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Calculational rules for finite temperature reaction rates
Volume 247, number 2,3
PHYSICS LETTERS B
13 September 1990
Calculational rules for finite temperature reaction rates A. N i 6 g a w a Department of Physics, Osaka City University, Osaka 558, Japan Received 27 May 1990
Systematic calculational rules are found for evaluating the reaction rate of a generic process taking place in a thermal reservoir in equilibrium. The rules are formulated in a diagrammatic representation within the framework of a real-time thermal field theory. It becomes clear how each "forward" diagram can be cut; thus the finite temperature generalization of the Cutkosky (or cutting) rules is settled. The relationship is clarified between our diagrammatic rules and those for finding the imaginary part of the relevant "forward" amplitude established by Kobes and Semenoff; the latter is shown to contain different reaction rates.
1. Recent e x p e r i m e n t a l studies on q u a r k - g l u o n p l a s m a f o r m a t i o n in heavy ion collisions have stimulated theoretical studies o f various reactions taking place in a heat bath in equilibrium [ 1 ]. At zero t e m p e r a t u r e T-- 0, any reaction rate can be calculated through the imaginary part o f the relevant forward amplitude, which can be conveniently calculated by employing the Cutkosky or cutting rules. On the other hand, at T ¢ 0, up until now, although the d i a g r a m m a t i c rules o f finding the imaginary part o f an a m p l i t u d e in real-time thermal field theory ( R T F T ) [2,3 ] have been settled by Kobes and Semenoff [4,5 ], the relation between reaction rate (s) and the imaginary part o f the corresponding " f o r w a r d " a m p l i t u d e is not yet clear except in the simplest case o f heavy particle decay [ 4 - 9 ]. In this letter we clarify the relationship between them, which provides us with an efficient calculational scheme o f finite t e m p e r a t u r e reaction rates. We take the following intuitive approach: Let us consider a set o f systems forming a grand canonical ensemble and calculate the reaction rate for the process under consideration taking place in one system belonging to the set, and then take an average over the ensemble. Through this procedure we can arrive at the d i a g r a m m a t i c rules for calculating the reaction rate: The reaction rate o f a particular process at T ¢ 0 is given by a physically apparent subset o f the set o f " c i r c l e d " diagrams introduced in ref. [5] (referred to as KS diagrams hereafter) which as a whole represent diagrammatically, through the discontinuity formula, the imaginary part o f the relevant forward R T F T a m p l i t u d e (with physical external legs). The imaginary part as a whole is shown to contain different reaction rates. A n i n t e r p r e t a t i o n o f any " f o r w a r d " KS d i a g r a m in terms o f reaction a m p l i t u d e s (cutting rules) is given. We also briefly present a n o t h e r m o r e elegant but less intuitive derivation o f the same result. 2. F o r definiteness, we e x a m i n e the decay ~1 o f a Higgs scalar tb (mass M ) i m m e r s e d in a heat bath o f thermal light neutral scalars ~ (mass m ) . The techniques and results, however, are general a n d can be a p p l i e d to any reaction. (See sections 3 and 4 below. ) The interaction lagrangian o f the system is Lint = 24 3 / 3 ! + ½g02tb up to a counter lagrangian. It is convenient, for our purpose, to restrict the space to the interior o f a large cube with v o l u m e V. We pick up one system out o f those forming a grand canonical ensemble; its initial state is specified by a Fock state ~J The decay processes themselves have already been treated extensively at the one- and two-loop levels. See, e.g., refs. [5-11 ] and earlier references cited therein. 0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )
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Volume 247, number 2,3
PHYSICS LETTERS B
13 September 1990
I n , = 1; {nl} ) in, where nl stands for the number of ¢ with the plane wave mode l ( = 0, _+ 1, _+2 .... ). Final states are specified similarly. Constructing the transition probability W ( n ~ = 0; {n} } I ne = 1; {nz} ) between these two states, summing over final states and averaging over the systems belonging to the grand canonical ensemble, we get, for the decay rate Fd of of, Fd = 1 2t,,} exp(--flX,niEi)~¢n~) W(n~ =0; {n~} [ n ~ = 1; {nt}) 21r5(0) 3~{,,} exp(--flZ~ngE,)Z{,~} W(n~ =0; {n~} I n ~ = 0 ; {nt}) '
(1)
where 3"'s indicate the sum with symmetry factors being taken into account, and W in the numerator is given by ~2
W=S*S,
(2)
S=(iK,,,)
l=max(O,
,,-,~) (nl-jt)! j~! jfl
..
n'=l
(iK~,,,) ... (iK,,,,) ( 0 i T n={
\,,=1
¢,,' I-I 0,,~ n=l
10),
(3)
and W i n the denominator in eq. ( 1 ) is defined similarly. In eq. (3),
j~ = n ~ - n l + j t , Kt,~...¢~=
(4)
2E,,/~tVZ~ ' I daxexp(--ipl'x)(VT~+m2)'"O(x)'
etc.
(5)
Now we consider fig. 1 (a) depicting W i n eq. (2): the LHS ( R H S ) of fig. la depicts S (S*) in W=S*S, and SL a n d / o r S~ in fig. 1 may contain disconnected parts, but SL®S~ as a whole is connected. Now, we pay attention to the 0 with the mode i, indicated explicitly, and assume that there are no additional ¢'s with the same mode i in the initial and final states of SL N S~. Opposite cases are treated shortly. Then, from eqs. ( 2 ) and (3), we get ~3, with an obvious notation, n,+l
W= 02) ~ - ~
( - i 2 )F~ ( E~,p~) ,
(6)
where only variables depending on the mode i are retained. Taking an average over the ensemble (c.f. eq. ( 1 ) ), we get 1
{I+nB(E,)}(-i2)F,(E,,p,)
(7)
nB(E,) = 1/ [exp(flE,) - 1 ] .
(8)
The sum over i yields f
F~a)= ~ Fd'~ -----~v-+~J ~
d4p
02){1 + n ~ ( [P0 I)}27~5(p2-rn2)O(Po)(-i2)F~(P)
•
(9)
~z It should be noted that S, eq. (3), contains, in general, several disconnected parts and spectator ¢'s. #3 As usual Z ; ~ is cancelled out by the counter diagrams.
--5"-q (a)
352
~
; °R r - - % (b)
Fig. 1. The diagrams contributing to the decay of a qb with mom e n t u m P. The solid (dashed) lines denote ¢ (qb).
Volume 247, number 2,3
PHYSICS LETTERSB
13 September 1990
Fig. lb, obtained from fig. la by a crossing substitution of two ¢ 's with the mode i, can be treated similarly. Summing the contributions from figs. la and lb, we arrive at f d4p E t ') = d ~ (i2)D + ( p ) ( - i 2 ) F , ( p ) ,
(10)
where D + (p) is the "propagator" from a so-called uncircled vertex toward a circled vertex, introduced by Kobes and Semenoff [ 5 ]: O + (p) = [0(po) +nB( IPol ) ] 2~zr(P 2 - m 2 ) •
( 11 )
The propagator from a circled vertex toward an uncircled one, D - (p), is obviously given by D - (p) = D + ( - p ) in the present neutral scalar theory. It is worth mentioning that, according to ref. [ 5 ], to an uncircled (circled) vertex a factor i2 ( - i 2 ) is attached, which is in accord with eq. (10) where i2 ( - i 2 ) comes from SL ( S ~ ) in fig. 1. In an analogous manner we get for fig. 2 F~ 2) = f ~d4p
(i2)D(p)(i2)F2(p),
(12)
i
D ( p ) - p2_m2 + ie +2nnB( ]Pol )6(pZ-m z) ,
(13)
where D(p) is the propagator between two uncircled vertices [ 5 ], which is just the physical ~ propagator in RTFT [ 2 ]. The diagrams obtained from fig. 2 by an interchange, left *--, right or S*--,S*, give the complex conjugate to eq. ( 12 ), in which D* (p) is the propagator between two circled vertices [5 ] and is nothing but the thermal ghost propagator in RTFT [ 2 ]. Through repeated applications of the above argument, we arrive at the KS diagram shown in fig. 3a which represents diagrammatically the decay rate/~d (uP to an overall factor). In fig. 3a, the external vertex from which the incoming (outgoing) • originates is of uncircled (circled) type and the sum is over all possible ways of circling the internal vertices. It should be stressed again that the uncircled and circled vertices come, respectively, from S and S* in eq. (2). To make the above derivation complete, the following three problems should be resolved. (i) "Mode overlapping" within SL®S~. The case where several additional ~ 's with the same mode i are present in the initial a n d / o r final states of SL®S~ in figs. 1 and 2. We can show that the contribution from this type of configuration can be neglected because it is suppressed by an inverse power of V relative to the corresponding mode non-overlapping case discussed above. This corresponds to the fact that as V--,oo the contribution in question takes up only a measure zero region in the momentum integrals inherent in the diagrams shown in fig. 3. (ii) "Disconnected reactions". The case where several ~'s in the heat bath participate in several "isolated" reactions which are diagrammatically disconnected from the "main body" SL®S~ (see, e.g., fig. 1 ). Suppose a configuration where there is no mode overlapping between the q~ 's participating in the isolated reactions and those in the Se ®S~. Then the contribution of this configuration is cancelled by the denominator, the "vacuum
int (a)
(b)
(c)
Fig. 2. The diagrams contributing to the decay of a ~.
int (a)
(b)
Fig. 3. The diagrams for the calculation Of Fa and Fp. The external energy Po is positive.
353
Volume 247, number 2,3
PHYSICS LETTERSB
13 September 1990
+
(ci)
•
i
•
i
T-
-';, (a)
fk
~
"
-*~
-
;
-
~
-'F C,C.
~r-
-~(b)
-p~-
-~(c)
Fig. 4. Examplesof the diagramswith isolated "self-energy"type reactions, c.c. denotes complexconjugate.
P
P
Z
~-
"~
,o, P
P (b)
• --Y~-~(c)
Fig. 5. The KS diagrams corresponding to fig. 4 together with their relatives. The external energies Pro,/°20and Po are positive.
bubble", in eq. ( 1 ). On the other hand, this cancellation does not take place for a mode-overlapping configuration, in which the isolated reactions and the "main body" are not literally disconnected from each other because the initial and final states are Bose symmetrized. For almost all such cases, however, we can find a connected diagram (with no "disconnected reactions" involved) such that the configuration under consideration takes up only a measure zero portion of this connected diagram and can be neglected. The only exception to this argument is the case where the "isolated" reactions are of "generalized self-energy" type and its iterations ~4, as illustrated in fig. 4. (In essential portions are not drawn. ) (iii) "Generalized self-energy" type disconnected reactions. Now we consider the one-loop diagrams, fig. 4, together with their many relatives (not shown ). The actual analysis by using eqs. ( 1 ) - ( 3 ) is rather lengthy and is described elsewhere. Here we only present the result: Figs. 4a, 4b and 4c together with their relatives lead to the KS diagrams depicted, respectively, in figs. 5a-c. Similarly, the diagrams, obtained from fig. 4 by replacing their main bodies including ~ ' s (dashed lines) by those in figs. 2a and 2b, and their relatives lead to the corresponding KS diagrams (not explicitly shown). In obtaining these results, the interplay of fig. 4 (and related diagrams) and the "vacuum bubbles" plays an important role. Analysis of configurations with multiloop "self energies" and with "beads diagrams" also leads to corresponding KS diagrams. This completes the derivation of the relation ~5 between the decay rate Fd and the KS diagram depicted in fig. 3a. As noted before, it is now obvious that, in any given KS diagram, the uncircled (circled) vertices come from the interaction vertices within S (S*) in eq. (2), thus the meaning of the KS diagram in terms of reaction amplitudes becomes clear: Any KS diagrams can be cut. More precisely, by tracing back the above procedure, any KS diagram can be cut into two elements S and S* in eq. (2). In this regard, we would like to mention the following: A somewhat bizarre diagram, fig. 5c, is regarded so far as playing only the "passive" role of eliminating the [ 6(p ~ - rn 2) ] 2 singularity [ 5,7-9] and has been thought to be non-cuttable. It is obvious, however, from the present analysis that fig. 5c plays the same "active" role as others, i.e., it is in actual "existence" representing the physical process fig. 4c and its relatives obtained from fig. 4c by different crossing replacements. 3. In much the same way as above, for a production rate Fp of a Higgs • with momentum P produced in a heat bath, we get (see footnote 5) a KS diagram shown in fig. 3b and Fp=exp( - f l P o ) F a (see refs. [5,8,9] ). According to ref. [5] the imaginary part of Xll, the RTFT self-energy part of a Higgs ~, is given by the sum Fd + Fp. This is, however, of no direct physical importance. What we want to know is the net decay rate F,] et, which can be evaluated by simply taking the difference between Fd and F o, Fc] et - - F d - F p . So far, the physical meaning of each KS diagram has not been clear and an evaluation of F~ e' from the knowledge of Im L'11 has needed to introduce a "diagonalized" or a quasiparticle self-energy 27 as an intermediate device [ 5,7,9 ]. ,4 Figs. 4b and 4c have a nonvanishing contribution only when m = 0 and a dimensional (infrared) regularization is employed [ 8 ]. ,5 The relations depicted in figs. 3a and 3b are inferred from a different context in ref. [ 12].
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Volume 247, number 2,3
PHYSICS LETTERS B
13 September 1990
4. Generalization to more general processes is straightforward. Any reaction rate can be described by a set of "forward" KS diagrams. For example, consider the following reaction taking place in a heat bath; two Higgs scalars • (with m o m e n t a P~ a n d / ' 2 ) -" one Higgs qb ( m o m e n t u m P). The reaction rate is given by a set of KS diagrams depicting (P~) + (P2) + ( - P) -" (P~) + (P2) + ( - P): Every KS diagram belonging to this set has three uncircled (circled) external vertices from which originate the three external ~'s, with m o m e n t a P1, P2 and - P , entering (leaving) the diagram, and have internal vertices being either circled or uncircled. The set as a whole contains all possible combinations o f circled and uncircled internal vertices. It is worth mentioning in passing that the imaginary part o f the above forward three body R T F T amplitude is represented by a set o f KS diagrams [ 5 ] which contains the set derived above as a subset, and represents the sum o f different reaction rates. Conversely, any forward KS diagram can be interpreted in terms of (squares o f ) the corresponding reaction amplitudes. 5. Finally we sketch very briefly another more elegant but less intuitive derivation of the same result. (Details will be presented elsewhere. ) Again we take the decay of a qb as an example. We introduce the so-called closedtime-path [2,3]; the path in a complex-time plane, which consists of three segments C~, C2 and C3; C1 = ( - 0 9 - , + 0 9 ) , C2= ( + 0 9 - " - 0 9 ) and C3= ( - 0 9 - " - 0 9 - i f l ) . One can show that, in the interaction representation, the decay rate Fd is written as 1 Fd -- - [ - i K T ( P ) ] [iK2 (P) ]Tr [ e x p ( - f l H o ) Tc(q~, ~2 U) ] / T r [ e x p ( - f l H o ) U]. 2n~(0)
(14)
The time argument o f ~1 ( qb2 ) lies on C L (C2), Tc indicates taking the time-path ordered product and U stands for the time-evolution operator (along the time path, - 0 9 -" + 09-" - 0 9 -" - 0 9 - ifl) in the interaction representation [ 3 ]. Following the standard method [2,3 ] we arrive at (perturbative) diagrammatic rules within the framework of RTFT. The emerging R T F T is a two-component theory and the bare 2 × 2 matrix propagator consists of just D(p), D*(p) and D ± (p) introduced in eqs. ( 1 1 ) and (13) (and its complex conjugate), the Fd, eq. (14), also goes to the KS diagram in fig. 3a. The same physical interpretation of each KS diagram is also possible within this approach per se. Whether or not the infrared and mass singularities survive in reaction rates is undoubtedly an important issue. Analyses at some lower orders indicate the absence o f these singularities. (See, e.g., refs. [ 7,8,10,13 ] and references therein. ) In the light o f the present approach, this is now under investigation. I would like to thank H. Nakkagawa, H. Yokota and N. Ashida for helpful discussions. I sincerely thank H. Nakkagawa also for a critical reading of the manuscript.
References [ 1] For a review on quark-gluon plasma see J. Cleymans, R.V. Gavai and E. Suhonen, Phys. Rep. 130 (86) 217; see also G. Baym, P. Brown-Munzingerand S. Nagamiya, Proc. Quark Matter '88, Nucl. Phys. A 498 (1989) 1. [2] N.P. Landsman and Ch.G. van Weert, Phys. Rep. 145 (1987) 141, and references therein. [ 3 ] R. Mills, Propagators for many particle systems (Gordon and Breach, New York, 1969). [4] R.L. Kobes and G.W. Semenoff, Nucl. Phys. B 260 (1985) 714. [5] R.L. Kobes and G.W. Semenoff, Nucl. Phys. B 272 (1986) 329. [6] H.A. Weldon, Phys. Rev. D 28 (1983) 2007. [ 7 ] R. Baier, B. Pire and D. Schiff, Phys. Rev. D 38 ( 1988) 2814. [ 8 ] T. Altherr, P. Aurenche and T. Becherrawy,Nucl. Phys. B 315 (1989) 436. [9] W. Keil, Phys. Rev. D 40 (1989) 1176. 355
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[ 10] J.F. Donoghue and B.R. Holstein, Phys. Rev. D 28 (1983 ) 340; T. Grandou, M. Le Bellac and J.-L. Meunier, Z. Phys. C 43 (1989) 575. [ 11 ] K. Ahmed and S. Saleen, Phys. Rev. D 35 (1987) 1861; T. Altherr and P. Aurenche, Z. Phys. C 45 (1989) 99. [ 12 ] R. Kobes, University of Winnipeg preprint (1990). [ 13 ] A.E.I. Johansson, G. Peressutti and B.-S. Skagerstam, Nucl. Phys. B 278 ( 1986 ) 324; J. Cleymans and I. Dadic, Z. Phys. C 42 (1989) 133.
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! 3 September 1990
PHYSICS LETTERS B
13 September 1990
Calculational rules for finite temperature reaction rates A. N i 6 g a w a Department of Physics, Osaka City University, Osaka 558, Japan Received 27 May 1990
Systematic calculational rules are found for evaluating the reaction rate of a generic process taking place in a thermal reservoir in equilibrium. The rules are formulated in a diagrammatic representation within the framework of a real-time thermal field theory. It becomes clear how each "forward" diagram can be cut; thus the finite temperature generalization of the Cutkosky (or cutting) rules is settled. The relationship is clarified between our diagrammatic rules and those for finding the imaginary part of the relevant "forward" amplitude established by Kobes and Semenoff; the latter is shown to contain different reaction rates.
1. Recent e x p e r i m e n t a l studies on q u a r k - g l u o n p l a s m a f o r m a t i o n in heavy ion collisions have stimulated theoretical studies o f various reactions taking place in a heat bath in equilibrium [ 1 ]. At zero t e m p e r a t u r e T-- 0, any reaction rate can be calculated through the imaginary part o f the relevant forward amplitude, which can be conveniently calculated by employing the Cutkosky or cutting rules. On the other hand, at T ¢ 0, up until now, although the d i a g r a m m a t i c rules o f finding the imaginary part o f an a m p l i t u d e in real-time thermal field theory ( R T F T ) [2,3 ] have been settled by Kobes and Semenoff [4,5 ], the relation between reaction rate (s) and the imaginary part o f the corresponding " f o r w a r d " a m p l i t u d e is not yet clear except in the simplest case o f heavy particle decay [ 4 - 9 ]. In this letter we clarify the relationship between them, which provides us with an efficient calculational scheme o f finite t e m p e r a t u r e reaction rates. We take the following intuitive approach: Let us consider a set o f systems forming a grand canonical ensemble and calculate the reaction rate for the process under consideration taking place in one system belonging to the set, and then take an average over the ensemble. Through this procedure we can arrive at the d i a g r a m m a t i c rules for calculating the reaction rate: The reaction rate o f a particular process at T ¢ 0 is given by a physically apparent subset o f the set o f " c i r c l e d " diagrams introduced in ref. [5] (referred to as KS diagrams hereafter) which as a whole represent diagrammatically, through the discontinuity formula, the imaginary part o f the relevant forward R T F T a m p l i t u d e (with physical external legs). The imaginary part as a whole is shown to contain different reaction rates. A n i n t e r p r e t a t i o n o f any " f o r w a r d " KS d i a g r a m in terms o f reaction a m p l i t u d e s (cutting rules) is given. We also briefly present a n o t h e r m o r e elegant but less intuitive derivation o f the same result. 2. F o r definiteness, we e x a m i n e the decay ~1 o f a Higgs scalar tb (mass M ) i m m e r s e d in a heat bath o f thermal light neutral scalars ~ (mass m ) . The techniques and results, however, are general a n d can be a p p l i e d to any reaction. (See sections 3 and 4 below. ) The interaction lagrangian o f the system is Lint = 24 3 / 3 ! + ½g02tb up to a counter lagrangian. It is convenient, for our purpose, to restrict the space to the interior o f a large cube with v o l u m e V. We pick up one system out o f those forming a grand canonical ensemble; its initial state is specified by a Fock state ~J The decay processes themselves have already been treated extensively at the one- and two-loop levels. See, e.g., refs. [5-11 ] and earlier references cited therein. 0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )
351
Volume 247, number 2,3
PHYSICS LETTERS B
13 September 1990
I n , = 1; {nl} ) in, where nl stands for the number of ¢ with the plane wave mode l ( = 0, _+ 1, _+2 .... ). Final states are specified similarly. Constructing the transition probability W ( n ~ = 0; {n} } I ne = 1; {nz} ) between these two states, summing over final states and averaging over the systems belonging to the grand canonical ensemble, we get, for the decay rate Fd of of, Fd = 1 2t,,} exp(--flX,niEi)~¢n~) W(n~ =0; {n~} [ n ~ = 1; {nt}) 21r5(0) 3~{,,} exp(--flZ~ngE,)Z{,~} W(n~ =0; {n~} I n ~ = 0 ; {nt}) '
(1)
where 3"'s indicate the sum with symmetry factors being taken into account, and W in the numerator is given by ~2
W=S*S,
(2)
S=(iK,,,)
l=max(O,
,,-,~) (nl-jt)! j~! jfl
..
n'=l
(iK~,,,) ... (iK,,,,) ( 0 i T n={
\,,=1
¢,,' I-I 0,,~ n=l
10),
(3)
and W i n the denominator in eq. ( 1 ) is defined similarly. In eq. (3),
j~ = n ~ - n l + j t , Kt,~...¢~=
(4)
2E,,/~tVZ~ ' I daxexp(--ipl'x)(VT~+m2)'"O(x)'
etc.
(5)
Now we consider fig. 1 (a) depicting W i n eq. (2): the LHS ( R H S ) of fig. la depicts S (S*) in W=S*S, and SL a n d / o r S~ in fig. 1 may contain disconnected parts, but SL®S~ as a whole is connected. Now, we pay attention to the 0 with the mode i, indicated explicitly, and assume that there are no additional ¢'s with the same mode i in the initial and final states of SL N S~. Opposite cases are treated shortly. Then, from eqs. ( 2 ) and (3), we get ~3, with an obvious notation, n,+l
W= 02) ~ - ~
( - i 2 )F~ ( E~,p~) ,
(6)
where only variables depending on the mode i are retained. Taking an average over the ensemble (c.f. eq. ( 1 ) ), we get 1
{I+nB(E,)}(-i2)F,(E,,p,)
(7)
nB(E,) = 1/ [exp(flE,) - 1 ] .
(8)
The sum over i yields f
F~a)= ~ Fd'~ -----~v-+~J ~
d4p
02){1 + n ~ ( [P0 I)}27~5(p2-rn2)O(Po)(-i2)F~(P)
•
(9)
~z It should be noted that S, eq. (3), contains, in general, several disconnected parts and spectator ¢'s. #3 As usual Z ; ~ is cancelled out by the counter diagrams.
--5"-q (a)
352
~
; °R r - - % (b)
Fig. 1. The diagrams contributing to the decay of a qb with mom e n t u m P. The solid (dashed) lines denote ¢ (qb).
Volume 247, number 2,3
PHYSICS LETTERSB
13 September 1990
Fig. lb, obtained from fig. la by a crossing substitution of two ¢ 's with the mode i, can be treated similarly. Summing the contributions from figs. la and lb, we arrive at f d4p E t ') = d ~ (i2)D + ( p ) ( - i 2 ) F , ( p ) ,
(10)
where D + (p) is the "propagator" from a so-called uncircled vertex toward a circled vertex, introduced by Kobes and Semenoff [ 5 ]: O + (p) = [0(po) +nB( IPol ) ] 2~zr(P 2 - m 2 ) •
( 11 )
The propagator from a circled vertex toward an uncircled one, D - (p), is obviously given by D - (p) = D + ( - p ) in the present neutral scalar theory. It is worth mentioning that, according to ref. [ 5 ], to an uncircled (circled) vertex a factor i2 ( - i 2 ) is attached, which is in accord with eq. (10) where i2 ( - i 2 ) comes from SL ( S ~ ) in fig. 1. In an analogous manner we get for fig. 2 F~ 2) = f ~d4p
(i2)D(p)(i2)F2(p),
(12)
i
D ( p ) - p2_m2 + ie +2nnB( ]Pol )6(pZ-m z) ,
(13)
where D(p) is the propagator between two uncircled vertices [ 5 ], which is just the physical ~ propagator in RTFT [ 2 ]. The diagrams obtained from fig. 2 by an interchange, left *--, right or S*--,S*, give the complex conjugate to eq. ( 12 ), in which D* (p) is the propagator between two circled vertices [5 ] and is nothing but the thermal ghost propagator in RTFT [ 2 ]. Through repeated applications of the above argument, we arrive at the KS diagram shown in fig. 3a which represents diagrammatically the decay rate/~d (uP to an overall factor). In fig. 3a, the external vertex from which the incoming (outgoing) • originates is of uncircled (circled) type and the sum is over all possible ways of circling the internal vertices. It should be stressed again that the uncircled and circled vertices come, respectively, from S and S* in eq. (2). To make the above derivation complete, the following three problems should be resolved. (i) "Mode overlapping" within SL®S~. The case where several additional ~ 's with the same mode i are present in the initial a n d / o r final states of SL®S~ in figs. 1 and 2. We can show that the contribution from this type of configuration can be neglected because it is suppressed by an inverse power of V relative to the corresponding mode non-overlapping case discussed above. This corresponds to the fact that as V--,oo the contribution in question takes up only a measure zero region in the momentum integrals inherent in the diagrams shown in fig. 3. (ii) "Disconnected reactions". The case where several ~'s in the heat bath participate in several "isolated" reactions which are diagrammatically disconnected from the "main body" SL®S~ (see, e.g., fig. 1 ). Suppose a configuration where there is no mode overlapping between the q~ 's participating in the isolated reactions and those in the Se ®S~. Then the contribution of this configuration is cancelled by the denominator, the "vacuum
int (a)
(b)
(c)
Fig. 2. The diagrams contributing to the decay of a ~.
int (a)
(b)
Fig. 3. The diagrams for the calculation Of Fa and Fp. The external energy Po is positive.
353
Volume 247, number 2,3
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13 September 1990
+
(ci)
•
i
•
i
T-
-';, (a)
fk
~
"
-*~
-
;
-
~
-'F C,C.
~r-
-~(b)
-p~-
-~(c)
Fig. 4. Examplesof the diagramswith isolated "self-energy"type reactions, c.c. denotes complexconjugate.
P
P
Z
~-
"~
,o, P
P (b)
• --Y~-~(c)
Fig. 5. The KS diagrams corresponding to fig. 4 together with their relatives. The external energies Pro,/°20and Po are positive.
bubble", in eq. ( 1 ). On the other hand, this cancellation does not take place for a mode-overlapping configuration, in which the isolated reactions and the "main body" are not literally disconnected from each other because the initial and final states are Bose symmetrized. For almost all such cases, however, we can find a connected diagram (with no "disconnected reactions" involved) such that the configuration under consideration takes up only a measure zero portion of this connected diagram and can be neglected. The only exception to this argument is the case where the "isolated" reactions are of "generalized self-energy" type and its iterations ~4, as illustrated in fig. 4. (In essential portions are not drawn. ) (iii) "Generalized self-energy" type disconnected reactions. Now we consider the one-loop diagrams, fig. 4, together with their many relatives (not shown ). The actual analysis by using eqs. ( 1 ) - ( 3 ) is rather lengthy and is described elsewhere. Here we only present the result: Figs. 4a, 4b and 4c together with their relatives lead to the KS diagrams depicted, respectively, in figs. 5a-c. Similarly, the diagrams, obtained from fig. 4 by replacing their main bodies including ~ ' s (dashed lines) by those in figs. 2a and 2b, and their relatives lead to the corresponding KS diagrams (not explicitly shown). In obtaining these results, the interplay of fig. 4 (and related diagrams) and the "vacuum bubbles" plays an important role. Analysis of configurations with multiloop "self energies" and with "beads diagrams" also leads to corresponding KS diagrams. This completes the derivation of the relation ~5 between the decay rate Fd and the KS diagram depicted in fig. 3a. As noted before, it is now obvious that, in any given KS diagram, the uncircled (circled) vertices come from the interaction vertices within S (S*) in eq. (2), thus the meaning of the KS diagram in terms of reaction amplitudes becomes clear: Any KS diagrams can be cut. More precisely, by tracing back the above procedure, any KS diagram can be cut into two elements S and S* in eq. (2). In this regard, we would like to mention the following: A somewhat bizarre diagram, fig. 5c, is regarded so far as playing only the "passive" role of eliminating the [ 6(p ~ - rn 2) ] 2 singularity [ 5,7-9] and has been thought to be non-cuttable. It is obvious, however, from the present analysis that fig. 5c plays the same "active" role as others, i.e., it is in actual "existence" representing the physical process fig. 4c and its relatives obtained from fig. 4c by different crossing replacements. 3. In much the same way as above, for a production rate Fp of a Higgs • with momentum P produced in a heat bath, we get (see footnote 5) a KS diagram shown in fig. 3b and Fp=exp( - f l P o ) F a (see refs. [5,8,9] ). According to ref. [5] the imaginary part of Xll, the RTFT self-energy part of a Higgs ~, is given by the sum Fd + Fp. This is, however, of no direct physical importance. What we want to know is the net decay rate F,] et, which can be evaluated by simply taking the difference between Fd and F o, Fc] et - - F d - F p . So far, the physical meaning of each KS diagram has not been clear and an evaluation of F~ e' from the knowledge of Im L'11 has needed to introduce a "diagonalized" or a quasiparticle self-energy 27 as an intermediate device [ 5,7,9 ]. ,4 Figs. 4b and 4c have a nonvanishing contribution only when m = 0 and a dimensional (infrared) regularization is employed [ 8 ]. ,5 The relations depicted in figs. 3a and 3b are inferred from a different context in ref. [ 12].
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4. Generalization to more general processes is straightforward. Any reaction rate can be described by a set of "forward" KS diagrams. For example, consider the following reaction taking place in a heat bath; two Higgs scalars • (with m o m e n t a P~ a n d / ' 2 ) -" one Higgs qb ( m o m e n t u m P). The reaction rate is given by a set of KS diagrams depicting (P~) + (P2) + ( - P) -" (P~) + (P2) + ( - P): Every KS diagram belonging to this set has three uncircled (circled) external vertices from which originate the three external ~'s, with m o m e n t a P1, P2 and - P , entering (leaving) the diagram, and have internal vertices being either circled or uncircled. The set as a whole contains all possible combinations o f circled and uncircled internal vertices. It is worth mentioning in passing that the imaginary part o f the above forward three body R T F T amplitude is represented by a set o f KS diagrams [ 5 ] which contains the set derived above as a subset, and represents the sum o f different reaction rates. Conversely, any forward KS diagram can be interpreted in terms of (squares o f ) the corresponding reaction amplitudes. 5. Finally we sketch very briefly another more elegant but less intuitive derivation of the same result. (Details will be presented elsewhere. ) Again we take the decay of a qb as an example. We introduce the so-called closedtime-path [2,3]; the path in a complex-time plane, which consists of three segments C~, C2 and C3; C1 = ( - 0 9 - , + 0 9 ) , C2= ( + 0 9 - " - 0 9 ) and C3= ( - 0 9 - " - 0 9 - i f l ) . One can show that, in the interaction representation, the decay rate Fd is written as 1 Fd -- - [ - i K T ( P ) ] [iK2 (P) ]Tr [ e x p ( - f l H o ) Tc(q~, ~2 U) ] / T r [ e x p ( - f l H o ) U]. 2n~(0)
(14)
The time argument o f ~1 ( qb2 ) lies on C L (C2), Tc indicates taking the time-path ordered product and U stands for the time-evolution operator (along the time path, - 0 9 -" + 09-" - 0 9 -" - 0 9 - ifl) in the interaction representation [ 3 ]. Following the standard method [2,3 ] we arrive at (perturbative) diagrammatic rules within the framework of RTFT. The emerging R T F T is a two-component theory and the bare 2 × 2 matrix propagator consists of just D(p), D*(p) and D ± (p) introduced in eqs. ( 1 1 ) and (13) (and its complex conjugate), the Fd, eq. (14), also goes to the KS diagram in fig. 3a. The same physical interpretation of each KS diagram is also possible within this approach per se. Whether or not the infrared and mass singularities survive in reaction rates is undoubtedly an important issue. Analyses at some lower orders indicate the absence o f these singularities. (See, e.g., refs. [ 7,8,10,13 ] and references therein. ) In the light o f the present approach, this is now under investigation. I would like to thank H. Nakkagawa, H. Yokota and N. Ashida for helpful discussions. I sincerely thank H. Nakkagawa also for a critical reading of the manuscript.
References [ 1] For a review on quark-gluon plasma see J. Cleymans, R.V. Gavai and E. Suhonen, Phys. Rep. 130 (86) 217; see also G. Baym, P. Brown-Munzingerand S. Nagamiya, Proc. Quark Matter '88, Nucl. Phys. A 498 (1989) 1. [2] N.P. Landsman and Ch.G. van Weert, Phys. Rep. 145 (1987) 141, and references therein. [ 3 ] R. Mills, Propagators for many particle systems (Gordon and Breach, New York, 1969). [4] R.L. Kobes and G.W. Semenoff, Nucl. Phys. B 260 (1985) 714. [5] R.L. Kobes and G.W. Semenoff, Nucl. Phys. B 272 (1986) 329. [6] H.A. Weldon, Phys. Rev. D 28 (1983) 2007. [ 7 ] R. Baier, B. Pire and D. Schiff, Phys. Rev. D 38 ( 1988) 2814. [ 8 ] T. Altherr, P. Aurenche and T. Becherrawy,Nucl. Phys. B 315 (1989) 436. [9] W. Keil, Phys. Rev. D 40 (1989) 1176. 355
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[ 10] J.F. Donoghue and B.R. Holstein, Phys. Rev. D 28 (1983 ) 340; T. Grandou, M. Le Bellac and J.-L. Meunier, Z. Phys. C 43 (1989) 575. [ 11 ] K. Ahmed and S. Saleen, Phys. Rev. D 35 (1987) 1861; T. Altherr and P. Aurenche, Z. Phys. C 45 (1989) 99. [ 12 ] R. Kobes, University of Winnipeg preprint (1990). [ 13 ] A.E.I. Johansson, G. Peressutti and B.-S. Skagerstam, Nucl. Phys. B 278 ( 1986 ) 324; J. Cleymans and I. Dadic, Z. Phys. C 42 (1989) 133.
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