Báo cáo "Quantum kinetic equation in the quantum hadrondynamics (QHD-I) model "

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Within the framework of the Walecka model (QHD-I)[1] the renormalized effective Dirac equation and the kinetic equation for fermion are presented. In fact, the fermion propagator in the medium is dramatically different from that in the vacuum. The main feature is the treating of the fermion distribution in non equilibrium, which depends on the interaction rate involving temperature.

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VNU Journal of Science, Mathematics - Physics 24 (2008) 145-154 Quantum kinetic equation in the quantum hadrondynamics (QHD-I) model Phan Hong Lien* 100 Hoang Quoc Viet Street, Hanoi, Vietnam Received 8 August 2008; received in revised form 30 August 2008 Abstract. Within the framework of the Walecka model (QHD-I)[1] the renormalized effective Dirac equation and the kinetic equation for fermion are presented. In fact, the fermion propa- gator in the medium is dramatically different from that in the vacuum. The main feature is the treating of the fermion distribution in non equilibrium, which depends on the interaction rate involving temperature. Keyworks:field Theory, nuclear Theory. 1. Introduction At present the theoretical study of quantum fields at finite temperature and density turns out to be more and more important for description of wide variety of physical effects in medium: in condensed matter [2,3], in stellar astrophysics [3,4], in a QED or QCD plasma [5,6]. The effective real-time Dirac equation in medium and the kinetic equation not only may provide an approximation beyond two-loop calculations, but also can be treated the correlation effects- those that are extremely important for physical processes near equilibrium [8]. In this paper we focus on the QHD-I model of non-zero density. We investigate in detail the fermion propagator and its relaxation and thermalization through the interaction with scalar and neutral vector mesons in the matter. Furthermore, the kinetic equation for fermion in the real time is shown the relation between the fermion distribution and the interaction rate. The paper is organized as follows. In Sec. II the QHD-I and the real time formalism are presented. Sec. III is devoted to considering the renormalized effective Dirac equation. In Sec. IV we carry out the quantum kinetic equation for fermion in the QHD-I. The discussion and conclusion are given in Sec. V. ∗ Tel.: (84-4) 38230469 E-mail: pnhue2000@yahoo.com 145
P.H. Lien / VNU Journal of Science, Mathematics - Physics 24 (2008) 145-154 146 2. Formalism 2.1. The Quantum Hadrondynamics (QHD-I) We start with the Lagrangian density Lo =Ψ (iγ µ∂µ − M0 − g Φ − gω γ µVµ ) Ψ ¯ (1) λ2 1µ 1 1 ∂ Φ∂µ Φ − m2 Φ − Φ4 + m2 V µVµ − Fµν Fµν , + 0 W 2 4 2 4 where Ψ, Φ and Vµ are the field operators of fermion, scalar and vector meson, respectively, and Fµν = ∂ µ V ν − ∂ ν V µ. In the medium of finite density (the nuclear matter), the symmetry of the ground state |F yields ¯ F Ψ F = 0; F | Ψ | F = 0, (2) F |Φ| F = v ; F |Vµ| F = δ0µ ωµ , where v and ω are the independent of space - time coordinate owing to the homogeneity of nuclear matter. Adding to (2.1) a term, for example L0 → L = L0 + c Φ , which leads to an explicit chiral symmetry breaking. By shifting the scalar and vector fields Φ and Vµ respectively Φ ⇒ v + Φ; Vµ ⇒ δ0µ ωµ + Wµ , (3) the Lagrangian density (1) in the presence of external sources now takes the form L =Ψ (iγ µ∂µ − MN − g Φ − gω γ µ Wµ ) Ψ ¯ λ2 1 (∂µ Φ)2 − m2 Φ2 − λ2 v Φ(Φ2 + v 2 ) − (Φ4 + v 4 ) + 2 4 (4) 12 1 µν + mW Wµ Wµ − gω Ψγ µδ0µ ωµ Ψ − F Fµν ¯ 2 4 + η Ψ + Ψη + Jφ Φ + J µ Wµ + c(v + Φ), ¯ ¯ where m2 = m2 + 3 λ 2 v 2 MN = M0 + gv ; (5) φ 0 are the masses of nucleons and scalar meson in the medium. The QHD - I is a renormalizable model. In the exact chiral limit, the parameter c = 0. Now, we write the bare fields and sources in terms of the renormalized quantities (referred to a subscript r by introducing the wave function renormalization constants Zψ , Zφ, Zω , the vertex renormalization Zg , Zgω , Zλ and the mass counter-terms δM, δm, δmW ) as follows 1/2 1/2 Ψ =Zψ Ψr , Φ = Zφ Φr , (6) 1/2 ¯ ¯ (Wµ, ω µ ) = Zω/2 (Wr , ωr ), 1 µ µ Ψ =Zψ Ψr ,
P.H. Lien / VNU Journal of Science, Mathematics - Physics 24 (2008) 145-154 147 the external sources −1/2 −1/2 η =Zψ ηr , Jφ = Zφ Jr , (7) −1/2 Lµ = Zω 1/2 Jr , µ − η =Zψ ¯ ηr , ¯ the coupling constants 1/2 gω = gω Zgω 2 , r 1/ gr =gZφ Zψ /Zg , (8) 1/2 λ2 =λ2 Zφ /Zλ , 2 cr = cZφ , r and the masses MN = (M + δM )/Zψ , m2 = (m2 + δm2 )/Zφ , m2 = (m2 + δm2 )/Zω , (9) ω W W φ where M, m2 and m2 are the renormalized masses. ω With the above definitions, the Lagrangian (4) can be rewritten as (we have suppressed the subscript r for notational simplicity) LQHD−I = LM F + LL + Lsource + LSB , (10) where m2 + δm2 1 LM F = − λ2 Zλ v 2 + W Wµ Wµ W (11) 4 2 LL =Ψ [iZψ γ µ∂µ − (M + δµ ) − gZg Φ] Ψ ¯ m2 + δm2 − gω Zgω Ψγ µ (ωµ + Wµ )Ψ − (ω µ ωµ + 2Wµ Wµ ) W W ¯ (12) 2 1 Zφ (∂µ Φ)2 − (m2 + δm2 )Φ2 − λ2 Zλ v Φ(Φ2 + v 2 ) + 2 1 1 − λ2 Zλ (Φ4 + v 4 ) − Zω Fµν Fµν + c(Φ + v ) 4 4 ¯ η + Jφ Φ + J µ Wµ =¯Ψ + Ψ η (13) Lsource or, equivalently 1 LQHD−I = Ψ (iγ µ∂µ − M ) Ψ + (∂µ Φ)2 − m2 Φ2 ¯ 2 1 − g ΨΦΨ + gω Ψγ µ(ωµ + Wµ )Ψ − m2 (ωµ + Wµ )2 ¯ ¯ 2W 1 1 1 − λ2 v 2 − (Φ4 + v 4 ) − Fµν Fµν 4 4 4 1 (14) δφ (∂µ Φ) − δm Φ + Ψ [iδψ γ µ∂µ − δM ] Ψ 2 22 ¯ + 2 1 1 1 − gδg ΨΦΨ − λ2 δλ v 2 − δλ (Φ4 + v 4 ) − δω Fµν Fµν ¯ 4 4 4 1 − gω δgω Ψγ (ωµ + Wµ )Ψ − δµν (ωµ + Wµ )2 µ ¯ 2 ¯ + Jφ Φ + J µ Wµ + c(Φ + v ), + ηΨ + Ψ ¯
P.H. Lien / VNU Journal of Science, Mathematics - Physics 24 (2008) 145-154 148 where g, gω and λ are the renormalized Yukawa couplings, and the terms δψ = Zψ − 1, δφ = Zφ − 1, δω = Zω − 1, (15) δM = Zψ MN − M, δm2 = Zφ m2 − m2 , δm2 = Zω m2 − m2 , (16) W ω W φ δg = Zg − 1, δgω = Zgω − 1, δλ = Zλ − 1, (17) δm2 δm2 δM , Zgω = Zψ 1 + 2ω Zg = 1 + , Zλ = 1 + . (18) m2 M mω The renormalization conditions for the self - energies ¯ ¯ ¯ Σ(k) =Σ(k) + δM − γk kµ (Zψ − 1) = Σ(k) + δM − δψ k, (19) ¯ ¯ Π(k) =Π(k) + δm2 − k2 (Zφ − 1) = Π(k) + δm2 − δφ k2 , (20) ¯ Πµν (k) =Πµν (k) − gµν δm2 − (kµkν − k2 gµν )(Zω − 1) (21) W ¯ µν (k) − gµν δm2 − δω (kµ kν − k2 gµν ), =Π W ¯¯¯ where Σ, Π, Πµν are ”unrenormalized” self-energies the (spinor) fermion, scalar meson and vector meson, respectively, and in (18) we introduced Zg ¯ k µ = k µ − gω ω W µ . (22) Zψ The renormalization conditions are imposed on the self - energies as follows ∂Σ Σ(k = µN ) =0, / (k = µN ) = 0, / (23) ∂k/ ∂Π 2 Π(k2 = µ2 ) =0, (k = µ2 ) = 0, (24) φ φ ∂k2 ∂ Πµν 2 Πµν (k2 = µ2 ) =0, (k = µ2 ) = 0, (25) ω ω ∂k2 here µN , µφ and µω are the renormalization points. The set of Dyson equations for propagators take the form S =So + So ΣS, (26) G =Go + Go ΠG, (27) µ µν Doλ Πλρ Dρ , ν Dµν =Do + (28) where kv kµ D µν (k) = , (29) m2 + δm2 W W ¯ kµ Πµν (k) = − kv δm2 , kµ Πµν (k) = 0 (30) W 2.2. The equation of motion for scalar and vector mesons From the Lagrangian (1), one gets δL = Zφ 2 + (m2 + δm2 ) Φ = Jφ = j − iSφ , (31) φ φ δΦ
P.H. Lien / VNU Journal of Science, Mathematics - Physics 24 (2008) 145-154 149 where δ LCT jφ = = c, (32) δΦ δ LL ¯ = gZg ΨΨ + λ2 Zλ (2v Φ2 + v 2 + Φ3 ), iSφ = (33) δΦ and δL = Zω 2 + (m2 + δm2 )ω µ = Jω = −jω + iSω , µ µ µ (34) ω ω δωµ where δ LCT jω =W µ (m2 + δm2 ) = µ , (35) W W δW µ δ LL ¯ µ iSω == gω Zgω Ψγµ Ψ. (36) δW µ µ Eqs.(31) and (34) determine ”unrenormalized” source Jφ and Jω . The conditions F |ω µ | F = 0 F |Φ| F = 0, (37) imply µ µ µ Jφ = jφ − iSφ = 0, −Jω = jω − iSω = 0 (38) or, equivalently gω Zgω µ −Jω = W µ (m2 + δm2 ) − µ J = 0, (39) W W Zψ B µ where JB = (ρ, jB ) is the baryon current in the medium δm2 gω Zψ 1 + W gω µ m2 µ Wµ = W JB = J (40) m2 B Zψ m2 + δm2 ω W W 2.3. The free real-time Green’s functions in momentum space 1. Scalar propagators in the real-time formalism are defined as G++ (k, t, t′) =G> (k, t, t′)θ(t − t′ ) + G< (k, t, t′)θ(t′ − t), (41) o o o =G> (k, t, t′)θ(t′ + G< (k, t, t′)θ(t − t′ ), G−− (k, t, t′) − t) (42) o o o −+ > Go (k, t, t′) =Go (k, t, t′), (43) G+− (k, t, t′) =G< (k, t, t′), (44) o o G> (k, t, t′) =i d3 xe−ikx Φ(x, t)Φ(0, t′) o (45) i ′ ′ [1 + nB (ωk )] e−iωk (t−t ) + nB (ωk )eiωk (t−t ) = 2ωk G< (k, t, t′) =i d3 xe−ikx Φ(0, t′ )Φ(x, t) o (46) i nB (ωk )e−iωk (t−t ) + [1 + nB (ωk )] eiωk (t−t ) ′ ′ = 2ωk √ 1 k2 + m2 , and nB (ω ) = where ωk = is the Bose - Einstein distribution. eβω −1
P.H. Lien / VNU Journal of Science, Mathematics - Physics 24 (2008) 145-154 150 2. Fermion propagators (zero fermion chemical potential) are defined by ++ > < So (k, t, t′) =So (k, t, t′ )θ(t − t′ ) + So (k, t, t′)θ(t′ − t), (47) > < So (k, t, t′) −− =So (k, t, t′ )θ(t′ So (k, t, t′)θ(t − t′ ), − t) + (48) So + (k, t, t′) > − =So (k, t, t′ ), (49) + < So − (k, t, t′) =So (k, t, t′ ), (50) ¯ > d3 xe−ikx Ψ(x, t)Ψ(0, t′ ) So (k, t, t′) = − i i (γ o ωk − γ k + M ) [1 − nF (¯k )] e−iωk (t−t ) ¯ ′ =− ¯ ω (51) 2¯k ω ′ + (γ oωk + γ k − M ) nF (¯k )eiωk (t−t ) ¯ ¯ ω ¯ < d3 xe−ikx Ψ(0, t′ )Ψ(x, t) So (k, t, t′) =i i ′ (γ oωk + γ k − M ) nF (¯k )e−iωk (t−t ) ¯ = ¯ ω (52) 2¯k ω ′ + (γ oωk + γ k − M ) [1 − nF (¯k )] eiωk (t−t ) ¯ ¯ ω √ 1 k2 + M 2 , and nF (ω ) = where ωk = ¯ is the Fermi - Dirac distribution. eβω +1 These free propagators given in Eqs.(45), (46) and (51), (52) are thermal because the initial state in chosen to be in thermal equilibrium and the interaction in assumed to be turned on adiabatically. 3. The renormalized effective diracequation We aim our effort at the relaxation of inhomogeneous fermion mean field ψ (x, t) = Ψ(x, t) induced by external source that is adiabatically switched on at t = −∞. At usually, the fermion field is shifted by Ψ± (x, t) = ψ (x, t) ± ϕ± (x, t), (53) with ϕ± (x, t) = 0. 3.1. The Initial Value Problem The effective real time Dirac equation for the mean field of momentum k reads [(iγ o∂t − γ k − M ) + δψ (iγ o∂t − γ k) − δM ] Ψ(k, t) t (54) dt′ Σ(k, t − t′ )Ψ(k, t′ ) = −η (k, t), − −∞ ∂ Σ(k, t − t′ ) is the retarded fermion self - energy and where ∂t ≡ ∂t , d3 xe−ikx Ψ(x, t) Ψ(k, t) ≡ (55) The source is taken to be switched on adiabatically from t = −∞ and switched off at t = 0 to provide the initial condition Ψ(k, t = 0) = Ψ(k, 0); Ψ(k, t < 0) = 0. (56)
P.H. Lien / VNU Journal of Science, Mathematics - Physics 24 (2008) 145-154 151 Introducing an auxiliary quantity χ(k, t − t′ ) defined as Σ(k, t − t′ ) = ∂t′ χ(k, t − t′ ), (57) and imposing η (k, t > 0) = 0, we obtain the following equation of motion for t > 0 (iγ o∂t − γ k − M ) +δψ (iγ o∂t − γ k) − χ(k, 0) − δM Ψk (t) (58) t dt′ χk (t − t′ )Ψk (t′ ) = 0. + 0 This equation of motion can be solved by Laplace transform as befits an initial value problem. The Laplace transformed equation of motion is given by ˜ [iγ os − γ k − M + δψ (iγ os − γ k) − δM − χ(k, 0) + sχ(s, k)] Ψ(s, k) ˜ (59) = [iγ o + iδψ γ o + χ(s, k)] Ψ(k, 0), ˜ where ∞ ∞ ˜ dte−st Ψ(k, t); χ(s, k) ≡ dte−st χ(k, t), Ψ(s, k) ≡ ˜ (60) 0 0 with Res > 0. We can write χ(k, t − t′ ) as χ(k, t − t′ ) = iγ oχ(0) (k, t − t′ ) + γ kχ(1)(k, t − t′ ) + χ(2)(k, t − t′ ). (61) A straightforward calculation leads to the ultraviolet divergences g2 g 2M Λ g2 Λ Λ χ(1) (k, 0) ≃ ln , χ(2)(k, 0) ≃ − 2 ln , χ(0)(s, k) ≃ ˜ ln , (62) 2 2 16π µ 8π µ 16π µ where χ(i) (s, k), (i = 0, 1, 2) are the Laplace transform of χ(i) (k, t), Λ is an ultraviolet momentum ˜ cutoff, µ is an arbitrary renormalization scale. From Eq.(59), one gets g2 Λ g 2M Λ ˜ iγ os − γ k − M +δψ iγ os − δψ γ k − δM − γ k ln − ln + sχ(s, k) Ψ(s, k) ˜ 16π 2 µ 8π 2 µ (63) = [iγ o + iδψ γ o + χ(s, k)] Ψ(k, 0). ˜ The counter-terms δψ and δM are chosen as g2 g 2M Λ Λ δψ = − ln + ﬁnite , δM = ln + ﬁnite , (64) 16π 2 µ 8π 2 µ and the components of the self-energy are rendered finite χ(k, 0) + γ kδψ + δM =ﬁnite , (65) χ(s, k) + iγ oδψ =ﬁnite . ˜ 3.2. Renormalized effective Dirac equation Hence, we obtain the renormalized effective Dirac equation in the medium and the corresponding initial value problem for the fermion mean field ˜ ˜ iγ os − γ k − M − Σ(s, k) Ψ(s, k) = [iγ o + χ(s, k)] Ψ(k, 0). ˜ (66) ˜ Compare with (59), it is easy to derive the form of Σ(s, k) ˜ Σ(s, k) = χ(k, 0) − sχ(s, k), ˜ (67)
P.H. Lien / VNU Journal of Science, Mathematics - Physics 24 (2008) 145-154 152 This is the Laplace transform of the renormalized retarded fermion self-energy , which can be written in its most general form ˜ Σ(s, k) = iγ osε(0) (s, k) + γ kε(1)(s, k) + M ε(2)(s, k). ˜ ˜ ˜ (68) The solution of Eq. (66) is given by 1 ˜ ˜ Ψ(s, k) = 1 + S (s, k) γ k + M + Σ(0, k) Ψ(k, 0), (69) s where S (s, k) is the fermion propagator in terms of the Laplace variable s −1 ˜ S (s, k) = iγ os − γ k − M − Σ(s, k) (70) iγ os 1 − ε(0)(s, k) − γ k 1 + ε(1)(s, k) + M 1 + ε(2)(s, k) ˜ ˜ ˜ =− . 2 2 2 s2 1 − ε(0) (s, k) + k2 1 + ε(1)(s, k) + M 2 1 + ε(0) (s, k) ˜ ˜ ˜ The square of the denominator in eq.(70) is being ˜ det iγ os − γ k − M − Σ(s, k) . (71) The real-time evolution of Ψ(k, t) is obtained by performing this inverse Laplace transform in the complex s-plane along contour parallel to the imaginary axis. The denominator can be rewritten in the form ω 2 − ωk − P (ω, k) , where 2 P (ω, k) = − 2 ω 2 ε(0) (ω, k) + k2 ε(1)(ω, k) + M 2 ε(2)(ω, k) (72) 1 0 = Tr γ ω − γ k + M Σ(ω, k) . 2 It is just the lowest order term of effective self-energy imaginary part of P (ω, k) evaluated on the fermion mass shell. 4. Quantum kinetic equation for fermion in QHD - I Let us denote the distribution function for fermion of momentum k and spin s by ns,k (t). Since ¯ for a fixed spin component the matrix elements for transition probabilities are rather cumbersome, we 1 study the spin - averaged fermion distribution function as nk (t) = 2 s ns,k (t). ¯ ¯ For a small departure from thermal equilibrium, one can approximate Φ2 and W2 by their thermal equilibrium values d3 k Φ2 = nB (ωk ), (73) (2π )3ωk d3 k nµν (ωk ). W 2 = δµν W µ W ν =δµν (74) (2π ) k B 3ω To two - loop order, the Feynman diagrams that contribute to the kinetic equation is shown in Fig.1
P.H. Lien / VNU Journal of Science, Mathematics - Physics 24 (2008) 145-154 153 The kinetic equation can be derived directly basing on [6] d3 q ωk ωq − kq − M 2 ¯¯ d nk (t) = πg 2 ¯ δ (¯k + ωq − ωp ) ω ¯ 3 dt (2π ) 2¯q ωk ωp ω¯ (75) × [nB (ωp ) (1 − nk (t)) (1 − nq (t)) − (1 + nB (ωp )) nk (t)¯ q (t)] ¯ ¯ ¯ n + δµν nµν (ωp ) (1 − nk (t)) (1 − nq (t)) − 1 + δµν nµν (ωp ) nk (t)¯ q (t) ¯ ¯ ¯ n , B B where p = k + q. f f t t w t” t’ t” t’ k k Φ −igω γ µ −igω γ ν -ig -ig f f Fig. 1. The Feynman diagrams contribute to the kinetic equation for fermion’s interaction up to two loop order. The bold solid line is the fermion propagator S, the only solid line is the scalar propagator G and the dashed line is the omega propagator Dµν It is easy to find that the above equation has an equilibrium solution given by nk (t) = nF (¯k ) ¯ ω for all momentum k nF (t) = nF (¯k ) + δ nk (t), ¯ ω ¯ (76) δ n (t) ¯ k where nF (¯ k )
P.H. Lien / VNU Journal of Science, Mathematics - Physics 24 (2008) 145-154 154 + − with q ∈ (q − , q +), qW ∈ qW , qW are the support of δ (¯k − ωp + ωq ) for fixed k. ω ¯ The kinetic analysis is implemented directly in real-time and clearly establishes the relation between the interaction rate in the relaxation time approximation and the damping rate of the mean field. 5. Discussion and conclusion In the above mentioned sections the real time formalism was used to study the fermion propagator in the matter modeled by the QHD-I model. It could eventually be used in other problems and non equilibrium processes in the medium of finite density and temperature. We have presented and solved the renormalized effective Dirac equation by Laplace transform. The formulation of the initial value problem yields unambiguous separation of the vacuum and in-medium effects. We obtained the kinetic equation for fermion in the QHD-I model, including the fermion’s interaction with the neutral scalar and vector mesons. The fermion distribution in non equilibrium is investigated. It is proportional to the interaction rate, whose inverse characterizes the time scalar for the fermion distribution to approach equilibrium. Our next paper is intended to be devoted to the quantum kinetic equation for scalar, pseudoscalar and vector meson in the QHD.II model. Acknowledgements. The authors would like to thank Prof. Tran Huu Phat for helpful discussions. References [1] B.D.Serot and Walecka, The relativistic nuclear many body problem, Adv. Nucl. Phys. (Plenum Press), (1974) 190. [2] J.P. Blaizot, E. Iancu, Parwani, Phys. Rev. D 52 (1995) 2543. [3] T. Kunihiro, T. Muto, T. Takasuka, R. Tamagaki, T. Tatsumi, Various Phases in High- Density Nuclear Matter and Neutron Stars, Prog. Theor. Phys. Supplement (1993) 112. [4] G.G. Raffelt, Stars as Laboratories for Fundamental Physics, University of Chicago Press (1996). [5] D. Boyanovsky, H.J. de Vega, S.Y.Wang, Dynamical Renormalization Group Approach to Quantum Kinetics in Scalar and Gauge Theories, Phys .Rev. D 61 (2000) 065006. [6] S.Y.Wang, D. Boyanovsky,H.J. de Vega, D.S. Lee, Real-time Non equilibrium Dynamic in Hot QED Plasmas: Dynamical Renormalization Group Approach, Phys. Rev. D62(2000) 105026. [7] R. Baier, R.Kober, Phys. Rev. D50 (1994) 5944. [8] H. Umezawa, Equilibrium and non equilibrium thermal physics, in Banff/Cap Workshop on Thermal Field Theory, eds. F.C. Khanna, R. Kobes, G.Kunstatter and H.Umezawa. World Scientific Singapore (1994). [9] M.Le Bellac, Thermal Field Theory, Cambridge University Press (1996). [10] S.Y. Wang, Preprint hep-ph/0007340. [11] H.A. Weldon, Phys. Rev. D28 (2007) 1983.