High-temperature field theory in curved spacetime

NUCLEAR PHYSICS B ELSEVIER

Nuclear Physics B 439 (1995) 131-143

High-temperature field theory in curved spacetime J. F r e n k e l a, E.A. G a f f n e y b, J.C. T a y l o r b a Instituto de F{sica, Universidade de Sdo Paulo, Sao Paulo, Brasil b DAMTP, University of Cambridge, Cambridge, UK

Received3 October 1994;accepted 10 January 1995

Abstract

We establish, to all orders in Newton's constant, the form of the effective actions describing hard-thermal matter loops in curved spacetime, both with extemal gluon lines (T2 effects) and without (T4 effects). In the latter case, the energy-momentum tensor corresponding to the effective action action has ToO> 0.

1. Introduction

In thermal QCD, some attention has been paid recently [ 1,2,3,4] to the effective action which generates the leading behaviour at high-temperature (T 2 effects) of oneloop thermal field theory diagrams ('hard thermal loops'). This effective action is a functional of a classical gluon field (assumed to tend to zero at infinity). It is relevant to physics in which the masses, frequencies and wave-numbers are much smaller than the temperature T. The corresponding field equation is a nonabelian form of the Vlasov transport equation in a collisionless plasma [4,5]. Although it would be rather complicated to calculate directly all n-gluon loops for all values of n, the effective action can be written down by appeal to a few general properties (most notably, gauge-invariance) which can be abstracted from the thermal field theory. There is also a rather simple physical interpretation in terms of the phasespace distribution for a plasma of classical coloured particles [6]. A closely related problem is to find the effective action describing hard thermal loops in a gravitational field, that is in a curved spacetime. Temperature may be defined, at least if the spacetime is asymptotically conformally fiat [7.8]. There are two cases to consider: the part of the effective action which depends upon the metric g/~ only, and the part which depends upon the gluon field A~ as well as the metric. The former is of order T 4 and the latter of order T 2. Elsevier ScienceB.V. 0550-3213 (94)00581-8

SSDI

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From the effective action, one may derive, by differentiation with respect to the metric, the energy-momentum tensor T ~ of the hard thermal matter. For the T4 part, we find that TOOis manifestly positive. In previous papers [8,9], we have conjectured the forms of these effective actions, by writing down the simplest expressions we could think of which had all the requisite properties. The main advance in the present paper is to prove that these properties do indeed fix the effective actions uniquely. (In [9], we claimed to check uniqueness to 1graviton order. In fact, because of an algebraic error in equation (B.3), this verification was incomplete [ 11 ] .) The results of hard thermal loop calculations may be written in the form [3] /

~

m

(2"n') 31

m

/d4Qr(Qz)O(Qo)N(Qo)j(Q )

(1.1)

where

1 N(Qo) = eQ0/r 4-1

(1.2)

for fermions or bosons respectively, and J is a (spin and colour summed) forward scattering amplitude. It is in fact the case that J is a divergence,

aw,,

(2~r)-3J = OQ~ '

(1.3)

which enables (1.1) to be re-written (see Section 5 of Ref. [3] )

F=

f d4Qr(Q2)O(Qo)N'(Qo)Wo-2 f d4Q 6~(Q2)O(Qo)N(Qo)QaWa. (1.4)

But in this paper we shall use only the form (1.1) in which J is a Lorentz invariant function of Q and of the external fields. In (1.1), J(Q) has the following properties: (a) It is nonlocal, but the nonlocality has the special form of line integrals in the direction of Q. (b) It is a homogeneous function of Q,,, of degree 2 for the part depending only upon the metric, and degree 0 for the part depending upon gluon fields (or quark fields) as well. (This clearly gives the T-dependence stated above.) (c) It has the dimensions of (energy) -2, so that (1.1) is dimensionless. (d) It is Lorentz-invariant, under Lorentz transformations in the asymptotic fiat space. (e) It is gauge-invariant (not just BRST invariant) under QCD gauge transformations. This is discussed in the Appendix A. (f) It is invariant under coordinate transformations (preserving the asymptotic Minkowski metric). This is certainly true for thermal matter loops. We have not proved it for thermal graviton loops (though it is probably true in this case also); so we confine ourselves to matter loops in this paper. (g) There is a form of axial gauge (both for the gluon field, equation (2.1), and for

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the metric, (3.2)), defined by the light-like vector Q in (1.1), in which all but a few low-order terms in J are zero. We shall call these gauges 'Q-gauges'. Properties (a), (b) , (c) and (d) are apparent from the derivation of hard thermal loops in terms of forward scattering amplitudes [3]. Property (f) follows from the invariance of the zero-temperature action. Property (g) is the new one which we make use of and establish in the present paper. Having found J in the special gauge, we can write down the gauge-invariant form for J which reduces to this in the special gauge. This, inserted into ( 1.1 ), is then the required result for F, to all orders. In the Q-gauges, J is, in spite of property (a), local. Because of this, the dimensional property (b) controls J much more directly than in a general gauge. The problem of enumerating all allowed terms in J is, in the Q-gauge, much more like finding all allowed counter-terms in a renormalizable field theory. We would like to point out the reason why direct calculations of hard thermal loops to higher orders are complicated. This is that individual graphs have T-dependences which are higher than the T4 or T 2 dependence of complete sets of graphs. We call these higher powers of T the 'apparent' T-dependence. For QCD in curved spacetime, a graph with n external gravitons and any number of external gluons has an apparent T 3+n dependence. Cancellations between different graphs reduce this to T4 (when there are no external gluons) or T 2 (when there are some external gluons). The large number of cancellations is presumably a consequence of the Ward identities which express gauge and coordinate invariance. In the 'Q-gauge', the apparent T-dependence is not higher than the true leading T-dependence. We begin in Section 2 by illustrating our type of argument for the case of QCD in fiat space, although the results are already known [ 1,2,3,4,5,6]. In Section 3 we treat the case where there are external gluon fields as well as external gravitons (when the leading temperature dependence is T2). Finally, in Section 4, we treat the case when the only external lines are gravitons (T4). We note that the energy-momentum tensor has TOO> 0, and this is apparent before doing the Q-integration. In this paper, we do not consider graphs with external quark lines. It should be possible to extend the method of Sections 2 and 3 to cover the case when there is one quark line and one antiquark line, in addition to gluon and graviton lines.

2. QCD in fiat spacetime In this paper, we use QCD as an example of a gauge field theory. Our results are easily extended to any other (renormalizable) gauge field theory. If we go to an axial gauge, denoted by ,4, in which Q.,4=0,

(2.1)

(of course, A depends upon Q, hut we shall not write this explicitly) any appearance of Q in the numerator must be in a factor Q • k, where k denotes a generic linear combination of external gluon momenta (remembering that in (1.1) Q2 = 0). By properties (a) and (b), such factors must be balanced by an equal number of Q • k

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factors in the denominator. By property (a), there are no other factors of k in the denominator. By the dimensional property (c), the number of gluon fields is at most two, and when it is two there are no further factors of k in the numerator. So factors involving Q and k can appear solely in a scalar factor of the form ( Q . k(l))...(Q • k(r)) (Q" k,(l))...(Q, kt(r))

(2.2)

,

where the k (i) are a set of linear combinations of the external momenta and the k t(i) are another such set. But we now show that the only possibility for (2.2) is the trivial one where (in the complete amplitude) the factors in (2.2) cancel between numerator and denominator. To show this, go to the static case in which nb(i) 0 = kt(oi) = O, and all the vectors in (2.2) reduce to 3-vectors. It is known [ 1,2,3] that F is local in the static case. But (2.2) is certainly not local. It has singularities where k t(i) = O. And it is not identically zero, even after the Q-integration, as can be seen for example from the special case where all the external momenta k are parallel. (In Appendix B, there is a simple example of the cancellation of factors like (2.2).) Thus, in the Q-gauge (2.1), J is local, and by property (c) has dimensions (energy)-2. The only possible such term involves two gluon fields and no derivatives, and it is known [ 1,2,3] to be J(Q; .4) = g2c / d4x/~(x)/~a/X(X)

,

(2.3)

where a is the gluon colour index, and c is a numerical factor related to the number of independent physical states in the thermal loop, i.e. for a gauge group SU(N) with N f quark flavours c = 2Nf

for quark loops,

c = 2N for gluons loops.

(2.4)

What remains is to construct the form for J in (1.1) which is gauge-invariant (property (e)) and which reduces to (2.3) in the gauge (2.1). To do this, we first define the Green function GQ(X;A) to satisfy (Q . o )abGbaC(x; A) = ~ ( x ) t ~ c ,

(2.5)

where D denotes the covariant derivative. In Minkowski space, GQ is not uniquely specified by (2.5), and the effective action for hard thermal loops strictly does not exist in Minkowski space (there is Landau damping). We shall go to Euclidean space, in which a~' = (iIQI,Q) .

(2.6)

Then there is a unique Green function which tends to zero at large (Euclidean) distances [5]. This may for example be given as a perturbation series in the form G~2b(x) =GOQab(x) + g f d4x' GOQac(x-- x ' ) Q . ACd(x')GO~db(x ') + ....

(2.7)

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where ~Q is the zero-order Green function

18 (XT)

=or

Q*.x

Q.xQ* x + 8 2 '

(2.8)

where Q . Xz = 0. With the above Green function, define f ~a( x , Q,• A) = f d4x ' G~b(x - x ' )a ~F ~~ ( x ' ),

(2.9)

This is a gauge-covariant expression, and it is easily checked that

f~(x, Q; .4) = .4ag(x) .

(2.10)

ThUS the required unique gauge-invariant expression for J is

J(Q) = g2c / d4x f~(x, Q; A) fa#(x, Q; A) .

(2.11 )

The effective action F is got by putting (2.11) into (1.1). The result (2.11) is well known [ 1,2,3 ]. The purpose in deriving it in this way here is to have a form of argument which we can use in the gravitational cases.

3. Hard thermal loops with gluon and graviton external lines In this section, we generalize the argument in the previous one to derive the effective action giving the hard thermal loop contribution to diagrams with any number of graviton external lines and also at least two gluon external lines. The leading temperature dependence for these graphs is T 2. We employ the zero-temperature action for gluons in curved space, as written down for example in [9] Appendix A. We write gU~ = ~u~ + h ~ ,

(3.1)

and define perturbation theory as an expansion in powers of h u~. Here ~/~ is the Minkowski metric (or, after going to Euclidean space, the Euclidean metric). We define a 'Q-gauge', ht,~ for the metric by ^

Quh ~ = 0.

(3.2)

We also use the gauge (2.1) for the external gluon fields. Just as in Section 2, we can argue that, with these gauges, J is independent of Q and is local. It follows by dimensions that J is quadratic in the gluon field, contains any number of graviton fields h ~ , and no derivatives (in this section we are not discussing the part independent of the gluon field). Thus J(~, ,~) is (the spacetime integral of) an algebraic function of h ~ and A,~, ^a being quadratic in the latter, and is Lorentz invariant. Since J ( h , ,4) contains no derivatives of the metric, we can determine it by working with a constant metric. This requires only small modifications of the flatspace calculation.

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We first find J ( h , A ) , in a general gauge for A, to order g2. We use Feynman gaugefixing for the internal gluons. We take the Feynman rules from Appendix A of [9], specialising to a constant metric. (There are of course ghost loops as well as gluon ones.) The factors of v ~ cancel between the propagators and the vertices. The only modification to be made in the fiat-space results is that scalar products are to be taken with the constant metric ~ instead of the Lorentz metric ~ (momenta and the gluon field all having indices down). If a Q~ is involved in a scalar product, we may use (3.2). Having found J ( h , A ) in this way, we specialize to J(h,,~), using (2.1). It is now clear how this is obtained from the fiat-space result (2.3). We have simply

j(Q; ,~, ~) = g2c f d4x ~t~ (x),~a (x),~a(x) ,

(3.3)

and (1.1) is unchanged. Of course, (3.3) is not invariant under coordinate transformations, because we have broken this invafiance by going to the Q-gauge (3.2). The above argument may not be very rigorous, but it merely summarises cancellations which occur between different Feynman graphs contributing to J. A complete verification in terms of graphs is rather cumbersome. In Appendix B, we give a simple example of such a cancellation, which we hope increases confidence in the argument above. It also serves as a simple example of the cancellation of the factors in (2.2). Our next task is to write down a form for J which is gauge and coordinate invariant and which reduces to (3.3) in the Q-gauges. Before we do this, we need some preliminary definitions. First we define a quantity Ya(x;g). This is a function of x and the metric, and is invariant under coordinate transformations, but transforms as a Lorentz vector under Lorentz transformations in the asymptotic flat Minkowski space. We define it indirectly by first defining the inverse function xu(Y; g). This is defined to be the solution of the equation

aY"

-

a

g

a

(3.4)

with the boundary condition

x~(Y) ~ Y~

for Y ---, c~

(3.5)

which of course implies

Y'~(x) ,,~ x ~' and

(Q. -a~ ) x ot ,~ Q~.

(3.6)

These equations may for example be solved iteratively in powers of h ~'~, defined in (3.1). The process requires the inverse ( Q . ~ )-1. Provided we go to Euclidean space, this may be defined like (2.8), and then the boundary condition (3.5) may be satisfied, since the first term x = Y + ... satisfies it and all higher-order terms tend to zero at infinity. The interpretation of Y'~ is that they parametrize a family of null geodesics. And any one of these geodesics has the form x(Y + QO) for fixed Y with 8 being a parameter along the geodesic.

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Next we define ay 'Y c~y~ G'~# (Y; g) - ~gx~ ax ~gUp (x(Y) ) .

(3.7)

It is easy to see that in the Q-gauge (3.2) we have x(Y;~) = Y,

(3.8)

G~#(Y; ~) = ~'~/~.

(3.9)

and

It was shown in [8] that, in any gauge,

G~I3QI3 = Qa .

(3.10)

Clearly equations (3.2), (3.9) and (3.10) are consistent. With these definitions, the required invariant form of (3.3) is

J(Q)

gZc f d4yG,~a(y;g) f a ( y a ;

A)f#(Y,Q; A ) ,

(3.11)

where f a is defined by a modification of (2.9):

f :"( , gQ ; A )

=

d4y!G~b[y_ y,;A]Q/3aOyX: 3x l~.b ( x ( Y ' ~ 8ytp- ~ . . . . •

(3.12)

Here the Green function G a is defined as in (2.5) and (2.7), but now all the variables are Y's. Eqs. (3.11) and (3.12) are clearly invariant under coordinate transformations, because they are expressed entirely in terms of invariants, Also, in the gauge (3.2), x = Y and so (3.12) reduces to (2.9) which then reduces to (2.10) in the gauge (2.1). Finally (3.9) completes the reduction of (3.11) to (3.3). It is therefore the unique required expression. The effective action is found by inserting (3.11) into (1.1). In [9] we had previously conjectured a form (equation (25)) for J. This contained an auxiliary quantity A ( x ) . Comparison with (3.11) shows that our conjecture was wrong to higher orders with A as defined in (21) of [9]. The correct definition of A was given in equation (2.22) of [11].

4. External gravitons only When there are no external gluons and quarks, J is homogeneous in Q of degree 2. Again we evaluate J ( Q ) first in the Q-gauge (3.2). The same argument we used at the beginning of Section 3 seems to show that J ( Q ; ~ ) is local and independent of Q. This contradicts the homogeneity property (b), and so seems to imply that J(Q; ~) = O. We shall try to explain this paradox later. In the meantime, we shall abandon trying to determine J and focus instead on 8J wg~(Q,x;g) =_ 8 g ~ ( x ) .

(4.1)

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This tensor W is related to the stress-energy tensor 8F x/gT~,v(x; g) = 2 8glZV(x)

(4.2)

by functional differentiation of (1.1). W~,~ is also homogeneous in Q of degree 2, but it has dimensions (energy) 2. We try to determine W in the Q-gauge. Just the same sort of arguments as used in Section 3 apply, and show that W must have the form W ~ ( Q, x; ~) = Q~Q~w(~( x) ) ,

(4.3)

where w is a local, dimensionless (so involving no derivatives) function of ~ " (x). We may now find w in the same sort of way as we found (3.3) in the last section. We repeat the flat-space calculation of the lowest-order graph (the 'tadpole' graph), but using the constant metric ~'" in place of the Minkowski metric. We need the zero-temperature energy-momentum tensor to 2-gluon order: 1[

~ ~a13

oa a oa~ +ga~axtZ ax v

1^ ^,~l~^agoaa aA~] ~g~,vg g ~x a Ox~j.

(4.4)

(Omitting ghost contributions.) The spin sum (as in J in (1.1)) involves ~a#- In the tadpole graph, all derivatives give factors of Q, and we may simplify using (3.2). In fact, only the second term in (4.4) contributes at all, and even this contribution is independent of ~a~. Thus the result is simply W ~ ( Q , x; ~) = cQ~Q~,

(4.5)

where the number c was defined in (2.4). The next task is to find a second rank pseudo-tensor which reduces to (4.5) in the Q-gauge. With the function Y(x;g) defined in (3.4), this is easily done. The required tensor is ay ,~ aya W~,v(x; g) = c I det( aY/ax) IQaQ# Ox~ ax ~ (4.6) Using (3.2), (3.8) and (3.9), Wm,(x; ~) = cQ~Qv

(4.7)

as required. In passing, as a check of (4.6), we verify that it has the expected transformation property under a Weyl transformation g~'" --* e2~(x) g~'~. (We expect Weyl invariance because masses are neglected in the high-temperature limit.) Then Y ~ Yt, where yU, = y~ + Q ~ f ( y ) ,

(4.8)

where e -2~(x(r')) = 1 + Q~ o f ( Y ) .

aYe,

(4.9)

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Consequently

Q,,yt,~ = Q,~y,~,

8 = Q . (~-ff) 8 e-2a(x(r')) Q • (~--yT)

(4.10)

and e 2'~(x(r')) det(aY'/ax)

= det(OY/ax).

(4.11)

These equations show that, under the Weyl transformation, e 2~(x(r')) W~ = Wu~,

(4.12)

as required from the definition (4.1) and the Weyl invariance of J. We now return to the quest for an effective action J satisfying (4.1). We will need an alternative form of (4.6): a

where we have used (3.7) and (3.10). Then we use the alternative form of (4.1):

W"~'(Q'x;g) -

6J

6g~,(x) '

(4.14)

which follows from (4.1) and (4.13). Now consider the action

](g,x(Y) ) = -c f dgY [g~,~(x(Y) ) { (Q " j-~)x~(Y) } { (Q " ~-~)x~(Y) } -2(Q. j-~)(Q~xa(Y) ) 1 .

(4.15)

Consider the Euler-Lagrange equation got by varying x(Y) in (4.15) (for given g(x) ). The second term in (4.15), being a derivative, does not affect this equation, except for the integrated term

-2c f d4y (Q. ff--~) [{gu~(x(Y)) ((Q. ~--~)x~(Y)) -Q~,}tx~'(Y)] .

(4.16)

Assuming that this can be discarded, the Euler-Lagrange equation is just the geodesic equation (3.4). Denote the solution of (3.4), with the boundary conditions (3.5), (3.6), by x ~ = sc~(Y;g) •

(4.17)

Then we claim that the required action is

J(g) = ](g, sC(Y)).

(4.18)

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To verify this, use 8 J ( g ) = [ Y(g + 8g, ~) - J(g, ~) ] + [ j ( g , ~ + 6() - J(g, ~) ] ,

(4.19)

where 8( = ~:(Y; g + 8g) - ((Y; g ) .

(4.20)

But, because sc is defined to make ar stationary, the second square bracket in (4.19) is zero. Thus, using (4.15), we get (4.13) from (4.19) as required. It remains to justify the neglect of (4.16), when 8~: in (4.20) is substituted for 8x in (4.16). Because the metric is assumed to be asymptotically Minkowskian, and because of the assumed boundary conditions (3.5), (3.6), we believe that (4.16) is zero. If the second, derivative, term in (4.15) had been omitted, the integrated part could not have been neglected. Because the geodesic (4.17) is null, the first term in (4.15) actually does not contribute to (4.18). The action (4.15) has been previously conjectured in equation (37) of [8]. We can now return to the problem identified at the beginning of this section, that the effective action appears to be zero in the Q-gauge. We see that (4.15) would indeed be zero if the second, derivative, term could be neglected. The reason it cannot be is that ((Y) - Y does not tend to zero fast enough asymptotically. This means that the coordinate transformation to the Q-gauge does not tend to unity fast enough asymptotically to transform J to the Q-gauge; though there is no such problem with Wu~. The action (4.15) may be thought of as a local version of J, produced by introducing an auxiliary 'field' x u (Y). In this light, it may be compared and contrasted with previous local, auxiliary field actions [4, 5, 10, 11]. A feature of (4.15) is that the auxiliary field has a physical interpretation. Finally, we comment on the positivity of Too defined from Woo in (4.13). To do this, we must return from Euclidean to Minkowski spacetime (thus getting rid of the complex fields resulting from (2.6)). Then the question arises what Green function to choose in (2.5) and (2.7). Fortunately, for the purpose of defining T ~ ' ( x ) , one may for example choose all Green function to be retarded with respect to x. Then everything is real and well-defined, and we may use the fact that Woo in (4.13) is manifestly positive. Writing the usual expression for the total energy, d3x [ToO + t °°]

(4.21)

where #'~ is the gravitational part, we see that the contribution from the hot matter is positive.

Acknowledgements We are grateful to Fernando Brandt for discussions. J.F. thanks CNPq for support. E.A.G and J.C.T. acknowledge the support of the EU Programme "Human Capital and Mobility", Network "Physics at High Energy Colliders", contract CHRX-CT93-0357 (DG 12 COMA), and of PPARC.

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Appendix A In this Appendix, we discuss property (e) of Section 1: the gauge invariance of the hard thermal loops. This follows provided that hard thermal loops with external ghost lines do not have a leading T 2 dependence. In Section 3 of Ref. [ 3 ], we gave an example of this in fiat space. The reason is that there is one factor of external momentum k~ where a ghost line leaves a Feynman graph. This reduces the maximum apparent dependence from T 3 to T 2. But the maximum apparent dependence always cancels between the different forward-scattering graphs corresponding to a single Feynman graph, by eikonal identities like (23) of [3]. We need to generalize this argument to cases with external graviton lines as well as gluons and ghosts. Now the maximum apparent dependence is T n+2 ( for n gravitons, 1 ghost and 1 antighost; it is less if there are more ghosts and antighosts). However, we know we have coordinate invariance (we are not treating the case of an internal graviton line, so there is no question of graviton ghosts). Therefore we may go to the gauge (3.2) (though not of course to (2.1) because we do not yet know we have QCD gauge invariance). Then there can be no Q~ factors contracted with the external gravitons. This reduces the maximum apparent dependence to T2, and the argument proceeds as before.

Appendix B Here we give one simple example of how the cancellations which lead to (3.3) work out in terms of Feynman graphs. Take the forward-scattering graphs in Fig. B.1. Here gluon lines are shown explicitly as solid lines, but clusters of external graviton lines (defining a graviton by h ~ in (3.1)) are denoted just by little circles. These clusters appear in the combination ~b~#(x) = X / ~ ~# - ~/~

(B.1)

in the graviton-2-gluon vertex (see (4.4)). The contributions from each graph contain the common factor X~~ ( 2 ga# gar _ ga~ g#r _ gar g#~) [r/~v~bVar/ap~bmrr/ov].

(B.2)

The remaining contributions from these three graphs are (omitting a common colour factor) Q . (Q + k ) ( Q + k ) . ( Q + k + k') Q . (Q + k ' ) ( Q + k ' ) . ( Q + k + k') + 4Q . kQ . ( k + k') 4Q . k'Q . ( k + k') Q . (Q + k ) Q . (Q - k') + O(k) = 1 + O(k). 4Q . kQ . k'

(B.3)

(This is an example of the cancellation expected from the argument following (2.2).) With this coefficient, the square bracket in (B.2) gives just the second-order term in the expansion in terms of ~b of the inverse of (~/~a + ~bra), that is of

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J. Frenkel et al./Nuclear Physics B 439 (1995) 131-143

Q r

I~

v

k'

k

0

0

X

p

o

(a)

k'

k 0

O--O v

X

p

Q r

o

#

(b)

k

0

o

p

~ o

k' r

/*

~

0

O

X

(c)

Fig. B.1. Forward scattering graphs contributing to equations (B.2) and (B.3). The lines represent gluons, and the small circles denote clusters of graviton lines. The Greek letters are Lorentz indices of gluons, k and k' are momenta of clusters of graviton lines.

/ a

o

~)

/ o

o

~

o

o~o

Fig. B.2. An example of another type of forward scattering graph, which also contributes to (3.3). ct and/~ are the Lorentz indices of the two external gluon fields. (g) --l/2gr/z.

(B.4)

Thus, summing to all orders in ~b and replacing the square bracket by (B.4), (B.2) gives just 6g ~a, as in (3.3). Note that in (B.3) terms containing factors like

Qr(Q + k)8hr~

(B.5)

have been omitted because o f (3.2). A similar type o f argument works for diagrams like that in Fig. B.2, which also contribute to (3.3).

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