# Functional Renormalization Group analysis of the quark-condensation pattern
on the Fermi surface: A simple effective-model approach

Kie Sang Jeong<EMAIL_ADDRESS>Fabrizio Murgana
<EMAIL_ADDRESS>Ashutosh Dash<EMAIL_ADDRESS>Dirk H.
Rischke<EMAIL_ADDRESS>Institute for Theoretical Physics,
Goethe University, Max-von-Laue-Str.1, D-60438 Frankfurt am Main, Germany

###### Abstract

A simple effective model for the intermediate-density regime is constructed
from the high-density effective theory of quantum chromodynamics (QCD). In the
effective model, under a renormalization-group (RG) scaling towards low
momenta, the original QCD interactions lead to four-quark contact interactions
for the relevant quark and hole modes around the Fermi surface. The contact
interaction in the scalar channel can be traced back to zero-sound-type
collinear quark scattering near the Fermi surface in an instanton background.
The quark and hole states in opposite directions of a given Fermi velocity
form the collective scalar bosonic mode $\sigma$. The magnitude of $\sigma$ is
investigated via the non-perturbative Functional Renormalization Group (FRG)
evolution of the effective average action from the ultraviolet (UV) to the
infrared (IR). In the mean background-field approximation for $\sigma$,
nontrivial minima ($\bar{\sigma}\neq 0$) are found in the IR limit of the
effective average action. A nonvanishing $\bar{\sigma}$ corresponds to
condensation of quark and hole states in opposite directions of a given Fermi
velocity, in a thin shell-like structure in momentum space around the Fermi
surface. This looks similar to the shell-like baryon distribution in momentum
space assumed in the quarkyonic-matter concept. However, when including a
dynamic bosonic $\sigma$-mode in the RG flow, we find that its diffusive
nature destroys the quark-hole condensate, i.e., the IR potential does not
show any minima beyond the trivial one.

##  I Introduction

The scalar quark condensate $\left\langle\bar{\psi}\psi\right\rangle$ plays an
important role in the descripton of the low-energy ground state of QCD since
it is the order parameter related to spontaneous chiral symmetry breaking. The
dynamically generated constituent quark mass in a hadronic state is determined
by the chiral order parameter in low-energy effective models of QCD
Nambu:1961tp ; Nambu:1961fr ; Gell-Mann:1960mvl ; Gross:1974jv . Also, in QCD
sum-rule analyses of hadronic states Shifman:1978bx ; Shifman:1978by , the
ground-state properties are mainly determined by the quark condensates
appearing in the operator-product expansion of the correlation function of the
hadron interpolating field. At low nuclear matter densities, the attractive
scalar self-energy of the quasi-nucleon state leads to a reduced magnitude of
the in-medium quark condensate Drukarev:1988ib ; Cohen:1994wm . If one assumes
the quasi-particle picture and, correspondingly, a homogeneous condensation
amplitude in the nuclear medium like in vacuum, such a decreasing tendency
Cohen:1991nk ; Fiorilla:2012bc  implies that, above some density, the quark
condensate vanishes and a phase transition to quark matter occurs.

On the other hand, it was speculated that an inhomogeneous configuration can
emerge in dense nuclear matter, where translation invariance of the system is
broken, rather than having a direct phase transition to quark matter. Density
waves in dense nuclear matter were discussed by Overhauser Overhauser:1960zz
and a series of studies about inhomogeneous pion condensation was reported in
Refs. Weise:1975tk ; Migdal:1978az ; Migdal:1973zm . Recently, the quarkyonic-
matter concept McLerran:2007qj  emerged from large-$N_{c}$ QCD tHooft:1973alw
and has been applied to dense nuclear matter McLerran:2018hbz ; Jeong:2019lhv
; Zhao:2020dvu ; Duarte:2020xsp ; Duarte:2020kvi ; Koch:2022act ; Xia:2023omv
; Poberezhnyuk:2023rct ; Dey:2024lco . The shell structure of quarkyonic
matter in momentum space implies a periodic distribution of baryon number in
configuration space. If the quasi-particle distribution is inhomogeneous in
configuration space, the corresponding quark-condensation pattern should have
a non-trivial inhomogeneity as well. The possibility of inhomogeneous quark
condensation was first discussed in the large-$N_{c}$ limit Deryagin:1992rw .
Subsequent studies investigated whether inhomogeneous quark condensation can
compete with the color-superconducting phase Shuster:1999tn ; Park:1999bz ;
Rapp:2000zd . Discussions within effective models have been reported as well
Nakano:2004cd ; Schnetz:2004vr ; Nickel:2009ke ; Nickel:2009wj ; Kojo:2009ha ;
Basar:2009fg ; Buballa:2014tba ; Buballa:2018hux ; Lenz:2020bxk ; Lenz:2020cuv
; Koenigstein:2021llr .

In the cold, dense limit ($\mu_{q}\gg\Lambda_{\textrm{QCD}}$,
$T\ll\Lambda_{\textrm{QCD}}$), the (3+1)-dimensional QCD Lagrangian can be
reduced to a longitudinal high-density effective theory (HDET) Hong:1998tn ;
Hong:1999ru ; Nardulli:2002ma ; Schafer:2003jn . At intermediate densities
($\mu_{q}\gtrsim\Lambda_{\textrm{QCD}}$), where non-perturbative gauge-field
interactions become dominant, one may construct an effective model analogous
to the Gross-Neveu (GN) model Gross:1974jv  in terms of the relevant quark and
hole modes in the collinear direction appearing in HDET. In this effective
model, QCD interactions lead to instanton-mediated four-quark interaction
terms on the Fermi surface. In this work, we investigate the possible
condensation pattern of quark modes around the Fermi surface via bosonization
of the four-quark interaction. The general form of the four-quark interaction
defined around the Fermi surface can be expressed as follows Nardulli:2002ma ;
Schafer:2003jn :

$\displaystyle\mathcal{L}_{\textrm{4q}}$
$\displaystyle=\sum_{\vec{v}_{i}}\sum_{\Gamma,\Gamma^{\prime}}\frac{c^{\Gamma\Gamma^{\prime}}(\vec{v}_{1},\vec{v}_{2},\vec{v}_{3},\vec{v}_{4})}{\mu^{2}}\left[\bar{\psi}_{+}(\vec{v}_{3},x)\Gamma\psi_{+}(\vec{v}_{1},x)\right]\left[\bar{\psi}_{+}(\vec{v}_{4},x)\Gamma^{\prime}\psi_{+}(\vec{v}_{2},x)\right]+\ldots\;,$
(1)

where $\Gamma$ and $\Gamma^{\prime}$ denote elements of the Clifford basis for
the quark bilinears. The fields $\psi_{+}(\vec{v}_{i},x)$ denote the quark
modes around the Fermi surface, where $\vec{v}_{i}$ is the Fermi velocity,
whose magnitude is given as
$|\vec{v}_{i}|=|\vec{p}_{Fi}|/\sqrt{\vec{p}_{Fi}^{\,2}+m_{i}^{2}}$. Two types
of collinear scattering processes become marginal in the Wilsonian sense when
scaling the momentum exchange towards the Fermi surface: the zero-sound
channel, cf. Fig. 1(a), and the Bardeen-Cooper-Schrieffer channel, cf. Fig.
1(b), both of which satisfy the collinear scattering conditions
Nardulli:2002ma . In this work, we consider the Overhauser-type scalar
pairing, for which
$\vec{v}_{1}\simeq-\vec{v}_{3}\simeq-\vec{v}_{2}\simeq\vec{v}_{4}$. This
channel gets contributions from both the zero-sound and BCS-type diagrams, and
the zero-sound type vector pairing. The BCS-type channel leading to color
superconductivity will be left for future work.

Figure 1: (a) Zero-sound-type diagram, (b) BCS-type diagram, and (c) effective
four-quark interaction diagram, which gets contribution from both (a) and (b).
In (a) and (b), the incoming momenta are $\vec{p}_{1}$ and $\vec{p}_{2}$,
while the outgoing momenta are $\vec{p}_{3}$ and $\vec{p}_{4}$. These diagrams
become dominating when the fermion lines in the loop hit the singular point at
the Fermi surface simultaneously. For (a), when the transferred momentum is
minimal ($\vec{k}\simeq 0$), both internal lines become singular at
$|\vec{q}|\rightarrow|\vec{p}_{F}|$. When scaling to the Fermi surface, the
collinear scattering ($\vec{p}_{1}\simeq\vec{p}_{3}$,
$\vec{p}_{2}\simeq\vec{p}_{4}$, $\vec{p}_{1}\simeq\pm\vec{p}_{2}$,
$|\vec{p}_{i}|=|\vec{p}_{F}|$) becomes marginal in the Wilsonian sense. For
(b), both internal lines become singular at
$|\vec{q}|\rightarrow|\vec{p}_{F}|$ when the incoming momenta are on opposite
sides of the Fermi surface ($\vec{p}_{1}\simeq-\vec{p}_{2}$). For the
effective description (c), if the incoming $\vec{p}_{2}$ line and the outgoing
$\vec{p}_{4}$ line are conjugated, then the process can be understood as the
forward scattering of an incoming particle with momentum $\vec{p}_{1}$ and a
hole with momentum $-\vec{p}_{4}$ to an outgoing particle with momentum
$\vec{p}_{3}$ and a hole with momentum $-\vec{p}_{2}$ with minimal momentum
transfer ($\vec{k}\simeq 0$). For Overhauser-type scalar quark pairing, the
corresponding Fermi velocities in Eq. (1) should be
$\vec{v}_{1}\simeq-\vec{v}_{3}\simeq-\vec{v}_{2}\simeq\vec{v}_{4}$, since
$\Gamma,\Gamma^{`}=I$ allows only for such a configuration.

Since we define the effective model with non-perturbative four-quark
interactions at intermediate densities, the effective average action of the
system will be investigated in the Functional Renormalization Group (FRG)
framework Wetterich:1991be ; Wetterich:1992yh ; Ellwanger:1993mw ;
Morris:1993qb ; Morris:1994ki ; Berges:2000ew ; Dupuis:2020fhh , instead of a
renormalization-group (RG) analysis of the couplings at a given finite order
of the perturbative expansion. In the FRG analysis, quantum and thermal
fluctuations are integrated out successively from an energy or momentum scale
in the UV to the IR. The RG flow of the effective average action corresponds
to a successive functional integration over the quantum fields. It can be
formulated via an exact flow equation for the effective average action at a
given scale $k$. The flow equation describes the evolution of the effective
average action in “RG time” $t\equiv-\ln{k/\Lambda_{\textrm{UV}}}$, where
$\Lambda_{\textrm{UV}}$ is the UV scale. In this approach, the effective four-
quark interactions can be expressed in terms of bosonic collective fields via
a Hubbard-Stratonovich transformation Stratonovich (1957); Hubbard:1959ub ,
where the bosonic fields have the quantum numbers of the mesons, which are the
dominant degrees freedom in the IR limit. The bosonized Lagrangian corresponds
to the UV limit of the effective average action and its RG flow to the IR
limit can be obtained in both the mean background-field approximation and for
a dynamical scalar boson field $\sigma$ within the FRG framework. Considering
a dynamical $\sigma$-mode, the derivative of the exact flow equation with
respect to $\sigma$ is a partial differential equation of advection-diffusion
type known from fluid dynamics Grossi:2021ksl ; Stoll:2021ori ;
Koenigstein:2021rxj ; Steil:2021cbu ; Murgana:2023xrq ; Murgana:2023pyx .
Thus, methods well-known from fluid dynamics can be utilized to solve the flow
equation. If the IR potential develops a non-trivial minimum $\bar{\sigma}\neq
0$ in a momentum shell around the Fermi surface, quark and hole states with
opposite momenta on the Fermi surface form a scalar condensate, which breaks
the chiral symmetry among the relevant quark modes. The broken phase in the
momentum shell around the Fermi surface is related to an exotic baryon-number
distribution in the chiral density-wave picture Overhauser:1960zz ;
Deryagin:1992rw  and the quarkyonic-matter concept McLerran:2007qj .

This paper is organized as follows: In Sec. II, the definition of quark and
hole states appearing in HDET will be introduced. The possible condensation
patterns in an instanton background and the QCD origin of the contact
interactions in the effective model will be discussed as well. In Sec. III,
the RG flow of the average effective action of the model will be analyzed in
the FRG framework: first using the mean-field approximation for the
$\sigma$-mode, including both the homogeneous and the inhomogeneous quark-hole
condensation, and later solving the FRG flow equation for a dynamical
$\sigma$-mode Stoll:2021ori ; Koenigstein:2021rxj ; Steil:2021cbu ;
Murgana:2023xrq ; Murgana:2023pyx . Concluding remarks are given in Sec. IV.

##  II High-density effective Lagrangian and Fermi-surface modes

In this section, we introduce the quark and hole modes which are excited in
the direction longitudinal to the Fermi momentum appearing in HDET. The
possible condensation patterns between the relevant surface quark modes, which
can be considered explicitly in an effective approach analogous to the GN
model, will be introduced briefly. The possibility of surface zero-mode
correlations in an instanton background will be discussed as well, since the
contact four-quark interaction appearing in a simple model (which will be
introduced in Sec. III) can be traced back to the four-quark interaction
between the surface quark modes in an instanton background.

###  II.1 Positive- and negative-energy mode decomposition

Our starting point is isospin-symmetric, high-density QCD matter, at a quark
chemical potential $\mu\gg m_{q},\Lambda_{\textrm{QCD}}$, such that we can
take quarks to be massless and weakly interacting. The Lagrangian of this
system is

$\displaystyle\mathcal{L}_{\textrm{QCD}}$
$\displaystyle=-\frac{1}{4}\mathcal{F}^{a}_{~{}\mu\nu}\mathcal{F}^{a\mu\nu}+\bar{\psi}(i\,/D+\mu\gamma^{0})\psi\;,$
(2)

where $\mathcal{F}_{\mu\nu}=(-i/g)[D_{\mu},D_{\nu}]$, with
$D_{\mu}=\partial_{\mu}+igA_{\mu}$, and a summation over the $N_{f}$ quark
flavors is implied. At low temperatures, low-energy quark and hole states
around the Fermi surface are excited mainly in the direction of the Fermi
velocity $\vec{v}=\vec{p}_{F}/|\vec{p}_{F}|$, because excitations tangential
to the Fermi surface are forbidden due to the Pauli principle. The momentum of
a low-energy mode near the Fermi surface can then be written as
$\vec{p}\equiv\vec{p}_{F}+\vec{l}$, with $\vec{p}_{F}=\mu\vec{v}$, and a
residual momentum $\vec{l}$, with $|\vec{l}\,|\lesssim|\vec{p}_{F}|$
Hong:1998tn ; Hong:1999ru . We then Fourier-decompose the quark spinors
$\psi(x)$ in terms of Fermi-surface modes $\psi(\vec{v},x)$,

$\displaystyle\psi(x)$
$\displaystyle=\sum_{\vec{v}}e^{i\mu\vec{v}\cdot\vec{x}}\psi(\vec{v},x)=\sum_{\vec{v}}e^{i\mu\vec{v}\cdot\vec{x}}\left[\psi_{+}(\vec{v},x)+\psi_{-}(\vec{v},x)\right]\;,$
(3a) $\displaystyle\psi_{\pm}(\vec{v},x)$
$\displaystyle=\int\frac{d^{4}l}{(2\pi)^{4}}e^{-il\cdot
x}\psi_{\pm}(\vec{v},l)\;.$ (3b)

where the sum over $\vec{v}$ runs over all possible values of the Fermi
velocity on the Fermi surface and $|l_{0}|\lesssim\mu$. The Fermi-surface mode
$\psi(\vec{v},x)$ has been decomposed into a slow $(+)$ and a fast $(-)$
eigenmode of the Hamiltonian $\mathcal{H}$ corresponding to the Lagrangian (2)
in the non-interacting limit:

$\displaystyle\mathcal{H}\psi_{\pm}(\vec{v},x)=\left(\vec{p}\cdot\vec{\alpha}-\mu\right)\psi_{\pm}(\vec{v},x)=E_{\pm}\psi_{\pm}(\vec{v},x)=\left(\pm|\vec{p}\,|-\mu\right)\psi_{\pm}(\vec{v},x)\;,$
(4)

with $\vec{\alpha}=\gamma_{0}\vec{\gamma}$. Each mode satisfies the following
relations:

$\displaystyle\vec{p}\cdot\vec{\alpha}\,\psi_{\pm}(\vec{v},x)=\pm\left|\vec{p}\right|\psi_{\pm}(\vec{v},x)=\pm\left|\mu\vec{v}+\vec{l}_{\parallel}+\vec{l}_{\bot}\right|\psi_{\pm}(\vec{v},x)\;,$
(5)

where $\vec{l}_{\parallel}=\vec{v}(\vec{v}\cdot\vec{l})$ and
$\vec{l}_{\bot}=\vec{l}-\vec{v}(\vec{v}\cdot\vec{l})$. In the limit where the
momentum approaches the Fermi surface ($|\vec{l}\,|\ll|\vec{p}_{F}|$,
$|l_{0}|\ll\mu$), the quark modes are almost on-shell and the following
projection becomes exact:

$\displaystyle\psi_{\pm}(\vec{v},x)$ $\displaystyle\equiv
P_{\pm}\psi(\vec{v},x)\equiv\frac{1\pm\vec{\alpha}\cdot\vec{v}}{2}\psi(\vec{v},x)\;.$
(6)

Since the surface modes have momenta $|\vec{p}\,|\simeq\mu$, the slow mode
$\psi_{+}(\vec{v},x)$ requires a small excitation energy $|E_{+}|\ll\mu$,
while the fast mode $\psi_{-}(\vec{v},x)$ requires a large excitation energy
$|E_{-}|\simeq 2\mu$. Using the algebraic relations (2.9) of Ref. Hong:1999ru
for the decomposition of the high-density QCD Lagrangian, i.e.,

$\displaystyle\bar{\psi}_{+}(\vec{v},x)P_{-}\gamma^{\mu}P_{+}\psi_{+}(\vec{v},x)$
$\displaystyle=V^{\mu}\bar{\psi}_{+}(\vec{v},x)\gamma^{0}\psi_{+}(\vec{v},x)\;,$
$\displaystyle\bar{\psi}_{-}(\vec{v},x)P_{+}\gamma^{\mu}P_{-}\psi_{-}(\vec{v},x)$
$\displaystyle=\bar{V}^{\mu}\bar{\psi}_{-}(\vec{v},x)\gamma^{0}\psi_{-}(\vec{v},x)\;,$
$\displaystyle\bar{\psi}_{-}(\vec{v},x)P_{+}\gamma^{\mu}P_{+}\psi_{+}(\vec{v},x)$
$\displaystyle=\bar{\psi}_{-}(\vec{v},x)\gamma_{\bot}^{\mu}\psi_{+}(\vec{v},x)\;,$
$\displaystyle\bar{\psi}_{+}(\vec{v},x)P_{-}\gamma^{\mu}P_{-}\psi_{-}(\vec{v},x)$
$\displaystyle=\bar{\psi}_{+}(\vec{v},x)\gamma_{\bot}^{\mu}\psi_{-}(\vec{v},x)\;,$
(7)

where $V^{\mu}=(1,\vec{v})$, $\bar{V}^{\mu}=(1,-\vec{v})$, and
$\gamma_{\bot}^{\mu}=\gamma^{\mu}-\gamma_{\parallel}^{\mu}$ with
$\gamma_{\parallel}^{\mu}=(\gamma^{0},\vec{v}(\vec{v}\cdot\vec{\gamma}))$, the
Lagrangian (2) can be expressed in terms of the slow and fast eigenmodes as

$\displaystyle\mathcal{L}_{\textrm{QCD}}$
$\displaystyle=-\frac{1}{4}\mathcal{F}^{a}_{~{}\mu\nu}\mathcal{F}^{a\mu\nu}$
$\displaystyle\quad+\sum_{\vec{v}}\big{[}\bar{\psi}_{+}(\vec{v},x)\gamma^{0}(iV^{\mu}D_{\mu})\psi_{+}(\vec{v},x)+\bar{\psi}_{-}(\vec{v},x)\gamma^{0}(2\mu+i\bar{V}^{\mu}D_{\mu})\psi_{-}(\vec{v},x)$
$\displaystyle\qquad\quad\quad+\bar{\psi}_{+}(\vec{v},x)\;i\,/D_{\bot}\psi_{-}(\vec{v},x)+\bar{\psi}_{-}(\vec{v},x)\;i\,/D_{\bot}\psi_{+}(\vec{v},x)\big{]}\;.~{}$
(8)

The fast mode $\psi_{-}(\vec{v},x)$ can be integrated out, which effectively
corresponds to replacing it by using its equation of motion
($\delta\mathcal{S}_{\textrm{QCD}}/\delta\bar{\psi}_{-}(\vec{v},x)=0$):

$\displaystyle\psi_{-}(\vec{v},x)=-\frac{\gamma^{0}}{2\mu+i\bar{V}^{\mu}D_{\mu}}\;i\,/D_{\bot}\psi_{+}(\vec{v},x)\;.~{}$
(9)

Then, the high-density QCD Lagrangian (2) can be reduced to that of the so-
called high-density effective theory (HDET) Hong:1998tn ; Hong:1999ru ;
Schafer:2003jn :

$\displaystyle\mathcal{L}_{\textrm{QCD}}$
$\displaystyle\simeq-\frac{1}{4}\mathcal{F}^{a}_{~{}\mu\nu}\mathcal{F}^{a\mu\nu}+\sum_{\vec{v}}\left[\bar{\psi}_{+}(\vec{v},x)\gamma^{0}(iV^{\mu}D_{\mu})\psi_{+}(\vec{v},x)-\bar{\psi}_{+}(\vec{v},x)\frac{\gamma^{0}}{2\mu+i\bar{V}^{\mu}D_{\mu}}(/D_{\bot})^{2}\psi_{+}(\vec{v},x)\right]+\ldots\;.$
(10)

One can calculate perturbatively the scattering processes of the relevant
modes around a given small patch on the Fermi surface by using HDET. In the
limit $\mu\gg\Lambda_{\textrm{QCD}}$, asymptotic freedom applies and the
effective form of the four-quark interaction channels (1) can be constructed
by summing up ladder diagrams and matching them to the corresponding channel
of QCD. However, as non-perturbative effects such as instanton contributions
may become important at $\mu\simeq\mathcal{O}(\Lambda_{\textrm{QCD}})$,
fermion contact interactions as considered in NJL-like models will be
investigated as a first step.

###  II.2 Surface zero-modes and instanton background

Figure 2: (Color online) The Dirac cone for a chemical potential $\mu\neq 0$.
A quark (filled circle) can be paired with hole state in the Dirac sea (empty
circle) or the Fermi sea (empty star). Each hole state plays the role of an
antiparticle state (filled star). An analogous, detailed discussion can be
found in Ref. Kojo:2009ha . Note that the pairing occurs in a shell of
thickness $\bar{\mu}$ around the Fermi surface, i.e., for chemical potentials
$\in[\mu_{q},\mu]$, where $\mu=\mu_{q}+\bar{\mu}$.

####  II.2.1 GN-like model and scalar condensation

At intermediate densities, where
$\mu\simeq\mathcal{O}(\Lambda_{\textrm{QCD}})$, it is not adequate to describe
the dynamics in terms of perturbative HDET. However, the effective form of the
marginal four-quark interaction can be expressed in terms of the relevant slow
modes of HDET. In the following, we consider the scalar-isoscalar quark
bilinear and its corresponding condensation pattern. With the help of Eq. (3),
the scalar-isoscalar quark condensate can be expressed as follows:

$\displaystyle\langle\bar{\psi}(x)\psi(x)\rangle=\sum_{\vec{v},\vec{v}^{\prime}}e^{i\mu(\vec{v}-\vec{v}^{\,\prime})\cdot\vec{x}}\langle\bar{\psi}(\vec{v}^{\,\prime},x)\psi(\vec{v},x)\rangle\;.$
(11)

The interaction between quarks becomes marginal only if the collinear
scattering occurs between the slow and fast modes (e.g., quark and anti-quark
states) at the same side of the Fermi surface,
$\vec{v}\simeq\vec{v}^{\,\prime}$, or between the slow modes (e.g., quark and
hole states) at opposite sides of the Fermi surface,
$\vec{v}\simeq-\vec{v}^{\,\prime}$ Deryagin:1992rw . If
$\vec{v}\simeq\vec{v}^{\,\prime}$, the respective contribution to Eq. (11) can
be written as)

$\displaystyle\sum_{\vec{v}}\left[\langle\bar{\psi}_{+}(\vec{v},x)\psi_{-}(\vec{v},x)\rangle+\langle\bar{\psi}_{-}(\vec{v},x)\psi_{+}(\vec{v},x)\rangle\right]\;,$
(12)

because
$\langle\bar{\psi}_{\pm}(\vec{v},x)\psi_{\pm}(\vec{v},x)\rangle=\langle\bar{\psi}_{\pm}(\vec{v},x)P_{\mp}P_{\pm}\psi_{\pm}(\vec{v},x)\rangle=0$.
The remaining condensates in Eq. (12) are quark-antiquark condensates, where
it costs at least an energy $\epsilon\simeq 2\mu$ to excite the fast mode.
Therefore, the contribution from Eq. (12) will be neglected in the following.
On the other hand, for the case of $\vec{v}\simeq-\vec{v}^{\,\prime}$, the
quark condensate becomes inhomogeneous and the respective contribution to Eq.
(11) reads

$\displaystyle\sum_{\vec{v}}e^{2i\mu\vec{v}\cdot\vec{x}}\left[\langle\bar{\psi}_{+}(-\vec{v},x)\psi_{+}(\vec{v},x)\rangle+\langle\bar{\psi}_{-}(-\vec{v},x)\psi_{-}(\vec{v},x)\rangle\right]\;,$
(13)

since according to Eq. (6),
$\psi_{\pm}(-\vec{v},x)\equiv\psi_{\mp}(\vec{v},x)$. The remaining condensates
cost minimal energy ($\epsilon\simeq 0$) for the excitation of the slow modes,
and therefore $\langle\bar{\psi}(x)\psi(x)\rangle$ will be approximated by the
term in Eq. (13) in the following.

According to Eq. (9), the fast modes are suppressed by a factor
$\sim\mathcal{O}(1/\mu)$ compared to the slow modes. Therefore, the fast-mode
condensate in Eq. (13) is suppressed by a factor $\sim\mathcal{O}(1/\mu^{2})$
compared to the slow-mode condensate, and can be neglected. The inhomogeneous
quark condensate can be expressed in terms of quark and hole modes with
opposite Fermi velocities:

$\displaystyle\langle\bar{\psi}(x)\psi(x)\rangle=\sum_{\vec{v}}e^{2i\mu\vec{v}\cdot\vec{x}}\langle\bar{\psi}_{+}(-\vec{v},x)\psi_{+}(\vec{v},x)\rangle\quad\Longrightarrow\quad
4\pi\cos(2\mu|\vec{x}|)\sigma\mathit{\Delta}_{\bot}\;,$ (14)

where we assumed that the system is isotropic, such that
$\langle\bar{\psi}_{+}(-\vec{v},x)\psi_{+}(\vec{v},x)\rangle=\langle\bar{\psi}_{+}(\vec{v},x)\psi_{+}(-\vec{v},x)\rangle=\sigma\mathit{\Delta}_{\bot}$.
Here, $\mathit{\Delta}_{\bot}$ denotes the mean contribution from a small
transverse patch on the Fermi surface (with dimension $\textrm{MeV}^{2}$),
while $\sigma$ is the condensation amplitude of the longitudinal modes (with
dimension MeV). Adding a condensate in the pseudoscalar channel, this form can
be cast into a chiral density-wave form.

In the Wilsonian scaling of the fermion action to the Fermi surface, the
collinear four-quark interactions with minimal momentum transfer
($\vec{p}_{1}\simeq\pm\vec{p}_{2}$, $|\vec{p}_{i}|=|\vec{p}_{F}|$) becomes
marginal (see Section 1.3 of Ref. Nardulli:2002ma  for details). Thus, if one
considers the effective form of the scalar interaction given in Eq. (1), for
the Overhauser-type pairing the Fermi velocities assigned to the quark fields
should fulfill
$\vec{v}_{1}\simeq-\vec{v}_{3}\simeq-\vec{v}_{2}\simeq\vec{v}_{4}$. Therefore,
one can make the following Ansatz for the scalar four-quark interaction
channel in a GN-like model, where the quarks couple to the scalar-meson state:

$\displaystyle\mathcal{L}_{4qs}$
$\displaystyle=\left[\bar{\psi}(x)\psi(x)\right]^{2}\Longrightarrow\sum_{\vec{v}}\frac{1}{\mu^{2}}\frac{g_{s}}{n_{D}}\left[\bar{\psi}_{+}(-\vec{v},x)\psi_{+}(\vec{v},x)\right]\left[\bar{\psi}_{+}(\vec{v},x)\psi_{+}(-\vec{v},x)\right]\;,$
(15)

where $n_{D}\equiv N_{f}N_{c}$ denotes the degeneracy factor and the phase
factors $e^{\pm 2i\mu\vec{v}\cdot\vec{x}}$ cancel between the two terms in Eq.
(15). The four-fermion interaction strength in the scalar channel is denoted
as $g_{s}$. One can find the explicit quark-scalar meson coupling channel
through bosonizing the four-quark interaction via a Hubbard-Stratonovich
transformation Stratonovich (1957); Hubbard:1959ub . To make the connection of
this channel with QCD, one should consider the surface zero-mode correlation
in an instanton-background field, which is subject of the next subsection.

####  II.2.2 HDET and the surface zero-modes

Instantons are topologically non-trivial solutions of the classical Yang-Mills
equations of motion in Euclidean space tHooft:1976rip ; Belavin:1975fg . These
special configurations minimize the gauge action and correlate topologically
different vacua, where the number of left- and right-handed quark zero modes
($I=n_{R}-n_{L}$) differs by an integer (details are given in Appendix B).

The QCD Lagrangian (2) in the presence of a quark-number chemical potential
$\mu$ can be written in Euclidean space as

$\displaystyle\mathcal{L}^{\textrm{E}}_{\textrm{QCD}}$
$\displaystyle=\frac{1}{4}\mathcal{F}^{a}_{\mu\nu}\mathcal{F}^{a}_{\mu\nu}-\bar{\psi}(i\,/D_{\textrm{E}}+\mu\gamma^{0})\psi\;,$
(16)

where the covariant derivative in Euclidean space is
$D_{\textrm{E}\mu}=\partial_{\mu}-igA_{\mu}$. The instanton field-strength
tensor will be denoted as $\mathcal{G}_{\mu\nu}$. From the matter part of the
Lagrangian (16), one can define a $\mu$-dependent Dirac operator as
$i\,/D_{\textrm{E}}(\mu)\equiv i/D_{\textrm{E}}+\mu\gamma^{0}$, which has
following properties:

$\displaystyle
i\,/D_{\textrm{E}}(\mu)\psi_{\lambda}=\lambda(\mu)\psi_{\lambda}\;,\quad\left[i\,/D_{\textrm{E}}(\mu)\right]^{\dagger}=-i\,/D_{\textrm{E}}(-\mu)\;,\quad\lambda^{*}(\mu)=-\lambda(-\mu)\;,$
(17)

where the gluon field in the covariant derivative corresponds to the instanton
background field. Since $\\{i\,/D_{\textrm{E}}(\mu),\gamma_{5}\\}=0$, the
eigenmodes $\psi_{\lambda}$ always exist in pairs with $\psi_{-\lambda}$
($\lambda\neq 0$). In the vacuum limit ($\mu\rightarrow 0$),
$i\,/D_{\textrm{E}}(0)$ becomes anti-Hermitian. Due to the definite anti-
hermiticity of $i\,/D_{\textrm{E}}(0)$, the eigenmodes satisfy an
orthonormality relation, and one can easily show from the eigenmode
decomposition of the quark propagator in an instanton background that only
zero modes contribute to the scalar quark condensate (12) (see Appendix C for
details). At finite density ($\mu\simeq O(\Lambda_{\textrm{QCD}})$), the
definite anti-hermiticity of the Dirac operator is lost and one should find a
new set of basis states for the orthonormality relation Bruckmann:2013rpa :

$\displaystyle
i\,/D_{\textrm{E}}(\mu)\psi_{m}(\mu)=\lambda_{m}(\mu)\psi_{m}(\mu)\;,\quad\psi^{\dagger}_{n}(-\mu)i\,/D_{\textrm{E}}(\mu)=\lambda_{n}(\mu)\psi^{\dagger}_{n}(-\mu)\;,\quad\textrm{STr}\left[\psi^{\dagger}_{n}(-\mu)\psi_{m}(\mu)\right]=\delta_{nm}\;,$
(18)

where ‘STr’ denotes the supertrace including a summation over color-spin
indices and a space-time integration. From the bi-orthonormal relationship
(18) between $\psi^{\dagger}_{n}(-\mu)$ and $\psi_{m}(\mu)$, one expects that
the quark zero modes for the scalar condensation should be found in pairs for
$\pm\mu$ and the proper index theorem can be written as follows
Bruckmann:2013rpa ; Kanazawa:2011tt :

$\displaystyle I=n_{R}(\mu)-n_{L}(-\mu)=n_{R}(-\mu)-n_{L}(\mu)\;.$ (19)

As $\psi_{\lambda}(-\mu)$ can be understood as an antiparticle mode of the
system with chemical potential $\mu$, one may recall the decomposition (3) and
investigate the quark field again.

The standard decomposition of the quark field in terms of annihilation
operators $b_{s}(\vec{p})$ and antiquark creation operators
$d^{\dagger}_{s}(\vec{p})$ for (anti-)quarks with spin $s$ and momentum
$\vec{p}$ reads

$\displaystyle\psi(x)$
$\displaystyle=\sum_{s}\int\frac{d^{4}p}{(2\pi)^{4}}\left[b_{s}(\vec{p})u_{s}(\vec{p})e^{-ip\cdot
x}+d^{\dagger}_{s}(\vec{p})v_{s}(\vec{p})e^{ip\cdot x}\right]\;.$ (20)

This decomposition can be mapped to the decomposition (3), noting that, at
finite chemical potential, the energy $p^{0}$ of a particle mode is measured
relative to $\mu$, such that $p^{0}\equiv l^{0}$. Furthermore, employing
$\vec{p}=\mu\vec{v}+\vec{l}$, we immediately deduce that $\int
d^{4}p\equiv\sum_{\vec{v}}\int d^{4}l$. A direct comparison of Eqs. (3) and
(20) then yields

$\displaystyle\psi_{+}(\vec{v},x)$
$\displaystyle=\sum_{s}\int\frac{d^{4}l}{(2\pi)^{4}}b_{s}(\vec{v},l)u_{s}(\vec{v},l)e^{-il\cdot
x}\;,$ (21a) $\displaystyle\psi_{-}(\vec{v},x)$
$\displaystyle=\sum_{s}\int\frac{d^{4}l}{(2\pi)^{4}}d^{\dagger}_{s}(\vec{v},-l)v_{s}(\vec{v},-l)e^{-2i\mu\vec{v}\cdot\vec{x}}e^{-il\cdot
x}\;.$ (21b)

The origin of the $2\mu$-scale of the fast modes is obvious from Eq. (21b).

The matter part of the decomposition (8) can be rewritten in the following
matrix form:

$\displaystyle\mathcal{L}^{\textrm{E}}_{\bar{\psi}D\psi}$
$\displaystyle=-\sum_{\vec{v}}\left(\bar{\psi}_{-}(\vec{v},x)\,,\;\bar{\psi}_{+}(\vec{v},x)\right)\left({\begin{array}[]{cc}i\,/D_{\textrm{E}\bot}&i\,/D_{\textrm{E}\parallel}+2\mu\gamma^{0}\\\
i\,/D_{\textrm{E}\parallel}&i\,/D_{\textrm{E}\bot}\\\
\end{array}}\right)\left({\begin{array}[]{c}\psi_{+}(\vec{v},x)\\\
\psi_{-}(\vec{v},x)\end{array}}\right)\;,$ (26)

where
$i\,/D_{\textrm{E}\parallel}\psi_{+}(\vec{v},x)=\gamma^{0}(iV_{\mu}D_{\textrm{E}\mu})\psi_{+}(\vec{v},x)$
and
$i\,/D_{\textrm{E}\parallel}\psi_{-}(\vec{v},x)=\gamma^{0}(i\bar{V}_{\mu}D_{\textrm{E}\mu})\psi_{-}(\vec{v},x)$.
For the sake of convenience, we abbreviate the $8\times 8$ matrix in Eq. (26)
by $i\,/D_{\textrm{E}}(\mu)$ in the following algebraic calculations. Since
the zero modes satisfy $i\,/D_{\textrm{E}}(\mu)\psi_{0}(\vec{v})=0$, the
relation $[i\,/D_{\textrm{E}}(\mu)]^{2}\psi_{0}(\vec{v})=0$ reads:

$\displaystyle\left({\begin{array}[]{cc}(i\,/D_{\textrm{E}\parallel}+2\mu\gamma^{0})i\,/D_{\textrm{E}\parallel}+(i\,/D_{\textrm{E}\bot})^{2}&(i\,/D_{\textrm{E}\parallel}+2\mu\gamma^{0})i\,/D_{\textrm{E}\bot}+i\,/D_{\textrm{E}\bot}(i\,/D_{\textrm{E}\parallel}+2\mu\gamma^{0})\\\
i\,/D_{\textrm{E}\bot}\;i\,/D_{\textrm{E}\parallel}+i\,/D_{\textrm{E}\parallel}\;i\,/D_{\textrm{E}\bot}&i\,/D_{\textrm{E}\parallel}(i\,/D_{\textrm{E}\parallel}+2\mu\gamma^{0})+(i\,/D_{\textrm{E}\bot})^{2}\\\
\end{array}}\right)\left({\begin{array}[]{c}\psi_{+0}(\vec{v},x)\\\
\psi_{-0}(\vec{v},x)\end{array}}\right)$ (31)
$\displaystyle\simeq\left({\begin{array}[]{cc}(i\,/D_{\textrm{E}\parallel}+2\mu\gamma^{0})i\,/D_{\textrm{E}\parallel}+(i\,/D_{\textrm{E}\bot})^{2}&0\\\
0&i\,/D_{\textrm{E}\parallel}(i\,/D_{\textrm{E}\parallel}+2\mu\gamma^{0})+(i\,/D_{\textrm{E}\bot})^{2}\\\
\end{array}}\right)\left({\begin{array}[]{c}\psi_{+0}(\vec{v},x)\\\
\psi_{-0}(\vec{v},x)\end{array}}\right)=0\;,$ (36)

where we have used
$\psi_{0}(\vec{v})=\left({\begin{array}[]{c}\psi_{+0}(\vec{v},x)\\\
\psi_{-0}(\vec{v},x)\end{array}}\right)$. The second line of Eq. (36) follows
from the fact that the off-diagonal elements are suppressed by
$\mathcal{O}(1/\mu)$, since

$\displaystyle(i\,/D_{\textrm{E}\bot}\;i\,/D_{\textrm{E}\parallel}+i\,/D_{\textrm{E}\parallel}\;i\,/D_{\textrm{E}\bot})\psi_{+0}(\vec{v},x)=ig\gamma_{\bot\beta}\gamma_{\parallel\alpha}\mathcal{G}_{\alpha\beta}\psi_{+0}(\vec{v},x)=igV_{\bot\beta}V_{\alpha}\mathcal{G}_{\alpha\beta}\psi_{+0}(\vec{v},x)\;,$
$\displaystyle(i\,/D_{\textrm{E}\bot}\;i\,/D_{\textrm{E}\parallel}+i\,/D_{\textrm{E}\parallel}\;i\,/D_{\textrm{E}\bot})\psi_{-0}(\vec{v},x)=ig\gamma_{\bot\beta}\gamma_{\parallel\alpha}\mathcal{G}_{\alpha\beta}\psi_{-0}(\vec{v},x)=ig\bar{V}_{\bot\beta}\bar{V}_{\alpha}\mathcal{G}_{\alpha\beta}\psi_{-0}(\vec{v},x)\;,$
(37a)

where $V_{\bot\beta}=(0,\vec{l}_{\bot}/|\mu\vec{v}+\vec{l}|)$ and
$\bar{V}_{\bot\beta}=(0,-\vec{l}_{\bot}/|\mu\vec{v}+\vec{l}|)$. Each element
of the column vector resulting from the last line of Eq. (36) can be simply
expressed as

$\displaystyle\left[2\mu(iV_{\mu}D_{\textrm{E}\mu})+(i\,/D_{\textrm{E}\parallel})^{2}+(i\,/D_{\textrm{E}\bot})^{2}\right]\psi_{\pm
0}(\vec{v},x)\simeq\left[2\mu(iV_{\mu}D_{\textrm{E}\mu})+D_{\textrm{E}}^{2}+\frac{g}{2}\frac{1\pm^{\prime}\gamma_{5}}{2}\sigma_{\mu\nu}\mathcal{G}_{\mu\nu}\right]\psi_{\pm
0}(\vec{v},x)=0\;,$ (38)

where
$\sigma_{\alpha\beta}\gamma_{5}=\frac{1}{2}\epsilon_{\alpha\beta\mu\nu}\sigma_{\mu\nu}$
and the following (anti-)self-duality relation of the (anti-)instanton field-
strength tensor has been used:

$\displaystyle\mathcal{G}_{\mu\nu}=\pm^{\prime}\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}\mathcal{G}_{\rho\sigma}\;.$
(39)

Here, $\pm^{\prime}$ denotes the instanton and anti-instanton field strength,
respectively (while the $\pm$ at $\psi_{\pm 0}(\vec{v},x)$ denotes the
slow/fast mode). Since we are investigating phenomena near the Fermi surface,
$|D_{\textrm{E}\mu}|\lesssim\Lambda_{\textrm{QCD}}$ by assuming a weak
coupling to the instanton background. In the vacuum limit $\mu\rightarrow 0$,
$2\mu(iV_{\mu}D_{\textrm{E}\mu})+D_{\textrm{E}}^{2}\rightarrow
D_{\textrm{E}}^{2}$ becomes negative definite. Since for left-handed modes the
right-handed projector $(1+\gamma_{5})/2$ eliminates the last term in Eq.
(38), while the left-handed projector $(1-\gamma_{5})/2$ does the same for
right-handed modes, the amplitude of the left-handed (right-handed) zero-mode
$\psi_{L(R)0}(\vec{v},x)$ is required to vanish. However, the right-handed
(left-handed) zero mode $\psi_{R(L)0}(\vec{v},x)$ does not need to vanish,
meaning that there is scalar quark condensation which correlates topologically
different vacua ($I=\pm 1$). As long as we are in the low-$\mu$ region, this
conclusion will remain valid, i.e., scalar quark condensation of the quark and
antiquark modes via instanton interactions persist.

Figure 3: A simple diagram depicting the transverse perturbation in a small
patch on the Fermi surface. The correlation of the instanton field strength on
both sides of the Fermi sphere can correlate the quark and hole state around
the patch in both directions of a given Fermi velocity.

On the other hand, if one considers a state $\psi_{\lambda}(\mu)$ near the
Fermi surface with a given Fermi velocity $\vec{v}$, another candidate for
$\psi_{\lambda}(-\mu)$ is a quark state on the other side of the Fermi
surface, in the direction $-\vec{v}$, since it also costs an energy $2\mu$ to
flip a surface quark state to the opposite side of Fermi sphere. The
antiparticle creation operator $d^{\dagger}_{s}(\vec{v},-l)$ in the fast mode
(21) is indeed the annihilation operator acting on an occupied state
($-s,-\mu\vec{v}+\vec{l}$) in the Dirac sea. If one separates the contribution
$\sim d^{\dagger}_{s}(\vec{v},-l)$ by a proper effective approach, then the
separated slow mode describes the $d$-type particle state obtained by
annihilation of the ($-s,-\mu\vec{v}+\vec{l}$) state below the reference
ground level where the effective dynamics is defined. From a technical point
of view, this operator can describe the hole state obtained by annihilation of
the ($-s,-\mu\vec{v}+\vec{l}$) state in the Fermi sphere if the reference
ground level is set as the Fermi surface in the direction $-\vec{v}$. By
denoting $d^{\dagger}_{s}(\vec{v},-l)$ as $b_{-s}(-\vec{v},l)$, which
annihilates an occupied state ($-s,-\mu\vec{v}+\vec{l}$) in the Fermi sea, the
$d$-type slow modes can be utilized to describe dynamics of the $b$-type slow
modes at the opposite side of the Fermi sphere. One can decompose the quark
field in a symmetric way to obtain the second version of the effective
description for the relevant slow modes moving in both directions of a given
Fermi velocity. If one writes the quark field as

$\displaystyle\psi(x)$
$\displaystyle=\sum_{s}\int\frac{d^{4}p}{(2\pi)^{4}}\left[b_{s}(\vec{p})u_{s}(\vec{p})e^{-ip\cdot
x}+d^{\dagger}_{s}(\vec{p})v_{s}(\vec{p})e^{ip\cdot
x}\right]\quad\Longrightarrow\quad\sum_{\vec{v}}\left[e^{i\mu\vec{v}\cdot\vec{x}}\psi_{p}(\vec{v},x)+e^{-i\mu\vec{v}\cdot\vec{x}}\psi_{n}(\vec{v},x)\right]\;,$
(40)

the positive- and negative-energy modes with residual momentum
$|l_{\mu}|\lesssim\mu$ can be identified as follows:

$\displaystyle\psi_{p}(\vec{v},x)$
$\displaystyle=\sum_{s}\int\frac{d^{4}l}{(2\pi)^{4}}b_{s}(\vec{v},l)u_{s}(\vec{v},l)e^{-il\cdot
x}\;,$ (41) $\displaystyle\psi_{n}(\vec{v},x)$
$\displaystyle=\sum_{s}\int\frac{d^{4}l}{(2\pi)^{4}}d^{\dagger}_{s}(\vec{v},-l)v_{s}(\vec{v},-l)e^{-il\cdot
x}\;.$ (42)

Then, one can express the matter part of the Lagrangian density (16) in
Euclidean space as follows:

$\displaystyle\mathcal{L}^{\textrm{E}}_{\bar{\psi}D\psi}$
$\displaystyle=-\sum_{\vec{v}}\big{[}\bar{\psi}_{p}(\vec{v},x)\gamma^{0}(iV_{\mu}D_{\textrm{E}\mu})\psi_{p}(\vec{v},x)+\bar{\psi}_{n}(\vec{v},x)\gamma^{0}(i\bar{V}_{\mu}D_{\textrm{E}\mu})\psi_{n}(\vec{v},x)$
$\displaystyle\qquad\quad+e^{-2i\mu\vec{v}\cdot\vec{x}}\bar{\psi}_{p}(\vec{v},x)\;i\,/D_{\textrm{E}\bot}\psi_{n}(\vec{v},x)+e^{2i\mu\vec{v}\cdot\vec{x}}\bar{\psi}_{n}(\vec{v},x)\;i\,/D_{\textrm{E}\bot}\psi_{p}(\vec{v},x)\big{]}\;,$
(43)

where the algebraic relations (7) have been used. This matter part can be
reduced further to the effective form:

$\displaystyle\mathcal{L}^{\textrm{E}}_{\bar{\psi}D\psi}$
$\displaystyle=-\sum_{\vec{v}}\Bigg{[}\bar{\psi}_{p}(\vec{v},x)\gamma^{0}(iV_{\mu}D_{\textrm{E}\mu})\psi_{p}(\vec{v},x)-\bar{\psi}_{p}(\vec{v},x)\frac{\gamma^{0}}{2\mu+i\bar{V}_{\mu}D_{\textrm{E}\mu}}(/D_{\textrm{E}\bot})^{2}\psi_{p}(\vec{v},x)$
$\displaystyle\qquad\qquad\quad+\bar{\psi}_{n}(\vec{v},x)\gamma^{0}(i\bar{V}_{\mu}D_{\textrm{E}\mu})\psi_{n}(\vec{v},x)-\bar{\psi}_{n}(\vec{v},x)\frac{\gamma^{0}}{2\mu+iV_{\mu}D_{\textrm{E}\mu}}(/D_{\textrm{E}\bot})^{2}\psi_{n}(\vec{v},x)\Bigg{]}\;,$
(44)

where the following equations of motion
$\delta\mathcal{S}_{\textrm{QCD}}/\delta\bar{\psi}_{p/n}(\vec{v},x)=0$ were
used:

$\displaystyle i\,/D_{\textrm{E}\parallel}\psi_{n}(\vec{v},x)$
$\displaystyle=-e^{2i\mu\vec{v}\cdot\vec{x}}i\,/D_{\textrm{E}\bot}\psi_{p}(\vec{v},x)\;,$
$\displaystyle i\,/D_{\textrm{E}\parallel}\psi_{p}(\vec{v},x)$
$\displaystyle=-e^{-2i\mu\vec{v}\cdot\vec{x}}i\,/D_{\textrm{E}\bot}\psi_{n}(\vec{v},x)\;.$
(45)

From the effective Lagrangian (44), one can see that the negative-energy slow
mode $\psi_{n}(\vec{v},x)$ follows the same longitudinal dynamics as the
positive-energy slow mode moving in the opposite direction
$\psi_{p}(-\vec{v},x)$, since $\bar{V}_{\mu}=(i,-\vec{v})$. Considering that
the $d$-type particle state with quantum numbers ($s,\vec{v},-l$) is obtained
by annihilating the occupied ($-s,-\vec{v},l$) state below the ground level,
one may match $\psi_{n}(\vec{v},x)$ as follows:

$\displaystyle\psi_{n}(\vec{v},x)$
$\displaystyle=\sum_{s}\int\frac{d^{4}l}{(2\pi)^{4}}d^{\dagger}_{s}(\vec{v},-l)v_{s}(\vec{v},-l)e^{-il\cdot
x}\quad\Longrightarrow\quad\psi_{p}(-\vec{v},x)=\sum_{s}\int\frac{d^{4}l}{(2\pi)^{4}}b_{-s}(-\vec{v},l)u_{-s}(-\vec{v},l)e^{-il\cdot
x}\;.$ (46)

Thus, the conjugated field $\bar{\psi}_{n}(\vec{v},x)$ describes the same
dynamics as a hole state in the Fermi sea in the $-\vec{v}$ direction
($\bar{\psi}_{p}(-\vec{v},x)$) and the condensate
$\left\langle\bar{\psi}_{+}(-\vec{v},x)\psi_{+}(\vec{v},x)\right\rangle$ can
be investigated in the $\\{\psi_{p}(\vec{v},x),\psi_{n}(\vec{v},x)\\}$ basis.
By denoting $\psi_{n}(\vec{v},x)$ as $\psi_{+}(-\vec{v},x)$, the matrix form
of the matter part (43) can be expressed accordingly as

$\displaystyle\mathcal{L}^{\textrm{E}}_{\bar{\psi}D\psi}$
$\displaystyle=-\sum_{\vec{v}}\left(\bar{\psi}_{+}(-\vec{v},x)\,,\;\bar{\psi}_{+}(\vec{v},x)\right)\left({\begin{array}[]{cc}e^{2i\mu\vec{v}\cdot\vec{x}}i\,/D_{\textrm{E}\bot}&i\,/D_{\textrm{E}\parallel}\\\
i\,/D_{\textrm{E}\parallel}&e^{-2i\mu\vec{v}\cdot\vec{x}}i\,/D_{\textrm{E}\bot}\\\
\end{array}}\right)\left({\begin{array}[]{c}\psi_{+}(\vec{v},x)\\\
\psi_{+}(-\vec{v},x)\end{array}}\right)\;,$ (51)

and $[i\,/D_{\textrm{E}}(\mu)]^{2}\psi_{0}=0$ can be reduced as follows:

$\displaystyle\left({\begin{array}[]{cc}(i\,/D_{\textrm{E}\parallel})^{2}+e^{2i\mu\vec{v}\cdot\vec{x}}i\,/D_{\textrm{E}\bot}e^{2i\mu\vec{v}\cdot\vec{x}}i\,/D_{\textrm{E}\bot}&e^{2i\mu\vec{v}\cdot\vec{x}}i\,/D_{\textrm{E}\bot}\;i\,/D_{\textrm{E}\parallel}+i\,/D_{\textrm{E}\parallel}e^{-2i\mu\vec{v}\cdot\vec{x}}i\,/D_{\textrm{E}\bot}\\\\[2.84544pt]
e^{-2i\mu\vec{v}\cdot\vec{x}}i\,/D_{\textrm{E}\bot}\;i\,/D_{\textrm{E}\parallel}+i\,/D_{\textrm{E}\parallel}e^{2i\mu\vec{v}\cdot\vec{x}}i\,/D_{\textrm{E}\bot}&(i\,/D_{\textrm{E}\parallel})^{2}+e^{-2i\mu\vec{v}\cdot\vec{x}}i\,/D_{\textrm{E}\bot}e^{-2i\mu\vec{v}\cdot\vec{x}}i\,/D_{\textrm{E}\bot}\\\
\end{array}}\right)\left({\begin{array}[]{c}\psi_{+0}(\vec{v},x)\\\
\psi_{+0}(-\vec{v},x)\end{array}}\right)$ (56)
$\displaystyle\simeq\left({\begin{array}[]{cc}(i\,/D_{\textrm{E}\parallel})^{2}-i\,/D_{\textrm{E}\parallel}\;i\,/D_{\textrm{E}\bot}\frac{\gamma^{0}}{i\,/D_{\textrm{E}\parallel}+2\mu\gamma^{0}}i\,/D_{\textrm{E}\bot}&0\\\
0&(i\,/D_{\textrm{E}\parallel})^{2}-i\,/D_{\textrm{E}\parallel}\;i\,/D_{\textrm{E}\bot}\frac{\gamma^{0}}{i\,/D_{\textrm{E}\parallel}+2\mu\gamma^{0}}i\,/D_{\textrm{E}\bot}\\\
\end{array}}\right)\left({\begin{array}[]{c}\psi_{+0}(\vec{v},x)\\\
\psi_{+0}(-\vec{v},x)\end{array}}\right)=0\;,$ (61)

where the off-diagonal elements are suppressed as $\mathcal{O}(1/\mu)$ by the
relation (37) and the constraints (45). The last line of Eq. (61) can be
simplified as

$\displaystyle\left[(i\,/D_{\textrm{E}\parallel})^{2}-i\,/D_{\textrm{E}\parallel}\;i\,/D_{\textrm{E}\bot}\frac{\gamma^{0}}{i\,/D_{\textrm{E}\parallel}+2\mu\gamma^{0}}i\,/D_{\textrm{E}\bot}\right]\psi_{+0}(\pm\vec{v},x)$
$\displaystyle=\left[D_{\textrm{E}\parallel}^{2}+\frac{g}{2}\sigma_{\mu\nu}\mathcal{G}_{\parallel\mu\nu}+\frac{i\,/D_{\textrm{E}\parallel}}{i\,/D_{\textrm{E}\parallel}+2\mu\gamma^{0}}\left(D_{\bot}^{2}+\frac{g}{2}\sigma_{\mu\nu}\mathcal{G}_{\bot\mu\nu}\right)\right]\psi_{+0}(\pm\vec{v},x)=0\;.$
(62)

As seen in the fast-slow mode condensation case, an isotropic scalar quark
condensate can be formed only in the vacuum limit $\mu\rightarrow 0$ (the
negative slow mode becomes the negative-energy mode in vacuum) since the self-
dual instanton field strength (39) guarantees the vanishing amplitude of
$\psi_{L0}$. In the limit that the momentum goes to the Fermi surface with
$\mu\gg\Lambda_{\textrm{QCD}}$, Eq. (62) reduces to
$\left(D_{\textrm{E}\parallel}^{2}+\frac{g}{2}\sigma_{\mu\nu}\mathcal{G}_{\parallel\mu\nu}\right)\psi_{+0}(\pm\vec{v},x)=0$.
Although the isotropic quark condensation is not possible at
$\mu\simeq\mathcal{O}(\Lambda_{\textrm{QCD}})$, the velocity-oriented quark
and hole condensation along a given Fermi-velocity axis is possible. Consider
a small patch on the Fermi surface whose area is
$\sim\mathcal{O}(l_{\bot}^{2})<\Lambda_{\textrm{QCD}}^{2}$ and a small
transverse perturbation in the patch: $\vec{v}^{\prime}=\vec{v}+\delta\vec{v}$
with $|\delta\vec{v}|\equiv l_{\bot}/\mu\ll 1$ and
$\vec{v}\cdot\delta\vec{v}=0$ (Fig. 3). Then, $i\,/D_{\textrm{E}\parallel}$
can be expressed as

$\displaystyle i\,/D_{\textrm{E}\parallel}$ $\displaystyle\simeq
i\left[\gamma_{4}D_{\textrm{E}4}+(\vec{v}\cdot\vec{\gamma})\vec{v}\cdot\vec{D}_{\textrm{E}}+(\delta\vec{v}_{T}\cdot\vec{\gamma})\delta\vec{v}_{T}\cdot\vec{D}_{\textrm{E}}\right]\;,$
(63) $\displaystyle(i\,/D_{\textrm{E}\parallel})^{2}$ $\displaystyle\simeq
D_{\vec{v}}^{2}+D_{\vec{v}_{T}}^{2}+\frac{g}{2}\sigma_{\mu\nu}\left(\mathcal{G}_{\vec{v}\mu\nu}+\mathcal{G}_{\vec{v}_{T}\mu\nu}\right)\;,$
(64)

where
$\delta\vec{v}_{T}=\delta\vec{v}/\sqrt{\delta\vec{v}\cdot\delta\vec{v}}$,
$D_{\vec{v}\mu}=(D_{\textrm{E}4},\vec{v}(\vec{v}\cdot\vec{D}_{\textrm{E}}))$,
and
$D_{\vec{v}_{T}\mu}=(0,\delta\vec{v}_{T}(\delta\vec{v}_{T}\cdot\vec{D}_{\textrm{E}}))$.
The definition for the velocity-oriented field strength is understood from the
definition of the covariant derivative. In the momentum scaling to the Fermi
surface, $|D_{\vec{v}\mu}|\simeq|D_{\vec{v}_{T}\mu}|\ll\Lambda_{\textrm{QCD}}$
and the self-duality relationship of the velocity-oriented instanton field
strength can be satisfied, which leads to the condensation of the surface
zero-mode of each helicity basis.

###  II.3 Scalar and vector channel in the instanton background

Consider the interaction terms of HDET which lead to marginal interactions in
the low-energy effective model. The simplest four-quark interaction can be
written as follows:

$\displaystyle\mathcal{L}^{\textrm{E}}_{(\bar{\psi}\Gamma\psi)^{2}}$
$\displaystyle\simeq-\frac{1}{4\mu^{2}}\left[\bar{\psi}_{+}(\vec{v})\gamma^{0}(i\,/D_{\vec{v}_{T}})^{2}\psi_{+}(\vec{v})\right]\left[\bar{\psi}_{+}(-\vec{v})\gamma^{0}(i\,/D_{-\vec{v}_{T}})^{2}\psi_{+}(-\vec{v})\right]$
$\displaystyle=\frac{1}{4\mu^{2}}\frac{1}{16}\left[\bar{\psi}^{c}_{+}(-\vec{v})\Gamma^{A}\psi^{a}_{+}(\vec{v})\right]\left[\bar{\psi}^{b}_{+}(\vec{v})\Gamma^{B}\psi^{d}_{+}(-\vec{v})\right]\left[\Gamma^{A}\gamma^{0}(i\,/D_{\vec{v}_{T}})^{2}_{ba}\Gamma^{B}\gamma^{0}(i\,/D_{-\vec{v}_{T}})^{2}_{cd}\right]_{\alpha\alpha}\;,$
(65)

where the rectangular brackets denote the trace over the spinor ($\alpha$) and
color ($a$) indices. By choosing $\Gamma^{A}=\Gamma^{B}=1$ and color-singlet
combinations, the scalar four-quark interaction can be expressed as

$\displaystyle\mathcal{L}^{\textrm{E}}_{(\bar{\psi}\psi)^{2}}$
$\displaystyle\Rightarrow-\frac{1}{4\mu^{2}}\frac{1}{16}\left(\frac{1}{3}\right)^{2}\left(\frac{g}{4}\right)^{2}[\gamma^{0}\alpha^{k}\gamma^{0}\alpha^{n}]_{\alpha\alpha}[\tau^{C}\tau^{D}]_{aa}\epsilon^{ijk}\epsilon^{lmn}$
$\displaystyle\qquad\quad\times\left[\bar{\psi}_{+}(-\vec{v})\psi_{+}(\vec{v})\right]\left[\bar{\psi}_{+}(\vec{v})\psi_{+}(-\vec{v})\right]\left\langle\mathcal{G}^{C}_{\vec{v}_{T}ij}\mathcal{G}^{D}_{-\vec{v}_{T}lm}\right\rangle$
$\displaystyle=-\frac{\pi}{3}\frac{1}{256}\frac{\alpha_{s}}{\mu^{2}}\left\langle{\mathcal{G}_{\vec{v}_{T}}}^{2}\right\rangle\left[\bar{\psi}_{+}(-\vec{v})\psi_{+}(\vec{v})\right]\left[\bar{\psi}_{+}(\vec{v})\psi_{+}(-\vec{v})\right]\;,$
(66)

where the following relations for the expectation value of the instanton field
strength on both sides of the Fermi sphere are used:

$\displaystyle(i\,/D_{\vec{v}_{T}})^{2}$
$\displaystyle=D_{\vec{v}_{T}}^{2}-\frac{g}{2}\sigma_{\mu\nu}\mathcal{G}_{\vec{v}_{T}\mu\nu}=D_{\vec{v}_{T}}^{2}+\frac{g}{4}\epsilon^{ijk}\alpha^{k}\mathcal{G}_{\vec{v}_{T}ij}\;,$
(67)
$\displaystyle\left\langle\mathcal{G}^{C}_{\vec{v}_{T}ij}\mathcal{G}^{D}_{-\vec{v}_{T}lm}\right\rangle$
$\displaystyle=-\frac{\delta^{CD}}{6}\left(\delta_{il}\delta_{jm}-v_{i}v_{l}\delta_{jm}-v_{j}v_{m}\delta_{il}-\delta_{im}\delta_{jl}+v_{i}v_{m}\delta_{jl}+v_{j}v_{l}\delta_{im}\right)\left\langle{\mathcal{G}_{\vec{v}_{T}}}^{2}\right\rangle\;,$
(68)

where $C,D=\\{1,2,3\\}$ denotes the adoint color index for the instanton field
strength. In the perturbative regime ($\mu\gg\Lambda_{\textrm{QCD}}$), the
strong coupling constant suppresses the contribution (66). However, at
$\mu\simeq\mathcal{O}(\Lambda_{\textrm{QCD}})$, multi-instanton configurations
Diakonov:1983hh ; Diakonov:1985eg ; Shuryak:1989cx  and an increasing
$\alpha_{s}$ in the low-momentum scaling to the Fermi surface can enhance the
contribution of this channel. This kind of interaction is of similar form as
the effective interaction (15), which will be adopted as a scalar interaction
in the simple model for the FRG analysis in Sec. III.

If repulsive vector interactions near the Fermi surface become dominant, the
non-trivial scalar channel could be suppressed in the RG flow to the IR limit.
Such a vector interaction can, for instance, be constructed from the vector
current of the surface modes:

$\displaystyle\bar{\psi}(x)\gamma^{\mu}\psi(x)\Longrightarrow$
$\displaystyle\sum_{\vec{v}}\bar{\psi}(\vec{v},x)\gamma^{\mu}\psi(\vec{v},x)\simeq\bar{\psi}_{+}(\vec{v},x)\gamma^{0}\psi_{+}(\vec{v},x)\;,$
(69)

where the relation
$\vec{\alpha}\psi_{\pm}(\vec{v},x)=\pm\vec{v}\psi_{\pm}(\vec{v},x)$ was used.
For the zero-sound type interaction, we have two possible configurations for
the Fermi velocities, (i)
$\vec{v}_{1}+\vec{v}_{2}\simeq\vec{v}_{3}+\vec{v}_{4}\simeq 0$, with
$\vec{v}_{1}\simeq\vec{v}_{3}$ ($\vec{v}_{2}\simeq\vec{v}_{4}$), and (ii)
$\vec{v}_{1}\simeq\vec{v}_{2}\simeq\vec{v}_{3}\simeq\vec{v}_{4}$ (cf. caption
of Fig. 1(a)). For case (i), the interaction Lagrangian can be constructed as

$\displaystyle\mathcal{L}_{(\bar{\psi}\gamma\psi)^{2}}$
$\displaystyle=\left[\bar{\psi}(x)\gamma^{\mu}\psi(x)\right]\left[\bar{\psi}(x)\gamma_{\mu}\psi(x)\right]$
$\displaystyle\Rightarrow\sum_{\vec{v}}\frac{1}{\mu^{2}}\frac{g_{v}}{n_{D}}\left[\bar{\psi}_{+}(\vec{v},x)\gamma^{\mu}\psi_{+}(\vec{v},x)\right]\left[\bar{\psi}_{+}(-\vec{v},x)\gamma_{\mu}\psi_{+}(-\vec{v},x)\right]$
$\displaystyle\simeq\sum_{\vec{v}}\frac{2}{\mu^{2}}\frac{g_{v}}{n_{D}}\left[\bar{\psi}_{+}(\vec{v},x)\gamma^{0}\psi_{+}(\vec{v},x)\right]\left[\bar{\psi}_{+}(-\vec{v},x)\gamma^{0}\psi_{+}(-\vec{v},x)\right]\;.$
(70)

The QCD origin of this channel can be found not only from the instanton
configuration but from the perturbative gluon exchange appearing in the
following interaction:

$\displaystyle\mathcal{L}^{\textrm{E}}_{(\psi^{\dagger}\psi)^{2}}$
$\displaystyle\simeq-\frac{1}{4\mu^{2}}\left[\bar{\psi}_{+}(\vec{v})\gamma^{0}D_{\textrm{E}}^{2}\psi_{+}(\vec{v})\right]\left[\bar{\psi}_{+}(-\vec{v})\gamma^{0}D_{\textrm{E}}^{2}\psi_{+}(-\vec{v})\right]\;.$
(71)

The second case (ii), which corresponds to collinear forward scattering,
actually does not appear at leading order since

$\displaystyle\left[\bar{\psi}_{+}(\vec{v},x)\gamma^{\mu}\psi_{+}(\vec{v},x)\right]\left[\bar{\psi}_{+}(\vec{v},x)\gamma^{\mu}\psi_{+}(\vec{v},x)\right]\simeq
V^{2}\left[\bar{\psi}_{+}(\vec{v},x)\gamma^{0}\psi_{+}(\vec{v},x)\right]^{2}=0\;,$
(72)

but can appear at next-to-leading order. Because this interaction is
constructed from the vector current, it becomes vanishingly small when the
outer shell of the Fermi sphere becomes infinitesimally thin (Fig. 2). One can
guess the transverse vector interaction from Eq. (65) but further analysis of
such a kind of channel will be deferred to future work.

##  III FRG analysis of effective model

In this section, we construct a simple effective model by considering scalar-
and vector-type four-quark contact interactions. The effective Lagrangian is
bosonized via a Hubbard-Stratonovich transformation, and this effective form
is taken as UV initial condition in the FRG analysis. First, we investigate
the quark condensation amplitude in the mean-field approximation, assuming a
homogeneous amplitude $\sigma$ in the FRG flow of the action. The case of an
inhomogeneous amplitude is investigated by analyzing the scalar two-point
function when allowing for spatial fluctuations. Finally, we investigate the
case of a dynamical $\sigma$ by solving the FRG flow equation.

###  III.1 Effective model Lagrangian and Hubbard-Stratonovich transformation

For the general description, one may consider $\mu=\mu_{q}+\bar{\mu}$,
$\mu_{q}\gg\bar{\mu}$ and write the effective Lagrangian in terms of the
surface modes. The parameter $\bar{\mu}$ can be interpreted as the thickness
of the momentum shell where one expects quark condensation, cf. Fig. 2. Then,
the effective Lagrangian of the matter part can be expressed as follows
Schafer:2003jn :

$\displaystyle\mathcal{L}_{q}$
$\displaystyle\simeq\sum_{\vec{v}}\left[\bar{\psi}_{+}(\vec{v},x)\gamma^{0}(iV^{\mu}D_{\mu}+\bar{\mu})\psi_{+}(\vec{v},x)-\bar{\psi}_{+}(\vec{v},x)\frac{\gamma^{0}}{2\mu_{q}+\bar{\mu}+i\bar{V}^{\mu}D_{\mu}}/D_{\bot}^{2}\psi_{+}(\vec{v},x)\right]$
$\displaystyle\qquad+\sum_{\vec{v}_{i}}\sum_{\Gamma\Gamma^{\prime}}\frac{d^{\Gamma\Gamma^{\prime}}(v_{1},v_{2},v_{3},v_{4})}{\mu_{q}^{2}}\left[\bar{\psi}_{+}(\vec{v}_{3},x)\Gamma\psi_{+}(\vec{v}_{1},x)\right]\left[\bar{\psi}_{+}(\vec{v}_{4},x)\Gamma^{\prime}\psi_{+}(\vec{v}_{2},x)\right]+\ldots\;.$
(73)

From this point, we will only keep the effective four-quark interactions
considered in the previous section and omit the subscript ‘$+$’ on the
positive quark modes. To describe the interaction between the quark and holes
on the patches in opposite directions of the Fermi velocity, it is convenient
to double the Fermi velocity sums and divide by 2. Converting Eq. (73) into
Euclidean space we then obtain

$\displaystyle\mathcal{L}_{\textrm{E}}=\frac{1}{2}\sum_{\vec{v}}$
$\displaystyle\bigg{\\{}\bar{\psi}(\vec{v},x)\left(-i{/\partial}_{\textrm{E}}-\bar{\mu}\gamma^{0}\right)\psi(\vec{v},x)+\bar{\psi}(-\vec{v},x)\left(-i{/\partial}_{\textrm{E}}-\bar{\mu}\gamma^{0}\right)\psi(-\vec{v},x)$
$\displaystyle\quad-\frac{1}{\mu_{q}^{2}}\frac{g_{s}}{n_{D}}\left[\bar{\psi}(-\vec{v},x)\psi(\vec{v},x)\right]\left[\bar{\psi}(\vec{v},x)\psi(-\vec{v},x)\right]-\frac{1}{\mu_{q}^{2}}\frac{g_{v}}{n_{D}}\left[\bar{\psi}(\vec{v},x)\gamma_{\mu}\psi(\vec{v},x)\right]\left[\bar{\psi}(-\vec{v},x)\gamma_{\mu}\psi(-\vec{v},x)\right]\bigg{\\}}\;.$
(74)

Employing a Hubbard-Stratonovich transformation Stratonovich (1957);
Hubbard:1959ub  we obtain the bosonized form of this Lagrangian. Explicitly,
we introduce auxiliary fields $\xi^{s},\xi^{v}$ into the partition function of
the model,

$\displaystyle\mathcal{Z}$
$\displaystyle=\textrm{const.}\int\mathcal{D}[\xi^{s},\xi^{v},\bar{\psi},\psi]\,e^{-S_{E}}\,\exp\left(-\frac{\mu^{2}_{q}}{2g_{s}}\sum_{\vec{v}}\xi^{s}_{\vec{v}}\xi^{s}_{-\vec{v}}-\frac{\mu^{2}_{q}}{2g_{v}}\sum_{\vec{v}}\xi^{v}_{\vec{v},\mu}\xi^{v}_{-\vec{v},\mu}\right)\;,$
(75)

where $S_{E}\equiv\int d^{4}x_{\textrm{E}}\,\mathcal{L}_{E}$. Then, we
substitute $\xi^{s}_{\vec{v}}$ and $\xi^{v}_{\vec{v}}$ by the shifted
auxiliary fields $\sigma^{\pm}$, $\omega_{\mu}^{\pm}$ via

$\displaystyle\xi^{s}_{\pm\vec{v}}$
$\displaystyle=h_{s}\sigma^{\pm}(x)+\frac{1}{\mu^{2}_{q}}\frac{g_{s}}{\sqrt{n_{D}}}\bar{\psi}(\mp\vec{v},x)\psi(\pm\vec{v},x)\;,$
(76) $\displaystyle\xi^{v}_{\pm\vec{v},\mu}$
$\displaystyle=h_{v}\omega_{\mu}^{\pm}(x)+\frac{1}{\mu^{2}_{q}}\frac{g_{v}}{\sqrt{n_{D}}}\bar{\psi}(\pm\vec{v},x)\gamma_{\mu}\psi(\pm\vec{v},x)\;,$
(77)

to obtain the bosonized partition function

$\mathcal{Z}=\textrm{const.}\int\mathcal{D}[\sigma,\omega,\bar{\psi},\psi]e^{-S_{b\textrm{E}}}\;,$
(78)

where $S_{b\textrm{E}}$ is the action corresponding to the bosonized
Lagrangian

$\displaystyle\mathcal{L}_{b\textrm{E}}=\frac{1}{2}\sum_{\vec{v}}$
$\displaystyle\left[\bar{\Psi}_{\vec{v}}\left(-i\partial_{\mu}\Gamma_{\mu}-\bar{\mu}\Gamma^{0}+\frac{h_{s}}{\sqrt{n_{D}}}\varphi+\frac{h_{v}}{\sqrt{n_{D}}}\omega_{\mu}\Gamma_{\mu}\right)\Psi_{\vec{v}}+\frac{\mu_{q}^{2}h^{2}_{s}}{g_{s}}\sigma^{-}(x)\sigma^{+}(x)+\frac{\mu_{q}^{2}h^{2}_{v}}{g_{v}}\omega^{-}_{\mu}(x)\omega^{+}_{\mu}(x)\right]\;.$
(79)

Here, we introduced the matrix representations

$\displaystyle\Psi_{\vec{v}}$
$\displaystyle=\left({\begin{array}[]{c}\psi_{+}(\vec{v},x)\\\
\psi_{+}(-\vec{v},x)\end{array}}\right),\quad\Gamma_{\mu}=\left({\begin{array}[]{cc}0&\gamma_{\mu}\\\
\gamma_{\mu}&0\\\
\end{array}}\right),\quad\varphi=\left({\begin{array}[]{cc}\sigma^{-}(x)I&0\\\
0&\sigma^{+}(x)I\\\
\end{array}}\right),\quad\omega_{\mu}=\left({\begin{array}[]{cc}\omega^{+}_{\mu}(x)I&0\\\
0&\omega^{-}_{\mu}(x)I\\\ \end{array}}\right),$ (88)

where $\sigma^{-}(x)=(\sigma^{+}(x))^{\dagger}$ follows from Eq. (76) and
where
$\omega^{\pm}_{\mu}(x)=\left(\omega^{\pm}_{4}(x),\vec{v}[\vec{v}\cdot\vec{\omega}^{\pm}(x)]\right)$.
One should note that the independent spin degrees of freedom have been reduced
by a factor of 2, since half of the helicity space corresponds to the
negative-energy modes, which have been integrated out. The quark mode along
the opposite Fermi velocity, i.e., $\psi({-\vec{v}},x)$, plays the role of the
negative-energy modes of the vacuum theory.

###  III.2 The exact RG flow equation in the mean-field approximation for the
bosonic fields

We first investigate the simplest configuration where the bosonic fields are
approximated by constant mean fields:

$\displaystyle\varphi\Rightarrow\left({\begin{array}[]{cc}\sigma I&0\\\
0&\sigma I\\\
\end{array}}\right)\;,\quad\omega_{\mu}\Rightarrow\left({\begin{array}[]{cc}\omega^{+}_{\mu}I&0\\\
0&\omega^{-}_{\mu}I\\\ \end{array}}\right)\;,$ (93)

where $\sigma$ is a real-valued background field
($\sigma^{-}=(\sigma^{+})^{\dagger}=\sigma^{+}=\sigma$) and
$\omega^{\pm}_{\mu}=\left(\omega_{4},\pm\vec{v}\omega_{\parallel}\right)$ with
$\omega^{\pm}_{4}=\omega_{4}$ and
$\omega_{\parallel}\equiv\vec{v}\cdot\vec{\omega}^{+}=-\vec{v}\cdot\vec{\omega}^{-}$.
Then, to leading order the effective average action corresponding to the
bosonized Lagrangian density (79) reads

$\displaystyle\Gamma^{m}_{k}=\int
d^{4}x_{\textrm{E}}\bigg{[}\frac{1}{2}\sum_{\vec{v}}\bar{\Psi}_{\vec{v}}\left(-i\Gamma_{\mu}\partial_{\mu}-\bar{\mu}\Gamma^{0}+\frac{h_{s}}{\sqrt{n_{D}}}\varphi+\frac{h_{v}}{\sqrt{n_{D}}}\omega_{\mu}\Gamma_{\mu}\right)\Psi_{\vec{v}}+2\pi\mu_{q}^{2}\frac{h^{2}_{v}}{g_{v}}(\omega_{4}^{2}-\omega_{\parallel}^{2})+U_{\sigma}(k,\sigma)\bigg{]}\;,$
(94)

where $U_{\sigma}(k,\sigma)$ is the RG time-dependent potential for the
$\sigma$-field, which has the UV initial condition

$U_{\sigma}(k=\mu_{q},\sigma)=2\pi\mu_{q}^{2}\frac{h_{s}^{2}}{g_{s}}\sigma^{2}\;.$
(95)

The coarse graining of $\Gamma^{m}_{k}$ with respect to the RG time can be
described by including the regulator term:

$\displaystyle\Gamma^{f}_{k}$
$\displaystyle=\Gamma^{m}_{k}+\Delta\Gamma^{f}_{k},$ (96)
$\displaystyle\Delta\Gamma^{f}_{k}$
$\displaystyle=\frac{1}{2}\sum_{\vec{v}}\int\frac{d^{4}p_{\textrm{E}}}{(2\pi)^{4}}\bar{\Psi}_{\vec{v}}(-p_{\parallel})R^{f}_{\vec{v}}(k,p)\Psi_{\vec{v}}(p_{\parallel}).$
(97)

The corresponding exact RG flow equation Wetterich:1991be ; Wetterich:1992yh ;
Ellwanger:1993mw  is

$\displaystyle\partial_{t}\Gamma^{f}_{k}=-\textrm{STr}\left\\{\left[\partial_{t}R^{f}_{\vec{v}}(k,p)\right]\left[\Gamma^{m(2)}_{\bar{\Psi}\Psi}(k,\vec{v})+R^{f}_{\vec{v}}(k,p)\right]^{-1}\right\\}\;,$
(98)

where $\Gamma^{m(2)}_{\bar{\Psi}\Psi}(k,\vec{v})$ denotes the second
functional derivative of $\Gamma^{m}_{k}$ with respect to the fermionic fields
evaluated at $\bar{\Psi}_{\vec{v}}=\Psi_{\vec{v}}=0$. ‘STr’ denotes the
supertrace including the summation over the spin, color, and flavor indices
and the integration over $p_{\parallel}$ and $\vec{v}$. The fermionic
regulator $R^{f}_{\vec{v}}(k,p)$ can be derived from the well-known Litim
regulator Litim:2000ci ; Litim:2001up  as

$\displaystyle
R^{f}_{\vec{v}}(k,p)=\frac{\beta\delta_{n^{\prime}n}\delta^{(2)}(\vec{v}^{\,\prime}-\vec{v})\,2\pi\delta(p_{\parallel}^{\prime}-p_{\parallel})}{2}\left({\begin{array}[]{cc}0&ip_{\parallel}\gamma_{4}\\\
-ip_{\parallel}\gamma_{4}&0\\\ \end{array}}\right)r_{f}(k,p_{\parallel})\;,$
(101)

where $p_{\parallel}=\vec{v}\cdot\vec{p}$, $k=\Lambda e^{-t}$, and
$r_{f}(k,p_{\parallel})$ is given as

$\displaystyle 1+r_{f}(k,p_{\parallel})$
$\displaystyle=\sqrt{1+r_{b}(k,p_{\parallel})}\;,$ (102) $\displaystyle
r_{b}(k,p_{\parallel})$
$\displaystyle\equiv\left(\frac{k^{2}}{p_{\parallel}^{2}}-1\right)\Theta\left(\frac{k^{2}}{p_{\parallel}^{2}}-1\right)\;,$
(103)

and the integration measure contained in the supertrace in Eq. (98) is
understood as

$\displaystyle\int\frac{d^{4}p_{\textrm{E}}}{(2\pi)^{4}}\beta\delta_{n^{\prime}n}(2\pi)^{3}\delta^{(3)}(\vec{p}^{\,\prime}-\vec{p})$
$\displaystyle\equiv\frac{1}{\beta}\sum_{n}\sum_{\vec{v}}\int_{-\mu_{q}}^{\mu_{q}}\frac{dp_{\parallel}}{2\pi}\beta\delta_{n^{\prime}n}\delta^{(2)}(\vec{v}^{\,\prime}-\vec{v})\,2\pi\delta(p_{\parallel}^{\prime}-p_{\parallel})\;,$
(104)

where we used
$\delta^{(2)}(\vec{p}_{\bot})=(1/\mu_{q}^{2})\delta^{(2)}(\vec{v})$ and
$d^{2}p_{\bot}=\mu_{q}^{2}d\vec{v}$. Note the following useful relations:

$\displaystyle
2\partial_{t}r_{f}(k,p_{\parallel})\left[1+r_{f}(k,p_{\parallel})\right]$
$\displaystyle=\partial_{t}r_{b}(k,p_{\parallel})=-\frac{2k^{2}}{p_{\parallel}^{2}}\left[\Theta\left(\frac{k^{2}}{p_{\parallel}^{2}}-1\right)+\left(\frac{k^{2}}{p_{\parallel}^{2}}-1\right)\delta\left(\frac{k^{2}}{p_{\parallel}^{2}}-1\right)\right]\;,$
(105) $\displaystyle\partial_{t}r_{f}(k,p_{\parallel})$
$\displaystyle=\frac{\partial_{t}r_{b}(k,p_{\parallel})}{2\sqrt{1+r_{b}(k,p_{\parallel})}}=\bigg{\\{}{\begin{array}[]{cc}0&|p_{\parallel}|>k\\\
-\frac{k}{p_{\parallel}}&|p_{\parallel}|\leq k\\\ \end{array}}\;.$ (108)

$\Gamma^{m(2)}_{\bar{\Psi}\Psi}(k,\vec{v})+R^{f}_{\vec{v}}(k,p)$ can be
expressed explicitly as follows:

$\displaystyle\Gamma^{m(2)}_{\bar{\Psi}\Psi}(k,\vec{v})$
$\displaystyle+R^{f}_{\vec{v}}(k,p)$
$\displaystyle=\frac{\beta\delta_{n^{\prime}n}\delta^{(2)}(\vec{v}^{\,\prime}-\vec{v})\,2\pi\delta(p_{\parallel}^{\prime}-p_{\parallel})}{2}\Bigg{[}\left({\begin{array}[]{cc}0&\gamma_{4}\left[p_{4}+i\bar{\mu}+ip_{\parallel}(1+r_{f}(k,p_{\parallel})\right]\\\
\gamma_{4}\left[p_{4}+i\bar{\mu}-ip_{\parallel}(1+r_{f}(k,p_{\parallel})\right]&0\\\
\end{array}}\right)$ (111)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\frac{h_{s}}{\sqrt{n_{D}}}\left({\begin{array}[]{cc}\sigma
I&0\\\ 0&\sigma I\\\
\end{array}}\right)+\frac{h_{v}}{\sqrt{n_{D}}}\left({\begin{array}[]{cc}0&\gamma_{4}\left(\omega_{4}+i\omega_{\parallel}\right)\\\
\gamma_{4}\left(\omega_{4}+i\omega_{\parallel}\right)&0\\\
\end{array}}\right)\Bigg{]}\;,$ (116)

and the inverse can be obtained as

$\displaystyle\left[\Gamma^{m(2)}_{\bar{\Psi}\Psi}(k,\vec{v})+R^{f}_{\vec{v}}(k,p)\right]^{-1}$
$\displaystyle\qquad\qquad=\frac{2\beta\delta_{n^{\prime}n}\delta^{(2)}(\vec{v}^{\,\prime}-\vec{v})\,2\pi\delta(p_{\parallel}^{\prime}-p_{\parallel})}{{\bar{p}_{4}}^{2}+{\bar{p}_{\parallel}}^{2}+\frac{h^{2}_{s}}{n_{D}}\sigma^{2}}\left[\frac{h_{s}}{\sqrt{n_{D}}}\left({\begin{array}[]{cc}\sigma
I&0\\\ 0&\sigma I\\\
\end{array}}\right)-\left({\begin{array}[]{cc}0&\gamma_{4}\left(\bar{p}_{4}+i\bar{p}_{\parallel}\right)\\\
\gamma_{4}\left(\bar{p}_{4}-i\bar{p}_{\parallel}\right)&0\\\
\end{array}}\right)\right]\;,$ (121)

where $\bar{p}_{4}$ and $\bar{p}_{\parallel}$ denote

$\displaystyle\bar{p}_{4}$
$\displaystyle=p_{4}+i\left[\bar{\mu}+\frac{h_{v}}{\sqrt{n_{D}}}\left(\omega_{0}+\omega_{\parallel}\right)\right]\;,$
(122) $\displaystyle\bar{p}_{\parallel}$
$\displaystyle=p_{\parallel}\left[1+r_{f}(k,p_{\parallel})\right]\;.$ (123)

Then, after taking the trace on the other indices (considering an additional
factor 1/2), Eq. (98) can be expressed as

$\displaystyle\partial_{t}\Gamma^{f}_{k}$
$\displaystyle=-\beta(2\pi)^{3}\delta^{(3)}(0)\frac{1}{\beta}\sum_{n}\frac{\mu_{q}^{2}}{(2\pi)^{2}}\sum_{\vec{v}}\int_{-\mu_{q}}^{\mu_{q}}\frac{dp_{\parallel}}{2\pi}\frac{4n_{D}p_{\parallel}^{2}[1+r_{f}(k,p_{\parallel})]\partial_{t}r_{f}(k,p_{\parallel})}{\left\\{p_{4}+i\left[\bar{\mu}+\frac{h_{v}}{\sqrt{n_{D}}}(\omega_{0}+\omega_{\parallel})\right]\right\\}^{2}+\left[p_{\parallel}(1+r_{f}(k,p_{\parallel}))\right]^{2}+\frac{h^{2}_{s}}{n_{D}}\sigma^{2}}$
$\displaystyle=\beta(2\pi)^{3}\delta^{(3)}(0)\frac{2n_{D}\mu_{q}^{2}}{\pi^{2}}\frac{k(t)^{3}}{E_{k}(\sigma)}\left\\{1-n_{\textrm{f}}\left[\beta\left(E_{k}(\sigma)+\tilde{\mu}(\omega_{\parallel\mu})\right)\right]-n_{\textrm{f}}\left[\beta\left(E_{k}(\sigma)-\tilde{\mu}(\omega_{\parallel\mu})\right)\right]\right\\}\;,$
(124)

where $n_{\textrm{f}}[x]=1/(e^{x}+1)$,
$E_{k}(\sigma)=\left[k(t)^{2}+\frac{h^{2}_{s}}{n_{D}}\sigma^{2}\right]^{\frac{1}{2}}$,
and $(2\pi)^{3}\delta^{(3)}(0)$ denotes the system’s volume. In mean-field
approximation,
$\frac{h_{v}}{\sqrt{n_{D}}}\left(\omega_{0}+\omega_{\parallel}\right)$ can be
absorbed into the shell parameter $\bar{\mu}$, as there is no kinetic
fluctuation of the bosonic fields:
$\tilde{\mu}(\omega_{\parallel\mu})=\bar{\mu}+\frac{h_{v}}{\sqrt{n_{D}}}\left(\omega_{0}+\omega_{\parallel}\right)$.
We will assume the condition $\tilde{\mu}(\omega_{\parallel\mu})\rightarrow 0$
in the limit $\bar{\mu}\rightarrow 0$ to make the medium contribution from the
momentum shell vanish when the parameter $\bar{\mu}\rightarrow 0$.

###  III.3 Analytic investigation and numerical results in the mean-field
approximation for the bosonic fields

As the quantum fluctuation from the bosonic part is missing in the mean-field
approximation, if one neglects the wave function renormalization of the
fermionic fields, the RG-time dependence of the effective action comes only
from the bosonic potential term:

$\displaystyle\partial_{t}\Gamma^{f}_{k}$
$\displaystyle=\beta(2\pi)^{3}\delta^{(3)}(0)\partial_{t}U(k(t),\sigma,\omega_{\parallel\mu})\;,$
(125)

with the UV initial condition

$\displaystyle U(k=\mu_{q},\sigma,\omega_{\parallel\mu})$
$\displaystyle=2\pi\mu_{q}^{2}\left[\frac{h^{2}_{s}}{g_{s}}\sigma^{2}+\frac{h^{2}_{v}}{g_{v}}\left(-\omega_{0}^{2}-\omega_{\parallel}^{2}\right)\right]\;.$
(126)

Then, the flow equation (124) can be integrated to give the following result:

$\displaystyle U(0,\sigma,\omega_{\parallel\mu})$
$\displaystyle=U(\mu_{q},\sigma,\omega_{\parallel\mu})+\frac{2n_{D}\mu_{q}^{2}}{\pi^{2}}\int^{\mu_{q}}_{0}dk\frac{k^{2}}{E_{k}(\sigma)}\bigg{\\{}1-n_{\textrm{f}}\left[\beta\left(E_{k}(\sigma)+\tilde{\mu}(\omega_{\parallel\mu})\right)\right]-n_{\textrm{f}}\left[\beta\left(E_{k}(\sigma)-\tilde{\mu}(\omega_{\parallel\mu})\right)\right]\bigg{\\}}\;,$
(127)

where $k(t)$ flows from $k(0)=\mu_{q}$ to $k(\infty)=0$. Since we are
considering the situation where $\bar{\mu}\ll\mu_{q}$ and $T\ll\mu_{q}$, the
contribution from the momentum shell around the quark Fermi sphere
($\bar{\mu}>0$) will be treated as the “medium correction” to the thin-shell
limit ($T\rightarrow 0,~{}\bar{\mu}\rightarrow 0$) of the potential (see Fig.
2). The IR potential in the thin-shell limit can be identified as follows:

$\displaystyle U_{\textrm{TS}}(0,\sigma,\omega_{\parallel\mu})$
$\displaystyle=2\pi\mu_{q}^{2}\left[\frac{h^{2}_{s}}{g_{s}(\mu_{q})}\sigma^{2}+\frac{h^{2}_{v}}{g_{v}(\mu_{q})}\left(-\omega_{0}^{2}-\omega_{\parallel}^{2}\right)\right]+\frac{2n_{D}\mu_{q}^{2}}{\pi^{2}}\left[\mu_{q}\left(\mu_{q}^{2}+\frac{h_{s}^{2}}{n_{D}}\sigma^{2}\right)^{\frac{1}{2}}-\int_{0}^{\mu_{q}}dk\left(k^{2}+\frac{h_{s}^{2}}{n_{D}}\sigma^{2}\right)^{\frac{1}{2}}\right]\;.$
(128)

We investigate the scalar potential first. If one assumes a nontrivial minimum
$\sigma_{0}\neq 0$, the corresponding $2\pi/g_{s}(\mu_{q})$ can be found from
the constraint $\partial_{\sigma}U_{\textrm{TS}}|_{\sigma=\sigma_{0}}=0$:

$\displaystyle\frac{2\pi}{g_{s}(\mu_{q})}\bigg{|}_{\textrm{TS}}$
$\displaystyle=-\frac{1}{\pi^{2}}\left\\{\left(1+\frac{h_{s}^{2}}{\mu_{q}^{2}}\frac{\sigma_{0}^{2}}{n_{D}}\right)^{-\frac{1}{2}}-\ln\left[\frac{\frac{h_{s}}{\mu_{q}}\frac{\sigma_{0}}{\sqrt{n_{D}}}}{\left(1+\frac{h_{s}^{2}}{\mu_{q}^{2}}\frac{\sigma_{0}^{2}}{n_{D}}\right)^{\frac{1}{2}}-1}\right]\right\\}\;.$
(129)

Keeping $h_{s}$ fixed and taking the large-$\mu_{q}$ limit, the following form
can be found:

$\displaystyle\frac{2\pi}{g_{s}(\mu_{q})}\bigg{|}_{\textrm{TS}}$
$\displaystyle=-\frac{1}{\pi^{2}}\left[1-\ln\left(\frac{2\sqrt{n_{D}}}{\sigma_{0}}\frac{\mu_{q}}{h_{s}}\right)\right]+\mathcal{O}\left(\frac{h_{s}^{2}}{\mu_{q}^{2}}\right)+\ldots\;,$
(130)

where $g_{s}(\mu_{q})\rightarrow 0$ in the limit of
$\mu_{q}/h_{s}\rightarrow\infty$ as discussed in Refs. Stoll:2021ori ;
Wolff:1985av . A possibly appearing divergence of
$U_{\textrm{TS}}(0,\sigma,\omega_{\parallel\mu})$ in the limit
$\mu_{q}/h_{s}\rightarrow\infty$ is regularized by Eq. (129).

Figure 4: (Color online) $U(0,\sigma,\bar{\omega}_{\parallel\mu})$ for
$\mu=400~{}\textrm{MeV}$ and $T=0.05\sigma_{0}$. From left to right,
$\bar{\mu}=0.65\sigma_{0}$, $\bar{\mu}=0.67\sigma_{0}$, and
$\bar{\mu}=0.85\sigma_{0}$. In this low-$T$ regime, the system develops a
local minimum at $\sigma=0$ (left spinodal point at
$\bar{\mu}=0.65\sigma_{0}$), which becomes the global minimum at the
transition (first-order phase transition). Around $\bar{\mu}=0.85\sigma_{0}$,
the local minima (which were the global minima before the transition)
disappear and only the global minimum remains (right spinodal point).

Since a nontrivial $\bar{\omega}_{\parallel\mu}$-dependence only comes from
the shell contribution, in the thin-shell limit ($T\rightarrow
0,~{}\bar{\mu}\rightarrow 0$)
$\bar{\omega}_{\parallel\mu}=(\bar{\omega}_{0},\bar{\omega}_{\parallel})\Rightarrow(0,0)$
and the corresponding $2\pi/g_{v}(\mu_{q})\rightarrow 0$ . Considering the
shell contribution, one can assume
$\bar{\omega}_{\parallel\mu}=(\bar{\omega}_{0},\bar{\omega}_{\parallel})\neq(0,0)$
and find the corresponding $2\pi/g_{v}(\mu_{q})\neq 0$ from the constraints
$\partial_{\omega_{0}}U_{\textrm{shell}}|_{\omega_{0}=\bar{\omega}_{0}}=0$ and
$\partial_{\omega_{\parallel}}U_{\textrm{shell}}|_{\omega_{\parallel}=\bar{\omega}_{\parallel}}=0$:

$\displaystyle\frac{2\pi}{g_{v}(\mu_{q})}$
$\displaystyle=\frac{1}{\pi^{2}}\frac{\sqrt{n_{D}}}{h_{v}\bar{\omega}_{\parallel}}\Bigg{(}\mu_{q}\left\\{n_{\textrm{f}}\left[\beta\left(E_{\mu_{q}}(\sigma)-\tilde{\mu}(\bar{\omega}_{\parallel\mu})\right)\right]-n_{\textrm{f}}\left[\beta\left(E_{\mu_{q}}(\sigma)+\tilde{\mu}(\bar{\omega}_{\parallel\mu})\right)\right]\right\\}$
$\displaystyle\qquad\qquad\qquad-\int_{0}^{\mu_{q}}dk\left\\{n_{\textrm{f}}\left[\beta\left(E_{k}(\sigma)-\tilde{\mu}(\bar{\omega}_{\parallel\mu})\right)\right]-n_{\textrm{f}}\left[\beta\left(E_{k}(\sigma)+\tilde{\mu}(\bar{\omega}_{\parallel\mu})\right\\}\right]\right)\Bigg{)}\;,$
(131)

from which one can check that $2\pi/g_{v}(\mu_{q})\leq 0$ for
$\bar{\omega}_{\parallel}\geq 0$ and the asymptotic behavior
$2\pi/g_{v}(\mu_{q})\rightarrow 0$ in the thin-shell limit ($T\rightarrow
0,~{}\bar{\mu}\rightarrow 0,~{}\bar{\omega}_{\parallel\mu}\rightarrow 0$),
which guarantees the repulsive nature of the $\omega^{2}$-potential in
$U(\mu_{q},\sigma,\omega_{\parallel\mu})$ and implies the absence of an
$\omega^{2}$-potential in the thin-shell limit, respectively. The thin-shell
limit of the IR potential can be summarized as follows:

$\displaystyle U_{\textrm{TS}}(0,\sigma,\omega_{\parallel\mu})$
$\displaystyle=-\frac{\mu_{q}^{2}(h_{s}\sigma)^{2}}{\pi^{2}}\left\\{\left(1+\frac{h_{s}^{2}}{\mu_{q}^{2}}\frac{\sigma_{0}^{2}}{n_{D}}\right)^{-\frac{1}{2}}-\ln\left[\frac{\frac{h_{s}}{\mu_{q}}\frac{\sigma_{0}}{\sqrt{n_{D}}}}{\left(1+\frac{h_{s}^{2}}{\mu_{q}^{2}}\frac{\sigma_{0}^{2}}{n_{D}}\right)^{\frac{1}{2}}-1}\right]+\ln\left[\frac{\frac{h_{s}}{\mu_{q}}\frac{\sigma}{\sqrt{n_{D}}}}{\left(1+\frac{h_{s}^{2}}{\mu_{q}^{2}}\frac{\sigma^{2}}{n_{D}}\right)^{\frac{1}{2}}-1}\right]\right\\}$
$\displaystyle\qquad+\frac{\mu_{q}^{4}n_{D}}{\pi^{2}}\left(1+\frac{h_{s}^{2}}{\mu_{q}^{2}}\frac{\sigma^{2}}{n_{D}}\right)^{\frac{1}{2}},$
(132)

which, from a technical point of view, corresponds to the IR vacuum potential
discussed in Refs. Stoll:2021ori ; Wolff:1985av . The full IR potential (127)
can be evaluated as follows:

$\displaystyle U(0,\sigma,\omega_{\parallel\mu})$
$\displaystyle=U_{\textrm{TS}}(0,\sigma,\omega_{\parallel\mu})$
$\displaystyle\quad-\frac{\mu_{q}^{2}(h_{s}\sigma)^{2}}{\pi^{2}}\int_{0}^{\mu_{q}}dk\Bigg{(}\frac{k^{2}}{E_{k}(\sigma_{0})^{3}}\left\\{n_{\textrm{f}}\left[\beta\left(E_{k}(\sigma_{0})+\tilde{\mu}(\omega_{\parallel\mu})\right)\right]+n_{\textrm{f}}\left[\beta\left(E_{k}(\sigma_{0})-\tilde{\mu}(\omega_{\parallel\mu})\right)\right]\right\\}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\frac{\beta
k^{2}}{E_{k}(\sigma_{0})^{2}}\left\\{n_{\textrm{f}}\left[\beta\left(E_{k}(\sigma_{0})+\tilde{\mu}(\omega_{\parallel\mu})\right)\right]^{2}+n_{\textrm{f}}\left[\beta\left(E_{k}(\sigma_{0})-\tilde{\mu}(\omega_{\parallel\mu})\right)\right]^{2}\right\\}\Bigg{)}$
$\displaystyle\quad+\frac{\mu_{q}^{2}}{\pi^{2}}\frac{h_{v}\sqrt{n_{D}}}{\bar{\omega}_{\parallel}}\left(-\omega_{0}^{2}-\omega_{\parallel}^{2}\right)\Bigg{(}\mu_{q}\left\\{n_{\textrm{f}}\left[\beta\left(E_{\mu_{q}}(\sigma)-\tilde{\mu}(\bar{\omega}_{\parallel\mu})\right)\right]-n_{\textrm{f}}\left[\beta\left(E_{\mu_{q}}(\sigma)+\tilde{\mu}(\bar{\omega}_{\parallel\mu})\right)\right]\right\\}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\int_{0}^{\mu_{q}}dk\left\\{n_{\textrm{f}}\left[\beta\left(E_{k}(\sigma)-\tilde{\mu}(\bar{\omega}_{\parallel\mu})\right)\right]-n_{\textrm{f}}\left[\beta\left(E_{k}(\sigma)+\tilde{\mu}(\bar{\omega}_{\parallel\mu})\right)\right]\right\\}\Bigg{)}$
$\displaystyle\quad-\frac{2n_{D}\mu_{q}^{2}}{\pi^{2}}\int_{0}^{\mu_{q}}dk\frac{k^{2}}{E_{k}(\sigma)}\left\\{n_{\textrm{f}}\left[\beta\left(E_{k}(\sigma)+\tilde{\mu}(\omega_{\parallel\mu})\right)\right]+n_{\textrm{f}}\left[\beta\left(E_{k}(\sigma)-\tilde{\mu}(\omega_{\parallel\mu})\right)\right]\right\\}\;.$
(133)

Figure 5: (Color online) $U(0,\sigma,\bar{\omega}_{\parallel\mu})$ for
$\mu=400~{}\textrm{MeV}$ and $\bar{\mu}=0.05\sigma_{0}$. From left to right,
$T=0.3\sigma_{0}$, $T=0.5\sigma_{0}$, and $T=0.6\sigma_{0}$. In this
low-$\bar{\mu}$ regime, the system does not develop another local minimum and
the nontrivial global minima merge at the second-order phase transition.

To investigate the nontrivial minima of $U(0,\sigma,\omega_{\parallel\mu})$,
we fix some of the parameters as follows:
$h_{s}=\sqrt{6},~{}h_{v}=\sqrt{6},~{}n_{D}=6$. (Since $h_{s}$ and $h_{v}$ can
be absorbed into the bosonic fields, these Yukawa parameters can be adjusted
freely.) As one can see from the potential (133), the qualitative behavior of
the contribution from the momentum shell is determined by the relative size of
$\bar{\mu}$ and $T$ compared to the nontrivial minimum $\sigma_{0}$. Here we
fix the scale $\sigma_{0}\ll\mu_{q}$ and express $\bar{\mu}$ and $T$ in units
of $\sigma_{0}$. Considering that
$\langle\psi^{\dagger}(\vec{v})\psi(\vec{v})\rangle$ corresponds to the quark
number density of the outer shell, the scale of $\bar{\omega}_{0}$ can be
estimated from Eq. (77):

$\displaystyle\bar{\omega}_{0}\simeq$
$\displaystyle\frac{1}{\mu_{q}^{2}}\frac{g_{v}}{\sqrt{n_{D}}}\langle\psi^{\dagger}(\vec{v})\psi(\vec{v})\rangle=\frac{g_{v}\sqrt{n_{D}}}{\pi^{2}}\bar{\mu}\;,$
(134)

where
$\langle\psi^{\dagger}(\vec{v})\psi(\vec{v})\rangle=(n_{D}/\pi^{2})\mu_{q}^{2}\bar{\mu}$
in the massless limit. As
$\langle\bar{\psi}(\vec{v})\gamma^{\mu}\psi(\vec{v})\rangle=V^{\mu}\langle\psi^{\dagger}(\vec{v})\psi(\vec{v})\rangle$,
in the mean-field approximation we will assume
$\bar{\omega}_{\parallel}=\bar{\omega}_{0}$. We will ignore the RG-time
dependence of $g_{v}$ and set $g_{v}\sqrt{n_{D}}=0.3$ for the sake of
simplicity.

We first investigate the two different limits ($T\rightarrow 0,\bar{\mu}\neq
0$) and ($T\neq 0,\bar{\mu}\rightarrow 0$). As one can observe in Fig. 4, the
system undergoes a first-order phase transition around ($T\rightarrow
0,\bar{\mu}=0.65\sigma_{0}$), where the nontrivial global minimum
$\bar{\sigma}$ vanishes. In this direction, the system develops a local
minimum at $\sigma=0$, which becomes the global minimum at the phase boundary.
The two local minima which were the nontrivial global minima before the phase
transition persist until around $\bar{\mu}=0.85\sigma_{0}$ (right spinodal
point).

In the other limit ($T\neq 0,\bar{\mu}\rightarrow 0$) a second order phase
transition occurs around ($T\simeq 0.6\sigma_{0},\bar{\mu}\rightarrow 0$), cf.
Fig. 5. In this direction, the system does not develop any additional local
minimum and the nontrivial minima disappear simultaneously at the phase
boundary. This qualitative behavior is summarized in the phase diagrams shown
in Fig. 6.

On the left-hand side of Fig. 6 we show the magnitude of $\bar{\sigma}$ at the
nontrivial global minimum. On the right-hand side, we show the magnitude of
$\bar{\sigma}$ at the nontrivial local minimum corresponding to the right
spinodal boundary, which agrees with the global minimum to the left of a
(tri-)critical point ($T=0.25\sigma_{0}$, $\bar{\mu}=0.63\sigma_{0}$). This
means there is a second-order phase transition in the high-$T$,
low-$\bar{\mu}$ region and a first-order phase transition in the low-$T$,
high-$\bar{\mu}$ region, separated by the tricritical point at
($T=0.25\sigma_{0}$, $\bar{\mu}=0.63\sigma_{0}$). The phase diagram is very
similar to the standard one of the GN-model in mean-field approximation and
assuming a spatially homogeneous condensate, except that the role of the
ordinary chemical potential $\mu_{q}$ is now assumed by the shell-thickness
parameter $\bar{\mu}$.

Figure 6: (Color online) Contour plot of the magnitude of the condensate
$\bar{\sigma}$ at the local or global minima in the ($\bar{\mu},T$)-plane for
$\mu=400~{}\textrm{MeV}$. Left: the magnitude of $\bar{\sigma}$ at the
nontrivial global minimum, which shows a first-order phase boundary in the
low-$T$ regime. Right: the magnitude of $\bar{\sigma}$ at the nontrivial local
minimum corresponding to the right spinodal boundary, which agrees with the
global minimum to the left of the tricritical point ($T=0.25\sigma_{0}$,
$\bar{\mu}=0.63\sigma_{0}$). As the shell thickness $\bar{\mu}$ increases in
the low-$T$ regime, the system undergoes a first-order phase transition (Fig
4). The local minimum corresponding to the right spinodal exists until around
$\bar{\mu}\simeq 0.85\sigma_{0}$.

###  III.4 Analysis of the bosonic two-point function for investigation of an
inhomogeneous $\sigma(x)$ configuration

The next step is to consider a spatially inhomogeneous fluctuation
$\delta\sigma(x)$ around a homogeneous mean-background field $\sigma$
Schnetz:2004vr ; Nickel:2009ke ; Nickel:2009wj ; Buballa:2014tba ;
Buballa:2018hux ; Koenigstein:2021llr :

$\displaystyle\varphi\Rightarrow\left({\begin{array}[]{cc}[\sigma+\delta\sigma(x)]I&0\\\
0&[\sigma+\delta\sigma(x)]I\\\ \end{array}}\right)\;,$ (137)

where the fluctation $\delta\sigma(x)$ can be related to the momentum
difference between the quark and hole modes in the outer shell of the Fermi
sphere (cf. Figs. 2 and 7). If the coefficient of the term quadratic in
$\delta\sigma(x)$ in the effective average action, evaluated on the
homogeneous background $\sigma$, becomes negative in some region of momentum
$q_{\parallel}$, the homogeneous ground state is unstable and there is a more
stable inhomogeneous configuration of the field $\sigma(x)$. The coefficient
of the second-order term in the $\delta\sigma(x)$ fluctuation is the bosonic
two-point function

$\displaystyle\Gamma^{m(2)}_{(\delta\sigma)^{2}}(q_{\parallel})=\beta(2\pi)^{3}\delta^{(3)}(0)\left\\{\frac{2\pi\mu_{q}^{2}}{g_{s}}h^{2}_{s}+\frac{1}{8}\frac{h_{s}^{2}}{n_{D}}\sum_{\vec{v}}\frac{1}{\beta}\sum_{n}\int\frac{d^{2}p_{\bot}}{(2\pi)^{2}}\int\frac{dp_{\parallel}}{2\pi}\textrm{Tr}\left[S_{\vec{v}}(p+q)S_{\vec{v}}(p)\right]\right\\}\;,$
(138)

where the momentum-space propagator $S_{\vec{v}}(p)$ in a homogeneous
background can be found as

$\displaystyle
S_{\vec{v}}(p)=\frac{2}{{\bar{p}_{4}}^{2}+{p_{\parallel}}^{2}+\frac{h^{2}_{s}}{n_{D}}\sigma^{2}}\left[\frac{h_{s}}{\sqrt{n_{D}}}\left({\begin{array}[]{cc}\sigma
I&0\\\ 0&\sigma I\\\
\end{array}}\right)-\left({\begin{array}[]{cc}0&\gamma_{4}\left(\bar{p}_{4}+ip_{\parallel}\right)\\\
\gamma_{4}\left(\bar{p}_{4}-ip_{\parallel}\right)&0\\\
\end{array}}\right)\right]\;.$ (143)

The trace over spinor and internal indices can be evaluated as follows:

$\displaystyle\textrm{Tr}\left[S_{\vec{v}}(p+q)S_{\vec{v}}(p)\right]$
$\displaystyle=-16n_{D}\left\\{\frac{1}{{\bar{p}_{4}}^{2}+E_{p+q}(\sigma)^{2}}+\frac{1}{{\bar{p}_{4}}^{2}+E_{p}(\sigma)^{2}}-\frac{1}{[{\bar{p}_{4}}^{2}+E_{p+q}(\sigma)^{2}][{\bar{p}_{4}}^{2}+E_{p}(\sigma)^{2}]}\left(q_{\parallel}^{2}+4\frac{h_{s}^{2}}{n_{D}}\sigma^{2}\right)\right\\}\;,$
(144)

where an additional factor $1/2$ has been taken into account due to the
double-counting of spinor degrees of freedom. After summation over Matsubara
frequencies, the bosonic two-point function (138) can be expressed as follows:

$\displaystyle\frac{\Gamma^{m(2)}_{(\delta\sigma)^{2}}(q_{\parallel})}{\beta(2\pi)^{3}\delta^{(3)}(0)}$
$\displaystyle=\frac{2\pi\mu_{q}^{2}}{g_{s}}h_{s}^{2}-\frac{\mu_{q}^{2}h_{s}^{2}}{\pi}\int\frac{dp_{\parallel}}{2\pi}\Bigg{(}\frac{1}{2E_{p}(\sigma)}\left\\{1-n_{\textrm{f}}\left[\beta\left(E_{p}(\sigma)+\tilde{\mu}(\omega_{\parallel\mu})\right)\right]-n_{\textrm{f}}\left[\beta\left(E_{p}(\sigma)-\tilde{\mu}(\omega_{\parallel\mu})\right)\right]\right\\}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad~{}~{}+\frac{1}{2E_{p+q}(\sigma)}\left\\{1-n_{\textrm{f}}\left[\beta\left(E_{p+q}(\sigma)+\tilde{\mu}(\omega_{\parallel\mu})\right)\right]-n_{\textrm{f}}\left[\beta\left(E_{p+q}(\sigma)-\tilde{\mu}(\omega_{\parallel\mu})\right)\right]\right\\}\Bigg{)}$
$\displaystyle-\frac{\mu_{q}^{2}h_{s}^{2}}{\pi}\left(q_{\parallel}^{2}+4\frac{h_{s}^{2}}{n_{D}}\sigma^{2}\right)$
$\displaystyle\qquad\qquad\times\int\frac{dp_{\parallel}}{2\pi}\frac{1}{q_{\parallel}(p_{\parallel}+q_{\parallel})}\Bigg{(}-\frac{1}{2E_{p}(\sigma)}\left\\{1-n_{\textrm{f}}\left[\beta\left(E_{p}(\sigma)+\tilde{\mu}(\omega_{\parallel\mu})\right)\right]-n_{\textrm{f}}\left[\beta\left(E_{p}(\sigma)-\tilde{\mu}(\omega_{\parallel\mu})\right)\right]\right\\}$
$\displaystyle\qquad\qquad\qquad\qquad\quad~{}+\frac{1}{2E_{p+q}(\sigma)}\left\\{1-n_{\textrm{f}}\left[\beta\left(E_{p+q}(\sigma)+\tilde{\mu}(\omega_{\parallel\mu})\right)\right]-n_{\textrm{f}}\left[\beta\left(E_{p+q}(\sigma)-\tilde{\mu}(\omega_{\parallel\mu})\right)\right]\right\\}\Bigg{)}\;.$
(145)

In the limit $\mu_{q}\gg q_{\parallel}$, the two vacuum terms in the first
integral in Eq. (145) can be regarded as identical and regularized like in the
thin-shell limit of the potential (129):

$\displaystyle\frac{2\pi\mu_{q}^{2}}{g_{s}}h_{s}^{2}-\frac{\mu_{q}^{2}h_{s}^{2}}{\pi^{2}}\int_{0}^{\mu_{q}}dp_{\parallel}\frac{1}{E_{p}(\sigma)}=\frac{\mu_{q}^{2}h_{s}^{2}}{\pi^{2}}\ln\left[\frac{\sigma_{0}}{\sigma}\frac{\left(1+\frac{h_{s}^{2}}{\mu_{q}^{2}}\frac{\sigma^{2}}{n_{D}}\right)^{\frac{1}{2}}-1}{\left(1+\frac{h_{s}^{2}}{\mu_{q}^{2}}\frac{\sigma_{0}^{2}}{n_{D}}\right)^{\frac{1}{2}}-1}\right]\;,$
(146)

where the constant terms with a $\sigma_{0}$-dependence are omitted in the
limit $\sigma_{0}/\mu_{q}\ll 1$. Now one can summarize the result for the two-
point function as follows

$\displaystyle\frac{\Gamma^{m(2)}_{(\delta\sigma)^{2}}(q_{\parallel})}{\beta(2\pi)^{3}\delta^{(3)}(0)}$
$\displaystyle\simeq\frac{\mu_{q}^{2}h_{s}^{2}}{\pi^{2}}\left\\{\ln\left[\frac{\frac{h_{s}}{\mu_{q}}\frac{\sigma_{0}}{\sqrt{n_{D}}}}{-1+\left(1+\frac{h_{s}^{2}}{\mu_{q}^{2}}\frac{\sigma_{0}^{2}}{n_{D}}\right)^{\frac{1}{2}}}\right]-\ln\left[\frac{\frac{h_{s}}{\mu_{q}}\frac{\sigma}{\sqrt{n_{D}}}}{-1+\left(1+\frac{h_{s}^{2}}{\mu_{q}^{2}}\frac{\sigma^{2}}{n_{D}}\right)^{\frac{1}{2}}}\right]\right\\}$
$\displaystyle\qquad\quad+\frac{\mu_{q}^{2}h_{s}^{2}}{\pi^{2}}\int_{0}^{\mu_{q}}dp_{\parallel}\frac{1}{E_{p}(\sigma)}\left\\{n_{\textrm{f}}\left[\beta\left(E_{p}(\sigma)+\tilde{\mu}(\omega_{\parallel\mu})\right)\right]+n_{\textrm{f}}\left[\beta\left(E_{p}(\sigma)-\tilde{\mu}(\omega_{\parallel\mu})\right)\right]\right\\}$
$\displaystyle\qquad\quad-\frac{\mu_{q}^{2}h_{s}^{2}}{\pi}\left(q_{\parallel}^{2}+4\frac{h_{s}^{2}}{n_{D}}\sigma^{2}\right)$
$\displaystyle\qquad\qquad~{}\times\int\frac{dp_{\parallel}}{2\pi}\frac{1}{q_{\parallel}(p_{\parallel}+q_{\parallel})}\Bigg{(}-\frac{1}{2E_{p}(\sigma)}\left\\{1-n_{\textrm{f}}\left[\beta\left(E_{p}(\sigma)+\tilde{\mu}(\omega_{\parallel\mu})\right)\right]-n_{\textrm{f}}\left[\beta\left(E_{p}(\sigma)-\tilde{\mu}(\omega_{\parallel\mu})\right)\right]\right)\\}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\frac{1}{2E_{p+q}(\sigma)}\left\\{1-n_{\textrm{f}}\left[\beta\left(E_{p+q}(\sigma)+\tilde{\mu}(\omega_{\parallel\mu})\right)\right]-n_{\textrm{f}}\left[\beta\left(E_{p+q}(\sigma)-\tilde{\mu}(\omega_{\parallel\mu})\right)\right]\right\\}\Bigg{)}\;.$
(147)

Figure 7: (Color online) Left: Diagrammatic representation of the
$\delta\sigma(q_{\parallel})$ fluctuation. Each momentum is defined with
respect to the Fermi surface ($|\vec{p}_{F}|=\mu_{q}$). A single line denotes
the quark propagation and a double line denotes the hole propagation along the
$\vec{v}$ direction. Right: Illustration of a possible configuration of the
quark ($\psi_{+}(\vec{v},q_{\parallel}=2\bar{\mu})$) and the hole
($\bar{\psi}_{+}(-\vec{v},q_{\parallel}\simeq 0)$) states forming
$\delta\sigma(q_{\parallel})$ near the Fermi surface ($\bar{\mu}\geq
0.7\sigma_{0}$, $T\leq 0.22\sigma_{0}$). The empty star near the Fermi surface
at $-\mu_{q}\vec{v}$ denotes the hole and the (dark blue) filled star at
$\mu_{q}\vec{v}$ denotes its realization as an anti-particle state with
respect to the ground state. The (dark blue) filled circle denotes the excited
quark state. Since the hole state should lie inside the Fermi sphere, the
anti-particle state can carry at most a momentum $\mu_{q}\vec{v}$
($q_{\parallel}=0$). The excited quark state can carry a momentum up to
$2\mu_{q}\vec{v}$ ($q_{\parallel}=\mu_{q}$) in this effective model, but when
the momentum difference between the anti-particle and quark becomes
$q_{\parallel}=2\bar{\mu}$, the oscillating mode with the dominant wave vector
$2\bar{\mu}\vec{v}$ would be an even more stable configuration.  Figure 8:
(Color online) $\Gamma^{m(2)}_{(\delta\sigma)^{2}}(q_{\parallel})$ as a
function of $q_{\parallel}$ (in units of $\sigma_{0}$) for
$\mu=600~{}\textrm{MeV}$ and various temperatures:
$T=0.01\sigma_{0}\;(\textrm{blue solid}),~{}0.25\sigma_{0}\;(\textrm{yellow
dashed}),~{}0.5\sigma_{0}\;(\textrm{black
dotted}),~{}0.8\sigma_{0}\;(\textrm{orange dot-dashed})$. Left:
$\bar{\mu}=0.1\sigma_{0}$. Middle: $\bar{\mu}=0.65\sigma_{0}$. Right:
$T=0.01\sigma_{0}\;(\textrm{blue solid}),0.025\sigma_{0}\;(\textrm{yellow
dashed}),0.05\sigma_{0}\;(\textrm{black
dotted}),0.22\sigma_{0}\;(\textrm{orange dot-dashed})$, for
$\bar{\mu}=0.7\sigma_{0}$. The non-trivial minimum $\bar{\sigma}$ obtained
with for a given ($\bar{\mu},T$) is used for each case.

First, we investigate the behavior of
$\Gamma^{m(2)}_{(\delta\sigma)^{2}}(q_{\parallel})$ in the
small-$q_{\parallel}$ region. As one can see in Fig. 8, if one investigates
the regime where the amplitude of the nontrivial minimum $\bar{\sigma}$ varies
smoothly ($\bar{\mu}=0.1\sigma_{0}$,
$T=0.01\sigma_{0},0.25\sigma_{0},0.5\sigma_{0},0.8\sigma_{0}$),
$\Gamma^{m(2)}_{(\delta\sigma)^{2}}(q_{\parallel})$ shows a monotonically
increasing behavior. On the other hand, in the regime where $\bar{\sigma}$
varies strongly ($\bar{\mu}=0.65\sigma_{0}$,
$T=0.01\sigma_{0},0.25\sigma_{0},0.5\sigma_{0},0.8\sigma_{0}$),
$\Gamma^{m(2)}_{(\delta\sigma)^{2}}(q_{\parallel})$ exhibits a minimum near
the phase boundary ($T=0.25\sigma_{0}$). Near the first-order transition
regime, e.g., for $\bar{\mu}=0.7\sigma_{0}$ and $T\simeq 0.01\sigma_{0}$,
$\Gamma^{m(2)}_{(\delta\sigma)^{2}}(q_{\parallel})$ becomes negative in a
certain range of $q_{\parallel}$, which persists until $T\lesssim
0.22\sigma_{0}$ (cf. right plot of Fig. 8).

To investigate the first-order transition regime, one fixes the temperature
below $T=0.4\sigma_{0}$ and varies $\bar{\mu}$. As one anticipates from the
phase diagram, $\Gamma^{m(2)}_{(\delta\sigma)^{2}}(q_{\parallel})$ does not
show a clear instability at $T=0.2\sigma_{0}$ but it shows
$\Gamma^{m(2)}_{(\delta\sigma)^{2}}(q_{\parallel})\simeq 0$ at
$\bar{\mu}=0.8\sigma_{0}$ (cf. middle panel of Fig. 9). As the temperature
decreases further ($T=0.01\sigma_{0}$),
$\Gamma^{m(2)}_{(\delta\sigma)^{2}}(q_{\parallel})$ shows an instability just
to the right of the phase boundary ($\bar{\mu}>0.7\sigma_{0})$.

Figure 9: (Color online) $\Gamma^{m(2)}_{(\delta\sigma)^{2}}(q_{\parallel})$
as a function of $q_{\parallel}$ (in units of $\sigma_{0}$). We fix
$\mu=600~{}\textrm{MeV}$ and show various values of
$\bar{\mu}=0.2\sigma_{0}\;(\textrm{blue
solid}),~{}0.4\sigma_{0}\;(\textrm{yellow
dashed}),~{}0.6\sigma_{0}\;(\textrm{black
dotted}),~{}0.8\sigma_{0}\;(\textrm{orange dot-dashed})$. From left to right,
the plots correspond to $T=0.2\sigma_{0}$, $T=0.1\sigma_{0}$, and
$T=0.01\sigma_{0}$, respectively. The value of non-trivial minimum
$\bar{\sigma}$ obtained with for a given ($\bar{\mu},T$) is used for each
case.

The most unstable point, which becomes singular in the limit $T\rightarrow 0$,
appears at $q_{\parallel}\simeq 2\bar{\mu}$. As the shell-thickness
$\bar{\mu}$ becomes larger in the low-$T$ regime, the system undergoes a
first-order phase transition and the homogeneous background field
$\bar{\sigma}$ vanishes. The momentum $q_{\parallel}$ can be understood as the
momentum difference between a quark and a hole state when the spatial
fluctuation is allowed. A possible configuration is given in Fig. 7: if the
hole state is located just below the Fermi surface, $q_{\parallel}$
corresponds to the momentum carried by the excited quark. Once the first-order
phase transition occurs in the low-$T$ regime, ($\bar{\sigma}\rightarrow 0$,
as one can see from Fig. 10), the wave vector $2\bar{\mu}\vec{v}$ is related
to the appearance of another oscillating mode Buballa:2018hux ;
Koenigstein:2021llr :

$\displaystyle\langle\bar{\psi}(x)\psi(x)\rangle=\sum_{\vec{v}}e^{2i\mu_{q}\vec{v}\cdot\vec{x}}[\bar{\sigma}+\delta\sigma(x)]\mathit{\Delta}_{\bot}\quad\Longrightarrow\quad\sum_{\vec{v}}e^{2i(\mu_{q}+\bar{\mu})\vec{v}\cdot\vec{x}}\kappa\mathit{\Delta}_{\bot}=4\pi\cos(2\mu|\vec{x}|)\kappa\mathit{\Delta}_{\bot}\;,$
(148)

if one approximates $\delta\sigma(x)$ as follows:

$\displaystyle\delta\sigma(x)=\int\frac{dq_{\parallel}}{2\pi}e^{iq_{\parallel}\vec{v}\cdot\vec{x}}\delta\sigma(q_{\parallel})\mathit{\Delta}_{\bot}\simeq
e^{2i\bar{\mu}\vec{v}\cdot\vec{x}}\kappa\mathit{\Delta}_{\bot}\;,$ (149)

where
$\delta\sigma(q_{\parallel})\simeq\delta(q_{\parallel}-2\bar{\mu})\kappa$. The
regime where a homogeneous $\bar{\sigma}$ condensate becomes unstable is shown
in Fig. 10. This phase diagram looks rather similar to the one of the standard
GN model Schnetz:2004vr ; Koenigstein:2021llr , except that the ordinary
chemical potential $\mu_{q}$ is replaced by the shell-thickness parameter
$\bar{\mu}$.

Figure 10: (Color online) Left: contour plot of the minimum value of
$\Gamma^{m(2)}_{(\delta\sigma)^{2}}(q_{\parallel})$ in the
$(\bar{\mu},T)$-plane for $\mu=600~{}\textrm{MeV}$. In the white region at
small $T$,
$\Gamma^{m(2)}_{(\delta\sigma)^{2}}(q_{\parallel})\rightarrow-\infty$. Here, a
spatial oscillation of the condensate with wave vector $2\bar{\mu}\vec{v}$ is
expected. Right: contour plot of the $\bar{\sigma}$ condensate. The regime
where $\Gamma^{m(2)}_{(\delta\sigma)^{2}}(q_{\parallel})<0$, and thus an
inhomogeneous order parameter is expected, is shown as blue-dotted region.

###  III.5 Incorporating $\sigma$ fluctuations via FRG

In the previous subsections, we only considered the bosonic degrees of freedom
as a mean background field or the boson mode generated by quantum fluctuations
of the quark Fermi surface. Now we investigate the case where quantum
fluctuations of the $\sigma$-mode are accounted for. Thus, we add the
following kinetic term to the effective average action,

$\displaystyle\Gamma^{b}=\frac{1}{2}\sum_{\vec{v}}\int
d^{4}x_{\textrm{E}}\partial_{\mu}\sigma\partial_{\mu}\sigma\;.$ (150)

The effective average action underlying the RG-time running of the effective
potential $U_{\sigma}(k,\sigma)$ reads as follows:

$\displaystyle\Gamma_{k}=\int d^{4}x_{\textrm{E}}\bigg{\\{}$
$\displaystyle\frac{1}{2}\sum_{\vec{v}}\bigg{[}\bar{\Psi}_{\vec{v}}\left(-i\Gamma_{\mu}\partial_{\mu}-\bar{\mu}\Gamma^{0}+\frac{h_{s}}{\sqrt{n_{D}}}\varphi+\frac{h_{v}}{\sqrt{n_{D}}}\omega_{\mu}\Gamma_{\mu}\right)\Psi_{\vec{v}}+\partial_{\mu}\sigma\partial_{\mu}\sigma\bigg{]}+2\pi\frac{\mu_{q}^{2}h^{2}_{v}}{g_{v}}(\omega_{4}^{2}-\omega_{\parallel}^{2})+U_{\sigma}(k,\sigma)\bigg{\\}}$
$\displaystyle+\Delta\Gamma^{f}_{k}+\Delta\Gamma^{b}_{k}\;,$ (151)

where $U_{\sigma}(k,\sigma)$ is assumed to depend only on the amplitude of
$\sigma^{+}$ field ($\sigma=\sqrt{(\sigma^{+})^{\dagger}\sigma^{+}}$). The
bosonic regulator term $\Delta\Gamma^{b}_{k}$ is given as

$\displaystyle\Delta\Gamma^{b}_{k}=\frac{1}{2}\sum_{\vec{v}}\int\frac{d^{4}p_{\textrm{E}}}{(2\pi)^{4}}\sigma(-p_{\parallel})R^{b}_{\vec{v}}(k,p)\sigma(p_{\parallel})\;.$
(152)

This approximation corresponds to the general form of the local potential
approximation (LPA) where the RG running of the kinetic terms are ignored
Morris:1994ki ; Hasenfratz:1985dm ; Zumbach:1994kc . Since the LPA can be
understood as keeping the lowest-order contribution of the derivative
expansion, this truncation scheme is suitable for systems where small
momentum-exchange interactions are dominant. Here we assume that the bosonic
fluctuation is absent in the UV limit ($\Gamma_{k}=\Gamma^{m}$ for
$k\rightarrow\mu_{q}$) and is generated during the RG running towards the IR
limit. Thus, one can impose the following UV initial condition
$U_{\sigma}(k\rightarrow\mu_{q},\sigma)$:

$\displaystyle U_{\sigma}(\mu_{q},\sigma)$
$\displaystyle=\frac{2\pi\mu_{q}^{2}(h_{s}\sigma)^{2}}{g_{s}}=-\frac{\mu_{q}^{2}(h_{s}\sigma)^{2}}{\pi^{2}}\left\\{\left(1+\frac{h_{s}^{2}}{n_{D}}\frac{\sigma_{0}^{2}}{\mu_{q}^{2}}\right)^{-\frac{1}{2}}-\ln\left[\frac{\frac{h_{s}}{\sqrt{n_{D}}}\frac{\sigma_{0}}{\mu_{q}}}{\left(1+\frac{h_{s}^{2}}{n_{D}}\frac{\sigma_{0}^{2}}{\mu_{q}^{2}}\right)^{\frac{1}{2}}-1}\right]\right\\}\;,$
(153)

where the UV coupling (129) from the mean-field approximation has been used.

The exact RG flow equation including the contribution from the dynamical
$\sigma$ mode reads as follows:

$\displaystyle\partial_{t}\Gamma_{k}=-\textrm{STr}\left\\{\frac{\partial}{\partial
t}R^{f}_{\vec{v}}(k,p)\left[\Gamma^{m(2)}_{\bar{\Psi}\Psi}(k,\vec{v})+R^{f}_{\vec{v}}(k,p)\right]^{-1}\right\\}+\textrm{STr}\left\\{\frac{\partial}{\partial
t}R^{b}_{\vec{v}}(k,p)\left[\Gamma^{b(2)}_{\sigma^{2}}(k,\vec{v})+R^{b}_{\vec{v}}(k,p)\right]^{-1}\right\\}\;,$
(154)

where $\Gamma^{b(2)}_{\sigma^{2}}(k,\vec{v})$ denotes the second-order
functional derivative of $\Gamma_{k}$ with respect to $\sigma$. The bosonic
Litim regulator $R^{b}_{\vec{v}}(k,p)$ is

$\displaystyle
R^{b}_{\vec{v}}(k,p)=\frac{1}{2}\beta\delta_{n^{\prime}n}\delta^{(2)}(\vec{v}^{\,\prime}-\vec{v})\,2\pi\delta(p_{\parallel}^{\prime}-p_{\parallel})p^{2}_{\parallel}r_{b}(k,p_{\parallel})\;,$
(155)

which leads to

$\displaystyle\left[\Gamma^{b(2)}_{\sigma^{2}}(k,\vec{v})+R^{b}_{\vec{v}}(k,p)\right]^{-1}=\frac{2\beta\delta_{n^{\prime}n}\delta^{(2)}(\vec{v}^{\,\prime}-\vec{v})\,2\pi\delta(p_{\parallel}^{\prime}-p_{\parallel})}{p_{4}^{2}+p_{\parallel}^{2}[1+r_{b}(k,p_{\parallel})]+\frac{\mu_{q}^{2}h^{2}_{s}}{g_{s}}}\;.$
(156)

The $\sigma$-part of the exact RG flow equation can be obtained as follows:

$\displaystyle\textrm{STr}\left\\{\frac{\partial}{\partial
t}R^{b}_{\vec{v}}(k,p)\left[\Gamma^{b(2)}_{\sigma^{2}}(k,\vec{v})+R^{b}_{\vec{v}}(k,p)\right]^{-1}\right\\}=-\beta(2\pi)^{3}\delta^{(3)}(0)\frac{\mu_{q}^{2}}{\pi^{2}}\frac{k(t)^{3}}{2E^{b}_{k}(\partial^{2}_{\sigma}U_{\sigma})}\left\\{1+\frac{2}{\exp\left[\beta\left(E^{b}_{k}(\partial^{2}_{\sigma}U_{\sigma})\right)\right]-1}\right\\}\;,$
(157)

where
$E^{b}_{k}(\partial^{2}_{\sigma}U_{\sigma})=\left[k(t)^{2}+\partial^{2}_{\sigma}U_{\sigma}\right]^{\frac{1}{2}}$.
Equation (150) can then be written as

$\displaystyle\partial_{t}\Gamma_{k}$
$\displaystyle=\beta(2\pi)^{3}\delta^{(3)}(0)\frac{2n_{D}\mu_{q}^{2}}{\pi^{2}}\frac{k(t)^{3}}{E_{k}(\sigma)}\left\\{1-n_{\textrm{f}}\left[\beta\left(E_{k}(\sigma)+\tilde{\mu}(\omega_{\parallel\mu})\right)\right]-n_{\textrm{f}}\left[\beta\left(E_{k}(\sigma)-\tilde{\mu}(\omega_{\parallel\mu})\right)\right]\right\\}$
$\displaystyle\qquad-\beta(2\pi)^{3}\delta^{(3)}(0)\frac{\mu_{q}^{2}}{\pi^{2}}\frac{k(t)^{3}}{E^{b}_{k}(\partial^{2}_{\sigma}U_{\sigma})}\left\\{1+2n_{\textrm{b}}\left[\beta\left(2E^{b}_{k}(\partial^{2}_{\sigma}U_{\sigma})\right)\right]\right\\}\;,$
(158)

where $n_{\textrm{b}}[x]=1/(e^{x}-1)$.

Considering that the only scale-dependent quantity in the ansatz (151) is the
effective potential $U_{\sigma}(k,\sigma)$, plugging Eq. 151 into the left-
hand side of Eq. 158 we obtain

$\displaystyle\partial_{t}U_{\sigma}(k,\sigma)$
$\displaystyle=\frac{2n_{D}\mu_{q}^{2}}{\pi^{2}}\frac{k(t)^{3}}{E_{k}(\sigma)}\left\\{1-n_{\textrm{f}}\left[\beta\left(E_{k}(\sigma)+\tilde{\mu}(\omega_{\parallel\mu})\right)\right]-n_{\textrm{f}}\left[\beta\left(E_{k}(\sigma)-\tilde{\mu}(\omega_{\parallel\mu})\right)\right]\right\\}$
$\displaystyle\qquad-\frac{\mu_{q}^{2}}{\pi^{2}}\frac{k(t)^{3}}{E^{b}_{k}(\partial^{2}_{\sigma}U_{\sigma})}\left\\{1+2n_{\textrm{b}}\left[\beta\left(2E^{b}_{k}(\partial^{2}_{\sigma}U_{\sigma})\right)\right]\right\\}\;.$
(159)

This flow equation closely resembles the one which is derived for the GN model
in the (1+1)-dimensional case Stoll:2021ori . In particular, we can follow
this analogy and use a similar approach to the one of Ref. Stoll:2021ori  for
the solution of the flow equation.

We first observe that the RG-time evolution of the effective potential, given
by the right-hand side of Eq. (159), is independent of the effective potential
itself, while it depends on the second derivative of the effective potential.
This suggests to use its derivative with respect to the $\sigma$-field as a
new variable. Thus, we define the new variable

$\displaystyle u(k,\sigma)\equiv\partial_{\sigma}U_{\sigma}(k,\sigma)\;,$
(160)

and analogously

$\displaystyle
u^{\prime}(k,\sigma)\equiv\partial_{\sigma}u(k,\sigma)\equiv\partial^{2}_{\sigma}U_{\sigma}(k,\sigma)\;.$
(161)

In order to express Eq. (159) in terms of the newly defined variables, we now
take the derivative with respect to $\sigma$ of Eq. (159) and obtain

$\displaystyle\partial_{t}u(k,\sigma)$
$\displaystyle=\frac{d}{d\sigma}\left(\frac{2n_{D}\mu_{q}^{2}}{\pi^{2}}\frac{k(t)^{3}}{E_{k}(\sigma)}\left\\{1-n_{\textrm{f}}\left[\beta\left(E_{k}(\sigma)+\tilde{\mu}(\omega_{\parallel\mu})\right)\right]-n_{\textrm{f}}\left[\beta\left(E_{k}(\sigma)-\tilde{\mu}(\omega_{\parallel\mu})\right)\right]\right\\}\right)$
$\displaystyle\qquad-\frac{d}{d\sigma}\left(\frac{\mu_{q}^{2}}{\pi^{2}}\frac{k(t)^{3}}{E^{b}_{k}(u^{\prime}(k,\sigma))}\left\\{1+2n_{\textrm{b}}\left[\beta\left(2E^{b}_{k}(u^{\prime}(k,\sigma))\right)\right]\right\\}\right)\;,$
(162)

which we can rewrite as

$\displaystyle\partial_{t}u(k,\sigma)+\frac{d}{d\sigma}g(t,u^{\prime}(k,\sigma))=\frac{d}{d\sigma}S(t,\sigma)\;.$
(163)

Equation (163) is a non-linear diffusion equation, like the heat equation,
with an additional source term induced by the fermions. If we let us then
guide by the interpretation of the FRG flow equation as a diffusion equation,
$t$ can be considered as a time variable for the FRG flow, while $\sigma$ can
be interpreted as a spatial variable. In this specific case $u(k,\sigma)$
plays the role of a fluid “density”, whose evolution properties are governed
by the two contributions we introduced in Eq. 163 and that we are about to
detail. It is clear that, once the association between the FRG flow equation
and a diffusion equation has been demonstrated, we are then allowed to exploit
the wide and well-established toolbox of powerful numerical methods that have
been developed to solve hydrodynamic equations. In particular, we will use the
so-called Kurganov and Tadmor (KT) scheme Kurganov:2000ovy .

First of all, the radial $\sigma$-mode produces the diffusion term

$\displaystyle
g(t,u^{\prime})=\frac{\mu_{q}^{2}}{\pi^{2}}\frac{k(t)^{3}}{E^{b}_{k}(u^{\prime}(k,\sigma))}\left\\{1+2n_{\textrm{b}}\left[\beta\left(2E^{b}_{k}(u^{\prime}(k,\sigma)\right)\right]\right\\}\;,$
(164)

since it depends on the curvature mass $u^{\prime}(k,\sigma)$. From a fluid-
dynamical perspective, diffusion leads to a smearing of the solution, since it
depends on the gradient of the solution itself. This implies that the
conserved quantity is transported from regions in the spatial domain where it
is more concentrated, i.e., where the solution significantly differs from a
constant one, to regions where it is less concentrated, i.e., where it is
closer to a constant one, which by definition is left unchanged by diffusion.
According to these features, the diffusion term plays a fundamental role in
the dynamics of symmetry restoration.

On the other hand, the fermionic loop gives rise to a time- and
$\sigma$-dependent source term

$\displaystyle
S(t,\sigma)=\frac{2n_{D}\mu_{q}^{2}}{\pi^{2}}\frac{k(t)^{3}}{E_{k}(\sigma)}\left\\{1-n_{\textrm{f}}\left[\beta\left(E_{k}(\sigma)+\tilde{\mu}(\omega_{\parallel\mu})\right)\right]-n_{\textrm{f}}\left[\beta\left(E_{k}(\sigma)-\tilde{\mu}(\omega_{\parallel\mu})\right)\right]\right\\}\;,$
(165)

which is independent of the conserved quantity $u(k,\sigma)$. According to
this observation, the fermionic contribution turns out to be completely
independent of the effective potential and of its time evolution, so it
receives no feedback from the bosonic sector.

If one considers the case $\tilde{\mu}=0$, and performs the
$\sigma$-derivative of Eq. (165), as indicated in Eq. (163), one finds a
source-like, positive contribution to the flow equation for $\sigma<0$ and a
sink-like negative contribution for $\sigma>0$. This is not trivially true for
$\tilde{\mu}\neq 0$, and also some high peaks and shocks in the field space
may develop, especially in the low-temperature case Stoll:2021ori .

Figure 11: (Color online) Scaled condensate (position of the minimum)
$\sigma_{\mathrm{min}}(t)/\sigma_{0}$ as a function of $t=-\ln(k/\mu_{q})$
with $\mu_{q}=40\sigma_{0}$. Left: at a given constant temperature
$T=10^{-4}\sigma_{0}$ but for different values of shell thickness $\bar{\mu}$.
Right: in the thin-shell limit $\bar{\mu}=0.0$, but for different values of
temperature $T$. Other relevant parameters are: $n_{D}=6$,
$h_{s}=h_{v}\equiv\sqrt{n_{D}}$, $\omega_{0}=0.0$,
$\omega_{\parallel}=0.2\bar{\mu}/\pi^{2}$, and $\sigma_{0}=10$ MeV,
respectively.

The results obtained within the FRG approach show that the effective potential
develops a nonzero minimum at a finite RG scale, which breaks the
$\mathbf{Z}_{2}$ symmetry of the potential. However, at a later RG scale the
symmetry is restored, i.e., the $\sigma$-condensate vanishes. The symmetry
restoration is independent on both the shell thickness and the temperature,
and increasing these parameters results in a higher RG scale at which the
symmetry is restored. From a fluid-dynamical point of view, this occurrence
can be interpreted as follows: in the early stages of the RG-time evolution, a
finite condensate is produced by the action of the sink-like fermionic term.
At later RG times, however, the action of the bosonic contribution becomes
dominant, implying that the diffusion term smears out the potential up to the
point where the symmetry is restored. This result is in agreement with the
previous findings in the context of the GN model in (1+1) dimensions
Stoll:2021ori . This result means that, if one considers the bosonic
$\sigma$-mode as dynamical degree of freedom, restoration of the
$\mathbf{Z}_{2}$ symmetry is inevitable and the nontrivial shell-like
structure where $\bar{\sigma}\neq 0$ is not present in the IR limit, in
contrast to the mean-field analysis of the simple bosonized model 94. However,
we cannot discard the possibility that higher-order truncations beyond LPA, or
including fluctuations of the vector meson $\omega_{\mu}$ in LPA, may lead to
different results. So far, our conclusion is thus that the inclusion of
bosonic fluctuations leads to an inevitable restoration of the
$\mathbf{Z}_{2}$ symmetry, implying that, in contrast to the mean-field
results, no finite condensate is present in the IR.

##  IV Conclusion

In this work, we constructed a simple effective model where the collinear
four-quark interactions in high-density effective theory (HDET) are further
reduced to contact interactions analogous to the GN model. The contact
interaction can be related to QCD via the collinear four-quark interaction
around the Fermi surface in an instanton background. By considering the
fluctuation in the direction transverse to a given Fermi velocity, transitions
between two topological different vacua ($I=\pm 1$) are possible via the
surface quark and hole mode on opposite sides of the Fermi sphere. The
corresponding scalar channel can form an Overhauser-type surface quark-hole
condensate $\sigma$, which corresponds to the amplitude of the inhomogeneous
scalar quark condensation:

$\displaystyle\langle\bar{\psi}(x)\psi(x)\rangle=\sum_{\vec{v}}e^{2i\mu_{q}\vec{v}\cdot\vec{x}}\langle\bar{\psi}_{+}(-\vec{v},x)\psi_{+}(\vec{v},x)\rangle\quad\Longrightarrow\quad
4\pi\cos(2\mu_{q}|\vec{x}|)\sigma\mathit{\Delta}_{\bot}\;.$ (166)

The vector channel was included to check the role of a repulsive interaction
in the scalar-condensation process. In the construction of the effective
model, the relevant slow modes are defined with respect to the ground level
defined at $\mu_{q}=\mu-\bar{\mu}$, and the condensation pattern is analyzed
as a function of the shell thickness parameter $\bar{\mu}$ and the temperature
$T$. The non-perturbative RG flow of the effective average action is first
analyzed in the mean-field approximation, i.e., where the RG flow equation can
be integrated analytically.

In the background mean-field approximation for the collective bosonic fields,
the phase diagram for a homogeneous condensate is found in the ($\bar{\mu}$,
$T$)-plane (see Fig. 6). To the left side of a tricritical point at
($\bar{\mu}=0.63\sigma_{0}$, $T=0.25\sigma_{0}$), a continuous variation of
the order parameter is found, which goes to zero in a second-order phase
transition. To the right of this point, the transition is of first order. At
larger values of $\bar{\mu}$, beyond the phase boundary for the first-order
transition, one can find a regime where an inhomogeneous configuration of
$\sigma(x)$ is energetically favored over the configuration with a homogeneous
condensate. This conclusion follows by analyzing the second-order coefficient
of the effective average action, which becomes negative (Fig. 10). If one
assumes the fluctuation $\delta\sigma(x)$ to have the main contribution from
the wave vector $2\bar{\mu}\vec{v}$, as one may expect from Fig. 8, the
inhomogeneous scalar quark condensation varies spatially with a frequency
$2\mu\equiv 2(\mu_{q}+\bar{\mu})$ (Eq. (148)),

$\displaystyle\langle\bar{\psi}(x)\psi(x)\rangle\quad\Longrightarrow\quad\sum_{\vec{v}}e^{2i(\mu_{q}+\bar{\mu})\vec{v}\cdot\vec{x}}\kappa\mathit{\Delta}_{\bot}=4\pi\cos(2\mu|\vec{x}|)\kappa\mathit{\Delta}_{\bot}\;.$
(167)

In other words, if one assumes a temperature $T$ corresponding to a state
which, in the mean-field approximation, corresponds to the homogeneously
broken phase, and one then increases $\bar{\mu}$, the homogeneous condensation
amplitude $\bar{\sigma}$ persists until the momentum shell reaches a thickness
of about $\bar{\mu}\simeq 0.7\sigma_{0}$. Above this value, as the
condensation amplitude has a dominant oscillation frequency $2\bar{\mu}$, the
quark condensation becomes periodic with the main frequency $2\mu$. This
configuration looks like quarkyonic matter McLerran:2007qj , where the
explicit baryon shell structure is deduced from large-$N_{c}$ QCD.

However, if one regards the collective bosonic mode $\sigma$ as a quasi-
particle field by assuming the dynamical $\sigma(x)$ mode, the aforementioned
nontrivial minimum $\sigma$ does not appear. The bosonic and fermionic part of
the FRG flow equation correspond to a diffusion and source term, respectively.
In the FRG flow, a nontrivial condensation amplitude is formed by the
fermionic sink-like term but, as the bosonic diffusion term becomes dominant
in the later RG evolution, the condensate vanishes eventually. While it is
then questionable whether an inhomogeneous configuration exists at all, the
possibility that it exists is not ruled out and needs to be explored in future
work.

Other extensions of the current work consist of extending the FRG analysis to
the LPA’ approximation, where a non-trivial wavefunction renormalization for
the fields is considered Dupuis:2020fhh , or by including fluctuations of the
vector mesons in LPA and beyond. This may potentially change the conclusions
of the present work.

###### Acknowledgements.

The authors gratefully acknowledge fruitful discussions with L. Kiefer, A.
Koenigstein, and R.D. Pisarski. K.S. acknowledges support of the Alexander von
Humboldt foundation. The authors acknowledge support by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) through the CRC-TR
211 ”Strong-interaction matter under extreme conditions” – project number
315477589 – TRR 211. The work is supported by the State of Hesse within the
Research Cluster ELEMENTS (Project ID 500/10.006).

##  Appendix A Euclidean convention

In this work, we follow the notations given in Ref. Bellac:2011kqa . From
Minkowski space to Euclidean space, the time components are transformed as
follows: $t\rightarrow-i\tau$, $\gamma_{0}\rightarrow-i\gamma_{4}$,
$p_{0}\rightarrow-ip_{4}=i\omega_{n}$, where $\omega_{n}$ denotes Matsubara
frequencies. Note following subsequent transformations:

$\displaystyle
g_{\mu\nu}\rightarrow-\delta_{\mu\nu},~{}A_{\mu}B^{\mu}\rightarrow-
A_{\mu}B_{\mu},~{}/p\rightarrow-/p_{E}=-p_{4}\gamma_{4}-\vec{\gamma}\cdot\vec{p},~{}\partial^{\mu}A_{\mu}=\partial_{\mu}A_{\mu}\;.$
(168)

The partition function is defined as follows:

$\displaystyle\mathcal{Z}$
$\displaystyle\equiv\textrm{const.}\int\mathcal{D}[\bar{\psi},\psi,\cdots]\exp\left[-S_{\textrm{E}}\right],\quad
S_{\textrm{E}}=\int
d^{4}x_{\textrm{E}}\mathcal{L}_{\textrm{E}}(\bar{\psi},\psi,\cdots)\;.$ (169)

##  Appendix B Instanton solution

The Euclidean SU(2) Yang-Mills action can be written as follows:

$\displaystyle S^{\textrm{E}}_{\textrm{YM}}=\frac{1}{4}\int
d^{4}x_{\textrm{E}}\mathcal{G}^{a}_{\mu\nu}\mathcal{G}^{a}_{\mu\nu}=\frac{1}{4}\int
d^{4}x_{\textrm{E}}\left[\pm\mathcal{G}^{a}_{\mu\nu}\tilde{\mathcal{G}}^{a}_{\mu\nu}+\frac{1}{2}\left(\mathcal{G}^{a}_{\mu\nu}\mp\tilde{\mathcal{G}}^{a}_{\mu\nu}\right)^{2}\right]\;,$
(170)

where $\pm$ denotes the instanton and anti-instanton field strength,
respectively and
$\tilde{\mathcal{G}}^{a}_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}\mathcal{G}^{a}_{\rho\sigma}$
with $\epsilon_{1234}=i$. This action can be minimized if the field-strength
tensor is (anti-)self-dual:
$\mathcal{G}^{a}_{\mu\nu}=\pm\tilde{\mathcal{G}}^{a}_{\mu\nu}$ (where $\pm$
denotes self-duality and anti-self-duality, respectively). Under the
(anti-)self-dual field-strength configuration, the minimized action
$S^{\textrm{E}}_{\textrm{YM}}=8\pi^{2}|Q|/g^{2}$ is determined by the
topological invariant $Q$:

$\displaystyle Q$ $\displaystyle=\frac{g^{2}}{32\pi^{2}}\int
d^{4}x_{\textrm{E}}\mathcal{G}^{a}_{\mu\nu}\tilde{\mathcal{G}}^{a}_{\mu\nu}=\int
d^{4}x_{\textrm{E}}\partial_{\mu}K_{\mu}\;,$ (171) $\displaystyle K_{\mu}$
$\displaystyle=\frac{1}{16\pi^{2}}\epsilon_{\alpha\beta\gamma\delta}\left(A^{a}_{\beta}\partial_{\gamma}A^{a}_{\delta}+\frac{1}{3}\epsilon^{abc}A^{a}_{\beta}A^{b}_{\gamma}A^{c}_{\delta}\right)\;.$
(172)

To have a finite action, the gauge field should be pure gauge at the surface
of a large sphere in 4D Euclidean space: $A_{\mu}\rightarrow
iU\partial_{\mu}U$ as $|x_{\mu}|\rightarrow\infty$. The gauge transformation
$U$ links the isospin to the spatial 3-sphere and can be classified by the
winding number $n_{\textrm{CS}}$. If the gauge field vanishes rapidly in the
large-sphere limit, the topological invariant can be understood as

$\displaystyle Q$ $\displaystyle=\int
d^{4}x_{\textrm{E}}\partial_{\mu}K_{\mu}=\int
d^{4}x_{\textrm{E}}\partial_{4}K_{4}=i\int dt\partial_{0}\int
d^{3}xK_{0}=n_{\textrm{CS}}(t=\infty)-n_{\textrm{CS}}(t=-\infty)\;,$ (173)

where one can find the transition between topologically different vacua
($Q\neq 0$) by the instanton solution. By taking the large-sphere limit of
$A_{\mu}\rightarrow iU\partial_{\mu}U$ with
$U=i(x_{\mu}/|x_{\mu}|)\tau^{+}_{\mu}$, where $\tau^{+}_{\mu}=(\mp
i,\vec{\tau})$, one can obtain the asymptotic form
$A^{a}_{\mu}=2\eta_{a\mu\nu}x_{\nu}/x^{2}$ for the $Q=1$ configuration. The
group structure $\eta_{a\mu\nu}$ known as ‘t Hooft symbol tHooft:1976rip  can
be found as

$\displaystyle\eta_{a\mu\nu}=$
$\displaystyle\Bigg{\\{}{\begin{array}[]{cc}\epsilon_{a\mu\nu},&\mu,\nu=1,2,3\\\
\delta_{a\mu},&\nu=4,\\\
-\delta_{a\nu},&\mu=4,\end{array}},\quad\bar{\eta}_{a\mu\nu}=\Bigg{\\{}{\begin{array}[]{cc}\epsilon_{a\mu\nu},&\mu,\nu=1,2,3\\\
-\delta_{a\mu},&\nu=4,\\\ \delta_{a\nu},&\mu=4,\end{array}},$ (180)

where $\bar{\eta}_{a\mu\nu}$ stands for the group structure of the anti-
instanton solution ($Q=-1$), which is embedded in another $\textrm{SO}(3)$
subgroup of $\textrm{O}(4)$. One can find a general solution of the self-
duality relation by using the ansatz
$A^{a}_{\mu}=2\eta_{a\mu\nu}x_{\nu}f(x^{2})/x^{2}$ with the boundary
constraint $f(x^{2})\rightarrow 1$ as $|x_{\mu}|\rightarrow\infty$. The
solution known as Belavin-Polyakov-Schwartz-Tyupkin instanton Belavin:1975fg
can be found as follows:

$\displaystyle A^{a}_{\mu}=\frac{2\eta_{a\mu\nu}x_{\nu}}{x^{2}+\rho^{2}}\;,$
(181)

where $f(x^{2})=x^{2}/(x^{2}+\rho^{2})$ and $\rho$ is the scale parameter for
the instanton size. A comprehensive explanation can be found in Refs.
Schafer:1996wv ; Coleman:1985rnk .

##  Appendix C Zero-mode contributions in vacuum

The physical meaning of the topological charge (173) can be understood from
the axial anomaly relation
$\partial_{\mu}j^{5}_{\mu}=(N_{f}g^{2}/16\pi^{2})\mathcal{G}^{a}_{\mu\nu}\tilde{\mathcal{G}}^{a}_{\mu\nu}$.
The space-time integration of the divergence of the axial current becomes

$\displaystyle\int d^{4}x\partial_{\mu}j^{5}_{\mu}$ $\displaystyle=\Delta
Q_{5}=Q_{5}(t=\infty)-Q_{5}(t=-\infty)=-iN_{f}\int
d^{4}x_{\textrm{E}}\partial_{\mu}\textrm{Tr}\left[S(x,x)\gamma_{5}\gamma_{\mu}\right]\;,$
(182)

where the quark propagator $S_{\textrm{E}}(x,y)=\left\langle
x|(i/D(\mu\rightarrow 0))^{-1}|y\right\rangle$ can be decomposed into
eigenmodes as follows:

$\displaystyle
S_{\textrm{E}}(x,y)=-\sum_{\lambda}\frac{\psi_{\lambda}(x)\psi^{\dagger}_{\lambda}(y)}{\lambda}\;,$
(183)

where each eigenmode is normalized and $\lambda$ is the pure imaginary
eigenvalue defined from $i/D(\mu\rightarrow
0)\psi_{\lambda}=\lambda\psi_{\lambda}$ (Eq. (17)). By using the anti-
hermiticity of $i/D(\mu\rightarrow 0)$, $\Delta Q_{5}$ can be evaluated as

$\displaystyle\Delta Q_{5}$ $\displaystyle=iN_{f}\int
d^{4}x_{\textrm{E}}\textrm{Tr}\left[\sum_{\lambda}\frac{\psi_{\lambda}(x)\psi^{\dagger}_{\lambda}(x)}{\lambda}2\lambda\gamma_{5}\right]=2N_{f}(n_{R}-n_{L})\;,$
(184)

where only the zero-modes $\psi_{(L/R)0}$ contribute to the summation since
$\psi_{\lambda}$ and $\gamma_{5}\psi_{\lambda}=\psi_{-\lambda}$ are orthogonal
due to the anti-hermiticity of the Dirac operator. This relation is known as
index theorem Atiyah:1970ws ; Callias:1977kg . The proper version of the index
theorem at $\mu\neq 0$ (Eq. (19)) is derived in Ref. Kanazawa:2011tt , and it
is shown that the following relations are satisfied in most of the gauge
configurations:

$\displaystyle I$
$\displaystyle=n_{R}(\mu)-n_{L}(\mu)=n_{R}(-\mu)-n_{L}(-\mu)=n_{R}(\mu)-n_{L}(-\mu)=n_{R}(-\mu)-n_{L}(\mu)\;.$
(185)

One can find the zero-mode contribution to the quark condensate in the vacuum
limit from the decomposition (183):

$\displaystyle\left\langle\bar{\psi}\psi\right\rangle=-\textrm{Str}[S_{\textrm{E}}(x,x)]=\int
d^{4}x_{\textrm{E}}\sum_{\lambda}\frac{\psi_{\lambda}(x)\psi^{\dagger}_{\lambda}(x)}{\lambda+m}=\sum_{\lambda}\frac{1}{\lambda+m}=-\sum_{\lambda\geq
0}\frac{2m}{-\lambda^{2}+m^{2}}\;,$ (186)

where $m$ denotes the small mass which will be eventually taken to zero,
$m\rightarrow 0$. If one takes the continuous (thermodynamic) limit first,
then the eigenvalue summation can be written as continuous integration with
the spectral density
$\rho(\bar{\lambda})=\sum_{\lambda}\delta(\bar{\lambda}-\lambda)$:

$\displaystyle\left\langle\bar{\psi}\psi\right\rangle=-\int_{-\infty}^{\infty}d\bar{\lambda}\rho(\bar{\lambda})\frac{m}{-\bar{\lambda}^{2}+m^{2}}=-\pi\rho(\bar{\lambda}=0)\;,$
(187)

where $\pi\delta(x)=\lim_{\epsilon\rightarrow 0}\epsilon/(x^{2}+\epsilon^{2})$
is used. One can see that only the zero-modes are correlated in the quark
condensate. Equation (187) is known as Banks-Casher relation Banks:1979yr .

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