# Holographic model with power-law Maxwell field for color superconductivity

Cao H. Nam<EMAIL_ADDRESS>Phenikaa Institute for Advanced
Study and Faculty of Fundamental Sciences, Phenikaa University, Yen Nghia, Ha
Dong, Hanoi 12116, Vietnam

###### Abstract

Studying the color superconductivity (CSC) phase is important to understand
the physics in the core of the neutron stars which is the only known context
where the CSC phase might appear due to the gravitational force squeezing the
matter to a sufficiently high density. We propose a simple holographic dual
description of the CSC phase transition in the realistic Yang-Mills theory
with the power-law Maxwell field. We find the CSC phase transition with the
large color number in the deconfinement phase, which is not found in the case
of the usual Maxwell field, if the power parameter characterizing for the
power-law Maxwell field is sufficiently lower than one but above $1/2$ and the
chemical potential is above a critical value. However, the power parameter is
not arbitrary below one because when this parameter is sufficiently far away
from one it leads to the occurrence of the CSC state in the confinement phase
which is not compatible with a nonzero vacuum expectation value of the color
nonsinglet operator.

##  I Introduction

In analogy to the electron Cooper pairs in condensed matter physics, at
sufficiently high density (or the chemical potential) and low-temperature
quarks with the attractive interaction in the color antitriplet channel are
expected to form the Cooper pairs (associated with the diquark operator) near
the Fermi surface. The quark Cooper pairs are not color singlets and hence
their condensation would break spontaneously the $SU(3)_{C}$ local gauge
symmetry of quantum chromodynamics (QCD) by which the gluons acquire a mass
(corresponding to color Meissner effects) through the Higgs mechanism where
the quark Cooper pairs play the role of the Higgs particles. This phase is
well-known as the color superconductivity (CSC) phase or the Higgs phase of
QCD, suggested first by the authors in Collins1975 . For a nice review see
Alford2008 .

The only known conditions of the density and temperature where the CSC phase
might appear are in the core of the neutron stars where the matter density can
reach up to ten times of the nuclear saturation density. Therefore, the
investigation of the CSC phase is important to understand the physics of
neutron stars as well as compact stars. Here, the effects on the observations
of the neutron stars caused by the presence of the CSC phase could allow us to
resolve the deviations between the experimental measurements and the
theoretical predictions.

Due to the asymptotic freedom, it is possible to study the CSC phase
transition at very high densities where QCD becomes weakly coupled. Since the
condensation pattern can be exactly calculated using the perturbative approach
Son1999 . However, for sufficiently high densities relevant to the real
context such as the core of the neutron stars which are the orders of the
strong coupling scale, perturbative QCD is not able to be applied although
there is the existence of the deconfined quarks. Most QCD investigations in
the nonperturbative regime are numerically performed by lattice gauge theory
which is difficult due to the sign problem of the fermion determinant.111Note
that, the sign problem can be avoided by restricting the calculations to the
imaginary chemical potential Roberge1986 ; Bilgici2008 ; Bilgici2010 ;
Ghoroku2020 .

In the last two decades, the gauge/gravity duality or the holographic approach
has provided a powerful tool to explore the strongly coupled quantum field
theories using the weakly coupled gravitational dual theories Maldacena ;
Witten ; Gubser . Based on this approach, a bottom-up holographic model has
been constructed to study the CSC phase transition in the Yang-Mills (YM)
theory Ghoroku2019  which extended the work in Fadafan2018  with considering
the backreaction of the matter part on the spacetime geometry and the work in
Basu2011  with the complex scalar field. The holographic investigations of the
CSC phase transition have lately received attention in the presence of the
higher derivative corrections Nam2021 ; Fadafan2021  and with including a
dilute gas of instantons which is introduced to study the nuclear matter
Ghoroku2021 .

It was pointed out by the authors in Ghoroku2019  that the CSC phase
transition can occur for $N_{c}$ (where $N_{c}$ is the color number of quark)
to be small. But, the CSC phase transition could be not found in the case of
$N_{c}\geq 2$ and hence this holographic model does not provide a
gravitational dual description for the CSC phase transition of the realistic
YM theory (i.e. $N_{c}\geq 2$). This restriction can be resolved by
considering Einstein-Gauss-Bonnet (EGB) gravity Nam2021 . Unfortunately, in
order to indicate the CSC phase transition for $N_{c}\geq 2$, the magnitude of
the GB coupling parameter is required to be rather large, which leads to the
violation of the causality bound of the boundary field theory Camanho2010 ;
Escobedo2010  and is beyond the region of the classical gravity.

The goal of the present work is to construct a holographic model with a power-
law Maxwell field which is simple but allows us to investigate the CSC phase
transition in the realistic YM theory. It should be noted here that the power-
law Maxwell electrodynamics is motivated by the conformal invariance of the
action of the $U(1)$ gauge field in arbitrary dimensions Hassaine2007  and is
a type of nonlinear electrodynamics which can lead to the regular black hole
solutions Ayon-Beato98 ; Ayon-Beato1999a ; Ayon-Beato1999b ; Bronnikov2001 ;
Dymnikova2004 ; Schee2015 ; Nam2018 ; Singh2018 ; SGGosh2019 ; Nam2019  and
also are used to study holographic superconductors Pan2011 ; Gangopadhyay2012
; Lai2015 ; Salahi2016 ; Ghorai2016 ; Jiang2016 ; Ghazanfari2018 ; Nam2019b ;
TZhang2022 . In Sect. II, we discuss the main obstacle in attempting to study
the CSC phase transition based on the holographic approach and indicate a
situation where this obstacle can be avoided due to $SU(3)_{C}$ gauge symmetry
of QCD behaving as just a global symmetry. In Sect. III, we propose the
gravitational dual model of the CSC phase transition which is given by the
system of Einstein gravity coupled minimally to the power-law Maxwell field
and a charged scalar field in the asymptotically anti-de-Sitter (AdS)
spacetime. The action of the system and the equations of motion in the
geometric configurations dual to the confinement and deconfinement phases are
presented in detail. The CSC phase appears when the quark Cooper pairs have
the nonzero vacuum expectation value (VEV) or the scalar field in the
gravitational dual theory condenses around the event horizon of the planar AdS
black hole. In Sect. IV, we point out that if the power parameter (which
characterizes for the power-law Maxwell field) is sufficiently smaller than
one then the CSC state would appear in the deconfinement phase of the
realistic YM theory as the chemical potential is above a critical value. In
Sect. V, with the critical line associated with the CSC phase transition found
by solving numerically the equations of motion near the critical chemical
potential and the free energy of the background configurations from computing
the Euclidean on-shell action of the system, we obtain the phase diagram in
the $\mu-T$ plane. The summary is given in the final section.

##  II  $SU(3)_{C}$ color symmetry as a global symmetry

The CSC phase appears when the diquark operator of the formal form $\langle
qq\rangle$ condenses, which leads to the spontaneous breaking of $SU(3)_{C}$
color symmetry. In fact, the $\langle qq\rangle$ operator is colored or not
gauge-invariant and hence this has been the main obstacle for building a
holographic model of the CSC phase transition because the bulk fields in the
gravity side are always dual to the gauge-invariant operators in the boundary
field theory side. However, the authors in Ref. Fadafan2018  have pointed out
a situation where $SU(3)_{C}$ color symmetry would appear as a global or
flavor symmetry due to the fact that the gluonic degrees of freedom are gapped
by the Debye screening as well as the effects of Landau damping and thus they
are integrated out. In this way, one can construct a holographic model of the
CSC phase transition which describes the breaking of the global symmetries
which consist of $SU(3)_{C}$ color symmetry, the usual $SU(N_{f})$ flavor
symmetry, and $U(1)_{B}$ symmetry.

In the strong coupling regime of QCD which is relevant to the core of neutron
stars, a quark-gluon plasma contains both quarks of color-electric charge and
composite scalars of color-magnetic charge. Their presence would thus generate
the Debye screening masses $\left(m^{2}_{E}\right)_{ab}$ and
$\left(m^{2}_{M}\right)_{ab}$ through the loop diagrams for the electric
gluons $A^{a}_{0}$ and the magnetic gluons $A^{a}_{i}$, respectively. The
electric and magnetic screening masses are determined as the static limit
(i.e. $q_{0}=0$ and $\textbf{q}\rightarrow 0$ with $(q_{0},\textbf{q})\equiv
q$ to be the four-momentum of gluons) of the temporal and spatial components
of the gluon self-energy tensor $\Pi^{\mu\nu}_{ab}(q_{0},\textbf{q})$,
respectively, and given as follows Freedman1977 ; Kapusta1989 ; Bellac1996

$\displaystyle\left(m^{2}_{E}\right)_{ab}$ $\displaystyle=$
$\displaystyle-\Pi^{00}_{ab}\left(0,\textbf{q}\rightarrow 0\right)\sim
g^{2}\left(T^{2}+\mu^{2}\right),$ $\displaystyle\left(m^{2}_{M}\right)_{ab}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(\delta^{ij}-\textbf{q}^{i}\textbf{q}^{j}\right)\Pi^{ij}_{ab}\left(0,\textbf{q}\rightarrow
0\right)\sim g^{2}\left(T^{2}+\mu^{2}\right),$ (1)

where $g$, $T$, and $\mu$ are the strong coupling, the temperature of the
quark-gluon plasma, and the chemical potential, respectively. In the regime of
the CSC phase transition where the temperature is low enough and the quark
matter density is high enough, the contribution of temperature is negligible
compared to the contribution of quark matter density. Hence, the Debye
screening masses of the electric and magnetic gluons are in order of the
$g\mu$ scale. In the intermediate density regime of QCD, the strong coupling
is $g\sim 4\pi$ which is large, thus the superconductivity gap scale which is
estimated as tens of MeV is much lower than the screening scale which is the
hundreds of MeV. Hence, the electric and magnetic gluons are gapped or are
Debye screened in the CSC phase which is much below the screening scale.

It should be noted that the QCD plasma at the weak coupling regime does not
contain the composite scalars of color-magnetic charge Freedman1977  and since
the Debye screening mass for the magnetic gluons can not be generated. This
means that the magnetic gluons would lead to the dominant interaction in the
momentum regime below the screening scale $g\mu$. However, the magnetic gluons
can be gapped due to the effects of the Landau damping Son1999  where their
self-energy behaves as $\Pi\sim g^{2}\mu^{2}|q_{0}|/|q|$ with the gap of
electric gluons to be the source for the mass of the magnetic gluons.

In summary, all gluonic degrees of freedom which are color-electric and color-
magnetic are screened at the momentum regime below the screening scale $g\mu$.
Due to the large separation between the screening scale and the chemical
potential scale, the gluons can be integrated out, and as a result, quarks
exist as the sole degrees of freedom below the screening scale. On the other
hand, in this momentum regime $SU(3)_{C}$ color symmetry can be realized as a
global or a flavor symmetry.

##  III  Gravitational dual model

Before introducing a gravitational dual model which is consistent with the CSC
phase transition in the boundary field theory, let us briefly discuss the CSC
condensation pattern which shall be considered in this work. A quark Cooper
pair or diquark operator is formed from two quarks each of which carries the
spin, color, and flavor degrees of freedom. For the formation of scalar
spin-$0$ condensation, the combination of spins in the diquark operator would
be antisymmetric. As mentioned in the introduction, in order to form the
diquark operator we need to have the attractive interaction which appears in
the color antitriplet channel. This means that the flavor combination in the
diquark operator must be antisymmetric due to the Pauli principle which
requires the wave function of the diquark operator to be antisymmetric under
the exchange of two quarks. In the present work, we are interested in the
presence of two lightest quark flavors (up and down quarks) participating in
the Cooper pairing, whereas the strange quark is taken to be infinitely
massive and hence it is absent. The corresponding CSC phase is well-known as
the two-flavor CSC phase Alford2008 . In this way, the expectation value
$\Delta$ of the diquark operator is given as

$\displaystyle\Delta=\left\langle
q^{T}C\gamma_{5}\tau_{2}\lambda_{2}q\right\rangle,$ (2)

which is antisymmetric in terms of the Dirac, color, and flavor indices that
have been dropped. Here, $C\equiv i\gamma^{2}\gamma^{0}$, $\tau_{2}$, and
$\lambda_{2}$ are the charge conjugation operator, the Pauli matrix, and the
Gell-Mann matrix which act in the Dirac, flavor, and color spaces,
respectively. In addition, the condensation of the diquark operator is
triggered by the quark chemical potential or quark number density associated
with the global $U(1)_{B}$ symmetry. Therefore, in the holographic model, we
need to introduce a complex scalar field that is dual to $\Delta$ and carries
a charge corresponding to a local $U(1)$ symmetry dual to the global
$U(1)_{B}$ symmetry.

The gravitational dual model for the CSC phase transition is Einstein gravity
coupled minimally to a power-law Maxwell field and a complex scalar field in
the asymptotically six-dimensional AdS spacetime. The action of the system is
given as follows

$S_{\text{bulk}}=\frac{1}{2\kappa^{2}_{6}}\int
d^{6}x\sqrt{-g}\left[R-2\Lambda+\beta\left(-F_{\mu\nu}F^{\mu\nu}\right)^{s}-|(\nabla-
iqA)\psi|^{2}-m^{2}|\psi|^{2}\right],$ (3)

where $\Lambda=-\frac{10}{l^{2}}$ is the cosmological constant with $l$ being
the asymptotic AdS radius, $\beta$ is a constant which is considered to be
$1/4$ in the present work without loss of generality, $s$ is the power
parameter characterizing for the power-law Maxwell field and in particular
when $s=1$ the power-law Maxwell electrodynamics would reduce to the usual
Maxwell case, and $\psi$ is the complex scalar field which has the mass $m$
and carries the charge $q$ under the gauge symmetry $U(1)$. It is important to
note here that the gauge symmetry $U(1)$ and the complex scalar field in the
bulk are dual to the baryon symmetry $U(1)_{B}$ and the diquark operator in
the boundary field theory, respectively. Therefore, the $U(1)$ charge of the
complex scalar field is expressed in terms of the color number of quarks as
$q=2/N_{c}$.

It is important to note that the four-dimensional gauge field theory possesses
a confinement scale and hence the asymptotic behavior of spacetime geometry
must manifest a corresponding scale. In the simplest setting Basu2011 , the
gravity dual of the boundary field theory can be considered in six dimensions
rather than five dimensions where one direction of spacetime is compactified
on a circle whose size is identified as the confinement scale. In this way,
the usually four-dimensional field theory can be obtained from the
compactification of the five-dimensional field theory on a circle.

The equations of motion are found by varying the action (3) with respect to
the metric, vector, and scalar fields, given by

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R-\frac{10}{l^{2}}g_{\mu\nu}$
$\displaystyle=$ $\displaystyle T_{\mu\nu},$
$\displaystyle\nabla_{\mu}\left[F^{\mu\nu}\left(-F_{\rho\sigma}F^{\rho\sigma}\right)^{s-1}\right]$
$\displaystyle=$
$\displaystyle\frac{iq}{s}\left[\psi^{*}(\nabla^{\nu}-iqA^{\nu})\psi-\psi(\nabla^{\nu}+iqA^{\nu})\psi^{*}\right],$
$\displaystyle(\nabla_{\mu}-iqA_{\mu})(\nabla^{\mu}-iqA^{\mu})\psi$
$\displaystyle=$ $\displaystyle m^{2}\psi,$ (4)

where the energy-momentum tensor $T_{\mu\nu}$ reads

$\displaystyle T_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{s}{2}\left(-F_{\rho\sigma}F^{\rho\sigma}\right)^{s-1}F_{\mu\lambda}{F_{\nu}}^{\lambda}+\frac{1}{2}\left[(\nabla_{\nu}-iqA_{\nu})\psi(\nabla_{\mu}+iqA_{\mu})\psi^{*}+\mu\leftrightarrow\nu\right]$
(5)
$\displaystyle+\frac{1}{2}g_{\mu\nu}\left[\frac{1}{4}\left(-F_{\mu\nu}F^{\mu\nu}\right)^{s}-|(\nabla-
iqA)\psi|^{2}-m^{2}|\psi|^{2}\right].$

By solving these equations of motion in the spacetime geometries dual to the
confinement and deconfinement phases, we shall find whether or not the CSC
phase transition can appear in these phases. The CSC phase transition appears
when the scalar field condenses where the $U(1)$ gauge symmetry is
spontaneously broken. In the canonical ensemble, the condensation of the
scalar field or the CSC phase transition is triggered by the chemical
potential and for the chemical potential above a critical value $\mu_{c}$ the
CSC phase transition would appear. In this work, we study the CSC phase
transition near the critical chemical potential $\mu_{c}$ and since the
condensate value of the scalar field is near zero. This suggests that the
backreaction of the scalar field on the spacetime geometry can be ignored
compared to the contribution of the vector field.

The spacetime geometry which is dual to the deconfinement phase is the high-
temperature solution of Eq. (4) known as the planar AdS black hole. Ansatz for
the line element of the planar AdS black hole is given as follows

$\displaystyle ds^{2}_{\text{BH}}$ $\displaystyle=$ $\displaystyle
r^{2}\left(-f(r)dt^{2}+h_{ij}dx^{i}dx^{j}+dy^{2}\right)+\frac{dr^{2}}{r^{2}f(r)},$
(6)

where $h_{ij}dx^{i}dx^{j}=dx^{2}_{1}+dx^{2}_{2}+dx^{2}_{3}$ refers to the line
element of the $3$-dimensional planar hypersurface and $y$ is the compactified
coordinate with the circle radius $R_{y}$. Whereas, the vector and scalar
fields are described by the following ansatz

$\displaystyle A_{\mu}dx^{\mu}=\phi(r)dt,\ \ \ \ \psi=\psi(r).$ (7)

The equations of motion corresponding to the geometric configuration of the
planar AdS black hole are found as

$\displaystyle rf^{\prime}(r)+5f(r)-5+(2s-1)2^{s-4}\phi^{\prime}(r)^{2s}$
$\displaystyle=$ $\displaystyle 0,$ (8)
$\displaystyle\phi^{\prime\prime}(r)+\frac{4}{(2s-1)r}\phi^{\prime}(r)-\frac{q^{2}\psi^{2}(r)\phi^{\prime}(r)^{2(1-s)}}{2^{s-2}s(2s-1)r^{2}f(r)}\phi(r)$
$\displaystyle=$ $\displaystyle 0,$ (9)
$\displaystyle\psi^{\prime\prime}(r)+\left[\frac{f^{\prime}(r)}{f(r)}+\frac{6}{r}\right]\psi^{\prime}(r)+\frac{1}{r^{2}f(r)}\left[\frac{q^{2}\phi^{2}(r)}{r^{2}f(r)}-m^{2}\right]\psi(r)$
$\displaystyle=$ $\displaystyle 0.$ (10)

In order for the matter fields to behave regularly at the event horizon, we
impose the following boundary condition

$\displaystyle\phi(r_{+})=0,\ \ \ \
\psi(r_{+})=r^{2}_{+}\frac{f^{\prime}(r_{+})\psi^{\prime}(r_{+})}{m^{2}}.$
(11)

In the asymptotic region $r\rightarrow\infty$, the matter fields behave as

$\displaystyle\phi(r)$ $\displaystyle=$
$\displaystyle\mu-\frac{\bar{d}^{\frac{1}{2s-1}}}{r^{\frac{5-2s}{2s-1}}},$
$\displaystyle\psi(r)$ $\displaystyle=$
$\displaystyle\frac{J_{C}}{r^{\Delta_{-}}}+\frac{C}{r^{\Delta_{+}}}.$ (12)

where $\mu$ and $\bar{d}$ are regarded as the baryon chemical potential and
the baryon charge density of the boundary field theory, respectively. The
overall coefficients $J_{C}$ and $C$ which both are normalizable modes are
identified as the external source and the condensate value of the diquark
operator, respectively. In order to guarantee that the $U(1)$ gauge symmetry
is spontaneously broken, the external source must vanish and hence we impose a
boundary condition as $J_{C}=0$. The conformal dimensions of $C$ and $J_{C}$
are

$\displaystyle\Delta_{\pm}=\frac{1}{2}\left(5\pm\sqrt{25+4m^{2}}\right),$ (13)

from which we see that the squared mass of the scalar field must satisfy the
Breitenlohner-Freedman (BF) bound Freedman1982 ; Breitenlohner1982  as

$m^{2}\geq-\frac{25}{4}.$ (14)

Because $C$ is identified as the condensate value of the diquark operator, the
conformal dimension of $C$ should be $\Delta_{+}=2\times(d-2)/2$ leading to
$\Delta_{+}=4$ for the six-dimensional case. This means that the squared mass
of the scalar field is $m^{2}=-4$. In addition, we note that, from the
asymptotic expression of the scalar potential $\phi(r)$, in order for
$\phi(r)$ to be finite as $r\rightarrow\infty$, the power parameter $s$ must
satisfy the following condition

$\displaystyle\frac{1}{2}<s<\frac{5}{2}.$ (15)

In the limit that the chemical potential approaches the critical value
$\mu_{c}$, we obtain

$\displaystyle f(r)$ $\displaystyle=$ $\displaystyle
1-\left(\frac{r_{+}}{r}\right)^{5}-\frac{2^{s-4}(2s-1)^{2}}{5-2s}\left[\frac{(5-2s)\mu}{(2s-1)r_{+}}\right]^{2s}\left(\frac{r_{+}}{r}\right)^{5}\left[1-\left(\frac{r_{+}}{r}\right)^{\frac{5-2s}{2s-1}}\right],$
$\displaystyle\phi(r)$ $\displaystyle=$
$\displaystyle\mu\left[1-\left(\frac{r_{+}}{r}\right)^{\frac{5-2s}{2s-1}}\right].$
(16)

The Hawking temperature of the planar AdS black hole thus is

$\displaystyle
T=\frac{r_{+}}{4\pi}\left\\{5-2^{s-4}(2s-1)\left[\frac{(5-2s)\mu}{(2s-1)r_{+}}\right]^{2s}\right\\}.$
(17)

The non-negative condition of the Hawking temperature leads to the condition
for the ratio of the chemical potential to the event horizon radius as

$\displaystyle
0\leq\frac{\mu}{r_{+}}\leq\frac{2s-1}{5-2s}\left[\frac{80}{2^{s}(2s-1)}\right]^{\frac{1}{2s}}.$
(18)

The spacetime geometry which is dual to the confinement phase is the low-
temperature solution of Eq. (4) known as the AdS soliton solution222The low-
temperature solution of AdS soliton was also confirmed in the extensions of
Einstein gravity, for example, in EGB gravity Cai-Kim2007 . Horowitz1998
described by the following ansatz

$\displaystyle
ds^{2}_{\text{Sol.}}=r^{2}\left(-dt^{2}+h_{ij}dx^{i}dx^{j}+f(r)dy^{2}\right)+\frac{dr^{2}}{r^{2}f(r)},$
(19)

where the function $f(r)$ vanishes at the tip $r=r_{0}$ of the AdS soliton
where the canonical singularity is removed by requiring the periodicity for
the coordinate $y$. The equations of motion corresponding to this geometric
configuration are

$\displaystyle
rf^{\prime}(r)+5f(r)-5+(2s-1)2^{s-4}f(r)^{s}\phi^{\prime}(r)^{2s}$
$\displaystyle=$ $\displaystyle 0,$ (20)
$\displaystyle\phi^{\prime\prime}(r)+\frac{1}{2s-1}\left[s\frac{f^{\prime}(r)}{f(r)}+\frac{4}{r}\right]\phi^{\prime}(r)-\frac{q^{2}\psi^{2}(r)\phi^{\prime}(r)^{2(1-s)}}{2^{s-2}s(2s-1)r^{2}f(r)^{s}}\phi(r)$
$\displaystyle=$ $\displaystyle 0,$ (21)
$\displaystyle\psi^{\prime\prime}(r)+\left[\frac{f^{\prime}(r)}{f(r)}+\frac{6}{r}\right]\psi^{\prime}(r)+\frac{1}{r^{2}f(r)}\left[\frac{q^{2}\phi^{2}(r)}{r^{2}}-m^{2}\right]\psi(r)$
$\displaystyle=$ $\displaystyle 0.$ (22)

The boundary condition for the matter fields is

$\displaystyle\phi^{\prime}(r_{0})$ $\displaystyle=$
$\displaystyle\left[\frac{q^{2}\psi^{2}(r_{0})}{2^{s-2}s^{2}r^{2}_{0}f(r_{0})^{s-1}f^{\prime}(r_{0})}\phi(r_{0})\right]^{\frac{1}{2s-1}},$
(23) $\displaystyle\psi^{\prime}(r_{0})$ $\displaystyle=$
$\displaystyle-\frac{1}{r^{2}_{0}f^{\prime}(r_{0})}\left[\frac{q^{2}\phi^{2}(r_{0})}{r^{2}_{0}}-m^{2}\right]\psi(r_{0}).$
(24)

The asymptotic behavior of the matter fields is the same as in Eq. (12). Near
the critical chemical potential where the value of the scalar field approaches
zero, the AdS soliton solution is determined by the line element (19) with the
function $f(r)$ given as

$\displaystyle f(r)=1-\frac{r^{5}_{0}}{r^{5}},\ \ \ \ r_{0}=\frac{2}{5R_{y}},$
(25)

and the potential of the $U(1)$ gauge field reads

$\displaystyle\phi(r)=\mu=\text{constant}.$ (26)

##  IV  CSC phase transition

In this section, we will study the CSC phase transition of the realistic YM
theory in the background configurations dual to the confinement and
deconfinement phases by examining the breaking of the BF bound and solving
numerically the equations of motion near the critical chemical potential. We
point out that the CSC phase transition with $N_{c}\geq 2$ in the
deconfinement phase appears above the critical chemical potential
corresponding to the condensation of the scalar field, which is not found in
the case of the usual Maxwell electrodynamics if the power parameter $s$ is
sufficiently smaller than one. Furthermore, the value of the power parameter
$s$ is not arbitrary below one (but above $1/2$). This is because when $s$ is
sufficiently far away from one it leads to the condensation of the scalar
field in the confinement phase, i.e. the occurrence of the CSC state in the
confinement phase, which suggests the breakdown of the holographic
description.

###  IV.1 In the deconfinement phase

In order to see that the CSC phase transition for $N_{c}\geq 2$ can appear in
the holographic model with the power-law Maxwell field for the appropriate
value of the power parameter $s$, first let us study the condition which
breaks the BF bound and thus makes the condensation of the scalar field
appearing. The condition of the BF bound breaking is given as

$\displaystyle m^{2}_{\text{eff}}<-\frac{25}{4}\ \ \ \ \Rightarrow\ \ \ \
\frac{9}{4}<\frac{q^{2}\phi^{2}(r)}{r^{2}f(r)}\equiv
q^{2}\mathcal{F}(z,\hat{\mu},s),$ (27)

where $m^{2}_{\text{eff}}=m^{2}-\frac{q^{2}\phi^{2}(r)}{r^{2}f(r)}$ is the
effective squared mass of the scalar field and

$\displaystyle\mathcal{F}(z,\hat{\mu},s)=\frac{z^{2}\left(1-z^{\frac{5-2s}{2s-1}}\right)^{2}\hat{\mu}^{2}}{1-z^{5}-\frac{2^{s-4}(2s-1)^{2}}{5-2s}\left[\frac{(5-2s)\hat{\mu}}{(2s-1)}\right]^{2s}z^{5}\left(1-z^{\frac{5-2s}{2s-1}}\right)},$
(28)

with $z\equiv r_{+}/r$ and $\hat{\mu}\equiv\mu/r_{+}$. We show the behavior of
the function $\mathcal{F}(z,\hat{\mu},s)$ as a function of $z$ for various
values of $\hat{\mu}$ and $s$ in Fig. 1. From this figure, we realize that
$\mathcal{F}(z,\hat{\mu},s)$ grows with the increasing of the scaled chemical
potential $\hat{\mu}$ and the decreasing of the power parameter $s$.

---
Figure 1: Plots describing the behavior of $\mathcal{F}(z,\hat{\mu},\alpha)$
in terms of $z$, $\hat{\mu}$ and $\alpha$, where $\hat{\mu}_{\text{max}}$ is
given in Eq. (29).

Furthermore, we find that the function $\mathcal{F}(z,\hat{\mu},s)$ gets the
maximum value when $z\rightarrow 1$ and at

$\displaystyle\hat{\mu}=\frac{2s-1}{5-2s}\left[\frac{80}{2^{s}(2s-1)}\right]^{\frac{1}{2s}}\equiv\hat{\mu}_{\text{max}}.$
(29)

From this result along with Eq. (27), we obtain an upper bound for the color
number, which depends on the power parameter $s$, as

$\displaystyle N_{c}<\frac{4}{3}\sqrt{\mathcal{F}(z\rightarrow
1,\hat{\mu}_{\text{max}},s)}\equiv N_{c,\text{max}}.$ (30)

The upper bound $N_{c,\text{max}}$ for the color number of quarks as a
function of the power parameter $s$ is depicted in Fig. 2.

---
Figure 2: The upper bound for the color number of quarks is plotted in the
power parameter $s$. The dashed black lines refer to the case of the usual
Maxwell electrodynamics.

We see here that $N_{c,\text{max}}$ increases when decreasing $s$, and
$N_{c,\text{max}}$ in the power-law Maxwell electrodynamics with $s<1(>1)$ is
larger(smaller) than that ($N_{c}<4\sqrt{2}/3\simeq 1.89$ Ghoroku2019 ) in the
usual Maxwell electrodynamics. In particular, we find that the upper bound for
the color number of quarks is much larger than $2$ for the appropriate value
of the power parameter $s$ belonging to the region $1/2<s<1$. This clearly
suggests that the power-law Maxwell electrodynamics with $s<1$ can support the
CSC phase transition for the realistic YM theory which has $N_{c}\geq 2$.

In order to determine the critical chemical potential, the critical
temperature, and the slope of the critical line $T_{c}=T_{c}(\mu_{c})$
associated with the CSC phase transition in the deconfinement phase, we solve
numerically Eqs. (8), (9), and (10) by using the shooting method. The
numerical results are given in Tables 1 and 2 for $N_{c}=2$ and $N_{c}=3$,
respectively.

$s$ | $\mu_{c}/r_{+}$ | $T_{c}/r_{+}$ | $T_{c}/\mu_{c}$
---|---|---|---
$\ \ \ \ \ \ 4/5\ \ \ \ \ \ $ | $\ \ \ \ \ \ 2.5413\ \ \ \ \ \ $ | $\ \ \ \ \ \ 0.0271\ \ \ \ \ \ $ | $\ \ \ \ \ \ 0.0107\ \ \ \ \ \ $
$\ \ \ \ \ \ 3/4\ \ \ \ \ \ $ | $\ \ \ \ \ \ 2.6192\ \ \ \ \ \ $ | $\ \ \ \ \ \ 0.0696\ \ \ \ \ \ $ | $\ \ \ \ \ \ 0.0266\ \ \ \ \ \ $
$\ \ \ \ \ \ 5/7\ \ \ \ \ \ $ | $\ \ \ \ \ \ 2.6299\ \ \ \ \ \ $ | $\ \ \ \ \ \ 0.1101\ \ \ \ \ \ $ | $\ \ \ \ \ \ 0.0418\ \ \ \ \ \ $
Table 1: The numerical values for the scaled critical chemical potential and temperature, and the slope of the critical line $T_{c}=T_{c}(\mu_{c})$ with various values of $s$ at $N_{c}=2$. $s$ | $\mu_{c}/r_{+}$ | $T_{c}/r_{+}$ | $T_{c}/\mu_{c}$
---|---|---|---
$\ \ \ \ \ \ 2/3\ \ \ \ \ \ $ | $\ \ \ \ \ \ 3.4058\ \ \ \ \ \ $ | $\ \ \ \ \ \ 0.0680\ \ \ \ \ \ $ | $\ \ \ \ \ \ 0.0200\ \ \ \ \ \ $
$\ \ \ \ \ \ 5/8\ \ \ \ \ \ $ | $\ \ \ \ \ \ 3.4375\ \ \ \ \ \ $ | $\ \ \ \ \ \ 0.1329\ \ \ \ \ \ $ | $\ \ \ \ \ \ 0.0387\ \ \ \ \ \ $
$\ \ \ \ \ \ 3/5\ \ \ \ \ \ $ | $\ \ \ \ \ \ 3.3839\ \ \ \ \ \ $ | $\ \ \ \ \ \ 0.1750\ \ \ \ \ \ $ | $\ \ \ \ \ \ 0.0517\ \ \ \ \ \ $
Table 2: The numerical values for the ratios $\mu_{c}/r_{+}$, $T_{c}/r_{+}$,
and $T_{c}/\mu_{c}$ with various values of $s$ at $N_{c}=3$.

These tables suggest that there exist the proper values of the power parameter
$s$ which are sufficiently low and below one (but still belonging to the
physical region) for which the CSC phase transition with $N_{c}\geq 2$ can
appear. In addition, we observe that for the event horizon radius $r_{+}$ kept
fixed the critical temperature grows when the power parameter $s$ decreases.
This means that the condensation of the scalar field gets easier to form in
the power-law Maxwell electrodynamics with the appropriately low power
parameter.

Let us interpret why the power-law Maxwell electrodynamics with the
appropriately low power parameter can lead to the occurrence of the CSC phase
transition for $N_{c}\geq 2$. Recall that the CSC phase of the realistic YM
theory which corresponds to $N_{c}\geq 2$ is not found in both the
deconfinement and the confinement phases with respect to the holographic model
which consists of the Einstein-Maxwell system coupled to the complex scalar
field Ghoroku2019 . When the color number of quarks increases the charge of
the complex scalar field would reduce. As a result, the electrostatic
repulsion becomes weak and since it would not be strong enough to overcome the
gravitational attraction for the scalar hair formed. However, when we consider
the power-law Maxwell electrodynamics instead of the usual Maxwell
electrodynamics, the gravitational attraction would become weak with the
decrease of the power parameter $s$. This can be seen in Fig. 3 where we
observe that with the same mass the event horizon of the planar AdS black hole
grows with the decreasing of the power parameter $s$.

---
Figure 3: The mass $m$ of the planar AdS black hole which can be inferred from
the terms proportional to $1/r^{5}$ in Eq. (III) in terms of the event horizon
radius $r_{+}$ for various values of the power parameter $s$.

On the other hand, the gravitational attraction around the event horizon is
sufficiently weak with the properly low power parameter, by which the
electrostatic repulsion can overcome the gravitational attraction to allow the
formation of the scalar hair corresponding to the CSC phase transition even
with the low charge of the complex scalar field or $N_{c}\geq 2$.

###  IV.2 In the confinement phase

We arrive at the investigation of whether or not the CSC state or the
condensation of the scalar field appears in the confinement phase. It is easy
to find a necessary condition for the appearance of the CSC phase transition
in this phase as $q\mu/r_{0}>1.5$, which corresponds to the breaking of the BF
bound for the effective squared mass
$m^{2}_{\text{eff}}=m^{2}-q^{2}\phi^{2}/r^{2}<-25/4$. Of course, in order to
determine a sufficient condition for the appearance of the CSC phase
transition in the confinement phase, we need to solve Eq. (22) in the
background configuration of the AdS soliton determined by Eqs. (19), (25), and
(26). By solving numerically Eq. (22), we find the condition of the chemical
potential for the appearance of the CSC phase transition in the confinement
phase as $\mu>1.505N_{c}$. However, in order to confirm that the CSC phase
transition can appear in the confinement phase when the chemical potential is
above the critical value $1.505N_{c}$, we need to check whether or not this
critical value belongs to the region of the confinement phase. By using the
results in the next section, we show the maximal chemical potential
$\mu_{\text{max}}$ which the confinement phase can reach as well as the
critical chemical potential $\mu_{c}$ for the occurrence of the CSC state in
the confinement phase for various values of the power parameter $s$ at
$N_{c}=2$ and $N_{c}=3$ in Table 3.

$N_{c}=2$ | $N_{c}=3$
---|---
$s$ | $\mu_{\text{max}}$ | $\mu_{c}$ | $s$ | $\mu_{\text{max}}$ | $\mu_{c}$
$\ \ \ \ 4/5\ \ \ \ $ | $\ \ \ \ 2.337\ \ \ \ $ | $\ \ \ \ 3.01\ \ \ \ $ | $\ \ \ \ 2/3\ \ \ \ $ | $\ \ \ \ 3.630\ \ \ \ $ | $\ \ \ \ 4.515\ \ \ \ $
$\ \ \ \ 3/4\ \ \ \ $ | $\ \ \ \ 2.672\ \ \ \ $ | $\ \ \ \ 3.01\ \ \ \ $ | $\ \ \ \ 5/8\ \ \ \ $ | $\ \ \ \ 4.492\ \ \ \ $ | $\ \ \ \ 4.515\ \ \ \ $
$\ \ \ \ 5/7\ \ \ \ $ | $\ \ \ \ 3.007\ \ \ \ $ | $\ \ \ \ 3.01\ \ \ \ $ | $\ \ \ \ 3/5\ \ \ \ $ | $\ \ \ \ 5.230\ \ \ \ $ | $\ \ \ \ 4.515\ \ \ \ $
Table 3: The numerical values for the maximal chemical potential
$\mu_{\text{max}}$ above which the confinement phase cannot appear and the
critical chemical potential $\mu_{c}$ above which the CSC phase transition
happens for various values of the power parameter $s$ at $N_{c}=2$ and
$N_{c}=3$.

Note that, the maximal chemical potential $\mu_{\text{max}}$ can be found from
Eq. (34) with $T(r_{+})=0$. Clearly, this table indicates that the critical
chemical potential $\mu_{c}=3.01$ and $\mu_{c}=4.515$ corresponding to
$N_{c}=2$ and $N_{c}=3$, respectively, are outside the confinement phase for
most values of the power parameter $s$ under consideration. This means that
the CSC phase transition cannot occur in the confinement phase except in the
case of $s=3/5$ corresponding to $N_{c}=3$. Also, we observe that the maximal
chemical potential $\mu_{\text{max}}$ grows with the decreasing of the power
parameter $s$ and as a result $\mu_{\text{max}}$ would be larger than the
critical chemical potential $\mu_{c}$ at the sufficiently low values of the
power parameter $s$. This suggests that the CSC state can exist even in the
confinement phase which would not be compatible with the nonzero VEV of the
color nonsinglet operator. On the other hand, the present holographic model
with sufficiently low values of the power parameter $s$ does not provide a
suitable holographic model for the CSC phase transition of the realistic YM
theory.

##  V  Phase diagram

In this section, we will obtain the phase diagram of the present holographic
model in the plane of the temperature $T$ and the chemical potential $\mu$,
which shows the information about the region of $T$ and $\mu$ where the
thermodynamically distinct phases (such as the deconfinement, confinement, and
CSC phases which we are interested in this work) appear. In addition, the
phase diagram shows the location of the phase boundaries or the critical
curves and when there is a phase transition between the different phases.

First, we determine the phase structure which describes the confinement and
deconfinement phases, which corresponds to the vanishing of the scalar field.
In order to do this, we need to compute the free energy of the AdS soliton
configuration and the planar AdS black hole configuration from the Euclidean
on-shell action of the system and then compare them to find which one is the
preferred configuration. By using the general result obtained in Miskovic2011
, one can write the free energy of Einstein gravity coupled minimally to the
power-law Maxwell field in the situation without the condensation of the
scalar field as

$\displaystyle S_{E}$ $\displaystyle=$
$\displaystyle\left[r^{4}\left(r^{2}f\right)^{\prime}\Big{|}^{\infty}_{r_{+}}-r^{4}f^{2}\left(r^{2}f\right)^{\prime}\Big{|}^{\infty}-s2^{s-1}\int^{\infty}_{r_{+}}drr^{4}\phi^{\prime}(r)^{2s}\right]\frac{4\pi}{5r_{0}}\frac{V_{3}}{T}$
(31) $\displaystyle=$
$\displaystyle\left[r^{4}\left(r^{2}f\right)^{\prime}\Big{|}^{\infty}_{r_{+}}-r^{4}f^{2}\left(r^{2}f\right)^{\prime}\Big{|}^{\infty}-s2^{s-1}r^{4}\phi\phi^{\prime
2s-1}\Big{|}^{\infty}_{r_{+}}\right]\frac{4\pi}{5r_{0}}\frac{V_{3}}{T},$

where $V_{3}=\int dx_{1}dx_{2}dx_{3}$ and in the second line we have used the
equation of motion for $\phi(r)$ as follows

$\displaystyle\frac{d}{dr}\left[r^{4}\phi^{\prime}(r)^{2s-1}\right]=0.$ (32)

Then, we obtain explicitly the free energy for the planar AdS black hole
configuration and the AdS soliton configuration as

$\displaystyle\Omega_{\text{BH}}$ $\displaystyle=$
$\displaystyle-r^{5}_{+}\left[1+\frac{2^{s-4}(2s-1)^{2}}{(5-2s)}\left(\frac{(5-2s)\mu}{(2s-1)r_{+}}\right)^{2s}\right]\frac{4\pi}{5r_{0}}V_{3},$
$\displaystyle\Omega_{\text{Sol.}}$ $\displaystyle=$
$\displaystyle-r^{5}_{0}\frac{4\pi}{5r_{0}}V_{3}.$ (33)

The free energy of the planar AdS black hole and AdS soliton is the same, i.e.
$\Omega_{\text{BH}}=\Omega_{\text{Sol.}}$, along the critical curve which
defines the phase boundary between the confinement and deconfinement phases.
From this relation, we find the parameter equation in terms of the event
horizon radius $r_{+}$, which determines this critical curve in the $\mu-T$
plane as

$\displaystyle\mu(r_{+})$ $\displaystyle=$
$\displaystyle\left[\frac{2^{4-s}(1-r^{5}_{+})}{(2s-1)r^{5-2s}_{+}}\right]^{1/2s}\left(\frac{5-2s}{2s-1}\right)^{(1-2s)/2s},$
$\displaystyle T(r_{+})$ $\displaystyle=$
$\displaystyle\frac{r_{+}}{4\pi}\left[5-\frac{(5-2s)}{(2s-1)}\frac{(1-r^{5}_{+})}{r^{5}_{+}}\right],$
(34)

where we have set $r_{0}$ to be unity without loss of generality. The
confinement phase corresponds to the region in the $\mu-T$ plane where the AdS
soliton has the free energy lower than that of the planar AdS black hole and
since the geometric configuration of the AdS soliton is thermodynamically
favored. On the contrary, the region where the free energy of the planar AdS
black hole is lower than that of the AdS soliton represents the deconfinement
phase.

From the results derived above, we can show the phase diagram of the present
holographic model for the realistic YM theory in the $\mu-T$ plane for various
values of the power parameter $s$ and the color number $N_{c}$ in Fig. 4.

---
Figure 4: The phase diagram is shown for various values of the power parameter
$s$ and the color number $N_{c}$. Top-left panel: $s=4/5$ and $N_{c}=2$. Top-
right panel: $s=5/7$ and $N_{c}=2$. Bottom-left panel: $s=2/3$ and $N_{c}=3$.
Bottom-right panel: $s=5/8$ and $N_{c}=3$.

The red curve refers to the critical curve corresponding to the
deconfinement/confinement phase transition. The region below the red curve
corresponds to the confinement phase due to its gravitational dual being
thermodynamically preferred. The region above the red curve and the blue line
represents the (normal) deconfinement phase. The CSC phase appears in the
region of the high chemical potential and the low temperature. Hence, the
region below the blue line which refers to the critical line associated with
the CSC phase transition represents the CSC phase. It is found that the region
of the CSC phase becomes larger when the power parameter $s$ decreases, which
corresponds to the growth of the slope of the blue line as seen in Tables 1
and 2. This suggests that the CSC phase is more stable in the regime of the
small power parameter such that it is larger than $1/2$ and it should not lead
to the appearance of the CSC state in the confinement phase. In particular,
this phase diagram shows that the present holographic model with the power-law
Maxwell field for the reasonable values of the power parameter $s$ can provide
a gravitational dual description for the CSC phase transition of the realistic
YM theory which appears in the deconfinement phase but not in the confinement
phase.

##  VI  Summary

We have studied the color superconductivity (CSC) phase transition in the
realistic Yang-Mills theory (i.e. $N_{c}\geq 2$ where $N_{c}$ is the color
number of quarks) using a simple holographic model. This holographic model is
constructed by Einstein gravity coupled minimally to the matter part which
consists of a power-law Maxwell field and a complex scalar field which are
dual to the baryon symmetry and the diquark operator, respectively. By
analyzing the breaking of the Breitenlohner-Freedman bound and solving
numerically the equations of motion in the configuration of the planar AdS
black hole dual to the deconfinement phase, we have pointed out that the
scalar field of the small charge ($q\leq 1$) can condense around the event
horizon of the planar AdS black hole if the power parameter characterizing for
the power-law Maxwell field is sufficiently lower than one but above $1/2$.
The condensation of the scalar field with the small charge in the
gravitational dual theory means that the corresponding diquark operator
develops a nonzero vacuum expectation value (VEV) and hence the CSC phase
transition with the large color number ($N_{c}\geq 2$) appears in the
deconfinement phase, which is not found in the gravitational dual description
with the usual Maxwell field. Furthermore, we have shown that the power
parameter $s$ is not arbitrary below one (but above $1/2$) because when $s$ is
sufficiently far away from one it leads to the condensation of the scalar
field in the configuration of the AdS soliton or the nonzero VEV of the
diquark operator in the confinement phase which would not be compatible with
the nonzero VEV of the color nonsinglet operator. Finally, we have obtained
the phase diagram in the plane of the temperature and the chemical potential.

The current work also claims that the condensate of the scalar field is
actually easier to form for the smaller power parameter, which has been
previously indicated in the investigation of the holographic superconductors
with the power-law Maxwell field in the probe limit Pan2011 ; Salahi2016  and
with the backreaction Jiang2016 . The critical values of the temperature and
chemical potential grow with the decrease of the power parameter, as seen in
Fig. 4. However, there are main differences between these works and the
current work. First, the gravitational dual model of the 4D usual
superconductors is studied in five dimensions. Whereas, the gravitational dual
description of the 4D color superconductors is considered in six dimensions
with one dimension compactified on a circle $S^{1}$. This is because the
boundary gauge field theory possesses a confinement scale which is identified
with the inverse radius of the $S^{1}$. Second, besides studying the effect of
the power-law Maxwell field on the critical temperature like in holographic
superconductors, the current world studies the relevant effects on the color
number of quarks (or the $U(1)_{B}$ charge of the condensate operator) and the
critical chemical potential which are the important parameters to trigger the
CSC phase transition. Third, unlike the holographic model of the usual
superconductors which describes the low- and high-temperature phases
corresponding to the superconducting and normal states, the holographic model
of the color superconductor represents three different phases: they are the
confinement, normal deconfinement, and CSC phases corresponding to the low $T$
and $\mu$, the high $T$ and high $\mu$, and the low $T$ and high $\mu$,
respectively.

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