arXiv:0704.1404v1  [nucl-th]  11 Apr 2007
Neutron Matter from Low-Momentum Interactions
Bengt Friman1,∗), Kai Hebeler1,∗∗), Achim Schwenk2,∗∗∗) and Laura Tolós3,†)
1GSI, Planckstr. 1, D-64291 Darmstadt, Germany
2TRIUMF, 4004 Wesbrook Mall, Vancouver, BC, Canada, V6T 2A3
2 FIAS, J.W. Goethe Universität, D-60438 Frankfurt am Main, Germany
We present a perturbative calculation of the neutron matter equation of state based
on low-momentum two- and three-nucleon interactions. Our results are compared to the
model-independent virial equation of state and to variational calculations, and we provide
theoretical error estimates by varying the cutoff used to regulate nuclear interactions. In ad-
dition, we study the dependence of the BCS 1S0 superfluid pairing gap on nuclear interactions
and on the cutoff. The resulting gaps are well constrained by the nucleon-nucleon scattering
phase shifts, and the cutoff dependence is very weak for sharp or sufficiently narrow smooth
regulators with cutoffs Λ > 1.6 fm−1.
§1. Introduction
The determination of a reliable equation of state of nucleonic matter plays a
central role for the physics of neutron stars1) and core-collapse supernovae.2), 3) Fur-
thermore the superfluidity and superconductivity of neutrons and protons is an im-
portant phenomenon in nuclear many-body systems,4), 5) in particular for the cooling
of neutron stars.6) In this contribution, we present calculations of the neutron mat-
ter equation of state at finite temperature and of the 1S0 superfluid gap in the BCS
approximation based on low-momentum interactions.
Renormalization group methods coupled with effective field theory (EFT) offer
the possibility for a systematic approach to the equation of state. By evolving
nuclear forces to low-momentum interactions Vlow k
7)–9) with cutoffs around 2 fm−1,
the model-dependent short-range repulsion is integrated out and the resulting low-
momentum interactions are well constrained by the nucleon-nucleon (NN) scattering
data. Furthermore, the corresponding leading-order three-nucleon (3N) interactions
(based on chiral EFT) become perturbative in light nuclei for Λ . 2 fm−1.10)
With increasing density, Pauli blocking eliminates the shallow two-nucleon bound
and nearly-bound states, and the contribution of the particle-particle channel to bulk
properties becomes perturbative in nuclear matter.8) The Hartree-Fock approxima-
tion is then a good starting point for many-body calculations with low-momentum
NN and 3N interactions, and perturbation theory (in the sense of a loop expansion)
around the Hartree-Fock energy converges at moderate densities. This can be under-
stood quantitatively based on the behavior of the Weinberg eigenvalues as a function
of the cutoff and density.8), 9)
∗) E-mail address: b.friman@gsi.de
∗∗) E-mail address: k.hebeler@gsi.de
∗∗∗) E-mail address: schwenk@triumf.ca
†) E-mail address: tolos@fias.uni-frankfurt.de
typeset using PTPTEX.cls 〈Ver.0.9〉
http://arxiv.org/abs/0704.1404v1
2 B. Friman, K. Hebeler, A. Schwenk and L. Tolós
Some uncertainty remained concerning a possible dependence of the 1S0 pairing
gap on the input NN interaction in low-density neutron matter (kF < 1.6 fm
We address this point and explore the dependence of 1S0 superfluidity on nuclear
interactions at the BCS level in detail. We find that the BCS gap is well constrained
by the NN phase shifts. Therefore, any uncertainties are due to polarization (induced
interaction), dispersion and three-nucleon interaction effects.
§2. Equation of State of Neutron Matter
Using the Kohn-Luttinger-Ward theorem,11), 12) the perturbative expansion of
the free energy (at finite temperature) can be formulated as a loop expansion around
the Hartree-Fock (HF) energy. In this work, we include the first-order NN and 3N
contributions, as well as normal and anomalous second-order NN diagrams. Other
thermodynamic quantities are computed using standard thermodynamic relations.
0 0.05 0.1
ρ [fm-3]
virial
T=3 MeV
T=6 MeV
T=10 MeV
0 0.05 0.1 0.15
ρ [fm-3]
Hartree-Fock (NN+3N) HF + 2nd-order NN
Fig. 1. Energy per particle E/N as a function of the density ρ at first order (left panel) and
including second-order NN contributions (right panel).13)
The resulting energy per particle E/N as a function of the density ρ is shown
in Fig. 1 for a cutoff Λ = 2.1 fm−1 and temperatures T = 3, 6 and 10MeV.13) The
results presented in the left panel are the first-order NN and 3N contributions, and
those in the right panel includes all second-order diagrams with NN interactions.
For T = 6MeV, we also give a band spanned between Λ = 1.9 fm−1 (lower line) and
Λ = 2.5 fm−1 (upper line). The inclusion of second-order contributions significantly
reduces the cutoff dependence of the results. The model-independent virial equation
of state14) and the variational calculations of Friedman and Pandharipande (FP)15)
are displayed for comparison.
The inclusion of second-order correlations lowers the energy below the variational
Neutron Matter from Low-Momentum Interactions 3
0 1 2 3
Λ [fm
sharp
=0.8 fm
=0.4 fm
=1.35 fm
Fig. 2. The neutron-neutron 1S0 superfluid pairing gap ∆ as a function of the cutoff Λ for three
densities and different smooth exponential regulators, as well as for a sharp cutoff.18) The
low-momentum interactions are derived from the N3LO chiral potential of Ref.19)
results for densities ρ . 0.05 fm−3, and we observe a good agreement for E/N with
the T = 10MeV virial result when the second-order contributions are included. In
the virial equation of state these contributions are included via the second-order
virial coefficient, while in the variational calculation the state dependence of such
correlations is only partly accounted for.16) Furthermore, the generic enhancement
of the effective mass at the Fermi surface leads to an enhancement of the entropy at
low temperatures above the variational and HF results.13), 16), 17)
§3. BCS gap in the 1S0 channel
We solve the BCS gap equation in the 1S0 channel
∆(k) = −
dp p2
Vlow k(k, p)∆(p)
ξ2(p) +∆2(p)
, (3.1)
with the (free-space) low-momentum NN interaction Vlow k(k, k
′). Here ξ(p) ≡ ε(p)−
µ, ε(p) = p2/2 and µ = k2F/2 (c = ~ = m = 1).
We find that the neutron-neutron BCS gap is practically independent of the NN
interaction.18) Consequently, 1S0 superfluidity is strongly constrained by the NN
scattering phase shifts. The maximal gap at the BCS level is ∆ ≈ 2.9− 3.0MeV for
kF ≈ 0.8−0.9 fm
−1. For the neutron-proton 1S0 case, we find somewhat larger gaps,
reflecting the charge dependence of realistic nuclear interactions.18)
In Fig. 2 we show the dependence of the neutron-neutron 1S0 superfluid pair-
ing gap on the cutoff starting from the N3LO chiral potential of Ref.19) for three
representative densities.18) We employed different smooth exponential regulators
f(k) = exp[−(k2/Λ2)n], as well as a sharp cutoff. As long as the cutoff is large com-
4 B. Friman, K. Hebeler, A. Schwenk and L. Tolós
pared to the dominant momentum components of the bound state (Λ > 1.2kF), the
gap depends very weakly on the cutoff. This shows that the 1S0 superfluid pairing
gap probes low-momentum physics. Below this scale, which depends on the density
and the smoothness of the regulator, the gap decreases, since the relevant momentum
components of the Cooper pair are then partly integrated out.
§4. Conclusions
In summary, we have studied the equation of state at finite temperature including
many-body contributions in a systematic approach. We have found good agreement
with the virial equation of state in the low-density–high-temperature regime. Ana-
lyzing the cutoff dependence of our results provides lower bounds for the theoretical
uncertainties. The possibility of estimating theoretical errors plays an important
role for reliable extrapolations to the extreme conditions reached in astrophysics.
In addition, we have shown that the 1S0 superfluid pairing gap in the BCS
approximation is practically independent of the choice of NN interaction, and there-
fore well constrained by the NN scattering data. This includes a very weak cutoff
dependence with low-momentum interactions Vlow k for sharp or sufficiently narrow
smooth regulators with Λ > 1.6 fm−1. At lower densities, it is possible to lower the
cutoff further to Λ > 1.2kF. Furthermore, the pairing gap clearly reflects the charge
dependence of nuclear interactions. The weak cutoff dependence indicates that, in
the 1S0 channel, the contribution of 3N interactions is small at the BCS level.
Acknowledgements
This work was supported in part by the Virtual Institute VH-VI-041 of the
Helmholtz Association, NSERC and US DOE Grant DE–FG02–97ER41014. TRI-
UMF receives federal funding via a contribution agreement through NRC.
References
1) J.M. Lattimer and M. Prakash, Astrophys. J. 550 (2001), 426.
2) A. Mezzacappa, Annu. Rev. Nucl. Part. Sci. 55 (2005), 467.
3) H.T. Janka, R. Buras, F.S. Kitaura Joyanes, A. Marek and M. Rampp, astro-ph/0405289.
4) Yu.A. Litvinov et al., Phys. Rev. Lett. 95 (2005), 042501.
5) F. Sarazin et al., Phys. Rev. C 70 (2004), 031302(R).
6) D.G. Yakovlev and C.J. Pethick, Ann. Rev. Astron. Astrophys. 42 (2004), 169.
7) S.K. Bogner, T.T.S. Kuo and A. Schwenk, Phys. Rep. 386 (2003), 1.
8) S.K. Bogner, A. Schwenk, R.J. Furnstahl and A. Nogga, Nucl. Phys. A 763 (2005), 59.
9) S.K. Bogner, R.J. Furnstahl, S. Ramanan and A. Schwenk, Nucl. Phys. A 773 (2006), 203.
10) A. Nogga, S.K. Bogner and A. Schwenk, Phys. Rev. C 70 (2004), 061002(R).
11) W. Kohn and J.M. Luttinger, Phys. Rev. 118 (1960), 41.
12) J.M. Luttinger and J.C. Ward, Phys. Rev. 118 (1960), 1417.
13) L. Tolós, B. Friman and A. Schwenk, nucl-th/0611070; and to be published.
14) C.J. Horowitz and A. Schwenk, Phys. Lett. B 638 (2006), 153.
15) B. Friedman and V.R. Pandharipande, Nucl. Phys. A 361 (1981), 502.
16) S. Fantoni, B.L. Friman and V.R. Pandharipande, Nucl. Phys. A 399 (1983), 51.
17) S. Fantoni, V.R. Pandharipande and K.E. Schmidt, Phys. Rev. Lett. 48 (1982), 878.
18) K. Hebeler, A. Schwenk and B. Friman, nucl-th/0611024, Phys. Lett. B (in press).
19) D.R. Entem and R. Machleidt, Phys. Rev. C 68 (2003), 041001(R).

  We present a perturbative calculation of the neutron matter equation of state
based on low-momentum two- and three-nucleon interactions. Our results are
compared to the model-independent virial equation of state and to variational
calculations, and we provide theoretical error estimates by varying the cutoff
used to regulate nuclear interactions. In addition, we study the dependence of
the BCS $^1$S$_0$ superfluid pairing gap on nuclear interactions and on the
cutoff. The resulting gaps are well constrained by the nucleon-nucleon
scattering phase shifts, and the cutoff dependence is very weak for sharp or
sufficiently narrow smooth regulators with cutoffs $\lm > 1.6 \fmi$.

Introduction
The determination of a reliable equation of state of nucleonic matter plays a
central role for the physics of neutron stars1) and core-collapse supernovae.2), 3) Fur-
thermore the superfluidity and superconductivity of neutrons and protons is an im-
portant phenomenon in nuclear many-body systems,4), 5) in particular for the cooling
of neutron stars.6) In this contribution, we present calculations of the neutron mat-
ter equation of state at finite temperature and of the 1S0 superfluid gap in the BCS
approximation based on low-momentum interactions.
Renormalization group methods coupled with effective field theory (EFT) offer
the possibility for a systematic approach to the equation of state. By evolving
nuclear forces to low-momentum interactions Vlow k
7)–9) with cutoffs around 2 fm−1,
the model-dependent short-range repulsion is integrated out and the resulting low-
momentum interactions are well constrained by the nucleon-nucleon (NN) scattering
data. Furthermore, the corresponding leading-order three-nucleon (3N) interactions
(based on chiral EFT) become perturbative in light nuclei for Λ . 2 fm−1.10)
With increasing density, Pauli blocking eliminates the shallow two-nucleon bound
and nearly-bound states, and the contribution of the particle-particle channel to bulk
properties becomes perturbative in nuclear matter.8) The Hartree-Fock approxima-
tion is then a good starting point for many-body calculations with low-momentum
NN and 3N interactions, and perturbation theory (in the sense of a loop expansion)
around the Hartree-Fock energy converges at moderate densities. This can be under-
stood quantitatively based on the behavior of the Weinberg eigenvalues as a function
of the cutoff and density.8), 9)
∗) E-mail address: b.friman@gsi.de
∗∗) E-mail address: k.hebeler@gsi.de
∗∗∗) E-mail address: schwenk@triumf.ca
†) E-mail address: tolos@fias.uni-frankfurt.de
typeset using PTPTEX.cls 〈Ver.0.9〉
http://arxiv.org/abs/0704.1404v1
2 B. Friman, K. Hebeler, A. Schwenk and L. Tolós
Some uncertainty remained concerning a possible dependence of the 1S0 pairing
gap on the input NN interaction in low-density neutron matter (kF < 1.6 fm
We address this point and explore the dependence of 1S0 superfluidity on nuclear
interactions at the BCS level in detail. We find that the BCS gap is well constrained
by the NN phase shifts. Therefore, any uncertainties are due to polarization (induced
interaction), dispersion and three-nucleon interaction effects.
§2. Equation of State of Neutron Matter
Using the Kohn-Luttinger-Ward theorem,11), 12) the perturbative expansion of
the free energy (at finite temperature) can be formulated as a loop expansion around
the Hartree-Fock (HF) energy. In this work, we include the first-order NN and 3N
contributions, as well as normal and anomalous second-order NN diagrams. Other
thermodynamic quantities are computed using standard thermodynamic relations.
0 0.05 0.1
ρ [fm-3]
virial
T=3 MeV
T=6 MeV
T=10 MeV
0 0.05 0.1 0.15
ρ [fm-3]
Hartree-Fock (NN+3N) HF + 2nd-order NN
Fig. 1. Energy per particle E/N as a function of the density ρ at first order (left panel) and
including second-order NN contributions (right panel).13)
The resulting energy per particle E/N as a function of the density ρ is shown
in Fig. 1 for a cutoff Λ = 2.1 fm−1 and temperatures T = 3, 6 and 10MeV.13) The
results presented in the left panel are the first-order NN and 3N contributions, and
those in the right panel includes all second-order diagrams with NN interactions.
For T = 6MeV, we also give a band spanned between Λ = 1.9 fm−1 (lower line) and
Λ = 2.5 fm−1 (upper line). The inclusion of second-order contributions significantly
reduces the cutoff dependence of the results. The model-independent virial equation
of state14) and the variational calculations of Friedman and Pandharipande (FP)15)
are displayed for comparison.
The inclusion of second-order correlations lowers the energy below the variational
Neutron Matter from Low-Momentum Interactions 3
0 1 2 3
Λ [fm
sharp
=0.8 fm
=0.4 fm
=1.35 fm
Fig. 2. The neutron-neutron 1S0 superfluid pairing gap ∆ as a function of the cutoff Λ for three
densities and different smooth exponential regulators, as well as for a sharp cutoff.18) The
low-momentum interactions are derived from the N3LO chiral potential of Ref.19)
results for densities ρ . 0.05 fm−3, and we observe a good agreement for E/N with
the T = 10MeV virial result when the second-order contributions are included. In
the virial equation of state these contributions are included via the second-order
virial coefficient, while in the variational calculation the state dependence of such
correlations is only partly accounted for.16) Furthermore, the generic enhancement
of the effective mass at the Fermi surface leads to an enhancement of the entropy at
low temperatures above the variational and HF results.13), 16), 17)
§3. BCS gap in the 1S0 channel
We solve the BCS gap equation in the 1S0 channel
∆(k) = −
dp p2
Vlow k(k, p)∆(p)
ξ2(p) +∆2(p)
, (3.1)
with the (free-space) low-momentum NN interaction Vlow k(k, k
′). Here ξ(p) ≡ ε(p)−
µ, ε(p) = p2/2 and µ = k2F/2 (c = ~ = m = 1).
We find that the neutron-neutron BCS gap is practically independent of the NN
interaction.18) Consequently, 1S0 superfluidity is strongly constrained by the NN
scattering phase shifts. The maximal gap at the BCS level is ∆ ≈ 2.9− 3.0MeV for
kF ≈ 0.8−0.9 fm
−1. For the neutron-proton 1S0 case, we find somewhat larger gaps,
reflecting the charge dependence of realistic nuclear interactions.18)
In Fig. 2 we show the dependence of the neutron-neutron 1S0 superfluid pair-
ing gap on the cutoff starting from the N3LO chiral potential of Ref.19) for three
representative densities.18) We employed different smooth exponential regulators
f(k) = exp[−(k2/Λ2)n], as well as a sharp cutoff. As long as the cutoff is large com-
4 B. Friman, K. Hebeler, A. Schwenk and L. Tolós
pared to the dominant momentum components of the bound state (Λ > 1.2kF), the
gap depends very weakly on the cutoff. This shows that the 1S0 superfluid pairing
gap probes low-momentum physics. Below this scale, which depends on the density
and the smoothness of the regulator, the gap decreases, since the relevant momentum
components of the Cooper pair are then partly integrated out.
§4. Conclusions
In summary, we have studied the equation of state at finite temperature including
many-body contributions in a systematic approach. We have found good agreement
with the virial equation of state in the low-density–high-temperature regime. Ana-
lyzing the cutoff dependence of our results provides lower bounds for the theoretical
uncertainties. The possibility of estimating theoretical errors plays an important
role for reliable extrapolations to the extreme conditions reached in astrophysics.
In addition, we have shown that the 1S0 superfluid pairing gap in the BCS
approximation is practically independent of the choice of NN interaction, and there-
fore well constrained by the NN scattering data. This includes a very weak cutoff
dependence with low-momentum interactions Vlow k for sharp or sufficiently narrow
smooth regulators with Λ > 1.6 fm−1. At lower densities, it is possible to lower the
cutoff further to Λ > 1.2kF. Furthermore, the pairing gap clearly reflects the charge
dependence of nuclear interactions. The weak cutoff dependence indicates that, in
the 1S0 channel, the contribution of 3N interactions is small at the BCS level.
Acknowledgements
This work was supported in part by the Virtual Institute VH-VI-041 of the
Helmholtz Association, NSERC and US DOE Grant DE–FG02–97ER41014. TRI-
UMF receives federal funding via a contribution agreement through NRC.
References
1) J.M. Lattimer and M. Prakash, Astrophys. J. 550 (2001), 426.
2) A. Mezzacappa, Annu. Rev. Nucl. Part. Sci. 55 (2005), 467.
3) H.T. Janka, R. Buras, F.S. Kitaura Joyanes, A. Marek and M. Rampp, astro-ph/0405289.
4) Yu.A. Litvinov et al., Phys. Rev. Lett. 95 (2005), 042501.
5) F. Sarazin et al., Phys. Rev. C 70 (2004), 031302(R).
6) D.G. Yakovlev and C.J. Pethick, Ann. Rev. Astron. Astrophys. 42 (2004), 169.
7) S.K. Bogner, T.T.S. Kuo and A. Schwenk, Phys. Rep. 386 (2003), 1.
8) S.K. Bogner, A. Schwenk, R.J. Furnstahl and A. Nogga, Nucl. Phys. A 763 (2005), 59.
9) S.K. Bogner, R.J. Furnstahl, S. Ramanan and A. Schwenk, Nucl. Phys. A 773 (2006), 203.
10) A. Nogga, S.K. Bogner and A. Schwenk, Phys. Rev. C 70 (2004), 061002(R).
11) W. Kohn and J.M. Luttinger, Phys. Rev. 118 (1960), 41.
12) J.M. Luttinger and J.C. Ward, Phys. Rev. 118 (1960), 1417.
13) L. Tolós, B. Friman and A. Schwenk, nucl-th/0611070; and to be published.
14) C.J. Horowitz and A. Schwenk, Phys. Lett. B 638 (2006), 153.
15) B. Friedman and V.R. Pandharipande, Nucl. Phys. A 361 (1981), 502.
16) S. Fantoni, B.L. Friman and V.R. Pandharipande, Nucl. Phys. A 399 (1983), 51.
17) S. Fantoni, V.R. Pandharipande and K.E. Schmidt, Phys. Rev. Lett. 48 (1982), 878.
18) K. Hebeler, A. Schwenk and B. Friman, nucl-th/0611024, Phys. Lett. B (in press).
19) D.R. Entem and R. Machleidt, Phys. Rev. C 68 (2003), 041001(R).
