PHYSICAL REVIEW D VOLUME 46, NUMBER 8 15 OCTOBER 1992 Finite-temperature radiative corrections to Compton scattering S. Varma Department of Physics and Center for Particle Physics, The University of Texas at Austin, Austin, Texas 78712 (Received 24 February 1992} We calculate the finite-temperature a' corrections to the differential and total Compton scattering cross sections. %e find that it is necessary to include soft photon absorption and emission diagrams to cancel the infrared divergences in the cross section. The cross sections are evaluated for the case where the initial electron is at rest. PACS number(s}: 12.20.Ds I. INTRODUCTION to p =(mo+mr) because the error induced is higher order than u . There have been many calculations of finite- We have a choice of formalisms to work in as, at the temperature corrections to various processes because, as one-loop level, the real-time and the imaginary-time forexperiments and observations have become better and malisms are equivalent. In the imaginary time formalism produced more accurate results, it has become corre- the energy is treated as a discrete variable and one sums spondingly necessary to make more and more exact cal- over energy and integrates over momenta without modiculations for these processes. In addition, theoretical in- fying the propagators. In the real time formalism energy vestigations of the early Universe have also necessitated is a continuous variable and one integrates as in zerothe calculation of finite-temperature corrections. There- ternperature field theory but with a modification to the fore, it is important to calculate at what temperatures T- propagators. We shall work in the real time formalism dependent corrections become of sufficient magnitude to because the finite-temperature pieces are explicitly cause appreciable corrections to conventional zero- separated out from the zero-temperature ones. This has ternperature results, and how large these corrections are. the added advantage that the ultraviolet divergences will For example, some authors have calculated the finite- be contained in the T =0 part of any diagram. Since the temperature corrections to the renorrnalization constants T =0 parts of the diagrams are already known, we need of QED [1—4], some have calculated helium abundance to only calculate the TAO parts of these diagrams and in the early Universe [5,6], and some have calculated the will thus not need to worry about ultraviolet divergences. finite-temperature corrections to the anomalous magnetic In the real time formalism, the tree-level electron propmoment of the electron [2,7 —8]. agator in momentum space is However, there have not been many calculations of the corrections to fundamental scattering processes. Such &(p) = l —2tr5(p m)(gf+m)n—F(E ), (1.1) processes are interesting, not only in themselves, but also P —m +i@ as parts of other phenomenona. In this paper we shall where calculate the T-dependent O(a ) corrections for one such 1 process, namely, Compton scattering, for which the nF(E )= (1.2) zero-temperature corrections were calculated by Brown e~@'+ 1 and Feynman [9]. The Compton scattering cross section, is the Fermi-Dirac distribution whose effects can be apart from its intrinsic interest, is useful in the calcula- neglected, and E~ = +p + m . We are using units of tion of several processes such as calculating stellar opaci- temperature where Boltzmann's constant is set equal to ty [10], and calculating relic radiation [11]. unity and the metric is (+ ———). The tree-level photon For simplicity we shall work at temperatures propagator in the Landau gauge is T= 1/P((mo, where mo is the electron mass at zero temperature. This means that we can neglect al1 FermiD~p (q)= —g"' +2tr5(q )ns(E ) (1.3) —Pmo factors because they wi11 be suppressed by a factor Dirac ' g +I E e as compared to the Bose-Einstein factors. Neglecting these factors leads to the results being gauge where nonin variant but, as these effects are exponentially suppressed, one need not be concerned about them. Note nit(E )= PE (1.4) that at these temperatures there are no finite-temperature e ' —l corrections of order a because the only possible correc- is the Bose-Einstein distribution and E = q ~. We shall ~ tions could come from the fermion propagators. Further- adopt this gauge for a11 the following calculations. The more, we shall take the on-shell mass condition to be Feynman rules and diagrams here are exactly the same as p =m o. For the purposes of this paper, this is equivalent in the zero-temperature case, but with the replacement of 3630 1992 The American Physical Society FINITE-TEMPERATURE RADIATIVE CORRECTIONS TO. . . 3631 these propagators for the zero-temperature ones. Note that the propagators explicitly split into zerotemperature and finite-temperature pieces making it easy to separate out the finite-temperature contributions. Note also that the Bose factor in Eq. (1.3) leads to extra infrared divergences. These are canceled, as usual, by the three-body diagrams but with the proviso that both emis- p-k sion and absorption diagrams be included. To avoid excessive tedium we shall not show intermediate results but FIG. 1. The electron self-energy. ™ will collect them all together at the end. The organization of the paper is as follows. In Sec. II we describe the various corrections to the cross section. In Sec. III we calculate the finite-temperature corrections to the mass and wave-function renormalization constants. 2e (2n') ~, fd k (p —k) —m na(k)5(k ) . (32) In Sec. IV we calculate the corrections due to the finite- Since we wish to calculate the mass shift 5m&, we evalutemperature renormalization of the a scattering cross ate P —m — Xr =0 and thus must put Eq. (3.2) on shell to section. Section V deals with the internal insertion diaget grams. In Sec. VI we calculate the three-body cross section necessary to cancel the infrared divergences appearing in these corrections. We present and discuss our results in Sec. VII. Xr=, fd k5(k )na(k) pk =I (2n ) (3.3} from which one obtains II. CORRECTIONS TO THE CROSS SEt.flON i0( ) 7TQ p + Ipl (3.4a) 1 ' 6O'Ipl p'-Ipl There are several corrections of order a that need to ~m' ~m'p' p'+lI be considered. They are the following. I I (3.4b) (1) Corrections to the Klein-Nishina cross section due 3O'I pl' 6O'Ip I' p' —Ipl to finite-temperature mass and wave-function renormal- Therefore, ization. These are due to the external line insertion of the electron self-energy diagram, Fig. 1, which produces a = (p'}' —lpl'= m,0'+ m'— (3.5) corrections to the cross-section. (2) Internal insertion diagrams shown in Fig. 3. and (3} Inelastic, three-body (bremsstrahlung} processes. We must include these to cancel the infrared divergences 5m (3.6) occurring in the first two sets of corrections. The key 3P m point to note here is that electrons can be emitted to and absorbed from the heat bath. Thus we must include both Note that the mass shift is independent of momentum. absorption and stimulated emission diagrams, and these This is a somewhat surprising result since blackbody rawill successfully cancel the infrared divergences [3]. diation defines a preferred frame; that it obtains is due to Stimulated emission modifies the density-of-final states our neglect of the Fermi-Dirac factors in the propagafactor [3] to tors. A detailed calculation shows that momentum dependence occurs only through those terms [3]. We define the finite-temperature correction to the (2n ) 2' [1+na(Eq )] (2. 1) wave-function 5Z2 =d Xr ldgf ~& renormalization This gives u.s constant Zz as for bosons. Again, there is no need to include finitetemperature mass or wave-function renormalization effects. 5Z2r= 2m f d k 5(k )n~(k) .k) ( (3.7) e (3.8) III. MASS AND WAVE-FUNCTION 2' RENORMALIZATION where write Fig. 1. We Let us consider the electron self-energy, X=X0+Xz-, where X0 is the usual T=O contribu- I„= f0~dk n~(k) . (3.9) tion. The expression for X is —iX=( ie) f d4k —y"S(p k)y"D" (k) . (2m } (3.1) — IV. CORREiwONS TO THE KLEIN-NISHINA CROSS SECTION If we neglect the Fermi-Dirac factor, the TAO part of X Figure 2 shows the Klein-Nishina diagrams. To sim1S plify the calculation, we define the quantities 3632 S. VARMA a shift in the renormalization factors; i.e. , we have doKN~doKN(Z2~Z2+5ZO m~ 0+5m@) . (4.9) In the Klein-Nishina cross section we need (Z2) . But Z2 has already been taken care of by the zeroP, P, P, temperature renormalization so we have (1+5Z2 ) =1+25Zz multiplying do. ~N. The total correction to do. ~N is then d [do'KN(m0)) d~KN=«KN(mo)+ 5mp dmp + 25Z qd o KN( m } (4.10} P, P, P, giving us two temperature corrections: FIG. 2. The Klein-Nishina diagrams. ma d [do KN(m0)] (4. 11) 3p m0 dm0 (4. 1) and ~=P j '02 =P2 q» (4.2) e I„ d CTKN( m ) . (4. 12) &=P&'P2 (4.3) To this order we can put m = m0 in Eq. (4. 12). ~ P3=P&+q& =P2+q2 ~ (4.4) P4 Pi q2 P2 (4.5) V. INTERNAL INSERTION DIAGRAMS In terms of these quantities the matrix elements are — =R Figure 3 shows the diagrams obtained from insertions 2K &e u(pz)y"(P +3m)y u(p, )— (4.6) on internal lines. To get a correction, we will take the interference of these diagrams with the Klein-Nishina di- —eI. agrams. However, since we are not including the u (p2 ) y "(A/4+ m )y"u (p, ) =— S (4.7) modification to the electron propagator, we need only 2r consider the interference of these diagrams with the and, summing and averaging over polarizations and zero-temperature Klein-Nishina diagrams. To this order, spins, we get we can again put m =mp and Z2=Z2. The expressions M„N =-,' IR +Sl' . (4.8) for the interference of these diagrams with the KleinNishina diagrams are (after summing and averaging over The temperature corrections to do KN are produced by spins and polarizations) I 3 2Re[J|&(R*+S')]= 16m f d k 5(k )na(k) XTr (g/z+m)y (gf2 —'+m)y"(P3 —g+m)y "(gf& —k'+m)y (gf&+m) It, X —y (f3+m)y„+ — 1 1 y„(g/4+m)y„ (5. 1a) K 3 2Re[L, &(R*+S')]= 16m f d k 5(k )na(k) (p3. k+~) a XTr (f2+m)y"($3+m)y (p3 —k'+m)y ($3+m)y'(P&+m) X K 1 „+— y(gf3+ m )y— 7. y„(f4+ )y, m 1 (5. 1b) 46 FINITE-TEMPERATURE WQ)IATIVE CORRECTIONS TO. . . 3633 2Re[KI &(R +S )]= CX Jtd k5(k )nz(k) K 16m pl k p3'k+K XTr (pz+m)y"(f3+m)y (p3 —k+m)y"(p, —I+m)y (p, +m) —y, (p'3+ )yq+ — 1 1 X m 7. y„(p'g+ m )y„ (5.1c) IC 2 Re[K", &(R '+S')]= f d k 5(k )nz(k) Kp2 k 16m p3 k+K XTr (P'z+m)y (P2 —k+m)y"(P3 —k+m)y (f3+m)y "(P', +m) X —y„(g/3+m)y„+ 1 y— 1 „(/~+ m)y„ (5. 1d) K (b) P, P, P, P, P, P, P, P, P P, '( I/ 1 P P, P, :42' P, P, P, FIG. 3. The interna1 insertion diagrams. 3634 S. VARMA We now let p3~p4 and ~+ iin Eqs. (5. 1) to account Let us investigate the range of validity of the approxifor the flipped diagrams in which emission takes place be- mations (5.2) and (5.3). First, we define the dimensionless fore absorption. The total contribution is then the sum of variables kp= Ilail I'mo coo=~/~0 and coo'=co'/mo, all these diagrams. In practice, one can just evaluate Eqs. where cu and co' denote the incoming and outgoing pho- (5. 1) and do the substitutions at the end. ton momenta, respectively. %'e shall assume that cop is To evaluate the integrals, we expand for large ~ and ~ large and choose it to be cop~100. We wish to 5nd the in the following manner: maximum value of the variable kp such that the approximation is still valid. Writing p3-k as p, k+q, k, inp3 k (5.2) tegrating out the 5 function, changing to dimensionless p3 k+K K variables, and applying the conservation of four1 1 5'4'k momentum we get (5.3) p4 k+r COp and also go into the rest frame of the electron where kp &( (5.4) 1+coo —co& cos[8] p, =m and p, =0. (a) (b) k P, p-k 1 p-k 3 P, P, P, p+k P, {c) P, p-k P, P, p-k 4 P, (e) P, P, p+k P, P, p-k 4 P, q q FIG. 4. The inelastic diagram&. FINITE-TEMPERATURE RADIATIVE CORRECTIONS TO. . . 3635 and similarly, writing p4. k as p, k —qz k we can derive which exactly cancels all the infrared divergences and I leaves us with finite integrals. COO ko (5.5) 1 + ci)o +coo cos[6}] VII. RESULTS where 8 is the scattering angle. For the worst case of It is important to note that we have only calculated the (5.2) we find from Eq. (5.4) that ko «0. 495 for coo= 100. finite-temperature corrections to Compton scattering to The worst case scenario for (5.3) for coo large is diFerent. order a . These corrections tend to reduce both the This is because we have the relation co'=moco/ differential and total cross sections. But, as the tempera- [rno+co(1 —cos[8])]. Putting this relation into Eq. (5.5) ture goes up, the a terms become more and more imporand using the worst case scenario we find that, for the ap- tant and, at a certain point, will start to increase the cross proximation to be valid, we must have kp«0. 254. section again. If one was to forget about these terms enThus, in general, this condition must be satisfied for the tirely, one would find that the cross section would seemapproximations (5.2} and (5.3} to be valid together. How- ingly become negative at higher temperatures because the ever, as long as the temperature T ((mo then, as soon as finite-temperature a terms would completely overwhelm the zero-temperature terms. Therefore, we have cut off kp attains any appreciable value, the integral will become exponentially suppressed. Therefore, if we restrict our- our calculations when the a terms reach 20%%uo of the selves to temperatures T «0 (mo }, any values of ko Klein-Nishina terms. At this point the a terms are still quite small being a few percent of the Klein-Nishina above or close to 0.254 will be inconsequential, and the approximations (5.2) and (5.3) will be valid over all angles terms. 8 and for all values of coo & 100. The explicit expression for the correction to the differential cross section is given in the Appendix. Here we plot it for interesting values of the temperature T, VI. INELASTIC DIAGRAMS momentum cutoff E;„, and incoming photon energy co. Figure 4 shows the inelastic scattering diagrams. Let We also calculate the correction to the total cross section us consider these terms for small k. In this case, the only by numerically integrating the correction to the contributions are from the terms I, II, V, and VI because differential cross section. This is plotted in the same way the terms III and IV are negligible. Thus, with k small, as above. However, the expression has the usual forward we get scattering divergence. Therefore, we exclude a cone of angle 0.01 radians in the forward direction. (I)= ep& 6 (k)R, (6.1a) In each plot the corrections are expressed as a fraction pi k of the zero-temperature Klein-Nishina results. The ternperatures may be expressed in terms of degrees kelvin us- (II)= ep& 6 (k)S, (6. 1b) ing the relation m0=5. 93X10 K. p, k Figures 5 and 6 show the corrections to the differential —ep~ cross section versus the cosine of the scattering angle at (V)= e (k)R, (6. 1c) temperatures T =10 mo and T=10 mo, respectively, p2 k for various values of co. As co increases, we get a larger ep2 (VI)= e (k)S . (6. 1d) p2. k cos 8 For these diagrams, we calculate the square of the mau = 500mp trix element and sum over spins and polarizations. Then we let k~ —k to account for the spontaneous emission u = 1000mp process and add this back to the original matrix element. Finally we integrate the resulting expression over the finite-temperature phase space, u = 1500mp fo min (2~)3 2k [I+n~(k)], where E;„ is the energy resolution of the experiment. ur = 2000mp We choose E;„((mo and thus, as it is much smaller than all the other momenta in the problem, we neglect its efFect on the momentum balance; i.e., we assume that the soft photon is emitted independent of q2. The result for ~ = 2500mo the three-body contribution then becomes -0.2" dcrT/d~ der 0/d~ d (2~)3 fo 2k 4 ~R+S~ p, .k p2-k FIG. 5. Corrections to the differential cross section for X [1+n&(k)] (6.2) T 10 Woy Emin = 10 mo ~ 3636 S. VARMA cog g og10 ( ) 47 = 1ppp 0. 02 0. 04. -O. p6. ~T -o. o8 = 2000mo -0, 1. 0. 12 = 2500m, -0. ]4.. I 1000 500 10p 5p 10 differentia] FI& 6.' o rectjons tp th cross ion for FIa 8. orrecti1pns to to taf te sectipn for ]Q 4 I;„=j Q — 0 ~ i T= jQ for va r1ous tern peratures T( Q ~i ' &,~ c 1pn as a funct' of the inccpm jng ph oton ener 'gy, and ar ger correction re aching a b ogt 9% for 5ppm 0 at T = «p —3m o ~e also e]ear«y see that ' method fo r ca«cu«at jn cprrecti o"s go as T2 Further 1t 1S jn teresting th at the ~ cprrections b ecome more and m «e import ntasone goes to backseat tering. ' exa - ' 1gures7 9 s&pw th e cprrections t the gota cross sec- « on versus ]og&o{~/'m ) t various t emperatgres with t jpn and 8 shoow a jon with tern peratgre th an those for differeent ways of ch oQsjng Em~& Fjggre 7 sho s the ca e . 1g. 9 where Em&n. = «P 2 . the mom entum cu tPff 1s fixe ' d f th etem ePerature or th e incpmjnng momen- CKNQWZ. ED GMENTS turn. F1gure g sh ows the case whe re ™~~~T=101e., ' the m mentum Qff jis temper a t ure de pendent and The a gthQr wQ «ike to thank . som e «aetio&n of the th erma1 ener'gy. F1ggre 9 ows the - 'e, case where Em jg. = «p 1 0~ . . the mo entum cutoff; ran Ye'hudaj ' agthor . ~ . show»comjn Some fr ~~tion of th "gures pendence on th e chojce gy These gg gest, o f constrgct, F ( G Sudars cr1t1c1 and 10g1( ( 4J ) g10( —) -0. p2. -0. p2. -0. p4. -0. 04. ~T -0. 06. ~T -0. 06. ~cr -o. os. - o. os. -0, 1. -0. 1. o ~ 12. -0. 12. -o. 14 -0. 14. 50 10 12 108 6 4 ' FI~ Correc 1pnS tp 1 section FIG — tp t e tot Correct 1pnS cross — section =]Q2it]0 fpr var&ous tern I eratures T( &Q o ), as a func- &~;„=)Q io M for 1ous tern eratures 7( ~Q 6 t1pnofthe e1ncomjng h energy ~ t1on of thee incom1ng p h oton energy, co ~ 46 FINITE-TEMPERATURE RADIATIVE CORRECTIONS TO. . . 3637 APPENDIX The explicit expression for the difFerential cross section is a[ — 4aoio —Sa, i1+2a2(i2+i2)+2a2(i4+i5) 4a4(i6+i7) 4a5(i8+i9) — 4a6(i, o+i11)+2a7(i12+i,2)+4aIIi, 4 +4a9i, 5 4— a, z(i 20+i 2, )+4a 14i 22+4a, 5i — a, oi, 6+ 8a»i, 7+4 a, 2(i, 8+i » ) 4— 4a 16(i 24+i 25 ) —4a, z(i26+i 27 ) ], 2& where 4 4 2 2 4 7m 02 ap= 3E 2E' 2 3+ 4+ 4+ 3+ 3 —7mp3 —4emp3+ 4mp3+ 3g 2mp 2mp 3& 2&2 K Km 0 Km 0 K K T T KT T KT KT ICT 2e 3m o 2e 3E 4am 0 3mo 4mo + moT KT K T K T KT moT K 7 K 'T K 'T —5g 2 2 4~m 02 + 2p + 3p + &m 0 +-&m 0 56 4E 36 2E m Km K 7 K7 m 07 Km 07 K7 a2 = — 2 —4 K 2 +—+ 3e Kmo 2 T 2 KT KT mo 2 + 3e2 + KT IC T 6 mOT 4 2 KmOT 4e 2 mo K T 2 6e 12 8e 2 14m p 4mp 3p 5m p 4g ]5p 2 K3 + K2 K2m 2 2 K3 + K4 K2T KT 0 m0 K7 KT K T 8emp 15mp 8mp + 7 K mp7 Km@7 K 7 K 7 K 7 a4=-1 1 a5= 1 a6= 1 —— 1 K Km 0 Km 0 T mT 4m o4 14m 0 8pm 0 8m 0 4e2 + ]2 + ]5g 8E' 6E' a7= + 6g + + 'T 7 K7 T KT KT T KT moT KmoT 15mp 2 2 2~ 5m 0 KT K 7 K7 mp'T Kmp'T K 'T 7 3p 2 2mp 4E 3 2E' 2E' m0 K4 K4m 02 K3m 02 K5 K7 K7 Km@7 Km@7 K7 26 —1 —+ 6 2mp 2 —1 2e 2e a9= K3 K2 E K3m 2 + K2m6 2 K27 + K27 a]0 0 0 ICT KmoT K T K moT 2 2 mo g 2 p2 mo 1 1 1 ICT ICmoT K7 K T KT K m OT K T KmoT Ic mO'T K T m o2 7 4e 3e 2e 2 o 2 3 2 2E' 14 4 2 4 75 7 K74 2 4 m07 Km07 K7 KT m 11T Km 2 2E' ] p p 2mp a, 5= + 73 m 02 73 KT moT KT moT KT F, 2' g) lp=, = 2~' l( = 2mFiIi 3p2 molp2I * m 3P'lp, m. l ' ' 2mp 3P'm, i4= 3p + 3p mo 0 3 2+3 ~ e2Fi 3 7T P2 q2 p2F, 4m mo(mo+q, ) 3p' 3p'Ip, l 3P2lp l2 41T'ps(mo+q1)' 4~'p z q11' I 81T'(m, + q01 )p2. q, 10 6(q 1 'q2 'Ql q2) 15P-mp 45p mp 45p mp 3638 S. VARMA 3 2& g2 11 7(q 1 'q 2 ql q2) 12 3P mo —qzo ql~ —qz) 47r 5 47r 5 113 o 15(ql~ i14= (mo+q1o )z + Iqll z 15 45 7r3(mo+qol)2F, 47r (mo+q, ) 27r (mo+q, )pzF, 115 3&'Ipzlmo 3&'molpzl' 3&'Ipzl'mo lqll 7r Fl 7r pz 7r (pz) Fl m F1 2 2 2 2 3 6p Ipzlmll 2p Ipzl mll 4p Ipzl mll 12P'Ipzlmo 3 0 3(+0 )2F ~3F + (P2 ql) 2 4 + 2 2 2P Ipzl mo 4P pz mo I I 5 12P pz 3 mo I I 47r5pzmo 47r pzqol 47r5pz q, 47r q, mo 47r (q, ) 4n lq, l 16 15P' 15P' 45P 15P' 15P 45P' i = 2n + mo~F, + 2~pe 1.i—— (q ~ — —1 i — o q,o q —q), 2m 117r 2 3P 3Pzlp I 3P m 5 4~5(poz)' 4~'Ipzlz 4~'Jl zq 1 4~'pz ql 4~'Jp2(q 1 )' 4~'ps lqll' L19— 1 sp 45p 15P 45P' 1 sp mll 45P'mo 45P'mll 15P'mo 45P'mo 45P'mll 47r qz(mo+q, ) 47r qzlq, I 87r (mo+q, )q, qz l 20 114 15P'mll 45p mo 45P'mll ~3P2'q1 P2OF Z21 15P' 15P' 45P 15P SP 3P pz I I 3P'Ipzl' Ipzl 2~'(q l )'F 1 4~'(q l )'P2 ql 2~'(q 1 )'Pz qd 2F 1 15P'Ipzl SP'Ipzlz sP'Ip, l' 27r'F, 27r p 7r (p ) F, 7r F, 4SP Ip I 1SP Ip I 15P'Ip I 45P'Ip I —27r5pz 7r (pz) F, n' Fl +(Pz ~1)' 5P'Ip, l' SP'Ipzl' 1sP'Ip, l' 2~5 ~'i 2F S 5(+0 )2 27r pz I I 7r'(p 2 ) F, 5lpzl' 15P' 15P'Ipzl 2lp, l' 25P' 25P' 15P 45P' 15P'Ip, l 5(pz ql)' 27r'I pzl' ~'p 2 lpzlF1 27r'(@ 2 )' 27r'lpzl' 7r'(p2)'F 1 Ipzl' 25P' 25P' 15P' 45P' 15P'Ipzl ~5(P02)2F, 15P'Ip I ' ' 45P'Ip I 15P'Ip I' 15P'Ip I' 45P'Ip, l —27r pz 7r (p ) F m5F, SP'Ipzl" SP'Ipzl' 15P'Ip, l' 0 16m 5 moq1p2-q1 5 0 0 Sm moq1p2. q1p2F1 15P Ip I 15P'Ip I 0 0 ~ ~ 0 0 22 14(q 1 q2 ql qz) 23 15(q 1 0 0 0 0 ql~ ~ ~ 24 8(q 1 qz qz) 125 i9(q 1 q2 ql qz) FINITE-TEMPERATURE RADIATIVE CORRECTIONS TO. . . 3639 0 0 ~ ~ 0 0 26 20(q 1 92 ql q2) 27 21 01 '92 ql l2) in which we have defined P', = —In[(Pz+ ~Pz~)/(Pzo —(Pz~ )] and I= f o" (dklk)ntt(k) .Note that the infrared divergence in integral I, is canceled by Eq. 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