Diagrammatic algorithm for evaluating finite-temperature reaction rates

ANNALS

OF PHYSICS

215, 315-371

(1992)

Diagrammatic Algorithm for Evaluating Finite-Temperature Reaction

Rates

NAOKI ASHIDA Department of Physics, Osaka City University. Sumiyoshi-ku, Osaka 558. Japan

HISAO NAKKAGAWA* AND AKIRA NII~GAWA' Fakultiit

,fiir Physik, Unirersitiit Bielqfeld, D-4800 Bielefeld, German?

AND HIROSHI YOKOTA Institute for Natural Science, Nura University, 1500 Misasagi-rho, Nara 631, Japan Received

July

25. 1991; revised

October

24, 1991

In this paper, by following the procedure of statistical mechanics we present the systematic calculational rules for evaluating the reaction rate of a generic dynamical process taking place in a heat bath. These rules are formulated within the framework of real-time thermal field theory (RTFT), in terms of the Feynman-like diagrams, the so-called circled diagrams. With the machinery developed in this paper we can establish the finite temperature generalization of the Cutkosky, or the cutting rules in quantum field theory at zero temperature. We have also studied the relation between the imaginary part of forward RTFT amplitude and the reaction rates; the imaginary part consists of various reaction rates. This is a finite temperature generalization of the optical theorem. ( 1992 Academic Press. Inc.

1. INTRODUCTION Theoretical developments in elementary particle physics have enabled us to draw a rough scenario on the history of our whole universe, especially in its early hot and dense era, up to 1O-“4 s after the Big Bang Cl, 23. However, in order to understand * On leave of absence from Institute of Natural 631, Japan. ’ Current address: Department of Physics, Osaka

Science, City

Nara

University,

University,

1500 Misasagi-cho,

Sumiyoshi-ku,

Osaka

Nara

558, Japan.

315 0003-49

16/92 $9.00

Copyrvght C 1992 by Academic Press. Inc. All rights ol reproduction in any form reserved.

316

ASHIDA

ET AL.

more accurately the precise history of the early universe, including the physical phenomena such as the baryogenesis and the nucleosynthesis, etc., that took place in a hot and dense era as well as the several steps of the vacuum transitions, it is urgent and important to develop some theoretical prescriptions that make it possible to study rigorously the evolution of physical processes in a hot and dense environment. On the other hand, the heavy ion collision experiments have recently made great progress and will have soon realized the formation of quark gluon plasma (QGP) in the laboratory system, and thus have made the physics of QGP, namely physics in the hot QCD environment, real [2, 31. This experimental progress has greatly stimulated the theoretical studies of QGP (or the hot QCD), especially those of various dynamical processes taking place in a hot and dense environment. Theoretical progress in this field is quite important in the sense that it will have achieved not only the understanding of the hot QCD, but also the precise understanding of the history of the early universe as mentioned above. Within our present wisdom, in studying various reactions taking place in a heat bath in equilibrium, the real-time method [4-61 of thermal field theory is much more convenient than the imaginary time method [6, 71. This is because in the imaginary time method the analytic continuation involved can be practically difficult if one wishes to study n-point functions with n > 3. Then for the study of such reactions what is necessary is a set of efficient calculational rules within the framework of real-time thermal field theory (RTFT) to evaluate the reaction rates. The main purpose of this paper is to present such rules. At zero temperature and density any reaction rate is neatly related to the imaginary or the absorptive part of the corresponding forward Feynman amplitude, which can be conveniently evaluated through the Cutkosky or the cutting rules [S]. At finite temperature and density, up until now, the diagrammatic rules to find the imaginary part of a given physical Feynman amplitude (i.e., an amplitude with all external legs being physical fields) in the RTFT are given by Kobes and Semenoff [9] (see also [lo]) as a finite-temperature generalization of the cutting equation at zero temperature and density [ 111. However, the relation between the reaction rate of a given process at finite temperature and the imaginary part of the relevant forward amplitude in RTFT is not yet clear, except in the simplest case of the heavy particle decay [9, 12-171, the production of lepton pairs in QGP [ 18-211, and the deep-inelastic lepton scattering off of quarks and gluons in QGP [22]. Due to this lack of knowledge, the physical meaning of an individual diagram in the sum of those involved in finding the imaginary part of a given amplitude remains unclear. We shall fill this gap of knowledge with our findings in this paper. The purpose of this paper is to establish a diagrammatic algorithm [23] to evaluate finite-temperature reaction rates, an algorithm which plays the role of the Cutkosky rules [S] in vacuum theory. Our approach is to faithfully follow the very notion in statistical mechanics: Let us imagine a grand canonical ensemble consisting of a huge number of identical systems and consider a dynamical process taking place in one of the systems belonging to that grand canonical ensemble.

ALGORITHM

FOR

TEMPERATURE

REACTION

RATES

317

Evaluate the reaction rate (or transition probability) of the above process in theory, then we take the average over systems forming the grand canonical ensemble, and we obtain the “thermal average” of the reaction rate of the process, taking place in a heat bath in equilibrium. We find that the reaction rate of a generic process can be represented in terms of a set of “circled diagrams” introduced in Ref. [9] and that each circled diagram represents the corresponding reaction process in a heat bath. It becomes clear that, in the light of this finding, the imaginary part of any forward amplitude in RTFT, in contrast to the zero-temperature (T= 0) case, does not represent a specific single reaction rate, but consists of reaction rates of several different reactions in a heat bath, those reactions which are related with each other through a kind of crossing substitutions. This can be regarded as an optical theorem generalized to the T # 0 case. Throughout this paper we take the theory with real scalars as the simplest example unless otherwise stated. Inclusion of other fields is straightforward, on which we will mention only briefly. This paper is organized as follows. In the next section, as the simplest example to illustrate our procedure, we study the decay of a non-thermal heavy scalar particle immersed in a heat bath of thermal light neutral scalar particles, and obtain the formula to represent the decay rate. Then in Section 3, through careful analysis of the formula obtained in Section 2, we arrive at the systematic rules, akin to those of the Feynman diagrammatic rules, for evaluating the decay rate, which is codified by referring to the circled diagrams. Section 4 is devoted to the study of the production of heavy scalar particles, the inverse process to that studied in Sections 2 and 3, and the diagrammatic rules to evaluate the corresponding production rate are presented. The results presented in Sections 3 and 4 are already known [9, 13, 16, 17, 19-21, 23, 241. Our approach just reproduces them. In Section 5 we extend our analysis to a more general reaction taking place in a heat bath in equilibrium and establish the diagrammatic rules to evaluate the reaction rate. Through the machinery developed in the above sections we can find a finite-temperature generalization of the Cutkosky rules or the cutting rules in quantum field theory at zero temperature, which is summarized in Section 6. Some miscellaneous topics, i.e., inclusion of chemical potential and of fermions, are briefly commented on in Section 7. Section 8 is devoted to the summary of the results disclosed in this paper. In Appendix A we present the result of a complete analysis of the “generalized self-energy” type diagrams, which are graphically disconnected to the “main-body” diagram that represents the physical reaction in consideration, taking place in a heat bath. Finally, in Appendix B we present a complete analysis of the “bead diagram” of the above generalized self-energy type diagrams, i.e., the one-particle reducible diagram constructed by linking an arbitrary number of the generalized self-energy type diagrams. uacuum

318

ASHIDA ET AL.

2. BASIC IDEAS

For our present purpose it is convenient to consider a system enclosed inside a large cube with volume V= L3. The fields are subject to periodic boundary conditions. At the final stage, we take the limit V --i cc. This system is considered to be the one that constitutes a grand canonical ensemble. For simplicity of presentation, we study here, as the simplest example, the decay of a heavy neutral scalar @ immersed in a heat bath of thermal light neutral scalars d’s, where @ is approximated to be nonthermal. The techniques and results, however, are general enough and they can be straightforwardly applied to any reaction in any theory, which we discuss in Section 5. The Lagrangian of the system is

40=-+,rl +M2)@-$12+m2)~+$5~+pD~+~.,

(2.1)

where M/m is the mass of the heavy/light scalar @N, and g. is the counter Lagrangian. We are to adopt the renormalization condition (01 4(x) IO) =O. Therefore, strictly speaking, it is necessary to include terms linear in 4 in 9. Although we simply ignore these terms to write down explicitly in Eq. (2.1), inclusion of them does not obstruct our following derivation, so that our conclusion is not spoilt. The ultraviolet renormalization does not ruin at all our results in this paper. Incidentally, the following arguments do not depend on the explicit form of Y. For simplicity, here we study the decay of @ taking place, only through the lowest order with respect to g. Undoing the lowest order approximation is immediate and is discussed in Section 3.6, where CDis also thermalized. As our machinery we employ vacuum field theory. The thermal effects enter through the statistical or ensemble average with the statistical weight describing the heat bath in thermal equilibrium (see Eqs. (2.10) and (2.11) below). We start with an expansion of the incoming and outgoing fields, CJ~~“(X) and bO,t(x), in terms of the plane-wave basis,

+ u:(p) e i(EPropP’x)}

(5 =in, out),

where E, = Jp2 + m2 is the energy of dc with momentum p+

zi=0,+1,*2,

... (i=x,

(2.2)

p, y, z),

(2.3)

ALGORITHM

FOR

TEMPERATURE

REACTION

RATES

319

and ~~(p)/~~(p) in Eq. (2.2) is the usual annihilation~creation operator of a quantum of 4e with the mode f (= (~/2~)p~. The propagator is evaluated as usual, (2.4)

Similar treatment also applies for Qin and Qout. The heat bath is a system where numbers of & are excited, namely any mode f is in general occupied by a number of (n quanta. Let us consider the decay of a heavy Sp taking place in this heat bath. An initial state of this system is represented by the Fock state characterized by the momentum P of QI and the number n, of # quanta in a plane-wave mode f (Zj= 0, f 1, &‘2, ..; i = IY. y, 2); In&P) = 1; {n,~)i” &J;“(p)

I-J MJ2xf’L)~“’ (I) &!

,o>

0.5)

with b%(P) the creation operator of Cp, with momentum P. Since we are considering the lowest non-trivial order in g (Eq. (2.1)), emerging final states take the form of In,(P’) = 1; {n;j)ou, which is defined in a manner similar to Eq. (2.5). The parent cf with momentum P immersed in a heat bath changes its momentum to P’ through the interaction with 4, and thus the parent Qi can be said to “decay out.” We follow the standard procedure of arriving at the reduction formuia for the S-matrix element. In the present case, however, care must be taken of the fact that any mode f/f’ in the initial/final state is in general occupied by a number (n,/n;.) of 4 quanta, subject to the Bose-Einstein statistics. Then we obtain

0.6)

where .j;=n;-n,+

j,,

G.@ ..-C,~~~jd’le.~.-~~O.~+M’)...~(r),

(2.7 I

(2.8af

320

ASHIDA

ET AL.

It should be noted that S in Eq. (2.6) contains, in general, several disconnected parts and “spectator” 4’s. In Eqs. (2.8), P is the four momentum of @ (PO= E > 0, the energy), p, is the four momentum of Q in a mode 1 (plo = E,> 0), and Z,/Z, is the wave-function renormalization constant of Q/d. In Eq. (2.6) j, and j; are the “active” numbers of 6s which directly participate in the decay of ~0; the factor JkjT$/(nI-jI)! j;! j,! comes from the bosonic nature of 4 (in the mode I). The transition probability W is given by w= (&!q2= s*s.

(2.9)

After summing W (Eq. (2.9)) over all the possible final states and then taking an average over all the systems forming the statistical ensemble containing a @ with momentum P in the initial state, we have the expression for the desired decay rate of CD:

1;(4 I n,(P)=1;(4,) r,=- 1 a,~, tn;)WMP’)= 27aO) CC{*;,W(n, =o; {n;} 1 n, =o; {n,))) where the statistical or the ensemble average of a generic quantity {n,}, say A({n,}) with irre . 1evant arguments deleted, is defined by (A)=

c expi-PZn,E,}A(ln,})lt I (n/l

in/l

exp{-P~d}

’

(2.10)

depending on

(2.11)

with T=fl-’ the temperature. In the above equations, (2.10) and (2.11), c stands for the summation with the symmetry factors being respected. The denominator in Eq. (2.10) is the counterpart of the familiar vacuum bubble in the vacuum (T = 0) field theory, which we refer to as the “thermal vacuum bubble” hereafter. Also in Eq. (2.10), tr- fi = 2&(O) ( = co) is the time interval during which a “measurement” is made. Thus the reaction rate rd represents the decay probability per unit time, or the decay rate. Here a comment on the adiabatic switching off of the interactions should be made. We started from the vacuum theory, and then introduced, as usual, in- and out-states (Eq. (2.2)) through the application of adiabatic switching off of the interactions. Nevertheless, we keep the concept of temperature even at t = ti = -CC through Eq. (2.10) with Eq. (2.1 l), following the standard assumption of statistical mechanics. We adopt this as our working hypothesis, although we do not know at the present moment how one can give a sound basis for this hypothesis.

3. HEAVY SCALAR DECAY-SIMPLE EXAMPLE In this section, on the basis of the formalism outlined in the last section, we derive a set of calculational rules for the decay rate I-, of a heavy scalar @, which is already available [9, 13, 161. General thermal reactions are treated in Section 5.

ALGORITHM

FOR

TEMPERATURE

REACTION

RATES

321

3.1. Preliminary

Here we define several terminologies appearing in the following arguments. For brevity let us consider the case in which only one kind of field 4 exists. Generalization to the case with more than one field is obvious. An example of reactions taking place in a system belonging to the grand canonical ensemble is given in Fig. 1. This figure depicts the contribution to the transition probability W= S*S, Eq. (2.9), of the reaction 1{nj”, ny’, n:“)) (= I{nj”, ni4’, rz:“})) -+ ){nil’, nj2)‘, nj”‘}): The 1.h.s. of the vertical cutting line (dash-dotted line) in Fig. 1 depicts the S-matrix element S, while the r.h.s. does S*; i.e., all the fields (particles) crossing this vertical cutting line are assumed to be the final state on-shell fields (particles). This specimen, Fig. 1, includes two graphically isolated “reactions” and a group of non-interacting $‘s (at the top of Fig. 1). Let us focus our attention to the middle part of the Fig. 1: With. this graphical representation, it is to be understood that f’,‘/fg’* may contain graphically disconnected parts, but f I”‘@ f g’* as a whole is graphically connected. (For an example, see Fig. 2) The same comment applies to the reaction in the lower part of Fig. 1. Now we define the following terminologies: (1) The 4 propagating freely is said to be “spectator” 4. We distinguish two types of spectators if necessary: Type 1 for those such as the group of 6s at the top of Fig. 1, and type 2 for those such as the &s at the bottom/top of the left/right side of Fig. 2. (2) f(L)@ f (R)*(i= a, b) is said to be strongly connected.

FIG. 1. An example of a graph representing the vertical cutting line (dash-dotted line) represents the state that crosses this vertical cutting line is the final that the state specified by {PZ~‘} + {n:31} and the one figure represents W as defined in Eq. (2.9).

transition probability W=S*S. The 1.h.s. of the Smatrix element S, while the r.h.s. does S*. The state of the reaction considered. It should bc noted specilied by { nj”} + {PZ:“} are identical. since this

322

ASHIDA

FIG. 2. An example describes the graphically

of the graphically connected disconnected process.

ET AL.

“reaction.”

in which

the S*-matrix

element

itself

(3) Two “reactions” in Fig. 1, or more precisely, Sf’Ofg’* and ff’Of‘$‘* are said to be disconnected, if all the modes of 6s participating in the final as well as in the initial states of the “reaction” (a) are different from those in the “reaction” (b). The opposite case is characterized by saying that two reactions are weakly connected. More precisely, the “reactions” (a) and (b) are said to be weakly connected if there exists at least one common mode which participates both in “reactions” (a) and (b). Evidently, a disconnected case can be realized only when these two reactions are “diagonal reactions,” i.e., (IZ:“‘} = {ni”} and hence also {nj”)

= {no”).

It should be mentioned in passing that in Fig. 1 some modes of Q’s participating in the two reactions and some modes of the spectator 4’s can be, of course, the identical modes. We can easily generalize the above definitions to graphs containing more than two “reactions.” (4) StrongI-y connected and weakl-v connected are collectively designated as connected. In ‘the following, a strongly connected part is schematically expressed by an ellipse such as in Fig. 1, while a connected part is expressed by a rectilinear

diagram (see, e.g., Fig. 5 below). In closing this subsection we give a remark on the uniqueness of the graphical representation of the reaction probability. Looking at Fig. 3(a), which consists of two weakly-connected “reactions,” we realize that this is equivalent to Fig. 3(b), which consists of a strongly-connected part and two type-l spectators. We adopt Fig. 3(b) to express this “reaction.” By generalizing this construction, we always draw a diagram for a given W= S*S such that the number of type-2 spectators becomes a minimum.

ALGORITHM

FOR

TEMPERATURE

REACTION

RATES

323

FIG. 3. An example of mutually equivalent “reactions.”

3.2. Analysis

Now let us study in detail how to calculate in the present formalism the reaction rate of the heavy Qi decay taking place in a heat bath. For the brevity of presentation here the contribution from the counter-Lagrangian 5CCin Eq. (2.1) is not taken into account explicitly, though its inclusion is straightforward. When we list the calculational rules in Section 3.5, we will present a completely general statement, taking account of the contribution from PC as well. We analyze the set of diagrams as depicted in Fig. 4, which represents the contribution to the transition probability W of heavy Q, decay, Eq. (2.9). This figure consists of a connected part involving the parent heavy scalar Q, {which is referred to as the main ho& hereafter) and spectator 0;‘s.~‘~(fff in Fig. 4 stands for (the complex conjugate of) the amplitude evaluated at some definite perturbative order in the vacuum theory. Incidentally, the connected part is either in fact strongly convected by itself or contains t)~o strongly’-connected parts, one with the heavy @ in the s-matrix element (1.h.s. of Fig. 4) and the other with the @ in the S*-matrix element (r.h.s. of Fig. 4). It is not hard to see that, in the latter case, these two strongly connected parts, together with those parts which are connected weakly to them, in fact, form a single weakly connected part. It is obvious from the definition of W= S*S (Eq. (2.9)) that any mode appears as a pair when participating in the diagram that represents W with the type-2 spec-

FIG. 4. Configurations in some definite perturbative order with connected main-body diagram and with bundles of spectator Ql’s,which contribute to W in Eq. (2.9). The summation symbol means to take all possible diagrams in the given perturbative order.

324

ASHIDA

ET AL.

tators removed (cf. Figs. 5, 6, and 8 below). We assume that all the pairs of modes participating in the main-body diagram in Fig. 4 with the type-2 spectators removed are different; i.e., for every mode no more than one pair of modes participate. Opposite cases are treated in Section 3.3. From Eq. (2.11), we see that manipulation of the statistical average factorizes with respect to the modes, so that we can handle each mode involved in Fig. 4 independently. Now among the diagrams in Fig. 4 we pick a diagram having the structure as shown in Fig. 5: a C$in the mode i (abbreviated as #i hereafter) is emitted from a definite vertex in fL to the final state and then absorbed into a definite vertex in f z. Strictly speaking the fL 0 f g in Fig. 5 is different from that in Fig. 4: The former is obtained from the latter by taking out the di lines shown in Fig. 5. Here and in the following, however, we always use the same notation f,O f z to avoid too many suffixes. Now we pay attention to di in the final state. According to the above assumption, there are no additional ii’s in the initial and final states of the main body in Fig. 5 and then jr’ = 1 andji = 0 in Eq. (2.6). The S-matrix element, S,, for the “scattering amplitude” represented in the 1.h.s. of Fig. 5 can be expressed as

x6

(

p;Pi+CP/-CP; f

j

>

.ff.({Pi>, CPj)),

(3.1)

where 6(...; ... ) denotes the Kronecker’s &symbol, E = dm and Ei = Jm are, respectively, the energies of the parent @ and the c+$,{Ei} and {p,} stand for the set of energies and three-momenta of a bundle of 4’s flowing from the left into fL, and {E,} and {pr} stand for those of a @ and a bundle of 4’s in the final state (i.e., those @ and 4’s crossing the vertical dot-dashed line) flowing out from fL and going to the right half of the figure, i.e., tofg. To the lowest order in

FIG. 5. Configuration with a bundle of spectator 4s and with a connected which a q5 in the mode i bridges from the S-matrix element side to the S*-matrix the vertical cutting line is along the final state.

main-body diagram, in element side. Note that

ALGORITHM

FOR

TEMPERATURE

REACTION

325

RATES

g, i.e., @ -+ di + @ (which can occur only when m = 0), the function fL in Eq. (3.1) is (3.2) We can obtain the similar expression for the S*-matrix element, Sz, representing the r.h.s. of the figure and thus obtain the transition probability W= SZS, (Eq. (2.9)). Taking the statistical (ensemble) average of Wand then taking the sum over the mode i, as well as over (P/} (cf. Eqs. (2.10) and (2.1 l)), we finally obtain

Fi(P)=2n6(0) ‘*l(E,P;Ei,Pi)=g*

(C

C w =CFi(p), i i. ipj: >i ~” 1 + n,(Ei) 1 2EV zF*ltE7 2E,

fZ({Pi},

{P,-})f~((pi},

(3.3) (3.4)

p; Ei, Pi)?

0))

(P/j

1 Ef-1

EC+ Pie-P,

Cp1--Cpi;

P-pi

(3.5)

(3.6) where ci, jp,) (clp,)) stands for the summation over i and {p,.} ({p,.}) with the Bose symmetry of 6s being respected. The factor N is defined formally by (3.7) which represents the ratio of the Bose symmetry factor with respect to the state specified by (P,~} to that specified by {i, p,.>. In order to evaluate the contribution to the decay rate rd, we substitute Eqs. (3.3)-(3.6) into the numerator of Eq. (2.10). In the present case, the contribution from the thermal vacuum bubble (the denominator of Eq. (2.10)) is the unit “matrix.” Then taking the limit I/-+ cc;, we obtain the contribution from Fig. 5 to the decay rate rd,

326

ASHIDA ET AL,

with pi0 = E,. It should be noted here that the V-+ co limit of the function Fz, (Eq. (3.5)) in Eq. (3.9) definitely exists, namely FZ,, Eq. (3.9), is a V-independent function. To the lowest order in g we can use Eq. (3.2) and obtain F2121(p3

Pi)=F*ICp,

g' =&--$H(P,o)

Pi)

d”P,

(3.10)

2ns(p:-‘M2)(27E)4S4(Pi+Pf-Pf.

All the singular functions associated with the propagator functions, such as the 6 functions and the principal parts, are assumed to be regularized with the finite c-prescription [6] (cf. Eq. (2.4)). Next, we pick up a diagram as depicted in Fig. 6 from Fig. 4, which is obtained from Fig. 5 by a “crossing substitution” with respect to the two 4:s. Here and in the following, by the term “crossing substitution” we mean the substitution of #i flowing from/into the initial state into fJfrom fg, for pi flowing into/from the final state from .fJinto fg, and vice versa. Then, fL/f 2 in Fig. 6 is the same function as fL/f $ in Fig. 5, defined in a different kinematical region. A similar procedure as above yields rtFig.6) =- 1 d4pi (-u)n,(~Pjo~)2n~(p~-I~Z)B(Pio)(il)~~~(P,-pi) tf 2E s m ‘I&]$$

Contributions

(-~~)~~(jPj~l)2~~(p~-~‘)~{-pi*)(i~)~2t(P~

Pi)*

(3.11)

from Figs. 5 and 6 add up to give a compact result, (3.12)

where D,,(p)

FIG. 6. Configuration qS;s.,see text.

= -27EiC@(Pof f dl

PO/)I

w

- m21.

(3.13)

obtained from Fig. 5 with the “crossing substitution” with respect to the two

ALGORITHM

FOR TEMPERATURE

REACTION

RATES

327

Each contribution from Fig. 5 or 6 builds the p. > 0 or p. < 0 sector of Eq. (3.12) with Eq. (3.13). As is obvious from our derivation the D,,(p), defined here, is a “propagator” from a vertex inf, (the S-matrix element part) toward a vertex in f z (the S* part). Here and in the following we use suffix l(2) to designate that the corresponding vertex has come from the S- (S*-) matrix element side; then the meaning of Dzl is clear. It is worth noting that the Dzl(p) is nothing but the “propagator” from a so-called uncircled (type-l) vertex toward a so-called circled (type-2) vertex, introduced by Kobes and Semenoff [9], and it is graphically depicted in Fig. 7. It is also obvious that to the vertex of type l(2), i.e., the one in the S- (S*-) matrix element side, a factor ii (- 2) should be attached. This is in accord with the circling rules in Ref. [9]. The above derivation directly discloses the fact that D,,(p) with p. > 0 represents both the spontaneous and induced emissions of &s, while for p. < 0, Dz, (p) represents the induced absorption of 4’s. It is worth pointing out, in passing, the following fact [23, 25-J: The circled diagram rules [9] are equivalent to Feynman rules in the real-time thermal field theory formulated on the basis of the sort of “closed-time path’ C= C, + C2 + C, in the complex-time plane [4] (see also [26]); C, runs from -co to + co, Cz runs back from + cc to -co, and then C3 runs from -cc to -00 - ip. The uncircled (circled) vertex corresponds to the vertex of physical (thermal-ghost) fields which lie on the contour segment C, (C,). Then the “propagator” D,, corresponds to the propagator from a vertex of physical fields to a vertex of thermal-ghost fields. Let us return to our main subject. In the above we have shown that the field 4 in the mode i, di, participating in the main body with the special configuration depicted in Figs. 5 and 6, actually turns out to be the “propagator” (3.13) with p. > O/p0 < 0 when having taken the sum over the final states as well as the statistical average (see Eqs. (3.3)-(3.5)). Obviously any other field 4 in Figs. 5 and 6, if any, having the same configuration as the one above, turns out, through the sum over the final states and the statistical average in Eq. (3.5), to be the same “propagator” (3.13) with p. > O/p, < 0. As we have noted, in the class of configurations depicted in Fig. 4, the configuration of the type of Fig. 5 and its crossing counterpart Fig. 6 always appear as a pair, and two contributions coming from these two configurations always add up to the “propagator” D2,, Eq. (3.13).

-+r

=

(+iX)i D,,(P)(+iX)

=

(-iX)i

7

=

(+iX)iD,JP)(-iX)

7

=

C-9.) iDz#)(-iA)

l

FIG.

59S/21 S/2-7

7.

;P

Propagators

Q

in the circled

l&j(P)(+iN

diagram

prescription.

328

ASHIDA

ET AL.

Next let us choose as one set the three diagrams shown in Fig. 8. The fg’s in Figs. 8(af-(c) are exactly the same function, and thejL’s in these figures represent the same function but are defined in different kinematical regions: If we write fL(po, p) for thef, in Fig. 8(c) which includes the propagator (with momentum p) in vacuum theory (cf. Eq. (2.4)), then those fL’s in Fig. 8(a) and (b) can be

expressedasfL(po = -Es, -~i)/f~(po = Ei, pi)* First we analyze Figs. 8(a), (b). Since no further &‘s are involved in the main body (i.e., the connected part involving the two parent G’s) in Figs. 8(a) and (b) as was assumed before, we obtain, by following a similar procedure as above, (3.14)

FIG. carrying

8.

Configurations where both the two &‘s are joined to the S-matrix element side. momentum p in (c) stands for the propagator in the vacuum theory (cf. Eq. (2.4)).

The

line

A~G~R~~M

FOR TEMPERATURE

REA~TK~~

RATES

329

where F,, represents the main body with &‘s removed. Summing the contributions from Fig. S(c) to those from Figs. g(a) and (b), i.e., Eq. (3.14), we obtain (3.15) where

1 D,,(P) = p2-?&i-k

- 2nin,(lp,l)

6(p2 -m2).

(3.16)

This If,,(p) is the “propagator” between the two uncircled vertices and is nothing but the propagator of the physical field # in any type of RTFT (see Fig. 7). If there are additional field #‘s in P,,, Eq. (3X), and in pzzi, Eq. (3.12) (cf. Eqs. (3.9) and (3.5)) which have the same con~guratio~s as those of Figs. 8(a), g(b), and 8(c), then when we have taken the sum over the final states and the statistical average in Eq. (3.5), they have given, respectively, the second term of the “propagator” f), , , Eq. (3.16), with p. < O(p, > 0) and the first term. The generic diagram in Fig. 4 always contains the three ~on~gurations Figs. g(a)-(c) as a set; the three cont~butions from these con~gurations add up, without fail, to the ‘“propagator” L>1I, Eq. (3.16). Finally, let us pick, as a set, yet another class of three diagrams obtained from Fig. 8 by interchanging the left and the right sides, So S*. ~ont~butions from these diagrams, through the same mechanism as above, add up to give the form (3.17)

The I,, in Eq. (3.18) is the “propagator” between the two circled vertices and ii equal to the pro~gator of the thermal ghost field associated with 4 in RTFT [4-61 (see Fig. 7). Th is mechanism applies to all the fields C#‘Sparticipating in the same con~gurations of this type. In the case of the present neutral scalar model, above arguments exhaust all the possible ~on~gurations of the incoming- and outgoing-# lines as well as those &s propagating inside the fL or fi involved in the main body of the original diagram, Fig. 4: Consider a set of diagrams having the generic structure as depicted in Fig. 4, which represent the transition probability W (Eq. (2.9)) evaluated in some definite ~rturbative order in vacuum theory. If we pick several related diagrams and study them simultaneously as a set (from the definition of W, Eq. (2.9), this procedure is guaranteed to be possible with neither excess nor shortage of diagrams); then, after having taken the sum over the final states and the statistical average in Eq. (2.10), the contributions from such a set of diagrams add up to the form of “propagators”

330

ASHIDA

ET AL.

(3.13), (3.16), and (3.18). The type-l spectators in Figs. 5, 6, and 8 reflect only on the Bose factors in Eqs. (3.13), (3.16), and (3.18). In case of the complex scalar model, however, another class of lines exist. In this case the c+$‘sin Fig. 5 represent particles 4’s in the mode i, then the &‘s in Fig. 6 (Figs. 5 and 6 are mutually the “crossing” counterparts to each other) represent antiparticles d*‘s in the mode i. These configurations lead to Eq. (3.12). The configurations that are obtained by interchanging the roles of particle and antiparticle lead to Eq. (3.12) with F?:,, replaced by P,,(P, p) = &,(P, -p), and with (-U) i&,(p)(i1) replaced by (U) iDJp)( -iA), D,*(P) = DX-P) = -2nire

-Po) + nB(lpcll )I d(P2 - m2),

(3.19)

which corresponds to the “propagator” from a circled vertex toward an uncircled one (see Fig. 7). It should be noted that the direction of the propagation of an antiparticle d* is, as usual, opposite to that indicated by the arrow on the 4 line. If the lines in Figs. 5, 6, and 8, to which we explicitly paid attention, are @ lines, trivial modification to the above derivation should be made. Now let us recall the present approximations: Heavy scalar @ is not thermalized; i.e., the heat bath is constituted only by light scalars #‘s, and the lowest nontrivial order with respect to g (see Eq. (2.1)) is taken. Then, Figs. 6 and 8(a) are absent. In place of Eqs. (3.13), (3.16), (3.18), and (3.19), we obtain, respectively, the vacuum-@ “propagators,” “DF, ,” “Dyl ,” “D$ ,” and “Of2 ,” which are obtained from &s counterparts with n,=O and m-+M. Through the analyses made so far the meaning of the four kinds of “propagators,” introduced in Ref. [9] and depicted in Fig. 7, becomes clear. In particular, it should be stressed again that the uncircled and circled vertices represent, respectively, those within S and S* in Eq. (2.9). 3.3. Mortaring

the Cracks-Two

Issues Left Unattended

In this subsection we investigate the following subsection.

two issues reserved in the last

(i) “Mode overlapping” withinf, 0 f E. Section 3.2 treated the case where the number of ~i’s (6s with the definite mode i) participating in the connected parts f,Q f X in Figs. 4, 5, 6, and 8(a) and (b) are two. As noted above, any diagram representing W= S*S with type-2 spectators removed includes even number of respective modes. What happens when several additional pairs of 4’s in the same mode i participate? (ii) “Disconnected reactions.” How about the case where several 6s in the heat bath participate in the “reactions” which are graphically disconnected from the main body? (see the third paragraph in Section 3.2). First, we consider the issue (i). It is not difficult to see that the configuration,

in

ALGORITHM FOR rEMPERA~RE

REACTION RATES

331

which more than one pair of the incoming and/or outgoing #‘s, being in the identical mode, participates in the main body, gives a contribution which is suppressed by some powers of l/V and thus can be ignored in the limit I’--+ co. An example of such configurations is shown in Fig. 9, the contribution of which is suppressed by l/V. In fact, if the mode overlapping does not exist in Fig. 9, i.e., if the mode i (in the final state) depicted at the lower part in Fig. 9 is i’ (#i), its contribution to fC, in Eq. (2.10) is, with obvious notation,

1 c j,ia.j(2Ei V)(2Ei’ V)(2Ej Y) Fi’i”j

(3.20)

While the contribution

of Fig. 9 to rd is

ci,j (2Ei V)’ l(2Ej V) F’,j =

1s

F

d3p d’q (27c)6(2E$2Eq

f”(p, q )3

(3.21)

and thus it is suppressed by a factor of l/V. It is obvious that this suppression reflects the fact that, in the V-+ co limit, the phase space region in Eq. (3.20) that gives this contribution reduces to a measure-zero integration region in the momentum space. Second, we turn to the issue (ii). If the main body and the graphically-disconnected reactions are mutually disconnected, then in calculating r, by Eq. (2.10) this configuration is canceled out by the contribution from the denominator (thermal vacuum bubble). Thus, we should study only the case where they are mutually deafly-confected. Almost all of these types of con~gurations are shown, just as in the issue (i) above, to give contributions to r,, which are suppressed by some powers of l/V and thus can be ignored.

FIG. 9. An example of configuration diagram.

where more than one pair of &‘s participate in the main-body

332

ASHIDA

ET AL.

+ c. c.

FIG. 10. Con~g~rations where the first-kind one-loop generalized self-energy type graph is connected to the ~~~-~~~ diagram. fn this and the ~~~~o~ing figures of that kind, the specta~ar
The only exception in which the above argument does not work is the case where a generalized self-energy type graph (the meaning of which becomes clear shortly) or its iterated one is weakly connected to the main body. Several examples are shown in Figs. 10, 12-14, 31, and 33. Some of the “self-energy” parts in those figures (e.g., Figs. IO(a)-(c), (f), (i), and {j)) survive only when m=O with the dimensional regularization method employed to regularize the infrared divergence [17,20]. We will analyze this case in the following subsection and in Appendices A and 8.

ALGORITHM

FOR

TEMPERATURE

REACTION

RATES

(h)

W FIG.

3.4. Graphically-Disconnected

IO-Continued

Generalized Self-Energy Parts

First consider the diagrams which include the graphically-disconnected generalized seIf-energy parts. We can show that these diagrams, together with their relatives, neatly reproduce the corresponding circled diagrams. Details are given in Appendix A, and here only the results are quoted. We study the simplest case of the one-loop “self-energy” part. Generalization to the arbitrary loop order is immediate and mentioned in Appendix A. We first examine the cases where the “self-energy” parts are of the first kind: One of the

334

ASHIDA

ET AL.

would-be legs of the two-point function of 4 with the generalized (one-loop) “selfenergy” blob is in the S-matrix element part, and the other is in the S*-matrix element part. Some of the relevant diagrams of the first kind are depicted in Fig. 10 (and also in Fig. 12). Note that in these figures the type-l spectator qYs are not shown explicitly and those initial state 4’s without special interest are shown collectively without mode indices, which are regarded to be different from those depicted explicitly. The contributions from the counter-Lagrangian are also not explicitly depicted. Figures 10(a)-(c), (d)-(i), and (j), together with all their respective “crossing relatives” (obtained through various crossing substitutions with respect to &‘s and/or dj’s and/or q&‘s) lead to the circled diagrams Figs. 1l(a), (b), and (c), respectively. The diagrams obtained from Figs. IO(d)-(i) by interchanging the left and right sides, i.e., S c* S*, together with all their crossing relatives, lead to the circled diagram Fig. 11(d), the self-energy part of which is the complex conjugate to that of Fig. 11(b). Figure 12 shows an example of one of the crossing relatives of Fig. 10(i), which is obtained through the crossing substitutions of ~i’s and ~j’s. Figure IO(i) is “absorbed” into the piO, Pjo, pkO>O region of the circled diagram, 11(b), while Fig. 12 is into the pie, Pjo < 0, and pkO > 0 region. It is worth mentioning here that the diagrams of the first kind together reproduce all possible circled diagrams, in which di outflows from the uncircled vertex and inflows into the circled vertex. Next let us consider the second-kind diagrams; both would-be legs of the twopoint function of q4with the generalized “self-energy” blob are either in the S-matrix element part or in the S*-matrix element part. Examples of the case where both legs

(a)

FIG. 11. Circled diagrams Fig. 10 and all their “crossing

reproduced relatives.”

lb)

through

the present

formalism,

starting

from

configurations

in

ALGORITHM

FOR

TEMPERATURE

REACTION

RATES

335

3 i

Ik i

FIG. 12. The crossing-relative diagram obtained from Fig. IO(i) through the crossing substitutions of 4;s and q$‘s.

are within the S-matrix element side are depicted in Figs. 13 and 14. One can show that Figs. 13(a), (b), and (c), together with all their crossing relatives, lead to the circled diagrams, as in Figs. 15(a), (b), and (c), respectively. The diagrams which, together with all their crossing relatives, lead to the circled diagram, Fig. 15(d), are those depicted in Figs. 14 and those obtained from Figs. 10(d)-(f) by shifting the

+ (a)

+ (b)

FIG. 13. Configurations where the second-kind connected to the main-body diagram.

one-loop generalized self-energy type graph is

FIG. 14. Same as Fig. loop generalized self-energy

13, but in these configurations both the two interaction type graph are inside the S-matrix element part.

336

vertices

in the one

ALGORITHM FOR TEMPERATURE REACTION RATES

337

(b)

(d)

(cl

FIG. 15. Circled diagrams derived through the present formalism from con~gurations where the second-kind one-loop generalized self-energy type graph is connected to the main-body. e.g., those in Figs. 13 and 14.

interaction vertex in the main body of the S*-matrix element side, at which the Q;i participates, into the main body of the S-matrix element side. All the diagrams being mirror symmetric to those discussed in the last paragraph, which lead to Fig. 15, reproduce the circled diagrams in Fig. 15 with the two uncircled vertices in the F’s, the inflow- and the outflow-points of the momentum ,Di, being replaced by the circled ones. Here we should remark on the important interplay between diagrams such as those depicted in Fig. 10 and the corresponding thermal vacuum-bubble contributions. Let us, for example, look at Fig. 10(b). Since this figure is composed of two gruphically disconnected parts, at first glance, the self-energy subdiagram seems to be canceled out in Eq. (2.10) by the thermal vacuum bubble in the denominator. The truth is, however, that this is not the case. Look at the diagram shown in Fig. 16, which is akin to Fig. 10(b). The contribution from this diagram to the numerator of rd, Eq. (2.10), can be written as Fi;‘i(Pfof

(i f 0,

(3.22)

where only labels corresponding to the relevant modes are explicitly written (modes vl represents the thermal-vacuum-bubbIe function, which is graphically expressed in Fig. 17, which we shall discuss in detail in Appendix A (see Eq. (A.4) and a related remark). On the other hand, for Fig. 10(b) it can be written as (modes j, k are summed)

j’, k’ are assumed to be already summed up). The function

Fi(P) Ui.

(3.23)

338

ASHIDA ET AL.

FIG. 16. An example of configuration where the generalized self-energy type graph coexists with, but is disconnected to, the main-body diagram.

The important point here is that ~i(i(p) # N_F,(P)I where N is a symmetry factor introduced in Eq. (3.7). Taking into account the contribution from the thermal vacuum bubble (Fig. 17) coming from the denominator of Eq. (2.101, we can obtain as the net contribution to r, (cf. Eqs. (A.6) and (A.7))

= C (pi(P)- NF,(P))Vi*

(3.24)

This fact is taken into account in the above discussion. When the generalized self-energy parts in Figs. 10, 12-14, 16, and 17 and all their crossing relatives are beyond the one-loop order, they still lead to the circled diagrams described by Figs. 11 and 15, with the “self-energy” subdiagrams of (b with momentum pi replaced by those in the corresponding loop-order. (See Appendix A.) Remaining subtleties yet to be discussed are the diagrams including graphicallydisconnected “bead” diagrams, which are constructed by linking together the generalized self-energy parts by 4 lines. We can show that these diagrams also lead to the corresponding circled diagrams, full account of which is given in Appendix B. 3.5. The Calculational

Rules

After the long-haul machinery developed above, we arrive at the formulas for the decay rate fd of a heavy scalar @ with four-momentum P (PO = E the energy), (3.25)

FIG. 17. An example of the thermal vacuum bubble diagrams.

339

ALGORITHM FOR TEMPERATURE REACTION RATES

where the “self-energy” part C2i(P) of @ is evaluated through the circled diagrams depicted in Fig. 18, where the blob in the figure is understood to be one-particle irreducible. Note that inclusion of the contributions coming from the counterLagrangian TC in Eq. (2.1) is immediate and does not spoil our goal, Eq. (3.25) with Fig. 18. As is clear from our derivation performed above, we can summarize the construction rules of Fig. 18 as follows (We are considering the lowest order analysis in g, 0( g2) (see Eq. (2.1)). We undo this approximation in the next subsection.): (i) Draw all topologically different one-particle irreducible Feynman diagrams contributing to the self-energy part of a heavy scalar Cp in the vacuum theory described by the Lagrangian (2.1). Then take the sum over all those diagrams, which is expressed by the summation symbol &,,, in Fig. 18. (ii) In every diagram, draw a circle on the external vertex from which d> flows out as shown in Fig. 18, leave the other external vertex into which @ flows intact. Exhaust all possible circling of the internal vertices so that the resulting diagrams cover all possible combinations of the circled and uncircled internal vertices, including the ones where none and all the internal vertices are circled. (iii) Reverse the sign of a vertex factor for a circled vertex. Note that, among the vertices, we include the “counter” vertices, coming from the counter-Lagrangian Z. (iv} Replace the 4; propagators with the appropriate ones displayed in Fig, 7. More precisely, calling uncircled (circled) vertices type-l(2) vertices, we assign iDjk (see Fig. 7) to the propagator from a type-k vertex forward a type-j vertex. Note that the blob in Fig. 18 contains one Q, propagator, the form of which is given by Eq. (3.13) with m + M and n,((p,l) -+ 0 (see the second paragraph after Eq. (3.19) in Section 3.2). (v) Sum up all these circled diagrams constructed in the above processes (ii)--( which is denoted by the summation symbol Cint in Fig. 18. It should be noted that the terminologies used here are nothing but those representing the “circling rules” in Ref. [9]. As already mentioned, in the present case of a heavy scalar decay process at finite temperature the caiculational rules thus obtained for the decay rate r; are already known [9, 13, 16, 23, 24-J. In this sense, what we have done up to this stage is to have succeeded in reproducing the known results -and adding some physical insight into them through our own approach. However, as we noted repeatedly, generalization of the above calcula-

-iE,,tPf

FIG.

595,215;2-8

=

18. Circled diagram~tic

x top

x int

;

@jjj)

!P

representation of the “self-energy” part ,Zzl (p).

340

ASHIDAE-l-AL.

tional rules to those evaluating the reaction rate of a generic dynamical process taking place in a heat bath is straightforward and is given in Section 5. 3.6. Decay

of Thermal

@j’s

Analyses presented so far have been carried out under the following two approximations: (i) The heat bath is constituted only by the thermal light 4 fields, namely the heavy @ field is not thermalized, and (ii) only the non-trivial lowest order process with respect to the coupling g (associated with the interaction vertex @‘@, see Eq. (2.1)) is considered. Here we briefly study the general situation undoing the above two rest~ctions, namely, a heat bath is composed of heavy Q’s as well as light #‘s in thermal equilibrium, and decay processes proceed through arbitrary orders in the coupling g as well as in ;1 (Eq. (2.1)). In the analyses hitherto presented, because of restriction (ii) above, there has been no need to refer explicitly to the ultraviolet reno~alization, whereas now there is the need to study the effect of the renormalization [6, 12, 27, 281 explicitly. For the present purpose we should renormalize the theory at the finite renormalization temperature T= j-’ through the “on-shell” renormalization scheme [ 12,271. With the above remarks the reader now can easily recognize the following: The decay rate r, of heavy scalar Q’s with momentum P, thermally distributed in the heat bath in equilibrium, can be expressed, in place of Eq. (3.25) as

c/=&(s)

Z;,,(P)

(E= Po),

(3.26)

where n,(E), the Bose-distribution factor (Eq. (3.6)), is the average number of Q’s in the heat bath under consideration, The C,,(P) in Eq. (3.26) again can be represented graphically in Fig. 18. Needless to say, however, the blob in the figure now represents a one-particle irreducible diagram with a number of aj2& as well as #3, vertices. The form of the four types of @-propagators is nothing but that of #‘s with the mass M (Fig. 7), and the vertex factor ig/-ig is assigned to the uncircled/ circled-@4 vertex. The same is true for the “counter” vertices contained in the counter-Lagrangian gC. The rate r, of the decay of thermal heavy ~0% taking place in the heat bath coincides exactly to the rate r, of producing the same Q’s in the heat bath, hence the term thermal equilibrium (see the next Section 4 for the detaiIs). The result itself in this subsection also has been known already [9, 13, 16, 241.

4. HEAVY SCALAR PRODUCTION Here we consider the production of heavy scalar Q’s in a heat bath in which both &‘s and Q’s are in thermal equilibrium, as in Section 3.6, and reproduce the known

ALGORITHM

-i FIG.

19.

E ,,(P) Circled

FOR

TEMPERATURE

REACTION

=

c

c

;

top

int

diagrammatic

representation

341

RATES

P

of the “self-energy”

part

Z,z cp).

result [9, 13, 16, 19-21, 23, 241. Through an argument completely parallel to the case of the decay process, we can show that the production rate r, is given by I-&

{l +n,(E)}

CiJP)

(E = P,),

(4.1)

with P the four-momentum of the produced @. The “self-energy” part C&P) in Eq. (4.1) is evaluated through the circled diagram depicted in Fig. 19. We illustrate one example; a configuration depicted in Fig. 20 leads to the circled diagram in Fig. 21 with pi0 < 0 and pi0 > 0. It should be noted that the factor n,(E) in the curly bracket represents that this part comes from the stimulated emission. From the circling rules established in Sections 3.5 and 3.6, we can easily prove [9, 13, 161 the relation Z,,(P)=e-~BPoG,,(P),

substitution

(4.2)

of which into Eq. (4.1) yields

rp=2 n,(P)C,,(P).

(4.3 1

Comparing Eq. (4.3) with Eq. (3.26) we can see that the decay rate of Q’s that constitute a heat bath is exactly equal to the production rate of Q’s in that heat bath. Thus the Q’s in question are in thermal equilibrium, as they should be. Now let us consider the case with special physical interest, where the distribution of Q’s in a heat bath shows a small but non-trivial deviation from that in thermal equilibrium. In this case the net decay rate ryt of those @‘s in the heat bath can be expressed as [ 13, 16,241

=$

FIG. 20. heavy G’s

An example

of “two-loop”

Cf(P)G(P) - c1+f(P)l c;,(p)l,

configuration

giving

contribution

to the thermal

(4.4)

production

of

342

ASHIDA

FIG. 21. Two-loop pkO > 0) comes from

circled diagram, the configuration

whose partial in Fig. 20.

ET AL.

region

of the loop-momenta

(pjO>

0, pie. and

where f(P) denotes the average number of Q’s with momentum P. C;(P) in (4.4) are functions that depend in a complicated manner on the explicit form of the distribution of Q’s in the heat bath, and one cannot construct simple calculational rules such as those summarized in Section 3.5 by referring to the circled diagrams. In the case with a small departure from equilibrium, however, the r?’ itself is small, and thus (as the first approximation) one can use [ 13, 16,241 the exact equilibrium distributions in evaluating Zl(P) and Z’,,(P) in Eq. (4.4). Then one can delete the primes from L$(P)‘s. By using the relation (4.2) we obtain C’-~

(4.5)

{f(P)-n,(P,)}CC2,(P)-I,,].

Thus we can recognize that the physically relevant quantity that governs the thermalization process is the difference, Z,,(P) -C,,(P). Finally let us recall the fact that, when applying the circled diagram rules [9] to evaluate the imaginary part of an arbitrary RTFT diagram to the physical component C,,( = Z, 1) of the RTFT self-energy matrix of a heavy scalar @, we can obtain the formula

where C,, and C,, are those in Eq. (4.5) (Figs. 18 and 19). With Eqs. (3.26) and (4.1) we obtain Im .Z@,( P) = - -

(P,=E>O).

the use of

(4.7)

This relation may be called the generalized optical theorem if we want. However, the quantity itself represented by Eq. (4.7) is not of direct physical interest (cf. Eqs. (4.5) and (4.7)). So far the extraction of the physical quantity, ryt, from the knowledge of Im z’,, alone has required us to introduce a “diagonalized” or quasiparticle selfenergy C (cf. Eq. (B.30)) as an intermediate device [ 15, 16, 191, or else to study the self-energy in the imaginary-time formalism [ 133 or the retarded RTFT self-energy [24]. In our approach, ,?YZ1and L’,, (and then ryt) can be evaluated directly from the first principle.

ALGORITHM FOR TEMPERATURE REACTION RATES

343

5. GENERAL PROCESS Extension of the arguments presented so far to more general cases is totally straightforward. With the correct treatment of a corresponding “main-body” diagram, the derivation of the reaction-rate formula proceeds completely parallel to the preceding analyses of decay and production processes, all other details of the analyses being unchanged. Therefore we shall not repeat here the full analysis, but rather present only the resulting reaction-rate formula, leaving the straightforward derivation to the readers’ exercise. Then, some new features that are not present in Sections 3 and 4 are discussed. Let us consider finite-temperature reactions of the following generic type, taking place in a heat bath of thermal light and heavy scalars #‘s and Q’s that are governed by the Lagrangian (2.1): I incident “particles” + heat bath -+ II detected “particles” + anything.

(5.1)

Here the “particles” are general enough; i.e., they can be either the on-shell constituent particles of the heat bath, or the foreigners to the constituents such as virtual C$‘S(c$*‘s) or the new heavy scalar H. As a first example of deriving the reaction-rate formula we study the virtual Q (qS*) process with the generic structure (5.1): I virtual q%*‘swith incident momenta p,, .... pl are injected into the heat bath and, after that, n virtual d*‘s with outcoming momenta q,, .... q,, are detected, q%*(p,) + ... + qS*(p,) + heat bath -d*(ql)+

... + d*(q,,) + anything.

(5.2)

This type of process is obviously of physical interest: (i) The case with I= 1 and n = 0 represents inelastic “lepton” scattering off of thermal Q’s and 4s in a heat bath, (ii) the case with I= 0 and n = 1 represents the thermal Drell-Yan process in a plasma, and (iii) the case with I = 1 and n = 1 represents the process: “lepton” + heat bath + “lepton” pairs + anything, etc. For this reaction process the derivation of the reaction rate formula proceeds totally parallel to that of heavy scalar decay (Section 3), but with the main-body diagram (f,@fz) having the following structure (cf. Fig. 22): the 1 virtual d*‘s with momenta p, (i= l-I), together with numbers of thermal 6s and Q’s flow into f, from the left and flow out from f z to the right; the n virtual q+*‘s with momenta qj (j= l-n), together with numbers of thermal d’s and Q’s flow out from fL to the right, cross the vertical cutting line, and then flow into f f from the left. As an example, the main body in this reaction which corresponds to that of Fig. 5 in the heavy scalar decay is shown in Fig. 22. We should bear in mind, however, that here the main body can consist of several disconnected diagrams. This causes no problem in the course of deriving the reaction-rate formula.

344

ASHIDA

ET AL.

FIG. 22. Main-body diagram in the virtual-4 process (5.2), which is the counterpart for the main body in heavy @ decay process, Fig. 5. Wavy lines stand for virtual d*‘s, while the solid lines for therma! (1s and 0’s. In this figure disconnected fL (fi) may be included.

As is evident from the analyses in Section 3, the replacement of the main-body diagram does not cause any change to obtain the final formula. Thus we can easily arrive at the following formula for the reaction rate r4* for this process (V is, as before, the very large volume of the heat bath),

= (Jj -2..-) A,*(py’, .... py’, q’ll’,...)q:l’; PiI’,*..>P!l’,4:? .... S!?)~

(5.3)

where A,. can be evaluated through the circled-diagram depicted in Fig. 23. In Fig. 23, only the circling arrangement of the external vertices is explicitly shown, the sum over all possible topologies of diagrams and all possible arrangements of circlings with respect to the internal vertices are assumed. It should be noted that Fig. 23 may include disconnected diagrams. A,, in Eq. (5.3) actually stands for the RTFT amplitude for the forward process; MPl)

+ ... +G%+YP,~+M41)+ -+ d:(Pl)

FIG.

23.

.‘. +Mqn)

+ ... +cwP,)+d:(91)+

Circled

diagram

repiesenting

... +eY4n),

A,.

in Eq. (5.3).

(5.4)

ALGORITHM

FOR

TEMPERATURE

REACTION

RATES

345

where by df/$$ we mean that the vertex it attaches (the external vertex explicitly shown on the large circle in Fig. 23f is uncircled/circled. Completely the same analysis applies for the process of new heavy scalars H’s of the type (5.2) in which H’s are substituted for d*‘s. We only remark that if the mass of H is much larger than the temperature of the heat bath, M, > T, then we can use the zero-temperature propagator for H in evaluating -4, (the counterpart of A,*). Next let us study a more complicated process, i.e., the process among the on-shell scalars Q’s with generic structure (5.1). This process is a pure thermal process taking place inside a heat bath (of &s and Q’s), a process which keeps the heat bath in thermal equilibrium. In this case it is not so straightforward to measure the reaction rate, which makes this kind of process physically not very interesting but of pure academic interest. We can, however, imagine some Cedanken experiments to measure the reaction rate of this process. In studying this type of process we should take note of the fact that particles involved in the reaction are nothing but the particles themselves constituting the heat bath. With this fact in mind and then repeating the preceding analysis, we arrive at the reaction-rate formula

x A&l’i’,

.,.) $2, &A’, ...) qf;‘“‘; p?), ...) p,(ii)) q1”‘, *.., 4fp’)>

(5.5)

where the product of Kronecker’s 6 symbols represents a spontaneous emission of !D, while N(p,)/N(qb) is the induced absorption/emission of rp’s. In Eq. (5.5) A, represents the RTFT amplitude for the forward process

(5.6)

where @r/Q, denotes that the vertex (the external vertex) it attaches is uncircled/ circled. Of course, the amplitude A, involves diagrams with all the possible topologies that represent the process (5.6) and, for each topology, it includes diagrams with all the possible arrangements of circlings with respect to the internal vertices. In Eq. (5.5) the summation symbol C’ stands for summing over all the possible arrangements of circlings with respect to the external vertices (at which external Q-lines attach), with the proviso that the following arrangements are excluded: at least one of the external vertices, at which the incoming momenta, say, p, (qb) attaches, is circled (uncircled) and, at the same time, the corresponding external vertex, at which the outgoing momenta pa (qh) attaches, is uncircled (circled). The reason for this exception becomes clear below.

346

FIG.

24.

A circled

ASHIDA

ET AL.

diagram

contained

in A,, in Eq. (5.5).

The reader may wonder why these come about the contribution coming from the circled diagram, e.g., as shown in Fig. 24, in order to get A,. If we take note of the fact that the reaction rate is given by Wz S*S (Eq. (2.9)), then it is not difficult to see that this circled diagram, Fig. 24, actually comes from the main body of the type as shown in Fig. 25. The meaning of Fig. 25 is obvious, and we only remark that the initial I @‘s in the S-matrix element side are type-2 spectators. It should be emphasized that in the reactions among particles, which are themselves the thermal constituents of the heat bath, in general there appears this kind of non-interacting pedestrian particles (type-2 spectators) that actually play an important role in evaluating the reaction rate. The readers now convince themselves that the configurations which are not included in Eq. (5.5) (as explained above, after Eq. (5.5)) do not correspond at all to the process under consideration. In the processes among virtual #*‘s and new heavy scalars H's the configurations in which the #*‘s and the H's are type-2 spectators play no role. The reason is that

p2

’

.

! .. . s

s 2* 91

.

.

-q2

.

. q2

9" fL

FIG. 25. Main-body diagram which momenta are not assigned

that corresponds explicitly stand

.

*

91

(4"

* fR

to the circled diagram, for thermal 4s and @‘s.

Fig. 24. The

solid

lines

on

ALGORITHM

FOR

TEMPERATURE

REACTiON

RAT!3

347

the virtual particles and heavy scalar particles have a finite lifetime and thus cannot be a pedestrian. This is the reason why their reaction rates are given by Eq. (5.3). Finally, considering the following “inclusive” @ production taking place in the heat bath, let us present the reaction-rate formula and give further discussion on the results including a finite-temperature generalization of the optical theorem. The process we take up is: An incident heavy scalar CD(if possible to be arranged) with momentum P coming into the heat bath from outside causes a reaction inside the heat bath to produce 4s and Q’s, and then one of Q’s, with momentum P’, is “detected.” We assume that the incident ai beam is weak enough so that, as the first approximation, to evaluate the reaction rate, we can use [ 13, 16,241 the equilibrium distributions for the thermal @‘p’sand 4s (cf. the discussion in Section 4 leading to Eq. (4.5)). Then this process is no more than the special case with 1= n = 1 of the generic on-shell heavy @ process discussed above, having, however, general enough structure for the purpose of illustrating the general structure of the thermal reaction rate. As in the general cases, taking note of the presence of the pedest~an particle that actually plays an important role, we can obtain the reaction rate for this process per one incident rf, as 2E’ dpfl:;n)l=~l(l+n,(E’)}.4(F,P’)+n,(C’)3(P,P’)].

(5.7)

where E/E’ stands for the energy of the incident/dete~ted @. The functions A and 3 in Eq. (5.7) are evaluated through the circled diagrams depicted in Figs. 26(a) and (b). Note that the contributions from Q’s (with momentum P) which are in thermal equilibrium in the heat bath are not included in Eq. (5.7). Now we examine the physical contents of the circled diagrams in Fig. 26, which can be disclosed through the procedure of deriving Eq. (5.7). For example, the first/second set of diagrams in Fig. 26(a) comes from the configurations shown in Fig. 27{a)~(b), which graphically represents a part of W- SS* in Eq. (2.9). It should be noted that the last two diagrams in Fig. 26(b) come from the con~gurations in which S* and S are trivial in the respective orders. These processes really take place in a heat bath. Experimentally the thermal Q’s in the heat bath are also detected, whose contribution should be subtracted off when confrontation with Eq. (5.7) is made. It should be remarked, in passing, that Fig. 27(b) also contributes to the rate of the following “forward inclusive” decay taking place in a heat bath; a heavy scalar @ (with momentum P) -+ two Q’s (with momenta P and P’). Now turn our attention to the imaginary part of a forward RTFT amplitude. In this respect it is worth mentioning that the circled diagrams in Fig. 26 form a part of those representing as a whole the imaginary part of a 2-body forward amplitude in RTFT. In fact, according to Kobes and Semenoff [IS] the imaginary part of the 2-body @@ forward amplitude can be represented by the set of circled diagrams depicted in Fig. 28, where C,,, denotes to take the sum over all distinct topologies and Cint over all possible ways of circling the internal vertices. The symbol c,,,

348

ASHIDA

ET AL.

A@?‘) =&; {w +m + P

P (a)

(b) FIG. 26.

Circled

diagram

representations

for the reaction

rate of the “inclusive”

(a) FIG. 27. Physical “reaction” diagrams in Fig. 26(a).

lb) configurations

c(cc ext

FIG. 28. Circled amplitude in RTFT.

diagram

@-production.

top

W=

S*S

leading

to the first

P’

int

representation

Xl of

the

and second

sets of circled

P’

P

P imaginary

part

of

the

two-body

00

forward

ALGORITHM FOR TEMPERAT~REREA~TIO~

EE

top

int

FIG. 29. A part of the contributions CpIzlforward amplitude in RTFT.

RATES

349

P

P

to the imaginary part, represented in Fig. 28, of the two-body

means to take summation over all those diagrams with both circled and un~ir~led external vertices (to which external lines attach). Namely a subset of circled diagrams inside the large curly brackets in Fig. 28 consists of all the circled diagrams with definite circling of the external vertices, physical content of which may now be easily analyzed. As an example, we consider a set of circled diagrams depicted in Fig. 29. One can see that Fig. 29, with P,, Pb> 0, contributes to the reaction rate, f2& P, P'), of two heavy scalars immersed simultaneously in a heat bath, which is, of course, not a simple product of each decay rate (3.26). Figure 29, however, does not exhaust all the contributions to f 29( P, P’). In fact, also contributing to fzQ(P, P’) is the following; first four circled diagrams reproducing B in Fig. 26(b), a set of circled diagrams with all but the one external vertex from which momentum P’ flows out uncircled (cf. Fig. 28), and a set of those with all but the one external vertex into which P’ Rows circled. Thus the correspondence between a definite set of circled diagrams (each set of which is enclosed within the curly brackets) in Fig. 28 and a definite type of reaction rate is not one to one. Generalization to more general forward RTFT amplitudes is straightforward and one can see that the imaginary part of any forward RTFT amplitude, say A, evaluated through the circled-diagram prescriptions [9] consists of contributions from various transition rates T,‘s corresponding to different reactions that are mutually related by, as it were, the generalized crossing substitutions:

ImA=Cf,L

(5.8)

where f:s stand for kinematical factors. Thus the imaginary part itself may not be of direct physical interest, as in the simplest case (4.7).

6. GENERALIZED

CUTKOSKY

RULES

Through the analyses developed above on the basis of Eqs. (2.6), (2.9), and (2.10), etc., the following fact is disclosed. In any “forward” circled diagram the circled (uncircled) vertices come from the interaction vertices in the S-(S*-)matrix element in Eq. (2.9). With this fact we can clarify the physical content implicated by the forward circled diagram, i.e., the relation between the forward circled

350

ASHIDA

ET AL.

diagram and the set of reaction amplitudes that represent corresponding dynamical processes taking place in a heat bath: For any given forward circled diagram, by tracing back the similar procedure as above of deriving the circled diagrams, we can write out the corresponding physical process taking place in a heat bath and thus can cut any forward circled diagram into diagram segments; one belongs to the S-matrix element and the other to the S*-matrix element (cf. Eq. (2.9)). In this sense, any forward circled diagram can be cut (or cuttahle). This can be regarded as a finite temperature generalization of the Cutkosky rules [8] in vacuum held theories. For completeness, here we enumerate in order the procedures to be summarized as the cutting rules at finite temperature: (i) For a given forward circled diagram, rearrange the external lines, without changing the topological structure of the diagram, so that the energies flow into the diagram from the left direction and flow out to the right direction. (ii) Perform continuous deformation without changing the topological structure of the diagram so that all the uncircled/circled vertices are laid within the left/right half of the diagram. (iii) Cut all the (internal) propagators linking an uncircled vertex to a circled one, i.e., linking the left half of the diagram to the right half. (iv) Rearrange each piece of the cut internal propagator with momentum p, which is the integration variable (i.e., the loop momentum of the original circled diagram) reassigned to flow from the uncircled vertex in the left half of the diagram to the circled vertex in the right half of the diagram, as follows: After the rearrangement the positive energy always flows from the left to right direction. Namely, in the integration region where the energy of the cut propagator pO is positive leave the two pieces of the cut propagator intact, while in the integration region where pO is negative rotate each piece of the lines of the cut propagator in a 180” arc round the corresponding vertex to which it attaches, so that the positive energy, now -pO, flows from the left to right direction. (v) For each propagator linking between two uncircled vertices, both of which have been rearranged to lie in the left half of the diagram, Fig. 30(a), construct another two corresponding diagrams with different configurations shown in Figs. 30(b) and (c), and obtain three mutually relative diagrams in total. (Note that in Figs. 30 only the left half of the diagram is shown explicitly, with the appropriate right half assumed.) After this operation, now the propagator with momentum pi in Fig. 30(a) is to be the propagator in the oacuum field theory, and the on-shell particle with the mode i (momentum pi) in Fig. 30(c) is to be the antiparticle of the corresponding particles in Figs. 30(a) and (b). It is to be noted that Fig. 30(b)/(c) contributes as a part to build up the positive/negative pi0 part of the thermal propagator with momentum pi in the original circled diagram. (vi) For each propagator linking two circled vertices both laid in the right half of the diagram, perform the same operation as in (v) and obtain three mutually relative diagrams.

ALGORITHM

FOR

TEMPERATURE

REACTION

RATES

351

I

Jf \

.

"2

"1

7

Ieft-side

i

. I

cutting line

(b)

(a)

(cl FIG. 30. Three mutually relative diagrams with two uncircled vertices I/, and Vz to which #J’S in interest attaches. These figures represent only the Left half (i.e., the S-matrix element) of the fufi “reaction” conftguration W= S*S.

(vii) In each diagram thus obtained, add appropriate numbers of type-2 spectators so that the right-most “state” of the left-half diagram will be identical to the left-most “state” of the right-half diagram. It is, then, obvious that, because we start from the “forward” circled diagram, in every diagram constructed through the above procedures the left-most “state” of the left-half diagram is idetntical to the right-most “state” of the right-half diagram, as it should be. (viii) Add appropriate numbers of type-l spectators. Procedures (i)-(viii) complete the cutting rules in finite temperature field theories. As an example of the above cutting rules, we examine the circled diagram Fig. 11(c) with pie, pi0 0, and get Fig. 31, from which we can easily recognize what kind of reaction really takes place, together with the corresponding thermal vacuum bubble diagram. i Ik i

FIG. 31. Configuration obtained by applying the “cutting rules” to the circled diagram Fig. 1l(c) with pi”, pi0 < 0, and pkO> 0.

352

ASHIDA ET AL.

It is worth mentioning the following: A somewhat bizarre diagram Fig. 11 (c) has been regarded so far only as playing a “passive” role of eliminating the {S(pf--m*))* singularity [9, 15, 16, 19, 201 and has been thought to be noncuttable. It is now obvious, however, that it really represents a set of physical processes, such as those depicted in Figs. 10(j), 31, and their crossing relatives (related to Fig. 10(j) by crossing substitutions), and thus can be cut in the sense defined above. It is to be pointed out that Kobes [24] made a different finite-temperature generalization of the Cutkosky rules in vacuum theory. 7. MISCELLANEA 7.1. Chemical Potential

Here we briefly discuss the problem of introducing the chemical potential in the present formalism. For definiteness let us consider the system that consists of the self-interacting complex scalar fields 4 and d* with the conserved charge

Q = j- dx 4*(x) d(x).

(7.1)

In order to describe the system whose total charge can freely change its value, we introduce a grand canonical ensemble of the above system and introduce the chemical potential p being conjugate to the charge Q. Then the heat bath is constituted by those particles (positively charged) and antiparticles (negatively charged), both of which are in thermal equilibrium. Thus, as a straightforward generalization of Eq. (2.11) in the case of neutral scalars, the statistical average of a generic quantity A now can be defined by

(A)=

~l,(+,,,t-,l

exp(-axO= f C,n:“‘(E,-a~)} A({n:+‘, C{n)+~.n:-~jexp(-BCrr= +- C,nj”‘(E,-0P))

tii-‘})

’

(7.2)

where n:*) stands for the numbers of particles/antiparticles in the mode 1, and c is the summation with the symmetry factor being respected, as it was in Eq. (2.11). If there are other fields being coupled to 4 and 1+5*,which are also constituting the heat bath, then, needless to say, the corresponding statistical sum over their degrees of freedom should be included in Eq. (7.2). Through the analysis completely parallel to the neutral scalar case, we can easily derive the following expressions for the four types of propagators, the building blocks of perturbation analysis:

D,,(p)= p2-mm?+k l - 2~indlA - P~P/I~~I ) d(p*-m*h D21(P)

=

-2N&po)

+ ndlp,l

- pop/lpol 11 &p* -m*),

(7.3a) (7.3b)

ALGORITHMFORTEMPERATUREREACTION

353

RATES

D21(P) = --27%0( -Po) + ndlp,l - PoP/lPol )I m* - m2), b*(P)= -cD,,(PIl* -1 =p’-m2-iE

- 27%(lP,l

- PodlPol) Q’-

m2),

(7.3c)

(7.3d)

which are the counterparts to Eqs. (3.13), (3.16), (3.18), and (3.19) in the neutral scalar case. Every analysis in the neutral scalar case can be repeated also in the present case, with a straightforward manner. 7.2. Fermions

In the case of fermion systems, when evaluating the transition rate of a reaction considered we should take notice of the anticommutativity of fermion fields. Thus in taking the sum over final states of the reaction considered, as in Eq. (2.10), and also in taking the statistical average over systems constituting the grand canonical ensemble, as in Eqs. (2.11) and (7.2), we should restrict the occupation number in each mode to two values, 0 or 1. Every analysis in the case of fermionic systems, or of systems including fermions, can be carried out with the above remarks. In this paper, however, we do not go into further details.

8. SUMMARY

In this paper we have found a set of systematic rules that enables us to evaluate any finite temperature reaction rate, i.e., the transition probability of any reaction taking place in a heat bath in equilibrium. The calculational rules presented here are stated in terms of the so-called circled diagram, a Feynman-like diagram introduced by Kobes and Semenoff [9]. which is equivalent [23,25] to the Feynman rules in the real-time thermal field theory (RTFT) formulated on the basis of the kind of “closed-time path” in the complex-time plane [4, 261. The rules in the neutral scalar theory in evaluating the reaction rate of some generic physical process in a heat bath can be summarized in short as follows: (i) Identify a relevant forward amplitude that represents the reaction rate in consideration. For details, see Section 5. (ii) Draw all topologically distinct Feynman diagrams contributing to the above forward amplitude in the vacuum field theory. Then take the sum over all these diagrams. (iii) In every diagram identify the appropriate set of circlings of the external vertices so that the resulting circled diagrams can contribute to the reaction rate in question (For details, see again Section 5). Then in every diagram exhaust all possible circlings of the internal vertices (including, of course, no circling at all).

354

ASHIDA

ET AL.

(iv) Reverse the sign of the vertex factor for each circled vertex. (v) Repiace the propagators with the appropriate thermal propagators shown in Fig. 7. More precisely, naming the uncircled (circled) vertices the type-l(2) vertices, we assign iDjk (see Fig. 7) to the propagator from a type-,4 vertex toward a type-,j vertex. (vi) Sum up all the above circled diagrams. The precise meaning of these rules can be easily recognized in referring to the analyses hitherto made in this paper, especially those in Sections 3.5 and 5. Having these rules in hand we can make a rigorous and thorough analysis of any reaction taking place in a heat bath. Thus we obtain a theoretical tool to attack the physics of hot QCD, or the quark-gluon plasma [2,3], and of evolution of the universe [ 1,2 J in its hot and dense era. Our machinery of derivation having arrived at the systematic caIculationa1 rules has enabled us to obtain the following important fact: For any circled diagram belonging to a set of circled diagrams which represent a generic reaction rate, there exists without exception a corresponding physical process taking place in a heat bath. More precisely, after some topological deformation and rearrangement any circled diagram, say T, contributing to some reaction rate can be cut off into the sum of two diagram-segments (one of which is constituted of only the uncircled vertices, the other of only the circled ones). Note that each of the resulting two diagram-segments is not necessarily a connected diagram. Then we find that the diagram segment S,‘s (Sf’s) constituted of only the uncirded (circled) vertices represent the S(S*)-matrix elements for the corresponding reaction process taking place in a heat bath, T-C Sz @SL. All the 5’2 OS, are shown to be mutually related through appropriate crossing substitutions. The above finding helps us to picture a visible image of the dynamical process taking place in a heat bath. The “cutting,” briefly described above, of the circled diagram can be considered as a finite temperature generalization of the Cutkosky, or the cutting, rules [8] in quantum field theory at zero temperature (for more details, see Section 6). Finally we briefly comment on the following fact. Kobes and Semenoff have established [9] an efficient method to evaluate the imaginary part of any physical amplitude, i.e., an amplitude with all the external lines restricted to the physical (non-ghost) lines, in the RTFT, which stimulated many elaborate studies in this area. As discussed in Section 5, the imaginary part of a “forward” RTFT amplitude evaluated through their method is shown, in the light of the calculational machinery developed in this paper, to consist of contributions from different reaction rates mutually related through the generalized crossing substitution and thus may not be the quantity of direct physical interest by itself.

ALGORITHMFOR

APPENDIX

TEMPERATUREREACTION

A: GENERALIZED

SELF-ENERGY

RATES

355

TYPE DIAGRAMS

In this appendix we study in detail the configuration, as a part of which there exists the generalized self-energy type diagram that is graphically disconnected from, but is in fact weakly connected to, the “main-body” diagram in which the parent heavy scalar @ participates. The purpose is to show that contributions from the above type diagrams to the decay rate of a heavy @, together with those contributions from their main-body diagrams and from the thermal vacuum bubble diagrams, correctly reproduce the corresponding circled diagrams as presented in Section 3.4 in the text. Let us study them in due order. (1) First let us start by considering two configurations shown in Fig. 32, which contribute to the S-matrix element in W= S*S, Eq. (2.9). In these figures spectator 6s are not shown explicitly, and the symbol being denoted by {pt.) in the final state of the S-matrix element, stands for several 4s each with momentum p,-, and also the initial state d’s, each with momentum pi, are collectively shown as (p,}. Note that, as explained in Section 3.3 in the text, we can assume that there are no other 4 in the definite modes i, j, and k, than those expdlicitly shown in the figures. (Contributions from the opposite cases are suppressed by some powers of l/V in the limit V-+ a.) Then, with the use of the reduction formula (2.6) we obtain the following expression for the S-matrix element represented by Fig. 32,

’ 2n v6(Pi0 - Ej - Ek) 6(Pi; Pj + Pk 1

{Pi’,. {PA L

(A.1)

FIG. 32. Configurations representing the S-matrix element, where a segment of generalized selfenergy type diagram is connected to the main-body diagram. In these figures, only the Smatrix element part (i.e., the left half of the “reaction” configuration) is depicted.

595.?15!2-9

356

ASHIDA

ET AL.

where a(...; ... ) denotes the Kronecker’s S-symbol as before, nj is the number of #;s (4’s in the mode i), E = +“m is the energy of the parent Qi; the other notations are now obvious. As stated in the text, all the singular functions (including the 6 function) associated with the propagator functions, which appear in the course of our analysis, are assumed to be regularized with the help of finite-c: prescription [6] until arriving at the final result. With the $*-matrix element, S& obtained from Eq. (A.l) by replacing fL by fR and taking the complex conjugate of the resulting expression, we obtain the corresponding transition probability W= S,* S,, which consists now of four distinct contributions whose graphical representations are depicted in Figs. 10(a)-(c). Carrying out the sum over the modes i, j, k and over (~~3, then taking the statistical average of W thus obtained, we have (A.21

2Pi0 ’ n13flPiOl ui(PiOl

-

2R6f0)

F21(E>

p;

1

-22 nB(lPiOl

2

ZP~O

1

r: j.k

PiO*

PiI9

1 +nB(Ej)

2Ei V

(A.31 1 + nJ3(Ek)

2E, v (A-4)

where F2,(E* P; Pi03 Pi)

x2nV6

Cpf-Cpi;P-pi I /

(A-5)

It should be noted that the function F2, in Eq. (A.5) is nothing but the function defined previously in Eq. (3.5) in the text and thus has a definite V-independent limit as V+ co. The factor $ in Eq. (A.2) is the Bose symmetry factor in the sum over the modes j and k, and the summation symbol ZIPS) in Eqs. (A.2) and (AS) denotes as before the sum over momenta (ps> with the Bose symmetry of #s constitu~ing (prf being respected. It should also be noted that the function Ui of pi0 in Eq. (A.4) expresses that 46 (4 in the mode i) is on the mass-shell, i.e., pi0 = Ei =

ALGORITHM

FOR

TEMPERATURE

REACTION

357

RATES

dpw, the contribution from the thermal vacuum bubble represented by Fig. 17 (obviously I should be replaced by i). Now substitute Eqs. (3.31, (3.22), and (A.21, which express, respectiveiy, the contributions from Fig. 5, Fig. 16 with 1 fi, and Figs. 10(a)-(c), altogether in the numerator of Eq. (2.10) (the formula to give decay rate) and also substitute the function Orof Ei, i.e., Eq. (A.4) for the on-shell QIit which expresses the contribution from the thermal vacuum bubble in Fig. 17, in the denominator of Eq. (2.10), and we obtain the following expression for the contribution to the decay rate rd.

1 r

F

F,(P)

f

i;,

FiIp>

u,(Elf

+

--

C I

1%

Fi(p;

PiO)

%lPiO) 9

d - 2716(O)

1 +

1

(A&a)

ui(Ei) I

=.$‘o’+~d+

. . .

d

Tit----

1

&iO

2740~ zC( j

(A.6b)

>

~x

v

f;{P; PiO) ni(piO) - NFS(p) ui(Ei)

I

3

(A.71

where N is formally defined by Eq. (3.7) in the text, i.e.,

(‘4.8)

In Eq. (A.6b), from Fig. 5. It is worth Ei =,/‘~:+Pz~, related remark

rio’ stands for r>F’g.5) in Eq. (3.8), which expresses the contribution noting that the S-function factor involved in the function ai of i.e., t’r(Er) in Eq. (A.7), can be rewritten as (see Eq. (A.4) and the above),

Zns(O) 6(Ej - Ej - E,) = 2n

s

~~io(2~io)2 s*(pt - m2) e( Pie) 6( Pi0 - Ej - E,).

(A.9)

With this rewriting, pd, Eq. (A.7), becomes

(A.ll)

358

ASHIDA

ET AL.

Then with Eqs. (A.3), (A.4) and with Eq. (3.4) in the text, we finally obtain {l +nB(Ej))(l

+n13(Ek)l

2Ej V 2E, V

1 “-rz

d”p, (-i%) I (2n)4

iD**(pi){

-,~~‘+

+ (pi)}

iD,,(pi)(U)

(A.13)

xF2zl(p-Pi),

where

x (kiA)

iDZl(Pj)

i”21(Pk)(iA),

(A.14)

and P, = E is the energy of the parent @. In the above equations the superscript + + on .Z stands for the fact that the corresponding C represents the contribution from the “region” pjo > 0 and pkO > 0. The decay rate r, given by Eq. (A.13) with Eq. (A.14) is represented, with some inessential factors, by the circled diagram depicted in Fig. 1l(a) with all relevant momenta restricted to the positive-energy region, i.e., {piO, pjO, pkO > 0) and, of course, P, > 0. Figures 32, or Figs. 10(a)-(c) yield various new diagrams through various crossing substitutions with respect to di, dj, and &. We can handle all such “crossing-relative” diagrams in a similar manner as above. For example, let us consider the diagrams obtained from Figs. 32 by replacing ij emitted into the final state with dj absorbed from the initial state (a crossing substitution with respect to 4j). These diagrams lead to (A.13) with C$:)-+ in place of Z\:)++, which corresponds to the (piO, pkO > 0, pjO < 0) sector of Fig. 11 (a). Now it is easy to understand that the diagrams in Fig. 32 (or equivalently, Figs. IO(a)-(c)), together with all their “crossing relatives,” yield the contributions to the decay rate (Eq. (A.13) with (A.14)), which as a whole exhaust the entire energy-region of relevant energies - cc < piO, pi,,, pkO < + cc in cooperation with all its relative contributions such as those explained above in Eqs. (A.6). These contributions to the decay rate can be expressed collectively by the (2211 )-component of ryw 3 ry 3

(a, b, y, 6 = 1 or 2, and not summed),

(AX)

ALGORITHM

FOR TEMPERATURE

REACTION

359

RATES

where

x (-2)

iD,,j(q) iDz&r)(ii),

(A.16)

and P, = E. Evidently ri2i1 is represented diagrammatically by the “entire” circled diagram depicted in Fig. 1 l(a). In the above, for the brevity of presentation, we have studied the generalized selfenergy type diagrams within one-loop order. We can show, however, that the above analysis works to arbitrary loop orders of the complete interaction Lagrangian including the counter terms and that the result can be summarized by the quantity ry”, Eq. (A.15), in which the one-loop “self-energy” part Ck:‘( p) should be replaced by its counterpart C,,(p) in the corresponding loop order. This can be easily understood by noticing the following fact: The generalized self-energy type diagram that we are studying now is the “generalized one-particle irreducible” diagram with obvious meaning. Thus it is sufficient to consider the case where any C$that participates in the generalized self-energy part of tii is in the mode different from i; otherwise the discussion in Section 3.3 can be applied to show that the configuration in which there are several 1+5’sin the same mode i gives only a vanishing contribution. (If it did really contribute, then it would yield statistical factors different from those in (A.3) in taking the statistical average.) In passing it is worth mentioning the following: C,, (p) is nothing but the selfenergy matrix of d-field, evaluated in the arbitrary loop order through the circled diagram rules. The diagonal elements, C,, and C 22, coincide [6], respectively, with the self-energy part of the physical &field, C,,, and that of the thermal ghost of $-field (usually denoted as $), CW in the framework of the RTFT: E,,(P) = I,,,

(A.17a)

222(P)= - [I~I,(P)l* = I,.

(A.17b)

As for the off-diagonal elements, it is easy to show [29] that the following relations hold between C,,(p) and the corresponding quantity C,&) (incident [ and outcoming 5, both with momentum p; [, 5 = 4 or 6) in the standard’ RTFT (cf. Eq. (4.2)): C,,(P)( = -G,(P))

= eopo~‘,2(P) =e ~po’2C~&2) = ePPo”Z,m(p).

’ By the standard [30], -co-+ +m-+ are equivalent [29]

RTFT we mean thermo-field dynamics [S] or the RTFT +cc-ij/2+-cr,-i/l/2 + --ic -i/?, in the complex-time as far as perturbative expansions are concerned.

(A.18)

based on the time path plane, both of which

360

ASHIDA

ET AL.

(2) By following the analyses completely parallel to those in (1) we can show the following: Contributions to the decay rate I-, from diagrams in Figs. 10(d)-(i), together with all of their “crossing relatives,” collectively yield, thanks to the interplay with the relevant thermal vacuum bubbles, the (2111)-component of Cifiy”, Eq. (A.15) with (A.16), i.e., ryi’, whose diagrammatic representation is given in Fig. 1 l(b). Yet other conligurations represented by the mirror images of those in Figs. 10(d)-(i) lead to r?j22’ (cf. Eqs. (A.15) and (A.16)). This is essentially the complex-conjugate to f 5”’ obtained from the original configurations, Figs. 10(d)-(i), and is represented diagrammatically in Fig. 1l(d). Generalization to the generalized self-energy part beyond one-loop orders is immediate as in (1). (3) Next let us study the contributions from the diagram in Fig. 10(j) and all of its “crossing-relative” diagrams, together with those from the relevant thermal vacuum bubble diagrams. They contribute to the decay rate r, collectively summarized in the form r:lrl (cf. Eq. (A.15)), whose diagrammatic representation is depicted in the circled diagram 11 (c). Generalization beyond the one-loop order is again immediate. (4) Finally let us study the configurations, in which both the ~i’s (d’s in the mode i) in question are joined to the fL in the S-matrix element part (one of which flows out from fL, the other flows into fL) and no di in question is in the f,$ in the S*-matrix element part. Several examples of such configuration are shown in Figs. 13 and 14. We can analyze these diagrams in the same way as above and can show that, by taking into account correctly the contribution from the relevant thermal vacuum bubbles, Figs. 13 and 14, together with their many “crossing-relative” diagrams, lead to the corresponding circled diagrams in Fig. 15, which represent diagrammatically rAp)‘i (p, y= 1, 2) (cf. Eqs. (A.15) and (A.16)). Note that F,, appearing in Eq. (A.15) in the present case is equal to pi, appearing in Eq. (3.15). The mirror symmetric cases to the above, namely, the conligurations in which both the &‘s in question are joined to thefz in the S*-matrix element part, can be analyzed similarly, leading to Tifi2 (/?. y = 1, 2), which correspond to the circled diagrams Fig. 15 with Pi, replaced by F22 and two uncircled vertices in F,, at which the 4 lines with the momentum pi attach to the F1, replaced by the circled vertices. Generalization beyond the one-loop order is immediate.

APPENDIX REDUCIBLE

B: “GENERALIZED ONE-PARTICLE SELF-ENERGY” TYPE DIAGRAMS

In this appendix we study the configurations where there exist some number of improper (i.e., reducible) generalized self-energy type diagrams, which are mutually graphically disconnected from each other and also from the main-body diagrams, but these diagrams and the main-body diagrams as a whole are weakly connected.

ALGORITHM

FOR TEMPERATURE

REACTION

361

RATES

We remind the reader that the main body is a part of the diagrams that involves the parent heavy scalar @ whose “decay” process we are considering. What we show here is that the contributions from such configurations to the decay rate rd of the heavy scalar @ can be represented exactly in terms of the corresponding circled diagrams. Let us start our analysis by studying the configuration depicted in Fig. 33, a generic configuration that contributes to the thermal vacuum bubble, namely the thermal vacuum transition rate W= S*S. In this figure both in the initial state (i.e., the left-most state in the figure, which is identical to the right-most state in the figure) and in the final state (i.e., the state along the vertical dot-dashed cutting line in the center of the figure), we are focusing our attention to the &field in the mode i, &, which is subject to the statistical ensemble average and with which the whole configuration is weakly connected, other 4’s without special interest not being shown explicitly in the figure. (Contributions from 8s in any other mode j ( #i) are totally canceled out between numerator and denominator in Eq. (2.10) when calculating the decay rate r,.) As for the bottom group in Fig. 33 the left-most

Nil

N22

N21

FIG. 33. A generic contiguration that contributes to the thermal vacuum bubble. The blobs in the figure represent f’op’s, the “improper generalized self-energy” parts, and are used solely in Appendix B. The #‘s, except those in the mode i of interest, are not shown explicitly. The number N: can be smaller than N,, which means that Nlz is larger than N,,.

362

ASHIDA

ET AL.

state (i.e., substate of the initial state of the S-matrix element) and the right-most state (i.e., substate of the initial state of the S*-matrix element) are the identical state, as is the case in the whole diagram, Fig. 33. The bundle of 1,4’son the top of Fig. 33 is spectator 4’s, and N&/N: denotes the number of them in the S/S*-matrix element. N,,/NZ2 denotes the number of oneblob graphs in the S/S*-matrix element in the upper second group in Fig. 33, and Nz, and N,2 represent respectively the numbers of one-blob graphs in the third and the fourth (i.e., the bottom) groups. Now we explain the four types of blobs in Fig. 33, denoted by - ifra (~1,/I= 1,2), with the proviso that -ii-,, appearing in the bottom group denotes the complete blob shaped by stitching the pair of half-blobs in the left and right ends of the diagram. These four types of blobs represent the sorts of “improper (reducible) generalized self-energy” type diagrams constructed by linking the generalized one-particle irreducible self-energy parts, which have been already constructed in Appendix A, with propagators in the vacuum theory,

-i&(p)=

[

-S(p)

f

1UP

(iD’“‘(p)(-iZ(p)))’

= -it,,(p)

+ C(-it(p))

(B.la)

,

I=0

i~‘“‘(p)( -if@ ))La

(48=1,2), (B.lb)

where p = pi, D$)(p) is the 2 x 2 matrix defined by

03.2)

and ,TE,(p) is the one-particle irreducible self-energy matrix introduced in Appendix A. Zz,(p) can be evaluated through the circled diagram rules and its expression in the one-loop order is given in Eq. (A.16). Note that D”‘(p) in Eq. (B.2) is not subject to the statistical average. With the relations (A.17b) and (A.18) we can derive the same relations among 27’,p’s, i.e.,

The contribution

i‘*,(P)

= - (G,(P))**

&2(P)

= e-@~%(p)(

from the configuration

(B.3a) = - (z,,(p))*).

(B.3b)

depicted in Fig. 33 to the reaction

ALGORITHM

probability as w,

3

FOR

TEMPERATURE

REACTION

RATES

363

W= S*S, W,, is then expressed, with the use of Eq. (2.6) without @,

s*s

(n- N2,+ N,,)! n! =(n-N,,-Nzl)!

It’ N,,! %.,j=l

(n-N22-N21)!

[ -27-S + .TzJp)l NyB, ^

U3.4)

where 6, =6(p2-m2)8(po)

(B.5)

and n-ni=N,y+

N,, +NzI = N:+ Nz2+ N2,

03.6)

is the number of &‘s in the initial state (i.e., the left-most state = right-most state in Fig. 33). Taking the sum over final states in Eq. (B.4) and averaging over the statistical ensemble, then after some manipulations we obtain (see, Eqs. (2.10) and (2.11))

=c rmax c (N,,+N,d! l! (N,, {NQ)

/=O

(N,,+N,,)! (C;,+, N,,-l)! + N,2 - l)! (N,, + N,, - I)! (B.7a)

L,, = Min{N,, + N,,, Nz2 + N,,),

(B.7b)

where we used the shorthand notations

and

Aa&) = -277L3 + &l(P)

(a, B= 1,2).

03.8)

Though we have not yet succeeded to get a closed compact form of (1 W,), Eq. (B.7) (N.B. the partial sum over N,, is easily carried out), we would like to guess that Eq. (B.7) takes the form

( > 1 W,

=(e-1)[e-1-{A,,+A,,fA,,A22+A2,+eA,2-A,~A2,}]~1.

(B.9)

As for the validity of Eq. (B.9), we have proved that when expanding Eq. (B.9) in powers of A%B)sthe expansion coefficients coincide exactly with the corresponding

364

ASHIDA

ET AL.

coefficients in Eq. (B.7) in the following eight cases (parameters not indicated are completely arbitrary); (i) N,, = 0, (ii) Nz2 = 0, (iii) N,, = 0, (iv) I= 0, (v) I = 1, (vi) I= 2, (vii) I= 3, (viii) I = 4. Also the validity of Eq. (B.9) is checked at several values of I and Nlg’s, extracted under the random sampling method. Next let us study the configuration where the parent heavy scalar @ that is the “decaying” particle in question directly participates. First we consider the configuration shown in Fig. 34(a). In this configuration no thermal pi participates, and thus its contribution to the decay rate r, takes a simple expression similar to Eq. (A.15), 1 rd=2E,

J-d4p (2rc)4

(-iA)

@2;,(P)(iA)

F21(P,

(B.lO)

Ph

where E@=P, is the energy of the parent CD (E in Eq. (3.12)), F,, is given in Eq. (3.5) or (A.5), and (B.ll)

s&(p) L 3;;“. 34’a’(P) = %21(P),

Lqj(p) = [D’o’(p){ 1 + (-L&J)) ~~‘“‘(P))l,l,

(4P=42),

(B.12)

with D(O) and 2 being given in Eqs. (B.2) and (B.l). Then we consider the configuration where the main-body diagram shown in Fig. 34(b), in which parent Q’s participate, coexists with the thermal vacuum bubble diagram depicted in Fig. 33. The rectangular blob in Fig. 34(b) represents the “two-point function” including its free part, namely, gg$J = (@‘(p)),’

+ ( - j f&g)

(a; not summed),

(B.13)

with c1= 1 and 2. As in the above thermal vacuum bubble calculation we obtain the contribution from this “coexisting” configuration, Fig. 33OFig. 34(b) to (C W) = (C SS) as

x Cl + i@?(p)( -i~,,(p))l XC t&p) x(e-l)‘-’

i2F2,(P, p)

‘ma~1(N,,+N,~+1)!(N22+N,2+1)!(~~,P~, f!(N,,+N,z-I+l)!(Nz,+N,,-I+l)! I=0 Nzp fi - 1 ‘&7(P) ___ =.fl=, Nm,! e- 1

[ 1

N,,-l+l)!

(B.14a)

(B.14b)

ALGORITHM FOR TEMPERATURE REACTION RATES

365

FIG. 34. Configurations (together with &heir “crossed” counterparts (cf., the text)) giving rise to the “full propagator” $,(p) of f$.

where p. = E, and l,,, and (C IV,> are given in Eqs. (B.7) or (B.9). Use Eq. (13.9) in (B.14b) and divide the resulting (C W} by the thermal vacuum bubble term Z~~(O}(~ IV,). Then after taking the sum over the mode i in interest and also taking the limit V --f 00, we obtain, for the contribution to the decay rate f,, Eq. (B.10) in proviso with

x El +~~:0,‘(PN4*(P))]

e--421(P) 1

e-

(B.15)

366

ASHIDA

ET AL.

We can go similarly to the configuration where the thermal vacuum bubble diagram, Fig. 33, coexists with the main-body diagrams shown in Fig. 34(c) or (d), and obtain, for its contribution to the decay rate r,, the same form as Eq. (B.10) in proviso with 3*,(p)=9;y34(c)(p) = -27&Y+ [( -i.&,(p))

jade)]

x [l + iD;;‘(p)( -i&,(p))]

l +e:;,,,

f C.C.

(B.16)

or

We can proceed, with the same analyses, to study configurations shown in Figs. 35(a)-(e). By studying the coexisting configurations of the thermal vacuum bubble diagram, Fig. 33, and the main-body diagram, Figs, 35(a)-(e), we obtain as corresponding contributions to the decay rate r,, (B.18) where F, I has already appeared in Eq. (A. 15) and is equal to P, , in Eq. (3.15 ). The function yl(p) takes the following form respective to the configurations in Figs. 35(a)-(e): (B.19)

3~lig.35(ayp) = B,,(p), $;;g.35’b’(p)=

-27cniii+ [l -iz’,,(p)iD~~‘(p)] (B.20)

gy35(c)(p)=

--27&j+ [( -i/f&))

zq’(p)]

x Cl+~~s+G(p))J 1-I,* e- 1 l c w,) I x CW(p)( -ifdp))l

e- AZ(P) e_ 1

(B.21)

(B.22)

ALGORITHM

q3yp)

= -2nG+[

FOR TEMPERATURE

-i&(p)

REACTION

367

RATES

iDy$l)]

x r-@?(P)~-~&&4)1

1+A,,@) e- 1 1 w, . ( >

(B.23)

Those configurations that can be obtained from the configurations depicted in Fig, 33, Figs. 34(b)-(d), and Figs. 35(b)-(e), by shifting &‘s from the initial/final state to the final/initial state without changing the other part of the diagram (which are referred to as the “crossed” diagrams hereafter), give respectively Eq. (B.9), Eqs. (B.15)-(B.17), and Eqs. (B.20)-(B.23) with the proviso that, in every equation, 6 + should be replaced with 6. , s

= S(p2 - nz2) 8( -po).

(B.24)

Thus by adding the original contributions to those from the “crossed” diagrams, we obtain, as the contributions to the decay rate, Eqs. (B.15))(B.17) and Eqs. (B.20)-(B.23) with “complete” S-function 6 = 6(p2 - m2) in place of 6 +. Configurations that can be obtained from those in Figs. 35, by exchanging the

(d)

(e) FIG.

propagator

35.

Configurations ,. g,,,(p) of 4.

(together

with

their

“crossed”

counterparts)

giving

rise

to

the

“full

368

ASHIDA

ET AL.

left half (S-matrix element side) for the right half (S*-matrix contributions with the form

element side), give

(B.25a)

%2(P)= -V,(P),

(B.25b)

where Fz2 has already appeared in Eq. (A.15). In deriving Eqs. (B.25) use has been made of Eqs. (B.3). The configurations depicted in Figs. 33 and 34, with the c+$‘sreplaced by their antiparticles 4:‘s (4: = pi in the present neutral scalar theory), give contributions to the decay rate with the form

(iA)i%2(p)( -iA) F12(P, P), %(P) = %I( -PI,

(B.26a) (B.26b)

where C?&takes the corresponding expression, depending on the original configurations Figs. 33 and 34(a)-(d). It is not difficult to see that ‘C$ and $i satisfy the relation (B.27)

S2(~) = e-p~oYl(p).

Now we have in hand the necessary four kinds of “full propagators”: (i) c+Qin Eq. (B.lO), which is the sum of Eqs. (B.ll) and (B.15)-(B.17) with 6, replaced by 6; (ii) gii in Eq. (B.18), which is the sum of Eqs. (B.19)-(B.23) with 6, replaced by 6; (iii) CC&in Eq. (B.25); and (iv) c!& in Eq. (B.26). Then let us find the compact functional forms of them. We can solve Eq. (B.lb) in terms of f(p) to find -if,,(p)

= &

{4P)(-GP))),/?

(a, P= 1,2)?

(B.28)

where n(p) is the 2 x 2 matrix 1+ NP) =

C,*(P) p2-m2-iE C,,(P)

i

p2-m’+iE

C,,(P) -p2--,2-ic 1-

Al, p2-m2+k

’

(B.29a)

i

D(P) = det A(p).

It is to be noted that all of the four components of Z‘,, (p) (a, F=L2)

(B.29b) are not

ALGORITHM

FOR TEMPERATURE

REACTION

RATES

369

independent quantities because of the relations (A.17) and (A.18) in the present case. Let us now introduce the quantity Z(p) by JmJ) = ill

+ Cl2(Pl

= - C22(P) - ~2’21(P),

(B30a) (B.30b)

where the second equality is proved to hold with the use of Eqs. (4.6), (A.17), and (A.18) that also hold for the present self-energy part of 4. With two independent quantities Z(p) and T;,,(p) we can express C,,(p) (n, /I= 1,2) in Eq. (B.28); then we substitute this Eq. (B.28) everywhere in the expression of C!&(P) and obtain, after some manipulations, (B.31) where 5& (CT,i = I$, $) are defined as

and

q&Jol)= q&ol)=

(1 +~B(lPol)yI

(B.33a)

U,6(lPo/)=

bB(lPOlfP2~

(B.33b)

q&$(lPolf=

14(p)=(p2--2--(p)+i~)--1,

(B.34)

with ng the Bose distribution function. From Eq. (B.31) supplemented with Eqs. (B.33) and (B.34) we see [6] that gC1 (5, [ = #, 7) are nothing but the full propagators in the standard RTFT, such as thermo-meld dynamics (TFD) [S]. In fact, ;I: defined by Eq. (B.30) is nothing but the diagonalized or quasiparticle self-energy [6] in the standard RTFT defined by

-Q) i

0

(B.35)

This proposition is proved by virtue of Eqs. (A.1 7) and (A. 18). By noting the fact that Eqs. (B.32) are the relations which hold between the full propagators 55& (5, [ = 4, d;) in the standard RTFT and Sgfl (tc, fi = 1,2) calculated in the circled diagram rules, we conclude that our Yes (GI,j = 1,2) (appearing in Eqs. (B.lO), (B.l8), (B.25), and (B.26) are the full propagators in the “real time thermal field theory” based on the circled diagram prescriptions.

370

ASHIDA ET AL. ACKNOWLEDGMENT

Two of us (H.N. and A.N.) would like to thank Professor R. Baier for the warm hospitality he extended to them. REFERENCES 1. See, for example, J. DIAS DE DEUS AND S. COSTA RAMOS(Eds.), “The Physics of the Quark-GIuon Plasma” World Scientific, Singapore, 1988; B. SINHA AND S. RAHA (&Is.), “Physics and Astrophysics of Quark-Gluon Plasma,” World Scientific, Singapore, 1988; M. S. TURNER, in “Cosmology and Elementary Particles” (D. R. Altschuler and J. F. Nieves, Ed%), World Scientific. Singapore, 1989. 2. B. SINIIA. S. PAL, AND S. RAHA (Eds.), “Quark-GIuon Plasma,” Springer-VerIag, Berlin/Heidelberg, 1990. 3. For a review on quark-gtuon plasma, see J. CLEYMANS, R, V. GAVAI. AND E. SUHONEN,P~Jv. Rep. 130 (19863. 217; see also G. A. BAYM, P. BRO~~~-MUNZIN~ER, AND S. NAGAM~YA (Eds.), “Proceedings, Quark Matter ‘88,” Nucl. P&s. A. Vol. 498, p. 1, North-Holland, Amsterdam, 1989; M. JACOB, invited contribution to the Heavy-Ion Session, 25th International Conference on High Energy Physics, Singapore, Aug., 1990, CERN Preprint CERN-TH5848, Aug. 1990. 4. See, for example, R. MILLS. “Propagators for Many-Particle Systems,” Gordon & Breach, New York, 1969; K.-C. CHOV, Z.-B. Su, B.-L. HAO, AND L. Yu, Pb.~s. Rep. 118 (1985), 1. 5. H. UMEZAWA, H. MATS~~OTO, AND M. TACHIKI. “Therm0 Field Dynamics and Condensed States,” North Holland, Amsterdam, 1982. 6. N. P. LANDSMAN AND CH. G. VAN WEERT, Phys. Rep. 145 (1987). 141. 7. See. for example, A. A. ABRIKOSOV. L. P. GOR’KOV. AND I. YE. DZYALOSHINSKII,“Quantum Field Theoretical Methods in Statistical Physics.” 2nd ed., Pergamon, Oxford, 1965: J. 1. KAPUSTA, “Finite-Temperature Field Theory,” Cambridge Univ. Press, Cambridge, UK, 1989. 8. R. E. CUTKOSKY, J. Math. Phys. 1 ( 1960) 429. 9. R. L. KOBE~ AND G. W. SEMENOFF.Nurl. Phw. 3 272 (t986), 329. 10. R. L. KOBE~ AND G. W. SEMENOFF,Nucl. Phys. B 260 (t986), 714. 11. G. ‘T H~OFT AND M. VELTMAN, “Diagrammar,” CERN Yellow Report 73-9. 12. J. F. DONOGHUEANU 8. R. HOLSTEIN,Phys. Reo. D 28 (1983). 340: Erratum. Ph,w. RED. D 29 (1984) 3004. 13. H. A. WELDON. Phys. Ret). D 28 (1983), 2001. 14. K. AHMED AND S. SALEEM. Phys. Rea. D 35 (1986). 1861; T. GRANDOU, M. LE BELLAC, AND J.-L. MEUNIER, Z. P&w. C 43 (1989), 575. 15. R. BAIER, E. PIMN, B. PIRE, AND D. SCHIFF, Nucl. Phys. B 336 (1990), 157. 16. W. KEIL, Phyx Ret D 40 (1989), 1176. 17. T. GRANDOU, M. LE BELLAC, AND D. POIZAT, Phvs. Len. B 249 (1990), 478; Nucl. Phys. B 358 (1991), 408. 18. L. D. MCLERRAN AND T. TOIMELA, Phys. Reo. D 31 (1985), 545; E. BRAATEN, R. D. PISARSKI, AND T. C. YUAN, Phys. Ret;. Lett. 64 (1990). 2242. 19. R. BAIER. B. PIRE, AND D. SCHIFF,Phys. Rev. D 38 (1988), 2814. 20. T. ALTHERR ANV P. AURENCHE, Z. Phys. C 45 (1989), 99; T. ALTHERR, P. AURENCHE, AND T. BECHERRAWY,NucI. Ph,vs. B 315 (1989), 436; T. ALTHERR ANI) T. BECHERRAWY,Nucl. Phys. E 330 (1990), 174. 21. A. H. WELDON, Phys. REV. D 42 (1990), 2384. 22. A. E. I. JOHANSSON, G. PERESSUTTI, AND B.-S. SKAGERSTAM, Nucl. Phys. B 278 (1986). 324; J. CLFYMA~ AND I. DADIC’, Z. Phys. C 42 [1989), 133. 23. A. NI~C~AWA,Phvs. Left. B 247 (1990), 3.51; in “Proceedings, Second Workshop on Thermal Field Theories and Their Applications. Tsukuba, 1990” (H. Ezawa, T. Arimitsu, and Y. Hashimoto, Eds.), Elsevier, Amsterdam, 1991; N. ASHIDA. H. NAKKAGAWA, A. NI~GAWA, AND H. YOKOTA, Osaka City Univ. Preprint OCU-137, 1991, to be published in Phys. Rev. D.

ALGORITHM

FOR

TEMPERATURE

REACTION

RATES

371

24. R. KOBES, Phys. Rev. D 43 (1991), 1269; see also Ph.r,s. Ret). D 42 (1990), 562. 25. R. KOBES. Feynman rules for linear and non-linear response functions at thermal equilibrium, University of Winnipeg preprint, 1991. 26. J. SCHWINGER, J. Math. Phys. 2 (1961), 407; L. V. KELDYSH. Zh. E/up. Tear. Fiz. 47 (1964). 1515 (Sm. Phys. JETP 20 (1965), 1018); R. A. CRAIG, J. Marh. Phys. 9 (1968), 605. 27. H. MATSUMOTO, I. OJIMA. AND H. UMEZAWA, Ann. Phys. (N.Y.) 152 (1984), 348; H. MATSUMOTO. Y. NAKANO, AND H. UMEZAWA. Phys. Rec. D 29 (1984). 1116. 28. A. J. NIEMI AND G. W. SEMENOFF. Nucl. Phw. B 230 [FSlO] (1984), 181. 29. H. MATSUMOTO, Y. NAKANO, H. UMEZAWA, F. MANCINI. AND M. MARINARO, Prog. Them. PhJs. 70 (1983) 599; H. MATSUMOTO, Y. NAKANO, AND H. UMEZAWA, J. Math. Phys. 25 (1984), 3076. 30. A. J. NIEMI AND G. W. SEMENOFF. Ann. Phys. (N.Y.) 152 (1984), 105.

595:x5/2-10