Chiral phase transition from the Dyson-Schwinger equations in a finite spherical volume
Ya-Peng Zhao Rui-Rui Zhang Han Zhang Hong-Shi Zong

zhaoyapeng2013@hotmail.com

zhanghan@nju.edu.cn
zonghs@nju.edu.cn
^1 Department of Physics, Nanjing University, Nanjing 210093, China
^2Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
^3State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, CAS, Beijing, 100190, China
^4Joint Center for Particle, Nuclear Physics and Cosmology, Nanjing, 210093, China



 Within the framework of Dyson-Schwinger equations and by means of Multiple Reflection Expansion, we study the finite volume effects on the chiral phase transition in a sphere, especially discuss its influence on the location of the possible critical end point (CEP).
According to our calculations, when we take the sphere instead of cube as a research, the influence of finite volume effects on phase transition is not as significant as previously calculated.
 For instance, as the radius of spherical volume decreases from infinite to 2 fm, at zero chemical potential and finite temperature, the critical temperature T_c has only a slight drop. And at finite chemical potential and finite temperature, the location of CEP shifts toward smaller temperature and higher chemical potential, but the amplitude of variation does not exceed 20%.
 So we find that not only the size of the volume, but also the shape of the volume will have a considerable impact on the phase transition.


Key-words: finite volume effects, Dyson-Schwinger equations, chiral phase transition


PACS Number(s): 12.38.Mh, 11.10.Wx, 64.60.an



12.38.Mh, 12.39.-x, 25.75.Nq




Chiral phase transition from the Dyson-Schwinger equations in a finite spherical volume
    Hong-Shi Zong^1,3,4
    December 30, 2023
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§ INTRODUCTION

It is widely believed that with the increases of temperature or/and chemical potential, the strongly-interacting matter undergoes a phase transition from hadronic matter to quark-gluon plasma (QGP), which now can be reproduced by relativistic heavy-ion collisions (RHIC) at CERN (France/Switzerland), BNL (USA), and GSI (Germany) <cit.>. Theoretically, on the one hand, ab initio lattice QCD <cit.> simulation found the transition is a crossover at low chemical potential. On the other hand, QCD effective model <cit.> calculations generally indicate that the phase transition is a first order at high chemical potential, and in the middle of the chemical potential, there exist a critical end point (CEP) where the first order phase transition ends. One important goal in the RHIC is to determine the existence and the location of the CEP. For this purpose, the second phase of the beam energy scan at RHIC will be performed between 2019 and 2021 <cit.>.

It should be noted that many previous calculations about the location of CEP are based on infinite thermodynamics systems. However, the QGP system produced in RHIC has a finite volume undoubtedly. The volume of homogeneity before freeze-out for Au-Au and Pb-Pb collisions ranges between approximately 50∼250 fm^3 <cit.> based on the UrQMD transport approach <cit.>. And the smallest quark-gluon plasma (QGP) system produced at RHIC could be as low as (2 fm)^3 as the Ref. <cit.> estimated. Therefore, theoretically studying whether the finite volume effects has a significant impact on CEP is important for RHIC experiment and it has been studied within the Nambu-Jona-Lasinio (NJL) model <cit.>, Polyakov-Nambu-Jona-Lasinio (PNJL) model <cit.>, quark-meson model <cit.> and Dyson-Schwinger equations (DSEs) <cit.>.

It is well known that when the volume of the strongly interacting system we studied is small enough, not only its size but also its shape will have an important impact on the QCD phase transition. However, it should be noted that in most of the previous calculations, for the sake of convenience, people usually use cube to simulate the fireball produced in RHIC, which ignores the influence of different shapes on the phase transition. Therefore, in order to get closer to the shape of the fireball produced by the RHIC experiment, we use the sphere and by means of the MRE <cit.> to study the finite volume phase transition in the framework of DSEs.
Compared to other effective models, the DSEs take the quarks and gluons as the fundamental degrees of freedom, have both confinement and dynamical chiral symmetry breaking (DCSB) effects. And it has provided many insights into the QCD phase diagram, for instance, the chiral and deconfinement phase transition <cit.>.

This paper is organized as follows: In Sec. II, we give a brief introduction to the quark gap equation in a finite spherical volume at finite temperature and finite chemical potential. In Sec. III, we study the finite volume effects on the chiral phase transition, especially its influence on the behaviour of the CEP. Finally, we will give a brief summary in Sec. IV.


§ QUARK GAP EQUATION IN A FINITE SPHERICAL VOLUME

The DSEs is a suitable QCD-connected non-perturbative method and it is widely used to study hadron physics <cit.> and QCD phase diagram. At zero temperature and zero chemical potential, the DSE of the quark propagator, namely the quark gap equation reads <cit.> [Here we work in Euclidean space, take the N_f=2 and N_c=3. Moreover, renormalization is actually unnecessary because of we employ a ultra-violet finite model.]

    S(p)^-1=S_0(p)^-1+4/3∫d^4q/(2π)^4g^2D_μν(p-q)γ_μS(q)Γ_ν,

where S(p)^-1 is the inverse of the dressed quark propagator and S_0(p)^-1 is the inverse of the free one. g is the coupling constant of strong interaction, D_μν(p-q) is the dressed gluon propagator, and Γ_ν is the one-particle-irreducible quark-gluon vertex.
According to the Lorentz structure analysis, S(p)^-1 can generally be decomposed as <cit.>

    S(p)^-1=ipA(p^2)+B(p^2),

where A(p^2) and B(p^2) are scalar functions of p^2. Note that for the free quark propagator S_0(p)^-1, scalar functions A=1, B=m.

Next we extend the quark gap equation to finite T and μ and it reads

    S(p⃗,ω̃_n)^-1   =   S_0(p⃗,ω̃_n)^-1+4/3T∫∑ g^2
          × D_μν(k⃗,Ω_nl)γ_μS(q⃗,ω̃_l)Γ_ν,

where

    S_0(p⃗,ω̃_n)^-1=iγ⃗·p⃗+iγ_4ω̃_n+m,

k⃗=p⃗-q⃗, Ω_nl=ω_n-ω_l, ω̃_n=ω_n+iμ, ω_n=(2n+1)π T, n∈𝐙 and  ∫∑ denotes ∑_l∫d^3q⃗/(2π)^3.
Nevertheless, due to the breaking of O(4) symmetry down to O(3) symmetry, the Lorentz structure of S(p⃗,ω̃_n)^-1 now have to be decomposed as

    S(p⃗,ω̃_n)^-1   =   ip⃗A(p⃗,ω̃_n)+1B(p⃗,ω̃_n)
          +iγ_4ω̃_nC(p⃗,ω̃_n)+p⃗γ_4ω̃_nD(p⃗,ω̃_n)

where p⃗=γ⃗·p⃗, γ⃗=(γ_1, γ_2, γ_3), and the four scalar functions F=A, B, C, D are complex and satisfy the condition

    F(p⃗,ω̃_n)^∗=F(p⃗,ω̃_-n-1),

which can be used to verify the accuracy of your numerical calculations.
In the next calculations we ignore the function D because it is power-law suppressed in the ultra-violate region. And at zero T but finite μ, D vanishes exactly as <cit.> shows because the corresponding tensor structure has the wrong transformation properties under time reversal. So, the widely used structure of S(p⃗,ω̃_n)^-1 as follows

    S(p⃗,ω̃_n)^-1   =   ip⃗A(p⃗,ω̃_n)+1B(p⃗,ω̃_n)
          +iγ_4ω̃_nC(p⃗,ω̃_n),


Now, we are ready to introduce the quark gap equation in a finite spherical volume, and for taking the finite volume effects into account we consider the MRE formalism <cit.> which modifies the density of states as follows

    ρ_MRE(p,m,R)=1+6π^2/pRf_S+12π^2/(pR)^2f_C,

where f_S denote the surface contribution to the density of states

    f_S=-1/8π(1-2/πarctanp/m),

and the curvature contribution is given by Madsen^,s ansatz <cit.>

    f_C=1/12π^2[1-3p/2m(π/2-arctanp/m)],

which takes the finite quark mass contribution into account. It should be noted that there have different interpretations of m in the MRE formula when it is applied to nonperturbative calculations, for example, (P)NJL model, see Refs. <cit.>. Here, in this paper, we treat m as current quark mass as Refs. <cit.> instead of constitute quark mass <cit.>.

For m≠ 0, the main problem with the MRE is that it predict a negative density of states at samll p where in reality there are no states <cit.>. Therefore we remove it by introducing an IR cutoff (Λ_IR) in momentum space as Refs. <cit.>. Actually for anti-periodic boundary conditions of spatial directions, in a cubic box of size L, we have p⃗^2=4π^2/L^2∑_i=1^3(n_i+1/2)^2, n_i=0, ±1, ±2 ···. The minimum momentum |p_min|=π/L, similar to Λ_IR. So in the next calculations, the following replacement must be performed.

    ∫_0^Λ,∞ d^3p⃗/(2π)^3⋯→∫_Λ_IR^Λ,∞ d^3p⃗/(2π)^3ρ_MRE⋯,

where the Λ_IR is the largest solution of the equation ρ_MRE(p,m,R)=0 with respect to the momentum p.
Thus in a finite spherical volume, the quark gap equation becomes

    S(p⃗,ω̃_n)_F^-1   =   S_0(p⃗,ω̃_n)^-1+4/3T∫∑_fg^2ρ_MRE
          × D_μν(k⃗,Ω_nl)γ_μS(q⃗,ω̃_l)Γ_ν,

where now  ∫∑_f denotes ∑_l∫_Λ_IRd^3p⃗/(2π)^3.

To solve the quark gap equation, truncations are inevitably. Here we employ the Rainbow truncation <cit.>

    Γ_μ(p_n,q_l)=γ_μ,

which is widely used in studies of hadron physics and QCD phase diagram.
We also employ the widely used gluon propagator model as Refs. <cit.>, which has the form

    g^2D_μν(k_Ω)   =   𝒢(k_Ω^2)(δ_μν-k_Ω^μk_Ω^μ/k_Ω^2),

where

    𝒢(k_Ω^2)=4π^2/ω^6D_0k_Ω^2e^-k_Ω^2/ω^2,

and k_Ω=(k⃗,Ω_nl), δ_μν=diag{+1,+1,+1,+1}.

The related parameters, D_0 and ω are usually fixed by observables in hadron physics: the pion mass m_π=0.139 GeV and the pion decay constant f_π=0.095 GeV. Here we use the typical values, that is ω=0.5 GeV, D_0=1.0 GeV^2 <cit.> and the current quark mass m=0.005 GeV.


§ FINITE VOLUME EFFECTS ON THE QCD CHIRAL PHASE DIAGRAM

In this section, we study the chiral phase transition in a finite spherical volume, especially discuss its influence on the location of the CEP. We first solve the quark gap equation. The procedure is to insert Eqs. (<ref>,<ref>,<ref>,<ref>) into Eqs. (<ref>), multiply each side by -ip⃗, -iγ_4ω̃_n and 1_4 respectively, and then take traces on both sides. Note that S(p⃗,ω̃_n) and S(p⃗,ω̃_n)^-1 have the same Lorentz structure. The coupled non-linear equations about scalar functions A, B, C can then be obtained as follows






    A_pn   =   1+4T/3p⃗^2∫∑_fρ_MRE𝒢(k_Ω^2)/q⃗^2A^2_ql+ω̃_l^2C^2_ql+B^2_ql{A_ql[p⃗·q⃗+2(k⃗·p⃗)(k⃗·q⃗)/k^2]+C_ql2ω̃_lΩ_nl(k⃗·p⃗)/k^2}


    C_pn   =   1+4T/3ω̃_n∫∑_fρ_MRE𝒢(k_Ω^2)/q⃗^2A^2_ql+ω̃_l^2C^2_ql+B^2_ql{A_ql2Ω_nl(k⃗·q⃗)/k^2+C_qlω̃_l(1+2Ω_nl^2/k^2)}


    B_pn   =   m+4T/3∫∑_fρ_MRE3𝒢(k_Ω^2)B_ql/q⃗^2A^2_ql+ω̃_l^2C^2_ql+B^2_ql


These coupled equations can be numerically solved by iteration.
In Fig. <ref>, we show B(0,ω̃_0^2) as a function of T for three different volumes. Firstly, B(0,ω̃_0^2) decreases continuously with increasing temperature, which means the transition is a crossover.
Secondly, B(0,ω̃_0) decreases as the volume decreases. For example, at T=0.05 GeV, we find that B(0,ω̃_0) is reduced from 0.525 GeV to 0.473 GeV , which means the DCSB effect becomes weak. This is consist with other model calculations <cit.>.

The crossover behavior can be further studied by the chiral susceptibility which is defind as <cit.>

    χ_m(T,μ)=∂ B(0,ω̃_0^2)/∂ m

and the volume dependence are plotted in Fig. <ref>. We find that the critical temperature T_c shows a slight volume dependence and it only decreases from 0.14 GeV to 0.135 GeV. So, the conclusion is that at zero chemical potential and finite temperature, the influence of finite volume effects on chiral phase transition is not so obvious.
In Fig. <ref>, we plot the B(0,ω̃_0^2) as a function of μ at T=0.09 GeV for three different volumes. At infinite volume, B(0,ω̃_0^2)  appears a sudden discontinuity at a critical value μ_c=0.275 GeV, which indicates the first order phase transition happens. But when the radius is reduced to 2 fm, B(0,ω̃_0^2) changes continuity and a crossover happens. This indicate that under T=0.09 GeV, the finite size effect has a significant impact on the phase diagram, that is, on the location of the CEP.

Therefore, we plot the volume dependence of the CEP in Fig. <ref>. First of all, when the radius of spherical volume is larger than 5 fm, the chiral phase diagram is almost the same as the infinite system. In the next place, as the radius of spherical volume decreases, we note that the CEP shifts toward smaller temperature and higher chemical potential.
Thus, for the CEP search in the RHIC experiments, according to our calculation, when the radius is below 3 fm, the finite volume effects may not be negligible.
However, even if the radius is as small as 2 fm, the location of the CEP shifts from (μ_E,T_E)=(0.237 GeV,0.101 GeV) to (μ_E,T_E)=(0.264 GeV,0.087 GeV), and the variation does not exceed 20%.
It is interesting here to compare our results with other DSE calculations <cit.>. The main difference is that when we study the sphere instead of cube, we find that the influence of finite volume effects on phase transitions is much weaker.
For instance, in Ref. <cit.>, the CEP moves from (μ_E,T_E)=(0.24 GeV,0.10 GeV) at L=∞ to (μ_E,T_E)=(0.37 GeV,0.025 GeV) at L=2.1  fm, and the variation more than 60%. In Ref. <cit.>, the variation more than 70%.
This reflects a fact that when we study the influence of finite volume effects on QCD phase transition, we should consider not only the size, but also the shape of the volume.
Finally, it is a little different from the PNJL model <cit.> in which the μ of CEP is almost a constant value.


§ SUMMARY AND CONCLUSION

Within the framework of DSEs, for the first time, we try to consider the influence of the finite volume effects on the chiral phase transition in a sphere. For taking the finite volume effects into account we consider the MRE formalism in which it also properly incorporate the surface and curvature effects. And it has been used to study thermodynamic quantities in the PNJL model <cit.> and color superconducting in the NJL model <cit.>.
Our main conclusion is that not only the size of the volume, but also the shape of the volume will have a sizable impact on the phase transition.
Relatively speaking, we find that the influence of finite volume effects on chiral phase transitions in the sphere is not as significant as in the cube. For example, at zero chemical potential and finite temperature, the T_c remains almost the same value as the volume decreases. At finite chemical potential and finite temperature, our results show that, as the radius of spherical volume decreases, the location of CEP shifts toward smaller temperature and higher chemical potential. But the amplitude of variation does not exceed 20%. Therefore, we should use a shape closer to the QGP fireball produced by RHIC in order to better study the finite volume phase transition.
Finally, in the next step, we intend to go beyond the bare vertex approximation <cit.> and see if it will make a difference.

This work is supported by the National Natural Science Foundation of China (under Grants No. 11475085, No. 11535005, No. 11690030 and No. 11574145).


