# On thermodynamically consistent quasiparticle model at finite chemical
potential111This article is dedicated to the memory of the late Professor Ru-
Keng Su.

Wei-Liang Qian1,2,3 Hong-Hao Ma4,5 Shao-Yu Yin6 Ping Wang7 1 Escola de
Engenharia de Lorena, Universidade de São Paulo, 12602-810, Lorena, SP, Brazil
2 Faculdade de Engenharia de Guaratinguetá, Universidade Estadual Paulista,
12516-410, Guaratinguetá, SP, Brazil  3 Center for Gravitation and Cosmology,
College of Physical Science and Technology, Yangzhou University, Yangzhou
225009, China  4 Department of Physics, Guangxi Normal University, Guilin
541004, China  5 Guangxi Key Laboratory of Nuclear Physics and Technology,
Guangxi Normal University, Guilin 541004, China  6 Institute for Theoretical
Physics & Cosmology, Zhejiang University of Technology, Hangzhou 310032, China
7 Institute of High Energy Physics, CAS, P. O. Box 918(4), Beijing 100049,
China

(Dec. 26th, 2022)

###### Abstract

We explore the quasiparticle model at finite chemical potential related to Ru-
Keng Su’s distinguished contributions to the topic. Besides, we discuss recent
developments in the model, and in particular, one argues that the effective
mass of the quasiparticle might attain a specific form as a function of
momentum, in addition to its dependence on temperature and chemical potential.
Unlike the approaches based on the properties of underlying symmetry or
renormalization group, the momentum dependence emerges as a special solution
to an integro-differential equation resulting from the underlying
thermodynamic consistency. Moreover, this special solution to the problem is
shown to be more general than previously explored in the literature. Instead
of fitting to the lattice QCD data at vanishing chemical potential, in this
work, we adopt a “bottom-up” approach by assuming some analytic ansatzes that
are manifestly thermodynamically consistent. The remaining physical quantities
are subsequently derived, and possible implications are also addressed.

##  I Introduction

The transition between the quark-gluon plasma (QGP) and hadronic phases
constitutes one of the most prominent problems in high-energy nuclear physics.
In the vicinity of such a region, the underlying dynamics are essentially
nonperturbative, through which the system undergoes a dramatic change in the
number of degrees of freedom. Moreover, even in the QGP phase, the system’s
thermodynamic properties deviate significantly from those of a non-interacting
ideal gas of quarks and gluons. For instance, the lattice quantum
chromodynamics (QCD) calculations showed [1] that the system’s pressure and
energy density undershoot the Stefan-Boltzmann limit by about 15-20% even at
temperatures $T\gtrsim 3T_{c}$. Also, the speed of sound extracted from
lattice QCD is found to be smaller than that of a massless ideal gas. In
particular, as the system approaches the transition region, it is observed
that the speed of sound varies non-monotonically [2]. The above properties are
crucial for adequately establishing the equation of state (EoS), which plays a
central role in providing an appropriate description of the hydrodynamic
evolution of the hot and dense system that emerged in the relativistic heavy
ion collisions [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15].

In the literature, the density-dependent quark mass was suggested by Fowler,
Raha, and Weiner [16] to address the transition between nuclear and quark
matter, and the thermal partonic quasiparticle was initially proposed by
Peshier, Kampfer, Pavlenko, and Soff [17] to accommodate the numerical
experiments from the lattice QCD. Primarily inspired by its counterparts in
other fields of physics, the notion of the quasiparticle is a phenomenological
approach aimed at capturing the bulk thermodynamic properties of QGP. The
model can be viewed as an effective and simplified imitation of the essence of
many existing theoretical efforts, namely, lattice QCD [1], dimensional
reduction [18], hard thermal loop [19], Polyakov-loop model [20], as well as
other hadronic-degree based approaches [21]. It has also been speculated that
the success of the notion of quasiparticle degree of freedom will further give
rise to novel effective theories from a more fundamental perspective while
properly incorporating the nonperturbative aspects of the QCD.

The quasiparticle model interprets the system as a composition of non-
interacting quanta which carry the same quantum numbers of quarks and gluons.
The medium-dependent quasiparticle mass implements the strong interactions
between the elementary degrees of freedom. For the description of gluon
plasma, the quasiparticle mass was initially assumed to be merely temperature
dependent [17]. As the concept flourished in literature, a crucial feature of
the model was elaborated in a seminal paper by Gorenstein and Yang [22] with
respect to its thermodynamic consistency. The authors solved the issue
elegantly via the requirement of an exact cancelation between the additional
contributions from the temperature-dependent quasiparticle mass and those from
the bag constant.

Subsequently, various relevant aspects of the topic were discussed and further
developed by, among others [23, 24, 25, 26, 27], Ru-Keng Su in collaboration
with his students and collaborators [28, 29, 30, 31, 32, 33, 34, 35, 36, 37,
38, 39, 40, 41, 42, 43, 44]. In [28], the role of an additional contribution
to the thermopotential and its consequential effect on the strange quark
matter were explored. A series of studies regarding the quark mass density-
and temperature-dependent (QMDTD) model was performed in [29, 29, 30, 31, 32].
The temperature dependence of the stable radius of a strangelet was discussed
in [29]. The temperature dependence of the bag constant $B$ was explored and
shown to cure the divergence that occurred at vanishing baryon density in the
phase diagram for the bulk strange quark matter of the original QMDTD model
[30]. A systematic analysis regarding the stability of strangelet was
performed in [31] in the framework of the QMDTD model. It was observed that
stable strangelets are more likely to be encountered in the region with a
sizeable negative electric charge and significant strangeness. The analysis
was then extended to the dibaryon systems [32] regarding different decay
channels, and the results were found in good agreement with those obtained by
the chiral SU(3) quark model. The QMDTD setup was then applied to the context
of Friedberg-Lee soliton bag [45, 46, 47] nonlinear coupled to the sigma [34]
as well as omega [35] mesons. The model was further extended to investigate
the properties of deconfinement [38, 36] and nuclear matter [37]. As an
alternative approach to address the thermodynamic consistency, an additional
fictitious degree of freedom was introduced [41, 42] to elaborate a
generalized version of the first law of thermodynamics.

From the field theory perspective, the mass of a particle can be defined
either by the pole of the effective propagator or via the Debye screen mass
extracted at small momentum, provided the question of gauge invariance is
adequately dealt with. In particular, the calculations with hard thermal loop
approximation show that the gluon screen mass extracted from the above
pictures is consistent [19, 48]. The derived quasiparticle mass, in turn, is a
function of temperature and chemical potential. As a result, the above
dependence calls for a generalization scheme for thermodynamic consistency.
Further developments by Peshier et al. gives rise to a flow equation [23, 24,
25, 49]. The latter is a partial differential equation, and its boundary
condition is chosen at vanishing baryon density, adapted to the lattice QCD
data. It was shown that the thermodynamic properties obtained from such a
framework agree well with the lattice calculations at finite baryon chemical
potential.

Following [22], one takes the grand partition function of the system ${\cal
Z}$ as the starting point, which reads

$\displaystyle{\cal
Z}\left(V,T,\mu\right)=\mathrm{Tr}\left[e^{-\beta\left(H-\mu
N\right)}\right],$ (1)

and $\beta$ is the reciprocal of the temperature $T$, $\mu$ represents the
chemical potential, $V$ is the volume of the system, $H$ and $N$ are the
Hamiltonian and conserved number operators. In order to derive the remaining
thermodynamic quantities (such as pressure, energy density, and conserved
number density) in a consistent fashion, the two following identities need to
be valid,

$\displaystyle\frac{\partial{\cal
Z}\left(V,T,\mu\right)}{\partial\beta}=-\mathrm{Tr}\left[\left(H-\mu
N\right)e^{-\beta\left(H-\mu N\right)}\right],$ (2)

and

$\displaystyle\frac{\partial{\cal
Z}\left(V,T,\mu\right)}{\partial\mu}=\beta\mathrm{Tr}\left[Ne^{-\beta\left(H-\mu
N\right)}\right].$ (3)

The conditions Eqs. (2) and (3) are manifestly satisfied when the Hamiltonian
is not medium dependent. As an example, for the quasiparticle model proposed
in [17],

$\displaystyle
H=\tsum\slimits@_{\mathbf{k}}\omega(\mathbf{k},T,\mu)a_{\mathbf{k}}^{\text{\textdagger}}a_{\mathbf{k}}+VB,$
(4)

where $B$, the bag constant, is attributed to the vacuum energy, mostly viewed
as a constant, and

$\displaystyle\omega(\mathbf{k},T,\mu)=\omega(\mathbf{k},m)=\sqrt{k^{2}+m^{2}},$
(5)

where $k=|\mathbf{k}|$ and $m=m(T)$ is an explicit function of the
temperature. The latter adds an additional contribution to the partial
derivative in Eq. (2), associated with $H$. The recipe by Gorenstein and Yang
is derived from the proposal that $B$ should also be medium dependent, namely,
$B=B(T)$, whose entire purpose is to identically cancel out the undesirable
contribution coming from the temperature-dependent quasiparticle mass. To be
explicit, it is not difficult to show [22, 23, 44] that the above requirement
dictates the relation

$\displaystyle\frac{dB}{dT}=\left.\frac{\partial
p_{\mathrm{id}}(T,\mu,m)}{\partial m}\right|_{T,\mu}\frac{dm}{dT},$ (6)

where the pressure of ideal gas is an intensive property given by the standard
statistical mechanics,

$\displaystyle p_{\mathrm{id}}=\left.\frac{T}{V}\ln{\cal
Z}\left(V,T,\mu\right)\right|_{B=0},$ (7)

whose specific form is given below in Eq. (13).

Since $m=m(T)$ and $B=B(T)$, we have $B=B(m)$. In other words, Eq. (6) implies

$\displaystyle\frac{dB}{dm}=\left.\frac{\partial
p_{\mathrm{id}}(T,\mu,m)}{\partial m}\right|_{T,\mu},$ (8)

where the bag constant $B$ is understood to be a function of the particle mass
$m$ only.

Similarly, if the quasiparticle mass is chemical potential dependent, it seems
that the above scheme can be readily applied. Specifically, one replaces the
temperature derivative in Eq. (6) with the chemical-potential derivative,
while Eq. (8) remains unchanged. Moreover, if the mass function depends on
both the temperature and chemical potential, namely, $m=m(T,\mu)$, Eq. (8)
seemly serves the purpose. However, though it might not be apparent at first
glimpse, one can argue [44] that Gorenstein and Yang’s scheme cannot be
applied straightforwardly to such a case. This can be understood as follows.
To be precise, one needs to solve for $B=B(m)$ for an arbitrarily given form
$m=m(T,\mu)$, using Eq. (8). Observing the l.h.s. of Eq. (8), one concludes
that the dependence of temperature and chemical potential can be entirely
“packed” into the quasiparticle mass $m$. On the other hand, since the form
$m=m(T,\mu)$ is arbitrary, one can always redefine this function so that the
r.h.s. of Eq. (8) cannot be written as a function of $m$.

We note that the above considerations do not necessarily invalidate Eq. (8).
Instead, it indicates the existence of some additional constraint when finite
chemical potential is involved. In [23], Peshier et al. derived a flow
equation giving a further constraint for the mass function $m=m(T,\mu)$. In
[44], some of us derived an integro-differential equation, which is shown to
fall back to the flow equation under certain circumstances. Moreover, it was
demonstrated that there are also other possibilities, and in particular, the
quasiparticle mass can be a function of the momentum.

In the present study, we proceed further to explore the topic by adopting a
“bottom-up” approach. Specifically, instead of numerically adjusting the model
parameters to the lattice QCD data, we choose a straightforward but analytical
form for the mass function at vanishing chemical potential. By adopting the
analytic function, one can scrutinize the different branches of the mass
function in the temperature-chemical potential parameter plane. The remainder
of the present paper is organized as follows. In the next section, we review
the relevant elements regarding the thermodynamic consistency in the
quasiparticle model. The resulting integro-differential equation is presented
and discussed. Sec. III focuses on the novel type of solution. In particular,
we explore a mathematically simple form of the mass function at vanishing
chemical potential. It is shown that such a choice will not entirely determine
the mass function in the temperature-chemical potential parameter plane.
Different possibilities are then investigated numerically. The last section is
devoted to further discussions and concluding remarks.

##  II The generalized condition for thermodynamical consistency

This section discusses the formal constraints for the thermodynamic
consistency in the quasiparticle model. For the present study, the term
consistency implies the following three essential aspects. First, all the
thermodynamic quantities can be derived using the standard formulae once the
grand partition function $\cal Z$ is given. Second, these thermodynamic
quantities possess an interpretation in accordance with the ensemble average
in statistics. Last but not least, most thermodynamic identities, for
instance, those based on the first law of thermodynamics (c.f. Eq. (19)) and
extensive properties (c.f. Eq. (17)), remain unchanged. To our knowledge, the
scheme proposed by Gorenstein and Yang is the only one that meets all three
above requirements.

As discussed in [22], once Eqs. (2) and (3) are satisfied, and the energy
density and particle number density derived either from the ensemble average
or from the partial derivative of the grand partition function possess
identical forms. These lead to the following forms of the energy density

$\displaystyle\varepsilon=\frac{\langle
E\rangle}{V}=-\frac{1}{V}\frac{\partial\ln{\cal
Z}}{\partial\beta}=\epsilon_{\mathrm{id}}+B,$ (9)

with

$\displaystyle\epsilon_{\mathrm{id}}=\frac{g}{2\pi^{2}}\int_{0}\frac{k^{2}dk\omega(k,T,\mu)}{\exp[(\omega(k,T,\mu)-\mu)/T]\mp
1}+\mathrm{c.t.}\ ,$ (10)

where $g$ indicates possible degeneracy, “” corresponds to boson and fermion,
and the counter term “$\mathrm{c.t.}$” indicates contributions from anti-
particles obtained by the substitution $\mu\rightarrow-\mu$ in the foregoing
term. We have also considered the isotropic case
$m(\mathbf{k},T,\mu)=m(k,T,\mu)$.

To derive the above equation, we have assumed the validity of Eq. (2), namely,
the contribution from the temperature dependence of quasiparticle mass has
precisely been canceled out with the temperature dependence of $B$. By writing
it out explicitly, one finds

$\displaystyle\frac{\partial B}{\partial
T}=-\frac{g}{2\pi^{2}}\int_{0}\frac{k^{2}dk}{\omega(k,T,\mu)}\frac{1}{\exp[(\omega(k,T,\mu)-\mu)/T]\mp
1}m\frac{\partial m}{\partial T}+\mathrm{c.t.}$ (11)

In statistical mechanics, the pressure is interpreted as a “general force”,
which reads

$\displaystyle p=\frac{1}{\beta}\frac{\partial\ln{\cal Z}}{\partial
V}=\frac{1}{\beta}\frac{\ln{\cal Z}}{V}=p_{\mathrm{id}}-B,$ (12)

where

$\displaystyle p_{\mathrm{id}}$ $\displaystyle=$ $\displaystyle\frac{\mp
g}{2\pi^{2}}\int_{0}k^{2}dk\ln\left\\{1\mp\exp\left[\left(\mu-\omega(k,T,\mu)\right)/T\right]\right\\}+\mathrm{c.t.}$
(13) $\displaystyle=$
$\displaystyle\frac{g}{12\pi^{2}}\int_{0}\frac{k^{3}dk}{\exp[(\omega(k,T,\mu)-\mu)/T]\mp
1}\left.\frac{\partial\omega(k,T,\mu)}{\partial
k}\right|_{T,\mu}+\mathrm{c.t.}$

Also, as an ensemble average, the number density is found to be

$\displaystyle n=\frac{\langle N\rangle}{V}=-\frac{1}{V}\frac{\partial\ln{\cal
Z}}{\partial\alpha}=n_{\mathrm{id}},$ (14)

where

$\displaystyle
n_{\mathrm{id}}=\frac{g}{2\pi^{2}}\int_{0}\frac{k^{2}dk}{\exp[(\omega(k,T,\mu)-\mu)/T]\mp
1}-\mathrm{c.t.}$ (15)

Again, we have assumed the condition Eq. (3), which states that the
contribution from the chemical-potential dependence of quasiparticle mass in
the ideal gas term and that from the bag constant $B$ cancel out each other.
The above condition can be specified to give

$\displaystyle\frac{\partial
B}{\partial\mu}=-\frac{g}{2\pi^{2}}\int_{0}\frac{k^{2}dk}{\omega(k,T,\mu)}\frac{1}{\exp[(\omega(k,T,\mu)-\mu)/T]\mp
1}m\frac{\partial m}{\partial\mu}+\mathrm{c.t.}$ (16)

The well-known thermodynamic identity

$\displaystyle\epsilon=T\frac{\partial p}{\partial T}-p+\mu n,$ (17)

essentially comes from the first law of thermodynamics and its extensive
properties. As a matter of fact, following the procedure elaborated in the
standard textbook [50], it is not difficult to verify that the total
derivative of $q=\ln{\cal Z}$ gives

$\displaystyle dq=-\langle N\rangle d\alpha-\langle E\rangle d\beta-\beta
pdV.$ (18)

By comparing the above expression with the first law of thermodynamics,
namely,

$\displaystyle d\langle E\rangle=TdS-pdV+\mu d\langle N\rangle,$ (19)

we have the mapping

$\displaystyle\beta=\frac{1}{k_{B}T},$
$\displaystyle\alpha=-\frac{\mu}{k_{B}T},$ $\displaystyle q+\alpha N+\beta
E=\frac{S}{k_{B}}.$ (20)

The validity of Eq. (17) is readily verified.

Now, we proceed to discuss the implications of the conditions Eqs. (11) and
(16). By taking partial derivative of Eq. (11) in $\mu$ and compare with the
partial derivative of Eqs. (16) in $T$, one arrives at the following integro-
differential equation [44]

$\displaystyle\llangle m\frac{\partial m}{\partial T}\rrangle_{-}=\llangle
m\frac{\partial m}{\partial\mu}\rrangle_{+},$ (21)

where

$\displaystyle\llangle O\rrangle_{-}$
$\displaystyle\equiv\int_{0}k^{2}dk\left\\{\frac{\exp[(\omega-\mu)/T]}{(\exp[(\omega-\mu)/T]\mp
1)^{2}\omega T}-\mathrm{c.t.}\right\\}O(k),$ (22) $\displaystyle\llangle
O\rrangle_{+}$
$\displaystyle\equiv\int_{0}k^{2}dk\left\\{\frac{\exp[(\omega-\mu)/T](\omega-\mu)}{(\exp[(\omega-\mu)/T]\mp
1)^{2}\omega T^{2}}+\mathrm{c.t.}\right\\}O(k).$

The solution of Eq.(21), $m=m(k,T,\mu)$, is in general a function also of the
momentum $k$. In turn, the bag constant $B$ is obtained by integrating Eqs.
(11) and (16) on the parameter plane. It can be viewed as a functional of
$m(k,T,\mu)$ besides being a function of $T$ and $\mu$. It is noted that the
above discussions can be straightforwardly generalized to the case where the
system is not isotropic, where $m=m(\mathbf{k},T,\mu)$. As pointed out in
[44], if one simplifies and considers the momentum-independent case, namely,
$m(\mathbf{k},T,\mu)=m(T,\mu)$, one readily falls back to the flow equation
derived in Ref. [23]. In this case, $B$ also simplifies to a function of $T$
and $\mu$. We are, however, more interested in exploring the momentum-
dependent case, which will be elaborated further in the following section.

##  III Bottom up toy-model approaches

An apparent momentum-dependent solution to Eq. (21) can be obtained by
“factoring out” the momentum integration $\int k^{2}dk$ and assuming the
integrand vanishes. In other words,

$\displaystyle\left\\{\frac{\exp[(\omega-\mu)/T]T}{(\exp[(\omega-\mu)/T]\mp
1)^{2}}-\mathrm{c.t.}\right\\}\frac{\partial m}{\partial
T}=\left\\{\frac{\exp[(\omega-\mu)/T](\omega-\mu)}{(\exp[(\omega-\mu)/T]\mp
1)^{2}}+\mathrm{c.t.}\right\\}\frac{\partial m}{\partial\mu}.$ (23)

The above equation can be solved by using the method of characteristics [51].
To be specific, for a given $k$, the solution is the surface tangent to the
vector field

$\left(a(T,\mu,m),b(T,\mu,m),0\right),$ (24)

where

$\displaystyle a(T,\mu,m)$
$\displaystyle=\frac{\exp[(\omega-\mu)/T]T}{(\exp[(\omega-\mu)/T]\mp
1)^{2}}-\mathrm{c.t.},$ (25) $\displaystyle b(T,\mu,m)$
$\displaystyle=-\frac{\exp[(\omega-\mu)/T](\omega-\mu)}{(\exp[(\omega-\mu)/T]\mp
1)^{2}}-\mathrm{c.t.}.$

Its formal solution is the characteristic curves obtained by the integration

$\displaystyle\frac{dT}{d\lambda}$ $\displaystyle=a(T,\mu,m),$ (26)
$\displaystyle\frac{d\mu}{d\lambda}$ $\displaystyle=b(T,\mu,m),$

where $\lambda$ is an intermediate variable, for given $k$, $m$, and thus
$\omega$.

An interesting scenario that gives rise to an analytic solution occurs when
one ignores anti-particles’ contributions. By taking $\omega,T,\mu$ as the
three independent variable, the method of characteristics gives [44] the
following formal solution

$\displaystyle m=f\left(\frac{T\omega}{\omega-\mu},k\right),$ (27)

where an arbitrary function $f$ furnishes the boundary condition at vanishing
chemical potential, namely, $f(T)\equiv f(T,0)=m(T,\mu=0,k=0)$, where we
assume that the mass is independent of the momentum at $\mu=0$. We note that
the resultant mass function is a function of $k,T$, and $m$, and therefore the
solution of the form Eq. (27) serves as a simple but non-trivial example.

In [44], the freedom in $f$ was employed to perform a numerical fit to the
lattice QCD results for $N_{f}=2+1$ flavor QCD system [52, 53, 54, 55, 56] at
vanishing chemical potential. Then the relevant physical quantities, such as
the trace anomaly, sound velocity, and particle number susceptibility, were
evaluated and compared to the lattice data. Instead, we adopt a “bottom-up”
approach for the present study. Specifically, we consider two cases where one
assumes a simple ansatz for $f$ posteriorly adapted to the lattice results and
proceeds analytically to a large extent.

Figure 1: The mass of $u$ and $d$ quarks at vanishing chemical potential
derived from the lattice data [52]. It can be readily extracted for vanishing
chemical potential using Gorenstein and Yang’s scheme [22]. The curve is then
fit to analytic form Eq. (28) discussed in the text.

Case 1: Our first choice is a simple linear fit. Based on the lattice data
[52] shown in Fig. 1, there are two regions where the mass of the
quasiparticle is primarily a linear function in temperature. In other words,

$\displaystyle\left.f\right|_{\mu=0}=c_{1}T+c_{2},$ (28)

which gives

$\displaystyle
f\left(\frac{T\omega}{\omega-\mu},k\right)=f\left(\frac{T\omega}{\omega-\mu}\right)=\frac{c_{1}T\omega}{\omega-\mu}+c_{2}.$
(29)

Despite its simple form, Eq. (28) might be plagued by the pole on its
denominator. To avoid the pole at $\omega=\mu$ for an arbitrary momentum $k$
indicates the condition

$\displaystyle\omega>\mu,$ (30)

that is, $\omega\ge m>\mu$, by considering the definition Eq. (5). Otherwise,
if one requires $\omega<\mu$, it is always possible to find a momentum $k$
large enough to violate the condition.

Substituting Eq. (29) into Eq. (27) gives

$\displaystyle\omega(m-c_{1}T-c_{2})=\mu(m-c_{2}),$ (31)

for which Eq. (30) dictates

$\displaystyle c_{1}>0,$ (32)

while given $T>0$ and $\mu>0$.

By substituting Eq. (5) into Eq. (31) and squaring both sides, one finds a
fourth-degree polynomial equation for $m$. This equation possesses four roots,
where complex roots always appear in pairs. The physically relevant solution
must sit on the positive real axis.

From this point on, we proceed numerically. One extracts the values $c_{1}$
and $c_{2}$ from the region shown in Fig. 1 satisfying $c_{1}>0$, and finds
$c_{1}=0.44$ and $c_{2}=0.13$. The fourth-degree polynomial contains a pair of
complex roots, which are subsequently discarded. One of the remaining two real
roots is extraneous, owing to the fact that we have squared both sides of Eq.
(31). The resultant mass function $m(k,T,\mu)$ and the bag constant $B(T,\mu)$
are shown in Fig. 2. The left and middle plots give the mass as a function of
$T$ and $\mu$ at given $k=1$ GeV and that of $k$ and $\mu$ at given $T=0.25$
GeV. The resulting bag constant is obtained by numerical integration of Eqs.
(11) and (16). The dependence of the bag constant on the temperature $T$ or
chemical potential $\mu$ is presented in the right plot of Fig. 2. The mass
function and the bag constant are found to be moderate in $T$ and $\mu$. As
$k\to 0$, according to the middle plot of Fig. 2, the quasiparticle mass
increases significantly. It is noted that the obtained bag constant $B$ is
manifestly path independent. For instance, one evaluates $B(T,\mu)$ by using
two following integration paths on the $T-\mu$ plane. The integration for $B$
is carried out from $(T_{0}=0.25$ GeV $,\mu_{0}=0)$ to
$(T_{1}=0.45,\mu_{1}=0.3)$, where path 1 is defined by
$(T_{0},\mu_{0})\rightarrow(T_{1},\mu_{0})\rightarrow(T_{1},\mu_{1})$, while
path 2 is through
$(T_{0},\mu_{0})\rightarrow(T_{0},\mu_{1})\rightarrow(T_{1},\mu_{1})$. One
finds

$\displaystyle\left.\left[B(T_{1},\mu_{1})-B(T_{0},\mu_{0})\right]\right|_{\rm{path~{}1}}=-0.606049=\left.\left[B(T_{1},\mu_{1})-B(T_{0},\mu_{0})\right]\right|_{\rm{path~{}2}}.$

We also note that based on the above discussions, the fit to the region
$c_{1}<0$, where the mass of the quasiparticle decreases with increasing
temperature in Fig. 1, is doomed to fail. A numerical attempt reveals path-
dependent values, which signals that those obtained by straightforward
integration do not yield mathematically well-defined results. This is due to
the undesirable pole at $\omega=\mu$ in the denominator of the first term on
the r.h.s. of Eq. (29). In order to handle the region where $c_{1}<0$, we
proceed to consider the second case.

Figure 2: The derived quasiparticle mass in the parameter space according to
the form given by Eq. (29) and the fit shown in Fig. 1. (a) The quasiparticle
mass $m$ as a function of $T$ and $\mu$ at $k=1$ GeV. (b) The quasiparticle
mass $m$ as a function of $k$ and $\mu$ at $T=0.25$ GeV. (c) The bag constant
$B$ as a function of $T$ and $\mu$.

Case 2: The second choice involves a linear function in the reciprocal of the
argument of Eq. (27). To be specific, we consider the ansatz,

$\displaystyle\left.f\right|_{\mu=0}=\frac{1}{c_{3}+c_{4}T},$ (33)

which gives

$\displaystyle
f\left(\frac{T\omega}{\omega-\mu},k\right)=\frac{\omega-\mu}{c_{3}(\omega-\mu)+c_{4}T\omega}.$
(34)

In this case, to avoid the pole in the denominator, one considers the
following constraint

$\displaystyle\omega(c_{4}T+c_{3})>\mu c_{3}.$ (35)

Substituting Eq. (34) into Eq. (27) gives

$\displaystyle\omega(c_{4}Tm+mc_{3}-1)=\mu(mc_{3}-1).$ (36)

Given $T>0$ and $m>0$, Eq. (35) implies

$\displaystyle\omega>\mu,\ \ c_{4}>0,$ (37)

and moreover, Eq. (36) further indicates

$\displaystyle c_{3}<\frac{1}{m},\ \ c_{4}Tm+mc_{3}<1.$ (38)

Otherwise, if $c_{3}\ge 1/m$, Eq. (36) can no longer hold.

To proceed, one substitutes Eq. (5) into Eq. (36) and squares both sides, and
one again finds a fourth-degree polynomial equation for $m$. Similarly, this
equation possesses four roots, where complex roots appear in pairs.

We proceed numerically from this point on. The values $c_{3}$ and $c_{4}$ are
extracted from a fit to the lattice QCD data shown in Fig. 3. One finds
$c_{3}=-1.84$ and $c_{4}=28.55$, which affirms the second choice above. By
discussing a pair of complex roots and an extraneous root, the physically
relevant solution is eventually singled out from the two sit on the positive
real axis.

The resultant mass function $m(k,T,\mu)$ and the bag constant $B(T,\mu)$ are
shown in Fig. 4. The left and middle plots give the mass as a function of $T$
and $\mu$ at given $k=1$ and as a function of $k$ and $\mu$ at given $T=0.12$.
Again, the bag constant can be obtained by numerical integration of Eqs. (11)
and (16). The dependence of the bag constant on the temperature $T$ or
chemical potential $\mu$ is presented in the right plot of Fig. 4. The mass
function and the bag constant are found to be moderate in $T$ and $\mu$,
mainly in accordance with the existing results [44]. Different from Fig. 2, as
$k\to 0$, the mass of the quasiparticle does not modify significantly. Again,
the obtained bag constant $B$ is manifestly path independent.

Before closing this section, we present in Fig. 5 a few resulting
thermodynamic quantities evaluated using the toy model proposed in case 2. In
the left plot, we show the pressure, energy density, and entropy density as a
function of temperature at vanishing chemical potential. The right plot gives
the difference in pressure between the states with finite and vanishing
chemical potential. It is noted in the calculations, in accordance with the
simplified scenario, one only takes into account the $u$ and $d$ quarks but
does not include the $s$ quarks, gluons, or the anti-particles. By comparing
the results with those obtained using more sophisticated approaches [26, 57,
44], one is led to the following observations. The tendency of the temperature
dependence is mainly correct, while the magnitudes of the calculated
thermodynamic quantities consistently underestimate the existing results. This
is because the simplified models do not consider the contributions from the
remaining degrees of freedom, including those from the anti-particles.
Moreover, the order of magnitude for these quantities can be roughly
recuperated by multiplying a factor of two. The latter effectively compensates
for the contributions missing from the anti-particles. We note, nonetheless,
that the main objective of the present approach is to explore the analytic
properties of the mass function from a bottom-up perspective rather than
reproduce the lattice data numerically by employing some sophisticated
approximate function.

Figure 3: The mass of $u$ and $d$ quarks at vanishing chemical potential
derived from the lattice data [52], which is fit to analytic form Eq. (33)
discussed in the text.

Figure 4: The derived quasiparticle mass in the parameter space according to
the form given by Eq. (34) and the fit shown in Fig. 3. (a) The quasiparticle
mass $m$ as a function of $T$ and $\mu$ at $k=1$ GeV. (b) The quasiparticle
mass $m$ as a function of $k$ and $\mu$ at $T=0.12$ GeV. (c) The bag constant
$B$ as a function of $T$ and $\mu$.

Figure 5: The derived thermodynamic quantities by considering the $u$ and $d$
quarks, where the $s$ quarks, gluons, and anti-particles are not explicitly
taken into account. (a) The pressure $3p/T^{4}$, energy density
$\epsilon/T^{4}$, and entropy density $3s/(4T^{3})$ as functions of the
temperature $T/T_{c}$ at vanishing chemical potential, where $T_{c}=1.5$ GeV
in accordance with lattice QCD data. (b) The difference in pressure between
the states with finite and vanishing chemical potential is shown as a function
of temperature $T/T_{c}$.

##  IV Concluding remarks

To summarize, in this work, we reviewed the topic of the quasiparticle model
closely related to Ru-Keng Su’s distinguished contributions in the past years.
Moreover, we explore the approach applied to scenarios with finite chemical
potential. Different from the standard recipe in the literature, we explored
the possibility that the effective mass of the quasiparticle might be a
function of the momentum, in addition to the dependence on temperature and
chemical potential. It was shown that such a scenario emerges as a special
solution to an integro-differential equation derived from the thermodynamic
consistency. We pointed out that the special solution in question is
essentially a generalization to those previously explored in the literature.
Instead of fitting to the lattice QCD data at vanishing chemical potential, we
performed a “bottom-up” approach by assuming two analytic ansatzes. The
remaining physical quantities were subsequently derived and discussed. We note
that the momentum-dependent quanta mass has also been addressed by some
authors from the QCD perspective, where the analyses were closely related to
the symmetry of the underlying system. In terms of the Gribov-Zwanziger
framework, results on the gluon [58, 59, 60, 61] and quark propagator [62]
indicated that the pole masses are functions of the momentum. Besides,
calculations using the Schwinger-Dyson equation showed momentum-dependence for
both gluon [63] and quark [64, 65] dynamic masses.

The current approach’s main objective is to explore the analytic properties of
the mass function. It is primarily motivated as one might distinguish the
various roots deriving from the thermodynamical consistency condition. As
observed and discussed in the main text, these different roots are somehow
separated by the pole of the relevant equation, which is not apparent if a
numerical scheme were utilized in the first place. The calculations primarily
employ Eq. (27). It is a simplified approach as it ignores anti-particles’
contributions and is only utilized to fit to accommodate the u and d quarks.
On the other hand, a numerical approach directly based on Eq. (21) was carried
out in a previous study [44], where the cancelations warranted by Eqs. (11)
and (16) take place for individual particles, as well as their anti-particles.
Nonetheless, the present study gives rise to the following speculations.
First, we have attempted to avoid the singularity of the mass function by
entirely evading its poles by imposing the conditions, Eqs. (30) and (35). The
resultant physical quantities are, in turn, manifestly analytic on the $T$ and
$\mu$ parameter space. Curiously, from a theoretical perspective, one expects
a curve of first-order phase transition on the parameter plane, which entails
some discontinuity. In other words, the discontinuity avoided in the present
study might be utilized in our favor. Specifically, a pole in the mass
function indicates an infinite mass, which can be viewed as a natural and
benign outcome when a degree of freedom can hardly be excited. We plan to
address these aspects in further studies.

## Acknowledgements

This work is supported by the National Natural Science Foundation of China.
This work is partially supported by the Central Government Guidance Funds for
Local Scientific and Technological Development, China (No. Guike ZY22096024).
We also gratefully acknowledge the financial support from Brazilian agencies
Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), Fundação de
Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ), Conselho Nacional de
Desenvolvimento Científico e Tecnológico (CNPq), and Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior (CAPES).

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