#  EoS for strange quark matter: linking the NJL model to pQCD

Marcus Benghi Pinto<EMAIL_ADDRESS>Departamento de Física,
Universidade Federal de Santa Catarina, Florianópolis, SC 88040-900, Brazil

###### Abstract

Neutron star constraints and ab initio pQCD evaluations require the EoS
representing cold quark matter to be stiff at intermediate baryonic densities
and soft at high-$n_{B}$. Here, it is suggested that the three flavor NJL
model with a density dependent repulsive coupling, $G_{V}(\mu)$, can generate
an EoS which interpolates between these two regimes. Such an interpolation
requires repulsion to start decreasing with the chemical potential just after
chiral transition takes place. The conjecture behind this mechanism is that
repulsion should be necessary only as long as the quark condensates, which
dress the effective masses, have non-vanishing values. This assumption
guarantees that an initially hard EoS suffers a conspicuous change of slope at
${\cal E}\simeq 0.7\,{\rm GeVfm^{-3}}$ converging to the pQCD results at
higher energy densities. Then, the speed of sound naturally reaches a non-
conformal maximum at $n_{B}=3.23\,n_{0}=0.52\,{\rm fm}^{-3}$ while the trace
anomaly remains positive for all densities, in agreement with recent
investigations. These non-trivial results cannot be simultaneously obtained
when $G_{V}$ vanishes or has a fixed value. Therefore, the simple model
proposed here may help us to link the (non-perturbative) region of
intermediate densities to the region where pQCD becomes reliable.

##  I Introduction

Effective quark models, such as the Nambu–Jona-Lasinio model (NJL) njl  and
the MIT bag model mit1 ; mit2 ; mit3  capture some of the most representative
characteristics of quantum chromodynamics (QCD), like confinement and chiral
symmetry respectively buballa . As a consequence, they are widely used to
describe the thermodynamics of strongly interacting matter in regions of the
phase diagram which are currently unaccessible to ab initio evaluations.
Nowadays, the corner of low densities and high temperatures can be well
described by first principle evaluations based on lattice QCD simulations
(LQCD). However, due to the well documented sign problem, LQCD is not yet in
position to describe the corner of low temperatures and finite baryonic
densities which concerns neutron stars (NSs). In this case the QCD equation of
state (EoS) describing cold and dense strongly interacting matter can be
reliably evaluated only in regimes where the baryonic density ($n_{B}$) is
very low or extremely high. In the limit of low densities, chiral effective
theory (CET) cet1 ; cet2 provides an accurate EoS up to about
$n_{B}\lesssim\;2\,n_{0}\equiv n_{\rm CET}$ ($n_{0}=0.16\,{\rm fm}^{-3}$) so
that the region composed by hadronic matter may be well described. At the
other extremum, perturbative QCD (pQCD) pqcd1 ; pqcd2 ; pqcd3  gives a
reliable equation for $n_{B}\gtrsim 40\,n_{0}\equiv n_{\rm pQCD}$, when quarks
and gluons represent the relevant degrees of freedom kojo . However, at the
intermediate range $2n_{0}\lesssim n_{B}\lesssim 8n_{0}$, which concerns NSs,
$\alpha_{s}$ is still high so that non-perturbative techniques and/or model
approximations are generally employed. Within this region the presence of
quark matter in massive NSs was recently found nature  to be linked to the
behavior of the speed of sound, $V_{s}$. The investigation performed in Ref.
nature  suggests that if the conformal bound $V_{s}^{2}\leq 1/3$ is not
strongly violated massive neutron stars should have sizable quark-matter
cores. Moreover, the recent discovery of NSs whose estimated masses are about
twice the value of the solar mass astro1 ; astro2 ; astro3  and the
theoretical predictions on the maximum (gravitational) mass performed in Refs.
measure1 ; measure2 ; measure3 ; measure4 ; measure5  favor an stiff EoS with
$V_{s}^{2}>1/3$ at $n_{B}>n_{0}$. In this case, recent simulations
sinansimulation ; michal ; weisesimulation  indicate that $V_{s}^{2}$ is a
non-monotonic function of $n_{B}$ , which in turn suggests the existence of at
least one local maximum where $V_{s}^{2}>1/3$. Together, all of these findings
constrain the EoS to be initially stiff (so that $V_{s}^{2}>1/3$) before
softening, at intermediate densities, to finally meet the pQCD predictions at
high-$n_{B}$. As it is well known, when effective quark models are being
employed the inclusion of a repulsive vector channel, parametrized by $G_{V}$,
generates a harder EoS in most cases fukushima1 ; fukushima2 . However, a
drawback is that such an equation remains stiff at higher densities so that
the conformal limit, observed by pQCD, cannot be attained. On the other hand,
when $G_{V}=0$, asymptotic convergence to pQCD is observed but the EoS is far
too soft to cope with NSs constraints at lower densities.

One way to circumvent this problem is to assume that $G_{V}$ is density
dependent as recently proposed in Ref. letter , where the two flavor NJL model
has been considered. There, it has been suggested that the repulsion among
(dressed) quarks is important only up to the point where the chiral transition
occurs so that repulsion among (bare) quarks should be negligible. In Ref.
letter , the running of $G_{V}$ was modeled by a simple ansatz which
interpolates between a regime where repulsion is high (the EoS is stiff) and a
regime where repulsion low (the EoS is soft). Thanks to this property the two-
flavor NJL model with a running $G_{V}(\mu)$ predicted letter  a non-monotonic
behavior for $V^{2}_{s}$ implying that the existence of a peak, at
$n_{B}\simeq 3.25n_{0}$, can be conciliated with pQCD predictions at
asymptotically high baryonic densities. Physically, these results indicate
that repulsion should be necessary only as long as the quark condensates
(which are directly related to the NJL quark self energies) exist.

Since strangeness may play an important role when describing more realistic
situations the present work contemplates an extension to the case where this
degree of freedom is present. With this purpose, the three flavor NJL model
with a repulsive channel will be considered here as a prototype to describe
cold strange quark matter. As we shall see, also in this case a density
dependent $G_{V}$ allows us to obtain a non-conformal bump in $V_{s}^{2}$ at
$n_{B}=3.23\,n_{0}=0.52\,{\rm fm}^{-3}$ (in agreement with Ref. michal ) while
the trace anomaly remains positive. This rather non-trivial result supports a
recent claim fukushimatrace  which states that the presence of a non-conformal
peak in $V_{s}^{2}$ is not necessarily in tension with the trace anomaly being
positive for all densities. Concerning the EoS another important result
obtained here predicts a prominent change of slope taking place at ${\cal
E}\simeq 0.7{\rm GeVfm^{-3}}$, in agreement with what is observed in Refs.
nature ; michal . These predictions indicate that the modified three flavor
NJL model discussed in this investigation may contribute to describe the QCD
EoS at intermediate baryonic densities.

The paper is organized as follows. In the next section the basic results for
the three flavor NJL model are reviewed. The possible density dependence of
the repulsive vector interaction is presented in Sec. III. Numerical results
associated with the relevant thermodynamical quantities are generated and
discussed in Sec. IV. The conclusions are presented in Sec. V.

##  II The $N_{f}=2+1$ NJL model: standard results

In the presence of a repulsive vector channel the standard three-flavor
version of the NJL model can be written as fukushima1 ; fukushima2 ; campoB

$\mathcal{L}\ =\
{\bar{\psi}}(i\gamma_{\mu}\partial^{\mu}-m)\psi+G_{S}\sum_{a=0}^{8}\left[({\bar{\psi}}\lambda^{a}\psi)^{2}+(\bar{\psi}i\gamma_{5}\lambda^{a}\psi)^{2}\right]-K\left\\{{\rm
det}_{f}[\bar{\psi}(1+\gamma_{5})\psi]+{\rm
det}_{f}[\bar{\psi}(1-\gamma_{5})\psi]\right\\}-G_{V}({\bar{\psi}}\gamma^{\mu}\psi)^{2}\
,$ (1)

where $\psi=(u,d,s)^{T}$ denotes a quark field with three flavors (and three
colors), and $m={\rm diag}_{f}(m_{u},m_{d},m_{s})$ is the corresponding mass
matrix. Setting $m_{u}=m_{d}\equiv m\neq m_{s}$ implies that isospin symmetry
is observed while the $SU(3)$ flavor symmetry is explicitly broken. The eight
Gell-Mann matrices are represented by $\lambda^{a}$ ($a=1,...,8$) and
$\lambda^{0}=\sqrt{2/3}\,\bm{I}$. In $3+1\,d$ the NJL is composed by
irrelevant operators so that the couplings $G_{S}$ and $K$ respectively have
canonical dimensions [-2] and [-5] implying that the model is non-
renormalizable. Here, the (ultra violet) divergent integrals will be
regularized by a sharp non-covariant cut-off, $\Lambda$, whose numerical value
is set by phenomenological inputs. For the numerical analysis let us adopt the
parameter values of Ref.rkh  which are $m=5.5\,{\rm MeV}$, $m_{s}=140.7\,{\rm
MeV}$, $G\Lambda^{2}=1.835$, $K\Lambda^{5}=12.36$, and $\Lambda=602.3\,{\rm
MeV}$. Then, at $T=0$ and $\mu_{f}=0$, one reproduces $f_{\pi}=92.4\,{\rm
MeV}$, $m_{\pi}=135\,{\rm MeV}$, $m_{K}=497.7\,{\rm MeV}$, and
$m_{\eta^{\prime}}=960.8\,{\rm MeV}$. For the quark condensates one obtains
$\sigma_{u}=\sigma_{d}=-(241.9\,{\rm MeV})^{3}$, and $\sigma_{s}=-(257.7\,{\rm
MeV})^{3}$. Fixing $G_{V}$ poses and additional problem since this quantity
should be determined by considering the $\rho$ meson mass which, in general,
happens to be higher than the maximum energy scale set by $\Lambda$. In this
situation, most authors adopt values between $0.25G_{S}$ and $0.5G_{S}$ (see
Ref. tulio  for more details). Here, the value $G_{V}=G_{S}/3$ will be adopted
when dealing with a fixed vector coupling tulio ; sugano . Note that to assure
rotational invariance only the zeroth component of the vector channel
contributes so that, at the mean field level, the chemical potential gets
shifted as buballa ; fukushima1 ; fukushima2

${\tilde{\mu}}_{f}=\mu_{f}-2G_{V}\sum_{f}n_{f}\,,$ (2)

with $n_{f}$ representing the quark number density per flavor buballa ;
fukushima1 ; fukushima2 ; tulio . At $T=0$, a standard mean field
approximation (MFA) evaluation yields the following result buballa ; tulio

$n_{f}=\frac{N_{c}}{3\pi^{2}}p_{F,f}^{3}\;,$ (3)

where the effective Fermi momentum is just
$p_{F,f}=\sqrt{{\tilde{\mu}}_{f}^{2}-M_{f}^{2}}$. The quark effective masses
are given by buballa

$M_{f}=m_{f}-4G_{S}\sigma_{f}+2K\sigma_{j}\sigma_{k}\,,$ (4)

where $\sigma_{f}=\langle{\overline{\psi}}\psi\rangle_{f}$ represents the
quark condensate for a given flavor

$\sigma_{f}=-\frac{N_{c}}{2\pi^{2}}M_{f}\left[\Lambda
p_{\Lambda,f}-M_{f}^{2}\ln\left(\frac{\Lambda+p_{\Lambda,f}}{M_{f}}\right)\right]+\frac{N_{c}}{2\pi^{2}}M_{f}\left[{\tilde{\mu}}_{f}p_{F,f}-M_{f}^{2}\ln\left(\frac{{\tilde{\mu}}_{f}+p_{F,f}}{M_{f}}\right)\right]\,,$
(5)

where $p_{\Lambda,f}=\sqrt{\Lambda^{2}+M_{f}^{2}}$. The effective Fermi
momentum, $p_{F,f}$, is then determined by solving Eqs. (2) and (4)
simultaneously.

Having the quark number density, $n=\sum_{f}n_{f}$, one can express the
squared speed of sound in terms of the baryonic number susceptibility,
$\chi_{B}=dn_{B}/d\mu_{B}$, as

$V_{s}^{2}=\frac{n_{B}}{\mu_{B}\chi_{B}}\,,$ (6)

where $\mu_{B}=\sum_{f}\mu_{f}$ and $n_{B}=n/3$. For simplicity, the present
application concerns the case of symmetric strange quark matter only so that
one can now set $\mu_{u}=\mu_{d}=\mu_{s}\equiv\mu$.

Then, at finite chemical potential and zero temperature, the pressure versus
chemical potential relation for quark matter can be obtained from fukushima2 ;
zong

$P(\mu)=P(0)+\int_{0}^{\mu}n(\nu)d\nu\,,$ (7)

where $P(0)=$ is the vacuum pressure. From $P(\mu)$ one can determine the
energy density, ${\cal E}=-P+\mu_{B}n_{B}$, the trace anomaly, $\Delta={\cal
E}-3P$, as well as the conformal measure, ${\cal C}=\Delta/{\cal E}$.

Figure 1:  Running repulsive coupling $G_{V}(\mu)$, in units of $G_{S}$, as a
function of the quark chemical potential, $\mu$. The Fermi momentum
corresponding to each distinct region is shown for reference.

Figure 2: A pictorial representation of the physical conjecture driving the
running of $G_{V}(\mu)$. Quarks with large effective masses (represented by
the gray area) tend to strongly repel each other as compression increases.
After the chiral transition occurs (at $\mu\sim M(0)$), and the masses tend to
their bare values, quarks can be further compressed without repelling each
other. Therefore, repulsion should be necessary only as long as the quark
condensates (which dress the NJL masses) exist ($\mu\lesssim\Lambda$).

Figure 3: (Light flavor) quark condensate, $\sigma_{u,d}(\mu)$ normalized by
$\sigma_{u,d}(0)$, as a function of $\mu/\mu_{c}$. For the cases
$G_{V}=G_{S}/3$ and $G_{V}(\mu)$ the coexistence quark chemical potential is
$\mu_{c}=0.368\,{\rm GeV}$ while for $G_{V}=0$ it reads $\mu_{c}=0.361\,{\rm
GeV}$.

Figure 4: Effective Fermi momentum for light quarks,
$p_{F}^{*}=\sqrt{{\tilde{\mu}}^{2}-M^{2}}$, normalized by
$p_{F}^{0}=\sqrt{\mu^{2}-m^{2}}$, as a function of $\mu/\mu_{c}$. For the
cases $G_{V}=G_{S}/3$ and $G_{V}(\mu)$ the coexistence quark chemical
potential is $\mu_{c}=0.368\,{\rm GeV}$ while for $G_{V}=0$ it reads
$\mu_{c}=0.361\,{\rm GeV}$.

##  III Interpolating between softness and stiffness

Let us now discuss how to tune $G_{V}(\mu)$ so as to obtain an EoS which
interpolates between the stiff and the soft regimes. When dealing with
symmetric quark matter one can further simplify the notation by setting
$M_{u}=M_{d}\equiv M$ as previously done for $m_{u}$ and $m_{d}$. Using these
definitions and taking $G_{V}=0$ one can write the Fermi momentum for a light
flavor, in symmetric matter, as $\sqrt{\mu^{2}-M^{2}}$. As compression
increases the quark condensates decrease and the chiral transition sets in
($M\to m$) so that the Fermi momentum changes as $\sqrt{\mu^{2}-M^{2}}\to
p_{F}^{0}=\sqrt{\mu^{2}-m^{2}}$, where $p_{F}^{0}$ represents the case of free
(bare) quarks considered within pQCD. Now, when $G_{V}$ is fixed chiral
symmetry (partial) restoration implies that
$\sqrt{{\tilde{\mu}}^{2}-M^{2}}\to\sqrt{{\tilde{\mu}}^{2}-m^{2}}$ and since
the quark number density grows with $\mu$ the Fermi momentum $p_{F}^{0}$
cannot be reached, preventing the NJL results to converge to the pQCD
predictions at arbitrarily high baryonic densities. Nevertheless, as proposed
in Ref. letter , one can assure ${\tilde{\mu}}\to\mu$ (and
$\sqrt{{\tilde{\mu}}^{2}-M^{2}}\to p_{F}^{0}$) by requiring $G_{V}(\mu)\to 0$
after the chiral transition takes place according to

$G_{V}(\mu)=\frac{G_{V}(0)}{1+e^{(\mu-\mu_{0})/\delta}}\;,$ (8)

where $G_{V}(0)=G_{S}/3$ sugano ; tulio ; letter . Considering the
parametrization adopted here one has $M(0)=367.7\,{\rm MeV}$ buballa  and
$\Lambda=602.3\,{\rm MeV}$ so that $\mu_{0}=[M(0)+\Lambda]/2\equiv 485\,{\rm
MeV}$. The “thickness” $\delta=10\,{\rm MeV}$ assures that the drop starting
at $\mu=M(0)$ terminates at $\mu=\Lambda$ just as in the $N_{f}=2$ case letter
. It is obvious from Eq. (8) that such running coupling interpolates between
the two extrema, $G_{V}=0$ and $G_{V}=G_{S}/3$, which respectively give a
softer and a stiffer EoS buballa ; fukushima1 ; fukushima2 . Fig. 1 shows the
running of $G_{V}(\mu)$ and also illustrates how it affects the Fermi
momentum. From the physical point of view it is important to notice that the
ansatz assumes that after chiral symmetry gets (partially) restored the
repulsion among the (bare) quarks decreases as the density increases. In other
words, it is assumed that quarks with large effective masses tend to strongly
repel each other as compression increases and the quark condensates decrease.
After the chiral transition occurs, and the effective masses tend to their
bare values, quarks can be further compressed without repelling each other
indicating that repulsion should be necessary only as long as the quark
condensates, $\sigma_{f}$, are non-zero. A pictorial representation of the
physical process driving the running of $G(\mu)$ is presented in 2. Also,
remark that $\delta$ was chosen so as to give a smooth transition within a
narrow $10\,{\rm MeV}$ width since taking $\delta\to 0$ could lead to
discontinuities in $V_{s}^{2}$ which do not seem to be observed in the
simulations of Refs. sinansimulation ; michal ; weisesimulation . Note that in
order for $\sqrt{{\tilde{\mu}}^{2}-M^{2}}\to\sqrt{{\mu}^{2}-m^{2}}$ it is not
compulsory that $G_{S}$ and $K$ run with $\mu$ since the quark condensates,
multiplying these parameters in Eq. (4), naturally decrease with $\mu$. In
summary, for a given flavor i, $G_{S}$ and $K$ always appear in combinations
such as $G_{S}\sigma_{i}$ and $K\sigma_{j}\sigma_{k}$ (see Eq. (4)) which tend
to vanish at high-$\mu$ while $G_{V}$ appears in combinations such as $G_{V}n$
(see Eq. (2)) which always give a finite high-$\mu$ contribution when $G_{V}$
is fixed.

Finally, it must be pointed out that the idea of considering $G_{V}$ to depend
on a control parameter, as proposed here, is not new. A similar course of
action was originally taken by Kunihiro kunihiro , who considered $G_{V}$ to
be temperature dependent in order to evaluate quark susceptibilities at
high-$T$ (see also Ref. lorenzo ).

##  IV Numerical results

Let us now analyze the effect of $G_{V}(\mu)$ on some relevant thermodynamical
observables starting with the quark condensates for the light flavors. Fig. 3
shows the results for
$\sigma_{u,d}=\langle{\overline{u}}u\rangle=\langle{\overline{d}}d\rangle$
obtained with different $G_{V}$ values. Around $\mu=1.3\,\mu_{c}$ the
$G_{V}(\mu)$ curve, which initially agrees with the $G_{V}=G_{S}/3$ result,
converges towards the one predicted by using $G_{V}=0$. In the same spirit
Fig. 4 compares the dressed Fermi momentum for light quarks,
$p_{F}^{*}=\sqrt{{\tilde{\mu}}^{2}-M^{2}}$, with its bare counterpart,
$p_{F}^{0}=\sqrt{\mu^{2}-m^{2}}$, reproducing the pictorial view (shown in
Fig. 2) from a quantitative perspective. Next, let us examine the baryonic
number density, $n_{B}$, which in the present work represents the fundamental
thermodynamical quantity. The result obtained with $G_{V}(\mu)$ is presented
in Fig. 5 together with the predictions from the $G_{V}=0$ and $G_{V}=G_{S}/3$
cases. The results from $G_{V}(\mu)$ and $G_{V}=G_{S}/3$ agree up to
$\mu_{B}\approx 1.4\,{\rm GeV}$ when the former starts to converge towards the
$G_{V}=0$ curve. Fig. 6 shows the speed of sound squared for the three
relevant cases. The $G(\mu)$ curve peaks at $n_{B}\simeq
3.23\,n_{0}=0.52\,{\rm fm}^{-3}$ (corresponding to ${\cal E}=0.59\,{\rm
GeVfm^{-3}}$) producing the non-conformal result $V_{s}^{2}\simeq 0.38$ (note
these numerical values are consistent with those reported in Ref michal ).

Figure 5: Baryonic number density, in units of $n_{0}=0.16\,{\rm fm}^{-3}$, as
a function of the baryonic chemical potential, $\mu_{B}=3\mu$. The
$G_{V}(\mu)$ result interpolates between those predicted by $G_{V}=0$ and
$G_{V}=G_{S}/3$. The chiral first order phase transition takes place at
$\mu_{B}=1.083\,{\rm GeV}$ for $G_{V}=0$ and at $\mu_{B}=1.104{\rm GeV}$ for
the other two cases.

Figure 6: Speed of sound (squared) as a function of $n_{B}/n_{0}$. The running
coupling predicts a non-conformal peak, $V_{s}^{2}=0.38$, at
$n_{B}=3.23n_{0}$. The light band corresponds to the pQCD results eduardo
when the $\overline{\rm MS}$ renormalization scale varies from the central
value, $2\mu$ (bottom edge), to $4\mu$ (top edge). The thin dotted line
represents the conformal result, $V_{s}^{2}=1/3$.

The curve then dives into the sub-conformal region reaching $V_{s}^{2}\simeq
0.08$ at $n_{B}\simeq 7.50\,n_{0}$ before converging to the conformal result
as $n_{B}$ further increases. The onset of strangeness occurs at $n_{B}\simeq
5.50\,n_{0}$ so that the three EoS become softer due to the inclusion of one
more degree of freedom. As a consequence of this softening the $G_{V}=G_{S}/3$
also predicts a non-conformal peak at $n_{B}\simeq 5.50\,n_{0}$ and
$V_{s}^{2}\simeq 0.43$ while $G_{V}=0$ predicts a peak at $n_{B}\simeq
5.50\,n_{0}$ and $V_{s}^{2}\simeq 0.33$. However, at higher $n_{B}$ values the
use of a fixed coupling prevents convergence towards to conformal result, as
the figure shows. This result is not unexpected since, as already discussed,
the Fermi momentum for this case does not converge to its pQCD counterpart,
$p_{F}^{0}$. It is important to remark that the conjectured coupling running
predicts that after peaking at the super-conformal region, $V_{s}^{2}$
approaches the conformal value from below, like pQCD. It should be also
emphasized that the shape of the curve generated with $G_{V}(\mu)$ resembles
some of those recently predicted in Refs. sinansimulation ; michal ;
weisesimulation .

Figure 7: Pressure, normalized by $P_{0}$ (see text), as a function of
$\mu_{B}$. The gray band represents the region where $\mu=\mu_{B}/3>\Lambda$.
The light band corresponds to the pQCD results eduardo  when the
$\overline{\rm MS}$ renormalization scale covers the range from $\mu$ (bottom
edge), to $4\mu$ (top edge). The dotted line represents the pQCD predictions
at the central $\overline{\rm MS}$ scale, $2\mu$.

Figure 8: EoS for the three cases considered. The light band corresponds to
the pQCD results eduardo  when the $\overline{\rm MS}$ renormalization scale
varies from $\mu$ (bottom edge) to $4\mu$ (top edge). The softening of the
$G_{V}(\mu)$ curve takes place at ${\cal E}\simeq 0.7\,{\rm GeVfm^{-3}}$ in
agreement with Refs. nature ; michal . For completeness, the region which
concerns CET has also been indicated.

Figure 9:  Conformal measure, ${\cal C}=\Delta/{\cal E}$, as a function of
$n_{B}/n_{0}$. The running coupling predicts a change of slope of high
amplitude, at $n_{B}=4-10\,n_{0}$. A fixed $G_{V}$ leads to a negative $\cal
C$ at $n_{B}/n_{0}\simeq 14$. The light band corresponds to the pQCD results
eduardo  when the $\overline{\rm MS}$ renormalization scale covers the range
from $\mu$ (top edge) to $4\mu$ (bottom edge).

The NJL pressure together with the pQCD results for the $N_{f}=2+1$ case,
obtained from Ref. eduardo , is displayed in Fig. 7. The pQCD results were
obtained by varying the $\overline{\rm MS}$ renormalization scale from $\mu$
to $4\mu$ while the Fermi-Dirac limit for free massless quarks, used to
normalize the pressure in Fig. 7, reads

$P_{0}=\frac{N_{c}N_{f}}{12\pi^{2}}\left(\frac{\mu_{B}}{3}\right)^{4}\,.$ (9)

Fig. 8 displays the NJL EoS as well as the pQCD result (generated from Ref.
eduardo ). The figure clearly shows that the predictions coming from
$G_{V}(\mu)$ and $G_{V}=0$ agree with pQCD at high energies while
$G_{V}=G_{S}/3$ does not. Of utmost importance is the fact that at ${\cal
E}\simeq 0.7\,{\rm GeVfm^{-3}}$ a sudden change of slope takes place producing
the softening of the EoS produced by $G_{V}(\mu)$, in accordance with Refs.
nature ; michal . A second change of slope happens at ${\cal E}\simeq 2\,{\rm
GeVfm^{-3}}$ redressing the $G_{V}(\mu)$ curve so that it smoothly joins the
pQCD band.

Finally, let us use our model in order to examine the possibility that the
conformal measure, $\cal C$, remains positive at all densities. This important
question has been recently addressed in Ref. fukushimatrace  where the authors
have considered the trace anomaly, $\Delta$, which trivially relates to $\cal
C$ via ${\cal C}=\Delta/{\cal E}$. Fig. 9 shows that the fixed $G_{V}=G_{S}/3$
produces a maximally stiff EoS which yields a negative $\cal C$ for
$n_{B}\gtrsim 14\,n_{0}$. When repulsion is absent, the EoS is softer causing
${\cal C}\to 0$ as $n_{B}\to\infty$ in conformity with the pQCD predictions
(generated from Ref. eduardo ). At the same time, our running coupling
predicts a change of slope of high amplitude, at $n_{B}=4-10\,n_{0}$,
preventing $\cal C$ from becoming negative. In summary, $G(\mu)$ shifts the
high-$n_{B}$ behavior of the trace anomaly which then approaches zero while
remaining positive, supporting the hypothesis advanced in Ref. fukushimatrace
.

##  V Conclusions

The three-flavor NJL model with a repulsive vector channel, parametrized by
$G_{V}$, has been considered in the evaluation of the EoS describing symmetric
cold quark matter. The work extends the application performed in Ref. letter ,
where the two-flavor version has been considered in the presence of density
dependent repulsive coupling, $G_{V}(\mu)$. Here, it has been shown that the
presence of strangeness does not affect the main physical properties displayed
by key thermodynamical quantities evaluated with $G_{V}(\mu)$. The advantage
of such a model is that one is then able to interpolate between a regime where
repulsion is high (the EoS is stiff) and a regime where repulsion low (the EoS
is soft). In this way the NJL model can simultaneously observe astrophysical
constraints, which require the EoS to be stiff at lower densities, while
producing results which agree with pQCD at arbitrarily high densities. For
instance, considering the moderate value $G_{V}(0)=G_{S}/3$ this work shows
that it is possible to describe a non-conformal peak at $V_{s}^{2}=0.38$ and
$n_{B}=3.23\,n_{0}=0.52\,{\rm fm}^{-3}$ (corresponding to ${\cal E}=0.59\,{\rm
GeVfm^{-3}}$). These numerical values are in good agreement with some of the
values quoted in Ref. michal . We have also seen that, as the density
increases, the interpolating model predicts that $V_{s}^{2}$ approaches the
pQCD (conformal) prediction, $V_{s}^{2}\to 1/3$ from below, as expected.
Another important result obtained here shows that our model can produce a
noticeable change of slope in an initially hard EoS so that it will soften and
join the pQCD predictions at higher energy densities. Interestingly enough
this change happens at ${\cal E}\simeq 0.7\,{\rm GeVfm^{-3}}$, in conformity
with predictions made in Refs. nature ; michal ; new . The model also allows
us to conclude that a non-conformal peak in $V_{s}^{2}$ is not in tension with
the trace anomaly being positive for all densities, a result which agrees with
a scenario proposed in Ref. fukushimatrace . As explicitly shown here, these
findings cannot be reproduced if one naively uses $G_{V}=0$ (the EoS is far
too soft at low-$n_{B}$), or if one fixes $G_{V}$ to a finite value (the EoS
is far too hard at high-$\mu_{B}$). At first sight it seems remarkable that
with a simple modification the model is able to reproduce such highly non-
trivial results, which were originally obtained through the use of more
sophisticated approaches nature ; michal ; fukushimatrace . However, it should
be clear that the simple modification encoded within the $G_{V}(\mu)$ running
has physical consequences which in turn imply that the fundamental concept of
repulsion should be reviewed. More precisely, the results obtained here
suggest that quarks with bare masses do not tend to repel each other when
compressed, in opposition to the behavior displayed by quarks with effective
masses. Obviously the simple ansatz proposed here is not unique so that one is
free to consider alternative forms (such as gaussian, skewed gaussian,etc) as
well as the use of other parametrizations provided that $G_{V}$ decreases with
the density after the chiral transition takes place (keeping in mind that this
is the main ingredient driving the crucial change of slope observed in the
corresponding EoS). In principle, the mechanism described in this work can be
generalized to any model which contains a repulsive channel. Possible
extensions include the consideration of non symmetric quark matter in
$\beta$-equilibrium, in order to describe quark stars, as well as the
inclusion of a diquark interaction channel, in order to explore the high-
density region of QCD, among others. In future applications one could also
consider replacing the popular pQCD predictions with those furnished by the
renormalization group optimized perturbation theory, since this resummation
technique generates results which are less sensitive to scale changes rgopt1 ;
rgopt2 ; rgopt3 .

###### Acknowledgements.

The author is partially supported by Conselho Nacional de Desenvolvimento
Científico e Tecnológico (CNPq), Grant No 307261/2021-2 and by CAPES - Finance
Code 001. This work has also been financed in part by Instituto Nacional de
Ciência e Tecnologia de Física Nuclear e Aplicações (INCT-FNA), Process No.
464898/2014-5.

## References

  * (1) Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961); ibid. 124, 246 (1961).
  * (2) A. Chodos, R.L. Jaffe, K. Johnson, C.B. Thorn and V.F. Weisskopf, Phys. Rev. D 9, 3471 (1974).
  * (3) A. Chodos, R.L. Jaffe, K. Johnson and C.B. Thorn, Phys. Rev. D 10, 2599 (1974).
  * (4) T. DeGrand, R.L. Jaffe, K. Johnson and J. Kiskis, Phys. Rev. D 12, 2060 (1975).
  * (5) M. Buballa, Phys. Rept. 407, 205 (2005), [arXiv:hep-ph/0402234 ].
  * (6) S. Gandolfi, A. Y. Illarionov, K. E. Schmidt, F. Pederiva and S. Fantoni, Phys. Rev. C 79, 054005 (2009), [arXiv:0903.2610].
  * (7) I. Tews, T. Krüger, K. Hebeler and A. Schwenk, Phys. Rev. Lett. 110, 032504 (2013), [arXiv:1206.0025].
  * (8) B. A. Freedman and L. D. McLerran, Phys. Rev. D 16, 1169 (1977).
  * (9) A. Kurkela, P. Romatschke and A. Vuorinen, Phys. Rev. D 81, 105021 (2010), [arXiv:0912.1856].
  * (10) T. Gorda, A. Kurkela, P. Romatschke, M. Säppi and A. Vuorinen, Phys. Rev. Lett. 121, 202701 (2018), [arXiv:1807.04120].
  * (11) T. Kojo, APPS Bull. 31, 11 (2021), [arXiv:2011.10940].
  * (12) E. Annala, T. Gorda, A. Kurkela, J. Nättilä and A. Vuorinen, Nature Phys. 16, 907 (2020), [arXiv:1903.09121].
  * (13) J. Antoniadis et al., Science  340, 6131 (2013), [arXiv:1304.6875].
  * (14) H. T. Cromartie et al. (NANOGrav), Nature Astron. 4, 72 (2019), [arXiv:1904.06759].
  * (15) E. Fonseca et al., Astrophys. J. Lett. 915, L12 (2021), [arXiv:2104.00880].
  * (16) B. Margalit and B. D. Metzger, Astrophys. J. Lett. 850, L19 (2017), [arXiv:1710.05938].
  * (17) L. Rezzolla, E. R. Most, and L. R. Weih, Astrophys. J. Lett. 852, L25 (2018), [arXiv:1711.00314].
  * (18) M. Ruiz, S. L. Shapiro, and A. Tsokaros, Phys. Rev. D 97, 021501 (2018), [arXiv:1711.00473].
  * (19) M. Shibata, E. Zhou, K. Kiuchi, and S. Fujibayashi, Phys. Rev. D 100, 023015 (2019), [arXiv:1905.03656].
  * (20) A. Nathanail, E. R. Most and L. Rezzolla, Astrophys. J. Lett. 908, L28 (2021), [arXiv:2101.01735].
  * (21) S. Altiparmak, C. Ecker and L. Rezzolla, [arXiv:2203.14974].
  * (22) M. Marczenko, L. McLerran, K. Redlich and C. Sasaki, [arXiv:2207.13059].
  * (23) L. Brandes, W. Weise and N. Kaiser, [arXiv:2208.03026].
  * (24) K. Fukushima, Phys. Rev. D 77, 114028 (2008), [erratum: Phys. Rev. D 78, 039902 (2008)], [arXiv:0803.3318 ].
  * (25) K. Fukushima, Phys. Rev. D 78, 114019 (2008), [arXiv:0809.3080].
  * (26) M. B. Pinto, [arXiv:2208.06911].
  * (27) Y. Fujimoto, K. Fukushima, L. D. McLerran and M. Praszalowicz, [arXiv:2207.06753].
  * (28) D. P. Menezes, M. B. Pinto, L. B. Castro, P. Costa and C. Providência, Phys. Rev. C 89, 055207 (2014), [arXiv:1403.2502].
  * (29) P. Rehberg, S. P. Klevansky and J. Hüfner, Phys. Rev. C 53, 410 (1996), [arXiv:9506436].
  * (30) T. E. Restrepo, J. C. Macias, M. B. Pinto and G. N. Ferrari, Phys. Rev. D 91, 065017 (2015), [arXiv:1412.3074].
  * (31) J. Sugano, J. Takahashi, M. Ishii, H. Kouno and M. Yahiro, Phys. Rev. D 90, 037901 (2014), [arXiv:1405.0103].
  * (32) H.-S. Zong and W.-M. Sun, Int. J. Mod. Phys. A 23, 3591 (2008).
  * (33) T. Kunihiro, Phys. Lett. B 271, 395 (1991).
  * (34) L. Ferroni and V. Koch, Phys. Rev. C 83, 045205 (2011), [arXiv:1003.4428].
  * (35) E. S. Fraga and P. Romatschke, Phys. Rev. D 71, 105014 (2005), [arXiv:hep-ph/0412298].
  * (36) O. Komoltsev, [arXiv:2208.03086].
  * (37) J. L. Kneur, M. B. Pinto and T. E. Restrepo, Phys. Rev. D 100, 114006 (2019), [arXiv:1908.08363].
  * (38) J. L. Kneur, M. B. Pinto and T. E. Restrepo, Phys. Rev. D 104, L031502 (2021), [arXiv:2101.02124].
  * (39) J. L. Kneur, M. B. Pinto and T. E. Restrepo, Phys. Rev. D 104, 034003 (2021), [arXiv:2101.08240].