# Bubble nucleation in the two-flavor quark-meson model

Junrong Wang  Ziwan Yu  Hong Mao<EMAIL_ADDRESS>(corresponding author) 1\.
School of Physics, Hangzhou Normal University, Hangzhou 311121, China

###### Abstract

We investigate the dynamics of a first-order quark-hadron transition via
homogeneous thermal nucleation in the two-flavor quark-meson model. The
contribution of fermionic vacuum loop in the effective thermodynamics
potential and phase diagram together with the location of critical end point
(CEP) have been obtained in the temperature and chemical potential plane. By
taking the temperature as variable, the critical bubble profiles, the
evolutions of the surface tension and the saddle-point action in the presence
of a nucleation bubble are calculated in detail when fixing the chemical
potentials at $\mu=306\mathrm{MeV}$ and $\mu=309\mathrm{MeV}$. Our results
show that the system could be trapped in the metastable state for a long time
as long as the temperature is between the up and low spinodal lines, and the
surface tension at criticality will rise up to about $4\mathrm{MeV/fm^{2}}$
when the chemical potential is very high. Such a small value of the surface
tension would favor a mixed phase in the cores of compact stars and may have
an important implication in astrophysics.

##  I Introduction

It is widely believed that the hadronic matter characterized by confinement
and chiral-symmetry at low net baryon densities undergoes a phase transition
into a deconfined and chirally symmetric quark-gluon-plasma (QGP) through a
smooth crossover as the temperature is increased. At large densities, one
expects a first-order phase transition line separating the hadronic matter
from the QGP and possible more exotic phases. At end of this line, there
should exist a so-called critical endpoint (CEP) where the transition should
be a continuous second-order one. To investigate and identify phase diagram is
one of the most challenging problems in high energy physics and astrophysics
[1, 2, 3], and the study is experimentally supported by the heavy-ion
collision experiments, such as the Relativistic Heavy Ion Collider (RHIC) at
Brookhaven National Laboratory and the Large Hadron Collider (LHC) at CERN.
These conducted experiments provide us with the opportunity to inspect and
reveal fundamental properties of the strong interaction. Moreover, to explore
a wider range of the QCD phase diagram up to several times the normal nuclear-
matter density, the new Facility for Antiproton and Ion Research at Darmstadt,
the Nuclotron-based Ion Collider Facility at the Joint Institute for Nuclear
Research in Dubna, and the Japan Proton Accelerator Research Complex at Japan
Atomic Energy Research Institute and Japan’s National Laboratory for High
Energy Physics have been scheduled and planed, and the CEP can be explored in
phase II of Beam Energy Scan program at RHIC and in upcoming experiments [4,
5].

From a theoretical point of view, Quantum Chromodynamics (QCD), the gauge
theory describing strong interactions in elementary particle physics, is
applicable for determining the properties of strongly interacting matter at
finite temperature and density. However, due to the fermion sign problem, an
ab initio approach, Lattice Field Theory, is severely hampered by the failure
of importance sampling if a chemical potential is involved [6]. In order to
describe the low-energy nonperturbative phenomena in the framework of QCD
theory, an alternative approach is effective models with possessing two
salient features of QCD, i.e. chiral symmetry and confinement. To mention a
few, these effective models, which are successfully utilized for many decades,
are the Nambu-Jona-Lasinio (NJL) model [7, 8], the linear sigma model (LSM)
[9] and their modernized extensions, the Polyakov Nambu-Jona-Lasinio model
(PNJL) [10, 14] and the Polyakov Quark Meson Model (PQM) [11, 12, 13].

Recently, after the discovery of gravitational waves by LIGO Collaboration
[15], the subject of cosmological first-order phase transition has gathered
increasingly interest due to the stochastic gravitational wave background
which would be produced [16, 17, 18]. The stochastic gravitational wave
background could be detected by current and near future detectors. Once this
signal is to be observed, it would provide us with the earliest known probe of
the universe. Moreover, aside from the early universe phenomena in primordial
first-order phase transition, the observation of gravitational waves also shed
light on the field of astrophysics, future gravitational wave observations
related to a first-order quark-hadron phase transition would not only enable
the probing of the equation of state for matter under extreme circumstances,
but also give a constrain on the quark-hadron surface tension. In combination
with other observations, astrophysics has now entered the multimessenger era
[19, 20, 21, 22, 23, 24]. Therefore, to understand the dynamics of the first-
order phase transition is crucial and important. With upcoming gravitational
waves experiments, there is a need to explore the anticipated phenomenology
tightly connected with its underlining fundamental mechanism.

It is well known that the dynamics of first-order phase transition in the
early universe and heavy-ion collisions at ultrarelativistic energies can be
applied through the homogeneous nucleation theory [25, 26]. The modern theory
pioneered by Langer in the late 1960s in the context of classical statistical
mechanics [27, 28] has been extended to relativistic quantum field theory by
Callan and Coleman for zero temperature [29, 30, 31] and by Affect [32] and
Linde [33, 34] for finite temperature. The remarkable goal of a nucleation
theory is to calculate the nucleation rate of a bubble or droplet of a stable
(true) vacuum inside a metastable (false) vacuum near the critical
temperature. Suppose that a system nearby its critical temperature, due to the
thermal and quantum fluctuations of any thermodynamic systems, bubbles of the
stable vacuum created by fluctuations may grow or shrink inside the
homogeneous false vacuum depending on its energy budget with regard to the
false vacuum. If a droplet is too small, the free energy gain from the phase
transition of the bulk is less than the energy cost in the creation of an
interface between two vacua. The total free energy is positive, the droplet
will shrink and evaporate. On the other hand, if the droplet is large, a bulk
free energy gain is relatively large and the surface energy cost is
negligible, the droplet will tend to grow and eventually occupy the whole
system, completing the phase conversion.

For a strong first-order phase transition, which is usually characterized by
an effective potential with a zero-temperature potential barrier, the dynamics
of the quark-hadron phase conversion based on the Friedberg-Lee (FL) model
[35] has been studied numerically [36], and the found has been also compared
to its analytic results obtained with the thin-wall approximation [37]. Since
the FL model is lack of chiral symmetry, the model only predicts a first-order
phase transition in whole QCD phase diagram, this, of course, conflicts with
other studies based on Lattice simulations or chiral models, so that the model
can merely serve as the prototypical toy model for current interests. To fixed
this problem, it is necessary to introduce a chiral symmetry in the FL model
in order to provide a proper description of hadron-quark phase transition
beyond the first-order transition. Then, the quark meson model treated as an
upgrade to the FL model seems fulfill the requirements in both the studies of
the static nucleon properties and the QCD phase transition [38, 39]. In the
framework of the quark meson model, homogeneous bubble nucleation has been
initially investigated both in numerical and analytic methods in Ref [40], but
the study was constrained on the temperature below the critical temperature
and an unphysical coupling constant was chosen in order to enhance the
strength of the first-order phase transition. Furthermore, with the thin-wall
approximation, bubble nucleations at low temperature but high density and the
strong magnetic field have been previously investigated in Refs.[41, 42]. In a
first-order phase transition, a key ingredient is the surface tension, by
using the analytical method or the thin-wall approximation, surface tensions
and phase diagram have been obtained in the quark meson model with the
Polyakov-loop in Refs [43, 44]. Since the thin-wall approximation can not be
applied when the temperature is far from the critical temperature, in this
work, by adopting the realistic coupling constant, we will carry on systematic
study on the first-order quark-hadron phase transition through the exact
numerical method when the temperature is below the critical one as well as the
temperature is above the critical value, also the fermion vacuum fluctuation
is included in the present study. Furthermore, we will compare our results
with the recently found for the quark meson model within the thin-wall
approximation [45], in order to discover some nontrivial but important results
missing in the analytical method.

The structure of the paper is as follows. In the next section we briefly
describe the quark meson model. After that we will discuss the effective
potential at finite temperatures and densities and present a phase diagram of
the QCD phase transition. In Sec. IV, we give detailed description of
homogeneous nucleation and the methods used for both numerical and analytic
computations of the critical bubble profiles. Our results and discussions are
presented in Sec. V, while in the last section we give our conclusions.

##  II The Model

In terms of chiral fields, the Lagrangians of two massless noninteracting
quarks $u$ and $d$ are invariant under the global $SU(2)_{L}\times SU(2)_{R}$
chiral phase transformations

$\psi_{L,R}\rightarrow\psi_{L,R}^{\prime}=U_{L,R}\psi_{L,R},$ (1)

where $\psi^{T}_{L,R}=(u,d)_{L,R}$ and
$U_{L,R}=\exp(-i\vec{\theta}_{L,R}\cdot\frac{\vec{\tau}}{2})$ . However this
chiral symmetry does not appear in the low energy particle spectrum and the
strong interaction theory exhibits the phenomenon of spontaneous symmetry
breaking. Consequently, three Goldstone bosons appear and the constituent
quarks become massive at low energy. In describing the symmetries of the
Lagrangian, it is useful to introduce three pion mesons $\vec{\pi}$ and a
$\sigma$ meson in terms of a matrix field as

$\Phi=\sigma\frac{\tau^{0}}{2}+i\vec{\pi}\cdot\frac{\vec{\tau}}{2},$ (2)

where $\tau^{0}$ is the unity matrix and $\vec{\tau}$ are the three Pauli
matrices. Under the $SU(2)_{L}\times SU(2)_{R}$ chiral symmetry
transformations, $\Phi$ transforms as

$\Phi\rightarrow\Phi^{\prime}=U_{L}\Phi U_{R}^{+}.$ (3)

Then the renormalizable effective Lagrangian of the two-flavors quark meson
model is defined as [9, 46]

${\mathcal{L}}={\mathcal{L}}_{\Phi}+{\mathcal{L}}_{q},$ (4)

where

${\mathcal{L}}_{\Phi}=\textrm{Tr}[(\partial_{\mu}\Phi)^{+}(\partial^{\mu}\Phi)]-\lambda[\textrm{Tr}(\Phi^{+}\Phi)-\frac{{\vartheta}^{2}}{2}]^{2}-H\textrm{Tr}[\Phi],$
(5)

and

${\mathcal{L}}_{q}=\overline{\psi}_{L}i\partial\\!\\!\\!/\psi_{L}+\overline{\psi}_{R}i\partial\\!\\!\\!/\psi_{R}-2g\overline{\psi}_{L}\Phi\psi_{R}+h.c..$
(6)

Here we have introduced a flavor-blind Yukawa coupling $g$ of the left-handed
and right-handed quark fields to interact with the $\Phi$ field.

The parameters of the Lagrangian $\mathcal{L}$ are chosen by the requirement
that the chiral symmetry $SU(2)_{L}\times SU(2)_{R}$ is spontaneously broken
down to $SU(2)_{L+R}$ in the vacuum while the $\sigma$ field takes on a non-
vanishing vacuum expectation value $\langle\sigma\rangle={\it
f}_{\pi}=93\mathrm{MeV}$. It results in a massive $\sigma$ meson and three
massless Goldstone bosons $\vec{\pi}$ mesons in chiral limit, as well as
giving an effective mass $m_{q}=gf_{\pi}$ to the constituent quarks.
Furthermore, the chiral symmetry is explicitly broken by adding the last term
in Eq.(5) due to the finite current quark masses. With this additional term,
the vector isospin $SU(2)$ symmetry remains exact but the axial $SU(2)$
transformation is no longer invariant. Accordingly, the constant $H$ is to be
fixed by the partially conserved axial vector current relation which gives
$H=f_{\pi}m_{\pi}^{2}$, where the pion mass is taken as $m_{\pi}=138$ MeV.
Moreover, the dimensionless coupling constant $g$ in the model is determined
by the constituent quark mass in vacuum, which is about $1/3$ of the nucleon
mass and gives $g\simeq 3.3$. The another dimensionless coupling constant
$\lambda$ is usually fixed by the sigma mass
$m_{\sigma}^{2}=m^{2}_{\pi}+2\lambda f^{2}_{\pi}$, here we set to $500$ MeV in
accord with the most recent compilation of the Particle Data Group [47].
Finally, the quantity $\vartheta$ is actually not a free parameter and can be
formally expressed as $\vartheta^{2}=f^{2}_{\pi}-m^{2}_{\pi}/\lambda$.

##  III Effective potential and phase structure

A convenient framework of studying phase transitions and the restoration of
the chiral symmetry at extreme high energy is the thermal field theory [18,
48]. Within this framework, the effective potential is one of the important
and powerful theoretical tool, and the standard approach for dealing with the
thermodynamics of various observables of interest relies on the grand
canonical ensemble. To make things lucid, we start with a spatially uniform
system in thermodynamical equilibrium at temperature $T$ and quark chemical
potential $\mu$, from here and henceforward, we will use the chemical
potential to represent the quark chemical potential. In general, the grand
partition function is commonly given in a schematic form

$\displaystyle\mathcal{Z}$ $\displaystyle=$
$\displaystyle\mathrm{Trexp}[-(\hat{\mathcal{H}}-\mu\hat{\mathcal{N}})/T]$ (7)
$\displaystyle=$
$\displaystyle\int\prod_{a}\mathcal{D}\sigma\mathcal{D}\pi_{a}\int\mathcal{D}\psi\mathcal{D}\bar{\psi}\mathrm{exp}\left[\int_{X}(\mathcal{L}+\mu\bar{\psi}\gamma^{0}\psi)\right],$

where $\int_{X}\equiv\int^{\beta}_{0}d\tau\int d^{3}x$, the inverse
temperature $\beta=1/T$, and $\mu=\mu_{B}/3$ for the homogeneous background
field.

In the mean-field approximation, the meson fields in the Lagrangian are
replaced by their expectation values, whereas the quark and antiquark fields
are still retained as quantum fields. This implies that the one-loop
correction to the effective potential from the quark fields are taken into
account, but treats the mesonic degrees of freedom at tree level. Following
this scheme, the integration over the fermions yields a determinant which can
be calculated by standard procedures [26, 50], it will generate an effective
potential for the mesons. Finally, the effective potential of the model can be
obtained exactly in a closed form as

$\displaystyle\Omega(T,\mu)=\frac{-T\mathrm{ln}\mathcal{Z}}{V}=U(\sigma,\vec{\pi})+\Omega_{\bar{\psi}\psi},$
(8)

where the classical potential for the $\sigma$ and $\vec{\pi}$ is rewritten as

$U(\sigma,\vec{\pi})=\frac{\lambda}{4}\left(\sigma^{2}+\vec{\pi}^{2}-{\vartheta}^{2}\right)^{2}-H\sigma,$
(9)

and the contribution of quarks and antiquarks are given by

$\displaystyle\Omega_{\bar{\psi}\psi}$ $\displaystyle=$
$\displaystyle\Omega_{\bar{\psi}\psi}^{\mathrm{v}}+\Omega_{\bar{\psi}\psi}^{th}$
(10) $\displaystyle=$ $\displaystyle-\nu\int\frac{d^{3}\vec{p}}{(2\pi)^{3}}E$
$\displaystyle-\nu
T\int\frac{d^{3}\vec{p}}{(2\pi)^{3}}\left\\{\mathrm{ln}\left[1+e^{-(E-\mu)/T}\right]+\mathrm{ln}\left[1+e^{-(E+\mu)/T}\right]\right\\}.$

Here, $\nu=2N_{f}N_{c}=12$ and $E=\sqrt{\vec{p}^{2}+m_{q}^{2}}$ is the valence
quark and antiquark energy for $u$ and $d$ quarks, and the minus sign is the
consequence of Fermi-Dirac statistics. The constituent quark (antiquark) mass
is set to $m_{q}=g\sigma$.

The first term of Eq.(10) denotes the fermions vacuum one-loop contribution
which is ultraviolet divergent and can only be evaluated in the presence of a
regulator. The divergence in Eq.(10) then can be appropriately renormalized by
using the dimensional regularization scheme [49, 51, 52]. After taking into
account the vacuum fluctuations and therefore renormalization issues, the
renormalized fermion vacuum one-loop contribution reads

$\displaystyle\Omega_{\bar{\psi}\psi}^{\mathrm{v}}=\Omega_{\bar{\psi}\psi}^{\mathrm{reg}}=-\frac{N_{c}N_{f}}{8\pi^{2}}m_{q}^{4}\mathrm{ln}(\frac{m_{q}}{\Lambda}),$
(11)

where $\Lambda$ denotes the arbitrary renormalization scale. It is worth
pointing out that dimensional regularization really introduces an arbitrary
renormalization scale parameter, nevertheless, at least in the one-loop
approximation, the thermodynamic potential and all physical observable are
independent of the choice of $\Lambda$, and the scale dependence can be neatly
cancelled out after the rearrangement of parameters in the model [39, 51, 52,
53].

Equipped with the above effective potential, we can explore the phase diagram
of the model at finite temperature and density by minimizing the
thermodynamical potential in equation (8) with respect to the order parameter
$\sigma$. Then an equation of motion is given by

$\displaystyle\frac{\partial\Omega(T,\mu)}{\partial\sigma}=0.$ (12)

The solution of the equation of motion determines the behavior of the chiral
order parameters $\sigma$ as a function of $T$ and $\mu$, as well as the phase
diagram of the model. As we know, the thermodynamical state of equilibrium is
given by the values of the order parameter at the global minimum of the
effective potential, once the order parameter for each given $T$ and $\mu$ is
obtained, any thermodynamical quantity of equilibrium, such as the pressure,
the entropy density, the energy density, the speed of sound, et al., can be
described and calculated.

Figure 1:  (Color online) The phase diagram in the $T-\mu$ plane for the two-
flavor quark meson model. The dashed lines are the critical line for
conventional chiral phase transition in the region of crossover. The solid
lines indicates the first-order phase transitions, and the solid circle
indicates the critical end points for chiral phase transitions of $u$ and $d$
quarks. The dashed-dotted line and the dashed-doted-dotted line are the lower
and upper spinodal lines.

In figure 1, we have presented the phase diagram in calculation with the
fermion vacuum fluctuation for the two-flavor quark meson model. The
temperature behavior of the chiral condensate $\sigma$ shows that the system
experiences a smooth crossover transition at low chemical potential, while
there is a first-order phase transition for larger chemical potential due to
the fact that the chiral order parameter makes a jump across the gap of the
condensate near the critical temperature $T_{c}$. Normally, the temperature
derivative of the chiral condensate $\sigma$ for quarks has a peak at some
specific temperature, which is established as the critical temperature for the
chiral phase transition. Because the temperature derivative of the chiral
condensate has simply one peak, we can not tell when and where the crossover
phase transition would convert to a first-order one at the critical end point
(CEP) with a second order phase transition [45, 46]. In order to locate the
CEP in the phase diagram, the quark number susceptibility
$\chi_{q}=\partial^{2}\Omega(T,\mu)/\partial^{2}\mu$ is to be introduced, and
it is believed to be divergent at the CEP [4, 5].

Aside from calculation of the quark number susceptibility $\chi_{q}$, in the
present work, we prefer to use the shapes of the effective potential at
various temperatures and chemical potentials to decide the position of the
CEP. In the case of the first-order phase transition, along the critical line
with the temperature $T\simeq T_{c}$, the thermodynamical potential
$\Omega(T,\mu)$ has two minima of equal depth separated by a potential
barrier. With the reduction of the chemical potential, the height of the
barrier decreases and finally disappears at the CEP, in which the phase
transition is of second order. In our calculation the corresponding CEP is
located at $(T_{E},\mu_{E})\simeq(30,301)$ MeV in Fig.1.

Figure 2: (Color online) (a) The grand canonical potentials $\Omega$ as a
function of the chiral order parameter $\sigma$ for $\mu=306$ MeV at various
temperatures. (b)The grand canonical potentials $\Omega$ as a function of the
chiral order parameter $\sigma$ for $\mu=309$ MeV at various temperatures.

As shown in Fig.2, in the region of the first-order phase transition, a
typical effective potential commonly displays a local minimum at a low sigma
$\sigma_{l}$ which is separated by a potential barrier from another local
minimum at a relative larger sigma $\sigma_{h}$. When a critical temperature
$T_{c}$ is reached, these two minima are degenerate. For $T<T_{c}$, the
minimum of the effective potential at $\sigma=\sigma_{h}$ is the absolute or
global minimum, which is regarded as the stable (true) vacuum, whereas the
minimum at $\sigma=\sigma_{l}$ is treated as the metastbale (false) vacuum. In
this case the chiral symmetry is broken so that the constituent quarks become
massive. On the contrary, when the temperature $T$ goes across above the
critical value $T_{c}$, these two vacua will flip over, the global minimum is
now at $\sigma=\sigma_{l}$, and the local minimum is at $\sigma=\sigma_{h}$.
Since the chiral symmetry is approximately restored and the quarks become
almost massless, the system for $T>T_{c}$ is then considered as the quark
phase. The previous case for $T<T_{c}$ is taken as the hadron phase, therefore
the critical lines divide the whole phase diagram into two categories: hadron
phase and the quark phase.

Normally, apart from the critical temperature $T_{c}$, there have two other
temperatures of interests in a first-order phase transitions. These two
temperatures $T_{c1}$ and $T_{c2}$ are named as the lower and upper spinodal
critical points, respectively. A typical example is shown in Fig.2, where the
evolutions of the potential for several temperatures when fixed a chemical
potential at $\mu=306$ MeV and $\mu=309$ MeV are exhibited. For the left panel
in Fig.2 as $\mu=306$ MeV, when the temperature is around $T_{c}\simeq 20.6$
MeV, the shape of the potential exhibits two degenerate minima. However, with
the increase of the temperature, the second minimum of the potential at
$\sigma=\sigma_{h}$ disappears at a higher temperature $T_{c2}\simeq 23.1$
MeV. In the meanwhile, when the temperature is to fall below the critical
temperature $T_{c}$, the first minimum of the potential at $\sigma=\sigma_{l}$
tends to wipe out around $T_{c1}\simeq 14.7$ MeV. Between these two specific
temperatures, metastable states or false vacuum exist, and the system can
exhibit supercooling or superheating.

For $\mu=309\mathrm{MeV}$, one can also observe the characteristic pattern of
a first order phase transition: two minima corresponding to phases of restored
and broken chiral symmetry are separated by a potential barrier and they will
become degenerate when the temperature is at $T_{c}\simeq 13.3$ MeV. Chiral
symmetry is approximately restored for $T>T_{c}$, where the minimum at false
vacuum $\sigma=\sigma_{l}$ becomes the absolute minimum as shown in the right
panel in Fig.2. Similarly to the previous case for $\mu=306$ MeV, when the
temperature $T$ goes across up the critical line and rises further, the
potential barrier between two minima will start to decrease gradually and
shrink to zero at the moment when the second minimum of the potential at
$\sigma=\sigma_{h}$ vanishes at a spinodal temperature $T_{c2}\simeq 18.8$
MeV. On the other hand, when $T<T_{c}$, the shapes of the effective potential
at various low temperatures display quite different behaviors in comparison
with those of the previous case as $\mu=306$ MeV. The barrier between two
minima of the effective potential is to maintain even when the temperature $T$
is very close to zero. This means that the first minimum of the effective
potential at $\sigma=\sigma_{l}$ always could exist in hadron phase.
Therefore, the phase transition could be identified as a strongly first-order
phase transition which is usually induced by an effective potential with a
nonvanishing zero-temperature potentia barrier.

To give out a complete description of a first-order phase transition, two
particular lines of the spinodal points which constrain the regions of
spinodal instability for the first-order phase transition at high density are
illustrated in Fig.1. Similarly to the critical line in Fig.1, both the lower
and upper spinodal lines grow up with the reduction of the chemical potential
$\mu$, but the gaps of these two spinodal lines become small and small, in the
end, two spindoal lines and the critical line will terminate by the same point
at the CEP. Moreover, since the lower spinodal line will end up at some point
$\mu_{c}\simeq 308$ MeV on the vertical axis of the chemical potential, the
area of the first-order phase transition can be technically split into a
weakly first-order phase transition and a strongly first-order phase
transition according to the above discussion.

Therefore, for a weakly first-order chiral phase transition as the chemical
potential is $\mu<\mu_{c}$ as $T<T_{c}$, the thermodynamic potential exhibits
a local minimum aside from the global minimum, when the temperature decreases
from $T_{c}$ to a specific value $T_{c1}$. The local minimum gradually
disappears at a point of the inflection known as spinodal instability.
Whereas, for $\mu>\mu_{c}$, the chiral phase transition is to be considered as
a strongly first-order one due to the fact that the local minimum remains for
the temperature is at $T<T_{c}$ and there is no spinodal temperature anymore.
The critical chemical potential for a transition from a weak first-order phase
transition to that of a strong one is then identified as the critical chemical
potential at $\mu_{c}\simeq 308$ MeV in hadron phase [36, 54].

##  IV Homogeneous thermal nucleation

If a first-order phase transition takes place, its dynamics is non-trivial due
to the fact that the discontinuity in energy density known as the latent heat
and the basic mechanism for the first-order phase transition can be described
by the bubble nucleation. In terms of the effective potential, the general
setting is illustrated as shown in Fig.2. For a first-order phase transition,
the effective potential has at least two minima which are separated by a
potential barrier. When the temperature is around its critical value, the
effective potential exhibits degenerate minima. However, as long as the
temperature $T$ deviates from its critical value $T_{c}$, one of the minimum
of the effective potential becomes a local minimum, whereas the other turns
into a global minimum. Traditionally, the local minimum is referred to a
metastable state or a false vacuum, and the global minimum is taken as the
stable state or a physical vacuum. The false vacuum would be classically
stable, but quantum mechanically it is only a metastable state and can decay
via the nucleation of bubbles of the stable state inside the unstable state.
This decay can be triggered by either quantum or thermal fluctuations,
depending on what kind of physics we are interested in. In the following
discussion, we will be mostly concerned with the regime in which thermal
fluctuations are much larger than quantum fluctuations.

The mechanism of the nucleation theory can be used to study the probability
that a bubble or droplet of the stable vacuum in a system initially trapped in
the metastable vacuum near the critical temperature $T_{c}$. For a pure
system, the formation of bubbles originates from intrinsic thermodynamic
fluctuations, this kind of nucleation mechanism is commonly called as
homogeneous nucleation. On the contrary, when impurities cause the formation
of bubbles or droplets, such a mechanism of the nucleation theory is known as
heterogeneous nucleation. In the everyday world, the external agents would
play the role of nucleating centers, such as dust or ions in the atmosphere,
leading to a much more efficient increase of the nucleation rate.
Nevertheless, for the physical interests related to our study, homogeneous
nucleation theory is appropriate and we will use this basic theoretical
apparatus to describe the decay of the a metastable vacuum of a system
interacting with a heat bath at temperature $T$.

A quantum field theoretical description of metastable vacuum decay was
initiated with the work of Coleman and collaborators in the late 1970s [29,
30]. These methods built upon earlier work of Langer [27, 28] were
subsequently extended to finite temperature by Linde [33, 34] and Affleck
[32]. Based on the framework of the homogeneous thermal nucleation, in the
limit that thermal fluctuations dominate quantum fluctuations, the nucleation
rate per unit time per unit volume is given by the form of

$\Gamma=\mathcal{P}\exp\left[-\frac{S_{3}}{T}\right],$ (13)

where $T$ is the temperature of the system in equilibrium with the thermal
bath, $S_{3}$ is the three-dimensional action associated with the
$O(3)$-symmetric critical bubble or droplet and $\mathcal{P}$ is the
exponential prefactor. For the mechanism of the bubble nucleation, to the
leading order, the nucleation rate is controlled by the exponent of the three-
dimensional action evaluated on the critical bubble. The sub-leading
corrections to the leading-order bubble action are included in the prefactor
$\mathcal{P}$, which can be technically expressed as

$\mathcal{P}=\frac{\omega_{-}}{2\pi}\left(\frac{S_{3}}{2\pi
T}\right)^{3/2}\left[\frac{\mathrm{Det}(-\nabla^{2}+\Omega^{\prime\prime}_{FV})}{\mathrm{Det}^{\prime}(-\nabla^{2}+\Omega^{\prime\prime}_{B})}\right]^{1/2}.$
(14)

Here $\omega_{-}$ is the eigenvalue of the negative mode, the terms
$\Omega^{\prime\prime}_{FV}$ and $\Omega^{\prime\prime}_{B}$ are abbreviations
for $\Omega^{\prime\prime}$ evaluated in the false vacuum and the critical
bubble, the prime in the determinant signifies that the zero eigenvalues
associated with the translation symmetry of the bubble are omitted.
$\Omega^{\prime\prime}$ is the second derivative of the effective potential
$\Omega(T,\mu)$ with respect to the order parameter $\sigma$ which is actually
represented a field in describing the extremum of the three-dimensional
Euclidean action, more specifically, the critical bubble or the bounce.
Usually, the calculation and evaluation of this prefactor is a nontrivial
matter, a rough estimate of their ratios can be obtained by dimensional
analysis and it can be approximately expressed as $T^{4}$ or $T^{4}_{c}$ for
simplicity [25, 40].

The result represented in the above equation (13) is a semi-classical
contribution based on a saddle point approximation around the bounce solution.
By taking the scalar field $\sigma$ as the order parameter, at finite-
temperature field theory, an Euclidean action we are interested in is

$S_{E}(\sigma)=\int_{0}^{\beta}d\tau\int
dr^{3}\left[\frac{1}{2}\left(\frac{\partial\sigma}{\partial\tau}\right)^{2}+\frac{1}{2}\left(\nabla\sigma\right)^{2}+\Omega(\sigma;T,\mu)\right],$
(15)

in which the subscript $E$ denotes Euclidean and the integral is over
Euclidean space. For the sake of convenience, in the following discussions, we
will keep the $\sigma$ field in the effective potential $\Omega$ in Eq.(8)
explicitly. As argued by Linde [34], for sufficiently high temperature as
length scales large compared to $\beta$, the relevant number of dimensions is
$d=3$, and the Euclidean action becomes

$S_{E}(\sigma)\equiv\frac{S_{3}}{T},$ (16)

where $S_{3}$ is the three-dimensional saddle-point action associated with the
formation of a critical-sized bubble or droplet, in what follows it is to be
called as the saddle-point action for abbreviation. Therefore, the bounce is
an $O(3)$ symmetric solution to the classical equation of motion that
extremizes the Euclidean action $S_{3}$. In particular, for a scalar field
$\sigma$, the bounce satisfies a nonlinear ordinary differential equation,

$\frac{d^{2}\sigma(r)}{dr^{2}}+\frac{2}{r}\frac{d\sigma(r)}{dr}=\frac{\partial\Omega(\sigma;T,\mu)}{\partial\sigma},$
(17)

with boundary conditions $\lim\limits_{r\to\infty}\sigma(r)=\sigma_{FV}$ and
$\frac{d\sigma(r)}{dr}|_{r=0}=0$. The first boundary condition is because that
the bubbles are embedded in the homogeneous false vacuum, outside the bubble,
the $\sigma$ field should arrive at its false vacuum at
$\sigma\simeq\sigma_{FV}$. While the second one is set by the requirement of
finite energy at the origin. The solution for this equation of motion with the
above proper boundary conditions is a saddle point solution or a bounce
$\sigma_{b}$. It is an $O(3)$ non-trivial field configuration that starts in
the false vacuum and reaches the other side of the potential barrier with zero
velocity. In this work we will use AnyBubble package [55] to determine the
bounce.

Once the solution $\sigma_{b}$ is obtained, the $S_{3}$ exponent in Eq.(13)
can be evaluated on the bounce solution $\sigma_{b}$ as

$S_{3}=\int
dr^{3}\left[\frac{1}{2}\left(\nabla\sigma\right)^{2}+\Omega(\sigma;T,\mu)\right],$
(18)

and the surface tension of the nucleation bubble interface between the false
vacuum and the true vacuum is defined accordingly as

$\Sigma=\int
dr\left[\frac{1}{2}\left(\frac{d\sigma}{dr}\right)^{2}+\Omega(\sigma;T,\mu)\right].$
(19)

It is worth noting that in the practical calculations, if the false vacuum has
a non-zero potential energy, an additional term $-\Omega(\sigma_{FV};T,\mu)$
should be included in the $S_{3}$ action and the surface tension $\Sigma$.

For a generic effective potential, the equation of motion of the bounce with
boundary conditions usually cannot be obtained analytically, we should rely on
numerical methods to do the computation. However, when the system is very
close to the critical coexistence line, the bubble radius $R$ is much larger
than the wall thickness ($\triangle R\sim m_{\sigma}^{-1}$), hence the damping
force $2\sigma^{\prime}/r$ in the field equation has become negligible, the
thin-wall approximation is applicable and the equation of motion (17) reduces
to the field equation for a typical one-dimensional soliton

$\frac{d^{2}\sigma(r)}{dr^{2}}=\frac{d\Omega}{d\sigma}.$ (20)

This static field equation implies that

$\frac{d\sigma(r)}{dr}=\pm\sqrt{2\Omega}.$ (21)

Integrating Eq.(21) yields

$\displaystyle r$ $\displaystyle=$
$\displaystyle\int_{\sigma}^{\sigma_{v}}\frac{d\sigma}{\sqrt{2\Omega}}.$ (22)

Therefore an approximate solution for the bounce can be obtained for arbitrary
potential with two or more degenerate minima. Moreover, within the thin-wall
approximation, the surface tension of the bubble can be calculated as

$\Sigma_{\mathrm{tw}}=\int_{0}^{\infty}dr\left[\frac{1}{2}\left(\frac{d\sigma}{dr}\right)^{2}+\Omega\right]=\int_{0}^{\sigma_{v}}d\sigma\sqrt{2\Omega},$
(23)

and the saddle-point action $S_{3}$ is given by

$S_{3}=\frac{16\pi}{3}\frac{\Sigma_{\mathrm{tw}}^{3}}{\varepsilon}.$ (24)

The quantity $\varepsilon=\Omega(\sigma_{f};T,\mu)-\Omega(\sigma_{t};T,\mu)$
is the difference between the values of the effective potential at the false
vacuum and at the true vacuum.

##  V Results and discussion

In this section, we will numerically solve the equation of motion in Eq.(17)
with some boundary conditions, $\sigma\rightarrow\sigma_{FV}$ as
$r\rightarrow\infty$ and $\sigma^{\prime}(0)=0$. Since a first-order phase
transition necessitates a discontinuity in the scalar field $\sigma$, the
transition does not take place exactly at the critical temperature $T_{c}$
along to the critical coexistence line in phase diagram in figure 1.
Consequently, the false vacuum $\sigma_{FV}$ should be addressed properly
before we obtain the exact numerical solutions of the equation of motion.

As discussed in Sec.III, for the first-order phase transition, the typical
potential of the $\sigma$ field exhibits three distinct extrema. One is the
maximum of the effective potential, which represents the barrier between the
quark and hadron phases, while others are two minima of the effective
potential corresponding to the false and true vacuum. In order to clarify
these two vacua, in the former section, we have denoted these two local minima
as $\sigma_{l}$ and $\sigma_{h}$ respectively. Here, $\sigma_{l}$ is the value
of the $\sigma$ field at one minimum of the potential close to the origin,
whereas $\sigma_{h}$ represents another minimum of the potential at a relative
lager sigma. Traditionally, $\sigma_{l}$ is referred as the quark phase, due
to the fact that the chiral symmetry is approximately restored and the quarks
throw away most of their constituent masses as $\sigma=\sigma_{l}$. On the
other hand, $\sigma_{h}$ stands for the hadron phase, because the chiral
symmetry is spontaneously broken and the quarks have a larger constituent
masses. So that the quark phase is a stable vacuum and the hadron phase is a
metastable vacuum as $T>T_{c}$. On the contrary, when $T<T_{c}$, the hadron
phase becomes stable and the original quark phase becomes a metastable one,
and these two vacua will get exchanged at $T=T_{c}$.

In order to get an intuitional description of the false vacuum decay at finite
temperature and density, the dynamics of a first-order phase transition at
metastability is to be classified into two categories: top-down and bottom-up.
More specifically, by “top-down”, we mean that we will study the false vacuum
decay at finite temperature and density for the temperature $T<T_{c}$ and the
initial false vacuum of the system is well prepared as quark phase at
$\sigma=\sigma_{l}$ for $T\simeq T_{c}$ when the system is cooling down from
very high temperature. On the other side, by “bottom-up”, we mean that the
original false vacuum for the temperature $T\simeq T_{c}$ is defined as the
hadron phase at $\sigma=\sigma_{h}$, and we will study the decay of the hadron
phase for the temperature $T>T_{c}$ in the event that the system is heating up
from the low energy state.

###  V.1 Top-down

Figure 3: (Color online) (a) Critical bubble profiles for different
temperatures when fixing the chemical potential at $\mu=306$ MeV for
$T<T_{c}$. From left to right, the curves correspond to $T=15$, $16$, $17$,
$18$, $19$ and $20$ MeV. (b) Critical bubble profiles for different
temperatures when fixing the chemical potential at $\mu=309$ MeV for
$T<T_{c}$. From left to right, the curves correspond to $T=1$, $4$, $6$, $8$,
$10$ and $12$ MeV.

As the system is cooling down from very high temperature. When $T\simeq
T_{c}$, the initial false vacuum can be causally set as the quark phase since
the first-order transition does not happen exactly at $T_{c}$ but upon lowing
the temperature. The appearance of a bubble of the hadron phase (the stable
state) inside the quark phase (the metastable state) is a natural consequence
of the thermal fluctuations of the thermodynamical system sufficiently close
to the coexistence line in phase diagram. To study the dynamics of a first-
order phase transition in this “top-down” scenario as $T<T_{c}$, we
numerically solve the equation of motion in Eq.(17) with the specific boundary
conditions as $\sigma\rightarrow\sigma_{l}$ as $r\rightarrow\infty$ and
$d\sigma(0)/dr=0$. Here, the false vacuum $\sigma_{l}$ is temperature-
dependent as a matter of fact that the local minimum of the effective
potential varies with increasing of the temperature when fixing the chemical
potential.

For $\mu=306$ MeV, the exact numerical solutions by taking the temperatures as
$T=15$, $16$, $17$, $18$, $19$, $20$ MeV are plotted in the left panel of
Fig.3. It is shown that with the temperature decreasing from $T_{c}\simeq
20.6$ MeV, all curves approach to their false vacuum $\sigma_{l}$ when the
radius $r$ is large, whereas $\sigma(r)$ at the center of the bubble deviates
from its stable vacuum value at $\sigma=\sigma_{v}$ significantly. When the
temperature is sufficiently close to the critical temperature, the $\sigma$
field at the center of the bubble only slightly deviates from its stable
vacuum value at $\sigma=\sigma_{h}$, however, as soon as $T<T_{c}$ MeV, the
$\sigma(0)$ field is visibly different from its stable vacuum value. Such a
deviation can be qualitatively explained by an “overshoot-undershoot” argument
due to Coleman [29]. According to this argument, the equation of motion (17)
is reinterpreted as the equation for a particle moving in an “upside-down”
potential energy $\Omega$, the second term in the field equation is taken as a
damping force. When the system is very close to the critical coexistence line,
the bubble radius $R$ is very larger, so that the damping force can be
neglected, the field $\sigma$ can start to roll down at the top of the
effective potential $\Omega$ around $\sigma\simeq\sigma_{h}$ to rest at its
false vacuum $\sigma_{l}$. However, with the temperature descends, the radius
of the bubble decreases accordingly, the damping force in the field equation
becomes important, consequently, the field $\sigma(0)$ will deviate from its
true vacuum value more and more dramatically. In other words, the thin-wall
approximation mentioned above is expected to be invalid, and any further
extension of the thin-wall approximation to the temperatures deviation from
$T_{c}$ should be checked very carefully.

A similar discussion can be applied to the second case when the chemical
potential is fixed at $\mu=309$ MeV. The critical bubble profiles at different
temperatures are illustrated in the right panel of Fig.3, where the
temperatures are taken as $T=1$, $4$, $6$, $8$, $10$, $12$ MeV from left to
right. The evolution of $\sigma(r)$ for different temperatures tells that the
typical radius of the critical bubble should increase as well as the increase
of the temperature, and $\sigma(r)$ approaches to its false vacuum value at
$\sigma=\sigma_{l}$ as $r\rightarrow\infty$. Also, from this figure, as long
as the temperature is lower than the critical temperature $T_{c}$, $\sigma(0)$
will depart from its stable vacuum value at $\sigma=\sigma_{v}$ significantly,
this nontrivial behavior of the $\sigma(r)$ in the center of the bubble can
also be interpreted as a limit to the applicability of the thin-wall
approximation.

Figure 4: (Color online) (a) Surface tension as a function of temperature $T$
for $T\leq T_{c}$ at $\mu=306$ MeV. (b) Surface tension as a function of
temperature $T$ for $T\leq T_{c}$ at $\mu=309$ MeV.

Once the bubble profiles have been solved, from the definition in Eq.(19), the
surface tension of the nucleation bubble interface between the false vacuum
and the stable vacuum as a function of the temperature can be obtained, and
the results are shown in Fig.4 when the chemical potentials are taken as
$\mu=306$ MeV and $\mu=309$ MeV. For both cases, with increasing of the
temperature, the surface tension $\Sigma(T)$ starts to grow firstly and
reaches a maximum at a certain temperature, then it will inflect and slope
downwards. These nontrivial behaviors of $\Sigma(T)$ were also reported in a
weak first-order phase transition [54] and a strong first-order phase
transition [36] by using the exact numerical method, but these nontrivial
properties seem to be completely destroyed by the thin-wall approximation. As
shown in the left panel of Fig.2, for a weak first-order phase transition,
there exists a spinodal temperature $T_{c1}$ where a small barrier between the
two minima in the potential will disappear. As a consequence, there is no
bubble solution anymore as $T<T_{c1}$ and the surface tension decreases
rapidly to zero when $T\rightarrow T_{c1}$. But for a strong first-order phase
transition, according to a standard criterion to guarantee the existence of
the stable bounce or soliton, it is indispensable for the potential of the
order parameter $\sigma$ field to exhibit three distinct extrema. Hence, we
can have a nontrivial solution to the equation of motion (17) for any
temperature as $T<T_{c}$. The surface tension will then decline to a nonzero
value as the decreasing of the temperature as depicted at the right panel of
Fig.4. This is the main difference between a strong first-order phase
transition and a weak one, this is the reason why we have separated the area
of the first-order phase transition in phase diagram into two categories: a
strong one and a weak one in the previous discussion.

Figure 5: (Color online) (a) The saddle-point action as a function of
temperature $T$ for $T\leq T_{c}$ at $\mu=306$ MeV. (b) The saddle-point
action as a function of temperature $T$ for $T\leq T_{c}$ at $\mu=309$ MeV.

We are now to determine the $S_{3}/T$ exponent in Eq.(13), which is the
saddle-point action evaluated on the bounce solution. Because the decay rate
per unit volume is what we are interested, the argument of the exponential
$S_{3}/T$ is more important in comparison with the prefactor $\Gamma$ if the
$S_{3}/T$ is larger than the unity $1$. As shown in the following discussions,
for the most studies in the present work, an estimate for the prefactor based
on dimensional analysis is sufficient. To show the saddle-point action due to
the appearance of the critical bubble and its crucial role played in the
nucleation rate for the first-order phase transition, the $S_{3}/T$ exponent
as a function of the temperature $T$ at different chemical potentials are
plotted in Fig.5. As shown in the left panel in Fig.5, for a weak first-order
phase transition as $\mu=306$ MeV, $S_{3}/T$ will start from zero when the
temperature is at the spinodal temperature $T_{c1}$ because the barrier is to
disappear and there is only a trivial solution for the field equation of the
$\sigma$ field. Then it will climb up very quickly with the increase of the
temperature and tend to diverge near the critical temperature $T_{c}$. By the
exponential form of equation (13), $\Gamma$ will be strongly suppressed by the
saddle-point action and the system is likely to stay in the metastable vacuum
for a relatively long time as long as $S_{3}/T>1$. Therefore, for a weak
first-order phase transition, the system can be trapped in quark phase even
when the temperature is below than the critical temperature $T_{c}$, until to
the temperature such that $S_{3}/T\simeq 1$. After that, the exponential
factor is unimportant and the probability of a false vacuum decay through the
barrier is essentially enhanced by the thermodynamical fluctuation. From the
left panel in Fig.5, for $\mu=306$ MeV, the remarkable temperature is about
$T\simeq 16$ MeV when $S_{3}/T\simeq 1$, it is very close to the spinodal
critical temperature $T_{c1}\simeq 14.7$ MeV. This indicates that the quark
phase could survive safely up to the temperature nearby the low spinodal
temperature $T_{c1}$ for a weak first-order phase transition.

When fixing the chemical potential at $309$ MeV, the resulting plot of the
$S_{3}/T$ exponent as a function of the temperature $T$ are shown in the right
panel in Fig.5. In this case, the $S_{3}/T$ firstly decreases with the
increase of the temperature, then it reaches a minimum point and restarts to
grow up very quickly. As the temperature is close to the critical temperature
$T_{c}$, it will become divergent. In comparison with the former case of a
weak first-order phase transition, for a strong one, the saddle-point action
as a function of temperature $T$ usually exhibits a non-monotonic behavior
with increasing of the temperature, this can be taken as one of the salient
properties of the strong first-order phase transition [36]. Besides the non-
trivial property of the saddle-point action, there is another obvious
difference from that of the first-order phase transitions. For $T<T_{c}$, no
matter what the temperature we are taken, the $S_{3}/T$ is always larger than
the unity $1$. Such that, for a strong first-order phase transition in the
present model, it seems the conversion of quark phase to hadron phase is
exponentially suppressed for $T\leq T_{c}$ and $\mu\geq\mu_{c}$ with
$\mu_{c}\simeq 308$ MeV, and the system is likely to stay in quark phase
rather than hadron phase, this could in turn induce various exotic structures
of the phase of the strong-interaction matter in high density and low
temperature due to the presence of the free quarks.

###  V.2 Bottom-up

By “bottom-up”, we mean that the system is heating up from low energy to very
high energy and the starting point is set as hadron phase when $T\simeq
T_{c}$. With increasing of the temperature further, while $T>T_{c}$, the
appearance of a bubble of the quark phase (the stable vacuum) inside the
hadron phase (the false vacuum) is also treated as a natural consequence of
the thermal fluctuations of the thermodynamical system sufficiently nearby the
coexistence line in phase diagram. Therefore in the “bottom-up” scenario when
$T>T_{c}$, we should numerically solve the equation of motion in Eq.(17) with
the specific boundary conditions as $\sigma\rightarrow\sigma_{h}$ as
$r\rightarrow\infty$ and $d\sigma(0)/dr=0$ because the false vacuum is located
at the hadron phase and the quark phase is now the stable state.

Figure 6: (Color online) (a) Critical bubble profiles for different
temperatures when fixing the chemical potential at $\mu=306$ MeV for
$T>T_{c}$. From right to left, the curves correspond to $T=20.7$, $21$, $22$
and $23$ MeV. (b) Critical bubble profiles for different temperatures when
fixing the chemical potential at $\mu=309$ MeV for $T>T_{c}$. From right to
left, the curves correspond to $T=14$, $15$, $16$ and $17$ MeV.

In what follows, we begin by showing the critical bubble profiles obtained
from the exact numerical solution of Eq.(17) with the boundary conditions
$\sigma(\infty)=\sigma_{h}$ and $\sigma^{\prime}(0)=0$ when fixing chemical
potentials at $\mu=306$ MeV and $\mu=309$ MeV in Fig.6. This is unlike the
results presented in the “top-down” scenario when $T<T_{c}$, in which there
have two different first-order phase transitions. However, for $T>T_{c}$, the
results for both panels in Fig.6 exhibit some quite similar features. For both
cases, with the increase of the temperature from the critical temperature to
the up spinodal line at $T=T_{c2}$, the typical size of the bubble which can
be approximately estimated by the maximal value of the quantity
$\left|\sigma^{\prime}(r)\right|$ will decrease rapidly to zero because the
barrier between the two minima in the potential disappears and there is no
stable bubble solution anymore as $T\geq T_{c2}$. Furthermore, the structures
of the critical bubbles also share alike properties. When the temperature is
close to the critical temperature $T_{c}$, the critical bubble has an obvious
“core” structure with $\sigma\simeq\sigma_{l}$ separated by a relatively thin
wall from the region $\sigma\simeq\sigma_{h}$. On the other side, when the
temperature comes up to another critical point at $T\simeq T_{c2}$, the
critical bubble usually becomes “coreless” structure due to the fact that the
thickness of the critical bubble has the same order as the radius and the
field at the original point $\sigma(r=0)$ departs from its true vacuum
$\sigma_{l}$ largely. At last, the curves in Fig.6 can be explained
qualitatively according to the “overshoot-undershoot” argument given by
Coleman. When the temperature is very close to the critical temperature
$T_{c}$, the potential has two degenerate vacua, and the damping force is
neglectful small, the field at escape point $\sigma(r=0)$ starts at the top of
the effective potential around $\sigma\simeq\sigma_{l}$. On the contrary, with
the temperature rising up, two degenerate vacua get decoupled and the damping
force takes effect, the field $\sigma(r=0)$ will deviate from its vacuum value
dramatically, especially as $T\rightarrow T_{c2}$. In other words, the thin-
wall approximation is not expected to be valid, any further extension of the
thin-wall approximation to higher temperatures should be checked very
carefully particularly when the temperature is close to the up spinodal line.

Figure 7: (Color online) (a) Surface tension as a function of temperature $T$
for $T\geq T_{c}$ at $\mu=306$ MeV. (b) Surface tension as a function of
temperature $T$ for $T\geq T_{c}$ at $\mu=309$ MeV.

From the definition of the surface tension in Eq.(19), we plot the temperature
dependence of the surface tensions for $T\geq T_{c}$ when fixing the chemical
potential at $\mu=306$ MeV and $\mu=309$ MeV in Fig.7. For $\mu=306$ MeV, the
values of the surface tension are between about $1.6$ MeV and $0$ MeV, while
for a relatively larger chemical potential $\mu=309$ MeV, they are among $\sim
2.8$ MeV to zero. Then, we can find the biggest values of the surface tension
occur near the critical line since this domain is characterized by large
barriers and a small energy difference between the true and false vacua.
Besides, the result in Fig.7 also implies that the surface tension nearby the
critical line at $T\simeq T_{c}$ will increase accordingly with the increase
of the chemical potential. Moreover, for both cases, the surface tensions will
continuously decline to zero as long as the temperature approaches to the up
spinodal line at $T\simeq T_{c2}$. Therefore, in the bottom-up scenario,
$\Sigma(T)$ is a monotonically decreasing function of $T$, whereas in the top-
down scenario it is a non-monotonic function and has a nontrivial behavior.
Sometimes, the nontrivial evolution of $\Sigma(T)$ suggests that the
temperature dependent surface tension has a maximum value at a specific
temperature and it can be taken as a limit to the applicability of the thin-
wall approximation [36, 54]. Thus, for the bottom-up scenario, we need to
develop an alternative method to estimate a scope in which the thin-wall
approximation is valid.

Figure 8: (Color online) (a) The saddle-point action as a function of
temperature $T$ for $T\geq T_{c}$ at $\mu=306$ MeV. (b) The saddle-point
action as a function of temperature $T$ for $T\geq T_{c}$ at $\mu=309$ MeV.

To study the dynamics of a first-order phase transition, the last important
quantity to be evaluated is the saddle-point action $S_{3}/T$ due to the
activation of a nucleation bubble, which is an essential ingredient for the
nucleation rate per unit time per unit volume in Eq.(13). In Fig.8, $S_{3}/T$
is plotted as a function of temperature $T$ for $T\geq T_{c}$ when fixing the
chemical potentials at $\mu=306$ MeV and $\mu=309$ MeV. From this figure, as
the temperature approaches from above to the critical temperature $T_{c}$,
$S_{3}/T$ will increase every quickly and diverges near the critical
temperature $T_{c}$. In the opposite direction, when the temperature is close
to the spinodal critical temperature $T_{c2}$, $S_{3}/T$ will decrease rapidly
to zero, during this procedure, what we are interested in is the moment when
$S_{3}/T$ is about a unity one because if $S_{3}/T>1$ the nucleation rate
$\Gamma$ will be strongly suppressed by the exponential factor and the system
is likely to stay in the false vacuum for a very long time. From the left
panel of Fig.8, for $\mu=306$ MeV, when the temperature is about $22.8$ MeV,
$S_{3}/T\simeq 1$, so that the system is likely to remain in hadron phase
until the temperature is less than $22.8$ MeV. For $\mu=309$ MeV, this
specific temperature is approximately $18.4$ MeV when $S_{3}/T\simeq 1$, this
indicates that hadron phase could also maintain its existence as long as the
temperature is below $18.4$ MeV. Since these two specific temperatures are
very close to their spinodal critical temperatures $T_{c2}$, to a rough
estimation, we can simply taken the spinodal critical line in phase diagram as
a phase boundary for the stable existence of the false vacuum in the first-
order phase transition.

##  VI Conclusion

In this work we have computed the effective potential for two-flavor quark-
meson model at finite temperature and density in the presence of a fermionic
vacuum term. Having the in-medium effective potential, the phase diagram
together with the critical end point has been given and the up and low
spinodal lines have been calculated explicitly for the first-order hadron
quark phase transition. For the low spinodal line, the first-order phase
transition can further be divided into a strong and weak ones in phase diagram
when the temperature is below the critical coexistence line. The critical
chemical potential is taken as $\mu\simeq 308$ MeV as the low spinodal line
terminates at this point. So that as $T<T_{c}$, for $\mu<308$ MeV, it is a
weak first-order phase transition, but for $\mu>308$ MeV, it belongs to a
strong first-order phase transition.

Provided by the temperature-dependent effective potential, the problem of
homogeneous nucleation of bubble in a first-order phase transition can be
investigated accordingly. For convenience, we have separated our discussions
into two scenarios: the top-down case and the bottom-up case. By “top-down”,
we consider the quark phase as a metastable (false) vacuum and the hadron
phase as a stable (true) vacuum when $T<T_{c}$. On the contrary, by “bottom
-up”, we mean that the false vacuum is the hadron phase whereas the true
vacuum is the quark phase as $T>T_{c}$. Then for the former case, the boundary
condition at infinite radius is $\sigma=\sigma_{h}$, whereas it is
$\sigma=\sigma_{l}$ for the latter case. With these specific boundary
conditions, a saddle point solution of the field equation has been solved and
the exact bubble profiles were obtained. Usually, when the temperature is
close to the critical temperature $T_{c}$, the bubble profile shows a “core”
structure with the sigma field at true vacuum $\sigma\simeq\sigma_{T}$
separated by a relatively thin wall from the false vacuum at
$\sigma\simeq\sigma_{FV}$. However, when the temperature approaches to the
spinodal critical temperature, the bubble profile exhibits a coreless
structure since now the thickness of the critical bubble has the order of the
radius and the $\sigma$ field inside the bubble departs from its true vacuum
value significantly.

Figure 9:  (Color online) Surface tension as a function of a quark chemical
potential when $T\simeq T_{c}$. The solid line is for the “top-down” scenario
and the dashed line is for the “bottom-up” scenario.

The calculation for the surface tension between a quark phase and hadron phase
were also presented in these two different scenarios. In the “top-down”
context, the surface tension first increases to a maximum value and then
decrease with the increase of the temperature. The top of the surface tension
could be taken as a limit on the reliability of the thin-wall approximation
because the bubble profile at this point hints of a largest distortion of that
of the thin-wall approximation. On the other side, for the “bottom-up”
context, the surface tension demonstrates a monotonic property with the growth
of the temperature, it will descend continually from its top value at the
critical temperature $T_{c}$ to zero as $T\rightarrow T_{c2}$. As we known,
the surface tension plays an important role in the field of nuclear physics
and astrophysics and has attracted much attention recently. To provide a
comprehensive consultation of the relevant research, the surface tension along
the critical coexistence line has been laid out in Fig.9, we believe the
remaining value of the surface tension in the domain of the first-order phase
transition can be easily extracted and estimated by using the method in this
work. Consequently, the present model predicts a surface tension of
$\Sigma\sim 0-4$ $\mathrm{MeV/fm^{2}}$, our results are very close to the ones
recently found for the same model in the thin-wall approximation in Ref [45]
since the thin-wall approximation is a reliable tool as $T\simeq T_{c}$. Note
that most effective models also give a predict for a small values as
$\Sigma\leq 30$ $\mathrm{MeV/fm^{2}}$, such as the MIT bag model[56], NJL
model[57, 58], three-flavor PQM model[43], the nucleon-meson model[59] and the
Friedberg-Lee model [36]. Such a small value of the surface tension would lead
to a quark-matter formation during core-collapse supernova explosion and
favors a mixed phase in the cores of compact stars, so that this reasonably
small surface tension could provide an observable signal of the first-order
phase transition within compact stars and play an important role in
astrophysics.

For a weak first-order phase transition, our results show a rapid decrease
with temperature in the saddle-point action of critical bubbles from infinity
at the critical temperature to zero at the spinodal critical temperature. This
implies that we can always have the moment at which $S_{3}/T\simeq 1$ for the
weak first-order phase transition. However, for a strong first-order phase
transition, the saddle-point action of critical bubbles shows some quite
different characterizations. When the temperature increases, it first
decreases to a minimum value, then it rises up and diverges as the temperature
is close to the critical temperature $T_{c}$. This non-monotonic behavior of
the $S_{3}/T$ with the increase of the temperature also reported in a previous
study based on the Friedberg-Lee model [36], thus it can be taken as a salient
feature of a strong first-order phase transition in comparison with the weak
ones. Another interesting character of the $S_{3}/T$ is that the saddle-point
action never gets chance to reach the unity $1$ for a strong first-order phase
transition. The result can be roughly interpreted with the former study in
Fig.6 in Ref.[36], where the evolution of the $S_{3}/T$ as a function of
chemical potential shows that for a fixed temperature the saddle-point action
will increase and go across unity $1$ quickly when the chemical potential
rises up to $\sim 231$ MeV. Hence the $S_{3}/T$ is believed to satisfy the
condition $S_{3}/T>1$ in the present study since we have a lower temperature
and larger chemical potential.

Given the exponential dependence of $\Gamma$ on $S_{3}/T$, the decay of the
false vacuum is to be exponentially suppressed and the system is likely to
stay in the metastable state for a relatively long time when $S_{3}/T>1$. So
that the false vacuum could survive and exist as a metastable state as long as
the temperature lies between the up and low spinodal lines, only if the
temperature is close to the spinodal critical line, the $S_{3}/T$ will descend
and go through the unity $1$. More specifically, a “conventional” hadron
matter among the critical coexistence line and the low spinodal line could be
potentially treated as a quark matter, then the exotic structures of the
strong-interaction matter, such as the quarkyonic matter [60, 61], pion
superfluidity [62] and color superconductors [63, 64], could be there too.

###### Acknowledgements.

We thank Ang Li, Jinshuang Jin, Ken D. Olum and Xinjian Wen for valuable
comments and discussions. This work is supported in part by National Natural
Science Foundation of China (NSFC) under No.11675048.

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