Robust features of QCD phase diagram through a Contact Interaction model for quarks: A view from the effective potential

Quantum Chromodynamics (QCD) is the theory that describes the color interactions among quarks and gluons. It plays a crucial role in understanding the symmetric and peculiar nature of the universe. Asymptotic freedom [1, 2] and quark confinement [3] are the two fundamental features of QCD. In the high-energy domain, or asymptotic freedom realm, quarks interact weakly at short distances inside hadrons. However, at larger distances, or in the low-energy domain, quarks are strongly interacting and cannot exist in isolation (i.e., the quark are confined inside hadrons). In addition to quark confinement, low-energy QCD also exhibits another important phenomenon, which is the dynamical breaking of chiral symmetry, and its consequence, the dynamical mass generation of constituent quarks. Studying low-energy QCD at finite temperature $T$ and density parametrized by a quark or baryon chemical potential $\mu$ has significant implications for understanding the phase transition that took place in the early universe a few microseconds after the Big Bang. The transition from hadronic matter to quark-gluon plasma [4], quarkyonic matter [5, 6], neutron star formation [7], and the color-flavor locked (CFL) region [8, 9, 10] are the subjects of current interest. The advancements in technology and experimental facilities have enabled us to study the properties of QCD at high temperatures and relatively low densities. The recent upgrades in detectors at various research facilities such as the sPHENIX detector and complementary STAR upgrades at RHIC, as well as upgraded detectors at ALICE, ATLAS, CMS, and LHCb, have paved the way for a multimessenger era for hot QCD (see for detail in recent review [11]), which is based on the combined constraining power of low-energy hadrons, jets, thermal electromagnetic radiation, heavy quarks, and exotic bound states. The increased luminosity at the LHC and other experimental facilities such as RHIC and the Compact Baryonic Matter (CBM) experiments, along with the facilities under construction in Fair [12] and NICA [13] will also provide a unique opportunity to study the phase transition from hadronic matter to quark-gluon plasma and the related phenomena.

At temperatures close to zero, the color-singlet hadrons are widely believed to be the fundamental degrees of freedom of low-energy QCD. However, once the temperature exceeds a critical value $T_{c}$, the interaction weakens, leading hadrons to melt into a new phase. In this phase, quarks and gluons become the new degrees of freedom, and the chiral symmetry is restored and quarks become deconfined. Several methods, including Lattice QCD calculations [14, 15, 16, 17, 18, 19, 20], Schwinger-Dyson equations [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], and other effective models of low-energy QCD [32, 33, 34, 35, 25, 36, 37, 38], have all suggested that the transition at hand is a cross-over when a finite current quark mass $m$ is taken into account; otherwise, chiral limit calculations show a second-order phase transition. In any case, as the quark chemical potential $\mu$ is raised, the same physical behavior persists ¹¹1The recent experimental measurement of the spectrum of pionic ¹²¹Sn atoms and the study of the interaction between the pion and the nucleus indicate that the chiral symmetry is partially restored due to the extremely high density of the nucleus [39]., but the nature of the phase transition changes from a cross-over to a first-order transition at the critical end point (CEP) in the QCD phase diagram, which is typically illustrated on the $T-\mu$ plane. The exact location of the critical endpoint is not predicted so far and has become a hot scientific topic which has deserved the design of experiments to observe it. However, if we represent the position of this point as  $(\mu_{E}/T_{c},T_{E}/T_{c})$, where $T_{c}$ denotes the temperature at which chiral symmetry restoration occurs for $\mu=0$, then it is reasonable to claim that model calculations [40, 41, 42, 43, 44, 45, 46, 23, 25, 38] locate this point in the region  $(\mu_{E}/T_{c}=1.0-2.0,T_{E}/T_{c}=0.4-0.8)$ considering $N_{f}=2$ flavors. However, mathematical extensions of lattice techniques [47, 48, 49, 50] predict that the point is located around approximately between $(\mu_{E}/T_{c}=1.0-1.4,T_{E}/T_{c}=0.9)$ for simulations including the strange quark. Background electromagnetic fields also affect the QCD phase diagram, since the critical temperature $T_{c}$ decreases with the increase of electric and magnetic fields [31, 27, 30, 51, 52, 53, 54]. Furthermore, the QCD phase transitions under consideration are also sensitive to the number of light quark flavors, as highlighted, for instance, in Ref. [31], where an increase in the number of light quark flavors leads to a suppression of the critical line between chiral symmetry broken and restoration phases in the phase diagram.

Our main goal in this work is to study the chiral symmetry breaking-restoration and confinement-deconfinement phase transitions, and chart out the QCD phase diagram. To achieve this, we use a confining regularization version of the Nambu-Jonal-Lasinio (NJL) model which consists in a symmetry-preserving vector-vector contact interaction model of quarks, in the Schwinger-Dyson equations framework in the Landau gauge with a Schwinger optimal time regularization scheme, at finite temperature $T$ and chemical potential $\mu$. As it is well-known, the NJL model [32, 55, 33] is quite successful in providing the properties of low-energy mesons. The most significant aspect of this model is that it exhibits spontaneous chiral symmetry. Unfortunately, the NJL model does not support quark confinement and suffers from an unphysical quark-antiquark threshold. To construct the confining version of the NJL model, the first attempt was made by [56] to regularize it by introducing an infrared cut-off $\tau_{ir}=1/\Lambda{ir}$ along with an ultraviolet cut-off $\tau_{uv}=1/\Lambda_{uv}$. The infrared cut-off removes the unphysical quark-antiquark production threshold, and by introducing a finite value for the infrared cut-off, confinement is implemented. Later on, a similar idea was promoted by [57, 58, 59], by removing the poles from the quark propagator, which implies that the excitation represented by a pole-less propagator is confined. With the particular choice of values for $\tau_{ir}=(0.24{\rm GeV})^{-1}$ and $\tau_{uv}=(0.905{\rm GeV})^{-1}$, along with other model parameters, they studied the properties of $\pi$ and $\rho$ mesons, and their diquark partners, with approximately up to a $10$ percent error with the experimentally measured values. Here, we are promoting the regularization procedure outlined in references [59, 60] to explore the QCD phase diagram at finite temperature and density. Under similar assumptions, the confining scale $\tau_{ir}(T)$ has been used as an order parameter for the confinement-deconfinement transition at finite temperature $T$ [61], as well as in the presence of background electromagnetic fields $\tau_{ir}(T,eB,eE)$ [62, 30, 31]. In our considerations, the gap equation obtained from this model can be integrated with respect to the dynamically generated mass, hence defining the effective thermodynamic potential of the present contact interaction model [63]. Here, we are interested in analyzing the traits of the chiral symmetry breaking and restoration and confinement-deconfinement transitions at finite temperature and density from this effective potential. In the present work, we consider the confining length as temperature and chemical potential dependent, and denote it as $\tau_{ir}(T,\mu)$. It is important to note that in this model, chiral symmetry restoration and deconfinement occur simultaneously [64, 62, 30, 31].

The remaining of this manuscript is organized as follows: In Section 2, we present the general formalism for the contact interaction model, the QCD gap equation, and the effective potential. In Section 3, we discuss the gap equation and effective potential at finite $T$ and $\mu$. Section 4 presents the numerical solutions for the gap equation and the effective potential, and we sketch the phase diagram in the $T-\mu$ plane. In Section 5 we explore the screening effects of the medium by including vacuum polarization effects in the coupling due to the quarks in the high-$T$ domain. In the last section, Section 6, we provide a summary and future perspectives of our work.

In this Section, we discuss the numerical solution of the gap equation and the effective potential at finite temperature and chemical potential. We also sketch the phase diagram in $T-\mu$ plane. We use the following set of parameters: $\tau_{ir}=(240{\rm MeV})^{-1}$, $\tau_{uv}=(905{\rm MeV})^{-1}$, $\alpha_{\rm eff}=5.739\times 10^{-5}{\rm MeV}^{-2}$, and a bare quark mass $m_{u/d}=7{\rm MeV}$. The values of these parameters are taken from [57], which were determined by fitting them to the appropriate values for the $\pi$ ans $\rho$ mesons properties. The solution of the gap Eq. (13) in the chiral limit gives a dynamical quark mass $M=0.358$ MeV and with current quark mass $m=7$ MeV, it is $M=0.368$. The vacuum effective potential, Eq. (15), as a function of $M$ is shown in Fig. 1, in the chiral limit ($m=0$) and with bare quark mass ($m=7$ MeV). In this case one obtains that the minima of $\Omega_{vac}$ are precisely located at the solutions to the gap equation, while the trivial solution $M=0$ is a maximum in both cases. The minimum value for positive mass corresponds to the global minimum of effective potential, so it is the stable dynamical quark mass.

Next, we solve the gap equation Eq. (19) and effective potential Eq. (20), at finite temperature $T$ but keeping $\mu=0$. The dynamical mass as a function of $T$ is plotted in Fig. 2(2(a)) which decreases with the increase in $T$. The thermodynamic potential as a function of $M$ for various values of the temperature $T$ is depicted in Fig. 2(2(b)) which shows that the global minima are shifted toward lower values of the dynamical mass $M$ upon increasing the temperature and, at some critical temperature $T=T_{c}=208$ MeV, the thermodynamic potential has minimum values at the lowest values of the effective mass $M$, which gives a clear signal about chiral symmetry restoration: the dynamical mass vanishes and the bare mass survives. The nature of the transition is smooth cross-over. It is also clear from Fig. 2(2(c)) that the thermal gradient of the effective mass $M$ peaks at $T_{c}=208$ MeV.

The thermodynamic potential at  $T=0$ for different values of the chemical potentials $\mu$ as a function of the effective quark mass $M$ is shown in Fig. 3(3(a)). This is a first order phase transition, where the thermodynamic potential has multiple minima and the global minimum switches from one minimum to another at and above $\mu_{c}$. In Fig. 3(3(b)), we plotted the effective mass $M$ i.e., the solution of Eq. (22) as a function of chemical potential $\mu$ which clearly indicates the chiral symmetry restoration discontinuously at and above $\mu_{c}\approx 380$ MeV.

As for the dynamical $M$ as a function of temperature $T$ for various values of the chemical potential $\mu$, results are shown in Fig. 4 (4(a)). The dynamical mass $M$ decreases monotonically with the temperature and suppresses as we increase the chemical potential $\mu$. At and above a critical value of the chemical potential $\mu_{c}\approx 350$ MeV the mass plateau shows a discontinuity. The critical temperature $T_{c}(\mu)$ at various values of the chemical potential can be obtained from the thermal gradient of the dynamical mass, which is shown in Fig. 4 (4(b)). The Critical End Point (CEP), where the cross-over phase transition ends and the first order phase transition begins, is therefore obtained from the divergence of the mass gradient at particular chemical potential and temperature. In this case, it happens at $(\mu^{E}_{c}\approx 350{\rm MeV},T^{E}_{c}\approx 90{\rm MeV})$. The effective potential at the CEP is depicted in the Fig. 5.

The confinement scale and its thermal gradient as a function of temperature for various values of the chemical potential is shown in Fig. (6). The critical temperature $T_{c}$ for the confinement-deconfinement transition for various values of the chemical potential $\mu$ can be obtained from the peak of the thermal gradient. It is clear from the thermal gradients of both the dynamical mass $M$ and the confinement scale $\hat{\tau}^{-1}_{ir}$ that the chiral symmetry breaking-restoration and confinement-deconfinement transition occur simultaneously. We have constructed a phase diagram the $T-\mu$ plane, which reveals that at finite temperature $T$ but zero chemical potential $\mu$, chiral symmetry is broken for temperatures $T\leq T_{c}\approx 208$ MeV. However, above this threshold, the symmetry is restored and deconfinement occurs. The order of the phase transition is a cross-over. On the other hand, when we consider finite chemical potential $\mu$ and zero temperature $T$, we find that chiral symmetry is broken and confinement settles below a critical chemical potential $\mu_{c}\approx 380$ MeV, while above this value, it is restored and deconfinement occurs, but the transition is of first-order. Based on Fig. 7, we confirmed that the cross-over line in the phase diagram begins at a finite point on the temperature-axis when $\mu=0$ but does not reach a finite point on the chemical potential-axis when $T=0$. As a result, a CEP exists where the cross-over transition shifts to a first-order transition. We have successfully determined that the coordinates of this are ($T^{E}_{c}/T_{c}\approx 0.43,~{}\mu^{E}_{c}/T_{c}\approx 1.67$).

We further consider the screening effects of the medium which dilute the strength of the effective coupling are by including the vacuum polarization contribution due to quarks at high temperatures into the framework with slight modification in Eq. (6) as [72]

With this modification, we obtain the critical temperature $T_{c}\approx 132$ MeV at $\mu=0$, which shows the effect of the medium in diluting the coupling. At $T=0$, $\mu_{c}\approx 380$ MeV. For the CEP, the model yields $T_{E}\approx 65$ MeV and $\mu_{E}\approx 340$ MeV such that ($T^{E}_{c}/T_{c}\approx 0.57,~{}\mu^{E}_{c}/T_{c}\approx 2.6$) in the phase diagram, as shown in Fig. 8.

In the next section, we summarize our findings and draw the conclusions.

In the present manuscript we have studied the dynamical chiral symmetry breaking/restoration and confinement-deconfinement phase transitions at finite temperature $T$ and chemical potential $\mu$ and sketched the QCD Phase diagram in the $T-\mu$ plane. We used a flavor dependent symmetry preserving vector-vector confining contact interaction model of quarks and Schwinger optimal proper time regularization scheme. We have developed an expression for effective thermodynamic potential and identify the signals of chiral symmetry breaking-restoration from the peaks of the thermal gradient of the dynamical mass. The confinement-deconfinement transition is pinpointed from the peaks of the thermal gradient of the confining scale at various quark chemical potential. At finite temperature $T$ and $\mu=0$, our analysis shows that the chiral symmetry is restored and deconfinement occurs when $T$ reaches a critical value $T_{c}\approx 208$ MeV and the order of the transition is a cross-over. In the presence of quark chemical potential $\mu$ and $T=0$, the dynamical chiral symmetry is restored and deconfinement occurs when $\mu$ exceeds its critical value $\mu_{c}\approx 350$ MeV. At the end, we have sketched the phase diagram in the $T-\mu$ plane. Our results in this scenario show that the cross-over transition line in the phase diagram started from the finite $T$-axis continued along the finite $\mu$-axis until the CEP $(\mu^{E}_{c}/T_{c}\approx 1.6$, $T^{E}_{c}/T_{c}\approx 0.43)$ where above, the transition changes to the first-order till at $T=0$ along $\mu$-axis. It should be noted that the exact location of the CEP is not yet exactly known. However some effective model calculations  [40, 41, 42, 43, 44, 45, 46, 23, 25, 38], Schwinger-Dyson equations [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31] and mathematical extensions of lattice techniques  [47, 48, 49, 50] set the range between $(\mu_{E}/T_{c}=1.0-2.0,T_{E}/T_{c}=0.4-0.9)$. When considering screening effects of the medium, which dilute the strength of the effective coupling, by including the vacuum polarization contribution due to quarks at high temperatures, it yields the location of the CEP at $(\mu^{E}_{c}/T_{c}\approx 2.6,T^{E}_{c}/T_{c}\approx 0.57)$, which hints for a deeper analysis of screening effects on models of this kind, which is currently been carried out. In the near future, we also plan to extend this work to study the electromagnetic effects and other areas of the hot and dense QCD, the light hadrons properties under extreme conditions etc., using the contact interaction model.

A. A. and M. A thank Adnan Bashir for valuable suggestions and guidance during the completion of this manuscript and also to the colleagues of Institute of Physics of Gomal University for their encouragement. A. R. acknowledges Saúl Hernández-Ortiz for enlightening discussions. He also acknowledges support from Consejo Nacional de Humanidades, Ciencia y Tecnología (México) under grant CF-2023-G-433.