Chiral phase transition in linear sigma model with non-extensive statistical mechanics

From the non-extensive statistical mechanics, we investigate the chiral phase transition at finite temperature $T$ and baryon chemical potential $\mu_B$ in the framework of the linear sigma model. The corresponding non-extensive distribution, based on Tsallis' statistics, is characterized by a dimensionless non-extensive parameter, $q$, and the results in the usual Boltzmann-Gibbs case are recovered when $q\to 1$. The thermodynamics of the linear sigma model and its correspodning phase diagram are analysed. At high temperature region, the critical temperature $T_c$ is shown to decrease with increasing $q$ from the phase diagram in the $(T,~\mu)$ plane. However, larger values of $q$ causes the rise of $T_c$ at low temperature but high chemical potential. Moreover, it is found that $\mu$ different from zero corresponds to a first-order phase transition while $\mu=0$ to a crossover one. The critical endpoint (CEP) carries higher chemical potential but lower temperature with $q$ increasing due to the non-extensive effects.


I. INTRODUCTION
Quantum Chromodynamics (QCD) is a basic theory of describing the strong interactions among quarks and gluons, the fundamental constituents of matter. More and more attentions have already been attracted to the QCD phase transition both theoretically and experimentally [1][2][3][4][5][6][7][8]. Though experimental studies and lattice Monte-Carlo simulations have made it possible to research on the phase diagram quantitatively, there still remains uncertainty at high baryon density region [9]. Consequently, the phase transition is also a vital topic in high energy physics, where the thermal vacuum created by heavy-ion collisions differs from the one at zero temperature and chemical potential [7]. In order to study certain essential features of it, the linear sigma model has been proposed to illuminate the restoration of chiral symmetry and its spontaneous breaking [8].
Near the phase transition boundary one should be cautious when using the Boltzmann-Gibbs (BG) statistics for the appearance of critical fluctuations due to a large correlation length [10]. It is of interest to investigate the phase transition in the formalism beyond conventional BG statistical mechanics. Recently a nonextensive statistics firstly proposed in Ref. [11] has attracted a lot of attention and discussions [12]. In this formalism, instead of the exponential function, a generalized q−exponential function is defined as [11,12], where the parameter q is called the non-extensive parameter, which accounts for all possible factors violating assumptions of the usual BG case. Its inverse function is * Electronic address: shenkm@mails.ccnu.edu.cn † Electronic address: mr.zhanghui@mails.ccnu.edu.cn ‡ Electronic address: bwzhang@mail.ccnu.edu.cn also listed [11,12], Both of them return to the usual exponential and logarithm function with q → 1. The purpose of this paper is to clarify the nonextensive effects on physical quantities of the chiral phase transition in the generalized linear sigma model. We focus on the situations where both of temperature and chemical potential are not vanished, which then indicates the influence of the Tsallis distribution on the whole phase diagram in the (T, µ) plane. Whereas the nonextensive parameter q is still a phenomenological parameter [12], not only the case of q > 1 but q < 1, in this work, are computed. For comparisons, we have presented discussions on this issue for finite temperature but vanishing chemical potential [13]. Given the consistency of non-extensive generalizations with the initially BG approaches, we also list the results of q = 1 which were investigated [14]. We close our researches with the comparisons to the Non-extensive Nambu Jona-Lasinio Model (q−NJL model) [15] of the critical endpoint (CEP), whose location is still the hot topic for experiments as well as its theoretical researches [3,5].
This paper is organized as follows. In Section II we introduce the theoretical framework, where the nonextensive q−linear sigma model is stated. Their consequences for various thermodynamic quantities with different non-extensive parameters, q, are explored in Section III; more detailed discussions on the results are also contained. Section IV is our brief summary and outlook.

II. THEORETICAL FRAMEWORK
Within the linear sigma model, the chiral effective Lagrangian with quark degrees of freedom reads [14,16,17] L =ψ[iγ µ ∂ µ − g(σ + iγ 5 τ · π)]ψ where ψ = (u, d) stands for the spin− 1 2 two flavors light quark fields, the scalar field σ and the pion field π = (π 1 , π 2 , π 3 ) together form a chiral field Φ = (σ, π), with its potential Considering the obvious symmetry breaking term Hσ = 0, L is invariant under chiral SU (2) L × SU (2) R transformations. The chiral symmetry is spontaneously broken in the vacuum with the expectation values: σ = f π and π = 0, where f π = 93 MeV is the pion decay constant. By the partially conserved axial vector current (PCAC) relation [14] the quantity v 2 = f 2 π − m 2 π /λ 2 with the constant H = f π m 2 π where m π = 138 MeV is the pion mass. The coupling constant λ 2 is fixed as 20 by m 2 σ = 2λ 2 f 2 π + m 2 π where m σ = 600 MeV is the sigma mass. Another coupling constant g is usually determined by the requirement of the constituent quark mass in vacuum, M vac = gf π , which is about 1/3 of the nucleon mass, leading to g ≈ 3.3. [14] In order to investigate the temperature T and the chemical potential µ dependence in this model, let us consider a system of both quarks and antiquarks in the thermodynamical equilibrium. Here quark chemical potential µ ≡ µ B /3. And the grand partition function goes like Thus the grand canonical potential can be obtained with the (anti)quark contribution is where ν = 12 is the internal degrees of freedom of quarks and E = p 2 + M 2 is the valence (anti)quark energy, with the mass of constituent (anti)quark defined as Here the first divergent term of E is absorbed in the coupling constant in the results which comes from the negative energy states of the Dirac sea.
It is inadequate to apply naively the BG statistics in such a system critical fluctuations of energy and particle numbers will appear as well as a large correlation. In order to investigate the phase transition of systems departing from the classical thermal equilibrium, the nonextensive statistics [11] is introduced. The so-called Tsallis entropy and density matrix are given, respectively, as where k B is the Boltzmann constant (set to 1 for simplicity next), q describes the degree of non-extensivity and Z q is the corresponding generalized partition function.
Recently this generalized statistics has been of great interest theoretically [18][19][20] and widely applied in many fields [21][22][23][24]. In the following we investigate the linear sigma model within the non-extensive statistics. Firstly we re-write the partition function of Eq.(5) as where the q−exponential is seen in Eq.(1). Considering the q−thermodynamics [25] we have Before carrying it out, we give out the generalized identities with respect to q−sums and integrals, and the generalized sum over n, in our assumptions, 12) where E n = (2n + 1)πT is used and the index 2 − q appears because of the duality, Integrating over θ and dropping terms that are independent of β and µ, we finally obtain In our calculations within the mean-field approximation, we follow Refs. [26] and [14] where the expectation value of the pion field is set to zero, π = 0. By solving the gap equation,

∂ ∂σ
Ωψ ψ (T, µ, q)| σ=σV = 0 (15) the value for constituent (anti)quark mass M = gσ V can be determined, which will be also affected from different non-extensive parameters, q. Here we have replaced σ and π in the exponent by their expectation values in the mean-field approximation.
With such a q−thermal effective potential, we then explore the non-extensive effects on the physical quantities, as well as the phase transition, in the linear sigma model. The numerical results will be shown in the next section.

III. RESULTS AND DISCUSSION
In virtue of it there still exist fierce controversies over the possible interpretations of the non-extensive parameter q, we shall discuss the non-extensive effects in the q−linear sigma model for both the q > 1 case and q < 1. Meanwhile, we give out the result of q = 1 as the baseline for better understanding.
Worthy to note that the value of non-extensive parameter q cannot be much smaller than 1 since in the expression of Eq. (14), whose corresponding generalized exponential where the part of the base should be larger than 0 Thus some upper limitation of energy of the integral in Eq. (7) should be given in case of divergence when q < 1. On the other hand, too much smaller values of q are not necessary to be computed physically during our investigation on the non-extensive effects on the phase transition. Therefore, here we just list the results of q = 1.1, 1.05, 0.95 and q = 1 as baseline. We start our discussions with presenting in Fig.1 the resulting thermodynamical potential Ω as a function of M , the constituent (anti)quark mass. Different q evidently results in a large change of the thermodynamical potential which shows the effects caused by nonextensivity are quite strong whether the quark chemical potential vanishes or not.
For the upper panel, the potentials with different q are shown for T = 148 MeV and µ = 0. Locations of its minimum become smaller when q gets larger. This means that in the case of high temperature and low density, correlations with the non-extensive q−version, shift the chiral condensation toward smaller M . On the other hand (for the lower one), at low temperature but high The upper plot, at low density, indicates the temperature dependence of M for q = 1 is quite similar to the case of q = 1, the usual BG situation. Both the minimum and maximum of M keep the same values for different q. Nevertheless, the behaviour of all curves tells us high temperature is required to restore the chiral symmetry for small q, which agrees with the results of [13].
At the same time, for the low temperature case, the lower panel of Fig.2 illustrates the µ−dependence of the constituent (anti)quark mass in the non-extensive linear sigma model for different non-extensive parameter q, which is not done in Ref. [13]. One easily observes an analogous pattern characteristic to the above, while for the µ−dependence, increasing q will also increase the value of phase transition chemical potential when T = 40 MeV is fixed. Moreover, for both of the cases, it is deserved to be mentioned that only the system near the chiral phase transition is well affected by non-extensive statistics.
In statistical physics, the critical properties of a thermodynamic system can be explored by studying the fluctuations of various observables. Particularly, the fluctuations of the order parameter probe the order of the phase transition and the position of a possible critical end point.
Then the negative partial derivative of M with respect to temperature T holding chemical potential µ constant, the susceptibility χ, is also investigated in this nonextensive linear sigma model, which describes the fluctuation of constituent (anti)quark mass. From the results seen in Fig.3, one can expect that, at the low density (µ = 0), the location of peak of susceptibility χ, as well as its own value, moves to the lower values of temperature T for larger q. This indicates that with larger q, the critical temperature T c gets smaller, which supports it that the non-extensive parameter q describes the departure of system from the conditions of BG situation.
Here we add a few remarks to better understand the results. Non-extensive dynamics develops the linear sigma model through the (anti)quark number distribution functions. These functions are connected with the thermal potential Ωψ ψ (T, µ, q) of Eq. (14), which modifies the fluctuations of fermions. The q−dependent chiral condensation can be obtained after solving out the gap equation Eq. (15). From the upper panel of Fig.2 we can see its shape with respect to T is strongly affected by the nonextensive parameter q. More specifically, q introduces differences of system itself from the usual BG one which decrease the values of critical temperature T c , seen in Fig.3.
In order to explore the chiral phase transition in the q−linear sigma model more specifically, we also present the phase diagrams (seen in Fig.4) based upon the analysis above. Easily seen that, indeed, at high temperature and low density region it exhibits a crossover transition in the (T − µ) plane for different non-extensive parameters of q, with smaller non-extensive parameter q expanding the relative values of critical temperature and chemical potential. Meanwhile, a first-order phase transition is shown at low temperature but high density region. And all the critical lines correspondingly develop differently, where larger q increases the position of T c at the same µ c .
As for the critical endpoint (CEP), which locates between the two kinds of phase transition, larger q occurs to higher chemical potential but lower temperature, which is also seen in the results of q−NJL model [15]. This is because systems from fewer particles will encounter a larger value of q, whose phase transition takes place with higher number density in turn.

IV. SUMMARY AND OUTLOOK
To summarize, we have calculated the non-extensive thermodynamics of the chiral phase transition in the lin- ear sigma model, to account for the sensitivity of the mean field theory of the linear sigma model to the departure from the usual BG statistics. By the q−version we have obtained generalized relations of the grand canonical potential Ω, the chiral condensation M and susceptibility χ. Before that we also analysis the values of nonextensive parameter q and reasonably consider the cases of q = 1.1, 1.05, 0.95 as well as q = 1.
Furthermore, we have investigated two scenarios, µ = 0 and µ = 0, respectively, which, as mentioned, correspond to different physical situations: a first-order and a crossover transition. For the studies of µ = 0, it is found to be in agreement with the results obtained in [13]. Besides, we discover it that different values of q only influence the quantities near the phase boundary. This also proves that it is valuable and desirable to discuss the non-extensive effects on the chiral phase transition.
As expected, the observed non-extensive effects of both the potential Ω and the mass M lead to it that higher values of q shift all to a earlier state with other parameters fixed. In another word, the internal divergence from the classical thermal equilibrium really impacts on the chiral phase transition. This is more illuminated in the phase diagrams of (T, µ) plane correspondingly.
The CEP (see Fig.4) reveals a clear variation with different non-extensive parameters of q, namely, holds higher chemical potential but lower temperature with q increasing, which agrees with Ref. [15]. As for the critical line in the diagram, as shown in Fig.4, q−effects derive different trends of it on the first-order and crossover phase transitions, whose physical mechanism needs us more attentions and investigations next.
Finally, It is worthy to mention that since CEP is still indistinct experimentally, our work may provide a possible intensively study of locating the CEP in high-energy physics [6]. Meanwhile, by comparing the results with experimental data, our researches could be of help to deeply understand the physical explanation of the Tsallis non-extensive parameter q, which is also what we will pay attention to in the future.