Finite volume effects on QCD chiral phase transition with finite size dependent Nambu-Jona-Lasinio model

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Yonghui Xia, Qingwu Wang, Hongtao Feng and Hongshi Zong. Finite volume effects on QCD chiral phase transition with finite size dependent Nambu-Jona-Lasinio model[J]. Chinese Physics C. doi: 10.1088/1674-1137/43/3/034101
Yonghui Xia, Qingwu Wang, Hongtao Feng and Hongshi Zong. Finite volume effects on QCD chiral phase transition with finite size dependent Nambu-Jona-Lasinio model[J]. Chinese Physics C.  doi: 10.1088/1674-1137/43/3/034101 shu
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Finite volume effects on QCD chiral phase transition with finite size dependent Nambu-Jona-Lasinio model

  • 1. Department of physics, Nanjing University, Nanjing 210093, China
  • 2. Department of physics, Sichuan University, Chengdu 610064, China
  • 3. School of physics, Southeast University, Nanjing 210093, China
  • 4. Joint Center for Particle, Nuclear Physics and Cosmology, Nanjing 210093, China

Abstract: In principle, the effective Lagrangian of the finite volume system should depend on the system size. To this end, in the framework of Nambu-Jona-Lasinio (NJL) model, by considering the influence of quark feedback on the effective coupling, we obtain a modified NJL model and it's Lagrangian depends on the volume. Based on the modified NJL model, we study the influence of the finite volume effect on the chiral phase transition at finite temperature, and find that the pseudo-critical temperature of crossover is much lower than the corresponding pseudo-critical temperature obtained by the normal NJL model. This clearly reflects that the volume dependent effective Lagrangian plays an important role in the chiral phase transitions at finite temperature.

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    1.   Introduction
    • Dynamical Chiral Symmetry Breaking (DCSB) is a very important feature of Quantum chromodynamics (QCD). With the increases of temperature or/and chemical potential, there undergoes a phase transition from hadronic matter to a new phase, the quark-gluon plasma (QGP) and this phase transition may occur in the relativistic heavy ion collisions (RHIC) [1, 2] at the Brookhaven National Laboratory (BNL). It is believed that the phase transition is a crossover at high temperature and small chemical potential while it is a first order phase transition at low temperature and large chemical potential. Thus, there exists a possible critical end point (CEP) at finite temperature and chemical potential (see Refs. [3-6] and reference therein). Determining the existence of CEP and it's location in QCD phase diagram is one of main goals in the heavy ion collisions experiments. On the one hand, on the experimental side, for this purpose, the second phase of the beam energy scan at RHIC will be performed between 2019 and 2020 [7, 8] and we are optimistic that the experimental results can answer these questions. On the other hand, on the theoretical side, people try various non-perturbative QCD models and theories and consider the influence of various factors in the real environment of the possible QGP produced in the heavy ion collisions experiments on the QCD phase diagram. For example, one needs to consider the effects of finite temperature, finite density, and strong magnetic fields generated in the experiment on the possible CEP at finite temperature and chemical potential [3-5, 9-13]. It should be pointed out here that many previous calculations about QCD phase diagrams are based on infinite thermodynamical systems, without considering the QGP generated by laboratory, i.e, the so-called fireball is very limited in space. Such as, the volume of the smallest fireball produced in RHIC could be as small as (2 fm)3 [14], although it's volume before freezing out for Au-Au and Pb-Pb collisions ranges from 50 fm3 to 250 fm3 [14-16]. Since the fireballs produced in the laboratory are of a finite size, the finite size effect on CEP must be considered in the second phase of the beam energy scan at RHIC.

      The finite volume effects has attracted lots of theoretical attentions in the past decades [17-20]. The current study of finite small volume effects is based on the assumption that the effective Lagrangian of a finite small volume physical system is the same as the corresponding infinite physical system. Based on this assumption, different models are adopted to perform calculation to study the finite volume effects on QCD phase transition, for example, Random Matrix Theory (RMT) [21-25], Quark-meson-model [26-31], (Polyakov-loop extended) Nambu-Jona-Lasinio (NJL) model [32-38], Dyson-Schwinger equations (DSEs) [39, 40] and so on. Since the spatial size of the physical system is limited, it is necessary to select appropriate spatial boundary conditions to study the finite volume effects. Usually, periodic boundary condition and anti-periodic boundary condition are adopted [20], which causes the discretized momenta in spatial direction as the case of Matsubara frequency in imaginary time temperature field theory. Here is an example of fermion, at finite volume, the quark momentum is discretized and the integral over all spatial momenta is replaced by a sum over discrete momentum modes. The discretized momentum depends on the selected anti-periodic boundary condition (APBC)

      where L is the cubic volume size. It can be easily seen from the above equation that we must consider the contribution of all frequency mode when we consider the finite volume effect. Once the infrared cutoff with three-dimensional momentum is used, it means that the contribution of low-frequency modes is ignored. Similarly, if three-dimensional ultraviolet momentum cutoff is used, it means that the contribution of all high-frequency modes to the finite small volume effect is neglected and thus in principle is problematic. In addition, if the same boundary conditions in both spatial directions and temporal (temperature) direction are chosen in Euclidean space, namely anti-periodic boundary conditions for fermion fields and periodic boundary conditions for boson fields, the discretization in the spatial direction and the discretization in temporal direction are physically equivalent [20].

      As shown in finite temperature field theory, the running coupling depends on temperature in Quantum electrodynamics (QED) [41, 42] and QCD [43] at finite temperature. Now, there is a question that naturally arises: in principle, the above assumptions are problematic and the effective Lagrangian of finite volume system should depend on the system size as that of finite temperature system depends on temperature, so how to get an effective Lagrangian that reasonably reflects the finite volume effects? And this is the motivation for us to write this paper.

      Before we properly introduce the effective Lagrangian of finite volume system with the spatial size effect, let us first briefly review how we have previously introduced a temperature dependent effective Lagrangian in a finite temperature system. As we know, the quark propagator and gluon propagator satisfy their respective DSEs and they are coupled with each other, therefore, the gluon propagator should depend on the temperature and/or chemical potential, which is also shown clearly in lattice simulations. However, in the framework of the usual DSEs, the gluon propagator is only used as a phenomenological input, and does not consider the coupling between the gluon propagator and the quark propagator, and does not take into account the effects of the temperature and chemical potential on the gluon propagator. More specifically, here we take the NJL model as an example. In NJL model, the coupling constant can be regarded as the inverse of "static" gluon propagator and in principle the effect of the quark propagator on the "static" gluon propagator should be reasonably considered. The authors of the Refs. [9, 11, 13, 44] considered the influence of quark feedback on the "static" gluon propagator through the operator product expansion (OPE) method, and obtained a modified NJL model considering the quark feedback. In the modified NJL model described by Refs. [9, 11, 13, 44], the coupling constant G in NJL model is modified into $ G_1+G_2\langle\bar\psi\psi\rangle $ , here, G2 weighs the quark's feedback influence to the "static" gluon propagator (The physical mechanism for quark's feedback is referred to the appendix of Ref. [44] for details). Although in the modified NJL model, we introduce a parameter more than the normal NJL model, it better reflects the relationship between the quark propagator and the gluon propagator, which is the basic requirement of QCD. More importantly, in the modified NJL model [44] we can better fit dozens of lattice data at finite temperature, which reflects the fact that the introduction of quark feedback to study the chiral phase transition at finite temperature is physically reasonable. Because the discretization of spatial direction and temporal direction are physically equivalent in Euclidean space. Similar to the imaginary temperature field theory, the effective coupling is temperature dependent, and we also construct an effective coupling that depends on the finite volume in this paper. Obviously, since the quark condensate depends on temperature and/or chemical potential, the effective coupling of the modified NJL model depends on temperature and/or chemical potential naturally. Countering the physical equivalence of spatial direction and temporal direction in Euclidean space, we can introduce the finite volume dependent effective Lagrangian with the same approach that we used in the finite temperature and finite chemical potential system.

      This paper is organized as follows: In section 2, the modified NJL model with finite volume dependent coupling is introduced, then chiral phase with the modified NJL model is shown in section 3 and the conclusion is summarized in section 4.

    2.   The modified Nambu-Jona-Lasinio model with finite volume dependent coupling
    • It is well known, the NJL model is a low energy effective theory of QCD, which can describe the main feature of non-perturbative QCD, e.g., Dynamical Chiral Symmetry Breaking. The Lagrangian of the usual two-flavor model is given by

      where m denotes current quark mass of two-flavor light quark and G denotes the coupling constant. Then, the gap equation is shown as

      here, M denotes constituent quark mass. As a unrenormalizable theoretical model, a method of regularization is necessary. The commonly used NJL model generally uses three-momentum cutoff in the momentum space to regularize the amount of the divergence. It is important to note here that this ultraviolet (UV) regularization of (P)NJL model is not suitable for studying limited small volume effects. The Eq. (1) shows clearly that all frequency modes in momentum space contribute to the finite volume effect. If the UV cutoff (for example, three- momentum cutoff NJL model) was adopted, the contribution of high frequency modes would be ignored. If the simple infrared cutoff was adopted [34, 36, 38], the contribution of low frequency modes would be ignored. In this case, we have to abandon the common used three-momentum cutoff and instead use the proper-time regularization to study the finite size effects. This is because the proper-time regularization is not plagued by the interruption of UV momentum.

      In this paper, the scheme of proper time regularization proposed by J. S. Schwinger [45] is adopted

      where $ \Lambda $ denotes the ultraviolet cut-off. Thus, the gap equation (3) is given by

      At zero temperature, the normal NJL model reproduce the property of hadron well. However, in the case of finite temperature, the result can not recover the results of lattice simulations. To fit the lattice simulations at finite temperature, Within the OPE method proposed in Refs. [10, 11, 13, 44], the coupling in Lagrangian (2) is modified into

      Here, there are four parameters, G1, G2, m and $ \Lambda $ to fit the results of lattice simulation at zero temperature and finite temperature. Here, we stress that the G1, G2 and $ \Lambda $ can reproduce the pion mass and pion decay constant and $ G= G_1+G_2(-\langle\bar\psi\psi\rangle) $ at zero temperature. It is found that, one more parameter makes fit the lattice results well at finite temperature [46, 50]. Therefore, the gap equation (3) is modified into

      According to the finite temperature field theory, the gap equation at finite temperature is given by

      Here, $ \theta_2(x)= 2 x^{1/4}\sum_{n=0}^{\infty}x^{n(n+1)} $ . Thus, the coupling depends on temperature naturally. Considering the equivalence of spatial direction and temporal direction in Euclidean space, the coupling constant should depend on finite volume in the finite volume system. We can directly extend the calculation of the above infinite thermodynamic system to a finite temperature and finite volume system.

      In principle, the thermodynamical properties of finite volume system depends on not only the size of system but also the geometric shape of system. In this paper, as an interesting attempt, the geometric shape of system is a cubic box. Considering the system in a finite size cubic box with the edge length L and the anti-periodic boundary condition in all spatial directions for fermion is adopted, e.g.,

      Thus the momentum is replaced by the discrete momenta $ (p_1,p_2,p_3,p_4)\rightarrow ((2n_1+1)\pi/L,(2n_2+1)\pi/L,(2n_3+1)\pi/L, $ $(2n_4+1)\pi T) $ , here $ n_1,n_2,n_3,n_4 \in (-\infty,\infty) $ , and momentum integral is replaced by sum of discrete momenta with

      Then the gap equation in finite volume (cubic box) at finite temperature is given by

      From Eq. (9), if we use anti-periodic boundary condition for both spatial direction and temporal direction in Euclidean space, it is easy to see that the discretization of time and space is completely equivalent. More specifically, the effects of 1/L and T on the quark gap equation are mathematically equivalent. Thus, similar as the increasing of temperature, with the decreasing of system size L, it is expected that the chiral symmetry will restore partially.

    3.   Chiral phase transition in finite size cubic box at finite temperature
    • As the increase of temperature, there undergoes a chiral phase transition. The quark condensate is adopted as an order parameter to describe the chiral phase transition,

      As it is known, there undergoes a second order phase transition at finite temperature in the chiral limit and a crossover beyond the chiral limit (m≠0). To locate the pesudcritical temperature of the crossover, the thermal susceptibility is adopted

      and the pesudocritical temperature is located at the peak of the susceptibility [47, 48].

      In this paper, to perform numerical calculation, the parameters are chosen as follows: $ \Lambda $ =990 MeV, m = 5.5 MeV, which are obtained by fitting the pion mass $ m_{\pi}=138 {\rm MeV} $ and pion decay constant $ f_{\pi}= 93 {\rm MeV} $ [49]. Moreover, G1 = 22.3 GeV−2 and G2 = 0.5003 GeV−5, which are obtained by fitting the results of lattice simulation [50] in the infinite volume thermodynamical system and the result is shown in Fig. 1 [11]. $ \sigma_n=(\langle\bar\psi\psi\rangle_T/\langle\bar\psi\psi\rangle_0) $ in Fig. 1 denotes the normalized quark condensate. From Fig. 1, it is found that the results fit the lattice simulation well at finite temperature, although only one more parameter is adopted. Because the quark condensate depends on both temperature and the size of finite volume system, therefore, the modified coupling depends on the temperature and size naturally.

      Figure 1.  (color online) The result of modified NJL model compared with lattice simulation from Ref. [50].

      The Fig. 2 shows the quark condensate as a function of temperature with different size. It is easy to find that the condensate decreases with the L decreasing at low temperature interval. As the equivalence of spatial direction and temporal direction, the decreases of size means the increases of temperature, which enlarge the fluctuation and leads to the restoration of DCSB. With the temperature increases, due to the thermal fluctuation, the DCSB is partially restored and the condensates of different size trend to the same. when L = 4 fm, the condensate is almost the same as it in the case of infinite thermodynamical system, which implies that the system can be regarded as an infinite thermodynamical system, as L > 4 fm. This result agrees with the results of different model calculation qualitatively [ 27, 39].

      Figure 2.  (color online) The quark condensate as a function of temperature within the coupling dependence NJL model.

      As is known, the effective coupling decreases in the perturbative regime with the increasing of temperature by the perturbative theory [43]. However, in the strong coupling regime, it is not very clear. In this paper, the influence of quark's feedback is countered in the modified NJL model and the effective coupling is divided into two parts: one part G1 is independent of system size and temperature and the other part $ G_2 \langle\bar\psi\psi\rangle $ depends on the system size and temperature. The Fig. 3 shows the part of system size dependent coupling as a function of system size at zero temperature. Obviously, the effective coupling decreases with the size decreasing. Because of the equivalence of spatial direction and temporal direction in Euclidean space, the decreasing of system size means the enlargement of fluctuation, therefore, the effective coupling should decrease with the system size decreasing in principle. It is known that, once the coupling constant is less than the critical coupling constant, one can not find the nontrivial solution of gap equation in the NJL model [51], which means that the DCSB is partially restored. The Fig. 4 shows the quark condensate as a function of system size at zero temperature. With the decreasing of system size, the quark condensate also decreases, thus, the chiral symmetry will partially restore. With the increase of system size L, the condensate trends to a constant when L > 4 fm, which means that when L is greater than 4 fm, the finite volume system at this time can be regarded as an infinite thermodynamical system. This is consistent with the conclusion that we have just analyzed the effective coupling as a function of the spatial size.

      Figure 3.  The part of system size dependent coupling as a function of size L at zero temperature.

      Figure 4.  Quark condensate as a function of system size L at zero temperature.

      Now let's go back to the calculation of the thermal susceptibility. Because the effective coupling decreases with the size of system decreasing, comparing with normal NJL model, the modified NJL should cause the chiral phase transition easily at same size L. The Fig. 5 and Fig. 6 show the susceptibilities as function with system size L = 3 fm and L = 2 fm respectively. It is found that the susceptibilities have a smooth peak at some temperature, which means the transition is a crossover at finite temperature. At fixed size L = 3 fm, the pseudo-critical temperature Tc = 156 MeV in modified NJL model is smaller than that in normal NJL model, which is Tc = 181 MeV. As the size decreases, e.g., L = 2 fm, pseudo-critical temperature also decreases: in modified NJL model, Tc = 149 MeV and Tc = 180 MeV in normal NJL model. As the system size decreases, the pseudo-critical temperature decreases in both models. These results agrees with other involved models calculations [34, 35, 38] The transition temperature with modified NJL model is smaller than that with NJL model, which means that the quark's feedback enlarges the fluctuation and lowers the transition temperature. In order to study the contributions of quark's feedback and the anti-periodic boundary condition to the effect on the pseudo-critical temperature, the susceptibilities of modified and normal NJL model are both shown in Fig. 6. From Fig. 6, it is easy to find that the pseudo-critical temperature of the chiral phase transition with decreasing size is due to both the modified effective coupling and the anti-periodic boundary condition, and the effect of the modified effective coupling is more pronounced, which further demonstrates the importance of introducing effective coupling related to the spatial sizes. Specifically, for the case of infinite volume, the modified NJL model reduces the critical temperature from Tc = 183 MeV to Tc = 156 MeV, and this critical temperature is even smaller than that of normal NJL model with L = 2 fm (Tc = 180 MeV). It is also found that the modified NJL model with the anti-periodic boundary condition further reduces the critical temperature (Tc = 149 MeV) The above calculations clearly show that the transition temperature obtained by considering the size-dependent effective Lagrangian is much smaller than the corresponding transition temperature obtained without considering the size-dependent Lagrangian. We believe this should be considered in the second phase of the energy scan of RIHC in the future.

      Figure 5.  The thermal susceptibility as a function of temperature with the size L = 3 fm.

      Figure 6.  (color online) The thermal susceptibilities of the modified and normal NJL model in infinite volume and the modified and normal NJL model with the size L = 2 fm (with anti-periodic boundary condition) as a function of temperature.

    4.   Conclusion
    • To summarize, in principle, the effective Lagrangian of finite volume thermodynamical system should be different from that of infinite thermodynamical system. However, people usually study the finite volume effect by the effective Lagrangian of infinite thermodynamical system with different boundary conditions [20]. In this paper, we try to introduce a effective Lagrangian naturally, which depends on the finite volume size. Specifically under the usual NJL model framework, we first obtained a modified NJL model with finite volume size dependence by considering the quark feedback to the "static" gluon propagator. Then based on the modified NJL model, we study the chiral phase transition in finite volume and finite temperature and compare the results with that of normal NJL model calculation. It is found that the influence of quark's feedback can lower the temperature of crossover obviously at fixed system size. This impact of quark's feedback is very large. To our knowledge, this is the first time that people have used the effective Lagrangian reliance on a finite volume size to analyze the effect of finite volume size on chiral phase transitions in finite temperature and finite volume system. Because of the equivalence of spatial direction and temporal direction in Euclidean space, the decreases of system size is equivalent to the increases of temperature. The effective coupling of modified NJL model shows a decreasing behavior as the decreasing of the volume size. In this paper, we only study the chiral phase transition at finite temperature. To locate the CEP, this model must be extended to the case of finite chemical potential, which is our next step work in the future.

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