# Chiral separation effect catalyzed by heavy impurities

Daiki Suenaga<EMAIL_ADDRESS>Research Center for Nuclear Physics,
Osaka University, Ibaraki, 567-0048, Japan  Yasufumi Araki
<EMAIL_ADDRESS>Advanced Science Research Center, Japan Atomic
Energy Agency (JAEA), Tokai 319-1195, Japan  Kei Suzuki
<EMAIL_ADDRESS>Advanced Science Research Center, Japan
Atomic Energy Agency (JAEA), Tokai 319-1195, Japan  Shigehiro Yasui
<EMAIL_ADDRESS>Research and Education Center for Natural Sciences,
Keio University, Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521, Japan  RIKEN
iTHEMS, Wako, Saitama 351-0198, Japan

###### Abstract

We investigate the influence of Kondo effect, namely, the nonperturbative
effect induced by heavy impurities, on the chiral separation effect (CSE) in
quark matter. We employ a simple effective model incorporating the Kondo
condensate made of a light quark and a heavy quark, and compute the response
function of axial current to the magnetic field in static limit and dynamical
limit. As a result, we find that the Kondo effect catalyzes the CSE in both
the limits, and particularly the CSE in dynamical limit can be enhanced by a
factor of approximately three. Our findings clearly show that the presence of
heavy impurities in quark matter can play an important role in the transport
phenomena of light quarks induced by a magnetic field.

##  I Introduction

Investigation of currents induced by an external magnetic field is one of the
most important subjects in association with heavy-ion collision (HIC) and
neutron star (NS) physics. For example, it was proposed that a magnetic field
applied in chirality imbalanced matter gives rise to a vector current, which
is called the chiral magnetic effect (CME) Kharzeev (2006); Kharzeev and
Zhitnitsky (2007); Kharzeev  _et al._ (2008); Fukushima  _et al._ (2008) (see
also Ref. Vilenkin (1980a); Nielsen and Ninomiya (1983) for earlier works).
Another example is the chiral separation effect (CSE) Son and Zhitnitsky
(2004); Metlitski and Zhitnitsky (2005); Newman and Son (2006) which refers to
an axial current induced by a magnetic field in ordinary baryonic or quark
matter. The CME has been mostly studied with respect to high-energy HICs in
which a chirality imbalanced environment may be created by metastable states
related to the $\theta$-vacua Kharzeev  _et al._ (1998); Kharzeev and Pisarski
(2000); Kharzeev  _et al._ (2002). The CSE is expected to play an important
role in low-energy HICs in which finite baryon density can be created, as well
as in NSs.

In quark matter, in addition to light quarks ($u,d$) with the Fermi surface,
heavy quarks ($c$) may exist as impurities.111In low-energy HICs, heavy quarks
can be created by hard processes of gluons, and they do not participate in the
equilibration of light quarks. In such a situation, light quarks near the
Fermi surface can be correlated with the heavy quarks in a nonperturbative way
due to the non-Abelian interaction with $SU(N_{c})$ symmetry ($N_{c}$ is the
number of colors) of quantum chromodynamics (QCD). As a result, a condensate
made of a light quark and a heavy quark is created, which is the so-called
Kondo condensate. The emergence of Kondo condensate and phenomena induced by
the condensate are called Kondo effect.222The Kondo effect driven in QCD is
sometimes called QCD Kondo effect. While the Kondo effect was originally found
in metal including impurities in the context of condensed-matter physics Kondo
(1964); Hewson (1993); Yosida (1996); Yamada (2004); Coleman (2015), it has
been theoretically shown to emerge in quark matter Yasui and Sudoh (2013);
Hattori  _et al._ (2015) (See Refs. Ozaki  _et al._ (2016); Yasui  _et al._
(2019); Yasui (2017); Kanazawa and Uchino (2016); Kimura and Ozaki (2017);
Yasui and Ozaki (2017); Suzuki  _et al._ (2017); Yasui  _et al._ (2017);
Kimura and Ozaki (2019); Fariello  _et al._ (2019); Hattori  _et al._ (2019);
Suenaga  _et al._ (2020a, b); Kanazawa (2020); Araki  _et al._ ; Araki  _et
al._ (2020); Kimura (2020) for related discussions).333The Kondo effect in
high-energy physics emerges not only in quark matter with heavy quarks but
also in nuclear matter with heavy hadrons Yasui and Sudoh (2013, 2017); Yasui
(2016); Yasui and Miyamoto (2019). While the former is caused by the QCD
interaction, the latter is caused by spin and/or isospin ones.

The Kondo effect is known to change the transport phenomena
characteristically. For example, this effect tends to suppress the electric
conductivity of charged particles as we lower the temperature. Similarly, we
expect that the Kondo effect also influences the transport phenomena induced
by a magnetic field. Based on this expectation, in this paper we discuss
modification of CSE due to the Kondo effect in quark matter, by focusing on
the response function of the axial current to a magnetic field. In the
ordinary quark matter where the Kondo condensate is absent, the magnitude of
CSE was found to be slightly modified by perturbative corrections in quantum
electrodynamics (QED) Gorbar  _et al._ (2013). On the other hand, in this
paper, we investigate modification of the CSE, not from perturbative effects
of QED but from nonperturbative corrections of QCD driven by the Kondo effect.
Hence we expect a significant modification of the CSE, and in fact this is the
case as we will see later.

To focus on the impact of Kondo effect on the CSE, we employ a simple
effective model which contains light quarks with a finite chemical potential
and heavy quarks defined within the heavy quark effective theory (HQET)
Eichten and Hill (1990); Georgi (1990); Neubert (1994); Manohar and Wise
(2000), in the presence of Kondo condensate. For evaluation of the response
function of CSE, we make use of the linear response theory Kapusta and Gale
(2011). This method yields the CSE in two types of limits which are the so-
called static limit and dynamical limit. The former (latter) describes an
induced current under an external magnetic field whose time dependence is
slower (faster) than the equilibration of the spatial part of the system, and
in general these two limits give different values of induced current Kharzeev
and Warringa (2009); Hou  _et al._ (2011); Son and Yamamoto (2013);
Landsteiner  _et al._ (2014); Kharzeev  _et al._ (2017). Thus, we investigate
the response function of CSE in both the static limit and dynamical limit.

This paper is organized as follows. In Sec. II we present a model employed in
our analysis and derive the Green’s function of fermions modified by the Kondo
condensate. In Sec. III we explain briefly the linear response theory and show
our strategy to evaluate the CSE. In Sec. IV, we present numerical results of
response function of the CSE in both static limit and dynamical limit. In Sec.
V we discuss gauge invariance of our computation, and in Sec. VI we conclude
the present work.

##  II Model

Here, we introduce an effective Lagrangian toward the investigation of CSE in
the presence of Kondo condensate. In the present work we are particularly
interested in the response of axial current to an external magnetic field, so
that we do not derive the Kondo condensate in a concrete model, such as the
Nambu–Jona-Lasinio–type (NJL-type) model which is powerful for the study of
phase structure as done in the literatures Yasui  _et al._ (2019, 2017).
Alternatively, we simply assume the presence of the Kondo condensate in a
reasonable form, for the clarity of discussion.

The kinetic terms of massless light quarks at finite density and heavy quarks
put as impurities are given by

$\displaystyle{\cal L}_{\rm
kin}=\bar{\psi}(i{\ooalign{\hfil/\hfil\crcr$D$}}+\mu\gamma_{0})\psi+\Psi_{v}^{\dagger}iD_{0}\Psi_{v}\
.$ (3)

In this Lagrangian, while the light quark field $\psi$ is given as an ordinary
Dirac fermion, the heavy quark field $\Psi_{v}$ is given within the HQET
Eichten and Hill (1990); Georgi (1990); Neubert (1994); Manohar and Wise
(2000), i.e., $\Psi_{v}$ is related to the ordinary Dirac field $\Psi$ as
$\Psi_{v}=\frac{1+\gamma_{0}}{2}{\rm e}^{iM_{Q}t}\Psi$ ($M_{Q}$ is the mass of
$\Psi$ and $t$ describes the time of the system) in the rest frame of the
heavy quark, which allows us to take into account only the particle component
of the heavy quark together with the energy and momentum of order
$\Lambda_{\rm QCD}$. $\mu$ is a chemical potential related to the density of
light quarks.

In Eq. (3), we have introduced covariant derivatives for the quarks to
incorporate a magnetic field as

$\displaystyle D_{\mu}\psi$ $\displaystyle=$
$\displaystyle(\partial_{\mu}+ie_{q}A_{\mu})\psi\ ,$ $\displaystyle
D_{0}\Psi_{v}$ $\displaystyle=$
$\displaystyle(\partial_{0}+ie_{Q}A_{0})\Psi_{v}\ $ (4)

[$A^{\mu}=(A_{0},{\bm{A}})$)]. In Eq. (4), $e_{q}$ and $e_{Q}$ are the
electric charges of light and heavy quarks, respectively. We note that the
magnetic coupling between the heavy quark and the gauge field is absent in the
Lagrangian (3) because of the heavy-quark spin-symmetry of the HQET Eichten
and Hill (1990); Georgi (1990); Neubert (1994); Manohar and Wise (2000). This
is explicitly shown by the lack of spatial derivative of $\Psi_{v}$ in Eq. (3)
in the rest frame. In other words, the heavy quark do not couple with the
magnetic field directly.

Field-theoretically, the Kondo effect can be described by a condensate formed
by a light quark and a heavy quark, referred to as the Kondo condensate, which
is analogous to the diquark condensate for color superconductivity Alford  _et
al._ (1998); Rapp  _et al._ (1998); Alford  _et al._ (1999, 2008) or the
chiral condensate for spontaneous chiral-symmetry breaking Nambu and Jona-
Lasinio (1961). One of the most concise effective terms incorporating the
Kondo condensate may be given as

$\displaystyle{\cal L}_{\Delta}$ $\displaystyle=$
$\displaystyle\Delta_{S}(\bar{\psi}\Psi_{v})+{\bm{\Delta}}_{V}\cdot(\bar{\psi}{\bm{\gamma}}\Psi_{v})+({\rm
c.c.})\ .$ (5)

Here we have included the vector condensate ${\bm{\Delta}}_{V}$ in addition to
the scalar one $\Delta_{S}$ as naturally suggested by the chiral partner
structure of HQET Bardeen and Hill (1994); Nowak  _et al._ (1993). Namely,
when the chiral symmetry is significantly restored at finite density, the
masses of scalar and vector heavy-light modes tend to degenerate with the
heavy-quark spin symmetry Suenaga  _et al._ (2015); Harada  _et al._ (2017);
Suenaga  _et al._ (2017), which suggests that those modes play comparable
roles in quark matter. We note that pseudoscalar and axial-vector condensates
could also be present in Eq. (5), but we have not included them due to the
assumption of parity invariance of the ground state. According to Refs. Yasui
_et al._ (2019, 2017) based on the NJL-type analysis, the condensates in
momentum space may take the ansatz of

$\displaystyle\Delta_{S}=\Delta\ ,$
$\displaystyle{\bm{\Delta}}_{V}=\Delta\hat{\bm{p}}\ ,$ (6)

($\hat{\bm{p}}={\bm{p}}/|{\bm{p}}|$) where $\Delta$ is a constant providing an
order-parameter in the ground state. In particular, the second line in Eq. (6)
is called the hedgehog ansatz.

In what follows, we assume that the light and heavy quarks carry the identical
electric charge as

$\displaystyle e\equiv e_{q}=e_{Q}\ ,$ (7)

which allows us to avoid complexity caused by the spontaneous breakdown of the
electromagnetic $U(1)_{\rm EM}$ symmetry and the appearance of the so-called
Nambu-Goldstone (NG) mode. For example, if we regard the heavy quark as the
charm quark, then the light quark is identified as the up quark, leading to
$e\equiv e_{q}=e_{Q}=+\frac{2}{3}\hat{e}$, where $\hat{e}$ is the elementary
charge. For the bottom quark and down quark, $e\equiv
e_{q}=e_{Q}=-\frac{1}{3}\hat{e}$.

Next, we derive a Green’s function of the fermions incorporating $\Delta$ in
the absence of the external gauge field ${\bm{A}}$. The effective Largangian
employed in this study is given by the sum of Eqs. (3) and (5):

$\displaystyle{\cal L}={\cal L}_{\rm kin}+{\cal L}_{\Delta}\ .$ (8)

Thus, by reading quadratic terms of the light and heavy quarks from the
Lagrangian (8) without ${\bm{A}}$ in momentum space, we can easily get the
Green’s function of the fermions as

$\displaystyle\tilde{\cal G}_{0}(p_{0},{\bm{p}})$ $\displaystyle=$
$\displaystyle\frac{i}{(p_{0}-E^{\rm
a}_{\bm{p}})(p_{0}-E_{\bm{p}}^{+})(p_{0}-E_{\bm{p}}^{-})}$ (12)
$\displaystyle\times\left(\begin{array}[]{ccc}p_{0}(p_{0}+\mu)-|\Delta|^{2}&-\left(|{\bm{p}}|p_{0}+|\Delta|^{2}\right)\hat{\bm{p}}\cdot{\bm{\sigma}}&-\Delta^{*}(p_{0}+|{\bm{p}}|+\mu)\\\
\left(|{\bm{p}}|p_{0}+|\Delta|^{2}\right)\hat{\bm{p}}\cdot{\bm{\sigma}}&-p_{0}(p_{0}+\mu)+|\Delta|^{2}&-\Delta^{*}(p_{0}+|{\bm{p}}|+\mu)\hat{\bm{p}}\cdot{\bm{\sigma}}\\\
-\Delta(p_{0}+|{\bm{p}}|+\mu)&\Delta(p_{0}+|{\bm{p}}|+\mu)\hat{\bm{p}}\cdot{\bm{\sigma}}&(p_{0}-|{\bm{p}}|+\mu)(p_{0}+|{\bm{p}}|+\mu)\\\
\end{array}\right)\ ,$

in the Dirac representation, with the dispersion relations for the fermions

$\displaystyle E_{\bm{p}}^{+}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(|{\bm{p}}|-\mu+\sqrt{(|{\bm{p}}|-\mu)^{2}+8|\Delta|^{2}}\right)\
,$ $\displaystyle E_{\bm{p}}^{-}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(|{\bm{p}}|-\mu-\sqrt{(|{\bm{p}}|-\mu)^{2}+8|\Delta|^{2}}\right)\
,$ $\displaystyle E^{\rm a}_{\bm{p}}$ $\displaystyle=$
$\displaystyle-|{\bm{p}}|-\mu\ .$ (13)

We note that the inverse of the Green’s function is a $6\times 6$ matrix
because the light quark field $\psi$ is a four-component Dirac spinor while
the heavy quark field $\Psi_{v}$ is essentially two-component one carrying
only its particle component. Throughout this paper, the modes carrying the
dispersion of $E_{\bm{p}}^{+}$, $E_{\bm{p}}^{-}$ and $E_{\bm{p}}^{\rm a}$ are
denoted by “q.p.” (Kondo quasiparticle), “q.h.” (Kondo quasihole) and “a.p.”
(light antiparticle), respectively. For later use, we plot a schematic
behavior of these dispersions in Fig. 1. In this figure, solid red, dashed
blue, and dotted purple curves correspond to $p_{0}=E_{\bm{p}}^{+}$ (q.p.),
$p_{0}=E_{\bm{p}}^{-}$ (q.h.), and $p_{0}=E_{\bm{p}}^{\rm a}$ (a.p.),
respectively. The gray horizontal line indicates $p_{0}=0$ corresponding to
the Fermi level. We find that $E_{\bm{p}}^{+}$ is always positive while
$E_{\bm{p}}^{-}$ and $E_{\bm{p}}^{\rm a}$ are always negative as long as
$\Delta$ is finite.

Figure 1: A schematic behavior of the dispersions in Eq. (13)
($p=|{\bm{p}}|$). Solid red, dashed blue, and dotted purple curves correspond
to $p_{0}=E_{\bm{p}}^{+}$ (q.p.), $p_{0}=E_{\bm{p}}^{-}$ (q.h.), and
$p_{0}=E_{\bm{p}}^{\rm a}$ (a.p.), respectively. The gray horizontal line
indicates $p_{0}=0$ corresponding to the Fermi level.

Equation (12) is the Green’s function found in the previous works Yasui  _et
al._ (2019, 2017). In the present paper, we rewrite Eq. (12) into a more
convenient form which shows its physical meaning manifestly. Namely, we
transform Eq. (12) into the following form:

$\displaystyle\tilde{\cal
G}_{0}(p_{0},{\bm{p}})=\left(\begin{array}[]{cc}\tilde{\cal
G}_{0}^{\bar{\psi}\psi}(p_{0},{\bm{p}})&\tilde{\cal
G}_{0}^{\bar{\psi}\Psi_{v}}(p_{0},{\bm{p}})\\\ \tilde{\cal
G}_{0}^{{\Psi}_{v}^{\dagger}\psi}(p_{0},{\bm{p}})&\tilde{\cal
G}_{0}^{{\Psi}_{v}^{\dagger}\Psi_{v}}(p_{0},{\bm{p}})\\\ \end{array}\right)\
,$ (16)

where the elements are given as

$\displaystyle\tilde{\cal G}_{0}^{\bar{\psi}\psi}(p_{0},{\bm{p}})$
$\displaystyle=$ $\displaystyle
i\left[\frac{U_{+}({\bm{p}})}{p_{0}-E_{\bm{p}}^{+}}+\frac{U_{-}({\bm{p}})}{p_{0}-E_{\bm{p}}^{-}}\right]\Lambda_{\rm
p}$ $\displaystyle+i\frac{U_{\rm a}({\bm{p}})}{p_{0}-{E}_{\bm{p}}^{\rm
a}}\Lambda_{\rm a}\,,$ $\displaystyle\tilde{\cal
G}_{0}^{\bar{\psi}\Psi_{v}}(p_{0},{\bm{p}})$ $\displaystyle=$ $\displaystyle
i\left[\frac{V^{*}_{+}({\bm{p}})}{p_{0}-E_{\bm{p}}^{+}}+\frac{V^{*}_{-}({\bm{p}})}{p_{0}-E_{\bm{p}}^{-}}\right]\Lambda_{{\rm
p}{\rm H}}\ ,$ $\displaystyle\tilde{\cal
G}_{0}^{{\Psi}_{v}^{\dagger}\psi}(p_{0},{\bm{p}})$ $\displaystyle=$
$\displaystyle
i\left[\frac{V_{+}({\bm{p}})}{p_{0}-E_{\bm{p}}^{+}}+\frac{V_{-}({\bm{p}})}{p_{0}-E_{\bm{p}}^{-}}\right]\Lambda_{{\rm
H}{\rm p}}\ ,$ $\displaystyle\tilde{\cal
G}_{0}^{{\Psi}_{v}^{\dagger}\Psi_{v}}(p_{0},{\bm{p}})$ $\displaystyle=$
$\displaystyle
i\left[\frac{W_{+}({\bm{p}})}{p_{0}-E_{\bm{p}}^{+}}+\frac{W_{-}({\bm{p}})}{p_{0}-E_{\bm{p}}^{-}}\right]{\bm{1}}\
.$ (17)

In these expressions, $\tilde{\cal G}_{0}^{\bar{\psi}\psi}$, $\tilde{\cal
G}_{0}^{\bar{\psi}\Psi_{v}}$, $\tilde{\cal G}_{0}^{{\Psi}_{v}^{\dagger}\psi}$,
and $\tilde{\cal G}_{0}^{{\Psi}_{v}^{\dagger}\Psi_{v}}$ are $4\times 4$,
$4\times 2$, $2\times 4$, and $2\times 2$ matrices, respectively. Also,

$\displaystyle\Lambda_{\rm
p}\equiv\frac{1+\hat{\bm{p}}\cdot{\bm{\alpha}}}{2}\gamma_{0}\ ,\ \
\Lambda_{\rm a}\equiv\frac{1-\hat{\bm{p}}\cdot{\bm{\alpha}}}{2}\gamma_{0}$
(18)

(${\bm{\alpha}}=\gamma_{0}{\bm{\gamma}}$) are the projection operators for
positive- and negative-energy states of the light quark, respectively.
Similarly

$\displaystyle\Lambda_{{\rm p}{\rm H}}\equiv\left(\begin{array}[]{c}1\\\
\hat{\bm{p}}\cdot{\bm{\sigma}}\\\ \end{array}\right)\ ,\ \ \Lambda_{{\rm
H}{\rm p}}\equiv\left(\begin{array}[]{cc}1&-\hat{\bm{p}}\cdot{\bm{\sigma}}\\\
\end{array}\right)$ (22)

are the operators mixing positive-energy states of the light quark and the
heavy quark.444One can understand these operators easily by
$\displaystyle\Lambda_{\rm p}\Lambda_{\rm H}=\left(\begin{array}[]{cc}1&0\\\
\hat{\bm{p}}\cdot{\bm{\sigma}}&0\\\ \end{array}\right)\ ,\ \ \Lambda_{\rm
H}\Lambda_{\rm
p}=\left(\begin{array}[]{cc}1&-\hat{\bm{p}}\cdot{\bm{\sigma}}\\\ 0&0\\\
\end{array}\right)$ (27) with $\Lambda_{\rm H}=(1+\gamma_{0})/2$. ${\bm{1}}$
is a $2\times 2$ unit matrix. The weight factors $U({\bm{p}})$’s,
$V({\bm{p}})$’s, and $W({\bm{p}})$’s in Eq. (17) are

$\displaystyle
U_{+}({\bm{p}})=\frac{2(|\Delta|^{2}+|{\bm{p}}|E_{\bm{p}}^{+})}{(E_{\bm{p}}^{-}-E_{\bm{p}}^{+})(E_{\bm{p}}^{\rm
a}-E_{\bm{p}}^{+})}\ ,$ $\displaystyle
U_{-}({\bm{p}})=\frac{2(|\Delta|^{2}+|{\bm{p}}|E_{\bm{p}}^{-})}{(E_{\bm{p}}^{+}-E_{\bm{p}}^{-})(E_{\bm{p}}^{\rm
a}-E_{\bm{p}}^{-})}\ ,$ $\displaystyle U_{\rm a}({\bm{p}})=1\ ,$
$\displaystyle V_{+}({\bm{p}})=\frac{\Delta}{E_{\bm{p}}^{-}-E_{\bm{p}}^{+}}\
,\ \ V_{-}({\bm{p}})=\frac{\Delta}{E_{\bm{p}}^{+}-E_{\bm{p}}^{-}}\ ,$
$\displaystyle
W_{+}({\bm{p}})=\frac{E_{\bm{p}}^{+}-|{\bm{p}}|+\mu}{E_{\bm{p}}^{+}-E_{\bm{p}}^{-}}\
,\ \
W_{-}({\bm{p}})=\frac{E_{\bm{p}}^{-}-|{\bm{p}}|+\mu}{E_{\bm{p}}^{-}-E_{\bm{p}}^{+}}\
.$ (28)

Equations (17) and (28) clearly show that only positive-energy components of
the light quark correlate with the heavy quark. Namely $U_{\rm a}({\bm{p}})$
in Eq. (28) is not changed from unity. In what follows we assume $\Delta$ to
be real.

##  III Analysis

###  III.1 Linear response theory

In Sec. II we introduced an effective Lagrangian with an appropriate ansatz
describing the Kondo condensate, and derived the Green’s function of fermions.
Here, we show our strategy to investigate the CSE in the presence of Kondo
condensate based on them. In the following calculation, we set $A_{0}=0$ since
the magnetic field is not generated by $A_{0}$.

The axial current is defined by $j_{5}^{i}=\bar{\psi}\gamma^{i}\gamma_{5}\psi$
($i=1,2,3$) including only the light-quark degrees of freedom. Its expectation
value can be given by

$\displaystyle\langle
j_{5}^{i}(t,{\bm{x}})\rangle_{\beta}=\frac{1}{\Omega}\lim_{q_{0},{\bm{q}}\to
0}\langle\tilde{j}_{5}^{i}(q_{0},{\bm{q}})\rangle_{\beta}\ ,$ (29)

where $\langle\cdots\rangle_{\beta}$ stands for a thermodynamic expectation
value and $\Omega=\beta V$ ($\beta=1/T$ is the inverse of temperature, and $V$
is the three-dimensional volume), and $\tilde{j}_{5}^{i}(q_{0},{\bm{q}})$ is
the Fourier transformation of $j_{5}^{i}(t,{\bm{x}})$. Thus, first we need to
evaluate the expectation value of axial current in momentum space. In the
present study, we are particularly interested in quark matter under a magnetic
field with magnitude smaller than that of the Fermi energy: $\sqrt{eB}\ll\mu$.
Therefore we can evaluate the CSE within the framework of the linear response
theory Kapusta and Gale (2011).555The axial current can be calculated
directly, but the computation is rather complicated in general. See Appendix A
for another derivation of the static limit of the axial current in case of
$e=e_{q}=e_{Q}$. The linear response theory tells us that the response
function to an external field takes the form of the retarded one. The retarded
function could be given by an analytic continuation of the one computed within
the imaginary-time formalism. Thus
$\langle\tilde{j}_{5}^{i}(q_{0},{\bm{q}})\rangle_{\beta}$ in the right-hand
side (RHS) of Eq. (29) is provided by

$\displaystyle\langle\tilde{j}_{5}^{i}(q_{0},{\bm{q}})\rangle_{\beta}=\langle\tilde{j}_{5}^{i}(i\bar{\omega}_{n},{\bm{q}})\rangle_{\beta}|_{i\bar{\omega}_{n}\to
q_{0}+i\eta}\ ,$ (30)

with $\eta$ an infinitesimal positive number, in which
$\langle\tilde{j}_{5}^{i}(i\bar{\omega}_{n},{\bm{q}})\rangle_{\beta}$ is given
by

$\displaystyle\langle\tilde{j}_{5}^{i}(i\bar{\omega}_{n},{\bm{q}})\rangle_{\beta}$
(31) $\displaystyle\approx$ $\displaystyle
eN_{c}T\sum_{m}\int\frac{d^{3}p}{(2\pi)^{3}}{\rm
tr}\Big{[}\Gamma^{i}\Gamma_{5}\tilde{\cal
G}_{0}(i{\omega}_{m}+i\bar{\omega}_{n},{\bm{p}}+{\bm{q}})$
$\displaystyle\times\tilde{A}^{j}(i\bar{\omega}_{n},{\bm{q}})\Gamma^{j}\tilde{\cal
G}_{0}(i{\omega}_{m},{\bm{p}})\Big{]}\ .$

Here $6\times 6$ gamma matrices $\Gamma^{i}$ and $\Gamma_{5}$ are defined by

$\displaystyle\Gamma^{i}=\left(\begin{array}[]{cc}\gamma^{i}&0\\\ 0&0\\\
\end{array}\right)=\left(\begin{array}[]{ccc}0&\sigma^{i}&0\\\
-\sigma^{i}&0&0\\\ 0&0&0\\\ \end{array}\right)\ ,$ (37)
$\displaystyle\Gamma_{5}=\left(\begin{array}[]{cc}\gamma_{5}&0\\\ 0&0\\\
\end{array}\right)=\left(\begin{array}[]{ccc}0&1&0\\\ 1&0&0\\\ 0&0&0\\\
\end{array}\right)\ ,$ (43)

respectively, which act on only light quark components. Note that
$\bar{\omega}_{n}=2\pi n/\beta$ is the Matsubara frequency for bosons which
shows the periodic boundary condition for the imaginary-time direction, while
${\omega}_{n}=(2n+1)\pi/\beta$ is for fermions which shows the anti-periodic
boundary condition. $N_{c}$ is the number of colors, and we set $N_{c}=3$
throughout this paper. $\tilde{\cal G}_{0}(i{\omega}_{m},{\bm{p}})$ in Eq.
(31) can be obtained by simply replacing $p_{0}$ as $p_{0}\to i{\omega}_{m}$
in Eq. (16).

Before moving on to the detailed calculation of Eq. (31), we mention the zero-
momentum limit in Eq. (29). In medium, due to the lack of Lorentz invariance,
it is possible to take the following two types of zero momentum limits:

  * •

static limit: $\langle j_{5}^{i}\rangle^{\rm
sta}_{\beta}\equiv\frac{1}{\Omega}\lim_{{\bm{q}}\to
0}\langle\tilde{j}_{5}^{i}(0,{\bm{q}})\rangle_{\beta}$ ,

  * •

dynamical limit: $\langle j_{5}^{i}\rangle^{\rm
dyn}_{\beta}\equiv\frac{1}{\Omega}\lim_{q_{0}\to
0}\langle\tilde{j}_{5}^{i}(q_{0},{\bm{0}})\rangle_{\beta}$ .

The static limit (dynamical limit) describes the response to a magnetic field
whose time dependence is slower (faster) than the equilibration of the spatial
part of the system. These two limits yield different results in general
Kharzeev and Warringa (2009); Hou  _et al._ (2011); Son and Yamamoto (2013);
Landsteiner  _et al._ (2014); Kharzeev  _et al._ (2017). Therefore in the
following analysis, we will investigate the CSE in both the limits.

###  III.2 Calculation of
$\langle\tilde{j}_{5}^{i}(i\bar{\omega}_{n},$$q$$)\rangle_{\beta}$

In Sec. III.1, we showed our strategy to evaluate the axial current $\langle
j_{5}^{i}(t,{\bm{x}})\rangle_{\beta}$ within the framework of the linear
response theory and presented a rather general expression of the axial current
$\langle\tilde{j}_{5}^{i}(i\bar{\omega}_{n},{\bm{q}})\rangle_{\beta}$ in
momentum space in Eq. (31) in the imaginary-time formalism. In this
subsection, we further proceed the analytical evaluation of
$\langle\tilde{j}_{5}^{i}(i\bar{\omega}_{n},{\bm{q}})\rangle_{\beta}$.

By inserting Eq. (16) into Eq. (31), the trace with respect to the Dirac
indices in Eq. (43) is reduced to the one for the light quark as

$\displaystyle\langle\tilde{j}_{5}^{i}(i\bar{\omega}_{n},{\bm{q}})\rangle_{\beta}$
$\displaystyle=$ $\displaystyle
eN_{c}T\sum_{m}\int\frac{d^{3}p}{(2\pi)^{3}}{\rm
tr}\Big{[}\gamma^{i}\gamma_{5}\tilde{\cal
G}^{\bar{\psi}\psi}_{0}(i{\omega}_{m}^{\prime},{\bm{p}}^{\prime})$ (44)
$\displaystyle\times\tilde{A}^{j}(i\bar{\omega}_{n},{\bm{q}})\gamma^{j}\tilde{\cal
G}^{\bar{\psi}\psi}_{0}(i{\omega}_{m},{\bm{p}})\Big{]}$

(${\bm{p}}^{\prime}={\bm{p}}+{\bm{q}}$ and
$i{\omega}_{m}^{\prime}=i{\omega}_{m}+i\bar{\omega}_{n}$). This reduction
could be done because the axial current is defined with respect to only the
light-quark degrees of freedom, and the magnetic field only couples with them
now. Recalling the first line in Eq. (17), the trace calculation is
straightforwardly carried out, which yields

$\displaystyle\langle\tilde{j}_{5}^{i}(i\bar{\omega}_{n},{\bm{q}})\rangle_{\beta}$
$\displaystyle=$ $\displaystyle-
ieN_{c}\epsilon^{ijk}\tilde{A}^{k}(i\bar{\omega}_{n},{\bm{q}})T\sum_{m}\int\frac{d^{3}p}{(2\pi)^{3}}$
$\displaystyle\sum_{\zeta,\zeta^{\prime}=+,-,{\rm
a}}(\epsilon_{\zeta^{\prime}}\hat{p}^{\prime
j}-\epsilon_{\zeta}\hat{p}^{j})\frac{U_{\zeta^{\prime}}({\bm{p}}^{\prime})U_{\zeta}({\bm{p}})}{(i{\omega}_{m}^{\prime}-E_{{\bm{p}}^{\prime}}^{\zeta^{\prime}})(i{\omega}_{m}-E_{\bm{p}}^{\zeta})}\
.$

In Eq. (LABEL:J5Zeta), the “sign function” $\epsilon_{\zeta}$ is defined by

$\displaystyle\epsilon_{+}=\epsilon_{-}=+1\ ,$ (46)

while

$\displaystyle\epsilon_{\rm a}=-1\ ,$ (47)

with the subscripts “$+$”, “$-$”, and “a” corresponding to the ones in Eq.
(13). In deriving Eq. (LABEL:J5Zeta), we have used an identity

$\displaystyle{\rm
tr}[\gamma^{i}\gamma_{5}\Lambda^{\prime}_{\xi^{\prime}}\gamma_{0}\gamma^{j}\Lambda_{\xi}\gamma_{0}]=-i\epsilon^{ijk}(\epsilon_{\xi^{\prime}}\hat{p}^{\prime
k}-\epsilon_{\xi}\hat{p}^{k})\ ,$ (48)

with the help of the formula ${\rm
tr}[\gamma_{5}\gamma^{0}\gamma^{i}\gamma^{j}\gamma^{k}]=-4i\epsilon^{ijk}$,
where $\xi,\xi^{\prime}={\rm p},{\rm a}$. In Eq. (48), the “sign function” is

$\displaystyle\epsilon_{\rm p}=+1\ ,\epsilon_{\rm a}=-1\ ,$ (49)

with the subscripts “p” and “a” corresponding to the ones in Eq. (18).

Figure 2: Diagrammatic interpretation of each term in Eq. (57). The arrows are
put such that the corresponding modes carry their positive energies.

Equation (LABEL:J5Zeta) includes an infinite sum with respect to the Matsubara
frequency. This summation can be done by transforming the infinite series into
a contour integral in the complex plane. Namely, we make use of the Abel-Plana
formula Kapusta and Gale (2011)

$\displaystyle T\sum_{m}F(p_{0}=i{\omega}_{m})$ (50) $\displaystyle=$
$\displaystyle\frac{1}{2\pi
i}\int_{-i\infty}^{i\infty}dp_{0}\frac{1}{2}\Big{[}F(p_{0})+F(-p_{0})\Big{]}$
$\displaystyle-\frac{1}{2\pi
i}\int_{-i\infty+\eta}^{i\infty+\eta}dp_{0}\Big{[}F(p_{0})+F(-p_{0})\Big{]}f_{F}(p_{0})\
,$

where $F(p_{0})$ is an arbitrary function which is analytic on the imaginary
axis, and $f_{F}(p_{0})=\frac{1}{{\rm e}^{\beta p_{0}}+1}$ is the Fermi-Dirac
distribution function. The calculation can be carried out without difficulty
by means of the Cauchy’s residue theorem, yielding

$\displaystyle\langle\tilde{j}_{5}^{i}(i\bar{\omega}_{n},{\bm{q}})\rangle_{\beta}=ie\epsilon^{ijk}\tilde{A}^{k}(i\bar{\omega}_{n},{\bm{q}})\tilde{\Pi}_{5}^{j}(i\bar{\omega}_{n},{\bm{q}})\
,$ (51)

in which we have defined

$\displaystyle\tilde{\Pi}_{5}^{j}(i\bar{\omega}_{n},{\bm{q}})\equiv-
N_{c}\int\frac{d^{3}p}{(2\pi)^{3}}\sum_{\zeta,\zeta^{\prime}={+},{-},{\rm
a}}(\epsilon_{\zeta^{\prime}}\hat{p}^{\prime
j}-\epsilon_{\zeta}\hat{p}^{j})\frac{U_{\zeta^{\prime}}({\bm{p}}^{\prime})U_{\zeta}({\bm{p}})}{i\bar{\omega}_{n}-E_{{\bm{p}}^{\prime}}^{\zeta^{\prime}}+E_{\bm{p}}^{\zeta}}\left[f_{F}(E_{\bm{p}}^{\zeta})-f_{F}(E_{{\bm{p}}^{\prime}}^{\zeta^{\prime}})\right]\
.$ (52)

In Eq. (51), $\tilde{\Pi}_{5}^{j}(i\bar{\omega}_{n},{\bm{q}})$ must be
proportional to $q^{j}$ due to its Lorentz structure. Thus, by defining
$\tilde{\Pi}_{5}(i\bar{\omega}_{n},{\bm{q}})$ in terms of

$\displaystyle\tilde{\Pi}_{5}^{j}(i\bar{\omega}_{n},{\bm{q}})\equiv\tilde{\Pi}_{5}(i\bar{\omega}_{n},{\bm{q}})q^{j}\
,$ (53)

now Eq. (51) reads

$\displaystyle\langle\tilde{j}_{5}^{i}(i\bar{\omega}_{n},{\bm{q}})\rangle_{\beta}$
$\displaystyle=$ $\displaystyle
ie\epsilon^{ijk}q^{j}\tilde{A}^{k}(i\bar{\omega}_{n},{\bm{q}})\tilde{\Pi}_{5}(i\bar{\omega}_{n},{\bm{q}})$
(54) $\displaystyle=$ $\displaystyle
e\tilde{B}(i\bar{\omega}_{n},{\bm{q}})\tilde{\Pi}_{5}(i\bar{\omega}_{n},{\bm{q}})\
.$

In the second line of Eq. (54), we have made use of
$\tilde{B}^{i}(i\bar{\omega}_{n},{\bm{q}})=i\epsilon^{ijk}q^{j}\tilde{A}^{k}(i\bar{\omega}_{n},{\bm{q}})$
[$\tilde{B}^{i}(q_{0},{\bm{q}})$ is defined by the Fourier transformation of
$B^{i}(t,{\bm{x}})$]. We note that
$\tilde{\Pi}_{5}(i\bar{\omega}_{n},{\bm{q}})$ can be “solved” as

$\displaystyle\tilde{\Pi}_{5}(i\bar{\omega}_{n},{\bm{q}})=\frac{1}{|{\bm{q}}|^{2}}\tilde{\Pi}_{5}^{j}(i\bar{\omega}_{n},{\bm{q}})q^{j}$
(55)

from Eq. (53).

Equation (54) shows that $\tilde{\Pi}_{5}(i\bar{\omega}_{n},{\bm{q}})$ is the
response function of axial current to a magnetic field. Concretely, from Eqs.
(52) and (55), the response function is expressed as

$\displaystyle\tilde{\Pi}_{5}(i\bar{\omega}_{n},{\bm{q}})=\sum_{\zeta,\zeta^{\prime}=+,-,{\rm
a}}\tilde{\Pi}^{\zeta^{\prime}\zeta}_{5}(i\bar{\omega}_{n},{\bm{q}})$ (56)

with

$\displaystyle\tilde{\Pi}^{\zeta^{\prime}\zeta}_{5}(i\bar{\omega}_{n},{\bm{q}})\equiv-\frac{N_{c}}{|{\bm{q}}|^{2}}\int\frac{d^{3}p}{(2\pi)^{3}}\Big{[}\epsilon_{\zeta^{\prime}}(\hat{\bm{p}}^{\prime}\cdot{\bm{q}})-\epsilon_{\zeta}(\hat{\bm{p}}\cdot{\bm{q}})\Big{]}\frac{U_{\zeta^{\prime}}({\bm{p}}^{\prime})U_{\zeta}({\bm{p}})}{i\bar{\omega}_{n}-E_{{\bm{p}}^{\prime}}^{\zeta^{\prime}}+E_{\bm{p}}^{\zeta}}\left(f_{F}(E_{\bm{p}}^{\zeta})-f_{F}(E_{{\bm{p}}^{\prime}}^{\zeta^{\prime}})\right)\
.$ (57)

After performing the analytic continuation to the real-time formalism and
taking zero limit of the external momentum ($q_{0},{\bm{q}}$) properly, static
and dynamical limits for $\langle j_{5}^{i}\rangle_{\beta}$ can be evaluated
from Eq. (54) as

$\displaystyle\langle j_{5}^{i}(t,{\bm{x}})\rangle_{\beta}^{\rm
sta}=eB\tilde{\Pi}_{5}^{\rm sta},$ $\displaystyle\langle
j_{5}^{i}(t,{\bm{x}})\rangle_{\beta}^{\rm dyn}=eB\tilde{\Pi}_{5}^{\rm dyn}\ ,$
(58)

respectively from Eqs. (29) and (30), with the definitions of

$\displaystyle\tilde{\Pi}_{5}^{\rm
sta}\equiv\lim_{{\bm{q}}\to{\bm{0}}}\tilde{\Pi}_{5}(0,{\bm{q}})\ ,$
$\displaystyle\tilde{\Pi}_{5}^{\rm dyn}\equiv\lim_{q_{0}\to
0}\tilde{\Pi}_{5}(q_{0},{\bm{0}})\ .$ (59)

We also used $\tilde{B}(q_{0}=0,{\bm{q}}={\bm{0}})=\Omega B$. By carrying out
the angular integral in Eq. (56) in two limits, the response function can be
analytically obtained apart from the $|{\bm{p}}|$ integral. We note that a
rather detailed discussion on the response function in static and dynamical
limits is given in Appendix B.

At the end of this subsection, we explain the details of the nine
contributions in Eq. (57) obtained by setting $\zeta,\zeta^{\prime}=+,-,{\rm
a}$. Diagrammatically, these contributions can be understood as in Fig. 2. In
this figure, the arrows are put in such a way that the corresponding modes
carry their positive energies. We note that the momentum integrals of the
loops in Fig. 2 do not include any ultraviolet divergences. According to Eqs.
(187) and (188) in Appendix B, each contribution in Eq. (57) in static limit
and dynamical limit after the analytic continuation is found to satisfy

$\displaystyle\lim_{{\bm{q}}\to{\bm{0}}}\tilde{\Pi}_{5}^{\zeta\zeta}(0,{\bm{q}})$
$\displaystyle=$ $\displaystyle-N_{c}\int\frac{d^{3}p}{(2\pi)^{3}}{\cal
I}_{\zeta}({\bm{p}})\frac{\partial
f_{F}(E_{\bm{p}}^{\zeta})}{\partial|{\bm{p}}|}\ ,$
$\displaystyle\lim_{q_{0}\to 0}\tilde{\Pi}_{5}^{\zeta\zeta}(q_{0},{\bm{0}})$
$\displaystyle=$ $\displaystyle 0\ ,$ (60)

[${\cal I}_{\zeta}({\bm{p}})$ is a function of ${\bm{p}}$ 666The expression of
${\cal I}_{\zeta}({\bm{p}})$ may be intricate, so that we do not show it
explicitly here.] for $\tilde{\Pi}_{5}^{++}$, $\tilde{\Pi}_{5}^{--}$ and
$\tilde{\Pi}_{5}^{\rm{aa}}$, while

$\displaystyle\lim_{{\bm{q}}\to{\bm{0}}}\tilde{\Pi}_{5}^{\zeta^{\prime}\zeta}(0,{\bm{q}})=\lim_{q_{0}\to
0}\tilde{\Pi}_{5}^{\zeta^{\prime}\zeta}(q_{0},{\bm{0}})$ (61)

for the remaining ones $(\zeta\neq\zeta^{\prime})$. In other words, only the
diagrams of (i), (ii), and (iii) in Fig. 2, which correspond to the scattering
processes of the identical particle by the magnetic field: ${\rm q.p.}\to{\rm
q.p.}$, ${\rm q.h.}\to{\rm q.h.}$, and ${\rm a.p.}\to{\rm a.p.}$ (or intraband
scattering processes in the terminology of condensed matter physics), can
generate the difference between the results in static limit and dynamical
limit. Such a difference between the static and dynamical limits was also
found in Ref. Araki  _et al._ (2020). In this reference, we investigated the
spin-orbital crossed susceptibility, namely the response of spin polarization
to a magnetic field, in which a relativistic (linear) band and a
nonrelativistic band coexist.

Figure 3: The $\mu$ dependence of response function in static limit
($\tilde{\Pi}_{5}^{\rm sta}$) with $\Delta=0.02$ GeV (red dashed-dotted),
$\Delta=0.05$ GeV (purple dashed), and $\Delta=0.1$ GeV (blue solid). The
black dashed curve indicates the response function in the normal matter given
in Eq. (62).

Figure 4: The $\mu$ dependence of response function in dynamical limit
($\tilde{\Pi}_{5}^{\rm dyn}$) with $\Delta=0.02$ GeV (red dashed-dotted),
$\Delta=0.05$ GeV (purple dashed), and $\Delta=0.1$ GeV (blue solid). The
black dashed curve indicates the response function in the normal matter given
in Eq. (63).

Figure 5: The gap dependence of response function in static limit
($\tilde{\Pi}_{5}^{\rm sta}$) (upper panel) and in dynamical limit
($\tilde{\Pi}_{5}^{\rm dyn}$) (lower panel) at $\mu=0.5$ GeV and at several
temperatures. The black dashed line indicates the response function in the
normal matter given in Eq. (62). Blue filled or open circle in dynamical limit
mean the discontinuous behavior at $T=0$ (see the main text).

Figure 6: The $\mu$ dependence at $T=0$ and $\Delta=0.1$ GeV of each process
contributing to the response function (upper panel), and the $\Delta$
dependence at $T=0$ and $\mu=0.5$ GeV of them (lower panel). We compare the
total value of response function (blue solid), the sum of pair creation and
pair annihilation processes of ${\rm q.p.}+{\rm q.h.}$ (magenta dashed), and
the sum of pair creation and pair annihilation processes of ${\rm q.p.}+{\rm
a.p.}$ (brown dashed-dotted).

##  IV Numerical results

In Sec. III we showed the detailed procedure to compute the axial current
induced by a magnetic field in the presence of Kondo condensate in static and
dynamical limits. In this section, we present numerical results of Eq. (59)
and examine the impact of Kondo effect on CSE. For the sake of clarity, here
we do not solve the gap equation to determine the magnitude, and its density
dependence of $\Delta$ and simply regard it as a constant parameter.

###  IV.1 Density dependence of the CSE

First, we show the density dependence of the response function of CSE obtained
in the present work.

In Fig. 3 and Fig 4, the $\mu$ dependences of response function
$\tilde{\Pi}_{5}^{\rm sta}$ in static limit and $\tilde{\Pi}_{5}^{\rm dyn}$ in
dynamical limit with $\Delta=0.02$ GeV (red), $\Delta=0.05$ GeV (purple), and
$\Delta=0.1$ GeV (blue) are plotted, respectively. The black dashed line
indicates the response function in the normal matter ($\Delta=0$), i.e.,

$\displaystyle\tilde{\Pi}^{\rm sta}_{5}|_{\Delta=0}$ $\displaystyle=$
$\displaystyle\frac{N_{c}}{2\pi^{2}}\mu\ ,$ (62)
$\displaystyle\tilde{\Pi}^{\rm dyn}_{5}|_{\Delta=0}$ $\displaystyle=$
$\displaystyle\frac{N_{c}}{6\pi^{2}}\mu\ ,$ (63)

at any temperatures. We have put this line so as to compare the results with
and without Kondo effect more clearly. We have shown the results in the range
of $0.3\,{\rm GeV}<\mu<0.6\,{\rm GeV}$ in which the Kondo condensate is
expected to appear significantly within reasonable model parameters, as
suggested by the previous works Yasui  _et al._ (2019, 2017).

Figures 3 and 4 show that the Kondo effect enhances the CSE in both static
limit and dynamical limit. In particular, the response function in dynamical
limit can be enhanced by a factor of approximately three. Besides, those
figures show that the results in static limit and dynamical limit at $T=0$
coincide with each other: $\tilde{\Pi}^{\rm sta}_{5}=\tilde{\Pi}^{\rm
dyn}_{5}$. Note that the magnitude of $\Delta$ was found to be $\Delta\approx
0.085$ GeV at $\mu=0.5$ GeV in the previous work based on the NJL-type
analysis Yasui  _et al._ (2019, 2017). Therefore, we can conclude that the
influence of Kondo effect on CSE with a reasonable value of $\Delta$ is
significantly large, showing that the heavy impurities in quark matter is
expected to play an important role in the transport phenomena induced by a
magnetic field. The large enhancement of the response function in dynamical
limit is consistent with the finding in Ref. Araki  _et al._ (2020).777In Ref.
Araki  _et al._ (2020) we have seen that the response function to a magnetic
field in dynamical limit gets enhanced for general dispersions with a
condensate. The large enhancement found in the present study can be understood
as its extreme case in the heavy-quark limit, i.e., the heavy-quark dispersion
is flat in momentum space.

###  IV.2 $\Delta$ dependence of the CSE

Next, we study the $\Delta$ dependence of the CSE to examine the influence of
Kondo condensate $\Delta$ on the CSE in more detail.

In Fig. 5 we display the $\Delta$ dependence of the response function in
static limit (upper panel) and dynamical limit (lower panel) at $\mu=0.5$ GeV
and at several temperatures. The black dashed line indicates the response
function in the normal matter given in Eq. (62). The lower panel tells us
that, $\tilde{\Pi}_{5}^{\rm dyn}$ for small $\Delta$ with $\Delta\neq 0$ is
enhanced largely due to an effect generated by the Kondo condensate, although
it converges on the result of normal matter in dynamical limit provided in Eq.
(63) at $\Delta=0$. Such a sudden and large enhancement around $\Delta=0$
stems from the fact that the effect of $\Delta$ is nonperturbative around the
Fermi level, which can be understood as follows. The CSE is mostly generated
by the response of fermions in the vicinity of the Fermi level
($|{\bm{p}}|\approx\mu$) in the momentum integrals of the loops. However, the
factor of $\sqrt{(|{\bm{p}}|-\mu)^{2}+8\Delta^{2}}$ appearing in
$E_{\bm{p}}^{\pm}$ prohibits us from expanding $E_{\bm{p}}^{\pm}$ with respect
to $\Delta$ around $|{\bm{p}}|=\mu$ now, because the magnitudes of
$(|{\bm{p}}|-\mu)^{2}$ and $8\Delta^{2}$ are comparable order in this case. As
a result, the Taylor expansion of the response function with respect to
$\Delta$ should fail, and hence the modification of $\tilde{\Pi}_{5}^{\rm
dyn}$ at small $\Delta$ is significant.

The signature of the nonperturbative effect by $\Delta$ can be seen
prominently at $T=0$ in the lower panel of Fig. 5. Namely, at $T=0$,
$\tilde{\Pi}_{5}^{\rm dyn}$ at $\Delta\neq 0$ varies continuously, but only at
$\Delta=0$ it jumps to the result of normal matter in dynamical limit
discontinuously. Here, we provide some mathematical explanations for the
origin of the discontinuity. As shown in Eqs. (60) and (61), the difference
between the results in static and dynamical limits originates from the
scattering process of the identical particles, i.e., from the diagrams (i),
(ii), and (iii) in Fig. 2. Below we focus on the diagram (i). At zero
temperature, from Eq. (60), the response functions from the diagram (i) in
static and dynamical limits are of the forms

$\displaystyle\lim_{{\bm{q}}\to{\bm{0}}}\tilde{\Pi}_{5}^{++}(0,{\bm{q}})\overset{T=0}{=}-N_{c}\int\frac{d^{3}p}{(2\pi)^{3}}{\cal
I}_{+}({\bm{p}})\frac{E_{\bm{p}}^{+}}{E_{\bm{p}}^{+}-E_{\bm{p}}^{-}}\delta(E_{\bm{p}}^{+})\
,$ $\displaystyle\lim_{q_{0}\to
0}\tilde{\Pi}_{5}^{++}(q_{0},{\bm{0}})\overset{T=0}{=}0\ ,$ (64)

respectively. These results coincide with each other, even when $\Delta$ is
infinitesimally small, since $E_{\bm{p}}^{+}$ and
$E_{\bm{p}}^{+}-E_{\bm{p}}^{-}$ are always positive in this case, resulting in

$\displaystyle\lim_{{\bm{q}}\to{\bm{0}}}\tilde{\Pi}_{5}^{++}(0,{\bm{q}})\overset{T=0}{=}0\
.$ (65)

Hence, $\tilde{\Pi}^{\rm sta}_{5}=\tilde{\Pi}^{\rm dyn}_{5}$ holds for
$\Delta\neq 0$. On the other hand, when $\Delta$ is exactly zero, the upper
equation of Eq. (64) is no longer zero because
$E_{\bm{p}}^{+}|_{\Delta=0}=\theta(|{\bm{p}}|-\mu)(|{\bm{p}}|-\mu)$ and
$E_{\bm{p}}^{-}|_{\Delta=0}=\theta(\mu-|{\bm{p}}|)(|{\bm{p}}|-\mu)$ can reach
zero, yielding $\tilde{\Pi}^{\rm sta}_{5}\neq\tilde{\Pi}^{\rm dyn}_{5}$. As a
result, the discontinuous behavior of the blue curve of lower panel in Fig. 5
at $\Delta\to 0$ emerges. The similar explanation can be also applied to (ii)
and (iii). The discontinuity at $\Delta=0$, however, should not be taken too
seriously, because in realistic measurements, this discontinuity would be
smeared by the finite relaxation time.

In association with the sudden and large enhancement at $T\neq 0$ (and the
discontinuity at $T=0$) in dynamical limit, in Ref. Satow and Yee (2014), it
was found that, in the normal matter the damping rate of free quark enhances
the CSE in dynamical limit to coincide with the one in static limit. Thus, by
focusing on the fact that nonzero $\Delta$ causes the sudden and large
enhancement of the response function in dynamical limit resulting in the
similar behavior to static limit, as shown in Fig. 5, we speculate that there
would be an analogy between the inclusion of damping rate in the normal matter
and the presence of Kondo condensate $\Delta$. A detailed comparison between
the present analysis and the one in Ref. Satow and Yee (2014) is left for
future study.

###  IV.3 Separate contributions to the CSE

Finally, we give a physical interpretation of how the each diagram in Fig. 2
contributes to the response function.

Again for the sake of clarity, we restrict ourselves to vanishing temperature.
At $T=0$, the Pauli principle forces the intermediate states of the loops to
excite above the Fermi level, i.e., at least one of the modes inside the loop
must be ${\rm q.p.}$ because only this mode lies above the Fermi level as
shown in Fig. 1. In fact we can confirm that only the diagrams (iv) and (v)
for ${\rm q.p.}+{\rm q.h.}$ and (viii) and (ix) for ${\rm q.p.}+{\rm a.p.}$ in
Fig. 2 contribute to the CSE through the pair creation and pair annihilation
processes. We note that the contribution from the diagram (i) vanishes at zero
temperature as explained around Eqs. (64) and (65) in detail.

In Fig. 6 we show the $\mu$ dependence of each process contributing to the
response function at $T=0$ and $\Delta=0.1$ GeV (upper panel), and the
$\Delta$ dependence of them at $T=0$ and $\mu=0.5$ GeV (lower panel). We note
that the point for $\Delta=0$ is excluded in the lower panel to avoid the
discontinuity explained in Sec. IV.2. In Fig. 6, the blue curve indicates the
total value of response function, while the magenta curve corresponds to the
sum of ${\rm q.p.}+{\rm q.h.}$ pair creation and pair annihilation, and the
brown one corresponds to the sum of ${\rm q.p.}+{\rm a.p.}$ pair creation and
pair annihilation. Only these processes contribute to the response function
due to the reason stated above. Note that at zero temperature the response
functions in static and dynamical limits coincide: $\tilde{\Pi}_{5}^{\rm
sta}=\tilde{\Pi}_{5}^{\rm dyn}$ as found in Sec. IV.1. Figure 6 shows that the
contribution from ${\rm q.p.}+{\rm q.h.}$ process and that of ${\rm q.p.}+{\rm
a.p.}$ are approximately in the ratio of $2:1$ for any values of $\mu$ or
$\Delta$. This ratio is easily understood because the former process includes
two modes (both ${\rm q.p.}$ and ${\rm q.h.}$) lying in the vicinity of the
Fermi level while the latter includes only one mode (only ${\rm q.p.}$) there.

At finite temperature, the above picture does not change significantly as long
as the temperature is smaller than the chemical potential $\mu$. In fact this
is the case in the present work so that the contributions from diagrams (i),
(ii), (iii), (vi), and (vii) in Fig. 2 remain small.

##  V Gauge invariance

Here we discuss the gauge invariance in our calculations. If we assume a
charged Kondo condensate with $e_{q}\neq e_{Q}$, then the condensate carries
an electric charge and leads to the spontaneous breakdown of the
electromagnetic $U(1)_{\rm EM}$ invariance. In this case, we have to correct
the vertex for the coupling between the quarks and the magnetic field in order
to recover the gauge invariance, leading to the appearance of the NG mode.
This treatment is an analogue of the derivation of NG mode appearing in the
theory of superconductivity Schrieffer (1999), or that of pion in chiral
symmetry breaking Cheng and Li (1994). On the other hand, even when we assume
the charge-neutral Kondo condensate as in the present study [recall Eq. (5)],
the nontrivial structure of the hedgehog ansatz (6) still requires to correct
the vertex for the gauge invariance. In actuality, such a vertex correction is
found not to affect our results considerably as it will be explained later.
Rigorous arguments on the gauge invariance are, however, important and useful
in the study of response to the external gauge field.

The underlying theory of the effective model (8), i.e., QCD and QED, must
possess the electromagnetic $U(1)_{\rm EM}$ gauge invariance. Thus the Ward-
Takahashi identity together with the Dyson-Schwinger equation with respect to
vector-axialvector current correlator tells us that the vertex $\Gamma^{\mu}$
between the gauge field and the quarks must obey the following identity Nambu
(1960)

$\displaystyle q_{\mu}\Gamma^{\mu}=-i{\cal
G}^{-1}(p_{0}^{\prime},{\bm{p}}^{\prime})Q+Qi{\cal G}^{-1}(p_{0},{\bm{p}})$
(66)

($p^{\prime}=p+q$), where $\Gamma^{\mu}$ is the $6\times 6$ vertex matrix

$\displaystyle\Gamma^{\mu}\equiv\left(\begin{array}[]{cc}\Gamma^{\mu}_{A\bar{\psi}\psi}&\Gamma^{\mu}_{A\bar{\psi}\Psi_{v}}\\\
\Gamma^{\mu}_{A\Psi_{v}^{\dagger}\psi}&\Gamma^{\mu}_{A\Psi_{v}^{\dagger}\Psi_{v}}\\\
\end{array}\right).$ (69)

Here, for example, the subscript of $\Gamma^{\mu}_{A\bar{\psi}\psi}$
represents the type of gauge interaction such as $A^{\mu}\bar{\psi}\psi$. $Q$
is a charge matrix

$\displaystyle Q=\left(\begin{array}[]{ccc}e_{q}&0\\\ 0&e_{Q}\\\
\end{array}\right)\ ,$ (72)

where we have assumed the electric charges of light and heavy quarks are not
identical for a general discussion. $i{\cal G}^{-1}(p_{0},{\bm{p}})$ is the
inverse of the full Green’s function in general. Within the present work, it
can be read by the Lagrangian (8) [where we do not necessarily assume Eq.
(7)]:

$\displaystyle i{\cal G}^{-1}(p_{0},{\bm{p}})=\left(\begin{array}[]{cc}i{\cal
G}_{\bar{\psi}\psi}^{-1}(p_{0},{\bm{p}})&i{\cal
G}_{\bar{\psi}\Psi_{v}}^{-1}(p_{0},{\bm{p}})\\\ i{\cal
G}_{\Psi_{v}^{\dagger}\psi}^{-1}(p_{0},{\bm{p}})&i{\cal
G}_{\Psi_{v}^{\dagger}\Psi_{v}}^{-1}(p_{0},{\bm{p}})\\\ \end{array}\right)$
(75)

with

$\displaystyle i{\cal G}_{\bar{\psi}\psi}^{-1}(p_{0},{\bm{p}})$
$\displaystyle=$ $\displaystyle{\ooalign{\hfil/\hfil\crcr$p$}}+\mu\gamma_{0}\
,$ (78) $\displaystyle i{\cal G}_{\bar{\psi}\Psi_{v}}^{-1}(p_{0},{\bm{p}})$
$\displaystyle=$ $\displaystyle\Delta\left(\begin{array}[]{c}1\\\
-\hat{\bm{p}}\cdot{\bm{\sigma}}\\\ \end{array}\right)\ ,$ (81) $\displaystyle
i{\cal G}_{\Psi_{v}^{\dagger}\psi}^{-1}(p_{0},{\bm{p}})$ $\displaystyle=$
$\displaystyle\Delta\left(\begin{array}[]{cc}1&\hat{\bm{p}}\cdot{\bm{\sigma}}\\\
\end{array}\right)\ ,$ (83) $\displaystyle i{\cal
G}_{\Psi_{v}^{\dagger}\Psi_{v}}^{-1}(p_{0},{\bm{p}})$ $\displaystyle=$
$\displaystyle p_{0}{\bm{1}}\ .$ (84)

Thus, each component of the Ward-Takahashi identity in Eq. (66) reads

$\displaystyle q_{\mu}\Gamma^{\mu}_{A\bar{\psi}\psi}$ $\displaystyle=$
$\displaystyle-e_{q}{\ooalign{\hfil/\hfil\crcr$q$}}\ ,$ (87) $\displaystyle
q_{\mu}\Gamma^{\mu}_{A\bar{\psi}\Psi_{v}}$ $\displaystyle=$
$\displaystyle\Delta\left(\begin{array}[]{c}e_{q}-e_{Q}\\\
e_{Q}\hat{\bm{p}}^{\prime}\cdot{\bm{\sigma}}-e_{q}\hat{\bm{p}}\cdot{\bm{\sigma}}\\\
\end{array}\right)\ ,$ (90) $\displaystyle
q_{\mu}\Gamma^{\mu}_{A\Psi_{v}^{\dagger}\psi}$ $\displaystyle=$
$\displaystyle\Delta\left(\begin{array}[]{cc}e_{Q}-e_{q}&-e_{q}\hat{\bm{p}}^{\prime}\cdot{\bm{\sigma}}+e_{Q}\hat{\bm{p}}\cdot{\bm{\sigma}}\\\
\end{array}\right)\ ,$ (92) $\displaystyle
q_{\mu}\Gamma^{\mu}_{A\Psi_{v}^{\dagger}\Psi_{v}}$ $\displaystyle=$
$\displaystyle-e_{Q}q_{0}{\bm{1}}\ .$ (93)

For the diagonal components, $A^{\mu}\bar{\psi}\psi$ and
$A^{\mu}\Psi_{v}^{\dagger}\Psi_{v}$, we can safely use their bare vertices:

$\displaystyle\Gamma^{\mu}_{A\bar{\psi}\psi}=-e_{q}\gamma^{\mu}\ ,\ \
\Gamma^{\mu}_{A\Psi_{v}^{\dagger}\Psi_{v}}=-e_{Q}\delta^{\mu 0}{\bm{1}}\ .$
(94)

On the other hand, for the off-diagonal components,
$A^{\mu}\bar{\psi}\Psi_{v}$ and $A^{\mu}\Psi_{v}^{\dagger}\psi$, we cannot use
their bare vertices, and they need to be corrected in such a way that they
satisfy the identities in Eq. (93).

When $e_{q}\neq e_{Q}$, the second and third identities in Eq. (93) suggest an
existence of collective mode (NG mode) coupling with the vertex, because the
RHS in the limit of $q_{0},{\bm{q}}\to 0$ does not vanish in this case Nambu
(1960). The appearance of NG mode can be also understood by the fact that
$\Delta\sim\langle\Psi_{v}^{\dagger}\psi\rangle$ with $e_{q}\neq e_{Q}$
carries its nonzero electric charge leading to the spontaneous breakdown of
$U(1)_{\rm EM}$ symmetry. This collective mode (NG mode) is analogous to the
one in the theory of superconductivity Schrieffer (1999).

On the other hand, when $e\equiv e_{q}=e_{Q}$ as we have assumed in the
present study, the NG mode related to the spontaneous breakdown of $U(1)_{\rm
EM}$ symmetry does not appear. However, modifications of the vertices are
still required, as seen by the RHS of second and third identities in Eq. (93):

$\displaystyle q_{\mu}\Gamma^{\mu}_{A\bar{\psi}\Psi_{v}}$ $\displaystyle=$
$\displaystyle e\Delta\left(\begin{array}[]{c}0\\\
(\hat{\bm{p}}^{\prime}-\hat{\bm{p}})\cdot{\bm{\sigma}}\\\ \end{array}\right)\
,$ (97) $\displaystyle q_{\mu}\Gamma^{\mu}_{A\Psi_{v}^{\dagger}\psi}$
$\displaystyle=$ $\displaystyle
e\Delta\left(\begin{array}[]{cc}0&-(\hat{\bm{p}}^{\prime}-\hat{\bm{p}})\cdot{\bm{\sigma}}\\\
\end{array}\right)\ .$ (99)

Obviously the modifications originate from the nontrivial momentum alignment
caused by hedgehog ansatz (6). Equation (99) can be solved with respect to
$\Gamma^{\mu}_{A\bar{\psi}\Psi_{v}}$ and
$\Gamma^{\mu}_{A\Psi_{v}^{\dagger}\psi}$ by imposing some assumptions. For
example, if we restrict ourselves to the vicinity of $q^{\mu}=0$, then the
Ward-Takahashi identities for $\Gamma^{\mu}_{A\bar{\psi}\Psi_{v}}$ and
$\Gamma^{\mu}_{A\Psi_{v}^{\dagger}\psi}$ are solved, and we obtain

$\displaystyle\Gamma^{\mu}_{A\bar{\psi}\Psi_{v}}$ $\displaystyle\approx$
$\displaystyle-e\Delta\delta^{i\mu}\left(\begin{array}[]{c}0\\\
\frac{\sigma^{i}-(\hat{\bm{p}}\cdot{\bm{\sigma}})\hat{p}^{i}}{|{\bm{p}}|}\\\
\end{array}\right)\ ,$ (102)
$\displaystyle\Gamma^{\mu}_{A\Psi_{v}^{\dagger}\psi}$ $\displaystyle\approx$
$\displaystyle
e\Delta\delta^{i\mu}\left(\begin{array}[]{cc}0&\frac{\sigma^{i}-(\hat{\bm{p}}\cdot{\bm{\sigma}})\hat{p}^{i}}{|{\bm{p}}|}\\\
\end{array}\right)\ .$ (104)

The vertex corrections in Eq. (104) can give rise to additional contributions
to the axial current as

$\displaystyle\langle\delta\tilde{j}_{5}^{i}(i\bar{\omega}_{n},{\bm{q}})\rangle_{\beta}$
(107) $\displaystyle\approx$ $\displaystyle-
N_{c}T\sum_{m}\int\frac{d^{3}p}{(2\pi)^{3}}{\rm
tr}\Big{[}\Gamma^{i}\Gamma_{5}\tilde{\cal
G}_{0}(i{\omega}_{m}+i\bar{\omega}_{n},{\bm{p}}+{\bm{q}})$
$\displaystyle\times\tilde{A}^{j}(i\bar{\omega}_{n},{\bm{q}})\left(\begin{array}[]{cc}0&\Gamma_{A\bar{\psi}\Psi_{v}}^{j}\\\
\Gamma_{A\Psi_{v}^{\dagger}\psi}^{j}&0\\\ \end{array}\right)\tilde{\cal
G}_{0}(i{\omega}_{m},{\bm{p}})\Big{]}\,,$

[compare to Eq. (31)]. Here, we note that the RHS of Eq. (107) is proportional
to $\Delta^{2}$, because Eq. (104) is proportional to $\Delta$ and computation
of the trace in Eq. (107) picks up another $\Delta$ in $V_{+}({\bm{p}})$ or
$V_{-}({\bm{p}})$ in Eq. (17). Namely the additional contributions from the
vertex corrections may be small in a realistic situation where $\Delta$ is
always smaller than $\mu$.

The above discussion suggests that, while the method in deriving the response
functions in Sec. IV seem to violate the $U(1)_{\rm EM}$ gauge invariance due
to lack of the vertex corrections, such a violation is expected to be
relatively small and gives only a subleading effect on our findings as long as
we assume $e\equiv e_{q}=e_{Q}$. On the other hand, when we take into account
the difference between the electric charges of light and heavy quarks, the
effects from the vertex correction would become rather important because of
the appearance of NG mode Nambu (1960). Therefore, the study of CSE in the
presence of Kondo condensate with other choices of electric charges would be
of interest, and we leave such an examination for future publication.

##  VI Conclusions

In this paper we studied the axial current induced by a magnetic field, i.e.,
the chiral separation effect (CSE) with the Kondo effect. To demonstrate the
impact of Kondo effect on the CSE in a transparent way, we employed a simple
effective Lagrangian incorporating a condensate between a light quark and a
heavy quark, which is the so-called Kondo condensate.

When we evaluate the CSE by means of the linear response theory, we can define
the axial current in the two types of limits: static limit and dynamical
limit. The former (latter) describes an induced current under an external
magnetic field whose time dependence is slower (faster) than the equilibration
of the spatial part of the system, and in general they yield different
results. Thus we studied the response function of the axial current to the
magnetic field in those two limits. We found that the Kondo effect enhances
the CSE in both the static limit and dynamical limit, and particularly the CSE
in dynamical limit can be increased by a factor of approximately three in the
range of the reasonable value of Kondo condensate. Those results clearly show
that the Kondo effect arising from heavy impurities in quark matter, can play
an important role in the transport phenomena of light quarks induced by a
magnetic field.

The results obtained in this paper are expected to contribute to a better
understanding of transport phenomena induced by a magnetic field in quark
matter with heavy impurities. In condensed matter physics, our findings are
related to the electron spin polarization (spin magnetization) in the response
to a magnetic field in Dirac/Weyl semimetals having a band structure including
both relativistic and nonrelativistic degrees of freedom, since the spin
polarization is related to axial current by spin-momentum locking Araki  _et
al._ (2020). In particular, the response function of the CSE to a magnetic
field is equivalent to the so-called spin-orbital crossed susceptibility
Tserkovnyak  _et al._ (2015); Koshino and Hizbullah (2016); Nakai and Nomura
(2016); Suzuura and Ando (2016); Ando and Suzuura (2017); Ominato  _et al._
(2019) which constitutes a part of the magnetic susceptibility, as pointed out
in Ref. Araki  _et al._ (2020).

In what follows, we give discussions related to the present study. While we
investigated the CSE with Kondo effect at vanishing frequency of magnetic
field in this paper, our analysis can be applied to the one at finite
frequency as in Refs. Kharzeev and Warringa (2009); Yee (2009); Landsteiner
_et al._ (2014); Satow and Yee (2014); Kharzeev  _et al._ (2017) for the CME
and in Ref. Landsteiner  _et al._ (2014) for the CSE. Such a study will be
important for a more realistic situation such as (low-energy and peripheral)
heavy-ion collisions where the strength of a magnetic field produced after a
nucleus-nucleus collision evolves with time.

Moreover, our analysis can be easily applied to the study of the chiral
magnetic effect (CME) Kharzeev (2006); Kharzeev and Zhitnitsky (2007);
Kharzeev  _et al._ (2008); Fukushima  _et al._ (2008) and the chiral vortical
effect (CVE) Vilenkin (1979, 1980b); Erdmenger  _et al._ (2009); Banerjee  _et
al._ (2011); Son and Surowka (2009); Landsteiner  _et al._ (2011); Stephanov
and Yin (2012); Abramchuk  _et al._ (2018) with the Kondo effect. The CME
describes the vector current induced by a magnetic field in chirality
imbalanced matter Kharzeev (2006); Kharzeev and Zhitnitsky (2007); Kharzeev
_et al._ (2008); Fukushima  _et al._ (2008), whereas the CVE describes the
vector or axial current induced by a magnetic field with rotation in ordinary
and/or chirality-imbalanced matter Vilenkin (1979, 1980b); Erdmenger  _et al._
(2009); Banerjee  _et al._ (2011); Son and Surowka (2009); Landsteiner  _et
al._ (2011); Stephanov and Yin (2012); Abramchuk  _et al._ (2018),
respectively. Without the Kondo effect, while the CSE and CVE were found to be
affected by a quark mass Metlitski and Zhitnitsky (2005); Gorbar  _et al._
(2013); Guo and Lin (2017); Flachi and Fukushima (2018); Lin and Yang (2018)
or quantum corrections Gorbar  _et al._ (2013), the CME may be protected by
the axial anomaly. Thus, the study of CVE and CME in addition to CSE toward
the comprehensive understanding of chiral transport phenomena with Kondo
effect, particularly from the viewpoint of axial anomaly, would be also of
great interest.

## Acknowledgements

D. S. wishes to thank Keio University and Japan Atomic Energy Agency (JAEA)
for their hospitalities during his stay there. Y. A. is supported by the
Leading Initiative for Excellent Young Researchers (LEADER). K. S. is
supported by Japan Society for the Promotion of Science (JSPS) KAKENHI (Grant
Nos. JP17K14277 and JP20K14476). S. Y. is supported by JSPS KAKENHI (Grant No.
JP17K05435), by the Ministry of Education, Culture, Sports, Science
(MEXT)-Supported Program for the Strategic Research Foundation at Private
Universities Topological Science (Grant No. S1511006) and by the
Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS) at
RIKEN.

##  Appendix A Another derivation of CSE in static limit

Here, we show another derivation of CSE for $e=e_{q}=e_{Q}$ in the presence of
Kondo condensate in static limit. It should be noted that the CSE calculated
in this appendix does not include any vertex corrections.

###  A.1 Green’s function in a magnetic field

To show the CSE in static limit in an alternative way, we go back to the
original Lagrangian (8):

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\bar{\psi}(i{\ooalign{\hfil/\hfil\crcr$D$}}+\mu\gamma_{0})\psi+\Psi_{v}^{\dagger}iD_{0}\Psi_{v}$
(110) $\displaystyle+$
$\displaystyle\Delta\left[\bar{\psi}(1-i{\bm{\gamma}}\cdot\hat{\bm{\nabla}})\Psi_{v}\right]+({\rm
c.c.})\ ,$ (111)

and derive the Green’s function of quasiparticle in a magnetic field by a
different treatment from the one done in main text. In Eq. (111),
$D_{\mu}\psi=(\partial_{\mu}+ieA_{\mu})\psi$ is a covariant derivative with
respect to the magnetic field and the differential operator
$-i\hat{\bm{\nabla}}\equiv(-i{\bm{\nabla}})/|-i{\bm{\nabla}}|$ corresponds to
the hedgehog ansatz defined by the second line of Eq. (6). The presence of a
magnetic field breaks the translational invariance perpendicular to the
direction of magnetic field, which suggests that it is not straightforward to
obtain the Green’s function in momentum space. For this reason, we first
consider the Green’s function in coordinate space. In what follows, we choose
$z$ direction as the direction of magnetic field.

The effective Lagrangian (111) can be written as

$\displaystyle{\cal L}=\big{(}\bar{\psi},\Psi_{v}^{\dagger}\big{)}i{\cal
G}^{-1}\left(\begin{array}[]{c}\psi\\\ \Psi_{v}\\\ \end{array}\right)\ ,$
(114)

where

$\displaystyle i{\cal G}^{-1}=i{\cal G}_{0}^{-1}+{\cal V}$ (115)

is the inverse of the Green’s function of the fermions (incorporating the
gauge field) in coordinate space, with the free quasiparticle part

$\displaystyle i{\cal
G}_{0}^{-1}=\left(\begin{array}[]{ccc}i{\partial}_{0}+\mu&i{\bm{\nabla}}\cdot{\bm{\sigma}}&\Delta^{*}\\\
-i{\bm{\nabla}}{\bm{\sigma}}&-i{\partial}_{0}-\mu&\Delta^{*}i\hat{{\bm{\nabla}}}\cdot{\bm{\sigma}}\\\
\Delta&-\Delta i\hat{\bm{\nabla}}\cdot{\bm{\sigma}}&i{\partial}_{0}\\\
\end{array}\right)$ (119)

and the contributions from magnetic field

$\displaystyle{\cal
V}=\left(\begin{array}[]{ccc}0&e{\bm{A}}\cdot{\bm{\sigma}}&0\\\
-e{\bm{A}}\cdot{\bm{\sigma}}&0&0\\\ 0&0&0\\\ \end{array}\right).$ (123)

Now the Landau gauge ${\bm{A}}=(-By,0,0)$ is taken. In this paper we assume
the magnetic field is so weak that the inverse of the Green’s function (115)
can be approximated up to the first order of $eB$ as

$\displaystyle{\cal G}=i\big{(}i{\cal G}_{0}^{-1}+{\cal
V}\big{)}^{-1}\approx{\cal G}_{0}+{\cal G}_{1}\ ,$ (124)

with ${\cal G}_{1}\equiv{\cal G}_{0}i{\cal V}{\cal G}_{0}$. In coordinate
space, the free quasiparticle part ${\cal G}_{0}$ is straightforwardly
expressed by means of the Fourier transformation thanks to the translational
invariance as

$\displaystyle{\cal
G}_{0}(r,r^{\prime})=\int\frac{d^{4}p}{(2\pi)^{4}}\tilde{\cal
G}_{0}(p_{0},{\bm{p}}){\rm e}^{-ip\cdot(r-r^{\prime})}\ ,$ (125)

where the one in momentum space is of the form

$\displaystyle\tilde{\cal G}_{0}(p_{0},{\bm{p}})$ $\displaystyle=$
$\displaystyle\frac{i}{(p_{0}-{E}^{\rm
a}_{\bm{p}})(p_{0}-E_{\bm{p}}^{+})(p_{0}-E_{\bm{p}}^{-})}$ (129)
$\displaystyle\times\left(\begin{array}[]{ccc}p_{0}(p_{0}+\mu)-|\Delta|^{2}&-\left[|{\bm{p}}|p_{0}+|\Delta|^{2}\right]\hat{\bm{p}}\cdot{\bm{\sigma}}&-\Delta^{*}(p_{0}+|{\bm{p}}|+\mu)\\\
\left[|{\bm{p}}|p_{0}+|\Delta|^{2}\right]\hat{\bm{p}}\cdot{\bm{\sigma}}&-p_{0}(p_{0}+\mu)+|\Delta|^{2}&-\Delta^{*}(p_{0}+|{\bm{p}}|+\mu)\hat{\bm{p}}\cdot{\bm{\sigma}}\\\
-\Delta(p_{0}+|{\bm{p}}|+\mu)&\Delta(p_{0}+|{\bm{p}}|+\mu)\hat{\bm{p}}\cdot{\bm{\sigma}}&(p_{0}-|{\bm{p}}|+\mu)(p_{0}+|{\bm{p}}|+\mu)\\\
\end{array}\right)\ ,$

with the dispersion relations of the quasiparticles being

$\displaystyle E_{\bm{p}}^{+}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(|{\bm{p}}|-\mu+\sqrt{(|{\bm{p}}|-\mu)^{2}+8|\Delta|^{2}}\right)\
,$ $\displaystyle E_{\bm{p}}^{-}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(|{\bm{p}}|-\mu-\sqrt{(|{\bm{p}}|-\mu)^{2}+8|\Delta|^{2}}\right)\
,$ $\displaystyle E^{\rm a}_{\bm{p}}$ $\displaystyle=$
$\displaystyle-|{\bm{p}}|-\mu\ .$ (130)

These equations coincide with Eqs. (12) and (13). In what follows we assume
$\Delta$ to be real.

Unlike ${\cal G}_{0}$, the ${\cal G}_{1}$ part in Eq. (124) is rather
complicated due to the violation of the translational invariance. In fact, by
defining a $6\times 6$ matrix

$\displaystyle{\mathscr{V}}=\left(\begin{array}[]{ccc}0&-\sigma^{1}&0\\\
\sigma^{1}&0&0\\\ 0&0&0\\\ \end{array}\right)$ (134)

for convenience, the ${\cal G}_{1}$ part in Eq. (124) can be evaluated as

$\displaystyle{\cal G}_{1}(r,r^{\prime})$ $\displaystyle=$ $\displaystyle
ieB\int
d^{4}r^{\prime\prime}\int\frac{d^{4}p}{(2\pi)^{4}}\frac{d^{4}p^{\prime}}{(2\pi)^{4}}{\cal
G}_{0}(p){\rm e}^{-ip\cdot(r-r^{\prime\prime})}{\mathscr{V}}{\cal
G}_{0}(p^{\prime})y^{\prime\prime}{\rm
e}^{-ip^{\prime}\cdot(r^{\prime\prime}-r^{\prime})}$ (135) $\displaystyle=$
$\displaystyle-eB\int
d^{4}r^{\prime\prime}\int\frac{d^{4}p}{(2\pi)^{4}}\frac{d^{4}p^{\prime}}{(2\pi)^{4}}{\cal
G}_{0}(p){\mathscr{V}}{\cal
G}_{0}(p^{\prime})\left[\partial_{p_{y}-p^{\prime}_{y}}{\rm
e}^{i(p-p^{\prime})\cdot r^{\prime\prime}}\right]{\rm e}^{-ip\cdot
r+ip^{\prime}\cdot r^{\prime}}$ $\displaystyle=$ $\displaystyle
eB\int\frac{d^{4}p}{(2\pi)^{4}}\frac{d^{4}p^{\prime}}{(2\pi)^{4}}\partial_{p_{y}-p^{\prime}_{y}}\left[\frac{1}{{\cal
G}_{0}(p)}{\mathscr{V}}\frac{1}{{\cal G}_{0}(p^{\prime})}{\rm e}^{-ip\cdot
r+ip^{\prime}\cdot r^{\prime}}\right](2\pi)^{4}\delta^{4}(p-p^{\prime})$
$\displaystyle=$
$\displaystyle\frac{eB}{2}\int\frac{d^{4}p}{(2\pi)^{4}}\Bigg{\\{}\Big{[}\partial_{p_{y}}{\cal
G}_{0}(p)\Big{]}{\mathscr{V}}{\cal G}_{0}(p)-{\cal
G}_{0}(p){\mathscr{V}}\Big{[}\partial_{p_{y}}{\cal
G}_{0}(p)\Big{]}+i(y+y^{\prime}){\cal G}_{0}(p){\mathscr{V}}{\cal
G}_{0}\Bigg{\\}}{\rm e}^{-ip\cdot(r-r^{\prime})}$

with the help of Eq. (125). To obtain the third line in Eq. (135), we have
made use of the integration by parts. The last term in the fourth line in Eq.
(135), which is proportional to $y+y^{\prime}$, obviously breaks the
translational invariance, which stems from the so-called Schwinger phase
Schwinger (1951). Here we ignore such a term Gorbar  _et al._ (2013); Miransky
and Shovkovy (2015), and Eq. (135) takes the form of

$\displaystyle{\cal
G}_{1}(r,r^{\prime})=\int\frac{d^{4}p}{(2\pi)^{4}}\tilde{\cal
G}_{1}(p_{0},{\bm{p}}){\rm e}^{-ip\cdot(r-r^{\prime})}\ ,$ (136)

where the counterpart in momentum space is given by

$\displaystyle\tilde{\cal G}_{1}(p_{0},{\bm{p}})$ $\displaystyle=$
$\displaystyle-\frac{eB}{2}\frac{{\cal I}}{|{\bm{p}}|^{3}(p_{0}-E^{\rm
a}_{\bm{p}})^{2}(p_{0}-E_{\bm{p}}^{+})^{2}(p_{0}-E_{\bm{p}}^{-})^{2}}$

with

$\displaystyle{\cal I}\equiv\left(\begin{array}[]{cccccc}{\cal I}_{1}&{\cal
I}_{2}&{\cal I}_{3}&0&{\cal I}_{4}&{\cal I}_{5}\\\ -{\cal I}_{2}&-{\cal
I}_{1}&0&{\cal I}_{3}&-{\cal I}_{5}&{\cal I}_{4}^{*}\\\ -{\cal I}_{3}&0&-{\cal
I}_{1}&-{\cal I}_{2}&-i{\cal I}_{6}&{\cal I}_{7}\\\ 0&-{\cal I}_{3}&{\cal
I}_{2}&{\cal I}_{1}&{\cal I}_{7}^{*}&-i{\cal I}_{6}^{*}\\\ -{\cal
I}_{4}^{*}&{\cal I}_{5}&i{\cal I}_{6}^{*}&{\cal I}_{7}&{\cal I}_{8}&{\cal
I}_{9}\\\ -{\cal I}_{5}&-{\cal I}_{4}&{\cal I}_{7}^{*}&i{\cal I}_{6}&-{\cal
I}_{9}&-{\cal I}_{8}\\\ \end{array}\right)\ ,$ (144)

and the matrix elements are

$\displaystyle{\cal
I}_{1}=-2i\Big{[}\Delta^{2}-p_{0}(p_{0}+\mu)\Big{]}\Big{[}\Delta^{2}(p_{x}^{2}+p_{z}^{2})+|{\bm{p}}|^{3}p_{0}\Big{]}\
,$ $\displaystyle{\cal
I}_{2}=-2p_{y}p_{z}\Delta^{2}\Big{[}\Delta^{2}-p_{0}(p_{0}+\mu)\Big{]}\ ,$
(146) $\displaystyle{\cal
I}_{3}=-2ip_{z}|{\bm{p}}|(\Delta^{2}+|{\bm{p}}|p_{0})^{2}\ ,$ (147)
$\displaystyle{\cal I}_{4}$ $\displaystyle=$
$\displaystyle-\Delta\Bigg{\\{}|{\bm{p}}|p_{y}(2\Delta^{2}+p_{0}|{\bm{p}}|)(p_{x}-ip_{y})$
(148)
$\displaystyle+ip_{0}\Big{[}|{\bm{p}}|^{2}(p_{0}+|{\bm{p}}|+\mu)^{2}+ip_{y}(p_{0}+\mu)^{2}(p_{x}+ip_{y})\Big{]}$
$\displaystyle+2p_{x}p_{y}\Delta^{2}(p_{0}+\mu)\Bigg{\\}}\ ,$
$\displaystyle{\cal
I}_{5}=-p_{y}p_{z}\Delta\Big{\\{}2\Delta^{2}|{\bm{p}}|+p_{0}\big{[}(p_{0}+\mu)^{2}+|{\bm{p}}|^{2}\big{]}\Big{\\}}\
,$ $\displaystyle{\cal I}_{6}$ $\displaystyle=$ $\displaystyle 2\Delta
p_{z}\Big{\\{}|{\bm{p}}|(p_{0}+|{\bm{p}}|+\mu)\big{(}\Delta^{2}+|{\bm{p}}|p_{0}\big{)}$
(150) $\displaystyle-
ip_{x}p_{y}\big{[}\Delta^{2}-p_{0}(p_{0}+\mu)\big{]}\Big{\\}}\ ,$
$\displaystyle{\cal I}_{7}=2\Delta
p_{y}\Big{[}p_{z}^{2}+ip_{y}(p_{x}-ip_{y})\Big{]}\Big{[}\Delta^{2}-p_{0}(p_{0}+\mu)\Big{]}\
,$ $\displaystyle{\cal
I}_{8}=2i\Delta^{2}(p_{x}^{2}+p_{z}^{2})(p_{0}+|{\bm{p}}|+\mu)^{2}\ ,$ (152)
$\displaystyle{\cal I}_{9}=2\Delta^{2}p_{y}p_{z}(p_{0}+|{\bm{p}}|+\mu)^{2}\ .$
(153)

Summarizing the above derivation, the Green’s function of the quasiparticle in
a weak magnetic field in momentum space $\tilde{\cal G}(p_{0},{\bm{p}})$ can
be written as

$\displaystyle\tilde{\cal G}(p_{0},{\bm{p}})\approx\tilde{\cal
G}_{0}(p_{0},{\bm{p}})+\tilde{\cal G}_{1}(p_{0},{\bm{p}})$ (154)

with $\tilde{\cal G}_{0}(p_{0},{\bm{p}})$ and $\tilde{\cal
G}_{1}(p_{0},{\bm{p}})$ being given by Eqs. (129) and (LABEL:G1Momentum),
respectively, apart from the Schwinger phase.

Note that when the Kondo condensate is absent, i.e., when we take $\Delta=0$
in Eq. (154), the Green’s function is reduced to

$\displaystyle\tilde{\cal
G}(p_{0},{\bm{p}})\to\left(\begin{array}[]{cc}\tilde{\cal
G}_{\psi}(p_{0},{\bm{p}})&0\\\ 0&\tilde{\cal G}_{\Psi_{v}}(p_{0})\\\
\end{array}\right)\ ,$ (157)

with

$\displaystyle\tilde{\cal G}_{\psi}(p_{0},{\bm{p}})$ $\displaystyle=$
$\displaystyle\frac{i({\ooalign{\hfil/\hfil\crcr$p$}}+\mu\gamma^{0})}{(p_{0}+\mu)^{2}-|{\bm{p}}|^{2}}$
(161)
$\displaystyle+eB\gamma^{1}\gamma^{2}\frac{(p_{0}+\mu)\gamma^{0}-p_{z}\gamma^{3}}{\big{[}(p_{0}+\mu)^{2}-|{\bm{p}}|^{2}\big{]}^{2}}\
,$ $\displaystyle\tilde{\cal G}_{\Psi_{v}}(p_{0})$ $\displaystyle=$
$\displaystyle\frac{i}{p_{0}}\ .$ (162)

Our result of Eq. (161) is consistent with the one found in Appendix A of Ref.
Miransky and Shovkovy (2015).

###  A.2 Calculation of the axial current

In this subsection, we calculate the axial current by making use of the
Green’s function obtained in Appendix A.1.

From Eq. (154), by employing the imaginary-time formalism the axial current is
now given by

$\displaystyle\langle
j_{5}^{i}(t,{\bm{x}})\rangle=-iN_{c}T\sum_{m}\int\frac{d^{3}p}{(2\pi)^{3}}{\rm
tr}\left[\Gamma^{i}\Gamma_{5}\tilde{\cal G}(i{\omega}_{m},{\bm{p}})\right]\ ,$

[${\omega}_{m}=(2m+1)\pi T$] with

$\displaystyle\Gamma^{i}=\left(\begin{array}[]{cc}\gamma^{i}&0\\\ 0&0\\\
\end{array}\right)=\left(\begin{array}[]{ccc}0&\sigma^{i}&0\\\
-\sigma^{i}&0&0\\\ 0&0&0\\\ \end{array}\right)\ ,$ (169)
$\displaystyle\Gamma_{5}=\left(\begin{array}[]{cc}\gamma_{5}&0\\\ 0&0\\\
\end{array}\right)=\left(\begin{array}[]{ccc}0&1&0\\\ 1&0&0\\\ 0&0&0\\\
\end{array}\right)\ .$ (175)

Here the nonzero component is given only for $i=3$ which is of the form

$\displaystyle\langle j_{5}^{i=3}\rangle$ $\displaystyle=$ $\displaystyle
4eN_{c}BT\sum_{m}\int\frac{d^{3}p}{(2\pi)^{3}}$ $\displaystyle\times$
$\displaystyle\frac{\Big{[}\Delta^{2}-i{\omega}_{m}(i{\omega}_{m}+\mu)\Big{]}\Big{[}\frac{2}{3}\Delta^{2}|{\bm{p}}|^{2}+i{\omega}_{m}|{\bm{p}}|^{3}\Big{]}}{|{\bm{p}}|^{3}(i{\omega}_{m}-E^{\rm
a}_{\bm{p}})^{2}(i{\omega}_{m}-E_{\bm{p}}^{+})^{2}(i{\omega}_{m}-E_{\bm{p}}^{-})^{2}}\
.$ (176)

Therefore, the Abel-Plana formula (50) enables us to carry out the Matsubara
summation straightforwardly. The resultant expression is lengthy and not
illuminating so that we do not show explicitly here, but it was found to be
the same as the one obtained in static limit within the procedure employed in
the main text.

##  Appendix B General discussions on each contribution of response function
in Eq. (57)

In this appendix, we give general discussions on each contribution of response
function in Eq. (57) in more detail, and explain the difference between static
limit and dynamical limit. For convenience, we again show Eq. (57) here:

$\displaystyle\tilde{\Pi}^{\zeta^{\prime}\zeta}_{5}(i\bar{\omega}_{n},{\bm{q}})$
$\displaystyle=$
$\displaystyle-\frac{N_{c}}{|{\bm{q}}|^{2}}\int\frac{d^{3}p}{(2\pi)^{3}}\Big{[}\epsilon_{\zeta^{\prime}}(\hat{\bm{p}}^{\prime}\cdot{\bm{q}})-\epsilon_{\zeta}(\hat{\bm{p}}\cdot{\bm{q}})\Big{]}$
$\displaystyle\times$
$\displaystyle\frac{U_{\zeta^{\prime}}({\bm{p}}^{\prime})U_{\zeta}({\bm{p}})}{i\bar{\omega}_{n}-E_{{\bm{p}}^{\prime}}^{\zeta^{\prime}}+E_{\bm{p}}^{\zeta}}\left(f_{F}(E_{\bm{p}}^{\zeta})-f_{F}(E_{{\bm{p}}^{\prime}}^{\zeta^{\prime}})\right)$

with $\zeta,\zeta^{\prime}=+,-,{\rm a}$.

With respect to Eq. (LABEL:Pi5EachApp), in order to evaluate the response
function properly, we need to take $q_{0},{\bm{q}}\to 0$ limit after the
analytic continuation as seen from Eq. (59) together with Eq. (56). Here, such
a limit suggests that the nine contributions in Eq. (LABEL:Pi5EachApp) are
mathematically classified into three cases of

(I)

$\zeta=\zeta^{\prime}$ case ($\tilde{\Pi}_{5}^{++}$, $\tilde{\Pi}_{5}^{--}$,
$\tilde{\Pi}_{5}^{\rm aa}$),

(II)

$\zeta\neq\zeta^{\prime}$ and $\epsilon_{\zeta}=\epsilon_{\zeta^{\prime}}$
case ($\tilde{\Pi}_{5}^{+-}$, $\tilde{\Pi}_{5}^{-+}$),

(III)

$\zeta\neq\zeta^{\prime}$ and $\epsilon_{\zeta}\neq\epsilon_{\zeta^{\prime}}$
case ($\tilde{\Pi}_{5}^{+{\rm a}}$, $\tilde{\Pi}_{5}^{{\rm a}+}$,
$\tilde{\Pi}_{5}^{-{\rm a}}$, $\tilde{\Pi}_{5}^{{\rm a}-}$).

Recall that the physical meaning of these processes is given in Fig. 2.

First, in the case (I), we find

$\displaystyle\epsilon_{\zeta}(\hat{\bm{p}}^{\prime}\cdot{\bm{q}})-\epsilon_{\zeta}(\hat{\bm{p}}\cdot{\bm{q}})=C_{\zeta}^{(2)}|{\bm{q}}|^{2}+{\cal
O}({\bm{q}}^{3})\ ,$ $\displaystyle
q_{0}-E_{{\bm{p}}^{\prime}}^{\zeta}+E_{\bm{p}}^{\zeta}=q_{0}+D_{\zeta}^{(1)}|{\bm{q}}|+{\cal
O}({\bm{q}}^{2})\ ,$ (178)

where $C_{\zeta}^{(2)}$ and $D_{\zeta}^{(1)}$ are coefficients independent of
${\bm{q}}$ but dependent on ${\bm{p}}$ (the number in the parenthesis
corresponds to the order of ${\bm{q}}$). In this appendix, we do not show the
explicit expressions of coefficients, because they are not important, and it
is enough to consider only the order of expansion with respect to ${\bm{q}}$.

From Eq. (178), the real part of $\tilde{\Pi}_{5}^{\zeta^{\prime}\zeta}$ at
small $(q_{0},{\bm{q}})$ takes the form of

$\displaystyle\tilde{\Pi}_{5}^{\zeta\zeta}(q_{0},{\bm{q}})$
$\displaystyle\approx$
$\displaystyle-\frac{N_{c}}{|{\bm{q}}|^{2}}\int\frac{d^{3}p}{(2\pi)^{3}}C_{\zeta}^{(2)}|{\bm{q}}|^{2}\frac{U_{\zeta}({\bm{p}})U_{\zeta}({\bm{p}})}{q_{0}+D_{\zeta}^{(1)}|{\bm{q}}|}$
(179) $\displaystyle\times$
$\displaystyle\left[-(\hat{\bm{p}}\cdot{\bm{q}})\frac{\partial
f_{F}(E_{\bm{p}}^{\zeta})}{\partial|{\bm{p}}|}\right]$ $\displaystyle=$
$\displaystyle-
N_{c}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{N^{(1)}_{\zeta}|{\bm{q}}|}{q_{0}+D^{(1)}_{\zeta}|{\bm{q}}|}\frac{\partial
f_{F}(E_{\bm{p}}^{\zeta})}{\partial|{\bm{p}}|}$

by the Taylor expansion. In Eq. (179), we have defined
$N_{\zeta}^{(1)}\equiv-(\hat{\bm{p}}\cdot\hat{\bm{q}})C_{\zeta}^{(2)}U_{\zeta}({\bm{p}})U_{\zeta}({\bm{p}})$
($\hat{\bm{q}}\equiv{\bm{q}}/|{\bm{q}}|$). Therefore, Eq. (179) yields

$\displaystyle\lim_{{\bm{q}}\to{\bm{0}}}\tilde{\Pi}_{5}^{\zeta\zeta}(0,{\bm{q}})$
$\displaystyle=$ $\displaystyle-
N_{c}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{N^{(1)}_{\zeta}}{D^{(1)}_{\zeta}}\frac{\partial
f_{F}(E_{\bm{p}}^{\zeta})}{\partial|{\bm{p}}|}$ $\displaystyle\lim_{q_{0}\to
0}\tilde{\Pi}_{5}^{\zeta\zeta}(q_{0},{\bm{0}})$ $\displaystyle=$
$\displaystyle 0\ ,$ (180)

which clearly shows that the results in static limit and dynamical limit
differ due to a singularity caused by the factor
$1/(q_{0}+D_{\zeta}^{(1)}|{\bm{q}}|)$ in the $q_{0},{\bm{q}}\to 0$ limit.

Next, in the case (II), we find

$\displaystyle\epsilon_{\zeta^{\prime}}(\hat{\bm{p}}^{\prime}\cdot{\bm{q}})-\epsilon_{\zeta}(\hat{\bm{p}}\cdot{\bm{q}})=C^{(2)}_{\zeta^{\prime}\zeta}|{\bm{q}}|^{2}+{\cal
O}({\bm{q}}^{3})\ ,$ $\displaystyle
q_{0}-E_{{\bm{p}}^{\prime}}^{\zeta^{\prime}}+E_{\bm{p}}^{\zeta}=q_{0}+D_{\zeta^{\prime}\zeta}^{(0)}+{\cal
O}({\bm{q}}^{1})\ ,$ (181)

so that the same expansion as in Eq. (179) reads

$\displaystyle\tilde{\Pi}_{5}^{\zeta^{\prime}\zeta}(q_{0},{\bm{q}})$
$\displaystyle\approx$
$\displaystyle-\frac{N_{c}}{|{\bm{q}}|^{2}}\int\frac{d^{3}p}{(2\pi)^{3}}C^{(2)}_{\zeta^{\prime}\zeta}|{\bm{q}}|^{2}\frac{U_{\zeta^{\prime}}({\bm{p}})U_{\zeta}({\bm{p}})}{q_{0}+D_{\zeta^{\prime}\zeta}^{(0)}}$
(182) $\displaystyle\times$
$\displaystyle\left(f_{F}(E_{\bm{p}}^{\zeta})-f_{F}(E_{{\bm{p}}}^{\zeta^{\prime}})\right)$
$\displaystyle\equiv$ $\displaystyle-
N_{c}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{N^{(0)}_{\zeta^{\prime}\zeta}}{q_{0}+D^{(0)}_{\zeta^{\prime}\zeta}}$
$\displaystyle\times$
$\displaystyle\left(f_{F}(E_{\bm{p}}^{\zeta})-f_{F}(E_{{\bm{p}}}^{\zeta^{\prime}})\right)$

[$N^{(0)}_{\zeta^{\prime}\zeta}\equiv
C_{\zeta^{\prime}\zeta}^{(2)}U_{\zeta^{\prime}}({\bm{p}})U_{\zeta}({\bm{p}})$].
Namely, in this case $D^{(0)}_{\zeta^{\prime}\zeta}$ in the denominator acts
as a regulator preventing us from getting the singularity, resulting in that
there is no difference between static and dynamical limits:

$\displaystyle\lim_{{\bm{q}}\to{\bm{0}}}\tilde{\Pi}_{5}^{\zeta^{\prime}\zeta}(0,{\bm{q}})=\lim_{q_{0}\to
0}\tilde{\Pi}_{5}^{\zeta^{\prime}\zeta}(q_{0},{\bm{0}})$
$\displaystyle=-N_{c}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{C^{(0)}_{\zeta^{\prime}\zeta}}{D^{(0)}_{\zeta^{\prime}\zeta}}\left(f_{F}(E_{\bm{p}}^{\zeta})-f_{F}(E_{{\bm{p}}}^{\zeta^{\prime}})\right)\
.$ (183)

Finally, in the case (III), we get

$\displaystyle\epsilon_{\zeta^{\prime}}(\hat{\bm{p}}^{\prime}\cdot{\bm{q}})-\epsilon_{\zeta}(\hat{\bm{p}}\cdot{\bm{q}})=C_{\zeta^{\prime}\zeta}^{(1)}|{\bm{q}}|+C_{\zeta^{\prime}\zeta}^{(2)}|{\bm{q}}^{2}|+{\cal
O}({\bm{q}}^{3})\ ,$ $\displaystyle
q_{0}-E_{{\bm{p}}^{\prime}}^{\zeta^{\prime}}+E_{\bm{p}}^{\zeta}=q_{0}+D_{\zeta^{\prime}\zeta}^{(0)}+{\cal
O}({\bm{q}}^{1})\ ,$ (184)

so that the same expansion as in Eq. (179) leads to a slightly complicated
form:

$\displaystyle\tilde{\Pi}_{5}^{\zeta^{\prime}\zeta}(q_{0},{\bm{q}})$
$\displaystyle\approx$
$\displaystyle-\frac{N_{c}}{|{\bm{q}}|^{2}}\int\frac{d^{3}p}{(2\pi)^{3}}\left(C_{\zeta^{\prime}\zeta}^{(1)}|{\bm{q}}|+C_{\zeta^{\prime}\zeta}^{(2)}|{\bm{q}}^{2}|\right)$
$\displaystyle\times$
$\displaystyle\frac{\left[U_{\zeta^{\prime}}({\bm{p}})+(\hat{\bm{p}}\cdot{\bm{q}})\frac{\partial
U_{\zeta^{\prime}}({\bm{p}})}{\partial|{\bm{p}}|}\right]U_{\zeta}({\bm{p}})}{q_{0}+D^{(0)}_{\zeta^{\prime}\zeta}}$
$\displaystyle\times$
$\displaystyle\left[f_{F}(E_{\bm{p}}^{\zeta})-f_{F}(E_{{\bm{p}}}^{\zeta^{\prime}})-(\hat{\bm{p}}\cdot{\bm{q}})\frac{\partial
f_{F}(E_{\bm{p}}^{\zeta^{\prime}})}{\partial|{\bm{p}}|}\right]$
$\displaystyle\approx$ $\displaystyle-
N_{c}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{\frac{1}{|{\bm{q}}|}{N}_{\zeta^{\prime}\zeta}^{(-1)}+{N}^{(0)}_{\zeta^{\prime}\zeta}}{q_{0}+D^{(0)}_{\zeta^{\prime}\zeta}}$
$\displaystyle\times$
$\displaystyle\left[f_{F}(E_{\bm{p}}^{\zeta})-f_{F}(E_{{\bm{p}}}^{\zeta^{\prime}})-(\hat{\bm{p}}\cdot{\bm{q}})\frac{\partial
f_{F}(E_{\rm p}^{\zeta^{\prime}})}{\partial|{\bm{p}}|}\right]\ .$

Here we have defined $N_{\zeta^{\prime}\zeta}^{(-1)}\equiv
C_{\zeta^{\prime}\zeta}^{(1)}U_{\zeta^{\prime}}({\bm{p}})U_{\zeta}({\bm{p}})$
and
$N_{\zeta^{\prime}\zeta}^{(0)}\equiv(\hat{\bm{p}}\cdot\hat{\bm{q}})C_{\zeta^{\prime}\zeta}^{(1)}\frac{\partial
U_{\zeta^{\prime}}({\bm{p}})}{\partial|{\bm{p}}|}U_{\zeta}({\bm{p}})+C_{\zeta^{\prime}\zeta}^{(2)}U_{\zeta^{\prime}}({\bm{p}})U_{\zeta}({\bm{p}})$.
Equation (LABEL:Pi5Genaral3) seemingly causes a divergence in the ${\bm{q}}\to
0$ limit due to the $1/|{\bm{q}}|$ in the numerator. However, such a
problematic contribution will be removed after the angular integral. Namely in
both static limit and dynamical limit, Eq. (LABEL:Pi5Genaral3) converges,
leading to the same result as

$\displaystyle\lim_{{\bm{q}}\to{\bm{0}}}\tilde{\Pi}_{5}^{\zeta^{\prime}\zeta}(0,{\bm{q}})=\lim_{q_{0}\to
0}\tilde{\Pi}_{5}^{\zeta^{\prime}\zeta}(q_{0},{\bm{0}})$
$\displaystyle=-N_{c}\int\frac{d^{3}p}{(2\pi)^{3}}\Bigg{\\{}\frac{N^{(0)}_{\zeta^{\prime}\zeta}}{D^{(0)}_{\zeta^{\prime}\zeta}}\left(f_{F}(E_{\bm{p}}^{\zeta})-f_{F}(E_{{\bm{p}}}^{\zeta^{\prime}})\right)$
$\displaystyle-(\hat{\bm{p}}\cdot\hat{\bm{q}})\frac{N_{\zeta^{\prime}\zeta}^{(-1)}}{D^{(0)}_{\zeta^{\prime}\zeta}}\frac{\partial
f_{F}(E_{\rm p}^{\zeta^{\prime}})}{\partial|{\bm{p}}|}\Bigg{\\}}\ .$ (186)

Summarizing the above discussions, we have confirmed that

$\displaystyle\lim_{{\bm{q}}\to{\bm{0}}}\tilde{\Pi}_{5}^{\zeta\zeta}(0,{\bm{q}})$
$\displaystyle=$ $\displaystyle-N_{c}\int\frac{d^{3}p}{(2\pi)^{3}}{\cal
I}_{\zeta}({\bm{p}})\frac{\partial
f_{F}(E_{\bm{p}}^{\zeta})}{\partial|{\bm{p}}|}$ $\displaystyle\lim_{q_{0}\to
0}\tilde{\Pi}_{5}^{\zeta\zeta}(q_{0},{\bm{0}})$ $\displaystyle=$
$\displaystyle 0$ (187)

for the case (I) from Eq. (180), while

$\displaystyle\lim_{{\bm{q}}\to{\bm{0}}}\tilde{\Pi}_{5}^{\zeta^{\prime}\zeta}(0,{\bm{q}})=\lim_{q_{0}\to
0}\tilde{\Pi}_{5}^{\zeta^{\prime}\zeta}(q_{0},{\bm{0}})$ (188)

for the case (II) and case (III) from Eqs. (183) and (186), respectively. In
Eq. (187), we have defined ${\cal I}_{\zeta}({\bm{p}})\equiv
N_{\zeta}^{(1)}/D_{\zeta}^{(1)}$ to simplify the expression. Equations (187)
and (188) show that the difference between the response function in static
limit and dynamical limit can be generated in only case (I) which corresponds
to the scattering process of ${\rm q.p.}\to{\rm q.p.}$, ${\rm q.h.}\to{\rm
q.h.}$ or ${\rm a.p.}\to{\rm a.p.}$.

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