# Mass spectrum of spin-one hadrons in dense two-color QCD:
Novel predictions by extended linear sigma model

Daiki Suenaga<EMAIL_ADDRESS>Few-body Systems in Physics Laboratory,
RIKEN Nishina Center, Wako 351-0198, Japan  Research Center for Nuclear
Physics, Osaka University, Ibaraki 567-0048, Japan  Kotaro Murakami
<EMAIL_ADDRESS>Department of Physics, Tokyo Institute of
Technology, 2-12-1 Ookayama, Megro, Tokyo 152-8551, Japan  Interdisciplinary
Theoretical and Mathematical Sciences Program (iTHEMS), RIKEN, Wako 351-0198,
Japan  Etsuko Itou<EMAIL_ADDRESS>Interdisciplinary Theoretical
and Mathematical Sciences Program (iTHEMS), RIKEN, Wako 351-0198, Japan
Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502,
Japan  Kei Iida<EMAIL_ADDRESS>Department of Mathematics and Physics,
Kochi University, 2-5-1 Akebono-cho, Kochi 780-8520, Japan

###### Abstract

We construct an extended version of the linear sigma model in such a way as to
describe spin-$1$ hadrons as well as spin-$0$ hadrons in two-color QCD (QC2D)
by respecting the Pauli-Gürsey $SU(4)$ symmetry. Within a mean-field
approximation, we therefrom examine a mass spectrum of the spin-$1$ hadrons at
finite quark chemical potential ($\mu_{q}$) and zero temperature. Not only
mean fields of scalar mesons and scalar-diquark baryons but also of vector
mesons and vector-diquark baryons are incorporated. As a result, we find that,
unless all of those four types of mean fields are taken into account, neither
lattice result for the critical $\mu_{q}$ that corresponds to the onset of
baryon superfluidity nor for $\mu_{q}$ dependence of the pion mass can be
reproduced. We also find that a slight suppression of the $\rho$ meson mass in
the superfluid phase, which was suggested by the lattice simulation, is
reproduced by subtle mixing effects between spin-$0$ and spin-$1$ hadrons.
Moreover, we demonstrate the emergence of an axialvector condensed phase and
possibly of a vector condensed phase by identifying the values of $\mu_{q}$ at
which the corresponding hadron masses vanish. The possible presence of iso-
triplet $1^{-}$ diquarks that may be denoted by a tensor-type quark bilinear
field is also discussed.

††preprint: YITP-23-172, RIKEN-iTHEMS-Report-23

##  I Introduction

Revealing the characteristics of hadrons in cold dense matter stands as a
significant pursuit in quantum chromodynamics (QCD), since these particles
serve as good probes to explore medium modifications of QCD symmetry
properties, such as chiral symmetry restoration. For this reason, thus far,
tremendous theoretical and experimental effort has been devoted to shed light
on the hadronic properties in dense nuclear matter Holt  _et al._ (2016);
Metag  _et al._ (2017). First-principles lattice Monte Carlo simulations,
however, are difficult to apply to such a cold and dense regime due to the so-
called sign problem Aarts (2016); Nagata (2022). Hence, our understanding of
how the hadronic properties are modified in cold dense matter is limited as
compared to the case of hot QCD matter.

Although lattice simulations remain to be effective in three-color QCD at
finite quark chemical potential ($\mu_{q}$), exceptionally for two-color QCD
(QC2D) with even numbers of quark flavors, the cumbersome sign problem
disappears and indeed the simulations turn out to be applicable at nonzero
$\mu_{q}$ Muroya  _et al._ (2003a). So far, many lattice simulations have been
performed toward the delineation of hadron modifications as well as of the
phase structure in cold dense QC2D Hands  _et al._ (1999); Kogut  _et al._
(2001); Hands  _et al._ (2001); Muroya  _et al._ (2003b); Chandrasekharan and
Jiang (2006); Hands  _et al._ (2006, 2008, 2010); Cotter  _et al._ (2013);
Hands  _et al._ (2012); Boz  _et al._ (2013); Braguta  _et al._ (2016); Puhr
and Buividovich (2017); Boz  _et al._ (2019); Astrakhantsev  _et al._ (2019);
Iida  _et al._ (2020); Wilhelm  _et al._ (2019); Buividovich  _et al._
(2021a); Iida  _et al._ (2021); Astrakhantsev  _et al._ (2020); Bornyakov  _et
al._ (2020); Buividovich  _et al._ (2020, 2021b); Iida and Itou (2022);
Murakami  _et al._ (2023a) (see Ref. Braguta (2023) and references therein).
In concert with those numerical experiments, theoretical examinations have
been made for qualitative understanding by using hadronic and microscopic
models Kogut  _et al._ (1999, 2000); Lenaghan  _et al._ (2002); Splittorff
_et al._ (2002); Ratti and Weise (2004); Sun  _et al._ (2007); Fukushima and
Iida (2007); Brauner  _et al._ (2009); Kanazawa  _et al._ (2009); Harada  _et
al._ (2010); Andersen and Brauner (2010); Zhang  _et al._ (2010); He (2010);
Strodthoff  _et al._ (2012); Imai  _et al._ (2013); Strodthoff and von Smekal
(2014); Khan  _et al._ (2015); Duarte  _et al._ (2016); Chao (2020); Adhikari
_et al._ (2018); Contant and Huber (2020); Suenaga and Kojo (2019); Khunjua
_et al._ (2020, 2022); Kojo and Suenaga (2021); Suenaga and Kojo (2021); Kojo
and Suenaga (2022); Suenaga  _et al._ (2023); Kawaguchi and Suenaga (2023).

In QC2D, diquarks, i.e., bound states of two quarks, emerge as color-singlet
hadrons thanks to the pseudoreality of $SU(2)_{c}$ color group, unlike in
three-color QCD. As a consequence, diquarks and mesons can be embedded into
single multiplets and described collectively. Moreover, the pseudoreality
allows us to extend $SU(2)_{L}\times SU(2)_{R}$ chiral symmetry to the so-
called Pauli-Gürsey $SU(4)$ symmetry Kogut  _et al._ (1999, 2000), and then,
the chiral condensate induces the symmetry breaking of $SU(4)\to Sp(4)$. Thus,
chiral models in QC2D are constructed based on this symmetry-breaking pattern
and, additionally, small violation of the Pauli-Gürsey $SU(4)$ symmetry to
account for a finite pion mass.

Another noteworthy feature of QC2D is emergence of the diquark condensed
phase: Diquarks are bosonic hadrons carrying the quark number in QC2D so that
they start to form a Bose-Einstein condensate (BEC) at certain $\mu_{q}$ Kogut
_et al._ (1999, 2000). This distinctive phase violates $U(1)_{B}$ baryon-
number symmetry spontaneously, and hence, the diquark condensed phase is also
referred to as the baryon superfluid phase. Meanwhile, the stable phase at
smaller $\mu_{q}$, which no longer contains BECs, is simply called the
hadronic phase. In the latter phase all thermodynamic quantities show no
$\mu_{q}$ dependence at zero temperature, and such a salient property is
called the Silver-Blaze property.

Recently, a mass spectrum of the low-lying spin-$0$ hadrons carrying negative
and positive parities was simulated on lattice at finite $\mu_{q}$ Murakami
_et al._ (2023b, ). The simulation result indicates that $\eta$ mesons (iso-
singlet $0^{-}$ mesons) are lighter than pions in the superfluid phase, which
is in contrast to our naive expectation that $\eta$ mesons are heavier than
pions due to the $U(1)_{A}$ anomaly effects. Motivated by this characteristic
mass inversion, in Ref. Suenaga  _et al._ (2023) we constructed the linear
sigma model (LSM) based on the (approximate) Pauli-Gürsey $SU(4)$ symmetry,
which is capable of describing not only $0^{-}$ mesons and $0^{+}$
(anti)diquark baryons but also $0^{+}$ mesons and $0^{-}$ (anti)diquark
baryons. Based on the LSM, indeed, we succeeded in explaining the mass
inversion by showing that the $\eta$ mass is sufficiently suppressed in the
superfluid phase owing to mixing with $0^{-}$ (anti)diquark baryons, which is
triggered by the $U(1)_{B}$ baryon-number violation.

In this paper, we extend the LSM by newly incorporating spin-$1$ hadrons,
i.e., $1^{\pm}$ mesons and (anti)diquark baryons, but still respecting the
Pauli-Gürsey $SU(4)$ symmetry. Then, we demonstrate the importance of mixing
effects between spin-$0$ and spin-$1$ hadrons for the $\mu_{q}$ dependence of
physical quantities such as the diquark condensate and the quark-number
density. Besides, we present the predicted masses of spin-$1$ hadrons in cold
matter and possible novel phases triggered by mass-vanishing spin-$1$ hadrons
such as axialvector and vector condensed phases.

The predictions given by the present study on how the spin-$1$ hadrons have
their masses modified in cold matter are expected to be checked by future
lattice QC2D simulations. Ultimately, our comprehensive model, which allows us
to simultaneously describe the spin-$0$ and spin-$1$ hadrons without
incorporating the quark degrees of freedom explicitly even at high densities,
could serve as a guideline on how to use a hadronic model in cold matter. In
addition, as diquarks themselves are observable, QC2D possesses an advantage
over three-color QCD where diquark dynamics can be solely seen through, e.g.,
singly heavy baryons (SHBs) made of one heavy quark and one diquark Kawakami
and Harada (2018, 2019); Harada  _et al._ (2020); Kim  _et al._ (2020);
Kawakami  _et al._ (2020); Suenaga and Hosaka (2021, 2022); Suenaga and Oka
(2023); Takada  _et al._ (2023). The SHBs are now under intensive
investigation in accordance with the recent development of experimental
techniques. In this regard, our findings on the diquarks in QC2D would also
serve as good references for understanding the SHBs dynamics from chiral
symmetry.

This article is organized as follows. In Sec. II we introduce quark-bilinear
fields for spin-$0$ and spin-$1$ hadrons and construct an effective model
describing these hadrons based on the Pauli-Gürsey $SU(4)$ symmetry. After
explaining, in Sec. III, our procedure to fix model parameters for later
numerical calculations, we show, in Sec. IV, $\mu_{q}$ dependence of the mean
fields within the present model and, in Sec. V, our main results, namely, the
hadron mass spectrum at finite $\mu_{q}$. Besides, the chiral partner
structure for the spin-$1$ hadrons is demonstrated in Sec. VI. Sections VII
and VIII are devoted to discussions and conclusions, respectively.

##  II Model construction

In this section, we construct our effective model describing both the spin-$0$
and spin-$1$ hadrons based on the linear realization of the Pauli-Gürsey
$SU(4)$ symmetry.

###  II.1 Spin-0 hadron fields

For the purpose of constructing the effective Lagrangian, we first introduce a
useful building block of the spin-$0$ hadrons whose $SU(4)$ symmetry
properties are manifest, following the previous work Suenaga  _et al._ (2023).

In two-flavor QC2D, $SU(2)_{L}\times SU(2)_{R}$ chiral symmetry is extended to
the Pauli-Gürsey $SU(4)$ symmetry due to the pseudoreal property of
$SU(2)_{c}$ gauge group, as shown in Appendix A. The extended symmetry enables
us to treat mesons and diquark baryons in a unified way. Then, as shown in
Ref. Suenaga  _et al._ (2023), it is useful to introduce a $4\times 4$ matrix
$\Sigma$ corresponding to both the spin-$0$ mesons and diquarks baryons as
defined by

$\displaystyle\Sigma=\sum_{a=0}^{5}(\mathscr{S}^{a}+i\mathscr{P}^{a})X^{a}E\
.$ (1)

In this equation, $X^{a=0}=\frac{1}{2\sqrt{2}}{\bm{1}}_{4\times 4}$ and
$X^{a=1-5}$ are generators belonging to the Lie algebra of $SU(4)/Sp(4)$, the
expressions for which are given by Eq. (103) in Appendix B. Besides, $E$ is
the $4\times 4$ symplectic matrix defined by

$\displaystyle E=\left(\begin{array}[]{cc}0&{\bm{1}}_{f}\\\ -{\bm{1}}_{f}&0\\\
\end{array}\right)\ .$ (4)

In Eq. (1) $\mathscr{S}^{a}$ and $\mathscr{P}^{a}$ represent a set of the
spin-$0$ hadron fields as

$\displaystyle\sigma=\mathscr{S}^{0}\ ,\ \ a_{0}^{0}=\mathscr{S}^{3}\ ,\ \
a_{0}^{\pm}=\frac{\mathscr{S}^{1}\mp i\mathscr{S}^{2}}{\sqrt{2}}\ ,$
$\displaystyle\eta=\mathscr{P}^{0}\ ,\ \ \pi^{0}=\mathscr{P}^{3}\ ,\ \
\pi^{\pm}=\frac{\mathscr{P}^{1}\mp i\mathscr{P}^{2}}{\sqrt{2}}\ ,$
$\displaystyle B=\frac{\mathscr{S}^{5}-i\mathscr{S}^{4}}{\sqrt{2}}\ ,\ \
\bar{B}=\frac{\mathscr{S}^{5}+i\mathscr{S}^{4}}{\sqrt{2}}\ ,$ $\displaystyle
B^{\prime}=\frac{\mathscr{P}^{5}-i\mathscr{P}^{4}}{\sqrt{2}}\ ,\ \
\bar{B}^{\prime}=\frac{\mathscr{P}^{5}+i\mathscr{P}^{4}}{\sqrt{2}}\ ,$ (5)

where $\sigma$, $a_{0}$, $\eta$, and $\pi$ are mesons while $B$ and
$B^{\prime}$ ($\bar{B}$ and $\bar{B}^{\prime}$) are (anti)diquark baryons. The
quantum numbers carried by those hadrons are summarized in Table 1. With the
correspondence (5), the $4\times 4$ matrix $\Sigma$ reads

$\displaystyle\Sigma=\frac{1}{2}\left(\begin{array}[]{cccc}0&-B^{\prime}+iB&\frac{\sigma-i\eta+a^{0}-i\pi^{0}}{\sqrt{2}}&a^{+}-i\pi^{+}\\\
B^{\prime}-iB&0&a^{-}-i\pi^{-}&\frac{\sigma-i\eta-a^{0}+i\pi^{0}}{\sqrt{2}}\\\
-\frac{\sigma-i\eta+a^{0}-i\pi^{0}}{\sqrt{2}}&-a^{-}+i\pi^{-}&0&-\bar{B}^{\prime}+i\bar{B}\\\
-a^{+}+i\pi^{+}&-\frac{\sigma-i\eta-a^{0}+i\pi^{0}}{\sqrt{2}}&\bar{B}^{\prime}-i\bar{B}&0\\\
\end{array}\right)\ .$ (10)

In terms of the quark doublet operator $\psi=(u,d)^{T}$, the hadrons are
denoted by

$\displaystyle\sigma\sim\bar{\psi}\psi\ ,\ \
a_{0}^{\pm}\sim\frac{1}{\sqrt{2}}\bar{\psi}\tau_{f}^{\mp}\psi\ ,\ \
a_{0}^{0}\sim\bar{\psi}\tau_{f}^{3}\psi\ ,$
$\displaystyle\eta\sim\bar{\psi}i\gamma_{5}\psi\ ,\ \
\pi^{\pm}\sim\frac{1}{\sqrt{2}}\bar{\psi}i\gamma_{5}\tau_{f}^{\mp}\psi\ ,\ \
\pi^{0}\sim\bar{\psi}i\gamma_{5}\tau_{f}^{3}\psi\ ,$ $\displaystyle
B\sim-\frac{i}{\sqrt{2}}\psi^{T}{\cal
C}\gamma_{5}\tau_{c}^{2}\tau_{f}^{2}\psi\ ,\ \
B^{\prime}\sim-\frac{1}{\sqrt{2}}\psi^{T}{\cal C}\tau_{c}^{2}\tau_{f}^{2}\psi\
,$ $\displaystyle\bar{B}\sim-\frac{i}{\sqrt{2}}\psi^{\dagger}{\cal
C}\gamma_{5}\tau_{c}^{2}\tau_{f}^{2}\psi^{*}\ ,\ \
\bar{B}^{\prime}\sim\frac{1}{\sqrt{2}}\psi^{\dagger}{\cal
C}\tau_{c}^{2}\tau_{f}^{2}\psi^{*}\ ,$ (11)

where ${\cal C}=i\gamma^{2}\gamma^{0}$ is the charge-conjugation Dirac matrix
and $\tau_{f}^{\pm}=\tau_{f}^{1}\pm i\tau_{f}^{2}$. Using Eqs. (5) and (11),
one can see that, in terms of the four-component quark field $\Psi$ defined by
Eq. (82), the $4\times 4$ matrix $\Sigma$, Eq. (10), reads

$\displaystyle\Sigma_{ij}\sim\Psi_{j}^{T}\sigma^{2}\tau_{c}^{2}\Psi_{i}\ .$
(12)

Thus, under the $SU(4)$ transformation, $\Sigma$ transforms as

$\displaystyle\Sigma\to U\Sigma U^{T}$ (13)

with $U\in SU(4)$. This symmetry property plays a significant role in
constructing the effective Lagrangian describing the spin-$0$ hadrons.

Hadron | $J^{P}$ | Quark number | Isospin
---|---|---|---
$\sigma$ | $0^{+}$ | $0$ | $0$
$a_{0}$ | $0^{+}$ | $0$ | $1$
$\eta$ | $0^{-}$ | $0$ | $0$
$\pi$ | $0^{-}$ | $0$ | $1$
$B$ ($\bar{B}$) | $0^{+}$ |  $+2$ ($-2$) | $0$
$B^{\prime}$ ($\bar{B}^{\prime}$) | $0^{-}$ |  $+2$ ($-2$) | $0$
Table 1: Quantum numbers of the spin-$0$ hadrons.

Here, it is well known that the sigma field ($\sigma$) can acquire its mean-
field value: $\sigma_{0}\equiv\langle\sigma\rangle$, to mimic the chiral
condensate which results in the chiral symmetry breaking. Within such a mean-
field level, the matrix (10) is reduced to

$\displaystyle\Sigma\to\Sigma_{0}\equiv\frac{\sigma_{0}}{2\sqrt{2}}E\ ,$ (14)

where $E$ is the symplectic matrix, Eq. (4). In general, $\Sigma_{0}$ is not
invariant under $SU(4)$: $\Sigma_{0}\to U\Sigma_{0}U^{T}$. Only when the
element $U$ is generated by $h$ satisfying

$\displaystyle hEh^{T}=E\ ,$ (15)

however, $\Sigma_{0}$ turns out to be invariant. Equation (15) shows that the
spontaneous symmetry-breaking pattern triggered by the chiral condensate is
$SU(4)\to Sp(4)$ in QC2D.

Before moving on to the spin-$1$ hadrons, we comment on the properties of
$\mathscr{S}^{a}$ and $\mathscr{P}^{a}$. With the correspondence (5), for
instance, one can see that the pseudoscalar mesons ($0^{-}$) and scalar
(anti)diquaks ($0^{+}$), parities of which are opposite, are collectively
denoted by $\mathscr{P}^{a}$, i.e., these hadrons belong to the same multiplet
of the $SU(4)$ algebra. The difference of parities are understandable from
their intrinsic parities; the pseudoscalar mesons contain one quark and one
antiquark while the scalar (anti)diquarks contain two (anti)quarks. Also, from
this consideration it can be understood that those hadrons are the ground
states in the hadronic phase because of their $S$ wave nature in a quark-model
description. Likewise, the scalar mesons ($0^{+}$) and pseudoscalar
(anti)diquarks ($0^{-}$) are collectively represented by $\mathscr{S}^{a}$,
while they belong to the same multiplet. Those hadrons can be identified as
$P$-wave excited states.

Hadron | $J^{P}$ | Quark number | Isospin
---|---|---|---
$\omega$ | $1^{-}$ | $0$ | $0$
$\rho$ | $1^{-}$ | $0$ | $1$
$f_{1}$ | $1^{+}$ | $0$ | $0$
$a_{1}$ | $1^{+}$ | $0$ | $1$
$B_{S}$ ($\bar{B}_{S}$) | $1^{+}$ |  $+2$ ($-2$) | $1$
$B_{AS}$ ($\bar{B}_{AS}$) | $1^{-}$ |  $+2$ ($-2$) | $0$
Table 2: Quantum numbers of the spin-$1$ hadrons.

###  II.2 Spin-1 hadron fields

In Sec. II.1, we have introduced the $4\times 4$ matrix $\Sigma$ corresponding
to the spin-$0$ mesons and diquark baryons. Next, in this subsection we
consider another $4\times 4$ matrix $\Phi^{\mu}$, which is essential to
describe the spin-$1$ hadrons toward construction of our effective Lagrangian.

In two-flavor QC2D, the relevant low-lying spin-$1$ mesons and diquark baryons
are

$\displaystyle\omega^{\mu}\sim\bar{\psi}\gamma^{\mu}\psi\ ,\ \
f_{1}^{\mu}\sim\bar{\psi}\gamma_{5}\gamma^{\mu}\psi\ ,$
$\displaystyle\rho^{0,\mu}\sim\bar{\psi}\tau_{f}^{3}\gamma^{\mu}\psi\ ,\ \
\rho^{\pm,\mu}\sim\frac{1}{\sqrt{2}}\bar{\psi}\tau_{f}^{\mp}\gamma^{\mu}\psi\
,$ $\displaystyle
a_{1}^{0,\mu}\sim\bar{\psi}\tau_{f}^{3}\gamma_{5}\gamma^{\mu}\psi\ ,\ \
a_{1}^{\pm,\mu}\sim\frac{1}{\sqrt{2}}\bar{\psi}\tau_{f}^{\mp}\gamma_{5}\gamma^{\mu}\psi\
,$ (16)

and

$\displaystyle B_{S}^{I_{z}=0,\mu}\sim-\frac{i}{\sqrt{2}}\psi^{T}{\cal
C}\gamma^{\mu}\tau_{c}^{2}\tau_{f}^{1}\psi$ $\displaystyle B_{S}^{I_{z}=\pm
1,\mu}\sim-\frac{i}{2}\psi^{T}{\cal
C}\gamma^{\mu}\tau_{c}^{2}({\bm{1}}_{f}\pm\tau_{f}^{3})\psi\ ,$ $\displaystyle
B_{AS}^{\mu}\sim-\frac{1}{\sqrt{2}}\psi^{T}{\cal
C}\gamma_{5}\gamma^{\mu}\tau_{c}^{2}\tau_{f}^{2}\psi$
$\displaystyle\bar{B}_{S}^{I_{z}=0,\mu}=(B_{S}^{I_{z}=0,\mu})^{\dagger}\ ,\ \
\bar{B}_{S}^{I_{z}=\pm 1,\mu}=(B_{S}^{I_{z}=\mp 1,\mu})^{\dagger}\ ,$
$\displaystyle\bar{B}_{AS}^{\mu}=(B_{AS}^{\mu})^{\dagger}\ ,$ (17)

respectively, in terms of the quark operator. The quantum numbers for those
states are summarized in Table 2. Here, for the spin-$1$ diquark baryons, the
subscripts $S$ and $AS$ denote the “symmetric” and “anti-symmetric” structure
of the flavor contents, respectively, that is, the former is iso-triplet while
the latter is iso-singlet. Besides, the superscript $I_{z}=0,\pm 1$ for
$B_{S}$ ($\bar{B}_{S}$) stands for the eigenvalues of the isospin “$z$
components.” We note that the axialvector and vector (anti)diquark baryons are
iso-triplet and iso-singlet, respectively, as dictated by the Pauli
principle.111In three-color QCD, the diquarks are translated into SHBs.
Indeed, the axialvector and vector diquarks, $B_{S}$ and $B_{AS}$, would
correspond to $\Sigma_{c}(2455)$ [and its heavy-quark spin partner
$\Sigma_{c}(2520)$] and $\Lambda_{c}(2595)$ [and $\Lambda_{c}(2625)$],
respectively.

For spin-$1$ hadrons, it would be convenient to introduce a $4\times 4$ matrix
$\Phi^{\mu}$ by the following assignment:

$\displaystyle\Phi^{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{cccc}\frac{\omega+\rho^{0}-(f_{1}+a_{1}^{0})}{\sqrt{2}}&\rho^{+}-a_{1}^{+}&\sqrt{2}B_{S}^{I_{z}=+1}&B_{S}^{I_{z}=0}-B_{AS}\\\
\rho^{-}-a_{1}^{-}&\frac{\omega-\rho^{0}-(f_{1}-a_{1}^{0})}{\sqrt{2}}&B_{S}^{I_{z}=0}+B_{AS}&\sqrt{2}B_{S}^{I_{z}=-1}\\\
\sqrt{2}\bar{B}_{S}^{I_{z}=-1}&\bar{B}_{S}^{I_{z}=0}+\bar{B}_{AS}&-\frac{\omega+\rho^{0}+f_{1}+a_{1}^{0}}{\sqrt{2}}&-(\rho^{-}+a_{1}^{-})\\\
\bar{B}_{S}^{I_{z}=0}-\bar{B}_{AS}&\sqrt{2}\bar{B}_{S}^{I_{z}=+1}&-(\rho^{+}+a_{1}^{+})&-\frac{\omega-\rho^{0}+f_{1}-a_{1}^{0}}{\sqrt{2}}\\\
\end{array}\right)^{\mu}\ .$ (22)

In fact, in terms of the four-component quark field $\Psi$, Eq. (82), and the
interpolating fields, Eqs. (16) and (17), $\Phi^{\mu}$ can be simply rewritten
in the form of

$\displaystyle\Phi_{ij}\sim\Psi_{j}^{\dagger}\sigma^{\mu}\Psi_{i}\ ,$ (23)

where $\sigma^{\mu}=({\bm{1}},\sigma^{i})$ ($\sigma^{i}$ is the Pauli matrix
in the two-component spinor space). Thus, the matrix (22) fulfills the
following homogeneous $SU(4)$ transformation law:

$\displaystyle\Phi^{\mu}\to U\Phi^{\mu}U^{\dagger}\ ,$ (24)

which allows us to construct an effective Lagrangian straightforwardly. For
this reason, in what follows we employ $\Phi^{\mu}$ as a building block of the
spin-$1$ hadrons. We note that, with the help of $U(4)$ generators $X^{a}$ and
$S^{i}$, Eqs. (103) and (97), the spin-$1$ hadron matrix $\Phi^{\mu}$ can be
expressed as

$\displaystyle\Phi^{\mu}=\left(\sum_{i=1}^{10}V^{i}S^{i}-\sum_{a=0}^{5}V^{\prime
a}X^{a}\right)^{\mu}\ ,$ (25)

where

$\displaystyle\omega=V^{0}\ ,\ \ \rho^{\pm}=\frac{V^{1}\mp iV^{2}}{\sqrt{2}}\
,\ \ \rho^{0}=V^{3}\ ,$ $\displaystyle f_{1}=V^{\prime 0}\ ,\ \
a_{1}^{\pm}=\frac{V^{\prime 1}\mp iV^{\prime 2}}{\sqrt{2}}\ ,\ \
a_{1}^{0}=V^{\prime 3}\ ,$ $\displaystyle
B_{S}^{I_{z}=0}=\frac{V^{9}+iV^{10}}{\sqrt{2}}\ ,\ \
\bar{B}_{S}^{I_{z}=0}=\frac{V^{9}-iV^{10}}{\sqrt{2}}\ ,$ $\displaystyle
B_{S}^{I_{z}=\pm 1}=\frac{(V^{5}+iV^{6})\pm(V^{7}+iV^{8})}{2}\ ,$
$\displaystyle\bar{B}_{S}^{I_{z}=\pm
1}=\frac{(V^{5}-iV^{6})\mp(V^{7}-iV^{8})}{2}\ ,$ $\displaystyle
B_{AS}=\frac{V^{\prime 5}-iV^{\prime 4}}{\sqrt{2}}\ ,\ \
\bar{B}_{AS}=\frac{V^{\prime 5}+iV^{\prime 4}}{\sqrt{2}}\ .$ (26)

The reduced form (25) is useful to see symmetry properties of the spin-$1$
hadrons. For instance, from Eq. (25) one can see that the vector mesons
($1^{-}$) and axialvextor (anti)diquarks ($1^{+}$), parities of which are
opposite, belong to the $Sp(4)$ algebra proportional to $S^{i}$ and hence to
the same multiplet. Likewise, the axialvector mesons ($1^{+}$) and vector
(anti)diquarks ($1^{-}$) are the elements of the remaining algebra. The
difference of parities between the mesons and (anti)diquarks in a single
multiplet can be understood from their intrinsic parities as in the case of
the spin-$0$ hadrons. We note that $V^{i}$ and $V^{\prime a}$ are the ground
and excited states, respectively, in the hadronic phase, since the former and
latter are identifiable as $S$-wave and $P$-wave states, respectively.

###  II.3 Extended linear sigma model

In Sec. II.1 and Sec. II.2, the $4\times 4$ matrices corresponding to the
spin-$0$ and spin-$1$ hadrons in two-flavor QC2D, $\Sigma$ and $\Phi^{\mu}$,
have been introduced. In this subsection, by making the most of these building
blocks, we construct an effective Lagrangian to describe interactions among
those hadrons.

The $SU(4)$ transformation laws for $\Sigma$ and $\Phi^{\mu}$ are given by
Eqs. (13) and (24). Toward construction of the effective Lagrangian, in
addition to the $SU(4)$ properties it is necessary to examine the discrete
symmetries: parity and charge conjugation invariance. Those discrete
transformation laws of $\Sigma$ and $\Phi^{\mu}$ can be read off from the
interpolating fields (12) and (23). From Eq. (82) the four-component quark
field $\Psi$ transforms as

$\displaystyle\Psi(x)\overset{\cal P}{\to}\
\Omega\tau_{c}^{2}\sigma^{2}{\Psi}^{*}(x_{P})\ ,\ \ \Psi\overset{\cal
C}{\to}iE^{T}\tau^{2}\Psi\ ,$ (27)

under parity and charge conjugation with $x_{P}=(x^{0},-{\bm{x}})$, where $E$
is the symplectic matrix (4) and $\Omega$ is defined by

$\displaystyle\Omega=\left(\begin{array}[]{cc}0&{\bm{1}}_{f}\\\
{\bm{1}}_{f}&0\\\ \end{array}\right)\ .$ (30)

Thus, the resultant transformation laws of $\Sigma$ and $\Phi^{\mu}$ read

$\displaystyle\Sigma(x)\overset{\cal
P}{\to}\Omega\Sigma^{\dagger}(x_{P})\Omega$ ,
$\displaystyle\Sigma\overset{\cal C}{\to}E^{T}\Sigma E\ ,$
$\displaystyle\Phi^{\mu}(x)\overset{\cal
P}{\to}-\Omega\Phi^{T}_{\mu}(x_{P})\Omega$ ,
$\displaystyle\Phi^{\mu}\overset{\cal C}{\to}E^{T}\Phi^{\mu}E\ .$ (31)

Using the transformation laws given by Eqs. (13), (24) and (31), one can
construct the following effective Lagrangian in such a way as to preserve the
Pauli-Gürsey $SU(4)$ symmetry as well as parity and charge-conjugation
invariance:

$\displaystyle{\cal L}_{\rm eLSM}$ $\displaystyle=$ $\displaystyle{\rm
tr}[D_{\mu}\Sigma^{\dagger}D^{\mu}\Sigma]-m_{0}^{2}{\rm
tr}[\Sigma^{\dagger}\Sigma]-\lambda_{1}\big{(}{\rm
tr}[\Sigma^{\dagger}\Sigma]\big{)}^{2}-\lambda_{2}{\rm
tr}[(\Sigma^{\dagger}\Sigma)^{2}]+{\rm
tr}[H^{\dagger}\Sigma+\Sigma^{\dagger}H]+c({\rm det}\Sigma+{\rm
det}\Sigma^{\dagger})$ $\displaystyle-$ $\displaystyle\frac{1}{2}{\rm
tr}[\Phi_{\mu\nu}\Phi^{\mu\nu}]+m_{1}^{2}{\rm
tr}[\Phi_{\mu}\Phi^{\mu}]+ig_{3}{\rm
tr}\big{[}\Phi_{\mu\nu}[\Phi^{\mu},\Phi^{\nu}]\big{]}+h_{1}{\rm
tr}[\Sigma^{\dagger}\Sigma]{\rm tr}[\Phi_{\mu}\Phi^{\mu}]+h_{2}{\rm
tr}[\Sigma\Sigma^{\dagger}\Phi_{\mu}\Phi^{\mu}]$ $\displaystyle+$
$\displaystyle h_{3}{\rm
tr}[\Phi_{\mu}^{T}\Sigma^{\dagger}\Phi^{\mu}\Sigma]+g_{4}{\rm
tr}[\Phi_{\mu}\Phi_{\nu}\Phi^{\mu}\Phi^{\nu}]+g_{5}{\rm
tr}[\Phi_{\mu}\Phi^{\mu}\Phi_{\nu}\Phi^{\nu}]+g_{6}{\rm
tr}[\Phi_{\mu}\Phi^{\mu}]{\rm tr}[\Phi_{\nu}\Phi^{\nu}]+g_{7}{\rm
tr}[\Phi_{\mu}\Phi_{\nu}]{\rm tr}[\Phi^{\mu}\Phi^{\nu}]\ .$

In this Lagrangian,

$\displaystyle\Phi_{\mu\nu}$ $\displaystyle\equiv$ $\displaystyle
D_{\mu}\Phi_{\nu}-D_{\nu}\Phi_{\mu}$ (33)

is the field strength of $\Phi^{\mu}$, and the covariant derivatives read

$\displaystyle D_{\mu}\Sigma\equiv\partial_{\mu}\Sigma-iG_{\mu}\Sigma-i\Sigma
G^{T}_{\mu}-ig_{1}\Phi_{\mu}\Sigma-ig_{2}\Sigma\Phi_{\mu}^{T}\ ,$
$\displaystyle
D_{\mu}\Phi_{\nu}\equiv\partial_{\mu}\Phi_{\nu}-i[G_{\mu},\Phi_{\nu}]\ ,$ (34)

where $G_{\mu}$ is an external field, the transformation law of which is
$G_{\mu}\to UG_{\mu}U^{\dagger}-i\partial_{\mu}UU^{\dagger}$. In the present
analysis, $G_{\mu}$ is replaced by the quark chemical potential $\mu_{q}$ to
access a finite-density system in a systematic way. The replacement can be
performed by

$\displaystyle G_{\mu}\to\mu_{q}\delta_{\mu 0}J$ (35)

with

$\displaystyle{J}\equiv\left(\begin{array}[]{cc}{\bm{1}}_{f}&0\\\
0&-{\bm{1}}_{f}\\\ \end{array}\right)\ ,$ (38)

[see Eq. (86)]. Besides, in constructing the model, we have included
contributions allowed by the relevant symmetries up to fourth order in
$\Sigma^{(\dagger)}$ and $\Phi^{\mu}$. We note that the $4\times 4$ matrix $H$
in Eq. (LABEL:ELSMLag) is responsible for explicit breaking of the Pauli-
Gürsey $SU(4)$ symmetry to yield the finite pion mass, which takes the form of

$\displaystyle H=h_{q}E\ .$ (39)

The Lagrangian (LABEL:ELSMLag) can be understood as an extension of the
previous LSM established in Ref. Suenaga  _et al._ (2023) where only spin-$0$
hadrons are treated. For this reason we call this model the extended linear
sigma model (eLSM). We note that this eLSM is the QC2D version of the one
invented for three-color QCD by the Frankfurt group Parganlija  _et al._
(2013) and applied to the case of finite density by one of the present authors
Suenaga and Lakaschus (2020).

In the eLSM Lagrangian (LABEL:ELSMLag), with the help of the antisymmetric
property: $\Sigma^{T}=-\Sigma$, the kinetic term for $\Sigma$ can be expanded
as

$\displaystyle{\rm tr}[D_{\mu}\Sigma^{\dagger}D^{\mu}\Sigma]$ $\displaystyle=$
$\displaystyle{\rm
tr}[\partial_{\mu}\Sigma^{\dagger}\partial^{\mu}\Sigma]+(g_{1}+g_{2}){\rm
tr}[\Sigma\Sigma^{\dagger}(\Phi_{\mu}G^{\mu}$ (40) $\displaystyle+$
$\displaystyle G_{\mu}\Phi^{\mu})]+2(g_{1}+g_{2}){\rm
tr}[\Phi_{\mu}^{T}\Sigma^{\dagger}G^{\mu}\Sigma]$ $\displaystyle+$
$\displaystyle i(g_{1}+g_{2}){\rm
tr}[\Phi_{\mu}(\partial^{\mu}\Sigma\Sigma^{\dagger}-\Sigma\partial^{\mu}\Sigma^{\dagger})]$
$\displaystyle+$ $\displaystyle(g_{1}^{2}+g_{2}^{2}){\rm
tr}[\Sigma\Sigma^{\dagger}\Phi_{\mu}\Phi^{\mu}]$ $\displaystyle+$
$\displaystyle 2g_{1}g_{2}{\rm
tr}[\Phi_{\mu}^{T}\Sigma^{\dagger}\Phi^{\mu}\Sigma]\ .$

The trace structure of the last two pieces in the right-hand side (RHS) is
equivalent to the $h_{2}$ and $h_{3}$ terms in Eq. (LABEL:ELSMLag), while the
remaining interactions are proportionally dependent on the combination of
$g_{1}+g_{2}$ alone. Hence, the four parameters $g_{1}$, $g_{2}$, $h_{2}$, and
$h_{3}$ can be absorbed into three new combinations

$\displaystyle C_{1}$ $\displaystyle\equiv$ $\displaystyle g_{1}+g_{2}\ ,$
$\displaystyle C_{2}$ $\displaystyle\equiv$ $\displaystyle
g_{1}^{2}+g_{2}^{2}+h_{2}\ ,$ $\displaystyle C_{3}$ $\displaystyle\equiv$
$\displaystyle 2g_{1}g_{2}+h_{3}\ .$ (41)

When the spectrum includes spin-$1$ hadrons, it is well known that the Zweig
rule, i.e., the large $N_{c}$ suppression of the interactions, works
phenomenologically. In other words, diagrams that are not linked by a single
quark line are not expected to play significant roles as far as the spin-$1$
hadrons are concerned. In the following analysis, therefore, we will leave
only terms including a single trace for $\Phi^{\mu}$’s, which allows us to
work with the following reduced eLSM:

$\displaystyle{\cal L}^{\rm red.}_{\rm eLSM}$ $\displaystyle=$
$\displaystyle{\rm tr}[D_{\mu}\Sigma^{\dagger}D^{\mu}\Sigma]-m_{0}^{2}{\rm
tr}[\Sigma^{\dagger}\Sigma]-\lambda_{1}\big{(}{\rm
tr}[\Sigma^{\dagger}\Sigma]\big{)}^{2}$ (42) $\displaystyle-$
$\displaystyle\lambda_{2}{\rm tr}[(\Sigma^{\dagger}\Sigma)^{2}]+{\rm
tr}[H^{\dagger}\Sigma+\Sigma^{\dagger}H]$ $\displaystyle+$ $\displaystyle
c({\rm det}\Sigma+{\rm det}\Sigma^{\dagger})-\frac{1}{2}{\rm
tr}[\Phi_{\mu\nu}\Phi^{\mu\nu}]$ $\displaystyle+$ $\displaystyle m_{1}^{2}{\rm
tr}[\Phi_{\mu}\Phi^{\mu}]+ig_{3}{\rm
tr}\big{[}\Phi_{\mu\nu}[\Phi^{\mu},\Phi^{\nu}]\big{]}$ $\displaystyle+$
$\displaystyle h_{2}{\rm
tr}[\Sigma\Sigma^{\dagger}\Phi_{\mu}\Phi^{\mu}]+h_{3}{\rm
tr}[\Phi_{\mu}^{T}\Sigma^{\dagger}\Phi^{\mu}\Sigma]$ $\displaystyle+$
$\displaystyle g_{4}{\rm
tr}[\Phi_{\mu}\Phi_{\nu}\Phi^{\mu}\Phi^{\nu}]+g_{5}{\rm
tr}[\Phi_{\mu}\Phi^{\mu}\Phi_{\nu}\Phi^{\nu}]\ .$

As we will see, the masses of the hadrons can be read off from quadratic terms
of the corresponding fields on top of the appropriate mean fields.

##  III Inputs

In Sec. II.3 we have constructed the eLSM to describe both the spin-$0$ and
spin-$1$ hadrons at arbitrary $\mu_{q}$. In this section, before numerical
investigation of $\mu_{q}$ dependence of the mean fields and the hadron
masses, we explain our procedure to determine various parameters of the
reduced eLSM.

The reduced eLSM (42) includes $12$ parameters: $m_{0}^{2}$, $m_{1}^{2}$,
$\lambda_{1}$, $\lambda_{2}$, $h_{q}$, $c$, $g_{3}$, $g_{4}$, $g_{5}$,
$C_{1}$, $C_{2}$, and $C_{3}$. In this exploratory study, we try to reduce the
number of the parameters as much as possible to avoid unnecessary complexities
in the following numerical analysis. First, as discussed in Ref. Suenaga  _et
al._ (2023), contributions from the $\lambda_{1}$ and $c$ terms, which only
affect the mass spectrum of the spin-$0$ hadrons, are expected to be small
from the $N_{c}$ counting. Thus, we take $\lambda_{1}=c=0$. The main aim of
the present paper is to delineate the behavior of the spin-$1$ hadrons at
finite $\mu_{q}$, so that this simplification does not affect the following
arguments considerably. Next, as for the couplings among the spin-$1$ hadrons,
there is no a priori way to determine all of the parameters due to the
currently limited lattice data. When we derive the present eLSM from the
${\cal O}(p^{2})$ hidden-local-symmetry (HLS) Lagrangian Harada  _et al._
(2010), however, it is expected that the interactions among the spin-$1$
hadrons satisfy the following “gauge-principle parametrization:”

$\displaystyle g_{3}=g_{\Phi}\ ,\ \ g_{4}=-g_{5}=g_{\Phi}^{2}\ .$ (43)

From this consideration, we assume Eq. (43) to employ only a single parameter
$g_{\Phi}$ instead of $g_{3}$, $g_{4}$, and $g_{5}$. In addition, one can see
from Eq. (41) that the roles of $C_{1}$, $C_{2}$, and $C_{3}$ are essentially
the same; these three parameters control the interaction strength between the
spin-$0$ and spin-$1$ hadrons. For this reason we assume

$\displaystyle C_{1}=C_{2}\equiv C$ (44)

for simplicity. In Eq. (44) we have not included $C_{3}$ to define the common
coupling, since $C_{3}$ is uniquely fixed by inputs as will be explained
below.

Now the number of the model parameters is reduced to seven: $m_{0}^{2}$,
$m_{1}^{2}$, $\lambda_{2}$, $h_{q}$, $g_{\Phi}$, $C$, and $C_{3}$. The present
study is devoted to unveiling properties of the spin-$1$ hadrons, and hence,
we take $C$ and $g_{\Phi}$, which control couplings related to the spin-$1$
hadrons, as free parameters. In order to determine the remaining five
parameters, as the first four inputs, we use the masses of $\pi$,
$B^{\prime}(\bar{B}^{\prime})$, $\rho$, and $a_{1}$ at vanishing $\mu_{q}$
simulated in Refs. Murakami  _et al._ (2023b, ). The input mass values
read222The physical scale is fixed such that the pseudo-critical temperature
of the chiral phase transition reads $T_{c}=200$ MeV at $\mu_{q}=0$ Iida  _et
al._ (2021).

$\displaystyle m_{\pi}^{(\rm H)}=738\,{\rm MeV}\ ,\ \
m_{B^{\prime}(\bar{B}^{\prime})}^{(\rm H)}\big{|}_{\mu_{q}=0}=1611\,{\rm MeV}\
,$ $\displaystyle m^{(\rm H)}_{\rho}=908\,{\rm MeV}\ ,\ \
m_{a_{1}}^{\rm(H)}=1614\,{\rm MeV}\ .$ (45)

As for the last input, following Ref. Suenaga  _et al._ (2023) we take the
mean field value of $\sigma$ as

$\displaystyle\sigma_{0}^{\rm(H)}=250\,{\rm MeV}\ ,$ (46)

which gives a typical strength of the chiral symmetry breaking. Here, in Eqs.
(45) and (46), the superscript $({\rm H})$ is attached to emphasize that the
quantities are defined in the hadronic phase where no diquark condensates
emerge. Using analytic expressions for the hadron masses derived in Appendix
D, together with Eqs. (45) and (46), the remaining parameters can finally be
determined as

$\displaystyle m_{0}^{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left[\big{(}m_{a_{0}}^{\rm(H)}\big{)}^{2}-3Z_{\pi}^{-2}\big{(}m_{\pi}^{\rm(H)}\big{)}^{2}\right]\
,$ $\displaystyle m_{1}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\big{(}m_{\rho}^{\rm(H)}\big{)}^{2}+\big{(}m_{a_{1}}^{\rm(H)}\big{)}^{2}\right]-\frac{C}{8}\big{(}\sigma_{0}^{(\rm
H)}\big{)}^{2}\ ,$ $\displaystyle\lambda_{2}$ $\displaystyle=$
$\displaystyle\frac{2}{\big{(}\sigma_{0}^{\rm(H)}\big{)}^{2}}\left[\big{(}m_{a_{0}}^{\rm(H)}\big{)}^{2}-Z_{\pi}^{-2}\big{(}m_{\pi}^{\rm(H)}\big{)}^{2}\right]\
,$ $\displaystyle h_{q}$ $\displaystyle=$ $\displaystyle\frac{\sigma_{0}^{(\rm
H)}}{2\sqrt{2}}Z_{\pi}^{-2}\big{(}m_{\pi}^{(\rm H)}\big{)}^{2}\ ,$
$\displaystyle C_{3}$ $\displaystyle=$
$\displaystyle\frac{4}{\big{(}\sigma_{0}^{(\rm
H)}\big{)}^{2}}\left[\big{(}m_{a_{1}}^{\rm(H)}\big{)}^{2}-\big{(}m_{\rho}^{\rm(H)}\big{)}^{2}\right]$
(47)

for a given $C$, where the renormalization factor $Z_{\pi}$ is of the form

$\displaystyle Z_{\pi}=\left(1-\frac{C^{2}\big{(}\sigma_{0}^{(\rm
H)}\big{)}^{2}}{8\big{(}m_{a_{1}}^{(\rm H)}\big{)}^{2}}\right)^{-1/2}\ .$ (48)

We note that the analytic expression for $h_{q}$, which links the magnitude of
explicit breaking of chiral symmetry to the pion mass, can be derived from a
stationary condition of the effective potential with respect to
$\sigma_{0}^{(\rm H)}$.

In the following analysis, we will regard $g_{\Phi}$ and $C$ as free
parameters to explore how the mean fields as well as hadron masses behave at
finite $\mu_{q}$. Recall that $g_{\Phi}$ is the coupling constant that
controls the interaction strength among only spin-$1$ hadrons. Meanwhile, $C$
is particularly responsible for the transition between the spin-$0$ and
spin-$1$ hadrons through derivative couplings, as indicated by Eq. (40). Thus,
$C$ can be regarded as a parameter that measures the magnitude of the spin-$0$
and spin-$1$ mixing effect. For instance, in the hadronic phase, nonzero $C$
induces $\pi$ \- $a_{1}$ mixing and $\eta$ \- $f_{1}$ mixing, which can be
captured by the renormalization constants, $Z_{\pi}$ and $Z_{\eta}$, as
derived in Appendix D.

Figure 1: $\mu_{q}$ dependence of the mean fields: $\sigma_{0}$, $\Delta$,
$\bar{\omega}$, and $\bar{V}$, with $C=12$. The value indicated by the arrow
represents the asymptotic constant value of $\Delta$: $\Delta\sim
0.34m_{1}=1.74\sigma_{0}^{(\rm H)}$.  Figure 2: $\mu_{q}$ dependence of
$\sigma_{0}$ and $\Delta$ (top) and of the quark-number density
$\tilde{n}_{q}$ (bottom) with and without spin-$0$ and spin-$1$ mixing, $C=12$
and $C=0$.

##  IV Mean fields

In this section, employing a mean-field approximation, we numerically explore
$\mu_{q}$ dependence of the mean fields at zero temperature from the reduced
eLSM (42) with the inputs presented in Sec. III.

At finite $\mu_{q}$, not only the sigma meson $\sigma$ but also the
(anti)diquark baryon $B$ ($\bar{B}$) can acquire a nonzero mean field value,
resulting in the appearance of the baryon superfluid phase Kogut  _et al._
(1999, 2000). Following Ref. Suenaga  _et al._ (2023) we take
$\Delta\equiv\langle B^{5}\rangle$ to express the mean field of the
(anti)diquark. In addition to those spin-$0$ hadrons, violation of the Lorentz
invariance yields a mean field of the $\omega$ meson Walecka (1974). Assuming
the parity invariance, only the time component of $\omega$ can have a nonzero
mean-field value: $\bar{\omega}\equiv\langle\omega_{\mu=0}\rangle$.
Furthermore, in the baryon superfluid phase, one can expect that the $\omega$
meson mixes with the vector (anti)diquark $B_{AS}$ ($\bar{B}_{AS}$) due to the
baryon-number violation. Thus, these diquarks are also capable of acquiring
nonzero mean-field values. When the phase of $\Delta$ is chosen according to
$\Delta=\langle B^{5}\rangle$, only $\bar{V}\equiv\langle V^{\prime
4}_{\mu=0}\rangle$ becomes nonzero.333In this phase choice, one can indeed
prove that $\langle V^{\prime 5}_{\mu=0}\rangle$ must be always zero by
solving the stationary conditions explicitly. Inversely, if we choose the
phase of $\Delta$ such that $\langle B^{4}\rangle\neq 0$ but $\langle
B^{5}\rangle=0$, then $\langle V^{\prime 4}_{\mu=0}\rangle=0$ but $\langle
V_{\mu=0}^{\prime 5}\rangle\neq 0$ is obtained for $\bar{V}$. To summarize, in
the present analysis we take into account the following four mean fields
without loss of generality:

$\displaystyle\sigma_{0}=\langle\sigma\rangle\ ,\ \ \Delta=\langle
B^{5}\rangle\ ,$ $\displaystyle\bar{\omega}=\langle\omega_{\mu=0}\rangle\ ,\ \
\bar{V}\equiv\langle V^{\prime 4}_{\mu=0}\rangle\ .$ (49)

The $\mu_{q}$ dependence of these mean fields can be determined by solving the
corresponding stationary conditions.

Depicted in Fig. 1 is the resultant $\mu_{q}$ dependence of the mean fields:
$\sigma_{0}$, $\Delta$, $\bar{\omega}$, and $\bar{V}$, normalized by
$\sigma_{0}^{(\rm H)}$. In obtaining this plot, the strength of the spin-$0$
and spin-$1$ mixing effect is chosen as $C=12$. As shown in Sec. V.2, this
value leads to a slight reduction of the $\rho$ mass in the superfluid phase,
a feature consistent with the lattice data Murakami  _et al._ (2023b, ). We
note that the stationary conditions hold independently of $g_{\Phi}$, so that
the resultant $\mu_{q}$ dependence of the mean fields is not affected by
$g_{\Phi}$. Figure 1 implies that the baryon superfluid phase is triggered by
the onset of nonzero $\Delta$ at a critical chemical potential
$\mu_{q}=\mu_{q}^{\rm cr}$ with

$\displaystyle\mu_{q}^{\rm cr}\equiv m_{\pi}^{(\rm H)}/2\ ,$ (50)

which is irrespective of whether or not the spin-$1$ hadrons are present. This
critical value is universal in the sense that it is also derived from chiral
effective models involving only spin-$0$ hadrons Kogut  _et al._ (1999, 2000);
Ratti and Weise (2004); Suenaga  _et al._ (2023). At the same time as the
onset, both $\bar{V}$ and $\bar{\omega}$ begin to acquire nonzero values.
Detailed analyses in the vicinity of the phase transition will be provided in
Sec. VII.1. At asymptotically high $\mu_{q}$, $\sigma_{0}$ and $\bar{V}$
vanish. Meanwhile, $\Delta$ converges to a constant (see the arrow in Fig. 1)
whose value can be evaluated as $\Delta/\sigma_{0}^{(\rm H)}\approx
0.34\big{(}m_{1}/\sigma_{0}^{(\rm H)}\big{)}=1.74$ for the present values of
$C$ and $C_{3}$, whereas $\bar{\omega}$ grows in the negative direction with a
power of $\mu_{q}$.

Here, we note that the onsets of nonzero $\bar{\omega}$ and $\bar{V}$ do not
necessarily coincide with the critical chemical potential (50), since those
mean fields are unphysical, in other words, they are gauge-dependent
quantities within the gauge-field description. In fact, the mean field of
$\omega$ was found to be proportional to $\mu_{q}$ in the hadronic phase
within the HLS formalism Harada  _et al._ (2010).

Figure 3: Schematic picture of corrections to the hadron masses from various
mixings.

The converging behavior of $\Delta$ is distinct from the result within the
conventional LSM in the absence of spin-$1$ hadrons Suenaga  _et al._ (2023).
To take a closer look at this difference, we depict the $\mu_{q}$ dependence
of $\sigma_{0}$ and $\Delta$ with and without the spin-$0$ and spin-$1$ mixing
effect, $C=12$ and $C=0$, in the top panel of Fig. 2. The figure indicates
that $\sigma_{0}$ does not have its $\mu_{q}$ dependence corrected
considerably by the mixing while $\Delta$ has its $\mu_{q}$ dependence
significantly modified by the mixing; the asymptotically converging behavior
of $\Delta$ is induced only when the spin-$0$ and spin-$1$ mixing takes
effect. In fact, the asymptotic constant value of $\Delta$ with the mixing
effect is essentially determined by the bare mass of the spin-$1$ hadrons,
$m_{1}$, together with a factor stemming from the mixing strength $C$ and
$C_{3}$. The diverging behavior of $\Delta$ in the absence of the mixing is
lost, but instead the negatively diverging growth of $\bar{\omega}$ appears as
indicated in Fig. 1. We note that spin-$1$ mean fields $\bar{\omega}$ and
$\bar{V}$ are always zero when $C=0$:

$\displaystyle\bar{\omega}=0\ \ {\rm and}\ \ \bar{V}=0\ \ \ ({\rm at\ any}\
\mu_{q}\ {\rm for}\ C=0)\ ,$ (51)

as will be argued in Sec. VII.1. We also note that, when we take $C$ to be
negative, the signs of the induced $\bar{\omega}$ and $\bar{V}$ in the
superfluid phase become positive and negative, respectively.

The significant corrections due to the spin-$0$ and spin-$1$ mixing effect are
also reflected by $\mu_{q}$ dependence of the quark-number density as
displayed in the bottom panel of Fig. 2. In this figure we have plotted the
normalized density

$\displaystyle\tilde{n}_{q}=\frac{n_{q}}{16f_{\pi}^{2}m_{\pi}^{(\rm H)}}\ ,$
(52)

where $n_{q}$ is the ordinary quark-number density given by

$\displaystyle n_{q}$ $\displaystyle=$ $\displaystyle\frac{\partial{\cal
L}^{\rm red.}_{\rm eLSM}}{\partial\mu_{q}}\Big{|}_{\rm mean\,field}$ (53)
$\displaystyle=$ $\displaystyle
4\Delta^{2}\mu_{q}+\frac{C\Delta}{\sqrt{2}}(\Delta\bar{\omega}-\bar{V}\sigma_{0})$

with the mean fields (49), and $f_{\pi}=\sigma_{0}^{\rm(H)}/\sqrt{2}$ is the
pion decay constant. The figure shows that the asymptotic growth that is
proportional to $\mu_{q}^{3}$ as can be derived in the absence of the mixing
is changed into the $m_{1}^{2}\mu_{q}$ dependence due to the mixing. As a
result, increment in the density gets milder. It should be noted that the
Silver-Blaze property for the quark-number density is obvious since $n_{q}$ is
proportional to $\Delta$.444Although the present eLSM predicts
$n_{q}\propto\mu_{q}$ for larger $\mu_{q}$, at some point, such a hadronic
description would be violated and quark matter would appear, which leads
eventually to $n_{q}\propto\mu_{q}^{3}$.

Before moving on to evaluation of the hadron masses at finite $\mu_{q}$, we
give comments on roles of the spin-$1$ mean fields $\bar{\omega}$ and
$\bar{V}$ in controlling the onset of the baryon superfluid phase. As
indicated in Fig. 1, the superfluid phase emerges once $\mu_{q}$ reaches the
critical chemical potential (50), which is exactly what lattice simulations
suggest Hands  _et al._ (2001); Braguta  _et al._ (2016); Iida  _et al._
(2020). Note, however, that this is only when we include all the four mean
fields: $\sigma_{0}$, $\Delta$, $\bar{\omega}$, and $\bar{V}$. If any of the
mean fields were dropped, the critical chemical potential would not coincide
with Eq. (50) in the presence of the spin-$0$ and spin-$1$ mixing effect. We
have checked this property by choosing various parameter sets.555In Sec.
VII.1, we, indeed, analytically prove that Eq. (50) holds as long as all of
$\sigma_{0}$, $\Delta$, $\bar{\omega}$, and $\bar{V}$ are included based on a
certain assumption on their critical behaviors. In fact, if $\bar{\omega}$ or
$\bar{V}$ is neglected while keeping $C=12$, then the critical chemical
potential is found to change to approximately $0.379m_{\pi}^{(\rm H)}$, which
is lower than $\mu_{q}^{\rm cr}=m_{\pi}^{\rm(H)}/2$ suggested by lattice
simulations. Accordingly, one can show that the pion mass in the superfluid
phase is not given by $m_{\pi}=2\mu_{q}$ that other chiral effective models
commonly predict Kogut  _et al._ (1999, 2000); Ratti and Weise (2004); Suenaga
_et al._ (2023). Furthermore, the Nambu-Goldstone (NG) boson associated with
the breakdown of $U(1)_{B}$ symmetry does not emerge in this case. From these
observations, we conclude that $\bar{\omega}$ and $\bar{V}$ as well as
$\sigma_{0}$ and $\Delta$ play important roles in cold and dense QC2D matter
when the spin-$0$ and spin-$1$ mixing effect is present.

##  V Mass spectrum

In this section we examine a mass spectrum of the spin-$1$ hadrons at finite
$\mu_{q}$ by expanding the Lagrangian (42) on top of the mean fields obtained
in the previous section.

###  V.1 General properties

Before showing numerical results for the $\mu_{q}$ dependence of the spin-$1$
hadron masses, we start with general properties of such hadrons in a medium.
In what follows, we consider the rest frame of the medium.

The quantum numbers carried by (axial)vector mesons are identical to those of
(pseudo)scalar mesons with a derivative, e.g., $a_{1}^{a,\mu}$ and
$\partial^{\mu}\pi^{a}$, so that such two kinds of mesons can mix with each
other even in the hadronic phase with nonzero $C$. Similarly, (anti)baryons
can mix with the corresponding spin-$1$ (anti)baryons through a derivative. In
the rest frame of the hadronic medium, therefore, the following four mixings
appear:

$\displaystyle(\partial_{0}\pi,a_{1}^{t})\,,\ (\partial_{0}\eta,f_{1}^{t})\,,\
(\partial_{0}B,B_{AS}^{t})\,,\ (\partial_{0}\bar{B},\bar{B}_{AS}^{t})\ ,$ (54)

where each bracket represents the mixing partners and the isospin indices are
suppressed for simplicity. Here, the superscript “$t$” stands for the time
component ($\mu=0$) of the respective spin-$1$ hadrons. The time components of
the spin-$1$ hadrons are unphysical, so the mixings in Eq. (54) only lead to
modifications of $\pi$, $\eta$, $B$, and $\bar{B}$, while the physical
components, i.e., the spatial components of the spin-$1$ hadrons remain
unaffected. We note that the remaining spin-$0$ hadrons: $\sigma$, $a_{0}$,
$B^{\prime}$, and $\bar{B}^{\prime}$, are not contaminated by any mixing.

Meanwhile, in the baryon superfluid phase the mixings get more complicated due
to the $U(1)_{B}$ baryon-number violation as well as the creation of a
baryonic medium. Taking into account the unbroken $SU(2)_{I}$ isospin
symmetry, one can see that the following six mixings are possible:

$\displaystyle(\partial_{0}a_{0},\rho^{t})\ ,$
$\displaystyle(\partial_{0}B^{\prime},\partial_{0}\bar{B}^{\prime},\partial_{0}\eta,f_{1}^{t})\
,$ $\displaystyle(\partial_{0}\pi,B_{S}^{t},\bar{B}_{S}^{t},a_{1}^{t})\,,\
(B_{S}^{s},\bar{B}_{S}^{s},a_{1}^{s})$
$\displaystyle(\partial_{0}B,\partial_{0}\bar{B},\partial_{0}\sigma,B_{AS}^{t},\bar{B}^{t}_{AS},\omega^{t})\,,\
(B_{AS}^{s},\bar{B}_{AS}^{s},\omega^{s})\ ,$ (55)

where each bracket again represents the mixing partners. That is, all of the
spin-$0$ hadronic states are corrected by the corresponding unphysical
spin-$1$ states. Besides, the physical components of $B_{S},\bar{B}_{S},a_{1}$
and of $B_{AS},\bar{B}_{AS},\omega$ are also corrected by the mixings. A
schematic picture of the mixings among the hadrons both in the hadronic and
superfluid phases are shown in Fig. 3.

In the present paper, the hadron masses are evaluated at tree level in the
presence of the mean fields (49). The analysis is straightforward but is
complicated and lengthy since, particularly in the superfluid phase, we need
to find pole positions of the respective propagator matrix for the mixed
states in Eq. (55). For this reason, we leave the detailed procedure to
compute the hadron masses to Appendix C.

---
Figure 4: $\mu_{q}$ dependence of the masses of negative-parity (left) and
positive-parity (right) spin-$1$ hadrons for $(g_{\Phi},C)=(10,12)$. In this
figure the masses are normalized by $m_{\pi}^{(\rm H)}$.
---
Figure 5: $\mu_{q}$ dependence of the masses of positive-parity (left) and
negative-parity (right) spin-$0$ hadrons for $(g_{\Phi},C)=(10,12)$. In this
figure the masses are normalized by $m_{\pi}^{(\rm H)}$.
---
Figure 6: Same as Fig. 4 but for $(g_{\Phi},C)=(10,16)$.
---
Figure 7: Same as Fig. 4 but for $(g_{\Phi},C)=(10,8)$.

###  V.2 Numerical results for the hadron masses at nonzero $\mu_{q}$

In this subsection, based on the expressions derived in Appendix C we
numerically elucidate $\mu_{q}$ dependence of the masses of spin-$1$ hadrons.
As for the free parameters $g_{\Phi}$ and $C$, we first take
$(g_{\Phi},C)=(10,12)$ to draw a typical prediction of the mass spectrum of
the spin-$1$ hadrons. Next, we vary the value of $C$ with $g_{\Phi}$ kept
fixed to unveil how the spin-$0$ and spin-$1$ mixing affects the mass
spectrum. More concretely we take $(g_{\Phi},C)=(10,16)$ and
$(g_{\Phi},C)=(10,8)$. Finally, we also choose $(g_{\Phi},C)=(5,12)$ to see
influence of the coupling $g_{\Phi}$ on the mass spectrum.

Depicted in Fig. 4 is the resultant $\mu_{q}$ dependence of the spin-$1$
hadron masses at $(g_{\Phi},C)=(10,12)$. As can be seen from this figure, at
vanishing chemical potential, the masses of $\omega$, $\rho$, $B_{S}$, and
$\bar{B}_{S}$ degenerate and so do those of $f_{1}$, $a_{1}$, $B_{AS}$, and
$\bar{B}_{AS}$. In the hadronic phase, as $\mu_{q}$ increases, all the meson
masses do not change and the (anti)baryon masses simply modified linearly.
These stable behaviors are reminiscent of the Silver-Blaze property. Here, the
slopes of the linear decrement (increment) of diquark (antidiquark) baryons
can be understood by their baryon numbers. More explicitly, their mass
formulas are given by Eqs. (217) and (218). In the baryon superfluid phase, on
the other hand, due to the $U(1)_{B}$ baryon-number violation, $\omega$ \-
$B_{AS}$ \- $\bar{B}_{AS}$ mixing and $a_{1}$ \- $B_{S}$ \- $\bar{B}_{S}$
mixing take place, leading to non-monotonic $\mu_{q}$ dependence of the masses
of the resulting mixed states. Most remarkably, the $a_{1}$ \- $B_{S}$ \-
$\bar{B}_{S}$ mixing has been observed by the recent lattice simulation
Murakami  _et al._ . Meanwhile, the $\rho$ and $f_{1}$ mesons are not
contaminated by any mixing so that those masses depend on $\mu_{q}$ fairly
monotonically. We note that the present parameter set yields a slight
reduction of the $\rho$ meson mass in the superfluid phase, which is
consistent with the lattice data Murakami  _et al._ (2023b, ).

In the presence of the spin-$0$ and spin-$1$ mixing effect, it is worth
examining the mass spectrum of the spin-$0$ hadrons in addition to that of the
spin-$1$ hadrons as depicted in Fig. 4. In Fig. 5, we thus draw the resultant
$\mu_{q}$ dependence of the spin-$0$ hadron masses at $(g_{\Phi},C)=(10,12)$.
In this figure, the hadron masses are again shown to depend on $\mu_{q}$
monotonically in the hadronic phase, which is consistent with the Silver-Blaze
property. Besides, in the superfluid phase, the left panel indicates that a
massless mode emerges in the $\sigma$ \- $B$ \- $\bar{B}$ mixed state. This
mode corresponds to a NG boson associated with the spontaneous breakdown of
$U(1)_{B}$ baryon-number symmetry. Moreover, from the right panel, the pion
mass is found to increase linearly in the superfluid phase; this numerical
result can be reproduced by a simple formula,

$\displaystyle m_{\pi}=2\mu_{q}\ ,$ (56)

which is consistent with other chiral models Kogut  _et al._ (1999, 2000);
Ratti and Weise (2004); Suenaga  _et al._ (2023). We again emphasize that all
these reasonable results stem from the present correct treatment of the four
mean fields (49) within the eLSM.

Figure 8: Critical chemical potential for the axialvector condensation as a
function of mixing strength $C$. We take $g_{\Phi}=10$.
---
Figure 9: Same as Fig. 4 but for $(g_{\Phi},C)=(5,12)$.

###  V.3 $C$ dependence of the mass spectrum

The mass spectrum of the spin-$1$ hadrons presented in Fig. 4 is just a
typical example. In this case, the value of $C$ is fixed such that the slight
reduction of the $\rho$ meson mass in the superfluid phase suggested by
lattice simulations is successfully reproduced. Next, we change the value of
$C$ while keeping $g_{\Phi}=10$. When we take $C$ to be larger, e.g., $C=16$,
the $\mu_{q}$ dependence of the spin-$1$ hadron masses is obtained as depicted
in Fig. 6. From this figure one can find that the mass spectrum in the
hadronic phase is identical to the one with $(g_{\Phi},C)=(10,12)$, Fig. 4, as
long as all the other parameters are the same. In the superfluid phase, the
$\rho$ meson mass slightly increases with $\mu_{q}$, which is clearly
different from the lattice result. This suggests that the relevant value of
the parameter $C$ that controls the magnitude of the spin-$0$ and spin-$1$
mixing effect could not be so large. We note that, since the renormalization
factor $Z_{\pi}$ (or $Z_{\eta}$), Eq. (225), must be real, the inputs
presented in Sec. III already constrains $C$ as $C\lesssim 18.3$.

When we take $C$ to be smaller, e.g., $C=8$, the mass spectrum is evaluated as
in Fig. 7. The right panel indicates that, in this parameter choice, the mass
of the lowest-lying $a_{1}$ \- $B_{S}$ \- $\bar{B}_{S}$ mixed state reaches
zero at $\mu_{q}\approx 0.88m_{\pi}^{(\rm H)}$, as the chemical potential
increases in the superfluid phase. Since this mode includes an iso-triplet
axialvector component, it is reasonable to argue that above this chemical
potential, an axialvector condensed phase where $SU(2)_{I}$ isospin symmetry
is broken emerges on top of the baryon superfluidity Lenaghan  _et al._
(2002), as exhibited by the shaded area in Fig. 7. Thus, the true mass
spectrum in this phase is obscure although we have still plotted the numerical
result. For a self-consistent analysis, however, it would be necessary to
include another mean field that is responsible for the axialvector condensed
phase.

Above the critical chemical potential $\mu_{q}\approx 0.88m_{\pi}^{(\rm H)}$,
the mass of the lowest-lying $\omega$ \- $B_{AS}$ \- $\bar{B}_{AS}$ mixed
state also converges to zero at $\mu_{q}\approx 0.95m_{\pi}^{(\rm H)}$ as
indicated in the left panel of Fig. 7. Besides, the $\rho$ meson mass also
becomes zero at $\mu_{q}\approx 1.1m_{\pi}^{(\rm H)}$. These critical chemical
potentials lie in the axialvector condensed phase, so that more precise
determination of their values would require extension of the present
exploratory analysis. From those findings, however, at least one could expect
the existence of the vector condensed phase. We note that the axialvector
condensation occurs prior to the vector condensation, reflecting the fact that
the $B_{S}$ mass is invariably lighter than the $B_{AS}$ one at
$\mu_{q}=\mu_{q}^{\rm cr}$, since $B_{S}$ is an $S$-wave state.

From the above analysis, one can infer that when $C$ is small enough, the
appearance of the axialvector (and vector) condensate in the superfluid phase
is favored. To see this tendency more clearly, in Fig. 8, we plot the critical
chemical potential for the appearance of the axialvector condensed phase as a
function of the mixing strength $C$. In this figure, the pure baryon
superfluid phase lies below the curve, while in the above shaded area the
axialvector condensate emerges in the superfluid phase. The figure indeed
indicates that the smaller value of $C$ triggers the axialvector condensation
at lower $\mu_{q}$. In other words, the spin-$0$ and spin-$1$ mixing
controlled by $C$ acts as a stabilizer to avoid emergence of the axialvector
condensate, i.e., onset of the Bose-Einstein condensation of parity-even
spin-$1$ hadrons, in the low density regime. We note that, even when we take a
value of $C$ as large as possible, we find the critical chemical potential for
the axialvector condensation at a certain value, which could be too high for
the present model to be valid.

Figure 10: Critical chemical potential for the axialvector condensation as a
function of the coupling $g_{\Phi}$. We take $C=12$.

###  V.4 $g_{\Phi}$ dependence of the mass spectrum

Thus far we have only varied the value of $C$, while keeping $g_{\Phi}=10$, to
focus on the spin-$0$ and spin-$1$ mixing effects on the hadron mass spectrum.
Let us now examine effects of the coupling $g_{\Phi}$. When we employ
$(g_{\Phi},C)=(5,12)$, the mass spectrum is obtained as in Fig. 9. This figure
exhibits appearance of the axial vector condensate and possibly of the vector
condensate similarly to Fig. 7. One obvious distinction is the qualitative
behavior of the $\rho$ meson mass: The $\mu_{q}$ dependence of the $\rho$
meson mass in Fig. 9 does not change from that in Fig. 4 in the sense that
both show the same gradual decrease with $\mu_{q}$, whereas the $\rho$ meson
mass in Fig. 7 decreases rapidly to zero. This characteristic behavior can be
understood by the fact that the $\rho$ meson mass has no dependence on
$g_{\Phi}$, as shown in Eq. (109). Thus, $g_{\Phi}$ plays a role in changing
the $\mu_{q}$ dependence of the $\omega$ \- $B_{AS}$ \- $\bar{B}_{AS}$ mixed
states and the $a_{1}$ \- $B_{S}$ \- $\bar{B}_{S}$ mixed states, particularly
the lowest one for each. Detailed consideration of th $\rho$ meson mass in the
superfluid phase will be done in Sec. VII.2.

Depicted in Fig. 10 is the $g_{\Phi}$ dependence of the critical chemical
potential for the axialvector condensation. In this figure the axialvector
condensed phase is indicated by the shaded area. Figure 10 implies that the
smaller value of $g_{\Phi}$ leads to the appearance of the axialvector
condensate at lower $\mu_{q}$. Moreover, one can see that reentrant
axialvector condensation occurs in a regime of large $g_{\Phi}$ in such a way
that the intervening pure superfluid region shrinks with increasing $\mu_{q}$.

###  V.5 Signs of $C$ and $g_{\Phi}$

We conclude this section by giving comments on the signs of $C$ and
$g_{\Phi}$. Throughout the above numerical analysis, we have assumed $C>0$ and
$g_{\Phi}>0$. When the signs are taken to be $C<0$ and $g_{\Phi}<0$, we can
obtain qualitatively similar results for the mass spectrum although the
detailed numerical values are slightly changed. For instance, a negatively
larger value of $C$ acts to prevent the axialvector condensation and possible
vector condensation from occurring in the superfluid phase. On the other hand,
when we take $C>0$ and $g_{\Phi}<0$ or $C<0$ and $g_{\Phi}>0$, the resultant
mass spectrum always exhibits both types of condensation in the range of
$m_{\pi}^{\rm(H)}/2<\mu_{q}\lesssim 2m_{\pi}^{(\rm H)}$.

##  VI Chiral partner structure

From the numerical analysis in Sec. V, we have succeeded in gaining insights
into roles of the mixing strength $C$ and the coupling $g_{\Phi}$ in
determining the spin-$1$ hadron masses at finite $\mu_{q}$. In this section,
by focusing on a high $\mu_{q}$ regime where chiral symmetry is sufficiently
restored, we demonstrate the so-called chiral partner structure of the
spin-$1$ hadrons by identifying the pairs of $1^{+}$ and $1^{-}$ hadrons that
are degenerate in mass.

Figure 11: $\mu_{q}$ dependence of all the $1^{+}$ and $1^{-}$ hadron masses
for $(g_{\Phi},C)=(10,16)$. The chiral partner structures are clearly shown by
the degeneracy.

At sufficiently high $\mu_{q}$, the four mean fields would asymptotically
behave as $\sigma_{0}\to\sigma_{\infty}\mu_{q}^{-2}$,
$\Delta\to\Delta_{\infty}$, $\bar{\omega}\to\bar{\omega}_{\infty}\mu_{q}$, and
$\bar{V}\to\bar{V}_{\infty}\mu_{q}^{-1}$, as expected from the numerical
results in Fig. 1, where $\sigma_{\infty}$, $\Delta_{\infty}$,
$\bar{\omega}_{\infty}$, and $\bar{V}_{\infty}$ are constants. Using these
asymptotic behaviors and the mass formulas derived in Appendix C, first, one
can easily show $m_{\omega}=m_{f_{1}}$ and $m_{\rho}=m_{a_{1}}$ at
$\mu_{q}\to\infty$, where $\omega$ and $a_{1}$ mesons are decoupled from the
respective mixings with the (anti)diquark baryons.666Here, the superscript
“$s$” for the hadron masses representing the spatial component is omitted for
simplicity. The same abbreviation applies to Eq. (68) to refer to the $\rho$
meson mass. That is, ($\omega,f_{1}$) and ($\rho,a_{1}$) can be regarded as
the chiral partners even in the presence of the mean fields $\Delta$ and
$\bar{\omega}$ in a high density region. Next, as for the remaining
(anti)diquark baryons, $B_{S}$, $\bar{B}_{S}$, $B_{AS}$, and $\bar{B}_{AS}$,
the asymptotic behaviors of the mean fields tell that the mixing structures
for the $V_{9}$ \- $V_{10}$ system and the $V^{\prime}_{4}$ \-
$V_{5}^{\prime}$ system become identical while the remaining bare masses
satisfy $m_{V_{9}}=m_{V_{4}^{\prime}}$ and $m_{V_{10}}=m_{V_{5}^{\prime}}$ at
$\mu_{q}\to\infty$, as can be seen from Appendix C. Thus, $(B_{S},B_{AS})$ and
$(\bar{B}_{S},\bar{B}_{AS})$ are also regarded as the chiral partners.

The above analytic consideration of the chiral partner structure is, indeed,
numerically confirmed as shown in Fig. 11. In this figure we have taken
$(g_{\Phi},C)=(10,16)$ to see the mass degeneracy clearly. Figure 11 indicates
that the mass degeneracy occurs between the following $1^{+}$ and $1^{-}$
hadrons, $(B_{S},B_{AS})$, $(\rho,a_{1})$, $(\omega,f_{1})$, and
$(\bar{B}_{S},\bar{B}_{AS})$ from below, at high $\mu_{q}$. We note that the
chiral partner structure for the spin-$0$ hadrons in the absence of the
$U(1)_{A}$ anomaly was examined in Ref. Suenaga  _et al._ (2023), where mass
degeneracy was demonstrated for $(B,B^{\prime})$, $(\sigma,\pi)$,
$(a_{0},\eta)$, and $(\bar{B},\bar{B}^{\prime})$.

##  VII Discussions

###  VII.1 Analytic derivation of $\mu_{q}^{\rm cr}=m_{\pi}^{(\rm H)}/2$ in
the eLSM

In Sec. IV, we have numerically seen that the phase transition from the
hadronic phase to the baryon superfluid phase takes place just when $\mu_{q}$
coincides with $m_{\pi}^{(\rm H)}/2$, similarly to other chiral models
ignoring spin-$1$ hadrons Kogut  _et al._ (1999, 2000); Ratti and Weise
(2004); Suenaga  _et al._ (2023). In this section we analytically prove such a
universal property within our present eLSM based on a reasonable assumption.

The $\mu_{q}$ dependence of the mean fields, $\sigma_{0}$, $\Delta$,
$\bar{\omega}$, and $\bar{V}$, has been determined by stationary conditions in
Sec. IV. These conditions can be derived from Eq. (42) as

$\displaystyle\sigma_{0}:\frac{2\sqrt{2}h_{q}}{\sigma_{0}}-\frac{C}{\sqrt{2}\sigma_{0}}\mu_{q}\Delta\bar{V}-\frac{C_{3}}{4\sigma_{0}}\Delta\bar{V}\bar{\omega}+\frac{C_{3}}{8}(\bar{V}^{2}-\bar{\omega}^{2})+\frac{C}{8}(\bar{V}^{2}+\bar{\omega}^{2})-m_{0}^{2}-\frac{\lambda_{2}}{4}(\sigma_{0}^{2}+\Delta^{2})=0\
,$ (57)
$\displaystyle\Delta:-\frac{C}{\sqrt{2}\Delta}\mu_{q}\bar{V}\sigma_{0}-\frac{C_{3}}{4\Delta}\bar{V}\sigma_{0}\bar{\omega}+4\mu_{q}^{2}+\sqrt{2}C\mu_{q}\bar{\omega}-\frac{C_{3}}{8}(\bar{V}^{2}-\bar{\omega}^{2})+\frac{C}{8}(\bar{V}^{2}+\bar{\omega}^{2})-m_{0}^{2}-\frac{\lambda_{2}}{4}(\sigma_{0}^{2}+\Delta^{2})=0\
,$ (58)
$\displaystyle\bar{\omega}:\frac{C}{\sqrt{2}\bar{\omega}}\mu_{q}\Delta^{2}-\frac{C_{3}}{4\bar{\omega}}\Delta\bar{V}\sigma_{0}-\frac{C_{3}}{8}(\sigma_{0}^{2}-\Delta^{2})+\frac{C}{8}(\sigma_{0}^{2}+\Delta^{2})+m_{1}^{2}=0\
,$ (59)

and

$\displaystyle\bar{V}:-\frac{C}{\sqrt{2}\bar{V}}\mu_{q}\Delta\sigma_{0}-\frac{C_{3}}{4\bar{V}}\Delta\sigma_{0}\bar{\omega}+\frac{C_{3}}{8}(\sigma_{0}^{2}-\Delta^{2})+\frac{C}{8}(\sigma_{0}^{2}+\Delta^{2})+m_{1}^{2}=0\
.$ (60)

The numerical solutions imply that emergence of $\bar{\omega}$ and $\bar{V}$
is accompanied by nonzero $\Delta$, i.e., by the onset of the baryon
superfluidity and that the phase transition is of second order. Let us now
suppose that the critical exponents of $\Delta$ and $\bar{V}$ are $+1/2$ while
that of $\bar{\omega}$ is $+1$. Indeed, these exponents are suggested by the
numerical result. Then, $\Delta$ and $\bar{V}$ take the form of

$\displaystyle\Delta\sim\Delta^{\rm cr}(\mu_{q}-\mu_{q}^{\rm cr})^{1/2}\ ,$
$\displaystyle\bar{V}\sim\bar{V}^{\rm cr}(\mu_{q}-\mu_{q}^{\rm cr})^{1/2}\ ,$
$\displaystyle\bar{\omega}\sim\bar{\omega}^{\rm cr}(\mu_{q}-\mu_{q}^{\rm
cr})^{1}\ ,$ (61)

in the vicinity of the phase transition, where $\Delta^{\rm cr}>0$,
$\bar{V}^{\rm cr}>0$, and $\bar{\omega}^{\rm cr}<0$. From these exponents, one
finds that in the limit of $\mu_{q}\to\mu_{q}^{\rm cr}$,
$\Delta^{2}/\bar{\omega}$, $\Delta\bar{V}/\bar{\omega}$, and $\Delta/\bar{V}$
approach nonzero values, while $\Delta\bar{\omega}/\bar{V}\to 0$. Thus, the
first and second terms of Eq. (59) and the first term in Eq. (60), which
remain finite at the critical chemical potential, act as catalyzers to yield
nonzero $\bar{\omega}$ and $\bar{V}$ above $\mu_{q}=\mu_{q}^{\rm cr}$,
respectively.

Given the critical behavior (61), at $\mu_{q}=\mu_{q}^{\rm cr}$, the
stationary conditions, Eqs. (57) - (60), are reduced to

$\displaystyle\frac{2\sqrt{2}h_{q}}{\sigma_{0}^{\rm(H)}}-Z_{\pi}^{-2}\big{(}m_{\pi}^{(\rm
H)}\big{)}^{2}=0\ ,$ (62) $\displaystyle-\frac{C\mu^{\rm
cr}_{q}\sigma_{0}^{\rm(H)}}{\sqrt{2}x}+4(\mu_{q}^{\rm
cr})^{2}-Z_{\pi}^{-2}\big{(}m_{\pi}^{(\rm H)}\big{)}^{2}=0\ ,$ (63)
$\displaystyle\frac{C\mu_{q}^{\rm
cr}y}{\sqrt{2}}-\frac{C_{3}\sigma_{0}^{\rm(H)}y}{4x}+\big{(}m_{\rho}^{(\rm
H)}\big{)}^{2}=0\ ,$ (64)

and

$\displaystyle-\frac{C\mu^{\rm
cr}_{q}\sigma_{0}^{\rm(H)}x}{\sqrt{2}}+(m_{f_{1}}^{(\rm H)})^{2}=0\ ,$ (65)

respectively, where we have used the hadron mass formulas in Appendix D and
defined

$\displaystyle x\equiv\frac{\Delta^{\rm cr}}{\bar{V}^{\rm cr}}>0\ ,\ \
y\equiv\frac{\big{(}\Delta^{\rm cr}\big{)}^{2}}{\bar{\omega}^{\rm cr}}<0\ .$
(66)

From Eqs. (63) and (65) as well as the renormalization factor (225),
therefore, one can analytically prove

$\displaystyle\mu_{q}^{\rm cr}=\frac{m_{\pi}^{(\rm H)}}{2}\ ,$ (67)

which was numerically confirmed in Fig. 1. We note that $C$ cannot be zero
from the stationary condition for $\bar{V}$ at $\mu_{q}=\mu_{q}^{\rm cr}$, Eq.
(65), as long as Eq. (61) holds. In other words, $\bar{V}$ always vanishes
when $C=0$, and similarly, from Eq. (64) one can see $\bar{\omega}=0$ at any
$\mu_{q}$ when $C=0$.

###  VII.2 Comments on the $\rho$ mass reduction

Here we provide comments on the $\rho$ meson mass reduction in the superfluid
phase.

As derived in Eq. (109), the $\rho$ meson mass is evaluated as

$\displaystyle
m_{\rho}^{2}=m_{1}^{2}+\frac{C-C_{3}}{8}(\sigma_{0}^{2}+\Delta^{2})\ ,$ (68)

in both the hadronic and superfluid phases. This universal structure stems
from the fact that the $\rho$ meson is not contaminated by any mixing with
other hadrons, even in the presence of the superfluidity. Then, if the
combination $\sigma_{0}^{2}+\Delta^{2}$ is enhanced in the superfluid phase
while $C<C_{3}$, the $m_{\rho}$ reduction observed on the lattice can be
reproduced within the present eLSM. From Fig. 2, however, one can see that the
enhancement of the combination $\sigma_{0}^{2}+\Delta^{2}$ in the superfluid
phase gets mild as the spin-$0$ and spin-$1$ mixing effect becomes prominent.
That is why $m_{\rho}$ is relatively hard to change for a larger value of $C$.

Since $\mu_{q}$ dependence of the combination $\sigma_{0}^{2}+\Delta^{2}$ in
the superfluid phase is rather monotonic as can be inferred from Fig. 2, the
resultant $m_{\rho}$ also changes almost linearly as a function of $\mu_{q}$.
On the other hand, the lattice data would imply a rather abrupt reduction of
$m_{\rho}$ just above $\mu_{q}=\mu_{q}^{\rm cr}$, although there remain error
bars; $\mu_{q}$ dependence of $m_{\rho}$ measured by the lattice simulation
would look convex downward Murakami  _et al._ (2023b, ). One promising
mechanism to yield such a downward-convex behavior of $m_{\rho}$ could be
additional mixing with other states that are associated with the superfluidity
but have yet to be considered in the present analysis. The $\rho$ meson, which
is an iso-triplet state carrying $J^{P}=1^{-}$, may strongly mix with an iso-
triplet and $J^{P}=1^{-}$ diquark
$\tilde{B}^{i}\sim\epsilon^{ijk}[\psi^{T}C\Sigma^{jk}\tau_{c}^{2}\psi]_{\rm
sym.}$, where $\Sigma^{\mu\nu}=\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}]$ is the
antisymmetric tensor and the subscript “sym.” means the flavor symmetric
structure. Inclusion of this new diquark state, however, requires us to
introduce another quark bilinear operator
$\tilde{\Phi}_{ij}^{\mu\nu}\sim\Psi_{j}^{T}\sigma^{2}\bar{\sigma}^{\mu\nu}\tau_{c}^{2}\Psi_{i}$,
where
$\bar{\sigma}^{\mu\nu}=\frac{i}{2}(\bar{\sigma}^{\mu}\sigma^{\nu}-\bar{\sigma}^{\nu}\sigma^{\mu})$
with $\bar{\sigma}^{\mu}=({\bm{1}},-\sigma^{i})$, which is beyond the scope of
the present study. Thus, we leave detailed examination of the $\tilde{B}^{i}$
\- $\rho$ mixing in the superfluid phase for future study.

##  VIII Conclusions

In summary, for dense QC2D at zero temperature, we have constructed the
extended linear sigma model, eLSM, in such a way as to respect the Pauli-
Gürsey $SU(4)$ symmetry and to describe both the spin-$0$ and spin-$1$
hadrons. Then, based on the eLSM, richness of the mass spectrum of the
spin-$1$ hadrons in dense QC2D has been explored.

In the baryon superfluid phase where the diquark condensate emerges, we have
found that not only the scalar meson and scalar diquark baryon but also the
time components of the vector meson and vector diquark baryon possess their
mean field values, in the presence of spin-$0$ and spin-$1$ mixing. These mean
fields are induced by violation of the Lorentz invariance as well as of $U(1)$
baryon-number conservation. Besides, we have analytically shown that the onset
condition of superfluidity corresponds to $\mu_{q}=m_{\pi}^{(\rm H)}/2$
($m_{\pi}^{(\rm H)}$ is the pion mass in the hadronic phase) and that the pion
mass in the superfluid phase reads $m_{\pi}=2\mu_{q}$. Moreover, the
appearance of the NG boson associated with $U(1)_{B}$ violation has been
confirmed. Those self-consistent properties, which were numerically indicated
by lattice simulations, are derived only when the above four types of mean
fields are included.

Inclusion of the vector-meson and vector-diquark mean fields has led to
suppression of the otherwise substantial increase of the scalar-diquark mean
field in the superfluid phase via the spin-$0$ and spin-$1$ mixing.
Simultaneously, it has been found that the slight reduction of the $\rho$
meson mass in the superfluid phase suggested by the lattice data is
successfully reproduced in the presence of the significant spin-$0$ and
spin-$1$ mixing. Furthermore, by varying the magnitude of this mixing and of
coupling among the spin-$1$ hadrons, we have demonstrated the emergence of the
axialvector condensate and the possible vector condensate. Those novel
condensates are induced when the masses of the corresponding modes reach zero.
Given such condensates, therefore, it is inevitable to investigate the in-
medium masses of the spin-$1$ hadrons from the first-principles lattice
calculation of dense QC2D toward further delineation of the phase structures.

In addition, we have also discussed a possible existence of an iso-triplet
$1^{-}$ diquark, motivated by the possible downward-convex behavior of the
$\rho$ meson mass reduction in the superfluid phase as suggested by the
lattice simulations Murakami  _et al._ (2023b, ). This diquark is denoted by a
tensor-type quark bilinear field, while no examination has been done so far.
Therefore, it would be challenging to pursue properties of such a new diquark
state from both effective theories and first-principles numerical studies.

## Acknowledgment

The authors thank Masayasu Harada for useful comments on the HLS formalism for
the spin-$1$ hadrons. D.S. was supported by the RIKEN special postdoctoral
researcher program and by the Japan Society for the Promotion of Science
(JSPS) KAKENHI Grants No. 23K03377. K. M. is supported in part by JST SPRING
with Grant Number JPMJSP2110, by Grants-in-Aid for JSPS Fellows (Nos.
JP22J14889, JP22KJ1870), and by JSPS KAKENHI with Grant No. 22H04917. The work
of K. I. is supported by JSPS KAKENHI with Grant Numbers 18H05406 and
23H01167. The work of E. I. is supported by JSPS KAKENHI with Grant Number
23H05439, JST PRESTO Grant Number JPMJPR2113, JSPS Grant-in-Aid for
Transformative Research Areas (A) JP21H05190, JST Grant Number JPMJPF2221 and
also supported by Program for Promoting Researches on the Supercomputer
“Fugaku” (Simulation for basic science: from fundamental laws of particles to
creation of nuclei) and (Simulation for basic science: approaching the new
quantum era), and Joint Institute for Computational Fundamental Science
(JICFuS), Grant Number JPMXP1020230411.The work of E. I is supported also by
Center for Gravitational Physics and Quantum Information (CGPQI) at YITP.

##  Appendix A The Pauli-Gürsey $SU(4)$ symmetry in two-flavor QC2D

In this appendix, we briefly show emergence of the Pauli-Gürsey $SU(4)$
symmetry in QC2D with two flavors Kogut  _et al._ (1999, 2000).

The QC2D Lagrangian for massless $u$ and $d$ quarks is of the form

$\displaystyle{\cal L}_{\rm
QC_{2}D}=\bar{\psi}i{\ooalign{\hfil/\hfil\crcr$D$}}\psi\ ,$ (71)

where $\psi=(u,d)^{T}$ is the quark doublet and
$D_{\mu}\psi=\partial_{\mu}\psi-ig_{c}A_{\mu}^{a}T_{c}^{a}\psi$ is the
covariant derivative describing interactions between the quarks $\psi$ and
gluons $A_{\mu}^{a}$. The $2\times 2$ matrix $T_{c}^{a}=\tau_{c}^{a}/2$ is the
$SU(2)_{c}$ generator ($\tau_{c}^{a}$ is the Pauli matrix for colors).
Adopting the Weyl representation for the Dirac matrices, the Lagrangian (71)
can be expressed in terms of left-handed and right-handed quarks as

$\displaystyle{\cal L}_{\rm QC_{2}D}$ $\displaystyle=$
$\displaystyle\psi^{\dagger}_{R}i\partial_{\mu}\sigma^{\mu}\psi_{R}-g_{c}\psi_{R}^{\dagger}A_{\mu}^{a}T_{c}^{a}\sigma^{\mu}\psi_{R}$
(72) $\displaystyle+$
$\displaystyle\psi^{\dagger}_{L}i\partial_{\mu}\bar{\sigma}^{\mu}\psi_{L}-g_{c}\psi_{L}^{\dagger}A_{\mu}^{a}T_{c}^{a}\bar{\sigma}^{\mu}\psi_{L}.$

In this Lagrangian, $u=(u_{R},u_{L})^{T}$ and $d=(d_{R},d_{L})^{T}$ in the
Weyl representation, and the $2\times 2$ matrices in the spinor space are
defined by $\sigma^{\mu}=({\bm{1}},\sigma^{i})$ and
$\bar{\sigma}^{\mu}=({\bm{1}},-\sigma^{i})$ with the Pauli matrix
$\sigma^{i}$. Here, we make use of the pseudoreal property of the Pauli
matrix. Namely, using relations

$\displaystyle T_{c}^{a}=-\tau_{c}^{2}(T_{c}^{a})^{T}\tau_{c}^{2}\ ,\ \
\sigma^{i}=-\sigma^{2}(\sigma^{i})^{T}\sigma^{2}\ ,$ (73)

and accordingly introducing the “conjugate quark fields”

$\displaystyle\tilde{\psi}_{R}\equiv\sigma^{2}{\bf\tau_{c}^{2}}\psi_{R}^{*}\
,\ \ \tilde{\psi}_{L}\equiv\sigma^{2}{\bf\tau_{c}^{2}}\psi_{L}^{*}\ ,$ (74)

the Lagrangian (72) can be expressed in a unified form as

$\displaystyle{\cal L}_{\rm
QC_{2}D}=\Psi^{\dagger}i\partial_{\mu}\sigma^{\mu}\Psi-g\Psi^{\dagger}A_{\mu}^{a}\sigma^{\mu}\Psi\
.$ (75)

Here, we have described the quark fields by using a four-component column
vector defined as

$\displaystyle\Psi\equiv\left(\begin{array}[]{c}\psi_{R}\\\
\tilde{\psi}_{L}\\\ \end{array}\right)=\left(\begin{array}[]{c}u_{R}\\\
d_{R}\\\ \tilde{u}_{L}\\\ \tilde{d}_{L}\\\ \end{array}\right)\ .$ (82)

The Lagrangian (75) is obviously invariant under an $SU(4)$ transformation of

$\displaystyle\Psi\to U\Psi\ \ \ {\rm with}\ \ \ U\in SU(4)\ ,$ (83)

rather than $SU(2)_{L}\times SU(2)_{R}$ chiral transformation. Such an
extended symmetry is sometimes referred to as the Pauli-Gürsey $SU(4)$
symmetry Kogut  _et al._ (1999, 2000). As can be seen from Eq. (82), the
Pauli-Gürsey $SU(4)$ symmetry is realized by treating $\psi$ and
$\tilde{\psi}$ in a single multiplet, reflecting the fact that mesons and
diquark baryons can be described in a unified way in two-flavor QC2D. It
should be noted that the $U(1)_{B}$ baryon-number transformation is generated
by

$\displaystyle\Psi\to{\rm e}^{-i\theta_{q}{J}}\Psi\ \ \ {\rm with}\ \ \
{J}\equiv\left(\begin{array}[]{cc}{\bm{1}}&0\\\ 0&-{\bm{1}}\\\
\end{array}\right)\ ,$ (86)

where ${\rm e}^{-i\theta_{q}{\bm{J}}}$ belongs to a subgroup of the Pauli-
Gürsey $SU(4)$ group.

##  Appendix B Generators of $U(4)$ Lie algebra

In this appendix, we list the generators of $U(4)$ Lie algebra.

The number of the $U(4)$ generators is $4\times 4=16$. It is convenient to
separate these $16$ generators into two sets $S^{i}$ ($i=1$ \- $10$) and
$X^{a}$ ($a=0$ \- $5$) that satisfy

$\displaystyle E(S^{i})^{T}=-S^{i}E\ ,\ \ E(X^{a})^{T}=X^{a}E\ ,$ (87)

with the symplectic matrix

$\displaystyle E=\left(\begin{array}[]{cc}0&{\bm{1}}_{f}\\\ -{\bm{1}}_{f}&0\\\
\end{array}\right)\ .$ (90)

That is, the elements generated by $S^{i}$, $h={\rm
e}^{-i\theta_{S}^{i}S^{i}}$, exhibit the following relation:

$\displaystyle hEh^{T}=E\ .$ (91)

This relation means that $h$ belongs to the $Sp(4)$ group, which is the
subgroup of the original $U(4)$ group.

More concretely, the generators $S^{i}$ belonging to the Lie algebra of
$Sp(4)$ read

$\displaystyle
S^{i=1-4}=\frac{1}{2\sqrt{2}}\left(\begin{array}[]{cc}\tau_{f}^{i}&0\\\
0&-(\tau_{f}^{i})^{T}\\\ \end{array}\right)\ ,$ (94) $\displaystyle
S^{i=5-10}=\frac{1}{2\sqrt{2}}\left(\begin{array}[]{cc}0&B^{i}\\\
(B^{i})^{\dagger}&0\\\ \end{array}\right)\ ,$ (97)

with $\tau_{f}^{4}={\bm{1}}_{f}$, $B^{5}={\bm{1}}_{f}$, $B^{6}=i{\bm{1}}_{f}$,
$B^{7}=\tau_{f}^{3}$, $B^{8}=i\tau_{f}^{3}$, $B^{9}=\tau_{f}^{1}$, and
$B^{10}=i\tau_{f}^{1}$. Meanwhile, the remaining generators belonging to the
algebras of $U(1)$ and $SU(4)/Sp(4)$ are given by

$\displaystyle
X^{a=0-3}=\frac{1}{2\sqrt{2}}\left(\begin{array}[]{cc}\tau_{f}^{a}&0\\\
0&(\tau_{f}^{a})^{T}\\\ \end{array}\right)\ ,$ (100) $\displaystyle
X^{a=4,5}=\frac{1}{2\sqrt{2}}\left(\begin{array}[]{cc}0&D^{a}\\\
(D^{a})^{\dagger}&0\\\ \end{array}\right)\ ,$ (103)

where $\tau_{f}^{a=0}={\bm{1}}_{f}$ is the $2\times 2$ unit matrix and
$\tau_{f}^{a=1-3}$ are the Pauli matrices in the flavor space. Besides,
$D^{4}=\tau_{f}^{2}$ and $D^{5}=i\tau_{f}^{2}$.

##  Appendix C Mass formulas

In this appendix, we derive hadron mass formulas from the reduced eLSM
Lagrangian (42).

The mass formulas are derived by picking up quadratic terms of the hadron
fields in the Lagrangian (42) on top of the mean fields $\sigma_{0}$,
$\Delta$, $\bar{\omega}$, and $\bar{V}$ defined by Eq. (49). The eLSM
describes the mass spectrum of 16 hadrons in total, namely, eight spin-$0$
hadrons, $\eta$, $\pi$, $B$, $\bar{B}$, $\sigma$, $a_{0}$, $B^{\prime}$, and
$\bar{B}^{\prime}$, and eight spin-$1$ hadrons, $\omega$, $\rho$, $B_{S}$,
$\bar{B}_{S}$, $f_{1}$, $a_{1}$, $B_{AS}$, and $\bar{B}_{AS}$. As explained in
Sec. V.1, those 16 hadrons are separated into the following four systems due
to different mixing patterns:

  1. 1.

$a_{0}$ \- $\rho$ system,

  2. 2.

$\eta$ \- $B^{\prime}$ \- $\bar{B}^{\prime}$\- $f_{1}$ system,

  3. 3.

$\pi$ \- $a_{1}$ \- $B_{S}$ \- $\bar{B}_{S}$ system,

  4. 4.

$\sigma$ \- $B$ \- $\bar{B}$ \- $\omega$ \- $B_{AS}$ \- $\bar{B}_{AS}$ system.

Then, in what follows we show the mass formulas for these four systems
separately.

###  C.1 $a_{0}$ \- $\rho$ system

As for the $a_{0}$ \- $\rho$ system, the relevant Lagrangian is obtained as

$\displaystyle{\cal L}_{(1)}={\cal L}_{a_{0}}+{\cal L}_{\rho}+{\cal
L}_{a_{0}\rho}\ ,$ (104)

where each term reads

$\displaystyle{\cal
L}_{a_{0}}=\frac{1}{2}\partial_{\mu}a_{0}\partial^{\mu}a_{0}-\frac{m_{a_{0}}^{2}}{2}a_{0}^{2}\
,$ (105) $\displaystyle{\cal L}_{\rho}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}(\partial_{\mu}\rho_{\nu}-\partial_{\nu}\rho_{\mu})^{2}+\frac{(m^{t}_{\rho})^{2}}{2}(\rho^{0})^{2}+\frac{(m_{\rho}^{s})^{2}}{2}\rho_{i}\rho^{i}\
,$

and

$\displaystyle{\cal
L}_{a_{0}\rho}=\frac{C-C_{3}}{4}(\Delta\bar{V}_{4}+\sigma_{0}\bar{\omega})a_{0}\rho^{0}\
.$ (107)

In these terms, we have suppressed the isospin indices for simplicity, and
have defined the mass parameters by

$\displaystyle
m_{a_{0}}^{2}=m_{0}^{2}+\frac{3\lambda_{2}}{4}(\sigma_{0}^{2}+\Delta^{2})-\frac{C-C_{3}}{8}(\bar{V}^{2}+\bar{\omega}^{2})\
,$ (108)

and

$\displaystyle(m_{\rho}^{t})^{2}=(m_{\rho}^{s})^{2}=m_{1}^{2}+\frac{C-C_{3}}{8}(\sigma_{0}^{2}+\Delta^{2})\
,$ (109)

where the superscripts “t” and “s” are attached to distinguish between time-
component (unphysical) and spatial-component (physical) masses.

Thus, from Eq. (107) one can see that the $a_{0}$ meson mixes with the time
component of the $\rho$ meson due to the violation of the Lorentz invariance.
For this reason, the mass of the $a_{0}$ meson does not coincide with
$m_{a_{0}}$ provided by Eq. (108) but is determined by a pole of the
propagator matrix for the $a_{0}$ \- $\rho^{0}$ system at vanishing momentum
${\bm{p}}={\bm{0}}$. The inverse of the propagator matrix $i{\cal
D}_{(1)}^{-1}(p_{0},{\bm{0}})$ can be derived as

$\displaystyle i{\cal
D}_{(1)}^{-1}(p_{0},{\bm{0}})=\left(\begin{array}[]{cc}{\cal
M}_{a_{0}a_{0}}&{\cal M}_{a_{0}\rho}\\\ {\cal M}_{\rho a_{0}}&{\cal
M}_{\rho\rho}\\\ \end{array}\right)\ ,$ (112)

with

$\displaystyle{\cal M}_{a_{0}a_{0}}=p_{0}^{2}-m_{a_{0}}^{2}\ ,$ (113)
$\displaystyle{\cal M}_{a_{0}\rho}={\cal M}_{\rho
a_{0}}=\frac{C-C_{3}}{4}(\Delta\bar{V}+\sigma_{0}\bar{\omega})\ ,$ (114)

and

$\displaystyle{\cal M}_{\rho\rho}=(m_{\rho}^{t})^{2}\ .$ (115)

Therefore, the mass of $a_{0}$ at arbitrary $\mu_{q}$ is evaluated by
numerically solving ${\rm det}\left[i{\cal
D}_{(1)}^{-1}(p_{0},{\bm{0}})\right]=0$. Meanwhile, the spatial components of
the $\rho$ meson, i.e., the physical states of $\rho$ do not join any mixing,
and hence, the $\rho$ meson mass is identical to $m_{\rho}^{s}$, Eq. (109).

###  C.2 $\eta$ \- $B^{\prime}$ \- $\bar{B}^{\prime}$ \- $f_{1}$ system

Employing a similar procedure demonstrated in Appendix C.1, the mass spectrum
of the $\eta$ \- $B^{\prime}$ \- $\bar{B}^{\prime}$ \- $f_{1}$ system can be
evaluated. In this case, $\eta$, $B^{\prime}$, $\bar{B}^{\prime}$ (or
$B^{\prime 4}$, $B^{\prime 5}$) and the time component of $f_{1}$ can mix. The
inverse propagator of these four states at vanishing momentum
${\bm{p}}={\bm{0}}$: $i{\cal D}_{(2)}^{-1}(p_{0},{\bm{0}})$, is given by

$\displaystyle i{\cal
D}_{(2)}^{-1}(p_{0},{\bm{0}})=\left(\begin{array}[]{cccc}{\cal
M}_{\eta\eta}&{\cal M}_{\eta B_{4}^{\prime}}&{\cal M}_{\eta
B_{5}^{\prime}}&{\cal M}_{\eta f_{1}}\\\ {\cal M}_{B_{4}^{\prime}\eta}&{\cal
M}_{B_{4}^{\prime}B_{4}^{\prime}}&{\cal
M}_{B_{4}^{\prime}B_{5}^{\prime}}&{\cal M}_{B_{4}^{\prime}f_{1}}\\\ {\cal
M}_{B_{5}^{\prime}\eta}&{\cal M}_{B_{5}^{\prime}B_{4}^{\prime}}&{\cal
M}_{B_{5}^{\prime}B_{5}^{\prime}}&{\cal M}_{B_{5}^{\prime}f_{1}}\\\ {\cal
M}_{f_{1}\eta}&{\cal M}_{f_{1}B_{4}^{\prime}}&{\cal
M}_{f_{1}B_{5}^{\prime}}&{\cal M}_{f_{1}f_{1}}\\\ \end{array}\right)\ ,$ (120)

where each matrix element reads

$\displaystyle{\cal
M}_{\eta\eta}=p_{0}^{2}-m_{\eta}^{2}+\frac{C^{2}}{8}\bar{V}^{2}\ ,$ (122)
$\displaystyle{\cal M}_{\eta B_{4}^{\prime}}=-{\cal
M}_{B_{4}^{\prime}\eta}=-i\frac{C}{\sqrt{2}}\bar{V}p_{0}\ ,$ (123)
$\displaystyle{\cal M}_{\eta B_{5}^{\prime}}={\cal
M}_{B_{5}^{\prime}\eta}=-\frac{2\lambda_{2}}{4}\sigma_{0}\Delta+\frac{C}{\sqrt{2}}\bar{V}\mu_{q}+\frac{C_{3}}{4}\bar{V}\bar{\omega}\
,$ $\displaystyle{\cal M}_{\eta f_{1}}=-{\cal
M}_{f_{1}\eta}=-i\frac{C}{2\sqrt{2}}\sigma_{0}p_{0}\ ,$ (125)
$\displaystyle{\cal
M}_{B_{4}^{\prime}B_{4}^{\prime}}=p_{0}^{2}-m_{B_{4}^{\prime}}^{2}+\frac{C^{2}}{8}\bar{V}^{2}+\left(2\mu_{q}+\frac{C}{2\sqrt{2}}\bar{\omega}\right)^{2}\
,$ $\displaystyle{\cal M}_{B_{4}^{\prime}B_{5}^{\prime}}=-{\cal
M}_{B_{5}^{\prime}B_{4}^{\prime}}=2i\left(2\mu_{q}+\frac{C}{2\sqrt{2}}\bar{\omega}\right)p_{0}\
,$ (127) $\displaystyle{\cal M}_{B_{4}^{\prime}f_{1}}={\cal
M}_{f_{1}B_{4}^{\prime}}=\frac{C+C_{3}}{4}(\sigma_{0}\bar{V}-\Delta\bar{\omega})-\sqrt{2}C\Delta\mu_{q}\
,$ $\displaystyle{\cal
M}_{B_{5}^{\prime}B_{5}^{\prime}}=p_{0}^{2}-m_{B_{5}^{\prime}}^{2}+\left(2\mu_{q}+\frac{C}{2\sqrt{2}}\bar{\omega}\right)^{2}\
,$ (129) $\displaystyle{\cal M}_{B_{5}^{\prime}f_{1}}=-{\cal
M}_{f_{1}B_{5}^{\prime}}=i\frac{C}{2\sqrt{2}}\Delta p_{0}\ ,$ (130)

and

$\displaystyle{\cal M}_{f_{1}f_{1}}=(m_{f_{1}}^{t})^{2}\ ,$ (131)

with the mass parameters

$\displaystyle m_{B_{4}^{\prime}}^{2}$ $\displaystyle=$ $\displaystyle
m_{0}^{2}+\frac{3\lambda_{2}}{4}(\sigma_{0}^{2}+\Delta^{2})$ (132)
$\displaystyle-\frac{C-C^{2}+C_{3}}{8}(\bar{V}^{2}+\bar{\omega}^{2})\ ,$
$\displaystyle m_{B_{5}^{\prime}}^{2}$ $\displaystyle=$ $\displaystyle
m_{0}^{2}+\frac{\lambda_{2}}{4}(3\sigma_{0}^{2}+\Delta^{2})$ (133)
$\displaystyle-\frac{C-C^{2}+C_{3}}{8}\bar{\omega}^{2}-\frac{C-C_{3}}{8}\bar{V}^{2}\
,$ $\displaystyle m_{\eta}^{2}$ $\displaystyle=$ $\displaystyle
m_{0}^{2}+\frac{\lambda_{2}}{4}(\sigma_{0}^{2}+3\Delta^{2})$ (134)
$\displaystyle-\frac{C-C_{3}}{8}\bar{\omega}^{2}-\frac{C-C^{2}+C_{3}}{8}\bar{V}^{2}\
,$

and

$\displaystyle(m_{f_{1}}^{t})^{2}$ $\displaystyle=$ $\displaystyle
m_{1}^{2}+\frac{C}{4}(\sigma_{0}^{2}+\Delta^{2})\ .$ (135)

In the presence of the mixing, the mass spectrum of the $\eta$ \-
$B^{\prime}$\- $\bar{B}^{\prime}$ system is evaluated by numerically solving
${\rm det}\left[i{\cal D}_{(2)}^{-1}(p_{0},{\bm{0}})\right]=0$. Meanwhile, the
space components of $f_{1}$ are decoupled, which allows us to simply obtain
the physical $f_{1}$ meson mass as

$\displaystyle(m_{f_{1}}^{s})^{2}$ $\displaystyle=$ $\displaystyle
m_{1}^{2}+\frac{C+C_{3}}{8}(\sigma_{0}^{2}+\Delta^{2})\ .$ (136)

###  C.3 $\pi$ \- $a_{1}$ \- $B_{S}$ \- $\bar{B}_{S}$ system

Here, we derive the mass formulas for the $\pi$ \- $a_{1}$ \- $B_{S}$ \-
$\bar{B}_{S}$ system. In this case, $\pi$ and the time components of $a_{1}$,
$B_{S}$, $\bar{B}_{S}$ (or $V^{9}$, $V^{10}$) can mix. In addition, the space
components of $a_{1}$, $B_{S}$, $\bar{B}_{S}$ (or $V^{9}$, $V^{10}$) also mix
with each other.

First, we consider the mixing among $\pi$ and the time components of $a_{1}$,
$B_{S}$, and $\bar{B}_{S}$. The inverse propagator of these four states at
vanishing momentum ${\bm{p}}={\bm{0}}$, $i{\cal
D}_{(3)}^{-1}(p_{0},{\bm{0}})$, is given by

$\displaystyle i{\cal
D}_{(3)}^{-1}(p_{0},{\bm{0}})=\left(\begin{array}[]{cccc}{\cal
M}_{\pi\pi}&{\cal M}_{\pi a_{1}}&{\cal M}_{\pi V_{9}}&{\cal M}_{\pi V_{10}}\\\
{\cal M}_{a_{1}\pi}&{\cal M}_{a_{1}a_{1}}&0&{\cal M}_{a_{1}V_{10}}\\\ {\cal
M}_{V_{9}\pi}&0&{\cal M}_{V_{9}V_{9}}&0\\\ {\cal M}_{V_{10}\pi}&{\cal
M}_{V_{10}a_{1}}&0&{\cal M}_{V_{10}V_{10}}\\\ \end{array}\right),$ (141)

where each matrix element reads

$\displaystyle{\cal M}_{\pi\pi}=p_{0}^{2}-m_{\pi}^{2}\ ,$ (143)
$\displaystyle{\cal M}_{\pi a_{1}}=-{\cal
M}_{a_{1}\pi}=-i\frac{C}{2\sqrt{2}}\sigma_{0}p_{0}\ ,$ (144)
$\displaystyle{\cal M}_{\pi V_{9}}={\cal
M}_{V_{9}\pi}=\frac{C_{3}}{4}\sigma_{0}\bar{V}-\frac{C}{\sqrt{2}}\Delta\mu_{q}-\frac{C_{3}}{4}\Delta\bar{\omega}\
,$ (145) $\displaystyle{\cal M}_{\pi V_{10}}=-{\cal
M}_{V_{10}\pi}=i\frac{C}{2\sqrt{2}}\Delta p_{0}\ ,$ (146) $\displaystyle{\cal
M}_{a_{1}a_{1}}=(m_{a_{1}}^{t})^{2}\ ,$ (147) $\displaystyle{\cal
M}_{a_{1}V_{10}}={\cal M}_{V_{10}a_{1}}=-\frac{C_{3}}{4}\sigma_{0}\Delta\ ,$
(148) $\displaystyle{\cal M}_{V_{9}V_{9}}=(m_{V_{9}}^{t})^{2}\ ,$ (149)

and

$\displaystyle{\cal M}_{V_{10}V_{10}}=(m_{V_{10}}^{t})^{2}\ ,$ (150)

with the mass parameters

$\displaystyle
m_{\pi}^{2}=m_{0}^{2}+\frac{\lambda_{2}}{4}(\sigma_{0}^{2}+\Delta^{2})-\frac{C-C_{3}}{8}(\bar{V}^{2}+\bar{\omega}^{2})\
,$ (151)
$\displaystyle(m^{t}_{a_{1}})^{2}=m_{1}^{2}+\frac{C}{8}(\sigma_{0}^{2}+\Delta^{2})+\frac{C_{3}}{8}(\sigma_{0}^{2}-\Delta^{2})\
,$ (152)
$\displaystyle(m_{V_{9}}^{t})^{2}=m_{1}^{2}+\frac{C-C_{3}}{8}(\sigma_{0}^{2}+\Delta^{2})\
,$ (153)

and

$\displaystyle(m_{V_{10}}^{t})^{2}=m_{1}^{2}+\frac{C}{8}(\sigma_{0}^{2}+\Delta^{2})-\frac{C_{3}}{8}(\sigma_{0}^{2}-\Delta^{2})\
.$ (154)

In the presence of the mixing, the pion mass is evaluated by numerically
solving ${\rm det}\left[i{\cal D}_{(3)}^{-1}(p_{0},{\bm{0}})\right]=0$.

Next, we consider the mixing among the spatial components of $a_{1}$, $B_{S}$,
and $\bar{B}_{S}$. In this case, the corresponding inverse propagator at
vanishing momentum ${\bm{p}}={\bm{0}}$, $i\tilde{\cal
D}_{(3)}^{-1}(p_{0},{\bm{0}})$, is given by

$\displaystyle i\tilde{\cal
D}_{(3)}^{-1}(p_{0},{\bm{0}})=\left(\begin{array}[]{ccc}\tilde{\cal
M}_{a_{1}a_{1}}&\tilde{\cal M}_{a_{1}V_{9}}&\tilde{\cal M}_{a_{1}V_{10}}\\\
\tilde{\cal M}_{V_{9}a_{1}}&\tilde{\cal M}_{V_{9}V_{9}}&\tilde{\cal
M}_{V_{9}V_{10}}\\\ \tilde{\cal M}_{V_{10}a_{1}}&\tilde{\cal
M}_{V_{10}V_{9}}&\tilde{\cal M}_{V_{10}V_{10}}\\\ \end{array}\right)\ ,$ (158)

where each matrix element reads

$\displaystyle\tilde{\cal
M}_{a_{1}a_{1}}=-p_{0}^{2}-\frac{g_{\Phi}^{2}}{2}\bar{V}^{2}+(m_{a_{1}}^{s})^{2}\
,$ (160) $\displaystyle\tilde{\cal M}_{a_{1}V_{9}}=-\tilde{\cal
M}_{V_{9}a_{1}}=\sqrt{2}ig_{\Phi}\bar{V}p_{0}\ ,$ (161)
$\displaystyle\tilde{\cal M}_{a_{1}V_{10}}=\tilde{\cal
M}_{V_{10}a_{1}}=-\frac{C_{3}}{4}\sigma_{0}\Delta-\sqrt{2}g_{\Phi}\bar{V}\mu_{q}-\frac{g_{\Phi}^{2}}{2}\bar{V}\bar{\omega},$
$\displaystyle\tilde{\cal
M}_{V_{9}V_{9}}=-p_{0}^{2}-\frac{g_{\Phi}^{2}}{2}\bar{V}^{2}-\left(2\mu_{q}+\frac{g_{\Phi}}{\sqrt{2}}\bar{\omega}\right)^{2}+(m_{V_{9}}^{s})^{2},$
$\displaystyle\tilde{\cal M}_{V_{9}V_{10}}=-\tilde{\cal
M}_{V_{10}V_{9}}=-2i\left(2\mu_{q}+\frac{g_{\Phi}}{\sqrt{2}}\bar{\omega}\right)p_{0}\
,$ (164)

and

$\displaystyle\tilde{\cal
M}_{V_{10}V_{10}}=-p_{0}^{2}-\left(2\mu_{q}+\frac{g_{\Phi}}{\sqrt{2}}\bar{\omega}\right)^{2}+(m_{V_{10}}^{s})^{2}\
,$ (165)

with the mass parameters

$\displaystyle(m^{s}_{a_{1}})^{2}=m_{1}^{2}+\frac{C}{8}(\sigma_{0}^{2}+\Delta^{2})+\frac{C_{3}}{8}(\sigma_{0}^{2}-\Delta^{2})\
,$ (166)
$\displaystyle(m_{V_{9}}^{s})^{2}=m_{1}^{2}+\frac{C-C_{3}}{8}(\sigma_{0}^{2}+\Delta^{2})\
,$ (167)

and

$\displaystyle(m_{V_{10}}^{s})^{2}=m_{1}^{2}+\frac{C}{8}(\sigma_{0}^{2}+\Delta^{2})-\frac{C_{3}}{8}(\sigma_{0}^{2}-\Delta^{2})\
.$ (168)

In the presence of the mixing, the mass spectrum for the $a_{1}$ \- $B_{S}$ \-
$\bar{B}_{S}$ system is evaluated by numerically solving ${\rm
det}\left[i\tilde{\cal D}_{(3)}^{-1}(p_{0},{\bm{0}})\right]=0$.

###  C.4 $\sigma$ \- $B$ \- $\bar{B}$ \- $\omega$ \- $B_{AS}$ \-
$\bar{B}_{AS}$ system

Here, we derive the mass formulas for the $\sigma$ \- $B$ \- $\bar{B}$ \-
$\omega$ \- $B_{AS}$ \- $\bar{B}_{AS}$ system. In this case, $\sigma$, $B$,
$\bar{B}$ (or $B^{4}$, $B^{5}$) and the time components of $\omega$, $B_{AS}$,
$\bar{B}_{AS}$ (or $V^{\prime 4}$, $V^{\prime 5}$) can mix. In addition, the
space components of $\omega$, $B_{AS}$, $\bar{B}_{AS}$ (or $V^{\prime 4}$,
$V^{\prime 5}$) also mix with each other.

First, we consider the mixing among $\sigma$, $B$, $\bar{B}$ (or $B^{4}$,
$B^{5}$) and the time components of $\omega$, $B_{AS}$, $\bar{B}_{AS}$. The
inverse propagator of these six states at vanishing momentum
${\bm{p}}={\bm{0}}$, $i{\cal D}_{(4)}^{-1}(p_{0},{\bm{0}})$, is given by

$\displaystyle i{\cal
D}_{(4)}^{-1}(p_{0},{\bm{0}})=\left(\begin{array}[]{cccccc}{\cal
M}_{\sigma\sigma}&{\cal M}_{\sigma B_{4}}&{\cal M}_{\sigma B_{5}}&{\cal
M}_{\sigma\omega}&{\cal M}_{\sigma V_{4}^{\prime}}&{\cal M}_{\sigma
V_{5}^{\prime}}\\\ {\cal M}_{B_{4}\sigma}&{\cal M}_{B_{4}B_{4}}&{\cal
M}_{B_{4}B_{5}}&{\cal M}_{B_{4}\omega}&{\cal M}_{B_{4}V_{4}^{\prime}}&{\cal
M}_{B_{4}V_{5}^{\prime}}\\\ {\cal M}_{B_{5}\sigma}&{\cal M}_{B_{5}B_{4}}&{\cal
M}_{B_{5}B_{5}}&{\cal M}_{B_{5}\omega}&{\cal M}_{B_{5}V_{4}^{\prime}}&{\cal
M}_{B_{5}V_{5}^{\prime}}\\\ {\cal M}_{\omega\sigma}&{\cal M}_{\omega
B_{4}}&{\cal M}_{\omega B_{5}}&{\cal M}_{\omega\omega}&{\cal M}_{\omega
V_{4}^{\prime}}&0\\\ {\cal M}_{V_{4}^{\prime}\sigma}&{\cal
M}_{V_{4}^{\prime}B_{4}}&{\cal M}_{V_{4}^{\prime}B_{5}}&{\cal
M}_{V_{4}^{\prime}\omega}&{\cal M}_{V_{4}^{\prime}V_{4}^{\prime}}&0\\\ {\cal
M}_{V_{5}^{\prime}\sigma}&{\cal M}_{V_{5}^{\prime}B_{4}}&{\cal
M}_{V_{5}^{\prime}B_{5}}&0&0&{\cal M}_{V_{5}^{\prime}V_{5}^{\prime}}\\\
\end{array}\right)\ ,$ (175)

where each matrix element reads

$\displaystyle{\cal
M}_{\sigma\sigma}=p_{0}^{2}+\frac{C^{2}}{8}\bar{V}^{2}-m_{\sigma}^{2}\ ,$
(176) $\displaystyle{\cal M}_{\sigma B_{4}}=-{\cal
M}_{B_{4}\sigma}=i\frac{C}{\sqrt{2}}\bar{V}p_{0}\ ,$ (177) $\displaystyle{\cal
M}_{\sigma B_{5}}={\cal
M}_{B_{5}\sigma}=-\frac{C}{\sqrt{2}}\bar{V}\mu_{q}-\frac{\lambda_{2}}{2}\sigma_{0}\Delta-\frac{C_{3}\bar{\omega}}{4}\bar{V}\
,$ $\displaystyle{\cal M}_{\sigma\omega}={\cal
M}_{\omega\sigma}=\frac{C-C_{3}}{4}\bar{\omega}\sigma_{0}-\frac{C_{3}}{4}\Delta\bar{V}\
,$ (179) $\displaystyle{\cal M}_{\sigma V_{4}^{\prime}}={\cal
M}_{V_{4}^{\prime}\sigma}=\frac{C+C_{3}}{4}\sigma_{0}\bar{V}-\frac{C}{\sqrt{2}}\Delta\mu_{q}-\frac{C_{3}}{4}\Delta\bar{\omega}\
,$ $\displaystyle{\cal M}_{\sigma V_{5}^{\prime}}=-{\cal
M}_{V_{5}^{\prime}\sigma}=i\frac{C}{2\sqrt{2}}\Delta p_{0}\ ,$ (181)
$\displaystyle{\cal
M}_{B_{4}B_{4}}=p_{0}^{2}+\frac{C^{2}}{8}\bar{V}^{2}+\left(2\mu_{q}+\frac{C}{2\sqrt{2}}\bar{\omega}\right)^{2}-m_{B_{4}}^{2}\
,$ $\displaystyle{\cal M}_{B_{4}B_{5}}=-{\cal
M}_{B_{5}B_{4}}=2i\left(2\mu_{q}+\frac{C}{2\sqrt{2}}\bar{\omega}\right)p_{0}\
,$ (183) $\displaystyle{\cal M}_{B_{4}\omega}=-{\cal M}_{\omega
B_{4}}=i\frac{C}{2\sqrt{2}}\Delta p_{0}\ ,$ (184) $\displaystyle{\cal
M}_{B_{4}V_{4}^{\prime}}=-{\cal
M}_{V_{4}^{\prime}B_{4}}=-i\frac{C}{2\sqrt{2}}\sigma_{0}p_{0}\ ,$ (185)
$\displaystyle{\cal M}_{B_{4}V_{5}^{\prime}}={\cal
M}_{V_{5}^{\prime}B_{4}}=\frac{C_{3}}{4}(\Delta\bar{V}_{4}+\sigma_{0}\bar{\omega})+\frac{C}{\sqrt{2}}\sigma_{0}\mu_{q}\
,$ $\displaystyle{\cal
M}_{B_{5}B_{5}}=p_{0}^{2}+\left(2\mu_{q}+\frac{C}{2\sqrt{2}}\bar{\omega}\right)^{2}-m_{B_{5}}^{2}\
,$ (187) $\displaystyle{\cal M}_{B_{5}\omega}={\cal M}_{\omega B_{5}}$
$\displaystyle=$
$\displaystyle\sqrt{2}C\mu_{q}\Delta+\frac{C+C_{3}}{4}\Delta\bar{\omega}$
(188) $\displaystyle-\frac{C_{3}}{4}\sigma_{0}\bar{V}\ ,$ $\displaystyle{\cal
M}_{B_{5}V_{4}^{\prime}}={\cal M}_{V_{4}^{\prime}B_{5}}$ $\displaystyle=$
$\displaystyle\frac{C-C_{3}}{4}\Delta\bar{V}-\frac{C_{3}}{4}\sigma_{0}\bar{\omega}$
(189) $\displaystyle-\frac{C}{\sqrt{2}}\sigma_{0}\mu_{q}\ ,$
$\displaystyle{\cal M}_{B_{5}V_{5}^{\prime}}=-{\cal
M}_{V_{5}^{\prime}B_{5}}=-i\frac{C}{2\sqrt{2}}\sigma_{0}p_{0}\ ,$ (190)
$\displaystyle{\cal M}_{\omega\omega}=(m_{\omega}^{t})^{2}\ ,$ (191)
$\displaystyle{\cal M}_{\omega V_{4}^{\prime}}={\cal
M}_{V_{4}^{\prime}\omega}=-\frac{C_{3}}{4}\sigma_{0}\Delta\ ,$ (192)
$\displaystyle{\cal
M}_{V_{4}^{\prime}V_{4}^{\prime}}=(m_{V_{4}^{\prime}}^{t})^{2}\ ,$ (193)

and

$\displaystyle{\cal
M}_{V_{5}^{\prime}V_{5}^{\prime}}=(m_{V_{5}^{\prime}}^{t})^{2}\ ,$ (194)

with the mass parameters

$\displaystyle m_{\sigma}^{2}$ $\displaystyle=$ $\displaystyle
m_{0}^{2}+\frac{\lambda_{2}}{4}(3\sigma_{0}^{2}+\Delta^{2})$ (195)
$\displaystyle-\frac{C-C_{3}}{8}\bar{\omega}^{2}-\frac{C-C^{2}+C_{3}}{8}\bar{V}^{2}\
,$ $\displaystyle m_{B_{4}}^{2}$ $\displaystyle=$ $\displaystyle
m_{0}^{2}+\frac{\lambda_{2}}{4}(\sigma_{0}^{2}+\Delta^{2})$ (196)
$\displaystyle-\frac{C-C^{2}+C_{3}}{8}(\bar{V}^{2}+\bar{\omega}^{2})\ ,$
$\displaystyle m_{B_{5}}^{2}$ $\displaystyle=$ $\displaystyle
m_{0}^{2}+\frac{\lambda_{2}}{4}(\sigma_{0}^{2}+3\Delta^{2})$ (197)
$\displaystyle-\frac{C-C^{2}+C_{3}}{8}\bar{\omega}^{2}-\frac{C-C_{3}}{8}\bar{V}^{2}\
,$
$\displaystyle(m_{\omega}^{t})^{2}=m_{1}^{2}+\frac{C}{8}(\sigma_{0}^{2}+\Delta^{2})-\frac{C_{3}}{8}(\sigma_{0}^{2}-\Delta^{2})\
,$ (198)
$\displaystyle(m_{V^{\prime}_{4}}^{t})^{2}=m_{1}^{2}+\frac{C}{8}(\sigma_{0}^{2}+\Delta^{2})+\frac{C_{3}}{8}(\sigma_{0}^{2}-\Delta^{2})\
,$ (199)

and

$\displaystyle(m_{V^{\prime}_{5}}^{t})^{2}=m_{1}^{2}+\frac{C+C_{3}}{8}(\sigma_{0}^{2}+\Delta^{2})\
.$ (200)

In the presence of the mixing, the mass spectrum for the $\sigma$ \- $B$ \-
$\bar{B}$ system is evaluated by numerically solving $i{\cal
D}_{(4)}^{-1}(p_{0},{\bm{0}})=0$.

Next, we consider the mixing among the spatial components of $\omega$,
$B_{AS}$ and $\bar{B}_{AS}$. In this case, the corresponding inverse
propagator at vanishing momentum ${\bm{p}}={\bm{0}}$, $i\tilde{\cal
D}_{(4)}^{-1}(p_{0},{\bm{0}})$, is given by

$\displaystyle i\tilde{\cal
D}_{(4)}^{-1}(p_{0},{\bm{0}})=\left(\begin{array}[]{ccc}\tilde{\cal
M}_{\omega\omega}&\tilde{\cal M}_{\omega V_{4}^{\prime}}&\tilde{\cal
M}_{\omega V_{5}^{\prime}}\\\ \tilde{\cal
M}_{V_{4}^{\prime}\omega}&\tilde{\cal
M}_{V_{4}^{\prime}V_{4}^{\prime}}&\tilde{\cal
M}_{V_{4}^{\prime}V_{5}^{\prime}}\\\ \tilde{\cal
M}_{V_{5}^{\prime}\omega}&\tilde{\cal
M}_{V_{5}^{\prime}V_{4}^{\prime}}&\tilde{\cal
M}_{V_{5}^{\prime}V_{5}^{\prime}}\\\ \end{array}\right)\ ,$ (204)

where each matrix element reads

$\displaystyle\tilde{\cal
M}_{\omega\omega}=-p_{0}^{2}-\frac{g_{\Phi}^{2}}{2}\bar{V}^{2}+(m_{\omega}^{s})^{2}\
,$ (206) $\displaystyle\tilde{\cal M}_{\omega V_{4}^{\prime}}=\tilde{\cal
M}_{V_{4}^{\prime}\omega}=-\frac{C_{3}}{4}\sigma_{0}\Delta+\sqrt{2}g_{\Phi}\bar{V}\mu_{q}+\frac{g_{\Phi}^{2}}{2}\bar{V}\bar{\omega}\
,$ $\displaystyle\tilde{\cal M}_{\omega V_{5}^{\prime}}=-\tilde{\cal
M}_{V_{5}^{\prime}\omega}=\sqrt{2}ig_{\Phi}\bar{V}p_{0}\ ,$ (208)
$\displaystyle\tilde{\cal
M}_{V_{4}^{\prime}V_{4}^{\prime}}=-p_{0}^{2}-\left(2\mu_{q}+\frac{g_{\Phi}}{\sqrt{2}}\bar{\omega}\right)^{2}+(m_{V_{4}^{\prime}}^{s})^{2}\
,$ (209) $\displaystyle\tilde{\cal
M}_{V_{4}^{\prime}V_{5}^{\prime}}=-\tilde{\cal
M}_{V_{5}^{\prime}V_{4}^{\prime}}=-2i\left(2\mu_{q}+\frac{g_{\Phi}}{\sqrt{2}}\bar{\omega}\right)p_{0}\
,$ (210)

and

$\displaystyle\tilde{\cal
M}_{V_{5}^{\prime}V_{5}^{\prime}}=-p_{0}^{2}-\frac{g_{\Phi}^{2}}{2}\bar{V}^{2}-\left(2\mu_{q}+\frac{g_{\Phi}}{\sqrt{2}}\bar{\omega}\right)^{2}+(m_{V_{5}^{\prime}}^{s})^{2}\
,$

with the mass parameters

$\displaystyle(m_{\omega}^{s})^{2}=m_{1}^{2}+\frac{C}{8}(\sigma_{0}^{2}+\Delta^{2})-\frac{C_{3}}{8}(\sigma_{0}^{2}-\Delta^{2})\
,$ (212)
$\displaystyle(m_{V^{\prime}_{4}}^{s})^{2}=m_{1}^{2}+\frac{C}{8}(\sigma_{0}^{2}+\Delta^{2})+\frac{C_{3}}{8}(\sigma_{0}^{2}-\Delta^{2})\
,$ (213)

and

$\displaystyle(m_{V^{\prime}_{5}}^{s})^{2}=m_{1}^{2}+\frac{C+C_{3}}{8}(\sigma_{0}^{2}+\Delta^{2})\
.$ (214)

In the presence of the mixing, the mass spectrum for the $\omega$ \- $B_{AS}$
\- $\bar{B}_{AS}$ system is evaluated by numerically solving ${\rm
det}\left[i\tilde{\cal D}_{(4)}^{-1}(p_{0},{\bm{0}})\right]=0$.

##  Appendix D Masses in the hadronic phase

As derived in Appendix C, in general, the hadron masses are evaluated by pole
positions of the appropriate propagator matrices, which would be obtained
numerically. When we focus on the hadronic phase, the mass formulas can be
evaluated analytically. In this appendix, we exhibit the resultant mass
formulas for all hadrons.

Such formulas can simply be obtained by taking $\Delta=0$ and accordingly
$\bar{\omega}=\bar{V}=0$, while keeping $\sigma_{0}\neq 0$, in the mass
formulas in Appendix C. In this limit, one can find

$\displaystyle m_{\omega}^{(\rm H)}=m_{\rho}^{(\rm
H)}=\sqrt{m_{1}^{2}+\frac{C-C_{3}}{8}\big{(}\sigma_{0}^{(\rm H)}\big{)}^{2}}\
,$ (215) $\displaystyle m_{f_{1}}^{(\rm H)}=m_{a_{1}}^{(\rm
H)}=\sqrt{m_{1}^{2}+\frac{C+C_{3}}{8}\big{(}\sigma_{0}^{(\rm H)}\big{)}^{2}}\
,$ (216) $\displaystyle m_{B_{S}}^{(\rm H)}$ $\displaystyle=$ $\displaystyle
m_{\omega}^{(\rm H)}-2\mu_{q}\ ,$ $\displaystyle m_{\bar{B}_{S}}^{(\rm H)}$
$\displaystyle=$ $\displaystyle m_{\omega}^{(\rm H)}+2\mu_{q}\ ,$ (217)
$\displaystyle m_{B_{\rm AS}}^{(\rm H)}$ $\displaystyle=$ $\displaystyle
m_{f_{1}}^{(\rm H)}-2\mu_{q}\ ,$ $\displaystyle m_{\bar{B}_{\rm AS}}^{(\rm
H)}$ $\displaystyle=$ $\displaystyle m_{f_{1}}^{(\rm H)}+2\mu_{q}\ ,$ (218)

for the spin-$1$ hadrons, while

$\displaystyle m_{\pi}^{(\rm
H)}=Z_{\pi}\sqrt{m_{0}^{2}+\frac{\lambda_{2}}{4}\big{(}\sigma_{0}^{(\rm
H)}\big{)}^{2}}\ ,$ (219) $\displaystyle m_{\eta}^{(\rm
H)}=Z_{\eta}\sqrt{m_{0}^{2}+\frac{\lambda_{2}}{4}\big{(}\sigma_{0}^{(\rm
H)}\big{)}^{2}}\ ,$ (220) $\displaystyle m_{a_{0}}^{(\rm
H)}=\sqrt{m_{0}^{2}+\frac{3\lambda_{2}}{4}\big{(}\sigma_{0}^{(\rm
H)}\big{)}^{2}}\ ,$ (221) $\displaystyle m_{\sigma}^{(\rm
H)}=\sqrt{m_{0}^{2}+3\frac{\lambda_{2}}{4}\big{(}\sigma_{0}^{(\rm
H)}\big{)}^{2}}\ ,$ (222) $\displaystyle m_{B}^{(\rm H)}$ $\displaystyle=$
$\displaystyle m_{\pi}^{(\rm H)}-2\mu_{q}\ ,$ $\displaystyle m_{\bar{B}}^{(\rm
H)}$ $\displaystyle=$ $\displaystyle m_{\pi}^{(\rm H)}+2\mu_{q}\ ,$ (223)
$\displaystyle m_{B^{\prime}}^{(\rm H)}$ $\displaystyle=$ $\displaystyle
m_{a_{0}}^{(\rm H)}-2\mu_{q}\ ,$ $\displaystyle m_{\bar{B}^{\prime}}^{(\rm
H)}$ $\displaystyle=$ $\displaystyle m_{a_{0}}^{(\rm H)}+2\mu_{q}\ ,$ (224)

for the spin-$0$ hadrons. In Eqs. (220) and (219), the renormalization factors
$Z_{\pi}$ and $Z_{\eta}$ are defined by

$\displaystyle Z_{\pi}$ $\displaystyle=$
$\displaystyle\left(1-\frac{C^{2}\big{(}\sigma_{0}^{(\rm
H)}\big{)}^{2}}{8\big{(}m_{a_{1}}^{(\rm H)}\big{)}^{2}}\right)^{-1/2}\ ,$
$\displaystyle Z_{\eta}$ $\displaystyle=$
$\displaystyle\left(1-\frac{C^{2}\big{(}\sigma_{0}^{(\rm
H)}\big{)}^{2}}{8\big{(}m_{f_{1}}^{(\rm H)}\big{)}^{2}}\right)^{-1/2}\ ,$
(225)

which stems from the $\pi$ \- $a_{1}$ mixing and the $\eta$ \- $f_{1}$ mixing,
respectively. It should be noted that $Z_{\pi}=Z_{\eta}$ follows from
$m_{f_{1}}^{(\rm H)}=m_{a_{1}}^{(\rm H)}$. These types of mixing originate
from the spontaneous breakdown of chiral symmetry since $Z_{\pi}=Z_{\eta}=1$
when $(\sigma_{0}^{(\rm H)})=0$, as in the three-color eLSM Parganlija  _et
al._ (2013).

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