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1 Quantum Field Theories in Nonextensive Tsallis Statistics Hiroaki Kohyama∗) and Akira Niégawa∗∗) Graduate School of Science, Osaka City University, Osaka 558-8585, Japan WithintheframeworkofTsallisstatisticswithq≃1,weconstructaperturbationtheory fortreatingrelativisticquantumfieldsystems. Wefindthatthereappearinitialcorrelations, 6 which donot exist in the Boltzmann-Gibbs statistics. Applyingthis framework to a quark- 0 0 gluon plasma, we find that the so-called thermal masses of quarks and gluons are smaller 2 than in the case of Boltzmann-Gibbs statistics. n a §1. Introduction J 8 Itisexpectedthataquark-gluonplasmawillsoonbeproducedinultrarelativistic 1 heavy-ion collision experiments at the BNL Relativistic Heavy Ion Collider (RHIC). 1 The CERN Large Hadron Collider (LHC) will also soon be ready for experiments. v A few observations regarding quark-gluon plasmas are in order. 0 2 1) According to hot QCD (statistical QCD at high temperature),1) in the chromo- 1 electric sector of gluons in a quark-gluon plasma, a Debye screening mass devel- 1 ops, and as a consequence the chromoelectric-gluon exchange interaction is short 0 6 range. By contrast, a Debye-like mass does notappear (at least to one-loop order) 0 in the chromomagnetic sector of gluons, and therefore the chromomagnetic-gluon h/ exchange interaction is long range.∗∗∗) t - p 2) The size of a region containing a quark-gluon plasma produced in the collision of e heavy ions in RHIC experiments is not very large. In fact, the radius of heavy h ions of mass number 200 is R 7.9 10−15 m. At the highest RHIC energy, : v ∼ ∼ × √s = 200 GeV, the Lorentz-contraction factor is 1/100, and therefore the i NN ∼ X longitudinal size of the system just after the collision is L 7.9 10−17 m, and l ∼ × r it increases due to the expansion of the system. Contrastingly, the chromoelectric a Debye mass, given by1) m 1.2gT (with g the QCD coupling constant and T D ≃ the temperature of the plasma) for three quark flavors, is 2.8 GeV for T = 1 ∼ GeV (where g2/4π 0.43), which corresponds to the Debye screening length†) ≃ l 6.9 10−17 m. As mentioned above, the chromomagnetic “Debye mass”, D ∼ × if it exists, is at most of O(g2T). Thus the “Debye screening length” in the chromomagnetic sector is much larger than l . D ∗) E-mail: [email protected] ∗∗) E-mail: [email protected] ∗∗∗) Itisworthmentioningherethat,accordingtohotQED,themagneticmassdoesnotappearin the“transverse-photon”sectortoallordersinperturbationtheory.2) Therefore,amagnetic-photon exchange interaction in an electron-positron-photon plasma is long range. †) The quark counterpart1) of the Debye screening length is much longer, specifically, lD ∼ 2.1×10−16 m for T =1 GeV. 2 H. Kohyama and A. Niégawa 3) The single-particle transverse momentum distribution of hadrons, which are produced by heavy ion collisions, exhibits a power-law tail. Boltzmann-Gibbs statistics is valid for the systems that have the following properties: (i)comparedtothesizeofthesystem,boththeinteractions andthememories are of short range, and (ii) the spacetime in which the system evolves is nonfractal. From the above observations 1) and 2), we find it doubtful that Boltzmann-Gibbs statistics can be applied in a strict sense to a quark-gluon plasma, at least in the early stages after a collision. It has been argued that the above observation 3) may be a sign of the fact that the object (a quark-gluon plasma or a hadronic fire ball) produced just after a heavy ion collision does not obey Boltzmann-Gibbs statistics (see, e.g., Ref. 3) and references therein). For a realistic description of a systems with long-range interactions, long-range memories and/or fractal structure, nonextensive generalization of Boltzmann-Gibbs statistics is essential. Such a generalization was proposed by Tsallis4) seventeen years ago (so called Tsallis statistics). Since then, a variety of works have appeared concerning theoretical aspects of this form as well as its applications to various nonextensive systems. The most important problem among the former is to de- termine the extent to which Tsallis statistics is unique among other nonextensive statistics. A short review of this problem is given in 2.1. Tsallis statistics has § been successfully applied to a number of nonextensive systems. Examples∗) are Lévy-type anomalous diffusion,6) Euler turbulence,7) the specific heat of the hydro- gen atom,8) peculiar velocities in galaxies,9) and self-gravitating systems and related matters.7),10) Some theoretical frameworks in Boltzmann-Gibbs statistics have been generalized to the case of Tsallis statistics, e.g., linear response theory,11) the Green function method12) and path integrals.13) Some approximation schemes, such as the (1 q) expansion,14),15) factorization approximation,15),16) perturbation theory17) − and the semi-classical expansion.18) Tsallis statistics contains a (real) parameter q, which is a measure of nonexten- siveness of the system, or more precisely, the range of interactions acting among the constituents of the system.19) For short-range interactions, standard Boltzmann- Gibbs statistics is realized, which corresponds to Tsallis statistics with q = 1. With the above motivation, in this paper we treat quantum field systems on the basis of Tsallis statistics assuming that q differs from 1 by a very small amount. More precisely, our region of interest is that satisfying 1 q O(1/(VT3)2) 1, (1.1) | − | ≤ ≪ where T is the “temperature” and V is the volume of the system.∗∗) (For 1 q | − | expansions in differentcontexts, see, e.g., Refs. 14) and 15), together with references therein.) ∗) For a comprehensivelist of references, see Ref.5). ∗∗) The volume of a heavy ion with mass number ∼ 200 is V ≃ 2.7×10−4 (MeV/~c)−3. (As mentioned above, at early stages after the collision, due to the Lorentz contraction, the volume of the produced quark-gluon plasma is much smaller than this V.) For T =200, 500 and 1000 MeV, VT3≃2.1×103,3.3×104 and 2.7×105,respectively. Quantum Field Theories in Nonextensive Tsallis Statistics 3 In 2, brief introductions to Tsallis statistics and the closed time-path (CTP) § formalism1),20),21) fortreating quantumfieldsystems aregiven. In 3, wededucethe § form for the CTP propagators in Tsallis statistics. In 4, we discuss physical impli- § cations of theresultsandpresentaprocedurefor computinghigher-order corrections to the propagators. §2. Preliminaries 2.1. Tsallis statistics Throughout this paper, we use units in which k = ~ = c = 1. Tsallis postu- B lates4) the following form for the generalized entropy (the Tsallis entropy): (1 Trρˆq) S [ρˆ] = − , (2.1) q q 1 − Here q is a real-number parameter and ρîs the density operator (Tr ρˆ= 1). In the limit q 1, Eq. (2.1) reduces to the standard Boltzmann-Gibbs-Shannon entropy → S = Trρˆlnρˆ. However, contrast to the case of Boltzmann-Gibbs statistics, the 1 − form for the entropy in the case we consider presently cannot be uniquely deduced. The Tsallis entropy (2.1) meets the requirement of “concavity”, the requirement which any entropy should satisfy.22) Since the introduction of the Tsallis entropy, dos Santos23) has demonstrated the uniqueness of the form appearing in Eq. (2.1) assuming pseudo-additivity, S = S + S + (1 q)S S (where ‘A’ and ‘B’ A+B A B A B − represent the systems in question, and ‘A + B’ represents the composite system of ‘A’ and ‘B’), together with several other conditions. Hotta and Joichi24) have shown that Eq. (2.1) can be “derived” from less restrictive requirements, namely, the composability condition, S = Ω(S ,S ) (with Ω being some function), and A+B A B the ansatz S[ρˆ]= C+Trφ(ρˆ) (with C a constant and φ some function of ρˆ), together with a few other conditions. Modified Tsallis entropies have also been proposed by several authors (see, e.g., Refs. 24) – 27)). The connection between S and the q theory of quantum groups has been pointed out and discussed, e.g., in Refs. 26) and 28). TheformofρîsdeterminedbygeneralizingtheprocedureemployedinBoltzmann- Gibbs statistical mechanics. There had been some disagreement regarding the defi- nition of the expectation value of an operator Aˆ, but this issue has been settled, and it is now known that this expectation value is given by the following:29) TrAˆρˆq Aˆ = , (2.2) h i Trρˆq which is called the q-expectation value and preserves various desirable properties. The form of the density operator ρîs determined by maximizing S [ρˆ] with the con- q straints Trρˆ= 1 and Hˆ = E, where Hˆ is the Hamiltonian. Introducing Lagrange h i multipliers, we easily carry out the maximization and obtain 1/(1−q) ρˆ= Z−1 1 (1 q)Hˆ/T , (2.3) q − − h i 4 H. Kohyama and A. Niégawa 1/(1−q) Z = Tr′ 1 (1 q)Hˆ/T . (2.4) q − − Here, Tr′ means that Tr is taken ohver the energy eigienstates with 1 (1 q)E/T 0 with E the eigenvalue of Hˆ. For 1 q < 0, this restriction obviousl−y do−es not ap≥ply. − As mentioned in 1, q = 1 corresponds to Boltzmann-Gibbs statistics, ρˆ e−βHˆ § ∝ with T 1/β the temperature. In the following, we simply refer to T in Eq. (2.4) ≡ as the temperature. (For a thorough discussion of the temperature of nonextensive systems, see, e.g., Ref. 30) and references therein.) Before moving on, for convinience, we rewrite the formula (2.2) for the q- expectation value as σˆq Aˆ = Tr′ρˆ′Aˆ, ρˆ′ = , (2.5) h i Tr′σˆq q/(1−q) (1−ǫ)/ǫ σˆq = 1 (1 q)Hˆ/T = 1 ǫHˆ/T (ǫ = 1 q) − − − − h 1/ǫ˜ i h i = 1 ǫ˜β˜Hˆ (2.6) − ǫ˜ h ǫ , i β˜ (1 ǫ)/T . (2.7) ≡ 1 ǫ ≡ − − 2.2. Closed time-path formalism For treating quantum field systems, we employ the closed time-path (CTP) formalism.1),20),21) In the single-time representation of the CTP formalism, every field becomes two fields: φ (φ ,φ ). Here, φ (i = 1,2) is called the type-i field, 1 2 i → and φ is called the physical field. Then, the n-point Green function consists of 2n 1 components. At the very end of calculation φ and φ are set equal. 1 2 In the following, we employ the complex scalar field theory governed by the Lagrangian density = φ†(∂2 +m2)φ+ with = (λ/4)(φ†φ)2. Gener- int int L − L L − alization to other field theories is straightforword. We restrict our consideration to the case in which the density operator is electrically neutral. 2.2.1. Single-time representation The 2n-point Green function, which consists of 22n components, is defined by G (x , ,x ;y , ,y ) i1···in;j1···jn 1 ··· n 1 ··· n n n = i( )nTr T φ(H)(x ) φ(H)†(y ) ρˆ , (2.8) − " c il l jm m ! # l=1 m=1 Y Y where the operators φ(H) and φ(H)† are the Heisenberg field operators, and ρˆ is the density operator. Note that, for Tsallis statistics, we have ρˆ = ρˆ′ [Eq. (2.5)]. Here, T is the “ordering” operator with the following properties: i) move the type-2 c (H) fields to the left of the type-1 fields, ii) rearrange the operators φ according to 1 (H) a time-ordering (T T), and iii) rearrange the operators φ according to an anti-time-ordering c(T→ T¯). 2 c → Here we summarize the Feynman rules for computing G perturbatively. The rules are the same as in the vacuum theory, except that the (bare) propagators and vertices take the following forms. Quantum Field Theories in Nonextensive Tsallis Statistics 5 1. Propagators (bare two-point functions) 1a) Two-point propagators: † i∆ (x y) = T φ (x)φ (y) , 11 − h 1 1 i φ1≡φ (cid:16) (cid:17) † i∆ (x y) = T φ (x)φ (y) , 22 − h 2 2 i φ2≡φ †(cid:16) (cid:17) i∆ (x y) = φ (y)φ (x) , 12 − h 2 1 i φ1=φ2≡φ i∆ (x y) = φ (x)φ†(y) , (2.9) 21 − h 2 1 i φ1=φ2≡φ where φ and φ† are the interaction-picture fields and Tr ρˆ with 0 h···i ≡ ··· ρˆ the “bare” density operator. 0 1b) “2n-point propagators” (2 n): The 2n-point propagators consist of 22n ≤ components, all of which are the same: (x , ,x ;y , ,y ), n 1 n 1 n C ··· ··· : φ(x ) φ(x )φ†(y ) φ†(y ): ,(2 n) (2.10) 1 n 1 n c ≡ h ··· ··· i ≤ where : : represents the operation of taking take a normal product and ··· ‘c’ stands for the contribution from the connected diagrams. is called the n C initial correlation. 2. Vertices The vertex factor for a vertex at which type-1 fields meet is the same as in the vacuum theory, i.e. iλ, while the vertex factor at which type-2 fields meet is − iλ. 2.2.2. Physical representation Let us introduce φ and φ (and their hermitian conjugates) through the rela- c ∆ tions 1 φ = (φ +φ ), φ = φ φ . c 1 2 ∆ 1 2 2 − Using these relations, the single-time representation outlined above can be trans- † † formed into the representation written in terms of φ , φ , φ and φ , which is called c ∆ c ∆ the physical representation. The 2n-point Green function is defined by∗) G˜ (x , ,x ;y , ,y ) c···c∆···∆;c···c∆···∆ 1 n 1 n ··· ··· m1 n m2 n = i( )nTr T φ(H)(x ) φ(H)(x ) φ(H)†(y ) φ(H)†(y ) ρˆ , −  c c i ∆ j c k ∆ l   Yi=1 j=Ym1+1 kY=1 l=Ym2+1   (2.11) The propagators and vertices in the Feynman rules are as follows. ∗) The normalization of G˜ hereis different from that of its counterpart in Ref. 21). 6 H. Kohyama and A. Niégawa 1. Propagators 1a) Two-point propagators: i∆˜ = 0, ∆∆ i∆˜ = i(∆ +∆ )/2 i∆ /2, cc 11 22 c ≡ i∆˜ = i(∆ ∆ ) i∆ , ∆c 11 21 A − ≡ i∆˜ = i(∆ ∆ ) i∆ , (2.12) c∆ 11 12 R − ≡ where ∆ is the retarded (advanced) propagator, which, in momentum R(A) space, reads 1 ∆ (P) = . (2.13) R(A) P2 m2 ip 0+ 0 − ± 1b)Initialcorrelations: Amongthe22n componentsofthe2n-pointpropagators ˜ (2 n), only the (cc c;cc c)-components are nonzero; n C ≤ ··· ··· ˜ (x , ,x ;y , ,y ) = (x , ,x ;y , ,y ),(2.14) n 1 n 1 n n 1 n 1 n C c···c;c···c ··· ··· C ··· ··· (cid:16) (cid:17) where is as in Eq. (2.10). n C 2. Vertices † † † † † † † † Thevertexfactorsfortheverticesφ φ φ φ ,φ φ φ φ ,φ φ φ φ andφ φ φ φ c c c ∆ c ∆ c c c ∆ ∆ ∆ ∆ ∆ c ∆ are iλ, iλ, iλ/4 and iλ/4, respectively. All other vertices vanish. − − − − 2.2.3. Forms of the propagators From this point, we restrict our attention to the case ρ = ρ (Hˆ ), with Hˆ the 0 0 0 0 free Hamiltonian. Then, the system under consideration is spacetime-translation invariant, and therefore can go to momentum space. To obtain the forms of the propagators, we first construct single-particle wave functions by adopting a fixed volume (V) quantization with discrete momenta, p = 2π/V1/3 n, where n = (n ,n ,n ), 1 2 3 with n ,n ,n integers. The complex scalar fields φ(x) are decomposed by using the 1 2 3 (cid:0) (cid:1) plane-wave basis constructed in this manner, 1 φ(x) = a e−i(Epx0−p·x)+b†ei(Epx0−p·x) , (2.15) (2E V)1/2 p p p p X h i a ,a† = b ,b† = δ , (2.16) p p′ p p′ p,p′ h i h i where E = p2+m2 is the single-particle energy. Then, Hˆ becomes p 0 p Hˆ = E a†a +b†b . (2.17) 0 p p p p p p X (cid:16) (cid:17) The form of ∆ (P) is obtained by substituting Eq. (2.15) and its hermitian ij conjugate for φ and φ†, respectively, into Eq. (2.9). Then, taking the large-volume limit, V , we obtain (after taking the Fourier transform) → ∞ (0) ∆ (P) = ∆ (P)+∆ (P), ij ij β Quantum Field Theories in Nonextensive Tsallis Statistics 7 ∗ 1 (0) (0) ∆ (P) = ∆ (P) = , 11 − 22 P2 m2+i0+ (cid:16) (cid:17) − ∆(0) (P) = 2πiθ( p )δ(P2 m2), 12(21) − ∓ 0 − ∆ (P) = 2πiN(p )δ(P2 m2), (2.18) β 0 − − where P2 = p2 p2 and N(p ) is the number-density function, 0− 0 N(p ) = lim θ(p )Tra†a ρˆ +θ( p )Trb†b ρˆ 0 0 p p 0 0 p p 0 V→∞ − TrNˆ(ph ). i (2.19) 0 ≡ Similarly, we obtain the form for [Eq. (2.10)] as follows: n C n d4P n l n = i 2πδ(P2)θ(p0) Nˆ(p ) Nˆ( p ) Cn (2π)4 i i h i0 − j0 ic ! Z i=1 l=0 i=1 j=l+1 Y X Y Y e−iP1·(xi1−yj1) e−iPl·(xil−yjl)+perms. × ··· i1<X···<il j1<X···<jlhn o eiPl+1·(xi′1−yj1′) eiPn·(xi′n−l−yjn′−l)+perms. . (2.20) × ··· (cid:26) (cid:27)(cid:21) Here, P (x y) = p (x y ) p (x y) and ( )isthesummation · − 0 0− 0 − · − i1<···<il j1<···<jl over all possible choices of i i (j j ) from among the values 1,2, n,subject 1 l 1 l ··· ··· P P ··· to the conditins i < i < < i (j < j < < j ). i′ < i′ < < i′ 1 2 ··· l 1 2 ··· l 1 2 ··· n−l (j′ < j′ < < j′ ) is obtained from 1,2, ,n by removing i < i < < i 1 2 ··· n−l ··· 1 2 ··· l (j < j < < j ). The first ‘perms.’ indicates that all permutations among 1 2 l ··· (j ,j , ,j ) are taken and the second ‘perms.’ indicates that all permutations 1 2 n ··· among (j′,j′, ,j′ ) are taken. 1 2 ··· n−l For Tsallis statistics [cf. Eq. (2.5) and (2.6)], we have 1/ǫ˜ q 1 ǫ˜β˜Hˆ σˆ 0 ρˆ = 0 = − . (2.21) 0 Tr′σˆq h i 1/ǫ˜ 0 Tr′ 1 ǫ˜β˜Hˆ 0 − h i This is invariant under charge conjugation, and therefore we have N(p ) = N(p ). 0 0 | | Equation (2.20) is simplified as n d4P n = i 2πδ(P2) Nˆ(p ) Cn (2π)4 i h | i0| ic ! Z i=1 i=1 Y Y e−iP1·(x1−y1) e−iPn·(xn−yn)+perms. , (2.22) × ··· h i where ‘perms.’ indicates that all permutations among (y ,y , ,y ) are taken. 1 2 n ··· 2.2.4. Gibbs ensemble AstandardGibbsensemblewithtemperatureT (= 1/β) andvanishingchemical potential is described by ρˆ = e−βHˆ0/Tre−βHˆ0. The initial correlation (2 n), 0 n C ≤ given in Eq. (2.10), vanishes. 8 H. Kohyama and A. Niégawa From Eq. (2.17) with Eq. (2.16), in the limit V , we obtain → ∞ Tre−βHˆ0 eP0βV ≡ −2 = lim 1 e−βEp V→∞ − p Y(cid:16) (cid:17) d3p = exp 2 V ln 1 e−βEp , (2.23) − × (2π)3 − (cid:20) Z (cid:16) (cid:17)(cid:21) Tra†pape−βHˆ0 Trb†pbpe−βHˆ0 N(p ) = lim = lim | 0| |p0|=Ep V→∞ Tre−βHˆ0 V→∞ Tre−βHˆ0 eβEp 1 −1 eP0βV = − eP0βV (cid:0) (cid:1) 1 = N (E ). (2.24) eβEp 1 ≡ BE p − Here isthepressureofthefreecomplexscalarfieldsystemandN isthefamiliar 0 BE P Bose distribution function. The factor of 2 in Eq. (2.23) corresponds to the number of degrees of freedom of the complex scalar field. A straightforward manipulation of Eq. (2.23) yields, for mβ << 1, Tre−βHˆ0 = e2×CBEV/(3β3), (2.25) π2 15 C = 1 (mβ)2 + . (2.26) BE 30 − 4π2 ··· (cid:18) (cid:19) More generally, for a system that consists of single or several kinds of bosons and/or fermions, we have Tre−βHˆ0 = exp n(i)C(i) + n(j)C(j) V/(3β3) , (2.27)  df BE df FD  i j X X    where i and j label the kinds of bosons and fermions, respectively. The quantity (i) (j) n (n ) is the number of degrees of freedom of the ith kind of boson (jth kind of df df fermion) and π2 15 C(i) = 1 (m β)2+ , BE 30 − 4π2 i ··· (cid:18) (cid:19) π2 7 15 C(j) = (m β)2+ . (2.28) FD 30 8 − 8π2 j ··· (cid:18) (cid:19) It is worth mentioning that when we replace Tr in Eq. (2.23) with Tr′ (for ··· ··· ǫ > 0) [cf. Eq. (2.21)], terms of O(e−1/ǫ/ǫ3) and O((βm/ǫ)2e−1/ǫ) appear in Eqs. (2.26) and (2.28). Such terms can safely be ignored for ǫ 1 [cf. Eq. (1.1)]. ≪ §3. Computation of the propagators In this section, we compute the propagators on the basis of Tsallis statistics. Quantum Field Theories in Nonextensive Tsallis Statistics 9 3.1. Preliminaries σˆq, appearing in Eq. (2.21), is expanded as follows: 0 1 σˆq = exp ln 1 ǫ˜β˜Hˆ (3.1) 0 ǫ˜ − 0 (cid:20) (cid:21) (cid:16) (cid:17) = e−β˜Hˆ0exp ∞ ǫ˜l(β˜Hˆ0)l+1 − l+1 " # l=1 X = e−β˜Hˆ0 ∞ ∞ 1 ǫ˜l(β˜Hˆ0)l+1 n . (3.2) n! − l+1 " ! # l=1 n=0 Y X The region of interest is that defined by Eq. (1.1). In the following, we compute field propagators up to and including O(ǫ) terms, with ǫ O(1/(VT3)2). For this q ≤ purpose, for σˆ , we should employ the form 0 ∞ 1 ǫ˜ l 2l ǫ˜2 2l+3 ˜ǫ4 2l+6 σˆq = β˜Hˆ β˜Hˆ + β˜Hˆ 0 l! −2 0 − 3 0 18 0 Xl=0 (cid:18) (cid:19) (cid:20)(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) ǫ˜3 β˜Hˆ 2l+4+ e−β˜Hˆ0 0 − 4 ··· (cid:21) (cid:16) (cid:17) = ∞ 1 ǫ˜ lβ˜2l ∂2l + ˜ǫ2β˜3∂2l+3+ ˜ǫ4β˜6∂2l+6 ǫ˜3β˜4∂2l+4+ e−β˜Hˆ0 l! −2 β˜ 3 β˜ 18 β˜ − 4 β˜ ··· l=0 (cid:18) (cid:19) (cid:20) (cid:21) X e−β˜Hˆ0 (3.3) ≡ Dβ˜ and for βN∂N Tre−βHˆ0 /∂βN, the form (cid:16) (cid:17) ∂N ∂N βN Tre−βHˆ0 = βN e2CBEV/(3β3) ∂βN ∂βN ( )N = − (2C V)N +2N(N 1)(2C V)N−1β3 β3N BE − BE (cid:20) 2N(N 1)(N 2)(3N 4)(2C V)N−2 + − − − BE β6+ e2CBEV/(3β3). 3 ··· (cid:21) (3.4) Here, the terms represented by “ ” yield at most O(ǫ3/2) contribution to the prop- ··· agators. In the reminder of the paper, for simplicity, we restrict ourselves to the massless case, m = 0, and therefore we have C = π2/30 and E = p = p. Generalization BE p | | to the massive case is straightforward. Equation (3.3) and the remark made at the end of 2 guarantee that Tr′σˆq Trσˆq. Then, using Eqs. (3.3), (2.25) and (3.4), we get § 0 ≃ 0 Trσˆq = Tre−β˜Hˆ0 0 Dβ˜ 10 H. Kohyama and A. Niégawa ∞ 1/2 1 ǫ˜ = ( y˜)l 1+4 2l(l 1)+l l! − { − } 2y˜ " l=0 (cid:18) (cid:19) X 8 ˜ǫ + l(l 1) 6(l 2)(l 3)+23(l 2)+12 3 − { − − − }y˜ 4y˜ ǫ˜y˜ 1/2 8y˜ 4y˜3 2l(l 1)+7l+3 ǫ˜+ ǫ˜ y˜2ǫ˜+ e2CBEV/(3β˜2) − 3 2 − 3 { − } 9 − ··· # (cid:18) (cid:19) 2 100 131 = 1+ 2y˜ǫ˜(5y˜ 3)+ǫ˜y˜ y˜2 y˜+24 + e−y˜e2CBEV/(3β˜2), 3 − 9 − 3 ··· (cid:20) (cid:18) (cid:19) (cid:21) p (3.5) where 2 y˜ 2C2 ǫ˜ VT˜3 . (T˜ = 1/β˜) (3.6) ≡ BE (cid:16) (cid:17) 3.2. Two-point propagator ∆ˆ (i,j = 1,2) i,j The propagator ∆ (P) is given by Eq. (2.18), with N(p ) = N(p) for ij 0 | | p=|p0| the number-density function N(p ). Then, from Eqs. (2.23) and (2.24), we have 0 lim Tra†a e−βHˆ0 = lim Trb†b e−βHˆ0 p p p p V→∞ V→∞ 1 = e2CBEV/(3β3) = N (p)e2CBEV/(3β3). (3.7) eβp 1 BE − Using Eqs. (3.3), (2.25) with m = 0 and (3.7), we obtain lim Tra†a σˆq p p 0 V→∞ = N˜e2CBEV/(3β˜3) Dβ˜ ∞(cid:16) (cid:17) 1/2 1 ˜ǫ = N˜ ( y˜)l 2lβ˜p(1+N˜) l! − 2y˜ l=0 (cid:20) (cid:18) (cid:19) X 1 ǫ˜ + 8l(l 1)(2l 1)β˜p(1+N˜)+l(2l 1)(β˜p)2(1+N˜)(1+2N˜) 2 − − − y˜ n o 2(2l+3)β˜p(1+N˜)ǫ˜y˜+ e2CBEV/(3β˜2)+N˜Trσˆq − 3 ··· 0 (cid:21) 1/2 ˜ǫy˜ = N˜e−y˜ 2β˜p(1+N˜) +4β˜p(1+N˜)y˜(3 2y˜)ǫ˜ − 2 − (cid:20) (cid:18) (cid:19) + (β˜p)2(2N˜2+3N˜ +1)(2y˜ 1)ǫ˜ 2β˜p(1+N˜)y˜(3 2y˜)ǫ˜+ e2CBEV/(3β˜3) 2 − − 3 − ··· (cid:21) +N˜Trσˆq, (3.8) 0