Studies On High-order Perturbation Effects In Quark-Gluon Plasma Free Energy Density

Quantum chromodynamics(QCD)is the theory of the strong interaction between quarks and gluons.The distance between quarks decreases as the typical momentum transfer of the high-energy process increases.Based on the "asymptotic freedom",the corresponding coupling constant would be smaller than 1 so that we can estimate the true value of the physical quantity according to the pertubative QCD which expanding the quantity by the strong coupling constant.And there would be divergence if the high order of pQCD is calculated,the corresponding renormalization is required so that we could obtain a reliable theoretical estimate.Besides,the distance between quarks increases as the typical momentum transfer of the high-energy process decreases so that the interactions between quarks becomes larger.When the distance between quarks is close to the scale of hadron,the strong interaction between quarks will become infinite,leading to quark confinement.This is the reason why the free quarks have still not been found experimentally.However,under extreme conditions such as high temperature or high pressure,the hadrons might transform into the quark-gluon plasma(QGP).The free energy density is a basic thermodynamic quantity of QGP,and studying the free energy density is very helpful for us to understand the early evolutionary nature of the universe.In this paper,we will focus on the free energy density of pQCD in the framework of effective field theory.Usually,according to the renormalization group invariance,the physical quantity is independent with the renormalization scale and renormalization scheme.While in the case of finite-order perturbation calculations,the conventional scale-setting method causes the mismatch between the perturbation coefficients and the strong coupling constant,resulting in the theoretical prediction relying on the selection of the renormalization scale and the renormalization scheme.In order to solve the problem,the Principle of Maximum Conformality(PMC)proposed in recent years provides us with a method for systematically setting renormalization scales.According to the renormalization group equation of the strong coupling constant,PMC uses the non-conformal ?-term to set the renormalization scales.Since these ?-term will be absorbed into strong coupling constants finally,the expressions will be a conformal series that is independent with the ?-term and the renormalization scheme.In the past,PMC usually used ?s(?)as the basis to the related processes.Here we use gs(?)as the basic and we can further explain the equivalence.Moreover,by taking the perturbation quantity R e+e-up to four-loop level as an example,we also illustrate the equivalence between these.In the present paper,we obtain the prediction of the free energy density which is independent with the renormalization scheme and the renormalization scale at the high temperature by applying the PMC scale-setting method.And our results show that if we take the factorization scale ?E around the effective parameter mE,the PMC prediction would be consistent with the Lattice QCD prediction at the low temperature.Contrary to this,the prediction under the conventional scale-setting method is smaller than the lattice QCD theory prediction.And it indicates the importance of setting the renormalization scales.