#  Charged pion condensation and color superconductivity phenomena in chirally
asymmetric dense quark matter

T. G. Khunjua 1, K. G. Klimenko 2, and R. N. Zhokhov 2,3 1 The University of
Georgia, GE-0171 Tbilisi, Georgia  2 State Research Center of Russian
Federation – Institute for High Energy Physics, NRC ”Kurchatov Institute”,
142281 Protvino, Moscow Region, Russia  3 Pushkov Institute of Terrestrial
Magnetism, Ionosphere and Radiowave Propagation (IZMIRAN), 108840 Troitsk,
Moscow, Russia

###### Abstract

In this paper, the question of the influence of color superconductivity (CSC)
on the formation of a phase with condensation of charged pions in dense
chirally asymmetric quark matter is studied. We consider it within the
framework of the massless NJL model with a diquark interaction channel at zero
temperature, but in the presence of baryon $\mu_{B}$, isospin $\mu_{I}$,
chiral $\mu_{5}$ and chiral isospin $\mu_{I5}$ chemical potentials. It has
been shown in the mean-field approximation that in the presence of chiral
imbalance, when $\mu_{5}\neq 0$ and/or $\mu_{I5}\neq 0$, CSC phenomenon does
not hinder the generation of charged pion condensation in dense quark matter
even at largest baryon densities attainable in heavy ion collisions
experiments and in cores of neutron stars and its mergers. So charged pion
condensation in dense quark matter with chiral imbalance predicted earlier is
not in a bit suppressed by the presence of CSC, highlighting the resilience of
this phase in dense quark matter. This makes phase structure of dense quark
matter even more rich, multifaceted and interesting, and shows that pion
condensation in dense quark matter is viable phenomenon to be explored, for
example, in intermediate energy heavy ion collisions experiments.

Nambu–Jona-Lasinio model; dense quark matter; color superconductivity; chiral
asymmetry

##  I Introduction

Quantum chromodynamics (QCD) is the fundamental theory of strong interactions,
which govern the behavior of quarks and gluons, the fundamental constituents
of matter. Dense quark matter, hypothetical state of matter which could exist
in the cores of compact stars or arise in heavy-ion collisions, is a subject
of intense research. However, perturbative QCD methods are not applicable to
this regime due to the large value of the strong coupling constant, and as a
result effective QCD-like models, such as the Nambu–Jona-Lasinio (NJL) model,
are often used to study its properties njl ; buballa ; buballa2 ; zhokhov1 ;
zhokhov2 .

The NJL model is based on the assumption of four-fermion interactions among
quarks, which effectively account for the effects of quark-gluon interactions
at low energies. In order to investigate the properties of quark matter with
nonzero baryon $n_{B}$ and nonzero isospin $n_{I}$ densities (which are
characteristic properties of the dense matter found in neutron stars), one
extends this model and incorporates the terms with baryon $\mu_{B}$ and
isospin $\mu_{I}$ chemical potentials. These chemical potentials are nothing
more than quantities that are thermodynamically conjugate to baryon and
isospin densities, respectively. And in terms of $\mu_{B}$ and $\mu_{I}$ the
task of studying the phase structure of low-energy quark matter is
significantly simplified son ; he ; ak ; ekkz ; Mammarella:2015pxa ;
Andersen:2018nzq ; Ayala ; kyll .

In addition to isospin asymmetry (when $n_{I}\neq 0$), recently the possible
emergence of chiral asymmetry (or chiral imbalance) in quark matter, and its
manifestations has been drawing more attention. Usually chiral asymmetry, i.
e. unequal densities $n_{L}$ and $n_{R}$ of all left- and all right-handed
quarks of a system, is characterized by the quantity $n_{5}$, called the
chiral density, $n_{5}\equiv n_{R}-n_{L}$. Chiral imbalance can be generated
dynamically due to the Adler-Bell-Jackiw anomaly and the interaction of quarks
with gauge field configurations, called sphalerons, in high temperature
environment, for example in heavy ion collisions andrianov ; cao ; braguta ;
braguta2 ; Farias . Let us note that in the most general case, the chiral
densities of $u$ and $d$ quarks separately, namely $n_{u5}=n_{uR}-n_{uL}$ and
$n_{d5}=n_{dR}-n_{dL}$, are not necessarily equal. In this case, an additional
chiral isospin density emerges, denoted by $n_{I5}=n_{u5}-n_{d5}$. This chiral
isospin density reflects the unequal densities of left- and right-handed $u$
and $d$ quarks, contributing to the overall chiral asymmetry of quark matter.
The presence of $n_{I5}$ density further enriches the phase structure of dense
quark matter, encompassing the effects of both chiral and isospin imbalances.
It is also important to note that, as it was discussed in Ref. Khunjua:2019lbv
, the chiral imbalance can be created not only in the medium with high
temperature, but strong magnetic fields can also induce the emergence of
chiral and chiral isospin densities even in cold (with zero or not so large
values of temperatures) dense quark matter, for example, in magnetars, i.e.
neutron stars, which have exceptionally strong magnetic fields.

The presence of baryon density ($n_{B}\neq 0$), isospin asymmetry ($n_{I}\neq
0$), as well as two types of chiral imbalance, when $n_{5}\neq 0$ and/or
$n_{I5}\neq 0$, introduces four additional parameters beyond temperature that
govern the behavior of dense quark matter. As a consequence, baryonic matter
composed of $u$ and $d$ quarks can be characterized by four additional
physical parameters: baryon chemical potential $\mu_{B}$, isospin chemical
potential $\mu_{I}$, chiral chemical potential $\mu_{5}$, and chiral isospin
chemical potential $\mu_{I5}$. Then the most general
$(\mu_{B},\mu_{I},\mu_{5},\mu_{I5})$-phase diagram of the simplest NJL model
with massless quarks, in which the phenomenon of color superconductivity was
not taken into account, has been explored in the mean-field approximation in
Refs. kkz18 ; kkz18-2 . (The predictions of this simplest NJL model are
equivalent to considering physical processes within the framework of QCD, but
only in the region of rather small energies and/or baryon densities, i.e. at
$\mu_{B}\lesssim 1$ GeV.) This investigation demonstrated that only two
nontrivial phases are allowed for dense quark matter in this scenario: (i)
chiral symmetry breaking (CSB) phase and (ii) charged pion condensation (PC)
one. Furthermore, it was established that in this case there is duality
between CSB and charged PC phenomena. A more detailed investigation of the
impact of this dual symmetry on the phase structure of dense quark matter has
been performed in the framework of the NJL model in Refs. zhokhov1 ; zhokhov2
; kkz18 ; kkz18-2 ; khunjua ; khunjua2 ; khkz . These investigations have
unveiled a curious capability of duality to streamline and simplify the study
of the phase structure of quark matter, particularly when it is exposed to
multiple external conditions, such as different chemical potentials.

At rather high baryon densities, i.e. at $\mu_{B}\gtrsim 1$ GeV, quarks are
packed so closely together that they can pair up and form bound states called
diquarks. These diquarks can then condense, leading to a new state of matter
called color superconductivity (CSC). This phase can be obtained in the
framework of NJL model with four-fermion interaction terms responsible for the
diquark interaction channel (see, e.g., the reviews alford ). Let us note that
in the papers zhokhov1 ; zhokhov2 ; kkz18 ; kkz18-2 ; khunjua ; khunjua2 ;
khkz  duality has been established in the framework of simplest NJL model,
which does not contain diquark interaction channel, i.e. under assumption that
color superconductivity is absent. Then it was shown in terms of the NJL model
approximation that the duality between CSB and charged PC is respected by
color superconductivity phenomenon as well CSC .

Recall that the idea of PC phenomenon in baryon matter has been discussed for
the first time in the early seventies in terms of nuclear matter, i.e. in the
case when baryon density is much less than in dense quark matter. It was also
understood that pion condensation phenomenon could be a significant factor in
the physical processes of dense nuclear matter pionnuclear  (see also, for
example, the reviews pionnuclearVoskresensky ). Pion condensation in nuclear
matter still draws attention Voskresensky:2022gts ; Yasuda ; VoskresenskyPion
; Dohi:2021lbu . Though let us note that there are some subtleties, for
example, the s-wave charged PC is considered highly unlikely to be realized in
nuclear matter Ohnishi:2008ng , but it was argued that p-wave charged PC is
possible EricsonWeise . However, quite recently the s-wave pion condensation
in the isospin-symmetric nuclear matter was revisited in terms of several
models in Voskresensky:2022gts . Then charged pion condensation in quark
matter with isospin imbalance has been considered thoroughly in son ; he ; ak
; ekkz ; Mammarella:2015pxa ; Andersen:2018nzq ; kyll  and, in particular, it
was shown that charged PC appears when isospin chemical potential $\mu_{I}$ is
greater than pion mass. The research was performed in terms of different
effective models, lattice QCD, chiral perturbation theory, etc. The pion
condensation phenomenon got into spotlight recently concerning various
physical systems such as Kuznietsov:2021lax ; Brandt:2018bwq ;
Andersen:2018nzq ; Stashko:2023gnn  and neutron star mergers Vijayan:2023qrt .
The possibility of pion condensation in the early Universe and its
implications for gravitational wave signatures has been considered in Refs.
Vovchenko:2020crk ; Cao:2021gfk ; Wygas:2018otj . Moreover, it was shown that
rotation of the media could lead to charged PC phenomenon as well Liu:2017spl
; Cao:2019ctl .

As for quark matter, the charged PC phenomenon has been discussed in son ; he
; ak ; ekkz ; kyll , where it was shown that charged pion condensate could be
generated in quark matter with nonzero baryon density and isospin imbalance.
So charged PC has been predicted at nonzero baryon density (below, to
emphasize this fact, instead of PC we will sometimes use the notation PCd for
the phenomenon, or phase, with pion condensation in quark medium with a
nonzero baryon density). Though later it became clear that in the physical
point the region occupied by this phase is rather small and its prediction is
not robust, and additionally it vanishes when one requires that the medium is
in beta equilibrium and electrically neutral kyll , i.e. in the case
interesting in context of matter in cores of neutron stars. Later there have
been shown that there exist several conditions that could lead to charged pion
condensation in dense quark matter. There are a lot of details discussed,
e.g., in zhokhov1 ; zhokhov2 . One of these parameters is chiral imbalance in
the medium and, as it has been shown in the recent papers khkz ; khunjua2 , it
is a factor that hugely promotes charged PC phenomenon in dense quark matter
even under neutron star conditions. In these papers, the prediction of charged
PCd phase of chirally asymmetric quark matter was made at a rather wide range
of baryon chemical potential $\mu_{B}$. However, it must be especially
emphasized that in all the papers concerning charged PC the possibility of the
color superconductivity phenomenon and quark-quark condensation was not taken
into account, which significantly undermined confidence in the results of
these papers, especially when $\mu_{B}>1$ GeV. First of all, it was done due
to simplicity reasons. Surely, there were already a lot of condensates and
chemical potentials in the consideration beside CSC, and it was reasonable as
a first step. Further, although one knows that CSC phenomenon takes place at
large baryon density and it is probably a dominating phase in that region of
QCD phase diagram, there was an expectation that at least not all regions of
charged PCd phase would be replaced by color superconductivity. Although there
were absolutely no solid grounds to guarantee this, ignoring color
superconductivity in Refs. zhokhov1 ; zhokhov2 ; son ; he ; ak ; ekkz ; kyll
was made for reasons of simplicity and easier considerations of the charged
PCd phase in chirally imbalanced media. And taking into account color
superconductivity could easily change all the results of such an analysis in
the region of high baryon densities. So omitting the role of color
superconductivity leads to an incomplete understanding of PCd phase generation
phenomenon. So it is reasonable to revise the generation of charged pion
condensation in dense chirally asymmetric quark matter, taking into account
such an important phenomenon as color superconductivity at large baryon
densities. It is this problem that is the subject of this work, in which, for
this purpose, we study the phase structure of the NJL model with an additional
diquark interaction channel and in the presence of several chemical
potentials, $\mu_{B},\mu_{I},\mu_{5}$ and $\mu_{I5}$. We could go forward a
bit and say that it turned out that CSC does not spoil the prediction of
charged PC generation in dense quark matter. And surprisingly enough CSC phase
in a way shies away from charged PCd phase and they realize in different
regions of chemical potentials. The chemical potential space is so huge that
charged pion condensation in dense quark matter and color superconductivity
phenomenon has its own niche in the phase diagram and does not bother each
other. That shows that the bigger the space of chemical potentials, i.e. the
more diverse conditions the quark matter has, the more interesting and replete
the phase diagram is.

The paper is organized as follows. In Sec. II a (3+1)-dimensional NJL model
with two massless quark flavors ($u$ and $d$ quarks) that includes four kinds
of chemical potentials, $\mu_{B},\mu_{I},\mu_{I5},\mu_{5}$, are introduced. In
addition to usual quark-antiquark channels, the model contains the diquark
interaction one and is intended to describe the phenomenon of color
superconductivity in a dense quark medium. Furthermore, the symmetries of the
model are discussed and its thermodynamic potential (TDP) is presented in the
mean-field approximation. In Sec. III the phase structure of this NJL model is
investigated in the chiral limit. In Sec. IV summary and conclusions are
given.

##  II The model and its thermodynamic potential

Our investigation employs NJL type model with two quark flavors with
Lagrangian containing the interaction of quarks both in the quark–antiquark
and scalar diquark channels,

$\displaystyle
L=\bar{q}\Big{[}\gamma^{\nu}i\partial_{\nu}-m\Big{]}q+G\Big{[}(\bar{q}q)^{2}+(\bar{q}i\gamma^{5}\vec{\tau}q)^{2}\Big{]}+H\sum_{A=2,5,7}[\overline{q^{c}}i\gamma^{5}\tau_{2}\lambda_{A}q][\bar{q}i\gamma^{5}\tau_{2}\lambda_{A}q^{c}].$
(1)

Here, $q$ is a flavor doublet of $u$ and $d$ quarks, color triplet as well as
a four-component Dirac spinor (the corresponding indexes are omitted in Eq.
(1)). The charge-conjugated spinor $q^{c}$ is defined as $q^{c}=C\bar{q}^{T}$,
where $C=i\gamma^{2}\gamma^{0}$ is the charge conjugation matrix. Current
(bare) mass of the $u$ and $d$ quarks is denoted by $m$, Pauli matrices
$\tau_{a}$ act in flavor space, while Gell-Mann matrices $\lambda_{A}$ act in
color space.

The Lagrangian (1) is invariant with respect to transformations from the color
SU(3)c and baryon U(1)B groups. At $m=0$, it is also invariant under the
chiral SU(2)${}_{L}\times$ SU(2)R group. However, when $m\neq 0$, this
symmetry group is reduced to a diagonal isospin subgroup SU(2)I with
generators $I_{k}=\tau_{k}/2$, where $k=1,2,3$. The electric charge $Q$ and
baryon charge $B$ are also conserved quantities in this model. The electric
charge is given by $Q=I_{3}+B/2$, where $I_{3}$ is the third component of the
isospin operator, and $B$ is the baryon charge operator. These quantities are
unit matrices in color space, but in flavor space they are $Q={\rm
diag}(2/3,-1/3)$, $I_{3}={\rm diag}(1/2,-1/2)$, and $B={\rm diag}(1/3,1/3)$.
In the absence of more rigorous theoretical or experimental prescriptions we
will employ the following value for diquark coupling $H\approx 0.75G$. That
could be obtained from QCD in one-gluon exchange approximation and using Fierz
transformation in color current interaction buballa ; alford .

To effectively utilize the model (1) for investigating the properties of dense
quark matter, it is necessary to augment Lagrangian (1) with terms that
incorporate chemical potentials,

$\displaystyle L_{dense}=L+\bar{q}{\cal M}\gamma^{0}q\equiv
L+\bar{q}\left[\frac{\mu_{B}}{3}+\frac{\mu_{I}}{2}\tau_{3}+\frac{\mu_{I5}}{2}\gamma^{5}\tau_{3}+\mu_{5}\gamma^{5}\right]\gamma^{0}q,$
(2)

where the chemical potential matrix ${\cal M}$ encompasses four distinct
chemical potential terms, denoted by $\mu_{B}$, $\mu_{I}$, $\mu_{5}$, and
$\mu_{I5}$, that allow for the description of quark matter with non-zero
baryon, isospin, chiral, and chiral-isospin densities, respectively. In the
case of all nonzero chemical potentials in Eq. (2), $SU(2)_{I}$ at $m\neq 0$
is not the symmetry group of $L_{dense}$. In this case, due to the presence of
the $\mu_{I}$ term, the Lagrangian (2) instead exhibits symmetry under the
flavor $U(1)_{I_{3}}$ group, represented by
$q\rightarrow\exp(\mathrm{i}\alpha\tau_{3}/2)q$. Nonetheless, in the chiral
limit ($m=0$), an additional symmetry arises,
$U(1)_{AI_{3}}:q\rightarrow\exp(\mathrm{i}\alpha\gamma^{5}\tau_{3})q$.

In order to investigate the phase diagram of the system (2), we need to derive
its thermodynamic potential, for example, within the mean-field approximation.
To achieve this, let us consider the linearized version, denoted as ${\cal
L}$, of Lagrangian (2) that involves auxiliary bosonic fields $\sigma(x)$,
$\pi_{a}(x)$ and $\Delta_{A}(x)$,

$\displaystyle{\cal L}\displaystyle$ $\displaystyle=$
$\displaystyle\bar{q}\Big{[}\gamma^{\nu}i\partial_{\nu}+{\cal
M}\gamma^{0}-\sigma-
m-i\gamma^{5}\vec{\pi}\vec{\tau}\Big{]}q-\frac{1}{4G}\Big{[}\sigma^{2}+\pi_{a}^{2}\Big{]}$
(3) $\displaystyle-$
$\displaystyle\frac{1}{4H}\Delta^{*}_{A}\Delta_{A}-\frac{\Delta^{*}_{A}}{2}[\overline{q^{c}}i\gamma^{5}\tau_{2}\lambda_{A}q]-\frac{\Delta_{A^{\prime}}}{2}[\bar{q}i\gamma^{5}\tau_{2}\lambda_{A^{\prime}}q^{c}].$

In Eq. (3) summation over repeated indices $a=1,2,3$ and $A,A^{\prime}=2,5,7$
is assumed implicitly. It is straightforward to demonstrate the equivalence of
Lagrangians (2) and (3) by employing the equations of motion for auxiliary
bosonic fields, which are given by

$\displaystyle\sigma(x)=-2G(\bar{q}q),~{}~{}\pi_{a}(x)=-2G(\bar{q}i\gamma^{5}\tau_{a}q),~{}~{}\Delta_{A}(x)\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!-2H(\overline{q^{c}}i\gamma^{5}\tau_{2}\lambda_{A}q),~{}~{}\Delta^{*}_{A}(x)=-2H(\bar{q}i\gamma^{5}\tau_{2}\lambda_{A}q^{c}).$
(4)

As it is evident from (4), the mesonic fields $\sigma(x)$ and $\pi_{a}(x)$ are
real, i. e. satisfying the condition $(\sigma(x))^{\dagger}=\sigma(x)$ and
$(\pi_{a}(x))^{\dagger}=\pi_{a}(x)$, where the superscript $\dagger$ denotes
hermitian conjugation. In contrast, the diquark fields $\Delta_{A}(x)$ are
complex scalars, satisfying $(\Delta_{A}(x))^{\dagger}=\Delta_{A}^{*}(x)$. The
real fields $\sigma(x)$ and $\pi_{a}(x)$ form color singlets, while the scalar
diquarks $\Delta_{A}(x)$ constitute a color antitriplet $\bar{3}_{c}$ under
the group SU(3)c. Notably, the auxiliary bosonic field $\pi_{3}(x)$
corresponds to the real $\pi^{0}(x)$ meson, while the physical $\pi^{\pm}(x)$
mesons are formed by combining other composite fields from equation (4),
$\pi^{\pm}(x)=(\pi_{1}(x)\mp i\pi_{2}(x))/\sqrt{2}$. There occurs a
spontaneous breakdown of the color symmetry in the model (2) when any of the
scalar diquark fields acquires a nonzero expectation value in the ground
state, $\langle\Delta_{A}(x)\rangle\neq 0$.

In the mean-field approximation the effective action ${\cal
S}{\rm{eff}}(\sigma,\pi_{a},\Delta_{A},\Delta^{*}_{A^{\prime}})$ of the model
(2) can be expressed using the path integral over quark fields:

$\displaystyle\exp(i{\cal
S}_{\rm{eff}}(\sigma,\pi_{a},\Delta_{A},\Delta^{*}_{A^{\prime}}))=N^{\prime}\int[d\bar{q}][dq]\exp\Bigl{(}i\int{\cal
L}\,d^{4}x\Bigr{)},$ (5)

where

$\displaystyle{\cal
S}_{\rm{eff}}(\sigma,\pi_{a},\Delta_{A},\Delta^{*}_{A^{\prime}})=-\int
d^{4}x\left[\frac{\sigma^{2}+\pi^{2}_{a}}{4G}+\frac{\Delta_{A}\Delta^{*}_{A}}{4H}\right]+\tilde{\cal
S}_{\rm{eff}},$ (6)

and $N^{\prime}$ is a normalization constant. The quark contribution to the
effective action, represented by the term $\tilde{\cal S}_{\rm{eff}}$ in (6),
is expressed as follows:

$\displaystyle\exp(i\tilde{\cal
S}_{\rm{eff}})=N^{\prime}\int[d\bar{q}][dq]\exp\Bigl{(}i\int\Big{[}\bar{q}Dq-\frac{\Delta^{*}_{A}}{2}[\overline{q^{c}}i\gamma^{5}\tau_{2}\lambda_{A}q]-\frac{\Delta_{A^{\prime}}}{2}[\bar{q}i\gamma^{5}\tau_{2}\lambda_{A^{\prime}}q^{c}]\Big{]}d^{4}x\Bigr{)},$
(7)

The notation has been used, where $1\\!\\!{\rm I}_{3_{c}}$ is the unit
operator in the tree-dimensional color space,

$\displaystyle D=\big{(}\gamma^{\nu}i\partial_{\nu}+{\cal
M}\gamma^{0}-\sigma(x)-m-i\gamma^{5}\vec{\pi}(x)\vec{\tau}\big{)}\cdot
1\\!\\!{\rm I}_{3_{c}},$ (8)

Starting from Eqs. (6) and (7), one can define the thermodynamic potential
(TDP) $\Omega(\sigma,\pi_{a},\Delta_{A},\Delta^{*}_{A^{\prime}})$ of the model
(2). Indeed, this quantity is defined by the following relation,

${\cal
S}_{\rm{eff}}~{}\bigg{|}_{~{}\sigma,\pi_{a},\Delta_{A},\Delta^{*}_{A^{\prime}}=\rm{const}}=-\Omega(\sigma,\pi_{a},\Delta_{A},\Delta^{*}_{A^{\prime}})\int
d^{4}x.$ (9)

The ground state expectation values of the bosonic fields (4), i.e. the
quantities
$\langle\sigma(x)\rangle,~{}\langle\pi_{a}(x)\rangle,~{}\langle\Delta_{A}(x)\rangle,~{}\langle\Delta^{*}_{A^{\prime}}(x)\rangle$,
are solutions obtained by minimizing the thermodynamic potential
$\Omega(\sigma,\pi_{a},\Delta_{A},\Delta^{*}_{A^{\prime}})$. They satisfy the
gap equations for the TDP (usually are the coordinates of the global minimum
point (GMP) of $\Omega$ as a function of
$\sigma,\pi_{a},\Delta_{A},\Delta^{*}_{A^{\prime}}$)

$\displaystyle\frac{\partial\Omega}{\partial\pi_{a}}=0,~{}~{}~{}~{}~{}\frac{\partial\Omega}{\partial\sigma}=0,~{}~{}~{}~{}~{}\frac{\partial\Omega}{\partial\Delta_{A}}=0,~{}~{}~{}~{}~{}\frac{\partial\Omega}{\partial\Delta^{*}_{A^{\prime}}}=0.$
(10)

At nonzero bare quark mass, $m\neq 0$, the chiral condensate
$\langle\sigma(x)\rangle$ is always nonzero 111See, e.g., the gap equations
(13) and (14) of Ref. Ebert , which at $m\neq 0$ have single solution
$\langle\sigma(x)\rangle\neq 0$ and $\langle\pi_{3}(x)\rangle=0$.. In
contrast, in the chiral limit, $m=0$, it is possible that chiral condensate
vanishes, leading to the chiral symmetric phase. However, at $m=0$ the model
has a more complex phase structure, because its phase diagram can contain both
a region with $\langle\sigma(x)\rangle=0$, which corresponds to a chiral
symmetrical phase, and a region with $\langle\sigma(x)\rangle\neq 0$, and in
the last case chiral symmetry is broken spontaneously. In addition, both at
$m=0$ and $m\neq 0$, the global minimum point of the TDP with
$\langle\pi^{\pm}(x)\rangle\neq 0$ is allowed, which corresponds to the
charged pion condensation (PC) phase. Finally, recall that if one of the
scalar diquark fields acquires a nonzero ground state expectation value,
$\langle\Delta_{A}(x)\rangle\neq 0$, a new phase emerges with spontaneously
broken color $SU(3)_{c}$ symmetry, known as the color superconducting (CSC)
phase.

The Lagrangian (2) and the effective action (6) are both invariant under the
color $SU(3)_{c}$ group. This invariance implies that the TDP (9) depends on
the combination
$\Delta_{2}\Delta^{*}_{2}+\Delta_{5}\Delta^{*}_{5}+\Delta_{7}\Delta^{*}_{7}=\Delta^{2}$,
where $\Delta$ is a real quantity. In the chiral limit, where the model
exhibits a $U_{I_{3}}(1)\times U_{AI_{3}}(1)$ invariance, the TDP (9)
effectively depends on the combinations $\sigma^{2}+\pi_{3}^{2}$ and
$\pi_{1}^{2}+\pi_{2}^{2}$, in addition to $\Delta$. It is reasonable to assume
that $\pi_{2}=\pi_{3}=0$, this allows us to study the TDP as a function of
only three variables: $\sigma$, $\pi_{1}$, and $\Delta$. Therefore,
$\Omega(\sigma,\pi_{1},\Delta)$ can be derived by setting
$\Delta_{2}=\Delta_{2}^{*}=\Delta$, $\Delta_{5}=\Delta_{7}=0$, and
$\pi_{2}=\pi_{3}=0$ in Eqs. (6) and (7). For simplicity, we also assume that
all auxiliary bosonic fields are independent of the space-time coordinates.

It can be shown that the contribution of the blue $q_{b}$ quarks in the
expression (7) is factorized, i.e.

$\displaystyle\exp(i\tilde{\cal S}_{\rm{eff}})$ $\displaystyle=$
$\displaystyle
N^{\prime}\int[d\bar{q}_{b}][dq_{b}]\exp\Bigl{(}i\int\Big{[}\bar{q}_{b}D^{+}q_{b}\Big{]}\Bigr{)}$
(11) $\displaystyle\times$
$\displaystyle\int[d\overline{Q}][dQ]\exp\Bigl{(}i\int\Big{[}\overline{Q}\big{(}D^{+}\cdot
1\\!\\!{\rm
I}_{2_{c}}\big{)}Q-\frac{\Delta}{2}[\overline{Q^{c}}i\gamma^{5}\tau_{2}\sigma_{2}Q]-\frac{\Delta}{2}[\overline{Q}i\gamma^{5}\tau_{2}\sigma_{2}Q^{c}]\Big{]}d^{4}x\Bigr{)},$

where quark fields $q_{b}$ and $Q$ represents the flavor doublet of blue
quarks and the flavor ($u$ and $d$) and color (red and green) quark doublet,
respectively. And operator $1\\!\\!{\rm I}_{2_{c}}$ denotes the unit operator
in the two-dimensional (of red and green quarks) color space. Additionally,
the notation $D^{+}$ is used to represent the operator that is explicitly
defined below in Eq. (14).

After carrying out the functional integrations in Eq. (11) (the integration
over the blue quarks $q_{b}$ is straightforward, and the one for quarks $Q$
can be found, e.g., in Appendix of Ref. CSC ) one can get

$\displaystyle\exp(i\tilde{\cal S}_{\rm{eff}})$ $\displaystyle=$
$\displaystyle N^{\prime}\det D^{+}\cdot{\det}^{1/2}(Z),$ (12)

where

$Z=\left(\begin{array}[]{cc}D^{+}\cdot 1\\!\\!{\rm I}_{2_{c}},&-K\\\
~{}-K~{}~{}~{}~{},&D^{-}\cdot 1\\!\\!{\rm I}_{2_{c}}\end{array}\right),$ (13)

and

$\displaystyle D^{+}=i\gamma^{\nu}\partial_{\nu}-m+{\cal
M}\gamma^{0}-\Sigma,~{}~{}~{}\Sigma=\sigma+i\gamma^{5}\pi_{1}\tau_{1},$
$\displaystyle D^{-}=i\gamma^{\nu}\partial_{\nu}-m-\gamma^{0}{\cal
M}-\Sigma,~{}~{}~{}K=i\Delta\gamma^{5}\tau_{2}\sigma_{2}.$ (14)

It is important to note that $Z$-matrix elements in Eqs. (13) and (14) are
operators that act both in the two-dimensional color and flavor spaces, as
well as in the four-dimensional spinor and coordinate spaces. Then, one can
get from Eqs. (6) and (12) the expression for effective action

${\cal S}_{\rm{eff}}(M,\pi_{1},\Delta)=-\int
d^{4}x\left[\frac{(M-m)^{2}+\pi^{2}_{1}}{4G}+\frac{\Delta^{2}}{4H}\right]-\frac{i}{2}\ln\det(Z)-i\ln\det(D^{+}),$
(15)

where the gap $M\equiv\sigma+m$. The last term of Eq. (15), which does not
depend on $\Delta$, was calculated in our recent paper kkz18  (see there Eqs.
(16)-(21)),

$\displaystyle
i\ln\det(D^{+})=i\int\frac{d^{4}p}{(2\pi)^{4}}\ln\Big{[}\big{(}\eta^{4}-2a_{+}\eta^{2}+b_{+}\eta+c_{+}\big{)}\big{(}\eta^{4}-2a_{-}\eta^{2}+b_{-}\eta+c_{-}\big{)}\Big{]}\int
d^{4}x,$ (16)

where $\eta=p_{0}+\mu$, $|\vec{p}|=\sqrt{p_{1}^{2}+p_{2}^{2}+p_{3}^{2}}$ and

$\displaystyle a_{\pm}$ $\displaystyle=$ $\displaystyle
M^{2}+\pi_{1}^{2}+(|\vec{p}|\pm\mu_{5})^{2}+\nu^{2}+\nu_{5}^{2};~{}~{}b_{\pm}=\pm
8(|\vec{p}|\pm\mu_{5})\nu\nu_{5};$ $\displaystyle c_{\pm}$ $\displaystyle=$
$\displaystyle
a_{\pm}^{2}-4\nu^{2}\left(M^{2}+(|\vec{p}|\pm\mu_{5})^{2}\right)-4\nu_{5}^{2}\left(\pi_{1}^{2}+(|\vec{p}|\pm\mu_{5})^{2}\right)-4\nu^{2}\nu_{5}^{2}$
(17)

(we also use in Eqs. (16), (17) and below the notations $\mu=\mu_{B}/3$,
$\nu=\mu_{I}/2$ and $\nu_{5}=\mu_{I5}/2$). The next undefined term of Eq. (15)
is the following

$\displaystyle\frac{i}{2}\ln\det(Z)=i\int\frac{d^{4}p}{(2\pi)^{4}}\ln\det
L(p)\int d^{4}x,$ (18)

where $L(p)$ is a matrix acting in 4-dimensional spinor space. Its matrix
elements were obtained in our previous work CSC  (the same matrix $L(p)$ also
arises when studying the phase structure of the two-color NJL model 2color ).
Here we only need the eigenvalues $\tilde{\lambda}_{i}(p)$ of $L(p)$, where
$i=1,...,4$, with the help of which we can find its determinant,

$\displaystyle\det
L(p)=\tilde{\lambda}_{1}(p)\tilde{\lambda}_{2}(p)\tilde{\lambda}_{3}(p)\tilde{\lambda}_{4}(p),$
(19)

appearing on the right side of equality (18). Then, taking into account the
definition (9), one can get the expression for TDP of the model,

$\displaystyle\Omega(M,\pi_{1},\Delta)$ $\displaystyle=$
$\displaystyle\left[\frac{(M-m)^{2}+\pi^{2}_{1}}{4G}+\frac{\Delta^{2}}{4H}\right]+i\int\frac{d^{4}p}{(2\pi)^{4}}\ln\big{[}\tilde{\lambda}_{1}(p)\tilde{\lambda}_{2}(p)\tilde{\lambda}_{3}(p)\tilde{\lambda}_{4}(p)\big{]}$
(20) $\displaystyle+$ $\displaystyle
i\int\frac{d^{4}p}{(2\pi)^{4}}\ln\Big{[}\big{(}\eta^{4}-2a_{+}\eta^{2}+b_{+}\eta+c_{+}\big{)}\big{(}\eta^{4}-2a_{-}\eta^{2}+b_{-}\eta+c_{-}\big{)}\Big{]}.$

Here the notations (17) are used and

$\displaystyle\widetilde{\lambda}_{1,2}(p)=\lambda_{1,2}(p)\Big{|}_{|\vec{p}|\to|\vec{p}|-\mu_{5}},~{}~{}\widetilde{\lambda}_{3,4}(p)=\lambda_{3,4}(p)\Big{|}_{|\vec{p}|\to|\vec{p}|+\mu_{5}}$
(21)

where

$\displaystyle\lambda_{1,2}(p)=N_{1}\pm
4\sqrt{K_{1}},~{}~{}\lambda_{3,4}(p)=N_{2}\pm 4\sqrt{K_{2}},$ (22)
$\displaystyle
N_{2}=N_{1}+16\mu\nu\nu_{5}|\vec{p}|,~{}~{}K_{2}=K_{1}+8\mu\nu\nu_{5}|\vec{p}|p_{0}^{4}-8\mu\nu\nu_{5}|\vec{p}|p_{0}^{2}\big{(}M^{2}+\pi_{1}^{2}+|\Delta|^{2}+|\vec{p}|^{2}+\mu^{2}+\nu^{2}-\nu_{5}^{2}\big{)},$
(23) $\displaystyle
K_{1}=\nu_{5}^{2}p_{0}^{6}-p_{0}^{4}\Big{[}2\nu_{5}^{2}\big{(}|\Delta|^{2}+\pi_{1}^{2}+M^{2}+|\vec{p}|^{2}+\nu^{2}+\mu^{2}-\nu_{5}^{2}\big{)}+4\mu\nu\nu_{5}|\vec{p}|\Big{]}+p_{0}^{2}\Big{\\{}\nu_{5}^{6}+2\nu_{5}^{4}\big{(}M^{2}-|\Delta|^{2}-\pi_{1}^{2}$
$\displaystyle-\nu^{2}-\mu^{2}-|\vec{p}|^{2}\big{)}+4\mu^{2}\nu^{2}\big{(}M^{2}+|\vec{p}|^{2}\big{)}+4|\vec{p}|\mu\nu\nu_{5}\big{(}|\Delta|^{2}+\pi_{1}^{2}+M^{2}+|\vec{p}|^{2}+\nu^{2}+\mu^{2}-\nu_{5}^{2}\big{)}$
$\displaystyle+\nu_{5}^{2}\Big{[}\big{(}|\Delta|^{2}+\pi_{1}^{2}+|\vec{p}|^{2}+\nu^{2}+\mu^{2}\big{)}^{2}+2|\vec{p}|^{2}M^{2}+M^{4}+2M^{2}\big{(}|\Delta|^{2}-\nu^{2}+\pi_{1}^{2}-\mu^{2}\big{)}\Big{]}\Big{\\}},$
(24) $\displaystyle N_{1}$ $\displaystyle=$ $\displaystyle
p_{0}^{4}-2p_{0}^{2}\Big{[}|\Delta|^{2}+\pi_{1}^{2}+M^{2}+|\vec{p}|^{2}+\nu^{2}+\mu^{2}-3\nu_{5}^{2}\Big{]}+\nu_{5}^{4}-2\nu_{5}^{2}\Big{[}|\Delta|^{2}+\pi_{1}^{2}+|\vec{p}|^{2}+\nu^{2}+\mu^{2}-M^{2}\Big{]}$
(25) $\displaystyle-$ $\displaystyle
8\mu\nu\nu_{5}|\vec{p}|+\left(|\vec{p}|^{2}+M^{2}+\pi_{1}^{2}+|\Delta|^{2}-\mu^{2}-\nu^{2}\right)^{2}-4\left(\mu^{2}\nu^{2}-\pi_{1}^{2}\nu^{2}-|\Delta|^{2}\mu^{2}\right).$

Previous studies of the phase structure of two-color QCD (see, for example,
Ref. 2color ) have yielded exact expressions for the eigenvalues
$\tilde{\lambda}_{i}(p)$. Using Eq. (20) for the TDP, we are ready now to
discuss in the chiral limit, i.e. at $m=0$, the phase structure of the NJL
model (1)-(2) in the mean-field approximation.

Throughout the paper we use in numerical investigations of the TDP (20) the
soft cut-off regularization scheme when $d^{4}p\equiv dp_{0}d^{3}\vec{p}\to
dp_{0}d^{3}\vec{p}f_{\Lambda}(\vec{p})$. Here the cut-off function is

$\displaystyle
f_{\Lambda}(\vec{p})=\sqrt{\frac{\Lambda^{2N}}{\Lambda^{2N}+|\vec{p}|^{2N}}},$
(26)

and the parameter fit used is $G=4.79$ GeV-2, $\Lambda=638.8$ MeV and $N=5$.

##  III Charged pion condensation at non-zero baryon density

As discussed above in the Introduction, the PC phenomenon in dense quark
matter has been discussed in various conditions, for example in nuclear matter
pionnuclear ; pionnuclearVoskresensky , in matter with isospin imbalance, in
quark/baryonic matter with baryon density and isospin imbalance, then it was
shown that charged PC could be generated in quark matter with nonzero baryon
density he . But this prediction was not robust since later it was shown that
in the physical point (and in the absence of chiral asymmetry) charged PC
occupies only a rather small region of phase diagram and, additionally it
vanishes when one considers electrically neutral medium in beta equilibrium,
i.e. the matter in cores of neutron stars kyll . Then several scenarios where
PC is generated in the dense quark matter has been discussed. Among them there
has been shown that chiral imbalance, when $\mu_{5}$ and/or $\mu_{I5}$ are
nonzero, leads to generation of charged pion condensation in dense quark
matter, even under the neutron stars conditions khkz ; khunjua2 . Pion
condensation discussed in terms of nuclear matter takes place at rather low
baryon density and it cannot be challenged by CSC phenomenon, since the
densities are not high enough. Charged PC predicted in dense quark matter
takes place at higher baryon densities, in the realm of color
superconductivity phenomenon, and it is different story. However, in all the
papers concerning charged PC in dense quark matter the possibility of the CSC
phenomenon and quark-quark condensation was not considered just for
simplicity. Surely, there were a lot of condensates and chemical potentials in
the consideration beside it, and it was reasonable as a first step. Since at
large baryon densities the CSC phenomenon is believed to be an important
feature of the QCD phase diagram, it is of great interest to explore the
situation in more detail.

Figure 1: $(\mu_{5},\mu)$-phase diagrams of the model at $H=0.75G$. Left
panel: The case of $\nu_{5}=-0.35$ GeV and $\nu=0$. Right panel: The case of
$\nu_{5}=0.3$ GeV and $\nu=-0.29$ GeV. Here PCd denotes the charged pion
condensation phase, CSB and CSC mean respectively the chiral symmetry breaking
and color superconducting phases, “sym“ is the symmetric phase.

Figure 2: Left panel: $(\nu,\mu)$-phase diagram of the model at $\nu_{5}=0.3$
GeV, $\mu_{5}=0$ and $H=0.75G$. Right panel: $(\nu,\nu_{5})$-phase diagram of
the model at $\mu=0.35$ GeV, $\mu_{5}=0.14$ GeV and $H=0.75G$. All the
notations are the same as in Fig. 1.

Figure 3:  $(\nu,\mu)$-phase diagrams of the model. Left panel: The case of
$\nu_{5}=0.3$ GeV and $\mu_{5}=-0.25$ GeV at $H=G$. Right panel: The case of
$\nu_{5}=0.3$ GeV and $\mu_{5}=0.2$ GeV at $H=0.75G$. All the notations are
the same as in Fig. 1.

Let us discuss the phase diagram and charged pion condensation in chirally
asymmetric dense quark matter in the light of color superconductivity
phenomenon. Upon tedious investigations of the TDP (20) (we study only the
case $m=0$), it turned out that in most cases the charged PCd phase 222To
especially emphasize that we are dealing with the phenomenon of charged PC in
medium with nonzero baryonic density, we will below use the abbreviation PCd
of this phase in all phase portraits of the model. is not affected by color
superconductivity at all, as one can see in Fig. 1 (left) where the
$(\mu_{5},\mu)$-phase diagram at $\nu_{5}=-0.35$ GeV and $\nu=0$ is drawn.
Note that all $(\mu_{5},\mu)$-diagrams are very similar for various fixed
values of $\nu_{5}$ and $\nu$. And it is clear that in order for the charged
PCd phase to be generated, $|\nu_{5}|$ should be larger than $0.1$-$0.15$ GeV
(see, e.g., in Figs. 2 and 3), does not matter positive or negative, otherwise
the CSB phase would prevail in the phase structure in this case, but it is not
the issue with color superconductivity, since the same is of course true in
the framework of NJL model without diquark interaction channel (see, e.g., in
Refs. kkz18 ; kkz18-2 ).

One can see from different cross-sections of the full
$(\mu,\nu,\mu_{5},\nu_{5})$-phase diagram that in the rather extensive range
of chemical potential space the CSC phase does not interfere with charged PCd
phase at all (see in Figs. 2 (left) and 2 (right) or 3 (right)), where the
($\nu$, $\mu$)-phase diagrams are presented. One can see in Fig. 2 (left) that
the CSC phase resembles an oval shaped region, but it occupies a different
from PCd phase region of chemical potential space. And does not meddle in
charged pion condensation phenomenon.

Let us try to see when it is possible that CSC phase could in principle
intervene in the charged PC phenomenon at nonzero baryon density. One can
explore the cross-sections of the full phase diagram similar to the one in
Fig. 1 (left) at various fixed values of $\nu$ and $\nu_{5}$ and behold, as we
pointed out earlier, that the pictures are very similar. But one could note
that in the case of $\nu\approx|\nu_{5}|$ there appears the band of CSC phase
that crosses the PCd phase, as one could see in Fig. 1 (right). Nonetheless,
one can see that in the region, where they interfere and compete, the charged
PCd phase is always more energetically favorable and the shape and the size of
charged PCd phase stays intact. Also let us note here that it is a rather
peculiar choice of chemical potentials, very fine-tuned one. When
$\nu=|\nu_{5}|$, the CSB and charged PCd phases are almost degenerate, it is a
region of the transition of these phases. And even in this specially chosen
chemical potential case the CSC phase does not undermine the stability of the
charged PCd phase in a bit. The whole charged PCd phase predicted in zhokhov1
; zhokhov2 ; kkz18 ; kkz18-2 ; khunjua ; khunjua2  previously, stays intact
even in this case.

The fact that at $\nu\approx|\nu_{5}|$ there is a region of phase transition
between charged PCd and CSB phases could be easily illustrated by another
cross-section of the full phase diagram, namely by the $(\nu,\nu_{5})$ one
depicted in Fig. 2 (right). Indeed, as it can be easily seen in Fig. 2
(right), the band of CSC phase crosses the region of charged PCd phase in a
rather small domain, where in addition the transition from CSB to charged PCd
phase, and vise versa, occurs, i.e. in the region where these two phases are
almost degenerate. It complies with Fig. 1 (right) where the same region is
depicted but in another cross-section of the full
$(\mu,nu,\mu_{5},\nu_{5})$-phase diagram of the model. So the main charged PCd
phase region in Fig. 2 (right), the region at smaller $\nu$, lies beyond and
is unbothered by the CSC phenomenon.

One can see from another angle that in the rather extensive range of chemical
potential space the CSC phase does not interfere with charged PCd phase at
all. And in those exceptional cases when the CSC and charged PCd phases could
occupy the same region in the space of chemical potentials, the charged PCd
phase is still realized in the ground state of the system. This property of
the phase structure of the model can be most successfully manifested in terms
of its various $(\nu,\mu)$ cross-sections. For example, contemplating the
figures (see Figs. 3 (right) or 2 (left)) one could see that at $\mu_{5}>0$
CSC phase peacefully shares the phase diagram with charged PCd phase and they
lie in different regions. (Remark that in Fig. 3 (right) CSC phase lies far
beyond the region occupied by charged PCd phase and even does not present at
the phase diagram. It lies in the region of $\mu>550$ MeV, i.e. outside the
scope of the figure.) But one can find values of $\mu_{5}$ when these phases
could go into clash with each other, e.g., for some values of $\mu_{5}<0$ the
regions occupied by CSC and charged PCd phases can slightly overlap. The
overlap region is still not extensive and these phases are elongated along
different directions, i.e. axes with different chemical potentials. Having
shown that there are such regions, it could be demonstrated that even if they
overlap, what is more important, the charged PCd phase is nowhere undermined
by CSC phase and turns out to be more energetically favorable in their
intersection. This property of the model is demonstrated by diagram in Fig. 3
(left). One should also notice that in Fig. 3 (left) in order to stress that
it is a quite universal prediction, the value of $H$ was chosen to be $G$, and
at this value we have the largest CSC phase and the biggest diquark condensate
values (though this value of the coupling constant $H$ cannot be now
determined precisely, it is believed to be the largest physically motivated
value for coupling constant of the quark-quark interaction channel). So in
this case diquark condensation is the most favourable and CSC phase is the
largest and it was chosen specially to show that even in this case the charged
PCd phase survives in full extent and could be realized without any
disturbance from CSC.

Let us briefly conclude this section:

  * •

Charged PC is generated by chiral imbalance and it is shown to be unbothered
by color superconductivity phenomenon in the whole range of baryon density.

  * •

Charged PC in dense quark matter and color superconductivity phenomena as a
rule occupies different regions of phase diagram, so one can say that they
peacefully coexist.

  * •

Even if one fine-tunes the values of chemical potentials in order to reach the
maximum crossing of charged PCd and CSC phases in the phase diagram and choose
the largest possible coupling constant $H$ in the quark-quark interaction
channel, the CSC phase does not substitute charged PCd phase in any region of
phase diagram and charged PCd phase stays intact even in this specially chosen
case.

##  IV Summary and conclusions

Pion condensation in dense quark matter has been investigated in several
studies, including, e.g., Refs. son ; he ; ak ; ekkz ; kyll . These studies
demonstrated that charged pion condensation could arise in quark matter with
nonzero baryon density and isospin imbalance. However, subsequent studies
revealed that the region occupied by the charged PCd phase is negligibly small
under real physical conditions (in the physical point, i. e. physical quark
masses), and it vanishes completely when imposing beta-equilibrium and
electrical neutrality conditions, as relevant to matter in the cores of
neutron stars.

Subsequent investigations (see, e.g., in Refs. kkz18 ; kkz18-2 ) revealed
several conditions that could facilitate charged PC in dense quark matter. One
of them, namely the chiral imbalance of dense quark medium, when $\mu_{5}\neq
0$ and/or $\mu_{I5}\neq 0$, is a key factor promoting this phenomenon at a
wide range of baryon chemical potentials. However, in these studies, the
possibility of color superconductivity and quark-quark condensation was often
disregarded for the sake of clarity and simplicity. This simplification was
reasonable given the complexity of the problem, which required consideration
of multiple condensates and chemical potentials.

It was initially assumed that charged PCd phase would not be entirely replaced
by CSC one. However, there was no guarantee of this, and the survival of the
charged CPd phase was primarily based on the argument that it encompasses a
substantial portion of the phase diagram. But strictly speaking color
superconductivity was neglected in the previous studies solely due to the
simplicity reasons, and the prediction of charged PCd phase at nonzero baryon
density in reality could be fully swept away by color superconductivity,
especially at larger baryon densities. Hence, it was reasonable and crucial to
revise the generation of charged pion condensation in chirally asymmetric
dense quark matter, when the appearance of color superconductivity is also
allowed in it.

In this paper, we investigate the phase structure of a generalized massless
Nambu-Jona-Lasinio model (1) in the presence of four chemical potentials:
baryon $\mu_{B}=3\mu$, isospin $\mu_{I}=2\nu$, chiral isospin
$\mu_{I5}=2\nu_{5}$, and chiral $\mu_{5}$ chemical potentials and at zero
temperature. The model describes interactions between quarks in quark-
antiquark and diquark channels and its primary objective is to provide a
theoretical framework for studying the properties of dense quark matter.

It has been shown that color superconductivity phenomenon, the phenomenon
believed to be dominant at large baryon density, does not hinder the
generation of charged pion condensation in dense quark matter by chiral
imbalance. If $\mu_{5}\neq 0$ and/or $\mu_{I5}\neq 0$, then charged PCd phase
in reality is never replaced by CSC phase even at large baryon density and
even if one takes values of coupling constant $H$ in the whole range of
physically motivated values. Color superconductivity phenomenon and charged
pion condensation in dense quark matter dwell in different regions of chemical
potential space and almost do not interfere with each other. It is rather
unexpected finding. It was expected that probably some regions occupied by
charged PCd phase survive the inclusion of color superconductivity phenomenon
in the consideration. But the expectation was that pion condensation phase
would considerably dwindle and would be vastly replaced by CSC phase at least
for rather large baryon density. One can say that charged pion condensation in
dense quark matter with chiral imbalance is shown to exist in a wide range of
baryon densities and does not compete in a way with color superconductivity
phenomenon. These two phenomena peacefully dwell in the QCD phase diagram and
make it even more rich and interesting. It is shown in the framework of
effective model, needless to say of course it would be great to probe this
phenomenon from first principles but due to sign problem it is out of range of
current capabilities of lattice QCD simulations.

We expect that our results may shed light on the intricate phase structure of
dense quark matter with isospin and chiral (isospin) imbalances, potentially
providing valuable insights into the physics of intermediate-energy heavy ion
collisions, as well as neutron star cores and its mergers. However, to make
the predictions more realistic, we plan in the future to consider the problem
taking into account the nonzero quark mass, as well as the electrical
neutrality of the quark medium being in beta equilibrium.

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