###### Abstract

QCD with the isospin chemical potential, $\mu_{I}$, is a useful laboratory to
delineate the microphysics in dense QCD. To study the quark-hadron-continuity
we use a quark-meson model that interpolates hadronic and quark matter physics
at microscopic level. The equation of state is dominated by mesons at low
density but taken over by quarks at high density. We extend our previous
studies with two-flavors to the three-flavors case to study the impact of the
strangeness which may be brought by kaons $(K_{+},K_{0})=(u\bar{s},s\bar{d})$
and the UA(1) anomaly. In the normal phase the excitation energies of kaons
are reduced by $\mu_{I}$ in the same way as hyperons in nuclear matter at
finite baryon chemical potential. Once pions condense, kaon excitation
energies increases as $\mu_{I}$ does. Moreover, strange quarks become more
massive through the UA(1) coupling to the condensed pions. Hence at zero and
low temperature the strange hadrons and quarks are highly suppressed. The
previous findings in two-flavor models, sound speed peak, negative trace
anomaly, gaps insensitve to $\mu_{I}$, persist in our three-flavor model and
remain consistent with the lattice results to $\mu_{I}\sim 1$ GeV. We discuss
the non-perturbative power corrections and quark saturation effects as
important ingredients to understand the crossover equations of state measured
on the lattice.

###### keywords:

equations of state; pion condensation; quark-hadron continuity; power
corrections; quark saturation

1 1 0 https://doi.org/ Isospin QCD as a laboratory for dense QCD Title Toru
Kojo 1, Daiki Suenaga 2,3 and Ryuji Chiba 1 Firstname Lastname, Firstname
Lastname and Firstname Lastname Kojo, T.; Suenaga, D.; Chiba, R.
Correspondence<EMAIL_ADDRESS>
##  1 Introduction

Recently there have been increasing attentions to two-color QCD (QC2D) Kogut
et al. (1999, 2000); Iida et al. (2024, 2020, 2021); Boz et al. (2020, 2013);
Cotter et al. (2013); Hands et al. (2011); Astrakhantsev et al. (2020);
Bornyakov et al. (2020); Muroya et al. (2003); Suenaga et al. (2023a, b);
Kawaguchi and Suenaga (2024); Sun et al. (2007); Brauner et al. (2009);
Strodthoff and von Smekal (2014); Strodthoff et al. (2012) or QCD at finite
isospin but zero baryon densities (isospin QCD, QCDI in short) Abbott et al.
(2024, 2023); Brandt et al. (2023); Son and Stephanov (2001a, b); Splittorff
et al. (2001); Gómez Nicola and Vioque-Rodríguez (2022); Lu et al. (2020). In
these theories lattice simulations are doable without the sign problem.
Confronting theories with lattice results should provide us with useful
insights into QCD matter at high baryon density ($n_{B}$), see, e.g., reviews
Fukushima and Hatsuda (2011); Oertel et al. (2017); Baym et al. (2018); Kojo
(2021); Vuorinen (2024).

Equations of state (EOS) of dense matter have one-to-one correspondence with
the mass-radius ($M$-$R$) relations of neutron stars. One of important
indications from neutron star observations and nuclear constraints is that QCD
matter is soft around nuclear saturation density $n_{0}\simeq 0.16\,{\rm
fm}^{-3}$, but rapidly becomes stiff around $n_{B}=2$-$4n_{0}$ Drischler et
al. (2021); Han et al. (2023). This density for this stiffening is smaller
than the density where baryons of the radius $0.5$-$0.8$ fm spatially overlap.
Meanwhile nuclear many-body calculations become problematic for $n_{B}\gtrsim
1.5$-$2n_{0}$. Observationally rapid stiffening is supported by small
variation (or even increase) in radii from $1.4M_{\odot}$ to $2.1M_{\odot}$
neutron stars. If EOSs stiffen only gently as in typical hadronic EOS with
many-body forces, the radii of $\simeq 2M_{\odot}$ neutron stars would be
substantially smaller than $1.4M_{\odot}$ neutron stars by $\sim 1$ km Kojo et
al. (2022). Such contrast in $1.4M_{\odot}$ to $2.1M_{\odot}$ radii may be
also studied by gravitational waves from binary neutron star mergers, see,
e.g., Refs. Huang et al. (2022); Fujimoto et al. (2023); Kedia et al. (2022);
Bauswein et al. (2019).

To study the interplay between hadronic and quark matter, it is crucial to
understand the rapid stiffening at microscopic level. Several theoretical
studies McLerran and Reddy (2019); Jeong et al. (2020); Kojo (2021); Fujimoto
et al. (2024) suggest that such rapid stiffening around $2$-$4n_{0}$ is
triggered by quark degrees of freedom, whether or not quarks are confined or
deconfined. Assuming a model in which quarks remain confined in baryons,
quarks can still occupy states with a certain probability and eventually
affect baryons through the quark Pauli blocking Kojo (2021); Fujimoto et al.
(2024). Such quark constraint becomes substantial even before baryons overlap.
After quark states at low momentum are saturated (“quark saturation” Kojo
(2021)), the quark Fermi sea begins to form with the diffused Fermi surface
whose thickness is $\simeq 200$-$300$ MeV. The quark saturation forces matter
of non-relativistic baryons to change into that of relativistic baryons or
quarks, driving the rapid growth in pressure but modest change in the energy
density Kojo (2021). The quark momentum distribution at finite density has
been manifestly computed for QCD in two-dimensional QCD Hayata et al. (2023).

The quark saturation and the associated stiffening may occur also in QC2D and
QCDI, although baryons are replaced with diquark baryons and mesons,
respectively. In these theories the Bose-Einstein Condensation of diquarks or
mesons occur at the onset of matter; (composite) bosons occupy the zero
momentum state. At finite chemical potential exceeding the mass threshold, the
amplitude of condensates would grow indefinitely unless some sort of repulsive
forces temper the growth of the amplitude. For theories of elementary bosons
we do not have definite rationals why such repulsion should exist, while, for
theories of bosons made of fermions, indefinite growth of boson amplitudes
would violate the Pauli exclusion principle. Hence, irrespective to the
details of interactions, effective repulsions among bosons must emerge.

For isospin QCD there have been many studies, see Ref. Mannarelli (2019) for
recent good summary to 2019. Recently isospin QCD attracts renewed attention
as a laboratory to study concepts proposed for neutron star physics Chiba and
Kojo (2024); Chiba et al. (2024); Fujimoto (2024) or to derive some
constraints on neutron star EOS Cohen (2003); Navarrete et al. (2024);
Fujimoto and Reddy (2024); Moore and Gorda (2023). In Refs. Chiba and Kojo
(2024); Chiba et al. (2024), we used a two-flavor quark-meson model to discuss
the rapid stiffening and related microphysics. This model is renormalizable
and includes mesons and quarks Adhikari et al. (2017); Ayala et al. (2023). At
low density the EOS is dominated by mesons while, at high density, quarks
dominate. In quark matter region mesonic degrees of freedom as condensates
remain near the quark Fermi surface.

In this paper, after adding some supplements to our previous two-flavor
studies, we then extend the analyses to three-flavors. Although a strange
quark does not have isospin, hadronic strange particles, such as kaons,
$\eta,\eta^{\prime}$, and so on, contain $u$,$d$-quarks and can be affected by
the isospin density. Indeed the excitation energies of
$(K_{+},K_{0})=(u\bar{s},s\bar{d})$ decrease with increasing $\mu_{I}$ and
would eventually condense unless other particles in the system repel such
kaons. Another effect of interest is the impact of UA(1) anomaly that affects
the strangeness in the Dirac sea Gao et al. (2022); Minamikawa et al. (2023).
At finite density, the reduction of the effective $u$,$d$ quark masses softens
the chiral symmetry breaking in the strange quark sector, while the appearance
of the pion condensates can enhance the strange quark mass by few hundred MeV.
This continues until the medium screening cutoff such effects. Similar
coupling, diquark-to-chiral order parameters, has been discussed in the
context of quark-hadron-crossover Hatsuda et al. (2006); Yamamoto et al.
(2007); Hatsuda et al. (2008); Zhang et al. (2009). Thus the strange particles
are useful probes to diagnose various medium effects in dense matter.

In this paper we use a quark-meson model to discuss a quark-hadron crossover
Schäfer and Wilczek (1999); Masuda et al. (2013a, b, 2016); Tajima et al.
(2023); Ma and Rho (2020). In Sec. 2 we first discuss a picture behind this
model to inform readers of what physics we try to describe. We will use the
$1/N_{\rm c}$ expansion to classify the effects caused by different degrees of
freedom ’t Hooft (1974); Witten (1979). In Sec. 3 we present a quark-meson
model including the strangeness and the UA(1) anomaly effects. In Sec. 4 we
study the effective potential and meson spectra at tree level. In Sec. 5 we
include quarks to change the structure of the theory and discuss how it leads
to the trend consistent with the lattice results. In Sec. 6 we mention our
parameter fixing through meson poles in vacuum with quark-loop corrections. In
Sec. 7 we examine EOS at zero temperature and its relationship with the
microphysics. Sec. 8 is devoted to summary.

##  2 A quark-meson model at finite density: outline

Figure 1: (upper) Some examples of the correspondence between quark-gluon,
color line graphs, and hadronic graphs. The coupling between quark bilinear
operators and mesonic state must be $\sim N_{\rm c}^{1/2}$ for the consistency
with the color line graphs. Meson three point vertices must be $\sim N_{\rm
c}^{-1/2}$. (lower) The quark-meson vertices in quark-gluon, color-line, and
quark-meson representations. The quark-meson coupling must be $\sim N_{\rm
c}^{-1/2}$ to be consistent with the color line representation.

We begin with the $N_{\rm c}$ counting of a quark-meson coupling. A
diagrammatic representation is given in Fig. 1. To derive the $N_{\rm c}$
counting of hadronic parameters we match the $N_{\rm c}$ counting in a color
line graph and a hadronic graph. For instance, from a mesonic correlator with
the magnitude of $N_{\rm c}$, one can conclude that a quark bilinear operator
couples to a mesonic state with the strength $N_{\rm c}^{1/2}$. Using this
estimate and repeating the similar matching, one can deduce that the meson-
meson interaction is $\sim N_{\rm c}^{-1/2}$ for three meson vertices and
$\sim N_{\rm c}^{-1}$ for four meson vertices, and so on. Similarly, for a
quark meson coupling graph with the amplitude of $\sim g_{s}^{2}N_{\rm c}\sim
N_{\rm c}^{0}$, one can conclude that the quark-meson coupling is $N_{\rm
c}^{-1/2}$. While we consider only a single gluon exchange, including more
gluons introduces $(g_{s}^{2}N_{\rm c})^{n}\sim 1$ so that the counting is not
affected.

Figure 2:  The mass self-energy graphs with (left) gluon and (right) meson
loops. The gluon loop graph is $\sim g_{s}^{2}N_{\rm c}\sim N_{\rm c}^{0}$.
The meson loop graph is $\sim g^{2}\sim g_{s}^{4}N_{\rm c}\sim N_{\rm c}^{-1}$
except when the meson condenses. With meson condensates of
$\langle\sigma\rangle\sim f_{\pi}\sim N_{\rm c}^{1/2}$, the graphs with
condensate can represent (some part of) the gluon loop graphs in the leftmost
panel.

Below we assume that this quark-meson coupling, $g$, is dominated by soft
gluons for which the quark-gluon coupling $g_{s}$ is large and characterizes
the size of the quark-meson coupling. The soft gluon exchanges occur
indefinitely within the produced mesons. This sort of process cannot be
represented by perturbative treatments at finite order. The purpose of
manifestly including mesonic degrees of freedom is to replace or parametrize
these process in a book keeping manner.

The quark self-energy with gluon loops (leftmost panel, Fig. 2) are
$g_{s}^{2}N_{\rm c}\sim N_{\rm c}^{0}$ yielding the mass of $\sim\Lambda_{\rm
QCD}$. The factor $N_{\rm c}$ amplification occurs as there is a color loop in
color line representations. In contrast, the self-energy with mesonic loops
are $g^{2}\sim g_{s}^{4}N_{\rm c}\sim N_{\rm c}^{-1}$ and suppressed
(rightmost panel, Fig. 2). There is an exception, however. When mesons form
condensates, the self-energies originating from meson loops can be amplified
to $N_{\rm c}^{0}$. The condensate contains indefinite numbers of $q\bar{q}$
pairs. In the case of chiral condensates, condensed $\bar{q}_{L}q_{R}$ or
$\bar{q}_{R}q_{L}$ pairs transform $q_{L}(q_{R})$ into $q_{R}(q_{L})$
generating the mass self-energies. If we neglect changes in the number of
$\bar{q}q$ pairs in the condensate, a quark propagator with meson loops can be
factorized into a product of quark propagators with a background meson
condensate. Since quark propagators with the meson background already contain
graphs with many soft gluons, to avoid double counting we do not manifestly
include the quark self-energy graphs with soft gluon loops. Meanwhile graphs
with hard gluons may be added separately for more complete exploration of the
phase space in the intermediate states.

The reduction of meson masses can be important even before mesons condense.
Such precursory effects are important near the phase boundaries and have been
studied in the context of color-superconductivity Nishimura et al. (2022);
Kitazawa et al. (2005, 2002).

We include “mesons” as a representative of $q\bar{q}$ propagating graph with
indefinite numbers of soft gluon exchanges. Using meson degrees of freedom at
high density might look unnatural, but models manifestly including mesons can
be dynamical reduced to pure quark descriptions; the dissociation and
structural changes of these mesons into quarks can be described by inserting
quark loops in the meson propagators. Thus unwanted contributions can be
canceled by including quark loops Blaschke et al. (2014).

At high density, quarks typically have large momenta but the excitation
energies can be small, $\sim E-\mu\ll\mu$ ($\mu$: chemical potential) near the
Fermi surface. These soft quarks and soft gluons may keep quark-meson
couplings as strong as in vacuum unless screening processes cut off soft
gluons. For S-wave condensates, a quark with $\vec{p}$ and an antiquark with
$-\vec{p}$ make a pair in which soft gluons are exchanged. For a pion
condensed phase in isospin QCD, these soft gluons should be largely unscreened
because the condensate is color-singlet and colored excitations are gapped.
These unscreened gluon propagators are supported by lattice simulations for
two-color QCD Boz et al. (2019); Bornyakov et al. (2020); Kojo and Baym
(2014); Suenaga and Kojo (2019); Kojo and Suenaga (2021); Contant and Huber
(2020).

Figure 3:  The gap created by soft gluons. The self-energies caused by soft
gluons are not very sensitive to the size of the Fermi surface because quarks
fluctuate only within small domain. The produced gap is characterized by soft
gluons and is supposed to be $\sim\Lambda_{\rm QCD}$, until hard gluon
processes dominate over soft gluon processes.

With soft gluons being unscreened, the size of the gap is $\sim\Lambda_{\rm
QCD}$ or can be even larger. To see this, we parametrize the soft gluon
exchange forces as

$\displaystyle
D_{g}(q)=\Theta\big{(}\Lambda-|\vec{q}|\big{)}\frac{\,c\,}{\,\Lambda^{2}\,}\,,~{}~{}~{}~{}~{}~{}c\sim
O(1)\,,$ (1)

where $c/\Lambda^{2}$ is the typical strength for a soft momentum transfer
$|\vec{q}|<\Lambda$. This model has been used for quarkyonic chiral spirals
Kojo et al. (2010a, b, 2012) and QCD in a magnetic field Kojo and Su (2013,
2014); Hattori et al. (2016) to yield the gap equation local in momentum space
(Fig. 3). The resultant gap is $\sim\Lambda$, and we expect the same happens
for the soft gluonic part in isospin QCD.

Now the gap equation reads

$\displaystyle\Delta~{}\sim~{}\frac{\,c\,}{\,\Lambda^{2}\,}\int_{\vec{l}}\Theta\big{(}\Lambda^{2}-|\vec{l}-\vec{p}|^{2}\big{)}\frac{\,\Delta\,}{\,\sqrt{\big{(}E(\vec{l})-\mu_{I}\big{)}^{2}+\Delta^{2}\,}\,}\,.$
(2)

Decomposing the loop momentum as $\vec{l}=\big{(}|\vec{p}|+\delta
l_{\parallel}\big{)}{\bf e}_{p}+\vec{l}_{\perp}$ with
$|\vec{p}|\simeq\mu_{I}$, the conservative estimate for the range is $|\delta
l_{\parallel}|\lesssim\Lambda$ and $|\vec{l}_{\perp}|\lesssim\Lambda$. Then
one can simplify the gap equation into ($v_{F}$: Fermi velocity)

$\displaystyle
1~{}\sim~{}\frac{\,c\,}{\,(2\pi)^{3}\Lambda^{2}\,}\int_{-\Lambda}^{\Lambda}\mathrm{d}\delta
l_{\parallel}\int_{|\vec{l}_{\perp}|\lesssim\Lambda}\mathrm{d}\vec{l}_{\perp}\frac{\,1\,}{\,\sqrt{\big{(}v_{f}\delta
l_{\parallel})^{2}+\frac{\,\vec{l}_{\perp}^{2}\,}{\,2\mu_{I}\,}+\Delta^{2}\,}\,}\,.$
(3)

When $\Lambda\ll\mu_{I}$ holds, we can omit the $\vec{l}_{\perp}^{2}/\mu_{I}$
term in the range of integral to factorize the $\vec{l}_{\perp}$ integral. At
this stage the equation loses the $\mu_{I}$-dependence and so does the
solution. Further calculation reads

$\displaystyle
1~{}\sim~{}\frac{\,c\,}{\,2\pi^{2}\,}\int_{0}^{\Lambda}\mathrm{d}\delta
l_{\parallel}\frac{\,1\,}{\,\sqrt{\big{(}v_{f}\delta
l_{\parallel})^{2}+\Delta^{2}\,}\,}~{}\sim~{}\frac{\,c\,}{\,2\pi^{2}\,}\ln\frac{\,\Lambda\,}{\,\Delta\,}\,.$
(4)

Thus the gap is $\Delta\sim\Lambda\mathrm{e}^{-2\pi^{2}/c}$. For a stronger
coupling the gap $\Delta$ grows to $\Lambda$.

The present arguments show that, even at large $\mu$ or large quark Fermi
momenta, non-perturbative phenomena characterized by $\Lambda_{\rm QCD}$ are
still possible. This continues until screening effects set in. When $\mu_{I}$
is sufficiently large, we should also add hard gluon contributions which can
be treated within weak coupling method; this hard contributions are sensitive
to the phase space around the Fermi surface and hence to $\mu_{I}$. As we see
later the gap (which is insensitive to $\mu_{I}$) adds the power corrections,
$\sim+\mu_{I}^{2}\Delta^{2}\sim\mu_{I}^{2}\Lambda_{\rm QCD}^{2}$, to the
pressure $P(\mu_{I})$. This power corrections, not capable within perturbative
computations, have played important roles in QCD phenomenology Novikov et al.
(1978); Shifman et al. (1979a, b) and also seem essential to describe the
lattice simulation data at large density Chiba and Kojo (2024); Chiba et al.
(2024).

##  3 A quark-meson model: mean field treatments

Now we consider a practical description of a three-flavor quark-meson model.
The Lagrangian is

$\displaystyle\mathcal{L}=\mathcal{L}_{\rm Y}+\mathcal{L}_{\rm kin}^{M}-V_{\rm
sym}-V_{\rm SB}-V_{\rm anom}\,.$ (5)

The quark with the Yukawa interaction takes the form

$\displaystyle\mathcal{L}_{\rm Y}$
$\displaystyle=\bar{q}\big{[}\mathrm{i}{\ooalign{\hfil/\hfil\crcr$\partial$}}+\mu_{I}\tau_{3}\gamma_{0}-g\big{(}\sigma_{a}+\mathrm{i}\gamma_{5}\pi_{a}\big{)}\lambda_{a}\big{]}q$
(8)
$\displaystyle=\bar{q}\big{[}\mathrm{i}{\ooalign{\hfil/\hfil\crcr$\partial$}}+\mu_{I}\tau_{3}\gamma_{0}\big{]}q-g\phi_{a}\big{[}\bar{q}_{L}\lambda_{a}q_{R}\big{]}-g\phi^{*}_{a}\big{[}\bar{q}_{R}\lambda_{a}q_{L}\big{]}\,,$
(11)

where $\phi_{a}=\sigma_{a}+\mathrm{i}\pi_{a}$. We have scalar and pseudoscalar
flavor nonets. In our notation the isospin chemical potential is
$\mu_{I}=\mu_{u}=-\mu_{d}$, coupled to $\bar{q}\gamma_{0}\tau_{3}q$ rather
than the isospin density operator $\bar{q}\gamma_{0}\frac{\,\tau_{3}\,}{2}q$
in the conventional sense.

The mesonic Lagrangian with the isospin chemical potential is

$\displaystyle\mathcal{L}^{M}_{\rm kin}=$
$\displaystyle\,\sum_{a=0,3,8}\frac{1}{\,2\,}|\partial_{\mu}\phi_{a}|^{2}$
$\displaystyle+(\partial_{\mu}+2\mathrm{i}\mu_{I}\delta_{\mu}^{0})a_{0}^{+}(\partial^{\mu}-2\mathrm{i}\mu_{I}\delta^{\mu}_{0})a_{0}^{-}+(\partial_{\mu}+2\mathrm{i}\mu_{I}\delta_{\mu}^{0})\pi^{+}(\partial^{\mu}-2\mathrm{i}\mu_{I}\delta^{\mu}_{0})\pi^{-}$
$\displaystyle+(\partial_{\mu}+\mathrm{i}\mu_{I}\delta_{\mu}^{0})\kappa^{+}(\partial^{\mu}-\mathrm{i}\mu_{I}\delta^{\mu}_{0})\kappa^{-}+(\partial_{\mu}+\mathrm{i}\mu_{I}\delta_{\mu}^{0})K^{+}(\partial^{\mu}-\mathrm{i}\mu_{I}\delta^{\mu}_{0})K^{-}$
$\displaystyle+(\partial_{\mu}+\mathrm{i}\mu_{I}\delta_{\mu}^{0})\bar{\kappa}^{0}(\partial^{\mu}-\mathrm{i}\mu_{I}\delta^{\mu}_{0})\kappa^{0}+(\partial_{\mu}+\mathrm{i}\mu_{I}\delta_{\mu}^{0})\bar{K}^{0}(\partial^{\mu}-\mathrm{i}\mu_{I}\delta^{\mu}_{0})K^{0}\,,$
(12)

where iso-triplet $(\sigma_{a=1-3},\pi_{a=1-3})=(a_{0},\pi)$ and two iso-
doublets $(\sigma_{a=4-7},\pi_{a=4-7})=(\kappa,K)$ couple to the isospin
chemical potential. The field normalization for isospin charged fields are
$a_{0}^{\pm}=(\sigma_{1}\pm\mathrm{i}\sigma_{2})/\sqrt{2}$, and so on. Since
$\tau_{3}$ commutes with the isospin rotation operator around the
$I_{3}$-axis, the U(1)${}_{I_{3}}$ symmetry is preserved at the level of the
Lagrangian. This symmetry is broken when pions condense.

The mesonic potential consists of three pieces

$\displaystyle V_{M}(\phi,\phi^{*})=V_{\rm sym}+V_{\rm SB}+V_{\rm anom}\,,$
(13)

with a U(3) symmetric potential

$\displaystyle V_{\rm sym}=-\frac{\,m_{M}^{2}\,}{\,4\,}{\rm
tr}\big{[}\mathcal{M}\mathcal{M}^{\dagger}\big{]}+\frac{\,\lambda\,}{\,48\,}{\rm
tr}\big{[}\big{(}\mathcal{M}\mathcal{M}^{\dagger}\big{)}^{2}\big{]}\,,~{}~{}~{}~{}~{}~{}~{}\mathcal{M}=\phi_{a}\lambda_{a}\,,$
(14)

and a symmetry breaking term associating the current quark masses,

$\displaystyle V_{\rm SB}=-\frac{\,c\,}{\,2\,}{\rm
tr}\big{[}\hat{m}_{q}^{\dagger}\mathcal{M}+\hat{m}_{q}\mathcal{M}^{\dagger}\big{]}\,,~{}~{}~{}~{}~{}~{}~{}~{}\hat{m}_{q}={\rm
diag.}(m_{l},m_{l},m_{s})\,,$ (15)

and the Kobayashi-Maskawa-’t Hooft (KMT) interaction

$\displaystyle V_{\rm
anom}=-\frac{\,K\,}{2}\big{[}\det\mathcal{M}+\det\mathcal{M}^{\dagger}\big{]}\,.$
(16)

which is responsible for the U${}_{A}(1)$ breaking.

Increasing $\mu_{I}$, the excitation energies of mesons including $u$ or
$\bar{d}$ are reduced, i.e., the excitation energies of
$\sigma^{+},\pi^{+},a_{0}^{+},K^{+},\cdots$ decrease linearly as a function of
$\mu_{I}$. As we see shortly, the tree level analyses show that the lightest
pion condenses first, and then the reduction of energies for the other mesons
is stopped; no other mesons condense. At tree level this is interpreted as the
effective repulsion between condensed pions and the other mesons. We assume
this holds even after including loop corrections. Thus we take the mean field
as

$\displaystyle\mathcal{M}_{\rm
MF}=\begin{bmatrix}~{}\langle\sigma\rangle&\mathrm{i}\langle\pi_{1}\rangle&0~{}\\\
~{}\mathrm{i}\langle\pi_{1}\rangle&\langle\sigma\rangle&0~{}\\\
~{}0&0&\langle\sigma_{s}\rangle~{}\end{bmatrix}\,,~{}~{}~{}~{}~{}\sigma=\sqrt{\frac{2}{\,3\,}}\sigma_{0}+\sqrt{\frac{1}{\,3\,}}\sigma_{8}\,,~{}~{}~{}~{}~{}\sigma_{s}=\sqrt{\frac{2}{\,3\,}}\sigma_{0}-2\sqrt{\frac{1}{\,3\,}}\sigma_{8}\,.$
(17)

Here we could choose $\pi_{1}$ fields for condensed fields without loss of
generality since the Lagrangian has the U(1)${}_{I_{3}}$ symmetry. The mean
field Yukawa term now takes the form $(q_{l}=u,d)$

$\displaystyle\mathcal{L}_{\rm Y}^{\rm
MF}=\bar{q}_{l}\big{[}\mathrm{i}{\ooalign{\hfil/\hfil\crcr$\partial$}}+\mu_{I}\tau_{3}\gamma_{0}-M_{l}-\mathrm{i}\gamma_{5}\lambda_{1}\Delta\big{]}q_{l}+\bar{s}\big{[}\mathrm{i}{\ooalign{\hfil/\hfil\crcr$\partial$}}-M_{s}\big{]}s\,,$
(22)

where we wrote $M_{l}=g\langle\sigma_{l}\rangle$,
$M_{s}=g\langle\sigma_{s}\rangle$, and $\Delta=g\langle\pi_{1}\rangle$.
Substituting the mean fields into the potentials, one finds

$\displaystyle V_{\rm sym}^{\rm
MF}=-\frac{\,m_{M}^{2}\,}{\,2g^{2}\,}(M_{l}^{2}+\Delta^{2})+\frac{\,\lambda\,}{\,24g^{4}\,}\big{(}M_{l}^{2}+\Delta^{2}\big{)}^{2}-\frac{\,m_{M}^{2}\,}{\,4g^{2}\,}M_{s}^{2}+\frac{\,\lambda\,}{\,48g^{4}\,}M_{s}^{4}$
(23)

and ($h_{l}=2cm_{l},h_{s}=2cm_{s}$)

$\displaystyle V_{\rm SB}^{\rm
MF}=-\frac{\,h_{l}\,}{g}M_{l}-\frac{\,h_{s}\,}{2g}M_{s}\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}V_{\rm
anom}^{\rm
MF}=-\frac{\,K\,}{\,g^{3}\,}\big{(}M_{l}^{2}+\Delta^{2}\big{)}M_{s}\,.$ (24)

Below we first examine the effective potential at tree level which includes
only the mesonic degrees of freedom. The quark dynamics manifestly enters only
after including quark loops. As we see later, the quark loops change the
structure of theories and impose important constraints on the meson mean
fields.

##  4 Purely hadronic descriptions: tree level analyses

###  4.1 Gap equations at tree level

We first study the tree level potential including only meson degrees of
freedom. Combining the potentials with the terms including chemical potential
in $\mathcal{L}^{M}_{\rm kin}$, the effective potential at tree level is

$\displaystyle\Omega_{0}=-\frac{\,2\mu_{I}^{2}\,}{g^{2}}\Delta^{2}+V_{\rm
sym}^{\rm MF}+V_{\rm SB}^{\rm MF}+V_{\rm anom}^{\rm MF}\,.$ (25)

The gap equations are

$\displaystyle\frac{\,\partial\Omega_{0}\,}{\,\partial M_{l}\,}$
$\displaystyle=\frac{\,M_{l}\,}{\,g^{2}\,}\bigg{[}-m_{M}^{2}+\frac{\,\lambda\,}{\,6g^{2}\,}\big{(}M_{l}^{2}+\Delta^{2}\big{)}-\frac{\,2K\,}{\,g\,}M_{s}-\frac{\,gh_{l}\,}{\,M_{l}\,}\bigg{]}=0\,,$
$\displaystyle\frac{\,\partial\Omega_{0}\,}{\,\partial M_{s}\,}$
$\displaystyle=\frac{\,M_{s}\,}{\,2g^{2}\,}\bigg{[}-m_{M}^{2}+\frac{\,\lambda\,}{\,6g^{2}\,}M_{s}^{2}-\frac{\,2K\,}{\,g\,}\frac{\,M_{l}^{2}\,}{\,M_{s}\,}-\frac{\,gh_{s}\,}{\,M_{s}\,}\bigg{]}=0\,,$
$\displaystyle\frac{\,\partial\Omega_{0}\,}{\,\partial\Delta\,}$
$\displaystyle=\frac{\,\Delta\,}{\,g^{2}\,}\bigg{[}-4\mu_{I}^{2}-m_{M}^{2}+\frac{\,\lambda\,}{\,6g^{2}\,}\big{(}M_{l}^{2}+\Delta^{2}\big{)}-\frac{\,2K\,}{\,g\,}M_{s}\bigg{]}=0\,.$
(26)

We can show that $gh_{l}/M_{l}^{\rm vac}=m_{\pi}^{2}$. We combine the first
and third equations to derive a simple relation,

$\displaystyle\frac{\,\partial\Omega_{0}\,}{\,\partial\Delta\,}=\frac{\,\Delta\,}{\,g^{2}\,}\bigg{[}-4\mu_{I}^{2}+\frac{\,gh_{l}\,}{\,M_{l}\,}\bigg{]}=0\,,~{}~{}~{}~{}~{}~{}~{}\frac{\,\partial^{2}\Omega_{0}\,}{\,(\partial\Delta)^{2}\,}=\frac{\,1\,}{\,g^{2}\,}\bigg{[}-4\mu_{I}^{2}+\frac{\,gh_{l}\,}{\,M_{l}\,}+\frac{\,\lambda\,}{\,3g^{2}\,}\Delta^{2}\bigg{]}\,.$
(27)

The curvature with respect to $\Delta$ is positive for small $\mu_{I}$ so that
$\Delta=0$ is favored. Then the gap equation is the same as in the vacuum
case; in this domain $M_{l}=M_{l}^{\rm vac}$ and $gh_{l}/M_{l}^{\rm
vac}=m_{\pi}^{2}$ holds. The curvature becomes negative for $m_{\pi}\geq
2\mu_{I}$, leading to the scaling $M_{l}\sim 1/\mu_{I}^{2}$. This in turn
leads to $\Delta^{2}\sim 1/M_{l}\sim\mu_{I}^{2}$ from the condition
$\partial\Omega_{0}/\partial M_{l}=0$. The strange quark is affected by
isospin density only through the anomaly term.

###  4.2 Meson kinetic and mass matrices at tree level

We now quickly review the properties of mesons in a medium of condensed pions.
We first take a look at the excitation energies of charged mesons in the low
density domain. Then the Lagrangian has the following structure
($E=\sqrt{\vec{p}^{2}+m^{2}}$)

$\displaystyle\frac{1}{\,2\,}[\phi_{+},\phi_{-}]\begin{bmatrix}~{}0~{}&~{}(p_{0}+N_{I}\mu_{I})^{2}-E^{2}~{}\\\
~{}(p_{0}+N_{I}\mu_{I})^{2}-E^{2}&0~{}\end{bmatrix}\begin{bmatrix}\phi_{+}\\\
\phi_{-}\end{bmatrix}\,,$ (28)

where $N_{I}=2$ for iso-triplet and $N_{I}=1$ for iso-doublet. The spectra are
found to be

$\displaystyle\omega_{\pm}(\vec{p})=E(\vec{p})\pm
N_{I}\mu_{I}\,.~{}~{}~{}~{}~{}~{}({\rm in~{}normal~{}phase})$ (29)

Mesons with positive isospins have energy reduction. This continues until one
of those mesons condenses. The pions are the lightest and have the largest
isospin $N_{I}=2$ so they condense first, at $\mu_{I}=m_{\pi}/2$.

Before the pion condensation, a pair of positive and negative charged mesons
together maintains the $\mu_{I}$-independence of the thermodynamics. The
thermodynamic potential from a (non-interacting) meson, for $N_{I}\mu_{I}\leq
E(\vec{p}=0)$, is

$\displaystyle\Omega=\frac{1}{\,2\,}\int_{\vec{p}}\big{(}|E+N_{I}\mu_{I}|+|E-N_{I}\mu_{I}|\big{)}~{}\rightarrow~{}\frac{1}{\,2\,}\int_{\vec{p}}\big{(}E+N_{I}\mu_{I}+E-N_{I}\mu_{I}\big{)}=\int_{\vec{p}}E\,,$
(30)

after the $\mu_{I}$ terms cancel. The resulting energy is the zero point
energy same as in the vacuum. The expression is valid until $N_{I}\mu_{I}$
reaches the meson mass.

After pions condense, mesons having energy reduction in the normal phase are
now subject to effective repulsions and hence the gap remains in the
excitation energies. In addition condensed pions supply isospin and parity
violating sources so that various mesons mix. At tree level the mixing is
caused through the quartic and the KMT interaction. We read off the meson mass
matrices masses by looking at the quadratic order of the potential

$\displaystyle V_{M}^{\rm
quad}=\frac{\,\partial^{2}\tilde{V}_{0}\,}{\,\partial\phi_{a}\partial\phi^{*}_{b}\,}\phi_{a}\phi_{b}^{*}+\frac{1}{\,2\,}\frac{\,\partial^{2}\tilde{V}_{0}\,}{\,\partial\phi_{a}\partial\phi_{b}\,}\phi_{a}\phi_{b}+\frac{1}{\,2\,}\frac{\,\partial^{2}\tilde{V}_{0}\,}{\,\partial\phi_{a}^{*}\partial\phi_{b}^{*}\,}\phi_{a}^{*}\phi_{b}^{*}+\cdots$
$\displaystyle=\frac{\,\partial^{2}\tilde{V}_{0}\,}{\,\partial\phi_{a}\partial\phi^{*}_{b}\,}\big{[}\sigma_{a}\sigma_{b}+\pi_{a}\pi_{b}\big{]}+{\rm
Re}\frac{\,\partial^{2}\tilde{V}_{0}\,}{\,\partial\phi_{a}\partial\phi_{b}\,}\big{(}\sigma_{a}\sigma_{b}-\pi_{a}\pi_{b}\big{)}-{\rm
Im}\frac{\,\partial^{2}\tilde{V}_{0}\,}{\,\partial\phi_{a}\partial\phi_{b}\,}\big{(}\sigma_{a}\pi_{b}+\pi_{a}\sigma_{b}\big{)}$
(31)

where we evaluated the second derivative at the mean field values, and also
used the fact that
$\frac{\,\partial^{2}\tilde{V}_{0}\,}{\,\partial\phi_{a}\partial\phi^{*}_{b}\,}\big{|}_{{\rm
MF}}$ is real. The mass matrices for $\sigma$, $\pi$ are

$\displaystyle
m_{\sigma_{a}\sigma_{b}}^{2}=2\bigg{[}\frac{\,\partial^{2}\tilde{V}_{0}\,}{\,\partial\phi_{a}\partial\phi^{*}_{b}\,}+{\rm
Re}\frac{\,\partial^{2}\tilde{V}_{0}\,}{\,\partial\phi_{a}\partial\phi_{b}\,}\bigg{]}\,,~{}~{}~{}~{}m_{\pi_{a}\pi_{b}}^{2}=2\bigg{[}\frac{\,\partial^{2}\tilde{V}_{0}\,}{\,\partial\phi_{a}\partial\phi^{*}_{b}\,}-{\rm
Re}\frac{\,\partial^{2}\tilde{V}_{0}\,}{\,\partial\phi_{a}\partial\phi_{b}\,}\bigg{]}\,,$
(32)

and the parity breaking $\sigma$-$\pi$ couplings induced by the pion
condensate are

$\displaystyle m_{\sigma_{a}\pi_{b}}^{2}=-2{\rm
Im}\frac{\,\partial^{2}\tilde{V}_{0}\,}{\,\partial\phi_{a}\partial\phi_{b}\,}~{}~{}~{}(\propto\Delta)\,.$
(33)

The pion condensate $\pi_{1}\sim u\bar{d}+d\bar{u}$ induces the conversion
$u\leftrightarrow d$ and $\bar{u}\leftrightarrow\bar{d}$, and also causes the
conversion between the $\sigma$\- and $\pi$-sectors. For light flavors all
channels but $\pi_{3},\sigma_{3}\sim u\bar{u}-d\bar{d}$ mix. The exceptional
$\pi_{3}$ and $\sigma_{3}$ are protected from the mixing because of the
G-parity Lee and Yang (1956) type symmetries, as we discuss shortly. All kaons
are mixed one another. But the iso-doublets do not mix with iso-scalar nor
iso-vector since the condensed $\pi_{1}$ supply integer isospins.

To understand the decoupling of $\pi_{3}$ and $\sigma_{3}$ from the others, we
consider unitary transformations induced by

$\displaystyle P_{I_{3}}(\theta)=P\mathrm{e}^{\mathrm{i}\theta
I_{3}}\,,~{}~{}~{}~{}~{}~{}~{}C_{I_{1}}(\theta)=C\mathrm{e}^{\mathrm{i}\theta
I_{1}}\,,$ (34)

where $P$ and $C$ are parity and charge conjugation, respectively. We define
$C^{-1}\pi_{+}C=\pi_{-}$ and $C^{-1}\pi_{3}C=\pi_{3}$ so that the G-parity is
$-1$ for all pions. In this definition $C^{-1}\pi_{1}C=\pi_{1}$ and
$C^{-1}\pi_{2}C=-\pi_{2}$.

The $P_{I_{3}}$ symmetry holds in our U${}_{I_{3}}$ symmetric Lagrangian for
any $\theta$. The question is whether this symmetry is spontaneously broken or
not. The $\pi_{1}$ condensate under this symmetry transforms as

$\displaystyle P_{I_{3}}^{-1}(\theta)\pi_{1}P_{I_{3}}(\theta)\sim
P^{-1}\big{(}\mathrm{e}^{-\mathrm{i}\theta}\bar{u}\mathrm{i}\gamma_{5}d+\mathrm{e}^{\mathrm{i}\theta}\bar{d}\mathrm{i}\gamma_{5}u\big{)}P=-\big{(}\mathrm{e}^{-\mathrm{i}\theta}\bar{u}\mathrm{i}\gamma_{5}d+\mathrm{e}^{\mathrm{i}\theta}\bar{d}\mathrm{i}\gamma_{5}u\big{)}\,,$
(35)

which in general is the mixture of $\pi_{1}$ and $\pi_{2}$. But setting
$\theta=\pi$ and writing $\mathcal{P}_{I_{3}}\equiv P_{I_{3}}(\pi)$, we
conclude that $\mathcal{P}_{I_{3}}^{-1}\pi_{1}\mathcal{P}_{I_{3}}=\pi_{1}$.
Thus the $\mathcal{P}_{I_{3}}$-parity is conserved in the presence of the
$\pi_{1}$ condensates. We also note that $\mathcal{P}_{I_{3}}$-parity is also
not violated by $\sigma$ and $\sigma_{s}$ condensates. Thus
$\mathcal{P}_{I_{3}}$ can be used to classify various mesons. Under the
$\mathcal{P}_{I_{3}}$ transformation, the isoscalar and isovectors transform
as

$\displaystyle\mathcal{P}_{I_{3}}^{-1}\big{(}\sigma_{0,3,8},\pi_{1,2}\big{)}\mathcal{P}_{I_{3}}=+\big{(}\sigma_{0,3,8},\pi_{1,2}\big{)}\,,~{}~{}~{}~{}~{}\mathcal{P}_{I_{3}}^{-1}\big{(}\sigma_{1,2},\pi_{0,3,8}\big{)}\mathcal{P}_{I_{3}}=-\big{(}\sigma_{1,2},\pi_{0,3,8}\big{)}\,.$
(36)

Next we consider $C_{I_{1}}(\theta)$. First we note the isospin density is
invariant for $\mathcal{C}_{I_{1}}\equiv C_{I_{1}}(\pi)$,

$\displaystyle
C_{I_{1}}^{-1}(\theta)\big{(}\bar{u}u-\bar{d}d\big{)}C_{I_{1}}(\theta)=-C^{-1}\big{(}\bar{u}u-\bar{d}d\big{)}C=\bar{u}u-\bar{d}d\,,$
(37)

where we used $\mathrm{e}^{-\mathrm{i}\pi I_{1}}(u,d)\mathrm{e}^{\mathrm{i}\pi
I_{1}}=(d,u)$. The Lagrangian is also invariant. The condensates $\pi_{1}$,
$\sigma$ and $\sigma_{s}$ are also invariant. Under the $\mathcal{C}_{I_{1}}$
transformation, the isoscalar and isovectors transform as

$\displaystyle\mathcal{C}_{I_{1}}^{-1}\phi_{0,1,2,8}\mathcal{C}_{I_{1}}=+\phi_{0,1,2,8}\,,~{}~{}~{}~{}~{}\mathcal{C}_{I_{1}}^{-1}\phi_{3}\mathcal{C}_{I_{3}}=-\phi_{3}\,.$
(38)

Combining these symmetries, the isoscalar and isovector sectors can be
decomposed into

$\displaystyle\sigma_{3}\,,~{}~{}~{}~{}~{}~{}\pi_{3}\,,~{}~{}~{}~{}~{}~{}(\sigma_{0},\sigma_{8};\pi_{1},\pi_{2})\,,~{}~{}~{}~{}~{}~{}~{}(\pi_{0},\pi_{8};\sigma_{1},\sigma_{2})\,.$
(39)

Meanwhile, for the isodoublet sectors

$\displaystyle\big{(}\sigma_{4}\,,\sigma_{5}\,,\sigma_{6}\,,\sigma_{7}\,;\pi_{4}\,,\pi_{5}\,,\pi_{6}\,,\pi_{7}\,\big{)}\,$
(40)

are all mixed.

####  4.2.1 The spectra of $\sigma_{3}$, $\pi_{3}$, and
$(\sigma_{0},\sigma_{8};\pi_{1},\pi_{2})$.

Here we display some analytic expressions for $\sigma_{3}$, $\pi_{3}$, and
$(\sigma_{0},\sigma_{8};\pi_{1},\pi_{2})$. The $\sigma_{3}$ and $\pi_{3}$
decouple from the other modes and the mass matrices are

$\displaystyle m^{2}_{\pi_{3}}$
$\displaystyle=-m_{M}^{2}+\frac{\,\lambda\,}{\,6g^{2}\,}(M_{l}^{2}+\Delta^{2})-\frac{\,2K\,}{\,g\,}M_{s}=4\mu_{I}^{2}\,,$
$\displaystyle m^{2}_{\sigma_{3}}$
$\displaystyle=-m_{M}^{2}+\frac{\,\lambda\,}{\,2g^{2}\,}(M_{l}^{2}+\Delta^{2})+\frac{\,2K\,}{\,g\,}M_{s}=4\mu_{I}^{2}+\frac{\,\lambda\,}{\,3g^{2}\,}(M_{l}^{2}+\Delta^{2})+\frac{\,4K\,}{\,g\,}M_{s}\,,$
(41)

where for $\pi_{3}$ we have used the gap equation. For these neutral modes the
excitation energies directly coincide with the mass,
$\omega_{\pi_{3},\sigma_{3}}=m_{\pi_{3},\sigma_{3}}$.

Next we look into the charged sector. The pion mass matrices are

$\displaystyle m^{2}_{\pi_{1}}$
$\displaystyle=-m_{M}^{2}+\frac{\,\lambda\,}{\,6g^{2}\,}(M_{l}^{2}+3\Delta^{2})-\frac{\,2K\,}{\,g\,}M_{s}=4\mu_{I}^{2}+\frac{\,\lambda\,}{\,3g^{2}\,}\Delta^{2}\,,$
$\displaystyle m^{2}_{\pi_{2}}$
$\displaystyle=-m_{M}^{2}+\frac{\,\lambda\,}{\,6g^{2}\,}(M_{l}^{2}+\Delta^{2})-\frac{\,2K\,}{\,g\,}M_{s}=4\mu_{I}^{2}\,,$
(42)

and the neutral $\sigma$-sector has the mass matrices

$\displaystyle m^{2}_{\sigma_{0}}$
$\displaystyle=-m_{M}^{2}+\frac{\,\lambda\,}{\,18g^{2}\,}\big{[}6M_{l}^{2}+3M_{s}^{2}+2\Delta^{2}\big{]}-\frac{\,4K\,}{\,3g\,}\big{[}2M_{l}+M_{s}\big{]}\,,$
$\displaystyle m^{2}_{\sigma_{8}}$
$\displaystyle=-m_{M}^{2}+\frac{\,\lambda\,}{\,18g^{2}\,}\big{[}3M_{l}^{2}+6M_{s}^{2}+2\Delta^{2}\big{]}+\frac{\,2K\,}{\,3g\,}\big{[}4M_{l}-M_{s}\big{]}\,.$
(43)

The mixing is induced by

$\displaystyle
m_{\sigma_{0}\pi_{1}}^{2}=\frac{\,\sqrt{2}K\,}{\,\sqrt{3}g\,}\Delta\,,~{}~{}~{}~{}~{}~{}~{}m^{2}_{\sigma_{0}\sigma_{8}}=\frac{\,\lambda\sqrt{2}\,}{\,12g^{2}\,}\Delta^{2}-\frac{\,\lambda\sqrt{2}\,}{\,36g^{2}\,}\delta
M_{s}^{2}+\frac{\,\sqrt{2}K\,}{\,3g\,}\delta M_{s}\,.$ (44)

The coupling between the $\sigma_{0,8}$ and $\pi_{1,2}$ sectors is induced
through the convolution of U${}_{A}(1)$ anomaly and the pion condensate.
Meanwhile, the mixing between $\sigma_{0}$ and $\sigma_{8}$ is due to the
explicit SU(3) breaking which appears through $\delta M_{s}=M_{s}-M_{l}$ and
the meson condensates existing only in the $u,d$ sector.

The isospin chemical potentials in the kinetic terms and mass terms together
yield the massless modes. For a simple illustration, we consider the case
where $K$ is small. In this approximation $\pi_{1,2}$ and $\sigma_{0,8}$
decouple. Now the terms quadratic in $\pi_{1,2}$ are

$\displaystyle\begin{bmatrix}~{}p_{0}^{2}-m_{\pi_{1}}^{2}+4\mu_{I}^{2}~{}&~{}4\mathrm{i}\mu_{I}p_{0}~{}\\\
~{}-4\mathrm{i}\mu_{I}p_{0}~{}&~{}p_{0}^{2}-m_{\pi_{2}}^{2}+4\mu_{I}^{2}~{}\end{bmatrix}=\begin{bmatrix}~{}p_{0}^{2}-\frac{\,\lambda\,}{\,3g^{2}\,}\Delta^{2}~{}&~{}4\mathrm{i}\mu_{I}p_{0}~{}\\\
~{}-4\mathrm{i}\mu_{I}p_{0}~{}&~{}p_{0}^{2}~{}\end{bmatrix}\,.$ (45)

We have used the expression of $m^{2}_{\pi_{1}}$ and $m^{2}_{\pi_{2}}$. The
determinant of the matrix becomes zero at $p_{0}=0$, reflecting that there is
a massless mode associated with the spontaneous breakdown of the U${}_{I_{3}}$
symmetry.

####  4.2.2 The spectra of $(\sigma_{4-7};\pi_{4-7})$.

Next we discuss the isodoublet (kaon) sector. The diagonal part is

$\displaystyle m^{2}_{\pi_{4-7}}$
$\displaystyle=-m_{M}^{2}+\frac{\,\lambda\,}{\,6g^{2}\,}(M_{l}^{2}+M_{s}^{2}-M_{l}M_{s}+\Delta^{2})-\frac{\,2K\,}{\,g\,}M_{l}$
$\displaystyle=4\mu_{I}^{2}+\frac{\,\lambda\,}{\,6g^{2}\,}M_{l}(M_{s}-M_{l})+\frac{\,2K\,}{\,g\,}(M_{s}-M_{l})\,,$
$\displaystyle m^{2}_{\sigma_{4-7}}$
$\displaystyle=-m_{M}^{2}+\frac{\,\lambda\,}{\,6g^{2}\,}(M_{l}^{2}+M_{s}^{2}+M_{l}M_{s}+\Delta^{2})+\frac{\,2K\,}{\,g\,}M_{l}$
$\displaystyle=4\mu_{I}^{2}+\frac{\,\lambda\,}{\,6g^{2}\,}M_{s}(M_{s}+M_{l})+\frac{\,2K\,}{\,g\,}(M_{s}+M_{l})\,,$
(46)

where we have used the gap equation for the pion sector. The difference in
mass matrices between the pion and kaon sectors come from the effective quark
mass. The mixing matrices are

$\displaystyle
m^{2}_{\sigma_{4},\sigma_{7}}=-m^{2}_{\sigma_{5},\sigma_{6}}=-m^{2}_{\pi_{4},\pi_{7}}=m^{2}_{\pi_{5},\pi_{6}}=\frac{\,\lambda\,}{\,24g^{2}\,}\Delta\delta
M_{s}\,,$ $\displaystyle
m_{\sigma_{4}\pi_{6}}^{2}=m_{\sigma_{5}\pi_{7}}^{2}=m_{\pi_{4}\sigma_{6}}^{2}=m_{\pi_{5}\sigma_{7}}^{2}=\frac{\,\lambda\,}{\,24g^{2}\,}\Delta\big{(}M_{l}+M_{s}\big{)}-\frac{\,K\,}{\,g\,}\Delta\,.$
(47)

These form a $8\times 8$ matrix and need numerical analyses for the
determination of the spectra.

To get analytic insights we consider the case where $M_{l},M_{s}\ll\mu_{I}$
for which the mixing is neglected compared to the diagonal part. Combining the
mass matrices with the kinetic term, (we write $m_{K}^{2}=m_{4-7}^{2}$)

$\displaystyle\begin{bmatrix}~{}p_{0}^{2}-m_{K}^{2}+\mu_{I}^{2}~{}&~{}2\mathrm{i}\mu_{I}p_{0}~{}\\\
~{}-2\mathrm{i}\mu_{I}p_{0}~{}&~{}p_{0}^{2}-m_{K}^{2}+\mu_{I}^{2}~{}\end{bmatrix}\sim\begin{bmatrix}~{}p_{0}^{2}-3\mu_{I}^{2}~{}&~{}2\mathrm{i}\mu_{I}p_{0}~{}\\\
~{}-2\mathrm{i}\mu_{I}p_{0}~{}&~{}p_{0}^{2}-3\mu_{I}^{2}~{}\end{bmatrix}\,.$
(48)

For kaons being the isodoublet, the energy reduction associated with the
chemical potential is weaker than in the pion, and hence $\mu_{I}$ terms do
not cancel. The excitation energies manifestly depend on $\mu_{I}$,

$\displaystyle\omega_{K}~{}\sim~{}\mu_{I}\,,~{}~{}3\mu_{I}\,.$ (49)

The same also holds for the scalar meson $\kappa$.

##  5 Quark descriptions

Now we include loops made of mean-field quark propagators. We keep only one
loop with the leading $N_{\rm c}$ contributions; quark loops for the vertex
corrections are neglected. In counting of loops the quark contributions appear
as radiative “corrections” to the tree level, but in fact the quark
contributions should be regarded as leading order contributions at finite
density Chiba and Kojo (2024); indeed as $\mu_{I}$ increases the quark
contributions dominate over hadronic contributions and change the structure of
gap equations and EOS.

###  5.1 The structure of the effective potential

Beyond tree calculations, the parameters in the effective potential must be
also reinterpreted as bare parameters which are split into the renormalized
parameters and counter terms. First we attach an index $B$ to fields $\phi$,
$q$, $m_{M}$, and so on, and factor out the renormalized parameters and fields
Chiba and Kojo (2024),

$\displaystyle\phi_{B}$ $\displaystyle=$ $\displaystyle
Z_{\phi}^{1/2}\phi\,,~{}~{}~{}~{}~{}~{}q_{B}=Z_{q}^{1/2}q\,,$ $\displaystyle
g_{B}$ $\displaystyle=$
$\displaystyle\tilde{Z}_{g}Z_{q}^{-1}Z_{\phi}^{-1/2}g=Z_{g}g\,,$
$\displaystyle(m_{M}^{2})_{B}$ $\displaystyle=$
$\displaystyle\tilde{Z}_{m^{2}}Z_{\phi}^{-1}m_{M}^{2}=Z_{m^{2}}m_{M}^{2}\,,$
$\displaystyle\lambda_{B}$ $\displaystyle=$
$\displaystyle\tilde{Z}_{\lambda}Z_{\phi}^{-2}\lambda=Z_{\lambda}\lambda\,,$
$\displaystyle(h_{l})_{B}$ $\displaystyle=$
$\displaystyle\tilde{Z}_{h_{l}}Z_{\phi}^{-1/2}h=Z_{h_{l}}h_{l}\,,$
$\displaystyle(h_{s})_{B}$ $\displaystyle=$
$\displaystyle\tilde{Z}_{h_{s}}Z_{\phi}^{-1/2}h=Z_{h_{s}}h_{s}\,.$ (50)

We also use $\delta Z_{i}=Z_{i}-1$ with $i=\phi,q,g$, and so on. The
$\tilde{Z}_{i}$ represents the radiative corrections without those for the
external lines. The loop corrections to the quark self-energies and quark-
meson vertices appear only through meson-loops and hence

$\displaystyle Z_{\psi}=1+O(1/N_{\rm c})\,,~{}~{}~{}\tilde{Z}_{g}=1+O(1/N_{\rm
c})\,.$ (51)

Meanwhile, the meson self-energies and tadpole contain quark loops of
$O(N_{\rm c})$ which are combined with $g^{2}\sim 1/N_{\rm c}$ vertices to
yield

$\displaystyle Z_{\phi}=1+O(g^{2}N_{\rm
c})\,,~{}~{}~{}\tilde{Z}_{m^{2}}=1+O(g^{2}N_{\rm c})\,,$
$\displaystyle\tilde{Z}_{h}=1+O(g^{2}N_{\rm
c})\,,~{}~{}~{}\tilde{Z}_{\lambda}=1+O(g^{4}N_{\rm c}/\lambda)\,.$ (52)

and hence one must keep these corrections. It is useful to note that the
relation

$\displaystyle g_{B}\phi_{B}=g\phi\,,~{}~{}~{}~{}~{}Z_{g}=Z_{\phi}^{-1/2}\,,$
(53)

in the large $N_{\rm c}$ limit. With the first relation the dynamically
generated quark mass and gap are RG invariant. The second relation tells that
the running of $g^{2}$ can be studied by examining the meson propagators.

The effective potential up to one-loop consists of three type of terms

$\displaystyle\Omega=\Omega_{0}+\Omega_{\rm c.t.}+\Omega_{q}\,.$ (54)

The $\Omega_{0}$ takes the same form as in the tree level but parameters are
to be interpreted as renormalized ones. The $\Omega_{\rm c.t.}$ includes the
counter terms originating from the splitting, e.g.,
$\lambda_{B}=\lambda+\delta\lambda$,

$\displaystyle\Omega_{c.t.}$ $\displaystyle=-\delta
Z_{\phi}\frac{\,2\mu_{I}^{2}\,}{g^{2}}\Delta^{2}-\frac{\,\delta
m_{M}^{2}\,}{\,2g^{2}\,}(M_{l}^{2}+\Delta^{2})+\frac{\,\delta\lambda\,}{\,24g^{4}\,}\big{(}M_{l}^{2}+\Delta^{2}\big{)}^{2}-\frac{\,\delta
h_{l}\,}{g}M_{l}$ (55) $\displaystyle-\frac{\,\delta
m_{M}^{2}\,}{\,4g^{2}\,}M_{s}^{2}+\frac{\,\delta\lambda\,}{\,48g^{4}\,}M_{s}^{4}-\frac{\,\delta
h_{s}\,}{2g}M_{s}-\frac{\,\delta
K\,}{\,g^{3}\,}\big{(}M_{l}^{2}+\Delta^{2}\big{)}M_{s}\,.$

The quark contribution is

$\displaystyle\Omega_{q}=-2N_{\rm
c}\int_{\vec{p}}\big{(}\xi_{+}+\xi_{-}+E_{s}\big{)}\,,$ (56)

where $E_{l}=\sqrt{\vec{p}^{2}+M_{l}^{2}}$ and
$E_{s}=\sqrt{\vec{p}^{2}+M_{s}^{2}}$, and

$\displaystyle\xi_{-}=\xi_{u}=\xi_{\bar{d}}=\sqrt{\big{(}E_{l}-\mu_{I}\big{)}^{2}+\Delta^{2}}\,,~{}~{}~{}~{}~{}~{}~{}\xi_{+}=\xi_{d}=\xi_{\bar{u}}=\sqrt{\big{(}E_{l}+\mu_{I}\big{)}^{2}+\Delta^{2}}\,.$
(57)

In the $u,d$-quark sector quarks have the energy dispersion of the BCS type.

For small $\mu_{I}\leq m_{\pi}/2\leq M_{l}$, the pion condensate is absent.
Then the chemical potential disappears from the $\Omega_{q}$,

$\displaystyle\sqrt{\big{(}E_{l}+\mu_{I}\big{)}^{2}+\Delta^{2}}+\sqrt{\big{(}E_{l}-\mu_{I}\big{)}^{2}+\Delta^{2}}~{}\rightarrow~{}(E_{l}+\mu_{I})+(E_{l}-\mu_{I})=2E_{l}\,,$
(58)

reflecting the Sliver Blaze property that the isospin density $n_{I}$ begins
to change only when either $\mu_{I}\geq M_{l}$ or $\Delta\neq 0$ are
satisfied. In the present case $\Delta\neq 0$ is the driving force.

The $\Omega_{q}$ is UV divergent. This divergence is cancelled by the counter
terms from the masses, couplings, and field normalization factor from the
mesonic Lagrangian. We isolate the divergences as Andersen et al. (2019)

$\displaystyle\Omega_{q}=\Omega_{l}^{\rm R}-2N_{\rm
c}\int_{\vec{p}}\bigg{[}2\sqrt{E_{q}^{2}+\Delta^{2}}+\frac{\,\mu_{I}^{2}\Delta^{2}\,}{\,(E_{q}^{2}+\Delta^{2}\,)^{3/2}}+E_{s}\bigg{]}$
(59)

where the twice subtracted energy density, including only $u,d$-quark
contributions, is

$\displaystyle\Omega_{l}^{\rm R}=-2N_{\rm
c}\int_{p}\bigg{[}\xi_{+}+\xi_{-}-2\sqrt{E_{q}^{2}+\Delta^{2}}-\frac{\,\mu_{I}^{2}\Delta^{2}\,}{\,(E_{q}^{2}+\Delta^{2}\,)^{3/2}}\bigg{]}\,.$
(60)

The $\Omega_{l}^{R}$ is UV finite and vanishing when $\mu_{I}=0$ and
$\Delta=0$. At large density, the subtracted energy scales as
$\Omega_{l}^{R}\simeq\mu_{I}^{4}$, dominating the thermodynamic potential.

To understand the physical meaning of the subtracted potential
$\Omega_{l}^{R}$ at large $\mu_{I}$, it is instructive to consider
$\Delta\rightarrow 0$ limit and decompose the integral into the
$E_{l}\leq\mu_{I}$ and $E_{l}\geq\mu_{I}$ domains. For $E_{l}\leq\mu_{I}$,

$\displaystyle\int_{\vec{p}}\Theta(\mu_{I}-E_{l})\big{(}\,|E_{l}+\mu_{I}|+|E_{l}-\mu_{I}|-2E_{l}\,\big{)}=2\int_{\vec{p}}\Theta(\mu_{I}-E_{l})\,\big{(}\mu_{I}-E_{l}\big{)}\,,$
(61)

and for $E_{I}\geq\mu_{l}$,

$\displaystyle\int_{\vec{p}}\Theta(E_{l}-\mu_{I})\big{(}\,|E_{l}+\mu_{I}|+|E_{l}-\mu_{I}|-2E_{l}\,\big{)}=0\,.$
(62)

Thus, $\Omega_{l}^{R}$ at $\Delta=0$ is nothing but the standard quasi-
particle contributions to the thermodynamic potential.

The divergent piece in $\Omega_{q}$ is evaluated in the $\overline{MS}$
scheme,

$\displaystyle\Omega_{q}^{\rm UV}$ $\displaystyle=\frac{\,2N_{\rm
c}\,}{\,(4\pi)^{2}\,}(M_{l}^{2}+\Delta^{2})^{2}\bigg{[}\frac{1}{\,\epsilon\,}+\frac{\,3\,}{2}+\ln\frac{\,\Lambda^{2}\,}{\,M_{l}^{2}+\Delta^{2}\,}\bigg{]}-\frac{\,8N_{\rm
c}\,}{\,(4\pi)^{2}\,}\mu_{I}^{2}\Delta^{2}\bigg{[}\frac{1}{\,\epsilon\,}+\ln\frac{\,\Lambda^{2}\,}{\,M_{l}^{2}+\Delta^{2}\,}\bigg{]}$
$\displaystyle~{}~{}~{}+\frac{\,N_{\rm
c}\,}{\,(4\pi)^{2}\,}M_{s}^{4}\bigg{[}\frac{1}{\,\epsilon\,}+\frac{\,3\,}{2}+\ln\frac{\,\Lambda^{2}\,}{\,M_{s}^{2}\,}\bigg{]}\,,$
(63)

where $\Lambda$ is the renormalization scale. In the $\overline{MS}$ scheme,
the counter terms are used to absorb only the $1/\epsilon$ terms. Then the
counter terms are fixed as

$\displaystyle\delta
m_{M}^{2}=0\,,~{}~{}~{}~{}~{}~{}\delta{h}_{l,s}=0\,,~{}~{}~{}~{}~{}~{}\delta\lambda=-\frac{\,48N_{\rm
c}g^{4}\,}{\,(4\pi)^{2}\epsilon\,}\,,~{}~{}~{}~{}~{}~{}\delta
Z_{\phi}=-\frac{\,4N_{\rm
c}g^{2}\,}{\,(4\pi)^{2}\epsilon\,}\,,~{}~{}~{}~{}~{}~{}\delta K=0\,.$ (64)

In our previous studies we further discussed the on-mass-shell renormalization
to tune the pole positions of $\pi$ and $\sigma$. We do this when we fix
parameters of our model.

Putting everything together, the effective potential
$\Omega=\Omega_{0}+\Omega_{\rm c.t.}+\Omega_{q}$ are reorganized as

$\displaystyle\Omega=\Omega_{l}+\Omega_{s}+\Omega_{\rm mix}$ (65)

where the contributions from $u,d$-quarks are

$\displaystyle\Omega_{l}$
$\displaystyle=\Omega_{l}^{R}-\frac{\,m_{M}^{2}\,}{\,2g^{2}\,}(M_{l}^{2}+\Delta^{2})+\frac{\,\lambda\,}{\,24g^{4}\,}\big{(}M_{l}^{2}+\Delta^{2}\big{)}^{2}-\frac{\,h_{l}\,}{g}M_{l}$
$\displaystyle+\frac{\,2N_{\rm
c}\,}{\,(4\pi)^{2}\,}(M_{l}^{2}+\Delta^{2})^{2}\bigg{[}\frac{\,3\,}{2}-\ln\frac{\,M_{l}^{2}+\Delta^{2}\,}{\,M_{0}^{2}\,}\bigg{]}+2\mu_{I}^{2}\Delta^{2}\bigg{[}-\frac{\,1\,}{g^{2}}+\frac{\,4N_{\rm
c}\,}{\,(4\pi)^{2}\,}\ln\frac{\,M_{l}^{2}+\Delta^{2}\,}{\,M_{0}^{2}\,}\bigg{]}$
(66)

and from $s$-quarks

$\displaystyle\Omega_{s}=\frac{1}{\,2\,}\bigg{(}-\frac{\,m_{M}^{2}\,}{\,2g^{2}\,}M_{s}^{2}+\frac{\,\lambda\,}{\,24g^{4}\,}M_{s}^{4}-\frac{\,h_{s}\,}{g}M_{s}+\frac{\,2N_{\rm
c}\,}{\,(4\pi)^{2}\,}M_{s}^{4}\bigg{[}\frac{\,3\,}{2}-\ln\frac{\,M_{s}^{2}\,}{\,M_{0}^{2}\,}\bigg{]}\bigg{)}$
(67)

and the $(u,d)$-$s$ quark mixing induced by the KMT interaction,

$\displaystyle\Omega_{\rm
mix}=-\frac{\,K\,}{\,g^{3}\,}\big{(}M_{l}^{2}+\Delta^{2}\big{)}M_{s}\,.$ (68)

We note that, the strange quark sector shows the substantial $\mu_{I}$
dependence because of the UA(1) induced coupling, $\sim\Delta^{2}M_{s}$; the
$M_{s}$ increases as $\Delta$ does. The similar effects have been studied in
the context of color-superconductivity Hatsuda et al. (2006); Yamamoto et al.
(2007); Hatsuda et al. (2008); Zhang et al. (2009).

The expression of $\Omega$ is the renormalization group (RG) invariant for a
given field values, provided that the renormalized masses, couplings, etc.
follow the RG evolution. We have set the renormalization scale $\Lambda$ to be
the vacuum light quark mass, $M_{0}\equiv M_{l}^{\rm vac}$, and then the
couplings etc should be interpreted as those evaluated at $\Lambda=M_{0}$.

Finally we mention that the expression is invalid for very large
$g^{2}\ln\big{(}M^{2}+\Delta^{2}\big{)}$ Coleman and Weinberg (1973); as is
well-known, the one-loop effective potential is unbound for large field
limits; there we need higher orders in loops.

###  5.2 The structure of the gap equations

The quark loops have added the logarithmic terms which change the structure of
the effective potential, especially at large $\mu_{I}$. Let us see this
assuming $M_{l},M_{s}\ll\mu_{I}$. At tree level the effective potential
behaves as

$\displaystyle\Omega_{\rm
tree}~{}\rightarrow~{}-\frac{\,2\mu_{I}^{2}\,}{\,g^{2}\,}\Delta^{2}+\frac{\,\lambda\,}{\,24g^{4}\,}\Delta^{4}\,.$
(69)

The gap equation for $\Delta$ balances the first and second terms, so
$\Delta_{\rm tree}\sim\mu_{I}$ inevitably follows. Substituting back the
solution $\Delta_{\rm tree}^{2}\sim 24\mu_{I}^{2}g^{2}/\lambda$ into the
$\Omega_{\rm tree}$, we find

$\displaystyle\Omega_{\rm
tree}~{}\sim~{}-\frac{\,24\,}{\,\lambda\,}\mu_{I}^{4}\,.$ (70)

The hadronic parameter $\lambda$ entirely fixes the coefficient of
$\mu_{I}^{4}$ term.

This trend changes after including quark contributions. We assume that the gap
equation leads to $M_{l,s},\Delta\ll\mu_{I}$, and then check the consistency
of this assumption. The large $\mu_{I}$ behavior of $\Omega$ is

$\displaystyle\Omega~{}\rightarrow~{}\Omega_{l}^{R}+2\mu_{I}^{2}\Delta^{2}\bigg{[}-\frac{\,1\,}{g^{2}}+\frac{\,4N_{\rm
c}\,}{\,(4\pi)^{2}\,}\ln\frac{\,\Delta^{2}\,}{\,M_{0}^{2}\,}\bigg{]}\,,$ (71)

where the tree level terms, except $\mu_{I}^{2}/g^{2}$ term, are suppressed
compared to $\mu_{I}^{2}$ and $\mu_{I}^{4}$ terms. We note that the
logarithmic contribution changes the sign at $M_{0}$; it becomes attractive
contributions for $\Delta\lesssim M_{0}$ but repulsive for $\Delta\gtrsim
M_{0}$. At large density the latter stops the growth of $\Delta$ induced by
the tree contribution before $\lambda\Delta^{2}$ terms are activated.

It turns out that $\Omega_{l}^{R}$ is well saturated by $\mu_{I}^{4}$ terms
and weakly depend on $\Delta$. Then the gap equation is dominated by the
coefficients of $\mu_{I}^{2}$ term as

$\displaystyle\frac{\,\partial\Omega\,}{\,\partial\Delta^{2}\,}~{}\rightarrow~{}2\mu_{I}^{2}\bigg{[}-\frac{1}{\,g^{2}\,}+\frac{\,4N_{\rm
c}\,}{\,(4\pi)^{2}\,}+\frac{\,4N_{\rm
c}\,}{\,(4\pi)^{2}\,}\ln\frac{\,\Delta^{2}\,}{\,M_{0}^{2}\,}\bigg{]}+O(M_{0}^{2})~{}\simeq~{}0\,.$
(72)

This form with the logarithm is typical for low dimensional systems; in the
present case the Fermi surface plays the role for the dimensional reduction.
Unlike the tree level solution, at $M_{0}/\mu_{I}\rightarrow 0$ the gap does
not manifestly contain the $\mu_{I}$-dependence,

$\displaystyle\Delta^{2}\simeq
M_{0}^{2}\,\mathrm{e}^{-1+\frac{\,4\pi^{2}\,}{\,N_{\rm c}g^{2}\,}}\,.$ (73)

The $g$ dependence is a bit puzzling; the usual BCS type calculations have the
form of $\mathrm{e}^{-\\#/G}$ with a coupling constant $G$, meaning that a
stronger $G$ increases the size of the gap. In contrast, a large $g$ reduces
the gap in our model. Our interpretation is the following: the condensed
mesons at tree level, where quarks inside are neglected, overreact to the
increase of $\mu_{I}$ and get overpopulated, as one can see from the scaling
$\Delta_{\rm tree}\sim\mu_{I}$. In particular the quark Pauli blocking is
neglected. In light of this viewpoint, the explicit inclusion of quarks with
the coupling to mesons should cancel the unwanted contributions and should
temper the growth of $\Delta$. This effect is stronger for a larger $g$. For a
very small $g$, apparently $\Delta$ blows up, but then $\lambda\Delta^{4}$
terms become dominant and the situation goes back to the tree level model or
purely mesonic models. The philosophy here is in line with Ref. Duarte et al.
(2021) that introduced ghosts to cancel double counted contributions.

Substituting back the $\Delta$ into $\Omega$, the thermodynamic potential at
very large $\mu_{I}$ scales as

$\displaystyle\Omega~{}\sim~{}\Omega_{l}^{R}-\frac{\,N_{\rm
c}\,}{\,2\pi^{2}\,}\mu_{I}^{2}\Delta^{2}\,,$ (74)

Note that, in $\Omega$ at large $\mu_{I}$, the mesonic parameters $m_{M}^{2}$
and $\lambda$ no longer play important roles, while the Yukawa coupling $g$ is
now hidden into the gap parameter $\Delta$.

Figure 4:  The effective quark masses $M_{l},M_{s}$ and the BCS gap $\Delta$
as functions of $\mu_{I}$ in the quark-meson model with the set B in Table. 1.
The growth of the strange quark mass comes from the UA(1) anomaly through
which strange quarks couple to the pion condensate.

##  6  Meson poles at one-loop: parameter fixing

Now we have all expressions needed to discuss EOS but still the parameters in
our model remain to be fixed. We use meson nonets in vacuum to fix our
parameters. Here the meson self-energies induced by quark loops must be taken
into account for the consistency with the gap equations, Meanwhile in this
work we do not address in-medium self-energies to meson masses.

###  6.1 Meson poles

As we have constructed the effective potential at one-loop, we need to upgrade
the tree level expressions of meson masses. This is essential to reproduce the
masses of the Goldstone bosons which arise at the minima of the effective
potential at a given order of approximation. The analyses go as in the Nambu-
Jona-Lasinio model Hatsuda and Kunihiro (1994).

An inverse of meson propagator with loops takes the form

$\displaystyle-p^{2}+m^{2}=-p^{2}+m^{2}_{\rm
tree}+\Sigma^{q}(m^{2})+\Sigma^{c.t.}(m^{2})\,,$ (75)

where $\Sigma^{q}$ comes from quark loops while $\Sigma^{c.t.}$ from the
counter terms. In case of pions, we have

$\displaystyle-p^{2}+m_{\pi}^{2}=\bigg{(}-m_{M}^{2}+\frac{\,\lambda\,}{\,2g^{2}\,}M_{0}^{2}-2\frac{\,K\,}{\,g\,}M_{s}\bigg{)}+\Sigma_{\pi}^{q}(m_{\pi}^{2})-p^{2}\delta
Z_{\phi}+\frac{\,\delta\lambda\,}{\,2g^{2}\,}M_{0}^{2}\,.$ (76)

We obtained the counter terms from the tree level bare parameters.

It is convenient to define

$\displaystyle\Sigma_{fg}^{q}(p^{2})=\mathrm{i}g^{2}\int_{l}{\rm tr}_{\rm
c,D}\big{[}S_{f}(l)\mathrm{i}\gamma_{5}S_{g}(p+l)\mathrm{i}\gamma_{5}\big{]}$
$\displaystyle=\frac{\,2g^{2}N_{\rm
c}\,}{\,(4\pi)^{2}\epsilon\,}\bigg{[}M_{f}^{2}+M_{g}^{2}+\big{[}\,(M_{f}-M_{g})^{2}-p^{2}\,\big{]}\bigg{]}$
$\displaystyle~{}~{}~{}+\frac{\,2g^{2}N_{\rm
c}\,}{\,(4\pi)^{2}\,}\bigg{[}M_{f}^{2}+M_{g}^{2}+M_{f}^{2}\ln\frac{\,\Lambda^{2}\,}{\,M_{f}^{2}\,}+M_{g}^{2}\ln\frac{\,\Lambda^{2}\,}{\,M_{g}^{2}\,}+\big{[}\,(M_{f}-M_{g})^{2}-p^{2}\,\big{]}\mathcal{B}_{fg}(p^{2})\bigg{]}\,,$
(77)

with a function including the Feynman parameter,

$\displaystyle\mathcal{B}_{fg}(p^{2})\equiv\int_{0}^{1}\\!\mathrm{d}x~{}\ln\frac{\,\Lambda^{2}\,}{\,(1-x)M_{f}^{2}+xM_{g}^{2}-p^{2}x(1-x)\,}\,.$
(78)

This logarithm in the integral can be negative for $p^{2}>(M_{f}+M_{g})^{2}$
and $p^{2}<(M_{f}-M_{g})^{2}$. (Actually the latter regime does not occur for
$0\leq x\leq 1$.) In such case the imaginary part appears as
$\ln(-y-\mathrm{i}\epsilon)\rightarrow\ln|y|-\mathrm{i}\pi$. The imaginary
part describes the decay of mesons into constituent quarks, but this is
supposed to be an artifact of not taking account the quark confinement.
Realistically a meson decays into mesons and the width is typically smaller
than the mass, except for some of scalar mesons. For simplicity, in this work
we estimate a meson mass as the location where the real part becomes zero.
Also we discuss the mixing between the flavor singlet and octet using the real
part only.

Remember that we chose $\Lambda=M_{0}$ for the effective potential so we do
the same in the following. For the pseudoscalar channels,

$\displaystyle\Sigma_{\pi}^{q}=2\Sigma^{q}_{ll}\,,~{}~{}~{}~{}~{}~{}\Sigma_{K}^{q}=2\Sigma^{q}_{ls}\,,~{}~{}~{}~{}~{}~{}$
$\displaystyle\Sigma_{00}^{q}=\frac{2}{\,N_{\rm
f}\,}\big{(}2\Sigma_{ll}^{q}+\Sigma_{ss}^{q}\big{)}\,,~{}~{}~{}~{}~{}~{}\Sigma_{88}^{q}=\frac{2}{\,N_{\rm
f}\,}\big{(}\Sigma_{ll}^{q}+2\Sigma_{ss}^{q}\big{)}\,,~{}~{}~{}~{}~{}~{}\Sigma_{08}^{q}=\frac{\,2\sqrt{2}\,}{\,N_{\rm
f}\,}\big{(}\Sigma_{ll}^{q}-\Sigma_{ss}^{q}\big{)}\,,$ (79)

Moreover, to obtain the similar expressions for the scalar channel, we can
simply replace $(M_{f}-M_{g})^{2}$ with $(M_{f}+M_{g})^{2}$.

Let us see some examples. For pions

$\displaystyle\Sigma_{\pi}^{q}(p^{2})$ $\displaystyle=\frac{\,4g^{2}N_{\rm
c}\,}{\,(4\pi)^{2}\epsilon\,}\big{(}2M_{0}^{2}-p^{2}\,\big{)}+\frac{\,4g^{2}N_{\rm
c}\,}{\,(4\pi)^{2}\,}\bigg{(}2M_{0}^{2}-p^{2}\,\mathcal{B}_{ll}(m_{\pi}^{2})\bigg{)}\,.$
(80)

Combining this expression with the counter term $\Sigma_{\pi}^{c.t.}$, the UV
divergence proportional to $M_{0}^{2}$ is cancelled by the previously
determined $\delta\lambda$. The divergence coupled to $p^{2}$ can be canceled
by choosing

$\displaystyle\delta Z_{\phi}=-\frac{\,4g^{2}N_{\rm
c}\,}{\,(4\pi)^{2}\epsilon\,}\,.$ (81)

Below we write $\Sigma=\Sigma^{q}+\Sigma^{c.t.}$ for the renormalized self-
energy. The pole condition for pions is $m_{\pi}^{2}=\big{(}m_{\pi}^{\rm
tree}\big{)}^{2}+\Sigma_{\pi}(m_{\pi}^{2})$ with

$\displaystyle\Sigma_{\pi}(m_{\pi}^{2})=\frac{\,4g^{2}N_{\rm
c}\,}{\,(4\pi)^{2}\,}\bigg{(}2M_{0}^{2}-m_{\pi}^{2}\,\mathcal{B}_{ll}(m_{\pi}^{2})\bigg{)}\,.$
(82)

For kaons, the above chosen counter terms cancel the divergence in the kaon
self-energy, leaving the condition $m_{K}^{2}=\big{(}m_{K}^{\rm
tree}\big{)}^{2}+\Sigma_{K}(m_{K}^{2})$ with

$\displaystyle\Sigma_{K}(m_{K}^{2})=\frac{\,4g^{2}N_{\rm
c}\,}{\,(4\pi)^{2}\,}\bigg{(}M_{0}^{2}+M_{s0}^{2}+M_{s0}^{2}\ln\frac{\,M_{0}^{2}\,}{\,M_{s0}^{2}\,}+\big{[}(M_{0}-M_{s0})^{2}-m_{K}^{2}\big{]}\mathcal{B}_{ls}(m_{K}^{2})\bigg{)}\,.$
(83)

We do the same for $00$, $88$, and $08$ channels which are to be diagonalized
to yield the $\eta$ and $\eta^{\prime}$ spectra. The $\eta$ and
$\eta^{\prime}$ masses are found by searching $p^{2}$ satisfying

$\displaystyle\det\begin{bmatrix}~{}-p^{2}+(m_{00}^{\rm
tree})^{2}+\Sigma_{00}(p^{2})~{}&~{}(m_{08}^{\rm
tree})^{2}+\Sigma_{08}(p^{2})~{}~{}\\\ ~{}(m_{80}^{\rm
tree})^{2}+\Sigma_{80}(p^{2})~{}&~{}-p^{2}+(m_{88}^{\rm
tree})^{2}+\Sigma_{88}(p^{2})~{}~{}\end{bmatrix}=0\,.$ (84)

The mixing is induced by the flavor breaking associated with $M_{l}-M_{s}$. It
is worth mentioning that the mixing angle depends on the energy, not a
constant Hatsuda and Kunihiro (1994).

###  6.2 Parameter fixing

We have $(g,m_{M}^{2},\lambda,K,h_{l},h_{s},M_{0},M_{s0})$ for our parameters.
We treat $M_{0}$ and $M_{s0}$ as if our input parameters, and set

$\displaystyle M_{0}=0.3\,{\rm GeV}\,,~{}~{}~{}~{}M_{s0}=0.5\,{\rm GeV}\,.$
(85)

Next, we fix $h_{l}$ and $h_{s}$ so that the minima of the effective potential
in vacuum are found at $M_{l}=M_{0}$ and $M_{s}=M_{s0}$,

$\displaystyle\frac{\,\partial\Omega\,}{\,\partial
M_{l}\,}\bigg{|}_{M_{l}=M_{0},\,M_{s}=M_{s0}}=\frac{\,\partial\Omega\,}{\,\partial
M_{s}\,}\bigg{|}_{M_{l}=M_{0},\,M_{s}=M_{s0}}=0\,.$ (86)

The leftover parameters are $(g,m_{M}^{2},\lambda,K)$. We have four meson
spectra,
$(m_{\pi},m_{K},m_{\eta},m_{\eta^{\prime}})\simeq(0.14,0.50,0.55,0.96)$ GeV,
so the four parameters can be fixed. Here we use the pseudo-scalar mesons for
the parameter fixing instead of using $\sigma$ mesons since the nature of
$\sigma$-mesons are more uncertain than the pseudo-scalar mesons. The results
are summarized in Table. 1 and 2 including spectra of scalar mesons. The
$\sigma$ and $\eta^{\prime}$ masses are sensitive to the value of $K$;
$m_{\sigma}$ is reduced while $m_{\eta^{\prime}}$ is enhanced by the anomaly.

Table 1: The parameters in the quark-meson model ($M_{0},M_{s0}$ are given in GeV unit). The parameters are chosen to reproduce the masses of the pseudo scalar nonet. We use the set B unless otherwise stated.  set | $M_{0}$ | $M_{s0}$ | $g$ | $\lambda$ | $m_{M}^{2}\,[{\rm GeV}^{2}]$ |  $h_{l}\,[{\rm GeV}]$ | $h_{s}\,[{\rm GeV}]$ | $K\,[{\rm GeV}]$
---|---|---|---|---|---|---|---|---
A | 0.27 | 0.50 | 3.0 | 38.1 | $-0.269$ | $1.82\cdot 10^{-3}$ | $4.16\cdot 10^{-2}$ | 1.2
B | 0.30 | 0.50 | 3.3 | 42.4 | $-0.298$ | $1.84\cdot 10^{-3}$ | $4.19\cdot 10^{-2}$ | 1.6
C | 0.33 | 0.50 | 3.6 | 60.0 | $-0.278$ | $1.85\cdot 10^{-3}$ | $4.32\cdot 10^{-2}$ | 2.0
Table 2: The masses of the pseudo scalar and scalar nonets with the masses indicated in the parenthesis. (Experimentally the $\sigma$ and $\kappa$ have the width of $\sim 500$ MeV.)  set | $m_{\pi}(138)$ | $m_{K}(496)$ | $m_{\eta}(548)$ | $m_{\eta^{\prime}}(958)$ | $m_{\sigma}(500^{*})$ | $m_{a_{0}}(980)$ | $m_{\kappa}(700^{*})$
---|---|---|---|---|---|---|---
A | 0.14 | 0.50 | 0.52 | 0.97 | 0.51 | 0.67 | 0.75
B | 0.14 | 0.50 | 0.54 | 0.97 | 0.54 | 0.73 | 0.80
C | 0.14 | 0.50 | 0.55 | 0.95 | 0.64 | 0.81 | 0.88

##  7 Equations of state at zero temperature

Figure 5:  The isospin density $n_{I}/n_{0}(=\tilde{n}_{I}/2n_{0})$ as a
function of $\mu_{I}$. The results of quark-meson model for $m_{\pi}=0.14$ GeV
and $m_{\pi}=0.17$ GeV are compared with the lattice data by Brandt et al.
Brandt et al. (2023) and Abbott et al. Abbott et al. (2023), respectively. The
overlap density $n_{\pi}^{\rm overlap}\simeq 5.2n_{0}$ which is achieved at
$\mu_{I}\simeq 0.256$ GeV is shown as a guide. The red and blue dotted curves
are from the leader order ChPT for $m_{\pi}=0.14$ and $0.17$ GeV,
respectively.  Figure 6:  The sound speed as a function of $\mu_{I}$. The
results of quark-meson model for $m_{\pi}=0.14$ GeV and $m_{\pi}=0.17$ GeV are
compared with the lattice data by Brandt et al. Brandt et al. (2023) and
Abbott et al. Abbott et al. (2023), respectively. The conformal value
$c_{s}^{2}=1/3$ and the ChPT results (dotted curves) are also shown as guides.
Figure 7:  The trace anomaly $\Delta_{\rm tr}$ for the same setup as Fig. 6.
The conformal limit is $\Delta_{\rm tr}=0$.

We examine equations of state. Substituting the solutions of gap equations
into $\Omega$, the thermodynamic potential is $P=-\Omega$. The isospin density
is

$\displaystyle
n_{I}\equiv\frac{\,\tilde{n}_{I}\,}{\,2\,}=\frac{\,1\,}{\,2\,}\frac{\,\partial
P\,}{\,\partial\mu_{I}\,}\,.$ (87)

As we mentioned before, we defined $\mu_{I}(=\mu_{u}=-\mu_{d})$ in
unconventional way to simplify the notation. For physical interpretation we
should use $n_{I}$, rather than $\tilde{n}_{I}$ directly derivable from the
derivative of $P$. The energy density is $\varepsilon=\mu_{I}\tilde{n}_{I}-P$.

At low density the condensate is driven by pions with $I_{3}=1$; as far as
isospin density is saturated by those pions, one can interpret the isospin
density as the pion density, $n_{I}\simeq n_{\pi}$. A good measure to estimate
the density where pions overlap is

$\displaystyle n_{\pi}^{\rm overlap}\equiv\bigg{(}\frac{\,4\pi
r_{\pi}^{3}\,}{3}\bigg{)}^{-1}\simeq 0.83\,{\rm fm}^{-3}\simeq 5.2n_{0}\,,$
(88)

where we introduced $n_{0}=0.16\,{\rm fm}^{-3}$ as a unit frequently used in
nuclear physics. The estimate utilizes the root-mean-square radius of a pion,
$r_{\pi}\simeq 0.66$ fm, in vacuum Tanabashi et al. (2018); Ananthanarayan et
al. (2017); Koponen et al. (2016); Wang et al. (2021). The size of a pion are
comparable to that of a nucleon; the nucleon charge root-mean-square radius is
$\simeq 0.84$ fm, while its mass radius based on the gravitational form factor
Polyakov (2003) is $\simeq 0.55$ fm Kharzeev (2021) or the “valence quark core
radius” is $\simeq 0.5$ fm Kaiser and Weise (2024). The relation between
$\mu_{I}$ and $n_{I}$ is shown in Fig. 5. The overlap density $n_{\pi}^{\rm
overlap}$ is reached at low chemical potential, $\mu_{I}\simeq 0.256$ GeV.

A useful quantity characterizing the rapid stiffening is the sound speed. A
matter is called stiff (soft), i.e., $P$ is large (small) at a given
$\varepsilon$, and its variation is reflected in $c_{s}^{2}$. Writing the
susceptibility as $\tilde{\chi}=\partial^{2}P/\partial\mu_{I}^{2}$, the sound
speed is given by

$\displaystyle
c_{s}^{2}=\frac{\,\mathrm{d}P\,}{\,\mathrm{d}\varepsilon\,}\bigg{|}_{s=0}=\frac{\,\tilde{n}_{I}\mathrm{d}\mu_{I}\,}{\,\mu_{I}\mathrm{d}\tilde{n}_{I}\,}=\frac{\,\tilde{n}_{I}\,}{\,\mu_{I}\tilde{\chi}_{I}\,}\,.$
(89)

The result of the quark-meson model is shown in Fig. 6. The $c_{s}^{2}$ grows
quickly just after the onset of the pion condensation, makes a peak before
$n_{I}$ reaching $n_{\pi}^{\rm overlap}$, and slowly relaxes to the conformal
limit $1/3$. The result is compared with the lattice results of Brandt et al.
Brandt et al. (2023) which have more focus on low density and those of Abbott
et al. Abbott et al. (2023) which study the global structure for wider range
of $\mu_{I}$. The results are consistent. Later we delineate the behavior of
$c_{s}^{2}$ at low and high densities from microphysics viewpoint.

Recently the trace anomaly of EOS attracts attention Fujimoto et al. (2022);
Marczenko et al. (2022); Ma and Rho (2020); Tajima et al. (2024). It is one of
hot topics for forthcoming electron ion collider experiments which will
explore the nucleon structure in great detail. The trace anomaly normalized by
$3\varepsilon$ Fujimoto et al. (2022),

$\displaystyle\Delta_{\rm tr}\equiv\frac{\,\langle
T^{\mu}_{\mu}\rangle\,}{\,3\varepsilon\,}=\frac{1}{\,3\,}-\frac{\,P\,}{\,\varepsilon\,}$
(90)

is a useful measure for the deviation from the conformal limit. The result is
shown in Fig. 7. In the non-relativistic limit, $P\ll\varepsilon$, the
$\Delta_{{\rm tr}}=1/3$. In the conformal limit $\Delta_{\rm tr}=0$. When
$\Delta_{\rm tr}<0$, there should be strong correlation effects. Below we
discuss the power corrections as candidates to drive $\Delta_{\rm tr}$ to the
negative value.

As a guide for the low density behaviors, it is useful to quote the results
from the chiral perturbation theory (ChPT),

$\displaystyle
P=2f_{\pi}^{2}\mu_{I}^{2}\bigg{(}1-\frac{\,m_{\pi}^{2}\,}{\,2\mu_{I}\,}\bigg{)}^{4}\,,~{}~{}~{}~{}~{}$
(91)

which leads to

$\displaystyle
n_{I}=\frac{\,\tilde{n}_{I}\,}{\,2\,}=2f_{\pi}^{2}\mu_{I}\bigg{[}1-\bigg{(}\frac{\,m_{\pi}\,}{\,2\mu_{I}\,}\bigg{)}^{4}\bigg{]}\,,~{}~{}~{}~{}~{}~{}~{}~{}c_{s}^{2}=\frac{\,(2\mu_{I})^{4}-m_{\pi}^{4}\,}{\,(2\mu_{I})^{4}+3m_{\pi}^{4}\,}\,.$
(92)

At large density the sound speed asymptotically approaches $c_{s}^{2}=1$. Here
the scaling $\varepsilon\sim n_{I}^{2}$ holds; the system is as if dominated
by repulsive two-body forces.

###  7.1  High density regime: conformal behaviors

Let us first consider the high density regime in our model and the effective
degrees of freedom. In the conformal limit with no mass scale other than
$\mu_{I}$, the dimensional analyses lead to

$\displaystyle\frac{\,\partial(P/\mu_{I}^{4})\,}{\,\partial\mu_{I}\,}=0~{}~{}\rightarrow~{}~{}P=\frac{\,\varepsilon\,}{3}\,.$
(93)

This relation is approximately satisfied in the high density limit where the
kinetic energy dominates over mass and interaction energies. (Strictly
speaking, there are self-energies that do not necessarily die out at high
energy; in QCD the asymptotic freedom guarantees the self-energies become
weaker at high energy.)

The conformal limit by itself does not tell which degrees of freedom is
relevant. Indeed, supposing $P\simeq a\mu_{I}^{4}$, the conformal limit is
obtained for whatever value of $a$; we have to specify $a$ to tell which
degrees of freedom dominates. As we saw, the tree level pressure including
only mesonic degrees of freedom, and the one-loop pressure (dominated by quark
degrees of freedom) scales as

$\displaystyle P_{\rm
tree}\sim\frac{\,24\,}{\,\lambda\,}\mu_{I}^{4}\,,~{}~{}~{}~{}~{}~{}~{}~{}P_{\rm
1-loop}\sim\frac{\,N_{\rm f}^{\rm eff}N_{\rm
c}\,}{\,12\pi^{2}\,}\mu_{I}^{4}\,,$ (94)

where $\mu_{I}=\mu_{u}=-\mu_{d}$ and we should substitute $N_{\rm f}^{\rm
eff}=2$. In order for the former to reproduce the latter, we need
$\lambda\simeq 474$ which is unacceptably large. For $P_{\rm tree}$, the
$\Delta_{\rm tree}\sim\mu_{I}g/\sqrt{\lambda}$ plays an essential role for the
conformal scaling and its magnitude is determined by details of interactions.
In stark contrast, the conformal scaling of $P_{\rm 1-loop}$ is determined by
the number of quark degrees of freedom and robust to the interactions.

###  7.2  High density regime: perturbative and power corrections

Next we consider corrections to the conformal limit. The most frequently and
rigorously discussed is the perturbative corrections at weak coupling regime.
The perturbative corrections modify the coefficient of $\mu_{I}^{4}$ term by
adding series of $\alpha_{s}(\bar{\Lambda})$ where $\bar{\Lambda}$ is the
renormalization scale to be set to the natural scale for a given problem. At
large density $\bar{\Lambda}\sim\mu_{I}$ is the natural choice. The intrinsic
scale of QCD, $\Lambda_{\rm QCD}\simeq 200$-$300$ MeV, appearing only through
$\alpha_{s}$; it shows up in the form of $\sim\ln(\mu_{I}/\Lambda_{\rm QCD})$.
To the one-loop $\beta$-function,

$\displaystyle\frac{\,\partial\alpha_{s}(\mu_{I})\,}{\,\partial\ln\mu_{I}\,}=-\frac{\,11N_{\rm
c}-2N_{\rm f}\,}{\,24\pi\,}\alpha_{s}^{2}\,,~{}~{}~{}~{}~{}~{}N_{\rm f}=3\,.$
(95)

This reduces $P$, $n_{I}$, and $c_{s}^{2}$ from the conformal value. At
asymptotic density, the $c_{s}^{2}$ approaches the conformal limit from below.

In addition, in QCD there are power corrections in powers of $\Lambda_{\rm
QCD}/\mu_{I}$. This sort of terms cannot be expressed by powers of series of
$\alpha_{s}$,

$\displaystyle\Lambda_{\rm QCD}^{\rm
1-loop}\simeq\mu_{I}\mathrm{e}^{-2\pi/\beta_{0}\alpha_{s}(\mu_{I})}\,,~{}~{}~{}~{}~{}~{}~{}\beta_{0}=11-2N_{\rm
f}/3\,.$ (96)

and hence is non-perturbative. For momentum transfer of $\sim\Lambda_{\rm
QCD}$, quarks are supposed to strongly interact.

Actually, whether such soft interactions are important or not depends on the
presence of the color screening. If the screening is strong in the infrared
soft interactions are cut off. In the case of pion condensed phase, the
condensate is color-singlet and produces a gap for a quark-hole excitation.
This suppresses the Debye and Meissner screening. Hence the electric and
magnetic color interactions can be as strong as in the vacuum until the phase
space factor $\sim 4\pi\mu_{I}^{2}$ overwhelms the suppression factor by the
gap.

Adding the power corrections can change the qualitative behavior of
$c_{s}^{2}$ Kojo et al. (2015); Kojo and Suenaga (2022); Chiba and Kojo
(2024); Geißel et al. (2024); Braun et al. (2024); Leonhardt et al. (2020). In
particular the power corrections adding positive contributions to the pressure
favors the trend opposite to those from perturbative corrections, and hence
can be thought of a clear indicator for non-perturbative physics in the high
density domain. For a simple parametrization Chiba and Kojo (2024)

$\displaystyle P=a\big{(}\mu_{I}^{4}+c_{2}\Delta^{2}\mu_{I}^{2}\big{)}-B\,,$
(97)

with $a,c_{2}>0$ being dimensionless constants and $B$ included for
normalization, we find

$\displaystyle\tilde{n}_{I}=2a\big{(}2\mu_{I}^{3}+c_{2}\Delta^{2}\mu_{I}\big{)}\,,~{}~{}~{}~{}~{}\tilde{\chi}_{I}=2a\big{(}6\mu_{I}^{2}+c_{2}\Delta^{2}\big{)}\,,$
(98)

with which

$\displaystyle
c_{s}^{2}=\frac{\,2\mu_{I}^{2}+c_{2}\Delta^{2}\,}{\,6\mu_{I}^{2}+c_{2}\Delta^{2}\,}=\frac{1}{\,3\,}+\frac{2}{\,3\,}\frac{\,c_{2}\Delta^{2}\,}{\,6\mu_{I}^{2}+c_{2}\Delta^{2}\,}\,,$
(99)

varying between 1/3 and 1. In our quark-meson model $a=N_{\rm c}/6\pi^{2}$,
$c_{2}\simeq 3$, and $\Delta\simeq 0.25$-0.3 GeV Chiba and Kojo (2024). The
power corrections of $O(10\%)$ to the pressure occurs when

$\displaystyle 3(\Delta/\mu_{I})^{2}\sim 0.1~{}\rightarrow~{}\mu_{I}\sim
1.6\,{\rm GeV}\times\frac{\,\Delta\,}{\,0.3\,{\rm GeV}\,}\,,$ (100)

where the corrections increase $c_{s}^{2}$ as $0.333\rightarrow 0.344$ for
$\Delta\simeq 0.3$ GeV. The non-perturbative effects would survive at
relatively high density of $\mu_{I}\sim 1$ GeV. These analyses, although still
crude, indicate that the convergence of $\alpha_{s}$ expansion at high
density, which seems to be universal, may not be sufficient to judge the
relevance/irrelevance of the non-perturbative physics.

The lattice results for EOS by Abbott et al. Abbott et al. (2023) to
$\mu_{I}\sim 1.7$ GeV ($\mu_{I}$ defined in our work is a half of theirs) show
qualitative deviation from perturbative QCD predictions. One of plausible
sources is the power corrections from the $\sim\mu_{I}^{2}\Delta^{2}$ terms
where $\Delta$ is non-analytic in the strong coupling constant $g_{s}$. This
indicates that non-perturbative physics can survive to very high density even
where $\alpha_{s}(\mu_{I})$ is reasonably small Chiba and Kojo (2024).
Recently Ref. Abbott et al. (2023) used the deviation between the perturbative
EOS and lattice results to estimate the size of the BCS gap in the weak
coupling region, $\mu_{I}\in[0.750,1.625]$ GeV Abbott et al. (2024). The
expression used for the EOS is Fujimoto (2024)

$\displaystyle\delta P=P(\Delta)-P(\Delta=0)=\frac{\,N_{\rm
c}\,}{\,2\pi^{2}\,}\mu_{I}^{2}\Delta^{2}\bigg{(}1+\frac{\,g_{s}\,}{\,6\,}\bigg{)}\,.$
(101)

The lattice results show that $\Delta$ increases toward the low density
region. Meanwhile the gap estimated by the weak coupling method is

$\displaystyle\Delta=\tilde{b}\mu_{I}\exp\bigg{[}-\frac{\,\pi^{2}+4\,}{16}\bigg{]}\exp\bigg{[}-\frac{\,3\pi^{2}\,}{\,2g_{s}\,}\bigg{]}\,,~{}~{}~{}~{}~{}~{}~{}~{}\tilde{b}=512\pi^{4}g_{s}^{-5}\,,$
(102)

which is sensitive to the choice of the running $g_{s}(\bar{\Lambda})$. The
estimates with $\bar{\Lambda}=\mu_{I}$ and $2\mu_{I}$ covers the range
consistent with those estimated from the lattice results and Eq. (101). On the
other hand, the $\mu_{I}$ dependence looks opposite; the gap of Eq. (102)
decreases toward the low density region. We conjecture that the physics of
soft-gluons and quarks enhances the gap and cures the discrepancy.

The power corrections also play important roles in the trace anomaly. In the
parametrization Eq. (97), the energy density is

$\displaystyle\varepsilon=a\big{(}3\mu_{I}^{4}+c_{2}\Delta^{2}\mu_{I}^{2}\big{)}+B\,,$
(103)

so that

$\displaystyle\langle
T_{\mu}^{\mu}\rangle=\varepsilon-3P=-2ac_{2}\Delta^{2}\mu_{I}^{2}+4B\,.$ (104)

Remarkably the power corrections yield negative contributions. Meanwhile the
normalization (bag) constant measures the energy difference between the
conformal and non-perturbative vacuum and should be $\sim\Lambda_{\rm
QCD}^{4}$ and positive. Perturbative corrections not written here also give
positive contributions Chiba and Kojo (2024). Hence the negative trace anomaly
can be regarded as a good indicator for the importance of the non-perturbative
physics in dense matter.

Figure 8:  The occupation probability $f_{u,\bar{d}}(p)$ as a function of $p$
for the parameter set B in Table. 1. Red curves represent curves for $n_{0}$
increments, $n_{0},2n_{0},\cdots$ while blue curves for $5n_{0}$ increments.
To $5n_{0}$ we also show thin light blue curves for $0.2n_{0}$ increments. The
figure covers to $n_{I}=20n_{0}$.  Figure 9:  The occupation probability at
zero momentum, $f_{u,\bar{d}}(p=0)$ at various $n_{I}/n_{0}$. With the
parameter sets A-C, we vary $M_{0}(=M_{u,d}^{\rm vac})$ to change the “binding
energies” for quarks to form pions; a larger $M_{0}$ has the stronger binding,
$\sim 2M_{0}-m_{\pi}$, and leads to a more spatially compact state. The dashed
curves are linear extrapolation of the low density behavior,
$f_{u,\bar{d}}^{\rm id}(n_{I})\equiv n_{I}\partial f_{u,\bar{d}}(p=0)/\partial
n_{I}\big{|}_{n_{I}=0}$, which characterizes the $f_{u,\bar{d}}$ for non-
interacting pions. We also show $n_{\pi}^{\rm overlap}$ as a guide.  Figure
10:  Behaviors of $M_{l}$ (dashed lines) and $\Delta$ (solid lines) for the
same parameter set as Fig. 9. The vertical lines with various colors represent
the $n_{I}^{\rm id}$ defined in Eq. (107) for the set A, B, and C.  Figure 11:
The sound speed as a function of $n_{I}/n_{0}$ for the same parameter set as
Fig. 9. When pions are more spatially compact ($2M_{0}-m_{\pi}$ is larger),
the repulsion among pions or quark saturation effects set in at larger
densities.

###  7.3  Quark saturation

Finally we try to characterize EOS through quarks inside of pions. At
sufficiently low density quarks should show up only as constituents of pions.
Pion condensation generates BCS gaps for quarks with which quark contributions
can become finite even for $\mu_{I}(=\mu_{u}=-\mu_{d})<M_{l}$, see the
structure of Eq. (58). In the pion condensed phase the occupation probability
of $u,\bar{d}$ quark states can be written as

$\displaystyle f_{Q}(p;n_{I})\equiv
f_{u,\bar{d}}(p;n_{I})=\frac{1}{\,2\,}\bigg{(}1-\frac{\,E_{l}-\mu_{I}\,}{\,\sqrt{(E_{l}-\mu_{I})^{2}+\Delta^{2}}\,}\bigg{)}\,,$
(105)

which becomes $\Theta(\mu_{I}-E_{l})$ for $\Delta\rightarrow 0$. The behavior
of $f_{Q}(p)$ as a function of $p$ for various $n_{I}$ is shown in Fig. 8 for
the parameter set B.

As quarks are bound to a compact object, the quark momentum distribution is
broad in momentum. In particular the occupation probability at $p=0$ is
substantially smaller than 1. As the density increases, the magnitude of
$f_{Q}$ becomes larger. If we neglect the interactions and the structural
changes of pions, we would find the scaling

$\displaystyle f_{Q}^{\rm id}(p;n_{I})\equiv n_{I}\frac{\,\partial
f_{Q}(p;n_{I})\,}{\,\partial n_{I}\,}\bigg{|}_{n_{I}=0}\,,$ (106)

where pions condense into zero momentum state; each pion gives the same quark
contribution and hence simply scale as $\sim n_{I}$. Obviously extrapolating
this expression would violate the Pauli exclusion principle for quarks,
$f_{Q}(p)\leq 1$ for any $p$. We define the “(pseudo-)quark-saturation”
density $n^{\rm id}_{I}$ as

$\displaystyle 1=f_{Q}^{\rm id}(p=0;n_{I}^{\rm id})\,,$ (107)

and use $n^{\rm id}_{I}$ as the characteristic measure where either pion
interactions or quark saturation constraints become important.

Figure 9 shows $f_{Q}(p=0;n_{I})$ as a measure of the quark saturation. We
also show the $f_{Q}^{\rm id}$ with dashed lines to examine $n_{I}^{\rm id}$.
To examine how $n_{I}^{\rm id}$ depends on the compactness, we vary $M_{0}$ to
change the binding energy of a pion, $2M_{0}-m_{\pi}$, while maintaining
spectra of pseudo-scalar nonets within reasonable range. Concretely we compare
the set A-C in Table. 1. With a larger $M_{0}$ the $n_{I}^{\rm id}$ is larger
as the pion is spatially more compact and has broader quark momentum
distribution. For $M_{0}=(0.27,0.30,0.33)$ GeV, we found $n_{I}^{\rm
id}\simeq(2.2,2.4,2.6)n_{0}$, respectively. These densities are substantially
smaller than $n_{\pi}^{\rm overlap}\simeq 5.2n_{0}$.

In principle, by specifying $f_{Q}$ and gaps $(M_{l}\,,\Delta\,,M_{s})$ at a
given $\mu_{I}$, one can reconstruct the corresponding EOS as

$\displaystyle
P(\mu_{I})=\int_{0}^{\mu_{I}}\mathrm{d}\mu^{\prime}n_{I}(\mu^{\prime})=\int_{0}^{\mu_{I}}\mathrm{d}\mu^{\prime}\int_{p}\sum_{f}\big{[}f_{Q}(p;\mu^{\prime})-f_{\bar{Q}}(p;\mu^{\prime})\big{]}\,,$
(108)

hence the evolution of $f_{Q}$ (and $f_{\bar{Q}}$ for anti-particles) contains
sufficient information to study the EOS.

Shown in Fig. 11 are the behaviors of $M_{l}$ and $\Delta$ at several
densities and $M_{0}$. For a greater $M_{0}$ the gap $\Delta$ is naturally
larger. As guides we display $n_{I}^{\rm id}$ using several vertical lines.
The general tendency is that, beyond $n_{I}^{\rm id}$, the gap becomes
insensitive to $\mu_{I}$ reflecting that the gap equations are dominated by
soft gluon and soft quark contributions. The parameter set A-C leads to the
gap of $\sim 0.23-0.28$ GeV around $\mu_{I}\sim 1$ GeV. Hard gluon
contributions, which are omitted in this work, should further enhance the size
of the gap and introduce the stronger $\mu_{I}$ dependence as it is sensitive
to the phase space around the Fermi surface.

Finally we examine the sound speed for varying $M_{0}$. For smaller $M_{0}$
rising of $c_{s}^{2}$ occurs at lower density. This is natural as pions are
less compact objects. Increasing $M_{0}$ delays rising of $c_{s}^{2}$. The
structure of a hadron, its valence core size and quantum fluctuations around
it Fukushima et al. (2020), has the direct relevance to stiffening of matter.
The relation between the baryon structure and nuclear matter has been
discussed in, e.g., Refs. Saito et al. (2007); Geesaman et al. (1995); Koch et
al. (2024); McLerran and Miller (2024).

##  8 Summary

In this paper we study the EOS of isospin QCD and its relationship to the
microphysics. We used a quark-meson model that interpolates the hadronic and
quark sector at microscopic level. The model is renormalizable so that it can
be used to cover from low to high densities, at least formally. Although the
model does not cover the aspects of QCD caused by hard gluons, it captures
some aspects of the physics caused by soft gluons in a parametrized manner.

One of important issues in dense QCD is how non-perturbative effects relevant
in hadron physics die out. Recent lattice results for EOS Abbott et al. (2023)
to $\mu_{I}\sim 1.7$ GeV ($\mu_{I}$ defined in our work is a half of theirs)
indicate the importance of the power corrections from the
$\sim\mu_{I}^{2}\Delta^{2}$ terms with $\Delta$ being non-analytic in the QCD
coupling constant $g_{s}$, even at high density where $\alpha_{s}(\mu_{I})$ is
reasonably small Chiba and Kojo (2024). Concerning the size of the gap, there
are several questions to be answered. The first question is at which density
the evaluation of the gap is closed within the weak coupling regime; in the
weak coupling estimates we assume that hard momentum transfer processes
dominate the gap equation because of the large phase space. This mechanism is
sensitive to the density and at low density soft gluons should become
important. How the transition between these two regimes occur is directly
related to the reliability of various estimates. The second question is to
what extent the extrapolation from isospin QCD to QCD at finite baryon density
can be valid. The mechanism for the emergence of the BCS gap is the diquark
condensates in the color-superconductivity. The typical estimate for the gap,
mostly based on the weak coupling picture, is $\Delta\lesssim 100$ MeV; the
coefficient of $1/g_{s}$ in the weak coupling expression of gap is larger than
in the isospin QCD so that the gap is smaller. Analyses predicting a greater
gap is mostly based on effective models constrained by hadron physics. It is
important to fill the gap between two regimes.

The quarks in pions seem important before reaching the overlap density for
pions, $n_{\pi}^{\rm overlap}\simeq 5.2n_{0}$, which is inferred from the size
of a pion in vacuum. Even if we neglect the structural change of pions such as
swelling or dissociation, we cannot go much beyond $\sim 0.5n_{\pi}^{\rm
overlap}\simeq 2.6n_{0}$ neglecting constraints from the quark Pauli blocking.
The sound speed peak is also found around $\sim 0.5n_{\pi}^{\rm overlap}$. If
the quark exchange interactions (or meson exchange) among pions effectively
increase the size of pions, quark states at low momenta get saturated more
quickly, inducing the Pauli blocking effects even earlier. The quark Pauli
blocking near hadronic matter should also give insights on many-body forces
among hadrons. In the context of neutron star EOS, it is typical to utilize
two- and three-body repulsion to satisfy the two-solar mass constraints. But
in such descriptions there always remains a question of how to handle the
convergence of many-body forces. We need an organizing principle. We guess
that the quark saturation effects do the job.

Analyses in this paper left several important problems. In methodology we
should improve one-loop results. Another important topic is the meson spectra
at finite density, including quark loops. We compute the meson spectra in
vacuum for the parameter fixing but have not performed analyses at finite
density. The latter is necessary to answer to interesting questions such as
how mesons dissociate and change the structure. How mesons or quarks in medium
contribute to the entropy is also important to understand the color
confinement at finite density, a question originally posed in the quarkyonic
matter hypothesis McLerran and Pisarski (2007). The analyses toward this issue
is in progress.

T.K. was supported by JSPS KAKENHI Grant No. 23K03377 and No. 18H05407, and by
the Graduate Program on Physics for the Universe (GPPU) at Tohoku university;
D.S. by JSPS KAKENHI Grant No. 23H05439.

###### Acknowledgements.

We thank Drs. Brandt and Endrodi for kindly providing us with their lattice
data in Ref. Brandt et al. (2023), and Dr. Abbott and his collaborators for
their kindness of sending the lattice data in Ref. Abbott et al. (2023). T.K.
thanks Drs. Yuki Fujimoto and Larry McLerran for discussions on the quark
saturation effects. The authors thank the Yukawa Institute for Theoretical
Physics at Kyoto University and RIKEN iTHEMS. Discussions during the workshop
(YITP-T-23-05) on “Condensed Matter Physics of QCD 2024” were useful to
complete this work. The authors declare no conflicts of interest.
Abbreviations The following abbreviations are used in this manuscript:
QC2D | two-color quantum chromodynamics
---|---
QCDI | isospin QCD
EOS | equations of state
$\Lambda_{\rm QCD}\simeq 200$-300 MeV | non-perturbative scale in QCD
$n_{0}\simeq 0.16\,{\rm fm}^{-3}$ | nuclear saturation density
ChPT | Chiral perturbation theory
References

## References

  * Kogut et al. (1999) Kogut, J.B.; Stephanov, M.A.; Toublan, D.  On two color QCD with baryon chemical potential.  Phys. Lett. B 1999, 464, 183–191, [hep-ph/9906346].  https://doi.org/10.1016/S0370-2693(99)00971-5.
  * Kogut et al. (2000) Kogut, J.B.; Stephanov, M.A.; Toublan, D.; Verbaarschot, J.J.M.; Zhitnitsky, A.  QCD - like theories at finite baryon density.  Nucl. Phys. B 2000, 582, 477–513, [hep-ph/0001171].  https://doi.org/10.1016/S0550-3213(00)00242-X.
  * Iida et al. (2024) Iida, K.; Itou, E.; Murakami, K.; Suenaga, D.  Lattice study on finite density QC2D towards zero temperature 2024.  [arXiv:hep-lat/2405.20566].
  * Iida et al. (2020) Iida, K.; Itou, E.; Lee, T.G.  Two-colour QCD phases and the topology at low temperature and high density.  JHEP 2020, 01, 181, [arXiv:hep-lat/1910.07872].  https://doi.org/10.1007/JHEP01(2020)181.
  * Iida et al. (2021) Iida, K.; Itou, E.; Lee, T.G.  Relative scale setting for two-color QCD with $N_{f}$=2 Wilson fermions.  PTEP 2021, 2021, 013B05, [arXiv:hep-lat/2008.06322].  https://doi.org/10.1093/ptep/ptaa170.
  * Boz et al. (2020) Boz, T.; Giudice, P.; Hands, S.; Skullerud, J.I.  Dense two-color QCD towards continuum and chiral limits.  Phys. Rev. D 2020, 101, 074506, [arXiv:hep-lat/1912.10975].  https://doi.org/10.1103/PhysRevD.101.074506.
  * Boz et al. (2013) Boz, T.; Cotter, S.; Fister, L.; Mehta, D.; Skullerud, J.I.  Phase transitions and gluodynamics in 2-colour matter at high density.  Eur. Phys. J. A 2013, 49, 87, [arXiv:hep-lat/1303.3223].  https://doi.org/10.1140/epja/i2013-13087-6.
  * Cotter et al. (2013) Cotter, S.; Giudice, P.; Hands, S.; Skullerud, J.I.  Towards the phase diagram of dense two-color matter.  Phys. Rev. D 2013, 87, 034507, [arXiv:hep-lat/1210.4496].  https://doi.org/10.1103/PhysRevD.87.034507.
  * Hands et al. (2011) Hands, S.; Kenny, P.; Kim, S.; Skullerud, J.I.  Lattice Study of Dense Matter with Two Colors and Four Flavors.  Eur. Phys. J. A 2011, 47, 60, [arXiv:hep-lat/1101.4961].  https://doi.org/10.1140/epja/i2011-11060-1.
  * Astrakhantsev et al. (2020) Astrakhantsev, N.; Braguta, V.; Ilgenfritz, E.; Kotov, A.; Nikolaev, A.  Lattice study of thermodynamic properties of dense QC2D.  Phys. Rev. D 2020, 102, 074507, [arXiv:hep-lat/2007.07640].  https://doi.org/10.1103/PhysRevD.102.074507.
  * Bornyakov et al. (2020) Bornyakov, V.G.; Braguta, V.V.; Nikolaev, A.A.; Rogalyov, R.N.  Effects of Dense Quark Matter on Gluon Propagators in Lattice QC2D.  Phys. Rev. D 2020, 102, 114511, [arXiv:hep-lat/2003.00232].  https://doi.org/10.1103/PhysRevD.102.114511.
  * Muroya et al. (2003) Muroya, S.; Nakamura, A.; Nonaka, C.  Behavior of hadrons at finite density: Lattice study of color SU(2) QCD.  Phys. Lett. B 2003, 551, 305–310, [hep-lat/0211010].  https://doi.org/10.1016/S0370-2693(02)03065-4.
  * Suenaga et al. (2023a) Suenaga, D.; Murakami, K.; Itou, E.; Iida, K.  Probing the hadron mass spectrum in dense two-color QCD with the linear sigma model.  Phys. Rev. D 2023, 107, 054001, [arXiv:hep-ph/2211.01789].  https://doi.org/10.1103/PhysRevD.107.054001.
  * Suenaga et al. (2023b) Suenaga, D.; Murakami, K.; Itou, E.; Iida, K.  Mass spectrum of spin-one hadrons in dense two-color QCD: Novel predictions by extended linear sigma model 2023.  [arXiv:hep-ph/2312.17017].
  * Kawaguchi and Suenaga (2024) Kawaguchi, M.; Suenaga, D.  Sound velocity peak induced by the chiral partner in dense two-color QCD 2024.  [arXiv:hep-ph/2402.00430].
  * Sun et al. (2007) Sun, G.f.; He, L.; Zhuang, P.  BEC-BCS crossover in the Nambu-Jona-Lasinio model of QCD.  Phys. Rev. D 2007, 75, 096004, [hep-ph/0703159].  https://doi.org/10.1103/PhysRevD.75.096004.
  * Brauner et al. (2009) Brauner, T.; Fukushima, K.; Hidaka, Y.  Two-color quark matter: U(1)(A) restoration, superfluidity, and quarkyonic phase.  Phys. Rev. D 2009, 80, 074035, [arXiv:hep-ph/0907.4905].  [Erratum: Phys.Rev.D 81, 119904 (2010)], https://doi.org/10.1103/PhysRevD.81.119904.
  * Strodthoff and von Smekal (2014) Strodthoff, N.; von Smekal, L.  Polyakov-Quark-Meson-Diquark Model for two-color QCD.  Phys. Lett. B 2014, 731, 350–357, [arXiv:hep-ph/1306.2897].  https://doi.org/10.1016/j.physletb.2014.03.008.
  * Strodthoff et al. (2012) Strodthoff, N.; Schaefer, B.J.; von Smekal, L.  Quark-meson-diquark model for two-color QCD.  Phys. Rev. D 2012, 85, 074007, [arXiv:hep-ph/1112.5401].  https://doi.org/10.1103/PhysRevD.85.074007.
  * Abbott et al. (2024) Abbott, R.; Detmold, W.; Illa, M.; Parreño, A.; Perry, R.J.; Romero-López, F.; Shanahan, P.E.; Wagman, M.L.  QCD constraints on isospin-dense matter and the nuclear equation of state 2024.  [arXiv:hep-lat/2406.09273].
  * Abbott et al. (2023) Abbott, R.; Detmold, W.; Romero-López, F.; Davoudi, Z.; Illa, M.; Parreño, A.; Perry, R.J.; Shanahan, P.E.; Wagman, M.L.  Lattice quantum chromodynamics at large isospin density: 6144 pions in a box 2023.  [arXiv:hep-lat/2307.15014].
  * Brandt et al. (2023) Brandt, B.B.; Cuteri, F.; Endrodi, G.  Equation of state and speed of sound of isospin-asymmetric QCD on the lattice.  JHEP 2023, 07, 055, [arXiv:hep-lat/2212.14016].  https://doi.org/10.1007/JHEP07(2023)055.
  * Son and Stephanov (2001a) Son, D.T.; Stephanov, M.A.  QCD at finite isospin density.  Phys. Rev. Lett. 2001, 86, 592–595, [hep-ph/0005225].  https://doi.org/10.1103/PhysRevLett.86.592.
  * Son and Stephanov (2001b) Son, D.T.; Stephanov, M.A.  QCD at finite isospin density: From pion to quark - anti-quark condensation.  Phys. Atom. Nucl. 2001, 64, 834–842, [hep-ph/0011365].  https://doi.org/10.1134/1.1378872.
  * Splittorff et al. (2001) Splittorff, K.; Son, D.T.; Stephanov, M.A.  QCD - like theories at finite baryon and isospin density.  Phys. Rev. D 2001, 64, 016003, [hep-ph/0012274].  https://doi.org/10.1103/PhysRevD.64.016003.
  * Gómez Nicola and Vioque-Rodríguez (2022) Gómez Nicola, A.; Vioque-Rodríguez, A.  Effective Lagrangian at nonzero isospin chemical potential.  Phys. Rev. D 2022, 106, 114017, [arXiv:hep-ph/2205.14609].  https://doi.org/10.1103/PhysRevD.106.114017.
  * Lu et al. (2020) Lu, Z.Y.; Xia, C.J.; Ruggieri, M.  Thermodynamics and susceptibilities of isospin imbalanced QCD matter.  Eur. Phys. J. C 2020, 80, 46, [arXiv:hep-ph/1907.11497].  https://doi.org/10.1140/epjc/s10052-020-7614-6.
  * Fukushima and Hatsuda (2011) Fukushima, K.; Hatsuda, T.  The phase diagram of dense QCD.  Rept. Prog. Phys. 2011, 74, 014001, [arXiv:hep-ph/1005.4814].  https://doi.org/10.1088/0034-4885/74/1/014001.
  * Oertel et al. (2017) Oertel, M.; Hempel, M.; Klähn, T.; Typel, S.  Equations of state for supernovae and compact stars.  Rev. Mod. Phys. 2017, 89, 015007, [arXiv:astro-ph.HE/1610.03361].  https://doi.org/10.1103/RevModPhys.89.015007.
  * Baym et al. (2018) Baym, G.; Hatsuda, T.; Kojo, T.; Powell, P.D.; Song, Y.; Takatsuka, T.  From hadrons to quarks in neutron stars: a review.  Rept. Prog. Phys. 2018, 81, 056902, [arXiv:astro-ph.HE/1707.04966].  https://doi.org/10.1088/1361-6633/aaae14.
  * Kojo (2021) Kojo, T.  QCD equations of state and speed of sound in neutron stars.  AAPPS Bull. 2021, 31, 11, [arXiv:nucl-th/2011.10940].  https://doi.org/10.1007/s43673-021-00011-6.
  * Vuorinen (2024) Vuorinen, A.  Particle-theory Input for Neutron-star Physics.  Acta Phys. Polon. B 2024, 55, 4–A4, [arXiv:hep-ph/2405.01141].  https://doi.org/10.5506/APhysPolB.55.4-A4.
  * Drischler et al. (2021) Drischler, C.; Han, S.; Lattimer, J.M.; Prakash, M.; Reddy, S.; Zhao, T.  Limiting masses and radii of neutron stars and their implications.  Phys. Rev. C 2021, 103, 045808, [arXiv:nucl-th/2009.06441].  https://doi.org/10.1103/PhysRevC.103.045808.
  * Han et al. (2023) Han, M.Z.; Huang, Y.J.; Tang, S.P.; Fan, Y.Z.  Plausible presence of new state in neutron stars with masses above 0.98MTOV.  Sci. Bull. 2023, 68, 913–919, [arXiv:astro-ph.HE/2207.13613].  https://doi.org/10.1016/j.scib.2023.04.007.
  * Kojo et al. (2022) Kojo, T.; Baym, G.; Hatsuda, T.  Implications of NICER for Neutron Star Matter: The QHC21 Equation of State.  Astrophys. J. 2022, 934, 46, [arXiv:astro-ph.HE/2111.11919].  https://doi.org/10.3847/1538-4357/ac7876.
  * Huang et al. (2022) Huang, Y.J.; Baiotti, L.; Kojo, T.; Takami, K.; Sotani, H.; Togashi, H.; Hatsuda, T.; Nagataki, S.; Fan, Y.Z.  Merger and Postmerger of Binary Neutron Stars with a Quark-Hadron Crossover Equation of State.  Phys. Rev. Lett. 2022, 129, 181101, [arXiv:astro-ph.HE/2203.04528].  https://doi.org/10.1103/PhysRevLett.129.181101.
  * Fujimoto et al. (2023) Fujimoto, Y.; Fukushima, K.; Hotokezaka, K.; Kyutoku, K.  Gravitational Wave Signal for Quark Matter with Realistic Phase Transition.  Phys. Rev. Lett. 2023, 130, 091404, [arXiv:astro-ph.HE/2205.03882].  https://doi.org/10.1103/PhysRevLett.130.091404.
  * Kedia et al. (2022) Kedia, A.; Kim, H.I.; Suh, I.S.; Mathews, G.J.  Binary neutron star mergers as a probe of quark-hadron crossover equations of state.  Phys. Rev. D 2022, 106, 103027, [arXiv:gr-qc/2203.05461].  https://doi.org/10.1103/PhysRevD.106.103027.
  * Bauswein et al. (2019) Bauswein, A.; Bastian, N.U.F.; Blaschke, D.B.; Chatziioannou, K.; Clark, J.A.; Fischer, T.; Oertel, M.  Identifying a first-order phase transition in neutron star mergers through gravitational waves.  Phys. Rev. Lett. 2019, 122, 061102, [arXiv:astro-ph.HE/1809.01116].  https://doi.org/10.1103/PhysRevLett.122.061102.
  * McLerran and Reddy (2019) McLerran, L.; Reddy, S.  Quarkyonic Matter and Neutron Stars.  Phys. Rev. Lett. 2019, 122, 122701, [arXiv:nucl-th/1811.12503].  https://doi.org/10.1103/PhysRevLett.122.122701.
  * Jeong et al. (2020) Jeong, K.S.; McLerran, L.; Sen, S.  Dynamically generated momentum space shell structure of quarkyonic matter via an excluded volume model.  Phys. Rev. C 2020, 101, 035201, [arXiv:nucl-th/1908.04799].  https://doi.org/10.1103/PhysRevC.101.035201.
  * Kojo (2021) Kojo, T.  Stiffening of matter in quark-hadron continuity.  Phys. Rev. D 2021, 104, 074005, [arXiv:nucl-th/2106.06687].  https://doi.org/10.1103/PhysRevD.104.074005.
  * Fujimoto et al. (2024) Fujimoto, Y.; Kojo, T.; McLerran, L.D.  Momentum Shell in Quarkyonic Matter from Explicit Duality: A Dual Model for Cold, Dense QCD.  Phys. Rev. Lett. 2024, 132, 112701, [arXiv:nucl-th/2306.04304].  https://doi.org/10.1103/PhysRevLett.132.112701.
  * Hayata et al. (2023) Hayata, T.; Hidaka, Y.; Nishimura, K.  Dense $\textrm{QCD}_{2}$ with matrix product states 2023.  [arXiv:hep-lat/2311.11643].
  * Mannarelli (2019) Mannarelli, M.  Meson condensation.  Particles 2019, 2, 411–443, [arXiv:hep-ph/1908.02042].  https://doi.org/10.3390/particles2030025.
  * Chiba and Kojo (2024) Chiba, R.; Kojo, T.  Sound velocity peak and conformality in isospin QCD.  Phys. Rev. D 2024, 109, 076006, [arXiv:hep-ph/2304.13920].  https://doi.org/10.1103/PhysRevD.109.076006.
  * Chiba et al. (2024) Chiba, R.; Kojo, T.; Suenaga, D.  Thermal effects on sound velocity peak and conformality in isospin QCD 2024.  [arXiv:hep-ph/2403.02538].
  * Fujimoto (2024) Fujimoto, Y.  Enhanced contribution of the pairing gap to the QCD equation of state at large isospin chemical potential.  Phys. Rev. D 2024, 109, 054035, [arXiv:hep-ph/2312.11443].  https://doi.org/10.1103/PhysRevD.109.054035.
  * Cohen (2003) Cohen, T.D.  QCD inequalities for the nucleon mass and the free energy of baryonic matter.  Phys. Rev. Lett. 2003, 91, 032002, [hep-ph/0304024].  https://doi.org/10.1103/PhysRevLett.91.032002.
  * Navarrete et al. (2024) Navarrete, P.; Paatelainen, R.; Seppänen, K.  Perturbative QCD meets phase quenching: The pressure of cold Quark Matter 2024.  [arXiv:hep-ph/2403.02180].
  * Fujimoto and Reddy (2024) Fujimoto, Y.; Reddy, S.  Bounds on the equation of state from QCD inequalities and lattice QCD.  Phys. Rev. D 2024, 109, 014020, [arXiv:nucl-th/2310.09427].  https://doi.org/10.1103/PhysRevD.109.014020.
  * Moore and Gorda (2023) Moore, G.D.; Gorda, T.  Bounding the QCD Equation of State with the Lattice.  JHEP 2023, 12, 133, [arXiv:nucl-th/2309.15149].  https://doi.org/10.1007/JHEP12(2023)133.
  * Adhikari et al. (2017) Adhikari, P.; Andersen, J.O.; Kneschke, P.  On-shell parameter fixing in the quark-meson model.  Phys. Rev. D 2017, 95, 036017, [arXiv:hep-ph/1612.03668].  https://doi.org/10.1103/PhysRevD.95.036017.
  * Ayala et al. (2023) Ayala, A.; Bandyopadhyay, A.; Farias, R.L.S.; Hernández, L.A.; Hernández, J.L.  QCD equation of state at finite isospin density from the linear sigma model with quarks: The cold case 2023.  [arXiv:hep-ph/2301.13633].
  * Gao et al. (2022) Gao, B.; Minamikawa, T.; Kojo, T.; Harada, M.  Impacts of the U(1)A anomaly on nuclear and neutron star equation of state based on a parity doublet model.  Phys. Rev. C 2022, 106, 065205, [arXiv:nucl-th/2207.05970].  https://doi.org/10.1103/PhysRevC.106.065205.
  * Minamikawa et al. (2023) Minamikawa, T.; Gao, B.; Kojo, T.; Harada, M.  Chiral restoration of nucleons in neutron star matter: studies based on a parity doublet model 2023.  [arXiv:nucl-th/2302.00825].
  * Hatsuda et al. (2006) Hatsuda, T.; Tachibana, M.; Yamamoto, N.; Baym, G.  New critical point induced by the axial anomaly in dense QCD.  Phys. Rev. Lett. 2006, 97, 122001, [hep-ph/0605018].  https://doi.org/10.1103/PhysRevLett.97.122001.
  * Yamamoto et al. (2007) Yamamoto, N.; Tachibana, M.; Hatsuda, T.; Baym, G.  Phase structure, collective modes, and the axial anomaly in dense QCD.  Phys. Rev. D 2007, 76, 074001, [arXiv:hep-ph/0704.2654].  https://doi.org/10.1103/PhysRevD.76.074001.
  * Hatsuda et al. (2008) Hatsuda, T.; Tachibana, M.; Yamamoto, N.  Spectral Continuity in Dense QCD.  Phys. Rev. D 2008, 78, 011501, [arXiv:hep-ph/0802.4143].  https://doi.org/10.1103/PhysRevD.78.011501.
  * Zhang et al. (2009) Zhang, Z.; Fukushima, K.; Kunihiro, T.  Number of the QCD critical points with neutral color superconductivity.  Phys. Rev. D 2009, 79, 014004, [arXiv:hep-ph/0808.0927].  https://doi.org/10.1103/PhysRevD.79.014004.
  * Schäfer and Wilczek (1999) Schäfer, T.; Wilczek, F.  Continuity of quark and hadron matter.  Phys. Rev. Lett. 1999, 82, 3956–3959, [hep-ph/9811473].  https://doi.org/10.1103/PhysRevLett.82.3956.
  * Masuda et al. (2013a) Masuda, K.; Hatsuda, T.; Takatsuka, T.  Hadron-Quark Crossover and Massive Hybrid Stars with Strangeness.  Astrophys. J. 2013, 764, 12, [arXiv:nucl-th/1205.3621].  https://doi.org/10.1088/0004-637X/764/1/12.
  * Masuda et al. (2013b) Masuda, K.; Hatsuda, T.; Takatsuka, T.  Hadron–quark crossover and massive hybrid stars.  PTEP 2013, 2013, 073D01, [arXiv:nucl-th/1212.6803].  https://doi.org/10.1093/ptep/ptt045.
  * Masuda et al. (2016) Masuda, K.; Hatsuda, T.; Takatsuka, T.  Hyperon Puzzle, Hadron-Quark Crossover and Massive Neutron Stars.  Eur. Phys. J. A 2016, 52, 65, [arXiv:nucl-th/1508.04861].  https://doi.org/10.1140/epja/i2016-16065-6.
  * Tajima et al. (2023) Tajima, H.; Tsutsui, S.; Doi, T.M.; Iida, K.  Density-Induced Hadron–Quark Crossover via the Formation of Cooper Triples.  Symmetry 2023, 15, 333, [arXiv:nucl-th/2211.14194].  https://doi.org/10.3390/sym15020333.
  * Ma and Rho (2020) Ma, Y.L.; Rho, M.  Towards the hadron–quark continuity via a topology change in compact stars.  Prog. Part. Nucl. Phys. 2020, 113, 103791, [arXiv:nucl-th/1909.05889].  https://doi.org/10.1016/j.ppnp.2020.103791.
  * ’t Hooft (1974) ’t Hooft, G.  A Planar Diagram Theory for Strong Interactions.  Nucl. Phys. B 1974, 72, 461.  https://doi.org/10.1016/0550-3213(74)90154-0.
  * Witten (1979) Witten, E.  Baryons in the 1/n Expansion.  Nucl. Phys. B 1979, 160, 57–115.  https://doi.org/10.1016/0550-3213(79)90232-3.
  * Nishimura et al. (2022) Nishimura, T.; Kitazawa, M.; Kunihiro, T.  Anomalous enhancement of dilepton production as a precursor of color superconductivity.  PTEP 2022, 2022, 093D02, [arXiv:hep-ph/2201.01963].  https://doi.org/10.1093/ptep/ptac100.
  * Kitazawa et al. (2005) Kitazawa, M.; Koide, T.; Kunihiro, T.; Nemoto, Y.  Pre-critical phenomena of two-flavor color superconductivity in heated quark matter: Diquark-pair fluctuations and non-Fermi liquid behavior.  Prog. Theor. Phys. 2005, 114, 117–155, [hep-ph/0502035].  https://doi.org/10.1143/PTP.114.117.
  * Kitazawa et al. (2002) Kitazawa, M.; Koide, T.; Kunihiro, T.; Nemoto, Y.  Precursor of color superconductivity in hot quark matter.  Phys. Rev. D 2002, 65, 091504, [nucl-th/0111022].  https://doi.org/10.1103/PhysRevD.65.091504.
  * Blaschke et al. (2014) Blaschke, D.; Buballa, M.; Dubinin, A.; Roepke, G.; Zablocki, D.  Generalized Beth–Uhlenbeck approach to mesons and diquarks in hot, dense quark matter.  Annals Phys. 2014, 348, 228–255, [arXiv:hep-ph/1305.3907].  https://doi.org/10.1016/j.aop.2014.06.002.
  * Boz et al. (2019) Boz, T.; Hajizadeh, O.; Maas, A.; Skullerud, J.I.  Finite-density gauge correlation functions in QC2D.  Phys. Rev. D 2019, 99, 074514, [arXiv:hep-lat/1812.08517].  https://doi.org/10.1103/PhysRevD.99.074514.
  * Kojo and Baym (2014) Kojo, T.; Baym, G.  Color screening in cold quark matter.  Phys. Rev. D 2014, 89, 125008, [arXiv:hep-ph/1404.1346].  https://doi.org/10.1103/PhysRevD.89.125008.
  * Suenaga and Kojo (2019) Suenaga, D.; Kojo, T.  Gluon propagator in two-color dense QCD: Massive Yang-Mills approach at one-loop.  Phys. Rev. D 2019, 100, 076017, [arXiv:hep-ph/1905.08751].  https://doi.org/10.1103/PhysRevD.100.076017.
  * Kojo and Suenaga (2021) Kojo, T.; Suenaga, D.  Thermal quarks and gluon propagators in two-color dense QCD.  Phys. Rev. D 2021, 103, 094008, [arXiv:hep-ph/2102.07231].  https://doi.org/10.1103/PhysRevD.103.094008.
  * Contant and Huber (2020) Contant, R.; Huber, M.Q.  Dense two-color QCD from Dyson-Schwinger equations.  Phys. Rev. D 2020, 101, 014016, [arXiv:hep-ph/1909.12796].  https://doi.org/10.1103/PhysRevD.101.014016.
  * Kojo et al. (2010a) Kojo, T.; Hidaka, Y.; McLerran, L.; Pisarski, R.D.  Quarkyonic Chiral Spirals.  Nucl. Phys. A 2010, 843, 37–58, [arXiv:hep-ph/0912.3800].  https://doi.org/10.1016/j.nuclphysa.2010.05.053.
  * Kojo et al. (2010b) Kojo, T.; Pisarski, R.D.; Tsvelik, A.M.  Covering the Fermi Surface with Patches of Quarkyonic Chiral Spirals.  Phys. Rev. D 2010, 82, 074015, [arXiv:hep-ph/1007.0248].  https://doi.org/10.1103/PhysRevD.82.074015.
  * Kojo et al. (2012) Kojo, T.; Hidaka, Y.; Fukushima, K.; McLerran, L.D.; Pisarski, R.D.  Interweaving Chiral Spirals.  Nucl. Phys. A 2012, 875, 94–138, [arXiv:hep-ph/1107.2124].  https://doi.org/10.1016/j.nuclphysa.2011.11.007.
  * Kojo and Su (2013) Kojo, T.; Su, N.  The quark mass gap in a magnetic field.  Phys. Lett. B 2013, 720, 192–197, [arXiv:hep-ph/1211.7318].  https://doi.org/10.1016/j.physletb.2013.02.024.
  * Kojo and Su (2014) Kojo, T.; Su, N.  The quark mass gap in strong magnetic fields.  Nucl. Phys. A 2014, 931, 763–768, [arXiv:hep-ph/1407.7925].  https://doi.org/10.1016/j.nuclphysa.2014.08.004.
  * Hattori et al. (2016) Hattori, K.; Kojo, T.; Su, N.  Mesons in strong magnetic fields: (I) General analyses.  Nucl. Phys. A 2016, 951, 1–30, [arXiv:hep-ph/1512.07361].  https://doi.org/10.1016/j.nuclphysa.2016.03.016.
  * Novikov et al. (1978) Novikov, V.A.; Okun, L.B.; Shifman, M.A.; Vainshtein, A.I.; Voloshin, M.B.; Zakharov, V.I.  Charmonium and Gluons: Basic Experimental Facts and Theoretical Introduction.  Phys. Rept. 1978, 41, 1–133.  https://doi.org/10.1016/0370-1573(78)90120-5.
  * Shifman et al. (1979a) Shifman, M.A.; Vainshtein, A.I.; Zakharov, V.I.  QCD and Resonance Physics. Theoretical Foundations.  Nucl. Phys. B 1979, 147, 385–447.  https://doi.org/10.1016/0550-3213(79)90022-1.
  * Shifman et al. (1979b) Shifman, M.A.; Vainshtein, A.I.; Zakharov, V.I.  QCD and Resonance Physics: Applications.  Nucl. Phys. B 1979, 147, 448–518.  https://doi.org/10.1016/0550-3213(79)90023-3.
  * Lee and Yang (1956) Lee, T.D.; Yang, C.N.  Charge Conjugation, a New Quantum Number $G$, and Selection Rules Concerning a Nucleon Anti-nucleon System.  Nuovo Cim. 1956, 10, 749–753.  https://doi.org/10.1007/BF02744530.
  * Andersen et al. (2019) Andersen, J.O.; Adhikari, P.; Kneschke, P.  Pion condensation and QCD phase diagram at finite isospin density.  PoS 2019, Confinement2018, 197, [arXiv:hep-ph/1810.00419].  https://doi.org/10.22323/1.336.0197.
  * Coleman and Weinberg (1973) Coleman, S.R.; Weinberg, E.J.  Radiative Corrections as the Origin of Spontaneous Symmetry Breaking.  Phys. Rev. D 1973, 7, 1888–1910.  https://doi.org/10.1103/PhysRevD.7.1888.
  * Duarte et al. (2021) Duarte, D.C.; Hernandez-Ortiz, S.; Jeong, K.S.; McLerran, L.D.  Quarkyonic effective field theory, quark-nucleon duality, and ghosts.  Phys. Rev. D 2021, 104, L091901, [arXiv:nucl-th/2103.05679].  https://doi.org/10.1103/PhysRevD.104.L091901.
  * Hatsuda and Kunihiro (1994) Hatsuda, T.; Kunihiro, T.  QCD phenomenology based on a chiral effective Lagrangian.  Phys. Rept. 1994, 247, 221–367, [hep-ph/9401310].  https://doi.org/10.1016/0370-1573(94)90022-1.
  * Tanabashi et al. (2018) Tanabashi et al., M.  Review of Particle Physics.  Phys. Rev. D 2018, 98, 030001.  https://doi.org/10.1103/PhysRevD.98.030001.
  * Ananthanarayan et al. (2017) Ananthanarayan, B.; Caprini, I.; Das, D.  Electromagnetic charge radius of the pion at high precision.  Phys. Rev. Lett. 2017, 119, 132002, [arXiv:hep-ph/1706.04020].  https://doi.org/10.1103/PhysRevLett.119.132002.
  * Koponen et al. (2016) Koponen, J.; Bursa, F.; Davies, C.T.H.; Dowdall, R.J.; Lepage, G.P.  Size of the pion from full lattice QCD with physical u , d , s and c quarks.  Phys. Rev. D 2016, 93, 054503, [arXiv:hep-lat/1511.07382].  https://doi.org/10.1103/PhysRevD.93.054503.
  * Wang et al. (2021) Wang, G.; Liang, J.; Draper, T.; Liu, K.F.; Yang, Y.B.  Lattice Calculation of Pion Form Factor with Overlap Fermions.  Phys. Rev. D 2021, 104, 074502, [arXiv:hep-ph/2006.05431].  https://doi.org/10.1103/PhysRevD.104.074502.
  * Polyakov (2003) Polyakov, M.V.  Generalized parton distributions and strong forces inside nucleons and nuclei.  Phys. Lett. B 2003, 555, 57–62, [hep-ph/0210165].  https://doi.org/10.1016/S0370-2693(03)00036-4.
  * Kharzeev (2021) Kharzeev, D.E.  Mass radius of the proton.  Phys. Rev. D 2021, 104, 054015, [arXiv:hep-ph/2102.00110].  https://doi.org/10.1103/PhysRevD.104.054015.
  * Kaiser and Weise (2024) Kaiser, N.; Weise, W.  Sizes of the Nucleon 2024.  [arXiv:nucl-th/2404.11292].
  * Fujimoto et al. (2022) Fujimoto, Y.; Fukushima, K.; McLerran, L.D.; Praszalowicz, M.  Trace Anomaly as Signature of Conformality in Neutron Stars.  Phys. Rev. Lett. 2022, 129, 252702, [arXiv:nucl-th/2207.06753].  https://doi.org/10.1103/PhysRevLett.129.252702.
  * Marczenko et al. (2022) Marczenko, M.; McLerran, L.; Redlich, K.; Sasaki, C.  Reaching percolation and conformal limits in neutron stars 2022.  [arXiv:nucl-th/2207.13059].
  * Tajima et al. (2024) Tajima, H.; Iida, K.; Liang, H.  Nonrelativistic trace anomaly and equation of state in dense fermionic matter.  Phys. Rev. C 2024, 109, 055203, [arXiv:hep-ph/2402.04960].  https://doi.org/10.1103/PhysRevC.109.055203.
  * Kojo et al. (2015) Kojo, T.; Powell, P.D.; Song, Y.; Baym, G.  Phenomenological QCD equation of state for massive neutron stars.  Phys. Rev. D 2015, 91, 045003, [arXiv:hep-ph/1412.1108].  https://doi.org/10.1103/PhysRevD.91.045003.
  * Kojo and Suenaga (2022) Kojo, T.; Suenaga, D.  Peaks of sound velocity in two color dense QCD: Quark saturation effects and semishort range correlations.  Phys. Rev. D 2022, 105, 076001, [arXiv:hep-ph/2110.02100].  https://doi.org/10.1103/PhysRevD.105.076001.
  * Geißel et al. (2024) Geißel, A.; Gorda, T.; Braun, J.  Pressure and speed of sound in two-flavor color-superconducting quark matter at next-to-leading order 2024.  [arXiv:hep-ph/2403.18010].
  * Braun et al. (2024) Braun, J.; Geißel, A.; Schallmo, B.  Speed of sound in dense strong-interaction matter.  SciPost Phys. Core 2024, 7, 015, [arXiv:nucl-th/2206.06328].  https://doi.org/10.21468/SciPostPhysCore.7.2.015.
  * Leonhardt et al. (2020) Leonhardt, M.; Pospiech, M.; Schallmo, B.; Braun, J.; Drischler, C.; Hebeler, K.; Schwenk, A.  Symmetric nuclear matter from the strong interaction.  Phys. Rev. Lett. 2020, 125, 142502, [arXiv:nucl-th/1907.05814].  https://doi.org/10.1103/PhysRevLett.125.142502.
  * Fukushima et al. (2020) Fukushima, K.; Kojo, T.; Weise, W.  Hard-core deconfinement and soft-surface delocalization from nuclear to quark matter.  Phys. Rev. D 2020, 102, 096017, [arXiv:hep-ph/2008.08436].  https://doi.org/10.1103/PhysRevD.102.096017.
  * Saito et al. (2007) Saito, K.; Tsushima, K.; Thomas, A.W.  Nucleon and hadron structure changes in the nuclear medium and impact on observables.  Prog. Part. Nucl. Phys. 2007, 58, 1–167, [hep-ph/0506314].  https://doi.org/10.1016/j.ppnp.2005.07.003.
  * Geesaman et al. (1995) Geesaman, D.F.; Saito, K.; Thomas, A.W.  The nuclear EMC effect.  Ann. Rev. Nucl. Part. Sci. 1995, 45, 337–390.  https://doi.org/10.1146/annurev.ns.45.120195.002005.
  * Koch et al. (2024) Koch, V.; McLerran, L.; Miller, G.A.; Vovchenko, V.  Might Normal Nuclear Matter be Quarkyonic? 2024.  [arXiv:nucl-th/2403.15375].
  * McLerran and Miller (2024) McLerran, L.; Miller, G.A.  The Quark Pauli Principle and the Transmutation of Nuclear Matter 2024.  [arXiv:nucl-th/2405.11074].
  * McLerran and Pisarski (2007) McLerran, L.; Pisarski, R.D.  Phases of cold, dense quarks at large N(c).  Nucl. Phys. A 2007, 796, 83–100, [arXiv:hep-ph/0706.2191].  https://doi.org/10.1016/j.nuclphysa.2007.08.013.