# Neutron Star Matter as a Relativistic Fermi Liquid

Bengt Friman  GSI Helmholtzzentrum für Schwerionenforschung, D-64291
Darmstadt, Germany  ExtreMe Matter Institute (EMMI) at GSI, D-64291 Darmstadt,
Germany  Wolfram Weise  Physics Department, Technical University of Munich,
D-85748 Garching, Germany  ExtreMe Matter Institute (EMMI) at GSI, D-64291
Darmstadt, Germany  Department of Physics, The University of Tokyo, 7-3-1
Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

###### Abstract

The equation of state (EoS) of neutron star matter, constrained by the
existence of two-solar-mass stars and gravitational wave signals from neutron
star mergers, is analysed using the Landau theory of relativistic Fermi
liquids. While the phase diagram of dense and cold QCD matter is still open
for scenarios ranging from hadronic to quark matter in the center of neutron
stars, a Fermi-liquid treatment is motivated by a microscopic approach
starting from a chiral nucleon-meson field theory combined with
nonperturbative functional renormalization group methods. In this scheme
effects of multipionic fluctuations and repulsive nuclear many-body
correlations suggest that the transition to chiral symmetry restoration is
shifted to densities above those typically encountered in the neutron star
core. Under such conditions a Fermi-liquid description in terms of nucleon
quasiparticles appears to be justified. The leading Landau parameters are
derived and discussed. Our results are contrasted with a well-known Fermi
liquid, namely liquid 3He.

##  I Introduction

The physics of neutron stars entered a new era Watts2016 ; Ozel2016  with the
observation of massive (two-solar-mass) pulsars Demorest2010 ; Fonseca2016 ;
Antoniadis2013 , which set stringent constraints on the stiffness of the
equation of state (EoS) of cold and dense baryonic matter, further stressed by
the recent Shapiro delay measurement Cromartie2019  of a neutron star with
mass $M\approx 2.17M_{\odot}$. The detection of gravitational wave signals
from two merging neutron stars Abbott2017  has furnished additional important
information on the EoS, by providing limits for the tidal deformability and
neutron star radii Most2018 ; De2018 .

The composition and properties of strongly interacting matter at high
densities and temperatures is a topic of continuing interest. Lattice QCD
provides information on the properties of QCD matter at high temperatures and
vanishing or small baryon chemical potential. However, owing to the sign
problem, lattice QCD cannot at present reliably address the physics of highly
compressed cold matter as it is realized in the central region of neutron
stars. In this context, a broad variety of options are under discussion. A
viable hypothesis invokes the description of matter in the core of neutron
stars in terms of hadronic degrees of freedom (baryons and mesons) with strong
many-body correlations APR1998 . Alternative descriptions involve a (possibly
smooth) transition from hadronic matter to some form of quark matter Baym2018
; McLerran2018 . The former option suggests that neutron star matter can be
viewed as a relativistic Fermi liquid Landau1 ; Landau2 ; Landau3 , composed
of neutron quasiparticles plus a small fraction of protons. This is the basic
theme of the present study, in which we explore how neutron star matter
behaves as a strongly coupled fermionic many-body system and how it compares
with textbook Fermi systems Abrikosov1959 ; BP1991 , such as liquid 3He.
Relativistic Landau Fermi-liquid theory BC1976  is an appropriate framework to
address such questions.

In the present study we are guided by a microscopic equation of state of
neutron star matter derived from a chiral nucleon-meson (ChNM) field theory
within the nonperturbative functional renormalization group (FRG) framework
DW2015 ; DW2017 . The low-density neutron star crust region is parametrized by
the Skyrme-Lyon (SLy) EoS DH2001 . The ChNM-FRG equation of state takes over
at baryon densities $\rho>0.3~{}\rho_{0}$ (with $\rho_{0}=0.16$ fm-3, the
equilibrium density of normal nuclear matter) and extends into the inner core
region of the neutron star. Beta equilibrium conditions with both electrons
and muons are routinely incorporated and imply a proton fraction of about 5%
in the core. The resulting EoS satisfies all empirical constraints from
astrophysical observations. It is also constructed such that properties of
symmetric and asymmetric nuclear matter are consistent with nuclear physics
constraints (e.g., the binding energy and equilibrium density of $N=Z$ nuclear
matter, the compressibility modulus, the critical temperature and density of
the liquid-gas phase transition, etc.) DW2017 .

For neutron star matter, the presence of hyperons (in particular $\Lambda$
hyperons) would soften the equation of state such that a maximum mass around
$2\,M_{\odot}$ could not possibly be reached Djapo:2008au ; Lonardoni:2014bwa
. Several recent and ongoing investigations Lonardoni:2014bwa ;
Haidenbauer:2016vfq  point out that repulsive hyperon-nuclear three-body
forces may be capable of preventing the appearance of hyperons in neutron
stars alltogether and thus maintain the necessary stiffness of the EoS.

Given its nonperturbative nature, the ChNM-FRG equation of state is in
principle not limited to low densities, in contrast to the EoS derived from
(perturbative) chiral effective field theory (ChEFT). The two schemes are
mutually consistent in the low-density range, $\rho\lesssim 2\,\rho_{0}$. The
window of applicability for the ChNM-FRG model potentially extends
considerably beyond these densities. Its limitation is ultimately determined
by the transition from the (spontaneously broken) Nambu-Goldstone phase to the
(restored) Wigner-Weyl realization of chiral symmetry. In the presence of
multipion fluctuations and nuclear many-body correlations, all treated
nonperturbatively by solving the FRG equations, it turns out that this
transition from spontaneously broken to restored chiral symmetry is, in this
model, shifted to baryon densities well above five times $\rho_{0}$. For
comparison, the central density of a $2\,M_{\odot}$ neutron star, computed
with such a model, does not exceed about $5\,\rho_{0}$. In such an approach
the neutron star core is thus composed entirely of “nonexotic” (nucleonic and
pionic) degrees of freedom in the presence of strongly repulsive correlations.

The input for the effective action of the ChNM-FRG model DW2015 ; DW2017  is
prepared such that it is consistent with well-known ground state properties of
nuclear matter, nuclear thermodynamics including the critical point of the
liquid-gas phase transition Elliot2013 , and ab initio neutron matter
computations using realistic nucleon-nucleon interactions Gandolfi2014 ;
Heb2013 ; HK2017 ; LH2018 ; Dri2019 . With isospin-dependent interactions that
reproduce the empirical asymmetry energy of $(32\pm 2)$ MeV at nuclear
saturation density Baldo2016 , the resulting EoS does indeed yield stable
neutron stars that satisfy the $2\,M_{\odot}$ constraint. Moreover, it is
located well within the band of equations of state, $P({\cal E})$ [pressure as
a function of energy density] that have been extracted in refs. Annala2018 ;
Vuorinen2018 . It is also consistent with the EoS band deduced by the LIGO and
Virgo Collaborations from the gravitational wave signals generated by the
neutron star merger event GW170817 Abbott2018 , as well as with the EoS
recently deduced from neutron star data using neural network techniques
FFM2019 .

The emerging picture of a neutron star with such a generic chiral EoS is that
of a relativistic liquid of nucleons (primarily neutrons) correlated by strong
multipion fields and short-distance repulsive forces. A description in terms
of fermionic quasiparticles thus suggests a treatment using Landau Fermi
liquid theory. The (small) proton fraction in neutron star matter plays a non-
negligible role in quantitative astrophysical considerations. It is of minor
importance, however, for our purpose of studying Fermi-liquid properties which
can therefore be focused primarily on pure neutron matter. The aim of the
present work is therefore to identify quasiparticle properties of the neutrons
as the dominant Fermionic degrees of freedom in the microscopic chiral FRG
equation of state DW2015 ; DW2017  that satisfies all observational
constraints, and to study and interpret the corresponding Fermi liquid
properties.

In the following sections we prepare the framework for such a Fermi-liquid
description. We then proceed to deduce and interpret the lowest Landau
parameters of the spin-independent quasiparticle interaction. These quantify
the strength of the correlations and characterize the bulk properties of
matter in the deep interior of neutron stars.

##  II Landau theory of relativistic Fermi liquids

###  II.1 Reminder of Landau Fermi-liquid theory

It is useful to start by first recalling the nonrelativistic theory. For
nuclear many-body systems, this limit is realized at low baryon densities
$\rho$ where the Fermi momentum, $p_{F}=(6\pi^{2}\rho/\nu)^{1/3}$ (with spin-
isospin degeneracy $\nu=2$ for neutron matter and $\nu=4$ for symmetric
nuclear matter), is small compared to the quasiparticle effective mass.

Most of our discussion will be restricted to vanishing temperatures, i.e.,
$T=0$. Superfluidity is ignored since the EoS in the density range of interest
is practically unaffected by pairing. In Landau’s theory of normal Fermi
liquids Landau1 ; Landau2 ; Landau3 ; Abrikosov1959 ; BP1991 , the variation
of the energy of the system with changes of the quasiparticle occupation
numbers is given by

$\delta E=\sum_{p}\varepsilon_{p}\,\delta
n_{p}+\frac{1}{2\,V}\,\sum_{p,p^{\prime}}{\cal F}_{pp^{\prime}}\,\delta
n_{p}\,\delta n_{p^{\prime}}~{}.$ (1)

Here $V$ is the volume, $\varepsilon_{p}$ is the quasiparticle energy, ${\cal
F}_{pp^{\prime}}$ is the quasiparticle interaction and $\delta
n_{p}=n_{p}-n^{(0)}_{p}$ is the deviation of the quasiparticle distribution
function from the ground state distribution

$n^{(0)}_{p}=\left\\{\begin{array}[]{cc}1,&~{}~{}~{}\varepsilon_{p}<\mu{}\\\
0,&~{}~{}~{}\varepsilon_{p}>\mu~{},\end{array}\right.$ (2)

where $\mu$ is the chemical potential, or equivalently the energy of a
quasiparticle on the Fermi surface. Consequently, in the ground state of a
uniform system, the distribution function equals unity for quasiparticle
momenta below the Fermi momentum $p_{F}$ and vanishes for momenta above
$p_{F}$. The energy of a quasiparticle with momentum $p$ is given by the first
variation of the energy with respect to the occupation number $n_{p}$

$\varepsilon_{p}=\frac{\delta E}{\delta n_{p}}\,,$ (3)

while the quasiparticle interaction is determined by the second variation

${\cal F}_{pp^{\prime}}=V\frac{\delta^{2}E}{\delta n_{p}\,\delta
n_{p\prime}}=V\frac{\delta\varepsilon_{p}}{\delta n_{p^{\prime}}}\,.$ (4)

For the low-lying excitations of interest in Fermi-liquid theory, the relevant
quasiparticle states are near the Fermi surface. Hence, in the quasiparticle
interaction one in general can set
$|\boldsymbol{p}|=|\boldsymbol{p}^{\prime}|=p_{F}$. The velocity of a
quasiparticle on the Fermi surface is given by

$v_{F}=\left(\frac{\partial\varepsilon_{p}}{\partial p}\right)_{p=p_{F}},$ (5)

and defines the quasiparticle effective mass $m^{*}$ through

$v_{F}=\frac{p_{F}}{m^{*}}\,.$ (6)

Thus, near the Fermi surface, the quasiparticle energy takes the form

$\varepsilon_{p}=\mu+v_{F}(p-p_{F})\,.$ (7)

The density of quasiparticle states at the Fermi surface is given by

$N(0)=\frac{1}{V}\sum_{p}\delta(\varepsilon_{p}-\mu)\,.$ (8)

Replacing the sum over $p$ by an integral (including the sums over spin and
isospin degrees of freedom), one finds:

$N(0)=\frac{\nu\,m^{*}p_{F}}{2\pi^{2}}\,.$ (9)

In particular, for neutron matter, $N(0)=m^{*}p_{F}/\pi^{2}$.

We now focus on pure neutron matter. For simplicity we ignore noncentral
forces (e.g. spin-orbit interactions). These are non-leading effects in
neutron matter which contribute only in $p$\- and higher partial waves. With
these restrictions the spin-dependent quasiparticle interaction is of the form

${\cal
F}_{pp^{\prime}}=f_{pp^{\prime}}+g_{pp^{\prime}}\,\boldsymbol{\sigma}\cdot\boldsymbol{\sigma^{\prime}}.$
(10)

The momentum dependence of the functions $f_{pp^{\prime}}$ and
$g_{pp^{\prime}}$ is expanded in Legendre polynomials according to

$f_{pp^{\prime}}=\sum_{\ell=0}^{\infty}f_{\ell}\,P_{\ell}(\cos\theta)\,,~{}~{}~{}g_{pp^{\prime}}=\sum_{\ell=0}^{\infty}g_{\ell}\,P_{\ell}(\cos\theta)~{},$
(11)

where $\theta$ is the angle between the two momenta $\boldsymbol{p}$ and
$\boldsymbol{p}^{\prime}$. The coefficients in this expansion are the Landau
Fermi-liquid parameters. It is useful to define dimensionless Landau
parameters:

$F_{\ell}=N(0)\,f_{\ell}~{},~{}~{}~{}G_{\ell}=N(0)\,g_{\ell}~{}.$ (12)

In the present context, as we focus on the ground state of spin-saturated
neutron matter, only the spin-independent parameters $F_{\ell}$ are of prime
interest.

Basic properties of the Fermi liquid can be expressed in terms of the first
few Landau parameters. For example, the quasiparticle effective mass, $m^{*}$,
is given by the spin-independent Landau parameter $F_{1}$ which is a measure
of the velocity dependence of the quasiparticle interaction:

$\frac{m^{*}}{M_{0}}=1+\frac{F_{1}}{3}\,,$ (13)

with the free (vacuum) particle mass $M_{0}$. The specific heat of a Fermi
liquid at low temperatures is determined by the quasiparticle effective mass
as follows:

$c_{V}=\frac{m^{*}p_{F}}{3}\,T.$ (14)

Finally, the incompressibility of the Fermi liquid is given by

$K=9\rho\,\frac{\partial^{2}{\cal
E}}{\partial\rho^{2}}=6\frac{p_{F}^{2}}{2m^{*}}(1+F_{0})=\frac{3\,p_{F}^{2}}{M_{0}}\,\,\frac{1+F_{0}}{1+F_{1}/3},$
(15)

where ${\cal E}=E/V$ is the energy density, and the speed of (first) sound in
the system is given by

$c_{1}^{2}=\frac{p_{F}^{2}}{3\,M_{0}^{2}}\,\,\frac{1+F_{0}}{1+F_{1}/3}.$ (16)

This is the nonrelativistic sound speed with $c_{1}\ll 1$.

###  II.2 Relativistic Fermi liquids

Baryonic matter at the high densities encountered in the core of neutron stars
makes a relativistic treatment mandatory. For example, the Fermi momentum in
neutron matter at $\rho=5\,\rho_{0}$ is $p_{F}\approx 0.57$ GeV, i.e., of a
magnitude comparable to the effective mass. In a relativistic Fermi liquid
BC1976 , the speed of sound is given by

$c_{1}^{2}=\frac{\partial P}{\partial{\cal
E}}=\frac{\rho}{\mu}\frac{\partial\mu}{\partial\rho},$ (17)

where $P$ is the pressure, ${\cal E}$ the energy density, $\mu$ the baryon
chemical potential, and we have used $dP=\rho\,d\mu$ and $d{\cal
E}=\mu\,d\rho$. With

$\rho=\frac{1}{V}\sum_{p}\theta(\mu-\varepsilon_{p}),$ (18)

one finds that

$\frac{\partial\rho}{\partial\mu}=\frac{1}{V}\sum_{p}\delta(\mu-\varepsilon_{p})\left(1-\frac{\partial\varepsilon_{p}}{\partial\mu}\right).$
(19)

Now, at zero temperature we have

$\frac{\partial\varepsilon_{p}}{\partial\mu}=\frac{\partial\varepsilon_{p}}{\partial\rho}\frac{\partial\rho}{\partial\mu}.$
(20)

In order to compute $\partial\varepsilon_{p}/\partial\rho$, we introduce a
variation of the quasiparticle occupation number, which is spin independent
and spherically symmetric, i.e., $\delta n_{p}=\eta\,\delta(p-p_{F})$, and
satisfies

$\delta\rho=\frac{1}{V}\sum_{p}\delta n_{p}=\frac{\nu}{(2\pi)^{3}}\int
d^{3}p\,\eta\,\delta(p-p_{F})=\frac{\nu\,\eta\,p_{F}^{2}}{2\pi^{2}}.$ (21)

From (4) and (11) it follows that

$\delta\varepsilon_{p}=\frac{1}{V}\sum_{p^{\prime}}{\cal
F}_{pp^{\prime}}\,\delta
n_{p^{\prime}}=\frac{\nu\,\eta\,p_{F}^{2}}{2\pi^{2}}f_{0}~{},$ (22)

and consequently that

$\frac{\partial\varepsilon_{p}}{\partial\rho}=f_{0}~{}.$ (23)

Inserting (23) and (20) in (19) and solving for $\partial\rho/\partial\mu$,
one finds

$\frac{\partial\rho}{\partial\mu}=\frac{N(0)}{1+N(0)f_{0}}=\frac{p_{F}\,m^{*}}{\pi^{2}(1+F_{0})}\,.$
(24)

We note that the definiton of the Landau effective mass,
$m^{*}=p_{F}\left(\frac{\partial\varepsilon_{p}}{\partial
p}\right)^{-1}_{p=p_{f}}$ remains unchanged in the relativistic formulation,
while the corresponding effective mass relation is given by BC1976

$\frac{m^{*}}{\mu}=1+F_{1}/3\,.$ (25)

Finally the squared speed of sound becomes

$c_{1}^{2}=\frac{p_{F}^{2}}{3\,\mu\,m^{*}}(1+F_{0})=\frac{p_{F}^{2}}{3\,\mu^{2}}\,\,\frac{1+F_{0}}{1+F_{1}/3}\,,$
(26)

and the incompressibility is

$K=6\,\frac{p_{F}^{2}}{2\mu}\,\frac{1+F_{0}}{1+F_{1}/3}=9\mu\,c_{1}^{2}\,.$
(27)

One notes that in these expressions, the relativistic treatment simply
replaces the Fermion mass $M_{0}$ in the nonrelativistic forms by the baryon
chemical potential

$\mu=\frac{\partial{\cal E}}{\partial\rho}=\varepsilon_{p=p_{F}}\,.$ (28)

###  II.3 Relativistic quasiparticles

Central to Landau Fermi-liquid theory is the notion of quasiparticles dressed
by their interactions with the surrounding many-body system. Consider first a
simplified example with fermions moving in a scalar field. This scalar field
modifies the fermion mass at nonzero densities. The quasiparticle energy is
given by

$\varepsilon_{p}=\sqrt{p^{2}+M^{2}}~{},$ (29)

where $M(\rho)$ is a function of the scalar field and thus of the density,
with $M(\rho=0)\equiv M_{0}$, the vacuum mass. The variation of the energy of
the system is again given by (1). Using (29) and (4) we find

$\delta E=\sum_{p}\sqrt{p^{2}+M^{2}}\,\delta
n_{p}+\frac{1}{2}\sum_{pp^{\prime}}\frac{\delta\sqrt{p^{2}+M^{2}}}{\delta
n_{p^{\prime}}}\,\delta n_{p}\,\delta n_{p^{\prime}}~{}.$ (30)

Thus, in this model the quasiparticle interaction is

${\cal F}_{pp^{\prime}}=V\,\frac{M}{\sqrt{p^{2}+M^{2}}}\,\frac{\delta
M}{\delta n_{p^{\prime}}}~{}.$ (31)

Using

$\frac{\delta M}{\delta n_{p^{\prime}}}=\frac{\partial
M}{\partial\rho}\,\frac{\delta\rho}{\delta n_{p^{\prime}}}~{}.$ (32)

together with the first equality in (21),

$\frac{\delta\rho}{\delta n_{p^{\prime}}}=\frac{1}{V}~{},$ (33)

one finds:

${\cal
F}_{pp^{\prime}}=\frac{M(\rho)}{\sqrt{p_{F}^{2}+M^{2}(\rho)}}\,\frac{\partial
M}{\partial\rho}~{},$ (34)

where we have put the quasiparticles on the Fermi surface. Since the
quasiparticle interaction is independent of the angle between $\boldsymbol{p}$
and $\boldsymbol{p}^{\prime}$ and independent of spin, the only nonzero Fermi-
liquid parameter is $f_{0}$. This means that, in particular, $F_{1}=0$ and
that the effective mass at the Fermi surface is equal to the chemical
potential:

$m^{*}=\mu=\sqrt{p_{F}^{2}+M^{2}(\rho)}.$ (35)

Hence, the dimensionless Fermi-liquid parameter is given by

$F_{0}=\frac{p_{F}\,M}{\pi^{2}}\,\frac{\partial
M}{\partial\rho}=\frac{M}{p_{F}}\,\frac{\partial M}{\partial p_{F}}$ (36)

and the squared speed of sound is

$c_{1}^{2}=\frac{p_{F}^{2}}{3\,(p_{F}^{2}+M^{2})}\left(1+\frac{p_{F}\,M}{\pi^{2}}\frac{\partial
M}{\partial\rho}\right)\,.$ (37)

The in-medium nucleon mass $M(\rho)$ is usually a decreasing function of
density. With $\partial M/\partial\rho<0$, the Landau parameter $F_{0}$ stays
negative. The squared speed of sound will always be $c_{1}^{2}<1/3$ and
approach the limit of a free ultra-relativistic gas from below. As we shall
see, the equation of state of neutron star matter implies instead
$c_{1}^{2}>1/3$ starting from some intermediate density, reflecting
increasingly strong repulsive correlations between the quasiparticles as the
density increases.

In a relativistic theory, a repulsive short-distance interaction is most
naturally viewed as being mediated by a vector field. Consider therefore
quasiparticles interacting with a vector field $V^{\mu}$ in addition to the
scalar field. The vector field is in turn assumed to be generated by the
baryon four-current $j^{\mu}=(\rho,\boldsymbol{j})$:

$V^{\mu}=(V_{0},\boldsymbol{V})=h\,j^{\mu}\,.$ (38)

In the mean-field approximation the vector field is a linear function of the
current, i.e., $h$ is a constant. Effects beyond the mean-field limit imply a
nonlinear behavior that can be incorporated by a generalized ansatz:

$V^{\mu}=h(j^{2})\,j^{\mu}\,,$ (39)

with $h$ now assumed to be a function of the Lorentz invariant
$j^{2}=j_{\mu}\,j^{\mu}$. In the rest frame of a system in its ground state,
only the zeroth component of the baryon current, $j^{0}=\rho$, is nonzero. The
current in the frame of an observer moving relative to the system is obtained
by a Lorentz transformation.

In a general frame, the quasiparticle energy in this model can be written:

$\varepsilon_{p}=\sqrt{(\boldsymbol{p}-\boldsymbol{V})^{2}+M^{2}}+V_{0}~{},$
(40)

where $\boldsymbol{p}-\boldsymbol{V}$ is the kinetic momentum. Thus, the
velocity of a quasiparticle is given by

$\boldsymbol{v}_{p}=\frac{\boldsymbol{p}-\boldsymbol{V}}{\sqrt{(\boldsymbol{p}-\boldsymbol{V})^{2}+M^{2}}}~{}.$
(41)

In the rest frame of the system, (40) reduces to

$\varepsilon_{p}=\sqrt{\boldsymbol{p}^{2}+M^{2}}+V_{0}~{},$ (42)

In general, a variation of the occupation number leads to nonzero spatial
components of the baryon current,

$\boldsymbol{j}=\frac{1}{V}\sum_{p}\boldsymbol{v}_{p}\,n_{p}~{}.$ (43)

The quasiparticle interaction is again obtained by varying the quasiparticle
energy with respect to the occupation number:

${\cal F}_{pp^{\prime}}=V\,\frac{\delta\,\varepsilon_{p}}{\delta
n_{p^{\prime}}}~{}.$ (44)

The following discussion is similar to that of Ref. Matsui1981 , although our
assumptions are more general. Given the quasiparticle energy (40), one thus
finds:

$\frac{f_{pp^{\prime}}}{V}=\frac{M}{\sqrt{(\boldsymbol{p}-\boldsymbol{V})^{2}+M^{2}}}\left(\frac{\delta
M}{\delta n_{p^{\prime}}}\right)+\frac{\delta V_{0}}{\delta
n_{p^{\prime}}}-\frac{\boldsymbol{p}-\boldsymbol{V}}{\sqrt{(\boldsymbol{p}-\boldsymbol{V})^{2}+M^{2}}}\left(\frac{\delta\boldsymbol{V}}{\delta
n_{p^{\prime}}}\right)~{}.$ (45)

At this level there are only spin-independent contributions to the
quasiparticle interaction. In the rest frame of the fluid, the first term in
(45) reduces to the result already found in Eq. (34),

$f_{pp^{\prime}}(1)=\frac{M}{\sqrt{\boldsymbol{p}^{2}+M^{2}}}\,\frac{\partial
M}{\partial\rho}~{}.$ (46)

The variation of the zeroth component of the vector field yields

$\frac{\delta V_{0}}{\delta n_{p^{\prime}}}=\frac{\partial
V_{0}}{\partial\rho}\,\frac{\delta\rho}{\delta n_{p^{\prime}}}+\frac{\partial
V_{0}}{\partial\boldsymbol{j}}\,\frac{\delta\boldsymbol{j}}{\delta
n_{p^{\prime}}}~{}.$ (47)

The second term in (47) vanishes in the rest frame of the system, since

$\frac{\partial
V_{0}}{\partial\boldsymbol{j}}=-h^{\prime}(j^{2})\,\boldsymbol{j}\,\rho~{},$
(48)

and $\boldsymbol{j}=0$ in that frame. Hence, using (33), we find

$\frac{\delta V_{0}}{\delta n_{p^{\prime}}}=\frac{1}{V}\,\frac{\partial
V_{0}}{\partial\rho}~{},$ (49)

and for the contribution of the second term to the quasiparticle interaction
(45):

$f_{pp^{\prime}}(2)=\frac{\partial V_{0}}{\partial\rho}~{}.$ (50)

The detailed derivation of the last term on the right-hand side. of Eq. (45)
is relegated to Appendix A. Its contribution to the quasiparticle interaction
is:

$f_{pp^{\prime}}(3)=-h(j^{2})\frac{\boldsymbol{p}\cdot\boldsymbol{p}^{\prime}}{\mu\,\sqrt{p_{F}^{2}+M^{2}}}~{},$
(51)

Now, collecting all pieces, the resulting Landau Fermi-liquid parameters are
given by

$\displaystyle F_{0}$ $\displaystyle=$
$\displaystyle\frac{p_{F}\,}{\pi^{2}}\,\left(M(\rho)\,\frac{\partial
M}{\partial\rho}+\sqrt{p_{F}^{2}+M^{2}(\rho)}\,\,\frac{\partial
V_{0}}{\partial\rho}\right)\,~{},$ (52) $\displaystyle F_{1}$ $\displaystyle=$
$\displaystyle-3\,\frac{h(j^{2})\,\rho}{\mu}=-\frac{3V_{0}(\rho)}{\mu}~{}.$
(53)

An equivalent way of deriving the result for $F_{0}$ proceeds directly through
the derivative of the chemical potential with the assumed ansatz

$\mu=\sqrt{p_{F}^{2}+M^{2}(\rho)}+V_{0}(\rho)\equiv
m^{*}(\rho)+V_{0}(\rho)\,.$ (54)

Observing that

$\frac{\partial\mu}{\partial\rho}=\frac{1}{m^{*}}\left[\frac{\pi^{2}}{p_{F}}+M\frac{\partial
M}{\partial\rho}\right]+\frac{\partial V_{0}}{\partial\rho}\,,$ (55)

leads to

$1+F_{0}(\rho)=\frac{p_{F}\,m^{*}}{\pi^{2}}\left(\frac{\partial\mu}{\partial\rho}\right)\,,$
(56)

[see also Eq. (24)], while the relation for $F_{1}$,

$1+\frac{F_{1}(\rho)}{3}=1-\frac{V_{0}(\rho)}{\mu}=\frac{m^{*}}{\mu}\,,$ (57)

is consistent with the nonrelativistic Eq. (13).

##  III Neutron Star Equation of State and Fermi-Liquid Theory

###  III.1 Chiral FRG equation of state

The starting point is now an equation of state, pressure $P({\cal E})$ as a
function of energy density ${\cal E}$, derived from a chiral field theory of
nucleons and mesons (the ChNM model), based on a linear sigma model with a
nonlinear effective potential. The basic Lagrangian involves the isospin-
doublet field of the nucleon, $N=(p,n)^{\top}$, and the chiral boson field
$\phi=(\sigma,\boldsymbol{\pi})$ composed of a heavy scalar $\sigma$ and the
pseudoscalar Nambu-Goldstone boson $\boldsymbol{\pi}$ of spontaneously broken
chiral $SU(2)_{L}\times SU(2)_{R}$ symmetry:

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\bar{N}\left[i\gamma_{\mu}\partial^{\mu}-g(\sigma+i\gamma_{5}\,\boldsymbol{\tau\cdot\pi})\right]N$
(58) $\displaystyle+$
$\displaystyle{\frac{1}{2}}\left(\partial_{\mu}\sigma\partial^{\mu}\sigma+\partial_{\mu}\boldsymbol{\pi}\cdot\partial^{\mu}\boldsymbol{\pi}\right)-{\cal
U}(\sigma,\boldsymbol{\pi})+\Delta{\cal L}~{}.$

The $\Delta{\cal L}$ term of this Lagrangian represents short-distance
dynamics expressed in terms of isoscalar and isovector vector fields coupled
to nucleons, corresponding to contact interactions in chiral effective field
theory (ChEFT). The potential ${\cal U}(\sigma,\boldsymbol{\pi})$ is written
as a polynomial up to fourth order in the chiral invariant,
$\chi\equiv\phi^{\dagger}\phi=\sigma^{2}+\boldsymbol{\pi}^{2}$, plus a
symmetry breaking piece proportional to $m_{\pi}^{2}\sigma$. This potential is
constructed such as to be consistent with pion-nucleon data and selected
ground state properties of nuclear matter. (For details see Refs. DW2015 ;
DW2017  and a brief overview in Appendix B.)

The action $S=\int d^{4}x\,{\cal L}$ of the ChNM Lagrangian (58) serves as
input for a functional renormalization group (FRG) calculation starting at a
UV scale (the chiral symmetry breaking scale $4\pi f_{\pi}\approx 1$ GeV, with
the pion decay constant $f_{\pi}\approx 0.09$ GeV). This is then evolved down
to the full effective action at the low momentum (IR) limit using the FRG flow
equations Wet1993 . The grand-canonical potential $\Omega(T,\mu)$ is
constructed DW2015 ; DW2017 , from which the pressure $P=-\Omega$ and the
energy density ${\cal E}=-P+\mu\rho$ (at $T=0$) are derived, resulting in the
equation of state $P({\cal E})$.

Figure 1 shows this equation of state, including an estimated band of
uncertainties. These uncertainties arise primarily from the input parameters
of the ChNM model, in particular from varying the nuclear asymmetry energy,
$A_{S}$, in the empirical range between 30 and 34 MeV. The solid curve with
$A_{S}=32$ MeV serves as our prototype EoS, referred to as the chiral FRG
equation of state in the following. A rough estimate of uncertainties in the
pressure $P$ amounts to $\pm 15\,\%$. For practical purposes, an accurate Padé
fit and a table of $P({\cal E})$ are given in Appendix C.

The squared velocity of (first) sound (17) is shown in Fig. 2. Notably, the
sound speed exceeds its canonical value for a noninteracting ultrarelativistic
Fermi gas $c_{1}^{2}=1/3$ at densities $\rho\gtrsim 4\,\rho_{0}$, mainly as a
consequence of repulsive multi-nucleon correlations which govern the stiffness
of the EoS. A similar behavior of $c_{1}^{2}$ is reported in FFM2019  and also
discussed in Tews2018 .

Important input for the chiral FRG EoS is set by nuclear physics constraints
at baryon densities $\rho\lesssim 2\rho_{0}$. The nonperturbative chiral FRG
approach is designed to be consistent with perturbative chiral effective field
theory calculations Heb2013 ; HK2017 ; LH2018 ; Dri2019  for both symmetric
nuclear matter and neutron matter in this density range. In fact, whereas the
chiral FRG starts from a Lagrangian based on a linear sigma model, the pion
sector of the ChEFT is built on a nonlinear sigma model, where the heavy
scalar $\sigma$ field has been eliminated. While the linear and nonlinear
sigma models are not equivalent at any perturbative level, resummations to all
orders in the nonperturbative FRG treatment of the ChNM model should yield
results that match those of the perturbative ChEFT at sufficiently low
momentum scales and densities. This is indeed demonstrated by explicit
calculations in Refs. DW2015 ; DW2017 .

Figure 3 displays the resulting EoS, $P({\cal E})$, in comparison with the
families of equations of state obtained in Refs. Annala2018 ; Vuorinen2018 .
The latter interpolate between the low-density ChEFT EoS and the high-density
EoS of perturbative QCD and satisfy the astrophysical constraints from neutron
star masses and the tidal deformability deduced from the recently observed
neutron star merger GW170817. Clearly the ChNM-FRG EoS fits well into the
allowed band.

Figure 1: The equation of state of neutron star matter in beta equilibrium
(pressure $P$ as a function of energy density ${\cal E}$) derived from chiral
nucleon-meson field theory combined with functional renormalization group
equations DW2015 ; DW2017 . The shaded band gives an uncertainty estimate when
varying the symmetry energy in the range $30-34$ MeV. In the limit of very low
densities ($\rho<0.3\,\rho_{0}$ corresponding to ${\cal E}<0.045$ GeV/fm3),
the EoS is matched to the Skyrme-Lyon (SLy) parametrization DH2001 . Figure 2:
The squared speed of sound, $c_{1}^{2}=\partial P({\cal E})/\partial{\cal E}$,
derived from the neutron star matter equation of state, Fig. 1.  Figure 3:
Pressure as a function of energy density for neutron star matter with
constraints from perturbative QCD and nuclear EFT calculations at the upper
and lower ends of the density scale, adapted from Refs.Annala2018 ;
Vuorinen2018 . The area between the upper and lower limits marks the range of
acceptable equations of state subject to empirical conditions on neutron star
maximum mass and tidal deformability, the latter from LIGO and Virgo
gravitational wave analysis. The solid curve represents the chiral FRG
equation of state $P({\cal E})$ as in Fig. 1.

###  III.2 Connection to relativistic Fermi-liquid theory and quasiparticles

The observation that chiral nucleon-meson field theory in combination with FRG
appears to successfully describe neutron star matter at core densities
suggests a study of the Fermi-liquid properties of neutron matter in a
relativistic framework BC1976 .

Given the neutron star EoS of the chiral FRG model, the aim is to deduce the
spin-independent Fermi-liquid parameters $F_{0}$ and $F_{1}$ and examine their
density dependence. The speed of sound (26) involves a combination of these
Landau parameters. In order to determine $F_{0}(\rho)$ and $F_{1}(\rho)$
separately, additional information on the quasiparticle properties is
required. In the core of neutron stars the relevant quasiparticles are
dominantly neutrons dressed by the strong interactions with the surrounding
matter, plus a small fraction of protons. Once again, we assume that the small
proton fraction in neutron star matter can be neglected in the analysis of the
spin-independent Fermi liquid properties.

The single-particle motion in a relativistic theory of neutron matter is
described by the in-medium Dirac equation Brockmann:1996xy

$\left[\not{p}-M_{0}-\Sigma_{s}(p;\rho)-\not{\Sigma}_{v}(p;\rho)\right]u(\mathbf{p},s)=0,$
(59)

where in standard notation $\not{p}=\gamma_{\mu}\,p^{\mu}$,
$p^{\mu}=(p_{0},\mathbf{p})$, $\Sigma_{s}$ is the scalar,
$\Sigma_{v}^{\mu}=(\Sigma_{0},\mathbf{\Sigma})$ is the vector neutron self
energy, and $u(\mathbf{p},s)$ is the corresponding Dirac spinor. The
quasiparticle energies (in the rest frame of the system) are defined by the
self-consistent solutions of the equation vanDalen:2005ns

$\varepsilon_{p}=\sqrt{\mathbf{p}^{2}+[M_{0}+\Sigma_{s}(\mathbf{p},\varepsilon_{p};\rho)]^{2}}+\Sigma_{0}(\mathbf{p},\varepsilon_{p};\rho).$
(60)

Here, the spatial component of the vector self-energy $\mathbf{\Sigma}$
vanishes as in the mean-field treatment discussed above. In order to obtain a
tractable scheme, we set $|\mathbf{p}|=p_{F}$ in the self-energies and use the
following ansatz for the quasiparticle energy

$\varepsilon_{p}=\sqrt{\mathbf{p}^{2}+M(\rho)^{2}}+U(\rho),$ (61)

with a density-dependent Fermion (nucleon) mass
$M(\rho)=M_{0}+\Sigma_{s}(p_{F},\varepsilon_{F};\rho)$ and an effective vector
potential $U(\rho)=\Sigma_{0}(p_{F},\varepsilon_{F};\rho)$. In Sec. IV we
discuss possible implications of this approximation.

Consider again the Fermi velocity BC1976 ,

$v_{F}=\left(\frac{\partial\varepsilon_{p}}{\partial
p}\right)_{p=p_{F}}=\frac{p_{F}}{\mu\left(1+\frac{1}{3}F_{1}\right)}~{}.$ (62)

Thus the Fermi velocity as a function of baryon density $\rho$ yields
$F_{1}(\rho)$. The parameter $F_{0}(\rho)$ is then obtained by insertion into
Eq. (26) (cf. Eq. (10) in Landau2 ):

$F_{0}(\rho)=\frac{3\mu}{p_{F}\,v_{F}}\,c_{1}^{2}(\rho)-1~{}.$ (63)

A key quantity for further steps is obviously the baryon chemical potential
$\mu(\rho)$ as a function of baryon density. This is obtained by first
constructing the energy density ${\cal E}(\rho)$ as a function of $\rho$. The
(zero temperature) thermodynamic relation

${\cal E}+P({\cal E})=\rho\frac{\partial{\cal E}}{\partial\rho}~{},$ (64)

yields the density as a function of ${\cal E}$

$\rho({\cal E})=\rho^{(0)}\exp\left[\int_{{\cal E}^{(0)}}^{\cal E}\frac{d{\cal
E}^{\prime}}{{\cal E}^{\prime}+P({\cal E}^{\prime})}\right]~{}.$ (65)

Here the lower limit of the integral is chosen at a very low density
$\rho^{(0)}$, where the energy density is well approximated by the mass
density,

${\cal E}^{(0)}\equiv{\cal E}(\rho^{(0)})=M_{0}\,\rho^{(0)}~{}.$

By inverting Eq. (65) with $P({\cal E})$ as input, we obtain the energy
density ${\cal E}(\rho)$ and also the baryon chemical potential,
$\mu=\partial{\cal E}/\partial\rho$, as functions of the density. With our
ansatz (61) for the quasiparticle energy, the Fermi energy is given by

$\mu=\varepsilon_{p=p_{F}}=\sqrt{p_{F}^{2}+{M}{{}^{2}}(\rho)}+U(\rho)~{}.$
(66)

It follows that

$v_{F}(\rho)=\frac{p_{F}}{\sqrt{p_{F}^{2}+{M}{{}^{2}}(\rho)}}~{}.$ (67)

In terms of the energy per particle, $E(\rho)/A$, the energy density is given
by:

${\cal E}(\rho)=\left(M_{0}+\frac{E(\rho)}{A}\right)\rho~{},$ (68)

and the chemical potential by

$\mu=M_{0}+\left(1+\rho\frac{\partial}{\partial\rho}\right)\frac{E(\rho)}{A}~{}.$
(69)

Equating the two expressions for $\mu$, we find

$M_{0}+\left(1+\rho\frac{\partial}{\partial\rho}\right)\frac{E(\rho)}{A}=m^{*}(\rho)+U(\rho)~{}.$
(70)

where we have identified the Landau effective mass

$m^{*}(\rho)=\sqrt{p_{F}^{2}+{M}{{}^{2}}(\rho)}.$ (71)

Thus, the density dependence of $E(\rho)/A$ is reflected in the density-
dependent mass $M(\rho)$ and the potential $U(\rho)$.

In the chiral FRG model, the in-medium nucleon mass $M(\rho)$ scales with the
expectation value of the scalar $\sigma$ field, which is identified with the
in-medium pion decay constant $f_{\pi}^{*}(\rho)$, associated with the time
component of the axial current. This plays the role of a chiral order
parameter. Consequently, in this model we have:

$M(\rho)=M_{0}\,\frac{\langle\sigma\rangle}{f_{\pi}}=M_{0}\,\frac{f_{\pi}^{*}(\rho)}{f_{\pi}}~{}.$
(72)

The proportionality of the nucleon mass to the pion decay constant originates
from current algebra and the Goldberger-Treiman relation. The chiral FRG
approach maintains this property for a nucleon in a dense medium.

Finally, given $M(\rho)$ and $U(\rho)$ one can return to Eqs. (52,53) or
(56,57) and determine the Landau parameters $F_{0}$ and $F_{1}$.

##  IV Results: Quasiparticles and Fermi-Liquid parameters

###  IV.1 Bulk and quasiparticle properties

Using Eq. (65), with the chiral FRG equation of state $P({\cal E})$ as input,
we compute the energy density, ${\cal E}(\rho)$, and the energy per particle,
shown in Figs. 4 and 5. The corresponding baryon chemical potential
$\mu(\rho)=\partial{\cal E}/\partial\rho$ is shown in Fig. 6.

Figure 4: Energy density of neutron star matter from chiral FRG calculations
DW2015 ; DW2017  as a function of density given in units of $\rho_{0}=0.16$
fm-3. Figure 5: Energy per particle deduced from the chiral FRG equation of
state as a function of baryon density given in units of $\rho_{0}=0.16$ fm-3.
Figure 6: Chemical potential of neutron matter from ChNM-FRG calculations
DW2015 ; DW2017  as a function of density given in units of $\rho_{0}=0.16$
fm-3.

The next step is the decomposition of the baryon chemical potential (69) in
terms of the density-dependent quasiparticle mass and effective potential. As
mentioned, the mass $M(\rho)$ scales with the density dependence of the chiral
order parameter $\langle\sigma\rangle=f_{\pi}^{*}(\rho)$. Even at $\rho\approx
5\,\rho_{0}$, i.e., at densities encountered in the inner core region of
neutron stars, the in-medium nucleon mass is still almost half of its vacuum
value. This indicates that the spontaneous breaking of the chiral symmetry
remains strong at such large densities. Here the nonperturbative treatment of
fluctuations beyond the mean-field approximation in the chiral FRG approach is
of crucial importance DW2015 ; DW2017 . In a mean-field calculation one finds
instead a first-order chiral phase transition at densities as low as
$\rho\approx 3\,\rho_{0}$. With inclusion of fluctuating fields and, in
particular, many-body correlations featuring repulsive Pauli effects that
become increasingly active with increasing density, the transition to chiral
symmetry restoration is shifted to densities beyond $5\,\rho_{0}$.

Once $M(\rho)$ is determined (see Appendix C for numerical details), the
effective potential is obtained by

$U(\rho)=\mu-m^{*}(\rho)~{},$ (73)

with the Landau effective mass $m^{*}(\rho)$, Eq. (71). Results for the in-
medium mass $M(\rho)$ and the effective potential $U(\rho)$ are summarized in
Fig. 7. The deviations in $M(\rho)$ from linear density dependence reflect
correlations and fluctuations beyond the mean field approximation. The Landau
effective mass, $m^{*}(\rho)$, is shown in Fig. 8. This is the quantity that
determines the Landau parameter $F_{1}$, see Eq. (57). At large densities, the
neutrons at the Fermi surface become relativistic, with an effective mass
$M(\rho)$ comparable to or smaller than $p_{F}$. Nevertheless, at the
densities relevant to neutron stars, $M(\rho)$ is still large compared to
values expected when approaching chiral symmetry restoration.

It is important to point out that $M(\rho)$ and $U(\rho)$ behave quite
differently in comparison with the corresponding quantities of standard
relativistic mean-field (RMF) models, which yield much stronger scalar and
vector mean fields. This difference comes primarily from the explicit
treatment of multipion exchange processes in the chiral FRG theory. For
example, we have $U(\rho=\rho_{0})\approx 0.14$ GeV whereas the corresponding
RMF vector potential would be almost three times as large. Similarly, the
density-dependent mass $M(\rho)$ in the chiral FRG approach drops to
$M(\rho=\rho_{0})\approx 0.83\,M_{0}$ at normal nuclear matter density. The
equivalent attractive scalar potential is $U_{s}(\rho=\rho_{0})\equiv
M(\rho_{0})-M_{0}\approx-0.16$ GeV, whereas the typical scalar potential in
RMF models is about twice as strong. In the chiral FRG scheme, much of the
intermediate-range attraction between nucleons in the medium is generated by
two-pion exchange processes treated explicitly, with inclusion of Pauli
effects.

Figure 7: Neutron quasiparticle mass $M(\rho)$ and effective potential
$U(\rho)$ as functions of density. The effective mass is given in units of the
vacuum neutron mass, $M_{0}=939.57$ MeV.  Figure 8: Landau effective mass
$m^{*}(\rho)=\sqrt{p_{F}^{2}+M^{2}(\rho)}$ as a function of density in units
of $\rho_{0}=0.16$ fm-3. Figure 9: Landau Fermi liquid parameters $F_{0}$ and
$F_{1}$ for neutron star matter as a function of density $\rho/\rho_{0}$ in
units of $\rho_{0}=0.16$ fm-3.

###  IV.2 Landau parameters $F_{0}$ and $F_{1}$

The input is now prepared to evaluate the dimensionless Fermi liquid
parameters $F_{0}$ and $F_{1}$ as functions of density, using Eqs. (52 and
53). The results for these Landau parameters are plotted in Fig. 9.

At low densities, $\rho<\rho_{0}$, the parameter $F_{0}$ starts out negative
and then turns positive around $\rho_{0}$. This behavior is consistent with a
perturbartive ChEFT calculation for neutron matter HKW2013 . With next-to-
next-to-next-to-leading order ($N^{3}LO$) chiral NN interactions and $N^{2}LO$
three-body forces, a second-order many-body calculation gives
$f_{0}(\rho=0.2\,\rho_{0})=-1.25$ fm-2 and $f_{0}(\rho=\rho_{0})=0.7$ fm-2.
The resulting dimensionless $F_{0}=N(0)f_{0}$ with the density of states
$N(0)=m^{*}p_{F}/\pi^{2}$ is indeed close to the one deduced from the chiral
FRG EoS at low densities.

On the other hand the Landau effective mass $m^{*}$ from the ChEFT calculation
HKW2013  does not decrease as fast as the one found in the relativistic chiral
FRG model. The resulting $F_{1}(\rho=\rho_{0})$ is close to zero but with a
negative slope indicating a decreasing effective mass at higher density, as
observed in the chiral FRG result.

Historically, the behavior of the nucleon effective mass close to the Fermi
surface has in fact been subject to many detailed investigations. A
representative overview can be found in Ref. Mahaux1985 . While $m^{*}$ in
mean-field calculations is generally less than the free mass, correlations
involving particle-hole excitations tend to increase the effective mass near
the Fermi surface. Thus, at least at densities $\rho\lesssim\rho_{0}$, the
resulting Landau effective mass in neutron matter remains close to its vacuum
value HKW2013 ; Ismail:2019rjg . Our parametrization of the quasiparticle
energy (61) is presumably not able to fully capture these detailed correlation
effects. Consequently, at low densities we find that the Landau effective
mass, and consequently also $F_{1}$, is somewhat smaller than obtained in
ChEFT HKW2013 .

A more detailed treatment of the correlations near the Fermi surface, e.g.,
retaining the explicit momentum dependence of the neutron self-energies in
(60), is expected to yield an $F_{1}$ somewhat larger than in the present
analysis vanDalen:2005ns ; HKW2013 . Consequently the $F_{0}(\rho)$ shown in
Fig. 9 is presumably a lower bound.

The overall strong increase of $F_{0}$ at high densities reflects the growing
importance of repulsive many-body correlations as the matter gets more and
more compact. Part of this effect is due to the action of the Pauli principle
on nucleons fluctuating around the Fermi surface. Such repulsive correlations
are at the same time responsible for the increase of the sound velocity at
high densities beyond its canonical value, $c_{1}>1/\sqrt{3}$.

###  IV.3 Upper bound for $F_{0}$

So far we have assumed that the density-dependent neutron mass, $M(\rho)$, is
proportional to the in-medium pion decay constant, $f_{\pi}^{*}(\rho)$, and
decreases continuously with increasing density. In order to assess
uncertainties possibly implied by this assumption, it is instructive to
examine a limiting case, replacing $M(\rho)$ by the constant free neutron mass
$M_{0}$ at all densities. Given the speed of sound (26) constrained by
astrophysical observations, such an extreme limit reduces the magnitude of
(negative) $F_{1}<0$, balanced by a corresponding increase of $F_{0}$. The
resulting $F_{0}$ can be considered as an upper limit that provides an
estimate of uncertainties in the determination of the Landau parameters.

With the constant (vacuum) mass $M_{0}=gf_{\pi}$ as input, the chemical
potential is

$\mu=\sqrt{p_{F}^{2}+M_{0}^{2}}+V_{0}(\rho)\,,$ (74)

where the previously used vector potential $U(\rho)$ in Eq. (66) is now
replaced by $V_{0}(\rho)$ subject to the condition that $\mu=\partial{\cal
E}/\partial\rho$ remains unchanged, given by the values listed in Table 1 and
shown in Fig. 6. This limiting vector potential is plotted in Fig. 10. Note
that $V_{0}(\rho)$ is weaker in magnitude than $U(\rho)$ because the
attraction, previously manifest in the decreasing $M(\rho)$, is now
effectively transferred to $V_{0}$. Also shown in Fig. 10 is the Landau
effective mass divided by the chemical potential,
$m^{*}/\mu=\sqrt{p_{F}^{2}+M_{0}^{2}}/\mu=1-V_{0}/\mu$. This ratio is close to
unity at densities up to about $2\rho_{0}$. Hence $F_{1}$ stays close to zero
within that density range, in qualitative agreement with the results obtained
in ChEFT HKW2013 .

Figure 10: The effective vector potential $V_{0}(\rho)$ and the ratio of
Landau effective mass over chemical potential, $m^{*}/\mu$, for the limiting
case $M(\rho)\equiv M_{0}$, as functions of baryon density in units of
$\rho_{0}=0.16$ fm-3. The dashed line indicates the range of densities, where
the ansatz (74) yields $V_{0}<0$, at variance with the model assumptions (see
main text).  Figure 11: Landau parameters $F_{0}$ and $F_{1}$ as functions of
baryon density in units of $\rho_{0}=0.16$ fm-3. The upper and lower bounds of
the shaded areas correspond to the following limiting cases for choices of the
baryon mass: (a) lower boundary lines:
$M(\rho)/M_{0}=\langle\sigma\rangle(\rho)/f_{\pi}$ as in Fig. 9; (b) upper
boundary lines: $M(\rho)\equiv M_{0}$ at all densities.

The Landau parameters are given by [see Eqs. (52) and (53)]:

$F_{0}(\rho)={p_{F}\over\pi^{2}}\sqrt{p_{F}^{2}+M_{0}^{2}}~{}{\partial
V_{0}(\rho)\over\partial\rho}~{}~{},~{}~{}~{}~{}F_{1}(\rho)=-{3V_{0}(\rho)\over\mu(\rho)}~{}.$
(75)

They are shown in Fig. 11 together with the ones previously computed. The
areas between the upper and lower boundary curves for $F_{0}$ and $F_{1}$ can
be considered as uncertainty measures covering the two limiting cases,
$M(\rho)\propto f_{\pi}^{*}(\rho)$ (lower bounds) and $M(\rho)\equiv M_{0}$
(upper bounds).

We note that at low densities, where the effective vector potential is
negative, the ansatz (74) is strictly speaking not consistent with the model
assumptions, since the interaction between two baryons mediated by the
exchange of an isoscalar vector boson is repulsive.111This inconsistency would
presumably not occur in a more refined approach, where the momentum dependence
of the potential $V_{0}$ is accounted for (see, e.g., vanDalen:2005ns ).
Nevertheless, as already indicated, the resulting values for the Landau
effective mass at low densities are consistent with ChEFT. We can therefore
conclude that our estimate of the upper bound on the Landau parameters remains
valid also at low densities.

###  IV.4 Zero sound

Cold Fermi liquids can develop a sound-like collective mode, zero sound
Landau2 . The velocity of zero sound, $c_{0}=\omega/q$ (in terms of frequency
and wave vector of the mode), is yet another characteristic property of the
fluid that is linked to $F_{0}$ and $F_{1}$. In particular, for the model
considered, where $F_{\ell}=0$ for $\ell\geq 2$, the zero-sound velocity is
determined by real solutions of the equation Abrikosov1959 ; BP1991

$\left(F_{0}+{F_{1}\over 1+F_{1}/3}\,s^{2}\right)\Omega_{00}(s)=-1~{},$ (76)

where

$\Omega_{00}(s)=1+{s\over 2}\ln\left({s-1\over s+1}\right)$ (77)

is the long-wavelength limit of the Lindhard function Lindhard:1954va  222Note
that in Ref.Matsui1981  $\Phi(s)=-\Omega_{00}(s)$ is used. and
$s=c_{0}/v_{F}$, the velocity of the sound mode divided by the Fermi velocity.

The zero sound velocity in units of the velocity of light is shown in Fig. 12.
A comparison is once again made between the standard case with density
dependent mass $M(\rho)=g\langle\sigma\rangle$ and the limit $M(\rho)\equiv
M_{0}$. With the former choice it turns out that Eq. (76) permits real
solutions of $c_{0}$ only in a restricted range of densities $\rho$. Outside
this range the zero-sound velocity is complex. Its imaginary part indicates
Landau damping.

From Fig. 12 one concludes that, while neutron star matter becomes a
relativistic fluid at high densities, it strictly satisfies the causality
constraint, $c_{0}<c$. This is implied by the strong repulsion encoded in the
vector field that grows linearly at high density and contributes to both
$F_{0}$ and $F_{1}$. Notably, setting $F_{1}=0$ would not be consistent as it
would lead to a superluminal velocity of zero sound at high densities.

Figure 12: Velocity $c_{0}$ (in units of the speed of light) of the zero-sound
collective mode, as a function of baryon density in units of $\rho_{0}=0.16$
fm-3. The two curves correspond to different choices of quasiparticle masses.
Curve (a): baryon mass $M(\rho)\propto f_{\pi}^{*}(\rho)$; curve (b): using
$M(\rho)\equiv M_{0}$. At densities below and above the end points indicated
in curve (a) the zero-sound velocity is complex.

###  IV.5 Comparison with liquid helium-3

By their magnitudes, the Landau parameters are measures of the correlation
strength in the Fermi liquid. At this point a comparison with another example
of a strongly correlated fermionic many-body system, liquid 3He, is quite
instructive Vollhardt1984 ; Leggett2016 . Normal 3He at low temperature is a
high-density liquid in which the average distance between the helium atoms is
of the same order as the atomic diameter. Their interaction has an attractive
van der Waals part with a range of a few angstroms, and a strongly repulsive
short-range core. Apart from a re-scaling of distances by a factor of
$10^{-5}$ between Å and fm, this is qualitatively reminiscent of the situation
in neutron matter at densities $\rho\approx 5\,\rho_{0}$, where the average
distance between baryons is of the same order as the diameter of their compact
valence quark cores, and the interaction is also characterized by the
combination of intermediate-range attraction and strong short-range repulsion.
The dimensionless Landau parameters $F_{0}$ and $F_{1}$ can thus give, by
comparison, an impression of how strong the correlations are in these two
systems.

A qualitative difference between liquid 3He and neutron matter is seen in the
quasiparticle effective masses. The Landau effective mass $m^{*}$ in liquid
3He is a factor of approximately 3 to 6 larger (depending on pressure) than
the mass of an isolated 3He atom Wheatley1975 ; Greywall1986 , indicating the
presence of strongly repulsive correlations. This implies that $F_{1}$ for
liquid 3He is positive and increasing with pressure, in contrast to the much
more moderate modifications of $m^{*}$ with increasing pressure in neutron
matter. Another qualitative distinction between the matter in neutron star
cores and liquid 3He is the highly nonrelativistic nature of the latter.
Characteristic Fermi velocities $v_{F}$ for 3He reported in a wide range of
pressures from zero up to about 35 bar Wheatley1975  are on the order of
$10^{-7}$ in units of the speed of light and actually decrease with increasing
pressure as a consequence of the increasing effective mass. On the other hand,
the sound velocity $c_{1}$ in liquid 3He is typically an order of magnitude
larger than $v_{F}$, and their ratio

$\left({c_{1}\over v_{F}}\right)^{2}={1\over 3}(1+F_{0})(1+F_{1}/3)~{},$

is then reflected in large values of the Landau parameters. The following
values are reported for liquid 3He (cf. Appendix C of Ref. BP1991 ): at zero
pressure, $F_{0}\approx 9.3$ and $F_{1}\approx 5.4$; at a pressure of $P=27$
bar, $F_{0}\approx 68.2$ and $F_{1}\approx 12.8$. Understanding such large
Fermi liquid parameters requires resummations to all orders and inclusion of
collective modes in the quasiparticle interaction (the induced interaction
Babu-Brown1978 ). We note that the neutron matter parameters $F_{0}$ and
$F_{1}$ shown in Figs. 9 and 11 turn out to be strikingly smaller in magnitude
than those for liquid 3He.

The reason for this qualitative difference can be traced to the scales at work
in the respective quasiparticle interactions. Both neutron star matter and
liquid 3He are governed by repulsive interactions with characteristic ranges,
$r_{c}$. This correlation scale is to be compared with the average distance
between Fermions in the medium $d\propto p_{F}^{-1}$. Their ratio
$r_{c}/d\propto r_{c}\,p_{F}$ is a parameter that measures the growing
importance of the repulsive forces as the density increases BM1969 . For
neutron matter, even at densities several times larger than $\rho_{0}$, this
parameter is still much smaller than in liquid 3He over a broad range of
pressures. Thus, although the repulsive correlations in dense neutron matter
are sufficiently strong to support two-solar-mass neutron stars against
gravitational collapse, they appear as relatively moderate in comparison with
those in liquid 3He. In this perspective, the dense baryonic matter
encountered in the core of neutron stars is perhaps not as extreme as
sometimes imagined.

##  V Summary and Concluding Remarks

This work has focused on the Fermi liquid properties of dense baryonic matter
as it may be realized in the center of neutron stars. While the detailed
composition of strongly interacting matter at densities encountered in neutron
star cores continues to be open to discussions of different scenarios ranging
from conventional hadronic matter to quark matter, our starting point in this
work has been an equation of state based on a chiral nucleon-meson field
theory in combination with a nonperturbative approach using functional
renormalization group methods.

The input of the effective Lagrangian is tuned to pion-nucleon and pion-pion
interactions in vacuum and to nuclear physics observables at densities around
the equilibrium density of normal nuclear matter, $\rho_{0}\approx 0.16$ fm-3.
The output is consistent with state-of-the-art chiral effective field theory
calculations at baryon densities $\rho\lesssim 2\rho_{0}$. Moreover, the
resulting EoS of neutron star matter is in agreement with observations,
including the existence of two-solar mass stars and information from the
gravitational wave signals produced by the merger of two neutron stars.

It turns out that dense matter at zero temperature described by this EoS
remains in the hadronic phase characterized by spontaneously broken chiral
symmetry up to baryon densities even beyond five times the density of normal
nuclear matter. This shifting of the chiral transition to extremely high
densities, even above those encountered in neutron star cores, is a
consequence of the important role played by fluctuations and correlations
beyond mean-field approximation that are treated nonperturbatively in the FRG
framework.

A special feature of this EoS is its stiffness, produced by repulsive multi-
nucleon correlations that become increasingly important with increasing baryon
density and drive the sound velocity significantly beyond the massless Fermi
gas limit, $c_{1}^{2}=1/3$. Repulsive short-range correlations and the action
of the Pauli principle in the dense medium are important ingredients behind
this mechanism. It is then interesting to explore the quasiparticle properties
of the dense Fermi system under such conditions.

Relativistic Fermi-liquid theory is applied to deduce the Landau parameters
$F_{0}$ and $F_{1}$ and to investigate their density dependence. A
relativistic treatment is mandatory because, at high densities, the Fermi
momentum becomes large and comparable to the quasiparticle mass. The density-
dependent neutron mass decreases in the compressed medium but still remains at
about half of its vacuum value at densities typically reached in the neutron
star core region. The magnitudes of the dimensionless Landau Fermi-liquid
parameters then provide a measure of the many-body correlations as they grow
continuously in strength with rising density. Remaining uncertainties in
$F_{0}$ are correlated with uncertainties in $F_{1}$, i.e., the quasiparticle
effective mass and its density dependence. Thus, an estimate of the upper
limit for $F_{0}$ is obtained by setting the in-medium nucleon mass equal to
the free mass at all densities. Such a choice reduces the magnitude of $F_{1}$
and consequently increases $F_{0}$ under the condition of leaving the sound
velocity unchanged. At densities up to $2\,\rho_{0}$, the resulting $F_{1}$
remains close to zero, in agreement with ChEFT results HKW2013 . On the other
hand, at higher baryon densities one may expect a more strongly negative
$F_{1}$ if the chiral order parameter (the in-medium pion decay constant) is
reduced from its vacuum value.

The results for the dimensionless parameters $F_{0}(\rho)$ and $F_{1}(\rho)$
display the typical behavior of a strongly correlated Fermi system. However,
it is interesting to observe by comparison with Landau parameters in a system
such as liquid 3He, that the correlations in neutron star matter are still
fairly moderate. For example, while one finds $F_{0}\approx 3$ at baryon
densities $\rho\approx 0.8$ fm-3 in neutron matter, the $F_{0}$ in normal
liquid 3He is already more than three times larger even at zero pressure, and
it strongly increases further with increasing pressure. Even in the extreme
case of setting the in-medium baryon mass equal to its mass in vacuum, this
Landau parameter does not exceed $F_{0}\approx 5$ at $\rho\approx
5\,\rho_{0}$, the densities characteristic of neutron star cores.

Extrapolations of the chiral FRG equation of state DW2015 ; DW2017  to
densities beyond the range realized in neutron stars indicate the existence of
a continuous chiral crossover transition around $\rho\gtrsim 8\,\rho_{0}$.
Under such conditions the valence quark cores of the nucleons begin to
overlap. It is then an interesting issue how to describe a possible hadron-
quark continuity region in an actual model, such as the one developed in Ref.
Baym2018 . The sound velocity is again a quantity of prime interest in this
context.

While the present work deals with neutron matter, extensions to symmetric and
asymmetric nuclear matter in order to study isospin-dependent Fermi-liquid
parameters at high densities are certainly of interest. At densities close to
equilibrium nuclear matter, such investigations have previously been performed
with nuclear interactions based on chiral effective field theory in
conjunction with many-body perturbation theory Holt:2011yj . The
nonperturbative chiral FRG approach DW2015 ; DW2017  used in the present work,
is in fact designed to be compatible with in-medium ChEFT at low densities,
$\rho\lesssim 2\rho_{0}$, and thus suitable for extrapolations to higher
densities. Such considerations might motivate future studies.

## APPENDICES

### A Vector potential contribution to the quasiparticle energy

Here we refer back to Eq. (45) and present some explicit steps in the
derivation of the third term on the right hand side of that equation.

Consider the variation of the spatial components of the vector field:

$\frac{\delta\boldsymbol{V}}{\delta
n_{p^{\prime}}}=\frac{\partial\boldsymbol{V}}{\partial\rho}\,\frac{\delta\rho}{\delta
n_{p^{\prime}}}+\frac{\partial\boldsymbol{V}}{\partial\boldsymbol{j}}\,\frac{\delta\boldsymbol{j}}{\delta
n_{p^{\prime}}}~{}.$

The first term becomes:

$\frac{\partial\boldsymbol{V}}{\partial\rho}=h^{\prime}(j^{2})\,\rho\,\boldsymbol{j}~{},$

and vanishes in the rest frame where $\boldsymbol{j}=0$. The derivative in the
second term is given by

$\frac{\partial V_{k}}{\partial
j_{m}}=h(j^{2})\delta_{km}-h^{\prime}(j^{2})\,j_{m}\,j_{k}~{},$

where again the latter part vanishes in the rest frame of the system.
Consequently, we find

$\frac{\delta\boldsymbol{V}}{\delta
n_{p^{\prime}}}=h(j^{2})\,\frac{\delta\boldsymbol{j}}{\delta
n_{p^{\prime}}}~{}.$

Using Eqs. (41), (43), and (A Vector potential contribution to the
quasiparticle energy) the variation of the current yields:

$\displaystyle\frac{\delta\boldsymbol{j}}{\delta n_{p^{\prime}}}$
$\displaystyle=$
$\displaystyle\frac{1}{V}\,\boldsymbol{v}_{p^{\prime}}-\frac{h(j^{2})}{V}\sum_{p}n_{p}\,\left(\frac{1}{\sqrt{\boldsymbol{p}^{2}+M^{2}}}\,\frac{\delta\boldsymbol{j}}{\delta
n_{p^{\prime}}}-\frac{\boldsymbol{p}}{(\sqrt{\boldsymbol{p}^{2}+M^{2}})^{3}}\,\boldsymbol{p}\cdot\frac{\delta\boldsymbol{j}}{\delta
n_{p^{\prime}}}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{V}\,\boldsymbol{v}_{p^{\prime}}-\frac{h(j^{2})}{V}\sum_{p}n_{p}\,\left(\frac{1}{\sqrt{\boldsymbol{p}^{2}+M^{2}}}-\frac{1}{3}\frac{\boldsymbol{p}^{2}}{(\sqrt{\boldsymbol{p}^{2}+M^{2}})^{3}}\right)\,\frac{\delta\boldsymbol{j}}{\delta
n_{p^{\prime}}}$ $\displaystyle=$
$\displaystyle\frac{1}{V}\,\boldsymbol{v}_{p^{\prime}}-\frac{h(j^{2})}{V}\sum_{p}n_{p}\,\frac{\frac{2}{3}\boldsymbol{p}^{2}+M^{2}}{(\sqrt{\boldsymbol{p}^{2}+M^{2}})^{3}}\,\frac{\delta\boldsymbol{j}}{\delta
n_{p^{\prime}}}\,.$

The sum in the last line can be evaluated Matsui1981 :

$\frac{1}{V}\sum_{p}n_{p}\,\frac{\frac{2}{3}\boldsymbol{p}^{2}+M^{2}}{(\sqrt{\boldsymbol{p}^{2}+M^{2}})^{3}}=\nu\int\frac{d^{3}p}{(2\pi)^{3}}\frac{\frac{2}{3}\boldsymbol{p}^{2}+M^{2}}{(\sqrt{\boldsymbol{p}^{2}+M^{2}})^{3}}\,n_{p}=\frac{\rho}{\sqrt{p_{F}^{2}+M^{2}}}~{}.$

Thus, the variation of the current is given by

$\frac{\delta\boldsymbol{j}}{\delta
n_{p^{\prime}}}=\frac{\frac{1}{V}\,\boldsymbol{v}_{p^{\prime}}}{1+\frac{h(j^{2})\rho}{\sqrt{p_{F}^{2}+M^{2}}}}=\frac{\frac{1}{V}\,\boldsymbol{v}_{p^{\prime}}}{1+\frac{V_{0}}{\sqrt{p_{F}^{2}+M^{2}}}}~{}.$

Finally the contribution of the last term in (45) to the quasiparticle
interaction is

$f_{pp^{\prime}}(3)=-h(j^{2})\frac{\boldsymbol{p}\cdot\boldsymbol{p}^{\prime}}{\sqrt{p_{F}^{2}+M^{2}}}\frac{1}{\sqrt{p_{F}^{2}+M^{2}}+V_{0}}=-h(j^{2})\frac{\boldsymbol{p}\cdot\boldsymbol{p}^{\prime}}{\mu\,\sqrt{p_{F}^{2}+M^{2}}}~{},$

the result quoted in Eq. (51).

### B Chiral Nucleon-Meson Field Theory and Functional Renormaliztion Group

The present study starts from a chiral field theory of mesons and nucleons,
based on a linear sigma model with a nonlinear effective potential. The
Lagrangian, written in Eq. (58), is used as UV input for the FRG flow
equations at the chiral symmetry breaking scale $\Lambda_{\chi}\approx 1$ GeV.

The thermodynamics of this model is developed in detail and summarized in
Refs. DW2015 ; DW2017 . Finite temperatures and chemical potentials are
treated using the Matsubara formalism. Minkowski space-time is Wick rotated to
Euclidean space. Time components are transformed as $x^{0}\rightarrow-i\tau$.
The $\tau$ dimension is compactified on a circle, such that $\tau$ is
restricted to $[0,\beta]$ with the inverse temperature $\beta=1/T$. Time
integrals are replaced by $-i\int_{0}^{\beta}d\tau$. Boson and fermion fields
are periodic or anti-periodic, respectively, under
$\tau\rightarrow\tau+\beta$. The Minkowski-space action $S=\int
d^{4}x\;\mathcal{L}$ is replaced by the Euclidean action $S_{\rm
E}=\int_{0}^{\beta}d\tau\int d^{3}x\;{\cal L}_{\rm E}$.

The FRG scheme proceeds as follows. An effective action, $\Gamma_{k}[\Phi]$
depending on a renormalization scale $k$, is introduced, where $\Phi$ stands
for the set of all chiral boson and nucleon fields. The action $S_{\rm
E}=\int_{0}^{\beta}d\tau\int d^{3}x\;{\cal L}_{\rm E}=\Gamma_{k\rightarrow
k_{UV}}$ serves as the initialization of $\Gamma_{k}$ at the UV scale,
$k_{UV}\approx\Lambda_{\chi}=4\pi f_{\pi}$. The flow of $\Gamma_{k}$ is
determined in such a way that it interpolates between the UV action and the
full quantum effective action $\Gamma_{\rm eff}=\Gamma_{k=0}$ in the infrared
limit, $k\rightarrow 0$. The evolution of $\Gamma_{k}$ as a function of $k$ is
given by Wetterich’s flow equation Wet1993 , schematically written as

$\displaystyle\begin{aligned} k\,\frac{\partial\Gamma_{k}[\Phi]}{\partial
k}=&\begin{aligned}
\includegraphics[width=43.36464pt]{wetterich_fermion}\end{aligned}\\\
=&\frac{1}{2}{\rm Tr}\left[k\frac{\partial R_{k}}{\partial
k}\cdot\Big{(}\Gamma_{k}^{(2)}[\Phi]+R_{k}\Big{)}^{-1}\right]~{}~{}.\end{aligned}$
(78)

The trace Tr stands for all relevant sums and integrations. A scale regulator,
$R_{k}(p)$, is introduced in order to restrict momenta $p$ in loop integrals
to $p^{2}<k^{2}$. The derivative $\partial_{k}R_{k}$ has maximum weight at
$p^{2}\approx k^{2}$. The matrix $\Gamma_{k}^{(2)}$ involves second functional
derivatives of the effective action with respect to chiral and nucleon fields.
It collects the full inverse propagators of all particles involved. In the
pictorial illustration of the flow equation (78) these full propagators are
marked by the dot on the loop line while the $k$ regulator is symbolized by
the crossed circle.

The welcome feature of the FRG system of equations is its capability to
generate fluctuations to all orders beyond mean-field approximation. The
“soft” degrees of freedom that contribute most prominently to these
fluctuations are the pion field with its small mass and low-energy nucleon-
hole excitations. Both these types of excitations enter non-perturbatively
through their full propagators in the flow equation (78).

For the treatment of a dense and thermal medium with inclusion of fluctuations
it is useful to compute the flow of the difference between the effective
action at given values of temperature and chemical potential,
$\Gamma_{k}(T,\mu)$, as compared to the potential at a reference point for
which we choose equilibrium nuclear matter at zero temperature,
$\Gamma_{k}(0,\mu_{0})$ with $\mu_{0}=M_{N}+E_{0}/A=923$ MeV. The flow of the
difference, $\bar{\Gamma}_{k}=\Gamma_{k}(T,\mu)-\Gamma_{k}(0,\mu_{0})$,
satisfies the FRG equation

$\displaystyle\begin{aligned} \frac{k\,\partial\bar{\Gamma}_{k}}{\partial
k}(T,\mu)&=\begin{aligned}
\vspace{1cm}\includegraphics[width=43.36464pt]{wetterich_fermion}\end{aligned}\vspace{-1cm}\Bigg{|}_{T,\mu}-\begin{aligned}
\vspace{1cm}\includegraphics[width=43.36464pt]{wetterich_fermion}\end{aligned}\Bigg{|}_{\begin{subarray}{l}T=0\\\
\mu=\mu_{0}\end{subarray}}.\end{aligned}$ (79)

The actual computational work involves some simplifying assumptions and
approximations: the effective action is treated in leading order of the
derivative expansion and we work in the local potential approximation,
neglecting (small) wave function renormalization effects on the chiral boson
fields and possible higher order derivative couplings. Moreover, the $k$
running of the Yukawa coupling $g$ is ignored; the dependence of the nucleon
mass on temperature and chemical potential scales with that of the in-medium
pion decay constant, $f_{\pi}^{*}(T,\mu)=\langle\sigma\rangle(T,\mu)$.

The nucleon mass $M=g\sigma$ is coupled dynamically to the scalar field. Its
expectation value, $\langle\sigma\rangle$ is normalized to the pion decay
constant $f_{\pi}$ in the vacuum so that the mass of the free nucleon
satisfies the Goldberger-Treiman relation, $M_{0}=g\,f_{\pi}$. The coupling
$g\approx 10$ stands for the ratio of pion-nucleon coupling constant, $g_{\pi
N}$, and the axial vector coupling constant, $g_{A}$, of the nucleon:
$g=g_{\pi N}/g_{A}$. In general, $\langle\sigma\rangle$ acts as an order
parameter for spontaneously broken chiral symmetry. The region of temperatures
$T$ and densities $\rho$ where this chiral FRG framework can be applied is
defined by nonzero $\langle\sigma\rangle(T,\rho)\equiv f^{*}_{\pi}(T,\rho)$.
The full FRG calculations DW2015 ; DW2017  point out that the domain with
$\langle\sigma\rangle\neq 0$ covers the broad range $0<T\lesssim 100$ MeV and
$0<\rho\lesssim 7\,\rho_{0}$. In contrast, using mean-field approximation
would generate a first-order chiral phase transition around $\rho\approx
3\,\rho_{0}$.

### C Chiral FRG equation of state and quasiparticle properties: numerical
details

####  .0.1 Pressure

For practical purposes and applications, the chiral EoS of neutron star
matter, $P(\cal E)$ shown in Fig. 1, is given here in a parametrized form of
an accurate Padé fit in the baryon density range $0.1\leq\rho/\rho_{0}\leq 6$
(with $\rho_{0}=0.16$ fm-3):

$\frac{P(z)}{\mathrm{MeV/fm^{3}}}=\frac{\sum_{n=3...5}a_{n}z^{n}}{1+\sum_{m=1}^{5}b_{m}z^{m}}~{},~{}~{}~{}~{}~{}~{}z=\frac{{\cal
E}}{\mathrm{GeV/fm^{3}}}~{},$ (80)

with the parameters

$\displaystyle a_{3}$ $\displaystyle=$ $\displaystyle
1270.28~{}~{}~{}~{}~{}a_{4}=7236.08~{}~{}~{}~{}~{}a_{5}=5.99$ $\displaystyle
b_{1}=1.7857~{}~{}~{}~{}~{}b_{2}$ $\displaystyle=$ $\displaystyle
26.4331~{}~{}~{}~{}~{}b_{3}=-3.0629~{}~{}~{}~{}~{}b_{4}=11.8366~{}~{}~{}~{}~{}b_{5}=-3.9431~{}.$
(81)

####  .0.2 Energy density

A further useful quantity to work with is the energy density ${\cal E}(\rho)$
as a function of baryon density, also presented in terms of a Padé fit for
practical computations:

$\frac{{\cal
E}(y)}{\mathrm{GeV/fm^{3}}}=\frac{c_{1}y+c_{2}y^{2}}{1+\sum_{n=1}^{4}d_{n}y^{n}}~{},~{}~{}~{}~{}~{}y=\rho/\rho_{0}~{},~{}~{}~{}~{}~{}(\rho_{0}=0.16\,{\mathrm{f}m^{-3}})~{},$
(82)

with the coefficients:

$\displaystyle c_{1}$ $\displaystyle=$ $\displaystyle
0.1509~{}~{}~{}~{}~{}~{}~{}c_{2}=0.0711$ $\displaystyle
d_{1}=0.4857~{}~{}~{}~{}~{}d_{2}$ $\displaystyle=$
$\displaystyle-0.0343~{}~{}~{}~{}~{}d_{3}=0.0020~{}~{}~{}~{}~{}d_{4}=6.5\times
10^{-5}~{}.$ (83)

####  .0.3 Quasiparticle mass

The density-dependent baryon mass $M(\rho)$ is a key ingredient of the
quasiparticle energy and chemical potential,

$\varepsilon_{p=p_{F}}=\mu=\sqrt{p_{F}^{2}+M^{2}(\rho)}+U(\rho)~{}.$

An excellent numerical fit to $M(\rho)$, resulting from the chiral FRG
calculation of DW2015 ; DW2017 , is found using a Padé representation as
follows:

$\frac{M(y)}{M_{0}}=\frac{\langle\sigma\rangle}{f_{\pi}}=\frac{1+\sum_{n=1}^{3}\alpha_{n}\,y^{n}}{1+\sum_{n=1}^{3}\beta_{n}\,y^{n}}~{},~{}~{}~{}~{}~{}y=\frac{\rho}{\rho_{0}}~{},$

with

$\displaystyle\alpha_{1}$ $\displaystyle=$ $\displaystyle
0.6170~{},~{}~{}~{}\alpha_{2}=-0.1462~{},~{}~{}~{}\alpha_{3}=0.0052~{},~{}~{}~{}$
$\displaystyle\beta_{1}$ $\displaystyle=$ $\displaystyle
0.8699~{},~{}~{}~{}\beta_{2}=-0.0821~{},~{}~{}~{}\beta_{3}=-0.0076~{}.$

Table 1 collects numerical values of $P(\cal E)$ together with selected
quasiparticle properties for baryon densities in the range $(0.08-0.96)$fm-3.

Table 1: Chiral FRG equation of state DW2015 ; DW2017  for neutron star matter at $T=0$: baryon density $\rho$ in units of $\rho_{0}=0.16$ fm-3, energy density ${\cal E}$, pressure $P$, baryon chemical potential $\mu=\frac{\partial{\cal E}}{\partial\rho}$, density-dependent quasiparticle mass $M(\rho)$ [with $M_{0}=M(\rho=0)=939.6$ MeV], and Landau effective mass $m^{*}(\rho)=\sqrt{p_{F}^{2}+M^{2}(\rho)}$.  $~{}~{}\rho/\rho_{0}~{}~{}~{}~{}$ |  ${\cal E}$ [GeV/fm3] |  $P$ [MeV/fm3] |  $\mu(\rho)$ [GeV] |  $M(\rho)$ [MeV] |  $m^{*}(\rho)$ [MeV]
---|---|---|---|---|---
0.5 | 0.076 | 0.62 | 0.9509 | 845.9 | 885.9
1.0 | 0.153 | 4.51 | 0.9829 | 779.1 | 846.6
1.5 | 0.233 | 13.15 | 1.0259 | 724.2 | 817.5
2.0 | 0.317 | 26.84 | 1.0746 | 676.0 | 794.6
2.5 | 0.405 | 45.53 | 1.1266 | 632.1 | 775.8
3.0 | 0.497 | 69.14 | 1.1805 | 591.0 | 760.1
3.5 | 0.594 | 97.89 | 1.2356 | 551.7 | 746.7
4.0 | 0.695 | 131.39 | 1.2914 | 513.3 | 735.0
4.5 | 0.801 | 169.89 | 1.3477 | 474.6 | 724.3
5.0 | 0.911 | 212.99 | 1.4044 | 433.9 | 713.7
5.5 | 1.025 | 260.73 | 1.4616 | 388.0 | 702.0
6.0 | 1.145 | 314.23 | 1.5195 | 329.0 | 686.2

ACKNOWLEDGEMENTS

One of the authors (W.W.) gratefully acknowledges fruitful periods as visiting
professor at the ExtreMe Matter Institute (EMMI) of GSI-FAIR in Darmstadt and
at the Physics Department of the University of Tokyo. He thanks Kenji
Fukushima for his kind hospitality and many stimulating discussions. Helpful
comments and suggestions by Norbert Kaiser are also much appreciated.

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