# Delineating chiral separation effect in two-color dense QCD

Daiki Suenaga<EMAIL_ADDRESS>Research Center for Nuclear Physics,
Osaka University, Ibaraki, 567-0048, Japan  Toru Kojo<EMAIL_ADDRESS>Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle
Physics, Central China Normal University, Wuhan 430079, China

###### Abstract

We study the chiral separation effect (CSE) in two-color and two-flavor QCD
(QC2D) to delineate quasiparticle pictures in dense matter from low to high
temperatures. Both massless and massive quarks are discussed. We particularly
focus on the high density domain where diquarks form a color singlet
condensate with the electric charge $1/3$. The condensate breaks baryon number
and $U(1)_{A}$ axial symmetry, and induces the electromagnetic Meissner
effects. Within a quark quasiparticle picture, we compute the chiral
separation conductivity at one-loop. We have checked that Nambu-Goldstone
modes, which should appear in the improved vertices as required by the Ward-
Takahashi identities, do not contribute to the chiral separation conductivity
due to their longitudinal natures. In the static limit, the destructive
interferences in the particle-hole channel, as in usual Meissner effects,
suppress the conductivity (in chiral limit, to $1/3$ of the normal phase’s).
This locally breaks the universality of the CSE coefficients, provided
quasiparticle pictures are valid in the bulk matter.

##  I Introduction

The quantum anomaly Adler (1969); Bell and Jackiw (1969); Alvarez-Gaume and
Witten (1984) and topological properties of matter have been attracting much
attention in wide area of physics, due to their universal natures valid for
abruptly different energy scales Vilenkin (1980); Kharzeev  _et al._ (2008);
Fukushima  _et al._ (2008); Son and Surowka (2009); Son and Yamamoto (2012);
Stephanov and Yin (2012); Yamamoto (2016). The non-renormalization theorem for
the anomaly relation Adler and Bardeen (1969); Adler (2005); Golkar and Son
(2015), as well as the robustness of topology under local perturbations, are
the key concepts for such universality.

The chiral conductivities are constrained by such universal relations,
especially in the chiral limit (see, e.g. a review Landsteiner (2016)). One
can also use the relations to constrain or enrich our pictures on the
nonperturbative dynamics Itoyama and Mueller (1983); Sannino (2000); Gaiotto
_et al._ (2015, 2017); Tanizaki  _et al._ (2018); Tanizaki (2018); Wan and
Wang (2020); Tanizaki and Ünsal (2020) which have important implications to
the physics of neutron stars (for a review, e.g., Baym  _et al._ (2018)). With
this perspective in mind, in this paper we study the conductivity for chiral
separation effect (CSE) Metlitski and Zhitnitsky (2005), chiral separation
(CS) conductivity, in two-color QCD with two-flavor (QC2D) at high baryon
density. The relation is

$\displaystyle
J_{i}^{5}=\bar{q}\gamma_{i}\gamma_{5}q=\frac{\,C\,}{\,2\pi^{2}\,}\mu B_{i}\,,$
(1)

with $q$ quark fields, $\mu$ quark chemical potential, and $C$ being some
constant related to the anomaly. The dense QC2D matter Kogut  _et al._ (2000,
1999) can be studied in lattice Monte-Carlo simulations Boz  _et al._ (2020);
Iida  _et al._ (2020); Astrakhantsev  _et al._ (2020); Boz  _et al._ (2019)
and hence can be used as laboratories to test the concepts and methodologies
of dense QCD calculations Kojo and Suenaga (2021); Suenaga and Kojo (2019);
Kojo and Baym (2014). The results may be extended to our main target, three-
color dense QCD.

The CSE in QC2D was studied for hadronic and quark-gluon-plasma phase
Buividovich  _et al._ (2020), and was found to agree with theoretical
predictions including explicit chiral symmetry breaking. Meanwhile, for cold
dense matter the CSE has not been studied yet. At $\mu\geq m_{\pi}/2$
($m_{\pi}$: pion mass), the dense matter has condensations of diquarks, $\sim
ud$, which are color-singlet, $J^{P}=0^{+}$, and have the electric charges
$1/3$. We call the phase diquark condensed phase. A number of new ingredients
appear here, modifying qualitative pictures conventionally used to explain the
CSE Metlitski and Zhitnitsky (2005); (i) A quark acquires the mass gap around
the Fermi surface and arguments based on the lowest Landau level no longer
directly apply; (ii) Diquarks are charged and should induce the
electromagnetic Meissner effects, expelling magnetic fields from the bulk;
(iii) The spontaneous symmetry breaking demands the improved vertices to
fulfill the Ward-Takahashi identity (WTI), and the vertices contain the poles
of the Nambu-Goldstone (NG) modes Nambu (1960), both in the axial vector and
the vector vertices.

We will address these qualitative issues within one-loop calculations in the
linear response regime, as done for the chiral magnetic conductivity Kharzeev
and Warringa (2009). We assume the quasiparticle pictures for quarks with the
gap $\Delta$, in a similar manner as in our previous calculations Kojo and
Suenaga (2021); Suenaga and Kojo (2019); Kojo and Baym (2014). The CS
conductivity in the static and dynamic limits will be analyzed for various
temperatures. We found that the dependence on temperatures and the orders of
static and dynamic limits can be understood through the particle-hole
dynamics; in the chiral limit and at $T=0$, the static CS conductivity in the
diquark condensed (normal) phase acquires $0$ ($2/3$) of the anomaly
coefficient from the particle-hole contributions and $1/3$ ($1/3$) from
particle-antiparticle contributions. Thus the static CS conductivity in the
diquark condensed phases provides only $1/3$ expected from the universal
relation.111In contrast, the CS conductivity was found to be enhanced by heavy
impurities in Refs. Araki  _et al._ (2021); Suenaga  _et al._ (2021).

Inclusion of the NG modes through the improved vertices does not help as they
are longitudinal modes and hence decouple. This suggests that, if the
coefficient of the CS conductivity is indeed constrained by the anomaly, we
should go beyond our quasiparticle descriptions or the linear response regime,
e.g., by manifestly treating vortices with fermion zero modes or zero modes on
the boundaries of the diquark condensed phase.

This paper is structured as follows. In Sec.II we set up our frameworks and
introduce quasiparticle propagators in the diquark condensed phase. In Sec.III
we summarize the expressions for the CS conductivity within the linear
response regime. In Sec.IV we present our numerical results for various
chemical potential, temperature, quark mass, and diquark gap. In Sec.V we
address the contributions not included in our quasiparticle calculations in
the bulk. This paper is closed with summary in Sec.VI.

##  II Model

Here we introduce an effective model to investigate the CSE in QC2D. Previous
works by lattice QCD Boz  _et al._ (2020); Iida  _et al._ (2020);
Astrakhantsev  _et al._ (2020); Boz  _et al._ (2019) and model calculations
such as the Nambu–Jona-Lasinio model Sun  _et al._ (2007); He (2010); Andersen
and Brauner (2010); Andersen  _et al._ (2015) and the low-energy effective
theory based on the nonlinear realization Kogut  _et al._ (2000, 1999) imply
the existence of the diquark condensed phase, or the diquark condensate phase,
in a region of $\mu\geq m_{\pi}/2$ ($\mu$ is a quark chemical potential and
$m_{\pi}$ is a pion mass) at zero temperature in QC2D. The diquark condensed
phase is described by $S$-wave, color-singlet and flavor-singlet diquark
condensate
$\Delta\sim\langle\psi^{T}C\gamma_{5}\sigma^{2}\tau^{2}\psi\rangle$. Here,
$\psi=(\psi_{u},\psi_{d})^{T}$ is a quark spinor in two-color and two-flavor
system, $C=i\gamma^{2}\gamma^{0}$ is the charge-conjugation operator, and
$\sigma^{2}$ and $\tau^{2}$ are the anti-symmetric Pauli matrices for color
and flavor spaces, respectively.

In the present study, in order to incorporate effects from the diquark
condensate but investigate the CSE in the most transparent way, we start with
the following concise effective model:

$\displaystyle{\cal L}_{\rm
eff}=\bar{\psi}(i{\ooalign{\hfil/\hfil\crcr$D$}}+\mu\gamma^{0}-m)\psi+\psi^{T}{\bm{\Delta}}\psi\
.$ (4)

In this Lagrangian a covariant derivative
$D_{\mu}\psi=(\partial_{\mu}-ieQA_{\mu})\psi$ with $Q={\rm
diag}(+\frac{2}{3},-\frac{1}{3})$ the charge matrix has been included to
describe interactions with the external magnetic field. $m$ is a quark mass
and ${\bm{\Delta}}=\gamma_{5}\sigma^{2}\tau^{2}\Delta$ is responsible for the
diquark condensate. For convenience we rewrite Eq. (4) into the Nambu-Gorkov
basis as

$\displaystyle{\cal L}_{\rm eff}$ $\displaystyle=$
$\displaystyle\bar{\Psi}\left(\begin{array}[]{cc}i{\ooalign{\hfil/\hfil\crcr$\partial$}}+\mu\gamma_{0}-m&\bar{\bf\Delta}\\\
{\bf\Delta}&i{\ooalign{\hfil/\hfil\crcr$\partial$}}-\mu\gamma_{0}-m\\\
\end{array}\right)\Psi$ (18)
$\displaystyle+e\bar{\Psi}\left(\begin{array}[]{cc}Q{\ooalign{\hfil/\hfil\crcr$A$}}&0\\\
0&-Q{\ooalign{\hfil/\hfil\crcr$A$}}\\\ \end{array}\right)\Psi\ ,$

where

$\displaystyle\Psi\equiv\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}\psi\\\
\psi_{c}\\\ \end{array}\right)\ ,\
\bar{\Psi}\equiv\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}\bar{\psi}\,,&\bar{\psi}_{c}\end{array}\right)$
(22)

are the Nambu-Gor’kov spinors ($\psi_{c}=C\bar{\psi}^{T}$). From the first
line of Eq. (18), the inverse of the fermion propagator
$S^{-1}(p_{0},{\bm{p}})$ is read as

$\displaystyle
iS^{-1}(p_{0},{\bm{p}})=\left(\begin{array}[]{cc}{\ooalign{\hfil/\hfil\crcr$p$}}+\mu\gamma^{0}-m&\bar{\bf\Delta}\\\
{\bf\Delta}&{\ooalign{\hfil/\hfil\crcr$p$}}-\mu\gamma^{0}-m\\\
\end{array}\right)\ ,$ (29)

with $\bar{\bm{\Delta}}\equiv\gamma^{0}{\bm{\Delta}}^{\dagger}\gamma^{0}$. By
taking the inverse of $S^{-1}(p_{0},{\bm{p}})$, the fermion propagator is
obtained as

$\displaystyle
S(p_{0},{\bm{p}})=\left(\begin{array}[]{cc}S_{11}(p_{0},{\bm{p}})&\sigma^{2}\tau^{2}S_{12}(p_{0},{\bm{p}})\\\
\sigma^{2}\tau^{2}S_{21}(p_{0},{\bm{p}})&S_{22}(p_{0},{\bm{p}})\\\
\end{array}\right)\ ,$ (32)

where each component has the Dirac structure as

$\displaystyle S_{11}(p_{0},{\bm{p}})$ $\displaystyle=$
$\displaystyle\sum_{\xi={\rm p},{\rm
a}}S^{\xi}_{11}(p_{0},{\bm{p}})\Lambda_{\xi}\gamma^{0}\ ,$ $\displaystyle
S_{12}(p_{0},{\bm{p}})$ $\displaystyle=$ $\displaystyle\sum_{\xi={\rm p},{\rm
a}}S^{\xi}_{12}(p_{0},{\bm{p}})\Lambda_{\xi}\gamma_{5}\ ,$ $\displaystyle
S_{21}(p_{0},{\bm{p}})$ $\displaystyle=$ $\displaystyle\sum_{\xi={\rm p},{\rm
a}}S^{\xi}_{21}(p_{0},{\bm{p}})\Lambda^{C}_{\xi}\gamma_{5}\ ,$ $\displaystyle
S_{22}(p_{0},{\bm{p}})$ $\displaystyle=$ $\displaystyle\sum_{\xi={\rm p},{\rm
a}}S^{\xi}_{22}(p_{0},{\bm{p}})\Lambda^{C}_{\xi}\gamma^{0}\ ,$ (33)

with

$\displaystyle S^{\xi}_{11}(p_{0},{\bm{p}})$ $\displaystyle=$ $\displaystyle
i\left(\frac{|u_{\xi}({\bm{p}})|^{2}\,}{\,p_{0}-\eta_{\xi}\epsilon_{\xi}({\bm{p}})\,}+\frac{\,|v_{\xi}({\bm{p}})|^{2}\,}{\,p_{0}+\eta_{\xi}\epsilon_{\xi}({\bm{p}})\,}\right)\
,$ $\displaystyle S^{\xi}_{12}(p_{0},{\bm{p}})$ $\displaystyle=$
$\displaystyle-i\left(\frac{\,u_{\xi}^{*}({\bm{p}})v_{\xi}^{*}({\bm{p}})\,}{\,p_{0}-\epsilon_{\xi}({\bm{p}})\,}-\frac{\,u_{\xi}^{*}({\bm{p}})v_{\xi}^{*}({\bm{p}})\,}{\,p_{0}+\epsilon_{\xi}({\bm{p}})\,}\right)\
,$ $\displaystyle S^{{\xi}}_{21}(p_{0},{\bm{p}})$ $\displaystyle=$
$\displaystyle
i\left(\frac{\,u_{\xi}({\bm{p}})v_{\xi}({\bm{p}})\,}{\,p_{0}-\epsilon_{\xi}({\bm{p}})\,}-\frac{\,u_{\xi}({\bm{p}})v_{\xi}({\bm{p}})\,}{\,p_{0}+\epsilon_{\xi}({\bm{p}})\,}\right)\
,$ $\displaystyle S^{\xi}_{22}(p_{0},{\bm{p}})$ $\displaystyle=$
$\displaystyle
i\left(\frac{\,|v_{\xi}({\bm{p}})|^{2}\,}{\,p_{0}-\eta_{\xi}\epsilon_{\xi}({\bm{p}})\,}+\frac{\,|u_{\xi}({\bm{p}})|^{2}\,}{\,p_{0}+\eta_{\xi}\epsilon_{\xi}({\bm{p}})\,}\right)\,.$
(34)

Here we introduced $\eta_{\rm p}=+1$ and $\eta_{\rm a}=-1$. In these
expressions, we have defined the positive-energy and negative-energy
projection operators $\Lambda_{\rm p}$ and $\Lambda_{\rm a}$ by

$\displaystyle\Lambda_{\xi}$ $\displaystyle=$
$\displaystyle\gamma_{0}\frac{\,E_{\bm{p}}\gamma_{0}+\eta_{\xi}\big{(}m+{\bm{\gamma}}\cdot{\bm{p}}\big{)}\,}{2E_{\bm{p}}}\,.$
(35)

with $E_{\bm{p}}=\sqrt{|{\bm{p}}|^{2}+m^{2}}$, and $\Lambda_{{\rm p}({\rm
a})}^{C}=\Lambda_{{\rm a}({\rm p})}$, and

$\displaystyle\epsilon_{\xi}({\bm{p}})$ $\displaystyle=$
$\displaystyle\sqrt{(E_{\bm{p}}-\eta_{\xi}\mu)^{2}+|\Delta|^{2}}\,,$ (36)

are the dispersion relations for quasiparticles. The factors $u_{\rm
p}({\bm{p}})$, $v_{\rm p}({\bm{p}})$, $u_{\rm a}({\bm{p}})$, and $v_{\rm
a}({\bm{p}})$ satisfy relations

$\displaystyle|u_{\xi}({\bm{p}})|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{\,2\,}\left(1+\frac{\,E_{\bm{p}}-\eta_{\xi}\mu\,}{\,\epsilon_{\xi}({\bm{p}})\,}\right)\
,$ $\displaystyle|v_{\xi}({\bm{p}})|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{\,2\,}\left(1-\frac{\,E_{\bm{p}}-\eta_{\xi}\mu\,}{\,\epsilon_{\xi}({\bm{p}})\,}\right)\
,$ (37)

and

$\displaystyle\\!\\!\\!\\!|u_{\xi}({\bm{p}})|^{2}+|v_{\xi}({\bm{p}})|^{2}=1\,,~{}~{}u_{\xi}({\bm{p}})v_{\xi}({\bm{p}})=\frac{\Delta}{2\epsilon_{\xi}({\bm{p}})}\,.$
(38)

By using the propagator of the quasiparticles in Eq. (32) together with Eqs.
(33) and (34), and with the help of linear response theory, the CSE in the
diquark condensed phase under a weak magnetic field can be evaluated.

##  III CS conductivity

In the present work we investigate the CSE in the diquark condensed phase
within the linear response theory. Our calculations are similar to that for
the chiral magnetic conductivity Kharzeev and Warringa (2009), but our
calculations are more complicated due to the presence of diquark condensates.
The axial current is defined by $j_{5}^{i}=\bar{\psi}\gamma^{i}\gamma_{5}\psi$
($i=1,2,3$). In the linear response theory we measure the difference between
the current with and without external electromagnetic vector potential $A$,

$\displaystyle\langle j_{5}^{i}\rangle\equiv\langle
j_{5}^{i}\rangle_{A}-\langle j_{5}^{i}\rangle_{A=0}\,.$ (39)

We note that the $U(1)_{A}$ axial current couples to the gluon topological
density as $\partial^{\mu}j_{\mu}^{A}\sim\tilde{G}_{\mu\nu}G^{\mu\nu}$. Below
we neglect the impact of $A$ on the color gauge dynamics, i.e.,
$\langle\tilde{G}_{\mu\nu}G^{\mu\nu}\rangle_{A}=\langle\tilde{G}_{\mu\nu}G^{\mu\nu}\rangle_{A=0}$.
Then we focus on the anomalous contributions coupled to the $\mu$ and $A$.

In the following calculation, we will make use of the imaginary time formalism
to incorporate finite temperature effects. In momentum space the axial current
induced by the external electromagnetic field is evaluated as the retarded
correlators between the axial vector and vector currents,
($\int_{\bm{p}}\equiv\int d^{3}p/(2\pi)^{3}$)

$\displaystyle\langle j_{5}^{i}(i\bar{\omega}_{n},{\bm{q}})\rangle$
$\displaystyle\equiv$ $\displaystyle\int_{0}^{\beta}d\tau\int
d^{3}x\,\langle\bar{\psi}(x)\gamma^{i}\gamma_{5}\psi(x)\rangle{\rm
e}^{-i\bar{\omega}_{n}\tau+i{\bm{q}}\cdot{\bm{x}}}$ (40) $\displaystyle=$
$\displaystyle\frac{\,e\,}{2}\int_{\bm{p}}T\sum_{m}{\rm
Tr}\Big{[}\Gamma^{i}\Gamma_{5}{S}(i\omega_{m},{\bm{p}})$ $\displaystyle\times$
$\displaystyle\big{(}\Gamma^{{\rm
bare}}_{\bar{\psi}A\psi}\big{)}^{\mu}{S}(i\omega_{m}^{\prime},{\bm{p}}^{\prime})\Big{]}{A}_{\mu}(i\bar{\omega}_{n},{\bm{q}})\,,$

with ${\bm{p}}^{\prime}={\bm{p}}+{\bm{q}}$,
$i\omega_{m}^{\prime}=i\omega_{m}+i\bar{\omega}_{n}$ and $N_{c}=2$, where we
have defined the vertices in the Nambu-Gor’kov space as

$\displaystyle\Gamma^{i}\Gamma_{5}=\left(\begin{array}[]{cc}\gamma^{i}\gamma_{5}&0\\\
0&\gamma^{i}\gamma_{5}\\\ \end{array}\right)\ ,\big{(}\Gamma^{{\rm
bare}}_{\bar{\psi}A\psi}\big{)}^{\mu}=\left(\begin{array}[]{cc}Q\gamma^{\mu}&0\\\
0&-Q\gamma^{\mu}\\\ \end{array}\right)\ .$ (45)

Here $\omega_{m}=(2m+1)\pi T$ and $\bar{\omega}_{n}=2n\pi T$ ($n$, $m$ are
integers) are the Matsubara frequencies for fermions and bosons, respectively.
The symbol “Tr” stands for the trace with respect to the color, flavor,
spinor, and Nambu-Gor’kov spaces. (We note that, in the presence of symmetry
breaking, the use of the bare vertices violates the conservation law and one
must improve the vertices to recover it. We will come back to this point
later, just mentioning that such improvements will not change the main
conclusion of the simplest one loop results.)

We will leave only spatial components of the external gauge field
$A_{\mu}(i\bar{\omega}_{n},{\bm{q}})$ because the magnetic field is solely
generated by them. The real time axial current $\langle
j_{5}^{i}(q_{0},{\bm{q}})\rangle$ is given by the analytic continuation as

$\displaystyle\langle j_{5}^{i}(q_{0},{\bm{q}})\rangle=\langle
j_{5}^{i}(i\bar{\omega}_{n},{\bm{q}})\rangle\Big{|}_{i\bar{\omega}_{n}\to
q_{0}+i\eta}\ ,$ (47)

with $\eta$ an infinitesimal positive number.

Performing the trace with respect to the color, spinor, and Nambu-Gorkov
spaces in Eq. (40), the axial current is reduced to the form

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\langle
j_{5}^{i}(i\bar{\omega}_{n},{\bm{q}})\rangle$ $\displaystyle=$ $\displaystyle-
ie\,{\rm tr}[Q]\epsilon^{ijk}A^{j}(i\bar{\omega}_{n},{\bm{q}})\sigma_{\rm
CSE}^{k}(i\bar{\omega}_{n},{\bm{q}})$ (48) $\displaystyle=$
$\displaystyle-e\,{\rm tr}[Q]B^{i}\sigma_{\rm
CSE}(i\bar{\omega}_{n},{\bm{q}})$

where we used that $\sigma^{k}_{\rm
CSE}(i\bar{\omega}_{n},{\bm{q}})=q^{k}\sigma_{\rm
CSE}(i\bar{\omega}_{n},{\bm{q}})$ because of the Lorentz structure, and
$B^{i}=\epsilon^{imn}(-iq^{m})A^{n}$. Hence we can extract the conductivity
$\sigma_{\rm CSE}(i\bar{\omega}_{n},{\bm{q}})$ as

$\displaystyle\sigma_{\rm
CSE}(i\bar{\omega}_{n},{\bm{q}})=\frac{\,q^{k}\,}{\,|{\bm{q}}|^{2}\,}\,\sigma^{k}_{\rm
CSE}(i\bar{\omega}_{n},{\bm{q}})\,.$ (49)

The function $\sigma^{k}_{\rm CSE}$ is computed using propagators decomposed
into the particle and antiparticle pieces with normal and anomalous
components, as in Eq.(33). Accordingly the function $\sigma_{\rm CSE}$ is also
decomposed as

$\displaystyle\sigma_{\rm
CSE}(i\bar{\omega}_{n},{\bm{q}})=\sum_{\xi,\xi^{\prime}={\rm p},{\rm
a}}\sigma^{\xi\xi^{\prime}}_{\rm CSE}(i\bar{\omega}_{n},{\bm{q}})\,.$ (50)

The set $(\xi,\xi^{\prime})=({\rm p},{\rm p})$ and $({\rm a},{\rm a})$
correspond to the particle-hole and antiparticle-antihole contributions, while
$({\rm p},{\rm a})$ or $({\rm a},{\rm p})$ are from the particle-antiparticle
contributions. Explicitly,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\sigma^{\xi\xi^{\prime}}_{\rm
CSE}(i\bar{\omega}_{n},{\bm{q}})=\frac{\,N_{c}q^{k}\,}{|{\bm{q}}|^{2}}\int_{\bm{p}}{\cal
F}^{\xi\xi^{\prime}}(\bar{\omega}_{n},{\bm{q}};{\bm{p}}){\cal
K}^{k}_{\xi\xi^{\prime}}({\bm{q}};{\bm{p}})\,,$ (51)

with the kinematic factor

$\displaystyle{\cal
K}^{k}_{\xi\xi^{\prime}}({\bm{q}};{\bm{p}})=-\eta_{\xi}\frac{p^{k}}{E_{\bm{p}}}+\eta_{\xi^{\prime}}\frac{(p^{\prime})^{k}}{E_{{\bm{p}}^{\prime}}}\,,$
(52)

which are independent of the diquark condensates, and the propagator part

$\displaystyle{\cal
F}^{\xi\xi^{\prime}}(\bar{\omega}_{n},{\bm{q}};{\bm{p}})=T\sum_{m}\Big{[}S_{11}^{\xi}(i\omega_{m},{\bm{p}})S_{11}^{\xi^{\prime}}(i\omega^{\prime}_{m},{\bm{p}}^{\prime})-S_{12}^{\xi}(i\omega_{m},{\bm{p}})S_{21}^{\xi^{\prime}}(i\omega^{\prime}_{m},{\bm{p}}^{\prime})$
$\displaystyle-
S_{21}^{\xi}(i\omega_{m},{\bm{p}})S_{12}^{\xi^{\prime}}(i\omega^{\prime}_{m},{\bm{p}}^{\prime})+S_{22}^{\xi}(i\omega_{m},{\bm{p}})S_{22}^{\xi^{\prime}}(i\omega^{\prime}_{m},{\bm{p}}^{\prime})\Big{]}\,,$
(53)

which carries the effects from the diquark condensates. In ${\cal
F}^{\xi\xi^{\prime}}$ we can carry out the Matsubara summation using an
identity

$\displaystyle
T\sum_{m}\frac{1}{(i\omega_{m}-\epsilon_{1})(i\omega_{m}^{\prime}-\epsilon_{2})}=\frac{\tilde{f}(\epsilon_{1})-\tilde{f}(\epsilon_{2})}{i\bar{\omega}_{n}-\epsilon_{2}+\epsilon_{1}}\
,$ (54)

where $\tilde{f}(\epsilon)=1/({\rm e}^{\beta\epsilon}+1)$ is the Fermi-Dirac
distribution function. Using this identity, the function ${\cal
F}^{\xi\xi^{\prime}}$ can be written as

$\displaystyle{\cal F}^{\xi\xi^{\prime}}(\bar{\omega}_{n},{\bm{q}};{\bm{p}})$
$\displaystyle={\cal C}_{1}^{\xi\xi^{\prime}}({\bm{q}};{\bm{p}})\,{\cal
G}_{1}^{\xi\xi^{\prime}}(q;p)+{\cal
C}_{2}^{\xi\xi^{\prime}}({\bm{q}};{\bm{p}})\,{\cal
G}_{2}^{\xi\xi^{\prime}}(q;p)\,,$ (55)

where the second term is nonzero only at finite temperature. The factors
${\cal C}^{\xi\xi^{\prime}}$ and $\tilde{{\cal C}}^{\xi\xi^{\prime}}$ are the
coherence factors which carry the information about the wavefunctions
$u_{\xi}$ and $v_{\xi}$,

$\displaystyle{\cal C}_{1,2}^{{\rm p}{\rm p}}({\bm{q}};{\bm{p}})$
$\displaystyle=\frac{1}{2}\left[1\mp\frac{\,(E_{\bm{p}}-\mu)(E_{{\bm{p}}^{\prime}}-\mu)+|\Delta|^{2}\,}{\epsilon_{\rm
p}({\bm{p}})\epsilon_{\rm p}({\bm{p}}^{\prime})}\right]\,,$
$\displaystyle{\cal C}_{1,2}^{{\rm a}{\rm a}}({\bm{q}};{\bm{p}})$
$\displaystyle=\frac{1}{2}\left[1\mp\frac{\,(E_{\bm{p}}+\mu)(E_{{\bm{p}}^{\prime}}+\mu)+|\Delta|^{2}\,}{\epsilon_{\rm
a}({\bm{p}})\epsilon_{\rm a}({\bm{p}}^{\prime})}\right]\,,$
$\displaystyle{\cal C}_{1,2}^{{\rm p}{\rm a}}({\bm{q}};{\bm{p}})$
$\displaystyle=\frac{1}{2}\left[1\pm\frac{\,(E_{\bm{p}}-\mu)(E_{{\bm{p}}^{\prime}}+\mu)-|\Delta|^{2}\,}{\epsilon_{\rm
p}({\bm{p}})\epsilon_{\rm a}({\bm{p}}^{\prime})}\right]\,,$
$\displaystyle{\cal C}_{1,2}^{{\rm a}{\rm p}}({\bm{q}};{\bm{p}})$
$\displaystyle=\frac{1}{2}\left[1\pm\frac{\,(E_{\bm{p}}+\mu)(E_{{\bm{p}}^{\prime}}-\mu)-|\Delta|^{2}\,}{\epsilon_{\rm
a}({\bm{p}})\epsilon_{\rm p}({\bm{p}}^{\prime})}\right]\,.$ (56)

The upper (lower) sign in the right-hand side (RHS) corresponds to the
subscript $1$ ($2$) in the left-hand side (LHS). The factors ${\cal
G}_{1,2}^{\xi\xi^{\prime}}$ are the propagator factors which carry the
information about the excitation spectra,

$\displaystyle{\cal
G}_{1}^{\xi\xi^{\prime}}(q;p)=\frac{\,1\,}{2}\big{[}\,1-\tilde{f}\big{(}\epsilon_{\xi}({\bm{p}})\big{)}-\tilde{f}\big{(}\epsilon_{\xi^{\prime}}({\bm{p}}^{\prime})\big{)}\,\big{]}$
$\displaystyle\times\bigg{(}\frac{1}{\,i\bar{\omega}_{n}+\epsilon_{\xi}({\bm{p}})+\epsilon_{\xi^{\prime}}({\bm{p}}^{\prime})\,}+\frac{1}{\,-i\bar{\omega}_{n}+\epsilon_{\xi}({\bm{p}})+\epsilon_{\xi^{\prime}}({\bm{p}}^{\prime})\,}\bigg{)}\,,$

and

$\displaystyle{\cal
G}_{2}^{\xi\xi^{\prime}}(q;p)=-\frac{\,1\,}{2}\big{[}\,\tilde{f}\big{(}\epsilon_{\xi}({\bm{p}})\big{)}-\tilde{f}\big{(}\epsilon_{\xi^{\prime}}({\bm{p}}^{\prime})\big{)}\,\big{]}$
$\displaystyle\times\bigg{(}\frac{1}{\,i\bar{\omega}_{n}+\epsilon_{\xi}({\bm{p}})-\epsilon_{\xi^{\prime}}({\bm{p}}^{\prime})\,}+\frac{1}{\,-i\bar{\omega}_{n}+\epsilon_{\xi}({\bm{p}})-\epsilon_{\xi^{\prime}}({\bm{p}}^{\prime})\,}\bigg{)}\,.$

Below we examine how the contributions in different sectors affect the CSE
conductivity.

###  III.1 Particle-hole

The coherence factors and propagators behave very differently for the normal
and condensed phases. We consider the low temperature case $T\ll\Delta$ where
quantum effects are important. The contributions which are most sensitive to
the condensates are particle-hole contributions near the edge of the Fermi sea
with $|{\bm{p}}|\simeq|{\bm{p}}^{\prime}|\simeq p_{F}$. Below we discuss the
particle-hole contributions for the $|{\bm{q}}|\ll\Delta$ and
$|{\bm{q}}|\gg\Delta$ cases. The latter can be seen as the results in the
normal phase.

####  III.1.1  $|{\bf q}|\ll\Delta$

Depending on the size of ${\bm{q}}$, the coherence factors appear
constructively or destructively. To see this, first we set ${\bm{q}}=0$ as a
special case of $|{\bm{q}}|\ll\Delta$. Then we find

$\displaystyle{\cal C}_{1}^{{\rm p}{\rm
p}}({\bm{q}}=0;{\bm{p}})=0\,,~{}~{}~{}~{}~{}{\cal C}_{2}^{{\rm p}{\rm
p}}({\bm{q}}=0;{\bm{p}})=1\,.$ (59)

For ${\cal C}_{1}^{{\rm p}{\rm p}}$, the correction starts with $O({\bm{q}})$.
With this ${\cal F}^{{\rm p}{\rm p}}\sim{\cal G}_{2}^{{\rm p}{\rm p}}$ for
small ${\bm{q}}$. For ${\cal G}_{2}^{{\rm p}{\rm p}}$, we focus on
$|{\bm{p}}|\simeq|{\bm{p}}^{\prime}|\simeq p_{F}$, where $\epsilon_{\rm
p}(p_{F})=\Delta$ and $\tilde{f}\sim{\rm e}^{-\Delta/T}\ll 1$. We expand

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\tilde{f}\big{(}\epsilon_{\rm
p}({\bm{p}})\big{)}-\tilde{f}\big{(}\epsilon_{\rm
p}({\bm{p}}^{\prime})\big{)}\simeq-\frac{\,\partial\tilde{f}(x)\,}{\,\partial
x\,}\bigg{|}_{x=\epsilon_{\rm p}}\frac{\,\partial\epsilon_{\rm
p}\,}{\,\partial{\bm{p}}\,}\cdot{\bm{q}}\,,$ (60)

and get

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!{\cal G}_{2}^{{\rm p}{\rm
p}}\sim\frac{\,\partial\tilde{f}(x)\,}{\,\partial
x\,}\bigg{|}_{x=\epsilon_{\rm p}}\frac{\,\big{(}\frac{\,\partial\epsilon_{\rm
p}\,}{\,\partial{\bm{p}}\,}\cdot{\bm{q}}\big{)}^{2}\,}{\,\bar{\omega}_{n}^{2}+\big{(}\frac{\,\partial\epsilon_{\rm
p}\,}{\,\partial{\bm{p}}\,}\cdot{\bm{q}}\big{)}^{2}\,}~{}\sim~{}\frac{1}{T}{\rm
e}^{-\Delta/T}\,.$ (61)

For $T\rightarrow 0$, $\partial\tilde{f}/\partial x\rightarrow 0$ at $x>0$,
and ${\cal G}_{2}^{{\rm p}{\rm p}}$ is vanishing for $\Delta\neq 0$ as ${\rm
e}^{-\Delta/T}/T\rightarrow 0$.

Finally we look at the kinematic factor couples to $q^{k}$,

$\displaystyle q^{k}{\cal K}^{k}_{{\rm p}{\rm
p}}({\bm{q}};{\bm{p}})=-\frac{{\bm{q}}\cdot{\bm{p}}}{E_{\bm{p}}}+\frac{{\bm{q}}\cdot({\bm{p}}+{\bm{q}})}{E_{{\bm{p}}+{\bm{q}}}}$
$\displaystyle\simeq\frac{\,{\bm{q}}^{2}\,}{\,E_{\bm{p}}\,}-\frac{\,({\bm{p}}\cdot{\bm{q}})^{2}\,}{\,E_{\bm{p}}^{3}\,}~{}\sim~{}\frac{\,{\bm{q}}^{2}\,}{\,E_{\bm{p}}\,}\bigg{(}1-\frac{1}{3}\frac{\,|{\bm{p}}|^{2}\,}{\,E_{\bm{p}}^{2}\,}\bigg{)}\,,$
(62)

where in the last expression we averaged
$({\bm{p}}\cdot{\bm{q}})^{2}\rightarrow{\bm{p}}^{2}{\bm{q}}^{2}/3$ in the
integral over ${\bm{p}}$. Then the CSE conductivity from the particle-hole can
be written as

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\sigma^{{\rm p}{\rm p}}_{\rm
CSE}(i\bar{\omega}_{n},{\bm{q}})~{}\sim~{}N_{c}\int_{\bm{p}}{\cal G}_{2}^{{\rm
p}{\rm
p}}\frac{\,1\,}{\,E_{\bm{p}}\,}\bigg{(}1-\frac{1}{3}\frac{\,|{\bm{p}}|^{2}\,}{\,E_{\bm{p}}^{2}\,}\bigg{)}\,.$
(63)

for small ${\bm{q}}$. Note that the thermal factor makes the expression
Eq.(63) UV finite.

At zero temperature the particle-hole contributions vanish by the presence of
the diquark gap as ${\cal G}_{2}^{{\rm p}{\rm p}}\sim{\rm
e}^{-\Delta/T}/T\rightarrow 0$. This suppression of particle-hole is what
happens for the Meissner effect in a superconductor. In contrast, in a normal
conductor the particle-hole induces the paramagnetic effects which cancel with
the diamagnetic contributions from particle-antiparticle contributions. The
absence of particle-hole contributions or paramagnetic effects, as found in
our calculations for the diquark condensed phase, makes the material
diamagnetic. The particle-hole pairs contribute only as thermal excitations.

####  III.1.2  $|{\bf q}|\gg\Delta$ or normal phase

For $|{\bm{q}}|\gg\Delta$, the above expression is qualitatively modified.
Here it is important to note that
$(E_{\bm{p}}-\mu)(E_{{\bm{p}}^{\prime}}-\mu)<0$ for the particle-hole
contributions; one of the excitations has $E>\mu$ and the other $E<\mu$. Then,
neglecting $\Delta$, one finds

$\displaystyle\frac{\,(E_{\bm{p}}-\mu)(E_{{\bm{p}}^{\prime}}-\mu)+|\Delta|^{2}\,}{\epsilon_{\rm
p}({\bm{p}})\epsilon_{\rm p}({\bm{p}}^{\prime})}~{}\simeq~{}-1\,,$ (64)

so that

$\displaystyle{\cal C}_{1}^{{\rm p}{\rm
p}}({\bm{q}};{\bm{p}})~{}\simeq~{}1\,,~{}~{}~{}~{}~{}{\cal C}_{2}^{{\rm p}{\rm
p}}({\bm{q}};{\bm{p}})~{}\simeq~{}0\,.$ (65)

With this ${\cal F}^{{\rm p}{\rm p}}\sim{\cal G}_{1}^{{\rm p}{\rm p}}$ for
$|{\bm{q}}|\gg\Delta$. This function is evaluated by focusing on the domain
$|{\bm{p}}|\simeq|{\bm{p}}^{\prime}|\simeq p_{F}$. For
$E_{\bm{p}}-\mu\gg\Delta$ and $\mu-E_{{\bm{p}}^{\prime}}\gg\Delta$, we use

$\displaystyle 1-\tilde{f}\big{(}\epsilon_{\rm
p}({\bm{p}})\big{)}-\tilde{f}\big{(}\epsilon_{\rm
p}({\bm{p}}^{\prime})\big{)}$ $\displaystyle~{}\sim~{}$
$\displaystyle\tilde{f}\big{(}E_{{\bm{p}}^{\prime}}-\mu\big{)}-\tilde{f}\big{(}E_{{\bm{p}}}-\mu\big{)}$
(66) $\displaystyle~{}\sim~{}$
$\displaystyle\frac{\,\partial\tilde{f}(x)\,}{\,\partial
x\,}\bigg{|}_{x=E_{\bm{p}}-\mu}\frac{\,\partial
E_{\bm{p}}\,}{\,\partial{\bm{p}}\,}\cdot{\bm{q}}\,,$
$\displaystyle\epsilon_{\xi}({\bm{p}})+\epsilon_{\xi^{\prime}}({\bm{p}}^{\prime})~{}\sim~{}E_{\bm{p}}-E_{{\bm{p}}^{\prime}}~{}\simeq~{}-\frac{\,\partial
E_{\bm{p}}\,}{\,\partial{\bm{p}}\,}\cdot{\bm{q}}\,,$

to get

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!{\cal G}_{1}^{{\rm p}{\rm
p}}\,\sim\,-\frac{\,\partial\tilde{f}(x)\,}{\,\partial
x\,}\bigg{|}_{x=E_{\bm{p}}-\mu}\times\frac{\,\big{(}\frac{\,\partial E_{\rm
p}\,}{\,\partial{\bm{p}}\,}\cdot{\bm{q}}\big{)}^{2}\,}{\,\bar{\omega}_{n}^{2}+\big{(}\frac{\,\partial
E_{\rm p}\,}{\,\partial{\bm{p}}\,}\cdot{\bm{q}}\big{)}^{2}\,}\,.$ (67)

For $T\rightarrow 0$, the contribution entirely comes from $|{\bm{p}}|=p_{F}$
where $\partial\tilde{f}/\partial x$ yields $-\delta(|{\bm{p}}|-p_{F})$, as in
the usual Debye screening calculations. Including the kinematic factor as
before, we find

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\sigma^{{\rm p}{\rm p}}_{\rm
CSE}(i\bar{\omega}_{n},{\bm{q}})~{}\sim~{}N_{c}\int_{\bm{p}}{\cal G}_{1}^{{\rm
p}{\rm
p}}\frac{\,1\,}{\,E_{\bm{p}}\,}\bigg{(}1-\frac{1}{3}\frac{\,|{\bm{p}}|^{2}\,}{\,E_{\bm{p}}^{2}\,}\bigg{)}\,,$
(68)

for $|{\bm{q}}|\gg\Delta$. The expression Eq.(68) is UV finite and the
integrand is dominated by the contributions from $|{\bm{p}}|\simeq p_{F}$.

As mentioned before, the result for $|{\bm{q}}|\gg\Delta$ can be taken as a
result for the normal phase with $\Delta=0$. In the normal phase the particle-
hole contributions are large at $T=0$ as they are gapless, in contrast to
fermions in the diquark condensed phase. In the massless limit, we find, for
the static limit,

$\displaystyle\sigma^{{\rm p}{\rm p}({\rm normal})}_{\rm CSE;\,{\it
m}=0}\big{|}^{\omega=0}_{{\bm{q}}\rightarrow
0}~{}\rightarrow~{}\frac{\,N_{c}\mu\,}{\,3\pi^{2}\,}\,.$ (69)

and for the dynamic limit

$\displaystyle\sigma^{{\rm p}{\rm p}({\rm normal})}_{{\rm CSE};\,{\it
m}=0}\big{|}^{\omega\rightarrow 0}_{{\bm{q}}=0}~{}\rightarrow~{}0\,.$ (70)

###  III.2 Particle-antiparticle

Next we discuss the particle-antiparticle contributions. These contributions
are not regulated by thermal factors and are potentially UV divergent. Our
main concern is the UV finiteness and for this purpose it is sufficient to
study $T\sim 0$.

The coherence factors in the particle-antiparticle contributions are not
dominated by the contributions from $|{\bm{p}}|\sim|{\bm{p}}^{\prime}|\simeq
p_{F}$, but come from large phase space with momenta much larger than
$\Delta$. For this reason we set $\Delta\rightarrow 0$ and focus on the
leading order contributions. Then for $E_{\bm{p}}-\mu>0$,

$\displaystyle\frac{\,(E_{\bm{p}}-\mu)(E_{{\bm{p}}^{\prime}}+\mu)-|\Delta|^{2}\,}{\epsilon_{\rm
p}({\bm{p}})\epsilon_{\rm a}({\bm{p}}^{\prime})}~{}\rightarrow~{}1\,,$ (71)

Accordingly,

$\displaystyle{\cal C}_{1}^{{\rm p}{\rm
a}}({\bm{q}};{\bm{p}})~{}\rightarrow~{}1\,,~{}~{}~{}~{}~{}{\cal C}_{2}^{{\rm
p}{\rm a}}({\bm{q}};{\bm{p}})~{}\rightarrow~{}0\,,$ (72)

and we find

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!{\cal F}^{{\rm p}{\rm
a}}\rightarrow\,{\cal G}^{{\rm p}{\rm
a}}_{1}=\big{[}\,1-\tilde{f}\big{(}E_{\bm{p}}-\mu\big{)}-\tilde{f}\big{(}E_{{\bm{p}}^{\prime}}+\mu\big{)}\,\big{]}G_{{\rm
p}{\rm a}},$ (73)

where

$\displaystyle G_{{\rm p}{\rm
a}}=\frac{\,E_{\bm{p}}+E_{{\bm{p}}^{\prime}}\,}{\,\bar{\omega}_{n}^{2}+\big{(}E_{\bm{p}}+E_{{\bm{p}}^{\prime}}\big{)}^{2}\,}\,.$
(74)

Likewise

$\displaystyle\\!\\!\\!\\!{\cal F}^{{\rm a}{\rm p}}\rightarrow{\cal G}^{{\rm
a}{\rm
p}}_{1}=\big{[}\,1-\tilde{f}\big{(}E_{\bm{p}}+\mu\big{)}-\tilde{f}\big{(}E_{{\bm{p}}^{\prime}}-\mu\big{)}\,\big{]}G_{{\rm
p}{\rm a}}.$ (75)

Meanwhile the kinematic factor is

$\displaystyle{\cal K}^{k}_{{\rm p}{\rm a}}=-{\cal K}^{k}_{{\rm a}{\rm
p}}=-\bigg{(}\frac{p^{k}}{E_{\bm{p}}}+\frac{(p^{\prime})^{k}}{E_{{\bm{p}}^{\prime}}}\bigg{)}\sim-2\frac{\,p^{k}\,}{\,E_{\bm{p}}\,}\,,$
(76)

for small ${\bm{q}}$. While each of (pa) and (ap) contributions is UV
divergent, the sum cancels the UV divergence.

$\displaystyle{\cal F}^{{\rm p}{\rm a}}{\cal K}^{k}_{{\rm p}{\rm a}}+{\cal
F}^{{\rm a}{\rm p}}{\cal K}^{k}_{{\rm a}{\rm p}}\,\sim\,{\cal K}^{k}_{{\rm
p}{\rm a}}\big{(}{\cal F}^{{\rm p}{\rm a}}-{\cal F}^{{\rm a}{\rm p}}\big{)}$
$\displaystyle\,\sim\,-2\frac{\,p^{k}\,}{\,E_{\bm{p}}\,}\,G_{{\rm p}{\rm
a}}\big{[}\,\tilde{f}\big{(}E_{{\bm{p}}^{\prime}}-\mu\big{)}-\tilde{f}\big{(}E_{\bm{p}}-\mu\big{)}\,\big{]}-(\mu\leftrightarrow-\mu)$
$\displaystyle\sim-2\frac{\,p^{k}\,}{\,E_{\bm{p}}\,}\frac{\,({\bm{p}}\cdot{\bm{q}})\,}{\,E_{\bm{p}}\,}\,G_{{\rm
p}{\rm a}}\sum_{\xi={\rm p},{\rm
a}}\eta_{\xi}\frac{\,\partial\tilde{f}(x)\,}{\,\partial
x\,}\bigg{|}_{x=E_{\bm{p}}-\eta_{\xi}\mu}$
$\displaystyle\sim-\frac{\,2q^{k}\,}{3}\frac{\,{\bm{p}}^{2}\,}{\,E_{\bm{p}}^{2}\,}\,G_{{\rm
p}{\rm a}}\sum_{\xi={\rm p},{\rm
a}}\eta_{\xi}\frac{\,\partial\tilde{f}(x)\,}{\,\partial
x\,}\bigg{|}_{x=E_{\bm{p}}-\eta_{\xi}\mu}\,,$ (77)

where we took the angular average for ${\bm{p}}$. The CS conductivity is

$\displaystyle\\!\\!\\!\\!\sigma^{{\rm p}{\rm a}+{\rm a}{\rm p}}_{\rm
CSE}(i\bar{\omega}_{n},{\bm{q}})$
$\displaystyle\\!\\!\\!\\!\sim\,-N_{c}\int_{\bm{p}}\frac{\,2\,}{3}\frac{\,{\bm{p}}^{2}\,}{\,E_{\bm{p}}^{2}\,}\,G_{{\rm
p}{\rm a}}\sum_{\xi={\rm p},{\rm
a}}\eta_{\xi}\frac{\,\partial\tilde{f}(x)\,}{\,\partial
x\,}\bigg{|}_{x=E_{\bm{p}}-\eta_{\xi}\mu}\,.$ (78)

Because of $\partial\tilde{f}/\partial x$, the integrand is dominated by
states with $E_{\bm{p}}\pm\mu$. In the massless limit, we find, for the static
limit,

$\displaystyle\sigma^{{\rm p}{\rm p}({\rm normal})}_{{\rm CSE};\,{\it
m}=0}\big{|}^{\omega=0}_{{\bm{q}}\rightarrow
0}~{}\rightarrow~{}\frac{\,N_{c}\mu\,}{\,6\pi^{2}\,}\,.$ (79)

and for the dynamic limit

$\displaystyle\sigma^{{\rm p}{\rm p}({\rm normal})}_{{\rm CSE};\,{\it
m}=0}\big{|}^{\omega\rightarrow
0}_{{\bm{q}}=0}~{}\rightarrow~{}\frac{\,N_{c}\mu\,}{\,6\pi^{2}\,}\,.$ (80)

That is, the static and dynamic limits are the same, as the propagators do not
have the sensitivity to the limiting order.

###  III.3 Antiparticle-antihole

The computations of antiparticle-antihole proceed in the very similar way as
the particle-hole case. The contributions are suppressed by thermal factor as
$\sim{\rm e}^{-(\mu+\Delta)/T}$.

##  IV Numerical results

In this section, we present the numerical results of the CS conductivity for
plural density, quark mass, and temperature. We discuss a normalized CS
conductivity,

$\displaystyle R^{\xi\xi^{\prime}}\equiv\sigma^{\xi\xi^{\prime}}_{\rm
CSE}/\sigma_{0}\,,~{}~{}~{}~{}~{}R=\sum_{\xi,\xi^{\prime}={\rm p},{\rm
a}}R^{\xi\xi^{\prime}}\,,$ (81)

with a $\mu$ dependent normalization factor

$\displaystyle\sigma_{0}=\frac{\,N_{c}\mu\,}{\,2\pi^{2}\,}\,,$ (82)

where $\sigma_{0}$ is the conductivity at massless, zero temperature, and
static limits in the normal phase.

To understand the results in this section, it is useful to use the massless
and zero temperature limit in the normal phase as a baseline. The static
conductivity is

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\big{(}R^{{\rm p}{\rm p}}+R^{{\rm p}{\rm
a}+{\rm a}{\rm p}}+R^{{\rm a}{\rm a}}\big{)}^{\omega=0}_{{\bm{q}}\rightarrow
0}\,\rightarrow\,\frac{2}{\,3\,}+\frac{1}{\,3\,}+0=1\,,$ (83)

and the dynamic conductivity is

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\big{(}R^{{\rm p}{\rm p}}+R^{{\rm p}{\rm
a}+{\rm a}{\rm p}}+R^{{\rm a}{\rm a}}\big{)}^{\omega\rightarrow
0}_{{\bm{q}}=0}\,\rightarrow\,0+\frac{1}{\,3\,}+0=\frac{1}{\,3\,}\,.$ (84)

The difference comes from the response of particle-hole to the external
fields.

To delineate the CS conductivity we found it convenient to discuss the normal
phase as a guideline. We discuss the CS conductivity including the mass
effects, $\mu$ dependence, and temperature dependence. Finally we include the
diquark gap. We focus on the $\omega=0$ case; the impact of $\omega$ can be
readily seen from the propagator factors.

In this section we present the CS conductivity pretending that quark matter
exists from $\mu\geq m$. This picture is not realistic at $\mu\sim m$ for QC2D
where a hadronic phase should exist at $\mu<m_{\pi}/2$ and diquarks with the
mass $m_{\pi}$ condense at $\mu\geq m_{\pi}/2$. At low density there are more
appropriate presentations based on the chiral Lagrangian Avdoshkin  _et al._
(2018), and we will not repeat those discussions. Hence, only our results at
$\mu\gg m$ can be taken at its face value. Here the chiral symmetry is
supposed to get restored and $m$ will be regarded as the current quark mass.
As for comparisons with the lattice simulation setup, we note that the pion
mass used is about $700$ MeV, and we should regard the current quark mass as
rather heavy. We take $m\sim 100$ MeV as a typical value.

###  IV.1 Normal phase

Figure 1: The $\mu$ dependence of the CS conductivity for the quark masses
$m=0,100,200,$ and 300 MeV, where $\omega,T,\Delta$ are set to zero.

First we examine the current quark mass dependence. A finite current quark
mass breaks the chiral symmetry explicitly, blurring qualitative pictures for
chiral conductivities whose descriptions are usually based on massless quarks.
Whether the mass effects enter as a quark mass or the mass of NG modes depend
on the phase structure of the QC2D, but we consider the former case as a guide
for the domain of $\mu\gg m$. Shown in Fig.1 is the (normalized) static CS
conductivity at zero temperature. It sets the overall size. The cases with
various quark masses $m$ are displayed. Clearly the mass characterizes the
onset of quark density. We have checked the scaling

$\displaystyle\sigma_{\rm CSE}\big{|}^{\omega=0}_{{\bm{q}}\rightarrow
0}=\frac{\,N_{c}\sqrt{\mu^{2}-m^{2}\,}\,}{\,2\pi^{2}\,}=\frac{\,N_{c}p_{F}\,}{\,2\pi^{2}\,}\,.$
(85)

where $p_{F}$ is the Fermi momentum in the normal phase, related to the baryon
density as $p_{F}=(3\pi^{2}n_{B}/N_{f})^{1/3}$. This expression is useful to
understand that the CS conductivity is sensitive to the number density, rather
than the chemical potential. Below we set $m=0$.

Figure 2:  The $\mu$ dependence of the CS conductivity as a function of
spatial momenta ${\bm{q}}$. As an eye-guide we show a $R=0.5$ line which is
found at $|{\bm{q}}|=2\mu=2p_{F}$.

Next we examine the $\mu$ dependence of the normalized CS conductivity $R$ at
various external momenta $|{\bm{q}}|$, shown in Fig.2. The $R$ in low momentum
limit approaches $1$ but damps at larger momenta. Its size becomes half at
$|{\bm{q}}|=2\mu=2p_{F}$, but after that the damping proceeds rather slowly.

Figure 3:  The $T$ dependence of the CS conductivity as a function of spatial
momenta ${\bm{q}}$.

Shown in Fig.3 is the temperature dependence of $R$. We fixed $\mu=500$ MeV.
The $R$ at ${\bm{q}}\rightarrow 0$ limit stays constant at a larger
temperature. The growth is more evident at high momentum modes.

###  IV.2 Diquark condensed phase: schematic setup

Figure 4:  The $\Delta$ dependence of the CS conductivity. We set
$T=m=\omega=0$ and $\mu=500$ MeV. The values of $\Delta$ are $0,1,10,50,100,$
and $200$ MeV. We show $R=1/3$ and 1 for eye-guides.  Figure 5:  The $T$
dependence of the CS conductivity. We set $m=\omega=0$, $\mu=500$ MeV, and
$\Delta=200$ MeV. The values of $T$ are $1,50,100,150$, and $200$ MeV.

Now we turn on the diquark gap. The major roles of $\Delta$ is to distinguish
the regimes, $|{\bm{q}}|<\Delta$ and $|{\bm{q}}|>\Delta$, and to determine the
abundance of thermal quarks which is controlled by a thermal factor
$\tilde{f}\sim{\rm e}^{-\Delta/T}$. Its impact is substantial only for the low
momentum or low temperature behaviors, $|{\bm{q}}|<\Delta$ or $T<\Delta$. In
the other domains the results are similar to the normal phase.

Shown in Fig.4 is the $\Delta$ dependence of the static CS conductivity as a
function of spatial momenta ${\bm{q}}$. The values of $\Delta$ are
$0,1,10,50,100,150,$ and $200$ MeV. We set $T=m=\omega=0$ and $\mu=500$ MeV.
It is clear that the size of $\Delta$ determines the domain of ${\bm{q}}$
where the CS conductivity deviates from that in the normal phase. With a
finite $\Delta$, the static conductivity is $R=1/3$, reflecting the
suppression of the particle-hole contributions by diquark gaps.

The temperature dependence of the CS conductivity can be also understood by
the suppression of particle-holes. Shown in Fig.5 are the temperature
dependence of the static CS conductivity at fixed gap, $\Delta=200$ MeV (more
realistic treatments of $\Delta$ will be discussed in the next section). At
low temperature $R=1/3$. As temperature increases from $T=0$ to
$T\simeq\Delta$, thermal quarks contribute as in the normal phase, and $R$
increases from $R=1/3$ to the value $R=1$ of the normal phase.

###  IV.3 Diquark condensed phase: realistic setup

Figure 6:  The $T$ and $\mu$ dependence of the static CS conductivity with
$\Delta$ prepared for comparisons with the lattice simulations. We terminate
the lines for low $\mu$ as our descriptions based on quarks are valid only for
$\mu\geq\mu_{c}$ and $T<T_{\rm SF}$. The low density and temperature domain
should be calculated with hadronic degrees of freedom.

Finally we consider a parameter set consistent with lattice simulations for
QC2D, and consider the static CS conductivity at $\omega=0$ and
${\bm{q}}\rightarrow 0$ for various $\mu$ and $T$. Most of lattice simulations
have been done for relatively heavy pion mass $\simeq 700$ MeV and $\rho$
meson mass of $\sim 900$ MeV. The onset chemical potential $\mu_{c}$ for the
baryon density is $\mu_{c}=m_{\pi}/2\simeq 350$ MeV. For comparisons of our
analytic results with the future lattice simulations in the diquark condensed
phase, we use a relatively large current quark mass of $m=100$ MeV as in our
previous works Kojo and Suenaga (2021); Suenaga and Kojo (2019), and assume
$m_{\pi}=700$ MeV in this setup.

Also, at this stage we introduce the $\mu$ and $T$ dependence of the gap. For
the $\mu$ dependence, we assume the zero temperature gap of the form Kogut
_et al._ (2000)

$\displaystyle\Delta_{0}(\mu)=\bar{\Delta}_{0}\sqrt{1-(\mu_{c}/\mu)^{4}\,}\,\theta(\mu-\mu_{c})\,,$
(86)

as predicted by the chiral effective theories. The $\Delta_{0}$ vanishes at
$\mu=\mu_{c}$, while approaches $\bar{\Delta}_{0}$ at high density. As for the
temperature dependence, we assume the Bardeen-Cooper-Schrieffer (BCS) formulas
as our baseline Schrieffer (1999)

$\displaystyle T_{\rm SF}(\mu)\simeq 0.57\Delta_{0}(\mu)\,,$ (87)

where $T_{\rm SF}(\mu)$ is the critical temperature of the diquark condensed
phase at given $\mu$. The gap depends on $T$ as

$\displaystyle\Delta(\mu,T)\simeq\Delta_{0}(\mu)\,\big{(}1-T/T_{\rm
SF}\big{)}^{1/2}\,.$ (88)

For the high density value of the diquark gap, we take $\bar{\Delta}_{0}=200$
MeV which, according to the BCS formulas, leads to $T_{\rm SF}\simeq 114$ MeV.
This value is consistent with the lattice results $T_{\rm SF}=90-120$ MeV in
the BCS domain of QC2D matter for $\mu\gtrsim 500$ MeV, where the lattice
results showed that $T_{\rm SF}$ depends on $\mu$ weakly. See Ref.Kojo and
Suenaga (2021) for more detailed considerations on the applicability of the
formulas.

Shown in Fig.6 is the static CS conductivity for various $T$ and $\mu$. We
display results only for $\mu\geq\mu_{c}$ and $T<T_{\rm SF}$; at lower $\mu$
and $T$ we should use the hadronic degrees of freedom. The normalized CS
conductivity $R$ is close to $1/3$ at low temperature and approach $R=1$ at
higher temperature as expected for the normal phase. The growth of $R$ is much
quicker than in Fig.5 because now we are including the temperature dependence
of gaps. For the same reason, the rapid change in $R$ at low $\mu$ happens
because the $T_{\rm SF}$ is low, accordingly $\Delta$ is smaller and thermal
quarks can more easily contribute to the conductivity.

We emphasize that our results on the temperature dependence are based on the
assumption that thermal quarks behave as in an ideal gas of (quasi)particles;
that is, once particles and holes are released from diquark condensates, they
behave individually. But the validity of this picture is not immediately
obvious except at very high density. The thermally excited particles and holes
may need to form hadronic objects to avoid the energetic cost of the color
electric flux from isolated quarks. If this is the case, the thermal factor
necessary for excitations are $\sim{\rm e}^{-2\Delta/T}$ of a particle-hole
pair, instead of $\sim{\rm e}^{-\Delta/T}$ for an isolated quark Kojo and
Suenaga (2021); the thermal corrections would be much smaller than we
computed. In this respect the lattice simulations in diquark condensed phase
at low density are important to delineate the properties of thermal
excitations in dense matter.

##  V Discussions

Our calculations of the CS conductivity rely on the validity of quasiparticle
pictures in the diquark condensed phase. While our results on the normal phase
lead to the static CS conductivity at ${\bm{q}}\rightarrow 0$ limit as
expected from the anomaly arguments, our calculations for the diquark
condensed phase lead to only the one-third; the coefficient of the CSE is not
universal even in chiral limit.

This conclusion puzzled us. The triangle graph for the A-V-V type (A: axial-
vector, V: vector) arises when we expand the A-V correlator by the term
$\mu\gamma_{0}$ in the quark propagators as $S\simeq
S_{0}+S_{0}(-\mu\gamma_{0})S_{0}+\cdots$ where $S_{0}$ is the propagator at
$\mu=0$. However, it turned out that such expansion does not yield the same
result as the usual triangle calculations. In fact, the static CS conductivity
at vanishing temperature in the diquark condensed phase can be expanded with
respect to $\mu$ as

$\displaystyle\sigma_{\rm CSE}\big{|}^{\omega=0}_{{\bm{q}}\rightarrow
0}=\frac{N_{c}}{6\pi^{2}}\frac{|\Delta|^{2}}{m^{2}+|\Delta|^{2}}\mu+{\cal
O}(\mu^{2})\ \ \ {\rm at}\ T=0\ .$ (89)

Here we have checked that the higher-order contributions ${\cal O}(\mu^{2})$
are proportional to $|\Delta|^{2}m^{2}$. In this way, the conductivity is
shown to be given by $S_{0}(-\mu\gamma_{0})S_{0}$ part solely in chiral limit
($m\to 0$), resulting in

$\displaystyle\sigma_{\rm CSE}\big{|}^{\omega=0}_{{\bm{q}}\rightarrow
0}\to\frac{C_{\rm uiv.}}{3}\mu\ ,$ (90)

with $C_{\rm univ.}=N_{c}/(2\pi^{2})$ the universal value. It should be noted
that, in Eq. (89), the dimensionless expansion parameter is found to be
$\mu/\sqrt{|\Delta|^{2}+m^{2}}$, hence $m\to 0$ limit to obtain (90) has been
safely taken due to a regulator played by $\Delta$. On the other hand, the
other limit $\Delta\to 0$ in Eq. (89) leads to an incorrect answer:
$\sigma_{\rm CSE}\big{|}^{\omega=0}_{{\bm{q}}\rightarrow 0}\to 0$, showing the
diquark gap $\Delta$ changes the analytic properties of the conductivity in
the infrared.

The above violation of the universality does not seems to be unnatural to us.
The $\mu\gamma_{0}$ vertex in $S_{0}(-\mu\gamma_{0})S_{0}$ introduces no
momenta; the only available momentum, after integrating out the loop momenta,
is the external momentum $q$. Thus the possible form from the expansion of the
A-V correlator in obtaining Eq. (90) should be

$\displaystyle\langle
j^{i}_{5}(q)j_{em}^{j}(-q)j_{B}^{0}(0)\rangle^{\mu=0}_{A=0}~{}\mu
A_{j}\sim\epsilon^{0ijk}\mu q^{j}A^{k}\,.$ (91)

Contracting with the external momentum $q^{i}$, the spatial divergence of
axial current vanishes, showing that our results do not affect
$\partial_{\mu}j^{\mu}_{5}$ and hence the anomaly relation. From this
viewpoint it is not clear to us why the CS coefficient should be constrained
by the anomaly. The possible exceptions for this discussion are the cases when
$A^{k}$ is singular Son and Zhitnitsky (2004); Son and Stephanov (2008), as in
the Dirac string on which
$(\partial_{i}\partial_{j}-\partial_{j}\partial_{i}){\bm{A}}\neq 0$, but this
sort of external fields are presumably not compatible with the linear response
regime utilized in this paper. The radiative corrections to chiral transport
coefficients have been discussed in Refs.Jensen (2012); Gorbar  _et al._
(2013); Feng  _et al._ (2019).

On the other hand, our one-loop calculations shown in the previous sectors
have not taken into account the vertex correction, and hence would miss some
qualitatively important effects. In particular, we must treat the
quasiparticle propagators and the vertices consistently in order to keep the
conservation laws. Our quasiparticle propagators include diquark mean fields
which break the $U(1)_{A}$ axial, baryon number, and electric charge
conservations. The improved vertices produce the poles of NG modes that
recover the conservation laws.

Below we give a brief discussion on the structure of the improved vertices and
will conclude that they do not influence the conclusions in the simplest
quasiparticle computations.

The vertex corrections appear both in the axial-vector current and the vector
current coupled to background electromagnetic fields. The WTI for the axial-
vector vertex leads to

$\displaystyle
q^{\mu}\bar{{\bf\Gamma}}_{\mu}^{A}-2m\Gamma_{5}=iS^{-1}(p+q)\gamma_{5}+\gamma_{5}iS^{-1}(p)\,,$
(92)

where the improved vertex is the sum of the bare and the correction,
$\bar{{\bf\Gamma}}_{\mu}^{A}=(\Gamma_{\rm
bare})_{\mu}^{5}+\delta\Gamma_{\mu}^{A}$, with

$\displaystyle
q^{\mu}\delta\Gamma_{\mu}^{A}=2\left(\begin{array}[]{cc}\,0&\bar{{\bm{\Delta}}}\\\
\,{\bm{\Delta}}&0\\\ \end{array}\right)=Z_{A}\,.$ (95)

[Reminder: ${\bm{\Delta}}=\gamma_{5}\sigma^{2}\tau^{2}\Delta$ and
$\bar{\bf\Delta}=-\gamma_{5}\sigma^{2}\tau^{2}\Delta^{*}$.] Similarly, the WTI
with respect to the $U(1)_{\rm em}$ gauge symmetry leads to

$\displaystyle q^{\mu}\bar{{\bf\Gamma}}^{V}_{\mu}$ $\displaystyle=$
$\displaystyle{\cal Q}i{S}^{-1}(p+q)-i{S}^{-1}(p){\cal Q}\,,$ (96)

where ${\cal Q}={\rm diag}(Q,-Q)$ and $\bar{{\bf\Gamma}}^{V}_{\mu}=\Gamma^{\rm
bare}_{\mu}+\delta\Gamma^{V}_{\mu}$ with

$\displaystyle
q^{\mu}\delta\Gamma_{\mu}^{V}=\left(\begin{array}[]{cc}0&Q\bar{\bm{\Delta}}+\bar{\bm{\Delta}}Q\\\
-Q{\bm{\Delta}}-{\bm{\Delta}}Q&0\\\ \end{array}\right)=Z_{V}\,.$ (99)

For Eqs.(95) and (99) to be valid for $q\rightarrow 0$, the correction of the
vertices ($\delta\Gamma_{\mu}^{A},\delta\Gamma_{\mu}^{V}$) must contains
poles; otherwise the LHS would vanish in spite of the finite RHS. The general
form of solutions is (assuming the linear dispersion between $q_{0}$ and
${\bm{q}}$)

$\displaystyle\delta\Gamma_{\mu}^{s}=\frac{\,\delta^{\mu
0}q_{0}+v_{s}\delta^{i\mu}q_{i}\,}{\,q_{0}^{2}-v^{2}_{s}{\bm{q}}^{2}\,}\,Z_{s}\,,$
(100)

where $s=V,A$ and $v_{s}$ is the medium velocity of the NG mode. From this
expression it is clear the corrections to the vertices are proportional to the
external momenta ${\bm{q}}$. Therefore it yields a term of the form
$\sim\epsilon^{0ijk}q_{i}q_{j}A_{k}$, and drop off from the evaluation of
$\langle j_{5}^{i}\rangle$ for a regular $A_{k}$. Thus, while the NG modes
join the anomalous processes, they do not contribute to the coefficients of
the anomaly relations.

With these discussions, it seems plausible to us that the CS conductivity in
chiral limit depends on the phase structure. We note that our calculations
correspond to the bulk part of matter. If the coefficient of the CSE is indeed
universal, it likely requires the manifest treatments of the boundaries of
matter which in turn introduce space variation of the chemical potential. We
expect that such boundaries accommodate surface zero modes that contribute to
the CS conductance as a global quantity, rather than the conductivity as a
local quantity.

##  VI Conclusions

In this paper we have delineated the CS conductivity within the quasiparticle
picture. The CS conductivity has been calculated in lattice simulations but
the result was for the domain other than the diquark condensed phase
Buividovich  _et al._ (2020). We hope our considerations in this paper to be
tested in near future.

The results depend on the particle-hole contributions which are sensitive to
the phase structure of QC2D. The particle-hole contributions are suppressed in
the presence of diquark condensates. As a consequence, the static CS
conductivity at low momentum limit leads to only the one-third of the normal
phase. We note that this particle-hole suppression is also the origin of the
electromagnetic Meissner mass, relating the CS conductivity to the Meissner
effects. As temperature increases, the particles-holes come back to enhance
the CS conductivity to the size of the normal phase.

In more general context, the nature of thermal corrections carry important
information on the properties of matter other than the diquark condensates.
Thermal excitations on top of the diquark condensed Fermi surface may be
hadronic. This sort of picture has been suggested in the quarkyonic matter
conjecture where the quark matter has the baryonic Fermi surface McLerran and
Pisarski (2007); Kojo  _et al._ (2010a, b); Kojo (2012); Kojo  _et al._
(2012). We studied this point of view by examining gluon propagators in the
diquark condensed phase Kojo and Suenaga (2021), and found indications that
thermal quarks, rather than thermal hadrons, induce too strong thermal
corrections to gluons. In order to derive definite conclusions, however, we
need more lattice data points at low temperature and should also examine the
systematics in our calculations. The CS conductivity may provide us
information complementary with those from gluon propagators.

In the astrophysical aspect, the understandings of excitations in dense QC2D
have important applications to the physics of neutron stars Kojo (2021). In
this context, recently the picture of quark-hadron continuity is actively
discussed Baym  _et al._ (2018); Kojo  _et al._ (2015); Kojo (2016); Baym  _et
al._ (2019); Zhao and Lattimer (2020); Fukushima  _et al._ (2020); Jeong  _et
al._ (2020); Duarte  _et al._ (2020a, b); Ma and Rho (2020, 2021)) to account
for the interplay between the low density nuclear physics Tews  _et al._
(2018), the neutron star radii for 1.4- Miller  _et al._ (2019); Riley  _et
al._ (2019) and 2-solar mass ($M_{\odot}$) neutron stars NIC ; Miller  _et
al._ (2021); Riley  _et al._ (2021); Raaijmakers  _et al._ (2021), and the
maximum mass $\simeq 2.08\pm 0.07M_{\odot}$ Fonseca  _et al._ (2021). In
particular the latest result by the NICER NIC ; Miller  _et al._ (2021); Riley
_et al._ (2021); Raaijmakers  _et al._ (2021) shows that the radii of
$1.4M_{\odot}$ and $2M_{\odot}$ neutron stars are close ($12-13$ km for both),
disfavoring strong first order transitions from the nuclear to quark matter
domain, and implying that there should be rather stiffening with the sound
velocity exceeding the conformal value222To the best of our knowledge, this
behavior first appeared in Ref.Masuda  _et al._ (2013a, b) in the context of
quark-hadron crossover model. A more general argument based on nuclear physics
and neutron star observations was given in Ref.Bedaque and Steiner (2015), and
a microscopic description was proposed in Ref.McLerran and Reddy (2019) in the
quakyonic matter context. , $\sqrt{1/3}$ Baym  _et al._ (2018); Masuda  _et
al._ (2013a, b); Bedaque and Steiner (2015); McLerran and Reddy (2019);
Hippert  _et al._ (2021). These continuity discussions are basically for
neutron star matter at zero temperature. In order to expand them into the
level of finite temperature and general charge chemical potentials, we need to
have more detailed insights on the excitations in dense QCD Kojo  _et al._
(2020). QC2D is an ideal laboratory to delineate these issues and further
studies are called for.

## Acknowledgement

D. S. wishes to thank Naoki Yamamoto for useful comments. Also, the authors
thank Yoshimasa Hidaka and Noriyuki Sogabe for fruitful discussions and
comments. T. K. is supported by NSFC grant No. 11875144.

## References

  * Adler (1969) Stephen L. Adler, “Axial vector vertex in spinor electrodynamics,” Phys. Rev. 177, 2426–2438 (1969).
  * Bell and Jackiw (1969) J. S. Bell and R. Jackiw, “A PCAC puzzle: $\pi^{0}\to\gamma\gamma$ in the $\sigma$ model,” Nuovo Cim. A 60, 47–61 (1969).
  * Alvarez-Gaume and Witten (1984) Luis Alvarez-Gaume and Edward Witten, “Gravitational Anomalies,” Nucl. Phys. B 234, 269 (1984).
  * Vilenkin (1980) A. Vilenkin, “EQUILIBRIUM PARITY VIOLATING CURRENT IN A MAGNETIC FIELD,” Phys. Rev. D 22, 3080–3084 (1980).
  * Kharzeev  _et al._ (2008) Dmitri E. Kharzeev, Larry D. McLerran, and Harmen J. Warringa, “The Effects of topological charge change in heavy ion collisions: ’Event by event P and CP violation’,” Nucl. Phys. A 803, 227–253 (2008), arXiv:0711.0950 [hep-ph] .
  * Fukushima  _et al._ (2008) Kenji Fukushima, Dmitri E. Kharzeev, and Harmen J. Warringa, “The Chiral Magnetic Effect,” Phys. Rev. D 78, 074033 (2008), arXiv:0808.3382 [hep-ph] .
  * Son and Surowka (2009) Dam T. Son and Piotr Surowka, “Hydrodynamics with Triangle Anomalies,” Phys. Rev. Lett. 103, 191601 (2009), arXiv:0906.5044 [hep-th] .
  * Son and Yamamoto (2012) Dam Thanh Son and Naoki Yamamoto, “Berry Curvature, Triangle Anomalies, and the Chiral Magnetic Effect in Fermi Liquids,” Phys. Rev. Lett. 109, 181602 (2012), arXiv:1203.2697 [cond-mat.mes-hall] .
  * Stephanov and Yin (2012) M. A. Stephanov and Y. Yin, “Chiral Kinetic Theory,” Phys. Rev. Lett. 109, 162001 (2012), arXiv:1207.0747 [hep-th] .
  * Yamamoto (2016) Naoki Yamamoto, “Chiral transport of neutrinos in supernovae: Neutrino-induced fluid helicity and helical plasma instability,” Phys. Rev. D 93, 065017 (2016), arXiv:1511.00933 [astro-ph.HE] .
  * Adler and Bardeen (1969) Stephen L. Adler and William A. Bardeen, “Absence of higher order corrections in the anomalous axial vector divergence equation,” Phys. Rev. 182, 1517–1536 (1969).
  * Adler (2005) Stephen L. Adler, “Anomalies to all orders,” in  _50 years of Yang-Mills theory_, edited by G. ’t Hooft (2005) arXiv:hep-th/0405040 .
  * Golkar and Son (2015) Siavash Golkar and Dam T. Son, “(Non)-renormalization of the chiral vortical effect coefficient,” JHEP 02, 169 (2015), arXiv:1207.5806 [hep-th] .
  * Landsteiner (2016) Karl Landsteiner, “Notes on Anomaly Induced Transport,” Acta Phys. Polon. B 47, 2617 (2016), arXiv:1610.04413 [hep-th] .
  * Itoyama and Mueller (1983) Hiroshi Itoyama and Alfred H. Mueller, “The Axial Anomaly at Finite Temperature,” Nucl. Phys. B 218, 349–365 (1983).
  * Sannino (2000) Francesco Sannino, “A Note on anomaly matching for finite density QCD,” Phys. Lett. B 480, 280–286 (2000), arXiv:hep-ph/0002277 .
  * Gaiotto  _et al._ (2015) Davide Gaiotto, Anton Kapustin, Nathan Seiberg, and Brian Willett, “Generalized Global Symmetries,” JHEP 02, 172 (2015), arXiv:1412.5148 [hep-th] .
  * Gaiotto  _et al._ (2017) Davide Gaiotto, Anton Kapustin, Zohar Komargodski, and Nathan Seiberg, “Theta, Time Reversal, and Temperature,” JHEP 05, 091 (2017), arXiv:1703.00501 [hep-th] .
  * Tanizaki  _et al._ (2018) Yuya Tanizaki, Yuta Kikuchi, Tatsuhiro Misumi, and Norisuke Sakai, “Anomaly matching for the phase diagram of massless $\mathbb{Z}_{N}$-QCD,” Phys. Rev. D 97, 054012 (2018), arXiv:1711.10487 [hep-th] .
  * Tanizaki (2018) Yuya Tanizaki, “Anomaly constraint on massless QCD and the role of Skyrmions in chiral symmetry breaking,” JHEP 08, 171 (2018), arXiv:1807.07666 [hep-th] .
  * Wan and Wang (2020) Zheyan Wan and Juven Wang, “Higher anomalies, higher symmetries, and cobordisms III: QCD matter phases anew,” Nucl. Phys. B 957, 115016 (2020), arXiv:1912.13514 [hep-th] .
  * Tanizaki and Ünsal (2020) Yuya Tanizaki and Mithat Ünsal, “Modified instanton sum in QCD and higher-groups,” JHEP 03, 123 (2020), arXiv:1912.01033 [hep-th] .
  * Baym  _et al._ (2018) Gordon Baym, Tetsuo Hatsuda, Toru Kojo, Philip D. Powell, Yifan Song, and Tatsuyuki Takatsuka, “From hadrons to quarks in neutron stars: a review,” Rept. Prog. Phys. 81, 056902 (2018), arXiv:1707.04966 [astro-ph.HE] .
  * Metlitski and Zhitnitsky (2005) Max A. Metlitski and Ariel R. Zhitnitsky, “Anomalous axion interactions and topological currents in dense matter,” Phys. Rev. D 72, 045011 (2005), arXiv:hep-ph/0505072 .
  * Kogut  _et al._ (2000) J. B. Kogut, Misha A. Stephanov, D. Toublan, J. J. M. Verbaarschot, and A. Zhitnitsky, “QCD - like theories at finite baryon density,” Nucl. Phys. B 582, 477–513 (2000), arXiv:hep-ph/0001171 .
  * Kogut  _et al._ (1999) J. B. Kogut, Misha A. Stephanov, and D. Toublan, “On two color QCD with baryon chemical potential,” Phys. Lett. B 464, 183–191 (1999), arXiv:hep-ph/9906346 .
  * Boz  _et al._ (2020) Tamer Boz, Pietro Giudice, Simon Hands, and Jon-Ivar Skullerud, “Dense two-color QCD towards continuum and chiral limits,” Phys. Rev. D 101, 074506 (2020), arXiv:1912.10975 [hep-lat] .
  * Iida  _et al._ (2020) Kei Iida, Etsuko Itou, and Tong-Gyu Lee, “Two-colour QCD phases and the topology at low temperature and high density,” JHEP 01, 181 (2020), arXiv:1910.07872 [hep-lat] .
  * Astrakhantsev  _et al._ (2020) N. Astrakhantsev, V.V. Braguta, E.M. Ilgenfritz, A.Yu. Kotov, and A.A. Nikolaev, “Lattice study of thermodynamic properties of dense QC2D,” Phys. Rev. D 102, 074507 (2020), arXiv:2007.07640 [hep-lat] .
  * Boz  _et al._ (2019) Tamer Boz, Ouraman Hajizadeh, Axel Maas, and Jon-Ivar Skullerud, “Finite-density gauge correlation functions in QC2D,” Phys. Rev. D 99, 074514 (2019), arXiv:1812.08517 [hep-lat] .
  * Kojo and Suenaga (2021) Toru Kojo and Daiki Suenaga, “Thermal quarks and gluon propagators in two-color dense QCD,” (2021), arXiv:2102.07231 [hep-ph] .
  * Suenaga and Kojo (2019) Daiki Suenaga and Toru Kojo, “Gluon propagator in two-color dense QCD: Massive Yang-Mills approach at one-loop,” Phys. Rev. D 100, 076017 (2019), arXiv:1905.08751 [hep-ph] .
  * Kojo and Baym (2014) Toru Kojo and Gordon Baym, “Color screening in cold quark matter,” Phys. Rev. D 89, 125008 (2014), arXiv:1404.1346 [hep-ph] .
  * Buividovich  _et al._ (2020) P. V. Buividovich, D. Smith, and L. von Smekal, “Numerical Study of the Chiral Separation Effect in Two-Color QCD at Finite Density,” (2020), arXiv:2012.05184 [hep-lat] .
  * Nambu (1960) Yoichiro Nambu, “Quasiparticles and Gauge Invariance in the Theory of Superconductivity,” Phys. Rev. 117, 648–663 (1960).
  * Kharzeev and Warringa (2009) Dmitri E. Kharzeev and Harmen J. Warringa, “Chiral Magnetic conductivity,” Phys. Rev. D 80, 034028 (2009), arXiv:0907.5007 [hep-ph] .
  * Araki  _et al._ (2021) Yasufumi Araki, Daiki Suenaga, Kei Suzuki, and Shigehiro Yasui, “Spin-orbital magnetic response of relativistic fermions with band hybridization,” Phys. Rev. Res. 3, 023098 (2021), arXiv:2011.00882 [cond-mat.mes-hall] .
  * Suenaga  _et al._ (2021) Daiki Suenaga, Yasufumi Araki, Kei Suzuki, and Shigehiro Yasui, “Chiral separation effect catalyzed by heavy impurities,” Phys. Rev. D 103, 054041 (2021), arXiv:2012.15173 [hep-ph] .
  * Sun  _et al._ (2007) Gao-feng Sun, Lianyi He, and Pengfei Zhuang, “BEC-BCS crossover in the Nambu-Jona-Lasinio model of QCD,” Phys. Rev. D 75, 096004 (2007), arXiv:hep-ph/0703159 .
  * He (2010) Lianyi He, “Nambu-Jona-Lasinio model description of weakly interacting Bose condensate and BEC-BCS crossover in dense QCD-like theories,” Phys. Rev. D 82, 096003 (2010), arXiv:1007.1920 [hep-ph] .
  * Andersen and Brauner (2010) Jens O. Andersen and Tomas Brauner, “Phase diagram of two-color quark matter at nonzero baryon and isospin density,” Phys. Rev. D 81, 096004 (2010), arXiv:1001.5168 [hep-ph] .
  * Andersen  _et al._ (2015) Jens O. Andersen, Tomas Brauner, and William Naylor, “Confronting effective models for deconfinement in dense quark matter with lattice data,” Phys. Rev. D 92, 114504 (2015), arXiv:1505.05925 [hep-ph] .
  * Avdoshkin  _et al._ (2018) A. Avdoshkin, A. V. Sadofyev, and V. I. Zakharov, “IR properties of chiral effects in pionic matter,” Phys. Rev. D 97, 085020 (2018), arXiv:1712.01256 [hep-ph] .
  * Schrieffer (1999) J.R. Schrieffer,  _Theory Of Superconductivity_, Advanced Books Classics (Avalon Publishing, 1999).
  * Son and Zhitnitsky (2004) D. T. Son and Ariel R. Zhitnitsky, “Quantum anomalies in dense matter,” Phys. Rev. D 70, 074018 (2004), arXiv:hep-ph/0405216 .
  * Son and Stephanov (2008) D. T. Son and M. A. Stephanov, “Axial anomaly and magnetism of nuclear and quark matter,” Phys. Rev. D 77, 014021 (2008), arXiv:0710.1084 [hep-ph] .
  * Jensen (2012) Kristan Jensen, “Triangle Anomalies, Thermodynamics, and Hydrodynamics,” Phys. Rev. D 85, 125017 (2012), arXiv:1203.3599 [hep-th] .
  * Gorbar  _et al._ (2013) E. V. Gorbar, V. A. Miransky, I. A. Shovkovy, and Xinyang Wang, “Radiative corrections to chiral separation effect in QED,” Phys. Rev. D 88, 025025 (2013), arXiv:1304.4606 [hep-ph] .
  * Feng  _et al._ (2019) Bo Feng, De-Fu Hou, and Hai-Cang Ren, “QED radiative corrections to chiral magnetic effect,” Phys. Rev. D 99, 036010 (2019), arXiv:1810.05954 [hep-ph] .
  * McLerran and Pisarski (2007) Larry McLerran and Robert D. Pisarski, “Phases of cold, dense quarks at large N(c),” Nucl. Phys. A 796, 83–100 (2007), arXiv:0706.2191 [hep-ph] .
  * Kojo  _et al._ (2010a) Toru Kojo, Yoshimasa Hidaka, Larry McLerran, and Robert D. Pisarski, “Quarkyonic Chiral Spirals,” Nucl. Phys. A 843, 37–58 (2010a), arXiv:0912.3800 [hep-ph] .
  * Kojo  _et al._ (2010b) Toru Kojo, Robert D. Pisarski, and A. M. Tsvelik, “Covering the Fermi Surface with Patches of Quarkyonic Chiral Spirals,” Phys. Rev. D 82, 074015 (2010b), arXiv:1007.0248 [hep-ph] .
  * Kojo (2012) Toru Kojo, “A (1+1) dimensional example of Quarkyonic matter,” Nucl. Phys. A 877, 70–94 (2012), arXiv:1106.2187 [hep-ph] .
  * Kojo  _et al._ (2012) Toru Kojo, Yoshimasa Hidaka, Kenji Fukushima, Larry D. McLerran, and Robert D. Pisarski, “Interweaving Chiral Spirals,” Nucl. Phys. A 875, 94–138 (2012), arXiv:1107.2124 [hep-ph] .
  * Kojo (2021) Toru Kojo, “QCD equations of state and speed of sound in neutron stars,” AAPPS Bull. 31, 11 (2021), arXiv:2011.10940 [nucl-th] .
  * Kojo  _et al._ (2015) Toru Kojo, Philip D. Powell, Yifan Song, and Gordon Baym, “Phenomenological QCD equation of state for massive neutron stars,” Phys. Rev. D 91, 045003 (2015), arXiv:1412.1108 [hep-ph] .
  * Kojo (2016) Toru Kojo, “Phenomenological neutron star equations of state: 3-window modeling of QCD matter,” Eur. Phys. J. A 52, 51 (2016), arXiv:1508.04408 [hep-ph] .
  * Baym  _et al._ (2019) Gordon Baym, Shun Furusawa, Tetsuo Hatsuda, Toru Kojo, and Hajime Togashi, “New Neutron Star Equation of State with Quark-Hadron Crossover,” Astrophys. J. 885, 42 (2019), arXiv:1903.08963 [astro-ph.HE] .
  * Zhao and Lattimer (2020) Tianqi Zhao and James M. Lattimer, “Quarkyonic Matter Equation of State in Beta-Equilibrium,” (2020), arXiv:2004.08293 [astro-ph.HE] .
  * Fukushima  _et al._ (2020) Kenji Fukushima, Toru Kojo, and Wolfram Weise, “Hard-core deconfinement and soft-surface delocalization from nuclear to quark matter,” Phys. Rev. D 102, 096017 (2020), arXiv:2008.08436 [hep-ph] .
  * Jeong  _et al._ (2020) Kie Sang Jeong, Larry McLerran, and Srimoyee Sen, “Dynamically generated momentum space shell structure of quarkyonic matter via an excluded volume model,” Phys. Rev. C 101, 035201 (2020), arXiv:1908.04799 [nucl-th] .
  * Duarte  _et al._ (2020a) Dyana C. Duarte, Saul Hernandez-Ortiz, and Kie Sang Jeong, “Excluded-volume model for quarkyonic Matter: Three-flavor baryon-quark Mixture,” Phys. Rev. C 102, 025203 (2020a), arXiv:2003.02362 [nucl-th] .
  * Duarte  _et al._ (2020b) Dyana C. Duarte, Saul Hernandez-Ortiz, and Kie Sang Jeong, “Excluded-volume model for quarkyonic matter. II. Three-flavor shell-like distribution of baryons in phase space,” Phys. Rev. C 102, 065202 (2020b), arXiv:2007.08098 [nucl-th] .
  * Ma and Rho (2020) Yong-Liang Ma and Mannque Rho, “Towards the hadron–quark continuity via a topology change in compact stars,” Prog. Part. Nucl. Phys. 113, 103791 (2020), arXiv:1909.05889 [nucl-th] .
  * Ma and Rho (2021) Yong-Liang Ma and Mannque Rho, “The sound speed and core of massive compact stars: A manifestation of hadron-quark duality,” (2021), arXiv:2104.13822 [nucl-th] .
  * Tews  _et al._ (2018) Ingo Tews, Joseph Carlson, Stefano Gandolfi, and Sanjay Reddy, “Constraining the speed of sound inside neutron stars with chiral effective field theory interactions and observations,” Astrophys. J. 860, 149 (2018), arXiv:1801.01923 [nucl-th] .
  * Miller  _et al._ (2019) M.C. Miller _et al._ , “PSR J0030+0451 Mass and Radius from $NICER$ Data and Implications for the Properties of Neutron Star Matter,” Astrophys. J. Lett. 887, L24 (2019), arXiv:1912.05705 [astro-ph.HE] .
  * Riley  _et al._ (2019) Thomas E. Riley _et al._ , “A $NICER$ View of PSR J0030+0451: Millisecond Pulsar Parameter Estimation,” Astrophys. J. Lett. 887, L21 (2019), arXiv:1912.05702 [astro-ph.HE] .
  * (69) “NICER Collaboration,Press release(2021),” https://www.nasa.gov/feature/goddard/2021/.
  * Miller  _et al._ (2021) M. C. Miller _et al._ , “The Radius of PSR J0740+6620 from NICER and XMM-Newton Data,” (2021), arXiv:2105.06979 [astro-ph.HE] .
  * Riley  _et al._ (2021) Thomas E. Riley _et al._ , “A NICER View of the Massive Pulsar PSR J0740+6620 Informed by Radio Timing and XMM-Newton Spectroscopy,” (2021), arXiv:2105.06980 [astro-ph.HE] .
  * Raaijmakers  _et al._ (2021) G. Raaijmakers, S. K. Greif, K. Hebeler, T. Hinderer, S. Nissanke, A. Schwenk, T. E. Riley, A. L. Watts, J. M. Lattimer, and W. C. G. Ho, “Constraints on the dense matter equation of state and neutron star properties from NICER’s mass-radius estimate of PSR J0740+6620 and multimessenger observations,” (2021), arXiv:2105.06981 [astro-ph.HE] .
  * Fonseca  _et al._ (2021) E. Fonseca _et al._ , “Refined Mass and Geometric Measurements of the High-Mass PSR J0740+6620,” (2021), arXiv:2104.00880 [astro-ph.HE] .
  * Masuda  _et al._ (2013a) Kota Masuda, Tetsuo Hatsuda, and Tatsuyuki Takatsuka, “Hadron-Quark Crossover and Massive Hybrid Stars with Strangeness,” Astrophys. J. 764, 12 (2013a), arXiv:1205.3621 [nucl-th] .
  * Masuda  _et al._ (2013b) Kota Masuda, Tetsuo Hatsuda, and Tatsuyuki Takatsuka, “Hadron–quark crossover and massive hybrid stars,” PTEP 2013, 073D01 (2013b), arXiv:1212.6803 [nucl-th] .
  * Bedaque and Steiner (2015) Paulo Bedaque and Andrew W. Steiner, “Sound velocity bound and neutron stars,” Phys. Rev. Lett. 114, 031103 (2015), arXiv:1408.5116 [nucl-th] .
  * McLerran and Reddy (2019) Larry McLerran and Sanjay Reddy, “Quarkyonic Matter and Neutron Stars,” Phys. Rev. Lett. 122, 122701 (2019), arXiv:1811.12503 [nucl-th] .
  * Hippert  _et al._ (2021) Maurício Hippert, Eduardo S. Fraga, and Jorge Noronha, “Insights on the peak in the speed of sound of ultradense matter,” (2021), arXiv:2105.04535 [nucl-th] .
  * Kojo  _et al._ (2020) Toru Kojo, Defu Hou, Jude Okafor, and Hajime Togashi, “Phenomenological QCD equations of state for neutron star dynamics: nuclear-2SC continuity and evolving effective couplings,” (2020), arXiv:2012.01650 [astro-ph.HE] .