Abstract
In this work, we study the quantum chromodynamics phase diagram at finite temperature and nonzero chemical potential in the framework of the SU(2) flavor of the Polyakov–Nambu – Jona–Lasinio model (PNJL). Applying multiple reflection expansion to cubic and spherical finite volumes of different sizes and boundary conditions, the chiral phase transition and deconfinement of strongly interacting matter are analyzed. We give special attention to find and locate the critical endpoint and locating the critical endpoint, if it exists, as a function of the volume size and its shape, and its dependence of chemical potential and temperature.
1 Introduction
Confinement and the spontaneous breaking of chiral symmetry are two of the main characteristics of quantum chromodynamics (QCD) [1,2,3]. A system of strongly interacting particles undergoes a phase transformation when it is subject to high temperatures and/or high densities. Quarks deconfinement and/or restoration of chiral symmetry can occur under these extreme conditions [4,5, 6,7]. These phase transformations are the subject of an intense research activity at present time due to the importance of this phenomenon in modern physics. The study of QCD phase diagrams has attracted a lot of attention in recent years due to the experimental projects in progress and the improved lattice QCD techniques and nonperturbative methods at finite temperature and chemical potential [8,9].
The topic of QCD phase transition is important in several fields of modern physics. In cosmology, for example, it is involved in the processes that happened in the early Universe. In astrophysics, it is important to understand the states of matter in the compact stars [10]. In highenergy physics and nuclear physics, the knowledge of the processes that occur in relativistic heavy ion collisions requires the study of strongly interacting systems subject to conditions of very high temperatures and density [11].
The exotic state of matter can be formed in heavy ion collisions. One of the main investigation lines in theoretical and experimental physics is to understand the properties of these new states of matter [12] subject to extreme conditions, studying their phase diagrams at finite temperature and chemical potentials. This is a topic of increasing interest being researched in experiments such as the ones taking place in the relativistic heavy ion collider and in the large hadron collider, from which they expected to get important information about the phases of strongly interacting matter. There are several important questions in the study of QCD phase transitions that we address in this work. One of them is to determine the existence and the location of the critical endpoint (CEP) in the chiral phase diagram [11,13,14]. For the case of current quark masses different from zero, the CEP is a point in the temperature–density
In a confinement/deconfinement phase diagram, a CEP is expected to appear in the point where the line of a firstorder transition undergoes a change to a crossover region. We have found that the location of the CEP is sensible to several conditions such as the size of the system, the geometry of the volume enclosing it, and the boundary conditions used in the model [15].
In our study of the phase transition between hadronic matter and quarkgluon plasma, as well as the chiral phase transition, the influence of finite volume in the location of CEP and in the quark effective mass is studied for a cubic volume with multiple reflection expansion (MRE) approximation.
The Polyakov–Nambu – Jona–Lasinio (PNJL) model involves chiral symmetry breaking and confinement at low energies, which are two of the main characteristic of QCD. We calculate the thermodynamic potential at finite volume on the framework of the SU(2) flavor version of the PNJL model. To take into account the effects of finite volumes in the model, we calculate the density of states by using the multiple reflection expansion approximation [16,17,18, 19,20,21].
In ref. [17], this method was used to study the Nambu–Jona–Lasinio (NJL) model in SU(2) with a spherical volume of radius
2 PNJL model
The NJL model is a good prospect to study systems of strongly interacting matter at finite temperature and chemical potential. One of the drawbacks of this model is that it does not take into account the property of confinement. The interaction between quarks is introduced as pointlike interactions and does not include gluon exchange. A way to go around this limitation is by introducing the Polyakov loop to construct a more complete model that takes into account the confinement of quarks at low energies [22,23,24].
We start with the two flavor quark version of the NJL model including the Polyakov loop for three colors with the Lagrangian given by ref. [24]
where
and then, the thermodynamic potential after using mean field approximation is expressed as follows:
where
is the Wilson line, a matrix in color space which, when written as a diagonal matrix, represents a complex field called the Polyakov loop [1,2,21,25,26]:
The expression
where
with parameters
The field
The constituent quark mass
where the functions
and the mass
In the same way, imposing equilibrium conditions minimizing the thermodynamic potential, equations for
it is possible to obtain the mass and the expectation values of the Polyakov loop with
3 Finite volume
To determine the effects of a finite volume in the QCD phase structure in the PNJL model, we follow the idea proposed by Kiriyama et al. [17] who used the MRE approximation to include the density of states for a spherical volume in a NJL model for SU(2). In this section, we extend this case to the PNJL model in the same SU(2) framework to study how the states of chiral symmetry and confinement behave in a finite volume.
3.1 Spherical volume
Following ref. [17], we redefine the density of states
where both the surface and curvature terms are a function of the momentum
To handle definitions (15) and (16) numerically, Dirichlet and Neumann boundary conditions are imposed on the
For Dirichlet conditions, we make
For Neumann boundary conditions, we establish
In this way, the final expression of the integral in Eq. (8) can be written as follows:
where
The density of states (14) has an intrinsic physical limit for momentum values since it is quadratic and therefore has a range of negative momentum values that are not physically acceptable. This imposes a lower limit
3.2 Cubic volume
There are several ways to study the problem with finite volume. In some of them, it is solved by establishing boundary conditions so that the shape of the region with finite volume is not relevant; in others, the geometry is important, and we can study the effect, for example, of using spherical or cubic volumes. In the previous analysis, the density of MRE states imposes the condition of finite volume and in particular that of a form of spherical droplets.
Adding boundary conditions to the PNJL model makes it more expensive, computationally speaking, so considering that the MRE approach should be simple to implement, and although we know that MRE approximation is designed to describe spheres, in this section, we will extend it in a rough approximation to a cubic box shape by making two simple changes to the
As mentioned in Eq. (14), the density of states is composed of three terms, and since in Eq. (3) the integrand is spherically symmetric, we can use the relation
By analogy with the volumetric density of states, the surface density of states is
Setting
where
4 Model parameters
This PNJL model was regularized by using the trimomentum cutoff scheme, and like any other regularization scheme, it needs two parameters: the coupling strength
5 Zero
μ
5.1 Order parameters: sphere and cubic box
At the chiral limit,
Unlike the cube, which shows similar values in the constituent mass
On the other hand, the Polyakov loop, at the right of Figure 1, shows a rapid phase transition for both sphere and cube, and practically no difference is observed when the volume size or the geometry are modified.
When the current mass of quarks is
Table 1 summarizes the constituent masses calculated in the chiral limit for different volumes of a sphere, with both Dirichlet and Neumann boundary conditions, and different volumes of a cube, and in Table 1, the constituent masses for the same geometries when the current quark mass is
PNJL

PNJL

PNJL MRE cube  











50  288.78  20  309.81  40  301.74 
20  255.19  10  308.63  15  287.30 
10  193.18  6  305.88  13  283.68 
6  77.35  2  272.13  11  278.69 
0.9  68.40  9  271.39  
5  237.63 
PNJL

PNJL

PNJL MRE cube  











50  305.33  20  325.15  40  317.52 
40  300.21  10  324.04  15  303.94 
38  298.86  8  323.22  13  300.55 
20  274.15  6  321.46  11  295.90 
10  219.16  4  316.49  9  289.11 
6  138.1  3  309.61  5  258.20 
2  290.12  
0.9  141.55 
6 Finite
μ
6.1 Order parameters
As expected, when the chemical potential
6.2 Susceptibilities
Susceptibility is the response of the potential to changes in the order parameters [24,36]. It can be used to study phase transitions, as divergences in susceptibility are associated with phase changes.
As ref. [26] says, when establishing the PNJL model, we have a model based on a chiral symmetry that compensates for the lack of confinement through the Polyakov loop. With this PNJL model, we can compare the temperatures at which deconfinement and restoration of chiral symmetry occur, and assuming that both processes occur at the same temperature, a susceptibility matrix can be defined as follows:
where its components are the second derivatives of the thermodynamic potential
We define
6.3 Chiral and loop susceptibilities for a finite volume
Peaks in susceptibility indicate changes in the phase of the model. These occur at
We are interested in finding, if it exists, a point at which the chiral phase transition takes place. The behavior of order parameters as functions of the temperature for
PNJL

PNJL

PNJL MRE cube  











50  251  20  257  40  253 
20  242  10  257  15  251 
10  223  6  256  13  250 
6  161  2  247  11  249 
0.9  163  9  247  
5  237 
PNJL

PNJL

PNJL MRE cube  











50  263  20  269  40  267 
40  260  10  269  15  263 
38  262  8  269  13  260 
20  255  6  268  11  261 
10  240  4  267  9  259 
6  216  3  265  5  251 
2  260  
0.9  221 
The susceptibilities help identifying phase transitions and therefore determining the existence, if any, of the CEP. In order of finding evidence of the existence of this CEP, the behavior of the maximum chiral susceptibility, varying the volume sizes and for values of
A more detailed analysis of the data was made to find the CEP. Systematical observation of the chiral susceptibility curves were performed for each
Top left panel of Figure 7 shows the evolution of chiral susceptibility when increasing the chemical potential
Figure 8 shows the chiral susceptibility in the
7 Phase diagrams
Figure 9 shows the phase diagram for different sizes of a sphere and a cube. For reference, the phase diagram of PNJL for an infinite volume, that is, without MRE, is shown as a black curve for each geometry. For the PNJL model in a finite volume with MRE approximation, we observe that smaller volume sizes, that is, smaller radius or edge values, correspond to smaller values for the critical temperature. In the same way, the position of the CEP is found each time at increasingly lower temperatures as the volume decreases, until it vanishes. In a sphere with
PNJL

PNJL

PNJL MRE cube  


CEP


CEP


CEP







50 

20 

40 

40 

10 

15 

38 

8 

13 


—  6 


— 
4 


3 



— 
8 Discussion
The effects of considering finite volumes were studied using the approximation MRE in the PNJL model in SU(2) in the chiral limit
In particular, chiral and deconfinement phase transitions in a finite volume were studied. Phase diagrams were obtained and special interest was taken in location, if any, of a CEP. For a spherical shape, it was found that there is a significant difference between Dirichlet and Neumann conditions. In
To establish the location of the CEP, we use a divergence criterion in a detailed analysis of the data obtained for the chiral susceptibility as a function of the temperature
Following these criteria, we found that the CEP exists in MRE
In all three cases, the temperatures at which the CEP occurs decrease with the size of the volume. The chemical potential values
Although the approximation is limited, it yields results for symmetry restoration temperatures, deconfinement, and/or the location of the CEP consistent with other publications of the NJL/PNJL model with and without volume constraints.
Acknowledgments
This work was partially supported by Consejo Nacional de Ciencia y Tecnología (Conacyt), PhD fellowships and SNI Mexico, and the authors thank the UJED for the facilities granted in computing time.

Funding information: The authors state no funding involved.

Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

Conflict of interest: The authors state no conflict of interest.

Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.
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