A Study of the Effect of the Area and Curvature Terms in the Quarks and Gluons Density of States on the Transition Temperature ^†

Abstract

This study investigates the effects of considering the area and curvature terms with the volume term in the quarks and gluons density of states on the transition temperature at which the deconfinement phase transition occurs. The analysis begins with the computation of the partition function of the Quark–Gluon Plasma (QGP) projected onto the SU(3) color-singlet representation, using the projection method, within the density of states given by the Multiple Reflection Expansion (MRE) approximation. We consider the transition between a hadronic gas (HG) phase, consisting of massive pions, and a QGP phase, which includes gluons, massless up and down quarks, and massive strange quarks along with their antiquarks, in the framework of the Bag model. We then examine the variations in the pressure of both phases, in the aim of determining the transition temperature in cases of different contributing terms in the quarks’ and gluons’ density of states, considering the contributions of the volume term, volume and area terms, volume and curvature terms, and the combined volume, area and curvature terms.

1. Introduction

In the hadronic phase, quarks and gluons, the fundamental constituents of matter, are confined within hadrons due to their strong interaction [1,2]. However, at high temperature and/or density, a deconfining phase transition occurs, leading to the formation of the Quark–Gluon Plasma (QGP). In this state of matter, quarks and gluons are no longer confined and can move freely [3,4]. The QGP is theorized to have been the dominant state of matter in the early universe, existing within the first 10⁻⁵ s after the Big Bang, when temperatures and densities were exceptionally high [5].

The QCD phase transition has been widely studied using both first-principles and effective approaches. Lattice QCD provides the most reliable description of the QCD thermodynamics at vanishing chemical potential [6,7], while effective models such as the Nambu–Jona-Lasinio (NJL) [8], Polyakov–Nambu–Jona-Lasinio (PNJL) [9,10], and Polyakov Quark Meson (PQM) models have been extensively used to investigate the QCD phase diagram and the interplay between chiral symmetry restoration and deconfinement [11].

In this work, we study the deconfinement phase transition from a hadronic gas phase, which consists of massive pions, to a Quark–Gluon Plasma (QGP) phase, made up of gluons, massless up and down quarks, and massive strange quarks along with their antiquarks, within the framework of the Bag model, which incorporates the bag constant B [12,13] representing the vacuum pressure exerted on the perturbative vacuum. We introduce the color-singletness condition for the QGP by means of the projection method [14,15] that we use to calculate the partition function of the QGP projected onto the SU(3) color-singlet representation. For this purpose, we consider the quarks and gluons density of states, given by the Multiple Reflection Expansion (MRE) approximation [16], with the contribution of the volume, area and curvature terms. We aim to investigate the impact of adding the area and curvature terms in the quarks and gluons density of states on the effective transition temperature in a finite volume, denoted $T_c\left(V\right)$; to determine the latter, we analyze the variations in the pressure of both hadronic gas and QGP phases, with temperature T at zero chemical potential, for various system sizes V, in different cases of contributing terms in the density of states.

2. Partition Functions of Hadronic Gas and QGP Phases

As a first step, we need to compute the partition function for each of the hadronic gas and QGP phases.

For the hadronic phase that consists of massive pions (m_π = 138 MeV), its partition function $Z_{HG}\left(T,V\right)$, in a volume V, at temperature T, can be expressed, after calculation, as

$$Z_{HG}\left(T,V\right)=\exp \left({\displaystyle \frac{d_{\pi }V}{2{\pi }^{2}T}}{\int }_0^{+\infty }{\displaystyle \frac{k^{4}dk}{\sqrt{k^2+m_{\pi }^2}\left(e^{\beta \sqrt{k^2+m_{\pi }^2}}-1\right)}}\right)$$

(1)

where $k$ is the momentum, $d_{\pi }$ the pion degeneracy factor, and $\beta =1/T$.

For the QGP phase, which contains gluons, massless up and down quarks, and massive strange quarks with their antiquarks, and assuming non-interaction between them, we carry out the calculation of the partition function $Z_{QGP}\left(T,V\right)$, using the $MRE$ approximation [16] for the quarks’ and gluons’ density of states ${\rho }_i\left(i=Q,G\right)$, which includes volume, area and curvature terms:

$${\rho }_i\left(k,V\right)={\displaystyle \frac{V{k}^2}{2{\pi }^2}}+f_{A,i}4\pi R^{2}k+f_{C,i}8\pi R+\dots \dots \dots \dots ,i=Q,G$$

(2)

where $R$ represents the radius of the bag defined by a spherical volume $V={\displaystyle \frac{4}{3}}\pi R^3$, in the Bag model [12,13]; $f_{A,i}$ and $f_{C,i}$ are the area and curvature coefficients respectively. In [17], the area coefficient for the quarks, $f_{A,Q}$, was given as

$$f_{A,Q}=-{\displaystyle \frac{1}{8\pi }}\left(1-{\displaystyle \frac{2}{\pi }}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\left({\displaystyle \frac{k}{m}}\right)\right)\to f_{A,Q}=0whenm=0$$

(3)

The curvature coefficient, $f_{C,Q}$, for massive quarks was introduced in [18],

$$f_{C,Q}={\displaystyle \frac{1}{12{\pi }^2}}\left\{1-{\displaystyle \frac{3k}{2m}}\left({\displaystyle \frac{\pi }{2}}-\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\left({\displaystyle \frac{k}{m}}\right)\right)\right\}$$

(4)

and it takes the value $-{\displaystyle \frac{1}{24{\pi }^2}}$ for non-massive quarks [19]. The area and curvature coefficients for the gluons $f_{A,G}$ and $f_{C,G}$ were provided in [16] by

$$f_{A,G}=0,f_{C,G}=-{\displaystyle \frac{1}{6{\pi }^2}}$$

(5)

In a first case, we consider a QGP with no condition on its global color, and the partition function of such a QGP, in volume V, at zero chemical potential µ = 0, and at temperature T, is given by

$$Z_{QGP}\left(T,V,\mu =0\right)=e^{-{\displaystyle \frac{BV}{T}}}{e}^{\left(X_1+X_2+X_3\right)}$$

(6)

After calculation and simplification, $X_1$, $X_2$ and $X_3$ correspond, respectively, to the volume, area and curvature contributions in the density of states of quarks and gluons, and are expressed as

$$X_1=V\left\{d_Q\left[{\displaystyle \frac{2{\pi }^2{T}^3}{90}}+{\displaystyle \frac{{\left(m_{S}T\right)}^{\frac{3}{2}}}{{\sqrt{2}\pi }^{\frac{3}{2}}}}{e}^{\left(-{\displaystyle \frac{m_S}{T}}\right)}\right]+{\displaystyle \frac{d_G{\pi }^2{T}^3}{90}}\right\}$$

(7)

$$X_2={{\displaystyle \frac{d_Q}{2}}\left({\displaystyle \frac{3V}{4\pi }}\right)}^{{\displaystyle \frac{2}{3}}}\left(\sqrt{{\displaystyle \frac{2m_S}{\pi }}}{T}^{{\displaystyle \frac{3}{2}}}-m_{S}T\right)e^{\left(-{\displaystyle \frac{m_S}{T}}\right)}$$

(8)

$$X_3={\left({\displaystyle \frac{3V}{4\pi }}\right)}^{{\displaystyle \frac{1}{3}}}\left\{\left[d_Q(-{\displaystyle \frac{\pi T}{18}})+\left({\displaystyle \frac{2}{3}}\sqrt{{\displaystyle \frac{2T}{\pi m_S}}}-T+\sqrt{{\displaystyle \frac{2}{\pi m_S}}}{T}^{{\displaystyle \frac{3}{2}}}\right)e^{\left(-{\displaystyle \frac{m_S}{T}}\right)}\right]-{\displaystyle \frac{2d_G\pi }{9}}T\right\}$$

(9)

$m_S$ is the strange quark mass, while $d_Q$ and $d_G$ represent the spin–isospin degeneracy factors for quarks–antiquarks and gluons, respectively.

In a second case, we consider the color-singletness condition for the QGP, and the partition function of a QGP projected on the color-singlet SU(3) representation, in a volume V, at temperature T, and at zero chemical potential (µ = 0), calculated using the group-theoretical projection method of Turko and Redlich [14,15], as in [20,21,22] and as in the works of one of us [23,24] where only the volume term was used in the density of states, is given now, considering the curvature and area terms additionally to the volume term, by

$$Z_{QGP}\left(T,V,\mu =0\right)={\displaystyle \frac{4}{9{\pi }^2}}{e}^{-{\displaystyle \frac{BV}{T}}}{\displaystyle \underset{-\pi }{\overset{+\pi }{\int }}}{\displaystyle \underset{-\pi }{\overset{+\pi }{\int }}}d\phi d\psi M\left(\phi ,\psi \right)e^{\left(Y_1+Y_2+Y_3\right)}$$

(10)

with $M\left(\phi ,\psi \right)$ being the weight function (Haar measure) given by [20,21,22]:

$$M\left(\phi ,\psi \right)={\left(\mathrm{S}\mathrm{i}\mathrm{n}\left[{\displaystyle \frac{1}{2}}\left(\psi +{\displaystyle \frac{\phi }{2}}\right)\right]\mathrm{S}\mathrm{i}\mathrm{n}\left[{\displaystyle \frac{\phi }{2}}\right]\mathrm{S}\mathrm{i}\mathrm{n}\left[{\displaystyle \frac{1}{2}}\left(\psi -{\displaystyle \frac{\phi }{2}}\right)\right]\right)}^2$$

(11)

$Y_1$, $Y_2$ and $Y_3$ correspond, respectively, to the volume, area and curvature contributions in the density of states of quarks and gluons, and are expressed as

$$\begin{gathered} Y_1=d_{Q}V\sum _{q=r,b,g}\left\{{\displaystyle \frac{{\pi }^2}{6}}{T}^3\left[{\displaystyle \frac{7}{30}}-{\left({\displaystyle \frac{{\theta }_q}{\pi }}\right)}^2+{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{{\theta }_q}{\pi }}\right)}^4\right]+{\displaystyle \frac{{\left(m_{S}T\right)}^{\frac{3}{2}}}{{\sqrt{2}\pi }^{\frac{3}{2}}}}\left(1-{\displaystyle \frac{{\theta }_q^2}{2!}}+{\displaystyle \frac{{\theta }_q^4}{4!}}\right)e^{\left(-{\displaystyle \frac{m_S}{T}}\right)}\right\} \\ +d_{G}V{T}^3{\displaystyle \sum _{G=1}^4}{\displaystyle \frac{{\pi }^2}{12}}\left[-{\displaystyle \frac{7}{30}}+{\left({\displaystyle \frac{{\theta }_G-\pi }{\pi }}\right)}^2-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{{\theta }_G-\pi }{\pi }}\right)}^4\right] \end{gathered}$$

(12)

$$Y_2=d_Q{\left({\displaystyle \frac{3V}{4\pi }}\right)}^{{\displaystyle \frac{2}{3}}}\left(\sqrt{{\displaystyle \frac{2m_S}{\pi }}}{T}^{{\displaystyle \frac{3}{2}}}-m_{S}T\right)\sum _{q=r,b,g}\left(1-{\displaystyle \frac{{\theta }_q^2}{2!}}+{\displaystyle \frac{{\theta }_q^4}{4!}}\right)e^{\left(-{\displaystyle \frac{m_S}{T}}\right)}$$

(13)

$$\begin{gathered} Y_3=d_Q{\left({\displaystyle \frac{3V}{4\pi }}\right)}^{{\displaystyle \frac{1}{3}}}\sum _{q=r,b,g}\left\{\left[{\displaystyle \frac{\pi }{3}}T\left(-{\displaystyle \frac{1}{3}}+{\left({\displaystyle \frac{{\theta }_q}{\pi }}\right)}^2\right)\right]+\left({\displaystyle \frac{2}{3}}\sqrt{{\displaystyle \frac{2T}{{\pi m}_S}}}-T\right.+\left.\sqrt{{\displaystyle \frac{2}{{\pi m}_S}}}{T}^{{\displaystyle \frac{3}{2}}}\right)\left(1-{\displaystyle \frac{{\theta }_q^2}{2!}}\right.\right. \\ \left.+\left.{\displaystyle \frac{{\theta }_q^4}{4!}}\right)e^{\left({\displaystyle \frac{-m_S}{T}}\right)}\right\}+d_G{\left({\displaystyle \frac{3V}{4\pi }}\right)}^{{\displaystyle \frac{1}{3}}}{\displaystyle \sum _{G=1}^4}{\displaystyle \frac{2\pi }{3}}T⌊{\displaystyle \frac{1}{3}}-{\left({\displaystyle \frac{{\theta }_G-\pi }{\pi }}\right)}^2⌋ \end{gathered}$$

(14)

where the angles ${\theta }_q(q=r,b,g)$ and ${\theta }_G\left(G=1,\dots ,4\right)$ in $SU\left(3\right)$ are given by [20,21]

$${\theta }_r={\displaystyle \frac{\phi }{2}}+{\displaystyle \frac{\psi }{3}},{\theta }_g=-{\displaystyle \frac{\phi }{2}}+{\displaystyle \frac{\psi }{3}},{\theta }_b={\displaystyle \frac{2\psi }{3}}$$

(15)

$${\theta }_1={\theta }_r-{\theta }_g,{\theta }_2={\theta }_g-{\theta }_b,{\theta }_3={\theta }_b-{\theta }_r,{\theta }_4=0$$

(16)

3. Study of the Effect of the Area and Curvature Terms in the Density of States on the Transition Temperature

To achieve our main goal, which is the study of the impact of including the area and curvature terms alongside the volume term in the quarks and gluons density of states on the deconfinement phase transition, by studying their impact on the transition temperature $T_c\left(V\right)$ in a finite size system V, we need to examine the behavior of the pressure for both hadronic gas (HG) and QGP phases, given by the well-known relation

$$P_i\left(T,V\right)=T{\displaystyle \frac{\partial ln{Z}_i}{\partial T}},i=HG,QGP$$

(17)

Thus, we derive the expression of the HG pressure from Equation (1) and that of the QGP without the color-singletness condition from Equation (6); then, by using a numerical integration method, we calculate the integrals involved in Equation (10), and derive the pressure of the color-singlet QGP. First, in Figure 1, we illustrate the variations of the HG and QGP pressures, with temperature T, at zero chemical potential (μ = 0) and with a Bag constant value B^1/4 = 225 MeV, for a large volume of V = 10,000 fm³, which effectively simulates an infinite system.

Figure 1. Variations of the pressure as a function of temperature T for the HG of pions (black solid line) and QGP phase, with the contribution of the volume term only (red dashed line), volume and area terms (blue dotted line), volume and curvature terms (magenta dashed dotted line), and volume, area and curvature terms (green dashed dotted dotted line), at µ = 0 and with B^1/4 = 225 MeV.

The obtained graphics reveal an intersection between the pressure curve of the HG phase (black solid line) and the pressure of the QGP phase, which is represented by four curves corresponding to different contributions to the quarks and gluons density of states: the contribution of the volume term only (red dashed line), volume and area terms (blue dotted line), volume and curvature terms (magenta dashed dotted line), and volume, area, and curvature terms (green dashed dotted dotted line).The intersection point of the HG and QGP pressure curves defines the infinite-volume transition temperature $T_c\left(ꝏ\right)$, as set by the Gibbs criterion and found in [25]. It can clearly be noted that this true transition temperature $T_c\left(ꝏ\right)$ varies from one case of contributing terms in the density of states to another, since $T_c\left(ꝏ\right)$ takes its maximum value with the contribution of all three terms (volume, area, and curvature) and its minimum value with the volume term only contributing to the quarks and gluons density of states, as shown in Table 1.

Table 1. True transition temperature $T_c\left(ꝏ\right)$ values with the various contributions to the density of states of quarks and gluons.

Contributing Terms	${\mathit{T}}_{\mathit{c}}\mathbf{\left(}\mathit{ꝏ}\mathbf{\right)}\mathbf{\left(}\mathbf{M}\mathbf{e}\mathbf{V}\mathbf{\right)}$
volume term	158,522
volume and area terms	158,551
volume and curvature terms	158,607
volume, area and curvature terms	158,636

In the following, we investigate the variations in the HG and QGP pressures with temperature T, for various system sizes V, in the case of the color-singlet QGP phase, i.e., by first deriving the QGP pressure using the partition function of a QGP projected on the color-singlet representation computed previously (Equations (10)–(16)). Figure 2 illustrates such thermal variations for two small volumes, (a) ${V=100\mathrm{f}\mathrm{m}}^3$ and (b) ${V=200\mathrm{f}\mathrm{m}}^3$, and Figure 3 for two large volumes, (c) ${V=1000\mathrm{f}\mathrm{m}}^3$ and (d) ${V=2000\mathrm{f}\mathrm{m}}^3$, both at a Bag constant value B^1/4 = 225 MeV.

Figure 2. Plots of the pressure of a HG of pions and a color-singlet QGP as a function of temperature T, with the four contributions in the quarks and gluons density of states, at µ = 0 and B^1/4 = 225 MeV, for a system volume (a) ${V=100\mathrm{f}\mathrm{m}}^3$ and (b) ${V=200\mathrm{f}\mathrm{m}}^3$.

Figure 3. Plots of the pressure of a HG of pions and a color-singlet QGP as a function of temperature T, with the four contributions in the quarks and gluons density of states, at µ = 0 and with B^1/4 = 225 MeV, for a system volume of (a) ${V=1000\mathrm{f}\mathrm{m}}^3$ and (b) ${V=2000\mathrm{f}\mathrm{m}}^3$.

From Figure 2 and Figure 3, one can clearly observe that, for each system volume, the curves corresponding to the different contributions to the quarks and gluons density of states exhibit a very similar behavior. In addition, it is evident that variations in the system size have a significant impact on the effective transition temperature $T_c\left(V\right)$ in a finite volume. Specifically, under the color-singletness constraint, for smaller volumes, the effective transition temperature in a finite volume $T_c\left(V\right)$ is systematically shifted from the true transition temperature (in infinitevolume) $T_c\left(ꝏ\right)$ towards higher values, reflecting finite-size effects that become increasingly important as the system size V decreases [25]. This shows that the phase transition characteristics are highly sensitive to finite-volume effects.

It can also be noted from the obtained graphics that there is a considerable difference between the effective transition temperatures in a volume V for each case of contributing terms to the density of states, as summarized in Table 1 and Table 2. This suggests that the inclusion of each of the area and curvature terms in the quarks and gluons density of states has a noticeable impact on the transition temperature value, especially within the context of a small system volume.

Table 2. Effective transition temperature $T_c\left(V\right)$ values for various volumes, with the four cases of contributing terms to the density of states.

System Volume  $\mathbf{\left(}{\mathbf{f}\mathbf{m}}^3\mathbf{\right)}$	${\mathbf{T}}_{\mathbf{c}}^{\mathbf{\left(}\mathbf{V}\mathbf{o}\mathbf{l}\mathbf{\right)}}\mathbf{\left(}\mathbf{V}\mathbf{\right)}$	${\mathbf{T}}_{\mathbf{c}}^{\mathbf{(}\mathbf{V}\mathbf{o}\mathbf{l}\mathbf{+}\mathbf{A}\mathbf{r}\mathbf{e}\mathbf{a}\mathbf{)}}\mathbf{\left(}\mathbf{V}\mathbf{\right)}$	${\mathbf{T}}_{\mathbf{c}}^{\mathbf{(}\mathbf{V}\mathbf{o}\mathbf{l}\mathbf{+}\mathbf{C}\mathbf{u}\mathbf{r}\mathbf{v}\mathbf{)}}\mathbf{\left(}\mathbf{V}\mathbf{\right)}$	${\mathbf{T}}_{\mathbf{c}}^{\mathbf{(}\mathbf{V}\mathbf{o}\mathbf{l}\mathbf{+}\mathbf{A}\mathbf{r}\mathbf{e}\mathbf{a}\mathbf{+}\mathbf{C}\mathbf{u}\mathbf{r}\mathbf{v}\mathbf{)}}\mathbf{\left(}\mathbf{V}\mathbf{\right)}$
	$\left(\mathbf{M}\mathbf{e}\mathbf{V}\right)$	$\left(\mathbf{M}\mathbf{e}\mathbf{V}\right)$	$\left(\mathbf{M}\mathbf{e}\mathbf{V}\right)$	$\left(\mathbf{M}\mathbf{e}\mathbf{V}\right)$
100	159,240	159,509	161,087	161,345
150	159.103	159,340	160,521	160,758
200	159,050	159,265	160,221	160,434
300	158.994	159,181	159,889	160,073
1000	158.541	158,666	158,937	159,063
2000	158,540	158,640	158,790	158,889

4. Conclusions

In the present work, we investigated the influence of the area and curvature terms in the density of states of quarks and gluons, used in the calculation of the QGP partition function, on the deconfinement phase transition. To achieve this, we focused on the transition temperature ${\mathit{T}}_{\mathit{c}}\left(\mathit{V}\right)$, determined according to the Gibbs criterion from the mechanical equilibrium point between the hadronic and QGP phases, i.e., from the intersection point of the pressures of the two phases. The plots of these pressures have been illustrated vs temperature T, at zero chemical potential (µ = 0), and with B^1/4 = 225 MeV, for various sizes V and in four different cases: with the contributions of the volume term, volume and area terms, volume and curvature terms, and volume, area and curvature terms in the quarks and gluons density of states. Subsequently, by analyzing the value of the transition temperature ${\mathit{T}}_{\mathit{c}}\left(\mathit{V}\right)$, we found that the inclusion of the area and curvature terms, additionally to the volume term in the density of states, increases the value of ${\mathit{T}}_{\mathit{c}}\left(\mathit{V}\right)$, especially in small volumes. Furthermore, we noticed that ${\mathit{T}}_{\mathit{c}}\left(\mathit{V}\right)$ reaches its highest value when all three contributions are included, then it slightly decreases when only the volume and curvature terms contribute, while it decreases more with the volume and area contribution, and ${\mathit{T}}_{\mathit{c}}\left(\mathit{V}\right)$ reaches its lower value when only the volume term contributes.

It can also be concluded that for large volumes, the transition temperatures $T_c\left(V\right)$ differ slightly, depending on the contributing terms to the quarks and gluons density of states, whereas for small volumes, the difference is relatively considerable, without exceeding 2 MeV, indicating that the difference itself is not particularly significant. Rather, our objective was to investigate the effect of the surface and curvature terms on the critical behavior of the shift in the transition temperature.