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An Effective Formulation on Quantum Hadrodynamics at Finite An effective formulation on quantum hadrodynamics at finite temperatures and densities B. X. Sun1,∗ X. F. Lu2, L. Li3, P. Z. Ning3, P. N. Shen4, E. G. Zhao5 1College of Applied Sciences, Beijing University of Technology, Beijing 100022, China 2Department of Physics, Sichuan University, Chengdu 610064, China 3Department of Physics, Nankai University, Tianjin 300071, China 4Institute of High Energy Physics, The Chinese Academy of Sciences, P.O.Box 918(4), Beijing 100039, China 5Institute of Theoretical Physics, The Chinese Academy of Sciences, Beijing 100080, China November 4, 2018 Abstract According to Wick’s theorem, the second order self-energy corrections of hadrons in the hot and dense nuclear matter are calculated. Furthermore, the Feynman rules are summarized, and an effective formulation on quantum hadrodynamics at finite temperatures and densities is evaluated. As the strong couplings between nucleons are considered, the self-consistency of this method is discussed in the framework of relativistic mean-field approximation. Debye screening masses of the scalar and vector mesons in the hot and dense nuclear matter are calculated with this method arXiv:nucl-th/0209041v5 10 May 2006 in the relativistic mean-field approximation. The results are different from those of thermofield dynamics and Brown-Rho conjecture. Moreover, the effective masses of the photon and the nucleon in the hot and dense nuclear matter are discussed. Key words : quantum hadrodynamics, Wick’s theorem, Debye screening masses PACS numbers : 21.65.+f, 24.10.Cn, 24.10.Jv, 24.10.Pa ∗Corresponding author. E-mail address: [email protected]. The project supported by the Foun- dations of Beijing University of Technology and Ren-Cai-Qiang-Jiao Foundation of Beijing Municipal Education Commission 1 1 Introduction The quantum field theory, introduced to solve the many-body problems since the late fifties, has proved to be highly successful in studying the ground state, the equilibrium and non-equilibrium properties at finite temperatures[1, 2, 3, 4, 5, 6, 7, 8, 9]. These theories to solve many-body problems are based on the conjecture that all of the effects of the medium at finite temperatures and densities would change the propagators of particles. When the propagators in the mediums are obtained, the properties of particles and the equation of state of the medium can be calculated easily. The propagators of particles in the medium are different from those in vacuum correspondingly, so this conjecture means the redefinition of the vacuum, and the calculation procedures are all carried out in this new vacuum. In a series of our previous papers[10, 11, 12], according to Wick’s theorem, we have evaluated an effective formulation on quantum hadrodynamics to solve the nuclear many- body problems. In this formulation, the Feynman propagators in vacuum are adopted, and the second order self-energies of particles in the nuclear matter are calculated, while the effects of the nuclear matter are treated as the condensations of nucleons and mesons. It shows the same results as the method of quantum hadrodynamics[1, 2, 3]. With this new method, we have studied the effective masses of the photon and mesons[10, 11], and constructed the density-dependent relativistic mean-field model[11]. In this paper we will generalize this new method to the situation at finite temperatures and densities. The organization of this paper is as follows. The second order self-energy corrections of the nucleon and the meson in the nuclear matter are calculated from Wick expansion in the framework of quantum hadrodynamics 1 in Sec. 2, then the Feynman rules and self-consistency for this method at finite temperatures and densities are summarized in Sec. 3. The results of screening masses of mesons are presented in Sec. 4. In Sec. 5, the effective mass of the photon in the hot nuclear matter is discussed. The summary is given in Sec. 6. 2 The self-energies of hadrons in the hot and dense nuclear matter According to Walecka-1 model, the Lagrangian density in nuclear matter can be written as 1 1 1 1 L = ψ¯ (iγ ∂µ − M ) ψ + ∂ σ∂µσ − m2 σ2 − ω ωµν + m2 ω ωµ µ N 2 µ 2 σ 4 µν 2 ω µ µ −gσψσψ¯ − gωψγ¯ µω ψ, (1) where ψ is the field of the nucleon, MN is the nucleon mass, and ωµν = ∂µων − ∂νωµ, (2) is the field tensor of the vector meson. 2 In the hot nuclear matter, the expectations of normal products of the creation opera- tor and corresponding annihilation operator are relevant to the distribution functions of particles. † 3 ~′ ′ h N, β |Ap′λ′ Apλ| N, β i = nF δ (p − ~p)δλ λ, (3) † 3 ~′ ′ h N, β |Bp′λ′ Bpλ| N, β i =n ¯F δ (p − ~p)δλ λ, (4) ′ where λ and λ denote spins of fermions, nF andn ¯F are the distribution functions of the nucleon and antinucleon, respectively. 1 n = , (5) F exp [(E(p) − µ)/T ]+1 1 n¯ = , (6) F exp [(E(p)+ µ)/T ]+1 2 2 where E(p) = ~p + MN , and µ is the chemical potential of the nucleon. The relation of chemical potentialq and the number density of nucleons ρB is 2 3 ρB = 3 d p [nF − n¯F ] , (7) (2π) Z † 3 ~′ ~ h N, β |ak′ ak| N, β i = nσδ (k − k), (8) † 3 ~′ ~ ′ h N, β |bk′δ′ bkδ| N, β i = nωδ (k − k)δδ δ, (9) ′ where δ and δ denote the spins of the vector meson, nσ and nω are the boson distribution functions of the scalar and vector mesons, respectively. 1 nα = , α = σ, ω, (10) exp [|Ωα|/T − 1] ~ 2 2 where Ωα = k + mα. Since the meson number is not conserved in nuclear matter, there is not the contributionq of the meson chemical potential in the boson distribution function of Eq. (10). The finite-temperature state of nuclear matter could be understood as the state that there are a great number of coupling nucleons, antinucleons and mesons in the perturba- tion vacuum, so the noninteracting propagators in vacuum are used in the calculation of the self-energies of particles. The momentum-space noninteracting propagators of the scalar meson, vector meson and nucleon in perturbation vacuum | 0 i follow as: −1 i∆0(p)= 2 2 , (11) p − mσ + iε µν µν g iD0 (p)= 2 2 , (12) p − mω + iε 3 αβ −1 iG0 (p)= µ . (13) γµp − MN + iε Since the vector meson couples to the conserved baryon current, the longitudinal part in the propagator of the vector meson will not contribute to physical quantities[3, 13]. Therefore, only the transverse part in the propagator of the vector meson is written in Eq. (12). According to Eq. (1), the interaction Hamiltonian can be expressed as µ HI = gσψσψ¯ + gωψγ¯ µω ψ. (14) In the second order approximation, only 2 (−i) 4 4 Sˆ2 = d x1 d x2T [HI (x1)HI (x2)] (15) 2! Z Z in the S-matrix should be calculated in order to obtain the self-energy corrections of the nucleon and mesons. The coupling constants gσ, gω, the masses of mesons mσ, mω and the nucleon mass MN are supposed to have been renormalized, then the contributions of the nucleon-loop diagrams are not needed to be considered[13, 14]. Only the terms with one contraction of two fields in the Wick expansions of Sˆ2 should be calculated. Thus with the similar procedure of Ref.[10, 11, 12], the scalar meson self-energy Σσ, the vector meson self-energy + − Σω, the nucleon self-energy Σ and the antinucleon self-energy Σ in the hot and dense nuclear matter are obtained. The second order self-energy of the scalar meson is d3p M Σ = (−ig )2 N σ σ (2π)3 E(p) λX=1,2 Z nF U¯(p,λ)(iG0(p − k)+ iG0(p + k)) U(p,λ) h − n¯F V¯ (p,λ)(iG0(−p − k)+ iG0(−p + k)) V (p,λ) d3p M i = g2 N σ (2π)3 E(p) λX=1,2 Z 1 1 nF U¯(p,λ) + U(p,λ) " /p − /k − MN /p +/k − MN ! 1 1 − n¯F V¯ (p,λ) + V (p,λ) , (16) −/p − /k − MN −/p +/k − MN ! # where U(p,λ) and V (p,λ) are the Dirac spinors of the nucleon and the antinucleon, respectively, and U¯(p,λ) and V¯ (p,λ) are their conjugate spinors, respectively. /p + M U(p,λ)U¯(p,λ)= N , (17) 2MN λX=1,2 4 −/p + M −V (p,λ)V¯ (p,λ)= N . (18) 2MN λX=1,2 The second order self-energy of the vector meson is d3p M −g Σ = (−ig )2 N µν ω ω (2π)3 E(p) λX=1,2 Z nF U¯(p,λ)(γνiG0(p − k)γµ + γµiG0(p + k)γν) U(p,λ) h − n¯F V¯ (p,λ)(γνiG0(−p − k)γµ + γµiG0(−p + k)γν) V (p,λ) d3p M i = g 2 N ω (2π)3 E(p) λX=1,2 Z 1 1 nF U¯(p,λ) γν γµ + γµ γν U(p,λ) " /p − /k − MN /p +/k − MN ! 1 1 − n¯F V¯ (p,λ) γν γµ + γµ γν V (p,λ) .(19) −/p − /k − MN −/p +/k − MN ! # The second order self-energy of the nucleon is + + + + Σ = Σs,1 + Σs,2 + Σs,3 , (20) s=σ,ω X in which d3p M Σ+ = (−ig )2 N i∆ (0) σ,1 σ (2π)3 E(p) 0 λX=1,2 Z nF U¯(p,λ)U(p,λ) − n¯F V¯ (p,λ)V (p,λ) h 2 i gσ = − 2 ρS, (21) mσ where ρS is the scalar density of protons or neutrons, 2 3 MN ρS = d p (nF +¯nF ) , (22) (2π)3 2 2 Z ~p + MN q d3p M Σ+ = g2 N σ,2 σ (2π)3 E(p) λX=1,2 Z nF U(p,λ) i∆0(k − p) U¯(p,λ) − n¯F V (p,λ) i∆0(k + p) V¯ (p,λ) (23) h 3 i 2 d p MN /p + MN 1 /p − MN 1 = −gσ 3 nF 2 2 − n¯F 2 2 , Z (2π) E(p) " 2MN (k − p) − mσ 2MN (k + p) − mσ # 5 3 + 2 d k Σσ,3 = (−igσ) 3 nσ (iG0(p − k)+ iG0(p + k)) Z 2Ωσ(2π) 3 2 d k 1 1 = gσ 3 nσ + , (24) Z 2Ωσ(2π) /p − /k − MN /p +/k − MN ! d3p M Σ+ = (−ig )2 N γ iDµν(0) ω,1 ω (2π)3 E(p) µ 0 λX=1,2 Z nF U¯(p,λ)γνU(p,λ) − n¯F V¯ (p,λ)γνV (p,λ) h 2 i gω = γ0 2 ρB, (25) mω where ρB defined in Eq.
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