s are greater than @xmath33 , or to consider non spherical halos . another way to understand the observed events is to assume that they are due to self - lensing , in which case it is important to study their spatial distribution on the face of the . in that respect , it is worth noting that our limit is derived from more than 17 million stars spread over 43 square degrees , in comparison with the experiment that monitored 9 millions stars covering 11 square degrees of the bar ( alcock et al . finally , more exotic microlensing events ( parallax effect , binary lens ... ) would allow us to locate precisely some lenses , and so to test the self - lensing hypothesis . afonso , c. et al . 1998 , a&a , 337 , l17 . afonso , c. et al . 1999 , ( , / , , , coll . ) , astro - ph/9907247 , submitted to apj . alcock , c. et al . 1997a , apj , 486 , 697 . alcock , c. et al . 1997b , apj , 491 , l11 . alcock , c. et al . ( & coll . ) 1998 , , 499 , 9 . ansari , r. et al . 1996 , a&a , 314 , 94a . gould , a. 1995 , apj , 441 , 77 . paczynski , b. 1986 , apj , 304 , 1 . palanque - delabrouille , n. et al . 1998 a&a , 332 , 1 . sahu , k. c. 1994 , nat . 370 , 275 . udalski , a. et al . 1994 , apj , 435 , 66 .

2 is a second generation microlensing experiment operating since mid-1996 at the european southern observatory ( eso ) at la silla ( chile ) . we present the two year analysis from our microlensing search towards the small magellanic cloud ( ) , and report on the intensive observation of the caustic crossing event -98 - 1 and the limit derived on the location of the lens . we also give preliminary results from our search towards the large magellanic cloud ( ) ; 25 square degrees are being analyzed and two candidates have been found . this allows us to set another limit on the halo mass fraction comprised of compact objects .

zero - field hall effect in chiral @xmath0-wave superconductors ( scs ) has drawn much attention in literature recently . @xcite because of the nature of broken time reversal ( @xmath5 ) symmetry , a nonzero hall conductivity can be possible in a chiral @xmath0-wave sc . indeed , it has already been shown that spontaneous hall effect could arise from the intrinsic angular momentum of cooper pairs @xcite as well as from the spontaneous surface current . @xcite more recently , hall conductivity due to impurity effect @xcite or to multiband sc structure @xcite was also studied , which could give possible explanation to the observed polar kerr effect in the superconducting state of sr@xmath6ruo@xmath7 . @xcite in this work , we address the zero - field hall effect in a chiral @xmath0-wave sc originating from another mechanism , namely the vortex dynamics near kosterlitz - thouless ( kt ) transition . in two - dimensional ( 2d ) superfluid ( sf ) or sc films , quantized vortices are realized as topological defects in the condensates , whose dynamics has been one of the key ingredients in understanding 2d phase transition phenomena . @xcite a few decades ago , kosterlitz and thouless @xcite suggested a static theory to relate a phase transition observed in superfluid @xmath8he film @xcite to vortex - antivortex pair unbinding process across a transition temperature @xmath9 . in this picture , the logarithmic vortex - antivortex interaction is screened by smaller pairs and is renormalized to @xmath10 of its bare value @xmath11 for temperature @xmath12 . the length - dependent dielectric constant @xmath13 is used to describe the static screening of pair interaction . when @xmath14 , there exists a finite pair size @xmath15 such that the interaction becomes vanishingly small . consequently , the pair unbinds and free vortices emerge ; superfluidity is then destroyed . soon after that , ambegaokar , halperin , nelson , and siggia ( ahns ) @xcite combined this static theory with hall and vinen s dynamical description of vortex motion @xcite to give an analysis of the dynamical effect on the phase transition . concisely speaking , the renormalization process in the static theory @xcite is truncated by vortex dynamics with a characteristic length @xmath16 instead of going to its completion . @xcite here @xmath17 is the diffusivity of vortex movement and @xmath18 is the driving frequency . this results in broadening transition observed in @xmath8he sf films @xcite as well as in charged fermi systems such as high - temperature scs . @xcite it is instructive to explore any physical consequence stemming from this kt transition in a broken @xmath5 symmetry state . possible experimental candidates of broken @xmath5 symmetry state could be superconducting sr@xmath6ruo@xmath7 ( ref . ) or @xmath19he - a phase thin film , @xcite in which pairing of chiral @xmath0-wave type is expected . indeed , in literature some theoretical works have been done to investigate new features specific to scs with pairing of this type near kt transition . @xcite in this work , we consider a 2d @xmath0-wave pairing state with @xmath20-vector @xmath21 where @xmath22 is the unit vector normal to film surface , @xmath23 and @xmath24 denote the @xmath25 and @xmath26 component of the relative momentum @xmath27 of a cooper pair , and @xmath28 is the fermi momentum . assuming isotropic fermi surface , two kinds of pairing fields can be obtained : @xmath29 and @xmath30 with asymptotic behavior at large @xmath31 being @xmath32 , @xmath33 , @xmath34 , and @xmath35 . @xcite here @xmath36 is the spatial coordinate and @xmath37 is the modulus of the energy gap in the bulk . from these pairing fields , we can identify two types of integer vortices called @xmath38vortex and @xmath39vortex respectively . because of spontaneously broken @xmath5 symmetry , these two types of vortices are not equivalent . @xcite in particular , their hall and vinen coefficients @xcite do not share the same value , i.e. , @xmath40 and @xmath41 ( see section ii a ) . this results in a nonzero convective " term in a vortex pair polarization fokker - planck equation in addition to the conventional diffusive terms , while in its @xmath42-wave counterpart such convective motion does not enter the dynamics . @xcite the relative strength of convection is quantified by a convective ratio @xmath43 in this paper . it is due to such distinct feature that pair polarization transverse to the driving force field becomes possible even without applied magnetic field . a nonzero vortex - dynamics - induced hall conductivity @xmath44 then follows naturally . the main result of this work is that in the bound pair dynamics description , we obtain nonvanishing hall conductivity @xmath44 and ac conductivity @xmath45 near the kt transition . one of the interesting features in the hall conductivity is that strong positive peak and sign changes in @xmath46 are observed at suitable frequency region above the transition temperature , as well as above @xmath9 in temperature domain at fixed frequencies . on the other hand , @xmath47 is shown to have similar features as in ahns s results . we note that the shapes of two length - dependent response functions @xmath48 and @xmath49 , which corresponds respectively to the longitudinal and transverse response of bound pairs with separation @xmath50 to external perturbation with frequency @xmath18 , play a determining role on the behavior of @xmath44 and @xmath45 . we also discuss the contribution of free vortex motion and the resulting total conductivity tensor . the paper is organized as follows : in section ii , we generalize ahns s vortex dynamics in the chiral @xmath0-wave context . to describe the vortex - antivortex bound pair dynamics , the above - mentioned response functions @xmath48 and @xmath49 are derived from the fokker - planck equation governing the pair motion . together with the free vortex contribution , we arrive at a matrix dielectric function @xmath51 which describes the total screening effect under time - dependent perturbation . in section iii , we investigate the frequency and temperature dependence of the conductivity tensor @xmath52 constructed from @xmath51 , treating the bound pair and the free vortex contribution separately . the behavior of total conductivity @xmath4 is also discussed . a summary and remark are given in section iv . finally , we discuss analytic expression of @xmath3 , @xmath53 , and @xmath54 in opposite limit of the convective ratio @xmath55 and @xmath56 in appendix a. to construct a matrix dielectric function , we consider a neutral sf film system resembling that employed in ahns s dynamical theory , with film thickness of order the superconducting coherence length and linear dimension @xmath57 ( @xmath58 ) along @xmath25 ( @xmath26 ) direction . @xmath57 is very large and @xmath58 is large but finite . vortex core motion relative to the local superfluid velocity leads to a magnus force @xmath59 @xcite @xmath60 in the above equation , @xmath61 and @xmath62 are the velocity of the @xmath63-th vortex core and the local superfluid flow excluding the diverging self - field of the @xmath63-th vortex respectively . @xmath64 is the vorticity of the @xmath63-th vortex . @xmath65 is the bare areal superfluid mass density , which is defined as the three - dimensional superfluid density integrated across the film thickness . @xmath66 is the mass of the constituting particle , which is equal to the mass of a cooper pair . the sf film is driven by a vibrating substrate . a vortex core moving relatively to the substrate experiences a vorticity - dependent drag force @xmath67 @xcite @xmath68 where @xmath69 is the moving substrate velocity . @xmath70 and @xmath71 denote the vorticity - dependent drag coefficients originating from interactions with the substrate and with thermally excited quasiparticles and collective modes . here @xmath72 ( @xmath73 ) @xmath74 ( @xmath75 ) for @xmath76 or @xmath77 ( @xmath78 ) for @xmath79 . these quantities have been obtained for a three - dimensional clean @xmath42-wave sf / sc with isotropic fermi surface , @xcite and are related to the relaxation time @xmath80 for the caroli - degennes - matricon mode @xcite in the vortex core . due to broken @xmath5 symmetry , the relaxation time @xmath81 for the mode in the @xmath39vortex and @xmath38vortex core , and thus the values of their drag coefficients , are different , resulting in the vorticity - dependent drag force @xmath67 on the vortex core . these drag coefficients for 2d clean sf / scs with cylindrical fermi surface can be inferred from the three - dimensional result easily . when two forces balance @xmath82 , the @xmath63-th vortex velocity is expressed by @xmath83 where @xmath84 here @xmath85 with @xmath86 being the boltzmann constant . @xmath87 are fluctuating gaussian noise sources incorporated to bring the vortices to equilibrium . their components satisfy @xmath88 . from ref . , the local superfluid velocity @xmath89 is related to the spatial average superfluid velocity @xmath90 and the positions of individual vortices @xmath91 by @xmath92 . here @xmath93 is the green function satisfying @xmath94 and boundary condition @xmath95 on the edges . the function @xmath96 is localized in a region around @xmath97 with radius of the order of coherence length ( we could say that the function @xmath96 is a delta function in the coarse - grained scale ) . far away from the edges , @xmath98 . the spatial average of @xmath99 turns out to be zero and thus @xmath100 represents the spatial average of @xmath89 . finally , the time evolution of @xmath100 obeys @xmath101 reflecting the fact that the average superfluid velocity in the @xmath25 direction changes by @xmath102 when a vortex with @xmath103 moves across a strip with width @xmath104 . in a charged system , if we follow kopnin s description , @xcite the driving force on the vortex due to a transport current @xmath105 is a lorentz force @xmath106/c $ ] where @xmath107 and @xmath108 is the electron charge . this force is balanced by the force from environment @xmath109 . if we set @xmath110 in eqs . ( [ fd ] ) and ( [ one_vortex ] ) , the results derived in a neutral system can be carried over to a charged system by the translation @xmath111 and @xmath112 . we consider the polarization of a test vortex - antivortex pair whose constituting vortices interact via a screened logarithmic interaction . the pair is under the influence of an infinitesimal oscillating external field @xmath113 . the langevin equation for their relative coordinate @xmath114 can be obtained by subtracting eq . ( [ one_vortex ] ) from each other for opposite vorticity @xmath115 here @xmath116 and @xmath117 . @xmath118 is one - half the dimensionless potential energy of the pair and the gaussian noise now satisfies @xmath119 . the potential energy is given by @xmath120 in the above equation , the first term on the right hand side describes the logarithmic interaction screened by the kosterlitz dielectric constant @xmath121 . @xmath122 in the second term is related to the energy required to create a pair with separation @xmath123 . in the last term , @xmath124 has dimension of velocity and acts as the perturbation . in the integration limit @xmath123 is a length scale related to the size of a vortex core , and @xmath50 is the pair separation . we can see in eq . ( [ pair_eom ] ) that in addition to the conventional diffusive terms depending on @xmath125 , a convective term proportional to @xmath126 also enters the dynamical equation . while the strength of the former is proportional to the _ average _ of @xmath127 , that of the latter is related to the _ difference _ of @xmath128 between opposite vorticity . we emphasize here that such a convective pair motion is one of the special features for a system with unequal opposite vortices , and is thus absent in an @xmath42-wave sf / sc since @xmath129 in that case . given this nonzero @xmath126 , the pair polarization is tilted away from the direction of the force field @xmath130 , and has both longitudinal and transverse components even without applied magnetic field . the fokker - planck equation corresponding to eq . ( [ pair_eom ] ) is given by @xmath131 where @xmath132 is the density of pairs per unit area of separation . we take the time - independent state @xmath133 to be @xmath134 where @xmath135 . now we follow the standard procedure , @xcite letting @xmath136 and keeping terms to first order in @xmath137 . in frequency space , we obtain @xmath138 \nonumber\\ & & + 2 \overline c \nabla \cdot \left [ \left ( \frac{m^\ast}{\hbar } \hat z \times \delta \boldsymbol e \gamma_0 \right ) - \delta \gamma \hat z \times \nabla u_0 \right ] + 2 \overline d \nabla^2 \delta \gamma.\end{aligned}\ ] ] we employ the expansion @xmath139 , where @xmath140 is the angle measured from @xmath141 to @xmath142 in anti - clockwise sense . only @xmath143 with @xmath144 and @xmath145 are coupled to the external field . we define a convective ratio @xmath146 , which measures the relative strength between convection and diffusion , and introduce an ansatz @xmath147 @xmath148 . together with an approximation @xcite @xmath149 ( which is valid near the transition ) and a change of variable @xmath150 , the equations of @xmath151 in the ansatz corresponding to angular momentum @xmath152 are given by @xmath153 we are then able to write down the change of distribution function @xmath154 in terms of @xmath155 explicitly @xmath156 , \label{d_gamma}\end{aligned}\ ] ] where @xmath157 , \quad\quad \mathcal{g}_\perp(r ) = -\frac{i}{2}\left [ g_+ ( r,\omega ) - g_- ( r,\omega ) \right ] , \label{g_g}\end{aligned}\ ] ] and @xmath158 . in eq . ( [ d_gamma ] ) , @xmath159 takes the role of pair - size - dependent response longitudinal to the driving field @xmath137 and @xmath160 is the transverse response function . in the limit of vanishing @xmath43 , @xmath159 reduces to the @xmath42-wave sf / sc result @xmath161 while @xmath160 becomes identically zero . in eq . ( [ g_g ] ) , we find that @xmath162 depend on two quantities @xmath163 which describe pair motion with angular momentum @xmath164 respectively . ( left column ) and @xmath160 ( right column ) are plotted as a function of @xmath165 for @xmath166 ( a ) @xmath167 , ( b ) @xmath168 , and ( c ) @xmath169 respectively . the real and imaginary part are indicated by blue dashed lines and red solid lines respectively . @xmath159 is qualitatively similar to the @xmath42-wave sf / sc result . @xmath160 is a new feature with a peak in the real part and a dip - recouping shape in the imaginary part around @xmath170.,scaledwidth=80.0% ] we describe @xmath162 first . in fig . [ mathcalg ] we plot @xmath159 and @xmath160 as a function of @xmath165 for fixed frequency @xmath171 for ( a ) @xmath172 , ( b ) @xmath173 , ( c ) @xmath174 . blue dashed lines and red solid lines represent their real and imaginary parts respectively . in the left column of fig . [ mathcalg ] , the longitudinal response function @xmath159 is qualitatively similar to the @xmath42-wave sf / sc result @xmath175 even for finite @xmath43 . it has a step - function - like real part and a delta - function - like imaginary part concentrating near @xmath170 ( @xmath170 marks the position where @xmath176 changes sign ; see appendix a ) . the response function @xmath177 is a new feature in this model . in the right column of fig . [ mathcalg ] , @xmath178 $ ] has a peak structure . this means that neither smaller nor larger pair gives response in transverse direction . only pairs with characteristic pair size @xmath179 can give rise to the hall effect . besides , @xmath180 $ ] shows negative dip shape for @xmath181 , and then recoups when @xmath182 . it turns out in later section that such a dip - recouping antisymmetric shape in @xmath176 about @xmath170 plays a determining role in many features of the hall conductivity . these features become more significant when @xmath43 increases . ( left column ) and @xmath183 ( right column ) corresponding to angular momentum @xmath152 motion are plotted as a function of @xmath165 for @xmath166 ( a ) @xmath167 , ( b ) @xmath168 , and ( c ) @xmath169 respectively . the real and imaginary part are indicated by blue dashed lines and red solid lines respectively . asymmetry between the two columns becomes more prominent when @xmath43 increases.,scaledwidth=80.0% ] in eq . ( [ g_g ] ) , we can see that @xmath184 depends on the average of @xmath163 , i.e. , the average response with @xmath185 and @xmath186 angular momentum , and @xmath187 is related to the _ difference _ between them . this means that any asymmetry between @xmath152 motion gives rise to nonzero transverse response . in fig . [ capitalg ] we plot @xmath163 as a function of @xmath188 using the same set of @xmath18 and @xmath43 as in fig . [ mathcalg ] . again , blue dashed lines and red solid lines represent their real and imaginary parts respectively . we observe that asymmetry between @xmath152 motion grows with increasing @xmath43 . for small @xmath43 in fig . [ capitalg](a ) , the motion described by @xmath189 and @xmath190 is only slightly asymmetric , and they look like the familiar curve @xmath191 . when @xmath43 increases in fig . [ capitalg](b ) and ( c ) , the asymmetry between the left and right column becomes more and more significant . it is worth noting that , when we subtract from each other the real part of the functions for opposite @xmath192 , we can obtain the dip - recouping shape of @xmath193 . equation ( [ eqt_for_g ] ) can be solved exactly to give the length - dependent response functions , but here we can employ approximate solution to @xmath194 in a manner similar to ref . by neglecting @xmath195 and @xmath196 in eq . ( [ eqt_for_g ] ) . then @xmath197 become @xmath198 where @xmath199 is a factor to fit the exact curve . it is selected to be @xmath200 when @xmath201 and is of order @xmath202 for a wide range of @xmath43 . from eqs . ( [ app_g1 ] ) and ( [ app_g2 ] ) we can see that for a small convective ratio @xmath203 , there are poles given by @xmath204 . this means that the pair size with which a pair gives a strong response differs from the standard result @xmath205 by a length of order @xmath206 for motion with @xmath152 , creating the asymmetry demonstrated in fig . [ capitalg ] . for a large convective ratio @xmath207 , such pole is removed for @xmath145 therefore , a broad and flat curve is expected [ fig . [ capitalg](c ) right column ] . in this subsection , we introduce a susceptibility matrix @xmath208 for the bound pair polarization @xmath209 whose components are defined as @xmath210 they are so defined that the real parts of @xmath211 and @xmath212 are positive when @xmath213 . using eqs . ( [ d_gamma ] ) and ( [ chib ] ) , we arrive at the expression @xmath214 the infinitesimal external field redistributes the pair polarization by an amount @xmath215 according to @xmath216 . the definition in eq . ( [ chib ] ) means the external field tilts the polarization in clockwise direction when @xmath213 . it should also be noted that the integration are performed from @xmath217 to @xmath15 where @xmath15 is a coherence length with behavior @xmath218 for @xmath12 and @xmath219 $ ] for @xmath14 . @xmath220 is a non - universal positive constant of order unity . from eq . ( [ chibb ] ) it is now clear that while the longitudinal polarization @xmath221 takes the role of the function @xmath222 in ahns s theory , the transverse polarization @xmath223 has no simple analogy in @xmath42-wave sf / scs and is specific to systems with finite convective ratios . integrating the two pair - size - dependent response functions with the weight function @xmath224 gives the susceptibility matrix elements from the bound pair contribution . as for the free vortex contribution , from eq . ( [ one_vortex ] ) we can obtain the equation for polarization @xmath225 for plasma of free vortices under the spatial averaged driving field @xmath226 . the total free vortex density is given by @xmath227 with @xmath228 being the total number of free vortices and @xmath229 being the area of the film . in frequency space , we have @xmath230.\end{aligned}\ ] ] if we define the susceptibility for free vortices as @xmath231 , we have @xmath232 where @xmath233 . on the other hand , the free vortex density is related to the coherence length by @xmath234 where @xmath235 is a positive constant of order unity . @xcite together with the bound pair contribution , the total dielectric function can be obtained as @xmath236 and its inverse reads @xmath237 where @xmath238 $ ] and @xmath239 $ ] . we are now in position to present our results on the hall conductivity and power dissipation due to this dielectric function . it was discussed that the conductivity tensor @xmath4 in a charged system was related to the inverse dielectric function @xmath240 discussed above . @xcite we can understand the relation by considering the total current under the influence of a driver coil electric field @xmath241 . @xcite the vortex - modified total current @xmath242 is related to the field by @xmath243 where @xmath244 is the sheet kinetic inductance . this relation is generalized to our model with a matrix dielectric function @xmath245 . if we use the sign convention that the normal state electron hall conductivity with magnetic field pointing in @xmath22 direction is positive , @xmath52 and @xmath246 is related by @xmath247 in particular , we may write down @xmath248 , @xmath249 , and @xmath250 . among these expressions , the first two are associated with the real and imaginary part of the hall conductivity , while the last one is related to the power dissipation @xmath251 because @xmath252 . in the following subsections , we investigate the frequency and temperature dependence of @xmath253 , @xmath254 , and @xmath255 respectively . as a function of ( a ) @xmath256 and ( b ) @xmath257 . the other parameters used are @xmath172 , @xmath258 and @xmath259 . in ( a ) , above the transition temperature , @xmath260 has peak structures and sign reversals . @xmath261 has a sharp increase when frequency increases . in ( b ) , @xmath260 shows peak structure and sign reversal above @xmath262 and @xmath261 has a peak . the features broaden and move to higher temperature when frequency increases.,scaledwidth=80.0% ] we consider bound pair contribution to the hall conductivity first by ignoring the terms @xmath263 and @xmath264 in @xmath240 . plots of the negative real part ( blue circles ) and imaginary part ( red squares ) of @xmath265 versus ( a ) @xmath256 and ( b ) @xmath266 are presented in fig . [ hall ] . here , @xmath267 is the scaled frequency , and @xmath257 is the reduced temperature . in this and the following figures , we use the renormalization group flow equations @xmath268 and @xmath269,@xcite with initial conditions @xmath270/(1+t)$ ] and @xmath258 related to the bare interaction strength @xmath11 and chemical potential @xmath271 respectively . at transition temperature , @xmath272 which is obtained numerically . here , it suffices to notice that @xmath273 is given by @xmath274 where @xmath275 is the renormalized interaction strength , and @xmath276 is related to the the number of bound pairs found with pair size @xmath277 . also , we use the convective ratio @xmath172 in the plots for illustration purpose . we first focus on the discussion of the frequency dependence of @xmath265 at some fixed temperatures in fig . [ hall](a ) . in the low temperature phase @xmath278 , both @xmath261 and @xmath260 are positive and increase steadily with frequency . when temperature increases to the high temperature phase @xmath279 , @xmath260 shows a positive - valued peak followed by sign reversal at higher frequency indicated by the red horizontal double - headed bar . meanwhile , @xmath261 increases sharply around the peak of @xmath260 . these features move to higher frequency side when temperature increases further ( @xmath280 ) . for temperature dependence at fixed frequencies in fig . [ hall](b ) , the sign reversal and peak structure in @xmath260 are also observed when temperature varies above @xmath9 . meanwhile , @xmath261 shows a simple peak structure across @xmath9 . when frequency increases , the structure broadens and moves to higher temperature region . sign anomaly in hall conductivity has been observed in various superconducting systems such as high - temperature scs , @xcite and it is known that single vortex upstream " motion or change of sign of charge carriers could give rise to such phenomenon.@xcite here in our model the sign change in @xmath260 stems from the vortex - antivortex pair unbinding process . for a vortex - antivortex pair moving downstream " with same speed , total transverse electric field is canceled and no hall effect can be observed . now that the constituting vortices are not equivalent to each other , they can respond differently under the influence of transport current , and a net transverse electric field follows . under transport current @xmath281 with convective ratio @xmath213 . the black solid arrow is the transport current @xmath281 . the blue and red solid arrow represent the pair polarization of pair with size @xmath181 and @xmath182 respectively . the dashed arrows show the electric field generated by pair motion in the respective cases . pairs with size @xmath181 contribute to positive @xmath260 and vice verse . the net @xmath282 determines the overall sign of @xmath260 . , scaledwidth=80.0% ] from the dip - recouping shape of @xmath283 $ ] in fig . [ mathcalg ] , the dip ( recouping ) region with pair size @xmath181 ( @xmath182 ) contributes to positive ( negative ) @xmath260 , since @xmath284/r$ ] . the function @xmath285 is proportional to the number of pairs present in a ring with radius @xmath50 and width @xmath286 . therefore , the sign of @xmath260 depends on whether there are more pairs with pair size smaller or larger than @xmath170 . an interesting point to mention is that , upstream vortex motion is not required for negative @xmath260 ; instead , it is the asymmetric pair motion in opposite angular directions in fig . [ capitalg ] that brings about the hall anomaly . we interpret such result to be that the direction of electric field @xmath282 generated by a vortex pair is also pair - size - dependent . this interpretation is illustrated in fig . [ polarization ] . the pair with @xmath181 polarizes in a such a way that @xmath282 points in the direction giving positive hall signal and vice verse . the final sign of @xmath260 depends on the net @xmath282 caused by all the pairs . versus @xmath188 for @xmath287 , @xmath288 , and @xmath289 . the vertical arrow marks @xmath170 , the position where @xmath290 changes sign . @xmath291 . at @xmath292 , @xmath293 is indicated by the red double - headed arrow . three representative situations concerning the sign issue of @xmath260 are @xmath294 increasing at @xmath170 , @xmath294 decreasing at @xmath170 , and @xmath295.,scaledwidth=80.0% ] the weight function @xmath294 , which describes the probability of a pair having pair size @xmath277 , is plotted in fig . [ y2 ] at different temperatures . the other parameters used are @xmath296 , @xmath172 , and @xmath258 . for @xmath287 , @xmath294 is slightly decreasing at @xmath170 , so the dip - recouping shape in @xmath283 $ ] almost cancels each other in the integral in eq . ( [ chibb ] ) , giving a weak positive @xmath260 . if the temperature increases , for example @xmath297 , @xmath294 becomes increasing at @xmath170 . more pairs are present in the recouping region and the sign change of @xmath260 thus occurs . if the temperature increases further to @xmath298 , the coherence length @xmath15 characterizing the maximum pair size becomes comparable to @xmath170 . as a result , only the dip part is integrated because the recouping part lies outside of the integration limit in eq . ( [ chibb ] ) . a strong positive - valued peak then replaces the sign reversal . in summary , temperature controls the pair size distribution , which then determines the sign of @xmath260 . since the positive slope of @xmath294 at @xmath179 only occurs at the high temperature phase , the claim that the sign reversal of @xmath260 is related to kt transition is justified . to close this subsection , we discuss the hall conductivity in the static limit . for low temperature phase @xmath278 , hall conductivity diverges in zero - frequency limit . log - log plots of @xmath260 and @xmath261 versus @xmath299 using linear fitting show that @xmath300 and @xmath301 with both @xmath302 and @xmath303 greater than zero but smaller than unity for the temperature range @xmath304 down to @xmath305 and frequency range @xmath306 to @xmath307 . this mean that @xmath308 decreases slower than @xmath18 when frequency approaches zero , and as a result @xmath309 diverges at @xmath310 . as a function of ( a ) @xmath256 and ( b ) @xmath257 . @xmath172 and @xmath258 . in ( a ) , @xmath311 increases with frequency when @xmath278 and is suppressed at small frequency when @xmath312 . in ( b ) , @xmath311 has peak structure around @xmath9 . the peak broadens and moves to higher temperature when frequency increases.,scaledwidth=80.0% ] in this subsection , we discuss the frequency and temperature dependence of @xmath3 . again , we consider bound pair contribution here . in fig . [ diss ] , plots of @xmath311 versus ( a ) @xmath256 and ( b ) @xmath266 are presented . the other parameters used are the same as those in fig . [ hall ] . in fig . [ diss](a ) , @xmath311 increases steadily with frequency at low temperature phase @xmath278 . when @xmath312 , it is suppressed at small frequency . in fig . [ diss](b ) , @xmath311 first increases with temperature , and is suppressed above @xmath9 , resulting in a peak structure . at higher frequency , the peak is broadened and shifts to higher temperature . these features can be understood under the standard ahns theory : above @xmath9 , the coherence length @xmath15 becomes finite , and the integration limits can not cover the peak structure in @xmath313 at large @xmath188 . thus , @xmath311 is suppressed at low frequency for @xmath312 in fig . [ diss](a ) , and above some temperatures in fig . [ diss](b ) . indeed , it is not surprising that the result behaves similarly to its @xmath42-wave counterpart when we notice that the shape of @xmath314 is qualitatively similar to that of @xmath315 even for finite @xmath43 . in zero - frequency limit in the low temperature phase , @xmath316 decreases slower than @xmath18 when frequency approaches zero and thus the @xmath317 diverges . this is found in log - log plots of @xmath311 versus @xmath18 that @xmath318 with @xmath319 but smaller than unity using linear fitting from @xmath304 to @xmath305 and @xmath320 to @xmath307 . however , we can not interpret it to be the divergence of power dissipation because the dissipation also depends on the magnitude of the electric field in the superconducting bulk which is supposed to be vanishingly small in the static limit . as a function of ( a ) @xmath256 and ( b ) @xmath266 . the red squares and the blue circles represent the real part and imaginary part of @xmath321 respectively . @xmath322 and @xmath323 for illustration . the other parameters used are the same as those in fig . [ hall].,scaledwidth=80.0% ] as a function of ( a ) @xmath256 and ( b ) @xmath266 . the other parameters used are the same as those in fig . [ totalhall].,scaledwidth=80.0% ] having discussed the result due to bound pair dynamics , we study the total contribution from both bound pair and free vortex dynamics in this subsection . in order to discern the contribution of bound pair and free vortex dynamics to the total hall conductivity and power dissipation , we plot the total conductivity tensor @xmath321 and @xmath324 in figs . [ totalhall ] and [ totaldiss ] as a function of ( a ) @xmath256 and ( b ) @xmath266 , as well as the @xmath325 , @xmath254 , and @xmath253 due purely to the free vortex motion in fig . [ figfree ] . we can compare these figures with those which take only the bound pair dynamics into account ( figs . [ hall ] and [ diss ] ) . for the sake of illustration , we use @xmath322 and @xmath323 for the weight of free vortex contribution . for the frequency dependence in figs . [ totalhall ] and [ totaldiss ] at temperature @xmath279 , we can see free vortex signal emerging at small frequency region . if the temperature increases further at @xmath280 , the free vortex signal can merge with and outweigh the bound pair signal [ compared with figs . [ hall ] and [ figfree](a ) @xmath280 ] . for @xmath278 ( not shown ) , since there is no free vortex contribution , we expect the curves are the same as those in low temperature phase in figs . [ hall ] and [ diss](a ) . as for the temperature dependence in figs . [ totalhall ] and [ totaldiss](b ) , we can see for @xmath320 the free vortex signal appears at temperature very close to the bound pair signal and they almost merge into each other . when the frequency increases ( @xmath327 ) , this extra strong signal moves to higher temperature and becomes distinguishable from the bound pair signal . the magnitude of the free vortex signal can be seen in fig . [ figfree ] . as discussed in the previous two subsections , @xmath328 , @xmath329 , and @xmath330 diverge when @xmath310 . however , it is only true in low temperature phase . in the case where @xmath312 , free vortex contribution dominates in the static limit @xmath331 . with such condition , we have @xmath332 which is purely imaginary . from eq . ( [ free ] ) , the diagonal part of @xmath333 and the off - diagonal part of @xmath334 are given by @xmath335 $ ] and @xmath336 $ ] respectively , and therefore both of them are proportional to @xmath337 . this shows that @xmath338 is purely real , constant in frequency , and diverges when @xmath339 . as a result , in the high temperature phase , both the hall conductivity and dissipation are protected from divergence in the zero - frequency limit but have strong signal when approaching @xmath9 from above . , @xmath254 , and @xmath253 as a function of ( a ) @xmath256 and ( b ) @xmath266 . the green triangles represent @xmath325 . the blue circles and the red squares represent the negative real part and imaginary part of @xmath265 respectively . the insets show the full plot range of @xmath325 . the other parameters used are the same as those in fig . [ totalhall].,scaledwidth=80.0% ] finally , fig . [ figfree ] shows @xmath325 , @xmath254 , and @xmath253 as a function of ( a ) @xmath256 and ( b ) @xmath266 including only the free vortex but not the bound pair contribution . the frequency and temperature range are chosen to match those in figs . [ totalhall ] and [ totaldiss ] . by eq . ( [ free ] ) , we see that important features in the figure appear around the crossover from @xmath340 to the opposite limit @xmath341 . in fig . [ figfree](a ) on the right panel , when observing from small frequency region @xmath340 , we see that all three quantities increase with frequency . when frequency increases further , @xmath253 changes sign and then all the quantities decrease in magnitude with frequency , which means that it starts to be inefficient for the free vortex to respond to the perturbation when frequency is high . in fig . [ figfree](b ) , similar features can be observed in the temperature domain . besides , we can see that the main features in the curves emerge at a temperature closer to @xmath9 for small frequency ( left panel ) than for high frequency ( right panel ) . although the sign change in @xmath253 can also be observed in free vortex picture , it does not arise from the response function @xmath342 mentioned before . instead , it is attributed to the pole structure @xmath343 . in this work , we have generalized ahns s vortex dynamics in the chiral @xmath0-wave superconducting state . the behavior of the conductivity tensor near kt transition is investigated . we show that the hall conductivity can be nonzero arising from the vortex - antivortex pair unbinding process or from free vortex motion even in the absence of magnetic field . power dissipation is also predicted in this dynamical picture . in low temperature phase , both the @xmath44 and @xmath344 diverge in the static limit . but in high temperature phase , the contribution from free vortex motion gives finite results in the static limit . we can distinguish the bound pair and free vortex contribution to the total conductivity tensor by comparing the figures from different contribution . sign reversal and strong positive peak in @xmath46 are illustrated in some suitable frequency close to transition , as well as above the transition temperature at fixed frequencies . on the other hand , the @xmath47 behaves in a fashion similar to that of the @xmath42-wave case . in bound pair description , the induced hall conductivity is strongly influenced by the dip - recouping shape of transverse response function @xmath290 , which stems from the asymmetric angular response of pairs to the driving field . the convective term in the fokker - planck equation , which originates from the broken @xmath5 symmetry nature of a chiral @xmath0-wave sc , is essential to this asymmetry . all these results depend solely on the convective ratio @xmath43 , and are valid even without applied magnetic field . in our work , temperature dependence of the drag coefficients is omitted . theoretical calculation of the drag coefficients has been performed microscopically for scs @xcite and for sf @xmath19he ( refs . ) at temperatures considerably lower than the superconducting transition temperature by the use of quasiclassical theory . generalization of such calculation in chiral @xmath0-wave pairing state around @xmath9 will be useful to estimate the strength of the convective ratio . indeed , if we use the formulas of drag coefficients which only take into account the caroli - degennes - matricon mode at very low temperature @xmath345 $ ] and @xmath346 $ ] , @xcite we can arrive at the result @xmath347 and @xmath348 , which results in @xmath201 . however , we believe that the effect of temperature and delocalized excitations can contribute to the convective ratio . in this appendix , we apply certain approximation to obtain analytic expression of @xmath311 , @xmath260 , and @xmath261 in the bound pair dynamics picture . we first obtain analytic expression for susceptibility matrix elements @xmath211 and @xmath212 by employing an approximate solution @xmath349 and taking @xmath350 to be a constant function of @xmath50 . we rewrite eq . ( [ chibb ] ) in the form @xmath351 , \label{chi_para0 } \\ \chi_\text{b}^{\perp } & = & -i\int_{a_0}^{\xi_+ } dr \frac{d \widetilde \epsilon}{dr } \left [ \left ( \frac { g_+ - g_-}{2 } \right ) + ix_0 \left ( \frac { g_+ + g_-}{2 } \right ) \right ] , \label{chi_perp0}\end{aligned}\ ] ] and further approximate the two functions appearing in the above integrands by @xmath352 \nonumber\\ & & + i \left[r i_2 \delta(r - r_2 ) - x_0 r i_1 \delta(r - r_1)\right ] , \\ \frac { g_+ - g_-}{2 } & \approx & r i_1 \delta(r - r_1 ) + i r i_3 \delta(r - r_3 ) \nonumber\\ & & -ix_0 \left [ \left ( \frac { 1 } { 1+x_0 ^ 2}\right ) \theta ( r_4 - r ) + \text{sgn}(x_0 ) r i_1 \delta(r - r_1 ) \right],\end{aligned}\ ] ] where @xmath353 ^ 2 } - \frac{1 } { 1 + [ \omega r^2 / ( \lambda \overline d)+ x_0]^2 } \right ) \nonumber\\ & = & \frac { x_0 \pi } { 4 + 4 x_0 ^ 2 } , \\ i_2 & \equiv & \frac{1}{2 } \int_0^\infty dr \frac{1}{r } \left ( \frac{\omega r^2 / ( \lambda \overline d ) } { 1 + [ \omega r^2 / ( \lambda \overline d)- x_0]^2 } + \frac{\omega r^2 / ( \lambda \overline d ) } { 1 + [ \omega r^2 / ( \lambda \overline d)+ x_0]^2 } \right ) \nonumber\\ & = & \frac { \pi } { 4 } , \\ i_3 & \equiv & \frac{1}{2 } \int_0^\infty dr \frac{1}{r } \left ( \frac{\omega r^2 / ( \lambda \overline d ) } { 1 + [ \omega r^2 / ( \lambda \overline d)- x_0]^2 } - \frac{\omega r^2 / ( \lambda \overline d ) } { 1 + [ \omega r^2 / ( \lambda \overline d)+ x_0]^2 } \right ) \nonumber\\ & = & \frac{1}{4 } \left [ \cot^{-1 } \frac{2 x_0}{-1+x_0 ^ 2 } + \frac{\pi}{2 } \text{sgn}(x_0 ) \right].\end{aligned}\ ] ] in the above expression , @xmath354 , @xmath355 , @xmath356 , and @xmath357 are some characteristic lengths defined as follows : @xmath354 , @xmath355 , and @xmath356 are the peak position of the integrand of @xmath358 , @xmath359 , and @xmath360 respectively . @xmath357 is chosen such that @xmath361 ^ 2 \ } = 1/ [ 2(1 + x_0 ^ 2 ) ] $ ] . they are given respectively by @xmath362 , @xmath363 , and @xmath364 . one more characteristic length worth mentioning is the pair size @xmath365 at which @xmath290 changes its sign . within the approximation described above , we find that @xmath366 . then we can obtain the quantity @xmath367 mentioned in the main text . finally , using @xmath368 , we obtain analytic expression for the components of the susceptibility matrix for @xmath369 @xmath370 where some shorthand notations are introduced : @xmath371 and @xmath372 . from the expression above , we notice that @xmath373 ( @xmath374 ) is ( anti- ) symmetric in @xmath43 as it should be . we can also identify the contribution from the dip and recouping part of @xmath290 to be @xmath375 and @xmath376 in the imaginary part of eq . ( [ chi_perp ] ) . [ [ analytic - expression - of - im - epsilon_parallel-1-imepsilon_perp-1-and - re - epsilon_perp-1 ] ] analytic expression of @xmath311 , @xmath260 , and @xmath261 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ in this subsection , @xmath311 , @xmath260 , and @xmath261 for nonzero @xmath43 are calculated in two limiting cases @xmath377 and @xmath378 . for @xmath377 , we take @xmath379 , @xmath380 , and @xmath381 at the opposite limit @xmath382 , we take @xmath383 , @xmath384 , and @xmath385 besides , we neglect the @xmath43 dependence of @xmath386 and @xmath387 in the expansion . such approximation is valid for @xmath388 and all temperature near @xmath9 because @xmath389 is almost independent of @xmath43 . it is also valid for @xmath382 in low temperature phase because the renormalization almost go to completion at large @xmath43 and @xmath387 can be treated as independent of @xmath390 . but it is not valid for @xmath382 in high temperature phase for two reasons : first , @xmath387 is increasing at large @xmath43 and second , @xmath389 are very likely to be greater than @xmath15 so that the integrations in eqs . ( [ chi_para0 ] ) and ( [ chi_perp0 ] ) give the trivial results @xmath391 and @xmath392 . expanding using small @xmath43 , the imaginary part of @xmath393 is approximately given by @xmath394 where we have used @xmath395 in the last step . the presence of the second term acts as a suppression of dissipation when @xmath43 is switched on . if we neglect the correction term proportional to @xmath396 , the expression reduces to the @xmath42-wave sf / scs result . in the opposite limit @xmath378 , the imaginary part of @xmath393 to the leading order in the smallness of @xmath397 is given by @xmath398 assuming @xmath399 . surprisingly , the leading order of @xmath311 at large @xmath43 has the same form as its small @xmath43 counterpart . the only difference is that the characteristic length is changed from @xmath400 to @xmath401 . this can be understood if we recall that the shape of @xmath402 has no qualitative change for finite @xmath43 . following similar procedure we can obtain analytic expression for @xmath403 and @xmath404 . for small @xmath43 , we obtain @xmath403 to the first order in @xmath43 @xmath405 } { ( \widetilde \epsilon_4 ^ 2 + k_0 ^ 2 \pi^8 y_4 ^ 4)^2 } \nonumber\\ & \approx & x_0 \widetilde \epsilon_4^{-2 } k_0 \pi^4 \left [ \left(y_1 ^ 2 - y_4 ^ 2\right ) + 4 \widetilde \epsilon_4^{-1 } k_0 \pi^3 y_4 ^ 4 \right].\end{aligned}\ ] ] we have used the condition @xmath406 in the last line . from the expression , the behavior depends on whether @xmath407 or @xmath408 dominates . from numerical results , @xmath407 dominates at a wide range of temperature in the low temperature phase . only when approaching the transition temperature from below and away from the vicinity @xmath409 , the @xmath410 becomes more significant . on the other hand , @xmath411 we have used @xmath412 in the second line . for large @xmath43 , we assume @xmath413 and obtain to the leading order @xmath414 and @xmath415 it has to be mentioned that , although we have used the condition @xmath406 extensively , it is only valid in the temperature range @xmath416 . above such temperature , numerical results show that @xmath417 can be greater than @xmath386 at small frequency region .

we discuss hall effect and power dissipation in chiral @xmath0-wave superconductors near kosterlitz - thouless transition in the absence of applied magnetic field . in bound pair dynamics picture , nonzero hall conductivity emerges when vortex - antivortex bound pair polarization has a component transverse to the direction of external perturbation . such effect arises from the broken time reversal symmetry nature of a chiral @xmath0-wave superconducting state and does not require an applied magnetic field . a frequency - dependent matrix dielectric function @xmath1 is derived to describe the screening effect due to the pair polarization . quantities related to the hall conductivity and power dissipation , denoted as @xmath2 and @xmath3 , are investigated in frequency and temperature domain . the imaginary part of the former can show peak structure and sign reversal as a function of frequency close to transition temperature , as well as in the temperature domain at various fixed frequencies . the latter shows peak structure near transition temperature . these features are attributed to pair - size - dependent longitudinal and transverse response function of bound pairs . consequences due to free vortex dynamics and the resulting total conductivity tensor @xmath4 are also discussed .

character varieties of @xmath1-manifold groups provide a useful tool in understanding the geometric structures of manifolds and notably the presence of essential surfaces . in this paper we wish to investigate @xmath2-character varieties of symmetric hyperbolic knots in order to pinpoint specific behaviours related to the presence of free or periodic symmetries . we will be mostly concerned with symmetries of odd prime order and we will concentrate our attention to the subvariety of the character variety which is invariant by the action of the symmetry ( see section [ s : invariantch ] for a precise definition of this action and of the invariant subvariety ) . as already observed in @xcite , the excellent component of the character variety containing the character of the holonomy representation is invariant by the symmetry , since the symmetry can be chosen to act as a hyperbolic isometry of the complement of the knot . hilden , lozano , and montesinos also observed that the invariant subvariety of a hyperbolic symmetric ( more specifically , periodic ) knot can be sometimes easier to determine than the whole variety . this follows from the fact that the invariant subvariety can be computed using the character variety of a two - component hyperbolic link . such link is obtained as the quotient of the knot and the axis of its periodic symmetry by the action of the symmetry itself . indeed , the link is sometimes much simpler " than the original knot , in the sense that its fundamental group has a smaller number of generators and relations , making the computation of its character variety feasible . this is , for instance , the case when the quotient link is a @xmath3-bridge link : hilden , lozano , and montesinos studied precisely this situation and were able to recover a defining equation for the excellent components of several periodic knots up to ten crossings . in what follows we will be interested in the structure of the invariant subvariety itself and we will consider not only knots admitting periodic symmetries but also free symmetries . our main result shows that the invariant subvariety has in general a different behaviour according to whether the knot admits a free or periodic symmetry . [ thm : main ] if @xmath4 has a periodic symmetry of prime order @xmath5 , then @xmath6 contains at least @xmath7 components that are curves and that are invariant by the symmetry . on the other hand , for each prime @xmath5 , there is a knot @xmath8 with a free symmetry of order @xmath9 such that the number of components of the invariant character variety of @xmath8 is bounded , independently of @xmath9 . the main observation here is that the invariant subvariety for a hyperbolic symmetric knot , or more precisely the zariski - open set of its irreducible characters , can be seen as a subvariety of the character variety of a well - chosen two - component hyperbolic link , even when the symmetry is free . to make the second part of our result more concrete , in section [ s : examples ] we study an infinite family of examples all arising from the two - component @xmath3-bridge link @xmath10 in rolfsen s notation ( with @xmath3-bridge invariant @xmath11 ) . our construction provides infinitely many knots with free symmetries such that the number of irreducible components of the invariant subvarieties of the knots is universally bounded . the invariant subvarieties of periodic knots over fields of positive characteristic exhibit a peculiar behaviour . it is well - known that for almost all odd primes @xmath9 the character variety of a finitely presented group resembles the character variety over @xmath12 . for a finite set of primes , though , the character variety over @xmath9 may differ from the one over @xmath13 , in the sense that there may be jumps " either in the dimension of its irreducible components or in their number . in this case we say that _ the variety ramifies at @xmath9_. the character varieties of the knots studied in @xcite provide the first examples in which the dimension of a well - defined subvariety of the character variety is larger for certain primes . here we give an infinite family of periodic knots for which the invariant character variety ramifies at @xmath9 , where @xmath9 is the order of the period . in this case , the ramification means that the number of @xmath14-dimensional components of the invariant subvariety decreases in characteristic @xmath9 . this gives some more insight in the relationship between the geometry of a knot and the algebra of its character variety , namely the primes that ramify . the paper is organised as follows : section [ s : quotientlink ] is purely topological and describes how one can construct any symmetric knot starting from a well - chosen two - component link . section [ s : chvar ] provides basic facts on character varieties and establishes the setting in which we will work . in section [ s : invariantch ] we introduce and study invariant character varieties of symmetric knots . the first part of theorem [ thm : main ] on periodic knots is proved in section [ s : periodic ] while in section [ s : free ] we study properties of invariant character varieties of knots with free symmetries . the proof of theorem [ thm : main ] is achieved in section [ s : examples ] , where an infinite family of free periodic knots with the desired properties is constructed . finally , in section [ s : modp ] we describe how the character varieties of knots with period @xmath9 may ramify @xmath0 . let @xmath4 be a knot in @xmath15 and let @xmath16 be a finite order diffeomorphism of the pair which preserves the orientation of @xmath15 . if @xmath17 acts freely we say that @xmath17 is a _ free symmetry of @xmath4_. if @xmath17 has a global fixed point then , according to the positive solution to smith s conjecture @xcite , the fixed - point set of @xmath17 is an unknotted circle and two situations can arise : either the fixed - point set of @xmath17 is disjoint from @xmath4 , and we say that @xmath17 is a _ periodic symmetry of @xmath4 _ , or it is not . in the latter case @xmath17 has order @xmath3 , its fixed - point set meets @xmath4 in two points , and @xmath17 is called a _ strong inversion of @xmath4_. in all other cases @xmath17 is called a _ semi - periodic symmetry of @xmath4_. note that if the order of @xmath17 is an odd prime , then @xmath17 can only be a _ free _ or _ periodic symmetry _ of @xmath4 . we start by recalling some well - known facts and a construction that will be central in the paper . let @xmath18 be a hyperbolic two - component link in the @xmath1-sphere such that @xmath19 is the trivial knot . let @xmath20 be an integer and assume that @xmath21 and the linking number of @xmath19 and @xmath22 are coprime . we can consider the @xmath21-fold cyclic cover @xmath23 of the solid torus @xmath24 which is the exterior of @xmath19 and contains @xmath22 . the lift of @xmath22 in @xmath25 is a ( connected ) simple closed curve @xmath26 . let @xmath27 be a meridian - longitude system for @xmath19 on @xmath28 and let @xmath29 be its lift on @xmath30 . the slopes @xmath31 , for @xmath32 , on @xmath30 are equivariant by the action of the cyclic group @xmath33 of deck transformations and the manifold @xmath34 obtained after dehn filling along @xmath35 is @xmath15 . the action of the group of deck transformations @xmath33 on @xmath25 extends to an action on @xmath34 which is free if @xmath36 is prime with @xmath21 and has a circle of fixed points if @xmath37 . for all other values of @xmath38 , the action is semi - periodic , that is a proper subgroup of @xmath33 acts with a circle of fixed points . for a fixed @xmath38 , the image of @xmath26 in @xmath34 is a knot that we will denote by @xmath4 admitting a periodic or free symmetry of order @xmath21 according to whether @xmath37 or prime with @xmath21 . for @xmath21 large enough , the resulting knot @xmath4 is hyperbolic because of thurston s hyperbolic dehn surgery theorem @xcite , e.g. ( * ? ? ? [ r : surgery ] of course , the above construction can be carried out for arbitrary integer values of @xmath38 . however , it is not restrictive to require the value of @xmath38 to be @xmath39 and @xmath40 . indeed , assume that @xmath41 where @xmath42 . the knot @xmath4 resulting from @xmath43 surgery along @xmath25 coincides with the knot @xmath44 obtained the same manner but starting from a different link @xmath45 and choosing @xmath46 as dehn filling slope . the link @xmath45 is obtained from @xmath47 by dehn surgery of slope @xmath48 along @xmath19 . the following proposition shows that periodic and free - symmetric knots can always be obtained this way . [ p : quotientlink ] let @xmath4 be a hyperbolic knot admitting a free or periodic symmetry of order @xmath21 . then there exist a two - component hyperbolic link @xmath18 with @xmath19 the trivial knot , and an integer @xmath49 such that the knot @xmath4 can be obtained by the above construction . the statement is obvious if the symmetry is periodic : in this case the link @xmath47 consists of the image @xmath19 of the axis of the symmetry and the image @xmath22 of the knot @xmath4 in the quotient of @xmath15 by the action of the symmetry . hyperbolicity of the link is a straightforward consequence of the hyperbolicity of @xmath4 and the orbifold theorem . if the symmetry is free , some extra work is necessary . the quotient of @xmath15 by the action of the free symmetry is a lens space containing a hyperbolic knot @xmath22 , image of @xmath4 . consider the cores of the two solid tori of a genus-@xmath14 heegaard splitting for the lens space induced by an invariant genus-@xmath14 splitting of @xmath15 . up to small isotopy one can assume that @xmath22 misses one of them , say @xmath50 . note that the free homotopy class of @xmath50 is non trivial both in the lens space and in the complement of @xmath22 . observe , moreover , that the exterior of @xmath50 is a solid torus . let @xmath51 denote the lift of @xmath50 . if @xmath52 is a hyperbolic link , then we are done by taking @xmath53 . otherwise we will modify the choice of @xmath54 . first of all , note that the link @xmath52 is not split . this is a consequence of the equivariant sphere theorem and the fact that @xmath54 is invariant , hence @xmath55 is irreducible and boundary irreducible . in addition @xmath56 is not seifert fibered , because a dehn filling on @xmath57 yields @xmath58 , which is hyperbolic . thus the only obstruction to hyperbolicity is that @xmath59 could be toroidal . assume that its jsj - decomposition is nontrivial and let @xmath60 be the piece of this splitting that is closest to @xmath4 . in particular @xmath60 is invariant by the action of the symmetry . the boundary of @xmath60 consists of @xmath61 , some tori @xmath62 , @xmath63 , and possibly a torus @xmath64 that separates @xmath60 from @xmath57 . we shall modify @xmath57 so that @xmath65 and @xmath37 , which will yield hyperbolicity . by hyperbolicity of @xmath4 , for @xmath66 , each @xmath67 either bounds a solid torus in @xmath58 or it is contained in a ball in @xmath58 . notice that @xmath64 must bound a solid torus in @xmath58 , because @xmath57 is not contained in a ball else the link @xmath52 would be split . in addition , none of the @xmath68 can bound a solid torus in @xmath58 , by nontriviality of the jsj - decomposition . first we modify @xmath57 so that @xmath69 . let @xmath25 be the solid torus bounded by @xmath64 . then @xmath70 and @xmath25 must be equivariant . in addition @xmath25 is not knotted , because @xmath57 is the trivial knot but also a satellite with companion @xmath71 . then the modification consists in replacing @xmath54 by the core of @xmath25 . this makes @xmath72 boundary parallel , and hence inessential . finally , we get rid of the tori @xmath68 . let @xmath73 denote the @xmath1-ball containing @xmath67 , for @xmath74 . on each ball there is a proper arc @xmath75 such that @xmath76 is a knot exterior with boundary ( parallel to ) @xmath77 . replace equivariantly each @xmath78 by a solid torus . this does not change @xmath4 , because the balls @xmath79 which are disjoint from @xmath4 are replaced again by balls . on the other hand , this may change @xmath54 to @xmath80 , but since every knot exterior has a degree - one map onto the solid torus , we find a degree - one map from @xmath81 onto @xmath82 , and since @xmath54 is unknotted , so is @xmath80 . note that for a given @xmath4 the choice of @xmath47 is not unique . indeed , links are not determined by their complements , and there are infinitely many slopes on the boundary of a solid torus such that performing dehn filling along them gives the @xmath1-sphere ( see also remark [ r : surgery ] ) . note that if @xmath4 admits a semi - free symmetry , then either all powers of the symmetry that act as periods have the same fixed - point set or the union of their fixed - point sets consists of two circles forming a hopf link . in the first situation a hyperbolic link @xmath47 can be constructed as in the case of periodic knots . in the second situation , one can construct @xmath47 by choosing one of the two components of the hopf link , but @xmath47 will not be hyperbolic in general . since we only consider symmetries of odd prime order in the following , we are not going to analyse this situation further . let @xmath83 be a finitely presented group . given a representation @xmath84 , its character is the map @xmath85 defined by @xmath86 , @xmath87 . the set of all characters is denoted by @xmath88 . given an element @xmath89 , we define the map @xmath90 [ proposition : x(g ) ] the set of characters @xmath88 is an affine algebraic set defined over @xmath91 , which embeds in @xmath92 with coordinate functions @xmath93 for some @xmath94 . the affine algebraic set @xmath88 is called the _ character variety _ of @xmath83 : it can be interpreted as the algebraic quotient of the variety of representations of @xmath83 by the conjugacy action of @xmath95 . note that the set @xmath96 in the above proposition can be chosen to contain a generating set of @xmath83 . for @xmath83 the fundamental group of a knot exterior , we will then assume that it always contains a representative of the meridian . a careful analysis of the arguments in @xcite shows that proposition [ proposition : x(g ) ] still holds if @xmath97 is replaced by any algebraically closed field , provided that its characteristic is different from @xmath3 . let @xmath98 denote the field with @xmath9 elements and @xmath99 its algebraic closure . we have : [ proposition : x(g)fp ] let @xmath100 be an odd prime number . the set of characters @xmath101 associated to representations of @xmath83 over the field @xmath99 is an algebraic set which embeds in @xmath102 with the same coordinate functions @xmath93 seen in proposition [ proposition : x(g ) ] . moreover , @xmath101 is defined by the reductions mod @xmath9 of the polynomials over @xmath91 which define @xmath103 . let @xmath104 be an algebraically closed field of characteristic different from @xmath3 . a representation @xmath105 of @xmath83 in @xmath106 is called _ reducible _ if there is a @xmath14-dimensional subspace of @xmath107 that is @xmath108-invariant ; otherwise @xmath105 is called _ irreducible_. the character of a representation @xmath105 is called _ reducible _ ( respectively _ irreducible _ ) if so is @xmath105 . the set of reducible characters coincides with the set of characters of abelian representations . such set is zariski closed and moreover is a union of irreducible components of @xmath88 that we will denote @xmath109 @xcite . assume now that @xmath83 is the fundamental group of a link in the @xmath1-sphere with @xmath110 components . in this case , @xmath111 is an @xmath110-dimensional variety that coincides with the character variety of @xmath112 , i.e. the homology of the link . in the case where @xmath113 , that is the link is a knot , @xmath111 is a line parametrised by the trace of the meridian . when @xmath114 , that is the link has two components , @xmath111 is parametrised by the traces @xmath115 of the two meridians and that , @xmath116 , of their product subject to the equation @xmath117 . the subvariety of abelian characters is well - understood for the groups that we will be considering . hence , in the rest of the paper , we will only consider the irreducible components of @xmath88 that are not contained in the subvariety of abelian characters . [ notation ] we will denote by @xmath118 the zariski closed set which is the union of of the irreducible components of @xmath88 that are not contained in the subvariety of abelian characters . if @xmath83 is the fundamental group of a manifold or orbifold @xmath60 we will write for short @xmath119 instead of @xmath118 . similarly if @xmath83 is the fundamental group of the exterior of a link @xmath47 we shall write @xmath120 instead of @xmath118 . notice that if @xmath83 is the fundamental group of a finite volume hyperbolic manifold then @xmath118 is non empty for it contains the character of the hyperbolic holonomy . assume now that @xmath121 is in @xmath122 . the automorphism @xmath121 induces an action on both @xmath88 and @xmath118 defined by @xmath123 . this action only depends on the class of @xmath121 in @xmath124 since traces are invariant by conjugacy . moreover , the action on the character varieties is realised by an algebraic morphism defined over @xmath125 . it follows readily that the set of fixed points of the action is zariski closed and itself defined over @xmath125 . as a consequence , the defining relations of the variety of characters that are fixed by the action considered over a field of characteristic @xmath9 , an odd prime number , are just the reduction @xmath0 of the given equations with integral coefficients . in this section we define and study the invariant subvariety of @xmath4 , where @xmath4 is a hyperbolic knot admitting a free or periodic symmetry of order an odd prime @xmath9 . let @xmath17 denote the symmetry of @xmath4 of order @xmath9 and let @xmath126 be the associated link as defined section [ s : quotientlink ] . denote by @xmath127 the space of orbits of the action of @xmath17 on the exterior @xmath58 of the knot @xmath4 . recall that @xmath127 is obtained by a ( possibly orbifold ) dehn filling on the component @xmath19 of the link @xmath47 . we have @xmath128 which splits if and only if @xmath17 is periodic . note that if @xmath17 is free then the quotient group @xmath129 can also be seen as the fundamental group of the lens space quotient . in any case , we see that @xmath17 defines an element @xmath130 of the outer automorphism group of @xmath131 . remark now that , since @xmath127 is obtained by dehn filling a component of @xmath47 , the exterior @xmath132 of the link @xmath47 is naturally embedded into @xmath127 . let @xmath133 be an element of @xmath134 corresponding to the image of a meridian of @xmath19 via this natural inclusion : it maps to a generator of @xmath129 . let @xmath135 be the automorphism of @xmath131 induced by conjugacy by @xmath133 . note that @xmath121 is a representative of @xmath130 . thus the symmetry @xmath17 induces an action on the character variety @xmath6 of the exterior of @xmath4 as defined in the previous section . we have seen that the fixed - point set of this action is an algebraic subvariety of @xmath6 . we will denote by @xmath136 the union of its irreducible components that are not contained in @xmath137 . note that @xmath136 is non empty for the character of the holonomy is fixed by the action . remark also that each irreducible component of @xmath136 contains at least one irreducible character by definition . indeed , each irreducible component of @xmath136 contains a whole zariski - open set of irreducible characters . we shall call @xmath136 _ the invariant subvariety of @xmath4_. let us now consider how the different character varieties of @xmath4 and @xmath47 are related . it is straightforward to see that the character variety @xmath138 of the quotient of the exterior @xmath58 of @xmath4 by the action of the symmetry injects into the character variety @xmath120 of the exterior of @xmath47 . indeed the ( orbifold ) fundamental group of @xmath127 is a quotient of the fundamental group of @xmath47 , induced by the dehn filling along the @xmath19 component of @xmath47 . on the other hand , there is a natural map from @xmath138 to the invariant submanifold @xmath136 of @xmath4 , induced by restriction in the short exact sequence above . assume now that @xmath139 is a character in @xmath136 associated to an irreducible representation @xmath105 of @xmath4 . we will show that @xmath105 extends in a unique way to a ( necessarily irreducible ) representation of @xmath127 giving a character in @xmath138 ( observe that here we only use that @xmath9 is odd ) . this proves that the above natural map is one - to - one and onto when restricted to the zariski - open set of irreducible characters . note that if @xmath105 is a representation of @xmath131 that extends to a representation of @xmath134 then , necessarily , its character must be fixed by the symmetry @xmath17 , for the action of @xmath133 on @xmath131 is by conjugacy and can not change the character of a representation . the idea is to extend @xmath105 to @xmath134 by defining @xmath140 in such a way that the action of @xmath133 by conjugacy on the normal subgroup @xmath131 coincides with the action of the automorphism @xmath121 . we know that @xmath141=[\rho\circ f]$ ] . since @xmath105 is irreducible , @xmath142 acts transitively on the fibre of @xmath139 so that there exists an element @xmath143 such that @xmath144 @xcite . the element @xmath60 is well - defined , up to multiplication times @xmath145 , i.e. up to an element in the centre of @xmath142 . the fact that @xmath17 has odd order implies that there is a unique way to choose the sign and so that @xmath146 is well - defined . note that in some instances @xmath140 can be the identity . we have thus proved the following fact . [ prop : invsubvar ] let @xmath4 be a hyperbolic knot admitting a symmetry @xmath17 of prime odd order . the restriction map from the @xmath17-invariant subvariety of @xmath4 to the character variety of @xmath127 induces a bijection between the zariski - open sets consisting of their irreducible characters . proposition [ prop : invsubvar ] holds more generally for hyperbolic knots admitting either a free or a periodic symmetry of odd order and for character varieties over fields of positive odd characteristic . let @xmath4 be a hyperbolic knot admitting a periodic symmetry @xmath17 of odd prime order @xmath9 . let @xmath18 be the associated quotient link . denote by @xmath147 the coordinate of the variety @xmath120 corresponding to the trace of @xmath133 . proposition [ prop : invsubvar ] implies at once that @xmath136 is birationally equivalent to a subvariety of @xmath148 , where @xmath149 and @xmath150 . note that since @xmath9 is odd , the set @xmath151 equals @xmath152 . in particular this includes a lift to @xmath153 of the holonomy of @xmath127 , when @xmath154 ; observe that this means that the image of the meridian is conjugate to @xmath155 a rotation of angle @xmath156 that has order @xmath9 in @xmath153 . [ prop : periodic ] the variety @xmath157 contains at least @xmath7 irreducible curves @xmath158 , each of which contains at least one irreducible character . as a consequence , all these components are birationally equivalent to a subvariety @xmath159 of @xmath136 . furthermore , the curves @xmath159 are irreducible components of the whole @xmath6 , not only the invariant part . first of all , remark that the intersection of @xmath120 with the hyperplane @xmath160 contains the holonomy character @xmath161 of the hyperbolic orbifold structure of @xmath127 . in particular , a component of @xmath162 is an irreducible curve @xmath163 containing @xmath161 , the so called excellent or distinguished component . this is the curve that , viewed as a deformation space , allows to prove thurston s hyperbolic dehn filling theorem @xcite , e.g. ( * ? ? ? the character @xmath161 takes values in a number field @xmath164 containing the subfield @xmath165 of degree @xmath166 . the galois conjugates of @xmath161 are contained in @xmath167 for some @xmath168 . as @xmath152 is precisely the set of galois conjugates of @xmath169 , this yields the @xmath166 components defined by @xmath170 , @xmath171 ( though the number of conjugates may be larger , depending on the degree of the number field @xmath164 ) . to prove the assertion that these curves are irreducible components of @xmath6 , notice that the restriction @xmath172 is the holonomy of the hyperbolic structure of @xmath58 . therefore , by calabi - weil rigidity , the zariski tangent space of @xmath173 at @xmath172 is one dimensional . this space equals the cohomology group of @xmath58 with coefficients in the lie algebra @xmath174 twisted by the adjoint of the holonomy , cf . @xcite . using for instance simplicial cohomology , the dimension of this cohomology can be established by the vanishing or not of certain polynomials ( with integer coefficients ) in the entries of the representation . in particular the same dimension count is true for its galois conjugates . this zariski tangent space gives an upper dimension bound that establishes the final claim . remark that @xmath136 may contain other components than the ones described above . in particular , if @xmath22 is itself hyperbolic , there is at least one extra component whose characters correspond to representations that map @xmath133 to the trivial element , that is the lift of the excellent component of @xmath175 . let @xmath4 be a hyperbolic knot which is periodic of prime order @xmath176 . then @xmath6 contains at least @xmath177 irreducible components which are curves . in addition there is an extra irreducible component when @xmath22 itself is hyperbolic . by considering the abelianisation @xmath178 of the fundamental group of the orbifold @xmath127 , it is not difficult to prove that @xmath179 consists of @xmath180 lines . on the other hand , the abelianisation of the fundamental group of the exterior of @xmath4 consists in a unique line which is fixed pointwise by the action induced by @xmath17 on @xmath181 . it follows that , in general , the fixed subvariety of the whole character variety of @xmath4 is not birationally equivalent to the whole character variety of the orbifold . for this reason we have restricted our attention to @xmath136 . let @xmath4 be a hyperbolic knot admitting a free symmetry @xmath17 of odd prime order @xmath9 . let @xmath18 be the associated link as defined in section [ s : quotientlink ] ( see in particular proposition [ p : quotientlink ] ) . in this case , the irreducible characters of @xmath136 are mapped inside the subvariety of @xmath120 obtained by intersection with the hypersurface defined by the condition that its characters correspond to representations that send @xmath182 to the trivial element . note that in @xmath134 one has @xmath183 . we write : @xmath184 thus the representations of @xmath127 must satisfy @xmath185 . this provides a motivation to look at the restriction to the peripheral subgroup @xmath186 generated by @xmath133 and @xmath187 : @xmath188 when this restriction has finiteness properties , we are able to find uniform bounds on the number of components of @xmath189 : [ prop : bound ] assume that is a finite map . then there is a constant @xmath26 depending only on @xmath120 such that the number of components of @xmath189 is @xmath190 . notice that the components of @xmath120 have dimension at least two @xcite , the hypothesis in proposition [ prop : bound ] implies in particular that they are always surfaces . we give in the next section an example of a link for which is a finite map . as a consequence we have : [ c : freebound ] there exists a sequence of hyperbolic knots @xmath8 parametrised by infinitely many prime numbers @xmath9 such that @xmath8 has a free symmetry @xmath17 of order @xmath9 but @xmath191 is bounded , uniformly on @xmath9 . since @xmath9 and @xmath38 are coprime , there exist @xmath192 and @xmath193 such that the elements @xmath194 and @xmath195 generate the fundamental group of @xmath196 . the character variety @xmath197 is a surface in @xmath198 with coordinates @xmath199 , @xmath200 , and @xmath201 , defined by the equation @xmath117 . the equations @xmath202 and @xmath203 determine a line @xmath204 contained in the surface @xmath197 which corresponds to the subvariety of characters of representations that are trivial on @xmath194 . to count the components of @xmath136 it is enough to count the components of @xmath205 . the map @xmath206 being finite , there is a zariski open subset of each irreducible component of @xmath120 on which the map is finite to one . as a consequence there is a finite number @xmath207 of curves in @xmath120 which are mapped to points of @xmath197 . it follows that the number of irreducible components @xmath205 is bounded above by @xmath208 where @xmath209 is the cardinality of the generic fibre of @xmath206 . consider the two - component @xmath3-bridge link @xmath10 pictured in figure [ f : link ] . -bridge link @xmath10 and the generators of its fundamental group.,title="fig:",height=151 ] for each prime @xmath210 and each @xmath211 one can construct a symmetric knot @xmath4 as described in section [ s : quotientlink ] . since the absolute value of the linking number of the two components of @xmath47 is @xmath1 , the construction does not give a knot for @xmath212 , which must thus be excluded . using wirtinger s method one can compute a presentation of its fundamental group : @xmath213 where the generators @xmath133 and @xmath214 are shown in figure [ f : link ] . having chosen the meridian @xmath133 , the corresponding longitude is @xmath215 . an involved but elementary computation gives the following defining equation for @xmath120 @xmath216 where @xmath50 , @xmath217 , and @xmath218 represent the traces of @xmath133 , @xmath214 , and @xmath219 respectively . the equation can also be found in @xcite . note that the variety consists of two irreducible components , the first one being that of the abelian characters . a similar computation gives an expression for the trace of @xmath187 in terms of @xmath50 , @xmath217 , and @xmath218 : @xmath220 we want to understand the generic fibre of the restriction map @xmath221 , where @xmath120 is a surface contained in @xmath222 with coordinates @xmath50 , @xmath217 , and @xmath218 and @xmath197 is also a surface contained in @xmath222 but with coordinates @xmath223 , @xmath224 , and @xmath225 . for each fixed point @xmath226 in @xmath227 , the fibre of @xmath206 consists of the points @xmath228 which satisfy @xmath229 once @xmath50 is replaced by its value @xmath223 , the points we are interested in correspond to the intersection of two curves in @xmath230 with coordinates @xmath231 . we see immediately that , for generic values of @xmath224 each point of @xmath232 is the image of at most a finite number of points in @xmath120 and such finite number is bounded above by the product of the degrees of the two polynomials in @xmath217 and @xmath218 , i.e. @xmath233 . this shows that proposition [ prop : bound ] applies to this link and corollary [ c : freebound ] holds . let @xmath126 be a hyperbolic link with two components such that @xmath19 is trivial . assume that @xmath234 . for each odd prime number @xmath9 that does not divide the linking number @xmath235 , the knot @xmath22 lifts to a knot @xmath8 in the @xmath9-fold cyclic cover of @xmath15 branched along @xmath19 . by construction ( see section [ s : quotientlink ] ) , @xmath8 is periodic of period @xmath9 , realised by @xmath17 , and the invariant subvariety @xmath191 contains at least @xmath7 irreducible components of dimension @xmath14 . these components of @xmath191 are constructed in proposition [ prop : periodic ] as the intersection of the character variety @xmath120 with a family of @xmath7 parallel hyperplanes . these parallel planes correspond to a hypersurface which is the vanishing locus of the minimal polynomial for @xmath236 in the variable @xmath223 . such polynomial can be easily computed from the @xmath9th cyclotomic polynomial and is defined over @xmath125 . the characters of @xmath191 correspond to representations of the orbifold @xmath237 . note that @xmath238 may have further components besides those provided by proposition [ prop : periodic ] , since the orbifold may admit irreducible representations that are trivial on @xmath133 . these irreducible representations correspond to characters for which @xmath239 . in any case , @xmath238 contains at least @xmath7 components of dimension @xmath14 . if we consider the character variety of @xmath237 in characteristic @xmath9 , we have that , since the only elements of order @xmath9 are parabolic , the entire character variety must be contained in the hyperplane defined by @xmath239 . we note that if @xmath9 is not a ramified prime for @xmath237 , then it must contain as many @xmath14-dimensional irreducible components as the one over @xmath12 , that is at least @xmath7 . let us now turn our attention to the subvariety of @xmath120 which consists in the intersection of @xmath120 with the hyperplane @xmath239 . we remark that it is non - empty since it must contain the character of the holonomy representation of @xmath47 . we are interested in its irreducible components of dimension @xmath14 . these are in finite number , say @xmath207 , depending on @xmath47 only , and constitute an affine variety of dimension @xmath14 that we shall denote @xmath240 . standard arguments of algebraic geometry show that for almost all ( odd ) primes @xmath192 , the character variety @xmath120 as well as its subvariety @xmath240 have the same properties over an algebraically closed field of characteristic @xmath192 they have over the complex numbers . in particular @xmath240 has @xmath207 irreducible components . we start by considering the invariant variety @xmath191 and show that this variety ramifies at @xmath9 if @xmath9 is large enough . indeed , if this were not the case , the above discussion implies that the number of irreducible curves of @xmath191 should be at least @xmath7 on one hand and at most @xmath207 on the other . it follows readily that @xmath191 ramifies at @xmath9 . now , since @xmath7 curves of the invariant variety @xmath191 are also irreducible components of @xmath241 and since @xmath191 is defined over @xmath125 , the character variety of @xmath8 ramifies at @xmath9 , too . the polynomial equations defined over @xmath125 of the character variety of the orbifold @xmath237 generate a non radical ideal when considered @xmath242 , since the minimal polynomial of @xmath243 is not reduced when considered @xmath0 . john w. morgan and hyman bass , editors . , volume 112 of _ pure and applied mathematics_. academic press , inc . , orlando , fl , 1984 . papers presented at the symposium held at columbia university , new york , 1979 .

we study character varieties of symmetric knots and their reductions @xmath0 . we observe that the varieties present a different behaviour according to whether the knots admit a free or periodic symmetry . _ ams classification : _ primary 57m25 ; secondary 20c99 ; 57m50 . _ keywords : _ character varieties , hyperbolic knots , symmetries .

an investigation of the properties of square - free monomial ideals in polynomial rings led francisco et al . @xcite to an interesting question about replication in colour - critical graphs that we answer in the present paper . in the area of graph colourings , constructions and properties of colour - critical graphs are a classical subject ( see , e.g. , ( * ? ? ? * section 14.2 ) ) . the replication of a set of vertices , whose definition we will recall shortly , is a natural operation in this context . it is also of central importance for the theory of perfect graphs ( cf . * chapter 65 ) ) . for the terminology and notation of graph theory , we follow bondy and murty @xcite . we deal with graphs without parallel edges and loops . the vertex set and the edge set of a graph @xmath4 are denoted by @xmath5 and @xmath6 , respectively . a graph @xmath4 is _ @xmath0-chromatic _ if its chromatic number is @xmath0 . if @xmath4 is @xmath0-chromatic and @xmath7 is @xmath8-colourable for each vertex @xmath9 of @xmath4 . furthermore , @xmath4 is _ @xmath0-edge - critical _ if @xmath4 is @xmath0-chromatic and every proper subgraph of @xmath4 is @xmath8-colourable . _ replicating _ ( also _ duplicating _ ) a vertex @xmath10 means adding a copy ( or _ clone _ ) @xmath11 of @xmath3 and making it adjacent to @xmath3 and all its neighbours . to replicate a set @xmath12 , we replicate each vertex @xmath13 in sequence . the resulting graph @xmath14 is independent of the order in which the individual vertices are replicated . francisco et al . @xcite posed the following conjecture : [ conj : main ] for any positive integer @xmath0 and any @xmath0-critical graph @xmath4 , there is a set @xmath12 such that @xmath14 is @xmath1-critical . in section [ sec : counterexample ] of the present paper , we disprove the conjecture by showing that each member of an infinite family of 4-critical graphs constructed by gallai @xcite is a counterexample . in section [ sec : algebra ] , we discuss the algebraic properties of the smallest member of this family and show that it also answers two open questions concerning square - free monomial ideals in polynomial rings . thus , the result provides a nice example of interplay and useful exchange between algebra and combinatorics . gallai s construction @xcite of an infinite family of 4-regular 4-edge - critical graphs provided the first example of a @xmath0-edge - critical graph without vertices of degree @xmath15 . the definition can be expressed as follows . for a positive integer @xmath16 , let @xmath17}$ ] denote the set @xmath18 . let @xmath19 be a path with vertex set @xmath17}$ ] , with vertices in the increasing order along @xmath19 . let @xmath20 be the complete graph whose vertex set is the group @xmath21 . for @xmath22 , we define @xmath23 as the graph obtained from the cartesian product @xmath24 by adding the three edges joining @xmath25 to @xmath26 for @xmath27 . ( see figure [ fig : graph]a . ) the 4-regular graphs @xmath23 are interesting in various ways ; for instance , they embed in the klein bottle as quadrangulations ( cf . figure [ fig : graph]b ) . in this section , we show that gallai s graphs are counterexamples to conjecture [ conj : main ] : [ t : main ] for any @xmath22 and any @xmath28 , the graph @xmath29 is not @xmath30-critical . it is interesting to note that by ( * ? ? ? * theorem 1.3 ) , conjecture [ conj : main ] holds for graphs @xmath4 satisfying @xmath31 , where @xmath32 denotes the chromatic number and @xmath33 denotes the fractional chromatic number ( see , e.g. , ( * ? ? ? * definition 3.8 ) for the definition ) . since the graphs @xmath23 are @xmath34-chromatic and their fractional chromatic number equals @xmath35 , they show that the bound in theorem 1.3 of @xcite can not be improved . we will divide the proof of theorem [ t : main ] into two parts . first , we show that for certain sets @xmath2 , the chromatic number of @xmath29 is at least 5 , but @xmath29 is not 5-critical ( lemma [ l : five ] ) . we then prove that for any other set @xmath2 , @xmath29 is 4-chromatic ( proposition [ p : four ] ) . let @xmath36}$ ] and @xmath27 . the _ @xmath37-th column _ of @xmath23 is the set @xmath38 . similarly , the _ @xmath39-th row _ of @xmath23 is @xmath40 } \times { \left\{{j}\right\}}$ ] . the vertex in @xmath41 is denoted by @xmath42 . in accordance with the notation introduced above , the clone of @xmath43 in @xmath29 is denoted by @xmath44 . we introduce notation for certain subgraphs of @xmath29 . let @xmath36}$ ] . we define @xmath45 as the clique in @xmath29 on the vertices in @xmath46 and their clones . furthermore , @xmath47 is the induced subgraph of @xmath29 on @xmath48 ( addition modulo @xmath16 ) . [ l : five ] let @xmath22 and let @xmath28 . in each of the following cases , the graph @xmath29 has chromatic number at least @xmath30 and is not @xmath30-critical : 1 . there is some @xmath36}$ ] such that the set @xmath49 has size at least 2 , 2 . @xmath2 contains at least @xmath50 vertices of @xmath51 and @xmath16 is odd , 3 . the induced subgraph of @xmath23 on @xmath52 contains a path with at least @xmath16 vertices and @xmath16 is even . \(a ) suppose that @xmath53 has size at least 2 , so @xmath54 . since @xmath29 contains the clique @xmath45 as a proper subgraph , it is neither 4-colourable nor 5-critical . \(b ) without loss of generality , assume that @xmath2 contains @xmath55 . furthermore , suppose that @xmath16 is odd . for contradiction , let @xmath56 be a 4-colouring of @xmath29 . by symmetry , the vertices @xmath57 and @xmath58 may be assumed to have colours 1 and 2 in @xmath56 . this forces the pairs of colours assigned to @xmath59 and @xmath60 alternate between @xmath61 and @xmath62 as @xmath37 increases . hence , @xmath63 has neighbours of all four colours , a contradiction which shows that @xmath29 is not 4-colourable . because the argument involves only vertices in @xmath51 and their clones , it implies that , say , @xmath64 is not 4-colourable . it follows that @xmath29 is not 5-critical . \(c ) suppose that @xmath16 is even and the induced subgraph of @xmath52 contains a path with at least @xmath16 vertices . by symmetry , we may assume that @xmath65 . we prove that @xmath29 is not 4-colourable . suppose the contrary and consider a 4-colouring of @xmath29 . an argument similar to the one used in part ( b ) implies that the vertices @xmath66 , @xmath67 , @xmath68 and @xmath69 have distinct colours . since they have a common neighbour @xmath70 , we obtain a contradiction . in the same manner as above , it follows that @xmath29 is not 5-critical . [ l : max ] if @xmath71 satisfies none of the conditions @xmath72@xmath73 in lemma [ l : five ] , then there is a set @xmath74 such that @xmath75 , @xmath74 contains exactly one vertex from each @xmath46 ( @xmath36}$ ] ) and @xmath74 still satisfies none of @xmath72@xmath73 . since @xmath2 does not satisfy condition ( a ) , it contains at most one vertex from each set @xmath46 ( @xmath36}$ ] ) . suppose that @xmath76 for some @xmath37 . we claim that conditions ( a)(c ) are still violated for the set @xmath77 , for some @xmath78 . if @xmath79 satisfies any of the conditions , it must be condition ( b ) , which means that @xmath16 is odd . in that case , @xmath80 trivially fails to satisfy the conditions . by adding further vertices in this way , we arrive at a set @xmath74 with the desired properties . before we embark on the proof of proposition [ p : four ] , it will be convenient to introduce some terminology . assume that @xmath28 is a set which satisfies none of the conditions in lemma [ l : five ] . in addition , we will assume that @xmath81}$ ) in exactly one vertex.}\ ] ] for each @xmath36}$ ] , we will define @xmath82 to be the unique element of @xmath21 such that @xmath83 . ( in the proof of proposition [ p : four ] below , we will ensure condition by appealing to lemma [ l : max ] . ) we will encode the set @xmath2 into a sequence of signs , defined as follows . sign sequence _ @xmath84 is a sequence of elements of @xmath21 . we will often write ` @xmath85 ' for the element 1 and ` @xmath86 ' for the element 2 ( which coincides with @xmath87 ) . thus , the sign sequence @xmath88 stands for the sequence @xmath89 . to the set @xmath2 , we assign the sign sequence @xmath90 , where each @xmath91 is defined as @xmath92 the change of sign in the latter case reflects the fact that the vertex @xmath93 is adjacent to @xmath94 rather than @xmath95 . it may be helpful to view @xmath23 as the graph obtained from the cartesian product @xmath96 by identifying the vertex @xmath25 with @xmath97 for each @xmath27 . it is then natural to define @xmath98 , in which case @xmath99 is precisely @xmath100 . to describe a 4-colouring of the clique @xmath45 in @xmath29 ( @xmath36}$ ] ) , we introduce the notion of a _ pattern_. this is a cyclically ordered partition of the set @xmath101 into three _ parts _ , with one part of size 2 and the remaining parts of size 1 . the two colours contained in the part of size 2 are _ paired_. two patterns differing only by a cyclic shift of the parts are regarded as identical . given a 4-colouring @xmath56 of @xmath45 , the corresponding _ pattern at @xmath45 _ is @xmath102 we use a more concise notation for patterns : for instance , instead of writing @xmath103 we write just @xmath104 . note that a pattern does not determine the colouring uniquely since it does not specify the order of the paired colours . we now determine the possible combinations of patterns at @xmath45 and at @xmath105 in a valid colouring of @xmath47 . suppose that @xmath106 is a colouring of @xmath107 with pattern @xmath104 , and let @xmath108 . consider first the case that @xmath109 . it is routine to check that for any valid extension of @xmath106 to @xmath110 , the pattern at @xmath111 is @xmath104 , @xmath112 or @xmath113 ( cf . figure [ fig : patt ] ) . conversely , each of these patterns determines a valid extension . 645 considering the other possibilities for @xmath114 , we find that the sets of patterns at @xmath111 corresponding to valid extensions of @xmath106 are as follows : [ cols="^,^,^ , < " , ] @xmath115 . applying a suitable symmetry of the graph @xmath23 , and using the fact that @xmath2 does not satisfy conditions ( b ) , ( c ) in lemma [ l : five ] , we may assume that @xmath116 . in view of lemma [ l : symm ] , it may further be assumed that @xmath117 . the sequence @xmath118 is good and we have @xmath119 . consequently , @xmath120 , and by symmetry , @xmath121 . in particular , none of @xmath122 is the symbol @xmath85 and at least one of @xmath122 is different from @xmath86 . it follows that @xmath123 . choose the least @xmath39 such that @xmath124 and @xmath125 we claim that there is @xmath126 such that @xmath127 . suppose the contrary . since the sum of all @xmath128 ( @xmath36}$ ] ) is @xmath129 we find that there are two possibilities : either @xmath130 and @xmath131 , or @xmath132 and @xmath133 . in the first case , however , @xmath2 would satisfy condition ( b ) in lemma [ l : five ] , while in the second case , condition ( c ) would be satisfied , a contradiction . let us choose the least @xmath0 such that @xmath126 and @xmath134 . assume first that @xmath135 . this implies that @xmath136 is odd , since otherwise @xmath137 and as we have seen , this would mean that @xmath138 is good . however , if @xmath136 is odd , then @xmath139 and we get a contradiction with lemma [ l : sub](ii ) as @xmath140 is reversing ( cf . table [ tab : reversing ] ) and @xmath141 . it remains to consider the possibility that @xmath142 . if @xmath136 is odd , then for the reversing sequence @xmath143 we have @xmath144 and we obtain a contradiction with lemma [ l : sub](ii ) again . thus , @xmath136 is even . in this case , we find @xmath145 . as we can see from table [ tab : good ] , @xmath146 is good . furthermore , @xmath147 , so @xmath138 is good by lemma [ l : sub](i ) , a contradiction . the discussion of case 3 , as well as the proof of proposition [ p : four ] , is complete . theorem [ t : main ] is now an immediate consequence of lemma [ l : five ] and proposition [ p : four ] . we conclude this section by pointing out that the graph @xmath148 is the only counterexample to conjecture [ conj : main ] among edge - critical graphs on up to 12 vertices , as was shown by a computer search using a list of edge - critical graphs provided in @xcite . as mentioned in section [ sec : intro ] , conjecture [ conj : main ] was motivated by questions arising from commutative algebra . it turns out that the graph @xmath148 serves as a counterexample for two other problems on the properties of square - free monomial ideals which we state in this section . for the terms not defined here , as well as for more information on commutative algebra and its relation to combinatorics , see @xcite . monomial ideals are the subject of the monograph @xcite . let @xmath149 be a commutative noetherian ring and @xmath150 an ideal . a prime ideal @xmath151 is _ associated _ to @xmath152 if there exists an element @xmath153 such that @xmath154 ( the ideal quotient of @xmath152 and @xmath155 ) . the _ set of associated prime ideals _ ( associated primes ) is denoted by @xmath156 . brodmann @xcite showed that @xmath157 for all sufficiently large @xmath114 . the ideal @xmath152 is said to have the _ persistence property _ if @xmath158 for all @xmath159 . let @xmath0 be a fixed field and @xmath160 $ ] a polynomial ring over @xmath0 . an ideal in @xmath149 is _ monomial _ if it is generated by a set of monomials . a monomial ideal is _ square - free _ if it has a generating set of monomials where the exponent of each variable is at most @xmath161 . the question that motivated francisco et al . @xcite to pose conjecture [ conj : main ] is the following one ( see ( * ? ? ? * question 3.28 ) , ( * ? ? ? * question 4.16 ) or @xcite ) : [ prob : persistence ] do all square - free monomial ideals have the persistence property ? francisco et al . @xcite proved that if conjecture [ conj : main ] holds , then the answer to problem [ prob : persistence ] is affirmative . while our counterexample to conjecture [ conj : main ] does not necessarily imply a negative answer to problem [ prob : persistence ] , the cover ideal of @xmath148 does in fact show that the answer is negative . given a graph @xmath4 , a _ transversal _ ( or _ vertex cover _ ) of @xmath4 is a subset @xmath162 such that every edge of @xmath4 has an end vertex in @xmath163 . if @xmath164 , we can associate each @xmath165 with the variables in the polynomial ring @xmath166 $ ] . the _ cover ideal _ @xmath167 is the ideal generated by all inclusion - wise minimal transversals of @xmath4 . let @xmath168 denote this cover ideal in the polynomial ring @xmath169 $ ] , where @xmath148 is the graph on 12 vertices defined in section [ sec : counterexample ] . using the commutative algebra program macaulay2 @xcite , we can compute the set of associated primes of @xmath170 and @xmath171 . by comparing the output , one finds that @xmath172 where @xmath173 is the maximal ideal of @xmath149 . in particular : [ t : persistence ] the cover ideal @xmath174 does not have the persistence property . the second question concerns the depth function of monomial ideals . if @xmath152 is an ideal in @xmath149 , then the _ depth function _ of @xmath152 is the function @xmath175 defined by @xmath176 where @xmath177 is the depth of a ring as defined , e.g. , in ( * ? ? ? * chapter 6 ) . herzog and hibi @xcite noted that the depth function of most monomial ideals is non - increasing , but they constructed examples where this is not the case ( for instance , one where the depth function is non - monotone ) . they asked the following question : [ prob : depth ] do all _ square - free _ monomial ideals have a non - increasing depth function ? ( see also @xcite . ) as noted in @xcite , the question of problem [ prob : depth ] is a natural one since a monomial ideal @xmath152 satisfies the persistence property if all monomial localisations of @xmath152 have a non - increasing depth function . according to @xcite , a positive answer was ` expected ' . however , the cover ideal of @xmath148 again provides a counterexample . using macaulay2 we find that @xmath178 so we have the following : [ t : depth ] the depth function of the cover ideal @xmath174 is not non - increasing . we would like to thank chris francisco , ti h , and adam van tuyl for their many helpful comments and suggestions , and particularly for pointing out that the cover ideal of @xmath148 provides negative answers to problems [ prob : persistence ] and [ prob : depth ] .

motivated by questions about square - free monomial ideals in polynomial rings , in 2010 francisco et al . conjectured that for every positive integer @xmath0 and every @xmath0-critical ( i.e. , critically @xmath0-chromatic ) graph , there is a set of vertices whose replication produces a @xmath1-critical graph . ( the replication of a set @xmath2 of vertices of a graph is the operation that adds a copy of each vertex @xmath3 in @xmath2 , one at a time , and connects it to @xmath3 and all its neighbours . ) we disprove the conjecture by providing an infinite family of counterexamples . furthermore , the smallest member of the family answers a question of herzog and hibi concerning the depth functions of square - free monomial ideals in polynomial rings , and a related question on the persistence property of such ideals .

the phenomenology of nuclear photoabsorption is governed by two characteristic features . first , all nuclei with mass numbers @xmath5 ranging from 10 to more than 200 obey the same fundamental curve @xmath6 for the total photoabsorption cross section devided by @xmath5 as a function of the photon energy @xmath7 . second , the @xmath1-isobar excitation of the nucleon is responsible for the main properties of this curve in the energy region between 200 and 400 mev . models , which focus on the behaviour of the @xmath1-isobar in a nuclear environment , namely the @xmath1-hole calculations @xcite , have proven to be highly successful in explaining the experimental findings for pion scattering processes @xcite . indeed , on grounds of the @xmath1-hole formalism a wide variety of pion - nucleus reactions can be described within one consistent framework @xcite . in the case of photonuclear reactions , however , serious descrepancies remain , which partially have been accounted for by including non - resonant background terms @xcite . nevertheless , such a procedure , in particular for nuclear photoabsorption , either leads to contradictions with previous @xmath1-hole results or lacks the complete agreement with experimental data @xcite . the question arises , whether theoretical ingredients other than in - medium @xmath1-hole propagations can lead to a similar degree of accuracy . therefore it is natural to address the situation from a different point of view asking to what extend the absorption process can be described , when only a very simple @xmath1-nucleon interaction is used and all additional effects are accounted for in a purely diagrammatic approach . when combined with a simple form of nucleon momentum distribution inside the nucleus , namely a fermi gas model , this leads to analytical expressions , in which different corrections to this lowest - order approximation can be studied . this is the aim of the present article . in our formalism we follow closely wakamatsu and matsumoto @xcite . however , we do not introduce a phenomenological potential to account for the deviation of the nucleon wave functions from plane waves , but study the influence of such corrections in a perturbative way , similar to our previous work @xcite . in addition , our focus is on the energy - dependence of the total photoabsorption , rather than on the differential cross section as a function of the momentum of the outgoing proton . a characteristic feature of wakamatsu s and matsumoto s approach is the technically equal treatment of the ( @xmath8,pn ) and the ( @xmath8,pp ) knock - out process , which allowed them to obtain a natural explanation for the supression of the two - proton knock - out . this characteristic property is also present in our calculation , where it is related to a vanishing trace in spin space . in the last years the ratio ( @xmath8,pp)/(@xmath8,pn ) has been investigated in depth within different formalisms . in an extension of wakamatsu s and matsumoto s work by boato and giannini @xcite , finite - size effects have been calculated and , more recently , a combination of pion exchange and shell - model wave functions was used to investigate this quantity @xcite . currently , two complementary approaches for the description of nuclear knock - out reactions exist . carrasco and oset @xcite used a diagram - oriented many - body expansion in a fermi gas . the evaluation of self - energy diagrams leads to an accuracy for medium effects high enough to study knock - out reactions in great detail . with their primary goal being thus different from ours , their formalism does not yield isolated expressions for the resonant and non - resonant parts of the mechanisms of nuclear photoabsorption a different approach is used by the gent group @xcite , where the main emphasis lies in the construction of realistic shell - model wave functions , rather than on a microscopic description fully based on the evaluation of feynman diagrams . as the nuclear photoabsorption is almost insensitive to structural differences between nuclei , the quality of their approach becomes obvious in the investigation of differential cross sections for nucleon knock - out , rather than of photoabsorption . nuclear photoabsorption provides an interesting tool to study the interplay between one - nucleon and two - nucleon contributions . we obtained analytical expressions for these contributions , as well as for their resonant and non - resonant parts . this set of results can be used as a starting point to include ( and test ) further nuclear or nucleonic effects . in section 2 the basic notations are listed , as well as the interaction terms and the most important model assumptions and approximations , which are present in this calculation . the main results and their most prominent properties , e.g. the effects of relativistic corrections and nuclear structure , are discussed in section 3 , where also the following possible extension of such an approach is considered : as the angular dependence of the two - nucleon process is not very strong ( cf . @xcite ) , the different mechanisms contributing to the photoabsorption curve can also be used to understand qualitative features of experimental data for nucleon knock - out processes as a function of the photon energy . in section 4 some concluding remarks are made , with an emphasis on the applicability of the partial cross sections , whose analytical forms are given in the appendix . the starting point of our investigation is the static hamiltonian for the pion - nucleon interaction , @xmath9 together with a minimal coupling to the photon field . in eq.([s1eq1 ] ) @xmath10 is the pion mass . for all coupling constants we use the same notation and values as given in @xcite , in particular @xmath11=0.08 . in eq.([s1eq1 ] ) underlined symbols denote vectors in ( cartesian ) isospin space , while an arrow indicates a vector in coordinate space . in the static limit the interactions with the @xmath1-isobar excitation of the nucleon are determined by the following hamiltonians ( see e.g. @xcite ) : @xmath12 and @xmath13 with the hermitian conjugate to be added in both cases . here @xmath14 and @xmath15 are the 1/2-to-3/2 transition operators in spin space and isospin space , respectively ; @xmath16 is the proton charge , @xmath17 . for the coupling constants we have @xmath18=2 and @xmath19=0.35 . in all cases , where high momentum transfers occur at the pion - nucleon vertices we regularize the vertex functions by introducing dipole form factors @xmath20 as was also done e.g. in @xcite . the value for the cut - off parameter @xmath21 has been taken to be 800 mev . this value gives the best agreement of our predictions with the experimental data . in addition , a similar value has been used in @xcite . the general expression for the total cross section @xmath22 of the photoabsorption with one nucleon outside the fermi sphere and one pion in the final state ( one - nucleon process ) is of the form @xmath23\ ; . \label{s1eq5}\end{aligned}\ ] ] for the total cross section @xmath24 of the process with two free nucleons in the final state it is given by @xmath25\left [ { 1-n(\vec p_4)\ ; } \right ] \label{s1eq6}\ ; .\end{aligned}\ ] ] the notation for the external momenta is shown in fig.([figa ] ) . the function @xmath26 is the occupation number , @xmath27 is the step function . furthermore , @xmath28 is the nuclear volume , @xmath29 is the fermi momentum , @xmath30 is the mass of the proton , @xmath28 is the nuclear volume , @xmath31 , and @xmath32 is the energy of the outgoing pion . both amplitudes @xmath33 and @xmath34 consist of non - resonant and resonant parts , @xmath35 . diagrammatically this decomposition is shown in fig.([figb ] ) . the second ( crossed ) contribution to the resonant part is small due to the big energy denominator and will be skipped in the following . in the energy @xmath36-function in eq.([s1eq5 ] ) and eq.([s1eq6 ] ) we will usually not consider the smallest term connected with the kinetic energy of the incoming nucleon , as it is smaller than @xmath37mev . indeed , as we expect its contribution to introduce only a small modification of the actual @xmath38-dependence in the integrand in ( [ s1eq5 ] ) , we substitute it by its average value @xmath39 , which results in an overall shift of the absorption cross section . for the one - nucleon process we find first - order relativistic corrections to be essential for a successful treatment of the absorption process . in the case of the resonant contribution , such corrections are accounted for by making the following substitutions in the vertices @xcite : @xmath40 where @xmath41 is the @xmath42-isobar mass and @xmath43 . for the non - resonant part , the corrections give amplitude @xmath33 of the following form : @xmath44-{{2i(\vec \sigma \cdot ( \vec q-\vec k))(\vec \varepsilon \cdot \vec q)}\over { ( \vec q-\vec k)^2+m^2-(\varepsilon _ q-\omega ) ^2 } } \nonumber \\ & -&{{2i(\vec \sigma \cdot \vec q)(\vec \varepsilon \cdot ( \vec k+\vec p-\vec q))}\over { \,\left [ { ( \vec k+\vec p-\vec q)^2 - 2m\omega -(\vec p-\vec q)^2}\right]}}\left . { \matrix{{}\cr { } \cr } } \right\ } , \label{s1eq9}\end{aligned}\ ] ] which corresponds to the production of a @xmath45 . in comparison with @xcite we neglected those terms in ( [ s1eq9 ] ) , which modify the result by less than 2 per cent . it should be noted that no free parameters are present in our approach . a certain model dependence exists , however , in the selection of diagrams . we neglect all those diagrams , which are suppressed by some mechanism . in fig.([figaa ] ) two examples for suppression mechanisms are given . for fig.([figaa]a ) the contribution is small because in the case of infinite nuclear matter due to momentum conservation the four - momentum of the photon should be equal to that of the outgoing pion , which is impossible . for fig.([figaa]b ) let us consider the case , where the upper two nucleons ( incoming and outgoing ) are identified . furthermore , let us select a @xmath1-isobar as an intermediate state . then , one has a vanishing trace in spin space : @xmath46\,=\ , 0\ ] ] at @xmath47 . therefore , the contribution in this case vanishes in the non - relativistic limit considered here . as a result of applying such methods to the various diagrams , we have obtained that only those displayed in fig.([figa ] ) should be taken into acount . explicit evaluation of the diagrams shown in fig.([figa ] ) leads to amplitudes for the one- and two - nucleon contribution to nuclear photoabsorption . squaring these amplitudes and summing over spin and isospin states of the nucleons by using trace methods as previously @xcite one finds the expressions @xmath48 and @xmath49 , which enter into eqs.([s1eq5 ] ) and ( [ s1eq6 ] ) . it has turned out to be convenient to investigate the resonant and non - resonant parts of each of these contributions separately . this can be done by neglecting the interference terms , which are highly suppressed ( cf . fig.([figh ] ) ) . shown here exemplary for the resonant parts , one obtains the following expressions , which serve as a starting point for the integrations with respect to nucleon momenta : @xmath50-q^2-m^2 } \right)\times \nonumber \\ & & n(\vec p-\vec k)\;\left [ { 1-n(\vec p-\vec q ) } \right]\,{{3\left [ { ( \vec q\times \vec k)\cdot \vec \varepsilon } \right]^2+q^2(\vec k\times \vec \varepsilon ) ^2 } \over { \left ( { \omega -\delta - p^2/2 m } \right)^2 + \gamma^2 / 4 } } , \label{s2eq1}\end{aligned}\ ] ] where @xmath51 and @xmath52\left [ { 1-n(\vec p_4)\ ; } \right]\times \nonumber \\ & & \delta \left ( { \omega -{{p_3 ^ 2+p_4 ^ 2 } \over { 2 m } } } \right)\,\delta ( \vec p_1+\vec p_2+\vec k-\vec p_3-\vec p_4)\times \nonumber \\ & & \left\ { { { { 2a^2\left [ { a^2 + 3(\vec \varepsilon \cdot \vec a)^2 } \right ] } \over { ( a^2+m^2)^2}}\,g_\pi ^4(a)+{{2\omega ^2(\vec \varepsilon \cdot \vec a)(\vec \varepsilon \cdot \vec b ) } \over { ( a^2+m^2)(b^2+m^2)}}\,g_\pi ^2(a)g_\pi ^2(b ) } \right\ } \label{s2eq2}\end{aligned}\ ] ] with @xmath53 the analytical expressions for the resulting partial cross sections are given in the appendix . for the total absorption cross section @xmath54 a comparison with experimental data is shown in fig.([figc ] ) . the result of this model calculation compares favorably with the data . the one - nucleon and two - nucleon parts of the cross section are equally important at energies around 250 mev . as can be seen in this plot , significant features of the data , e.g. the position of the peak , are only obtained due to the interplay between the two mechanisms . as mentioned before , each of the contributions to the full curve in fig.([figc ] ) has a resonant and a non - resonant part . in fig.([figd ] ) and ( [ fige ] ) this decomposition is shown for the one - nucleon and the two - nucleon mechanism , respectively . here it is clearly seen that the non - resonant parts give an important contribution at lower energies . in the two - nucleon case the non - resonant part decreases with energy , while in the one - nucleon process it remains almost constant . it is interesting to see , in what way the one - nucleon partial cross sections are affected by the use of only the static limit of the interaction . neglecting the first - order relativistic corrections in the current one has @xmath55}}\,\int\limits_{q_{min}}^{q_{max } } { q\,dq}\,\left ( { 1+s_0(q ) } \right)\times & & \nonumber \\ \left\ { { \left [ { \omega ^2-m^2-{{\omega q^2 } \over m } } \right]+{3 \over 2 } \left ( { q^2-\left [ { q^2\left ( { 1+{\omega \over m } } \right)+m^2 } \right]^2{1 \over { 4\omega ^2 } } } \right ) } \right\ } & & \label{s2eq3}\end{aligned}\ ] ] and @xmath56 ^ 2 } \right\}\left . { \matrix{{}\cr { } \cr } } \right ] , \label{s2eq4}\end{aligned}\ ] ] where the integration limits in both cases are given by @xmath57 and the integrand contains the function @xmath58 in fig.([figf ] ) the corresponding cross sections are compared with those resulting from eq.([s2eq1 ] ) and its non - resonant counterpart . the relativistic corrections lead mainly to a shift of the one - nucleon curve . this effect is essential for obtaining a good agreement with the experimental data . although no relativistic corrections in the current are included , note that in ( [ s2eq3 ] ) and ( [ s2eq4 ] ) terms of that order have been kept in the kinematical contributions to the integrand . as the two - nucleon mechanism gives a comparatively small contribution at higher energies , we neglect relativistic corrections in @xmath59 . this has also been done in @xcite . a more difficult problem is the influence , which nucleon correlations inside the nucleus can have on the nuclear photoabsorption process . we investigate this aspect for the non - relativistic forms ( [ s2eq3 ] ) and ( [ s2eq4 ] ) of the one - nucleon case . the main reason for doing so is the fact that due to the square of the amplitudes we can express part of the integrand via the standard lowest - order central correlation function of a fermi gas ( see e.g. @xcite ) . the object , which is obtained by diagrammatically squaring the one - nucleon contribution of fig.([figa]a ) , can be coupled to an additional nucleon . in the incoherent case of the diagrammatical square also a further pion exchange can be allowed . the modifications of the correlation function , which occur due to such effects , have been investigated analytically in @xcite , where a corrected central correlator @xmath60 has been constructed . the function @xmath61 is given in fig.([figj ] ) . by making the substitution @xmath62 these further correlations can be incorporated effectively . in fig.([figg ] ) the one - nucleon contribution resulting from @xmath61 is compared with the original form , in which @xmath63 has been used . it can be seen that such medium effects modify the result by about 15 per cent . naturally , the use of this substitution technique can only be used to obtain an estimate for such a medium - induced modification . a full examination should involve the inclusion of the additional two - and three - nucleon diagrams in eq.([s1eq5 ] ) . the limit of our approach is certainly reached , when a comparison with differential cross sections is attempted . an extreme case is the comparison with @xmath64he , for which a recent measurement of both , the one - nucleon and the two - nucleon channel exists @xcite . we compare the calculated average cross section with the data for the differential cross section in c.m . frame , as we expect the angular dependence not to be strong in that frame . the corresponding plot is shown in fig.([figi ] ) . although no full agreement is obtained , it is interesting to note that the general features of the two cross sections are well reproduced , such as the peak positions and the relative size of the two processes . by these means it is possible to unambiguously identify the physical mechanisms behind the data points . a similar degree of agreement is obtained for other data @xcite . in the present paper we have developed a diagrammatical description of the nuclear photoabsorption process . the main result of our investigation is that the total photoabsorption cross section can be fully understood in terms of a simple physical picture , where point - like nucleons and @xmath65-isobars interacting via pion exchange are the relevant degrees of freedom . due to the diagram - oriented formalism and the fermi gas model as an approximate description of the nucleons in momentum space , we could obtain analytical expressions for all the relevant contributions to the photoabsorption curve . in this way a flexible and efficient description has been obtained , which can be used as a starting point for the investigation of additional effects . especially in the low - energy part of our calculated curve , the agreement with experiment comes about as a non - trivial interplay between the one - nucleon and two - nucleon contributions . it is worth noting that , as long as a comparatively low cut - off parameter in the vertex form factor is used , there seems to be no need for an explicit diagrammatical inclusion of the @xmath66-meson as an additional mechanism of the nucleon - nucleon interaction . we found relativistic corrections in the case of the one - nucleon process to be crucial for obtaining a good agreement . the aspect of additional nucleon correlations , which can be accounted for as a deviation of the nucleon wave functions from plane waves , deserves some further attention in future investigations . we could estimate the overall effect to be of the order of 15 per cent . we are most grateful to a.i . lvov for useful comments and discussions . one of us ( m.t.h . ) wishes to thank the budker institute , novosibirsk , for the kind hospitality accorded him during his stay , when part of this work was done . here we present the explicit expressions for the four contributions to the photoabsorption curve , which have been used to obtain the figures shown in section 3 . as was mentioned earlier , the interference terms between the resonant and the non - resonant contributions are small ( cf . fig.([figh ] ) ) . therefore , we can write the absorption cross section as a sum of four parts , @xmath67 . first , we deal with the one - nucleon case . in the case of the non - resonant contribution the absorption cross section can be represented in the following form : @xmath68\times \nonumber\\ & \displaystyle g(p , q,\omega)\,\theta(2q\omega - m^2-q^2-\frac{\omega}{m}p^2 ) , \label{app1}\end{aligned}\ ] ] where @xmath69 \label{app4}\ ] ] and @xmath70 $ ] , @xmath71 , @xmath72 . in eq.([app1 ] ) the form factor @xmath73 is the same as in ( [ s1eq4 ] ) . note that the integration with respect to the variable @xmath38 can easily be performed , but the result is too lengthy to be given here explicitely . the resonant part of the one - nucleon process has the following form : @xmath74 ^ 2 + \gamma^2/4}\ ; h(p , y,\omega ) \label{app6}\end{aligned}\ ] ] where @xmath75 and the integrand is given by @xmath76 + \nonumber\\ & \displaystyle [ ( x+1)a_1-{1 \over 8}(x - x^3)a_2]\theta(f-|p - p_f|)\,\theta(p+p_f - f ) \ , , \end{aligned}\ ] ] with @xmath77^{1/2}\ ; , \ ; x=\frac{p^2+f^2-p_f^2}{2pf}\ , , \nonumber \\ & \displaystyle a_1=\omega^2 - 2\omega ay+a^2\ ; , \ ; a_2=\omega^2(1 - 3y^2)+4\omega ay-2a^2\ ; , \nonumber \\ & \displaystyle a=\frac{p\delta}{m}\left(\frac{1+\omega / m}{1+\delta / m}\right)\ , .\end{aligned}\ ] ] the mass difference @xmath1 between the proton and the @xmath78-excitation is @xmath79 mev , the width @xmath80 of the @xmath42-isobar has been taken to be 115 mev . again , the integration with respect to @xmath81 in eq.([app6 ] ) can be performed analytically , but due to its length the result is not presented here . for the partial cross sections of the two - nucleon case , we obtained the following result : @xmath82+\nonumber \\ & \displaystyle \theta(\omega -5\varepsilon ) \,\theta ( 9\varepsilon - \omega ) \,\int\limits_{l_1(q)}^{l_3(q)}\phi(\beta_1,\ , q ) \biggr\}g_\pi ^4(p)\left[g^{(nr)}(p,\omega ) + g^{(r)}(p,\omega ) \right]dp , \nonumber \label{app10}\end{aligned}\ ] ] where @xmath83 , @xmath84 , @xmath85 the elementary function @xmath86 is @xmath87\ , + \nonumber \\ & \displaystyle \sqrt 8pq\,[\frac{1}{3}(2q^2+p^2-p_f^2)\cos ^3x -\frac{2}{5}q^2\cos ^5x\,]\biggr\}\ , \biggl.\biggr|_{x=\beta}^{x=\pi /2 -\beta}\ , . \label{app13}\end{aligned}\ ] ] the functions in the integrand of eq.([app10 ] ) , which characterize the resonant and non - resonant part , are given by @xmath88 and @xmath89 \frac{p^2}{(p^2+m^2)^2 } \label{app15}\ ] ] respectively . in eq.([app10 ] ) the integration limits are @xmath90 and @xmath91 99 m. hirata , j. koch , f. lenz and e. moniz , ann.phys.120 ( 1979 ) 205 e. oset and w. weise , nucl.phys.a329 ( 1979 ) 365 c. garcia - recio , l. salcedo , e. oset , d. strottman and m. lopez - santodomingo , nucl.phys.a526 ( 1991 ) 685 l. salcedo , e. oset , m. vicente - vacas and c. garcia - recio , nucl.phys.a484 ( 1988 ) 557 j. koch , e.j . moniz and n. ohtsuka , ann.phys.154 ( 1984 ) 99 e. oset and w. weise , nucl.phys.a368 ( 1981 ) 375 m. wakamatsu and k. matsumoto , nucl.phys.a392 ( 1983 ) 323 m .- th . htt and a.i . milstein , nucl.phys.a in press l. boato and m. giannini , j.phys.g15 ( 1989 ) 1605 j. ryckebusch , m. vanderhaeghen , l. machenil , m. waroquier , nucl.phys.a568 ( 1994 ) 828 r.c . carrasco and e. oset , nucl.phys.a536 ( 1992 ) 445 m. vanderhaeghen , k. heyde , j. ryckebusch , m. waroquier , nucl.phys.a595 ( 1995 ) 219 j. ryckebusch , l. machenil , m. vanderhaeghen , v. van der sluys and m. waroquier , phys.rev.c49 ( 1994 ) 2704 j. ryckebusch , l. machenil , m. vanderhaeghen , v. van der sluys and m. waroquier , phys.lett.b291 ( 1992 ) 213 s. homma et al . , phys.rev.c36 ( 1987 ) 1623 t. ericson , w. weise , `` pions and nuclei '' , oxford univ . press 1988 d.o . riska , phys.rep.181 ( 1989 ) 207 t. sasakawa , s. ishikawa , y. wu and t. saito , phys.rev.lett.68 ( 1992 ) 3503 a. molinari , phys.rep.64 ( 1980 ) 283 r. wichmann et al . , submitted to phys.lett.b s. homma et al . , phys.rev.lett.53 ( 1984 ) 2538 j. ahrens , nucl.phys.a446 ( 1985 ) 229c [ figa ] notation for three - momenta of the external particles a ) for the one - nucleon process and b ) for the two - nucleon reaction . the wavy lines denote photons and dashed lines denote pions . a circle indicates a bound , an arrow a free nucleon . [ figb ] diagrammatical forms of the resonant part @xmath92 ( first two terms ) and the non - resonant part @xmath93 ( last term ) of the amplitude @xmath94 . these contributions to the @xmath95-interaction enter into the diagrams shown in fig.([figa ] ) . [ figc ] comparison of the calculated curve @xmath96 for nuclear photoabsorption with the experimental data . the dotted curve is the one - nucleon contribution @xmath97 , while the dashed curve represents the two - nucleon mechanism @xmath97 . the data are taken from ref . the empty ( full ) circles correspond to @xmath98pb ( @xmath99c ) data , while the squares represent data on @xmath100o . [ figd ] resonant ( dashed ) and non - resonant ( dotted ) contribution to the one - nucleon reaction ( full line ) . the corresponding analytical expressions @xmath101 , @xmath102 and @xmath97 , respectively , can be found in the appendix . [ figh ] contribution of interference terms . the full ( dashed ) curve corresponds to the one- ( two- ) nucleon case . as can be seen from the overall scale , both for @xmath106 and @xmath107 this contribution is highly suppressed . [ figg ] effect of two- and three - nucleon correlation in the case of the non - relativistic one - nucleon part of the photoabsorption cross section . the dashed curve contains only @xmath63 , while in the full curve @xmath61 ( cf . fig.([figj ] ) ) has been included . [ figi ] approximate description of differential cross sections for helium . the differential cross section with respect to the direction of the outgoing proton is shown as a function of the energy of the incoming photon . data points are taken from @xcite . the full curve and filled circles correspond to the one - nucleon process , while the dashed curve and empty circles represent the two - nucleon case .

the universal curve @xmath0 of nuclear photoabsorption is investigated within a fermi gas model of nuclear matter . an energy range from pion threshold up to 400 mev is considered . the interactions between nucleon , pion , @xmath1-isobar and photon are considered in the non - relativistic approximation with corrections of the order @xmath2 taken into account with respect to proton mass . analytical expressions are obtained , in which the influence of nuclear correlations , two - nucleon contributions and relativistic corrections is studied explicitely . an extension of the model calculation to nucleon knock - out reactions is discussed . contribution of real and virtual pions to nuclear photoabsorption at intermediate energies + m .- th.htt@xmath3 , a.i.milstein@xmath4 and m.schumacher@xmath3 + ( a ) ii . institut der universitt gttingen , gttingen , germany \(b ) budker institute of nuclear physics , 630090 novosibirsk , russia 0em 1.5ex plus 0.5ex minus 0.5ex _ pacs code : _ 25.20.-x _ keywords : _ photoabsorption , mesonic exchange currents , nuclear correlation functions

the measurement of the rotation curves ( rcs ) of disk galaxies is a powerful tool to investigate the nature of dark matter ( dm ) , including its content relative to the baryonic components and their distributions . in particular , dwarf galaxies are good candidates to reach this aim as their kinematics are generally dominated by the dark component , down to small galactocentric radii @xcite . this leads to a reliable measurement of the dynamical contribution of the dm to the rc and hence of its density profile . therefore , a dwarf galaxy like the orion dwarf provides us with an important test as to whether dm density profiles arising in @xmath3 cold dark matter ( @xmath3cdm ) numerical simulations @xcite are compatible with those detected in actual dm halos around galaxies . let us comment that nfw profile arises from pure n - body dm simulations . it is well known that , as effect of the baryonic infall in the cosmological dm halos and of the subsequest process of stellar disk formation , shallower profiles of the dm halo may arise ( see @xcite ) . recent studies of the rcs of dwarf galaxies have tested the nfw scenario . it is now clear that kinematic data are better fitted by a dm halo with a constant density core ( e.g. @xcite ) , than by one that is centrally peaked . one specific example is ddo 47 , whose velocity field is clearly best fitted if the dm halo is cored ; moreover , its ( small ) detected non - circular motions can not account for the discrepancy between data and the nfw predictions @xcite . + the present investigation examines the dm content of the orion dwarf galaxy . this nearby system harbors an extended disk , and thus provides us with an important test of the above paradigm . as we show below , the orion dwarf is one of the few known galaxies whose kinematics _ unambiguously _ point towards a cored profile . this system is thus critically important for investigating the nature of the dm particle and of the evolution of dm halos . + mond accounts for the evidence that rcs of spiral galaxies are inconsistent with the corresponding distribution of the luminous matter @xcite . rather than postulating the existence of a dark halo made by massive collisionless elementary particles , this scenario advocates that the gravitational force at low accelerations leaves the standard newtonian regime to enter a very different one . historically mond has generally been successful in reproducing the rcs of spiral galaxies with only the ( observed ) luminous matter ( e.g. @xcite ) . however , cases of tension between data and the mond formalism do exist @xcite . + it is important to stress that in order to derive the dm density profile or to test the mond formalism , we must know the distribution of the ordinary baryonic components , as well as have reliable measurements of the gas kinematics . for the orion dwarf , 21-cm surface brightness and kinematics have recently been published @xcite : their analysis provides a high quality , high resolution rc , that , in addition , can be easily corrected for asymmetric drift and tested for non - circular motions . this galaxy is a very useful laboratory in that a simple inspection of the rc ensures us that it shows a large mass discrepancy at all radii . moreover , the baryonic components are efficiently modeled ( i.e. , no stellar bulge is evident and the stellar disk shows a well - behaved exponential profile , see @xcite ) . the distance to the galaxy , which is critical for an unambiguous test of mond @xcite , is estimated to be 5.4@xmath41.0 mpc @xcite . it is important to stress that the distance of the orion dwarf remains a significant source of uncertainty . @xcite estimate the distance using the brightest stars method . the intrinsic uncertainty in this technique may allow a distance ambiguity much larger than the formal errors estimated by @xcite , because in their work this method yields a scatter as large as @xmath5 in distance . finally , the system s inclination ( 47@xmath1 ) is kinematically measured ( see section ( [ 3.1 ] ) ) and is high enough to not affect the estimate of the circular velocity . the properties described above make the orion dwarf galaxy an attractive candidate to determine the underlying gravitational potential of the galaxy . this paper is organized as follows . in sec . 2 we present the stellar surface photometry . in sec . 3 , the surface density and kinematics data are presented and discussed ; we also provide the analysis of possible non - circular motions of the neutral gas . in sec . 4 we model the rc in the stellar disk using a cored / cusped halo framework . in sec . 5 we test the orion kinematics against the mond formalism . our conclusions are given in sec . following the discussion in @xcite , the underlying stellar mass in the orion dwarf is estimated using the near - infrared ( ir ) photometry ( j and bands ) presented by @xcite . those authors find ( j@xmath6 ) @xmath0 @xmath20.80 and a total magnitude of @xmath210.90 . when comparing to models ( see below ) we assume that the color difference between k and is negligible ; further , we assume l@xmath73.33 @xcite . accounting for extinction , the total k - band luminosity of the orion dwarf is @xmath83.5@xmath910@xmath10 l@xmath11 . the mass of the stellar component was estimated by @xcite to be ( 3.7@xmath41.5)@xmath910@xmath12 . the stellar surface brightness profile is well fitted by an exponential thin disk , with a scale length of @xmath13= 25 @xmath4 1 ( equivalent to 1.33 @xmath4 0.05 kpc at the adopted distance ) . moreover , there are no departures from an exponential profile that would be indicative of a prominent central bulge . spectral line imaging was acquired with the _ very large array _ and presented in @xcite . we refer the reader to that work for a full discussion of the data handling , and we summarize salient details here . the final data cubes have a circular beam size of 20 , with a 3@xmath14 column density sensitivity of n@xmath15= 1.5 @xmath910@xmath16@xmath17 . the first three moment maps ( i.e. the integrated intensity , the velocity field , and the velocity dispersion ) are shown in figure ( [ figcap3 ] ) . + the neutral gas disk of the orion dwarf shows rich morphological and kinematic structure at this physical resolution . the outer disk contains tenuous gas , but column densities rise above the 5@xmath910@xmath18 @xmath17 level at intermediate radii . there is plentiful high - column density ( @xmath1910@xmath20 @xmath17 ) throughout the disk . the more or less parallel iso - velocity contours at inner radii are indicative of linear rotation ( although almost certainly not solid body ) and the curving of the outer contours suggests that the outer rotation curve has a fairly constant velocity . the outer disk contours show no evidence for a decrease in rotational velocity at large radii . in the central regions of the disk , however , some `` holes '' or `` depressions '' manifest a pronounced kink in these contours ( consider the contours at 370@xmath420 ) . the intensity weighted velocity dispersion averages to @xmath87 - 8 throughout the disk , although the innermost regions show dispersions above 10 . + the total flux integral , proportional to the disk mass , was found to be 50.3@xmath45.1 jykm s@xmath21 , a value somewhat lower than the single - dish flux measure of 80.6@xmath47.72 jykm s@xmath21 by @xcite ; the difference may arise from the lack of short interferometric spacings that provide sensitivity to diffuse structure . the total mass is found to be m@xmath15 @xmath0 ( 3.5@xmath40.5)@xmath910@xmath22 . after applying the usual 35% correction for helium and molecular material , we adopt m@xmath23 @xmath0 ( 4.7@xmath40.7)@xmath910@xmath22 as the total gas mass . + in figure ( [ reswri ] ) we plot the 10 /20 resolution surface density , throughout the gas disk . a simple fit ( valid out to the last measured point and for the scope of this work ) yields : _ m_/pc^2 , where @xmath24 is in kpc . the related fitting uncertainty on @xmath25 is about @xmath26 . figure ( [ reswri ] ) shows that the surface density rises from the center of the galaxy , reaches a maximum , and then declines exponentially . at the last measured point , i.e. out to @xmath8 7 kpc , the profile has almost ( though not completely ) reached the edge of the disk and rapidly converges to zero . note that , in newtonian gravity , the outer gaseous disk contributes in a negligible way to the galaxy total gravitational potential . the channel maps of the orion dwarf provide evidence of well - ordered rotation throughout the disk ( see cannon et al . the intensity - weighted - mean velocity field ( figure ( [ figcap3]b ) ) exhibits symmetric structure in the outer disk . twisted iso - velocity contours at inner radii coincide with the holes near the centre of the disk . the disk is therefore dominated by circular motion . the rc of the galaxy was derived by fitting a tilted ring model to the intensity - weighted - mean velocity field using the gipsy task rotcur . the routine carries out a least - squares fit to @xmath27 , the line of sight - velocity . to derive the best - fitting model , an iterative approach was adopted in which the various combinations of the parameters were fitted . the final rc was extract by fixing all other parameters . the receding and approaching sides of the galaxy were fitted separately . the best fitting parameters are @xmath28 , @xmath29 , @xmath30 km / s , and ( @xmath31,@xmath32 ) = ( 05:45:01.66 , 05:03:55.2 ) for the dynamical centre . we have realized that the inclination is not dependent on the radius , and the fit is shown in figure ( [ inclination ] ) . its weighted value is @xmath33 . notice that because the errors reported by gipsy / rotcur include only errors on the fits and systematics are not included , the @xmath34 error estimate comes from attempting the rotcur fits in various orders ( e.g. , holding each variable fixed in turn ) . + the resulting rc is shown in figure ( [ asymm ] ) . notice that in this object the disk inclination is determined kinematically and therefore it is quite accurate . no result of this paper changes by adopting different values of @xmath35 , inside the quoted errorbar . the second - order moment map for the galaxy is shown in figure ( [ figcap3]c ) . throughout most of the disk , the velocity dispersion is roughly constant at @xmath14 @xmath36 7@xmath42 km / s , with a more complex behaviour near the galaxy centre and at the outermost radii . this velocity dispersion estimate allows us to derive the asymmetric drift correction to the rc yielded by the tilted ring model . the observed rotation velocity , @xmath37 , is related to the circular velocity @xmath38 via v^2_c(r)= v^2_rot(r ) - ^2(r ) . [ circ ] from an examination of figure ( [ asymm ] ) it is clear that the @xmath37 and @xmath38 profiles differ by less than 1@xmath39 . throughout this paper , we use the latter for the purposes of mass modelling . we notice that in very small dwarfs this correction is not negligible ( @xmath40 ) and it introduces an uncertainty in the analysis , e.g. @xcite . in summary , the orion dwarf rc has a spatial resolution of 0.26 kpc ( i.e. 0.2 @xmath13 ) , and extends out to 5.1 r@xmath41 . the uncertainties on the rc are few km / s and the error on the rc slope @xmath42 . is the circular velocity given by eq . ( [ circ ] ) a proper estimate of the gravitational field ? to further investigate the presence of non - circular motions within the disk that jeopardize the kinematics , we carried out a harmonic decomposition of the intensity - weighted velocity field to search for any significant non - circular components . this test is necessary in that the undetected presence of non - circular motions can lead to incorrect parametrization of the total mass distribution . + following @xcite , the line - of - sight velocities from the velocity field are decomposed into harmonic components up to order @xmath43 according to @xmath44 where @xmath45 is the systemic velocity , @xmath46 and @xmath47 are the magnitudes of the harmonic components , @xmath48 the harmonic number , and @xmath49 the azimuthal angle in the plane of the galaxy . the gipsy task reswri was used to carry out the decomposition by fitting a purely circular model to the velocity field , subtracting it from the data , and then determining from the residual the magnitudes of the non - circular components . the tilted ring model fitted by reswri had its kinematic centre fixed to that of the purely circular tilted ring model used to derive the rc above . the position angles and inclinations were fixed to constant values of 20 and @xmath50 , respectively . the parameters of the best - fitting model are shown in figure ( [ reswri4 ] ) . adjacent points are separated by a beam width in order to ensure that they are largely independent of one another . we argue that because the standard tilted ring model has fewer free parameters than the model incorporating the higher order fourier components , it is not as essential to space the points on the rotation curve by a full beam width . then in this model only @xmath51 points are considered instead of the @xmath52 points used in fitting the rc . + at inner radii the inferred non - circular motions are not negligible , but this is almost certainly due to the fact that the distribution over this portion of the disk is irregular , being dominated by the large central under - densities . the harmonic components of the outer disk are , instead , reliable and demonstrate the gas flow to be dominated by circular kinematics . the circular velocity so obtained well matches that found by means of the tilted ring model presented above . the amplitudes of @xmath53 and @xmath54 are too small to hide a cusp inside an apparently solid body rc ( as suggested by @xcite ) . these results provide further decisive support for the use of @xmath37 of the orion dwarf as a tracer of its mass distribution . we model the orion dwarf as consisting of two `` luminous '' components , namely the stellar and the gaseous disks , embedded in a dark halo . the stellar component is modelled as an exponential thin disk @xcite with a scale length of 1.33 kpc . any bulge component is assumed to be negligible in terms of mass . the dynamical contribution of the gas to the observed rc is derived from the total intensity map . a scaling factor of 1.33 is incorporated to account for the presence of helium and other elements . for the dark halo we consider two different parametrizations of the mass distribution : an nfw profile @xcite and the cored profile of the halo universal rotation curve ( urch ) @xcite . it is well known that the nfw profile is one outcome in numerical simulations of cold dark matter structure formation , whereas the cored profile ( an empirical result ) , by design , fits the broad range of rc shapes of spiral galaxies . the rc is modelled as the quadrature sum of the rcs of the individual mass components : @xmath55 for the cored halo parametrization we adopt the urch profile : v^2_urch(r)&=&6.4 , where the disk mass , the core radius @xmath56 and the central halo density @xmath57 are free parameters . it is evident that this model yields a total rc that fits the data extremely well ( see figure ( [ urcnfw ] ) left panel ) , with best - fitting parameters of @xmath58 kpc , m@xmath59 m@xmath60 and @xmath61g/@xmath62 . more accurate statistics is not necessary ; the mass model predicts all the @xmath63 data points within their observational uncertainty . notice that the derived value of the disk mass agrees with the photometric estimate discussed above . the corresponding virial mass and radius of the dm halo are m@xmath64 m@xmath60 ( see eq . 10 in @xcite ) and r@xmath65 kpc @xcite , respectively . we note that the orion dwarf has a mass 20 times smaller than that of the milky way , with the dm halo dominating the gravitational potential at all galactocentric radii . the baryonic fraction is @xmath66 , while the gas fraction is @xmath67 . the rc for the nfw dark matter profile is v^2_nfw(r)=v^2_vir , where @xmath68 , @xmath69^{-1}$ ] and @xmath70 is the concentration parameter ( see @xcite ) . we fitted the rc of the orion dwarf by adjusting m@xmath71 and m@xmath72 . the resulting best - fit values are m@xmath73 m@xmath60 and m@xmath74 m@xmath60 , but since @xmath75 , i.e. the fit is unsuccessful , the best - fit values of the free parameters and those of their fitting uncertainties do not have a clear physical meaning . we plot the results in the right panel of figure ( [ urcnfw ] ) . the nfw model , at galactocentric radii r@xmath762 kpc , overestimates the observed circular velocity ( see figure ( [ urcnfw ] ) right panel ) . an alternative to newtonian gravity was proposed by @xcite to explain the phenomenon of mass discrepancy in galaxies . it was suggested that the true acceleration @xmath77 of a test particle , at low accelerations , is different from the standard newtonian acceleration , @xmath78 : a= , where @xmath79 is an interpolation function and @xmath80 cm s@xmath81 is the critical acceleration at which the transition occurs ( see @xcite ) . for this we adopt the following form of the interpolation function ( see @xcite ) : ( a / a_0)=. in this framework the circular velocity profile can be expressed as a function of @xmath82 and of the standard newtonian contribution of the baryons to the rc , v@xmath83=(v@xmath84+v@xmath85)@xmath86 , obtaining for it v^2_mond = v^2_bar(r ) ( ) , [ mond ] ( see @xcite ) ( [ mond ] ) shows that in the mond framework the resulting rc is similar to the no - dm standard newtonian one , with an additional term that works to mimic and substitute for the dm component @xcite . no result of this paper changes by adopting the `` standard '' mond interpolation function ( see @xcite ) . the best - fitting mond mass model is shown in figure ( [ mond1 ] ) . the model total rc ( cyan line ) completely fails to match the observations . we fix the stellar mass m@xmath41 at m@xmath87 . if we let the disk mass becomes higher , covering the mass range estimated in @xcite , the fit is not even able to reproduce the rc at inner radii . note that in the mond formalism , the distance of the galaxy and the amount of gaseous mass are both crucial in deriving the model rc . to quantify the discrepancy of these observations with the mond formalism , note that only if the orion dwarf were 1.9 times more distant than the current estimate we would obtain a satisfactory fit to the rc ( see figure ( [ mond2 ] ) ) . the orion dwarf galaxy is representative of a population of dwarfs with a steep inner rc that gently flattens at the edge of the gas disk . the observed kinematics imply the presence of large amounts of dm also in the central regions . we have used new observations of the orion dwarf to analize its kinematics and derive the mass model . the derived rc is very steep and it is dominated by dm at nearly all galactocentric radii . baryons are unable to account for the observed kinematics and are only a minor mass component at all galactic radii . we have used various mass modeling approaches in this work . using the nfw halo , we find that this model fails to match the observed kinematics ( as occurs in other similar dwarfs ) . we show that non - circular motions can not resolve this discrepancy . then we modelled the galaxy by assuming the urch parametrization of the dm halo . we found that this cored distribution fits very well the observed kinematics . orion is a typical dwarf showing a cored profile of the dm density and the well - known inability of dm halo cuspy profiles to reproduce the observed kinematics . finally , we find that the mond model is discrepant with the data if we adopt the literature galaxy distance and gas mass . the kinematic data can be reproduced in the mond formalism if we allow for significant adjustments of the distance and/or value of the gas mass . let us point out that the present interferometric observations may miss some of the objects flux , although this may be limited in that the cubes do not have significant negative bowls . obviously , for bigger values of the mass , the distance at which the baryon components would well fit the data will also somewhat decrease . + it is worth stressing that there is a galaxy distance ( albeit presently not - favoured ) for which mond would strike an extraordinary success in reproducing the observed kinematics of the orion dwarf . the orion dwarf has a favorable inclination , very regular gas kinematics , a small asymmetric drift correction , a well - understood baryonic matter distribution , and a large discrepancy between luminous and dynamical mass . all of these characteristics make this system a decisive benchmark for the mond formalism and a promising target for further detailed studies . of particular value would be a direct measurement of the distance ( for example , infrared observations with the hubble space telescope would allow a direct distance measurement via the magnitude of the tip of the red giant branch ) . the authors would like to thank the referee , gianfranco gentile , for his very fruitful comments that have increased the level of presentation of the paper .

dwarf galaxies are good candidates to investigate the nature of dark matter , because their kinematics are dominated by this component down to small galactocentric radii . we present here the results of detailed kinematic analysis and mass modelling of the orion dwarf galaxy , for which we derive a high quality and high resolution rotation curve that contains negligible non - circular motions and we correct it for the asymmetric drift . moreover , we leverage the proximity ( d @xmath0 5.4 kpc ) and convenient inclination ( 47@xmath1 ) to produce reliable mass models of this system . we find that the universal rotation curve mass model ( freeman disk @xmath2 burkert halo @xmath2 gas disk ) fits the observational data accurately . in contrast , the nfw halo + freeman disk @xmath2 gas disk mass model is unable to reproduce the observed rotation curve , a common outcome in dwarf galaxies . finally , we attempt to fit the data with a modified newtonian dynamics ( mond ) prescription . with the present data and with the present assumptions on distance , stellar mass , constant inclination and reliability of the gaseous mass , the mond `` amplification '' of the baryonic component appears to be too small to mimic the required `` dark component '' . the orion dwarf reveals a cored dm density distribution and a possible tension between observations and the canonical mond formalism . [ firstpage ] dark matter ; galaxy : orion dwarf ; mass profiles

in recent years considerable progress has been achieved in the partonic interpretations of diffractive processes in @xmath1 collisions ( see e.g. @xcite ) , most of diffractive studies performed up to now at hera have been based on the characteristic presence of a rapidity gap in the diffractive final state . the precision of this method is limited by the uncertainty related to the presence of dissociated proton background events . the only precise and unambiguous way of studying diffraction is by tagging the diffracted proton and measuring its four momentum by means of a proton spectrometer . such devices have been used by the h1 and zeus collaborations and have delivered interesting results , but their acceptances are small , with the result that the collected statistics are limited and large systematic errors affect the measurements . to fully profit from the hera luminosity upgrade in the study of diffraction after the year 2003 , a very forward proton spectrometer ( vfps ) which identifies and measures the momentum of the diffracted proton with a high acceptance has therefore been installed by h1 . this contribution reports the vfps tagging performance using diffractive events collected during hera running at high energy ( 27.5 gev for the electron / positron beam and 920 gev for the proton ) in 2006 and 2007 . the vfps @xcite is a set of two `` roman pots '' located at 118 m and 222 m downstream of the h1 interaction point . each pot consists of an insert into the beam pipe , allowing two tracking detectors equipped with scintillating fibres to be moved very close to the proton beam . + many aspects of the design of the roman pots , including the stainless plunger vessel and the scintillating fiber detectors , are adaptations of the fps proton spectrometer @xcite , installed and operational in h1 since 1994 . both detectors of each roman pot consists of two planes of scintillating fibres oriented at @xmath2 w.r.t . the horizontal plane and moving perpendicularly to the beam line direction . each detector allows to reconstruct of the position of one impact point of the scattered proton trajectory with a precision of about 100 @xmath3 m . for triggering purposes each detector is sandwiched between 2 scintillating planes which are connected to different pm s . a trigger signal , corresponding to a activity in at least 3 planes out of four , is delivered separately for each station at the first trigger level . the vfps have been installed at the very end of 2003 . radiation damage of the optical readout fiber prohibited data taking during 2004 . hence data available for physics r0.5 analysis started in 2005 . the bulk of data were taken in 2006 and 2007 , they correspond to and integrated luminosity of 140 @xmath4 . from an operational point of view , the vfps was into data taking position for 70% of the luminosity collected by h1 . + the @xmath5 beam orbit has been modified in april 2006 to increase the vfps acceptance . a large fraction of protons with a energy loss above 2% w.r.t . the beam energy are hitting the beam pipe around 200 m when the nominal orbit is used . the orbit has been changed moving the @xmath5 outwards hera by 6 mm at about 200 m from the interaction point . this procedure leads to an increase of the rate of events tagged by the vfps at @xmath6 ( see fig . [ fig : kick ] ) . to study the description of the beam optics and of the vfps system by our simulation , first , vfps tagged events , i.e. with a fired trigger , are compared to the full sample of diffractive events selected using the information from the main detector using the rapidity gap method ( see e.g. @xcite ) . the full event sample is selected asking for an electromagnetic cluster in the backward ( lepton beam direction ) calorimeter spacal of more than 10 gev ( corresponding to the scattered electron candidate ) , a reconstructed vertex and that the most forward particle in the main detector has pseudo - rapidity of less than 2.5 ( this latest condition is equivalent to asking for a rapidity gap ) . additionally the forward muon detector should not have recorded a signal above the noise level . one can then look to what fraction of this sample largely dominated by diffractive events is tagged by vfps . this selection was applied to produce the fig . [ fig : kick ] discussed in the previous section . over the 140 @xmath4 collected , 880,000 events are tagged by the vfps . if a kinematic cut of @xmath7 gev@xmath8 is applied , 215,000 events remain . + this sample is compared to the sum of diffractive and background contributions as estimated by monte carlo . in fig . [ fig : dis ] data corresponding to about 1 month of running in @xmath9 mode with a 6 mm bump applied ( 24 @xmath4 ) are compared to monte carlo predictions ( see figure caption for details ) . + ( 100,120 ) ( -10,-10 ) . * right : * as a function of @xmath0 . the monte carlo simulation contains contributions from pomeron exchange with elastically scattered proton ( ep ip ) , reggeon exchange with elastically scattered proton ( ep ir ) , pomeron and reggeon exchange with proton dissociation ( pdiss ip+ir ) . among them simulated events tagged by the vfps are shown ( mc vfps tag).,title="fig : " ] ( 206,-10 ) . * right : * as a function of @xmath0 . the monte carlo simulation contains contributions from pomeron exchange with elastically scattered proton ( ep ip ) , reggeon exchange with elastically scattered proton ( ep ir ) , pomeron and reggeon exchange with proton dissociation ( pdiss ip+ir ) . among them simulated events tagged by the vfps are shown ( mc vfps tag).,title="fig : " ] ( 120,69 ) . * right : * as a function of @xmath0 . the monte carlo simulation contains contributions from pomeron exchange with elastically scattered proton ( ep ip ) , reggeon exchange with elastically scattered proton ( ep ir ) , pomeron and reggeon exchange with proton dissociation ( pdiss ip+ir ) . among them simulated events tagged by the vfps are shown ( mc vfps tag).,title="fig : " ] a good agreement is found between the full data sample and the monte carlo ( normalized to the data luminosity ) . the trend in @xmath0 of vfps tagged sample is described by the monte carlo . a more precise understanding of the vfps acceptance is needed and will lead to a better description in @xmath0 . the analysis of diffractive dijets in dis regime is based on 42.6@xmath10 and 54.7@xmath11 @xmath4 taken in 2006 . additionally to the selection applied in the previous section , a requirement of at least two jets ( using the @xmath12 algorithm ) is asked , with a minimal transverse momentum in the photon - proton frame of @xmath13 and @xmath14 respectively for the first and the second jet . the jets are asked to be well contained in the main detector , by requirering @xmath15 $ ] . + the @xmath0 distribution is shown in fig . [ fig : jets]a comparing the full dijet sample and vfps tagged dijet sample . this plot illustrates the well suited acceptance of the vfps for the dijets production in diffraction . in fig . [ fig : jets]b the transverse momentum of the first jet in the laboratory frame is shown . here again the full dijet sample is compared to the vfps tagged dijet sample . ( 100,80 ) ( 0,0 ) distribution of dijet diffractive events in dis . the highest histogram corresponds to the full dijet sample and the lowest one to the vfps tagged dijet sample . * b ) * highest transverse momentum of jets in the laboratory frame of dijet diffractive events in dis . the highest histogram corresponds to the full dijet sample and the lowest one to the vfps tagged dijet sample . * c ) * highest transverse momentum of jets in the laboratory frame of dijet vfps tagged diffractive events in photoproduction . , title="fig : " ] ( 140,0 ) distribution of dijet diffractive events in dis . the highest histogram corresponds to the full dijet sample and the lowest one to the vfps tagged dijet sample . * b ) * highest transverse momentum of jets in the laboratory frame of dijet diffractive events in dis . the highest histogram corresponds to the full dijet sample and the lowest one to the vfps tagged dijet sample . * c ) * highest transverse momentum of jets in the laboratory frame of dijet vfps tagged diffractive events in photoproduction . , title="fig : " ] ( 280,0 ) distribution of dijet diffractive events in dis . the highest histogram corresponds to the full dijet sample and the lowest one to the vfps tagged dijet sample . * b ) * highest transverse momentum of jets in the laboratory frame of dijet diffractive events in dis . the highest histogram corresponds to the full dijet sample and the lowest one to the vfps tagged dijet sample . * c ) * highest transverse momentum of jets in the laboratory frame of dijet vfps tagged diffractive events in photoproduction . , title="fig : " ] ( 22,58)a ) ( 162,58)b ) ( 300,65)c ) to record with a high efficiency diffractive dijet events in photoproduction tagged by the vfps , a special trigger has been developed . it allows to lower the threshold in jet transverse momentum down to 5 gev . a luminosity of 23.7 @xmath4 has been collected in 2006 and 2007 with that trigger corresponding to a selected sample of 6000 events . the selection criteria are the same as in the dijet dis case except that the scattered electron escapes undetected , at small angle , in the beam pipe . figure [ fig : jets]c shows the transverse momentum of the first jet in the laboratory frame ( equivalent to the photon - proton frame for the transverse direction in the present photoproduction case ) . the distribution of vfps tagged events can not be compared to a full dijet sample as no trigger allowed to keep efficiently those events down to a transverse momentum of 5 gev . the vfps has run successfully collecting a luminosity of 140 @xmath4 . the observed acceptance is high ( above 60% ) in a region of @xmath0 around @xmath16 . the trend in @xmath0 of diffractive events tagged by the vfps is described by the monte carlo simulation . important statistics have been collected ( 880,000 diffractive dis events , 800 dijets diffractive dis events and 6000 dijets diffractive events in photoproduction ) for diffractive structure function measurement and qcd factorisation tests . the proton momentum reconstruction based on vfps fiber information is still in progress . 99 l. favart , _ experimental review of diffractive phenomena _ , in proceedings of the 10@xmath17 international baryons conference ( baryons 2004 ) , [ hep - ex/0501052 ] . , proposal submitted to the physics research committee , * prc-01/00 * , h1 note * h1 - 05/00 - 582 * , http://www-h1.desy.de/h1det/tracker/vfps/

the very forward proton spectrometer ( vfps ) of the h1 experiment at hera is collecting data since 2005 . the fiber detectors in the roman pots located at 218 and 222 m downstream from the h1 interaction point , tag and measure diffractively scattered protons with a high acceptance in the @xmath0 range [ 0.01 , 0.025 ] . the experimental set up and the spectrometer tagging performance using diffractive events collected during 2006 and 2007 are discussed .

it is by now well - established that neutrinos are massive and mixed , and that these properties lead to the oscillations observed in measurements of neutrinos produced in the sun @xcite@xcite , in the atmosphere @xcite , by accelerators @xcite , and by reactors @xcite . the mixing model predicts not only neutrino oscillations in vacuum , but also the effects of matter on the oscillation probabilities ( the ` msw ' effect ) @xcite . to date , the effects of matter have only been studied in the solar sector , where the neutrinos passage through the core of both the sun and the earth can produce detectable effects . the model predicts three observable consequences for solar neutrinos : a suppression of the @xmath6 survival probability below the average vacuum value of @xmath16 for high - energy ( @xmath4b ) neutrinos , a transition region between matter - dominated and vacuum - dominated oscillations , and a regeneration of @xmath6s as the neutrinos pass through the core of the earth ( the day / night effect ) . in addition to improved precision in the extraction of the total flux of @xmath4b neutrinos from the sun , an advantage of the low energy threshold analysis ( leta ) presented here is the enhanced ability to explore the msw - predicted transition region and , in addition , more stringent testing of theories of non - standard interactions that affect the shape and position of the predicted rise in survival probability @xcite@xcite . we present in this article a joint analysis of the data from the first two data acquisition phases of the sudbury neutrino observatory ( sno ) , down to an effective electron kinetic energy of @xmath0 mev , the lowest analysis energy threshold yet achieved for the extraction of neutrino signals with the water cherenkov technique . the previous ( higher threshold ) analyses of the two data sets have been documented extensively elsewhere @xcite , and so we focus here on the improvements made to calibrations and analysis techniques to reduce the threshold and increase the precision of the results . we begin in section [ sec : detector ] with an overview of the sno detector and physics processes , and provide an overview of the data analysis in section [ sec : anal_overview ] . in section [ sec : dataset ] we briefly describe the sno phase i and phase ii data sets used here . section [ sec : montecarlo ] describes changes to the monte carlo detector model that provides the distributions used to fit our data , and section [ sec : hitcal ] describes the improvements made to the hit - level calibrations of pmt times and charges that allow us to eliminate some important backgrounds . sections [ sec : recon]- [ sec : beta14 ] describe our methods for determining observables like position and energy , and estimating their systematic uncertainties . section [ sec : cuts ] describes the cuts we apply to our data set , while section [ sec : treff ] discusses the trigger efficiency and section [ sec : ncap ] presents the neutron capture efficiency and its systematic uncertainties . we provide a detailed discussion of all background constraints and distributions in section [ sec : backgrounds ] . section [ sec : sigex ] describes our ` signal extraction ' fits to the data sets to determine the neutrino fluxes , and section [ sec : results ] gives our results for the fluxes and mixing parameters . sno was an imaging cherenkov detector using heavy water ( @xmath10h@xmath17o , hereafter d@xmath17o ) as both the interaction and detection medium @xcite . sno was located in vale inco s creighton mine , at @xmath18 n latitude , @xmath19 w longitude . the detector was 1783 m below sea level with an overburden of 5890 meters water equivalent , deep enough that the rate of cosmic - ray muons passing through the entire active volume was just 3 per hour . one thousand metric tons ( tonnes ) of d@xmath17o was contained in a 12 m diameter transparent acrylic vessel ( av ) . cherenkov light produced by neutrino interactions and radioactive backgrounds was detected by an array of 9456 hamamatsu model r1408 20 cm photomultiplier tubes ( pmts ) , supported by a stainless steel geodesic sphere ( the pmt support structure or psup ) . each pmt was surrounded by a light concentrator ( a ` reflector ' ) , which increased the effective photocathode coverage to nearly @xmath20% . the channel discriminator thresholds were set to 1/4 of a photoelectron of charge . over seven kilotonnes ( 7@xmath21 kg ) of h@xmath17o shielded the d@xmath17o from external radioactive backgrounds : 1.7 kt between the av and the psup , and 5.7 kt between the psup and the surrounding rock . extensive purification systems were used to purify both the d@xmath17o and the h@xmath17o . the h@xmath17o outside the psup was viewed by 91 outward - facing 20 cm pmts that were used to identify cosmic - ray muons . an additional 23 pmts were arranged in a rectangular array and suspended in the outer h@xmath17o region to view the neck of the av . they were used primarily to reject events not associated with cherenkov light production , such as static discharges in the neck . the detector was equipped with a versatile calibration - source deployment system that could place radioactive and optical sources over a large range of the @xmath22-@xmath23 and @xmath24-@xmath23 planes ( where @xmath23 is the central axis of the detector ) within the d@xmath17o volume . deployed sources included a diffuse multi - wavelength laser that was used to measure pmt timing and optical parameters ( the ` laserball ' ) @xcite , a @xmath25n source that provided a triggered sample of 6.13 mev @xmath26s @xcite , and a @xmath4li source that delivered tagged @xmath27s with an endpoint near 14 mev @xcite . in addition , 19.8 mev @xmath26s were provided by a @xmath28 ( ` pt ' ) source @xcite and neutrons by a @xmath29cf source . some of the sources were also deployed on vertical lines in the h@xmath17o between the av and psup . ` spikes ' of radioactivity ( @xmath30na and @xmath31rn ) were added at times to the light water and d@xmath17o volumes to obtain additional calibration data . table [ tbl : cal_sources ] lists the primary calibration sources used in this analysis .

results are reported from a joint analysis of phase i and phase ii data from the sudbury neutrino observatory . the effective electron kinetic energy threshold used is @xmath0 mev , the lowest analysis threshold yet achieved with water cherenkov detector data . in units of @xmath1 @xmath2 s@xmath3 , the total flux of active - flavor neutrinos from @xmath4b decay in the sun measured using the neutral current ( nc ) reaction of neutrinos on deuterons , with no constraint on the @xmath4b neutrino energy spectrum , is found to be @xmath5 these uncertainties are more than a factor of two smaller than previously published results . also presented are the spectra of recoil electrons from the charged current reaction of neutrinos on deuterons and the elastic scattering of electrons . a fit to the sno data in which the free parameters directly describe the total @xmath4b neutrino flux and the energy - dependent @xmath6 survival probability provides a measure of the total @xmath4b neutrino flux @xmath7 . combining these new results with results of all other solar experiments and the kamland reactor experiment yields best - fit values of the mixing parameters of @xmath8 degrees and @xmath9 ev@xmath10 . the global value of @xmath11 is extracted to a precision of @xmath12% . in a three - flavor analysis the best fit value of @xmath13 is @xmath14 . this implies an upper bound of @xmath15 ( 95% c.l . ) .

the origin and nature of the dark energy @xcite is one of the most difficult challenges facing physicists and cosmologists now . among all the proposed models to tackle this problem , a scalar field is perhaps the most popular one up to now . the scalar field , denoted by @xmath1 , might only interact with other matter species through gravity , or have a coupling to normal matter and therefore producing a fifth force on matter particles . this latter idea has seen a lot of interests in recent years , in the light that such a coupling could potentially alleviate the coincidence problem of dark energy @xcite and that it is commonly predicted by low energy effective theories from a fundamental theory . nevertheless , if there is a coupling between the scalar field and baryonic particles , then stringent experimental constraints might be placed on the fifth force on the latter provided that the scalar field mass is very light ( which is needed for the dark energy ) . such constraints severely limit the viable parameter space of the model . different ways out of the problem have been proposed , of which the simplest one is to have the scalar field coupling to dark matter only but not to standard model particles , therefore evading those constraints entirely . this is certainly possible , especially because both dark matter and dark energy are unknown to us and they may well have a common origin . another interesting possibility is to have the chameleon mechanism @xcite , by virtue of which the scalar field acquires a large mass in high density regions and thus the fifth force becomes undetectablly short - ranged , and so also evades the constraints . study of the cosmological effect of a chameleon scalar field shows that the fifth force is so short - ranged that it has negligible effect in the large scale structure formation @xcite for certain choices of the scalar field potential . but it is possible that the scalar field has a large enough mass in the solar system to pass any constraints , and at the same time has a low enough mass ( thus long range forces ) on cosmological scales , producing interesting phenomenon in the structure formation . this is the case of some @xmath2 gravity models @xcite , which survives solar system tests thanks again to the chameleon effect @xcite . note that the @xmath2 gravity model is mathematically equivalent to a scalar field model with matter coupling . no matter whether the scalar field couples with dark matter only or with all matter species , it is of general interests to study its effects in cosmology , especially in the large scale structure formation . indeed , at the linear perturbation level there have been a lot of studies about the coupled scalar field and @xmath2 gravity models which enable us to have a much clearer picture about their behaviors now . but linear perturbation studies do not conclude the whole story , because it is well known that the matter distribution at late times becomes nonlinear , making the behavior of the scalar field more complex and the linear analysis insufficient to produce accurate results to confront with observations . for the latter purpose the best way is to perform full @xmath0-body simulations @xcite to evolve the individual particles step by step . @xmath0-body simulations for scalar field and relevant models have been performed before @xcite . for example , in @xcite the simulation is about a specific coupled scalar field model . this study however does not obtain a full solution to the spatial configuration of the scalar field , but instead simplifies the simulation by assuming that the scalar field s effect is to change the value of the gravitational constant , and presenting an justifying argument for such an approximation . as discussed in @xcite , this approximation is only good in certain parameter spaces and for certain choices of the scalar field potential , and therefore full simulations tare needed to study the scalar field behaviour more rigorously . recently there have also appeared @xmath0-body simulations of the @xmath2 gravity model @xcite , which do solve the scalar degree of freedom explicitly . however , embedded in the @xmath2 framework there are some limitations in the generality of these works . as a first thing , @xmath2 gravity model ( no matter what the form @xmath3 is ) only corresponds to the couple scalar field models for a specific value of coupling strength @xcite . second , in @xmath2 models the correction to standard general relativity is through the modification to the poisson equation and thus to the gravitational potential as a whole @xcite , while in the coupled scalar field models we could clearly separate the scalar fifth force from gravity and analyze the former directly @xcite . also , in @xmath2 models , as well as the scalar - tensor theories , the coupling between matter and the scalar field is universal ( the same to dark matter and baryons ) , while in the couple scalar field models it is straightforward to switch on / off the coupling to baryons and study the effects on baryonic and dark matter clusterings respectively ( as we will do in this paper ) . correspondingly , the general framework of @xmath0-body simulations in coupled scalar field models could also handle the situation where the chameleon effect is absent and/or scalar field only couples to dark matter , and thus provide a testbed for possible violations of the weak equivalence principle . in this paper we shall go beyond @xcite and consider the case where the chameleon scalar field couples differently to different species of matter . to be explicit , we consider two matter species , and let one of them have no coupling to the scalar field . because it is commonly believed that normal baryons , being observable in a variety of experiments , should have extremely weak ( if any ) coupling to scalar fields , we call the uncoupled matter species in our simulation `` baryons '' . it is however reminded here that this matter species is not really baryonic in the sense that it does not experience normal baryonic interactions . the inclusion of true baryons will make the investigation more complicated and is thus beyond the scope of the present work . the paper is organized as follows : in [ sect : eqns ] we list the essential equations to be implemented in the @xmath0-body simulations and describe briefly the difference from normal lcdm simulations . [ sect : simu ] is the main body of the paper , in which [ subsect : simu_detail ] gives the details about our simulations , such as code description and parameter set - up , [ subsect : simu_result ] displays some preliminary results for visualization , such as baryon / cdm distribution , potential / scalar field configuration and the correlation between the fifth force ( for cdm particles ) and gravity ; [ subsect : simu_ps ] quantifies the nonlinear matter power spectrum of our model , especially the difference from lcdm results and the bias between cdm and baryons ; [ subsect : simu_mf ] briefly describes the essential modifications one must bear in mind when identifying _ virialized _ halos from the simulation outputs and shows the mass functions for our models ; [ subsect : simu_prof ] we pick out two halos from our simulation box and analyzes their total internal profiles , as well as their baryonic / cdm density profiles . we finally summarize in [ sect : con ] . in this section we first describe the method to simulate structure formation with two differently coupled matter species and the appropriate equations to be used . those equations for a single matter species have been discussed in details previously in @xcite , but the inclusion of different matter species requires further modifications and we list all these for completeness . the lagrangian for our coupled scalar field model is @xmath4 + v(\varphi ) - c(\varphi)\mathcal{l}_{\mathrm{cdm } } + \mathcal{l}_{\mathrm{s}}\ \\end{aligned}\ ] ] where @xmath5 is the ricci scalar , @xmath6 with @xmath7 newton s constant , @xmath1 is the scalar field , @xmath8 is its potential energy and @xmath9 its coupling to dark matter , which is assumed to be cold and described by the lagrangian @xmath10 . @xmath11 includes all other matter species , in particular our _ baryons_. the contribution from photons and neutrinos in the @xmath0-body simulations ( for late times , _ i.e. _ , @xmath12 ) is negligible , but should be included when generating the matter power spectrum from which the initial conditions for our @xmath0-body simulations are obtained ( see below ) . the dark matter lagrangian for a point - like particle with bare mass @xmath13 is @xmath14 where @xmath15 is the coordinate and @xmath16 is the coordinate of the centre of the particle . from this equation it can be easily derived that @xmath17 also , because @xmath18 where @xmath19 is the four velocity of the dark matter particle , the lagrangian could be rewritten as @xmath20 which will be used below . ( [ eq : dmemt_particle ] ) is just the energy momentum tensor for a single dark matter particle . for a fluid with many particles the energy momentum tensor will be @xmath21 in which @xmath22 is a volume microscopically large and macroscopically small , and we have extended the 3-dimensional @xmath23 function to a 4-dimensional one by adding a time component . here @xmath19 is the averaged 4-velocity of the collection of particles inside this volume , and is not necessarily the same as the @xmath24-velocity of the observer . meanwhile , using @xmath25 it is straightforward to show that the energy momentum tensor for the scalar field is given by @xmath26.\end{aligned}\ ] ] so the total energy momentum tensor is @xmath27\nonumber\\ & & + c(\varphi)t^{\mathrm{cdm}}_{ab } + t^{\mathrm{s}}_{ab}\end{aligned}\ ] ] where @xmath28 , @xmath29 is the energy momentum tensor for all other matter species including baryons , and the einstein equation is @xmath30 where @xmath31 is the einstein tensor . note that due to the coupling between the scalar field @xmath1 and the dark matter , the energy momentum tensors for either will not be conserved , and we have @xmath32 where throughout this paper we shall use a @xmath33 to denote the derivative with respect to @xmath1 . finally , the scalar field equation of motion ( eom ) from the given lagrangian is @xmath34 where @xmath35 . using eq . ( [ eq : dmlagrangian2 ] ) it can be rewritten as @xmath36 eqs . ( [ eq : emt_tot ] , [ eq : einsteineq ] , [ eq : dm_energy_conservation ] , [ eq : phieom ] ) summarize all the physics that will be used in our analysis . we will consider a special form for the scalar field potential , @xmath37^{\mu}},\end{aligned}\ ] ] where @xmath38 and @xmath39 are dimensionless constants while @xmath40 has mass dimension four . as has been discussed in @xcite , @xmath41 to evade observational constraints and @xmath39 can be set to @xmath42 without loss of generality , since we can always rescale @xmath1 as we wish . meanwhile , the coupling between the scalar field and dark matter particle is chosen as @xmath43 where @xmath44 is yet another dimensionless constant characterizing the strength of the coupling . as discussed in @xcite , the two dimensionless parameters @xmath38 and @xmath45 have clear physical meanings : roughly speaking , @xmath38 controls the time when the scalar field becomes important in cosmology while @xmath45 determines how important the scalar field would ultimately be . in fact , the potential given in eq . ( [ eq : potential ] ) is partly motivated by the @xmath2 cosmology @xcite , in which the extra degree of freedom behaves as a coupled scalar field in the einstein frame . as we can see from eq . ( [ eq : potential ] ) , the potential @xmath46 when @xmath47 while @xmath48 when @xmath49 . in the latter case , however , @xmath50 , so that the effective total potential @xmath51 has a global minimum at some finite @xmath1 . if the total potential @xmath52 is steep enough around this minimum , then the scalar field becomes very heavy and thus follows its minimum dynamically , as is in the case of the chameleon cosmology ( see _ e.g. _ if @xmath53 is not steep enough at the minimum , however , the scalar field will experience a more complicated evolution . these two different cases can be obtained by choosing appropriate values of @xmath45 and @xmath38 : if @xmath45 is very large or @xmath38 is small then we run into the former situation and if @xmath45 is small and @xmath38 is large we have the second . in reality , the situation can get even more complicated because when @xmath45 , which characterizes the coupling strength , increases , the cdm evolution could also get severely affected , which in turn has back - reactions on the scalar field itself . the @xmath0-body simulation only probes the motion of particles at late times , and we are not interested in extreme conditions such as black hole formation / evolution , which mean that taking the non - relativistic limit of the above equations should be a sufficient approximation for our purpose . the existence of the scalar field and its ( different ) couplings to matter particles lead to the following changes to the @xmath54cdm model : firstly , the energy momentum tensor has a new piece of contribution from the scalar field ; secondly , the energy density of dark matter in gravitational field equations is multiplied by the function @xmath9 , which is because the coupling to scalar field essentially renormalizes the mass of dark matter particles ; thirdly , dark matter particles will not follow geodesics in their motions as in @xmath54cdm , but rather the total force on them has a contribution , the fifth force , from the exchange of scalar field quanta ; finally , cdm particles must be distinguished from baryons so that the fifth force only acts on the former and these two species only interact gravitationally . this last point is one main difference between the present work and a previous one @xcite . these imply that the following things need to be modified or added : 1 . the scalar field @xmath1 equation of motion , which determines the value of the scalar field at any given time and position ; 2 . the poisson equation , which determines the gravitational potential ( and thus gravity ) at any given time and position , according to the local energy density and pressure , which include the contribution from the scalar field ( as obtained from @xmath1 equation of motion ) ; 3 . the total force on the dark matter particles , which is determined by the spatial configuration of @xmath1 , just like gravity is determined by the spatial configuration of the gravitational potential ; 4 . the cdm and baryonic particles must be tagged respectively so that the code knows to assign forces correctly to different species . we shall describe these one by one now . for the scalar field equation of motion , we denote @xmath55 as the background value of @xmath1 and @xmath56 as the scalar field perturbation . then eq . ( [ eq : phieom ] ) could be rewritten as @xmath57 by subtracting the corresponding background equation from it . here @xmath58 is the covariant spatial derivative with respect to the physical coordinate @xmath59 with @xmath60 the conformal coordinate , and @xmath61 . @xmath58 is essentially the @xmath62 , but because here we are working in the weak field limit we approximate it as @xmath63 by assuming a flat background ; the minus sign is because our metric convention is @xmath64 instead of @xmath65 . for the simulation here we will also work in the quasi - static limit , assuming that the spatial gradient is much larger than the time derivative , @xmath66 ( which will be justified below ) . thus the above equation can be further simplified as @xmath67,\nonumber\end{aligned}\ ] ] in which @xmath68 is with respect to the conformal coordinate @xmath60 so that @xmath69 , and we have restored the factor @xmath70 in front of @xmath71 ( the @xmath1 here and in the remaining of this paper is @xmath72 times the @xmath1 in the original lagrangian unless otherwise stated ) . note that here @xmath22 and @xmath73 both have the dimension of _ mass _ density rather than _ energy _ density . next look at the poisson equation , which is obtained from the einstein equation in weak - field and slow - motion limits . here the metric could be written as @xmath74 from which we find that the time - time component of the ricci curvature tensor @xmath75 , and then the einstein equation @xmath76 gives @xmath77 where @xmath78 and @xmath79 are respectively the total energy density and pressure . the quantity @xmath80 can be expressed in terms of the comoving coordinate @xmath60 as @xmath81 where we have defined a new newtonian potential @xmath82 and used @xmath83 . thus @xmath84\nonumber\end{aligned}\ ] ] where in the second step we have used eq . ( [ eq : einsteineqn ] ) and the raychaudhrui equation , and an overbar means the background value of a quantity . because the energy momentum tensor for the scalar field is given by eq . ( [ eq : phiemt ] ) , it is easy to show that @xmath85 $ ] and so @xmath86\right\}\nonumber\\ & & + 4\pi ga^{3 } \left\{\bar{\rho}_{\mathrm{cdm}}c(\bar{\varphi})+ \bar{\rho}_{\mathrm{b}}+ 2\left[\dot{\bar{\varphi}}^{2}-v(\bar{\varphi})\right]\right\}.\nonumber\end{aligned}\ ] ] now in this equation @xmath87 in the quasi - static limit and so could be dropped safely . so we finally have @xmath88\nonumber\\ & & + 4\pi ga^{3 } \left[\rho_{\mathrm{b}}-\bar{\rho}_{\mathrm{b}}\right ] - 8\pi ga^{3}\left[v(\varphi)-v(\bar{\varphi})\right].\ \\end{aligned}\ ] ] finally , for the equation of motion of the dark matter particle , consider eq . ( [ eq : dm_energy_conservation ] ) . using eqs . ( [ eq : dmemt_particle ] , [ eq : dmlagrangian2 ] ) , this can be reduced to @xmath89 obviously the left hand side is the conventional geodesic equation and the right hand side is the new fifth force due to the coupling to the scalar field . note that because @xmath90 is the projection tensor that projects any 4-tensor into the 3-space perpendicular to @xmath19 , so @xmath91 is the spatial derivative in the 3-space of the observer and perpendicular to @xmath92 ; consequently the fifth force @xmath93 has no component parallel to @xmath92 ( the time component ) , indicating that the energy density of cdm will be conserved and only the particle trajectories are modified , as mentioned in @xcite . remember that @xmath92 in eq . ( [ eq : dmeom ] ) is the 4-velocity of individual particles , but from eq . ( [ eq : wfphieom ] ) we see that @xmath94 is computed in the fundamental observer s frame ( where density perturbation is calculated ) , so if we also want to work on eq . ( [ eq : dmeom ] ) in the fundamental observer s frame ( so that we can use the @xmath94 from eq . ( [ eq : wfphieom ] ) directly ) , then we must rewrite eq . ( [ eq : dmeom ] ) by substituting @xmath95 up to first order in perturbations , in which @xmath96 is the 4-velocity of the fundamental observer and @xmath97 is the peculiar velocity of the particle . then the first term in the above expression is the gradient of @xmath94 observed by the fundamental observer ( rather than an observer comoving with the particle ) and the second term is a velocity dependent acceleration @xcite . in @xcite it is claimed that the second term is of big importance ; in our simulations , however , this term will be neglected ( from here on ) because it depends on @xmath98 , which is very small due to the chameleon nature of the model . we have checked in a linear perturbation computation that removing this term only changes the matter power spectrum by less than 0.0001% . now in the non - relativistic limit the spatial components of eq . ( [ eq : dmeom ] ) can be written as @xmath99 where @xmath100 is the physical time coordinate . if we instead use the comoving coordinate @xmath60 , then this becomes @xmath101 where we have used eq . ( [ eq : newphi ] ) . the canonical momentum conjugate to @xmath60 is @xmath102 so we have now @xmath103 in which eq . ( [ eq : wfdpdtcomov ] ) is for cdm particle and eq . ( [ eq : wfdpdtcomovb ] ) is for baryons . note that according to eq . ( [ eq : wfdpdtcomov ] ) the quantity @xmath104 $ ] acts as a new piece of potential : the potential for the fifth force . this is an important observation and we will come back to it later when we calculate the escape velocity of cdm particles within a virialized halo . ( [ eq : wfphieom ] , [ eq : wfpoisson ] , [ eq : wfdxdtcomov ] , [ eq : wfdpdtcomov ] , [ eq : wfdpdtcomovb ] ) will be used in the code to evaluate the forces on the dark matter particles and evolve their positions and momenta in time . in our numerical simulation we use a modified version of mlapm ( @xcite , see [ subsect : simu_detail ] ) , and we will have to change our above equations in accordance with the internal units used in that code . here we briefly summarize the main features . mlapm code uses the following internal units ( with subscript @xmath105 ) : @xmath106 in which @xmath107 is the present size of the simulation box and @xmath108 is the present hubble constant , and @xmath109 , with subscript , can represent the density for either cdm ( @xmath110 ) or baryons ( @xmath111 ) . using these newly - defined quantities , it is easy to check that eqs . ( [ eq : wfdxdtcomov ] , [ eq : wfdpdtcomov ] , [ eq : wfpoisson ] , [ eq : wfphieom ] ) could be rewritten as @xmath112,\\ \label{eq : intpoisson}\nabla^{2}\phi_{c } & = & \frac{3}{2}\omega_{\mathrm{cdm}}\bar{c } \left(\rho_{c,\mathrm{cdm}}\frac{c}{\bar{c}}-1\right)\nonumber\\ & & + \frac{3}{2}\omega_{\mathrm{b}}\left(\rho_{c,\mathrm{b}}-1\right ) - \kappa\frac{v-\bar{v}}{h^{2}_{0}}a^{3},\end{aligned}\ ] ] and @xmath113 where @xmath114 is the present cdm fractional energy density , we have again restored the factor @xmath70 and again the @xmath1 is @xmath72 times the @xmath1 in the original lagrangian . note that in eq . ( [ eq : intdpdtcomov ] ) the term in the bracket on the right hand side only applies to cdm but not to baryons . also note that from here on we shall use @xmath115 unless otherwise stated , for simplicity . we also define @xmath116 to be used below . making discretized version of the above equations for @xmath0-body simulations is non - trivial task . for example , the use of variable @xmath117 instead of @xmath1 ( appendix [ appen : discret ] ) helps to prevent @xmath118 , which is unphysical , but numerically possible due to discretization . we refer the interested readers to appendix [ appen : discret ] to the whole treatment , with which we can now proceed to do @xmath0-body runs . [ subsect : eqn_nonrel ] the full name of mlapm is multi - level adaptive particle mesh code . as the name has suggested , this code uses multilevel grids @xcite to accelerate the convergence of the ( nonlinear ) gauss - seidel relaxation method @xcite in solving boundary value partial differential equations . but more than this , the code is also adaptive , always refining the grid in regions where the mass / particle density exceeds a certain threshold . each refinement level form a finer grid which the particles will be then ( re)linked onto and where the field equations will be solved ( with a smaller time step ) . thus mlapm has two kinds of grids : the domain grid which is fixed at the beginning of a simulation , and refined grids which are generated according to the particle distribution and which are destroyed after a complete time step . one benefit of such a setup is that in low density regions where the resolution requirement is not high , less time steps are needed , while the majority of computing sources could be used in those few high density regions where high resolution is needed to ensure precision . some technical issues must be taken care of however . for example , once a refined grid is created , the particles in that region will be linked onto it and densities on it are calculated , then the coarse - grid values of the gravitational potential are interpolated to obtain the corresponding values on the finer grid . when the gauss - seidel iteration is performed on refined grids , the gravitational potential on the boundary nodes are kept constant and only those on the interior nodes are updated according to eq . ( [ eq : gs ] ) : just to ensure consistency between coarse and refined grids . this point is also important in the scalar field simulation because , like the gravitational potential , the scalar field value is also evaluated on and communicated between multi - grids ( note in particular that different boundary conditions lead to different solutions to the scalar field equation of motion ) . in our simulation the domain grid ( the finest grid that is not a refined grid ) has @xmath119 nodes , and there are a ladder of coarser grids with @xmath120 , @xmath121 , @xmath122 , @xmath123 , @xmath124 nodes respectively . these grids are used for the multi - grid acceleration of convergence : for the gauss - seidel relaxation method , the convergence rate is high upon the first several iterations , but quickly becomes very slow then ; this is because the convergence is only efficient for the high frequency ( short - range ) fourier modes , while for low frequency ( long - range ) modes more iterations just do not help much . to accelerate the solution process , one then switches to the next coarser grid for which the low frequency modes of the finer grid are actually high frequency ones and thus converge fast . the mlapm solver adopts the self - adaptive scheme : if convergence is achieved on a grid , then interpolate the relevant quantities back to the finer grid ( provided that the latter is not on the refinements ) and solve the equation there again ; if convergence becomes slow on a grid , then go to the next coarser grid . this way it goes indefinitely except when converged solution on the domain grid is obtained or when one arrives at the coarsest grid ( normally with @xmath125 nodes ) on which the equations can be solved exactly using other techniques . for our scalar field model , the equations are difficult to solve anyway , and so we truncate the coarser - grid series at the @xmath124-node one , on which we simply iterate until convergence is achieved . furthermore , we find that with the self - adaptive scheme in certain regimes the nonlinear gs solver tends to fall into oscillations between coarser and finer grids ; to avoid such situations , we then use v - cycle @xcite instead . for the refined grids the method is different : here one just iterate eq . ( [ eq : gs ] ) until convergence , without resorting to coarser grids for acceleration . as is normal in the gauss - seidel relaxation method , convergence is deemed to be achieved when the numerical solution @xmath126 after @xmath127 iterations on grid @xmath128 satisfies that the norm @xmath129 ( mean or maximum value on a grid ) of the residual @xmath130 is smaller than the norm of the truncation error @xmath131\end{aligned}\ ] ] by a certain amount , or , in the v - cycle case , the reduction of residual after a full cycle becomes smaller than a predefined threshold ( indeed the former is satisfied whenever the latter is ) . note here @xmath132 is the discretization of the differential operator eq . ( [ eq : diffop ] ) on grid @xmath128 and @xmath133 a similar discretization on grid @xmath134 , @xmath135 is the source term , @xmath136 is the restriction operator to interpolate values from the grid @xmath128 to the grid @xmath134 . in the modified code we have used the full - weighting restriction for @xmath136 . correspondingly there is a prolongation operator @xmath137 to obtain values from grid @xmath134 to grid @xmath128 , and we use a bilinear interpolation for it . for more details see @xcite . mlapm calculates the gravitational forces on particles by centered difference of the potential @xmath138 and propagate the forces to locations of particles by the so - called triangular - shaped - cloud ( tsc ) scheme to ensure momentum conservation on all grids . the tsc scheme is also used in the density assignment given the particle distribution . the main modifications to the mlapm code for our model are : 1 . we have added a parallel solver for the scalar field based on eq . ( [ eq : u_phi_eom ] ) . the solver uses a nonlinear gauss - seidel method and the same criterion for convergence as the ( linear ) gauss - seidel poisson solver . 2 . the solved value of @xmath117 is then used to calculate local mass density and thus the source term for the poisson equation , which is solved using fast fourier transform . 3 . the fifth force is obtained by differentiating the @xmath117 just like the calculation of gravity . 4 . the momenta and positions of particles are then updated taking in account of both gravity and the fifth force . there are a lot of additions and modifications to ensure smooth interface and the newly added data structures . for the output , as there are multilevel grids all of which host particles , the composite grid is inhomogeneous and thus we choose to output the positions , momenta of the particles , plus the gravity , fifth force and scalar field value _ at the positions _ of these particles . we can of course easily read these data into the code , calculate the corresponding quantities on each grid and output them if needed . as mentioned above , the most important difference of the present work from @xcite is the inclusion of baryons - the particles which do not couple to the scalar field . the baryons do not contribute to the scalar field equation of motion and are not affected by the scalar fifth force , at least directly , so that it is important to make sure that they do not mess up the physics . in the modified code we distinguish baryons and cdm particles by tagging all of them . we consider the situation where 20% of all matter particles are baryonic and 80% are cdm . at the beginning of each simulation , we loop over all particles and for each particle we generate a random number from a uniform distribution in @xmath139 $ ] . if this random number is less than 0.2 then we tag the particle as baryon , and otherwise we tag it as cdm . once these tags have been set up they will never been changed again , and the code then determines whether the particle contributes to the scalar field evolution and feels the fifth force or not according to its tag . all the simulations are started at the redshift @xmath140 . in principle , modified initial conditions ( initial displacements and velocities of particles which is obtained given a linear matter power spectrum ) need to be generated for the coupled scalar field model , because the zeldovich approximation @xcite is also affected by the scalar field coupling @xcite . in practice , however , we have found in our linear perturbation calculation @xcite that the effect on the linear matter power spectrum is negligible ( @xmath141 ) for our choices of parameters @xmath142 . another way to see that the scalar field has really negligible effects on the matter power spectrum at early times is to look at fig . [ fig : figure2 ] below , which shows that at those times the fifth force is just much weaker than gravity and therefore its impact ignorable . considering these , we simply use the @xmath54cdm initial displacements / velocities for the particles in these simulations , which are generated using grafic2 @xcite . the physical parameters we use in the simulations are as follows : the present - day dark energy fractional energy density @xmath143 and @xmath144 , @xmath145 km / s / mpc , @xmath146 , @xmath147 . the simulation box has a size of @xmath148 mpc , where @xmath149 . we simulate 4 models , with parameters @xmath150 equal to @xmath151 , @xmath152 , @xmath153 and @xmath154 respectively ( such parameters are chosen so that the deviation from @xmath54cdm will be neither too small to be distinguishable or too large to be realistic ) . for each model we make 5 runs with exactly the same initial condition , but different seeds in generating the random number to tag baryons and cdm particles ; all the 4 models use the same 5 seeds so that results can be directly compared . we hope the average of the results from 5 runs could reduce the scatter . in all those simulations the mass resolution is @xmath155 , the particle number is @xmath156 , the domain grid is a @xmath157 cubic and the finest refined grids have 16384 cells on one side , corresponding to a force resolution of @xmath158kpc . we also make a run for the @xmath54cdm model using the same parameters ( except for @xmath159 , which are not needed now ) and initial condition . in table [ tab : table1 ] we have listed some of the main results for the 20 runs we have made , from which we could obtain some rough idea how the motions of baryons and cdm particles differ from each other . we see that for the model @xmath160 the cdm particles could be up to @xmath161 times faster than baryons , thanks to the enhancement by the fifth force . we will come back to this point later when we argue for the necessity of a modified strategy of identifying virialized halos . [ cols="^,^,^,^,^",options="header " , ] in fig . [ fig : figure1 ] we have shown some snapshots of the distribution of baryonic and cdm particles , to give some idea about the hierachical structure formation and for comparisons with other figures below . it shows clearly how some clustering objects develop , with filaments connecting them together . the baryons roughly follow the clustering of cdm particles , but in some low density regions they become slightly separated . to understand how the motion of the particles is altered by the coupling to the scalar field , in fig . [ fig : figure2 ] we have shown the correlation between the magnitudes of the fifth force and gravity on the cdm particles ( remember that baryons do not feel the fifth force ) . ref . @xcite has made a detailed qualitatively analysis about the general trend of this correction , and here we just give a brief description : from eqs . ( [ eq : intpoisson ] , [ eq : intphieom ] ) we could see that , _ when the scalar field potential _ , _ i.e. _ , _ the last term of eqs . ( [ eq : intpoisson ] , [ eq : intphieom ] ) could be neglected _ , then the scalar field @xmath1 is simply proportional to the gravitational potential @xmath138 and as a result eq . ( [ eq : intdpdtcomov ] ) tells us that the strength of the fifth force is just @xmath162 times that of gravity ; in other words , the effect of the scalar field is a rescaling of the gravitational constant by @xmath163 . this is because in this situation , the effective mass of the scalar field , which is given by @xmath164 , where @xmath165 is the effective total potential , is light and the fifth force is long - range , like gravity . for comparison , in fig . [ fig : figure2 ] we also plot this @xmath162 proportion between the two forces , as a straight line : @xmath166 , where @xmath167 denote respectively the magnitudes of the fifth force and gravity , and the factor 0.8 in front of @xmath168 comes from the fact that only 80% of the particles are cdm ( and thus contribute to the fifth force ) . this scaling relation actually sets an upper limit on how strong the fifth force could be relative to gravity , should it not be suppressed by other effects . in contrast , when the value of @xmath1 is small , the last term of eqs . ( [ eq : intpoisson ] , [ eq : intphieom ] ) is not negligible and the scalar field acquires a heavy mass , making it short - ranged . as a result , a particle outside a high density region might not feel the fifth force exerted by particles in that region , even it is quite close to the region . but because it can feel gravity from that region , so the total fifth force on the particle becomes less than the @xmath162 scaling . in general , the value of @xmath1 is determined by @xmath169 as well as its background value @xmath55 ( which sets the boundary condition to solve the interior value ) . at early times @xmath55 is very close to 0 and @xmath73 is high everywhere , making @xmath1 small everywhere too , and suppressing the fifth force so that it is significantly below the @xmath162-scaling ( first row of fig . [ fig : figure2 ] ) . at later times , @xmath55 increases and @xmath73 decreases , weakening the above effect so that the fifth force becomes `` saturated '' ( _ i.e. _ , approaches the @xmath170 prediction ) and the points in the figure hit the straight lines ( last two rows ) . because decreasing @xmath38 and increasing @xmath45 have the same effects of making @xmath1 small , in the models with @xmath171 the fifth force saturates later than in the models with @xmath172 . in addition , because high @xmath73 tends to decrease @xmath1 and increase the scalar field mass , so in high density regions ( where gravity is stronger ) the fifth force also saturates later . the agreement between the numerical solution of the fifth force and the @xmath162-scaling relation in cases of weak - chameleon effect serves as an independent check of our numerical code . [ fig : figure3 ] plots the spatial configuration for the gravitational potential at the same position and output times as in fig . [ fig : figure1 ] . as expected , the potential is significantly deeper where there is significant clustering of matter [ cf . [ fig : figure1 ] ] . we also show in fig . [ fig : figure4 ] the spatial configuration for the scalar field @xmath1 at the same output position and times . at early times when @xmath1 is small and the scalar field mass is heavy , the fifth force is so short - ranged that @xmath1 only depends on the local density . this means that the spatial configuration of @xmath1 in this situation could well reflect the underlying dark matter distribution , a fact which could be seen clearly in the first row . as time passes by , the mass of the scalar field decreases on average and @xmath55 increases , the value of @xmath1 at one point is more and more influenced by the matter distribution in neighboring regions , and such an `` averaging '' effect weakens the contrast and makes the plots blurring ( last two rows ) . furthermore , obviously decreasing @xmath38 and increasing @xmath45 could increase the scalar field s mass , shorten the range of the fifth force , make @xmath1 less dependent on its value in neighboring regions , and thus strengthen the contrast in the figures . to have a more quantitative description about how the matter clustering property is modified with respect to the @xmath54cdm model , we consider the matter power spectra @xmath173 in our simulation boxes . the nonlinear matter power spectrum in the present work is measured using powmes @xcite , which is a public available code based on the taylor expansion of trigonometric functions and yields fourier modes from a number of fast fourier transforms controlled by the order of the expansion . we also average the results from the 5 runs for each model and calculate the variance . in fig . [ fig : figure5 ] shown are the fractional differences between the @xmath173 for our 4 models and for @xmath54cdm , at two different output times . at early times ( @xmath174 , upper solid curves in each panel ) , the difference is generally small , but still the 2 models with @xmath172 show up to @xmath175 deviation from @xmath54cdm prediction . this is because for larger @xmath38 the scalar field is lighter and the fifth force less suppressed , its influence in the structure formation therefore enhanced . notice that on small scales the deviation from @xmath54cdm decreases , which is a desirable property of chameleon models which are designed to suppress the fifth force on small scale high density regions . the lower solid curves in each panel of fig . [ fig : figure5 ] display the same quantities at @xmath176 ( late times ) . we can see the trend of increasing deviation from @xmath54cdm for all 4 models , because fifth force is essentially unsuppressed at the late epoch [ cf . [ fig : figure2 ] ] . for example , the deviation of the model @xmath177 is significantly larger than that of the model @xmath178 as navely expected , thanks to the lack of suppression of fifth force in both models ( the @xmath179 model obviously has a smaller saturated fifth force ) . for comparison we also plot the @xmath180 that is predicted by the linear perturbation theory for the 4 models under consideration ( the dashed curves ) . as can be seen there , at large scales ( small @xmath128 ) where linear perturbation is considered as a good approximation , the linear and nonlinear results agree pretty well ( especially for the @xmath174 case ) . the largest scale we can probe is limited by the size of our simulation boxes ( @xmath181 ) and as a result we can not make plot beyond the point @xmath182 , where nonlinearity is expected to first become significant . however , in the case of @xmath176 , we can see the clear trend of the linear and nonlinear results merging towards @xmath183 at vanishing @xmath180 . similar results can be found in fig . 2 of @xcite for f(r ) gravity . because we have two species of matter particles , one uncoupled to the scalar field , we are also interested in their respect power spectrum and the bias between them . these are displayed in fig . [ fig : figure6 ] . the results could be understood easily : because a cdm particle always feel stronger total force than a baryon at the same position , so the clustering of the former is identically stronger than the latter as well . this could result in a significant bias between these two species at the present time , especially for the models with @xmath172 , where the fifth force is less suppressed . we identify halos in our @xmath0-body simulations using mhf ( mlapm halo finder ) @xcite , which is the default halo finder for mlapm . mhf optimally utilizes the refinement structure of the simulation grids to pin down the regions where potential halos reside and organize the refinement hierarchy into a tree structure . because mlapm refines grids according to the particle density on them , so the boundaries of the refinements are simply isodensity contours . mhf collect the particles within these isodensity contours ( as well as some particles outside ) . it then performs the following operations : ( i ) assuming spherical symmetry of the halo , calculate the escape velocity @xmath184 at the position of each particle , ( ii ) if the velocity of the particle exceeds @xmath184 then it does not belong to the virialized halo and is removed . ( i ) and ( ii ) are then iterated until all unbound particles are removed from the halo or the number of particles in the halo falls below a pre - defined threshold , which is 20 in our simulations . note that the removal of unbound particles is not used in some halo finders using the spherical overdensity ( so ) algorithm , which includes the particles in the halo as long as they are within the radius of a virial density contrast . another advantage of mhf is that it does not require a pre - defined linking length in finding halos , such as the friend - of - friend procedure . when it comes to our coupled scalar field model , the mhf algorithm also needs to be modified . the reason is that , as we mentioned above , the scalar field @xmath1 behaves as an extra `` potential '' ( which produces the fifth force ) and so the cdm particles experience a deeper total `` gravitational '' potential than what they do in the @xmath54cdm model . consequently , the escape velocity for cdm particles increases compared with the newtonian prediction . this is indeed important to bear in mind because , as we have seen in table [ tab : table1 ] , the cdm particles are typically much faster than what they are in the @xmath54cdm simulation , and so if we underestimate @xmath184 then some particles which should have remained in the halo would be incorrectly removed by mhf . in general , similar things should be taken care of in other theories involving modifications to gravity in the non - relativistic limit , such as mond and @xmath2 gravity . an exact calculation of the escape velocity in the coupled scalar field model is obviously difficult due to the complicated behaviour of the scalar field , and thus here we introduce an approximated algorithm , which is based on the mhf default method @xcite , to estimate it . mhf works out @xmath184 using the newtonian result @xmath185 in which @xmath138 is the gravitational potential . under the assumption of spherical symmetry , the poisson equation @xmath186 could be integrated once to give @xmath187 which is just the newtonian force law . this equation can be integrated once again to obtain @xmath188 where @xmath189 is an integration constant and can be fixed @xcite by requiring that @xmath190 as @xmath191 in which @xmath192 is the virial radius of the halo and @xmath193 is the mass enclosed in @xmath192 . when the fifth force acts on cdm particles , the force law eq . ( [ eq : ahf_newton_law ] ) is modified and these particles feel a larger total `` gravitational '' potential . to take this into account , we need to have the knowledge about how the force law is modified and a simple rescaling of gravitational constant @xmath7 has been shown to be not physical in certain regimes . to solve the problem , we notice that in the mhf code eq . ( [ eq : ahf_newton_law ] ) is used in the numerical integrations to obtain both @xmath194 and @xmath189 [ cf . eqs.([eq : ahf_phi ] , [ eq : ahf_phi0 ] ) ] . more explicitly , the code loops over all particles in the halo in ascending order of the distance from the halo centre , and when a particle is encountered its mass is uniformly distributed into the spherical shell between the particle and its previous particle ( the thickness of the shell is then the @xmath195 of the integration ) . obviously when the fifth force is added into eq . ( [ eq : ahf_newton_law ] ) we could use the same method to compute the total `` gravitational '' potential which includes the contribution from @xmath1 ( call this contribution @xmath196 ; because @xmath197 , so from eq . ( [ eq : wfdpdtcomov ] ) we can easily see @xmath198 ) . but now there is a subtlety here : not all particles are cdm while only cdm particles contribute to @xmath196 . so in the modified mhf code we calculate @xmath199 and @xmath200 separately , using all particles for the former and only cdm for the latter . finally @xmath201 is our estimate of the escape velocity . because we have recorded the components of gravity and the fifth force for each particle in the simulation , so the fifth - force - to - gravity ratio can be computed at the position of each particle , which is approximated to be @xmath202 at that position . in this way we have , at least approximately , taken into account the environment dependence of the fifth force and thus of @xmath196 . to see how our modification of the mhf code affects the final result on the mass function , in fig . [ fig : figure7 ] we compare the mass functions for the model @xmath177 calculated using three methods : the modified mhf , the original mhf ( by default for @xmath54cdm simulations ) and another modified mhf in which we set @xmath184 to be very large so that no particles will ever escape from any halo ( this is similar to the spherical overdensity algorithm mentioned above ) . it could be seen that the difference between these methods can be up to a few percent for small halos where the potential is shallow , particle number is small and the result is sensitive to whether more particles are removed . as expected , the second method gives least halos because there @xmath184 is smallest and many particles are removed , while the third method gives most halos because no particles are removed at all . [ fig : figure8 ] displays the mass functions for the 4 models as compared to the @xmath54cdm result . as expected , the fifth force enhances the structure formation and thus produces more halos in the simulation box . as our simulations have reached a force resolution of @xmath203 kpc , while the typical size of the large halos in the simulations is of order mpc , we could ask what the internal profiles of the halos look like and how they have been modified by the coupling between cdm particles and the scalar field . we have selected 2 typical halos from each simulation . halo i is centred on @xmath204 mpc , which is slightly different for different simulations , and has a virial mass @xmath205 ; halo ii is centred on @xmath206 mpc , which is also slightly different for different simulations , and has a virial mass @xmath207 ( note here that the virial masses are for the _ @xmath54cdm simulations _ , and for scalar field simulations they can be , and generally are , slightly different ) . the largest halos , such as halo i , generally reside in the higher density regions where the scalar field has a heavier mass and shows stronger chameleon effect . consequently the fifth force inside them is severely suppressed so that we can expect small deviation from the @xmath54cdm halo profile . on the other hand , the intermediate and small halos ( such as halo ii ) are mostly in relatively low density regions in which scalar field has a light mass and the fifth force tends to saturate ; this means that they should generally have a higher internal density than the same halos in the @xmath54cdm simulation due to the enhanced growth by the fifth force . this above analysis is qualitatively confirmed by fig . [ fig : figure9 ] , where we can see that for the cases of @xmath171 ( stronger chameleon ) the difference between the predictions of the coupled scalar field models and the @xmath54cdm paradigm is really small for halo i. furthermore , this figure also shows some new and more interesting features for the cases of @xmath172 . considering halo i in the models @xmath178 and @xmath177 for example : in the former case , the coupled scalar field model produces an obviously consistent higher internal density profile than @xmath54cdm , from the inner to the outer regions of the halo ; while for the latter case , the density profile of the coupled scalar field model is lower in the inner region but higher in the outer region ! we plan to make a detailed analysis of the complexities arising here regarding the effects of a coupled scalar field on the internal density profiles for halos in a future work , and in this work we will only give a brief explanation for the new feature observed above : fig . [ fig : figure11 ] shows the distributions of particle velocities and fifth - force - to - gravity ratio at the positions of the particles in halo i for the two models @xmath178 and @xmath177 . as we can see there , for the model @xmath177 the large value of @xmath45 makes the chameleon effect strong so that the fifth force is generally much smaller than gravity in magnitude , while at the same time the velocities of particles are more concentrated towards the high end , implying a significantly higher mean speed than @xmath54cdm ( as we have checked numerically ) ; as a result in the central region of the halo the particles have higher kinetic energy than in @xmath54cdm but the potential is not significantly deeper , so that particles tend to get far away from the centre of the halo , producing a lower inner density profile . as for the model with @xmath178 , the situation is just the opposite : the fifth force is saturated and of the same order as gravity due to the weak chameleon effect , so that the total potential is greatly deeper than its @xmath54cdm counterpart , while at the same time the particles are not as fast as those in @xmath177 : the consequence is that the halo accretes more particles towards its centre . one might also have interests in how the cdm part and the baryonic part of the halo profile differ from each other , and this is given in fig . [ fig : figure10 ] , in which we compare the cdm and baryon density profiles of halos i and ii . obviously the cdm density is higher than baryons everywhere , again thanks to the boost from the fifth force . for smaller @xmath38 and larger @xmath45 ( which help strengthen the chameleon effect and suppress the fifth force ) , the difference between the two is smaller ( compare the halo i in the models @xmath178 and @xmath208 ) . however , in the situations where the fifth force is already saturated or unsuppressed ( such as in halo ii ) , increasing @xmath45 increases the saturated value of the fifth force , and thus can instead magnify the difference ( compare the halo ii in the models @xmath178 and @xmath208 ) . the above results again show the complexity of the chameleon scalar field model as compared to other coupled scalar field models . we will study the environment and epoch dependence of the halo density profiles in a upcoming work . to summarize , in this paper we have investigated into a model where cdm and baryons couple differently to a chameleon - like scalar field , and performed full @xmath0-body simulations , by directly solving the spatial distribution of the scalar field , to study the nonlinear structure formation under this setup . the new complexity introduced here compared to @xcite is that we must distinguish between baryons and cdm so that we know how to calculate the force upon each particle . we do this by tagging the initial almost homogeneously distributed particle randomly so that 80% of all particles are tagged as cdm . we then only use cdm particles to calculate the scalar field and only applies the fifth scalar force on cdm particles . the coupling function ( characterized by the coupling strength @xmath45 ) and bare potential ( characterized by the parameter @xmath38 which controls its steepness ) of the scalar field are chosen to be the same as in @xcite . as discussed there , the coupling of the scalar field to cdm particles acts on the latter a fifth force . when @xmath38 is small and @xmath45 is large , the chameleon effect becomes stronger , which gives the scalar field a heavy mass , making the fifth force short - ranged and the scalar field dependent only on the local matter density . other ways to strengthen the chameleon effect includes increasing the density and decreasing the background value of the scalar field , which itself is equivalent to increasing the background cdm energy density . we have displayed in figs . [ fig : figure2 ] , [ fig : figure4 ] how changing the determining factors change the scalar field configuration and the strength of the fifth force . we have also measured the nonlinear matter power spectrum from the simulation results and compared them with the @xmath54cdm prediction . depending on the values of @xmath142 as well as the background cdm density , the former can be up to @xmath209 larger than the latter . nonetheless , when the chameleon effect is set to be strong , the deviation gets suppressed and in particular decreases towards small scales , showing the desirable property of chameleon models that they evade constraints on small scales . the bias between cdm and baryons power spectra follows the same trend . to identify virialized halos from the simulations , we have modified mhf , mlapm s default halo finder , so that the calculation of the escape velocity includes the effect of the scalar field . we find that such a modification leads to up to a - few - percent enhancement on the mass function compared with what is obtained using the default mhf code , because in the latter case the escape velocity is underestimated and some particles are incorrectly removed from the virialized halos . we find that the mass function in the coupled scalar field models is significantly larger than the @xmath54cdm result , because of the enhanced structure growth induced by the fifth force . finally , we have analyzed the internal profiles of ( the same ) two halos selected from each simulation . we find that when chameleon effect is strong and fifth force is suppressed , our result is very close to the @xmath54cdm prediction ; this is the case for very large halos ( which generally reside in higher density regions ) and @xmath171 . for large halos and @xmath172 , the situation is more complicated because the competition between two effects of the coupled scalar field , namely the speedup of the particles and the deepening of the total attractive potential , has arrived a critical point : if the former wins , such as in the model @xmath177 , then the inner density profile can be lower than that in @xmath54cdm ; if the latter wins , such as in the model @xmath177 , then we expect the opposite . for smaller halos which locate in lower density regions , the fifth force is not suppressed that much and causes faster growth of the structure , so the halo is more concentrated and has a higher internal density . meanwhile , the bias between baryons and cdm density profiles also increase as the fifth force becomes less suppressed , which is as expected . our results already show that new features can be quantitatively studied with the @xmath0-body method and the improving supercomputing techniques , and that the chameleon model has rather different consequences from other coupled scalar field models . the enhancement in the structure formation due to the fifth force is significant for some of our parameter space , which means other observables , such as weak lensing , could place new constraints on the model . also one might be interested in how the halo profiles would be like at different epoch of the cosmological evolution and in different environments . these will be left as future work . the work described here has been performed under the hpc - europa project , with the support of the european community research infrastructure action under the fp8 `` structuring the european research area '' programme . the @xmath0-body simulations are performed on the huygens supercomputer in the netherlands , and the post - precessing of data is performed on cosmos , the uk national cosmology supercomputer . we thank john barrow , kazuya koyama , andrea maccio and gong - bo zhao for helpful discussions relevant to this work , and lin jia for assistance in plotting the figures . b. li is supported by a research fellowship in applied mathematics at queens college , university of cambridge and the stfc rolling grant in damtp . natexlab bibnamefont bibfnamefont citenamefont url urlprefix [ 2]#2 [ 2][]#2 , and , * * , ( ) . , * * , ( ) . and , * * , ( ) . and , * * , ( ) . and , * * , ( ) . and , * * , ( ) . , , , and , * * , ( ) , , and , * * , ( ) . and , * * , ( ) . and , * * , ( ) . and , * * , ( ) . and , * * , ( ) . , , and , * * , ( ) . , * * , ( ) . and , * * , ( ) . , , and , * * , ( ) . and , * * , ( ) . and , * * , ( ) ; * * , ( ) and , * * , ( ) . , and ( ) , arxiv:0902.3452 [ astro - ph ] . , , , and , * * , ( ) . , , and ( ) , arxiv:0812.3901 [ astro - ph ] . and , * * , ( ) . , , , and , submitted ; arxiv:0910.3207 [ astro-ph.co ] . , * * , ( ) . , and , * * , ( ) . , and , * * , ( ) . and , * * , ( ) . and , * * , ( ) . , and , * * , ( ) . , * * , ( ) . , and , * * , ( ) . , * * , ( ) . , , and , _ numerical recipes in c. the art of scientific computing _ ( cambridge university press , cambridge 1992 ) , second ed . , and , _ a multigrid tutorial _ ( society for industrial and applied mathematics , philadelphia 2000 ) , second ed . , * * , ( ) . , , and , * * , ( ) . ) , arxiv : astro - ph/9506070 . , and , * * , ( ) . , , and , * * , ( ) . , and , * * , ( ) . and , * * , ( ) . in the mlapm code the partial differential equation eq . ( [ eq : intpoisson ] ) is ( and in our modified code eq . ( [ eq : intphieom ] ) will also be ) solved on discretized grid points , and as such we must develop the discretized versions of eqs . ( [ eq : intdxdtcomov ] - [ eq : intphieom ] ) to be implemented into the code . but before going on to the discretization , we need to address a technical issue . as the potential is highly nonlinear , in the high density regime the value of the scalar field @xmath210 will be very close to 0 , and this is potentially a disaster as during the numerical solution process the value of @xmath210 might easily go into the forbidden region @xmath118 @xcite . one way of solving this problem is to define @xmath211 in which @xmath212 is the background value of @xmath213 , as in @xcite . then the new variable @xmath117 takes value in @xmath214 so that @xmath215 is positive definite which ensures that @xmath216 . however , since there are already exponentials of @xmath213 in the potential , this substitution will result terms involving @xmath217 $ ] , which could potentially magnify any numerical error in @xmath117 . instead , we can define a new variable @xmath117 according to @xmath218 by this , @xmath117 still takes value in @xmath214 , @xmath219 and thus @xmath220 which ensures that @xmath213 is positive definite in numerical solutions . besides , @xmath221^{\beta}$ ] so that there will be no exponential - of - exponential terms , and the only exponential is what we have for the potential itself . @xmath222 above . then the poisson equation becomes @xmath223\nonumber\\ & & + \frac{3}{2}\omega_{\mathrm{b}}\left(\rho_{c,\mathrm{b}}-1\right ) - \frac{3\omega_{v_{0}}a^{3}}{\left[1-\left(1+e^{u}\right)^{\beta}\right]^{\mu } } + 3\bar{\omega}_{v}a^{3},\ \ \ \ \\end{aligned}\ ] ] where we have defined @xmath224 which is determined by background cosmology , the quantity @xmath225 is also determined solely by background cosmology . these background quantities should not bother us here . the scalar field eom becomes @xmath226^{\mu+1}}\nonumber\\ & & - 3\gamma\omega_{\mathrm{cdm}}e^{\gamma\sqrt{\kappa}\bar{\varphi } } - \frac{3\mu\beta\omega_{v_{0}}a^{3}e^{\beta\sqrt{\kappa}\bar{\varphi } } } { \left[1-e^{\beta\sqrt{\kappa}\bar{\varphi}}\right]^{m+1}}\end{aligned}\ ] ] in which we have used the fact that @xmath227 , and moved all terms depending only on background cosmology ( the source terms ) to the right hand side . so , in terms of the new variable @xmath117 , the set of equations used in the @xmath0-body code should be @xmath228\end{aligned}\ ] ] plus eqs . ( [ eq : u_poisson ] , [ eq : u_phi_eom ] ) . these equations will ultimately be used in the code . among them , eqs . ( [ eq : u_poisson ] , [ eq : u_dpdt ] ) will use the value of @xmath117 while eq . ( [ eq : u_phi_eom ] ) solves for @xmath117 . in order that these equations can be integrated into mlapm , we need to discretize eq . ( [ eq : u_phi_eom ] ) for the application of newton - gauss - seidel iterations . to discretize eq . ( [ eq : u_phi_eom ] ) , let us define @xmath229 . the discretization involves writing down a discretion version of this equation on a uniform grid with grid spacing @xmath230 . suppose we require second order precision as is in the standard poisson solver of mlapm , then @xmath231 in one dimension can be written as @xmath232 where a subscript @xmath233 means that the quantity is evaluated on the @xmath234-th point . of course the generalization to three dimensions is straightforward . @xmath240\nonumber\\ & & + \frac{1}{h^{2}}\left[b_{i , j+\frac{1}{2},k}u_{i , j+1,k } - u_{i , j , k}\left(b_{i , j+\frac{1}{2},k}+b_{i , j-\frac{1}{2},k}\right ) + b_{i , j-\frac{1}{2},k}u_{i , j-1,k}\right]\nonumber\\ & & + \frac{1}{h^{2}}\left[b_{i , j , k+\frac{1}{2}}u_{i , j , k+1 } - u_{i , j , k}\left(b_{i , j , k+\frac{1}{2}}+b_{i , j , k-\frac{1}{2}}\right ) + b_{i , j , k-\frac{1}{2}}u_{i , j , k-1}\right].\end{aligned}\ ] ] @xmath242\nonumber\\ & & + \frac{1}{h^{2}}\left[b_{i , j+\frac{1}{2},k}u_{i , j+1,k } - u_{i , j , k}\left(b_{i , j+\frac{1}{2},k}+b_{i , j-\frac{1}{2},k}\right ) + b_{i , j-\frac{1}{2},k}u_{i , j-1,k}\right]\nonumber\\ & & + \frac{1}{h^{2}}\left[b_{i , j , k+\frac{1}{2}}u_{i , j , k+1 } - u_{i , j , k}\left(b_{i , j , k+\frac{1}{2}}+b_{i , j , k-\frac{1}{2}}\right ) + b_{i , j , k-\frac{1}{2}}u_{i , j , k-1}\right]\nonumber\\ & & -\frac{\left(h_{0}b\right)^{2}}{ac^{2}}\left[3\gamma\omega_{\mathrm{cdm}}\rho^{\mathrm{cdm}}_{c , i , j , k } \left(1+e^{u_{i , j , k}}\right)^{\gamma } + \frac{3\mu\beta\omega_{v_{0}}a^{3}\left(1+e^{u_{i , j , k}}\right)^{\beta } } { \left[1-\left(1+e^{u_{i , j , k}}\right)^{\beta}\right]^{\mu+1}}\right]\nonumber\\ & & + \frac{\left(h_{0}b\right)^{2}}{ac^{2}}\left [ 3\gamma\omega_{\mathrm{cdm}}e^{\gamma\sqrt{\kappa}\bar{\varphi } } + \frac{3\mu\beta\omega_{v_{0}}a^{3}e^{\beta\sqrt{\kappa}\bar{\varphi } } } { \left[1-e^{\beta\sqrt{\kappa}\bar{\varphi}}\right]^{\mu+1}}\right].\end{aligned}\ ] ] then the newton - gauss - seidel iteration says that we can obtain a new ( and often more accurate ) solution of @xmath117 , @xmath243 , using our knowledge about the old ( and less accurate ) solution @xmath244 as @xmath245 the old solution will be replaced by the new solution to @xmath246 once the new solution is ready , using the red - black gauss - seidel sweeping scheme . note that @xmath247\nonumber\\ & & -\frac{1}{2h^{2}}\left[b_{i+1,j , k}+b_{i-1,j , k}+b_{i , j+1,k } + b_{i , j-1,k}+b_{i , j , k+1}+b_{i , j , k-1}+6b_{i , j , k}\right]\nonumber\\ & & -\frac{\left(h_{0}b\right)^{2}}{ac^{2}}3\gamma^{2}\omega_{\mathrm{cdm}}\rho^{\mathrm{cdm}}_{c , i , j , k } \left(1+e^{u_{i , j , k}}\right)^{\gamma}b_{i , j , k}\nonumber\\ & & -\frac{\left(h_{0}b\right)^{2}}{ac^{2 } } \frac{3\mu\beta^{2}\omega_{v_{0}}a^{3}\left(1+e^{u_{i , j , k}}\right)^{\beta } } { \left[1-\left(1+e^{u_{i , j , k}}\right)^{\beta}\right]^{\mu+1}}b_{i , j , k } \left[1+(\mu+1)\frac{\left(1+e^{u_{i , j , k}}\right)^{\beta}}{1-\left(1+e^{u_{i , j , k}}\right)^{\beta}}\right].\end{aligned}\ ] ] in principle , if we start from a high redshift , then the initial guess of @xmath246 could be such that the initial value of @xmath213 in all the space is equal to the background value @xmath212 , because anyway at this time we expect this to be approximately true . for subsequent time steps we could use the solution for @xmath246 at the previous time step as our initial guess ; if the time step is small enough then we do not expect @xmath117 to change significantly between consecutive times so that such a guess will be good enough for the iteration to converge fast .

in this paper we present the results of @xmath0-body simulations with a scalar field coupled differently to cold dark matter ( cdm ) and baryons . the scalar field potential and coupling function are chosen such that the scalar field acquires a heavy mass in regions with high cdm density and thus behaves like a chameleon . we focus on how the existence of the scalar field affects the formation of nonlinear large - scale structure , and how the different couplings of the scalar field to baryons and cdm particles lead to different distributions and evolutions for these two matter species , both on large scales and inside virialized halos . as expected , the baryon - cdm segregation increases in regions where the fifth force is strong , and little segregation in dense regions . we also introduce an approximation method to identify the virialized halos in coupled scalar field models which takes into account the scalar field coupling and which is easy to implement numerically . it is find that the chameleon nature of the scalar field makes the internal density profiles of halos dependent on the environment in a very nontrivial way .

despite a few important successes ( e.g. , bean et al . 2007 , and references therein ) , astrometric measurements with mas precision have so far proved of limited utility when employed as either a follow - up tool or to independently search for planetary mass companions orbiting nearby stars ( see for example sozzetti 2005 , and references therein ) . in several past exploratory works ( casertano et al . 1996 ; lattanzi et al . 1997 , 2000 ; sozzetti et al 2001 , 2003 ) , we have shown in some detail what space - borne astrometric observatories with @xmath0as - level precision , such as gaia ( perryman et al . 2001 ) , can achieve in terms of search , detection and measurement of extrasolar planets of mass ranging from jupiter - like to earth - like . in those studies we adopted a qualitatively correct description of the measurements that each mission will carry out , and we estimated detection probabilities and orbital parameters using realistic , non - linear least squares fits to those measurements . those exploratory studies , however , need updating and improvements . in the specific case of planet detection and measurement with gaia , we have thus far largely neglected the difficult problem of selecting adequate starting values for the non - linear fits , using perturbed starting values instead . the study of multiple - planet systems , and in particular the determination of whether the planets are coplanar within suitable tolerances is incomplete . the characteristics of gaia have changed , in some ways substantially , since our last work on the subject ( sozzetti et al 2003 ) . last but not least , in order to render the analysis truly independent from the simulations , these studies should be carried out in double - blind mode . we present here a substantial program of double - blind tests for planet detection with gaia ( preliminary findings were recently presented by lattanzi et al . ( 2005 ) ) , with the three - fold goal of obtaining : a ) an improved , more realistic assessment of the detectability and measurability of single and multiple planets under a variety of conditions , parametrized by the sensitivity of gaia ; b ) an assessment of the impact of gaia in critical areas of planet research , in dependence on its expected capabilities ; and c ) the establishment of several centers with a high level of readiness for the analysis of gaia observations relevant to the study of exoplanets . we carry out detailed simulations of gaia observations of synthetic planetary systems and develop and utilize in double - blind mode independent software codes for the analysis of the data , including statistical tools for planet detection and different algorithms for single and multiple keplerian orbit fitting that use no a priori knowledge of the true orbital parameters of the systems . overall , the results of our earlier works ( e.g. , lattanzi et al . 2000 ; sozzetti et al . 2001 , 2003 ) are essentially confirmed , with the fundamental improvement due to the successful development of independent orbital fitting algorithms applicable to real - life data that do not utilize any a priori knowledge of the orbital parameters of the planets . in particular , the results of the t1 test ( planet detection ) indicate that planets down to astrometric signatures @xmath1 @xmath0as , corresponding to @xmath2 times the assumed single - measurement error , can be detected reliably and consistently , with a very small number of false positives ( depending on the specific choice of the threshold for detection ) . the results of the t2 test ( single - planet orbital solutions ) indicate that : 1 ) orbital periods can be retrieved with very good accuracy ( better than 10% ) and small bias in the range @xmath3 yrs , and in this period range the other orbital parameters and the planet mass are similarly well estimated . the quality of the solutions degrades quickly for periods longer than the mission duration , and in particularly the fitted value of @xmath4 is systematically underestimated ; 2 ) uncertainties in orbit parameters are well understood ; 3 ) nominal uncertainties obtained from the fitting procedure are a good estimate of the actual errors in the orbit reconstruction . modest discrepancies between estimated and actual errors arise only for planets with extremely good signal ( errors are overestimated ) and for planets with very long period ( errors are underestimated ) ; such discrepancies are of interest mainly for a detailed numerical analysis , but they do not touch significantly the assessment of gaia s ability to find planets and our preparedness for the analysis of perturbation data . the results of the t3 test ( multiple - planet orbital solutions ) indicate that 1 ) over 70% of the simulated orbits under the conditions of the t3 test ( for every two - planet system , periods shorter than 9 years and differing by at least a factor of two , @xmath5 , @xmath6 ) are correctly identified ; 2 ) favorable orbital configurations ( both planet with periods @xmath7 yr and astrometric signal - to - noise ratio @xmath8 , redundancy of over a factor of 2 in the number of observations ) have periods measured to better than 10% accuracy @xmath9 of the time , and comparable results hold for other orbital elements ; 3 ) for these favorable cases , only a modest degradation of up to @xmath10 in the fraction of well - measured orbits is observed with respect to single - planet solutions with comparable properties ; 4 ) the overall results are mostly insensitive to the relative inclination of pairs of planetary orbits ; 5 ) over 80% of the favorable configurations have @xmath11 measured to better than 10 degrees accuracy , with only mild dependencies on its actual value , or on the inclination angle with respect to the line of sight of the planets ; 6 ) error estimates are generally accurate , particularly for fitted parameters , while modest discrepancies ( errors are systematically underestimated ) arise between formal and actual errors on @xmath11 . g dwarf primary at 200 pc , while the blue curves are for a 0.5-@xmath12 m dwarf at 25 pc . the radial velocity curve ( pink line ) is for detection at the @xmath13 level , assuming @xmath14 m s@xmath15 , @xmath16 , and 10-yr survey duration . for transit photometry ( green curve ) , @xmath17 milli - mag , @xmath18 , @xmath19 @xmath12 , @xmath20 @xmath21 , uniform and dense ( @xmath22 datapoints ) sampling . black dots indicate the inventory of exoplanets as of october 2007 . transiting systems are shown as light - blue filled pentagons . jupiter and saturn are also shown as red pentagons.,scaledwidth=75.0% ] [ nplan ] in figure [ detmeas ] we show gaia s discovery space in terms of detectable and measurable planets of given mass and orbital separation around stars of given mass at a given distance from earth ( see caption for details ) . from the figure , one would then conclude that gaia could discover and measure massive giant planets ( @xmath23 @xmath24 ) with @xmath25 au orbiting solar - type stars as far as the nearest star - forming regions , as well as explore the domain of saturn - mass planets with similar orbital semi - major axes around late - type stars within 30 - 40 pc . these results can be turned into a number of planets detected and measured by gaia , using galaxy models and the current knowledge of exoplanet frequencies . by inspection of tables [ nplan ] and [ nmult ] , we then find that gaia could measure accurately thousands of giant planets , and accurately determine coplanarity ( or not ) for a few hundred multiple systems with favorable configurations . in conclusion , gaia s main strength continues to be the ability to measure actual masses and orbital parameters for possibly thousands of planetary systems . the gaia data have the potential to a ) significantly refine our understanding of the statistical properties of extrasolar planets : the predicted database of several thousand extrasolar planets with well - measured properties will allow for example to test the fine structure of giant planet parameters distributions and frequencies , and to investigate their possible changes as a function of stellar mass with unprecedented resolution ; b ) help crucially test theoretical models of gas giant planet formation and migration : for example , specific predictions on formation time - scales and the role of varying metal content in the protoplanetary disk will be probed with unprecedented statistics thanks to the thousands of metal - poor stars and hundreds of young stars screened for giant planets out to a few aus ; c ) improve our comprehension of the role of dynamical interactions in the early as well as long - term evolution of planetary systems : for example , the measurement of orbital parameters for hundreds of multiple - planet systems , including meaningful coplanarity tests will allow to discriminate between various proposed mechanisms for eccentricity excitation ; d ) aid in the understanding of direct detections of giant extrasolar planets : for example , actual mass estimates and full orbital geometry determination for suitable systems will inform direct imaging surveys about where and when to point , in order to estimate optimal visibility , and will help in the modeling and interpretation of giant planets phase functions and light curves ; e ) provide important supplementary data for the optimization of the target selection for darwin / tpf : for example , all f - g - k - m stars within the useful volume ( @xmath26 pc ) will be screened for jupiter- and saturn - sized planets out to several aus , and these data will help probing the long - term dynamical stability of their habitable zones , where terrestrial planets may have formed , and maybe found .

in this paper , we first summarize the results of a large - scale double - blind tests campaign carried out for the realistic estimation of the gaia potential in detecting and measuring planetary systems . then , we put the identified capabilities in context by highlighting the unique contribution that the gaia exoplanet discoveries will be able to bring to the science of extrasolar planets during the next decade .

radiative decays of vector mesons have traditionally been a good laboratory for various tests of the quark model and su(3 ) symmetry @xcite . a recent discovery of the @xmath5 decay by the cmd-2 group @xcite has been the last link in the otherwise complete picture of radiative magnetic dipole transitions between light vector and pseudoscalar mesons . this observation was later confirmed by the snd group @xcite . both experiments suffered from a low number of observed events , resulting in large uncertainties in the determined branching ratio and making comparison to theory difficult . in this paper we report on the improved measurement of the rate of the @xmath5 decay based upon the total data sample accumulated with cmd-2 in the @xmath6-meson energy range . it includes 3.1 pb@xmath1 of data collected in 1992 1996 in our first measurement which used only photons observed in the csi barrel calorimeter , and about 11.4 pb@xmath1 collected in 1997 1998 . in addition , this analysis uses photons detected in either the csi barrel or the bgo endcap calorimeters for both data samples providing better detection efficiency than before . the general purpose detector cmd-2 operating at the high luminosity @xmath7 collider vepp-2 m in novosibirsk has been described in detail elsewhere @xcite . it consists of a drift chamber and proportional z - chamber used for trigger , both inside a thin ( 0.4 @xmath8 ) superconducting solenoid with a field of 1 t. the barrel calorimeter placed outside the solenoid consists of 892 csi crystals of @xmath9 @xmath10 size and covers polar angles from @xmath11 to @xmath12 . the energy resolution for photons is about 9% in the energy range from 50 to 600 mev . the end - cap calorimeter placed inside the solenoid consists of 680 bgo crystals of @xmath13 @xmath10 size and covers forward - backward polar angles from 16@xmath14 to 49@xmath14 and from 131@xmath14 to 164@xmath14 . the energy and angular resolution are equal to @xmath15 and @xmath16 radians respectively . the luminosity was determined from the detected @xmath17 events @xcite . since @xmath0 is a two - body decay and @xmath18 is a narrow state , the momentum of the recoil photon is fixed and approximately equals 60 mev . to study this decay we searched for the decay chain @xmath19 , @xmath20 . the photons are ordered by decreasing energy ( @xmath21 ) . in these events the softest photon must be a monochromatic recoil photon with the energy @xmath22 mev at the @xmath6 meson peak , while the energies of the harder ones range from 170 to 440 mev . the invariant mass of the two harder photons @xmath23 . the main source of background for this study is the decay mode @xmath24 giving the same final state with two charged pions and three photons via the decay chain @xmath25 , @xmath26 . here the hardest photon is monochromatic with @xmath27 mev and the invariant mass of two others is @xmath28 . this decay can be used as a monitoring process and the branching ratio @xmath29 will be calculated relative to @xmath30 . due to similar kinematics and detection efficiency dependence on detector parameters some systematic errors will cancel in such a ratio . events with two tracks and three photons were selected using the following criteria : * one vertex is found in the event * two tracks with opposite charges are reconstructed from this vertex and there are no other tracks * the angles of both tracks with respect to the beam are limited by @xmath31 to match the optimal drift chamber coverage * the number of photons detected in the csi and bgo calorimeters is three . the cluster in the calorimeter is accepted as a photon when it does not match any charged track and its energy is more than 30 mev in the csi calorimeter or more than 40 mev in the bgo calorimeter . * the distance from each track to the beam @xmath32 cm * the distance from the vertex to the interaction point along the beam direction @xmath33 cm * the space angle between the tracks @xmath34 * the angle between the tracks in the r-@xmath35 plane @xmath36 * the total energy of the charged particles ( assuming that both particles are charged pions ) @xmath37 mev . the events thus selected were subject to the kinematical reconstruction assuming energy - momentum conservation . events with good quality of the reconstruction were selected by the following criteria : * @xmath38 * the ratio of the photon energy measured in the calorimeter @xmath39 to that from the constrained fit @xmath40 is @xmath41 * @xmath42 mev vs hardest photon energy @xmath43 . a ) simulation of @xmath44 ; b ) simulation of @xmath45 at the @xmath6-meson energy ; c ) simulation of @xmath46 ; d ) experimental data.,scaledwidth=80.0% ] events surviving after all above criteria mostly come from the process @xmath47 , @xmath48 and @xmath45 , as illustrated by fig . [ fig : w1m23 ] showing the scatter plot of the invariant mass @xmath49 versus the hardest photon energy @xmath50 . the data are shown in fig . [ fig : w1m23]d . the region around @xmath51 mev and @xmath27 mev is densely populated with @xmath52 events . simulated events of this process are presented in fig . [ fig : w1m23]a . to determine the number of @xmath47 events we count the number of events inside the ellipse - like region : @xmath53 for our data this number is @xmath54 . determination of the number of @xmath55 events for simulation gives the detection efficiency @xmath56 . figure [ fig : w1m23]b presents the simulation of @xmath45 , where a densely populated region is also observed at large values of @xmath50 . comparison of these distributions with that for the data ( fig . [ fig : w1m23]d ) confirms that the dominant contribution to selected events comes from these two processes . the same distribution for the simulation of the process under study is shown in fig . [ fig : w1m23]c . for a ) simulation of @xmath44 ; b ) simulation of @xmath45 at the @xmath6-meson energy ; c ) simulation of @xmath46.,scaledwidth=80.0% ] to search for the rare decay @xmath57 we need to suppress the events from @xmath47 and @xmath58 . to this end a cut on the energy of the hardest photon is applied : @xmath59 mev . the @xmath50 distributions for the simulation of @xmath57 and background processes are shown in fig . [ fig : wr1 ] . vs softest photon energy @xmath60 . points present the simulation of @xmath61 , triangles data after all the selections.,scaledwidth=80.0% ] [ fig : finsum ] although this cut causes a decrease of efficiency for the @xmath62 decay ( see fig . [ fig : wr1]c ) , the suppression of the background processes is rather good . one more cut suppressing the background from the @xmath63 and @xmath64 decays is : @xmath65 mev . after all the cuts the scatter plot of the invariant masses for two hardest photons @xmath66 versus the weakest photon energy @xmath67 was studied . figure [ fig : finsum ] presents the data ( black triangles ) together with simulation of @xmath0 ( points ) . the simulation points show the region of the plot which should be populated by the events of @xmath57 and experimental points are densely covering this region . the lower part of the figure contains obvious background events which can be suppressed by imposing the additional cut @xmath68 mev . to determine the number of events the one - dimensional distribution of @xmath69 ( projection of the plot in fig . [ fig : finsum ] to the axis perpendicular to the correlation line ) was studied . such projection is shown in fig . [ fig : etpr]c for the data . the same projection for 10000 simulated events of @xmath70 is shown in fig . [ fig : etpr]b , and the fit of this distribution fixes the signal shape and gives the detection efficiency @xmath71 . the background distribution in this parameter determined from the data before applying the last two cuts ( @xmath59 mev and @xmath65 mev ) is shown in fig . [ fig : etpr]a . the fit of this distribution fixes the background shape . finally , the data were fit using the background shape fixed from fig . [ fig : etpr]a together with that of the signal from simulation in fig . [ fig : etpr]b . the result of the fit is @xmath72 . using the number of events from the fit , one can calculate the relative branching ratio : @xmath73 + where the values of the branching ratios of @xmath74 and @xmath75 were taken from @xcite . a separate analysis of the normalizing decay @xmath24 has recently been published @xcite , with a branching ratio of @xmath76 consistent with previous measurements @xcite and thus giving confidence in the analysis presented here . in the above calculation of the relative branching ratio common systematic errors such as luminosity determination cancel exactly , while others such as detector inefficiency and evaluation of radiative corrections cancel approximately . finally , using the value of @xmath77 from @xcite , one obtains : @xmath78 . the last error is our estimate of the systematic uncertainty . the sources of systematic errors are the following : * uncertainties in the ratio @xmath79 caused by different energy spectra for final photons and pions - 10% ; * uncertainties in the branching ratios @xmath80 , @xmath81 , @xmath82 and @xmath83 - 6.3% ; * determination of the background shape 5% ; * different resonance shape caused by different energy dependence of the phase space - 2% ; the total systematic error obtained by adding separate contributions quadratically is 13% . the results of our analysis have higher statistical significance than before since they are based on a data sample of 21 selected events compared to 6 events in our previous work @xcite and 5 events observed by snd @xcite . the obtained value of the branching ratio @xmath84 @xmath85 . + agrees with our previous result based on part of the whole data sample @xcite @xmath86 + as well as with the result of the snd group @xcite @xmath87 + and is more precise . within experimental accuracy it is also consistent with the preliminary result of cmd-2 based on other decay modes of the @xmath88 meson ( @xmath89 ) with four charged pions and two or more photons in the final state @xcite : @xmath90 . analysis of the available data sample of the produced @xmath6 mesons by both cmd-2 and snd and full use of other decay modes of the @xmath18 and @xmath88 mesons will further improve the statistical error . much larger increase can be expected from the da@xmath91ne @xmath6-factory where one plans to accumulate the number of @xmath6 mesons by at least two orders of magnitude higher than ours . let us briefly discuss theoretical predictions for the decay under study . usual methods of the description of radiative decays are based on the nonrelativistic quark model @xcite . various ways of incorporating effects of su(3 ) breaking have been suggested leading to the values of @xmath29 in the range @xmath92 @xcite . the value of the branching ratio studied in our work is also of interest for the problem of @xmath93 mixing which has been a subject of intense investigation for a long time @xcite . it is sensitive to the structure of the @xmath18 wave function or , in other words , to the contribution of various @xmath94 states as well as the possible admixture of glue in it @xcite . according to @xcite , a branching ratio @xmath95 would indicate a substantial glue component in the @xmath18 , while the expected branching ratio is less than @xmath96 for a pure gluonium . even smaller values were obtained in @xcite assuming a specific model of qcd violation . the revival of interest to the problem of the @xmath18 structure and possible contents of glue in it ( see @xcite and references therein ) was partially due to two recent observations by cleo involving the @xmath18 meson : in @xcite it was shown that the transition form factor of the @xmath18 studied in the two photon processes strongly differs from those for the @xmath75 and @xmath88 mesons and in @xcite the unexpectedly high magnitude of the rate of @xmath97 was observed . however , in a recent paper @xcite it is claimed that it is impossible to disentangle the effects of the nonet symmetry breaking and those of glue inside the @xmath18 . most of the models mentioned above are able to describe the data reasonably well in terms of some number of free parameters which , unfortunately , can not be determined from first principles . an attempt to overcome this drawback was made in @xcite where radiative decays of light vector mesons are considered in the approach based on qcd sum rules @xcite and the value @xmath98 is obtained for the branching ratio of @xmath5 decay . one can summarize that the variety of theoretical approaches to the problem of the description of the @xmath6 meson radiative decay to @xmath99 is rather broad and more theoretical insight into the problem is needed . using an almost five times bigger data sample than in the first measurement the cmd-2 group confirmed the observation of the rare radiative decay @xmath100 . the measured branching ratio is : @xmath78 . + its value is consistent with most of the theoretical predictions based on the quark model and assuming a standard quark structure of the @xmath18 . it rules out exotic models suggesting a high glue admixture @xcite or strong qcd violation @xcite . further progress in this field can be expected after the dramatic increase of the number of produced @xmath6 mesons expected at the da@xmath91ne @xmath6-factory and refinement of theoretical models of radiative decays . the authors are grateful to m.benayoun and v.n.ivanchenko for useful discussions .

a new measurement of the rare decay @xmath0 performed with the cmd-2 detector at novosibirsk is described . of the data sample corresponding to the integrated luminosity of 14.5 pb@xmath1 , twenty one events have been selected in the mode @xmath2 , @xmath3 . the following branching ratio was obtained : b(@xmath4 .

models of cyclic dominance are traditionally employed to study biodiversity in biologically inspired settings @xcite . the simplest such model is the rock - paper - scissors game @xcite , where rock crashes scissors , scissors cut paper , and paper wraps rock to close the loop of dominance . the game has no obvious winner and is very simple , yet still , it is an adequate model that captures the essence of many realistic biological systems . examples include the mating strategy of side - blotched lizards @xcite , the overgrowth of marine sessile organisms @xcite , genetic regulation in the repressilator @xcite , parasitic plant on host plant communities @xcite , and competition in microbial populations @xcite . cyclical interactions may also emerge spontaneously in the public goods game with correlated reward and punishment @xcite , in the ultimatum game @xcite , and in evolutionary social dilemmas with jokers @xcite or coevolution @xcite . an important result of research involving the rock - paper - scissors game is that the introduction of randomness into the interaction network results in global oscillations @xcite , which often leads to the extinction of one species , and thus to the destruction of the closed loop of dominance that sustains biodiversity . more precisely , in a structured population where the interactions among players are determined by a translation invariant lattice , the frequency of every species is practically time - independent because oscillations that emerge locally can not synchronize and come together to form global , population - wide oscillations . however , if shortcuts or long - range interactions are introduced to the lattice , or if the original lattice is simply replaced by a small - world network @xcite , then initially locally occurring oscillations do synchronize , leading to global oscillations and to the accidental extinction of one species in the loop , and thus to loss of biodiversity @xcite . if the degree distribution of interaction graph is seriously heterogeneous , however , then such kind of heterogeneity can facilitate stable coexistence of competing species @xcite . interestingly , other type of randomness , namely the introduction of mobility of players , also promotes the emergence of global oscillations that jeopardize biodiversity @xcite . interestingly , however , although long - range interactions and small - world networks abound in nature , and although mobility is an inherent part to virtually all animal groups , global oscillations are rarely observed in actual biological systems . it is thus warranted to search for universal features in models of cyclic dominance that work in the opposite way of the aforementioned types of randomness . the questions is , what is the missing ingredient that would prevent local oscillations to synchronize across the population to form global oscillations ? preceding research has already provided some possible answers . for instance peltomki and alava observed that global oscillations do not occur if the total number of players is conserved @xcite . mobility , for example , then has no particular impact on biodiversity because oscillations are damped by the conservation law . however , the consequence of the conservation law does not work anymore if a tiny fraction of links forming the regular lattice is randomly rewired @xcite . zealots , on the other hand , have been identified as a viable means to suppress global oscillations in the rock - paper - scissors game in the presence of both mobility and interaction randomness @xcite . in addition to these examples , especially in the realm of statistical physics , there is a wealth of studies on the preservation and destruction of biodiversity in models of cyclic dominance @xcite . here we wish to extend the scope of this research by considering a partly overlooked property , namely the consideration of site - specific heterogeneous invasion rates . importantly , we wish to emphasize an important distinction to species - specific heterogeneous invasion rates , which have been considered intensively before . in the latter case , different pairs of species are characterized by different invasion rates , but these differences are then applied uniformly across the population . in case of spatially variable invasion rates , these could be site - specific , and hence particular pairs of species may have different invasion rates even though they are of the same type . such a setup has many analogies in real life , ranging from differing resources , quality or quantity wise , to variations in the environment , all of which can significantly influence the local success rate of the governing microscopic dynamics . notably , this kind of heterogeneity was already studied in a two - species lotka - volterra - like system @xcite , and in a three - species cyclic dominant system where a lattice has been used as the interaction network @xcite . the latter work concluded that the invasion heterogeneity in spatial rock - paper - scissors models has very little effect on the long - time properties of the coexistence state . in this paper , we go beyond the lattice interaction topology , exploring the consequences of quenched and annealed randomness being present in the interaction network . in the latter case , as we will show , it could be a decisive how heterogeneity is introduced into the invasion rate because annealed randomness does not change the oscillation but quenched heterogeneity can mitigate the global oscillation effectively . in what follows , we first present the main results and discuss the implications of our research , while details concerning the model and the methodology are described in the methods section . we first consider results obtained with species - specific invasion rates . indeed , it is possible to argue that it is too idealistic to assume homogenous invasion rates between different species , and that it would be more realistic to assume that these invasion rates are heterogeneous . but as results presented in fig . [ suppressed ] show , this kind of generalization does not bring about a mechanism that would suppress global oscillations . these oscillations clearly emerge for homogeneous species - specific invasion rates , as soon as the fraction of rewired links of the square lattice @xmath0 exceeds a threshold . if we then assume that species - specific invasion rates are heterogeneous , say @xmath1 , @xmath2 , and @xmath3 ( here @xmath4 denotes the invasion rates of @xmath5 transition where @xmath6 runs from @xmath7 to @xmath8 in a cyclic manner ) , it can be observed that nothing really changes . in fact , the threshold in @xmath0 remains much the same , and the order parameter @xmath9 ( the area of the limit cycle in the ternary diagram ) reaches the same close to @xmath10 plateau it does when these invasion rates are homogenous . further along this line , we can even adopt invasion rates that are chosen uniformly at random from the unit interval at each particular instance of the games . more precisely , we still keep the original @xmath5 direction of invasion , but the strength of the invasion rate @xmath4 is chosen randomly in each particular case . but no matter the fact that this rather drastically modifies the microscopic dynamics , the presence of shortcuts will still trigger global oscillations ( marked random in fig . [ suppressed ] ) . we thus arrive at the same conclusion that was already pointed out in @xcite , which is that heterogeneous invasion reaction rates have very little effect on the dynamics and the long - time properties of the coexistence state . having established the ineffectiveness of heterogeneous species - specific invasion rates to prevent local oscillations to synchronize across the population to form global oscillations , we next consider site - specific heterogeneous interaction rates , denoted as @xmath11 and applied to each site @xmath12 . here @xmath11 determines the probability that a neighbor will be successful when trying to invade player @xmath12 according to the original @xmath5 rule . as such , different values of @xmath11 influence the success of microscopic dynamics locally . moreover , these invasion rates are determined once at the start of the game and can be drawn from different distributions . the simplest case is thus to consider values drawn uniformly at random from the unit interval . as results in fig . [ suppressed ] show ( see quenched random ) , this modification of the rock - paper - scissors game clearly blocks the emergence of global oscillations regardless of the value of @xmath0 . indeed , even if the square lattice is , through rewiring , transformed into a regular random graph , the order parameter @xmath9 still remains zero . even if the uniform distribution is replaced by a simple discrete double - peaked distribution ( practically it means that half of the players has @xmath13 for example , while the other half retains @xmath14 ) , the global oscillations never emerge ( see quenched double in fig . [ suppressed ] ) . the coordination effect leading up to global oscillations is thus very effectively disrupted by heterogeneous site - specific invasion rates , and this regardless of the distribution from which these rates are drawn . to illustrate the dramatically contrasting consequences of different types of randomness , we show in fig . [ ternary ] representative time evolutions for both cases . the comparison reveals that , as deduced from the values of the order parameter @xmath9 displayed in fig . [ suppressed ] , time - varying invasion rates fail to suppress global oscillations , the emergence of which is supported by the small - world properties of the interaction network ( ternary diagram and the time course on the left ) . the limit cycle denoted black in the ternary diagram and the large - amplitude oscillations of the densities of species in the corresponding bottom panel clearly attest to this fact . this stationary state is robust and is reached independently of the initial mixture of competing strategies . conversely , quenched heterogeneous interaction rates drawn from a uniform distribution clearly suppress global oscillations ( ternary diagram and the time course on the right ) . here , the system will always evolve into the @xmath15 state , central point of the diagram , even if we launched the evolution from a biased initial state . thus , if heterogeneities are fixed in space , just like in several realistic biological systems , then this effectively prohibits global oscillation by disrupting the organization of a coordinated state , i.e. , synchronization of locally occurring oscillations across the population . as demonstrated previously @xcite , the type of randomness in the interaction network responsible for the emergence of global oscillations plays a negligible role . be it quenched through the one - time rewiring of a fraction @xmath0 of links forming the original translation invariant lattice , or be it annealed through the random selection of far - away players to replace nearest neighbors as targets of invasion with probability @xmath16 , there exist a critical threshold in both where global oscillations emerge if invasion rates are homogeneous . accordingly , it makes sense to test whether heterogeneous site - specific invasions rates are able to suppress such oscillations regardless of the type of randomness that supports them . to that effect , we make use of the discrete double - peaked distribution , where the fraction of sites @xmath17 having a lower invasion rate @xmath13 then the rest of the population at @xmath14 can be a free parameter determining the level of heterogeneity . evidently , at @xmath18 we retain the traditional rock - paper - scissors game with homogeneous invasion rates ( all sites have @xmath14 ) , while for @xmath19 the fraction of sites having @xmath13 , and thus the level of heterogeneity , increases . at the other extreme , for @xmath20 , we of course again obtain a homogeneous population where everybody has @xmath13 , but we do not explore this option since it is practically identical , albeit the evolutionary process is much slower . by introducing heterogeneity into the system gradually , we can monitor how it influences the stationary state . in fig . [ nu ] , we present representative results for both quenched and annealed randomness of interaction graph ( see legend ) . the first observation is that only a minute fraction of suppressed nodes ( less than @xmath21 ) suffices to fully suppress global oscillations , and this regardless of the applied high @xmath0 and @xmath16 values that practically ensure an optimal support for local oscillations to synchronize across the population into global oscillations . moreover , it can be observed that both transitions to the oscillation - free state are continuous . in other words , there does not exist a sharp drop in the value of @xmath9 at a particular value of @xmath17 . instead , the suppression of global oscillations is gradual as the level of site - specific invasion heterogeneity in the population increases . similar in spirit , another way to introduce invasion heterogeneity gradually into the population is to use a fixed fraction of nodes with a lower invasion rate , but vary the difference to @xmath14 . accordingly , we have a fraction @xmath22 of nodes , which instead of @xmath14 have the invasion rate @xmath23 . here @xmath24 becomes the free parameter , which for zero returns the traditional rock - paper - scissors game with homogeneous invasion rates , while for @xmath25 the distance in the peaks of the discrete double - peaked distribution , and thus the level of heterogeneity in the population , increases . representative results obtained with this approach are shown in fig . [ dif ] for both quenched and annealed randomness of interaction graph ( see legend ) . in comparison with results presented in fig . [ nu ] , it can be observed that increasing @xmath24 has somewhat different consequences than increasing @xmath17 . in the former case , when @xmath24 is small , the slight heterogeneity has no particular influence on the stationary state and global oscillations persist well beyond @xmath26 for annealed randomness and @xmath27 for quenched randomness . but if the difference reaches a sufficiently large value , global oscillations disappear in much the same gradual way as observed before in fig . [ nu ] , although the transition for annealed randomness is more sudden . to sum up our observations thus far , different versions of the same concept reveal that spatially quenched heterogeneity in site - specific invasion rates is capable to effectively suppress global oscillations that would otherwise be brought about by either annealed or quenched randomness in the interaction network . however , there is yet another possible source or large - amplitude global oscillations in the population , namely mobility . as is well - known , mobility can give rise to global oscillation that jeopardizes biodiversity @xcite . although subsequent research revealed that global oscillations due to mobility do not emerge if the total number of competing players is conserved @xcite , more recently it was shown that , if in addition to a conservation law also either quenched or annealed randomness is present in the interaction network , then mobility still induces global oscillations @xcite . in particular , if the site exchange is intensive then only a tiny level of randomness in the host lattice suffices to evoke global oscillations . lastly , we thus verify if heterogeneity in site - specific invasion rates is able to suppress global oscillations brought about by mobility . as results in fig . [ mob ] show , the impact of quenched invasion heterogeneity is very similar to the above - discussed cases . it is worth noting that conceptually similar behavior can be observed when biological species are hosted in a turbulent flow of fluid environment @xcite . in fact , as a general conclusion , neither randomness in the interaction network nor the mobility of players can compensate for the detrimental impact of spatial invasion heterogeneity on global oscillations , thus establishing the latter as a very potent proponent of biodiversity in models of cyclic dominance . we have studied the impact of site - specific heterogeneous invasion rates on the emergence of global oscillations in the spatial rock - paper - scissors game . we have first confirmed that species - specific heterogeneous invasion rates , either fixed or varying over time , fail to disrupt the synchronization of locally emerging oscillations into a global oscillatory state on a regular small - world network . on the contrary , we have then demonstrated that site - specific heterogeneous invasion rates , determined once at the start of the game , successfully hinder the emergence of global oscillations and thus preserves biodiversity . we have shown this conclusion to be valid independently of the properties of the distribution that determines the invasion heterogeneity , specifically demonstrating the failure of coordination for uniformly and double - peak distributed site - specific invasion rates . moreover , our research has revealed that quenched site - specific heterogeneous invasion rates preserve biodiversity regardless of the type of randomness that would be responsible for the emerge of global oscillations . in particular , we have considered quenched and annealed randomness in the interaction network , as well as mobility . regardless of the type of randomness that would promote local oscillations to synchronize across the population to form global oscillations , site - specific heterogeneous invasion rates were always found to be extremely effective in suppressing the emergence of global oscillations . drawing from the colloquial expression used to refer to alcohol that is consumed with the aim of lessening the effects of previous alcohol consumption , the introduction of randomness in the form of site - specific heterogeneous invasion rates lessens , in fact fully suppresses , the effects of other types of randomness hence the `` hair of the dog '' phenomenon . our setup takes into account heterogeneities that are inherently present in virtually all uncontrolled environments , ranging from bacterial films to plant communities . examples include qualitative and quantitative variations in the availability of nutrients , local differences in the habitat , or any other factors that are likely to influence the local success rate of the governing dynamics @xcite . the consideration of site - specific heterogeneous invasion rates and their ability to suppress global oscillations joins the line of recent research on the subject , showing for example that the preservation of biodiversity is promoted if a conservation law is in place for the total number of competing players @xcite , or if zealots are introduced to the population @xcite . notably , previously it was shown that zealotry can have a significant impact on the segregation in a two - state voter model @xcite , and research in the realm of the rock - paper - scissors game confirmed such an important role of this rather special uncompromising behavior . in general , since global , population - wide oscillations are rarely observed in nature , it is of significance to determine key mechanisms that may explain this , especially since factors that promote such oscillations , like small - world properties , long - range interactions , or mobility , are very common . in this sense , site - specific heterogeneous invasion rates fill an important gap in our understanding of the missing ingredient that would prevent local oscillations to synchronize across the population to form global oscillations . there is certainly no perfect spatial system where microscopic processes would unfold identically across the whole population . these imperfections are elegantly modeled by the heterogeneous host matrix that stores the individual invasion rates of each player . as we have shown , the coordination of species evolution is highly sensitive on such kind of heterogeneities when they are fixed in space . ultimately , this prevents the synchronization of locally emerging oscillations , and gives rise to a `` hair of the dog''-like phenomenon , where one type of randomness is used to mitigate the adverse effects of other types of randomness . we hope that these theoretical explorations will help us to better understand the rare emergence of global oscillations in nature , as well as inspire further research , both experimental and theoretical , along similar lines . the spatial rock - paper - scissors game evolves on a @xmath28 square lattice with periodic boundary conditions , where each site @xmath12 is initially randomly populated by one of the three competing species . for convenience , we introduce the @xmath5 notation , where @xmath6 runs from @xmath7 to @xmath8 in a cyclic manner . hence , species @xmath29 ( for example paper ) invades species @xmath30 ( rock ) , while species @xmath30 invades species @xmath31 ( scissors ) , which in turn invades species @xmath29 to close the loop of dominance . the evolution of species proceeds in agreement with a random sequential update , where during a full monte carlo step ( @xmath32 ) we have chosen every site once on average and a neighbor randomly . in case of different players the invasion was executed according to the rock - scissors - paper rule with probability @xmath33 . in the simplest , traditional version of the game , all invasion rates between species are equal to @xmath34 . species - specific heterogeneous invasion rates can be introduced through the parameter @xmath35 , which is simply the probability for the @xmath5 invasion to occur when given a chance . the values @xmath36 , @xmath37 and @xmath38 can be determined once at the start of the game , or they can be chosen uniformly at random from the unit interval at each particular instance of the game . on the other hand , site - specific heterogeneous invasion rates , which we denote as @xmath11 , apply to each site @xmath12 in particular , and determine the probability that a neighbor will be successful when trying to invade player @xmath12 according to the @xmath5 rule . this rule can be considered as `` prey - dependent '' because the @xmath39 value at the prey s position determines the probability of invasion . as an alternative rule , we can consider the @xmath39 value of predator s position that determines the invasion probability . lastly , we can assume that the @xmath39 values of both the predator and prey s positions influence the invasion rate via their @xmath40 product . while the time dependence of the evolution will be different in the mentioned three cases but the qualitative behavior is robust . therefore we restrict ourself to the first mentioned `` prey - dependent '' rule . these invasion rates are determined once at the start of the game and can be drawn uniformly at random from the unit interval , or from any other distribution . here , in addition to uniformly distributed @xmath11 , we also consider site - specific heterogeneous invasion rates drawn from a discrete double - peaked distribution , where a fraction @xmath17 of sites have @xmath23 , while the remaining @xmath41 have @xmath14 . to test the impact of site - specific heterogeneous invasion rates under different circumstances , we consider interaction randomness in the form of both quenched and annealed randomness . quenched randomness is introduced by randomly rewiring a fraction @xmath0 of the links that form the square lattice whilst preserving the degree @xmath42 of each site . this is done only once at the start of the game . this procedure returns regular small - world networks for small values of @xmath0 and a regular random network in the @xmath43 limit @xcite . annealed randomness , on the other hand , is introduced so that at each instance of the game a potential target for an invasion is selected randomly from the whole population with probability @xmath16 , while with probability @xmath44 the invasion is restricted to a randomly selected nearest neighbor @xcite . this procedure returns well - mixed conditions for @xmath45 , while for @xmath46 only short - range invasions as allowed by the original square lattice are possible . we also consider the impact of site - specific heterogeneous invasion rates in the presence of mobility . the latter is implemented so that during each instance of the game we choose a nearest - neighbor pair randomly where players exchange their positions with probability @xmath47 . oppositely , with probability @xmath48 , the dominant species in the pair invades the other in agreement with the rules or the rock - paper - scissors game . the parameter @xmath47 hence determines the intensity of mobility while the number of players is conserved . technically , however , the strategy exchange between neighboring players is determined not only by the level of mobility @xmath47 , but it also depends on the individual @xmath11 and @xmath49 values characterizing the neighboring sites @xmath12 and @xmath50 . in this way , we can consider the fact that different sites may be differently sensitive to the change of strategy , and the success of mutual change is then practically determined by the site that is more reluctant to change its state . accordingly , when the strategy exchange is supposed to be executed , then this happens only with the probability @xmath51 that is equal to the smaller of @xmath11 and @xmath49 values ( all the other details of the model remain the same as above ) . global oscillations are characterized with the order parameter @xmath9 , which is defined as the area of the limit cycle in the ternary diagram @xcite . this order parameter is zero when each species occupies one third of the population , and becomes one when the system terminates into an absorbing , single - species state . we have used lattices with up to @xmath52 sites , which was large enough to avoid accidental fixations when the amplitude of oscillations was large , and which allowed an accurate determination of strategy concentrations that are valid in the large population size limit . naturally , the relaxation time depends sensitively on the model parameters and the system size , but @xmath53 mcs was long enough even for the slowest evolution that we have encountered during this study .

global , population - wide oscillations in models of cyclic dominance may result in the collapse of biodiversity due to the accidental extinction of one species in the loop . previous research has shown that such oscillations can emerge if the interaction network has small - world properties , and more generally , because of long - range interactions among individuals or because of mobility . but although these features are all common in nature , global oscillations are rarely observed in actual biological systems . this begets the question what is the missing ingredient that would prevent local oscillations to synchronize across the population to form global oscillations . here we show that , although heterogeneous species - specific invasion rates fail to have a noticeable impact on species coexistence , randomness in site - specific invasion rates successfully hinders the emergence of global oscillations and thus preserves biodiversity . our model takes into account that the environment is often not uniform but rather spatially heterogeneous , which may influence the success of microscopic dynamics locally . this prevents the synchronization of locally emerging oscillations , and ultimately results in a phenomenon where one type of randomness is used to mitigate the adverse effects of other types of randomness in the system .

simple models for the interaction between fermions and bosons continue to be fascinating , as often very non - trivial results can be obtained from even the most primitive hamiltonians . exactly solvable models for the interaction of photons or phonons with electrons in quantum dots @xcite or quasi - one - dimensional systems @xcite provide the best examples , as they often provide a deep insight into rich and complex physics . in this contribution , we re - consider a simple model for a single electron of mass @xmath0 in one dimension that interacts with a delta - barrier through a coupling parameter that itself is a dynamical quantity . the hamiltonian is written as @xmath1\right\}+\omega a^{\dagger}a.\end{aligned}\ ] ] here , @xmath2 creates a boson of frequency @xmath3 and @xmath4 $ ] is a dynamical contribution on top of the static coupling constant @xmath5 . the constant zero point energy is omitted since it merely shifts the energy scale by @xmath6 . the lattice version of this model was originally introduced by gelfand , schmitt - rink and levi @xcite years ago in the study of tunneling in presence of phonons , and was shortly afterwards compared to a corresponding time - dependent classical hamiltonian @xcite , the continuous version of which reads @xmath7 @xmath8 is obtained as the interaction picture hamiltonian of eq.([hamiltonian ] ) with respect to @xmath9 , after replacing the boson operators by @xmath10 . in its time - dependent version , eq.([hamiltonianc ] ) has subsequently been used as a model for scattering in quasi - one - dimensional quantum wires by bagwell and co - workers @xcite , who found fano - type resonances in the transmission coefficient as a function of the energy of an incident electron . it soon turned out that the scattering properties of this hamiltonian are quite intriguing as they very much depend on the relative sign and strength of the two coupling parameters @xmath5 and @xmath11 . the interplay between evanescent modes , quasibound states @xcite , and the behaviour of the transmission amplitude in the complex energy plane @xcite have been studied recently . our focus here is on the quantum version eq . ( [ hamiltonian ] ) of the model and its peculiarities in comparison with @xmath12 . it turns out that beside transmission zeroes , there are points of perfect transparency in the fano resonance that only appear in the model @xmath13 but not in @xmath12 . perfect transmission and fano resonances have been found recently in the transmission of phonons through non - linear chains without delta impurities @xcite . although not discussed in detail here , these results indicate that there still is rich and non - trivial behavior to be discovered from models like eq.([hamiltonian ] ) . the total wave function @xmath14 of the coupled electron - boson system can be expanded in the oscillator basis @xmath15 as @xmath16 with wave function coefficients @xmath17 depending on the position @xmath18 of the electron . we solve the stationary schrdinger equation at total energy @xmath19 , implying a scattering condition for the electron part of the wave function in demanding that there is no electron incident from the right . for @xmath20 , the @xmath17 are superpositions of plane waves if @xmath21 is above the threshold for the @xmath22 boson energy , @xmath23 whereas normalizabale evanescent modes occur if @xmath21 is below the threshold , @xmath24 here and in the following we set @xmath25 . we impose the condition that the boson is in its ground state for an electron incoming from the left , @xmath26 where we set the corresponding amplitude @xmath27 to unity . continuity of @xmath17 at @xmath28 yields @xmath29 for all @xmath30 , whereas the jump in derivative of @xmath17 across the delta barrier leads to a recursion relation for the transmission amplitudes @xmath31 , @xmath32 where the @xmath33 are real ( imaginary ) above ( below ) the boson energy @xmath34 , @xmath35 the total transmission coefficient @xmath36 is obtained from the sum over all _ propagating _ modes , @xmath37}\frac{k_n(e)}{k_0(e)}|t_n(e)|^2,\end{aligned}\ ] ] where the sum runs up to the largest @xmath30 such that @xmath38 remains real . although eq.([transmission ] ) is a finite sum , its evaluation requires the solution of the _ infinite _ recursion relation eq.([recursion ] ) due to the fact that the propagating modes are coupled to all evanescent modes . the transmission amplitudes can be determined from the linear equation @xmath39 numercally , this can easily been solved by truncation of the matrix @xmath40 . alternatively , one can solve eq.([matrix ] ) recursively which actually is numerically more efficient . for example , the result for the zero - channel transmission amplitude @xmath41 can be written in a very intuitive form : defining the ` greens function ' @xmath42 by @xmath43^{-1},\end{aligned}\ ] ] one can write @xmath41 with the help of a recursively defined ` self energy ' @xmath44 , @xmath45 in fact , using @xmath46 , the self energy @xmath47 can be represented as a continued fraction @xmath48 this demonstrates that @xmath41 depends on @xmath11 only through @xmath49 . truncating the matrix @xmath40 to a @xmath50 matrix corresponds to the approximation that sets @xmath51 and recursively solves eq . ( [ selfenergy ] ) for @xmath52 down to @xmath47 . for example , truncating at @xmath53 we obtain the approximation @xmath54 an important observation can be made with respect to the stability of our theory for large coupling constants @xmath11 . in fact , the truncation at @xmath55 is only consistent if the truncated self energy @xmath44 is a small correction to the inverse ` free propagator ' , @xmath56 which by use of eq . ( [ g0def ] ) at large @xmath57 implies @xmath58 or @xmath59 . the tridiagonal form of the matrix , eq . ( [ matrix ] ) , actually implies that the method based on the recursion eq . ( [ recursion ] ) is perturbative in the coupling @xmath11 to the boson . we conjecture that for @xmath11 above the critical value , the perturbation based on the oscillator basis @xmath15 used here breaks down . a similar breakdown of numerical approaches that start from a weak coupling regime in single boson hamiltonians is known from the rabi hamiltonian @xcite , i.e. the coupling of a single boson mode to a spin @xmath60 . we mention that the lattice version of the present model would be a natural starting point for a more detailed analysis of the strong coupling ( small polaron ) limit . transmission coefficient through a dynamical one - dimensional delta barrier with repulsive ( @xmath61 , left ) and attractive ( @xmath62 , right ) static part , cf . ( [ hamiltonian ] ) and ( [ hamiltonianc ] ) . @xmath21 is the energy of the incident particle.,title="fig:",scaledwidth=50.0% ] transmission coefficient through a dynamical one - dimensional delta barrier with repulsive ( @xmath61 , left ) and attractive ( @xmath62 , right ) static part , cf . ( [ hamiltonian ] ) and ( [ hamiltonianc ] ) . @xmath21 is the energy of the incident particle.,title="fig:",scaledwidth=50.0% ] the recursion relation corresponding to eq . ( [ recursion ] ) for the classical time - dependent hamiltonian , eq . ( [ hamiltonianc ] ) , was derived and discussed by bagwell and lake @xcite , @xmath63 here , @xmath31 is the coefficient of the time - dependent electron wave function in photon side - band @xmath30 , where @xmath30 runs through positive _ and negative _ integers @xmath30 . in further contrast to the recursion relation eq . ( [ recursion ] ) , there are no factors @xmath64 and @xmath65 multiplying the coupling constant @xmath11 . this latter fact is an important difference to the quantum case where these terms lead to the factors @xmath57 that multiply @xmath49 in the self energies @xmath44 , eq . ( [ selfenergy ] ) . this difference is eventually responsible for the breakdown of the perturbative approach for large @xmath11 in the quantum case . a continued fraction representation of @xmath41 for the classical case has been derived recently by martinez and reichl @xcite . the corresponding matrix defining the transmission amplitudes @xmath66 in the classical case is the infinite tridiagonal matrix @xmath67 with @xmath68 on the diagonal and @xmath11 on the lower and upper diagonals , @xmath69 fig . ( [ delta1.eps ] ) shows a comparison between the transmission coefficient @xmath36 , eq . ( [ transmission ] ) , for the quantum and the classical barrier . in the repulsive case with @xmath70 , the dynamical part of the barrier is essentially a weak perturbation to the unperturbed @xmath71 case . additional structures ( cusps ) appear at the boson ( photo side - band ) energies @xmath34 although the overall @xmath36-curve resembles the @xmath71 case . the more interesting case occurs for barriers with an attractive static part , @xmath62 ( fig . ( [ delta1.eps ] ) , right ) . a fano type resonance appears below the first threshold @xmath72 where the transmission coefficient has a zero in both the classical and the quantum case . in the classical case , this is a well - known phenomenon @xcite : the transmission zero for weak coupling ( small @xmath11 ) shows up when the fano resonance condition @xmath73 is fulfilled.there , the energy of the electron in the first side channel ( @xmath74 ) coincides with the bound state of the attractive delta barrier potential , @xmath75 . in the quantum case , the self energy in eq.([selfenergy ] ) diverges at the zeros of @xmath36 , @xmath76^{-1}=0.\end{aligned}\ ] ] for @xmath77 , @xmath78 , cf . eq.([t0approx ] ) , and the two conditions eq.([fano1 ] ) and eq.([fano1a ] ) coincide . the most interesting feature in the scattering properties of the dynamical quantum barrier is the appearance of an energy close to the first channel @xmath79 threshold where perfect transmission @xmath80 occurs . this is clearly visible in the vanishing of the reflection coefficient , @xmath81 , in the logarithmic plot fig . ( [ delta3.eps ] ) . for a repulsive static part , @xmath82 , this occurs at an energy below the energy where the reflection coefficient comes close to unity , and above that energy if the static part is attractive ( @xmath83 ) . in contrast , in the classical case the reflection coefficient never reaches zero in neither the repulsive nor the attractive case . this contrast becomes even more obvious in a two - dimensional plot where the zeros in @xmath84 correspond to ` ridges ' in the @xmath5-@xmath21 plane , cf . ( [ delta4.eps ] ) . logarithmic plot of reflection coefficient @xmath85 for dynamical delta barrier with static repulsive ( @xmath61 , left ) and attractive ( @xmath62 , right ) core . , title="fig:",scaledwidth=45.0% ] logarithmic plot of reflection coefficient @xmath85 for dynamical delta barrier with static repulsive ( @xmath61 , left ) and attractive ( @xmath62 , right ) core . , title="fig:",scaledwidth=45.0% ] density plot of @xmath86 ( reflection coefficient ) for the quantum delta barrier at @xmath87 . exact solution from eq . ( [ selfenergy ] ) ( left ) , from the @xmath53 truncation eq . ( [ t0approx ] ) ( center ) , and from the classical model eq.([hamiltonianc ] ) ( right ) . the light ` ridges ' correspond to curves of perfect transmission @xmath88 , cf . ( [ perfecta ] ) and eqs . ( [ perfectb]).,scaledwidth=95.0% ] perfect transparency ( @xmath89 ) can be understood by considering the transmission amplitude @xmath41 which determines the total transmission below the first sideband threshold . recalling that @xmath90 , in the quantum case the transmission coefficient becomes unity when @xmath91 our exact continued fraction expression for the self energy , eq.([sigmacontinued ] ) , implies that for @xmath92 , @xmath47 is real because @xmath93 is real for @xmath94 . the condition eq.([perfect ] ) then means that the self energy exactly renormalizes the static part @xmath5 of the scattering potential to zero . for small @xmath11 , we can use our perturbative expression corresponding to truncating the matrix @xmath40 , eq.([matrix ] ) , to a two - by - two matrix . the perfect transparency condition eq.([perfect ] ) then becomes @xmath95 which determines the position of the perfect transmission energy . the solution of eq.([perfecta ] ) defines two curves in the @xmath21@xmath5-plane with perfect transmission for @xmath92 , @xmath96 these two curves can be clearly identified in the logarithmic density plots of the reflection coefficient @xmath97 , cf . ( [ delta4.eps ] ) . the @xmath53 approximation to the transmission amplitude , eq.([t0approx ] ) , turns out to reproduce these features quite well even at moderate coupling constants @xmath11 . the above analysis of the two models eq . ( [ hamiltonian ] ) and eq . ( [ hamiltonianc ] ) has revealed some interesting differences between scattering properties of simple electron - boson models and their classical counter - part . the strong coupling limit of the quantum model and its extension to more complicated situations like multi - channel scattering remain to be explored .

we discuss electron scattering in a one - dimensional delta barrier potential with either time - dependent coupling constant ( classical model ) or a coupling constant that is linear in a boson coordinate ( quantum model ) . we find an exact continued fraction solution and fano like resonances in the transmission coefficient . in the quantum model , energies for perfect transmission exist below the first sideband threshold .

we study the algorithmic properties of infinite graphs that result from a natural unfolding operation applied to finite graphs . the unfolding process always produces infinite graphs of finite degree . moreover , the class of resulting graphs is a subclass of the class of automatic graphs . as such , any element of this class possesses all the known algorithmic and algebraic properties of automatic structures . an equivalent way to describe these graphs employs automata over a unary alphabet ( see theorem [ thm : gsigma ] ) . therefore , we call this class of graphs _ unary automatic graphs of finite degree_. in recent years there has been increasing interest in the study of structures that can be presented by automata . the underlying idea in this line of research consists of using automata ( such as word automata , bchi automata , tree automata , and rabin automata ) to represent structures and study logical and algorithmic consequences of such presentations . informally , a structure @xmath0 is _ automatic _ if the domain @xmath1 and all the relations @xmath2 , @xmath3 , @xmath4 of the structure are recognized by finite automata ( precise definitions are in the next section ) . for instance , an automatic graph is one whose set of vertices and set of edges can each be recognized by finite automata . the idea of automatic structures was initially introduced by hodgson @xcite and was later rediscovered by khoussainov and nerode @xcite . automatic structures possess a number of nice algorithmic and model - theoretic properties . for example , khoussainov and nerode proved that the first - order theory of any automatic structure is decidable @xcite . this result is extended by adding the @xmath5 ( there are infinitely many ) and @xmath6 ( there are @xmath7 many mod @xmath8 ) quantifiers to the first order logic @xcite . blumensath and grdel proved a logical characterization theorem stating that automatic structures are exactly those definable in the following fragment of the arithmetic @xmath9 , where @xmath10 and @xmath11 have their usual meanings and @xmath12 is a weak divisibility predicate for which @xmath13 if and only if @xmath14 is a power of @xmath15 and divides @xmath16 @xcite . automatic structures are closed under first - order interpretations . there are descriptions of automatic linear orders and trees in terms of model theoretic concepts such as cantor - bendixson ranks @xcite . also , khoussainov , nies , rubin and stephan have characterized the isomorphism types of automatic boolean algebras @xcite ; thomas and oliver have given a full description of finitely generated automatic groups @xcite . some of these results have direct algorithmic implications . for example , isomorphism problem for automatic well - ordered sets and boolean algebras is decidable @xcite . there is also a body of work devoted to the study of resource - bounded complexity of the first order theories of automatic structures . for example , on the one hand , grdel and blumensath constructed examples of automatic structures whose first - order theories are non - elementary @xcite . on the other hand , lohrey in @xcite proved that the first - order theory of any automatic graph of bounded degree is elementary . it is worth noting that when both a first - order formula and an automatic structure @xmath17 are fixed , determining if a tuple @xmath18 from @xmath17 satisfies @xmath19 can be done in linear time . most of the results about automatic structures , including the ones mentioned above , demonstrate that in various concrete senses automatic structures are not complex from a logical point of view . however , this intuition can be misleading . for example , in @xcite it is shown that the isomorphism problem for automatic structures is @xmath20-complete . this informally tells us that there is no hope for a description ( in a natural logical language ) of the isomorphism types of automatic structures . also , khoussainov and minnes @xcite provide examples of automatic structures whose scott ranks can be as high as possible , fully covering the interval @xmath21 $ ] of ordinals ( where @xmath22 is the first non - computable ordinal ) . they also show that the ordinal heights of well - founded automatic relations can be arbitrarily large ordinals below @xmath22 . in this paper , we study the class of unary automatic graphs of finite degree . since these graphs are described by the unfolding operation ( definition [ dfn : unfolding ] ) on the pair of finite graphs @xmath23 , we use this pair to represent the graph . the size of this pair is the sum of the sizes of the automata that represent these graphs . in the study of algorithmic properties of these graphs one directly deals with the pair @xmath23 . we are interested in the following natural decision problems : * * connectivity problem*. given an automatic graph @xmath24 , decide if @xmath24 is connected . * * reachability problem*. given an automatic graph @xmath24 and two vertices @xmath14 and @xmath16 of the graph , decide if there is a path from @xmath14 to @xmath16 . if we restrict to the class of finite graphs , these two problems are decidable and can be solved in linear time on the sizes of the graphs . however , we are interested in infinite graphs and therefore much more work is needed to investigate the problems above . in addition , we also pose the following two problems : * * infinity testing problem*. given an automatic graph @xmath24 and a vertex @xmath14 , decide if the component of @xmath24 containing @xmath14 is infinite . * * infinite component problem*. given an automatic graph @xmath24 decide if @xmath24 has an infinite component . unfortunately , for the class of automatic graphs all of the above problems are undecidable . in fact , one can provide exact bounds on this undecidability . the connectivity problem is @xmath25-complete ; the reachability problem is @xmath26-complete ; the infinite component problem is @xmath27-complete ; and the infinity testing problem is @xmath25-complete @xcite . since all unary automatic structures are first - order definable in @xmath28 ( the monadic second - order logic of the successor function ) , it is not hard to prove that all the problems above are decidable @xcite . direct constructions using this definability in @xmath28 yield algorithms with non - elementary time since one needs to transform @xmath28 formulas into automata @xcite . however , we provide polynomial - time algorithms for solving all the above problems for this class of graphs . we now outline the rest of this paper by explaining the main results . we comment that these polynomial - time algorithms are based on deterministic input automata . section 2 introduces the main definitions needed , including the concept of automatic structure . section 3 singles out unary automatic graphs and provides a characterization theorem ( theorem [ thm : characterization ] ) . section 4 introduces unary automatic graphs of finite degree . the main result is theorem [ thm : gsigma ] that explicitly provides an algorithm for building unary automatic graphs of finite degree . this theorem is used throughout the paper . section 5 is devoted to deciding the infinite component problem . the main result is the following : * theorem [ thm : infinite component ] * _ the infinite component problem for unary automatic graph of finite degree @xmath24 is solved in @xmath29 , where @xmath8 is the number of states of the deterministic finite automaton recognizing @xmath24 . _ in this section , we make use of the concept of oriented walk for finite directed graphs . the subsequent section is devoted to deciding the infinity testing problem . the main result is the following : * theorem [ thm : inftest ] * _ the infinity testing problem for unary automatic graph of finite degree @xmath24 is solved in @xmath29 , where @xmath8 is the number of states of the deterministic finite automaton @xmath17 recognizing @xmath24 . in particular , when @xmath17 is fixed , there is a constant time algorithm that decides the infinity testing problem on @xmath24 . _ the fact that there is a constant time algorithm when @xmath17 is fixed will be made clear in the proof . the value of the constant is polynomial in the number of states of @xmath17 . the reachability problem is addressed in section 7 . this problem has been studied in @xcite,@xcite , @xcite via the class of * pushdown graphs*. a pushdown graph is the configuration space of a pushdown automaton . unary automatic graphs are pushdown graphs @xcite . in @xcite it is proved that for a pushdown graph @xmath24 , given a node @xmath30 , there is an automaton that recognizes all nodes reachable from @xmath30 . the number of states in the automaton depends on the input node @xmath30 . this result implies that there is an algorithm that decides the reachability problem on unary automatic graphs of finite degree . however , there are several issues with this algorithm . the automata constructed by the algorithm are not uniform in @xmath30 in the sense that different automata are built for different input nodes @xmath30 . moreover , the automata are nondeterministic . hence , the size of the deterministic equivalent automata is exponential in the size of the representation of @xmath30 . section 7 provides an alternative algorithm to solve the reachability problem on unary automatic graphs of finite degree uniformly . this new algorithm constructs a deterministic automaton @xmath31 that accepts the set of pairs @xmath32 there is a path from @xmath33 to @xmath34 . the size of @xmath31 only depends on the number of states of the automaton @xmath8 , and constructing the automaton requires polynomial - time in @xmath8 . the practical advantage of such a uniform solution is that , when @xmath31 is built , deciding whether node @xmath30 is reachable from @xmath33 by a path takes only linear time ( details are in section 7 ) . the main result of this section is the following : * theorem [ thm : reachability ] * _ suppose @xmath24 is a unary automatic graph of finite degree represented by deterministic finite automaton @xmath17 of size @xmath8 . there exists a polynomial - time algorithm that solves the reachability problem on @xmath24 . for inputs @xmath35 , the running time of the algorithm is @xmath36 . _ finally , the last section solves the connectivity problem for @xmath24 . * theorem [ thm : connectivity ] * _ the connectivity problem for unary automatic graph of finite degree @xmath24 is solved in @xmath29 , where @xmath8 is the number of states of the deterministic finite automaton recognizing @xmath24 . _ the authors would like to thank referees for comments on improvement of this paper . a * finite automaton * @xmath17 over an alphabet @xmath37 is a tuple @xmath38 , where @xmath39 is a finite set of * states * , @xmath40 is the * initial state * , @xmath41 is the * transition table * and @xmath42 is the set of * final states*. a * computation * of @xmath17 on a word @xmath43 ( @xmath44 ) is a sequence of states , say @xmath45 , such that @xmath46 and @xmath47 for all @xmath48 . if @xmath49 , then the computation is * successful * and we say that automaton @xmath17 * accepts * the word . the * language * accepted by the automaton @xmath17 is the set of all words accepted by @xmath17 . in general , @xmath50 is * fa recognizable * , or * regular * , if @xmath51 is the language accepted by some finite automaton . in this paper we always assume the automata are deterministic . for two states @xmath52 , the * distance * from @xmath53 to @xmath54 is the minimum number of transitions required for @xmath17 to go from @xmath53 to @xmath54 . to formalize the notion of a relation being recognized by an automaton , we define synchronous @xmath8-tape automata . such an automaton can be thought of as a one - way turing machine with @xmath8 input tapes . each tape is semi - infinite having written on it a word in the alphabet @xmath37 followed by a succession of @xmath55 symbols . the automaton starts in the initial state , reads simultaneously the first symbol of each tape , changes state , reads simultaneously the second symbol of each tape , changes state , etc . , until it reads @xmath55 on each tape . the automaton then stops and accepts the @xmath8-tuple of words if and only if it is in a final state . more formally , we write @xmath56 for @xmath57 where @xmath55 is a symbol not in @xmath37 . the * convolution * of a tuple @xmath58 is the string @xmath59 of length @xmath60 symbol is @xmath61 where @xmath62 is the @xmath63 symbol of @xmath64 if @xmath65 is the relation @xmath66 formed as the set of convolutions of all the tuples in @xmath67 . an @xmath8-ary relation @xmath68 is * fa recognizable * , or * regular * , if its convolution @xmath69 is recognizable by a finite automaton . a * structure * @xmath70 consists of a countable set @xmath51 called the * domain * and some relations and operations on @xmath51 . we may assume that @xmath70 only contains relational predicates since operations can be replaced with their graphs . we write @xmath71 where @xmath72 is an @xmath73-ary relation on @xmath51 . the relation @xmath74 are sometimes called basic or atomic relations . we assume that the function @xmath75 is always a computable one . a structure @xmath70 is * automatic over alphabet @xmath37 * if its domain @xmath50 is finite automaton recognizable , and there is an algorithm that for each @xmath76 produces an @xmath77-tape automaton recognizing the relation @xmath78 . a structure is called * automatic * if it is automatic over some alphabet . if @xmath79 is isomorphic to an automatic structure @xmath70 , then we call @xmath70 an * automatic presentation * of @xmath79 and say that @xmath79 is _ * automatically presentable*_. an example of an automatic structure is the word structure @xmath80 , where for all @xmath81 , @xmath82 , @xmath83 , @xmath84 if and only if @xmath85 , and @xmath86 is the lexicographical order . the configuration graph of any turing machine is another example of an automatic structure . examples of automatically presentable structures are @xmath87 , @xmath88 , @xmath89 , the group @xmath90 , the order on the rational @xmath91 , and the boolean algebra of finite and co - finite subsets of @xmath92 . consider the first - order logic extended by @xmath93 ( there exist infinitely many ) and @xmath6 ( there exist @xmath8 many mod @xmath7 , where @xmath8 and @xmath7 are natural numbers ) quantifiers . we denote this logic by @xmath94 . we will use the following theorem without explicit reference to it . @xcite let @xmath17 be an automatic structure . there exists an algorithm that , given a formula @xmath19 in @xmath95 , produces an automaton that recognizes exactly those tuples @xmath18 from the structure that make @xmath96 true . in particular , the set of all sentences of @xmath95 which are true in @xmath17 is decidable . we now turn our attention to the subclass of the automatic structures which is the focus of the paper . a structure @xmath17 is * unary automatic * if it has an automatic presentation whose domain is @xmath97 and whose relations are automatic . examples of unary automatic structures are @xmath98 and @xmath99 . some recent work on unary automatic structures includes a characterization of unary automatic linearly ordered sets , permutation structures , graphs , and equivalence structures @xcite . for example , unary automatic linearly ordered sets are exactly those that are isomorphic to a finite sum of orders of type @xmath100 , @xmath101 ( the order of negative integers ) , and finite @xmath8 . a * unary automatic graph * is a graph @xmath102 whose domain is @xmath97 , and whose edge relation @xmath103 is regular . we use the following example to illustrate that this class of graphs are the best possible . consider the class of graphs with all vertices being of the form @xmath104 for some alphabet @xmath105 . at first sight , graphs of this form may have an intermediate position between unary and general automatic graphs . however , the infinite grid @xmath106 can be coded automatically over @xmath104 by @xmath107 , and @xmath108 is not decidable @xcite . in particular , counter machines can be coded into the grid , so the reachability problem is not decidable . to eliminate bulky exposition , we make the following assumptions in the rest of the paper . * the automata under consideration are viewed as deterministic . hence , when we write automata , we mean deterministic finite automata . * all structures are infinite unless explicitly specified . * the graphs are undirected . the case of directed graphs can be treated in a similar manner . let @xmath109 be an automatic graph . let @xmath17 be an automaton recognizing @xmath103 . we establish some terminology for the automaton @xmath17 . the general shape of @xmath17 is given in figure [ fg : standard ] . all the states reachable from the initial state by reading inputs of type @xmath110 are called * @xmath110-states*. a * tail * in @xmath17 is a sequence of states linked by transitions without repetition . a * loop * is a sequence of states linked by transitions such that the last state coincides with the first one , and with no repetition in the middle . the set of @xmath110-states is a disjoint union of a tail and a loop . we call the tail the * @xmath110-tail * and the loop the * @xmath110-loop*. let @xmath111 be a @xmath110 state . all the states reachable from @xmath111 by reading inputs of type @xmath112 are called * @xmath112-states*. this collection of all @xmath112-states is also a disjoint union of a tail and a loop ( see the figure ) , called the * @xmath112-tail * and the * @xmath112-loop * , respectively . the * @xmath113-tails * and * @xmath113-loops * are defined in a similar matter . we say that an automaton is * standard * if the lengths of all its loops and tails equal some number @xmath114 , called the * loop constant*. if @xmath17 is a standard automaton recognizing a binary relation , it has exactly @xmath115 @xmath110-states . on each of these states , there is a @xmath116-tail and a @xmath117-tail of length exactly @xmath114 . at the end of each @xmath112-tail and @xmath113-tail there is a @xmath112-loop and @xmath118-loop , respectively , of size exactly @xmath114 . therefore if @xmath8 is the number of states in @xmath17 , then @xmath119 . [ lemma : standard ] let @xmath17 be an @xmath8 state automaton recognizing a binary relation @xmath103 on @xmath97 . there exists an equivalent standard automaton with at most @xmath120 states . let @xmath114 be the least common multiple of the lengths of all loops and tails of @xmath17 . an easy estimate shows that @xmath114 is no more than @xmath121 . one can transform @xmath17 into an equivalent standard automaton whose loop constant is @xmath114 . hence , there is a standard automaton equivalent to @xmath17 whose size is bounded above by @xmath122 . we can simplify the general shape of the automaton using the fact that we consider undirected graphs . indeed , we need only consider transitions labelled by @xmath123 . to see this , given an automaton with only @xmath118 transitions , to include all symmetric transitions , add a copy of each @xmath118 transition which is labelled with @xmath116 . we recall a characterization theorem of unary automatic graphs from @xcite . let @xmath124 and @xmath125 be finite graphs . let @xmath126 be subsets of @xmath127 , and @xmath128 be subsets of @xmath129 . consider the graph @xmath130 followed by @xmath100 many copies of @xmath79 , ordered as @xmath131 . formally , the vertex set of @xmath132 is @xmath133 and we write @xmath134 for @xmath135 and @xmath136 . the edge set @xmath137 of @xmath132 consists of all pairs @xmath138 such that @xmath139 . we define the infinite graph , * @xmath140 * , as follows : @xmath141 the vertex set is @xmath142 ; @xmath143 the edge set contains @xmath144 as well as the following edges , for all @xmath145 , @xmath146 , and @xmath147 : * @xmath148 when @xmath149 , and @xmath150 when @xmath151 , * @xmath152 when @xmath153 , and @xmath154 when @xmath155 . [ thm : characterization ] @xcite a graph is unary automatic if and only if it is isomorphic to @xmath140 for some parameters @xmath79 , @xmath130 , and @xmath156 . moreover , if @xmath17 is a standard automaton representing @xmath24 then the parameters @xmath157 can be extracted in @xmath158 ; otherwise , the parameters can be extracted in @xmath159 , where @xmath8 is the number of states in @xmath17 . a graph is of * finite degree * if there are at most finitely many edges from each vertex @xmath30 . we call an automaton @xmath17 recognizing a binary relation over @xmath160 a * one - loop automaton * if its transition diagram contains exactly one loop , the @xmath110-loop . the general structure of one - loop automata is given in figure [ fg : type1auto ] . we will always assume that the lengths of all the tails of the one - loop automata are not bigger than the size of the @xmath110-loop . the following is an easy proposition and we omit its proof . let @xmath109 be a unary automatic graph , then @xmath24 is of finite degree if and only if there is a one - loop automaton @xmath17 recognizing @xmath103 . by lemma [ lemma : standard ] , transforming a given automaton to an equivalent standard automaton may blow up the number of states exponentially . however , there is only polynomial blow up if @xmath17 is a one - loop automaton . if @xmath17 is a one - loop automaton with @xmath8 states , there exists an equivalent standard one - loop automaton with loop constant @xmath161 . let @xmath162 be the length of the loop in @xmath17 and @xmath163 be the length of the longest tail in @xmath17 . let @xmath114 be the least multiple of @xmath162 such that @xmath164 . it is easy to see that @xmath165 . one can transform @xmath17 into an equivalent standard one - loop automaton whose loop constant is @xmath114 . note that the equivalent standard automaton has @xmath115 ( 1,1)-states . from each of them there is a @xmath112-tail of length @xmath114 and a @xmath118-tail of length @xmath114 . hence the automaton has @xmath166 states . by the above lemma , we always assume the input automaton @xmath17 is standard . in the rest of the paper , we will state all results in terms of the loop constant @xmath114 instead of @xmath8 , the number of states of the input automaton . since @xmath161 , for any constant @xmath167 , an @xmath168 algorithm can also be viewed as an @xmath169 algorithm . given two unary automatic graphs of finite degree @xmath170 and @xmath171 ( where we recall the convention that the domain of each graph is @xmath97 ) , we can form the * union graph * @xmath172 and the * intersection graph * @xmath173 . automatic graphs of finite degree are closed under these operations . indeed , let @xmath174 and @xmath175 be one - loop automata recognizing @xmath176 and @xmath177 with loop constants @xmath178 and @xmath179 , respectively . the standard construction that builds automata for the union and intersection operations produces a one - loop automaton whose loop constant is @xmath180 . we introduce another operation : consider the new graph @xmath181 , where the set @xmath182 of edges is defined as follows ; a pair @xmath183 is in @xmath184 if and only if @xmath185 and @xmath186 . the relation @xmath182 is recognized by the same automaton as @xmath176 , modified so that all @xmath113-states that are final declared non - final , and all the @xmath113-states that are non - final declared final . thus , we have the following proposition : if @xmath187 and @xmath188 are automatic graphs of finite degree then so are @xmath189 , @xmath190 , and @xmath191 . now our goal is to recast theorem [ thm : characterization ] for graphs of finite degree . our analysis will show that , in contrast to the general case for automatic graphs , the parameters @xmath79 , @xmath130 , and @xmath156 for graphs of finite degree can be extracted in linear time . [ dfn : unfolding ] let @xmath192 and @xmath193 be finite graphs . consider the finite sets @xmath194 consisting of all mappings @xmath195 , and @xmath196 consisting of all mappings @xmath197 . any infinite sequence @xmath198 where @xmath199 and @xmath200 for each @xmath76 , defines the infinite graph @xmath201 as follows : * @xmath202 . * @xmath203 . thus @xmath204 is obtained by taking @xmath130 together with an infinite disjoint union of @xmath205 such that edges between @xmath130 and the first copy of @xmath205 are put according to the mapping @xmath206 , and edges between successive copies of @xmath205 are put according to @xmath62 . figure [ fg : gsigma ] illustrates the general shape of a unary automatic graph of finite degree that is build from @xmath130 , @xmath205 , @xmath206 , and @xmath207 , where @xmath208 is the infinite word @xmath209 . [ thm : gsigma ] a graph of finite degree @xmath109 possesses a unary automatic presentation if and only if there exist finite graphs @xmath210 and mappings @xmath211 and @xmath212 such that @xmath24 is isomorphic to @xmath213 . let @xmath109 be a unary automatic graph of finite degree . let @xmath17 be an automaton recognizing @xmath103 . in linear time on the number of states of @xmath17 we can easily transform @xmath17 into a one - loop automaton . so , we assume that @xmath17 is a one - loop automaton with loop constant @xmath114 . we construct the finite graph @xmath130 by setting @xmath214 , where @xmath53 is the starting state , @xmath215 are all states on the @xmath110-tail such that @xmath216 is reached from @xmath217 by reading @xmath110 for @xmath218 ; and for @xmath219 , @xmath220 iff there is a final state @xmath221 on the @xmath113-tail out of @xmath216 , and the distance from @xmath216 to @xmath221 is @xmath222 . we construct the graph @xmath205 similarly by setting @xmath223 where @xmath224 are all states on the @xmath110-loop . the edge relation @xmath225 is defined in a similar way as @xmath226 . the mapping @xmath227 is defined for any @xmath228 by putting @xmath229 in @xmath230 if and only if there exists a final state @xmath231 on the @xmath118-tail out of @xmath232 , and the distance from @xmath232 to @xmath231 equals @xmath233 . the mapping @xmath234 is constructed in a similar manner by reading the @xmath118-tails out of the @xmath110-loop . it is clear from this construction that the graphs @xmath24 and @xmath235 are isomorphic . conversely , consider the graph @xmath213 for some @xmath236 and @xmath237 . assume that @xmath238 , @xmath239 . a one - loop automaton @xmath17 recognizing the edge relation of @xmath240 is constructed as follows . the @xmath110-tail of the automaton is formed by @xmath241 and the @xmath110-loop is formed by @xmath242 , both in natural order . the initial state is @xmath53 . if for some @xmath243 , @xmath244 , then put a final state @xmath221 on the @xmath118-tail starting from @xmath216 such that the distance from @xmath216 to @xmath221 is @xmath222 . if @xmath245 , then repeat the process but make the corresponding distance @xmath246 . the set of edges @xmath225 and mapping @xmath234 are treated in a similar manner by putting final states on the @xmath118-tails from the @xmath110-loop . again , we see that @xmath17 represents a unary automatic graph that is isomorphic to @xmath213 . the proof of the above theorem also gives us the following corollary . if @xmath24 is a unary automatic graph of finite degree , the parameters @xmath130 , @xmath205 , @xmath234 and @xmath206 can be extracted in @xmath247 time , where @xmath114 is the loop constant of the one - loop automaton representing the graph . furthermore , @xmath248 . recall the graphs are undirected . a * component * of @xmath24 is the transitive closure of a vertex under the edge relation . the * infinite component problem * asks whether a given graph @xmath24 has an infinite component . [ thm : infinite component ] the infinite component problem for unary automatic graph of finite degree @xmath24 is solved in @xmath249 , where @xmath114 is the loop constant of the automaton recognizing @xmath24 . by theorem [ thm : gsigma ] , let @xmath250 . we observe that it is sufficient to consider the case in which @xmath251 ( hence @xmath252 ) since @xmath213 has an infinite component if and only if @xmath253 has one . let @xmath254 be the @xmath255 copy of @xmath205 in @xmath24 . let @xmath256 be the copy of vertex @xmath14 in @xmath257 . we construct a finite directed graph @xmath258 as follows . each node in @xmath259 represents a distinct connected component in @xmath205 . for simplicity , we assume that @xmath260 and hence use @xmath14 to denote its own component in @xmath205 . the case in which @xmath261 can be treated in a similar way . for @xmath262 , put @xmath263 if and only if @xmath264 for some @xmath265 and @xmath266 that are in the same component as @xmath14 and @xmath16 , respectively . constructing @xmath267 requires finding connected components of @xmath205 hence takes time @xmath247 . to prove the above theorem , we make essential use of the following definition . see also @xcite . [ dfn : edge - path ] an * _ oriented walk _ * in a directed graph @xmath268 is a subgraph @xmath269 of @xmath268 that consists of a sequence of nodes @xmath270 such that for @xmath271 , either @xmath272 or @xmath273 is an arc in @xmath268 , and for each @xmath274 , exactly one of @xmath275 and @xmath276 belongs to @xmath269 . an oriented walk is an * _ oriented cycle _ * if @xmath277 and there are no repeated nodes in @xmath278 . in an oriented walk @xmath269 , an arc @xmath279 is called a * forward arc * and @xmath280 is called a * backward arc*. the * net length * of @xmath269 , denoted @xmath281 , is the difference between the number of forward arcs and backward arcs . note the net length can be negative . the next lemma establishes a connection between oriented cycles in @xmath267 and infinite components in @xmath24 . [ lm : cycle ] there is an infinite component in @xmath24 if and only if there is an oriented cycle in @xmath282 such that the net length of the cycle is positive . suppose there is an oriented cycle @xmath269 from @xmath14 to @xmath14 in @xmath282 of net length @xmath283 . for all @xmath284 , @xmath269 defines the path @xmath285 in @xmath24 from @xmath286 to @xmath287 where @xmath285 lies in @xmath288 . therefore , for a fixed @xmath284 , all vertices in the set @xmath289 belong to the same component of @xmath24 . in particular , this implies that @xmath24 contains an infinite component . conversely , suppose there is an infinite component @xmath51 in @xmath24 . since @xmath205 is finite , there must be some @xmath14 in @xmath290 such that there are infinitely many copies of @xmath14 in @xmath51 . let @xmath286 and @xmath291 be two copies of @xmath14 in @xmath51 such that @xmath292 . consider a path between @xmath286 and @xmath291 . we can assume that on this path there is at most one copy of any vertex @xmath293 apart from @xmath14 ( otherwise , choose @xmath291 to be the copy of @xmath14 in the path that has this property ) . by definition of @xmath253 and @xmath267 , the node @xmath14 must be on an oriented cycle of @xmath267 with net length @xmath222 . by the equivalence in lemma [ lm : cycle ] , it suffices to provide an algorithm that decides if @xmath282 contains an oriented cycle with positive net length . notice that the existence of an oriented cycle with positive net length is equivalent to the existence of an oriented cycle with negative net length . therefore , we give an algorithm which finds oriented cycles with non - zero net length . for each node @xmath14 in @xmath267 , we search for an oriented cycle of positive net length from @xmath14 by creating a labeled queue of nodes @xmath294 which are connected to @xmath14 . ` alg : oriented - cycle ` 1 . pick node @xmath295 for which a queue has not been built yet . initially the queue @xmath294 is empty . let @xmath296 , and put @xmath14 into the queue . mark @xmath14 as _ unprocessed_. if queues have been built for each @xmath295 , stop the process and return _ let @xmath16 be the first _ unprocessed _ node in @xmath294 . if there are no _ unprocessed _ nodes in @xmath294 , return to ( 1 ) . 3 . for each of the nodes @xmath297 in the set @xmath298 , do the following . 1 . if @xmath299 , set @xmath300 ; if @xmath301 , set @xmath302 . ( if both hold , do steps ( a ) , ( b ) , ( c ) first for @xmath303 and then for @xmath304 . ) 2 . if @xmath305 , then set @xmath306 , put @xmath297 into @xmath294 , and mark @xmath297 as _ unprocessed_. 3 . if @xmath307 then 1 . if @xmath306 , move to next @xmath297 , 2 . if @xmath308 , stop the process and return _ yes_. 4 . mark @xmath16 as _ processed _ and go back to ( 2 ) . an important property of this algorithm is that when we are building a queue for node @xmath14 and are processing @xmath297 , both @xmath309 and @xmath310 represent net lengths of paths from @xmath14 to @xmath297 . we claim that the algorithm returns _ yes _ if and only if there is an oriented cycle in @xmath267 with non - zero net length . suppose the algorithm returns _ yes_. then , there is a base node @xmath14 and a node @xmath297 such that @xmath308 . this means that there is an oriented walk @xmath269 from @xmath14 to @xmath297 with net length @xmath309 and there is an oriented walk @xmath311 from @xmath14 to @xmath297 with net length @xmath310 . consider the oriented walk @xmath312 , where @xmath313 is the oriented walk @xmath311 in reverse direction . clearly this is an oriented walk from @xmath14 to @xmath14 with net length @xmath314 . if there are no repeated nodes in @xmath312 , then it is the required oriented cycle . otherwise , let @xmath16 be a repeated node in @xmath312 such that no nodes between the two occurrences of @xmath16 are repeated . consider the oriented walk between these two occurrences of @xmath16 , if it has a non - zero net length , then it is our required oriented cycle ; otherwise , we disregard the part between the two occurrences of @xmath297 and make the oriented walk shorter without altering its net length . conversely , suppose there is an oriented cycle @xmath315 of non - zero net length where @xmath316 . however , we assume for a contradiction that the algorithm returns _ no_. consider how the algorithm acts when we pick @xmath317 at step ( 1 ) . for each @xmath318 , one can prove the following statements by induction on @xmath76 . 1 . @xmath319 always gets a label @xmath320 2 . @xmath320 equals the net length of the oriented walk from @xmath317 to @xmath319 in @xmath269 . by the description of the algorithm , @xmath317 gets the label @xmath321 . suppose the statements holds for @xmath319 , @xmath322 , then at the next stage , the algorithm labels all nodes in @xmath323 . in particular , it calculates @xmath324 . by the inductive hypothesis , @xmath324 is the net length of the oriented walk from @xmath317 to @xmath325 in @xmath269 . if @xmath325 has already had a label @xmath326 and @xmath327 , then the algorithm would return _ yes_. therefore @xmath328 . by assumption on @xmath269 , @xmath329 . however , since @xmath330 , the induction gives that @xmath331 . this is a contradiction , and thus the above algorithm is correct . in summary , the following algorithm solves the infinite component problem . suppose we are given an automaton ( with loop constant @xmath114 ) which recognizes the unary automatic graph of finite degree @xmath24 . recall that @xmath114 is also the cardinality of @xmath290 . we first compute @xmath267 , in time @xmath247 . then we run ` oriented - cycle ` to decide whether @xmath267 contains an oriented cycle with positive net length . for each node @xmath14 in @xmath267 , the process runs in time @xmath247 . since @xmath267 contains @xmath114 number of nodes , this takes time @xmath332 . note that lemma [ lm : cycle ] holds for the case when @xmath333 deciding the infinity testing problem ------------------------------------- we next turn our attention to the * infinity testing problem * for unary automatic graphs of finite degree . recall that this problem asks for an algorithm that , given a vertex @xmath30 and a graph @xmath24 , decides if @xmath30 belongs to an infinite component . we prove the following theorem . [ thm : inftest ] the infinity testing problem for unary automatic graph of finite degree @xmath24 is solved in @xmath332 , where @xmath114 is the loop constant of the automaton @xmath17 recognizing @xmath24 . in particular , when @xmath17 is fixed , there is a constant time algorithm that decides the infinity testing problem on @xmath24 . for a fixed input @xmath286 , we have the following lemma . [ lm : infinite ] if @xmath286 is connected to some @xmath334 such that @xmath335 , then @xmath286 is in an infinite component . suppose such a @xmath334 exists . take a path @xmath336 in @xmath24 from @xmath286 to @xmath334 . since @xmath114 is the cardinality of @xmath290 , there is @xmath337 such that @xmath338 and @xmath339 appear in @xmath336 with @xmath340 . therefore all nodes in the set @xmath341 are in the same component as @xmath286 . let @xmath342 . to decide if @xmath286 and @xmath334 are in the same component , we run a breadth first search in @xmath24 starting from @xmath286 and going through all vertices in @xmath343 . the algorithm is as follows : ` alg : finitereach ` 1 . let @xmath344 . 2 . initialize the queue @xmath345 to be empty . put the pair @xmath346 into @xmath345 and mark it as _ unprocessed_. 3 . if there are no _ unprocessed _ pairs in @xmath345 , stop the process . otherwise , let @xmath347 be the first _ unprocessed _ pair . for arcs @xmath348 of the form @xmath304 or @xmath303 in @xmath349 , do the following . 1 . if @xmath348 is of the form @xmath304 , let @xmath350 ; if @xmath348 is of the form @xmath303 , let @xmath351 . 2 . if @xmath352 and @xmath353 is not in @xmath345 , then put @xmath353 into @xmath345 and mark @xmath353 as _ unprocessed_. 4 . mark @xmath347 as _ processed _ , and go to ( 2 ) . note that any @xmath334 is reachable from @xmath286 on the graph @xmath24 restricted on @xmath354 if and only if after running ` finitereach ` on the input @xmath286 , the pair @xmath355 is in @xmath345 . when running the algorithm we only use the exact value of the input @xmath76 when @xmath356 ( we set @xmath357 whenever @xmath284 ) , so the running time of ` finitereach ` is bounded by the number of edges in @xmath24 restricted to @xmath358 . therefore the running time is @xmath332 . let @xmath359 . [ lm : inftest - b ] let @xmath360 . @xmath286 is in an infinite component if and only if @xmath361 . suppose a vertex @xmath362 , then there is a path from @xmath286 to @xmath363 . by lemma [ lm : infinite ] , @xmath286 is in an infinite component . conversely , if @xmath286 is in an infinite component , then there must be some vertices in @xmath364 reachable from @xmath286 . take a path from @xmath286 to a vertex @xmath363 such that @xmath363 is the first vertex in @xmath364 appearing on this path . then @xmath362 . we assume the input vertex @xmath286 is given by tuple @xmath365 . the above lemma suggests a simple algorithm to check if @xmath286 is in an infinite component . ` alg : infinitetest ` 1 . run ` finitereach ` on vertex @xmath256 , computing the set @xmath366 while building the queue @xmath345 . 2 . for every @xmath362 , check if there is edge @xmath299 . return @xmath367 if one such edge is found ; otherwise , return @xmath368 . running ` finitereach ` takes @xmath332 and checking for edge @xmath304 takes @xmath247 . the running time is therefore @xmath332 . since @xmath14 is bounded by @xmath114 , if @xmath17 is fixed , checking whether @xmath286 belongs to an infinite component takes constant time . suppose @xmath24 is a unary automatic graph of finite degree represented by an automaton with loop constant @xmath114 . the * reachability problem * on @xmath24 is formulated as : given two vertices @xmath369 in @xmath24 , decide if @xmath286 and @xmath334 are in the same component . we prove the following theorem . [ thm : reachability ] suppose @xmath24 is a unary automatic graph of finite degree represented by an automaton @xmath17 of loop constant @xmath114 . there exists a polynomial - time algorithm that solves the reachability problem on @xmath24 . for inputs @xmath35 , the running time of the algorithm is @xmath370 . we restrict to the case when @xmath371 . the proof can be modified slightly to work in the more general case , @xmath372 . since , by theorem [ thm : inftest ] , there is an @xmath332-time algorithm to check if @xmath256 is in a finite component , we can work on the two possible cases separately . we first deal with the case when the input @xmath286 is in a finite component . by lemma [ lm : infinite ] , @xmath286 and @xmath334 are in the same ( finite ) component if and only if after running ` finitereach ` on the input @xmath286 , the pair @xmath355 is in the queue @xmath345 . [ cr : finite reach ] if all components of @xmath24 are finite and we represent @xmath373 as @xmath374 , then there is an @xmath332-algorithm deciding if @xmath286 and @xmath334 are in the same component . now , suppose that @xmath286 is in an infinite component . we start with the following question : given @xmath375 , are @xmath286 and @xmath376 in the same component in @xmath24 ? to answer this , we present an algorithm that computes all vertices @xmath375 whose @xmath255 copy lies in the same @xmath24-component as @xmath286 . the algorithm is similar to ` finitereach ` , except that it does not depend on the input @xmath76 . line(3b ) in the algorithm is changed to the following : ( 3b ) if @xmath377 and @xmath353 is not in @xmath345 , then put @xmath353 into @xmath345 and mark @xmath353 as _ unprocessed_. we use this modified algorithm to define the set @xmath378 . intuitively , we can think of the algorithm as a breadth first search through @xmath379 which originates at @xmath380 . therefore , @xmath381 if and only if there exists a path from @xmath380 to @xmath382 in @xmath24 restricted to @xmath379 . [ lm : reach_reach ] suppose @xmath286 is in an infinite component . the vertex @xmath376 is in the same component as @xmath286 if and only if @xmath376 is also in an infinite component and @xmath383 . suppose @xmath376 is in an infinite component and @xmath383 . if @xmath284 , then the observation above implies that there is a path from @xmath286 to @xmath376 in @xmath384 . so , it remains to prove that @xmath286 and @xmath376 are in the same component even if @xmath385 . since @xmath381 , there is a path @xmath336 in @xmath24 from @xmath380 to @xmath382 . let @xmath386 be the least number such that @xmath387 . if @xmath388 , then it is clear that @xmath286 and @xmath376 are in the same component . thus , suppose that @xmath389 . let @xmath297 be such that @xmath390 . then @xmath336 is @xmath391 where @xmath392 is a path from @xmath380 to @xmath393 and @xmath394 is a path from @xmath393 to @xmath382 . since @xmath286 is in an infinite component , it is easy to see that @xmath380 is also in an infinite component . there exists an @xmath395 such that all vertices in the set @xmath396 are in the same component . likewise , there is an @xmath397 such that all vertices in @xmath398 are in the same component . consider @xmath399 and @xmath400 . analogous to the path @xmath392 , there is a path @xmath401 from @xmath399 to @xmath402 . similarly , there is a path @xmath403 from @xmath402 to @xmath400 . we describe another path @xmath404 from @xmath380 to @xmath382 as follows . @xmath404 first goes from @xmath380 to @xmath399 , then goes along @xmath405 from @xmath399 to @xmath400 and finally goes to @xmath382 . notice that the least @xmath406 such that @xmath407 must be larger than @xmath386 . we can iterate this procedure of lengthening the path between @xmath380 and @xmath382 until @xmath408 , as is required to reduce to the previous case . to prove the implication in the other direction , we assume that @xmath286 and @xmath376 are in the same infinite component . then @xmath376 is , of course , in an infinite component . we want to prove that @xmath409 . let @xmath410 . suppose there exists a path @xmath336 in @xmath24 from @xmath286 to @xmath376 which stays in @xmath411 . then , indeed , @xmath381 . on the other hand , suppose no such path exists . since @xmath286 and @xmath376 are in the same component , there is some path @xmath336 from @xmath286 to @xmath376 . let @xmath412 be the largest number such that @xmath413 . let @xmath414 be the least number such that @xmath415 . we are in one of two cases : @xmath416 or @xmath417 . we will prove that if @xmath416 then there is a path @xmath404 from @xmath286 to @xmath376 such that @xmath418 and @xmath419 . the case in which @xmath417 can be handled in a similar manner . without loss of generality , we assume @xmath420 since otherwise we can change the input @xmath14 and make @xmath421 . let @xmath297 be a vertex in @xmath205 such that @xmath422 . then @xmath336 is @xmath391 where @xmath392 is a path from @xmath286 to @xmath423 and @xmath394 is a path from @xmath423 to @xmath376 . since @xmath424 , there must be some @xmath425 and @xmath426 in @xmath392 such that @xmath427 . for the same reason , there must be some @xmath428 and @xmath429 in @xmath394 such that @xmath430 . therefore , @xmath336 contains paths between any consecutive pair of vertices in the sequence @xmath431 . consider the following sequence of vertices : @xmath432 it is easy to check that there exists a path between each pair of consecutive vertices in the sequence . therefore the above sequence describes a path @xmath404 from @xmath286 to @xmath376 . it is easy to see that @xmath433 . also since @xmath421 , @xmath434 . therefore @xmath404 is our desired path . in the following , we abuse notation by using @xmath435 and @xmath234 on subsets of @xmath436 . we inductively define a sequence @xmath437 such that each @xmath438 is a subset of @xmath436 . let @xmath439 and for @xmath427 , we define @xmath440 . the following lemma is immediate from this definition . [ lm : reach_closure ] suppose @xmath286 is in an infinite component , then @xmath286 and @xmath334 are in the same component if and only if @xmath334 is also in an infinite component and @xmath441 . we can use the above lemma to construct a simple - minded algorithm that solves the reachability problem on inputs @xmath442 . ` alg : navereach ` 1 . check if each of @xmath286 , @xmath334 are in an infinite component of @xmath24 ( using the algorithm of theorem [ thm : inftest ] ) . 2 . if exactly one of @xmath286 and @xmath334 is in a finite component , then return _ if both @xmath286 and @xmath334 are in finite components , then run ` finitereach ` on input @xmath286 and check if @xmath355 is in @xmath345 . if both @xmath286 and @xmath334 are in infinite components , then compute @xmath443 . if @xmath441 , return _ yes _ ; otherwise , return _ no_. we now consider the complexity of this algorithm . the set @xmath444 can be computed in time @xmath332 . given @xmath445 , we can compute @xmath438 in time @xmath332 by computing @xmath446 for any @xmath447 . therefore , the total running time of ` navereach ` on input @xmath286 , @xmath334 is @xmath448 . we want to replace the multiplication with addition and hence tweak the algorithm . from lemma [ lm : inftest - b ] , @xmath286 is in an infinite component in @xmath24 if and only if ` finitereach ` finds a vertex @xmath363 connecting to @xmath286 . now , suppose that @xmath286 is in an infinite component . we can use ` finitereach ` to find such a @xmath16 , and a path from @xmath286 to @xmath363 . on this path , there must be two vertices @xmath449 with @xmath450 . let @xmath451 . note that @xmath452 can be computed from the algorithm . it is easy to see that all vertices in the set @xmath453 belong to the same component . [ lm : reach_repeat ] @xmath454 . by definition , @xmath455 if and only if @xmath380 and @xmath382 are in the same component of @xmath24 . suppose that there exists a path in @xmath24 from @xmath380 to @xmath382 . then there is a path from @xmath456 to @xmath457 . since @xmath380 and @xmath456 are in the same component of @xmath24 , @xmath380 and @xmath457 are in the same component . hence @xmath458 . for the reverse inclusion , suppose @xmath459 . then there exists a path from @xmath380 to @xmath457 . therefore , @xmath456 and @xmath457 are in the same component . since @xmath460 , @xmath380 and @xmath382 are in the same component . using the above lemma , we define a new algorithm ` reach ` on inputs @xmath286 , @xmath334 by replacing line ( 4 ) in ` navereach ` with \(4 ) if @xmath286 and @xmath334 belong to infinite components , then compute @xmath461 . if @xmath462 for @xmath463 such that @xmath464 , return _ yes _ ; otherwise , return _ no_. say input vertices are given as @xmath286 and @xmath334 . by lemma [ lm : reach_closure ] and lemma [ lm : reach_repeat ] , the algorithm ` reach ` returns _ yes _ if and only if @xmath286 and @xmath334 are in the same component . since @xmath460 , calculating @xmath461 requires time @xmath465 . therefore the running time of ` reach ` on input @xmath286 , @xmath334 is @xmath466 . notice that , in fact , the algorithm produces a number @xmath467 such that in order to check if @xmath286 , @xmath334 ( @xmath468 ) are in the same component , we need to test if @xmath469 and if @xmath470 . therefore if @xmath24 is fixed and we compute @xmath471 for all @xmath14 beforehand , then deciding whether two vertices @xmath33 , @xmath30 belong to the same component takes linear time . the above proof can also be used to build an automaton that decides reachability uniformly : [ cr : reach_aut ] given a unary automatic graph of finite degree @xmath24 represented by an automaton with loop constant @xmath114 , there is a deterministic automaton with at most @xmath472 states that solves the reachability problem on @xmath24 . the time required to construct this automaton is @xmath473 . for all @xmath474 , @xmath475 , let string @xmath476 represent vertex @xmath286 in @xmath24 . suppose @xmath477 , we construct an automaton @xmath31 that accepts @xmath478 if and only if @xmath286 and @xmath334 are in the same component in @xmath24 . 1 . @xmath31 has a @xmath110-tail of length @xmath479 . let the states on the tail be @xmath480 , where @xmath53 is the initial state . these states represent vertices in @xmath481 . 2 . from @xmath482 , there is a @xmath110-loop of length @xmath114 . we call the states on the loop @xmath483 . these states represent vertices in @xmath484 . 3 . for @xmath485 , there is a @xmath118-tail from @xmath486 of length @xmath487 . we denote the states on this tail by @xmath488 . these states represent vertices in @xmath489 . 4 . for @xmath490 , if @xmath286 is in an infinite component , then there is a @xmath118-loop of length @xmath491 from @xmath492 . the states on this loop are called @xmath493 . these states represent vertices in @xmath494 . 5 . for @xmath495 , if @xmath380 is in a finite component , then there is a @xmath118-tail from @xmath496 of length @xmath479 . these states are denoted @xmath497 and represent vertices in @xmath498 . if @xmath380 is in an infinite component , from @xmath496 , there is a @xmath118-loop of length @xmath491 . we write these states as @xmath499 . the final ( accepting ) states of @xmath31 are defined as follows : 1 . states @xmath500 are final . 2 . for @xmath385 , if @xmath286 is in a finite component , run the algorithm ` finitereach ` on input @xmath286 and declare state @xmath501 final if @xmath502 . 3 . for @xmath385 , if @xmath286 is in an infinite component , compute @xmath503 . 1 . make state @xmath501 final if @xmath504 is in an infinite component and @xmath505 . 2 . make state @xmath506 final if @xmath505 4 . if @xmath380 is in a finite component , run the algorithm ` finitereach ` on input @xmath380 and make state @xmath507 final if @xmath502 . if @xmath380 is in an infinite component , compute @xmath503 . declare state @xmath508 final if @xmath505 . one can show that @xmath31 is the desired automaton . to compute the complexity of building @xmath31 , we summarize the computation involved . 1 . for all @xmath286 in @xmath509 , decide whether @xmath286 is in a finite component . this takes time @xmath473 by theorem [ thm : inftest ] . 2 . for all @xmath286 in @xmath509 such that @xmath286 is in a finite component , run ` finitereach ` on input @xmath286 . this takes time @xmath473 by corollary [ cr : finite reach ] . 3 . for all @xmath360 such that @xmath380 is in an infinite component , compute the sets @xmath510 @xmath511 . this requires time @xmath473 by theorem [ thm : reachability ] . therefore the running time required to construct @xmath31 is @xmath473 . finally , we present a solution to the * connectivity problem * on unary automatic graphs of finite degree . recall a graph is * connected * if there is a path between any pair of vertices . the construction of @xmath31 from the last section suggests an immediate solution to the connectivity problem . ` alg : naveconnect ` 1 . construct the automaton @xmath31 . 2 . check if all states in @xmath31 are final states . if it is the case , return @xmath367 ; otherwise , return @xmath512 . the above algorithm takes time @xmath473 . note that @xmath31 provides a uniform solution to the reachability problem on @xmath24 . given the `` regularity '' of the class of infinite graphs we are studying , it is reasonable to believe there is a more intuitive algorithm that solves the connectivity problem . it turns out that this is the case . [ thm : connectivity ] the connectivity problem for unary automatic graph of finite degree @xmath24 is solved in @xmath332 , where @xmath114 is the loop constant of the automaton recognizing @xmath24 . observe that if @xmath24 does not contain an infinite component , then @xmath24 is not connected . therefore we suppose @xmath24 contains an infinite component @xmath513 . [ lm : infinite - connect ] for all @xmath514 , there is a vertex in @xmath257 belonging to @xmath513 . since @xmath513 is infinite , there is a vertex @xmath286 and @xmath515 such that all vertices in @xmath516 belong to @xmath513 and @xmath76 is the least such number . by minimality , @xmath517 . take a walk along the path from @xmath518 to @xmath256 . let @xmath519 be the first vertex in @xmath520 that appears on this path . it is easy to see that @xmath521 must also be in @xmath513 . therefore @xmath513 has a non - empty intersection with each copy of @xmath205 in @xmath24 . pick an arbitrary @xmath522 and run ` finitereach ` on @xmath523 to compute the queue @xmath345 . set @xmath524 . suppose @xmath24 contains an infinite component , then @xmath24 is connected if and only if @xmath525 . suppose there is a vertex @xmath526 . then there is no path in @xmath24 between @xmath523 to @xmath521 . otherwise , we can shorten the path from @xmath523 to @xmath521 using an argument similar to the proof of lemma [ lm : reach_reach ] , and show the existence of a path between @xmath523 to @xmath521 in the subgraph restricted on @xmath527 . therefore @xmath24 is not connected . conversely , if @xmath528 , then every set of the form @xmath529 for @xmath530 equals @xmath290 . by lemma [ lm : infinite - connect ] , all vertices are in the same component . by the above lemma the following algorithm decides the connectivity problem on @xmath268 : ` alg : connectivity ` 1 . use the algorithm proposed by theorem [ thm : infinite component ] to decide if there is an infinite component in @xmath268 . if there is no infinite component , then stop and return @xmath512 . 2 . pick an arbitrary @xmath522 , run ` finitereach ` on @xmath523 to compute the queue @xmath345 . 3 . let @xmath531 . if @xmath532 , return @xmath367 ; otherwise , return @xmath512 . solving the infinite component problem takes @xmath332 by theorem [ thm : infinite component ] . running algorithm ` finitereach ` also takes @xmath332 . therefore ` connectivity ` takes @xmath332 . in this paper we addressed algorithmic problems for graphs of finite degree that have automata presentations over a unary alphabet . we provided polynomial - time algorithms that solve connectivity , reachability , infinity testing , and infinite component problems . in our future work we plan to improve these algorithms for other stronger classes of unary automatic graphs . we also point out that there are many other algorithmic problems for finite graphs that can be studied for the class of unary automatic graphs . these , for example , may concern finding spanning trees for automatic graphs , studying the isomorphism problems , and other related issues . j. r. bchi , _ on a decision method in restricted second - order arithmetic_. proc . international congress on logic , methodology and philosophy of science ( e. nagel , p. suppes , a. tarski , eds . ) , stanford university press , 1 - 11 , 1960 .

this paper studies infinite graphs produced from a natural unfolding operation applied to finite graphs . graphs produced via such operations are of finite degree and automatic over the unary alphabet ( that is , they can be described by finite automata over unary alphabet ) . we investigate algorithmic properties of such unfolded graphs given their finite presentations . in particular , we ask whether a given node belongs to an infinite component , whether two given nodes in the graph are reachable from one another , and whether the graph is connected . we give polynomial - time algorithms for each of these questions . for a fixed input graph , the algorithm for the first question is in constant time and the second question is decided using an automaton that recognizes reachability relation in a uniform way . hence , we improve on previous work , in which non - elementary or non - uniform algorithms were found .

the trihydrogen dication , @xmath1 , which consists of three protons and one electron , is among the simplest coulomb systems . its stability has been studied intensely in the sixties and early seventies . in a series of articles , conroy @xcite investigated the potential energy surfaces of the electronic ground state and the lowest excited states at linear and isosceles triangular configurations . he employed a variational approach in which the electronic trial wavefunction is expanded around the center of the nuclear charges . analyzing the contour plots conroy concluded that @xmath1 is not stable . schwartz and schaad @xcite , and somorjai and yue @xcite , who reported single - point calculations of the system @xmath2 at the supposed equilibrium equilateral triangular configuration of @xmath3 , did not address the stability problem . to assess conroy s results , berkowitz and stocker @xcite searched for this ion through charge stripping experiments on @xmath4 . they could not find evidence of stable @xmath1 . later , the issue was reconsidered also from the theoretical side , by shoucri and darling @xcite , who examined equilateral configurations with the variational linear combination of atomic orbitals ( lcao ) method , and by hernndes and carb @xcite , who studied two particular configurations with a more compact variational approach and obtained total energy values below those published before . no bound state has been determined in these calculations . johnson and poshusta @xcite reported another single - point calculation in the context of gaussian basis set optimization for some one - electron systems . about twenty years later ackermann _ et al . _ @xcite revisited the question about the existence of @xmath1 using the finite element method which provided much higher accuracy than previously achieved . the problem of the stability of @xmath1 was treated keeping the nuclear charge as a continuous parameter . critical values of the charges for the existence of stable or metastable equilateral triangular configurations were obtained as @xmath5 and @xmath6 , respectively . the authors excluded the possibility of stable @xmath1 in the electronic ground state . however , the explicit electronic energy data are reported only for one particular equilateral triangular configuration at the triangle size @xmath7 . in conclusion , accurate _ ab initio _ results on the basis of which the non - existence of @xmath1 can be demonstrated are scarce and not that convincing . this question is thus addressed once again in the present study . one of the motivations of our study is related to a fact that @xmath1 in equilateral triangular configuration may exist as metastable state in a magnetic field @xmath8 g @xcite . we study a coulomb system of one electron and three protons @xmath2 which form an equilateral triangle of size @xmath9 . the protons are assumed to be infinitely massive according to the born - oppenheimer approximation at zero order . the schrdinger equation for the system is written as @xmath10\psi({\mathbf r } ) = e\psi({\mathbf r } ) \ , \ ] ] where @xmath11 is the electron momentum , @xmath12 and @xmath13 are the distances from each proton to the electron and @xmath9 is the interproton distance , see figure [ trian ] . atomic units are used throughout ( @xmath14=@xmath15=@xmath16=1 ) , although energies are expressed in rydbergs ( ry ) . our goal is to study the stability of the molecular ion @xmath1 . if such an ion exists , it implies the existence of the ground state of the system @xmath2 . based on symmetry arguments it seems evident that the optimal geometry of @xmath2 in the case of existence of a bound state is the equilateral triangle . two methods are used to explore the system : ( i ) variational with physically relevant trial functions ( see e.g. @xcite ) which we will call _ specialized _ and ( ii ) _ standard _ variational based on using standard gaussian trial functions as implemented in _ ab initio _ quantum chemistry packages such as molpro @xcite . both methods lead to highly accurate quantitative results for total energy versus the size of the triangle . in the first variational approach , a trial function is taken in a form of linear superposition of six basis functions @xmath17 where @xmath18 are linear parameters . each function @xmath19 is chosen in such a way as to describe different physical characteristics of the system . in general , @xmath19 has the form of a symmetrized product of three coulomb orbitals @xmath20 let us give a brief description of each of them : @xmath21 : : : all @xmath22 s are chosen to be equal to @xmath23 , @xmath24 it is a heitler - london type function . this corresponds to _ coherent _ interaction between the electron and all protons . supposedly , it describes the system at small interproton distances and , probably , the equilibrium configuration . it might be verified a posteriori . @xmath25 : : : two @xmath22 s are equal to zero and the remaining one is set to be equal to @xmath26 , @xmath27 it is a hund - mulliken type function . this function possibly describes the system at large distances , where essentially the electron interacts with only one proton at a time thus realizing _ incoherent _ interaction . @xmath28 : : : one @xmath22 is equal to zero , two others are different from zero and equal to each other and to @xmath29 , @xmath30 it is assumed that this function describes the system @xmath31 plus proton when a triangle is of a sufficiently small size . in fact , it is the heitler - london function of @xmath31 symmetrized over protons . @xmath32 : : : one @xmath22 is equal to zero and two others are different from each other being equal to @xmath33 , respectively , @xmath34 it is assumed that this function describes the system @xmath31 plus one proton . in fact , it is the guillemin - zener function of @xmath31 symmetrized over protons . if @xmath35 , the function @xmath32 is reduced to @xmath28 . if @xmath36 , the function @xmath32 is reduced to @xmath25 . hence @xmath32 is a non - linear interpolation between @xmath25 and @xmath28 . it has to describe intermediate interproton distances . @xmath37 : : : two @xmath22 s are equal but the third one is different , @xmath38 it describes a `` mixed '' state of three hydrogen atoms . if @xmath39 , the function @xmath37 is reduced to @xmath21 . if @xmath40 , the function @xmath37 is reduced to @xmath25 . if @xmath41 , the function @xmath37 is reduced to @xmath28 . hence @xmath37 is a non - linear interpolation between @xmath21 , @xmath25 and @xmath28 . as function ( [ psi4 ] ) this is a type of guillemin - zener function and should describe intermediate interproton distances . @xmath42 : : : all @xmath22 s are different , @xmath43 this is a general non - linear interpolation of all functions @xmath44 . the total number of the parameters of the function ( [ trial ] ) is equal to 15 , where five of them are linear ones . note that @xmath45 can be fixed at @xmath46 . + in standard _ ab initio _ calculations , @xmath47 is most commonly expanded in terms of gaussian basis functions @xcite @xmath48 centered at atoms @xmath49 , @xmath50 whose coefficients @xmath51 are then determined variationally . the basis functions @xmath52 themselves are built up by primitive gaussians @xcite @xmath53 with contraction coefficients @xmath54 held fixed . our calculations were performed using the hartree - fock code implemented in the molpro suite of programs @xcite with the correlation consistent cc - pv6z and modified mcc - pv7z basis sets @xcite . the cc - pv6z basis set contains 91 contracted gaussians per atom , with @xmath55 quantum numbers up to @xmath56 , i.e. @xmath57 $ ] , yielding a total of 273 basis functions . the mcc - pv7z basis includes @xmath58 functions , leading to 140 contracted gaussians per atom , @xmath59 $ ] , or 420 basis functions in total . calculations were carried out for a range of equilateral triangular configurations using @xmath60 symmetry . in this point group , the lowest electronic state is @xmath61 . the total number of contracted gaussians of this symmetry is 168 for the cc - pv6z basis set and 255 for the mcc - pv7z basis set , respectively . the cc - pv6z results , which are not reported here explicitly , have been generated to assess the accuracy of this type of calculations . judging from such a comparison , we estimate the accuracy of the mcc - pv7z data to about @xmath62 over a large range of distances , deteriorating somewhat at short distances where the basis functions tend to become linearly dependent . in framework of the specialized variational method ( i ) some numerical computations were made . the minimization routine minuit @xcite from the cern - lib library was used as well as d01fcf routine from the nag - lib @xcite for three - dimensional numerical integration . numerical values of the total energy @xmath63 of the system @xmath2 for different values of the interproton distance @xmath9 were obtained , see table [ tmedel ] . the results from the molpro calculation with a huge standard - type basis set ( mcc - pv7z ) are given for comparison . a problem of the standard apprach is its slow convergence with respect to the angular momentum quantum number @xmath55 , requiring the use of large basis sets . the method based on the specially tailored trial function , eq . ( [ trial ] ) , leads to systematically lower variational energy values with considerably less terms . it should be noted that this method relies on a careful optimization of non - linear parameters . .variational results obtained with the specialized method ( [ trial ] ) and with a standard quantum chemistry method molpro for the total energy @xmath63 as a function of the internuclear distance @xmath9 for the system @xmath64 in the equilateral triangular geometry . for @xmath65 , in @xcite @xmath66 . [ tmedel ] [ cols="^ , > , > " , ] as a conclusion we have to state that the total energy @xmath63 as a function of the internuclear distance @xmath9 does not indicate either to a minimum or even slight non - adiabatic irregularities for finite @xmath9 . thus , the molecular ion @xmath1 does not exist in equilateral triangular configuration in the born - oppenheimer approximation . hmc expresses his deep gratitude to j.c . lpez vieyra for the valuable comments and for their interest in the present work , avt thanks universite libre de bruxelles for the hospitality extended to him where this work was completed . this work was supported in part by fenomec and papiit grant * in121106 * ( mexico ) . turbiner and j.c . lopez vieyra , _ phys.rev . * a66 * _ , 023409 ( 2002 ) h .- j . werner , p. j. knowles , r. lindh , f. r. manby , m. schtz , p. celani , t. korona , g. rauhut , r. d. amos , a. bernhardsson , a. berning , d. l. cooper , m. j. o. deegan , a. j. dobbyn , f. eckert , c. hampel , g. hetzer , a. w. lloyd , s. j. mcnicholas , w. meyer , m. e. mura , a. nicklass , p. palmieri , r. pitzer , u. schumann , h. stoll , a. j. stone , r. tarroni , and t. thorsteinsson , molpro , version 2006.1 , a package of ab initio programs ( 2006 ) , see http://www.molpro.net

it is shown that the molecular ion @xmath0 does not exist in a form of the equilateral triangle . to this end , a compact variational method is presented which is based on a linear superposition of six specially tailored trial functions containing non - linear parameters . careful optimization of a total of fifteen parameters gives consistently lower variational results for the electronic energy than can be obtained with standard methods of quantum chemistry even with huge basis sets as large as mcc - pv7z . _ dedicated to professor rudolf zahradnik on the occasion of his 80th birthday _

due to the exploding popularity of all things wireless , the demand for wireless data traffic increases dramatically . according to a cisco report , global mobile data traffic will increase 13-fold between 2012 and 2017 @xcite . this dramatic demand puts on pressure on mobile network operators ( mnos ) to purchase more spectrum . however , wireless spectrum is a scarce resource for mobile services . even if the continued innovations in technological progress relax this constraint as it provides more capacity and higher quality of service ( qos ) , the shortage of spectrum is still the bottleneck when the mobile telecommunications industry is moving toward wireless broadband services @xcite . to achieve a dominant position for future wireless services , thus , it is significant how new spectrum is allocated to mnos . since the spectrum is statically and infrequently allocated to an mno , there has been an ongoing fight over access to the spectrum . in south korea , for example , the korea communications commission ( kcc ) planed to auction off additional spectrum in both 1.8 ghz and 2.6 ghz bands . the main issue was whether korea telecom ( kt ) acquires the contiguous spectrum block or not . due to the kt s existing holding downlink 10 mhz in the 1.8 ghz band , it could immediately double the existing long term evolution ( lte ) network capacity in the 1.8 ghz band at little or no cost . this is due to the support of the downlink up to 20 mhz contiguous bandwidth by lte release 8/9 . to the user side , there is no need for upgrading their handsets . lte release 10 ( lte - a ) can support up to 100 mhz bandwidth but this requires the carrier aggregation ( ca ) technique , for which both infrastructure and handsets should be upgraded . if kt leases the spectrum block in the 1.8 ghz band , kt might achieve a dominant position in the market . on the other hand , other mnos expect to make heavy investments as well as some deployment time to double their existing lte network capacities compared to kt @xcite . thus , the other mnos requested the government to exclude kt from bidding on the contiguous spectrum block to ensure market competitiveness . although we consider the example of south korea , this interesting but challenging issue on spectrum allocation is not limited to south korea but to most countries when asymmetric - valued spectrum blocks are auctioned off to mnos . spectrum auctions are widely used by governments to allocate spectrum for wireless communications . most of the existing auction literatures assume that each bidder ( i.e. , an mno ) only cares about his own profit : what spectrum block he gets and how much he has to pay @xcite . given spectrum constraints , however , there is some evidence that a bidder considers not only to maximize his own profit in the event that he wins the auction but to minimize the weighted difference of his competitor s profit and his own profit in the event that he loses the auction @xcite . this strategic concern can be interpreted as a _ spite motive _ , which is the preference to make competitors worse off . since it might increase the mno s relative position in the market , such concern has been observed in spectrum auctions @xcite . in this paper , we study bidding and pricing competition between two competing / spiteful mnos with considering their existing spectrum holdings . given that asymmetric - valued spectrum blocks are auctioned off to them , we developed an analytical framework to investigate the interactions between two mnos and users as a three - stage dynamic game . in tage i , two spiteful mnos compete in a first - price sealed - bid auction . departing from the standard auction framework , we address the bidding behavior of the spiteful mno . in tage ii , two competing mnos optimally set their service prices to maximize their revenues with the newly allocated spectrum . in tage iii , users decide whether to stay in their current mno or to switch to the other mno for utility maximization . our results are summarized as follows : * _ asymmetric pricing structure _ : we show that two mnos announce different equilibrium prices to the users , even providing the same quality in services to the users . * _ different market share _ : we show that the market share leader , despite charging a higher price , still achieve more market share . * _ impact of competition _ : we show that the competition between two mnos leads to some loss of their revenues . * _ cross - over point between two mno s profits _ : we show that two mnos profits are switched . the rest of the paper is organized as follows : related works are discussed in ection ii . the system model and three - stage dynamic game are described in ection iii . using backward induction , we analyze user responses and pricing competition in ections vi and v , and bidding competition in ection vi . we conclude in section ii together with some future research directions . in wireless communications , the competition among mnos have been addressed by many researchers @xcite@xcite . yu and kim @xcite studied price dynamics among mnos . they also suggested a simple regulation that guarantees a pareto optimal equilibrium point to avoid instability and inefficiency . niyato and hossain @xcite proposed a pricing model among mnos providing different services to users . however , these works did not consider the spectrum allocation issue . more closely related to our paper are some recent works @xcite@xcite . the paper @xcite studied bandwidth and price competition ( i.e. , bertrand competition ) among mnos . by taking into account mnos heterogeneity in leasing costs and users heterogeneity in transmission power and channel conditions , duan _ et al_. presented a comprehensive analytical study of mnos spectrum leasing and pricing strategies in @xcite . in @xcite , a new allocation scheme is suggested by jointly considering mnos revenues and social welfare . x. feng _ et al . _ @xcite suggested a truthful double auction scheme for heterogeneous spectrum allocation . none of the prior results considered mnos existing spectrum holdings even if the value of spectrum could be varied depending on mnos existing spectrum holdings . we consider two mnos ( @xmath0 and @xmath1 ) compete in a first - price sealed - bid auction , where two spectrum blocks @xmath2 and @xmath3 are auctioned off to them as shown in ig . 1 . note that @xmath2 and @xmath3 are the same amount of spectrum ( i.e. , 10 mhz spectrum block ) . without loss of generality , we consider only the downlink throughput the paper . note that both mnos operate frequency division duplex lte ( fdd lte ) in the same area . due to the mnos existing spectrum holdings ( i.e. , each mno secures 10 mhz downlink spectrum in the 1.8 ghz band ) , the mnos put values on spectrum blocks @xmath2 and @xmath3 asymmetrically . if mno @xmath4 leases @xmath2 , twice ( 2x ) improvements in capacity over his existing lte network capacity are directly supported to users . in third generation partnership project ( 3gpp ) lte release 8/9 , lte carriers can support a maximum bandwidth of 20 mhz for both in uplink and downlink , thereby allowing for mno @xmath4 to provide double - speed lte service to users without making many changes to the physical layer structure of lte systems @xcite . on the other hand , mno @xmath5 who leases @xmath3 should make a huge investment to double the capacity after some deployment time @xmath6 . without loss of generality , we assume that mno @xmath4 leases @xmath2 . to illustrate user responses , we define the following terms as follows . * definition 1 . * ( asymmetric phase ) _ assume that mno @xmath5 launches double - speed lte service at time @xmath6 . when @xmath7 , we call this period asymmetric phase due to the different services provided by mnos @xmath4 and @xmath5 . _ * definition 2 . * ( symmetric phase ) _ assume that @xmath8 denotes the expiration time for the mnos new spectrum rights . when @xmath9 , we call this period symmetric phase because of the same services offered by mnos @xmath4 and @xmath5 . _ we investigate the interactions between two mnos and users as a three - stage dynamic game as shown in ig . 2 . in tage i , two spiteful mnos compete in a first - price sealed - bid auction where asymmetric - valued spectrum blocks @xmath2 and @xmath3 are auctioned off to them . the objective of each mno is maximizing his own profit when @xmath2 is assigned to him , as well as minimizing the weighted difference of his competitor s profit and his own profit when @xmath3 is allocated to him . in tage ii , two competing mnos optimally announce their service prices to maximize their revenues given the result of tage i. the analysis is divided into two phases : asymmetric phase and symmetric phase . in tage iii , users determine whether to stay in their current mno or to switch to the new mno for utility maximization . to predict the effect of spectrum allocation , we solve this three - stage dynamic game by applying the concept of backward induction , from tage iii to tage i. each user subscribes to one of the mnos based on his or her mno preference . let us assume that mnos @xmath4 and @xmath5 provide same quality in services to the users so they have the same reserve utility @xmath10 before spectrum auction . each mno initially has 50% market share and the total user population is normalized to 1 . in asymmetric phase , the users in mnos @xmath4 and @xmath5 obtain different utilities , i.e. , @xmath11 where @xmath12 is a user sensitivity parameter to the double - speed lte service than existing one . it means that users care more about the data rate as @xmath13 increases . the users in mno @xmath5 have more incentive to switch to mno @xmath4 as @xmath13 increases . when they decide to change mno @xmath4 , however , they face switching costs , the disutility that a user experiences from switching mnos . in the case of higher switching costs , the users in mno @xmath5 have less incentive to switch . the switching cost varies among users and discounts over time . to model such users time - dependent heterogeneity , we assume that the switching cost is heterogeneous across users and uniformly distributed in the interval @xmath14 $ ] at @xmath15 , where @xmath16 denotes the discount rate @xcite . this is due to the fact that the pays for the penalty of terminating contract with operators decrease as time passes . now let us focus on how users churn in asymmetric phase . a user @xmath17 in mno @xmath5 , with switching cost , @xmath18 , observes the prices charged by mnos @xmath4 and @xmath5 ( @xmath19 and @xmath20 ) . a user @xmath17 in mno @xmath5 will switch to mno @xmath4 if and only if @xmath21 thus the mass of switching users from mno @xmath5 to @xmath4 is = -2mu = -0.5mu @xmath22 where @xmath23 is a uniform @xmath24 random variable and @xmath25 denotes the initial market share . since the market size is normalized to one , each mno s market share in asymmetric phase is as follows : = 1mu = 1mu @xmath26 given users responses ( 4 ) , mnos @xmath4 and @xmath5 set their service prices @xmath27 and @xmath28 to maximize their revenues , respectively , i.e. , @xmath29 the nash equilibrium in this pricing game is described in the following proposition . * proposition 1 . * _ when @xmath30 and @xmath31 , there exists a unique nash equilibrium , i.e. , @xmath32 _ * proof . * in asymmetric phase , two competing mnos try to maximize their revenues @xmath33 and @xmath34 , respectively , given users responses , i.e. , @xmath35 a nash equilibrium exists by satisfying and solving the following first order conditions with respect to @xmath19 and @xmath20 , i.e. , @xmath36 @xmath37 @xmath38 proposition 1 shows two mnos equilibrium prices in asymmetric phase . intuitively , @xmath27 increases as @xmath13 increases . with larger @xmath13 , users care more about the data rate . thus , mno @xmath4 increases his service price to obtain more revenue . on the other hand , @xmath39 decreases as @xmath13 increases . it means that mno @xmath5 tries to sustain the revenue margin by lowering the service price and holding onto market share . an interesting observation is that both mnos decrease their service prices as @xmath40 increases . due to the discount factor ( @xmath16 ) , the users in mno @xmath5 are not locked - in and tries to maximize their utilities by churning to mno @xmath4 as switching costs decrease over time . therefore , mno @xmath4 lowers his service price to maximize his revenue , which forces mno @xmath5 to decrease the service price . this phenomenon is consistent with the previous results @xcite , @xcite in that the reduction of switching costs intensifies the price - down competition between two mnos . if @xmath41 , then all users in mno @xmath5 churn to mno @xmath4 . however , it is an unrealistic feature of the mobile telecommunication industry so we add the constraint @xmath42 . next we will show how each mno s market share changes in asymmetric phase . inserting the equilibrium prices ( 6 ) into ( 4 ) , each mno s market share can be calculated as follows : @xmath43 intuitively , mno @xmath4 takes mno @xmath5 s market share more as @xmath40 increases or @xmath13 increases . to hold onto or take mno @xmath4 s market share , the time to launch double - speed lte service @xmath6 is of great importance to mno @xmath5 . when mno @xmath5 launches double - speed lte service at time @xmath6 , each mno s total revenue in asymmetric phase is given by = -0.5mu @xmath44 @xmath45 similar to the analysis of market share , equation ( 8) shows that mno @xmath5 should launch double - speed lte service as quickly as possible to narrow the revenue gap between mno @xmath4 and mno @xmath5 ( see the last term of the revenues ( 8) ) . since mno @xmath5 launches double - speed lte service in symmetric phase , we assume that the users in mnos @xmath4 and @xmath5 obtain same utility , i.e. , @xmath46 for better understanding of user responses in symmetric phase , we first discuss the effect of switching costs on market competition . given the same services offered by two mnos , an mno s current market share plays an important role in determining its price strategy . each mno faces a trade - off between a low price to increase market share , and a high price to harvest profits by exploiting users switching costs . the following emma examines this trade - off and characterizes each mno s price strategy , which is directly related to user responses in symmetric phase . * lemma 1 . * _ in a competitive market with switching costs , the market share leader ( i.e. , mno @xmath4 ) charges a high price to exploit its current locked - in users while the marker share followers ( i.e. , mno @xmath5 ) charge low prices to increase market share for revenue maximization , respectively , given the same services offered by them . _ * proof . * we prove emma 1 by contradiction . suppose that mno @xmath5 charges a higher price than mno @xmath4 ( i.e. , @xmath47 . the mass of switching users from mno @xmath5 to @xmath4 is = -1mu = 0mu = 0mu @xmath48 where @xmath49 is the market share of mno @xmath5 at the end of asymmetric phase . then , each mno s market share is given by @xmath50 @xmath51 where @xmath52 is the market share of mno @xmath4 at the end of asymmetric phase . following the same steps of the roposition 1 , we can find the nash equilibrium by satisfying and solving the following first order conditions with respect to @xmath19 and @xmath20 , i.e. , @xmath53 @xmath54 = 0,\nonumber\end{aligned}\ ] ] which yields the solution given as follows @xmath55 thus , this contradicts to our assumption , completing the proof . @xmath38 with emma 1 , let us illustrate the process of user churn in symmetric phase . the mass of switching users from mno @xmath4 to @xmath5 is = -1mu @xmath56 where @xmath57 is the market share of mno @xmath4 at the end of asymmetric phase . then each mno s market share in symmetric phase is given by @xmath58 @xmath59 where @xmath60 is the market share of mno @xmath5 at the end of asymmetric phase . as noted in emma 1 , mno @xmath5 charges a lower price than mno @xmath4 in symmetric phase . following the same procedure ( 5 ) , the nash equilibrium is described in the following proposition . * proposition 2 . * _ when @xmath61 , there exists a unique nash equilibrium , i.e. , = -0.5mu = 0mu @xmath62 _ * proof . * following the same steps of the roposition 1 , a nash equilibrium exists by satisfying and solving the following first order conditions with respect to @xmath19 and @xmath20 , i.e. , @xmath63 = 0 , \nonumber\end{aligned}\ ] ] @xmath64 @xmath38 , @xmath65 ) . other parameters are @xmath66 , @xmath67 , @xmath68 , and @xmath69.,width=326 ] , @xmath65 ) . other parameters are @xmath66 , @xmath67 , @xmath68 and @xmath69.,width=326 ] roposition 2 states the mnos equilibrium prices in symmetric phase . as described in emma 1 , mno @xmath4 , the market share leader announces a higher service price up to @xmath70 than mno @xmath5 . to further investigate the effect of competition under the same quality in services , let us calculate each mno s falling price level in the neighborhood of the point @xmath6 . from ( 6 ) and ( 15 ) , each mno s falling price level ( i.e. , @xmath71 and @xmath72 ) is = 0mu = 0mu = 0mu @xmath73 @xmath74 because @xmath75 , mnos @xmath4 and @xmath5 always decrease their prices up to @xmath71 and @xmath76 at the starting point of the symmetric phase , respectively . perhaps counter - intuitively , it shows that mno @xmath5 always lowers his price despite launching double - speed lte service at the starting point of the symmetric phase . it can be interpreted as follows . since mno @xmath5 loses his market share in asymmetric phase , mno @xmath5 attempts to maximize his revenue by lowering his service price and increasing his market share , which forces mno @xmath4 to drop the service price at the same time . this means that the mnos competition under the same quality in services lead to some loss of their revenues , which , known as a _ price war _ , is consistent with our previous work @xcite . 3 shows @xmath77 and @xmath78 as a function of @xmath40 under two different user sensitivities ( @xmath79 , @xmath65 ) . note that mno @xmath4 s falling price level is more sensitive to @xmath13 . next we show that how each mno s market share varies in symmetric phase . from ( 14 ) and ( 15 ) , each mno s market share is @xmath80 unlike the asymmetric phase , each mno s market share only depends on the deployment time of carrier aggregation @xmath6 in symmetric phase . an interesting observation is that the market share leader ( i.e. , mno @xmath4 ) , despite charging a higher price , still achieves more market share up to @xmath81 than mno @xmath5 . in terms of market share , mno @xmath4 always gains a competitive advantage over mno @xmath5 if mno @xmath5 was forced to lease less - valued spectrum block . this explains how critical new spectrum is allocated to the mnos , and how struggling they are over access to the spectrum for improving market competitiveness for future wireless services . 4 shows user responses as a function of @xmath40 under two different user sensitivities ( @xmath79 , @xmath65 ) . if the new spectrum rights expire at @xmath82 , each mno s total revenue in symmetric phase is = 2mu @xmath83 = -1mu=-1mu=-1mu @xmath84 @xmath85 @xmath86 using ( 8) and ( 18 ) , we examine the two mnos aggregate revenues when mno @xmath4 leases @xmath2 and mno @xmath5 leases @xmath3 . each mno s aggregate revenue at @xmath82 is given in ( 19 ) . when mno @xmath5 decides to launch double - speed lte service , the optimal deployment time of the carrier aggregation @xmath87 should be studied . the following lemma describes the mno @xmath5 s optimal deployment time . * lemma 2 . * _ the market share followers ( i.e. , mno @xmath5 ) should launch double - speed lte service as quickly as possible not only for maximizing their own revenues but also for minimizing the market leader s revenue . _ * proof . * by taking the derivative of the two mno s aggregate revenues @xmath88 and @xmath89 with respect to @xmath6 , respectively , it can be checked that @xmath90 and @xmath91 . we omit the details of the derivations here . @xmath38 emma 2 states that the revenue of mno @xmath5 is strictly decreasing over @xmath6 while the reverse is for mno @xmath4 . to gain more insight into the effect of the allocation of asymmetric - valued spectrum blocks , let us define the revenue gain as follows : @xmath92 ig . 5 shows the revenue gain as a function of @xmath13 under two different deployment times ( @xmath68 , @xmath93 ) . as expected , the revenue gain is strictly increasing over @xmath6 and @xmath13 . in terms of @xmath13 , it can be checked directly by following the same steps of the emma 2 . such result explains why each mno should spitefully bid in a first - price sealed - bid auction to achieve a dominant position or compensate the revenue gap , which we will discuss these points in the next section . in tage i , two spiteful mnos @xmath4 and @xmath5 compete in a first - price sealed - bid auction where asymmetric - valued spectrum blocks @xmath2 and @xmath3 are auctioned off to them . for fair competition , each mno is constrained to lease only one spectrum block ( i.e. , @xmath2 or @xmath3 ) . we assume that the governments set the reserve prices @xmath94 and @xmath95 to @xmath2 and @xmath3 , respectively . note that the reserve price is the minimum price to get the spectrum block . since @xmath2 is the high - valued spectrum block , we further assume that two spiteful mnos are only competing on @xmath2 to enjoy a dominant position in the market . mnos @xmath4 and @xmath5 bid @xmath2 independently as @xmath96 and @xmath97 , respectively . in this case , @xmath3 is assigned to the mno who loses in the auction as the reserve price @xmath95 . because the mno who leases @xmath3 should make huge investments to double the existing lte network capacity compared to the other mno , we also assume the only mno who leases @xmath3 incurs the investment cost @xmath98 . under two different times ( @xmath68 , @xmath93 ) . other parameters are @xmath66 , @xmath67 and @xmath69.,width=326 ] when asymmetric - valued spectrum blocks are allocated to the mnos , there is a trade - off between self - interest and spite . to illustrate this trade - off , we first restrict ourselves to the case where spite is not present . if mno @xmath4 is _ self - interested _ , his objective function is as follows . = 1mu=1mu=1mu @xmath99 \cdot i_{b_i \geqslant b_j } + \pi^{b}(t_1,t_2 ) \cdot i_{b_i < b_j } , \end{aligned}\ ] ] where @xmath100 is the indicator function and @xmath101 is the profit when leasing @xmath3 . this case is the standard auction framework in that mno @xmath4 maximizes his own profit without considering the other mno s profit . in the real world , however , there is some evidence that some mnos are _ completely malicious_. the german third generation ( 3 g ) spectrum license auction in 2000 is a good example @xcite . german telekom kept raising his bid to prevent his competitors from leasing spectrum . if mno @xmath4 is completely malicious , his objective function can be changed as follows . = -1mu=-1mu=-1mu @xmath99 \cdot i_{b_i \geqslant b_j } - \left [ { r^a(t_1,t_2 ) - b_j } \right ] \cdot i_{b_i < b_j } .\end{aligned}\ ] ] it means that mno @xmath4 gets disutility as much as the profit of mno @xmath5 when he loses the auction . the minus term in ( 22 ) implies this factor . to reflect this strategic concern , our model departs from the standard auction framework in that each spiteful mno concerns about maximizing his own profit when he leases @xmath2 , as well as minimizing the weighted difference of his competitor s profit and his own profit when he leases b. combining ( 21 ) and ( 22 ) , we define each mno s objective function as follows . * definition 3 . * _ assume that two spiteful mnos ( i.e. , @xmath102 and @xmath103 ) compete in a first - price sealed - bid auction . the objective function that each mno tries to maximize is given by : = 0.5mu=0.5mu=0mu @xmath104 \cdot i_{b_i \geqslant b_j } \nonumber\\ & + & \left [ { ( 1 - \alpha _ i ) \pi ^b ( t_1 , t_2 ) - \alpha _ i ( r^a ( t_1 , t_2 ) - b_j ) } \right ] \cdot i_{b_i < b_j } \nonumber\\\end{aligned}\ ] ] = 3mu=3mu=2mu where @xmath100 is the indicator function , @xmath105 is the mno s profit when leasing @xmath3 , and @xmath106 $ ] is a parameter called the spite ( or competition ) coefficient . _ as noted , mno @xmath4 is self - interested and only tries to maximize his own profit when @xmath107 . when @xmath108 , mno @xmath4 is completely malicious and only attempts to obtain more market share by forcing mno @xmath5 to lease the less - valued spectrum block . for given @xmath106 $ ] and @xmath109 $ ] , we can derive the optimal bidding strategies that maximize the objective function in efinition 3 as follows . * proposition 3 . * _ in a first - price sealed - bid auction , the optimal bidding strategy for a spiteful mno @xmath110 and @xmath111 is : = 0.5mu=0.5mu=0.5mu @xmath112 @xmath113 _ * proof . * without loss of generality , suppose that mno @xmath4 knows his bid @xmath96 . further , we assume that mno @xmath4 infer that the bidding strategy of mno @xmath5 on @xmath2 is drawn uniformly and independently from @xmath114 $ ] . the mno @xmath4 s optimization problem is to choose @xmath96 to maximize the expectation of = 0.5mu=0.5mu=0.5mu @xmath115 } { \rm { } } f(b_j ) db_j \nonumber \\ & & + \int\limits_{b_i } ^{r^a ( t_1 , t_2 ) } { \left [ { ( 1 - \alpha _ i ) ( \pi ^b ( t_1 , t_2 ) ) - \alpha _ i ( r^a ( t_1 , t_2 ) - b_j ) } \right ] } { \rm { } } f(b_j ) db_j . \nonumber \\\end{aligned}\ ] ] differentiating equation ( 25 ) with respect to @xmath96 , setting the result to zero and multiplying by @xmath116 give = 5mu=5mu=5mu @xmath117 since the same analysis can be applied to the mno @xmath5 , the proof is complete . @xmath38 roposition 3 states that the mnos equilibrium bidding strategies . intuitively , the more spiteful the mno is , the more aggressively the mno tends to bid . for consistency , we assume that @xmath118 . then we can now calculate mno @xmath4 s profit and mno @xmath5 s profit as follows = 2mu=2mu=2mu @xmath119 where @xmath120 is calculated by substraction of the bidding price @xmath121 of ( 24 ) from @xmath88 of ( 19 ) . under two different costs ( @xmath122 , @xmath123 ) . other parameters are @xmath66 , @xmath67 , @xmath65 , @xmath68 , @xmath69 , @xmath124 , and @xmath125.,width=326 ] under two different spite coefficients ( @xmath126 , @xmath127 ) . other parameters are @xmath66,@xmath67 , @xmath65 , @xmath69 , @xmath123 , @xmath124 , and @xmath125.,width=326 ] to get some insight into the properties of the mnos equilibrium profits , let us define @xmath128 is different from @xmath129 of ( 19 ) where @xmath130 is the revenue gain from @xmath2 relative to @xmath3 without considering any cost . ] , which can be interpreted as the profit gain from @xmath2 relative @xmath3 . when @xmath131 , the profit of mno @xmath4 is higher than that of mno @xmath5 . it implies that mno @xmath4 could gain a competitive advantage over mno @xmath5 in both market share and profit . when @xmath132 , the situation is reversed . mno @xmath5 could take the lead in the profit despite losing some market share to mno @xmath4 . if the role of the government is to ensure fairness in two mnos profits , the government may devise two different schemes : setting appropriate reserve prices and imposing limits on the timing of the double - speed lte services . according to the ofcom report , setting the reserve prices closer to market value might be appropriate @xcite . it indicates that the government set @xmath94 and @xmath95 by estimating the value asymmetries between spectrum blocks @xmath2 and @xmath3 ( i.e. , @xmath133 ) and the spite coefficient @xmath134 . ig . 6 shows the profit gain as a function of @xmath134 under two different reserve prices for @xmath2 ( i.e. , @xmath122 , @xmath123 ) . for example , if @xmath135 , the government should set the reserve prices @xmath123 , @xmath124 . on the other hand , the government should set the reserve prices @xmath122 , @xmath124 when @xmath136 . besides setting appropriate reserve prices , the government can impose limits on the timing of the double - speed lte service . in south korea , for instance , korea telecom ( kt ) who acquired the continuous spectrum spectrum is allowed to start its double - speed lte service on metropolitan areas immediately in september 2013 , other major cities staring next march , and nation - wide coverage starting next july @xcite . this scheme implies to reduces @xmath6 by limiting the timing of the double - speed lte service to the mno who acquires spectrum block @xmath2 . 7 shows the profit gain as a function of @xmath6 under two different spite coefficients ( i.e. , @xmath126 , @xmath127 ) . in this paper , we study bidding and pricing competition between two spiteful mnos with considering their existing spectrum holdings . we develop an analytical framework to investigate the interactions between two mnos and users as a three - stage dynamic game . using backward induction , we characterize the dynamic game s equilibria . from this , we show the asymmetric pricing structure and different market share between two mno . perhaps counter - intuitively , our results show that the mno who acquires the less - valued spectrum block always lowers his price despite providing double - speed lte service to users . we also show that the mno who acquires the high - valued spectrum block , despite charging a higher price , still achieves more market share than the other mno . we further show that the competition between two mnos leads to some loss of their revenues . with the example of south korea , we investigate the cross - over point at which two mnos profits are switched , which serves as the benchmark of practical auction designs . results of this paper can be extended in several directions . extending this work , it would be useful to propose some methodologies for setting reserve prices @xcite , @xcite . second , we could consider an oligopoly market where multiple mnos initially have different market share before spectrum allocation , where our current research is heading . m. shi , j. chiang , and b .- d . price competition with reduced consumer switching costs : the case of `` wirelss number portability '' in the cellular phone industry , " , vol . 1 , pp . 2738 , 2006 . dotecon and aetha , spectrum value of 800mhz , 1800mhz , and 2.6ghz , " a dotecon and aetha report , jul . 2012 . available : http://stakeholders.ofcom.org.uk/binaries/consultations/award800mhz/ statement / spectrum value.pdf .

we study bidding and pricing competition between two spiteful mobile network operators ( mnos ) with considering their existing spectrum holdings . given asymmetric - valued spectrum blocks are auctioned off to them via a first - price sealed - bid auction , we investigate the interactions between two spiteful mnos and users as a three - stage dynamic game and characterize the dynamic game s equilibria . we show an asymmetric pricing structure and different market share between two spiteful mnos . perhaps counter - intuitively , our results show that the mno who acquires the less - valued spectrum block always lowers his service price despite providing double - speed lte service to users . we also show that the mno who acquires the high - valued spectrum block , despite charing a higher price , still achieves more market share than the other mno . we further show that the competition between two mnos leads to some loss of their revenues . by investigating a cross - over point at which the mnos profits are switched , it serves as the benchmark of practical auction designs .

unlike water , a layer of sand will not flow unless its surface is inclined beyond a characteristic angle , known as the maximum angle of stability @xcite . this simple fact translates into a host of threshold phenomena wherever granular material is found . many such phenomena play a crucial role in the erosion of earth s surface , and very likely manifest themselves in the richness of the patterns exhibited by drainage networks . depending on geological , hydrological , and climatological properties , erosion by water is mainly driven either by overland flow or subsurface flow . the former case occurs when the shear stress imposed by a sheet flow exceeds a threshold @xcite . erosion in the latter case known as seepage erosion , or sapping occurs when a subsurface flow emerges on the surface . here the eroding stresses derive not only from the resulting sheet flow but also the process of seepage itself @xcite . the onset of erosion for both overland flow and seepage is threshold - dependent , but the additional source of stress in the case of seepage has the potential to create significantly different erosive dynamics . here we study the seepage case . whereas the case of horton overland flow has been extensively studied @xcite , seepage erosion has received less attention . @xcite suggests that erosive stresses due to seepage are more widespread in typical environments than commonly assumed . he also provides a detailed description of seepage erosion in the field , together with a discussion of the various factors that influence its occurrence . another focus of attention has been the controversial possibility that many erosive features on mars appear to have resulted from subsurface flows @xcite . although the importance of seepage stresses in erosion have been realized by @xcite and @xcite , comprehensive quantitative understanding is difficult to obtain . the complexity arises from the interdependent motion of the sediment and fluid the `` two - phase phenomenon '' @xcite which , of course , is common to _ all _ problems of erosion . to further understand seepage erosion , we proceed from experiments @xcite . questions concerning the origin of ancient martian channels have motivated considerable experimental work in the past @xcite . the process of seepage erosion has also been studied as an example of drainage network development @xcite . our experiments , following those of @xcite and others , are designed to enable us to construct a predictive , quantitative theory . consequently , they stress simplicity and completeness of information . although our setup greatly simplifies much of nature s complexity , we expect that at least some of our conclusions will improve general understanding , and therefore be relevant to real , field - scale problems . a previous paper by @xcite provided a qualitative overview of the phenomenology in our experiment . it described the main modes of sediment mobilization : channelization , slumping , and fluidization . here we provide quantitative understanding of the onset and transitions between these modes . our emphasis is on the threshold phenomena associated with the onset of erosion , which we will ultimately characterize in the same way that others @xcite have characterized the onset of dry granular flow beyond the maximum angle of stability . this involves a construction of a generalized shields criterion @xcite valid in the presence of seepage through an inclined surface . a major conclusion is that the onset of erosion driven by seepage is significantly different from the onset of erosion driven by overland flow . we find that there is a critical slope @xmath0 , significantly smaller than the maximum angle of stability , above which the threshold disappears . therefore any slope greater than @xmath0 is unstable to erosion if there is seepage through it . this result is similar to well - known conclusions for the stability to frictional failure of slopes with uniform seepage @xcite . an important distinction in our work , however , concerns the mode of sediment mobilization and its local nature . the existence of the critical slope for seepage erosion may provide a useful quantitative complement to the qualitative distinctions between seepage and overland flow that have already been identified @xcite . the remaining modes of sediment mobilization , fluidization and slumping , are modeled using well established ideas @xcite . the result of applying these ideas together with the generalized shields criterion provides a theoretical prediction of the outcomes of the experiment , i.e. , a phase diagram . agreement between theory and experiment is qualitative rather than quantitative . we nevertheless believe that our theoretical approach is fundamentally sound and that better agreement would follow from improved experimental procedures . in our experimental setup , first introduced by @xcite , a pile of identical cohesionless glass beads @xmath1 mm in diameter is saturated with water and compacted to create the densest possible packing . it is then shaped into a trapezoidal wedge inclined at an angle @xmath2 with slope @xmath3 as shown in fig.[fig : expt ] . the downslope length of the wedge is @xmath4 cm , its width across the slope is @xmath5 cm , and its height in the middle is approximately @xmath6 cm . water enters the sandpile underneath through a fine metal mesh and exits at the lower end of the pile through the same kind of mesh . a constant head at the inlet is maintained by keeping a constant water level @xmath7 in the reservoir behind the sandbox with the help of an outflow pipe . the slope @xmath8 of the pile and the water level @xmath7 are the control parameters of the experiment . the degree of packing of the granular pile is the variable most difficult to control . our particular method of feeding water into the sandpile , similar to that of @xcite , can be motivated in three ways . the most important justification is the fact that the amount of water flowing on the surface can be finely controlled in our geometry . this feature is essential in probing the onset of erosion . second , our setup allows us to access heads @xmath7 larger than the height of the pile , which therefore allows us to explore dynamic regimes unavailable if water enters the pile through a mesh in the back . third , a similar seepage water flow geometry can exist in the field wherever water travels beneath an impermeable layer that terminates . we have performed two types of experiments : steady and non - steady . for a fixed water level and in absence of sediment motion , water flow reaches steady state . by monitoring the total water flux through the system we estimate the time to reach steady state to be approximately ten minutes . to explore the onset of sediment motion , we raised the water level @xmath7 in small increments , waiting each time for steady state to be established . due to the particular shape of the bulk flow in our experiment , surface flow exists over a finite region of the surface . the width of this seepage face and therefore the depth of the surface flow can be tuned by changing @xmath7 . because of the finite extent of surface flow , its depth and therefore the viscous shear stress reaches a maximum at a certain location . thus , by increasing @xmath7 we can continuously tune the maximum shear stress experienced by the surface grains . the maximum shear stress reaches a critical value for the onset of sediment motion in a certain location on the slope . as we show below , we can compute where the maximum shear stress occurs and thus can reliably detect the onset of sediment motion visually because we focus our attention on this location . once sediment begins to move , channels form almost immediately . these channels grow in length , width , and depth . an example of the evolving channel network is shown in fig.[fig : channels ] . depending on the slope , as the channels deepen , the pile becomes unstable to fluidization or slumping . for slopes lower than approximately 0.05 , the fluidization threshold is reached before sediment is mobilized on the surface . m. the slope of the pile is @xmath9 and the water level @xmath10 cm.,width=316 ] we also explored the non - steady evolution of the bulk and surface water flow and resulting sediment motion by raising the water level @xmath7 to some higher value from zero . in this case one of three things can happen . the pile can be fluidized within a few seconds or fail by slumping as shown in fig.[fig : slump ] . if this does not occur , the water emerges on the surface just above the inlet . a sheet of water then washes down the slope of the pile . during this initial wash , sediment is mobilized and incipient channels form . these channels grow during subsequent relaxation of the bulk water flow towards steady state . because of the initial wash s erosive power , channels are able to form and grow for lower water pressures than in steady experiments . angle to the slope after the water flow has been stopped . the width of the imaged region is approximately @xmath11 m . slumping happens along a convex upward arc which looks darker because it is deeper and therefore wetter.,width=316 ] outcomes of a large number of non - steady experiments and several steady experiments for varying slope @xmath8 and the water level @xmath7 are summarized in the phase diagram in fig.[fig : phase ] . each symbol in the plot represents one experiment . the sediment is either immobile ( stable seepage ) , or it is mobilized on the surface where channels form ( channelization ) or in the bulk ( slumping or fluidization ) . in several experiments , slumping or fluidization happened after channels formed and grew . in the following sections we describe the computations that allow us to construct the theoretical boundaries between the three different modes of sediment mobilization in our experiment . ; those that produced channels are indicated by @xmath12 ; and those that produced fluidization and/or slumping within one hour of the beginning of the experiment are represented by @xmath13 . the straight line and gray - shaded curves are theoretical predictions for the boundaries separating the four regions indicated by their labels . the thickness of the lines indicates uncertainty in the theory . the boundary between the uneroded and channelized states is reasonably well approximated by our theory . the theoretical boundaries for fluidization and slumping , however , appear to overestimate the critical water level , possibly as a result of inhomogeneities , dynamic changes in the sandpile s shape , or from the assumption of a steady state.,width=316 ] whereas steady - state flow can be readily characterized quantitatively , non - steady flow characterization requires knowledge of the water - table dynamics . however , the theory of the water - table dynamics is less well established than that of the flow through the bulk of a porous medium . also , our steady - state experiments probe all aspects of sediment dynamics . we can therefore focus on the quantitative characterization of the steady - state flow . to study the onset of erosion quantitatively we need to be able to establish a correspondence between the experimentally measurable quantities such as the slope @xmath8 , the water level @xmath7 , the size of the seepage face , and the water fluxes . the seepage and surface fluxes are the most difficult to measure . in this section we set up their computation . the computation is designed to enable us to infer water fluxes indirectly by measuring the size of the seepage face . in the following sections we will use this computation to quantify the onset of erosion and to compute the slumping and liquefaction boundaries of the channelization phase diagram shown in fig.[fig : phase ] . fig.[fig : expt ] specifies the key quantities and coordinate systems we use in computing the fluxes . the flow profile is independent of the @xmath14-coordinate across the slope of the sandpile except near the side walls of the box . we therefore treat the box as if it were infinitely wide . flow is then two - dimensional and the specific discharge vector @xmath15 is in the @xmath16-@xmath17 plane . we will use two coordinate systems . as shown in fig.[fig : expt ] , the @xmath18 coordinate is measured vertically from the bottom of the box while the @xmath17 coordinate is the normal distance away from the surface of the pile . the flow is governed by darcy s law , @xmath19 where @xmath20 is the scalar hydraulic conductivity , and @xmath21 is the total hydraulic head of the pore water . both @xmath15 and @xmath20 have units of velocity while the scaled pore pressure @xmath22 has units of length . here @xmath23 is the density of water and @xmath24 is the magnitude of the acceleration of gravity . we have measured @xmath20 via a -tube relaxation experiment . to do so we created a water level difference @xmath25 between the two arms of a transparent -shaped tube of width @xmath26 partially filled with glass beads . the rate of change of @xmath25 is given by @xmath27 . by measuring the rate of change of @xmath25 we deduced the value of the hydraulic conductivity @xmath28 mm / s ( @xcite ) . hydraulic conductivity is sensitive to the packing of the grains and is the variable most difficult to control in our experiment . water incompressibility implies @xmath29 , therefore yielding laplace s equation , @xmath30 to compute the pore pressure @xmath31 , boundary conditions must be specified . the walls of the box are impenetrable . therefore the discharge vector is parallel to the walls . in other words , the flux @xmath32 in the direction @xmath33 normal to the walls vanishes . thus @xmath34 . because the glass beads in our experiment are small , capillarity is important . when a tube filled with glass beads is lowered into a reservoir of water , the porous bead - pack fully saturates in a region that extends above the surface of the water by a capillary rise @xmath35 . we measured @xmath36 mm for our material . the capillary rise is a measure of the average radius of the water menisci at the edge of the fully saturated zone . the pore pressure at the edge of the fully saturated zone is @xmath37 ( without loss of generality we set the atmospheric pressure to zero ) . water can rise above the fully saturated zone through the smaller pores and narrower throats . thus a partially saturated capillary fringe exists above the fully saturated zone . however , in this fringe the water is effectively immobile since it is confined to the smaller pores and narrower throats . since water flows only in the fully saturated zone , we define the water table to be at its edge . thus , the pore pressure at the water table is equal to the negative capillary rise @xmath38 . in steady state the discharge vector is parallel to the water table . this extra condition allows us to determine the location of the water table in steady state . we neglect the pressure drop across the inlet mesh . therefore , the pore pressure at the inlet mesh is @xmath39 . the boundary conditions at the surface of the sandpile and at the outlet mesh are more subtle . when no water seeps out , i.e. , when the discharge vector is parallel to the surface , the curvature of the water menisci between grains can freely adjust so that the pressure @xmath31 can vary between zero and @xmath37 . therefore when @xmath40 , no seepage occurs . otherwise , the pore pressure equals the atmospheric pressure @xmath41 ( we neglect the pressure exerted by the thin layer of water on the surface ) , and the discharge vector has a component normal to the surface , i.e. , there is either exfiltration or infiltration . , or the pressure at the water table when it is below the surface of the pile . note that seepage occurs only where the pore pressure reaches atmospheric pressure . slope is @xmath9 , water level @xmath42 cm.,width=316 ] to obtain the steady state location of the water table , we guess its position and solve laplace s equation with the @xmath43 boundary condition on the water table . we then move the water table in the direction of the local discharge vector by an amount proportional to its length . iteration of this procedure converges to the steady - state position of the water table . an example is shown in figure [ fig : table_pressure_overland ] . once the steady flow pattern is known , we can calculate the overland water flux @xmath44 by integrating the one - dimensional continuity condition which states that the downslope derivative of the overland flux is equal to the seepage flux : @xmath45 in this section we assume , based on direct observation , that the onset of channel incision coincides with the onset of erosion ( i.e. , we never observed a homogeneously eroding state ) . in other words , when the overland water flux becomes strong enough to carry grains , the flow of sediment becomes immediately unstable to perturbations transverse to the downslope direction and incipient channels form @xcite . using this assumption and the calculation of the overland water flux we can deduce the threshold condition for the onset of erosion . it is universally assumed after @xcite that the hydrodynamic stresses exerted on the sandpile by the fluid flowing on its surface determine whether cohesionless granular material is entrained . in the limit of laminar flow , the dominant hydrodynamic stress is the viscous shear stress @xmath46 . appropriately scaled this shear stress is termed the shields number @xcite , defined by @xmath47 where @xmath48 is the density of the granular material , @xmath49 is the grain diameter ( @xmath50 mm in our experiment ) , and the surface is not inclined . the conventional shields number is the ratio between the horizontal force exerted by the flow and vertical force due to grain s weight . to generalize the notion of the shields number to the situation with seepage through an inclined surface , we make two changes in eq . . we first add the tangential component of the seepage force density @xmath51 acting over a length @xmath49 to the numerator of . the numerator thus becomes @xmath52 . note that we did not include the tangential component of the grain s weight to the numerator . defined in this way , the generalized shields number measures the effect of the fluid : both the bulk as well as the surface flows . -axis of the resultant of the grain s weight and the seepage force both scaled by @xmath53.,width=220 ] second , we replace the denominator of eq . by the resultant ( vectorial sum ) of the seepage force on a grain and its submerged weight , as shown in fig.[fig : mod_shields ] , both scaled by @xmath53 ( @xmath54 to obtain stress as in the numerator and @xmath55 for agreement with the conventional shields number ) , projected onto the @xmath17-axis . according to @xcite the grains on the surface of the bed experience a seepage force roughly half as large as the grains several layers deep . consequently , we assume that the seepage force is reduced by a factor of @xmath56 ; therefore @xmath57 where @xmath2 is the inclination angle of the surface . the importance of the seepage stresses for the criterion for the onset of erosion was previously realized by @xcite . it can be shown that their equation ( 10 ) expressing marginal stability of a surface grain is equivalent to writing @xmath58 , the tangent of the angle of internal friction . the generalized shields number eq . is a measure of the relative importance of the tangential and normal forces acting on a grain at the surface of the sandpile . therefore , we expect @xmath59 to be a control parameter for erosion . in other words , there exists a critical shields number @xmath60 , such that when @xmath61 , surface grains are immobile , and when @xmath62 , sediment is mobilized . note that ( [ eq : shields ] ) reduces to the classical definition of the shields number for a flat surface without seepage . also note that since we did not include the tangential component of grain s weight , the critical shields number at the onset of sediment motion vanishes when the inclination angle @xmath2 reaches the maximum angle of stability . although we obtain the seepage force density @xmath63 as a result of computing the pore - water pressure , to calculate the boundary shear stress @xmath46 we must estimate the thickness of the surface water layer @xmath64 . since this thickness changes slowly in the downslope direction , we can approximate the surface flow by the steady flow of a uniform layer of viscous fluid . also , the surface water flux is small enough for turbulence to be of no importance . the thickness @xmath65 of laminar surface flow for a given flux @xmath66 is @xcite @xmath67 where @xmath68 is the viscosity , while the viscous shear stress exerted on the sandpile is @xmath69 the particle reynolds number can then be calculated using the bottom shear stress and shear velocity @xmath70 as @xmath71 where @xmath72 is the kinematic viscosity of water . we estimate that in our experiments , this particle reynolds number varies between 5 and 20 depending on the slope @xmath8 of the pile and the water level @xmath7 . we verify this estimate of the reynolds number by a direct measurement of the thickness of the surface flow . we find that this thickness is several grain diameters . this justifies the laminar flow assumption used in obtaining eq . . using , the shields number can now be conveniently rewritten as @xmath73 this expression can be further simplified by noting that @xmath41 along the seepage face . therefore @xmath74 at the surface wherever there is overland flow . we arrive at the final expression for the modified shields number which depends on the surface flow thickness @xmath65 , the normal component @xmath75 of the seepage force density at the surface , and the seepage force reduction factor @xmath76 @xmath77 in our geometry , both the surface and the seepage water fluxes reach a maximum somewhere along the slope . therefore the shields number has a maximum value as well . below we calculate this maximum shields number in steady state for a given slope @xmath8 and water level @xmath7 . for a pile of slope @xmath9 . at @xmath78 water first seeps through the surface and the shields number jumps to a nonzero value . afterwards it increases rapidly as @xmath79 ( solid line ) . inset : corresponding size of the computed seepage face.,width=316 ] we now explore the consequences of seepage for the phenomenology of the onset of erosion . because of the additional force on the surface grains , seepage flow is more erosive than overland flow . this notion is reflected quantitatively in the generalized shields number . let us examine how the maximum shields number varies with the water level @xmath7 in our experiment . a representative graph of the maximum shields number versus the water level is shown in fig.[fig : shields_h ] . below a water level @xmath80 that is a function of the slope , no water seeps out to the surface of the pile . even though the water table may be at the surface , the pressure at the water table is below atmospheric pressure and capillarity prevents seepage . when @xmath81 , i.e. , exactly at the onset of seepage , the pressure reaches @xmath41 at some point on the surface . since the seepage flux is still zero , @xmath82 along the wet part of the surface . therefore , just above the seepage onset , when the water layer thickness and the seepage flux are both infinitesimally small , the maximum shields number is @xmath83 in contrast to overland flow , the consequence of seepage is that as soon as the water emerges on the surface , the maximum shields number is some non - zero value which depends on the slope . this also implies that there exists a critical slope @xmath0 such that @xmath84 is equal to the critical shields number @xmath60 , i.e. , @xmath85 for slopes greater than @xmath0 seepage is always erosive . note that for low - density particles this critical slope can be arbitrarily small . the expression for the critical slope for seepage erosion in eq . is analogous to well - known formulas for stability of slopes to coulomb failure due to uniform seepage @xcite . our result applies locally to the point where non - uniform seepage first emerges on the surface . in this situation , the pile is generally stable to coulomb failure and the sediment is eroded only locally on the surface . as a function of the downslope coordinate @xmath16 at the onset of seepage @xmath86 and just above.,width=220 ] we now show that above @xmath87 , the maximum shields number increases rapidly as a @xmath88 power of the water level excess @xmath89 . at the onset of seepage , i.e. , when @xmath81 , the pressure at the water table reaches atmospheric pressure @xmath41 at some point @xmath90 located at @xmath91 and @xmath92 on the surface . even though the water table is at the surface , there is no seepage anywhere , i.e. , @xmath93 . because the pressure is smooth , it can be approximated by a quadratic function near this point so that @xmath94 , where @xmath95 and @xmath96 are constants with appropriate dimensions . when the water level is raised by a small increment @xmath97 , the lowest order change in the head at the water table is an increase of @xmath98 with the exception of the region where this increase would lead to a positive pressure . as illustrated in fig.[fig : increment ] , in this region the pore pressure @xmath31 is set to @xmath99 , and thus this region becomes the seepage face . the width of the seepage face @xmath100 scales like the square root of @xmath98 , i.e. , @xmath101 as seen in the inset of fig.[fig : shields_h ] . the seepage flux can be estimated by noting that the hydraulic head is modified by an amount @xmath98 over a vertical region of order @xmath100 . therefore we obtain @xmath102 . the total surface flux therefore scales like the product of the seepage flux and the width @xmath100 of the seepage face , i.e. , @xmath103 . the lowest order change in the maximum shields number is due to the change of the surface flow depth @xmath104 . thus as we claimed above , just above the water level @xmath87 for the onset of seepage , @xmath105 where the constant @xmath106 is a function of the slope . variation with water level of the computed maximum shields number shown in fig.[fig : shields_h ] is consistent with eq . . in the previous sections we have detailed the way of calculating the bulk and surface water fluxes in our experiment and the resulting maximum generalized shields number . in this section we use this calculation to examine the onset of the sediment flow and channelization . our first goal is to measure the threshold or critical shields number required for the mobilization of sediment . we then use this measured value of the critical shields number to predict the outcome of steady - state experiments for various values of the slope and the water level and thus compute the channelization boundary in the phase diagram in fig.[fig : phase ] . the actual maximum shields number in the experiment differs from the quantity calculated in eq . ( [ eq : shields - prefinal1 ] ) . in addition to random errors in the measurements of the pile dimensions and water level , there are several sources of systematic error . for example , the pressure drop across the inlet mesh results in a lower effective hydraulic head . also , our measurement of the capillary rise @xmath35 is dependent on a visual estimate of the fully saturated zone and thus can be a source of systematic error . we indeed find that the size of the seepage face calculated for a particular water level @xmath7 is greater than measured in the experiment . however , the size of the seepage face translates directly into the surface water flux and therefore the maximum shields number . the inset of figure [ fig : shields_h ] shows the typical dependence of the size of the seepage face on the water level . the variation of the maximum shields number with the size of the seepage face is shown in fig.[fig : shields_seep ] for three different slopes . we use this computed correspondence between the size of the seepage face and the maximum shields number to infer the maximum shields number in the experiment by measuring the size of the seepage face . to measure the critical shields number we raise the water level @xmath7 in increments of a few millimeters at a time . each time the water level is increased , the seepage flow is allowed to reach a steady state . in each of these steady states we measure the seepage face size and infer the corresponding maximum shields number . eventually , sediment is mobilized and we record the size of the seepage face and compute the corresponding maximum shields number . this number is an upper bound on the critical shields number for our granular material at that particular slope . the lower bound on the critical shields number is obtained from the largest seepage face at which no sediment is moving or sediment motion is only transient . averaging over several experiments with slope @xmath9 we estimate the critical shields number to be @xmath107 it is not obvious that the generalized critical shields number for the onset of seepage driven erosion should coincide with the critical shields number for overland flow . however our measured value of the critical generalized shields number is within the scatter of the existing data for overland flow summarized in @xcite . our measurement the critical generalized shields number is equivalent to measuring the angle of internal friction due to the correspondence of our definition of @xmath59 and howard and mclane s equation ( 10 ) . deviations from flatness of the pile s surface result in the fluctuations of the thickness of the surface water film . as a result , the maximum bottom shear stress in the experiment is systematically greater than that calculated at a given size of the seepage face . thus the shields number calculated for a particular size of the seepage face is the lower bound on the actual shields number in the experiment . in principle , the critical shields number should vary with the slope of the pile . evidence for this is the fact that at the maximum angle of stability any additional forcing from the water flowing over the bed mobilizes sediment . since it is reasonable to assume that the critical shields number is continuous and monotonic , we arrive at the notion that it decreases monotonically with slope and vanishes at the maximum angle of stability . for small slopes the critical shields number is expected to decrease as the cosine of the inclination angle since this is the lowest order change in the stabilizing effect of gravity . for most slopes in our experiments , @xmath108 is within a few percent of unity and thus we can ignore the variation of the critical shields number with slope . this assumption allows us to predict the water level at which erosion and therefore channelization should commence in our experiment . in fig.[fig : phase ] a boundary is drawn between regions where sediment is expected to mobilize and remain immobile . to obtain this line we computed for each slope the water level @xmath109 at which the shields number is equal to the critical shields number . below this water level , i.e. , when @xmath110 , the maximum shields number is below critical and thus sediment is immobile . conversely , for @xmath111 , the maximum shields number is above critical and thus sediment is mobilized and channels form . the channelization boundary is widened because of the uncertainty in the critical shields number . qualitative agreement of the channelization boundary with experiments is perhaps due to the opposite action of two effects . first , channelization occurs for lower water levels in non - steady experiments . this happens because in non - steady experiments the maximum shields number overshoots its steady - state value . the overshoot is greatest for small slopes . second , a pressure drop across the inlet mesh and the compacted region of sand close to it has an opposite effect which increases the water level needed for channelization . these two effects , though small , could together affect the accuracy of our predictions . since these effects act in opposite ways , our the predictions of the calculated channelization water level @xmath109 agree qualitatively with the experiments . having computed the channelization boundary in the phase diagram , we now pursue a quantitative description of the other two modes of sediment mobilization exhibited by our sandpile . higher water pressures can cause the sandpile to fail in one of two ways . first , an upward seepage force can lift sand and result in a fluidization or quicksand instability . second , the pile can become unstable to slipping , slumping , or sliding . both failure mechanisms have been discussed by a number of studies , e.g. , those of @xcite or @xcite . and height @xmath26 and compare it with its weight.,width=144 ] fluidization occurs when at some point @xmath90 in the sandpile the pore pressure is larger than the total hydrostatic pressure due to the weight of the sand and the water above @xmath90 . to see this we compute the total seepage force acting on a slice of sand of width @xmath112 between point @xmath90 and point @xmath113 on the surface of the pile directly above @xmath90 . the vertical component of this force is ( see fig.[fig : fluidize ] ) @xmath114 where @xmath26 is the height of the slice . when this force exceeds the submerged weight @xmath115 of the slice , the slice is lifted and the bed is fluidized . here @xmath116 is the total density of the saturated sand , which for our sand is approximately 2 g/@xmath117 . thus fluidization occurs when there exists points @xmath90 and @xmath113 on the surface directly above @xmath90 such that @xmath118 for uniform seepage this condition is equivalent to those in @xcite and @xcite . to construct the fluidization boundary in the phase diagram ( fig.[fig : phase ] ) , we find the water level @xmath119 above which there exists at least one point in the pile for which condition is satisfied . below this fluidization water level this condition is not satisfied for any point in the pile . in addition to fluidization the sandpile can fail by slumping . this can happen in one of two ways . frictional failure can occur in the bulk of the pile due to the seepage stresses . alternatively , surface avalanching can occur . to establish an upper bound on the water level at which the sandpile slumps via either mechanism we use the criterion developed by @xcite for determining when a slope is destabilized by uniform groundwater seepage . essentially it requires calculating the vectorial sum of the seepage and gravity forces acting on a small element of soil near the surface . when the angle between this total force and the downward normal to the surface , which we will call the effective inclination angle , exceeds the maximum angle of stability , the surface grains are destabilized . we measured the maximum angle of stability to be @xmath120 for dry glass beads . the slumping boundary in the phase diagram ( fig.[fig : phase ] ) is constructed by computing the effective inclination angle along the surface of the pile and noting the water level @xmath121 , at which the effective inclination angle reaches the maximum angle of stability @xmath122 at some point of the surface . figure [ fig : phase ] shows the critical water level at which fluidization and slumping should occur according to the criteria above . failure occurs at systematically lower water levels in the experiment . there are several effects which can account for this difference . first , any irregularities in the construction of the pile such as voids or surface height fluctuations make the pile more unstable to fluidization and slumping . second , we compute the instability of an uneroded pile , whereas in most experiments , the pile failed after erosion had changed the shape of the pile . the decrease of pile s height due to erosion increases the head gradient in the bulk and thus makes the pile more prone to slumping and/or fluidization . and three different water levels . for the highest water level , a region on the surface has an effective angle above the maximum angle of stability and thus the slope is unstable to slumping . the inset shows the plateau value of the effective inclination angle as a function of slope . when the plateau value reaches the maximum angle of stability , even a small amount of seepage destabilizes the pile to slumping.,width=316 ] at @xmath123 , a jump in the slumping water level is observed in both the experiment and the model . this jump is a purely geometric effect . slumping occurs when , somewhere along the slope , the effective inclination angle , which includes the effect of the seepage force , exceeds the maximum angle of stability . as shown in fig.[fig : slump_jump ] , for slopes smaller than @xmath124 , the effective inclination angle is flat and develops a peak under the water inlet as the water pressure is increased . when the top of this peak crosses the value of the maximum angle of stability , the pile slumps . when the slope exceeds @xmath124 , however , the value of the plateau in the effective inclination angle is above the maximum angle of stability . therefore , for these slopes , the pile will be unstable to slumping as soon as the water emerges on the surface . this article reports on our progress in understanding seepage erosion of a simple non - cohesive granular material in a laboratory - scale experiment introduced in @xcite . our ultimate goal is to construct a quantitative predictive theory of the onset and growth of the channel network observed in this experiment . this goal requires a complete sediment transport model as well as the calculation of the relevant water fluxes . here we obtain the latter and focus on the onset of erosion . prediction of the onset of erosion based on the generalized shields conjecture explains qualitatively the channelization boundary in the experimental phase diagram . by invoking well established simple ideas we also roughly explain the fluidization and slumping boundaries in the phase diagram . greater discrepancy with the experiment for these boundaries indicates that better understanding of the slumping / fluidization mechanisms particular to our experiment is needed . the central result of our exploration is the introduction of the generalized shields criterion for seepage erosion . as a consequence of seepage forces on the surface grains , the threshold for the onset of erosion driven by seepage is slope dependent . the threshold disappears at a critical slope @xmath0 determined by the critical shields number for overland flow and the density contrast between the granular material and water . in most cases this critical slope is significantly smaller than the maximum angle of stability . we find , therefore , that slopes above this critical slope are unstable to any amount of seepage . as a consequence , slopes that sustain seepage must be inclined at an angle smaller than the critical angle for seepage erosion . this behavior contrasts strongly with the threshold phenomena in erosion by overland flow , and therefore provides a mechanistic foundation for distinguishing the two types of erosion . this work was supported by a doe grants de - fg02 - 99er15004 and de - fg02 - 02er15367 . aharonson , o. , m. t. zuber , d. h. rothman , n. schorghofer , and k. x. whipple , drainage basins and channel incision on mars , _ proceedings of the national academy of sciences usa _ , _ 99 _ , 17801783 , 2002 . baker , v. r. , spring sapping and valley network development , in _ groundwater geomorphology : the role of subsurface water in earth - surface processes and landforms _ , edited by c. higgins and d. coates , chap . 11 , geological society of america , boulder , colorado , 1990 . buffington , j. m. , and d. r. montgomery , a systematic analysis of eight decades of incipient motion studies , with special reference to gravel - bedded rivers , _ water resources research _ , _ 33_(8 ) , 19932029 , 1997 . dunne , t. , k. x. whipple , and b. f. aubry , microtopography and hillslopes and initiation of channels by horton overland flow , in _ natural and anthropogenic influences in fluvial geomorphology _ , pp . 2744 , american geophysical union , 1995 . howard , a. d. , groundwater sapping experiments and modelling , in _ sapping features of the colorado plateau , a comparative planetary geology field guide _ , edited by a. d. howard , r. c. kochel , and h. e. holt , pp . 7183 , nasa scientific and technical information division , washington , d.c . , 1988 . kochel , r. c. , a. d. howard , and c. mclane , channel networks developed by groundwater sapping in fine - grained sediments : analogs to some martian valleys , in _ models in geomorphology _ , edited by m. j. woldenberg , pp . 313341 , allen and unwin , boston , 1985 . kochel , r. c. , d. w. simmons , and j. f. piper , groundwater sapping experiments in weakly consolidated layered sediments : a qualitative summary , in _ sapping features of the colorado plateau , a comparative planetary geology field guide _ , edited by a. d. howard , r. c. kochel , and h. e. holt , pp . 8493 , nasa scientific and technical information division , washington , d.c . , 1988 . shields , a. , _ anwendung der hnlichkeitsmechanik und der turbulenzforschung auf die geschiebebewegung _ , heft 26 , mitteilung der preussischen versuchsanstalt fr wasserbau und schiffbau , berlin , germany , ( in german ) , 1936 .

we study channelization and slope destabilization driven by subsurface ( groundwater ) flow in a laboratory experiment . the pressure of the water entering the sandpile from below as well as the slope of the sandpile are varied . we present quantitative understanding of the three modes of sediment mobilization in this experiment : surface erosion , fluidization , and slumping . the onset of erosion is controlled not only by shear stresses caused by surfical flows , but also hydrodynamic stresses deriving from subsurface flows . these additional forces require modification of the critical shields criterion . whereas surface flows alone can mobilize surface grains only when the water flux exceeds a threshold , subsurface flows cause this threshold to vanish at slopes steeper than a critical angle substantially smaller than the maximum angle of stability . slopes above this critical angle are unstable to channelization by any amount of fluid reaching the surface .

when meteoroids enter the earth s atmosphere , they create long and narrow trails of ionized gas , which can scatter radio waves . the meteor radio echo theory finds its roots in studies of the ionosphere made toward the end of the twenties ( skellet @xcite ) . but only after the second world war , and the development of military radar , the correlation between radio echoes and meteor trails became clear ( hey & stewart @xcite ) . the first experiments explicitly devoted to meteor studies were carried out by pierce ( @xcite ) , who observed draconids during the night of 9 to 10 october 1946 . the first theories on the interaction of radio waves with meteors were due to lovell & clegg ( @xcite ) , kaiser & closs ( @xcite ) , herlofson ( @xcite ) . thereafter , between 1950 and 1960 , a lot of efforts in this field were undertaken , but after 1960 the interest quickly decayed . the state of knowledge was well exposed in a classical paper by sugar ( @xcite ) and the book by mckinley ( @xcite ) . toward the end of the eighties , the advent of digital technology renewed the interest in forward scattering as a useful tool for communication channels over the horizon ( see weitzen & ralston @xcite ) . today , meteor radars are widely used , even by amateur astronomers , because of low cost ( jenniskens et al . @xcite , yrjl & jenniskens @xcite ) . actually , the two main theories for radio echoes are due to poulter & baggaley ( @xcite ) , who deal with back scattering , and jones & jones ( @xcite , @xcite , @xcite ) , who deal with forward scattering . we can note that both theories _ do not consider the meteor as a plasma _ , but as a simply ionized gas , with a negligible collision frequency . this is an assumption quite common in work on radio echoes from meteors , except for herlofson ( @xcite ) , who first considered the meteor as a plasma . concerning the difference between back scattering and forward scattering , it is worth noting that the two systems use different types of radio waves . a forward scattering radar uses a continuous sine wave , while a back scattering one uses a pulsed wave . this influences the mathematical approach to the problem , but also physical theories . the pulse shape of the back scattering radar can be represented by a sum of several components with different frequencies . it follows that , during the propagation , the various components tend to change phase with respect to one another , which leads to a change in shape of the pulse . the dispersion relation thus depends on the frequency and can be expressed in a taylor series . a detailed treatment of the pulse propagation in dielectrics is not the subject of this paper : the interested reader can find useful mathematical tools in , for example , oughstun & sherman ( @xcite ) . here we want only to underline that we can not simply consider the signal emitted by a back scattering radar as sinusoidal : thus , the extension of the back scattering echo theory to forward scattering is not correct . the purpose of this paper is to settle some basic concepts in meteor physics and we leave a `` cook book '' approach to other articles . the interaction of sine waves ( for the sake of simplicity ) in the radio frequency range with meteoric plasma is investigated . we will see that the common assumption about meteors as a collisionless ionized gas does not have any physical ground . as a meteoroid enters the earth s atmosphere , it collides with air molecules . at the heights where most meteors ablate , the mean free path of the air molecules is about @xmath0 m. on the other hand , common meteoroid dimensions are of the order of @xmath1 m. this means that there is no hydrodynamic flow around the meteoroid and single air molecules impact on the body . if we consider a meteoroid a typical geocentric speed of 40 km / s , it can be found that air molecules impinge on the body with the same speed . the kinetic energy is about @xmath2 j ( 8 ev ) per nucleon : a nitrogen molecule then has an energy of about @xmath3 j or 230 ev . the impact energy is readily transformed into heat , which makes atoms evaporate from the meteoroid . the collisions between free atoms and air molecules produce heat , light and ionization , i.e. a meteor . since this transformation occurs throughout the flight , the meteoroid atoms are dispersed in a cylindrical channel along the path . the electron line density is proportional to initial mass of the meteoroid , because the air mass involved is negligible when compared to the meteoroid mass . after the escape , the first collisions of meteoroid atoms with air molecules take place at a distance of about one mean free path from the meteoroid path . it is useful to consider only the first collision to be important for ionization . this explains why the radio echo quickly rises to maximum amplitude and then slowly decays . at the moment of creation , all electrons are thus located inside a cylinder with a radius of about one mean free path . it is possible to calculate the debye length , a parameter that allows us to establish if the meteor is a plasma or simply an ionized gas . if the debye length is small when compared with meteor characteristic dimensions , then it is possible to speak of _ plasma _ , i.e. a gas where the electrostatic energy exceeds the thermal energy . in this case , if the thermal energy produces deviations from charge neutrality , a strong electric field arises in order to restore the charge neutrality . on the other hand , if the meteor characteristic dimensions are small compared to the debye length , this means that the thermal energy exceeds the electrostatic energy and there is no charge neutrality . in this case we have a simple ionized gas and we can not speak of a plasma . it is important to underline the difference between a plasma and an ionized gas : a plasma has some macroscopic properties , such as the langmuir frequency , which are absent in ionized gas . as known , the debye length is obtained by equating the thermal energy to the electrostatic energy and equals ( see mitchner & kruger @xcite ) : @xmath4 \label{e : debye}\ ] ] where @xmath5 [ f@xmath6 m@xmath7 is the vacuum dielectric constant , @xmath8 [ j@xmath6 k@xmath7 the boltzmann constant , @xmath9 [ c ] the elementary electric charge , @xmath10 [ k ] the temperature and @xmath11 [ m@xmath12 the electron volume density . for the sake of simplicity , in this paper , the electron volume density is used , although in radar studies on meteors the electron line density is used . however , it is easy to obtain the electron line density noting that the meteor trail is like a long circular cylinder . common radio meteors are characterized by electron volume densities between @xmath13 m@xmath14 and about @xmath15 m@xmath14 ( sugar @xcite ) and temperatures between 1000 and 5000 k ( bronshten @xcite , borovika @xcite , borovika & zamorano @xcite ) . by substituting these values in eq . ( [ e : debye ] ) it is possible to calculate the maximum value of the debye length , which is about 0.01 m. comparing it to the lower value of the meteor initial radius ( 0.1 m ) , _ it is possible to say that the meteor is a plasma and not an ionized gas_. since any slight distortion of the plasma from a condition of electrical neutrality gives rise to strong restoring forces , we have to consider how fast these forces act . writing the equation of motion for the electrons , it is possible to find that they oscillate with a characteristic angular frequency ( see mitchner & kruger @xcite ) : @xmath16 \label{e : pfreq}\ ] ] where @xmath17 [ kg ] is the electron mass . it is possible to calculate eq . ( [ e : pfreq ] ) for any type of charged particle , but as electrons move faster than ions , they give the main contribution to plasma frequency and other contributions can be neglected . the characteristic angular frequency is often called the langmuir frequency . now , it is possible to understand the different behaviour of radio echoes . already kaiser & closs ( @xcite ) noted a different behaviour of meteors due to electron line density . they introduced the names _ underdense _ and _ overdense _ to characterize meteors with electron line densities , respectively , below or above the value @xmath18 m@xmath19 . in underdense meteors , the electron density is sufficiently weak to allow the incident wave to propagate along the ionized gas and the scattering is done independently by each individual electron . on the other hand , when the meteor is overdense , the electron density is sufficient to reflect totally the incident wave . it has been usual to resort to a simple model in which the trail is regarded as a totally reflecting cylinder . now , comparing the electromagnetic wave frequency to the plasma frequency ( fig . [ fig1 ] ) we can distinguish two regions depending on whether the plasma frequency is higher or lower than the radar frequency . in the first case , the charges in the plasma have sufficient time to rearrange themselves so as to shield the interior of the plasma from the electromagnetic field ( overdense meteor ) . in the second case , since the radar frequency is higher the than plasma frequency , the incident wave can propagate along the plasma ( underdense meteor ) . moreover , space charges can appear . thus , a first method to distinguish underdense from overdense meteors , is by equating plasma frequency to radar frequency . we stress that this is valid for forward scattering radar , which use continuous waves , while back scattering radar see the meteors in a different way , because they use short pulses . in a plasma the magnetic induction @xmath20 and the magnetic field strenght @xmath21 are almost always treated by the same relationship as in free space . in order to account for polarizability associated with bound electrons of neutral particles and ions , it is necessary to calculate the dielectric constant of the plasma . however it is possible to see that , in the radar frequency range , the contribution due to polarizability is negligible . then , the relation between the electric field strenght @xmath22 and electric induction @xmath23 can also be treated as in free space . the generalized ohm law serves to relate @xmath24 and @xmath22 : @xmath25 where @xmath26 [ s@xmath7 is the mean collision frequency and @xmath27 [ s@xmath6 m@xmath7 is the electric conductivity . the meteoric plasma can be considered an isotropic medium since the electron gyrofrequency is much less than the radio wave frequency and thus the effect of the geomagnetic field can be neglected . as stressed in sect . 1 , forward scattering radars use continuous waves : taking into account that the time dependence of the electromagnetic field is @xmath28 , and using basic equations and standard identities , it is possible to combine maxwell s equations in the following form : @xmath29 @xmath30 these are equations for the electric and magnetic fields in a meteoric plasma . it must be noted that eqs . ( [ e : e1 ] ) and ( [ e : h1 ] ) are useful in the underdense region , where there are space charges . on the other hand , in the overdense region , when the plasma frequency is higher than the radar frequency , charge neutrality is present and then , the right side term of eq . ( [ e : e1 ] ) vanishes , according to gauss law , when the net charge density is zero : @xmath31 the solutions of eq . ( [ e : e2 ] ) are nonuniform harmonic plane waves of this type : @xmath32 where the wave vector in eq . ( [ e : solu ] ) has the form : @xmath33 the two real vectors @xmath34 and @xmath35 generally point in different directions . if we substitute eqs . ( [ e : solu ] ) and ( [ e : wv ] ) in eq . ( [ e : e2 ] ) , it is possible to find that : @xmath36 taking into account eqs . ( [ e : pfreq ] ) and ( [ e : ohm ] ) and recalling that @xmath37 ( @xmath38 : light velocity in vacuum ) , it is possible to rearrange eq . ( [ e : wave1 ] ) : @xmath39\right\ } = \\[0.5 cm ] = \frac{\omega^{2}}{c^{2}}(\kappa_{r}+i \kappa_{i } ) \end{array } \label{e : wave2}\ ] ] the quantity @xmath40 is often identified as the _ complex dielectric constant _ for the medium . from eqs . ( [ e : wv ] ) and ( [ e : wave2 ] ) , when vectors @xmath34 and @xmath35 point to the same direction , it is possible to define the real numbers @xmath41 and @xmath42 as the _ attenuation _ and _ phase _ constants respectively : @xmath43 @xmath44 if the collision frequency is negligible , i.e. @xmath45 , then eq . ( [ e : wave2 ] ) can be reduced to : @xmath46 that is , the equation used until now . if @xmath47 , then @xmath8 is a real number and the incident wave propagates without attenuation . on the other hand , if @xmath48 , then @xmath8 is a purely imaginary number and the incident wave is totally reflected . in all work on the theory of radio echoes from meteor trails , the collision frequency is considered 2 or 3 orders of magnitude lower than the radar frequency ( a detailed analysis of past radio echo theories can be found in foschini @xcite ) . this is the typical collision frequency of electrons with air molecules at the heights where meteors ablate . work on collisions in meteor trails generally deals with ionization and excitation , in order to know processes during trail formation ( massey & sida @xcite , sida @xcite , baggaley @xcite ) . other authors are interested in diffusion and thus study attachment , recombination and other chemical reactions between the atmosphere and meteoric plasma ( baggaley @xcite , baggaley & cummack @xcite , baggaley @xcite , jones & jones @xcite ) . in fig . [ fig2 ] a typical overdense radio echo from a meteor trail is plotted . when the `` flat top '' ( p ep ) is reduced to zero , we have an underdense echo . collision processes during the trail formation ( from start to peak , s p ) are ionization and excitation , caused by air molecules impinging on the meteoroid . during echo decay , from end peak ( ep ) to end decay ( ed ) , attachment and recombination are dominant processes , owing to the diffusion of the plasma in the surrounding atmosphere . during the _ plateau _ ( p ep ) the trail can be considered in _ local thermodynamic equilibrium _ , when matter is in equilibrium with itself , but not with photons ( mitchner & kruger @xcite ) . the plasma in the trail is transparent to radiation at some optical frequencies , which escapes from the meteor and form the light we observe . for a plasma in local thermodynamic equilibrium it is still possible to employ the boltzmann and saha equations ( see mitchner & kruger @xcite ) , even though complete thermodynamic equilibrium does not prevail . now , thermal ionization becomes the main process and metals , with low ionization energy , drive this phase . it must be noted that in meteoroids there are some few per cent of alkaline and alkaline earth metals , such as na , k , ca and mg . these data are obtained from studies on meteorites ( mason @xcite , millman @xcite , wasson @xcite ) and on bright fireball spectra ( bronshten @xcite , borovika @xcite , borovika & zamorano @xcite ) . because of their low ionization energy , at typical radar meteor temperatures ( 10005000 k ) , these metals easily ionize and thus , they are the main contributors to thermal ionization . a small percentage of one of these metals suffices to produce a huge amount of electrons . in the calculation of collision frequency of electrons with other particles , it must be taken into account that , because of the electrostatic field , ions have a cross section larger than that of atoms and molecules . thus , electron ion collisions must be considered , and not electron air molecule collisions , since the former are more frequent than any other . moreover , in calculating the electron ion collision frequency we have considered potassium , because it has a low ionization energy ( 4.34 ev ) . this choice could be questionable because potassium is scarcely present in meteor spectra ( bronshten @xcite ) or even absent ( borovika @xcite , borovika & zamorano @xcite ) . however , this metal is present in almost all meteoritic specimens ( mason @xcite ) and other studies on meteor shower have reported its presence . specifically goldberg & aikin ( @xcite ) , using a rocket borne ion mass spectrometer , detected the presence of potassium ion in @xmath42taurids . the absence of potassium in meteor spectra is due to the low efficiency of light emission processes in this type of atom : it is well known that the most intense spectral line of potassium is in the infrared range ( 766.49 nm ) . it is worth noting that also iron and sodium , two of the commonest elements in meteor spectra , could be absent in some cases ( see millman @xcite ) . therefore , we can consider a potassium percentage of about 1% of meteoroid mass . it is very difficult to find experimental values for electron ion collision cross sections of various species , in the temperature range of meteors . experimental studies are mainly devoted to excitation and ionization cross sections , in order to compare results with data from spectra ( neff @xcite , boitnott & savage @xcite , @xcite , @xcite , savage & boitnott @xcite ) . rosa ( @xcite ) gives collision cross sections of k , k@xmath49 and some atmospheric gases , such as o@xmath50 , in the temperature range from 2000 k to 3500 k. it is important to note that the k@xmath49 cross section is about 3 ( _ three _ ) orders of magnitude larger than those of other species . we have further supposed that only a single ionization is possible , i.e. we have a reaction such as : @xmath51 we would like to stress that we are only interested in the analysis of meteoric plasma in a steady state condition ( during the _ plateau _ ) and we do not actually consider echo decay ( recombination and other chemical processes ) or trail formation ( collisional ionization and excitation ) . thus , it is possible to use a simplified model of collision frequency ( mitchner & kruger @xcite ) , viz . : @xmath52 where @xmath53 is the electron ion collision cross section and @xmath54 is the electron mean velocity with respect to ions ( it is assumed to be about equal to the electron mean thermal velocity ) , and @xmath55 is the ion volume density . in our case , owing to eq . ( [ e : reaction ] ) , @xmath56 . now , it is possible to observe that @xmath57 can not be negligible anymore , especially for a high electron volume density ( fig . [ fig3 ] ) . if we consider the contributions of other species , it is necessary to sum the various collision frequencies obtained using eq . ( [ e : collf ] ) . now , in the overdense regime , we have to consider the dispersion relation in eq . ( [ e : wave2 ] ) and not in eq . ( [ e : wave3 ] ) . it is worth noting that eq . ( [ e : wave3 ] ) is still valid in the underdense regime because at high electron densities only the collisions are not negligible . it is possible to define , from eq . ( [ e : wave2 ] ) , a _ critical electron density _ in order to separate the underdense from the overdense regime and this occurs when : @xmath58 in order to make some example , we can assume a mean temperature of 3000 k. if we consider the cnr radar facility ( cevolani et al . @xcite ) , which has a radar frequency of 42.7 mhz , we obtain a critical electron density of @xmath59 m@xmath14 . on the other hand , if we consider the m. de meyere s radar ( see c. steyaert , radio meteor obs . bulletin , ftp://charlie.luc.ac.be/pub/icaros/rmob/ ) , which has a frequency of 66.51 mhz , we obtain a critical electron density of @xmath60 m@xmath14 . below this critical density ( underdense meteors ) , the wave number is real and the incident wave propagates into the meteoric plasma with negligible attenuation . we obtain well known results about underdense meteors . above this value , waves which were previously excluded ( see eq . ( [ e : wave3 ] ) ) , can now propagate , but are strongly attenuated ( figs . [ fig4 ] and [ fig5 ] ) . from fig . [ fig5 ] , it is possible to see that the presence of a collision frequency comparable with the radar frequency determines a rise of the real part of the wave vector modulus . physically , this behaviour may be understood on the basis that collisions subtract energy from plasma oscillations , allowing the incident wave to penetrate , even if strongly attenuated . we can observe two types and two sub types of behaviour . we have two main classes of meteors , _ overdense _ and _ underdense _ , according to whether the plasma frequency is higher or lower than the radar frequency respectively , depending on the initial electron volume density . moreover , overdense meteors are divided into two sub classes ( see fig . [ fig5 ] ) : for an electron volume density between the critical density and about @xmath61 m@xmath14 , the real part of the wave vector is negligible and the incident wave is totally reflected ( _ overdense i _ ) . for a higher electron volume density , collisions allow the propagation of the incident wave , even if with a strong attenuation ( _ overdense ii _ ) . total reflection , in the overdense i range , allow us to make some useful approximations in calculating the attenuation of radio waves after the forward scattering . if the plasma had a definite boundary and a uniform electron density , then reflection at its surface would be simple because the gas would act as a dielectric with a complex dielectric constant . but a meteor trail does not have a definite boundary and thus , the incident wave penetrates a little into the plasma before reaching the density necessary to allow total reflection . reflection occurs gradually , as in a mirage . taking into account this fact , it is possible to make some approximations . we can consider a simple geometry , as shown in fig . [ fig6 ] , and then use the definition of the attenuation @xmath62 in decibel units : @xmath63 \label{e : decib}\ ] ] where subscripts @xmath64 and @xmath65 stand for incident and reflected wave . we substitute eqs . ( [ e : solu ] ) and ( [ e : wv ] ) in eq . ( [ e : decib ] ) and , taking into account that the amplitude of a totally reflected wave is equal to the amplitude of the incident wave , we can obtain an attenuation value of about @xmath66 , where @xmath67 is the path of the wave into the plasma . from fig.[fig6 ] we can see that : @xmath68 where @xmath69 is the penetration depth and @xmath70 is the incidence angle . now , if we consider something similar to the `` skin effect '' in metals , we have @xmath71 . then , eq . ( [ e : decib ] ) becomes : @xmath72 \label{e : decib1}\ ] ] the attenuation is simply a function of the angle of incidence and this is compatible with results obtained by forsyth & vogan ( @xcite ) , who predicted an attenuation proportional to @xmath73 . it is necessary to stress that eq . ( [ e : decib1 ] ) is valid only for type i overdense meteors , where total reflection is allowed . in order to make some comparison with experimental data , we can consider a sample of radio echoes obtained from the cnr forward scattering radar facility ( e.g. foschini et al . @xcite , poruban et al . attenuation of reflected waves , in the overdense regime , ranges from -67 to -47 db . by using the radar equation ( kingsley & quegan @xcite ) , attenuation values range from -155 to -125 db , which gives a strong discrepancies with what is observed . to obtain from eq . ( [ e : decib1 ] ) the attenuation values requested , it is necessary to use incidence angles from 68@xmath74 to 75@xmath74 . this is a very good approximation because the cnr radar has the main beam with about 15@xmath74 elevation angle : then 75@xmath74 is just the complementary angle . for overdense type i meteors , i.e. those with a _ plateau _ , eq . ( [ e : decib1 ] ) can be used to calculate the meteor height . taking into account the cnr radar geometry ( transmitter receiver distance is about 700 km , see cevolani et al . @xcite ) , it is possible to obtain a range of overdense type i meteor heights from about 94 to 141 km . this is very important , because the height is obtained without taking into account the diffusion coefficient , which is very uncertain . this requires further investigations , which are currently being carried out . in this paper , we settled some basic concepts in meteor physics . we have dealt with the interaction of sine waves , in the radio frequency range , with meteoric plasma . attention is drawn to some macroscopic characteristics of a meteoric plasma and it is shown that the electron ion collision frequency is not negligible , as commonly thought . it is possible to define two meteor classes ( overdense and underdense ) according to whether the plasma frequency is higher or lower than the radar frequency respectively . overdense meteors are divided into two sub classes ( i and ii ) , depending on the ratio between collision frequency and radar frequency . taking into account that the meteoric plasma does not have a definite boundary , a simple formula for the calculations of radio wave attenuation after the forward scattering is also presented . this formula allows us to calculate the meteor height , without taking into account the diffusion coefficient . further questions can be put : in this work , potassium ion is considered , but other studies can be carried out in order to know the impact of other alkaline and alkaline earth metals , such as na , ca and mg .

in this paper , a meteoric plasma is analyzed from a physical viewpoint , with particular emphasis on its interaction with radio waves . the attention is drawn to some macroscopic characteristics of a meteoric plasma and it is shown that the electron ion collision frequency is not negligible , as commonly thought .

considerable progress have been made over the last ten years or so , in our understanding of the spin structure of the nucleon . this is essentially due to a better determination of the polarized structure functions @xmath0 , from polarized deep - inelastic - scattering ( _ dis _ ) on different targets ( hydrogen , deuterium , helium-3 ) . however these fixed polarized targets experiments @xcite , performed at _ cern _ , _ desy _ and _ slac _ , cover only a limited kinematic region , that is @xmath1 , with the corresponding average @xmath2 between @xmath3 and @xmath4 . in spite of the constant progress realized in the accuracy of the data , they can still be described , non uniquely , in terms of several sets of polarized parton distributions . in particular , sea quark , antiquark and gluon distributions remain highly ambiguous . the restricted @xmath5 range accessible by the data makes also rather difficult , sensible tests of the @xmath5 evolution , predicted by recent higher order _ qcd _ calculations . moreover it is not possible to obtain a good flavor separation , to isolate the contribution of each quark to the nucleon spin . polarized hadronic collisions , which are another way to investigate this research field , have accomplished little progress due to the scarcity of the data in appropriate kinematic regions , and a low energy range , so far accessible to very few dedicated experiments . let us recall that the highest energy for studying polarized @xmath6 ( @xmath7 ) collisions has been obtained at fermilab by the _ e704 _ experiment @xcite with a @xmath8 polarized proton ( antiproton ) beam on a fixed target , that is @xmath9 . this situation will change drastically soon when the _ rhic _ facility at _ bnl _ will start running , by 1999 or so , part of the time as a polarized @xmath6 collider . a vast spin programme will be undertaken by the two large detectors _ phenix _ and _ star _ , which will operate at _ rhic _ and also by the @xmath10 experiment , dedicated to @xmath6 elastic and total cross sections . before we go on and explain very briefly what will be done , let us recall three key parameters , which will be crucial to answer some of the very challenging questions . the proton beam polarization @xmath11 will be maintained at the level of 70% , in both _ longitudinal _ and _ transverse _ directions , the center - of - mass energy @xmath12 will be ranging between @xmath13 and @xmath14 and at its maximum value , the luminosity is expected to reach _ l_=@xmath15 . the _ siberian snakes _ magnets which preserve the degree of polarization in the _ rhic _ main rings and the _ spin rotators _ which select the beam spin direction , are under construction thanks to a substantial financial contribution from the japanese institute _ riken _ in collaboration with _ bnl_. the high luminosity will allow very copious effective yields for different reactions ( @xmath16 , jet , @xmath17 production , etc ... ) and therefore the measurement of the corresponding spin asymmetries will be made to a very good level of accuracy , in the kinematic regions relevant for _ qcd _ spin tests . the spin programme at _ rhic _ will provide answers to fundamental questions which will be listed now in turn . in the next section we will recall some basic definitions of the helicity asymmetries . section 3 will be devoted to prompt photon production and jet production , which will allow the first direct determination of the gluon helicity distribution @xmath18 inside a polarized nucleon . next we will show in section 4 , how antiquark helicity distributions @xmath19 can be isolated in @xmath17 production , which leads also to the _ u - d _ flavor separation . this has been done , rather inaccurately , in semi - inclusive _ dis_. from transversely polarized proton beams , as we will see in section 5 , it is possible to make the first measurement of the transversity distributions @xmath20 in drell - yan lepton pair production . finally , in section 6 we will indicate the relevance of the parity violating asymmetry in single jet production . it might provide a clean signature for new physics and , as an example , we will consider the possible effects of a quark - quark contact interaction . fundamental interactions at short distances which are explored in high energy hadronic collisions , involve hard scattering of quarks , antiquarks and gluons . let us consider the general hadronic reaction @xmath21 where @xmath22 , in the cases we will consider below , is either a photon , a @xmath23 , a @xmath17 or a single - jet . in the hard scattering kinematic region , the cross section describing ( 1 ) reads in the _ qcd _ parton model , provided factorization holds , as @xmath24 . % \end{array}\ ] ] the summation runs over all contributing parton configurations , the @xmath25 s are the parton distributions , directly extracted from _ dis _ for quarks and antiquarks and indirectly for gluons . here d@xmath26 is the cross section for the interaction of two partons @xmath27 and @xmath28 which can be calculated perturbatively , some of which , at the lowest order , are given in ref.@xcite . if we consider the reaction ( 1 ) with _ both _ initial hadrons , @xmath29 and @xmath30 longitudinally polarized , one useful observable is the _ double _ helicity asymmetry @xmath31 defined as @xmath32 when we assume parity conservation , i.e. d@xmath33 = d@xmath34 . its explicit expression , assuming factorization , is given by @xmath35 , \ ] ] where d@xmath36 is given by eq.(2 ) and @xmath37 denotes the corresponding subprocess double asymmetry for initial partons @xmath27 and @xmath28 . the @xmath38 s are defined as @xmath39 where @xmath40 are the parton distributions in a polarized hadron with helicity either parallel ( + ) or antiparallel ( - ) to the parent hadron helicity . recall that the unpolarized distributions are @xmath41 and @xmath38 measures how much the parton @xmath42 `` remembers '' the parent hadron helicity . if the subprocess involves parity violating interactions , one can consider another interesting observable which requires only _ one _ initial hadron polarized , that is the _ single _ helicity asymmetry @xmath43 , defined as @xmath44 in addition , if both @xmath29 and @xmath30 are polarized one can also have two _ double _ helicity parity violating asymmetries defined as @xmath45 which can be simply related to @xmath43 @xcite . several sets of polarized parton densities @xmath46 ( @xmath47 ) have been proposed in the recent literature [ 5 - 10 ] . using some of these parametrizations to calculate helicity asymmetries for various processes , we will show how the _ rhic - bnl _ spin programme will be able , in particular , to pin down the polarized helicity distributions which remain badly constrained by polarized _ _ experiments . the cross section for direct photon production on @xmath6 collisions at high @xmath48 is considered as one of the cleanest probe of the unpolarized gluon distribution @xmath49 . this is partly due to the fact that the photon originates in the hard scattering subprocess and is detected without undergoing fragmentation . moreover in @xmath6 collisions the quark - gluon compton subprocess @xmath50 dominates largely and the quark - antiquark annihilation subprocess @xmath51 can be neglected . consequently the double helicity asymmetry @xmath52 ( see eq.(4 ) ) , which involves in this case only one subprocess , becomes particularly simple to calculate and is expected to be strongly sensitive to the sign and magnitude of @xmath18 . for the compton subprocess , @xmath53 whose expression at the lowest order is given in ref.@xcite , is always positive and such that @xmath54 where @xmath55 is the center of mass ( c.m . ) angle in the subprocess . in fig . 1 we show a complete comparison of the results we obtained for the @xmath56 distribution of @xmath57 , at pseudo rapidity @xmath58 , using the different sets of polarized parton distributions we have mentioned in the previous section , with a leading order @xmath5 evolution . actually we find that the smallest predictions are obtained from the sets ref.@xcite and ref.@xcite which have the smallest @xmath59 . the predictions differ substantially at large @xmath48 , which corresponds to the region , say around @xmath60 or so , where the distributions @xmath61 have rather different shapes . we have also indicated the expected statistical errors based on an integrated luminosity @xmath62 pb@xmath63 at @xmath64 gev , for three months running time . we have evaluated the event rates in the pseudo - rapidity gap @xmath65 , assuming a detector efficiency of 100% and for a @xmath56 acceptance @xmath66 gev / c . we see that up to @xmath67 gev / c or so , @xmath52 will be determined with an error less than 5% which therefore will allow to distinguish between these different possible @xmath68 . for very large @xmath56 , the event rate drops too much to provide any sensitivity in the determination of @xmath61 . the pseudo - rapidity distribution of @xmath69 at a fixed @xmath56 value has been also calculated in ref.(@xcite ) and it shows a systematic increase of @xmath52 for higher @xmath70 . inclusive jet production is also a physics area where one can learn a lot about parton densities and , considering the vast amount of unpolarized existing data , it has been regarded as an important qcd testing ground . event rates are much larger than for prompt photon production , but there is a drawback because many subprocesses are involved , unlike in the previous case . in principle one should take into account gluon - gluon ( @xmath71 ) , gluon - quark ( @xmath72 ) and quark - quark ( @xmath73 ) scatterings . although these subprocesses cross sections are not so much different , after convolution with the appropriate parton densities ( see eq.(2 ) ) , they lead to very distinct contributions to the hadronic spin - average cross section . here we will restrict ourselves to the double helicity asymmetry @xmath74 for single jet production and in order to clarify the interpretation of our results below , let us recall some simple dynamical features . in the very low @xmath75 region , say @xmath76 gev / c or so , @xmath71 scattering dominates by far , but its contribution drops down very rapidly with increasing @xmath75 . in the medium @xmath75 range , say 20 gev / c @xmath77 gev / c or so , @xmath72 scattering dominates and then decreases for large @xmath75 , to be overcome by @xmath73 scattering . of course these are rough qualitative considerations and accurate numerical estimates for the relative fractions of these different contributions depend strongly on the parton densities one uses.let us look at @xmath78 and , from the above discussion , we see that in the medium @xmath75 range where @xmath72 scattering dominates , @xmath78 should have a trend similar to @xmath79 , with perhaps a magnitude reduced by a factor two , since about half of the jet cross section is due to @xmath71 and @xmath73 scatterings . this is what we see approximately in fig.2 , where we present the numerical results for @xmath78 at @xmath80 gev , which should be compared to fig.1 . we have also indicated the statistical errors which are extremely small in this case , because of the huge event rates . let us now consider , for the reaction @xmath83 , the parity violating single helicity asymmetry @xmath43 defined in eq.(6 ) . in the standard model , the @xmath84 gauge boson is a purely left - handed object and this asymmetry reads simply for @xmath17 production @xmath85 assuming the proton @xmath29 is polarized . here we have @xmath86 , @xmath87 and @xmath88 . for @xmath89 production the quark flavors are interchanged ( @xmath90 ) . the calculation of these asymmetries is therefore very simple and the results are presented in figs.3,4 at @xmath80 gev , for different sets of distributions . as first noticed in ref.@xcite , the general trend of @xmath43 can be easily understood as follows : at @xmath91 one has @xmath92 evaluated at @xmath93 , for @xmath94 one has @xmath95 evaluated at @xmath96 and for @xmath97 one has @xmath98 evaluated at @xmath99 . therefore these measurements will allow a fairly clean flavor separation , both for quarks and antiquarks , for some interesting ranges of @xmath100 values . we see in fig.3 that @xmath101 , which is driven by the @xmath102 and @xmath103 polarizations , leads to similar predictions for all cases . this is mainly due to our knowledge of @xmath104 , except for @xmath105 where it comes out to be larger for the set proposed in ref.@xcite . in fig.4 , for @xmath106 which is sensitive to the @xmath107 and @xmath108 polarizations , we see that the various predictions lead to the same general trend , with some differences in magnitude due to a large uncertainty in the determination of @xmath109 . also in ref.@xcite , one has assumed a larger negative @xmath110 , which is reflected in the behaviour near @xmath94 . the statistical errors have been calculated with a rapidity resolution @xmath111 and taking into account only the events from the leptonic decay modes . they are smaller for @xmath112 production which has larger event rates . the existence of @xmath113 for quarks ( @xmath114 for antiquarks ) was first detected in a systematic study of the drell - yan process with polarized beams @xcite and some of its relevant properties were discussed later in various papers @xcite . pretty much like @xmath115 and @xmath116 , the transversity distributions @xmath117 are of fundamental importance for our understanding of the nucleon structure and they are all leading - twist distributions . due to scaling violations , these quark distributions depend also on the scale @xmath118 and their @xmath5-behavior is predicted by the _ qcd _ evolution equations . they are different in the three cases but , due to lake of time , we will not discuss here this important question@xcite . we recall that @xmath119 ( or @xmath120 ) are not simply accessible in _ dis _ because they are in fact chiral - odd distributions , contrarely to @xmath121 and @xmath81 which are chiral - even @xcite . however they can be extracted from polarized drell - yan processes with two transversely polarized proton beams . for lepton pair production @xmath122 @xmath123 mediated by a virtual photon @xmath124 , the double transverse - spin asymmetry @xmath125 defined , similarly to eq.(3 ) , as @xmath126 , reads @xmath127 where @xmath128 is the partonic asymmetry calculable in perturbative _ qcd _ and @xmath129 is the dilepton mass . the rapidity @xmath130 of the dilepton is @xmath131 , and for @xmath91 one has @xmath132 . note that this is a leading - order expression , which can be used to get a first estimate of @xmath133 from different theoretical results for @xmath134 and @xmath135 . if the lepton pair is mediated by a @xmath23 gauge boson , one has a similar expression for @xmath136 @xcite , namely @xmath137 where @xmath138 and @xmath139 are the vector and axial couplings of the flavor @xmath140 to the @xmath23 . however in the case of @xmath17 production one expects @xmath141 , because the @xmath84 gauge boson is a pure left - handed object ( _ i.e. _ , @xmath142 ) , which does not allow a left - right interference effect associated to the existence of @xmath143 @xcite . we show in fig.5 some predictions , at leading and next to leading orders , for @xmath144 , together with some expected statistical errors , where one sees a characteristic effect in the @xmath23 mass region . the asymmetry is only a few percents , due to the smallness of @xmath114 , but this is a real challenge for the experiment . let us consider again one - jet inclusive production . as discussed in section 3 , the cross section is dominated by the pure _ qcd _ subprocesses @xmath71 , @xmath72 and @xmath73 scatterings , but the existence of the electroweak ( _ ew _ ) interaction , via the effects of the @xmath145 gauge bosons , adds to it , a small contribution . consequently , the parity violating asymmetry @xmath146 defined in eq.(7 ) and resulting from the _ qcd - ew _ interference is non - zero , as shown in fig.6 ( see curve _ sm _ ) and has a small structure near @xmath147 . now if we introduce a new contact interaction , belonging purely to the quark sector and normalized to a certain compositeness scale @xmath148 , under the form @xmath149 where @xmath150 is a quark doublet and @xmath151 . if parity is not conserved @xmath152 and we show in fig.6 , how the _ sm _ prediction will be affected by such a new interaction assuming @xmath153 . as expected , the errors are large in the high @xmath154 region , but if the observation of such a signal is confirmed , it will be extremely important . 99 g. altarelli , these proceedings . e.g. _ , fnal - e704 collaboration , a. bravar _ et al _ , phys.rev.lett.*77 * , 2626 ( 1996 ) and references therein . c. bourrely , j. soffer , f.m . renard and p. taxil , phys . reports , * 177 * , ( 1989 ) 319 , and the errata preprint cpt - 87/p.2056 ( january 1991 ) ; see also p. taxil , riv . nuovo cimento , vol.16 ( 1993 ) , no . c. bourrely and j. soffer , phys . b314 * , 132 ( 1993 ) . c. bourrely and j. soffer , nucl . b445 * , 341 ( 1995 ) . c. bourrely , f. buccella , o. pisanti , p. santorelli and j. soffer , preprint cpt-96/pe.3327 , dsf - t96/17 , to appear in the proceedings of the fundamental structure of matter , ouranoupoulis ( greece ) may 28 - 31 ( 1997 ) ( editor a. nicolaidis ) . t. gehrmann and w.j . stirling , phys . rev . * d53 * , 6100 ( 1996 ) . m. glck , e. reya , m. stratmann and w. vogelsang , phys . rev . * d53 * , 4775 ( 1996 ) . cheng and h.h . liu , phys . rev . * d53 * , 2380 ( 1996 ) . g. ridolfi , these proceedings . j. soffer and j.m . virey , nucl . b509 * , 297 ( 1998 ) . ralston and d.e . soper , nucl . phys . , * b152 * , 109 ( 1979 ) . x. artru and m. mekhfi , z. phys . , * c45 * , 669 ( 1990 ) . cortes , b. pire and j.p . ralston , z. phys . * c55 * , 409 ( 1992 ) . jaffe and x. ji , phys . lett . , * 67 * , 552 , ( 1991 ) ; nucl . * b375 * , 527 ( 1992 ) . c. bourrely , j. soffer and o.v . teryaev , preprint cpt-97/p.3538 , ( phys.lett . * b * , to appear ) . o. martin , a. schfer , m. stratmann and w. volgelsang , preprint cern - th/97 - 270 ( october 1997 ) . c.bourrely and j. soffer , nucl . b423 * , 329 ( 1994 ) . p. taxil and j.m . virey , phys . lett . * b364 * , 181 ( 1995 ) .

the _ rhic _ facility at _ bnl _ will be operating soon , part of the year , as a polarized proton - proton collider . this will allow the undertaking of a vast spin physics programme , mainly by the two large detectors _ phenix _ and _ star_. we review some theoretical aspects of this research programme which will allow , firstly to improve our present knowledge on polarized quark , gluon and sea distributions in a nucleon , secondly to perform novel _ qcd _ spin tests and finally , perhaps , to uncover some new physics .

the recent demonstration of artificial materials ( metamaterials ) with the left oriented triplet of electric @xmath0 , magnetic @xmath1 and wave vector @xmath2 of electromagnetic field @xcite stimulated studies of nonlinear optical phenomena in such materials @xcite . nonlinear dynamics of extremely short optical pulses in left - handed materials was the subject of particular interest in several recent papers @xcite . the first experimental realization of the left - handed property based on the resonant response of the artificial material to both electric and magnetic fields was described in @xcite . to mention just one of the latest experimental achievements , valentine et al @xcite were able to observe the negative refractive index in the balk material in the _ optical _ range . a theoretical description of the electromagnetic wave interaction with such double resonance materials ( drm ) was considered in @xcite . presence of two frequency intervals with different orientation of @xmath3 triplets is a characteristic feature of such materials . most of the studies of electromagnetic pulse propagation in drm has been conducted in the slowly varying envelope approximation . on the other hand , there is a broad area of nonlinear optical phenomena taking place in the limit of extremely short pulses , when the slowly varying envelope approximation is not valid @xcite . the case of extremely short electromagnetic pulses offers a new type of nonlinear interaction , when different frequency components of electromagnetic pulses have different orientations of the @xmath4 triplets . the design of currently available drm is based upon the use of embedded metallic structures whose size is on the same order as the spatial size of an extremely short electromagnetic pulse . therefore a theoretical and numerical investigation of the currently existing drm would require 3d computer simulation on maxwell s equations that takes into account the strong inhomogeneity of composite materials . recently , there have been introduced some qualitatively different approaches to design of drm , including the use of multilevel atoms @xcite ; the latter gives rise to a spatially homogeneous medium . possibilities of experimental realizations of such an approach were recently discussed in @xcite . as a first step in the theoretical investigation of electrodynamics of homogeneous drm in this paper we study a simple model of a homogeneous doubly - resonant medium . even under such simplification , dynamics of extremely short pulses turn out to be quite complex . the system of equations that describe interaction of coherent light with a medium consisting of molecules ( considered as harmonic oscillators ) is known as the maxwell - lorentz model @xcite . in this work we use a version of the maxwell - lorentz system that is extended to account for simultaneous magnetic and electric resonances , with the magnetic susceptibility being linear , while the electric polarization being nonlinear . consider the general form maxwell s equations : @xmath5 for simplicity , we consider transverse electromagnetic plane waves propagating along the @xmath6-axis with the electric field @xmath7 and the magnetic field @xmath8 then the maxwell equations transform to the scalar form : @xmath9 which leads to @xmath10 the system ( [ maxwell ] ) must be closed by two additional equations describing the interaction of the electric and magnetic fields with the dr medium . as usual , it is convenient to avoid the @xmath11-factors by changing the units for @xmath12 and @xmath13 : @xmath14 @xmath15 in the sequel we drop the tildes over @xmath12 and @xmath16 assume that the medium polarization is defined by the plasma oscillation of electron density , @xmath17 here @xmath18 is an effective parameter characterizing polarizability of the medium ; in the case of metallic nanostructures it would be the effective plasma frequency . to account for the dimensional quantization due to the confinement of the plasma in nanostructures one should include the additional term @xmath19 , where @xmath20 is the frequency of dimensional quantization . we take into account nonlinearity in the lowest order of @xmath13 , which is @xmath21 . a more accurate analysis , based on a quantum mechanical approach @xcite and experimental measurements @xcite confirms validity of this assumption . therefore we consider the modeling equation for the medium polarization dynamics in the following form@xmath22 where @xmath23 is a constant of anharmonisity . to account for magnetic resonances we use the standard model @xcite @xmath24 here @xmath25 is a constant characterizing magnetization . we represent equations ( [ maxwell ] ) , ( [ polarization ] ) and ( [ magnetization ] ) in a dimensionless form by introducing @xmath26 ( @xmath27 is the characteristic time ) , @xmath28 ( @xmath29 is the characteristic distance ) , @xmath30 ( @xmath31 is the maximal achievable medium polarization ) . it is convenient to normalize remaining variables as follows : @xmath32 , @xmath33 , @xmath34 . the system of dimensionless equations then takes the following form : @xmath35 where @xmath36 , @xmath37 , @xmath38 . the system possesses the following conserved quantity:@xmath39 ^{2}\right ] d\eta=0\nonumber\end{gathered}\ ] ] which is positive - definite for @xmath40 for the traveling - wave solutions the conservation relation ( [ conserve ] ) yields conservation of electromagnetic energy @xmath41 ( see @xcite @xmath42for details ) . a natural question arises is whether the system in ( [ dimensionless : system ] ) possesses any solitary - wave solutions . we address this issue in the following section . consider a traveling wave solution of ( [ dimensionless : system ] ) , i.e. , a solution that is a function of the variable @xmath43 then the pdes in ( [ dimensionless : system ] ) become odes , and one obtains the following system : @xmath44 upon the integration of equations ( [ ode : basic : system1 ] ) and ( [ ode : basic : system2 ] ) once , we get the algebraic conservation relations @xmath45 we are interested in a traveling - wave solution _ on the zero background _ , hence @xmath46 at @xmath47 therefore the constants of integration @xmath48 this yields the following expressions for @xmath49 and @xmath50 @xmath51 where @xmath52 we insert expressions ( [ ode : reduced : system1 ] ) and ( [ ode : reduced : system2 ] ) for @xmath49 and @xmath50 into the equations ( [ ode : basic : system3 ] ) and ( [ ode : basic : system4 ] ) for @xmath53 and @xmath54 and obtain the following system of second order equations : @xmath55 this system can be diagonalized with respect to the second derivatives @xmath56 by the means of the transformation @xmath57{l}% q\\ m \end{array } \right ] = \left [ \begin{array } [ c]{ll}% 1 & 0\\ \dfrac{-\beta a_{2}}{1+\beta a_{1 } } & \dfrac{\omega_{2}\sqrt{\beta}}{1+\beta a_{1}}% \end{array } \right ] \left [ \begin{array } [ c]{l}% q\\ m \end{array } \right]\ ] ] the matrix @xmath58 in ( [ diagonalized : form ] ) is symmetric @xmath59 @xmath60{ll}% \omega_{1}^{2}-a_{1}+\dfrac{\beta a_{2}^{2}}{1+\beta a_{1 } } & -\dfrac { a_{2}\omega_{2}\sqrt{\beta}}{1+\beta a_{1}}\\ -\dfrac{a_{2}\omega_{2}\sqrt{\beta}}{1+\beta a_{1 } } & \dfrac{\omega_{2}^{2}% } { 1+\beta a_{1}}% \end{array } \right ] \label{matr_a}%\ ] ] instead of the second order system ( [ diagonalized : form ] ) we will consider the following equivalent @xmath61 first order system @xmath62{c}% q\\ m\\ q_{1}\\ m_{1}% \end{array } \right ] = \left [ \begin{array } [ c]{cccc}% 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ -a_{11 } & -a_{12 } & 0 & 0\\ -a_{21 } & -a_{22 } & 0 & 0 \end{array } \right ] \left [ \begin{array } [ c]{c}% q\\ m\\ q_{1}\\ m_{1}% \end{array } \right ] -\left [ \begin{array } [ c]{c}% 0\\ 0\\ \gamma q^{3}\\ 0 \end{array } \right ] \label{first}%\ ] ] obviously @xmath63 $ ] ( the zero background ) is the only equilibrium solution ( the critical point ) of the system . the pulse solutions are the trajectories of the system ( [ first ] ) that start and end at the equilibrium ( homoclinic orbits ) . thus , the investigation of solitary pulses is mathematically equivalent to studying homoclinic solutions . cross - section of the potential energy landscape @xmath64 . the newtonian particle trajectory corresponds to a one - hump solution presented in the right figure.,width=287 ] cross - section of the potential energy landscape @xmath64 . the newtonian particle trajectory corresponds to a one - hump solution presented in the right figure.,width=279 ] to investigate the structure of homoclinic solutions , we linearize the system in ( [ first ] ) near the critical point @xmath65 : @xmath62{c}% q\\ m\\ q_{1}\\ m_{1}% \end{array } \right ] = \left [ \begin{array } [ c]{cccc}% 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ -a_{11 } & -a_{12 } & 0 & 0\\ -a_{21 } & -a_{22 } & 0 & 0 \end{array } \right ] \left [ \begin{array } [ c]{c}% q\\ m\\ q_{1}\\ m_{1}% \end{array } \right ] : = \tilde{a}\left [ \begin{array } [ c]{c}% q\\ m\\ q_{1}\\ m_{1}% \end{array } \right ] \label{a - tilde}%\ ] ] the characteristic equation of the matrix @xmath66 on the right - hand side is given by @xmath67 therefore , the values of @xmath68 coincide with the eigenvalues of the matrix @xmath69 it is easy to see that@xmath70 thus , the condition @xmath71 makes @xmath72 causing @xmath58 to have eigenvalues of opposite signs , which is a necessary condition for the existence of homoclinic orbits . indeed , in the case of @xmath58 having eigenvalues of the opposite signs , the @xmath61 matrix @xmath66 has two pure imaginary eigenvalues ( square roots of the negative eigenvalue of @xmath73 ) and one negative , and one positive eigenvalues . therefore the nonlinear system has one - dimensional stable and unstable manifolds , and a two - dimensional center manifold ( corresponding to the imaginary eigenvalues ) . it was first noticed in @xcite that the nonlinear system in ( [ first ] ) has hamiltonian structure . if the kinetic , @xmath74 and potential , @xmath75 energies and the hamiltonian , @xmath76 are introduce as follows @xmath77 then the system ( [ first ] ) takes the form @xmath78 since the hamiltonian is a conserved quantity , @xmath79 any trajectory issued from the critical point @xmath80 $ ] stays on the zero energy level surface @xmath81 for all time@xmath82 note that the surface @xmath81 is a 3d manifold in @xmath83 the intersection of this 3d hypersurface with the hyperplanes @xmath84 and @xmath85 is a curve @xmath86 in the @xmath87-plane @xmath88 ( the figure - eight shaped curve on the left in fig . [ fig2 ] and [ fig3 ] ) . if for a given @xmath89 there exits a homoclinic trajectory of ( [ first ] ) , then on this trajectory @xmath90 and since @xmath91 , necessarily @xmath92 . at the extrema of @xmath93 its gradient is zero : @xmath94 by eliminating @xmath12 from the equations above , we obtain the cubic equation @xmath95 whose roots are easily found : @xmath96 thus , @xmath97 at the points @xmath98 which are real if @xmath99 thus producing the figure eight level curves . we already encountered this inequality above , see ( [ necessary ] ) . after some algebra it can be rewritten as the following constraints on the traveling wave velocity : @xmath100 thus , a possible velocity of the propagating pulse is bounded below . the nonlinear system ( [ first ] ) has the time reversal symmetry ; therefore , if @xmath101 @xmath102(t):=\mathbf{u}(t)$ ] is a homoclinic orbit , @xmath103(-t)$ ] also is ( recall that @xmath104 and @xmath105 are time derivatives of @xmath106 and @xmath12 ) . a priori it is not clear why any homoclinic solution would possess this symmetry , and it is quite likely that there exist non - symmetric homoclinic orbits ; we plan to investigate them elsewhere . the characteristic property of a time reversal orbit is that at the symmetry point @xmath107 and consequently the kinetic energy @xmath108 must be zero ; i.e. , the symmetry point lies on the curve @xmath109 see ( [ gamma ] ) . moreover at the symmetry point the trajectory is orthogonal to @xmath86 ( for an illustration , see the @xmath87 diagrams on the left of fig . [ fig2 ] and fig . [ fig3 ] ) . and @xmath110 . the left and right figures illustrate four - hump and eight - hump solitary wave solutions . , title="fig:",scaledwidth=49.0% ] and @xmath110 . the left and right figures illustrate four - hump and eight - hump solitary wave solutions . , title="fig:",scaledwidth=49.0% ] our search algorithm for finding solitary - wave solutions is based on the following minimization idea . if for a given value of the propagation velocity @xmath89 there exists a homoclinic orbit with the time - reversal symmetry , then at some point both the kinetic and potential energies are zero . the algorithm takes the initial condition @xmath111 from a domain @xmath112 on the zero - energy surface , near the critical point @xmath113 and in the direction close to that of the unstable eigenvector of the linearized problem . then the following optimization problem is posed : determine @xmath114 \label{optimization}\ ] ] r0.5 where @xmath108 is the kinetic energy , @xmath115 is the solution of ( [ first ] ) with the initial condition @xmath116 recall that @xmath117.$ ] the parameter @xmath118 is the expected width of the pulse . since @xmath119 at @xmath120 we take @xmath121 to obtain a nontrivial solution for the energy minimization problem . computation of any particular value of @xmath122 the search of the optimal initial datum is stochastic and is organized via a version of simulated annealing @xcite . on each step the initial datum is obtained by sampling a random distribution with the density determined by the results of the previous step ( see @xcite for more detail ) . if @xmath123 then there exists a homoclinic solutions with velocity @xmath124 when the kinetic energy possesses several local minima along the trajectory the corresponding solitary wave has the multi - hump structure . figures [ fig2 ] and [ fig3 ] illustrate this phenomenon . the figure - eight shaped curves on the left correspond to the @xmath119 cross - section of the potential energy landscapes ; the curves inside the domains represent the newtonian particle trajectories in the @xmath87 configuration space . the graphs on the right show the profiles of the corresponding solitary wave solutions . [ fig2 ] illustrates a typical one - hump solution . in contrast , the trajectory shown in fig . [ fig3 ] has a point of the nearest approach to the boundary where the kinetic energy attains a local minimum . the resulting solution has a two - hump structure . multi - hump solutions correspond to more complicated trajectories . each of these trajectories has the return point at which it has the normal incidence with the @xmath119 contour . for the fixed set of physical parameter values , the shape of the potential energy landscape is controlled by the pulse velocity @xmath89 via the coefficients @xmath125 and @xmath126 in ( [ coefficients : a1:a2 ] ) . we investigated numerically the set @xmath127 of values of @xmath89 which give rise to homoclinic orbits ; in some sense one might think of these @xmath89s as the spectrum of the problem . for numerous applications with soliton - like solutions the velocity value is known to change continually ( a continuous spectrum ) . however , for the maxwell - duffing model under consideration our numerical investigation demonstrates that the spectrum @xmath127 contains both an interval of a continuous spectrum and a discrete subset of parameter values @xmath89 for which a wave solution exists . one of the principal issues is to understand the correspondence between types of solitary wave solutions and values of @xmath128 . we first investigated numerically the distribution density of the values of @xmath89 , which give rise to homoclinic orbits . for all numerical computations of this section we adopted the following values of the nondimensional physical parameters : @xmath129 for @xmath130 the allowable range of values of @xmath89 from ( [ v - bounds ] ) is given by @xmath131 the plot in fig . [ fig4 ] illustrates the density distribution of @xmath128 on the interval @xmath132 $ ] . the search algorithm tested potential values of @xmath89 on the grid @xmath133 the plot depicts the number of `` successful '' homoclinic orbits per velocity interval @xmath134 ( a `` bin '' ) ; in this particular case the value has been chosen as @xmath135 . and @xmath136 ( left ) ; an eight - hump soliton with @xmath137 and a phase - inverted soliton with @xmath138 ( right).,title="fig:",scaledwidth=49.0% ] and @xmath136 ( left ) ; an eight - hump soliton with @xmath137 and a phase - inverted soliton with @xmath138 ( right).,title="fig:",scaledwidth=49.0% ] our numerical computations show that on a rather small interval @xmath139 $ ] at the low end of the spectrum , _ every _ attempt of computing a homoclinic orbit was successful ( 20 orbits per bin ) . these results stay consistent with the refinement of the computational grid size @xmath140 all the solitary wave solutions in @xmath141 are of the one - hump variety ; note , however that the single - hump solitons are not exclusively confined to the lower end of @xmath127 . elsewhere the spectrum density is very low , and the solitons are mostly of a multi - hump kind . somewhat arbitrarily , we define a hump as a local maximum of the electric field @xmath142 which is at least 50% of the global maximum . next we studied the distribution of the different type of solitary wave solutions on the interval of velocities @xmath132 $ ] . the figure ( fig . [ thehumpdistribution ] ) gives a very clear idea of the placement of solitons according to the number of humps , which ranges from one to ten . some typical soliton profiles for four- and eight - hump solutions with @xmath143 and @xmath144 respectively are collected in fig . [ profiles:1 - 4 ] . different types of solitary wave solutions have different energy values . because of the multi - hump nature of these solutions it is convenient to introduce the energy of the electromagnetic field per one hump . we analyzed dependence of the electromagnetic field energy @xmath145 per one hump versus velocity of solitary wave , see fig . [ energy : per : single : hump ] . here @xmath145 is defined as @xmath146 dt,\ ] ] where @xmath147 is the number of humps . as follows from this figure , in the log - log coordinates the energy increase is very well approximated by a linear function . the least square fit of the data from this figure shows that the energy increases approximately as a polynomial of fifth degree in @xmath124 in this section through direct numerical simulations we study evolution of waves as well as wave interactions . we consider formation of solitary wave solutions from arbitrary initial - boundary condition , stability of traveling waves under small perturbations and stability under strong perturbations due to wave collisions . in all numerical simulations of this section we use the same values of physical parameters ( [ params ] ) as in sec . [ num_waves ] . numerically we solve the signaling problem for ( [ dimensionless : system ] ) . in other words we give boundary conditions on either one or both ends of the spatial interval @xmath148 ; as initial conditions we assign zero values for all the variables , which corresponds to propagation in a quiescent medium . for solving the initial - boundary value problem for the system in ( [ dimensionless : system ] ) we devised a simple fractional step numerical method . because the first two equations in ( [ dimensionless : system ] ) are hyperbolic _ pdes _ while the rest are _ odes _ the choice of the fractional steps is extremely natural : on the first half - step we propagate the pde part of the governing equations , and on the second half - step we march according to the system of odes . the resulting ode system is solved by using the midpoint rule , while the pdes are solved by the explicit mccormack method @xcite . the midpoint rule and the mccormack method are both second order accurate . to increase the accuracy of the fractional step method we utilize the strang split approximation @xcite , which results in the second order convergence of the final numerical scheme . for many of the solitary - wave solutions discussed in sec [ num_waves ] we ran direct numerical simulations on the model with these solitary waves as input pulses . all the waves tested , even the ones of a rather intricate shape , propagate with constant speed and without any shape distortion . see , for example , fig . [ fig : ev6prop ] where propagation of an eight - hump soliton is depicted . a `` sharp '' gaussian quickly evolves into a near - solitary wave , leaving some disturbance in the wake . the speed of the near - solitary wave is significantly higher than the speed of propagation of the radiation ; thus the wave quickly leaves the disturbance behind , scaledwidth=60.0% ] this suggests that the solitary waves are ( nonlinearly ) stable with respect to numerical perturbations . we remark that although because of the scale of fig . [ fig : ev6prop ] , the pulses appear rather singular , they are in fact completely smooth and numerically resolved . the numerical resolution of this computation is @xmath149 mesh intervals per unit length , which provides about 60 mesh points per each hump of the traveling wave . similarly , fine computational meshes are employed in all the simulations below . the issue of stability can be addressed analytically by studying the linearization of the system of partial differential equations ( [ dimensionless : system ] ) about arbitrary traveling wave solutions and analyzing the corresponding linear evolution operator . our analysis showed that this operator is skew - hermitian in @xmath150 with the appropriate norm . therefore the spectrum of the evolution operator is pure imaginary and the traveling wave solutions are neutrally linearly stable ( see @xcite for detail ) . a medium size gaussian evolves into two waves . the velocity of the smaller wave is on the order of the velocity of radiation . , scaledwidth=55.0% ] to further elucidate the issue of stability we consider stability with respect to a finite - size perturbation in the initial wave shape . this situation is illustrated in fig . [ fig6 ] . to the two - hump numerical soliton we add a rather substantial perturbation and employ the thus obtained functions as boundary data for the system of partial differential equations ( [ dimensionless : system ] ) . as the result of evolution , the solution relaxes to the solitary wave shape followed by a low - amplitude `` continuous radiation '' . stability with respect to strong perturbations due to collision of two traveling wave solutions is illustrated in fig . we take two solitary waves obtained by numerical solutions of odes and use these solutions as the boundary conditions for the pdes . the left part of the fig . [ fig7 ] shows collision of two - hump solitary wave solutions . the right part of the figure shows collision of eight - hump and one - hump solitary waves . in both cases collision of solitary waves leads to formation of the steady state solutions . the collisions are followed by emission of a small amplitude continuous radiation and a residual phase shift . a soliton nature of solutions of ( [ dimensionless : system ] ) is further confirmed by the set of numerical simulations we present next . we consider propagation of solutions with the pulses in fig . [ fig : energyic ] given as a series of boundary conditions at the @xmath151 boundary . the soliton of velocity @xmath152 ( see fig . [ fig2 ] ) propagates in a stable fashion , while its least - squares approximation by a gaussian @xmath153 approaches the soliton shape after shedding a small amount of residual continuous radiation . these time evolutions are not included for space saving ( the energy dissipation curves for these cases show conservation of electromagnetic energy , see fig . [ fig : energy ] ) . r[ht ] gaussian . large gaussian quickly breaks up into four near - solitary waves , leaving some disturbance in the wake . the waves become more separated over the time , since the near solitary waves with higher amplitude have higher velocities . , scaledwidth=55.0% ] in the next three figures ( fig . [ fig : goldgauss3d ] -[fig : widegauss3d ] ) we present evolutions of the larger gaussian pulses from fig . [ fig : energyic ] . evolution of the sharpest gaussian ( @xmath154 is displayed in fig . [ fig : goldgauss3d ] . very fast the solution forms a solitary wave that moves with constant velocity with no shape change . it is followed by low magnitude oscillations whose leading edge also moves with constant speed . during the evolution , the oscillatory part disperses more and more . this part of the solutions appears to be of a nonlinear nature ; it will be studied separately . we note that although because of the scale of the figure , the pulse appears very sharp , it is in fact completely smooth with `` width '' about @xmath155 and about @xmath156 computational mesh points within the pulse . the evolution of a wider gaussian , @xmath157 ( see fig . [ fig : sharpgauss3d ] ) is similar with a very interesting distinction . now the leading soliton is trailed by a slower low amplitude soliton . the latter is followed by low amplitude oscillations that again lag behind and disperse . the waves become more separated over time because the solitary waves with higher amplitude have higher velocities . finally , the widest gaussian , @xmath158 develops into a train of four solitons , see fig . [ fig : widegauss3d ] . to characterize the energy exchange between the propagating pulse and the medium , in fig . [ fig : energy ] we present plots of the total electromagnetic energy as a function of time for all the input profiles from fig . [ fig : energyic ] . for the soliton solution there is a dynamic equilibrium between the energy stored in the medium and the electromagnetic energy of the pulse . in case of the input impulse being not a soliton , the balance between the medium and the pulse is violated , which leads to the dissipation of the electromagnetic energy into the medium . in this paper we considered propagation of extremely short pulses in a nonlinear medium , which is characterized by both electric and magnetic resonance responses . interaction of the electromagnetic field with the medium was described in the framework of the maxwell - duffing model . in particular we employed the classical maxwell - lorenz model for describing the magnetic resonance , @xcite . for describing the interaction of the electric field component with the medium we used a generalized maxwell - lorentz model which takes into account cubic anharmonism of the polarization response ( i.e. , the maxwell - duffing system ) . our findings demonstrate that the model supports a wide array of traveling - wave solutions . we investigated the structure and properties of these solutions through a combination of analysis and numerical modeling . we determined that the family of traveling - wave solutions is parameterized by one parameter , which is the velocity of a steady wave solution , normalized by the speed of light in vacuum . the spectrum @xmath127 contains both an interval of a continuous spectrum and a discrete subset of parameter values for which a traveling - wave solution exists . computer modeling demonstrated a multi - hump structure of these solutions . their multi - hump nature suggests to characterize solitary wave solutions by a number of humps ( types ) . all types are determined by not overlapping sets of velocities . direct numerical simulations showed that solitary - wave solutions are dynamically stable . this dynamical stability is consistent with the analysis of the system linearized about solitary wave solutions @xcite . stability of these solutions with respect to strong perturbations was studied by means of solitary wave collisions . computer simulations indicated nearly elastic nature of scattering followed by a residual excessive radiation and a phase shift . in addition to traveling - wave solutions , numerical simulations demonstrated presence of another type of nonlinear oscillatory solutions with extended tail . frenkel s work was partially supported by the nsf emsw21-rtg grant no . part of this work is based on his ph.d . thesis @xcite . this work was partially supported by nsf ( grant dms-0509589 ) , aro - muri award 50342-ph - mur , the state of arizona ( proposition 301 ) , and by the russian foundation for basic research through grant 06 - 02 - 16406 . roytburd s work was partially supported by the national science foundation , while working at the foundation . part of his work was performed during a sabbatical leave at the lawrence berkeley national laboratory . the authors would like to thank m. stepanov for the enlightening discussions and for the valuable help in preparation of this manuscript .

propagation of extremely short electromagnetic pulses in a homogeneous doubly - resonant medium is considered in the framework of the total maxwell - duffing - lorentz model , where the duffing oscillators ( anharmonic oscillators with cubic nonlinearities ) represent the dielectric response of the medium , and the lorentz harmonic oscillators represent the magnetic response . the wave propagation is governed by the one - dimensional maxwell equations . it is shown that the model possesses a one - parameter family of traveling - wave solutions with the structure of single or multiple humps . solutions are parametrized by the velocity of propagation . the spectrum of possible velocities is shown to be continuous on a small interval at the lower end of the spectrum ; elsewhere the velocities form a discrete set . a correlation between the number of humps and the velocity is established . the traveling - wave solutions are found to be stable with respect to weak perturbations . numerical simulations demonstrate that the traveling - wave pulses collide in a nearly elastic fashion .

in this supplementary material , we compare the value of the penetration depth obtained from experiments @xcite with the prediction from homes law ; for the latter , we use a combination of the experimental data obtained from optical - conductivity and dc transport . for each value of the doping ( @xmath8 ) , we estimate the ( approximate ) dc resistivity ( @xmath80 ) by extrapolating the curves to @xmath9 , from the transport data in fig.1(b ) of ref.@xcite . we estimate the value of @xmath81 , where @xmath23 is the superconducting gap , from the data for optical conductivity in the superconducting state , as shown in fig . 3(b ) of ref . @xcite . since @xmath7 remains relatively unchanged as a function of @xmath8 in the vicinity of optimal doping , we assume @xmath82 to be independent of @xmath8 such that @xmath83@xmath84s@xmath85 . then , in the dirty limit , _ s = _ . in order to obtain the penetration depth , we need to restore various dimensionful constants such that , _ l^2(0)= , where @xmath86 m / s ) is the speed of light and @xmath87 f / m ; 1 f=1 @xmath88s ) is the permitivity of free space . the values obtained are shown in the table below and have been presented in fig . 2 of the main text , along with a comparison to the experimental data @xcite .

we present a theory for the large suppression of the superfluid - density , @xmath0 , in bafe@xmath1(as@xmath2p@xmath3)@xmath1 in the vicinity of a putative spin - density wave quantum critical point at a p - doping , @xmath4 . we argue that the transition becomes weakly first - order in the vicinity of @xmath5 , and disorder induces puddles of superconducting and antiferromagnetic regions at short length - scales ; thus the system becomes an electronic micro - emulsion . we propose that frustrated josephson couplings between the superconducting grains suppress @xmath0 . in addition , the presence of ` normal ' quasiparticles at the interface of the frustrated josephson junctions will give rise to a highly non - trivial feature in the low - frequency response in a narrow vicinity around @xmath6 . we propose a number of experiments to test our theory . _ introduction.- _ an important focus of the study of high temperature superconductivity ( sc ) has been on the role of antiferromagnetism ( afm ) and its relation to sc @xcite . there is clear evidence across many different families of compounds that sc appears in close proximity to an afm phase @xcite ; these families include the iron - pnictides , the electron - doped cuprates and the heavy - fermion superconductors . moreover , the optimal transition temperature ( @xmath7 ) of the sc is often situated where the normal state afm quantum critical point ( qcp ) would have been located , in the absence of superconductivity . the experimental detection of the qcp is often challenging in the normal state , and more so in the superconducting state . recently , a number of measurements were reported in a member of the pnictide family , bafe@xmath1(as@xmath2p@xmath3)@xmath1 , as a function of the isovalent p - doping , @xmath8 . the experiments show a phase transition involving onset of spin - density wave ( sdw ) order in the normal state above @xmath7 , which extrapolates to a @xmath9 sdw qcp ( see @xcite and references therein ) . these experiments include : ( _ i _ ) a sharp enhancement in the effective mass , @xmath10 , upon approaching a critical doping from the overdoped side , as obtained from de haas - van alphen oscillations @xcite and from the jump in the specific - heat at @xmath7 @xcite , and , ( _ ii _ ) a vanishing curie - weiss temperature ( @xmath11 ) , extracted from the @xmath12 measurements using nmr . as we will review below , a number of puzzling results have appeared from experiments investigating whether the sdw qcp actually survives `` under the sc dome . '' here we propose a resolution of these puzzles by postulating a weakly first - order transition for the onset of sdw order in the presence of sc order ( see fig . [ ph]a ) . our results are independent of the specific microsopic mechanism responsible for rendering the transition weakly first - order @xcite . it is well known that ` random bond ' disorder has a strong effect on symmetry - breaking first - order transitions @xcite , and ultimately replaces them with a disorder - induced second order transition in two dimensional systems . our main claim is that the inhomogeneities associated with these highly relevant effects of disorder can resolve the experimental puzzles . the possiblity of a qcp within the sc state was investigated by measurements @xcite of the zero temperature london penetration depth , @xmath13 ( @xmath14 superfluid - density ) , as a function of @xmath8 . a sharp peak in @xmath15 was observed at @xmath16 and interpreted as evidence for a qcp @xcite . however , this interpretation is at odds with general theoretical considerations @xcite concerning a qcp associated with the onset of sdw order in the presence of a superconductor with gapped quasiparticle excitations @xcite . these considerations suggest that such systems will display a _ monotonic _ variation in @xmath15 across the qcp , rather than a sharp peak ( see dashed - blue / solid - red curves in fig . [ ph]b ) @xcite . as a first step toward resolving this discrepancy , it is useful to place measurements of @xmath0 in the context of what is known about the normal state conductivity of the bafe@xmath1(as@xmath2p@xmath3)@xmath1 system , as these quantities are intimately related through a sum rule . the low temperature superfluid density of a spatially homogeneous superconductor can be estimated from the missing area " relation , _ s_0 ^ 2/ ( z)dz , [ homese ] where @xmath17 is the elastic scattering rate and @xmath18 . in the dirty limit where @xmath19 , the above relation yields homes law @xcite , @xmath20 , whereas in the clean limit @xmath21 where @xmath22 is the conductivity spectral weight in the normal state . eqn . [ homese ] is particularly useful when the normal state resistivity data can reasonably be extrapolated to @xmath9 . by combining dc transport data as a function of @xmath8 @xcite and a measurement of 2@xmath23 from optical conductivity @xcite , eq . [ homese ] provides a lower bound on @xmath15 ( with the assumption that @xmath23 is independent of @xmath8 ) . fig.[homes ] shows @xmath15 as a function of @xmath8 obtained under this assumption ( details of the procedure are presented as supplementary information ) . the decrease of superfluid density on the underdoped side reflects the growth in residual resistivity that begins as @xmath8 drops below about 0.33 . the values of @xmath15 estimated from eq . [ homese ] form a baseline for comparison with the experimental results presented in ref . @xcite . on the same graph in fig . [ homes ] , we show the experimentally measured @xmath15 @xcite . the data generally reflect the trend expected from the variation in the residual resistivity , with the exception of the sample with @xmath24 , in which the condensate spectral weight is suppressed by about 40% from the homes law estimate . given the constraints imposed by the sum rule , there are two possible sources of this discrepancy : ( _ i _ ) the quasiparticle mass could be renormalized at this value of @xmath8 , corresponding to an intrinsic decrease in @xmath22 , or , ( _ ii _ ) a considerable fraction of the ( unrenormalized ) @xmath22 could fail to contribute to the low temperature superfluid density . the latter possibility is suggested within the scenario that we develop here . we analyze the above experiments by assuming a weakly first - order transition @xcite , and argue that the presence of quenched disorder leads to formation of a _ micro - emulsion _ at small scales @xcite . the system consists of sc puddles , where some of the puddles additionally have sdw order ( see fig . [ ph]a inset ) . the sdw(+sc ) regions , which have a locally well - developed antiferromagnetic moment but no long - range orientational order , act as barriers between the different sc grains . upon moving deeper into the ordered side of the transition , the sdw(+sc ) regions start to percolate and crossover to a state with long - range sdw order ; this is the regime with a microscopically coexistent sc+sdw . as a function of decreasing @xmath8 , the micro - emulsion is therefore a transitional state ( shown as grey region in fig . [ ph]a ) between a pure sc and a coexistent sc+sdw . recent experiments in the vicinity of optimal doping using neutron - scattering and nmr have found results broadly consistent with our proposed phase diagram @xcite . we note that the granular nature of superconductivity should have no effect on the bulk @xmath7 in the presence of percolating sc channels . _ model.- _ when the system is well described in the vicinity of @xmath5 by a micro - emulsion as explained above , the phase fluctuations associated with the sc grains ( shown as purple regions in fig . [ ph]a inset ) , can be modeled by the following effective theory , h_= - _ a , bj_ab(_a-_b ) , where @xmath25 represent the josephson junction ( jj ) couplings between grains ` @xmath26 ' and ` @xmath27 ' . we have ignored the capacitive contributions . the josephson current across the junction will be given by @xmath28 , and @xmath25 may therefore be interpreted as the lattice version of the local superfluid density , @xmath29 , i.e. @xmath30 , with @xmath31 representing the superfluid - current and velocity respectively . having a frustrated jj ( also known as a @xmath32junction ) with a negative value of @xmath25 leads to a local suppression in @xmath0 . similar ideas have been discussed in the past in a variety of contexts ( see refs . @xcite for a specific example ) , though the mechanism considered here will be different . we shall now propose an explicit scenario under which a suppression in @xmath0 arises in the vicinity of putative magnetic qcps , utilizing the sc gap structure in the material under question . the basic idea is as follows : suppose that the tunneling of electrons between the two grains is mediated by the sdw moment in the intervening region @xcite , and is accompanied by a transfer of finite momentum that scatters them from a hole - like to an electron - like pocket . because the sc gaps on the two pockets have a relative phase - difference of @xmath33 , the jj coupling will be frustrated @xcite . let us first focus on a single grain . in order to capture the multi - band nature of the scs , we introduce two superconducting order parameters , @xmath34 with @xmath35 to model the @xmath36 state on the two pockets . microscopically , these belong to regions in the grain having different momenta , @xmath37 , parallel to the junction . the gaps are related to the microscopic degrees of freedom @xcite via the following relation , _ i(z)=___i v__,____- _ , where @xmath38 creates an electron at position @xmath39 with momentum @xmath37 parallel to the junction and spin @xmath40 . @xmath41 is the pairing interaction in the cooper channel and @xmath39 is the coordinate perpendicular to the junction with area @xmath42 . the regions @xmath43 are defined as , @xmath44 and @xmath45 , where @xmath46 is an arbitrary momentum scale chosen such that @xmath47 ( see fig . [ jj ] for an illustration ) . we ll assume that such a prescription is valid for each grain , with possibly different values of @xmath46 . let us then write down a model for the two coupled sc grains with an intervening proximity coupled sdw that has a well developed moment , @xmath48 . our notation is as follows : we use @xmath49 to denote the grain index and @xmath35 to denote the band index within each grain . from now on , we relabel @xmath37 as @xmath50 . we introduce the nambu spinor , @xmath51 , where now @xmath52 creates an electron with momentum @xmath50 parallel to the junction and at a position @xmath39 ( label suppressed ) , which belongs to a region of band @xmath53 " within grain @xmath54 " . the effective hamiltonian is given by , [ heff ] h_&=&h_+h_t , + h_&=&_,i , _ i,,^_i,,^ , + h_t&=&g_k ( ^a_+,,[_^0]_-,,^b + & & + ^a_-,,[_^0]_+,,^b ) + , where @xmath55 is the tunneling matrix element , @xmath56 @xmath57 act in nambu space and @xmath58 @xmath57 act in spin space . in the above , @xmath59 corresponds to the bare pairing hamiltonian written for the @xmath60 bands within each of the two grains . @xmath61 represents the sdw moment mediated hopping of electrons from one grain to the other ( represented by the @xmath62 superscripts ) and simultaneously scattering from one band to the other ( represented by the @xmath60 subscripts ) . therefore , @xmath48 imparts a finite momentum ( along the interface ) to the electrons when it scatters them from the electron ( hole ) pocket on one grain to the hole ( electron ) pocket on the other grain ( shown as the black arrows in fig . [ jj ] ) . _ results.- _ using the ambegaokar - baratoff relation @xcite , we can write the josephson coupling ( at @xmath9 ) between the two grains as , j_ab= where @xmath63 and @xmath64 represent the band indices on the different grains . since @xmath65 , the coupling @xmath66 . note that the specific nature of the frustrated tunneling arises from the same spin - fluctuation mediated mechanism that is predominantly responsible for the @xmath67 pairing symmetry @xcite . however , there will also be a direct tunneling term ( not included in eqn . [ heff ] ) in the hamiltonian , which does not scatter the electrons from one pocket to the other , as they hop across the junction . the contribution to the jj coupling from this term will be unfrustrated ( i.e. @xmath68 ) . the ratio of the tunneling amplitudes in the two different channels is non - universal and depends on various microscopic details . in particular , the emulsion is associated with a distribution of josephson - couplings , @xmath69 , with a mean coupling strength , @xmath70 . if a substantial fraction of the jj couplings become negative due to the mechanism proposed above , @xmath71 will be small , and the superfluid density will be suppressed ( see green curve in fig.[ph]b ) . we now propose a resolution as to the fate of the uncondensed spectral weight ( highlighted in fig . [ homes ] ) , which can potentially be tested by measurements of the low frequency optical conductivity . frustrated @xmath32junctions host gapless states at the interface between the two grains @xcite , giving rise to a finite density of states around zero energy ( see fig [ sigw ] inset ) . as a result of the gapless ` normal'-fluid component at the interface , a fraction @xmath72 of the spectral weight will be displaced from the superfluid - density to non - zero frequencies ( shaded region in fig . [ sigw ] ) . given that the weight of the condensate is proportional to @xmath73 , the 40% suppression in @xmath0 for bafe@xmath1(as@xmath2p@xmath3)@xmath1 in the vicinity of the putative qcp corresponds to @xmath74 . our proposed optical conductivity , @xmath75 , in the vicinity of penetration depth anomaly is shown in in fig [ sigw ] . the spectrum shows clearly that the connection between normal state conductivity and superfluid density implied by eq . [ homes ] will break down . in particular , @xmath76 ( which is a property of the normal state ) , could vary monotonically with isovalent - doping across @xmath6 , while the abundance of low - energy excitations in the immediate vicinity of @xmath6 would give rise to a non - monotonic variation in the superfluid density . this allows for an unusual way of rearranging spectral weight in the _ superconducting _ state below the gap , without violating optical sum - rules . the above scenario will give rise to a number of interesting low temperature thermodynamic and transport properties , as we now discuss . first of all , there should be a striking enhancement in the low - temperature thermal conductivity and specific - heat , as a function of @xmath8 in the narrow vicinity of @xmath6 , due to the ` normal'-component . it is important to recall that this material has loop - like nodes on the electron - pockets @xcite . however , the geometry of the electron - pockets and the magnitude of the gap do not change substantially in the vicinity of @xmath6 , and therefore it is unlikely that the contribution to the above quantities from the nodal - quasiparticles will have a drastic modificiation . it should therefore be relatively straightforward to disentangle the contribution arising from the nodal versus the ` normal ' quasiparticles . studying the nmr - spectra as a function of decreasing temperature ( across @xmath7 ) and down to sufficiently low temperatures in the vicinity of @xmath6 should also reveal the spatial inhomogeneity associated with the sdw regions . a large residual density of states in the superconducting state has been detected at a particular p - doping via the power - law temperature dependence of @xmath77 @xcite . within our scenario , there should be a striking enhancement in this quantity as a function of doping around @xmath6 . finally , we note that a promising direction for future studies would be to measure the magnetic - field distribution due to the propagating currents in the emulsion using nv - based magnetometers @xcite . _ discussion.- _ the theoretical study in this paper was motivated by a number of remarkable experiments carried out in bafe@xmath1(as@xmath2p@xmath3)@xmath1 , as a function of @xmath8 in the normal and superconducting phases . our primary objective was to provide an explanation for the striking enhancement of the london penetration depth in the vicinity of a putative sdw qcp in the sc state . we developed a scenario based on the idea that true sdw criticality is masked by a weak first - order phase transition in the superconducting state at @xmath9 . in this picture , quenched disorder naturally gives rise to an _ emulsion _ at small length scales with puddles of sc and sdw(+sc ) . it is then , in principle , possible for sdw moments at the interface of the sc grains to generate frustrated josephson couplings , which deplete the local superfluid - density . our proposed scenario naturally calls for a number of experimental tests that should be carried out in the near future , which should directly look for both the spatial inhomogeneities associated with the emulsion @xcite , and probe the gapless excitations using thermodynamic probes , as explained above . in addition to experiments on bafe@xmath1(as@xmath2p@xmath3)@xmath1 , it should be important to further investigate the contrasting behavior of the electron - doped system , ba(fe@xmath2co@xmath3)@xmath1as@xmath1 , where @xmath78 behaves monotonically as a function of @xmath8 across the putative qcp @xcite . electron - doping leads to significantly higher amounts of disorder compared to the isovalently - doped case , and would therefore lead to puddles with typically much smaller size @xcite . our proposed mechanism for the strong suppression of the superfluid - density in the isovalently - doped material relies on the existence of an emulsion with puddles of appreciable size , in the presence of an optimal amount of disorder . a comparison of the nmr spectra in the narrow vicinity of the putative qcp in the electron and isovalently doped materials would shed light on these microscopic differences between the two families . finally , though we have hypothesized that the sdw onset transition _ inside _ the sc is , in the absence of disorder , a weak first order transition , we emphasize that the normal state properties are consistent with the presence of a hidden " qcp around optimal doping @xcite . it is plausible that in the normal state , different experimental techniques are probing the critical fluctuations associated with not one , but distinct qcps as a function of @xmath8 . for instance , @xmath10 extracted from high - field quantum oscillations is dominated by the vicinity of ` hot - spots ' , where quasiparticles are strongly damped due to coupling to the sdw fluctuations @xcite . on the other hand , strong critical fluctuations associated with the nematic order - parameter @xcite , that couple to the entire fermi - surface , would dominate @xmath10 extracted at zero - field from the jump in the specific heat at @xmath7 . _ acknowledgements.- _ we thank a. carrington , a. chubukov , n. curro , j.c . davis , r. fernandes , k. ishida , m .- h . julien , s. kivelson , y. matsuda , a. millis and a. vishwanath for useful discussions . we thank k. hashimoto and y. matsuda for providing us with the data shown in fig.[homes ] . dc is supported by the harvard - gsas merit fellowship and acknowledges the boulder summer school for condensed matter physics - modern aspects of superconductivity " , where some preliminary ideas for this work were formulated . dc and ss were supported by nsf under grant dmr-1360789 , the templeton foundation , and muri grant w911nf-14 - 1 - 0003 from aro . ts was supported by department of energy desc-8739- er46872 , and partially by a simons investigator award from the simons foundation . jo acknowledges the office of basic energy sciences , materials sciences and engineering division , of the u.s . department of energy under contract no . de - ac02 - 05ch11231 for support . part of this work was completed when jo was visiting mit as a moore visitor supported by grant gbmf4303 . research at perimeter institute is supported by the government of canada through industry canada and by the province of ontario through the ministry of research and innovation . 99 l. taillefer , ann . rev . cond . mat . phys . * 1 * , 51 ( 2010 ) ; s. sachdev , science * 336 * , 1510 ( 2012 ) . d.j . scalapino , rev . mod . phys . * 84 * , 1383 ( 2012 ) . t. shibauchi , a. carrington and y. matsuda , ann . rev . cond . mat . phys . * 5 * , 113 ( 2014 ) . h. shishido et al . , phys . rev . lett . * 104 * , 057008 ( 2010 ) . p. walmsley et al . , phys . rev . lett . * 110 * , 257002 ( 2013 ) . r. fernandes , s. maiti , p. wolfle and a. chubukov , phys . rev . lett . * 111 * , 057001 ( 2013 ) ; j. wu , q. si and e. abrahams arxiv:1406.5136 . y. imry and m. wortis , phys . rev . b * 19 * , 7 ( 1979 ) ; k. hui and a. nihat berker , phys . rev . lett . * 62 * , 21 ( 1989 ) . k. hashimoto et al . , science * 336 * , 1554 ( 2012 ) . a. levchenko , m.g . vavilov , m. khodas and a.v . chubukov , phys . rev . lett . * 110 * , 177003 ( 2013 ) . d. chowdhury , b. swingle , e. berg and s. sachdev , phys . rev . lett . * 111 * , 157004 , ( 2013 ) . there is reason to believe that there are accidental ( loop - like ) nodes on the electron - pockets in the s@xmath79 state in this particular material ( see refs . @xcite ) ; this feature does nt affect most of the qualitative features of the computation as long as the nodes do not coincide with the sdw hot - spots " . k. hashimoto _ et al . _ , phys . rev . b * 81 * , 220501(r ) ( 2010 ) ; t. shimojima et al . , science * 332 * , 564 ( 2011 ) ; y. zhang et al . , nat . phys . * 8 * , 371 ( 2012 ) ; y. mizukami _ et al . _ , nat . comms . * 5 * , 5657 ( 2014 ) . c.c . homes et al . , nature * 430 * , 539 ( 2004 ) ; s.v . dordevic , d.n . basov and c.c . homes , sci . rep . * 3 * , 1713 ( 2013 ) . s. kasahara et al . , phys . rev . b * 81 * , 184519 ( 2010 ) . s.j . moon et al . , phys . rev . b * 90 * , 014503 ( 2014 ) . d. hu et al . , phys . rev . lett . * 114 * , 157002 ( 2015 ) b.i . spivak and s.a . kivelson , phys . rev . b * 43 * , 3740 ( 1991 ) ; j.a . van dam , y.v . nazarov , e.p.a.m . bakkers , s.defranceschi and l.p . kouwenhoven , nature * 442 * , 667 ( 2006 ) . p.w . anderson , phys . rev . lett . * 17 * , 95 ( 1966 ) . v. ambegaokar and a. baratoff , phys . rev . lett . * 10 * , 486 ( 1963 ) . e. berg , n.h . lindner and t. pereg - barnea , phys . rev . lett . * 106 * , 147003 ( 2011 ) . r. jackiw and c. rebbi , phys . rev . d * 13 * , 12 ( 1976 ) . c - r hu , phys . rev . lett . * 72 * , 10 ( 1994 ) ; y. tanaka and s. kashiwaya , phys . rev . lett . * 74 * , 17 ( 1995 ) . y. nakai et al . , phys . rev . b * 81 * , 020503(r ) , 2010 . s. hong , m.s . grinolds , l.m . pham , d.l . sage , l. luan , r.l . walsworth and a. yacoby , mrs bulletin * 38 * , 155 ( 2013 ) . n. curro , private communication and http://meetings.aps.org/meeting/mar15/session/t5.7 r.t . gordon _ et al . _ , phys . rev . b * 82 * , 054507 ( 2010 ) . a. p. dioguardi _ et al . _ , phys . rev . lett . * 111 * , 207201 ( 2013 ) . j. analytis _ et al . _ , nat . phys . * 10 * , 194 ( 2014 ) . t. senthil , arxiv:1410.2096 . r. fernandes , a.v . chubukov and j. schmalian , nat . phys . * 10 * , 97 ( 2014 ) .

12 url # 1#1urlprefix[2][]#2

we study the coupling of magneto - acoustic waves to alvn waves using 2.5d numerical simulations . in our experiment , a fast magnetoacoustic wave of a given frequency and wavenumber is generated below the surface . the magnetic field in the domain is assumed homogeneous and inclined . the efficiency of the conversion to alfvn waves near the layer of equal acoustic and alfven speeds is measured calculating their energy flux . the particular amplitude and phase relations between the oscillations of magnetic field and velocity help us to demonstrate that the waves produced after the transformation and reaching upper atmosphere are indeed alfvn waves . we find that the conversion from fast magneto - acoustic waves to alfvn waves is particularly important for the inclination @xmath0 and azimuth @xmath1 angles of the magnetic field between 55 and 65 degrees , with the maximum shifted to larger inclinations for lower frequency waves . the maximum alfvn flux transmitted to the upper atmosphere is about 23 times lower than the corresponding acoustic flux . conversion from fast - mode high-@xmath2 magneto - acoustic waves ( analog of @xmath3 modes ) to slow - mode waves in solar active regions is relatively well studied both from analytical theories and numerical simulations ( e.g. , @xcite ) , see @xcite for a review . in a two - dimensional situation , the transformation from fast to slow magnetoacoustic modes is demonstrated to be particularly strong for a narrow range of the magnetic field inclinations around 2030 degrees to the vertical . however , no generalized picture exists so far for conversion from magneto - acoustic to alfvn waves in a three - dimensional situation . studies of this conversion were initiated by cally & goossens @xcite , who found that the conversion is most efficient for preferred magnetic field inclinations between 30 and 40 degrees , and azimuth angles between 60 and 80 degrees , and that alfvnic fluxes transmitted to the upper atmosphere can exceed acoustic fluxes in some cases . newington & cally @xcite studied the conversion properties of low - frequency gravity waves , showing that large magnetic field inclinations can help transmitting an important amount of the alfvnic energy flux to the upper atmosphere . time - height variations of the three projected velocity components corresponding to @xmath4 ( alfven wave , left ) , @xmath5 ( fast wave , middle ) and @xmath6 ( slow wave , right ) for @xmath7 mhz in a simulation with @xmath8 inclined by @xmath9 and @xmath10 . the solid line marks the position @xmath11 , and the dashed line marks the cut - off layer @xmath12 . the colour scaling is the same in all panels . the amplitudes are scaled with @xmath13 ( first two panels ) @xmath14 ( last panel).,width=566 ] motivated by these recent studies , here we attack the problem by means of 2.5d numerical simulations . the purpose of our study is to calculate the efficiency of the conversion from fast - mode high-@xmath2 magneto - acoustic waves to alfvn and slow waves in the upper atmosphere for various frequencies and wavenumbers as a function of the field orientation . we limit our study to a plane parallel atmosphere permeated by a constant inclined magnetic field , to perform a meaningful comparison with the work of cally & goossens @xcite . numerical simulation will allow generalization to more realistic models in our future work . we numerically solve the non - linear equations of ideal mhd assuming all vectors in three spatial directions and all derivatives in two directions ( i.e. 2.5d approximation , see @xcite ) , though perturbations are kept small to approximate the linear regime . an acoustic wave of a given frequency and wave number is generated at @xmath15 mm below the solar surface in a standard model atmosphere permeated by a uniform inclined magnetic field . the top boundary of the simulation box is 1 mm above the surface , and 0.8 mm above the layer where the acoustic speed , @xmath16 , and the alfvn speed , @xmath17 , are equal . we consider frequencies @xmath18 and 5 mhz and wave numbers @xmath19 mm@xmath20 and @xmath21 . the simulation grid covers field inclinations @xmath0 from 0@xmath22 to 80@xmath22 and field azimuths @xmath1 from 0@xmath22 to 160@xmath22 . the field strength is kept at @xmath23 g. to separate the alfvn mode from the fast and slow magneto - acoustic modes in the magnetically dominated atmosphere we use velocity projections onto three characteristic directions : @xmath24 ; \nonumber\\ { \hat\mathbf{e}}_{\rm perp } & = & [ - \cos\phi \sin^2\theta \sin\phi , \ , 1-\sin^2\theta \sin^2\phi , \ , - \cos\theta \sin\theta \sin\phi ] ; \\ \nonumber { \hat\mathbf{e}}_{\rm trans } & = & [ -\cos\theta , \ , 0 , \ , \cos\phi \sin\theta].\end{aligned}\ ] ] to measure the efficiency of conversion to alfvn waves near and above the @xmath11 equipartition layer , we calculate acoustic and magnetic energy fluxes , averaged over time : @xmath25 figure [ fig : modes ] shows an example of the projected velocities in our calculations as a function of space and time . in this representation the larger inclination of the ridges mean lower propagation speeds and vice versa . note , that by projecting the velocities , we are able to separate the modes only in the magnetically dominated atmosphere , i.e. above the solid line in fig . [ fig : modes ] . the figure shows how the incident fast mode wave propagates to the equipartition layer and then splits into several components . the alfvn wave is produced by mode conversion above 0.2 mm ( left panel ) and propagates upwards with the ( rapid ) alfvn speed , confirmed by almost vertical inclination of the ridges . conversely , the essentially magnetic fast - mode low-@xmath2 wave produced in the upper atmosphere ( middle panel ) is reflected , and its velocity variations in the upper layers vanish with height . the ( acoustic ) slow - mode low-@xmath2 wave escapes to the upper atmosphere tunnelling over the cut - off layer due to the field inclination of @xmath9 . the amplitudes of the velocity variations of the alfvn wave are comparable to those of the slow wave . left panel : log@xmath26 of the ratio @xmath27 to @xmath28 for projected velocities and magnetic field variations , averaged over all @xmath1 , as a function of @xmath0 . black line : fast mode ( @xmath5 projection ) ; red line : alfvn mode ( @xmath29 ) ; blue line : slow mode ( @xmath6 ) . right panel : phase shift between the projected variations of @xmath30 and @xmath27 , as a function of @xmath0 for selected @xmath1 . red lines : alfvn mode ; black lines : fast mode.,title="fig : " ] left panel : log@xmath26 of the ratio @xmath27 to @xmath28 for projected velocities and magnetic field variations , averaged over all @xmath1 , as a function of @xmath0 . black line : fast mode ( @xmath5 projection ) ; red line : alfvn mode ( @xmath29 ) ; blue line : slow mode ( @xmath6 ) . right panel : phase shift between the projected variations of @xmath30 and @xmath27 , as a function of @xmath0 for selected @xmath1 . red lines : alfvn mode ; black lines : fast mode.,title="fig : " ] to confirm the alfvn nature of the transformed waves , as revealed by the projection calculations , we checked the amplitude and phase relations for all three modes reaching the upper atmosphere . for the alfvn mode the magnetic field @xmath27 and velocity variations @xmath30 should be in equipartition ( i.e. @xmath31 ) , and both magnitudes should oscillate in phase ( see priest @xcite ) . figure [ fig : phases ] presents the calculations of the amplitude ratio @xmath32 and temporal phase shift between @xmath27 and @xmath30 , where both velocity and magnetic field variations are projected in the corresponding characteristic direction for each mode ( eq . [ eq : directions ] ) . this calculation confirms that , indeed , for all magnetic field orientations @xmath0 and @xmath1 , the amplitude ratio for the alfvn mode ( @xmath29 projection ) is around one ( left panel ) . this is clearly not the case for the slow and fast modes . for the fast mode , the amplitude ratio is two orders of magnitude larger , and for the slow mode , it is two orders of magnitude lower than one . for the alfvn mode the phase shifts group around zero for all @xmath1 , unlike the case of the fast mode ( right panel ) . we did not calculate the phase shifts for the slow mode as the variations of the magnetic field are negligible . thus , we conclude that the properties of the simulated alfvn mode separated by the projection correspond to those expected for a classical alfvn mode . examples of the height dependence of the magnetic ( solid line ) and acoustic ( dashed line ) vertical fluxes , defined by eq . [ eq : fluxes ] , for @xmath7 mhz and several @xmath0 and @xmath1 . solid vertical line marks the position @xmath11 , dashed vertical line marks the cut - off layer @xmath12 . , title="fig : " ] examples of the height dependence of the magnetic ( solid line ) and acoustic ( dashed line ) vertical fluxes , defined by eq . [ eq : fluxes ] , for @xmath7 mhz and several @xmath0 and @xmath1 . solid vertical line marks the position @xmath11 , dashed vertical line marks the cut - off layer @xmath12 . , title="fig : " ] examples of the height dependence of the magnetic ( solid line ) and acoustic ( dashed line ) vertical fluxes , defined by eq . [ eq : fluxes ] , for @xmath7 mhz and several @xmath0 and @xmath1 . solid vertical line marks the position @xmath11 , dashed vertical line marks the cut - off layer @xmath12 . , title="fig : " ] an example of the height variations of the acoustic and magnetic fluxes is given in figure [ fig : fluxes2 ] . the total vertical flux ( dotted line ) is conserved in the simulations except for the limitations caused by the finite grid resolution not resolving slow small - wavelength waves in the deep layers ( see fig . [ fig : modes ] ) . both acoustic and magnetic fluxes show strongest variations near the conversion layer and become constant above it between 0.5 and 1 mm height . the fluxes reaching the upper atmosphere depend crucially on the orientation of the field . in this example , the acoustic flux decreases with @xmath0 whilst the magnetic flux increases with @xmath0 and becomes larger than the acoustic fluxes for @xmath33 . as the fast wave is already reflected in the upper atmosphere ( see fig . [ fig : modes ] ) , the magnetic flux at these heights is due to the propagating alfvn wave . vertical fluxes measured at the top of the atmosphere at 1 mm for waves with @xmath7 mhz ( left panels ) and 3 mhz ( right panels ) . upper panels give magnetic fluxes and lower panels give acoustic fluxes . , title="fig:",width=226 ] vertical fluxes measured at the top of the atmosphere at 1 mm for waves with @xmath7 mhz ( left panels ) and 3 mhz ( right panels ) . upper panels give magnetic fluxes and lower panels give acoustic fluxes . , title="fig:",width=226 ] finally , figure [ fig : fluxes ] gives the time averages of the vertical magnetic and acoustic fluxes at the top of the atmosphere as a function of the field orientation . as proven above , the magnetic flux at 1 mm corresponds to the alfvn mode . at @xmath7 mhz , the maximum of the magnetic flux corresponds to @xmath34 and @xmath35 . this maximum is shifted to larger inclinations @xmath36 for waves with @xmath18 mhz . the presence of the sharp maximum of the alfvnic flux transmission agrees well with the conclusions made previously by cally & goossens @xcite , though the exact position of the maximum is shifted to somewhat larger inclinations . the maximum of the transmitted acoustic flux corresponds to inclinations @xmath37 for @xmath7 mhz waves , and to @xmath38 for @xmath18 mhz waves , again , in agreement with previous calculations @xcite . the absolute value of the fluxes is about 30 times lower for 3 mhz compared to 5 mhz . at some angles the afvn magnetic flux transmitted to the upper atmosphere is larger than the acoustic flux . however , at angles corresponding to the maximum of the transmission , the alfvn flux is 2 - 3 times lower than the corresponding acoustic flux . it is important to realize that quantitatively simulating mode transformation numerically is a challenge , as any numerical inaccuracies are amplified in such second - order quantities as wave energy fluxes . the tests presented in this paper prove the robustness of our numerical procedure and offer an effective way to separate the alfvn from magneto - acoustic modes in numerical simulations . this will allow us in future to study the coupling between magneto - acoustic and alfvn waves in more realistic situations resembling complex solar magnetic structures .

classical electrodynamics has been extremely successful for the past 150 years or more . however , whenever it dealt with point charged particles , the results were disappointing : a divergent electromagnetic energy , the infamous @xmath0 problem of the electromagnetic mass in the abraham lorentz theory , and the runaway solutions of the classical lorentz dirac equation of motion are all symptoms of a deeper maladie . in our view , there is no really satisfactory way to solve these issues entirely within the classical context . all of these problems occur in a very small length scale , in which classical electrodynamics is not supposed to work properly . therefore , any solution to these problems will have to involve contributions from processes that take place in the quantum realm . despite its impressive record , quantum electrodynamics is also plagued by the same type of problems encountered in many linear quantum theories @xcite . in fact , any quantum theory will feature runaway solutions if its classical counterpart also have it @xcite . again , the problem is that the introduction of point particles in a theory leads us directly into length , time , and energy scales in which strong , weak , and even gravitational phenomena are expected to play a significant role . owing to regularization and renormalization techniques , which summed up all these high energy contributions in an effective way , accurate results were achieved in quantum electrodynamics . in the context of the quantum theory of non - relativistic electrons , it was found that the interaction of a point electron with its own electromagnetic field induces an effective cutoff of the order of the electron reduced compton wavelength @xmath1 @xcite . this cutoff owes its existence to _ zitterbewegung _ , the jittery motion caused by the never ending creation and annihilation of virtual electron positron pairs around the point particle , effectively spreading its charge over a region of length comparable to @xmath2 . therefore , classical electrodynamics is a theory valid at a length scale in which quantum phenomena are not very important , a few dozen bohr radius , for instance . however , in order to describe some of the physical phenomena that take place in a length scale comparable to the electron compton wavelength , we must extend classical electrodynamics , treating it as an effective theory in which a cutoff owes its existence to quantum phenomena at small distance . in sections [ sec : podolsky_electrodynamics ] and [ sec : field_and_potential ] , we present a brief review of some aspects of the podolsky regularized electrodynamics in a classical context , in which a second - order derivative term that introduces a cutoff @xmath3 to the electromagnetic interaction is added to the maxwell lagrangian density in order to allow us to describe a range of phenomena in which vacuum polarization is important . classical electrodynamics is a linear theory . although interesting , attempts to formulate a nonlinear electrodynamics have not gained enough traction @xcite . in order to preserve the linear structure of classical electrodynamics , and still allow for a cutoff @xmath3 into the theory in a lorentz and gauge invariant way , a term involving second order derivatives of the electromagnetic potential @xmath4 may be introduced in the lagrangian density for the electromagnetic field . in this case , the lagrangian density reads @xmath5 where , as usual , @xmath6 are the components of the electromagnetic field tensor @xmath7 , and @xmath8 is the current . the middle extra term was proposed long ago in an effort to regularize quantum electrodynamics @xcite . at about the same time , a number of equivalent proposals were made @xcite . recently , it was shown that podolsky lagrangian is the only linear second - order gauge - invariant generalization of maxwell electrodynamics @xcite . regarding quantum electrodynamics , podolsky proposal to generalize electrodynamics is akin to pauli villars regularization procedure @xcite . in the pauli - villars regularization of the electron self - energy , an extra term is introduced in the lagrangian density , corresponding to a heavy auxiliary particle . the mass of this particle is related to a cutoff @xmath3 , which tames the infinities of the theory , by @xmath9 . as the cutoff goes to zero , the mass of the auxiliary particle tends to infinity and disappears from the theory . nowadays , higher order derivatives appears in attempts to regularize various gauge theories @xcite . the good ultraviolet behavior of podolsky quantum electrodynamics comes at the cost of introducing a non - tachyonic ghost in the theory @xcite . therefore , podolsky quantum electrodynamics may be viewed as an effective field theory as this kind of ghost may lead to non - unitary evolution in a quantum theory @xcite . despite that , it was pointed out that magnetic monopoles and massive photons may coexist in podolsky quantum electrodynamics @xcite . in fact , this coexistence is not ruled out by the analysis performed in finite - range electrodynamics @xcite owing to the fact that podolsky quantum electrodynamics is a truly long - range electrodynamics with a massless excitation accompanied by a massive one . however , it may be argued that the massive photon of podolsky quantum electrodynamics is not physically sound @xcite . however , when dealing with podolsky regularized electrodynamics as an effective theory aiming at introducing some quantum effects in a otherwise classical realm , these troubles are avoided . at the same time , we may achieve a more vivid description of the physical phenomena . in podolsky regularized classical electrodynamics , it was possible to solve the infamous @xmath0-problem @xcite , and to eliminate runaway solutions from the lorentz dirac equation of motion @xcite . requiring that the correction to the hydrogen ground state energy be smaller than the relative experimental uncertainty , it was shown that the cutoff @xmath10 @xcite . hence , the cutoff length scale is well within the range of quantum phenomena such as pair creation and annihilation , and _ in podolsky electrodynamics , the potential @xmath4 obeys the equations @xmath11 where @xmath12 is the usual dalembert differential operator . these equations for the potential lead to electromagnetic field equations @xmath13 that are of fourth order in the field @xmath7 in contrast to the usual maxwell equations . as @xmath14 are still the components of an antisymmetric tensor , bianchi identities @xmath15 still hold . we see that the regulator modifies only coulomb gauss and ampre maxwell laws , altering the relationship between the electromagnetic field @xmath7 and its sources @xmath8 only at small distances . in order to see that , consider a point particle at rest . it is easy to show that the electric potential @xmath16 in regularized electrodynamics tends to the usual @xmath17 at large distances @xmath18 from the particle , and to the finite @xmath19 at small distances . therefore , podolsky regularized electrodynamics alters the behavior of the field only at small distances from its source . at large distances , we recover maxwell electrodynamics . to see that this modification is generally effective only at small distances , we write eq . [ eq:069 ] in the lorenz gauge , @xmath20 and represent the potential @xmath21 as as a linear combination of propagating plane waves @xmath22 while the current density @xmath23 is expanded accordingly as @xmath24 it is , then , straightforward to show that @xmath25 where @xmath26 . from eq . [ eq:076 ] , we easily arrive at @xmath27 and discover that @xmath28 is reduced to the usual maxwell expression @xmath29 at large distances ( small @xmath30 ) while @xmath28 tends to a distinctive @xmath31 form , @xmath32 at small distances ( large @xmath30 ) . these results hint that the regulator also affects the free space wave propagation , allowing for the coexistence of propagating and evanescent modes even in vacuum . setting @xmath33 in eq . [ eq:076 ] , we find @xmath34 as @xmath35 , we may rewrite eq . [ eq:087 ] as @xmath36 from which we can derive two different dispersion relations for the propagation of an electromagnetic wave in free space . the first dispersion relation is the familiar @xmath37 of maxwell electrodynamics . this linear dispersion relation corresponds to non - dispersive wave propagation with phase velocity @xmath38 . the second dispersion relation , @xmath39 may be more familiar in the form @xmath40 where @xmath41 if @xmath42 . [ eq:090 ] describes electromagnetic wave propagation through a colisionless plasma @xcite with a very small attenuation length @xmath43 . in fact , from eq . [ eq:089 ] , we see that the wavenumber @xmath44 is real only if @xmath45 . otherwise , @xmath44 is a pure imaginary number , leading to evanescent wave modes as shown in figure [ fig:1 ] . associated with evanescent modes . ] we can determine phase and group velocity for the dispersive modes . while phase velocity is determined by the relation @xmath46 group velocity is determined by @xmath47 as suggested by the plasma - like vacuum analogy we are pursuing , @xmath48 and @xmath49 for the propagating modes , for which @xmath44 is real . on the other hand , @xmath50 and @xmath51 for the evanescent modes , for which @xmath44 is imaginary . charged particle concentration in this plasma - like vacuum is given by @xmath52 if @xmath42 . this huge particle density , corresponding to approximately @xmath53 charged particles popping in and out of existence in a sphere of radius @xmath2 , may be interpreted as the cause of the electron jittery motion induced by pair creation and annihilation . in a quantum theory , these results were interpreted as a sign of the existence of two excitations in podolsky electrodynamics @xcite , corresponding to the two kinds of wave we have found : a massless photon , and a massive neutral boson with mass in the range of the @xmath54 boson @xcite . we analyzed wave propagation in the vacuum of podolsky regularized electrodynamics , discovering two different waves : the usual non - dispersive wave with @xmath37 , and a dispersive wave with a propagating mode for high frequency ( @xmath55 ) , and an evanescent mode for low frequency ( @xmath56 ) . in a classical effective theory framework , we interpret this result as arising from a plasma - like behavior of the vacuum induced by quantum vacuum polarization . while high - energy photons ( @xmath57 ) may produce real electron - positron pairs , low - energy photons ( @xmath58 ) can only yield virtual pairs . these virtual particles , living in an otherwise classical vacuum , act like a plasma , disturbing wave propagation . therefore , podolsky regularized electrodynamics inserts some features of pair creation and annihilation into the classical domain . interaction of free - space electromagnetic waves with these virtual particles is equivalent to wave propagation in an effective medium , the classical vacuum plus quantum vacuum polarization , that behaves like a plasma . 42 r. e. norton and w. k. r. watson , _ phys . rev . _ * 116 * , 1597 ( 1959 ) . s. coleman , _ phys . rev . _ * 125 * , 1422 ( 1962 ) . e. j. moniz , d. h. sharp , _ phys d _ * 10 * , 1133 ( 1974 ) . e. j. moniz , d. h. sharp , _ phys d _ * 15 * , 2850 ( 1977 ) . h. levine , e. j. moniz , d. h. sharp , _ am . j. phys . _ * 45 * , 75 ( 1977 ) . a. proca , _ compt . rend . _ * 190 * , 1377 ( 1930 ) . m. born , l. infeld , _ proc . a _ * 144 * , 425 ( 1934 ) . b. podolsky , _ phys . * 62 * , 68 ( 1942 ) . b. podolsky , c. kikuchi , _ phys . rev . _ * 65 * , 228 ( 1944 ) . b. podolsky , c. kikuchi , _ phys . rev . _ * 67 * , 184 ( 1945 ) . b. podolsky , p. schwed , _ rev . phys . _ * 20 * , 40 ( 1948 ) . f. bopp , _ ann . physik _ * 38 * , 345 ( 1940 ) . a. land , _ phys . _ * 60 * , 121 ( 1941 ) . a. land , l. h. thomas , _ phys . _ * 60 * , 514 ( 1941 ) . f. bopp , _ ann . phys . _ * 42 * , 573 ( 1943 ) . a. land , l. h. thomas , _ phys . rev . _ * 65 * , 175 ( 1944 ) . r. p. feynman , _ phys . rev . _ * 74 * , 939 ( 1948 ) . r. r. cuzinatto , c. a. m. de melo , p. j. pompeia , _ ann . phys . _ * 322 * , 1211 ( 2007 ) . w. pauli , f. villars , _ rev . * 21 * , 434 ( 1949 ) . a. a. slavnov , _ nucl . phys . b _ * 31 * , 301 ( 1971 ) . a. a. slavnov , _ theor . math . phys . _ * 13 * , 1064 ( 1972 ) . a. a. slavnov , _ theor . _ * 33 * , 997 ( 1978 ) . l. d. faddeev , a. a. slavnov , _ gauge fields : an introduction to quantum theory _ edition ( addison - wesley , 1991 ) . v. a. rubakov , _ classical theory of gauge fields _ ( princeton university press , 2002 ) . b. grinstein , d. oconnell , m. b. wise , _ phys . d _ * 77 * , 025012 ( 2008 ) . m. a. namazie , _ a _ * 11 * , 713 ( 1980 ) . p. west , _ nucl . b _ * 268 * , 113 ( 1986 ) . a. m. polyakov , _ nucl . phys . b _ * 268 * , 406 ( 1986 ) . j. barcelos - neto , c. a. p. galvo , c. p. natividade , _ z. phys . c _ * 52 * , 559 ( 1991 ) . r. r. cuzinatto , c. a. m. de melo , l. g. medeiros , p. j. pompeia , _ eur . j. c _ * 53 * , 99 ( 2008 ) . a. accioly , h. mukai , _ b _ * 112 * , 1061 ( 1997 ) . c. bloch , _ kgl . danske videnskab selskab mat .- * 27 * , no . 8 , 2 ( 1952 ) . j. j. sakurai , _ advanced quantum mechanics _ ( addison - wesley , 1967 ) . t. matthews , _ math . cambridge phil . soc . _ * 45 * , 441 ( 1949 ) . m. v. s. fonseca , a. v. paredes , _ braz . j. phys . _ * 40 * , 319 ( 2010 ) . a. y. ignatiev , g. c. joshi , _ phys . d _ * 53 * , 984 ( 1996 ) . s. i. kruglov , _ a : math . theor _ * 43 * , 245403 ( 2010 ) . j. frenkel , _ phys . rev . * 54 * , 5859 ( 1996 ) . j. frenkel , r. b. santos , _ int . b _ * 13 * , 315 ( 1999 ) . r. r. cuzinatto , c. a. m. de melo , l. g. medeiros , p. j. pompeia , arxiv:0810.4106v2 [ quant - ph ] ( 2009 ) . m. a. lieberman , a. lichtenberg , _ principles of plasma discharges and materials processing _ ( wiley , 1994 ) . j. d. jackson , _ classical electrodynamics _ edition ( wiley , 1998 ) . a. accioly , e. scatena , _ mod . a _ * 25 * , 1115 ( 2010 ) .

we analyze wave propagation in the vacuum of podolsky regularized electrodynamics . two kinds of waves were found in the theory : the traditional non - dispersive waves of maxwell electrodynamics , and a dispersive wave reminiscent of wave propagation in a collisionless plasma . charged particle concentration was determined , and found to be huge in this vacuum . we interpret the results in terms of vacuum polarization effects induced in an otherwise classical theory .

since the pioneering papers by watts and strogatz on small - world networks @xcite and barabsi and albert on scale - free networks @xcite , complex networks , which describe many systems in nature and society , have become an area of tremendous recent interest @xcite . in the last few years , modeling real - life systems has attracted an exceptional amount of attention within the physics community . while a lot of models have been proposed , most of them are stochastic @xcite . however , because of their advantages , deterministic networks have also received much attention @xcite . first , the method of generating deterministic networks makes it easier to gain a visual understanding of how networks are shaped , and how do different nodes relate to each other @xcite ; moreover , deterministic networks allow to compute analytically their properties : degree distribution , clustering coefficient , average path length , diameter , betweenness , modularity and adjacency matrix whose eigenvalue spectrum characterizes the topology @xcite . the first model for deterministic scale - free networks was proposed by barabsi _ _ in ref . @xcite and was intensively studied in ref . @xcite . another elegant model , called pseudofractal scale - free web ( psw ) @xcite , was introduced by dorogovtsev , goltsev , and mendes , and was extended by comellas _ . @xcite . based on a similar idea of psw , jung _ et al . _ presented a class of recursive trees @xcite . additionally , in order to discuss modularity , ravasz _ et al . _ proposed a hierarchical network model @xcite , the exact scaling properties and extensive study of which were reported in refs . @xcite and @xcite , respectively . recently , in relation to the problem of apollonian space - filing packing , andrade _ et al . _ introduced apollonian networks @xcite which were also proposed by doye and massen in ref . @xcite and have been intensively investigated @xcite . in addition to the above models , deterministic networks can be created by various techniques : modification of some regular graphs @xcite , addition and product of graphs @xcite , edge iterations @xcite and other mathematical methods as in refs . @xcite . as mentioned by barabsi _ _ , it would be of major theoretical interest to construct deterministic models that lead to scale - free networks @xcite . here we do an extensive study on pseudofractal scale - free web @xcite . the psw can be considered as a process of edge multiplication . in fact , a clique ( edge is a special case of it ) can also reproduce new cliques and the number of the new reproduction may be different at a time . motivated by this , in a simple recursive way we propose a general model for psw by including two parameters , with psw as a particular case of the present model . the deterministic construction of our model enables one to obtain the analytic solutions for its structure properties . by adjusting the parameters , we can obtain a variety of scale - free networks . before introducing our model we give the following definitions on a graph ( network ) . the term _ size _ refers to the number of edges in a graph . the number of nodes in a graph is called its _ order_. when two nodes of a graph are connected by an edge , these nodes are said to be _ adjacent _ , and the edge is said to join them . complete graph _ is a graph in which all nodes are adjacent to one another . thus , in a complete graph , every possible edge is present . the complete graph with @xmath1 nodes is denoted as @xmath2 ( also referred in the literature as @xmath1-_clique _ ) . two graphs are _ isomorphic _ when the nodes of one can be relabeled to match the nodes of the other in a way that preserves adjacency . so all @xmath1-cliques are isomorphic to one another . and @xmath3 . only the first three steps are shown.,width=491 ] the network is constructed in a recursive way . we denote the network after @xmath4 steps by @xmath5 , @xmath6 ( see fig . [ recursive ] ) . then the network at step @xmath4 is constructed as follows : for @xmath7 , @xmath8 is a complete graph @xmath9 ( or @xmath10-clique ) consist of @xmath11 @xmath1-cliques ) , and @xmath8 has @xmath11 nodes and @xmath12 edges . for @xmath13 , @xmath5 is obtained from @xmath14 by adding @xmath15 new nodes for each of its existing subgraphs isomorphic to a @xmath1-clique , and each new node is connected to all the nodes of this subgraph . in the special case @xmath16 and @xmath3 , it is reduced to the pseudofractal scale - free web described in ref . @xcite . in the limiting case of @xmath16 , we obtain the same networks as in ref . @xcite . however , our family is richer as @xmath15 can take any natural value . there is an interpretation called ` aggregation ' @xcite for our model . as an example , here we only explain them for the case of @xmath16 and @xmath3 . figure [ pseudofractal ] illustrates the growing process for this particular case , which may be accounted for as an ` aggregation ' process described in detail as follows . first , three of the initial triangle ( @xmath7 ) are assembled to form a new unit ( @xmath17 ) . then we assemble three of these units at the hubs ( the nodes with highest degree ) in precise analogy with the step leading from @xmath7 to @xmath17 to form a new cell ( @xmath18 ) ( see fig . [ aggregation ] ) . this process can be iterated an arbitrary number of times . moreover , an alternative explanation of our model which is often useful is that of ` miniaturization ' ( see ref . @xcite ) . and @xmath3 ) , exhibiting the first three steps.,width=453 ] to @xmath19 , which is obtained by adjoining of three copies of @xmath20 at the hubs.,width=453 ] below we will find that the tunable parameters @xmath15 and @xmath1 control some relevant characteristics of the network @xmath5 . because @xmath3 is a particular case , for conveniences , we treat @xmath3 and @xmath21 separately . _ order and size . _ in the case of @xmath3 , we denote @xmath5 by @xmath22 . let us consider the total number of nodes @xmath23 and total number of edges @xmath24 in @xmath22 . denote @xmath25 as the number of nodes created at step @xmath4 . note that the addition of each new node leads to two new edges . by construction , for @xmath26 , we have @xmath27 and @xmath28 considering the initial condition @xmath29 and @xmath30 , it follows that @xmath31 then the number of nodes increases with time exponentially and the total number of nodes present at step @xmath4 is @xmath32 thus for large @xmath4 , the average degree @xmath33 is approximately @xmath34 . _ degree distribution . _ let @xmath35 be the degree of node @xmath36 at step @xmath4 . then by construction , it is not difficult to find following relation : @xmath37 which expresses a preference attachment @xcite . if node @xmath36 is added to the network at step @xmath38 , @xmath39 and hence @xmath40 therefore , the degree spectrum of the network is discrete . it follows that the degree distribution is given by @xmath41 and that the cumulative degree distribution @xcite is @xmath42 substituting for @xmath38 in this expression using @xmath43 gives @xmath44 so the degree distribution follows the power law with the exponent @xmath45 . for the particular case of @xmath16 , eq . ( [ gamma1 ] ) recovers the result previously obtained in ref . @xcite . _ second moment of degree distribution . _ let us calculate the second moment of degree distribution @xmath46 . it is defined by @xmath47^{2},\end{aligned}\ ] ] where @xmath48 is the degree of a node at step @xmath4 , which was generated at step @xmath38 . this quality expresses the average of degree square over all nodes in the network . it has large impact on the dynamics of spreading @xcite and the onset of percolation transitions @xcite taking place in networks . when @xmath46 is diverging , the networks allow the onset of large epidemics whatever the spreading rate of the infection @xcite , at the same time the networks are extremely robust to random damages , in other words , the percolation transition is absent @xcite . substituting eqs . ( [ nv1 ] ) , ( [ nt1 ] ) and ( [ ki1 ] ) into eq . ( [ ki21 ] ) , we derive @xmath49\nonumber\\ & \approx&\frac{8\,(m+1)^{2t+1}}{m(2m+1)}\rightarrow\infty \qquad\hbox{for large $ t$.}\end{aligned}\ ] ] in this way , second moment of degree distribution @xmath50 has been calculated explicitly , and result shows that it diverges as an exponential law . so the networks are resilient to random damage and are simultaneously sensitive to the spread of infections . _ degree correlations . _ as the field has progressed , degree correlation @xcite has been the subject of particular interest , because it can give rise to some interesting network structure effects . an interesting quantity related to degree correlations is the average degree of the nearest neighbors for nodes with degree @xmath51 , denoted as @xmath52 @xcite . when @xmath52 increases with @xmath51 , it means that nodes have a tendency to connect to nodes with a similar or larger degree . in this case the network is defined as assortative @xcite . in contrast , if @xmath53 is decreasing with @xmath51 , which implies that nodes of large degree are likely to have near neighbors with small degree , then the network is said to be disassortative . if correlations are absent , @xmath54 . we can exactly calculate @xmath55 for the networks using eq . ( [ ki1 ] ) to work out how many links are made at a particular step to nodes with a particular degree . except for three initial nodes generated at step 0 , no nodes born in the same step , which have the same degree , will be linked to each other . all links to nodes with larger degree are made at the creation step , and then links to nodes with smaller degree are made at each subsequent steps . this results in the expression @xmath56 for @xmath57 . here the first sum on the right - hand side accounts for the links made to nodes with larger degree ( i.e.@xmath58 ) when the node was generated at @xmath38 . the second sum describes the links made to the current smallest degree nodes at each step @xmath59 . substituting eqs . ( [ nv1 ] ) and ( [ ki1 ] ) into eq . ( [ knn1 ] ) , after some algebraic manipulations , eq . ( [ knn1 ] ) is simplified to @xmath60^{t_i}-\frac{2(m+1)}{m}+\frac{2m}{m+1}\,(t - t_i).\end{aligned}\ ] ] thus after the initial step @xmath55 grows linearly with time . writing eq . ( [ knn2 ] ) in terms of @xmath51 , it is straightforward to obtain @xmath61^{t}\,\left ( \frac{k}{2}\right)^{-\frac{\ln\left [ \frac{(m+1)^{2}}{2m+1}\right ] } { \ln(m+1)}}\nonumber\\ \qquad\qquad\qquad\qquad-\frac{2(m+1)}{m}+\frac{2m}{m+1}\,\frac{\ln(\frac{k}{2})}{\ln(m+1)}.\end{aligned}\ ] ] therefore , @xmath52 is approximately a power law function of @xmath51 with negative exponent , which shows that the networks are disassortative . note that @xmath52 of the internet exhibit a similar power - law dependence on the degree @xmath62 , with @xmath63 @xcite . _ clustering coefficient . _ the clustering coefficient defines a measure of the level of cohesiveness around any given node . by definition , the clustering coefficient @xcite @xmath64 of node @xmath36 is the ratio between the number of edges @xmath65 that actually exist among the @xmath66 neighbors of node @xmath36 and its maximum possible value , @xmath67 , i.e. , @xmath68 . the clustering coefficient of the whole network is the average of all individual @xmath69 . next we will compute the clustering coefficient of every node and their average value . obviously , when a new node @xmath36 joins the network , its degree @xmath70 and @xmath71 is @xmath72 and @xmath73 , respectively . each subsequent addition of a link to that node increases both @xmath70 and @xmath71 by one . thus , @xmath71 equals to @xmath74 for all nodes at all steps . so one can see that , there is a one - to - one correspondence between the degree of a node and its clustering . for a node with degree @xmath51 , the exact expression for its clustering coefficient is @xmath75 . therefore , the clustering coefficient spectrum of nodes is discrete . using this discreteness , it is convenient to work with the cumulative distribution of clustering coefficient @xcite as @xmath76 it is worth noting that for the special case of @xmath16 , this result has been obtained previously @xcite . the clustering coefficient of the whole network at arbitrary step @xmath4 can be easily computed , @xmath77}.\end{aligned}\ ] ] in the infinite network size limit ( @xmath78 ) , @xmath79 . thus the clustering @xmath80 is high and increases with @xmath15 . moreover , similarly to the degree exponent @xmath81 , @xmath80 is tunable by choosing the right value of parameter @xmath15 : in particular , @xmath80 ranges from @xmath82 ( in the special case of @xmath16 @xcite ) to limit of 1 when @xmath15 becomes very large . _ diameter . _ the diameter of a network is defined as the maximum of the shortest distances between all pairs of nodes , which characterizes the longest communication delay in the network . small diameter is consistent with the concept of small - world and it is easy to compute for our networks . below we give the precise analytical computation of diameter of @xmath22 denoted by @xmath83 . it is easy to see that at step @xmath84 ( resp . @xmath85 ) , the diameter is equal to 1 ( resp . 2 ) . at each step @xmath86 , one can easily see that the diameter always lies between a pair of nodes that have just been created at this step . in order to simplify the analysis , we first note that it is unnecessary to look at all the nodes in the networks in order to find the diameter . in other words , some nodes added at a given step can be ignored , because they do not increase the diameter from the previous step . these nodes are those that connect to edges that already existed before step @xmath87 . indeed , for these nodes we know that a similar construction has been done in previous steps , so we can ignore them for the computation of the diameter . let us call `` outer '' nodes the nodes which are connected to a edge that did not exist at previous steps . clearly , at each step , the diameter depends on the distances between outer nodes . at any step @xmath86 , we note that an outer node can not be connected with two or more nodes that were created during the same step @xmath88 . indeed , we know that from step @xmath72 , no outer node is connected to two nodes of the initial triangle @xmath89 . thus , for any step @xmath86 , any outer node is connected with nodes that appeared at pairwise different steps . now consider two outer nodes created at step @xmath86 , say @xmath90 and @xmath91 . then @xmath90 is connected to two nodes , and one of them must have been created before or during step @xmath92 . we repeat this argument , and we end up with two cases : ( 1 ) @xmath93 is even . then , if we make @xmath15 jumps " , from @xmath90 we reach the initial triangle @xmath89 , in which we can reach any @xmath91 by using an edge of @xmath89 and making @xmath15 jumps to @xmath91 in a similar way . thus @xmath94 . ( 2 ) @xmath95 is odd . in this case we can stop after @xmath15 jumps at @xmath96 , for which we know that the diameter is 2 , and make @xmath15 jumps in a similar way to reach @xmath91 . thus @xmath97 . it is easily seen that the bound can be reached by pairs of outer nodes created at step @xmath4 . more precisely , those two nodes @xmath90 and @xmath91 share the property that they are connected to two nodes that appeared respectively at steps @xmath87 , @xmath92 . hence , formally , @xmath98 for any @xmath99 . note that @xmath100 , thus the diameter is small and scales logarithmically with the number of network nodes . in these cases , the analysis is a little difficult than those of the last subsection . an alternative approach has to be adopted , although it may also holds true for the first case in some situations . the method of the last subsection is relatively easy to generalize to these cases , and below we will address it , focusing on order , size , degree distribution , clustering coefficient and diameter . _ order and size . _ let @xmath25 , @xmath101 be the number of nodes and edges created at step @xmath4 , respectively . denote @xmath102 as the total number of @xmath1-cliques in the whole network at step @xmath4 . note that the addition of each new node leads to @xmath1 new @xmath1-cliques and @xmath1 new edges . by construction , we have @xmath103 , @xmath104 and @xmath105 . thus one can easily obtain @xmath106 ( @xmath99 ) , @xmath107 ( @xmath108 ) and @xmath109 ( @xmath108 ) . from above results , we can easily compute the order and size of the networks . the total number of nodes @xmath23 and edges @xmath24 present at step @xmath4 is @xmath110}{q}\end{aligned}\ ] ] and @xmath111 respectively . for infinite @xmath4 , the average degree @xmath33 is approximately @xmath112 . _ degree distribution . _ when a new node @xmath36 is added to the graph at step @xmath38 , it has degree @xmath1 and forms @xmath1 new @xmath1-cliques . let @xmath113 be the total number of @xmath1-cliques at step @xmath4 that will created new nodes connected to the node @xmath36 at step @xmath114 . so at step @xmath38 , @xmath115 . by construction , we can see that in the subsequent steps each new neighbor of @xmath36 generates @xmath116 new @xmath1-cliques with @xmath36 as one node of them . let @xmath35 be the degree of @xmath36 at step @xmath4 . it is not difficult to find following relations for @xmath117 : @xmath118 and @xmath119 from the above two equations , we can derive @xmath120n_q(i , t-1)$ ] . considering @xmath115 , we obtain @xmath121^{t - t_i}$ ] and @xmath122^{t - t_i-1}$ ] . then the degree @xmath123 of node @xmath36 at time @xmath4 is @xmath124^{t - t_i}+q^{2}-2q}{q-1}.\end{aligned}\ ] ] since the degree of each node has been obtained explicitly as in eq . ( [ ki ] ) , we can get the degree distribution via its cumulative distribution @xcite , i.e. @xmath125 , where @xmath126 denotes the number of nodes with degree @xmath127 . the analytic computation details are given as follows . for a degree @xmath51 @xmath128^{t - j}+q^{2}-2q}{q-1},\ ] ] there are @xmath129 nodes with this exact degree , all of which were born at step @xmath130 . all nodes with birth time at @xmath130 or earlier have this and a higher degree . so we have @xmath131}{q}. \nonumber\\\ ] ] as the total number of nodes at step @xmath4 is given in eq . ( [ nt ] ) , we have @xmath132^{t - j}+q^{2}-2q}{q-1}\right]^{1-\gamma}\nonumber\\ = \frac{\frac{(q+1)[(mq+1)^{j}+q-1]}{q}}{\frac{(q+1)[(mq+1)^{t}+q-1]}{q}}.\nonumber\\\ ] ] therefore , for large @xmath4 we obtain @xmath133^{t - j}\right ] ^{1-\gamma}=(mq+1)^{j - t}\ ] ] and @xmath134}.\ ] ] for the special case @xmath16 , eq . ( [ gamma ] ) recovers the results previously reported in ref . @xcite . _ clustering coefficient_. the analytical expression for clustering coefficient @xmath135 of the individual node with degree @xmath51 can be derived exactly . when a node is created it is connected to all the nodes of a @xmath1-clique whose nodes are completely interconnected . its degree and clustering coefficient are @xmath1 and 1 , respectively . in the following steps , if its degree increases one by a newly created node connecting to it , then there must be @xmath116 existing neighbors of it attaching to the new node at the same time . thus for a node of degree @xmath51 , we have @xmath136 which depends on degree @xmath51 and @xmath1 . for @xmath137 , the @xmath135 is inversely proportional to node degree . the scaling @xmath138 has been found for some network models @xcite , and has also observed in several real - life networks @xcite . using eq . ( [ ck ] ) , we can obtain the clustering @xmath139 of the networks at step @xmath4 : @xmath140 where the sum is the total of clustering coefficient for all nodes and @xmath141^{t - r}+q^{2}-2q}{q-1}$ ] shown by eq . ( [ ki ] ) is the degree of the nodes created at step @xmath142 . on @xmath1 and @xmath15.,width=340 ] it can be easily proved that for arbitrary fixed @xmath15 , @xmath139 increases with @xmath1 , and that for arbitrary fixed @xmath1 , @xmath139 increases with @xmath15 . in the infinite network order limit ( @xmath143 ) , eq . ( [ ac ] ) converges to a nonzero value @xmath144 . when @xmath3 , for @xmath16 , 2 , 3 and 4 , @xmath144 equal to 0.8000 , 0.8571 0.8889 and 0.9091 , respectively . when @xmath145 , for @xmath3 , 3 , 4 and 5 , @xmath144 are 0.8571 , 0.9100 , 0.9348 and 0.9490 , respectively . therefore , the clustering coefficient of our networks is very high . moreover , similarly to the degree exponent @xmath81 , clustering coefficient @xmath144 is determined by @xmath1 and @xmath15 . figure [ cc ] shows the dependence of @xmath144 on @xmath1 and @xmath15 . _ diameter . _ in what follows , the notations @xmath146 and @xmath147 express the integers obtained by rounding @xmath148 to the nearest integers towards infinity and minus infinity , respectively . now we compute the diameter of @xmath5 , denoted @xmath149 for @xmath150 ( @xmath3 is a particular case that is treated separately in the last subsection ) : _ step 0_. the diameter is @xmath73 . _ steps 1 to @xmath151_. in this case , the diameter is 2 , since any new node is by construction connected to a @xmath1-clique forming a @xmath10-clique , and since any @xmath10-clique during those steps contains at least @xmath152 ( @xmath1 even ) or @xmath152 + 1 ( @xmath1 odd ) nodes from the initial @xmath10-clique @xmath8 obtained after step 0 . hence , any two newly added nodes @xmath153 and @xmath154 will be connected respectively to sets @xmath155 and @xmath156 , with @xmath157 and @xmath158 , where @xmath159 is the node set of @xmath8 ; however , since @xmath160 ( @xmath1 even ) and @xmath161 + 1 ( @xmath1 odd ) , where @xmath162 denotes the number of elements in set @xmath163 , we conclude that @xmath164 , and thus the diameter is 2 . _ steps @xmath152 to @xmath1_. in any of these steps , some newly added nodes might not share a neighbor in the original @xmath10-clique @xmath8 ; however , any newly added node is connected to at least one node of the initial @xmath10-clique @xmath8 . thus , the diameter is equal to 3 . _ further steps_. similar to the case of @xmath3 , we call `` outer '' nodes the nodes which are connected to a @xmath1-clique that did not exist at previous steps . clearly , at each step , the diameter depends on the distances between outer nodes . now , at any step @xmath165 , an outer node can not be connected with two or more nodes that were created during the same step @xmath166 . moreover , by construction no two nodes that were created during a given step are neighbors , thus they can not be part of the same @xmath1-clique . therefore , for any step @xmath165 , some outer nodes are connected with nodes that appeared at pairwise different steps . thus , if @xmath167 denotes an outer node that was created at step @xmath4 , then @xmath167 is connected to nodes @xmath168s , @xmath169 , where all the @xmath36s are pairwise distinct . we conclude that @xmath167 is necessarily connected to a node that was created at a step @xmath170 . if we repeat this argument , then we obtain an upper bound on the distance from @xmath167 to the initial @xmath10-clique @xmath8 . let @xmath171 , where @xmath172 . then , we see that @xmath167 is at distance at most @xmath173 from a node in @xmath5 . hence any two nodes @xmath167 and @xmath174 in @xmath8 lie at distance at most @xmath175 ; however , depending on @xmath176 , this distance can be reduced by 1 , since when @xmath177 , we know that two nodes created at step @xmath176 share at least a neighbor in @xmath8 . thus , when @xmath178 , @xmath179 , while when @xmath180 , @xmath181 . one can see that these bounds can be reached by pairs of outer nodes created at step @xmath4 . more precisely , those two nodes @xmath167 and @xmath174 share the property that they are connected to @xmath1 nodes that appeared respectively at steps @xmath182 . based on the above arguments , one can easily see that for @xmath183 , the diameter increases by 2 every @xmath1 steps . more precisely , we have the following result , for any @xmath150 and @xmath13 ( when @xmath7 , the diameter is clearly equal to 1 ) : @xmath184 where @xmath185 if @xmath186 , and 1 otherwise . when @xmath4 gets large , @xmath187 , while @xmath188 , thus the diameter grows logarithmically with the number of nodes . it is easy to see that these cases of @xmath21 have very similar topological properties to the case @xmath3 . additionally , for the cases of @xmath21 , the networks will again be disassortative with respect to degree because of the lack of links between nodes with the same degree ; the second moment of degree distribution @xmath50 will also diverge , which is due to the fat tail of the degree distribution . to sum up , we have proposed and investigated a deterministic network model , which is constructed in a recursive fashion . our model is actually a tunable generalization of the growing deterministic scale - free networks introduced in ref . @xcite . aside from their deterministic structures , the statistical properties of the resulting networks are equivalent with the random models that are commonly used to generate scale - free networks @xcite . we have obtained the exact results for degree distribution and clustering coefficient , as well as the diameter , which agree well with large amount of real observations @xcite . the degree exponent can be adjusted , the clustering coefficient is very large , and the diameter is small . therefore , out model may perform well in mimicking a variety of scale - free networks in real - life world . moreover , our networks consist of cliques , which has been observed in variety of the real - world networks , such as movie actor collaboration networks , scientific collaboration networks and networks of company directors @xcite . this research was supported in part by the national natural science foundation of china ( nnsfc ) under grant nos . 60373019 , 60573183 , and 90612007 . lili rong gratefully acknowledges partial support from nnsfc under grant nos . 70431001 and 70571011 . the authors thank the anonymous referees for their valuable comments and suggestions .

we propose a general geometric growth model for pseudofractal scale - free web , which is controlled by two tunable parameters . we derive exactly the main characteristics of the networks : degree distribution , second moment of degree distribution , degree correlations , distribution of clustering coefficient , as well as the diameter , which are partially determined by the parameters . analytical results show that the resulting networks are disassortative and follow power - law degree distributions , with a more general degree exponent tuned from 2 to @xmath0 ; the clustering coefficient of each individual node is inversely proportional to its degree and the average clustering coefficient of all nodes approaches to a large nonzero value in the infinite network order ; the diameter grows logarithmically with the number of network nodes . all these reveal that the networks described by our model have small - world effect and scale - free topology . complex networks , scale - free networks , disordered systems , networks

recent investigations of the large scale distribution of galaxies in the sloan digital sky survey ( sdss ; @xcite ) have revealed a complex relationship between the properties of galaxies , ( such as color , luminosity , surface brightness , and concentration ) and their environments ( @xcite ) . these and other investigations using the sdss ( @xcite ) and the two - degree field galaxy redshift survey ( @xcite ) have found that galaxy clustering is a function both of star formation history and of luminosity . for low luminosity galaxies , clustering is a strong function of color , while for luminous galaxies clustering is a strong function of luminosity . for red galaxies , clustering is a non - monotonic function of luminosity , peaking at both high and low luminosities . although galaxy clustering correlates also with surface brightness and concentration , @xcite and @xcite show that galaxy environment is independent of these properties at fixed color and luminosity . thus , color and luminosity measures of star formation history appear to have a more fundamental relationship with environment than do surface brightness and concentration measures of the distribution of stars within the galaxy . some of the investigations above have explored the scale dependence of these relationships . studies of the correlation function , such as @xcite and @xcite , can address this question , but do not address directly whether the density on large scales is related to galaxy properties _ independent _ of the relationships with density on small scales . if only the _ masses _ of the host halos of galaxies strongly affect their properties , then we expect no such independent relationship between galaxy properties and the large scale density field . thus , it is important to examine this issue in order to test the assumptions of the `` halo model '' description of galaxy formation and of semi - analytic models that depend only on the properties of the host halo ( _ e.g. _ , @xcite ) . recent studies of this question have come to conflicting conclusions . for example , @xcite have concluded from their analysis of sdss and 2dfgrs galaxies that the equivalent width of h@xmath4 is a function of environment measured on scales of 1.1 @xmath2 mpc and 5.5 @xmath2 mpc independently of each other . on the other hand , @xcite find that at fixed density at scales of 1 @xmath2 mpc , the distribution of d4000 ( a measure of the age of the stellar population ) is not a strong function of density on larger scales . here we address the dependence on scale of the relative bias of sdss galaxies . section [ data ] describes our data set . section [ results ] explores how the relationship between the color , luminosity , and environments of galaxies depends on scale . section [ bluefrac ] resolves the discrepancy noted in the previous paragraph between @xcite and @xcite , finding that only small scales are important to the recent star formation history of galaxies . section [ summary ] summarizes the results . where necessary , we have assumed cosmological parameters @xmath5 , @xmath6 , and @xmath7 km s@xmath8 mpc@xmath8 with @xmath9 . the sdss is taking @xmath10 ccd imaging of @xmath11 of the northern galactic sky , and , from that imaging , selecting @xmath12 targets for spectroscopy , most of them galaxies with @xmath13 ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? automated software performs all of the data processing : astrometry @xcite ; source identification , deblending and photometry @xcite ; photometricity determination @xcite ; calibration @xcite ; spectroscopic target selection @xcite ; spectroscopic fiber placement @xcite ; and spectroscopic data reduction . an automated pipeline called idlspec2d measures the redshifts and classifies the reduced spectra ( schlegel et al . , in preparation ) . the spectroscopy has small incompletenesses coming primarily from ( 1 ) galaxies missed because of mechanical spectrograph constraints ( 6 percent ; * ? ? ? * ) , which leads to a slight under - representation of high - density regions , and ( 2 ) spectra in which the redshift is either incorrect or impossible to determine ( @xmath14 percent ) . in addition , there are some galaxies ( @xmath15 percent ) blotted out by bright galactic stars , but this incompleteness should be uncorrelated with galaxy properties . for the purposes of computing large - scale structure and galaxy property statistics , we have assembled a subsample of sdss galaxies known as the nyu value added galaxy catalog ( nyu - vagc ; @xcite ) . one of the products of that catalog is a low redshift catalog . here we use the version of that catalog corresponding to the sdss data release 2 ( dr2 ) . the low redshift catalog has a number of important features which are useful in the study of low luminosity galaxies . most importantly : 1 . we have checked by eye all of the images and spectra of low luminosity ( @xmath16 ) or low redshift ( @xmath17 ) galaxies in the nyu - vagc . most significantly , we have trimmed those which are `` flecks '' incorrectly deblended out of bright galaxies ; for some of these cases , we have been able to replace the photometric measurements with the measurements of the parents . for a full description of our checks , see @xcite . 2 . for galaxies which were shredded in the target version of the deblending , the spectra are often many arcseconds away from the nominal centers of the galaxy in the latest version of the photometric reductions . we have used the new version of the deblending to decide whether these ( otherwise non - matched spectra ) should be associated with the galaxy in the best version . we have estimated the distance to low redshift objects using the @xcite model of the local velocity field ( using @xmath18 ) , and propagated the uncertainties in distance into uncertainties in absolute magnitude . for the purposes of our analysis below , we have matched this sample to the results of @xcite , who measured emission line fluxs and equivalent widths for all of the sdss spectra . below , we use their results for the h@xmath4 equivalent width . the range of distances we include is @xmath19 @xmath2 mpc , making the sample volume limited for galaxies with @xmath20 . the total completeness - weighted effective area of the sample , excluding areas close to tycho stars , is 2220.9 square degrees . the catalog contains 28,089 galaxies . @xcite have investigated the luminosity function , surface brightness selection effects , and galaxy properties in this sample . we will be studying the environments of galaxies as a function of their luminosity and color below . to give a sense of the morphological properties of galaxies with various luminosities and colors , figure [ color_mag ] shows galaxies randomly selected in bins of color and luminosity . each image is 40 @xmath2 kpc on a side . red , high luminosity galaxies are classic giant ellipticals . lower luminosity red galaxies tend to be more flattened and less concentrated . blue , high luminosity galaxies have well - defined spiral structure and dust lanes . lower luminosity blue galaxies have less well - defined bulges and fewer spiral features . in order to evaluate the environments of galaxies in our sample , we perform the following procedure . first , for each given galaxy in the sample , we count the number of other galaxies @xmath21 with @xmath22 outside a projected radius of 10 @xmath2 kpc and within some outer radius @xmath23 , which we will vary below , and within @xmath24 km s@xmath8 in the redshift direction . this trace catalog is volume - limited within @xmath25 . in order to make a more direct comparison to @xcite , we will also use a trace catalog containing only galaxies with @xmath26 . second , we calculate the mean expected number of galaxies in that volume as : @xmath27 where @xmath28 is the sampling fraction of galaxies in the right ascension ( @xmath4 ) and declination ( @xmath29 ) direction of each point within the volume . we perform this integral using a monte carlo approach , distributing random points inside the volume with a density modulated by the sampling fraction @xmath28 . in order to calculate the mean density around galaxies in various classes , we will simply calculate : @xmath30 as the density with respect to the mean . when one calculates the mean density around galaxies , it is necessary to have a fair sample of the universe . for the most luminous galaxies in our sample ( @xmath31 ) the sample is volume - limited out to our redshift limit of @xmath32 and constitutes the equivalent of a 60 @xmath2 mpc radius sphere , which constitutes a fair sample for many purposes ( @xmath33cdm predicts a variance in such a sphere to be about 0.13 ) . however , the lower luminosity galaxies can only be seen in the fraction of this volume which is nearby , and below a certain luminosity the sample is no longer fair . for example , consider figure [ check_rho_converge ] , which shows the cumulative mean density around galaxies with @xmath34 in spheres of larger and larger radius around the milky way . the mean overdensity does not converge until a volume which corresponds to approximately @xmath35 . thus , it is not really safe to evaluate the mean density around galaxies that are too low luminosity to be observed out that far in redshift , which is to say , less luminous than @xmath36 . however , for the moment let us consider figure [ biden_all ] . the greyscale and contours show the mean density relative to the mean as a function of color and luminosity , using a projected radius of @xmath37 @xmath2 mpc . the mean is calculated in a sliding box with the width shown . if the sliding box contains fewer than 20 galaxies , the result is ignored and colored pure white . here we show the results for the entire sample . our statistical uncertainties are well - behaved down to about @xmath38 , but we are likely to be cosmic variance limited for @xmath39 , as indicated by the solid vertical line . thus , the apparent decline in the mean overdensity for red galaxies lower luminosity than @xmath40 is probably spurious . despite that limitation , we note that there is a strong relationship between environment and color even at @xmath41 . we note in passing that we can still use the variation of the density within @xmath42 to study the properties of galaxies as a function of density down to low luminosity . just because the _ mean _ density of galaxies in that volume has not converged does _ not _ imply that there is insufficient variation of density to study the variation of galaxy properties with environment . for our fair sample of galaxies with @xmath43 , figure [ biden_scales ] shows the dependence of overdensity on luminosity and color for six different projected radii : 0.2 , 0.5 , 1 , 2 , 4 , and 6 @xmath2 mpc . we only show results for @xmath44 , for which we have a fair sample . obviously , the density contrast decreases with scale ; on the other hand , the qualitative form of the plot does not change . our results remain similar to those shown in @xcite and @xcite . the results here demonstrate that the environments of low luminosity , red galaxies do continue to become denser as absolute magnitude increases down to absolute magnitudes of @xmath45 ( about two magnitudes less luminous than explored by our previous work ) . figure [ biden_ratios ] shows the ratio of the overdensity @xmath29 at each scale relative to that at the largest scale of @xmath46 @xmath2 mpc . this ratio is a measure of the steepness of the cross - correlation between galaxies of a given color and absolute magnitude with _ all _ galaxies in our volume - limited sample ( @xmath47 ) . interestingly , the contours in steepness are qualitatively similar to the contours in overdensity in figure [ biden_scales ] . this similarity implies that for each class of galaxy , the strength of the correlation on large scales always is associated with a _ steeper _ correlation function . another way of looking at similar results is to ask , as a function of environment , what fraction of galaxies are blue . we split the sample into `` red '' and `` blue '' galaxies using the following , luminosity - dependent cut : @xmath48 blue galaxies thus have @xmath49 . we then sort all the galaxies with @xmath50 into bins of density on three different scales : @xmath51 , @xmath52 , and @xmath53 @xmath2 mpc . in each bin we calculate the fraction of blue galaxies . figure [ lowz_fracblue ] shows this blue fraction as a function of density . in all cases , the blue fraction declines as a function of density , as one would expect based on figure [ biden_scales ] above , and from the astronomical literature ( a highly abridged list of relevant work would include @xcite ) . if we divide the sample into bins of luminosity , we find that higher luminosities have smaller blue fractions ( of course ) but that the dependence of blue fraction on density does not change . the question naturally arises : _ which _ scales are important to the process of galaxy formation ? is the local environment within 0.5 @xmath2 mpc the only important consideration ? or is the larger scale environment also important ? for example , consider figure [ denvden ] , which shows the conditional dependence of the three density estimators at the three scales on each other . the diagonal plots simply show the distribution within our sample of each density estimator . the off - diagonal plots show the conditional distribution of the quantity on the @xmath54-axis given the quantity on the @xmath55-axis . as an example , the lower right panel shows @xmath56 shows the fraction of blue galaxies as a function of two density estimates : one with @xmath51 @xmath2 mpc and one with @xmath57 @xmath2 mpc . in this case it is clear that the blue fraction is a function of both densities . that is , even at a fixed density on scales of @xmath37 @xmath2 mpc , the density outside that radius matters to the blue fraction ; in addition , at a fixed density on scales of @xmath52 @xmath2 mpc , the distribution of galaxies within that radius appears to affect the blue fraction as well . on the other hand , consider figure [ lowz_fracblue2_1.0 - 6.0 ] , which is the same as figure [ lowz_fracblue2_0.5 - 1.0 ] , but now showing the densities at scales of @xmath57 and @xmath53 @xmath2 mpc . in figure [ lowz_fracblue2_1.0 - 6.0 ] the contours are vertical , indicating that the density between @xmath52 and @xmath53 @xmath2 mpc has very little effect on galaxy properties . at a fixed value of the density at the smaller scale , the larger scale environment appears to be of little importance . @xcite found that these contours were not vertical when he looked at the fraction of galaxies for which the h@xmath4 equivalent width was @xmath58 . their result appears in conflict with that of the previous paragraph . on the other hand , the emission lines measure a more recent star formation rate than does the color ; it is possible in principle that the more recent star formation rate depends more strongly on large - scale environment . to rule out this possibility , figure [ lowz_frachalpha2_1.0 - 6.0 ] shows the same result as figure [ lowz_fracblue2_1.0 - 6.0 ] , but now showing the fraction of galaxies with h@xmath4 equivalent widths ( as measured by @xcite ) greater than 4 . again , for strong emission line fraction as for the blue fraction , the smaller scales are important , but the 6 @xmath2 mpc scales are not , in contradiction with @xcite . why , then , did @xcite conclude that large scales _ were _ important ? there are a number of differences between our study and theirs . first , their contouring method differs ; instead of measuring the blue fraction in bins of fixed size , at each point they measure the star - forming fraction among the nearest 500 galaxies in the plane of @xmath59 and @xmath60 . we have found that this procedure creates a _ slight _ bias in the contouring in the sense that near the edges of the distribution vertical contours will become diagonal . however , this effect is not strong enough to explain the differences between our results and those of @xcite . second , to estimate the density in their sample they used a spherical gaussian filter , whereas here we use the overdensity in cones . we have not investigated what effect this difference has . finally , they use tracer galaxies with a considerably lower mean density than ours . their effective absolute magnitude limit is @xmath61 ; such galaxies have a mean density of @xmath62 @xmath63 mpc@xmath64 . our tracers ( @xmath22 ) have a mean density of @xmath65 @xmath63 mpc@xmath64 , almost six times higher . figure [ lowz_frachalpha2_m-20.5_1.0 - 6.0 ] shows our results when we restrict our tracer sample to @xmath61 . the contours in this figure are very diagonal , similar to the results of @xcite . this result suggests that one of two possible mechanisms are causing the differences between our results and those of @xcite . first , the higher luminosity galaxies with @xmath61 might be yielding fundamentally different information about the density field than our lower luminosity tracers . second , the lower mean density of the galaxies with @xmath61 might be effectively introducing `` noise '' in the measurement on small scales . remember that the large scale and small scale densities are intrinsically correlated . so if the small scale measurement is noisy enough , the higher signal - to - noise ratio large scale measurement could actually be adding extra information about the environment on _ small _ scales . such an effect would make the contours in figure [ lowz_frachalpha2_m-20.5_1.0 - 6.0 ] diagonal . we have performed a simple test to distinguish these possibilities , which is to remake figure [ lowz_frachalpha2_1.0 - 6.0 ] using the low luminosity tracers ( @xmath66 ) but subsampling them to the same mean density as the high luminosity tracers ( @xmath67 ) . this test yields diagonal contours , meaning one can understand the diagonal contours of figure [ lowz_frachalpha2_m-20.5_1.0 - 6.0 ] and of @xcite as simply reflecting the low signal - to - noise ratio of the density estimates . we explore the relative bias between galaxies as a function of scale , finding the following . 1 . the dependence of mean environment on color persists to the lowest luminosities we explore ( @xmath68 ) . red , low luminosity galaxies tend to be in overdense regions , down at least to @xmath69 . this result extends those found by @xcite and @xcite towards lower luminosities by about 2 magnitudes . 3 . at any given point of color and luminosity , a correlation function with a stronger amplitude implies correlation function with a steeper slope . 4 . in regions of a given overdensity on small scales ( @xmath70 @xmath2 mpc ) , the overdensity on large scales ( @xmath46 @xmath2 mpc ) does not appear to relate to the recent star formation history of the galaxies . the last point above deserves elaboration . first , it contradicts the results of @xcite . we have found that their results are probably due to the low mean density of the tracers they used . this explanation underscores the importance of taking care when using low signal - to - noise quantities . galaxy environments are difficult to measure , in the sense we use tracers that do not necessarily trace the `` environment '' perfectly , meaning neither with low noise nor necessarily in an unbiased manner . we claim here that our higher density of tracers marks an improvement over previous work , but it is worth noting the limitations of assuming that the local galaxy density fairly and adequately represents whatever elements of the environment affect galaxy formation . second , if the galaxy density field is an adequate representation of the environment , the result has important implications regarding the physics of galaxy formation . in simulations whose initial conditions are constrained by cosmic microwave background observations and galaxy large - scale structure observations , virialized dark matter halos do not extend to sizes much larger than @xmath71@xmath72 @xmath2 mpc . thus , our results are consistent with the notion that only the masses of the host halos of the galaxies we observe are strongly affecting the star formation of the galaxies . in addition , @xcite find that only the star formation histories , not the azimuthally - averaged structural parameters , are directly related to environment . for these reasons , it is likely that we can understand the process of galaxy formation by only considering the properties of the host dark matter halos . our results therefore encourage the `` halo model '' description of galaxy formation and the pursuit of semi - analytic models which depend only on the properties of the host halo ( _ e.g. _ , @xcite ) . thanks to eric bell and george lake for useful discussions during this work . thanks to guinevere kauffmann for encouraging us to pursue this question . thanks to christy tremonti and jarle brinchmann for the public distribution of their measurements of sdss spectra . funding for the creation and distribution of the sdss has been provided by the alfred p. sloan foundation , the participating institutions , the national aeronautics and space administration , the national science foundation , the u.s . department of energy , the japanese monbukagakusho , and the max planck society . the sdss web site is http://www.sdss.org/. the sdss is managed by the astrophysical research consortium ( arc ) for the participating institutions . the participating institutions are the university of chicago , fermilab , the institute for advanced study , the japan participation group , the johns hopkins university , los alamos national laboratory , the max - planck - institute for astronomy ( mpia ) , the max - planck - institute for astrophysics ( mpa ) , new mexico state university , university of pittsburgh , princeton university , the united states naval observatory , and the university of washington . , m. , eke , v. , miller , c. , lewis , i. , bower , r. , couch , w. , nichol , r. , bland - hawthorn , j. , baldry , i. k. , baugh , c. , bridges , t. , cannon , r. , cole , s. , colless , m. , collins , c. , cross , n. , dalton , g. , de propris , r. , driver , s. p. , efstathiou , g. , ellis , r. s. , frenk , c. s. , glazebrook , k. , gomez , p. , gray , a. , hawkins , e. , jackson , c. , lahav , o. , lumsden , s. , maddox , s. , madgwick , d. , norberg , p. , peacock , j. a. , percival , w. , peterson , b. a. , sutherland , w. , & taylor , k. 2004 , , 348 , 1355

we investigate the relationship between the colors , luminosities , and environments of galaxies in the sloan digital sky survey spectroscopic sample , using environmental measurements on scales ranging from @xmath0 to @xmath1 @xmath2 mpc . we find : ( 1 ) that the relationship between color and environment persists even to the lowest luminosities we probe ( @xmath3 ) ; ( 2 ) at luminosities and colors for which the galaxy correlation function has a large amplitude , it also has a steep slope ; and ( 3 ) in regions of a given overdensity on small scales ( 1 @xmath2 mpc ) , the overdensity on large scales ( 6 @xmath2 mpc ) does not appear to relate to the recent star formation history of the galaxies . of these results , the last has the most immediate application to galaxy formation theory . in particular , it lends support to the notion that a galaxy s properties are related only to the mass of its host dark matter halo , and not to the larger scale environment .

it is well established that the magnetic field that permeates the solar corona has a highly complex structure . although it is very difficult to measure directly the magnetic field vector in the corona , this complexity can be inferred from observations of the line - of - sight magnetic field at the photosphere . with each new satellite mission that is launched , we observe photospheric magnetic flux concentrations on ever smaller scales ( that seem to exhibit a power - law distribution with size , * ? ? ? magnetic field extrapolations based on these observed photospheric polarity distributions exhibit an often bewildering degree of complexity . understanding the evolution of such a complex magnetic field structure is a major challenge . in recent years , significant progress has been made in developing tools with which to characterise the coronal magnetic field . one approach involves segregating the photospheric magnetic field into discrete flux patches . this then allows the corona to be divided into distinct domains , each defined by the flux connecting pairs of these patches . between these coronal flux domains are _ separatrix surfaces _ , that emanate from magnetic null points . the intersection of two such separatrix surfaces forms a _ separator _ field line a field line that connects two null points and lies at the intersection of four flux domains . indeed , magnetic field extrapolations reveal the presence of a web of null points , separatrix surfaces , and separators that form a _ skeleton _ based upon which the magnetic connectivity of the coronal field may be understood ( e.g. * ? ? ? * ; * ? ? ? the separatrix surfaces of this skeleton represent locations at which the mapping between boundary points via the magnetic field lines exhibits discontinuities . also of interest are layers in which this field line mapping exhibits strong ( but finite ) gradients . these are known as _ quasi - separatrix layers _ ( qsls ) , being regions at which the _ squashing factor _ , @xmath0 @xcite , is large . null points , separators , and qsls , at which the field line mapping is either discontinuous or varies rapidly , are of interest not only in analysing the structure of the coronal magnetic field , but for understanding its dynamics . this is because these locations are prime sites for the formation of current layers at which magnetic reconnection may occur , releasing stored magnetic energy ( * ? ? ? * and references therein ) . in particular , they have been implicated in the formation of current sheets associated with solar flares , jets , and coronal mass ejections ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? one major piece of supporting evidence is the coincidence of flare ribbons with footpoints of separatrix and qsl field lines in coronal field extrapolations @xcite . one particular location at which the magnetic field line mapping is discontinuous is at the interface between closed and open magnetic flux , i.e. the boundary between magnetic field lines that are anchored at both ends at the photosphere , and those that extend out into the heliosphere . magnetic reconnection at this open - closed flux boundary is one of the principal mechanisms proposed to explain the properties of the slow , non - steady , solar wind ( e.g. * ? ? ? . the slow solar wind is characterised by strong fluctuations in both velocity and plasma composition , the latter of which is consistent with the wind being composed of some component of closed - flux coronal plasma ( e.g. * ? ? ? reconnection specifically at the open - closed flux boundary is also implicated in the generation of impulsive _ solar energetic particle ( sep ) _ events , due to the observed ion abundances of these events @xcite . typically , computational models of the sun s global magnetic field exclude the outflowing plasma of the solar wind , but include its effect by imposing a magnetic field that is purely radial at some height above the photosphere ( termed the ` source surface ' ) . excluding all contributions to the solar magnetic field other than the global dipole , the coronal field is characterised by two polar coronal holes of open magnetic field lines and a band of closed flux around the equator , these two being separated by separatrix surfaces that meet the base of the heliospheric current sheet ( hcs ) at the source surface . the question arises : when the full complexity of the coronal field is introduced , what is the nature of the boundary between open and closed flux ? in a series of papers , fisk and co - workers ( e.g. * ? ? ? ? * ; * ? ? ? * ) developed a model for the dynamics of the sun s open magnetic flux , that was also used to explain the acceleration of the solar wind mediated by reconnection between open and closed field lines ( termed _ interchange reconnection _ by * ? ? ? * ) . in their model , open field lines can freely mix with and diffuse through the closed field regions , and indeed it is predicted that this open flux component should become uniformly distributed throughout the ( predominantly ) closed field region @xcite . while noting that such a scenario requires the presence of current sheets in the corona between open and closed flux , these studies do not address the magnetic field structure in detail . indeed , the topological admissibility of such free mixing of open and closed flux has since been questioned @xcite , making it difficult to reconcile the interchange reconnection solar wind acceleration mechanism with the broad observed latitudinal extension of the slow solar wind streams ( up to @xmath1 , especially at solar minimum ) ( e.g. * ? ? ? nonetheless , recent modeling of the global coronal magnetic field has suggested a resolution to the apparent contradiction that plasma that appears to originate in the closed corona is observed far from the hcs at large radii . it has been demonstrated that additional regions of open flux that are disconnected from the polar coronal holes ( at the photosphere ) may indeed exist . the distinct photospheric regions of open magnetic flux are partitioned by multi - separatrix structures associated with multiple nulls points , typically comprising a dome - shaped separatrix enclosing the closed flux between the two open field regions , intersecting with a vertical _ separatrix curtain _ @xcite . even when coronal holes are not disconnected , there may exist very narrow channels of open magnetic flux at the photosphere connecting two larger open flux regions . in this case the narrow channel is associated with a qsl curtain . both the qsl and separatrix curtains extend out into the heliosphere , and have been shown to map out a broad latitudinal band around the hcs , termed the _ s - web _ @xcite . the corresponding arc structures at the source surface in global models are associated with pseudo - streamers in the observations , and there is growing evidence that these structures are associated with slow solar wind outflow ( e.g. * ? ? ? the above studies have revealed that the open - closed flux boundary has a complex topological structure involving null points and their associated separatrices , separators and qsls . moreover , it has been recently demonstrated that when reconnection occurs in astrophysical plasmas , the 3d topological complexity can dramatically increase beyond that of the equilibrium field ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? in this paper we use simple static magnetic field models to investigate the implications of reconnection for the magnetic field connectivity at the open - closed flux boundary when the reconnecting current layer exhibits a fragmented structure , expected to be the typical case in the corona . the paper is structured as follows . in section [ 3dtearsec ] we summarise recent relevant results on current layer instabilities . in sections [ domesec ] and [ curtainsec ] we investigate the topological effect of reconnection in configurations defined by an isolated separatrix dome and a separatrix curtain . in sections [ discusssec ] and [ concsec ] , respectively , we present a discussion of the results and our conclusions . in recent years a major advance in our understanding of magnetic reconnection has been the realisation that the reconnection rate can be substantially enhanced when the current layer breaks up in response to a tearing instability . while the linear phase of the classical tearing mode is slow , non - linear tearing in two dimensions ( 2d ) via the _ plasmoid instability _ can grow explosively , and lead to a reconnection rate that is only weakly dependent on the resistivity @xcite . in 2d as the instability proceeds a myriad of magnetic islands are formed as current layers fragment into chains of x - type and o - type nulls . the conditions for onset of the instability are that the ( inflow ) lundquist number be @xmath2 , and the current layer aspect ratio be @xmath3 . it has recently been demonstrated that the plasmoid instability also occurs in three - dimensional ( 3d ) current layers . @xcite performed 3d particle - in - cell simulations of an initially planar , infinite current layer . they noted that magnetic ` flux ropes ' formed in place of the magnetic islands from the 2d picture with flux often threading in and out of multiple flux ropes . @xcite by contrast studied mhd simulations of a 3d magnetic null point undergoing external shear driving . they observed the initial formation of a laminar current layer centred on the null , which was found to become unstable at a threshold similar to the 2d case ( lundquist number @xmath4 , aspect ratio @xmath5 ) . this threshold onset condition is very likely to be exceeded for typical current sheets formed in the corona . in this study , it was demonstrated that the onset of tearing leads to the creation of new 3d null points in bifurcation processes . in particular , 3d spiral nulls are formed that are the analogue of 2d islands , and the spine lines of each of these nulls forms the axis of a pair of magnetic flux ropes , as shown in figure [ wptop ] . crucially , and in contrast to the 2d case , these flux ropes are open structures they are not surrounded by flux surfaces as 2d islands are ( this is required on topological grounds due to the variation of @xmath6 along the direction of the flux rope axis and the solenoidal condition on @xmath6 ) . as a result , no new isolated domains of magnetic connectivity are formed . rather , the flux ropes are composed of a mixture of flux from the two connectivity domains ( flux located above and below the separatrix of the single null prior to the instability ) , wrapped around each other . the result is that , when the field line connectivity is analysed , a layer is found in which magnetic flux from the two connectivity domains is rather efficiently mixed in spiral patterns associated with the multiple flux rope structures . our intention here is to investigate the effect on the global connectivity when the fragmented , reconnecting current layer is embedded in some generic coronal field structures . ( a ) ) , and ( b ) after the addition of one flux ring ( state 1a ) sample field lines in the flux ropes are coloured red . green boxes outline the regions in which connectivity and @xmath0 maps are calculated in figures [ dome_oc_map ] , [ dome_q_map].,title="fig:",width=264 ] ( b ) ) , and ( b ) after the addition of one flux ring ( state 1a ) sample field lines in the flux ropes are coloured red . green boxes outline the regions in which connectivity and @xmath0 maps are calculated in figures [ dome_oc_map ] , [ dome_q_map].,title="fig:",width=264 ] we first examine the simplest generic coronal configuration containing a 3d null point the isolated dome topology ( see figure [ dometop](a ) ) . such a configuration is always present , for example , when a photospheric region of one magnetic polarity is embedded in a region of opposite polarity . such 3d nulls are preferred sites of current sheet formation @xcite , and the significance of reconnection in an isolated dome configuration has been considered , for example , by @xcite . as demonstrated by @xcite , spine - fan reconnection in such a dome topology is characterised by a transfer of flux in one side of the separatrix dome and out the other side , which can be driven dynamically or occur during a relaxation process as the coronal field seeks a minimum energy state . in order to examine the effect of tearing of the null point current sheet on the field structure we consider the following simple model . the fields that we construct are not equilibrium fields ( e.g. are not force - free ) however this is of no importance since we do not consider here any dynamical processes . rather , our purpose is solely to examine the field topology / geometry that results from breakup of a reconnecting current layer . we consider the ( dimensionless ) field @xmath7 where @xmath8 . this magnetic field contains a null point at @xmath9 above a photospheric plane represented by @xmath10 , see figure [ dometop](a ) . the isolated dome topology appears over a wide range of scales in the corona . hereafter we discuss any length scales in terms of a characteristic ` macroscopic ' length scale of the overall structure , that we denote @xmath11 . in our model field both the separatrix dome diameter at @xmath10 and the null point height are of order 1 , so for the model field @xmath12 . from magnetic field extrapolations it is observed that in the corona this scale @xmath11 of the dome separatrix structure can be as large as a few hundred mm ( usually in the vicinity of active regions , e.g. * ? ? ? * ; * ? ? ? * ) , and at least as small as a few tens of km ( in quiet - sun regions , where the lowest null point height in extraploations is likely limited by the magnetogram resolution , e.g. * ? ? ? * ) . onto the ` background ' field ( [ bdome ] ) we super - impose a magnetic flux ring to simulate the topological effect of magnetic reconnection occurring at some particular location in the volume . this method is motivated and described in detail in @xcite . the field of this flux ring is taken of the form @xmath13 we begin by super - imposing a single flux ring of this form onto the field of equation ( [ bdome ] ) , centred at the null ( i.e. we set @xmath14 , @xmath15 ) . @xmath16 and @xmath17 are the characteristic size of the flux ring in the @xmath18-plane ( plane of @xmath19 ) and along @xmath20 , respectively . for @xmath21 small , the effect of adding the flux ring is to collapse the spine and fan of the null point towards one another , as described by @xcite . this has the effect of transferring flux in one side of the dome and out of the other , and is consistent with the topology of a single laminar reconnection layer at the null @xcite . however , for larger values of @xmath21 the field becomes elliptic at @xmath22 as the flux ring field dominates over the hyperbolic background field . the result is a bifurcation of the original null into three null points as described above in section [ 3dtearsec ] see figure [ wptop ] and the generation of a pair of flux ropes ( see figure [ dometop](b ) ) . this models the magnetic topology when the current layer undergoes a spontaneous tearing instability as observed by @xcite . parameters for the magnetic field denoted state 1b are presented in table [ tbl ] . in order to visualise the new field structure created after tearing onset we first plot a connectivity map of field lines from the lower boundary . that is , we trace field lines from a grid of footpoints on the lower boundary and distinguish field lines that are closed ( return to the lower boundary @xmath10 ) and open ( exit through the upper boundary @xmath23 ) . the resulting map can be seen in figure [ dome_oc_map ] . while the flux ropes are approximately circular near the apex of the dome , they are compressed towards the separatrix and stretched in the azimuthal direction by the global field geometry , and thus appear as flattened spiral structures in the connectivity map ( figure [ dome_oc_map](a ) ) . in order to more clearly visualise this structure we reproduce the map in a polar coordinate system in figure [ dome_oc_map](b ) . the observed spiral pattern of mixing of open and closed flux reproduces the behaviour in the dynamic mhd simulations see figure 8 of @xcite . ( a ) coordinates centred at @xmath24 . ( c ) state 1c containing three flux rings.,title="fig:",width=264 ] ( b ) coordinates centred at @xmath24 . ( c ) state 1c containing three flux rings.,title="fig:",width=264 ] ( c ) coordinates centred at @xmath24 . ( c ) state 1c containing three flux rings.,title="fig:",width=264 ] one can include the effect of a further breakup of the current layer through the inclusion of additional flux rings . adding two such flux rings centred at the non - spiral nulls of state 1a leads each of these nulls to undergo a bifurcation , resulting in a total of seven nulls . this naturally introduces additional spiral structures in the field line mapping , as show in figure [ dome_oc_map](c ) ( state 1c , for parameters see table [ tbl ] ) , and if one were to iterate this procedure by adding more flux ropes a mapping with complexity of the order of that seen in the mhd simulations of @xcite could be obtained . ( a ) on the top boundary @xmath25 , for ( a ) state 1a , ( b ) state 1b , and ( c ) state 1c.,title="fig:",width=264 ] ( b ) on the top boundary @xmath25 , for ( a ) state 1a , ( b ) state 1b , and ( c ) state 1c.,title="fig:",width=264 ] ( c ) on the top boundary @xmath25 , for ( a ) state 1a , ( b ) state 1b , and ( c ) state 1c.,title="fig:",width=264 ] we now turn to consider the characteristics of the open flux that exits the domain through the top boundary at @xmath23 . since all of this flux is open , a connectivity map does not reveal this structure . however , we can use for example the _ squashing factor _ , @xmath0 , to visualise the field line mapping from @xmath10 to @xmath25 . here we plot @xmath0 on the surface @xmath25 . the distribution of @xmath0 is obtained by integrating field lines from a rectangular grid of typically around @xmath26 footpoints and then calculating the required derivatives using finite differences over this grid . @xmath0 is formally infinite on spine and fan field lines since they represent discontinuities in the field line mapping . however , calculating @xmath0 numerically as we do here they show up only as sharp points and lines , respectively , with very high values of @xmath0 . one should therefore not attach physical meaning to the maximum value of @xmath0 in the plots ( attained at the separatrix / spine footpoints ) as it is determined entirely by the resolution of the field line grid . in the background dome topology of equation ( [ bdome ] ) a single spine line intersects the @xmath25 boundary , and the @xmath0-map displays a single maximum at the origin . when a null point bifurcation occurs during reconnection , the topological structure changes to that shown in figure [ wptop ] . in this case a vertical separatrix extends up to the top boundary , bounded on either side by a pair of spine lines . examining the @xmath0 distribution on @xmath25 for state 1a ( figure [ dome_q_map]a ) , the separatrix footprint is clearly in evidence ( horizontal line of high @xmath0 ) . increasing the strength of the flux ring in the model ( state 1b ) leads to a lengthening of this separatrix due to the increased separation of the nulls ( figure [ dome_q_map]b ) . a further breakup of the current sheet leads to the appearance of multiple vertical separatrices , as seen in figure [ dome_q_map](c ) . there are two additional noteworthy features of the @xmath0 distributions . first , note the arcs of high and low @xmath0 that run parallel to the separatrix footprint . these become more pronounced and numerous as the flux ring strength is increased ( compare figures [ dome_q_map](a , b ) ) . their origin can be understood as follows . consider field lines traced down from the top boundary that enter one of the flux ropes . some local bundles of field lines will spiral around the rope axis and then ` leave ' the flux rope at its top or bottom ( in @xmath27 ) with a range of values of @xmath28 but roughly constant @xmath27 values ( @xmath28 being the azimuthal angle in the @xmath29-plane ) . due to their range of @xmath28 values on leaving the rope they diverge in the azimuthal direction as they are traced onwards to the lower boundary and therefore exhibit relatively high @xmath0 values . by contrast , adjacent field lines that leave the flux rope along its sides at approximately equal @xmath28 values but differing @xmath27 values are naturally squeezed in towards the fan as they approach the photosphere they do not diverge with the null point fan geometry owing to their close alignment in the @xmath28 direction . this leads to a lower @xmath0 . for a stronger flux ring ( more substantial flux rope ) field lines have the opportunity to spiral multiple times around the rope axis , leading to multiple @xmath0 stripes . the second feature to note are the high-@xmath0 ridges emanating from each spine footpoint in the @xmath20-direction , that are present for the following reason . adding the flux rings naturally generates a strong field component in the @xmath30 plane . therefore the two non - spiral nulls have quite asymmetric fan eigenvalues ( their ratio is around 2.5 in state 1a ) . the weak field direction corresponds to the @xmath20-direction , and it is natural that @xmath0 is largest in this weak - field region of diverging fan field lines . to test the robustness of the structures in @xmath0 described above , two @xmath0-maps were calculated using magnetic fields taken from the dynamic mhd simulation of @xcite . to avoid discontinuities in @xmath0 brought on by the corners of the domain we calculated @xmath0 using the foot points of field lines traced from a fixed grid on the top boundary to a cylindrical surface defined by @xmath31 , red and grey surfaces in fig . [ sim_q_map](c ) respectively . [ sim_q_map](a ) shows @xmath0 on the top `` open '' boundary soon after tearing has occurred in the layer see also fig . 5 of @xcite . note that @xmath32 is the vertical direction in the simulation domain and that the spine - fan collapse occurs in the @xmath10 plane , fig [ sim_q_map](c ) . at this time there is a single pair of flux ropes within the current layer . despite significant fine structure ( likely resulting from turbulent dynamics in the outflow region ) the structures in our simple model described above are clearly evident also in the dynamic mhd simulation . a short high-@xmath0 line corresponding to a vertical separatrix surface is apparent near @xmath33 , whilst a number of parallel stripes of @xmath0 can be seen extending to either side of it . additionally , two high-@xmath0 ridges emanate from the ends of this separatrix surface . the observed closed loop of high @xmath0 results from the flux rope pair being located in the reconnection outflow having detached from the open - closed boundary , see the discussion of @xcite . this splits the field lines that connect from @xmath34 to @xmath35 into two bundles : those that connect directly from the top boundary to the side and those that loop first around the back of the flux rope pair . the foot points of the latter are found within the loop of high @xmath0 . note that the field line connectivity changes continuously around the boundary of this loop , so the value of @xmath0 is large but finite . at the later time two pairs of flux ropes are present in the outflow region of the current layer resulting in an additional separatrix footprint being present , fig . [ sim_q_map](b ) . the gap observed between the pair of vertical separatrix footprints ( in contrast to fig . [ dome_q_map](c ) ) is again a result of the detachment of the nulls from the open - closed boundary . ) . red and blue spheres correspond to nulls with topological degree of @xmath36 and @xmath37 respectively , whilst the two separators are shown in green and purple . green boxes outline the regions in which connectivity maps are calculated.,width=302 ] we conclude that tearing of the reconnecting current layer at an isolated coronal null separatrix dome leads to the formation of an envelope around the initial dome structure in which magnetic flux from inside and outside the dome is efficiently mixed together . additionally , vertical separatrix curtains are formed during each null point bifurcation . the implications of these results will be discussed in section [ discusssec ] . isolated separatrix dome structures associated with a single null as considered in the previous section separate small pockets of closed flux from the open flux in the polar regions ( as well as being prevalent in closed flux regions ) . however , it is also typical to have much more complicated separatix configurations separating open and closed flux . in particular , in global field extrapolations it is seen that vertical _ separatrix curtains _ lie between coronal holes that are of the same polarity but are disconnected at the photosphere @xcite . these curtains , together with qsls associated with narrow corridors of open flux , are associated with arc structures at the source surface in global models that are interpreted as being associated with _ pseudo - streamers _ @xcite . we consider here a simple model containing a vertical separatrix surface representing one of these curtains . this intersects a separatrix dome associated with three coronal null points along two separator lines . the separatrix curtain consists of the fan surface of one of these nulls ( see figure [ curtaintop ] ) . the magnetic field expression for our model is as follows @xmath38 again a characteristic length scale of the overall structure is of order 1 in the model field ( say the null point height or separation see figure [ curtaintop ] ) , that we refer to as @xmath11 . on the sun , @xmath11 is observed to be as large as a quarter of the solar radius ( @xmath39 mm , e.g. * ? ? ? * ) , and may be at least as small as tens of kilometers in quiet sun regions as discussed before . note that to find the separators in these models the `` progressive interpolation '' method @xcite was used , whereby field lines were traced from a ring encircling one of the associated null points on its fan plane to identify the approximate position of each separator , before using an iterative bisection procedure to find each separator to a desired accuracy . connecting the three null points along the top of the separatrix dome are a pair of separator field lines , fig . [ curtaintop ] . like 3d null points , these are known to be preferred sites for current sheet formation and magnetic reconnection ( e.g. * ? ? ? it has been previously observed that these sheets are prone to fragment , yielding a current layer containing multiple separators @xcite . we thus begin by considering the effect of super - imposing one then more flux rings to simulate the effect of tearing in a reconnecting current layer around the separator . such a model with a single flux ring was presented by @xcite using a background field with two nulls , both with initially planar fan surfaces . they noted that as they increased the strength of the flux ring new separators appeared , coinciding with the formation of distinct new domains of magnetic flux connectivity . by magnetic flux domain here and throughout we mean a volume within which there is a continuous change of field line connectivity . distinct flux domains are bounded by separatrix surfaces at which this connectivity change is discontinuous . we observe the same effect when adding a single flux ring on the separator state 2a ( see table [ tbl ] ) . tracing field lines from the photosphere ( @xmath10 ) and making a connectivity map as before , we observe the presence of a region of open flux nested within the closed field region , figure [ sep_oc_1rope_phot](a , b ) . increasing the strength of the flux ring , we observe progressively more open and closed flux volumes being created , nested within one another fig . [ sep_oc_1rope_phot](c ) , state 2b . the formation of these new flux domains corresponds to the formation of new pairs of separators joining the two associated nulls . figure [ sep_diagram](a ) demonstrates this for state 2b . whereas originally one separator joined the central and end nulls the formation of the three nested flux domains ( fig . [ sep_oc_1rope_phot](b ) ) corresponds to the birth of three additional pairs of separators , giving seven in total ( green and purple field lines ) , see below for a further discussion . on the surface @xmath40 , and ( b ) close - up connectivity map , both for state 2b . the black region contains footpoints of field lines that connect to the photosphere at @xmath41 ( on one side of the separatrix dome / curtain structure ) , in the white regions field lines connect to @xmath42 ( on the other side ) . the letters correspond to the letters marking the flux tubes in the 3d plot of figure 10 . ( c ) connectivity map for state 2c.,scaledwidth=45.0% ] we now turn to examine the connectivity of field lines that extend outwards into the heliosphere ( those that exit through the top boundary ) . throughout we consider the surface @xmath40 as being the ` top ' boundary field lines are close to vertical above this plane and so little deformation of the field line mapping occurs . in figure [ sep_oc_1rope_top](a ) a map of @xmath0 is plotted on the top boundary ( as calculated between the two surfaces @xmath10 and @xmath40 ) for state 2b . note that a colour scale is not shown since the maximum value is arbitrary , depending only on the resolution of the field line grid . we observe the imprint of the separatrix curtain , as well as additional nested loop structures that correspond to additional separatrix surfaces separating nested flux domains . in figure [ sep_oc_1rope_top](b ) a connectivity map is plotted field lines that intersect the black region connect to the photosphere at @xmath41 on one side of the separatrix dome / curtain structure , while field lines intersecting the white region connect to the photosphere on the other side of the dome , @xmath42 . embedding our structure in a global field these two different regions would correspond to open field regions of the same polarity that are disconnected at the photosphere ( see e.g. figure 5 of * ? ? ? * ) , and thus the figure shows that flux from the two disconnected coronal holes forms a mixed , nested pattern . these nested connectivity regions are entirely equivalent to those described in the photospheric connectivity maps above . it is expected that in a dynamic evolution there is continual reconnection of field lines within the current layer , and thus a mixing of plasma between all of the nested flux domains ( see e.g. * ? ? ? as such , field lines at large height in these nested flux regions will continually be reconnected with those from the closed flux region . the addition of further flux rings representing further plasmoid structures in the reconnecting current layer leads to the formation of additional adjacent sets of nested open / closed flux domains . figure [ sep_oc_1rope_top](c ) shows the connectivity map when three flux rings are present , state 2c . the complexity quickly becomes very high , with extremely thin layers of connectivity with characteristic thickness of order @xmath43 even though the flux rope structures and their collective footprint in the solar wind remain much larger , having diameters of order @xmath44 . the inclusion of further flux ropes would decrease the length scales in the mapping yet further . this complexity of field line mapping was observed in a related context by @xcite . the direct association between the newly - created nested flux domains and additional separators that form in the domain is demonstrated in the right - hand frame of figure [ sep_diagram](a ) . here we note that the seven separators lie at the intersections of the four different connectivity regions . the nested formation of flux regions and the associated pairs of separators may be understood as follows . consider a bundle of field lines passing in along the open spine of the null at @xmath45 . as the flux ring strength is increased some of these field lines are wrapped repeatedly around the axis of the flux rope before they reach the photosphere . field lines reach the photosphere near spine footpoints of the central null , either at @xmath46 or @xmath42 . they may do this directly , or by first winding once , twice , or more times around the flux rope axis . this is demonstrated in figure [ newfig ] , where flux tubes are plotted from each of the nested connectivity domains that intersect the upper boundary of state 2b ( marked ` a ' , ` b ' and ` c ' in figure [ sep_oc_1rope_top](b ) ) . each additional winding corresponds to a new flux domain . this is because as the strength of the flux ring is increased the separatrix surfaces of the two nulls ( red and cyan curves in the right panel of figure [ sep_diagram](a ) ) fold over to intersect with one another multiple times . each additional pair of intersections correspond to a pair of new flux domains ( bounded by portions of the separatrices ) and a pair of new separators ( see the right - hand image of figure [ sep_diagram](a ) and figures 3 and 4 of * ? ? ? field lines wind more times closer to the flux ring axis , and so new topological domains are formed within the previous ones along with a pair of separators . this explains the nested nature of the connectivity domains observed in fig . [ sep_oc_1rope_phot ] and why each new pair of separators have one half twist more than the preceding two , fig . [ sep_diagram](a ) . ( b).,scaledwidth=50.0% ] so far we modelled the case in which the current layer forms along the separator , with this current layer breaking up but the number of nulls remaining fixed that is , no null bifurcations occurred . however , another distinct possibility is that the current layer that forms in response to a dynamic driving of the system contains one or more of the coronal nulls . when such a current layer breaks up we would expect a bifurcation of the corresponding null point(s ) , which naturally should also coincide with the formation of additional separators indeed this may well occur even when the current is focussed away from the nulls , as observed by @xcite . we consider here two distinct cases , in the first of which the null point initially at @xmath47 is bifurcated into multiple null points , and in the second of which the central null ( initially at ( 0,0,1 ) ) is bifurcated . consider first the situation where the end null is bifurcated . first , adding a single flux ring of sufficient strength at the initial location of the null we obtain a bifurcation to form three nulls as in figure [ wptop ] ( state 3a ) see figure [ sep_diagram](b ) . let us now examine the result for the magnetic flux connectivity , considering first flux intersecting the photosphere . in figure [ sep_oc_1ropeen](a ) we see that the photospheric connectivity map appears as before : new nested open and closed flux domains are created in the vicinity of the spine footpoints of the central null . the connectivity map for open flux traced from the upper boundary is shown in figure [ sep_oc_1ropeen](c ) ( where the colours have the same meaning as before ) . as shown in the @xmath0-map ( figure [ sep_oc_1ropeen]b ) , the main separatrix curtain is diverted in the positive @xmath32-direction for negative @xmath20 . it terminates on the spine of one of the null points located in the vicinity of @xmath48 . there is then an additional separatrix footprint orthogonal to this bounded by the spines of the newly created nulls as in figures [ wptop ] , [ dome_q_map ] . interestingly though , the arcs of high @xmath0 emanating from this separatrix ( as in figure [ dome_q_map ] ) now form the boundaries of the flux domains that connect to opposite sides of the dome footprint . this can be understood by considering that each arc of high @xmath0 in fig . [ dome_q_map ] represents a further half turn of field lines along the axis of the flux rope , i.e. the outermost two arcs correspond to field lines exiting along one or other spine of the central null having wound up to once around the flux rope axis , the next pair to field lines that first wind between once and twice around the flux rope axis , and so on . when these field lines are mapped on to the photosphere as in the domed single null case this leads to the continuous but rapid change in connectivity denoted by the @xmath0 ridges . when such field lines separate along the spine of a distant null the change in connectivity becomes discontinuous , forming the nested flux domains , see also section [ subsubtop ] . as before , the connectivity maps quickly become significantly more complex when additional flux ropes are added . figures [ sep_oc_1ropeen](d , e ) show the photospheric and upper boundary connectivity maps when an additional two flux rings are added to generate a bifurcation to a state with three flux ropes pairs and seven nulls , state 3b . again , characteristic length scales of the mapping layers of order @xmath43 or below are observed within a mixed flux region with dimensions of order @xmath44 . ( a ) . ( b ) @xmath0 on the upper boundary . ( c ) close - up connectivity map on the upper boundary with colours as in figure [ sep_oc_1rope_top ] note that the @xmath32-direction is stretched in ( c ) for clarity.,title="fig:",height=188 ] + ( b ) . ( b ) @xmath0 on the upper boundary . ( c ) close - up connectivity map on the upper boundary with colours as in figure [ sep_oc_1rope_top ] note that the @xmath32-direction is stretched in ( c ) for clarity.,title="fig:",height=204 ] ( c ) . ( b ) @xmath0 on the upper boundary . ( c ) close - up connectivity map on the upper boundary with colours as in figure [ sep_oc_1rope_top ] note that the @xmath32-direction is stretched in ( c ) for clarity.,title="fig:",height=204 ] finally , suppose that the fragmentation of the coronal current layer leads to a bifurcation of the central null point . adding a single flux ring leads to a bifurcation to a state with three null points ( state 4 ) as before . analysis of the resulting topology reveals a situation that mirrors state 3a . as shown in figure [ sep_oc_1ropecn](a ) , this time the photospheric connectivity maps show adjacent crescent - shaped domains of open and closed flux , symmetric about @xmath49 since the null bifurcation now leads to a bifurcation of both of the initial separators . correspondingly , nested flux domains of alternating connectivity are now observed in the connectivity map for the upper boundary ( figure [ sep_oc_1ropecn](b ) ) , this time emanating from the footpoints of both of the vertical open spines . as shown in the above models , reconnection at the sun s open - closed flux boundary can result in that boundary taking on a highly non - trivial structure . in the presence of an isolated null point separatrix dome no new flux domains are created but an envelope forms around the initial dome structure in which magnetic flux from inside and outside the dome is efficiently mixed together in spiral patterns . as shown by @xcite magnetic flux is continually and recursively reconnected from open to closed and back again within this envelope ( i.e. is reconnected back and forth multiple times between open and closed regions * ? ? ? the result for the field at large heights is that a flux tube is present around the original spine line within which field lines are being continually reconnected with those from the closed region beneath the dome . if we consider a more complicated structure in which coronal separators are present , the breakup of the current layer leads to the formation of new flux domains . in particular , new open and closed magnetic flux domains form in nested structures , whose length scales become rapidly shorter ; even for the models considered here containing just three flux rope pairs characteristic length scales of the mapping layers of order @xmath43 or smaller are observed . the expectation is that in a dynamic evolution , continual transfer of flux / plasma between the narrow open and closed layers would occur . the new regions of flux are observed to form in the vicinity of the footpoints of spine field lines in the pre - reconnection field . together they cover a region of comparable scale to the distribution of current and flux rope structures , here of order @xmath44 . our results imply that in the vicinity of open spine structures and open separatrix curtain structures , an efficient mixing of open and closed magnetic flux , and the associated plasma , is likely to take place whenever reconnection occurs at the corresponding nulls or separatrices . this is an attractive ingredient for explaining observed properties of the slow solar wind by the interchange reconnection model . in particular , the slow solar wind is known to be highly fluctuating in both composition and velocity ( in both space and time ) , with the composition properties varying from close to those of the closed corona to nearly photospheric @xcite . contributing factors to this fluctuating , filamentary structure could be the bursty nature of the interchange reconnection , and the complex spatial structuring on large and small scales of the open - closed boundary . combining our results with those from simulations of current layer instabilities , it is clear that the reconnection process should lead to a highly dynamic magnetic topology in which regions of open and closed flux are born and evolve in a complex pattern . interchange reconnection models for solar wind acceleration share the common feature that they require regions of open flux at photospheric heights that are at least predominantly surrounded by closed flux . this is consistent with observations of significant components of solar wind outflow emanating from locations adjacent to active regions ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? the models of fisk and co - workers take a statistical approach to the evolution of the sun s open flux , in which they assume a random orientation for the closed loops and thus an isotropic diffusion of open field lines as they random walk through the closed flux region . one piece of observational evidence that is argued to support their model of open field dynamics is the coincidence of extended open field regions with minima in the local rate of flux emergence @xcite . interestingly , the models also predict that the random walk of open field lines will induce a braiding of field lines in the heliosphere , and this is also a prediction of the mhd simulations of @xcite ( see figure [ sim_q_map ] ) , though in our model the braiding is induced by turbulent dynamics in the reconnecting current layer . the s - web model , by contrast to the models of fisk and co - workers , seeks to identify explicitly the locations of possible interchange reconnection and thus outflow by analysing the detailed magnetic field topology . in a given magnetic field extrapolation open field channels and patches can be identified , however there are indications that the number of disconnected open field regions can greatly increase when the resolution of the photospheric magnetogram is increased @xcite . such an increase corresponds to an increase in the number of arcs in the s - web . in this paper , we have argued that when interchange reconnection occurs at a structure of the open - closed boundary and the full reconnection dynamics are included it will tend to generate a layer within which open and closed flux are mixed . so in this picture , each separatrix arc of the s - web becomes not a single line but a band within which the open - closed flux boundary is highly structured . this may partially mitigate against the concern that the static s - web is not space - filling and therefore can not provide a continuous slow solar wind outflow @xcite . the dimensions of this layer of mixed open - closed flux are of course crucial , but can not be readily estimated from the present approach . they will depend both on the size and geometry of the fragmented reconnecting current layer , and the overall global field geometry that connects this volume out to the heliosphere in any given situation . while the latter can be estimated from static models , elucidating the former will require a full , detailed dynamical understanding of the reconnection process . we note also that our results perhaps provide a ` bridge ' between the fisk et al . and s - web models , in the sense that the broad bands of mixed open - closed flux expected to form around the arcs of the s - web can be thought of as a regions within which the diffusive random walk envisaged by fisk and co - workers becomes highly efficient . the indication is then that the broadly uniform diffusion coefficient in those models should in fact be highly structured in both space and time . however , to determine the spatial and temporal distribution of these regions of efficient open - closed flux mixing and thus their overall effect will require a statistical study of the structures of the open - closed flux boundary , together with the results of the dynamical simulations suggested above . we now consider the implications of our results for signatures of particle acceleration in topologies involving coronal null points and separators . of course , explaining sep and flare ribbon observations requires knowledge of particle behaviour , and that remains to be studied . however , the above results allow us to make a number of predictions . consider first impulsive sep events . a series of recent observational studies has shown that impulsive sep sources are located in open field regions magnetically well - connected with the target ( * ? ? ? * and references therein ) . however , since their composition is more indicative of closed field regions , it has been proposed that they are accelerated directly during interchange reconnection in the low corona @xcite . the present results demonstrate that particles accelerated ( by some means ) during the reconnection process readily have access to open field lines , since all field lines are recursively reconnected from open to closed within the envelope of mixed flux described above . what s more , when open and closed flux are mixed into sufficiently thin layers , the distinction between the two is lost for the particles certainly the case if the layers are thinner than the larmor radius . furthermore , the dynamics in the current layer are most likely to be turbulent , and braiding of field lines is expected @xcite , both of which are known to lead to enhancement of cross - field particle transport ( e.g. * ? ? ? * and references therein ) . our results , combined with those of @xcite , also provide insights into the expected structure of flare ribbons in coronal null point and pseudo - streamer topologies . in particular , it has been shown that the separatrix and qsl footprints often map onto the locations of the flare ribbons . however , the flare ribbons usually exhibit additional structure , often bright kernel - like structures that move along the ribbons @xcite . these features could correspond to the footpoints of the flux rope structures formed during 3d current sheet fragmentation , these being associated with bundles of efficiently mixed open and closed flux . if this were the case , one would expect the motion of the bright features to be linked to the velocity of the outflow from the reconnection region , multiplied by some factor resulting from the geometry of the magnetic connection between the reconnection site and the photosphere . however , as shown by @xcite , the flux ropes exhibit a complicated dynamics as they kink and interact with one another in the reconnection region , so the motion of their footprint on the photosphere is expected to deviate significantly from a simple advection . we note that performing the same procedure as herein adding a flux ring to simulate the effect of reconnection for a background field defined by a qsl or hyperbolic flux tube leads not to more structure in the associated @xmath0-map , but simply to a break of the high-@xmath0 layer ( results not presented here ) . this may correspondingly imply that the signatures of bursty reconnection in such a field topology are different to the case when separatrices are present . one further specific conclusion that can be drawn regards the nature of the flare ribbons associated with the spine footpoints in the isolated dome topology . @xcite and @xcite reported observations of the flare ribbons in such a topology and noted that the ribbons that were postulated to be connected with the spine footpoints were extended structures with high aspect ratio ( rather than being circular ) . they proposed that this was related to the distribution of the squashing factor for field lines around the spine in the associated extrapolated equilibrium field . the results of section [ domesec ] provide us with an alternative hypothesis : the extended elliptical ribbons may mark out the imprint of the vertical separatrix curtains ( and surrounding arcs of high @xmath0 ) associated with a fragmented current layer . it may also be that both effects are important for the spine footpoint ribbon extension . of course it remains to be seen what accelerated particle distributions are expected during a dynamic reconnection event , and this will be pursued in a future study . magnetic reconnection in the solar corona is likely to occur in highly fragmented current layers , as demonstrated in recent 3d simulations @xcite . here we have used simple static magnetic field models to investigate the implications of the current layer fragmentation on the large - scale topology of representative solar coronal field structures . we have shown that this fragmentation can vastly increase the topological complexity beyond that of the equilibrium magnetic field . in particular , when the fragmenting current layer forms at the open - closed magnetic flux boundary , the structure of that boundary can become highly complex . the results , however , are also relevant for the studied topologies in the case where all flux is globally closed in this case some flux closes at some distant point on the photosphere ( but is ` locally open ' ) . we considered here two principal topologies ; the isolated null point dome and the separatrix curtain topology . both of these are observed over a broad range of characteristic scales in the corona , from hundreds of mm down to tens of km . in the presence of an isolated dome , non - linear tearing of the reconnecting current layer leads to the formation of an envelope of magnetic flux around the initial dome structure in which flux from inside and outside the dome is efficiently mixed together . magnetic flux is continually recursively reconnected from open to closed and back again within this envelope . the result for the field at large heights is that a flux tube is present around the original spine line within which field lines are being continually reconnected with those from the closed region beneath the dome . such isolated dome structures are typically found in abundance in coronal holes in magnetic field extrapolations . in the separatrix curtain ( ` pseudo - streamer ' ) topology of section [ curtainsec ] we saw that the breakup of the current layer leads to the formation of new flux domains . in particular , open and closed flux ( as well as flux from pairs of disconnected open field regions ) form in nested domains with very short length scales . the thickness of the adjacent open and closed flux domains can be many orders of magnitude smaller than the global length scale of the field structure or indeed of the flux ropes in the current layer in our models with only three flux ropes the mapping layers were two orders of magnitude smaller than the flux ropes . the expectation is that in a dynamic evolution , continual reconnection between the narrow layers of open and closed flux would occur within a flux envelope surrounding the new nested flux domains . our static models predict that in the corona immediately above the pseudo - streamer this envelope will cover a region of comparable scale to the distribution of current and flux rope structures . however , this would be expected to widen with height as the field strength reduces with radial distance from the sun and the field expands laterally . understanding particle acceleration in topologies such as those studied could help us comprehend both the release of impulsive seps to open field lines and the appearance of certain flare ribbons . in particular , the appearance of extended flare ribbons at spine footpoints in the dome topology could be related to the separatrix footprints that appear there when null point bifurcations occur as the current layer fragments . what s more , the efficient mixing of open and closed flux in the reconnection process provides a natural mechanism for accelerated particles to access the open field region . future studies of particle acceleration during the reconnection process will reveal much more . global magnetic field extrapolations are now revealing the huge complexity of the coronal field , and in particular the structure of the boundary between open and closed magnetic flux . regions of open flux that are either disconnected from the polar coronal holes at the photosphere or connected only by narrow open - flux corridors contribute arcs to the s - web @xcite . our results show that whenever reconnection occurs at a null point or separator of the open - closed boundary , the associated separatrix arc of the s - web becomes not a single line but a band of finite thickness within which the open - closed flux boundary is highly structured . the dimensions of this band are of course crucial , but can not be readily estimated from the present approach . the next step then , to determine the importance of this effect , requires dynamical mhd simulations of the process in order to quantify the dimensions of this band and the flux associated with it . dp acknowledges financial support from the uk s stfc ( grant number st / k000993 ) and the leverhulme trust . pw acknowledges support from an appointment to the nasa postdoctoral program at goddard space flight center , administered by oak ridge associated universities through a contract with nasa . , s. k. ( 1996 ) . . in balasubramaniam , k. s. , keil , s. l. , and smartt , r. n. , editors , _ solar drivers of the interplanetary and terrestrial disturbances _ , volume 95 of _ astronomical society of the pacific conference series _ , page 1 . cccccccccccc state & @xmath6-field & @xmath50 & @xmath51 & @xmath52 & @xmath53 & @xmath54 & @xmath55 & @xmath56 & @xmath57 & @xmath58 + 1a & dome & 0.03 & @xmath59 & 0.025 & 0.02 & & & & & + 1b & dome & 0.05 & @xmath59 & 0.025 & 0.02 & & & & & + 1c & dome & 0.08 & @xmath59 & 0.025 & 0.02 & 0.05 & @xmath60 & @xmath61 & 0.005 & 0.005 + 2a & curtain & 0.18 & @xmath62 & 0.1 & 0.167 & & & & & + 2b & curtain & 0.25 & @xmath62 & 0.1 & 0.167 & & & & & + 2c & curtain & 0.25 & @xmath62 & 0.1 & 0.167 & 0.06 & @xmath63 & @xmath64 & 0.005 & 0.167 + 3a & curtain & 0.22 & @xmath65 & 0.1 & 0.333 & & & & & + 3b & curtain & 0.3 & @xmath65 & 0.1 & 0.333 & 0.05 & @xmath66 & @xmath67 & 0.0025 & 0.333 + 4 & curtain & 0.2 & @xmath68 & 0.1 & 0.333 & & & & & +

global magnetic field extrapolations are now revealing the huge complexity of the sun s corona , and in particular the structure of the boundary between open and closed magnetic flux . moreover , recent developments indicate that magnetic reconnection in the corona likely occurs in highly fragmented current layers , and that this typically leads to a dramatic increase in the topological complexity beyond that of the equilibrium field . in this paper we use static models to investigate the consequences of reconnection at the open - closed flux boundary ( interchange reconnection " ) in a fragmented current layer . we demonstrate that it leads to efficient mixing of magnetic flux ( and therefore plasma ) from open and closed field regions . this corresponds to an increase in the length and complexity of the open - closed boundary . thus , whenever reconnection occurs at a null point or separator of this open - closed boundary , the associated separatrix arc of the so - called _ s - web _ in the high corona becomes not a single line but a band of finite thickness within which the open - closed boundary is highly structured . this has significant implications for the acceleration of the slow solar wind , for which the interaction of open and closed field is thought to be important , and may also explain the coronal origins of certain solar energetic particles . the topological structures examined contain magnetic null points , separatrices and separators , and include a model for a pseudo - streamer . the potential for understanding both the large scale morphology and fine structure observed in flare ribbons associated with coronal nulls is also discussed .

transverse spin fluctuations are gapless , low - energy excitations in the broken - symmetry state of magnetic systems possessing continuous spin - rotational symmetry . therefore at low temperatures they play an important role in diverse macroscopic properties such as existence of long - range order , magnitude and temperature - dependence of the order parameter , nel temperature , spin correlations etc . specifically in the antiferromagnetic ( af ) ground state of the half - filled hubbard model transverse spin fluctuations are important both in two and three dimensions , where antiferromagnetic long - range order ( aflro ) exists at @xmath4 . in the strong - coupling limit @xmath5 , where spin fluctuations are strongest , they significantly reduce the zero - temperature af order parameter in two dimensions to nearly @xmath6 of the classical ( hf ) value.@xcite similarly the nel temperature in three dimensions is reduced to nearly @xmath7 of the mean - field result @xmath8 , for the equivalent @xmath9 quantum heisenberg antiferromagnet ( qhaf).@xcite recently there has also been interest in spin fluctuations due to defects , disorder and vacancies in the quantum antiferromagnet . in the random-@xmath3 model , where @xmath3 is set to zero on a fraction @xmath10 of sites , the lattice - averaged af order parameter appears to be enhanced for small @xmath10 , as seen in qmc calculations,@xcite presumably due to an overall suppression of quantum spin fluctuations . on the other hand spin fluctuations are enhanced by strong disorder in the hubbard model with random on - site energies . in the strong disorder regime , overlap of the two hubbard bands leads to formation of essentially empty and doubly - occupied sites , which act like spin vacancies.@xcite the problem of spin vacancies in the quantum antiferromagnet is also relevant to the electron - doped materials like @xmath11 , where spin - dilution behavior is observed.@xcite while the problem of magnon energy renormalization due to spin vacancies has been addressed recently,@xcite these methods are limited to the low - concentration limit , and the vacancy - induced enhancement in transverse spin fluctuations has not been studied in the whole range of vacancy concentration . in this paper we describe a new method for evaluating transverse spin correlations and quantum spin - fluctuation corrections about the hf - level broken - symmetry state , in terms of magnon mode energies and spectral functions obtained in the random phase approximation ( rpa ) . the method is applicable in the whole @xmath0 range of interaction strength , and is illustrated with three applications involving the af ground state of the half - filled hubbard model ( i ) the pure model in @xmath12 , ( ii ) spin vacancies in the strong coupling limit in @xmath13 , and ( iii ) low-@xmath3 impurities in @xmath13 . this method for obtaining quantum correction to sublattice magnetization solely in terms of transverse spin correlations is parallel to the spin - wave - theory ( swt ) approach,@xcite and differs from the method involving self - energy corrections.@xcite the rpa approach has been demonstrated earlier to properly interpolate between the weak and strong coupling limits for the spin - wave velocity.@xcite by going beyond the rpa level within a systematic inverse - degeneracy expansion scheme , which preserves the spin - rotational symmetry and hence the goldstone mode order by order , it was also shown that in the strong coupling limit identical results are obtained for all quantum corrections , order by order , as from the swt approach for the qhaf.@xcite a renormalized rpa approach has also been devised recently to obtain the magnetic phase diagram for the three dimensional hubbard model in the whole @xmath0 range , and the @xmath14 vs. @xmath3 behaviour was shown to properly interpolate between the weak and strong coupling limits.@xcite the method is based on a convenient way to perform the frequency integral in order to obtain spin correlations from spin propagators , and we illustrate it here for transverse spin correlations . we write the time - ordered , transverse spin propagator for sites @xmath15 and @xmath16 , @xmath17|\psi_{\rm g}\rangle$ ] at the rpa level in frequency space as , @xmath18=\frac{[\chi^0(\omega)]}{1-u[\chi^0(\omega ) ] } = \sum_n \frac{\lambda_n(\omega)}{1-u\lambda_n(\omega ) } @xmath19 and @xmath20 are the eigenvectors and eigenvalues of the @xmath21 $ ] matrix . here @xmath21_{ij}=i\int ( d\omega'/2\pi ) g_{ij}^{\uparrow}(\omega')g_{ji}^{\downarrow}(\omega'-\omega)$ ] is the zeroth - order , antiparallel - spin particle - hole propagator , evaluated in the self - consistent , broken - symmetry state from the hf green s functions @xmath22 . spin correlations are then obtained from , @xmath23_{ij}\ ; e^{-i\omega ( t - t ' ) } \nonumber \\ & = & \pm \sum_n \frac{\phi_n ^i ( \omega_n)\phi_n ^j ( \omega_n ) } { u^2 ( d\lambda_n / d\omega)_{\omega_n } } e^{-i\omega_n ( t - t ' ) } \ ; , \end{aligned}\ ] ] where the collective mode energies @xmath24 are obtained from @xmath25 , and @xmath20 has been taylor - expanded as @xmath26 near the mode energies to obtain the residues . for convergence , the retarded ( advanced ) part of the time - ordered propagator @xmath27 , having pole below ( above ) the real-@xmath28 axis , is to be taken for @xmath29 ( @xmath30 ) . the frequency integral is conveniently replaced by an appropriate contour integral in the lower or upper half - plane in the complex-@xmath28 space for these two cases , respectively , which results in eq . we first illustrate this method for the half - filled hubbard model in two and three dimensions on square and simple - cubic lattices , respectively . in this case it is convenient to use the two - sublattice representation due to translational symmetry , and we work with the @xmath32 matrix @xmath33 $ ] in momentum space , which is given in terms of eigensolutions of the hf hamiltonian matrix.@xcite the @xmath34-summation is performed numerically using a momentum grid with @xmath35 and 0.05 , in three and two dimensions , respectively . equal - time , same - site transverse spin correlations are then obtained from eq . ( 2 ) by summing over the different @xmath36 modes , using a momentum grid with @xmath37 and @xmath38 in three and two dimensions , respectively . we consider @xmath39 , so that the retarded part is used , with positive mode energies . from spin - sublattice symmetry , correlations on a and b sublattice sites are related via @xmath40 . thus the transverse spin correlations are obtained from magnon amplitudes on a and b sublattices , and from eq . ( 2 ) we have @xmath41 from the commutation relation @xmath42=2s^z$ ] , the difference @xmath43 , of transverse spin correlations evaluated at the rpa level , should be identically equal to @xmath44 . this is becasue both the rpa and hf approximations are o(1 ) within the inverse - degeneracy expansion scheme@xcite in powers of @xmath45 ( @xmath46 is the number of orbitals per site ) , and therefore become exact in the limit @xmath47 , when all corrections of order @xmath45 or higher vanish . this is indeed confirmed as shown in figs . 1 and 2 . the deviation at small @xmath3 is because of the neglect in eq . ( 2 ) of the contribution from the single particle excitations across the charge gap , arising from the imaginary part of @xmath48 in eq . the sum @xmath49 yields a measure of transverse spin fluctuations about the hf state , and in the strong coupling for spin @xmath50 , one obtains @xmath51.@xcite using the identity , @xmath52 in this limit , the sublattice magnetization @xmath53 is then obtained from , @xmath54^{1/2 } .\ ] ] to order @xmath55 , this yields the correction to the sublattice magnetization of @xmath56=0.156 $ ] and 0.393 for @xmath9 in three and two dimensions , respectively . this same result at one loop level was also obtained from a different approach in terms of the electronic spectral weight transfer,@xcite and is in exact agreement with the swt result.@xcite as @xmath57 is the maximum ( classical ) spin polarization in the z - direction , and therefore also the maximum eigenvalue of the local @xmath58 operator , therefore for arbitrary @xmath3 , the hf magnitude @xmath57 plays the role of the effective spin quantum number , @xmath50 . the sublattice magnetization @xmath59 is therefore obtained from @xmath60 , where the first - order , quantum spin - fluctuation correction @xmath61 is obtained from eq . ( 4 ) with @xmath62 , @xmath63 in the strong coupling limit @xmath2 , @xmath64 for @xmath9 , so that the spin - fluctuation correction simplifies to , @xmath65 . for a site on the b sublattice , where @xmath66 , we have @xmath67 . the @xmath3-dependence of sublattice magnetization @xmath59 is shown in figs . 1 , and 2 for @xmath13 and @xmath68 , respectively . in both cases it interpolates properly between the weak and strong coupling limits , approaching the swt results 0.607 and 0.844 , respectively , as @xmath2 . a comparison of the @xmath59 vs. @xmath3 behaviour with earlier results is presented in fig . 3 for the well studied @xmath13 case . earlier studies have employed a variety of methods including the variational monte carlo ( vmc),@xcite self - energy corrections ( se),@xcite quantum monte carlo ( qmc),@xcite functional - integral schemes,@xcite the generalized linear spin - wave approximation ( glswa),@xcite and mapping of low - energy excitations to those of a qhaf with @xmath3-dependent , extended - range spin couplings.@xcite in addition to the two - sublattice basis , we have also used the full site representation in the strong coupling limit , in order to illustrate the scaling of the quantum correction with system size . here the @xmath69 matrix is evaluated and diagonalized in the site basis for finite lattices . results for lattice sizes with @xmath70 are shown in fig . 4 . a quadratic least - square fit is used to extrapolate to infinite system size , which yields @xmath71 , in agreement with the result from eq . the site representation has also been used to obtain transverse spin fluctuations for the problem of spin vacancies in the af in the limit @xmath2 . as mentioned already this method is applicable in the whole range of vacancy concentration , and allows determination of the critical vacancy concentration at which the af order parameter vanishes . for the vacancy problem we consider the following hamiltonian on a square lattice with nearest - neighbor ( nn ) hopping , @xmath72 where the hopping terms @xmath73 if sites @xmath15 or @xmath16 are vacancy sites , and @xmath74 otherwise . thus , for a vacancy on site @xmath15 , all hopping terms @xmath75 connecting @xmath15 to its nn sites @xmath16 are set to zero . the vacancy site is thus completely decoupled from the system . half - filling is retained by having one fermion per remaining site . we consider the @xmath2 limit , where the local moments are fully saturated , and the vacancy problem becomes identical to the spin - vacancy problem in the qhaf . this is also equivalent to the problem of non - magnetic impurities in the af in the limit of the impurity potential @xmath76.@xcite the structure of the @xmath48 matrix in the host af , and the modification introduced by spin vacancies has been considered earlier in the context of static impurities.@xcite as the goldstone mode is preserved , it is convenient to work with the matrix @xmath77 $ ] , inverse of which yields the spin propagator in eq . ( 1 ) near the mode energies . when expressed in units of @xmath78 , where @xmath79 is the af gap parameter , and @xmath80 , it has the following simple structure for the host af : @xmath81 vacancies introduce a perturbation in @xmath82 due to absence of hopping between the vacancy and nn sites , and we take @xmath83 to refer to this vacancy - induced perturbation , so that @xmath84 . if site @xmath15 is a vacancy site and @xmath16 the nn sites , then the matrix elements of @xmath85 are , @xmath86 and the magnitude of @xmath87 is irrelevant since the vacancy site @xmath15 is decoupled . thus @xmath88 , reflecting the decoupling of the vacancy , and the static part of the diagonal matrix elements @xmath89 on nn sites are reduced by 1/4 . for @xmath90 vacancies on nn sites , the static part is @xmath91 . this ensures that the goldstone mode is preserved . thus if a spin on site @xmath16 were surrounded by a maximum of four vacancies on nn sites , then the static part vanishes , and @xmath92 , representing an isolated spin , which yields a @xmath93 pole in the transverse spin propagator . this is an isolated single - spin cluster , and with increasing vacancy concentration , larger isolated spin clusters are formed . as these are decoupled from the remaining system , their spin - fluctuation contributions are not included , as the op vanishes for finite spin clusters . when @xmath94 exceeds the percolation threshold @xmath95 , the fraction of macroscopically large spin clusters in the system vanishes , and therefore no aflro is possible for @xmath96 . for a given vacancy concentration and system size , the appropriate number of vacancies are placed randomly across the lattice , and the matrix @xmath82 constructed accordingly . exact diagonalization of @xmath82 is carried out , and the eigensolutions are used to compute the transverse spin correlations from eq . ( 2 ) . the quantum , spin - fluctuation correction is then obtained from the strong - coupling limit of eq . the transverse spin correlation @xmath97 is averaged over all spins within the a sublattice . as mentioned already , only the spins in the macroscopic cluster spanning the whole lattice are considered , and contributions from spins in isolated spin clusters are excluded . configuration averaging over several realizations of the vacancy distribution is also carried out . the quantum spin - fluctuation correction vs. vacancy concentration is shown in fig . 5 for three lattice sizes , @xmath98 . best fits are obtained with an expression including a cubic term , @xmath99 . for the three lattice sizes @xmath1000.236 , 0.260 and 0.278 , respectively , and as shown in fig . 3 , it extrapolates to 0.39 as @xmath101 . the coefficient of the linear term is found to be nearly independent of system size , @xmath102 . and the cubic term @xmath103 takes values approximately 3.6 , 4.0 , and 4.4 for the three lattice sizes , and extrapolates to about 6.5 as @xmath101 . with these coefficients , the spin - fluctuation correction @xmath61 is nearly 1 for @xmath104 . beyond @xmath104 , the percolation limit , there is no single , macroscopically large spin cluster left in the system . therefore the point where the af order parameter vanishes and aflro is destroyed due to spin fluctuations nearly coincides with the percolation threshold . this is in agreement with results from series expansion@xcite and quantum monte carlo simulations@xcite of the qhaf with spin vacancies . we now consider a quenched impurity model with a random distribution of impurity sites characterized by a local coulomb interaction @xmath105 for the host sites . with h and i referring to the sets of host and impurity sites respectively , we consider the following hubbard model in the particle - hole symmetric form at half - filling and on a square lattice , @xmath106 the motivations for studying this impurity model are threefold . in view of the observed _ enhancement _ of magnetic order at low concentration of impurities,@xcite we shall analyze the suppression of quantum spin fluctuations to examine whether this is due to a local suppression at the low-@xmath3 sites . the rpa evaluation of transverse spin correlations is also extended to the case of site - dependent interactions . furthermore , at half - filling this model also provides a simplistic representation for magnetic impurity doping in an af . this may appear contradictory in view of the apparently nonmagnetic ( @xmath107 ) nature of the impurity sites . however , this feature is expressed only away from half - filling.@xcite the atomic limit @xmath108 provides a convenient starting point for further discussions . in the particle - hole symmetric form of eq . ( 9 ) , since not only local interaction terms , but the on - site energy terms are also modified ( from @xmath109 to @xmath110 ) at the impurity sites , therefore the energy levels for added hole and particle are @xmath111 and @xmath112 for host and impurity sites respectively . to order @xmath113 , this impurity model therefore canonically maps to the following @xmath9 heisenberg model @xmath114 where in the first term @xmath80 is the conventional exchange coupling between neighboring host spins , and @xmath115 is the exchange coupling between impurity spins and neighboring host spins . in writing eq . ( 10 ) we have assumed the dilute impurity limit , and discounted the possibility of two impurity spins occupying nn positions , in which case the impurity - impurity exchange coupling will be @xmath116 . therefore , in the strong correlation limit , this random-@xmath3 model also describes magnetic impurities in the af within an impurity - spin model . the magnetic - impurity doping is characterized by identical impurity and host spins @xmath117 , but with different impurity - host exchange coupling @xmath118 . both cases @xmath119 or @xmath120 are possible , and in this paper we have considered the two cases : ( i ) @xmath121 so that @xmath122 , and @xmath123 so that @xmath124 . while this model is easily generalized to other magnetic - impurity models represented by locally modified impurity - host hopping terms @xmath125 , and/or different impurity energy levels , in fact , the essential features are already contained here as the impurity exchange coupling @xmath126 is the relevant quantity in determining the spin fluctuation behavior . we recast the rpa expression for the transverse spin propagator in a form suitable for site - dependent interactions . in terms of a diagonal interaction matrix @xmath127 $ ] , with elements @xmath127_{ii}=u_i $ ] , the local coulomb interaction at site @xmath15 , the time - ordered transverse spin propagator at the rpa level can be rewritten , after simple matrix manipulations , as @xmath18= \frac{[\chi^{0}(\omega ) ] } { 1 - [ u ] [ \chi^{0}(\omega ) ] } = \frac{1}{[a(\omega ) ] } - \frac{1}{[u]}\ ] ] where @xmath128=[u ] - [ u][\chi^0 ( \omega)][u]$ ] is a symmetric matrix . as @xmath127 $ ] is non - singular , the singularities in @xmath129 $ ] , which yield the magnon modes , are then given completely by the vanishing of the eigenvalues of the matrix @xmath130 $ ] . in terms of @xmath131 and @xmath132 , the eigenvalues and eigenvectors of the matrix @xmath130 $ ] , we have @xmath128^{-1 } = \sum_n \lambda_n(\omega)^{-1 } magnon - mode energies @xmath24 are then given by @xmath133 , and in analogy with eq . ( 2 ) the transverse spin correlations are obtained from , @xmath134 as we are interested in the dilute behaviour , we examine the correction to sublattice magnetization due to two impurities , one on each sublattice for symmetry . since @xmath135 , corrections to both @xmath136 and @xmath61 are expressed in the dilute limit ( impurity concentration @xmath94 ) as @xmath137 , and @xmath138 . the overall @xmath139 behavior therefore depends on the relative magnitude of the coefficients @xmath140 and @xmath141 . for @xmath142 and @xmath143 we find , at the hf level , that @xmath144 and @xmath145 , both quite independent of system size . the ( site - averaged ) spin fluctuation correction @xmath61 is obtained from eq . ( 5 ) with and without impurities , and the impurity contribution extracted . for @xmath143 we find a net _ reduction _ in @xmath61 . divided by @xmath146 , the impurity concentration , this yields the coefficient @xmath141 defined above , and also the per - impurity contribution to the total spin fluctuation correction over the whole lattice . the spin - fluctuation correction @xmath61 for the pure case , and the per - impurity contribution @xmath147 are shown in fig . 6 for different lattice sizes , along with least - square fits . it is seen that in the infinite size limit , the per - impurity reduction is nearly 0.2 , which is more than half of the correction per site in the pure case ( 0.35 ) . thus , there is a substantial reduction in the averaged spin fluctuation correction due to the low@xmath1 impurities . as the two coefficients @xmath140 and @xmath141 are very nearly the same , the sublattice magnetization @xmath148 shows negligible concentration dependence . thus the ( relatively small ) reduction in the hf value due to the low-@xmath3 impurities is almost fully compensated by the ( relatively substantial ) reduction in the spin fluctuation correction . to first order in @xmath94 , we thus find that there is nearly no loss of af order due to the low-@xmath3 impurities . as mentioned earlier , even a slight enhancement in the af order was recently seen for the case @xmath149.@xcite we next examine the site - dependence of the local spin fluctuation corrections @xmath150 near the impurities . table i shows that spin fluctuation is actually _ enhanced _ on the low-@xmath3 impurity sites . the suppression of @xmath150 in the vicinity more than compensates for this local enhancement , resulting in an overall reduction on the average . on the other hand , for @xmath151 , we find that the correction is suppressed on the high-@xmath3 impurity site , while it is enhanced on the average . thus , to summarize , when the impurity spin is coupled more strongly ( weakly ) , the spin - fluctuation correction is enhanced ( suppressed ) locally at the impurity site , but the average correction to sublattice magnetization is suppressed ( enhanced ) . this local enhancement can be understood in terms of the correlations @xmath152 as follows . since the impurity spin is more strongly coupled to the neighboring spins , the nn matrix elements @xmath153 are enhanced . this puts more magnon amplitude @xmath154 on the impurity site , so that from eq . ( 2 ) the transverse spin correlations @xmath152 , and therefore the spin - fluctuation correction , are enhanced for low-@xmath3 impurities . the overall decrease in the averaged fluctuation correction , however , is due to the stiffening of the magnon spectrum in the important low - energy sector , following from the increased average spin coupling . .the local spin - fluctuation corrections @xmath155 for a @xmath156 system ( @xmath142 ) , with two impurities at ( 11,4 ) and ( 4,14 ) . for the two cases @xmath143 ( @xmath157 ) and @xmath151 ( @xmath158 ) , quantum corrections are enhanced / suppressed locally at the impurity sites ( indicated in boldfaces ) , but are suppressed / enhanced on the average . [ cols="^,^,^,^,^,^,^,^,^,^,^,^,^,^,^,^,^ " , ] 2 while these results also follow from the impurity - spin picture , the small charge gap at the impurity site does have an impact on the magnon spectrum . within the localized spin picture , the highest - energy magnon mode corresponding to a local spin deviation at the impurity site would cost energy @xmath159 . however , the highest energy in the magnon spectrum is actually seen to be @xmath160 , which is substantially smaller than @xmath161 . this shows the compression effect of the low charge gap ( @xmath162 ) on the magnon spectrum.@xcite in conclusion , using a convenient numerical method for evaluating transverse spin correlations at the rpa level , quantum spin - fluctuation corrections to sublattice magnetization are obtained for the half - filled hubbard model in the whole @xmath0 range . results in two and three dimensions are shown to interpolate properly between both the weak and strong correlation limits , and approach the swt results as @xmath2 . the method is readily extended to other situations of interest involving defects in the af , such as vacancies / impurities / disorder . numerical diagonalization for finite lattices , along with finite - size scaling tested with the pure hubbard model , allows for exact treatment of defects at the rpa level . this is illustrated with a study of the defect - induced enhancement / suppression in transverse spin fluctuations for spin vacancies and low-@xmath3 impurities in two dimensions . while the quantum spin fluctuation correction to sublattice magnetization is sharply enhanced by spin vacancies , it is strongly suppressed by the low-@xmath3 impurities , although the fluctuation correction is enhanced at the low@xmath1 sites . helpful conversations with d. vollhardt and m. ulmke , and support from the alexander von humboldt foundation through a research fellowship at the universitt augsburg are gratefully acknowledged .

a numerical method is described for evaluating transverse spin correlations in the random phase approximation . quantum spin - fluctuation corrections to sublattice magnetization are evaluated for the antiferromagnetic ground state of the half - filled hubbard model in two and three dimensions in the whole @xmath0 range . extension to the case of defects in the af is also discussed for spin vacancies and low@xmath1 impurities . in the @xmath2 limit , the vacancy - induced enhancement in the spin fluctuation correction is obtained for the spin - vacancy problem in two dimensions , for vacancy concentration up to the percolation threshold . for low-@xmath3 impurities , the overall spin fluctuation correction is found to be strongly suppressed , although surprisingly spin fluctuations are locally enhanced at the low@xmath1 sites . 2

zeus is a freely available mhd code that is widely used by the astrophysical community . although stone & norman ( 1992a , b ) give results for the sod problem ( sod 1978 ) and its mhd equivalent , the brio and wu problem ( brio & wu 1988 ) , zeus does not appear to have been tested on a wide range of riemann problems such as those described in e.g. dai & woodward ( 1994 ) , ryu & jones ( 1995 ) , falle , komissarov & joarder ( 1998)and balsara ( 1998 ) . since zeus is neither upwind for all characteristic fields nor conservative , we might expect it to perform significantly less well than upwind conservative codes ( e.g. brio & wu 1988 ; dai & woodward 1994 ; ryu & jones 1995 ; falle , komissarov & joarder 1998 ; balsara 1998 ; powell et al . as we shall see , this is indeed true in the sense that there are a number of simple problems for which the zeus solution contains significant errors that are absent in solutions calculated with an upwind conservative scheme . figures 1 and 2 show that zeus generates rarefaction shocks for both pure gas rarefactions and fast magnetosonic rarefactions , whereas the upwind conservative scheme described in falle , komissarov & joarder ( 1998 ) gives quite satisfactory results . in both cases , the zeus solutions are sensitive to the inertial frame and the rarefaction shocks can be removed by a galilean transformation that increases the x velocity sufficiently . these rarefaction shocks are steady structures whose width does not increase with time . since the effect of the nonlinear terms is to spread such structures , it is clear that the truncation errors in zeus must be anti - diffusive in these cases . the most obvious explanation for this is that zeus is second order in space , but first order in time since this can lead to an anti - diffusive term in the truncation error . for example , the upwind scheme can be made first order in time and second order in space by omitting the preliminary first order step and in that case it can be shown to be anti - diffusive and also produces rarefaction shocks . although zeus is second order in space and time for linear advection , the use of a partially updated velocity in the advection step means that it is only first order in time if the velocity is not constant . further evidence that this is the cause of the problem is provided by the sensitivity of the rarefaction shocks to the galilean frame and the fact that they disappear when the courant number is reduced from @xmath0 to @xmath1 , whereas they become much worse if the courant number is increased above @xmath0 . zeus has a facility for adding a linear artificial viscosity whose magnitude is determined by the parameter @xmath2 . the addition of such a viscosity removes the anti - diffusive terms by reducing the scheme to first order in space for everything except linear advection . for the gas rarefaction , @xmath3 cures the problem and seems to be optimal for a global courant number of @xmath0 , but it is too large if the local courant number associated with the wave is small . since the linear viscous term must balance an anti - diffusive term that scales like the timestep , it would be better if the viscous term that is implemented in zeus were multiplied by the local courant number associated with the wave that is causing the problem . since rarefaction shocks only arise for rarefactions associated with the sound wave with the largest speed relative to the grid , it is the smallest local courant number that is appropriate . in mhd the situation is even worse since , although the rarefaction shocks in the fast rarefaction can be removed by setting @xmath4 , this makes the scheme very diffusive for other waves . furthermore , the required value of @xmath2 depends on the particular problem . it might be possible to avoid such a large value of @xmath2 by adding an appropriate artificial resistivity , but the code has no facility for this . figures 3 and 4 show that , even for an initially smooth rarefaction wave , zeus is significantly less accurate than an upwind scheme . the results are for @xmath5 , but figure 4 shows that zeus is still first order even without this . in contrast , it is evident from figure 4 that the rate of convergence of the upwind scheme is second order . note that the upwind scheme also has an artificial viscosity as described in falle , komissarov & joarder ( 1998 ) , but since this is applied in the riemann solver , it does not reduce the order in smooth regions . incidentally , zeus performs even worse if one does not take the staggered grid into account in setting up the the initial solution . furthermore , for both codes , point samples were used to project the exact solution onto the grid , which is reasonable for zeus , but is somewhat unfair to a conservative scheme . the upwind scheme produces reasonable results at the lowest resolution , even though this corresponds to only @xmath6 cells in the rarefaction at the initial time , whereas zeus needs at least @xmath7 cells for the same accuracy . for a three dimensional calculation , this would require @xmath8 times the computing time and @xmath9 times the memory since zeus is about @xmath10 times the speed of the upwind scheme . the disparity in efficiency is actually greater than this because for both codes the courant number was set to the zeus default value of @xmath0 for all cases described in this paper . the upwind code can run at larger courant numbers than this , whereas even @xmath0 can be too large for zeus for some riemann problems . of course , the slower convergence of zeus also means that the situation would be even worse if greater accuracy were required . since zeus is not conservative , we expect it to generate errors at shocks which can not be reduced by increasing the resolution . as it turns out , these errors are small ( @xmath11 ) for pure gas dynamics and are entirely absent for an isothermal equation of state . however , they can be significant for adiabatic mhd . figure 5 shows that , for a nearly perpendicular fast shock , the post - shock gas pressure in the zeus solution is too low by a factor of 2 . in contrast , the conservative upwind scheme gets the solution exact to rounding . it is true that this is a somewhat extreme case since the plasma @xmath12 is negligible upstream of the shock and @xmath13 downstream . however , such low values of @xmath12 do occur in dense molecular clouds and protostellar discs ( e.g. crutcher 1999 ) . furthermore , even though @xmath12 is small , such errors in the gas pressure can have a significant effect on the dynamics because the gas pressure provides a force parallel to the field , whereas the lorentz force does not . finally , figure 6 shows that a relatively small error at a fast shock can be amplified by a slow shock following on behind . in this case the zeus solution has an error of @xmath14 in the density behind the slow shock travelling to the right . this is not caused by small @xmath12 since @xmath15 behind the fast shock , @xmath16 behind the slow shock and the error in the gas pressure is much smaller than in the density . like balsara ( 2001 ) , we find that zeus produces large post - shock oscillations for strong mhd shocks , but that these can be reduced substantially by adding the same linear artificial viscosity that removes gas dynamic rarefaction shocks . this is presumably because a quadratic viscosity leads to algebraic decay of these oscillations , whereas a linear viscosity gives exponential decay . the calculation shown in figure 6 used this value of the linear artificial viscosity and it can be seen that the amplitude of the post - shock oscillations is quite small . the calculations presented are all coplanar ( @xmath17 ) , but we have also looked at some non - coplanar problems in order to see whether the presence of alfvn waves causes any additional difficulties for zeus . this is not the case , at least for the problems that we have considered . it is evident from these results that , zeus can be made just about acceptable for pure gas dynamics if the linear artificial viscosity is multiplied by the smallest local courant number since the shock errors are small in this case . however , it is not satisfactory for adiabatic mhd , at least in its present form . the shock errors do not occur for an isothermal equation of state , but , since the rarefaction shocks do , zeus is also not reliable for isothermal mhd . it is possible that the rarefaction shocks in mhd waves can be removed without using an excessive linear artificial viscosity by the addition of an appropriate linear artificial resistivity . the shock errors might also be reduced by advecting the total energy rather than the internal energy . however , even with such improvements , the low order of accuracy makes zeus very inefficient compared with a modern upwind scheme . this should not be taken to mean that conservative upwind codes are in any sense perfect . for example , it is necessary to introduce some extra dissipation in the riemann solver to remove the serious errors discussed by quirk ( 1994 ) and some desirable properties , such as strict conservation , may have to be sacrificed in order to satisfy the constraint @xmath18 in multidimensional mhd ( see e.g. powell et al . 1999 ; balsara 2001 ) . these results obviously have implications for the reliability of the numerous calculations in the literature that have used zeus . although these effects are likely to be present in many cases , the associated errors are not necessarily so serious as to completely invalidate the calculations . whether or not they make any qualitative difference in any particular case can only be decided either by a thorough examination of the results to see whether any of these errors are present , or by repeating the calculations using a modern code . these calculations were performed with the version of zeus2d available from the ncsa website , but , since all versions of zeus appear to use the same algorithms , the results should not depend on the particular version . it is also worth pointing out that although we used the scheme described by falle , komissarov and joarder ( 1998 ) , similar results would probably have been obtained with any modern upwind code . balsara , d. s. 2001 , jcp , 174 , 614 balsara , d. s. 1998 , , 116 , 133 brio , m. , & wu , c. c. 1988 , jcp , 75 , 400 crutcher , r. m. 1999 , , 520 , 706 dai , w. , & woodward , p. r. 1994 , jcp , 111 , 354 falle , s. a. e. g. , komissarov , s. s. , & joarder , p. 1998 , , 297 , 265 powell , k. g. , roe , p. l. , linde , t. j. , gombosi , t .i . , de zeeuw , d. l. 1999 , jcp , 154 , 284 quirk , j. j. 1994 , int . j. numer . meth . , 18 , 555 ryu , d. , & jones , t. w. 1995 , , 442 , 228 sod , g. a. 1978 , jcp , 27 , 1 stone , j. m. , & norman , m. l. 1992a , , 80 , 753 stone , j. m. , & norman , m. l. 1992b , , 80 , 791

we show that there are simple one dimensional problems for which the mhd code , zeus , generates significant errors , whereas upwind conservative schemes perform very well on these problems .

concentrations of excess @xmath0 mg , the decay product of the short - lived radionuclide @xmath0al [ mean life = 1.03 myr @xcite ] , show that the solar system formed with @xmath1(@xmath0al)/@xmath1(@xmath2al ) = 5.2 @xmath3 10@xmath4 @xcite . although there is evidence that there may have been deviations from this canonical " ratio across the solar protoplanetary disk by as much as a factor of 2 @xcite , the overall concentration of @xmath0al in the solar disk was more than a factor of 10 greater than the current average value in the interstellar medium of 3.0 @xmath3 10@xmath5 @xcite . while some @xmath0al may have been produced within the early solar system , most of it was not @xcite ; there must have been a significant external source of this short - lived nuclide . commonly , the natal @xmath0al is taken as a signature of a nearby supernova that may have triggered the collapse of the molecular cloud from which the sun formed @xcite . alternatively , winds from massive stars may have supplied the bulk of the @xmath0al @xcite . a major consequence of large amounts of @xmath0al in the early solar system was substantial internal heating of young planetesimals which therefore melted and subsequently experienced igneous differentiation . iron meteorites are thought to be modern fragments of iron - rich cores formed during this era @xcite . if other planetary systems formed with considerably less @xmath0al , then their asteroids may not be differentiated . we can test this scenario by examining the elemental compositions of extrasolar minor planets . evidence is now compelling that some white dwarfs have accreted some of their own asteroids @xcite . in some instances , we have detected excess infrared emission from circumstellar disks composed of dust @xcite where gas also is sometimes evident @xcite . these disks lie within the tidal radius of the white dwarf and are understood to be the consequence of an asteroid having been shredded after its orbit was perturbed so it passed very close to the star @xcite . accretion from these disks supplies the orbited white dwarf s atmosphere with elements heavier than helium where they are normally not found because the gravitationally settling times are very short compared to the cooling age of the star . estimates of the amount of accreted mass argue that we are witnessing the long - lived evolution of ancient asteroid belts @xcite . in the most extreme case , the accreted parent body may have been as massive as ceres @xcite which has a radius near 500 km . however , the required mass more typically implies parent bodies with radii near 200 km @xcite . externally - polluted white dwarfs provide a means for placing the solar concentration of @xmath0al in context . as a first approximation , extrasolar asteroids resemble bulk earth being largely composed of oxygen , magnesium , silicon and iron and deficient in volatiles such as carbon and water @xcite as expected in simple models for planet formation from a nebular disk . when eight or more polluting elements are detected , it is possible to tightly constrain the history and evolution of the parent body @xcite . recent studies of such richly polluted stars have shown abundance patterns that can be best explained if the accreted planetesimal evolved beyond simple condensation from the nebula where it formed . for example , nltt 43806 is aluminum rich as would be expected if the accreted planetesimal largely was composed of a crust @xcite while pg 0843 + 516 is iron rich which can be explained by the accretion of a core @xcite . @xcite found that the abundance pattern of the object accreted onto gd 362 resembles that of a mesosiderite a rare kind of meteorite that is best understood as a blend of core and crustal material @xcite . here , we first revisit the current sample of extrasolar planetesimals with well - measured abundances and reconfirm that igneous differentiation is widespread @xcite . we then present a model to explain this result . finally , we consider our solar system from the perspective of extrasolar environments . the evidence for igneous differentiation among extrasolar planetesimals can be presented in a variety of ways @xcite . here , we display in figure 1 the abundance ratios by number , @xmath1(fe)/@xmath1(al ) vs. @xmath1(si)/@xmath1(al ) , for all seven externally - polluted white dwarf atmospheres where these three elements have been reported . we see that @xmath1(fe)/@xmath1(al ) varies by more than a factor of 100 , a much greater range than shown among main - sequence planet - hosting stars , solar system chondrites and even @xmath1(si)/@xmath1(al ) among these same polluted stars . the large range in @xmath1(fe)/@xmath1(al ) among extrasolar planetesimals must be the result of some powerful cosmochemical process . one possibility is that unlike in the solar system , some extrasolar planetesimals were formed largely of refractory elements @xcite resulting in low values of @xmath1(fe)/@xmath1(al ) because al is highly refractory . however , this scenario is not supported by available observations @xcite , and can not explain why some systems have relatively high values of @xmath1(fe)/@xmath1(al ) . because there is no viable nebular model to explain the observed range in @xmath1(fe)/@xmath1(al ) , the abundance variations must have been produced within the planetesimals themselves . abundance patterns in extrasolar planetesimals reproduce those in familiar rocks . the lowest value of @xmath1(fe)/@xmath1(al ) is comparable to the ratio in morb ( mid ocean ridge basalt ) , a characteristic crustal rock @xcite . the highest value of @xmath1(fe)/@xmath1(al ) exceeds that of dunite , a mantle rock , implying sampling of iron - rich core material @xcite . figure 1 shows that the range of @xmath1(fe)/@xmath1(al ) among extrasolar asteroids is even greater than the difference found between bulk moon @xcite and bulk mercury @xcite , two solar system objects which are understood as having a small and large iron core , respectively . we understand the variety of elemental compositions among extrasolar planetesimals as the consequence of a familiar three - step process . first , planetesimals form within the disk ; in this environment , volatiles such as water may be excluded . second , differentiation results in iron being concentrated in the core and aluminum being concentrated in the crust . third , collisions lead to stripping and blending of cores and crusts with a consequent dramatic variation in @xmath1(fe)/@xmath1(al ) in the end - product planetesimals , blends of different portions of core , mantle and crustal material . as with solar system asteroids , the heat source for igneous differentiation of extrasolar planetesimals most likely was from radioactive decay of @xmath0al @xcite . other possibilities do not seem viable . the gravitational potential energy released by forming a body of radius , @xmath6 , and mass , @xmath7 , can raise the temperature an amount , @xmath8 t given by : @xmath9 where @xmath10 is the specific heat ( j kg@xmath11 k@xmath11 ) and @xmath12 is the gravitational constant . for a typical object with @xmath6 @xmath13 200 km and @xmath7 @xmath13 1.0 @xmath3 10@xmath14 kg @xcite and with @xmath10 @xmath13 1000 j kg@xmath11 s@xmath11 @xcite , then @xmath15 @xmath13 20 k , much too small to be of importance . although mutual collisions can produce local heating , it seems unlikely that most of the material is melted during the period of planetesimal growth by collisions @xcite . within the average interstellar medium , @xmath16fe/@xmath17fe = 2.8 @xmath3 10@xmath18 @xcite , and if this ratio prevails within star forming regions , then heating from the radioactive decay of @xmath16fe can not be an important heating source within extrasolar planetesimals . it is possible that some stars form near supernovae that produce large amounts of @xmath16fe @xcite , and in these environments newly formed planetesimals could be significantly heated by radioactive decay of this radionuclide . however , because by number there is more @xmath0al than @xmath16fe within the entire galaxy @xcite and because @xmath0al is readily produced within massive stars and then injected into the local molecular interstellar medium where new stars form @xcite , it is probable that the majority of young stellar disks are similar to our own solar system where radioactive decay of @xmath0al was the dominant source of planetesimal heating . the usual expression @xcite governing the time ( @xmath19 ) variation of internal temperature , @xmath20 , as a function of radius , @xmath21 , of a spherical rocky body is : @xmath22 where @xmath23 ( m@xmath24 s@xmath11 ) is the thermal diffusivity and @xmath25 ( j kg@xmath11 s@xmath11 ) is the heating energy per unit mass per unit time . the typical timescale for the loss of internal heat is @xmath26 . for a 200 km radius object with a thermal diffusivity of 10@xmath5 m@xmath24 s@xmath11 @xcite , the outward diffusion of heat represented by the first term on the right hand side of equation ( 2 ) typically requires more than 1 gyr and has a negligible effect on the body s central temperature during the era of heating from @xmath0al . to compute the maximum internal temperature , @xmath27 , we integrate over a time scale much longer than the average decay time , @xmath28 , and consider the total released energy , @xmath29 , defined as : @xmath30 from the decay of @xmath0al . consequently , if the planetesimal originates at temperature , @xmath31 ( k ) , then : @xmath32 where @xmath33 is the initial fraction of the mass of the planetesimal that is @xmath0al . we assume that igneous differentiation is only possible if the internal temperature exceeds the solidus temperature , @xmath34 , @xcite and then derive the minimum aluminum isotope ratio by number , @xmath35al)/@xmath36al ) , required to achieve this temperature . we assume that the extrasolar planetesimal will have formed at some time , @xmath37 , after inheritance of @xmath0al from the molecular cloud . subsequently , no fresh @xmath0al enters the star - forming cloud ; instead , there is only radioactive decay with mean life , @xmath28 . validated by our detailed calculations not shown here , we take @xmath10 to be constant and independent of temperature . if @xmath38 is the fraction of mass of the planetesimal which is aluminum , and if @xmath31 @xmath39 @xmath34 , then : @xmath40 using cv chondrites with their relatively high al , thus providing a minimum for equation [ 5 ] , we take @xmath38 = 0.0175 @xcite . we adopt @xmath34 = 1500 k @xcite , and , for @xmath0al , we take @xmath29 = 1.2 @xmath3 10@xmath41 j kg@xmath11 @xcite . we assume two contributions to @xmath37 . first , there is free - fall gravitational collapse of a cloud core with an initial radius of 0.1 parsec that requires @xmath13 0.5 myr @xcite . second , planetesimals must assemble within the disk which , by analogy with the solar system , probably takes @xmath421 myr @xcite . adding both terms , @xmath37 = 1.5 myr . consequently , we compute from equation ( 5 ) that in extrasolar environments where planetesimals internally melted , @xmath35al)/@xmath36al ) @xmath43 3 @xmath3 10@xmath4 , approximately its value in the early solar system . this result is inexact . if , for example , we take @xmath38 = 0.0086 as found in ci chondrites @xcite , then the minimum values of @xmath35al)/@xmath36al ) should be doubled . as has been previously suggested qualitatively , a general enrichment of @xmath0al in protoplanetary disks might occur if this radionuclide is not distributed evenly throughout the milky way but , instead , is confined to regions of star formation @xcite . a plausible model to explain why @xmath0al would be so concentrated is that this species is largely injected into the interstellar medium from rotating massive stars @xcite . these massive stars are so short lived , that they all reside near their birth sites within molecular clouds . such a model can also explain why the solar system has a relatively high concentration of @xmath0al and a relatively low concentration of @xmath16fe @xcite . however , winds from wolf - rayet stars might shred cloud cores and prevent the formation of planets @xcite . while some recent models for supernova ejecta into molecular clouds also predict that solar mass stars commonly form with elevated amounts of @xmath0al @xcite , they do not naturally explain the solar system s simultaneously depressed value of @xmath16fe/@xmath17fe . in the entire milky way , the mass of hydrogen in h@xmath44 is 8.4 @xmath3 10@xmath45 m@xmath46 @xcite . the amount of interstellar @xmath0al is measured from the intensity of the @xmath47-ray line at 1.8 mev that results from its radioactive decay . including foreground emission , there is somewhere between 1.5 and 2.2 m@xmath46 of @xmath0al within the galaxy @xcite . if we assume the solar aluminum abundance of @xmath1(@xmath2al)/@xmath1(h ) = 3.5 @xmath3 10@xmath5 @xcite , and if all measured interstellar @xmath0al is confined only to molecular clouds , then in these locations @xmath35al)/@xmath36al ) @xmath13 2.0 - 3.0 @xmath3 10@xmath4 , nearly the same as the minimum ratio we infer for the birth environment of extrasolar planetesimals . consider not only the entire milky way but also observations of the orion region , the nearest molecular cloud where large numbers of high - mass stars currently are being formed . orion s @xmath47-ray line emission is explained with 5.8 @xmath3 10@xmath48 m@xmath46 of @xmath0al @xcite from a region where the total mass of h@xmath44 is approximately 2 @xmath3 10@xmath49 m@xmath46 @xcite . the implied value of @xmath35al)/@xmath36al ) in the orion star - forming region is therefore 3 @xmath3 10@xmath4 , again substantially elevated over the average interstellar value . remarkably , the apparent fraction of @xmath0al within star - forming molecular clouds agrees with the value required to explain the widespread occurrence of differentiated extrasolar planetesimals . it follows that the solar system s initial complement of @xmath0al was essentially normal . 99 bond , j. c. , obrien , d. p. , & lauretta , d. s. 2010 , , 715 , 1050 bonsor , a. , mustill , a. j. , & wyatt , m. c. 2011 , , 414 , 930 boss , a. p. & keiser , s. a. 2013 , , 770 , 51 brown , s. m. , & elkins - tanton , l. t. 2009 , e&psl , 286 , 446 castillo - rogez , j. , lee , m. h. , turner , n. j. , et al . 2009 , icar . , 204 , 658 davison , t . m. , collins , g. s. , & ciesla , f. j. 2010 , icar . , 208 , 468 debes , j. h. , & sigurdsson , s. 2002 , , 572 , 556 debes , j. h. , walsh , k. j. , & stark , c. 2012 , , 747 , 148 desch , s. j. , morris , m. a. , connolly , h. c. , & boss , a. p. 2010 , , 725 , 692 draine , b. t. 2011 , physics of the interstellar and intergalactic medium ( princeton : princeton univ . press ) dufour , p. , kilic , m. , fontaine , g. et al . , 2012 , , 749 , 6 duprat , j. & tatischeff , v. 2007 , , 671 , l69 farihi , j. , jura , m. , & zuckerman , b. 2009 , , 694 , 805 gaensicke , b. , t. , koester , d. , farihi , j. , et al . , 2012 , , 424 , 323 gaenscike , b t. , marsh , t. r. , southworth , j. , & rebassa - mansergas , a. 2006 , science , 314 , 1908 gaidos , e. , krot , a. n. , williams , j. p. , & raymond , s. n. 2009 , , 696 , 1854 genzel , r. , & stutzki , j. 1989 , ann . astrophys . , 27 , 41 ghosh , a. & mcsween , h. y. 1998 , icar . , 134 , 187 gilli , t. , israelian , g. , ecuvillon , a. , santos , n. c. , & mayor , m. 2006 , , 449 , 723 gounelle , m. , & meynet , g. 2012 , , 545 , a4 gritschneder , m. , lin , d.n.c . , murray , s. , d. , yin , q .- z . , & gong , m .- 2012 , , 745 , 22 hanghoj , k. keleman , p. b. , hassler , d. , & godard , m. 2010 , j. petrology , 51 , 201 hartmann , l. 2009 , accretion processes in star formation ( 2nd ed . ; cambridge : cambridge univ . press ) jacobsen , b. , yin , q .- z . , moynier , f. , et al . , e&psl , 272 , 353 jura , m. 2003 , , 584 , l91 jura , m. 2006 , , 653 , 613 jura , m. , & xu , s. 2012 , , 143 , 6 jura , m. , & xu , s. 2013 , , 145 , 30 jura , m. , xu , s. , klein , b. , koester , d. , & zuckerman , b. 2012 , , 750 , 69 jura , m. & young , e. d. 2014 , ann . earth planet . , 42 , in press klein , b. , jura , m. , koester , d. , zuckerman , b. , & melis , c. 2010 , , 709 , 650 larsen , k. , trinquier , a. , paton , c. , et al . 2011 , , 735 , l37 liu , m .- c . , chaussidon , m. , gopel , g. , & lee , t. 2012 , e&psl , 327 , 75 lodders , k. 2003 , , 591 , 1220 martin , p. j. , knoedlseder , j. , diehl , r. , & meynet , g. 2009 , , 506 , 703 mcsween , h. y. , & huss , g. r. 2010 , cosmochemistry ( cambridge : cambridge univ . press ) meyer , b. s. , & clayton , d. d. 2000 , space sci . , 92 , 133 pan , l. , desch , s. j. , scannapieco , e. , & timmes , f. x. 2012 , , 756 , 102 prantzos , n. 2004 , , 420 , 1033 presnall , d. c. , & hoover , j. d. in magmatic processes : physicochemical principles , the geochemical society special publications , 1 , 75 scott , e. r. d. , haack , h. , & love , s. g. 2001 , m&ps , 36 , 869 tang , h. , & dauphas , n. 2012 , e&psl , 359,248 turcotte , r. , & schubert , g. 2002 , geodynamics ( 2nd ed . ; cambridge : cambridge univ . press ) vasileiadis , a. , nordlund , a. , & bizarro , m. 2013 , , 769 , l8 voss , r. , diehl , r. , vink , j. s. , & hartmann , d. h. 2010 , , 520 , a51 wang , w. , harris , j. , diehl , r. , et al . , 2007 , , 469 , 1005 warren , p. h. 2005 , meteorit . , 40 , 477 wasson , j. t. , & kallemeyn , g. w. 1988 , phil . a , 325 , 535 xu , s. & jura , m. 2012 , , 745 , 88 xu , s. , jura , m. , koester , d. , klein , b. , & zuckerman , b. 2013 , , 766 , 132 zhou , q. , yin , q .- z . , young , e. d. , 2013 , gecoa , 110 , 152 zuckerman , b. , koester , d. , dufour , p. , et al . 2011 , , 739 , 101 zuckerman , b. , koester , d. , melis , c. , hansen , b. , & jura , m. 2007 , , 671 , 872 zuckerman , b. , melis , c. , klein , b. , koester , d. , & jura , m. 2010 , , 722 , 725

recently acquired evidence shows that extrasolar asteroids exhibit over a factor of 100 variation in the iron to aluminum abundance ratio . this large range likely is a consequence of igneous differentiation that resulted from heating produced by radioactive decay of @xmath0al with an abundance comparable to that in the solar system s protoplanetary disk at birth . if so , the conventional view that our solar system began with an unusually high amount of @xmath0al should be discarded .

in recent years , the study of nonequilibrium dynamics in quantum field theory has received much attention in various areas of physics , and particularly in cosmology . the work has been driven largely by inflation @xcite , the most successful known mechanism for explaining the large - scale homogeneity and isotropy of the universe _ and _ the small - scale inhomogeneity and anisotropy of the universe @xcite . with observations for the first time able to directly test the more detailed predictions of specific inflationary models , the efforts in understanding inflation and its dynamics have redoubled . one area of particular interest is the dynamics of multi - field models of inflation in which the inflaton is coupled to another dynamical field during inflation . these models can lead to a variety of features unavailable in the case of a single field . such multi - field scenarios include the well known hybrid inflation models @xcite . on top of the dynamics during inflation , the subsequent phase of energy transfer between the inflaton and other degrees of freedom leading to the standard picture of big bang cosmology has been the subject of intense study . the inflaton may decay through perturbative processes @xcite as well as non - perturbative parametric amplification @xcite . the latter can lead to explosive particle production and very efficient reheating of the universe . hybrid inflation and reheating models share an important common thread . they both involve the coupling of two or more dynamical , interacting scalar fields ( or higher spin fields @xcite ) . an important aspect of such systems is the possibility of mixing between the fields . in ref . @xcite for example the classical inflaton decay is investigated for a two field model by solving the non - linear equations of motions on a grid . in ref . @xcite , the authors treat the problem of coupled quantum scalar and fermion fields at the tree level . due to the small couplings involved in inflationary cosmology , such a tree level analysis is useful in a variety of physical situations . however , hybrid models as well as the dynamics of reheating typically include processes such as spinodal decomposition @xcite and parametric amplification which require one to go beyond the tree level by including quantum effects either in a perturbative expansion or by means of non - perturbative mean field techniques such as the hartree approximation or a large-@xmath0 expansion @xcite . going beyond tree level brings in the issue of renormalization . the problem of renormalization of time evolution equations in single field models was understood several years ago . in one of the first papers in this field , cooper and mottola showed in 1987 ( ref . @xcite , that it is possible to find a renormalization procedure which leads to counter terms independent of time and initial conditions of the mean field . they used a wkb expansion in order to extract the divergences of the theory . in a later paper cooper et al . also discussed a closely related adiabatic method in order to renormalize the @xmath1-theory in the large n approximation . also boyanovsky and de vega , ref . @xcite , used a wkb method in order to renormalize time dependent equations in one - loop order , later on boyanovsky et al . @xcite investigated a @xmath1 model in the large - n approximation and the hartree approximation , too . in 1996 baacke et al . , ref . @xcite , proposed a slightly different method in order to extract the divergences of the theory , which enabled them to use dimensional regularization . in contrast to the wkb ansatz this method can be extended for coupled system , which was demonstrated in ref . this procedure will be used also in this paper . we work in the context of a closed time path formalism @xcite appropriate to following the time - dependent evolution of the system . in this formalism , the _ in_-vacuum plays a predominant role , as quantities are tracked by their _ in - in _ expectation values ( in contrast to the _ in - out _ formalism of scattering theory ) . we construct the _ in_-vacuum by diagonalizing the mass matrix of the system at the initial time @xmath2 . however , because of the time - dependent mixing , a system initially diagonalized in this way will generally not be diagonalized at later times . one approach to this problem , taken in ref . @xcite , is to diagonalize the mass matrix at each moment in time through the use of a time - dependent rotation matrix . the cost of doing so is the appearance of time derivatives of the rotation matrix into the kinetic operators of the theory . while such a scheme is in principle workable beyond the tree level , the modified kinetic operators introduce complications into the extraction of the fluctuation corrections as well as the divergences that are to be removed via renormalization . we take an alternative approach where the mass matrix is allowed to be non - diagonal for all times @xmath3 and account for the mixing by expanding each of the fields in terms of _ all _ of the _ in_-state creation and annihilation operators . the cost of doing so in an @xmath0-field system is the need to track @xmath4 complex mode functions representing the fields instead of the usual @xmath0 . however , this allows standard techniques to be used to properly renormalize the system . for the two - field systems common in inflationary models , this effective doubling of the field content adds a relatively minor cost . for simplicity and clarity , we will work in minkowski space time and in a one - loop approximation . extensions both to friedmann - robertson - walker spacetimes and to simple non - perturbative schemes such as the hartree approximation , while more complicated than the present analysis , present no fundamental difficulties . we note that minkowski space time is a good approximation in the latter stages of certain hybrid inflation models , and it will also allow comparison with much of the original reheating literature @xcite which often neglects the effects of expansion , allowing us to directly determine the role played by the mixing of the fields in the dynamics . the outline of the paper is as follows . we begin by considering the lagrangian for @xmath0 coupled scalar fields and set up our formalism for the quantization of the system . this is followed by an outline of the renormalization procedure . we then provide a summary of the results for the two - field case . we demonstrate the formalism with two examples : a simple reheating model and a hybrid inflation model motivated by supersymmetry . in the reheating model we investigate two relevant regimes discussed in detail in the literature @xcite , viz . , the narrow resonance regime and the broad resonance regime these different regimes occur depending on the choice of initial conditions . usually in these models the mixing effects of the fields were neglected by choosing a vanishing initial value for one of the mean fields : we are now able to treat the full system and to investigate these mixing effects . for this purpose we concentrate on studying the behavior of the fluctuation integrals for the different fields and the time - dependent mixing angle . depending on the regime , as the mean fields evolve , the effects of the mixing can be quite different . in the narrow resonance regime the mixing angle is very small and plays a sub - dominant role , whereas in the broad resonance regime the mixing effects are very important . therefore , we emphasize that neglecting the mixing could lead to incomplete results . supersymmetric hybrid models are a special realization of general hybrid inflationary models ( see e.g. refs . @xcite ) . based on a softly broken supersymmetry potential , the special feature of these models is the occurrence of only one coupling constant , whereas in nonsupersymmetric hybrid models there are at least two different couplings . thus , in the supersymmetric case there is only one natural frequency of oscillation for the mean fields as long as fluctuations are neglected . this leads to efficient particle production during the preheating stage in the early universe . however , we show below that , by taking into account the fluctuations and investigating the full mixed system , this feature of supersymmetric hybrid models can be lost in some regimes . this is because the effective mass corrections for the two fields are different in these regimes , which leads to a chaotic trajectory for the renormalized field equations of motion in a phase space which mimics the situation of a nonsupersymmetric hybrid model . it appears , then , that supersymmetric hybrid models can lose some of their attractiveness compared to general hybrid models . we work with the following lagrangian for real scalar fields @xmath5 with @xmath6 : @xmath7 = \sum_{i=1}^n \frac12 \partial_\mu \phi_i(x ) \partial^\mu \phi_i(x ) - v[\phi_i(x ) ] \ ; , \ ] ] where the potential is @xmath8 note that @xmath9 , @xmath10 , and @xmath11 are symmetric in each index , but are generally non - diagonal resulting in the mixing of the different fields . in what follows , subscripts and superscripts @xmath12 run over the values @xmath13 and we use a convention in which summation is assumed over repeated lowered indices , but not raised indices . we will expand each field about their expectation values ( taken to be space translation invariant ) : @xmath14 expanding the equations of motion and keeping terms to quadratic order in the fluctuations yields a one loop approximation . the equations of motion for the zero modes @xmath15 are determined via the tadpole condition . we have @xmath16 to this order , the fluctuations obey the equation @xmath17 with the mass matrix @xmath18 as indicated in the introduction , the complication that arises is not the fact that the mass matrix ( [ mij ] ) contains mixing between the various fields , rather that the mixing changes with time as the @xmath15 evolve according to ( [ phiieq ] ) . this means that if we diagonalize the mass matrix at one time , it will not generally be diagonal at any other time . nonetheless , it is most convenient to quantize in terms of a diagonal system at the initial time @xmath2 . we define the matrix @xmath19 and the corresponding fluctuation fields @xmath20 where @xmath21 is an orthogonal rotation matrix . @xmath22 is diagonal at the initial time : @xmath23 without summation over the raised index @xmath24 . the @xmath25 obey the equations of motion @xmath26 we quantize the system by defining a set of creation and annihilation operators @xmath27 and @xmath28 where @xmath29 corresponds to the _ in_-state quanta of frequency @xmath30 as the mixing changes in time , each of the fields @xmath25 is expanded in terms of all of the _ in_-state operators . we have @xmath31 \ ; .\ ] ] the initial conditions for the @xmath4 complex mode functions are @xmath32 it is convenient to define the fluctuation integrals @xmath33 from which it is straightforward to determine the contributions appearing in the zero mode equations ( [ phiieq ] ) : @xmath34 it will also prove convenient to introduce the rotated couplings @xmath35 which allows us to write the zero mode equations as @xmath36 while the mode functions obey the equations @xmath37 in addition to the equations of motion , it is useful to have an expression for the energy density of the system . this is particularly true when one completes numerical simulations of the system , since energy conservation is a powerful check of the accuracy of the simulations . after once again decomposing the fields into their expectation values and fluctuations , the energy density to one loop order is @xmath38 where we ve defined the integrals @xmath39 the mode integrals in the equation of motion defined by eq . ( [ xijfluct ] ) and in the energy density defined by eqs . ( [ xidotfluct ] ) , ( [ xigradfluct ] ) are divergent and have t be regulated , allowing for a renormalization of the theory . we require a method of extracting the divergent terms appearing in the mode integrals , a nontrivial task , since the mode equations vary in time and they are coupled . our aim is now , to find counter terms , which are independent of the initial value of the mean fields in order to formulate a finite theory . the correct choice of the initial condition for the fluctuations guarantees that the theory is renormalizable . one way to extract the divergences of the mode integrals is due to a wkb method which allows for a high momentum expansion of the mode functions . however , when the fields are coupled , as in the present case , the usual formulation of the wkb expansion runs into difficulties which are yet to be resolved . an alternative method has been developed @xcite which relies on a formal perturbative expansion in the effective masses and time derivatives of the masses of the fields . as such , it results in a series expansion of the mode functions in powers of @xmath40 and @xmath41 , etc . the first few terms in the series include the divergent parts of the integrals that are to be removed via renormalization . we begin by introducing the following ansatz for the mode functions : @xmath42 the first term on the right hand side anticipates a quadratic divergence in the quantities @xmath43 . we define the following potential @xmath44 the equations of motion for the mode functions eqs . ( [ mod ] ) can be written in a suggestive form with the help of eqs . ( [ ansatz0],[ansatz1 ] ) @xmath45 the terms on the right hand side of this expression are treated as perturbations to write the @xmath46 order by order in @xmath47 , with the initial conditions @xmath48 . to first order in @xmath47 , we have the equations of motion : @xmath49 the corresponding integral solutions for the real part of the @xmath50 s are : @xmath51 while the imaginary part is of order @xmath52 and does not contribute to the divergences @xcite . using these results , we find quadratic and logarithmic divergences : @xmath53 which must be removed via some renormalization procedure while also providing finite corrections to the parameters of the theory . to make the renormalization scheme explicit , we adopt dimensional regularization . we define the following divergent integrals @xmath54 where @xmath55 is an arbitrary renormalization point and @xmath56 carries the infinite contributions . in dimensional regularization @xmath57 is given by @xmath58 the infinite part of @xmath59 is found to be simply @xmath60 this leads to mass and coupling constant counter terms of the following form @xmath61 \ ; , \\ \delta g_{ijk } & = & \frac32 i_{-3}(\mu ) g_{ilm } \lambda_{lmjk } \ ; , \\ \label{dell } \delta \lambda_{ijkl } & = & \frac32 i_{-3}(\mu ) \lambda_{ijmn } \lambda_{mnkl } \ ; . \end{aligned}\ ] ] it is important to notice , that these counterterms are independent of the initial conditions of the mean fields @xmath15 . in addition to these counterterms , there are finite corrections of the parameters coming from the finite parts of the integrals ( [ xixjdiv ] ) : @xmath62 from this , we extract the following finite contributions to the couplings and mass : @xmath63 \ ; , \\ \delta m_{ij } & = & - \frac{1}{32\pi^2 } \left [ \lambda_{ijkk } d^k + \left(g_{ikl}g_{klj } + \lambda_{ijkl } m_{kl}\right ) \ln \frac{d^k}{\mu^2 } \right ] \ ; , \\ \delta g_{ijk } & = & - \frac{1}{8\pi^2 } g_{ilm}\lambda_{lmjk } \ln \frac{d^m}{\mu^2 } \ ; , \\ \label{dellf } \delta \lambda_{ijkl } & = & - \frac{3}{32\pi^2 } \lambda_{ijmn } \lambda_{mnkl } \ln \frac{d^m}{\mu^2 } \ ; .\end{aligned}\ ] ] these finite corrections are also contributing to the energy density . in addition we find a finite part due to the cosmological constant renormalization . the full , finite equations of motion become @xmath64 = 0 \ ; . \label{phiieqxren}\end{aligned}\ ] ] the two - field case is often encountered , and the physical applications we present in the next section are both in this category . it is therefore worthwhile to pause to look at a few details of such systems . we begin with a system of two real scalar fields @xmath65 and @xmath66 @xmath67 with the potential @xmath68 this lagrangian has the same form as that studied in the preceding section with the identifications @xmath69 the remaining components of @xmath11 are determined by the fact that it is symmetric in each of its indices . the mass matrix @xmath70 is @xmath71 for two fields , the orthogonal rotation matrix can be written in terms of a single mixing angle @xmath72 . the matrix has the form @xmath73 where the mixing angle is determined by the @xmath2 mass matrix @xmath70 , eq . ( [ m_ij ] ) , through the relation @xmath74 \ ; . \label{thetaeqn}\ ] ] the eigenvalues of @xmath75 are the diagonal elements of the matrix @xmath76 , eq . ( [ dij ] ) , at the initial time : @xmath77 with the values @xmath78 \ ; , \\ d^2 & = & \frac12 \left [ { \cal m}_{11}(0 ) + { \cal m}_{22}(0 ) - \sqrt{\left({\cal m}_{22}(0 ) - { \cal m}_{11}(0 ) \right)^2 + 4 { \cal m}^2_{12}(0 ) } \right ] \ ; .\end{aligned}\ ] ] for general times , the mass matrix for the fields @xmath79 and @xmath80 , writing @xmath81 and @xmath82 , is @xmath83 the zero mode equations , before renormalization , read @xmath84 where @xmath85 the total energy density of the system including the fluctuations can be expressed as @xmath86 now we have to formulate finite equations of motion and a finite energy density . we adopt the renormalization procedure of section [ renorm ] for the @xmath0 field case . by using the identifications ( [ ident ] ) it is to derive the appropriate counterterms for the two field case from eqs . ( [ dela])-([dell ] ) . we find in particular @xmath87 of course we get also similar to eqs . ( [ delaf])-([dellf ] ) in the @xmath0 field case finite corrections to the masses and couplings of the form @xmath88 where @xmath89 as a result of these finite corrections eqs . ( [ fin0]-[fin4 ] ) , the total lagrangian eq . ( [ lag ] ) is also modified . this is exactly the renormalized lagrangian which we needed . we also find an additional finite contribution to the classical lagrangian given by @xmath90\right \}\ , , \nonumber \\\end{aligned}\ ] ] and , the final zero mode equations for @xmath91 and @xmath92 are given by @xmath93 and , @xmath94 after writing down the finite zero mode equations of motion we also have to renormalize the energy density . again by using the ansatz ( [ ansatz0 ] ) we can extract the divergent terms of the fluctuation integrals in eq . ( [ energy0 ] ) . in addition to the quadratic and logarithmic divergences we find a quartic divergence . this leads to a counter term which acts as a cosmological constant and has the form @xmath95 altogether the divergent part of the energy density reads : @xmath96 this expression leads of course to the same counter terms we found for the equations of motion , and therefore also to the same finite corrections to couplings and masses . therefore it is straightforward to formulate a finite energy expression . now , we are in a position to discuss the physical applications of our problem . this we shall do in the next section , but first , we introduce one more quantity that is convenient in discussing the degree to which the mixing plays a role in the dynamics . to better understand how the system evolves in time , it is useful to have a measure of how much the fields @xmath91 and @xmath92 mix at each moment , and how this mixing evolves with the system . to provide us with a measure of the mixing , we introduce a time - dependent mixing angle @xmath97 , which is defined in terms of the time - dependent mass matrix @xmath98 , eq . ( [ m_ij ] ) : @xmath99 \ ; . \label{thetateqn}\ ] ] after setting up the technical framework , we are now in a position to investigate some relevant cosmological multi - field models for inflation . we begin our analysis with a simple two - field model often used for studying the phase of parametric amplification . ( this phase occurs just after the completion of inflation in chaotic inflationary models @xcite . ) this model provides a useful context to analyze the effects due to field mixing . next we turn our attention to a supersymmetric hybrid inflationary model , which is of particular interest in cosmology . as discussed in the literature ( see for example ref . @xcite ) particle production ( and hence reheating ) in these models is much more efficient compared to the nonsupersymmetric hybrid models . until now the mixing effects in these models have not been treated fully including back reaction effects of the quantum fluctuations in the mean field approximation . this approximation does not take rescattering processes into account and therefore we can not address the problem of thermalization . the reheating phase in chaotic inflationary models is characterized by two different regimes , which depend on the chosen initial conditions : the first is the narrow resonance regime , while the second is the case of broad resonance . in order to investigate these regimes we examine two different parameter sets , where only the initial values for the zero modes are varied . we find significant differences in the behavior of the fluctuation integrals as well as the mixing angle in the two regimes . to make the analysis as simple as possible , we set @xmath100 as well as @xmath101 . for the remaining parameters , we set @xmath102 , which just acts as a unit of mass , and @xmath103 . with these parameters , the case usually studied in the literature , @xmath104 , does not introduce any mixing between the fields since the off - diagonal elements of the @xmath91-@xmath92 mass matrix are proportional to @xmath92 . however , taking @xmath104 may not always be the case . for instance , @xmath92 field could take a large vacuum expectation value during inflation , provided @xmath92 is treated as a field other than the inflaton . for the purpose of illustration we may consider a non - zero initial condition for @xmath92 which is of order its effective mass @xmath105 at the beginning of the preheating stage and examine the consequences . the initial condition for @xmath91 in the narrow resonance regime is fixed by the condition @xmath106 ( remember , @xmath107 is fixed to be @xmath108 ) and for the broad resonance regime by @xmath109 . if we take @xmath110 to be approximately the planck scale as appropriate to the end of inflation , this would correspond to @xmath111 and @xmath112 for the two respective cases . note that these parameters are chosen to depict the phenomena of interest during a time scale that can be accurately simulated . the results are , in any case , representative of two field mixing in the narrow and broad resonance regimes regardless of the precise parameter values in any particular model . 1 shows the log of the three fluctuation integrals @xmath113 , @xmath114 and @xmath115 for the narrow resonance case . these are seen to be dominated by the exponential growth of @xmath115 , while the other contributions grow more slowly . therefore , the evolution is characterized by production of @xmath80 particles . we now turn to the broad resonance regime , where things look quite different . here , each of the fluctuation integrals grows rapidly as shown in fig . 2 . significant mixing of the species occurs along with copious particle production . the behavior of the fluctuation integrals is consistent with the behavior of the time - dependent mixing angle @xmath116 . here , the mixing plays a sub - dominant role in the narrow resonance regime ( fig.3 ) with the mixing angle remaining near zero . this means that @xmath80 predominantly corresponds to the @xmath92 field , such that the process is one of @xmath92 particle production , which is as expected . concentrating on the time - dependent mixing angle in the broad resonance regime , fig . 4 , significant mixing between the fields is observed . the rapid variation in the mixing angle indicates that mixing between the fields plays a very important role in the evolution of preheating in the broad resonance regime . the large influence this has on the behavior of the system is clear from the evolution of the zero mode components @xmath117 in fig . 5 and @xmath118 in fig . we now consider a hybrid inflationary model where the finite coupling of two fields plays an interesting role in the termination of slow roll inflation @xcite . the particular model we study is based on softly broken supersymmetry @xcite with the potential @xmath119 the field @xmath120 plays the role of an inflaton during inflation while the field @xmath0 is trapped in a false vacuum @xmath121 . the inflaton rolls down the potential along the @xmath120 direction to reach a critical value @xmath122 . once @xmath120 reaches its critical value , the effective squared mass for the @xmath0 field becomes negative and consequently it rolls down from the false vacuum to its global minimum through the mechanism of spinodal instability @xcite . thus , inflation comes to an end and both the fields begin oscillations around their respective minima given by @xmath123 this is the onset of the preheating stage , which has been discussed in the literature @xcite . the difference between the supersymmetric hybrid potential and non - supersymmetric hybrid potentials lies in different coupling constants . in eq . ( [ pot1 ] ) , there is only single coupling parameter @xmath124 , while in the non - supersymmetric version there can be at least two different coupling constants . the above potential , except for the @xmath120 mass term , can be derived very easily from the superpotential for f - term spontaneously supersymmetry breaking : @xmath125 the appearance of a mass term for @xmath120 in eq . ( [ pot1 ] ) is due to the presence of soft supersymmetry breaking . its presence is essential for slow roll inflation to produce adequate density perturbations and also to provide a correct tilt in the power spectrum @xcite . the height of the potential during inflation is given by @xmath126 . similar potentials to eq . ( [ pot1 ] ) can also be derived from d - term supersymmetry breaking as discussed in refs . @xcite . in these models the critical value @xmath127 and the height of the potential energy are related to the fayet - illipoulus term coming from an anomalous u(1 ) symmetry . as in any inflationary model , hybrid inflation is constrained by cobe @xcite . this imposes the bound @xmath128 where @xmath129 is one of the slow roll parameters which determines the slope of the power spectral index @xcite . for our purpose we fix it to be @xmath130 . in order to discuss the details of the physics we mention here the equivalence between eq . ( [ pot1 ] ) and eq . ( [ pot ] ) . this helps us to establish direct relationship with our earlier analysis : @xmath131 notice , that @xmath132 is negative . an interesting feature of the hybrid model is that irrespective of the values of the parameters @xmath124 , @xmath127 , and @xmath133 , as long as they satisfy the cobe constraints , the behavior of the mean fields follow a common pattern once they begin to oscillate @xcite . first of all , the mass term for the @xmath120 field , @xmath133 , becomes less dominant compared to the effective frequency for the two fields , which is given by the effective mass for the two fields during oscillations @xmath134 hence , there is a single natural frequency of oscillation , thanks to supersymmetry . since the masses of the fields are the same at the global minima , there exists a particular solution of the equations of motion for the @xmath120 and @xmath0 fields . their trajectory follows a straight line towards their global minima : @xmath135 the maximum amplitude attained by the @xmath120 field is @xmath136 , while the other field attains a larger amplitude @xmath137 . ( we remind readers that @xmath138 corresponds to the point where the effective mass for @xmath0 field changes its sign . ) this is the point of instability which we need to discuss here . from eq . ( [ pot1 ] ) , we notice that prior to the oscillations of the fields , and during the oscillations , the effective mass square for @xmath120 is always positive . however , this is not the case for @xmath0 , and its mass square can be positive as well as negative even during the oscillations of the fields , provided the amplitudes are large enough . if the amplitudes for @xmath120 and @xmath0 are such that they satisfy eq . ( [ traj ] ) , then the effective mass square for the @xmath0 field is in fact always negative for @xmath139 . if the amplitudes are large enough such that after the second order phase transition the initial amplitude for @xmath140 , it is then quite possible that near the critical point the effective mass for the field @xmath0 vanishes completely . as far as the motion of the mean field without including fluctuations is concerned this does not provide any new insight . however , if the fields are quantized then the perturbations in the field , especially for @xmath141 , grow exponentially because @xmath142 in eq . ( [ zerofreq ] ) becomes negative for sufficiently small momentum @xmath143 . this shows that the vacuum is unstable near the critical point @xmath127 . another intuitive way to appreciate this point is to consider the adiabatic condition for the vacuum . the adiabatic evolution for the zero mode evolution for @xmath0 field is given by @xmath144 . this condition is maximally violated at the point where the effective mass square for @xmath0 becomes zero , and , violation in adiabatic evolution of the zero mode for @xmath0 suggests that many fluctuations of @xmath141 are produced during the finite period when the adiabaticity is broken @xcite . this explanation is quite naive because the overall production of particles and fluctuations depends also upon the global behavior of the zero mode fields . the effect of corrections due to fluctuations might affect the production of particles and this is the point we are going to emphasize in our numerical simulations . in some sense the hybrid model is quite different from chaotic inflationary models . in chaotic models , the inflaton field rolls down with an amplitude @xmath145 , where @xmath146 is the mass of the oscillating field . however , in the hybrid model the amplitude of the oscillations die down very slowly , allowing many oscillations of the @xmath120 and @xmath0 fields in one hubble time . thus , one could expect large amplitude oscillations of the fields for a long time . this crucially depends on the parameter @xmath127 . if @xmath147 , then we notice that effective masses for @xmath120 and @xmath0 fields during oscillations are much larger than the hubble parameter . the hubble parameter is given by @xmath148 during inflation , so , @xmath149 provided the scale of @xmath127 is quite small compared to the planck mass , we can effectively neglect the expansion of the universe . in the supersymmetric hybrid model there are two regimes of interest . just after the mass square of the @xmath0 field becomes negative , the fields begin to oscillate with an amplitude which decreases as @xmath150 . when the field amplitude drops below @xmath151 , the amplitude of the oscillations decreases as @xmath152 . in this regime , when the expansion of the universe is neglected , the amplitude of the oscillations remains constant and the oscillations are harmonic : @xmath153 the corresponding evolution equation for the @xmath120 field can be found from eq . ( [ traj ] ) . in this paper we are neglecting the expansion of the universe . we will to concentrate upon two regimes : one with large amplitude oscillations which leads to the following parameters @xmath154 and , the other with small amplitude oscillations with the parameters @xmath155 the coupling constants are dimensionless while the other dimensionful parameters denoted in planck units . we find below a marked difference in the zero mode behavior of the fields @xmath120 and @xmath0 in these two cases depending on whether the fluctuations are taken into account or neglected . in parameter set ( [ par3 ] ) , we study the features of the fields with a large amplitude . this can happen when the @xmath120 and @xmath0 fields begin their oscillations just after the end of inflation . as mentioned earlier , after the end of inflation the maximum amplitude attained by the mean fields can be quite large @xmath138 , and @xmath156 . this is precisely the initial condition we have chosen for the mean fields for our numerics , as shown in fig . the values for @xmath157 and @xmath127 can be evaluated from eq . ( [ relation ] ) , which yields @xmath158 we notice that the evolution for @xmath120 and @xmath0 fields without taking into account the fluctuations are anharmonic , see fig . 7 , and , their trajectories in the @xmath159 plane is a straight line , as shown in fig . however , switching on the fluctuations leads to a completely chaotic trajectory as shown in fig . the departure from the straight line trajectory is quite significant and it tells us that the renormalized zero mode equations have different contributions to the parameter @xmath160 and to the effective mass of the @xmath0 field . this mismatch in the frequencies of the zero mode equations for @xmath161 and @xmath0 leads to an irregular trajectory . the other way to interpret this behavior is to think in terms of different effective mass corrections to @xmath120 and @xmath0 fields , such that the effective frequencies of the oscillations for @xmath120 and @xmath0 do not match each other at the bottom of the potential . this is certainly a nontrivial result . nonetheless , the result is quite expected from the fact that the amplitude of the oscillations are quite large and the effective mass for the @xmath0 field is zero at each and every oscillation when @xmath122 . as we mentioned earlier , the frequencies of the oscillations of the zero modes are different , as can be noticed in figs . 10 and 11 . the zero mode of @xmath120 influenced by the fluctuations oscillates around its minimum @xmath162 with a more rapid frequency than when fluctuations are neglected . this suggests that the effective mass correction to the zero mode for @xmath120 is coming solely from the finite coupling contribution from the @xmath0 field . ( note , that we have already set the bare mass for @xmath163 . ) the oscillations maintain the regularity with increasing and decreasing amplitude . however , the story is not the same for the zero mode behavior for the @xmath0 field . the amplitude of @xmath0 increases gradually and the frequency of the oscillations varies . we mention here that the effective mass for the @xmath0 field can vanish at a critical point . as a result , the adiabatic condition for the @xmath0 field is violated at those instants and this is the reason why the amplitude of the @xmath0 field is enhanced rather than suppressed . the evolution of the energy density is shown in fig . 12 . at first instance it seems quite odd that the energy density of the fluctuations does not increase further . one would naively expect a larger contribution of the energy density of @xmath164 and @xmath141 . this is not the case here . the energy density for the mean fields and the fluctuations are equally shared . the reason is the correction due to the fluctuations . these corrections modify the effective mass of the @xmath0 field and induce corrections to the coupling constants , namely @xmath165 and @xmath107 . the coupling constants are modified in such a way that the trajectory of zero mode fields become irregular . usually the production of fluctuations is not efficient in this case . this is quite similar to the situation of preheating in non - supersymmetric hybrid models @xcite . even though we started with a supersymmetric hybrid model where at the bottom of the potential there is a single effective frequency , the situation changes completely if the fluctuations are taken into account . essentially the coupling constants get a large correction which does not preserve an effective single coupling constant for the evolution of the zero mode fields . this is precisely the reason why the zero mode trajectory becomes irregular and also why the production of @xmath166 and @xmath141 is so low . as a next example of the supersymmetric hybrid model we choose parameter set ( [ par4 ] ) , with a small coupling @xmath124 and small @xmath127 @xmath167 in this example , the coupling between the fields is quite small ; @xmath168 , and also the initial conditions for @xmath120 and @xmath0 have been chosen such that the fields oscillate around their respective minima . the maximum amplitude for @xmath169 and @xmath170 is much below the critical point @xmath127 . we remind the readers that the chosen initial conditions for the oscillations do not come naturally just after the end of inflation , therefore this example does not represent a real situation . in spite of this we study the particular situation in order to notice the contrast in the behavior of the zero modes and the energy densities in the fluctuations . this particular set of initial conditions for @xmath171 offers an alternative example where spinodal instability in the @xmath0 field does not take place . as a result the effective mass square for the @xmath0 field never crosses zero and the adiabatic condition for the @xmath0 field is not strongly broken . the oscillations of the mean fields @xmath120 and @xmath0 are harmonic in nature , as shown in figs . 13 and 14 by the dotted lines . the amplitudes are constant with a frequency given by eq . ( [ frequency ] ) . the oscillations of the mean fields is governed by eqs . ( [ traj ] ) and ( [ evolv ] ) . the trajectory in the @xmath172 plane is a straight line whose slope is governed by eq . ( [ traj ] ) . the effect of the fluctuations is also quite expected in this case . the amplitudes of the zero mode for @xmath120 and @xmath0 fields decreases after a while and , in contrast to the preceding example , the frequency of the oscillations do not change very dramatically ; see the behavior of zero mode in solid lines in figs . 13 and 14 . the trajectories for the zero mode evolution remain a straight line in this case as shown in fig . this is quite reasonable for the parameters we have chosen , but an important observation is that the effect of fluctuations does not alter the straight line trajectory for the zero mode fields . this suggests that for small amplitude oscillations the corrections to the coupling constants ; @xmath173 and @xmath174 are such that the zero mode equations still have a similar oscillating frequency . this can be seen in figs . 13 and 14 . the production of @xmath164 and @xmath141 is not very significant because the energy density stored in @xmath164 and @xmath141 does not grow rapidly . thus the energy transfer from the zero modes to the fluctuation modes is not favorable for such a small amplitude oscillations as can be seen in fig . we conclude this section by mentioning that preheating in this supersymmetric hybrid model is quite interesting . depending on the amplitude of the oscillations of the fields , the behavior of the zero mode can be quite different . as a new feature we noticed that if the amplitude of the oscillations is close to the critical value @xmath127 , the effective mass square for the @xmath0 field becomes negative and as a result the fluctuations of the field grows exponentially . however , the effect of fluctuations alters the coupling constants in such a way that the trajectory of the zero modes become irregular . even though the adiabatic conditions seem to be broken for the @xmath0 field near the critical value , the energy density transferred from the zero mode to the fluctuations is not sufficient . our study reveals some interesting messages which we briefly mention here . we emphasize the point that the departure from the straight line trajectory of the zero mode is an essential feature of a supersymmetric hybrid model if the fluctuations are taken into account . even though , we have not included the hubble expansion , the results we have obtained are quite robust because supersymmetric hybrid inflationary models have a unique behavior of the fields which allows a smaller inflationary scale compared to the effective masses of the fields around their global minima . this suggests that during the oscillations , the expansion is felt much later , on a time scale determined by the parameters . this behavior is not shared by models where inflation is governed by a single field as in chaotic inflationary models . this undermines the production of quanta from the vacuum fluctuations . in several ways this affects the post inflationary radiation era of the universe . supersymmetric , weakly interacting dark matter formation and generation of baryonic asymmetry in the universe during preheating are the two most important frontiers which due to our results may warrant a careful revaluation . in order to substantiate our claim that a due consideration of fluctuations after the end of inflation is an important feature of any supersymmetric hybrid model , we have chosen an unphysical example which serves the purpose of making a vivid distinction . we stress here that the spinodal instability which is actually responsible for producing an irregular trajectory of the zero mode of the fields in a phase space is completely lacking if the amplitudes of the oscillations for @xmath175 are small compared to the critical value @xmath127 . this acts as a comparative study and shows that after the end of inflation , in a supersymmetric hybrid inflationary model , due to the spinodal instability in a field , a proper renormalization of the masses and the coupling constant have to be taken into account . we have introduced a formalism to address the dynamics of @xmath0 nonequilibrium , coupled , time varying scalar fields . we have shown that the one - loop corrections to the mean field evolution can be renormalized by dimensional regularization . for the sake of clarity and simplicity we restricted ourselves to minkowski spacetime while deriving the renormalized equations of motion and the energy density of the system . we applied our formalism to a two field case where we study the behavior of the quantized mode functions and the effect of fluctuations on the zero mode equations of motion for various parameters including small and large amplitude oscillations and large and weak coupling between two scalar fields . the varied couplings and amplitudes illustrate various facets of the intertwined dynamics of the two fields which lead to a deeper understanding of the production of self quanta and transfer of energy density between the fields in a cosmological context . as a special example we have chosen a two field inflationary model which is genuinely motivated by supersymmetry and thus preserves the effective masses of the fields to be the same in their local minima . the model , as a paradigm , predicts inflation which comes to an end via a smooth phase transition , and robustness of the model is confirmed by a slightly tilted spectrum of scalar density fluctuations within the cobe limit . the model parameters can be adjusted to give an inflationary scale covering a wide range of energy scales from tev to @xmath176 gev . the phase transition leads to a spinodal instability in one of the fields which leads to a coherent oscillations of the fields around their global minima . the instability occurs in one of the fields which demands careful study of the back reaction to an otherwise growing mean field in an intertwined coupled bosonic system . an account of influence of the fluctuations gives rise to uneven contribution to the renormalized masses of the fields . this results in an irregular trajectory of the zero mode in a phase space , which breaks the coherent oscillations of the two fields . this prohibits an excessive production of particles from the vacuum fluctuations . this requires a careful revaluation of the successes of the production of weakly interacting massive particles and baryogenesis via out of equilibrium decay in supersymmetric hybrid inflationary models . our study implies that exciting higher spin particles from the vacuum fluctuations of the coherent oscillations of the fields in a supersymmetric hybrid inflationary model demands careful reconsideration . even though , we have neglected the effect of expansion in our calculation , our results are robust enough to claim that the fluctuations in a supersymmetric hybrid model do not grow if the back reaction of the fluctuations are taken into account in the mean field evolution . an extension of our formalism to an expanding universe deserves separate attention . the authors are thankful to mar bastero - gil and michael g. schmidt for helpful discussion . we thank salman habib for helpful comments on the manuscript . a.m. is partially supported by * the early universe network * hprn - ct-2000 - 00152 . a. albrecht , p. j. steinhardt , m. s. turner and f. wilczek , phys . lett . * 48 * , 1437 ( 1982 ) ; a. d. dolgov and a. d. linde , phys . b*116 * , 329 ( 1982 ) ; l. f. abbott , e. farhi and m. wise , phys . b*117 * , 29 ( 1982 ) ; l. kofman , a. linde and a. starobinsky , phys . rev . lett . * 73 * , 3195 ( 1994 ) ; phys . d * 56 * 3258 ( 1997 ) ; d. boyanovsky , m. dattanasio , h.j . de vega , r. holman , d .- s . lee , phys . d*52 * 6805 , ( 1995 ) ; d. boyanovsky , d. cormier , h. j. de vega , r. holman , a. singh , m. srednicki , phys . rev . d*56 * , 1939 ( 1997 ) . examples of reheating involving higher spin fields are included in j. baacke , k. heitmann and c. ptzold , phys . d * 58 * , 125013 ( 1998 ) ; a. l. maroto and a. mazumdar , phys . lett . * 84 * , 1655 ( 2000 ) . d. boyanovsky , d. cormier , h. j. de vega , r. holman and p. kumar phys . d*57 * , 2166 ( 1998 ) ; d. cormier and r. holman , phys . rev.d*60 * , 41301 ( 1999 ) ; phys . rev . d*62 * , 23520 ( 2000 ) ; phys . lett . * 84 * , 5936 ( 2000 ) . f. cooper , s. habib , y. kluger , and e. mottola , phys . rev . * d55 * , 6471 ( 1997 ) ; d. boyanovsky , h. j. de vega , r. holman , and j. salgado , phys . rev . * d59 * , 125009 ( 1999 ) ; j. baacke and k. heitmann , phys . rev . * d62 * , 105022 ( 2000 ) . e. halyo , phys . lett . b * 387 * , 43 ( 1996 ) ; p. bintruy and g. dvali , phys . b * 388 * , 241 ( 1996 ) . for earlier work on this subject see : j. a. casas and c. muoz , phys . b*216 * , 37 ( 1989 ) ; j. a. casas , j. moreno , c. muoz and m. quiros , nucl . b*328 * , 272 ( 1989 ) .

coupled , multi - field models of inflation can provide several attractive features unavailable in the case of a single inflaton field . these models have a rich dynamical structure resulting from the interaction of the fields and their associated fluctuations . we present a formalism to study the nonequilibrium dynamics of coupled scalar fields . this formalism solves the problem of renormalizing interacting models in a transparent way using dimensional regularization . the evolution is generated by a renormalized effective lagrangian which incorporates the dynamics of the mean fields and their associated fluctuations at one - loop order . we apply our method to two problems of physical interest : ( i ) a simple two - field model which exemplifies applications to reheating in inflation , and ( ii ) a supersymmetric hybrid inflation model . this second case is interesting because inflation terminates via a smooth phase transition which gives rise to a spinodal instability in one of the fields . we study the evolution of the zero mode of the fields and the energy density transfer to the fluctuations from the mean fields . we conclude that back reaction effects can be significant over a wide parameter range . in particular for the supersymmetric hybrid model we find that particle production can be suppressed due to these effects .

x - ray free - electron lasers ( fels ) offer a brilliant tool for science at atomic length and ultrafast time scales @xcite , and they have been realized with the operation of the free - electron laser in hamburg ( flash ) @xcite , the linac coherent light source ( lcls ) @xcite , and the spring-8 angstrom compact free electron laser ( sacla ) @xcite . the x - ray fel driving electron bunches are subject to several collective effects , e.g. , microbunching instabilities or coherent synchrotron radiation ( csr ) , which degrade the required high transverse and longitudinal beam brightness @xcite . these instabilities may not only result in significant deteriorations of the fel performance @xcite but also in coherent radiation effects @xcite such as coherent optical transition radiation ( cotr ) or csr in the optical wavelength range @xcite ( abbreviated as cosr ) . beam profile imaging dominated by coherent optical radiation leads to an incorrect representation of the transverse charge distribution @xcite and renders electron beam diagnostics with standard imaging screens , e.g. , otr screens , and all the related diagnostics such as emittance or bunch length diagnostics impossible . however , beam diagnostics with imaging screens are essential for single - shot measurements or in cases where two transverse dimensions are required , e.g. , in slice - emittance or longitudinal phase space measurements @xcite . microbunching instabilities associated with longitudinal electron bunch compression can be mitigated by introducing additional uncorrelated energy spread @xcite as successfully demonstrated by the operation of the laser heater system at the lcls @xcite . however , the microbunching gain suppression is not necessarily perfect , and the corresponding remaining small but existing level of cotr still hampers electron beam profile diagnostics using standard imaging screens ( e.g. , ref . the origin of coherent optical radiation effects is not only restricted to microbunching instabilities but can also be related to ultrashort spikes inside electron bunches or generated by intrinsically ultrashort electron bunches like at laser - plasma accelerators ( e.g. , ref . @xcite ) or at x - ray fels with ultra - low charge operation @xcite . transition radiation is emitted when a charged particle beam crosses the boundary between two media with different dielectric properties @xcite , hence transition radiation is emitted using any kind of imaging screen and thus precludes the stand - alone use of scintillation screens in the presence of coherent optical radiation effects ( e.g. , cotr ) . however , by using ( scintillation ) imaging screens in dedicated measurement configurations , cotr can be mitigated ( see , e.g. , ref . @xcite ) . in this paper , we discuss methods to suppress coherent optical radiation effects both by electron beam profile imaging in dispersive beamlines and by utilizing scintillation imaging screens in combination with several separation techniques . the experimental setup and observations of coherent optical radiation effects at flash are described in sec . [ sec : setup ] . in sec . [ sec : es ] we discuss the suppression of coherent optical emission in dispersive beamlines and present experimental results for cotr generated by a local ultrashort charge concentration . section [ sec : sep ] covers the suppression of coherent optical radiation effects by using scintillation screens in combination with separation techniques . the experimental results obtained with the temporal separation technique are presented in sec . [ sec : res ] , and a summary and conclusions are given in sec . [ sec : summary ] . the measurements presented in this paper have been carried out at flash , which is a self - amplified spontaneous emission ( sase ) fel @xcite for extreme - ultraviolet ( euv ) and soft x - ray radiation , driven by a superconducting radio - frequency ( rf ) linear accelerator @xcite . the schematic layout of flash is depicted in fig . [ fig : flash_1 ] , showing the injector , which is based on a laser - driven normal conducting rf gun , the superconducting accelerating structures , two magnetic bunch compressor chicanes , and the undulator magnet system . the positions of the experimental setups used for the measurements presented in this paper are indicated by green dots and arrows . the third - harmonic rf system ( denoted by l3 in fig . [ fig : flash_1 ] ) is dedicated to the linearization of the longitudinal phase space upstream of the first bunch compressor @xcite . in order to properly set up fel operation with applied third - harmonic rf linearizer , a lola - type @xcite transverse deflecting rf structure ( tds ) has been integrated in a dedicated setup for diagnosis of the longitudinal phase space @xcite close to the fel undulators . as depicted in fig . [ fig : flash_1 ] , the tds can either be operated in combination with imaging screens in the dispersive magnetic energy spectrometer or by using off - axis imaging screens operated with a fast kicker magnet in the non - dispersive main beamline during fel operation . technical details and performance measurements on the setup for longitudinal beam diagnostics can be found in refs . @xcite . transverse deflecting rf structures are widely used for electron bunch length and longitudinal profile measurements at present fels and provide high - resolution single - shot diagnostics @xcite . detailed descriptions of time - domain electron bunch diagnostics using a tds can be found in refs . @xcite . here we describe only the basic principles of longitudinal electron beam diagnostics that are required throughout this paper . the vertical betatron motion of an electron passing a vertical deflecting tds around the zero - crossing rf phase , neglecting intrinsic longitudinal - to - vertical correlations @xcite which are not relevant for the experiments presented throughout this paper , can be given by @xcite@xmath0 with the vertical shear ( streak ) function @xmath1 where @xmath2 is the angular - to - spatial element of the vertical beam transfer matrix from the tds at @xmath3 to any position @xmath4 , @xmath5 is the vertical beta function , @xmath6 is the vertical phase advance between @xmath3 and @xmath4 , and @xmath7 describes an intrinsic offset . the expression @xmath8 is the vertical kick strength with the peak deflection voltage @xmath9 in the tds , @xmath10 is the speed of light in vacuum , @xmath11 is the elementary charge , @xmath12 is the electron momentum , @xmath13 is the longitudinal position of the electron relative to the zero - crossing rf phase , and @xmath14 is the operating rf frequency . the expression in eq . ( [ eq : motion ] ) shows a linear mapping from the longitudinal to the vertical coordinate and allows longitudinal electron beam profile measurements by means of transverse beam diagnostics using imaging screens . the shear function @xmath15 determines the slope of this mapping and can be calibrated by measuring the vertical centroid offset of the bunch as a function of the tds rf phase . the electron bunch current is given by the normalized longitudinal bunch profile multiplied by the electron bunch charge . the bunch length ( duration ) is given by the root mean square ( r.m.s . ) value @xmath16 , where @xmath17 is the vertical r.m.s . beam size during tds operation , and @xmath18 is the intrinsic vertical r.m.s . beam size when the tds is switched off . both @xmath17 and @xmath18 can be determined by measurements , and the latter limits the achievable r.m.s . time resolution to @xmath19 @xcite . the screen stations in both the magnetic energy spectrometer and non - dispersive main beamline ( see fig . [ fig : flash_1 ] ) are each equipped with different imaging screens and a charge - coupled device ( ccd ) camera @xcite ( 1360@xmath201024 pixels with 12bit dynamic range and @xmath21 pixel size ) with motorized optics ( motorized macro lens with teleconverter mounted on a linear translation stage ) . the translation stage allows variable demagnification @xmath22 in the range between @xmath231.5 - 3 with spatial resolutions of better than @xmath24 . the imaging screen station in the energy spectrometer ( es - ccd in fig . [ fig : flash_1 ] ) is equipped with an otr screen ( aluminum coated silicon ) and two scintillation screens made of cerium - doped yttrium aluminum garnet ( yag : ce ) and bismuth germanate ( bgo ) , respectively . in the non - dispersive beamline , the screen station is operated with a fast kicker magnet ( k - ccd in fig . [ fig : flash_1 ] ) , which is able to deflect one bunch out of the bunch train at the bunch train repetition rate of flash @xcite of 10hz , and provides an otr screen and a cerium - doped lutetium aluminum garnet ( luag : ce ) scintillation screen . all screens are mounted at a 45@xmath25 angle ( the cameras at a 90@xmath25 angle ) with respect to the incoming electron beam . the scintillation screens have a thickness of @xmath26 . the experimental setup in the non - dispersive beamline is additionally equipped with a fast gated intensified ccd camera @xcite ( k - iccd in fig . [ fig : flash_1 ] , 1280@xmath201024 pixels with 12bit and @xmath27 pixel size ) , which has been used for the temporal separation technique ( see sec . [ sec : res ] ) . further technical details on the screen stations and camera systems can be found in refs . @xcite . microbunching instabilities at x - ray fels can lead to significant generation and amplification of density modulations in the optical wavelength range @xcite which may result in coherent optical radiation effects such as cotr . this has been observed by spectral measurements and characteristic ring - shaped light patterns at the lcls @xcite and flash @xcite , and renders accurate electron beam profile diagnostics using standard imaging screens impossible . first observations of cotr @xcite and microbunching in the frequency - domain ( coherent transition radiation around 10 @xmath28 @xcite ) at flash were made directly upstream of the collimator ( see fig . [ fig : flash_1 ] ) . electron beam profile imaging performed downstream of the collimator section @xcite , an achromatic bending system , resulted in considerably more prominent observation of coherent optical radiation effects and microbunching . the measurements presented in fig . [ fig : indi_2 ] show single - shot light patterns , generated by moderately compressed electron bunches , at the imaging screens in the non - dispersive main beamline at k - ccd directly upstream of the undulators . ring - shaped structures in the profiles , characteristic for cotr @xcite , are clearly visible in the images of figs . [ fig : indi_2_a ] and [ fig : indi_2_b ] , which have been recorded by using an otr and luag imaging screen , respectively . for both images a long - pass filter , blocking wavelengths below 780 nm , was used . the luminescence emission of the luag scintillation screen occurs below 700 nm @xcite and is thus well blocked by the 780-nm long - pass filter used during the measurements . hence , the light pattern in fig . [ fig : indi_2_b ] is due to cotr without contribution from scintillation light . complementary to the observation of cotr , the images in fig . [ fig : indi_3 ] show single - shot longitudinal phase space measurements in the magnetic energy spectrometer ( es - ccd ) . the measurements were done for accelerator settings typical for fel operation with applied third - harmonic rf linearizer system upstream of the bunch compressor chicanes , and they clearly indicate microbunching in the time - domain with modulation periods of about 25fs and 30fs , respectively . we note that a maximum modulation wavelength of @xmath29 ( @xmath30 ) was predicted theoretically in ref . @xcite and measured by spectroscopy of coherent transition radiation in ref . the energy - dependent beam trajectories in dispersive beamlines can be utilized as a magnetic energy spectrometer for charged particle beams . by combining such an energy spectrometer with the operation of a tds and using imaging screens to get two - dimensional transverse beam profiles , longitudinal phase space measurements ( see , e.g. , fig . [ fig : indi_3 ] ) with single - shot capability can be accomplished . the corresponding horizontal betatron motion , which should be perpendicular to the vertical shearing plane of the tds @xcite , can be written as @xmath31 with the intrinsic offset @xmath32 , the horizontal momentum dispersion @xmath33 and the relative momentum deviation @xmath34 . for relativistic electron beams with lorentz factors of @xmath35 , the electron beam energy is given by @xmath36 , and @xmath37 represents the relative energy deviation . the dispersion @xmath38 can be determined by measuring the horizontal centroid offset of the bunch as a function of the relative energy deviation . the dispersion in the magnetic energy spectrometer at es - ccd ( see fig . [ fig : flash_1 ] ) , which is generated by two subsequent dipole magnets with 5@xmath25 deflection each ( equivalent to a single dipole magnet with 10@xmath25 deflection ) , amounts to 750 mm ( nominal ) @xcite , whereas @xmath38 at k-(i)ccd due to the kicker magnet operation is negligible . in addition to the momentum dispersion introduced in the horizontal betatron motion , the longitudinal particle motion can be described by @xmath39 with the initial bunch length coordinate @xmath40 and the initial horizontal offset @xmath41 and slope @xmath42 . the transfer matrix elements @xmath43 describe the mapping from position @xmath3 to @xmath4 , i.e. , @xmath44 throughout the rest of this paper . the expression in eq . ( [ eq : longdisp ] ) does not affect the principle of longitudinal phase space diagnostics described by eqs . ( [ eq : motion ] ) and ( [ eq : disp ] ) , but results in the suppression of coherent optical emission as is shown in the following . the spectral and angular intensity distribution , denoted as @xmath45 with the three - dimensional wave vector @xmath46 , of transition ( synchrotron ) radiation emitted by an electron bunch with @xmath47 electrons and charge @xmath48 is given by ( e.g. , refs . @xcite ) @xmath49 where @xmath50 describes the intensity distribution of a single electron as a function of the transverse and longitudinal wavenumber @xmath51 and @xmath52 , respectively , and @xmath53 is the three - dimensional form factor of the electron bunch . the latter can be expressed by the fourier transform of the normalized charge density @xmath54 as @xmath55 where @xmath56 . normalized charge distributions without longitudinal - transverse correlations can be factorized as @xmath57 , and by taking into account @xmath58 , which is assumed in the following , we get @xmath59 with the transverse and longitudinal form factor @xmath60 and @xmath61 , respectively . for small observation angles @xmath62 ( small covered solid angles @xmath63 ) with respect to the central axis ( @xmath40-axis ) of the emitted radiation we have @xmath64 with the wavenumber @xmath65 , and the expression in eq . ( [ eq : spe ] ) reads @xmath66 the first term on the right - hand side is linear in @xmath67 and describes the contribution of incoherent radiation , whereas the second term scales with @xmath68 , which describes the coherent radiation part . in order to perform electron beam diagnostics with incoherent radiation , we demand that the total spectral radiation intensity in eq . ( [ eq : spec ] ) is dominated by the incoherent term , i.e. , @xmath69 . in following , we derive an analytical expression describing a general strong suppression of the longitudinal form factor at optical wavelengths in a magnetic energy spectrometer . a transverse form factor of @xmath70 , i.e. , full transverse coherence , at the imaging screens is assumed , which is the worst case scenario . the actual transverse form factor in the experiment will be reduced due to the finite beam size and observation angle @xcite . however , the suppression of the longitudinal form factor @xmath61 presented below is much stronger in the general case . a cutoff wavelength @xmath71 can be defined via @xmath72 , and beam diagnostics at wavelengths below @xmath73 becomes dominated by incoherent radiation . the cutoff wavelength initially depends on the charge distribution [ via eq . ( [ eq : form ] ) ] , and significant values of @xmath74 in the optical wavelength range can occur due to the existence of density modulations or charge concentrations at ultrashort length scales . however , following the analytical treatment of microbunching degradation in ref . @xcite , we show that the cutoff wavelength in magnetic energy spectrometers is entirely determined by the terms in eq . ( [ eq : longdisp ] ) with a corresponding strong suppression of coherent emission at optical wavelengths for common magnetic energy spectrometers used at present fels . the amount of density modulations in a normalized electron beam distribution @xmath75 with the phase space vector @xmath76 and @xmath77 can be quantified by a complex bunching factor @xmath78 as @xcite @xmath79 where @xmath65 is the wavenumber of the modulation . according to refs . @xcite , the evolution of the bunching factor @xmath80 $ ] along dispersive beamlines can be expressed by @xmath81 = b_0[k(s),s ] + \int_{s_0}^{s}{ds'\,k(s',s)b[k(s'),s ' ] } \ , , \label{eq : bu2}\ ] ] where @xmath82 $ ] is the bunching factor in the absence of collective beam interactions due to csr . the second term on the right - hand side of the integral equation with the kernel @xmath83 @xcite ( a complicated expression that is not relevant here ) describes the induced bunching due to csr interactions . as discussed in refs . @xcite and verified by numerical particle tracking simulations below , the bunching induced in a dipole magnet from the energy modulation generated in the same dipole magnet can be neglected with the kernel @xmath84 , and the bunching factor in eq . ( [ eq : bu2 ] ) becomes @xmath80 \approx b_0[k(s),s]$ ] . this is also the case in a magnetic energy spectrometer consisting of a single dipole magnet , and the resulting evolution of the total bunching factor for a given initial bunching @xmath85 $ ] can be expressed by @xcite @xmath86\approx&\ , b_0[k(s_0),s_0]\,\mathrm{exp}\left [ -\frac{k^2(s)\sigma_{\delta0}^2}{2 } r_{56}^2\right]\nonumber \\ & \times \mathrm{exp}\left [ -\frac{k^2(s)\varepsilon_0\beta_0}{2 } \left ( r_{51 } -\frac{\alpha_0}{\beta_0 } r_{52}\right)^2\right]\nonumber \\ & \times \mathrm{exp}\left[-\frac{k^2(s)\varepsilon_0}{2\beta_0}r_{52}^2 \right]\ , , \label{eq : bu3 } \end{aligned}\ ] ] where the motion in eq . ( [ eq : longdisp ] ) is taken into account , and an initial beam distribution @xmath87 that is uniform in @xmath40 and gaussian in @xmath41 , @xmath88 , and @xmath37 is assumed . the initial uncorrelated energy spread and geometrical horizontal emittance are denoted by @xmath89 and @xmath90 , respectively , and @xmath91 and @xmath92 are the initial horizontal lattice functions ( twiss parameters ) . the compression of the wavenumber by @xmath93^{-1}$ ] with the initial energy chirp @xmath94 can be neglected , i.e. , @xmath95 , since the @xmath96 generated by a single dipole magnet is rather small . in addition to the evolution of an initial bunching , energy modulations generated upstream of a magnetic energy spectrometer can initiate bunching and , according to ref . @xcite and by using eq . ( [ eq : bu3 ] ) , the induced bunching @xmath97 due to an initial energy modulation is given by @xmath98 where @xmath99 is the fourier amplitude of the initial energy modulation @xmath100 . fortunately , the bunching @xmath101 can be neglected due to the small @xmath96 ( see above ) and the additional suppression discussed in the following . equation ( [ eq : bu3 ] ) implies a suppression of initial bunching due to the coupling with the transverse phase space given in eq . ( [ eq : longdisp ] ) , and a suppression factor @xmath102 can be defined as @xmath103 where @xmath104 by comparing eqs . ( [ eq : form ] ) and ( [ eq : bun ] ) , and taking into account @xmath105=\rho(z)\rho[(x , x',\delta)]$ ] , the suppression factor can be expressed as @xmath106 ( cf . the analytical treatment in refs . @xcite ) , which describes the general suppression of coherent emission in a common magnetic energy spectrometer . assuming a maximum initial density modulation or an ultrashort electron bunch , both with @xmath107 , the cutoff wavelength ( defined via @xmath72 ) is given by [ cf . eq . ( [ eq : bu4 ] ) ] @xmath108 we note that the suppression for ultrashort electron bunches is simply given by the lengthening due to the transverse phase space parameters and longitudinal motion given in eq . ( [ eq : longdisp ] ) , which act like a low - pass filter . lcccc parameter & symbol & value & unit + beam energy & @xmath109 & 1000 & mev + lorentz factor & @xmath110 & 1957 & + electron bunch charge & @xmath111 & 150 & pc & + horizontal emittance ( normalized ) & @xmath112 & 1.0 & @xmath113 m + relative slice energy spread & @xmath89 & @xmath114 & + horizontal beta function & @xmath92 & 13.55 & m + horizontal alpha function & @xmath91 & 5.33 & + spatial - to - longitudinal coupling & @xmath115 & -0.174 & + angular - to - longitudinal coupling & @xmath116 & -0.089 & + momentum compaction factor & @xmath96 & 0.006 & m + [ tab : spec ] for initial density modulations . the theory curve ( solid red line ) is calculated for the full term in eq . ( [ eq : bu5 ] ) , and the approximation ( dashed green line ) is calculated for @xmath117 . the inset shows the wavelength range below @xmath118 on a logarithmic scale including the cutoff wavelength @xmath119 calculated for @xmath120 electrons . ] the analytical treatment has been verified by numerical simulations using the tracking code _ elegant _ @xcite with gaussian and uniform beam distributions ( @xmath121 particles ) including csr effects , and by using the parameters of the magnetic energy spectrometer at flash , summarized in table [ tab : spec ] . figure [ fig : supp_4 ] shows the suppression factor for both numerical simulations with initial density modulations ( @xmath122 peak amplitude ) and analytical calculations using eqs . ( [ eq : bu4 ] ) and ( [ eq : bu5 ] ) for the parameters of flash . the analytical calculations are in perfect agreement with the numerical simulations . the shown approximation is calculated by using @xmath117 , which is a good practical estimate ( @xmath123 for a single dipole magnet with bending angle @xmath124 ) . according to the full term in eq . ( [ eq : bu5 ] ) , the cutoff wavelength in the magnetic energy spectrometer at flash amounts to @xmath125 , which manifests a strong suppression of coherent optical emission . coherent emission does not only lead to intense radiation , which is described by means of the form factor @xmath74 in the intensity distribution given in eq . ( [ eq : spec ] ) , but also to an incorrect representation of the transverse charge distribution in beam profile imaging @xcite . the imaging of transverse beam distributions with optical systems , e.g. , by using an imaging screen , a lens , and a camera , is generally described by means of the intensity distribution of a point source in the image plane ( e.g. , ref . @xcite ) , which is the so - called point spread function . according to ref . @xcite , the image formation with optical transition radiation of a normalized three - dimensional charge distribution @xmath126 with @xmath67 electrons can be expressed by @xmath127 where @xmath128 describes the measured intensity distribution proportional to the absolute square of the total electric field @xmath129 evolved from the charge distribution , and @xmath130 corresponds to the imaged electric field of a single electron , which can be expressed by means of the fresnel - kirchhoff diffraction integral ( e.g. , ref . the second integral in eq . ( [ eq : field ] ) describes the coherent radiation part ( @xmath131 ) , and by taking into account @xmath132 with @xmath58 , the expression for image formation in eq . ( [ eq : field ] ) can be rewritten as [ cf . ( [ eq : spec ] ) ] @xmath133 the first integral in eq . ( [ eq : field2 ] ) simply describes the incoherent imaging as a convolution of the transverse charge distribution @xmath134 with the point spread function related term @xmath135 . in the case of a nonvanishing longitudinal form factor @xmath136 , the second integral in eq . ( [ eq : field2 ] ) contributes to the image formation and describes no longer a simple convolution with a point spread function , but rather takes into account the actual field distribution . thus , significant deviations in the measured transverse charge distribution can occur even with a small longitudinal form factor due to the second term @xmath137 in eq . ( [ eq : field2 ] ) , where @xmath138 . an example with initially inconspicuous cotr , impeding the electron beam diagnostics finally , is demonstrated in the following . figures [ fig : spike_5_a ] and [ fig : spike_5_b ] show single - shot images of longitudinal bunch profile measurements using the tds that were recorded in the non - dispersive main beamline at k - ccd and in the energy spectrometer at es - ccd , respectively . the images were measured under the same electron beam conditions with a bunch charge of @xmath139 and do not display any conspicuous features of cotr . however , as can be seen in fig . [ fig : spike_5_e ] , the corresponding longitudinal bunch profile taken at k - ccd comprises a much narrower spike with higher peak current . when increasing the bunch charge to @xmath140 , cotr emission became apparent at k - ccd [ fig . [ fig : spike_5_c ] ] , whereas the image in the energy spectrometer at es - ccd [ see fig . [ fig : spike_5_d ] ] did not show any coherent radiation effects . the cotr emission in fig . [ fig : spike_5_c ] ( we chose a single - shot image with low saturation of the ccd ) is clearly localized in the longitudinal electron bunch profile at a time coordinate of about 0.5ps . at the same time coordinate , the longitudinal phase space in fig . [ fig : spike_5_d ] exhibits a huge but narrow increase in energy spread ( the width in the time is limited by the tds resolution ) . from this we conclude that the single - shot image in fig . [ fig : spike_5_a ] already partially contains cotr as a consequence of a small but nonvanishing form factor @xmath74 [ cf . eqs . ( [ eq : spec ] ) and ( [ eq : field2 ] ) ] and that the cotr emission in fig . [ fig : spike_5_c ] seems most probably to be generated by a local ultrashort charge concentration such as a sharp spike inside the electron bunch . we note that the measurements presented in fig . [ fig : spike_5_e ] should give the same longitudinal electron bunch profiles , and the existing deviations can not be explained due to a worse resolution as is the case in sec . [ sec : lp ] . in order to demonstrate the local energy spread increase in figs . [ fig : spike_5_b ] and [ fig : spike_5_d ] with a reasonable signal - to - noise ratio ( snr ) , the longitudinal phase space measurements are presented with the yag imaging screen . the measurement performed with the otr imaging screen , presented in fig . [ fig : spike_5_f ] , shows the same strong cotr suppression ( but worse snr ) . as demonstrated in sec . [ sec : es ] , electron beam profile measurements can be accomplished in dispersive beamlines , such as magnetic energy spectrometers , with standard optical imaging systems as the emission of coherent optical radiation is strongly suppressed . however , linear accelerators consist mainly of beamlines which are in general designed to be dispersion - free , and imaging in energy spectrometers precludes measuring pure transverse beam profiles due to the dispersion . in this section , we discuss methods that suppress the impact of coherent radiation by separation from an incoherent radiation part . in the bunch compressors ( see ref . @xcite for experimental details ) . the spectral intensity of the incoherent part of transition radiation is indicated as dashed black line . ] the spectral intensity of transition ( synchrotron ) radiation emitted by an electron bunch consists of two terms that describe the incoherent ( @xmath141 ) and coherent ( @xmath142 ) radiation part [ cf . ( [ eq : spec ] ) or eq . ( [ eq : field2 ] ) ] . a spectral separation of these terms in electron beam profile imaging can be accomplished by restricting the imaging with wavelengths below the cutoff wavelength @xmath73 , i.e. , where the emission is dominated by incoherent radiation . spectral separation has been considered in ref . @xcite by using a scintillation screen in combination with a bandpass filter . however , this method requires a good knowledge and control of the expected spectra , and a vanishing form factor ( @xmath143 ) in the detectable wavelength range , which is not the general case as the spectra can vary strongly with the operation modes of a linear accelerator . this is demonstrated in fig . [ fig : spec_6 ] , in which spectral measurements of transition radiation in the visible and near - infrared wavelength range are presented for different compression settings at flash . the dashed black line represents the incoherent radiation part convoluted with the transmission of the optical setup . in contrast to the measurements presented in sec . [ sec : subseccotr ] , the measurements shown in fig . [ fig : spec_6 ] were performed upstream of the collimator section . we note that similar , reproducible measurements for uncompressed electron bunches , showing coherent radiation prominently at the micrometer scale , have been presented in ref . @xcite , and cotr for uncompressed bunches has been reported in ref . @xcite . in general , the probability of coherent emission decreases at shorter wavelengths , which is often not sufficiently reduced for optical wavelengths , and imaging with transition radiation in the euv region might be an option @xcite . in addition to the knowledge and control of the spectra , the imaging with euv radiation also requires dedicated detectors and optics , and a complete set - up in vacuum to prevent strong absorption in air . the luminescence of scintillation screens @xcite , which is a stochastic process , is inherently linear in the number of interacting electrons ( neglecting quenching and saturation effects ) , hence coherent radiation effects are not expected in pure scintillation light . however , transition radiation is also emitted at the boundary of vacuum and scintillator , and coherent optical radiation can still appear [ see , e.g. , fig . [ fig : indi_3_b ] ] . then , the total spectral and angular intensity distribution can be written as ( omitting the arguments @xmath144 in the intensity distributions @xmath145 ) @xmath146\mathcal{i}_o\ , , \label{eq : specfull}\ ] ] where @xmath147 and @xmath148 are related to scintillation light and transition radiation , respectively . as discussed in sec . [ sec : sepspec ] for otr imaging screens and with the same requirements and restrictions , spectral separation can also be applied when using scintillation screens ( @xmath149 ) . another method , particularly suited for scintillation screens , which have nearly isotropic emission , is to make use of the strong angular dependence of optical transition radiation ( e.g. , refs . @xcite ) and to perform electron beam profile imaging with radiation that is dominated by scintillation light , i.e. , @xmath150 $ ] in eq . ( [ eq : specfull ] ) . spatial separation can be achieved with imaging geometries having large angular or spatial offsets , e.g. , by using tilted imaging screens @xcite or central masks @xcite , where @xmath151 is suppressed sufficiently . however , just as for spectral separation , this method also requires good knowledge and control of the form factor , and dedicated imaging geometries . in addition , the resolution depends on the observation angle of the scintillation screen ( e.g. , ref . @xcite ) , which has to be taken into account in the layout of the imaging system . we note that an experiment on the spatial separation technique is currently being commissioned at flash . the fundamentally different light generation processes of scintillators and optical transition radiators result in clearly distinct temporal responses . the emission of transition radiation from relativistic electrons is instantaneous ( @xmath152 ) and prompt @xcite compared to the decay times ( @xmath153 ) of common scintillators ( e.g. , ref . accordingly , the temporal profiles of the otr pulses resemble the longitudinal electron beam profiles , whereas the temporal scintillation light pulses are fully dominated by the decay of the excited states in the scintillator . temporal separation makes use of the distinct temporal responses and allows to entirely eliminate otr , i.e. , the term @xmath148 in eq . ( [ eq : specfull ] ) which is time - dependent with @xmath154 , and , therewith , coherent optical radiation effects in electron beam profile imaging with scintillation screens when reading out a gated camera with a certain time delay after the prompt emission of otr . image recording with delayed readout ( e.g. , ref . @xcite ) can be accomplished with intensified ccd ( iccd ) cameras , where a control voltage in the intensifier between photocathode and micro - channel plate allows fast gating and exposure times of a few nanoseconds ( e.g. , refs . the experiments on the temporal separation technique at flash have been performed by using the iccd camera `` pco : dicam pro ( s20 ) '' @xcite in combination with the off - axis luag scintillation imaging screen in the non - dispersive main beamline at k - iccd , which has a decay time of @xmath155 @xcite . the cameras used for the presented measurements are able to readout images at the bunch train repetition rate of flash of 10hz , hence one bunch per bunch train can be measured with single - shot capability . further technical details on the equipment used for the measurements presented in the following can be found in sec . [ sec : screens ] and in refs . @xcite . the series of single - shot images in fig . [ fig : proof_7 ] present first proof - of - principle measurements on the temporal separation technique . the image shown in fig . [ fig : proof_7_a ] was recorded at k - iccd with an otr screen , whereas for figs . [ fig : proof_7_b ] and [ fig : proof_7_c ] a luag scintillation screen was used . the image shown in fig . [ fig : proof_7_c ] has been recorded with a time delay of @xmath156 , which is rather long compared to the emission time of otr but takes into account the large camera trigger - jitter that existed during the measurements . the image recorded with the otr screen and time delay simply showed background noise and is not presented here . the intensity distributions in fig . [ fig : proof_7 ] have been generated by moderately compressed electron bunches with a charge of 0.5nc and a beam energy of 700mev . figures [ fig : proof_7_a ] and [ fig : proof_7_b ] show a composite of cotr and cosr with a contribution of scintillation light in fig . [ fig : proof_7_b ] . the round - shaped light pattern on the right - hand side of figs . [ fig : proof_7_a ] and [ fig : proof_7_b ] is most probably due to synchrotron radiation generated upstream of the off - axis screens ( a polarizer was not available during the measurements ) , where the appearance in fig . [ fig : proof_7_b ] is reduced by the transparency of the luag screen . the image in fig . [ fig : proof_7_c ] , recorded with a time delay of @xmath156 , can be attributed purely to scintillation light allowing for a quantitative analysis of the transverse beam profiles . in contrast to spectral and spatial separation , the temporal separation technique provides a definite method to suppress coherent optical transition radiation without further relying on the wavelength - dependent longitudinal form factor . in addition , this technique inherently includes the suppression of secondary incoherent radiation sources such as synchrotron radiation generated from magnets directly upstream of the imaging screen or backward otr emitted from the second imaging screen boundary , whereas spectral components in the uv region or at shorter wavelengths may excite the scintillator , affecting the temporal separation . as is shown in ref . @xcite , however , potential synchrotron radiation sources can be identified and thus separated by adjusting the upstream magnets . furthermore , the coherent emission of otr at the second scintillator screen boundary is mitigated due to multiple scattering in the scintillator material as is described and demonstrated in refs . we note that the current implementation of the temporal separation technique presented throughout this paper utilizes fast iccd cameras , which are currently an order of magnitude more expensive than conventional ccd cameras . the proof - of - principle measurements on the temporal separation technique presented in fig . [ fig : proof_7 ] were carried out at k - iccd . however , a reference measurement to quantitatively prove this technique in terms of transverse beam profiles , as would be provided by a wire - scanner , which is insensitive to coherent effects , is not available at this position . in this section , we verify the method of temporal separation by investigations on the charge - dependent image intensities and comparisons with longitudinal bunch profiles recorded in the energy spectrometer at es - ccd . incoherent radiation is linear in the number of electrons contributing to the emission process ( cf . [ sec : supp ] ) , i.e. , linear in the electron bunch charge ( @xmath157 ) , and deviations caused by the nonlinear charge dependence of coherent radiation ( @xmath158 ) are ideally suited to verify the temporal separation technique . the integrated image intensities presented in fig . [ fig : charge_8 ] were measured for bunch charges between 0.13nc and 0.87nc at k - iccd for different imaging screen and readout configurations . each data point represents the average intensity of 20 background - corrected single - shot images and the error bars indicate the statistical r.m.s . image intensity fluctuations . up to an electron bunch charge of @xmath159 , the integrated intensity is linear ( solid black line ) in @xmath111 for all presented configurations . for higher bunch charges , deviations from the linear dependence appear in the configurations without delayed readout , i.e. , the form factor @xmath74 becomes significant in the visible wavelength range , which are caused by contributions from coherent optical radiation . the inset in fig . [ fig : charge_8 ] shows the bunch charge range from 0.55nc to 0.9nc more detailed . we note that the integrated intensity of the otr ( blue dots ) has actually been higher than presented for @xmath160 , because of camera saturation due to the strong optical emission and the corresponding underestimated integrated intensity . the large error bars , representing the r.m.s . jitter , indicate strong fluctuations due to the cotr . in the case of the luag imaging screen recorded with a time delay ( green diamonds ) , the dependence of the integrated intensity is entirely linear in the bunch charge , which verifies the power of the temporal separation technique . , the measurements in the magnetic energy spectrometer ( `` es - ccd ( yag ) '' ) are intended to provide an absolute reference measurement . ] as the emission of cotr is strongly suppressed in the magnetic energy spectrometer at flash ( see sec . [ sec : es ] ) , electron bunch profiles measured at the screen station es - ccd can serve as a reference for comparison with the temporal separation technique applied in the non - dispersive beamline at k - iccd . while the transverse bunch profiles can differ at both locations due to different twiss parameters and dispersion at es - ccd , longitudinal bunch compression does not take place in between , and longitudinal bunch profile measurements using the tds can be used for a direct comparison . the measurements presented in fig . [ fig : compression_9 ] show the mean r.m.s . electron bunch length of 20 single - shot images , including the statistical r.m.s . jitter indicated via error bars , for various acc1 rf phases measured at es - ccd and k - iccd by using the tds . the electron bunches were set up with an energy of 700mev and a bunch charge of 0.5nc . the rf phase of acc1 affects the energy chirp of the electron bunches upstream of the first bunch compressor and , accordingly , the final electron bunch lengths . the r.m.s . electron bunch lengths measured in the magnetic energy spectrometer at es - ccd ( black dots ) decrease almost linearly and do not possess large fluctuations . in contrast to the magnetic energy spectrometer at es - ccd , coherent optical emission is not suppressed in the non - dispersive beamline at k - iccd , leading to a sudden increase of the r.m.s . electron bunch lengths in combination with large fluctuations , represented by the large error bars ( statistical r.m.s . jitter ) , for acc1 rf phases @xmath161 measured with a luag screen without a certain time delay ( red squares ) , i.e. , without applied temporal separation . the electron bunch length measurements using an otr screen are omitted in fig . [ fig : compression_9 ] due to even larger deviations and fluctuations compared to the reference at es - ccd for acc1 rf phases @xmath161 . instead , the otr images ( single - shots ) for acc1 rf phases of 3.25deg and 3.75deg are presented in figs . [ fig : comp_10_a ] and [ fig : comp_10_b ] , respectively , with obvious coherent optical radiation effects in fig . [ fig : comp_10_b ] . due to the fact that the electron beam images shown in fig . [ fig : comp_10 ] are sheared vertically by means of the tds , the vertical coordinate implies time information ( see eq . [ eq : motion ] ) and the faint bunching visible in fig . [ fig : comp_10_a ] may be assigned to microbunching . figure [ fig : comp_10_c ] shows a single - shot image taken at k - iccd using a luag screen without time delay for an acc1 rf phase of 3.75deg . the image clearly shows contributions of coherent optical radiation similar to the image in fig . [ fig : comp_10_b ] . by imaging the luag screen with a time delay of 100ns , the obtained distribution shown in fig . [ fig : comp_10_d ] is acceptable without obvious contributions from coherent optical radiation . in addition , the corresponding electron bunch length measurements with applied temporal separation ( green diamonds ) in fig . [ fig : compression_9 ] are in perfect agreement with the reference measurements in the energy spectrometer at es - ccd ( black dots ) . the electron bunch durations for fel operation at flash are typically shorter than 150fs ( e.g. , ref . @xcite ) , and typical electron beam parameters are given in table [ tab : spec ] . the temporal separation technique , which has demonstrated accurate r.m.s . electron bunch length measurements in the presence of coherent optical radiation effects , gives confidence that single - shot measurements of longitudinal bunch profiles and , accordingly , electron bunch currents using temporal separation result in reliable results . the single - shot measurements presented in fig . [ fig : prof_11 ] ( cf . measurements shown in figs . [ fig : compression_9 ] and [ fig : comp_10 ] for the same acc1 rf phase settings ) have been recorded for an acc1 rf phases of 3.75deg in fig . [ fig : prof_11_a ] and for 4.05deg in fig . [ fig : prof_11_b ] , i.e. , in the presence of coherent optical radiation effects . the longitudinal electron bunch profiles taken in the non - dispersive beamline at k - iccd ( blue line ) with temporal separation show good agreement with the reference measurements at es - ccd ( red line ) , and the observed deviations are most probably due to slightly nonlinear amplification in the intensifier ( photocathode and micro - channel plate ) of the iccd camera . the reduced peak current with broadening in time in the case of k - iccd : time delay , which is apparent on the right - hand side ( @xmath162 ) of fig . [ fig : prof_11_b ] , can be explained by the different time resolutions of @xmath163 and @xmath164 achieved with the tds during the measurements for es - ccd and k - iccd , respectively . in order to compare the longitudinal bunch profiles with comparable resolution , a convolution has been applied for the measurement at es - ccd in fig . [ fig : prof_11_b ] by taking into account the actual time resolution . the longitudinal bunch profile after carrying out the convolution ( green dashed line ) is in good agreement with the bunch profile taken at k - iccd with applied temporal separation ( blue line ) . electron beam profile imaging is crucial for many applications in electron beam diagnostics at fels , and particularly required to perform single - shot diagnostics . however , the frequent appearance of coherent optical radiation effects , e.g. , cotr , in high - brightness electron beams impedes incoherent beam profile imaging with standard techniques . the theoretical considerations , numerical simulations , and experimental data presented in this paper show that coherent optical emission can be strongly suppressed by performing beam profile imaging in a magnetic energy spectrometer due to sufficient spatial - to - longitudinal coupling . however , energy spectrometers preclude measuring pure transverse beam profiles due to dispersion in the bending plane . for incoherent beam profile imaging in non - dispersive beamlines , we discussed methods to separate the incoherent radiation from scintillation screens and to simultaneously exclude coherent optical radiation from detection . in contrast to spectral and spatial separation , the temporal separation technique , utilizing an iccd camera , provides a definite method to suppress coherent optical transition radiation without knowledge of the longitudinal form factor . in terms of readout times and rates , iccd cameras have the same applicability as standard ccd cameras . by applying the temporal separation technique in the presence of coherent optical radiation , we demonstrated reliable measurements of longitudinal electron beam profiles , and measurements of r.m.s . electron bunch lengths in excellent agreement with reference measurements in a magnetic energy spectrometer . limitations may appear due to scintillator excitation by secondary coherent radiation sources . however , the presented experimental results prove the temporal separation technique as a promising method for future applications in beam profile diagnostics for high - brightness electron beams . we would like to thank the whole flash - team , and the engineers and technicians of the desy groups fla , mcs , and mvs for their great support . we also thank b. faatz , k. honkavaara , and s. schreiber for providing beam time , and y. ding and h. loos for fruitful discussions . in particular , we are deeply grateful to e.a . schneidmiller for careful reading of the manuscript and to z. huang for providing many helpful explanations . m. borland , y.c . chae , p. emma , j.w . lewellen , v. bharadwaj , w.m . fawley , p. krejcik , c. limborg , s.v . milton , h .- nuhn , r. soliday , and m. woodley , nucl . instrum . methods phys . res . , sect . a * 483 * , 268 ( 2002 ) . z. huang , a. brachmann , f .- j . decker , y. ding , d. dowell , p. emma , j. frisch , s. gilevich , g. hays , ph . hering , r. iverson , h. loos , a. miahnahri , h .- nuhn , d. ratner , g. stupakov , j. turner , j. welch , w. white , j. wu , and d. xiang , phys . rev . beams * 13 * , 020703 ( 2010 ) . h. loos , r. akre , a. brachmann , f .- j . decker , y. ding , d. dowell , p. emma , j. frisch , s. gilevich , g. hays , ph . hering , z. huang , r. iverson , c. limborg - deprey , a. miahnahri , s. molloy , h .- nuhn , d. ratner , j. turner , j. welch , w. white , and j. wu , proceedings of the 30th international free electron laser conference , gyeongju , korea , 2008 , thbau01 . r. akre , d. dowell , p. emma , j. frisch , s. gilevich , g. hays , ph . hering , r. iverson , c. limborg - deprey , h. loos , a. miahnahri , j. schmerge , j. turner , j. welch , w. white , and j. wu , phys . rev . beams * 11 * , 030703 ( 2008 ) . bane , f .- j . decker , y. ding , d. dowell , p. emma , j. frisch , z. huang , r. iverson , c. limborg - deprey , h. loos , h .- nuhn , d. ratner , g. stupakov , j. turner , j. welch , and j. wu , phys . rev . beams * 12 * , 030704 ( 2009 ) . y. ding , a. brachmann , f .- j . decker , d. dowell , p. emma , j. frisch , s. gilevich , g. hays , ph . hering , z. huang , r. iverson , h. loos , a. miahnahri , h .- nuhn , d. ratner , j. turner , j. welch , w. white , and j. wu , phys . 102 * , 254801 ( 2009 ) . z. huang , a. baker , c. behrens , m. boyes , j. craft , f .- j . decker , y. ding , p. emma , j. frisch , r. iverson , j. lipari , h. loos , and d. walz , , proceedings of the 24th particle accelerator conference , new york , usa , 2011 , thp183 . m. yan , c. behrens , ch . gerth , g. kube , b. schmidt , and s. wesch , proceedings of the 10th european workshop on beam diagnostics and instrumentation for particle accelerators , hamburg , germany , 2011 , tupd59 . sukhikh , g. kube , yu.a . popov , a.p . potylitsyn , d. krambrich , and w. lauth , proceedings of the 10th european workshop on beam diagnostics and instrumentation for particle accelerators , hamburg , germany , 2011 , weoa02 .

high - brightness electron beams with low energy spread at existing and future x - ray free - electron lasers are affected by various collective beam self - interactions and microbunching instabilities . the corresponding coherent optical radiation effects , e.g. , coherent optical transition radiation , impede electron beam profile imaging and become a serious issue for all kinds of electron beam diagnostics using imaging screens . furthermore , coherent optical radiation effects can also be related to intrinsically ultrashort electron bunches or the existence of ultrashort spikes inside the electron bunches . in this paper , we discuss methods to suppress coherent optical radiation effects both by electron beam profile imaging in dispersive beamlines and by using scintillation imaging screens in combination with separation techniques . the suppression of coherent optical emission in dispersive beamlines is shown by analytical calculations , numerical simulations , and measurements . transverse and longitudinal electron beam profile measurements in the presence of coherent optical radiation effects in non - dispersive beamlines are demonstrated by applying a temporal separation technique .

active galactic nuclei ( agn ) mapped in radio waves are an interesting probe of large - scale structure . they can be routinely detected out to very large redshift ( @xmath6 ) over wide areas of the sky and hence delineate the largest structures and their evolution over cosmic epoch . radio emission is insensitive to dust obscuration and radio agn are effective tracers of mass : they are uniformly hosted by massive elliptical galaxies and have been shown to trace both clusters ( hill & lilly 1991 ) and superclusters ( brand et al . 2003 ) . the current generation of wide - area radio surveys such as faint images of the radio sky at twenty centimetres ( first ; becker , white & helfand 1995 ) and the nrao vla sky survey ( nvss ; condon et al . 1998 ) contain radio galaxies in very large numbers ( @xmath7 ) and have allowed accurate measurements of the imprint of radio galaxy angular clustering . these patterns are considerably harder to detect in radio waves than in optical light due to the huge redshift range that is probed . whilst this provides access to clustering on the largest scales , it also washes out much of the angular clustering signal through the superposition of unrelated redshift slices . the angular correlation function was measured for first by cress et al . ( 1996 ) and magliocchetti et al . ( 1998 ) and for nvss by blake & wall ( 2002a ) and overzier et al . ( 2003 ) . the nvss radio survey , covering @xmath8 per cent of the sky , permits the measurement of fluctuations over very large angles . blake & wall ( 2002b ) detected the imprint of the cosmological velocity dipole in the nvss surface density , in a direction consistent with the cosmic microwave background ( cmb ) dipole . in this study we measure the angular power spectrum , @xmath0 , of the radio galaxy distribution ( baleisis et al . 1998 ) . this statistic represents the source surface - density field as a sum of sinusoidal angular density fluctuations of different wavelengths , using the spherical harmonic functions . the angular power spectrum is sensitive to large - angle fluctuations and hence complements the measurement of the angular correlation function , @xmath9 , at small angles . measurement of the @xmath0 spectrum has some advantages in comparison with @xmath9 . firstly , the error matrix describing correlations between multipoles @xmath10 has a very simple structure , becoming diagonal for a complete sky . this is not the case for the separation bins in a measurement of @xmath9 : even for a full sky , an individual galaxy appears in many separation bins , automatically inducing correlations between those bins . secondly , there is a natural relation between the angular power spectrum and the spatial power spectrum of density fluctuations , @xmath11 . this latter quantity provides a very convenient means of describing structure in the universe for a number of reasons . firstly its primordial form is produced by models of inflation , which prescribe the initial pattern of density fluctuations @xmath12 . furthermore , in linear theory for the growth of perturbations , fluctuations described by different wavenumbers @xmath13 evolve independently , enabling the model power spectrum to be easily scaled with redshift . the physics of linear perturbations are hence more naturally described in fourier space . in contrast , the angular correlation function is more easily related to the spatial correlation function @xmath14 , the fourier transform of @xmath11 . correlation functions more naturally serve to describe the real - space profile of collapsing structures evolving out of the linear regime . we emphasize that although the two functions @xmath0 and @xmath9 are _ theoretically _ equivalent linked by a legendre transform this is not true in an _ observational _ sense . for example , @xmath9 can only be successfully measured for angles up to a few degrees , but @xmath0 depends on @xmath9 at _ all _ angles . we derive the angular power spectrum using two independent methods . firstly we apply a direct spherical harmonic estimator following peebles ( 1973 ) . secondly , we use maximum likelihood estimation , commonly employed for deriving the angular power spectra of the cmb temperature and polarization maps . these two methods are described in section [ secmeth ] . we find that these two approaches yield very similar results ( section [ secres ] ) , which is unsurprising given the wide sky coverage of the nvss . in section [ secpk ] we interpret the nvss angular power spectrum in terms of the underlying spatial power spectrum of mass fluctuations and the radial distribution of radio sources . finally in section [ secbias ] we employ these models to derive the linear bias factor of nvss radio galaxies by marginalizing over the other model parameters . the 1.4 ghz nrao vla sky survey ( nvss ; condon et al . 1998 ) was performed at the very large array over the period 1993 to 1996 and covers the sky north of declination @xmath15 . the source catalogue contains @xmath16 entries and is 99 per cent complete at integrated flux density @xmath17 mjy . the full width at half - maximum of the synthesized beam is 45 arcsec ; the majority of radio sources are thus unresolved . the relatively broad nvss beam yields excellent surface brightness sensitivity and photometric completeness . before analyzing the survey for large - scale structure we imposed various angular masks . firstly we excluded catalogue entries within @xmath18 of the galactic plane , many of which are galactic in origin ( mostly supernova remnants and hii regions ) . the contribution of foreground galactic sources at latitudes @xmath19 is negligible . we also placed 22 masks around bright local extended radio galaxies contributing large numbers of catalogue entries , as described in blake & wall ( 2002a ) . figure [ fignvss ] plots the remaining sources with @xmath20 mjy . the remaining nvss geometry corresponds to 75 per cent of the celestial sphere . blake & wall ( 2002a ) demonstrated that the nvss suffers from systematic gradients in surface density at flux - density levels at which it is complete . these gradients corresponding to a @xmath21 per cent variation in surface density at a threshold of 3 mjy are entirely unimportant for the vast majority of applications of this catalogue . however , they have a significant influence on the faint imprint of large - scale structure . if left uncorrected a distortion of the measured angular power spectrum would result , because the harmonic coefficients would need to reproduce the systematic gradients as well as the fluctuations due to clustering . blake & wall ( 2002a ) found that these surface gradients are only significant at fluxes @xmath22 mjy . at brighter fluxes the survey is uniform to better than 1 per cent sufficient to allow the detection of the anticipated cosmological velocity dipole ( blake & wall 2002b ) . hence we simply restricted our @xmath0 analysis to fluxes @xmath23 mjy . the nvss source surface density at this threshold is @xmath24 deg@xmath25 . we note that the broadness of the radio luminosity function ensures that the projected clustering properties of radio galaxies are not a strong function of flux density in the range 3 mjy @xmath26 50 mjy , as verified by the correlation function analyses of blake & wall ( 2002a ) and overzier et al . ( 2003 ) . in this flux - density range , the redshift distribution of the radio galaxies does not change significantly . a distribution of galaxies on the sky can be generated in two statistical steps . firstly , a density field @xmath27 is created ; this may be described in terms of its spherical harmonic coefficients @xmath28 : @xmath29 where @xmath30 are the usual spherical harmonic functions . secondly , galaxy positions are generated in a poisson process as a ( possibly biased ) realization of this density field . the angular power spectrum @xmath0 prescribes the spherical harmonic coefficients in the first step of this model . it is defined over many realizations of the density field by @xmath31 the assumption of isotropy ensures that @xmath32 is a function of only @xmath10 , not @xmath33 . the angular power spectrum @xmath0 is theoretically equivalent to the angular correlation function @xmath9 as a description of the galaxy distribution . the two quantities are connected by the well - known relation @xmath34 where @xmath35 is the source surface density and @xmath36 is the legendre polynomial . however , the angular scales on which the signal - to - noise is highest are very different for each statistic . @xmath9 can only be measured accurately at small angles up to a few degrees ( blake & wall 2002a ) , beyond which poisson noise dominates . by contrast , @xmath0 for galaxies has highest signal - to - noise at small @xmath10 , corresponding to large angular scales @xmath37 . hence the two statistics are complementary , the @xmath0 spectrum probing fluctuations on the largest angular scales . peebles ( 1973 ) presented the formalism of spherical harmonic analysis of a galaxy distribution over an incomplete sky ( for refinements see e.g. wright et al . 1994 ; wandelt , hivon & gorski 2000 ) . for a partial sky , a spherical harmonic analysis is hindered by the fact that the spherical harmonics are not an orthonormal basis , which causes the measured coefficients @xmath28 to be statistically correlated , entangling different multipoles of the underlying @xmath0 spectrum . however , for the case of a survey covering @xmath38 of the sky , the repercussions ( discussed below ) are fairly negligible , implying shifts and correlations in the derived power spectrum that are far smaller than the error bars . we employed the original method of peebles ( 1973 ) with only one small correction for sample variance . in section [ secmaxlik ] we compare spherical harmonic analysis with the technique of maximum likelihood estimation and find that the two methods yield results in good agreement . the spherical harmonic coefficients of the density field may be estimated by summing over the @xmath39 galaxy positions @xmath40 : @xmath41 for an incomplete sky , these values need to be corrected for the unsurveyed regions , so that an estimate of @xmath0 is @xmath42 ( peebles 1973 equation 50 ) where @xmath43 and @xmath44 @xmath45 where the integrals are over the survey area @xmath46 , and are determined in our analysis by numerical integration . the final term in equation [ eqclpeeb ] corrects for the finite number of discrete sources : for a full sky ( @xmath47 ) we expect @xmath48 in the absence of clustering , i.e. this is the power spectrum of the shot noise . note that @xmath35 is the apparent source density @xmath49 , not the average over an imagined ensemble of catalogues . we determined the angular power spectrum for the @xmath10th multipole , @xmath50 , by averaging equation [ eqclpeeb ] over @xmath33 . because the density field is real rather than complex , @xmath51 , resulting in @xmath52 independent measurements of @xmath0 : @xmath53 there is no need for us to use the modified weighting formula of peebles ( 1973 equation 53 ) . in our case , @xmath54 does not vary significantly with @xmath33 . we verified that the modified weighting formula produced indistinguishable results . one consequence of the partial sky is to `` mix '' the harmonic coefficients such that the measured angular power spectrum at @xmath10 depends on a range of @xmath55 around @xmath56 : @xmath57 the angled brackets refer to an imagined averaging over many realizations of density fields generated by @xmath0 , in accordance with equation [ eqcldef ] . peebles showed that @xmath58 ; i.e. mixing does not spuriously enhance the measured power ( this is accomplished by the factor @xmath54 in equation [ eqclpeeb ] ) . for a complete sky , @xmath59 , where @xmath60 ( @xmath61 ) or 0 ( @xmath62 ) . for a partial sky , the matrix @xmath63 can be computed from the geometry of the surveyed region ( hauser & peebles 1973 ) . figure [ figrll ] illustrates the result for the nvss for @xmath64 ( computed using hauser & peebles 1973 equation 12 ) . the nvss covers a sufficiently large fraction of the sky ( 75 per cent ) that mixing only occurs at the @xmath65 per cent level and can be neglected because the underlying @xmath0 spectrum is smooth : @xmath66 we checked the nvss @xmath63 matrix for other multipoles @xmath10 and found very similar results . this argument ensures that the measured multipoles are statistically independent to a good approximation . the statistical error on the estimator of equation [ eqclpeeb ] is @xmath67 ( peebles 1973 equation 81 ) . there are two components of the error : * `` shot noise '' ( @xmath35 ) because the number of discrete objects is finite and therefore does not perfectly describe the underlying density field . * `` cosmic variance '' ( @xmath0 ) because even with perfect sampling of the density field , there are only a finite number of harmonics associated with the @xmath10th multipole . the error for the @xmath68 case in equation [ eqclsig ] is greater because @xmath69 is purely real , rather than complex . in the latter case , we are averaging over the real and imaginary parts of @xmath70 , two independent estimates of @xmath0 , which reduces the overall statistical error by a factor @xmath71 . for a partial sky equation [ eqclsig ] is an approximation , because the variance of multipoles of given @xmath10 depends on the underlying power spectrum at @xmath72 . as discussed above , this effect is negligible for the nvss . the averaging over @xmath33 ( equation [ eqclobs ] ) decreases the error in the observation . combining the errors of equation [ eqclsig ] , assuming estimates at different @xmath33 are statistically independent , the resulting error in @xmath50 is @xmath73 we used monte carlo simulations to verify that equation [ eqclerr ] produced results within 5 per cent of the true error for all relevant multipoles . our only addition to the formalism of peebles ( 1973 ) was to increase the total variance on the estimate of @xmath0 by a factor @xmath74 , where @xmath75 is the fraction of sky covered ( i.e. multiply equation [ eqclerr ] by @xmath76 ) . this correction factor was motivated by scott , srednicki and white ( 1994 ) as a fundamental property of sample variance for a partial sky , and is part of the standard cmb formalism ( e.g. bond , efstathiou & tegmark 1997 ) . for the nvss geometry , @xmath77 , thus this correction corresponds to a @xmath78 per cent increase in the error . radio sources have complex morphologies and large linear sizes ( up to and exceeding 1 mpc ) . a radio - source catalogue such as the nvss will contain entries which are different components of the same galaxy ( for example , the two radio lobes of a `` classical double '' radio galaxy ) . the broad angular resolution of the nvss beam leaves over 90 per cent of radio sources unresolved ; however , the remaining multiple - component sources have a small but measurable effect on the angular power spectrum . it is relatively simple to model the effect of multiple - component sources on the estimator for @xmath0 described in section [ secestharm ] . the relevant angular scales ( @xmath79 ) are much bigger than any component separation , and equation [ eqalm ] can be replaced by @xmath80 where @xmath81 is the total number of galaxies and @xmath82 is the number of components of the @xmath83th galaxy . thus the quantity @xmath84 is unchanged by the presence of multiple components ( @xmath85 denotes the average number of components per galaxy , and @xmath86 is the total number of catalogue entries , as in equation [ eqalm ] ) . but @xmath87 in equation [ eqclpeeb ] depends on @xmath88 multiple - component sources only affect the first term in this expression , producing an offset in the @xmath0 spectrum @xmath89 but @xmath90 from equation [ eqjlm ] , and this expression simplifies to an offset independent of @xmath10 : @xmath91 most multiple - component sources in the nvss catalogue are double radio sources . let a fraction @xmath92 of the radio galaxies be doubles . then @xmath93 and @xmath94 , thus the constant offset may be written @xmath95 we can deduce @xmath96 from the form of the nvss angular correlation function @xmath9 at small angles @xmath97 , where double sources dominate the close pairs ( see blake & wall 2002a and also section [ secalm ] ) . this correction was applied to the measured nvss @xmath0 spectrum and successfully removed the small systematic offset in @xmath0 at high @xmath10 . a sophisticated suite of analytical tools has been developed by the cmb community for deriving the angular power spectra of the observed cmb temperature and polarization maps . these methods can also be exploited to analyze galaxy data ( see for example efstathiou & moody 2001 , huterer , knox & nichol 2001 and tegmark et al . 2002 ) . in this approach the power spectrum is determined using an iterative maximum likelihood analysis , in contrast to the direct estimator discussed in section [ secestharm ] . the likelihood is a fundamental statistical quantity , and this analysis method permits straightforward control of such issues as edge effects , noise correlations and systematic errors . the starting point for maximum likelihood estimation ( mle ) is bayes theorem @xmath98 where @xmath99 are the parameters one is trying determine , @xmath100 is the data and @xmath101 is the additional information describing the problem . the quantity @xmath102 is the likelihood , i.e. the probability of the data given a specific set of parameters , while the left - hand side is the posterior , i.e. the probability of the parameters given the data . we will assume that the sky is a realization of a stationary gaussian process , with an angular power spectrum @xmath0 . we assume no cosmological information about the distribution of the @xmath0 . the rendition of the sky will be a pixelized map , created by binning the galaxy data in equal - area cells such that the count in the @xmath83th cell is @xmath103 , effectively constructing a `` temperature map '' of galaxy surface density . we performed this task using the healpix software package ( gorksi , hivon & wandelt 1999 ; http://www.eso.org/science/healpix ) . we chose the healpix pixelization scheme @xmath104 , which corresponds to 12,288 pixels over a full sky . the angular power spectrum may be safely extracted to multipole @xmath105 . we then defined a data vector : @xmath106 where @xmath107 is the mean count per pixel . figure [ fighist ] demonstrates that the data vector @xmath108 for the nvss sample is well - approximated by a gaussian distribution , as assumed in a maximum likelihood analysis . the covariance matrix @xmath109 due to primordial fluctuations is given by @xmath110 where @xmath36 is the legendre polynomial and @xmath111 is the angle between pixel pair @xmath112 . in order to apply a likelihood analysis we must also specify a noise covariance matrix @xmath113 . we modelled the noise as a gaussian random process with variance @xmath114 , uncorrelated between pixels , such that @xmath115 . the likelihood of the map , with a particular power spectrum @xmath0 , is given by @xmath116\ ] ] the goal of mle is to maximize this function , and the fastest general method is to use newton - raphson iteration to find the zeroes of the derivatives in @xmath117 with respect to @xmath0 . we used the madcap package ( borrill 1999 ; http://www.nersc.gov/@xmath118borrill/cmb/madcap ) to derive the maximum likelihood banded angular power spectrum from the pixelized galaxy map and noise matrix . madcap is a parallel implementation of the bond , jaffe & knox ( 1998 ) maximum - likelihood algorithms for the analysis of cmb datasets . we ran the analysis software on the supercomputer seaborg , administered by the national energy research scientific computing centre ( nersc ) at lawrence berkeley national laboratory , california . we again applied equation [ eqcldoub ] to the madcap results to correct the measured power spectrum for the influence of multiple - component sources . boughn & crittenden ( 2002 ) also performed a healpix analysis of the nvss as part of a cross - correlation analysis with the cmb searching for evidence of the integrated sachs - wolfe effect . from the pixelized map they derived an angular correlation function for the nvss , which they used to constrain theoretical models we compare their results with ours in section [ secbias ] . we tested the two methods , direct spherical harmonic and maximum likelihood estimation , by generating a dipole distribution of @xmath119 sources over the nvss geometry , where the model dipole possessed the same amplitude and direction as that detected in the nvss ( see blake & wall 2002b ) . in order to simulate multiple components , we added companion sources to @xmath120 per cent of the objects , with the same separation distribution as that measured in the nvss ( blake & wall 2002a ) . the two measurements of the @xmath0 spectrum , corrected for multiple components using equation [ eqcldoub ] , are plotted in figure [ figcltest ] and are consistent with zero . the @xmath0 measurements have been averaged into bands of width @xmath121 , starting from @xmath122 . the dipole term @xmath123 is of course spuriously high , but has a negligible effect on the measured harmonics at @xmath124 the galaxy dipole ( unlike the cmb dipole ) is only barely detectable in current surveys ( blake & wall 2002b ) . the normalization convention used in figure [ figcltest ] , and the remaining power spectrum plots , is to expand the surface overdensity @xmath125 in terms of spherical harmonics . to convert from the definition of @xmath0 in equations [ eqsigylm ] and [ eqcldef ] we simply divide @xmath70 by @xmath35 ( in units of sr@xmath126 ) and hence @xmath0 by @xmath127 . figure [ figcltest ] permits a first comparison of the two independent methods of estimating @xmath0 . at low @xmath10 , the variances are in excellent agreement . as @xmath10 approaches @xmath128 , the variance of the maximum likelihood method begins to exceed that of the spherical harmonic analysis . this occurs as the resolution of the pixelization scheme becomes important : the angular pixel size is no longer much less than the characteristic angular scale probed by the @xmath10th multipole . the variance of the high @xmath10 bins could be reduced by adopting a finer pixelization , such as @xmath129 , with the penalty of a rapidly increasing requirement of supercomputer time . given that the signal in the nvss angular power spectrum turns out to be confined to @xmath130 , our optimum pixelization remains @xmath104 . moreover , higher pixel resolution would decrease the mean pixel count @xmath107 in equation [ eqdata ] , rendering a gaussian distribution a poorer approximation for @xmath108 . figure [ figcl ] plots the nvss angular power spectrum measured for flux - density threshold @xmath131 mjy using both spherical harmonic analysis and maximum likelihood estimation . the constant offset due to double sources ( equation [ eqcldoub ] ) has been subtracted and the measurements are averaged into bands of width @xmath121 , starting from @xmath132 . measurements are plotted up to @xmath133 , although note that the variance of the maximum likelihood estimation is increased by pixelization effects above @xmath134 , as discussed in section [ sectest ] . table [ tabcl ] lists the plotted data . figure [ figclcmb ] displays the same data scaled by the usual cmb normalization factor , @xmath135 . .data table of banded nvss @xmath0 values plotted in figure [ figcl ] . the offset due to double sources is @xmath136 . [ cols="^,^,^,^ " , ] we assumed that the primordial matter power spectrum is a featureless power - law , @xmath137 . on very large scales , the only alteration to this spectrum in linear theory will be an amplitude change due to the growth factor . however , during the epoch of radiation domination , growth of fluctuations on scales less than the horizon scale is suppressed by radiation pressure . this process is described by the transfer function @xmath138 , such that the present - day linear matter power spectrum is given by @xmath139 accurate fitting formulae have been developed for the transfer function @xmath138 in terms of the cosmological parameters ( eisenstein & hu 1998 ) , which we employed in our analysis ( these fitting formulae assume adiabatic perturbations ) . in our fiducial cosmological model , we fixed the values of the cosmological parameters at @xmath140 , @xmath141 and @xmath142 ( spergel et al . 2003 , table 7 , column 3 ) . we also chose a primordial spectral index @xmath143 ( spergel et al . 2003 ) , which is close to the predictions of standard inflationary models . we consider the effect of variations in these parameter values in section [ secbias ] . we assumed that the universe is flat , with the remaining energy density provided by a cosmological constant @xmath144 . there are at least two independent ways of estimating the amplitude @xmath145 in equation [ eqpkmod ] . firstly we can use constraints on the number density of massive clusters at low redshift , expressed in terms of @xmath4 , the rms fluctuation of mass in spheres of radius @xmath146 mpc : @xmath147 ^ 2 \ , dk \label{eqsigsq}\ ] ] for example , viana & liddle ( 1999 ) determined the most likely value of @xmath4 using this method to be @xmath148 . alternatively , @xmath145 can be expressed in terms of the amplitude of fluctuations at the hubble radius , @xmath149 , and constrained by measurements of cmb anisotropies on large angular scales : @xmath150 for example , bunn & white ( 1997 ) give the best - fitting constraint on @xmath149 and @xmath151 for flat models ( @xmath152 ) based on results of the cobe dmr experiment : @xmath153 with a maximum @xmath154 statistical uncertainty of 7 per cent . for our fiducial cosmological parameters , a ( reasonably ) consistent cosmology is produced if @xmath155 ( i.e. @xmath156 ) . we assumed this fiducial normalization in our model , noting that the value of @xmath4 is in fact degenerate with the amplitude of a constant linear bias factor @xmath157 . equation [ eqpkmod ] is only valid ignoring non - linear effects , which will boost the value of @xmath11 on small scales as modes commence non - linear collapse . we incorporated non - linear corrections using the fitting formula provided by peacock & dodds ( 1996 ) . the resulting model power spectrum is shown in figure [ figpkmod ] . strictly , this modification violates the assumption of linear evolution implicit in equation [ eqdz ] . however , this is not significant in our analysis because the small scales for which non - linear evolution is important are only significant in the projection at @xmath158 . the projection of the spatial power spectrum onto the sky depends on the radial distribution of the sources under consideration , which may be deduced from their redshift distribution . we only need to know the probability distribution @xmath159 of the sources ( equation [ eqrad ] ) , the absolute normalization is not important . unfortunately , the radial distribution of mjy radio sources is not yet accurately known . the majority of radio galaxies are located at cosmological distances ( @xmath160 ) and their host galaxies are optically very faint . however , models exist of the radio luminosity function of agn ( i.e. the co - moving space density of objects as a function of radio luminosity and redshift ) , from which the redshift distribution at any flux - density threshold can be inferred . such models have been published by dunlop & peacock ( 1990 ) and willott et al . these luminosity function models are by necessity constrained by relatively bright radio sources ( @xmath161 mjy ) and the extrapolation to nvss flux - density levels must be regarded as very uncertain . the willott model is constrained by a larger number of spectroscopic redshifts , and the samples of radio sources used provide fuller coverage of the luminosity - redshift plane ( thus the required extrapolation to @xmath131 mjy is less severe ) . however , the willott samples are selected at low frequency ( 151 and 178 mhz ) , necessitating a large extrapolation to the nvss observing frequency of 1.4 ghz . the dunlop & peacock models are constrained at high frequencies , but treat steep - spectrum and flat - spectrum radio sources as independent populations , which is inconsistent with current ideas concerning the unification of radio agn ( e.g. jackson & wall 1999 ) . in addition , they were computed for cosmological parameters @xmath162 , @xmath163 rather than the currently favoured `` @xmath164cdm '' cosmology . furthermore , none of the aforementioned luminosity function models incorporate starburst galaxies , which contribute in significant numbers to the radio galaxy population mix at @xmath5 for flux - density threshold @xmath165 mjy . direct measurements of @xmath159 are currently only achievable at low redshifts ( @xmath166 ) , where comparison with large optical galaxy redshift surveys is possible ( sadler et al . 2002 ; magliocchetti et al . we matched the nvss 10 mjy catalogue with the final data release of the 2df galaxy redshift survey ( 2dfgrs , available online at http://msowww.anu.edu.au/2dfgrs/ , also see colless et al . 2001 ) , in order to estimate @xmath159 at low redshift . as our fiducial model , we fitted the resulting redshift histogram with the simplest possible function , a constant @xmath167 over the range @xmath168 . in section [ secbias ] we include the effect of variations in this model . we only considered the redshift range @xmath169 in this analysis because at @xmath170 , the ( very luminous ) optical counterparts of the 10 mjy nvss sources begin slipping below the 2dfgrs magnitude threshold , which we verified by plotting the magnitudes of matched 2dfgrs galaxies against redshift . having determined the value of @xmath171 , we created the full redshift distribution by assigning the remaining probability @xmath172 over the redshift range @xmath173 in proportion to the prediction of the dunlop & peacock ( 1990 ) luminosity function models . for this investigation we used the average of the seven models provided by dunlop & peacock . assuming the willott et al . ( 2001 ) radial distribution for @xmath173 made a negligible difference to the results ( because most of the contribution to the @xmath0 spectrum arises at low redshifts , see section [ secclpred ] ) . matching nvss catalogue entries brighter than @xmath174 mjy with the 2dfgrs database yielded @xmath175 identifications with redshifts @xmath169 , using matching tolerance 10 arcsec ( see sadler et al . we restricted the 2dfgrs sample to `` high quality '' spectra ( @xmath176 ) . an estimate of the probability of an nvss source being located at @xmath169 is @xmath177 where @xmath178 deg@xmath25 is the surface density of nvss sources brighter than 10 mjy , and @xmath179 is the 2dfgrs area under consideration , which ( owing to the varying angular completeness ) is not trivial to calculate . we followed sadler et al . ( 2002 ) by dividing the number of 2dfgrs galaxies contained in the nvss geometry by the 2dfgrs surface density @xmath180 deg@xmath25 , resulting in an effective area @xmath181 deg@xmath182 . the result , @xmath183 ( per unit redshift ) , is a significant underestimate for various reasons : * incompletenesses in the 2dfgrs input catalogue ( @xmath184 ) . * input catalogue galaxies unable to be assigned a 2df spectrograph fibre ( @xmath185 ) . * observed 2dfgrs spectra with insufficient quality ( @xmath186 ) to determine a redshift ( @xmath187 ) . * extended radio sources with catalogue entries located more than 10 arcsec from the optical counterpart ( @xmath188 ) . the estimated correction factors in brackets were obtained from colless et al . ( 2001 ) and from carole jackson ( priv . comm . ) . multiplying these corrections implies a total incompleteness of @xmath189 , and on this basis we increased the value of @xmath171 to @xmath190 . figure [ fignz ] plots 2dfgrs - nvss matches in redshift bins of width @xmath191 , together with the low - redshift fit described above . a constant @xmath159 is a fairly good approximation at low redshifts ( @xmath192 ) . this flat distribution arises because the overall redshift distribution is a sum of that due to agn and that due to starburst galaxies ; @xmath159 increases with @xmath193 for the agn , but decreases with @xmath193 for the starbursters . figure [ fignz ] also displays the predictions of the luminosity function models of dunlop & peacock ( 1990 ) and willott ( 2001 ) . as explained above , the large extrapolations involved render these models a poor fit to the redshift distribution at mjy flux levels , and their use without modification at low redshift would have caused significant error . we used equation [ eqpktocl ] to predict the @xmath0 spectrum from our fiducial models of the spatial power spectrum ( section [ secpkmod ] ) and the radial distribution of the sources ( section [ secnz ] ) . we found that a good match to the measured angular power spectrum resulted if the nvss sources were assigned a constant linear bias factor @xmath194 ( figure [ figclmod ] ) ; @xmath195 provides a very poor fit to the results . the bias factor of the radio galaxies is analyzed more thoroughly in section [ secbias ] . we note that these measurements of the radio galaxy @xmath0 spectrum at low @xmath10 are _ not _ directly probing the large - scale , small @xmath13 , region of the power spectrum @xmath11 . investigation of the integrands of equations [ eqclkapp ] and [ eqclxapp ] revealed that the majority of the signal is built up at low redshift , @xmath196 ( see figure [ figkz ] ) , where small - scale spatial power is able to contribute on large angular scales ( i.e. contribute to low multipoles ) . higher redshift objects principally serve to dilute the clustering amplitude . this is unfortunate : the potential of radio galaxies distributed to @xmath160 to probe _ directly _ the large - scale power spectrum is forfeited by projection effects . in order to realize this potential , we must measure redshifts for the nvss sources . a three - dimensional map extending to @xmath160 would directly yield @xmath11 on large scales , defining the `` turn - over '' sketched in figure [ figpkmod ] . the enhanced radio galaxy bias apparent in figure [ figclmod ] is consistent with the nature of agn host galaxies : optically luminous ellipticals inhabiting moderate to rich environments . similarly high radio galaxy bias factors have been inferred from measurements of the spatial power spectrum of low - redshift radio galaxies ( peacock & dodds 1994 ) and from deprojection of the nvss angular correlation function @xmath9 ( blake & wall 2002a ; overzier et al . 2003 ) . furthermore , boughn & crittenden ( 2002 ) cross - correlated nvss and cmb overdensities in a search for the integrated sachs - wolfe effect ( see also nolta et al . their analysis included fitting theoretical models to the nvss angular correlation function . they derived good fits for a slightly lower linear bias factor than ours , @xmath197 . this difference is due to the assumed redshift distribution . boughn & crittenden also used the dunlop & peacock ( 1990 ) average model , but we corrected this model using observational data at low redshifts @xmath169 . as can be seen from figure [ fignz ] , our correction reduces the number of low - redshift sources , necessitating a higher bias factor to recover the same angular correlations . in order to derive a formal confidence interval for the linear bias parameter @xmath3 we must incorporate the effects of uncertainties in the underlying model parameters ( by marginalizing over those parameters ) . with this in mind , we assumed gaussian priors for hubble s constant @xmath198 , for the matter density @xmath199 , and for the primordial spectral index @xmath200 . the widths of these priors were inspired by the cosmological parameter analysis combining the wmap satellite observations of the cmb and the 2dfgrs galaxy power spectrum ( spergel et al . 2003 , table 7 , column 3 ) and are a good representation of our current knowledge of the cosmological model . in addition , we considered variations in the model for the radial distribution of nvss sources at low redshift ( section [ secnz ] ) , using a more general fitting formula @xmath201 to describe the probability distribution for @xmath169 . for each pair of values of @xmath202 we derived the chi - squared statistic between the model and the observations , @xmath203 , which we converted into an ( unnormalized ) probability density @xmath204 . our model is thus specified by values of ( @xmath3 , @xmath205 , @xmath206 , @xmath151 , @xmath207 , @xmath208 ) from which we can calculate a model @xmath0 spectrum and hence a chi - squared statistic with the observations , @xmath209 , corresponding to a probability density @xmath210 . we used the spherical harmonic estimation of the @xmath0 spectrum as the observational data . after multiplying @xmath211 by the redshift distribution probability density @xmath212 and the gaussian prior probability densities for @xmath205 , @xmath206 and @xmath151 , we derived the probability distribution for @xmath3 by integrating over each of the other parameters . we do not marginalize over the normalization of the matter power spectrum , @xmath4 , because this quantity is degenerate with @xmath3 ( using equations [ eqrad ] , [ eqclxapp ] and [ eqsigsq ] : @xmath213 ) . the resulting normalized probability distribution for @xmath214 is displayed in figure [ figb0 ] , from which we determined a @xmath1 confidence region @xmath215 . when combined with the wmap determination of @xmath216 ( spergel et al . 2003 ) , we infer that @xmath217 . this investigation has measured the angular power spectrum of radio galaxies for the first time , yielding consistent results through the application of two independent methods : direct spherical harmonic analysis and maximum likelihood estimation . the nvss covers a sufficient fraction of sky ( @xmath8 per cent ) that spherical harmonic analysis is very effective , with minimal correlations amongst different multipoles . the form of the @xmath0 spectrum can be reproduced by standard models for the present - day spatial power spectrum and for the radial distribution of nvss sources provided that this latter is modified at low redshift through comparison with optical galaxy redshift surveys . the results strongly indicate that radio galaxies possess high bias with respect to matter fluctuations . a constant linear bias @xmath194 permits a good fit , and by marginalizing over the other parameters of the model we deduce a @xmath1 confidence interval @xmath215 where @xmath4 describes the normalization of the matter power spectrum . we find that the majority of the angular power spectrum signal is generated at low redshifts , @xmath5 . therefore , in order to exploit the potential of radio galaxies to probe spatial fluctuations on the largest scales , we require individual redshifts for the nvss sources . we thank jasper wall and steve rawlings for helpful comments on earlier drafts of this paper . we acknowledge valuable discussions with carole jackson concerning cross - matching the nvss and 2dfgrs source catalogues . we are grateful to sarah bridle for useful guidance on marginalizing over the cosmological model . baleisis a. , lahav o. , loan a.j . , wall j.v . , 1998 , mnras , 297 , 545 becker r.h . , white r.l . , helfand d.j . , 1995 , apj , 450 , 559 blake c.a . , wall j.v , 2002a , mnras , 329 , l37 blake c.a . , wall j.v , 2002b , nature , 416 , 150 bond j.r . , jaffe a.h . , knox l. , 1998 , phrvd , 57 , 2117 bond j.r . , efstathiou g. , tegmark m. , 1997 , mnras , 291 , 33 borrill j. , 1999 , in _ proceedings of the 5th european sgi / cray mpp workshop _ ( astro - ph/9911389 ) boughn s.p . , crittenden r.g . , 2002 , phrvl , 88 , 1302 brand k. , rawlings s. , hill g.j . , lacy m. , mitchell e. , tufts j. , 2003 , mnras , 344 , 283 bunn e.f . , white m. , 1997 , apj , 480 , 6 carroll s.m . , press w.h . , turner e.l . , 1992 , ara&a , 30 , 499 colless m. et al . , 2001 , mnras , 328 , 1039 condon j. , cotton w. , greisen e. , yin q. , perley r. , taylor g. , broderick j. , 1998 , aj , 115 , 1693 cress c. , helfand d. , becker r. , gregg m. , white r. , 1996 , apj , 473 , 7 dunlop j.s . , peacock j.a . , 1990 , mnras , 247 , 19 efstathiou g. , moody s. , 2001 , mnras , 325 , 1603 eisenstein d.j . , hu w. , 1998 , apj , 496 , 605 gorski k.m . , hivon e. , wandelt b.d . , 1999 , in _ proceedings of the mpa / eso cosmology conference `` evolution of large - scale structure '' _ , p.37 ( astro - ph/9812350 ) hauser m.g . , peebles p.j.e . , 1973 , apj , 185 , 757 hill g.j . , lilly s.j . , 1991 , apj , 367 , 1 huterer d. , knox l. , nichol r.c . , 2001 , apj , 555 , 547 jackson c.a . , wall j.v . , 1999 , mnras , 304 , 160 magliocchetti m. , maddox s. , lahav o. , wall j. , 1998 , mnras , 300 , 257 magliocchetti m. et al . , 2002 , mnras , 333 , 100 nolta et al . , 2003 , apj submitted ( astro - ph/0305097 ) overzier r.a . , rttgering h.j.a . , rengelink r.b . , wilman r.j . , 2003 , a&a , 405 , 53 peacock j.a . , dodds s.j . , 1994 , mnras , 267 , 1020 peacock j.a . , dodds s.j . , 1996 , mnras , 280 , 19 peebles p.j.e . , 1973 , apj , 185 , 413 sadler e.m . et al . , 2002 , mnras , 329 , 227 scott d. , srednicki m. , white m. , 1994 , apj , 421 , 5 spergel d.n . 2003 , apjs , 148 , 175 tegmark m. et al . , 2002 , apj , 571 , 191 viana p.t.p . , liddle a.r . , 1999 , mnras , 303 , 535 wandelt b.d . , hivon e. , gorski k.m . , 2001 , phrvd , 64 , 3003 willott c.j . , rawlings s. , blundell k.m . , lacy m. , eales s.a . , 2001 , mnras , 322 , 536 wright e.l . , smoot g.f . , bennett c.l . , lubin p.m. , 1994 , apj , 436 , 441

we measure the angular power spectrum @xmath0 of radio galaxies in the nrao vla sky survey ( nvss ) using two independent methods : direct spherical harmonic analysis and maximum likelihood estimation . the results are consistent and can be understood using models for the spatial matter power spectrum and for the redshift distribution of radio galaxies at mjy flux - density levels . a good fit to the angular power spectrum can only be achieved if radio galaxies possess high bias with respect to mass fluctuations ; by marginalizing over the other parameters of the model we derive a @xmath1 confidence interval @xmath2 , where @xmath3 is the linear bias factor for radio galaxies and @xmath4 describes the normalization of the matter power spectrum . our models indicate that the majority of the signal in the nvss @xmath0 spectrum is generated at low redshifts @xmath5 . individual redshifts for the nvss sources are thus required to alleviate projection effects and probe directly the matter power spectrum on large scales . large - scale structure of universe galaxies : active surveys

large - scale optical surveys show that the luminous qso number density peaks at @xmath4 , before which ( in cosmic time ) the qso number density grows rapidly and after which the density steadily decays until the present epoch ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? the first indication of a density decline ( with redshift ) at @xmath5 was reported in the pioneering work of @xcite . the palomar transit grism survey ( ptgs ) @xcite was designed to investigate this possible redshift cutoff " , and produced a sample of 90 @xmath6 qsos ; an analysis of the ptgs @xcite ( hereafter ssg95 ) revealed a very rapid growth , by a factor of @xmath7 , of the number density of luminous qsos from @xmath8 to @xmath9 . this result was verified by @xcite using a sample of @xmath10 quasars found in the sloan digital sky survey ( sdss ; * ? ? ? this redshift cutoff has also been found in the number densities of radio flat - spectrum qsos @xcite . in the x - ray regime , however , the size of a sample based on the combination of various _ rosat _ surveys , @xcite ( heraafter , sxlf1 ) was not sufficient to accurately probe this `` growth '' phase of the number density of x - ray - selected luminous ( @xmath11 ) qsos . while this study appears to show a flat number density in @xmath6 , the uncertainties produced by the small number of quasars limited the robustness of the conclusions . more recent studies including the results from _ chandra _ and _ xmm - newton _ surveys @xcite showed that the number density curve is luminosity - dependent and a trend that as luminosity goes lower , the density peak shifts to lower redshifts ( as some authors call an anti - hierarchical agn evolution and others call a down - sizing of the agn activity ) . however , these studies did not sample well the high - redshift , high - luminosity regime , in which optically selected qsos show a decline at @xmath12 . in a recent study based on an updated soft x - ray sample including _ chandra _ and _ xmm - newton _ surveys , @xcite ( hereafter sxlf3 ) also measured the number densities in various luminosity bins with a better accuracy over a large range in the redshift - luminosity space . the study revealed this early `` growth '' of agn number density ( or decline with increasing @xmath13 ) at z@xmath14 for agn / qso luminosities at ( @xmath15 44 - 45 ) . the results from `` champs '' survey , which was designed to optimally trace the high - redshift , high - luminosity ( @xmath16 ) regime with improved statistics , found that the density curves x - ray selected qsos ( @xmath16 ) declines with increasing redshift at @xmath12 . this decline is , however , shallower than that seen in optically - selected qsos . the dependence of the optical to x - ray flux ratio ( customarily expressed by the quantity @xmath1 , the effective spectral index between the rest - frame 2500 and 2 kev ) , where xx is any subscript to @xmath17 . ] on redshift and luminosity has been a key issue in x - ray observations of high - redshift qsos and has important implications for possible differences in the agn evolution traced by x - ray and optical samples . some authors ( e.g. * ? ? ? * ; * ? ? ? * ) found that @xmath1 strongly depends on luminosity , with @xmath18 , and no evidence for any evolution of the x - ray properties with redshift . @xcite found that variations in @xmath1 depends primarily on redshift . the dependence may be sensitive to the selection effects , including but not limited to whether the sample is optically - selected or x - ray selected . @xcite pointed out that such an apparent non - linearity of the luminosity correlation in two bands can arise from the difference in the luminosity variations in the two bands . in order to investigate the redshift dependence of the optical - to - x - ray luminosity ratios and its impact on the density curves of luminous qsos in x - ray and optically selected samples at @xmath19 , we have obtained _ chandra _ observations of six ptgs qsos with redshifts between 2.91 and 2.96 . this is the era of maximum number density of luminous qsos . there was practically no systematic observations in x - rays in this redshift regime before . thus our observations also serve to fill this observation gap . throughout this paper we adopt @xmath20},\omega_m,\omega_\lambda)= ( 70h_{70},0.3,0.7)$ ] and @xmath21 unless otherwise noted . the original motivation of the program was to compare the mean @xmath1 values of @xmath0 and @xmath22 qsos . in chandra cycle 4 , six @xmath0 qsos from the ptgs ( out of 15 qsos proposed ) have been observed . none of these qsos are broad absorption line ( bal ) qsos . table [ tab : log ] shows the observed targets , log of observations , optical ab magnitudes at the object s rest frame of 1450 . the core radio loudness @xmath23/f_\nu [ 4400 $ ] @xmath24 $ ] is also shown , where those with @xmath25 and @xmath26 are divided into radio - quiet qsos ( rrq ) and radio - loud qsos ( rlq ) respectively ( @xcite and references therein ) . the fluxes are in the object s rest frame , calculated assuming radio and optical spectral indices of @xmath27 and @xmath28 respectively . the radio data are from the nvss ( pc 0041 + 0024 * ? ? ? * ) or first ( all others * ? ? ? * ) surveys . only one qso ( pc 1035 + 4747 ) was detected in the radio band and for others , 3@xmath29 upper limits of @xmath30 are shown . the only radio - detected qso , pc 1035 + 4747 , has @xmath31 ; this object falls well into the rlq regime . two of the @xmath30 upper limits ( 1.5 for pc 0041 + 0214 and 1.2 for pc1000 + 4751 ) are above the rqq / rlq border , but their limits are well below the peak of the @xmath30 distribution of rlq . thus we tentatively classify them as rqqs . all observations have been made with the advanced ccd imaging spectrometer ( acis ; @xcite ) and the targets aimed at the default position of acis - s with the s3-chip . in all cases except pc 0947 + 5628 , x - ray counterparts have been found within @xmath32 of the cataloged optical center of the target qsos , consistent with the combined systematic error on the absolute astrometries of 1@xmath33 - 2@xmath33 in both the _ chandra _ data products and the ptgs survey . although the detected x - ray source closest to pc0947 + 5628 was @xmath34 away from the catalogued position of the qso , 6 other x - ray sources in the same acis observation also had optical counterparts at @xmath35 away with practically the same offset directions . thus we also identify the x - ray source with pc0947 + 5628 . for all six observations , we have extracted the pulse - height spectra using an extraction radius of 2@xmath33 . spectra , response matrices ( rmf ) and ancillary response files ( arf ) were created using the software package * ciao 3.0.2 * or later versions , in conjunction with the calibration database * caldb 2.26 * or later versions . these versions enabled construction of the response files which takes the time - dependent low energy efficiency degradation into account . due to low number counts of the involved objects , changes due to further updates of the calibration have negligible effects . the spectral analyses were made to the pulse - height channels corresponding to observed photon energies of 0.3 - 7 kev . background level is typically @xmath36 counts in the extraction radius and is thus negligible . the spectral analysis were made with * xspec 11.2*. in spite of small number of x - ray photons , the negligible background and the use of the * xspec * implementation of the @xcite c - statistics allowed placement of some constraints on the spectral indexes , although there are not sufficient number of photons in any individual spectrum to simultaneously constrain the intrinsic absorption column density . the results of the spectral fits with a single power - law with a photon index @xmath37 and the galactic absorption @xmath38 @xmath39}$ ] @xcite at the position of the qsos are shown in table [ tab : spec ] . we see that the only radio - detected qsos photon index of @xmath40 is constrained to be harder than the mean qso spectrum . the rest - frame 2 - 10 kev luminosities ( @xmath41 ; logarithm is base-10 ) are also shown as well as the source counts in 0.3 - 7 kev . the rest - frame 2 - 10 kev corresponds to observed frame 0.5 - 2.5 kev . the b - band absolute magnitudes ( @xmath42 ) , that have been recalculated using our default cosmology ( sect . [ sec : intr ] ) and the optical spectral index @xmath28 ( following * ? ? ? * ; * ? ? ? * ) , are also listed here . using @xmath43 @xcite increases @xmath42 by 0.35 . we also analyzed the summed spectrum from all six qsos . because of the small spread of the redsifts of these qsos , we can analyze the summed spectrum assuming a single redshift . the response matrix for the summed spectrum was constructed by a source - count weighted mean the 6 matrices . the galactic column densities @xmath38 of these six qsos are 0.9 , 1.0 , 1.0 , 1.3 , 2.0 and 2.8 . the fit was made to a model with the sum of three power - laws , with different galactic absorptions of @xmath44 and 2.8 respectively , where the four qsos with @xmath45 were represented by a single column density of 1.0 . the photon indices of all the three components were set to equal and the ratios of the 3 normalizations were fixed to those of the total source counts of the qsos with @xmath38 of 0.9 - 1.3 , 2.0 and 2.8 respectively . also an intrinsic absorption component @xmath46 is included with @xmath47 , which is a source - count weighted mean redshift of the sample . again , the c - statistics was used for the fit . the summed spectrum is well represented by a single power - law with @xmath48 and no intrinsic absorption , as shown in the last entry of table [ tab : spec ] . the pulse - height spectrum , folded best - fit power - law model , and fit residuals in terms of data - to - model ratio are shown in fig . [ fig : sum_spec ] , with confidence contours for the intrinsic absorption versus photon index space . removing the one rlq ( pc 1035 + 4747 ) from the analysis did not change the fitted parameters and errors to the smallest digits displayed in table [ tab : spec ] . this result is consistent with the mean slopes of rqqs and unabsorbed agns measured in the 2 - 10 kev in the rest frame over a wide range of redshift and luminosity ( e.g. * ? ? ? * ; * ? ? ? note , however , that the stacked spectrum is dominated by a few brightest sources , with 60% of the photons coming from the two brightest objects . the quoted error only includes the statistical error of photon counts . the sampling error is estimated by a bootstrapping method , where the 90% error range was determined by 500 bootstrap runs of a photon - count weighted mean best - fit @xmath37 values . the results were ( 90% bootstrap errors ) @xmath49 ( @xmath50 with the rlq removed ) . the optical ( rest - frame ultraviolet ) to x - ray flux ratio of a qso is customarily expressed in terms of the effective index @xmath1 between 2 kev and 2500 in the qsos rest frame . upon calculating @xmath1 , we assumed @xmath51 for all , which is the average qso spectral index . the result of the spectral analysis of all but one is consistent with the canonical spectral index of @xmath52 1.9 - 2.0 . the rlq pc 1035 + 4747 has @xmath53 and using @xmath54 for k - correction gives @xmath55 . other than this one , the main source of errors in @xmath1 is the x - ray flux . even for the source with the smallest source count ( pc 1035 + 4747 ) the 1@xmath29 error on @xmath1 is @xmath36 . a decrease of @xmath37 by 0.2 leads to an decrease of @xmath1 by @xmath56 at @xmath0 . using @xmath43 instead of @xmath57 , @xmath1 increases by 0.03 . this program is mainly focused on the systematic difference in @xmath1 between @xmath0 and @xmath2 . because @xmath1 values of qsos show a large scatter and we only sample a small number of qsos in the redshift - luminosity regimes of our interest , the sampling error is the dominant effect in the error budget of the _ mean value _ ( @xmath58 ) of qsos , which can be estimated by @xmath59 , where @xmath29 is the standard deviation of the @xmath1 distribution of @xmath60 qsos . note that the @xmath59 estimation of the _ standard deviation of the mean _ is also valid for small @xmath60 . it is well known that this estimator gives the exact confidence range of gaussian 1@xmath29 when the parent @xmath1 distribution is a gaussian and it is widely used in more general cases . the @xmath1 distributions in ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) are well characterized by a gaussian , which justifies the use of this estimator in our analysis . for our sample , we obtain @xmath61 ( @xmath62 , for the 5 rqqs only ) . our results are compared with those of @xmath22 qsos in table [ tab : aox ] . ideally , we would like to compare with @xmath2 qsos in the same luminosity range ( -27.0@xmath63 -26.3 ) . , who used @xmath64 , @xmath65 , @xmath66 with @xmath43 , corresponds to @xmath67 and -26.55 for our default cosmology and @xmath28 at @xmath68 and 5 respectively . ] unfortunately all but a few of the @xmath2 qsos observed previously with x - rays found in literature @xcite are more luminous ( @xmath69 ) than those in our sample . keeping this limitation in mind , we compare our result with the mean @xmath1 values in tables 3 and a1 in @xcite and table 3 in @xcite , who used the same methods and assumptions as our work in deriving fluxes and @xmath1 values . their table a1 includes recalculated @xmath1 values of @xmath2 qsos previously observed by _ chandra _ ( including those in @xcite ) using the same method . their recalculations also took the time - dependent low - energy degradation of acis - s quantum efficiency into account . balqsos sdss 1129 - 0142 and sdss 1605 - 0122 @xcite have been excluded from the following analysis , because balqso are known to be x - ray weak due to a heavy absorption . furthermore , we have derived x - ray fluxes of further three @xmath2 qsos from public archival _ acis - s data and calculated their @xmath1 values in the same way , as summarized in table [ tab : arch ] . the 1450 magnitudes of these three have been obtained from ssg95 ( pc 0910 + 5625 ) or the spectra from the sdss dr3 database after corrections for galactic extinction ( the others ) . these three have been included in our statistical analysis . the combined @xmath2 sample is somewhat heterogeneous , as it consists of data obtained by a variety of programs , each with different interests and strategies . in our statistical analysis , we have divided the sample into five groups ( group a - e , with group a being our @xmath0 sample ) as shown in table [ tab : aox ] . figure [ fig : aox](b ) shows the scatter diagram of the combined sample in the @xmath13@xmath42 plane with symbols showing the group membership . for each group , we have calculated the mean value @xmath58 and the standard deviation of the mean . the combined sample includes six upper limit @xmath1 values ( no x - ray detection ) and we have used the kaplan - meier estimator to calculate the mean and standard deviation of the mean using the asurv software@xcite , when necessary . figure [ fig : aox](a ) and ( c ) plots @xmath58 of each group against @xmath13 and @xmath42 respectively . figure [ fig : aox](b ) shows a rather incoherent scatter , reflecting the interests of individual original observing programs . as a result , @xmath13 and @xmath42 do not show strong monotonic @xmath13 versus luminosity correlation , unlike flux - limited samples . this is fortunate , because if such a correlations existed , it would be difficult to separate he redshift and luminosity dependences of @xmath1 . figs . [ fig : aox](a ) and ( c ) seems to support the conclusion by @xcite that the @xmath1 is anti - correlated with ultraviolet luminosity , while the correlation with @xmath13 is weak , if any . we made a linear regression analysis of the dependent parameter @xmath1 against the two independent parameters , namely @xmath13 and @xmath42 . we also made separate one - independent parameter regressions for @xmath13 or @xmath42 versus the dependent parameter @xmath1 . the results the analysis of the entire sample of 64 qsos using the em method available in asurv @xcite are shown below : @xmath70 the coefficients for the regression analysis with the buckley - james ( b - j ) and schmitt s methods ( if applicable ) that are included in asurv are consistent with those from the em method shown above . from either of eqs . [ eq : reg1 ] & [ eq : reg3 ] , we find a 3.5@xmath29 dependence of @xmath1 to @xmath42 . the slope coefficeint can be translated into @xmath71 with @xmath72 ( from eq . [ eq : reg3 ] ) , that is consistent with the recent extensive analysis including lower redshift qsos by @xcite . our primary goal of this study is to investigate the redshift dependence of @xmath1 between @xmath0 and @xmath22 to constrain the difference of number density behaviors of optically and x - ray selected qsos at @xmath73 . in order to achieve this goal , we make more careful comparison of groups a ( our sample ) and d , which are well - separated in the redshift range ( @xmath74 of 2.93 and 4.79 respectively ) and occupy approximately the same luminosity range ( fig . [ fig : aox](b ) ) . since the difference in the mean @xmath42 values for these two groups are small ( @xmath75 ) , the systematic difference of @xmath1 between these groups due to optical luminosity dependence is negligible ( @xmath76 in @xmath77 ) . the observed @xmath58 values of these groups are essentially the same . we have made monte - carlo simulations to find the probability distribution of @xmath78 , which is the _ difference _ between groups d and a to constrain the systematic change in @xmath1 with redshift at @xmath6 at this luminosity range . in order to make a parent distribution of @xmath1 from which the simulations draw objects , we used the all qsos in the current sample except for upper limits . also @xmath1 of qsos in each group have been shifted so that all groups have the same mean value . the standard deviation of these 58 @xmath1 values is consistent with those of groups a and d respectively . in each run , we randomly took 6 qsos to represent group a and 11 to represent d respectively . the mean of each of the 6 and 11 random @xmath1 values has been calculated and the distribution of the difference of these means ( @xmath78 ) for 2000 simulations have been investigated . as a result , the range where 90% of the simulations fall in were @xmath79 . this limits the systematic difference in @xmath80 between @xmath0 ( group a ) and @xmath81 ( group d ) of @xmath82 . for the slope of the luminosity function ( lf ) of @xmath83 , this corresponds to a number density difference of @xmath84 , where @xmath85 and @xmath86 are the number densities of qsos above optical luminosity and x - ray luminosities respectively . figure [ fig : ndens ] shows the evolution of comoving number density of soft x - ray selected qsos ( sxlf3 ) as a function of redshift plotted with those of optical qsos from ssg95 and @xcite at @xmath6 ( their original values are converted to our default cosmology and @xmath87 ) . the luminosity limit for the soft x - ray qsos is set at @xmath88 , where @xmath89 is the observed frame 0.5 - 2 kev in @xmath90 ) corresponding to approximately the @xmath91 kev luminosity at the rest frame . note that this is a rest - frame 0.5 - 2 kev luminosity , under the assumtion of @xmath51 power - law spectrum , which is representative of x - ray selected type 1 agns and sxlf3 treats the luminosities as such . we also show the results from @xcite s sample for the same observed 0.5 - 2 kev luminosity cut . this luminosity was selected such that the space density became equal to that of optical qsos at @xmath92 in thick lines and filled symbols . for reference , we show the number density curves from sxlf3 and @xcite ( open symbols with thin lines ) , in lower luminosity ranges ( see labels ) , which are representative of the respective samples . these show declines in @xmath19 . the large error bars in the @xmath88 data at @xmath6 show that the qsos in this regime is still underrepresented by current x - ray surveys . the limits of the number density curve of x - ray selected sample are shown in the shaded area in fig . [ fig : ndens ] , corresponding to @xmath93 . a limitation of the above investigation is the uncertainties in the effects of the sample selection and variability . the x - ray and optical luminosities have been measured in different epochs . thus the variability of agns have a net effect of increasing the variance of the @xmath1 distribution . our underlying assumption is that the variability does not cause a net systematic difference in its effect on the mean @xmath1 between sample a and sample d , separeted in redshift , but not in luminosity . both are optically selected samples and @xmath58 should be biased towards larger ( more optically luminous ) values than the `` true '' @xmath58 ( i.e. @xmath1 of time - averaged mean optical and x - ray luminosities ) , because optical selection is more likely to pick up the agn when it is more optically - bright , while the x - ray followup of the same object typically gives average x - ray luminosity of the source . as long as both are selected in the optical and followed up by x - ray , that the slope of the lfs are the same at both redshifts , and that there is no systematic difference in the variability amplitudes of agns with redshift , the effect of this `` variability '' bias should be the equal between sample a and sample d. thus we do not expect that the variabilty bias plays a major role in our analysis on the redshift dependence of @xmath58 . from a combination of our sample at @xmath0 and @xmath2 qsos observed by _ chandra _ , we have confirmed the apparent dependence of @xmath1 on optical luminosity . this result , however , should be treated with caution , because our sample is optically selected and subject to the variability effect of preferentially picking up optically brighter phase as described above . an x - ray selected sample covering a much larger regime in @xmath13-@xmath94 space rather showed @xmath95 @xcite this is , however subject to a similar effect working in the opposite sense . at this limited regime , on the other hand , there is a hint that our presently determined dependence might reflect the true behavior of the shift of @xmath1 with qso power . expressing the lf as @xmath96 at the luminosities of interest , the soft x - ray lf has @xmath97 and ( 90% error ) in @xmath98 ( from the same sample as sxlf3 ) . this can be compared with the optical lf of @xmath99 ( 1@xmath29 error ) by @xcite or @xmath100 by @xcite . the trend that the optical lf has a flatter slope than the soft x - ray counterpart is consistent with the relation @xmath101 within errors . this is also consistent with the comparison between x - ray and optically - selected agn luminosity functions by @xcite ( see their fig . 20 ) , where a conversion of their hard x - ray lf to the optical band assuming @xmath102 gave a good match to an observed optical qso lf at high luminosities . however , our most recent comparison of the sxlf @xcite ( high luminosity end ) and optical qso lf by @xcite at @xmath103 is more consistent with @xmath95 , thus a more study is needed to investigate the relationship between direct comparison of @xmath89 and @xmath104 and the conversion between the x - ray and optically selected qso lfs . we have made _ chandra _ acis - s observations of six qsos at @xmath0 , which marks the peak of luminous qso number density . these observatios fill a redshift gap in the x - ray coverage of luminous qsos . we found an average photon index of @xmath105 from the stacked spectrum and we also found @xmath106 . the @xmath58 value is essentially the same as those at @xmath2 in the similar uv luminosity range and thus we have found no systematic shift of x - ray to uv luminosity ratios with redshuft above @xmath68 . the density curves of @xmath107 optically selected qsos and @xmath108 soft x - ray selected qsos , giving the same densities at @xmath92 , are statistically consistent with each other within our limit of the systematic @xmath58 shift at @xmath12 . we note that this regime is still underrepresented by x - ray surveys . large - area moderarely - deep x - ray surveys are needed to trace the rise of number density of the most luminous qsos at @xmath12 in x - rays . this work has been supported by chandra general observer award go3 - 4153x , nasa ltsa grant nag5 - 10875 ( tm ) and nsf grant ast03 - 07582 ( dps ) . we thank john silverman for calculating the space density curves for our luminosity cuts and cosmological parameter choice . barger , a. j. , cowie , l. l. , mushotzky , r. f. , yang , y. , wang , w .- h . , steffen , a. t. , & capak , p. 2005 , , 129 , 578 bechtold , j. , et al . 2003 , , 588 , 119 becker , r. h. , white , r. l. , & helfand , d. j. 1995 , , 450 , 559 boyle , b. j. , shanks , t. , & peterson , b. a. 1988 , , 235 , 935 cash , w. 1979 , , 228 , 939 condon , j. j. , cotton , w. d. , greisen , e. w. , yin , q. f. , perley , r. a. , taylor , g. b. , & broderick , j. j. 1998 , , 115 , 1693 cowie , l. l. , barger , a. j. , bautz , m. w. , brandt , w. n. , & garmire , g. p. 2003 , , 584 , l57 croom , s. m. , smith , r. j. , boyle , b. j. , shanks , t. , miller , l. , outram , p. j. , & loaring , n. s. 2004 , , 349 , 1397 dickey , j. m. & lockman , f. j. 1990 , , 28 , 215 isobe , t. , feigelson , e. d. , & nelson , p. i. 1986 , , 306 , 490 fan , x. , et al . 2001 , , 121 , 54 feigelson , e. d. & nelson , p. i. 1985 , , 293 , 192 fiore , f. , et al . 2003 , , 409 , 79 garmire , g. p. , bautz , m. w. , ford , p. g. , nousek , j. a. , & ricker , g. r. 2003 , , 4851 , 28 hasinger , g. 2005 in growing black holes " eds a. merloni , s. nayakshin and r. sunyaev ( heidelberg : springer ) , p418 hasinger , g. , miyaji , t. , schmidt , m. 2005 , , 441 , 417 miyaji , t. , hasinger , g. & schmidt , m. 2000 , , 353 , 25 ( sxlf1 ) osmer , p. s. 1982 , , 253 , 28 schneider , d. p. , schmidt , m. , & gunn , j. e. 1994 , , 107 , 1245 schneider , d. p. , et al . 2001 , , 121 , 1232 schmidt , m. , schneider , d. p. , & gunn , j. e. 1995 , , 110 , 68 ( ssg95 ) shaver , p. a. , wall , j. v. , kellermann , k. i. , jackson , c. a. , & hawkins , m. r. s. 1996 , , 384 , 439 silverman , j. d. , et al . 2005 , , 624 , 630 strateva , i. v. , brandt , w. n. , schneider , d. p. , vanden berk , d. g. , & vignali , c. 2005 , , 130 , 387 ueda , y. , akiyama , m. , ohta , k. , & miyaji , t. 2003 , , 598 , 886 vignali , c. , brandt , w. n. , fan , x. , gunn , j. e. , kaspi , s. , schneider , d. p. , & strauss , m. a. 2001 , , 122 , 2143 vignali , c. , brandt , w. n. , schneider , d. p. , garmire , g. p. , & kaspi , s. 2003a , , 125 , 418 vignali , c. , brandt , w. n. , & schneider , d. p. 2003b , , 125 , 433 vignali , c. , et al . 2003c , , 125 , 2876 vignali , c. , brandt , w. n. , schneider , d. p. , & kaspi , s. 2005 , , 129 , 2519 wall , j. v. , jackson , c. a. , shaver , p. a. , hook , i. m. , & kellermann , k. i. 2005 , , 434 , 133 warren , s. j. , hewett , p. c. , & osmer , p. s. 1994 , , 421 , 412 wilkes , b. j. 2000 , in `` allen s astrophysical quantities , fourth edition '' ed . a. n. cox ( new york : springer ) , chap . 24 york , d. g. , et al . 2000 , , 120 , 1579 yuan , w. , brinkmann , w. , siebert , j. , & voges , w. 1998 , , 330 , 108 yuan , w. , siebert , j. , & brinkmann , w. 1998 , , 334 , 498 pc 0041 + 0215 & 4150/2003 sep 01 & 9.2 & 2.93 & 19.5 & @xmath109 + pc 0947 + 5628 & 4151/2003 jan 25 & 9.0 & 2.91 & 19.5 & @xmath110 + pc 1000 + 4751 & 4152/2002 dec 18 & 13.9 & 2.95 & 20.0 & @xmath111 + pc 1015 + 4752 & 4153/2003 jan 01 & 8.1 & 2.92 & 19.4 & @xmath112 + pc 1035 + 4747 & 4154/2003 mar 16 & 10.0 & 2.96 & 19.6 & 2.3 + pc 1447 + 4750 & 4155/2003 jul 28 & 7.0 & 2.93 & 19.3 & @xmath112 + cccccccccc pc 0041 + 0215&2.93 & 2.8&2.4(1.9;2.9 ) & 1.4(1.0;1.8)&0&36&45.0&@xmath113&@xmath114 + pc 0947 + 5628&2.91 & 1.0&1.5(0.7;2.3 ) & .31(.17;.52)&0&10&44.4&@xmath113&@xmath115 + pc 1000 + 4751&2.95 & 0.9&1.5(1.2;1.9 ) & 1.2(1.0;1.5)&0&60&45.0&@xmath116&@xmath117 + pc 1015 + 4752&2.92 & 1.0&2.4(1.8;3.0 ) & .94(.65;1.2)&0&24&44.9&@xmath118&@xmath119 + pc 1035 + 4747&2.96 & 1.3&0.6(@xmath120;1.4)&.17(.08;.34)&0 & 9&44.3&@xmath121&@xmath122 + pc 1447 + 4750&2.93 & 2.0&2.0(1.4;2.6 ) & 1.0(.64;1.4)&0&21&44.9&@xmath123&@xmath124 + @xmath1256 qsos@xmath126 & 2.93 & ... & 1.9(1.7;2.1 ) & ... & @xmath127 & & 160 & ... + ccrcccccrcc pc 0910 + 5625 & 4.04 & 4821/2004 mar 28 & 23 . & 2.9 & 1.7(1.0;2.7 ) & 20.7 & @xmath109&-26.2&-1.67 + sdss j235718.36 + 004350.3 & 4.36 & 4827/2003 nov 26 & 12 . & 3.3 & 5.0(3.3;7.2 ) & 20.2 & @xmath111&-26.9&-1.56 + sdss j144428.67 - 012344.1 & 4.17 & 4826/2004 jan 8 & 10 . & 4.0 & 1.8(0.7;3.4 ) & 19.8 & @xmath111&-27.1&-1.79 + cccccrr a & 2.9,3.0 & 2.93 & @xmath128 & @xmath129 & 6 & @xmath130.046 + b & 3.5,4.6 & 4.29 & @xmath131 & @xmath132&15 & @xmath133.042 + c & 3.5,4.6 & 4.26 & @xmath134 & @xmath135&17 & @xmath136.035 + d & 4.0,5.3 & 4.79 & @xmath137 & @xmath138&11 & @xmath139.031 + e & 4.6,6.4 & 5.13 & @xmath140 & @xmath141&14 & @xmath142.047 +

we report the results of our _ chandra _ observations of six qsos at @xmath0 from the palomer transit grism survey . our primary goal is to investigate the possible systematic change of @xmath1 between @xmath2 and @xmath0 , between which a rapid rise of luminous qso number density with cosmic time is observed . the summed spectrum showed a power - law spectrum with photon index of @xmath3 , which is similar to other unabsorbed agns . combining our @xmath0 qsos with x - ray observations of qsos at @xmath2 from literaure / archive , we find a correlation of @xmath1 with optical luminosity . this is consistent with the fact that the luminosity function slope of the luminous end of the x - ray selected qsos is steeper than that of optically - selected qsos . we discuss an upper limit to the redshift dependence of @xmath1 using a monte - carlo simulation . within the current statistical errors including the derived limits on the redshift dependence of @xmath1 , we found that the behaviors of the x - ray and optically - selected qso number densities are consistent with each other .

nowadays , in condensed matter physics and semiconductor microelectronics , two - dimensional ( 2d ) electron system is one of the main objects of detailed study . such a system is formed by , e.g. , surface - state electrons or electrons in semiconductor heterostructures . phenomenon that is observed in such systems and makes them of great interest , especially in context of spintronic applications , is spin - orbit interaction ( soi ) . this interaction arise from the structure inversion asymmetry of potential confining the electron system in directions perpendicular to the confinement plane ( the rashba spin - orbit interaction@xcite ) and the bulk inversion asymmetry that is present in semiconductor heterostructures based on materials with a zinc - blende structure ( the dresselhaus spin - orbit interaction@xcite ) . the dresselhaus interaction depends on semiconductor material and growth geometry , whereas the interaction strength of the rashba soi can be tuned via an externally applied electric field perpendicular to the confinement plane.@xcite as a result , one can controllably manipulate the spin in devices without recourse to an external magnetic field.@xcite . in order to efficiently exploit the mentioned phenomenon , a theoretical study of dynamics of electrons and holes in the 2d spin - orbit coupled electron systems is needed . the most discussed and studied processes concerning this problem are spin relaxation and spin dephasing.@xcite however , to our knowledge , such crucial quasiparticle property as the lifetime caused by inelastic electron - electron scattering remains still insufficiently studied . to all appearance the first attempt to analyze what effect the soi has on the quasiparticle lifetime has been made in ref . . in the work cited , a particular case of the 2d electron gas ( 2deg ) with the rashba soi was considered at the limit of @xmath1 , where @xmath2 is the fermi energy and @xmath3 with @xmath4 and @xmath5 being the interaction strength and the effective electron mass , respectively ( unless stated otherwise , atomic units are used throughout , i.e. , @xmath6 . ) . within the @xmath0 approximation , it has been shown that in a small vicinity of @xmath2 a modification of the lifetime due to the soi is insignificant and does not depend on the subband index of the spin - orbit split band . to go beyond the limits of ref . , in ref . the inelastic lifetime ( decay rate ) of quasiparticles in the 2deg with the rashba soi has been studied within a wide energy region . for material parameters typical for in@xmath7ga@xmath8as 2degs , it has been revealed that modifications induced by the soi and the dependence on the subband index become noticeable , when the decay channel due to plasmon emission appears . the first joint theoretical and experimental investigation of hole lifetimes in a 2d spin - orbit coupled electron system has been done in ref . . in addition to a demonstration of the weak influence of the soi on hole lifetimes by the case of the au(111 ) surface state , a hypothetical system , where the soi can have a profound effect , has been considered . in this work , we generalize the results on effect of the soi on the quasiparticle lifetime . within the @xmath0 approach with the screened interaction @xmath9 evaluated in the random phase approximation ( rpa ) , we study the inelastic decay rate of quasiparticles in a 2deg with the rashba and dresselhaus interactions linear in @xmath10magnitude of the electron 2d momentum @xmath11 . in our @xmath0-calculations , material parameters suitable for inas quantum wells are taken . we compare the inelastic decay rates calculated at different ratios between the interaction strengths of the mentioned spin - orbit interactions . we show that on the energy scale , for the taken material parameters , the main visible effect induced by the soi is modifications of the plasmon - emission decay channel via the extension of the landau damping region . we also consider a hypothetical small - density case , when in the 2d spin - orbit coupled electron system the fermi level is close to the band energy at @xmath12 . for such a system , we predict strong subband - index dependence and anisotropy of the inelastic decay rate for electrons and appearance of a plasmon decay channel for holes . and @xmath13 determining the spin - quantization axis with polar angles @xmath14 and @xmath15 and the rotation axis , respectively . ] we consider a 2deg described by the hamiltonian @xmath16 with @xmath17 and the spin - orbit contribution @xmath18 that includes both rashba and dresselhaus terms . the latter is written with the assumption that a quantum well grown in [ 001 ] direction is considered . in eq . ( [ hamiltonian ] ) , @xmath19 are the electron momenta along the [ 100 ] and [ 010 ] cubic axes of the crystal , respectively , @xmath20 are the pauli matrices , @xmath5 is the effective electron mass , @xmath4 and @xmath21 are the interaction strengths for the rashba and dresselhaus spin - orbit interactions . to bring the hamiltonian to a diagonal form , we perform the rotation in spin space generated by @xmath22 $ ] dependent on the momentum @xmath11 . the rotation is performed with the angle @xmath14 around the axis determined by @xmath23 . a positional relationship of the axis @xmath23 and the spin - quantization axis @xmath24 is shown in fig . [ fig1 ] . we suppose that we deal with the in - plane spin polarization , i.e. , @xmath25 . in the new , unitary transformed , spin basis the spin - orbit contribution has the form@xcite @xmath26\sigma_{z},\end{aligned}\ ] ] where the angle @xmath15 is related to the polar angle @xmath27 of the momentum @xmath11 as @xmath28 due to the diagonal form of @xmath29 , the energy bands are simply given by@xcite @xmath30\ ] ] and correspond to the wave functions @xmath31 with the subband index @xmath32 , where @xmath33 are the spin components in the new spin basis . this means that for the initial , untransformed , hamiltonian we have the following eigenstates @xmath34 . the spin orientation in @xmath11 space reads as ( see fig . [ fig2 ] ) @xmath35 note that the case with @xmath36 and @xmath37 ( pure rashba ) is characterized by the angle @xmath38 , whereas in the situation with @xmath39 and @xmath40 ( pure dresselhaus ) one has @xmath41 . in the special case of @xmath42 , the angle @xmath43 does not depend on @xmath27 . the inelastic decay rate ( inverse lifetime @xmath44 caused by inelastic electron - electron scattering ) is determined by the imaginary part of the matrix elements of the quasiparticle self - energy @xmath45 at the energy @xmath46 as @xmath47 . at the hartree - fock ( hf ) mean - field level , these elements are totally real and have the form @xmath48 where @xmath49 is the bare coulomb interaction with @xmath50 being the static dielectric constant . the factors @xmath51/2 $ ] come from @xmath52 and @xmath53 is the fermi factor . such a form ( [ hf_sigma ] ) is similar to the exchange contribution to the single - particle energies considered in refs . in the pure rashba case . however , in order to examine quasiparticle lifetimes , one has to go beyond the hf approximation . the simplest variant is the @xmath0 approximation ( for details about the approximation , we refer the reader to refs . and ) . within such an approximation , we arrive at the following expression for the imaginary part of the mentioned matrix elements @xmath54 when @xmath55 , and @xmath56\\ & \times&\mathrm{im}w^0(\mathbf{k}-\mathbf{q},\omega - e_{\mathbf{q}s'})\theta(\omega - e_{\mathbf{q}s'}),\nonumber\end{aligned}\ ] ] when @xmath57 . in these equations , @xmath58 is the step function and the screened interaction @xmath59^{-1}\ ] ] is defined by the rpa irreducible polarizability ( see also refs . and , where the retarded part of @xmath60 was examined ) @xmath61 using the example of an inas quantum well , we take the effective mass @xmath62 ( see , e.g. , ref . ) and the static dielectric constant @xmath63 ( see , e.g. , ref . ) . the interaction strength of the dresselhaus soi is chosen to be @xmath64 evm to simulate a quite narrow quantum well.@xcite at the ratio @xmath65 ( see , e.g. , ref . ) , we have the rashba interaction strength @xmath66 evm . the electron density is put at @xmath67 @xmath68 that corresponds to the fermi energy @xmath69 mev . [ fig3 ] shows our results on the inelastic decay rate @xmath70 obtained with the material parameters listed above . two main points caused by the spin - orbit splitting of the band make the considered 2deg different from that without the soi . these are an angle - dependent relative shift of @xmath71 and @xmath72 on the momentum scale and some smoothing of sharp forms of the peak caused by opening of the plasmon decay channel . the former reflects the fact that subbands of the split band reach the same energy at different momenta , while the latter originates from the extension of the landau damping region ( the region where plasmons decay into single - particle excitations@xcite ) due to appearance of inter - subband transitions ( for a detailed discussion of the screening properties of the 2deg with the soi we refer the reader to refs . and ) this extension varying with the polar angle @xmath27 leads to a nonzero plasmon linewidth , when the plasmon spectrum enters into the soi - induced damping region . in order to show what effect the soi has on the inelastic decay rate for different subbands , in the inset of fig . [ fig3 ] , by setting up a correspondence between @xmath70 and @xmath73 via the momentum @xmath11 , we plot the decay rate as a function of energy . on first glance , it may seem that we have an ordinary energy dependence of the decay rate as in a 2deg without the soi : the quadratic behavior with the logarithmic enhancement in the vicinity of the fermi energy with @xmath74 at @xmath2 and the jump above the fermi energy , which is caused by opening the plasmon decay channel for excited electrons.@xcite however , on examining the energy dependence of @xmath75 in detail , we can say that due to the finite plasmon linewidth the plasmon decay channel manifests itself at lower energies , when it occurs in a 2deg without the soi . the same reason leads to reduction in the jump . also , we can reveal distinctions between @xmath71 and @xmath72 , which become noticeable , when the plasmon - emission decay channel appears , and increase upon moving from @xmath76 to @xmath77 . an analysis of the inelastic mean - free path ( imfp ) @xmath78 as a function of energy has shown that , as a consequence of the distinctions between @xmath71 and @xmath72 , the imfp of electrons can vary with the subband index . for example , at @xmath79 for electrons this variation can reach , e.g. , @xmath80% . the obtained results can be understood by inspecting constant - energy contours shown in fig . [ fig2 ] with mental drawing of possible transitions selected by the factors @xmath81 of eqs . ( [ me_sigma_im_below_ef ] ) and ( [ me_sigma_im_above_ef ] ) at a given @xmath27 ( see also ref . ) . actually , for each subband ( @xmath82 ) one has a set of intra- and inter - subband transition momenta as arguments of @xmath83 . for the chosen material parameters and for @xmath76 , these momenta do not vary considerably with the subband index @xmath84 . for @xmath79 differences in both intra- and inter - subband transitions for @xmath85 and @xmath86 become already sensible for values of @xmath83 , especially in the vicinity of plasmon peaks of the latter . now , remaining @xmath5 , @xmath50 , @xmath87 , and @xmath2 unchanged , we consider the case of @xmath88 [ @xmath89 evm ] and the pure dresselhaus ( rashba ) case [ @xmath90 evm and @xmath91 . the case of equal interaction strengths , when the rashba and dresselhaus interactions can cancel each other , is distinguished by various significant effects reported in the literature ( see , e.g. , refs . ) . in this case , one has the 2d electron system with two uncoupled spin components ( see fig . [ fig2 ] ) , each of which demonstrates the properties peculiar to a 2deg without the soi.@xcite our results@xcite on the inelastic decay rate at @xmath88 are shown in fig . the sharp edges of the plasmon contribution are evidence of the fact that there is no modifications of the landau damping region induced by the soi . as is seen from the inset of the figure , due to the shifting property @xmath92 , where @xmath93 , the @xmath71 and @xmath72 curves coincide and have the form of that in a 2deg without the soi . in the pure rashba or pure dresselhaus cases ( see fig . [ fig4 ] ) , the resulting @xmath75 does not tell the difference between spin orientations in the momentum plane , which correspond to the rashba or dresselhaus soi ( see fig . [ fig2 ] ) . as well as before , we have the relative shift ( but angle - independent ) on the momentum scale and main modifications induced by the soi in the energy region , where a quasiparticle can decay into plasmons . all the considered cases meet the condition of @xmath94 , where @xmath95 is the measure of influence of the soi on the band structure . however , as is partly discussed in ref . , in two - dimensional electron systems with much greater @xmath96 as compared to @xmath2 the inelastic decay rate can substantially differ from that in the 2deg without the soi . a striking example of such a system is that formed by surface - state electrons in ordered surface alloys,@xcite which are very promising materials for spintronics applications . in order to predict how the inelastic decay rate can behave in a system , where @xmath97 , we consider the hypothetical case with the unchanged @xmath62 , @xmath63 , and @xmath64 evm , but with @xmath98 evm ( @xmath99 ) and @xmath100 @xmath68 , which give @xmath101 mev and @xmath102 mev . the obtained results are presented in fig . [ fig5 ] . the main feature we would like to note first is that for holes the decay rate @xmath72 as a function of @xmath10 has an `` outgrowth '' at @xmath103 , where @xmath104^{1/2}$ ] the momentum , at which @xmath105 has a minimum . a close analysis of the imaginary part of the screened interaction @xmath9 and the region of integration in eq . ( [ me_sigma_im_below_ef ] ) has shown that the outgrowth is caused by opening of the plasmon decay channel for transitions between the @xmath86 and @xmath106 subbands . it is important that in a 2deg without the soi such a channel is impossible for holes.@xcite as is evident from the figure , in this case on the energy scale we have a strong anisotropy of the inelastic decay rate . also the presented curves clearly demonstrate that the latter depends strongly on the subband index @xmath84 of the spin - orbit split band . keeping in mind that the index @xmath84 distinguishes spin components , we can say that the subband - index dependence reflects a spin asymmetry of the inelastic decay rate @xmath107 for a given direction . the asymmetry shows its worth most brightly in the @xmath79 direction . in fact , in that very direction there are significant distinctions in @xmath72 and @xmath71 as functions of the exciting energy and , as a consequence , in the corresponding imfp for electrons . for instance , the ratio @xmath107 is about 3 at @xmath108 mev and about 2 at @xmath109 mev . at further increasing of energy , the ratio continues to decrease . note that the imfp spin asymmetry makes a basis of the spin filter effect observed in hot electron transport through a ferromagnetic ( see , e.g. , refs . and ) . in the considered case of the quantum well , the spin asymmetry is not such big as in ferromagnetics ( see , e.g. , ref . ) , but , as distinct from the latters , values of the imfp spin symmetry depend strongly on direction and can be tuned by external electric field . in conclusion , we have presented a study of the inelastic decay rate of quasiparticles in a two - dimensional electron gas with the @xmath10-linear spin - orbit interaction that includes both rashba ( interaction strength @xmath4 ) and dresselhaus ( interaction strength @xmath21 ) contributions . in this study , the electron gas is characterized by material parameters suitable for [ 001]-grown inas quantum wells . we have considered the cases of @xmath110 , @xmath42 , @xmath40 ( @xmath111 ) , and @xmath37 ( @xmath112 ) . the cases meet the condition of @xmath94 , where @xmath95 is the measure of influence of the spin - orbit interaction on the band structure . as compared to a two - dimensional electron gas without the spin - orbit interaction , we have revealed a relative shift of the inelastic decay rates for different subbands of the spin - orbit split band on the momentum scale . also , except for the case of equal interaction strengths , we have found a some smoothing of sharp forms of the peak concerned with opening of the plasmon decay channel for electrons . we have shown that , on the energy scale , in this very region distinctions between the decay rates for different subbands become noticeable . these distinctions depend on the polar angle @xmath27 and cause the inelastic mean free path to be angle- and subband - dependent . as to the case of @xmath42 , due to the shifting property , the decay rate as a function of energy has the form of that in a two - dimensional electron gas without the spin - orbit interaction . in order to predict how the inelastic decay rate can behave in a system , where @xmath97 , we have considered the hypothetical case of small electron density . we have revealed that in such a system the decay rate demonstrates strong anisotropy and subband dependence within all the considered interval of momenta and exciting energies . since the subband dependence can be interpreted as a spin asymmetry of the decay rate in a given direction of @xmath11 , one can expect the spin - filter effect driven by externally applied electric field . also , we have found that in the system with @xmath97 holes can decay into plasmons , what is impossible in a two - dimensional electron gas without the spin - orbit interaction . we acknowledge partial support from the university of the basque country ( grant no . gic07it36607 ) and the spanish ministerio de ciencia y tecnologa ( grant no . fis2007 - 66711-c02 - 01 ) . calculations were partly performed on skif - cyberia supercomputer of tomsk state university . rashba , sov . solid state * 2 * , 1109 ( 1960 ) ; y.a . bychkov and e.i . rashba , jetp lett . * 39 * , 78 ( 1984 ) ; j. phys . c * 17 * , 6039 ( 1984 ) . g. dresselhaus , phys . rev . * 100 * , 580 ( 1955 ) . dyakonov and v.y . kacharovskii , sov . phys . semicond . * 20 * , 110 ( 1986 ) . m. studer , g. salis , k. ensslin , d.c . driscoll , and a.c . gossard , phys . lett . * 103 * , 027201 ( 2009 ) . s. datta and b. das , appl . . lett . * 56 * , 665 ( 1990 ) . j. nitta , t. akazaki , h. takayanagi , and t. enoki , phys . * 78 * , 1335 ( 1997 ) . i. uti , j. fabian , and s. das sarma , rev . mod . phys . * 76 * , 323 ( 2004 ) . d.s . saraga and d. loss , phys . b * 72 * , 195319 ( 2005 ) . i.a . nechaev and e.v . chulkov , phys . solid state * 51 * , 1772 ( 2009 ) . nechaev , m.f . jensen , e.d.l . rienks , v.m . silkin , p.m. echenique , e.v . chulkov , and ph . hofmann , phys . b * 80 * , 113402 ( 2009 ) . in the case of @xmath42 the hamiltonian @xmath113 in the new spin basis and the unitary transformation @xmath114 become exactly the same as @xmath115}}$ ] and @xmath116 of ref . , respectively . note that eq . ( [ energies ] ) can be cast into the form that is frequently used in the literature . actually , from @xmath117\right)^2 $ ] with the help of eq . ( [ phi_k_def ] ) we arrive at @xmath118 $ ] . the latter can be solved as @xmath119^{1/2}$ ] . however , in such an expression the subband index @xmath82 distinguishes the inner- and outer - branch and , e.g. , in the case of @xmath42 does not correspond to spin components . bernevig , j. orenstein , and s .- c . zhang , phys . lett . * 97 * , 236601 ( 2006 ) . g .- h . chen and m.e . raikh , phys . b * 60 * , 4826 ( 1999 ) . s. chesi and g.f . giuliani , phys . b * 75 * , 155305 ( 2007 ) . juri and p.i . tamborenea , phys . b * 77 * , 233310 ( 2008 ) . nechaev , i.yu . sklyadneva , v.m . silkin , p.m. echenique , and e.v . chulkov , phys . b * 78 * , 085113 ( 2008 ) . m. pletyukhov and v. gritsev , phys . b * 74 * , 045307 ( 2006 ) . badalyan , a. matos - abiague , g. vignale , and j. fabian , phys . b * 79 * , 205305 ( 2009 ) . w. knap , c. skierbiszewski , a. zduniak , e. litwin - staszewska , d. bertho , f. kobbi , j.l . robert , g.e . pikus , f.g . pikus , s.v . iordanskii , v. mosser , k. zekentes , yu . b. lyanda - geller , phys . b * 53 * , 3912 ( 1996 ) . o.g . lorimor and w.g . spitzer , j. appl . phys . * 36 * , 1841 ( 1965 ) . s. giglberger , l.e . golub , v.v . belkov , s.n . danilov , d. schuh , c. gerl , f. rohlfing , j. stahl , w. wegscheider , d. weiss , w. prettl , and s.d . ganichev , phys . b * 75 * , 035327 ( 2007 ) . h. bruus and k. flensberg , _ many - body quantum theory in condensed matter physics : an introduction _ ( oxford university press , oxford , 2004 ) . g. f. giuliani , g. vignale , _ quantum theory of the electron liquid _ ( cambridge university press , cambridge , 2005 ) . n.s . averkiev and l.e . golub , phys . b * 60 * , 15582 ( 1999 ) . j. schliemann , j.c . egues , and d. loss , phys . 90 * , 146801 ( 2003 ) . koralek , c.p . weber , j. orenstein , b.a . bernevig , s .- c . zhang , s. mack , d.d . awschalom , nature * 458 * , 610 ( 2009 ) . we have chosen the direction @xmath76 , in which we can reproduce the momentum dependence of the inelastic decay rate peculiar for a 2deg without the soi . for each spin components , such a dependence becomes isotropic if one shifts the origin in the momentum plane on the vector @xmath120 . ast , j. henk , a. ernst , l. moreschini , m.c . falub , d. pacil , p. bruno , k. kern , and m. grioni , phys . * 98 * , 186807 ( 2007 ) ; c.r . ast , d. pacil , l. moreschini , m.c . falub , m. papagno , k. kern , m. grioni , j. henk , a. ernst , s. ostanin , and p. bruno , phys . b * 77 * , 081407(r ) ( 2008 ) ; h. mirhosseini , j. henk , a. ernst , s. ostanin , c .- t . chiang , p. yu , a. winkelmann , and j. kirschner , phys . rev . b * 79 * , 245428 ( 2009 ) . v.p . zhukov and e.v . chulkov , phys . usp . * 52 * , 105 ( 2009 ) . i.a . nechaev and e.v . chulkov , phys . solid state * 51 * , 754 ( 2009 ) .

we present a study of the inelastic decay rate of quasiparticles in a two - dimensional electron gas with spin - orbit interaction . the study is done within the @xmath0 approximation . the spin - orbit interaction is taken in the most general form that includes both rashba and dresselhaus contributions linear in magnitude of the electron 2d momentum . spin - orbit interaction effect on the inelastic decay rate is examined at different parameters characterizing the electron gas and the spin - orbit interaction strength in it .

we are assisting at a booming expansion of nanoparticle research and technology . synthesis method especially make fast progresses@xcite . analysis methods , however , are not up to speed . a fundamental simple task as determining and controlling the size distribution of nanoparticles ( nps hereafter ) is currently a complex experimental work , involving electron microscopy and combined techniques . in this work we want to highlight the possibilities offered in this issue by a much less complex technique as powder diffraction . powder diffraction is a widespread technique with a great potential to meet the increasing demands of microstructural material characterization . the methods of powder diffraction data analysis have reached maturity for micrometer - sized polycrystalline materials . however , when the particle size falls much below 100 nm , specifically tuned methods of analysis are needed to extract meaningful information from powder diffraction patterns . in fact , nanoparticles ( nps hereafter ) present unique analytical challenges . in the most complex cases , non - crystallographic structures @xcite may occur . surface - related deformation fields @xcite are another challenge . in these extreme cases , the classical crystallographic formalism becomes quite useless . the debye scattering function@xcite ( that is , the direct evaluation of the np structure factor from the interatomic distances ) is the only choice in those cases . we are currently developing @xcite methods to increase the efficiency of such calculations and make them a practical tool . even for crystalline nps , however , the small size plays a decisive role . bragg peaks may be so much broadened that they can not be simply separated and many approximations , commonly accepted for micrometer size domains , fail . as we will show , also models specifically corrected for nps@xcite may fail for ultra - small nps ( say below 5 nm diameter , as it will be better specified ) . again for these ultra - small sizes the debye scattering function is the only choice for obtaining precise results , while the smaller number of atoms makes it extremely practical . the plan of the paper is the following . in sec . [ sec1 ] we discuss the shape - based method for calculating np powder patterns in relation to the surface structure and to its limits of validity at small sizes . application to full - pattern fit on a test - case ( 20-nm ceo@xmath1 ) is shown in sec . summary and conclusions are given in sec . scherrer s formula@xcite is the most known method for extracting size information from powder patterns ( namely , from the bragg peaks width ) . this is a simple method , but accurate only to the order of magnitude . however , since scherrer s work , line profile analysis has made enormous progress @xcite . theoretical progress on understanding the physical origin of peak broadening has been focused on the dislocation analysis , size broadening being considered as a side effect to be corrected for in order to determine the defect structure . nevertheless , today it is possible to determine the parameters of a ( log - normal ) size distribution of crystallites , together with information on type and concentration of dislocations . these methods are , however , complex and sophisticated , requiring a fairly high signal - to - noise ratio , low and flat background , a precise deconvolution of the instrumental broadening and especially well - isolated bragg peaks . full - pattern fitting methods ( _ cf . _ sec . [ sec2 ] ) are more direct and robust , especially when the target is the size analysis . firstly , they use all the experimental information , regardless of partial or total peak overlap , increasing redundancy and therefore precision and decreasing experimental requirement . furthermore , they allow the evaluation of a np - characteristic feature , namely the variation with size of the lattice parameter@xcite ( an effect that can be important below 20 nm ) . corrections for texture , microabsorption , anisotropic elastic peak shifts and instrumental broadening can also be implemented . an efficient and precise method to evaluate np diffraction patterns is needed to perform full - pattern fits . hereafter we discuss the shape - based method@xcite with a thorough analysis of its validity limits . we shortly recall some methods for the calculation of the powder diffraction intensity for a np with known periodic structure and definite size and shape . in the following the length of a vector @xmath2 will be denoted by @xmath3 . accordingly , @xmath4 will be the scattering vector of length @xmath5 , where @xmath6 is the scattering half - angle and @xmath7 the incident wavelength ; @xmath8 shall denote the scattering vector associated with a bragg peak , its length being @xmath9 . a np occupies a geometrical region of space @xmath10 . we recall @xcite the definition of a shape function @xmath11 , such that @xmath12 if @xmath13 lies inside @xmath10 , @xmath14 otherwise . we shall hereforth suppose that @xmath15 so that its fourier transform is real . however , defining the shape of a crystal means also to describe what happens to the atoms on the surface . these are increasingly important at very small sizes . in fact , there are different ways of interpreting the action of @xmath11 , the most meaningful ones being : * truncating sharply the scattering density ( the electron density for x - rays ) at the surface @xcite ; * selecting all whole unit cells whose origins are in @xmath10 and all whole atoms whose centres lie in the selected cells@xcite ; * selecting all whole atoms whose centres are in @xmath10 . useful illustrations are found in fig . 1 of ref . ( see figs . 1a , 1c and 1d , respectively for a , b , c ) . to evaluate the diffracted intensities , in cases b ) , c ) , one may utilize the debye function . in this way the chosen model is faithfully represented . it is possible , however , to proceed in a different way , that is , by the shape - function method . accordingly , we first evaluate the scattering amplitude @xmath16 . the explicit expressions@xcite are , for cases a , b , c : @xmath17 where @xmath18 is the reciprocal lattice ; @xmath19 is the fourier transform from the fourier amplitudes and @xmath20 from the related intensities , where @xmath21 is the unit cell volume . ] of @xmath11 , or @xmath22 and it satisfies @xmath23 because @xmath15 ; @xmath24 is the unit cell structure factor @xmath25 where the sum index @xmath26 runs on the atoms in the unit cell , which have form factors @xmath27 and position vectors ( relative to the cell origin ) @xmath28 ; @xmath29 is the same as the former but evaluated in @xmath4 ; and @xmath30 is the mixed expression @xmath31 it is evident that form a ) is simpler but by construction less reasonable - for electron and x - ray diffraction - than b ) and c ) . in fact , the sharp truncation of the electron density at the surface is unjustified . for neutron nuclear elastic scattering the atoms are point scatterers , therefore , construction a ) coincides with c ) . accordingly , in the neutron case , the atomic form factors are constant and @xmath32 . form b ) depends on an appropriate choice of the unit cell . clearly , it preserves the stoichiometric composition and symmetry . form c ) needs a careful implementation ( regarding the definition of @xmath10 ) to preserve stoichiometry , that is important for ionic compounds ; however , it is clearly more flexible . remark also that , in the case of monoatomic lattices , instead - as for simple - cubic , face - centered or body - centered cubic metals - construction b ) and c ) will be coincident and @xmath33 . squaring eqs . ( [ eq : ampla],[eq : amplb],[eq : amplc ] ) we obtain the intensities . supposing @xmath34 centrosymmetric and @xmath35 real , we have @xmath36 here , we have neglected cross - summations of the form @xmath37 where overbar stands for complex conjugate and , for x = a , b , c , respectively , it is @xmath38 , @xmath39 or @xmath40 . neglecting @xmath41 is , first of all , a question of convenience , because its evaluation - either analytical or numerical - is a nightmare . there are obvious reasons for neglecting @xmath41 for large particles . consider a spherical particle with cubic structure with lattice parameter @xmath42 and radius @xmath43 . @xmath19 is large only for @xmath44 , and decreases as @xmath45 for @xmath46 . as for any bragg peak @xmath8 it is @xmath47 , @xmath48 can be neglected . for smaller particles the situation is different . in refs . it is proposed that @xmath41 is negligible due to a certain statistical ` smearing ' of the np surface region on a thickness of the order of the lattice parameter @xmath42 . however , this hypothesis can not be accepted by default . firstly , the order at the surface strongly depends on the considered crystal phase and on the actual sample . consider that for a np of diameter @xmath49 , the fraction of atoms included in a layer of thickness @xmath42 is @xmath50 ( about 50% at @xmath51 , still 12% at @xmath52 ) . the structure of this large fraction should be carefully considered on a case - by - case basis . relaxations in the core due to a disordered layer of thickness @xmath42 should also be considered . secondly , supposing a default smearing of the np boundaries flattens the different construction principles of forms a , b , c. in fact , the differences among them regard the finest details of the np surface structure . we shall hereafter assess the effect of neglecting @xmath41 on the calculation of a powder diffraction pattern . [ appa ] we carry out some relevant calculations . evidently this will depend on the choice of form a , b , or c. examples are reported in the following section . for form @xmath53 it turns out that , even when @xmath41 is not negligible , it yields a contribution that is approximately proportional to the retained term @xmath53 of the scattered intensity . this means that the effect of neglecting @xmath41 may be just a small error on the global scale factor for samples composed of particles of equal size . however , as this effect is size - dependent , it may hamper the evaluation of size distribution when this is not very narrow . a size - related correction factor for the scale factor may - and should - be evaluated ( see app . [ appa ] ) in this case . this of course is an undesired complication . in cases a ) and c ) the neglected term @xmath41 depends on the crystal structure ( see app . [ appa ] ) . it is not a constant scale factor for all bragg peaks , and it may have a significant gradient in the bragg peak positions . at very small sizes the latter may induce a systematic error also in the lattice constant determination . however , in the x - ray case , for form a ) @xmath41 is larger - and has a larger gradient in the bragg peak neighbourhood - than the corresponding term for form c ) . to obtain a powder diffraction pattern , we must integrate @xmath54 ( x = a , b , c , see eqs . ( [ eq : ampla],[eq : amplb],[eq : amplc ] ) ) at constant @xmath55 . we write @xmath4 in polar coordinates as @xmath56 , where @xmath57 is the orientation defined by the pair @xmath58 . we have to integrate over the set of all orientations @xmath59 ( with @xmath60 ) , as @xmath61 in detail , considering the expressions for the different cases , we have @xmath62 the integration in case b ) is much more difficult and it can not generally be expressed in closed form even for simple shapes . therefore , as a careful implementation of form c ) is at least as good a description as form b ) , we shall disregard b ) in the following . suppose now that @xmath10 is a sphere of radius @xmath63 and volume @xmath64 , we have @xmath65_{y=2\pi q r } \label{eq : ip2}\ ] ] and , as @xmath66 , @xmath67 } \label{eq : ip3}\\ \text{with}&&{y=2\pi ( q^2+h^2 - 2qh\cos\psi)^{1/2 } r}.\nonumber\end{aligned}\ ] ] substituting in eqs . ( [ eq : intap],[eq : intcp ] ) yields @xcite @xmath68 now we consider the crystal s laue group @xmath69 so that we can extend the summation on the asymmetric part @xmath70 of the reciprocal lattice : @xmath71 where @xmath72 is the multiplicity of @xmath8 subject to @xmath69 . evaluation of @xmath73 is only slightly more complex than @xmath74 , and the gain in accuracy justifies the effort . we have computed test patterns to compare forms a ) and c ) , considering nps of diameter @xmath75 , being this the lower size limit of validity of the shape - based approach . we have considered au spherical nps of diameter 5 nm ( @xmath42=0.40786 nm , @xmath7=0.154056 nm , @xmath76 , lorentz correction and debye - waller factor @xmath77 , with @xmath78 nm@xmath79 ) . the powder pattern was calculated exactly by the debye sum @xcite and by eqs . ( [ eq : intap3],[eq : intcp3 ] ) . the profiles showed in fig . [ fig1]a are calculated on an absolute scale . they match quite well , but a maximum error @xmath80 is present in both cases a , c . the profile @xmath81 agreement index between @xmath82 and @xmath83 is 3.1% , between @xmath82 and @xmath84 is @xmath81=4.4% . the difference profiles ( fig . [ fig1]b ) show that @xmath85 has a similar shape to @xmath82 , while @xmath86 is quite different . accordingly , refining a scale factor between @xmath82 and @xmath83 lowers @xmath81 to 2.0% ( with featureless difference , fig . [ fig1]c ) , while a scale factor between @xmath82 and @xmath84 yields @xmath81=3.5% , with still a characteristic difference profile . furthermore , the peak positions result very little shifted ( @xmath87 ) between @xmath82 and @xmath83 , while they are shifted up to @xmath88 between @xmath82 and @xmath84 ( fig . [ fig1]d ) . then , we have considered znse spherical nps of diameter 4.8 nm ( @xmath42=0.5633 nm , @xmath7=0.154056 nm , @xmath89 , lorentz correction and debye - waller factor with @xmath78 nm@xmath79 ) . once more , the powder pattern was calculated exactly by the debye sum @xcite and by eqs . ( [ eq : intap3],[eq : intcp3 ] ) . the profiles - calculated on an absoulte scale ( fig . [ fig2]a ) - match quite well with a maximum error @xmath90 for both cases a , c . the profile agreement index @xmath81 between @xmath82 and @xmath83 is 1.8% , between @xmath82 and @xmath84 is @xmath81=3.1% . the difference profiles ( fig . [ fig2]b ) show again that @xmath85 has a similar shape to @xmath82 , while @xmath86 is quite different . accordingly , we have refined again a scale factor ( and this time also a different debye - waller factor @xmath91 ) between @xmath82 and @xmath83 . @xmath81 decreases to 1.6% with featureless difference ( fig . [ fig2]c ) . on the opposite , when refining scale factor and debye - waller factor between @xmath82 and @xmath84 the agreement index does not go below @xmath81=3.1% . also the difference profile is little changed ( fig . [ fig2]c ) . again , the peak positions result very little shifted ( @xmath92 ) between @xmath82 and @xmath83 , while peak shifts up to @xmath93 between @xmath82 and @xmath84 are visible ( fig . [ fig2]d ) . form c ) again turns out to be less affected than a ) by neglecting the cross - term @xmath94 . a small variation of the debye - waller factor ( from 0.005 to 0.0047 nm@xmath79 ) is due to the fact that the @xmath94-neglection error changes slightly the intensity ratios . this is however less troublesome than the peak shifts observed for form a ) . it results that at np diameters @xmath95 the errors in the shape - based diffraction pattern calculations , whatever form we choose , start to be evident . this approach should not be used below this threshold . also , form a ) - which is the standard choice for large particles - shows a much larger error and should be avoided in favor of c ) . there are several experimental and theoretical reasons@xcite to believe that np powders have a log - normal distribution of np size . the log - normal distribution of np radii is usually written in terms of its mode @xmath96 and width @xmath97 , as @xmath98 ^ 2}{2w_{r}^2}\right\}. \label{eq : lon0}\ ] ] the most direct information on a distribution is provided by the distribution - averaged np radius @xmath99 and the relevant standard deviation @xmath100 . for a log - normal , the latter parameters are related to the former by @xmath101 and @xmath102 we shall use a form depending directly on @xmath99 , @xmath100 @xcite . setting two adimensional parameters @xmath103 , @xmath104 , we have @xmath105 . \label{eq : lonx}\ ] ] volume- and area - averaged np diameters can be derived by @xmath106 x - ray powder diffraction patterns of a nanocrystalline 20-nm ceo@xmath107 sample , available for a round - robin@xcite , were downloaded ( ` http://www.boulder.nist.gov/div853/balzar/ ` , ` http://www.du.edu/~balzar/s-s_rr.htm ` ) . the np size is well inside the limits of validity of the shape - based method . among the available datasets , the selected raw data were collected at the nsls x3b1 beamline of the brookhaven national laboratory in flat - plate geometry , with a double - crystal si(111 ) monochromator on the incident beam ( @xmath108 , @xmath109 = 12@xmath110(0.01@xmath110)60@xmath110 ) and a ge(111 ) analyzer crystal on the diffracted beam . three data preprocessing stages have been accomplished . first , the instrumental function has been deconvoluted by an original advanced technique , including denoising and background subtraction , described in ref . . secondly , the pattern has been fitted by generic asymmetric voigt profiles so as to obtain information about peak positions and intensities . by comparing the intensities as evaluated from the fit with the theoretical ones a small correction for texture and/or microabsorption has been evaluated . the intensity corrections so obtained have then been stored and used in the subsequent stages . finally , the peak positions were found to be slightly anisotropically shifted . this has been attributed to a small residual stress , due _ e.g. _ to dislocations . to confirm this point , we have evaluated the average lattice spacing variations @xmath111_{{{\ensuremath{{\bm{h}}}}}}=-\frac{\pi}{360}\cot(\theta_{{{\ensuremath{{\bm{h}}}}}})\delta(2\theta_{{{\ensuremath{{\bm{h}}}}}})$ ] for all single reflections @xmath8 . then we have compared those values with a simple model of elastic anisotropy @xcite . they resulted in good agreement . in fig . [ fig3 ] we show the fit of @xmath112_{{{\ensuremath{{\bm{h}}}}}}$ ] s with eq . ( 28 ) of ref . . the magnitudes of the residual stress tensor components , at least for those which can be determined in this way , resulted to be in the range 110 mpa . the values of @xmath113 are below @xmath114 , and @xmath112_{{{\ensuremath{{\bm{h}}}}}}$ ] range in 17@xmath115 , which are quite small values . as the strain broadening is of the same order of magnitude of the peak shifts @xcite , we can confirm that strain broadening is rather small in the ceo@xmath1 sample and can be neglected , as in ref . . also the residual - stress peak shifts so obtained have been saved as fixed corrections for the subsequent stages . the total intensity diffracted by the powder np sample is described by the sum @xmath116 where @xmath117 , @xmath118 ; @xmath119 is @xmath73 of eq . ( [ eq : intcp3 ] ) evaluated at @xmath120 ; and @xmath121 is a polynomial modelling the background . the step @xmath122 is chosen so as to have an integer number of atoms in each @xmath123-th x - ray sphere of radius @xmath124 , while keeping the point density constant and preserving stoichiometry . it is evidently possible to use a size - dependent lattice parameter @xmath125 in the calculation of @xmath119 . for this sample this has been deemed unnecessary . indeed , for diameters of 20 nm , the lattice parameter of ceo@xmath0 has been found@xcite to be already equal to the bulk value . a least - square full - pattern refinement means minimizing the quantity @xmath126 here @xmath127 is the @xmath128-th point of the experimental pattern corresponding to the scattering vector @xmath129 , @xmath130 the number of experimental points and the weights @xmath131 are the estimated inverse variance of the observations . the refined parameters are : the average nps radius ( @xmath99 ) and the radius dispersion @xmath100 , the isotropic debye - waller factors @xmath91 for o and ce atoms , the cubic unit cell parameter @xmath42 and seven background coefficients . for the minimization , we have used ( for this work ) a modified simplex algorithm @xcite , which is robust but time - consuming ; however , computing times were reasonable . a derivative - based algorithm ( newton , in progress ) should give a handsome acceleration . the final results are given in tab . [ tab1 ] , together with the corresponding values of ref . . the debye - waller factors result to @xmath132 nm@xmath133 and @xmath134 nm@xmath133 . the calculated profile is plotted in fig . [ fig4 ] with the experimental pattern and the profile difference . the excellent fit quality and the final gof value ( 1.21 ) indicate the achievement of a reliable result . indeed , the estimated parameters are in good agreement with ref . . the slight discrepancy ( @xmath135 nm ) , larger than standard deviations , might be explained by the improved deconvolution method here applied and by the use of the whole pattern instead of a limited number of peaks as in ref . . .comparison of size distribution results . standard deviations are in brackets . units are nm . [ cols="<,^,^,^,^ " , ] [ tab1 ] the method of shape - convolution to calculate the diffraction pattern of np powders has been thoroughly discussed with respect to its limits of validity . concerns in applying this method below its optimal size range have been demonstrated theoretically and by simulated patterns . finally , the effectiveness of full - pattern powder data analysis based on the shape - convolution method was proved to obtain precise size distribution information on np powder samples with a log - normal distribution of spherical crystallites . 41 natexlab#1#1bibnamefont # 1#1bibfnamefont # 1#1citenamefont # 1#1url # 1`#1`urlprefix[2]#2 [ 2][]#2 , * * , ( ) . , * * , ( ) . , * * , ( ) . , * * , ( ) . , * * , ( ) . , * * , ( ) . , , , , * * , ( ) . , , , * * , ( ) . , * * , ( ) . , , , * * , ( ) . , , , , * * , ( ) . , , , , , , , , , , , * * , ( ) . , , , , , , , , , * * , ( ) . , , , , , , , , * * , ( ) . , _ _ ( , ) , chap . , pp . . , , ( ) , . , * * , ( ) . , * * , ( ) . , * * , ( ) . , * * , ( ) . , * * , ( ) . , _ _ ( , ) . , * * , ( ) . , * * , ( ) . , , , * * , ( ) . , , , * * , ( ) . , , , , * * , ( ) . , * * , ( ) . , , , , * * , ( ) . , * * , ( ) . , * * , ( ) . , * * , ( ) . , _ _ ( , ) , pp . . , , , , * * , ( ) . , , , , , , , , , , , * * , ( ) . , , , ( ) , . , * * , ( ) . , * * , ( ) . , , , , , , , * * , ( ) . , * * , ( ) . , , , , , , , _ _ ( , ) , chap . , pp . . of 5.0 nm diameter with fcc structure ( @xmath42=0.40786 nm ) has been constructed according to principle c ) of sec . [ sec2 ] . in this case , as the monoatomic fcc wigner - seitz unit cell contains one atom , principle c ) coincides with b ) . + a : the powder diffraction pattern : red , exact intensity @xmath136 calculated by the debye function @xcite ; blue dotted , @xmath74 calculated by approach a ) , eq . ( [ eq : intap3 ] ) ; green dashed , @xmath73calculation by approach c ) , eq . ( [ eq : intcp3 ] ) . all intensities have been calculated on an absolute scale and then scaled by the same factor . + b : lower line , red , difference @xmath137 ; middle line , green , difference @xmath138 ; upper line , blue , the exact powder pattern @xmath139 ( debye method ) for comparison . + c : lower line , red , difference @xmath140 after refining a scale factor @xmath141 ; middle line , green , difference @xmath142after refining a scale factor @xmath143 ; upper line , blue , the exact powder pattern @xmath139 ( debye method ) for comparison . note that the c)-type pattern difference is flattened while the a)-type retains sharp contributions . + d : detail around the ( 111 ) peak of the @xmath82 and @xmath84 patterns ( same coding as in part a ) after scaling , showing a significant peak shift for the @xmath84 pattern . , title="fig:",width=321 ] [ fig1 ] of 4.8 nm diameter with fcc structure ( @xmath42=0.5633 nm ) has been constructed according to principle c ) of sec . [ sec2 ] . in this case , as the fcc wigner - seitz unit cell contains two atoms , construction c ) differs from b ) . + a : the powder diffraction pattern : red , exact intensity @xmath136 calculated by the debye function @xcite ; blue dotted , @xmath74 calculated by approach a ) , eq . ( [ eq : intap3 ] ) ; green dashed , @xmath73calculation by approach c ) , eq . ( [ eq : intcp3 ] ) . again all intensities have been calculated on an absoulte scale . + b : lower line , red , difference @xmath137 ; middle line , green , difference @xmath138 ; upper line , blue , the exact powder pattern @xmath139 ( debye method ) for comparison . + c : lower line , red , difference @xmath140 after refining a scale factor @xmath144 and an overall - isotropic debye - waller factor @xmath145 ( @xmath82 has been evaluated with @xmath146 ) ; middle line , green , difference @xmath142 after refining a scale factor @xmath147 and @xmath148 ; upper line , blue , the exact powder pattern @xmath139 ( debye method ) for comparison . note again that the c)-type pattern difference is flattened while the a)-type retains sharp contributions . + d : detail around the ( 531 ) peak of the @xmath82 and @xmath84 patterns ( same coding as in part a ) after scaling , showing a significant peak shift for the @xmath84 pattern . , title="fig:",width=321 ] [ fig2 ] plotted against the relevant peaks diffraction angle . error bars have been evaluated assuming a constant error of 0.0006@xmath110 on the anisotropic angular peak shift . calculated values refer to the model of ref . where residual stress components have been refined . , title="fig:",width=321 ] [ fig3 ] powder pattern final fit . blue diamonds - the observed deconvoluted intensity ; red continuous line - the calculated intensity ; black continuous line , below - difference profile ( same scale ) . , title="fig:",width=321 ] [ fig4 ] assume to deal with particles of centrosymmetric shape @xmath149 and equivalent spherical radius @xmath63 ( _ i.e. _ , the radius of the sphere of equal volume . ) . the shape fourier transform @xmath19 is then a real even function : @xmath150 recall also that the gradient of an even function is odd : @xmath151 our aim is to evaluate - for the different forms a ) , b ) , c ) as introduced in sec . [ sec2 ] and carried out in sec . [ sec22 ] , sec . [ sec23 ] - the neglected residual intensity contribution @xmath41 of eq . ( [ eq : x ] ) with respect to the respective retained term ( _ cf . ( [ eq : inta],[eq : intb],[eq : intc ] ) ) in the immediate vicinity of a bragg peak . let @xmath152 the nearest bragg peak to @xmath4 . first note that , if @xmath153 , @xmath41 is of order @xmath154 , so we neglect it altogether . if @xmath4 is very close to @xmath152 , set @xmath155 ( so @xmath156 ) . we can drop in the sum over @xmath8 all terms with @xmath157 because they are @xmath158 and reorder the second sum , obtaining @xmath159 @xmath160 @xmath161 at the same time , for @xmath162 with @xmath156 , the intensities @xmath54 of eqs . ( [ eq : inta],[eq : intb],[eq : intc ] ) can be approximated by the @xmath152-th term of the rhs sum , neglecting terms of @xmath158 . furthermore , in general , @xmath163 . therefore , the ratios @xmath164 are given by @xmath165 @xmath166 @xmath167 note that , because of eqs . ( [ eq : even],[eq : oddg ] ) , we have @xmath168 we can immediately veryfy that in case b ) it results @xmath169 in the sum above the term with index @xmath170 is always accompanied by a term with index @xmath171 . setting also @xmath172 , and using eq . ( [ eq : oddg ] ) , we have @xmath173 where @xmath174 denotes an arbitrarily chosen half - space of the reciprocal lattice without the origin . now , expanding @xmath175 in taylor series at @xmath172 , we have @xmath176 note also in eq . ( [ eq : aar2 ] ) that @xmath177 does not depend on the considered bragg reflection @xmath178 . therefore , we can write @xmath179 and the proportionality constant can be evaluated by eq . ( [ eq : aar2 ] ) with @xmath172 . we can conclude that the effect of neglecting @xmath180 will be just a relative error on the global profile scale factor . this factor is size - dependent , however , therefore for size distribution analysis at small sizes it may be necessary to introduce a correction as from eq . ( [ eq : aar2 ] ) . cases a ) , c ) , are more complex . we are interested to powder diffraction , where @xmath181 is to be integrated at constant @xmath55 , therefore we shall consider @xmath182 \label{eq : topo}\ ] ] expanding @xmath183 in taylor series at @xmath172 , we have @xmath184 we shall now develop @xmath185 and @xmath186 in cases a , c. first , recall that the atomic form factors @xmath187 are constants for neutron scatering and monotonically decreasing smooth functions in the x - ray case . in the latter case , furthermore , the form factors of different elements have remarkably similar profiles . for a structure with @xmath188 atoms in the unit cell , it is then possible @xcite to approximate @xmath189 with @xmath190 appropriate constants . therefore the structure factor ratios appearing in eq . ( [ eq : aar3 ] ) can be simplified as @xmath191 independent of @xmath192 . note that @xmath193 now we can write explicitly @xmath194 using eqs . ( [ eq : aar3],[eq : topo ] ) and @xmath195 . \label{eq : xpl1}\end{aligned}\ ] ] splitting the sum , reordering @xmath196 in one part , using eq . ( [ eq : even2 ] ) and recombining , we have @xmath197 . \label{eq : xpl}\ ] ] again as in eq . ( [ eq : prb2 ] ) , we can pair terms with @xmath170 and @xmath171 . using eq . ( [ eq : taup ] ) , we obtain @xmath198 . \label{eq : xpl2}\end{aligned}\ ] ] define now the arbitrary half - lattice @xmath174 as that defined by a plane @xmath199 passing through the origin and containing @xmath152 . the origin is excluded . we have @xmath200 . \label{eq : xplf}\ ] ] then , evaluating the gradient in @xmath172 , using eq . ( [ eq : oddg ] ) , we have finally @xmath201 . \label{eq : xplg}\end{aligned}\ ] ] the gradient @xmath202 is a vector . we have to take its angular average to determine the effect on the powder pattern . this is done by simply taking the scalar product with @xmath203 : @xmath204 . \label{eq : palle}\end{aligned}\ ] ] for spherical shape , it will be @xmath205 ; therefore terms with @xmath206 will be zero and those with @xmath207 will be most important . both @xmath208 and @xmath209 are damped oscillatory functions with amplitude @xmath210 . as @xmath211 , the magnitudes of both @xmath212 and @xmath202 are of order @xmath213 . unfortunately , eq . ( [ eq : palle ] ) can not be estimated more in detail , because of the dependence from the ` reduced ' structure factors @xmath214 . however , we can assess that its importance would be smaller than the corresponding term for case a ) for x - ray scattering . in case a ) , we can trace the same steps as in case c ) but instead of the ` reduced ' structure factors @xmath214 we have to consider the ratios @xmath215 and in the analog sums of eq . ( [ eq : xplf ] ) and eq . ( [ eq : palle ] ) for @xmath216 and @xmath186 there will appear terms as @xmath217 . \label{eq : casea2}\ ] ] the most important terms for the powder pattern are again those with @xmath207 . the structure factors @xmath218 ( see eq . ( [ eq : strf ] ) ) depend on form factors @xmath219 , and for @xmath207 these will be strongly different . this in turn will amplify the differences @xmath220 . therefore it is likely that for case a ) the effect of the neglected term @xmath221 will be significantly larger than for case c ) . the examples reported in sec . [ sec23 ] show just that .

the increasing scientific and technological interest in nanoparticles has raised the need for fast , efficient and precise characterization techniques . powder diffraction is a very efficient experimental method , as it is straightforward and non - destructive . however , its use for extracting information regarding very small particles brings some common crystallographic approximations to and beyond their limits of validity . powder pattern diffraction calculation methods are critically discussed , with special focus on spherical particles with log - normal distribution , with the target of determining size distribution parameters . a 20-nm ceo@xmath0 sample is analyzed as example .

existence of dark matter ( dm ) has been established , and its thermal relic abundance has been determined by the wmap experiment @xcite . if the essence of dm is an elementary particle , the weakly interacting massive particle ( wimp ) would be a promising candidate . it is desired to have a viable candidate for the dark matter in models beyond the standard model ( sm ) . the wimp dark matter candidate can be accommodated economically by introducing only an inert scalar field @xcite , where we use `` inert '' for the @xmath1-odd property . the imposed @xmath1 parity ensures the stability of the dm candidate . phenomenology in such models have been studied in , e.g. , refs . @xcite . on the other hand , it has been confirmed by neutrino oscillation measurements that neutrinos have nonzero but tiny masses as compared to the electroweak scale @xcite . the different flavor structure of neutrinos from that of quarks and leptons may indicate that neutrino masses are of majorana type . in order to explain tiny neutrino masses , many models have been proposed . the seesaw mechanism is the simplest way to explain tiny neutrino masses , in which right - handed neutrinos are introduced with large majorana masses @xcite . another simple model for generating neutrino masses is the higgs triplet model ( htm ) @xcite . however , these scenarios do not contain dark matter candidate in themselves . in a class of models where tiny neutrino masses are generated by higher orders of perturbation , the dm candidate can be naturally contained @xcite . in models in refs . @xcite , the yukawa couplings of neutrinos with the sm higgs boson are forbidden at the tree level by imposing a @xmath1 parity . the same @xmath1 parity also guarantees the stability of the lightest @xmath1-odd particle in the model which can be the candidate of the dm as long as it is electrically neutral . in this paper , we consider an extension of the htm in which by introducing the @xmath1 parity @xmath6 is generated at the one - loop level and the dm candidate appears . in the htm , majorana masses for neutrinos are generated via the yukawa interaction @xmath7 with a nonzero vacuum expectation value ( vev ) of an @xmath2 triplet scalar field @xmath8 with the hypercharge of @xmath9 . the vev of @xmath8 is described by @xmath10 , where @xmath11 is the vev of the higgs doublet field @xmath12 and @xmath13 is the typical mass scale of the triplet field ; the dimensionful parameter @xmath14 breaks lepton number conservation at the trilinear term @xmath15 which we refer to as the @xmath14-term . as the simplest explanation for the smallness of neutrino masses , the mass of the triplet field is assumed to be much larger than the electroweak scale . on the other hand a characteristic feature of the htm is the fact that the structure of the neutrino mass matrix @xmath16 is given by that of the yukawa matrix , @xmath17 . the direct information on @xmath16 would be extracted from the decay @xmath18 @xcite if @xmath19 is light enough to be produced at collider experiments , where @xmath19 is the doubly charged component of the triplet field @xmath8 . at hadron colliders , the @xmath20 can be produced via @xmath21 @xcite and @xmath22 @xcite . the @xmath20 searches at the lhc put lower bound on its mass as @xmath23 @xcite , assuming that the main decay mode is @xmath24 . phenomenological analyses for @xmath20 in the htm at the lhc have also been performed in ref . triplet scalars can contribute to lepton flavor violation ( lfv ) in decays of charged leptons , e.g. , @xmath25 and @xmath26 at the tree level and @xmath27 at the one - loop level . relation between these lfv decays and neutrino mass matrix constrained by oscillation data was discussed in refs . @xcite . in order to explain the small @xmath0 with such a detectable light @xmath19 , the @xmath14 parameter has to be taken to be unnaturally much lower than the electroweak scale . therefore , it would be interesting to extend the htm in order to include a natural suppression mechanism of the @xmath14 parameter ( therefore @xmath0 ) in addition to the dm candidate . in our model , lepton number conservation is imposed to the lagrangian in order to forbid the @xmath14-term in the htm at the tree level while the triplet yukawa term @xmath28 exists . the vev of a @xmath1-even complex singlet scalar @xmath5 breaks the lepton number conservation by a unit . an @xmath2 doublet @xmath3 and a real singlet @xmath4 are also introduced as @xmath1-odd scalars in order to accommodate the dm candidate . then , the @xmath14-term is generated at the one - loop level by the diagram in which the @xmath1-odd scalars are in the loop . by this mechanism , the smallness of @xmath29 is realized , and the tiny neutrino masses are naturally explained without assuming the triplet fields to be heavy . the yukawa sector is then the same as the one in the htm , so that its predictions for the lfv processes are not changed . see refs . @xcite for some discussions about two - loop realization of the @xmath14-term - term in ref . @xcite is given with softly - broken @xmath30 symmetry , but the tree level @xmath14-term would be also accepted as a soft breaking term . the two - loop @xmath14-term in ref . @xcite is given with @xmath31 symmetry which is broken by a vev of a scalar @xmath32 , but the tree level @xmath33 seems allowed by the @xmath31 . ] . this paper is organized as follows . in sec . [ sec : htm ] , we give a quick review for the htm to define notation . in sec . [ sec:1-loop ] , the model for radiatively generating the @xmath14 parameter with the dark matter candidate is presented . some phenomenological implications are discussed in sec . [ sec : pheno ] , and the conclusion is given in sec . [ sec : concl ] . the full expressions of the higgs potential and mass formulae for scalar bosons in our model are given in appendix . in the htm , an @xmath34 triplet of complex scalar fields with hyperchage @xmath9 is introduced to the sm . the triplet @xmath8 can be expressed as @xmath35 where @xmath36 . the triplet has a new yukawa interaction term with leptons as @xmath37 where @xmath38 ( @xmath39 ) are the new yukawa coupling constants , @xmath40 [ @xmath41 are lepton doublet fields , a superscript @xmath42 means the charge conjugation , and @xmath43 ( @xmath44 ) denote the pauli matrices . lepton number ( @xmath45 ) of @xmath8 is assigned to be @xmath46 as a convention such that the yukawa term does not break the conservation . a vacuum expectation value @xmath47 [ @xmath48 breaks lepton number conservation by two units . the new yukawa interaction then yields the majorana neutrino mass term @xmath49 where @xmath50 . the scalar potential in the htm can be written as @xmath51 ^ 2 + \lambda_3\ , { { \text{tr}}}[(\delta^\dagger \delta)^2 ] \nonumber\\ & & { } + \lambda_4\ , ( \phi^\dagger \phi ) { { \text{tr}}}(\delta^\dagger \delta ) + \lambda_5\ , \phi^\dagger \delta \delta^\dagger \phi , \end{aligned}\ ] ] where @xmath52 [ @xmath53 is the higgs doublet field in the sm . the @xmath14 parameter can be real by using rephasing of @xmath8 . because we take @xmath54 , there is no nambu - goldstone boson for spontaneous breaking of lepton number conservation . the small triplet vev @xmath47 is generated by an explicit breaking parameter @xmath14 of the lepton number conservation as @xmath55 where @xmath11 ( @xmath56 ) is the doublet vev defined by @xmath57 . in order to obtain small neutrino masses in the htm , at least one of @xmath58 , @xmath38 , @xmath59 should be tiny . a small @xmath14 is an attractive option because @xmath60 can be small ( @xmath61 ) so that triplet scalars can be produced at the lhc . furthermore , large @xmath38 can be taken , which have direct information on the flavor structure of @xmath16 . there is , however , no reason why the @xmath14 parameter is tiny in the htm . in our model presented below , the @xmath14 parameter is naturally small because it arises at the one - loop level . since we try to generate the @xmath14-term in the htm radiatively , the term must be forbidden at the tree level . the simplest way would be to impose lepton number conservation to the lagrangian . the conservation is assumed to be broken by the vev of a new scalar field @xmath5 which is singlet under the sm gauge symmetry . notice that @xmath5 [ @xmath62 is a complex ( `` charged '' ) field with non - zero lepton number although it is electrically neutral . one might think that the vev of @xmath5 could be generated by using soft breaking terms of @xmath45 . however , the @xmath14-term is also a soft breaking term . therefore lepton number must be broken spontaneously in our scenario . one may worry about nambu - goldstone boson corresponds to the spontaneous breaking of the lepton number conservation ( the so - called majoron , @xmath63 ) . however the majoron which comes from gauge singlet field can evade experimental searches ( constraints ) because it interacts very weakly with matter fields @xcite . it is also possible to make it absorbed by a gauge boson by introducing the @xmath64 gauge symmetry to the model ( see , e.g. , ref . @xcite ) . in this paper we just accept the majoron without assuming the @xmath64 gauge symmetry for simplicity . if @xmath5 has @xmath65 , we can have a dimension-4 operator @xmath66 . this gives a trivial result @xmath67 at the tree level . although the dim.-4 operator could be forbidden by some extra global symmetries with extra scalars to break them , we do not take such a possibility in this paper . we just assume the @xmath5 has @xmath68 . then the lepton number conserving operator which results in the @xmath14-term is of dimension-5 as @xmath69 we consider below how to obtain the dim.-5 operator at the loop level by using renormalizable interactions of @xmath70 . ] . we restrict ourselves to extend only the @xmath71-singlet scalar sector in the htm because it seems a kind of beauty that the htm does not extend the fermion sector and colored sector in the sm . an unbroken @xmath1 symmetry is introduced in order to obtain dark matter candidates , and new scalars which appear in the loop diagram for the @xmath14-term are aligned to be @xmath1-odd particles . we emphasize that the unbroken @xmath72 symmetry is not for a single purpose to introduce dark matter candidates but utilized also for our radiative mechanism for the @xmath14-term . . list of particle contents of our one - loop model . [ cols="^,^,^,^,^,^,^",options="header " , ] we present the minimal model where the dim.-5 operator in eq . ( [ eq : dim5op ] ) is generated by a one - loop diagram with dark matter candidates . table [ tab:1-loop ] shows the particle contents . a real singlet scalar field @xmath4 and the second doublet scalar field @xmath3 [ @xmath73 are introduced to the htm in addition to @xmath5 . lepton numbers of @xmath4 and @xmath3 are 0 and @xmath74 , respectively . then @xmath75 conserves lepton number . in order to forbid the vev of @xmath3 , we introduce an unbroken @xmath1 symmetry for which @xmath4 and @xmath3 are odd . other fields are even under the @xmath1 . the yukawa interactions are the same as those in the htm . the higgs potential is given as @xmath76 here we show only relevant parts for radiative generation of the @xmath14-term . see appendix for the other terms . vacuum expectation values @xmath11 and @xmath77 [ @xmath78 are given by @xmath79 the @xmath1-odd scalars in this model are two cp - even neutral ones ( @xmath80 and @xmath81 ) , a cp - odd neutral one ( @xmath82 ) , and a charged pair ( @xmath83 ) . the cp - even scalars are defined as @xmath84 where @xmath85 and @xmath86 . squared masses of these scalars are given by @xmath87 notice that @xmath88 . we assume @xmath89 and then @xmath80 becomes the dark matter candidate . hereafter it is assumed that the mixing @xmath90 is small . the @xmath14-term is generated by the one - loop diagram . figure [ fig:1-loop ] is the dominant one in the case of small @xmath90 . then , the parameter @xmath14 is calculated as @xmath91 the one - loop induced @xmath14 parameter can be expected to be much smaller than @xmath92 . the suppression factor @xmath93 is estimated in sec . [ subsec : dm ] . we call it `` a. oryzae diagram '' @xcite . , title="fig : " ] -term . we call it `` a. oryzae diagram '' @xcite . , title="fig : " ] if @xmath94 , the dark matter candidate @xmath80 is given by @xmath95 approximately because we assume small mixing . see , e.g. , ref . @xcite for studies about the inert doublet scalar . let us assume @xmath96 and @xmath97 . as shown in ref . @xcite , these values satisfy constraints from the lep experiments @xcite and the wmap experiment @xcite . the mass splitting ( @xmath98 ) suppresses quasi - elastic scattering on nuclei ( @xmath99 mediated by the @xmath100 boson ) enough to satisfy constraints from direct search experiments of the dm @xcite . by using eqs . and , we obtain @xmath101 in order to be consistent with our assumption of small @xmath90 ( e.g. , @xmath102 ) , @xmath103 is required . the value in eq . results in @xmath104 for the greater value of @xmath105 , the larger @xmath106 is predicted . in particular , by taking @xmath105 to be the tev scale , we obtain @xmath107 , which yields @xmath108 for @xmath109 and @xmath60 to be at the electroweak scale . such a value for @xmath47 is suggested in the recent study of radiative corrections to the electroweak parameters @xcite . on the contrary , if we take @xmath110 which is allowed in a tiny region @xcite , values in eqs . and become 10 times smaller . we mention that the wmap constraint might be changed by a characteristic annihilation process @xmath111 where @xmath112 interaction is governed by @xmath109 ( not by a tiny @xmath14 ) . this additional process could sift allowed value of @xmath113 to lower one while @xmath114 due to the lep constraint . then , @xmath106 might become larger than the value in eq . because of larger @xmath115 . this undesired effect would be easily avoided if @xmath116 is away enough from @xmath117 . on the other hand , @xmath80 comes dominantly from @xmath4 if @xmath118 . see , e.g. , ref . @xcite for studies about the real inert singlet scalar . coupling @xmath119 of the @xmath120 interaction ( @xmath121 is the sm higgs boson ) determines annihilation cross section of @xmath80 and scattering cross section on nuclei . if we introduce the @xmath64 gauge symmetry , the scattering of @xmath4 on nuclei can be mediated also by the gauge boson @xmath122 . notice that the parameter @xmath123 ( and also the @xmath64 gauge coupling constant ) does not affect on @xmath14 parameter in eq . . let us estimate the magnitude of @xmath106 . in the usual htm , @xmath38 is expected to be @xmath124 for @xmath125 in order to suppress lfv processes . thus , we may accept @xmath126 as a value which is not too small . assuming @xmath127 for example gauge symmetry in order to eliminate the majoron , @xmath77 should be a little bit larger ( e.g. , @xmath128 ) due to constraint on the mass of @xmath122 . ] , we have a suppression factor as @xmath129 thus , even if the value of @xmath92 is in the tev scale , we can obtain @xmath130 although we need further suppression with @xmath131 to have @xmath132 . if we use @xmath133 , we obtain @xmath134 which can connect the tev scale @xmath92 to the ev scale @xmath6 . . the bosonic decay of @xmath135 contains information of @xmath92 indicated by a red blob . ] the characteristic feature of our model is that @xmath92 is much larger than @xmath14 . let us consider possibility to probe the large @xmath92 in collider experiments . a favorable process is shown in fig . [ fig : lhc ] for @xmath136 . for simplicity , we take @xmath137 which results in @xmath138 . recently , it was found in ref . @xcite that the electroweak precision test prefers @xmath139 in the htm where the electroweak sector is described by four input parameters . however , results in ref . @xcite might not be applied directly to our model is generated at the 1-loop level . ] because the scalar sector is extended . since @xmath140 is the most interesting decay in the htm , we assume @xmath141 in order to forbid @xmath142 . even in this case , the dm @xmath80 can be light enough ( @xmath143 ) so that @xmath1-even charged scalar @xmath144 ( @xmath145 ) can decay into @xmath146 via @xmath92-term which is indicated by a red blob in fig . [ fig : lhc ] . the partial decay width of @xmath147 is determined by @xmath148 while the width of @xmath149 is proportional to @xmath150 . taking @xmath151 , @xmath152 , @xmath153 , and @xmath154 for example , we have @xmath155 and @xmath156 . then , @xmath144 dominantly decays into @xmath146 . finally , @xmath157 decays into @xmath158 . therefore , from a production mechanism @xmath159 , we would have @xmath160 as a final state in fig . [ fig : lhc ] can be replaced with @xmath161 which decays into @xmath162 for @xmath136 . ] for which @xmath163 has the invariant mass @xmath164 at @xmath165 assuming that the value of @xmath165 has been known already . if @xmath166 , then @xmath144 decays via @xmath92-term into @xmath167 or @xmath168 followed by @xmath169 where a sizable @xmath170 is assumed is small , @xmath81 ( @xmath171 ) decays into @xmath172 . ] . because of @xmath173 through @xmath174 , we have again @xmath160 with @xmath175 from @xmath159 . in the usual htm in contrast , the final state with such @xmath163 is likely to include additional charged leptons ( @xmath176 from @xmath177 , @xmath178 from @xmath179 , etc . ) if @xmath20 decay dominantly into @xmath180 . therefore , our model would be supported if experiments observe final states which include jets and only two @xmath181 whose invariant mass gives @xmath175 . this potential signature might be disturbed by hadronic decays of @xmath182 because @xmath183 can result in @xmath160 with @xmath175 . realistic simulation is necessary to see the feasibility . we have presented the simple extension of the htm by introducing a @xmath1-even neutral scalar @xmath5 of @xmath68 , a @xmath1-odd neutral real scalar @xmath4 of @xmath184 , and a @xmath1-odd doublet scalar field @xmath3 of @xmath68 . the dm candidate @xmath185 in our model is made from @xmath4 and @xmath186 . the @xmath187 interaction which is the origin of @xmath0 ( and neutrino masses ) is induced at the one - loop level while the @xmath188 interaction exists at the tree level . because of the loop suppression for @xmath14 parameter , the model gives small neutrino masses naturally without using very heavy particles . for @xmath189 , the suppression factor @xmath190 is constrained by the dm relic abundance measured by the wmap experiment . we have shown that @xmath191 is possible . on the other hand , for @xmath192 , the suppression factor is somewhat free from experimental constraints on the dm . in our estimate , @xmath193 can be obtained as an example with @xmath126 . the characteristic feature of the model is that @xmath92 is not small while @xmath14 can be small . a possible collider signature which depends on @xmath92 would be @xmath160 with the invariant mass @xmath175 because more charged leptons are likely to exist in such final states in the usual htm . the work of s.k . was supported by grant - in - aid for scientific research nos . 22244031 and 23104006 . the work of h.s . was supported by the sasakawa scientific research grant from the japan science society and grant - in - aid for young scientists ( b ) no . the higgs potential of our model is given by @xmath194 where @xmath195 @xmath196 @xmath197 ^ 2 + \lambda_3\ , { { \text{tr}}}[(\delta^\dagger \delta)^2 ] \nonumber\\ & & { } + \lambda_{4\phi}\ , ( \phi^\dagger \phi)\ , { { \text{tr}}}(\delta^\dagger \delta ) + \lambda_{4\eta}\ , ( \eta^\dagger \eta)\ , { { \text{tr}}}(\delta^\dagger \delta ) \nonumber\\ & & { } + \lambda_{5\phi}\ , ( \phi^\dagger \delta \delta^\dagger \phi ) + \lambda_{5\eta}\ , ( \eta^\dagger \delta \delta^\dagger \eta ) \nonumber\\ & & { } + \lambda_{s1}\ , |s_1 ^ 0|^4 + \lambda_{s2}\ , ( s_2 ^ 0)^4 + \lambda_{s3}\ , |s_1 ^ 0|^2 ( s_2 ^ 0)^2 \nonumber\\ & & { } + \lambda_{s\phi 1}\ , |s_1 ^ 0|^2\ , ( \phi^\dagger \phi ) + \lambda_{s\phi 2}\ , ( s_2 ^ 0)^2\ , ( \phi^\dagger \phi ) \nonumber\\ & & { } + \lambda_{s\eta 1}\ , |s_1 ^ 0|^2\ , ( \eta^\dagger \eta ) + \lambda_{s\eta 2}\ , ( s_2 ^ 0)^2\ , ( \eta^\dagger \eta ) + \left\ { \lambda_{s\phi\eta}\ , s_1 ^ 0\ , s_2 ^ 0\ , ( \eta^\dagger \phi ) + \text{h.c . } \right\ } \nonumber\\ & & { } + \lambda_{s\delta 1}\ , |s_1 ^ 0|^2 { { \text{tr}}}(\delta^\dagger \delta ) + \lambda_{s\delta 2}\ , ( s_2 ^ 0)^2 { { \text{tr}}}(\delta^\dagger \delta ) .\end{aligned}\ ] ] all coupling constants are real because the phases of @xmath92 and @xmath174 can be absorbed by @xmath8 and @xmath5 , respectively . mass eigenstates of two @xmath1-even cp - even neutral scalars which are composed of @xmath198 and @xmath199 are obtained as @xmath200 their masses eigenvalues are given by @xmath201 where small contributions from @xmath47 are neglected . two @xmath1-even cp - odd neutral bosons ( @xmath202 and @xmath203 ) are nambu - goldstone bosons ; @xmath202 is absorbed by the @xmath100 boson , and @xmath203 is the majoron ( or absorbed by the @xmath122 boson ) . c. p. burgess , m. pospelov and t. ter veldhuis , nucl . b * 619 * , 709 ( 2001 ) ; g. cynolter , e. lendvai and g. pocsik , acta phys . b * 36 * , 827 ( 2005 ) ; c. bird , r. v. kowalewski and m. pospelov , mod . phys . a * 21 * , 457 ( 2006 ) ; s. profumo , m. j. ramsey - musolf and g. shaughnessy , jhep * 0708 * , 010 ( 2007 ) ; v. barger , p. langacker , m. mccaskey , m. j. ramsey - musolf and g. shaughnessy , phys . d * 77 * , 035005 ( 2008 ) ; j. march - russell , s. m. west , d. cumberbatch and d. hooper , jhep * 0807 * , 058 ( 2008 ) ; m. pospelov and a. ritz , phys . b * 671 * , 391 ( 2009 ) ; phys . d * 84 * , 113001 ( 2011 ) ; r. n. lerner and j. mcdonald , phys . d * 80 * , 123507 ( 2009 ) ; m. gonderinger , y. li , h. patel and m. j. ramsey - musolf , jhep * 1001 * , 053 ( 2010 ) ; c. s. kim , s. c. park , k. wang and g. zhu , phys . d * 81 * , 054004 ( 2010 ) ; x. g. he , t. li , x. q. li , j. tandean and h. c. tsai , phys . b * 688 * , 332 ( 2010 ) ; m. asano and r. kitano , phys . d * 81 * , 054506 ( 2010 ) ; c. arina , f. x. josse - michaux and n. sahu , phys . d * 82 * , 015005 ( 2010 ) ; a. badin and a. a. petrov , phys . d * 82 * , 034005 ( 2010 ) ; s. kanemura , s. matsumoto , t. nabeshima and n. okada , phys . d * 82 * , 055026 ( 2010 ) ; w. l. guo and y. l. wu , jhep * 1010 * , 083 ( 2010 ) . s. profumo , l. ubaldi and c. wainwright , phys . d * 82 * , 123514 ( 2010 ) ; a. abada , d. ghaffor and s. nasri , phys . d * 83 * , 095021 ( 2011 ) ; s. kanemura , s. matsumoto , t. nabeshima and h. taniguchi , phys . b * 701 * , 591 ( 2011 ) ; j. mcdonald , phys . d * 84 * , 103514 ( 2011 ) ; x. g. he and j. tandean , phys . d * 84 * , 075018 ( 2011 ) ; a. drozd , b. grzadkowski and j. wudka , arxiv:1112.2582 [ hep - ph ] ; a. djouadi , o. lebedev , y. mambrini and j. quevillon , arxiv:1112.3299 [ hep - ph ] ; a. abada and s. nasri , arxiv:1201.1413 [ hep - ph ] . r. barbieri , l. j. hall and v. s. rychkov , phys . d * 74 * , 015007 ( 2006 ) ; j. a. casas , j. r. espinosa and i. hidalgo , nucl . b * 777 * , 226 ( 2007 ) ; l. lopez honorez , e. nezri , j. f. oliver and m. h. g. tytgat , jcap * 0702 * , 028 ( 2007 ) ; m. gustafsson , e. lundstrom , l. bergstrom and j. edsjo , phys . * 99 * , 041301 ( 2007 ) ; t. hambye and m. h. g. tytgat , phys . b * 659 * , 651 ( 2008 ) ; e. m. dolle and s. su , phys . d * 80 * , 055012 ( 2009 ) ; m. gustafsson , pos c * harged2010 * , 030 ( 2010 ) ; l. lopez honorez and c. e. yaguna , jcap * 1101 * , 002 ( 2011 ) . x. g. he , t. li , x. q. li and h. c. tsai , mod . a * 22 * , 2121 ( 2007 ) ; x. g. he , t. li , x. q. li , j. tandean and h. c. tsai , phys . d * 79 * , 023521 ( 2009 ) ; h. s. goh , l. j. hall and p. kumar , jhep * 0905 * , 097 ( 2009 ) ; m. aoki , s. kanemura and o. seto , phys . b * 685 * , 313 ( 2010 ) ; b. grzadkowski , o. m. ogreid and p. osland , phys . rev . d * 80 * , 055013 ( 2009 ) ; g. k. yeghiyan , phys . d * 80 * , 115019 ( 2009 ) ; b. grzadkowski and p. osland , phys . d * 82 * , 125026 ( 2010 ) ; h. e. logan , phys . rev . d * 83 * , 035022 ( 2011 ) ; t. li and q. shafi , phys . d * 83 * , 095017 ( 2011 ) ; y. cai , x. g. he and b. ren , phys . d * 83 * , 083524 ( 2011 ) ; x. g. he , b. ren and j. tandean , arxiv:1112.6364 [ hep - ph ] . b. t. cleveland _ et al . _ , astrophys . j. * 496 * , 505 ( 1998 ) ; w. hampel _ et al . _ [ gallex collaboration ] , phys . b * 447 * , 127 ( 1999 ) ; j. n. abdurashitov _ et al . _ [ sage collaboration ] , j. exp . phys . * 95 * , 181 ( 2002 ) [ zh . fiz . * 122 * , 211 ( 2002 ) ] ; k. abe _ et al . _ [ super - kamiokande collaboration ] , phys . d * 83 * , 052010 ( 2011 ) ; b. aharmim _ et al . _ [ sno collaboration ] , arxiv:1109.0763 [ nucl - ex ] ; c. arpesella _ et al . _ [ the borexino collaboration ] , phys . rev . * 101 * , 091302 ( 2008 ) . m. h. ahn _ [ k2k collaboration ] , phys . d * 74 * , 072003 ( 2006 ) ; p. adamson _ et al . _ [ the minos collaboration ] , phys . rev . lett . * 106 * , 181801 ( 2011 ) ; k. abe _ et al . _ [ t2k collaboration ] , phys . lett . * 107 * , 041801 ( 2011 ) . p. minkowski , phys . b * 67 * , 421 ( 1977 ) ; t. yanagida , in proceedings of the _ `` workshop on the unified theory and the baryon number in the universe '' _ , tsukuba , japan , feb . 13 - 14 , 1979 , edited by o. sawada and a. sugamoto , kek report kek-79 - 18 , p. 95 ; prog . phys . * 64 * , 1103 ( 1980 ) ; m. gell - mann , p. ramond and r. slansky , in _ supergravity _ _ eds_. d. z. freedom and p. van nieuwenhuizen , ( north - holland , amsterdam , 1979 ) ; w. konetschny and w. kummer , phys . b * 70 * , 433 ( 1977 ) ; m. magg and c. wetterich , phys . b * 94 * , 61 ( 1980 ) ; t. p. cheng and l. f. li , phys . d * 22 * , 2860 ( 1980 ) ; j. schechter and j. w. f. valle , phys . rev . d * 22 * , 2227 ( 1980 ) ; g. lazarides , q. shafi and c. wetterich , nucl . b * 181 * , 287 ( 1981 ) . l. m. krauss , s. nasri and m. trodden , phys . d * 67 * , 085002 ( 2003 ) ; k. cheung and o. seto , phys . d * 69 * , 113009 ( 2004 ) . e. ma , phys . d * 73 * , 077301 ( 2006 ) ; phys . b * 662 * , 49 ( 2008 ) ; t. hambye , k. kannike , e. ma and m. raidal , phys . d * 75 * , 095003 ( 2007 ) ; e. ma and d. suematsu , mod . a * 24 * , 583 ( 2009 ) . v. d. barger , h. baer , w. y. keung and r. j. n. phillips , phys . d * 26 * , 218 ( 1982 ) ; j. f. gunion , j. grifols , a. mendez , b. kayser and f. i. olness , phys . rev . d * 40 * , 1546 ( 1989 ) ; j. f. gunion , c. loomis and k. t. pitts , econf * c960625 * , lth096 ( 1996 ) [ arxiv : hep - ph/9610237 ] ; m. muhlleitner and m. spira , phys . d * 68 * , 117701 ( 2003 ) ; t. han , b. mukhopadhyaya , z. si and k. wang , phys . rev . d * 76 * , 075013 ( 2007 ) ; k. huitu , j. maalampi , a. pietila and m. raidal , nucl . b * 487 * , 27 ( 1997 ) . p. fileviez perez , t. han , g. y. huang , t. li and k. wang , phys . d * 78 * , 015018 ( 2008 ) ; f. del aguila and j. a. aguilar - saavedra , nucl . b * 813 * , 22 ( 2009 ) ; s. t. petcov , h. sugiyama and y. takanishi , phys . rev . d * 80 * , 015005 ( 2009 ) ; a. g. akeroyd and c. w. chiang , phys . d * 80 * , 113010 ( 2009 ) . a. g. akeroyd , c. w. chiang and n. gaur , jhep * 1011 * , 005 ( 2010 ) ; a. g. akeroyd and h. sugiyama , phys . d * 84 * , 035010 ( 2011 ) ; a. g. akeroyd and s. moretti , phys . d * 84 * , 035028 ( 2011 ) ; a. melfo , m. nemevsek , f. nesti , g. senjanovic and y. zhang , arxiv:1108.4416 [ hep - ph ] ; m. aoki , s. kanemura and k. yagyu , arxiv:1110.4625 [ hep - ph ] ; a. arhrib , r. benbrik , m. chabab , g. moultaka and l. rahili , arxiv:1112.5453 [ hep - ph ] ; c. w. chiang , t. nomura and k. tsumura , arxiv:1202.2014 [ hep - ph ] . m. kakizaki , y. ogura and f. shima , phys . b * 566 * , 210 ( 2003 ) ; a. g. akeroyd , m. aoki and h. sugiyama , phys . d * 79 * , 113010 ( 2009 ) ; t. fukuyama , h. sugiyama and k. tsumura , jhep * 1003 * , 044 ( 2010 ) . s. khalil , j. phys . g * 35 * , 055001 ( 2008 ) ; s. iso , n. okada and y. orikasa , phys . b * 676 * , 81 ( 2009 ) ; phys . rev . d * 80 * , 115007 ( 2009 ) ; n. okada and o. seto , phys . d * 82 * , 023507 ( 2010 ) ; s. kanemura , o. seto and t. shimomura , phys . d * 84 * , 016004 ( 2011 ) ; l. basso , a. belyaev , s. moretti and c. h. shepherd - themistocleous , phys . d * 80 * , 055030 ( 2009 ) ; l. basso , a. belyaev , s. moretti , g. m. pruna and c. h. shepherd - themistocleous , eur . j. c * 71 * , 1613 ( 2011 ) ; l. basso , s. moretti and g. m. pruna , eur . j. c * 71 * , 1724 ( 2011 ) ; l. basso , arxiv:1106.4462 [ hep - ph ] ; m. lindner , d. schmidt and t. schwetz , phys . b * 705 * , 324 ( 2011 ) ; s. kanemura , t. nabeshima and h. sugiyama , phys . d * 85 * , 033004 ( 2012 ) .

we extend the higgs triplet model so as to include dark matter candidates and a simple suppression mechanism for the vacuum expectation value ( @xmath0 ) of the triplet scalar field . the smallness of neutrino masses can be naturally explained with the suppressed value of @xmath0 even when the triplet fields are at the tev scale . the higgs sector is extended by introducing @xmath1-odd scalars ( an @xmath2 doublet @xmath3 and a real singlet @xmath4 ) in addition to a @xmath1-even complex singlet scalar @xmath5 whose vacuum expectation value violates the lepton number conservation by a unit . in our model , @xmath0 is generated by the one - loop diagram to which @xmath1-odd particles contribute . the lightest @xmath1-odd scalar boson can be a candidate for the dark matter . we briefly discuss a characteristic signal of our model at the lhc .

to explore the interactions of young stellar objects ( ysos ) with their environments , we recently carried out mm - wavelength molecular line observations towards star forming regions ( sfrs ) with well defined and bright high - velocity components . in particular , we mapped the well known cepheus a ( cepa ; * ? ? ? * and references therein ) sfr in several shock - chemistry tracers such as h@xmath0s , so@xmath0 , and hdo @xcite . those results show that the group of b - type stars located in cepa - east producing a hot core @xcite , are also associated with multiple mass loss processes . in particular , beside the already known three flows pointing in the sw , ne , and se directions , a fourth outflow flowing towards the south has been detected thanks to the shock - chemistry tracers . cepa - east can thus be considered an ideal laboratory in which to study how outflow motions affect the gas , from both the kinematical and chemical points of view . @xcite have already presented a multi - species and multi - line mm - survey of the central region of cepa - east where the ysos are located . using the 30-m iram antenna , the authors detected emission in different transitions of 21 molecular species tracing a wide range of physical conditions . analysis of these spectra shows that different molecules exhibit different spectral behaviours and that three classes can be distinguished : ( i ) hot core molecules ( e.g. hc@xmath4o@xmath3 , ch@xmath1c@xmath0h ) emitting only at the velocity of the hot core ( 10.7 km s@xmath2 ) and with no line wings , ( ii ) outflow molecules ( e.g. cs , sio , h@xmath0s , so@xmath0 , and so ) spanning the whole range of observed outflowing velocities so that bright wings are added to the hot core emission , and ( iii ) four species ( ocs , h@xmath0cs , hdo , and ch@xmath1oh ) which are associated with wings and which , in addition , clearly show a redshifted spectral peak at 5.5 km s@xmath2 , well separated from the hot core peak . while the peak at 10.7 km s@xmath2 is tracing the high - density material hosting the ysos and the wings are tracing the multiple outflows , the origin of the redshifted spectral peak is unknown . the @xcite data did not allow us to clarify the spatial distribution of this spectral peak and to establish if it is tracing a small structure or it is related with an extended component . it is worth noting that , as far as we know , this is the first study to reveal duality in the line - wing profiles observed in outflows driven by ysos , i.e that ocs , h@xmath0cs , hdo , and ch@xmath1oh ( hereafter called double - peaked species ) have a different behaviour with respect to cs , sio , h@xmath0s , so@xmath0 , and so ( hereafter called single - peaked species ) . this suggests that the redshifted spectral peak could be tracing a different gas component with respect to the gas outflowing at the other velocities as well as to the hot core gas , and indicates that high angular resolution observations are required for a detailed analysis . in this letter we present observations which clarify the spatial distribution of the redshifted spectral peak at 5.5 km s@xmath2 . in addition , we produce chemical models suggesting that we are probably observing the first direct evidence of turbulent interfaces ( i ) where the outflow gas impinges on and detach dense gas , and ( ii ) in which dynamical mixing and diffusion are occurring between the mass ejected from a newly - formed massive yso and the interstellar cloud from which it was formed . in the following , the kinematical component at 5.5 km s@xmath2 will be referred to as the i - feature . the main goal of the observations was to determine the spatial distribution of the i - feature observed towards cepa - east . in order to select the best molecular tracers , following @xcite , we noted that a good compromise between high angular resolution , a simple spectral pattern , and an intense line emission was represented by h@xmath0cs , and in particular by its 6@xmath55@xmath6 transition at 202923.55 mhz ( @xmath7=47 k ) . thus , we mapped a region of @xmath8 1@xmath9@xmath102@xmath9 in h@xmath0cs(6@xmath55@xmath6 ) on 2004 june with the 30-m iram radiotelescope of pico veleta ( spain ) . we used a sampling of 10@xmath11 around the coordinates of hw2 yso , which is thought to be among the main drivers of the cepa - east outflows and it represents the center of the present map , whereas a spacing of 20@xmath11 was chosen to scan coarsely the outer part of the cloud . the system temperature , @xmath12 , was @xmath8 500 k , the hpbw was 12@xmath11 , while the pointing was accurate to within 2@xmath11 - 3@xmath11 . as spectrometer , we used an autocorrelator with a configuration providing a velocity resolution of 0.06 km s@xmath2 , successively smoothed to 0.92 km s@xmath2 . the spectra were calibrated with the standard chopper wheel method and reported here in units of main - beam brightness temperature ( @xmath13 ) : the average r.m.s . is @xmath8 20 mk . figure 1_a _ reports the velocity channel maps of the h@xmath0cs(6@xmath55@xmath6 ) emission . each panel shows the emission integrated over a velocity interval of 2 km s@xmath2 ; the ambient velocity ( @xmath14 ) is 10.7 km s@xmath2 ( e.g. * ? ? ? * ) . in summary , h@xmath0cs(6@xmath55@xmath6 ) emission is associated ( i ) with the central position , where the yso cluster is located and where a hot core has been detected @xcite , and ( ii ) with the four outflow directions , ne , se , s , and sw , confirming h@xmath0cs as a tracer of high - temperature and/or shocked regions . in particular , the new h@xmath0cs maps confirm that the spatial distribution of the i - feature is not limited to the central position tracing also the sw and southern outflows , as clearly shown by the 5.5 and 3.5 km s@xmath2 panels in fig . examples of h@xmath0cs(6@xmath55@xmath6 ) line profiles are shown in fig . 1_b _ which compares the spectra observed at the ( 0@xmath11,+10@xmath11 ) and ( 0@xmath11,10@xmath11 ) offsets with that observed at the ( 0@xmath11,0@xmath11 ) position @xcite . given the hpbw , the three spectra are sampling different regions of cepa - east . the i - feature is still present in the southern position and it is even more redshifted , approaching the 5 km s@xmath2 velocity , thus suggesting the presence of a velocity gradient . in order to make the same comparison with another tracer of the i - feature , we observed the ch@xmath1oh(5@xmath154@xmath15 ) transition at 241.8 ghz towards the ( 0@xmath11,+10@xmath11 ) and ( 0@xmath11,10@xmath11 ) positions . in this case , @xmath12 @xmath16 650 k , the hpbw = 10@xmath11 , and the resulting spectra were smoothed to 0.39 km s@xmath2 , with a r.m.s . @xmath8 80 mk . fig . 1_c _ compares the profiles due to the high excitation ( @xmath7=115 k ) ch@xmath1oh(5@xmath174@xmath17 a@xmath18 ) transition at 241806.51 mhz , which according to the ( 0@xmath11,0@xmath11 ) spectrum , can be considered as the best tracer of the i - feature among the lines of the ch@xmath1oh(5@xmath154@xmath15 ) pattern . also in this case , the i - feature is still present in the southern position and again it seems more redshifted , in agreement with the h@xmath0cs data . it is worth noting that the two continuous lines across the methanol ( 0@xmath11,0@xmath11 ) spectrum refer to the two velocity components of the ch@xmath1oh(5@xmath194@xmath19 e ) transition ( @xmath7=122 k ) : the 5.5 km s@xmath2 ( 5@xmath204@xmath19 e ) emission is blended with the 10.7 km s@xmath2 ( 5@xmath174@xmath17 a@xmath18 ) feature , producing a broad line . finally , fig . 1_d _ reports the hdo(1@xmath211@xmath22 ) ( @xmath7=47 k ) profiles that were also reported in the @xcite paper . since the angular resolution is definitely higher ( hpbw=31@xmath11 ) we show the spectra taken in the ( 0@xmath11,+20@xmath11 ) and ( 0@xmath11,20@xmath11 ) positions to sample different parts of cepa - east . the hdo data are well in agreement with the picture given above , showing how bright is the i - feature as observed towards the southern position . we conclude that there is a previously undetected structure extending toward the south from the central position giving rise to emission at 5.5 km s@xmath2 in lines of h@xmath0cs , ocs , hdo , and ch@xmath1oh . the structure appears to be at least 0.1 pc in length . in this section we consider possible origins for the extended i - feature at 5.5 km s@xmath2 . since cepa - east is a yso cluster the possibility that this emission comes from a second hot core seems , at first sight , likely . however , this can not be the case because : ( i ) the emission is extended and _ not _ compact as it would be if it came from a hot core ( see fig . 1 ) . in addition , ( ii ) figure 2 shows that what makes the i - feature peculiar is that some line ratios such as ocs / so , h@xmath0cs / so@xmath0 or ch@xmath1oh / h@xmath0s are definitely much higher than those measured towards the 10.7 km s@xmath2 component . in other words , although we can not exlude that conventional hot core tracers , including , so , so@xmath0 , and h@xmath0s , can have some weak emission at 5.5 km s@xmath2 , they are _ not _ tracing the i - feature . finally , ( iii ) we find that the column densities of the double peaked species in the i - feature ( ch@xmath1oh : 3 10@xmath23 @xmath24 ; ocs : 5 10@xmath25 ; h@xmath0cs : 1 10@xmath26 ; hdo : 6 10@xmath26 ; codella et al . 2005 ) , calculated assuming extended emission , are lower that those typically found in hot cores @xcite . we conclude that there are both morphological and chemical reasons for excluding a hot core as the origin of the i - feature . we have also investigated the possibility that the i - feature is simply the emission from a shocked molecular outflow . this possibility has been proposed in the @xcite paper , where the observations were compared with the theoretical calculations recently reported by @xcite . however , a closer inspection of such model suggests that the theoretical abundances of the single - peaked species are not reproduced and that both single and double peaked species abundances , once converted into column densities ( i.e. taking into consideration the geometry ) are more typical of hot cores densities and hence too high . in addition , it is difficult to understand why standard shock tracers like sio . so@xmath0 , and h@xmath0s do not show the same profile observed with the double - peaked species , as suggested by shock - chemistry models and as happens in the well - studied chemical rich outflows bhr71 and l1157 @xcite . a new explanation is therefore required . we now investigate the possibility that the i - feature arises from a molecular interface between the outflow gas and the ambient medium . highly turbulent interfaces should be present in all environments where jets or outflows interact with the surrounding molecular clouds . such interfaces have been shown to be characterized by a chemistry which is quite distinct in nature from that of typical dense cores and would not be achieved by any modification of conventional cold cloud chemistry . all chemical interface models , whether with turbulent mixing or diffusion , explore the consequences of mixing warm largely ionized gas with cold dense and mainly neutral gas ( e.g. * ? ? ? * ; * ? ? ? there is as yet no direct evidence of the existence of such interfaces in high - mass sfrs , while in low - mass sfrs , observations of enhanced hco@xmath3 towards several class 0 objects can be interpreted as coming from the walls of the outflow cavity @xcite . in order to test this hypothesis we make use of the @xcite interface model and attempt to reproduce the different trends displayed by the single - peaked and double - peaked species . note that since the model does not include the deuterium chemistry , we could not calculate the abundance of hdo . the main physical characteristics often adopted for an interface are : low visual extinctions ( @xmath27 1.5 mags ) ; high gas densities ( @xmath28 @xmath16 10@xmath29 - 10@xmath30 @xmath31 ) , high radiation fields compared to the mean interstellar radiation field ( here called one habing ) , short lifetimes ( @xmath27 100 yr ) and relatively high temperatures ( @xmath32 @xmath33 100 k ) due to non - dissociative shocks associated with the highly turbulent interface between the outflow and the molecular core . note that although timescales may be very short , the continual erosion of dense material by the wind or jet re - supplies the interface , so a near steady - state pertains . the temperature and density of the region , as deduced from the molecular observations @xcite , are @xmath8 200 k and 10@xmath3410@xmath30 @xmath31 , respectively . we have therefore run a small grid of models using these values and varying the visual extinction ( 0.5 - 1.5 mags ) and the radiation fields ( 5 - 100 habing ) . a further parameter that we investigate is the form that sulfur takes once depleted on the grains , before the mantles are evaporated . we have considered 100% in solid h@xmath0s , 100% in solid ocs and several mixtures in between . note that while it is often assumed that most simple species , once frozen out , react with hydrogen atoms to form saturated species , recently doubts have been raised regarding the form that sulfur takes once it depletes onto the grains . in particular , the detection of solid ocs @xcite and the non - detection of h@xmath0s ices @xcite have raised the possibility that ocs may be the main reservoir of sulfur on the grains . it is not unrealistic to assume that before evaporation most of the sulfur is in the form of ocs : studies of the effects of uv on grains ( e.g. * ? ? ? * ) have shown that species such as h@xmath0o for example , are easily dissociated by radiation ; once the oxygen is freed , it is believed to react with co , which is the third most abundant species on the grains , to form co@xmath0 . it is not unreasonable to assume that a similar process occurs with h@xmath0s : i.e. that once atomic sulfur is freed it reacts efficiently with co to form ocs . the aim of this exercise is not to match perfectly the observed column densities but to see whether there is an epoch when a molecular outflow - ambient interface could give rise to high abundances of ch@xmath1oh , ocs and h@xmath0cs while so , so@xmath0 and h@xmath0s remained low . figure 3 shows one of our best fit models , where , in general , the behaviour is as expected : the double - peaked species show a clear anti - correlation with respect to the single - peaked family . note that in any case the molecular abundances of all the species ultimately ( @xmath33 100 yr ) decline as the radiation field induces photodissociation , but , as mentioned earlier , the interface is turbulent and therefore a continuous erosion of material will constantly replenish the interface gas . a radiation field larger than 10 habing seems to destroy most of the doubly - peaked species too early . a density lower than 10@xmath34 @xmath31 does not produce enough methanol . but the most interesting constraint is the amount of sulfur ( at least 90% ) that needs to be in solid ocs before evaporation occurs . thus , these calculations suggest that the observations are consistent with ( but do not prove ) a model of an outflow - ambient turbulent interface with the following physical conditions : @xmath28 @xmath16 10@xmath3410@xmath30 @xmath31 , @xmath32 @xmath8 200 k , and a radiation field between 5 and 10 habing . according to this modelling , our observational data may be evidence of the presence of a molecular interface in high mass star forming regions . as well as indicating high abundance of the double - peaked species , our model : 1 . requires high abundances of solid ocs , and an entrainment of sufficient material into the interface on timescales of 10 - 50 years . while it is feasible that ocs on ices is abundant in an environment where the uv radiation field is strong , it is clear that experimental as well as observational studies on ices are desirable to confirm such picture ; 2 . predicts high abundances of other species , including hco@xmath3 , h@xmath0co , and nh@xmath1 . to confirm the presence of a molecular interface between outflow and ambient gas , we thus suggest further observations to map cepa - east in these species as well as ocs lines . bachiller , r. , & perz gutirrez , m. 1999 , , 487 , l93 codella , c. , bachiller , r. , benedettini , m. , & caselli , p. 2003 , , 341 , 707 codella , c. , bachiller , r. , benedettini , m. , caselli , p. , viti , s. , & wakelam , v. 2005 , , 361 , 244 garay , g. , ramrez , s. , rodrguez , l.f . , curiel , s. , & torrelles , j.m . 1996 , , 459 , 193 garay , g. , khnenkamp , i. , bourke , t.l . , rodrguez , & lehtinen , k.k . 1998 , , 509 , 768 hogerheijde , m.r . , van dishoeck , e.f . , blake , g.a . , & van langevelde , h.j . 1998 , , 502 , 315 jacq , t. , walmsley , c.m . , henkel , c. , baudry , a. , mauersberger , r. , & jewell , p.r . 1990 , , 228 , 447 lim , a.j . , rawlings , j.m.c . , & williams , d.a . 2001 , , 376 , 336 mannella , v. , palumbo , m.e . , & baratta , g.a . 2004 , , 615 , 1073 martn - pintado , j. , jimnez - serra , i. , rodrguez - franco , a. , martn , s. , & thum , c. 2005 , , 628 , l61 millar , t.j . , & hatchell , j. , 1998 , in chemistry and physics of molecules and grains in space , faraday discussions n. 109 , the faraday division of the royal society of chemistry , london , 15 palumbo , m.e . , geballe , t.r . , & tielens , a.g.g.m . 1997 , , 479 , 839 rawlings , j.m.c . , & hartquist , t.w . 1997 , , 487 , 672 rawlings , j.m.c . , taylor , s.d . , & williams , d.a . 2000 , , 313 , 461 redman , m.p . , rawlings , j.m.c . , yates , j.a . , & williams , d.a . 2004 , , 352 , 243 van dishoeck , e.f . , & blake , g.a . 1998 , , 36 , 317 viti , s. , natarajan , s. , & williams , d.a . 2002 , , 336 , 797 wakelam , v. , caselli , p. , herbst , e. , ceccarelli , c. , & castets , a. 2004 , , 422 , 159

we present new observations of the cepa - east region of massive star formation and describe an extended and dynamically distinct feature not previously recognised . this feature is present in emission from h@xmath0cs , ocs , ch@xmath1oh , and hdo at 5.5 km s@xmath2 , but is not traced by conventional tracers of star forming regions h@xmath0s , so@xmath0 , so , cs . the feature is extended up to at least 0.1 pc . we show that the feature is neither a hot core nor a shocked outflow . however , the chemistry of the feature is consistent with predictions of a model of an eroding interface between a fast wind and a dense core ; mixing between the two media occurs in the interface on a timescale of 10 - 50 years . if these observations are confirmed by detailed maps and by detections in species also predicted to be abundant ( e.g. hco@xmath3 , h@xmath0co , and nh@xmath1 ) this feature would be the first detection of such an interface in regions of massive star formation . an important implication of the model is that a significant reservoir of sulfur in grain mantles is required to be in the form of ocs .

for general notations and concepts in graph theory , we refer to @xcite , @xcite and @xcite . all graphs mentioned in this paper are simple , connected undirected and finite , unless mentioned otherwise . a _ hole _ of a simple connected graph @xmath0 is a chordless cycle @xmath3 , where @xmath4 , in @xmath0 . the _ girth _ of a simple connected graph @xmath0 , denoted by @xmath5 , is the order of the smallest cycle in @xmath0 . the following notions are introduced in @xcite . @xcite a _ primitive hole _ of a graph @xmath0 is a cycle of length @xmath1 in @xmath0 . the number of primitive holes in a given graph @xmath0 is called the _ primitive hole number _ of that graph @xmath0 . the primitive hole number of a graph @xmath0 is denoted by @xmath6 . @xcite the _ primitive degree _ of a vertex @xmath2 of a given graph @xmath0 is the number of primitive holes incident on the vertex @xmath2 and the primitive degree of the vertex @xmath2 in the graph @xmath0 is denoted by @xmath7 . some studies on primitive holes of certain graphs have been made in @xcite . the number of primitive holes in certain standard graph classes , their line graphs and total graphs were determined in this study . some of the major results proved in @xcite are the following . @xcite the number of primitive holes in a complete graph @xmath8 is @xmath9 . @xcite for any subgraph @xmath10 of a graph @xmath0 , we have @xmath11 . moreover , if @xmath0 is a graph on @xmath12 vertices , then @xmath13 . in this paper , we introduce the notion of set - graphs and study certain characteristics of set - graphs and also present a number of interesting results related to graph properties and invariants . a set - graph is defined as follows . [ d - sg ] let @xmath14 be a non - empty set and the @xmath15-th @xmath16-element subset of @xmath17 be denoted by @xmath18 . now consider @xmath19 . the _ set - graph _ corresponding to set @xmath17 , denoted @xmath20 , is defined to be the graph with @xmath21 and @xmath22 , where @xmath23 . it can be noted from the definition of set - graphs that @xmath24 and if @xmath25 is a singleton , then @xmath20 to be the trivial graph . hence , all sets we consider here are non - empty , non - singleton sets . let us now write the vertex set of a set - graph @xmath20 as @xmath26 , where @xmath16 is the cardinality of the subset @xmath27 of @xmath17 corresponding to the vertex @xmath28 . the following result is perhaps obvious , but an important property of set - graphs . if @xmath0 is a set - graph , then @xmath0 has odd number of vertices . let @xmath0 be a set - graph with respect to the set @xmath17 . it is to be noted the number of non - empty subsets of @xmath17 is @xmath29 . since every vertex of @xmath0 corresponds to a non - empty subset of @xmath17 , the number of vertices in @xmath0 must be @xmath29 , an odd integer . consider the set - graph with respect to the set @xmath30 . here we have the subsets of @xmath31 which are @xmath32 . then , the vertices of @xmath33 have the labeling as follows . @xmath34 . figure [ fig-1 ] depicts the above mentioned labeling procedure of the set - graph @xmath33 . [ t - sgdv ] let @xmath20 be a set - graph . then , the vertices @xmath35 of @xmath20 , corresponding to subsets @xmath36 and @xmath37 in @xmath38 of equal cardinality , have the same degree in @xmath20 . that is , @xmath39 . consider the set - graph @xmath40 . we begin by considering the vertices of @xmath0 corresponding to the @xmath12 singleton subsets of @xmath17 . let these vertices be denoted by @xmath41 , where @xmath42 . clearly , for all @xmath43 , we have @xmath44 . hence , by the definition of set - graphs , it follows that no edges are induced amongst the vertices @xmath45 . now , construct all the two element subsets of @xmath17 . now choose two arbitrary vertices @xmath46 and @xmath47 , where @xmath48 . then , here we have the subsets of @xmath17 of the form @xmath49 , for @xmath50 . it can be observed that the subsets of the form @xmath49 and @xmath51 are the elements of @xmath38 , where @xmath52 . moreover , @xmath53 for all @xmath54 . in a similar way , we can extend this argument for the sets @xmath55 and an arbitrary subset of @xmath17 containing the element @xmath56 . that is , the vertex @xmath41 is adjacent to those vertices of @xmath0 whose corresponding sets have @xmath57 elements including the common elements @xmath56 , for @xmath58 . therefore , @xmath59 . since the choice of @xmath15 is arbitrary , we have @xmath60 for all @xmath61 . therefore , the result holds for @xmath62 . now , assume that the result holds for @xmath63 , where @xmath64 is a positive integer . that is , we have @xmath65 for all @xmath66 . next , consider the vertices of @xmath0 corresponding to the @xmath67-element subsets of @xmath17 . let @xmath68 be a @xmath67-element subset of @xmath17 and let @xmath69 be the vertex of @xmath0 corresponding to the set @xmath68 . let @xmath70 be an arbitrary element of the set @xmath68 and let @xmath71 . then , the vertex @xmath69 is adjacent to the vertices of @xmath0 corresponding to the sets containing the element @xmath70 in addition to the vertices of @xmath0 corresponding to the proper subsets of @xmath68 and @xmath72 . hence , the difference between the number of edges incident on @xmath69 and the number of edges incident on the vertex @xmath73 corresponding to the set @xmath72 is equal to the number of subsets of @xmath17 containing the element @xmath70 , other than @xmath68 . this number is a constant for any set of @xmath67-element sets . therefore , @xmath74 for all @xmath75 . that is , the result is true for @xmath76 if it is true for @xmath63 . therefore , the theorem follows by induction . a question that arouses much interest in this context is what the degree of an arbitrary vertex of a set - graph @xmath20 . the following result provides a solution to this problem . let @xmath0 be a set - graph with respect to a non - empty set @xmath77 and let @xmath78 be an arbitrary vertex of @xmath0 corresponding to an @xmath64-element subset of @xmath17 . then , @xmath79 , where @xmath80 is an indexing set such that @xmath81 and @xmath82 is the collection of subsets of @xmath17 containing the element @xmath83 . let @xmath0 be a set - graph with respect to a non - empty set @xmath17 . without loss of generality , let @xmath84 be a @xmath64-element subset of @xmath17 , say @xmath85 and let @xmath78 be the vertex of @xmath0 corresponding to the set @xmath84 . therefore , the vertex @xmath78 is adjacent to the vertices of @xmath0 which correspond to the subsets of @xmath17 , containing the at least one element of @xmath84 . that is @xmath86 . but , by principle of inclusion and exclusion of sets , we have @xmath87 , where @xmath81 . therefore , @xmath79 , where @xmath81 . determining the degree of vertices of a set - graph is an important and interesting problem at this time . the following result determines a lower and upper limits for the degree of vertices of a given set - graph . [ t - sgdv1 ] for any vertex @xmath88 of a set - graph @xmath89 , we have @xmath90 . let @xmath89 be a set - graph with respect to a non - empty set @xmath17 . here , we need to consider the following two cases . it is to be noted that the vertices of @xmath0 corresponding to singleton subsets of @xmath0 have the minimum degree in @xmath0 . without loss of generality , let the vertex @xmath88 of @xmath0 corresponds to the set @xmath55 . then , @xmath88 should be adjacent to the vertices of @xmath0 corresponding to the subsets of @xmath17 , other than itself , containing the element @xmath56 . therefore , degree of the vertex @xmath88 is equal to the the number of @xmath57-element subsets of @xmath17 containing the element @xmath56 for @xmath91 . by binomial theorem , the total number of subsets of an @xmath12-element set , containing a particular element is @xmath92 . therefore , the minimum degree of a vertex in @xmath0 is @xmath93 . note that we need to consider @xmath94 of the subsets of @xmath17 only excluding @xmath95 . hence , the final vertex @xmath96 of the graph @xmath89 corresponding to the set @xmath17 in @xmath38 will be adjacent to all its preceding vertices . since @xmath0 has @xmath29 vertices , @xmath97 . no other vertices in @xmath0 can be adjacent to all other vertices of @xmath0 , the vertex @xmath96 has the maximum possible degree in @xmath0 . that is , the maximum degree of a vertex in @xmath0 is @xmath98 . the following results are immediate consequences of the above theorem . for any set - graph @xmath89 , @xmath99 . from the proof the above theorem , we have @xmath100 and @xmath101 . this completes the proof . [ c - udg ] there exists a unique vertex @xmath96 in a set - graph @xmath20 having degree @xmath102 . the proof follows from case-2 of theorem [ t - sgdv1 ] . the following result indicates the nature of the minimal and maximal degrees of the vertices of a set - graph . the maximal degree of vertex in a set - graph @xmath0 is always an even number and the minimal degree of a vertex in @xmath0 is always an odd number . let @xmath0 be a set - graph with respect to a non - empty set @xmath17 . then , by theorem [ t - sgdv1 ] , the maximum degree of a vertex in a set - graph @xmath0 is @xmath103 , which is always an even number and the minimal degree of a vertex @xmath0 is @xmath104 , which is always an odd number . we have already proved that the vertex of the set - graph @xmath0 corresponding to the set @xmath17 itself has the maximum degree @xmath105 in @xmath0 . analogous to this result , we propose the following result on the primitive degree of this vertex @xmath96 . for set - graph @xmath89 , the primitive degree of the vertex corresponding the set @xmath17 is @xmath106 . let @xmath89 be a set - graph with respect to the set @xmath17 . consider the subgraph @xmath107 . by theorem [ t - sgdv ] , we have @xmath108 and hence @xmath109 . furthermore , both ends ends of every edge @xmath110 are adjacent to vertex @xmath96 in @xmath20 . hence , each such edge @xmath111 in @xmath112 corresponds to a primitive hole @xmath113 in @xmath20 on the vertices @xmath114 . hence , @xmath115 . in the result given below , we describe a recursive formula to determine the number of edges of a set - graph . [ t - sgrd ] for a set - graph @xmath116 we have 1 . @xmath117 2 . @xmath118 . consider the set - graph @xmath20 . to extend it to @xmath116 , we proceed in five steps as explained below . 1 . replicate the vertices of @xmath20 as an _ edgeless _ graph and add the new element @xmath119 as an element to all subsets corresponding to all @xmath120 and label each _ replica vertex _ , @xmath121 . also add the new vertex @xmath122 corresponding to the single element subset @xmath123 . apply the definition of a set - graph to these new vertices . clearly , we obtain the complete graph @xmath124 . each vertex @xmath120 corresponding to the subset @xmath18 can be linked with its replica vertex corresponding to the new subset @xmath125 . we refer to _ parallel linkage _ and exactly @xmath126 such edges are added . 4 . for the ends of each edge in @xmath127 say @xmath128 and @xmath129 with corresponding subsets say , @xmath130 and @xmath131 we have @xmath132 and @xmath133 . so the edges @xmath134 and @xmath135 , with @xmath136 corresponding to subsets @xmath137 and @xmath138 respectively , exist . hence @xmath139 additional edges are linked . relabel the vertices according to the definition [ d - sg ] to obtain the set - graph @xmath116 . the summation of the edges added through steps ( i ) to ( v ) plus the existing edges of @xmath20 provides the result : @xmath117 . the second part of the theorem is an immediate consequence of the above proof of first part . then , the proof is complete . the following is a result related to the largest complete graphs found in a set - graph . [ p - sgkn ] the set - graph @xmath140 has exactly two largest complete graphs , @xmath141 . consider the set - graph @xmath142 and extend to @xmath20 . from step ( ii ) in the proof of theorem [ t - sgrd ] , we construct a largest complete graph amongst the replica vertices and the new vertex @xmath143 because no vertex of @xmath142 is linked to @xmath144 . however , the erstwhile vertex @xmath145 of @xmath142 is also linked to all the _ replica _ vertices hence , inducing the complete graph @xmath141 . clearly , another largest complete graph does not exist . the primitive hole number of a set - graph is determined recursively in the following theorem . for a set - graph @xmath146 we find the number of primitive holes through the recursive formula @xmath147 . consider the set - graph @xmath20 whose primitive hole number is denoted by @xmath148 . extending @xmath20 to @xmath116 will only increase the number of primitive holes . what we need here to determine the number of additional primitive holes formed on extending @xmath20 to @xmath116 . this calculation is done as follows . the set of replica vertices together with the vertex @xmath149 induce a complete subgraph @xmath150 and hence an additional @xmath151 primitive holes are added to the extended graph . finally , for each edge @xmath152 the vertices @xmath153 induce a @xmath154 subgraph and the number of primitive holes thus formed is @xmath155 . hence , a further @xmath156 primitive holes are added to the extended graph . this completes the proof . next , we introduce the following notions for a set - graph as follows . let @xmath89 be a set - graph on a non - empty set @xmath17 and let @xmath18 be an arbitrary subset of the set @xmath17 . characteristic function _ of a subset @xmath157 of @xmath17 with respect to @xmath18 , denoted by @xmath158 , is defined as @xmath159 the _ tightness number _ of a subset @xmath160 , denoted @xmath161 is the number of subsets distinct from @xmath130 for which the intersection with @xmath130 is non - empty . hence , @xmath162 . we note that in terms of the definition of a set - graph and for the vertex @xmath88 corresponding to the subset @xmath130 we have , @xmath163 . also , we have that @xmath164 . the next theorem enables us to employ a step - wise recursive formula to determine the tightness number of all non - empty subsets of @xmath165 if the the tightness number of all non - empty subsets of @xmath17 are known . consider a set - graph @xmath166 and its extended set - graph @xmath116 . then we have 1 . @xmath167 , 2 . for each erstwhile subset @xmath36 with @xmath168 in @xmath20 , we have @xmath169 in @xmath116 , 3 . for a replica vertex say , @xmath121 representing the new subset @xmath125 we have @xmath170 . let @xmath20 be a set - graph and @xmath116 be its extended graph obtained by introducing a new element , say @xmath119 , to the set @xmath17 . then , 1 . to generate the set - graph @xmath116 by extending the set - graph @xmath20 , we initially add the subsets @xmath123 and @xmath125 , for all applicable values of @xmath16 and @xmath15 . clearly , @xmath171 . so , @xmath172 . also , @xmath173 . therefore , we have @xmath174 . 2 . if @xmath175 in @xmath20 , the subset @xmath18 has non - zero intersections with exactly @xmath64 distinct subsets of @xmath17 . since in the replication , we have @xmath176 together with the subsets @xmath177 , for all @xmath64 , the result follows . the replica vertices are @xmath29 in number and induce a complete graph together with vertex @xmath122 . this partially represents @xmath94 non - zero intersections in respect of any subset say , @xmath176 corresponding to any replica vertex say , @xmath121 . clearly , @xmath178 and @xmath179 , for all @xmath180 . it implies that an additional @xmath67 non - zero intersections exist in respect of @xmath176 . hence , @xmath181 . this completes the proof . let @xmath0 be a given non - trivial finite graph . the _ chromatic number _ , denoted by @xmath182 , of @xmath0 is the minimum @xmath64 for which @xmath0 is @xmath64-colourable . the chromatic number of a set - graph @xmath20 is @xmath183 . it is easy to see that @xmath184 , @xmath185 , @xmath186 ( see figure [ fig-1 ] ) . assume the result holds for the set - graph @xmath187 therefore , we have @xmath188 . now consider the set - graph @xmath189 . from the steps to be followed to extend from @xmath187 to @xmath189 ( see proof of theorem [ t - sgrd ] ) , we have the erstwhile vertices of @xmath187 , and in addition , the replica vertices corresponding to the vertices of @xmath187 and one more vertex @xmath190 . from the proof of proposition [ p - sgkn ] we can notice that the replica vertices and vertex @xmath190 induce a largest complete subgraph , @xmath191 in the extended graph of @xmath20 . we also note that the replica vertices and vertex @xmath192 form a second largest complete subgraph , @xmath191 . since the vertices @xmath192 and @xmath193 are not adjacent in @xmath20 , both of them have the same colour , say @xmath194 and colour the replica vertices by the colours @xmath195 . since , no other largest complete graph exists it is always possible to find at least one pair of erstwhile - replica vertices which are non - adjacent . hence the erstwhile vertex can carry the colour of such a replica vertex . this can be done in such a way that two adjacent erstwhile vertices do not carry the same colour by using the colours @xmath196 accept for the colour of @xmath96 , exhaustively . so the result @xmath197 follows . hence , the main result follows by induction . an _ independent set _ of graph @xmath0 is a set of mutually non - adjacent vertices of @xmath0 . the _ independence number _ , denoted by @xmath198 , of @xmath0 is the cardinality of a maximal independent set of @xmath0 . the independence number of a set - graph is determined in the following theorem . the independence number of a set - graph @xmath20 is @xmath199 . let @xmath200 be a given set - graph . then , as explained in the proof of theorem [ t - sgdv ] , the vertices @xmath201 , corresponding to the singleton subsets of @xmath17 , are pairwise non - adjacent . hence , the set @xmath202 is an independent set . by the definition [ d - sg ] , we note that any vertex in @xmath203 is adjacent to at least one vertex in @xmath204 . therefore , @xmath204 is the maximal set of mutually non - adjacent vertices and hence is the maximal independent set in @xmath0 . hence , @xmath199 . a _ domianting set _ of a graph is a set of vertices @xmath205 such that every vertex of @xmath0 is either in @xmath205 or is adjacent to at least one vertex in @xmath205 . the _ domination number _ , denoted by @xmath206 , of a graph @xmath0 is the cardinality of the minimal dominating set of @xmath0 . the following discusses the domination number of a set - graph . the domination number of a set - graph@xmath20 is @xmath207 . let @xmath89 be a given set - graph . by corollary [ c - udg ] , the vertex @xmath96 , corresponding to the @xmath12-element set @xmath17 , is the unique vertex in the set - graph @xmath0 that is adjacent to all other vertices of the set - graph @xmath0 . therefore , the singleton set @xmath208 is the minimal set such that every vertex of @xmath0 is adjacent to the unique element in the set @xmath205 . therefore , @xmath209 . another parameter we consider here is the bondage number of a graph @xmath0 , which is denoted by @xmath210 defined as the minimum number of edges to be removed to increase the domination number @xmath206 by @xmath211 . the bondage number of a set - graph @xmath20 is @xmath212 . since @xmath208 is the minimal dominating set of the set - graph @xmath89 , the removal of any edge @xmath213 will increase the domination number by @xmath211 in the reduced graph @xmath214 . another parameter we are going to discuss here is the mcphersion number of undirected graphs . for this , let us now recall the definition mcphersion number , as given in @xcite . @xcite the _ mcpherson recursion _ is a series of _ vertex explosions _ such that on the first iteration a vertex @xmath215 explodes to arc ( directed edges ) to all vertices @xmath216 for which the edge @xmath217 , to obtain the mixed graph @xmath218 . now @xmath218 is considered on the second iteration and a vertex @xmath219 may explode to arc to all vertices @xmath220 if edge @xmath221 and arc @xmath222 or @xmath223 . the _ mcpherson number _ , denoted by @xmath224 , of a simple connected graph @xmath0 is the minimum number of iterative vertex explosions say @xmath225 , to obtain the mixed graph @xmath226 such that the underlying graph @xmath227 . the mcpherson number of a set - graph @xmath20 is determined in the following theorem . for a set - graph @xmath20 we have @xmath228 . consider the set - graph @xmath142 which has @xmath93 vertices . on extending the set - graph @xmath20 , the replica vertices , as explained in theorem [ t - sgrd ] , together with vertex @xmath96 induce a complete subgraph and hence no further vertex explosions are required to ensure complete induced by these vertices . however , at least all the erstwhile vertices require vertex explosions to ensure complete connectivity amongst themselves and the replica vertices . hence , @xmath229 . we have discussed particular types of graphs called set - graphs and studied certain characteristics and structural properties of these graphs . the study seems to be promising as it can be extended to certain standard graph classes and certain graphs that are associated with the given graphs . more problems in this area are still open and hence there is a wide scope for further studies . now , we have the notion of mla numbers as follows . finding other number theoretical results for _ mla numbers _ are also challenging problems which seems to be promising . all these facts indicate that there is a wide scope for further research in this area .

a _ primitive hole _ of a graph @xmath0 is a cycle of length @xmath1 in @xmath0 . the number of primitive holes in a given graph @xmath0 is called the primitive hole number of that graph @xmath0 . the primitive degree of a vertex @xmath2 of a given graph @xmath0 is the number of primitive holes incident on the vertex @xmath2 . in this paper , we introduce the notion of set - graphs and study the properties and characteristics of set - graphs . we also check the primitive hole number and primitive degree of set - graphs . interesting introductory results on the nature of order of set - graphs , degree of the vertices corresponding to subsets of equal cardinality , the number of largest complete subgraphs in a set - graph etc . are discussed in this study . a recursive formula to determine the primitive hole number of a set - graph is also derived in this paper . * key words : * set - graphs , primitive hole , primitive degree . * ms classification * : 05c07 , 05c38 , 05c78 .

general relativity ( gr ) is widely accepted as a fundamental theory to describe the geometric properties of spacetime @xcite . in an homogeneous and isotropic spacetime , the so - called friedmann - robertson - walker ( frw ) model , the einstein field equations give rise to the friedmann equations that describe the evolution of the universe . it describes the universe from around one second after the big bang to the present matter dominated era . this standard model s success is in part due to several of its predictions having been verified by observation . for example , the abundances helium with respect to other light elements observed in the universe agrees well with the predictions of this model . the period of recombination is strongly supported by the cmb which is arguably the strongest evidence supporting the standard model . a good amount of observational data indicate quite clearly that the present universe is in an accelerated expanding phase @xcite . the universe may enter in a so - called phantom era with an effective equation of state parameter @xmath7 less than @xmath8 . the simplest way to explain this phantom dark energy era is based on the introduction of a scalar field with negative kinetic energy @xcite . the main property of such a phantom field in the accelerating frw universe is the appearance of finite - time future singularity @xcite ( see @xcite for the classification of singularities ) . in turn , this can lead to bizarre consequences such as negative entropy of the universe @xcite . however , gr is not the only relativistic theory of gravity . in the last decades several generalizations of einstein field equations has been proposed @xcite . within these extended theories of gravity nowadays , a subclass known as @xmath9 theory are an alternative for classical problems , as the accelerated expansion of the universe , instead of the dark energy and quintessence models @xcite . then , an effective phantom phase can be realized without a scalar phantom . @xmath9 theories of gravity are basically extensions of the einstein - hilbert action with an arbitrary function of ricci scalar @xmath0 @xcite . there are three formalisms in deriving field equations from the action in f(r ) gravity . the first is the standard metric formalism in which the field equations are derived by the variation of the action with respect to the metric tensor @xmath10 . in this formalism the affine connection @xmath11 depends on @xmath10 . the second is the palatini formalism @xcite in which the metric and the connection are treated as independent variables when the action is varied . the third is the metric - affine formalism is which the gravitational action is a general function of the scalar curvature while the matter action is allowed to depend also on the connection @xcite . note that these approaches give different field equations in f(r ) gravity , while for the gr action they are identical with each other . the methodology leads to a boundary contribution which is usually dropped out setting null fluxes through gauss - stokes theorem @xcite . in this paper we will use the metric formalism for obtaining the field equations . as we mentioned above , in einstein gravity , when the dark energy is introduced for explaining the late acceleration of the universe , finite time singularities can appear . note that even in f(r ) gravity where we do not need to introduce the dark energy for explaining the late acceleration , finite time singularities can also appear in a background driven by a dark fluid . the question to be asked is : does particle creation can avoid these singularities or modify their nature ? particle production phenomenon in f(r ) is then analysed in this paper and its impact as quantum effects is checked at singularity time . particle production from the vacuum by the gravitational field using quantum field theory in an expanding universe has been introduced firstly by parker in the 1960s @xcite . one of the interesting results of this work is that in a radiative dominated expanding universe there is no massless particle creation due to the conformal invariance of the metric . latter , quantum process of particle production has been studied by several authors , done in the course of the cosmological expansion @xcite . recently , various investigations in the aim of dealing with singularities have been done . batista and collaborators @xcite studied the effects of particle creation when a massless scalar field is minimally coupled with the einstein gravity . they found that the energy density of created particles never dominates the phantom energy density . in the same way , quantum effects near the big rip are studied in @xcite where they used the @xmath6-wave regularization for calculating the energy density of particle creation and found that , in this case , it tends to infinity when the big rip is approached and becomes the dominant component of the universe . this means that big rip can be avoided . pavlov @xcite computed both the number density of created particles and the stress - energy tensor for a conformally coupled massive field for the case in which @xmath12 . he found that quantum effects are not important if the field mass is much smaller than the planck mass and the time left to the big rip is greater than the planck time . bates and anderson @xcite used a background field approach in which the energy densities of the quantized fields are computed in the background spacetime which contains the big rip singularity . they found that for fields in realistic states for which the energy density of the quantized fields is small compared to that of the phantom energy density at early times , and for spacetimes with realistic values of @xmath7 , there is no evidence that quantum effects become large enough to significantly affect the expansion of the spacetime until the spacetime curvature is of the order of the planck scale or larger , at which point the semi classical approximation breaks down . the calculations of particle production is usually done comparing the particle number at asymptotically early times , or with respect to vacuum state defined in two different frames . however , treating of quantizaton in curved space the main problem concerns the field theory interpretation in terms of particles . note that in curved spacetime there is no poincar group symmetry and the notion of vacuum becomes ambiguous . the problem may be solved using the diagonalization method of instantaneous hamiltonian by a bogoliubov transformation which leads to a finite results for the created particles number @xcite . particles are created because the modes with positive and negative frequency of the field become mixed during the universe expansion . particle production is directly connected with the curvature of the universe and then when the field equation is put in the form of harmonic oscillator equation without friction part , its effect appears in the effective mass . hence , in the radiative dominated universe with massless scalar field , either of zero or non - conformal coupling , the effective mass vanishes and the modes are not mixed in the course of the expansion due to conformal invariant of the metric and as consequence there is no creation of massless particles . note also that in the static universe where the scale factor is constant and the energy density of ordinary matter is null , even the field is massive and minimally or conformally coupled with einstein gravity , the effective mass vanishes and once again there is no particle production . however , even the density of ordinary matter is null and the modified gravity is taken into account , this could not hold . under some assumptions the scale in such a condition could not be constant and as consequence , particle could be created . in this paper we use the ansatz that the function @xmath13 and analyse the phenomenon of particle creation . in a first step we verify a known result that the particle production is the same as in einstein gravity and this is seen through the scale factor behaviour . we consider the main content of the universe as the dark fluid and analyse particle production phenomenon . note that with this dark fluid driven the background , the universe may present the big rip . quantum effects from particle creation is then checked near this singularity , comparing the classical energy density of the dark fluid with the renormalised energy density from the stress tensor corresponding to a massless minimally couple scalar field . we find that depending on the values of the parameter @xmath6 , the big rip can be avoided . then , we find that for @xmath14\frac{1}{2},1[$ ] the big rip can be removed whereas for @xmath4 it remains . in the second step an attention is devoted to the cardy - verlinde formula based on inhomogeneous equation of state . verlinde @xcite made an interesting proposal that cardy formula @xcite in two - dimensional conformal field theory can be generalized to arbitrary spacetime dimensions . verlinde further proposed that a closed universe has subextensive ( casimir ) contribution to its energy and entropy with the casimir energy conjectured to be bounded from above by the bekenstein - hawking energy and as consequence , one obtains a very deep relation between gravity and thermodynamics @xcite . within the context of the radiation dominated universe , such bound on the casimir energy is shown to lead to the hubble and the bekenstein entropy bounds respectively for the strongly and the weakly self - gravitating universes . the generalized entropy formula , called the cv formula , is further shown to coincide with the total entropy of the universe coming from the friedmann equations . these results were later generalized @xcite . our goal here is to analyse the equivalence between the cv formula and the total entropy coming from the generalized friedmann equations in f(r ) theory . hence , we find that for the inhomogeneous eos , the cv formula can always be recovered from the generalized entropy of the universe in einstein gravity but only for @xmath5 in f(r ) gravity . moreover we check the behaviour of the cv formula terms near the singularity and find particularly that the casimir energy decreases as the universe expands and vanishes near the singularity . the paper is organized as follows . in the second section a breve notion on particle production in expanding universe is addressed with the use of the diagonalization of instantaneous hamiltonian by bogoliubov transformation . in the third section we present the view of modified gravity about particle production and also quantum effects near the big rip are analysed . in the fourth section the equivalence between the total entropy of the universe and the cv formula is checked and an analysis of the quantum effects on cv formula terms near the singularity is also investigated . the conclusions and perspectives are presented in the final section . let @xmath15 denotes a scalar field of mass m , and @xmath16 the scale factor for spatially flat homogeneous and isotropic frw spacetime with conformal line element @xmath17 the lagrangian density for this massive scalar field minimally coupled with the gravitational field in a conformal spacetime reads @xmath18 where @xmath19 is minkowski metric . the corresponding field equation reads @xmath20 we can decompose the real scalar field into the modes as @xmath21\quad.\end{aligned}\ ] ] the mode functions now satisfy the equation @xmath22 where the prime denotes the derivative with respect to the conformal time @xmath23 . the modes @xmath24 satisfy to the wronskian relation @xmath25 the system is quantized in a standard fashion by treating the field @xmath26 as an operator , imposing the equal - time commutation relations @xmath27= i\delta^3(\vec{x}-\vec{x}^{\,\prime})\,\,,\end{aligned}\ ] ] where @xmath28 is the canonical momentum . then , the operators @xmath29 and @xmath30 satisfy the usual commutation relations @xmath31=\delta^3(\vec{k}-\vec{k}^{\,\prime})\,\quad,\quad[a_{\vec{k}},a_{\vec{k}^{\prime}}]=[a^{\dagger}_{\vec{k}},a^{\dagger}_{\vec{k}^{\prime}}]=0 \,\,.\end{aligned}\ ] ] the vacuum state is then defined as the state for which @xmath32 other states are built up from this by acting on it various combinations of creation operators @xmath33 . the hamiltonian corresponding to ( [ vincent2 ] ) is @xmath34+a^4m^2\phi^2\right\}\quad.\end{aligned}\ ] ] making use of ( [ vincent4 ] ) , the hamiltonian can be re - written in terms of creation and annihilation operators and the mode functions as @xmath35\quad,\end{aligned}\ ] ] where @xmath36 and @xmath37 are defined by @xmath38 with @xmath39 . in general , the field may be decomposed into many different complete set of modes and each of these has its own vacuum state . suppose we label one such set of modes as @xmath40 . then , in terms of these modes the field is @xmath41\quad,\end{aligned}\ ] ] and the vacuum state is defined by @xmath42 for all @xmath43 . the hamiltonian in this case is written as @xmath44\quad,\end{aligned}\ ] ] where @xmath45 and @xmath46 are similar expressions as in ( [ vincent12 ] ) and ( [ vincent13 ] ) respectively , replacing @xmath24 by @xmath47 . because of completeness , the two sets of modes @xmath48 and @xmath49 are related by the bogolubov transformations that diagonalize the hamiltonian i.e. , @xmath50 , satisfying the relation @xmath51 with the normalization condition @xmath52 , where @xmath53 and @xmath54 are constants and called bogolubov coefficients . one can compare the two vacuum states by noting that the number operator for the barred states is @xmath55 . taking its expectation value with respect to the unbarred vacuum , one finds @xmath56 thus , the number of barred particles in the unbarred vacuum in the mode @xmath43 is @xmath57 . similarly , the number of unbarred particles in the barred vacuum in the mode @xmath43 is @xmath57 . since the diagonalization imposes @xmath50 , using ( [ vincent12 ] ) with @xmath24 replaced by @xmath47 , we get @xmath58 substituting this result in the expression of @xmath59 , we get @xmath60 using ( [ vincent11 ] ) and ( [ vincent15 ] ) we obtain @xmath61 and consequently @xmath62 ) and ( [ vincent19 ] ) one can rewrite ( [ vincent21 ] ) as @xmath63 ) . now , to calculate @xmath64 we have to find @xmath65 from ( [ vincent5 ] ) and @xmath66 from ( [ vincent18 ] ) , but this will be done explicitly in the next section where we will analyse the particle creation phenomenon when the scalar field is minimally coupled with the modified gravity . with the metric ( [ vincent1 ] ) the einstein equation leads to friedmann ones as @xmath67 where @xmath68 and @xmath69 are the energy density and the pressure respectively , and @xmath70 . we suppose that they are related by a barotropic equation of state such that @xmath71 with @xmath7 the barotropic parameter . combining eqs . ( [ vincent23 ] ) , ( [ vincent24 ] ) and ( [ vincent25 ] ) , one obtains for the scale factor @xmath72 for the conformal time and @xmath73 for the cosmic time . our goal is to analyse particle creation phenomenon in f(r ) gravity and check its impact as quantum effects on a possible appearance of singularities in the classical background when this is essentially driven by a dark fluid . in this case the curvature @xmath0 in einstein hilbert action is replaced by a function f(r ) and the total action reads @xmath74\quad,\end{aligned}\ ] ] where @xmath75 is the gravitational constant , @xmath76 the determinant of the metric @xmath10 and @xmath77 the matter lagrangian which depends on the metric and the matter field @xmath15 . the field equation can be derived varying the action ( [ vincent26 ] ) with respect to @xmath10 , the so called metric formalism @xcite . the variation of the determinant is always : @xmath78 and for the ricci scalar , one has @xmath79 now , since @xmath80 is actually the difference of two connections , it should transform as a tensor . therefore , one can write it as @xmath81 and substituting in the equation ( [ vincent28 ] ) one finds @xmath82 hence , the variation in the action reads : @xmath83d^4x \nonumber\\ & = & \frac{1}{2\kappa^2}\int\left [ f(r)\delta r\sqrt{-g}-\frac{1}{2}\sqrt{-g}g_{\mu\nu}\delta g^{\mu\nu}f(r)\right]d^4x\nonumber\\ & = & \frac{1}{2\kappa^2}\int\sqrt{-g}\left[f(r)(r_{\mu\nu}\delta g^{\mu\nu}+g_{\mu\nu}\box\delta g^{\mu\nu}-\nabla_{\mu}\nabla_{\nu}\delta g^{\mu\nu})-\frac{1}{2}g_{\mu\nu}\delta g^{\mu\nu}f(r)\right]d^4x\label{vincent31}\end{aligned}\ ] ] where @xmath84 . doing integration by parts on the second and third terms of ( [ vincent31 ] ) , one gets @xmath85f(r)\right\}d^4x\end{aligned}\ ] ] demanding that action remains invariant under variation of the metric , i.e @xmath86 , one obtains the field equations as @xmath87 where @xmath88 is the energy momentum tensor of the matter fields defined by the variational derivative of @xmath77 with respect to the metric @xmath89 : @xmath90 this satisfies the continuity equation @xmath91 the trace of ( [ vincent33 ] ) gives @xmath92 where @xmath93 and @xmath94 . einstein gravity , without the cosmology constant , corresponds to @xmath95 and @xmath96 , so that the term @xmath97 in ( [ vincent36 ] ) vanishes . in this case one has @xmath98 and hence the ricci scalar is directly determined by the matter . in modified gravity the term @xmath97 does not vanish in ( [ vincent36 ] ) . the time - time and space - space components of the field equation ( [ vincent33 ] ) give respectively @xmath99 making use of ( [ vincent25 ] ) and combining ( [ vincent37 ] ) and ( [ vincent38 ] ) one obtains @xmath100=h\dot{f}-2f\dot{h}-\ddot{f}\,\,.\end{aligned}\ ] ] the difference with the modified gravity is that the usual matter energy density is not the effective one . then , even when there is no energy density for usual matter , there exists a kind of fluids called dark fluids @xcite . this appears explicitly when we put ( [ vincent37 ] ) and ( [ vincent38 ] ) in the followings forms , neglecting the contributions of any other kind of usual matter , @xmath101 comparing eqs . ( [ vincent40 ] ) and ( [ vincent41 ] ) with the standar friedmann equations ( @xmath102 and @xmath103 ) , the both right sides of these equations may be identified with the energy and pressure of a perfect fluid , in such a way that the barotropic parameter of the eos for this dark fluid is defined after simplification by @xmath104 and the corresponding eos may be written as follows : @xmath105 where the subscript @xmath106 means `` dark fluid '' . this inhomogeneous eos ( [ vincent43 ] ) for this dark fluid takes the form of the kind of dark fluids studying in several works @xcite one can put the function @xmath9 in the einstein - hilbert part and a function @xmath1 as @xmath107 , which includes the starobinsky s model @xcite as a specific case ( @xmath108 ) , @xmath109 being a constant . let us choose the scale factor as @xmath110 , where @xmath111 may be determined in terms of the barotropic parameter @xmath7 as usually done in the einstein gravity . then , using the curvature as @xmath112 , ( [ vincent39 ] ) takes a new form as @xmath113=0\,\,\,.\label{vincent44}\end{aligned}\ ] ] note that in the special case of @xmath114 and setting @xmath115 , one obtains the result of einstein gravity , that is @xmath116 . even for any @xmath117 , this results remains the same with the ansatz @xmath118 . remember that particle production is directly connected with the curvature of the universe . then , since the scale factor remains the same as is einstein gravity , it is obvious that the curvature of the universe may remain the same and also as the effective mass . this means that particle creation aspect is the same as in einstein gravity when ordinary matter is taken into account . this result does not agree with one found by pereira and collaborators @xcite . they used palatini formalism in the context of f(r ) gravity for analysing cosmological particle creation for a spatially flat universe and found that a conformal invariant metric does not forbid the creation of massless during the radiative era of the universe . their result is that , in the context of modified gravity , the scale factor in radiative universe behaves as whose of de - sitter universe in einstein gravity which would lead to particle creation and consequence implies that parker s result is valid only in the context of general relativity . note that in a radiative universe , the trace of the energy momentum vanishes and so , their equation ( 6 ) must lead to @xmath119 which means that their equation ( 7 ) would never hold . from our result in this paper , it is clear that the scale factor of a radiative universe in the context of f(r ) gravity remains the same as in einstein gravity . then , without any approximation , parker s result must also hold in f(r ) gravity . however , it is important to mention that models of the form @xmath107 , that admit the existence of a viable matter dominated epoch prior to a late time acceleration requires that the parameter @xmath2 belongs to @xmath120 0 , 1\right[$ ] , @xcite . let us consider that the main content of the universe is a dark fluid , so , neglecting any contribution of ordinary matter . in the absence of ordinary matter fluid @xmath121 , eq ( [ vincent37 ] ) gives @xmath122 the cosmic acceleration can be realized in the regime @xmath123 @xcite . under the approximation @xmath124 , by dividing eq ( [ vincent41 ] ) by @xmath125 , one obtains @xmath126 let us assume the power law expansion solutions for the scale factor , @xmath110 , under the restriction that @xmath127 . the particular case , @xmath128 , must be treated separately and will be presented later . with this power law expansion for the scale factor the hubble parameter behaves as @xmath129 and using ( [ vincent46 ] ) , we find @xmath130 the acceleration requires that we have @xmath131 , which means that @xmath132-\infty , \frac{1-\sqrt{3}}{2}[\ , \cup\ , ] \frac{1}{2 } , 1[\,\cup\,]\frac{1+\sqrt{3}}{2 } , 2[\,\cup\,]2 , \infty[\quad.\end{aligned}\ ] ] in terms of conformal time the scale factor behaves as @xmath133 as we need an expanding universe , one can distinguish two different interval for the conformal time . hence , for @xmath134 one has @xmath135 whereas for @xmath136 or @xmath137 , one has @xmath138 . we can now examine particle creation phenomenon in spatially flat spacetime for massless scalar field minimally coupled with the gravitational field , @xmath139 . then , the equation ( [ vincent5 ] ) becomes @xmath140\chi_k=0 \,\,\,.\end{aligned}\ ] ] this is essentially the same equation that governs the evolution of gravitational waves in an expanding universe @xcite . this equation admits solutions in terms of hankel functions : @xmath141\,\ , , \quad \quad \nu=|1/2-q|\,\,\,,\end{aligned}\ ] ] where @xmath142 and @xmath143 are constants to be determined . making use of the wronskian relation @xmath144 one obtains , imposing orthonormalization of the modes , @xmath145 for fixing the initial vacuum state we use the bunch - davies sate @xcite by the choice @xmath146 and @xmath147 , then the solution ( [ vincent51 ] ) becomes @xmath148 note that the solution ( [ vincent54 ] ) reduces to the minkowski one in which it does not occur particle creation phenomenon only for the time interval @xmath149 . now we can proceed to the calculation of @xmath64 through the expression ( [ vincent22 ] ) . since we are dealing with a minimally coupled massless scalar field , the expression ( [ vincent22 ] ) becomes @xmath150 ) into ( [ vincent55 ] ) and making the change @xmath151 , one we get @xmath152\nonumber\\ + \frac{k^2\eta}{4}\left[h^{(1)}_{\nu-1}(k\eta)h^{(2)}_{\nu-1}(k\eta)-h^{(1)}_{\nu-1}(k\eta)h^{(2)}_{\nu+1}(k\eta)-h^{(1)}_{\nu+1}(k\eta)h^{(2)}_{\nu-1}(k\eta)+h^{(1)}_{\nu+1}(k\eta)h^{(2)}_{\nu+1}(k\eta)\right]\bigg]\end{aligned}\ ] ] note that at the early time , @xmath153 , @xmath154 and using ( [ vincent56 ] ) , we obtain @xmath155 . we see clearly through ( [ vincent56 ] ) that particle production can be known at any time and also the initial vacuum condition is correctly reproduced at early time where there is no particle production . then , as the conformal time grows , particle creation becomes important . note that as the big rip time ( @xmath156 ) is approached the rate of particle production diverges . the evolution of the particle production rate for a @xmath14\frac{1}{2},1[$ ] is presented in fig 1 . we will see later that this behaviour of particle production rate has an effective impact on the singularity . an important problem to be put out here is that , as the conformal time @xmath23 approaches @xmath157 , the scale factor @xmath16 , the energy density @xmath158 and the pressure @xmath159 of the dark fluid , diverge : this corresponds to the big rip singularity . obviously , the question to be asked when there is appearance of singularity is : does this particle production , as quantum effects , can lead to the avoidance of such a singularity ? to answer to this question it is important to evaluate quantum energy density and pressure , and compare them with the classical ones of the dark fluid . the suitable expression for the quantum energy density is the renormalized one and this in perfectly known in the literature . for a minimally coupled scalar field , the renormalized energy density reads @xcite @xmath160\nonumber\\ + \frac{1}{69120\pi^2}\bigg(-168r_{\,;\mu\nu}+288\box rg_{\mu\nu}+24r_{\mu\sigma}r^{\sigma}_{\nu}\nonumber\\ -12r^{\alpha\beta}r_{\alpha\beta}g_{\mu\nu}-64rr_{\mu\nu}+63r^2g_{\mu\nu}\bigg ) -\frac{rg_{\mu\nu}}{192\pi^2c\eta^2}\,\,\quad.\end{aligned}\ ] ] since we are leading with a massless scalar field , we need to take a null mass in ( [ vincent57 ] ) . note that in this massless limit , we have to introduce an arbitrary length ( or inverse mass ) @xmath161 @xmath162 in a full gravitational dynamical theory , there would be a term @xmath163 on the left - hand side of the gravitational field equations which arises from the presence on an @xmath164 term in the generalised gravitational action . then , the final term of ( [ vincent58 ] ) is proportional to @xmath163 and so may be taken over the left - hand side of the field equations and absorbed in the renormalisation of the coupling constant of this term . the other term on the right side remains in @xmath165 and is essential to the conservation of that quantity . then , the energy momentum renormalised tensor of a massless minimally coupled scalar field is @xmath166\nonumber\\ + \frac{1}{69120\pi^2}\bigg(-168r_{\,;\mu\nu}+288\box rg_{\mu\nu}+24r_{\mu\sigma}r^{\sigma}_{\nu}\nonumber\\ -12r^{\alpha\beta}r_{\alpha\beta}g_{\mu\nu}-64rr_{\mu\nu}+63r^2g_{\mu\nu}\bigg ) -\frac{rg_{\mu\nu}}{192\pi^2c\eta^2}\,\,\quad.\end{aligned}\ ] ] with the scale factor ( [ vincent45 ] ) we see from ( [ vincent59 ] ) that in frw the renormalised energy density behaves as ( the same also occurs for the pressure ) @xmath167 for other hand , the energy density of the dark fluid is @xmath168 \propto \eta^{-2\alpha(q+1)}\quad.\end{aligned}\ ] ] the ratio of the energy densities is @xmath169 since we are leading with an expanding universe we have @xmath170 and two important cases can be distinguished : @xmath171 for @xmath172-\infty , \frac{1-\sqrt{3}}{2}[\,\cup\,]\frac{1+\sqrt{3}}{2 } , 2[\,\cup\,]2 , \infty[$ ] , the product @xmath173 is positive and when the big rip time is approached , that is @xmath174 , the ratio of energy densities goes to zero . this means that the dark fluid is dominant and then the big rip can not be avoided . @xmath171 for @xmath175\frac{1}{2 } , 1[$ ] , the product @xmath173 is negative and the ratio of energy densities diverges . this indicates that quantum effects may be dominant as the singularity is approached and the big rip may be avoided . note that this result agrees with that obtained in the @xcite where it has been considered an expanding universe dominated by a dark energy fluid in the einstein gravity context . we also mention here that this is a case for which there is existence of viable model @xmath107 , describing an early matter dominated and the late time acceleration universe . however , it is important to mention that , the real aspect of the avoidance can be observed far before the singularity time . so , the question to be asked is : how long time before the big rip the quantum mechanism becomes important enough to begin dominating the matter content of the universe ? it is a question that requires an analysis of many details of the evolution of the universe . but , we can answer this question in a semi - quantitative way . let us put the general energy density due to quantum effects in the form @xmath176 where @xmath177 is the energy density due to quantum effects today . on the other hand , the energy density of the dark fluid is given by @xmath178 . so , their ratio is @xmath179 where @xmath180 characterizes the present conformal time . the scale factor is normalized so that it equals @xmath181 at present . let us suppose that the present - time ratio between the two densities can not exceed the ratio of the total radiation observed today with respect to the total density , i.e. , @xmath182 . hence the energy densities due to quantum effects and the dark fluid become comparable when @xmath183 this may allow estimation of how much the scale factor has increased from today till the equality moment . normalizing the present scale factor to unity and re - expressing it in terms of the cosmic time , we have @xmath184 where @xmath185 is the present cosmic time and @xmath186 the singularity time . using the estimates made before , we find that the energy densities equal at @xmath187 where @xmath188 is the equality time . hence , typically , the energy density due to quantum effects begins to dominate a fraction of the present age of the universe before the singularity . we conclude that it is sufficient to lead to big rip avoidance . so , quantum effects prevent the divergence which would occur about the energy density and pressure of the dark fluid as the singularity is approached and make them finite . this can be interpreted as a change of dynamical regime . as we told above , the case @xmath108 , the starobinsky s model , must be analysed separately . then , using directly the equation ( [ vincent46 ] ) without any ansatz about the explicit expression of the scale factor , one obtains @xmath189 whose general solution can not be found analytically . however , one can observe two particular solutions , @xmath190 and @xmath191 , which can lead to a constant scale factor . with a constant hubble parameter , the energy density and the pressure of the dark fluid are finite and then , there is no finite time future singularity . this result agrees with that obtained by bamba and collaborators where they shown that the model @xmath192 naturally removes finite time singularity @xcite . this section is devoted to the application of the cv formula to the dark fluid characterising the main contain of the universe . note that the general approach to cv formula has already been done extensively for an ideal fluid in @xcite . they derived cv formula - like which relates the entropy of the closed frw universe to its energy , and casimir energy , for a multicomponent coupled fluid where the generalized fluid obeys an inhomogeneous equation of state , both for einstein and modified gravities . it has been explicitly shown as special result that when the equation of state is a linear one and the barotropic parameter @xmath193 ( the radiative universe ) , the cv formula is recovered from the total entropy of the universe . also , the recovery of the cv formula from friedmann equations has been investigated in our early work @xcite with einstein gravity context , where the inhomogeneous equation of state contains a variable cosmological terms . the interesting case to be investigated here is the recovery of cv formula from the total entropy of the universe with an ideal fluid with inhomogeneous equation of state . as we know that the classical background goes toward a big rip , an analysis will be done around the singularity about cv formula terms . looking at the equation ( [ vincent43 ] ) , it clear that it is an inhomogeneous one . then , as has been done in @xcite , one can put it in general on the form @xmath194 and the conservation law for the energy , @xmath195 where @xmath6 is the space dimension , becomes , using ( [ vincent69 ] ) @xmath196 the general solution of ( [ vincent71 ] ) is @xmath197 where @xmath198 is an integration constant . identifying ( [ vincent43 ] ) with ( [ vincent71 ] ) , one gets @xmath199 and consequently , @xmath200 . making use of ( [ vincent45 ] ) and setting @xmath201 , one gets @xmath202 hence , the total energy in the volume @xmath203 behaves as @xmath204 , which is also the behaviour of the extensive and sub - extensive energies . assuming the conformal invariance , one can write the extensive and sub - extensive ( the casimir energy ) parts of the total energy as @xmath205 where @xmath206 and @xmath207 , and @xmath208 the total entropy of the universe . then the total entropy of the universe is @xmath209^{-\frac{n}{g(\alpha)+1}}\ , . \end{aligned}\ ] ] then , we see that the cv formula is recovered for @xmath210 , which means that the dimension of the spacetime may be @xmath211 . then , @xmath212 has to take only discrete values . solving the equation @xmath213 , one finds two value for the parameter @xmath2 , @xmath214 then , it appears that cv formula may always be recovered from the total entropy of the universe in einstein gravity . however , this occurs in f(r ) gravity only for @xmath215 . on the other hand it is important to note that for either einstein or f(r ) gravity , if @xmath216 , cv formula can not be recovered . let us now analyse what happens about cv formula terms around the big rip singularity . note that the cv formula is reproduced for @xmath217 , then , the casimir energy is inversely proportional to the scale factor . we find then an interesting result that , in an expanding universe , the casimir energy decreases and as the singularity ( big rip ) time is approached , the scale factor diverges and consequently the casimir energy vanishes . this result is in agreement with the brevik and collaborators s result where they shown that either with viscous or non - viscous fluid the casimir effect fades away near the big rip @xcite . on the other hand , when we put @xmath218 on the form @xmath219 , it appears evidently that it is less that @xmath220 and then must belong to @xmath221 . hence the cv formula is reproduced only when the big rip can not be avoided . we studied particle creation aspect in f(r ) gravity in which we considered the assumption @xmath222 . note that particle creation is directly connected with the curvature of the universe . when the curvature vanishes , the modes are not affected by gravitational field and particle creation phenomenon can not hold . this aspect is realized when we considered a massless scalar field in a radiative universe . also in a static universe where the scalar remains constant at any time , the curvature vanished and particle creation can not be expected . note that these situation appears clearly when the field equation is put on the form of an harmonic oscillator equation . the frequency depends on the wave number @xmath223 and on the effective mass . then , the gravitational field effect through the curvature is incorporated in the effective mass . consequently , when the curvature grows , the effective mass does not vanish and the particle production can hold , at least when the mass is considered and/or the coupling is not conformal . it appears that particle creation consequently is linked with the behaviour of the scale factor . we found then that the scale factor does not change in the f(r ) gravity with respect to his form in einstein gravity and consequently the particle production aspect is the same as in einstein gravity . we considered that the main content of the universe is a dark fluid and then the universe may evolve toward a finite time singularity , the big rip . we then analysed quantum effects near this singularity comparing the classical energy density and the pressure of the dark fluid with the renormalized energy density and the pressure due to quantum effects . hence , we find that for @xmath224\frac{1}{2 } , 1[$ ] the ratio of the renormalized energy density due to quantum effects to the classical energy density of the dark energy diverges as the big rip is approached . we conclude that quantum effect are dominant near the singularity and the big rip may be avoided . for @xmath225 , the ratio goes do zero as the big rip is approached . this means that the dark fluid is the dominant component of the universe near the singularity and we conclude in this case that the big rip can not be avoided . the starobinsky s model has been analysed separately and reveals the absence of singularity . however , due to the fact that power law expansion for the scale factor presents restriction on starobinsky model , it would be interesting to use an exponential solution and incorporating conformal anomaly as quantum effects around the big rip . also , in the case for which the singularity is not avoided , it would be interesting to allow the cosmic fluid to possess viscosity for analysing the possible avoidance of this singularity . we propose to address these investigations in a future work . another interesting point to which we devoted our attention in this work is the equivalence between the cv formula and the generalized total entropy of the universe , coming from the inhomogeneous equation of state that appears in the f(r ) gravity . we found that the cv formula can always be recovered in einstein gravity while in f(r ) gravity this equivalence holds only for @xmath5 corresponding to the situation in which the big rip can not be avoided . we also analysed the behaviour of cv formula terms and found that as the universe expands , the casimir energy decreases and fades away as the big rip is approached . * acknowledgement : * the authors thank prof . s. d. odintsov for useful criticisms and comments . m. j. s. houndjo thanks cnpq ( brazil ) for partial financial support . a. v. monwanou and j. b. chabi orou thank iaea / ictp step program for financial support . c. m. will . theory and experiment in gravitational physics . cambridge university press , 1993 . d. n. spergel et al . , astrophys . * 170 * , 377 ( 2007 ) . grav.*39 * , 307 - 342 , ( 2007 ) . s. nesseris , l. perivolaropoulos . , jcap.*0701 * , 018 , ( 2007 ) . , h. jassal , j. bagla and t. padmanabhan , mon . . soc . * 405 * , 2639 - 2650 ( 2010 ) . , p.wu and h. yu . b * 643 * : 315 - 318 , ( 2006 ) . , z. huang , q. sun , w . fang and h. lu . a * 22 * : 3073 - 3082 , ( 2008 ) v. faraoni , phys . rev . d*68 * ( 2003 ) 063508 ; class . * 22 * ( 2005 ) 3235 . e. elizalde , s. nojiri and s. d. odintsov , phys . d * 70 * ( 2004 ) 043539 s. nojiri and s. d. odintsov , phys . b * 595 * ( 2004 ) 1 ; phys . lett . b*571 * ( 2003 ) 1 ; phys . lett . b*562 * ( 2003 ) 147 b. gumjudpai , t. naskar , m. sami and s. tsujikawa , jcap * 0506 * ( 2005 ) 007 s. nesseris and l. perivolaropoulos , phys . rev . d*70 * ( 2004 ) 123529 y. wei , mod . lett . a * 20 * ( 2005 ) 1147 p. gonzalez - diaz , tspu vestnik * 44n7 * ( 2004 ) 36 l. p. chimento and r. lazkoz , int . d * 14 * ( 2005 ) 587 s. tsujikawa and m. sami , phys . b * 603 * ( 2004 ) 113 m. alimohammadi and h. mohseni , phys . d * 74 * ( 2006 ) 043506 m. setare , phys . lett b * 641 * ( 2006 ) 130 s. tsujikawa , phys.rev . d * 73 * ( 2006 ) 103504 m. sami and a. toporensky , mod.phys.lett . a * 19 * ( 2004 ) 1509 p. wu and h. yu , int . d * 14 * ( 2005 ) 1873 . r. r. caldwell , m. kamionkowski and n. n. weinberg . lett.*91 * , 071301 , ( 2003 ) m. j. s. houndjo . , europhys lett,*92 * , 10004 ( 2010 ) nojiri s. , odintov s. d. and tsujikawa s. , phys . d , * 71 * ( 2005 ) 063004 . i. brevik , s. nojiri , s. d. odintsov and l. vanzo , phys . d*70 * , 043520 ( 2004 ) h - j . schmidt . class . quantum . grav.*7 * ( 1990 ) , 1023 - 1031 . l. querella . arxiv 9902044v1 s. nojiri and s. d. odintsov . j. phys . a*40 * ( 2007 ) , 6725 - 6732 ; arxiv : 1011 . 0544 s. capozziello and m. francaviglia . rel . grav.*40 * ( 2008 ) , 357 - 420 s. nojiri and s. d. odintsov , econf * c0602061*:06 , ( 2006 ) ; int . j. geom . phys . * 4*:115 - 146,2007 t. ruzmaikina and a. ruzmaikin , zh . teor . fiz.,*57 * , 680 , ( 1969 ) . jetp , * 30 * , 372 , ( 1970 ) . b. breizman , v. gurovich , and v. sokolov . , zh . eksp . teor . fiz.,*59 * , 288 , ( 1970 ) . phys . jetp,*32 * , 155 , ( 1971 ) . h. buchdahl . , mon . not . r. astron . soc.,*150 * , 1 - 8 , ( 1970 ) . a. palatini . , circ . mat . palermo,*43 * , 203 , ( 1919).g . d * 20 * , 413 - 462 ( 2011 ) s. m. caroll , spacetime and geometry : an introduction to general relativity . addison wesley , 2004 s. w. hawking and j. f. r. ellis . the large scale structure of space - time . cambridge university press , 1973 . l. parker . rev . lett.*21 * , 562 ( 1968 ) ; l. parker , phd thesis , harvard university , 1966 . + l. parker and d. j. toms.,_quantum field theory in curved spacetime : quantized fields and gravity _ , ( cambridge university press , cambridge , england , 2009 ) l. parker , phys . rev.*183 * ( 1969 ) 1057 ; l. parker , phys . rev . d*3 * ( 1971 ) 346 ; l. parker , phys . rev . lett.*28 * ( 1972 ) 705 ; l. parker , phys . rev . d*7 * ( 1973 ) 976 ; l. parker and a. raval . lett . * 86 * , 749 ( 2001 ) ; l. parker and a. raval . , phys . d * 62 * , 083503 ( 2000 ) , erratum - ibid . d * 67 * , 029903 ( 2003 ) ; l. parker and a. raval . , d * 60 * , 123502 ( 1999 ) , erratum - ibid d * 67 * , 029902 ( 2003 ) , l. parker and a. raval . d * 60 * , 063512 ( 1999 ) , erratum - ibid . d * 67 * , 029901 ( 2003 ) ; r. r. caldwell , l. parker and d. a. t. vanzella . d * 73 * , 023513 ( 2006 ) ; l. parker and d. a. t. vanzella . d * 69 * , 104009 ( 2004 ) n.d . birrell , p.c.w . davies , quantum fields in curved space , cambridge university press , cambridge , 1982 ; s.a . fulling , aspects of quantum field theory in curved spacetime , cambridge university press , cambridge , 1989 ; a.a . grib , s.g . mamayev , v.m . mostepanenko , vacuum quantum effects in strong fields , friedmann laboratory publishing , st . petersburg , 1994 . mukhanov , s. winitzki , introduction to quantum effects in gravity , cambridge university press , cambridge , 2007 . zeldovich , pisma zh . teor . fiz.*12 * ( 1970 ) 443 , english transl . jetp * 12 * ( 1970 ) 307 ; a.a . grib , s.g . mamayev , v.m . mostepanenko , gen . relativ . gravit.*7 * ( 1975 ) 535 ; a.a . grib , yu.v . pavlov , grav . cosmol.*11 * ( 2005 ) 119 ; a.a . grib , yu.v . pavlov , grav . cosmol.*12 * ( 2006 ) 159 . grishchuk , class . quantum grav.*10 * ( 1993 ) 2449 m.r.g . maia , phys . d * 48 * ( 1993 ) 647 ; m.r.g . maia , j.d . barrow , phys . * ( 1994 ) 6262 ; m.r.g . maia , j.a.s . lima , phys . rev . d*54 * ( 1996 ) 6111 . grib , s.g . mamayev , yad . fiz . * 10 * ( 1969 ) 1276 , english transl . j. nucl . phys.*10 * ( 1970 ) 722 ; flavio . g. alvarenga , a.b . batista , j.c . fabris and s. houndjo , grav.&cosmol.*16 * , 105 ( 2010 ) . pavlov yu . v. , gravit . , * 15 * ( 2009 ) 341 bates j. d. and anderson p. r. , phys . rev . d,*82 * ( 2010 ) 024018 . e. verlinde , hep - th/0008140 . cardy , nucl.phys . b*270 * , 186 ( 1986 ) . d. youm , phys . b * 531 * , 276 ( 2002 ) . s. nojiri and s.d . odintsov , int . j. mod . a*16 * , 3273 ( 2001 ) . s. nojiri and s.d . odintsov , class . * 18 * , 5227 ( 2001 ) . d. youm , mod . a*16 * ( 2001 ) 1327 . b. wang , e. abdalla and r.k . su , mod.phys.lett . a*17 * , 23 ( 2002 ) . s. nojiri , o. obregon , s.d . odintsov , h. quevedo and m.p . ryan , mod . lett . a*16 * ( 2001 ) 1181 . danielsson , jhep * 0203 * ( 2002 ) 020 . s. ogushi , phys.lett . * , 51 ( 2002 ) . a.c . petkou and g. siopsis , jhep*0202 * , 045 ( 2002 ) a. de felice and s. tsujikawa . , living rev . rel . * 13 * , 3 ( 2010 ) l. amendola , r. gannouji , d. polarski , and s. tsujikawa . d * 75 * , 083504 ( 2007 ) i. brevik , e. elizalde , o. gorbunova , and a. v. timoshkin . eur . phys . j. c , * 52 * : 223 - 228 , ( 2007 ) . i. brevik , o. g. gorbunova , and a. v. timoshkin . j. c , * 51 * : 179 - 183 , ( 2007 ) s. capozziello , v. f. cardone , e. elizalde , s. nojiri , and s. d. odintsov . phys . rev . d , * 73 * ( 4 ) : 043512 , ( 2006 ) . s. nojiri and s. d. odintsov . d , * 72 * ( 2):023003 , ( 2005 ) . s. nojiri and s. d. odintsov . b , * 639*:144150 , ( 2006 ) . l. p. grishchuk , phys . rev . d*48 * , 3513 ( 1993 ) . k. bamba , s. nojiri and s. d. odintsov , jcap * 0810 * ( 2008 ) 045 . t. s. bunch and p. c. w. davies , j. phys . * a11 * , 1315(1978 ) .

we investigate particle production in an expanding universe under the assumption that the lagrangian contains the einstein term @xmath0 plus a modified gravity term of the form @xmath1 , where @xmath2 is a constant . dark fluid is considered as the main content of the universe and the big rip singularity appears . quantum effects due to particle creation is analysed near the singularity and we find that for @xmath3\frac{1}{2 } , 1[$ ] , quantum effects are dominant and the big rip may be avoided whereas for @xmath4 the dark fluid is dominant and the singularity remains . the cardy - verlinde formula is also introduced and its equivalence with the total entropy of the universe is checked . it is found that this can always occur in einstein gravity while in f(r ) gravity , it holds only for @xmath5 , @xmath6 being the space dimension , corresponding to the situation in which the big rip can not be avoided . * quantum effects from particle production on background evolution and cardy - verlinde formula in f(r ) gravity + * m. j. s. houndjo , a . v. monwanouand jean b. chabi orou + pacs numbers : 04.50.kd , 98.80.cq , 02.40.xx , 05 .

since many decades , design codes ensure a very low probability that a building collapses under ordinary loads , like self weight , dead and live service load , or snow . nevertheless buildings still do collapse , from time to time . an extremely small fraction of collapses originates from unlikely combinations of intense ordinary load with very poor strength of the building . the majority of structural collapses are due to accidental events that are not considered in standard design . examples of such events are : gross design or construction errors , irresponsible disregard of rules or design prescriptions , and several rare load scenarios like e.g. earthquakes , fire , floods , settlements , impacts , or explosions @xcite . accidental events have low probability of occurrence , but high potential negative consequences . since risk is a combination of probability and consequences , the risk related to accidental events is generally significant @xcite . in 1968 a gas explosion provoked the partial collapse of the ronan point building in london . this event highlighted for the first time the urgency for _ robust _ structures , enduring safety in extraordinary scenarios @xcite . since then , interes was driven by striking catastrophic collapses @xcite , until in 2001 the tragic collapse of the world trade center renewed the attention to the topic ( see e.g. @xcite and @xcite ) . the last decades , several design rules aimed at improving structural robustness have been developed ( see e.g. @xcite ) . accidental events can be classified into _ identified _ and _ unidentified _ @xcite . identified events are statistically characterizable in terms of intensity and frequency of occurrence . examples are earthquakes , fire not fueled by external sources , gas explosions , and unintentional impacts by ordinary vehicles , airplanes , trains , or boats . specific design rules and even entire codes are devoted to specific identified accidental events . unidentified events comprise a wide variety of incidents whose intensity and frequency of occurrence can not be described statistically , e.g. terrorist attacks or gross errors . the risk related to unidentified accidental events can be mitigated both by structural and nonstructural measures @xcite . nonstructural measures such as barriers and monitoring can reduce the probability that an accidental event affects the structural integrity , others like a wise distribution of plants and facilities can minimize the negative consequences of eventual collapses . otherwise , structural measures can improve local resistance of structural elements to direct damage , e.g. the design of _ key elements _ for intense local load @xcite , or the application of the _ enhanced local resistance _ method @xcite . structural measures can also provide progressive collapse resistance , i.e. prevent spreading of local direct damage inside the structure to an extent that is disproportioned with respect to the initial event . usual strategies to improve progressive collapse resistance are compartmentalization of structures @xcite and delocalization of stress after local damage . stress delocalization can be obtained exploiting redundancy , plastic stress redistributions ( masoero , wittel et al . , 2010 ) , ties @xcite , and moment resisting connections @xcite . nowadays several design codes employ the conventinal _ alternate load path method ( alpm ) _ to evaluate progressive collapse resistance , e.g. @xcite and @xcite . the method consists in removing one key element , generally a column or a wall , and measuring the extent of subsequent collapse . if the final collapse is unacceptably wide , some of the previously listed measures have to be employed . hence structures are first designed and subsequently tested to be robust - they are not conceived _ a priori_. this course of action excludes optimizations of the basic structural topology and geometry , that actually play a key role in the response to local damage , considering as an example the very different behavior of redundant and statically determined structures . anti - seismic design @xcite already contains some prescriptions that should be considered before starting a new design , e.g. geometric regularity on the horizontal and on the vertical planes . furthermore , anti - seismic _ capacity design _ requires a hierarchy of the structural elements ensuring that earthquakes can only provoke ductile collapse of the horizontal beams , while failure of columns and brittle ruptures due to shear are inhibited . differently , for what concerns progressive collapse resistance , optimal overall geometric features are not known , except for the concepts of redundancy and compartmentalization . furthermore , the idea of hierarchically maximizing progressive collapse resistance is completely absent . in this paper , we make a first step to cover this deficiency , showing that progressive collapse resistance can be improved by hierarchy in the overall geometry ( _ topological _ hierarchy ) and in the relative strength and stiffness of horizontal and vertical structural elements ( _ mechanical _ hierarchy ) . our approach incorporates the simulation of progressive collapse of regular 2d frames made of reinforced concrete ( rc ) subjected to the sudden removal of structural elements , following the alpm framework . we first describe the analyzed frame structures and briefly sketch the approach that is based on the _ discrete element method ( dem)_. after the model description , we present the results of the simulations , with focus on the effect of geometry and hierarchy on the activated collapse mechanisms and , consequently , on progressive collapse resistance . we consider two representative sets of regular 2d framed structures in fig . [ fig_struct]-a . each set consists of three frames with identical total width @xmath1 and different topological _ hierarchical level _ @xmath2 , where @xmath3 is the number of structural cells in a frame . the horizontal beams , excluded those of the secondary structure , carry a uniform load per unit length @xmath4 . the frames are made of rc with typical mechanical parameters of concrete and steel , as shown in table [ tabmecpar ] . the total height @xmath5 of the structure is kept constant , and two different height - bay aspect ratios @xmath6 of the structural cells are considered ( @xmath7 and @xmath8 ) . @xmath9 & @xmath10 & @xmath11 + + specific weight & @xmath12 & kg / m@xmath13 & 2500 + young modulus & @xmath14 & n / m@xmath15 & 30@xmath16 + compressive strength ( high ) & @xmath17 & n / m@xmath15 & 35@xmath18 + compressive low ( low ) & @xmath17 & n / m@xmath15 & 0.35@xmath18 + ultimate shortening & @xmath19 & - & 0.0035 + + young s modulus & @xmath20 & n / m@xmath15 & 200@xmath16 + yield stress & @xmath21 & n / m@xmath15 & 440@xmath18 + ultimate strain & @xmath22 & - & 0.05 + there exist several ways of introducing hierarchy into the topology of framed structures ; here we call a structure hierarchical " if it has a primary structure , made of few massive structural elements , that supports a secondary one . the latter defines the living space and has negligible stiffness and strength compared to the primary structure . the frames with @xmath23 and @xmath24 can be seen as reorganizations of those with @xmath25 . in detail , each column of the frames with @xmath24 corresponds to two columns of the frames with @xmath25 , and the same is valid for the beams , disregarding the first floor beam of the frames with @xmath25 , which is simply deleted ( see fig . [ fig_struct]-a ) . analogously , the geometry of the frames with @xmath23 can be obtained starting from the frames with @xmath24 . the cross sections of columns are square ( see fig . [ fig_struct]-b ) , with edges @xmath26 proportional to @xmath27 with factor @xmath28 . the beams have rectangular cross section whose height @xmath29 is proportional to @xmath30 with factor @xmath31 , and whose base @xmath32 is proportional to @xmath29 with aspect ratio @xmath33 . the reinforcement is arranged as shown in fig . [ fig_struct]-b , with area @xmath34 proportional to the area of the cross section by factor @xmath35 for the columns ( i.e. 8@xmath3618 when n=11 ) , and @xmath37 for the beams ( i.e. 4@xmath3614 when n=11 ) . the damage areas , ( dotted in fig . [ fig_struct]-a ) contain the structural elements that are suddenly removed to represent an accidental damage event , following the alpm framework . the damage is identical for frames with same @xmath38 , and is defined by the breakdown of one third of the columns on a horizontal line . the columns and beams removed from frames with @xmath25 correspond to the structural elements removed from frames with @xmath24 and @xmath23 . this kind of damage is employed to represent accidental events with a given amount of destructive energy or spatial extent , like explosions or impacts . in this work we do not explicitly simulate very local damage events like gross errors , which would be better represented by the removal of single elements . nevertheless , we will generalize our results to consider also localized damage events . we employ the discrete element method ( dem ) to simulate the dynamics after a sudden damage @xcite . dem is based on a lagrangian framework , where the structure is meshed by massive elements interacting through force potentials . the equations of motion are directly integrated , in our case using a 5@xmath39 order gear predictor - corrector scheme , with time increments between 10@xmath40s and 10@xmath41s ( see masoero , wittel et al . , 2010 ) . dem is an equivalent formulation to finite elements , converging to the same numerical solution of the dynamics if identical force - displacement laws are implemented . a detailed description of the algorithm for 3d systems can be found in ( masoero , wittel et al . , 2010 ) ; @xcite , together with a discussion on the applicability . for this work , the code was restricted to 2d by allowing only two displacements and one rotation in the vertical plane . in ( masoero , vallini et al . , 2010 ) , the dem model is tested against dynamic energy - based collapse analyses of a continuous horizontal beam suddenly losing a support . in the appendix , we compare our dem results to experimental observations of a 2d frame undergoing quasi - static column removal @xcite . to the best of our knowlede , literature still lacks on experiments of dynamic collapse of framed structures due to accidental damage . in the following we will review only the essentials of our model , focusing on the details tha are relevant for the application to 2d frames . we assume simplified force - displacement laws for the beam element and for the hertzian contacts . predicting collapse of real structures would require more specialized interaction as compared to here , for example using the fiber approach for the cross sections . by contrast , we are interested in fundamental mechanisms of damage propagation within complex structural systems . in this research perspective , and according to a basic principle of statistical mechanics , minimizing the complexity of local interactions improves the interpretation of the systemic response . despite the strong assumptions , in the appendix we show that our model can match reasonably well with with experimental observations . in a first step , the structure needs to be assembled by discrete elements and beams . [ fig_2d_mesh]-a shows the four types of spherical discrete elements ( sde ) that we employed . columns and beams are made of 9 sdes , respectively with diameter @xmath42 and @xmath43 , slightly smaller than the distance between them to prevent contact form occurring before local rupture . constrained sdes and connection sdes have same diameter @xmath44 as column sdes . the constrained sdes are clamped to a plane that represents the ground by means of the hertzian contact model , discussed further in this section . pairs of sdes are connected by _ euler - bernoulli beam elements - ( ebe ) _ that , when deformed , transmit forces and moments to their edge nodes , locally labeled 0 and 1 ( see fig . [ fig_2d_mesh]-b ) . the mass @xmath45 of an sde is defined on the basis of the ebes connected to it . namely @xmath46 , where @xmath47 labels the generic ebe connected to sde , and @xmath48 is the cross sectional area of the structural element corresponding to the ebe . the external load @xmath4 is introduced adding a mass @xmath49 to the beam sdes . @xmath4 is not treated directly as a force to avoid downward accelerations of the sdes greater than gravity @xmath50 during free fall . for sufficiently small deformations , the ebes are linear elastic and exert a force @xmath51 proportional to the elongation @xmath52 and directed along the @xmath53 segment , a shear force @xmath54 proportional to the sum of the nodal rotations @xmath55 , and a bending moment @xmath56 proportional to the nodal effective rotations , defined as @xmath57 , and @xmath58 . furthermore , we introduce damping by forces and moments directed opposite to @xmath51 , @xmath54 , and @xmath56 , and proportional to the time derivative of @xmath52 with factor @xmath59ns / m , and of @xmath60,@xmath61 with factor @xmath62nms . geometric nonlinearity due to large displacements is considered by referring rotations and elongation to the @xmath53 segment . in the small deformations regime of our simulations , @xmath51 is with good approximation equal to the axial force inside the ebe , and thus perpendicular to @xmath54 . if @xmath51 overcomes a yield threshold in tension @xmath63 or under compression @xmath64 , the ideally plastic regime is entered and plastic axial strain @xmath65 is applied to maintain @xmath66 or @xmath67 . neglecting the contributions of concrete in tension and of steel in compression , we set the yield thresholds in terms of @xmath51 and @xmath52 to : @xmath68 ideally plastic regime in bending is entered when @xmath69 . we obtain the bending yield threshold @xmath70 and the corresponding yielding effective rotation @xmath71 , neglecting the strength contribution of concrete and assuming a lever arm between upper and lower reinforcement equal to the height @xmath72 of the cross section : @xmath73 @xmath74 is the cross sectional moment of inertia of the ebe , and @xmath75 is the fraction of reinforcement in tension ( @xmath76 for columns and @xmath77 for beams , as in fig . [ fig_struct]-b ) . @xmath78 considers the beneficial compression effect compression in the ebe . we set @xmath78 assuming bending carried by the reinforcement alone , and that the strain @xmath79 in the reinforcement put under tension by @xmath78 equals the compressive strain @xmath80 due to @xmath81 , namely : @xmath82 in this way , eventual tension @xmath83 inside the ebe produces negative @xmath78 , and thus reduces @xmath70 . when yielding in bending occurs , plastic rotations are added at the edge nodes of the ebe . if only @xmath84 , with @xmath85 , is greater than @xmath70 , then only @xmath86 is applied to restore @xmath87 . differently , if both @xmath88 and @xmath89 are greater than @xmath70 , both @xmath90 and @xmath91 are applied to restore @xmath92 . for the sake of simplicity , we assume yielding in bending uncoupled from yielding in axial direction . furthermore , we neglect yielding due to shear because small plastic deformations are generally associated with shear . we consider an ebe failed when excessive @xmath65 and @xmath93 are cumulated . for this purpose , the coupled breaking criterion : @xmath94 is employed . @xmath95 , @xmath96 , and @xmath97 are the maximum allowed plastic elongation , shortening , and rotation in uncoupled conditions . we consider high plastic capacity of the structural elements setting @xmath98 , @xmath99 , and @xmath100rad ( see table [ tabmecpar ] ) . failed ebes are instantly removed from the system . we neglect ruptures due to shear assuming that , in agreement with a basic principle of capacity design , a sufficient amount of bracings ensures the necessary shear strength . the hertzian contact model is employed for the sdes to consider collisions between structural elements . the model consist of repulsive forces between partially overlapping sdes , damped by additional forces proportional and opposite to the overlapping velocity . we also set tangential forces that simulate static and dynamic friction , as well as damping moments opposed to the relative rolling velocity . a similar hertzian contact model is also employed for sdes colliding with the ground plane . in the following simulations we employ contact parameters that can be found in @xcite . we do not transcribe them because impacts do not affect significantly the collapse mechanisms sudied here . nevertheless , in general simulation algorithms for progressive collapse should consider impacts , because initial damage located at upper stories generates falling debris , and because impacts can drive the transition from partial to total collapse ( see ( masoero , wittel et al . , 2010 ) and @xcite ) . in granular dynamics , the contact parameters are generally set referring to the material of the grains @xcite . in our model the sdes represent large heterogeneous portions of structural elements , for which there are not conventionally defined contact parameters so far . we emply parameters yielding a qualitatively realistic dynamics ( e.g. the elements do not rebound or pass through each other ) , and chosen from sets of possible one that were defined through preliminary studies . such studies also indicated that the collapse loads of a beam due to debris impact varies of less than 15% upon orders of magnitude change in the contact parameters . the simulations are organized into two steps : first the structure is equilibrated under the effect of @xmath4 and gravity , then the ebes inside the damage area are suddenly removed , and the subsequent dynamic response is simulated . our aim is to quantify three _ collapse loads _ : * @xmath101 : maximum static load that the intact structure can carry ; * @xmath102 : minimum _ critical load _ that causes dynamic collapse after damage . applied statically to the intact structure first , it is then kept constant during the post - damage dynamic response . * @xmath103 : minimum load corresponding to total collapse after damage . by definition , @xmath104 . in our dem model we do not have a straightforward unique measure of load , because the mass of the sdes depends on the external load @xmath4 and on the self weight of the structural elements . the mass of the beam sdes effectively acts as a distributed horizontal load . on the other hand , the columns at each story transmit vertical concentrated forces either to other columns at a lower story , or to the horizontal transfer beam over the damage area . therefore we introduce a load measure that we call _ equivalent load _ @xmath105 , applied to the massless structure and analytically related to the geometry , the mass , and the activated collapse mechanism of the frames in the simulations . namely , @xmath105 is defined to produce the same static effect as the various masses and concentrated forces of the simulation frames , at the critical points where collapse is triggered . the derivation of the analytical expressions used in this work is shown in @xcite . for each analyzed structure , we first apply the entire structural mass . in a subsequent step , the external load @xmath4 is increased until the intact structure collapses in static conditions . the collapse mechanism indicates what equivalent load expression should be used to compute @xmath101 . then we slightly decrease @xmath4 , equilibrate , introduce the damage , and calculate whether dynamic progressive collapse is triggered and to what an extent . performing several simulations with progressively smaller @xmath4 , the final extent of collapse changes from total to partial , and we employ again an adequate equivalent load to compute @xmath103 . if the structure collapses even when @xmath4 is reduced to zero , we start reducing the specific weight of the structural elements , i.e. the structural mass . when dynamic collapse does not occur anymore , an adequate equivalent load provides @xmath102 . once we obtain the collapse loads , we estimate the progressive collapse resistance referring to the _ residual strength fraction _ @xmath106 . actually , progressive collapse resistance is more directly related to @xmath102 , but the advantage of @xmath0 is that it can not be improved by simply strengthening the structural elements , which would increase both @xmath102 and @xmath101 . robustness - oriented structural optimization is required to increase @xmath0 , which therefore is a good indicator to compare different structural solutions . in our model , the bending yield threshold @xmath70 does not depend on the strength of concrete @xmath17 . therefore , setting the high value @xmath107n / mm@xmath15 , the mainly compressed columns get much stronger than the horizontal beams , that fail in bending ( see figs . [ fig_2d_bendcoll_before],[fig_2d_bendcoll ] ) . the resulting collapse mechanisms resemble triple - hinge and four hinges mechanisms , reflecting the large plastic capacity of the structural elements . s corresponds to the first breaking of an ebe . ] s corresponds to the application of the initial damage . ] if the initial damage triggers a bending mechanism , frames with @xmath23 undergo total collapse , while frames with lower hierarchical level @xmath2 initially suffer only a local collapse ( see fig . [ fig_par - tot - bend ] ) . the local collapse can nevertheless evolve to total collapse , if high applied load and plastic capacity cause the falling central part of the structure to dynamically drag down the lateral portions ( masoero , wittel et al . , 2010 ) . = 13kn / m ) and total ( @xmath108=26kn / m ) bending collapse after damage of a frame with very strong columns , @xmath109=11 , and @xmath8 . time @xmath110s corresponds to the application of the initial damage . ] the collapse loads , expressed in terms of equivalent loads @xmath105 , are summarized in fig . [ fig_mu - rsr - bend ] as a function of the hierarchical level @xmath2 , for different slenderness of the structural cells @xmath38 . in fig . [ fig_mu - rsr - bend ] , superscript @xmath111 indicates bending collapse mechanism . we employ equivalent loads referring to perfectly brittle or perfectly plastic bending failure ( see the appendix ) . the collapse loads decrease with @xmath38 , i.e. a slender structure seems weaker , and increase with @xmath2 , i.e. hierarchical frames are stronger . the residual strength fraction @xmath0 does not depend on @xmath38 , while hierarchical structures with low @xmath109 are more robust than homogeneous ones ( see fig . [ fig_mu - rsr - bend ] ) . in fact , the concentration of bending moment at the connection between a beam hanging above the damage area and the first intact column depends on the _ number _ of removed columns . in the simulations , we remove a constant fraction of one third of the columns on a horizontal line ( see fig . [ fig_struct ] ) . therefore homogeneous structures lose more columns and are less robust toward the bending collapse mechanisms . on the other hand , since the number of removed columns is decisive , we expect that the hierarchical level does not influence @xmath0 toward bending collapse in case of single column removal . finally we consider the 2d frame as part of a regular 3d structure and divide the collapse loads in fig . [ fig_mu - rsr - bend ] by @xmath30 , i.e.by the tributary length of the beams in the direction perpendicular to the frame . in this way , collapse loads per unit area are obtained in fig . [ fig_divl_mu - bend ] , showing that : @xmath38 does not influence @xmath112 and @xmath113 ; structures with slender cells are less likely to collapse entirely ; @xmath113 is independent from the hierarchical level ; @xmath112 is proportional to @xmath2 . for frames that undergo bending progressive collapse . ] divided by @xmath30 , considering the 2d frames as part of regular 3d structures . ] progressive compressive failure of the columns , also called pancake collapse , occurs when we set the compressive strength of concrete to a small value @xmath114n / mm@xmath15 ( see fig . [ fig_2d_pancake ] ) . this choice is unphysical but allows us to separate the effect of strength reduction from that of stiffness reduction in the columns . more realistic scenarios would involve columns with small cross section and highly reinforced , tall beams . the columns immediately next to the damage area are the first to fail under compression , and then progressive collapse spreads horizontally to the outside . we employ equivalent loads @xmath105 referring to the two limit cases of _ local _ and of _ global _ pancake collapse . local pancake collapse occurs when the bending stiffness of the beams is very low and when the compressive failure of the columns is very brittle . in this case , the overload after damage is entirely directed to the intact columns that are closer to the damage area , and collapse propagates by _ nearest neighbor _ interactions . on the other hand , high stiffness of the beams and large plastic capacity of the columns induce _ democratic _ redistribution of overload between the columns . consequently , the columns crush simultaneously triggering global pancake collapse . the collapse dynamics recorded in our simulations resembles global pancake . note that in the studied framed structures , all the columns have identical compressive strength without disorder . therefore , once the first two columns crush , pancake collapse can not be arrested . nevertheless , at some @xmath115n / mm@xmath15 our frames undergo partial collapse because the progressive failure of the columns can be arrested by the initiation of bending collapse . = 5 , @xmath7 , and @xmath109=11 , @xmath8 . ] [ fig_2d_mu - rsr_pank ] , where superscript @xmath116 indicates pancake collapse , shows that the collapse loads increase with the structural slenderness @xmath117 , because the columns have tributary area related to @xmath118 and compressive strength proportional to @xmath119 . furthermore , hierarchical structures with small @xmath109 appear to be stronger than homogeneous ones both in terms of @xmath101 and of @xmath102 . finally , the residual strength fraction @xmath0 toward pancake collapse is remarkably higher than that toward bending collapse ( cf . [ fig_mu - rsr - bend ] ) , and is neither influenced by the hierarchical level @xmath2 , nor by @xmath38 . in fact , @xmath0 toward global pancake mode is related to the _ fraction _ of columns that are initially removed at one story . in our simulations , we always remove one third of the columns at one story , and obtain the constant value @xmath120 , slightly smaller than a theoretical 2/3 because of dynamics . we showed how the dynamic strength after damage @xmath102 of 2d frames depends on the activated collapse mechanism . we can now drive a series of conclusions regarding the effect of damage extent , structural slenderness , and topological and mechanical hierarchy . bending collapse provokes a local intensification of bending moments at the connections between the transfer beams above the damage area and the first intact column . consequently , @xmath102 and the residual strength fraction @xmath0 decrease with the number of removed columns . in analogy with fracture mechanics , structures that are prone to bending collapse correspond to notch sensitive materials , and the number of removed columns corresponds to the crack width @xcite . if global pancake collapse is triggered , @xmath102 and @xmath0 decrease with the fraction of removed columns , which is analogous to plastic failure of materials that are not notch sensitive . consistently , @xmath0 corresponding to global pancake collapse is remarkably larger than that corresponding to bending collapse . the structural slenderness @xmath121 affects in general the collapse loads for both bending and pancake collapse modes . the effect of @xmath122 depends on the scaling of cross section and reinforcement of the structural elements , with the beam length @xmath123 and with the column height @xmath124 ( see the analytical results in @xcite , regarding the simulations in this paper ) . nevertheless @xmath125 turns out to be independent from @xmath122 , because @xmath125 is the ratio between two collapse loads with same scaling respect to @xmath123 and @xmath124 . considering structural topology , in case of bending collapse hierarchical structures are more robust toward initial damage with fixed spatial extent ( e.g. explosion , impact ) , and as robust as homogeneous structures toward single column removal ( e.g. design error ) . the reason is that @xmath0 toward bending collapse decreases with the number of removed columns at one story . this confirms the analogy with fracture mechanics , where notch sensitive hierarchical materials are tougher than homogeneous ones @xcite . on the other hand , considering global pancake collapse , structural hierarchy does not influence @xmath0 toward initial damage with fixed spatial extent , while hierarchical structures are more sensitive than homogeneous ones to single column removals . this is due to @xmath0 toward global pancake collapse decreasing with the fraction of removed columns . [ fig_2d_mu - rsr_pank ] shows that damaged frames undergoing global pancake collapse can carry the @xmath126 of the static ultimate load of the intact structure @xmath101 . since well designed structures can carry a @xmath101 remarkably greater than the environmental load expected when an accidental event occurs , @xmath0 related to global pancake collapse can ensure structural robustness for most of the practical cases @xcite . on the other hand , @xmath0 related to bending collapse is generally much smaller , making structures vulnerable to accidental damage . in this work we considered idealized structures , with simplified geometry and mechanical behavior of the elements . reducing local complexity enables a better interpretation of the coral system response to damage . this study provides a basis of knowledge preceding the incorporation of more details and degrees of freedom , to investigate further aspects of progressive collapse . shear failures can cause brittle ruptures and reduce the collapse resistance of large structural elements . different locations of the initial damage may activate different collapse mechanisms . for example , damaging the upper stories would cause debris impacts , while removing external columns reduces @xmath127 without producing significant lateral toppling @xcite . the dem algorithm was already applied to 3d structures in ( masoero , wittel et al . , 2010 ) , showing that the bending and pancake collapse mechanisms persist also in 3d . on the other hand , in 3d structures the horizontal floor slabs improve the horizontal redistribution of loads and the catenary action , increasing the strength toward bending collapse and impacting debris ( see the appendix and , e.g. , @xcite ) . it is worth noting that horizontal ties and diaphragms increase the strength both after and before damage , causing a compensation that limits the effect on @xmath125 . finally , future works can incorporate a detailed description of structural connections , which are crucial for energy dissipation , catenary effect , and compartmentalization . coming back to the central theme of structural hierarchy , our results already suggest that hierarchical structures are more robust toward accidental damage . an optimal solution would be to design : 1 ) a primary frame made of few large elements , with columns weaker than the beams , and 2 ) a secondary structure , made of many smaller elements , which defines the living space and follows traditional design rules . the primary frame would provide topological hierarchy , maximizing @xmath0 toward bending collapse and enabling new possible compartmentalization strategies . the strong beams and weak columns of the primary frame would favor pancake collapse over bending collapse , and improve the vertical compartmentalization of high - rise buildings against falling debris . on the other hand , in real structures , the beams generally fail before the columns , and imposing the opposite is expensive . nevertheless , designing a strong - beam weak - column behavior _ only for the primary frame _ can significantly limit the extra cost . hierarchical structures can be a novel and somehow counterintuitive feature of robustness - oriented capacity design . planning structural hierarchy requires understanding the complex system response to local damage , and should drive the design process since the very beginning . by contrast , traditional design is focused on local resistance against ordinary actions , and considers robustness toward accidents only at the end . this generally leads to non - hierarchical structures with strong columns and poorly understood system behavior . in addition , anti - seismic capacity design requires plastic failure of the beams to precede columns rupture ( see e.g. @xcite ) . overcoming these contradictions is a challenge toward optimizing structures against exceptional events . 22 [ 1]#1 [ 1]`#1 ` urlstyle [ 1]doi : # 1 s. alexander . new approach to disproportionate collapse . _ struct . eng . _ , 820 ( 23/24):0 1418 , 2004 . z.p . baant and y. zhou . why did the world trade center collapse ? - simple analysis . mech .- asce _ , 1280 ( 1):0 26 , 2002 . . actions on structures - part 1 - 7 : general actions - accidental actions . technical report en 1991 - 1 - 7 , bsi , 2004 . . design of structures for earthquake resistance . technical report , bsi , 2004 . design of steel framed buildings at risk from terrorist attack . _ _ , 820 ( 22):0 3138 , 2004 . a. calvi . il crollo delle torri gemelle : analisi dellevento e insegnamenti strutturali . master s thesis , politecnico di torino , 2010 . ( in italian ) . carmona , f.k . wittel , f. kun , and h.j . fragmentation processes in impact of spheres . e _ , 770 ( 5):0 243253 , 2008 . cherepanov and i.e. esparragoza . progressive collapse of towers : the resistance effect . _ , 143:0 203206 , 2007 . chiaia and e. masoero . analogies between progressive collapse of structures and fracture of materials . _ , 1540 ( 1 - 2):0 177193 , 2008 . technical report , department of defence , 2005 . technical report , gsa , 2003 . h. gulvanessian and t. vrouwenvelder . robustness and the eurocodes . eng . int . _ , 2:0 161171 , 2006 . r. hamburger and a. whittaker . design of steel structures for blast - related progressive collapse resistance . _ modern steel constr . _ , march:0 4551 , 2004 . r. lakes . materials with structural hierarchy . _ , 361:0 511515 , 1993 . e. masoero . _ progressive collapse and robustness of framed structures_. phd thesis , politecnico di torino , italy , 2010 . e. masoero , p. dar , and b.m . chiaia . progressive collapse of 2d framed structures : an analytical model . _ , 54:0 94102 , 2013 . c. pearson and n. delatte . ronan point apartment tower collapse and its effect on building codes . _ j. perf . fac . -asce _ , 190 ( 2):0 172177 , 2005 . t. pschel and t. schwager . _ computational granular dynamics_. springer - verlag gmbh , berlin , 2005 . u. starossek . progressive collapse of structures : nomenclature and procedures . _ , 160 ( 2):0 113117 , 2006 . val and e.g. val . robustness of framed structures . _ , 160 ( 2):0 108112 , 2006 . vlassis , b.a izzuddin , a.y . elghazouli , and d.a nethercot . progressive collapse of multi - storey buildings due to sudden column loss - part ii : application . _ , 300 ( 5):0 14241438 , 2008 . w .- j . yi , q .- f . he , y. xiao , and s.k . experimental study on progressive collapse - resistant behavior of reinforced concrete frame structures . _ aci structural journal _ , 1050 ( 4):0 433439 , 2008 . in this appendix , we compare the numerical predictions of our dem model with the experimental observations in @xcite . we also briefly discuss some effects of catenary actions on collapse resistance . the experimental setup in @xcite consists of a plane frame made of reinforced concrete ( see fig . [ fig_exper](a ) ) . columns are square in section ( 200x200 mm ) , beams are rectangular ( 200 mm tall , 100 mm wide ) . everywhere , the longitudinal reinforcement is symmetrically distributed within the cross section ( 4@xmath12812 steel bars ) . the strength and ultimate strain of concrete and steel are specified in @xcite , while the elastic moduli are not . the mid column at the first floor is replaced by jacks that provides an upward vertical force @xmath129 . in the middle of the top floor , a servo - hydraulic actuator applies a constant downward vertical force @xmath130=109kn , to represent the self weight of upper stories . initially @xmath131109kn , and then it is progressively reduced to reproduce quasi - static column loss , until a bending mechanism triggers collapse ( see fig . [ fig_exper](b ) ) . during the experiments , the increasing values of the midspan inflection @xmath132 is plotted against @xmath129 , to get the force - displacement reaction curve . the integral of the curve represents the energy dissipation capacity , which relates to the dynamic strength of the structure with respect to the activated collapse mechanism . our target is to capture the experimental reaction curve @xmath133 through dem simulations . the parametrization of our model , based on the geometry and mechanical data in @xcite , is straightforward . therefore , we focus on the discrepancies between model and experimental inputs , and a few necessary additional assumptions . regarding the overall geometry , we consider all the columns to be equally tall ( 1,100 mm ) , while in the experiments the columns at the first floor were taller ( 1,567 mm ) . this discrepancy should not have a significant effect on the collapse mechanism and the strength . the mechanical behavior of the real steel bars was strain hardening , with yielding at 416mpa , and rupture at 526mpa . in our model , we consider two limit cases of elastic - perfectly plastic behavior of the steel bars : _ weak steel _ `` ws '' with yielding threshold set at 416mpa , and _ strong steel _ `` ss '' yielding at 526mpa . @xcite provide two measures of the ultimate tensile strain @xmath134 of the steel bars . we employ @xmath135 , which was measured on a longer bar segment , because in our simulation the strain develops within relatively long euler - bernoulli beam elements , ebes . we assume young moduli @xmath136gpa for the steel , and @xmath137gpa for the concrete . in order to better understand the development of catenary actions , we consider two limit cases of cross section behavior under tension : _ fully reacting sections _ `` frs '' , where the concrete always contributes to the tensile stiffness , and _ partially reacting sections _ `` prs '' , where the concrete cracks and only the steel provides axial stiffness as soon as the cross section goes in tension . furthermore , in order to focus exclusively on ruptures due to tensile strain in the steel , we allow for an infinite rotation capacity of the cross sections . we subject our model frames to gravity , but remain in the quasi - static regime by adding a high viscous damping force proportional to the velocity of each spherical discrete element . we repeat numerous simulations with fixed @xmath138kn and @xmath129 , ranging from @xmath139kn to values that are small enough to cause the quasi - static rupture of at least one ebe . we track the midspan deflection @xmath140for comparison in fig . [ fig_exper](c ) . in the experimental results , as @xmath129 decreases , the system crosses several stages : ( i ) linear elastic @xmath141 mm , ( ii ) elasto - plastic @xmath142 mm , ( iii ) plastic hinges @xmath143 mm , ( iv ) catenary action @xmath144 mm , and ( v ) collapse . the transition from elastic to elasto - plastic is not evident from the curve , as well as that from plastic hinge to catenary action . by contrast , plastic hinges formation is clearly marked by a sudden change of slope at @xmath145 mm . our simulations do not capture the initial elasto - plastic stage because we do not model the non - linear elasto - plastic behavior of concrete . this leads to an overestimation of the stiffness @xmath146 before the formation of the plastic hinges . nevertheless , the additional strain energy produced by this approximation is negligible when compared to the energy dissipated in the subsequent stages , ie . the overestimation of the initial stiffness is irrelevant for the actual dynamic collapse . assuming weak steel ws , yielding at 416mpa , provides a good agreement with the experiment in terms of transition point to the plastic hinges stage . considering fractured concrete under tension yields the prs - ws curve , which underestimates the structural strength at large @xmath132 . the reason for this divergence can be that the steel hardens under strain , with reaction stress increasing from from 416mpa ( ws ) to 526mpa ( ss ) . this interpretation is supported by the fact that the prs - ws and prs - ss curves envelop the experimental one . in particular , the prs - ss curve reproduces well the last part of the experimental curve , as well as the collapse point . despite strong simplifying assumptions in the formulation , our dem model provides reasonably good quantitative predictions of the experimental results . for the simulations in the body of this paper , we always considered fully reactive cross sections with concrete that does not crack under tension . the frs - ws curve in fig . [ fig_exper](c ) shows that this assumption leads to an overestimation of the static collapse strength against column removal ( @xmath147% ) . let us conjecture that frs induce the same strength increase of + 70% in the structure without column removal , i.e. in @xmath148 . from a heuristic application of energy conservation , one can estimate the dynamic collapse load after sudden column removal by considering the mean @xmath129 in the catenary stage : @xmath149kn from the experiment , and @xmath150kn from the simulation with frs - ws . consequently , the increase in post - damage dynamic collapse strength @xmath127 due to frs is approximately @xmath151% . in conclusion , assuming fully reactive cross sections causes an _ underestimation _ of the residual strength fraction @xmath152 , which our example quantifies as @xmath153% . however this assumption does not affect the main statement of this work on hierarchical structures

in this paper , we study the response of 2d framed structures made of rectangular cells , to the sudden removal of columns . we employ a simulation algorithm based on the discrete element method , where the structural elements are represented by elasto - plastic euler bernoulli beams with elongation - rotation failure threshold . the effect of structural cell slenderness and of topological hierarchy on the dynamic residual strength after damage @xmath0 is investigated . topologically _ hierarchical _ frames have a primary structure made of few massive elements , while _ homogeneous _ frames are made of many thin elements . we also show how @xmath0 depends on the activated collapse mechanisms , which are determined by the mechanical hierarchy between beams and columns , i.e. by their relative strength and stiffness . finally , principles of robustness - oriented capacity design which seem to be in contrast to the conventional anti - seismic capacity design are addressed . * keywords : * frames , progressive collapse , robustness , hierarchy

we would like to thank f.s . navarra for fruitiful conversations . this work has been partly supported by fapesp and cnpq - brazil . for a review and references to original works , see e.g. , s. narison , _ qcd as a theory of hadrons , cambridge monogr . part . * 17 * , 1 ( 2002 ) [ hep - h/0205006 ] ; _ qcd spectral sum rules , world sci . notes phys . _ * 26 * , 1 ( 1989 ) ; acta phys . pol . * b26 * , 687 ( 1995 ) ; riv . * 10n2 * , 1 ( 1987 ) ; phys . rept . * 84 * , 263 ( 1982 ) .

we use the qcd sum rules to evaluate the mass of a possible scalar mesonic state that couples to a molecular @xmath0 current . we find a mass @xmath1 gev , which is in a excellent agreement with the recently observed @xmath2 charmonium state . we consider the contributions of condensates up to dimension eight , we work at leading order in @xmath3 and we keep terms which are linear in the strange quark mass @xmath4 . we also consider a molecular @xmath5 current and we obtain @xmath6 , around 200 mev above the mass of the @xmath7 charmonium state . we conclude that it is possible to describe the @xmath2 structure as a @xmath8 molecular state . there is growing evidence that at least some of the new charmonium states recently discovery in the b - factories are non conventional @xmath9 states . some possible interpretations for these states are mesonic molecules , tetraquarks , or / and hybrid mesons . some of these new mesons have their masses very close to the meson - meson threshold like the @xmath10 @xcite and the @xmath11 @xcite . therefore , a molecular interpretation for these states seems natural . the most recent aquisiton for this list of peculiar states is the narrow structure observed by the cdf collaboration in the decay @xmath12 . the mass and width of this structure is @xmath13 , @xmath14 @xcite . since the @xmath2 decays into two @xmath15 vector mesons , it has positive @xmath16 and @xmath17 parities . there are already some theoretical interpretations for this structure . its interpretation as a conventional @xmath9 state is complicated because , as pointed out by the cdf collaboration @xcite , it lies well above the threshold for open charm decays and , therefore , a @xmath9 state with this mass would decay predominantly into an open charm pair with a large total width . in ref . @xcite , the authors interpreted the @xmath2 as the molecular partner of the charmonium - like state @xmath7 , which was observed by belle and babar collaborations near the @xmath18 threshold @xcite . they concluded that the @xmath2 is probably a @xmath19 molecular state with @xmath20 or @xmath21 . in ref . @xcite they have interpreted the @xmath2 as an exotic hybrid charmonium with @xmath22 . in this work , we use the qcd sum rules ( qcdsr ) @xcite , to study the two - point function based on a @xmath8 current with @xmath20 , to see if the new observed resonance structure , @xmath2 , can be interpreted as such molecular state . in previous calculations , the hidden charm mesons @xmath23 and @xmath24 have been studied using the qcdsr approach as tetraquark or molecular states @xcite . in some cases a very good agreement with the experimental mass was obtained . the starting point for constructing a qcd sum rule to evaluate the mass of a hadronic state , @xmath25 , is the correlator function ( q)=id^4x e^iq.x0 |t[j_h(x)j_h^(0)]|0 , where the current @xmath26 creates the states with the quantum numbers of the hadron @xmath25 . a possible current describing a @xmath27 molecular state with @xmath28 is j=(|s_a_c_a)(|c_b^s_b ) , [ field ] where @xmath29 and @xmath30 are color indices . the qcd sum rule is obtained by evaluating the correlation function in eq . ( [ 2po ] ) in two ways : in the ope side , we calculate the correlation function at the quark level in terms of quark and gluon fields . we work at leading order in @xmath3 in the operators , we consider the contributions from condensates up to dimension eight and we keep terms which are linear in the strange quark mass @xmath4 . in the phenomenological side , the correlation function is calculated by inserting intermediate states for the @xmath27 molecular scalar state . parametrizing the coupling of the scalar state , @xmath31 , to the current , @xmath32 , in eq . ( [ field ] ) in terms of the parameter @xmath33 : [ eq : decay ] 0 | j|h=. [ lam ] the phenomenological side of eq . ( [ 2po ] ) can be written as ^phen(q^2)=^2m_h^2-q^2+_0^ds ^cont(s)s - q^2 , where the second term in the rhs of eq.([phe ] ) denotes higher scalar resonance contributions . it is important to notice that there is no one to one correspondence between the current and the state , since the current in eq . ( [ field ] ) can be rewritten in terms of sum a over tetraquark type currents , by the use of the fierz transformation . however , the parameter @xmath33 , appearing in eq . ( [ lam ] ) , gives a measure of the strength of the coupling between the current and the state . the correlation function in the ope side can be written as a dispersion relation : ^ope(q^2)=_4m_c^2^ds ^ope(s)s - q^2 , where @xmath34 is given by the imaginary part of the correlation function : @xmath35 $ ] . as usual in the qcd sum rules method , it is assumed that the continuum contribution to the spectral density , @xmath36 in eq . ( [ phe ] ) , vanishes bellow a certain continuum threshold @xmath37 . above this threshold , it is given by the result obtained with the ope . therefore , one uses the ansatz @xcite ^cont(s)=^ope(s)(s - s_0 ) , to improve the matching between the two sides of the sum rule , we perfom a borel transform . after transferring the continuum contribution to the ope side , the sum rules for the scalar meson , considered as a scalar @xmath38 molecule , up to dimension - eight condensates , using factorization hypothesis , can be written as : ^2e^-^2/m^2=_4m_c^2^s_0ds e^-s / m^2 ^ope(s ) , [ sr1 ] where ^ope(s)=^pert(s)+^(s ) + ^g^2(s)+^mix(s)+^^2(s)+^mix(s ) , with [ eq : pert ] & & ^pert(s)=32 ^ 9 ^6_^ d^3 _ ^1-d^2(1 - - ) ^3(-4m_cm_s ) , + & & ^(s)=32 ^ 5 ^ 4_^ d\{m_s(m_c^2-(1-)s)^21 - - m_c_^1-d . + & & . } , + & & ^g^2(s)=m_c^22 ^ 8 ^ 6_^ d^3_^1-d(1 - - ) , + & & ^mix(s)=-m_0 ^ 22 ^ 6 ^ 4\ { 3m_c_^d [ m_c^2-(1-)s ] -m_s(8m_c^2-s ) } , + & & ^^2(s)=m_c^28 ^ 2\ { ( 2m_c - m_s)-m_sm_c^2_0 ^ 1d ( s - m_c^2(1- ) ) } , [ dim6 ] where the integration limits are given by @xmath39 , @xmath40 , @xmath41 , and we have used @xmath42 . we have neglected the contribution of the dimension - six condensate @xmath43 , since it is assumed to be suppressed by the loop factor @xmath44 . we also include a part of the dimension-8 condensate contributions , related with the mixed condensate - quark condensate contribution : ^mix(s)&=&-m_cm_0 ^ 2 ^ 216 ^ 2_0 ^ 1 d ( s - m_c^2(1- ) ) . [ dim8 ] it is important to point out that a complete evaluation of the dimension-8 condensate , and higher dimension condensates contributions , require more involved analysis @xcite , which is beyond the scope of this calculation . to extract the mass @xmath45 we take the derivative of eq . ( [ sr ] ) with respect to @xmath46 , and divide the result by eq . ( [ sr ] ) . for a consistent comparison with the results obtained for the other molecular states using the qcdsr approach , we have considered here the same values used for the quark masses and condensates as in refs . @xcite : @xmath47 , @xmath48 , @xmath49 , @xmath50 , @xmath51 with @xmath52 , @xmath53 . the borel window is determined by analysing the ope convergence and the pole contribution . to determine the minimum value of the borel mass we impose that the contribution of the dimension-8 condensate should be smaller than 20% of the total contribution . in fig . [ figconv ] we show the contribution of all the terms in the ope side of the sum rule . from this figure we see that for @xmath54 gev@xmath55 the contribution of the dimension-8 condensate is less than 20% of the total contribution . therefore , we fix the lower value of @xmath56 in the sum rule window as @xmath57 gev@xmath55 . the maximum value of the borel mass is determined by imposing that the pole contribution must be bigger than the continuum contribution . in table i we show the values of @xmath58 . in our numerical analysis , we will consider the range of @xmath56 values from 2.3 @xmath59 until the one allowed by the pole dominance criterion given in table i. + [ cols="^,^",options="header " , ] taking into account the incertainties given above we finally arrive at = ( 4.140.09 ) , [ ymass ] in an excellent agreement with the mass of the narrow structure @xmath2 observed by cdf . one can also deduce , from eq . ( [ sr1 ] ) , the parameter @xmath33 defined in eq . ( [ lam ] ) . we get : = ( 4.220.83 ) 10 ^ -2 ^5 , [ la1 ] from the above study it is very easy to get results for the @xmath60 molecular state with @xmath20 . for this we only have to take @xmath61 and @xmath62 in eqs . ( [ dim6 ] ) , ( [ dim8 ] ) . this study was already done in ref . @xcite considering @xmath63 . although in the case of the @xmath64 scalar molecule we get a worse borel convergence than for the @xmath38 scalar molecule , as can be seen by fig . [ opedd ] , there is still a good ope convergence for @xmath65 . if we allow also for the @xmath60 molecule values of the continuum threshold in the range @xmath66 we get @xmath67 . therefore , from a qcd sum rule study , the difference between the masses of the states that couple with scalar @xmath8 and @xmath60 currents , is consistent with zero . the mass obtained with the @xmath60 scalar current is about 100 mev above the @xmath68 threshold . this could be an indication that there is a repulsive interaction between the two @xmath69 mesons . strong interactions effects might lead to repulsive interactions that could result in a virtual state above the threshold . therefore , this structure may or may not indicate a resonance . however , considering the errors , it is not compatible with the observed @xmath70 charmonium - like state . in fig . [ dif ] we show the relative ratio @xmath71 as a function of the borel mass for @xmath72 . from this figure we can see that the ratio is very stable as a function of @xmath56 and the difference between the masses is smaller than 0.5% . although the ratio is shown for @xmath72 , the result is indiscernible from the one shown in fig . [ dif ] for other values of the continuum threshold in the range @xmath73 . this result for the mass difference is completely unexpected since , in general , each strange quark adds approximately 100 mev to the mass of the particle . therefore , one would naively expect that the mass of the @xmath38 state should be around 200 mev heavier than the mass of the @xmath64 state . this was , for instance , the result obtained in ref . @xcite for the vector molecular states @xmath74 and @xmath75 , where the masses obtained were : @xmath76 and @xmath77 . for the value of the parameter @xmath33 we get : _ d^*d^ * = ( 4.200.96)10 ^ -2 ^5 . [ la2 ] therefore , comparing the results in eqs . ( [ la1 ] ) and ( [ la2 ] ) we conclude that the currents couple with similar strength to the corresponding states , and that both , @xmath8 and @xmath60 scalar molecular states have masses compatible with the recently observed @xmath2 narrow structure . however , since the @xmath2 was observed in the decay @xmath78 , the @xmath8 assignment is more compatible with its quark content . in conclusion , we have presented a qcdsr analysis of the two - point function for possible @xmath8 and @xmath60 molecular states with @xmath20 . our findings indicate that the @xmath2 narrow structure observed by the cdf collaboration in the decay @xmath12 can be very well described by using a scalar @xmath8 current . although the authors of ref . @xcite interpreted the @xmath2 as a @xmath27 molecular scalar state and the @xmath7 as a @xmath5 molecular scalar state , we have obtained similar masses for the states that couple with the scalars @xmath27 and @xmath5 currents . therefore , from a qcd sum rule point of view , the charmonium - like state @xmath7 , observed by belle and babar collaborations , has a mass around 200 mev smaller than the state that couples with a @xmath5 scalar current and , therefore , can not be well described by such a current . while this work has been finalized , a similar calculation was presented in ref . @xcite . however , the author of ref . @xcite arrived to a different conclusion .

the lhc run 1 results , and especially the measured higgs boson properties @xcite , reveal a sm - like picture of the particle physics at the electroweak scale . nevertheless this situation is compatible with a potentially very rich scalar sector , hidden from direct inspection due to some decoupling or alignment arguments . assuming that this is the situation at the tev scale , one can ask what are the best experimental shortcuts to this hidden sector , and how one should probe it at colliders . of course , the answer strongly depends on the higgs sector chosen , and hundreds of papers have investigated it in various specific circumstances . also , the full parameter space is usually huge which renders it impossible to systematically explore the model in its full complexity , neither with algebra nor with numerical methods . in these cases , one usually resorts to specific models in particular corners of the parameter space . without any direct clue from experiment , one usually tries to uncover all physically interesting situations within a given model and to check which of them offer the most attractive description of the data . this undertaking requires a systematic exploration of the entire parameter space of a model , which for most cases is beyond the reach of traditional methods . one of the main reasons is that , with multiple higgs fields , the scalar potential , even the renormalizable one , becomes very complicated . even an efficient description of the full parameter space is a challenge , let alone its investigation . the two - higgs - doublet model ( 2hdm ) @xcite is a hallmark example of the bsm higgs sector , whose phenomenology exhibits a variety of bsm effects and which , at the same time , is still amenable to systematic treatment in the entire parameter space , although with somewhat non - standard mathematical methods . for more involved higgs sectors , the complexity of the analysis skyrockets , making the systematic study impossible . as a result , extensions of the higgs sector beyond 2hdm , such as @xmath0-higgs - doublet models ( nhdm ) , are nowhere near in the detail of their investigation , despite hundreds of publications ( for a few early publications , see @xcite ) . it all makes systematic investigation of nhdm a task which is very challenging but worth pursuing . how should one attack this problem ? experience in conservative bsm model - building shows that models , which are phenomenolgically viable and theoretical attractive , often arise from additional symmetries , either exact or approximate . nhdms can have much richer ( discrete ) symmetries than 2hdm but usually these exact symmetries lead to either unrealistic or very sm - like phenomenology . it is therefore natural to systematically explore nhdms _ in the vicinity of large discrete symmetry group_. this task splits into two parts , each of them being challenging on its own . first , one needs to know which highly symmetric nhdms one can construct for a given @xmath0 and what are their phenomenological consequences . if we assume that higgs doublets transform under irredicible representation of a large symmetry group @xmath1 , then the renormalizable higgs potential takes the form @xmath2 with the quadratic part being symmetric under all linear unitary transformations of @xmath0 doublets and the quartic part @xmath3 encoding the desired symmetry . second , one should explicitly break this symmetry group in a way that does not lead to dramatic consequences in order not to violate existing data . a natural procedure is to introduce soft breaking terms in the quadratic potential , @xmath4 , keeping the quartic part unchanged . in this way , the symmetric model serves as a starting point , and one needs to systematically trace the evolution of phenomenology as one shifts away from the symmetry . the model still remains highly multidimensional , and the direct scans of the entire parameter space is of little use . i propose instead to search for analytically calculable _ robust quantities _ which would be applicable to a wide class of models and not be sensitive to specific numerical values of free parameters . it would be especially encouraging if these quantities are basis - invariant . below i will outline some directions in 3hdm , along which this task can be pursued . the first step is to understand the symmetric situations possible within a given scalar sector . here , we use for illustration the three - higgs - doublet model ( 3hdm ) , whose scalar symmetries have been recently investigated in much detail in @xcite . the effect of these symmetries and their breaking on the fermionic sector were investigated in @xcite for specific groups and , in general terms , in @xcite which completes the old analysis @xcite . the mere fact that we have at our disposal only three doublets , which interact via a renormalizable potential , restricts the list of symmetry groups @xmath1 which can be implemented in such scalar sectors . limiting ourselves only to discrete groups , one obtains the following list @xcite : @xmath5 imposing any other discrete symmetry group on the 3hdm scalar sector will unavoidably lead to an accidental continuous symmetry . some of these groups , namely , @xmath6 , @xmath7 , @xmath8 , @xmath9 , and @xmath10 , automatically lead to explicit @xmath11-conservation in the scalar sector ; the others are compatible with explicit @xmath11-violation . .the amount of residual symmetry possible after ewsb for each discrete symmetry group of the 3hdm scalar potential ( see text for details ) . @xmath12 signals the presence of a ( generalized ) @xmath11 symmetry in the model . [ cols=">,^,^,^,^",options="header " , ] all possible symmetry breaking patterns for each of these groups were listed in @xcite ; see also @xcite for results in specific groups . these findings are summarized in table [ table ] . the strongest symmetry breaking of a given group @xmath1 , in which we also include ( generalized ) @xmath11-symmetries when they are present , corresponds to the smallest residual symmetry group @xmath13 , and its order is denoted by @xmath14 . the weakest breaking corresponds to the largest residual symmetry group , with order @xmath15 . the groups in the upper block allow for all types of symmetry breaking : complete , partial , or no breaking at all . the groups in the middle block can remain intact at the global minimum , but if they are broken , their breaking is only partial . the last block contains groups which can neither remain unbroken nor break completely . they are always broken to a proper subgroup . the fact the large discrete symmetry groups can not be broken completely with tree - level potentials has several phenomenological consequences . in the last column of table [ table ] we indicate whether the @xmath11-symmetry present in the scalar sector can spontaneously break . one sees that highly symmetric models prevent not only explicit but also spontaneous @xmath11-violation in the scalar sector . it is curious to note that these two types of @xmath11-violations always come in pairs , at least in the 3hdm . whether this is just a coincidence or reveals a generic fact in nhdm is not yet known . incomplete symmetry breaking has also consequences for the fermion sector . working within pure nhdm ( no bsm fields beyond several higgs doublets ) and extending the symmetry group @xmath1 to the full theory , one can prove that the lack of complete breaking leads to unphysical quark sector . the presence of a residual symmetry among higgses coupled to quarks leads either to massless quarks , or to block - diagonal form of the ckm matrix , or to the absence of @xmath11-violation . this detrimental role of residual family symmetries were noted long ago @xcite but the accurate statement and its proof were presented only recently @xcite . nhdm with large symmetry groups contain many fields but very few free parameters , which leads to degeneracies among some of the physical higgses . examples of this situation appeared in many papers , and , for the charged higgs bosons , could be traced from general nhdm analysis @xcite . residual symmetries can also stabilize some of the higgs bosons making them dark matter candidates . this is a well - known feature of multi - scalar models , the inert doublet model @xcite and the simple extensions with ew singlets @xcite being the most studied examples . here we want to stress that , for large discrete groups , such a symmetry - protected stabilization can be automatic and does not rely on one s choice of free parameters . in short , nhdms equipped with large discrete symmetry groups have predictive phenomenological consequences , which are not sensitive to the exact numerical values of the free parameters . these consequences reflect robust structural properties of these models and especially of the scalar potential . it is true that some of nhdms with large discrete symmetry group are already incompatible with experiment , especially in the fermon sector . however if one wants to softly break these symmetries later , it is not an obstacle but rather a good starting point in understanding phenomenology . the second step is to softly break the large symmetry group and to track down how the phenomenological picture changes . certainly , for concrete models , one can perform this analysis numerically . however understanding in this way the entire spectrum of physical possibilities is a hopeless task due to the huge number of free parameters in multi - higgs sectors . if we want to gain some insight into these models in the entire parameter space , we need to devise a method beyond simple numerical scans or case - by - case investigation . a promising direction is to find robust and calculable quantities , which would not be too sensitive to the exact numerical values of the free parameters but which would reflect the structural features of the whole class of models . here i describe one example of such quantities which i call `` critical exponents '' . consider a model with large multi - dimensional parameter space @xmath16 , see fig . [ fig - space ] . models possessing ( large ) symmetry groups can be contructed by imposing relations among these free parameters ; thus , they occupy certain low - dimensional manifolds in this space . their phenomenology in the scalar and fermionic sector often features quantities which are zero . depending on the specific construction , they can reflect degenerate extra higgs bosons , their stability due to residual symmetry after ewsb , massless quarks , absence of the @xmath11-violation in the quark mixing matrix , etc . symmetry - based nhdm examples with these properties can be easily constructed . we call these symmetry - protected quantities `` order parameters '' and denote them generically as @xmath17 . in the multi - dimensional parameter space , symmetric models occupy lower - dimensional manifolds . these manifolds are embedded into one another dependeing on their symmetrry groups @xmath1 . the phenomenology of a non - symmetric model in the vicinity of a symmetric situation can bear resemblance to the symmetric case , up to corrections which are powerlike in the distance @xmath18 . ] now , a generic point @xmath19 in the parameter space does not correspond to any exact symmetry , hence it leads to non - zero order parameters @xmath20 . however this point can lie close to a symmetric manifold , within small distance @xmath18 , where `` small '' means comparable or smaller than typical sizes of the symmetric manifold structures . one can then expect the order parameters to exhibit a powerlike behaviour : @xmath21 . the index @xmath22 is called the critical exponent for the quantity @xmath17 . a set of critical exponents describes how the phenomenology evolves in the vicinity of a symmetric situation . there are two important aspects of these critical exponents . first , they are robust upon variation of free parameters and are , therefore , calculable . they do not depend on the exact position of the model in the parameter space , but only reflect its degrees of proximity to a certain symmetric situation . they can depend , though , on the symmetry group and on the geometric shape of its manifold in the parameter space . in short , critical exponents reveal certain structural properties of the whole family of models , and should be calculable analytically . second , having these critical exponents at hand will provide qualitative understanding of multi - parametric models in the vicinity of symmetries , and in particular of nhdm with large softly broken non - abelian groups . indeed , knowing whether certain critical exponent is @xmath23 and another is @xmath24 would immediately provide insight on which effects are phenomenologically important and how they can be related . in short , when working in hugely multidimensional parameter spaces , critical exponents can guide the search for models with desired phenomenology . as an illustration of critical exponents displaying non - trivial behaviour , consider the scalar sector in the general 2hdm in the vicinity of critical points in the parameter space at which the lightest higgs mass @xmath25 . the parameter space can be fully reconstructed @xcite , and locus of critical points corresponds to a certain ellipse in it . in its vicinity , the mass is non - zero and the critical exponent @xmath22 was found to be either @xmath26 or @xmath27 , depending on how one approaches the ellipse @xcite . these two cases differ by a residual discrete symmetry of the model . this example , not pretending to be phenomenologically relevant , provides a taste of how easily the things can become non - trivial even in rather simple situations . systematric derivation of critical exponents for all order parameters in nhdm will already provide a new insight into its phenomenology . 99 g. aad _ et al . _ [ atlas collaboration ] , science * 338 * , 1576 ( 2012 ) ; v. khachatryan _ et al . _ [ cms collaboration ] , arxiv:1412.8662 [ hep - ex ] . g. c. branco , p. m. ferreira , l. lavoura , m. n. rebelo , m. sher and j. p. silva , phys . rept . * 516 * , 1 ( 2012 ) . s. weinberg , phys . lett . * 37 * , 657 ( 1976 ) ; e. derman , phys . b * 78 * , 497 ( 1978 ) ; s. pakvasa and h. sugawara , phys . b * 73 * , 61 ( 1978 ) ; d. wyler , phys . d * 19 * , 3369 ( 1979 ) ; g. c. branco , phys . lett . * 44 * , 504 ( 1980 ) ; e. ma , phys . b * 96 * , 115 ( 1980 ) ; k. shizuya and s. h. h. tye , phys . d * 23 * , 1613 ( 1981 ) ; y. yamanaka , h. sugawara and s. pakvasa , phys . d * 25 * , 1895 ( 1982 ) ; g. c. branco , j. m. gerard and w. grimus , phys . b * 136 * , 383 ( 1984 ) . i. p. ivanov , v. keus and e. vdovin , j. phys . a * 45 * , 215201 ( 2012 ) . i. p. ivanov and e. vdovin , eur . j. c * 73 * , 2309 ( 2013 ) . a. degee , i. p. ivanov and v. keus , jhep * 1302 * , 125 ( 2013 ) . i. p. ivanov and c. c. nishi , jhep * 1501 * , 021 ( 2015 ) . r. gonzlez felipe , h. serdio and j. p. silva , phys . d * 87 * , 055010 ( 2013 ) ; phys . d * 88 * , 015015 ( 2013 ) . r. gonzlez felipe , i. p. ivanov , c. c. nishi , h. serdio and j. p. silva , eur . j. c * 74 * , 2953 ( 2014 ) . m. leurer , y. nir and n. seiberg , nucl . b * 398 * ( 1993 ) 319 . n. g. deshpande and e. ma , phys . d * 18 * , 2574 ( 1978 ) ; r. barbieri , l. j. hall and v. s. rychkov , phys . d * 74 * , 015007 ( 2006 ) ; l. lopez honorez , e. nezri , j. f. oliver and m. h. g. tytgat , jcap * 0702 * , 028 ( 2007 ) . t. robens and t. stefaniak , eur . j. c * 75 * , 104 ( 2015 ) .

conservative bsm models with rich scalar sector , such as multi - higgs - doublet models , can easily accommodate the sm - like properties of the 125 gev scalar observed at the lhc . possessing a variety of bsm signals , they are worth investigating in fuller detail . systematic study of these models is hampered by the highly multi - dimensional parameter space and by mathematical challenges . i outline some directions along which multi - higgs - doublet models in the vicinity of a large discrete symmetry can be systematically explored .

in healthy cells , a loopback mechanism involving the protein p53 is believed to cause growth arrest and apoptosis as a response to dna damage @xcite . mutations in the sequence of p53 that potentially interfere with this mechanism have been observed to lead to the upraise of cancer @xcite . under normal conditions the amount of p53 protein in the cell is kept low by a genetic network built of the mdm2 gene , the mdm2 protein and the p53 protein itself . p53 is produced at a essentially constant rate and promotes the expression of the mdm2 gene @xcite . on the other hand , the mdm2 protein binds to p53 and promotes its degradation @xcite , decreasing its concentration . when dna is damaged , a cascade of events causes phosphorylation of several serines in the p53 protein , which modifies its binding properties to mdm2 @xcite . as a consequence , the cell experiences a sudden increase in the concentration of p53 , which activates a group of genes ( e.g. , p21 , bax @xcite ) responsible for cell cycle arrest and apoptosis . this increase in p53 can reach values of the order of 16 times the basal concentration @xcite . a qualitative study of the time dependence of the concentration of p53 and mdm2 has been carried out in ref . approximately one hour after the stress event ( i.e. , the dna damage which causes phosphorylation of p53 serines ) , a peak in the concentration of p53 is observed , lasting for about one hour . this peak partially overlaps with the peak in the concentration of mdm2 , lasting from @xmath1 to @xmath2 hours after the stress event . another small peak in the concentration of p53 is observed after several hours . the purpose of the present work is to provide the simpest mathematical model which describes all the known aspect of the p53mdm2 loop , and to investigate how the loop is robust to small variations to the ingredients of the model . the `` weak points '' displayed by the system , namely those variations in some parameters which cause abrupt changes in the overall behaviour of the loop , are worth to be investigated experimentally because they can contain informations about how a cell bacomes tumoral . the model we suggest is described in fig . the total number of p53 molecules , produced at constant rate @xmath3 , is indicated with @xmath4 . the amount of the complexes built of p53 bound to mdm2 is called @xmath5 . these complexes cause the degradation of p53 ( through the ubiquitin pathway ) , at a rate @xmath6 , while mdm2 re enters the loop . furthermore , p53 has a spontaneous decay rate @xmath7 . the total number of mdm2 proteins is indicated as @xmath8 . since p53 activates the expression of the mdm2 gene , the production rate of mdm2 is proportional ( with constant @xmath9 ) to the probability that the complex p53/mdm gene is built . we assume that the complex p53/mdm2gene is at equilibrium with its components , where @xmath10 is the dissociation constant and only free p53 molecules ( whose amount is @xmath11 ) can participate into the complex . the protein mdm2 has a decay rate @xmath12 . the constants @xmath7 and @xmath12 describe not only the spontaneous degradation of the proteins , but also their binding to some other part of the cell , not described explicitely by the model . the free proteins p53 and mdm2 are considered to be at equilibrium with their bound complex pm , and the equilibrium constant is called @xmath0 . the dynamics of the system can be described by the equations @xmath13 in the second equation we allow a delay @xmath14 in the production of mdm2 , due to the fact that the transcription and translation of mdm2 lasts for some time after that p53 has bound to the gene . the choice of the numeric parameters is somewhat difficult , due to the lack of reliable experimental data . the degradation rate through ubiquitin pathway has been estimated to be @xmath15 @xcite , while the spontaneous degradation of p53 is @xmath16 @xcite . the dissociation constant between p53 and mdm2 is @xmath17 @xcite ( expressed as number of molecules , assuming for the nucleus a volume of @xmath18 ) , and the dissociation constant between p53 and the mdm2 gene is @xmath19 @xcite . in lack of detailed values for the protein production rates , we have used typical values , namely @xmath20 and @xmath21 . the degradation rate of mdm2 protein has been chosen of the order of @xmath22 to keep the stationary amount of mdm2 of the order of @xmath23 . the behaviour of the above model is independent on the volume in which we assume the reaction takes place . that is , multiplying @xmath3 , @xmath9 , @xmath10 and @xmath0 by the same constant @xmath24 gives exactly the same dynamics of the rescaled quantities @xmath25 and @xmath26 . futhermore , due to the fact that the chosen parameters put the system in the saturated regime , an increase in the producing rates @xmath3 and @xmath9 with respect to @xmath10 and @xmath0 will not affect the response . on the contrary , a decrease of @xmath3 and @xmath9 with respect to @xmath10 and @xmath0 can drive the system into a non saturated regime , inhibiting the response mechanism . in the case that the production of mdm2 can be regarded as instantaneous ( no delay , @xmath27 ) , the concentration of p53 is rather insensitive to the change of the dissociation constant @xmath0 . the stationary values of @xmath4 and @xmath8 are found as fixed points of the equations [ eq1 ] ( see appendix ) and in table i we list the stationary values @xmath28 of the amount of p53 molecules for values of @xmath0 spanning seven orders of magnitude around the basal value @xmath29 . moreover , transient oscillatory behaviour upon change in the dissociation constant @xmath0 is not observed . this is supported by the fact that the eigenvalues of the stationary points ( listed in table i ) have negative real parts , indicating stable fixed points , and rather small imaginary pats indicating absence of oscillations . more precisely , the variation @xmath30 of the stationary amount of p53 if the dissociation constant undergoes a change @xmath31 can be estimated , under the approximation that @xmath32 ( cf . the appendix ) , to be @xmath33 the fact that @xmath30 is approximately linear with @xmath31 with a proportionality constant which is at most of the order of @xmath34 makes this system rather inefficient as response mechanism . furthermore , it does not agree with the experimental data which show a peak of p53 followed , after several minutes , by a peak in mdm2 @xcite , and not just a shift of the two concentration to higher values . to check whether the choice of the system parameters affects the observed behaviour , we have repeated all the calculations varying each parameter of five orders of magnitude around the values used above . the results ( listed in table ii for @xmath3 and @xmath10 and not shown for the other parameters ) indicate the same behaviour as above ( negative real part and no or small imaginary part in the eigenvalues ) . consequently , the above results about the dynamics of p53 seem not to be sensitive to the detailed choice of parameters ( on the contrary , the amount of mdm2 is quite sensitive ) . the dynamics changes qualitatively if we introduce a nonzero delay in eqs . [ eq1 ] . keeping that the halflife of an rna molecule is of the order of 1200 s @xcite , we repeat the calculations with @xmath35 . the eqs . [ eq1 ] are solved numerically , starting from the conditions @xmath36 and @xmath37 and making use of a variable step adams algorithm . after the system has reached its stationary state under basal condition , a stress is introduced ( at time @xmath38 s ) by changing instantaneously the dissociation constant @xmath0 . in fig . [ fig2 ] we display a case in which the stress multiplies @xmath0 by a factor @xmath39 ( a ) , a case in which it divides it by a factor @xmath39 ( b ) and by a factor @xmath40 ( c ) . when @xmath0 is increased by any factor , the response is very similar to the response of the system without delay ( cf . e.g. fig . [ fig2]a ) . on the contrary , when @xmath0 is decreased the system displays an oscillatory behaviour . the height @xmath30 of the response peak is plotted in fig . [ fig3 ] as a function of the quantity which multiplies @xmath0 . if the multiplier is larger than @xmath41 the response is weak or absent . at the value @xmath41 the system has a marked response ( cf . also figs . [ fig2]b and c ) . the maximum of the first peak takes place approximately @xmath42s after the stress , which is consistent with the lag time observed in the experiment @xcite , and the peaks are separated from @xmath43s . although it has been suggested that the effect of the stress is to increase the dissociation constant between p53 and mdm2 @xcite , our results indicate that an efficient response take place if @xmath0 decreases of a factor @xmath44 ( cf . [ fig2]b ) . one has to notice that the conclusions of ref . @xcite have been reached from the analysis _ in vivo _ of the overall change in the concentration of p53 , not from the direct measurement of the binding constant after phosphorylation . our results also agree with the finding that p53asp20 ( a mutated form of p53 which mimicks phosphorylated p53 , due to the negative charge owned by aspartic acid ) binds mdm2 _ in vitro _ more tightly than p53ala20 ( which mimicks unphosphorylated p53 ) @xcite . this hypothesis is supported by molecular energy calculations made with classical force fields . even if this kind of force fields is not really reliable for the calculation of binding constants , it gives an estimate of the sign of the change in interaction among p53 and mdm2 upon phosphorylation . we have performed an energy minimization of the conformation of the system composed by the binding sites of p53 and mdm2 , starting form the crystallographic positions of ref . @xcite and using the force fields mm3 @xcite and mmff @xcite , for both the wild type system and for the system where serine 20 of p53 in phosphorylated . using mm3 we found that the phosphorylated system has an electrostatic energy @xmath45 kcal / mol lower than the wild type system , while this difference is @xmath46 kcal / mol using the mmff force field . our calculations suggest that phopshprylated p53 is more attracted by mdm2 due to the enhanced interaction of phosphorylated ser20 with lys60 , lys46 and lys70 of mdm2 , and consequently the dissociation constant is lowered . the robustness of the response mechanism with respect to the parameters of the system , which is typical of many biological systems ( cf . , e.g. , @xcite ) , has been checked both to assess the validity of the model and to search for weak points which could be responsible for the upraise of the disease . each parameter has been varied of five orders of magnitude around its basal quantity . the results are listed in table iii . one can notice that the response mechanism is quite robust to changes in the parameters @xmath6 , @xmath7 and @xmath9 . for low values of @xmath6 or @xmath9 the system no longer oscillates , but displays , in any case , a rapid increase in the amount of @xmath4 which can kill the cell . this is true also for large values of @xmath12 . what is dangerous for the cell is a decrease of @xmath12 or of @xmath10 , which would drop the amount of p53 and let the damaged cell survive . this corresponds either to an increase of the affinity between p53 and the mdm2 gene , or to an increase of mdm2 half life . to be noted that , unlike the case @xmath27 , the system with delay never displays damped oscillations as a consequence of the variation of the parameters in the range studied in the present work . this sharp behaviour further testifies to the robustness of the response mechanism . anyway , one has to keep in mind that the oscillating response produces the death of the cell , and consequently the long time behaviour is only of theoretical interest . the minimum value of the delay which gives rise to the oscillatory behaviour is @xmath47s . for values of the delay larger than this threshold , the amplitude of the response is linear with @xmath14 ( cf . 4 ) , a fact which is compatible with the explanation of the response mechanism of section iv . the lag time before the p53 response is around @xmath48s ( in accordance with the 1h delay observed experimentally @xcite and is independent on all parameters , except @xmath9 and @xmath14 . the dependence of the lag time on @xmath14 is approximately linear up to @xmath49s ( the longest delay analyzed ) . increasing @xmath9 the lag time increases to @xmath50s ( for @xmath51 ) and @xmath52s ( for @xmath53 ) . on the other hand , the period of oscillation depends only on the delay @xmath14 , being approximately linear with it . we have repeated the calculations squaring the variable @xmath4 in eqs . 1 , to keep into account the cooperativity induced by the fact that the active form of p53 is a dimer of dimers @xcite . the results display qualitative differences neither for non delayed nor for the delayed system . all these facts can be rationalized by analyzing the mechanism which produces the response . the possibility to trigger a _ rise _ in p53 as a dynamic response to an _ increased _ binding between p53 and mdm2 , relies on the fact that a sudden increase in p m binding diminishes the production of @xmath54 , and therefore ( subsequently ) diminishes the amount of @xmath5 . in other words , while the change in @xmath0 has no direct effect in the first of eqs . [ eq1 ] , it directly reduces mdm2 production by subtracting p53 from the gene which producese mdm2 . mathematically , the oscillations arise because the saturated nature of the binding @xmath5 imply that pm is approximately equal to the minimum between p and m. each time the curves associated with p and m cross each other ( either at a given time or @xmath14 instants before ) , the system has to follow a different set of dynamic equations than before , finding itself in a state far from stationarity . this gives rise to the observed peaks . to be precise , the starting condition ( before the stress ) is @xmath55 . the stress reduces the dissociation constant @xmath0 of , at least , one order of magnitude , causing a drop in @xmath4 , which falls below @xmath8 . for small values of @xmath0 ( to be precise , for @xmath56 ) , one can make the simplification @xmath57 , and consequently rewrite eqs . [ eq1 ] as @xmath58 each period after the stress can be divided in four phases . in the first one @xmath59 and @xmath60 , so that @xmath4 stays constant at its stationary value @xmath61 , while @xmath8 decreases with time constant @xmath62 towards zero ( not exactly zero , since the approximated equations do not hold for @xmath63 ) . in the second phase one has to consider the second ( @xmath60 ) and the third ( @xmath64 ) equations ( [ simply2 ] and [ simply3 ] ) . the new stationary value for @xmath4 is @xmath65 which is much larger than @xmath28 since @xmath66 . this boost takes place in a time of the order of @xmath67 , so if @xmath68 , as in the present case , @xmath4 has no time to reach the stationary state and ends in a lower value . in the meanwhile , @xmath8 remains in the low value given by eq . [ simply2 ] . at a time @xmath14 after the stress eq . [ simply2 ] gives way to eq . [ simply4 ] . the latter is composed by a positive term which is @xmath69 if @xmath70 and @xmath71 under the opposite condition . since @xmath72 ( it refers to the boost of @xmath4 ) , then the new stationary value of @xmath8 is @xmath73 . the raise of @xmath8 takes place in a time of the order of @xmath62 and causes the decrease of @xmath4 , whose production rate is ruled by @xmath74 . the fourth phase begins when @xmath4 approaches @xmath8 . now one has to keep eqs . [ simply1 ] and [ simply4 ] , so that @xmath4 returns to the basal value @xmath28 , while @xmath8 stays for a period of @xmath14 at the value @xmath73 reached in the third phase . after such period , eq . [ simply2 ] substitutes eq . [ simply4 ] and another peak takes place . the heigth of the p53 peak is given by @xmath75 if @xmath4 has time to reach its stationary state of phase two ( i.e. , if @xmath76 ) , or by @xmath77 if the passage to the third phase takes place before it can reach the stationary state . the width of the peak is @xmath78 and the spacing among the peaks @xmath78 , so that the oscillation period is @xmath79 . the necessary conditions for the response mechanism to be effective are 1 ) that @xmath80 , that is that the stationary value of @xmath4 just after the stress is much lower than the stationary value of @xmath8 , 2 ) that @xmath66 , in such a way that the stationary state of @xmath4 in the second phase is much larger than that in the first phase , in order to display the boost , 3 ) that @xmath81 , otherwise @xmath8 has not enough time to decrease in phase one and to increase in phase three . the failure of the response for low values of @xmath6 ( cf . table 3 ) is due to the fall of condition 2 ) , the failure for small @xmath9 is caused by condition 1 ) , the failure at small and large values of @xmath12 is associated with conditions 3 ) and 1 ) , respectively . at low values of @xmath10 the response does not take place because the positive term in eq . [ simply4 ] is always @xmath82 , and thus @xmath8 never decreases . in summa , we have shown that the delay is an essential ingredient of the system to have a ready and robust peak in p53 concentration as response to a damage stress . in order to have a peak which is comparable with those observed experimentally , the dissociation constant between p53 and mdm2 has to decrease of a factor @xmath39 . although it is widely believed that phosphorylation of p53 increases the dissociation constant , we observe an oscillating behaviour similar to the experimental one only if @xmath0 decreases . in this case the response is quite robust with respect to the parameters , except upon increaasing of the half life of mdm2 and upon decreasing of the dissociation constant between p53 and the mdm2 gene , in which cases there is no response to the stress . moreover , an increase in the production rate of mdm2 can delay the response and this can be dangerous to the cell as well . we hope that detailed experimental measurements of the physical parameters of the system will be made soon , in order to improve the model and to be able to make more precise predictions about the weak point of the mechanism , weak points which could be intimately connected with the upraise of cancer . the stationary condition for eqs . [ eq1 ] without delay can be found by the intersection of the curves @xmath83 which have been obtained by the conditions @xmath84 , explicitating @xmath5 from the first of eqs . [ eq1 ] and substituting it in the second and the third , respectively . to be noted that @xmath85 is linear in @xmath0 . the variation @xmath30 of the stationary value of p53 upon change @xmath31 in the dissociation constant can be found keeping that @xmath86 where the approximation @xmath32 has been used . consequently , @xmath87 which assumes the largest value when @xmath4 is smallest . using the parameters listed above , the proportionality constant is , at most , @xmath34 . 99 b. vogelstein , d. lane and a. j. levine , nature * 408 * ( 2000 ) 307310 m. d. shair , chemistry and biology * 4 * ( 1997 ) 791794 m. l agarwal , w. r. taylor , m. v. chernov , o. b. chernova and g. r. stark , j. biol . chem . * 273 * ( 1998 ) 14 r. l. bar or _ et al . _ , usa * 97 * ( 2000 ) 11250 m. s. greenblatt , w. p. bennett , m. hollstein and c. c. harris , cancer res . * 54 * ( 1994 ) 48554878 t. gottlieb and m. oren , biochim . acta * 1287 * ( 1996 ) 77102 y. haupt , r. maya , a. kazaz and m. oren , nature * 387 * ( 1997 ) 296299 m. h. g. kubbutat , s. n. jones and k. h. vousden , nature * 387 * ( 1997 ) 299393 t. unger , t. juven gershon , e. moallem , m. berger , r. vogt sionov , g. lozano , m. oren and y. haupt , embo journal * 18 * ( 1999 ) 18051814 w. el deiry , cancer biology * 8 * ( 1998 ) 345357 j. d. oliner , j. a. pietenpol , s. thiagalingam , j. gyuris , k. w. kinzler and b. vogelstein , nature * 362 * ( 1993 ) 857860 k. d. wilkinson , cell . . biol . * 11 * , 141 ( 2000 ) p. h. kussie _ et al . _ , science * 274 * ( 1996 ) 948 p. balagurumoortitiy _ et al . _ , usa * 92 * , 8591 ( 1995 ) b. alberts _ et al . _ , _ molecular biology of the cell _ , garland ( 1994 ) f. s. holstege _ _ , cell * 95 * , 717 ( 1995 ) j. h. lii and n. l. allinger , j. comput . * 12 * ( 1991 ) 186 t. a. halgren , j . comput . chem . * 17 * ( 1996 ) 490 m. a. savageau , nature * 229 * , 542 ( 1971 ) u. alon , m. g. surette , n. barkai , s. leibler , nature * 397 * , 168 ( 1999 ) m. g. mateu , m. m. sanchez del pino , a. r. fersht , nature struct . biol . * 6 * , 191 ( 1999 ) .stationary values @xmath28 and @xmath89 for the amount of p53 and mdm2 , respectively , calculated at @xmath27 . in the last column the eigenvalues of the linearized ( around the fixed points @xmath28 , @xmath89 ) dynamical matrix are displayed , by real and imaginary part . the real part of the eigenvalues is always negative and the imaginary part , when different from zero , is lower than the real part , indicating that the stationary values are always stable and the dynamics is overdamped . [ cols="^,^,^,^",options="header " , ]

a feedback mechanism that involves the proteins p53 and mdm2 , induces cell death as a controled response to severe dna damage . a minimal model for this mechanism demonstrates that the respone may be dynamic and connected with the time needed to translate the mdm2 protein . the response takes place if the dissociation constant @xmath0 between p53 and mdm2 varies from its normal value . although it is widely believed that it is an increase in @xmath0 that triggers the response , we show that the experimental behaviour is better described by a decrease in the dissociation constant . the response is quite robust upon changes in the parameters of the system , as required by any control mechanism , except for few weak points , which could be connected with the onset of cancer . pacs : 87.16.yc

this work is supported by the national sciences foundations of china under grant no . 10775148 , and by the cas knowledge innovation project kjcx3-syw - n2 . n. bianchi , a. fantoni , and s. liuti , phys . rev . d*69 * , 014505 ( 2004 ) ; a. fantoni and s. liuti , hep - ph/0511278 ; m. osipenko et al . b*609 * , 259 ( 2005 ) ; m. osipenko et al . rev . d*71 * , 054007 ( 2005 ) ; m. osipenko , w. melnitchouk , s. simula , s. kulagin , and g. ricco , nucl . a*766 * , 142 ( 2006 ) . p. l. anthony , et al . , [ e155 collaboration ] , phys . lett . b*553 * , 18 ( 2003 ) ; p. l. anthony et al . , [ e155 collaboration ] , phys . b*493 * , 19 ( 2000 ) ; p. l. anthony et al . b*463 * , 339 ( 1999 ) ; p. l. anthony et al . b*458 * , 529 ( 1999 ) ; p. l. anthony et al . , [ e142 collaboration ] , phys . rev d*54 * , 6620 ( 1996 ) . a. v. sidorov and d. b. stamenov , mod . a*21 * , 1991 ( 2006 ) ; e. leader , a. v. sidorov and d. b. stamenov , phys . rev . d*73 * , 034023 ( 2006 ) ; e. leader , a. v. sidorov and d. b. stamenov , phys . rev . d*75 * , 074027 ( 2007 ) . h. georgi and h. d. politzer , phys . d*14 * , 1829 ( 1976 ) ; s. wandzura , nucl . b*122 * , 412 ( 1977 ) ; s. matsuda and t. uematsu , nucl . b*168 * , 181 ( 1980 ) ; o. nachtmann , nucl . b*63 * , 237 ( 1975 ) .

target mass corrections to the twist-4 terms @xmath0 as well as to the leading - twist @xmath1 are discussed . pacs : 13.60.hb , 12.38.aw , 12.38cy ; 12.40.dh keywords : target mass corrections ; nachtmann moment ; higher - twist . + we know that different approaches [ 1 - 7 ] have been employed to study higher - twist effect to the nucleon structure functions . there were also several phenomenological analyses of the nucleon structure functions to study quark - hadron duality and to extract the higher - twist contributions ( like the ones of the twist-3 and twist-4 terms ) from experimental measurements [ 8 - 11 ] . those analyses are going to be more and more accurate since the more and more precise measurements of the nucleon spin structure functions @xmath2 and @xmath3 are becoming available [ 11 - 12 ] . the high precision data have been employed to study the validity of the quark - hadron duality for the nucleon structure function @xmath4 [ 13 ] and even for spin asymmetry @xmath5 by hermes [ 14 ] recently . several experiments to test the higher - twist effect on the nucleon spin structure functions are being carried out in the jefferson laboratory [ 9,15 ] . it has been pointed out , in the literature , that the target mass corrections ( tmcs ) should be considered in the studies of the nucleon structure functions [ 16 ] in a moderate @xmath6 region , and of the bloom - gilman quark - hadron duality [ 17 - 18 ] . therefore , only after the important target mass corrections are removed from the experimental data , one can reasonably extract the higher - twist effect [ 18 ] . there were several papers about the target mass corrections to @xmath7 and @xmath8 in the past [ 19 ] . recently , the target mass corrections to the nucleon structure functions for the polarized deep - inelastic scattering have been systematically studied [ 20 - 21 ] . in our previous work [ 22 ] , tmcs to the twist-3 matrix element in the nucleon structure functions are addressed . in this report , tmcs to the twist-4 terms @xmath0 as well as to the leading - twist @xmath1 will be discussed . consider the cornwall - norton ( cn ) moments @xmath9 , we know that the first cn moment of @xmath2 can be generally expanded in inverse powers of @xmath6 in operator production expansion ( ope ) [ 1 - 2 ] as @xmath10 with the coefficients @xmath11 relating to the nucleon matrix elements of operators of twist @xmath12 . in eq . ( 1 ) , the leading - twist ( twist-2 ) component @xmath13 is determined by the matrix elements of the axial vector operator @xmath14 , summed over various quark flavors . the coefficient of @xmath15 term , @xmath16 , contains the contributions from the twist-2 @xmath1 , twist-3 @xmath17 , and twist-4 @xmath18 , respectively . usually , @xmath17 is extracted from the third moments of the measured @xmath19 and @xmath20 by using @xmath21 . however , it is pointed out that this method for @xmath17 ignores the target mass corrections to the third moments of @xmath22 , and the target mass corrections play a sizeable role to @xmath17 [ 22 ] in a moderate @xmath6 region . to further estimate tmcs to the twist-4 of the nucleon spin structure functions , one may assume that the contributions from higher - twist term with @xmath23 can be ignored [ 23 ] or assume this term to be a constant ( neglecting any possible @xmath6-dependence ) [ 8 ] . based on the first assumption , we have @xmath24 when no tmcs are considered , @xmath1 and @xmath17 can be simply expressed by the cn moments of the nucleon spin structure functions , and we get @xmath25 when tmcs are considered , we have to employ the nachtmann moments @xmath26g_1(x , q^2)-y^2x^2\frac{4n}{n+2}g_2(x , q^2)\bigg \ } , \nonumber \\ m^{(n)}_2(q^2)&=&\int^1_0dx\frac{\xi^{n+1}}{x^2 } \bigg \{\frac{x}{\xi}g_1(x , q^2 ) + \big [ \frac{n}{n-1}\frac{x^2}{\xi^2}- \frac{n}{n+1}y^2x^2\big ] g_2(x , q^2)\bigg \},\end{aligned}\ ] ] where the nachtmann variable @xmath27 ( with @xmath28 ) , @xmath29 , and @xmath30 is the bjorken variable . the two nachtmann moments are simultaneously constructed by the two spin structure functions @xmath22 . if @xmath8 are replaced by the ones with tmcs ( see refs . [ 20 - 22 ] ) , one can easily expand the two nachtmann moments with respect to @xmath31 . the results are @xmath32 , and @xmath33 . the two expressions explicitly tell that , different from the cn moments , one can get the contributions of a pure twist-2 with spin - n and a pure twist-3 with spin-(n-1 ) operators from the nachtmann moments . the advantage of the nachtmann moments means that they contain only dynamical higher - twist , which are the ones related to the correlations among the partons . as a result , they are constructed to protect the moments of the nucleon spin structure functions from the target mass corrections . consequently , to extract the higher - twist effect , say twist-3 or twist-4 contribution , one is required to consider the nachtmann moments instead of the cn moments . we use the nachtmann moments to express @xmath34 and @xmath35 and obtain @xmath36\end{aligned}\ ] ] thus , tmcs to the twist-4 contribution , due to the two different moments , is @xmath37 . here , we employ the parametrization forms of the spin structure functions of the proton , neutron and deuteron [ 11 - 12 ] to estimate @xmath38 . note that the well - known wandzura and wilczek ( ww ) relation [ 24 ] @xmath39 is valid if only the leading - twist is considered , and tmcs to the twist-2 contribution do not break the ww relation . however , if the higher - twist operators , like twist-3 and twist-4 , are considered , the ww relation @xmath40 no longer preserves . thus , one may write @xmath41 [ 8,9 ] , where @xmath42 represents the violation of the ww relation . the non - vanishing value of @xmath42 just results from the higher - twist effect . one can calculate @xmath38 with the parametrizations of @xmath22 . the results are plotted in fig . 1 . we see that the typical values of the differences are in order of @xmath43 . there are several theoretical estimated values for the twist-4 term @xmath18 in the literature ( see table 1 ) , like the ones of the bag model [ 4 ] , of the qcd sum rule [ 5,6 ] , of the empirical analyses of the experimental measurements [ 8 , 23 ] , and of the instanton model [ 25 ] . comparing the estimated differences in fig . 1 to those estimated values displayed in table 1 , we conclude that tmcs to the twist-4 term @xmath18 are negligible ( less than 2% ) . we also find that @xmath38 of the proton and deuteron are always larger than that of the neutron . in addition , we check tmcs to the leading twist term ( with spin-3 ) @xmath1 . if no tmcs are considered , @xmath44 . when tmcs are taken into account , we get , from the nachtmann moments , @xmath45g_1(x , q^2 ) -\frac{12}{5}y^2x^2g_2(x , q^2)\bigg \}.\end{aligned}\ ] ] fig . 2 displays the @xmath6-dependence of the ratio @xmath46 for the proton , neutron and deuteron targets . the sizable effect of tmcs is clearly seen , since the ratios all diverge from unity obviously . when @xmath47 , the effect of tmcs is still about 10% for the proton and deuteron targets . in addition , the effect on the proton and deuteron targets is much larger than that on the neutron . here the @xmath6-dependences of the three ratios are similar to those of the twist-3 terms [ 22 ] . the sizeable effect tells that tmcs should be taken into account . therefore , to estimate the matrix element of @xmath1 , the nachtmann moments are required to be employed . the solid , dashed and dotted - dashed curves are the results of the proton , neutron and deuteron , respectively . , width=377,height=264 ] . the solid , dashed and dotted - dashed curves are the results of the proton , neutron and deuteron , respectively . , width=377,height=264 ] table 1 , the estimated values for @xmath18 in different approaches in the literature . + [ cols="^,^,^,^,^,^",options="header " , ] in summary , we have explicitly shown the target mass corrections to the twist-4 @xmath18 term and to the leading - twist one ( spin-3 ) @xmath1 . it is reiterated that in order to precisely and consistently extract the contributions of the leading - twist @xmath1 , of the twist-3 @xmath17 and of the twist-4 @xmath18 with a definite spin and with a moderate @xmath6 value , one is required to employ the nachtmann moments @xmath48 instead of the cn moments . our results show that tmcs play an evidently role to @xmath1 when @xmath6 is small . the above conclusion does not change if different parameterizations of the structure functions are employed . we also show that tmcs to the twist-4 term is much smaller than those to the twist-3 term and to the leading - twist term . finally , the expressions of the differences @xmath38 and @xmath49 between the cn and nachtmann moments are @xmath50 \nonumber \\ & & + \big [ 87g_2^{(5 ) } -258y^2g_2^{(7)}+798y^4g_2^{(9)}\big ] \bigg \}+{\cal o}(y^8),\nonumber \\ \delta a_2&=&\tilde a_2-\tilde a_2 ^ 0=2m^{(3)}_1 - 2g_1^{(3 ) } = y^2\bigg \ { \big [ -\frac{168}{25}g_1^{(5 ) } + \frac{108}{5}y^2g_1^{(7)}-\frac{352}{5}y^4g_1^{(9)}\big ] \nonumber \\ & & + \big [ -\frac{24}{5}g_2^{(5 ) } + \frac{96}{5}y^2g_2^{(7)}-\frac{336}{5}y^4g_2^{(9)}\big ] \bigg \ } + { \cal o}(y^8).\end{aligned}\ ] ] one sees that the two expressions mainly depend on the higher - moment of the nucleon spin structure functions , and therefore , on the spin structure function in the large - x region . in the most of the empirical analyses of the ellis - jaffe sum rule ( the first moment of @xmath2 ) , the contribution from the spin structure function in the large - x region is assumed to be trivial , since it behaves like @xmath51 . when the higher - moment of the spin structure function is considered , the effect of the spin structure functions in the large - x region becomes important . consequently , the measurement of the nucleon spin structure functions in the large - x region with a high precision is required .

phase transitions in ising spin systems driven entirely by quantum fluctuations have been getting a lot of attention recently [ 1 ] . the simplest of such systems is the ising model in a transverse field which can be exactly solved in one dimension . quantum fluctuations in ising systems with more complicated interactions which , for example , incorporate frustration and or disorder , give rise to novel and intriguing features . recently , the experimental realisation of some cases like the spin glass system in a transverse or tunnelling field , have added to the interest in such systems [ 1 ] . we apply the method of interfaces [ 2 ] in the ising model and the anisotropic next nearest neighbour ising ( annni ) model [ 3 ] in a transverse field at zero temperature to study the quantum fluctuation driven transitions . in the process , we also explore the scope of the so called twist method [ 2,4 ] which we have shown to have additional features apart from the ones already known . recently , it has been shown in a variety of spin systems how the interfaces caused by twisting a system is closely linked to the phase transition . apart from the application of the twist method to several classical models like ising spins systems , potts model and spin glasses [ 2 ] , very recently it has been used for quantum ground state problems also [ 4 ] . in this method , the interface free energy is generated by the excess free energy between systems with and without a twist . in general , twisting the system may be done by changing the boundary condition in one direction . the idea is that long range order produces stiffness . the interface free energy , which is the response to the stress generated by the twist provides direct information on the stiffness of the ordered state . for classical systems , i.e. , in a thermally driven phase transition , this method analyzes size ( @xmath0 ) and temperature ( @xmath1 the critical temperature ) dependence of the stiffness free energy ( which is the increment of free energy due to the change @xmath2 in boundary conditions ) defined by @xmath3 where @xmath4 and @xmath5 are the free energy with and without twist respectively . @xmath6 has the simple scaling form [ 5,2 ] @xmath7 where the stiffness exponent @xmath8 is a constant for @xmath9 , equal to zero for @xmath10 and negative for @xmath11 . hence the critical point can be obtained from @xmath12 . in ising spin systems with nearest neighbour interactions , @xmath13 where @xmath14 is the dimension of the system . for frustrated systems , @xmath15 may be nonintegral [ 2 ] . on the other hand , in phase transitions driven by quantum fluctuations at zero temperature , one needs to consider only the ground state energy ( which is equivalent to the free energy ) and here the interfacial free energy is expected to have a different stiffness exponent . we have applied the twist method in two quantum systems : first to reproduce the exact result of the ising chain in a transverse field [ 6 ] and then to the annni model in a transverse field [ 1 ] . in the latter , there are additional frustration effects which have to be taken under consideration . our results show that apart from the interfacial free energy , there are at least two other response functions which carry information of the phase transition and follow simple scaling laws . in section ii , we describe the method used to study the quantum ising models as well as the results . the results are discussed in section iii . the stiffness exponent for the quantum model at zero temperature is defined in the same way as in ( 2 ) , the role of temperature now being assumed by the transverse field such that @xmath16 @xmath17 the ising chain in a transverse field is described by the hamiltonian @xmath18 and the ferromagnetic to paramagnetic phase transition occurs at @xmath19 for @xmath20 . we take the basis states to be diagonal in the representation of @xmath21 . the twist is applied in the following way [ 2 ] : in one case we have fixed spins pointing parallely in the left and right boundaries which favours the ferromagnetic alignment and is called the favourable boundary condition ( fbc ) , while in the other case we have fixed spins at the boundaries antiparallely oriented ( unfavourable boundary condition or ubc ) . the latter generates an interface and hence the excess energy . the first spin also interacts with the extra spin ( fixed ) on its left and the last ( @xmath0th ) spin interacts with the extra ( @xmath22th ) spin ( fixed ) on its right . it needs to be clarified here that we have used open boundary conditions with two extra spins pointed either parallely or antiparallely at the edges . this , while generating the interface , will also introduce boundary effects ( finite size effects in a numerical study ) : the two effects are intermingled and difficult to separate . it might be possible to study the interface effect alone by using periodic and antiperiodic boundary conditions [ 7 ] , but that involves more complicated programming and computer time . therefore , we have both interface and boundary effects , and when we talk of interface effect in the rest of the paper , it essentially includes boundary effect , the latter diminishing with system size . we proceed to find out the ground state of a system of @xmath0 spins ( excluding the two at the boundary ) in a transverse field by using a lanczos algorithm for both kinds of boundary conditions ( fbc and ubc ) . apart from the interfacial energy defined in eq . ( 3 ) , we also investigate the behaviour of the interfacial cooperative energy and the interfacial magneitsation . these two quantities are defined in the following way : let @xmath23 = @xmath24 where @xmath25 is the term(s ) in the hamiltonian involving only the cooperative interaction energy and @xmath26 the ground state . for ( 4 ) , then the interfacial cooperative energy is given by @xmath28 . the interfacial magnetisation is similarly defined @xmath29 where @xmath30 is the magnetisation in the ground state with ( without ) twist . we have obtained results for system sizes @xmath0 = 6 to @xmath0 = 20 and studied the bahaviour of @xmath31 , @xmath32 and @xmath33 . all three scale in general as ( 3 ) giving the exact result @xmath34 and @xmath35 ( see fig . although the exact critical point is known for ( 4 ) , certain other features are available from our study which shows novel features of the stiffness exponent for quantum systems . we have discussed these scaling behaviour and commented about them in section iii . we next extend the study to the annni chain in a transverse field . the hamiltonian is described by @xmath36\eqno{(7)}\ ] ] here @xmath37 denotes the frustration parameter . the classical ground state without @xmath38 at zero temperature is exactly known : ferromagnetic for @xmath39 , antiphase for @xmath40 and highly degenerate phases exist at @xmath41 [ 3 ] . the quantum annni model , which is perhaps the simplest model incorporating both frustration and quantum fluctuation , has been studied extensively ( and the corresponding classical model ) in the last few years [ 1 ] . however , the nature of the ground state and the phase transition is yet to be understood clearly especially in the region @xmath40 . it is believed that a floating phase exists [ 1,8 ] close to the @xmath41 region which has also been found for the classical two dimensional model in the free fermion approximation [ 3 ] . all earlier studies indicate that there is a ferromagnetic to paramagnetic transition at @xmath42 . hence , the twist method is easily applicable here in the same manner as in the nearest neighbour ising case . in order to impose favourable and unfavourable boundary conditions , we fix two spins on the left and right end of the chain , and find the ground states . the spins on the boundaries interact with the extra fixed spins as in the ising case , with open boundary conditions prevailing . for @xmath42 , the fbc consists of parallel spins , and for ubc , it is antiparallel just like the nearest neighbour case . it maybe mentioned that one could do without bringing in two fixed spins but we keep this in order it is consistent with the ground states also at @xmath40 . we have applied here the twist method and found that it gives consistent results in the @xmath42 region where a ferromagnetic to paramagnetic transition occurs . again we find that @xmath31 , @xmath32 and @xmath33 have simple scaling forms and we get the critical field for any @xmath42 in this way . as an example , we have shown the scaling of the three quantities in fig . 2 for @xmath43 . in the @xmath40 region , we have no clear idea about what kind of a transition is taking place which is clear - cut ferromagnetic to paramagnetic in the @xmath42 region . therefore , all we have attempted to do here is to find out the phase boundary where the antiphase disappears by putting appropriate ubc and fbc for the antiphase . however , there still remains a problem . the frustration effects now become dominant and the ground state is no longer trivially degenerate . this generates not a single interface but maybe more than one . also , because of the structure of the degenarate ground states due to the presence of both nearest and next nearest neighbour interactions , the so called unfavourable boundary condition for one particular ground state may become favourable for another degenerate ground state , thus making it difficult to feel the effect of the field due to the twist . for example , if we set the two spins on the left boundary down and the two on the right up , then the state with minimum interaction energy is @xmath44 , a member of the set of the 4 degenerate ground states in the antiphase . setting all the boundary spins on the left and right down to provide the necessary twist , the new ground state should apparently have a structure @xmath45 , where we do not know how the spins in the interior are oriented . the cooperative energy contribution at the boundary to this state is @xmath46 . however , if we look at another antiphase state which is @xmath47 , then the energy contribution at the boundary is @xmath48 . hence it is possible that the latter is lower in energy compared to @xmath49 especially if @xmath50 or when @xmath51 hence , a second antiphase state becomes the ground state when the twist is applied therefore making the present method ineffective . however , with the quantum term also present , we observed from the numerical exercise that this problem disappears for @xmath52 where we find out the phase boundary . the interfacial magnetisation is of course not meaningful here . we have estimated the phase boundary where the @xmath53 phase disappears again from the best scaling plots for @xmath31 and @xmath54 ( the @xmath37 = 1.0 case is shown in fig . however , the data collapse is not so impressive as in the @xmath42 region . the resulting partial phase diagram is shown in fig . we have studied the behaviour of essentially three quantities and found that they carry information about the quantum phase transitions in the ising and annni models in the interface approach . of these , the behaviour of the total interface energy had been known earlier , but the scaling of the interfacial cooperative energy and interfacial magnetisation appear to be new results . however , there were earlier evidence that the cooperative energy contribution is significant in a study of quantum spin glasses [ 1,9 ] . in [ 4 ] , it was argued that one should look at the scaling behaviour of the quantity @xmath55 which is expected to have a stiffness exponent = 1 for the transverse ising chain ( the same as that of the 2-@xmath14 classical model ) . however , this is the same as saying @xmath31 scales as @xmath56 , and we do not find this behaviour ( except , of course , at @xmath57 , but we are interested in the scaling behaviour near the critical point ) . on the other hand , we do find that @xmath32 does have a stiffness exponent 0 , ( i.e. scales as @xmath56 ) while @xmath58 shows a scaling bahaviour with a stiffness exponent @xmath59 ( see figs . 1(a - c ) drawn with @xmath60 ) . now , in case of the classical systems , we have stiffness exponent = @xmath61 . of course for @xmath62 , there is no thermal phase transition and therefore the exponent @xmath63 is never encountered . but , here we do have a phase transition driven by quantum fluctuations and that may be the reason for obtaining an exponent @xmath15 = 0 for the interfacial cooperative energy . the interfacial magnetisation also scales with an exponent @xmath63 . the scaling function @xmath64 for the interfacial cooperative energy is also evidently of the following form @xmath65 @xmath66 where @xmath67 is a constant depending on @xmath68 . it maybe noted that the magnetisation depends not only on the number of interfaces but also their positions and it is apparent from the data that as the system size is increased , the interface caused by the twist moves towards the center of the chain . therefore , the exponent @xmath69 for the interfacial magnetisation is not surprising . one can say that the nontrivial exponent of @xmath70 obtained for the total interfacial energy is a novel feature of the quantum model . on the other hand , if one looks at the scaling functions in figs . 1 - 3 , it is obvious that they are different for @xmath55 and @xmath32 . the scaling functions for @xmath32 and @xmath33 are , however , similar . apparently the scaling function @xmath71 for @xmath72 has the following form the scaling behaviour of @xmath32 and @xmath31 are different but the quantities @xmath32 and @xmath76 have the same stiffness exponent . hence , there is an additional dimension @xmath0 in the total energy which may be related to the additional dimension which comes into play in quantum models . that the interface method is quite powerful is again proved . we obtain the exact critical point for the transverse ising chain and a phase diagram for the transverse annni model consistent with the previous studies . however , we did not venture to investigate the regime @xmath40 in the annni model fully because of the nontrivial nature of the transition to a possible floating phase . the phase boundary where the antiphase disappears is also not obtained for @xmath77 because of the difficulty in imposing conflicting boundary conditions . since in degenerate systems , there can be a number of ways to impose the fbc and the ubc we tried several combinations but faced the same difficulty . this is because of the very structure of the degenerate ground states as elaborated in section ii . it is true that the more interesting phase transitions for @xmath40 could not be obtained here , but we showed that estimating the boundary above which the antiphase vanishes is a nontrivial task itself . in fact , most of the analytical and numerical methods give an incomplete picture for @xmath40 . * acknowledgments * this work is supported by sfb341 . the author is grateful to subinay dasgupta for bringing ref . [ 4 ] to notice , and also very much to heiko rieger for useful discussions during the development of the program and for sending preprint of ref . [ 8 ] prior to publication . she is thankful to dietrich stauffer for discussions . see e.g. , b. k. chakrabarti , a. dutta and p. sen _ quantum ising phases and transitions in transverse ising models _ , lecture notes in physics , * m41 * , springer - verlag , heidelberg , 1996 , and the references therein . k. okamoto and y. ueno , j. phys . . jpn . * 64 * , 86 ( 1995 ) . m. e. fisher , _ critical phenomena proc . 51st enrico fermi summer school _ , ed . m. s. green ( academic , new york , 1971 ) ; m. n. barber , _ phase transitions and critical phenomena _ , * 8 * , ed . c. domb and j. l. lebowitz ( academic , new york , 1984 ) ; d. jasnow , rep . . phys . * 47 * , 1059 ( 1984 ) .

we investigate phase transitions in the ising model and the annni model in transverse field using the interface approach . the exact result of the ising chain in a transverse field is reproduced . we find that apart from the interfacial energy , there are two other response functions which show simple scaling behaviour . for the annni model in a transverse field , the phase diagram can be fully studied in the region where a ferromagnetic to paramagnetic phase transition occurs . the other region can be studied partially ; the boundary where the antiphase vanishes can be estimated . pacs nos . : 64.60.cn,75.10.jm,75.30.kz

complex system modeling and simulation often mandate global sensitivity analysis , which constitutes the study of how the global variation of input , due to its uncertainty , influences the overall uncertain behavior of a response of interest . most common approaches to sensitivity analysis are firmly anchored in the second - moment properties the output variance which is divvied up , qualitatively or quantitatively , to distinct sources of input variation @xcite . there exist a multitude of methods or techniques for calculating the resultant sensitivity indices of a function of independent variables : the random balance design method @xcite , the state - dependent parameter metamodel @xcite , sobol s method @xcite , and the polynomial dimensional decomposition ( pdd ) method @xcite , to name but four . a few methods , such as those presented by kucherenko , tarantola , and annoni @xcite and rahman @xcite , are also capable of sensitivity analysis entailing correlated or dependent input . implicit in the variance - driven global sensitivity analysis is the assumption that the statistical moments satisfactorily describe the stochastic response . in many applications , however , the variance provides a restricted summary of output uncertainty . therefore , sensitivity indicators stemming solely from the variance should be carefully interpreted . a more rational sensitivity analysis should account for the entire probability distribution of an output variable , meaning that alternative and more appropriate sensitivity indices , based on probabilistic characteristics above and beyond the variance , should be considered . addressing some of these concerns has led to a sensitivity index by exploiting the @xmath1 distance between two output probability density functions @xcite . such sensitivity analysis establishes a step in the right direction and is founded on the well - known total variational distance between two probability measures . there remain two outstanding research issues for further improvements of density - based sensitivity analysis . first , there is no universal agreement in selecting the total variational distance as the undisputed measure of dissimilarity or affinity between two output probability density functions . in fact , a cornucopia of divergence or distance measures exist in the literature of information theory . therefore , a more general framework , in the spirit of density - based measures , should provide diverse choices to sensitivity analysis @xcite . second , the density - based sensitivity indices in general are more difficult to calculate than the variance - based sensitivity indices . this is primarily because the probability density function is harder to estimate than the variance . moreover , nearly all estimation methods available today are very expensive due to the existence of the inner and outer integration loops . therefore , efficient computational methods for computing density - based sensitivity indices are desirable . the purpose of this paper is twofold . first , a brief exposition of the @xmath0-divergence measure is given in section 2 , setting the stage for a general multivariate sensitivity index , referred to as the @xmath0-sensitivity index , presented in section 3 . the section includes new theoretical results representing fundamental properties and important inequalities pertaining to the @xmath0-sensitivity index . second , section 4 introduces three distinct approximate methods for estimating the @xmath0-sensitivity index . the methods depend on how the probability densities of a stochastic response are estimated , including an efficient surrogate approximation commonly used for high - dimensional uncertainty quantification . numerical results from three mathematical functions , as well as from a computationally intensive stochastic mechanics problem , are reported in section 5 . finally , conclusions are drawn in section 6 . let @xmath2 , @xmath3 , @xmath4 , and @xmath5 represent the sets of positive integer ( natural ) , non - negative integer , real , and non - negative real numbers , respectively . for @xmath6 , denote by @xmath7 the @xmath8-dimensional euclidean space and by @xmath9 the @xmath8-dimensional multi - index space . these standard notations will be used throughout the paper . let @xmath10 be a measurable space , where @xmath11 is a sample space and @xmath12 is a @xmath13-algebra of the subsets of @xmath11 , satisfying @xmath14 and @xmath15 , and @xmath16 be a @xmath13-finite measure on @xmath10 . let @xmath17 be a set of all probability measures on @xmath18 , which are absolutely continuous with respect to @xmath16 . for two such probability measures @xmath19 , let @xmath20 and @xmath21 denote the radon - nikodym derivatives of @xmath22 and @xmath23 with respect to the dominating measure @xmath16 , that is , @xmath24 and @xmath25 . let @xmath26 $ ] be an extended real - valued function , which is 1 . continuous on @xmath27 and finite - valued on @xmath28 ; 2 . convex on @xmath27 , that is , @xmath29 for any @xmath30 and @xmath31 $ ] ; , @xmath32 , are excluded . ] strictly convex at @xmath33 , that is , @xmath34 for any @xmath35 and @xmath36 such that @xmath37 ; ; and @xmath38 evaluated on two sides of the point @xmath33 on the graph of @xmath0 lies above the function value @xmath39 . ] and 4 . equal to _ zero _ at @xmath33 , that is , @xmath40 . the @xmath0-divergence , describing the difference or discrimination between two probability measures @xmath22 and @xmath23 , is defined by the integral @xmath41 provided that the undefined expressions are interpreted by @xcite @xmath42 @xmath43 to define the @xmath0-divergence for absolutely continuous probability measures in terms of elementary probability theory , take @xmath11 to be the real line and @xmath16 to be the lebesgue measure , that is , @xmath44 , @xmath45 , so that @xmath20 and @xmath21 are simply probability density functions , denoted by @xmath46 and @xmath47 , respectively . then the @xmath0-divergence can also be defined by @xmath48 the divergence measures in ( [ 1 ] ) and ( [ 2 ] ) were introduced in the 1960s by csiszr @xcite , ali and silvey @xcite , and morimoto @xcite . similar definitions exist for discrete probability measures . vajda @xcite , liese and vajda @xcite , and sterreicher @xcite discussed general properties of the @xmath0-divergence measure , including a few axiomatic ones . the basic but important properties are as follows @xcite : + 1 . _ non - negativity and reflexivity : _ @xmath49 with equality if and only if @xmath50 . + 2 . _ duality : _ @xmath51 , where @xmath52 , @xmath53 , is the * -conjugate ( convex ) function of @xmath0 . when @xmath54 , @xmath0 is * -self conjugate . _ invariance : _ @xmath55 , where @xmath56 , @xmath57 . symmetry : _ @xmath58 if and only if @xmath59 , where @xmath52 , @xmath53 , and @xmath57 . when @xmath60 , the symmetry and duality properties coincide . range of values : _ @xmath61 , where @xmath62 and @xmath63 . the left equality holds if and only if @xmath50 . the right equality holds if and only if @xmath64 , that is , for mutually singular ( orthogonal ) measures , and is attained when @xmath65 . + the normalization condition @xmath40 is commonly adopted to ensure that the smallest possible value of @xmath66 is _ zero_. but fulfilling such condition by the class @xmath67 of convex functions @xmath0 is not required . this is because , for the subclass @xmath68 such that @xmath69 satisfies @xmath40 , the shift by the constant @xmath70 sends every @xmath71 to @xmath72 . indeed , some of these properties may still hold if @xmath73 or if @xmath0 is not restricted to the convexity properties . depending on how @xmath0 is defined , the @xmath0-divergence may or may not be a true metric . for instance , it is not necessarily symmetric in @xmath22 and @xmath23 for an arbitrary convex function @xmath0 ; that is , the @xmath0-divergence from @xmath22 to @xmath23 is generally not the same as that from @xmath23 to @xmath22 , although it can be easily symmetrized when required . furthermore , the @xmath0-divergence does not necessarily satisfy the triangle inequality . it is well known that @xmath74 has a versatile functional form , resulting in a number of popular information divergence measures . indeed , many of the well - known divergences or distances commonly used in information theory and statistics are easily reproduced by appropriately selecting the generating function @xmath0 . familiar examples of the @xmath0-divergence include the forward and reversed kullback - leibler divergences @xmath75 and @xmath76 @xcite , kolmogorov total variational distance @xmath77 @xcite , hellinger distance @xmath78 @xcite , pearson @xmath79 divergence @xmath80 @xcite , neyman @xmath79 divergence @xmath81 @xcite , @xmath82 divergence @xmath83 @xcite , vajda @xmath84 divergence @xmath85 @xcite , jeffreys distance @xmath86 @xcite , and triangular discrimination @xmath87 @xcite , to name a few , and are defined as @xmath88 d\xi , \label{3a}\ ] ] @xmath89 d\xi = : d_{kl}\left ( p_2 \parallel p_1\right ) , \label{3b}\ ] ] @xmath90 @xmath91 ^ 2 d\xi , \label{3d}\ ] ] @xmath92 d\xi , \label{3e}\ ] ] @xmath93 d\xi : = d_{p}\left ( p_2 \parallel p_1\right ) , \label{3f}\ ] ] @xmath94 , ~\alpha \in \mathbb{r } \setminus \{\pm 1\ } , \label{3g}\ ] ] @xmath95 @xmath96 \ln \left [ \dfrac{f_1(\xi)}{f_2(\xi ) } \right ] d\xi , \label{3i}\ ] ] @xmath97 ^ 2}{f_1(\xi)+f_2(\xi ) } d\xi . \label{3j}\ ] ] the definitions of some of these divergences , notably the two kullback - leibler and pearson - neyman @xmath79 divergences , are inverted when the @xmath0-divergence is defined by swapping @xmath22 and @xmath23 in ( [ 1 ] ) or ( [ 2 ] ) . there are also many other information divergence measures that are not subsumed by the @xmath0-divergence measure . see the paper by kapur @xcite or the book by taneja @xcite . nonetheless , any of the divergence measures from the class of @xmath0-divergences or others can be exploited for sensitivity analysis , as described in the following section . let @xmath98 be a complete probability space , where @xmath99 is a sample space , @xmath100 is a @xmath13-field on @xmath99 , and @xmath101 $ ] is a probability measure . with @xmath102 representing the borel @xmath13-field on @xmath103 , @xmath104 , consider an @xmath103-valued absolutely continuous random vector @xmath105 , describing the statistical uncertainties in all system and input parameters of a general stochastic problem . the probability law of @xmath106 , which may comprise independent or dependent random variables , is completely defined by its joint probability density function @xmath107 . let @xmath108 be a non - empty subset of @xmath109 with the complementary set @xmath110 and cardinality @xmath111 , and let @xmath112 , @xmath113 , be a subvector of @xmath106 with @xmath114 defining its complementary subvector . then , for a given @xmath115 , the marginal density function of @xmath116 is @xmath117 . let @xmath118 ) , a real - valued , continuous , measurable transformation on @xmath119 , define a general stochastic response of interest . define @xmath120 to be the associated output random variable . for global sensitivity analysis , suppose that the sensitivity of @xmath121 with respect to a subset @xmath122 , @xmath115 , of input variables @xmath106 is desired . as shown by a few researchers for univariate cases only @xcite , such a multivariate sensitivity measure can be linked to the divergence between the unconditional and conditional probability measures of @xmath121 . denote by @xmath123 and @xmath124 the probability measures and by @xmath125 and @xmath126 the probability density functions of random variables @xmath121 and @xmath127 , respectively , where @xmath127 stands for @xmath121 conditional on @xmath122 , which is itself random . setting @xmath128 , @xmath129 , @xmath130 , and @xmath131 in ( [ 2 ] ) , the @xmath0-divergence becomes @xmath132 as explained in the preceding section , @xmath133 characterizes the discrimination between @xmath123 and @xmath124 , but it is random because @xmath122 is random . a general multivariate @xmath0-sensitivity index of an output random variable @xmath121 for a subset @xmath116 , @xmath115 , of input random variables @xmath134 , denoted by @xmath135 , is defined as the expected value of the @xmath0-divergence from @xmath123 to @xmath124 , that is , @xmath136 , \label{4b}\ ] ] where @xmath137 is the expectation operator with respect to the probability measure of @xmath116 . from the definition of the expectation operator , the @xmath0-sensitivity index @xmath138 where @xmath139 and @xmath140 are the probability measure and probability density function , respectively , of @xmath121 conditional on @xmath141 , and @xmath142 is the joint probability density function of @xmath143 . the last equality in ( [ 5 ] ) is formed by the recognition that @xmath144 and is useful for calculating the sensitivity index , to be discussed in section 4 . for variance - based sensitivity analysis entailing independent random variables , there exists a well - known importance measure , namely , the sobol index . one way to explain the sobol index is the analysis - of - variance ( anova ) decomposition of a square - integrable function @xmath145 , expressed by the compact form @xcite @xmath146 [ 5a ] which is a finite , hierarchical expansion in terms of its input variables with increasing dimensions . here , @xmath147 is a @xmath148-variate component function describing a constant or the interactive effect of @xmath116 on @xmath145 when @xmath149 or @xmath150 . the summation in ( [ 5a1 ] ) comprises @xmath151 component functions , with each function depending on a group of variables indexed by a particular subset of @xmath109 , including the empty set @xmath152 . applying the expectation operator @xmath153 on @xmath154 and its square from ( [ 5a1 ] ) and recognizing the _ zero_-mean and orthogonal properties of @xmath155 , @xmath156 , in ( [ 5a3 ] ) @xcite , the variance @xmath157\right]^2= \sum_{\emptyset\ne u\subseteq\{1,\cdots , n\ } } { \sigma_u}^2 \label{5b}\ ] ] of @xmath121 splits into partial variances @xmath158= \mathbb{e}_{\mathbf{x}_{u}}\left[y_{u}^{2}(\mathbf{x}_{u})\right ] , ~\emptyset\ne u\subseteq\{1,\cdots , n\ } , \label{5c}\ ] ] of all non - constant anova component functions . henceforth , the sobol sensitivity index of @xmath121 for a subset of variables @xmath122 is defined as @xcite @xmath159 provided that @xmath160 . the sobol index is bounded between 0 and 1 and represents the fraction of the variance of @xmath121 contributed by the @xmath148-variate interaction of input variables @xmath122 . there exist @xmath161 such indices , adding up to @xmath162 . does the @xmath0-sensitivity index provide a more useful insight than the existing variance - based sobol index into the importance of input variables ? to answer this question , consider a purely additive function @xmath163 , where @xmath164 is an arbitrary real - valued constant and the input random variables @xmath165 , @xmath166 , have _ zero _ means and identical variances @xmath167=s^2 $ ] , @xmath168 , but otherwise follow independent and arbitrary probability distributions . then , from ( [ 5a ] ) through ( [ 5d ] ) , ( 1 ) the anova component functions @xmath169 , @xmath170 , @xmath166 , and @xmath171 for @xmath172 ; ( 2 ) the variances @xmath173 , @xmath174 , @xmath166 , and @xmath175 for @xmath172 ; and ( 3 ) the sobol indices @xmath176 , @xmath166 , and @xmath177 for @xmath178 . as all univariate sobol indices are the same , so are the contributions of input variables to the variance of @xmath121 . hence , according to the sobol index , all input variables are equally important , regardless of their probability distributions . this is unrealistic , but possible because the variance is just a moment and provides only a partial description of the uncertainty of an output variable . in contrast , the @xmath0-sensitivity indices will vary depending on the choice of the input density functions , therefore , providing a more rational measure of the influence of input variables . it is important to derive and emphasize the fundamental properties of the @xmath0-sensitivity index @xmath135 inherited from the @xmath0-divergence measure . the properties , including a few important inequalities , are described in conjunction with six propositions as follows . the @xmath0-sensitivity index @xmath135 of @xmath121 for @xmath122 , @xmath115 , is non - negative and vanishes when @xmath121 and @xmath122 are statistically independent . [ p1 ] since @xmath179 by virtue of the non - negativity property of the @xmath0-divergence and @xmath180 for any @xmath181 , the first line of ( [ 5 ] ) yields @xmath182 proving the first part of the proposition . if @xmath121 and @xmath122 are statistically independent , then @xmath183 for any @xmath184 , resulting in @xmath185 , owing to the reflexivity property or the range of values ( left equality ) of the @xmath0-divergence . in that case , @xmath186 proving the second part of the proposition . the range of values of @xmath135 is @xmath187 where @xmath62 and @xmath63 . [ p2 ] see the proof of proposition [ p1 ] for the left inequality . the right inequality is derived from the largest value of @xmath188 , which is @xmath189 , according to the range of values ( right equality ) of the @xmath0-divergence . therefore , ( [ 4b ] ) yields @xmath190 \le \mathbb{e}_{\mathbf{x}_u } \left [ f(0)+f^*(0 ) \right ] = f(0)+f^*(0),\ ] ] completing the proof . from proposition [ p2 ] , @xmath135 has a sharp lower bound , which is _ zero _ since @xmath40 . in contrast , @xmath135 may or may not have an upper bound , depending on whether @xmath189 is finite or infinite . if there is an upper bound , then the largest value @xmath189 is a sharp upper bound , and hence can be used to scale @xmath135 to vary between 0 and 1 . for instance , when @xmath191 , the result is the well - known variational distance measure @xmath192 and the upper bound of the associated sensitivity index @xmath193 ( say ) is @xmath194 . when @xmath195 or @xmath196 , then @xmath197 , meaning that the sensitivity index @xmath198 ( say ) or @xmath199 ( say ) , derived from the kullback - leibler divergence measure @xmath200 or @xmath201 , has no upper bound . no scaling is possible in such a case . the @xmath0-sensitivity index @xmath202 of @xmath121 for all input variables @xmath203 is @xmath204 where @xmath62 and @xmath63 . [ p3 ] the probability measure @xmath205 is a dirac measure , representing an almost sure outcome @xmath206 , where @xmath207 and @xmath208 . decompose @xmath4 into two disjoint subsets @xmath209 and @xmath210 and observe that @xmath211 therefore , the probability measures @xmath123 and @xmath205 are mutually singular ( orthogonal ) , that is , @xmath212 . consequently , @xmath213 , according to the range of values ( right equality ) of the @xmath0-divergence . finally , for @xmath214 , ( [ 4b ] ) yields @xmath215 = \mathbb{e}_{\mathbf{x } } \left [ f(0)+f^*(0 ) \right ] = f(0)+f^*(0).\ ] ] for the special case of @xmath191 , the index derived from the total variational distance @xmath216 . therefore , when normalized , @xmath217 , which is the same value reported by borgonovo @xcite . let @xmath218 . if @xmath121 and @xmath219 are statistically independent , then @xmath220 [ p4 ] in addition , if @xmath108 and @xmath221 are disjoint subsets , that is , @xmath222 , then @xmath223 [ p4b ] for any @xmath218 , observe that @xmath224 and @xmath225 . since @xmath121 is independent of @xmath219 , the probability measures @xmath226 and @xmath227 are the same , yielding @xmath228 . applying this condition to the expression of @xmath229 the @xmath0-sensitivity index of @xmath121 for @xmath230 in the first line of ( [ 5 ] ) and noting @xmath231 results in @xmath232 proving the first part of the proposition . here , the second equality is obtained by recognizing that @xmath233 does not depend on @xmath234 and @xmath235 . the third equality is attained by integrating out @xmath236 with respect to @xmath234 on @xmath237 , resulting in @xmath238 . the second part of the proposition results from the reduction , @xmath239 , when @xmath222 . as a special case , consider @xmath240 and @xmath241 , where @xmath242 , @xmath243 . then , according to proposition [ p4 ] , @xmath244 , meaning that there is no contribution of @xmath245 to the sensitivity of @xmath121 for @xmath246 if @xmath145 does not depend on @xmath245 . the @xmath0-sensitivity index @xmath135 of @xmath121 for @xmath122 , @xmath115 , is invariant under smooth and uniquely invertible transformations ( diffeomorphisms ) of @xmath121 and @xmath122 . [ p5 ] for @xmath115 , let @xmath247 and @xmath248 be smooth and uniquely invertible , that is , diffeomorphic maps of random variables @xmath249 and @xmath121 . from elementary probability theory , the probability densities of the transformed variables @xmath250 , @xmath251 , and @xmath252 are @xmath253 respectively , where @xmath254 \in \mathbb{r}^{|u|\times |u|}$ ] , @xmath255 , is the jacobian matrix of the transformation such that @xmath256 for any @xmath181 and @xmath257 . applying these relationships to the sensitivity index @xmath258 of @xmath251 for @xmath259 defined in the last line of ( [ 5 ] ) and noting @xmath260 and @xmath261 yields @xmath262 completing the proof . for a special case of @xmath240 , @xmath166 , corollary 4 of borgonovo et al . @xcite describes the monotonic invariance of a univariate sensitivity index derived from @xmath263 norm or @xmath0-divergence . in contrast , proposition [ p5 ] and its proof presented here are more general and different than those reported in the existing work @xcite . the invariance property of the @xmath0-sensitivity index described by proposition [ p5 ] does not hold in general for the variance - based sobol index @xcite . the latter index is invariant only under affine transformations . moreover , the @xmath0-sensitivity index , unlike the sobol index , is applicable to random input following dependent probability distributions . let @xmath218 be two disjoint subsets such that @xmath222 . for probability measures @xmath123 , @xmath124 , and @xmath264 , let @xmath0 be a select convex generating function , which produces metric @xmath0-divergences from @xmath123 to @xmath264 , from @xmath123 to @xmath265 , and from @xmath124 to @xmath264 , satisfying the triangle inequality @xmath266 then @xmath267 where @xmath268 $ ] is the conditional sensitivity index of @xmath269 for @xmath230 . furthermore , if @xmath122 and @xmath219 are statistically independent , then @xmath270 [ p6 ] applying the expectation operator @xmath271 on both sides of the triangle inequality yields @xmath272 since , for @xmath222 , @xmath273 does not depend on @xmath274 , the first integral on the right side of ( [ sr3 ] ) reduces to @xmath275 therefore , ( [ sr3 ] ) becomes @xmath276 recognizing the sensitivity indices @xmath229 , @xmath135 , and @xmath277 to be respectively the integral on the left side , the first integral on the right side , and the second integral on the right side of ( [ sr4 ] ) produces the upper bound in ( [ sr2 ] ) . in addition , observe that the sensitivity index @xmath277 is non - negative , represents the contribution of the divergence from @xmath124 to @xmath264 , and vanishes if and only if @xmath121 and @xmath219 are statistically independent . therefore , @xmath229 reaches the lower bound , which is @xmath135 , if and only if @xmath121 and @xmath219 are statistically independent . to obtain ( [ sr2b ] ) , use the last line of ( [ 5 ] ) to write @xmath278 where , by invoking the statistical independence between @xmath122 and @xmath219 , the numerator and denominator of the argument of @xmath0 become @xmath279 and @xmath280 respectively . applying ( [ sr4c ] ) and ( [ sr4d ] ) to ( [ sr4b ] ) results in @xmath281 which transforms ( [ sr2 ] ) to ( [ sr2b ] ) and hence completes the proof . as a special case , consider again @xmath240 and @xmath241 , where @xmath242 , @xmath243 . then , according to proposition [ p6 ] , applicable to sensitivity indices rooted in metric @xmath0-divergences only , @xmath282 which states the following : if @xmath121 depends on @xmath245 , then the contribution of @xmath245 to the sensitivity of @xmath121 for @xmath246 increases from @xmath283 , but is limited by the residual term @xmath284 . if @xmath121 and @xmath245 are statistically independent , then @xmath284 vanishes , resulting in @xmath285 . this agrees with proposition [ p4 ] , which , however , is valid whether or not the underlying @xmath0-divergence is a metric . in addition , if @xmath165 and @xmath245 are statistically independent , then @xmath286 , yielding @xmath287 . borgonovo @xcite derived the same bounds for a special case when the sensitivity index stems from the total variational distance . proposition [ p6 ] , by contrast , is a general result and applicable to sensitivity indices emanating from all metric @xmath0-divergences . a plethora of @xmath0-sensitivity indices are possible by appropriately selecting the convex function @xmath0 in ( [ 4b ] ) or ( [ 5 ] ) . listed in table [ table1 ] are ten such sensitivity indices derived from the forward and reversed kullback - leibler divergences , total variational distance , hellinger distance , pearson @xmath79 divergence , neyman @xmath79 divergence , @xmath82 divergence , vajda @xmath84 divergence , jeffreys distance , and triangular discrimination in ( [ 3a ] ) through ( [ 3j ] ) . three prominent sensitivity indices , for example , the mutual information @xcite @xmath288 f_{\mathbf{x}_u , y}(\mathbf{x}_u,\xi ) d{\mathbf{x}_u}d\xi = : h_{u , kl'}\ ] ] between @xmath122 and @xmath121 , the squared - loss mutual information @xcite @xmath289 ^ 2 f_{y}(\xi ) f_{\mathbf{x}_u}(\mathbf{x}_u ) d{\mathbf{x}_u}d\xi \\ & = & \int_{\mathbb{r}^{|u|}\times\mathbb{r } } \dfrac{f_{\mathbf{x}_u , y}(\mathbf{x}_u,\xi)}{f_{y}(\xi ) f_{\mathbf{x}_u}(\mathbf{x}_u ) } \left[1 - \left\ { \dfrac{f_{y}(\xi ) f_{\mathbf{x}_u}(\mathbf{x}_u)}{f_{\mathbf{x}_u , y}(\mathbf{x}_u,\xi ) } \right\}^2 \right ] f_{\mathbf{x}_u , y}(\mathbf{x}_u,\xi ) d{\mathbf{x}_u}d\xi \\ & = : & h_{u , n } \end{array}\ ] ] between @xmath122 and @xmath121 , and borgonovo s importance measure @xcite @xmath290 of @xmath122 on @xmath121 , are rooted in reversed kullback - leibler , neyman , and total variational divergences or distances , respectively . indeed , many previously used sensitivity or importance measures are special cases of the @xmath0-sensitivity index derived from the @xmath0-divergence . .ten special cases of the @xmath0-sensitivity index [ cols="<,<,<",options="header " , ] [ table8 ] table [ table8 ] presents the approximate univariate sensitivity indices @xmath291 ( total variational distance ) and @xmath292 ( reversed kullback - leibler divergence ) of the maximum von mises stress by the pdd - kde - mc method . the pdd expansion coefficients were estimated by @xmath293-variate dimension - reduction integration @xcite , requiring one- ( @xmath294 ) or at most two - dimensional ( @xmath295 ) gauss quadratures . the order @xmath296 of orthogonal polynomials and number @xmath297 of gauss quadrature points in the dimension - reduction numerical integration are @xmath298 and @xmath299 , respectively . the indices are broken down according to the choice of selecting @xmath300 and @xmath301 . in all pdd approximations , the sample size @xmath302 . the sensitivity indices by the pdd - kde - mc methods in table [ table8 ] quickly converge with respect to @xmath293 and/or @xmath296 . since fea is employed for response evaluations , the computational effort of the pdd - kde - mc method comes primarily from numerically determining the pdd expansion coefficients . the expenses involved in estimating the pdd coefficients vary from 25 to 33 fea for the univariate pdd approximation and from 277 to 481 fea for the bivariate pdd approximation , depending on the two values of @xmath296 . based on the sensitivity indices in table [ table8 ] , the horizontal boundary conditions ( @xmath303 and @xmath304 ) are highly important ; the vertical load ( @xmath305 ) , elastic modulus ( @xmath306 ) , and vertical boundary conditions ( @xmath307 and @xmath308 ) are slightly important ; and the horizontal load ( @xmath309 ) and poisson s ratio ( @xmath310 ) are unimportant in influencing the maximum von mises stress . it is important to recognize that the respective univariate and bivariate pdd solutions in this particular problem are practically the same . therefore , the univariate pdd solutions are not only accurate , but also highly efficient . this is because of a realistic example chosen , where the individual main effects of input variables on the von mises stress are dominant over their interactive effects . finally , this example also demonstrates the non - intrusive nature of the pdd - kde - mc method , which can be easily integrated with commercial or legacy computer codes for analyzing large - scale complex systems . a general multivariate sensitivity index , referred to as the @xmath0-sensitivity index , is presented for global sensitivity analysis . the index is founded on the @xmath0-divergence , a well - known divergence measure from information theory , between the unconditional and conditional probability measures of a stochastic response . the index is applicable to random input following dependent or independent probability distributions . since the class of @xmath0-divergence subsumes a wide variety of divergence or distance measures , numerous sensitivity indices can be defined , affording diverse choices to sensitivity analysis . several existing sensitivity indices or measures , including mutual information , squared - loss mutual information , and borgonovo s importance measure , are shown to be special cases of the proposed sensitivity index . a detailed theoretical analysis reveals the @xmath0-sensitivity index to be non - negative and endowed with a range of values , where the smallest value is _ zero _ , but the largest value may be finite or infinite , depending on the generating function @xmath0 chosen . the index vanishes or attains the largest value when the unconditional and conditional probability measures coincide or are mutually singular . unlike the variance - based sobol index , which is invariant only under affine transformations , the @xmath0-sensitivity index is invariant under nonlinear but smooth and uniquely invertible transformations . if the output variable and a subset of input variables are statistically independent , then there is no contribution from that subset of input variables to the sensitivity of the output variable . for a metric divergence , the resultant @xmath0-sensitivity index for a group of input variables increases from the unconditional sensitivity index for a subgroup of input variables , but is limited by the residual term emanating from the conditional sensitivity index . three new approximate methods , namely , the mc , kde - mc , and pdd - kde - mc methods , are proposed to estimate the @xmath0-sensitivity index . the mc and kde - mc methods are both relevant when a stochastic response is inexpensive to evaluate , but the methods depend on how the probability densities of a stochastic response are calculated or estimated . the pdd - kde - mc method , predicated on an efficient surrogate approximation , is relevant when analyzing high - dimensional complex systems , demanding expensive function evaluations . therefore , the computational burden of the mc and kde - mc methods can be significantly alleviated by the pdd - kde - mc method . in all three methods developed , the only requirement is the availability of input - output samples , which can be drawn either from a given computational model or from actual raw data . numerical examples , including a computationally intensive stochastic boundary - value problem , demonstrate that the proposed methods provide accurate and economical estimates of density - based sensitivity indices . , _ on the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling _ , phil . mag . , 50 ( 1900 ) , pp .

this article presents a general multivariate @xmath0-sensitivity index , rooted in the @xmath0-divergence between the unconditional and conditional probability measures of a stochastic response , for global sensitivity analysis . unlike the variance - based sobol index , the @xmath0-sensitivity index is applicable to random input following dependent as well as independent probability distributions . since the class of @xmath0-divergences supports a wide variety of divergence or distance measures , a plethora of @xmath0-sensitivity indices are possible , affording diverse choices to sensitivity analysis . commonly used sensitivity indices or measures , such as mutual information , squared - loss mutual information , and borgonovo s importance measure , are shown to be special cases of the proposed sensitivity index . new theoretical results , revealing fundamental properties of the @xmath0-sensitivity index and establishing important inequalities , are presented . three new approximate methods , depending on how the probability densities of a stochastic response are determined , are proposed to estimate the sensitivity index . four numerical examples , including a computationally intensive stochastic boundary - value problem , illustrate these methods and explain when one method is more relevant than the others . borgonovo s importance measure , @xmath0-sensitivity index , kernel density estimation , mutual information , polynomial dimensional decomposition , squared - loss mutual information .

the introduction of new quantum mechanical technologies promises to fundamentally alter the way we communicate . quantum key distribution ( qkd ) , for instance , will allow us to communicate in an intrinsically secure way @xcite . but new quantum communication technologies will require a new telecommunications infrastructure , one which is quantum - enabled . that is , this network must be able to properly accommodate the quantum properties that quantum communications inherently rely on . such a quantum network will contain many novel components , such as quantum memories @xcite , quantum repeaters @xcite , or , most generally , quantum channels . these components must each operate in a strictly quantum way . of course , no technology is perfect , and quantum technologies offer a new set of practical challenges . however , as we have learned from qkd , perfectly ideal devices are not a necessity . by shifting our efforts into classical post - processing of data , we can deal with imperfections in quantum technologies . the question then becomes , how much imperfection can be tolerated before a device is no longer operating in a sufficiently quantum way ? we can enforce a minimal quantum requirement on devices by insisting that they do not act as _ measure and prepare _ channels @xcite ( or , in the parlance of qkd , _ intercept and resend _ channels ) , since communication through such channels is equivalent to classical communication . indeed , this type of channel destroys any quantum correlations in bipartite states when one subsystem is sent through it . of course , this is just the minimum requirement . it is also important to quantify the quantum behaviour , as is done in the field of entanglement measures , or in qkd through the secret key rate . for quantum channels , we can ask , _ how well does the channel preserve quantum correlations in bipartite systems , when only one subsystem passes through it ? _ to study this question , we take a state with well - quantified quantum correlations , send one subsystem through the channel , and examine the output . we then compare the quantum correlations detectable in the output with the input correlations . in fact , as we shall see , we can test for these correlations in a so - called ` prepare and measure ' picture , bypassing the need to use actual bipartite states . a strong quantum channel is one which preserves all or nearly all of the quantum correlations . this idea corresponds to what we shall call the _ quantum throughput_. such a measure would allow us to characterize the suitability of devices for quantum communication tasks . the goal of this work is to illustrate that these ideas about device characterization via quantum throughput can be implemented in a meaningful way . although we will make specific choices regarding device types or quantification measures , the basic idea remains quite general , and our scheme can be extended and adapted to other methods as well . finally , if we picture a future quantum communications network consisting of many components , it should be evident that any device - testing procedure should be as experimentally practical as possible . ideally , we seek a testing scenario where a finite number of test states and a limited set of measurements are sufficient to understand the quantum throughput . the latter requirement is especially important for optical systems , which are perhaps the most natural choice of carrier for quantum information . in these systems , full tomography is not really a practical option because of the dimension of the hilbert space . we have previously examined quantum correlations in optical devices in a qualitative way @xcite ; in the present contribution , we will extend those results to provide a quantitative picture of optical devices . the rest of this paper is organized as follows . in sec . [ sec : quant ] we outline our quantitative device - testing scheme , focusing mainly on optical systems . we show how to estimate important parameters from homodyne measurements on the output , and how to use these estimates to make quantitative statements about the optical device . in sec . [ sec : results ] , we give the results of this quantification procedure for a wide class of optical channels , and examine the strength of our method . sec . [ sec : conclusion ] summarizes the paper , while appendices [ app : overlapbounds]-[app : offdiagbounds ] provide technical details and derivations . the quantum device testing procedure we employ is the same as the one found in @xcite . this protocol is based on the idea that a truly quantum channel should be distinguishable from those channels where the input quantum state is temporarily converted to classical data before a new quantum state is output , a so - called _ measure and prepare _ channel . measure and prepare channels are also called _ entanglement - breaking _ channels , as the two notions are equivalent @xcite . this provides a hint on how to quantify a channel s quantum throughput , namely by sending part of an entangled state through the channel and determining the amount of entanglement that still remains afterwards . to this end , imagine we have an entangled state of the form @xmath0\ ] ] where system @xmath1 is a qubit and system @xmath2 is an optical mode . we can assume , without loss of generality , that @xmath3 , so that @xmath4 and @xmath5 denote coherent states of opposite phase . this is an entangled state for all values @xmath6 , as can be seen by calculating the entropy of entanglement . keeping subsystem a isolated , an optical channel can be probed using subsystem b of this state , followed by local projective measurements @xmath7 by alice and homodyne measurements @xmath8 by bob . these expectation values , along with the knowledge of alice s reduced density matrix @xmath9 , can be used to determine just how much of the initial state s entanglement is remaining . of course , states like eq . ( [ eq : initialstate ] ) may be difficult to create and therefore not suited for practical device testing . however , notice that alice s reduced density matrix does not depend on what happens in the optical channel , nor on any of bob s measurement results . her expectation values can be completely determined from the initial state @xmath10 . indeed , alice s measurement results can be thought of as classical registers which merely record which mode state was sent through the device . this observation allows us to move from an entanglement - based ( eb ) picture to an equivalent ` prepare and measure ' ( pm ) scenario @xcite , in which alice s measurements are absorbed into the initial state preparation . in a pm scenario , we retain full knowledge of @xmath9 , in particular the off - diagonal coherence term @xmath11_{01}={\left\langle{\alpha}\right\vert { { \hspace{-0.1 em}}}{{\hspace{-0.1 em}}}\left . { -\alpha}\right\rangle}$ ] . we must insert this additional information _ by hand _ into the set of expectation values for @xmath12 . this distinguishes the expectation values from data which would come from using just a classical mixture of test states @xmath13 . other than this , the procedure is the same as the eb scenario described above . quantum correlations introduced in this way are referred to as ` effective entanglement . ' using this convenient theoretical trick , the testing protocol can be accomplished simply by probing the channel using a source which prepares one of the two conditional states @xmath14 with equal probability . if the measured expectation values , along with the inserted knowledge of @xmath9 , are not compatible with any separable qubit - mode state , then there is ( effective ) entanglement and the channel is certifiably quantum . exploiting the duality between the pm picture and the eb picture , we can quantify the quantum correlations remaining in the output state through a suitable entanglement measure . in turn , this can be compared to the entanglement of the state in eq . ( [ eq : initialstate ] ) to determine the quantum throughput . our main goal in this work is to give an estimate of the amount of effective entanglement observable in an optical system after transmission through an optical channel . our method is based on the following observation : when the two , initially pure , conditional states @xmath15 pass through the channel , they are subject to loss and noise , and evolve in general to mixed states @xmath16 on the infinite - dimensional mode hilbert space ; however , since we work with coherent states , this change in purity comes only from the noise . thus , for any loss value , if the noise introduced by the channel is not too high , then the output states @xmath17 and @xmath18 will still be nearly pure . in this case , most of the information about the state is still contained in a very small subspace of the full infinite dimensional hilbert space . estimating the ` most significant ' subspace on the mode system can therefore be quite useful . this subspace should contain as much information as possible about both conditional states . additionally , we will concentrate on the simplest non - trivial mode subspace , namely one of dimension 2 . writing the conditional output states @xmath17 and @xmath18 in terms of their eigenvectors , in order of descending eigenvalues , we have @xmath19 the most significant subspace is then the one formed using @xmath20 and @xmath21 as basis vectors . three parameters will be important to identify this subspace : @xmath22 , @xmath23 , and @xmath24 . we will estimate these parameters using homodyne detection . specifically , if @xmath25 is the annihilation operator for the mode at bob s detector , then a balanced homodyne detection scheme allows us to measure the field quadratures , here defined as @xmath26 we will use the mean values @xmath27 and the variances @xmath28 of the quadratures from both conditional states to estimate the three subspace parameters . exactly how this is done will be shown in the next part . with these parameters , we can build a @xmath29 density matrix @xmath30 , which corresponds to the projection of the full qubit - mode state @xmath12 onto the two - qubit subspace spanned by the basis @xmath31 . the idea is now to bound the entanglement of the full state @xmath12 using the entanglement of the projection @xmath30 . for this , we need to exploit the strong monotonicity under local operations and classical communication ( locc ) property found in many entanglement measures . specifically , if we perform a complete set of local measurements on a bipartite state @xmath32 , which yields ( perhaps using some classical communication ) the state @xmath33 with probability @xmath34 , then the strong monotonicity property captures the idea that the entanglement should not increase , on average , under this process . in other words , for a given measure @xmath35 , @xmath36 as the name implies , this is a stronger condition than just monotonicity under locc alone . in our case , the measurement consists of projecting the mode system onto the most significant subspace or onto the orthogonal complement . we denote the former projection by @xmath37 and the latter by @xmath38 . then , if we choose an entanglement measure @xmath35 with the strong monotonicity property , we have @xmath39 where @xmath40 and @xmath41 . for later practicality purposes , we would like to factor the probabilities through the entanglement measure , so that we work directly with unnormalized states . the unnormalized projected state is thus given by @xmath42 . we must be careful to choose an entanglement measure which , in addition to being a strong monotone , can be defined for unnormalized states and which permits a positive prefactor to be absorbed into the state . we will focus on the negativity @xcite in this work , for which this choice is justified . we will not attempt to estimate the second term in eq . ( [ eq : strongmono ] ) coming from the orthogonal projection ; we only note that it is non - negative , so that we have the bound @xmath43 in practice , the projected matrix @xmath30 will not be fully characterized and will contain open parameters . on the other hand , some constraints can be imposed on @xmath30 from our knowledge of the initial conditional states and the homodyne measurement results , as well as natural positivity ( @xmath44 ) and trace constraints @xmath45 for unnormalized @xmath30 ) . as a final step , we must determine the minimal entanglement of @xmath30 compatible with all allowed values of these open parameters , subject to the known constraints . we are left with the final relation @xmath46 which will be used as the basis for calculating bounds on @xmath47 . the next two subsections will cover how to estimate @xmath30 and how we minimize the entanglement over all compatible forms of @xmath30 . the first step in our method requires determining the projection @xmath30 of the full state @xmath12 onto the most - significant subspace . for this , we need to estimate the three parameters @xmath22 , @xmath23 and @xmath24 from the decomposition in eq . ( [ eq : eigendecomp ] ) . in ref . @xcite , which considers the related problem of effective entanglement verification using heterodyne measurements ( i.e. full knowledge of the @xmath48 function ) , several useful formulas for estimating these maximal eigenvalues and overlaps are given . these bounds are later refined in @xcite , where they are used to derive secret key rates for continuous variable quantum key distribution . here , we use these bounds as a starting point to build up a good estimate of the projected state for our quantification scheme . we will roughly follow the notation of @xcite in the following . first , since the conditional output states @xmath49 have unit trace , their maximal eigenvalues can be parameterized by @xmath50 , with @xmath51 $ ] . then eqs . ( 66 ) and ( 69 ) from @xcite give directly the following bound : @xmath52 = : u_j.\ ] ] this bound comes up several times , so it is denoted @xmath53 ( @xmath54 ) to make later equations more readable . importantly , the bound can be calculated using only the measured variances of the conditional states . estimating the overlap @xmath24 is more involved . we need to derive bounds on its magnitude based on our available information . again , we begin with bounds provided in refs . @xcite . with suitable relaxations , their bounds can be put into a specific form which will be more desirable for us later , as we would ultimately like to do a convex optimization . the specific details of this relaxation are straightforward , and are outlined in appendix [ app : overlapbounds ] . we will need an additional parameter , @xmath55 , which can be calculated directly using the measured first moments @xmath56 . defining two coherent states with the same means as the conditional states , @xmath57 the new parameter is given through the overlap of these coherent states , @xmath58 with this definition in place , we can give the relaxed bounds @xmath59 where @xmath60 and @xmath61 having these bounds , obtained purely through homodyne measurements , we can now move on to estimating the elements of the projected density matrix @xmath30 . we can already estimate matrix elements of the form @xmath62 using eq . ( [ eq : noisebound ] ) , but to build @xmath30 we also require bounds on the supplementary elements @xmath63 for @xmath64 . to get these , we first expand @xmath17 into its eigenbasis , eq . ( [ eq : eigendecomp ] ) . then , using the fact that @xmath65 $ ] for any normalized vector @xmath66 , we can easily derive the following bounds on the desired matrix element ( see appendix [ app : suppdiagbounds ] for full details ) : @xmath67 analogous bounds can be given for @xmath68 . finally , we need to estimate some elements of the off - diagonal blocks of @xmath30 , or else there would be no way to differentiate an entangeld state from a classical mixture of the conditional states . to this end , we label the off - diagonal block of the full density matrix @xmath12 by @xmath69 , so that it is naturally split into the form @xmath70 where the diagonal blocks correspond to the two conditional states . in the pm picture , we hold full knowledge of the alice s reduced density matrix @xmath71 where @xmath72 . each element in eq . ( [ eq : rhoa ] ) is the trace of the corresponding element in eq . ( [ eq : rhoblocks ] ) , so we can enforce the condition @xmath73 . using this as our starting point , and with an appropriate basis choice for system b , we can determine the following off - diagonal bounds which can be incorporated into @xmath30 : @xmath74 details on how to arrive at these inequalities can be found in appendix [ app : offdiagbounds ] . we now have sufficient information to construct a useful estimate of the projected state . to summarize , we have the quantities @xmath55 and @xmath75 , which can be calculated from measurements of the first moments and second moments , respectively . we want to determine @xmath30 , which is the projection of @xmath12 from eq . ( [ eq : rhoblocks ] ) onto the subspace spanned by @xmath31 . we have estimated some of the overlaps of @xmath12 with these basis vectors in eqs . ( [ eq : suppdiagbounds1]-[eq : suppdiagbounds2 ] ) and ( [ eq : offdiagbound1]-[eq : offdiagbound2 ] ) . these estimates depend only on the input parameter @xmath72 and on the output state quantities @xmath76 , @xmath77 , and @xmath78 . this last overlap quantity is itself bounded to a region defined by eqs . ( [ eq : overlapbounds]-[eq : bupper ] ) , which depends only on @xmath76 , @xmath77 and @xmath55 . hence , for a fixed input overlap @xmath79 and a fixed set of homodyne measurement results , we have a parameter region which forms a set of constraints on @xmath30 . this region must be searched to find the minimal entanglement compatible with @xmath30 . we will now move on to address the question of how to find the minimal entanglement compatible with our constraints . as mentioned earlier , we will choose the negativity as the entanglement measure for demonstrating our method . in principle , we would like to find the minimal entanglement using the methods of semidefinite programming . but we must make some simplifications and relaxations which will allow us to do so . first , we exploit the fact that local unitary operations can not change the quantity of entanglement . therefore , without loss of generality , we can assume that the overlap of the maximal eigenstates is real and positive ( since this can be accomplished by a relative change of phase on subsystem b ) . @xmath80 as well , we can perform local phase changes on subsystem a , which allows us to also make the restriction @xmath81 the other off - diagonal element of interest , @xmath82 , is in general still a complex number . the main problem is that eq . ( [ eq : offdiagbound2 ] ) is a _ non - convex _ constraint on @xmath83 . to use this constraint in a semidefinite program , we have to replace it with a set of convex constraints . we accomplish this by denoting the right - hand side of eq . ( [ eq : offdiagbound2 ] ) as @xmath84 and expanding our constraints to the region @xmath85 this new constraint still non - convex , but we can search for the minimum entanglement independently in each of the four quadrants , where the constraints are convex ( see fig . [ fig : convexregions ] ) , and take the minimum over these four searches . the final result will be a lower bound to the minimum entanglement in the region constrained by eq . ( [ eq : offdiagbound2 ] ) . we can extend this idea further , replacing the inscribed square from fig . [ fig : convexregions ] with any other inscribed polygon . with more sides , we can better approximate the non - convex constraint eq . ( [ eq : offdiagbound2 ] ) , but this will also increase the number of convex subregions which must be searched to find the overall minimum . numerical evidence indicates that the minimum entanglement is often , though not always , found at a point outside the circle . we tested with an inscribed octagon and it was not found to alter the final results significantly . the final hurdle comes from the overlap @xmath24 . since the maximal eigenstates will in general have a non - zero overlap ( indeed , for zero overlap , we will not find any entanglement in @xmath30 ) , we must construct an orthogonal basis in order to explicitly write down a matrix representing @xmath30 . doing so introduces matrix elements that are both linear and quadratic in the overlap @xmath24 . if the overlap is used as a parameter in the semidefinite programming , this non - linear dependence becomes problematic . fortunately , it turns out that to find the minimal entanglement we only need to consider the case where the overlap takes the largest allowed value , i.e. @xmath86 . the reason for this is that , for fixed values of @xmath22 , @xmath23 , and @xmath24 , there always exists a cptp map on the b subsystem which preserves the maximal eigenvalues while making the corresponding overlap larger . such a local map can not increase the entanglement , so indeed the minimal entanglement will be found at @xmath86 . this useful result will be shown in detail elsewhere @xcite . in the previous section , we outlined a method for calculating the effective entanglement in optical systems . this began with the observation that we can get bounds just by looking at the most significant two - qubit subsystem . the remainder of sec . [ sec : quant ] provided the necessary tools to allow us to calculate these bounds efficiently as a semidefinite program . now that all the pieces are in place , we can turn to applying our scheme . to illustrate our quantification method , we use data corresponding to the action of the optical channel on the field quadratures , which we assume to be symmetric for both signal states and for both quadratures . these symmetry assumptions are made solely to aid the graphical representation of our results , and our method does not rely on them . it is also important to note that , beyond the symmetry , we do not make any assumptions about how the channel works . in the absence of experimental data , we merely parameterize the channel s effect on the first quadrature moments by a loss parameter and on the second moments by the excess noise . specifically , if the means of the two conditional output states are denoted by @xmath87 from eq . ( [ eq : cohmean ] ) , then the loss is parameterized through the transmittivity @xmath88 and the symmetric excess noise ( expressed in shot noise units ) by @xmath89 the input states are characterized entirely by the overlap parameter @xmath90 . the quantification program was carried out using the negativity @xcite , @xmath91 this measure has all the properties demanded by our quantification method , but more importantly , the trace norm @xmath92 of a matrix can be computed efficiently as a semidefinite program @xcite . we have normalized the negativity so that a maximally entangled two - qubit state has @xmath93 . our calculations were done in matlab using the yalmip interface @xcite along with the solver sdpt3 @xcite . our main results are shown in fig . [ fig : mainresults ] , where the minimal negativity of @xmath30 compatible with the initial overlap @xmath90 and excess noise @xmath94 is given , for various values of the transmittivity @xmath95 . this quantity gives a lower bound on the negativity of the full state @xmath12 . the entanglement of the initial state , eq . ( [ eq : initialstate ] ) , is also shown as a function of the initial overlap in fig . ( [ fig : test1 ] ) . for figs . ( [ fig : test2]-[fig : test3 ] ) , the modification @xmath96 is made to eq . ( [ eq : initialstate ] ) for these comparisons . the initial entanglement can be compared with the calculated bounds to help understand the quantum throughput of a device . in the limit of zero excess noise and zero loss , our entanglement bound is tight with the initial entanglement . our bounds are quite high for very low noise , but they become lower as the measurement results get more noisy . at some point , a non - trivial entanglement bound can no longer be given , despite the fact that quantum correlations can still be proven for higher noise values ( cf . @xcite ) . as well , for larger loss values , the tolerance for excess noise is lower , and the region where non - trivial bounds can be given becomes smaller . the exact noise value where our bounds become trivial depends on the initial overlap and on the measured loss , but the highest tolerable excess noise is around 5% of the vacuum for @xmath97 . this shrinks to about 3% for a transmittivity of @xmath98 . though the quantification region is small , it is within the limits of current experimental technology @xcite . some entanglement degradation should be expected as the noise is increased , but , as mentioned earlier , entanglement can still be verified ( though not previously quantified ) under the same testing scenario up to much higher noise values than seen here @xcite . thus , our bounds do not provide the full picture . the weakening of the bounds with higher noise is mainly due to the estimation procedure . certain approximations become cruder ( though still valid ) as the noise increases . first , for higher noise , the conditional states become more mixed , spreading out into more of the infinite - dimensional mode hilbert space . this leads to additional information being lost when we truncate down from @xmath12 to @xmath30 . another problem stems from the bounds we use to estimate @xmath30 . higher noise leads to weaker bounds on the maximal eigenvalues from eq . ( [ eq : noisebound ] ) , which weakens all other inequalities . to examine the effects of these two approximations , we briefly consider a simple channel where the test state , eq . ( [ eq : initialstate ] ) , is mixed at a @xmath99 beam - splitter with a thermalized vacuum . the first moments reduce by a factor of @xmath100 , and the increased variances of the output optical states can be determined from the mean photon number @xmath101 of the thermal state . for @xmath102 , the conditional output states are displaced thermal states . the reason for studying this channel is that we can _ exactly _ determine the maximal eigenvalues @xmath22 , @xmath23 , and the overlap @xmath24 . this allows us to study our approximations independently , since we decouple the effects of the two - qubit projection from the homodyne parameter estimation ( in practice , of course , our quantification scheme must use both ) . in fig . ( [ fig : comparison ] ) we show the result of the quantification scheme , when this extra information is included . we see that the tolerable excess noise is @xmath103 of the vacuum , more than three times what it would be if we had to estimate the eigenvalues and overlap using homodyne results ( cf ( [ fig : test3 ] ) ) . also included in fig . ( [ fig : comparison ] ) is an entanglement verification curve , obtained using the methods of @xcite . any points with lower noise than this verification curve must come from entangled states . the two - qubit projection is tight to the entanglement verification curve for low overlaps . for higher values , the projection becomes weaker , only working to about half the noise value that the entanglement verification curve reaches . ideally , we want to be able to calculate non - trivial values for the entanglement wherever it be verified . this would give us a true quantitative complement to existing entanglement verification methods . one obvious extension to our method would be to truncate the mode subspace using the two largest eigenstates from each conditional state , or even more . in theory , this would strictly improve the estimates . however , in practice , this will increase the complexity of the quantification calculation , since some simplifying assumptions ( i.e. certain overlaps are real ) may no longer be valid . as well , the number of additional minimizations we have to do , as in our non - convex relaxation of eq . ( [ eq : offdiagbound2 ] ) , increases fourfold with each added dimension . another approach might therefore be necessary to overcome this problem . nevertheless , the quantification scheme outlined here is a useful method for characterizing the degree of quantumness of optical channels , especially when these channels introduce low noise . we have outlined a method for quantifying the effective entanglement in qubit - mode systems using only homodyne measurement results and knowledge of the initial preparation . this quantification method works particularly well if the mode subsystem exhibits low noise . by combining this quantification scheme with a device testing scenario which uses two nonorthogonal test states , one can examine how strongly an optical device or experiment is operating in the quantum domain . our scheme provides a useful tool for understanding the quantum nature of optical devices , especially the question of how well they preserve quantum correlations . in this appendix , we derive the bounds from eqs . ( [ eq : blower]-[eq : bupper ] ) for the absolute value of the overlap of the maximal eigenstates , @xmath104 . from @xcite , we have the following : _ overlap bounds . _ let the largest eigenvalue of @xmath105 be parameterized by @xmath106 and let the fidelity between the conditional states and the coherent states @xmath107 from eq . ( [ eq : cohmean ] ) be given by @xmath108 and let @xmath58 then the following holds : @xmath109 with @xmath110 and @xmath111 since we can not calculate @xmath112 or @xmath113 in practice , we now modify these bounds from the above form found in @xcite to one involving only the parameters @xmath55 ( calculated from first moments ) and the @xmath53 ( calculated from second moments ) . to do this , we make use only of the obvious inequality @xmath114 from this , we can easily derive the following auxiliary inequalities : @xmath115 it is important to note that the second and third inequalities only hold so long as @xmath116 . for symmetric noise , the value @xmath117 corresponds to @xmath118 , almost twice the vacuum variance . this value is far outside the region where our method gives non - trivial bounds , so it is not an issue . substituting the inequalities ( [ eq : aux1]-[eq : aux3 ] ) into eqs . ( [ eq : oldoverlaplowerbound ] ) and ( [ eq : oldoverlapupperbound ] ) , we arrive at the bounds given in eqs . ( [ eq : blower]-[eq : bupper ] ) . here we aim to bound the quantities @xmath63 for @xmath64 , as found in eqs . ( [ eq : suppdiagbounds1]-[eq : suppdiagbounds2 ] ) . an eigenbasis expansion of @xmath17 leads to @xmath119 a lower bound can be derived in a similar way : @xmath120 the bounds for @xmath121 follow by interchanging indices . this appendix outlines the derivation of the off - diagonal bounds from eqs . ( [ eq : offdiagbound1]-[eq : offdiagbound2 ] ) . we completely know @xmath9 , which constrains that we must have @xmath122 . first , we consider the full density matrix @xmath12 in the basis defined by @xmath123 for system @xmath1 and the eigenbasis of @xmath17 , @xmath124 , for system b. we can still write this in the block form of eq . ( [ eq : rhoblocks ] ) , where we denote the diagonal elements of the block @xmath18 by @xmath125 and the diagonal elements of the block @xmath69 by @xmath126 ( the diagonal elements of @xmath17 are its eigenvalues ) . using the triangle inequality , we have @xmath127 from positivity of @xmath12 , we find @xmath128 and from the cauchy - schwarz inequality , @xmath129 the first sum is just @xmath130 and the second is @xmath131 . now , using the bounds from appendix [ app : suppdiagbounds ] , we get @xmath132 which we can substitute above to obtain @xmath133 replacing @xmath134 with @xmath135 , we are led to the off - diagonal bound @xmath136 by applying the same arguments using the eigenbasis of @xmath18 , we can arrive at an analogous bound for @xmath137 .

quantum communication relies on optical implementations of channels , memories and repeaters . in the absence of perfect devices , a minimum requirement on real - world devices is that they preserve quantum correlations , meaning that they have some thoughput of a quantum mechanical nature . previous work has verified throughput in optical devices while using minimal resources . we extend this approach to the quantitative regime . our method is illustrated in a setting where the input consists of two coherent states while the output is measured by two homodyne measurement settings .

recently , within the framework of a coarse - grained nonlinear network model ( nnm ) , we have shown that dbs in proteins feature strongly site - modulated properties @xcite . more precisely , we have shown that spatially localized band - edge normal modes ( nm ) can be continued from low energies to db solutions centered at the same sites as the corresponding nms ( the nm sites ) . note that the latters lie , as a rule , within the stiffest regions of a protein @xcite . more generally , however , dbs display a gap in their excitation spectrum . as a consequence , they can `` jump '' to another site as their energy is varied , following spatial selection rules matching the pattern of dbs localized elsewhere @xcite . as a matter of fact , such jumps realize efficient _ energy transfers_. hereafter , we show that events of this kind , connecting with high yields even widely separated locations , can be triggered by a localized excitation , so long as its energy @xmath4 lies above a given threshold . energy transfer : all - site analysis . percentage of sites in subtilisin that transmit most of the kick energy to the band - edge nm site , val 177 ( black diamonds ) , or to the nm site of the second edge mode , met 199 ( stars ) . for a given kick energy , each site is kicked once , the most energetic nonlinear mode obtained is analyzed , and the site the most involved in this mode is recorded . when initial excitations are not imparted along the local stiffest direction , but are oriented at random , energy transfer towards val 177 is less likely ( open diamonds ) . ] -5 mm fig . [ ekept ] summarizes the outcome of one such experiment , where energy is initially either localized in nm ( m ) or in real ( r ) space . typically , the initial excitation is found to spark the formation of a discrete breather , pinning a variable amount of energy @xmath5 at a specific location . when less than 10 kcal / mole of kinetic energy is injected into the edge nm , nearly all this energy is kept by the db , whose overlap with the edge nm is large at low energies . increasing @xmath4 further , the frequency of the excited mode detaches from the linear band , while the excitation efficiency @xmath6 is eroded . in fact , as db localization builds up with energy ( see lower left panel ) , the spatial overlap with the edge nm diminishes , thus reducing excitation efficiency @xcite . the same db is also excited when the edge nm site is `` kicked '' along an _ appropriate _ direction , namely the maximum stiffness ( ms ) one @xcite ( see data marked ( r ) in fig . [ ekept ] ) . in this case , however , the excitation becomes more efficient as @xmath4 is increased , since the db asymptotically approaches a single - site vibration . for @xmath7 kcal / mole , the db looses its energy , which flows rapidly into the system . we find that the maximum strain direction invariably allows for the most efficient excitation of a nonlinear mode at a given site . [ eangle ] illustrates the efficiency of kicks given along the ms direction , with respect to kicks imparted along random directions . the correlation with the squared cosine of the angle between the kick and the ms unit vectors indicates that it is the amount of energy injected along the ms vector which is the dominant factor allowing for efficient excitation of a discrete breather . + interestingly , kicking away from the ms direction can promote energy transfer to another site . for instance , while a kick along the ms unit vector at the nm site of the band - edge mode invariably results in a db sitting at the same site , when the direction of the kick is picked at random discrete breathers localized elsewhere are also observed ( see again fig . [ eangle ] ) . in the following , we take advantage of the fact that ms directions can be easily calculated at any site in any structure @xcite in order to investigate energy transfer in a systematic manner . energy transfer as a function of distance from excitation site . the figure illustrates the outcome of an all - site kick experiment in myosin , a large 746 amino - acids enzyme involved in muscle contraction ( pdb code 1vom ) . the fraction of excitation energy found in the db is plotted versus the distance ( in units of links in the connectivity graph ) between the kicked site and the site where the nonlinear mode self - excites . the maximum amount of energy found in the db decreases with the number of links separating the feed and the target sites . for instance , when gln 246 is kicked , more than 40% of the energy ends up in a db localized at ala 125 ( the band - edge nm site ) . this amounts to four links , corresponding to a span of about 25 in real space . otherwise , when a kick is given to ile 351 , gln 246 or tyr 34 , 25 - 65% of the excitation energy flows either to ala 125 or leu 296 , the nm site of the third edge normal mode . in cases where more than 30% of the kick energy is transferred away , three sites turn out to be targeted half of the times , namely ala 125 ( 27% ) , leu 296 ( 13% ) and gly 451 ( 7% ) . when only long - range energy transfers are considered ( covering three or more links ) , the shares raise to 71 % and 18 % for ala 125 and leu 296 , respectively . in the remaining cases , the db is found either at leu 516 ( 7% , 14@xmath8 mode ) or at arg 80 ( 4% , 10@xmath8 mode ) . ] -5 mm when a given residue is kicked along the ms direction , a transfer event can occur when @xmath9 kcal / mol ( see an example in fig . [ etrans ] ) . at peak transfer , more than 75 % of such kicks excite a db localized at the band - edge nm site , while otherwise energy flows towards the nm site of another edge mode . conversely , when the kick is imparted along a random direction , energy transfer is found to be less efficient . + quite generally , a transfer event can be observed when almost any site is kicked , and in the majority of cases only a handful of well - defined sites are targeted . this means that energy transfer can occur between widely separated locations . indeed , as illustrated in fig . [ elost ] for myosin , only about 5 % of 55 kcal / mole kicks result in a db localized at the same location . for all other kicked sites , a transfer occurs to a db pinning a decreasing fraction of the excitation energy , one to eleven links away . note that all high - yield and long - range energy transfers aim at the nm sites of one of the edge nms , the nm site of the bande - edge mode being the most likely target . thus , energy systematically flows toward the stiffest regions of the structure . interestingly , this is where functionally relevant residues tend to be located @xcite . + in one occurrence , more than 20% of the kick energy ends up in a nonlinear mode localized more than five links away : following a kick at tyr 34 a remarkable nine - link stretch is covered up to leu 296 , making a jump of more than 60 . however , cases of ultra long - range energy transfer like this are more rare and , at the same time , less efficient . in fact , as a consequence of the rather small amount of energy transferred ( nearly 14 kcal / mole ) , the db that self - excites at the target site is poorly localized ( like in fig . [ ekept ] ) . site to site energy transfer in myosin . the local energies at sites ile 351 ( dotted line ) and leu 296 ( solid line ) are plotted as functions of time , after a 55 kcal / mole kick at ile 351 . the fluctuations occurring well before and after the transfer reflect the fact that the corresponding nonlinear modes are not perfectly localized on both sites . as a consequence , they exchange significant amounts of energy with their _ environs_. ] -5 mm a more efficient transfer event , covering two links ( about 11 ) , is analyzed in fig . [ edet ] . at first , a db is excited at the kicked site . however , due to interactions with the background , its energy slowly but steadily flows into the system . after approximately 1 ns , about 65 % of the excitation energy is still there . at @xmath10 ns , this amount of energy is rapidly and almost entirely transferred to leu 296 , marking the self - localization of another db . although the transfer itself is a quite complex process , involving several intermediate sites , it may well prove to be an example of _ targeted energy transfer _ @xcite . indeed , as the energy of the db at the the initial site drops , its frequency diminishes as well . this may allow for a transfer to occur if a resonance condition with the frequency of another db is met . the transmission should be irreversible , as a consequence of both dbs frequency drifts during energy exchange @xcite . note that , as the energy of the first db is eroded , the mode becomes also less and less localized @xcite . this , in turn , is likely to increase the overlap between the two db displacement patterns , thus allowing for more efficient energy channelling @xcite . to gain further understanding on the transfer mechanism , we investigated energy circulation in a dimeric form of rhodopsin . very few high - yield and long - range energy transfers were recorded between sites belonging to different monomers , the vast majority of transfer events being confined within the same domain . indeed , in less than 1% of the instances more than 30% of the kick energy ( 55 kcal / mole ) injected at one monomer is transmitted to the other . here , at variance with most protein dimers , the stiffest regions are located in monomer bulks , so that the edge nms are localized far away from the interface . this strongly suggests that energy transfers not only target stiff regions , but can couple any two sites efficiently only through rather stiff channeling pathways . on the other hand , when kicking one of the two ( almost ) equivalent sites of rhodopsin that are covalently linked to the retinal chromophore , up to about 50 % of the excitation energy ends up in a db localized at one of three specific sites , the targeted location depending upon where ( which monomer ) the kick is imparted and on the magnitude of the latter . interestingly , fig . [ k296 ] reveals that transfer efficiency is optimum in the narrow range 50 - 55 kcal / mole , _ i.e. _ exactly the energy of photons that can be absorbed by the retinal chromophore when it is embedded within rhodopsin ( @xmath11 nm ) . interestingly , the preferentially targeted residue in this energy range ( glu 113 ) is known to be involved in the early stages of the signaling cascade following rhodopsin activation @xcite . energy transfer in rhodopsin ( pdb code 3cap ) . the fraction of energy @xmath6 found in the discrete breather when kicking the site attached to the retinal chromophore ( lys 296 ) of monomer b is plotted versus the excitation energy . symbols indicate at which site the db self - localizes : glu 113 ( black diamonds ) , cys 185 ( open diamonds ) , met 86 ( open circle ) or another one ( stars ) . ] -5 mm in summary , despite its coarse - grained nature , the nnm framework is able to provide biologically sensible clues about energy circulation in proteins . high - yield and long - range energy transfers systematically pin energy at the sites the most involved in a small subset of band - edge linear modes , that is , within the stiffest parts of protein structures . these , in turn , are the regions preferentially hosting residues involved in catalytic mechanisms @xcite . thus , what our study suggests is that protein structures may have been designed , during the course of evolution , so as to allow energy to flow where it is needed , _ e.g. _ to , or close to catalytic sites , with the aim of lowering the energy barriers that have to be overcome during catalytic processes . interestingly , in view of the coarse - grained nature of the nnm scheme , the same site - specific , high - yield and long - range energy transfers observed in proteins are also likely to occur in other physical systems , possibly simpler to engineer and to handle , so long as they share with proteins both spatial and stiffness heterogeneity . proteins are modelled as networks of nodes of mass @xmath12 ( the @xmath13-carbons of their amino - acid residues ) linked by springs . specifically , in the nonlinear network model ( nnm ) @xcite , the potential energy of a protein , @xmath14 , has the following form : @xmath15 \notag\ ] ] where @xmath16 is the distance between particles @xmath17 and @xmath18 , @xmath19 their distance in the equilibrium structure ( as _ e.g. _ solved through x - ray crystallography ) and @xmath20 is a distance cutoff that specifies which pairs of nodes are interacting . note that @xmath21 corresponds to the widely used elastic network model ( enm ) @xcite , which has proven useful for quantitatively describing amino - acid fluctuations at room temperature @xcite , as well as for predicting and characterizing large - amplitude functional motions of proteins @xcite , in agreement with all - atom models @xcite , paving the way for numerous applications in structural biology @xcite , such as fitting atomic structures into low - resolution electron density maps @xcite , or providing templates for molecular replacement techniques @xcite . as in previous nnm studies @xcite , we take @xmath2210 , @xmath23 kcal / mol / @xmath24 and fix @xmath25 so that the low - frequency part of the linear spectrum matches actual protein frequencies , as calculated using realistic force fields @xcite . when @xmath26 a.m.u . ( the average amino - acid residue mass ) , this gives @xmath27 kcal / mol / @xmath28 . for each site in a given structure , the maximum - stiffness ( ms ) direction is computed through the sequential maximum strain algorithm @xcite . following an initial kinetic - energy impulse ( kick ) at a specific site along the local ms unit vector , a 2-ns microcanonical simulation is performed . after a 1-ns transient period during which a part of the excitation energy flows into the system , the velocity - covariance matrix is computed . its first eigenvector provides the pattern of correlated site velocities involved in the dominant ( most energetic ) nonlinear mode ( the db ) . accordingly , a transfer is recorded to the site at which the first principal mode ( pm1 ) is found localized . projecting the system trajectory on pm1 yields fair estimates of the db frequency and average energy @xcite . the localization index @xmath29 of a db centered at site @xmath30 is obtained from the weight of the latter in the normalized displacement pattern of the db , namely @xmath31 ^ 2 $ ] , where @xmath32 are the components at site @xmath30 of pm1 . ishikura , t , yamato , t ( 2006 ) energy transfer pathways relevant for long - range intramolecular signaling of photosensory protein revealed by microscopic energy conductivity analysis . _ chemical physics letters _ 432:533537 . dauxois , t , litvak - hinenzon , a , mackay , r , spanoudaki , a , eds ( 2004 ) _ energy localisation and transfer in crystals , biomolecules and josephson arrays . advanced series in nonlinear dynamics , vol.22 _ ( world scientific , singapore ) . perahia , d , mouawad , l ( 1995 ) computation of low - frequency normal modes in macromolecules : improvements to the method of diagonalization in a mixed basis and application to hemoglobin . _ 19:241246 . tama , f , miyashita , o , brooks iii , cl ( 2004 ) flexible multi - scale fitting of atomic structures into low - resolution electron density maps with elastic network normal mode analysis . _ 337:985999 .

proteins are large and complex molecular machines . in order to perform their function , most of them need energy , _ e.g. _ either in the form of a photon , like in the case of the visual pigment rhodopsin , or through the breaking of a chemical bond , as in the presence of adenosine triphosphate ( atp ) . such energy , in turn , has to be transmitted to specific locations , often several tens of away from where it is initially released . here we show , within the framework of a coarse - grained nonlinear network model , that energy in a protein can jump from site to site with high yields , covering in many instances remarkably large distances . following single - site excitations , few specific sites are targeted , systematically within the stiffest regions . such energy transfers mark the spontaneous formation of a localized mode of nonlinear origin at the destination site , which acts as an efficient energy - accumulating centre . interestingly , yields are found to be optimum for excitation energies in the range of biologically relevant ones . , 0 mm protein dynamics is encoded in their structures and is often critical for their function @xcite . since the early eighties , it is well known that vibrational non - harmonicity has to be accounted for to understand intra - structure energy redistribution @xcite . among nonlinear effects , localized modes were suggested to play a key role @xcite , including topological excitations , such as solitons @xcite as well as discrete breathers ( db ) @xcite . the latter , also known as intrinsic localized modes ( ilms ) , are spatially localized , time - periodic vibrations found generically in many systems as a combined effect of nonlinearity and spatial discreteness @xcite . notably , dbs are able to _ harvest _ from the background and pin down for long times amounts of energy much larger than @xmath0 . indeed , their ability to pump energy from neighboring sites is a distinctive signature of db self - excitation @xcite , _ e.g. _ observed as a consequence of surface cooling @xcite or due to modulational instability of band - edge waves in nonlinear lattices @xcite . therefore , provided such phenomena are compatible with cellular constraints , it is tempting to speculate that evolution has found a way to put such long - lived modes at work for lowering energy barriers associated with chemical reactions , _ e.g. _ for boosting enzyme efficiency during catalytic processes @xcite . optimum kick direction for exciting discrete breathers in dimeric citrate synthase ( pdb code 1ixe ) . percentage of the system energy found in a nonlinear mode as a function of the direction of the initial kick given to ser 213a , the nm site of the band - edge mode . the latter is measured by the angle @xmath1 between the kick direction and the ms unit vector . in all simulations , the ( kinetic ) energy of the kick is 55 kcal / mole and its direction is chosen at random , except when the maximum strain ( ms ) direction is picked instead ( black diamond at @xmath2 ) . filled circles : ser 213a is found to be the most energetic site during the analysis timespan . stars : it is another one . in one instance , while the kick was given in a direction close to the ms direction ( @xmath3 ) , the db jumped on a neighboring site ( namely , thr 208a ) . ] -5 mm

the derivation of low - energy hadronic observables , e.g. meson masses and decay constants , from the theory of the strong interaction ( qcd ) has so far proven to be impossible by means of analytical methods . because of this situation , numerical lattice qcd simulations , whereby the functional integral is evaluated numerically on a discretized space - time lattice , have developed into a major field of study . such simulations are , however , seriously hampered by difficulties in the simulation of dynamical sea quark effects . although much progress has been made recently , it is still impractical , for computational reasons , to simulate with sea quark masses that are close to the physical @xmath2 quark masses of a few mev . this situation , with sea quark masses of a few tens of mev , is referred to as partially quenched ( pq ) qcd . consequently , the physical values of the sea quark masses have to be reached by extrapolation from the partially quenched simulation results . a practical method for this extrapolation is provided by chiral perturbation theory ( @xmath0pt ) , which provides the correct quark mass dependences of the various physical quantities that are measured on the lattice . standard three - flavor @xmath0pt as introduced by weinberg , gasser and leutwyler in refs . @xcite , is valid in the ( unquenched ) qcd case of equal valence and sea quark masses . the generalization of @xmath0pt to the quenched case ( without sea quarks ) or to the partially quenched case ( sea quark masses different from the valence ones ) has been carried out by bernard and golterman in refs . the quark mass dependence of partially quenched chiral perturbation theory ( pq@xmath0pt ) is explicit , and thus the limit where the sea quark masses become equal to the valence quark masses can be taken . as a consequence , @xmath0pt is included in pq@xmath0pt and the free parameters , or low - energy constants ( lec : s ) , of @xmath0pt can be directly obtained from those of pq@xmath0pt @xcite . the calculation of charged pseudoscalar meson masses and decay constants to one loop ( nlo ) in pq@xmath0pt has been carried out in refs . @xcite , and first results for the mass of a charged pseudoscalar meson at two loops or next - to - next - to - leading order ( nnlo ) in pq@xmath0pt have already appeared , for degenerate sea quark masses , in ref . the need for such calculations is clear as nnlo effects have already been detected in lattice qcd simulations @xcite . a calculation of the pseudoscalar meson masses for nondegenerate sea quarks is in progress @xcite . this paper presents the first calculation of the decay constants of the charged , or flavor off - diagonal , pseudoscalar mesons in nnlo pq@xmath0pt , for three flavors of sea quarks ( @xmath3 ) . the results are characterized by the number of nondegenerate valence and sea quarks , denoted @xmath4 and @xmath5 , respectively . for the decay constants of the charged pseudoscalar mesons , the maximum number of nondegenerate valence quark masses is @xmath6 . the degree of quark mass degeneracy in each result is sometimes also referred to with the notation @xmath7 . the decay constant of the charged pion in the @xmath8 symmetric limit thus corresponds to the @xmath9 case . likewise , the decay constants of the charged and neutral kaons may be obtained from the @xmath6 results with @xmath5 = 2 . results are also presented for the case of @xmath5 = 1 ( all sea quark masses equal ) , and @xmath5 = 3 ( all sea quark masses different ) . an extension of the present work to the neutral pseudoscalar mesons is also planned . the analytical expressions for the nnlo shift of the decay constants are in general very long , but the expressions simplify considerably when pairs of sea or valence quark masses become degenerate . in view of this , the nnlo loop results are given separately for each case of @xmath10 considered . in the next sections , the technical background for the nnlo calculations , the full results for the decay constants of the charged pseudoscalar mesons and numerical results as a function of the input quark masses are given , along with a concluding discussion . most of the technical aspects that concern the calculation of the pseudoscalar meson decay constants to two loops , or nnlo , are identical to those of the calculation of the pseudoscalar meson mass , and have already been justified in ref . most significantly , the lagrangians of pq@xmath0pt at @xmath11 and @xmath1 may be directly obtained from the corresponding lagrangians of normal ( unquenched ) @xmath12 flavor @xmath0pt , provided that the traces and meson matrices are replaced with the supertraces and meson matrices relevant to the partially quenched theory @xcite . this can be argued from the replica method as in ref . @xcite , or by the fact that all the relations used to constrain the number of terms in ref . @xcite remain valid when traces are changed to supertraces . we work here in the version of pq@xmath0pt without the @xmath13 as discussed in ref . @xcite . all calculations in this paper have been performed with three flavors of valence quarks , three flavors of sea quarks and three flavors of bosonic ghost quarks . these may be viewed as the @xmath2 and @xmath14 quarks in the valence , sea and ghost sectors , respectively . the purpose of the ghost quarks is to remove the valence quark loops which are disconnected from the external legs . the input quark masses @xmath15 enter into the calculation in terms of the lowest order squared meson masses @xmath16 , which are defined as usual in @xmath0pt , by @xmath17 . in the present calculations , we thus have three valence inputs @xmath18 , three sea inputs @xmath19 , and three ghost inputs @xmath20 . in order for the disconnected valence quark loops to be canceled , the masses of the ghost quarks are always equal to those of the corresponding valence quarks , such that @xmath21 , @xmath22 and @xmath23 . explicitly , for @xmath24 , we have @xmath25 , for @xmath26 , we have @xmath27 and for @xmath28 , we have @xmath29 . similarly , for the sea quarks @xmath30 implies @xmath31 , while @xmath32 implies @xmath33 and finally @xmath34 that @xmath35 . the number of independent low - energy constants in unquenched and partially quenched @xmath0pt is slightly different , but the former are always linear combinations of the latter . for pq@xmath0pt , they are @xmath36 and @xmath37 at leading order , @xmath38 through @xmath39 at nlo and @xmath40 through @xmath41 at nnlo . in contrast , for three flavor unquenched @xmath0pt they are @xmath36 and @xmath37 at leading order , @xmath42 through @xmath43 at nlo and @xmath44 through @xmath45 at nnlo . note that the parameters @xmath46 and @xmath47 correspond to the usual @xmath48 . also , the parameters @xmath36 and @xmath37 are identical for unquenched and partially quenched @xmath0pt . at nlo , the relations between the low - energy constants are @xcite @xmath49 and the corresponding linear relations relevant for the nnlo parameters can be found in ref . @xcite . the calculation of the pseudoscalar meson mass in ref . @xcite was only performed for @xmath50 . however , the present calculation of the decay constants to nnlo is also concerned with the more general cases of @xmath4 and @xmath51 . this leads to much more involved expressions because of the appearance of the residues of the neutral meson propagators . we recall here the results of ref . @xcite which we have translated to minkowski space from the euclidean formalism used there . in minkowski space , the propagator @xmath52 of a charged , or flavor - off - diagonal meson with flavor structure @xmath53 in supersymmetric pq@xmath0pt , is given by @xcite @xmath54 the factor @xmath55 is defined in terms of the squared masses as @xmath56 . the sign vector @xmath57 is defined as @xmath58 for the fermionic valence and sea quarks ( @xmath59 ) and @xmath60 for the bosonic ghost quarks ( @xmath61 ) . the propagator of a flavor - neutral meson in supersymmetric pq@xmath0pt is more complicated , as it connects mesons of different flavor indices as well . the propagator for a meson with quantum numbers @xmath62 to one with quantum numbers @xmath63 may be written in minkowski space @xcite as @xmath64 the nontrivial part @xmath65 of the neutral meson propagator may be expressed , by means of partial fractioning , in terms of a sum of single and double poles in @xmath66 . for the most general case of @xmath67 , in terms of the single - pole residue @xmath68 , the double - pole residue @xmath69 and the auxiliary residue @xmath70 , the propagator @xmath65 is @xmath71 for uniformity of notation , the squared ( lowest order ) masses of the neutral pion and the eta meson in the sea quark sector have been denoted by @xmath72 and @xmath73 , respectively . they are functions of the sea quark masses , and in the case of @xmath67 they are given by the lowest order @xmath0pt result with @xmath74 mixing active @xcite . they can be obtained from the relations @xmath75 for @xmath76 , the @xmath77 pole in @xmath65 disappears , and in that case the ( lowest order ) eta meson mass is trivially given in terms of the remaining sea quark masses as @xmath78 . in this case the index @xmath79 has been suppressed in the residue notation . further , for @xmath80 , both the @xmath77 and @xmath81 poles disappear and consequently both the @xmath79 and @xmath81 indices have been suppressed . in this way we obtain a unique notation for the residues , from which the number of nondegenerate sea quarks is immediately apparent . the various residues @xmath68 of the propagators of the neutral , or flavor - diagonal mesons @xcite appear in the results , and are one reason why the pq@xmath0pt expressions are much more involved than the @xmath0pt results of ref . the use of the propagators ( [ npropij ] ) and ( [ npropii ] ) in the present form has the advantage of producing results in terms of standard loop integrals which can be treated with known methods . on the other hand , the various residues @xmath68 of the flavor - neutral meson propagator fulfill a very large number of relations and the direct output from the calculations of the diagrams consequently produces a large number of redundant terms . this problem does not yet manifest itself at the one - loop level , but becomes troublesome at the two - loop level when the mass degeneracies in the sea and valence quark sectors are lifted . with a major effort , the end result can be simplified , in some cases it has been compressed by more than an order of magnitude . the form of the single - pole residues @xmath82 and the double - pole residue @xmath83 , which appear in eqs . ( [ npropij ] ) and ( [ npropii ] ) , depends on the degree of degeneracy in the sea quark masses , which in turn is indicated by the number of indices in the single - pole residue @xmath68 . it is useful to define the more general quantities @xmath84 such that @xmath85 and so on . the @xmath86 notation is primarily useful for defining the residues of the flavor - neutral propagators , but it may also appear independently in the final result for the decay constant . in such cases , the @xmath86 have been generated by simplification procedures , as all the residues that are naturally generated by partial fractioning of the propagator @xmath65 are of the form given below . note that @xmath84 has the same dimension as @xmath16 for an even number of indices and is dimensionless for an odd number of indices . for the case of @xmath80 , all residues associated with the sea quark sector have reduced to numbers . some nontrivial residues still appear if the valence quarks are nondegenerate , according to @xmath87 for convenience , @xmath88 is also used for @xmath89 in order to maintain a notation similar to the more general cases . as noted in ref . @xcite , the residues also simplify for @xmath76 . in particular , the residue of the neutral pion pole in the sea quark sector ( @xmath90 ) vanishes , and the remaining ones satisfy a larger number of relations than for @xmath67 . the remaining nontrivial residues for @xmath76 may be expressed as @xmath91 it is also noteworthy that the above residue notation is highly redundant because of the trivial relation between @xmath92 and @xmath73 . this fact has been exploited in the simplification of the end results . for @xmath67 , the naturally generated residues are @xmath93 even in this case there are many relations between the various residues . some can be found in ref . @xcite but many more exist . finally , it should be noted that in the limit where the sea quark masses become equal to the valence quark masses , the propagator residues of pq@xmath0pt reduce so that the @xmath77 and @xmath81 meson propagators of unquenched @xmath0pt are recovered . the expression for the decay constant of a charged pseudoscalar meson depends , in nnlo pq@xmath0pt , on a number of one - loop and two - loop integrals . the finite parts of the chiral logarithms @xmath94 and @xmath95 are @xmath96 where the subtraction scale dependence has been moved into the loop integrals . we also define @xmath97 . note that in the limit @xmath98 , the integral @xmath99 reduces to @xmath100 the following combinations of finite one - loop integrals are also introduced , as they are naturally generated by the procedure of dimensional regularization : @xmath101 additionally , the upper middle and upper left diagrams in fig . [ decfig ] generate integrals of the form @xmath102 which are symmetric under the interchange of @xmath16 and @xmath103 . the above integrals have been introduced in order to make the symmetries in the end results more explicit . the finite two - loop integrals @xmath104 that are generated by the top right diagram of fig . [ decfig ] may be evaluated using the methods of ref . note that the corresponding primed integrals @xmath105 indicate differentiation with respect to @xmath106 . the notation used for the @xmath107 integrals in this paper is similar to that of ref . @xcite , except that an extra integer argument now indicates the propagator structure , such that e.g. @xmath108 the case of @xmath109 indicates that the integral consists of single propagators only , as in ref . @xcite , whereas indicates that the first propagator appears squared and that the second propagator appears squared . the cases with two double propagators that can appear in the calculations are , for which the first and second propagators appear squared , and for which the second and third propagators are squared . explicit expressions for can be found in ref . @xcite , and the other cases may be obtained by differentiation with respect to the masses of those expressions . a number of combinations of quark masses and propagator residues are naturally generated in the calculations , and have consequently been given special notations . the most common of these is @xmath110 of which @xmath111 is equal to the average sea quark mass @xmath112 defined in ref . other combinations that appear in the calculations consist of products of quark masses and propagator residues . for @xmath76 , these include @xmath113 for @xmath67 these become @xmath114 it should be noted that also in the case of @xmath76 , the sums run over all three sea quark flavors . furthermore , only some of the above @xmath112 functions appear in the results , as many of them can be reexpressed in terms of other @xmath68 or @xmath86 functions . the actual simplification of the end results is at first glance a formidable task , since the expression for @xmath115 as calculated from the diagrams in fig . [ decfig ] contains several thousand terms . especially for @xmath6 , the parts proportional to @xmath116 and @xmath117 alone are several hundred terms long , while in simplified form they typically contain only @xmath118 terms . however , these simplifications are not easily apparent and are consequently best accomplished by the employment of suitable software , such as ` maple ` or ` mathematica ` . even so , as the expressions can seldom be factored into a single term , considerable trial and error is usually required . the expressions for the decay constants are symmetric , within the sea and valence quark sectors , under the interchange of quark masses . consequently , the end results can be conveniently compactified by the introduction of summation conventions which exploit these symmetries . for example , the notation for the sea quark sector may be considerably compactified by the introduction of the summation indices @xmath14 and @xmath119 , which may appear for the quark masses @xmath0 , and among the indices of the functions @xmath68 and @xmath112 . these sea - quark summation indices should be interpreted as follows : if an index @xmath119 is present once or several times , then there will always be an occurrence of the @xmath14 index as well , and the term is then to be summed over all pairs of different sea quark indices . if the index @xmath14 is present but @xmath119 is not , then the entire term is to be summed over all sea quark indices . simple examples of terms are @xmath120 for terms consisting of products of several factors , the summation sign should always be inserted at the beginning of each relevant term , such that e.g. @xmath121 thus any contribution which is written in terms of the indices @xmath14 and @xmath119 explicitly fulfills the required symmetry properties in the sea quark sector . note that also for @xmath122 , the summation is over all three sea quark flavors , although the index @xmath119 has not been implemented in that case . further compactification of the results is possible , as the valence quark sector is symmetric under the exchange of the valence quark indices @xmath123 and @xmath124 . for this purpose , the summation indices @xmath125 and @xmath126 have been introduced . these are in the present paper only needed for the case of @xmath6 and then always occur for the squared valence quark masses @xmath127 and @xmath128 . if the index @xmath126 is present , there will always be an index @xmath125 and the resulting sum is over the pairs @xmath129 and @xmath130 . if only @xmath125 is present , the sum is over the indices @xmath123 and @xmath124 . an example is @xmath131.\end{aligned}\ ] ] any contribution written in terms of the @xmath132 notation is thus symmetric under the interchange of the valence quark masses @xmath127 and @xmath128 . for @xmath67 , there exists an additional symmetry , i.e. the results are symmetric under the interchange of the lowest order neutral meson masses , @xmath72 and @xmath73 , in the sea - quark sector . this is exploited by the summation indices @xmath133 and @xmath134 . if the index @xmath133 is present , there will always be an index @xmath134 and the corresponding sum is over the pairs @xmath135 and @xmath136 . an example is @xmath137.\end{aligned}\ ] ] the above summation conventions thus provide a means of eliminating a large number of terms in the expression for @xmath115 , which are of similar form . as an added bonus , it gives a possibility to conveniently check that the nnlo loop results fulfill the required symmetry relations in the sea and valence sectors . the decay constants @xmath115 of the pseudoscalar mesons are obtained from the definition @xmath138 in terms of the axial current operator @xmath139 . the diagrams that contribute to that operator at @xmath140 , or nnlo , are shown in fig . [ decfig ] . diagrams at @xmath141 and @xmath142 also contribute to eq . ( [ decdef ] ) via the renormalization of the pseudoscalar meson wave function @xmath143 . the results for charged pseudoscalar mesons so obtained depend on the @xmath1 low - energy constants @xmath144 through @xmath145 . or two - loop for the matrix element of the axial current operator . filled circles denote vertices of the @xmath146 lagrangian , whereas open squares and shaded diamonds denote vertices of the @xmath147 and @xmath148 lagrangians , respectively . ] the results are expressed in terms of the valence inputs @xmath149 and the sea inputs @xmath19 , which are defined in terms of the quark masses through @xmath150 , and the quantity @xmath151 , which corresponds to the lowest order charged meson mass . other parameters include the decay constant in the chiral limit ( @xmath36 ) , the quark condensate in the chiral limit , via @xmath152 , and the lec : s of @xmath153 and @xmath1 , i.e. the @xmath154 and the @xmath155 @xcite , respectively . the decay constants of the pseudoscalar mesons are given in the form @xmath156 , \label{delteq}\ ] ] where the @xmath11 and @xmath1 contributions have been separated . the nnlo contribution @xmath157 has been further split into the contributions from the chiral loops and from the @xmath1 counterterms . the superscripts ( v ) and ( s ) indicate the values of @xmath4 and @xmath5 , respectively . the nlo result for @xmath89 is fairly short , and will thus only be given for @xmath67 . the results for @xmath158 may readily be derived from that expression . the combined nlo result ( loops and counterterms ) for @xmath67 , which is in agreement with ref . @xcite , is @xmath159 at nnlo , the chiral loops form , by far , the largest contribution to the decay constant . as a straightforward derivation of the results for @xmath158 from the @xmath67 case is tedious and complicated , the expressions for the different cases are given separately below . as expected , the infinities in all expressions for the decay constant have canceled . the appearance of unphysical @xmath99 logarithms in the results is , in part , due to the partial quenching , and to the fact that the results have been expressed in terms of the lowest order masses rather than the full physical ones . consequently , not all of them correspond to the quenched chiral logarithms which are ill - behaved in the chiral limit . the contribution from the @xmath1 counterterms to the decay constant at nnlo is , for @xmath67 , @xmath160 from which the results for @xmath158 may be readily inferred . the nnlo contributions from the chiral loops for @xmath161 are , respectively , @xmath162 \:-\,2\,\pi_{16}\,l^r_{1}\,\chi_1 ^ 2 \:-\ : \pi_{16}\,l^r_{2}\,\left [ \chi_1 ^ 2 + 8\,\chi_4 ^ 2 \right ] \nonumber \\ & + & \pi_{16}\,l^r_{3}\,\left [ - 17/6\,\chi_1 \chi_4 + 5/4\,\chi_1 ^ 2 - 3/4\,\chi_4 ^ 2 \right ] \:+\ , \pi_{16}^2\,\left [ - 1/2\,\chi_1 \chi_4 + 1/64\,\chi_1 ^ 2 - 73/128\,\chi_4 ^ 2 \right ] \nonumber \\ & - & 48\,l^r_{4}l^r_{5}\,\chi_1 \chi_4 \:-\,72\,l^{r2}_{4}\,\chi_4 ^ 2 \:-\,8\,l^{r2}_{5}\,\chi_1 ^ 2 \:+\,4\,\bar{a}(\chi_1)\,l^r_{0}\,\left [ \chi_1 + r^d_{1 } \right ] \:-\,4\,\bar{a}(\chi_1)\,l^r_{1}\,\chi_1 \:-\,10\,\bar{a}(\chi_1)\,l^r_{2}\,\chi_1 \nonumber \\ & + & 4\,\bar{a}(\chi_1)\,l^r_{3}\,\left [ \chi_1 + r^d_{1 } \right ] \:-\,4/3\,\bar{a}(\chi_1)\,l^r_{5}\,\chi_1 \:-\,1/2\,\bar{a}(\chi_1 ) \bar{b}(\chi_{14},\chi_{14};0)\,\chi_{14 } \:+\,1/8\,\bar{a}(\chi_1;\varepsilon)\,\pi_{16}\,\chi_4 \nonumber \\ & - & \bar{a}(\chi_{14})\,\pi_{16}\,\left [ 3/4\,\chi_{14 } + 9/8\,\chi_4 \right ] \:-\,12\,\bar{a}(\chi_{14})\,l^r_{0}\,\chi_{14 } \:-\,30\,\bar{a}(\chi_{14})\,l^r_{3}\,\chi_{14 } \:-\,18\,\bar{a}(\chi_{14})\,l^r_{4}\,\chi_4 \nonumber \\ & + & 6\,\bar{a}(\chi_{14})\,l^r_{5}\,\chi_1 \:+\,9/4\,\bar{a}(\chi_{14};\varepsilon)\,\pi_{16}\,\chi_4 \:-\,64\,\bar{a}(\chi_4)\,l^r_{1}\,\chi_4 \:-\,16\,\bar{a}(\chi_4)\,l^r_{2}\,\chi_4 \:+\,32\,\bar{a}(\chi_4 ) l^r_{4}\,\chi_4 \nonumber \\ & + & \bar{a}(\chi_4;\varepsilon)\,\pi_{16}\,\chi_4 \:+\,4\,\bar{b}(\chi_1,\chi_1;0)\,l^r_{0}\,r^d_{1}\,\chi_1 \:+\,4\,\bar{b}(\chi_1,\chi_1;0)\,l^r_{3}\,r^d_{1}\,\chi_1 \:-\,4/3\,\bar{b}(\chi_1,\chi_1;0)\,l^r_{5}\,r^d_{1}\,\chi_1 \nonumber \\ & - & 1/8\,\bar{b}(\chi_1,\chi_1;0,\varepsilon)\,\pi_{16}\,r^d_{1}\,\chi_1 \:-\,36\,\bar{b}(\chi_{14},\chi_{14};0)\,l^r_{4}\,\chi_{14 } \chi_4 \:-\,12\,\bar{b}(\chi_{14},\chi_{14};0)\,l^r_{5}\,\chi_{14}^2 \nonumber \\ & + & 72\,\bar{b}(\chi_{14},\chi_{14};0)\,l^r_{6}\,\chi_{14 } \chi_4 \:+\,24\,\bar{b}(\chi_{14},\chi_{14};0)\,l^r_{8}\,\chi_{14}^2 \:-\,1/8\,h^{f}(1,\chi_1,\chi_{14},\chi_{14};\chi_1)\,\left [ \chi_1 - r^d_{1 } \right ] \nonumber \\ & - & h^{f}(1,\chi_{14},\chi_{14},\chi_4;\chi_1)\,\chi_4 \:+\,1/8\,h^{f}(2,\chi_1,\chi_{14},\chi_{14};\chi_1)\,r^d_{1}\,\chi_1 \:+\,5/18\,h^{f'}(1,\chi_1,\chi_1,\chi_1;\chi_1)\,\chi_1 ^ 2 \nonumber \\ & + & h^{f'}(1,\chi_1,\chi_{14},\chi_{14};\chi_1)\ , [ 1/8\,\chi_1 \chi_4 - 1/2\,\chi_1 ^ 2 ] \:+\ , h^{f'}(1,\chi_{14},\chi_{14},\chi_4;\chi_1)\,\chi_1 \chi_4 \nonumber \\ & + & 2/9\,h^{f'}(2,\chi_1,\chi_1,\chi_1;\chi_1)\,r^d_{1}\,\chi_1 ^ 2 \:+\,3/8\,h^{f'}(2,\chi_1,\chi_{14},\chi_{14};\chi_1)\,r^d_{1}\,\chi_1 ^ 2 \:+\,1/9\,h^{f'}(5,\chi_1,\chi_1,\chi_1;\chi_1)\,(r^{d}_{1})^2 \chi_1 ^ 2 \nonumber \\ & - & 2\,h^{f'}_1(3,\chi_{14},\chi_1,\chi_{14};\chi_1)\,r^d_{1 } \chi_1 ^ 2 \:+\,3/8\,h^{f'}_{21}(1,\chi_1,\chi_{14},\chi_{14};\chi_1)\,\chi_1 ^ 2 \:+\,3\,h^{f'}_{21}(1,\chi_4,\chi_{14},\chi_{14};\chi_1)\,\chi_1 ^ 2 \nonumber \\ & - & 3/8\,h^{f'}_{21}(2,\chi_1,\chi_{14},\chi_{14};\chi_1)\,r^d_{1}\,\chi_1 ^ 2 , \label{f0p611loop } \\ & & \nonumber \\ \delta^{(6)12}_{\mathrm{loops } } & = & \pi_{16}\,l^r_{0}\,\left [ 4/9\,\chi_\eta \chi_4 + 1/2\,\chi_1 ^ 2 - 13/3\,\bar\chi_{1 } \chi_1 - 35/18\,\bar\chi_{2 } \right ] \:-\,2\,\pi_{16}\,l^r_{1}\,\chi_1 ^ 2 \nonumber \\ & - & \pi_{16}\,l^r_{2}\,\left [ 11/3\,\chi_\eta \chi_4 + \chi_1 ^ 2 + 13/3\,\bar\chi_{2 } \right ] \:+\,\pi_{16}\,l^r_{3}\,\left [ 4/9\,\chi_\eta \chi_4 + 5/4\,\chi_1 ^ 2 - 17/6\,\bar\chi_{1 } \chi_1 - 43/36\,\bar\chi_{2 } \right ] \nonumber \\ & + & \pi_{16}^2\,\left [ - 15/64\,\chi_\eta \chi_4 + 1/64\,\chi_1 ^ 2 - 1/2\,\bar\chi_{1 } \chi_1 - 43/128\,\bar\chi_{2 } \right ] \:-\,48\,l^r_{4}l^r_{5}\,\bar\chi_{1}\,\chi_1 \:-\,72\,l^{r2}_{4}\,\bar\chi_{1}^2 \:-\,8\,l^{r2}_{5}\,\chi_1 ^ 2 \nonumber \\ & + & 4\,\bar{a}(\chi_\eta)\,l^r_{0}\,r^{\eta}_{11}\,\chi_\eta \:-\,8\,\bar{a}(\chi_\eta)\,l^r_{1}\,\chi_\eta \:-\,2\,\bar{a}(\chi_\eta)\,l^r_{2}\,\chi_\eta \:+\,4\,\bar{a}(\chi_\eta)\,l^r_{3}\,r^{\eta}_{11}\,\chi_\eta \:+\,4\,\bar{a}(\chi_\eta)\,l^r_{4}\,\chi_\eta \nonumber \\ & - & 4/3\,\bar{a}(\chi_\eta)\,l^r_{5}\,r^{\eta}_{11}\,\chi_1 \:-\,1/6\,\bar{a}(\chi_\eta ) \bar{b}(\chi_{1s},\chi_{1s};0)\,r^{\eta}_{1s}\,\chi_{1s } \:+\,1/8\,\bar{a}(\chi_\eta;\varepsilon)\,\pi_{16}\,r^c_{1}\,\chi_\eta \nonumber \\ & + & 4\,\bar{a}(\chi_1)\,l^r_{0}\,\left [ r^c_{1}\,\chi_1 + r^d_{1 } \right ] \:-\,4\,\bar{a}(\chi_1)\,l^r_{1}\,\chi_1 \:-\,10\,\bar{a}(\chi_1)\,l^r_{2}\,\chi_1 \:+\ , 4\,\bar{a}(\chi_1)\,l^r_{3}\,\left [ r^c_{1}\,\chi_1 + r^d_{1 } \right ] \nonumber \\ & - & 4/3\,\bar{a}(\chi_1)\,l^r_{5}\,r^c_{1}\,\chi_1 \:-\,1/6\,\bar{a}(\chi_1 ) \bar{b}(\chi_{1s},\chi_{1s};0)\,r^{1}_{s\eta}\,\chi_{1s } \:+\,\bar{a}(\chi_1;\varepsilon)\,\pi_{16}\,\left [ 1/4\,\chi_1 - 1/8\,r^c_{1 } \chi_1 - 1/8\,r^d_{1 } \right ] \nonumber \\ & - & 24\,\bar{a}(\chi_4)\,l^r_{1}\,\chi_4 \:-\,6\,\bar{a}(\chi_4)\,l^r_{2}\,\chi_4 \:+\,12\,\bar{a}(\chi_4)\,l^r_{4}\,\chi_4 \:+\,3/8\,\bar{a}(\chi_4;\varepsilon)\,\pi_{16}\,\chi_4 \:-\,32\,\bar{a}(\chi_{46})\,l^r_{1}\,\chi_{46 } \nonumber \\ & - & 8\,\bar{a}(\chi_{46})\,l^r_{2}\,\chi_{46 } \:+\,16\,\bar{a}(\chi_{46})\,l^r_{4}\,\chi_{46 } \:+\,1/2\,\bar{a}(\chi_{46};\varepsilon)\,\pi_{16}\,\chi_{46 } \:-\,\bar{a}(\chi_{1s})\,\pi_{16}\,\left [ 1/4\,\chi_{1s } + 3/8\,\bar\chi_{1 } \right ] \nonumber \\ & - & 4\,\bar{a}(\chi_{1s})\,l^r_{0}\,\chi_{1s } \:-\,10\,\bar{a}(\chi_{1s})\,l^r_{3}\,\chi_{1s } \:-\,6\,\bar{a}(\chi_{1s})\,l^r_{4}\,\bar\chi_{1 } \:+\,2\,\bar{a}(\chi_{1s})\,l^r_{5}\,\chi_1 \:+\,3/8\,\bar{a}(\chi_{1s};\varepsilon)\,\pi_{16}\,\left[\chi_s + \bar\chi_{1 } \right ] \nonumber \\ & + & 4\,\bar{b}(\chi_1,\chi_1;0)\,l^r_{0}\,r^d_{1}\,\chi_1 \:+\,4\,\bar{b}(\chi_1,\chi_1;0)\,l^r_{3}\,r^d_{1}\,\chi_1 \:-\,4/3\,\bar{b}(\chi_1,\chi_1;0)\,l^r_{5}\,r^d_{1}\,\chi_1 \nonumber \\ & - & 1/8\,\bar{b}(\chi_1,\chi_1;0,\varepsilon)\,\pi_{16}\,r^d_{1}\,\chi_1 \:-\,12\,\bar{b}(\chi_{1s},\chi_{1s};0)\,l^r_{4}\,\bar\chi_{1}\,\chi_{1s } \:-\,4\,\bar{b}(\chi_{1s},\chi_{1s};0)\,l^r_{5}\,\chi_{1s}^2 \nonumber \\ & + & 24\,\bar{b}(\chi_{1s},\chi_{1s};0)\,l^r_{6}\,\bar\chi_{1}\,\chi_{1s } \:+\,8\,\bar{b}(\chi_{1s},\chi_{1s};0)\,l^r_{8}\,\chi_{1s}^2 \:+\,1/24\,h^{f}(1,\chi_\eta,\chi_{1s},\chi_{1s};\chi_1)\,r^v_{\eta 1s}\,\chi_\eta \nonumber \\ & + & h^{f}(1,\chi_1,\chi_{1s},\chi_{1s};\chi_1)\,\left [ - 1/12\,r^{1}_{s\eta}\,\chi_1 + 1/24\,r^c_{1}\,\chi_1 + 1/24\,r^d_{1 } \right ] \:-\,3/8\,h^{f}(1,\chi_{14},\chi_{14},\chi_4;\chi_1)\,\chi_4 \nonumber \\ & - & 1/2\,h^{f}(1,\chi_{14},\chi_{16},\chi_{46};\chi_1)\,\chi_{46 } \:+\,1/24\,h^{f}(2,\chi_1,\chi_{1s},\chi_{1s};\chi_1)\,r^d_{1}\,\chi_1 \nonumber \\ & + & 1/9\,h^{f'}(1,\chi_\eta,\chi_\eta,\chi_1;\chi_1)\,(r^{\eta}_{11})^2 \chi_1 ^ 2 \:+\,2/9\,h^{f'}(1,\chi_\eta,\chi_1,\chi_1;\chi_1)\,r^{\eta}_{11 } r^c_{1}\,\chi_1 ^ 2 \nonumber \\ & - & h^{f'}(1,\chi_\eta,\chi_{1s},\chi_{1s};\chi_1)\,\left [ 1/6\,r^{\eta}_{11}\,\chi_1 ^ 2 + 1/24\,r^v_{\eta 1s}\,\chi_\eta \chi_1 \right ] \:+\,h^{f'}(1,\chi_1,\chi_1,\chi_1;\chi_1)\,\left [ 1/6\,\chi_1 ^ 2 \right . \nonumber \\ & + & 1/9 \left . ( r^{c}_{1})^2 \chi_1 ^ 2 \right ] \:+\,h^{f'}(1,\chi_1,\chi_{1s},\chi_{1s};\chi_1)\,\left [ - 1/4\,r^{1}_{s\eta}\,\chi_1 ^ 2 + 1/8\,r^c_{1}\,\chi_1 ^ 2 - 1/24\,r^d_{1}\,\chi_1 \right ] \nonumber \\ & + & 3/8\,h^{f'}(1,\chi_{14},\chi_{14},\chi_4;\chi_1)\,\chi_1 \chi_4 \:+\,1/2\,h^{f'}(1,\chi_{14},\chi_{16},\chi_{46};\chi_1)\,\chi_1 \chi_{46 } \nonumber \\ & + & 2/9\,h^{f'}(2,\chi_1,\chi_\eta,\chi_1;\chi_1)\,r^{\eta}_{11 } r^d_{1}\,\chi_1 ^ 2 \:+\,2/9\,h^{f'}(2,\chi_1,\chi_1,\chi_1;\chi_1)\,r^c_{1 } r^d_{1}\,\chi_1 ^ 2 \nonumber \\ & + & 1/8\,h^{f'}(2,\chi_1,\chi_{1s},\chi_{1s};\chi_1)\,r^d_{1}\,\chi_1 ^ 2 \:+\,1/9\,h^{f'}(5,\chi_1,\chi_1,\chi_1;\chi_1)\,(r^{d}_{1})^2\,\chi_1 ^ 2 \nonumber \\ & - & 1/3\,h^{f'}_1(1,\chi_\eta,\chi_{1s},\chi_{1s};\chi_1)\,r^{\eta}_{1s } r^z_{1s\eta}\ , \chi_1 ^ 2 \:-\,2/3\,h^{f'}_1(1,\chi_{1s},\chi_{1s},\chi_1;\chi_1)\,r^{\eta}_{1s } r^z_{1s\eta}\ , \chi_1 ^ 2 \nonumber \\ & - & 2/3\,h^{f'}_1(3,\chi_{1s},\chi_1,\chi_{1s};\chi_1)\,r^d_{1}\,\chi_1 ^ 2 \:-\,1/8\,h^{f'}_{21}(1,\chi_\eta,\chi_{1s},\chi_{1s};\chi_1)\,r^v_{\eta 1s}\,\chi_1 ^ 2 \nonumber \\ & + & h^{f'}_{21}(1,\chi_1,\chi_{1s},\chi_{1s};\chi_1)\,\left [ 1/4\,r^{1}_{s\eta}\,\chi_1 ^ 2 - 1/8\,r^c_{1}\,\chi_1 ^ 2 \right ] \:+\,9/8\,h^{f'}_{21}(1,\chi_4,\chi_{14},\chi_{14};\chi_1)\,\chi_1 ^ 2 \nonumber \\ & + & 3/2\,h^{f'}_{21}(1,\chi_{46},\chi_{14},\chi_{16};\chi_1)\,\chi_1 ^ 2 \:-\,1/8\,h^{f'}_{21}(2,\chi_1,\chi_{1s},\chi_{1s};\chi_1)\,r^d_{1}\,\chi_1 ^ 2 , \label{f0p612loop } \\ & & \nonumber \\ \delta^{(6)13}_{\mathrm{loops } } & = & \pi_{16}\,l^r_{0}\,\left [ 4/9\,\chi_\pi \chi_\eta + 1/2\,\chi_1 ^ 2 - 13/3\,\bar\chi_{1 } \chi_1 - 35/18\,\bar\chi_{2 } \right ] \:-\,2\,\pi_{16}\,l^r_{1}\,\chi_1 ^ 2 \nonumber \\ & - & \pi_{16}\,l^r_{2}\,\left [ 11/3\,\chi_\pi \chi_\eta + \chi_1 ^ 2 + 13/3\,\bar\chi_{2 } \right ] \:+\,\pi_{16}\,l^r_{3}\,\left [ 4/9\,\chi_\pi \chi_\eta + 5/4\,\chi_1 ^ 2 - 17/6\,\bar\chi_{1 } \chi_1 - 43/36\,\bar\chi_{2 } \right ] \nonumber \\ & + & \pi_{16}^2\,\left [ - 15/64\,\chi_\pi \chi_\eta + 1/64\,\chi_1 ^ 2 - 1/2\,\bar\chi_{1 } \chi_1 - 43/128\,\bar\chi_{2 } \right ] \:-\,48\,l^r_{4}l^r_{5}\,\bar\chi_{1 } \chi_1 \:-\,72\,l^{r2}_{4}\,\bar\chi_{1}^2 \:-\,8\,l^{r2}_{5}\,\chi_1 ^ 2 \nonumber \\ & + & 4\,\bar{a}(\chi_m)\,l^r_{0}\,r^{m}_{n11}\,\chi_m \:+\,8\,\bar{a}(\chi_m)\,l^r_{1}\,\bar\chi^{m}_{n0}\,\chi_m \:+\,2\,\bar{a}(\chi_m)\,l^r_{2}\,\bar\chi^{m}_{n0}\,\chi_m \:+\,4\,\bar{a}(\chi_m)\,l^r_{3}\,r^{m}_{n11}\,\chi_m \nonumber \\ & - & 4\,\bar{a}(\chi_m)\,l^r_{4}\,\bar\chi^{m}_{n1 } \:-\,4/3\,\bar{a}(\chi_m)\,l^r_{5}\,r^{m}_{n11}\,\chi_1 \:-\,1/6\,\bar{a}(\chi_m ) \bar{b}(\chi_{1s},\chi_{1s},0)\,r^{m}_{n1s}\,\chi_{1s } \nonumber \\ & + & \bar{a}(\chi_m;\varepsilon)\,\pi_{16}\,\left [ - 7/24\,\bar\chi^{m}_{n0}\,\chi_m + 1/6\,\bar\chi^{m}_{n1 } - 1/8\,r^{m}_{n11}\,\chi_m \right ] \:+\,4\,\bar{a}(\chi_1)\,l^r_{0}\,\left [ r^c_{1}\,\chi_1 + r^d_{1 } \right ] \nonumber \\ & - & 4\,\bar{a}(\chi_1)\,l^r_{1}\,\chi_1 \:-\,10\,\bar{a}(\chi_1)\,l^r_{2}\,\chi_1 \:+\,4\,\bar{a}(\chi_1)\,l^r_{3}\,\left [ r^c_{1}\,\chi_1 + r^d_{1 } \right ] \:-\,4/3\,\bar{a}(\chi_1)\,l^r_{5}\,r^c_{1}\,\chi_1 \nonumber \\ & - & 1/6\,\bar{a}(\chi_1 ) \bar{b}(\chi_{1s},\chi_{1s},0)\,r^{1}_{s\pi\eta}\,\chi_{1s } \:+\,\bar{a}(\chi_1;\varepsilon)\,\pi_{16}\,\left [ 1/4\,\chi_1 - 1/8\,r^c_{1}\,\chi_1 - 1/8\,r^d_{1 } \right ] \nonumber \\ & - & \bar{a}(\chi_{1s})\,\pi_{16}\,\left [ 1/4\,\chi_{1s } + 3/8\,\bar\chi_{1 } \right ] \:-\,4\,\bar{a}(\chi_{1s})\,l^r_{0}\,\chi_{1s } \:-\,10\,\bar{a}(\chi_{1s})\,l^r_{3}\,\chi_{1s } \:-\,6\,\bar{a}(\chi_{1s})\,l^r_{4}\,\bar\chi_{1 } \nonumber \\ & + & 2\,\bar{a}(\chi_{1s})\,l^r_{5}\,\chi_1 \:+\,3/8\,\bar{a}(\chi_{1s};\varepsilon)\,\pi_{16}\,\left [ \chi_s + \bar\chi_{1 } \right ] \:+\,\bar{a}(\chi_s)\,l^r_{1}\,\left [ - 8\,\chi_s + 8/3\,r^c_{s}\,\chi_s \right ] \nonumber \\ & - & \bar{a}(\chi_s)\,l^r_{2}\,\left [ 2\,\chi_s - 2/3\,r^c_{s}\,\chi_s \right ] \:+\,\bar{a}(\chi_s)\,l^r_{4}\,\left [ 4\,\chi_s - 4/3\,r^c_{s}\,\chi_s \right ] \:+\,\bar{a}(\chi_s;\varepsilon)\,\pi_{16}\,\left [ 1/8\,\chi_s - 1/24\,r^c_{s}\,\chi_s \right ] \nonumber \\ & - & 16\,\bar{a}(\chi_{st})\,l^r_{1}\,\chi_{st } \:-\,4\,\bar{a}(\chi_{st})\,l^r_{2}\,\chi_{st } \:+\,8\,\bar{a}(\chi_{st})\,l^r_{4}\,\chi_{st } \:+\,1/4\,\bar{a}(\chi_{st};\varepsilon)\,\pi_{16}\,\chi_{st } \nonumber \\ & + & 4\,\bar{b}(\chi_1,\chi_1;0)\,l^r_{0}\,r^d_{1}\,\chi_1 \:+\,4\,\bar{b}(\chi_1,\chi_1;0)\,l^r_{3}\,r^d_{1}\,\chi_1 \:-\,4/3\,\bar{b}(\chi_1,\chi_1;0)\,l^r_{5}\,r^d_{1}\,\chi_1 \nonumber \\ & - & 1/8\,\bar{b}(\chi_1,\chi_1;0,\varepsilon)\,\pi_{16}\,r^d_{1}\,\chi_1 \:-\,12\,\bar{b}(\chi_{1s},\chi_{1s};0)\,l^r_{4}\,\bar\chi_{1}\,\chi_{1s } \:-\,4\,\bar{b}(\chi_{1s},\chi_{1s};0)\,l^r_{5}\,\chi_{1s}^2 \nonumber \\ & + & 24\,\bar{b}(\chi_{1s},\chi_{1s};0)\,l^r_{6}\,\bar\chi_{1}\,\chi_{1s } \:+\,8\,\bar{b}(\chi_{1s},\chi_{1s};0)\,l^r_{8}\,\chi_{1s}^2 \:+\,1/24\,h^{f}(1,\chi_m,\chi_{1s},\chi_{1s};\chi_1)\,r^v_{mn1s}\,\chi_m \nonumber \\ & + & h^{f}(1,\chi_1,\chi_{1s},\chi_{1s};\chi_1)\,\left [ - 1/12\,r^{1}_{s\pi\eta}\,\chi_1 + 1/24\,r^c_{1}\,\chi_1 + 1/24\,r^d_{1 } \right ] \:+\,h^{f}(1,\chi_{1s},\chi_{1s},\chi_s;\chi_1)\,\left [ - 1/8\,\chi_s \right . \nonumber \\ & + & 1/24 \left . r^c_{s}\,\chi_s \right ] \:-\,1/4\,h^{f}(1,\chi_{1s},\chi_{1t},\chi_{st};\chi_1)\,\chi_{st } \:+\,1/24\,h^{f}(2,\chi_1,\chi_{1s},\chi_{1s};\chi_1)\,r^d_{1}\,\chi_1 \nonumber \\ & + & 1/9\,h^{f'}(1,\chi_m,\chi_m,\chi_1;\chi_1)\,(r^{m}_{n11})^2 \chi_1 ^ 2 \:+\,2/9\,h^{f'}(1,\chi_m,\chi_1,\chi_1;\chi_1)\,r^{m}_{n11 } r^c_{1}\,\chi_1 ^ 2 \nonumber \\ & - & h^{f'}(1,\chi_m,\chi_{1s},\chi_{1s};\chi_1)\,\left [ 1/6\,r^{m}_{n11}\,\chi_1 ^ 2 + 1/24\,r^v_{mn1s}\,\chi_m \chi_1 \right ] \nonumber \\ & + & 2/9\,h^{f'}(1,\chi_\pi,\chi_\eta,\chi_1;\chi_1)\,r^{\pi}_{\eta 11 } r^{\eta}_{\pi 11 } \,\chi_1 ^ 2 \:+\,h^{f'}(1,\chi_1,\chi_1,\chi_1;\chi_1)\,\left [ 1/6\,\chi_1 ^ 2 + 1/9\,(r^{c}_{1})^2 \chi_1 ^ 2 \right ] \nonumber \\ & + & h^{f'}(1,\chi_1,\chi_{1s},\chi_{1s};\chi_1)\,\left [ - 1/4\,r^{1}_{s\pi\eta}\,\chi_1 ^ 2 + 1/8\,r^c_{1}\,\chi_1 ^ 2 - 1/24\,r^d_{1}\,\chi_1 \right ] \:+\ , h^{f'}(1,\chi_{1s},\chi_{1s},\chi_s;\chi_1)\,\left [ 1/8\,\chi_1 \chi_s \right . \nonumber \\ & - & 1/24 \left . r^c_{s}\,\chi_1 \chi_s \right ] \:+\,1/4\,h^{f'}(1,\chi_{1s},\chi_{1t},\chi_{st};\chi_1)\,\chi_1 \chi_{st } \:+\,2/9\,h^{f'}(2,\chi_1,\chi_m,\chi_1;\chi_1)\,r^{m}_{n11 } r^d_{1}\,\chi_1 ^ 2 \nonumber \\ & + & 2/9\,h^{f'}(2,\chi_1,\chi_1,\chi_1;\chi_1)\,r^c_{1 } r^d_{1}\,\chi_1 ^ 2 \:+\,1/8\,h^{f'}(2,\chi_1,\chi_{1s},\chi_{1s};\chi_1)\,r^d_{1}\,\chi_1 ^ 2 \nonumber \\ & + & 1/9\,h^{f'}(5,\chi_1,\chi_1,\chi_1;\chi_1)\,(r^{d}_{1})^2\,\chi_1 ^ 2 \:-\,1/3\,h^{f'}_1(1,\chi_m,\chi_{1s},\chi_{1s};\chi_1)\,r^{m}_{n1s}\,r^z_{1sm}\,\chi_1 ^ 2 \nonumber \\ & - & 2/3\,h^{f'}_1(1,\chi_{1s},\chi_{1s},\chi_1;\chi_1)\,r^{m}_{n1s } r^z_{1sm}\,\chi_1 ^ 2 \:-\,2/3\,h^{f'}_1(3,\chi_{1s},\chi_1,\chi_{1s};\chi_1)\,r^d_{1}\,\chi_1 ^ 2 \nonumber \\ & - & 1/8\,h^{f'}_{21}(1,\chi_m,\chi_{1s},\chi_{1s};\chi_1)\,r^v_{mn1s}\,\chi_1 ^ 2 \:+\,h^{f'}_{21}(1,\chi_1,\chi_{1s};\chi_{1s};\chi_1)\,\left [ 1/4\,r^{1}_{s\pi\eta}\,\chi_1 ^ 2 - 1/8\,r^c_{1}\,\chi_1 ^ 2 \right ] \nonumber \\ & + & h^{f'}_{21}(1,\chi_s,\chi_{1s},\chi_{1s};\chi_1)\,\left [ 3/8\,\chi_1 ^ 2 - 1/8\,r^c_{s } \,\chi_1 ^ 2 \right ] \:+\,3/4\,h^{f'}_{21}(1,\chi_{st},\chi_{1s},\chi_{1t};\chi_1)\,\chi_1 ^ 2 \nonumber \\ & - & 1/8\,h^{f'}_{21}(2,\chi_1,\chi_{1s},\chi_{1s};\chi_1)\,r^d_{1}\,\chi_1 ^ 2 . \label{f0p613loop } \end{aligned}\ ] ] in general , the expressions for @xmath6 are longer than those for @xmath89 , as the number of independent integrals that can appear is significantly larger . the full result for @xmath67 at nnlo is very large , and has not yet been worked out . consequently , the largest number of nondegenerate sea quarks considered for @xmath6 is @xmath76 . the nlo result for @xmath6 and @xmath76 , which agrees with the result of ref . @xcite is @xmath163 \:+\,1/4\,\bar{a}(\chi_{ps } ) \:-\,1/12\,\bar{a}(\chi_\eta)\,r^v_{\eta 13 } \nonumber \\ & - & 1/12\,\bar{b}(\chi_p,\chi_p,0)\,r^d_{p}. \label{f0p422 } \end{aligned}\ ] ] at nnlo , the contribution to the decay constant from the @xmath1 counterterms for @xmath6 is similar to that for @xmath89 . for @xmath122 , that contribution is @xmath164 the nnlo contributions to the decay constant from the chiral loops are , for @xmath165 and @xmath76 , respectively , @xmath166 \:-\,2\,\pi_{16}\,l^r_{1}\,\chi_{13}^2 \:-\,\pi_{16}\,l^r_{2}\,\left [ \chi_{13}^2 + 8\,\chi_4 ^ 2 \right ] \nonumber \\ & + & \pi_{16}\,l^r_{3}\,\left [ - 7/12\,\chi_1 \chi_3 - 17/6\,\chi_{13 } \chi_4 + 11/6\,\chi_{13}^2 - 3/4\,\chi_4 ^ 2 \right ] \:+\,\pi_{16}^2\,\left [ - 59/384\,\chi_1 \chi_3 - 1/2\,\chi_{13 } \chi_4 + 65/384\,\chi_{13}^2 \right . \nonumber \\ & - & 73/128 \left . \chi_4 ^ 2 \right ] \:-\,48\,l^r_{4}l^r_{5}\,\chi_{13 } \chi_4 \:-\,72\,l^{r2}_{4}\,\chi_4 ^ 2 \:-\,8\,l^{r2}_{5}\,\chi_{13}^2 \:+\,\bar{a}(\chi_p)\,\pi_{16}\,\left [ 5/96\,\chi_p + 1/32\,\chi_q + 1/48\,\chi_4 \right . \nonumber \\ & - & 1/12 \left . r^{p}_{q}\,\chi_p - 1/8\,r^{p}_{q}\,\chi_4 \right ] \:+\,\bar{a}(\chi_p)\,l^r_{0}\,\left [ 2/3\,\chi_p + 8/3\,r^{p}_{q}\,\chi_p + 2/3\,r^d_{p } \right ] \:+\,\bar{a}(\chi_p)\,l^r_{3}\,\left [ 5/3\,\chi_p + 2/3\,r^{p}_{q}\,\chi_p \right . \nonumber \\ & + & 5/3 \left . r^d_{p } \right ] \:+\,\bar{a}(\chi_p)\,l^r_{4}\,\left [ \chi_4 - 2\,r^{p}_{q}\,\chi_4 \right ] \:+\,\bar{a}(\chi_p)\,l^r_{5}\,\left [ 1/6\,\chi_p - 1/6\,\chi_q - 1/3\,\chi_4 - 2/3\,r^{p}_{q } \,\chi_p \right ] \nonumber \\ & + & \bar{a}(\chi_p)^2\,\left [ 19/288 - 1/72\,r^{p}_{q } r^{q}_{p } \right ] \:-\,\bar{a}(\chi_p ) \bar{a}(\chi_{p4})\,\left [ 1/16 + 1/12\,r^{p}_{q } \right ] \:-\,\bar{a}(\chi_p ) \bar{a}(\chi_{q4})\,\left [ 3/16 + 1/12\,r^{p}_{q } \right ] \nonumber \\ & + & 1/8\,\bar{a}(\chi_p ) \bar{a}(\chi_{13 } ) \:+\,\bar{a}(\chi_p ) \bar{b}(\chi_p,\chi_p;0)\,\left [ 11/36\,\chi_p - 1/18\,r^{p}_{q}\,\chi_p - 1/72\,r^{p}_{q } r^d_{p } + 1/144\,r^d_{p } \right ] \nonumber \\ & + & \bar{a}(\chi_p ) \bar{b}(\chi_q,\chi_q;0)\,\left [ - 1/72\,r^{p}_{q } r^d_{q } + 1/144\,r^d_{q } \right ] \:-\,1/4\,\bar{a}(\chi_p ) \bar{b}(\chi_{p4},\chi_{p4};0)\,\chi_{p4 } \nonumber \\ & - & 1/18\,\bar{a}(\chi_p ) \bar{b}(\chi_1,\chi_3;0)\,r^{q}_{p}\,\chi_p \:+\,1/18\,\bar{a}(\chi_p ) \bar{c}(\chi_p,\chi_p,\chi_p;0)\,r^d_{p}\,\chi_p \:+\,1/8\,\bar{a}(\chi_p;\varepsilon)\,\pi_{16}\,\left [ - \chi_p + r^{p}_{q}\,\chi_4 \right ] \nonumber \\ & + & \bar{a}(\chi_{p4})\,\pi_{16}\,\left [ 3/16\,\chi_{p4 } - 9/16\,\chi_{q4 } - 9/16\,\chi_4 \right ] \:-\,6\,\bar{a}(\chi_{p4})\,l^r_{0}\,\chi_{p4 } \:-\,15\,\bar{a}(\chi_{p4})\,l^r_{3}\,\chi_{p4 } \:-\,9\,\bar{a}(\chi_{p4})\,l^r_{4}\,\chi_4 \nonumber \\ & + & 3\,\bar{a}(\chi_{p4})\,l^r_{5}\,\chi_{13 } \:-\,9/32\,\bar{a}(\chi_{p4})^2 \:+\,\bar{a}(\chi_{p4 } ) \bar{b}(\chi_p,\chi_p;0)\,\left [ 1/8\,\chi_p - 5/8\,\chi_{p4 } \right ] \:-\,1/16\,\bar{a}(\chi_{p4 } ) \bar{b}(\chi_q,\chi_q;0)\,r^d_{q } \nonumber \\ & + & 1/6\,\bar{a}(\chi_{p4 } ) \bar{b}(\chi_1,\chi_3;0)\,\chi_4 \:+\,1/3\,\bar{a}(\chi_{p4 } ) \bar{b}(\chi_1,\chi_3;0,k ) \:+\,9/8\,\bar{a}(\chi_{p4};\varepsilon)\,\pi_{16}\,\chi_4 \:+\,\bar{a}(\chi_1 ) \bar{a}(\chi_3)\,\left [ - 1/144 \right . \nonumber \\ & + & 1/36 \left . r^{1}_{3 } r^{3}_{1 } \right ] \:-\,4\,\bar{a}(\chi_{13})\,l^r_{1}\,\chi_{13 } \:-\,10\,\bar{a}(\chi_{13})\,l^r_{2}\,\chi_{13 } \:+\,1/8\,\bar{a}(\chi_{13})^2 \:-\,1/2\,\bar{a}(\chi_{13 } ) \bar{b}(\chi_1,\chi_3;0,k ) \nonumber \\ & + & 1/4\,\bar{a}(\chi_{13};\varepsilon)\,\pi_{16}\,\chi_{13 } \:+\,9/16\,\bar{a}(\chi_{14 } ) \bar{a}(\chi_{34 } ) \:-\,64\,\bar{a}(\chi_4)\,l^r_{1}\,\chi_4 \:-\,16\,\bar{a}(\chi_4)\,l^r_{2}\,\chi_4 \:+\,32\,\bar{a}(\chi_4)\,l^r_{4}\,\chi_4 \nonumber \\ & + & 2/9\,\bar{a}(\chi_4 ) \bar{b}(\chi_p,\chi_p;0)\,\chi_4 \:-\,4/9\,\bar{a}(\chi_4 ) \bar{b}(\chi_1,\chi_3;0)\,\chi_4 \:+\,\bar{a}(\chi_4;\varepsilon)\,\pi_{16}\,\chi_4 \:+\,\bar{b}(\chi_p,\chi_p;0)\,\pi_{16}\,\left [ 1/96\,r^d_{p}\,\chi_p \right . \nonumber \\ & + & 1/32 \left . r^d_{p}\,\chi_q + 1/16\,r^d_{p}\,\chi_4 \right ] \:+\,2/3\,\bar{b}(\chi_p,\chi_p;0)\,l^r_{0}\,r^d_{p}\,\chi_p \:+\,5/3\,\bar{b}(\chi_p,\chi_p;0)\,l^r_{3}\,r^d_{p}\,\chi_p \nonumber \\ & + & \bar{b}(\chi_p,\chi_p;0)\,l^r_{4}\,\left [ 2\,\chi_p \chi_4 - 4\,r^{p}_{q}\,\chi_p \chi_4 + 3\,r^d_{p}\,\chi_4 \right ] \:+\,\bar{b}(\chi_p,\chi_p;0)\,l^r_{5}\,\left [ - 2/3\,\chi_p \chi_4 + 4/3\,\chi_p^2 - 4/3\,r^{p}_{q}\,\chi_p \chi_{13 } \right . \nonumber \\ & - & 1/3 \left . r^d_{p}\,\chi_{13 } \right ] \:+\,\bar{b}(\chi_p,\chi_p;0)\,l^r_{6}\,\left [ 4\,\chi_4 ^ 2 - 8\,r^{q}_{p}\,\chi_p \chi_4 \right ] \:+\,4\,\bar{b}(\chi_p,\chi_p;0)\,l^r_{7}\,(r^{d}_{p})^2 \:+\,\bar{b}(\chi_p,\chi_p;0)\,l^r_{8}\,\left [ 4/3\,\chi_4 ^ 2 \right . & - & 8/3 \left . r^{q}_{p}\,\chi_p^2 \right ] \:+\,\bar{b}(\chi_p,\chi_p;0)^2\,\left [ 1/18\,r^{q}_{p } r^d_{p}\,\chi_p + 1/288\,(r^{d}_{p})^2 \right ] \:-\,1/18\,\bar{b}(\chi_p,\chi_p;0 ) \bar{b}(\chi_1,\chi_3;0)\,r^{q}_{p } r^d_{p}\,\chi_p \nonumber \\ & + & 1/18\,\bar{b}(\chi_p,\chi_p;0 ) \bar{c}(\chi_p,\chi_p,\chi_p;0)\,(r^{d}_{p})^2\,\chi_p \:-\,1/8\,\bar{b}(\chi_p,\chi_p;0,\varepsilon)\,\pi_{16}\,r^d_{p}\,\chi_{p4 } \nonumber \\ & - & 18\,\bar{b}(\chi_{p4},\chi_{p4};0)\,l^r_{4}\,\chi_{p4 } \chi_4 \:-\,6\,\bar{b}(\chi_{p4},\chi_{p4};0)\,l^r_{5}\,\chi_{p4}^2 \:+\,36\,\bar{b}(\chi_{p4},\chi_{p4};0)\,l^r_{6}\,\chi_{p4 } \chi_4 \nonumber \\ & + & 12\,\bar{b}(\chi_{p4},\chi_{p4};0)\,l^r_{8}\,\chi_{p4}^2 \:+\,1/144\,\bar{b}(\chi_1,\chi_1;0 ) \bar{b}(\chi_3,\chi_3;0)\,r^d_{1 } r^d_{3 } \:-\,8\,\bar{b}(\chi_1,\chi_3;0)\,l^r_{7}\,r^d_{1 } r^d_{3 } \nonumber \\ & - & 8/3\,\bar{b}(\chi_1,\chi_3;0)\,l^r_{8}\,r^d_{1 } r^d_{3 } \:+\,4\,\bar{c}(\chi_p,\chi_p,\chi_p;0)\,l^r_{4}\,r^d_{p}\,\chi_p \chi_4 \:+\,4/3\,\bar{c}(\chi_p,\chi_p,\chi_p;0)\,l^r_{5}\,r^d_{p}\,\chi_p^2 \nonumber \\ & - & 8\,\bar{c}(\chi_p,\chi_p,\chi_p;0)\,l^r_{6}\,r^d_{p}\,\chi_p \chi_4 \:-\,8/3\,\bar{c}(\chi_p,\chi_p,\chi_p;0)\,l^r_{8}\,r^d_{p}\,\chi_p^2 \:+\,h^{f}(1,\chi_p,\chi_p,\chi_{13};\chi_{13})\,\left[1/8\,\chi_p \right . & - & 5/72 \left . \chi_{13 } + 1/36\,r^{p}_{q } r^{q}_{p}\,\chi_{13 } \right ] \:+\,h^{f}(1,\chi_p,\chi_{14},\chi_{34};\chi_{13})\,\left [ - 1/16\,\chi_p + 1/16\,\chi_q - 1/8\,r^{p}_{q}\,\chi_4 \right ] \nonumber \\ & + & h^{f}(1,\chi_1,\chi_{13},\chi_3;\chi_{13})\,\left [ 1/72\,\chi_{13 } - 1/18\,r^{1}_{3 } r^{3}_{1}\,\chi_{13 } \right ] \:-\,1/8\,h^{f}(1,\chi_{13},\chi_{13},\chi_{13};\chi_{13})\,\chi_{13 } \nonumber \\ & - & h^{f}(1,\chi_{14},\chi_{34},\chi_4;\chi_{13})\,\chi_4 \:+\,h^{f}(2,\chi_p,\chi_p,\chi_{13};\chi_{13})\,\left [ 1/36\,r^{p}_{q } r^d_{p}\,\chi_{13 } - 1/72\,r^d_{p}\,\chi_{13 } \right ] \nonumber \\ & + & h^{f}(2,\chi_p,\chi_{13},\chi_q;\chi_{13})\,\left [ 1/36\,r^{q}_{p } r^d_{p}\,\chi_{13 } - 1/72\,r^d_{p}\,\chi_{13 } \right ] \:+\,1/8\,h^{f}(2,\chi_p,\chi_{14},\chi_{34};\chi_{13})\,r^d_{p}\,\chi_{p4 } \nonumber \\ & - & 1/144\,h^{f}(5,\chi_p,\chi_p,\chi_{13};\chi_{13})\,(r^{d}_{p})^2\,\chi_{13 } \:-\,1/72\,h^{f}(5,\chi_1,\chi_3,\chi_{13};\chi_{13})\,r^d_{1 } r^d_{3}\,\chi_{13 } \nonumber \\ & + & h^{f'}(1,\chi_p,\chi_p,\chi_{13};\chi_{13})\,\left [ - 1/8\,\chi_p \chi_{13 } - 5/24\,\chi_{13}^2 + 1/12\,r^{p}_{q}\,\chi_{13}^2 - 1/36\,(r^{p}_{q})^2\,\chi_{13}^2 \right ] \nonumber \\ & + & h^{f'}(1,\chi_p,\chi_{14},\chi_{34};\chi_{13})\,\left [ 3/16\,\chi_p \chi_{13 } - 1/16\,\chi_{13 } \chi_4 - 1/8\,r^{p}_{q } r^d_{p}\,\chi_{13 } + 3/8\,r^{q}_{p}\,\chi_{13}^2 \right ] \nonumber \\ & + & h^{f'}(1,\chi_1,\chi_{13},\chi_3;\chi_{13})\,\left [ - 1/72\,\chi_{13}^2 + 1/6\,r^{1}_{3 } r^{3}_{1}\,\chi_{13}^2 \right ] \:+\,1/8\,h^{f'}(1,\chi_{13},\chi_{13},\chi_{13};\chi_{13})\,\chi_{13}^2 \nonumber \\ & + & h^{f'}(1,\chi_{14},\chi_{34},\chi_4;\chi_{13})\,\chi_{13 } \chi_4 \:+\,h^{f'}(2,\chi_p,\chi_p,\chi_{13};\chi_{13})\,\left [ - 1/36\,r^{p}_{q } r^d_{p}\,\chi_{13}^2 + 5/72\,r^d_{p}\,\chi_{13}^2 \right ] \nonumber \\ & + & h^{f'}(2,\chi_p,\chi_{13},\chi_q;\chi_{13})\,\left [ - 1/36\,r^{q}_{p } r^d_{p}\,\chi_{13}^2 + 1/72\,r^d_{p}\,\chi_{13}^2 \right ] \:-\,1/8\,h^{f'}(2,\chi_p,\chi_{14},\chi_{34};\chi_{13})\,r^d_{p}\,\chi_{p4 } \chi_{13 } \nonumber \\ & + & 5/144\,h^{f'}(5,\chi_p,\chi_p,\chi_{13};\chi_{13})\,(r^{d}_{p})^2\,\chi_{13}^2 \:+\,1/72\,h^{f'}(5,\chi_1,\chi_3,\chi_{13};\chi_{13})\,r^d_{1 } r^d_{3}\,\chi_{13}^2 \nonumber \\ & + & h^{f'}_1(1,\chi_p,\chi_p,\chi_{13};\chi_{13})\,\left [ 10/9\,\chi_{13}^2 - 2/9\,r^{p}_{q } r^{q}_{p}\,\chi_{13}^2 \right ] \:-\,h^{f'}_1(1,\chi_p,\chi_{14},\chi_{34};\chi_{13})\,r^{q}_{p}\,\chi_{13}^2 \nonumber \\ & - & h^{f'}_1(1,\chi_{p4},\chi_{q4},\chi_p;\chi_{13})\,\chi_{13}^2 \:+\,h^{f'}_1(1,\chi_{13},\chi_1,\chi_3;\chi_{13})\,\left [ 1/9\,\chi_{13}^2 - 2/9\,r^{1}_{3 } r^{3}_{1}\,\chi_{13}^2 \right ] \nonumber \\ & + & h^{f'}_1(3,\chi_{13},\chi_p,\chi_p;\chi_{13})\,\left [ 1/9\,r^{p}_{q } r^d_{p}\,\chi_{13}^2 - 1/9\,r^d_{p}\,\chi_{13}^2 \right ] \:+\,1/9\,h^{f'}_1(3,\chi_{13},\chi_p,\chi_q;\chi_{13})\,r^{q}_{p } r^d_{p}\,\chi_{13}^2 \nonumber \\ & - & 1/18\,h^{f'}_1(7,\chi_{13},\chi_p,\chi_p;\chi_{13})\,(r^{d}_{p})^2\,\chi_{13}^2 \:-\,3/8\,h^{f'}_{21}(1,\chi_p,\chi_p,\chi_{13};\chi_{13})\,\chi_{13}^2 \nonumber \\ & + & 3/8\,h^{f'}_{21}(1,\chi_p,\chi_{14},\chi_{34};\chi_{13})\,r^{q}_{p}\,\chi_{13}^2 \:+\,3/8\,h^{f'}_{21}(1,\chi_{p4},\chi_p,\chi_{q4};\chi_{13})\,r^{p}_{q}\,\chi_{13}^2 \nonumber \\ & - & 3/8\,h^{f'}_{21}(1,\chi_{p4},\chi_q,\chi_{q4};\chi_{13})\,r^{p}_{q}\,\chi_{13}^2 \:+\,h^{f'}_{21}(1,\chi_{13},\chi_p,\chi_p;\chi_{13})\,\left [ 5/24\,\chi_{13}^2 - 1/12\,r^{p}_{q } r^{q}_{p}\,\chi_{13}^2 \right ] \nonumber \\ & + & h^{f'}_{21}(1,\chi_{13},\chi_1,\chi_3;\chi_{13})\,\left [ - 1/24\,\chi_{13}^2 + 1/6\,r^{1}_{3 } r^{3}_{1}\,\chi_{13}^2 \right ] \:+\,3/8\,h^{f'}_{21}(1,\chi_{13},\chi_{13},\chi_{13};\chi_{13})\,\chi_{13}^2 \nonumber \\ & + & 3\,h^{f'}_{21}(1,\chi_4,\chi_{14},\chi_{34};\chi_{13})\,\chi_{13}^2 \:-\,3/8\,h^{f'}_{21}(3,\chi_{p4},\chi_p,\chi_{q4};\chi_{13})\,r^d_{p}\,\chi_{13}^2 \nonumber \\ & + & h^{f'}_{21}(3,\chi_{13},\chi_p,\chi_p;\chi_{13})\,\left [ - 1/12\,r^{p}_{q } r^d_{p } \,\chi_{13}^2 + 1/24\,r^d_{p}\,\chi_{13}^2 \right ] \:+\,h^{f'}_{21}(3,\chi_{13},\chi_p,\chi_q;\chi_{13})\,\left [ - 1/12\,r^{q}_{p } r^d_{p } \,\chi_{13}^2 \right . \nonumber \\ & + & 1/24 \left . r^d_{p}\,\chi_{13}^2 \right ] \:+\,1/48\,h^{f'}_{21}(7,\chi_{13},\chi_p,\chi_p;\chi_{13})\,(r^{d}_{p})^2\,\chi_{13}^2 \:+\,1/24\,h^{f'}_{21}(7,\chi_{13},\chi_1,\chi_3;\chi_{13})\,r^d_{1 } r^d_{3}\,\chi_{13}^2 , \\ \label{f0p621loop } & & \nonumber \\ \delta^{(6)22}_{\mathrm{loops } } & = & \pi_{16}\,l^r_{0}\,\left [ 4/9\,\chi_\eta \chi_4 - 1/2\,\chi_1 \chi_3 + \chi_{13}^2 - 13/3\,\bar\chi_{1 } \chi_{13 } - 35/18\,\bar\chi_{2 } \right ] \:-\,2\,\pi_{16}\,l^r_{1}\,\chi_{13}^2 \nonumber \\ & - & \pi_{16}\,l^r_{2}\,\left [ 11/3\,\chi_\eta \chi_4 + \chi_{13}^2 + 13/3\,\bar\chi_{2 } \right ] + \pi_{16}\,l^r_{3}\,\left [ 4/9\,\chi_\eta \chi_4 - 7/12\,\chi_1 \chi_3 + 11/6\,\chi_{13}^2 - 17/6\,\bar\chi_{1 } \chi_{13 } - 43/36\,\bar\chi_{2 } \right ] \nonumber \\ & + & \pi_{16}^2\,\left [ - 15/64\,\chi_\eta \chi_4 - 59/384\,\chi_1 \chi_3 + 65/384\,\chi_{13}^2 - 1/2\,\bar\chi_{1 } \chi_{13 } - 43/128\,\bar\chi_{2 } \right ] \:-\,48\,l^r_{4}l^r_{5}\,\bar\chi_{1 } \chi_{13 } \:-\,72\,l^{r2}_{4}\,\bar\chi_{1}^2 \nonumber \\ & - & 8\,l^{r2}_{5}\,\chi_{13}^2 \:+\,\bar{a}(\chi_p)\,\pi_{16}\,\left [ - 1/24\,\chi_p + 1/48\,\bar\chi_{1 } - 1/8\,\bar\chi_{1}\,r^{p}_{q\eta } + 1/16\,\bar\chi_{1}\,r^c_{p } - 1/48\,r^{p}_{q\eta}\,\chi_p - 1/16\,r^{p}_{q\eta}\,\chi_q \right.\nonumber \\ & + & \left . 1/48\,r^{\eta}_{pp}\,\chi_\eta + 1/16\,r^c_{p}\,\chi_{13 } \right ] \:+\,\bar{a}(\chi_p)\,l^r_{0}\,\left [ 8/3\,r^{p}_{q\eta}\,\chi_p + 2/3\,r^c_{p}\,\chi_p + 2/3\,r^d_{p } \right ] \:+\,\bar{a}(\chi_p)\,l^r_{3}\,\left [ 2/3\,r^{p}_{q\eta}\,\chi_p \right . \nonumber \\ & + & \left . 5/3\,r^c_{p}\,\chi_p + 5/3\,r^d_{p } \right ] \:+\,\bar{a}(\chi_p)\,l^r_{4}\,\left [ - 2\,\bar\chi_{1 } \bar\chi^{pp}_{\eta\eta 0 } - 2\,\bar\chi_{1}\,r^{p}_{q\eta } + 3\,\bar\chi_{1}\,r^c_{p } \right ] \:+\ , \bar{a}(\chi_p)\,l^r_{5}\,\left [ - 2/3\,\bar\chi^{pp}_{\eta\eta 1 } - r^{p}_{q\eta}\,\chi_p \right . \nonumber \\ & + & \left . 1/3\,r^{p}_{q\eta}\,\chi_q + 1/2\,r^c_{p}\,\chi_p - 1/6\,r^c_{p}\,\chi_q \right ] \:+\ , \bar{a}(\chi_p)^2\,\left [ 1/16 + 1/72\,(r^{p}_{q\eta})^2 - 1/72\,r^{p}_{q\eta } r^c_{p } + 1/288\,(r^{c}_{p})^2 \right ] \nonumber \\ & + & \bar{a}(\chi_p ) \bar{a}(\chi_{ps})\,\left [ - 1/36\,r^{p}_{q\eta } - 5/72\,r^{p}_{s\eta } + 7/144\,r^c_{p } \right ] \:-\,\bar{a}(\chi_p ) \bar{a}(\chi_{qs})\,\left [ 1/36\,r^{p}_{q\eta } + 1/24\,r^{p}_{s\eta } + 1/48\,r^c_{p } \right ] \nonumber \\ & + & \bar{a}(\chi_p ) \bar{a}(\chi_\eta)\,\left [ - 1/72\,r^{p}_{q\eta } r^v_{\eta 13 } + 1/144\,r^c_{p } r^v_{\eta 13 } \right ] \:+\,1/8\,\bar{a}(\chi_p ) \bar{a}(\chi_{13 } ) \:+\,1/12\,\bar{a}(\chi_p ) \bar{a}(\chi_{46})\,r^{\eta}_{pp } \nonumber \\ & + & \bar{a}(\chi_p ) \bar{b}(\chi_p,\chi_p;0)\,\left [ 1/4\,\chi_p - 1/18\,r^{p}_{q\eta } r^c_{p}\,\chi_p - 1/72\,r^{p}_{q\eta } r^d_{p } + 1/18\,(r^{c}_{p})^2\,\chi_p + 1/144\,r^c_{p } r^d_{p } \right ] \nonumber \\ & + & \bar{a}(\chi_p ) \bar{b}(\chi_p,\chi_\eta;0)\,\left [ 1/18\,r^{\eta}_{pp } r^c_{p}\,\chi_p - 1/18\,r^{\eta}_{13 } r^c_{p}\,\chi_p \right ] \:+\,\bar{a}(\chi_p ) \bar{b}(\chi_q,\chi_q;0)\left [ - 1/72\,r^{p}_{q\eta } r^d_{q } + 1/144\,r^c_{p } r^d_{q } \right ] \nonumber \\ & - & 1/12\,\bar{a}(\chi_p ) \bar{b}(\chi_{ps},\chi_{ps};0)\,r^{p}_{s\eta}\,\chi_{ps } \:-\,1/18\,\bar{a}(\chi_p ) \bar{b}(\chi_1,\chi_3;0)\,r^{q}_{p\eta } r^c_{p}\,\chi_p \nonumber \\ & + & 1/18\,\bar{a}(\chi_p ) \bar{c}(\chi_p,\chi_p,\chi_p;0)\,r^c_{p } r^d_{p}\,\chi_p \:+\,\bar{a}(\chi_p;\varepsilon)\,\pi_{16}\,\left [ 1/8\,\bar\chi_{1 } r^{p}_{q\eta } - 1/16 \bar\chi_{1}\,r^c_{p } - 1/16\,r^c_{p}\,\chi_p - 1/16\,r^d_{p } \right ] \nonumber \\ & + & \bar{a}(\chi_{ps})\,\pi_{16}\,\left [ 1/16\,\chi_{ps } - 3/16\,\chi_{qs } - 3/16\,\bar\chi_{1 } \right ] \:-\,2\,\bar{a}(\chi_{ps})\,l^r_{0}\,\chi_{ps } \:-\,5\,\bar{a}(\chi_{ps})\,l^r_{3}\,\chi_{ps } \:-\,3\,\bar{a}(\chi_{ps})\,l^r_{4}\,\bar\chi_{1 } \nonumber \\ & + & \bar{a}(\chi_{ps})\,l^r_{5}\,\chi_{13 } \:+\,\bar{a}(\chi_{ps } ) \bar{a}(\chi_\eta)\,\left [ 7/144\,r^{\eta}_{pp } - 5/72\,r^{\eta}_{ps } - 1/48\,r^{\eta}_{qq } + 5/72\,r^{\eta}_{qs } - 1/36\,r^{\eta}_{13 } \right ] \nonumber \\ & + & \bar{a}(\chi_{ps } ) \bar{b}(\chi_p,\chi_p;0)\,\left[1/24\,r^{p}_{s\eta}\,\chi_p - 5/24\,r^{p}_{s\eta}\,\chi_{ps } \right ] \:+\,\bar{a}(\chi_{ps } ) \bar{b}(\chi_p,\chi_\eta;0)\,\left [ - 1/18\,r^{\eta}_{ps } r^z_{qp\eta}\,\chi_p \right . \nonumber \\ & - & 1/9 \left . r^{\eta}_{ps } r^z_{qp\eta}\,\chi_{ps } \right ] \:-\,1/48\,\bar{a}(\chi_{ps } ) \bar{b}(\chi_q,\chi_q;0)\,r^d_{q } \:+\,1/18\,\bar{a}(\chi_{ps } ) \bar{b}(\chi_1,\chi_3;0)\,r^{q}_{s\eta}\,\chi_s \nonumber \\ & + & 1/9\,\bar{a}(\chi_{ps } ) \bar{b}(\chi_1,\chi_3;0,k)\,r^{q}_{s\eta } \:+\,3/16\,\bar{a}(\chi_{ps};\varepsilon)\,\pi_{16}\,\left[\chi_s + \bar\chi_{1 } \right ] \:-\,1/8\,\bar{a}(\chi_{p4})^2 \:-\,1/8\,\bar{a}(\chi_{p4 } ) \bar{a}(\chi_{p6 } ) \nonumber \\ & + & 1/8\,\bar{a}(\chi_{p4 } ) \bar{a}(\chi_{q6 } ) \:-\,1/32\,\bar{a}(\chi_{p6})^2 \:+\,\bar{a}(\chi_\eta)\,\pi_{16}\,\left [ 1/16\,\bar\chi_{1}\,r^v_{\eta 13 } - 1/48\,r^v_{\eta 13}\,\chi_\eta + 1/16\,r^v_{\eta 13}\,\chi_{13 } \right ] \nonumber \\ & + & \bar{a}(\chi_\eta)\,l^r_{0}\,\left [ 4 r^{\eta}_{13}\,\chi_\eta + 2/3\,r^v_{\eta 13}\,\chi_\eta \right ] \nonumber \:-\,8\,\bar{a}(\chi_\eta)\,l^r_{1}\,\chi_\eta \:-\,2\,\bar{a}(\chi_\eta)\,l^r_{2}\,\chi_\eta \:+\,\bar{a}(\chi_\eta)\,l^r_{3}\,\left [ 4 r^{\eta}_{13}\,\chi_\eta + 5/3\,r^v_{\eta 13 } \chi_\eta \right ] \nonumber \\ & + & \bar{a}(\chi_\eta)\,l^r_{4}\,\left [ 4\,\chi_\eta + \bar\chi_{1}\,r^v_{\eta 13 } \right ] \:-\,\bar{a}(\chi_\eta)\,l^r_{5}\,\left [ 1/6\,r^{\eta}_{pp}\,\chi_q + r^{\eta}_{13}\,\chi_{13 } + 1/6\,r^v_{\eta 13}\,\chi_\eta \right ] \:+\,1/288\,\bar{a}(\chi_\eta)^2\,(r^v_{\eta 13})^2 \nonumber \\ & + & 1/12\,\bar{a}(\chi_\eta ) \bar{a}(\chi_{46})\,r^v_{\eta 13 } \:+\,\bar{a}(\chi_\eta ) \bar{b}(\chi_p,\chi_p;0)\,\left [ - 1/36\,\bar\chi^{pp\eta}_{\eta\eta 1 } - 1/18\,r^{p}_{q\eta } r^{\eta}_{pp}\,\chi_p + 1/18\,r^{\eta}_{pp } r^c_{p}\,\chi_p \right . \nonumber \\ & + & 1/144 \left . r^d_{p } r^v_{\eta 13 } \right ] \:+\,\bar{a}(\chi_\eta ) \bar{b}(\chi_p,\chi_\eta;0)\,\left [ - 1/18\,\bar\chi^{\eta p\eta}_{p\eta 1 } + 1/18\,\bar\chi^{\eta p\eta}_{q\eta 1 } + 1/18\,(r^{\eta}_{pp})^2 r^z_{qp\eta}\,\chi_p \right ] \nonumber \\ & - & 1/12\,\bar{a}(\chi_\eta ) \bar{b}(\chi_{ps},\chi_{ps};0)\,r^{\eta}_{ps}\,\chi_{ps } \:-\,\bar{a}(\chi_\eta ) \bar{b}(\chi_\eta,\chi_\eta;0)\,\left [ 1/216\,r^v_{\eta 13}\,\chi_4 + 1/27\,r^v_{\eta 13}\,\chi_6 \right ] \nonumber \\ & - & 1/18\,\bar{a}(\chi_\eta ) \bar{b}(\chi_1,\chi_3;0)\,r^1_{\eta\eta } r^3_{\eta\eta}\,\chi_\eta \:+\,1/18\,\bar{a}(\chi_\eta ) \bar{c}(\chi_p,\chi_p,\chi_p;0)\,r^{\eta}_{pp } r^d_{p}\,\chi_p \:+\,\bar{a}(\chi_\eta;\varepsilon)\,\pi_{16}\,\left [ 1/8\,\chi_\eta \right . \nonumber \\ & - & 1/16\,\bar\chi_{1}\,r^v_{\eta 13 } - 1/8\,r^{\eta}_{13}\,\chi_\eta - \left . 1/16\,r^v_{\eta 13}\,\chi_\eta \right ] \:+\,\bar{a}(\chi_1 ) \bar{a}(\chi_3)\left [ - 1/72\,r^{p}_{q\eta } r^c_{q } + 1/36\,r^{1}_{3\eta } r^{3}_{1\eta } + 1/144\,r^c_{1 } r^c_{3 } \right ] \nonumber \\ & - & 4\,\bar{a}(\chi_{13})\,l^r_{1}\,\chi_{13 } \:-\,10\,\bar{a}(\chi_{13})\,l^r_{2}\,\chi_{13 } \:+\,1/8\,\bar{a}(\chi_{13})^2 \:-\,1/2\,\bar{a}(\chi_{13 } ) \bar{b}(\chi_1,\chi_3;0,k ) \nonumber \\ & + & 1/4\,\bar{a}(\chi_{13};\varepsilon)\,\pi_{16}\,\chi_{13 } \:+\,1/4\,\bar{a}(\chi_{14 } ) \bar{a}(\chi_{34 } ) \:+\,1/16\,\bar{a}(\chi_{16 } ) \bar{a}(\chi_{36 } ) \:-\,24\,\bar{a}(\chi_4)\,l^r_{1}\,\chi_4 \:-\,6\,\bar{a}(\chi_4)\,l^r_{2}\,\chi_4 \nonumber \\ & + & 12\,\bar{a}(\chi_4)\,l^r_{4}\,\chi_4 \:+\,1/12\,\bar{a}(\chi_4 ) \bar{b}(\chi_p,\chi_p;0)\,(r^{p}_{4\eta})^2\,\chi_4 \:+\,1/6\,\bar{a}(\chi_4 ) \bar{b}(\chi_p,\chi_\eta;0)\,\left [ r^{p}_{4\eta } r^{\eta}_{p4}\,\chi_4 - \,r^{p}_{4\eta } r^{\eta}_{q4}\,\chi_4 \right ] \nonumber \\ & - & 1/24\,\bar{a}(\chi_4 ) \bar{b}(\chi_\eta,\chi_\eta;0)\,r^v_{\eta 13}\,\chi_4 \:-\,1/6\,\bar{a}(\chi_4 ) \bar{b}(\chi_1,\chi_3;0)\,r^{1}_{4\eta } r^{3}_{4\eta}\,\chi_4 \:+\,3/8\,\bar{a}(\chi_4;\varepsilon)\,\pi_{16}\,\chi_4 \nonumber \\ & - & 32\,\bar{a}(\chi_{46})\,l^r_{1}\,\chi_{46 } \:-\,8\,\bar{a}(\chi_{46})\,l^r_{2}\,\chi_{46 } \:+\,16\,\bar{a}(\chi_{46})\,l^r_{4}\,\chi_{46 } \:+\,\bar{a}(\chi_{46 } ) \bar{b}(\chi_p,\chi_p;0)\,\left [ 1/9\,\chi_{46 } + 1/12\,r^{\eta}_{pp } \,\chi_p \right . \nonumber \\ & + & 1/36\,r^{\eta}_{pp}\,\chi_4 + \left . 1/9\,r^{\eta}_{p4}\,\chi_6 \right ] \:+\,\bar{a}(\chi_{46 } ) \bar{b}(\chi_p,\chi_\eta;0)\,\left [ - 1/18\,r^{\eta}_{pp}\,\chi_4 - 1/9\,r^{\eta}_{p4}\,\chi_6 + 1/9\,r^{\eta}_{q4}\,\chi_6 + 1/18\,r^{\eta}_{13}\,\chi_4 \right ] \nonumber \\ & - & 1/6\,\bar{a}(\chi_{46 } ) \bar{b}(\chi_p,\chi_\eta;0,k)\,\left [ r^{\eta}_{pp } - r^{\eta}_{13 } \right ] \:+\,1/9\,\bar{a}(\chi_{46 } ) \bar{b}(\chi_\eta,\chi_\eta;0)\,r^v_{\eta 13}\,\chi_{46 } \:-\,\bar{a}(\chi_{46 } ) \bar{b}(\chi_1,\chi_3;0)\,\left [ 2/9\,\chi_{46 } \right . \nonumber \\ & + & 1/9\,r^{\eta}_{p4}\,\chi_6 + \left . 1/18\,r^{\eta}_{13}\,\chi_4 \right ] \:-\,1/6\,\bar{a}(\chi_{46 } ) \bar{b}(\chi_1,\chi_3;0,k)\,r^{\eta}_{13 } \:+\,1/2\,\bar{a}(\chi_{46};\varepsilon)\,\pi_{16}\,\chi_{46 } \nonumber \\ & + & \bar{b}(\chi_p,\chi_p;0)\,\pi_{16}\,\left [ 1/16\,\bar\chi_{1}\,r^d_{p } + 1/96\,r^d_{p } \,\chi_p + 1/32\,r^d_{p}\,\chi_q \right ] \:+\,2/3\,\bar{b}(\chi_p,\chi_p;0)\,l^r_{0}\,r^d_{p}\,\chi_p \nonumber \\ & + & 5/3\,\bar{b}(\chi_p,\chi_p;0)\,l^r_{3}\,r^d_{p}\,\chi_p \:+\,\bar{b}(\chi_p,\chi_p;0)\,l^r_{4}\,\left [ - 2\,\bar\chi_{1 } \bar\chi^{pp}_{\eta\eta 0 } \chi_p - 4\,\bar\chi_{1 } r^{p}_{q\eta}\,\chi_p + 4\,\bar\chi_{1 } r^c_{p}\,\chi_p + 3\,\bar\chi_{1 } r^d_{p } \right ] \nonumber \\ & + & \bar{b}(\chi_p,\chi_p;0)\,l^r_{5}\,\left [ - 2/3 \bar\chi^{pp}_{\eta\eta 1}\chi_p - 4/3\,r^{p}_{q\eta}\,\chi_p^2 + 4/3\,r^c_{p}\,\chi_p^2 + 1/2\,r^d_{p}\,\chi_p - 1/6\,r^d_{p}\,\chi_q \right ] \nonumber \\ & + & \bar{b}(\chi_p,\chi_p;0)\,l^r_{6}\,\left [ 4\,\bar\chi_{1 } \bar\chi^{pp}_{\eta\eta 1 } + 8\,\bar\chi_{1 } r^{p}_{q\eta}\,\chi_p - 8\,\bar\chi_{1 } r^c_{p}\,\chi_p \right ] \:+\,4\,\bar{b}(\chi_p,\chi_p;0)\,l^r_{7}\,(r^{d}_{p})^2 \nonumber \\ & + & \bar{b}(\chi_p,\chi_p;0)\,l^r_{8}\,\left [ 4/3\,\bar\chi^{pp}_{\eta\eta 2 } + 8/3\,r^{p}_{q\eta}\,\chi_p^2 - 8/3\,r^c_{p}\,\chi_p^2 \right ] \:+\,\bar{b}(\chi_p,\chi_p;0)^2\,\left [ - 1/18\,r^{p}_{q\eta } r^d_{p}\,\chi_p + 1/18\,r^c_{p } r^d_{p}\,\chi_p \right . \nonumber \\ & + & 1/288 \left . ( r^{d}_{p})^2 \right ] \:+\,1/18\,\bar{b}(\chi_p,\chi_p;0 ) \bar{b}(\chi_p,\chi_\eta;0)\,\left [ r^{\eta}_{pp } r^d_{p}\,\chi_p - r^{\eta}_{13 } r^d_{p}\,\chi_p \right ] \nonumber \\ & - & 1/18\,\bar{b}(\chi_p,\chi_p;0 ) \bar{b}(\chi_1,\chi_3;0)\,r^{q}_{p\eta } r^d_{p}\,\chi_p \:+\,1/18\,\bar{b}(\chi_p,\chi_p;0 ) \bar{c}(\chi_p,\chi_p,\chi_p;0)\,(r^{d}_{p})^2\,\chi_p \nonumber \\ & - & 1/16\,\bar{b}(\chi_p,\chi_p;0,\varepsilon)\,\pi_{16}\,\left [ \bar\chi_{1 } r^d_{p } + r^d_{p}\,\chi_p \right ] \:+\,8\,\bar{b}(\chi_p,\chi_\eta;0)\,l^r_{7}\,r^z_{qp\eta } r^d_{p } r^z_{\eta 46p } \nonumber \\ & + & 8/3\,\bar{b}(\chi_p,\chi_\eta;0)\,l^r_{8}\,r^z_{qp\eta } r^d_{p } r^z_{\eta 46p } \:-\,6\,\bar{b}(\chi_{ps},\chi_{ps};0)\,l^r_{4}\,\bar\chi_{1 } \chi_{ps } \:-\,2\,\bar{b}(\chi_{ps},\chi_{ps};0)\,l^r_{5}\,\chi_{ps}^2 \nonumber \\ & + & 12\,\bar{b}(\chi_{ps},\chi_{ps};0)\,l^r_{6}\,\bar\chi_{1 } \chi_{ps } \:+\,4\,\bar{b}(\chi_{ps},\chi_{ps};0)\,l^r_{8}\,\chi_{ps}^2 \:+\,2\,\bar{b}(\chi_\eta,\chi_\eta;0)\,l^r_{4}\,\bar\chi_{1 } r^v_{\eta 13}\,\chi_\eta \nonumber \\ & + & 2/3\,\bar{b}(\chi_\eta,\chi_\eta;0)\,l^r_{5}\,r^v_{\eta 13}\,\chi_\eta^2 \:-\,4\,\bar{b}(\chi_\eta,\chi_\eta;0)\,l^r_{6}\,\bar\chi_{1 } r^v_{\eta 13}\,\chi_\eta \:+\,4\,\bar{b}(\chi_\eta,\chi_\eta;0)\,l^r_{7}\,r^z_{311\eta\eta}(r^z_{\eta 461})^2 \nonumber \\ & - & \bar{b}(\chi_\eta,\chi_\eta;0)\,l^r_{8}\,\left [ 4/9\,r^v_{\eta 13}\,\chi_4 ^ 2 + 8/9\,r^v_{\eta 13}\,\chi_6 ^ 2 \right ] \:+\,1/144\,\bar{b}(\chi_1,\chi_1;0 ) \bar{b}(\chi_3,\chi_3;0)\,r^d_{1 } r^d_{3 } \nonumber \\ & - & 8\,\bar{b}(\chi_1,\chi_3;0)\,l^r_{7}\,r^d_{1 } r^d_{3 } \:-\,8/3\,\bar{b}(\chi_1,\chi_3;0)\,l^r_{8}\,r^d_{1 } r^d_{3 } \:+\,4\,\bar{c}(\chi_p,\chi_p,\chi_p;0)\,l^r_{4}\,\bar\chi_{1 } r^d_{p}\,\chi_p \nonumber \\ & + & 4/3\,\bar{c}(\chi_p,\chi_p,\chi_p;0)\,l^r_{5}\,r^d_{p}\,\chi_p^2 \:-\,8\,\bar{c}(\chi_p,\chi_p,\chi_p;0)\,l^r_{6}\,\bar\chi_{1 } r^d_{p}\,\chi_p \:-\,8/3\,\bar{c}(\chi_p,\chi_p,\chi_p;0)\,l^r_{8}\,r^d_{p}\,\chi_p^2 \nonumber \\ & + & h^{f}(1,\chi_p,\chi_p,\chi_{13};\chi_{13})\,\left [ 1/8\,\chi_p - 1/16\,\chi_{13 } - 1/36\,(r^{p}_{q\eta})^2\,\chi_{13 } + 1/36\,r^{p}_{q\eta } r^c_{p}\,\chi_{13 } - 1/144\,(r^{c}_{p})^2\,\chi_{13 } \right ] \nonumber \\ & + & h^{f}(1,\chi_p,\chi_{1s},\chi_{3s};\chi_{13})\,\left [ - 1/12\,r^{p}_{q\eta}\,\chi_{qs } + 1/24\,r^{p}_{q\eta}\,\chi_{13 } - 1/16\,r^{p}_{s\eta}\,\chi_p + 1/48\,r^{p}_{s\eta}\,\chi_q + 1/24\,r^c_{p}\,\chi_{ps } \right ] \nonumber \\ & - & 1/4\,h^{f}(1,\chi_{p4},\chi_{q6},\chi_{46};\chi_{13})\,\chi_{46 } \:+\,h^{f}(1,\chi_\eta,\chi_p,\chi_{13};\chi_{13})\,\left [ 1/36\,r^{p}_{q\eta } r^v_{\eta 13 } \,\chi_{13 } - 1/72\,r^c_{p } r^v_{\eta 13}\,\chi_{13 } \right ] \nonumber \\ & - & 1/144\,h^{f}(1,\chi_\eta,\chi_\eta,\chi_{13};\chi_{13})\,(r^v_{\eta 13})^2\,\chi_{13 } \:-\,1/48\,h^{f}(1,\chi_\eta,\chi_{1s},\chi_{3s};\chi_{13})\,\left [ r^{\eta}_{ps}\,\chi_p - r^{\eta}_{ps}\,\chi_q \right . \nonumber \\ & - & r^v_{\eta ps}\,\chi_\eta - r^v_{\eta 13 } \left . \chi_s \right ] \:+\,h^{f}(1,\chi_1,\chi_{13},\chi_3;\chi_{13})\,\left [ 1/36\,r^{p}_{q\eta } r^c_{q}\,\chi_{13 } - 1/18\,r^{1}_{3\eta } r^{3}_{1\eta}\,\chi_{13 } \right . \nonumber \\ & - & 1/72 \left . r^c_{1 } r^c_{3}\,\chi_{13 } \right ] \:-\,1/8\,h^{f}(1,\chi_{13},\chi_{13},\chi_{13};\chi_{13})\,\chi_{13 } \:-\,3/8\,h^{f}(1,\chi_{14},\chi_{34},\chi_4;\chi_{13})\,\chi_4 \nonumber \\ & + & h^{f}(2,\chi_p,\chi_p,\chi_{13};\chi_{13})\,\left [ 1/36\,r^{p}_{q\eta } r^d_{p}\,\chi_{13 } - 1/72\,r^c_{p } r^d_{p}\,\chi_{13 } \right ] \:-\,1/72\,h^{f}(2,\chi_p,\chi_\eta,\chi_{13};\chi_{13})\,r^d_{p } r^v_{\eta 13}\,\chi_{13 } \nonumber \\ & + & h^{f}(2,\chi_p,\chi_{13},\chi_q;\chi_{13})\,\left [ 1/36\,r^{q}_{p\eta } r^d_{p}\,\chi_{13 } - 1/72\,r^c_{q } r^d_{p}\,\chi_{13 } \right ] \:+\,1/24\,h^{f}(2,\chi_p,\chi_{1s},\chi_{3s};\chi_{13})\,r^d_{p}\,\chi_{ps } \nonumber \\ & - & 1/144\,h^{f}(5,\chi_p,\chi_p,\chi_{13};\chi_{13})\,(r^{d}_{p})^2\,\chi_{13 } \:-\,1/72\,h^{f}(5,\chi_1,\chi_3,\chi_{13};\chi_{13})\,r^d_{1 } r^d_{3}\,\chi_{13 } \nonumber \\ & + & h^{f'}(1,\chi_p,\chi_{1s},\chi_{3s};\chi_{13})\,\left [ - 5/24\,r^{p}_{q\eta}\,\chi_q \chi_{13 } + 1/12\,r^{p}_{q\eta}\,\chi_{ps } \chi_{13 } + 1/16\,r^{p}_{s\eta}\,\chi_q \chi_{13 } + 7/48\,r^{p}_{s\eta}\,\chi_{13 } \chi_s \right . \nonumber \\ & - & 1/24 \left . r^c_{p}\,\chi_{ps } \chi_{13 } \right ] \:+\,1/4\,h^{f'}(1,\chi_{p4},\chi_{q6},\chi_{46};\chi_{13})\,\chi_{13 } \chi_{46 } \:+\,h^{f'}(1,\chi_\eta,\chi_p,\chi_{13};\chi_{13})\left [ 1/9\,r^{p}_{q\eta } r^{\eta}_{13 } \,\chi_{13}^2 \right . \nonumber \\ & - & 1/36 \left . r^{p}_{q\eta } r^v_{\eta 13}\,\chi_{13}^2 + 1/18\,r^{\eta}_{pp } r^c_{p}\,\chi_{13}^2 + 1/72\,r^c_{p } r^v_{\eta 13}\,\chi_{13}^2 \right ] \:+\,h^{f'}(1,\chi_\eta,\chi_\eta,\chi_{13};\chi_{13})\,\left [ 1/9\,(r^{\eta}_{13})^2 \,\chi_{13}^2 \right . \nonumber \\ & + & 1/9 \left . r^{\eta}_{13 } r^v_{\eta 13}\,\chi_{13}^2 + 5/144\,(r^v_{\eta 13})^2\ , \chi_{13}^2 \right ] \:+\,h^{f'}(1,\chi_\eta,\chi_{1s},\chi_{3s};\chi_{13})\,\left [ - 1/16\,r^{\eta}_{ps}\ , \chi_p \chi_{13 } - 5/48\,r^{\eta}_{ps}\,\chi_q \chi_{13 } \right . \nonumber \\ & + & 1/6 \left . r^{\eta}_{1s } r^z_{\eta s3}\,\chi_{13}^2 - 1/48\,r^v_{\eta ps}\,\chi_\eta \chi_{13 } - 1/48\,r^v_{\eta 13}\,\chi_{13 } \chi_s \right ] \:+\,h^{f'}(1,\chi_1,\chi_{13},\chi_3;\chi_{13})\,\left [ - 1/36\,r^{p}_{q\eta } r^c_{q } \,\chi_{13}^2 \right . \nonumber \\ & + & 1/6 \left . r^{1}_{3\eta } r^{3}_{1\eta}\,\chi_{13}^2 + 1/72\,r^c_{1 } r^c_{3}\,\chi_{13}^2 \right ] \:+\,h^{f'}(1,\chi_{13},\chi_p,\chi_p;\chi_{13})\,\left [ - 1/8\,\chi_p \chi_{13 } - 3/16\,\chi_{13}^2 - 1/36\,(r^{p}_{q\eta})^2\,\chi_{13}^2 \right . \nonumber \\ & + & 1/12 \left . r^{p}_{q\eta } r^c_{p}\,\chi_{13}^2 - 1/48\,(r^{c}_{p})^2\,\chi_{13}^2 \right ] \:+\,1/8\,h^{f'}(1,\chi_{13},\chi_{13},\chi_{13};\chi_{13})\,\chi_{13}^2 \nonumber \\ & + & 3/8\,h^{f'}(1,\chi_{14},\chi_{34},\chi_4;\chi_{13})\,\chi_{13 } \chi_4 \:+\,h^{f'}(2,\chi_p,\chi_p,\chi_{13};\chi_{13})\,\left [ - 1/36\,r^{p}_{q\eta } r^d_{p}\ , \chi_{13}^2 + 5/72\,r^c_{p } r^d_{p}\,\chi_{13}^2 \right ] \nonumber \\ & + & h^{f'}(2,\chi_p,\chi_\eta,\chi_{13};\chi_{13})\,\left [ 1/18\,r^{\eta}_{pp } r^d_{p}\ , \chi_{13}^2 + 1/72\,r^d_{p } r^v_{\eta 13}\,\chi_{13}^2 \right ] \nonumber \\ & + & h^{f'}(2,\chi_p,\chi_{13},\chi_q;\chi_{13})\,\left [ - 1/36\,r^{q}_{p\eta } r^d_{p}\ , \chi_{13}^2 + 1/72\,r^c_{q } r^d_{p}\,\chi_{13}^2 \right ] \:-\,1/24\,h^{f'}(2,\chi_p,\chi_{1s},\chi_{3s};\chi_{13})\,r^d_{p}\,\chi_{ps } \chi_{13 } \nonumber \\ & + & 5/144\,h^{f'}(5,\chi_p,\chi_p,\chi_{13};\chi_{13})\,(r^{d}_{p})^2\,\chi_{13}^2 \:+\,1/72\,h^{f'}(5,\chi_1,\chi_3,\chi_{13};\chi_{13})\,r^d_{1 } r^d_{3}\,\chi_{13}^2 \nonumber \\ & + & h^{f'}_1(1,\chi_p,\chi_p,\chi_{13};\chi_{13})\,\left [ \chi_{13}^2 + 2/9\,(r^{p}_{q\eta})^2 \,\chi_{13}^2 - 2/9\,r^{p}_{q\eta } r^c_{p}\,\chi_{13}^2 + 1/9\,(r^{c}_{p})^2\,\chi_{13}^2 \right ] \nonumber \\ & + & 1/3\,h^{f'}_1(1,\chi_p,\chi_{1s},\chi_{3s};\chi_{13})\,r^{p}_{q\eta } r^z_{sqp}\,\chi_{13}^2 \:-\,1/3\,h^{f'}_1(1,\chi_{ps},\chi_{qs},\chi_p;\chi_{13})\,r^{p}_{s\eta } \,\chi_{13}^2 \nonumber \\ & - & 1/3\,h^{f'}_1(1,\chi_{ps},\chi_{qs},\chi_\eta;\chi_{13})\,r^{\eta}_{13 } r^z_{sp\eta}\,\chi_{13}^2 \:+\,1/9\,h^{f'}_1(1,\chi_{13},\chi_p,\chi_\eta;\chi_{13})\,\left [ r^{p}_{q\eta } r^v_{\eta 13}\,\chi_{13}^2 \right . \nonumber \\ & - & r^{\eta}_{pp } r^z_{qp\eta } r^c_{p } \left . \chi_{13}^2 \right ] \:-\,h^{f'}_1(1,\chi_{13},\chi_\eta,\chi_\eta;\chi_{13})\,\left [ 1/9\,r^{\eta}_{13 } r^v_{\eta 13}\,\chi_{13}^2 + 1/18\,(r^v_{\eta 13})^2\,\chi_{13}^2 \right ] \nonumber \\ & + & h^{f'}_1(1,\chi_{13},\chi_1,\chi_3;\chi_{13})\,\left [ 1/9\,r^{p}_{q\eta } r^c_{q}\ , \chi_{13}^2 - 2/9\,r^{1}_{3\eta } r^{3}_{1\eta}\,\chi_{13}^2 \right ] \:+\,1/9\,h^{f'}_1(3,\chi_{13},\chi_p,\chi_p;\chi_{13})\,\left [ r^{p}_{q\eta } r^d_{p } \,\chi_{13}^2 \right . \nonumber \\ & - & r^c_{p } r^d_{p } \left . \chi_{13}^2 \right ] \:+\,1/9\,h^{f'}_1(3,\chi_{13},\chi_p,\chi_q;\chi_{13})\,r^{q}_{p\eta } r^d_{p}\,\chi_{13}^2 \:+\,1/9\,h^{f'}_1(3,\chi_{13},\chi_p,\chi_\eta;\chi_{13})\,r^{\eta}_{13 } r^z_{pq\eta } r^d_{p}\,\chi_{13}^2 \nonumber \\ & - & 1/18\,h^{f'}_1(7,\chi_{13},\chi_p,\chi_p;\chi_{13})\,(r^{d}_{p})^2\,\chi_{13}^2 \:-\,3/8\,h^{f'}_{21}(1,\chi_p,\chi_p,\chi_{13};\chi_{13})\,\chi_{13}^2 \nonumber \\ & - & 1/8\,h^{f'}_{21}(1,\chi_p,\chi_{1s},\chi_{3s};\chi_{13})\,r^{p}_{q\eta } r^z_{sqp } \,\chi_{13}^2 \:+\,1/8\,h^{f'}_{21}(1,\chi_{ps},\chi_{qs},\chi_p;\chi_{13})\,\left [ r^{p}_{q\eta}\ , \chi_{13}^2 + r^{p}_{s\eta}\,\chi_{13}^2 - r^c_{p}\,\chi_{13}^2 \right ] \nonumber \\ & + & 1/8\,h^{f'}_{21}(1,\chi_{ps},\chi_{qs},\chi_q;\chi_{13})\,r^{q}_{p\eta } r^z_{spq}\,\chi_{13}^2 \:+\,1/8\,h^{f'}_{21}(1,\chi_{ps},\chi_{qs},\chi_\eta;\chi_{13})\,r^{\eta}_{13 } r^z_{pq\eta } r^z_{sp\eta}\,\chi_{13}^2 \nonumber \\ & - & 1/8\,h^{f'}_{21}(1,\chi_\eta,\chi_{1s},\chi_{3s};\chi_{13})\,r^{\eta}_{13 } r^z_{s1\eta } r^z_{s3\eta}\,\chi_{13}^2 \:+\,h^{f'}_{21}(1,\chi_{13},\chi_p,\chi_p;\chi_{13})\,\left [ 3/16\,\chi_{13}^2 + 1/12\,(r^{p}_{q\eta})^2\,\chi_{13}^2 \right . \nonumber \\ & - & 1/12 \left . r^{p}_{q\eta } r^c_{p}\,\chi_{13}^2 + 1/48\,(r^{c}_{p})^2\,\chi_{13}^2 \right ] \:+\,h^{f'}_{21}(1,\chi_{13},\chi_p,\chi_\eta;\chi_{13})\,\left [ - 1/12\,r^{p}_{q\eta } r^v_{\eta 13}\,\chi_{13}^2 + 1/24\,r^c_{p } r^v_{\eta 13}\,\chi_{13}^2 \right ] \nonumber \\ & + & 1/48\,h^{f'}_{21}(1,\chi_{13},\chi_\eta,\chi_\eta;\chi_{13})\,(r^v_{\eta 13})^2\,\chi_{13}^2 \:+\,h^{f'}_{21}(1,\chi_{13},\chi_1,\chi_3;\chi_{13})\,\left [ - 1/12\,r^{p}_{q\eta } r^c_{q } \,\chi_{13}^2 + 1/6\,r^{1}_{3\eta } r^{3}_{1\eta}\,\chi_{13}^2 \right . \nonumber \\ & + & 1/24 \left . r^c_{1 } r^c_{3 } \,\chi_{13}^2 \right ] \:+\,3/8\,h^{f'}_{21}(1,\chi_{13},\chi_{13},\chi_{13};\chi_{13})\,\chi_{13}^2 \:+\,9/8\,h^{f'}_{21}(1,\chi_4,\chi_{14},\chi_{34};\chi_{13})\,\chi_{13}^2 \nonumber \\ & + & 3/4\,h^{f'}_{21}(1,\chi_{46},\chi_{p4},\chi_{q6};\chi_{13})\,\chi_{13}^2 \:-\,1/8\,h^{f'}_{21}(3,\chi_{ps},\chi_p,\chi_{qs};\chi_{13})\,r^d_{p}\,\chi_{13}^2 \nonumber \\ & + & h^{f'}_{21}(3,\chi_{13},\chi_p,\chi_p;\chi_{13})\,\left [ - 1/12\,r^{p}_{q\eta } r^d_{p } \,\chi_{13}^2 + 1/24\,r^c_{p } r^d_{p}\,\chi_{13}^2 \right ] \:+\,h^{f'}_{21}(3,\chi_{13},\chi_p,\chi_q;\chi_{13})\,\left [ - 1/12\,r^{q}_{p\eta } r^d_{p } \,\chi_{13}^2 \right . \nonumber \\ & + & 1/24 \left . r^c_{q } r^d_{p}\,\chi_{13}^2 \right ] \:+\,1/24\,h^{f'}_{21}(3,\chi_{13},\chi_p,\chi_\eta;\chi_{13})\,r^d_{p } r^v_{\eta 13 } \,\chi_{13}^2 \:+\,1/48\,h^{f'}_{21}(7,\chi_{13},\chi_p,\chi_p;\chi_{13})\,(r^{d}_{p})^2\,\chi_{13}^2 \nonumber \\ & + & 1/24\,h^{f'}_{21}(7,\chi_{13},\chi_1,\chi_3;\chi_{13})\,r^d_{1 } r^d_{3}\,\chi_{13}^2 . \label{f0p622loop } \end{aligned}\ ] ] the analytical formulas for the nnlo contributions to the decay constants are very complicated , especially when the valence quark masses are nondegenerate . the practical usefulness of those results thus depends completely on the availability of convenient numerical implementations . in the near future , we will make such an implementation available @xcite . the results , as given in the preceding section , depend on nine @xmath11 and five @xmath1 lec : s . eventually , their proper values should be determined from lattice qcd by a fit of the nlo + nnlo formulas to the simulation data . at the present time , these input parameters have been taken from the continuum work in nnlo @xmath0pt of ref . the combination of lec : s used in this paper corresponds to fit 10 of ref . @xcite , which had @xmath167 mev and @xmath168 mev . the parameters which were not determined by that fit , namely @xmath169 and @xmath170 , have been set to zero , as have all the @xmath171 . this has been done at a scale of @xmath172 mev . it should be noted that @xmath38 can not be determined from experiment , but is obtainable from partially quenched simulations . some recent results on @xmath173 and @xmath170 may be found in ref . @xcite . the graphical presentation of multidimensional functions is a more immediate problem . in the most general case , the decay constant of a charged pseudoscalar meson in pq@xmath0pt is a function of two valence and three sea quark masses . the plots presented in this section should therefore be viewed as an attempt to cover the potentially most interesting parts of the parameter space . they also serve as a consistency check for the formulas in the preceding section , as the cases with fewer different masses should be obtained numerically as limits of the more general cases . to this end , the result for @xmath89 and @xmath80 has been plotted along different rays in the @xmath127-@xmath174 plane , characterized by an angle @xmath175 which is defined according to @xmath176 such that @xmath177 corresponds to the unquenched case of equal sea and valence quark masses . the typical situation in lattice qcd simulations , where sea quarks are heavier than their valence counterparts , is attained for @xmath178 . the shift in the decay constant due to the combined nlo + nnlo contributions has been plotted in fig . [ 11fig ] for different values of @xmath175 . the quantity plotted , @xmath179 , denotes the relative change in the decay constant and is defined in accordance with eq . ( [ delteq ] ) , as @xmath180 thus , when plotted along a curve of the form ( [ rayplot ] ) , @xmath179 is expected to vanish in the chiral limit , which is indeed borne out in all plots presented in this section . this shows that the nnlo shifts to the decay constant in pq@xmath0pt do not produce unphysical logarithms . on the other hand , the curve labeled a in fig . [ 11fig ] is plotted for a constant sea mass with @xmath181 , and in that case @xmath179 approaches a constant as @xmath182 . and @xmath80 , for values of @xmath175 between @xmath183 and @xmath184 . the curve a corresponds to @xmath181 . in order to reduce clutter , most of the lines in this plot , with the exception of the @xmath183 line , have not been drawn all the way to the origin.,width=294 ] and @xmath80 , for values of @xmath175 between @xmath183 and @xmath184 . the curve a corresponds to @xmath181.,width=294 ] , @xmath76 , @xmath185 , and a strange sea quark mass parameter @xmath186 between @xmath187 and @xmath188.,width=294 ] the results along the lines ( [ rayplot ] ) at nlo is shown in fig . [ 11bfig ] . the chiral expansion for the decay constants converges better than for the masses as can be seen by comparison of figs . [ 11fig ] and [ 11bfig ] with the corresponding ones in ref . note , however , that the dependence of @xmath179 on the numerical values of the lec : s is strong and can change the behavior of the nnlo results quite considerably . it should also be kept in mind that many of the curves plotted in fig . [ 11fig ] reach far beyond the expected radius of convergence of pq@xmath0pt . the physically more interesting case of @xmath76 , where the strange sea quark mass may deviate from that of the @xmath2 quarks , can then be accounted for by the introduction of an additional parameter @xmath186 , which is defined as @xmath189 the change in @xmath179 when the strange sea quark mass is allowed to vary is shown in fig . [ 12fig ] . in general , the nnlo effects appear to become larger as the strange sea quark mass is increased . [ 12fig ] also shows that eq . ( [ f0p612loop ] ) reduces numerically to the @xmath89 , @xmath80 result of eq . ( [ f0p611loop ] ) when all sea quark masses become equal , i.e. @xmath190 . , @xmath76 , @xmath191 and @xmath185 . the strange valence quark mass parameter @xmath192 varies between @xmath123 and @xmath193.,width=294 ] it is also instructive to investigate the nnlo correction to the decay constant for @xmath6 and @xmath76 . the behavior of @xmath179 when the strange valence quark mass deviates from that of the @xmath2 quarks is shown in fig . [ 22fig ] , where the results have been parameterized in terms of @xmath194 it is evident that @xmath179 changes rapidly with increasing @xmath192 for values of @xmath127 above about @xmath195 . also , eq . ( [ f0p622loop ] ) is seen to reduce correctly to the @xmath89 , @xmath76 result when @xmath196 . in the case of @xmath67 , @xmath72 and @xmath73 are non - trivially related to the sea quark masses , and consequently the sea quark sector is more complicated . the variation in the mass of the @xmath197 quark with respect to the @xmath198 quark , in the sea quark sector , has been parameterized by @xmath199 sample plots of @xmath179 for @xmath89 , with all three sea quark masses different , are shown in fig . [ 13fig ] . those plots also demonstrate numerically the consistency of eq . ( [ f0p613loop ] ) with the @xmath158 results . , @xmath67 , @xmath185 and @xmath191 . the mass parameter @xmath200 of the down quark in the sea sector ranges between @xmath123 and @xmath193.,width=294 ] a generic feature of the plots presented in this section is that the curves for @xmath179 show a pronounced dip around @xmath201 . this indicates a strong cancellation between the nlo and nnlo contributions to @xmath179 . it should be noted that such a feature is apparently not exhibited by the analogous expressions for the pseudoscalar meson mass @xcite . this cancellation also depends strongly on the choice of the lec : s . many more plots can of course be produced but those presented give a first indication of the size of the corrections and how they vary with the different quark masses used as input . we have not attempted any fit of our results to the available lattice data ; many simulations are performed with only two flavors of sea quarks , and in addition we need extrapolations to zero lattice spacing and infinite volume at each quark mass to apply the present formulas . work for the case with two sea quark flavors is in progress . in conclusion , we have calculated the decay constants of the charged , or off - diagonal , pseudoscalar mesons to nnlo in pq@xmath0pt and presented analytical as well as numerical results for a variety of different combinations of quark masses . the nnlo contributions were found , as expected from previous work in nnlo @xmath0pt @xcite , to be sizable even though there is a tendency toward cancellation with the nlo result . as the results depend on a number of largely unknown lec : s , statements about the convergence of the chiral expansion have to be postponed at this time . the nnlo effects are definitely non - negligible at presently used quark masses in lattice qcd simulations . the program ` form 3.0 ` has been used extensively in these calculations @xcite . this work is supported by the european union tmr network , contract no . hprn - ct-2002 - 00311 ( euridice ) . tl wishes to thank the thomas jefferson national accelerator facility ( tjnaf ) and the helsinki institute of physics ( hip ) , where part of this paper was completed , for their hospitality . tl also thanks the mikael bjrnberg memorial foundation for a travel grant . s. weinberg , physica a * 96 * ( 1979 ) 327 ; j. gasser and h. leutwyler , ann . * 158 * ( 1984 ) 142 ; j. gasser and h. leutwyler , nucl . b * 250 * , 465 ( 1985 ) . c. w. bernard and m. f. l. golterman , phys . d * 46 * , 853 ( 1992 ) [ arxiv : hep - lat/9204007 ] . c. w. bernard and m. f. l. golterman , phys . d * 49 * , 486 ( 1994 ) [ arxiv : hep - lat/9306005 ] . s. r. sharpe and n. shoresh , phys . d * 62 * , 094503 ( 2000 ) [ arxiv : hep - lat/0006017 ] . s. r. sharpe and n. shoresh , phys . d * 64 * , 114510 ( 2001 ) [ arxiv : hep - lat/0108003 ] . j. bijnens , n. danielsson and t. a. lhde , phys . d * 70 * , 111503 ( 2004 ) [ arxiv : hep - lat/0406017 ] . f. farchioni , i. montvay and e. scholz [ qq+q collaboration ] , eur . j. c * 37 * , 197 ( 2004 ) [ arxiv : hep - lat/0403014 ] . c. aubin _ [ milc collaboration ] , phys . d * 70 * , 114501 ( 2004 ) [ arxiv : hep - lat/0407028 ] . j. bijnens , n. danielsson and t. a. lhde , to be published . p. h. damgaard and k. splittorff , phys . d * 62 * , 054509 ( 2000 ) [ arxiv : hep - lat/0003017 ] . j. bijnens , g. colangelo and g. ecker , jhep * 9902 * , 020 ( 1999 ) [ arxiv : hep - ph/9902437 ] . j. bijnens , g. colangelo and g. ecker , ann . phys . * 280 * , 100 ( 2000 ) [ arxiv : hep - ph/9907333 ] . g. amors , j. bijnens and p. talavera , nucl . b * 568 * , 319 ( 2000 ) [ arxiv : hep - ph/9907264 ] . the analytical formulas are available via ` http://www.thep.lu.se/~bijnens/chpt.html ` . the programs are available from the authors . g. amors , j. bijnens and p. talavera , nucl . b * 602 * , 87 ( 2001 ) [ arxiv : hep - ph/0101127 ] . j. bijnens , p. dhonte and p. talavera , jhep * 0401 * , 050 ( 2004 ) [ arxiv : hep - ph/0401039 ] , jhep * 0405 * , 036 ( 2004 ) [ arxiv : hep - ph/0404150 ] . e. golowich and j. kambor , phys . d * 58 * , 036004 ( 1998 ) [ arxiv : hep - ph/9710214 ] . j. a. vermaseren , arxiv : math - ph/0010025 .

this paper presents a first study of the decay constants of the charged , or flavor - off - diagonal , pseudoscalar mesons to two loops for three flavors of sea quarks , in partially quenched chiral perturbation theory ( pq@xmath0pt ) . explicit analytical expressions up to @xmath1 in the momentum expansion are given . the calculations have been performed within the supersymmetric formulation of pq@xmath0pt . we also present some numerical results to indicate the size of the corrections . lu tp 05 - 1 + hep - lat/0501014 + january 2005 * decay constants of pseudoscalar mesons to two loops + in three - flavor partially quenched @xmath0pt * * johan bijnens and timo a. lhde * + department of theoretical physics , lund university , + slvegatan 14a , s 223 62 lund , sweden * abstract * : 12.38.gc , 12.39.fe , 11.30.rd

in both classical and quantum physics isolated systems can display unpredictable behavior , but the reasons for the unpredictability are quite different . in classical ( hamiltonian ) mechanics unpredictability is a consequence of chaotic dynamics , or exponential sensitivity to initial conditions , which makes it impossible to predict the phase - space trajectory of a system to a certain accuracy from initial data given to the same accuracy . this unpredictability , which comes from not knowing the system s initial conditions precisely , is measured by the kolmogorov - sinai ( ks ) entropy , which is the rate at which initial data must be supplied in order to continue predicting the coarse - grained phase - space trajectory @xcite . in quantum mechanics there is no sensitivity to initial conditions in predicting the evolution of a state vector , because the unitary evolution of quantum mechanics preserves the inner product between state vectors . the absence of sensitivity to initial conditions seems to suggest that there is no quantum chaos . yet quantum mechanics has an even more fundamental kind of unpredictability , which has nothing to do with dynamics : even if a system s state vector is known precisely , the results of measurements are generally unpredictable . to compare the unpredictability of classical and quantum dynamics , we first remove the usual sources of unpredictability from consideration and then introduce a new source of unpredictability that is the same in both classical and quantum dynamics . the first step is to focus in classical physics on the evolution of phase - space distributions , governed by the liouville equation , instead of on phase - space trajectories , and to focus in quantum physics on the evolution of state vectors , governed by the schrdinger equation . the liouville equation preserves the overlap between distributions , so there is no sensitivity to initial conditions in predicting the evolution of a phase - space distribution . by shifting attention from phase - space trajectories to distributions , we remove lack of knowledge of initial conditions as a source of unpredictability . moreover , by considering only schrdinger evolution of state vectors , i.e. , evolution uninterrupted by measurements , we eliminate the intrinsic randomness of quantum measurements as a source of unpredictability . the conclusion that there is no chaos in quantum evolution is now seen to be too facile . were things so simple , one would have to conclude that there is no chaos in classical liouville evolution either @xcite . having taken both classical and quantum unpredictability out of the picture , we introduce a new source of unpredictability to investigate chaos in the dynamics . we do this by adding to the system hamiltonian , either classical or quantum mechanical , a stochastic perturbation . we measure the unpredictability introduced by the perturbation in terms of the increase of system entropy . by gathering information about the history of the perturbation , one can make the increase of system entropy smaller . to characterize the resistance of the system to predictability , we compare the information gathered about the perturbation with the entropy reduction that this information purchases . we say that a system is _ hypersensitive to perturbation _ @xcite if the perturbation information is much larger than the associated system - entropy reduction , and we regard hypersensitivity to perturbation as the signature of chaos in liouville or schrdinger evolution ( see sec . [ sechyp ] ) . for classical systems we have shown that systems with chaotic dynamics display an _ exponential _ hypersensitivity to perturbation @xcite , in which the ratio of perturbation information to entropy reduction grows exponentially in time , with the exponential rate of growth given by the ks entropy . thus , for classical systems , we have established that exponential hypersensitivity to perturbation characterizes chaos in liouville evolution in a way that is exactly equivalent to the standard characterization of chaos in terms of the unpredictability of phase - space trajectories ( see sec . [ secclassical ] ) . for a variety of quantum systems we have used numerical simulations to investigate hypersensitivity to perturbation @xcite . the simulations suggest that hypersensitivity to perturbation provides a characterization of chaos in quantum dynamics : quantum systems whose classical dynamics is chaotic display a quantum hypersensitivity to perturbation , which comes about because the perturbation generates state vectors that are nearly randomly distributed in the system hilbert space , whereas quantum systems whose classical dynamics is not chaotic do not display hypersensitivity to perturbation ( see sec . [ secquantum ] ) . hypersensitivity to perturbation , in either classical or quantum mechanics , is defined in terms of information and entropy . the entropy @xmath0 of an isolated physical system ( gibbs entropy for a classical system , von neumann entropy for a quantum system ) does not change under hamiltonian time evolution . if the time evolution of the system is perturbed through interaction with an incompletely known environment , however , averaging over the perturbation typically leads to an entropy increase @xmath1 . throughout this paper , we make the simplifying assumption that the interaction with the environment is equivalent to a stochastic perturbation of the hamiltonian , a restriction we hope to be able to remove in the future . conditions under which this assumption is valid are discussed in @xcite . the increase of the system entropy can be limited to an amount @xmath2 , the _ tolerable entropy increase _ , by obtaining , from the environment , information about the perturbation . we denote by @xmath3 the minimum information about the perturbation needed , on the average , to keep the system entropy below the tolerable level @xmath2 . a formal definition of the quantities @xmath1 , @xmath2 , and @xmath3 can be found in @xcite for the classical case and in @xcite for the quantum case . entropy and information acquire physical content in the presence of a heat reservoir at temperature @xmath4 . if all energy in the form of heat is ultimately exchanged with the heat reservoir , then each bit of entropy , i.e. , each bit of _ missing information _ about the system state , reduces by the amount @xmath5 the energy that can be extracted from the system in the form of useful work . the connection between _ acquired _ information and work is provided by landauer s principle @xcite , according to which not only each bit of missing information , but also each bit of acquired information , has a free - energy cost of @xmath5 . this cost , the _ landauer erasure cost _ , is paid when the acquired information is erased . acquired information can be quantified by algorithmic information @xcite . we now define that a system is hypersensitive to perturbation if the information @xmath3 required to reduce the system entropy from @xmath6 to @xmath2 is large compared to the entropy reduction @xmath7 , i.e. , @xmath8 the information @xmath3 purchases a reduction @xmath9 in system entropy , which is equivalent to an increase in the useful work that can be extracted from the system ; hypersensitivity to perturbation means that the landauer erasure cost of the information is much larger than the increase in available work . hypersensitivity to perturbation means that the inequality ( [ eqhyp ] ) holds for almost all values of @xmath2 . the inequality ( [ eqhyp ] ) tends always to hold , however , for sufficiently small values of @xmath10 . the reason is that for these small values of @xmath2 , one is gathering enough information from the perturbing environment to track a particular system state whose entropy is nearly equal to the initial system entropy . in other words , one is essentially tracking a particular realization of the perturbation among all possible realizations . thus , for small values of @xmath2 , the information @xmath3 becomes a property of the perturbation ; it is the information needed to specify a particular realization of the perturbation . the important regime for assessing hypersensitivity to perturbation is where @xmath2 is fairly close to @xmath6 , and it is in this regime that one can hope that @xmath3 reveals something about the system dynamics , rather than properties of the perturbation . in this section we do not aim for rigor ; many statements in this section are without formal proof . instead , our objective here is to extract the important ideas from the rigorous analysis given in @xcite and to use them to develop a heuristic physical picture of why chaotic systems display exponential hypersensitivity to perturbation . for a simple illustration and a system where exact solutions exist , see @xcite . this section is an abbreviated version of the discussion section of @xcite . consider a classical hamiltonian system whose dynamics unfolds on a @xmath11-dimensional phase space , and suppose that the system is perturbed by a stochastic hamiltonian whose effect can be described as diffusion on phase space . suppose that the system is globally chaotic with ks entropy @xmath12 . for such a system a phase - space density is stretched and folded by the chaotic dynamics , developing exponentially fine structure as the dynamics proceeds . a simple picture is that the phase - space density stretches exponentially in half the phase - space dimensions and contracts exponentially in the other half of the dimensions . the perturbation is characterized by a perturbation strength and by correlation cells . we can take the perturbation strength to be the typical distance ( e.g. , euclidean distance with respect to some fixed set of canonical cordinates ) that a phase - space point diffuses under the perturbation during an @xmath13-folding time , @xmath14 , in a typical contracting dimension . the perturbation becomes effective ( in a sense defined precisely in ref . @xcite ) when the phase - space density has roughly the same size in the contracting dimensions as the perturbation strength . once the perturbation becomes effective , the effects of the diffusive perturbation and of the further exponential contraction roughly balance one another , leaving the _ average _ phase - space density with a constant size in the contracting dimensions . the correlation cells are phase - space cells over which the effects of the perturbation are well correlated and between which the effects of the perturbation are essentially uncorrelated . we assume that all the correlation cells have approximately the same phase - space volume . we can get a rough idea of the effect of the perturbation by regarding the correlation cells as receiving independent perturbations . moreover , the diffusive effects of the perturbation during an @xmath13-folding time @xmath14 are compressed exponentially during the next such @xmath13-folding time ; this means that once the perturbation becomes effective , the main effects of the perturbation at a particular time are due to the diffusion during the immediately preceding @xmath13-folding time . since a chaotic system can not be shielded forever from the effects of the perturbation , we can choose the initial time @xmath15 to be the time at which the perturbation is just becoming effective . we suppose that at @xmath15 the unperturbed density is spread over @xmath16 correlation cells , @xmath17 being the time when the unperturbed density occupies a single correlation cell . the essence of the ks entropy is that for large times @xmath18 the unperturbed density spreads over @xmath19 correlation cells , in each of which it occupies roughly the same phase - space volume . the exponential increase of @xmath20 continues until the unperturbed density is spread over essentially all the correlation cells . we can regard the unperturbed density as being made up of _ subdensities _ , one in each occupied correlation cell and all having roughly the same phase - space volume . after @xmath15 , when the perturbation becomes effective , the _ average _ density continues to spread exponentially in the expanding dimensions . as noted above , this spreading is not balanced by contraction in the other dimensions , so the phase - space volume occupied by the average density grows as @xmath21 , leading to an entropy increase @xmath22 just as the unperturbed density can be broken up into subdensities , so the average density can be broken up into _ average subdensities _ , one in each occupied correlation cell . each average subdensity occupies a phase - space volume that is @xmath21 times as big as the volume occupied by an unperturbed subdensity . the unperturbed density is embedded within the phase - space volume occupied by the average density and itself occupies a volume that is smaller by a factor of @xmath23 . we can picture a _ perturbed _ density crudely by imagining that in each occupied correlation cell the unperturbed subdensity is moved rigidly to some new position within the volume occupied by the _ average _ subdensity ; the result is a _ perturbed subdensity_. a _ perturbed density _ is made up of perturbed subdensities , one in each occupied correlation cell . all of the possible perturbed densities are produced by the perturbation with roughly the same probability . suppose now that we wish to hold the entropy increase to a tolerable amount @xmath2 . we must first describe what it means to specify the phase - space density at a level of resolution set by a tolerable entropy increase @xmath2 . an approximate description can be obtained in the following way . take an occupied correlation cell , and divide the volume occupied by the average subdensity in that cell into @xmath24 nonoverlapping volumes , all of the same size . aggregate all the perturbed subdensities that lie predominantly within a particular one of these nonoverlapping volumes to produce a _ coarse - grained subdensity_. there are @xmath25 coarse - grained subdensities within each occupied correlation cell , each having a phase - space volume that is bigger than the volume occupied by a perturbed subdensity by a factor of @xmath26 a _ coarse - grained density _ is made up by choosing a coarse - grained subdensity in each occupied correlation cell . a coarse - grained density occupies a phase - space volume that is bigger than the volume occupied by the unperturbed density by the factor @xmath27 of eq . ( [ cgvolume ] ) and hence represents an entropy increase @xmath28 thus to specify the phase - space density at a level of resolution set by @xmath29 means roughly to specify a coarse - grained density . the further entropy increase on averaging over the perturbation is given by @xmath30 what about the information @xmath3 required to hold the entropy increase to @xmath2 ? since there are @xmath31 coarse - grained subdensities in an occupied correlation cell , each produced with roughly the same probability by the perturbation , it takes approximately @xmath32 bits to specify a particular coarse - grained subdensity . to describe a coarse - grained density , one must specify a coarse - grained subdensity in each of the @xmath20 occupied correlation cells . thus the information required to specify a coarse - grained density and , hence , the information required to hold the entropy increase to @xmath2is given by @xmath33 corresponding to there being a total of @xmath34 coarse - grained densities . the entropy increase ( [ furtherincrease ] ) comes from counting the number of _ nonoverlapping _ coarse - grained densities that are required to fill the volume occupied by the average density , that number being @xmath31 . in contrast , the information @xmath3 comes from counting the exponentially greater number of ways of forming _ overlapping _ coarse - grained densities by choosing one of the @xmath35 nonoverlapping coarse - grained subdensities in each of the @xmath20 correlation cells . the picture developed in this section , summarized neatly in eq . ( [ picsum ] ) , requires that @xmath2 be big enough that a coarse - grained subdensity is much larger than a perturbed subdensity , so that we can talk meaningfully about the perturbed subdensities that lie predominantly _ within _ a coarse - grained subdensity . if @xmath2 becomes too small , eq . ( [ picsum ] ) breaks down , and the information @xmath36 , rather than reflecting a property of the chaotic dynamics as in eq . ( [ picsum ] ) , becomes essentially a property of the perturbation , reflecting a counting of the number of possible realizations of the perturbation . the boundary between the two kinds of behavior of @xmath3 is set roughly by the number @xmath37 of contracting phase - space dimensions . when @xmath38 , the characteristic scale of a coarse - grained subdensity in the contracting dimensions is a factor of @xmath39 larger than the characteristic size of a perturbed subdensity in the contracting dimensions . in this regime the picture developed in this section is at least approximately valid , because a coarse - grained subdensity can accommodate several perturbed subdensities in each contracting dimension . the information @xmath3 quantifies the effects of the perturbation on scales as big as or bigger than the finest scale set by the system dynamics . these effects , as quantified in @xmath3 , tell us directly about the size of the exponentially fine structure created by the system dynamics . thus @xmath3 becomes a property of the system dynamics , rather than a property of the perturbation . in contrast , when @xmath40 , we are required to keep track of the phase - space density on a very fine scale in the contracting dimensions , a scale smaller than the characteristic size of a perturbed subdensity in the contracting dimensions . subdensities are considered to be distinct , even though they overlap substantially , provided that they differ by more than this very fine scale in the contracting dimensions . the information @xmath36 is the logarithm of the number of realizations of the perturbation which differ by more than this very fine scale in at least one correlation cell . the information becomes a property of the perturbation because it reports on the effects of the perturbation on scales finer than the finest scale set by the system dynamics , scales that are , at the time of interest , irrelevant to the system dynamics . we are now prepared to put in final form the exponential hypersensitivity to perturbation of systems with a positive ks entropy : @xmath41 once the chaotic dynamics renders the perturbation effective , this exponential hypersensitivity to perturbation is essentially independent of the form and strength of the perturbation . its essence is that within each correlation cell there is a roughly even trade - off between entropy reduction and information , but for the entire phase - space density the trade - off is exponentially unfavorable because the density occupies an exponentially increasing number of correlation cells , in each of which it is perturbed independently . what about systems with regular , or integrable dynamics ? though we expect no universal behavior for regular systems , we can get an idea of the possibilities from the heuristic description developed in this section . hypersensitivity to perturbation requires , first , that the phase - space density develop structure on the scale of the strength of the perturbation , so that the perturbation becomes effective , and , second , that after the perturbation becomes effective , the phase - space density spread over many correlation cells . for many regular systems there will be no hypersensitivity simply because the phase - space density does not develop fine enough structure . regular dynamics can give rise to nonlinear shearing , however , in which case the density can develop structure on the scale of the strength of the perturbation and can spread over many correlation cells . in this situation , one expects the picture developed in this section to apply at least approximately : to hold the entropy increase to @xmath2 requires giving @xmath42 bits per occupied correlation cell ; @xmath3 is related to @xmath2 by eq . ( [ picsum ] ) , with @xmath20 being the number of correlation cells occupied at time @xmath18 . thus regular systems can display hypersensitivity to perturbation if @xmath20 becomes large ( although this behavior could be eliminated by choosing correlation cells that are aligned with the nonlinear shearing produced by the system dynamics ) , but they can not display _ exponential _ hypersensitivity to perturbation because the growth of @xmath20 is slower than exponential . a more direct way of stating this conclusion is to reiterate what we have explained in this section and shown in ref . @xcite : exponential hypersensitivity to perturbation is equivalent to the spreading of phase - space densities over an exponentially increasing number of phase - space cells ; such exponential spreading holds for chaotic , but not for regular systems and is quantified by a positive value of the kolmogorov - sinai entropy . the simplifying restriction on the interaction with the environment made in sec . [ sechyp ] means , for the quantum case , that the interaction with the environment is equivalent to a stochastic unitary time evolution . given this assumption , we can proceed as follows . at a given time , we describe the result of the perturbed time evolution by a list @xmath43 of @xmath44 vectors in @xmath45-dimensional hilbert space , with probabilities @xmath46 , each vector in the list corresponding to a particular realization of the perturbation , which we call a _ perturbation history_. averaging over the perturbation leads to a system density operator @xmath47 with entropy @xmath48 consider the class of measurements on the environment whose outcomes partition the list @xmath49 into @xmath50 groups labeled by @xmath51 . we denote by @xmath52 the number of vectors in the @xmath53th group ( @xmath54 ) . the @xmath52 vectors in the @xmath53th group and their probabilities are denoted by @xmath55 and @xmath56 , respectively . the measurement outcome @xmath53 , occurring with probability @xmath57 indicates that the system state is in the @xmath53th group . the system state conditional on the measurement outcome @xmath53 is described by the density operator @xmath58 we define the conditional system entropy @xmath59 the average conditional entropy @xmath60 and the average information @xmath61 we now describe nearly optimal measurements , i.e. , nearly optimal groupings , for which @xmath62 is a close approximation to @xmath3 , the minimum information about the environment needed , on the average , to keep the system entropy below a given tolerable entropy @xmath2 , as described in sec . [ sechyp ] . given @xmath2 , we want to partition the list of vectors @xmath49 into groups so as to minimize the information @xmath62 without violating the condition @xmath63 . to minimize @xmath62 , it is clearly favorable to make the groups as large as possible . furthermore , to reduce the contribution to @xmath64 of a group containing a given number of vectors , it is favorable to choose vectors that are as close together as possible in hilbert space . here the distance between two vectors @xmath65 and @xmath66 can be quantified in terms of the hilbert - space angle @xcite @xmath67 consequently , to find a nearly optimal grouping , we choose an arbitrary _ resolution angle _ @xmath68 ( @xmath69 ) and group together vectors that are less than an angle @xmath68 apart . more precisely , groups are formed in the following way . starting with the first vector , @xmath65 , in the list @xmath49 , the first group is formed of @xmath65 and all vectors in @xmath49 that are within an angle @xmath68 of @xmath65 . the same procedure is repeated with the remaining vectors to form the second group , then the third group , continuing until no ungrouped vectors are left . this grouping of vectors corresponds to a partial averaging over the perturbations . to describe a vector at resolution level @xmath68 amounts to averaging over those details of the perturbation that do not change the final vector by more than an angle @xmath68 . for each resolution angle @xmath68 , the grouping procedure described above defines an average conditional entropy @xmath70 and an average information @xmath71 . if we choose , for a given @xmath68 , the tolerable entropy @xmath72 , then to a good approximation , the information @xmath3 is given by @xmath73 . by determining the entropy @xmath74 and the information @xmath75 as functions of the resolution angle @xmath68 , there emerges a rather detailed picture of how the vectors are distributed in hilbert space . if @xmath75 is plotted as a function of @xmath74 by eliminating the angle @xmath68 , one obtains a good approximation to the functional relationship between @xmath3 and @xmath2 . as a further characterization of our list of vectors , we calculate the distribution @xmath76 of hilbert - space angles @xmath77 between all pairs of vectors @xmath78 and @xmath79 . for vectors distributed randomly in @xmath45-dimensional hilbert space , the distribution function @xmath76 is given by @xcite @xmath80 the maximum of this @xmath76 is located at @xmath81 ; for large - dimensional hilbert spaces , @xmath76 is very strongly peaked near the maximum , which is located at @xmath82 , very near @xmath83 . to investigate if a quantum map shows hypersensitivity to perturbation , we use the following numerical method . we first compute a list of vectors corresponding to different perturbation histories . then , for about 50 values of the angle @xmath68 ranging from 0 to @xmath83 , we group the vectors in the nearly optimal way described above . finally , for each grouping and thus for each chosen angle @xmath68 , we compute the information @xmath75 and the entropy @xmath74 . in addition , we compute the angles between all pairs of vectors in the list and plot them as a histogram approximating the distribution function @xmath76 . in this section , we present a typical numerical result for the quantum kicked top taken from @xcite , where more details can be found . we look at the time evolution of an initial hilbert - space vector @xmath84 at discrete times @xmath85 . after @xmath86 time steps , the unperturbed vector is given by @xmath87 is the unitary floquet operator @xcite @xmath88 and where @xmath89 is the angular momentum vector for a spin-@xmath90 particle evolving in @xmath91-dimensional hilbert space . depending on the initial condition , the classical map corresponding to the floquet operator ( [ eqqtop ] ) displays regular as well as chaotic behavior @xcite . following @xcite , we choose initial hilbert - space vectors for the quantum evolution that correspond to classical initial conditions located in regular and chaotic regions of the classical dynamics , respectively . for this purpose , we use _ coherent states _ @xcite . in this section , we consider two initial states . the first one is a coherent state centered in a regular region of the classical dynamics ; we refer to it as the _ regular initial state_. the second one , referred to as the _ chaotic initial state _ , is a coherent state centered in a chaotic region of the classical dynamics . the perturbation is modeled as an additional rotation by a small random angle about the @xmath92 axis . the system state after @xmath86 perturbed steps is thus given by @xmath93 where @xmath94 , with @xmath95 , is the unperturbed floquet operator ( [ eqqtop ] ) followed by an additional rotation about the @xmath92 axis by an angle @xmath96 , the parameter @xmath97 being the _ perturbation strength_. there are @xmath98 different perturbation histories obtained by applying every possible sequence of perturbed unitary evolution operators @xmath99 and @xmath100 for @xmath86 steps . we have applied the method described in sec . [ secdist ] to find numerically a nearly optimal grouping of the list @xmath49 of @xmath98 vectors generated by all perturbation histories . figure [ figtop ] shows results for spin @xmath101 and a total number of @xmath102 vectors after @xmath103 perturbed steps @xcite . we used a _ twist parameter _ @xmath104 and perturbation strength @xmath105 . for fig . [ figtop](a ) , the chaotic initial state was used . the distribution of hilbert - space angles , @xmath76 , is concentrated at large angles ; i.e. , most pairs of vectors are far apart from each other . the information @xmath62 needed to track a perturbed vector at resolution level @xmath68 is 12 bits at small angles , where each group contains only one vector . at @xmath106 the information suddenly drops to 11 bits , which is the information needed to specify one pair of vectors out of @xmath107 pairs , the two vectors in each pair being generated by perturbation sequences that differ only at the first step . the sudden drop of the information to 10 bits at @xmath108 similarly indicates the existence of @xmath109 quartets of vectors , generated by perturbation sequences differing only in the first two steps . figure [ figtop](a ) suggests that , apart from the organization into pairs and quartets , there is not much structure in the distribution of vectors for a chaotic initial state . the @xmath109 quartets seem to be rather uniformly distributed in a @xmath110-dimensional hilbert space ( see @xcite for a definition of the number of explored hilbert - space dimensions , @xmath111 ) . the inset in fig . [ figtop](a ) shows the approximate functional dependence of the information needed about the perturbation , @xmath3 , on the tolerable entropy @xmath2 , based on the data points @xmath112 and @xmath74 . there is an initial sharp drop of the information , reflecting the grouping of the vectors into pairs and quartets . then there is a roughly linear decrease of the information over a wide range of @xmath2 values , followed by a final drop with increasing slope down to zero at the maximum value of the tolerable entropy , @xmath113 . the large slope of the curve near @xmath113 can be regarded as a signature of hypersensitivity to perturbation . the linear regime at intermediate values of @xmath2 is due to the finite size of the sample of vectors : in this regime the entropy @xmath114 of the @xmath53th group is limited by @xmath115 , the logarithm of the number of vectors in the group . figure [ figtop](b ) shows data for @xmath116 vectors after 12 perturbed steps in the regular case . the distribution of perturbed vectors starting from the regular initial state is completely different from the chaotic initial condition of fig . [ figtop](a ) . the angle distribution @xmath76 is conspicuously nonrandom : it is concentrated at angles smaller than roughly @xmath117 , and there is a regular structure of peaks and valleys . accordingly , the information drops rapidly with the angle @xmath68 . the number of explored dimensions is @xmath118 , which agrees with results of peres @xcite that show that the quantum evolution in a regular region of the kicked top is essentially confined to a 2-dimensional subspace . the @xmath3 vs. @xmath2 curve in the inset bears little resemblance to the chaotic case . summarizing , one can say that , in the regular case , the vectors do not get far apart in hilbert space , explore only few dimensions , and do not explore them randomly . to obtain better numerical evidence for hypersensitivity in the chaotic case and for the absence of it in the regular case would require much larger samples of vectors , a possibility that is ruled out by restrictions on computer memory and time . the hypothesis most strongly supported by our data is the random character of the distribution of vectors in the chaotic case . in the following section we show that randomness in the distribution of perturbed vectors implies hypersensitivity to perturbation . guided by our numerical results we now present an analysis of hypersensitivity to perturbation for quantum systems based on the conjecture that , for chaotic systems , hilbert space is explored randomly by the perturbed vectors . we consider a hamiltonian quantum system whose classical phase - space dynamics is chaotic and assume the system is perturbed by a stochastic hamiltonian that classically gives rise to diffusion on phase space . we suppose that at time @xmath15 the system s state vector has a wigner distribution that is localized on phase space . we further assume that at @xmath15 the perturbation is just becoming effective in the classical sense described in sec . [ secclassical ] . our numerical analyses @xcite suggest the following picture . for times @xmath119 , the entropy @xmath1 of the average density operator @xmath120 ( [ eqrhos ] ) increases linearly with time . this is in accordance with an essentially classical argument given by zurek and paz @xcite . denoting the proportionality constant by @xmath121 , we have @xmath122 since the von neumann entropy of a density operator is bounded by the logarithm of the dimension of hilbert space , it follows that the realizations of the perturbation i.e . , the state vectors that result from the different perturbation histories explore at least a number @xmath123 of hilbert - space dimensions , which increases exponentially . our main conjecture now is that these dimensions are explored quasi - randomly , i.e. , that the realizations of the perturbation at time @xmath18 are distributed essentially like random vectors in a @xmath124-dimensional hilbert space . starting from this main conjecture , we will now derive an estimate of the information @xmath3 needed to keep the system - entropy increase below the tolerable amount @xmath2 . following the discussion on grouping vectors in sec . [ secdist ] , a tolerable entropy increase @xmath29 corresponds to gathering the realizations of the perturbation into hilbert - space spheres of radius @xmath68 . the state vectors in each such sphere fill it randomly ( since the perturbation is diffusive , there are plenty of vectors ) , so the entropy of their density operator which is the tolerable entropy is @xmath125 ( eq . ( b6 ) of @xcite ) . the number of spheres of radius @xmath68 in @xmath126-dimensional hilbert space is @xmath127 ( eq . ( 5.1 ) of @xcite ) , so the information needed to specify a particular sphere is @xmath128 the information @xmath129 consistently underestimates the actual value of @xmath3 , which comes from an optimal grouping of the random vectors ; the reason is that the perfect grouping into nonoverlapping spheres of uniform size assumed by eq . ( [ eqiphi ] ) does not exist . using eq . ( [ eqiphi ] ) to eliminate @xmath68 from eq . ( [ eqhphi ] ) gives an expression for @xmath2 as a function of @xmath129 , @xmath130 from which @xmath126 could be eliminated in favor of @xmath1 by invoking eq . ( [ eqd ] ) . the behavior of @xmath129 as a function of @xmath2 expressed in eq . ( [ eqhi ] ) is the universal behavior that we conjecture for chaotic systems , except for when @xmath10 is so close to @xmath1 that @xmath131 , as the spheres approximation used above breaks down for angles @xmath68 for which hilbert space can accommodate only one sphere . since @xmath2 increases and @xmath129 decreases with @xmath68 , @xmath132 increases as @xmath2 decreases from its maximum value of @xmath1 . to gain more insight into eq . ( [ eqhi ] ) , we calculate the derivative @xmath133 which is the marginal tradeoff between between information and entropy . for @xmath68 near @xmath83 , so that @xmath134 , the information becomes @xmath135 , and the derivative ( [ eqdi ] ) can be written as @xmath136 for @xmath137 , i.e. , when eq . ( [ eqhi ] ) is valid , the size of the derivative ( [ eqdiapprox ] ) is determined by @xmath138 , with a slowly varying logarithmic correction . this behavior , characterized by the typical slope @xmath124 , gives an _ exponential _ hypersensitivity to perturbation , with the classical number of correlation cells , @xmath20 , roughly replaced by the number of explored hilbert - space dimensions , @xmath124 . it is a remarkable fact that the concept of perturbation cell or perturbation correlation length ( see sec . [ secclassical ] ) did not enter this quantum - mechanical discussion . indeed , our numerical results suggest that our main conjecture holds for a single correlation cell , i.e. , for a perturbation that is correlated over all of the relevant portion of phase space . that we find this behavior indicates that we are dealing with an intrinsically quantum - mechanical phenomenon . what seems to be happening is the following . for tolerable entropies @xmath139 , where @xmath11 is the dimension of classical phase space as in sec . [ secclassical ] , we can regard a single - cell perturbation as perturbing a classical system into a set of nonoverlapping densities . in a quantum analysis these nonoverlapping densities can be crudely identified with orthogonal state vectors . the single - cell quantum perturbation , in conjunction with the chaotic quantum dynamics , seems to be able to produce arbitrary linear superpositions of these orthogonal vectors , a freedom not available to the classical system . the result is a much bigger set of possible realizations of the perturbation . this paper compares and contrasts hypersensitivity to perturbation in classical and quantum dynamics . although hypersensitivity provides a characterization of chaos that is common to both classical and quantum dynamics , the mechanisms for hypersensitivity are different classically and quantum mechanically . the classical mechanism has to do with the information needed to specify the phase - space distributions produced by the perturbation this is classical information whereas the quantum mechanism has to do with the information needed to specify the random state vectors produced by the perturbation this is quantum information because it relies on the superposition principle of quantum mechanics . captured in a slogan , the difference is this : _ a stochastic perturbation applied to a classical chaotic system generates classical information , whereas a stochastic perturbation applied to a quantum system generates quantum information_.

hypersensitivity to perturbation is a criterion for chaos based on the question of how much information about a perturbing environment is needed to keep the entropy of a hamiltonian system from increasing . in this paper we give a brief overview of our work on hypersensitivity to perturbation in classical and quantum systems . #

employing the surface brightness fluctuation signal of unresolved stars in distant galaxies is an effective and inexpensive new way to measure accurate distances to early - type ( dwarf ) galaxies . unlike other extragalactic distance indicators ( e.g. trgb , rr lyrae stars ) , this method does _ not _ require resolved stars therefore allowing distance measurements for early - type galaxies far beyond the practical limits of any of the classical distance indicators ( @xmath05mpc ) . with fourier analysis techniques , the sbf method quantifies the mean stellar flux per ccd pixel and rms variation due to poisson noise across a designated area in a dwarf galaxy . initially the sbf method was almost exclusively applied on nearby giant ellipticals and mw globular clusters ( e.g. tonry et al . 1989 , 1994 ) but was found to work equally well with dwarf elliptical ( de ) galaxies ( e.g. jerjen et al . 1998 , 2000 , 2001 , 2004 , and rekola et al . 2005 ) . as de galaxies are by far the most numerous galaxy type at the current cosmological epoch , the sbf method in combination with wide - field ccd imaging offers the opportunity for the first time to spatially locate des in vast numbers and thereby to map in 3d the densest environments of the local universe ( for first results see contributions by ct et al . , jerjen , jordan et al . , and rekola et al . in this volume ) . first sbf distances are published for des as distant as 15mpc ( using 2 m ground - based telescopes ) and 25mpc ( using 8 m vlt+fors and hst & acs ) . the _ minimal requirements _ for the sbf analysis of an early - type galaxy are : * galaxy morphology : the light distribution of the stellar system must be radially symmetric and have minimal structure . an overall elliptical shape of the galaxy is crucial as this is modelled and subtracted as part of the sbf analysis . * photometry : calibrated ccd images are required in two photometric bands , e.g. ( @xmath1 ) or ( @xmath2 , @xmath3 ) , as the fluctuation magnitude shows a colour dependency . * image quality : fwhm @xmath4/20 $ ] , where @xmath5 is the half - light radius of the galaxy . * integration time : @xmath6s / n@xmath7 , where @xmath8 is the mean surface brightness of the galaxy , @xmath9 the surface brightness of the sky background , @xmath10 the estimates distance modulus of the galaxy , @xmath11 the fluctuation luminosity of the underlying stellar population , and @xmath12 the magnitude of a star providing 1 count / sec on the ccd detector at the telescope . to give a general idea of these constraints , fig . [ fig1 ] illustrates the depth required for an image of a de at the distance of the fornax cluster observed with vlt+fors1 . the sbf amplitude above the shot noise level ( signal - to - noise ) in the power spectrum is shown as a function of integration time and mean effective surface brightness of the galaxy . a sbf distance can be determined when the s / n is approximately 0.5 , ( see fig . 8 in rekola et al . 2005 ) , but that depends largely on the image quality i.e. seeing . for example , to achieve a s / n@xmath02 in the galaxy power spectrum , the minimum exposure time required for a de with a mean surface brightness of 25 magarcsec@xmath13 is 1600s . it is interesting to note that this exposure time is by a factor of 20 shorter than the 32,000s of hst time spent by harris et al . ( 1998 ) to measure the trgb distance of a dwarf elliptical at a similar distance . previous sbf work has entailed individuals hand selecting regions in galaxy images for the analysis . to make the results as impartial as possible and data reduction more efficient we are developing a rapid , semi - automatic sbf analysis package named sapac that can process large numbers of galaxies . sapac is a software package that carries out a semi - automatic sbf analysis of any early - type galaxy for which ccd data meets the requirements as discussed above . for a detailed description of the fluctuation magnitude calibration and the individual reduction steps such as the modelling of the galaxy , foreground star removal , selection of sbf fields etc . we refer the reader to jerjen ( 2003 ) . sapac consists of perl scripts using and iraf module and uses a sophisticated graphical user interface , also written in perl . the average processing time for 10 sbf fields in a galaxy and measuring a distance is approximately 20 minutes . initially we have concentrated the pipeline on @xmath14 , @xmath15 images , but the implementation of calibration information for a wider range of commonly used filter sets for sbf work like @xmath16 of the sdss @xmath17 filters is in process . potential users of sapac who are interested in testing this package for calculating accurate distances of early - type dwarfs are welcome to contact laura dunn . this software package will be made available to the astronomical community soon .

large volumes of ccd imaging data that will become available from wide - field cameras at telescopes such as the cfht , subaru , vst , or vista in the near future are highly suitable for systematic _ distance surveys of early - type galaxies _ using the surface brightness fluctuation ( sbf ) method . for the efficient processing of such large data sets , we are developing the first semi - automatic sbf analysis pipeline named sapac . after a brief description of the sbf method we discuss the image quality needed for a successful distance measurement and give some background information on sapac .

in the bcs to bose - einstein condensation ( bec ) crossover @xcite , largely overlapping cooper pairs smoothly evolve into non - overlapping composite bosons as the fermionic attraction is progressively increased . these two physical situations ( cooper pairs vs composite bosons ) correspond to the weak- and strong - coupling limits of the theory , while in the interesting intermediate - coupling regime neither the fermionic nor the bosonic properties are fully realized . under these circumstances , the theory is fully controlled on the weak- and strong - coupling sides , while at intermediate coupling an interpolation scheme results ( as for all crossover approaches ) . these physical ideas are implemented , in practice , by allowing for a strong decrease of the chemical potential _ at a given temperature _ when passing from the weak- to the strong - coupling limit . the bcs - bec crossover can be considered both below ( broken - symmetry phase ) and above ( normal phase ) the superconducting critical temperature . in particular , in the normal phase preformed pairs exist in the strong - coupling limit up to a temperature @xmath0 corresponding to the breaking of the pairs , while coherence among the pairs is established when the temperature is lowered below the superconducting critical temperature @xmath1 . this framework could be relevant to the evolution of the properties of high - temperature cuprate superconductors from the overdoped ( weak - coupling ) to the underdoped ( strong - coupling ) regions of their phase diagram @xcite . the bcs - bec crossover can be also explicitly realized with ultracold fermionic atoms in a trap , by varying their mutual effective attractive interaction via a fano - feshbach resonance @xcite . the bcs - bec crossover has been studied extensively in the past , either at @xmath2 or for @xmath3 . at @xmath2 , the solution of the two coupled bcs ( mean - field ) equations for the order parameter @xmath4 and the chemical potential @xmath5 has been shown to cross over smoothly from a bcs weak - coupling superconductor with largely overlapping cooper pairs to a strong - coupling superconductor where tightly - bound pairs are condensed in a bose - einstein ( coherent ) ground state @xcite . for this reason , the bcs mean field has often been considered to be a reliable approximation for studying the whole bcs - bec crossover at @xmath2 . at finite temperature , the increasing importance in strong coupling of the thermal excitation of collective modes ( corresponding to noncondensed bosons ) was first pointed out by nozires and schmitt - rink @xcite . by their approach , the expected result that the superconducting critical temperature should approach the bose - einstein temperature @xmath6 in strong coupling was obtained ( coming from _ above @xmath1 _ ) via a ( first - order ) inclusion of the @xmath7-matrix self - energy in the fermionic single - particle green s function . the same type of @xmath7-matrix approximation ( also with the inclusion , by some authors , of self - consistency ) has then been widely adopted to study the bcs - bec crossover above @xmath1 , both for continuum @xcite and lattice models @xcite . despite its conceptual importance , a systematic study of the bcs - bec crossover in the temperature range @xmath8 is still lacking . a diagrammatic theory for the bcs - bec crossover that extends below @xmath1 the self - consistent @xmath7-matrix approximation was proposed some time ago by haussmann @xcite . the ensuing coupled equations for the order parameter and chemical potential were , however , solved explicitly only at @xmath1 , @xcite leaving therefore unsolved the problem of the study of the whole temperature region below @xmath1 . the work by levin and coworkers @xcite , on the other hand , even though based on a `` preformed - pair scenario '' , has focused mainly on the weak - to - intermediate coupling region , where the fermionic chemical potential remains inside the single - particle band . an extension of the self - consistent @xmath7-matrix approximation to the superconducting phase for a two - dimensional lattice model was considered in ref . . in that paper , however , the shift of the chemical potential associated with the increasing coupling strength was ignored , by keeping it fixed at the noninteracting value.@xcite the results of ref . are thus not appropriate to address the bcs - bec crossover , for which the renormalization of the chemical potential ( that evolves from the fermi energy in weak coupling to half the binding energy of a pair in strong coupling ) plays a crucial role @xcite . additional studies have made use of a fermion - boson model @xcite , especially in the context of trapped fermi gases @xcite . purpose of the present paper is to study the bcs - bec crossover in the superconducting phase over the whole temperature range from @xmath2 to @xmath9 , thus filling a noticeable gap in the literature . we will consider a three - dimensional continuum model , for which the fermionic attraction can be modeled by a point - contact interaction . as noted in refs . and , with this model the structure of the diagrammatic theory for the single - particle fermionic self - energy simplifies considerably , since only limited sets of diagrammatic structures survive the regularization of the contact potential in terms of the fermionic two - body scattering length @xmath10 . @xcite the dimensionless interaction parameter @xmath11 ( where the fermi wave vector @xmath12 is related to the density via @xmath13 ) then ranges from @xmath14 in weak coupling to @xmath15 in strong coupling . the crossover region of interest is , however , restricted in practice by @xmath16 . for this model , a systematic theoretical study of the evolution of the single - particle spectral function in the normal phase from the bcs to bec limits has been presented recently @xcite . like in ref . , also in ref . the coupling of a fermionic single - particle excitation to a ( bosonic ) superconducting fluctuation mode was taken into account by the @xmath7-matrix self - energy . this approximation embodies the physics of a dilute fermi gas in the weak - coupling limit and reduces to a description of independent composite bosons in the strong - coupling limit . in this way , single - particle spectra were obtained in ref . as functions of coupling strength and temperature . in the present paper , the @xmath7-matrix approximation for the self - energy is suitably extended below @xmath1 . in particular , the _ same _ superconducting fluctuations , that in refs . and were coupled to fermionic independent - particle excitations above @xmath17 , are now coupled to fermionic bcs - like single - particle excitations below @xmath17 . in the strong - coupling limit , it turns out that these superconducting fluctuations merge in a nontrivial way@xcite into a state of condensed composite bosons described by the bogoliubov theory , and evolve consistently into a state of independent composite bosons above @xmath17 ( as the bogoliubov theory for point - like bosons does @xcite ) . in this way , a direct connection is established between the structures of the single - particle fermionic self - energy above _ and _ below @xmath17 , as they embody the same kind of bosonic mode which itself evolves with temperature . a comment on the validity of the bogoliubov theory at finite temperature ( and , in particular , close to the bose - einstein transition temperature @xmath18 ) might be relevant at this point . a consistent theory for a _ dilute _ condensed bose gas was developed long ago in terms of a ( small ) gas parameter @xcite , of which the bogoliubov theory @xcite is only an approximate form valid at low enough temperatures ( compared with @xmath18 ) . that theory correctly describes also the dilute bose gas in the normal phase @xcite , whereas the bogoliubov theory ( when extrapolated above the critical temperature ) recovers the independent - boson form ( albeit in a non - monotonic way , with a discontinuous jump affecting the bosonic condensate @xcite ) . it would therefore be desirable to identify ( at least in principle ) a fermionic theory that , in the strong - coupling limit of the fermionic attraction , maps onto a more sophisticated bosonic theory , overcoming the apparent limitations of the bogoliubov theory . in practice , however , it should be considered already a nontrivial achievement of the present approach the fact that the bosonic bogoliubov approximation can be reproduced from an originally fermionic theory . for these reasons , and also because it is actually the intermediate - coupling ( crossover ) region to be of most physical interest , in the following we shall consider the bogoliubov approximation as a reasonable limiting form of our fermionic theory . as it is always the case for the bcs - bec crossover approach , implementation of the theory developed in the present paper rests on solving two coupled equations for the order parameter @xmath4 and the chemical potential @xmath5 . the equations here considered for @xmath4 and @xmath5 generalize the usual equations already considered at the mean - field level @xcite , by including fluctuation corrections . our equations reproduce the expected physics in the strong - coupling limit , at least at the level of approximation here considered . their solution provides us with the values of @xmath4 and @xmath5 as functions of coupling strength @xmath11 and temperature @xmath19 , thus extending results obtained previously at the mean - field level . in particular , the order parameter is now found to vanish at a temperature ( close to ) @xmath17 even in the strong - coupling limit , while it would had vanished close to @xmath20 at the mean - field level @xcite . the analytic continuation of the fermionic self - energy to the real frequency axis is further performed to obtain the single - particle spectral function @xmath21 , that we study in a systematic way as a function of wave vector @xmath22 , frequency @xmath23 , coupling strength @xmath11 , and temperature @xmath19 . in this context , two novel sum rules ( specific to the broken - symmetry phase ) are obtained , which provide compelling checks on the numerical calculations . in addition , the numerical calculations are tested against analytic ( or semi - analytic ) approximations obtained in the strong - coupling limit . the study of a dynamical quantity like @xmath24 enables us to attempt a comparison with the experimental arpes and tunneling spectra for cuprate superconductors below @xmath1 , for which a large amount of data exists showing peculiar features for different doping levels and temperatures . as in ref . above @xmath1 , this comparison concerns especially the experimental data about the m points in the brillouin zone of cuprates , where pairing effects are supposed to be stronger than along the nodal lines . our main results are the following . about thermodynamic quantities , we will show that fluctuation corrections over and above mean field are especially important at finite temperature @xmath25 when approaching the strong - coupling limit . at zero temperature , fluctuation corrections to thermodynamic quantities turn out to be of some relevance only in the intermediate - coupling region . this supports the expectation @xcite that the bcs mean field at zero temperature should describe rather well the bcs - bec crossover essentially for all couplings . regarding instead dynamical quantities like @xmath24 , our calculation based on a `` preformed - pair scenario '' reveals two distinct spectral features for @xmath26 . these features , which have different temperature and doping dependences , together give rise to a peak - dip - hump structure which is actively debated for the arpes spectra of cuprate superconductors . our results differ from those previously obtained by other calculations @xcite also based on a `` preformed - pair scenario '' , where a single feature was instead obtained in the spectral function for @xmath26 . an explanation of this discrepancy between the two calculations will be provided . it will also turn out from our calculation that the coherent part of @xmath24 for @xmath26 follows essentially a bcs - like behavior as far as its wave - vector dependence is concerned , albeit with a gap value which contains an important contribution from fluctuations at finite temperature . the same bcs - like behavior is not found , however , by our calculation for the dependence of the spectral weight of the coherent peak on temperature and coupling . this evidences a dichotomy in the behavior of @xmath24 , according to which of its dependences one is after . such a dichotomy is clearly observed in experiments on cuprate superconductors , in good qualitative agreement with the results obtained by our calculations . . a detailed quantitative comparison of our results with the experimental data on cuprates would , however , require a more refined theoretical model , as to include the quasi - two - dimensional lattice structure , the @xmath27-wave character of the superconducting gap , and also a fermionic attraction that depends effectively on doping ( and possibly on temperature ) . future work on this subject should address these additional issues . the present theory could be improved in several ways . in the present approach , the effective interaction between the composite bosons is treated within the born approximation . for a dilute system of composite bosons one knows how to improve on this result , as shown in ref . ( see also ref . ) . in addition , the bogoliubov description for the composite bosons could be also improved , for instance , by extending to the composite bosons the popov treatment for point - like bosons @xcite . finally , on the weak - coupling side of the crossover the bcs theory could be modified by including the contributions shown by gorkov and melik - barkhudarov @xcite to yield a finite renormalization of the critical temperature and of the gap function _ even _ in the extreme weak - coupling limit . work along these lines is in progress . the plan of the paper is as follows . in sec . ii we discuss our choice for the fermionic self - energy in the superconducting phase , from which the order parameter @xmath4 and the chemical potential @xmath5 are obtained as functions of temperature and coupling strength , and the spectral function @xmath21 also results . analytic results are presented in the strong - coupling limit , where the order parameter is shown to be connected with the bosonic condensate density of the bogoliubov theory . in addition , the analytic continuation of our expressions for the fermionic self - energy and spectral function is carried out in detail . in sec . iii we present our numerical calculations , and discuss the results for the single - particle spectral function in the context of the available experimental data for high - temperature cuprate superconductors . section iv gives our conclusions . in appendix a two sum rules are derived for the superconducting phase , which are used as checks of the numerical results . in this section , we discuss the choice of the fermionic single - particle self - energy in the superconducting phase for a ( three - dimensional ) continuum system of fermions mutually interacting via an attractive point - contact potential , with an @xmath28-wave order parameter . we shall place special emphasis to the strong - coupling limit of the theory , where composite bosons forms as bound fermion pairs . we extend in this way _ below _ @xmath17 an analogous treatment for the self - energy , made previously in the normal phase to calculate the single - particle spectral function.@xcite knowledge of the detailed form of the attractive interaction is not generally required when studying the bcs - bec crossover . accordingly , one may consider the simple form @xmath29 of a `` contact '' potential , where @xmath30 is a negative constant . this choice entails a suitable regularization in terms , e.g. , of a cutoff @xmath31 in wave - vector space . in three dimensions , this is achieved via the scattering length @xmath32 of the associated fermionic two - body problem , by choosing @xmath30 as follows @xcite : @xmath33 @xmath34 being the fermion mass . with this choice , the classification of the ( fermionic ) many - body diagrams is considerably simplified not only in the normal phase @xcite but also in the broken - symmetry phase @xcite , since only specific diagrammatic substructures survive when the limit @xmath35 ( and thus @xmath36 ) is eventually taken . in particular , the _ particle - particle ladder _ depicted in fig . 1(a ) survives the regularization of the potential.@xcite it is obtained by the matrix inversion : @xmath37^{-1 } \label{gamma - solution}\end{aligned}\ ] ] with the notation @xmath38 \label{a - definition}\\ \chi_{12}(q ) & = & \int \ ! \frac{d { \mathbf p}}{(2\pi)^{3 } } \ , \frac{1}{\beta } \ , \sum_{n } \ , { \mathcal g}_{12}(p+q ) \,{\mathcal g}_{21}(-p ) \,\,\ , . \label{b - definition}\end{aligned}\ ] ] in these expressions , @xmath39 and @xmath40 , where @xmath41 and @xmath42 are wave vectors , and @xmath43 ( @xmath44 integer ) and @xmath45 ( @xmath46 integer ) are bosonic and fermionic matsubara frequencies , respectively ( with @xmath47 , @xmath48 being the boltzmann s constant ) ; @xmath49 are the bcs single - particle green s functions in nambu notation , with @xmath50 and @xmath51 for an isotropic ( @xmath28-wave ) order parameter @xmath4 . [ hereafter , we shall take the order parameter to be real with no loss of generality . ] the expressions ( [ a - definition ] ) and ( [ b - definition ] ) for @xmath52 and @xmath53 considerably simplify _ in the strong - coupling limit _ ( that is , when @xmath54 and @xmath55 ) . in this limit , one then obtains for the matrix elements ( [ gamma - solution ] ) @xcite : @xmath56 and @xmath57 where @xmath58 has the form of the bogoliubov dispersion relation @xcite ( @xmath59 being the bosonic mass , @xmath60 the bosonic chemical potential , and @xmath61 the bound - state energy of the associated fermionic two - body problem ) . the above relation between the fermionic and bosonic chemical potentials holds provided @xmath62 ( cf . also sec . note that @xmath63 can be cast in the bogoliubov form @xmath64 where @xmath65 is the residual bosonic interaction @xcite and @xmath66 is the _ condensate density_. the relation ( [ pot - chim - bog ] ) is formally obtained already at the ( bcs ) mean - field level @xcite , albeit with an unspecified dependence of @xmath67 on temperature . within our fluctuation theory , the temperature dependence of @xmath67 will coincide in strong coupling with the expression given by the bogoliubov theory ( see sec . in particular , at zero temperature and at the lowest order in the residual bosonic interaction@xcite , @xmath68 reduces to the bosonic density @xmath69 and @xmath63 is given by @xmath70 . pauli matrices . only combinations with @xmath71 and @xmath72 occur owing to the regularization we have adopted for the potential . ( b ) fermionic self - energy diagram associated with the expression ( [ sigma - normal - phase ] ) in the normal phase . ( c ) fermionic self - energy diagram associated with the expressions ( [ sigma - broken-11 ] ) and ( [ sigma - broken-12 ] ) in the broken - symmetry phase . ( d ) bcs contribution ( 15 ) to the self - energy . ] note further that the above result for @xmath73 can be cast in the bosonic form @xmath74 with @xmath75 . the present theory thus describes the effective interaction between the composite bosons within the born approximation , while improved theories@xcite for @xmath76 would give smaller values for the ratio @xmath77 . these improvements will not be considered in the present paper . apart from the overall factor @xmath78 ( and a sign difference in the off - diagonal component @xcite ) , the expressions ( [ gamma-11-approx ] ) and ( [ gamma-12-approx ] ) coincide with the normal and anomalous non - condensate bosonic green s functions within the bogoliubov approximation @xcite , respectively . these expressions will be specifically exploited in sec . iid , where the strong - coupling limit of the fermionic self - energy will be analyzed in detail . in the normal phase , on the other hand , the bcs single - particle green s functions are replaced by the bare single - particle propagator @xmath79^{-1}$ ] , while for arbitrary coupling the particle - particle ladder acquires the form : @xmath80 \right\}^{-1 } \ , . \label{most - general - pp - sc}\end{aligned}\ ] ] in particular , in the strong - coupling limit the expression ( [ most - general - pp - sc ] ) reduces to @xmath81 which coincides ( apart again from the overall factor @xmath82 ) with the free - boson green s function . the above quantities constitute the essential ingredients of our theory for the fermionic self - energy and related quantities in the broken - symmetry phase . as shown in ref . , they also serve to establish a _ mapping _ between the fermionic and bosonic diagrammatic structures in the broken - symmetry phase , in a similar fashion to what was done in the normal phase @xcite . in a recent study @xcite of the single - particle spectral function in the normal phase based on the bcs - bec crossover approach , the fermionic self - energy was taken of the form : @xmath83 where @xmath84 is the quantization volume and @xmath85 is again a four - vector notation with wave vector @xmath86 and fermionic matsubara frequency @xmath87 ( @xmath28 integer ) . in this expression , @xmath88 is given by eq . ( [ most - general - pp - sc ] ) for arbitrary coupling and @xmath89 is the bare single - particle propagator . the self - energy diagram corresponding to the expression ( [ sigma - normal - phase ] ) is depicted in fig . the fermionic single - particle excitations are effectively coupled to a ( bosonic ) superconducting fluctuation mode , which reduces to a free composite boson in the strong - coupling limit . physically , the choice ( [ sigma - normal - phase ] ) for the self - energy entails the presence of a pairing interaction above @xmath17 , which can have significant influence on the single - particle ( as well as other ) properties . in the present paper , we choose the self - energy in the broken - symmetry phase below @xmath1 , with the aim of recovering the expression ( [ sigma - normal - phase ] ) when approaching @xmath1 from below and the bogoliubov approximation for the composite bosons in the strong - coupling limit . to this end , we adopt the _ simplest _ approximations to describe fermionic _ as well as _ bosonic excitations in the broken - symmetry phase , which reduce to bare fermionic and free bosonic excitations in the normal phase , respectively . these are the bcs single - particle green s functions ( [ bcs - green - function ] ) ( in the place of the bare single - particle propagator @xmath90 ) and the particle - particle ladder ( [ gamma - solution ] ) ( in the place of its normal - phase counterpart @xmath91 ) . by this token , the fermionic self - energy ( [ sigma - normal - phase ] ) is replaced by the following @xmath92 matrix : @xmath93 where the label @xmath94 refers to the particle - particle ladder . the corresponding self - energy diagram is depicted in fig . 1(c).@xcite the choice ( [ sigma - broken-11 ] ) and ( [ sigma - broken-12 ] ) for the self energy is made on physical grounds . a formal `` ab initio '' derivation of these expressions can also be done in terms of `` conserving approximations '' in the baym - kadanoff sense , that hold even in the broken - symmetry phase @xcite . in such a formal derivation , however , the single - particle green s functions entering eqs.([sigma - broken-11 ] ) and ( [ sigma - broken-12 ] ) ( also through the particle - particle ladder ( [ gamma - solution ] ) ) would be required to be self - consistently determined with the _ same _ self - energy insertions . in our approach , we take instead the single - particle green s functions to be of the bcs form ( [ bcs - green - function ] ) . the order parameter @xmath4 and chemical potential @xmath5 are obtained , however , via two coupled equations ( to be discussed in sec . iic ) that include the self - energy insertions ( [ sigma - broken-11 ] ) and ( [ sigma - broken-12 ] ) . in this way , we will recover the bogoliubov form ( [ gamma-11-approx ] ) and ( [ gamma-12-approx ] ) for the particle - particle ladder not only at zero temperature but also at finite temperatures ( and , in particular , close to the bose - einstein transition temperature ) . the choice ( [ sigma - broken-11 ] ) and ( [ sigma - broken-12 ] ) for the self energy is not exhaustive . in the broken - symmetry phase there , in fact , exists an additional self - energy contribution that survives the regularization ( [ v0 ] ) of the interaction potential in the limit @xmath35 , even though it does not contain particle - particle rungs@xcite . this additional self - energy diagram is the ordinary bcs contribution depicted in fig . 1(d ) , with the associated expression @xmath95 while the corresponding ( hartree - fock ) diagonal elements vanish with the regularization we have adopted . relating the expression ( [ sigma-12-bcs ] ) to the diagram of fig . 1(d ) rests on the validity of the bcs gap equation [ eq . ( [ bcs - gap_equation ] ) below ] , for _ arbitrary _ values of the chemical potential . for this , as well as for an additional reason ( cf . iid ) , we shall consistently consider that equation to hold for the order parameter @xmath4 . the choice ( [ sigma-12-bcs ] ) alone would be appropriate to describe the system in the weak - coupling ( bcs ) limit , where the superconducting fluctuation contributions ( [ sigma - broken-11 ] ) and ( [ sigma - broken-12 ] ) represent only small corrections . in the intermediate- and strong - coupling regions , on the other hand , both contributions ( [ sigma - broken-11])-([sigma - broken-12 ] ) _ and _ ( [ sigma-12-bcs ] ) might become equally significant ( depending on the temperature range below @xmath17 ) . we thus consider both contributions _ simultaneously _ and write the fermionic self - energy in the matrix form : @xmath96 in the following , however , we shall neglect @xmath97 in comparison to @xmath98 . it will , in fact , be proved in sec . iid that , in strong coupling , @xmath97 is subleading with respect to both @xmath98 and @xmath99 . inclusion of @xmath97 is thus not required to properly recover the bogoliubov description for the composite bosons in the strong - coupling limit . to summarize , the fermionic single - particle green s functions are obtained in terms of the bare single - particle propagator @xmath89 and of the self - energy ( [ sigma - broken-11 ] ) and ( [ sigma-12-bcs ] ) via the dyson s equation in matrix form : @xmath100 if only the bcs contribution ( [ sigma-12-bcs ] ) to the self - energy were retained , the fermionic single - particle green s functions @xmath101 ( @xmath102 ) would reduce to the bcs form ( [ bcs - green - function ] ) . upon including , in addition , the fluctuation contribution ( [ sigma - broken-11 ] ) to the self - energy , modified single - particle green s functions result , which we are going to study as functions of coupling strength and temperature . the choice of the self - energy ( [ sigma - broken-11 ] ) and ( [ sigma-12-bcs ] ) resembles the approximation for the self - energy introduced by popov@xcite for superfluid fermions in the dilute limit @xmath103 ( with @xmath104 ) . there is , however , an important difference between the popov fermionic approximation and our theory . we include in eq . ( [ sigma - broken-11 ] ) the full @xmath105 obtained by the matrix inversion of eq . ( [ gamma - solution ] ) ; popov instead neglects @xmath106 therein and approximate @xmath105 by @xmath107 , thus removing the feedback of the bogoliubov - anderson mode on the diagonal fermionic self - energy @xmath108 . retaining this mode is essential when dealing with the bcs - bec crossover , to describe the composite bosons in the strong - coupling limit by the bogoliubov approximation , as discussed in sec . iia . approaching the weak - coupling limit , on the other hand , the presence of the bogoliubov - anderson mode becomes progressively irrelevant and the self - energies coincide in the two theories . as a check on this point , we have verified that , in the weak - coupling limit and at zero temperature , @xmath108 obtained by our theory ( using the numerical procedures discussed in sec . iii ) reduces to @xmath109 , which is the expression obtained also with the popov approximation@xcite in the absence of the bogoliubov - anderson mode . there is another difference between the popov fermionic approximation and our theory as formulated in sec . iia , which concerns the off - diagonal fermionic self - energy @xmath110 . our expression ( [ sigma-12-bcs ] ) for @xmath110 was obtained from the diagram of fig . 1(d ) , where the single particle line represents the off - diagonal bcs green s function of eq . ( [ bcs - green - function ] ) with no insertion of the diagonal self - energy @xmath108 . within the popov approximation , on the other hand , @xmath110 is defined formally by the same diagram of fig . 1(d ) , but with the single - particle line being fully self - consistent ( and thus including @xmath108 ) . since @xmath108 turns out to approach a constant value @xmath111 in the weak - coupling limit ( as discussed above ) , inclusion of @xmath112 can be simply made by a shift of the chemical potential ( such that @xmath113 ) . this shift affects , however , the value of the gap function @xmath4 in a non - negligible way even in the extreme weak - coupling limit . neglecting this shift , in fact , results in a reduction by a factor @xmath114 of the bcs asymptotic expression @xmath115 $ ] for @xmath4 ( where @xmath116 ) . inclusion of the shift @xmath111 is thus important to recover the bcs value for @xmath4 in the ( extreme ) weak - coupling limit . the need to include the constant shift @xmath111 on the weak - coupling side of the crossover was also discussed in ref . while studying the spectral function @xmath24 in the normal phase with the inclusion of pairing fluctuations . in that context , inclusion of the shift @xmath111 proved necessary to have the pseudogap depression of @xmath24 centered about @xmath117 . inclusion of the shift @xmath111 in the broken - symmetry phase ( at least when approaching the critical temperature from below ) is thus also necessary to connect the spectral function @xmath24 with continuity in the weak - coupling side of the crossover . combining the above needs for @xmath4 and @xmath24 , we have introduced the constant shift @xmath111 for all temperatures below @xmath1 , by replacing @xmath5 with @xmath118 in the bcs green s functions ( [ bcs - green - function ] ) entering the convolutions ( [ a - definition ] ) and ( [ b - definition ] ) . the same replacement is made in the gap equation [ eq . ( [ bcs - gap_equation ] ) below ] . in the dyson s equation ( [ dyson - equation ] ) , however , @xmath5 is left unchanged since the constant shift @xmath111 is already contained in @xmath119 as soon as its @xmath120-dependence is irrelevant . accordingly , we have included this constant shift in the calculation of both thermodynamic and dynamical quantities in the weak - coupling side for @xmath121 , and neglected it for larger couplings when @xmath119 can no longer be approximated by a constant . ( in units of @xmath122 ) vs temperature @xmath19 ( units of @xmath1 ) and coupling @xmath123 . ] it turns out that the temperature dependence of @xmath111 is rather weak in the above coupling range . a plot of @xmath111 vs @xmath124 and @xmath123 is shown in fig . [ shift ] . here , the critical temperature @xmath1 is obtained by applying the thouless criterion from the normal phase as was done in ref . ( this procedure to obtain @xmath1 will be used in the rest of the paper ) . in this plot , the constant shift @xmath111 is obtained as @xmath125 , in analogy to what was also done in ref . . here , @xmath126 is the analytic continuation to the real frequency axis of the matsubara self - energy @xmath127 discussed in sec . iie . thermodynamic quantities , such as the order parameter @xmath4 and the chemical potential @xmath5 , are obtained directly in terms of the matsubara single - particle green s functions , without the need of resorting to the analytic continuation to the real frequency axis . quite generally , the order parameter @xmath4 is defined in terms of the `` anomalous '' green s function @xmath128 via @xmath129 [ cf . ( [ equation - of - motion ] ) ] , where the strength @xmath30 of the contact potential is kept to comply with a standard definition of bcs theory @xcite . one obtains : @xmath130 by the same token , the chemical potential @xmath5 can be obtained in terms of the `` normal '' green s function @xmath131 via the particle density @xmath46 : @xmath132 where @xmath133 . the two equations ( [ delta - g-12 ] ) and ( [ n - g-11 ] ) are coupled , since the green s functions depend on both @xmath4 and @xmath5 . the results of their numerical solution will be presented in the next section for various temperatures and couplings . in the following treatment , we shall deal with the two equations ( [ delta - g-12 ] ) and ( [ n - g-11 ] ) _ on a different footing_. specifically , we will enter in the density equation ( [ n - g-11 ] ) the expression for the normal green s function obtained from eq . ( [ dyson - equation ] ) , that includes both bcs _ and _ fluctuation contributions ( see eq . ( [ g-11-matsubara ] ) below ) . we will use instead in the gap equation ( [ delta - g-12 ] ) the bcs anomalous function ( [ bcs - green - function ] ) , that includes only the bcs self - energy ( [ sigma-12-bcs ] ) . in this way , the gap equation ( [ delta - g-12 ] ) reduces to the form @xmath134 \ , = \ , 0 \label{bcs - gap_equation}\ ] ] where the regularization of the contact potential in terms of the scattering length @xmath32 has been introduced . this equation has the same _ formal _ structure of the bcs gap equation , although the numerical values of the chemical potential entering eq . ( [ bcs - gap_equation ] ) differ from those obtained by the bcs density equation . this procedure ensures that the bosonic propagators ( [ gamma - solution ] ) in the broken - symmetry phase are _ gapless _ , as shown explicitly by the bogoliubov - type expressions ( [ gamma-11-approx ] ) and ( [ gamma-12-approx ] ) in the strong - coupling limit . in general , in fact , there is no _ a priori _ guarantee that a given ( conserving ) approximation for fermions would result into a `` gapless '' approximation @xcite for the composite bosons in the strong - coupling limit of the fermionic attraction . including fluctuation corrections to the bcs density equation as in eq . ( [ n - g-11 ] ) , on the other hand , results in the emergence of important effects in the strong - coupling limit of the theory , as discussed next . we proceed to show that the original fermionic theory , as defined by the dyson s equation ( [ dyson - equation ] ) , maps onto the bogoliubov theory for the composite bosons which form as bound - fermion pairs in the strong - coupling limit . to this end , we shall exploit the conditions @xmath135 and @xmath136 ( @xmath137 ) ( which _ define _ the strong - coupling limit ) in the ( matsubara ) expressions ( [ sigma - broken-11 ] ) and ( [ sigma - broken-12 ] ) for @xmath138 , thus also verifying that @xmath139 can be neglected . these expressions are calculated by performing the wave - vector and frequency convolutions with the approximate expressions ( [ gamma-11-approx ] ) and ( [ gamma-12-approx ] ) for the particle - particle ladder and the expressions ( [ bcs - green - function ] ) for the bcs single - particle green s functions . upon neglecting contributions that are subleading under the above conditions , we obtain in this way for the diagonal part of the self - energy : @xmath140 \ , . \label{sigma-11-strong - coupling}\end{aligned}\ ] ] in this expression , @xmath141 is the bcs dispersion of eqs . ( [ bcs - green - function ] ) , @xmath142 is the bogoliubov dispersion relation ( [ bogoliubov - disp ] ) , @xmath143^{-1}$ ] is the bose distribution , and @xmath144 are the standard bosonic factors of the bogoliubov transformation @xcite . in the numerators of the expressions within brackets in eq . ( [ sigma-11-strong - coupling ] ) , the bose functions are peaked at about @xmath145 and vary over a scale @xmath146 . similarly , the factors @xmath147 and @xmath148 are also peaked at about @xmath145 and vary over a scale @xmath149 . the denominators in the expression ( [ sigma-11-strong - coupling ] ) , on the other hand , vary over the much larger scale @xmath150 . for these reasons , we can further approximate the expression ( [ sigma-11-strong - coupling ] ) as follows : @xmath151 where @xmath152 \label{noncondensate - density}\ ] ] identifies the bosonic _ noncondensate density _ according to bogoliubov theory @xcite . note that in the normal phase ( when the condensate density @xmath153 of eq . ( [ pot - chim - bog ] ) vanishes ) , the noncondensate density ( [ noncondensate - density ] ) becomes the full bosonic density @xmath69 , and eq . ( [ sigma-11-n - prime ] ) reduces to the expression obtained in ref . directly from the form ( [ sigma - normal - phase ] ) of the fermionic self - energy . the off - diagonal self - energy @xmath154 can be analyzed in a similar way . since its magnitude is supposed to be the largest at zero temperature , we estimate it correspondingly for @xmath155 and @xmath156 as follows : @xmath157 at the leading order , we can neglect both @xmath4 and @xmath63 in the integrand , where the energy scale @xmath150 dominates . we thus obtain @xmath158 - see below - has been used ) . this represents a subleading contribution in the small dimensionless parameter @xmath159 with respect to both the bcs contribution @xmath160 and the diagonal fluctuation contribution @xmath161 . it can accordingly be neglected . within the above approximations , the inverse ( [ dyson - equation ] ) of the fermionic single - particle green s function reduces to : @xmath162 with the notation @xmath163 from eq . ( [ g - inverse - strong - coupling ] ) we get the desired expression for @xmath131 in the strong - coupling limit : @xmath164 where we have discarded a term of order @xmath165 with respect to @xmath150 . note that eq . ( [ g-11-strong - coupling ] ) has the same formal structure of the corresponding bcs expression ( [ bcs - green - function ] ) , with the replacement @xmath166 . we rewrite it accordingly as : @xmath167 with the modified bcs coherence factors @xmath168 . before making use of the asymptotic expression ( [ g-11-strong - coupling - figo ] ) in the density equation ( [ n - g-11 ] ) , it is convenient to manipulate suitably the gap equation ( [ bcs - gap_equation ] ) in the strong - coupling limit . expanding @xmath169 therein as @xmath170/\xi({\mathbf k})$ ] and evaluating the resulting elementary integrals , one obtains : @xmath171 setting further @xmath172 , one gets the relation @xmath173 quoted already after eqs . ( [ bogoliubov - disp ] ) and ( [ sigma-12-strong - coupling ] ) . let s now consider the density equation ( [ n - g-11 ] ) . with the bcs - like form ( [ g-11-strong - coupling - figo ] ) one obtains immediately : @xmath174 that holds for @xmath175 , at temperatures well below the dissociation threshold of the composite bosons . similarly to what was done to get the gap equation ( [ delta - mu - strong - coupling ] ) , in eq . ( [ n - g-11-strong - coupling ] ) one expands @xmath176 as @xmath177/\xi({\mathbf k})$ ] and evaluates the resulting elementary integrals , to obtain : @xmath178 recalling the definition ( [ delta - o ] ) for @xmath179 , as well as the expressions ( [ delta - mu - strong - coupling ] ) and ( [ pot - chim - bog ] ) for the order parameter , which we rewrite in the form @xmath180 in analogy to eq . ( [ delta - o ] ) , the result ( [ final - n ] ) becomes eventually : @xmath181 that holds asymptotically for @xmath182 . these results imply that , in the strong - coupling limit , the original fermionic theory recovers the bogoliubov theory for the composite bosons , not only at zero temperature but also _ at any temperature _ in the broken - symmetry phase . accordingly , the noncondensate density @xmath183 is given by the expression ( [ noncondensate - density ] ) , the bosonic factors @xmath148 and @xmath147 are given by eq . ( [ u - v - bogoliubov ] ) , and the dispersion relation @xmath142 is given by eq . ( [ bogoliubov - disp ] ) . in the strong - coupling limit , the present fermionic theory thus inherits all virtues and shortcomings of the bogoliubov theory for a weakly - interacting bose gas @xcite . the present fermionic theory at arbitrary coupling then provides an interpolation procedure between the bogoliubov theory for the composite bosons and the weak - coupling bcs theory plus pairing fluctuations . both these analytic limits will constitute important checks on the numerical calculations reported in sec . iii . note that inclusion of the off - diagonal fluctuation contribution @xmath184 to the self - energy is not required to recover the bogoliubov theory in strong coupling . for this reason , we will not consider @xmath184 altogether in the numerical calculations presented in sec . iii , as anticipated in eq . ( [ dyson - equation ] ) . the above analytic results enable us to infer the main features of the temperature dependence of the order parameter in the strong - coupling limit . in particular , the low - temperature behavior @xmath185 ( where @xmath186 is the sound velocity ) within the bogoliubov approximation , implies that @xmath187 decreases from @xmath188 with a @xmath189 behavior , in the place of the exponential behavior obtained within the bcs theory ( with an @xmath28-wave order parameter ) @xcite . in addition , in the present theory the order parameter vanishes over the scale of the bose - einstein transition temperature @xmath18 , while in the bcs theory it would vanish over the scale of the bound - state energy @xmath190 of the composite bosons . note finally that the fermionic quasi - particle dispersion @xmath191 , entering the expression ( [ g-11-strong - coupling - figo ] ) of the diagonal green s function in the strong - coupling limit , contains the sum @xmath192 instead of the single term @xmath193 of the bcs dispersion @xmath141 . we pass now to identify the form of the spectral function @xmath21 associated with the approximate choice of the matsubara self - energy of eq . ( [ dyson - equation ] ) . to this end , we need to perform the _ analytic continuation _ in the complex frequency plane , thus determining the _ retarded _ fermionic single - particle green s functions from their matsubara counterparts . the approach developed in this subsection holds specifically for the approximate choice for the self - energy of eq . ( [ dyson - equation ] ) . it thus differs from the general analysis presented in the appendix which holds for the exact green s functions , irrespective of any specific approximation . in general , the process of analytic continuation to the real frequency axis from the numerical matsubara green s functions proves altogether nontrivial , as it requires in practice recourse to approximate numerical methods such as , e.g. , the method of pad approximants @xcite . we then prefer to rely on a procedure whereby the analytic continuation to the real frequency axis is achieved by avoiding numerical extrapolations from the matsubara green s functions . the fermionic normal and anomalous matsubara single - particle green s functions are obtained at any given coupling from matrix inversion of eq . ( [ dyson - equation ] ) : @xmath194^{-1 } \label{g-11-matsubara } \\ & & g_{12}({\mathbf k},\omega_{s})= \delta [ \,(i\omega_{s}-\xi({\mathbf k})- \sigma_{11}({\mathbf k},\omega_{s}))\nonumber\\ & & \phantom{1111 } \times ( i\omega_{s}+\xi({\mathbf k})- \sigma_{22}({\mathbf k},\omega_{s } ) ) - \delta^{2}]^{-1}\ ; . \label{g-12-matsubara } \end{aligned}\ ] ] consider first the normal green s function ( [ g-11-matsubara ] ) , which we rewrite in the compact form @xmath195 with the short - hand notation @xmath196 to perform the analytic continuation of this expression , we look for a function @xmath197 of the complex frequency @xmath198 which satisfies the following _ requirements _ at any given @xmath22 : + ( i ) it is analytic off the real axis ; + ( ii ) it reduces to @xmath199 given by eq . ( [ sigma-11-compact ] ) when @xmath198 takes the discrete values @xmath200 on the imaginary axis ; + ( iii ) its imaginary part is negative ( positive ) for @xmath201 ( @xmath202 ) ; + ( iv ) it vanishes when @xmath203 along any straight line parallel to the real axis with @xmath204 . once the function @xmath197 is obtained , the expression @xmath205 ( @xmath206 being a positive infinitesimal ) represents the _ retarded _ ( @xmath207 ) single - particle green s function ( for real @xmath23 ) associated with the matsubara green s function ( [ g-11-compact ] ) , since it satisfies the requirements of the baym - mermin theorem @xcite for the analytic continuation from the matsubara green s function . the first step of the above program is to find the analytic continuation of @xmath208 ( and @xmath209 ) off the real axis in the complex @xmath198-plane . to this end , it is convenient to express @xmath208 via the spectral form : @xmath210 with the replacement @xmath211 , the spectral representation ( [ spect ] ) defines an analytic function @xmath212 off the real axis . in the case of interest with @xmath208 given by eq . ( [ sigma - broken-11 ] ) , the function @xmath213 of eq . ( [ spect ] ) reads : @xmath214 \nonumber \\ & + & v^2_{{\bf q}-{\bf k } } { \rm im}\,\gamma_{11}^r({\bf q } , \omega - e({\bf q}- { \bf k}))\nonumber\\ & \times & \left.\left[f(-e({\bf q}-{\bf k}))+b(\omega - e({\bf q}-{\bf k } ) ) \right ] \right\ } \label{imsig11}\end{aligned}\ ] ] where @xmath215^{-1}$ ] is the fermi distribution while @xmath216 and @xmath217 are the bcs coherence factors . to obtain the expression ( [ imsig11 ] ) , a spectral representation has been also introduced for @xmath105 entering eq . ( [ sigma - broken-11 ] ) , by writing : @xmath218 here , the spectral function @xmath219 is _ defined _ by @xmath220 , which is obtained from the definitions ( [ gamma - solution])-([b - definition ] ) with the replacement @xmath221 _ after _ the sum over the internal frequency @xmath222 has been performed therein . even in the absence of an explicit lehmann representation for @xmath105 , in fact , it can be shown that the spectral representation ( [ spectral - representation ] ) holds provided the function @xmath223 of the complex variable @xmath198 is analytic off the real axis . the crucial point is to verify that the denominator in eq . ( [ gamma - solution ] ) with the replacement @xmath224 never vanishes off the real axis . this property can be explicitly verified in the strong - coupling limit , as discussed below . for arbitrary coupling , we have checked it with the help of numerical calculations . for the validity of the expression @xmath225 , it is also required that @xmath226 vanishes for @xmath203 . this property can be proved directly from eqs . ( [ gamma - solution])-([b - definition ] ) , according to which @xmath226 has the asymptotic expression @xmath227 and thus vanishes for @xmath228 . once @xmath212 has been explicitly constructed according to the above prescriptions , @xmath229 is obtained as @xmath230 in accordance with eq . ( [ sigma - broken-11 ] ) . from the spectral representation ( [ spect ] ) for @xmath212 , it can be further shown that @xmath212 vanishes when @xmath203 along any straight line parallel to the real axis with @xmath231 . it can also be shown that @xmath232 ( @xmath233 ) when @xmath234 ( @xmath235 ) . this property follows from the spectral representation of @xmath212 , provided @xmath236 in eq . ( [ spect ] ) . for arbitrary coupling , we have verified that @xmath236 with the help of numerical calculations . in the strong - coupling limit , this condition can be explicitly proved , as discussed below . from these properties of @xmath212 ( and @xmath229 ) it can then be verified that the function @xmath237 satisfies the requirements ( i)-(iv ) stated after eq . ( [ sigma-11-compact ] ) . with the replacement @xmath238 , eq . ( [ g - r - compact ] ) follows eventually on the real frequency axis for the retarded green s function @xmath239 . for later convenience , we introduce the following notation on the real frequency axis : @xmath240 such that @xmath241 and @xmath242 from eq . ( [ spect ] ) it is also clear that @xmath243 , and that @xmath244 and @xmath245 are related by a kramers - kronig transform . as anticipated , the properties of the function @xmath246 , required above to obtain the retarded green s function ( [ g - r - compact ] ) on the real axis , can be explicitly verified in the strong - coupling limit without recourse to numerical calculations . in this case , the approximate expression ( [ gamma-11-approx ] ) can be used for @xmath105 . this can be cast in the form ( [ spectral - representation ] ) , with @xmath247\ ; . \label{imgamma}\end{aligned}\ ] ] entering the expression ( [ imgamma ] ) into eq . ( [ imsig11 ] ) and the resulting expression into eq . ( [ spect ] ) , one obtains for @xmath127 the sum of four terms : @xmath248 since in strong coupling @xmath249 , @xmath250 , and @xmath251 , the second and fourth term within braces on the right - hand side of the matsubara expression ( [ sig11 ] ) may be dropped . the simplified expression ( [ sigma-11-strong - coupling ] ) then results from eq . ( [ sig11 ] ) . in the strong - coupling limit , one would then be tempted to perform the analytic continuation @xmath252 directly from the expression ( [ sigma-11-strong - coupling ] ) . care must , however , be exerted on this point since _ the processes of taking the strong - coupling limit and performing the analytic continuation may not commute_. by performing the analytic continuation @xmath253 directly in eq . ( [ sig11 ] ) one , in fact , obtains two additional terms with respect to the analytic continuation of eq . ( [ sigma-11-strong - coupling ] ) . these two additional terms can not be dropped _ a priori _ by the presence of the small factor @xmath254 in the strong - coupling limit , because for real @xmath198 the corresponding energy denominators may vanish . retaining properly these two additional terms indeed affects in a qualitative way the spectral function @xmath24 in the strong - coupling limit , as discussed in sec . iii . with the expression obtained by the analytic continuation @xmath253 of eq . ( [ sig11 ] ) , one can prove explicitly that @xmath246 is analytic off the real axis and vanishes like @xmath255 along any straight line parallel to the real axis with @xmath256 , and that @xmath257= -{\rm sgn } [ { \rm im}\,\ , z]$ ] . in this way , the properties of the function @xmath246 , required to obtain the retarded green s function ( [ g - r - compact ] ) on the real axis , are explicitly verified in the strong - coupling limit . once the retarded green s function has been obtained in the form ( [ g - r - compact ] ) according to the above prescriptions , its imaginary part defines the spectral function @xmath258 which will be calculated numerically in sec . iii for a wide range of temperatures and couplings . in the appendix , it is shown at a formal level that @xmath21 satisfies the sum rule ( [ sum - rule - g - r ] ) . this sum rule will be considered an important test for the numerical calculations of sec . iii . to this end , it is necessary to prove that the sum rule ( [ sum - rule - g - r ] ) holds even for our approximate theory based on the dyson s equation ( [ dyson - equation ] ) . to prove the sum rule ( [ sum - rule - g - r ] ) for the approximate theory , it is sufficient that the approximate @xmath259 ( from which the retarded green s function ( [ g - r - compact ] ) results when @xmath260 ) behaves like @xmath255 for large @xmath261 . this property is verified by our theory , as shown above . as a consequence : @xmath262\nonumber\\ = - \frac{1}{\pi}\ , \mathrm{im } \left [ - \oint_{c } \ ! d\omega \ , g_{11}({\mathbf k},z ) \right ] = 1 \ ; \label{sum - rule - a}\end{aligned}\ ] ] where the contour @xmath263 is a half - circle in the upper - half complex plane with center in the origin , large radius ( such that the approximation @xmath264 is valid ) , and counterclockwise direction . finally , the analytic continuation of the anomalous matsubara single - particle green s function ( [ g-12-matsubara ] ) can be obtained by following the same procedure adopted for the normal green s function ( [ g-11-matsubara ] ) . one writes for the retarded anomalous green s function @xmath265^{-1}\ ; . \label{fret } \end{aligned}\ ] ] in the place of eqs . ( [ g - r - compact ] ) and ( [ sigma - tilde - real ] ) . in this case , the analytic properties of @xmath266 ( @xmath267 ) discussed above imply that @xmath268 asymptotically for large @xmath261 . as a consequence , the imaginary part of @xmath269 @xmath270 satisfies the two following sum rules : @xmath271 and @xmath272 these sum rules can be verified by introducing the contour @xmath263 as in eq . ( [ sum - rule - a ] ) . note again that these sum rules ( which are proved on general grounds in the appendix for the exact anomalous retarded single - particle green s function ) follow here from our approximate form of @xmath273 only on the basis of the properties of analyticity . verifying numerically the sum rules ( [ sum - rule - a ] ) , ( [ sum - rule - b-1 ] ) , and ( [ sum - rule - b-2 ] ) at any coupling and temperature will , in practice , constitute an important check on the validity of the above procedure for the analytic continuation . an additional numerical check on the validity of the whole procedure at intermediate - to - weak coupling will be provided by the merging of the results , obtained by calculating the spectral function @xmath21 when approaching @xmath17 from below , with the results previously obtained in the normal phase@xcite when approaching @xmath17 from above . in this section we present the numerical results based on the formal theory developed in sec . ii . specifically , in sec . iiia we present the results obtained by solving the coupled equations ( [ n - g-11 ] ) and ( [ bcs - gap_equation ] ) for the order parameter and chemical potential . section iiib deals instead with the numerical calculation of the spectral function ( [ akw ] ) in the broken - symmetry phase , over the whole coupling range from weak to strong . before presenting the numerical results for @xmath4 and @xmath5 , it is worth outlining briefly the numerical procedure we have adopted . at given temperature and coupling , the coupled equations ( [ n - g-11 ] ) and ( [ bcs - gap_equation ] ) for the unknowns @xmath4 and @xmath5 are solved via the newton s method . this requires knowledge of the self - energy @xmath274 of eq . ( [ sigma - broken-11 ] ) , with @xmath275 obtained from eqs . ( [ gamma - solution])-([b - definition ] ) . [ as anticipated , in the numerical calculations we neglect @xmath139 in comparison to @xmath276 , since inclusion of @xmath139 is not required to recover the bogoliubov results in the strong - coupling limit , as shown in sec . iid . ] to this end , the frequency sums in eqs . ( [ a - definition ] ) and ( [ b - definition ] ) are evaluated analytically , while the remaining wave - vector integral is calculated numerically by the gauss - legendre method . in particular , the radial wave - vector integral extending up to infinity is partitioned into an inner and an outer region , with the transformation @xmath277 exploited in the outer region . the bosonic frequency sum in eq . ( [ sigma - broken-11 ] ) requires special care , owing to its slow convergence and the lack of an intrinsic energy cutoff within our continuum model . we have accordingly partitioned this frequency sum into three regions , separated by the frequency scales @xmath278 and @xmath279 ( with @xmath280 . for @xmath281 , the frequency sum is calculated explicitly . for @xmath282 , the frequency sum is approximated with great accuracy by the corresponding numerical integral , owing to the slow dependence of @xmath105 on @xmath283 . finally , the tail of the frequency sum for @xmath284 ( where the asymptotic expression ( [ gammasy ] ) yields @xmath285 ) is evaluated analytically . typically , @xmath278 is taken of the order of the largest among the energy scales @xmath286 , and @xmath287 ; @xmath279 is then taken at least ten times @xmath278 . it turns out that it is most convenient to apply this procedure to the frequency sum in eq . ( [ sigma - broken-11 ] ) after the integration over the two angular variables of the wave vector @xmath288 has been performed analytically ; the remaining radial wave - vector integration is then performed numerically , with a cutoff much larger than the wave - vector scales @xmath289 , and @xmath290 . ( in units of @xmath122 ) vs temperature ( in units of @xmath1 ) for different values of the coupling @xmath123 . results obtained by the inclusion of fluctuations ( full lines ) are compared with mean - field results ( dashed lines ) . ] finally , the frequency sum in the particle number equation ( [ n - g-11 ] ) is evaluated by adding and subtracting the bcs green s function @xmath291 on the right - hand side of that equation , in order to speed up the numerical convergence . matsubara frequencies are here summed numerically up to a cutoff frequency , beyond which the sum is approximated by the corresponding numerical integral . the radial part of the wave - vector integral in eq . ( [ n - g-11 ] ) is also calculated numerically up to a cutoff scale beyond which a power - law decay sets in , so that the contribution from the tail can be calculated analytically . with the above numerical prescriptions , we have obtained the behavior of @xmath4 and @xmath5 vs temperature and coupling reported in figs . [ gapvst]-[mut0 ] . specifically , fig . [ gapvst ] shows the order parameter @xmath4 vs temperature for different couplings [ @xmath292 , from top to bottom ] , in the window @xmath293 where the crossover from weak to strong coupling is exhausted . comparison is made with the corresponding curves obtained within mean field ( dashed lines ) , when the bcs green s function @xmath294 enters eq . ( [ n - g-11 ] ) in the place of the dressed @xmath295 . in these plots , the temperature is normalized with respect to the critical temperature @xmath1 for the given coupling . this comparison shows that fluctuation corrections on top of mean field get progressively important at given coupling as the temperature is raised toward @xmath1 . close to @xmath1 , fluctuation corrections become even more important upon approaching the strong - coupling limit . near zero temperature , on the other hand , fluctuation corrections become negligible when approaching strong coupling . this confirms the expectation that , near zero temperature , the bcs mean field should be rather accurate both in the weak- and strong - coupling limits @xcite . note from fig . [ gapvst ] that @xmath4 jumps discontinuously close to the critical temperature when fluctuations are included on top of mean field . this jump becomes more evident as the coupling is increased . it reflects an analogous behavior of the condensate density near the critical temperature as obtained by the bogoliubov theory for point - like bosons @xcite . in the present theory this jump is carried over to the composite bosons , even at fermionic couplings [ as in the middle panel of fig . [ gapvst ] ] when the composite bosons are not yet fully developed . when the fermionic coupling increases beyond the values reported in fig . [ gapvst ] , however , the residual interaction between the composite bosons decreases further and the jump becomes progressively smaller . more refined theories for point - like bosons ( see , e.g. , ref . ) remove the jump of the bosonic condensate density , which thus should be considered as an artifact of the bogoliubov approximation . apart from this jump , note that when the temperature is decreased below @xmath1 the order parameter @xmath4 grows more rapidly with the inclusion of fluctuations than within mean field . ( in units of @xmath122 ) vs temperature ( in units of @xmath1 ) , for the same values of the coupling @xmath296 as in fig . [ gapvst ] . ] figure [ muvst ] shows the chemical potential @xmath5 vs temperature for the same coupling values of fig . [ gapvst ] . note that in weak coupling the chemical potential decreases slightly upon moving deep in the superconducting phase from @xmath1 to @xmath2 , in agreement with the bcs behavior . in strong coupling the opposite occurs , reflecting the behavior of the bosonic chemical potential @xmath297 within the bogoliubov theory . it should be , however , mentioned that with improved bosonic approximations @xcite , the bosonic chemical potential would rather decrease upon entering the condensed phase . at @xmath2 ( in units of @xmath122 ) vs the coupling @xmath296 . results obtained by the inclusion of fluctuations ( full line ) are compared with mean - field results ( dashed line ) . ] figure [ gapt0 ] shows the order parameter @xmath4 at zero temperature ( full line ) and the corresponding mean - field value ( dashed line ) vs the coupling @xmath123 . while @xmath4 increases monotonically in absolute value from weak to strong coupling ( as expected on physical grounds ) , the relative importance of the fluctuation corrections to the order parameter at zero temperature ( over and above mean field ) reaches a maximum in the intermediate - coupling region , never exceeding about 30% . this results confirms again that the bcs mean field is a reasonable approximation to the ground state for all couplings . figure [ mut0 ] shows the chemical potential @xmath5 at zero temperature vs the coupling parameter @xmath123 . the results obtained by the inclusion of fluctuations ( full lines ) are compared with mean field ( dashed lines ) . even for this thermodynamic quantity the fluctuation corrections to the mean - field results appear to be not too important at zero temperature . note , finally , that the values for @xmath4 and @xmath5 obtained from our theory at @xmath2 with the coupling value @xmath298 are in remarkable agreement with a recent quantum monte carlo calculation @xcite performed for the same coupling . our calculation yields , in fact , @xmath299 and @xmath300 , to be compared with the values @xmath301 and @xmath302 of ref . . [ in contrast , bcs mean field yields @xmath303 and @xmath304 . ] at @xmath2 ( in units of @xmath122 for @xmath305 and of @xmath306 for @xmath137 ) vs the coupling @xmath296 . results obtained by the inclusion of fluctuations ( full line ) are compared with mean - field results ( dashed line ) . ] in summary , the above results have shown that , for thermodynamic quantities like @xmath4 and @xmath5 , fluctuation corrections to mean - field values in the broken - symmetry phase are important only as far as the temperature dependence is concerned , while at zero temperature the mean - field results are reliable . for a generic value of the coupling , calculation of the imaginary part of the retarded self - energy @xmath307 ( with @xmath308 given by eq . @xmath309 ) requires us to obtain the imaginary part of the particle - particle ladder @xmath310 on the real - frequency axis , as determined by the formal replacement @xmath311 in the matsubara expressions ( [ gamma - solution])-([b - definition ] ) . after performing the frequency sum therein , the wave - vector integrals of eqs . ( [ a - definition ] ) and ( [ b - definition ] ) for the functions @xmath312 ( @xmath313 ) are evaluated numerically , by exploiting the properties of the delta function for the imaginary part and keeping a finite albeit small value of @xmath314 for the real part . of the pole of @xmath315 at @xmath2 ( full lines ) and boundary of the particle - particle continuum ( dashed lines ) for three characteristic couplings . ] direct numerical calculation of the imaginary part of the particle - particle ladder fails , however , when this part has the structure of a delta function for real @xmath23 at given @xmath288 . this occurs when the determinant in the denominator of eq . ( [ gamma - solution ] ) vanishes for real @xmath23 . to deal with this delta function , let s first consider the case @xmath2 for which three cases can be distinguished , according to : ( i ) @xmath316 and @xmath317 ( weak - to - intermediate coupling ) ; ( ii ) @xmath318 and @xmath317 ( intermediate coupling ) ; ( iii ) @xmath318 and @xmath137 ( intermediate - to - strong coupling ) . the curves @xmath319 where the ( analytic continuation of the ) determinant in the denominator of eq . ( [ gamma - solution ] ) vanishes are shown ( full lines ) for these three cases in figs . [ gamma ] ( a ) , ( b ) , and ( c ) , respectively . in these figures we also show the boundaries ( dashed lines ) delimiting the particle - particle continuum , where the imaginary part of the particle - particle ladder is nonvanishing and regular ( in the sense that it does not have the structure of a delta function ) . at finite temperature , the sharp boundary of the particle - particle continuum smears out , owing to the presence of fermi functions after performing the sum over the matsubara frequencies in eqs . ( [ a - definition ] ) and ( [ b - definition ] ) . the fermi functions produce , in fact , a finite ( albeit exponentially small with temperature ) imaginary part of the particle - particle ladder also below the ( dashed ) boundaries of fig . [ gamma ] , resulting in a landau - type damping of the bogoliubov - anderson mode @xmath319 . in addition , the finite imaginary part broadens the delta - function structure centered at the curves @xmath319 of fig . [ gamma ] , turning it into a lorentzian function . in practice , our numerical calculation takes advantage of this broadening occurring at finite temperature , and deals with smooth lorentzian functions instead of the delta - function peaks . @xcite as a further consistency check on our numerical calculations , we have sistematically verified that the three sum rules ( [ sum - rule - a ] ) , ( [ sum - rule - b-1 ] ) , ( [ sum - rule - b-2 ] ) are satisfied within numerical accuracy , for all temperatures and couplings we have considered . for @xmath320 vs frequency ( in units of @xmath122 ) at different temperatures for the coupling values @xmath321 -0.5 ( a ) , 0.1 ( b ) , and 0.5 ( c ) . ] the imaginary and real parts of the retarded self - energy @xmath322 obtained from eq . ( [ sigma - tilde - real ] ) are shown , respectively , in figs . [ sigmaim ] and [ sigmareal ] as functions of frequency at different temperatures and for different couplings ( about the crossover region of interest ) . the magnitude of the wave vector @xmath86 is taken in figs . [ sigmaim ] and [ sigmareal ] at a special value ( denoted by @xmath323 ) , which is identified from the behavior of the ensuing spectral function @xmath24 when performing a scanning over the wave vector ( see fig . 12 below ) . accordingly , @xmath323 is chosen to minimize the gap in the spectral function , in agreement with a standard procedure in the arpes literature . on the weak - coupling side ( when the the self - energy shift @xmath111 discussed in sec . iib is included in our calculation ) , @xmath323 coincides with @xmath324 . on the strong - coupling side ( when @xmath5 becomes negative ) one takes instead @xmath325 . for @xmath320 vs frequency ( in units of @xmath122 ) at different temperatures for the coupling values @xmath321 -0.5 ( a ) , 0.1 ( b ) , and 0.5 ( c ) . ] vs frequency ( in units of @xmath122 ) at different temperatures for the coupling values @xmath321 -0.5 ( a ) , 0.1 ( b ) , and 0.5 ( c ) . ] for all couplings here considered , the progressive evolution found in @xmath24 ( from the presence of a pseudogap about @xmath117 at @xmath1 to the occurrence of a superconducting gap near zero temperature ) stems from the interplay of the two contributions in eq . ( [ sigma - tilde - real ] ) to the imaginary part of @xmath326 about @xmath117 . specifically , for intermediate - to - weak coupling ( with @xmath317 ) the first term on the right - hand side of eq . ( [ sigma - tilde - real ] ) ( which is responsible for the pseudogap suppression in @xmath24 at @xmath1 ) would produce a narrow peak structure in @xmath327 about @xmath117 upon lowering @xmath19 , since @xmath328 vanishes while @xmath329 becomes progressively smaller . the presence of the second term on the right - hand - side of eq . ( [ sigma - tilde - real ] ) , however , gives rise to a narrow peak in @xmath330 about @xmath117 , as seen from fig . [ sigmaim ] ( a ) , resulting in a depression of @xmath327 about @xmath117 . [ this occurs barring a small temperature range close to @xmath1 , where the second term on the right - hand - side of eq . ( [ sigma - tilde - real ] ) is not yet well developed . ] at larger couplings ( when @xmath331 ) , the first term on the right - hand - side of eq . ( [ sigma - tilde - real ] ) would not produce a peak in @xmath332 about @xmath117 upon lowering the temperature , because @xmath333 does not correspondingly vanish in this case even though @xmath334 does . in addition , in this case the second term on the right - hand - side of eq . ( [ sigma - tilde - real ] ) does not produce a peak in @xmath332 about @xmath117 . figure [ spectt ] shows the resulting spectral function @xmath24 vs @xmath23 for @xmath320 at different temperatures and couplings . in all cases , at @xmath1 there occurs only _ a broad pseudogap feature _ both for @xmath335 and @xmath336 . [ for photoemission experiments only the case @xmath336 is relevant , so that we shall mostly comment on this case in the following . ] a _ coherent peak _ is seen to grow on top of this broad pseudogap feature as the temperature is lowered below @xmath1 . when zero temperature is eventually reached , the pseudogap feature is partially suppressed in favor of the coherent peak , which thus absorbs a substantial portion of the spectral intensity . this interplay between the broad pseudogap feature and the sharp coherent peak results in a characteristic _ peak - dip - hump structure _ , which is best recognized from the features for weak - to - intermediate coupling . generally speaking , this coherent peak ( and its corresponding counterpart at positive frequencies ) for intermediate - to - weak coupling is associated with the two dips in @xmath337 symmetrically located about zero frequency [ cf . figs . [ sigmaim ] ( a ) and ( b ) ] . in strong coupling , instead , the coherent peak results from a delicate balance between the real and imaginary parts of @xmath338 near the boundary of the region where @xmath339 . an interesting fact is that the weights of the negative and positive frequency parts of the spectrum turn out to be separately ( albeit approximatively ) constant as functions of temperature for given coupling , as shown in fig . [ weightneg ] for three characteristic couplings . this implies that , for a given coupling , the coherent peak for @xmath26 grows at the expenses of the accompanying broad pseudogap feature upon decreasing the temperature . the result that the total area for _ negative _ @xmath23 should be ( approximately ) constant as a function of temperature can be realized also from the analytic results in the extreme strong - coupling limit discussed in sec . iid . taking the analytic continuation of the matsubara green s function ( [ g-11-strong - coupling - figo ] ) ( which is appropriate in the strong - coupling limit as far as this total area is concerned , as it will be shown below ) results , in fact , in the total weight @xmath340 of the @xmath26 region being independent of temperature , since the combination @xmath341 entering the expression of @xmath342 is proportional to the total density in this limit [ cf . eq . ( [ final - n ] ) ] . returning to fig . [ spectt ] , it is also interesting to comment on the positions of the pseudogap feature and the coherent peak as functions of temperature for given coupling . the position of the coherent peak depends markedly on temperature , shifting progressively toward more negative frequencies as the temperature is lowered . in particular , for weak - to - intermediate coupling the position of the coherent peak about coincides with ( minus ) the value of the order parameter @xmath4 . in the strong - coupling region ( where @xmath343 ) , on the other hand , its position is about at @xmath344 . this remark entails the possibility of extracting two important quantities from the temperature evolution of the coherent peak in the spectral function : ( i ) the frequency position of this peak when approaching @xmath1 determines whether @xmath5 is positive ( when the peak position approaches @xmath117 ) or negative ( when the peak position approaches @xmath345 ) , corresponding to weak - to - intermediate coupling and strong coupling , respectively ; ( ii ) in both cases , the temperature dependence of the order parameter can be extracted from the frequency position of the coherent peak . the above results for the coherent peak contrast somewhat with the position of the pseudogap feature by decreasing temperature below @xmath1 , also determined from fig . [ spectt ] . the broad pseudogap feature does not depend sensitively on temperature for all couplings shown in this figure . this indicates that the broad pseudogap feature does not relate to the order parameter below @xmath1 . as far as the spectral function is concerned , one of the key results of our theory is thus the presence of _ two structures _ ( coherent peak and pseudogap ) , which behave rather independently from each other as functions of temperature and coupling . this result , which is also evidenced by the behavior of the experimental spectra in tunneling experiments on cuprates @xcite , originates in our theory from the presence of two distinct contributions to the self - energy , namely , the bcs and fluctuation contributions of eq . ( [ dyson - equation ] ) . while the broad pseudogap feature at @xmath346 develops with continuity from the only feature present at @xmath347 , the coherent peak _ per se _ would be present in a bcs approach even in the absence of the fluctuation contribution . this remark , of course , does not imply that the two contributions to the self - energy of eq . ( [ dyson - equation ] ) are totally independent from each other . they both depend , in fact , on the value of the order parameter @xmath4 which is , in turn , determined by both self - energy contributions via the chemical potential . for @xmath348 vs frequency ( in units of @xmath122 ) for the coupling value @xmath349 . ] a further important feature that can be extracted from our calculation of the spectral function is the evolution of the coherent peak for varying wave vector at fixed temperature and coupling . figure [ momentumevol ] reports @xmath24 vs @xmath23 for different values of the ratio @xmath350 about unity when @xmath349 and @xmath351 . here , @xmath352 identifies the underlying fermi surface that represents the `` locus of minimum gap '' . when @xmath353 , there is a strong asymmetry between the two coherent peaks at @xmath26 and @xmath354 , with the peak at @xmath26 absorbing most of the total weight . the situation is reversed when @xmath355 . when @xmath356 the spectrum is ( about ) symmetric between @xmath23 and @xmath357 . in addition , when following the position of the coherent peak at @xmath26 starting from @xmath353 , one sees that this position moves toward increasing @xmath23 , reaches a minimum distance from @xmath117 , and bounces eventually back to more negative values of @xmath23 . the value of the minimum distance from @xmath117 identifies an energy scale @xmath358 . at the same time , the weight of the coherent peak at @xmath26 progressively decreases for increasing @xmath350 starting from @xmath353 . when @xmath350 becomes larger than unity , the weight of the coherent peak is transferred from negative to positive frequencies . this situation is characteristic of the bcs theory , where only the coherent peaks are present without the accompanying broad pseudogap features . our calculation shows that this situation persists also for couplings values inside the crossover region , where the presence of the pseudogap feature is well manifest due to strong superconducting fluctuations . [ sufficiently far from the underlying fermi surface , the coherent peak and the pseudogap feature merge into a single structure , as it is evident from fig . [ momentumevol ] . in this case , the above as well as the following considerations apply to the structure as a whole and not to its individual components . ] ) vs the wave vector as extracted from fig . [ momentumevol ] . positive ( squares ) and negative ( circles ) branches are compared with bcs - like dispersions ( full lines ) , as explained in the text . ( b ) corresponding weights vs the wave vector , with particle - like ( full line ) and hole - like ( dashed line ) contributions . ] figure [ bcslike](a ) summarizes this finding for the dispersion of the coherent peaks , by showing the positions of the two coherent peaks as extracted from fig . [ momentumevol ] vs @xmath350 . these positions are compared with the two branches @xmath359 of a bcs - like dispersion , where @xmath358 is also identified from fig . [ momentumevol ] . [ the value of @xmath358 turns out to about coincide with the value of the order parameter @xmath4 at the same temperature , see below . ] the corresponding evolution of the weights of these peaks is shown in fig . [ bcslike](b ) , where the characteristic feature of an avoided crossing is evidenced . the dispersion of the positions and weights of the coherent peaks shown in fig . [ bcslike ] compare favorably with those recently obtained experimentally @xcite for slightly overdoped bi2223 samples below the critical temperature ( for @xmath360 ) . an additional outcome of our calculation is reported in fig . [ deltam ] , where the distance @xmath358 of the coherent peak in @xmath361 from @xmath117 at @xmath320 is compared at low temperature with the order parameter @xmath4 when @xmath362 and with @xmath363 when @xmath343 . this plot thus compares dynamical and thermodynamic quantities . the good agreement between the two curves confirms our identification of the coherent - peak position in @xmath24 with the minimum value of the excitations in the single - particle spectra according to a bcs - like expression ( where the value of the order parameter @xmath4 is , however , obtained by including also fluctuation contributions ) . ( in units of @xmath122 ) of the quasi - particle peak at @xmath364 vs the coupling @xmath296 ( full line ) . the dashed line corresponds to the value of the order parameter @xmath4 when @xmath362 and of @xmath365 when @xmath343 . ] finally , it is interesting to comment on the strong - coupling result ( [ g-11-strong - coupling ] ) for the diagonal green s function , with a characteristic double - fraction structure . the corresponding spectral function @xmath24 , obtained from that expression after performing the analytic continuation @xmath366 , shows only a _ single _ feature for @xmath26 , with a temperature - independent position . this contrasts the numerical results we have presented [ cf . in particular fig . [ spectt ] ] . this difference is due to the fact that , in our numerical calculation , the analytic continuation has been properly performed _ before _ taking the strong - coupling limit , as emphasized in sec . iie . with this procedure , in fact , the pseudogap structure and the coherent peak remain distinct from each other even in the strong - coupling limit , without getting lumped into a single feature . such a noncommutativity of the processes of taking the analytic continuation and the strong - coupling limit was noted already in a previous paper @xcite when studying the spectral function above @xmath1 . more generally , the occurrence of this noncommutativity is expected whenever one considers approximate expressions in the matsubara representation and takes the analytic continuation of these expressions to real frequency . to make evident the noncommutativity of the two processes , we show in fig . [ fignoncomm ] the spectral function @xmath332 for @xmath367 and @xmath368 , obtained by two alternative methods , namely : ( i ) using the analytic continuation of the expression ( [ sig11 ] ) for @xmath108 where @xmath369 ( full line ) ; ( ii ) taking the strong - coupling expression ( [ sigma-11-n - prime ] ) for @xmath108 , in which the analytic continuation @xmath370 is performed ( broken line ) . method ( i ) results in the presence of _ two _ distinct structures in @xmath24 for @xmath26 , corresponding to the coherent ( delta - like ) peak and the broad pseudogap feature . method ( ii ) gives instead a _ single _ delta - like peak . it is interesting to note that the total spectral weight of the two peaks for @xmath26 obtained by method ( i ) ( = 0.049 for the coupling of fig . [ fignoncomm ] ) about coincides with the weight of the delta - like peak ( = 0.044 ) obtained by method ( ii ) . [ we have verified that this correspondence between the spectral weights persists also at stronger couplings . ] and @xmath368 , obtained by taking alternatively the analytic continuation of @xmath108 from the expression ( [ sig11 ] ) ( full line ) or from the expression ( [ sigma-11-n - prime ] ) ( broken line ) . ] these remarks explain the occurrence of a single feature in the spectral function as obtained by a different theory based on a preformed - pair scenario @xcite . in that theory , a single - particle green s function with a double - fraction structure is considered in the matsubara representation for any coupling , and correspondingly a single feature in the spectral function is obtained for real frequencies @xcite . our theory shows instead the appearance of two distinct energy scales ( pseudogap and order parameter ) in the spectral function below @xmath1 . we are thus led to conclude that the occurrence of two distinct energy scales below @xmath1 in photoemission and tunneling spectra should not be necessarily associated with the presence of an `` extrinsic '' pseudogap due to additional non - pairing mechanisms , as sometimes reported in the literature @xcite . in this paper , we have extended the study of the bcs - bec crossover to finite temperatures below @xmath1 . this has required us to include ( pairing ) fluctuation effects in the broken - symmetry phase on top of mean field . our approximations have been conceived to describe both a system of superconducting fermions in weak coupling and a system of condensed composite bosons in strong coupling , via the simplest theoretical approaches valid in the two limits . these are the bcs mean field ( plus superconducting fluctuations ) in weak coupling and the bogoliubov approximation in strong coupling . to this end , analytic results have been specifically obtained in strong coupling from our general expression of the fermionic self - energy . results of numerical calculations have been presented both for thermodynamic and dynamical quantities . the latter have been defined by a careful analytic continuation in the frequency domain . in this context , a noncommutativity of the analytic continuation and the strong - coupling limit has been pointed out . results for thermodynamic quantities ( like the order parameter and chemical potential ) have shown that the effects of pairing fluctuations over and above the bcs mean field become essentially irrelevant in the zero - temperature limit , even in strong coupling . results for a dynamical quantity like @xmath24 have shown , in addition , that two structures ( a broad pseudogap feature that survives above @xmath1 and a strong coherent peak which emerges only below @xmath1 ) are present simultaneously , and that their temperature and coupling behaviors are rather ( even though not completely ) independent from each other . these features produced in the spectral function by our theory originate from a totally _ intrinsic _ effect , namely , the occurrence of a strong attractive interaction ( irrespective of its origin ) . additional features produced by other _ extrinsic _ effects could obviously be added on top of the intrinsic effects here considered . similar results have recently been obtained in ref . , using a boson - fermion model for precursor pairing below @xmath1 . in that reference , a two - peak structure for @xmath24 has also been obtained , although with a self - energy correction introduced by a totally different method . the attractive interaction adopted in this paper is the simplest one that can be considered , depending on a single parameter only . detailed comparison of the results of this theory with experiments on cuprates would then require one to specify the dependence of this effective parameter on temperature and doping . the simplified model that we have adopted in this paper should instead be considered realistic enough for studying theoretically the bcs - bec crossover for fermi atoms in a trap . the occurrence of this crossover in these systems is being rather actively studied experimentally at present . @xcite in this case , the calculation should also take into account the external trapping potential by considering , e.g. , a local version of our theory with local values of the density and chemical potential in the trap . @xcite . financial support from the italian miur under contract cofin 2001 prot.2001023848 is gratefully acknowledged . in this appendix , we extend _ below _ the critical temperature a standard procedure for obtaining _ at a formal level _ the fermionic retarded single - particle green s functions via analytic continuation from their matsubara counterparts . this is done in terms of the lehmann representation @xcite and of the baym - mermin theorem@xcite . in this context , besides the usual sum rule that holds also above the critical temperature @xcite , we will obtain two additional sum rules that hold specifically below the critical temperature . the results proved in this appendix hold _ exactly _ , irrespective of the approximations adopted for the matsubara self - energy . to satisfy the above three sum rules with an approximate choice of the self - energy , however , it is _ not _ required for the ensuing approximation to the fermionic single - particle green s functions to be `` conserving '' in the baym s sense @xcite . rather , it is sufficient that the analytic continuation from the matsubara frequencies to the real frequency axis is taken properly , as demonstrated in sec . iie with the specific choice ( [ total - self - energy ] ) of the self - energy . we begin by considering the fermionic `` normal '' and `` anomalous '' _ retarded _ single - particle green s functions in the broken - symmetry phase , defined respectively by @xmath371 here , @xmath372 is the unit step function , @xmath373 is the fermionic field operator with spin @xmath374 at position @xmath375 and ( real ) time @xmath7 ( such that @xmath376 with @xmath377 in terms of the system hamiltonian @xmath378 and the particle number @xmath379 ) , the braces represent an anticommutator , and @xmath380 stands for the grand - canonical thermal average . the matsubara counterparts of ( [ g - retarded ] ) and ( [ f - retarded ] ) are similarly defined by @xmath381 \rangle \label{g - matsubara } \\ f({\mathbf r},\tau;{\mathbf r'},\tau ' ) & = & - \langle t_{\tau } \left [ \psi_{\uparrow}({\mathbf r},\tau ) \psi_{\downarrow}({\mathbf r'},\tau ' ) \right ] \rangle \ , , \label{f - matsubara}\end{aligned}\ ] ] where now @xmath382 , @xmath383 , and @xmath384 is the time - ordering operator for imaginary time @xmath385 . the lehmann analysis for the normal function @xmath386 in the broken - symmetry phase proceeds along similar lines as for the normal phase @xcite . the result is that ( for a homogeneous system ) the wave - vector and ( real ) frequency fourier transform can be obtained by the spectral representation @xmath387 @xmath206 being a positive infinitesimal . here , the real and positive definite _ spectral function _ @xmath388 satisfies the sum rule @xmath389 for any given @xmath22 , as a consequence of the canonical anticommutation relation of the field operators . a similar analysis for the matsubara normal green s function leads to the spectral representation @xmath390 in terms of the _ same _ spectral function @xmath21 of eq . ( [ spectral - repres - g - r ] ) , where @xmath391 ( @xmath28 integer ) is a fermionic matsubara frequency and the diagonal nambu green s function has been introduced . the spectral representations ( [ spectral - repres - g - r ] ) and ( [ spectral - repres - g - matsubara ] ) , together with knowledge of the asymptotic behavior @xmath392 for large @xmath393 , are sufficient to guarantee that the retarded normal function is the correct analytic continuation of its matsubara counterpart in the upper - half of the complex frequency plane @xcite , in accordance with the baym - mermin theorem @xcite . the above lehmann analysis can be extended to the anomalous function ( [ f - retarded ] ) as well . one obtains @xmath394 in the place of eq . ( [ spectral - repres - g - r ] ) . the new spectral function @xmath395 vanishes for large @xmath393 but , in general , is no longer real and positive definite . [ one obtains for @xmath395 the same formal expression @xcite for @xmath21 in terms of the eigenstates @xmath396 of the operators @xmath378 and @xmath379 , apart from the replacement of @xmath397 with @xmath398 latexmath:[$\langle n ' readily verified that @xmath395 satisfies the sum rule @xmath400 which is again a consequence of the canonical anticommutation relation of the field operators . the above properties guarantee that @xmath273 vanishes faster than @xmath401 for large @xmath393 . by a similar token , considering the matsubara anomalous green s function leads to the spectral representation @xmath402 where the off - diagonal nambu green s function has been introduced . these considerations suffice again to guarantee that the retarded anomalous function is the correct analytic continuation of its matsubara counterpart in the upper - half complex frequency plane , in accordance with the baym - mermin theorem @xcite . finally , an additional sum rule for @xmath403 can be obtained by using the relation @xmath404 and exploiting the equation of motion for the field operator . for the contact potential we are considering throughout this paper , we write @xmath405 in terms of the order parameter @xmath4 . the expression ( [ 3rd - sum - rule - initial ] ) thus becomes : m. bartenstein , a. altmeyer , s. riedl , s. jochim , c. chin , j.h . denschlag , and r. grimm , phys . lett . * 92 * , 120401 ( 2004 ) ; c.a . regal , m. greiner , and d.s . jin , phys . lett . * 92 * , 040403 ( 2004 ) ; m.w . zwierlein , c.a . stan , c.h . schunck , s.m.f . raupach , a.j . kerman , and w. ketterle , phys . * 92 * , 120403 ( 2004 ) ; j. kinast , s.l . hemmer , m.e . gehm , a. turlapov , and j.e . thomas , phys . lett . * 92 * , 150402 ( 2004 ) ; t. bourdel _ et al . _ , cond - mat/0403091 . in the theory of ref . , the ladder propagator in the broken - symmetry phase was obtained for the lattice case by inverting a @xmath92 matrix , thereby neglecting the coupling of the fluctuations of the order parameter to the fluctuations of the particle density . this coupling has , however , been shown to be essential for a correct description of the strong - coupling limit of the ladder propagator [ s. de palo , c. castellani , c. di castro , and b.k . chakraverty , phys . b * 60 * , 564 ( 1999 ) ] . at the mean - field level the ( square of the ) order parameter is , in fact , proportional to the particle density of the composite bosons . the arrows attached to the nambu green s functions have the usual meaning of pointing from the second to the first argument of the green s functions . the distinction between particle - particle and particle - hole diagrams is , however , purely conventional since particle and hole modes get intimately interrelated in the broken - symmetry phase [ see j.r . schrieffer , _ theory of superconductivity _ ( w.a . benjamin , new york , 1964 ) ] . we shall nevertheless maintain the terminology used in the normal phase and refer to the diagrams of fig . 1(a ) as the `` particle - particle '' ladder diagrams . haussmann had originally omitted the self - energy contribution ( [ sigma-12-bcs ] ) in his treatment @xcite of the dilute superconducting fermi system . that contribution was later included in a revised treatment [ r. haussmann , _ self - consistent quantum - field theory and bosonization for strongly correlated electron systems _ ( springer - verlag , new york , 1999 ) ] . the boundary ( dashed ) lines in fig . [ gamma ] are determined by the conditions : ( i ) @xmath407 for all @xmath288 when @xmath343 and for @xmath408 when @xmath362 ; ( ii ) @xmath409 for @xmath410 when @xmath362 . note also that the bogoliubov - anderson mode ( full line ) terminates at @xmath411 in the upper panel of fig . [ gamma ] , while it remains below the particle - particle continuum ( dashed lines ) in the middle and lower panels of fig . [ gamma ] . besides the difference mentioned in the text about the relative order of performing the analytic continuation to real frequency and of considering approximate expressions for the self - energy in the matsubara representation , it is also worth mentioning some additional differences between our theory and the theory of ref . based on a `` preformed - pair scenario '' below @xmath1 . in our theory , the fluctuation propagator ( [ gamma - solution ] ) is built on the bcs green s functions ( [ bcs - green - function ] ) and is obtained via the inversion of a 2 @xmath412 2 matrix . in the theory of ref . , the corresponding propagator is instead built on a non - interacting green s function @xmath413 and on a dressed green s function with the functional form of eq . ( [ g-11-strong - coupling ] ) . we have verified that , in the extreme strong - coupling limit ( where the residual interaction between the composite bosons becomes irrelevant ) and for temperatures @xmath414 , the two theories give essentially the same results in the matsubara representation . differences show up , however , at weaker couplings , when the interaction between composite bosons matters . specifically , our theory for the composite bosons reproduces the bogoliubov results for point - like bosons with @xmath415 and @xmath416 , owing to the boson - boson interaction and the depletion of the condensate . in the theory of ref . , on the other hand , @xmath417 for the composite bosons is taken to vanish for @xmath418 and also @xmath419 for all couplings . more generally for any coupling , in our theory ( quantum ) fluctuation corrections to mean - field quantities survive even at @xmath2 while in the theory of ref . fluctuation corrections vanish identically at @xmath2 . quite generally , the function @xmath395 introduced through the lehmann representation ( [ spectral - repres - f - r ] ) can be cast in the form @xmath420/(2 \pi i)$ ] , in terms of the _ advanced _ ( @xmath421 ) and _ retarded _ ( @xmath207 ) `` anomalous '' functions . in particular , when @xmath422 for _ real _ @xmath23 , @xmath395 can be identified with @xmath423 . this is the case for the `` anomalous '' retarded and advanced green s functions resulting from the approximate expression ( [ g-12-matsubara ] ) of the text with a _ real _ order parameter @xmath4 .

the bcs - bec crossover is studied in a systematic way in the broken - symmetry phase between zero temperature and the critical temperature . this study bridges two regimes where quantum and thermal fluctuations are , respectively , important . the theory is implemented on physical grounds , by adopting a fermionic self - energy in the broken - symmetry phase that represents fermions coupled to superconducting fluctuations in weak coupling and to bosons described by the bogoliubov theory in strong coupling . this extension of the theory beyond mean field proves important at finite temperature , to connect with the results in the normal phase . the order parameter , the chemical potential , and the single - particle spectral function are calculated numerically for a wide range of coupling and temperature . this enables us to assess the quantitative importance of superconducting fluctuations in the broken - symmetry phase over the whole bcs - bec crossover . our results are relevant to the possible realizations of this crossover with high - temperature cuprate superconductors and with ultracold fermionic atoms in a trap .

over the past 83 years , the study of dipole moments of elementary particles has provided a wealth of information on subatomic physics . from the pioneering work of stern@xcite through the discovery of the large anomalous magnetic moments of the proton@xcite and neutron@xcite , the ground work was laid for the discovery of spin , of radiative corrections and the renormalizable theory of qed , of the quark structure of baryons and the development of qcd . a charged particle with spin @xmath2 has a magnetic moment @xmath3 where @xmath4 is the gyromagnetic ratio , @xmath5 is the anomaly , and the latter expression is what one finds in the particle data tables.@xcite the dirac equation tells us that for spin one - half point - like particles , @xmath6 for spin angular momentum , and is unity for orbital angular momentum ( the latter having been verified experimentally@xcite ) . for point particles , the anomaly arises from radiative corrections , two examples of which are shown in fig . [ fg : aexpan ] . the lowest - order correction gives the famous schwinger@xcite result , @xmath7 , which was verified experimentally by foley and kusch.@xcite the situation for baryons is quite different , since their internal quark structure gives them large anomalies . in general @xmath5 ( or @xmath8 ) is an expansion in @xmath9 , @xmath10 with 1 diagram for the schwinger ( second - order ) contribution , 5 for the fourth order , 40 for the sixth order , 891 for the eighth order . the qed contributions to electron and muon 2 have now been calculated through eighth order , @xmath11 and the tenth - order contribution has been estimated.@xcite .,scaledwidth=45.0% ] .transformation properties of the magnetic and electric fields and dipole moments . [ cols="^,^,^,^",options="header " , ] the magnetic and electric dipole moments can be represented as the real and imaginary parts of a generalized dipole operator @xmath12 , and the interaction lagrangian becomes @xmath13 \mu f_{\alpha \beta}\ ] ] with @xmath14 and @xmath15 . the electron anomaly is now measured to a relative precision of about four parts in a billion ( ppb),@xcite which is better than the precision on the fine - structure constant @xmath16 , and kinoshita has used the measured electron anomaly to give the best determination of @xmath16.@xcite the electron anomaly will be further improved over the next few years.@xcite the muon anomaly is measured to 0.5 parts per million ( ppm).@xcite the relative contributions of heavier particles to @xmath5 scales as @xmath17 , so the muon has an increased sensitivity to higher mass scale radiative corrections of about 40,000 over the electron . at a precision of @xmath18 ppm , the muon anomaly is sensitive to @xmath19 gev scale physics . the standard model value of @xmath20 has measurable contributions from three types of radiative processes : qed loops containing leptons ( @xmath21 ) and photons;@xcite hadronic loops containing hadrons in vacuum polarization loops;@xcite and weak loops involving the @xmath22 and @xmath23 weak gauge bosons ( the standard model higgs contribution is negligible),@xcite @xmath24 a significant difference between the experimental value and the standard model prediction would signify the presence of new physics . a few examples of such potential contributions are lepton substructure , anomalous @xmath25 couplings , and supersymmetry.@xcite the cern experiment@xcite observed the contribution of hadronic vacuum polarization shown in fig . [ fg : had](a ) at the 8 standard deviation level . unfortunately , the hadronic contribution can not be calculated directly from qcd , since the energy scale is very low ( @xmath26 ) , although blum@xcite has performed a proof of principle calculation on the lattice . fortunately dispersion theory gives a relationship between the vacuum polarization loop and the cross section for @xmath27 , @xmath28 where @xmath29 and experimental data are used as input . the factor @xmath30 in the dispersion relation , means that values of @xmath31 at low energies ( the @xmath32 resonance ) dominate the determination of @xmath33 . in principle , this information could be obtained from hadronic @xmath34 decays such as @xmath35 , which can be related to @xmath36 annihilation through the cvc hypothesis and isospin conservation.@xcite however , inconsistencies between information obtained from @xmath36 annihilation and hadronic tau decays , plus an independent confirmation of the cmd2 high - precision @xmath36 cross - section measurements by the kloe collaboration,@xcite have prompted davier , hcker , et al , to state that until these inconsistencies can be understood only the @xmath36 data should be used to determine @xmath33.@xcite conversion , showing the relevant slepton mixing matrix elements . the mdm and edm give the real and imaginary parts of the matrix element , respectively . ] the hadronic light - by - light contribution ( see fig . [ fg : had](e ) ) has been the topic of much theoretical investigation.@xcite unlike the lowest - order contribution , it can only be calculated from a model , and this contribution is likely to provide the ultimate limit to the precision of the standard - model value of @xmath20 . one of the very useful roles the measurements of @xmath20 have played in the past is placing serious restrictions on physics beyond the standard model . with the development of supersymmetric theories as a favored scheme of physics beyond the standard model , interest in the experimental and theoretical value of @xmath20 has grown substantially . susy contributions to @xmath20 could be at a measurable level in a broad range of models . furthermore , there is a complementarity between the susy contributions to the mdm , edm and transition moment for the lepton - flavor violating ( lfv ) process @xmath37 in the field of a nucleus . the mdm and edm are related to the real and imaginary parts of the diagonal element of the slepton mixing matrix , and the transition moment is related to the off diagonal one , as shown in fig . [ fg : susy ] . this reaction , along with the companion lfv decay @xmath38 , will be searched for in `` next generation '' experiments now under construction.@xcite from neutrino oscillations we already know that lepton flavor is violated , and this violation will be enhanced if there is new dynamics at the tev scale . this same new physics could also generate measurable effects in the magnetic and electric dipole moments of the muon as well.@xcite the method used in the third cern experiment and the bnl experiment are very similar , save the use of direct muon injection@xcite into the storage ring,@xcite which was developed by the e821 collaboration . these experiments are based on the fact that for @xmath39 the spin precesses faster than the momentum vector when a muon travels transversely to a magnetic field . the spin precession frequency @xmath40 consists of the larmor and thomas spin - precession terms . the spin frequency @xmath40 , the momentum precession ( cyclotron ) frequency @xmath41 , are given by @xmath42 the difference frequency @xmath43 is the frequency with which the spin precesses relative to the momentum , and is proportional to the anomaly , rather than to @xmath8 . a precision measurement of @xmath20 requires precision measurements of the muon spin precession frequency @xmath44 , and the magnetic field , which is expressed as the free - proton precession frequency @xmath45 in the storage ring magnetic field . the muon frequency can be measured as accurately as the counting statistics and detector apparatus permit . the design goal for the nmr magnetometer and calibration system was a field accuracy of 0.1 ppm . the @xmath46 which enters in eq . [ eq : omeganoe ] is the average field seen by the ensemble of muons in the storage ring . in e821 we reached a precision of 0.17 ppm in the magnetic field measurement . an electric quadrupole@xcite is used for vertical focusing , taking advantage of the `` magic '' @xmath47 at which an electric field does not contribute to the spin motion relative to the momentum . with both an electric and a magnetic field , the spin difference frequency is given by @xmath48 , \label{eq : tbmt}\ ] ] which reduces to eq . [ eq : omeganoe ] in the absence of an electric field . for muons with @xmath49 in an electric field alone , the spin would follow the momentum vector . the experimental signal is the @xmath50 from @xmath51 decay , which were detected by lead - scintillating fiber calorimeters.@xcite the time and energy of each event was stored for analysis offline . muon decay is a three - body decay , so the 3.1 gev muons produce a continuum of positrons ( electrons ) from the end - point energy down . since the highest energy @xmath50 are correlated with the muon spin , if one counts high - energy @xmath50 as a function of time , one gets an exponential from muon decay modulated by the @xmath52 precession . the expected form for the positron time spectrum is @xmath53 $ ] , however in analyzing the data it is necessary to take a number of small effects into account in order to obtain a satisfactory @xmath54 for the fit.@xcite the data from our 2000 running period is shown in fig . [ fg : wig00 ] the experimental results from e821 are shown in fig . [ fg : amu ] , with the average @xmath55 which determines the `` world average '' . the theory value@xcite @xmath56 is taken from hcker et al.,@xcite , which updates their earlier analysis@xcite with the kloe data;@xcite and from hagiwara , et al.,@xcite who use a different weighting scheme for the experimental data when evaluating the dispersion integral but do not include the kloe data . when this theory value is compared to the standard model value using either of these two analyses@xcite for the lowest - order hadronic contribution , one finds @xmath57 or a discrepancy of 2.7 standard deviations . . the strong interaction contribution is taken from references @xcite and @xcite . ] to show the sensitivity of our measurement of @xmath20 to the presence of virtual electroweak gauge bosons , we subtract off the electroweak contribution of @xmath58 from the standard model value , compare with experiment and obtain @xmath59 a 4.7 standard deviation discrepancy . this difference shows conclusively that e821 was sensitive to physics at the 100 gev scale . at present , it is inconclusive whether we see evidence for contributions from physics beyond the standard - model gauge bosons . with each data set , the systematic error was reduced , and for the final data set taken in 2001 the systematic error on @xmath60 was 0.27 ppm with a statistical error of 0.66 ppm . given the tantalizing discrepancy between our result and the latest standard - model value , and the fact that the hadronic error could be reduced by about a factor of two over the next few years,@xcite we submitted a new proposal to brookhaven to further improve the experimental measurement . the goal of this new experiment is @xmath61 ppm total error , with the goal of controlling the total systematic errors on the magnetic field and on the muon frequency measurement to 0.1 ppm each . our proposal@xcite was given enthusiastic scientific approval in september 2004 by the laboratory , and has been given the new number , e969 . negotiations are underway between the laboratory and the funding agencies to secure funding . a letter of intent ( loi ) for an even more precise 2 experiment was also submitted to j - parc.@xcite in that loi we proposed to reach a precision below 0.1 ppm . since it is not clear how well the hadronic contribution can be calculated , and whether the new brookhaven experiment e969 will go ahead , we will evaluate whether to press forward with this experiment at a later time . our loi at j - parc@xcite was predicated on pushing as far as possible at brookhaven before moving to japan . while the mdm has a substantial standard model value , the predicted edms for the leptons are unmeasurably small and lie orders of magnitude below the present experimental limits given in table [ tb : edm ] . an edm at a measurable level would signify physics beyond the standard model . susy models , and other dynamics at the tev scale do predict edms at measurable levels.@xcite a new experiment to search for a permanent edm of the muon with a design sensitivity of @xmath62 @xmath63-cm is being planned for j - parc.@xcite this sensitivity lies well within values predicted by some susy models.@xcite feng , et al.,@xcite have calculated the range of @xmath64 available to such an experiment , assuming a new physics contribution to @xmath20 of @xmath65 , @xmath66 where @xmath67 is a _ cp _ violating phase . this range is shown in fig . [ fg : phicp ] . available to a dedicated muon edm experiment.@xcite the two bands show the one and two standard - deviation ranges if @xmath20 differs from the standard model value by @xmath68 . ] of course one wishes to measure as many edms as possible to understand the nature of the interaction . while naively the muon and electron edms scale linearly with mass , in some theories the muon edm is greatly enhanced relative to linear scaling relative to the electron edm when the heavy neutrinos of the theory are non - degenerate.@xcite and @xmath69 . ] with an edm present , the spin precession relative to the momentum is given by @xmath70 \nonumber \\ & \qquad \ \ + & { e \over m}\left [ { \eta \over 2 } \left ( { \vec e \over c } + \vec \beta \times \vec b \right ) \right ] \label{eq : omegawedm}\end{aligned}\ ] ] where @xmath71 and @xmath72 . for reasonable values of @xmath73 , the motional electric field @xmath74 is much larger than electric fields that can be obtained in the laboratory , and the two vector frequencies are orthogonal to each other . the edm has two effects on the precession : the magnitude of the observed frequency is increased , and the precession plane is tipped relative to the magnetic field , as illustrated in fig . [ fg : omegaeta ] . e821 was operated at the magic @xmath75 so that the focusing electric field did not cause a spin precession . in e821 the tipping of the precession plane is very small , ( @xmath76 mrad ) if one uses the cern limit@xcite given in table [ tb : edm ] . this small tipping angle makes it very difficult to observe an edm effect in e821 , since the 2 precession ( @xmath44 ) is such a large effect . we have recently introduced a new idea which optimizes the edm signal , and which uses the motional electric field in the rest frame of the muon interacting with the edm to cause spin motion.@xcite the dedicated experiment will be operated off of the magic @xmath75 , for example at @xmath77 mev / c , and will use a radial electric field to stop the @xmath52 precession.@xcite then the spin will follow the momentum as the muons go around the ring , except for any movement arising from an edm . thus the edm would cause a steady build - up of the spin out of the plane with time . detectors would be placed above and below the storage region , and a time - dependent up - down asymmetry @xmath78 would be the signal of an edm , @xmath79 a simulation for @xmath80 cm is given in fig . [ fg : edmsig ] . or @xmath81 ] the figure of merit for statistics in the edm experiment is the number of muons times the polarization . in order to reach @xmath82 cm , the muon edm experiment would need @xmath83 , a flux only available at a future facility . while progress can still be made at brookhaven on @xmath20 , a dedicated muon edm experiment must be done elsewhere . measurements of the muon and electron anomalies played an important role in our understanding of sub - atomic physics in the 20th century . the electron anomaly was tied closely to the development of qed . the subsequent measurement of the muon anomaly showed that the muon was indeed a `` heavy electron '' which obeyed qed.@xcite with the sub - ppm accuracy now available for the muon anomaly,@xcite there may be indications that new physics is beginning to appear in loop processes.@xcite the non - observation of an electron edm is becoming an issue for supersymmetry , just as the non - observation of a neutron edm implies such a mysteriously ( some would say un - naturally ) small @xmath84-parameter for qcd . the search for edms will continue , and if one is observed , the motivation for further searches in other systems will be even stronger . the muon presents a unique opportunity to observe an edm in a second - generation particle , where the _ cp _ phase might be different from the first generation , or the scaling with mass might be quadratic rather than linear . if susy turns out to be _ the _ extension to the standard model , then there will be susy enhancements to @xmath20 to the muon edm and also to the amplitudes for lepton flavor violating muon decays . once the susy mass spectrum is measured , @xmath20 will provide a very clean measurement of @xmath85.@xcite if susy or other new dynamics at the tev scale are not found at lhc , then precision experiments , which are sensitive through virtual loops to much higher mass scales than direct searches for new particles , become even more important . experiments such as edm searches , 2 and searches for lepton flavor violation , all carried out at high intensity facilities , may provide the only way to probe these higher energy scales . opportunities at future high intensity facilities are actively being pursued , and both the theoretical and experimental situations are evolving . it is clear that the study of lepton moments and lepton flavor violation , along with neutron edm searches , will continue to be a topic of great importance in the first part of the 21st century . _ acknowledgments : i wish to thank my colleagues on the muon 2 experiment , as well as m. davier , j. ellis , e. de rafael , w. marciano and t. teubner for helpful discussions . special thanks to y. semertzidis for critically reading this manuscript . _ j.l . feng , k.t . matchev .y , shadmi , nucl * b 613 * , 366 ( 2001 ) , and phys . lett . * b555 * , 89 ( 2003 ) . r.s . van dyck et al . , phys . lett . , * 59 * , 26(1987 ) and in _ quantum electrodynamics _ , ( directions in high energy physics vol . 7 ) t. kinoshita ed . , world scientific , 1990 , p.322 . a. hcker , ichep04 , which can be found at http://ichep04.ihep.ac.cn/program.htm , also m. davier 8th international workshop on tau - lepton physics , september 2004,http://www.hepl.phys.nagoya - u.ac.jp / public / tau04/ r.m . carey , a. gafarov , i. logashenko , k.r . lynch , j.p . miller , b.l . roberts ( co - spokesperson ) , g. bunce , w. meng , w.m . morse ( resident spokesperson ) , y.k . semertzidis , d. grigoriev , b.i . khazin , s.i . redin , yuri m. shatunov , e. solodov , y. orlov , p. debevec , d.w . hertzog ( co - spokesperson ) , p. kammel , r. mcnabb , f. mlhauser , k.l . giovanetti , k.p . jungmann , c.j.g . onderwater , s. dhamija , t.p . gorringe , w. korsch , f.e . gray , b. lauss , e.p . sichtermann , p. cushman , t. qian , p. shagin , s. dhawan and f.j.m . farley , which can be found at : http://g2pc1.bu.edu/@xmath88roberts/

from the famous experiments of stern and gerlach to the present , measurements of magnetic dipole moments , and searches for electric dipole moments of `` elementary '' particles have played a major role in our understanding of sub - atomic physics . in this talk i discuss the progress on measurements and theory of the magnetic dipole moment of the muon . i also discuss a new proposal to search for a permanent electric dipole moment ( edm ) of the muon and put it into the more general context of other edm searches . these experiments , along with searches for the lepton flavor violating decays @xmath0 and @xmath1 , provide a path to the high - energy frontier through precision measurements .

the high energy emission from neutron stars and black holes in x - ray binaries is generally powered by accretion onto the compact object . the variability of x - ray light curve with different time scales from milliseconds to days is usually attributed to various characteristic time scales associated with accretion flow around the black hole or neutron star . in low - mass x - ray binaries ( lmxbs ) , where the central object is fed by an accretion disk , a few variability frequencies observed as quasi - periodic oscillation ( qpo ) peaks in addition to other broad - band features in the power spectra are common to both black hole and neutron star sources . although there are some phenomenological differences between qpos in black hole candidates and those observed in neutron star lmxbs , the similarities such as tight correlations of high and low frequency power spectral features in black hole and neutron star sources are remarkable ( van der klis 1994 ; psaltis , belloni , & van der klis 1999 ; wijnands & van der klis 1999 ) . any interpretation or model solely based on the existence of a magnetic field , a hard surface , which are neutron star like properties , or an innermost stable circular orbit ( isco ) as a black hole like property to produce qpos can not account for the correlations of timing properties among different sources . qpos were discovered in the x - ray power density spectrum of black hole transients with frequencies in the @xmath7 hz range during the very high spectral state of these sources ( motch et al . 1983 ; miyamoto & kitamoto 1989 ; miyamoto et al . 1991 ; morgan , remillard , & greiner 1997 ; wijnands , homan , & van der klis 1999 ; sobczak et al . 2000 ; strohmayer 2001 ; muno et al . 2001 ) . for several black hole sources , high - frequency qpos were observed as single peaks roughly around @xmath8 hz ( see remillard & mcclintock 2006 and references therein ) . the discovery of twin khz qpos in neutron star lmxbs with their peak frequencies within around the @xmath9 hz range and a peak separation in the @xmath10 hz range together with their tight correlations with the low frequency power spectral components which are also observed in black hole candidates strengthens the idea that qpos are produced in the inner regions of accretion disks around compact objects ( see van der klis 2000 and references therein ; mndez & belloni 2007 ) . the discoveries of twin hectohz qpos from black hole candidates such as gro j1655@xmath1140 , xte j1550@xmath11564 , and grs 1915@xmath12105 ( remillard et al . 2002 , 2003 ; remillard & mcclintock 2006 ) have almost certified the idea of unifying the interpretation of high - frequency qpos observed in black hole and neutron star sources within a single qpo model . one of the most striking differences between the high - frequency qpos of black holes and the khz qpos in neutron star lmxbs is the fact that the former do not show any significant correlation with the x - ray luminosity whereas the latter do . from one observation to another in a given source , the frequency shifts in the high - frequency qpos of black holes are negligible as compared to the variations in the frequencies of khz qpo peaks of accreting neutron stars . as compared to khz qpos detected in the power spectra of neutron stars , high - frequency qpos from black holes are weak features having relatively low - quality factors . the commonly observed property of both low and high - frequency qpos in black holes is the fact that these oscillations are strongest at photon energies above 6 kev when the power law component in the energy spectra dominates over the disk component ( remillard & mcclintock 2006 ) . though it is not conclusive regarding the number of black hole sources exhibiting twin qpos , the frequency ratio of the upper high - frequency qpo to the lower one is close to 3:2 in black hole candidates . there is no specific value for the ratio of two simultaneous khz qpos observed in neutron star sources ; the ratio of the upper qpo frequency to the lower one rather takes different values changing between one and three from one source to another ( belloni , mndez , & homan 2005 ) . beside similarities and tight correlations , the phenomenological differences between the high - frequency qpos observed from black hole candidates and those from neutron star lmxbs likely arise from the dominant effect of different boundary conditions imposed by a hard surface or a magnetic field and the isco as they might be more appropriate for a neutron star and a black hole , respectively . high - frequency qpos were detected in seven black hole sources among which three black hole candidates exhibited qpo pairs in their power spectra . in all the sources with qpo pairs of commensurate frequencies , the ratio of two qpo frequencies is very close to 1.5 in two black hole binaries and to 1.6 in the third one ( remillard et al . 2002 ; remillard et al . the common property of high - frequency qpos observed in black hole systems is based on the spectral state of a given source . all high - frequency black - hole qpos are usually observed in very high and high spectral states which are characterized by the most luminous states of the source ( see remillard & mcclintock 2006 and references therein ) . for such high luminosities , the accretion disk around the black hole is expected to be truncated at the radius of the isco according to the standard model ( novikov & thorne 1973 ; shakura & sunyaev 1973 ) . for radii less than the radius of the isco , the accreting gas is thought to plunge radially towards the black hole . in the recent mhd simulations by beckwith , hawley , & krolik ( 2008 ) , however , the innermost ring of the disk which emits significant radiation has been shown to lie inside where the standard model predicts . these simulations have modified the stress - free boundary condition of the standard model at the isco , but they have employed the classical relationship between magnetic stresses and energy dissipation and the assumption that the radial inflow time - scale of the accreting matter inside the isco is longer than the time - scales for thermalization and radiation of the dissipated heat , both of which may not be valid at all within the plunging region . we therefore anticipate the innermost part of the disk beyond the isco to be the most probable region near the black hole for the production of high - frequency qpos . it is very likely within this region that the frequencies of the unstable growing disk modes correspond to the frequency bands of qpos . the early attempts to interpret qpo frequencies in terms of disk modes were made by alpar et al . ( 1992 ) and alpar & yilmaz ( 1997 ) . the initial contributions to the theory of disk oscillations explored the role of trapped disk oscillations in the variability of the x - ray power spectra of black hole sources ( kato & fukue 1980 ; nowak & wagoner 1991 ; kato 2001 and references therein ) . the general relativistic test - particle frequencies were recognized and employed to construct models of qpos for both neutron stars and black holes ( stella , vietri , & morsink 1999 ; abramowicz et al . the model of psaltis & norman ( 2000 ) revealed the importance of hydrodynamic corrections to relativistic test - particle frequencies and the effect of hydrodynamic disk parameters on the correlations of qpos and broad - band noise component . alpar & psaltis ( 2008 ) noted that the radial epicyclic frequency would be the highest dynamical frequency in the inner region of an accretion disk . this conclusion is based upon the existence of a magnetohydrodynamic boundary region around the neutron star where orbital frequencies deviate from keplerian test - particle frequencies due to viscous and magnetic stresses ( see erkut & alpar 2004 ) . the recent analysis by erkut , psaltis , & alpar ( 2008 ) of global hydrodynamic modes in the boundary regions of neutron stars showed how important the hydrodynamic effects are in estimating the observational characteristics of high - frequency qpos . in this paper we apply the mode analysis , which has been developed by erkut , psaltis , & alpar ( 2008 ) for the boundary region model of khz qpos from neutron stars , to the inner region of an accretion disk around a kerr black hole . in order to account for the general relativistic effects of a kerr metric on the ratio of dynamical frequencies , we work with a new pseudo - newtonian potential that is appropriate for the analysis of temporal behavior of relativistic disks . this approach allows us to extend our recent study of global hydrodynamic modes of free oscillations to black holes as well . our aim is to identify the modes whose frequency bands correspond to the high - frequency qpo pairs usually detected with a frequency ratio close to 1.5 in black hole sources . most importantly , we provide a way to use the frequency ratio of these modes as a diagnostics of the spin parameter of a rotating black hole that exhibit high - frequency qpos . in section 2 we introduce our pseudo - newtonian approach . the basic equations and parameters related to the analysis of disk modes for a kerr black hole are presented in section 3 . in section 4 we come up with the mode analysis and identify the modes relevant to the commensurate high - frequency qpo pairs with a frequency ratio around 1.5 . we discuss the results and present our conclusions in section 5 . using pseudo - newtonian potentials in hydrodynamic simulations is an effective and easy method to incorporate relativistic effects into accretion flows ( e.g. , chan , psaltis , & zel 2009 ) . the pseudo - newtonian potential proposed by paczyski & wiita ( 1980 ) is successful in estimating the isco for schwarzschild geometry . it is therefore appropriate for a non - rotating black hole . recently , two modified newtonian force models have been introduced by mukhopadhyay & misra ( 2003 ) to approximate the dynamical frequencies of an accretion disk around a rotating black hole . in our current analysis , the ratio of two successive frequency bands of disk modes is of interest in identifying the high - frequency qpos from black holes . unlike the early studies we mentioned above our pseudo - newtonian treatment of dynamical disk frequencies keeps the ratio of the radial epicyclic frequency @xmath0 to the orbital frequency @xmath3 exactly the same as the corresponding ratio observed by a distant observer of a kerr black hole . the kerr expression for the ratio of the test - particle frequencies @xmath0 and @xmath3 is @xmath13where @xmath14 is the spin parameter of the black hole and @xmath15 with @xmath16 and @xmath17 being the mass of the black hole and the speed of light , respectively . the newtonian expression for the same ratio is@xmath18the ratio given by equation ( [ knew ] ) is valid for both the test - particles and the hydrodynamical fluids rotating in orbits . in the steady state of a geometrically thin disk , the hydrodynamical effects of pressure gradients on the test - particle frequencies are negligible . as we mention in section 3 , the radial momentum balance in a geometrically thin disk can be approximated by the test - particle orbits where the centripetal acceleration of each gas particle rotating with the frequency @xmath3 is due to the gravitational force . setting equations ( [ kker ] ) and ( [ knew ] ) equal to each other , we obtain within our pseudo - newtonian approach a differential equation for the orbital frequency @xmath19 . given a suitable pseudo - newtonian potential , the orbital frequency @xmath3 satisfies the radial momentum equation and mimics the effect of strong gravity on the gas - particle orbits by keeping the value of @xmath20 the same as the corresponding kerr value in the test - particle regime . in this sense , our approach is similar to the early pseudo - newtonian treatments of the dynamical frequencies in a geometrically thin accretion disk around a black hole . to find a solution for @xmath19 , we require that our pseudo - newtonian orbital frequency match the kerr orbital frequency in the outer disk . according to a distant observer , the kerr expression for the orbital frequency is@xmath21where @xmath22 is the keplerian frequency . for sufficiently large radii , that is for @xmath23 , the kerr orbital frequency assumes its keplerian value . using the same asymptotic boundary condition on the pseudo - newtonian orbital frequency , it follows from equations ( [ kker ] ) and ( [ knew ] ) that@xmath24 . \label{pnom}\ ] ] to illustrate our treatment of the radial epicyclic and orbital frequencies , we plot in figure @xmath25 the pseudo - newtonian frequencies @xmath0 and @xmath3 over a wide range of disk radii in comparison with the corresponding kerr frequencies . figure @xmath26 shows the radial profiles of @xmath0 and @xmath3 for a schwarzschild black hole for which the spin parameter @xmath27 . the radial profiles of the same frequencies for a kerr black hole with a spin parameter @xmath28 are shown in figure @xmath29 . the pseudo - newtonian frequencies @xmath0 and @xmath3 can be seen to slightly deviate from the kerr frequencies in the inner disk while they asymptotically match them in the outer disk . note , however , that the pseudo - newtonian frequency @xmath0 always yields the correct estimation for the isco as @xmath20 matches its kerr value exactly for all disk radii . in the next section , we present the basic equations for a geometrically thin disk and obtain within the current pseudo - newtonian approach the hydrodynamic parameters that are necessary for the analysis of global modes in the inner disk . the long - wavelength global hydrodynamic modes of free oscillations have been recently studied for the boundary regions of accretion disks around neutron stars ( see erkut , psaltis , & alpar 2008 , hereafter epa08 ) . in the mode analysis by epa08 the basic disk equations are perturbed for a geometrically thin disk in vertical hydrostatic equilibrium . the fourier decomposition of the linearized perturbation equations leads to the identification of the complex mode frequencies @xmath30 . the real parts of axisymmetric @xmath31 and nonaxisymmetric @xmath32 mode frequencies correspond to the frequency bands of qpos while the imaginary parts determine the growth rates of the oscillations . in the global mode analysis , both the frequency bands and their growth rates depend on several key parameters such as @xmath20 , the radial profile of the surface density , @xmath33 , the inverse timescale @xmath34 associated with the radial drift velocity , and the inverse timescale @xmath35 associated with the sound speed in the inner disk . these parameters are determined by the global structure of the unperturbed steady disk ( see epa08 ) . for sufficiently high mass accretion rates , e.g. , @xmath36 , where @xmath37 is the eddington mass accretion rate , black holes in lmxbs accrete matter through radiatively efficient accretion disks whose innermost regions are dominated by radiation pressure ( shakura & sunyaev 1973 , hereafter ss73 ) . the innermost truncation radius of such a disk around a kerr black hole is estimated by the radius of the isco , @xmath38 , which can be found as a solution of @xmath39 for @xmath40 ( see eq . [ [ kker ] ] ) . the unperturbed steady structure of a radiation pressure dominated inner disk is described by@xmath41where @xmath42 is the effective sound speed , @xmath43 is the average mass density , and @xmath44 is the radiation energy density . we write , for the vertical hydrostatic equilibrium in the disk,@xmath45where @xmath46 is the half - thickness of the disk . the average mass density can be expressed in terms of the surface mass density @xmath47 as@xmath48the vertical energy balance in the inner disk is satisfied for@xmath49where @xmath50 is the electron scattering opacity and @xmath51 is the energy dissipation rate per unit area of the disk ( see ss73 ) . the energy flux due to viscous energy dissipation is@xmath52here , @xmath53 is the kinematic viscosity for which the @xmath54-prescription ( ss73 ) can be written as@xmath55 for a geometrically thin disk , the radial momentum balance can be written to a good approximation as @xmath56 , where @xmath57 is the pseudo - newtonian potential that mimics the gravitational field of the black hole . in the radial momentum equation , the pseudo - newtonian force , @xmath58 , is the source of acceleration , @xmath59 , where @xmath3 is given by equation ( [ pnom ] ) . for the conservation of mass and angular momentum , we write @xmath60and @xmath61respectively , where @xmath62 is the radial drift velocity of the accreting matter in the inner disk and @xmath63 is an arbitrary constant of integration . we solve equation ( [ agm ] ) using torque - free boundary condition at the innermost disk radius , @xmath38 , which is appropriate for a disk around a black hole . the constant of integration can be determined as @xmath64 to satisfy the torque - free boundary condition . using equation ( [ pnom ] ) , it follows from equations ( [ rpd])([agm ] ) that @xmath65@xmath66@xmath67and@xmath68where @xmath69 is the typical value for the radiation flux and @xmath70 here , @xmath71 is a function of the spin parameter @xmath14 such that @xmath72 ^{1/2}-2\left ( 1-\eta \right ) } { 3\sqrt{3}\left [ 1-\left ( 1-\eta \right ) ^{2}\right ] } \label{earlt}\ ] ] for prograde accretion disks around rotating black holes ( see shapiro & teukolsky 1983 ) . in equations ( [ beta])([flx ] ) , the dimensionless factors arising from the boundary conditions and pseudo - newtonian corrections are @xmath73@xmath74@xmath75@xmath76and@xmath77with@xmath78 . \label{f}\ ] ] for illustrative purposes we display in figure @xmath79 the radial distributions of the outgoing radiation flux ( see eq . [ [ flx ] ] ) throughout the inner disk for two putative black holes with spin parameters @xmath27 and @xmath80 . in the following section , we use the ratio @xmath20 ( see eq . [ kker ] ) and the global hydrodynamic parameters @xmath33 , @xmath81 , and @xmath82 ( see eq . [ [ beta]][soo ] ) to identify the radial zone in the inner disk where the modes grow . as we will see , the hydrodynamic modes grow only within a limited range of radii in the innermost disk region out of which the radiation flux is maximum ( see fig . @xmath79 ) . when there are no external perturbations due to large - scale magnetic fields of the accreting star , the free oscillation modes in a boundary region or the inner disk are excited through the dynamical effect of the viscosity . in the limit of small hydrodynamic corrections , this can be seen from the growth rates of both axisymmetric and nonaxisymmetric modes for which im@xmath83 ( see epa08 ) . in the presence of viscosity and therefore of radial drift velocity , @xmath84 and the high - frequency modes can have positive growth rates only if @xmath85 . this is also valid for the global modes in a disk around a black hole . unlike neutron stars , the effect of a large - scale toroidal magnetic force in addition to viscosity on the excitation of global modes might be absent in the case of black holes ( see section 3 in epa08 ) . for a black hole disk , the presence of the isco with a torque - free boundary condition determines the radial profile of the surface density , @xmath33 , and thus the growth rates of the modes . note that @xmath86 for @xmath87 ( see eq . [ [ beta ] ] ) and we expect , in the regime of small hydrodynamic corrections , that the global hydrodynamic modes do not grow for sufficiently large radii in the inner disk . for the innermost disk region , however , the hydrodynamic corrections can be important to distinguish among the growth rates of different modes and to identify the set of radii at which these modes grow . in order to see the effects of hydrodynamic parameters on both the frequency bands and growth rates of the modes in the inner disk beyond the regime of negligible hydrodynamic corrections , we use equations ( [ beta])([soo ] ) together with equation ( [ kker ] ) in the full eigenfrequency solutions for axisymmetric and nonaxisymmetric perturbations given in the appendix of epa08 . figures @xmath88@xmath89 show the real and imaginary parts of the complex mode frequencies in units of the orbital frequency @xmath3 as functions of the radial distance in the inner disk . the real and imaginary parts represent the frequency bands and the growth rates of the modes , respectively . in figures @xmath88@xmath89 , we label the hydrodynamic mode frequencies and their growth rates with notation corresponding to the test - particle frequencies . this provides us with an easy identification and a simple designation of each mode without ambiguity and without loss of generality . in the limit of small hydrodynamic corrections , the frequencies of all hydrodynamic modes converge to the test - particle frequencies . we mark axisymmetric @xmath90 modes with the corresponding test - particle frequencies @xmath91 and @xmath92 . we use @xmath93 and @xmath94 as the appropriate labels to distinguish among nonaxisymmetric modes @xmath95 . figure @xmath96 exhibits the run of the mode frequencies in the inner disk of a schwarzschild black hole @xmath97 . we display the growth rates of the modes in figure @xmath98 . figure @xmath88 is obtained for @xmath99 and @xmath100 . note that the hydrodynamic modes do not grow in the inner disk for @xmath101 . the radial zone within which all modes grow covers only a limited range of radii around @xmath102 in the innermost disk region , as shown in figure @xmath98 . the hydrodynamic modes with frequency bands around @xmath3 and @xmath103 have relatively higher growth rates as compared to those around @xmath104 and @xmath0 bands . figure @xmath105 reveals how the mode frequencies and the growth rates are affected by the rotation of the black hole . for the same values of the viscosity parameter and the mass accretion rate , that is , for @xmath99 and @xmath106 , we plot the frequency profiles of the modes in figure @xmath107 and the corresponding growth rates in figure @xmath108 for a rotating black hole with a spin parameter @xmath80 . in comparison with figure @xmath98 , the growth rates of the modes are higher in figure @xmath108 . the range of radii at which the hydrodynamic modes grow is around @xmath109 . all modes decay for @xmath110 ( see fig . @xmath111 ) . we obtain figure @xmath112 and figure @xmath89 keeping the spin parameter of the black hole at @xmath80 , however , changing the viscosity parameter @xmath54 and the mass accretion rate @xmath113 . for @xmath114 and @xmath106 , we explore the run of the mode frequencies and the growth rates in figures @xmath115 and @xmath116 , respectively . as compared to figure @xmath107 , the radial profiles of the mode frequencies can be seen to be almost unaffected by a change in the viscosity parameter @xmath54 ( see fig . @xmath117 ) . we observe , in figure @xmath116 , that the growth rates are lower than those in figure @xmath108 by a factor around @xmath118 which , indeed , is the factor of decrease in @xmath54 . figure @xmath119 is obtained for @xmath120 while keeping the values of @xmath54 and the spin parameter @xmath14 the same as in figure @xmath112 . note that both the frequency bands and the growth rates of the modes are modified to some level at relatively high mass accretion rates . the greater the mass accretion rate @xmath113 , the higher are the growth rates of hydrodynamic modes ( see fig . @xmath121 ) . we note that the frequency bands that are related to @xmath122 , @xmath3 , and @xmath0 branches in the limit of small hydrodynamic corrections begin to deviate from the test - particle frequencies for sufficiently large mass accretion rates @xmath123 as shown in figure @xmath124 . the common property of figures @xmath88@xmath89 is that all the hydrodynamic modes grow within a limited region in the innermost part of the disk with characteristic radii in the @xmath125 range . the radiation flux emerging from the same region attains the highest values with maxima at @xmath126 and @xmath127 for the black holes with spin parameters @xmath27 and @xmath80 , respectively ( see fig . @xmath79 ) . it is interesting to deduce from figures @xmath88@xmath119 that the frequency ratio of the hydrodynamic modes we associate with @xmath128 and @xmath3 frequency bands is close to @xmath4 at radii in the @xmath129 range , where the disk flux is maximum . as mentioned above , the modes , however , grow throughout an extended region , with @xmath130 , of radii rather than being excited at a particular radius . moreover , the modes with frequencies around @xmath131 and @xmath0 bands also grow within the same region . to distinguish among the pairs of growing modes which can be regarded as plausible candidates for the high - frequency qpo pairs from black holes , we consider the mutual ratios of the flux weighted averages of the frequency bands for different modes . we define the flux weighted average of a frequency branch @xmath132 as@xmath133where @xmath134 is the critical radius beyond which the corresponding mode decays in the inner disk . using equation ( [ wav ] ) , we calculate the ratios of the flux weighted averages of the frequency bands @xmath2 , @xmath3 , @xmath131 , and @xmath0 for different values of the black hole spin parameter @xmath14 between @xmath103 and @xmath25 . for each model value of @xmath135 , we find a critical radius @xmath134 such that all the hydrodynamic modes grow for @xmath136 . unlike the growth rates , the mode frequencies and the width of the radial zone where the modes grow are sensitive to the spin parameter @xmath14 , but not sensitive to the viscosity parameter @xmath54 and the mass accretion rate @xmath113 ( see figs . @xmath88@xmath89 ) . in figure @xmath137 we display the run of @xmath138/@xmath6 , @xmath139/@xmath140 , and @xmath141/@xmath142 for @xmath143 . figure @xmath137 is obtained for the typical values , @xmath99 and @xmath144 . the values for @xmath5/@xmath6 are densely clustered around @xmath4 over a wide range of values for @xmath14 as shown in figure @xmath137 . for slow rotators @xmath145 , @xmath138/@xmath146 . the same ratio drops below @xmath4 as @xmath14 approaches @xmath25 for rapidly rotating black holes . the values of @xmath147/@xmath148 and @xmath6/@xmath149 , on the other hand , span a wide range as the spin parameter @xmath14 varies between @xmath103 and @xmath25 ( see fig . @xmath137 ) . our analysis suggests the hydrodynamic modes with frequency bands around @xmath128 and @xmath3 to be the plausible candidates for the high - frequency qpo pairs observed in black hole systems . note that our model estimation for the frequency ratio of high - frequency qpo pairs involves the two highest frequency modes with positive growth rates . we give examples for surface density perturbations of such global modes in figures @xmath150@xmath151 . we display the three dimensional profile of surface density perturbation @xmath152 in terms of background surface density @xmath153 in the innermost region ( @xmath154 ) of a disk around a rotating black hole with spin parameter @xmath80 for the typical values , @xmath99 and @xmath106 . the examples for nonaxisymmetric modes with frequencies @xmath3 ( fig . @xmath150 ) and @xmath155 ( fig . @xmath156 ) show the surface density perturbations at the time @xmath157 , where @xmath158 is the rotation period at the innermost disk radius . in figures @xmath150 and @xmath156 , the spiral like shapes of different iso - level contours plotted on the @xmath159-plane reveal the similar nonaxisymmetric nature of these modes . in the long run , such as for @xmath160 , the surface density perturbations of both axisymmetric and nonaxisymmetric modes grow in amplitude only for @xmath161 within the same domain . we illustrate this typical behavior in figure @xmath151 as compared to figure @xmath162 for the case of axisymmetric mode with frequency @xmath0 . note that the perturbations at the time @xmath163 ( see fig . @xmath162 ) are comparable in amplitude over the whole computational domain ( @xmath164 ) . the perturbations at the time @xmath165 , however , have large amplitudes only for @xmath166 whereas their amplitudes become negligible for @xmath167 ( see fig . @xmath151 ) . in the global three - dimensional magnetohydrodynamic simulations of black hole accretion disks , the innermost disk region near the isco was found to show qpos with frequency around the maximum of epicyclic frequency ( machida & matsumoto 2003 ) . in one of the recent simulations of the three - dimensional magnetohydrodynamic accretion flows around schwarzschild black holes ( kato 2004 ) , the structure of the flow has been changed at radial distances within the @xmath168 range , where @xmath169 is the schwarzschild radius . two pairs of qpos have been observed to be excited in that region with frequencies around the keplerian frequency and the sum of keplerian and epicyclic frequencies in the power spectra of these simulations . most importantly , the frequency ratio of these qpo features has been found to be near 1.5 . these results are in close agreement with the result of our mode analysis in the present work . we have probed the stability of the global modes in the inner region of a standard accretion disk around a black hole . our study is the application of the recently developed analysis of global hydrodynamic modes ( see epa08 ) to the identification of the high - frequency qpo pairs observed in black hole sources . the presence of the isco allows for effects of strong gravity on both the dynamical frequencies and the global hydrodynamic parameters . our pseudo - newtonian approach takes account of these effects to determine the frequency bands and the growth rates of the unstable modes in the inner disk . the disk is truncated at the radius of the isco , @xmath38 . the growth rates of the modes are negative for sufficiently large distances from the isco . we find that the modes grow in amplitude only within a narrow zone in the innermost disk region . for a non - rotating black hole @xmath170 the characteristic radii of the zone lie in the @xmath171 range . the modes grow within the @xmath172 range for a rotating black hole with spin @xmath173 . among the growing modes the growth rates of the frequency branches around @xmath3 and @xmath103 are higher as compared to those of the modes with frequency bands around @xmath104 and @xmath0 ( see figs . @xmath88@xmath89 ) . due to the effect of enhanced hydrodynamic corrections on the growth rates , the modes grow faster in an accretion regime with relatively high rate and viscosity ( see epa08 ) . the radiation flux due to viscous energy dissipation in the inner disk takes the highest values within the narrow region where the modes grow ( see fig . @xmath174 ) . we deduce from the radial profiles of the mode frequencies that the frequency ratio of the modes around @xmath2 and @xmath3 bands is very close to @xmath4 at the radius where the disk radiation is maximum . this value was observed for the frequency ratio of the high - frequency qpo pairs in black hole sources ( see remillard & mcclintock 2006 ) . instead of being excited at a particular radius in the disk , the hydrodynamic modes grow in a region of finite radial extension . to make an estimation for the expectation value of a frequency band and therefore for the frequency ratios of the relevant modes , we calculate the flux weighted averages of the frequency bands over the innermost disk region where the modes grow . scanning the ratios of the expected mode frequencies for all possible values of the spin parameter @xmath14 ( see fig . @xmath137 ) , we find that only the modes around @xmath2 and @xmath3 branches have a frequency ratio around @xmath4 . this ratio is slightly higher than @xmath4 if the black hole is a slow rotator @xmath175 . the same ratio falls below @xmath4 for fast rotating black holes @xmath176 . the frequency ratios of other modes significantly deviate from @xmath4 over a wide range of values for the spin parameter . relying on the observed values for the frequency ratio of the upper high - frequency qpo to the lower one , we conclude that the modes with frequency branches @xmath2 and @xmath177 are the most plausible candidates for the high - frequency qpos from black holes . the observations of high - frequency qpos can be used to determine the underlying mechanism that produces these oscillations and to measure the spin parameter of the black hole ( remillard & mcclintock 2006 ) . our analysis may provide a way to employ the observed frequency ratio of a high - frequency qpo pair in a given source to estimate the spin parameter @xmath14 . in this sense , figure @xmath137 comes out as an efficient tool for reading the spin parameter @xmath14 that corresponds to the value of @xmath138/@xmath6 to be interpreted as the frequency ratio of a high - frequency qpo pair observed in the x - ray power spectra of the black hole source . there are several reasons for expecting to see the fingerprint of global long wavelength modes in the form of high - frequency qpos observed in the x - ray power spectra of black hole sources in lmxbs . as compared to the neutron stars , there is little chance for accretion flows around the black holes in lmxbs to be affected by the dynamical action of a magnetic field of stellar origin . in the case of a black hole , instead of a direct feedback from the compact object , except gravity , the fluctuations in the mass transfer rate from the binary companion introduce perturbations with a broad band of frequencies including those of the inner disk . the disk modes which depend on global disk parameters become unstable in the innermost disk region and thus the disk oscillation frequencies are selectively amplified without any need for an external mechanism to force them to attain high amplitudes . furthermore , the observable luminosity variation in the x - ray light curve of a source due to global free oscillation modes of long wavelength , that is , of sufficiently large lengthscale is expected to be least affected by the mhd turbulent eddies of short wavelength . according to our present analysis , the higher the mass accretion rate @xmath178 and the greater the viscosity parameter @xmath54 , the higher are the growth rates of the modes . we therefore expect to observe these modes particularly in the state of high mass accretion rate and high viscosity . in such a state , the turbulent disk may also interact with its corona ( see tagger & varnire 2006 ) . this would lead to the formation of high - frequency qpos in a spectral state where the contribution from the power law component is important . in our present analysis we identify the relevant disk modes without deliberating the disk - corona interaction which we plan to consider in a future work . i would like to express my special thanks to m. a. alpar who carefully read the manuscript and contributed it through various suggestions and to d. psaltis for reading the manuscript and very useful discussions . i also thank u. ertan for his valuable comments . i would like to thank the anonymous referee whose suggestions lead me to improve this manuscript . i acknowledge support from tbitak ( the scientific and technical research council of turkey ) for a postdoctoral fellowship and the marie curie fp6 transfer of knowledge project astrons , mktd - ct-2006 - 042722 .

we apply the global mode analysis , which has been recently developed for the modeling of khz quasi - periodic oscillations ( qpos ) from neutron stars , to the inner region of an accretion disk around a rotating black hole . within a pseudo - newtonian approach that keeps the ratio of the radial epicyclic frequency @xmath0 to the orbital frequency @xmath1 the same as the corresponding ratio for a kerr black hole we determine the innermost disk region where the hydrodynamic modes grow in amplitude . we find that the radiation flux emerging from the inner disk has the highest values within the same region . using the flux weighted averages of the frequency bands over this region we identify the growing modes with highest frequency branches @xmath2 and @xmath3 to be the plausible candidates for the high - frequency qpo pairs observed in black hole systems . the observed frequency ratio around @xmath4 can therefore be understood naturally in terms of the global free oscillations in the innermost region of a viscous accretion disk around a black hole without invoking a particular resonance to produce black hole qpos . although the frequency ratio @xmath5/@xmath6 is found to be not sensitive to the black hole s spin which is good for explaining the high - frequency qpos it may work as a limited diagnostic of the spin parameter to distinguish black holes with very large spin from the slowly rotating ones . within our model we estimate the frequency ratio of a high - frequency qpo pair to be greater than @xmath4 if the black hole is a slow rotator . for fast rotating black holes , we expect the same ratio to be less than @xmath4 .

the research on nondiffracting accelerating optical beams has attracted considerable attention since the first study and observation of such beams reported in 2007 @xcite . these are optical wave packets that propagate along a curved trajectory while preserving its transverse amplitude structure . the first of such beams , known as the airy beam , traces back to the context of quantum mechanics @xcite , as a solution to the free - potential schrdinger equation . in optics @xcite , the airy beam is understood as a solution to the paraxial wave equation , which is a good approximation of the propagation dynamics when the beam trajectory is limited to small ( paraxial ) angles @xmath0 . in this case , the airy beam propagates along a parabolic trajectory while maintaining its intensity profile . the desirable feature of simultaneous shape - preserving and self - bending has invoked many intriguing applications including inducing curved plasma filaments @xcite , synthesizing versatile bullets of light @xcite , carrying out autofocusing and supercontinuum experiments @xcite , manipulating microparticles @xcite and so on . however the spatial acceleration of a beam will make the bending angle continue increasing , eventually , the wave packet falls off into the non - paraxial regime and no longer nondiffracting . in @xcite , the authors found solutions to the maxwell s equations in free space , that propagate along semicircular trajectories without losing the intensity of their main lobes after a large angle ( @xmath1 ) bending . roughly speaking , in the two dimensional transverse electric ( te ) or transverse magnetic ( tm ) polarized cases , they examine the spherical harmonic expansion for the solution to the helmholtz equation , splitting the integral for the bessel function into two parts corresponding to both forward and backward propagations . the non - diffracting accelerating beam is the forward bessel wave packets with apodization on the initial axis . the three - dimensional accelerating beams in @xcite are composed by a superposition of scalar solutions for the te /tm polarization , multiplied by a plane wave in the direction perpendicular to the plane the accelerating trajectory lies in . another approach to obtain 3d beams is to implement directly the splitting approach to the 3d spherical harmonic expansions in spherical coordinates ( instead of cylindrical coordinates ) , as indicated in @xcite . this was generalized in @xcite to find non - paraxial beams propagating along elliptic trajectories , that is , the magnitude of acceleration is no longer constant . it is also explained in the previously mentioned physics literatures that the nondiffracting accelerating wave packets in free space , e.g. , the airy beam , is not contradicting the ehrenfest s theorem , because the transverse intensity is not square integrable , hence the beam does not have a transverse center of mass . in experiments , see @xcite , the localized beams with finite energy are induced by applying an exponential truncation ( apodization ) . they still exhibit the key features over long distance propagation in spite of the fact that the center of gravity of these wave packets remains constant ( an outcome of ehrenfest s theorem ) and diffraction eventually takes over . non - diffracting accelerating beams are shown to exist in other type of media . in @xcite , these beams , as analytic solutions of maxwell s equations with linear or nonlinear losses , propagate in absorbing media while maintaining their peak intensity . while the power such beams carry decays during propagation , the peak intensity and the structure of their main lobe region are maintained over large distances . such loss - proof beams , when launched in vacuum or in lossless media , display exponential growth in peak intensity . this is achieved through the property of self - healing of non - diffracting beams , which allows energy transfer from the oscillating tail of the beam to the main lobe region . the self - healing properties , as a result of self transverse acceleration , is studied in @xcite . in @xcite , the idea is generalized to construct shape - preserving wave packets in curved space that propagate along non - geodesic trajectories . in this paper , we show existence of other accelerating and near - nondiffracting solutions for non - paraxial equations . more precisely , we construct the complex geometrical optics ( cgo ) solutions with nonlinear limiting carleman weights ( lcw ) . in the first approach , we consider the propagation in heterogeneous media . let @xmath2 be a bounded domain in @xmath3 with smooth boundary . we consider the time dependent maxwell s equations @xmath4 where @xmath5 for some positive constants @xmath6 and @xmath7 , and the corresponding time - harmonic maxwell s equations of @xmath8 given by @xmath9 where @xmath10 and @xmath11 is the angular frequency . the beam we construct is based on the so - called complex geometrical optics ( cgo ) solutions to maxwell s equations . these solutions were widely used in solving inverse problems arising from imaging modalities using electricity , electromagnetic waves and so on . taking electric impedance tomography ( eit ) for example , the inverse problem is to reconstruct the conductivity function @xmath12 in a conductivity equation @xmath13 in a medium @xmath2 from the boundary dirichlet - to - neumann ( voltage - to - current ) map @xmath14 . this is also known as the caldern problem and was generalized to an inverse problem for maxwell s equations , namely , to reconstruct the parameter set @xmath15 from boundary impedance map given by @xmath16 . here @xmath17 denotes the unit outer normal vector on the boundary @xmath18 . these inverse problems were studied extensively ( see @xcite for a detailed review ) while through all the analysis , the cgo solutions introduced in @xcite plays a key role . an example of such solutions to the conductivity equation @xmath13 is given by @xmath19 where @xmath20 ( @xmath21 is the spatial dimension ) satisfying @xmath22 and @xmath23 satisfies a decaying property with respect to @xmath24 . more precisely , @xmath25 let @xmath26 and @xmath27 . in @xmath28 , set @xmath29 ( note that @xmath22 and @xmath30 ) . assume that @xmath31 is independent of @xmath32 . the above decaying property implies that @xmath33 , with @xmath34 given by , is roughly a plane wave packet propagating nondiffractingly along @xmath35 direction for @xmath36 large . the transverse profile of the beam is oscillatory in @xmath37 and exponential in @xmath38 . ( compared to the airy beam , the tail of the cgo plane beam does not decay with respect to @xmath39 ) . note that the cgo solution is a high energy near - nondiffracting wave packet for @xmath36 is large . using the liouville transform @xmath40 , the conductivity equation @xmath13 is reduced to the schrdinger equation @xmath41 with potential @xmath42 . the exsitence of a global cgo solution @xmath34 of the form is equivalent to solving an equation of @xmath43 in @xmath44 given by @xmath45 the faddeev s kernel defines an inverse of @xmath46 by @xmath47 where @xmath48 and @xmath49 denote the fourier transform and its inverse , respectively . furthermore , it is shown in @xcite that for @xmath36 large enough , @xmath50 for @xmath51 , where @xmath52 is the closure of @xmath53 with respect to the weighted norm @xmath54 . then the existence of @xmath43 to and the estimate are corollary of and the fact that @xmath55 can be viewed as a compactly supported @xmath56 function . for maxwell s equations , by introducing two auxiliary scalar fields @xmath57 and @xmath58 and a liouville type rescaling , the first order system can be reduced to a dirac system @xmath59 . a solution @xmath60 of the dirac system gives a solution to the original maxwell system iff @xmath61 . the reduction to the schrdinger equation @xmath62 is then due to @xmath63 . ( detail of the reduction is outlined in section [ section3d ] . ) this reduction was first introduced in @xcite and became a standard step for construction of cgo for maxwell s equations ever since . once the cgo solutions are constructed for the schrdinger equation , a uniqueness result is required to show that the scalar potentials @xmath57 and @xmath58 are vanishing . as a result , one obtains a cgo solution to maxwell s equations of the form @xmath64 for @xmath36 sufficiently large , where @xmath65 and @xmath66 are two constant vectors in @xmath67 and bounded in @xmath36 . similar to the scalar case , under the assumption that the parameters are functions only depending on transverse variables , these solutions are high energy near - nondiffracting beams . in above construction , replacing the linear phase @xmath68 by a nonlinear one , denoted by @xmath69 , it is shown in @xcite that solutions of cgo type can be constructed to the schrdinger equations using carleman estimates , and also to the magnetic schrdinger equations as seen in @xcite . these admissible phases are called limiting carleman weights ( lcw ) . there are only handful lcw in three dimensions while all analytical functions in @xmath70 can be used as an lcw in two dimensions . with nonlinear phases , it allows to construct solutions propagating along a curved surface ( accelerate ) while almost preserving its intensity profile ( near - nondiffracting ) , usually due to a smallness estimate similar to . in @xcite , the cgo with the lcw @xmath71 was used to solve the caldern problem with partial measurements . in @xcite , the lcw @xmath72 was used to construct solutions on riemannian manifolds with a family of admissible metrics , modeling the anisotropic materials . in order to generalize this to construct cgo solutions to maxwell s equations with nonlinear lcw , it requires uniqueness of the solution to the reduced schrndinger equation . in @xcite , the uniqueness is obtained on a bounded domain by projecting to a function space with fixed boundary conditions . in @xcite , different approach is adopted to obtain the uniqueness , by applying the original fourier analysis in @xcite to @xmath73 direction . the cgo solutions with either @xmath71 or @xmath72 can be put into a unified framework in cylindrical coordinate @xmath74 . in particular , we show that the dependence of the cgo solutions on @xmath66 can be mainly @xmath75 , so that we obtain the bending electromagnetic beams . in the second framework , we adopt a very different approach that is based on a special transformation , known as the kelvin transform . in @xmath44 , the three - dimensional kevin transform @xmath76 is a reflection with respect to a sphere , which maps hyperplanes to spheres that pass the origin . we exploit the transformation law by @xmath76 . moreover , we show that a non - diffracting beam that propagates along straight lines , such as a cgo solution with a linear phase , can be pushed forward " to generate a beam that accelerates along the circular trajectory . the acceleration is strong enough to shift the energy from the tail to the main lobe , over - compensating the intensity of the first lobe while propagating . this effect is independent of the background medium , homogeneous or heterogeneous , lossless or lossy . the paper is organized as follows . in section 2 , we show the main steps to construct the cgo solutions to maxwell s equations based on carleman estimates . section 3 is devoted to the construction based on kelvin transform . both sections are complemented with the demonstration of corresponding solutions . * acknowledgements . * both authors thank professor gunther uhlmann for suggesting this problem and for useful discussions . the research of the first author is partly supported by the ams - simons travel grant . the research of the second author is supported by the nsf grant dms-1501049 . in this section , we aim to construct the almost diffraction - free beams for the inhomogeneous maxwell s equations in dimension three . we first discuss the construction of such beams with lcw @xmath77 . the second construction is to apply another nonlinear lcw @xmath78 . numerical demonstrations of these accelerating beams are also presented . for completeness , we first include a reduction , introduced in @xcite , that transforms the maxwell s system to the vectorial schrdinger equation . let @xmath2 be a bounded domain in @xmath3 with smooth boundary . we consider the time - harmonic maxwell s equations where @xmath79 for some positive constant @xmath6 and @xmath7 , @xmath80 where @xmath10 and @xmath11 in @xmath81 . we remark here that the asymptotic - to - constant assumption @xmath82 is without loss of generality since any smooth parameters @xmath83 in a bounded domain can be extended to satisfy it in a larger domain that contains the propagating region . from ( [ r : max ] ) , one has the following compatibility conditions for @xmath84 and @xmath85 @xmath86 there are eight equations in and for six unknowns , components of @xmath87 and @xmath88 . it allows us to augment two scalar potentials @xmath57 and @xmath58 to obtain @xmath89 where @xmath90 , @xmath91 ( the principal branch ) and @xmath92 . set the unknown to be the eight vector @xmath93 . we can write as @xmath94 where @xmath95 is a first order dirac operator given by @xmath96 and @xmath97 here @xmath98 is the @xmath99-identity matrix . note that @xmath100 . moreover , we mention a fact , later becoming very important , that maxwell s equations is equivalent to the dirac system if and only if @xmath61 . throughout the paper we also use the notation @xmath101 for all eight - vectors @xmath102 where lower cases @xmath103 are scalar and upper cases @xmath104 are three - vectors . now we apply a liouville type of rescaling by letting @xmath105 then we have @xmath106 where @xmath107 and @xmath108 with @xmath109 . one can easily verify that with the rescaling , the first order terms in @xmath110 cancel each other and we obtain @xmath111 where the matrix potential @xmath112 is an @xmath113-valued potential with compact support in @xmath2 , whose entries involve the parameters @xmath114 and their first and second derivatives . the form of @xmath112 is not crucial in this construction so we omit writing the explicit formula . for readers who are interesting in the expression of @xmath112 , it can be found , for example , in @xcite . we will also need the following relations @xmath115 where @xmath116 shares the same property as @xmath112 . an extra fact of @xmath116 we will take advantage of is that the first and the fifth rows are diagonal . here @xmath117 denotes the transpose of @xmath118 . in this part , we outline the construction steps of cgo solutions to the schrdinger equations based on the carleman estimate as in @xcite . however , the general construction does not provide uniqueness that is necessary for translating to solutions to maxwell s equations . therefore , our argument will bifurcate slightly for two different choices of limiting carleman weights in order to obtain uniqueness respectively . a real smooth function @xmath119 on an open set @xmath120 is a lcw if it has a non - vanishing gradient on @xmath120 and if it satisfies the condition @xmath121 roughly speaking , this is the hrmander condition that guarantees the solvability of the semi - classical conjugate operator @xmath122 for both @xmath119 and @xmath123 . the tool used to obtain it is the carleman estimate , see @xcite . we are looking for the complex geometrical optics solutions of the form @xmath124 where @xmath125 is neglectable compared to @xmath126 in the semi - classical sense . here @xmath43 satisfies the eikonal equation , read as @xmath127 with such @xmath119 and @xmath43 , we denote @xmath128 then we have @xmath129a.\end{aligned}\ ] ] for the remainder term @xmath125 to satisfy the decaying condition with respect to @xmath36 , we ask @xmath126 to satisfy the transport equation @xmath130a=0.\ ] ] suppose @xmath126 is a @xmath131 solution ( they exist as shown in the cases below ) . also , suppose @xmath132 and @xmath133 . then by the proposition 2.4 in @xcite , for @xmath36 large enough , there exists an @xmath134 satisfying @xmath135 for some constant @xmath136 independent of @xmath36 . now our attention switches to the phase @xmath137 and vector field @xmath126 in order to obtain near non - diffracting accelerating solutions in @xmath2 . we consider the phase in terms of @xmath138 where @xmath139 denotes the cylindrical coordinate for @xmath140 with @xmath141 , polar coordinates in the @xmath142 variable . set @xmath143 where @xmath144 is a @xmath145 function to be specified later . then the transport equation reads @xmath146 we can then choose @xmath147 where @xmath148 is a constant and @xmath149 is an arbitrary vector function of @xmath66 . then the dominant term of @xmath150 as @xmath151 is given by @xmath152 which propagates non - diffractingly and along a circle if we choose @xmath149 appropriately . similarly , one can choose @xmath153 . then reads @xmath154 and the solution is @xmath155 . + the choice of @xmath156 has to be such that @xmath119 and @xmath43 satisfy respective conditions of lcw and . in three dimensional euclidean space , it is mentioned in @xcite that there are only six lcws up to translation and scaling , among which @xmath72 and @xmath71 only depend on @xmath73 and @xmath157 , hence @xmath158 . in @xcite , @xmath159 is used . it is not hard to verify that @xmath160 satisfies the eikonal equation by writing the gradient as @xmath161 this is the case @xmath162 . in order to show uniqueness , an approach different from carleman estimate was taken in @xcite by taking advantage of that the phase is linear in @xmath73 . suppose @xmath163 . using the fourier decomposition of @xmath164 with imposed zero dirichlet condition on the transversal domain @xmath165 , this is reduced to solving @xmath166 for complex geometrical optics eigen - modes . since the phase is linear in @xmath73 , this can be achieved by simulating the direct analysis for the linear phase case as in @xcite , which carries a uniqueness result globally in @xmath73 . that is why the domain under consideration in this case is assumed to be cylindrical , for example , @xmath167 where @xmath168 denotes the disc in @xmath169 centered at @xmath170 with radius @xmath125 . then the space in the norm estimate is replaced by @xmath171 or @xmath172 defined using @xmath73-weighted norms @xmath173 we also define the spaces @xmath174 then we have the inverse of the conjugate operator @xmath175 @xmath176 satisfies @xmath177 for @xmath178 and @xmath179 . however , if @xmath180 in the choice of @xmath126 given by , then @xmath181 , the right hand side of the equation for @xmath125 , is not in @xmath171 ( not enough decay in @xmath73 ) . in @xcite , it is shown in proposition 5.1 that @xmath182 can be extended to include functions with special dependence in @xmath73 such as @xmath183 on the right hand side , which takes care of this problem . therefore , there exists a unique @xmath184 . moreover , it satisfies @xmath185 in the case that @xmath186 with compact support , we have @xmath187 . + on the other hand , in @xcite , the authors used @xmath71 for the case that @xmath2 does not contain the origin . for our purpose , we consider @xmath188 where @xmath189 denotes the upper half plane corresponding to @xmath190 . the corresponding @xmath191 can be chosen as the angle formed by the vector @xmath67 and @xmath73 axis . in terms of @xmath158 , we have @xmath192 therefore , @xmath193 . to address the uniqueness of the solution to the schrdinger equation , one can adopt the orthogonal projection technique in @xcite onto a subspace of @xmath194 with specified boundary condition adjusted to suit our case . then we have @xmath195 with @xmath196 where @xmath197 . [ rmk : linear ] here we would like to comment also on the case of linear phase @xmath198 in the cartesian coordinate @xmath199 with @xmath200 , corresponding to another lcw , @xmath201 . then the eikonal equation gives @xmath202 and @xmath203 , or equivalently @xmath204 . the transport equation ( [ eqn : csw_a ] ) becomes @xmath205 . hence @xmath126 is chosen to be a constant vector . the invertibility of the conjugate operator @xmath206 relies on the direct fourier analysis , as in @xcite , in the whole @xmath44 for uniqueness . the operator @xmath207 satisfies @xmath208 for @xmath178 and @xmath179 , where @xmath209 and @xmath210 are defined by the norms with @xmath211 replaced by @xmath212 . since our schrdinger equations here are more of the helmholtz type with wave number @xmath213 . the phase @xmath214 above can be replaced by @xmath215 with @xmath216 and @xmath217 . the cgo solutions constructed above are the exact solutions of the schrdinger equations in the inhomogeneous space . these solutions will be applied to construct the solutions to maxwell s equations below and more importantly , they possess near - nondiffracting property as @xmath218 is large . besides , these solutions have implications for the study of other wave system in nature . in this part , we compute the cgo solutions to the original maxwell s equations using the cgo solutions to the reduced schrdinger equation and their uniqueness for the two lcw in cylindrical coordinate , in order to obtain the near non - diffracting accelerating beams . now the corresponding cgo solutions to the dirac system @xmath59 are given by @xmath219 where @xmath220 we recall that for the corresponding @xmath221 to be the solution to maxwell s equations , one must have @xmath222 . in general , the strategy is to choose a proper @xmath223 such that the corresponding scalars of @xmath224 , given by @xmath225 satisfy @xmath226 . this is motivated by the fact that @xmath60 satisfies the other schrdinger equation @xmath227 which implies @xmath228 where @xmath229 and @xmath230 are diagonal components of the first and fifth row of @xmath116 , given by @xmath231 they are compactly supported in @xmath2 . to obtain the solution to maxwell s equations , we start with the case on the cylinder @xmath232 such that @xmath233 with @xmath234 . note that @xmath235 . then we have [ cgomax ] let @xmath236 , @xmath180 be constant and let @xmath237 . there exists @xmath238 such that when @xmath239 we have that given smooth functions @xmath240 and @xmath241 , there exists a solution to maxwell s equations in @xmath232 of the form @xmath242 where @xmath243 stands for a function whose @xmath244 , hence @xmath194 norm is of order @xmath245 as @xmath246 . since @xmath234 , we have @xmath247 . suppose @xmath126 is given by . denote @xmath248 the same convention of other eight - vectors . then @xmath249,\\ b^{(2)}=&\frac{e^{i\lambda z}}{\sqrt{2ir}}\left[\left(\begin{array}{c}i\tau+\lambda \\(-\tau+i\lambda+\frac i{2r})\cos\theta+\frac i r\sin\theta\partial_\theta \\(-\tau+i\lambda+\frac i{2r})\sin\theta-\frac i r\cos\theta\partial_\theta\end{array}\right)\cdot g^{(1)}(\theta)+kg^{(2)}(\theta)\right ] . \end{split}\ ] ] given @xmath250 we choose @xmath251 by direct computation , we obtain @xmath226 . from and , we can obtain @xmath252 . then @xmath253 is the only solution to @xmath254 by uniqueness result ( [ eqn : csw_r_t_smallness ] ) , we obtain @xmath255 . hence @xmath256 to obtain @xmath257 , observe that @xmath258 ( @xmath259 ) is bounded in @xmath36 and @xmath260 by the @xmath36-dependence of above choice of @xmath149 , it is easy to see that the leading term in @xmath261 norm of @xmath262 and @xmath263 are out of the first term @xmath264 respectively . this finishes the proof of . for the other case of nonlinear phase @xmath265 , although we are able to construct the cgo solution to the reduced schrdinger equation of the form ( [ eqn : csw_z_sch ] ) with uniqueness , the calculation suggests that we are unable to pick a proper @xmath126 such that the @xmath226 . however , the construction here is on a bounded domain @xmath2 and the assumption that the parameters @xmath266 and @xmath267 are asymptotic to constants @xmath268 is artificial , namely , we could extend the parameters to a larger domain @xmath269 such that @xmath270 and construct the solution there . as a result , @xmath271 in our calculation and the corresponding @xmath226 if we choose @xmath272 , by . meanwhile , @xmath273 is still bounded in @xmath274 with respect to @xmath36 . again , by uniqueness , we have @xmath255 . finally , for arbitrary @xmath145 functions @xmath240 and @xmath241 , we can have @xmath275 by letting @xmath276 and @xmath277 . together , this implies that we obtain the cgo solutions to maxwell s equations on @xmath120 given by @xmath278 note that these solutions are not oscillating with respect to @xmath157 . we conclude this section with more remarks discussing on the properties of the solutions we constructed above . these beams exhibit shape - preserving , nondiffracting acceleration . in ( [ cylinermax ] ) , if we choose @xmath279 for @xmath280 , then for @xmath36 large , the electromagnetic wave packets behave as @xmath281 to explain the non - diffracting property , when looking at the transverse profile of @xmath87 and @xmath88 ( without considering @xmath282 ) , we observe that both real part and imaginary part have * intensity * independent of @xmath66 if @xmath283 and @xmath266 are independent of @xmath66 . on the other hand , we observe that the first component @xmath284 electric wave is shape - preserving as shown in figure [ kelvinfig ] a ) while the second and the third components are not shape - preserving on their own , as shown in figure [ kelvinfig ] b ) and c ) . it suggests a power shift from @xmath285 component to @xmath286 component , which was also observed in @xcite and explained as the rotation of the fields to stay normal to the beam bending trajectory . for the spherical wave ( [ s_eh ] ) , we observe that there is no oscillation with respect to the radius @xmath157 as shown in figure [ plt_sphere_e1 ] . however , the intensity profile still preserves along every circle whenever @xmath73 is fixed . ( a ) and @xmath287 ( a ) the beam @xmath284 exhibiting shape - preserving . the trajectory is confined in a circle on the plane @xmath288 . ( b ) and ( c ) , the figures show that the propagating beams @xmath285 and @xmath286 are along a circular trajectory with varying intensity . ( d ) the intensity of @xmath289 decreases and oscillates when @xmath157 is growing . , title="fig:",width=2 ] ( b ) and @xmath287 ( a ) the beam @xmath284 exhibiting shape - preserving . the trajectory is confined in a circle on the plane @xmath288 . ( b ) and ( c ) , the figures show that the propagating beams @xmath285 and @xmath286 are along a circular trajectory with varying intensity . ( d ) the intensity of @xmath289 decreases and oscillates when @xmath157 is growing . , title="fig:",width=2 ] ( c ) and @xmath287 ( a ) the beam @xmath284 exhibiting shape - preserving . the trajectory is confined in a circle on the plane @xmath288 . ( b ) and ( c ) , the figures show that the propagating beams @xmath285 and @xmath286 are along a circular trajectory with varying intensity . ( d ) the intensity of @xmath289 decreases and oscillates when @xmath157 is growing . , title="fig:",width=2 ] ( d ) and @xmath287 ( a ) the beam @xmath284 exhibiting shape - preserving . the trajectory is confined in a circle on the plane @xmath288 . ( b ) and ( c ) , the figures show that the propagating beams @xmath285 and @xmath286 are along a circular trajectory with varying intensity . ( d ) the intensity of @xmath289 decreases and oscillates when @xmath157 is growing . , title="fig:",width=2 ] in ( [ s_eh ] ) when @xmath290 and @xmath291 on the plane @xmath292.,width=192 ] the reduction of maxwell s equations to the vectorial schrdinger equation was inspired ( see @xcite ) by the physically referred auxiliary hertz potentials ( also known as sommerfeld potentials ) introduced to handle equations in vacuum ( homogeneous ) background . that is to write the electric and magnetic fields as @xmath293 where the hertz vector potential @xmath294 satisfies the vector schrdinger equation @xmath295 note that these potentials are given by @xmath150 through @xmath296 when @xmath297 where @xmath298 is a scalar function independent of @xmath73 , and @xmath299 , we have @xmath300 where @xmath301 and @xmath302 are the corresponding radial and angular unit vectors . we recover the tm - polarized propagation mode . it was analyzed in @xcite that , for their nondiffracting accelerating beam , the radial @xmath303 component plays the dominant role and preserves the shape of propagation . we observe that there is a similar behavior for our solution , the radial component of also dominates the propagation direction of the waves . we would like to point out that the cgo solutions are constructed on a bounded domain with boundary conditions , although we do not restrict ourselves to a specific domain . in another word , this is not a freely propagating solution ( a half - space solution ) sent initially from a plane like @xmath304 physically as in @xcite . in this section , we apply kelvin transform to construct the solutions to maxwell s equations in the physical space , that is , the parameter are constants , which is the situation discussed in most physical papers . we show these beams also have shape - preserving property . first , we recall the transformation law for maxwell s equations . suppose @xmath305 is a diffeomorphism on @xmath306 , where @xmath307 and let @xmath308 be the solution to maxwell s equations @xmath309 if we define the _ push - forward _ of the electric and magnetic fields @xmath289 and @xmath310 by @xmath311 as @xmath312 where @xmath313 denotes the jacobian matrix of the transformation , then these push - forward fields @xmath314 and @xmath315 satisfy maxwell s equations in the space of @xmath316 @xmath317 with _ push - forward _ parameters @xmath318 this invariance is just another presentation of independence of choice of coordinates for maxwell s equations . in order to construct accelerating beams in the free space with constant @xmath6 and @xmath7 , which we call the _ physical space _ , we consider the push - forward system living in the space with @xmath319 and @xmath320 , which we call the _ virtual space _ , by a special @xmath311 , known as the kelvin transform . it is the following reflection map with respect to the sphere @xmath321 of radius @xmath322 , @xmath323 where @xmath324 . it maps any sphere through origin to a hyperplane and satisfies @xmath325 . a two dimensional configuration is shown in figure [ fig : k2 ] . let @xmath340 be a @xmath341 cut - off function with @xmath342 on @xmath343 and @xmath344 on @xmath345 $ ] . set @xmath346 then we want to construct cgo solutions to maxwell s equations with parameters @xmath347 in @xmath348 . note that now @xmath349 . following the steps in section [ section3d ] , one looks for the cgo solution to the schrdinger equation @xmath350 by remark [ rmk : linear ] , we have for @xmath215 satisfying @xmath22 and @xmath351 sufficiently large , @xmath352 for some constant vector @xmath353 of order @xmath354 , where @xmath355 . moreover , @xmath356 naturally , given @xmath150 as , we have @xmath360 where @xmath361 here @xmath362 denotes the matrix of the form but with @xmath363 replaced by @xmath364 . if we choose @xmath365 for arbitrary @xmath366 , it immediately gives @xmath367 where @xmath368 since the first and the fifth components of @xmath369 vanish , applying the uniqueness addressed in remark [ rmk : linear ] , we obtain that there exists a solution to maxwell s equations in @xmath2 given by @xmath370 here we denote @xmath371,\\ { \widetilde}{\mathbf{h}}(\tilde x ) & = e^{\frac{1}{2}(-\tau(\tilde{x}_1 - \tilde{x}_2 ) + i\sqrt{\tau^2-\rho^2}(\tilde{x}_1 + \tilde{x}_2))}e^{i\rho\tilde x_3}\left[(\widehat\zeta_0\cdot\mathbf b)\widehat\zeta_0+o_\tau(\tau^{-1})\right ] . \end{split}\ ] ] here @xmath372 denotes vector functions whose @xmath194 norm is bounded with respect to @xmath36 . in the annulus @xmath373 , from ( [ eqn : pf_mu_eps ] ) , we have for @xmath36 large @xmath374 note that @xmath375 and @xmath376 are the near planewave parts in the virtual space . the other observation is that both @xmath314 and @xmath315 are almost perpendicular to @xmath357 for @xmath36 large , due to the choice of @xmath377 . by and that @xmath329 , we obtain the solutions to the original maxwell s equations in free space @xmath378{\widetilde}{\mathbf{e}}(k(x)),\\ { \mathbf{h}}(x ) & = k_*{\widetilde}{\mathbf{h}}=-i\tau\mu_0^{-1/2}\frac{r^3}{|x|^3}\left[i-2\widehat r\widehat r^t\right]{\widetilde}{\mathbf{h}}(k(x ) ) . \end{split}\ ] ] the formula suggests the following properties of such a wave . 1 . in figure [ physical_e1 ] a ) , the transverse profile @xmath379 of the near plane - wave part at @xmath380 in the virtual space is shown . one notices that the peaks and valleys of the oscillation reside on lines @xmath381 ( while the exponential decay is along the perpendicular lines ) . in the virtual space , these peaks and valleys propagate straight in @xmath357 direction . these peak and valley planes @xmath382 are mapped to spheres passing through the origin in the physical space . in figure [ physical_e1 ] b ) , by using kelvin transform with respect to the sphere of radius @xmath383 , we depict the peak propagation due to the factor @xmath384 ( without multiplication by @xmath385 $ ] ) . note that the lobes " feature shrinks " as propagating away from the @xmath386-plane @xmath387 while the intensity increases due to @xmath388 . hence the intensity keeps shifting from the tail towards the main lobe " , suggesting an overly self - healing due to the wave acceleration . 2 . multiplication by the matrix @xmath389 $ ] , that is , we consider the solutions in ( [ 3:kelvinsol ] ) , which corresponds to a reflection of the electric and magnetic fields and does not change the norm of any real vector . the real and imaginary intensities of the fields are preserved . 3 . the construction applies to a lossy system where the conductivity @xmath31 is a positive constant , in which case we only need to replace @xmath7 by the complex number @xmath390 . above self - healing property still exists and compensates the energy loss during the propagation . r. schley , i. kaminer , e. greenfield , r. bekenstein , y. lumer , and m. segev , _ loss - proof self - accelerating beams and their use in non - paraxial manipulation of particles trajectories _ , nat . * 5:5189 * doi : 10.1038/ncomms6189 ( 2014 ) .

we show that new families of accelerating and almost nondiffracting beams ( solutions ) for maxwell s equations can be constructed . these are complex geometrical optics ( cgo ) solutions to maxwell s equations with nonlinear limiting carleman weights . they have the form of wave packets that propagate along circular trajectories while almost preserving a transverse intensity profile . we also show similar waves constructed using the approach combining cgo solutions and the kelvin transform .

experimental studies of neutrino oscillations have provided us with compelling evidence that neutrinos have masses and lepton flavors mix . among various theoretical models , the famous seesaw mechanism @xcite provides us with a very natural description of why the masses of the three known neutrinos are so small compared to the masses of the other standard model ( sm ) fermions . in the simplest type - i seesaw model , heavy right - handed neutrinos with a mass scale @xmath0 are introduced in addition to the sm particle content . in order to stabilize the masses of the light neutrinos around the sub - ev scale , @xmath1 is naturally expected , if the dirac mass @xmath2 between the left- and right - handed neutrinos is comparable with the mass of the top quark . the testability of conventional seesaw models is therefore questionable . furthermore , the heavy right - handed neutrinos potentially contribute to the hierarchy problem through loop corrections to the higgs potential , unless a supersymmetric framework is considered . the large hadron collider ( lhc ) will soon start to probe tev scale physics , and the question of whether we can find hints on the neutrino mass generation mechanism at the lhc or not is relevant and interesting . there are several indications that new physics will show up at the tev scale , in particular theories that are able to stabilize the higgs mass and to solve the gauge hierarchy problem . the geometric mean of the planck mass and the @xmath3 k background temperature also suggests that 1 tev is the maximum mass that any cosmologically stable perturbatively coupled elementary particle can have , otherwise the density of the universe exceeds its critical value @xcite . within the seesaw framework , for the purpose of lowering the seesaw scale without spoiling the naturalness criterion , some underlying symmetry preserving the lepton number , @xmath4 , is usually incorporated . for example , in the type - i seesaw with more than one heavy right - handed neutrino , contributions to the light - neutrino masses from different right - handed neutrinos may cancel each other due to the symmetry , which results in massless left - handed neutrinos after integrating out the heavy degrees of freedom from the theory @xcite . such a low - scale fermionic seesaw mechanism may not be able to stabilize the masses of the light neutrinos , since loop corrections may be unacceptably large . a possible way to avoid this problem of the type - i seesaw model is given by the inverse seesaw model , which contains a majorana insertion used to reduce the @xmath5 scale @xcite . in the type - ii seesaw model , extending the sm with an @xmath6 triplet higgs scalar @xcite , the coupling between the triplet and the sm higgs scalar breaks lepton number explicitly and is expected to be very small . thus , the masses of the light neutrinos are suppressed through the approximate symmetry . in general , the canonical leptogenesis mechanism @xcite , which provides a very attractive description of the origin of the observed baryon asymmetry of the universe , does not work for the low - scale seesaw mechanisms unless severe fine - tuning is invoked @xcite . in this paper , we employ the alternative framework of extra spacetime dimensions , where the fundamental grand unified scale and the planck scale are lowered in a natural way @xcite . we work exclusively within the context of flat extra dimensions . in our higher - dimensional seesaw model , a truncating scale restoring the renormalizability of the theory plays the role of breaking @xmath5 , so that the masses of the light neutrinos are suppressed , while the lower kaluza klein ( kk ) states can be searched for at the lhc . significant low - energy non - unitary leptonic mixing , due to integrating out the heavy kk states , could give observable phenomena in future neutrino oscillation experiments , such as a neutrino factory @xcite . in addition , resonant leptogenesis could possibly be achieved in this model . for earlier studies of the generation of small neutrino masses in the context of extra dimensions , see for example refs . a study of unitarity violation in scenarios with bulk gauge singlet neutrinos was performed in ref . an alternative higher - dimensional seesaw model was investigated in ref . @xcite . the remaining parts of the paper are organized as follows : first , in sec . [ sec : introduction ] , we present the general formalism of our model . then , in sec . [ sec : nu ] , we show explicitly how sizable non - unitarity effects emerge in the leptonic flavor mixing . section [ sec : lhc ] is devoted to the collider signatures and the discovery potential of the heavy kk modes at the lhc . we comment on the origin of baryon number asymmetry in our model in sec . [ sec : leptogenesis ] . finally , a summary and our conclusions are given in sec . [ sec : summary ] . we consider a brane world theory with a five - dimensional bulk , where the sm particles are confined to the brane . we also introduce three sm singlet fermions @xmath7 ( @xmath8 ) @xcite . being singlets , they are not restricted to the brane and can propagate in the extra spacetime dimensions . the action responsible for the neutrino masses is given by @xmath9 \nonumber \\ & & + \int_{y=0 } { \rm d}^4 x \left ( - \frac{1}{\sqrt{m_s } } \overline{\nu_{\rm l } } \hat m^c \psi - \frac{1}{\sqrt{m_s } } \overline{\nu^c_{\rm l } } \hat m \psi + { \rm h.c.}\right),\end{aligned}\ ] ] where @xmath10 is the coordinate along the extra compactified dimension and @xmath11 denotes the mass scale of the higher - dimensional theory . note that , although @xmath12 is defined in the same way as in four dimensions , it does not represent the charge conjugate of @xmath13 in five dimensions @xcite , and hence , the term @xmath14 is not a majorana mass term . however , in the four - dimensional theory , it leads to effective majorana mass terms for the kk modes of @xmath13 . due to the freedom in the choice of basis for the singlet fermion fields , one can always apply a unitary transformation in flavor space in order to diagonalize @xmath0 . without loss of generality , we will therefore work in a basis in which @xmath15 is real and diagonal . the dirac masses @xmath16 and @xmath17 could be generated by couplings of the bulk neutrinos to a brane - localized higgs boson receiving a vacuum expectation value . we decompose the spinors of the bulk singlet fermions into two two - component objects : @xmath18 , where @xmath19 . since the extra dimension is compactified on the @xmath20 orbifold , the kk modes of @xmath21 and @xmath22 are four - dimensional weyl spinors . we take @xmath21 to be even under the @xmath23 transformation @xmath24 , while @xmath25 is taken to be odd . thus , in eq . , the @xmath17 term corresponding to the coupling between @xmath26 and @xmath25 is not allowed . the kk expansions of @xmath21 and @xmath25 are given by @xmath27 in general , an extra - dimensional model must be viewed as an effective theory , since it is non - renormalizable . this means that the kk towers are expected not to be infinite , but truncated after a finite number of levels . the nature of this cutoff depends on the specific ultraviolet ( uv ) completion of the model , which is not known . here , we impose a truncation of the kk towers at a maximum kk index @xmath28 . a cutoff of this kind arises , for example , in deconstructed models of extra dimensions @xcite . in general , other kinds of truncation schemes are possible , but the one that we consider has the virtue of giving rise to a mechanism for generating small neutrino masses from the tops of the kk towers , as will be discussed below . inserting the above expansion into eq . and integrating over the compactified extra dimension , we arrive at the following form for the four - dimensional action @xmath29 \right . \nonumber \\ & & \phantom{\int d^4 x}-\left . { \rm i } \left(\nu_{\rm l}^t \sigma^2 m_{\rm d } \xi^{(0 ) } + \sqrt{2 } \sum^n_{n=1 } \nu_{\rm l}^t \sigma^2 m_{\rm d } \xi^{(n ) } + { \rm h.c . } \right ) \right\},\end{aligned}\ ] ] where , written in block - form , the mass matrix @xmath30 for the kk modes at the @xmath31th level takes the form @xmath32 the dirac mass term is then given by @xmath33 . for the purpose of simplicity in the following discussion , we define the linear combinations @xmath34 for @xmath35 . the full mass matrix in the basis @xmath36 then reads @xmath37 the scale of @xmath0 is not governed by the electroweak symmetry breaking , and hence , one can expect that @xmath38 holds . then , by approximately solving the eigenvalue equation of the matrix in eq . with respect to the small ratio @xmath39 , the light - neutrino mass matrix is found to be @xmath40 in refs . @xcite , the limit @xmath41 is considered , and the light - neutrino mass matrix is then given by @xmath42 the masses of the light neutrinos are suppressed only if @xmath43 in the denominator of eq . is very large . therefore , a severe fine - tuning between @xmath44 and @xmath45 has to be invoked , which appears quite unnatural . however , bare majorana masses of the form @xmath46 , where @xmath47 is an odd integer , emerge naturally from the sherk schwarz decomposition in string theory as a requirement of topological constraints , and hence , such relations do not suffer any fine - tuning problems ( see detailed discussions in ref . @xcite ) . with our chosen cutoff scheme , together with the above condition on @xmath48 , lepton number violation will be induced only at the top of the kk tower , as we will see shortly . there could , of course , be other lepton number violating processes at some intermediate point , but we choose to treat the simple scenario where the cutoff is the only source . one can easily prove that , in the simplest case @xmath49 , the light - neutrino mass matrix is given by @xmath50 instead of a large mass scale @xmath0 for the singlet fermions , the light - neutrino masses are suppressed by the large cutoff scale @xmath51 . we consider the interesting case where the scale of the uv completion is much larger than the scale of the extra dimension @xmath52 and the singlet fermion masses , _ i.e. _ , we assume @xmath53 to hold . in this limit , the neutrino mass matrix is simply given by @xmath54 , _ i.e. _ , the scale of the neutrino masses is determined by a high - energy scale associated with the fundamental theory underlying the effective extra - dimensional model . as for the heavy kk modes , from eq . , the masses of the @xmath31th excited kk modes are given by @xmath55 as we will discuss later , this implies that @xmath56 and @xmath57 ( as well as @xmath58 and @xmath59 ) form dirac pairs . thus , lepton number can be assigned to these pairs and the lepton number violating effects , such as neutrino masses , can only arise from the unpaired @xmath60 at the top of the kk tower . in order to compute the effective low - energy leptonic mixing , we first consider the light - neutrino mass matrix . generally , @xmath61 is a complex symmetric matrix , and can be diagonalized by means of a unitary matrix @xmath62 as @xmath63 where @xmath64 , with @xmath65 being the masses of the light neutrinos . note that , similarly to the ordinary fermionic seesaw mechanism , the light neutrinos mix with the heavy kk modes . thus , @xmath62 is not the exact leptonic mixing matrix entering into neutrino oscillations , even if one works in a basis where the charged - lepton mass matrix is diagonal . to see this point clearly , we can fully diagonalize eq . and then write down the neutrino flavor eigenstates in terms of the mass eigenstates @xmath66,\end{aligned}\ ] ] where @xmath67 denotes the mass eigenstates of the light neutrinos , and @xmath68 is the upper - left @xmath69 sub - matrix of the complete mixing matrix containing the light neutrinos as well as the full kk tower for the singlet fermions . furthermore , we have introduced the quantities @xmath70 which represent the mixing between the light neutrinos and the kk modes . the charged - current lagrangian in mass basis can be rewritten as @xmath71w^-_\mu + { \rm h.c.},\end{aligned}\ ] ] where @xmath72 is the @xmath6 coupling constant . due to the existence of the kk modes , the light - neutrino mixing matrix is no longer unitary . to a very good precision , we have @xmath73 assuming that @xmath74 , eq . can be approximated by @xmath75 compared to the conventional parametrization of non - unitarity effects @xmath76 @xcite , where @xmath77 is a hermitian matrix , we thus obtain @xmath78 an interesting feature of eq . arises immediately : the non - unitarity effects are dominated only by the combination @xmath79 . as a rough estimate , if we keep @xmath52 at the tev scale and @xmath80 , @xmath81 can be naturally expected . another typical feature is that , if @xmath82 holds , then both the neutrino mixing and the non - unitarity effects are determined by a single dirac mass matrix @xmath2 . therefore , in such a realistic low - scale extra - dimensional model , the non - unitarity effects are strongly correlated with the neutrino mixing matrix and the radius of the extra spacetime dimension . in our numerical computations , we adopt a convenient parametrization @xcite , and rewrite @xmath2 as @xmath83 with @xmath84 being an arbitrary complex orthogonal matrix . with this parametrization , eq . takes the form @xmath85 the present bounds at 90 % c.l . on the non - unitarity parameters are given by @xcite @xmath86 where the most severe constraint is that on the @xmath87 element , coming from the @xmath88 decay . however , in the case that @xmath0 lies below the electroweak scale , but above a few gev , the @xmath88 constraint is lost due to the restoration of the glashow iliopoulos maiani ( gim ) mechanism @xcite , and a less stringent bound of @xmath89 should be used . apart from resulting in non - unitarity effects in neutrino mixing , the heavy singlet fermions in the bulk will also contribute to the lepton flavor violating ( lfv ) decays of charged leptons , _ e.g. _ , @xmath88 and @xmath90 , through the loop exchange of kk modes @xcite . different from the standard type - i seesaw mechanism , the corresponding branching ratios are not dramatically suppressed by the light - neutrino masses , but only driven down by the factor @xmath91 defined in eq . . thus , appreciable lfv rates could be obtained . as shown in eq . , the heavy singlets @xmath59 , @xmath56 , and @xmath92 couple to the gauge sector of the sm , and thus , if kinematically accessible , they could be produced at hadron colliders . for a quantitative discussion , we now restrict ourselves to the simplest case @xmath93 . note that @xmath59 and @xmath58 are two - component majorana fields with equal masses but opposite cp parities @xcite . thus , they are equivalent to a single dirac field @xmath94 with @xmath95 $ ] , @xmath96 $ ] , and mass @xmath97 . similarly , @xmath98 can be combined with @xmath99 , and hence , forms a higher kk dirac mode with @xmath100 $ ] and mass @xmath101 . as a general result of the mass degeneracy , all the kk modes are paired together except for the highest mode @xmath60 with mass @xmath102 . actually , @xmath60 is now the sole source of lepton number violation , and thus , gives rise to the masses of the light neutrinos , which can also be seen from eq . . the structure of the singlet dirac and majorana fermions is schematically depicted in fig . [ fig : dirac ] . illustration of the construction of dirac particles from pairs of modes in the kk tower . two heavy kk majorana modes with equal masses , but opposite cp parities , can be grouped together , as shown with double lines , in order to form a dirac particle . in the case @xmath103 ( left column ) , the heaviest mode @xmath60 is left , while for the case @xmath104 ( right column ) , there are three modes left : @xmath105 , @xmath106 , and @xmath60.,width=604 ] the weak interaction lagrangian for the heavy states can now be rewritten as @xmath107w^-_\mu + { \rm h.c . } , \\ { \cal l}_{\rm nc } & = & \frac{g}{2 \cos \theta_{\rm w } } { \nu_{m\rm l}^\dagger } \bar{\sigma}^\mu v^\dagger \left [ \sqrt{2 } \sum^{n-1}_{n=0 } k^{(n ) } p^{(n)}_l + k^{(n ) } y^{(n ) } \right ] z_\mu+ { \rm h.c . } , \\ { \cal l}_{h } & = & \frac{-{\rm i } g}{\sqrt{2}m_{w } } { \nu_{m\rm l}^t } \sigma^2 v^t m_d \left [ \sqrt{2 } \sum^{n-1}_{n=0 } p^{(n)}_l + y^{(n ) } \right ] h + { \rm h.c.},\end{aligned}\ ] ] where @xmath108 denotes the weak mixing angle and @xmath109 is the mass of the @xmath110 boson . in the case @xmath111 ( where @xmath112 denotes the higgs mass ) , the heavy kk modes decay in the channels @xmath113 , @xmath114 , and @xmath115 . the corresponding partial decay widths are given by @xcite @xmath116 where @xmath117 and @xmath118 denote that masses of @xmath119 and @xmath120 , respectively . since the lower kk modes are dirac particles , and lepton number breaking occurs only at the top of the kk towers , we focus our attention on lepton number conserving channels mediated by the lightest kk modes . for example , an interesting channel is the production of three charged leptons and missing energy @xcite , _ i.e. _ , @xmath121 , which is depicted in fig . [ fig : lhc ] . feynman diagrams for the potentially interesting lhc signatures with three charged leptons and missing energy in the model under consideration.,width=302 ] another possible process is the pair production of charged leptons with different flavor and zero missing energy , _ i.e. _ , @xmath122 . however , it is difficult to make significant observations in this channel at the lhc , due to the large sm background @xcite . an analysis of the collider signatures of an extra - dimensional model similar to the one that we consider was performed in ref . it was found that the most promising channel for that model is three leptons and large missing energy . since taus are difficult to detect , due to their short lifetime , only electrons and muons in the final state were considered . the signals were combined into two classes , the @xmath123 signal , given by the sum of the @xmath124 and @xmath125 signals , where @xmath126 and @xmath127 denote both leptons and antileptons of the indicated flavors , and the @xmath128 signal , given by the sum of the @xmath129 and @xmath130 signals . for the case of normal neutrino mass hierarchy ( @xmath131 ) , it was found that the @xmath123 combination gives the most promising signal . in order to reduce the sm background , which mainly comes from decays of @xmath119 bosons , the following kinematic cuts , taken from ref . @xcite , were adopted : i ) the two like - sign leptons must each have a transverse momentum larger than 30 gev and ii ) the invariant masses from the two opposite - sign lepton pairs must each be separated from the mass of the @xmath119 boson by at least 10 gev . only the effects of the lowest kk level were considered , as it was concluded that the contributions from higher modes would be more than one order of magnitude smaller . we have calculated the @xmath123 as well as the @xmath128 signals for our model . the results , using an integrated luminosity of @xmath132 , are shown in fig . [ fig : lhcsignals ] . we have considered the normal neutrino mass hierarchy ( @xmath131 ) as well as the inverted hierarchy ( @xmath133 ) , and for each case , we have chosen the mass of the lightest neutrino to be equal to zero or @xmath134 ev , corresponding to the hierarchical or nearly degenerate neutrino mass spectrum , respectively . for the neutrino oscillation parameters , we have used the best - fit values from ref . @xcite , _ i.e. _ , @xmath135 , @xmath136 , @xmath137 , @xmath138 , and @xmath139 . we have put the dirac cp - violating phase to zero . for each case , we have set the value of the cutoff scale in order to maximize the signal , while respecting the non - unitarity bounds given in eq . . like ref . @xcite , we have only taken the lightest kk modes of the singlet fermions into account . the signals are dominated by the on - shell production of the internal gauge bosons and sterile fermions . since @xmath140 , on - shell production of the gauge bosons is not possible if @xmath141 , and in that case , the signals are suppressed by the off - shell propagators . hence , we have chosen @xmath142 as the lower bound in our figures . in the case that the lightest neutrino is massless , the @xmath123 signal is stronger than the @xmath128 signal by approximately one order of magnitude for the normal hierarchy , while the opposite is true for the inverted hierarchy . in the case of a nearly degenerate mass spectrum , _ i.e. _ , that the lightest neutrino has a non - zero mass equal to 0.1 ev , the two signals are almost identical , especially in the inverted hierarchy case . since the expected background , after the kinematic cuts have been imposed , is of the order of 100 events @xcite and none of the signals is stronger than @xmath143 events , we conclude that , for our model , the non - unitarity bounds are strong enough to rule out the part of the parameter space that could possibly be probed by the lhc . the expected number of events for the @xmath123 and @xmath128 signals at the lhc as functions of the inverse radius @xmath45 , for an integrated luminosity of @xmath144 . note that the masses of the lightest singlet fermions are equal to @xmath145 . for @xmath141 , on - shell production of the internal gauge bosons is not possible , and the signal is suppressed . the values of the neutrino oscillation parameters are given in the main text . left panel : normal neutrino mass hierarchy . right panel : inverted neutrino mass hierarchy.,title="fig : " ] the expected number of events for the @xmath123 and @xmath128 signals at the lhc as functions of the inverse radius @xmath45 , for an integrated luminosity of @xmath144 . note that the masses of the lightest singlet fermions are equal to @xmath145 . for @xmath141 , on - shell production of the internal gauge bosons is not possible , and the signal is suppressed . the values of the neutrino oscillation parameters are given in the main text . left panel : normal neutrino mass hierarchy . right panel : inverted neutrino mass hierarchy.,title="fig : " ] baryogenesis via leptogenesis is one of the main candidates for being the theory appropriately describing the production of a baryon asymmetry in the early universe , which is measured to be @xmath146 @xcite . in its most basic form , leptogenesis occurs in a type - i seesaw scenario , where a net lepton asymmetry is produced through the out - of - equilibrium decay of the heavy neutrinos and then partially converted to a baryon asymmetry through sphaleron processes . the sakharov conditions @xcite are fulfilled by the decays occurring out of equilibrium , the loop level cp - violation of the decays through complex yukawa couplings , and the baryon number violation of the sphalerons , respectively . usually , the net lepton number is produced by the decays of the lightest singlet fermions , since asymmetries produced by the heavier neutrinos will be washed out . however , in our scenario , the tower of dirac fermions can be given definite lepton number assignments and lepton number violation only occurs at the top of the tower through the unpaired @xmath147-states , which could take on the role of the singlet fermions in the basic scenario . it is important to note that for @xmath93 ( @xmath148 ) , there will be no net lepton number violation , since all of the three unpaired states will be degenerate in mass . however , if the @xmath47 are different , _ e.g. _ , @xmath149 and @xmath150 , @xmath151 , then @xmath152 ( see fig . [ fig : dirac ] ) will be the unique lightest majorana state and a net lepton asymmetry could be produced . since the mass splitting of @xmath52 between the @xmath147-states is expected to be very small compared to the masses , the model would have to be treated within the framework of resonant leptogenesis @xcite . furthermore , to accurately examine the prospects for leptogenesis in this model , one would have to properly take into account the effects of the dirac tower . even if the dirac fermions in the tower preserve lepton number , they do not participate in the sphaleron processes , since they are sm singlets , which could hide some part of the produced lepton number from the sphalerons if all dirac fermions do not decay before sphaleron processes become inactive . thus , a detailed analysis , which is beyond the scope of this paper , would be required to properly analyze the prospects for leptogenesis in this model . for earlier studies of leptogenesis in extra dimensions , see for example refs . in this work , we have studied a possible mechanism for generating small neutrino masses in the context of extra dimensions . in the model that we consider , the sm particles are confined to a four - dimensional brane , while three sm singlet fermions are allowed to propagate in an extra dimension , compactified on the @xmath153 orbifold . since extra - dimensional models are generally non - renormalizable , and can only be considered as effective theories , the kk expansions of the higher - dimensional fields are expected to be truncated at some cutoff scale . we have imposed a cut on the kk number , truncating the towers at @xmath28 . in the case that the bulk majorana mass term for the singlet fermions has the form @xmath154 , where @xmath155 is an odd integer , the kk modes of the singlet fermions pair to form dirac fermions . such a form for a majorana mass is motivated by , for example , the scherk schwarz mechanism . due to the truncation of the kk towers , a number of unpaired majorana fermions remain at the top of each kk tower , and these are the only sources of lepton number violation in this model . if the cutoff scale is large , small masses for the left - handed neutrinos are naturally generated . due to mixing between the light neutrinos and the kk modes of the singlet fermions , large non - unitarity effects can be induced . since the masses of the light neutrinos are generated by the top of each tower , these non - unitarity effects are not suppressed by the light - neutrino masses . current bounds on the non - unitarity parameters have constrained the parameter space of the model . finally , we have considered the prospects of observing the effects of the lowest kk modes of the singlet fermions at the lhc . in particular , we have considered the three leptons and large missing energy signal , which has previously been found to be promising for a similar model . we have found that , in contrast to the previous results in the literature , the potential of discovering such models at the lhc is actually pessimistic . in particular , the parts of the parameter space that could be probed at the lhc are ruled out by the bounds imposed by the stringent constraints on the effective low - energy leptonic mixing . however , the non - unitarity effects in neutrino oscillations could be observable at future neutrino factory experiments . therefore , future long baseline neutrino oscillation experiments could play a very complementary role in searching for hints of extra dimensions . we would like to thank steve blanchet for useful discussions . we acknowledge the hospitality and support from the nordita scientific program `` astroparticle physics a pathfinder to new physics '' , march 30 april 30 , 2009 during which parts of this study was performed . this work was supported by the european community through the european commission marie curie actions framework programme 7 intra - european fellowship : neutrino evolution [ m.b . ] , the royal swedish academy of sciences ( kva ) [ t.o . ] , the gran gustafsson foundation [ h.z . ] , and the swedish research council ( vetenskapsrdet ) , contract no . 621 - 2008 - 4210 [ t.o . ] .

we study the generation of small neutrino masses in an extra - dimensional model , where singlet fermions are allowed to propagate in the extra dimension , while the standard model particles are confined to a brane . motivated by the fact that extra - dimensional models are non - renormalizable , we truncate the kaluza klein towers at a maximal kaluza klein number . this truncation , together with the structure of the bulk majorana mass term , motivated by the sherk schwarz mechanism , implies that the kaluza klein modes of the singlet fermions pair to form dirac fermions , except for a number of unpaired majorana fermions at the top of each tower . these heavy majorana fermions are the only sources of lepton number breaking in the model , and similarly to the type - i seesaw mechanism , they naturally generate small masses for the left - handed neutrinos . the lower kaluza klein modes mix with the light neutrinos , and the mixing effects are not suppressed with respect to the light - neutrino masses . compared to conventional fermionic seesaw models , such mixing can be more significant . we study the signals of this model at the large hadron collider , and find that the current low - energy bounds on the non - unitarity of the leptonic mixing matrix are strong enough to exclude an observation .

last time the interest has sharply increased for searching the conditions for realization supersolidity phenomenon in solid @xmath1he @xcite , when the crystalline order combines with superfluidity . in spite of the great number of experimental and theoretical investigations in this area , the consensus has not been attained yet . for the present , it has been determined well that observing effects strongly depend on the growing conditions and annealing degree of helium crystals . the special modeling which was conducted from the first principles by monte - carlo method , showed that in the perfect hcp @xmath1he crystal the supersolidity effects can not appear @xcite . the most authors connect such effects in solid @xmath1he at low temperatures with the disorder in helium samples . possible kinds of the disorder may be the defects , grain boundaries @xcite , glass phase , or liquid inclusions @xcite . also , the possible interpretation @xcite of the experiments on flow the superfluid helium through the solid helium @xcite show the essential role of the liquid channels , which may exist in the solid helium up to the ultralow temperatures . in this connection , the experiments which allow to identify the kind of the disorder , for example , in rapidly grown helium crystals , interesting . these data can be obtained by nuclear magnetic resonance ( nmr ) . whereas for its realization the nuclei of @xmath0he are necessary , we deal hereafter with the samples of not pure @xmath1he but with dilute @xmath0he-@xmath1he mixture . since nmr technique allows to measure diffusion coefficient in different coexisting phases and difference of diffusion coefficients in liquid and solid helium are several orders of the magnitude then such an experiment may answer the question whether liquid inclusions are formed in solid helium under very rapid crystal growing . the aim of present work is to elucidate this problem . we detect , by nmr technique , the presence of liquid phase in solid helium samples grown in different conditions and also establish the influence of annealing effect on character of diffusion processes . the crystals were grown by the capillary blocking method from initial helium gas mixture with a 1% of @xmath0he concentration . the copper cell of cylindrical form with inner diameter of 8 mm and length of 18 mm has the nmr coil glued to the inner surface of the cell . the pressure and temperature variations of the sample in the cell were controlled by two capacitive pressure gauges fixed to the both cylinder ends and by two resistance thermometers attached to the cold finger of the cell with sensitivities about 1 mbar and 1 mk , respectively . two series of crystals under the pressure above 33 bar were studied . the first one ( `` low quality crystals '' ) was prepared by quick step - wise cooling from the melting curve down to the lowest temperature ( 1.27 k ) without any special thermal treatment . to improve the crystal quality of the second series ( `` high quality crystals '' ) a special three - stage thermal treatment was used : annealing at the melting curve , thermocycling in single phase regions and annealing in the hcp single phase region near the melting curve @xcite . the criterions of crystal quality are , first , constancy of the pressure with time under constant temperature which is closed to melting and , second , reaching the pressure minimum under thermal cycling . the spin diffusion coefficient was determined with the help of the pulsed nmr technique at a frequency of @xmath2 mhz . the carr - purcell ( @xmath3 ) spin - echo method @xcite was used with a 90@xmath4-@xmath5 - 180@xmath4 sequence of probe pulses as well as the method of stimulated echo ( @xmath6 ) with the sequence of three probes pulses 90@xmath4-@xmath7 - 90@xmath4-@xmath8 - 90@xmath4 were applied to the nuclear system of the sample . generally , if a few phases do coexist in the sample , the echo amplitude @xmath9 for @xmath3 is given by @xmath10 and for @xmath6 @xmath11 \label{2}\ ] ] where @xmath12 is the maximal amplitude of a echo amplitude at @xmath13 , @xmath14 is the magnetic field gradient , @xmath15 is a gyromagnetic ratio , index @xmath16 numerates coexisting phases with the diffusion coefficients @xmath17 , @xmath18 is the relative content of the @xmath16-th phase in the sample . one can choose duration parameters @xmath5 , @xmath7 , and @xmath8 in order to get the strongest @xmath19 dependence and to single out @xmath17 fitting parameter . it should be emphasized that spin - diffusion coefficient @xmath20 measurement was just the method to identify a thermodynamical phases by their typical @xmath20 value . neither contribution of @xmath0he atoms in a phase transition processes nor even the dynamics of different phase s ratio could be tracking because of too long spin - lattice relaxation times . the typical results of nmr measurements for diffusion coefficients in two - phase sample on the melting curve are presented in fig . [ fig_mc ] in @xmath19 scale . there are two slopes for the data obtained which correspond to two different diffusion coefficients . experimental data analysis according to eq . ( [ 1 ] ) gives for curve piece with sharp slope @xmath21 @xmath22/s which corresponds to diffusion in liquid phase @xcite and for curve piece with mildly slope @xmath23 @xmath22/s which corresponds to diffusion in hcp phase @xcite . the phase ratio is @xmath24 . then this sample was rapidly cooled down to 1.3 k in the hcp region . the results of nmr measurements are shown in fig . [ fig_quenched ] . the presence of significant contribution ( @xmath25 ) of phase with fast diffusion coefficient ( @xmath26 @xmath22/s ) was unexpected . this fact can be interpreted as existence of liquid - like inclusions in hcp matrix which were apparently quenched from the melting curve . such a situation was visually observed in pure @xmath1he in refs . [ 1,4,15,16].the liquid droplets formation was also observed by nmr technique in 1% @xmath0he-@xmath1he mixture under bcc and hcp phases coexistence @xcite . note that this effect was observed in all three low - quality samples studied . after that this crystal was heated up to melting curve and , after annealing procedure described above ( sec . [ method ] ) , to avoid a thermal shock , was slowly cooled down to 1.3 k ( the hcp region ) . the results are presented in fig . [ fig_good ] . both the absence of visible @xmath19 functional dependence ( see eq . ( [ 1 ] ) ) which should be characteristic feature for @xmath27 @xmath22/s under @xmath28 ms at @xmath29 gs/@xmath22 and the position ( 0 ; 0 ) of the intersection point of @xmath3 and @xmath6 data curves are the evidences of the liquid - like diffusion absence in the crystal . it also should be noted that monotonous pressure decrease was observed in low - quality samples with fast diffusion coefficient . the typical pressure relaxation times were about @xmath30 hour . after annealing of such samples along with fast diffusion process disappearing , monotonous pressure decreasing was also stopped . this relaxation indirectly confirms our speculation about liquid - like inclusions quenched from the melting curve in the samples without any annealing . detailed study of pressure relaxation in quenched samples is projected . it is shown that under rapidly cooling from the melting curve ( without annealing ) solid helium samples contain liquid - like inclusions identified by additional fast diffusion decay of echo - signal . subsequent annealing of these samples leads to fast diffusion disappearing which is connected with crystallization of liquid - like inclusions . coming out of these defects is accompanied by pressure relaxation in the system . we thank b.cowan for useful consultations and for applying of his nmr spectrometer . this work has also been partially supported by grant stcu # 3718 , program of cooperation in research and education in science and technology for the 2008 ukrainian junior scientist research collaboration , and the ministry of education and science of ukraine ( project m/386 - 2009 ) .

the study of phase structure of dilute @xmath0he - @xmath1he solid mixture of different quality is performed by spin echo nmr technique . the diffusion coefficient is determined for each coexistent phase . two diffusion processes are observed in rapidly quenched ( non - equilibrium ) hcp samples : the first process has a diffusion coefficient corresponding to hcp phase , the second one has huge diffusion coefficient corresponding to liquid phase . that is evidence of liquid - like inclusions formation during fast crystal growing . it is established that these inclusions disappear in equilibrium crystals after careful annealing . pacs numbers : 61.72.cc , 66.30.ma , 61.50.-f , 64.70.d- keywords : nmr , @xmath0he-@xmath1he solid mixture , diffusion , defects * * + _ ye.o . vekhov , a.p . birchenko , n.p . mikhin , and e.ya . rudavskii _ + _ _

nowadays , with the standard cosmology the famous fundamental question , where did it all come from ? " still it does not have a convincing answer , reason why a new description is necessary . cosmologists during long time have believed that quantum cosmology can shed light on this question @xcite but some issues are in controversy , e.g. the lack of an intrinsic time variable in the theory @xcite , the validity of the minisuperspace approximation , the problem of cosmological boundary conditions @xcite , to mention something . among the proposals trying to outline a possible answer to the fundamental question , the so - called brane world scenaries ( bws ) @xcite became a promising way to understand the birth and then the evolution of our universe . grounded on the proposal that our universe can be thought as a 4-dimensional spacetime object embedded in an n - dimensional spacetime , the main physical idea behind of bws is that the matter fields are confined to a 3-dimensional space ( brane ) while the gravitational fields can extend into a higher - dimensional space ( bulk ) , where the graviton can travel into the extra dimensions . originally proposed to resolve the hierarchy problem , bws has been applied to a great diversity of situations such as dark matter / energy , quientessence , cosmology , inflation and particle physics . on other hand , at the formal mathematical level , related applications of embedding theory such as generation of internal symmetries , quantum gravity and alternative kaluza - klein theories have been exploited @xcite . in the cosmology context there are predictions of these ideas , that could be tested by astronomical observations what constitutes one of the several reasons for which it is so attractive , so that it has predictive power @xcite . in these brane world programs , gravity on the brane can be recovered by compactifying the extra dimensions @xcite or by introducing an ads background spacetime @xcite . however , dvali , gabadadze and porrati @xcite ( dgp ) showed that , even in an asymptotically minkowski bulk , 4-dimensional gravity can be recovered if one includes a brane curvature term in the action . furthermore , dgp considered the @xmath0 reflection symmetry with respect to the brane getting that gravity , is 4-dimensional on smaller scales than a certain scale , or it is 5-dimensional on larger distances @xcite . it is noteworthy that reflection symmetry is not the only possibility in these models . with regard to the last , several works have been devoted to antisymmetric cases @xcite , for instance , when the brane is coupled to a 4-form field @xcite . in a pionner work , brown and teitelboim worked out the process of membrane creation by an antisymmetric field motivated by schwinger process of pair creation induced for the presence of a electric field @xcite . garriga @xcite has also studied the creation of membranes for this field in a ds background . others authors have been interested in brane world creation in ads spacetime or in other particular situations @xcite but , upon our knowledge , nobody has been devoted to the nucleation of brane world universes ( bwu ) induced by a 4-form field besides a brane curvature term included in the action . generally , bws are studied mostly for ads / ds as well as empty ( minkowski ) backgrounds . in this paper we are going to discuss the nucleation of bwu with a curvature term induced by a 4-form field in a ds background spacetime . we get the friedman like equation when 5-dimensional gravity is fixed and perform geometric hamiltonian analysis in order to obtain , by means of canonical quantization , the corresponding wheeler - dewitt equation . the setup for the induced brane production is as follows . there is an external homogeneous field that produces a brane ; then , the natural question there , is : what is the probability of such process ? in the present paper we calculate the creation probability for a brane universe embedded in a de sitter space , produced by a 4-form potential gauge field in the same way that the standard electromagnetic potential bears to a charged particle . in its quantum analisys we shall use a wkb approximation attaining the same results by an instanton method . we could try to answer the question of which one of the universes arose is the more probable universe produced in this model and if our universe is one of them , or could be a very special universe . parameters of this model must be constrained by cosmological requirements like nucleosynthesis @xcite . the paper is organized as follows . in sec . ii we present the equations of motion of a brane with matter and curvature term that lives in a ads / ds or minkowski bulk when there is no @xmath0 symmetry and , by means of a limit equivalent to the presence of a 4-form field in a fixed background the corresponding equations . a geometric hamiltonian approach is done in sec . iii , where the fundamental canonical structure is obtained and the canonical constraints are listed . the next step is specialize the general canonical analysis to the case of a spherical 3-brane floating in an @xmath1 background spacetime which is the issue of sec . the last provides the preamble to obtain the wdw equation in the canonical quantization context , which is done in sec . v. the creation probability is calculated in sec . vi by two methods , the first is an instanton approach and the other one by means of a wkb approach for barrier tunneling of the wdw equation . finally in sec . vii , we present our conclusions as well as some perspectives of our work . the effective action that we are interested in the brane world model corresponds to a 3-brane with a intrinsic curvature term considered from its worldsheet and no @xmath0 symmetry in the presence of a fixed background spacetime . we consider the following action @xmath2 where @xmath3 and @xmath4 stand for matter lagrangians for the bulk and the brane , respectively . in our case , we will consider those as cosmological constants . the constants @xmath5 and @xmath6 , where @xmath7 and @xmath8 are the brane plank and bulk masses . @xmath9 denotes the dimension of the bulk . the respective equations of motion for the brane are @xcite , @xmath10\gamma_{a b } - [ k_{ab}]&= & k{\cal t}_{ab } , \\ { \widetilde{t}}^{ab } < k_{a b } > & = & [ { \cal t}_{nn } ] , \\ \nabla_{a } ( t^a { } _ b ) & = & -[\widetilde{t}_{b n } ] .\end{aligned}\ ] ] where @xmath11 is the extrinsic curvature of the brane , @xmath12 denotes the worldsheet metric . @xmath13 , @xmath14 and @xmath15 are the projections onto the worldsheet of the bulk energy - momentum tensor . the square and angular brackets represent the difference and the average of the corresponding embraced quantity , on the two sides of the brane , respectively , i.e. , @xmath16 = k^+ _ { ab } - k^- _ { ab}$ ] and @xmath17 , where ` + ' and ` - ' denote the exterior and interior of the brane . taking into account that the bulk energy momentum tensor has the form @xmath18 and by means of the generalized birkhof theorem , the 5-dimensional frw metric can be written as @xmath19 where @xmath20 and @xmath21 denotes the metric of a 3-sphere , @xmath22 is the cosmic scale factor and @xmath23 is the mass . furthermore , in the cosmic time gauge the 4-dimensional metric on the brane reduces to @xmath24 using the junction conditions , and due to we have isotropy and homogeneity in ( [ eq : ds5 ] ) , matter can be parametrized completely via a perfect fluid brane energy - momentum tensor @xmath25 so the relevant equations of motion for the model are the following @xmath26 last equation represents the energy - momentum conservation on the brane . the former system was discussed in @xcite where several interesting cases were treated . suppose now @xmath27 , @xmath28 , and consider at the same time , the limits of fixed bulk gravity , @xmath29 and , @xmath30 but satisfying the following relation @xmath31 so , expanding the second term of the lhs of eq . ( [ eq : motion ] ) , this equation transform to @xmath32 in order to get the friedman like equation we define a @xmath33 quantity through its definition @xmath34 note that @xmath33 is only a function of @xmath22 and it is a solution of the following relation @xmath35 as we will see below , this approach is equivalent to a brane interacting with a 4-form field and propagating in a fixed background spacetime . the hamiltonian framework has been a fundamental prop in the study of the dynamics of field theories besides of appoint oneself a preliminary step towards canonical quantization in physical theories . knowingly of previous fact , canonical quantization is the oldest and most conservative approach to quantization which we would like to develop in order to attain the quantum cosmology emerged from our bwu model . to carry out the previous thing , we must begin by casting the theory in a canonical fashion , then we shall proceed to its quantization . to begin with , we are going to mimic the well known adm procedure for canonical gravity to get a hamiltonian description of the brane . we shall assume that the worldsheet @xmath36 admits a foliation , i.e. , we will begin with a time like 4-manifold @xmath36 topologically @xmath37 , equipped with a metric @xmath12 , such that @xmath36 is an outcome of the evolution of a space like 3-manifold @xmath38 , representing instants of time " , each of which is diffeomorphic to @xmath39 . then we shall proced to identify the several geometric quantities inherent to the hypersurface @xmath38 . the adm decomposition of the action , computation of the momenta as well as the recognition of the constraints are the succesive stages . leaning in results achieved in @xcite , we are going to display the standard procedure . we start considering the action @xmath40 where @xmath41 is the ricci scalar curvature of the worldsheet @xmath36 , @xmath42 and @xmath43 being the cosmological constant on the brane . @xmath44 is a gauge 4-form ramond - ramond field onto the bulk , @xmath45 . @xmath46 is an antisymmetric bulk tensor which can be expressed in terms of the worldsheet levi - civita tensor as @xmath47 , where @xmath48 denotes the tangent vectors to the worldsheet , @xmath49 . @xmath50 is the coupling constant between the brane and the antisymmetric tensor . before going on , we would like to glimpse onto the adm decomposition of some important geometric quantities defined onto the branes in our geometrical approach . in the appendix we have included notation and some important facts for embedding theories to have reference of the material useful through the paper . taking into account the gauss - codazzi relations for the embedding of @xmath38 in @xmath36 , eqs . ( [ eq : gw4 ] ) and ( [ eq : gw5 ] ) , up to a divergence term we have an equation involving the curvatures either extrinsic and intrinsic @xmath51 where @xmath52 denotes the intrinsic curvature}t^c = - { \cal r}_{abd}{}^c \,t^d$ ] @xcite ] of @xmath38 which does not have any dependence of the velocity and @xmath53 its extrinsic curvature associated with the unit timelike normal @xmath54 , given by @xmath55 besides of ( [ eq : ktensor ] ) , in @xmath38 we have another curvature tensor associated with the @xmath56th unit normal @xmath57 @xmath58 where @xmath59 denotes the background spacetime metric and @xmath60 . note that the configuration space consists of the embedding functions @xmath61 for the brane , instead of 3-metrics as is customary in the adm approach for general relativity . in order to simplify the computations below , the next relations will be more useful since the velocities appear explicitly @xmath62 for canonical purposes will be useful the next time derivative @xmath63 as before , we will need the derivatives of the extrinsic curvature @xmath64 where in the second line on the rhs we have used the gauss - weingarten equations ( [ eq : gw1 ] ) . the adm decomposed action ( [ eq : rmaction ] ) now looks like @xmath65 + \int_{\sigma_t } \int_r \frac{k_2}{3!}\ , a_{\mu \nu \rho \sigma } \dot{x}^\mu \epsilon^\nu{}_a \epsilon^\rho{}_b \epsilon^\sigma{}_c \ , \varepsilon^{abc}\ ] ] where we have defined @xmath66 and @xmath67 is the determinant of the hypersurface metric @xmath68 and @xmath69 is the @xmath38 levi - civita antisymmetric symbol . we define for convenience the following symmetric tensor which is independent of the velocities @xmath70 where @xmath71 denotes the trace of the curvature @xmath72 , i.e. , @xmath73 . this tensor will keep track of the dynamics of the theory as we will below . the tensor ( [ eq : tensor ] ) was previously defined in @xcite where a hamiltonian analysis for geodetic brane gravity was performed . we will have in mind some ideas of the classical approach developed there . some of the important properties we are interested from the tensor ( [ eq : tensor ] ) are the following @xmath74 we shall adopt the notation @xmath75 throughout the paper . taking advantage of the previous results we are able to rewrite the lagrangian density as follows @xmath76 + \frac{k_2}{3!}a_{\mu \nu \rho \sigma } \dot{x}^\mu \epsilon^\nu { } _ a \epsilon^\rho { } _ b \epsilon^\sigma { } _ c \ , \varepsilon^{abc}.\ ] ] using the tensor ( [ eq : tensor ] ) , the momenta associated to the embedding functions are the following @xmath77\ , \eta_\mu + \frac{2}{n } \theta_{\mu \nu } \dot{x}^\nu \right\ } + \frac{k_2}{3 ! } \,a_{\mu \alpha \beta \gamma}\ , \bar{\varepsilon}^{\,\alpha \beta \gamma } , \label{eq : p}\end{aligned}\ ] ] where we have defined the @xmath38-antisymmetric tangent tensor @xmath78 with normalization @xmath79 . due to we have in hands an invariant reparametrization theory , a natural question to ask is what its inherited primary constraints are . this is part of the chore for constrained field theories . according to the standard dirac - bergmann algorithm , we will get the constraints from the momenta ( [ eq : p ] ) . it is convenient for the computation , define the matrix @xmath80 where @xmath81 is a not dynamical field which is gauge dependent @xcite , to be found . if we assume that the form of momenta have the following pattern , @xmath82 we are free to compare both expressions ( [ eq : p ] ) and ( [ eq : p3 ] ) to get a condition to be satisfied @xmath83 this expression will metamorphose in a primary constraint after we express it in terms of phase space variables . profitable is the introduction of the field @xmath84 since we can solve eq.([eq : p3 ] ) for the timelike unit normal vector @xmath85 where we have defined @xmath86 , but we have to pay a price which is enlarge the number of constraints as we will see below . inserting this form of the unit time - like vector in the relation ( [ eq : relation ] ) , we get the main scalar primary constraint . in a similar way , inserting @xmath54 in its square relation , @xmath87 , we have another scalar constraint . the complete set of primary constraints we have in hand are the following @xmath88 where we have defined @xmath89 . the third constraint is the always inherited constraint to the parametrized theories while the last one came from the fact that @xmath90 is not a dynamical field , i.e. , its time derivative does not appear in the lagrangian . it is worthy mention that the constraint @xmath91 is a byproduct of @xmath92 taking advantage of the identity @xmath93 the main idea in this section is adapt the previous dynamical description to the case of a spherical brane immersed in a specific background spacetime in order to apply the quantum approach to our bws model . consider a 3-dimensional spherical brane evolving in a de sitter 5-dimensional background spacetime , @xmath94 , where @xmath95 is given by ( [ eq : a ] ) . the worldsheet generated by the motion of the brane can be described by the following embedding @xmath96 the line element induced on the worldsheet is given by @xmath97 where the dot stands for derivative with respect to cosmic time @xmath98 . for convenience in notation we define @xmath99 . the frecuently appealed cosmic gauge will be set up by @xmath100 . in order to evaluate the extrinsic curvature tensors involved in our approach , ( [ eq : ktensor ] ) and ( [ eq : ktensor ] ) , we need the orthonormal @xmath38 basis @xmath101 the only nonvanishing components for the extrinsic curvatures are @xmath102 it is a straightforward task compute the tensor ( [ eq : tensor ] ) for the present case , which give us @xmath103 the next task is compute the matrix @xmath104 so , in order to know @xmath104 is necessary evaluate @xmath90 . it is easily calculated from the relation ( [ eq : relation ] ) , given by @xmath105 this seems contradict the functional dependence for the field previously assumed , but we are free to implement an artistry to convert the velocity dependence to the right form by means of the generalized evolution equation , @xmath106 , avoiding any misunderstanding . we turn now to compute a first integral for our specific model . this is performed from ( [ eq : p ] ) by setting up @xmath107 proportional to the brane energy , @xmath108 . furthermore , since we have a homogeneous isotropic space in ( [ eq : metric ] ) , we can invoke the typical value @xmath109 for the gauge field , which is supported by some kind of cosmological solutions @xcite , where f is a constant and the corresponding gauge independent field tensor @xmath110}$ ] is expressed in terms of it @xmath111 . explicitly , we have @xmath112 now , taking into account the generalized evolution equation and @xmath113 being the cosmological constant on the brane , we find the desired result @xmath114 where @xmath115 is the cosmological constant living in the bulk appearing in eq . ( [ eq : a ] ) and we have used the cosmic gauge in the last step . note that ( [ eq : energyagain ] ) is in keep with eq . ( [ eq : first ] ) , confirming equivalence with the limit process developed in sect . we turn now in this section to develop the quantum description for our specific problem . the canonical quantization procedure is well known so , just remain apply the recipe in the matter of our case . we shall set @xmath116 in such a way that scalar constraints ( [ eq : c0 ] ) and ( [ eq : c00 ] ) transform into quantum equations @xmath117 where we have defined @xmath118 . specializing to the embedding ( [ eq : embedding ] ) and having in mind the matrix ( [ eq : psicosmicgauge ] ) in the cosmic gauge , we are able to get the inverse matrix @xmath119 } & 0 \\ 0 & 0 & n_{3 \times 3 } ^{-1 } \end{smallmatrix } \right ) , \ ] ] in such a way that ( [ eq:0quantum ] ) and ( [ eq : quantum ] ) transform in the pair of relations @xmath120 where we introduce the notation @xmath121 . taking into account the value @xmath122 $ ] expressed in the cosmic gauge , the couple of quantum relations can be rewritten as , @xmath123 ^ 2}{(1 - \frac{\lambda a^2}{6})^2}\,\psi \,.\end{aligned}\ ] ] at this time , we are more interested in identify the potential governing the dynamics of our model instead of solve exactly the wdw equation so , to get insight we propose the wave function of separable form , @xmath124 . the wdw equation adquires the form @xmath125 ^ 2 \left ( -1 + \upsilon h^2a^2 \right ) } { \left ( 1-\frac{\lambda a^2}{6}\right)^2}\psi , \label{eq : wdw}\ ] ] accompanied by the energy equation @xmath126 where , as before , we have assumed @xmath127 . at this stage , we are ready to compute the creation probability which the universe could be created . some simplifications are necessary due to the general problem itself is hard to solve . from wdw equation ( [ eq : wdw ] ) , is easily read off the potential which is subjected the model ( [ eq : rmaction ] ) @xmath128 ^ 2 ( 1 - \upsilon h^2a^2)}{(1 - \frac{\lambda a^2}{6})^2 } \ , . \label{eq : potential}\ ] ] note that this is a very hard expression to work out if one is interested in the general integration , specially if , in the cosmological context , creation probability is desire computed . recall that the last is written in terms of the potential extracted from the wdw equation , namely , @xmath129 in order to get some interesting results from the quantum approach , we shall consider some special cases . if @xmath130 from eq . ( [ eq : moreenergy ] ) then @xmath33 is just a constant given by @xmath131 the probability rate in this case is @xmath132 } , \label{eq : prob}\ ] ] where @xmath133 . now , if @xmath134 and , at first order the probality rate is @xmath135 this means that it is more probable to create a universe when @xmath136 than @xmath137 . we will comment about it below . now , we would like calculate the probability nucleation using the instanton method . the corresponding euclidean action in de sitter bulk can be found by complexifying the temporal coordinate and keeping the field strength @xmath138 fixed @xmath139 in euclidean space we have now closed worldsheets that split the desitter background spacetime of radius @xmath140 in two regions . this is the basic geometry of the instanton calculation . following @xcite , by using stoke s theorem we can transform ( [ eq : instanton ] ) to an instanton action that involves a volume of the spacetime enclosed by the brane @xmath141 for spherical worlsheets the former action is expressed through the radius @xmath142 of the brane @xmath143 where @xmath144 is the surface of a worldsheet of radius @xmath142 , and @xmath145 is the volume enclosed by the brane of radius @xmath142 and @xmath146 . extremizing ( [ eq : euclidean ] ) we find that the radius of the euclidean brane is a solution of @xmath147 where @xmath148 . the resulting euclidean action is @xmath149}{\upsilon \lambda } + 2(\upsilon -1 ) \left(\frac{6h}{\lambda}\right)^2 \left[1 -\frac{1}{x}\tan ^{-1 } x \right ] \right\ } , \ ] ] and the nucleation probability @xmath150 is in agreement with ( [ eq : prob ] ) modulo a normalizing factor . we now go back to the meaning of equation ( [ eq : prob1 ] ) . the behavior of strength field @xmath138 is the key , when @xmath151 the field decrease in the inside region with respect to its original value and corresponds to screening membrane discuss in @xcite . when @xmath152 correspond to antiscreening membrane and the field increase its value , and as it is expected , is less probable to produce such a universe . this situation is resembled in phenomena of vacuum decay , where ordinary transition from false to true vacuum corresponds to @xmath151 , and the decay of true vacuum , by means of false vaccum bubbles , corresponds to @xmath152 and @xmath153 represents the difference in energy density between the false and true vacuum . we proceed to calculate an approximate expression for the nucleation rate at first order , when both @xmath154 and @xmath155 are small . the potential is @xmath156 and the nucleation probability is @xmath157 in complete agreement with ( [ eq : prob1 ] ) when @xmath154 vanishies . the potential for case a , is plotted in figure ( [ fig:1 ] ) and the corresponding one for the case b is in figure ( [ fig:2 ] ) . using this kind of plots for the potential , we can deduce that creation probability is enhance when the nucleation process take place in de sitter background spacetime with small radius @xmath158 . we have calculated the nucleation probability of brane world universes induced by a totally antisymmetric tensor living in a ds fixed background spacetime . this was done by means of canonical quantum approach where the wheeler - dewitt equation was found . besides , we found for one specific case , the nucleation rate computing the corresponding instanton . when the energy of the brane @xmath159 in the bulk space and the coupling constant of the brane @xmath50 with the antisymmetric field is positive , the creation probability is enhanced with respect to no interaction of the brane with the 4-form . for @xmath160 the nucleation rate decresed as is expected . this situation is resembled in phenomena of vacuum decay , where ordinary transition from false to true vacuum corresponds to @xmath151 , and the decay of true vacuum by means of false vaccum bubbles corresponds to @xmath152 . furthermore , @xmath153 represents the difference in energy density between the false and true vacuum . for large expansion rate of the de sitter bulk we observed an increase nucleation rate . at this point we ask ourselves about possible brane collisions , and what the most important factor in this issue is . the branes will be driven apart by the exponential expansion of the bulk reducing brane collision but at the same time , there is an increase in nucleation rate . we expect now that the problem of old inflationary model of the universe is an advantage : bubbles may not be produced fast enough , to complete cover the bulk . once the brane universe was created it still could be hitting by stealth branes @xcite , that by means of constraining some parameters of the model reduce the rate of brane collisions to an acceptable level . we think that cosmological constraints can impose bounds on the values of @xmath153 and with this value one could try to answer the question : is our universe very special ? we benefited from germn mandujano for assistance . er would like to thank csar de la cruz , ral hernndez , carlos vargas and alfredo villegas for useful discussions and encourage the paper . we also thank to sni - mxico for partial support . consider a brane , @xmath39 , of dimension @xmath161 whose worldsheet , @xmath36 is an oriented timelike manifold living in a @xmath9-dimensional arbitrary fixed background spacetime @xmath162 with metric @xmath59 . for hamiltonian purposes , we shall foliate the worldsheet @xmath36 in spacelike leaves @xmath38 . taking advantage of the differential geometry for surfaces , as well as novelty variational techniques developed in @xcite we can write the gauss - weingarten equations associated with the embedding of @xmath38 in @xmath162 ( @xmath163 ) , i.e. , the gradients of the @xmath38 basis @xmath164 . these spacetime vectors can be decomposed with respect to the adapted basis to @xmath38 , as @xmath165 where @xmath166 are the christoffel coefficients of the background manifold and , @xmath167 is a piece of the generalized extrinsic twist potential and both @xmath53 and @xmath72 are the extrinsic curvatures of @xmath38 associated with the normals @xmath54 and @xmath168 , respectively . @xmath169 denotes the covariant derivative adapted to @xmath38 and @xmath170 is the covariant derivative that preserves invariance under rotations of the normals @xmath168 , i.e. , @xmath171 . in a similar way , we can write the gauss - weingarten equations associated with the embedding of @xmath38 in the worldsheet @xmath36 , ( @xmath172 ) , i.e. , the gradients of the @xmath38 basis @xmath173 . these worldsheet vectors can be decomposed with respect to the adapted basis to @xmath38 , as @xmath174 where @xmath175 is the gradient along the tangent basis , i.e. , @xmath176 , where @xmath177 is the covariant derivative compatible with @xmath12 . the time vector field , written in terms of the adapted basis of a leaf @xmath38 , is given by @xmath178 which represents the flow of time throughout spacetime . note that we are able to rewrite the previous time deformation vector as follows @xmath179 where , taking into account the well known notation , @xmath177 denotes the covariant derivative compatible with @xmath12 ( @xmath180 ; @xmath181 and @xmath182 ) . furthermore , from ( [ eq : timevector1 ] ) note that the following relations hold : @xmath183 in this appendix we write the full matrix @xmath104 for our embedding ( [ eq : embedding ] ) . taking into account the eq . ( [ eq : tensor - matrix ] ) as well as eq . ( [ eq : lambda ] ) we have @xmath184 & 0 & 0 & 0\\ 0 & \frac{a_{\pm}}{2a^2}\left [ 6 + \lambda_b a^2 + \frac{6\dot{a}^2}{(-\delta ) } + 12 a_{\pm } \right ] & 0 & 0 \\ 0 & 0 & \frac{1}{2a^4}\left [ 6 + \lambda_b a^2 + \frac{6\dot{a}^2}{(-\delta ) } \right ] & 0 \\ 0 & 0 & 0 & m_{2\times 2 } \end{smallmatrix } \right ) . \ ] ] the previous matrix , in the cosmic gauge , reduces to a more manageable form @xmath185 & 0 & 0 \\ 0 & 0 & - 3 a^{-2 } h^2 ( 1 - \upsilon ) & 0 \\ 0 & 0 & 0 & n_{2\times 2 } \end{smallmatrix } \right ) , \label{eq : psicosmicgauge}\ ] ] where @xmath186 and @xmath187 denote @xmath188 diagonal matrices .

the creation of brane universes induced by a totally antisymmetric tensor living in a fixed background spacetime is presented , where a term involving the intrinsic curvature of the brane is considered . a canonical quantum mechanical approach employing wheeler - dewitt equation is done . the probability nucleation for the brane is calculated taking into account both an instanton method and a wkb approximation . some cosmological implications arose from the model are presented .

the study of the gaugino sector of supersymmetry is a complex and important endeavour , which appears well suited to a linear collider of sufficient energy and luminosity . the main observables of interest are the masses of the @xmath1 and @xmath2 states and their production cross sections , including those with polarised beams . @xmath3 collisions offer two independent techniques for determining the mass of supersymmetric particles . these are the analysis of the energy spectrum of the sm particle produced in association with a lighter supersymmetric state in the two - body decays and the study of the pair production cross section near threshold . these techniques have already been extensively studied for lower centre - of - mass energies , @xmath4 , between 0.35 to 0.5 tev @xcite . in this note , we analyse the gaugino pair production and derive the statistical accuracy on their masses using both techniques and including the effects of initial state radiation ( isr ) , beamstrahlung ( bs ) and parton energy resolution for multi - tev @xmath3 collisions . we follow the evolution of these accuracies for fully hadronic final states from pure signal samples to realistic inclusive susy samples and validate the results obtained at generator level with analyses performed on fully simulated and reconstructed events . the study provides us with requirements on parton energy resolution which are complementary to those obtained from other processes , such as heavy susy higgs decays , since the kinematics of decays of gaugino pairs with large missing energy into pairs of escaping neutralinos does not benefit from the kinematic fits , which are instead applicable to processes where the full beam energy is deposited in the detector . the estimated mass accuracies can be compared in a next step to those required for the reconstruction of the gut scale susy parameters @xcite and the determination of the lightest neutralino contribution to the dark matter relic density in the universe @xcite . this comparison will provide us with well - motivated quantitative requirements on parton energy resolution in susy events . this study considers two scenarios in the constrained mssm ( cmssm ) model , which offer different experimental challenges . their parameters are given in table [ tab : modelpar ] . the first ( model i ) , adopted as a benchmark point for the clic cdr studies @xcite , has the lightest neutralino at 340 gev and the chargino and heavier neutralinos with masses in the range 640 to 917 gev ( see table [ tab : mass ] and the left panel of figure[fig : spectra ] ) . at @xmath4 = 3 tev all the gauginos are observables . the relatively low masses and the 3 tev centre - of - mass energy make cross sections sizable but the beamstrahlung effects more significant ( see table [ tab : modelpar ] ) . in the second ( model ii ) the lightest neutralino has a mass of 554 gev , while the other neutralinos and the charginos have masses in the range from 1064 to 1414 gev ( see table [ tab : mass ] and the right panel of figure[fig : spectra ] ) @xcite . at 3 tev , most gauginos are close to threshold for pair production and cross sections are small . this minimises the beamstrahlung effects , since the production cross section drops significantly when the beams lose energy due to radiation . the cross sections are given in table [ tab : xsec ] and figure [ fig : xsec ] . .parameters of the two cmssm models adopted in this study [ cols="<,^,^",options="header " , ] [ tab : scan ] we compute the cross section @xmath5 at various @xmath4 values for a set of closely spaced masses and obtain the derivative @xmath6 of the change of the cross section at each energy per unit of mass change . results are shown in figure [ fig : sens ] , which indicate that the maximum of the sensitivity to the mass is achieved near threshold . the number of scan points and the share of the statistics among them is optimised by studying the mass uncertainty obtained from the fit for different assumptions . we find that it is preferable to concentrate the luminosity in a small number of scan points . for example , the statistical accuracy on the mass of the @xmath7 in the model i varies from @xmath80.85 gev , obtained for a four - point scan ( 1310@xmath91950 gev ) , to @xmath80.45 gev , when the luminosity is split between just two points , one of which at the peak of the sensitivity ( @xmath4=1350 gev ) and the second close to threshold ( @xmath4=1310 gev ) . this confirms the findings of @xcite for lower sparticle masses and different luminosity spectrum . finally , we consider the option of operating the collider with polarised beams . results are summarised in table [ tab : scan ] . in all cases , except the @xmath10 , the mass accuracies obtained with a dedicated threshold scan improve on those resulting from the kinematic edge analysis at 3 tev by factors of 2 or more . the use of polarised beam further improves these accuracies , effectively compensating for the loss of sensitivity due to isr and bs . the determination of chargino and neutralino masses in high - mass susy scenarios with two - body decays into @xmath11 , @xmath12 and @xmath13 bosons provides us with a mean to quantify the effect of radiation , by isr and beamstrahlung , and parton energy resolution on the accuracy achievable in supersymmetric particle mass measurements at a multi - tev @xmath3 linear collider . in our analysis both fits to boson energy spectra and threshold scans are considered for fully hadronic final states . results from generator - level quantities are validated using fully simulated and reconstructed events in the @xmath14 and @xmath15 final states . not accounting for reconstruction efficiencies , estimated to be @xmath1660% in four jet final states , the mass of charginos and neutralinos can be determined from the kinematic edges of the boson energy in inclusive susy event samples to a relative accuracy in the range 0.3% to 1.0% ( 0.6% - 1.0% ) in absence of radiation and energy resolution effects to 0.8% to 1.7% ( 1.1% - 2.0% ) accounting for isr , bs and realistic energy resolution for the benchmark with particle masses in the range 600 - 900 gev ( @xmath171000 gev ) , respectively , with 2 ab@xmath18 of integrated luminosity at @xmath4 = 3 tev . the relative increase of the statistical uncertainty of the mass measurement is larger for the model i which has the sparticles masses far way from pair the production thresholds . however , in absolute terms the larger production cross sections in this model yield better statistical accuracy in the mass determination . by adopting the criterion that the degradation to the mass measurement statistical accuracy from the parton energy resolution should not exceed that induced by isr and bs , we derive the requirement of a relative energy resolution for jets , @xmath190.05 . if the accelerator can operate at energies below the nominal @xmath4 ( down to @xmath4=1310 gev for model i and @xmath4=2200 gev for model ii ) with comparable performance to collect about one third of the statistics at centre - of - mass energies close to the kinematic thresholds for sparticle pair production , the mass accuracies from these threshold scans improves by factors of 2 or more compared to those obtained from study of the kinematic edges at the maximum @xmath4 energy . the availability of polarised beam in the scan further improves these accuracies , effectively compensating for the loss of sensitivity due to the effect of isr and beamstrahlung . we are grateful to the colleagues who contributed to this study . in particular to jean - jacques blaising , sabine kraml and abdelhak djouadi for extensive discussion and their careful reading of the text . we are also thankful to by dieter schlatter for valuable suggestions on this note . y. li and a. nomerotski , arxiv:1007.0698 [ physics.ins-det ] . g. a. blair , a. freitas , h. u. martyn , g. polesello , w. porod and p. m. zerwas , acta phys . polon . b * 36 * ( 2005 ) 3445 [ arxiv : hep - ph/0512084 ] . e. a. baltz , m. battaglia , m. e. peskin and t. wizansky , phys . d * 74 * ( 2006 ) 103521 [ arxiv : hep - ph/0602187 ] . s. martin , private communication . m. battaglia , a. de roeck , j. r. ellis , f. gianotti , k. a. olive and l. pape , eur . j. c * 33 * ( 2004 ) 273 [ arxiv : hep - ph/0306219 ] . j. r. ellis , t. falk , g. ganis , k. a. olive and m. srednicki , phys . b * 510 * ( 2001 ) 236 [ arxiv : hep - ph/0102098 ] . a. djouadi , j. l. kneur and g. moultaka , comput . commun . * 176 * ( 2007 ) 426 [ arxiv : hep - ph/0211331 ] . m. muhlleitner , a. djouadi and y. mambrini , comput . commun . * 168 * ( 2005 ) 46 [ arxiv : hep - ph/0311167 ] . g. belanger , f. boudjema , a. pukhov and a. semenov , comput . commun . * 176 * ( 2007 ) 367 [ arxiv : hep - ph/0607059 ] . d. larson _ et al . _ , arxiv:1001.4635 [ astro-ph.co ] . t. sjostrand , s. mrenna and p. z. skands , jhep * 0605 * ( 2006 ) 026 [ arxiv : hep - ph/0603175 ] . f. e. paige , s. d. protopopescu , h. baer and x. tata , arxiv : hep - ph/0312045 . s. katsanevas , p. morawitz , comput . phys . commun . * 112 * ( 1998 ) 227 - 269 . [ hep - ph/9711417 ] . j. l. feng and d. e. finnell , phys . d * 49 * ( 1994 ) 2369 [ arxiv : hep - ph/9310211 ] . h. u. martyn and g. a. blair , arxiv : hep - ph/9910416 . f. james and m. roos , comput . commun . * 10 * ( 1975 ) 343 . h. braun _ et al . _ [ clic study team ] , clic - note-764 ( 2008 ) . m. skrzypek and s. jadach , z. phys . c * 49 * ( 1991 ) 577 . e. boos _ et al . _ [ comphep collaboration ] , nucl . instr . and meth . a * 534 * ( 2004 ) , 250 . s. catani , y. l. dokshitzer , m. olsson , g. turnock and b. r. webber , phys . b * 269 * ( 1991 ) 432 . m. a. thomson , nucl . instrum . meth . a * 611 * ( 2009 ) 25 [ arxiv:0907.3577 [ physics.ins-det ] ] . g. a. blair , econf c010630 ( 2001 ) e3019 .

this note reports the results of a study of the accuracy in the determination of chargino and neutralino masses in two high - mass supersymmetric scenarios through kinematic endpoints and threshold scans at a multi - tev @xmath0 collider . the effects of initial state radiation , beamstrahlung and parton energy resolution are studied in fully hadronic final states of inclusive susy samples . results obtained at generator level are compared to those from fully simulated and reconstructed events for selected channels .

the study of the role of individual nuclear reactions in stellar evolution has been an important field of research in the last few decades . as a star evolves with time it passes through burning in different ranges of nuclear mass . at the same time , different nuclear processes become important at different time periods of evolution . a comprehensive study of these processes sheds light on various astrophysical phenomena . there are certain astrophysical sites which are responsible for the production of heavier nuclei beyond iron through the rapid capture of protons on seed nuclides . in the mass region of our interest there are certain proton rich naturally occurring nuclei , which are not produced by the @xmath0-process or the @xmath1-process . these are called @xmath2-nuclei . proton capture reactions in certain astrophysical sites can account for the formation of some of these proton rich nuclides . for example x - ray bursters with a large proton flux in the peak temperature around 1 - 3 gk are suitable astrophysical sites for the production of certain nuclei . to find out the abundance of different nuclei as well as the evolution of the process in these sites a network calculation is necessary which involves a large number of reactions . it is thus imperative to calculate the rates and/or cross sections of these reactions in different mass ranges . our group has already calculated the cross sections and hence the astrophysical s - factors in the mass range @xmath3 @xcite . some implications of the new rates has also been investigated in the context of rp - process @xcite . in the present work , we extend our calculation to the @xmath4 region . the rp - process is sensitive to a number of reactions in this region . the most challenging aspect to look at in these scenarios is that most of the nuclei involved in those reactions are not produced in the laboratory . for example , parikh _ et al . _ @xcite have identified proton capture reactions on @xmath5ni and @xmath6cu targets as important in the rp - process in certain scenarios . however , experimental rates are not available for these reactions because stable targets do not occur in nature . hence , one has to depend on theoretical calculations in this domain . in explosive proton rich environments , such as x - ray bursts , proton capture has to compete with its inverse , _ i.e. _ photo - disintegration . this competition results in waiting points and causes delay of further nucleosynthesis . with temperature , pressure and proton mass fractions being different at different regions of these sites as well as being time - varying quantities , incorporation of all these physical conditions in the nuclear network is a big challenge . et al . _ @xcite have calculated the rates for various proton , neutron and @xmath7-particle induced reactions and their reverse reactions in hauser - feshbach formalism for targets with wide range of atomic numbers and masses and for a wide range of temperature . theoretical calculations in this mass region essentially utilize the hauser - feshbach formalism where , the optical model potential , a key ingredient , is often taken in a local or a global form . however , a more microscopic approach is also possible using an optical potential constructed utilizing nuclear densities . if the target is stable , nuclear density is available through electron scattering . however , in the absence of a stable target , theory remains our sole guide to describing the density . it is imperative to test the theoretical calculations , where experimental data are available , to verify its applicability . we aim to check the success of microscopic optical potentials based on mean - field densities in explaining the available reaction cross sections in this mass region . a good description depending essentially on theory will allow one to extend the present method to the critical reactions , which are beyond present day laboratory capabilities . a well defined nucleon - nucleon ( @xmath8 ) interaction is of major importance for microscopic calculation of nucleon - nucleus and nucleus - nucleus potentials used in the theoretical analysis of different reactions as well as scattering . the optical model potential is highly successful for explanation of different branches of nuclear reaction . it can reliably predict the basic observables such as total and partial cross sections , elastic scattering angular distributions , etc , even for those target nuclei and for those energy regions for which no experimental data exist . we have used the density dependent m3y interaction by folding the potential with target radial matter densities . this interaction has been used in many calculations and has given satisfactory results . the paper is organized as follows . in the next section , we outline our method of calculation . essentially we construct an optical model potential through folding an @xmath8 interaction with the theoretical density profile . for this purpose we use the relativistic mean field ( rmf ) theory to obtain the density profile of the targets . in sec . [ secresults ] the results of our work are discussed in detail . finally we summarize our work . the rmf approach has proved to be very successful in describing various nuclear properties such as binding energy of nuclei in ground states as well as excited states , nuclear density profile , rms charge radii , deformation , nuclear halo , moment of inertia , etc @xcite . it is considered to be the relativistic generalization of the non - relativistic models such as gogny force or skyrme force hartree - fock theory using effective mesonic degrees of freedom rather than instantaneous forces . the model is basically based upon two major approximations namely mean - field approximation and no - sea approximation @xcite . the starting point of rmf is a suitable lagrangian density that includes the coupling between the nucleon field and meson field as well as meson self couplings so that the lagrangian can successfully describe the properties of finite nuclei as well as the equation of state ( eos ) of nuclear matter . there are different variations of lagrangian density as well as different parameterizations . an accurately calibrated relativistic lagrangian density , fsugold @xcite , has been fitted to the charge radii of nuclei . it contains two additional parameters , compared to conventional rmf models , describing self coupling of vector - isoscalar meson and coupling between the vector - isovector meson and vector - isoscalar meson . these two additional parameters significantly affect the softening of the eos , the accurate determination of which is needed for the study of various nuclear properties such as charge radii , masses , etc . thus theoretical density profiles are extracted in the rmf approach considering the fsugold interaction . the charge density is obtained by convoluting the point proton density considering the finite size of the nucleus . @xmath9 where @xmath10 is the gaussian form factor given by , @xmath11 where @xmath12 is a constant whose value is assigned to 0.8 fm . using the nuclear density profile we have numerically obtained rms charge radii . while calculating the charge density or the radius , no attempt has been made to take the correction due to center of mass into account . calculations on harmonic oscillator wave functions show that the correction is small for heavier nuclei . for example , quentin has shown@xcite that the effect of inclusion of center of mass correction in the radius is given by @xmath13 . hence , we do not expect the density profile to be affected significantly due to this approximation . the m3y interaction @xcite is based on a realistic g - matrix which in turn is constructed in a harmonic oscillator representation averaging over a range of energies as well as densities . it has no explicit density dependence nor energy dependence . although in most cases these averages do not matter producing satisfactory results , in few cases it becomes necessary to incorporate explicit density dependence into m3y interaction and then it is named as density dependent m3y ( ddm3y ) effective interaction @xcite . low energy proton capture reactions are highly sensitive to nuclear radius as well as density . in the present work we have used density dependent m3y reid - elliot effective nucleon - nucleon interaction within a folding model prescription @xcite . the density dependence is incorporated in the same way as suggested in refs . @xcite . further , we have included a spin - orbit term into the potential considering scheerbaum prescription @xcite which has been coupled with the phenomenological complex potential depths . these depths are functions of energy which are assigned standard values as in lahiri _ et al_. @xcite . these values are kept unaltered throughout our present work . we have incorporated the density dependent m3y interaction within the talys1.4 code @xcite and performed a hauser - feshbach ( hf ) calculation . we have chosen goriely s microscopic level densities and hartree fock bogolyubov model for @xmath14 @xmath15-ray strength function . as seen in our previous calculations @xcite , these choices can explain the experimental results more accurately . all these options are available in the code . we have also included the effect of the width fluctuation correction which has a significant impact at low incident energies . up to 30 discrete levels are included for both target and residual nuclei , which are considered in hauser - feshbach decay and @xmath15-ray cascade . we also include a maximum of 30 discrete levels for the nuclei resulting from binary emission in hauser - feshbach decay and @xmath15-ray cascade . hf calculations are done with full @xmath16 coupling . we have incorporated the density data obtained from rmf approach to obtain the optical model potential . because of rapid variation of cross - section with energy in the low energy region , it is difficult to compare the theory and experiment . a standard alternative way is to compare another important quantity instead of cross - section , namely astrophysical s - factor @xcite . the proton capture reactions in astrophysical sites occur within a narrow energy window @xcite . this effective energy window approximately of gaussian shape around a peak ( known as gamow peak ) is known as gamow window . the expressions for the gamow peak and gamow width in a practical form are given respectively as , @xmath17 @xmath18 wherei @xmath19 is the reduced mass and @xmath20 denotes the temperature in gk . thus most of the astrophysically important reactions occur within a narrow energy window @xmath21 to @xmath22 . we see that for ( @xmath23 ) reactions on stable isotopes in the mass range 55 - 60 , the gamow window lies between 1 mev to 3 mev for temperature around 3 gk . hence , we have carried out our calculation in this low energy window and compared our results with the measured data where available . in calculating the gamow peak , gamow width and hence gamow window we have taken the masses from audi _ et al_. @xcite . because the optical model is dependent on the density profile of the nucleus , we calculate the density and the charge radii of nuclei in this mass region using rmf formalism . the theoretical density values are plotted as a function of radius and compared with available experimental values in fig . [ fig : den ] . as can be seen the agreement is extremely good . the experimental data are taken from wohlfahrt _ et al _ @xcite . we also compare all the available rms charge radii values with theoretical results in table [ tab : exp2 ] . the experimental values are taken from angeli _ et al_. @xcite . it can be seen that the rmf calculation has an excellent predictive power , the relative difference between theory and experiment in all cases being less than 0.5% . .charge radii of various nuclei extracted in the rmf approach compared with measured values from angeli _ et al_. @xcite [ cols="^,^,^,^,^ " , ] + we have tried to set a definite normalization for the optical model potential that fits all the reaction data in the concerned mass region . the potential obtained by folding has been multiplied by the normalization constant 2.0 to get the real part of the potential . the ddm3y interaction does not have any imaginary part . we have multiplied the folding potential by the normalization constant 1.4 to obtain the imaginary part of the optical potential . these final parameters have been obtained after many trials to ensure a reasonable agreement with experimental data for all the known low energy proton capture reactions in the mass region of our interest . although a single normalization can not reproduce the experimental data excellently for all reactions in the region , _ i.e. _ each individual reaction may have different normalization for best matching with measurement , it is necessary to consider a single definite normalization to extend the work to unknown nuclei in the mass region for which no experimental data exist . we note that the fitted parameters for the present mass range differ from the neighbouring mass region in our earlier calculation . this is possibly due to the fact that the mass selected in the present calculation is lighter than our previous regions . possibly , the larger depths of the potential is required to adjust for the low mass region . the comparison of s - factors obtained after incorporating ddm3y interaction using the above normalization constants with experimental data are shown in fig . [ fig : ddm3y ] . the numerical values of the s - factors and the reaction rates are given in supplemental material @xcite . the experimental data are taken from ref @xcite for @xmath24mn , @xmath25fe and @xmath26co , respectively . for @xmath25ni and @xmath27ni , experimental data are taken from refs . @xcite and refs . @xcite , respectively . in many cases the experimental data are very old . errors are also not available in some cases . for @xmath24mn , @xmath25fe and @xmath26co , circles represent the experimental data . for @xmath25ni , triangles , squares and circles represent the data from refs . @xcite , respectively . for @xmath27ni there are three different sources of data @xcite which are denoted by squares , circles and diamonds , respectively . in all cases the solid line denotes the theoretical ddm3y result . in @xmath24mn , there are certain ambiguities in experimental data , especially in the energy range between 1.3 to 1.6 mev . the experiment was done using ge(li ) detector by integrated beam current method more than three decades ago . however , errors are not associated with most of the data points . only four data points in the energy range of our interest have errors associated with those . our calculations give an excellent description of experimental data for @xmath25fe . the experiment for @xmath25fe was done using ge(li ) detector and the data was compared with statistical model predictions @xcite . for @xmath26co again there are large fluctuations in experimental data . et al . _ @xcite stated that they had observed several resonances in the reaction @xmath26co@xmath28ni but the resonances were too close to be resolved clearly . our calculation for @xmath25ni overpredicts the measurement of tingwell _ et al . _ @xcite by a factor of @xmath29 2.5 , approximately . this experiment was carried out by both beam current integrated method and single target irradiation method using ge(li ) detector . et al . _ also compared their data with statistical model calculations . they found that their statistical calculation overestimates the measurement by a factor of @xmath29 2.5 for @xmath25ni , which agrees with our results . _ @xcite also measured cross section for this reaction using the activation technique . except in the energy range @xmath29 1.4 - 1.8 mev , where the measurement itself has large discrepancies , the data agree more or less well with our theoretical calculations . for the reaction @xmath27ni@xmath30cu , tingwell _ et al . _ themselves compared the experimental results with the statistical model predictions and showed that normalizing the optical model imaginary well depth for ni isotopes by a factor of 1.5 leads to a better agreement between theory and experiment @xcite . our calculation , in the case of @xmath27ni , overpredicts the experimental data of tingwell _ et al . _ @xcite by a factor @xmath29 1.5 , whereas it underpredicts the data of krivonosov _ et al . _ @xcite by a factor @xmath29 0.35 . with the above normalization , we have calculated the rates for @xmath31 reactions identified as important by parikh _ et al_. @xcite . the calculated rates are compared with non - smoker @xcite rates . the non - smoker results are from a hf calculation based on masses from experimental measurements and calculation in the finite range droplet model @xcite . other details of the calculation can be obtained from the references . the results have been plotted in figure 3 . we see that in the range 1 - 4 gk , the non - smoker results differ from the present calculations significantly . for the @xmath5ni(@xmath32 reaction , although the results agree at low temperature , at higher temperature the non - smoker rates are larger . on the other hand , for the other two reactions , _ viz . _ @xmath33cu(@xmath32 and @xmath26cu(@xmath32 , our calculation predicts a significantly larger rate throughout the temperature range . it will be interesting to see the effects of these results on astrophysical scenarios low energy @xmath31 reactions are studied in a semi - microscopic approach in the hf formalism and compared with experiments in the mass region 55 - 60 . radial density profiles are obtained using the rmf approach and are folded with the ddm3y nn interaction to obtain semi - microscopic optical potentials . both the real and imaginary depths of the potential are normalized to obtain a good agreement between theory and experiment . the s - factors for @xmath31 reactions are evaluated in the gamow window corresponding to 3 gk . we have not modified the parameters to fit individual reactions as our aim is to construct a framework for calculation of astrophysical reactions involving unstable nuclei . rates for important astrophysical reactions calculated in the present approach differ significantly from non - smoker rates . the key feature of our work is that we have taken all nuclei in the same footing and same methodology has been used for all of them to avoid systematic error . the authors acknowledge the financial support provided by ugc(drs ) , dst and the university of calcutta . 99 g. gangopadhyay , phys . c * 82*,027603 ( 2010 ) . c. lahiri and g. gangopadhyay , eur . j. a * 47 * , 87 ( 2011 ) . c. lahiri and g. gangopadhyay , phys . c*84 * , 057601 ( 2011 ) . c. lahiri and g. gangopadhyay , phys . c * 86 * , 047601(2012 ) . c. lahiri and g. gangopadhyay , int . e * 21 * , 1250074 ( 2012 ) . c. lahiri and g. gangopadhyay , mod . a * 28 * , 1350076 ( 2013 ) . a. parikh , j. jos , f. moreno and c. iliadis , new astron.rev . * 52 * , 409 ( 2008 ) . t. rauscher , f. k. thielemann , at . data nucl . data tables * 75*,1 ( 2000 ) . t. rauscher , f. k. thielemann , at . data nucl . data tables * 79 * , 47 , ( 2000 ) p. ring , prog . part . . phys . * 37 * , 193 ( 1996 ) . horst mller and brian d. serot , nucl . phys . * a606 * , 508 ( 1996 ) . b. g. todd - rutel and j. piekarewicz , phys . * 95 * , 122501 ( 2005 ) . p. quentin , in nuclear self - consistent fields , edited by g. ripka and m. porneuf ( north - holland/ american elsevier , 1975 ) p 297 . g. bertsch , j. borysowicz , h. mcmanus , and w.g . love , nucl . phys . * a284 * , 399 ( 1977 ) . satchler and w.g . love , phys . rep . * 55 * , 183 ( 1979 ) . myers , nucl . phys . * a204 * , 465 ( 1973 ) . basu , p. roy chowdhury , c. samanta , phys . c * 72 * , 051601 ( 2005 ) . basu , j. phys . g : nucl . part . phys . * 30 * , b7 ( 2004 ) . r. r. scheerbaum , nucl . phys . * a257 * , 77 ( 1976 ) . a. j. koning s. hilaire , and m. duizvestijn , proceedings of the international conference on nuclear data for science and technology , april 22 - 27 , 2007 , nice , france , edited by o. bersillon , f. gunsing , e. bauge , r. jacgmin , s. leray ( edp sciences , 2008 ) p. 211 . c. e. rolfs and w. s. rodney , cauldrons in the cosmos , the university of chicago press , chicago and london , 1988 . g. audi and a. h. wapstra , nucl . phys . * a729 * , 129 ( 2003 ) . h. d. wohlfahrt , o. schwentker , g. fricke , h. g. anderson , e. b. shera , phys . c * 22 * , 264 ( 1980 ) . i. angeli , k.p . marinova , at . data . and nucl . data tables * 87 * , 185 ( 2004 ) . see supplemental material at [ url will be inserted by publisher ] for numerical values of calculated @xmath34-factors and reaction rates . l. w. mitchell , d. g. sargood , aust . j. phys . * 36 * , 1 ( 1983 ) . s. g. tims , a. f. scott , a. j. morton , v. y. hansper , and d. g. sargood , nucl.phys . * a563 * , 473 ( 1993 ) . j. w. butler , c. r. gossett , phys . rev . * 108 * , 1473 ( 1957 ) . cheng , j. d. king , can . * 58 * , 1677 ( 1980 ) . et al . _ , nauk sssr , ser . fiz . * 41 * , 2196(1977 ) . tingwell , l. w. mitchell , m. e. sevior and d. g. sargood , nucl.phys . * a439 * , 371(1985 ) . a. simon _ et al_. , phys . c * 87 * , 055802 ( 2013 ) . tingwell _ et al_. , nucl a496 * , 127 ( 1989 ) . http://nucastro.org/nonsmoker.html j.r . nix , w.d . myers , and w. swiatecki , at . data nucl . data tables * 59 * , 185 ( 1995 ) .

low energy proton capture reactions in the mass 55 - 60 region are studied in a microscopic optical model . nuclear density profile is calculated using the relativistic mean field theory . the ddm3y interaction is folded with the theoretical density to obtain the proton - nucleus optical potential . a definite set of normalization parameters has been obtained for the concerned mass region by comparing with all available experimental data in this mass region . these parameters have been used to obtain proton capture rates for astrophysically important reactions in this mass region .

in a recent paper li _ et al . _ @xcite presented a new design for an optomechanical system that consists of a microdisk resonator coupled to a waveguide . this design has several attractive features . besides its universality , it enables one to study the reactive effects @xcite in optomechanical coupling . the origin of the reactive coupling is well explained in ref . its origin lies in the mechanical motion dependence of the extrinsic losses of the disk resonator . further phase - dependent gradient forces lead to reactive coupling . have also argued that this design is more effective in achieving cooling of the system to its ground state . while cooling is desirable for studying quantum effects at the macroscopic scale @xcite , we examine other possibilities , which do not depend on the cooling of the system , to investigate the effects arising from strong reactive coupling . since optomechanical coupling effects are intrinsically nonlinear , we examine the nonlinear response of the microdisk resonator to pump probe fields . we report reactive - coupling - induced normal mode splitting . note that in previous works @xcite on normal mode splitting in optomechanical devices , only dispersive coupling was used . in this paper , we report on normal mode splitting due to reactive effects . the paper is organized as follows . in sec . ii , the physical system is introduced and the time evolutions of the expectation values of the system operators are given and solved . in sec . iii , the expectation value of the output fields is calculated , and the nonlinear susceptibilities for stokes and anti - stokes processes are obtained . in sec . iv , we discuss normal mode splitting in output fields with or without reactive coupling . we find that there is no normal mode splitting in output fields in the absence of reactive coupling . however , normal mode splitting occurs in output fields in the presence of reactive coupling . we consider the system shown in fig . [ fig1 ] , in which a microdisk cavity is coupled to a freestanding waveguide . a strong pump field with frequency @xmath0 and a weak stokes field with frequency @xmath1 enter the system through the waveguide . the waveguide will move along the @xmath2 direction under the action of the optical force exerted by the photons from the cavity . further , considering the dispersive coupling and reactive coupling between the waveguide and the cavity , displacement @xmath3 of the waveguide from its equilibrium position will change the resonant frequency of the cavity field and the cavity decay rate , represented by @xmath4 and @xmath5 , respectively . in a rotating frame at pump frequency @xmath0 , the hamiltonian of the system is given by @xcite @xmath6c^{\dag}c+\frac{p^2}{2m}+\frac{1}{2}m\omega_{m}^2q^2\vspace{0.1in}\\\hspace{0.3in}+\hbar\frac { l}{c}\tilde{n}_{g}(\omega_{l}\varepsilon_{l}^2+\omega_{s}|\varepsilon_{s}|^2)+i\hbar\sqrt{2\kappa_{e}(q)}\varepsilon_{l}(c^{\dag}-c)\vspace{0.1in}\\\hspace{0.3 in } + i\hbar\sqrt{2\kappa_{e}(q)}(\varepsilon_{s}e^{-i\delta t}c^{\dag}-\varepsilon^{*}_{s}e^{i\delta t}c ) . \end{array}\ ] ] the first term is the energy of the cavity field , whose annihilation ( creation ) operators are denoted @xmath7 . the second and third terms are the energy of the waveguide with mass @xmath8 , frequency @xmath9 , and momentum operator @xmath10 . the fourth term gives the interactions between the waveguide and the incident fields ( the pump field and the stokes field ) , @xmath11 is the length of the waveguide , @xmath12 is the speed of light in vacuum , @xmath13 is the group index of the waveguide optical mode @xcite , @xmath14 and @xmath15 are the amplitudes of the pump field and the stokes field , respectively , and they are related to their corresponding power @xmath16 and @xmath17 by @xmath18 and @xmath19 . the latter two terms describe the coupling of the cavity field to the pump field and the stokes field , respectively . and @xmath20 is the detuning between the stokes field and the pump field . we would study the physical effects by scanning the stokes laser . for a small displacement @xmath3 , @xmath4 and @xmath5 can be expanded to the first order of @xmath3 , @xmath21 thus the quantities @xmath22 and @xmath23 describe the cavity - waveguide dispersive and reactive coupling strength , respectively . further , note that the photons in the cavity can leak out of the cavity by an intrinsic damping rate @xmath24 of the cavity and by a rate of @xmath5 due to the reactive coupling between the waveguide and the cavity . in addition , the velocity of the waveguide is damped at a rate of @xmath25 . applying the heisenberg equation of motion and adding the damping terms , the time evolutions of the expectation values ( @xmath26 , and @xmath27 ) for the system can be expressed as @xmath28-\gamma_{m}\langle p\rangle,\vspace{0.2in}\\ \displaystyle\langle\dot{c}\rangle=-[\kappa+\langle q\rangle\kappa_{om}+i(\omega_{c}-\omega_{l}+\langle q\rangle\chi)]\langle c\rangle\vspace{0.2in}\\\hspace{0.4in}\displaystyle+\sqrt{\kappa}[1+\langle q\rangle\frac{\kappa_{om}}{\kappa}](\varepsilon_{l}+\varepsilon_{s}e^{-i\delta t } ) , \end{array}\ ] ] where we have used the mean field assumption @xmath29 , expanded @xmath5 to the first order of @xmath3 , and assumed @xmath30 , where @xmath31 is the half - linewidth of the cavity field . it should be noted that the steady - state solution of eq . ( [ 3 ] ) contains an infinite number of frequencies . since the stokes field @xmath32 is much weaker than the pump field @xmath14 , the steady - state solution of eq . ( [ 3 ] ) can be simplified to first order in @xmath32 only . we find that in the limit @xmath33 , each @xmath34,@xmath35 , and @xmath27 has the form @xmath36 where @xmath37 stands for any of the three quantities @xmath3 , @xmath10 , and @xmath12 . thus the expectation values @xmath38 , and @xmath27 ) oscillate at three frequencies ( @xmath0 , @xmath1 , and 2@xmath39 ) . substituting eq . ( [ 4 ] ) into eq . ( [ 3 ] ) , ignoring those terms containing the small quantities @xmath40 , @xmath41 , @xmath42 , and equating coefficients of terms with the same frequency , respectively , we obtain the following results @xmath43,\vspace{.1in}\\ c_{+}=\displaystyle \frac{1}{d(\delta)}[a(be+fj)-i\hbar\frac{\kappa_{om}}{\sqrt{\kappa}}c_{0}^{*}bf^{*}],\vspace{.1in}\\ c_{-}=\displaystyle \frac{f^{*}}{d^{*}(\delta)}(-aj+i\hbar\frac{\kappa_{om}}{\sqrt{\kappa}}c_{0}v),\vspace{0.1in}\\ q_{+}=\displaystyle \frac{b}{d(\delta)}(-aj^{*}-i\hbar\frac{\kappa_{om}}{\sqrt{\kappa}}c_{0}^{*}v^{*}),\vspace{0.1in}\\ q_{-}=(q_{+})^ { * } , \end{array}\ ] ] where @xmath44 @xmath45 and @xmath46 , @xmath47 , @xmath48 , @xmath49 , @xmath50 , @xmath51 . the approach used in this paper is similar to our earlier work @xcite which dealt with optomechanical systems with dispersive coupling only . to investigate the normal mode splitting of the output fields , we need to calculate their expectation value . it can be obtained by using the input - output relation @xcite @xmath52 . if we write @xmath53 as @xmath54 where @xmath55 is the response at the pump frequency @xmath0 , @xmath56 is the response at the stokes frequency @xmath1 , and @xmath57 is the field generated at the new anti - stokes frequency @xmath58 . then we have @xmath59 furthermore , whether there is normal mode splitting in the output fields is determined by the roots of the denominator @xmath60 of @xmath56 . here we examine the roots of @xmath60 given by eq . ( [ 7 ] ) numerically . the response of the system is expected to be especially significant if we choose @xmath1 corresponding to a sideband @xmath61 or @xmath62 , so we consider the case @xmath63 . the other parameters are chosen from a recent experiment focusing on the effect of the reactive force on the waveguide @xcite : the wavelength of the laser @xmath64 nm , @xmath65 mhz / nm , @xmath66 pg ( density of the silicon waveguide , 2.33 g/@xmath67 ; length , 10 @xmath68 m ; width , 300 nm ; height , 300 nm ) , @xmath69 , @xmath70 mhz , and the mechanical quality factor @xmath71 . in the following , we work in the stable regime of the system . figure [ fig2 ] shows the variation of the real parts of the roots of @xmath60 in the domain re@xmath72 with increasing pump power for no reactive coupling , @xmath73 , and for @xmath74 mhz / nm . for @xmath73 , the interaction of the waveguide with the cavity is purely dispersive ; the cavity decay rate does not depend on the displacement of the waveguide . in this case , the real parts of the roots of @xmath60 always have two equal values with increasing pump power . thus there is no splitting because the dispersive coupling is not strong enough . however , for @xmath74 mhz / nm , the system has both dispersive and reactive couplings , the cavity decay rate depends on the displacement of the waveguide , and the real parts of the roots of @xmath60 will change from two equal values to two different values with increasing pump power . and the difference between two real parts of the roots of @xmath60 in the domain re@xmath72 is increased with increasing pump power . therefore , the reactive coupling between the waveguide and the cavity can result in normal mode splitting of the output fields , and the peak separation becomes larger with increasing pump power . figure [ fig3 ] shows the variation of the imaginary parts of the roots of @xmath60 with increasing pump power for zero reactive coupling @xmath73 and nonzero reactive coupling @xmath74 mhz / nm . for @xmath73 , the imaginary parts of the roots of @xmath60 do not change with increasing pump power . however , for @xmath74 mhz , the imaginary parts of the roots of @xmath60 change with increasing pump power . we thus conclude that for the present microdisk resonator coupled to a waveguide the normal mode splitting is solely due to the reactive coupling . we now discuss how the output fields depend on the behavior of the roots of @xmath60 . for convenience , we normalize all quantities to the input stokes power @xmath17 . assuming that @xmath32 is real , we express the output power at the stokes frequency @xmath1 in terms of the input stokes power @xmath75 further , we introduce the two quadratures of the stokes component of the output fields by @xmath76 and @xmath77 . one can measure either the quadratures of the output by homodyne techniques or the intensity of the output . for brevity , we only show @xmath78 and @xmath79 as a function of the normalized detuning between the stokes field and the pump field @xmath80 for this model , without reactive coupling ( @xmath23=0 ) and with it ( @xmath74 mhz / nm ) , for different pump powers in figs . [ fig4][fig5 ] . for @xmath23=0 , it is found that @xmath78 has a lorentzian lineshape corresponding to the absorptive behavior . note that @xmath78 and @xmath79 exhibit no splitting when @xmath23=0 . however , for @xmath74 mhz / nm , it is clearly seen that normal mode splitting appears in @xmath78 and @xmath79 . therefore reactive coupling can lead to the appearance of normal mode splitting in the output stokes field . and the peak separation increases with increasing pump power @xcite . the dip at the line center exhibits power broadening . we also find that the stokes field can be amplified by the stimulated process . obviously the maximum gain @xmath79 for the stokes field depends on the system parameters . for a pump power @xmath81 @xmath68w , the maximum gain for the stokes field is about 1.3 . note that the nonlinear nature of the reactive coupling generates anti - stokes radiation . in a similar way , we define a normalized output power at the anti - stokes frequency @xmath58 as @xmath82 the plots of @xmath83 versus the normalized detuning between the stokes field and the pump field @xmath80 for this model , without reactive coupling ( @xmath23=0 ) and with it ( @xmath74 mhz / nm ) , for different pump powers are presented in fig . we can see that @xmath84 for @xmath23=0 . the reason is that the dispersive coupling constant @xmath22 is too small . however , for @xmath74 mhz / nm , @xmath83 is not equal to zero . this shows that the optomechanical system can generate an anti - stokes field with frequency @xmath85 due to the reactive coupling . for pump power @xmath81 @xmath68w , the maximum gain defined with reference to the input stokes power for the anti - stokes field is about 0.1 . in conclusion , we have observed normal mode splitting of output fields due to reactive coupling between the waveguide and the cavity . meanwhile , the separation of the peaks increases for larger pump powers . further , the reactive coupling can also cause four - wave mixing , which creates an anti - stokes component generated by the optomechanical system . 99 m. li , w. h. p. pernice , and h. x. tang , phys . lett . * 103 * , 223901 ( 2009 ) . f. elste , s. m. girvin , and a. a. clerk , phys . . lett . * 102 * , 207209 ( 2009 ) . m. li , w. h. p. pernice , and h. x. tang , nat . photon . * 3 * , 464 ( 2009 ) . m. j. hartmann and m. b. plenio , phys * 101 * , 200503 ( 2008 ) . m. bhattacharya and p. meystre , phys . rev . lett . * 99 * , 073601 ( 2007 ) . s. bose , k. jacobs , and p. l. knight , phys . a * 56 * , 4175 ( 1997 ) . s. huang and g. s. agarwal , new j. phys . * 11 * , 103044 ( 2009 ) . m. paternostro , d. vitali , s. gigan , m. s. kim , c. brukner , j. eisert , and m. aspelmeyer , phys . * 99 * , 250401 ( 2007 ) . d. vitali , s. gigan , a. ferreira , h. r. bhm , p. tombesi , a. guerreiro , v. vedral , a. zeilinger , and m. aspelmeyer , phys . * 98 * , 030405 ( 2007 ) . f. marquardt , j. p. chen , a. a. clerk , and s. m. girvin , phys . lett . * 99 * , 093902 ( 2007 ) . j. m. dobrindt , i. wilson - rae , and t. j. kippenberg , phys . * 101 * , 263602 ( 2008 ) . s. grblacher , k. hammerer , m. vanner , and m. aspelmeyer , nature ( london ) * 460 * , 724 ( 2009 ) . s. huang and g. s. agarwal , phys . a * 81 * , 033830 ( 2010 ) . [ this paper deals exclusively with designs where only dispersive optomechanical coupling occurs . ] w. h. p. pernice , m. li , and h. x. tang , opt . express * 17 * , 1806 ( 2009 ) . d. f. walls and g. j. milburn , _ quantum optics _ ( springer - verlag , berlin , 1994 ) . g. s. agarwal and s. huang , phys . rev . a * 81 * , 041803(r ) ( 2010 ) ; p. anisimov and o. kocharovskaya , j. mod . opt . * 55 * , 3159 ( 2008 ) .

we study the optomechanical design introduced by m. li _ et al . _ [ phys . rev . lett . * 103 * , 223901 ( 2009 ) ] , which is very effective for investigations of the effects of reactive coupling . we show the normal mode splitting that is due solely to reactive coupling rather than due to dispersive coupling . we suggest feeding the waveguide with a pump field along with a probe field and scanning the output probe for evidence of reactive - coupling - induced normal mode splitting .

ngc6712 is a small ( tidal radius @xmath10 ) and relatively loose ( @xmath11 ) and faint ( @xmath12 ; djorgovski @xcite ) galactic globular cluster ( gc ) that has not yet received much observational attention . its main claim to fame so far is due to the presence in its core of the high luminosity x - ray burster x1850086 whose optical counterpart may be a faint uv - excess object ( anderson et al . this fact presents somewhat of a puzzle since one would expect such an x - ray source to be located in a highly concentrated cluster where the stellar density favors its formation via tidal capture of a neutron star ( hertz & grindlay @xcite ) . most other sources of this type have indeed been found in high density core collapse clusters suggesting that , perhaps , ngc6712 has already undergone such an event in the past and is now in a state of re - expansion ( grindlay et al . @xcite ) . this unusual situation may also be connected in some way to its galactic orbit as computed recently by dauphole et al.(@xcite ) that is fairly well restricted to the vicinity of the disk and penetrates very deeply in the galactic bulge . this certainly means that one would expect this cluster to have undergone severe tidal shocking during the numerous encounters with both the disk and the bulge during its lifetime and the consequences on the dynamical status of the cluster to be significant and observable . a simple single - mass approximation of these effects was computed by gnedin & ostriker ( @xcite ) for both disk and bulge shocks under differing assumptions on the galactic model with a resultant time to destruction as small as @xmath13h@xmath14 . according to these calculations , then , the cluster should have evaporated long ago and at the very least may have lost a very substantial portion of its original mass during its lifetime . clearly , this catastrophe should be well impressed on its present day distribution of stars on the main sequence ( ms ) with its lowest mass members beyond the half - mass radius particularly vulnerable to escape . this effect may well have been detected already in m4 , another cluster at significant risk of tidal disruption ( kanatas et al . @xcite ; pulone et al . @xcite ) , but until this cluster s structural parameters are pinned down more accurately this remains still speculative . there is , therefore , much interest today in determining accurately the present day mass function ( pdmf ) of ngc6712 to look for the signature of such powerful effects . currently available observations of the color magnitude diagram ( cmd ) of this cluster , however , only reach to just above the ms turn - off ( cudworth @xcite ; anderson et al . @xcite ) and are , therefore , of limited use for this task . in order to push the observations well into the relevant part of the ms below the turn - off ( to ) , the vlt was used to probe deeply into this cluster with its unprecedented sensitivity and resolution . this paper describes the first results of these observations that give clear evidence that there is indeed a distortion of the mf of ngc6712 with respect to that of other dynamically much less disturbed clusters . the observational data used in this paper were collected during the science verification ( sv ; leibundgut & renzini @xcite ) phase of the first 8m - diameter very large telescope ( vlt ) at eso , using the vlt test camera ( vlt - tc ) . readers interested in the vlt and its instruments should consult eso s world - wide web at http://www.eso.org/paranal , while the scope of the vlt science verification is described in leibundgut , de marchi , & renzini ( @xcite ) . images of the globular cluster ngc6712 were taken with the vlt - tc in the v and r bands . with a @xmath15 square pixel detector and a plate scale of @xmath16pixel@xmath17 , the vlt - tc offers a field of view of @xmath18 . sv observations , however , were obtained with an electronically enforced @xmath19 binning of the ccd , so that the actual size of each pixel in these images is of @xmath20 on a side . observations of four regions of the cluster are available , located between one and two times the half - light radius ( @xmath21 ; djorgovski @xcite ) . the coordinates ( j2000 ) of the center of each field are given in table1 along with the total exposure time in each band . fields f1 and f2 were observed during the night of 1998 aug 23 , and f3 and f4 on 1998 aug 27 . .vlt sv observations of ngc6712 [ cols= " < , < , < , > , > , > " , ] the lfs measured in this way and corrected for photometric incompleteness are shown in figure3 as a function of the r - band magnitude . the lfs have been registered through a vertical shift in the logarithmic plane by imposing a least square fit in the range @xmath22 . the error bars associated with each point reflect the poisson statistics of the counting process ( and include the correction for incompleteness ) . these lfs can be directly compared to one another as they have all been measured at the same radial distance from the center ( @xmath23 or @xmath24 times the half - light radius @xmath25 ) . they show the same overall trend , i.e. an increase with decreasing luminosity up to a peak at @xmath26 ( close to the to luminosity ) , and from there they all flatten out and possibly drop with decreasing luminosity even after the incompleteness of our photometry has been accounted for . and indeed , to ensure that our lfs are robust , we have not included in figure3 any datapoints whose associated photometric completeness is worse than @xmath27 . stars brighter than @xmath28 have already evolved off the ms and , therefore , their lf provides no information on the underlying mf without uncertain corrections for evolution ( scalo @xcite ) . moreover , because of saturation at the bright end of our cmds , the brightest portion of our lfs is uncertain . for cluster stars which are still on their ms , however , the lfs in figure3 directly reflect the pdmf of the local population and immediately indicate a relative deficiency of low mass objects with respect to the stars with the to mass ( @xmath29m@xmath7 , ) , as we discuss below . indeed , the most important conclusion that one can draw from figure3 is that the shape of the lfs completely deviates from that of any other gc for which relatively deep photometric data are available near the half - mass radius . observations carried out with the wfpc2 on board the hst over the past few years ( paresce , de marchi & romaniello @xcite ; cool , piotto , & king @xcite ; elson et al . @xcite ; de marchi & paresce @xcite ; piotto , cool , & king @xcite ; pulone et al . @xcite ; king et al . @xcite ; de marchi @xcite ) have consistently revealed lfs that , near the cluster half - mass radius , increase with decreasing luminosity from the to magnitude all the way down to about @xmath30 ( @xmath31m@xmath7 , ) where they flatten out and drop at fainter luminosities . inverted lfs such as those shown in figure3 have been observed right in the core of high density gcs ( 47tuc , ngc6397 , m15 ) but in those cases a simple isothermal model of a cluster in equilibrium can easily explain this effect as being due to mass segregation ( paresce , de marchi , & jedrzejewski @xcite ; king , sosin , & cool @xcite ; de marchi & paresce @xcite ) . more complete multi - mass king michie models show , however , that thermal relaxation is much less efficient ( if at all ) at depleting low - mass stars near the half - mass radius ( see pulone , de marchi , & paresce @xcite ) , and we can not therefore trace the origin of the lfs that we observe back to the effects of mass segregation alone . to make it easier to compare the lf of ngc6712 with that of other clusters , we display it in figure4 as a function of the absolute r - band magnitude , assuming @xmath32 and @xmath33 or @xmath34 ( djorgovski @xcite ) . rather than showing the three individual lfs , we have combined them together into one single function by averaging their values in each magnitude bin , and have taken the standard deviation as a measure of the associated uncertainty ( error bars ) . the dashed line shown in figure4 corresponds to the lf of the low - metallicity cluster ngc6397 as measured by king et al . ( 1998 ) , while the dot - dashed line reproduces the lf of the metal rich cluster 47tuc from hesser et al . ( @xcite ) . both lfs have been translated into the r - band by using the m l relationship of baraffe et al . ( @xcite ) for the appropriate metallicity , i.e. the magnitude corresponding to each observed point in the lf has been converted into a mass which has then been used to read the corresponding magnitude in the r band from the appropriate m - l relation . the size of each magnitude bin has also been rescaled to reflect the difference in the slopes of the m - l relationships for different bands . we have selected ngc6397 and 47tuc as they both have accurate lf measurements at and below the to luminosity , where we have normalized them to our observations , and because the metal content of these clusters nicely brackets that of ngc6712 ( [ fe / h]@xmath35 ; zinn & west @xcite ) . it should , nevertheless , be clear that , due the uncertainties in the theoretical m - l relations and in the observed lfs , our comparison will only provide an indication of the true differences . the difference between these two lfs and that of ngc6712 is striking . while the lfs of ngc6397 , measured at @xmath36 , shows a steep increase starting from the to , the lf of ngc6712 sampled at @xmath37 slowly drops from the to all the way to the detection limit at @xmath38 . we would like to point out that the discrepancy is so large that to bring the two lfs into agreement would require us to have underestimated the photometric incompleteness by a factor of @xmath39 . the same reasoning holds true for the lf of 47tuc , which has been measured at @xmath40 . this difference must thus be physical and reflect the properties of the local stellar population . under the simple assumption that the mf should be represented by an exponential distribution in the mass range @xmath41m@xmath7 , ( a reasonable hypothesis given the narrow mass range ) , we have used the m l relationship of baraffe et al . ( @xcite ) appropriate for the metallicity of ngc6712 to reproduce the observed lf . we obtain a fairly reasonable fit to the observations with a power - law distribution of the type @xmath42 ( salpeter s imf would be @xmath43 ) , in which the number of objects decreases with mass ( solid line in figure4 ) . richer et al . ( @xcite ) and , more recently , de marchi & paresce ( 1997 ) , vesperini & heggie ( @xcite ) , and pulone et al . ( @xcite ) have convincingly shown that near the cluster half - mass radius the lf should closely reflect the imf , as dynamical modifications should leave these regions almost untouched . in fact , the internal relaxation mechanism governed by energy equipartition through two- and three - body encounters mostly affects the region within a few core radii , while the interaction with the galactic tidal field is expected to simply speed up the evaporation of light stars near the tidal boundary , but none of these processes should , in principle , significantly alter the properties of stars located close to the much safer half - mass radius area . if this were true for ngc6712 as well , one should conclude that this cluster is the only one so far to feature an inverse imf ( increasing with mass ) that has not been observed in any other environment . while this hypothesis can not be safely ruled out , there are far better reasons to believe that ngc6712 might have experienced a much stronger interaction with the galaxy than any other of the clusters studied so far . and indeed , with a perigalactic distance smaller than 300pc this cluster ventures so frequently and so deeply into the galactic bulge ( dauphole et al . @xcite ) that it is likely to have undergone severe tidal shocking during the numerous encounters with both the disk and the bulge during its lifetime . the latest galactic plane crossing could have happened as recently as @xmath44year ago ( cudworth @xcite ) which is much smaller than its half - mass relaxation time ( @xmath45yr ) . such an event might have imparted strong modifications on the mass distribution not only of the stars in the cluster periphery but also well into its innermost regions , perhaps even reaching the core where it could have triggered a premature collapse because of tidally induced relaxation ( see kundi & ostriker @xcite and gnedin & ostriker @xcite for a detailed description of this mechanism ) . as a result of such a catastrophe , it would be surprising if the present - day mf were still to bear any memory of its parent imf anywhere in the cluster , including the half - mass radius region . vesperini & heggie ( @xcite ) have estimated that these effects would substantially decrease the slope of a simple power law mf , much in the same way as we are observing here . we , therefore , conclude that the vlt has revealed the consequences of the strong tidal stripping that the galaxy ( and particularly its bulge ) exerts on gcs orbiting close to the center , and which might have contributed to the destruction of an initially much more numerous population of gcs ( aguilar , hut , & ostriker @xcite ; vesperini @xcite ) . although kanatas et al . ( @xcite ) and piotto et al . ( @xcite ) had speculated that similar events could have happened respectively in m4 and ngc6397 , the result that we show here is the first , clear , unambiguous detection of this mechanism . to characterize the strength and extent of these phenomena more accurately would require the investigation of the ms population outside the half - mass radius in many more clusters , and possibly at larger distance from the galactic center , so as to probe the intensity of the stripping process as a function of the depth of the galactic potential well . if the z component of the space velocity of ngc6712 is indeed appropriate for a halo cluster , as suggested by cudworth ( @xcite ) , then this violent stripping process might not be restricted only to objects orbiting the innermost galactic regions . grindlay , j. , bailyn , c. , mathieu , r. , & latham , d. 1988 , in the harlow shapley symposium on globular cluster systems in galaxies , iau symp . j. grindlay & a. davis philip ( dordrecht : kluwer ) , 659

the vlt on cerro paranal was used to observe four fields located at @xmath0 from the center of the galactic globular cluster ngc6712 in the v and r bands . the resulting color - magnitude diagram shows a well defined main sequence reaching down to the @xmath1 detection limit at @xmath2 , @xmath3 or approximately 4 magnitudes below the main sequence turn - off , the deepest obtained so far on this cluster . this yields a main sequence luminosity function that peaks at @xmath4 and drops down to the 50% completeness limit at @xmath5 . transformation to a mass function via the latest mass - luminosity relation appropriate to this object indicates that the peak of the luminosity function corresponds to @xmath6m@xmath7 , a value significantly higher than the @xmath8m@xmath7 , measured for most other clusters observed so far . since this object , in its galactic orbit , penetrates very deeply into the galactic bulge with perigalactic distance of @xmath9kpc , this result is the first strong evidence that tidal forces have stripped this cluster of a substantial portion of its lower mass star population all the way down to its half - light radius and possibly beyond .

quantum information processing ( qip ) often requires pure state as the initial state @xcite . shor s prime factorizing algorithm @xcite , grover search algorithm @xcite are few examples . creation of pure state in nmr is not easy due to small gaps between nuclear magnetic energy levels and demands unrealistic experimental conditions like near absolute zero temperature or extremely high magnetic field . this problem has been circumvented by creating a pseudo pure state ( pps ) . while in a pure state all energy levels except one have zero populations , in a pps all levels except one have equal populations . since the uniform background populations do not contribute to the nmr signal , such a state then mimics a pure state . several methods of creating pps have been developed like spatial averaging @xcite , logical labeling @xcite , temporal averaging @xcite , spatially averaged logical labeling technique ( sallt ) @xcite . however pseudo pure state , as well as pure states are not stationary and are destroyed with time as the spin system relaxes toward equilibrium . in qip there are also cases where one or more qubits are initialized to a suitable state at the beginning of the computation and are used as storage or memory qubits at the end of the computation performed on some other qubits@xcite . in these cases it is important for memory qubits to be in the initialized state till the time they are in use since deviation from the initial state adds error to the output result . since it is not possible to stop decay of a state which is away from equilibrium , alternate strategies like quantum error correction @xcite , noiseless subspace @xcite are being tried . recently sarthour et al.@xcite has reported a detailed study of relaxation of pseudo pure states and few other states in a quadrupolar system . here we experimentally examine the lifetime of various pseudo pure states in a weakly j - coupled two qubit system . we find that cross terms ( known as cross - correlation ) between different pathways of relaxation of a spin can retard the relaxation of certain pps and accelerate that of others . + + in 1946 bloch formulated the behavior of populations or longitudinal magnetizations when they are perturbed from the equilibrium @xcite . the recovery toward equilibrium is exponential for a two level system and for a complex system the recovery involves several time constants @xcite . for complex systems the von neumann - liouville equation @xcite describes mathematically the time evolution of the density matrix in the magnetic resonance phenomena . for system having more than one spin the relaxation is described by a matrix called the relaxation matrix whose elements are linear combinations of spectral densities , which in turn are fourier transforms of time correlation function @xcite of the fluctuations of the various interactions responsible for relaxation . there exist several different mechanisms for relaxation , such as , time dependent dipole - dipole(dd ) interaction , chemical shift anisotropy(csa ) , quadrupolar interaction and spin rotation interaction @xcite . the correlation function gives the time correlations between different values of the interactions . the final correlation function has two major parts , namely the ` auto - correlation ' part which gives at two different times the correlation between the same relaxation interaction and the ` cross - correlation ' part which gives the time correlation between two different relaxation interactions . the mathematics of cross correlation can be found in detail , in works of schneider @xcite , blicharski @xcite and hubbard @xcite . recently a few models have been suggested to study the decoherence of the quantum coherence , the off - diagonal elements in density matrix @xcite . it can be shown that in absence of r.f . pulses and under secular approximation the relaxation of the diagonal and the off - diagonal elements of the density matrix are independent @xcite . here we study the longitudinal relaxation that is the relaxation of the diagonal elements of the density matrix and the role of cross - correlations in it . in terms of magnetization modes the equilibrium density matrix of a two spin system is given by @xcite[fig.[eqlev ] ] , @xmath0 where @xmath1 and @xmath2 are gyro - magnetic ratios of the two spins @xmath3 and @xmath4 respectively . the density matrix of a general state can be written as , @xmath5 \label{general}\end{aligned}\ ] ] which for the condition @xmath6=@xmath7=@xmath8=k , corresponds to the density matrix of a pps given by @xcite , @xmath9 \label{pps}\end{aligned}\ ] ] where , k is a constant , the value of which depends on the method of creation of pps . + the first two terms in the right hand side in eq.[general ] and eq.[pps ] are the single spin order modes for the first and second spin respectively while the last term is the two spin order mode of the two spins @xcite . choosing properly the signs of the modes , the various pps of a two - qubit system are , @xmath10\nonumber \\ \chi_{pps}^{01 } = k[- i_{1z } + i_{2z } + 2i_{1z}i_{2z } ] \nonumber \\ \chi_{pps}^{10 } = k[+ i_{1z } - i_{2z } + 2i_{1z}i_{2z}]\nonumber \\ \chi_{pps}^{11 } = k[+ i_{1z } + i_{2z } -2i_{1z}i_{2z}]\end{aligned}\ ] ] the relative populations of the states for different pps are shown in fig . [ ppslev ] . as seen in eq.2 , in pps the coefficients of the all three modes are equal . on the other hand equilibrium density matrix does not contain any two spin order mode . to reach eq.[pps ] starting from eq.[eqd ] , the two spin order mode has to be created and at the same time the coefficients of all the modes have to be made equal . the equation of motion of modes m is given by @xcite , @xmath11 \label{magmode}\end{aligned}\ ] ] where @xmath12 is the relaxation matrix and @xmath13 is the equilibrium values of a mode . for a weakly coupled two - spin system relaxing via mutual dipolar interaction and the csa relaxation , the two dominant mechanism of relaxation of spin half nuclei in liquid state , the above equation takes the form , @xmath14 = \left [ \begin{array}{ccc } \rho_1 & \sigma_{12 } & \delta_{1,12}\\ \sigma_{12 } & \rho_2 & \delta_{2,12}\\ \delta_{1,12 } & \delta_{2,12 } & \rho_{12 } \end{array } \right ] \cdot \left [ \begin{array}{c } i_{1z}(0)-i_{1z}(\infty)\\ i_{2z}(0)-i_{2z}(\infty)\\ 2i_{1z}i_{2z } \end{array } \right ] \label{relax}\end{aligned}\ ] ] where @xmath15 is the self relaxation rate of the single spin order mode of spin @xmath16 , @xmath17 is the self relaxation rate of the two spin order mode of spin @xmath16 and @xmath18 , @xmath19 is the cross - relaxation ( nuclear overhouser effect , noe ) rate between spins @xmath16 and @xmath18 and @xmath20 is the cross - correlation term between csa relaxation of spin @xmath16 and the dipolar relaxation between the spins @xmath16 and @xmath18 . @xmath21 and @xmath22 involve only the auto - correlation terms and @xmath23 involves only the cross - correlation terms@xcite . magnetization modes of one order relaxes to other orders through cross - correlation and in absence of it the relaxation matrix becomes block diagonal within each order . the relaxation of modes are in general dominated by their self relaxation @xmath21 , but in case of samples having long @xmath24 , the cross - correlation terms become comparable with self - relaxation and play an important role in relaxation of the spins . + the formal solution of eq . [ magmode ] is given by , @xmath25 exp(-\hat{\gamma}t)\end{aligned}\ ] ] as time evolution of various modes are coupled , a general solution of the above equation requires diagonalization of the relaxation matrix . however , in the initial rate approximation eq.7 can be written ( for small values of t=@xmath26 ) as , @xmath27[1-\hat{\gamma}\tau ] \cr & = & \vec{m}(0 ) - \hat{\gamma}\tau[\vec{m}(0)-\vec{m}(\infty ) ] \label{initialapp}\end{aligned}\ ] ] this equation asserts that in the initial rate approximation ( for low @xmath26 ) , the decay or growth of a mode is linear with time and the initial slope is proportional to the corresponding relaxation matrix element . if the modes are allowed to relax for a longer time , their decay or growth deviates from the linear nature and adopts a multi - exponential behavior to finally reach the equilibrium@xcite . let a two qubit system be in @xmath28 pps at t=0 . @xmath29 \end{aligned}\ ] ] after time t it will relax to , @xmath30 where @xmath31,@xmath32 and @xmath33 are the time dependent deviations of respective modes from their initial values . the deviation of the two spin order can be measured from spectrum of either spin . eq.[e10 ] can also be written as , @xmath34 + ( \delta_1(t ) - \delta_{12}(t))i_{1z } + ( \delta_2(t ) - \delta_{12}(t))i_{2z } \label{chi001 } \label{chi00}\end{aligned}\ ] ] the first term is the pseudo pure state with the coefficient decreasing in time while the other two terms are the excesses of the single spin order modes with coefficients increasing in time . for other pseudo pure states eq.[chi001 ] becomes , @xmath35 + ( \delta_1(t ) + \delta_{12}(t))i_{1z } + ( \delta_2(t ) - \delta_{12}(t))i_{2z } \\ \chi^{10}(t ) & = & ( k + \delta_{12}(t))[i_{1z } - i_{2z } + 2i_{1z}i_{2z } ] + ( \delta_1(t ) - \delta_{12}(t))i_{1z } + ( \delta_2(t ) + \delta_{12}(t))i_{2z } \\ \chi^{11}(t ) & = & ( k - \delta_{12}(t))[i_{1z } + i_{2z } - 2i_{1z}i_{2z } ] + ( \delta_1(t ) + \delta_{12}(t))i_{1z } + ( \delta_2(t ) + \delta_{12}(t))i_{2z } \label{chi11}\end{aligned}\ ] ] in the initial rate approximation ( using eq.[initialapp ] ) we obtain for the @xmath28 pps , @xmath36 \label{d1}\\ \delta_2(\tau ) & = & \tau[\sigma_{12}(\gamma_1 -k ) + \rho_{1}(\gamma_2 -k ) -k \delta_{2,12 } ] \\ \delta_{12}(\tau ) & = & \tau[\delta_{1,12}(\gamma_1 -k ) + \delta_{2,12}(\gamma_2 -k ) -k \rho_{12 } ] \label{d2}\end{aligned}\ ] ] let the coefficients of the pps term and the two single spin order modes @xmath37 and @xmath38 in eq.[chi00 ] be called as @xmath39,@xmath40 and @xmath41 respectively . fig.[ppsmode ] schematically shows the time evolution of the coefficients @xmath39,@xmath40 and @xmath41 for @xmath28 pps . any coefficient for any pps at any instant , is simply the initial value plus the total deviation due to the auto and the cross - correlations . for example , @xmath42 for @xmath28 pps at time @xmath26 , is @xmath43 ) , where k is the initial value , and @xmath44 and @xmath45 are the deviations at @xmath26 due to auto - correlation and cross - correlation parts respectively . putting the values of the deviations of different modes obtained from eq.([d1]-[d2 ] ) in eq.([chi001]-[chi11 ] ) , we obtain the contribution only of auto - correlation terms to the deviation from initial value of the coefficients @xmath46,@xmath40 and @xmath41 under initial rate approximation ( at t=@xmath26 ) as , @xmath47 @xmath48\tau & ; & { \mathcal{b}}_{auto}^{01}(\tau ) = [ \rho_{1}(\gamma_1 + k)+ \sigma_{12 } ( \gamma_2 -k ) - k \rho_{12}]\tau \nonumber \\ { \mathcal{b}}_{auto}^{10}(\tau ) = [ \rho_{1}(\gamma_1 -k)+ \sigma_{12 } ( \gamma_2 + k ) + k \rho_{12}]\tau & ; & { \mathcal{b}}_{auto}^{11}(\tau ) = [ \rho_{1}(\gamma_1 -k)+ \sigma_{12 } ( \gamma_2 -k ) + k \rho_{12}]\tau \nonumber \\ { \mathcal{c}}_{auto}^{00}(\tau ) = [ \sigma_{12}(\gamma_1 -k)+ \rho_2 ( \gamma_2 -k ) + k \rho_{12}]\tau & ; & { \mathcal{c}}_{auto}^{01}(\tau ) = [ \sigma_{12}(\gamma_1 + k)+ \rho_2 ( \gamma_2 -k ) + k \rho_{12}]\tau \nonumber \\ { \mathcal{c}}_{auto}^{10}(\tau ) = [ \sigma_{12}(\gamma_1 -k)+ \rho_2 ( \gamma_2 + k ) - k \rho_{12}]\tau & ; & { \mathcal{c}}_{auto}^{11}(\tau ) = [ \sigma_{12}(\gamma_1 -k)+ \rho_2 ( \gamma_2 -k ) + k \rho_{12}]\tau\end{aligned}\ ] ] it is evident that in absence of cross - correlations the @xmath28 and @xmath49 pps relax at the same initial rate since @xmath50 , @xmath51 and @xmath52 . however the same is not true for @xmath53 and @xmath54 pps . the contribution by the cross - correlation terms is given by , @xmath55\tau \nonumber \\ { \mathcal{a}}^{01}_{cc}(\tau ) = + [ \delta_{1,12 } ( \gamma_1 + k)+ \delta_{2,12}(\gamma_2 -k)]\tau \nonumber \\ { \mathcal{a}}^{10}_{cc}(\tau ) = + [ \delta_{1,12 } ( \gamma_1 -k)+ \delta_{2,12}(\gamma_2 + k)]\tau \nonumber \\ { \mathcal{a}}^{11}_{cc}(\tau ) = -[\delta_{1,12 } ( \gamma_1 -k)+ \delta_{2,12}(\gamma_2 -k)]\tau \nonumber \\ \end{aligned}\ ] ] @xmath56\tau & ; & { \mathcal{b}}^{01}_{cc}(\tau ) = + [ \delta_{1,12 } \gamma_1 + \delta_{2,12}(\gamma_2 -k)]\tau \nonumber \\ { \mathcal{b}}^{10}_{cc}(\tau ) = - [ \delta_{1,12 } \gamma_1 + \delta_{2,12}(\gamma_2 + k)]\tau & ; & { \mathcal{b}}^{11}_{cc}(\tau ) = + [ \delta_{1,12 } \gamma_1 + \delta_{2,12}(\gamma_2 -k)]\tau\nonumber \\ { \mathcal{c}}^{00}_{cc}(\tau ) = - [ \delta_{1,12 } ( \gamma_1 -k ) + \delta_{2,12 } \gamma_2 ] \tau & ; & { \mathcal{c}}^{01}_{cc}(\tau ) = - [ \delta_{1,12 } ( \gamma_1 + k ) + \delta_{2,12 } \gamma_2 ] \tau\nonumber \\ { \mathcal{c}}^{10}_{cc}(\tau ) = + [ \delta_{1,12 } ( \gamma_1 -k ) + \delta_{2,12 } \gamma_2 ] \tau & ; & { \mathcal{c}}^{11}_{cc}(\tau ) = + [ \delta_{1,12 } ( \gamma_1 -k ) + \delta_{2,12 } \gamma_2 ] \tau \label{abc - cross } \end{aligned}\ ] ] the important thing is that the presence of cross - correlation can lead to differential relaxation of all pps . positive cross - correlation rates @xmath57 and @xmath58 , slow down the relaxation of all the three coefficients for @xmath28 pps since ( @xmath59 ) , while make the relaxation of all three coefficients faster for @xmath49 pps since ( @xmath60 ) . for @xmath53 and @xmath54 pps cross - correlations give a mixed effect since ( @xmath61 ) and ( @xmath62 ) . as the contributions of the auto - correlation part for @xmath28 and @xmath49 pps are equal , we have monitored the relaxation behavior only of @xmath28 and @xmath49 pps to study the effect of cross - correlations . + + for samples having long @xmath24 , where the cross - correlations becomes comparable with auto - correlation rates , the four pps relax with four different rates and the difference increases with the increased value of the cross - correlation terms . the three coefficients @xmath39,@xmath40 and @xmath41 ( normalized to the equilibrium line intensities ) in terms of proton and fluorine line intensities for @xmath28 pps are , @xmath63 @xmath64 and for @xmath49 pps are , @xmath65 @xmath66 where , @xmath67 and @xmath68 are intensities of the two proton transitions , when the fluorine spin is respectively in state @xmath69 and @xmath70 . similarly @xmath71 and @xmath72 are intensities of two fluorine transitions corresponding to the proton spin being respectively in the state @xmath69 and @xmath70 , as shown in fig.[eqlev ] and fig.[allspec ] . @xmath73 and @xmath74 give the @xmath67 line intensity respectively at time t and at equilibrium . thus by monitoring the intensities of the two proton and two fluorine transitions as a function of time , one can calculate the coefficient @xmath75 which is a measure of decay of pps . relaxation of the coefficients @xmath39,@xmath40 and @xmath41 have been simulated using matlab , for a weakly coupled @xmath76-@xmath77 system . the relaxation matrix used for the simulation is , @xmath78\end{aligned}\ ] ] fig.[asimu ] shows the decay of coefficient @xmath39 with time . @xmath79 and @xmath80 show no difference in decay rate in absence of cross - correlation rates . as @xmath57 and @xmath58 are increased more and more difference in decay rate is observed . fig.[bcsimu ] shows growth of coefficients @xmath40 and @xmath41 . as @xmath58 is taken smaller than @xmath57 , difference in decay rate between @xmath81 and @xmath82 is found to be less than between @xmath83 and @xmath84 . all the relaxation measurement were performed on a two qubit sample formed by one fluorine and one proton of 5-fluro 1,3-dimethyl uracil yielding an ax spin system with a j - coupling of 5.8 hz . longitudinal relaxation time constants for @xmath76 and @xmath85 are 6 and 7.2 sec respectively at room temperature ( 300k ) . all the experiments were performed in a bruker drx 500 mhz spectrometer where the resonance frequencies for @xmath76 and @xmath85 are 470.59 mhz and 500.13 mhz respectively . the pseudo - pure state was prepared by spatial averaging method using j - evolution @xcite . relaxation of all the three coefficients for @xmath28 and @xmath49 pps has been calculated . since auto - correlations contribute equally to the relaxation of these two pps , any difference in relaxation rate can be attributed to cross - correlation rates . + sample temperature was varied to change the correlation time and hence the cross - correlation rate @xmath23 . four different sample temperatures , 300k , 283k , 263k and 253k were used . fig.[allspec ] shows the proton and fluorine spectra obtained using recovery measurement at four different temperatures . the spectra correspond to the initial pps state and that after an interval of 2.5 sec . fig.[fht1 ] shows the longitudinal relaxation times ( @xmath24 ) of fluorine and proton as function of temperature obtained from initial part of inversion - recovery experiment . a steady decrease in @xmath24 with decreasing temperature indicates that the dynamics of the sample molecule is in the short correlation time limit @xcite . in this limit auto as well as cross - correlations increase linearly with decreasing temperature . + all the spectra were fitted to bi - lorentzian lines in matlab and various parameters were extracted using the origin software . fig.[aplot ] shows the decay of the coefficient @xmath39 calculated independently from proton and fluorine spectra . at 300k , @xmath79 and @xmath80 showed almost same rate of decay . as the temperature was gradually lowered , a steady increase in difference in decay rate was observed . this is due to the steady increase in cross - correlation rates with decreasing temperature , which is expected in the short correlation time limit . in fig.[bcplot ] the growths of the coefficients @xmath40 and @xmath41 are shown . similar to the coefficient @xmath39 , coefficients @xmath40 and @xmath41 also show differences in decay rate between @xmath28 and @xmath49 pps at lower temperatures . the difference between @xmath83 and @xmath84 at any temperature was found to be larger compared to between @xmath81 and @xmath82 . this is expected since , according to eq.[abc - cross ] the dominant cross - correlation factor in @xmath83 and @xmath84 is @xmath57 which is the cross - correlation between csa of fluorine with fluorine - proton dipolar interaction whereas in @xmath81 and @xmath82 the dominant factor is @xmath58 which is cross - correlation between csa of proton , which is much less than fluorine , with fluorine - proton dipolar interaction . thus it is found that at lower temperatures the @xmath28 pps decays slower than the @xmath49 pps . the dominant difference in the decay rates arises from the cross - correlations between the csa of the fluorine and the dipolar interaction between the fluorine and the proton spin . to the best of our knowledge this is the first study of its kind where the differential decay of the pps has been attributed to cross - correlations . we have demonstrated here that in samples having long @xmath24 cross - correlations plays an important role in determining the rate of relaxation of pseudo pure state . in qip sometimes one or more qubits having comparatively longer longitudinal relaxation are used as storage or memory qubits . recently levitt et al . have demonstrated a long living antisymmetric state arrived by shifting the sample from high to very low magnetic field , suggesting that this long living state could be used as memory qubit @xcite . in such cases fidelity of computation depends on how much the memory qubits have been deviated from the initialized state at the beginning of the computation till the time they are actually used . theoretically it is shown here that in presence of cross - correlations , all the four pps relax with different initial rates . for positive cross - correlations the @xmath28 pps relaxes significantly slower than @xmath49 pps . it is therefore important to choose a proper initial pseudo pure state according to the sample . we gratefully acknowledge prof . k. v. ramanathan for discussions and mr . rangeet bhattacharyya for his help in data processing . the use of drx-500 high resolution liquid state spectrometer of the sophisticated instrument facility , indian institute of science , bangalore , funded by department of science and technology ( dst ) , new delhi , is gratefully acknowledged . ak acknowledges `` dae - brns '' for `` senior scientist scheme '' , and dst for a research grant . ( a ) chemical structure of 5-fluro 1,3-dimethyl uracil . the fluorine and the proton spins ( shown by circles ) are used as the two qubits @xmath3 and @xmath4 respectively . ( b ) the energy level diagram of a two qubit system identifying the four states 00,01,10 and 11 . under high temperature and high field approximation @xcite the relative equilibrium deviation populations are indicated in the bracket for each level . assuming this to be a weakly coupled two spin system the deviation populations become proportional to the gyromagnetic ratios @xmath1 and @xmath2 . @xmath86 refers to the transition of the @xmath87 spin when the other spin is in state @xmath88 . thus @xmath67 means the proton transition when the fluorine is in state @xmath69 . + + figure 2 . population distribution of different energy levels of a two spin system in different pseudo - pure states . k is a constant whose value depends on the protocol used for the preparation of pps . ( a),(b),(c ) and ( d ) show respectively the @xmath28,@xmath53,@xmath54 and @xmath49 pps . + + figure 3 . schematic representation of decay of the coefficient @xmath39 and growth of the coefficients @xmath40 and @xmath41 . the magnetization modes are normalized to their respective equilibrium values . in each sub - figure the three bars correspond to the modes @xmath37,@xmath38 and 2@xmath89 from left to right . the amount of any mode present at any time is directly proportional to the height of the corresponding bar . the numbers provided in the rightmost column represent typical values of the modes . ( a ) thermal equilibrium . at thermal equilibrium only @xmath37 and @xmath38 exist . ( b ) @xmath28 pseudo - pure state just after creation , where all the three modes are equal in magnitude . for @xmath28 pps all modes are of same sign but this is not the case for other pps [ eq.3 ] . coefficient @xmath39 is the common equal amount of all the modes and it is maximum at t=0 . ( c ) the amount of magnetization modes ( schematic ) at time @xmath26 , after preparation of the pps at t=0 . the two single spin order modes increase and the two spin order mode decreases from their initial values . ( d ) the state of various modes at time @xmath26 , ( same as fig.c ) redrawn with filled bar to indicate the residual value of @xmath39 . all the three coefficients @xmath39,@xmath40 and @xmath41 are shown . @xmath39 ( shown by the filled bar ) , which is the measure of the pps , has come down by the same amount as the two spin order . @xmath40 ( shown by the empty bar ) and @xmath41 ( shown by the striped bar ) are the residual part of the single spin order modes @xmath37 and @xmath38 respectively.(e ) the values of various modes and coefficients after a delay @xmath90 . + + simulation of decay of coefficient @xmath39 . the boxes ( @xmath91 ) and circles ( @xmath92 ) correspond to the @xmath49 and @xmath28 pps respectively . in each plot deviation from initial value ( @xmath93)has been plotted . + + figure 5 . simulation of growth of coefficient @xmath40 and @xmath41 . the boxes ( @xmath91 ) and circles ( @xmath92 ) correspond to the @xmath49 and @xmath28 pps respectively . + + figure 6 . relaxation of pseudo pure state as monitored on ( a ) fluorine spin and ( b ) proton spin of the 5-fluro 1,3-dimethyl uracil at four different temperatures . the top row in ( a ) and ( b ) show the equilibrium spectrum at each temperature . with decrease in temperatures the lines broaden due to decreased @xmath94 . the second row in ( a ) and ( b ) show the spectra corresponding to the @xmath28 pps , prepared by spatial averaging method using j - evolution . the state of pps was measured by @xmath95 pulse at each spin . the third row in ( a ) and ( b ) show the spectra after an interval of 2.5 seconds after creation of the @xmath28 pps . the fourth row shows the spectra immediately after creation of @xmath49 pps and the fifth row , the spectra after 2.5 seconds . + + + figure 7 . longitudinal relaxation time @xmath24 of fluorine ( a ) and proton ( b ) as function of temperature , measured from the initial part of inversion recovery experiment for each spin . + + figure 8 . the deviation from initial value ( at t=0 ) of the coefficient @xmath39 of the pps term calculated from proton ( left column ) and fluorine ( right column ) at four different sample temperature . the empty ( @xmath92 ) and filled ( @xmath96 ) circles correspond to the @xmath28 and @xmath49 pps respectively . + + figure 9 . the growth of the coefficients @xmath40 and @xmath41 at different sample temperatures . @xmath40 was calculated from fluorine spectrum while @xmath41 was calculated from the proton spectrum . the empty ( @xmath92 ) and filled ( @xmath96 ) circles correspond to the @xmath28 and @xmath49 pps respectively . + + 99 j. preskill , lecture notes for physics 229 : quantum information and computation , http://theory.caltech.edu / people / preskill/. m.a . nielsen and i.l . chuang , quantum computation and quantum information , cambridge university press 2000 . p.w.shor , polynomial - time algorithms for prime factorization and discrete algorithms on quantum computer , siam rev . 41 ( 1999 ) 303 - 332 . grover , quantum mechanics helps in searching for a needle in a haystack , phys . 79 ( 1997 ) 325 . cory , a.f . fahmy and t.f . havel , ensemble quantum computing by nmr spectroscopy , proc.natl.acad.sci . usa , 94 ( 1997 ) 1634 . cory , m. d. price and t.f . havel , nuclear magnetic resonance spectroscopy : an experimentally accessible paradigm for quantum computing , physica d , 120 ( 1998 ) 82 . n. gershenfeld and i.l . chuang , bulk spin - resonance quantum computation , science , 275 ( 1997 ) 350 . chuang , n. gershenfeld , m.g . kubines and d.w . leung , bulk quantum computation with nuclear magnetic resonance , proc.roy.soc.lond . a , 454 ( 1998 ) 447 - 467 . kavita dorai , arvind and anil kumar , implementing quantum - logic operations , pseudopure states , and the deutsch - jozsa algorithm using noncommuting selective pulses in nmr , phys . a. 61 ( 2000 ) 042306 . kavita dorai , t.s.mahesh , arvind and anil kumar , quantum computations by nmr , current science . 79 ( 2000 ) 1447 - 1458 . e. knill , i. l. chuang and r. laflamme , effective pure states for bulk quantum computation , phys . a. 57 ( 2000 ) 3348 . mahesh and anil kumar , ensemble quantum - information processing by nmr : spatially averaged logical labeling technique for creating pseudopure states , phys . a. 64 ( 2001 ) 012307 . d. gottesman and i.l.chuang , demonstrating the viability of universal quantum computation using teleportation and single - qubit operations , nature ( london ) . 402 ( 1999 ) 390 . e.knill and r. laflamme , theory of quantum error correcting codes , phys . a. 55 ( 1997 ) 900 - 911 . e.knill , r. laflamme and l. viola , theory of quantum error correction for general noise , phys . 84 ( 2000 ) 2525 . l. viola , e. m. fortunato , m. a. pravia , e.knill , r. laflamme and d. g. cory , experimental realization of noiseless subsystems for quantum information processing , science . 293 ( 2001 ) 2059 - 2063 . r. s. sarthour , e. r. deazevedo , f. a. bonk , e. l. g. vidoto , t. j. bonagamba , a. p. guimar@xmath97es , j. c. c. freitas and i. s. oliveira , relaxation of coherent states in a two - qubit nmr quadrupolar system , phys . a. 68 ( 2003 ) 022311 . f. bloch , nuclear induction , phys . 70 ( 1946 ) 460 . a. g. redfield , the theory of relaxation processes , adv . res . 1 ( 1966 ) 1 . j. von neumann , measurement and reversibility and the measuring process , chapter v and vi in mathematische grund lagen der quantenmechanik , springer , berlin ( 1932 ) . english translation by r. t. beyer , mathematical foundations of quantum mechanics , princeton unv . press , princeton . a. abragam , principles of nuclear magnetic resonance , claredon press , oxford,1961 . anil kumar , r. c. r. grace , p. k. madhu , cross correlation in nmr , prog . in nucl . res . spec . 37 ( 2000 ) 191 - 319 . h. schneider , kernmagnetische relaxation von drei - spin - molek@xmath98len i m fl@xmath98ssign oder adsorbierten zustand.i , ann . ( 1964 ) 313 . h. schneider , kernmagnetische relaxation von drei - spin - molek@xmath98len i m fl@xmath98ssign oder adsorbierten zustand.ii , ann . ( 1965 ) 135 . j. s. blicharski , interference effect in nuclear magnetic relaxation , phys . ( 1967 ) 608 . p. s. hubbard , some properties of correlation functions of irreducible tensor operators , phys . 180 ( 1969 ) 319 . w. h. zurek , environment - induced superselection rules , phys . d. 26 ( 1982 ) 1862 . g. teklemariam , e. m. fortunato , c. c. lopez , j. emerson , j. p. paz , t. f. havel and d. g. cory , a method for modeling decoherence on a quantum information processor , phys . a. 67 ( 2003 ) 062316 . ersnt , g. bodenhausen , and a. wokaun , principles of nuclear magnetic resonance in one and two dimensions , clarendon press , oxford,1987 . j. a. jones , r. h. hansen and m. mosca , quantum logic gates and nuclear magnetic resonance pulse sequences , jl . of mag . res . 135 ( 1998 ) 353 . m. carravetta and m. h. levitt , long - lived nuclear spin states in high - field solution nmr , j. am . ( 2004 ) 6228 . m. carravetta , o. g. johannessen and malcolm h. levitt , beyond the @xmath24 limit : singlet nuclear spin states in low magnetic fields , phys . 92 ( 2004 ) 153003 .

in quantum information processing by nmr one of the major challenges is relaxation or decoherence . often it is found that the equilibrium mixed state of a spin system is not suitable as an initial state for computation and a definite initial state is required to be prepared prior to the computation . as these preferred initial states are non - equilibrium states , they are not stationary and are destroyed with time as the spin system relaxes toward its equilibrium , introducing error in computation . since it is not possible to cut off the relaxation processes completely , attempts are going on to develop alternate strategies like quantum error correction codes or noiseless subsystems . here we study the relaxation behavior of various pseudo pure states and analyze the role of cross terms between different relaxation processes , known as cross - correlation . it is found that while cross - correlations accelerate the relaxation of certain pseudo pure states , they retard that of others .

the third egret catalog ( hartman et al . @xcite ) contains 271 point sources detected at energies above 100 mev . the majority of these sources , @xmath1168 or @xmath162% , still remain unidentified . among them , there are 72 sources located at low galactic latitudes , having @xmath2@xmath310@xmath4 , which represents around 45% of the ues population . therefore , several of these objects are presumably of galactic nature . similar properties between some of these uess , indicate that there are at least three different groups of galactic populations ( romero et al . @xcite , grenier @xcite ) . the group of young stellar objects and star - forming regions ( romero @xcite ) , those sources forming a halo around the galactic center and a group of sources correlated with the gould belt ( grenier @xcite ) . based both on multiwavelength observations and theory , microquasars ( see mirabel & rodrguez @xcite for a review ) with massive companions have been proposed as possible counterparts of the first group of galactic uess by several authors ( paredes et al . @xcite , kaufman bernad et al . @xcite , romero et al . @xcite , bosch - ramon et al . @xcite ) . in sects . 2 and 3 of this paper we will briefly review the properties of the two well - known microquasars ls 5039 and ls i + 61 303 , typically associated with the first group of uess , while in sect . 4 we will present the possible association between the microquasar candidate ax j1639.0@xmath04642 and the ues 3eg j1639@xmath04702 . finally , in sect . 5 we will compare the available data of these 3 sources from radio to gamma - rays , and we will discuss on similarities pointing towards a population of hmxb with ns microquasars as counterparts of low - latitude uess . the high mass x - ray binary system ls 5039 ( paredes et al . @xcite ) is one of the @xmath115 confirmed galactic microquasars ( rib @xcite ) . ls 5039 is a bright @xmath5@xmath111.2 star with an on6.5v((f ) ) spectral type ( mcswain et al . @xcite ) . the binary system has a short orbital period of @xmath6 d , a high eccentricity of @xmath7 , and a low mass function @xmath8 @xmath9 , suggesting the presence of a ns as the compact object in the system ( mcswain et al . @xcite ) . observations conducted with the evn and merlin ( see fig . [ ls5039_evn_merlin ] ) confirmed the persistent nature of this mq , and revealed the presence of an asymmetric two - sided jet reaching up to 1000 au on the longest jet arm ( paredes et al . these observations also suggest a bending of the jets with increasing distance from the core and/or precession . the possibility that ls 5039 is a @xmath10-ray emitter was suggested by paredes et al . ( @xcite ) , who proposed the association of the system with the ues 3eg j1824@xmath01514 ( hartman et al . @xcite ) . we show in fig . [ 3egj1824 ] the location map of the @xmath10-ray source together with the nvss and bright / faint rosat sources . the only simultaneous x - ray / radio source within the statistical contours of 3eg j1824@xmath01514 is the microquasar ls 5039 . we note that this binary system is present in the batse earth occultation catalog of low - energy gamma - ray sources ( harmon et al . @xcite ) , with a positive detection of a few mcrab up to @xmath1100 kev . the source is not present in cumulative observations conducted with the integral satellite ( bird et al . @xcite ) , although it is expected to be detected when adding a few more months of data . we also point out that there is an unidentified comptel source with a position compatible with ls 5039 ( collmar @xcite ) . astrometric studies carried out by rib et al . ( @xcite ) , show that it is a runaway system with a systemic velocity of @xmath1150 km s@xmath11 that moves away from the galactic plane with a velocity of @xmath1100 km s@xmath11 . this result , combined with the possible lifetime of the donor star , indicates that it could reach a not - so - low galactic latitude of @xmath12 still behaving as a microquasar . bosch - ramon & paredes ( @xcite ) have recently developed a detailed numerical model to test whether this system can actually produce the emission detected by egret through inverse compton ( ic ) scattering . their numerical approach considers a population of relativistic electrons entrained in a cylindrical inhomogeneous jet , which interact with both the radiation and the magnetic fields . the computed spectrum is able to reproduce the observed spectral characteristics at very high ( gev ) energies . the be / x - ray binary system ls i + 61 303 is a well - studied object since it presents radio and x - ray variability linked to its @xmath126.5 d orbital period ( gregory @xcite ; paredes et al . @xcite ) . the donor star in this system is a rapidly rotating b0v star with variable mass loss ( hutchings & crampton @xcite ) . some properties of this system can be explained assuming that the unseen companion is a non - accreting young pulsar with a relativistic wind strongly interacting with the wind of the be star ( maraschi & treves @xcite ) . on the contrary , other properties of ls i + 61 303 fit better a model where the companion is accreting even with two episodes of super - critical accretion along the orbit ( mart & paredes @xcite ) . this x - ray binary system has been associated for long time with the gamma - ray source 2cg 135 + 01/3eg j0241@xmath136103 ( see fig . [ 3egj0241 ] ) , which displays variability on timescales of days ( tavani et al . @xcite , @xcite ; wallace et al . @xcite ) . during the last years , massi et al . ( @xcite , @xcite ) have revealed its mq nature through the discovery of a radio jet ( see fig . [ lsi_merlin ] ) extending 200 au at both sides of a central core , that appears to experience a fast precession , which could explain the short - term gamma - ray variability of 3eg j0241@xmath136103 ( as proposed by kaufman bernad et al . @xcite ) and the puzzling vlbi structures found in previous observations . this result points to the occurrence of accretion / ejection processes in this system , ruling out , in principle , the non - accreting young pulsar scenario . massi ( @xcite ) has recently studied the data acquired within the pointed egret observations of 3eg j0241@xmath136103 and claimed the detection of a periodicity of @xmath14 d , consistent with the orbital period of the binary system . if this is confirmed , the identification of ls i + 61 303 as the counterpart of 3eg j0241@xmath136103 would be unambiguous . in any case , an important point is that all the available data are compatible with an increase of @xmath10-ray emission around periastron , that can be tested with cherenkov telescopes and future satellites . this binary system is also present in the batse earth occultation catalog of low - energy gamma - ray sources ( harmon et al . @xcite ) , with a positive detection of a few mcrab up to @xmath1100 kev , although the detection is not as significant as in the case of ls 5039 . ls i + 61 303 is not present in cumulative observations conducted with the integral satellite ( bird et al . @xcite ) , although it is expected to be detected when adding a few more months of data , like in the case of ls 5039 . we note that there is a comptel source containing ls i + 61 303 and the quasar qso 0241 + 622 ( van dijk et al . @xcite ) . a numerical model to explain the egret emission of ls i + 61 303 has also been developed by bosch - ramon & paredes ( @xcite ) . aimed at discovering new mqs , combi et al . ( @xcite ) have recently carried out a multiwavelength study of the unidentified x - ray source ax j1639.0@xmath04642 . this object was discovered by the advanced satellite for cosmology and astrophysics ( asca ) observatory at the 0.710 kev energy range , and presented as a possible hmxb ( sugizaki et al . its measured flux was @xmath15= @xmath16 erg @xmath17 s@xmath11 , and it showed variable x - ray emission , with a confidence @xmath18 99 % . its spectrum was fitted with a power law with a very hard photon index @xmath19 and a poorly constrained hydrogen column density of @xmath20@xmath17 . combi et al . ( @xcite ) re - analyzed these data and found evidences for variability on timescales of hours . in searching for radio sources in the field of ax j1639.0@xmath04642 , combi et al . ( @xcite ) found that the molonglo galactic plane survey ( mgps ) at 843 mhz ( green et al . @xcite ) revealed a point - like radio source ( see fig . [ most ] ) , dubbed most j1639.0@xmath04642 , well within the error box of the x - ray source , with a flux density of @xmath21 mjy . at near infrared ( nir ) wavelengths combi et al . ( @xcite ) inspected the 2 micron all sky survey ( 2mass , cutri et al . @xcite ) , and found 10 sources in the 3@xmath22 error circle in position of most j1639.0@xmath04642 , some of them visible in the @xmath23-band image shown in fig . [ 2mass ] . at the far infrared part of the spectrum , from 12 to 100 microns , they found that the source iras 16353@xmath04636 lies inside the error box of the x - ray source . this source overlaps the southern part of the 3@xmath22 position error circle of most j1639.0@xmath04642 , and its location uncertainty ellipse contains several 2mass sources , as can be seen in fig . [ 2mass ] . the x - ray source ax j1639.0@xmath04642 has been recently re - discovered at higher energies with the ibis telescope onboard the integral satellite , dubbed igr j16393@xmath04643 ( malizia et al . @xcite ; bird et al . this source shows an average flux of @xmath24 erg @xmath17 s@xmath11 , and presents a factor of 23 flux variability on timescales of months . although there is no spectroscopic / photometric information of a nir / optical counterpart to derive a distance to the source , assuming that it is located in the scutum - crux or in the norma spiral arms , a range of distances between 3 and 13 kpc is obtained . combi et al . ( @xcite ) pointed out that ax j1639.0@xmath04642 lies inside the 95% location contour of the ues 3eg j1639@xmath04702 ( hartman et al . @xcite ) as can be seen in fig . [ 3egj1639 ] . its @xmath10-ray flux is @xmath25 photon @xmath17 s@xmath11 , presents a steep @xmath10-ray spectral index of @xmath26 and has a variability index of @xmath27 . although torres et al . ( @xcite ) found three radio pulsars inside the 95% confidence contour of the @xmath10-ray source , its possible variability and steep photon index do not seem to agree , in principle , with a pulsar origin . similarly , these properties would rule out an association with the three snrs found within the 95% confidence contour ( torres et al . @xcite ) . moreover , no identified blazar has been found within the @xmath10-ray contours . therefore , combi et al . ( @xcite ) suggested that the microquasar candidate ax j1639.0@xmath04642/most j1639.0@xmath04642 ( = igr j16393@xmath04643 ) is the counterpart of 3eg j1639@xmath04702 . observations with atca are in progress to unveil the nature of this source . as discussed above , the possibility of mqs being @xmath10-ray emitters was suggested by paredes et al . ( @xcite ) , who proposed the association between the hmxb ls 5039 and the ues 3eg j1824@xmath01514 . in their scenario ( paredes et al . @xcite , @xcite ) the @xmath10-rays are produced by ic upscattering of stellar ultraviolet ( uv ) photons by the non - thermal relativistic electron population that later on will produce the detected radio emission . recently , more detailed models considering precession ( kaufman bernad et al . @xcite ) , hadronic jets in windy microquasars ( romero et al . @xcite ) and all possible photon fields ( bosch - ramon et al . @xcite ) have been proposed to explain the high - energy gamma - ray emission from hmxb microquasars . on the other hand , as already stated , the x - ray binary system ls i + 61 303 has been associated with the ues 3eg j0241@xmath136103 , and massi et al . ( @xcite , @xcite ) have revealed its mq nature . if the mq nature of ax j1639.0@xmath04642 is confirmed , it could be the third mq source related to a ues . we quote the basic properties of these three @xmath10-ray sources in table [ table : egret ] , and the properties of the proposed x - ray counterparts in table [ table : xray ] . there are 3 observational facts that should be noted . the first one is that ls 5039 and ls i + 61 303 , and probably ax j1639.0@xmath04642 , have massive optical companions , which provide an intense stellar uv photon field . on the other hand , the compact object appears to be compatible with a neutron star in the cases of ls 5039 ( mcswain et al . @xcite ) and ls i + 61 303 ( hutchings & crampton @xcite ; casares et al , @xcite ; but see massi @xcite ) . finally , it is interesting to point out that the luminosities obtained in each spectral domain are very similar in all three sources , specially for the shorter distances to ax j1639.0@xmath04642 , giving support to the idea that all of them have similar emission processes . moreover , it should be noted that ls 5039 and ls i + 61 303 are the only microquasars having both a high - mass donor and ( possibly ) a ns as the compact object , and that there are no other microquasars ( containing low - mass donors and/or black holes ) located within the probability contours of unidentified egret sources . therefore , a strong statement can be made : hmxb / ns microquasars appear as good counterparts of low - latitude unidentified egret sources . this statement is followed by some natural questions . 1 ) may ls 5039 and ls i + 61 303 still contain black holes ( bhs ) ? although formally possible , this is unlikely , because regarding the properties of their x - ray emission they should be in the so - called low / hard state ( fender & maccarone @xcite ) , but they do not follow the empirical correlation between x - ray and radio flux found by gallo et al . ( @xcite ) , in the sense that they are clearly too radio loud . 2 ) why no hmxb microquasars with bhs ( e.g. , cygnus x-1 ) are present in the third egret catalog ? a possibility is that black hole state changes may play a role , preventing the detection of gamma - ray emission when the jet is not present during the high / soft state ( but this state is very rare in the case of cygnus x-1 and in fact , as discussed by romero et al . ( @xcite ) , a low high - energy cutoff of a few hundreds in the lorentz factor of the electrons is compatible with the data ) . 3 ) why no low mass x - ray binary ( lmxb ) microquasars are present in the third egret catalog ? one possibility is that the optical companions do not provide the necessary intense uv radiation fields needed for an effective ic process to produce high - energy gamma - rays . this possibility is strongly model dependent , and not valid if self synchrotron compton losses are dominant . the other possibility is that lmxbs are in general terms transient objects , that maybe were not active during the egret viewing periods , while hmxbs tend to be persistent systems . 4 ) of course , bh and lmxb microquasars could still emit high - energy gamma - rays and not be present in the third egret catalog because of the relatively poor sensitivity threshold of the instrument . in conclusion , persistent hmxbs containing nss not experiencing state changes are good candidates for the counterparts of the still unidentified high - energy gamma - ray sources at low - galactic latitudes ( approximately up to @xmath28 , as discussed for ls 5039 ) , and we consider that these objects may define a population among uess . observations with the future missions agile and glast will confirm or reject both the proposed associations between these microquasars and the corresponding high - energy gamma - ray sources , and the hypothesis discussed above . we thank sylvain chaty , paula benaglia , gustavo e. romero , josep m. paredes , josep mart , valent bosch - ramon and rob fender for useful discussions , and an anonymous referee for useful comments that helped to improve the paper . m.r . acknowledges support by a marie curie fellowship of the european community programme improving human potential under contract number hpmf - ct-2002 - 02053 . m.r . also acknowledges partial support by dgi of the ministerio de ciencia y tecnologa ( spain ) under grant aya2001 - 3092 , as well as partial support by the european regional development fund ( erdf / feder ) . j.a.c . is a researcher of the programme _ ramn y cajal _ funded by the spanish ministery of science and technology and the university of jan . was supported by conicet ( under grant pei 6384/03 ) . romero , g. e. 2001 , in proc . of the nature of unidentified galactic high - energy gamma - ray sources , a. carramiana , o. reimer , and d. j. thompson , assl series of kluwer academic publishers 51 , p. 65

the discovery of the microquasar ls 5039 well within the 95% conficence contour of the unidentified egret source ( ues ) 3eg j1824@xmath01514 was a major step towards the possible association between microquasars ( mqs ) and uess . the recent discovery of precessing relativistic radio jets in ls i + 61 303 , a source associated for long time with 2cg 135 + 01 and with the ues 3eg j0241 + 6103 , has given further support to this idea . finally , the very recently proposed association between the microquasar candidate ax j1639.0@xmath04642 and the ues 3eg j1639@xmath04702 points towards a population of high mass x - ray binary ( hmxb)/neutron star ( ns ) microquasars as counterparts of low - latitude unidentified egret sources . marc rib + service dastrophysique + cea saclay + bt . 709 , lorme des merisiers + f-91191 gif - sur - yvette , cedex + france

galaxy , group and cluster distributions probe matter clustering in the universe , not only over different scales , but also for different density contrasts . however , while galaxy and cluster clustering have been widely inspected , a measurement of group clustering meets several conceptual and technical difficulties and it is not surprising that its results are controversial and partially contradictory . in this note we report the result of an analysis of clustering properties of loose groups in the perseus pisces redshift survey ( hereafter pps ; see giovanelli , haynes , & chincarini 1986 ; haynes et al . 1988 ; giovanelli & haynes 1989 , 1991 , 1993 ) . through such analysis we believe that the reasons of previous discrepant results become clear . it is also worth soon mentioning that our error analysis , based on bootstrap criteria , detects a precise signal of clustering for loose groups above statistical noise . as is known , the 2point functions of galaxies and clusters are consistent with the power laws @xmath11 characterized by the same exponent @xmath12 , but by widely different amplitudes @xmath1 and @xmath13 . the detection of such difference ( bahcall @xmath14 soneira 1983 , klypin @xmath14 kopylov 1983 ) led kaiser ( 1984 ) and politzer @xmath14 wise ( 1984 ) to suggest the mechanism of biased galaxy formation . results are far less clear for galaxy groups . jing @xmath14 zhang ( 1988 , hereafter jz88 ) and maia @xmath14 dacosta ( 1990 , hereafter mdc90 ) claimed that the 2point function for groups is still consistent with a power law @xmath15 with @xmath16 and @xmath17 with @xmath18@xmath19 . on the contrary , ramella , geller , & huchra ( 1990 , hereafter rgh90 ) found @xmath20 and , although their analysis can not reject a value @xmath21 , the preferred value ranges around 1@xmath22 . according to rgh90 , the main contribution to @xmath23 comes from the the 2point function @xmath24 of galaxies members of groups . recently frederic ( 1995a&b , hereafter f95 ) determined @xmath25 for haloes and halo groups in cdm simulations by gelb ( 1992 ) . he found groups to be significantly more correlated than single halos , and interpreted this as contrasting with rgh90 s results for galaxies and galaxy groups ( but he also showed that the correlation strength depends on the prescription adopted for halo identification and illumination ) . in all the above mentioned studies , groups were identified with the adaptive friends of friends algorithms of huchra & geller ( 1982 ; hg82 hereafter ) or nolthenius & white ( 1987 ; hereafter nw87 ) . such algorithms require several input parameters . some ( the galaxy luminosity function @xmath8 and the magnitude limit @xmath7 ) are set by the data themselves . others ( the `` sky link '' @xmath6 and the `` redshift link '' @xmath10 ) must be decided by the user : @xmath6 can be related to the normalization @xmath26 of @xmath8 ( nw87 ) , while the choice of @xmath10 is more complex ( hg82 ; nw87 ; ramella , geller , & huchra 1989 , rgh89 hereafter ) . as already pointed out by nw87 , confirmed by rgh89 , and stressed by nolthenius , klypin , & primack ( 1994 , 1995 ; hereafter nkp94&95 ) , a delicate point in group analysis is the _ sensitivity _ of the results to the details of the adopted algorithm and/or data set . also forgetting possible intrinsical differences among the galaxy samples where groups were drawn from , the different search parameters of the algorithm used to identify galaxy groups could be at the origin of the above mentioned discrepancies . however , as we shall see below , this is actually more relevant for internal than for clustering properties ( f95 ; trasarti battistoni 1995 , 1996 tb96 hereafter ) . another problem is the high noise in the determination of @xmath27 , due to the limited extension of the group catalogs previously studied . loose groups in pps were systematically identified and analyzed in tb96 , who concentrated mainly on internal properties and their dependence on the adopted algorithm and/or data sample . differences between data samples are small but detectable , and the effect of the magnitude limit @xmath7 is to be properly taken into account . note that pps is wider than the cfa2 slices ( de lapparent , geller , & huchra 1986 , 1988 , 1989 dlgh86/88/89 hereafter ; geller & huchra 1989 ; huchra et al . 1990 , huchra , geller , & corwin 1995 ) used by rgh8990 and f95 , and is spatially disconnected from them as it lies in a different galactic hemisphere . it is also deeper than the redshift surveys cfa1 ( davis & huchra 1982 , davis et al . 1982 , huchra et al . 1983 ) and ssrs1 ( da costa et al . 1988 ) , where groups identified by geller & huchra ( 1983 ; hereafter gh83 ) and maia , dacosta , & latham ( 1989 ; hereafter mdcl89 ) were used by jz88 and mdc90 , respectively . in fact , the number of groups in pps is @xmath28-@xmath29 , while it is @xmath30-@xmath31 in the other samples , and this helps to reduce the above mentioned statistical noise . internal properties of groups have been used to constrain cosmological models and , in particular , the dark matter composition ( nkp94&95 ) . also group clustering has been suggested as a test for cosmological models , both on analytical bases ( e.g. , kashlinsky 1987 ) , or through the comparison with numerical n body simulations ( f95 ) . in the latter case , the key point is that galaxy groups can be identified automatically and _ exactly in the same way _ both from galaxy catalogs and from large ( @xmath32 ) n body simulations ( nw87 ; moore , frenk , & white 1993 ; nkp94&95 ; f95 ) . although such groups are basically expected to be physical objects this is no longer the basic requirement to have an effective comparison . once groups are suitably defined , then properties are compared to find out which simulation best matches the observations . there is a precise physical reason which favours loose groups over single galaxies ( and compact groups ) or rich clusters as a test of cosmological models . at intermediate separations ( @xmath33 ) , mass scales ( @xmath34 ) , and density contrast ( @xmath35@xmath36 ) typical of galaxy groups , gravitational evolution is still in the mildly non linear regime . therefore , lss keeps memory of the shape of the post recombination power spectrum @xmath37 . at larger scales linearity keeps the lss signal at a level too low in respect to the noise , so @xmath37 is not easily detectable , and the limited extension ( in volume and number of objects ) of available observational samples is often a problem . at smaller scales stronger non linear and non gravitational effects complicate everything . moreover , the most widely studied observational samples of rich clusters ( abell 1958 ; abell , corwin , olowin 1989 ) suffer of various biases ( sutherland 1988 ; see borgani 1995 for a review ) , mainly due to partially subjective criteria used in their compilation . compact groups and rich clusters were recently identified from observational samples also employing objective and automatic procedures ( prandoni , iovino , & macgillivray 1994 ; nichol et al . 1992 ; dalton et al . 1992 ; nichol , briel , & henry 1994 ) . however such procedures are difficult to reproduce on n body simulations . this is due to a combined need of high resolution ( to ease object identification ) , large sample volume ( to have a statistically meaningful number of objects ) , and computational speed ( to reach a statistically meaningful number of independent realizations of the same theoretical model ) . in the case of clusters , the latter two difficulties can be circumvented by using a combination of numerical and analytical approaches based on the zeldovich approximation ( e.g. , sahni & coles 1995 ; borgani et al . 1995 ) , but the identification of observational like clusters is still not an easy task . the plan of the paper is the following . in section [ sez : data ] we describe the galaxy data and the group catalogs , while section [ sez : xi ] describes the estimation of clustering properties . results are presented and discussed in section [ sez : resdis ] . we summarize our conclusions in in section [ sez : conclu ] . the pps database was compiled by giovanelli & haynes in the last decade ( see giovanelli & haynes 1991 , 1993 ; wegner , giovanelli , & haynes 1993 , and the references therein ) . the full redshift survey is magnitude limited down to @xmath38 , and now it covers the whole region @xmath39 and @xmath40 . as in tb96 , we restricted to the region @xmath41 and @xmath42 , to avoid regions of high interstellar extinction . magnitudes are anyway corrected as in burstein & heiles ( 1978 ) , and redshifts are corrected for galactic rotation and local group motion as in yahil , sandage , & tamman ( 1977 ) . the two subsamples pps1 and pps2 ( shown in fig . 1 , top panels ) are magnitude limited to @xmath43 and @xmath44 respectively , in analogy with cfa1 and cfa2 . this makes our comparison of data sample as clean as possible . in fact , the selection criteria of ssrs makes it qualitatively different from cfa or pps . on the other hand , the full cfa1 ( north+south ) and the first two cfa2 slices in the north are neither fully disjoint ( as pps2 and cfa2 slices ) nor one a subset of the other ( as pps1 of pps2 ) . the shape of the survey is also important . it is more difficult to identify groups in the proximity of the edges , so wide angle surveys are favoured over thin slices or pencil beams . to summarize , pps1 ( pps2 ) covers a solid angle @xmath45 , and consists of 769 ( 3030 ) galaxies with magnitude @xmath46 and redshift @xmath47 . for comparison , the previously analyzed subsamples of cfa1 n+s , ssrs1 , and cfa2 slices , are characterized respectively by : @xmath48 , @xmath49 , @xmath50 @xmath51 , @xmath43 , @xmath52 ( apparent diameter limited ) , @xmath44 , @xmath53 , @xmath54 , @xmath55 . the characteristics of all group catalogs are listed in table 1 . the two significative cases tb96@xmath56 and tb96@xmath57 ( in pps1 and pps2 , respectively ) are also shown in fig . 1 ( bottom panels ) . groups are identified with the friends of friends algorithms described in tb96 , both hg like ( hg82 ) and nw like ( nw87 ) . briefly , two galaxies closer then some specified transverse separation @xmath5 and radial separation @xmath9 in redshift space are friends of each other . friendship is transitive , and a galaxy group is an isolated set of friends . the two links are normalized by @xmath6 and @xmath10 at a given fiducial redshift ( here @xmath58 ) , and are then scaled up with @xmath59 , using the selection function @xmath60 here @xmath7 is the apparent magnitude limit of the sample , @xmath61 is the galaxy luminosity function , @xmath62 is the faintest absolute magnitude in the sample , while the dependence on @xmath63 and @xmath59 arises through @xmath64 to scale up the links , the original hg prescription ( based on simple arguments , monte carlo tested ) gives @xmath65 , while the nw recipes ( based and tested on n body simulations ) takes @xmath66 , and @xmath67 , where @xmath59 is the mean redshift of the pair of galaxies considered , and @xmath68 is a suitable constant ( for a detailed discussion of the reasons behind such different choices , see tb96 and the references therein ) . the value of @xmath6 corresponds to an effective density threshold ( in redshift space ) , given by @xmath69^{-1 } \ ] ] we adopt the galaxy luminosity function in the schechter ( 1976 ) form determined from pps2 by tb96 ( @xmath70 , @xmath71 ; here we take @xmath72 ) . the number of groups in all catalogs is approximately @xmath73 , i.e. @xmath74 in pps2 and @xmath75 in pps1 . for sake of comparison , @xmath76,@xmath77 , and @xmath78 in jz88 , mdc90 , and rgh90 respectively . it is clear that the characteristics of each groups catalogue are fixed , to some extent , by the choice of parameters in the identification algorithm . smaller values of the links @xmath6 and @xmath10 yield groups with higher density contrast and , within biased theories of galaxy formation , this is expected to cause a stronger spatial correlation . as outlined in tb96 , a similar effect arises from the sample depth . this subtler point requires some explanation . it is clear that the increase of the mean galaxy density , which occurs when the magnitude limit passes from @xmath79 to @xmath80 , is accounted for by a slower decrease of the selection function with @xmath59 . in fact , eq . [ eq : m_lim ] shows that @xmath81 , provided that @xmath82 ; accordingly , everything which happened at @xmath83 is now moved to @xmath84 , in the deeper sample . everything is therefore scaled , apart of @xmath85 , the redshift value where the sky link @xmath5 is normalized . normalizing at the _ same _ location @xmath86 for _ different _ @xmath7 s yields different density contrasts @xmath87 for the same value @xmath6 . groups identified with the same values of @xmath6 and @xmath10 are then expected to be more correlated in pps1 then in pps2 . an analogous effect is expected for the redshift link @xmath9 , though different for the hg and nw algorithms . in fact , rgh90 suggested that the different values jz88 and mdc90 adopted for @xmath10 could account for the discrepancies from their results , but they did not take into account the stronger effect presumably arising from the different @xmath7 . ( the effects of changing @xmath7 , @xmath6 , and @xmath10 , are shown in fig . 5 ; a much more detailed discussion will be provided in sect . [ sez : resdis ] . ) it is also importants to outline that giving the @xmath6 link is not immediately equivalent to providing a threshold density contrast . according to eq . ( [ eq : dn_n ] ) , @xmath88 , but @xmath26 variations by a factor of 2 can arise both from observational techniques and/or local physical conditions . the other parameters determining the schechter function , adopted by rgh89 and f95 , and worked out by dlgh88 ( @xmath89 , @xmath90 ) are almost consistent with those worked out from our sample . tb96 showed that , for similar changes ( @xmath91 and @xmath92 ) , the net effect on the internal properties of output groups is small . furthermore , the spatial distribution of groups is almost insensitive even to greater differences in the identification algorithm . as shown by dlgh88 and martinez et al . ( 1993 ) , estimates of @xmath25 are sensitive to the presence of large scale features in the samples . in the present analysis , we cut _ all _ samples within @xmath93 . the number of groups does not change in pps1 ( 767 galaxies ) , while it is reduced by @xmath94@xmath95 in pps2 ( 2693 galaxies ) . this cut off ensures that we are always dealing with the same physical structures , though differently sampled by different @xmath7 s . the same main lss features are present in both galaxy samples , and they are reflected on the spatial distribution of groups ( fig . 1 ) . let us now describe the procedure leading to the 2point function estimate . as usual , we center spherical shells of radius @xmath96 and width @xmath97 on each observed object in the data ( @xmath98 : galaxy / group ) sample . we then count neighbours in the data sample and in a random control sample , spatially uniform but with the same shape and radial selection function as the data . to do so , @xmath99 random points are taken from a spatially uniform distribution with the same shape of the data sample , but they are weighted down by the selection function @xmath100 . in particular , the total number of random points @xmath99 is replaced by the total weight @xmath101 , where @xmath102 is the weight of the @xmath103 random point . we then divide the number @xmath104 of ( observed ) neighbour objects within @xmath105 by the ( weighted down ) number @xmath106 of neighbour random points , and then average over all @xmath107 centers . our weighting scheme is analogous to that of the @xmath108 estimator discussed in dlgh88 ( see also martinez et al . 1993 ) . altogether , this amounts to estimating @xmath25 through the following formula , that we use both for groups and galaxies : @xmath109 we adopt the same selection functions for galaxies and groups , by using both in pps1 and pps2 the luminosity function @xmath110 evaluated in tb96 from pps2 ( in some cases , we use also @xmath61 from dlgh88 ) . in fact , the radial distribution of galaxies and groups substantially agree over the relevant redshift range ( see fig . 2 ) . errorbars are computed with 10 bootstrap resampling of the data ( barrow , bhavsar , & sonoda 1984 ) . bootstrap errors are expected to overestimate the true ensemble errors , in turn larger than formal poisson errors . the ratio between bootstrap and poisson errors is expected to be @xmath111@xmath112 ( martinez et al . 1993 ) , with a slight dependence on @xmath96 and data depth . on the other hand , weighting data by @xmath113bootstrap errorbar@xmath114 in the least squares fit of @xmath25 to @xmath115 , and simultaneously treating @xmath96-bins as independent , tends to compensate the bootstrap overestimate , yielding fair values for @xmath116 and @xmath117 ( e.g. , ling , frenk , & barrow 1986 ) . plots of 2point correlation functions for groups are given in figs . 3a and b. for the sake of comparison , in each figure , the galaxy 2point functions are from the same sample also plotted . here , groups are identified with @xmath118 and @xmath119 , as in rgh89 . bootstrap error bars are given for all points . a least square bootstrap weighted fit to @xmath120 is then performed . the best fit @xmath121 and @xmath122 and their errors are listed in tables 2 and 3 for pps1 and pps2 , respectively . for galaxies the fits can be extended from 1 to 31.7 @xmath4 . for groups , instead , the most reasonable distance interval is from 1.5 to 10 @xmath4 . for @xmath123 anti correlation due to the intrinsical size of groups is expected to ( and does ) take over . the plots also show that , for @xmath124 the signal is too noisy to be of any use . this distance interval is similar to those used in previous group correlation analyses . we however performed fits both for galaxies and groups in both kinds of distance intervals . from fig . 3 we see that , although ( 1 ) within 2@xmath125 bootstrap errorbars , @xmath126 , ( 2 ) @xmath27 is mostly _ higher _ than @xmath127 by a factor @xmath128 . accordingly , from tables 2 and 3 , we see that @xmath0 systematically exceeds @xmath1 . for the narrower and most reliable distance range , @xmath129 for pps1 , and @xmath130@xmath131 for pps2 . in this parameter fit , the amplitudes are different at the @xmath1322.5@xmath125 level , but their difference has the same sign anywhere , and can be suspected to be real . also the slope @xmath122 of groups is however greater than the corresponding slope of galaxies . let us just remind that the range of @xmath122 s found here is not unusual in the redshift space and , for galaxies , corresponds to steeper @xmath133 s in the range @xmath134@xmath135 in the real space ( e.g. , gramann , cen , & bahcall 1993 ) . the shift from redshift to real space is clearly expected to be weaker for groups . it is not clear to which extent this can justify the steeper @xmath122 found for groups ; this effect exceeds 2@xmath125 s for pps1 only , but is present anywhere . with a comparable level of confidence we also see that : ( 3 ) @xmath0 for pps1 exceeds @xmath0 for pps2 by more than 50@xmath136 , while , in the same ( narrower ) distance range @xmath1 for pps1 and pps2 are almost consistent within 1@xmath125 . however , for galaxies , we can exploit the wider distance range , and there we find a probable signal of luminosity segregation at the 3@xmath125 level . the point ( 1 ) was also outlined by rgh90 in cfa2 slices , while the point ( 2 ) is in contrast with rgh90 , mdc89 , and jz88 . let us however remark that jz88 and mdc89 used much larger @xmath6 s than us and rgh90 . adopting the same links and @xmath8 parameters as for jz88 s groups ( gh83 ) , we obtain much lower and noisier @xmath27 s , both for pps1 and pps2 . this is shown in fig . 4a , where , at variance with jz88 , error bars are also provided . the discrepancy with rgh90 , instead , is less pronounced . besides of using a different data set , rgh90 make also use of different @xmath8 parameters . if we make use of such parameters to detect groups in pps2 we obtain a lower @xmath27 , nearly overlapping @xmath127 ( compare fig . 3b and fig . 4b ) . an attractive interpretation of point ( 2 ) is a _ relative _ segregation of groups with respect to galaxies . a similar effect between halos and halo groups in n body simulations of flat unbiased cdm ( gelb 1992 ) was found by f95 , who however showed that this result can depend on the identification scheme . although bearing in mind such reserves , we agree with f95 that rgh90 s outputs conflict with flat unbiased cdm models . however , this conflict is not evident in our outputs . as already outlined , the obvious interpretation of the point ( 3 ) for galaxies is luminosity segregation : as first shown by davis et al . ( 1988 ) and convincingly demonstrated by park et al . ( 1994 ) using complete _ volume limited _ samples , brighter galaxies are more clustered than fainter one . this can be related to their greater luminosity either directly ( e.g. , hamilton 1988 ; hollosi & efstathiou 1988 ; davis et al . 1988 ; gramann & einasto 1992 ) or through the luminosity morphology connection ( e.g. , iovino et al . 1993 ; hasegawa & umemura 1993 ) . such effect is also predicted by theoretical scenarios ( e.g. , white et al . 1987 ; bonometto & scaramella 1988 ; valls gabaud , alimi , & blanchard 1989 ) and a similar effect of _ mass _ segregation has been found in large n body simulations ( e.g. , campos et al . 1995 ) . independently of the interpretation , it is clear that cutting both pps1 and pps2 within the _ same _ limit @xmath137 , pps1 contains all the brighter galaxies of pps2 because of the brighter @xmath7 . for these _ incomplete _ samples , limited both in @xmath59 and in @xmath138 , we expect @xmath127 to be greater for brighter @xmath7 . on the contrary , for _ complete _ apparent magnitude limited samples ( @xmath139 increasing with @xmath7 ) , we expect @xmath127 to be greater for fainter @xmath7 ( e.g. , hamilton 1988 ) , as the total number of intrinsically bright galaxies in such samples increases with fainter @xmath7 and dominates the sample . a possible interpretation of the point ( 3 ) for groups is that , in pps1 , we selected higher density contrast groups than in pps2 . in fact , using the same @xmath6 for both pps1 and pps2 , while the mean inter galaxy separation is smaller for fainter @xmath7 , can enhance the correlation in pps1 . but this effect might not be enough . in fact we checked that the change of @xmath6 in either pps1 or pps2 has smaller effects than passing from pps1 to pps2 . this is shown in fig . 5a and b , where we give @xmath27 for groups selected in 2 different ways both in pps1 and pps2 . using @xmath140 ( @xmath141 ) in pps2 ( pps1 ) we have the same @xmath142 that @xmath143 gives in pps1 ( pps2 ) . the changes of @xmath27 are however slight within either pps1 or pps2 , in spite of the change of @xmath6 . we explore further the dependence of @xmath27 on @xmath6 in fig . as expected , @xmath27 increases with @xmath142 within a given sample ( the figure refers to pps2 ) . in fig . 5d , instead , we test the dependence on @xmath10 and discover that , at variance with what could be expected , @xmath27 does not depend on @xmath10 , in spite of varying the redshift link over a greater range than @xmath6 . as a further check , we also used the nw scaling recipe , whose @xmath9 has a quite different shape from the hg scaling ( @xmath5 s are very similar ) . no effect on @xmath27 is however present . ( this agrees with f95 , and is consistent with tb96 who showed how for given self consistent normalizations and different scaling recipes the spatial distribution of hg like and nw like groups is very similar . ) it is difficult to comment on this point , which however seems to indicate that varying @xmath10 is not so effective to change the density contrast and that the redshift link is therefore less _ physical _ than the sky link . in fact , tb96 showed how a restrictive @xmath9 affect groups more as a cut off ( discarding high velocity dispersion systems ) than as a systematic variation of the properties of each single group . on the other hand , changing @xmath5 directly affects @xmath142 of the otput groups . from a physical point of view , this might suggest that clustering of galaxy systems is more directly related to their density contrast than to their velocity dispersion ( mass ? ) . note that groups are on average brighter in pps1 than in pps2 ( tb96 ) . a natural interpretation of point ( 3 ) could be luminosity ( mass ? ) segregation among _ groups _ themselves . but this could be a premature conclusion . in fact , in order to compare group luminosity the _ observed _ group luminosity @xmath144 should be implemented by @xmath145 for group members below @xmath7 . but this correction depends on several assumptions ( mezzetti et al . 1985 ; gourgoulhon , chamaraux , & fouqu 1992 ; moore et al . the enhancement of @xmath27 in pps1 could be just an effect of a higher correlation of group members . in fact , groups are brighter in pps1 than in pps2 because their _ members _ are . henceforth , luminosity segregation of _ galaxies _ , not of groups themselves . however , fig . 1 shows that the relatively nearby main ridge in pps at @xmath146 is equally well sampled in both pps1 and pps2 , while the more sparsely populated region between @xmath147 and @xmath148 in pps2 is completely devoid of groups in pps1 . from this point of view , groups _ are _ indeed more clustered in pps1 than in pps2 , but simply because of `` cosmographical '' reasons , not because of their higher luminosity . therefore , an explanation of effect ( 3 ) could be an intrinsic difference in the data sets pps1 and pps2 . previous analyses of group clustering ( jz88 , mdc90 , rgh90 ) led to apparently contradictory results . the main source of discrepancy lies in the different search parameters of the adaptive friends of - friends algorithms adopted by various authors . the greater difference is due to the transverse normalization @xmath6 . stricter @xmath5 s yield stronger correlations , thus explaining the discrepancy among jz88 on one side ( @xmath149 ) and rgh90 on the other ( @xmath150 ) . instead , contrary to simple expectation , no dependence on the radial link @xmath9 ( as suggested by rgh90 ) is detected . a further reason of difference is the impact on group properties of the different depth of the adopted samples . this effect ( partially considered by rgh89 and mdcl89 , but not by rgh90 and mdc90 ) requires a suitable handling of the `` parameter '' @xmath7 in the identification algorithm , and a self consistent normalization of @xmath5 . simply adopting the same value of @xmath6 leads to lower density contrasts in shallower samples , also contributing to the shallower @xmath27 of jz88 and mdc90 . the residual difference among samples of different depth is nevertheless more important . this is the case for the samples pps1 and pps2 , and an analogous effect was pointed out by rgh89 for cfa1 survey and cfa2 slices . a further reason of discrepancy between rgh90 and mdc90 could be the different physical nature of the cfa and ssrs samples . as shown by mdcl89 , groups in cfa1 and ssrs1 have similar physical properties when _ different _ values of the links are used . in fact , the former survey contains a higher percentage of early type galaxies , in general more clustered than the late type galaxies in the latter . the attention devoted by a number of authors to the clustering of loose groups is widely justified . galaxies and clusters are both clustered , but their clustering parameters are different . the discovery of such difference was the original motivation to introduce the biased theory of galaxy formation . an independent test of such theory as a whole and a further constraint on its parameters arise from a sufficiently precise determination of clustering parameters of groups . to pursue this program we need an unambiguous definition of loose groups and a sample wide enough to test their properties . in the literature there has been a complex debate on group individuation , which is technically provided by suitable sky link ( @xmath5 ) and redshift link ( @xmath9 ) , in transverse and radial direction , respectively . this is due to the need of exploiting the whole information contained in apparent magnitude limited samples , and overcoming the critical problem of distortions due to the passage from the redshift to the real space . this work shows that results on clustering parameters , obtained with different group individuation recipes , are essentially consistent . this does not mean that the values of the links do not matter . on the contrary , the clustering strength is critically dependent on how restrictive the selection is . the point is that @xmath5 has a stronger impact on clustering properties than @xmath9 . different criteria ( e.g. , nw vs. hg ) are essentially different as far as the radial link @xmath9 is concerned , but the effect on @xmath25 of fairly wide variations of @xmath9 is neglegible , while smaller variations of @xmath5 are reflected on the clustering of groups . to some extent , this is a natural outcome of working in _ redshift space_. the transverse link @xmath5 is directly related to the spatial separation among group members . in the radial direction , instead , peculiar velocities inside groups yield ambiguous apparent separations . relating @xmath9 to a value of @xmath151 is then not trivial and depends on the cosmological model ( e.g. , nw87 ) . this problem would be present even if we could select a complete volume limited galaxy sample , and then identify groups therein as suggested by ramella et al . ( 1995a , b ) . ( selecting volume limited subsamples of the group catalogs _ after _ group identification with the adaptive hg and nw algorithms ( zabludoff et al . 1993a , b ) , is not the ideally correct solution , as groups could be already biased by the magnitude limited nature of the galaxy data source . ) unfortunately , to be really efficient , this procedure requires a high number of measured redshifts . @xmath152 , even adopting the _ generous _ constant links @xmath153 , @xmath154 , volume limiting pps2 to @xmath155 ( @xmath156 ) as in guzzo et al . ( 1991 ) and bonometto et al . ( 1993 ) yields only @xmath157 groups , and it is not sure that this is enough . resorting then to _ apparent magnitude limited _ samples , two further but connected source of complication arise . first , the physical mixture of what we consider _ bright _ and _ faint _ galaxies varies with redshift . second , the mean separation among observed galaxies grows with @xmath59 , and group identification is harder at higher redshifts . to some extent , the traditional strategy of compensating the decrease of galaxy density by scaling up @xmath5 and @xmath9 , gives group properties doomed to be systematically dependent on @xmath59 ( and , to a lesser extent , to the scaling recipe and galaxy luminosity function ) . previous analyses of group clustering ( jz88 , mdc90 , rgh90 ) in cfa and ssrs surveys led to apparently contradictory results . in this paper , we have investigated the source of such discrepancies , finding satisfactory explanations for them . together with that , we have found a signal of group clustering , whose amplitude exceeds the amplitude of galaxy clustering . such excess is perhaps to be trusted more than what its formal probability ( @xmath158 s ) prescribes , as it persists through various density contrasts and link recipes ; it could be found mostly thanks to the wider extension of pps in respect to the sample used for previous analyses . & & & & tb96@xmath159 & 0.27 & 350 & 160,110 & 48 , 192 & & & & wide@xmath56 & 0.30 & 350 & 110 , ... & 48 , ... strict@xmath57 & 0.24 & 350 & ... , 160 & ... , 179high@xmath159 & 0.21 & 350 & 330 , 240 & 40 , 168low@xmath159 & 0.42 & 350 & 40 , 30 & 59 , 205cold@xmath159 & 0.27 & 150 & 160 , 110 & 43 , 186hot@xmath159 & 0.27 & 600 & 160 , 110 & 50 , 201tb96nw@xmath159 & 0.27 & 350 & 160 , 110 & 48 , 162rgh89@xmath159 & 0.27 & 350 & 120 , 80 & 49 , 192rgh83@xmath159 & 0.27 & 600 & 120 , 80 & 52 , 205dgh83@xmath159 & 0.52 & 600 & 16 , 10 & 60 , 185 & & & & * pps1 * & @xmath160&@xmath161&@xmath162tb96@xmath56 & @xmath163&@xmath164&@xmath165 & & & pps1 & @xmath166&@xmath167&@xmath168 & @xmath169&@xmath170&@xmath171wide@xmath56 & @xmath172&@xmath173&@xmath174 & & & high@xmath56 & @xmath175&@xmath176&@xmath177low@xmath56 & @xmath178&@xmath179&@xmath180 & & & nw@xmath56 & @xmath181&@xmath182&@xmath183cold@xmath56 & @xmath184&@xmath185&@xmath186hot@xmath56 & @xmath187&@xmath188&@xmath189 & & & rgh89@xmath56 & @xmath190&@xmath191&@xmath192rgh83@xmath56 & @xmath193&@xmath194&@xmath195dgh83@xmath56 & @xmath196&@xmath197&@xmath198 & & & & & & & @xmath199&@xmath200&@xmath201tb96@xmath57 & @xmath202&@xmath203&@xmath204 & & & pps2 & @xmath205&@xmath206&@xmath207 & @xmath208&@xmath209&@xmath210strict@xmath57 & @xmath211&@xmath212&@xmath213 & & & high@xmath57 & @xmath214&@xmath215&@xmath216low@xmath57 & @xmath217&@xmath218&@xmath219 & & & nw@xmath57 & @xmath220&@xmath221&@xmath222cold@xmath57 & @xmath223&@xmath224&@xmath225hot@xmath57 & @xmath226&@xmath227&@xmath228 & & & rgh89@xmath57 & @xmath229&@xmath230&@xmath231rgh83@xmath57 & @xmath232&@xmath233&@xmath234dgh83@xmath57 & @xmath235&@xmath236&@xmath237 & & & bahcall , n. a. , & soneira , r. m. , 1983 , apj 270 , 20 barrow , j. d. , bhavsar , s. p. , & sonoda , d.h . 1984 , mnras 210 , 19p bonometto , s. a. , & scaramella , r. 1988 , preprint , submitted to a&a bonometto , s. a. , iovino , a. , guzzo , l. , giovanelli , r. , & haynes , m. p. , 1993 , apj 419 , 451 borgani s. , 1995 , phys.rep . 251 , 1 borgani s. , et al . 1995 , mnras 277 , 1191 burstein , d. , & heiles , c. , 1978 , apj 225 , 40 da costa , l. n. , pellegrini , p. s. , sargent , w. l. w. , et al . 1988 , apj 327 , 544 dalton , g. b. , efstathiou , g. , maddox , s. j. , & sutherland , w. j. , 1992 , apj 390 , l1 davis , m. , & huchra , j. p. , 1982 , apj 254 , 437 davis , m. , & huchra , j. p. , latham , d. w. , & tonry , j. , 1982 , apj 253 , 423 davis , m. , meiksin , a. , strauss , m. a. , dacosta , l. n. & yahil , a. , 1988 , apj 333 , l9 de lapparent , v. , geller , m. j. , & huchra , j. p. , 1988 , apj 302 , l1 ( dlgh86 ) de lapparent , v. , geller , m. j. , & huchra , j. p. , 1988 , apj 332 , 88 ( dlgh88 ) de lapparent , v. , geller , m. j. , & huchra , j. p. , 1989 , apj 343 , 117 ( dlgh89 ) gelb , j. m. , 1992 , m.i.t . ph.d . dissertation geller , m. j. , & huchra , j. p. , 1983 , apjs 52 , 61 ( gh83 ) geller , m. j. , & huchra , j. p. , 1989 , sci 246 , 897 ghigna , s. , borgani , s. , bonometto , s. a. , et al . , 1994 , apj 437 , l71 giovanelli , r. , haynes , m. p. , & chincarini , g. , 1986 , apj 300 , 77 giovanelli , r. , & haynes , m. p. , 1989 , aj 97 , 633 giovanelli , r. , & haynes , m. p. , 1991 , ara&a 29 , 499 giovanelli , r. , & haynes , m. p. , 1993 , aj 105 , 1271 gourgoulhon , e. , chamaraux , p. , & fouqu , p. , 1992 , a&as 255 , 69 gramann , m. , & einasto , j. , 1992 , mnras 254 , 453 gramann , m. , cen , r. , & bahcall n.a . 1993 , apj 419 , 440 guzzo , l. , iovino , a. , chincarini , g. , giovanelli , r. , & haynes , m. p. , 1991 , apj 382 , l5 hamilton , a. j. s. , 1988 , apj 331 , l59 haynes , m. p. , giovanelli , r. , starosta , b. , & magri c.a . , 1988 , aj 95 , 607 hasegawa , t. , & umemura , m. , 1993 , mnras 263 , 191 hollosi , j. , efstathiou , g. , 1988 , `` large scale structure in the universe '' , springer verlag huchra , j. p. , & geller , m. j. , 1982 , apj 257 , 423 ( hg82 ) huchra , j. p. , davis , m. , latham , d. , & tonry , j. , 1983 , apjs 52 , 89 huchra , j. p. , geller , m. j. , de lapparent , v. , & corwin , h. g. jr . , 1990 , apjs 72 , 433 huchra , j. p. , geller , m. j. , & corwin , h. g. jr . , 1995 , apjs 99 , 391 kaiser , n. , 1984 , apj 284 , l9 kashlinsky , a. , 1987 , apj 317 , 19 klypin , a. a. , holtzmann , j. , primack , j. r. , & regs e. , 1993 , apj 416 , 1 klypin , a. a. , & kopylov , a. i. 1983 , sov.astron.lett . 9 , 41 klypin , a. a. , nolthenius , r. , & primack , j. r. , 1995 , apj submitted , astro ph 9502062 maia , m. a. g. , da costa , l. n. , & latham , d. w. , 1989 , apjs 69 , 809 ( mdcl89 ) maia , m. a. g. , & da costa , l. n. , 1990 , apj 349 , 477 ( mdc90 ) martinez , v. j. , portilla , m. , jones , b. j. t. , & paredes , s. 1993 , a&a 280 , 5 mezzetti , m. , pisani , a. , giuricin , g. , & mardirossian , f. 1985 , a&a 143 , 188 moore , b. , frenk , c. s. , & white , s. d. m. , 1993 , mnras 261 , 827 nichol , r. c. , collins , c. a. , guzzo , l. , & lumsden , s. l. 1992 , mnras 255 , 21 nichol , r. c. , briel , u. g. , henry , j. p. , 1994 , mnras 265 , 867 nolthenius , r. , klypin , a. a. , & primack , j. r. , 1994 , apj 422 , l25 ( nkp94 ) nolthenius , r. , klypin , a. a. , & primack , j. r. , 1995 , apj submitted , astro ph 9410095 ( nkp95 ) nolthenius , r. , & white , s. , 1987 , mnras 225 , 505 ( nw87 ) park , c. , vogeley , m. s. , geller , m. j. , & huchra , j. p. , 1994 , apj 431 , 569 peebles , p. j. e. , 1980 , `` the large scale structure of the universe '' , princeton university press politzer , h. d. , wise , m. d. 1984 , apj 285 , l1 prandoni , i. , iovino , a. , & macgillivray , h. t. , 1994 , aj 107 , 1235 ramella , m. , geller , m. j. , & huchra , j. p. , 1989 , apj 344 , 57 ( rgh89 ) ramella , m. , geller , m. j. , & huchra , j. p. , 1990 , apj 353 , 51 ( rgh90 ) ramella , m. , geller , m. j. , huchra , j. p. , & thorstensen , j. r. 1995a , aj 109 , 1458 ramella , m. , geller , m. j. , huchra , j. p. , & thorstensen j.r . 1995b , aj 109 , 1469

we investigate the clustering properties of loose groups in the perseus pisces redshift survey . previous analyses based on cfa and ssrs surveys led to apparently contradictory results . we investigate the source of such discrepancies , finding satisfactory explanations for them . furthermore , we find a definite signal of group clustering , whose amplitude @xmath0 exceeds the amplitude @xmath1 of galaxy clustering ( @xmath2 , @xmath3 for the most significant case ; distances are measured in @xmath4 ) . groups are identified with the adaptive friends of friends ( fof ) algorithms hg ( huchra & geller 1982 ) and nw ( nolthenius & white 1987 ) , systematically varying all search parameters . correlation strenght is especially sensitive to the sky link @xmath5 ( increasing for stricter normalization @xmath6 ) , and to the ( depth @xmath7 of the ) galaxy data . it is only moderately dependent on the galaxy luminosity function @xmath8 , while it is almost insensitive to the redshift link @xmath9 ( both to the normalization @xmath10 and to the scaling recipes hg or nw ) .

modulational instability ( mi ) refers to a process where a weak periodic perturbation of an intense continuous wave ( cw ) grows exponentially as a result of the interplay between dispersion and nonlinearity . mi constitutes one of the most basic and widespread nonlinear phenomena in physics , and it has been studied extensively in several different physical systems like water waves , plasmas , and optical devices @xcite . for a cubic nonlinearity , as the one occurring in the nonlinear schrdinger equation used to model optical fibers , the underlying physical mechanism can be understood in terms of four - wave mixing between the pump , signal and idler waves . however , the scalar four - wave interactions in a homogeneous fiber can be phase matched , and hence efficient , only in the anomalous group - velocity dispersion ( gvd ) regime . in the normal gvd regime , on the other hand , mi can occur in detuned cavities @xcite , thanks to constructive interference between the external driving and the recirculating pulse . alternatively mi with normal gvd can also arise in systems with built - in periodic dispersion @xcite , among which dispersion oscillating fibers ( dofs ) have recently attracted renewed attention @xcite . in this case , phase matching relies on the additional momentum carried by the periodic dispersion grating ( quasi - phase - matching ) . the occurrence of unstable frequency bands can then be explained using the theory of parametric resonance , a well - known instability phenomenon which occurs in linearized systems for which at least one parameter is varied periodically during the evolution @xcite . up to now , most experimental investigations realised in optical fibers have been performed with basic sinusoidal @xcite or amplitude modulated @xcite modulation formats . in this work , on the other hand , we study a radically different periodic modulation of the gvd , in the form of a periodic train ( or comb ) of dirac delta spikes . this is a fundamental and widespread modulation format , encountered in a variety of physical systems . in optics , delta combs have been exploited to model lumped amplification in long haul fiber optic transmission systems @xcite , or to model the power extraction in soliton based fiber lasers @xcite . moreover , comb - like dispersion - profiled fibers have been exploited to generate trains of solitons starting from a beat signal @xcite . at more fundamental level kicked systems are widely investigated as a paradigm for the emergence of chaos in perturbed hamiltonian systems , with the delta - kicked rotor being the most renowned example @xcite . its quantum version is described by a schrdinger equation forced by a dirac comb and has been extensively analyzed to study chaos in quantum systems @xcite . recirculating fiber loops have been used to reproduce the quantum kicked rotor with an optical system , to study chaos and anderson localization @xcite , and to illustrate how an optical system can be used to mimic other physical systems that are more difficult to reproduce experimentally . in the same vein , we hope the experimental setup we propose in this paper could be used as an experimental platform to investigate such phenomena in the presence of nonlinearities , a topic of much current interest . finally the approach that we propose to analyse mi in the fiber with delta - kicked gvd allows us , on one hand to enlighten the featuress of the parametric resonance that are not dependent on the specific format of the modulation , and on the other hand to compare and contrast the features of the ideal delta - kicked profile with other formats including non - ideal ( physically realizable ) kicking as well as widely employed profiles such as oscillating gvd . the paper is organized as follows . in section [ s : centralfreq ] we provide a simple argument allowing to determine the central frequencies of the unstable sidebands for general periodically modulated fibers . in section [ s : diraccomb ] , we then use floquet theory to analytically compute the width of the gain bands and as well as their maximum gain for dispersion - kicked fibers . in section [ s : approxdelta ] we investigate numerically the effect of the smoothing of the delta comb . in section [ s : expresults ] , we describe the experimental set - up and we compare the experimental results with theory and numerical simulations based on the generalized nonlinear schrdinger equation . we draw our conclusions in section [ s : conclusions ] . consider the nlse @xmath0 we will assume the dispersion @xmath1 and the nonlinearity coefficient @xmath2 are of the form @xmath3 where @xmath4 and @xmath5 are periodic functions of period @xmath6 such that @xmath7 , and @xmath8 . let @xmath9 be a stationary solution of . we consider a perturbation of @xmath10 in the form @xmath11 , where the perturbation @xmath12 satisfies @xmath13 . inserting this expression in , and retaining only the linear terms we find @xmath14 writing @xmath15 , with @xmath16 and @xmath17 real functions , we obtain the following linear system : @xmath18 finally , taking the fourier transform of this system in the time variable @xmath19 , leads to @xmath20 where we used the definiton @xmath21 . note that this is a hamiltonian dynamical system in a two - dimensional phase plane with canonical coordinates @xmath22 . analyzing the linear ( in)stability of the stationary solution @xmath10 therefore reduces to studying the solutions to for each @xmath23 . since the coefficients in the equation are @xmath24-periodic with period @xmath6 , floquet theory applies . this amounts to studying the linearized evolution over one period @xmath6 , to obtain the floquet map @xmath25 which in the present situation is the two by two real matrix defined by @xmath26 . as a result @xmath27 . note that @xmath28 necessarily has determinant one , since it is obtained by integrating a hamiltonian dynamics , of which we know that it preserves phase space volume . as a consequence , if @xmath29 is one of its eigenvalues , then so are both its complex conjugate @xmath30 and its inverse @xmath31 . this constrains the two eigenvalues of @xmath32 considerably : they are either both real , or lie both on the unit circle . now , the dynamics is unstable only if there is one eigenvalue @xmath29 satisfying @xmath33 , in which case both eigenvalues are real . we will write @xmath34 for the two eigenvalues of @xmath28 . we are interested in studying the gain , that is @xmath35 as a function of @xmath23 , @xmath36 and @xmath37 . it measures the growth of @xmath38 . the gain vanishes if the two eigenvalues lie on the unit circle . a contour plot of the gain in the @xmath39 plane , for the case of the delta comb dispersion modulation that is the main subject of this paper , can be found in fig . [ fig : arnoldtongues ] . the regions where the gain does not vanish are commonly referred to as arnold tongues . we will explain below that , whereas their precise form depends on the choice of @xmath40 , the position of their tips does not . since the system is not autonomous , it can not be solved analytically in general . nevertheless , the above observations will allow us to obtain some information about its ( in)stability for small @xmath36 , @xmath37 , and valid for all perturbations @xmath40 , whatever their specific form . to see this , we first consider the case @xmath41 . it is then straightforward to integrate the system . the linearized floquet map is then given by @xmath42 where @xmath43 here @xmath44 ( normal average dispersion ) , since we restrict our investigations to the defocusing nls . note that the matrix @xmath45 has determinant equal to @xmath46 , as expected . the eigenvalues of @xmath45 can be readily computed as @xmath47 what will happen if we now switch on the interaction terms @xmath48 and @xmath49 ? it is then no longer possible , in general , to give a simple closed form expression of the solution to , which is no longer autonomous , and hence of the linearized floquet map @xmath50 . nevertheless , we do know that , for small @xmath51 , the eigenvalues of @xmath50 must be close to the eigenvalues @xmath52 . we then have two cases to consider . @xmath53*. now @xmath54 , _ they are distinct _ , and they both lie on the unit circle , away from the real axis . they then must remain on the unit circle under perturbation since , for the reasons explained above , they can not move into the complex plane away from the unit circle . consequently , in this case , the stationary solution @xmath10 is linearly stable under a sufficiently small perturbation by @xmath55 and @xmath56 , and this statement does not depend on the precise form of @xmath48 or of @xmath49 . in fact , with growing @xmath36 and/or @xmath37 , the two eigenvalues will move along the unit circle until they meet either at @xmath57 or at @xmath58 for some critical value of the perturbation parameters . only for values of the latter above that critical value can the system become unstable . a pictorial description of this situation is shown in the left hand side of fig . [ eigenvalues ] . + @xmath59 , @xmath60 . * now @xmath61 is a doubly degenerate eigenvalue of @xmath62 . under a small perturbation , the degeneracy can be lifted and two real eigenvalues can be created , one greater than one , one less than one in absolute value . the system has then become unstable ! of course , it will now depend on the type of perturbation whether the system becomes unstable , remains marginally stable ( the two eigenvalues do nt move at all , but stay at @xmath46 or @xmath57 ) , or becomes stable ( the two eigenvalues move in opposite directions along the unit circle ) . a pictorial description of this situation is shown in the right hand side of fig . [ eigenvalues ] . for the dirac comb modulation of @xmath1 , which is our main object of study in this paper , the details are given in the next section . in conclusion , examining , one sees that only if @xmath63 , where @xmath64 can an infinitely small hamiltonian perturbation of @xmath62 lead to an unstable linearized dynamics near the fixed points @xmath10 considered . these values of @xmath23 therefore correspond to the tips of the arnold tongues , that is , to the positions of the ( centers of ) the unstable sidebands of the defocusing nls under a general periodic perturbation @xmath40 . this is illustrated for a dirac comb modulation of the gvd in fig . [ fig : arnoldtongues ] . one also observes in that figure that , for a value of @xmath23 close to some @xmath65 , the system becomes unstable only for a small but nonzero critical value of @xmath36 , that we shall compute below for the dirac delta comb gvd . and @xmath49 on the eigenvalues of the linearized floquet map ( [ floq_lin ] ) . black dots correspond to the unperturbed eigenvalues lying on the unit circle ( dashed line ) . coloured dots show the new position of the eigenvalues after switching on the perturbations , leading to a stable regime when @xmath66 and an unstable one when @xmath67 , width=340 ] equation ( [ omega_l ] ) was derived in @xcite by appealing to the theory of parametric resonance and poincar - lindstedt perturbation theory . our argument above is elementary and shows in a simple manner that the resonant frequencies @xmath65 do not at all depend on the form of @xmath68 or @xmath5 . note that , if @xmath69 and @xmath70 , a case considered in @xcite-@xcite , the system is equivalent to the equation of a harmonic oscillator of ( spatial ) frequency @xmath71 , sinusoidally modulated with period @xmath6 . in that case the system leads to a mathieu equation for which it is known that resonance occurs when the period of the modulation is a integer multiple of the half ( spatial ) period of the oscillator , which is @xmath72 . additional physical insight can be obtained by expanding equation ( [ omega_l ] ) for small power , i.e. assuming @xmath73 . at zero order we recover the well known quasi - phase - matching relation @xcite @xmath74 equation ( [ phasematching ] ) entails the conservation of the momentum , made possible thanks to the virtual momentum carried by the dispersion grating , of the four wave mixing interaction between two photons from the pump , going into two photons in the symmetric unstable bands at lower ( stokes ) and higher ( antistokes ) frequencies with respect to the pump . in the @xmath75 plane , for @xmath76 , @xmath77 , @xmath78 and @xmath79 . the dashed black lines corresponds to the tips of the arnold tongues ( [ omega_l ] ) at @xmath80 and @xmath81 . the solid red lines corresponds to the gain bandwidth , which can be computed from . ( b ) mi gain for @xmath82 ; red circles , estimates of maximum gain ( [ gmax ] ) ; black crosses , estimates of the bandwidth . ( c ) solid blue curve , mi gain for @xmath83 ; dashed red curve , approximation of maximum gain ( [ gmax ] ) . [ fig : arnoldtongues],title="fig:",width=302 ] in the @xmath75 plane , for @xmath76 , @xmath77 , @xmath78 and @xmath79 . the dashed black lines corresponds to the tips of the arnold tongues ( [ omega_l ] ) at @xmath80 and @xmath81 . the solid red lines corresponds to the gain bandwidth , which can be computed from . ( b ) mi gain for @xmath82 ; red circles , estimates of maximum gain ( [ gmax ] ) ; black crosses , estimates of the bandwidth . ( c ) solid blue curve , mi gain for @xmath83 ; dashed red curve , approximation of maximum gain ( [ gmax ] ) . [ fig : arnoldtongues],title="fig:",width=302 ] in the @xmath75 plane , for @xmath76 , @xmath77 , @xmath78 and @xmath79 . the dashed black lines corresponds to the tips of the arnold tongues ( [ omega_l ] ) at @xmath80 and @xmath81 . the solid red lines corresponds to the gain bandwidth , which can be computed from . ( b ) mi gain for @xmath82 ; red circles , estimates of maximum gain ( [ gmax ] ) ; black crosses , estimates of the bandwidth . ( c ) solid blue curve , mi gain for @xmath83 ; dashed red curve , approximation of maximum gain ( [ gmax ] ) . [ fig : arnoldtongues],title="fig:",width=302 ] we now turn our attention to the computation of the gain @xmath84 , in particular for values of @xmath23 close to the resonant frequencies . we concentrate on the special case where the gvd is a dirac delta comb : @xmath85 - 1,\qquad \gamma_m=0.\ ] ] since in the rest of this paper , @xmath86 , we will drop it from the notation . to compute the gain , we need to compute the linearized dynamics @xmath87 and determine the behaviour of its eigenvalues @xmath88 in the neighbourhood of @xmath89 and @xmath63 in the @xmath75-plane . in this case the linearized floquet map is easily seen to be explicitly given by @xmath90 where @xmath45 is defined by equation ( [ floq_lin ] ) , but now with @xmath91 $ ] , and @xmath92 the characteristic polynomial of @xmath93 is given by @xmath94 so that the eigenvalues of ( [ eq : linfloqbetam ] ) can be computed explicitly as : @xmath95 with @xmath96 and @xmath97 . a taylor expansion of @xmath98 about @xmath99 yields @xmath100,\ ] ] where @xmath101 the dependence in @xmath102 ( not in @xmath36 ) entails that the sign of the kick has no incidence in this regime , i.e. assuming @xmath103 . formula ( [ eq : arnoldtongueslin ] ) shows that @xmath104 is a saddle point for @xmath105 . if @xmath106 is even , @xmath107 occurs close to @xmath104 , and if @xmath106 is odd , @xmath108 close to @xmath104 . more precisely , @xmath109 from which we can find an estimate of the gain amplitude @xmath110 and of the bandwidth @xmath111 near the tips of the tongue at @xmath65 , as : @xmath112 @xmath113 note that the threshold value for @xmath36 above which instability occurs can be read off from the above by setting @xmath114 which corresponds to @xmath115 this confirms again , as expected , that an arbitrary small @xmath36 will generate instability right at @xmath63 . in fig . [ fig : arnoldtongues]a we show an example of the analytically computed mi gain , showing the first two arnold tongues . as can be seen , for a small enough strength of perturbation , let s say @xmath116 , the approximation ( [ band ] ) gives a good estimate of the width of the parametric resonance ( see red curves ) . this situation is detailed further in figs . [ fig : arnoldtongues]b , c , showing a section for @xmath82 and @xmath83 , respectively . finally , a straightforward calculation gives the asymptotic behaviour of the gain @xmath117 at @xmath65 for @xmath106 large and @xmath36 fixed , that is @xmath118 with @xmath119 whenever @xmath120 , and @xmath121 otherwise . as a function of @xmath106 ; red circles , estimated gain given by . parameter values are @xmath76 , @xmath77 , @xmath78 , @xmath79 , @xmath82 . , width=302 ] in fig . [ profile_gain_large ] , we show an example of the analytically computed mi gain at @xmath65 as a function of @xmath106 . we compare it to the approximation , which is very accurate , even for small @xmath106 ( see red circles ) . note in particular that the oscillating behaviour of the gain is well captured by which , for @xmath106 large enough and @xmath36 small , can be approximated by @xmath122 * summing up . * it is clear from the above that , precisely at the values @xmath65 , which only depend on @xmath6 and on @xmath123 , but not on the precise form of @xmath68 , any small perturbation can create an instability and hence a gain . at frequencies @xmath23 near these particular values , a minimal threshold strength of @xmath36 is needed to create an instability . this minimal value , and even the fact that an instability is indeed generated , does depend on the precise form of @xmath68 . for the dirac comb the explicit expression for the gain in this regime can be read off from . , with @xmath82 , calculated numerically at the first parametric resonant frequency @xmath124 , for two approximations of delta functions as a function of their inverse height @xmath125 . solid blue curve , gaussian approximation ; dashed red curve , rectangular pulse approximation ; dashed horizontal line , dirac delta limit ; dash - dotted horizontal line , sinusoidal modulation . parameter values are @xmath126 , @xmath77 , @xmath127 , @xmath79 . , in order to shed light on the dependence of the gain on the shape of the periodic modulation , and also with an eye towards the experimental realization of the dirac comb fiber , we now analyze what happens when the dirac comb is approximated by a train of physically realizable `` kicks '' . we thus consider a smoothened dirac comb described by @xmath128 where we normalize the positive function @xmath129 in order to have @xmath130 . for a rectangular pulse of width @xmath131 , we get @xmath132 for a gaussian function @xmath133 , the maximum amplitude of the kick can be calculated as @xmath134 that in the limit @xmath135 gives @xmath136 note that , in these models , we have ( @xmath137 ) @xmath138 hence @xmath139 corresponds to a rather symmetric situation where @xmath140 is close to the midpoint between @xmath141 and @xmath142 , so that @xmath1 fluctuates symmetrically about its average value . whereas @xmath143 corresponds to a very asymmetric situation where @xmath1 has a large abrupt peak . the parameter @xmath144 therefore controls the shape of the gvd modulation at fixed @xmath140 and @xmath145 ( or @xmath146 ) . as shown in the previous section , by changing the shape of the kick , we do not change the frequency of the parametric resonances . the smoothing of the delta function nevertheless does modify the characteristics of the mi by changing the value of the gain , as we now illustrate by computing the gain numerically at the resonant frequencies @xmath65 . an example of how the changing shape of the modulation @xmath68 modifies the first parametric resonance is illustrated in fig . [ delta_non_perf ] , that shows the gain @xmath147 at fixed @xmath124 and @xmath36 , as a function of the peak amplitude @xmath144 ( or , equivalently , the width @xmath148 ) of the kicks . we make the following observations . first , a good approximation of the gain given by the dirac comb is obtained for @xmath149 , both for the rectangular and gaussian pulses . second , for @xmath150 , the gain of the square pulse modulation is zero , as expected , since we are then in the limit case of a constant modulation ( and normal gvd ) . third , it is apparent that the dirac comb gives the highest possible gain , for a fixed area of the kicks and fixed @xmath36 and @xmath140 . finally , it is interesting to note that a sinusoidal modulation @xmath69 , with the same value of @xmath36 , gives a gain close to one half with respect to the delta case . indeed for a sinusoidal modulation , it has been shown that ( see equation ( 7 ) from ref . @xcite ) @xmath151 by expanding equation ( [ gmax_sin ] ) for small @xmath36 , we get @xmath152 at first order in @xmath36 . in conclusion , a large concentrated perturbation of the gvd about its average enhances the mi gain . it is well known that in homogeneous fibers , the gvd depends on the diameter of the fiber . one can therefore modulate the gvd by modulating the diameter of the fibers as a function of @xmath24 , as in @xcite . we manufactured three different microstructured optical fibers modulated by a series of gaussian pulses to approximate the ideal dirac delta comb studied in section [ s : diraccomb ] . the change of their outer diameters @xmath153 along the fiber is represented in fig . [ fig : fibres exp](a ) . as can be seen in the inset , their diameters have a gaussian shape with a standard deviation @xmath148 , which is the same for all three fibers , and very small ( @xmath154 m ) compared to the period of the comb ( 10 m ) . hence we can write @xmath155 where @xmath156 . the three fibers have a very similar minimum diameter @xmath157 , while their maximum values are different . we have @xmath158172 @xmath159 , @xmath160207 @xmath159 and @xmath161240 @xmath159 for fibers labelled a , b and c , respectively , corresponding to @xmath162 . to understand how the two experimental parameters @xmath148 and @xmath163 control the quality of the approximation of the delta function on the one hand , and the value of @xmath36 on the other hand , we proceed as follows . first , @xmath164 , so that @xmath165 . a first order taylor expansion of @xmath166 about @xmath167 yields @xmath168.\ ] ] comparing this to we find @xmath169 and @xmath170 hence , with the notation of section [ s : approxdelta ] , @xmath171 . this corresponds to @xmath172 proving that these gaussian pulses should induce a very similar parametric gain compared to ideal dirac delta functions ( see fig . [ delta_non_perf ] ) . furthermore , the height @xmath163 of the gaussian pulse controls @xmath36 , which will allow us to investigate the impact of this parameter on the first mi side lobe gain , as it was done in the theoretical study and illustrated in fig.[fig : arnoldtongues ] . and @xmath173 . [ fig : fibres exp],width=302 ] for all three fibers , the ratio of the diameter of the holes over @xmath174 ( the pitch of the periodic cladding ) is assumed to be constant along the fiber and estimated to about 0.48 from scanning electron microscope images . the diameter variations of the fibers are proportional to those of the pitch , with @xmath175 corresponding to the minimum value of the diameter ( @xmath176 , green line in fig . [ fig : fibres exp](a ) ) and @xmath177 and @xmath178 , blue , black and red curves respectively , for the maximum values . as an example , the group velocity dispersion ( gvd ) curve corresponding to the minimum pitch value has been calculated from ref.@xcite and is represented in fig . [ fig : fibres exp](b ) as a green curve . its zero dispersion wavelength ( zdw ) is located at 1055 nm while those of the gvd curves corresponding to the maximum values of the diameters of fibers a , b and c are red - shifted to 1110 nm , 1136 nm and 1168 nm respectively ( fig . [ fig : fibres exp](b ) ) . in order to give another illustration of the large dispersion variations induced in these fibers by varying their diameters , the maximum gvd values for fibers a , b and c have been calculated at a fixed wavelength ( 1052.5 nm ) and compared to the background value . as can be seen in fig . [ fig : fibres exp](b ) , an increase of the diameters by a factor of only 1.27 , 1.45 and 1.75 , leads to a one order of magnitude improvement on the gvd values : 21 , 29 and 35 respectively . under such large variation of the fiber diameter , the gvd can no longer be considered as proportional to the pitch value , as was the case in ref.@xcite for instance . this is illustrated in fig.[fig : fibres exp](c ) , where the evolution of the gvd calculated at 1052.5 nm as a function of the fiber diameter is represented . it can be seen to be well approximated by an affine function in the range between 140@xmath179 m and 180 @xmath179 m , but not beyond . as a consequence , the shape of the gvd variations will be slightly different from the one of the diameter , specifically for fiber c. however , we checked numerically that this can be considered as relatively weak distortions that do not significantly impact the gain of the mi process . we can still consider that the key parameters remain the different heights in gvd of the gaussian - like pulses in fibers a , b and c. in order to get a more complete picture of the impact of the fiber diameter variations on its guiding properties , the variation of the nonlinear coefficient is plotted as a function of the fiber diameter in fig . [ fig : fibres exp](d ) . the most important feature to note here is that the amplitude of variation is much smaller , and only the same order as the one of the diameter itself . hence , these variations are more than one order of magnitude lower than those of the gvd and we have checked numerically that their impact on the mi process is negligible . consequently , we can infer that these fibers represent a good prototype to validate our theoretical investigation in the previous sections , where only longitudinal gvd variations have been taken into account . , width=302 ] the experimental setup is schematized in fig . [ fig : experimental setup ] . the pump system is made of a continuous - wave tunable laser ( tl ) diode that is sent into an intensity modulator ( mod ) in order to shape 2 ns square pulses at 1 mhz repetition rate . they are amplified by two ytterbium - doped fiber amplifiers ( ydfas ) at the output of which two successive tunable filters are inserted to remove the amplified spontaneous emission in excess around the pump . these quasi - cw laser pulses have been launched along the birefringent axis of the fibers . the pump peak power has been fixed to 6.5 w and the pump wavelength at 1052.5 nm for fiber a. the output spectrum recorded at its output is represented as a blue curve in fig . [ fig : exp et simul ] ( a ) . two mi side lobes , located at + /-4.8 thz appear on both sides of the pump . these experimental results have been compared with numerical simulations performed by integrating the generalized nonlinear schrdinger equation . we used the gvd , @xmath180 and @xmath181 variations calculated from the measured diameter values ( see figs . [ fig : fibres exp ] ) . other parameters are extracted from experiments and are listed in the caption of fig . [ fig : exp et simul ] . note that we checked that except the longitudinal gvd variations , all other parameters can be assumed to be constant and equal to the average values . as can be seen in the blue curve in fig . [ fig : exp et simul](b ) , two symmetric mi side lobes also appear in the simulated spectrum , in a very good agreement with experiments . their positions have been compared with the predictions of equation [ omega_l ] , represented by green dashed lines in fig . [ fig : exp et simul](b ) ( calculated with @xmath182 ) . an excellent agreement is also obtained . in order to show that the mi gain is larger when the weight of the dirac delta function is increased , we performed similar experiments in fibers b and c where the areas of the gaussian pulses are larger than in fiber a. however , in experiments , due to the fact that the dirac comb has been approximated with a series of gaussian functions , changing their amplitudes also modifies the average value of the dispersion . as a consequence , mi side lobes would be generated at different frequency shifts . in order to keep constant the position of the mi side lobes , and then provide a correct comparison with the theoretical study , one have to take care to keep the average gvd value constant in all the fibers . to do so experimentally in fibers b and c , we slightly tuned the pump wavelength until the first mi side lobe is generated at 4.8 thz as in fiber a. as can be seen in fig . [ fig : exp et simul](a ) ( red and black curves ) , the position of the first mi side lobe in fibers b and c is indeed located at that frequency by tuning the pump wavelength to 1061.8 nm and 1067 nm , respectively . we can therefore consider that the average gvd values are very similar in the three fibers and hence that only the areas of the gaussian functions , i.e. the equivalent of the dirac weights , vary . the amplitudes of the first mi side lobes generated in fibers b and c are indeed larger compared to fiber a , as predicted by the theory . this is in pretty good agreement with numerical simulations ( fig.[fig : exp et simul](b ) ) , where the same procedure was used . we found that the average values in fibers b and c are @xmath183 and @xmath184 respectively . the small discrepancy between these values is attributed to spurious longitudinal fluctuations arising during the drawing process . indeed , as can be seen in fig . [ fig : fibres exp](a ) , the background over which gaussian pulses are superimposed is not perfectly flat , and in fibre c , it is not horizontal . to counterbalance these imperfections , it was necessary to adjust the average dispersion values . we checked that with a series of perfect gaussian pulses superimposed on a flat and horizontal background , the same average gvd value would be obtained . furthermore , we can note that in fibers b and c , additional mi side lobes are generated due to the periodic modulation of the gvd ( labelled mii in [ fig : exp et simul](a ) , up to 5 in fiber b ) . their positions are also well predicted by numerical simulations and by equation [ omega_l ] ( green lines in [ fig : exp et simul](b ) ) . this excellent agreement confirms that their positions indeed scale approximately as @xmath185 , @xmath186 being the side lobe order , that is the typical signature of the mi process occurring in dispersion oscillating fibers . it was already reported experimentally in refs.@xcite , with a sinusoidal variation of the gvd , modulated in amplitude or not and it is now illustrated in this paper with a dirac delta comb . moreover , in fibers b and c two symmetric side lobes that are not predicted by the theory appear around the pump at about 2.15 thz ( labelled spurious side lobes in fig.[fig : exp et simul](a ) ) . they result from a non - phase matched four - wave mixing process involving the pump , the first and second mi side lobes . the energy conservation relation involving these waves predicts a frequency shift of 2.2 thz from the pump for the fourth wave , that is in good agreement with the shift of 2.15 thz measured experimentally . .(a ) experiments and ( b ) numerical simulations . parameters : @xmath187 . [ fig : exp et simul],width=302 ] modulation instability has been investigated theoretically and experimentally in dispersion kicked optical fibers . an analytical expression of the parametric gain has been obtained allowing to predict the behavior of the mi process in such fibers . specifically , it was shown that increasing the weights of the dirac functions leads to larger mi gains for the first mi side lobe . we exploit the fact that the dirac delta comb can be well approximated by a series of short gaussian pulses in order to perform an experimental investigation using microstructured optical fibers . we then experimentally report , for the first time to our knowledge , multiple mi side lobes at the output of these dispersion kicked optical fibers . we demonstrate that they originate from the periodic variations of the dispersion . we also validate experimentally that increasing the height of the modulation leads to a larger gain for the first mi side lobe . this illustrates that optical fibers constitute an interesting platform to realize experimental investigations of fundamental physical phenomena . the present research was supported by the agence nationale de la recherche in the framework of the labex cempi ( anr-11-labx-0007 - 01 ) , equipex flux ( anr-11-eqpx-0017 ) , and by the projects topwave ( anr-13-js04 - 0004 ) , fopafe ( anr-12-js09 - 0005 ) and noawe ( anr-14-achn-0014 ) .

we study , both theoretically and experimentally , modulational instability in optical fibers that have a longitudinal evolution of their dispersion in the form of a dirac delta comb . by means of floquet theory , we obtain an exact expression for the position of the gain bands , and we provide simple analytical estimates of the gain and of the bandwidths of those sidebands . an experimental validation of those results has been realized in several microstructured fibers specifically manufactured for that purpose . the dispersion landscape of those fibers is a comb of gaussian pulses having widths much shorter than the period , which therefore approximate the ideal dirac comb . experimental spontaneous mi spectra recorded under quasi continuous wave excitation are in good agreement with the theory and with numerical simulations based on the generalized nonlinear schrdinger equation .