LFTC-23-1/74

#  In-medium properties of the light and heavy-light mesons
in a light-front quark model

Ahmad Jafar Arifi<EMAIL_ADDRESS>Asia Pacific Center for Theoretical
Physics, Pohang, Gyeongbuk 37673, Korea  Few-Body Systems in Physics
Laboratory, RIKEN Nishina Center, Wako 351-0198, Japan  Parada. T. P. Hutauruk
<EMAIL_ADDRESS>Department of Physics, Pukyong National University
(PKNU), Busan 48513, Korea  Department of Physics Education, Daegu University,
Gyeongsan 39453, Korea  Kazuo Tsushima<EMAIL_ADDRESS><EMAIL_ADDRESS>Laboratório de Física Teórica e Computacional-LFTC,
Universidade Cidade de São Paulo, 01506-000 São Paulo, SP, Brazil

###### Abstract

We investigate the in-medium properties of pseudoscalar and vector mesons with
the light-light and heavy-light quarks in a light-front quark model (LFQM),
using the in-medium quark properties computed by the quark-meson coupling
(QMC) model. Both models are constructed on an equal footing with the
constituent quark degree of freedom. Here, we particularly focus on the weak
decay constants and distribution amplitudes (DAs) of the mesons in symmetric
nuclear matter. We find that the weak decay constants decrease as nuclear
density increases for $\pi$, $K$, $D$, and $B$ pseudoscalar as well as $\rho$,
$K^{*}$, $D^{*}$, and $B^{*}$ vector mesons, where their properties in free
space have good agreement with the available experimental and lattice QCD
data. A larger reduction is found for the light-light quark pseudoscalar
mesons, while a smaller reduction is found for the heavy-light quark vector
mesons, in particular, with the bottom quark. We discuss the effect of the
vector potential on the weak decay constants, and present our predictions for
the in-medium modifications of DAs. Also, a comparison with the free space
lattice QCD data is made.

##  I Introduction

One of the most challenging problems in hadronic and nuclear physics is how
the properties and structure of hadron are modified in a nuclear medium, and
how such modifications would be reflected on the observables, such as cross-
sections and extracted form factors [1, 2, 3]. It was indicated experimentally
that the nucleon structure function is modified in nuclei, which is known as
the European Muon Collaboration (EMC) effect [1]. Thus, the properties of
hadrons are also expected to be modified in the nuclear medium, that may be
associated with the partial restoration of chiral symmetry. Experimental
evidence for this partial restoration of chiral symmetry has been confirmed
through the deeply bound pionic atoms [2], the low-energy pion-nucleus
scattering [4], and di-pion production in hadron-nucleus and photon-nucleus
reactions [5, 6]. In the deeply bound pionic atom experiment [2] and pion-
nucleus scattering [4], the analysis concluded that the pion decay constant
(associated with the temporal part) is reduced in the nuclear medium.

From quantum chromodynamics (QCD) in the Standard Model (SM), we know that
hadrons are composed of quarks and gluons. One can naturally expect that the
quark and gluon dynamics is modified when hadrons are immersed in a nuclear
medium. But, how are the internal structure of hadrons and the dynamics of the
quarks and gluons modified in the nuclear medium? These questions, however,
are still far remote to answer in terms of the first principle, QCD.
Therefore, more studies are strongly required to understand the medium
modifications of hadron structure and properties. Motivated by this, many
studies on the in-medium modifications of hadron properties and structure have
been made both in theoretically [3] and experimentally [1, 2] (references
therein). Until now, many theoretical studies have been made on the light and
heavy mesons in the nuclear medium using various models and approaches, such
as the QMC model [7, 8, 9, 10], the Dyson-Schwinger equation (DSE) based
approach [11], the holographic model [12], the QCD sum rules (QSR) [13, 14],
the Linear-sigma model (L$\sigma$M) [15], the Bethe-Salpeter equation-
Nambu–Jona-Lasinio (BSE-NJL) model [16, 17, 18, 19], the instanton liquid
model (ILM) [20, 21], and the hybrid light front-quark-meson coupling (LF-QMC)
models [22, 23, 24].

Among the models mentioned above, authors of Refs. [22, 23, 24] studied the
weak-decay constants, DAs, and electromagnetic elastic form factors (EFFs)
using the combined approach with the LFQM and the QMC models in the nuclear
medium. However, their studies focused only on the $\pi$ and $\rho$ mesons
with the light-light quark constituents. In this work, we extend the studies
in a more systematic manner for the light and heavy-light pseudoscalar as well
as vector mesons. It is worth noting that, in the nuclear medium, the mesons
with the heavy-light quark pair are subject to feel not only the scalar
potential but also the vector potential in the QMC model. In this study, we
employ a similar hybrid approach to that practiced in Refs. [22, 23, 24],
namely, the meson properties are computed in the LFQM, using the in-medium
quark properties simulated in the QMC model. Note that, in the present study,
we employ the Gaussian wave function in the LFQM, which is different from
Refs. [22, 23, 24], where they used the Bethe-Salpeter amplitude (BSA) with
the mass regulator to tame the divergence in the loop integral by the quark
propagators. Instead, here we introduce the quark potential in the Hamiltonian
that describes the quark-antiquark interaction inside the meson.

The highlight of the present study is to study the dynamics of the light-light
and heavy-light quark systems in the spin-0 pseudoscalar and spin-1 vector
mesons, where the quarks in both mesons compose different symmetries and
quantum numbers. The present study may also provide useful information on the
properties of quarkonium in the nuclear medium [8, 9, 10]. Such studies should
be important for understanding the more complicated quark dynamic systems like
heavy baryons with bottom and/or charm quarks, as well as exotic states in
free space and in the nuclear medium. These kinds of studies are relevant for
the ongoing or planned experiments in the modern, international facilities,
such as PANDA and CBM collaborations at FAIR [25, 26, 27, 28, 29], PHENIX
collaboration at RHIC [30], BELLE and BELLE II collaboration at KEK [31], LHCb
collaboration at CERN [32], and J-PARC [33].

This article is organized as follows. In Sec. II, we first briefly introduce
the effective Hamiltonian and light-front wave functions (LFWFs) used in the
present LFQM approach. Then, we describe how the model parameters are fixed
via the variational approach. Additionally, we give the expressions for
various properties of mesons, such as weak-decay constants, and DAs in free
space in the LFQM formalism. In Sec. III, we describe briefly the QMC model,
so that we can input the medium effect in the LFQM calculation. In Sec. IV, we
present the formulas for the vector potentials, weak decay constants, and DAs
for the light and heavy-light mesons in the nuclear medium. Section V presents
the results for the light and heavy-light meson properties in the nuclear
medium. Section VI is devoted to the summary and conclusion.

##  II Properties and structure of mesons in free space

In this section, we briefly describe the free space properties of the light
and heavy-light mesons by employing the LFQM that is based on the constituent
quark picture with light-front dynamics (LFD). Here, we first explain the key
ideas of the LFQM, starting from the Hamiltonian and LFWF. We then show the
expressions for the weak-decay constant and the DA in free space. We emphasize
again that the present approach differs from the BSA approach of Refs. [22,
23, 24], namely the present approach uses the vertex function that is
regulated by the Gaussian functional form, and the meson states are built
through the Bakamjian-Thomas (BT) construction [34, 35], which is a Poincaré
invariant and guarantees that it is independent of any specific kinematics of
any chosen frames [36].

###  II.1 LFWF and effective Hamiltonian

In the LFQM, a meson state is described as a bound state of the constituent
quark and antiquark pair in the noninteracting representation following the BT
construction, where the interaction is included in the meson mass operator to
satisfy the Poincaré group structure [34, 35]. Thus, the interaction is
encoded in the mass eigenfunction. In the present approach, we apply a
variational principle to deal with the mass eigenvalue problem, introducing a
trial wave function in the Gaussian basis including the QCD-motivated
effective Hamiltonian with linear confinement potential [37]. The LFQM has
been successfully applied to various studies of free space meson properties in
Refs. [38, 37, 39, 40, 41] and references therein.

The meson state $\ket{{\cal M}(P,J,J_{z})}$, as a bound state of the
constituent quark $q$ and antiquark $\bar{q}$ with meson momentum $P$ and
total angular momentum $(J,J_{z})$, can be written as

$\displaystyle\ket{{\cal M}(P,J,J_{z})}$ $\displaystyle=$
$\displaystyle\int\left[{\rm d}^{3}{\bf p}_{q}\right]\left[{\rm d}^{3}{\bf
p}_{\bar{q}}\right]2(2\pi)^{3}\delta^{3}\left({\bf P}-{\bf p}_{q}-{\bf
p}_{\bar{q}}\right)$ (1)
$\displaystyle\times\mbox{}\sum_{\lambda_{q},\lambda_{\bar{q}}}\Psi_{\lambda_{q}\lambda_{\bar{q}}}^{JJ_{z}}(x,{\bf
k}_{\perp})\ket{q_{\lambda_{q}}(p_{q})\bar{q}_{\lambda_{\bar{q}}}(p_{\bar{q}})},\quad\quad$

where ${\bf p_{i}}=(p_{i}^{+},{\bf p}_{i\perp})$ and $\left[{\rm d}^{3}{\bf
p}_{i}\right]\equiv{\rm d}p_{i}^{+}{\rm
d}^{2}\mathbf{p}_{i\perp}/[2(2\pi)^{3}]$. Here, we define
$(p_{q},\lambda_{q})$ and $(p_{\bar{q}},\lambda_{\bar{q}})$ as the momentum
and the helicity of quark $(i=q)$ and antiquark $(i={\bar{q}})$, respectively.
The LF internal variables $(x,{\bf k}_{\perp})$ are then denoted as
$x=p^{+}_{q}/P^{+}$ and ${\bf k}_{\perp}={\bf p}_{q\perp}-x{\bf P}_{\perp}$.

The LFWF of the ground state meson in momentum space is defined by

$\displaystyle\Psi^{JJ_{z}}_{\lambda_{q}\lambda_{\bar{q}}}(x,\mathbf{k}_{\bot})=\Phi(x,\mathbf{k}_{\bot})\
\mathcal{R}^{JJ_{z}}_{\lambda_{q}\lambda_{\bar{q}}}(x,\mathbf{k}_{\bot}),$ (2)

where $\Phi(x,\mathbf{k}_{\bot})$ and
$\mathcal{R}^{JJ_{z}}_{\lambda_{q}\lambda_{\bar{q}}}(x,\mathbf{k}_{\bot})$ are
respectively the radial and spin-orbit wave functions, where the latter
distinguishes the vector (V) and pseudoscalar (P) mesons. The
$\mathcal{R}^{JJ_{z}}_{\lambda_{q}\lambda_{\bar{q}}}$ is obtained from the
Melosh transformation [42] and has the covariant forms as

$\displaystyle\mathcal{R}^{JJ_{z}}_{\lambda_{q}\lambda_{\bar{q}}}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}\tilde{M}_{0}}\bar{u}_{\lambda_{q}}(p_{q})\Gamma_{\rm
M}v_{\lambda_{\bar{q}}}(p_{\bar{q}}),$ (3)

with ${\rm M}=$ P or V meson and
$\tilde{M}_{0}\equiv\sqrt{M_{0}^{2}-(m_{q}-m_{\bar{q}})^{2}}$, and the
invariant meson mass $M_{0}^{2}$ is defined as

$\displaystyle
M_{0}^{2}=\frac{\mathbf{k}_{\bot}^{2}+m_{q}^{2}}{x}+\frac{\mathbf{k}_{\bot}^{2}+m_{\bar{q}}^{2}}{1-x}.$
(4)

In the above, the vertices for the pseudoscalar $\Gamma_{\rm P}$ and vector
$\Gamma_{\rm V}$ are given by

$\displaystyle\Gamma_{\rm P}$ $\displaystyle=$ $\displaystyle\gamma_{5},$ (5)
$\displaystyle\Gamma_{\rm V}$ $\displaystyle=$
$\displaystyle-\not{\epsilon}(J_{z})+\frac{\epsilon\cdot(p_{q}-p_{\bar{q}})}{M_{0}+m_{q}+m_{\bar{q}}},$
(6)

where the polarization vectors
$\epsilon^{\mu}(J_{z})=(\epsilon^{+},\epsilon^{-},\bm{\epsilon}_{\perp})$ are
defined by

$\displaystyle\epsilon^{\mu}(\pm 1)$ $\displaystyle=$
$\displaystyle\left(0,\frac{2\bm{\epsilon}_{\perp}(\pm)\cdot{\bf
P}_{\perp}}{P^{+}},\bm{\epsilon}_{\perp}(\pm)\right),$
$\displaystyle\epsilon^{\mu}(0)$ $\displaystyle=$
$\displaystyle\frac{1}{M_{0}}\left(P^{+},\frac{-M^{2}_{0}+{\bf
P}^{2}_{\perp}}{P^{+}},{\bf P}_{\perp}\right),$ (7)

with $\bm{\epsilon}_{\perp}(\pm 1)=\mp\frac{1}{\sqrt{2}}\left(1,\pm i\right)$.
We note that the the spin-orbit wave function is normalized to unity.

In the meson ground state, the trial radial wave function in Gaussian basis is
given by

$\displaystyle\Phi_{1S}(x,\mathbf{k}_{\bot})$ $\displaystyle=$
$\displaystyle\frac{4\pi^{3/4}}{\beta^{3/2}}\sqrt{\frac{\partial
k_{z}}{\partial x}}e^{-{\bf k}^{2}/2\beta^{2}},$ (8)

with $\beta$ as the variational parameter that is related to the size of the
wave function, and the Jacobian factor in Eq. (8) is expressed by

$\frac{\partial k_{z}}{\partial
x}=\frac{M_{0}}{4x(1-x)}\left[1-\frac{(m_{q}^{2}-m_{\bar{q}}^{2})^{2}}{M_{0}^{4}}\right],$
(9)

which takes account of the variable transformation,
$(k_{z},\mathbf{k}_{\perp})$ to $(x,\mathbf{k}_{\perp})$, where
$k_{z}=\left(x-1/2\right)M_{0}+(m^{2}_{\bar{q}}-m^{2}_{q})/{2M_{0}}$. The LFWF
is normalized by

$\displaystyle\int\frac{{\rm d}x{\rm
d}^{2}\mathbf{k}_{\bot}}{2(2\pi)^{3}}\absolutevalue{\Psi(x,\mathbf{k}_{\bot})}^{2}=1.$
(10)

Following the BT construction, the quark and antiquark interactions are
included in the meson mass operator to compute the mass eigenvalue
$M_{q\bar{q}}$, namely,

$\displaystyle(H_{0}+V_{q\bar{q}})\ket{\Psi_{q\bar{q}}}=M_{q\bar{q}}\ket{\Psi_{q\bar{q}}},$
(11)

where $\Psi_{q\bar{q}}$ is the eigenfunction for the meson. In the
calculation, we use the relativistic kinetic energy of the quark and antiquark
given by $H_{0}=\sqrt{m_{q}^{2}+{\bf p}^{2}_{q}}+\sqrt{m_{\bar{q}}^{2}+{\bf
p}^{2}_{\bar{q}}}$, and the quark-antiquark potential is defined by
$V_{q\bar{q}}=V_{\rm Conf}+V_{\rm Coul}+V_{\rm Hyp}$, that consists of the
confining, Coulomb-like, and hyperfine potentials, given by

$\displaystyle V_{\rm Conf}$ $\displaystyle=$ $\displaystyle a+br,$ (12)
$\displaystyle V_{\rm Coul}$ $\displaystyle=$
$\displaystyle-\frac{4\alpha_{s}}{3r},$ (13) $\displaystyle V_{\rm Hyp}$
$\displaystyle=$
$\displaystyle\frac{32\pi\alpha_{s}\expectationvalue{\mathbf{S}_{q}\cdot\mathbf{S}_{\bar{q}}}}{9m_{q}m_{\bar{q}}}\delta^{3}(r),$
(14)

where $a$ and $b$ are parameters for the linear confining potential, and
$\alpha_{s}$ is the strong running coupling, which is taken as a constant
parameter in the free space. (Its in-medium modification will be explained
later.) The $\Braket{\mathbf{S}_{q}\cdot\mathbf{S}_{\bar{q}}}$ yields the
values of $1/4$ and $-3/4$ for the vector and pseudoscalar mesons,
respectively.

To obtain the mass and wave function of the meson ground state, we perform the
variational analysis. The mass eigenvalue of the meson can be computed by
$M_{q\bar{q}}=\Braket{\Psi_{q\bar{q}}}{H_{q\bar{q}}}{\Psi_{q\bar{q}}}=\Braket{\phi_{1S}}{H_{q\bar{q}}}{\phi_{1S}}$,
where the unity of the spin-orbit wave function has to be employed. We then
have an analytic mass formula [39],

$\displaystyle M_{q\bar{q}}$ $\displaystyle=$
$\displaystyle\frac{\beta}{\sqrt{\pi}}\sum_{i=q,{\bar{q}}}z_{i}e^{z_{i}/2}K_{1}\left(\frac{z_{i}}{2}\right)+a+\frac{2b}{\beta\sqrt{\pi}}$
(15)
$\displaystyle-\frac{8\alpha_{s}\beta}{3\sqrt{\pi}}+\frac{32\alpha_{s}\beta^{3}\left<\mathbf{S}_{q}\cdot\mathbf{S}_{\bar{q}}\right>}{9\sqrt{\pi}m_{q}m_{\bar{q}}},\quad$

where $z_{i}=m_{i}^{2}/\beta^{2}$, $K_{n}$ the modified Bessel function of the
second kind of order-$n$. The model parameters including the variational
parameters $\beta$ are obtained by performing the variational principle and by
imposing $\alpha_{s}$ to be the same for all mesons. The detailed procedures
can be found in Ref. [37]. Since our main purpose of the present work is to
investigate the in-medium modifications of the meson properties and structure,
we use the same parameters as those in free space, which are adapted from Ref.
[37]. The explicit values of the parameters used in the present LFQM will be
presented in Sec. V.

###  II.2 Weak-decay constant and distribution amplitude

Here, we review the properties and structure of the mesons in free space in
the LFQM, such as the weak-decay constants and the DAs for the pseudoscalar
and vector mesons [40]. The weak-decay constants are related to the
normalization of the leading twist quark DAs of the corresponding mesons that
dictate the probability of the valence-quark distributions, and they can be
extracted through the hard exclusive reaction processes [43, 44].

The weak-decay constants of the pseudoscalar meson $f_{\rm P}$ and vector
meson $f_{\rm V}$ are defined by

$\displaystyle\bra{0}\bar{q}\gamma^{\mu}\gamma_{5}q\ket{{\rm P}(P)}$
$\displaystyle=$ $\displaystyle if_{\rm P}P^{\mu},$
$\displaystyle\bra{0}\bar{q}\gamma^{\mu}q\ket{{\rm V}(P,J_{z})}$
$\displaystyle=$ $\displaystyle f_{\rm V}M_{\rm V}\epsilon^{\mu}(J_{z}),$ (16)

where $\epsilon^{\mu}(J_{z})$ and $M_{\rm V}$ are the polarization vector and
the mass of the vector meson, respectively. The explicit form of the decay
constants in the LFQM is given by

$\displaystyle f_{\rm M}$ $\displaystyle=$ $\displaystyle
2\sqrt{6}\int_{0}^{1}{\rm d}x\int\frac{{\rm
d}^{2}\mathbf{k}_{\bot}}{2(2\pi)^{3}}\frac{{\Phi}(x,\mathbf{k}_{\bot})}{\sqrt{\mathcal{A}^{2}+\mathbf{k}_{\bot}^{2}}}~{}\mathcal{O}_{\rm
M},\quad\quad$ (17)

where the operators $O_{\rm M}$ are defined by

$\displaystyle\mathcal{O}_{\rm P}$ $\displaystyle=$
$\displaystyle\mathcal{A},$ (18) $\displaystyle\mathcal{O}_{\rm V}$
$\displaystyle=$
$\displaystyle\mathcal{A}+\frac{2\mathbf{k}_{\bot}^{2}}{D_{0}},$ (19)

with $\mathcal{A}=(1-x)m_{q}+xm_{\bar{q}}$ and
$D_{0}=M_{0}+m_{q}+m_{\bar{q}}$.

The DAs of mesons are defined from the light-like separated gauge invariant
free space-to-meson matrix elements, which can be interpreted as the
probability amplitudes to find the hadron in a state with a minimum number of
the Fock constituents and small transverse momentum separation [40]. The
leading twist DAs for the pseudoscalar and vector mesons are obtained from the
plus component of the currents, which are respectively given by

$\displaystyle A^{+}_{P}$ $\displaystyle=$
$\displaystyle\bra{0}\bar{q}(z)\gamma^{+}\gamma_{5}q(-z)\ket{{\rm P}(P)},$
(20) $\displaystyle=$ $\displaystyle if_{\rm P}P^{+}\int^{1}_{0}{\rm d}x\ {\rm
e}^{i\zeta P\cdot z}\phi_{\rm P}(x)\biggr{|}_{z^{+}=z_{\perp}=0},$
$\displaystyle A_{V}^{+}$ $\displaystyle=$
$\displaystyle\bra{0}\bar{q}(z)\gamma^{+}q(-z)\ket{{\rm V}(P,0)},$ (21)
$\displaystyle=$ $\displaystyle f_{\rm V}M_{\rm
V}\epsilon^{+}(0)\int^{1}_{0}{\rm d}x\ {\rm e}^{i\zeta P\cdot z}\phi_{\rm
V}(x)\biggr{|}_{z^{+}=z_{\perp}=0},$

where $\zeta=2x-1$. In the LFQM, the $\phi_{\rm M}(x)$ can be obtained by the
transverse momentum integration of the LFWF:

$\displaystyle\phi_{\rm M}(x)$ $\displaystyle=$
$\displaystyle\frac{2\sqrt{6}}{f_{\rm M}}\int\frac{{\rm
d}^{2}\mathbf{k}_{\bot}}{2(2\pi)^{3}}\frac{{\Phi}(x,\mathbf{k}_{\bot})}{\sqrt{\mathcal{A}^{2}+\mathbf{k}_{\bot}^{2}}}~{}\mathcal{O}_{\rm
M}.\quad$ (22)

We note that the DAs are normalized as

$\int_{0}^{1}{\phi}_{\rm M}(x)\ {\rm d}x=1.$ (23)

##  III Quark properties in nuclear matter

Before proceeding to study the in-medium effect on the meson properties, we
briefly explain the QMC model, where the parameters are fixed to reproduce the
saturation properties of nuclear matter. The QMC model is a quark-based model
of nuclear matter and nuclei with the relativistic mean field approximation
[45, 46, 47, 48]. In this model, the Lorentz-scalar-isoscalar $\sigma$,
Lorentz-vector-isoscalar $\omega$, and Lorentz-vector-isovector $\rho$ mean
fields generated from the surrounding nuclear medium are directly coupled to
the confined light $u$ and $d$ valence quarks. In this regard, the mean fields
modify the light quark masses and energies, and thus modify the internal
structure of nucleons in the nuclear medium. We note that the QMC model is
different from the usual relativistic mean-field (RMF) approach, where the
latter mean-field couples to the point-like nucleons [49, 50]. Thus, in the
QMC model, the coupling constants between the light quarks and mean fields are
fixed to be the same for all the light quarks in any hadrons, once constrained
by the nuclear matter saturation properties.

###  III.1 Quark-based relativistic mean-field model: QMC model

The effective Lagrangian density for the symmetric nuclear matter (SNM) at the
hadronic level reads [45, 7, 46, 47, 48]

$\displaystyle\mathscr{L}_{\mathrm{QMC}}$ $\displaystyle=$
$\displaystyle\mathscr{L}_{\rm nucleon}+\mathscr{L}_{\rm
meson}+\mathscr{L}_{\rm int},$ (24)

and they are given by

$\displaystyle\mathscr{L}_{\rm nucleon}$ $\displaystyle=$
$\displaystyle\bar{\psi}[i\not{\partial}-m_{N}]\psi,$ (25)
$\displaystyle\mathscr{L}_{\rm meson}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\partial_{\mu}\hat{\sigma}\partial^{\mu}\hat{\sigma}-m_{\sigma}^{2}\hat{\sigma}^{2})$
(26)
$\displaystyle-\frac{1}{2}\left[\partial_{\mu}\hat{\omega}_{\nu}(\partial^{\mu}\hat{\omega}^{\nu}-\partial^{\nu}\hat{\omega}^{\mu})-m_{\omega}^{2}\hat{\omega}^{\mu}\hat{\omega}_{\mu}\right],\quad\quad$

where $\psi$, $\hat{\sigma}$, and $\hat{\omega}$ represent the nucleon,
$\sigma$, and $\omega$ field operators, respectively. In the present work, we
do not include the $\rho$ meson field, since we consider the isospin-symmetric
nuclear matter in which the contribution from the $\rho$ mean-field is
vanishing in the Hartree mean-field approximation. The corresponding
interaction Lagrangian density is given by

$\displaystyle\mathscr{L}_{\rm
int}=\tilde{g}_{\sigma}^{N}(\hat{\sigma})\bar{\psi}\psi\hat{\sigma}-g_{\omega}^{N}\hat{\omega}^{\mu}\bar{\psi}\gamma_{\mu}\psi,$
(27)

with the $\sigma$-field dependent $N\sigma$ coupling constant
$\tilde{g}_{\sigma}^{N}(\hat{\sigma})$ and the $N\omega$ coupling constant
$g_{\omega}^{N}$. We may also write the Lagrangian density as

$\displaystyle\mathscr{L}$ $\displaystyle=$
$\displaystyle\bar{\psi}[i\not{\partial}-m_{N}^{*}(\hat{\sigma})-g_{\omega}^{N}\hat{\omega}^{\mu}\gamma_{\mu}]\psi+\mathscr{L}_{\rm
meson},$ (28)

where the nucleon effective mass at a given density is defined by

$\displaystyle
m_{N}^{*}(\hat{\sigma})=m_{N}-\tilde{g}_{\sigma}^{N}(\hat{\sigma})\hat{\sigma}.$
(29)

The $\tilde{g}_{\sigma}^{N}(\hat{\sigma})$ coupling appears in the nucleon
effective mass and it modifies the nucleon mass in a nonlinear manner in
$\sigma$ field, while the $g^{N}_{\omega}$ coupling modifies the nucleon four-
momentum. It is important to emphasize that in the QMC model, the couplings of
the meson fields are applied directly to the light quarks and we solve the
Dirac equation of the light quarks in the presence of the meson fields
generated by the surrounded nuclear medium self-consistently. The quark-meson
coupling constant with the $\sigma$ field is encoded in
$\tilde{g}_{\sigma}^{N}(\hat{\sigma})$ at the hadron level.

In the mean-field approximation, we replace the meson field operators with
their constant mean field expectation values as
$\hat{\sigma}\to\sigma=\expectationvalue{\hat{\sigma}}$ and
$\hat{\omega}_{\mu}\to\delta_{\mu,0}\,\omega=\expectationvalue{\hat{\omega}_{\mu}}$,
where we consider the nuclear matter at rest so that only the time component
of $\omega_{\mu}$ survives.

At the nucleon level, the equations of motion of the meson fields can be
obtained as

$\displaystyle(\Box+m_{\sigma}^{2})\sigma$ $\displaystyle=$
$\displaystyle\left(-\frac{\partial
m_{N}^{*}(\sigma)}{\partial\sigma}\right)(\bar{\psi}\psi)=\tilde{g}_{\sigma}^{N}(\sigma)\rho_{s},\quad\quad$
(30) $\displaystyle(\Box+m_{\omega}^{2})\omega$ $\displaystyle=$
$\displaystyle
g_{\omega}^{N}(\bar{\psi}\gamma^{0}\psi)=g_{\omega}^{N}(\psi^{\dagger}\psi)=g_{\omega}^{N}\rho,$
(31)

where in nuclear matter, the d’Alembert operator $\Box\to 0$ is made, and
$\rho_{s}$ and $\rho$ are the nucleon scalar and vector (baryon) densities,
respectively. The Dirac equation for the nucleon is given by

$\displaystyle(i\not{\partial}-g_{\omega}^{N}\omega\gamma^{0}-m_{N}^{*}(\sigma))\psi=0.$
(32)

Note that the effective nucleon mass enters in this equation and Eq. (30) via,

$\displaystyle-\frac{\partial m_{N}^{*}(\sigma)}{\partial\sigma}$
$\displaystyle=$
$\displaystyle\tilde{g}_{\sigma}^{N}(\sigma)=g_{\sigma}^{N}C_{N}(\sigma),$
(33)

where

$\displaystyle C_{N}(\sigma)=\frac{S_{N}(\sigma)}{S_{N}(\sigma=0)}.$ (34)

The $C_{N}(\sigma)$ is the so-called scalar polarizability that describes the
nucleon response to the external scalar field [7]. If the nucleon is assumed
as a point-like particle, $C_{N}(\sigma)=1$. In the QMC model, the meson-
nucleon coupling $(g_{\sigma}^{N},g_{\omega}^{N})$ are defined by the quark-
meson coupling $(g_{\sigma}^{q}$, $g_{\omega}^{q})$ as

$\displaystyle g_{\sigma}^{N}$ $\displaystyle=$
$\displaystyle\tilde{g}_{\sigma}^{N}(\sigma=0)=3g_{\sigma}^{q}S_{N}(\sigma=0),$
(35) $\displaystyle g_{\omega}^{N}$ $\displaystyle=$ $\displaystyle
3g_{\omega}^{q},$ (36)

where $S_{N}(\sigma)$ is computed in the MIT bag model. In the above, factor
three reflects the fact that the nucleon is made of three light valence
quarks.

From Eqs. (30) and (31), the vector and scalar meson fields are respectively
calculated as

$\displaystyle\omega$ $\displaystyle=$
$\displaystyle\frac{g_{\omega}^{N}\rho}{m_{\omega}^{2}},$ (37)
$\displaystyle\sigma$ $\displaystyle=$
$\displaystyle\frac{g_{\sigma}^{N}\rho_{s}}{m_{\sigma}^{2}}C_{N}(\sigma),$
(38)

where the nuclear density $\rho$ and scalar density $\rho_{s}$ are given by

$\displaystyle\rho$ $\displaystyle=$ $\displaystyle\frac{4}{(2\pi)^{3}}\int
d^{3}\mathbf{k}\ \Theta(k_{F}-k)=\frac{2k_{F}^{3}}{3\pi^{2}},$ (39)
$\displaystyle\rho_{s}$ $\displaystyle=$
$\displaystyle\frac{4}{(2\pi)^{3}}\int d^{3}\mathbf{k}\
\Theta(k_{F}-k)\frac{m_{N}^{*}(\sigma)}{\sqrt{m_{N}^{*2}(\sigma)+k^{2}}},$
(40)

with $k=|\mathbf{k}|$ and $\Theta(k_{F}-k)$ being the step function that
guarantees the integral is performed up to the nucleon Fermi momentum $k_{F}$
that can be written in terms of the nuclear density $\rho$. The factor of four
represents spin and isospin degeneracy. As seen in Eq. (38), we solve the
self-consistent equation for the $\sigma$ mean-field to determine its value at
each nuclear density.

Once we obtain the $\sigma$ and $\omega$ mean fields, it is straightforward to
compute the total energy per nucleon, which is given by

$\displaystyle\frac{E_{\rm tot}}{A}$ $\displaystyle=$
$\displaystyle\frac{1}{\rho}\biggl{[}\frac{4}{(2\pi)^{3}}\int d^{3}\mathbf{k}\
\Theta(k_{F}-k)\sqrt{m_{N}^{*2}(\sigma)+k^{2}}$ (41)
$\displaystyle+\frac{1}{2}g_{\sigma}^{N}C_{N}(\sigma)\sigma\rho_{s}+\frac{1}{2}g_{\omega}^{N}\omega\rho\biggr{]}.$

Later, we determine the model parameters by the fit to the extracted nuclear
matter saturation properties at the saturation density $\rho_{0}=0.15$ fm-3
($k_{F}=1.305$ fm-1), such as the negative of binding energy ($E_{\rm
tot}/A-m_{N}$), $-15.7$ MeV.

###  III.2 MIT bag model

As discussed earlier, the nucleon-meson couplings are defined based on the
quark-meson couplings. In the standard QMC model [45, 46, 47, 48], this is
done by using the MIT Bag model for the nucleon (hadrons) and solving the
Dirac equations for the quarks and antiquarks with the presence of the meson
mean fields in the nuclear medium,

$\displaystyle V_{\sigma}^{q}$ $\displaystyle=$ $\displaystyle
g_{\sigma}^{q}\sigma,$ (42) $\displaystyle V_{\omega}^{q}$ $\displaystyle=$
$\displaystyle g_{\omega}^{q}\omega,$ (43)

where $\sigma$ and $\omega$ are the same constant mean fields as discussed in
Eqs. (37) and (38). Here we consider $q=u$ or $d$ and $Q=s,c,$ or $b$ quark
confined inside the bag of a hadron in symmetric nuclear matter at the
position $z=(t,\mathbf{r})$ with $r=|\mathbf{r}|\leq$ bag radius.

The Dirac equations for the quark and antiquark in the presence of the mean-
field potentials are given by

$\displaystyle\biggl{[}i\not{\partial}-(m_{q}-V_{\sigma}^{q})\mp\gamma^{0}V_{\omega}^{q}\biggr{]}\begin{pmatrix}\psi_{q}(z)\\\
\psi_{\bar{q}}(z)\\\ \end{pmatrix}=0,$ (44)
$\displaystyle\left[i\not{\partial}-m_{Q}\right]\begin{pmatrix}\psi_{Q}(z)\\\
\psi_{\bar{Q}}(z)\\\ \end{pmatrix}=0,$ (45)

where we assume the SU(2) symmetry for the light quarks, $m_{q}=m_{u}=m_{d}$.
While the vector potential shifts the quark energy in the nuclear medium, the
scalar potential modifies the quark mass as

$\displaystyle m_{q}^{*}=m_{q}-V_{\sigma}^{q},$ (46)

so that the quark effective mass is reduced by the scalar $\sigma$ potential
in the nuclear medium, and thus $m_{q}^{*}$ can be negative when a small free
space mass value $m_{q}$ is used, but it is nothing but the reflection of the
attractive potential, and thus a naive interpretation of “mass” should not be
applied. It should be noted again that the mean fields are coupled only to the
light quarks and antiquarks. In the QMC model, the masses of the quark
$Q=s,c,b$ are assumed to be the same in the nuclear medium as in free space
($m_{Q}^{*}=m_{Q}$) [but energies may be modified by the modifications of the
hadron bag radius when a light quark is contained in the hadron], since the
$\sigma$ mean-field is not coupled to the heavier quarks in the model, based
on the fact that the heavy quark chiral condensates are expected to be
modified only slightly in the cold nuclear medium, or, their masses are mostly
due to the Higgs mechanism. (See the introduction part of Ref. [51] for
detailed discussions and some evidence, as well as references on this issue.)

Here, we obtain the static solution for the ground state quark or antiquark,
where the Hamiltonian is time-independent, and the wave function can be
written as

$\displaystyle\psi(z)=\psi(r)\exp{-i\varepsilon^{*}t/R^{*}},$ (47)

with the in-medium bag radius $R^{*}$. The eigenenergies in units of $1/R^{*}$
are given by

$\displaystyle\begin{pmatrix}\varepsilon^{*}_{q}\\\
\varepsilon^{*}_{\bar{q}}\\\ \end{pmatrix}$ $\displaystyle=$
$\displaystyle\Omega_{q}^{*}\pm R^{*}V_{\omega}^{q},$ (48)

with

$\displaystyle\Omega^{*}_{q}$ $\displaystyle=$
$\displaystyle\sqrt{x_{q}^{*2}+(m_{q}^{*}R^{*})^{2}},$ (49)

where $x_{q}^{*}$ is the lowest mode bag eigenvalue. The normalized ground
state quark eigenfunction can be written as

$\displaystyle\psi(z)=\frac{N{\rm
e}^{-i\varepsilon^{*}t/R^{*}}}{\sqrt{4\pi}}\begin{pmatrix}j_{0}(x_{q}^{*}r/R^{*})\\\
i\beta_{q}^{*}j_{1}(x_{q}^{*}r/R^{*})\
{\bm{\sigma}}\cdot\hat{r}\end{pmatrix}\chi_{m},$ (50)

where $\chi_{m}$ is the spin function, the spherical Basel functions are given
by

$\displaystyle j_{0}(r)$ $\displaystyle=$ $\displaystyle\frac{\sin r}{r},$
(51) $\displaystyle j_{1}(r)$ $\displaystyle=$ $\displaystyle\frac{\sin
r}{r^{2}}-\frac{\cos r}{r}.$ (52)

The normalization $N$ is obtained from

$\displaystyle\int_{0}^{R^{*}}{\rm d}^{3}\mathbf{r}\
\psi^{\dagger}(r)\psi(r)=1,$ (53)

where the integration is up to the radius $R^{*}$ meaning that the quarks only
exist (confined) in the spherical cavity. We then obtain

$\displaystyle
N^{-2}=2R^{*3}j_{0}^{2}(x_{q}^{*})\frac{[\Omega_{q}^{*}(\Omega_{q}^{*}-1)+m_{q}^{*}R^{*}/2]}{x_{q}^{*2}}.$
(54)

The eigenfrequency of the quark is obtained by imposing the continuity of the
quark eigenfunction at the bag boundary $r=R^{*}$ which yields the relation,

$\displaystyle j_{0}(x_{q}^{*})=\beta_{q}^{*}j_{1}(x_{q}^{*}),$ (55)

with

$\displaystyle\beta_{q}^{*}=\sqrt{\frac{\Omega_{q}^{*}-m_{q}^{*}R^{*}}{\Omega_{q}^{*}+m_{q}^{*}R^{*}}}.$
(56)

By solving the above equation, we obtain the lowest positive eigenvalue of
$x_{q}^{*}$. The effective nucleon mass in the MIT bag model is computed as

$\displaystyle
m_{N}^{*}(\sigma)=\frac{3\Omega^{*}_{q}-Z_{N}}{R^{*}}+\frac{4\pi R^{*3}}{3}B,$
(57)

where $Z_{N}$ is the corrections from the gluon fluctuation and center-of-mass
motion [46] and $B$ is the density-independent bag pressure. It is worth
noting that the bag volume energy provides an inward pressure while the
kinetic energy gives an outward pressure. As a consequence, there is an
equilibrium where the nucleon mass is minimized and stabilized by

$\displaystyle\left.\frac{{\rm d}m_{N}^{*}(R^{*})}{{\rm
d}R^{*}}\right|_{R^{*}=R_{N}^{*}}=0,$ (58)

which determines the $R^{*}_{N}$ and mass of the nucleon, self-consistently
with $x_{q}^{*}$. We also clarify that the bag radius ($R^{*}_{N}$) is not
observable, which is different from the nucleon radius. So, to estimate the
nucleon radius (in the nuclear medium as well as in free space), one needs to
compute it from the quark wave function. Lastly, the quantity associated with
the scalar polarizability $C_{N}(\sigma)$,
$S_{N}(\sigma)=S_{N}(\sigma=0)\,C_{N}(\sigma)$ in the $\sigma$-dependent
$N\sigma$ coupling constant $\tilde{g}_{\sigma}^{N}(\sigma)$, is computed as

$\displaystyle S_{N}(\sigma)$ $\displaystyle=$
$\displaystyle\int_{0}^{R^{*}}{\rm d}^{3}\mathbf{r}\ \bar{\psi}(r)\psi(r),$
(59) $\displaystyle=$
$\displaystyle\frac{\Omega_{q}^{*}/2+m_{q}^{*}R^{*}(\Omega_{q}^{*}-1)}{\Omega_{q}^{*}(\Omega_{q}^{*}-1)+m_{q}^{*}R^{*}/2}.$

This provides one of the origins for the novel saturation properties of
nuclear matter starting from the quark degrees of freedom, basing on the quark
structure of the nucleon. With this ingredient, we do not have to introduce
the nonlinear couplings of the meson fields in the effective Lagrangian
density at the hadronic level to obtain a reasonable incompressibility value
in the range $K\simeq$ 200 - 300 MeV, which is usually adopted in many
sophisticated relativistic mean field models at the hadron level [49] for
obtaining a reasonable value $K$.

##  IV Properties and structure of meson in nuclear medium

In this section, we explore the mechanism of how the in-medium modified quark
properties simulated in the QMC model affect the meson properties in nuclear
medium, such as weak-decay constants and DAs described in the LFQM. Two
important inputs obtained from the QMC model applied are: (i) the in-medium
light quark effective mass modified by the scalar $\sigma$ mean-field, and
(ii) the in-medium light quark energy modified by the vector potential. Since
we work on the light and heavy-light mesons, the medium effects vary for each
type of meson depending on the quark constituents of mesons. We will explain
this in the following. In this work, the Gaussian parameter $\beta$ is not
assumed to be modified in the medium. This may be justified by the fact that
it is associated with a short-range scale in the meson wave function [23]. One
may also expect that $\beta$ decreases in the medium as the meson radius
increases. In that case, the decay constant will decrease much faster in the
medium as studied by one of us in the previous work [24].111 The parameter
$\beta$ is related to the regulator mass $m_{R}$ in Ref. [24]. A quick
decrease in the decay constant is due to the fact that it is more sensitive to
the origin of the wave function. However, such a density dependence is not
well understood either and will add only model ambiguities. Therefore, we
assume the density-independent $\beta$ in this exploratory study.

The light quark or light antiquark mass is modified by the scalar potential as

$\displaystyle m_{q}^{*}=m_{q}-V_{\sigma}^{q},$ (60)

while the light quark and light antiquark energies $p^{*0}_{i}$ are modified
by the vector potential:

$\displaystyle p^{*0}_{i}=\begin{cases}E_{q}^{*}+V_{\omega}^{q},&{\rm for\
light\ quark,}\\\ E_{\bar{q}}^{*}-V_{\omega}^{q},&{\rm for\ light\
antiquark,}\\\ \end{cases}$ (61)

with $i=q,\bar{q}$ and
$E^{*}_{q}=E^{*}_{\bar{q}}=\sqrt{m_{q}^{*2}+\mathbf{p}^{2}_{q}}$. Then, the
total energy of meson $P^{*0}$ is given by

$\displaystyle P^{*0}=\begin{cases}E_{M}^{*},&{\rm for\ }(q\bar{q}),\\\
E_{M}^{*}+V_{\omega}^{q},&{\rm for\ }(q\bar{Q}),\\\
E_{M}^{*}-V_{\omega}^{q},&{\rm for\ }(Q\bar{q}),\\\ \end{cases}$ (62)

where $E_{M}^{*}=\sqrt{M^{*2}+\mathbf{P}^{2}}$ with $M^{*}$ being the in-
medium meson mass, and the vector potential cancels in the case of
$(q\bar{q})$ mesons, and no vector potentials are acted for $Q$. The variable
$x$, defined by the plus-component ratio of the quark to meson momenta in free
space,

$\displaystyle
x=\frac{p^{+}_{q}}{P^{+}}=\frac{p^{0}_{q}+p^{3}_{q}}{P^{0}+P^{3}}=\frac{E_{q}+p^{3}_{q}}{E_{M}+P^{3}},$
(63)

will be modified in the medium by the scalar and vector potentials.

For the $q\bar{q}$ mesons, we define

$\displaystyle p_{q}^{*+}=p_{\bar{q}}^{*+}$ $\displaystyle\equiv$
$\displaystyle E_{q}^{*}+p^{*3}_{q},$ (64) $\displaystyle P^{*+}$
$\displaystyle\equiv$ $\displaystyle E^{*}_{M}+P^{*3},$ (65) $\displaystyle
x^{*}$ $\displaystyle\equiv$ $\displaystyle\frac{p^{*+}_{q}}{P^{*+}},$ (66)

and the longitudinal light quark ($q$) momentum is modified by

$\displaystyle x$ $\displaystyle\to$
$\displaystyle\tilde{x}^{*}=\frac{p_{q}^{*+}+V_{\omega}^{q}}{P^{*+}}=x^{*}+\frac{V_{\omega}^{q}}{P^{*+}},$
(67)

where the vector potentials for the $q$ and $\bar{q}$ cancel for the
$q\bar{q}$ mesons. With this new definition, we have to shift the longitudinal
quark momentum newly denoted by $x\to\tilde{x}^{*}$ because
$(p_{q}^{+},P^{+})\neq(p_{q}^{*+},P^{*+})$ in computing the weak decay
constant in the medium,

$\displaystyle\tilde{x}^{*}=x^{*}+\frac{V_{\omega}^{q}}{P^{*+}},$ (68)

and then, the weak-decay constant is calculated as

$\displaystyle f_{\rm M}^{*}$ $\displaystyle=$ $\displaystyle
2\sqrt{6}\int_{-\frac{V_{\omega}^{q}}{P^{*+}}}^{1-\frac{V_{\omega}^{q}}{P^{*+}}}{\rm
d}x^{*}\int\frac{{\rm d}^{2}\mathbf{k}_{\bot}}{2(2\pi)^{3}}$ (69)
$\displaystyle\times\frac{{\Phi}(\tilde{x}^{*},\mathbf{k}_{\bot})}{\sqrt{\mathcal{A}(\tilde{x}^{*})^{2}+\mathbf{k}_{\bot}^{2}}}~{}\mathcal{O}_{\rm
M}(\tilde{x}^{*},\mathbf{k}_{\perp}).\quad$

As seen in Eq. (69), the integration limits of $x^{*}$ are shifted by
${V_{\omega}^{q}}/{P^{*+}}$. We then evaluate the equation by integration over
$x^{*}$, which will give the same result as the decay constant expression in
Eq. (17). This can be understood because the upper limit of the integration
will cancel with the lower limit of integration when integrating over $x^{*}$.
So, the final expression for the decay constant in the medium, using the
variable $\tilde{x}^{*}$, is given by222 $\tilde{x}^{*}$ is a dummy variable
to emphasize the difference with $x$.

$\displaystyle f_{\rm M}^{*}$ $\displaystyle=$ $\displaystyle
2\sqrt{6}\int_{0}^{1}{\rm d}\tilde{x}^{*}\int\frac{{\rm
d}^{2}\mathbf{k}_{\bot}}{2(2\pi)^{3}}$ (70)
$\displaystyle\times\frac{{\Phi}(\tilde{x}^{*},\mathbf{k}_{\bot})}{\sqrt{\mathcal{A}(\tilde{x}^{*})^{2}+\mathbf{k}_{\bot}^{2}}}~{}\mathcal{O}_{\rm
M}(\tilde{x}^{*},\mathbf{k}_{\perp}).\quad$

This equation shows that the scalar potential modifies the weak decay constant
in the medium through the effective quark mass $m_{q}^{*}$ [22], while the
vector potential modifies the energies but they cancel each other between the
quark and antiquark as in Eq. (70). Note that, since $f_{\rm M}^{*}$ is
computed with the plus current $J^{+}=J^{0}+J^{3}$ in the LFQM, we cannot
separate the time and space components, as discussed in Refs. [52, 20].

For the $q\bar{Q}$ and $Q\bar{q}$ mesons, both the scalar and vector
potentials contribute explicitly to the meson four-momenta and weak-decay
constants. The longitudinal momenta for the $q$ and $\bar{q}$ in medium are
given by

$\displaystyle x$ $\displaystyle\to$
$\displaystyle\begin{cases}\tilde{x}^{*}=\dfrac{p_{q}^{*+}+V_{\omega}^{q}}{P^{*+}+V_{\omega}^{q}}=\dfrac{x^{*}+V_{\omega}^{q}/P^{*+}}{(1+V_{\omega}^{q}/P^{*+})},&{\rm
for\ }(q\bar{Q}),\\\ \\\
\tilde{x}^{*}=\dfrac{p_{q}^{*+}-V^{q}_{\omega}}{P^{*+}-V^{q}_{\omega}}=\par\dfrac{x^{*}-V^{q}_{\omega}/P^{*+}}{(1-V_{\omega}^{q}/P^{*+})},&{\rm
for\ }(Q\bar{q}).\end{cases}$

In such cases, we can define the variable of the integration by

$\displaystyle{\rm d}x^{*}=(1\pm V_{\omega}^{q}/P^{*+})\ {\rm
d}\tilde{x}^{*},$ (72)

and the meson weak decay constants for the $q\bar{Q}$ and $Q\bar{q}$ mesons in
the medium can be written as

$\displaystyle f_{\rm M}^{*}$ $\displaystyle=$ $\displaystyle
2\sqrt{6}\int_{0}^{1}{\rm d}\tilde{x}^{*}\int\frac{{\rm
d}^{2}\mathbf{k}_{\bot}}{2(2\pi)^{3}}\left(1\pm\frac{V_{\omega}^{q}}{P^{*+}}\right)$
(73)
$\displaystyle\times\frac{{\Phi}(\tilde{x}^{*},\mathbf{k}_{\bot})}{\sqrt{\mathcal{A}(\tilde{x}^{*})^{2}+\mathbf{k}_{\bot}^{2}}}~{}\mathcal{O}_{\rm
M}(\tilde{x}^{*},\mathbf{k}_{\perp}).\quad$

Our final expressions for the weak-decay constants for the heavy-light mesons
with vector potential in the medium are similar to those of the quark
distribution in the medium with vector potential obtained in Ref. [53].333The
quark distribution with and without vector potential differs by the factor of
$(\tilde{P}^{*+}/P^{*+})$ [53], which is equivalent to the $(1\pm
V_{q}^{\omega}/P^{*+}\to 1)$ in this work. In the meson rest frame, we have
$P^{*+}=E^{*}_{M}+P^{*3}=\sqrt{M^{*2}+\mathbf{P}^{2}}+P^{*3}=M^{*}$. In our
LFQM, the meson mass should be replaced by the interaction-independent
invariant mass $M_{0}^{*}$. However, in the case of the weak decay constant
for the $q\bar{Q}$ and $Q\bar{q}$ mesons, we obtain that the decay constants
are frame-dependent because of the $V^{q}_{\omega}/P^{*+}$ appearance in Eq.
(73). In fact, since $V^{q}_{\omega}$ is the time component of the vector mean
field obtained in the nuclear matter rest frame, this should also be Lorentz
transformed for the general frame with nonzero relative velocity case between
the frames.

Table 1: The quark masses and $\beta$ parameters used in the present work in units of GeV. The LFQM parameters are given by $a=-0.724$ GeV, $b=0.18$ GeV2, and $\alpha_{s}=0.313$ (this will be modified in medium, to be discussed later), which are adapted from Ref. [37]. $m_{q}$ | $m_{s}$ | $m_{c}$ | $m_{b}$ | $\beta_{q\bar{q}}$ | $\beta_{q\bar{s}}$ | $\beta_{q\bar{c}}$ | $\beta_{q\bar{b}}$
---|---|---|---|---|---|---|---
0.22 | 0.45 | 1.8 | 5.2 | 0.3659 | 0.3886 | 0.4679 | 0.5266
Table 2: The predicted masses and weak-decay constants of the light and heavy-light pseudoscalar and vector mesons in comparison with experimental data, adapted from Ref. [37]. The underlined experimental data represents the model’s input.  |  $M_{\rm expt}$ [MeV] |  $M_{\rm theo}$ [MeV] |  $f_{\rm expt}$ [MeV] |  $f_{\rm theo}$ [MeV]
---|---|---|---|---
$\pi$ | 135 | 135 | 130 | 130
$\rho$ | 770 | 770 | 216 | 247
$K$ | 498 | 478 | 156 | 162
$K^{*}$ | 892 | 850 | 217 | 256
$D$ | 1865 | 1836 | 206 | 197
$D^{*}$ | 2007 | 1998 | … | 239
$B$ | 5279 | 5235 | 188 | 171
$B^{*}$ | 5325 | 5315 | … | 186

In Section V, we will present the results by dropping the factor of $(1\pm
V_{\omega}^{q}/P^{*+})$ in the weak decay constants. This can be understood as
the average of the decay constant for the meson multiplet as
$f^{*}=(f_{q\bar{Q}}^{*}+f_{Q\bar{q}}^{*})/2.$ Analogously, this “average”
will also be practiced for the calculation of the in-medium modifications of
DAs.

##  V Numerical result

In this section, we present our numerical results for the in-medium
modifications of the weak-decay constants and DAs for the light and heavy-
light pseudoscalar and vector mesons in symmetric nuclear matter. The results
are shown in Figs. 4-11. Before discussing the obtained results, we first
explain the model parameters of the LFQM and the QMC model.

###  V.1 Meson mass and weak-decay constant
in free space

The model parameters of the LFQM (in free space) are determined by fitting the
mass spectra of the $\pi$ and $\rho$ mesons, as practiced in Ref. [37]. The
parameters determined are shown in Table 1. With these parameters, the
predicted masses and weak-decay constants for the light and heavy-light
pseudoscalar and vector mesons are in reasonable agreement with the data, as
shown in Table 2. We note that a further improvement of the model can be done
by modifying the trial wave function and effective Hamiltonian [38] and
reference therein. Using the same parameters given in Table 1, we study the
meson properties in symmetric nuclear matter with the help of the QMC model
inputs for the in-medium light quark properties.

###  V.2 Nuclear matter

In this section, we explain the QMC model parameters as well as those of the
MIT bag model. The quark-meson coupling constants are determined by the fit to
the equation of state (EoS) extracted by the empirical data.

####  V.2.1 Bag parameter

Figure 1:  Dependence of the nucleon mass $m_{N}$ on the nucleon bag radius $R_{N}$ for different values of the free space quark masses, $m_{q}=$ 5 and 220 MeV. The physical nucleon mass $m_{N}=939$ MeV (input) is achieved by the minimization condition, $\frac{{\rm d}m_{N}}{{\rm d}R}\bigr{|}_{R=R_{N}}=0$ with the input $R_{N}=0.8$ fm, by varying unknown values to determine $B^{1/4},Z_{N}$ and $x_{q}$. Once $B^{1/4}$ and $Z_{N}$ are determined by the minimization condition, $R_{N}$ dependence of $m_{N}$ is calculated with the fixed values of $B^{1/4}$ and $Z_{N}$, and with $x_{q}$ obtained by solving $j_{0}(x_{q})=\beta_{q}j_{1}(xq)$ ($R_{N}$ dependent), and all the relevant values into the MIT bag mass formula Eq. (57) but for the free space ($\sigma=0$).  Figure 2:  (a) The total energy per nucleon [$E_{\rm tot}/A-m_{N}$], minimized at the saturation density $\rho_{0}=0.15$ fm-3 with the negative of binding energy $-15.7$ MeV, (b) the effective nucleon mass $m_{N}^{*}$, (c) the mean-field potentials of $V_{\omega}^{q}$ and $-V_{\sigma}^{q}$, and (d) the light-quark effective mass with $m_{q}=$ 5 and 220 MeV. Table 3: The bag parameters are determined by reproducing the nucleon mass ($m_{N}=939$ MeV) and radius ($R_{N}=0.8$ fm) in free space for two different values of the quark masses. The lowest positive eigenvalue $x_{q}$ and a constant $S_{N}(\sigma=0)$, associated with the scalar polarizability and used to define the $g^{N}_{\sigma}$ coupling constant, are also quoted. (See also Eq. (36).)  $m_{q}$ [MeV] |  $B^{1/4}$ [MeV] | $Z_{N}$ | $x_{q}$ | $S_{N}(\sigma=0)$
---|---|---|---|---
5 | 170 | 3.295 | 2.052 | 0.483
220 | 148 | 4.327 | 2.368 | 0.609

In the QMC model, the bag model parameters $(Z_{N}$ and $B)$ are fitted to the
nucleon mass and the bag radius in free space,

$\displaystyle m_{N}$ $\displaystyle=$ $\displaystyle 939\ {\rm MeV},$ (74)
$\displaystyle R_{N}$ $\displaystyle=$ $\displaystyle 0.8\ {\rm fm},$ (75)

together with the mass stability/minimization condition which yield the bag
pressure $B$ and $Z_{N}$ values in free space. Here, we show the results for
two different values of the quark masses with $m_{q}=$ 5 MeV and 220 MeV. The
values of the obtained bag parameters are summarized in Table 3. The results
for the $R_{N}$ dependence of nucleon mass in free space for two different
quark masses are also shown in Fig. 1. It clearly shows that the nucleon mass
is minimized at a radius of 0.8 fm, as it should be by the
stability/minimization for the both quark mass values. It is worth noting that
the $m_{q}=$ 5 MeV is the standard input value in the QMC model [7]. In
addition, when we use the constituent quark mass, the values of the bag
parameters are insignificantly changing.

####  V.2.2 Equation of state

Here, we show the EoS of nuclear matter for $m_{q}=5$ and 220 MeV, where the
bag parameters are determined in free space. As shown in Table 4, the obtained
coupling constants $g_{\sigma}^{N}$ and $g_{\omega}^{N}$ for $m_{q}=220$ MeV
are slightly larger than those for $m_{q}=5$ MeV, where both cases are
determined by fitting the negative of binding energy $-15.7$ MeV at saturation
density $\rho_{0}=0.15$ fm-3 ($k_{F}=1.305$ fm-1).

Table 4: Coupling constants and the incompressibility $K$ calculated by the QMC model. The bag model parameters for each quark mass value are determined in free space. The coupling constants are determined by the negative of binding energy $-15.7$ MeV at the saturation density $\rho_{0}=$ 0.15 fm-3 ($k_{F}=$ 1.305 fm-1). (See also Eq. (36).)  $m_{q}$ [MeV] | $(g_{\sigma}^{N})^{2}/4\pi$ | $(g_{\omega}^{N})^{2}/4\pi$ |  $m_{N}^{*}$ [MeV] |  $K$ [MeV]
---|---|---|---|---
5 | 5.39 | 5.30 | 755 | 279
220 | 6.40 | 7.57 | 699 | 321

Figures 2 (a) and (b) display the total energy per nucleon and effective
nucleon mass versus nuclear density ratio $\rho/\rho_{0}$ for two different
quark mass values, respectively. For the larger quark mass value, the larger
incompressibility $K$ is obtained. In comparison with empirical values of $K$,
we find that the obtained values of $K$ for both cases are consistent with the
empirical range $K=200-300$ MeV [54], although the $K$ for $m_{q}=220$ MeV is
slightly larger about 7%. The density dependence of the energy per nucleon
[$(E_{\rm tot}/A)-m_{N}$] for both quark mass cases have very similar values
up to $\rho\simeq\rho_{0}$, and then they begin to show the difference as
nuclear density increases in the range $\rho\gtrsim\rho_{0}$. Figure 2 (b)
shows that the effective nucleon mass for $m_{q}=220$ MeV decreases faster as
nuclear density increases than that for 5 MeV. The values of incompressibility
and the effective nucleon mass at saturation density $\rho_{0}$ for two
$m_{q}$ values are given in Table 4.

Figure 3:  (a) Modifications of $\alpha_{s}^{*}$ in the nuclear medium with
two different forms, and (b) $K^{+}$ excitation energy $K^{+}$ at rest,
$\omega_{K^{+}}\equiv M^{*}_{K^{+}}+V^{q}_{\omega}$
($M^{*}_{K^{\pm}}=M^{*}_{K,\bar{K}}$) in the nuclear medium. We have
determined the parameters to reproduce the repulsive $K^{+}$ total potential,
$\omega_{K^{+}}=M^{*}_{K^{+}}+V^{q}_{\omega}=M_{K^{+}}+(20-30)$ MeV at
$\rho_{0}$.

Besides the energy per nucleon and effective nucleon mass, we also show the
mean-field potentials for the light quarks, which should also apply to the
light quarks in the mesons. The mean-field potentials, $V_{\omega}^{q}$ and
$-V_{\sigma}^{q}$ are depicted in Fig. 2 (c). One can notice that the
$-V_{\sigma}^{q}$ potentials have similar density dependence for both quark
mass values. This indicates that the slope of the effective quark mass
$m_{q}^{*}$ is rather similar, although the $m_{q}$ values in free space are
different as shown in Fig. 2 (d). In addition, the $V_{\omega}^{q}$ is
slightly larger for $m_{q}=220$ MeV than that for $m_{q}=$ 5 MeV, indicating
that the light quark with a larger $m_{q}$ value would experience more
repulsion. (For a larger value of e.g., $m_{q}=430$ MeV, the differences for
$m_{N}^{*}$ and the light quark mean field potentials from those of the
$m_{q}=5$ MeV become more evident [24].) Using the obtained mean-field
potentials for the light (anti)quarks and the effective (anti)quark masses, we
study the meson properties in symmetric nuclear matter.

Recall that the effective quark mass in the nuclear medium is nothing but the
reflection of the attractive scalar potential, and thus one should not regard
that it should be positive which applies to the usual physical mass of the
particle ($m_{q}$ and $m_{q}^{*}$ are not observables).

###  V.3 Modifications of $\alpha_{s}$ in nuclear medium

Before addressing the weak-decay constant and distribution amplitude in the
nuclear medium, we comment on the modifications of $\alpha_{s}$ in Eqs.
(13)-(15) in the medium.

As reported in Refs. [55, 56] (and references therein), it is important to
satisfy the constraint on the in-medium $K^{+}$ total potential that was
extracted by heavy-ion simulation and experimental data, which implies to be
20-30 MeV repulsive at $\rho_{0}$, or equivalently, the excitation energy
$\omega_{K^{+}}\equiv M^{*}_{K^{+}}+V^{q}_{\omega}$
($M^{*}_{K^{\pm}}=M^{*}_{K,\bar{K}}$) at $\rho_{0}$ to be
$\omega_{K^{+}}=M_{K^{+}}+(20-30)$ MeV with $M_{K^{+}}$ being the free mass.
This constraint should also be satisfied in the present approach. In the QMC
model, $\omega_{K^{+}}$ can be calculated directly [55], but the naive result
gives a small negative value for the $K^{+}$ total potential at $\rho_{0}$. To
satisfy the constraint, one may consider a density dependence of the bag
constant $B$ [57], and/or the scaling of the kaon vector potential. However,
in the present study, we consider the medium modifications of $\alpha_{s}$,
which may also play an important role to reproduce the small positive value of
the $K^{+}$ total potential at $\rho_{0}$. To see the behaviour of
$\alpha_{s}$ in the nuclear medium, we choose two naive functional forms of
the in-medium $\alpha^{*}_{s}$ for this exploratory study,

$\alpha_{s}^{*}\equiv\begin{cases}\alpha^{*}_{s(1)}(\rho)=\alpha_{s}\left(1-b_{1}\hat{\rho}\right),\\\
\\\ \alpha^{*}_{s(2)}(\rho)=\dfrac{\alpha_{s}}{1+b_{2}~{}{\rm
ln}(\hat{\rho}+1)},\end{cases}$ (76)

where $\hat{\rho}=\rho/\rho_{0}$ and dimensionless parameters $b_{1}=0.534$
and $b_{2}=1.650$ are obtained to satisfy the constraint on $\omega_{K^{+}}$.
The results for the $\alpha^{*}_{s}$ and
$\omega_{K^{+}}=M^{*}_{K^{+}}+V^{q}_{\omega}$ are shown in Fig. 3. To date, it
is not well-known how large the reduction of $\alpha^{*}_{s}$ is in the
medium, although it is expected that $\alpha^{*}_{s}$ should decrease as the
nuclear density increases, since the Fermi motion makes the (light) quarks
more energetic, and by the QCD’s asymptotic freedom. Using the two
$\alpha^{*}_{s}$ functional forms, we have determined the parameters
$(b_{1},b_{2})$ without scaling the vector potential for the $K^{+}$ meson
(automatically for $K$ and $\bar{K}$ mesons). If the empirical data are
available for the $K^{-}$ meson, i.e.,
$\omega_{K^{-}}=M^{*}_{K^{-}}-V^{q}_{\omega}$ ($M^{*}_{K^{-}}=M^{*}_{K^{+}}$),
we can deduce whether a scaling for $V^{q}_{\omega}$ for $K^{\pm}$ mesons are
necessary or not. Using the two different density dependencies, we simulate
the modifications of $\alpha^{*}_{s}$ in the nuclear medium as shown in Fig.
3. The results clearly show that more reduction of $\alpha^{*}_{s}$ results in
more repulsion for the total $K^{+}$ potential ($\omega_{K^{+}}$). Recall that
the $\alpha_{s}$ in free space dictates the strength of the attractive
Coulomb-like potential coming from the one-gluon exchange. We note that
$\alpha_{s}$ is also modified at finite temperature [58].

Moreover, in this study, instead of considering the meson loop mechanism
applied for the studies of mass shifts of quarkonia (with no light quarks) in
medium [9, 10], we apply the in-medium modifications of $\alpha^{*}_{s}$
“effectively” for all the quarks treated in this study, according to the quark
flavor blindness of QCD. A more rigorous analysis of the meson mass shift in
the medium with more elaborated density-dependent $\alpha_{s}$ and $\beta$ is
left for future studies.

Figure 4:  The ratios of weak-decay constants in-medium to free space,
$f^{*}_{\rm M}/f_{\rm M}$, versus $\rho/\rho_{0}$ for (a) pseudoscalar and (b)
vector mesons. For the unequal mass mesons ($q\bar{Q}$ and $Q\bar{q}$), the
averaged decay constants without the vector potential $V_{\omega}^{q}$ are
presented. (See section IV.) The effects of the vector potential are shown in
Fig. 5.  Figure 5:  Weak-decay constants in symmetric nuclear matter with the
vector potential $V_{\omega}^{q}$ versus $\rho/\rho_{0}$ for unequal quark
mass mesons ($q\bar{Q}$ and $Q\bar{q}$), $K,D,$ and $B$ mesons. The impact of
$V_{\omega}^{q}$ is in a similar order for the pseudoscalar and vector mesons
corresponding to the quark flavors $s,c$ and $b$ (and their respective
antiquarks). The effect is more suppressed for heavier flavor mesons.

###  V.4 Weak-decay constant in nuclear medium

Next, we discuss the medium modifications of the averaged weak-decay constants
of the light and heavy-light pseudoscalar and vector mesons, namely, without
the influence of the vector potentials for the ratio of $R_{\rm M}=f_{\rm
M}^{*}/f_{\rm M}$ with ${\rm M}={\rm P,V}$ mesons. The results are shown in
Fig. 4. On the effects of the vector potentials for the unequal quark mass
mesons ($q\bar{Q}$ and $Q\bar{q}$) such as $K^{(*)},D^{(*)},$ and $B^{(*)}$,
the results are shown in Fig. 5.

For the lightest pseudoscalar meson, the pion, the ratio of $R_{\pi}$ without
vector potential is around 0.5 at $\rho=\rho_{0}$, showing that the pion weak
decay constant in the medium becomes appreciably reduced relative to that in
free space, which is consistent with that computed in the BSA calculation with
the mass regulator [23, 22]. In comparison with the deeply bound pionic atom
experiment, our result is smaller than the empirical value $R_{\pi}^{2}\approx
0.64$ [2]. We also notice that the reduction is sensitive to the quark mass
value used. We separately check for the larger $m_{q}$ value and the less
reduction is obtained, which is consistent with the the BSE-NJL model results
in larger quark mass [16, 18]. However, overall, our predicted results are
consistent with other theoretical calculations, namely, the weak-decay
constant decreases as the nuclear density increases.

While results for the $R_{K}$, $R_{D},$ and $R_{B}$ without vector potential
are found to have smaller reductions, which are approximately about $0.8-0.9$
at $\rho=\rho_{0}$, as shown in Fig. 4 (a). Also, it shows a large reduction
of $R_{\pi}$, compared to the other pseudoscalar mesons. This can be
understood by the fact that the pion contains purely light quarks and
antiquark, and the two light quarks are subject to feel stronger medium
effects since the light quark chiral condensates reduce faster as the nuclear
density increases than those of the heavier quarks. In contrast, the quark
condensates for the $s$, $c$, and $b$ quarks change very slowly and a small
amounts in the nuclear medium. (See again the discussions made in the
introduction part of Ref. [51].) The in-medium weak decay constant ratios at
$\rho_{0}$ follow the order as

$\displaystyle
1>\frac{f_{B}^{*}}{f_{B}}>\frac{f_{D}^{*}}{f_{D}}>\frac{f_{K}^{*}}{f_{K}}>\frac{f_{\pi}^{*}}{f_{\pi}}.$
(77)

For the heavy-light mesons, again it is clearly shown that the reductions of
the weak-decay constants are less pronounced. This is because the medium
effect becomes smaller relative to the masses of the heavy quarks and they
weakly couple in the medium. (In the present approach, mean fields do not
couple directly to the heavy quarks at the lowest order.)

Now, we discuss the crucial role of the vector potential ($\pm V_{\omega}^{q}$
for $q$ and $\bar{q}$) on the in-medium weak-decay constants for the
positively and negatively charged states for $K^{\pm},D^{\mp}$ and $B^{\pm}$.
Figure 5 demonstrates that the effect of $V_{\omega}^{q}$ on the ratios of the
in-medium weak-decay constants obviously depends on the light quark contents
that are controlled by the term of $(1\pm V_{\omega}^{q}/M_{0}^{*})$. It shows
that the differences in the weak-decay constant ratios for the heavy-light
mesons with opposite charge states are getting suppressed as the mesons become
heavier, since the invariant meson mass $M_{0}^{*}$ appearing in the
denominator becomes larger, and thus the effect of the vector potential is
relatively suppressed by $M_{0}^{*}$. Therefore, the effect of
$V_{\omega}^{q}$ is more pronounced for the lighter mesons, and suppressed for
the heavier mesons.

For the light and heavy-light vector mesons, the ratios of the in-medium weak
decay constants are $R_{\rm V}\simeq 0.95$ at $\rho_{0}$ without the vector
potential. The order of the ratios for the in-medium vector meson decay
constants have opposite order to those for the pseudoscalar mesons:

$\displaystyle\frac{f_{B^{*}}^{*}}{f_{B^{*}}}<\frac{f_{D^{*}}^{*}}{f_{D^{*}}}<\frac{f_{K^{*}}^{*}}{f_{K^{*}}}<\frac{f_{\rho}^{*}}{f_{\rho}}<1.$
(78)

Also, it is found that the differences among the ratios of the different
vector mesons are rather small as can be seen in Fig. 4 (b). Moreover, the
effect of the $\pm V_{\omega}^{q}$ for the in-medium vector meson decay
constants have a similar tendency to that for the pseudoscalar mesons, as
shown in Fig. 5. However, the magnitude of the ratios is different from that
for pseudoscalar mesons, in particular for $K^{*\pm}$.

The different behavior of $R_{\rm M}$ between the pseudoscalar and vector
mesons can be qualitatively explained by the weak-decay constant formula. For
instance, the operators in Eqs. (18) and (19) for $\pi$ and $\rho$ mesons,
which are respectively given by

$\displaystyle\mathcal{O}_{\rm P}^{*}$ $\displaystyle=$ $\displaystyle
m_{q}^{*},$ (79) $\displaystyle\mathcal{O}_{\rm V}^{*}$ $\displaystyle=$
$\displaystyle
m_{q}^{*}+\frac{2\mathbf{k}_{\perp}^{2}}{M_{0}^{*}+2m_{q}^{*}}.$ (80)

If we assume SU(6) symmetry, the difference between the in-medium decay
constant of $\pi$ and $\rho$ mesons comes from the second term of
$\mathcal{O}_{\rm V}^{*}$. Since $m_{q}^{*}$ decreases in the medium as
density increases, the first term of $\mathcal{O}_{\rm V}^{*(1)}$ decreases,
while the second term $\mathcal{O}_{\rm V}^{*(2)}$ increases as density
increases. For this reason, the decay constant for the $\pi$ meson is
significantly reduced in the medium.

Figure 6:  Contributions of the first term $\mathcal{O}_{\rm
V}^{*(1)}=m_{q}^{*}$ and the second term $\mathcal{O}_{\rm
V}^{*(2)}=2\mathbf{k}_{\perp}^{2}/(M_{0}^{*}+2m_{q}^{*})$ of the operator
$\mathcal{O}_{\rm V}^{*}$ for the weak-decay constant of $\rho$ meson in
medium versus $\rho/\rho_{0}$.

The results for the weak-decay constants of $\rho$ meson as well as the other
vector mesons are found to be almost constant and nearly without the influence
of medium effect, due to the competing contributions between the two terms for
the total vector-meson decay constants as shown in Fig. 6. One can see that
the second term $\mathcal{O}_{\rm V}^{*(2)}$ in Eq. (80) dominates at
$\rho_{0}$, while the first and second terms contribute rather similarly to
the decay constant in free space. For heavy-light vector mesons, the
contribution from $\mathcal{O}_{\rm V}^{*(2)}$ is rather suppressed even in
free space due to heavy-flavor quark mass. As a result, the medium
modifications of $f_{\rm P}^{*}/f_{\rm P}$ and $f_{\rm V}^{*}/f_{\rm V}$ for
the heavy mesons are rather similar as shown in Figs. 4 (a) and (b).

Figure 7:  Ratios of $f_{\rho}^{*}/f_{\rho}$ for different quark mass values
versus $\rho/\rho_{0}$. The Gaussian parameter $\beta_{q\bar{q}}$ is fixed to
reproduce the decay constant $f_{\rho}^{\rm expt}$, when we use $m_{q}=$ 0.430
GeV.

Figure 8:  In-medium DAs of the pion (upper panel) and kaon (lower panel) with
several densities versus longitudinal momentum $x$. The lattice QCD data in
free space is represented by a green-shaded region, which is taken from Ref.
[59]. The predicted results for the DAs in the asymptotic region are
represented by the smallest dotted lines.

In the following, we study the effect of the different quark mass values
$m_{q}$ on the weak-decay constant $f_{\rho}^{*}$. To do so, we use
$m_{q}=0.430$ GeV so that we can compare our result with that obtained in Ref.
[24], although they calculated in the light-front constituent quark model
based on the BSA approach with a regulator mass, while, in the present work,
we employ the Gaussian parameter with $\beta_{q\bar{q}}=$ 0.330 GeV to
reproduce the empirical value $f_{\rho}^{\rm expt.}=269$ MeV. Results with the
two values of $m_{q}$, 0.220 and 0.430 GeV are shown in Fig. 7. The results of
the $f_{\rho}^{*}/f_{\rho}$ ratio with $m_{q}=$ 0.220 GeV smoothly decrease as
density increases, while the result with $m_{q}=$ 0.430 GeV slightly increases
as the density increases.

Similar behavior of the weak-decay constants can be found for the other vector
mesons in our approach. The increase of $f_{\rho}^{*}$ in medium with
$m_{q}=0.430$ GeV is also observed in Ref. [24], however, it shows an
oscillating behavior as nuclear density increases. Note that, the increasing
rate is smooth in the present approach, as seen in Fig. 7. The differences in
the $f_{\rho}^{*}$ increasing behavior with that of Ref. [24] may be
attributed to the use of the invariant mass $M_{0}$ in the present approach.
Overall, our result for the in-medium $\rho$ meson decay constant
$f_{\rho}^{*}$ with $m_{q}=$ 0.430 GeV is consistent with Ref. [24]. We argue
that the moderate increase of $f_{\rho}^{*}$ in the present approach can be
attributed to the use of the invariant mass $M_{0}$ in calculating the weak-
decay constant, which is a special aspect of the present LFQM, where it was
shown in Ref. [36] that the use of the invariant mass $M_{0}$ enables to
calculate the $\rho$-meson weak-decay constant from various light-front
current components and polarizations, and it can produce the weak-decay
constant self-consistently.

###  V.5 Distribution amplitude in nuclear medium

Figure 9:  In-medium DAs of the $\rho$ (upper panel) and $K^{*}$ (lower panel)
vector mesons versus $x$ for several densities. See also the caption of Fig.
8.  Figure 10:  Same as in Fig. 9, but for the DAs of $D^{(*)}$ and $B^{(*)}$
in medium.

Here, we present our final results for the in-medium DAs of the light and
heavy-light pseudoscalar and vector mesons. We set the longitudinal momentum
$x$ carried by the light quark $q$ in $(q\bar{Q})$ meson and by antiquark
$\bar{q}$ in $(Q\bar{q})$ meson. Furthermore, for simplicity, we present the
average, or equivalently, the results calculated without the vector potential
$V_{\omega}^{q}$. The in-medium DAs for the pion and kaon are presented in
Fig. 8, while those for the $\rho$ and $K^{*}$ are shown in Fig. 9, and those
for the heavy-light pseudoscalar and vector mesons are depicted in Fig. 10.

In Fig. 8, the calculated DAs for the pion (upper panel) and kaon (lower
panel) are compared with the predicted asymptotic ones (dotted line) as well
as the recent lattice QCD data (green shaded region) [59]. Very recently,
another lattice QCD result for pion DA is reported [60] and they find that
their result lies within the green shaded region that obtained in Ref. [59].
We observe that the pion DA in free space [40] is consistent with the recent
lattice QCD results. In the medium, the pion DA becomes flatter with the value
approaching unity at $\rho=\rho_{0}$, but the $x$ dependence of DA becomes
broadened. This is expected due to the reduction of the quark mass $m_{q}^{*}$
in the medium, reflecting the effect of the partial restoration of the chiral
symmetry.

Results for the kaon DA in free space as well as in a medium are shown in the
lower panel of Fig. 8, where the result in free space is calculated with the
parameters of Ref. [40], and has a reasonable agreement with the lattice QCD
data especially in the $x<0.5$ region. However, the kaon DA in free space
overestimates the lattice data in the $0.5<x<0.8$ region and underestimates
the asymptotic prediction in the small $x$ region. Compared with the lattice
QCD data, the kaon DA is consistent with lattice data in the near endpoint
region at $x\to 0$. However, at the near-end point $x\to$ 1, the result is
suppressed and shows different behavior from the lattice QCD data. It is worth
noting that the fast suppression of the Kaon DA near the endpoint $x\to$ 1 may
indicate that the kaon DA is sensitive to the choice of the radial wave
function. In Ref. [61], the authors compared the twist-3 DA of pion and kaon
with two different trial wave functions, i.e., the Gaussian and Power-law
radial wave function. One of the apparent differences is that the DA near-end
point with the Power-law wave function is more enhanced as compared with the
Gaussian one. Recall that the Gaussian radial wave function is employed in the
present work. Furthermore, the DA near endpoints cannot be computed by the
lattice QCD directly, the values are obtained from a phenomenological
extrapolation that may give some ambiguity [59]. Another possible source of
discrepancy is the SU(3) flavor symmetry-breaking effect. In this work, the
$(m_{q},m_{s})=(220,450)$ MeV is taken which shows a considerable difference.
A smaller quark mass difference may be favored as far as the lattice data are
concerned. Further analysis of the kaon DA in free space is needed by
modifying the trial wave function to get the best fit to the lattice QCD. As
for the kaon DA in the medium, it is found to have the largest reduction at
around $x=$ 0.2, whose peak slightly moves to the larger $x$ region, since the
(effective) mass difference between the light quark and the strange quark in
the medium is enhanced.

Results for the $\rho$ and $K^{*}$ DAs are shown in the upper and lower panels
of Fig. 9, respectively. The DA for the $\rho$ is consistent with that for the
asymptotic result. The $\rho$ DA is moderately modified in the medium.
However, the $K^{*}$ DA shows different behavior from the asymptotic results.
It is shifted to the smaller $x$ region, which is expected due to the SU(3)
flavor symmetry breaking in the quark masses. Similar to the $\rho$ DA, the
$K^{*}$ DA is moderately modified in the medium, as shown in the lower panel
of Fig. 9. Note that the small-medium modifications of the vector meson DAs
are in conformity with the small reduction of $f_{\rm V}^{*}$ in medium, as
shown in Fig. 4.

Our results for the heavy-light meson $B,D,D^{*}$, and $B^{*}$ DAs are shown
in the left and right panels of Fig. 10. The results indicate that the heavy-
light meson DAs are nearly unmodified in the medium. A similar reason for the
kaon with the strange quark content, can be understood by the heavy quark
contents of the heavy-light mesons, where the heavy quarks are not modified
directly in the medium.

Figure 11:  The difference of DAs defined by
$\Delta\phi_{M}(x)=\phi_{M}^{*}(x)-\phi_{M}(x)$ for various mesons.

Since the smaller in-medium DA modifications for the light and heavy-light
mesons cannot easily be seen from the results shown in Figs. 9 and 10,
respectively, we show the difference between the DAs in the medium [$\phi_{\rm
M}^{*}(x)$] and in free space [$\phi_{\rm M}(x)$], defined by

$\displaystyle\Delta\phi_{M}(x)=\phi_{\rm M}^{*}(x)-\phi_{\rm M}(x).$ (81)

The DA differences in free space and in the medium are shown in Fig. 11. The
effects of the in-medium modifications of DAs are more evident for each meson.
The results show that the effects of the in-medium modifications are not
negligibly small for the heavy-light mesons $B,D,D^{*}$, and $B^{*}$, in
particular, at small $x$. However, the medium effect for the $B$ and $B^{*}$
mesons is almost negligible at around $x>0.5$, and the medium effect for the
$D$ and $D^{*}$ mesons is also negligible at around $x>0.8$. Figure 11 clearly
shows that the medium effect on the DAs for the light pseudoscalar meson is
more pronounced compared with that for vector mesons. Also, the medium
modifications of DAs for the light pseudoscalar mesons are similarly larger
than those for the weak-decay constants compared with the other mesons.

##  VI Summary

To summarize, in the present work, we have investigated the in-medium
modifications of the weak-decay constants and distribution amplitudes (DAs) of
the light and heavy-light pseudoscalar and vector mesons. For this purpose, we
have constructed a combined model based on a light-front quark model (LFQM)
that describes the meson properties both in free space as well as in a nuclear
medium based on an equal footing, and the quark-meson coupling (QMC) model to
simulate the medium effects on the (light) quarks. The in-medium quark
properties are modified by the self-consistent scalar and vector mean fields
generated by the surrounding nucleons. It is evident that the light quark mass
is reduced in the medium, reflecting that the chiral symmetry is partially
restored. We note that the present approach is similar to that was practiced
in Refs. [22, 23, 24], but we use a different approach, a more QCD dynamics-
motivated light-front quark model with the Gaussian wave function.

We have found that the weak decay constants for the pseudoscalar mesons with
light-light quark contents decrease relatively faster in the medium as nuclear
density increases compared with those for the vector mesons with the light-
light and heavy-light quark contents (as in Fig. 4). For the less pronounced
in-medium modifications of the vector meson decay constants are, due to the
competition between the decreasing and increasing contributions of the
$\mathcal{O}_{\rm V}^{*(1)}$ and $\mathcal{O}_{\rm V}^{*(2)}$ operators in the
medium. We should note that the medium modifications of weak-decay constants
are sensitive to the choice of the free space quark mass value. The use of the
larger values of the quark mass value leads to a less reduction of the weak-
decay constant.

The role of the vector potential $V_{\omega}^{q}$ is also studied for the
mesons with heavy-light mesons (as in Fig. 5). We have found that the vector
potential modifies differently the decay constants within the meson multiplet
members, namely for the different charges (isospin) cases, by a factor $(1\pm
V_{\omega}^{q}/M_{0}^{*})$. Also, we have found that the trend for the effect
of the vector potential is similar for the pseudoscalar and vector mesons, but
the effect is much more suppressed for the heavy mesons due to the factor by
the invariant mass, $1/M_{0}^{*}$.

Interestingly, we have found that the pion and kaon DAs are more modified in
medium among all the mesons studied in the present study. The medium
modifications of DAs can be clearly visualized by calculating the DA
differences in free space and in a medium, namely by calculating
$\Delta\phi_{\rm M}(x)=\phi_{\rm M}^{*}(x)-\phi_{\rm M}(x)$ (as in Fig. 11).
It is found that the shape of DAs in medium has large enhancement in the
endpoint regions, while it is reduced in the moderate $x$ for the light
mesons. The enhancement near the endpoint at $x\to 1$ is smeared and shifted
to the lower $x$ region.

We expect that the present results may provide useful guidance for
constructing more sophisticated models based on the quark degrees of freedom
to investigate the in-medium properties of the light and heavy-light
pseudoscalar and vector mesons. Also, the present study will provide useful
information on the possible meson-nuclear bound states, that are planned to
measure in the modern experiment facilities [62, 63, 64]. Furthermore, the
experiment via the pionic or kaonic Drell-Yan reaction process with the heavy
nuclear targets can be a way to probe the modifications of DAs in nuclear
medium and nuclei [65]. In principle, our present approach can be applied to
investigate various meson properties in medium, for instance, in-medium meson
electromagnetic form factors. Such studies are underway.

## Acknowledgements

P.T.P.H. thanks Jian-Hui Zhang (Lattice Parton Collaboration) for providing us
with the recent lattice QCD data for the kaon and pion distribution
amplitudes. A.J.A. was supported by Special Postdoctoral Researcher (SPDR)
Program at RIKEN and the Young Scientist Training (YST) Program at the Asia
Pacific Center for Theoretical Physics (APCTP) through the Science and
Technology Promotion Fund and Lottery Fund of the Korean Government and also
by the Korean Local Governments-Gyeongsangbuk-do Province and Pohang City. The
work of P.T.P.H. was supported by the National Research Foundation of Korea
(NRF) grants funded by the Korean government (MSIT) Nos. 2018R1A5A1025563,
2022R1A2C1003964, and 2022K2A9A1A0609176. The work of K.T. was supported by
the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)
Process, No. 426150/2018-0, and No. 304199/2022-2, and Fundação de Amparo à
Pesquisa do Estado de São Paulo (FAPESP) Process, No. 2019/00763-0, and the
work was also part of the projects, Instituto Nacional de Ciência e Tecnologia
— Nuclear Physics and Applications (INCT-FNA), Brazil, Process. No.
464898/2014-5.

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