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Trace anomalies and λφ4 theory in curved space
ANNALS
OF PHYSICS
139, 136-197
(1982)
Trace Anomalies
and AQ4 Theory
in Curved
Space
S. J. HATHRELL* Department of Applied Mathematics and Theoretical University of Cambridge, Cambridge England Received
September
Physics,
7, 1981
It is shown for a conformally invariant A#’ theory in a weakly curved background, how to extend previous results to obtain full information about the trace anomaly in perturbation theory, including the “topological” term in the gravitational part of the anomaly. There is a strong connection among renormalisability of the curved space theory, finiteness of the energy-momentum tensor, and the role of normal products. Combined with a renormalisationgroup analysis this provides an efftcient means of calculating some terms in the anomaly to high orders of perturbation theory. In particular, the first I-dependent coefficient of the topological part of the anomaly appears at 0@“) and can be deduced from simple flat-space results without the calculation of any further Feynman diagrams. Some techniques based on an absorptive-part argument are developed in order to compute other anomalous coefftcients. and a direct 5-100~ calculation confirms the indirect renormalisation-group derivation of a non-vanishing R2 anomaly at O(n’). All the essential information can be obtained from the massless theory. The underlying ideas are applicable to other theories. and similar results for massless QED are obtained in a subsequent paper.
1.
INTRODUCTORY
REMARKS
It has been known for some time that anomalies in the trace of the energymomentum tensor are present in curved space even when the coupling constants for the quantum fields vanish [l-7],’ but more generally the values of the various terms in the anomaly will depend on the coupling constants [S, 91, and it is of interest to find out exactly how. In this paper we consider as a prototype the conformal selfinteracting 14” theory. In a recent paper Brown and Collins [lo] investigated the renormalisation of this theory in a curved background, and used a renormalisationgroup analysis combined with a study of certain functions of renormalised composite operators to deduce relationships between various quantities in the theory. They showed that some of the parameters describing the anomaly could be efficiently calculated from flat-space functions alone. We extend their analysis here to show that the L-dependenceof all the remaining parameters can also be determined in flat space in a similarly efficient manner, using only the masslesstheory throughout; and then
* Present
address:
’ Ref. 171 contains
UOC/13
Shell-Mex
a useful summary
House, of results
the Strand,
136 0003.4916/82/030136-62%05.00/O Copyright
.C: 1982 by Academic
Press, Inc.
London,
for non-interacting
England.
theories.
TRACE ANOMALIES
AND&d4
THEORY
137
we apply these general principles to calculate the higher order, I-dependent corrections. We use dimensional regularisation [ 1l-131 and renormalisation throughout, and this has the particular advantage in the case of a masslesstheory that many Feynman diagrams can readily be evaluated in terms of simple functions of the space-time dimension n. A subsidiary feature of the calculations presentedhere, though of wider applicability, is that some new results in the evaluation of multiloop integrals are obtained through an absorptive-part argument [ 14, 151 based on the unitarity of the theory. Some amplitudes can conveniently be expressed in closed form in terms of generalised hypergeometric functions, for which there is a useful body of standard mathematical literature [ 16-181, and the approach compares favourably with the use of Gegenbauer polynomials [ 191 in these cases. We illutrate this by calculating a particular siet of 5-100~ amplitudes which on the one hand verifies directly certain results obtained through the more powerful but indirect renormalisation-group arguments, and which on the other hand simultaneously enablescalculation of one of the parameters describing the anomaly up to O(1”). Other calculations also verify consistency in numerous instances. In Section 2 we define the theory in detail and introduce the parameters appearing in its description. In Sections 3-5 we review the analysis of Brown and Collins [lo], adapted somewhat for the exclusively masslesstheory. This involves a study or 2point functions of scalar, renormalised, composite operators in flat space, and establishes efficient routes of calculation for some of the parameters, including the coefficient of one of the “gravitational” terms. In Section 6 we extend their analysis to the study of 2-point functions of tensor operators, and thereby bring the second of the three gravitational terms into the same flat-space framework. In Section 7 we show that the third, “topological,” gravitational term can also be examined in flat space [5, 61 through a study of composite 3-point functions. Furthermore the renormalisat.ion-group structure of the theory entails a particular cancellation of the leading divergence of a certain 3-point function, and this enablesthe topological part of the anomaly to be determined to O(n”) without the necessity of calculating any new Feynman diagrams. In the subsequentsections of the paper we introduce the absorptive-part argument in a form suitble for the present purposes, and use it to calculate the specific values of the parameters using the results established. Some details of the calculations and some useful general results are given in the Appendices.
2. DESCRIPTION
OF THE THEORY
In a renormalisable quantum field theory, as Brown and Collins [IO] emphasize, one should start with a Lagrangian containing all possible terms of (mass-)dimension lessthan or equal to 4 which respect the symmetry of the regularized theory, because in general one can expect perturbation theory to introduce divergences proportional to those terms unless there is a clear reason why not. Here the dimensional
138
S. .I. HATHRELL
regularisation breaks the superficial conformal invariance of the bare theory in four dimensions, and so non-conformally invariant counterterms can be expected. On the other hand the full coordinate-invariance of the theory in the gravitational background is maintained. We therefore describe the theory by a generating functional W[J, g,,] for connected amplitudes: e iW =
1
(d$) e is
(2.1)
with the action S given by S=
i
d”x 69(x),
(2.2)
P=9g++@,
(2.3)
where (2.4 1 Pg = a,F + b,G + c,,H’.
P-5)
The theory is further defined by the requirement that W must generate finite Green’s functions. The nature of the terms appearing in (2.4) and (2.5) needs some explanation: (i)
The parameter r is defined by c=(n-2)/4(n-1),
(2.6)
this being the usual value in n dimensions necessary to render the scalar theory conformally invariant in the absence of any coupling constants. It is natural to regard the fl#’ term as part of the “kinetic” term in the Lagrangian. (ii) factor:
The quantity H is just the curvature H=R/(n-
scalar R, but scaled by an n-dependent 1).
(2.7)
The reason for this factor is that it absorbs the factor of (n - 1) which appears under a conformal scaling:
a’J(x - y) + curvature
terms.
(2.8)
TRACE
AND
ANOMALIES
/1d4 THEORY
139
The renormalisation process which involves minimal subtraction through expansion in inverse powers of (n - 4) is not then complicated by factors of (n - 1); although the choice is essentially arbitrary.2 It is also convenient to regard the H#’ term as being in the interaction part of P@, and separate from the t;R#’ term in the kinetic part. Note that a2 in (2.8) and everywhere subsequently is the covariant d’Alembertain, containing implicit dependence on the metric g,,.. The notation s/&2 will be used as a shorthand for the functional derivative with respect to the metric. (iii) The terms F and G in rt’, are, respectively, the square of the coliformal (Weyl) tensor in n dimensions, and a topological quantity (Gauss-Bonnet identity) related to the Euler number of the manifold. They are 4 F = RGL.poRL(L’Pa- ~ (n - 2) G = Rw,.poRL(L’P* - 4R,,.RU“
(2.9) f R2.
2; then contains independent linear combinations of the three possible coordinateinvariant terms of dimension 4 (up to total derivatives) containing only the metric tensor. (iv) J,(x) is a source term for the scalar field. It is renormalised by J” = z;yn,
n)J,
(2.11)
wave-function renormalisation factor. Functional where Z, is the flat-space derivatives of W with respect to J then generate renormalised connected Green’s functions, i.e., there is a factor of Z;’ for each external leg.3 We will always assume that ./ has been put to zero once the functional derivatives have been performed, and the role of J is otherwise unimportant. (v) There is no mass term for the scalar field in Ye, as indicated in the Introduction, because such a mass term is multiplicatively renormalised in the dimensional scheme and can consistently be taken to vanish [20, 211. There is also no cosmological term /i 0, or Einstein term K,,R in U,. This is a matter of choice in the case of a lmassless theory, because although such terms could be included, the parameters /i 0 and K,, carry dimensions 4 and 2, respectively, and their renormalisation therefore depends on a mass-parameter. So in the absence of a mass term their renormalisation must either be multiplicative or it must involve a mixing of the two parameters, and in either case both terms can consistently be taken to vanish. (The ‘t Hooft unit of mass ,LL, implicit in the theory through the renormalisation process, does not affect the argument because [22] it always appears in rational ‘This differs from Ref. [ 101, and changes the definitions of some functions by powers of 3. The definition here seems to be uniformly more convenient than using R instead of H. Moreover the convenience is not restricted to the scalar theory. 3 This follows the convention of Ref. [ 1C 1. The quantity Z, here is often defined to be Zi12 elsewhere in the literature, with a corresponding difference of a factor of 2 in the anomalous dimension.
140
S. J. HATHRELL
powers of r(l”-“.) Th’ IS is consistent with the explicit forms of renormalisation for these parameters derived by Brown and Collins [lo]. Since we are concerned in particular with the trace anomalies, and as the contributions to the trace from the cosmological, Einstein and mass terms are not “anomalous” in the usual sense, we have chosen to drop them. The theory then contains no intrinsic length-scale except through the renormalisation process. (vi) Finally, and this is fundamental to the whole argument, for each interaction term appearing in the action there is a coupling constant. The theory should therefore be regarded as a theory of not one but five bare coupling constants, A,, I]~, a,, 6, and c,,. The values of the four “new” coupling constants depend to some extent on the value of 1, (and on each other) because of the requirement that all the Green’s functions are finite, and it is not possible to ignore them. They are responsible for most of the anomaly. The energy-momentum tensor Bw” is obtained by functionally differentiating the action with respect to the metric: (2.12) and its trace 19= P,
is given by
- (n - 4)[a,F
+ b,G + coH2] + 4c,a*H.
(2.13)
Here E, is an opertor of dimension 4 related to the equation of motion of the scalar field [23]:
40 6s E”= --=-~oa’io+SR~:-~+~oH~~, fi slj,
(2.14)
whose properties are discussed in more detail later. We will use the term “operator” in a sense which includes the gravitational source as well as the scalar fields, although in the case of the gravitational source the operator is just an ordinary cnumber. From Eq. (2.13) it can be seen that the determination of the trace anomaly amounts to the determination of the way in which the various composite operators and the various coupling constants are renormalised in perturbation theory. It is therefore necessary (i) to examine the constraints imposed on the coupling constants by the requirement that the Green’s functions of the theory are finite, and (ii) to express the bare operators in terms of finite composite operators. By “finite” in the context of an operator we will always mean finite when inserted into finite Green’s functions containing only elementary renormalised fields at various points.‘For such a definition to make any sense it is necessary that an operator which
TRACE ANOMALIES
AND&d4
THEORY
141
is finite in one elementary Green’s function is also finite in all other elementary Green’s functions, and at all other momentum configurations. This is an exercise in renormalisation theory to justify [21, 24, 251 and is a property which will be assumedhere throughout. Since we have a masslesstheory, however, we must exclude a few exceptional on-shell momentum configurations from some Green’s functions in order to avoid the introduction of spurious infra-red divergences. The operations (i) and (ii) above were performed by Brown and Collins [lo], who then showed how some of the terms appearing in the renormalisation were related to 2-point composite operator Green’s functions, and they used this together with a renormalisation-group analysis as a powerful tool in the determination of the leading terms in the renormalisation of q0 and c,,. In the next three sections we summarise their analys,is in a manner adapted for the masslesstheory, and then in the following sections we extend their arguments to show how to find a, and b,.
3. RENORMALISATION
OF THE PARAMETERS
The arguments throughout the whole analysis rely on the use of the “renormalised action principle” [21, 261 to derive reltions between Green’s functions, and we use these to infer relations between composite operators. If we differentiate a finite Green’s function with respect to a finite quantity, whether functionally or otherwise, we must get another finite quantity. In particular, we know immediately that 0,” must be a finite operator because its insertions are obtained from metric derivatives of renormallisedelementary Green’s functions. Specifically, recalling Eq. (2.13) &
(#i . .. (5,V)= i(&$, .a. 4,) = (finite).
(3.1)
The line of argument in this section runs in two stages: (1) determine the form of renormalisation of ‘lo from the composite opertors of dimension 2; and (2), apply the action prinlciple to find the form of the renormalisation of a,, b,, c,,. Stage
(I)
Renormalisation theory tells us that we may construct a finite composite operator, or “normal product” [21, 23-251, of dimension 2 as a linear combination of all available composite operators of dimension less than or equal to 2 with the appropriate symmetries. The definitions are made unique by minima1 subtraction in poles in (n - 4), so that the finite operator reduces to the bare operator in the limit A + 0, and others are mixed in multiplied by coefficients which are Laurent seriesof poles in (n - 4). These definitions provide explicit rather than implicit expressionsfor the normal products, as pointed out by Brown [23]. In flat space there is only one such operator of dimension 2, and so we must have
[Pl=z2’(~,~)$&
(3.2)
142
S. .I. HATHRELL
where Z;’
= 1 + z (poles).
(3.3)
Z, would just be the mass renormalisation if there were a mass in the theory, although that is incidental to the present arguments. It can not depend on I?,,, a,, b,, or cO because these appear nowhere else in the insertions of [#‘I into renormalised elementary Green’s functions in flat space-they disappear from Yi and Yg when the curvature is set to zero. The notation [ ] in (3.2) will be used subsequently for other normal products. No distinction will be made between position and momentum space, as the meaning should be clear from the context. Consider now the Green’s functions containing one insertion of the flat-space 0, given by (2.13):
+q,3*&+
( bT ") E,.
(3.4)
These Green’s functions are in effect the first order terms in the expansion of the elementary Green’s functions in powers of the weak gravitational source, cf. Eq. (3.1) taken in the flat-space limit. We see that the term r,, 3’4: differs only by an external momentum factor from the insertion of q,,#i, and so we could modify q,, by an amount (vZ; ‘) and still obtain finite results, where v is any finite parameter. The other components in qO must be chosen to cancel the divergences caused by the other terms in (3.4), which only depend on A. Therefore choose4
(3.5
‘lo= (rl fLJG’~ where “,
Lq=iL,
Vi@>
(3.6 1
(n-4)"'
L, depends on A but not on v: it is a Laurent series of poles in (n - 4) with residues which are functions of ,l only, appropriate to the minimal subtraction scheme. We may determine L, by requiring that the insertion of 0 into a 2-point function is finite, though there is a simpler way which will become apparent soon. Notation similar to (3.6) will be used subsequently for all functions which are Laurent series of pure pole-terms in (n - 4), with residues which are functions of A only. Next we consider the form of [#‘I m curved space. There is only one other ’ An alternative is to modify 5 by a tinite function of (n ~ 4), starting at O((n by Collins, Ref. (271. The relationship between these two approaches is explained
- 4)‘). as pointed in Ref. [ 101.
out
TRACE ANOMALIESANDkdJTHEORY
coordinate-invariant therefore have
143
operator of dimension 2, the curvature term H. We must
[#‘I = Z;‘q5: +,unp4H(’
(poles)},
(3.7)
with ,U the ‘t Hooft unit of mass required to balance the physical dimensions. The poles are determined by requiring the finiteness of ([#I]) in curved space. So
& (lPl>= i(e[#‘l>+ ($ [$21)= (finite),
(3.8)
and this can now be taken in the flat-space limit. The term i(B[#‘]) is linear in ?I, so the coefficients of H in (3.7) must also be linear in II, and we can write I#‘]=
z;‘@; +$-4H{2L,
+ 4p5, I
(3.9)
defining’ L., and L, as Laurent seriesof poles like (3.6) i.e., (3.10)
and similarly for L,. They are determined in principle from their definition by equating coefficients of 17in (3.8) though again there is a simpler way which will become apparent. Although we have chosen to drop the cosmological term II, and the Einstein term K~R from the discussion, the notation of (3.9) reflects the fact that L,, and L, would also play a fundamental role in the renormalisation of A, and K~ if such terms were present. For more details see Ref. 1lo]. Stage (2) The action principle with a/&l on an elementary Green’s function produces an insertion of i as/all:
1
+F$+G?$.
!
(3.11)
There are no other terms becauseZ,, Z, and A, are all fixed when ,I and ,D are held fixed. The curvature terms F, G and HZ are independent in a general curved background,
’ The exaci relationship between the definitions here and those of Brown from this equation and its counterpart in Ref. [IO].
and Collins
can be deduced
S.J. HATHRELL
144
and the operator insertion must be finite, so we can compare with Eq. (3.9) and deduce a, =/r4(a
+ Id,),
(3.12)
6, =/r4(b
+ Lb),
(3.13)
co =pn--4(C
+ L, + VL, + q2L).
(3.14)
a, b and c are independent, dimensionless.finite parameters, and a, and b, are independent of v. L,, L, and L, are Laurent seriesof poles with residuesfunctions of ,I only, as in (3.6). (We could have had &,,/a~ finite and non-zero, but that would just be a special case of the above with a = a(v); likewise for b.)
4. DIMENSION
4 OPERATORS
AND THE RENORMALISATION
GROUP
The composite operators or normal products of dimension 4 are E,, #i, a’[@‘], H[$*], a*H, F, G and HZ: linear combinations of all the terms appearing in 9, plus two which are total derivatives. The only new ones which have to be considered are E, and #i, as the rest are finite already. In fact E, is also finite, which is most economically demonstrated in the path-integral formalism for Green’s functions by a functional “integration-by-parts.” Recalling the definition of E, in (2.14),
(4.1)
There is no term from &(y)/&(y) as this is dimensionally regularised to zero. The validity of (4.1) can easily be checked in a number of simple examples, and it is a straightforward consequenceof the Feynman rules for the insertion of E,. The righthand side is clearly finite, and we therefore write E, = [El. When integrated over all space with respect to y, i.e., at zero momentum, the insertion of [E] just multiplies the Green’s function by a factor iN, where N is the number of external legs [23, 261. [E] also gives zero when inserted into a Green’s function containing just one other composite operator: if P is some polynomial in the fields 4(x), not necessarily finite, then
(E,(Y) P(x))= -i j (4) ]f’(x)g =-+($(y)z)=O
& eis1 in flat space.
(4.2)
TRACE
ANOMALIES
AND
,$j’
THEORY
14.5
This vanishes in flat space because every term contains 6(x - y), and all such l-point functions are dimensionally regularised to zero for a massless theory. The nature of the normal product [#“I is deduced by using the action principle with the finite derivative a/~%. We therefore need to know how all the bare parameters, and the renormalisation factors Z, and Z,, vary with 1 when all the other finite parameters ,B, ‘I, a, b, and c are held constant. This information is provided by the renormalisation-group equations (RGEs) for the parameters [28, 291, and it shows why the finite renormalisation-group functions play a natural role in the study of normal products of dimension 4. To discuss RGEs we introduce the differential operator (4.3) and the equations are derived6 by requiring that all the bare quantities appearing in the action should be independent of ,u. First of all, from the renormalisation of ,I, in flat space &=iu
4-n’
1
“7 ‘YiPI ’ + ,&, tn - 4)’ i ’
(4.4)
we use DA,, = 0 and equate coefficients of (n - 4) to derive as usual DA =&I.
n) = (n - 4) /I +/?(A).
(4.5)
There are also relations between the pole residuesg,(L), and p is determined by the residue of the single pole (4.6) so that it is clearly O(L’) in leading order. The distinction between p^and p is a very important one, as we will frequently equate coefficients of powers of (n - 4). Applying D to Eqs. (2.11) and (3.2) we deduce (D+y)Z;‘=O,
(4.7)
(D+@Z;‘=O,
(4.8)
where y and 6 are the anomalous dimensions of d and 4’ (or the mass in a massive theory), and are finite functions of 2 only. We may apply D to an operator directly, as in (3.2), rather than just to Green’s functions containing that operator, becausethe ’ Renormalisation-group equations are sometimes taken to be those equations obtained by combining with the equations of dimensional analysis and then eliminating the terms in pa/&, thereby obtaining equations for the properties of Green’s functions under scaling of the external momenta. However. we will not be directly concerned with such scaling behaviour here, and will not use the equation of dimen sional analysis.
146
S. J. HATHRELL
action is independent of p, and because D only produces a finite variation on the elementary renormalised fields by virtue of Eq. (4.7). It is therefore consistent to argue that D[#2] is finite because [d’] is, and likewise for other operators. Next, applying D to (3.5) gives (4.9) and (4.10)
(D-b=P,,
where p,, is a finite function of L only, given by (4.9) in terms of the residue of the single pole of L,. In addition, higher pole residues of L, must obey consistency conditions. Applying D to Eqs. (3.12E(3.14) we then get (4.11) (4.12)
[(n-4)+D+26]L,=-/I,(A)=-$4,).
(4.13)
i(n-4)+D+61~,+2~,L,,=-~~(~)=~(1~,),
(4.14)
lb - 4) + Dl L, +P,L,
= -P,(l)
= $ (AC,),
D,=~c++rlr(l,+~~&~
(4.15)
(4.16)
fit = -(n - 4) c + P,(A). The last four equations follows from equating coefficients of r in (3.14). This must be done after applying D because the RGE for v, Eq. (4.10), has an inhomogeneous term which is a function of 2 only. The notation is systematic: all the functjons-/I, p,, /?,, Pb, PC, p,, p,, are finite functions of 1 only. The functions /3, p,, Pb, p, have an additional term linear in (n - 4). We will usually take advantage of this notation to suppress the functional dependence of the terms for the sake of brevity. We are now in a position to find the variation of the bare parameters with L when ,u, q, a, b, and c are held constant. From (4.3)-(4.5) we find
alo -d!y4)L a1
4
0’
(4.17)
TRACE
From (3.5), (4.8)-(4.10)
ANOMALIES
AND
Ai4
147
THEORY
we get (4.18)
From (3.12t(3.14)
and (4.11)-(4.16)
aa,- -(n -4) ia-- p ab,- -(n-4) a-- p %l -=. 81
L + pa i * (n-4) ! L + Pb
$-4
pn-4
b
-(n - 4)
we get
$-4
(n-4)
L
+
L [
+ A
@A
+r
(n-4)
I[
.+r2
(4.20)
I’
ca,+P,U
c li
(4.19)
1
L
+
cp,+~~,+2P&*)
[ K
1
(n - 4)
+2dLJ
(n-4)
1 (4.21)
I! *
Finally, we need to know how the renormalised elementary fields vary with 1. The bare fields are independent of A, so from (4.1) and (4.7) we have
The importance of the distinction between /3 and B is now clear. While p is 0(n’). and finite, b is O(A) and in perturbation theory its inverse begins with a pole,
p-‘=
1 (n-4)1
1 l+
PIA (n-4)
-’
(4.23)
i
and contains successive poles in higher orders. So (4.17) has the form nd all the other terms in (4.18F(4.22) begin with a single pole p’+“( 1 + :c (poles)}, a at least in the leading term.’ We can now derive the form of the normal product [#“I from the finiteness of @/an) ($1 . . . dK). It must have the general form ( 1 + C (poles)} 4: + poles multiplying all the other operators of dimension 4 listed earlier, and because of (4.17~(4.122) this is exactly the form produced at zero momentum by the derivative a/an. We (deduce that
’ The RGEs
ensure
that there
are no O(A
‘) terms
in lj-‘{L,
+ /lo/(n
- 4)/,
etc.
148
S. J. HATHRELL
+ cO,+r14 (n-4)
IH,@‘, 2
(e+L,)+rl(f+L,) B2H
-pn-4
(n - 4) (Lb+&)
+P4 fP
G
Pk LK+(n-4)
n-4
1)HZ. I The coefficients of a’[$‘] and a2H are not given in terms of known quantities by this argument because they vanish when integrated over all space, i.e., at zero momentum, but they must be pole series in the minimal subtraction scheme, and thejr are instead defined by (4.24). The particular form of these coefficients quoted in (4.24) needs a little further justification, however. First, if we go to flat space the expression for [4”] reduces to [E] _:dn-+‘fB’[m’]I.
(4.25)
The coefficient of+a2 [#‘I must therefore be independent of q, a, b and c because these only couple to curvature terms, and there are no curvature terms in the insertion of [#“I into a flat-space elementary Green’s function. The particular form of the coefficient is chosen for later convenience, and consistently with convention (3.6) because it turns out that the single pole residue d(A) plays a privileged role in the subsequent analysis and is more conveniently regarded as separate from the higher poles in L, (although they are related). The values of d and L, can be determined from the insertion of [$“I into a flat-space 2-point Green’s function. The leading contribution to d is required to cancel the divergences of the graphs in Fig. 1, and from these we find that d/B = O(l’)/(n - 4), and thus d(A) = O(13). This is sufficient information for the present. The actual value is needed when we come to computing the various quantities, and is given in Appendix B. Second, similar considerations apply to the coefficient of a2H in (4.24). This is needed to cancel the divergences for the insertion of [#“I into a one-graviton diagram, i.e., (S/SQ)((#“]) in flat space; so the coefficient is at most linear in r and does not depend on a, b or c. The particular values of the finite functions e(L) andf@)), and of the pole series L, and L,, can be obtained by equating coefficients of q in the calculation of this diagram; although we postpone determination of the leading behaviour until the end of the next section. It is at this stage that the rather roundabout approach to the discussion of the
TRACE
FIG.
1.
ANOMALIES
Flat-space
functions
AND
14’
149
THEORY
i < [@‘I $d) in leading
order
anomalies Ibegins to show dividends. The reasonis a consequenceof the form of [4” ] in (4.24), and in particular the overall factor ofp-‘. We now expressBLlrr,Eq. (2.13) in terms of the finite operators determined above. We also know that 8”, must finite, so any pole-terms multiplying finite operators mush vanish. Equations (2.13). (3.7) and (4.24) give’
and in addition the consistency conditions
L,=L.,
(4.27)
L,=4L,-2L,L,,
(4.28)
L,=2L,-4L,L,.
(4.29)
These coml: respectively from the coefficients of a’[@‘], a2H and q a*H. Stronger constraints come from a consideration of the RGE for 8. We apply p^-’ to (4.26) and use the fact that 6’ is independent of ,u, as evident from its expression in terms of bare quantities in (2.13). Furthermore ~P’D{~~4P”[#4]} must be finite because ]#“] is, and
(4.30)
’ The term
i?*H
is perhaps
more familiar
as the “ambiguous”
OR
term
in the anomaly.
S. J. HATHRELL
150
is also finite. Therefore the poles which must vanish, giving
which
/pD
/j P4-“[#“l 4! I
appear in the remaining
expression
I
(primes denote differentiation
with respect to A), together with the conditions &+6d=O,
(4.32) (4.33)
4P,-P,f+2dp,+~/~)e=O, 2p, + 4dP.4 + (/3/n - S)f = 0.
(4.34)
These last three equations constitute relations between finite functions of 1 only, valid to all orders of perturbation theory, and are a powerful source of information. For example, the anomalous dimension of [$2], 6(A), is given by (3.2) and (4.8) and is easily found to be O(A). The relevant diagrams are shown in Fig. 2. This and knowledge of d(A), the ,O(A”) function describing the mixing of 4: and a’[#‘] in flat space, give complete information about /?,, and hence qO. In lowest order we have p, = O(A”) and L, = O(ii”)/(n - 4). There is an important feature of this derivation which will recur frequently. Equations (4.32ti4.34) would have been lost in applying just D rather than p-‘D to Eq. (4.26). Since p-’ begins with a single pole, any statement about the finiteness of an expression involving inverse powers of p’ is correspondingly stronger than the same statement multiplied through by p. This point is particularly useful to the determination of p, to O(A5), which by indirect arguments relies on a simple 3-100~ calculation, whereas the direct verification of (4.33) to O(A4) requires a 5-100~ calculation. The 5-100~ calculation is performed later in Section 9; but we turn now
FIG.
2.
The flat-space
function
i(/@‘]
$4) in low order.
TRACE
ANOMALIES
AND
rid’
151
THEOR
to the study of 2-point functions of normal products to find efficient routes to the calculation of the terms appearing in the renormalisation of the HZ part of the anomaly, i.e., the terms in Eqs. (3.14) and (4.13~(4.16). 5. THE ~-POINT
FUNCTIONS
OF NORMAL
PRODUCTS
The values of the terms appearing in the renormalisation of c0 are obtained through a study of the 2-point functions of normal products defined by
r**=i
(pg),
(5. la)
r24= j [@‘I P4-“14”l 4! ( 2! )’ P”-“P”l P4-“[4”l r,, = i (
4!
4!
@lb) (5.lc)
)’
where [d’] and [d”] are given by the flat-space values (3.2) and (4.25). These functions are divergent, and in the case of r12 and r,, are divergent even when ,I = 0. The lowest order terms are shown in Figs. 3a-c and are readily calculated using formula (A.l) from the Appendices. The leading behaviors for Tz2, r24 and r44 are, respectively, 0( I/(n - 4)), O(A/(n - 4)), 0( l/(n - 4)). The renormalisation of r22 is easily derived from the action principle with J/&2 applied to (i[#‘]). We have
Tj- [@‘(x>] B(y)
=I
.(I)
FIGS.
3
$*
(ak(c)
Composite
+/r4(2L,
+ 417L/#qx
- J’),
.
(52)
)e
2-point
functions
in leading
order.
152
S. .I. HATHRELL
where 0 takes the flat-space value
e=-fip“-“[#“I 4,.
+ (l-;-Y)
PI +~vd)~2[~2]
(5.3)
given by the first set of terms in (4.26). The coefftcient of q in (5.2) therefore informs us that Tzz(p2) +,Pm4 2L, = (finite),
(5.4)
where we have gone to momentum space. This is a non-trivial statement about the renormalisation of rz2 because it implies that there are no pole-terms in (In(p2/p2)), and it is only valid using the renormalised operator [d’] in (5.la) rather than the bare operator 4:. It is now apparent that the single-pole residue of L, and the associated p-function /I,, are O(1) in leading order. We could also read off the form of renormalisation of r24 from the q-independent part of (5.2). However, this actually only provides information about Br24, and a more powerful result can be obtained instead from
(
6
WY)
P”-“M”(41
(
4!
_ . P”-“i4”w 4! 4
= (finite) )) flat 8(y)
- 2/P-4@*)*
6(x - y)
)
where use has been made of Eqs. (2.8), (3.9) and (4.24). The coefficient gives r24(P2)
+P*fl”-”
of v then
(5.6)
where
(5.7) =&+2&+4dL,}, using (4.27) and (4.29) in the last step. A factor of (n - 4)p^-’ has been extracted in the definition of L, for convenience. Using the fact that rz4 = O@)/(n - 4) in lowest order (see Fig. 3), we find L, = O(A’)/(n - 4), and f(A) = X,(A) = O(A 2).
(5.8)
TRACE
ANOMALIES
AND
id’
THEORY
153
This result would have been lost looking at the q-independent part of (5.2). We have already found that we can obtain L, , and hence /3,, , from calculation of rZ2. Using Eqs. (4.34), (5.8) and a calculation of rZ4 we can therefore find p,. Leading-order behaviour of the other terms in (4.34) shows that /I, = O(A3) at least. The general pattern of the argument has by now begun to emerge. The action principle with li/SQ and an examination of the coefficient of q give full information about an insertion of [$‘I; but it brings in [d”] with a multiplying factor of @ which gives weak results. -On the other hand the action principle with c?/c% gives insertions of [#“I without the /I factor, but only at zero momentum. So full information can not be deduced from the action principle. We use instead the fact that the divergences of r44 do not involve ln(p2/p2), and the subtractions take the form
+p”-4(p2)2 (^ ) n-4
r44
2
L, = (finite),
P
where L, contains “new” information. Returning for the moment to the trace anomaly as exhibited in Eq. (4.26) and focusing on the coefficients of HZ and a2H, we have found that the coefficient of q2 is determined from r22, and the coefficient of q from r24. It is natural, therefore, to expect to find e and /I, from r,,. This becomes much more obvious if we apply c?/cM to (8), which is best done using the bare form (2.13). We conclude that
i(flS) + 8c,(p2)’
= (finite),
evaluated in flat space. Recalling the expressions can equate coefficients of (q - d). (This is as because q is an arbitrary finite parameter.) The are consistent with Eqs. (5.4) and (5.7), and the
(5.10)
for c0 and 0 in (3.14) and (5.3). we good as equating coefficients of v coefficients of (q-d)’ and (v -d) remainder gives
8(L, + dL, + d2L,) - (n - 4)2 L,, = (finite).
(5.11)
Then L, and e can be deduced from (4.28) and (4.33). The lowest order pole contribution to r,, is O(l)/(n - 4), and so L,, must be O(A’)/(n - 4). To find the single pole of L,, and hence /I,, we must use an RGE to derive the leading term in the residue of the treble pole of L,, Y,(A). Superficially this would be expected at O(A4). The relevant RGE is obtained from applying p’-’ Dp^’ to Eq. (5.9) using the flat-space RGE for [$“I which can be read off from Eq. (4.31):
pD
p^P”-“P”l 4! I
= --d’3’[#‘]
- y’[E].
(5.12)
I
Thus L, = (finite),
(5.13)
S.J. HA1‘HRELL
154
where the term in rz4 has been eliminated using Eq. (5.6). The finite term on the right side of (5.13) must simply by (l/A”)(8/LU)(AY,) as this is the only finite term on the left. The equation may then be integrated iteratively, and Y3 is given by $(AY,)=-;/A~2Yl
$ (;)(d;)+~j~4d%f,(d~)+4d’U,.
(5.14)
0
X,(L) is the residue of the double pole in L, and is given from Eq. (5.7) by X2 = 2/c, + 4dA, = O(A3),
(5.15)
and hence by p, and /?,, . (It could also be obtained from X, by deriving an RGE for r,, similar to the above.) The three terms on the right-hand side of (5.14) are, respectively, O(A5), O(A”), O(A”). The O(A4) contribution has vanished from Y3, and hence from /3,, because the first term depends on (a/LU)@/A’). The O(A”) contribution depends on the O(A3), 2-100~ /?-function, and on the leading term in Y,(A), which is O(A’). This concludes the summary of the arguments of Brown and Collins [IO], adapted for the masslesscase. In the expression (4.26) for the trace anomaly, the functions p^, y and 6 are the usual functions appearing in flat-space #4-theory; d and /?, are computed from the mixing of a’[$‘] in the renormalisation of 4: ; e, f, p,, , /?, and /?, are computed from the functions r22, r24, and r44. The computations rely in particular on the relations (4.32)-(4,34), and enable low order flat-space calculations to be used to derive the terms appearing in the curved-space anomaly, in some cases in high orders of perturbation theory. Particularly striking is the result that j3, is O(A’), so that e is O(A4) following from (4.33), and in Section 9 we confirm this directly by an explicit 5-100~ calculation. So far, however, very little has been derived about the coefficients p, and pb of the other two “gravitational” terms appearing in the anomaly. In Sections 6 and 7 we show to extend the analyis of Brown and Collins to determine these also.
6. TENSOR OPERATORS
AND THE F TERM
Renormalisation-group equations for a, and 6, were derived in the last section, but nothing else was found out about them as they appeared to be unrelated to the other parameters of the theory or to the divergences of the functions r22, r24, and r,,. They appear in the action multiplying the F and G terms which are second order in the curvature, but these have the property (6.1) so that the functional derivative S/&2 produces quantities which are still second order in the curvature. Therefore a second application of d/&2 still produces terms which
TRACE
ANOMALIES
AND
iq3’
THEORY
are at least first order in the curvature, and vanish on going to the flat-space Contrast this with the behaviour of the non-conformal Hz term:
. -g - 6 bm.
HZ = -(n - 4) H* + 413*H.
155 limit.
(6.2)
The last term produces a non-zero quantity in flat space under a second application of 6/6s2. and the appearance of this non-conformal 8*H term is precisely why the coefficient e,, appears in the renormalisation of i(M), and hence the functions Tz2, I- 24, and rJ4. So to extend the investigation to include u,, and 6, two possibilities immediately present themselves: (i) look at the effect of -2 6/fi6gHL, rather than the traced version S/&2, and (ii) look at the effect of (S/&2)>‘. This section deals with (i), and the next with (ii). The action principle informs us that -2 d/\/--gag,, gives finite results. Applying it twice to the vacuum functional. taking the flat-space limit and transforming to momentum spacesgives PUU + a,A ‘API + cOC+”
= (finite),
(6.3)
where
correspond, respectively, to the F and H* terms in the action. There is still no csontribution at all from the G term becausein fact and
A
K.\UP,
CK.iUI’
+ 8(RU”R”, - Rv”a14R,o),
(6.5)
making use of the Bianchi identities, and this is second order in the curvature again. Therefore b, remains absent in this approach. Thus full expressionsfor AK-l”” and CKAfi” are
and
156
S. J. HATHRELL
The flat-space energy-momentum in the form
tensor P”(x)
is given by (2.12) and can be written
where
P” = -qi, fU”QO+ & (using (3.2) and (3.5))
(fz2 - @I+L,)) t”“[~‘l~
(6.9)
and
P” is a traceless differential operator [23], and (6.8) represents a decomposition of t!Y into irreducible components: c?’ is manifestly traceless in n dimensions as well as in 4. Furthermore both 0 and 0”” are finite operators, so @” must also be finite. We can therefore introduce the new finite composite operator [qW”$] and deduce its renormalisation from Eq. (6.9): [qh”“#] = q&P”q& + (c poles) t”“[#2],
(6.11)
where the poles-terms are just the q-independent pole-terms multiplying t”“[qi’] in (6.9). The q-dependent part is clearly finite also. The particular form of the pole-terms multiplying t”“[#‘] is unfortunately rather untidy because of the presence of the factors (n - I))’ and n/4 which mix up residues of different order poles of 2, and L,. This is an inevitable problem in the discussion of tensor operators in the minimal subtraction scheme [24], because any contractions implicitly introduce factors of n with the (flat) metric tensor q’*‘. The difficulty was avoided in earlier discussions by confining attention to scalar operators and keeping a factor of (n - 1) buried in H, but this factor re-emerges when the metric derivative is not traced. The resolution of the problem becomes a matter of defining just what tensor the “minimal” subtractions are made from. However, all of the Laurent series of poles with the form of (3.6) obey some kind of RGE which implies that each successive pole appears at O(A) below the leading order of the vanishing of the leading order previous pole (though there may be an “accidental” coefficient, as in the single pole of Lx--see later), so multiplication by some finite function of n brings no ambiguity in the leading k-dependence of each pole residue. In quite a few cases there is no ambiguity for some of the next-to-leading terms either. There is consequently still considerable value in discussing the renormalisation of the tensor quantities mentioned above. Equation (6.3) expressing the renormalisation of i(f?“‘P”) clearly contains some new information, but it is not immediately obvious just how much. A count of the
TRACE
ANOMALIES
AND
ii4
157
THEORY
form-factors is therfore quite illuminating. A quantity Pa”’ with symmetries under (K -A), (u - V) and (~1) - (u v ) contains five scalar form-factors-these multiply an assortment of tensor terms similar to those appearing in (6.6). In addition to the constraints imposed by these symmetries, however, there is a further constraint because the energy-momentum tensor obeys an equation of motion. Classically this is the vanishing of its divergence, and the operator version is analogous (we only need to consider flat space): ~3~,8”” = -Eb”, where Et is an equation-of-motion
operator
(6.12)
[23 ] similar to E, : (6.13)
Furthermore
Eg is finite, Eg = [E”],
and satisfies
W(x)
(6.14)
P(Y)) = 0
using exactly the same arguments as for E,. Therefore in momentum space, pJ+yp)
= 0
(6.15)
and this constraint reduces the number of independent form-factors from 5 to 2-and recall the appearance of just two parameters a, and c,, in Eq. (6.3). So it is hardly surprising that the study of 2-point functions should not give information about the third quantity b, as well. Notice also that PICA
KAUI = p, pw
= 0,
(6.16)
a consequence of the coordinate-invariance of F and HZ. The discussion of fnarc” may now be simplified to a discussion of the two formfactors by taking contractions. Contraction with v~.~I],,~,just reproduces Eq. (5.10) losing all information about @“ and the parameter a,. The latter disappears because AK“” also satisfies A”
I(
L(L’=AKk
u =
(6.17)
0,
consistent ,with Eq. (6.1). Contraction with v,,, vl, is more useful, however. Inserting the expressions (3.12) and (3.14) for the renormalisation of a, and cO, and the expansion for I!?’ in (6.8), it gives
+ (p2yy
I
4(n - 3)(n + I)&
+ &
(15, + ~5, + ry2L,)I
= (finite).
(6.18)
158
S. J. HATHRELL
Now the coefticients of q2 and q in i(@‘“g,,,,) involve the expressions t”‘t,,( [#‘I [#‘I) and P”( [#tMU#] [@‘I). Both of these can be expressed solely in terms of the functions I-,, and r,,. using symmetry arguments and Eq. (6.12) in the case of the latter. It can be verified that the coefftcients of q2 and q in (6.18) exactly reproduce the results for L, and L, found in the previous section-including cancellation between the various functions of n arising from contractions of tensor indices. There is therefore no ambiguity about the definitions of the subtractions in the minimal subtraction scheme for these terms, nor are there any new results. The new subtraction-term L, appears in the q-independent part of Eq. (6.18), and this depends on the “new” function i([qWr#][qb,l,#]). It is impossible to express this function in terms of previous functions using any kind of symmetry argument or results involving (6.12tagain a form-factor count shows why-and so it requires a separate Feynman-diagram evaluation. Given this fact, and in view of the rather clumsy definition of the renormalised operator [#t”“#] derived from Eq. (6.9) there seems to be no advantage in considering i([qW”~][~t,L,~]) rather than the more directly relevant function i(f?“#u,,). The same Feynman diagrams appear in the same orders of perturbation theory, and both involve Feynman integrals with the same complications from numerator factors in the integrand. Indeed the latter function has the advantage that @” has a simple expression in terms of bare operators. Both [#t”“$] and @“’ are finite, and [@““d] d oes not contain extra factors of BP’ to make it a source of more powerful results. We conclude, therefore, that the F part of the trace anomaly is determined by setting r = 0 and then using the flat-space result i(@‘“g,,,,> + (P’)”
pnP4
I
4(n - 3)(n + 1) L, + H(H8P 1) L,
1
= (finite).
(6.19)
The term i(l/n)(&J) h as b een translated into a pole-term proportional to L, using Eq. (5.10). Below O(L”) the L, term in pl’ does not contribute, and @” takes the simple form
(6.20) which is quite convenient for computation. L, does not contribute below O(A5). The evaluation of the relevant Feynman diagrams to the 3-100~ level is described later in Section 8. But we observe for the present that there is an O(1) contribution to the single pole of L,, and hence to p,, arising from the divergence of the l-loop diagram in i(@“$,,). This 0( 1) contribution reproduces the known, A-independent coefficient of the F term in the trace anomaly. The advantage here is that the reduction to flat space makes the discovery of the higher order corrections much more accessible than it would be from calculations based on a curved background [30, 311.
TRACE
7. THE
ANOMALIES
~-POINT
AND
FUNCTIONS
Ad4
AND
159
THEORY
THE
G TERM
It was shown in the last section that if the coefficient b, of the G term was to be determined by eventual reduction to flat space,then 3-point or higher functions would have to be considered. There is also a clue as to the outcome: in an earlier paper Drummond and Shore 191considered a conformal scalar theory with no q,,H#i or c,H* terms in the action, and they showed in a spherical background-for which the F term vanishes-that b, was not renormalised by A. becausethere were no primitive divergences. Introducing in particular the c,H* term would introduce divergences into the spherical vacuum functional, but only at O(A’), and this would affect the argument for b,. It is reasonable to infer, therefore, that in the current approach based on flat-space calculations, the first A-dependent terms in b, will appear at comparabl:y high orders of perturbtion theory, and will be dependent in some way on the terms in cO. This is indeed what happens, and can be discovered from flat-space considerations I.51 in spite of the fact that G is a topological quantity. The simplest Green’s function involving 6, is the 3-graviton function obtained with (s/&2)‘, so we apply this to the vacuum functional, invoking the action principle, and obtain
w VI
i*(B(x) O(y) O(z)) + i B(x) L aqz)
(
+
1
cm(x) sn(J,) sn(z)
)
de(z)
’ sn(x) + e(z) = + e(J) sfl(J’) )
= r, = (finite>,
(7.1)
which is taken in the flat-space limit. The individual matrix elements are of course symmetric in x, y and z becausethe functional derivatives 6/&J commute, and so
Wx) ---= WY)
d2S _ WY). a?(y) &2(x) &2(x)
(7.2)
Henceforth we will condense the notation slightly, deliberately suppressing the distinction between position and momentum-space, and refer to 0(x), M(y)/&(z) and 8*0(x)/&2(v) &2(z) as 0,, eYZand 8,,;. Similar labelling will be used for other operators. Note that the terms in By2involve (in position-space) the function S(y - z); and in 8,,, the function 6(x - y) S(.y -z), so that x, y and z coincide for this third operator. Labelling of this kind becomesnecessary for 3-point functions becausethey are in general functions of three invariant momenta: pi, p.z and pi. Such general functions can not usually be handled easily. but there are casesin which they can: (i) ifpL = 0, then p: = p: = p2, and the whole is a function of one variable p2. (ii) If pi = p,’ = 0 and p: = p2 # 0, then again the whole is a function ofp* only. These two prove quite adequate for the determination of 6,, though in fact (ii) may be relaxed to case (iii), when only pi = 0, and on dimensional grounds the Green’s functions must be a power of p: multiplied by a function of the dimensionless ratio (p-t/pi). The last case
160
S. .I. HATHRELL
becomes important for later calculations in Section 9. Some general formulae relevant to cases (ii) and (iii) are given in Appendix A. This plurality of momenta is not the only new complication arising with 3-point functions. Another becomes apparent if we consider as an example the leading, Aindependent contribution to the function i2&4-n/4!)3 ([$:I [#;I [#:I). The diagram for this is shown in Fig. 4, and with pi = pf = 0 it can easily be computed using the formulae (A.1) and (A.5). Note that putting these two momenta on shell does not introduce any infra-red singularities. The result is 2(n-4)
x
r(6 - 2n) r(2 - n/2) T(n/2 - 1)” Z-(3(n/2) W(n - 2)3 r(5(n/2) - 6) I
- 4)2 I
+ U@).
(7.3)
This has a double pole, so that the single pole must contain terms involving ln(p2/p2) which cannot be removed by a constant additive counterterm. (It also contains such undesirable objects as ln(4z) and Euler’s number.) Clearly the renormalisation of 3point composite functions must involve subtractions of other Green’s functions as well, and we will return to this a little later. The third complication involves the equation-of-motion operator [El. The individual contributions from this no longer vanish. If A(y) and B(z) are polynomials in the field 4, then with the usual functional integration-by-parts i2(~,A,B,)~=
i j (dq+) A$; I
=+I,
(-jL-z))-(B++$)).
(7.4)
The two functions on the right-hand side do not vanish in flat space. The 3-point
FIG.
4.
A composite
3-point
function.
TRACE
ANOMALIES
AND
Lb4
161
THEORY
functions involving [E] can at least be expressed in terms of 2-point functions, however. Furthermore we can use the result (4.2) to deduce the special cases (7.5) i’(E,E,E,)
= 0.
(7.6)
We also have
(7.7) and
Equations (7.2) and (7.4k(7.8) can now be used to simplify [E] appearing in Eq. (7.1). If we define
the contributions
fixxe8,-aE,,
(7.9)
where a is some parameter as yet unspecified. after a little simplification obtain I-= i’(B,B,gJ
from
we can insert this into Eq. (7.1) and
+ i(B,OYL + 8,8,; + OZ8,,.) +
(e,yz)=
(finite),
(7.10)
where (7.11) The third lmatrix element is insensitive to the choice of a because x, ~1and z coincide and so any terms involving the fields are dimensionally regularised to zero in the massless theory. The only contributions to it come from Yg and are obtained by two functional derivatives of the curvature terms in 0. Eq. (2.13). After going to momentum-space we find (7.12) where B = 2(n - 2)(n - 3m - 411 HP:)’ - 2IP2;P: + P:Pi c=
4{@ + wP:)2 + 41p?;: + P:Pl+
+ cp:>* +
+ PIP:115 + (P:)’ PIP:lI.
+
(Pi>‘1 (7.13)
(P5121 (7.14)
162
S. J. HATHRELL
This time it is the F term in Pg which gives no contribution. The reason is that F involves the square of the n-dimensional Weyl tensor C(“)aqy6, which has the property that it is invariant under a conformal scaling of the metric, i.e., (7.15) for any f(x). So any number of functional derivatives S/&L’ applied to F always leaves an expression which is second order in the curvature, and which therefore vanishes in the flat-space limit. We can now see how b, comes into the picture. It multiplies the 3-graviton vertex B given by (7.13) and appears as part of the Green’s function r in (7.10). The divergences of b,B are needed to make the rest of r finite. B itself involves a factor of (n - 4), so (7.10) determines the residues of the double and higher poles of Lb, but not the single pole. The single-pole residue b,(L) and the P-function Pb can then be deduced from the double-pole residue b,(A) using the RGE for 6, previously derived. Specifically,
This method can therefore be used to deduce everything except the leading, constant contribution to b, and Pb; but as this constant contribution is already known from elsewhere [7, 91 it does not matter that it escapes detection here. In fact the present approach is an efficient route to higher order corrections without involving the leading terms on the way. The other two matrix elements in r involve the parmeter ~1, so a value for this needs to be chosen. The two obvious possibilities are, (1 - n/2) to simplify 0 in terms of bare operators, from Eq. (3.4); or (1 - n/2 - 7) to simplify $ in terms of renormalised operators, from Eq. (5.3). Although the latter is perhaps the more obvious choice in keeping with the general approach here, the former is in fact more convenient and allows the problem to be tackled in the following way. We define the composite operator ( O4 } as the linear combination
(7.17)
so that with a = (1 - n/2) we have ( in flat space)
(7.18)
TRACE
ANOMALIES
AND
/zd’
163
THEORY
( 04) is the “natural” object in the present context, because although it is not actually finite, it behaves in many respects as though it were, and it has the following useful properties: (i) p^(O”) is finite. (ii)
It obeys the simple RGE (obtained from Eq. (5.12))
pD@{o”})
= -d’a’[qP],
(7.19)
which is also finite. (iii)
It arises naturally
with the use of the action principle, --= C3S d”x (O:}. .I I3
i.e., (7.20)
(a/an only produces insertions of [#“I into Green’s functions of renormalised elementary fields, and we are more concerned with functions containing other composite operators.) (iv) We may substitute { O4 } into the definitions of the functions r24 and r,, of Section 5 swithout altering their values. are the same because y(A) = O(A*). (v) In leading order (04} and ~‘~“[#~]/4! (vi) (0’) has the same renormalisation structure in 3-point functions as the finite operator p4P”[#4]/4!. The point of these remarks, and especially the justification of (vi), will become apparent when the renormalisation of the composite 3-point functions is determined. But before: passing on to that we note that, with the choice of a as specified, the operator tTYz can be obtained from Eqs. (2.13), (7.9) and (7.11), followed by expressing the result in terms of renormalised operators. In momentum-space the contribution relevant to the matrix element in (7.10) is given by 0)s: =: -2(n
- 2)&04}
+ (d + L,)((3n
- 10) p: - (n - 2)(p; + p;)) &iP]
- (rl - dN(n + 2) P: + (n - 2)(~: + pi)1 +[#‘I.
(7.21)
(This form assumes p, + py + pz = 0.) Thus the second of the matrix elements in (7.10) involves only linear combinations of the 2-point functions r44, r,, and Tzz met earlier, multiplied by various functions of A and polynomials in the momenta. r44r etc., come only as functions of the individual momenta pt., pz or pi. The remaining term in I- is the 3-point function i2(~~y~yf?z), with f? given by Eq. (7.18). It involves the functions which can be defined by r444(dr
P$ pi) = i’(~O~l~O~~~O4~)~
(7.22)
r244(d.
PC, PI) = i’GkC1~0~~~041>~
(7.23)
r4,,(p:~
P:, ~1) = i’(lOZl
(7.24)
rzz2(pf3 P$ PI) =
+[#:I SKI>,
i’(fhGl $:I %W,
(7.25)
164
S. J. HATHFCELL
and similar ones obtained by permutation. They are the natural extension of the 2point functions rz2, rz4, rd4 in Section 5 if these are regarded as involving { O4 } rather than ~~-“[$~]/4!, i.e., r24
=w”lP”~)~
(7.26) (7.27)
r4, =~w%W
These definitions are identical to the previous ones in (5.1) because [E] gives no contribution in 2-point functions. r22 is unchanged anyway. The advantage of doing this is that the large number of 3-point functions explicitly involving [E] do not now have to be considered, either for the purposes of calculation, because of the absence of [E] from both forms for # in Eq. (7.18); or in the derivation of the RGEs, because of Eq. (7.19). The divergences of r,,, are removed by subtractions of sums of poles in A multiplying the terms T,,(pf), pfr,,(pj), pfpjr,,(pi) and ,~“-“pfpj, suitably symmetrised. (The pole-terms are independent of q because n only couples to the curvature and we are now in flat space.) The justification of this statement involves considering first the subtractions necessary to renormalise the function i2@4Pn/4!)’ ([#~][(li~][#4]) constructed from finite operators, introduced in Eq. (7.3). The divergence associatedwith the coincidence of the points y and z is subtracted by pure pole-terms multiplying the functions remaining when these points coincide, and these latter functions are just the 2-point functions previously encountered (or can be taken as such without loss of generality). The leading pole coefftcient of T,,(pi) is O(1) to cancel the divergence of the graph in Fig. 4 when y and z coincide. The leading pole coefficient of T,,(pi) is O(1) to cancel the divergence of the graph in Fig. 5a, though r,, is itself O(A); and the leading pole coefficient of T,,(p:) is O(A2) to cancel the divergences of the graphs in Fig. 5b (associated with the mixing of a’[$‘] in 4:). Other terms appear from similar subtractions obtained by permutation of the spacetime labels. Finally there is an overall subtraction of pure pole-terms because the whose 3-point function is itself primitively divergent. Returning now to rjq4, involving (04) instead of [#” ], the pole-term (y/B)[E] in (04 1 gives contributions which are translated by means of Eq. (7.4) into pole-terms multiplying r,, and r,,.
FIG.
5a.
An O(A)
3-point
function.
TRACE
ANOMALIES
AND
&,b4
THEORY
165
t Cl -
5b.
O(,l*)
3-point
functions.
We see that these terms are just the same in structure-through with different pole coefficients-as the terms appearing in the subtractions for i2(@4-“[#4]/4!)3). This justifies the earlier statement (vi) about the use of (04}. Exactly the same arguments apply to r244 and r,,,. The divergences of r244 are subtracted by poles multiplying the functions r,,(pf), pfTt2(p,!) and ~“-~pf. The divergences of r422 are removed by poles multiplying T,,(pi), T,,(pf) and ,~~-j--but not T,,(p:) in this case because there are no diagrams in r,,, which have a primitive subdivergence associated with the coincidence of y and z. The leading poles in r4** are O(1) corresponding to the diagram in Fig. 6. Finally, Tzz2 is finite to all orders because it contains no primitive divergences, and all subdivergences are incorporated into the renormalisation of 1 and [$‘I. All this analysis is rather cumbersome, so let us summarise the argument so far. The renor:malisation of b, is determined through the residue of its double pole by requiring that the single pole of the 3-graviton function r should vanish. The contributions to r can be expressed in terms of the insertions of the operators (04) and I#‘] into 2- and 3-point functions, and all contributions from [E] can be absorbed into the functions in such a way that they do not have to be considered explicitly, either when calculating directly or when deriving RGEs. The renormalisation of a Green’s function containing insertions of the divergent operator {04} has the same general structure as the renormalisation of that function containing instead the finite operator p”-“[@“l/4!, and furthermore it involves only combinations
166
S. J. HATHRELL
FIG.
6.
Leading
contribution
to r4??.
of the f-point functions previously discussed. This fact and the removal of [E] constitute a major simplification. So far, however, we have only found the general form of the renormalisation of the composite 3-point functions. We can do much better than this and derive many of the pole coefficients explicitly, as well as verifying the general form, by once again using the action principle. As before, there are two basic methods: (i) find full information about insertions of [@‘I using the functional derivative S/S0 and looking at the coefficient of q; and (ii) find full information about insertions of ( O4 ) at zero momentum only, using the ordinary derivative a/aL This leaves just one quantity whose divergence can not in principle be expressed in terms of the divergences previously encountered, namely, I-,,, with none of the momenta vanishing. The non-vanishing of any of p,, pY, pL is actually implied by the conditions p-t = pz = 0, p.t # 0, so it will be sufficient to consider the case r444(p2, 0,O). Recall now Eqs. (7.10) and (7.12t(7.14) expressing the renormalisation of the 3graviton function r. It contains a large amount of information which can be organised into two categories: (i) it is an identity of cubic order in (II -d), and (ii) it is an identity in the various momenta. b, is independent of (q - d), and so it is determined by the corresponding part of Eq. (7.10)--which contains p^‘r,,,. Furthermore the vertex B which b, multiplies, corresponding to the topological term G in the action, has the property B(p*, p*, 0) = 0,
(7.28)
i.e., it vanishes when any one of the (vector) momenta is set to zero. On the other hand B(p*, 0,O) = 2(n - 2)(n - 3)(n - 4)(p*)*,
(7.29)
which does not vanish in n dimensions. Thus it is r444(p2, 0,O) which contains the “new” divergences, and it is precisely this quantity which is relevant to the determination of the double pole of b,. From Eq. (7.14) we also have C(p2,0,0)=4(n
+ 2)(P2)*,
(7.30)
TRACE
ANOMALIES
AND
,@’
THEORY
167
that the (r - d)-independent part of c,, also appears in the resultant equation for &-just as suggested by the remarks at the beginning of this section. The simplest composite 3-point example to consider is Tzz2. Its renormalisation is obtained by applying a/&2 to the corresponding equation for the renormalisation of rz2, and then taking the coefficient of II. To do this it is necessary to use the full curved-space version of Tzz first, rather than the flat-space version in Eq. (5.la), and allowance .must be made for the possible appearance of l-point functions containing 8(x -y) which only vanish in flat space. In effect we should really start with the full operator version of Eq. (5.4) in position-space. In the case of Tzz there are in fact no new terms because the only dimension-4 2-point operator involving 6(x-y) is 8(x -JI) itself, with an accompanying (fi)-’ for coordinate invariance. Therefore the operator ]O.:t] defined by SO
+fi”-42L* lo.3 = __ (2;)’[#.:][$;I
6(x-y) Gm
is a finite 2point composite operator in curved space, with [#.:I given by Eq. (3.9). Applying S/S0 to ([Of:]), taking the flat-space limit, and then taking the coefficient of q gives (7.32) i~“-~4L,,(f[#.z] ats(y -z) + $[#f] 8.:8(x -z)) + 28fTzzz = (finite). The first term vanishes in flat space, leaving
r,,,
= (finite)
(7.33)
just as argued above on general grounds. One way to obtain the renormalisation of r,,, is to apply S/SQ to the curved-space version of rz4. The operator version [O::] of the renormalised 2-point insertion the @“-“/4!)[&J t[&] th’is t’ime contains, in addition to the original cY’J(x -u)/\/-g, extra terms +[#.:I 6(x -JJ)/~ and ,u’-~H~(x -,v)/fi This appraoch fits into the schemeoutlined, but as the renormalisation of [O::,] was not fully investigated earlier it pays instead to examine r,,, by remaining in flat space and using the action principle with --a/a2 on rz2. Recalling (3.2), (4.8), (5.4) and (7.20) we get
i2 ((d”xiOZi ![#:I +ktl j + 7 i(ijp:l tl#l-I> - 2p”-46( WV - .z) $ L, = (finite).
(7.34)
This gives I’,,, at pr = 0, but in this instance we can use dimensional and symmetry arguments to deduce the full result:
-
2p-4
($L*1
= (finite).
(7.35)
168
S. J. HATHRELL
Then we could work backwards to deduce [O::]. The leading, L-independent contribution to rdZ2 is given in Fig. 6, and it is easy to verify by direct calculation that (7.35) is indeed correct using the values of the parameters 6, p^ and L, listed in Appendix B. In fact in this particular case the pure-pole term ((a/an) L,) must contain a double pole at O(1) but no single pole at O(1) because of the form of Fig. 6, and the absence of this single pole means that there is no O(L) contribution to /3,, (which is calculated to O(,I’) later in Section 8). We now have some consistency checks in the argument concerning the finiteness of the 3-graviton function r. The coefftcient of (v - d)3 in Eq. (7.10) appears only in the term i2(gX(?,,gZ) and just reproduces the result that r222 is finite. The coefficient of OI- 4’ can likewise be shown to reproduce Eq. (7.35), in a weaker version multiplied through by /? (This requires Eqs. (4.13) and (5.4).) There is an important point concerning infra-red (ir) divergences which is exhibited by Eq. (7.35). This equation only remains valid provided pz # 0, pz # 0 because it is a statement about the systematic subtraction of ultra-violet (uv) divergences only. T,,(pi) is a function containing uv poles, but it is regularised to zero in the dimensional scheme when pz = 0, thereby effectively introducing ir poles. These appear because the subtracted function contains ln( p’/,u’) which diverges when p2 + 0. Similarly the leading contribution to r,,,, (from Fig. 6) is regularised to zero when pz = 0, and the “finite” term containing ln(p2/p2) then diverges. On the other hand there are no ir divergences at pz = 0 in (7.35), and it remains valid for this. The “finite” terms involving ln(p:/p2) appear in the form p: ln(p:/p2) and remain finite as pi + 0. But if the whole of Eq. (7.35) is multiplied through by pi pz it then remains strictly true for all momenta; and this is just the equation for the renormalisation of the 3-point composite functions involving a’[d’] rather than [#‘I. The important point is that in the 3-graviton equation for r, l? only contains ]#‘I in the form of a’[#‘], so there are no ir divergences for this. Equation (7.10) remains valid at all momenta, and we are justified in equating poles for various zero-momentum or onshell configurations. The renormalisation of r,,, can be derived by a straightforward application of the arguments outlined, but we need not consider it as it only serves to provide a consistency check for the coefficient of (v - d). The useful result comes with rdd4, for which we apply the action principle with -a/a1 to Eq; (5.9). We also need the result (7.36) This approach only gives r444 at p, = 0, but since the terms in its renormalisation involving r44 are not multiplied by any factors of pf , we can use symmetry arguments to deduce this part of the renormalisation in full. Thus
r444(p:&d) + $- v-44(d)+r44(p:)+ r44(d)l + {pole-terms = (finite).
involving
p;r,,(p;),, pf pfr22(p:), ,unp4pfpf 1 (7.37)
TRACE
At pl =pi
ANOMALIES
AND
(p’)’
169
THEORY
= 0 it must therefore take the form
r444cp2v 0,0)+$ r44(p*) + I( In-4 )2Lfl +
,@’
/F4
P2L2(P2)
+ (-n-4P 13
uP2)*~22(P2) I
= (finite),
/(ni4,3L*f
where L,, L, and L, are unknown Laurent seriesof poles, and the various factors of (n - 4)//I? have been extracted their detinitions for convenience in their RGEs. The crucial feature is that the pole series multiplying r44 is known. Remembering that (04} differs from p”-“[$“]/4! only at O(A), it follows from the earlier arguments that the pole coefficients of r14 and Z-2* are 0( 1) and O(A) in leading order, so that the leading poles of L,, L, and L, are O(n’)/(n - 4), O(I’)/(n - 4) and O@“)/(n - 4)2. L, also contains a single pole which is O@‘)/(n - 4), but this is only next-to-leading in the sensethat it is tied by the general renormalisation structure to the 0(,14) double pole. It is now possible to deduce the value of the G term in the trace anomaly. Using the expressionsfor e,, gY2and BXyz given by Eqs. (7.12), (7.18) (7.21) (7.29) and (7.30), and the expansions of b, and c, given in (3.13) and (3.14) we set 17= d in Eq. (7.10) and obtain -p^9-444(p*, 0, 0) + 2(n - qP*r,,
- (3n - io)(d + L,)p*@-42
+ ( p2)2j.i”-4{ 2(n - 2)(n - 3)(n - 4) L, + 4(n + 2)(L, + L, + d*L,,)} = (finite). (7.39) This may be further simplified by removing p^‘r,, and bi-24, using Eqs. (4.28) (4.29) (5.6), (5.7) and (5.9)-(5.11): (~*)%“-4rw
-
w
-
3)(n - 4)L,i
-P3r444(p*, 0~0)
= (finite) + (3n - 10)(p2)* ,B’-~(~(L, + dL, + d2L,) - (d + L,)(2L, = (finite) + (p”)‘,u”-“(
(3~ - lO)(L, + dLf)].
+ 4 dL,)} (7.40)
All that remains is to find the poles in p3rqq4, and these can be determined from has a leading (7.38). r,,,q has a double pole at O(l), so superficially B’r,,, contribution to its single pole at O(A’), and this can be determined from the leading pole of r,,, with the aid of the appropriate RGE. We can seealready that the leading single-pole residue on the right-hand side of (7.40) is also O(,I’) from the contribution of L, (or L,,), so the double-pole residue of L,, b,(A), must be O@“) as well. It turns out that the renormalisation-group structure of r444(p2, 0,O) actually implies a cancellation in the 0(,15) leading pole of j?3r444, in much the sameway as the OQ4) single pole of p^‘r,, disappeared in Section 5. This is a non-trivial cancellation which would require 6-100~ calculations to verify by direct means.From
170
S. J. HATHRELL
now on we will confine attention to the leading terms to demonstrate this cancellation. The RGE relating the poles of r,,, is obtained by applying p-’ Dp^” to Eq. (7.38), and using the RGEs for [#*I and {04} given by Eqs. (3.2), (4.8) and (7.19). Thus pr44
n-4 ’ + (p2)2pn-4 --J( 1 l@-4)+Dlh
= (finite) + 0
A2
a.
(n _ 4)1 (n _ 4)2 ,... . 1 (
The contribution from the 3-point functions is removed from the leading order because of the O(A*) factor of d’ in Eq. (7.19). The significance of this is that the leading terms in the poles of L, are shown to depend only on the leading poles of I- 44, known from previous work, and have been deduced without the necessity of calculating any Feynman diagrams for r444. Multiplying (7.38) by p’ gives p3r444(p2, 0,O) + (p2)*pnP4(n
- 4)3 LT = (finite) + O(A’j){poles},
so the single pole of b3r444 is given by T,(A) + O@“). r,(A) substituting for r44 in Eq. (5.9) ,+4)+D]L,=P’T;
is obtained
by
(7.43)
LY + 0
(n-4)
(7.42)
and then integrating iteratively. The higher poles of L, are given in terms of the lowest pole Y,(A) using the RGE for L, derived earlier in Eq. (5.13). It is not necessary to know the actual values of the coefficients yet, just their order in A, so if we put
P(l) =p*lz* + O(k’), Yl(A) = cd* + O(P)
(7.44)
then (5.13) gives
Substitution
Y,(l)
= ap,A’ + O(A4),
Y’(A)
=
O(P).
(7.45)
of these into Eq. (7.43), with the known fact that
T,(A) = O(P)
(7.46a)
then gives T*(d) = gfp,~3 + O(A4), T’(A)
= fc&A4 + O(A5),
T,(n)
=
0(/l”).
(7.46b)
TRACE
ANOMALIES
AND
14”
171
THEORY
Conclusion: the leading pole of /?3T444(pz, 0,O) is not O(d’) after all, but O(L6). This is quite a striking prediction, which as mentioned above would require 6-100~ calculations to verify directly. There is, however, another prediction of this analysis which can be put to the test quite easily: the double-pole residue r,(L) is given by Eqs. (7.44) and (7.46b) solely in terms of the l-loop p-function and the leading pole of r,,. Substituting into the original equation for the renormalisation of l-444, Eq. (7.38), gives the leading order (4-100~) result L(P2*
2P* 03 0) + (n _ 4) L(P2) + (P*)*P”-~
I
5%
O(l)
cn _ 4j2 + cn _ 4j
i
= (finite)
+ WI2
(7.47)
a non-trivial prediction about the subtraction of both the leading, O(1) double pole and the ln( p’/p’) terms in the single pole of r,,,. Using the values listed in Appendix El9 and the value of r444(p2, 0,O) given by Eq. (7.3) this is indeed seento be correct. Finally, Eq. (7.40) shows that the leading contribution to b2(IZ) is after all given only by the leading contribution to cl(l), i.e., b,(l) = 2c,(A) + O(P).
(7.48)
(The mixing of pole residuescausedby the various factors of (n - 2), (3~ - IO), etc., only contributes at O(A6).) 6,(L) and p6 can then be recovered through Eqs. (7.16) and (4.11). c1(A) is determined by the analysis of Section 5 and rests on an explicit calculation of the leading pole in I-,,. The values of all these terms are listed in Appendix B. In view of the length of the argument in this section, the result in Eq. (7.48) is rather conspicuous in its simplicity. However, it only appears to be a feature of the leading order of the theory. In higher orders all the terms listed in (7.38) become relevant, and so do other terms on the right-hand side of (7.40), and so does the mixing of different pole residuescaused by functions of n. These considerations are mainly bound up with the mixing of a’[#‘] in the renormalisation of $i, and all have first effect in c, and b, at O(L6). It therefore seemsunlikely that the simplicity of the relationship between b, and c, remains beyond O@‘). One of the key points of the analysis was that it gave the leading behaviour of the double-pole residue b,, at 0(;1j). The single-pole residue and the P-function therefore have the form {constant + O(,J4)}, with the constant term remaining undetermined. We conjecture that this constant term could be found within the general flat-space approach of this paper by considering instead a l-loop, 3-point Green’s function analogous tO r, but using untraced metric derivatives -26/fi6g,, with the action 9 Calculation of rd4 is straightforward using only (A.]). The required expansion may, however, be read off from Table I as well. This calculation also provides the value of T,(L) as (-1 l/3456) (L3/(4n)8), which would be needed for the O(1”) term in 8,.
S. J. HATHRELL
172
principle instead of s/&2. It would involve functions like i2(8”B,,P,), and would avoid the (n - 4) factor multiplying b, in view of its absence from the untraced Eq. (6.5). Coupling constants could be ignored completely. This would be a rather clumsy approach, however, and it is better to take advantage of the fact that it is a lloop result and use a fully covariant position-space determination of the l-loop effective action, for example, either in the “proper-time” representation of Schwinger and Dewitt [4], or on a topologically non-trivial background [9]. The result has been obtained elsewhere [4,9, lo], and is incorporated in the values quoted at the end of the paper. This concludes the general analysis of the trace anomalies and their relationships to flat-space functions of normal products. The remainder of the paper concentrates on using the methods explained to calculate the actual values of the parameters.
8. ABSORPTIVE
PARTS AND THE CALCULATION
OFP~,P,
It was argued earlier in the paper that PA and /I, were both O(1) in leading order, and given by l-loop calculations. These leading-order, I-independent terms have been calculated by Brown and Collins [lo] in the case of /?,,, and by several authors in the case of /I,. lo In this section we determine the effect of introducing the scalar quartic self-interaction by calculating both these p-functions to the 3-100~ level and O(L’). Some of the Feynman integrals appearing at this level can not easily be evaluated directly, and for these we use a technique based on a calculation of the absorptive part of the integral [ 14, 151. The full Feynman amplitudes can be reconstructed from their absorptive parts, and this approach enables all the 3-100~ amplitudes to be obtained explicitly in closed form as a function of the space-time dimension n. In particular we find that the “difficult” amplitudes can be expressedrather simply in terms of generalised hypergeometric functions, a form which compares favourably with the infinite double sums appearing in calculations based on the use of Gegenbauer polynomials [ 191. As explained in Section 5, p,, is obtained from L, and the divergences of the 2point function Zz2. The relevant diagrams up to the 3-100~ level are shown in Fig. 7, and their values denoted Zi”‘-I?‘. Th ese include an overall factor of i, and symmetry ’ Generic operator insertions are denoted by factors, respectively, of 4, 7, ’ 8, ’ 5, i 7. double lines. Exactly the same set of diagrams appears in the calculation of /I,, but with the scalar operator insertion $[@‘I replaced by the tensor operator @“’ (cf. (6.19)); and their values denoted I,(a)-I, @. In both cases I,-I, are easily evaluated using only the result (A.l). I, is more difficult, and for this the absorptive-part approach is used. We consider Z,, by way of illustration; the argument for i(@‘“guL,) is identical. The argument is clearer if in fact all the contributions I,-I, are at first taken together. It follows from the form of Eq. (5.4) that the absorptive (imaginary) part of ” See Duff.
Ref. 17 1, for a summary.
TRACE
FIG.
ANOMALIES
7.
Diagrams
AND
Ad4
for /3, and
THEORY
173
8,.
Z,, is finite, and it also follows from the unitarity of the theory expressed in terms of a sum over intermediate states. Thus
that it can be
Im Z2*( p) = (finite)
where CN represents a complete sum over both phase-space and all intermediate particle states. The right-hand side corresponds to the various ways of cutting the diagrams in Fig. 7, replacing the cut propagators on the (zero) mass-shell, and complex-conjugating the amplitudes to the right of the cut. The result really only depends on the structure of the Feynman integrals, while the various symmetry factors or identical-particle factors match exactly, and it is actually true diagram by diagram, using the symmetry factors appropriate to that diagram. However, the tricky point concerns the ir divergences and the normalisation of the intermediate states IN). These states correspond to the cutting of the bare propagators (#0$0), and do not include factors of Z, for wave-function renormalisation. Also all diagrams containing self-energy insertions on external on-shell legs are regularised to zero. Although [d’] IS a finite operator, the states (0 I+[#‘][ N) are therefore not finite just to the extent that these factors of Z, are absent. But the integration over phase-space incorporated in C,,, contains ir divergences, and Eq. (8.1) shows that these must exactly balance the uv divergences of the unrenormalised matrix elements to give an overall finite answer. In the present case there is an ir divergence for the phase-space integral in the cut version of I, in Fig. 7, and this balances the uv divergences of the 2-particle states in some of the cuts of I,. (There is in principle a second category of ir divergence, though not present in the current examples, which may arise from putting the intermediate states on shell. For such divergences the cancellation comes instead between different cuts of the same diagram. Both categories do occur for massless QED, and the cancellation of divergences is illustrated in detail in Ref. [ 141.) For the purposes of diagram-by-diagram use of the absorptive-part analysis, however, it is sufficient just to calculate using the unrenormalised states.
174
S. J. HATHKELL
Furthermore it is only necessary to evaluate I, this way-the “direct” method suffices for the other diagrams. The full amplitudes can be reconstructed from their absorptive parts by means of a very simple dispersion relation. On dimensional grounds a k-loop amplitude Z(p) appearing as part of T,,(p) must have the form
I(p) = F(n, A)(-p’ - i&)k’“‘* - 2) +‘(,,,
A)
for some function F(n, A), so that Im I(p) Im I(p) = -F(n,
(8.2)
i-k(n-4)(~2)k(n/2-2),
= 0 for p* ( 0. Therefore
A) sin
P2 > 0,
(8.3)
and Z(p) is reconstructed from Im I(p) by the simple expedient of multiplying by ( -iCk”/sin(n/2 - 2) kn}. Th’ is clearly introduces one extra pole in each component. The correctness of the whole procedure can easily be verified for a wide variety of simple examples. As a preliminary, 2, and Z, are needed to O(L*). Z, is obtained as usual from the finiteness of the renormalised propagator (#), and
;1* ‘1
= ’ +
(4n)4
y(i.)=/?$lnZ,
1 24(n _ 4) + o(A3)y A2 =-I--. 12 (47r)
(8.5)
Z, is obtained from the finiteness of the insertion of f[#‘] into a flat-space 2-point function, for which the diagrams Vi-V, are exhibited in Fig. 8. The simple vertex V, and V, and V, are obtained using formula (A.1). When the scalar legs is just Z;‘, are on shell, qy = qi = 0, V, is easily obtained from (A.l) and (AS):
v, =z;’
f$
(p’)“-”
/I;
q2 - n/2) qn/2 - 1)’ f(4 - n) qn - 3) . 2(n - 3) r(3(n/2) - 4)
V* FIG.
v, 8.
Diagrams
for i(f[$‘]
44) to O(l.‘).
V4
(8.6)
TRACE
ANOMALIES
AND
Ad4
175
THEORY
Then Z, is fixed by incorporating the coupling constant (Appendix B), and requiring that Zf Ci Vi is finite. Hence
4 5 A2 (4x)2+-- 12 (471)
1+
1 (n-4)2
2L2 (4n)4
renormalisation
[ 1
+ O(i3).
(8.7)
and 5 /I2 A @I = (471)2 - 6(4n)4
+ W3).
There is once again a slightly tricky point involving ir divergences. The usual theory is that an off-shell Green’s function is renormalised either by including selfenergy insertions on external legs and multiplying each leg by Z;‘, or by excluding insertions ‘on external legs and multiplying each by Z,. But only the latter version remains finite if the external legs are put on the zero mass-shell. This is because the extra factor of Z;’ for each leg in the former case subtracts the extra poles in the self-energy insertions and gives finite terms in ln(q2/p2); but these logarithms diverge when q2 + 0. (An alternative view is that all the self-energy insertions are regularised to zero when q2 = 0 because they involve multiples of (q2)n’2-2, and so no extra factors of ,Z;’ are needed to subtract the poles.) The argument above gives the right values of ,Z;’ in (8.7) for all the ln(p2/p2) poles to be correctly cancelled from T2,(p2) at the 3-100~ level. Since the absence of ln(p2/p2) poles in Zz2(p2) is essential to the general argument, we will demonstrate it explicitly to O(L2). It is convenient to introduce the quantity w defined by n’2-2 Z(3 - n/2)Z-(n/2 - 1)2 = 1 + (n - 4)w. r(n - 2)
(8.9)
fr om the diagrams in Fig. 7 are easily evaluated using the The integrals II*‘-Zy’ result (A.l). Combining these with the value of Z;’ obtained from (8.7) and the renormalisation (321 of 1, from Appendix B gives
1 + (n _ 4)
[
/I2 A20 1 - 8(4n)4 + ---co (47c)4
2A2 (4r)z,
2 + O(1) + W3). I I
I, is calculated through its absorptive part. The cut diagrams Fig. 9. I,. , and I, . 2 are complex conjugates, and I,., and I,.,, each other. The 2-particle on-shell amplitude to the left of the V,, Eq. (8.6), and the 2-particle phase-space integrals [33,34]
(8.10)
I,.,-I,,, are shown in are real and equal to cut in Ii!,) is given by are given by formula
176
S. J. HATHRELL
-:-cp+=k5@+* I 5.1
I 5.3
I 5.2
FIG.
(A.6), with an additional Hence
9.
Cut diagrams
I 54
for I,.
symmetry factor of 4. (V, already included a factor of f.) Z(2 - n/2) Z(3 - n) T(n/2 - 1)” cos n?r 8(n - 3) Z(3(n/2) - 4) *
(8.11)
The 4-particle phase-space integrals in I:“‘: and Z::,j are obtained from the general result (A.lO), with an extra factor of a: T(n/2 - 1)4 B(n - 3) Z(3(n/2) - 3) Z(2n - 5) 1, (n - 2), (3w2) (3(n/2)-3),(2n-5)
The full amplitude Ii”’ ‘:^)
- 4) ; l
(8.12)
1*
can then be reconstructed from the argument of (8.2), (8.3):
= ‘;’
W-‘”
(&~3’“,2’
(P2 >3(n/2-6
w2
-
r(5
-
3(n/2))
2(n - 3)
X IT (2-+(3-n)cosnn-
(3n2;(;;;c;nL)5)
1, (n - 2), (3w2) - 4) ; 1 (3(n/2)3), (2n-5)
Expanding this in poles”
‘I3
II
(8.13)
*
and in terms of the function o introduced above gives
w’] + (finite)/
+ O(L’).
(8.14)
” The first four terms in the expansion (i.e., including the finite term) agree with the result quoted in Ref. 1191 based on the use of Gegenbauer polynomials. The first three terms only are required here and are easily extracted, but the fourth term is more easily obtained if the ,F, is first transformed to make it “Saalschutzian.” See Refs. [ 16-181.
TRACE
ANOMALIES
AND
A#”
177
THEORY
Adding this to the other terms in Eq. (8.10) shows that the poles in w do indeed disappear, and comparing with Eq. (5.4) gives 1 1 1 1 A2 1 1 L (n 4)2 -5(4n)4+-6 qi 4) 2 (4n)2 48 (47r)6 I LA = __ [ -+ I +
5 /I2 18 (4n) I (8.15)
and hence (8.16)
The higher poles in L, are consistent with the RGE (4.13). The absence of an O(L) termin /3, agrees with the prediction in Section 7. The calculation of L, and p, is essentially the same as that above, but with f[$‘] replaced b:y @” and therefore with the additional complication of numerator factors in the Feynman intregrals. The insertions into 2-point functions, epicted in Fig. 8, are now represented by the tensor functions Vy”-VgV. The simple vertex Vg’ is derived from Eq. (6.20) and is vyyq,
3q2) = ;2 vl’q,
.
92
- s:s”; - 93’;
+ (n - 2) ( p”p” - + p2f” 1 . 2(n- 1)
(8.17)
This has the properties v,“,
= 0
(8.18)
+d)-qY9:-@?:~
(8.19)
and
which vanishes when q:, q: are put on shell. It follows from the structure of the diagram V;" and Vy" in Fig. 8 that they must be functions of p” only, and so Eqs. (8.18) and (8.19) immediately imply that VP” 2
=
VU” 3
=
0
.
(8.20)
178
S. J. HATHRELL
Consequently
the corresponding
diagrams in Fig. 7 give vanishing integrals also: z(a) = ,y
= 0
2
The on-shell function
Vft” is obtained using formula (AS): 4;
C”(919
(8.21)
q2)
=
-(dryI x
ir2yp2y4
V’;“(Sl, 42)
Z-(4 - n/2) Z(n/2 - l)? Z(3 - n) Z(n - 3) 2Z-(3(n/2) - 2) I
(8.22)
Notice that these functions Vy” satisfy x4=, Zy Vy” = (finite), which is necessary because @‘” is a finite operator. Evaluation of the integrals Zy’ and Zy’ again uses only formula (A.l). Some details concerning the numerator factors in the integrand are given in Appendix C. The results are .-n z:a)
=
(417)n,2 (P
n,2 r(2
2
-
1
42)
=-my&i
I-&
(p2)3’“‘2’-41;
=-(P’)’=j T4n,2 1 + (n-4)
-
1)
WG)
-2) (8.23)
[+1+0(l)/.
.-3n
I$’ = (4;)3(“,2)
W/2
4(n - l)Z(n
Z-@I/~)~ Z(n/2 - 1) Z-@/2 - 2) Z-(4 - 3(n/2)) n(n - l)(n - 3) Z(2n - 4)
l [&&‘J
cn _ 412
-7
lb2
[ 216(4x)4+36
1
/I* -w] (47r)4
+ (finite)!
+ O(L”).
(8.24)
The evaluation of Zy) is again through its absorptive part and the cut diagrams in Fig. 9. The on-shell 2-particle state to the left of the cut in Zyi is given by Eq. (8.22), and the 2- and 4-particle phase-space integrals and the appropriate reduction of the integrands are obtained from Appendices A and C. The results for the absorptive part are 1:“; +I$=-
~;(p2)3’n/2’-4 3(n/2)-1
(471)
cos n?rZ(4 - n/2) Z-@/2 - 1)” Z(3 - n) , (8.25) 48(n - l)(n - 3) Z-(3(n/2) - 3)
TRACE
ANOMALIES
AND
@”
179
THEORY
z-@/2) - 1)4 48(n - l)(n - 3) r(3(n/2) - 3) r(2n - 4) 1on3 - 39n2 + 25n + 18 n
I
X
- + (n - 2)(2n - 5) 3FZ [ :;&;t
-
2(n
-
l>Cn
-
3, 3F2
[
n(n - l)(n - 3) (3n - 4)
;i;;)154) 3
; I]
I, cn - 2), (3w) - 3) (3tn/2) _ 2, (zn _ 4)
;’I
1, (n - 2), (3(n/2) - 2) (3(n/2)-1),(2n-3) ” Ii ’
(8.26)
(The relative complexity of this expression is a consequence of the complexity of the numerator factors in the integrand, rather than of the basic loop integral.) The full amplitude Zy) is regenerated as before by adding these two expressions together and multiplying by {-i-3”/sin 3z(n/2 - 2)}; and the expansion in poles is 1-m 5
(p*)*
p-4
=
(41r)Z 2 A2 ~04-~(4rr)4~
1
A2
1
tWte)+W’)
The doublle pole, and the single pole in w, cancel between Zip’ and I?‘. Eqs. (4.11) and (6.19) we deduce12 La=
1
120(47Q2
1 ~(n - 4)
!
[1--7
5 A2 108 (47~)
pa=-l 1-L A2 120(4x)*
1+W)i3
I .
(8.27) Recalling
(8.28) (8.29)
36 4(47~) + O(A3)
Unlike L,, L, contains no double or treble poles below O(A3). The difference is a consequence of the 26(A) term in the RGE for L,, and its absence from the otherwise identical R.GE for L,. The absence of these low order double and treble poles in L, means that the function of n premultiplying L, in Eq. (6.19) does not mix pole residues below O(A’), so there is no possible ambiguity in the value of 8, up to O(A2). In view of the fact that the double-pole residue u*(A) is O(A’) and depends only on the O@‘) term in a,(A), it might be supposed that a,(A) could be directly obtained by an argument analogous to that used for b,(A) in Section 7. This is indeed the case following a consideration of the 3-point function (B@‘“$,,), whose renormalisation ” A calculation result.
based on the use of the Gegenbauer
polynomial
formulae
of Ref. [ 191 gives the same
180
S.J. HATHRELL
involves the pole-terms in (n - 4) L,. However, there is no advantage to be gained this way because the leading, non-vanishing contribution to ((04} &‘#,,) is O(n) and dependson just those 3-100~ diagrams appearing in Fig. 7. It seemslikely that whatever route is adopted, at least in flat space, the determination of /I, ultimately rests on a calculation of these diagrams-the chief reason being that there are no factors of p^ which can be extracted from the problem using normal-product and renormalisation-group arguments.
9. A
~-LOOP
CALCULATION,AND
THE OTHER PARAMETERS
One of the most significant consequencesof the analysis of Brown and Collins [lo] was that the HZ term in the anomaly only appeared at O@‘) and not O(J”)neglecting, for the moment, the dependenceon q. The O(L”) term is obtained from Eq. (5.14), and it only depends on the 2-100~ p-function and the leading, 3-100~ divergence of r,,. A further consequenceof this, following from Eq. (4.33) is that e(L) is 0(14) and not O(L3). Since these results depend in no small part on the specific details of the renormalization-group analysis and the nature of the divergences of functions of normal products, and since the consistency of field theory in curved space is something of an open question, it is perhaps of interest that the results can be verified by direct calculation. To this end we calculated the set of amplitudes exhibited in Figs. 1Oaand b, J,-J,. Apart from showing the usefulnessof the absorptive-part approach in certain contexts, this 5-100~ calculation serves altogether three purposes. (i) With operator insertions of (n - 4) &$:/4! at each end it gives the qindependent poles of i(&?), and hence L,, to O@“). The term r10a2@iin 8 does not contribute below O(L6) becauseL, is O(L4). (ii) With an operator insertion of (n - 4)&#:/$4!) at one end and (n - 4)&#$‘4! at the other it gives the O(L4) contribution to e(A), the parameter which describes the q-independent part of the mixing of aZH in 4:. This is because the diagrams are just those appearing in the function (d/&2)@4P”[$4]/4!) in flat
FIG. 10a.
The 3- and 4-100~
integrals.
TRACE
ANOMALIES
AND
Ad4
181
THEORY
+=@= +=a++=e J,
J,
e==a= J,
“J
5
J,
b FIG.
lob.
The
space, and the finiteness of this function be precise, we have
5-100~
integrals.
provided the original definition
= (linite) + O(L”). Again the various terms in a’#’
of e(L). To
(9.1)
do not contribute
to the leading order.
(iii) With operator insertions of (n - 4) &#:/@4!) at both ends it gives the value of the function r,, up to O@*). Used in collaboration with the earlier renormalis,ation-group analysis and a 3-100~ calculation of j? from elsewhere [32], it provides the values of PC to O(L6), and e to O(L5). (In fact r,, is only needed up to O(n), the 41-100~ calculation involving only J, and J,.) To summarise, if we define the amplitudes Ji as for (i) with operator insertions of (n - 4) &$:/4! at both ends, we have 4- J.I + (p’)’
pLlnP48L, = (finite) + O@‘),
(9.2)
(e + L,) = (finite) t O(L4),
(9.3)
i=l
L
+ Ji + (p’)’
P
l
,P-4 +
iTi
6
p p,
J,
'
+
(p2)2$--4
^ (
n-4 P
2 1
L, = (finite) t O(L3).
(9.4)
182
S. J. HATHRELL
These are obtained from Eqs. (5.10), (9.1) and (5.9). The value of 8-r and the renormalisation of ,I, are given in Appendix B. The basic Sloop calculation of cs Ji therefore gives a direct determination of L,, (e + L,) and L, to O(d4) from their definitions, without assuming any relationships between them. Computation of the 3-100~ amplitude Jr, the 4-100~ amplitude J2, and the first five of the 5-100~ amplitudes J,J, is entirely straightforward, and uses only the standard TABLE Values
Coeff.of:
J,
1 ~ (n-4) 1
I
of J,-.I,
1
I4
16 (47r)’
1
(n-4)?
(n - 4)
J3
1
-72(4n)6
.k2
43204 I3 + (4n)6
35 i 864
I 24 wj
+O(IJ) J*
-3
A4
16 (47~)’
1
I’
I3
6(4n)
48(4n)6
-49
1
(576+12w
!
O(P)
+ O@“)
J,
1
/IJ
I”
40
(47?)8
(4n)8
J4
1
A4
60(4x)8
J,
0
J,
0
J,
0
-21
1
160+TW
1
W”)
m”)
W”)
O(l.“,
W”)
o(14)
W”)
W’)
W4)
O(i”)
/I4
216008 1
,I4
1080(4K)8 0
J*
1 /I4 60(4x)8
W”)
O@“)
J,
1 A4 rs(4n)8
O(A4)
OCR”,
Note.
A factor
of (-(P’)~
p”-‘/(4x)‘}
has been removed
from
each term.
TRACE
ANOMALIES
AND
liJ
THEORY
183
a FIG.
1 la.
The 4-100~
subdiagram
of J,.
integral (A.l). J, is also easily obtained using the result (8.13) for the 3-100~ subdiagram in it which has the form of I, (Fig. 7)-the remaining integrations depend onl:y on the momentum factor (p2)‘3’“‘2’P6’ in (8.13), and are again of type (A.l). The expansion of these amplitudes in poles is given in Table I. The term w appearing in the expansions is the same as that defined in the previous section, Eq. (8.9), and it contains all the “unwanted” dependence on ln( p’/p 2), Euler’s number and ln(4z). (Each amplitude includes an overall factor of i, and the symmetry factors for ,the nine diagrams are, respectively 3 L24 3 L8 9 J-16 5 r32 9 L36 3 L36 7 L36 7 18 and f.) J, is obtained through an absorptive-part calculation of the 4-100~ subdiagram in Fig. 1 la. The corresponding cut subdiagrams J,.,-J,,, are shown in Fig. 1 lb. The absorptive-part argument for this 4-100~ subdiagram works perfectly well even though the cut diagrams do not actually appear in their own right as “physical” on-shell amplitudes in the cuts of J, itself. It only matters that the Feynman propagators have the correct L-prescription, and that the various other parts of the diagram have the correct factors of i for unitarity. The symmetry factors are those appropriate to the diagram J, and not to the N-particle intermediate states of J,,,-J,.,. The loops of J,., are evaluated as usual with (A.l), and the remaining 3-particle
FIG.
595/139/l-13
1 lb.
Cut subdiagrams.
184
S. J. HATHRELL
phase-space integral factors the result is
is of type (A.8). Neglecting
A:
J,.,=
(4n)2”-, (P*)*“-’
for the moment
the symmetry
q3 - n/2)2 qn/2 - 1)’ (n - 3)* (n - 4)2 r(n - 2) z-(5(n/2) - 7) ’
(9.5)
J,., and J,., are complex conjugates. The on-shell amplitude to the left of the cut in J,,, (see Fig. 1 lc) is effectively a 3-point function, W, with only one leg on shell; i.e., kf = 0, k,, s k, + k,, and it is given in terms of ordinary hypergeometric functions
from (A.2):
4
2 ” ‘, 2r(3 - n)T(n/2 - 1)3 [if+,;;; ) - (n-4)Z-(3(n/2)-4) 2F1
w= ---2”(p (47c)”
(1 --$)I.
(9.6)
The 3-particle phase-space integrals can then be evaluated using (A.9), so that J,,,
+
Jg
3 =
2cosn~rr(3 -n)T(n/21)” (n - 4)(3n - 8)T(n - 2) r(3(n/2) - 4)*
‘itp2)2n-7
(47+-l
1, (4 -n>, (n - 2) ’ 3F2 (3 - n/2), (3(n/2) - 3) ’ ’ I
(9.7)
Note that this 3F2 has the simple expansion 1 - $(n - 4) + O(n - 4)2. Next, examination of the 5-particle phase-space integral J,., shows that there are no ir divergences, and as there are no closed loops either in the on-shell amplitudes, the whole integral must therefore be finite:
Jw = O(l).
k,
(9.8)
The value could be obtained in leading order in the expansion in (n - 4) by calculating in four rather than in n dimensions, but since it is only relevant to the third, single-pole term in the expansion of J, (J,.,-J,.3 each contain two more poles than J9.& we have left it undetermined. It would be necessary for a calculation of /I, to O(3L’) by the “indirect” route.
-c P
FIG.
1 Ic.
An on-shell
3-point
k2
function
in J,,,.
TRACE
ANOMALIES
AND
@”
185
THEORY
Finally, ,J9 is regenerated according to the arguments of the last section by adding the cut dliagrams together, multiplying the sum by {---““/sin 4n(n/2 - 2)}, performing the last trivial loop integration using (A.l), and including the overall symmetry factor of f. The appropriate expansion in poles is given in Table I along with the expansions fo_r the other diagrams. Adding all these together and multiplying by inverse powers of /3 give the results required for Eqs. (9.2)-(9.4):
;7 J,= _ (PZYP4 -
1
i=l
(W
+ (n - 4)3 O(A’) + O(L”)
i
+
P
Ji
=
-
1 +
, I
)I (9.9)
(lo’)’ Pn-4 (4n)4
iG
-1081 1 12,960 +16O
&j’ol+[&&+&
432(4a)2+
I 1
A2 (47r)4
(9.10)
1 I
+ (finite) + O(L3)
I
.
(9.11)
As required by the arguments of Sections 4 and 5, the O(L4) pole has vanished from (9.9), the O(L2) single pole and the O(L”) double pole have vanished from (9.10), and all the poles in w have vanished from all three expressions. Comparison with Eqs. (9.2)-(9.4) gives L, = O(#q,
(9.12)
186
S. J. HATHRELL
17 L4 e(A) = ----1. 12,960 (4n)
(9.13)
+ W%
L, = O(P),
A2
Yl(L) = J-
432 (4n)6
(9.14) 7 A3 - -7 + 0(/l”). 1728 (47~)
(9.15)
In addition, the pole residues in (9.11) satisfy the RGE (5.13). More than predicting (9.12), however, the arguments of Section 5 showed that L, and e were given in terms of Y, and the standard P-function. According to Eqs. (5.11) and (5.14), L, is given to O(L”) by the 3-100~ p-function [ 321, by Y, to O(L3), and by the leading values of d, X, and X,; then Eq. (4.33) gives e to O(A5) using also the leading values of /?,, p, and f: The only “new” information provided by the calculation in this section is the 4-100~ result contained in Eq. (9.15), and the 5-100~ calculation contained in Eq. (9.13) serves only to show consistency with thefirst term in the expansion in (9.15). Clearly the arguments [lo] of Section 5 provide a much more efficient route to the values of the terms in the anomaly than does direct calculation. The leading-order values of the terms d, Xi, X2, ,!?,, /?, and f are readily determined from the definitions and relations earlier in the section, the calculations requiring only the use of (A.l) and (A.5). In brief, d is determined by the requirement that the divergences of the graphs in Fig. 1 should cancel, yielding the result
dd-
A3
6 + O(A”). 36 (471)
Hence Eqs. (4.32) and (8.8) give
+ O(A’).
(9.17)
From the RGE (4.9) Ll=-L= From Eqs. (5.6), (see Fig. 3b),
(5.8)
1 Q-4)
1 yq$p
and a calculation
L4
1
+O(A5).
of the function
(9.18) r24 in leading order
f-=x&- 36 m ,I2 +O@“). So from Eq. (4.34), the RGE (4.14) for L,, and the results of Section 8 for p,, , L,: (9.20)
P, = W”), LK=
l-(n-4)
[O@“)]+ (n-4)2l
A4 [ T&p
1+O(k5).
(9.21)
TRACE ANOMALIES
AND&h'THEoRy
187
Thus Eq. (5.15) gives
(9.22) The fact that the double pole of L, is O(A”), and determined in leading order, whereas the single pole and j3, vanish to O(A3), suggeststhat the cancellation is “accidental,” unlike the other cancellations found so far. The values of Pbr /?, and e may now be found to high orders of A. First, using the 3-100~p-function and the values of the various other parameters now established,L, and hence /?, can be deducedI from Eqs. (5.11) and (5.14): c,@) =
17 A5 A6 - 14 103,680 m (471)
-17 A5
A6
(4n)
5191
4,354,560 5191
622,080
+
+ O(k’),
+1 C(3)) + O(A’>. 288
(9.23) (9.24)
From Eq. (4.33) we next obtain 17 II” e(l)=-y-7 12,960 (4n)
/I5
(9.25)
(471)
The first term of this is confirmed by the results of the direct calculation given in Eq. (9.13). Last of all, L, and Pb are found by combining the A-independent result from elsewhere 14, 7. 9, lo] with the results of Section 7. Eq. (7.48), and the RGE (4.11): 17 /I4 1 +288m+0’L5)
L,= + 2(n - 4)2 [&$]
P/7=360(4~)~ ’
+O(n”)j.
1 (9.26) (9.27)
I
10. SUMMARY
AND DISCUSSION
We have shown how to extend the analysis of Brown and Collins to include the determination of the coefficients of the F and G terms in the trace anomaly. With this extension all the terms can be found by relating them to flat-space functions of renorI’ In agreement
with
Ref. [lo],
but recall
footnote
2.
188
S. J. HATHRELL
malised composite operators, or “normal products,” in the massless theory. In particular the coefficient of the F term, a,, depends especially on the traceless symmetric part of the energy-momentum tensor of the scalar field, &I”“, and can be found in perturbation theory from the amplitude i(&“$,,,). The coefficient 6, of the topological G term can be found from a consideration of 3-point functions of normal products. Moreover the leading L-dependent terms in b, depend only on the leading poles in c,,, the coefficient of the Hz (i.e., the R*) term, and can be found without the necessity of calculating any 3-point functions. This is a consequence of the renormalisation-group structure of the theory, which ensures a cancellation in the O@“) poles of the 3-point function i*(&W), in much the same way as the 0(14) poles of the 2-point function i(&9) are cancelled. In order to calculate the actual values of a number of the coefficients in the anomaly, as well as to verify the consistency of the general arguments based on the renormalisation-group analysis, an absorptive-part argument was used and extended to provide analytic, closed-form expressions for certain classes of multiloop Feynman diagrams. These expressions involve only gamma functions and generalised hypergeometric functions of the space-time dimension n, and provide a convenient alternative to the double series appearing in evaluations based on the use of Gegenbauer polynomials. Combining these multiloop integrals with the general renormalisation arguments enabled calculation of higher order, l-dependent corrections to /I,, Pb, p,, p,, and e(A). A direct 5-100~ calculation of the leading term in e(L) agreed with the value obtained through the more efficient but indirect arguments, and the same calculation also directly confirmed the vanishing of the O(L”) term in the HZ anomaly. Higher order corrections have not been calculated for d(jL) or f(L), nor has the leading O@“) term in /I, been found. Although there seems to be no obvious reason for /I, to vanish, it would be interesting if this could be confirmedparticularly as in the massive theory it is relevant to the renormalisation of the Einstein gravitational action [lo]. The value appears to depend unavoidably on the O(L4) term in d(L), and the next-to-leading terms in the 2-point function r24. All three terms d, f and /I, are closely related to each other and to the mixing of the operator 8’(4’] in the renormalisation of 4:. We have found, then, that the trace anomaly is given in terms of manifestly finite quantities by the expression
e= -/P 4-“M41 4r. + I +,P-~{~,F+&G+
(1-~-r)[~l+(tl-d)~‘[~‘]~ &+@,
+v2Pn)H2
+/@,+@)+H[#‘]I + (4c-e-rlf)a*H).
(10.1)
The definitions of all these terms are given in Section 2 and 3, and the values of the coefficients are listed in Appendix B. The whole expression is independent of the unit of mass ,u by construction, and it is in a form suitable for taking the limit as n -+ 4. In this sense it is a regularisation-independent result, although the expression of the normal products in terms of the bare operators of the theory depends upon some
TRACE
ANOMALIES
AND
189
Ad4 THEORY
regularisation and renormalisation prescription. i4 Through the renormalisation-group equations the finite “anomalous” coefficients provide complete information about the renormalisation of the parameters describing the whole scalar theory on a general curved background. The expression in the first curly bracket is the remainder when the curvature is set to zero, and is then the ordinary flat-space trace anomaly for @” theory [35, 361. It is interesting to investigate the parameter q a little more closely. It was introduced in Section 3 as an independent finite parameter, but obeying the inhomogeneous RGE (10.2) CD - @7= P,(A). If we split 7 into two parts, II = W)
(10.3)
+ $9
where f obeys the corresponding homogeneous RGE, then we can solve (10.2) in the n = 4 limit. Using the values given in Appendix B we have -1 A3 - -+ O@“>, ’ = 288 (47r)6 t=
kP(1
+ O(l)),
(10.5)
where k is now a truly independent, arbitrary constant. The value of k must be determined in principle by experiment, but the non-perturbative aspect of 7 is clearly shown by (10.5) and it would be interesting to examine the consequences of this term for non-zero k in more depth. Finally, we can compare these results for the renormalisation of the R#’ term with the related results of Birrell [37], and Bunch and Panangaden [ 3 11. They obtained
si,= rR- ~[(5,-&$ (n_ 4) l + (n - 4)’ so that even if (corresponding to disagree with f= 0) appears
[2
+G-(4)&J
~R-&$]+o(i~)~
(10.6)
the renormalised coupling CR is chosen as b, the bare coupling & to (< + qO) here) contains a pole at O(L’). At first sight this appears the arguments in this section suggesting that the first pole in no (with at 0(14), but the disagreement is illusory: if we write
(n- 2) c-+I;lo=5R+ (4(np1)-5X+% 1 1 I4 Also the value of the 3-100~ p-function, and hence the values it, depends upon the renormalisation prescription. See Ref. [321.
of the other
parameters
derived
from
190
S.J.HATHRELL
then the poles appearing in Z,( (n - 2)/4(n - 1) - & + rO} to 0(3,‘)-which multiplies the finite operator R [$‘I-are just the poles appearing in (10.6). The O(I’) pole in C$for & = i therefore appears as a consequenceof regularising < to d rather than (n - 2)/4(n - l), and really only depends on Z,. There seems to be no advantage in breaking the conformal invariance of the L = 0 theory this way when it can be maintained using the regularisation (n - 2)/4(n - l), and then the introduction of L only forces this symmetry to be broken by a pole at O(1”). The “true” effect on the trace anomaly is a finite O(L”) contribution multiplying a’[$‘].
APPENDIX
A: FEYNMAN
INTEGRALS
All integrals or subintegrals involving only two invariant momenta in the integrand can be evaluated using the well-known result
i$
(k2)’((k+ s>‘>” - s- 42) e/2 + f-1w2 + $1 , (A.1) (42>n,2+r+s q--r I-(--r) z-(-s) II z-(n+r+s) ! I
where the usual is-prescription in Minkowski space is implied. The factor i’-” is particularly important to the absorptive-part analysis, and there is one such factor for each closed loop. The terms in curly brackets exhibit, respectively, the ultra-violet and the infra-red behaviour of the integral. Notice that any symmetry factors, vertex factors or factors of i for propagators have to be included separately as appropriate, both here and for subsequentformulae. The 3-point functions with one external momentum on shell can be evaluated using
ig
(k2)’ ((k +p>‘)” ((k + d2)’
(A-2) where p = k, + q and k: = 0. (A. 1) is a special caseof this when k, = 0. The function ,F, is a standard hypergeometric function with seriesexpansion 2
F1
(A.31
TRACE
Here (a), is Pochammer’s (a>, =
ANOMALIES
AND
Ad4
191
THEORY
symbol: r(u + m) qu) (A.4)
=a@+
I)***
(u+m-
1)
for
m>l.
The ,F, in (A.2) can be summed in a number of cases (including when s = t = -1). A special case of particular interest is when a second leg is put on shell, i.e., p=k,+k:t, k2-1- kg = 0. Then
Is
[(k - k2)2]’
[(k + k,)2]”
(k’)’
= -$+
(P~)~“+~+‘+’
x
l--r I
x
The 2-particle phase-space
- s - t - n/2) I-(-r) l--s) I
T(n/2+r+t)T(n/2+s+t) r(n + r + s + t) I
integrals can be performed
I-
using
(P2)“‘2-2 T(n/2 - 1) + (271)n J dp(2) = (4x)n/2- 1 4r(n _ 2) ) where the :Lorentz-invariant
(A.51
(A-6)
phase-space for N particles is given by [33, 341 (‘4.7)
and kf = c~f - kf = 0. The phase-space vanishes for p2 < 0, so the integrals carry an implicit step-function. The 3-particle phase-space integrals encountered here can be evaluated using either + (2~)”
I’ W)Kp - kA21’ [(p - kJ21S
qn/2+r-l)qn/2+s-l)qn/2-1) = (p2)r+s+n-3 4r(n - 2) T(r + s + 3(n/2) (4?r)“- ’
- 3)
’
(A.8)
or
= (P’Y’ (4x)“-’
T(n/2 - 1)’ * 4r(n - 2)I-(3(n/2)
(n - 2)nl - 3) * (3(n/2) - 3), *
64.9)
192
S.J.HATHRELL
A general result for 4-particle phase-space integrations, needed here, is + (2+ =
J dp(4)[(k,
+ k4)T
(p2)rtst1t3w2)-4
[(P - k)T
[(P -
which includes all those
k,)*i’
T(n/2-1)3z-(r+n/2-1)~(r+s+n-2) 4Z+r-2)T(r+3@/2)-3)T(r+s+2n-4)
3(x/2)--1
(47r)
3F2
-t,(n - 2),(r+s + 3(n/2)- 3) (A.lO) ' [ (r+s + 2n -4),(r+ 3(n/2)- 3) " I ' The function 3F2 is a generalised hypergeometric function with definition analogous to (A.3): 3F2
a, b, c = 5 (4, W, (4, zm [ 1 d,e
-. ( m!
;z
,zo
(4,
(4,
(A.1 1)
1
There are a number of standard results known [ 16-181 about this function, especially when z = 1. When either or both of s, t are non-negative integers the 3F2 in (A.lO) can be reduced to a product or (terminating) sum of products of gamma functions only. This corresponds to the case when the original Feynman integrals can be expressed in terms of integrals and subintegrals of type (A.l) only, and then the result (A.lO) combined with the rest of the absorptive-part argument reproduces the correct answers. For example, the value of J, in Fig. 10a can be obtained from (A.lO) when r = s = t = 0. By construction the right-hand side of (A.lO) must be symmetric in s and t, but the direct proof of this seems to be non-trivial and we have only been able to confirm it for some particular cases. APPENDIX
B: VALUES OF THE PARAMETERS
The new results obtained in this paper are 1 1 PA = - T .- (4n)*
1 A2 -+ I+ s (47r)4
O@‘)
I
,
(B.1) P.2) (B.3) 03.4) (B-5)
TRACE
ANOMALIES
AND
A#”
193
THEORY
Previous results gave only the first term of each expansion. For convenience we also list the values of the other finite parameters, and some of the pole series. A2 17 1’ + O(O, q + 121(3)) & p = 3 (47# -3(4x)4+ ( A
5 k2 6 (4~)
J=pj--4
+ O(L”>,
,I3 + O(J4), dL-36 (471)6
p = (n - 4)L + P(A), & = -(n - 4)a + P,(A), 6, = -(n - 4)b + B&L PC = -(n - 4) + P,(A), q, a, b, c are independent finite parameters.
1 1
d3*7+-T (4x)
+(n [ A1 _
zl=l+i
z;l
4)3
9
(4n)4
+
17 I 3 (4n)
1
W’),
,&$,+o@3,,
= l+
1 -I. (n - 4) [ (47q
5 + 12
A2 -]+&)[2&]
(47r)4
+w3)?
194
La=--
S.J.HATHRELL
1 1 1 ~ 120 (47c)2 ! (n-4)
5 I2 + W’) l--108 (47~) I [ + O(P)
A2 m
7 A3 -4(4n)6
I
7
1
+ O(A4) + O(A”> . I !
Other pole series can be constructed from the finite parameters above and the equations given earlier.
APPENDIX
C: SOME DETAILS
OF THE CALCULATIONS
The simplification of the numerator factors in the integrands of Ip’-Zy’ and appearing in i(P”B,,) can be derived from the following general case (see Figs. 7 and 9):
It&I:(f:
= 4;n--:) KP=)= + @:I=+ @:I=+ ((P- k,)=)’+ ((P- kJ=)= I - 2p=(k: + k: + (P -k,)’
+ (P - k,)*)l
+ (q*)=+p*q= - q=[k: + k: + (p -k,)* + (P - kA*] +q
[k;k;+k;(p-k,)=+k;(p-k,)=
+ 2(nn 1>[k:(p - k)= + k:(p - kJ=l 13
+(P+)=
(P--J’]
cc.11
TRACE ANOMALIES
195
AND,@'THEORY
where q”=p”--y-k@
(C.2)
23
and q“ is the momentum flowing “down the middle.” (Cl) is arranged so that the right-hand side contains only the invariant momenta which appear in the denominators of the Feynman integrals. The evaluation of Z:(2)and Zy’ requires (C. 1) with qw = 0, kz = (p - k,)“. The absorptive-part calculations in Z:‘fi and Ziyi also use (C.1) with qL1= 0, and in addition kf = k: = 0, leaving only the single term ((n - 2)/(4(n - l))(~~)~. The absorptive-part cuts in I,., and Z5,4use (C.1) and (C.2) again with the on-shell condition k: = kz = 0; and with q’ = k: + k:, k: = k: = 0. The evaluation of I,(‘) based instead upon the use of Gegenbauer polynomials requires all the terms in (C.l) as they stand.
APPENDIX
D: DEFINITIONS,
CONVENTIONS
AND SOME USEFUL RESULTS
The curvature tensor is defined by
so that
V,i:,,r- V,,:,, = RK,tw V, for a covariant vector V,. The metric tensor g,,,, has signature (+---),
and
rxr” = WY g&l,” + &?A”,@ - &“.il)’ The Ricci tensor and scalar are defined by R,” = R”,,,,, R = g”“R,“, and in the weak-field limit with g,,(x) = q,, + h,,(x) these curvature terms have the expansion
Functional derivatives with respect to g,, are most conveniently obtained by using Riemann normal coordinates, in which TX,,,, = 0, since while ZKLlcdoes not transform like a tensor, i3ZKctUdoes. Hence
196
S.J. HATHRELL
Also
In particular,
Two further standard results which prove useful here are -1 =-&.,(, sin(n/2 - 2) kn used in conjunction given by
(+)T(k(+)
with the absorptive-part
lnQl+c)=---YE+
g
+l),
analysis; and the expansion of r(l + E)
(-)m*Emm, C(m)
m=2
where y is Euler’s constant (zO.58), and
ACKNOWLEDGMENTS I am grateful to I. T. Drummond for many conversations and useful suggestions, and to the U. K. Science Research Council for supporting part of this work.
REFERENCES 1. D. M. CAPPER AND M. J. DUFF, Nuouo Cimenfo A 23 (1974), 173. 2. D. M. CAPPER, M. J. DUFF, AND L. HALPERN, Phys. Rev. D 10 (1974), 3. S. DESER, M. J. DUFF, AND C. J. ISHAM, Nucl. Phys. B 111 (1976), 45. 4. L. S. BROWN, Phys. Rev. D 15 (1977), 1469. 5. A. DUNCAN, Phys. Left. B 66(2) (1977), 170. 6. H. S. TSAO, Phys. Lett. B 68 (1977), 79. 7. M. J. DUFF, Nucl. Phys. B 125 (1977), 334. 8. N. D. BIRRELL
AND
P. C. W.
DAVIES,
Phys. Rev.
D 18
(1978), 4408.
461.
TRACE
9. I. T. DRUMMOND
AND
G. M.
ANOMALIES
Phys.
SHORE,
AND J. C. COLLINS, AND M. VELTMAN,
Princeton Nucl. Phys.
AND
,+$”
197
THEORY
Rev. D 19 (1979),
1134.
10. 11.
L. S. BROWN G. ‘THOOFT
12. 13.
C. G. BNDLLINI G. M. C’ICUTA
14. 15.
I. T. DRUMMOND T. CURTRIGHT.
16. 17. 18. 19.
Cambridge Univ. Press, Cambridge, L. J. SL.4TER, “Generalised Hypergeometric Functions,” Vol. 3, McGraw-Hill, New York, A. ERDELYI et al., “The Bateman Manuscript Project,” W. N. BAILEY, “Generalised Hypergeometric Series,” Cambridge Univ. Press, Cambridge. K. G. CHETYRKIN AND F. V. TKACHOV, preprint INR, II-125 (Moscow, 1979).
20. 21.
J. C. COLLINS. Phys. Rev. D lO(4) (1974), 1213. P. BREITENLOHNER AND D. MAISON, Commun. Math.
22. 23.
J. C. COLLINS, L. S. BROWN,
24. 25.
J. C. COLLINS. W. ZIMMERMAN,
26. 27.
Grisaru, and H. Pendleton, Eds.), Vol. I, MIT Press, Cambridge, J. H. LOWENSTEIN, Commun. Math. Phys. 24 (I 97 1 ), 1. J. C. COLLINS, PhJw. Rev. D 14(8) (1976), 1965.
28. 29. 30.
G. ‘T HOOFT, J. C. COLLINS, T. S. B~INCH,
1980. B 44 (1972), 189.
Phys. Lett. B 40 (1972), 566. Left. Nuovo Cimento 4 (1972), 329. AND S. J. HATHRELL. Phys. Rev. D 21(4) (1980), 958. Phys. Rev. D 21(6) (1980), 1543.
AND AND
J. J. GIAMBIAGI, E. MONTALDI,
Phys.
52 (1977).
11, 39, and
1966. 1955. 1935.
55.
Phys. B 80 (1974), 341. Ann. Phys. (N.Y.) 126(l) (1980), 135. Nucl. Phys. B 92(3) (1975), 477. Nucl.
“Lecture
on
Elementary
Particles
Nucl. Phvs. B 61 (1973), 455. Nuovo Cimenta A 25(l) (1975), P. PANANGADEN. AND L. PARKER,
3 1. T. S. BUNCH AND 32. A. A. VLADIMIROV,
P. PANANGADEN,
Theor.
Math.
J. Phys.
Phys.
33. 34.
R. GAslrM.&xs AND R. MEULDERMANS, W. J. MARCIANO AND A. SIRLIN. Nucl.
35. 36.
J. H. LOWENSTEIN, B. SCHROER. Lea.
37.
N.
D. BIRRELL.
preprint,
and
Quantum
47. J. Phys. A 13 (1980),
A 13 (1980),
919.
36 (1978), 732. Phys. B 63 (1973), Phys. B 88 (1975). 86.
Nurl.
Phys. Rev. D 4 (1971). 2281. Nuovo Cimento 2 (1971). 867. J. Phys. A 13 (1980). 569.
Field Mass.,
271.
Theory” 1970.
901.
(S.
Deser,
M.
OF PHYSICS
139, 136-197
(1982)
Trace Anomalies
and AQ4 Theory
in Curved
Space
S. J. HATHRELL* Department of Applied Mathematics and Theoretical University of Cambridge, Cambridge England Received
September
Physics,
7, 1981
It is shown for a conformally invariant A#’ theory in a weakly curved background, how to extend previous results to obtain full information about the trace anomaly in perturbation theory, including the “topological” term in the gravitational part of the anomaly. There is a strong connection among renormalisability of the curved space theory, finiteness of the energy-momentum tensor, and the role of normal products. Combined with a renormalisationgroup analysis this provides an efftcient means of calculating some terms in the anomaly to high orders of perturbation theory. In particular, the first I-dependent coefficient of the topological part of the anomaly appears at 0@“) and can be deduced from simple flat-space results without the calculation of any further Feynman diagrams. Some techniques based on an absorptive-part argument are developed in order to compute other anomalous coefftcients. and a direct 5-100~ calculation confirms the indirect renormalisation-group derivation of a non-vanishing R2 anomaly at O(n’). All the essential information can be obtained from the massless theory. The underlying ideas are applicable to other theories. and similar results for massless QED are obtained in a subsequent paper.
1.
INTRODUCTORY
REMARKS
It has been known for some time that anomalies in the trace of the energymomentum tensor are present in curved space even when the coupling constants for the quantum fields vanish [l-7],’ but more generally the values of the various terms in the anomaly will depend on the coupling constants [S, 91, and it is of interest to find out exactly how. In this paper we consider as a prototype the conformal selfinteracting 14” theory. In a recent paper Brown and Collins [lo] investigated the renormalisation of this theory in a curved background, and used a renormalisationgroup analysis combined with a study of certain functions of renormalised composite operators to deduce relationships between various quantities in the theory. They showed that some of the parameters describing the anomaly could be efficiently calculated from flat-space functions alone. We extend their analysis here to show that the L-dependenceof all the remaining parameters can also be determined in flat space in a similarly efficient manner, using only the masslesstheory throughout; and then
* Present
address:
’ Ref. 171 contains
UOC/13
Shell-Mex
a useful summary
House, of results
the Strand,
136 0003.4916/82/030136-62%05.00/O Copyright
.C: 1982 by Academic
Press, Inc.
London,
for non-interacting
England.
theories.
TRACE ANOMALIES
AND&d4
THEORY
137
we apply these general principles to calculate the higher order, I-dependent corrections. We use dimensional regularisation [ 1l-131 and renormalisation throughout, and this has the particular advantage in the case of a masslesstheory that many Feynman diagrams can readily be evaluated in terms of simple functions of the space-time dimension n. A subsidiary feature of the calculations presentedhere, though of wider applicability, is that some new results in the evaluation of multiloop integrals are obtained through an absorptive-part argument [ 14, 151 based on the unitarity of the theory. Some amplitudes can conveniently be expressed in closed form in terms of generalised hypergeometric functions, for which there is a useful body of standard mathematical literature [ 16-181, and the approach compares favourably with the use of Gegenbauer polynomials [ 191 in these cases. We illutrate this by calculating a particular siet of 5-100~ amplitudes which on the one hand verifies directly certain results obtained through the more powerful but indirect renormalisation-group arguments, and which on the other hand simultaneously enablescalculation of one of the parameters describing the anomaly up to O(1”). Other calculations also verify consistency in numerous instances. In Section 2 we define the theory in detail and introduce the parameters appearing in its description. In Sections 3-5 we review the analysis of Brown and Collins [lo], adapted somewhat for the exclusively masslesstheory. This involves a study or 2point functions of scalar, renormalised, composite operators in flat space, and establishes efficient routes of calculation for some of the parameters, including the coefficient of one of the “gravitational” terms. In Section 6 we extend their analysis to the study of 2-point functions of tensor operators, and thereby bring the second of the three gravitational terms into the same flat-space framework. In Section 7 we show that the third, “topological,” gravitational term can also be examined in flat space [5, 61 through a study of composite 3-point functions. Furthermore the renormalisat.ion-group structure of the theory entails a particular cancellation of the leading divergence of a certain 3-point function, and this enablesthe topological part of the anomaly to be determined to O(n”) without the necessity of calculating any new Feynman diagrams. In the subsequentsections of the paper we introduce the absorptive-part argument in a form suitble for the present purposes, and use it to calculate the specific values of the parameters using the results established. Some details of the calculations and some useful general results are given in the Appendices.
2. DESCRIPTION
OF THE THEORY
In a renormalisable quantum field theory, as Brown and Collins [IO] emphasize, one should start with a Lagrangian containing all possible terms of (mass-)dimension lessthan or equal to 4 which respect the symmetry of the regularized theory, because in general one can expect perturbation theory to introduce divergences proportional to those terms unless there is a clear reason why not. Here the dimensional
138
S. .I. HATHRELL
regularisation breaks the superficial conformal invariance of the bare theory in four dimensions, and so non-conformally invariant counterterms can be expected. On the other hand the full coordinate-invariance of the theory in the gravitational background is maintained. We therefore describe the theory by a generating functional W[J, g,,] for connected amplitudes: e iW =
1
(d$) e is
(2.1)
with the action S given by S=
i
d”x 69(x),
(2.2)
P=9g++@,
(2.3)
where (2.4 1 Pg = a,F + b,G + c,,H’.
P-5)
The theory is further defined by the requirement that W must generate finite Green’s functions. The nature of the terms appearing in (2.4) and (2.5) needs some explanation: (i)
The parameter r is defined by c=(n-2)/4(n-1),
(2.6)
this being the usual value in n dimensions necessary to render the scalar theory conformally invariant in the absence of any coupling constants. It is natural to regard the fl#’ term as part of the “kinetic” term in the Lagrangian. (ii) factor:
The quantity H is just the curvature H=R/(n-
scalar R, but scaled by an n-dependent 1).
(2.7)
The reason for this factor is that it absorbs the factor of (n - 1) which appears under a conformal scaling:
a’J(x - y) + curvature
terms.
(2.8)
TRACE
AND
ANOMALIES
/1d4 THEORY
139
The renormalisation process which involves minimal subtraction through expansion in inverse powers of (n - 4) is not then complicated by factors of (n - 1); although the choice is essentially arbitrary.2 It is also convenient to regard the H#’ term as being in the interaction part of P@, and separate from the t;R#’ term in the kinetic part. Note that a2 in (2.8) and everywhere subsequently is the covariant d’Alembertain, containing implicit dependence on the metric g,,.. The notation s/&2 will be used as a shorthand for the functional derivative with respect to the metric. (iii) The terms F and G in rt’, are, respectively, the square of the coliformal (Weyl) tensor in n dimensions, and a topological quantity (Gauss-Bonnet identity) related to the Euler number of the manifold. They are 4 F = RGL.poRL(L’Pa- ~ (n - 2) G = Rw,.poRL(L’P* - 4R,,.RU“
(2.9) f R2.
2; then contains independent linear combinations of the three possible coordinateinvariant terms of dimension 4 (up to total derivatives) containing only the metric tensor. (iv) J,(x) is a source term for the scalar field. It is renormalised by J” = z;yn,
n)J,
(2.11)
wave-function renormalisation factor. Functional where Z, is the flat-space derivatives of W with respect to J then generate renormalised connected Green’s functions, i.e., there is a factor of Z;’ for each external leg.3 We will always assume that ./ has been put to zero once the functional derivatives have been performed, and the role of J is otherwise unimportant. (v) There is no mass term for the scalar field in Ye, as indicated in the Introduction, because such a mass term is multiplicatively renormalised in the dimensional scheme and can consistently be taken to vanish [20, 211. There is also no cosmological term /i 0, or Einstein term K,,R in U,. This is a matter of choice in the case of a lmassless theory, because although such terms could be included, the parameters /i 0 and K,, carry dimensions 4 and 2, respectively, and their renormalisation therefore depends on a mass-parameter. So in the absence of a mass term their renormalisation must either be multiplicative or it must involve a mixing of the two parameters, and in either case both terms can consistently be taken to vanish. (The ‘t Hooft unit of mass ,LL, implicit in the theory through the renormalisation process, does not affect the argument because [22] it always appears in rational ‘This differs from Ref. [ 101, and changes the definitions of some functions by powers of 3. The definition here seems to be uniformly more convenient than using R instead of H. Moreover the convenience is not restricted to the scalar theory. 3 This follows the convention of Ref. [ 1C 1. The quantity Z, here is often defined to be Zi12 elsewhere in the literature, with a corresponding difference of a factor of 2 in the anomalous dimension.
140
S. J. HATHRELL
powers of r(l”-“.) Th’ IS is consistent with the explicit forms of renormalisation for these parameters derived by Brown and Collins [lo]. Since we are concerned in particular with the trace anomalies, and as the contributions to the trace from the cosmological, Einstein and mass terms are not “anomalous” in the usual sense, we have chosen to drop them. The theory then contains no intrinsic length-scale except through the renormalisation process. (vi) Finally, and this is fundamental to the whole argument, for each interaction term appearing in the action there is a coupling constant. The theory should therefore be regarded as a theory of not one but five bare coupling constants, A,, I]~, a,, 6, and c,,. The values of the four “new” coupling constants depend to some extent on the value of 1, (and on each other) because of the requirement that all the Green’s functions are finite, and it is not possible to ignore them. They are responsible for most of the anomaly. The energy-momentum tensor Bw” is obtained by functionally differentiating the action with respect to the metric: (2.12) and its trace 19= P,
is given by
- (n - 4)[a,F
+ b,G + coH2] + 4c,a*H.
(2.13)
Here E, is an opertor of dimension 4 related to the equation of motion of the scalar field [23]:
40 6s E”= --=-~oa’io+SR~:-~+~oH~~, fi slj,
(2.14)
whose properties are discussed in more detail later. We will use the term “operator” in a sense which includes the gravitational source as well as the scalar fields, although in the case of the gravitational source the operator is just an ordinary cnumber. From Eq. (2.13) it can be seen that the determination of the trace anomaly amounts to the determination of the way in which the various composite operators and the various coupling constants are renormalised in perturbation theory. It is therefore necessary (i) to examine the constraints imposed on the coupling constants by the requirement that the Green’s functions of the theory are finite, and (ii) to express the bare operators in terms of finite composite operators. By “finite” in the context of an operator we will always mean finite when inserted into finite Green’s functions containing only elementary renormalised fields at various points.‘For such a definition to make any sense it is necessary that an operator which
TRACE ANOMALIES
AND&d4
THEORY
141
is finite in one elementary Green’s function is also finite in all other elementary Green’s functions, and at all other momentum configurations. This is an exercise in renormalisation theory to justify [21, 24, 251 and is a property which will be assumedhere throughout. Since we have a masslesstheory, however, we must exclude a few exceptional on-shell momentum configurations from some Green’s functions in order to avoid the introduction of spurious infra-red divergences. The operations (i) and (ii) above were performed by Brown and Collins [lo], who then showed how some of the terms appearing in the renormalisation were related to 2-point composite operator Green’s functions, and they used this together with a renormalisation-group analysis as a powerful tool in the determination of the leading terms in the renormalisation of q0 and c,,. In the next three sections we summarise their analys,is in a manner adapted for the masslesstheory, and then in the following sections we extend their arguments to show how to find a, and b,.
3. RENORMALISATION
OF THE PARAMETERS
The arguments throughout the whole analysis rely on the use of the “renormalised action principle” [21, 261 to derive reltions between Green’s functions, and we use these to infer relations between composite operators. If we differentiate a finite Green’s function with respect to a finite quantity, whether functionally or otherwise, we must get another finite quantity. In particular, we know immediately that 0,” must be a finite operator because its insertions are obtained from metric derivatives of renormallisedelementary Green’s functions. Specifically, recalling Eq. (2.13) &
(#i . .. (5,V)= i(&$, .a. 4,) = (finite).
(3.1)
The line of argument in this section runs in two stages: (1) determine the form of renormalisation of ‘lo from the composite opertors of dimension 2; and (2), apply the action prinlciple to find the form of the renormalisation of a,, b,, c,,. Stage
(I)
Renormalisation theory tells us that we may construct a finite composite operator, or “normal product” [21, 23-251, of dimension 2 as a linear combination of all available composite operators of dimension less than or equal to 2 with the appropriate symmetries. The definitions are made unique by minima1 subtraction in poles in (n - 4), so that the finite operator reduces to the bare operator in the limit A + 0, and others are mixed in multiplied by coefficients which are Laurent seriesof poles in (n - 4). These definitions provide explicit rather than implicit expressionsfor the normal products, as pointed out by Brown [23]. In flat space there is only one such operator of dimension 2, and so we must have
[Pl=z2’(~,~)$&
(3.2)
142
S. .I. HATHRELL
where Z;’
= 1 + z (poles).
(3.3)
Z, would just be the mass renormalisation if there were a mass in the theory, although that is incidental to the present arguments. It can not depend on I?,,, a,, b,, or cO because these appear nowhere else in the insertions of [#‘I into renormalised elementary Green’s functions in flat space-they disappear from Yi and Yg when the curvature is set to zero. The notation [ ] in (3.2) will be used subsequently for other normal products. No distinction will be made between position and momentum space, as the meaning should be clear from the context. Consider now the Green’s functions containing one insertion of the flat-space 0, given by (2.13):
+q,3*&+
( bT ") E,.
(3.4)
These Green’s functions are in effect the first order terms in the expansion of the elementary Green’s functions in powers of the weak gravitational source, cf. Eq. (3.1) taken in the flat-space limit. We see that the term r,, 3’4: differs only by an external momentum factor from the insertion of q,,#i, and so we could modify q,, by an amount (vZ; ‘) and still obtain finite results, where v is any finite parameter. The other components in qO must be chosen to cancel the divergences caused by the other terms in (3.4), which only depend on A. Therefore choose4
(3.5
‘lo= (rl fLJG’~ where “,
Lq=iL,
Vi@>
(3.6 1
(n-4)"'
L, depends on A but not on v: it is a Laurent series of poles in (n - 4) with residues which are functions of ,l only, appropriate to the minimal subtraction scheme. We may determine L, by requiring that the insertion of 0 into a 2-point function is finite, though there is a simpler way which will become apparent soon. Notation similar to (3.6) will be used subsequently for all functions which are Laurent series of pure pole-terms in (n - 4), with residues which are functions of A only. Next we consider the form of [#‘I m curved space. There is only one other ’ An alternative is to modify 5 by a tinite function of (n ~ 4), starting at O((n by Collins, Ref. (271. The relationship between these two approaches is explained
- 4)‘). as pointed in Ref. [ 101.
out
TRACE ANOMALIESANDkdJTHEORY
coordinate-invariant therefore have
143
operator of dimension 2, the curvature term H. We must
[#‘I = Z;‘q5: +,unp4H(’
(poles)},
(3.7)
with ,U the ‘t Hooft unit of mass required to balance the physical dimensions. The poles are determined by requiring the finiteness of ([#I]) in curved space. So
& (lPl>= i(e[#‘l>+ ($ [$21)= (finite),
(3.8)
and this can now be taken in the flat-space limit. The term i(B[#‘]) is linear in ?I, so the coefficients of H in (3.7) must also be linear in II, and we can write I#‘]=
z;‘@; +$-4H{2L,
+ 4p5, I
(3.9)
defining’ L., and L, as Laurent seriesof poles like (3.6) i.e., (3.10)
and similarly for L,. They are determined in principle from their definition by equating coefficients of 17in (3.8) though again there is a simpler way which will become apparent. Although we have chosen to drop the cosmological term II, and the Einstein term K~R from the discussion, the notation of (3.9) reflects the fact that L,, and L, would also play a fundamental role in the renormalisation of A, and K~ if such terms were present. For more details see Ref. 1lo]. Stage (2) The action principle with a/&l on an elementary Green’s function produces an insertion of i as/all:
1
+F$+G?$.
!
(3.11)
There are no other terms becauseZ,, Z, and A, are all fixed when ,I and ,D are held fixed. The curvature terms F, G and HZ are independent in a general curved background,
’ The exaci relationship between the definitions here and those of Brown from this equation and its counterpart in Ref. [IO].
and Collins
can be deduced
S.J. HATHRELL
144
and the operator insertion must be finite, so we can compare with Eq. (3.9) and deduce a, =/r4(a
+ Id,),
(3.12)
6, =/r4(b
+ Lb),
(3.13)
co =pn--4(C
+ L, + VL, + q2L).
(3.14)
a, b and c are independent, dimensionless.finite parameters, and a, and b, are independent of v. L,, L, and L, are Laurent seriesof poles with residuesfunctions of ,I only, as in (3.6). (We could have had &,,/a~ finite and non-zero, but that would just be a special case of the above with a = a(v); likewise for b.)
4. DIMENSION
4 OPERATORS
AND THE RENORMALISATION
GROUP
The composite operators or normal products of dimension 4 are E,, #i, a’[@‘], H[$*], a*H, F, G and HZ: linear combinations of all the terms appearing in 9, plus two which are total derivatives. The only new ones which have to be considered are E, and #i, as the rest are finite already. In fact E, is also finite, which is most economically demonstrated in the path-integral formalism for Green’s functions by a functional “integration-by-parts.” Recalling the definition of E, in (2.14),
(4.1)
There is no term from &(y)/&(y) as this is dimensionally regularised to zero. The validity of (4.1) can easily be checked in a number of simple examples, and it is a straightforward consequenceof the Feynman rules for the insertion of E,. The righthand side is clearly finite, and we therefore write E, = [El. When integrated over all space with respect to y, i.e., at zero momentum, the insertion of [E] just multiplies the Green’s function by a factor iN, where N is the number of external legs [23, 261. [E] also gives zero when inserted into a Green’s function containing just one other composite operator: if P is some polynomial in the fields 4(x), not necessarily finite, then
(E,(Y) P(x))= -i j (4) ]f’(x)g =-+($(y)z)=O
& eis1 in flat space.
(4.2)
TRACE
ANOMALIES
AND
,$j’
THEORY
14.5
This vanishes in flat space because every term contains 6(x - y), and all such l-point functions are dimensionally regularised to zero for a massless theory. The nature of the normal product [#“I is deduced by using the action principle with the finite derivative a/~%. We therefore need to know how all the bare parameters, and the renormalisation factors Z, and Z,, vary with 1 when all the other finite parameters ,B, ‘I, a, b, and c are held constant. This information is provided by the renormalisation-group equations (RGEs) for the parameters [28, 291, and it shows why the finite renormalisation-group functions play a natural role in the study of normal products of dimension 4. To discuss RGEs we introduce the differential operator (4.3) and the equations are derived6 by requiring that all the bare quantities appearing in the action should be independent of ,u. First of all, from the renormalisation of ,I, in flat space &=iu
4-n’
1
“7 ‘YiPI ’ + ,&, tn - 4)’ i ’
(4.4)
we use DA,, = 0 and equate coefficients of (n - 4) to derive as usual DA =&I.
n) = (n - 4) /I +/?(A).
(4.5)
There are also relations between the pole residuesg,(L), and p is determined by the residue of the single pole (4.6) so that it is clearly O(L’) in leading order. The distinction between p^and p is a very important one, as we will frequently equate coefficients of powers of (n - 4). Applying D to Eqs. (2.11) and (3.2) we deduce (D+y)Z;‘=O,
(4.7)
(D+@Z;‘=O,
(4.8)
where y and 6 are the anomalous dimensions of d and 4’ (or the mass in a massive theory), and are finite functions of 2 only. We may apply D to an operator directly, as in (3.2), rather than just to Green’s functions containing that operator, becausethe ’ Renormalisation-group equations are sometimes taken to be those equations obtained by combining with the equations of dimensional analysis and then eliminating the terms in pa/&, thereby obtaining equations for the properties of Green’s functions under scaling of the external momenta. However. we will not be directly concerned with such scaling behaviour here, and will not use the equation of dimen sional analysis.
146
S. J. HATHRELL
action is independent of p, and because D only produces a finite variation on the elementary renormalised fields by virtue of Eq. (4.7). It is therefore consistent to argue that D[#2] is finite because [d’] is, and likewise for other operators. Next, applying D to (3.5) gives (4.9) and (4.10)
(D-b=P,,
where p,, is a finite function of L only, given by (4.9) in terms of the residue of the single pole of L,. In addition, higher pole residues of L, must obey consistency conditions. Applying D to Eqs. (3.12E(3.14) we then get (4.11) (4.12)
[(n-4)+D+26]L,=-/I,(A)=-$4,).
(4.13)
i(n-4)+D+61~,+2~,L,,=-~~(~)=~(1~,),
(4.14)
lb - 4) + Dl L, +P,L,
= -P,(l)
= $ (AC,),
D,=~c++rlr(l,+~~&~
(4.15)
(4.16)
fit = -(n - 4) c + P,(A). The last four equations follows from equating coefficients of r in (3.14). This must be done after applying D because the RGE for v, Eq. (4.10), has an inhomogeneous term which is a function of 2 only. The notation is systematic: all the functjons-/I, p,, /?,, Pb, PC, p,, p,, are finite functions of 1 only. The functions /3, p,, Pb, p, have an additional term linear in (n - 4). We will usually take advantage of this notation to suppress the functional dependence of the terms for the sake of brevity. We are now in a position to find the variation of the bare parameters with L when ,u, q, a, b, and c are held constant. From (4.3)-(4.5) we find
alo -d!y4)L a1
4
0’
(4.17)
TRACE
From (3.5), (4.8)-(4.10)
ANOMALIES
AND
Ai4
147
THEORY
we get (4.18)
From (3.12t(3.14)
and (4.11)-(4.16)
aa,- -(n -4) ia-- p ab,- -(n-4) a-- p %l -=. 81
L + pa i * (n-4) ! L + Pb
$-4
pn-4
b
-(n - 4)
we get
$-4
(n-4)
L
+
L [
+ A
@A
+r
(n-4)
I[
.+r2
(4.20)
I’
ca,+P,U
c li
(4.19)
1
L
+
cp,+~~,+2P&*)
[ K
1
(n - 4)
+2dLJ
(n-4)
1 (4.21)
I! *
Finally, we need to know how the renormalised elementary fields vary with 1. The bare fields are independent of A, so from (4.1) and (4.7) we have
The importance of the distinction between /3 and B is now clear. While p is 0(n’). and finite, b is O(A) and in perturbation theory its inverse begins with a pole,
p-‘=
1 (n-4)1
1 l+
PIA (n-4)
-’
(4.23)
i
and contains successive poles in higher orders. So (4.17) has the form nd all the other terms in (4.18F(4.22) begin with a single pole p’+“( 1 + :c (poles)}, a at least in the leading term.’ We can now derive the form of the normal product [#“I from the finiteness of @/an) ($1 . . . dK). It must have the general form ( 1 + C (poles)} 4: + poles multiplying all the other operators of dimension 4 listed earlier, and because of (4.17~(4.122) this is exactly the form produced at zero momentum by the derivative a/an. We (deduce that
’ The RGEs
ensure
that there
are no O(A
‘) terms
in lj-‘{L,
+ /lo/(n
- 4)/,
etc.
148
S. J. HATHRELL
+ cO,+r14 (n-4)
IH,@‘, 2
(e+L,)+rl(f+L,) B2H
-pn-4
(n - 4) (Lb+&)
+P4 fP
G
Pk LK+(n-4)
n-4
1)HZ. I The coefficients of a’[$‘] and a2H are not given in terms of known quantities by this argument because they vanish when integrated over all space, i.e., at zero momentum, but they must be pole series in the minimal subtraction scheme, and thejr are instead defined by (4.24). The particular form of these coefficients quoted in (4.24) needs a little further justification, however. First, if we go to flat space the expression for [4”] reduces to [E] _:dn-+‘fB’[m’]I.
(4.25)
The coefficient of+a2 [#‘I must therefore be independent of q, a, b and c because these only couple to curvature terms, and there are no curvature terms in the insertion of [#“I into a flat-space elementary Green’s function. The particular form of the coefficient is chosen for later convenience, and consistently with convention (3.6) because it turns out that the single pole residue d(A) plays a privileged role in the subsequent analysis and is more conveniently regarded as separate from the higher poles in L, (although they are related). The values of d and L, can be determined from the insertion of [$“I into a flat-space 2-point Green’s function. The leading contribution to d is required to cancel the divergences of the graphs in Fig. 1, and from these we find that d/B = O(l’)/(n - 4), and thus d(A) = O(13). This is sufficient information for the present. The actual value is needed when we come to computing the various quantities, and is given in Appendix B. Second, similar considerations apply to the coefficient of a2H in (4.24). This is needed to cancel the divergences for the insertion of [#“I into a one-graviton diagram, i.e., (S/SQ)((#“]) in flat space; so the coefficient is at most linear in r and does not depend on a, b or c. The particular values of the finite functions e(L) andf@)), and of the pole series L, and L,, can be obtained by equating coefficients of q in the calculation of this diagram; although we postpone determination of the leading behaviour until the end of the next section. It is at this stage that the rather roundabout approach to the discussion of the
TRACE
FIG.
1.
ANOMALIES
Flat-space
functions
AND
14’
149
THEORY
i < [@‘I $d) in leading
order
anomalies Ibegins to show dividends. The reasonis a consequenceof the form of [4” ] in (4.24), and in particular the overall factor ofp-‘. We now expressBLlrr,Eq. (2.13) in terms of the finite operators determined above. We also know that 8”, must finite, so any pole-terms multiplying finite operators mush vanish. Equations (2.13). (3.7) and (4.24) give’
and in addition the consistency conditions
L,=L.,
(4.27)
L,=4L,-2L,L,,
(4.28)
L,=2L,-4L,L,.
(4.29)
These coml: respectively from the coefficients of a’[@‘], a2H and q a*H. Stronger constraints come from a consideration of the RGE for 8. We apply p^-’ to (4.26) and use the fact that 6’ is independent of ,u, as evident from its expression in terms of bare quantities in (2.13). Furthermore ~P’D{~~4P”[#4]} must be finite because ]#“] is, and
(4.30)
’ The term
i?*H
is perhaps
more familiar
as the “ambiguous”
OR
term
in the anomaly.
S. J. HATHRELL
150
is also finite. Therefore the poles which must vanish, giving
which
/pD
/j P4-“[#“l 4! I
appear in the remaining
expression
I
(primes denote differentiation
with respect to A), together with the conditions &+6d=O,
(4.32) (4.33)
4P,-P,f+2dp,+~/~)e=O, 2p, + 4dP.4 + (/3/n - S)f = 0.
(4.34)
These last three equations constitute relations between finite functions of 1 only, valid to all orders of perturbation theory, and are a powerful source of information. For example, the anomalous dimension of [$2], 6(A), is given by (3.2) and (4.8) and is easily found to be O(A). The relevant diagrams are shown in Fig. 2. This and knowledge of d(A), the ,O(A”) function describing the mixing of 4: and a’[#‘] in flat space, give complete information about /?,, and hence qO. In lowest order we have p, = O(A”) and L, = O(ii”)/(n - 4). There is an important feature of this derivation which will recur frequently. Equations (4.32ti4.34) would have been lost in applying just D rather than p-‘D to Eq. (4.26). Since p-’ begins with a single pole, any statement about the finiteness of an expression involving inverse powers of p’ is correspondingly stronger than the same statement multiplied through by p. This point is particularly useful to the determination of p, to O(A5), which by indirect arguments relies on a simple 3-100~ calculation, whereas the direct verification of (4.33) to O(A4) requires a 5-100~ calculation. The 5-100~ calculation is performed later in Section 9; but we turn now
FIG.
2.
The flat-space
function
i(/@‘]
$4) in low order.
TRACE
ANOMALIES
AND
rid’
151
THEOR
to the study of 2-point functions of normal products to find efficient routes to the calculation of the terms appearing in the renormalisation of the HZ part of the anomaly, i.e., the terms in Eqs. (3.14) and (4.13~(4.16). 5. THE ~-POINT
FUNCTIONS
OF NORMAL
PRODUCTS
The values of the terms appearing in the renormalisation of c0 are obtained through a study of the 2-point functions of normal products defined by
r**=i
(pg),
(5. la)
r24= j [@‘I P4-“14”l 4! ( 2! )’ P”-“P”l P4-“[4”l r,, = i (
4!
4!
@lb) (5.lc)
)’
where [d’] and [d”] are given by the flat-space values (3.2) and (4.25). These functions are divergent, and in the case of r12 and r,, are divergent even when ,I = 0. The lowest order terms are shown in Figs. 3a-c and are readily calculated using formula (A.l) from the Appendices. The leading behaviors for Tz2, r24 and r44 are, respectively, 0( I/(n - 4)), O(A/(n - 4)), 0( l/(n - 4)). The renormalisation of r22 is easily derived from the action principle with J/&2 applied to (i[#‘]). We have
Tj- [@‘(x>] B(y)
=I
.(I)
FIGS.
3
$*
(ak(c)
Composite
+/r4(2L,
+ 417L/#qx
- J’),
.
(52)
)e
2-point
functions
in leading
order.
152
S. .I. HATHRELL
where 0 takes the flat-space value
e=-fip“-“[#“I 4,.
+ (l-;-Y)
PI +~vd)~2[~2]
(5.3)
given by the first set of terms in (4.26). The coefftcient of q in (5.2) therefore informs us that Tzz(p2) +,Pm4 2L, = (finite),
(5.4)
where we have gone to momentum space. This is a non-trivial statement about the renormalisation of rz2 because it implies that there are no pole-terms in (In(p2/p2)), and it is only valid using the renormalised operator [d’] in (5.la) rather than the bare operator 4:. It is now apparent that the single-pole residue of L, and the associated p-function /I,, are O(1) in leading order. We could also read off the form of renormalisation of r24 from the q-independent part of (5.2). However, this actually only provides information about Br24, and a more powerful result can be obtained instead from
(
6
WY)
P”-“M”(41
(
4!
_ . P”-“i4”w 4! 4
= (finite) )) flat 8(y)
- 2/P-4@*)*
6(x - y)
)
where use has been made of Eqs. (2.8), (3.9) and (4.24). The coefficient gives r24(P2)
+P*fl”-”
of v then
(5.6)
where
(5.7) =&+2&+4dL,}, using (4.27) and (4.29) in the last step. A factor of (n - 4)p^-’ has been extracted in the definition of L, for convenience. Using the fact that rz4 = O@)/(n - 4) in lowest order (see Fig. 3), we find L, = O(A’)/(n - 4), and f(A) = X,(A) = O(A 2).
(5.8)
TRACE
ANOMALIES
AND
id’
THEORY
153
This result would have been lost looking at the q-independent part of (5.2). We have already found that we can obtain L, , and hence /3,, , from calculation of rZ2. Using Eqs. (4.34), (5.8) and a calculation of rZ4 we can therefore find p,. Leading-order behaviour of the other terms in (4.34) shows that /I, = O(A3) at least. The general pattern of the argument has by now begun to emerge. The action principle with li/SQ and an examination of the coefficient of q give full information about an insertion of [$‘I; but it brings in [d”] with a multiplying factor of @ which gives weak results. -On the other hand the action principle with c?/c% gives insertions of [#“I without the /I factor, but only at zero momentum. So full information can not be deduced from the action principle. We use instead the fact that the divergences of r44 do not involve ln(p2/p2), and the subtractions take the form
+p”-4(p2)2 (^ ) n-4
r44
2
L, = (finite),
P
where L, contains “new” information. Returning for the moment to the trace anomaly as exhibited in Eq. (4.26) and focusing on the coefficients of HZ and a2H, we have found that the coefficient of q2 is determined from r22, and the coefficient of q from r24. It is natural, therefore, to expect to find e and /I, from r,,. This becomes much more obvious if we apply c?/cM to (8), which is best done using the bare form (2.13). We conclude that
i(flS) + 8c,(p2)’
= (finite),
evaluated in flat space. Recalling the expressions can equate coefficients of (q - d). (This is as because q is an arbitrary finite parameter.) The are consistent with Eqs. (5.4) and (5.7), and the
(5.10)
for c0 and 0 in (3.14) and (5.3). we good as equating coefficients of v coefficients of (q-d)’ and (v -d) remainder gives
8(L, + dL, + d2L,) - (n - 4)2 L,, = (finite).
(5.11)
Then L, and e can be deduced from (4.28) and (4.33). The lowest order pole contribution to r,, is O(l)/(n - 4), and so L,, must be O(A’)/(n - 4). To find the single pole of L,, and hence /I,, we must use an RGE to derive the leading term in the residue of the treble pole of L,, Y,(A). Superficially this would be expected at O(A4). The relevant RGE is obtained from applying p’-’ Dp^’ to Eq. (5.9) using the flat-space RGE for [$“I which can be read off from Eq. (4.31):
pD
p^P”-“P”l 4! I
= --d’3’[#‘]
- y’[E].
(5.12)
I
Thus L, = (finite),
(5.13)
S.J. HA1‘HRELL
154
where the term in rz4 has been eliminated using Eq. (5.6). The finite term on the right side of (5.13) must simply by (l/A”)(8/LU)(AY,) as this is the only finite term on the left. The equation may then be integrated iteratively, and Y3 is given by $(AY,)=-;/A~2Yl
$ (;)(d;)+~j~4d%f,(d~)+4d’U,.
(5.14)
0
X,(L) is the residue of the double pole in L, and is given from Eq. (5.7) by X2 = 2/c, + 4dA, = O(A3),
(5.15)
and hence by p, and /?,, . (It could also be obtained from X, by deriving an RGE for r,, similar to the above.) The three terms on the right-hand side of (5.14) are, respectively, O(A5), O(A”), O(A”). The O(A4) contribution has vanished from Y3, and hence from /3,, because the first term depends on (a/LU)@/A’). The O(A”) contribution depends on the O(A3), 2-100~ /?-function, and on the leading term in Y,(A), which is O(A’). This concludes the summary of the arguments of Brown and Collins [IO], adapted for the masslesscase. In the expression (4.26) for the trace anomaly, the functions p^, y and 6 are the usual functions appearing in flat-space #4-theory; d and /?, are computed from the mixing of a’[$‘] in the renormalisation of 4: ; e, f, p,, , /?, and /?, are computed from the functions r22, r24, and r44. The computations rely in particular on the relations (4.32)-(4,34), and enable low order flat-space calculations to be used to derive the terms appearing in the curved-space anomaly, in some cases in high orders of perturbation theory. Particularly striking is the result that j3, is O(A’), so that e is O(A4) following from (4.33), and in Section 9 we confirm this directly by an explicit 5-100~ calculation. So far, however, very little has been derived about the coefficients p, and pb of the other two “gravitational” terms appearing in the anomaly. In Sections 6 and 7 we show to extend the analyis of Brown and Collins to determine these also.
6. TENSOR OPERATORS
AND THE F TERM
Renormalisation-group equations for a, and 6, were derived in the last section, but nothing else was found out about them as they appeared to be unrelated to the other parameters of the theory or to the divergences of the functions r22, r24, and r,,. They appear in the action multiplying the F and G terms which are second order in the curvature, but these have the property (6.1) so that the functional derivative S/&2 produces quantities which are still second order in the curvature. Therefore a second application of d/&2 still produces terms which
TRACE
ANOMALIES
AND
iq3’
THEORY
are at least first order in the curvature, and vanish on going to the flat-space Contrast this with the behaviour of the non-conformal Hz term:
. -g - 6 bm.
HZ = -(n - 4) H* + 413*H.
155 limit.
(6.2)
The last term produces a non-zero quantity in flat space under a second application of 6/6s2. and the appearance of this non-conformal 8*H term is precisely why the coefficient e,, appears in the renormalisation of i(M), and hence the functions Tz2, I- 24, and rJ4. So to extend the investigation to include u,, and 6, two possibilities immediately present themselves: (i) look at the effect of -2 6/fi6gHL, rather than the traced version S/&2, and (ii) look at the effect of (S/&2)>‘. This section deals with (i), and the next with (ii). The action principle informs us that -2 d/\/--gag,, gives finite results. Applying it twice to the vacuum functional. taking the flat-space limit and transforming to momentum spacesgives PUU + a,A ‘API + cOC+”
= (finite),
(6.3)
where
correspond, respectively, to the F and H* terms in the action. There is still no csontribution at all from the G term becausein fact and
A
K.\UP,
CK.iUI’
+ 8(RU”R”, - Rv”a14R,o),
(6.5)
making use of the Bianchi identities, and this is second order in the curvature again. Therefore b, remains absent in this approach. Thus full expressionsfor AK-l”” and CKAfi” are
and
156
S. J. HATHRELL
The flat-space energy-momentum in the form
tensor P”(x)
is given by (2.12) and can be written
where
P” = -qi, fU”QO+ & (using (3.2) and (3.5))
(fz2 - @I+L,)) t”“[~‘l~
(6.9)
and
P” is a traceless differential operator [23], and (6.8) represents a decomposition of t!Y into irreducible components: c?’ is manifestly traceless in n dimensions as well as in 4. Furthermore both 0 and 0”” are finite operators, so @” must also be finite. We can therefore introduce the new finite composite operator [qW”$] and deduce its renormalisation from Eq. (6.9): [qh”“#] = q&P”q& + (c poles) t”“[#2],
(6.11)
where the poles-terms are just the q-independent pole-terms multiplying t”“[qi’] in (6.9). The q-dependent part is clearly finite also. The particular form of the pole-terms multiplying t”“[#‘] is unfortunately rather untidy because of the presence of the factors (n - I))’ and n/4 which mix up residues of different order poles of 2, and L,. This is an inevitable problem in the discussion of tensor operators in the minimal subtraction scheme [24], because any contractions implicitly introduce factors of n with the (flat) metric tensor q’*‘. The difficulty was avoided in earlier discussions by confining attention to scalar operators and keeping a factor of (n - 1) buried in H, but this factor re-emerges when the metric derivative is not traced. The resolution of the problem becomes a matter of defining just what tensor the “minimal” subtractions are made from. However, all of the Laurent series of poles with the form of (3.6) obey some kind of RGE which implies that each successive pole appears at O(A) below the leading order of the vanishing of the leading order previous pole (though there may be an “accidental” coefficient, as in the single pole of Lx--see later), so multiplication by some finite function of n brings no ambiguity in the leading k-dependence of each pole residue. In quite a few cases there is no ambiguity for some of the next-to-leading terms either. There is consequently still considerable value in discussing the renormalisation of the tensor quantities mentioned above. Equation (6.3) expressing the renormalisation of i(f?“‘P”) clearly contains some new information, but it is not immediately obvious just how much. A count of the
TRACE
ANOMALIES
AND
ii4
157
THEORY
form-factors is therfore quite illuminating. A quantity Pa”’ with symmetries under (K -A), (u - V) and (~1) - (u v ) contains five scalar form-factors-these multiply an assortment of tensor terms similar to those appearing in (6.6). In addition to the constraints imposed by these symmetries, however, there is a further constraint because the energy-momentum tensor obeys an equation of motion. Classically this is the vanishing of its divergence, and the operator version is analogous (we only need to consider flat space): ~3~,8”” = -Eb”, where Et is an equation-of-motion
operator
(6.12)
[23 ] similar to E, : (6.13)
Furthermore
Eg is finite, Eg = [E”],
and satisfies
W(x)
(6.14)
P(Y)) = 0
using exactly the same arguments as for E,. Therefore in momentum space, pJ+yp)
= 0
(6.15)
and this constraint reduces the number of independent form-factors from 5 to 2-and recall the appearance of just two parameters a, and c,, in Eq. (6.3). So it is hardly surprising that the study of 2-point functions should not give information about the third quantity b, as well. Notice also that PICA
KAUI = p, pw
= 0,
(6.16)
a consequence of the coordinate-invariance of F and HZ. The discussion of fnarc” may now be simplified to a discussion of the two formfactors by taking contractions. Contraction with v~.~I],,~,just reproduces Eq. (5.10) losing all information about @“ and the parameter a,. The latter disappears because AK“” also satisfies A”
I(
L(L’=AKk
u =
(6.17)
0,
consistent ,with Eq. (6.1). Contraction with v,,, vl, is more useful, however. Inserting the expressions (3.12) and (3.14) for the renormalisation of a, and cO, and the expansion for I!?’ in (6.8), it gives
+ (p2yy
I
4(n - 3)(n + I)&
+ &
(15, + ~5, + ry2L,)I
= (finite).
(6.18)
158
S. J. HATHRELL
Now the coefticients of q2 and q in i(@‘“g,,,,) involve the expressions t”‘t,,( [#‘I [#‘I) and P”( [#tMU#] [@‘I). Both of these can be expressed solely in terms of the functions I-,, and r,,. using symmetry arguments and Eq. (6.12) in the case of the latter. It can be verified that the coefftcients of q2 and q in (6.18) exactly reproduce the results for L, and L, found in the previous section-including cancellation between the various functions of n arising from contractions of tensor indices. There is therefore no ambiguity about the definitions of the subtractions in the minimal subtraction scheme for these terms, nor are there any new results. The new subtraction-term L, appears in the q-independent part of Eq. (6.18), and this depends on the “new” function i([qWr#][qb,l,#]). It is impossible to express this function in terms of previous functions using any kind of symmetry argument or results involving (6.12tagain a form-factor count shows why-and so it requires a separate Feynman-diagram evaluation. Given this fact, and in view of the rather clumsy definition of the renormalised operator [#t”“#] derived from Eq. (6.9) there seems to be no advantage in considering i([qW”~][~t,L,~]) rather than the more directly relevant function i(f?“#u,,). The same Feynman diagrams appear in the same orders of perturbation theory, and both involve Feynman integrals with the same complications from numerator factors in the integrand. Indeed the latter function has the advantage that @” has a simple expression in terms of bare operators. Both [#t”“$] and @“’ are finite, and [@““d] d oes not contain extra factors of BP’ to make it a source of more powerful results. We conclude, therefore, that the F part of the trace anomaly is determined by setting r = 0 and then using the flat-space result i(@‘“g,,,,> + (P’)”
pnP4
I
4(n - 3)(n + 1) L, + H(H8P 1) L,
1
= (finite).
(6.19)
The term i(l/n)(&J) h as b een translated into a pole-term proportional to L, using Eq. (5.10). Below O(L”) the L, term in pl’ does not contribute, and @” takes the simple form
(6.20) which is quite convenient for computation. L, does not contribute below O(A5). The evaluation of the relevant Feynman diagrams to the 3-100~ level is described later in Section 8. But we observe for the present that there is an O(1) contribution to the single pole of L,, and hence to p,, arising from the divergence of the l-loop diagram in i(@“$,,). This 0( 1) contribution reproduces the known, A-independent coefficient of the F term in the trace anomaly. The advantage here is that the reduction to flat space makes the discovery of the higher order corrections much more accessible than it would be from calculations based on a curved background [30, 311.
TRACE
7. THE
ANOMALIES
~-POINT
AND
FUNCTIONS
Ad4
AND
159
THEORY
THE
G TERM
It was shown in the last section that if the coefficient b, of the G term was to be determined by eventual reduction to flat space,then 3-point or higher functions would have to be considered. There is also a clue as to the outcome: in an earlier paper Drummond and Shore 191considered a conformal scalar theory with no q,,H#i or c,H* terms in the action, and they showed in a spherical background-for which the F term vanishes-that b, was not renormalised by A. becausethere were no primitive divergences. Introducing in particular the c,H* term would introduce divergences into the spherical vacuum functional, but only at O(A’), and this would affect the argument for b,. It is reasonable to infer, therefore, that in the current approach based on flat-space calculations, the first A-dependent terms in b, will appear at comparabl:y high orders of perturbtion theory, and will be dependent in some way on the terms in cO. This is indeed what happens, and can be discovered from flat-space considerations I.51 in spite of the fact that G is a topological quantity. The simplest Green’s function involving 6, is the 3-graviton function obtained with (s/&2)‘, so we apply this to the vacuum functional, invoking the action principle, and obtain
w VI
i*(B(x) O(y) O(z)) + i B(x) L aqz)
(
+
1
cm(x) sn(J,) sn(z)
)
de(z)
’ sn(x) + e(z) = + e(J) sfl(J’) )
= r, = (finite>,
(7.1)
which is taken in the flat-space limit. The individual matrix elements are of course symmetric in x, y and z becausethe functional derivatives 6/&J commute, and so
Wx) ---= WY)
d2S _ WY). a?(y) &2(x) &2(x)
(7.2)
Henceforth we will condense the notation slightly, deliberately suppressing the distinction between position and momentum-space, and refer to 0(x), M(y)/&(z) and 8*0(x)/&2(v) &2(z) as 0,, eYZand 8,,;. Similar labelling will be used for other operators. Note that the terms in By2involve (in position-space) the function S(y - z); and in 8,,, the function 6(x - y) S(.y -z), so that x, y and z coincide for this third operator. Labelling of this kind becomesnecessary for 3-point functions becausethey are in general functions of three invariant momenta: pi, p.z and pi. Such general functions can not usually be handled easily. but there are casesin which they can: (i) ifpL = 0, then p: = p: = p2, and the whole is a function of one variable p2. (ii) If pi = p,’ = 0 and p: = p2 # 0, then again the whole is a function ofp* only. These two prove quite adequate for the determination of 6,, though in fact (ii) may be relaxed to case (iii), when only pi = 0, and on dimensional grounds the Green’s functions must be a power of p: multiplied by a function of the dimensionless ratio (p-t/pi). The last case
160
S. .I. HATHRELL
becomes important for later calculations in Section 9. Some general formulae relevant to cases (ii) and (iii) are given in Appendix A. This plurality of momenta is not the only new complication arising with 3-point functions. Another becomes apparent if we consider as an example the leading, Aindependent contribution to the function i2&4-n/4!)3 ([$:I [#;I [#:I). The diagram for this is shown in Fig. 4, and with pi = pf = 0 it can easily be computed using the formulae (A.1) and (A.5). Note that putting these two momenta on shell does not introduce any infra-red singularities. The result is 2(n-4)
x
r(6 - 2n) r(2 - n/2) T(n/2 - 1)” Z-(3(n/2) W(n - 2)3 r(5(n/2) - 6) I
- 4)2 I
+ U@).
(7.3)
This has a double pole, so that the single pole must contain terms involving ln(p2/p2) which cannot be removed by a constant additive counterterm. (It also contains such undesirable objects as ln(4z) and Euler’s number.) Clearly the renormalisation of 3point composite functions must involve subtractions of other Green’s functions as well, and we will return to this a little later. The third complication involves the equation-of-motion operator [El. The individual contributions from this no longer vanish. If A(y) and B(z) are polynomials in the field 4, then with the usual functional integration-by-parts i2(~,A,B,)~=
i j (dq+) A$; I
=+I,
(-jL-z))-(B++$)).
(7.4)
The two functions on the right-hand side do not vanish in flat space. The 3-point
FIG.
4.
A composite
3-point
function.
TRACE
ANOMALIES
AND
Lb4
161
THEORY
functions involving [E] can at least be expressed in terms of 2-point functions, however. Furthermore we can use the result (4.2) to deduce the special cases (7.5) i’(E,E,E,)
= 0.
(7.6)
We also have
(7.7) and
Equations (7.2) and (7.4k(7.8) can now be used to simplify [E] appearing in Eq. (7.1). If we define
the contributions
fixxe8,-aE,,
(7.9)
where a is some parameter as yet unspecified. after a little simplification obtain I-= i’(B,B,gJ
from
we can insert this into Eq. (7.1) and
+ i(B,OYL + 8,8,; + OZ8,,.) +
(e,yz)=
(finite),
(7.10)
where (7.11) The third lmatrix element is insensitive to the choice of a because x, ~1and z coincide and so any terms involving the fields are dimensionally regularised to zero in the massless theory. The only contributions to it come from Yg and are obtained by two functional derivatives of the curvature terms in 0. Eq. (2.13). After going to momentum-space we find (7.12) where B = 2(n - 2)(n - 3m - 411 HP:)’ - 2IP2;P: + P:Pi c=
4{@ + wP:)2 + 41p?;: + P:Pl+
+ cp:>* +
+ PIP:115 + (P:)’ PIP:lI.
+
(Pi>‘1 (7.13)
(P5121 (7.14)
162
S. J. HATHRELL
This time it is the F term in Pg which gives no contribution. The reason is that F involves the square of the n-dimensional Weyl tensor C(“)aqy6, which has the property that it is invariant under a conformal scaling of the metric, i.e., (7.15) for any f(x). So any number of functional derivatives S/&L’ applied to F always leaves an expression which is second order in the curvature, and which therefore vanishes in the flat-space limit. We can now see how b, comes into the picture. It multiplies the 3-graviton vertex B given by (7.13) and appears as part of the Green’s function r in (7.10). The divergences of b,B are needed to make the rest of r finite. B itself involves a factor of (n - 4), so (7.10) determines the residues of the double and higher poles of Lb, but not the single pole. The single-pole residue b,(L) and the P-function Pb can then be deduced from the double-pole residue b,(A) using the RGE for 6, previously derived. Specifically,
This method can therefore be used to deduce everything except the leading, constant contribution to b, and Pb; but as this constant contribution is already known from elsewhere [7, 91 it does not matter that it escapes detection here. In fact the present approach is an efficient route to higher order corrections without involving the leading terms on the way. The other two matrix elements in r involve the parmeter ~1, so a value for this needs to be chosen. The two obvious possibilities are, (1 - n/2) to simplify 0 in terms of bare operators, from Eq. (3.4); or (1 - n/2 - 7) to simplify $ in terms of renormalised operators, from Eq. (5.3). Although the latter is perhaps the more obvious choice in keeping with the general approach here, the former is in fact more convenient and allows the problem to be tackled in the following way. We define the composite operator ( O4 } as the linear combination
(7.17)
so that with a = (1 - n/2) we have ( in flat space)
(7.18)
TRACE
ANOMALIES
AND
/zd’
163
THEORY
( 04) is the “natural” object in the present context, because although it is not actually finite, it behaves in many respects as though it were, and it has the following useful properties: (i) p^(O”) is finite. (ii)
It obeys the simple RGE (obtained from Eq. (5.12))
pD@{o”})
= -d’a’[qP],
(7.19)
which is also finite. (iii)
It arises naturally
with the use of the action principle, --= C3S d”x (O:}. .I I3
i.e., (7.20)
(a/an only produces insertions of [#“I into Green’s functions of renormalised elementary fields, and we are more concerned with functions containing other composite operators.) (iv) We may substitute { O4 } into the definitions of the functions r24 and r,, of Section 5 swithout altering their values. are the same because y(A) = O(A*). (v) In leading order (04} and ~‘~“[#~]/4! (vi) (0’) has the same renormalisation structure in 3-point functions as the finite operator p4P”[#4]/4!. The point of these remarks, and especially the justification of (vi), will become apparent when the renormalisation of the composite 3-point functions is determined. But before: passing on to that we note that, with the choice of a as specified, the operator tTYz can be obtained from Eqs. (2.13), (7.9) and (7.11), followed by expressing the result in terms of renormalised operators. In momentum-space the contribution relevant to the matrix element in (7.10) is given by 0)s: =: -2(n
- 2)&04}
+ (d + L,)((3n
- 10) p: - (n - 2)(p; + p;)) &iP]
- (rl - dN(n + 2) P: + (n - 2)(~: + pi)1 +[#‘I.
(7.21)
(This form assumes p, + py + pz = 0.) Thus the second of the matrix elements in (7.10) involves only linear combinations of the 2-point functions r44, r,, and Tzz met earlier, multiplied by various functions of A and polynomials in the momenta. r44r etc., come only as functions of the individual momenta pt., pz or pi. The remaining term in I- is the 3-point function i2(~~y~yf?z), with f? given by Eq. (7.18). It involves the functions which can be defined by r444(dr
P$ pi) = i’(~O~l~O~~~O4~)~
(7.22)
r244(d.
PC, PI) = i’GkC1~0~~~041>~
(7.23)
r4,,(p:~
P:, ~1) = i’(lOZl
(7.24)
rzz2(pf3 P$ PI) =
+[#:I SKI>,
i’(fhGl $:I %W,
(7.25)
164
S. J. HATHFCELL
and similar ones obtained by permutation. They are the natural extension of the 2point functions rz2, rz4, rd4 in Section 5 if these are regarded as involving { O4 } rather than ~~-“[$~]/4!, i.e., r24
=w”lP”~)~
(7.26) (7.27)
r4, =~w%W
These definitions are identical to the previous ones in (5.1) because [E] gives no contribution in 2-point functions. r22 is unchanged anyway. The advantage of doing this is that the large number of 3-point functions explicitly involving [E] do not now have to be considered, either for the purposes of calculation, because of the absence of [E] from both forms for # in Eq. (7.18); or in the derivation of the RGEs, because of Eq. (7.19). The divergences of r,,, are removed by subtractions of sums of poles in A multiplying the terms T,,(pf), pfr,,(pj), pfpjr,,(pi) and ,~“-“pfpj, suitably symmetrised. (The pole-terms are independent of q because n only couples to the curvature and we are now in flat space.) The justification of this statement involves considering first the subtractions necessary to renormalise the function i2@4Pn/4!)’ ([#~][(li~][#4]) constructed from finite operators, introduced in Eq. (7.3). The divergence associatedwith the coincidence of the points y and z is subtracted by pure pole-terms multiplying the functions remaining when these points coincide, and these latter functions are just the 2-point functions previously encountered (or can be taken as such without loss of generality). The leading pole coefftcient of T,,(pi) is O(1) to cancel the divergence of the graph in Fig. 4 when y and z coincide. The leading pole coefficient of T,,(pi) is O(1) to cancel the divergence of the graph in Fig. 5a, though r,, is itself O(A); and the leading pole coefficient of T,,(p:) is O(A2) to cancel the divergences of the graphs in Fig. 5b (associated with the mixing of a’[$‘] in 4:). Other terms appear from similar subtractions obtained by permutation of the spacetime labels. Finally there is an overall subtraction of pure pole-terms because the whose 3-point function is itself primitively divergent. Returning now to rjq4, involving (04) instead of [#” ], the pole-term (y/B)[E] in (04 1 gives contributions which are translated by means of Eq. (7.4) into pole-terms multiplying r,, and r,,.
FIG.
5a.
An O(A)
3-point
function.
TRACE
ANOMALIES
AND
&,b4
THEORY
165
t Cl -
5b.
O(,l*)
3-point
functions.
We see that these terms are just the same in structure-through with different pole coefficients-as the terms appearing in the subtractions for i2(@4-“[#4]/4!)3). This justifies the earlier statement (vi) about the use of (04}. Exactly the same arguments apply to r244 and r,,,. The divergences of r244 are subtracted by poles multiplying the functions r,,(pf), pfTt2(p,!) and ~“-~pf. The divergences of r422 are removed by poles multiplying T,,(pi), T,,(pf) and ,~~-j--but not T,,(p:) in this case because there are no diagrams in r,,, which have a primitive subdivergence associated with the coincidence of y and z. The leading poles in r4** are O(1) corresponding to the diagram in Fig. 6. Finally, Tzz2 is finite to all orders because it contains no primitive divergences, and all subdivergences are incorporated into the renormalisation of 1 and [$‘I. All this analysis is rather cumbersome, so let us summarise the argument so far. The renor:malisation of b, is determined through the residue of its double pole by requiring that the single pole of the 3-graviton function r should vanish. The contributions to r can be expressed in terms of the insertions of the operators (04) and I#‘] into 2- and 3-point functions, and all contributions from [E] can be absorbed into the functions in such a way that they do not have to be considered explicitly, either when calculating directly or when deriving RGEs. The renormalisation of a Green’s function containing insertions of the divergent operator {04} has the same general structure as the renormalisation of that function containing instead the finite operator p”-“[@“l/4!, and furthermore it involves only combinations
166
S. J. HATHRELL
FIG.
6.
Leading
contribution
to r4??.
of the f-point functions previously discussed. This fact and the removal of [E] constitute a major simplification. So far, however, we have only found the general form of the renormalisation of the composite 3-point functions. We can do much better than this and derive many of the pole coefficients explicitly, as well as verifying the general form, by once again using the action principle. As before, there are two basic methods: (i) find full information about insertions of [@‘I using the functional derivative S/S0 and looking at the coefficient of q; and (ii) find full information about insertions of ( O4 ) at zero momentum only, using the ordinary derivative a/aL This leaves just one quantity whose divergence can not in principle be expressed in terms of the divergences previously encountered, namely, I-,,, with none of the momenta vanishing. The non-vanishing of any of p,, pY, pL is actually implied by the conditions p-t = pz = 0, p.t # 0, so it will be sufficient to consider the case r444(p2, 0,O). Recall now Eqs. (7.10) and (7.12t(7.14) expressing the renormalisation of the 3graviton function r. It contains a large amount of information which can be organised into two categories: (i) it is an identity of cubic order in (II -d), and (ii) it is an identity in the various momenta. b, is independent of (q - d), and so it is determined by the corresponding part of Eq. (7.10)--which contains p^‘r,,,. Furthermore the vertex B which b, multiplies, corresponding to the topological term G in the action, has the property B(p*, p*, 0) = 0,
(7.28)
i.e., it vanishes when any one of the (vector) momenta is set to zero. On the other hand B(p*, 0,O) = 2(n - 2)(n - 3)(n - 4)(p*)*,
(7.29)
which does not vanish in n dimensions. Thus it is r444(p2, 0,O) which contains the “new” divergences, and it is precisely this quantity which is relevant to the determination of the double pole of b,. From Eq. (7.14) we also have C(p2,0,0)=4(n
+ 2)(P2)*,
(7.30)
TRACE
ANOMALIES
AND
,@’
THEORY
167
that the (r - d)-independent part of c,, also appears in the resultant equation for &-just as suggested by the remarks at the beginning of this section. The simplest composite 3-point example to consider is Tzz2. Its renormalisation is obtained by applying a/&2 to the corresponding equation for the renormalisation of rz2, and then taking the coefficient of II. To do this it is necessary to use the full curved-space version of Tzz first, rather than the flat-space version in Eq. (5.la), and allowance .must be made for the possible appearance of l-point functions containing 8(x -y) which only vanish in flat space. In effect we should really start with the full operator version of Eq. (5.4) in position-space. In the case of Tzz there are in fact no new terms because the only dimension-4 2-point operator involving 6(x-y) is 8(x -JI) itself, with an accompanying (fi)-’ for coordinate invariance. Therefore the operator ]O.:t] defined by SO
+fi”-42L* lo.3 = __ (2;)’[#.:][$;I
6(x-y) Gm
is a finite 2point composite operator in curved space, with [#.:I given by Eq. (3.9). Applying S/S0 to ([Of:]), taking the flat-space limit, and then taking the coefficient of q gives (7.32) i~“-~4L,,(f[#.z] ats(y -z) + $[#f] 8.:8(x -z)) + 28fTzzz = (finite). The first term vanishes in flat space, leaving
r,,,
= (finite)
(7.33)
just as argued above on general grounds. One way to obtain the renormalisation of r,,, is to apply S/SQ to the curved-space version of rz4. The operator version [O::] of the renormalised 2-point insertion the @“-“/4!)[&J t[&] th’is t’ime contains, in addition to the original cY’J(x -u)/\/-g, extra terms +[#.:I 6(x -JJ)/~ and ,u’-~H~(x -,v)/fi This appraoch fits into the schemeoutlined, but as the renormalisation of [O::,] was not fully investigated earlier it pays instead to examine r,,, by remaining in flat space and using the action principle with --a/a2 on rz2. Recalling (3.2), (4.8), (5.4) and (7.20) we get
i2 ((d”xiOZi ![#:I +ktl j + 7 i(ijp:l tl#l-I> - 2p”-46( WV - .z) $ L, = (finite).
(7.34)
This gives I’,,, at pr = 0, but in this instance we can use dimensional and symmetry arguments to deduce the full result:
-
2p-4
($L*1
= (finite).
(7.35)
168
S. J. HATHRELL
Then we could work backwards to deduce [O::]. The leading, L-independent contribution to rdZ2 is given in Fig. 6, and it is easy to verify by direct calculation that (7.35) is indeed correct using the values of the parameters 6, p^ and L, listed in Appendix B. In fact in this particular case the pure-pole term ((a/an) L,) must contain a double pole at O(1) but no single pole at O(1) because of the form of Fig. 6, and the absence of this single pole means that there is no O(L) contribution to /3,, (which is calculated to O(,I’) later in Section 8). We now have some consistency checks in the argument concerning the finiteness of the 3-graviton function r. The coefftcient of (v - d)3 in Eq. (7.10) appears only in the term i2(gX(?,,gZ) and just reproduces the result that r222 is finite. The coefficient of OI- 4’ can likewise be shown to reproduce Eq. (7.35), in a weaker version multiplied through by /? (This requires Eqs. (4.13) and (5.4).) There is an important point concerning infra-red (ir) divergences which is exhibited by Eq. (7.35). This equation only remains valid provided pz # 0, pz # 0 because it is a statement about the systematic subtraction of ultra-violet (uv) divergences only. T,,(pi) is a function containing uv poles, but it is regularised to zero in the dimensional scheme when pz = 0, thereby effectively introducing ir poles. These appear because the subtracted function contains ln( p’/,u’) which diverges when p2 + 0. Similarly the leading contribution to r,,,, (from Fig. 6) is regularised to zero when pz = 0, and the “finite” term containing ln(p2/p2) then diverges. On the other hand there are no ir divergences at pz = 0 in (7.35), and it remains valid for this. The “finite” terms involving ln(p:/p2) appear in the form p: ln(p:/p2) and remain finite as pi + 0. But if the whole of Eq. (7.35) is multiplied through by pi pz it then remains strictly true for all momenta; and this is just the equation for the renormalisation of the 3-point composite functions involving a’[d’] rather than [#‘I. The important point is that in the 3-graviton equation for r, l? only contains ]#‘I in the form of a’[#‘], so there are no ir divergences for this. Equation (7.10) remains valid at all momenta, and we are justified in equating poles for various zero-momentum or onshell configurations. The renormalisation of r,,, can be derived by a straightforward application of the arguments outlined, but we need not consider it as it only serves to provide a consistency check for the coefficient of (v - d). The useful result comes with rdd4, for which we apply the action principle with -a/a1 to Eq; (5.9). We also need the result (7.36) This approach only gives r444 at p, = 0, but since the terms in its renormalisation involving r44 are not multiplied by any factors of pf , we can use symmetry arguments to deduce this part of the renormalisation in full. Thus
r444(p:&d) + $- v-44(d)+r44(p:)+ r44(d)l + {pole-terms = (finite).
involving
p;r,,(p;),, pf pfr22(p:), ,unp4pfpf 1 (7.37)
TRACE
At pl =pi
ANOMALIES
AND
(p’)’
169
THEORY
= 0 it must therefore take the form
r444cp2v 0,0)+$ r44(p*) + I( In-4 )2Lfl +
,@’
/F4
P2L2(P2)
+ (-n-4P 13
uP2)*~22(P2) I
= (finite),
/(ni4,3L*f
where L,, L, and L, are unknown Laurent seriesof poles, and the various factors of (n - 4)//I? have been extracted their detinitions for convenience in their RGEs. The crucial feature is that the pole series multiplying r44 is known. Remembering that (04} differs from p”-“[$“]/4! only at O(A), it follows from the earlier arguments that the pole coefficients of r14 and Z-2* are 0( 1) and O(A) in leading order, so that the leading poles of L,, L, and L, are O(n’)/(n - 4), O(I’)/(n - 4) and O@“)/(n - 4)2. L, also contains a single pole which is O@‘)/(n - 4), but this is only next-to-leading in the sensethat it is tied by the general renormalisation structure to the 0(,14) double pole. It is now possible to deduce the value of the G term in the trace anomaly. Using the expressionsfor e,, gY2and BXyz given by Eqs. (7.12), (7.18) (7.21) (7.29) and (7.30), and the expansions of b, and c, given in (3.13) and (3.14) we set 17= d in Eq. (7.10) and obtain -p^9-444(p*, 0, 0) + 2(n - qP*r,,
- (3n - io)(d + L,)p*@-42
+ ( p2)2j.i”-4{ 2(n - 2)(n - 3)(n - 4) L, + 4(n + 2)(L, + L, + d*L,,)} = (finite). (7.39) This may be further simplified by removing p^‘r,, and bi-24, using Eqs. (4.28) (4.29) (5.6), (5.7) and (5.9)-(5.11): (~*)%“-4rw
-
w
-
3)(n - 4)L,i
-P3r444(p*, 0~0)
= (finite) + (3n - 10)(p2)* ,B’-~(~(L, + dL, + d2L,) - (d + L,)(2L, = (finite) + (p”)‘,u”-“(
(3~ - lO)(L, + dLf)].
+ 4 dL,)} (7.40)
All that remains is to find the poles in p3rqq4, and these can be determined from has a leading (7.38). r,,,q has a double pole at O(l), so superficially B’r,,, contribution to its single pole at O(A’), and this can be determined from the leading pole of r,,, with the aid of the appropriate RGE. We can seealready that the leading single-pole residue on the right-hand side of (7.40) is also O(,I’) from the contribution of L, (or L,,), so the double-pole residue of L,, b,(A), must be O@“) as well. It turns out that the renormalisation-group structure of r444(p2, 0,O) actually implies a cancellation in the 0(,15) leading pole of j?3r444, in much the sameway as the OQ4) single pole of p^‘r,, disappeared in Section 5. This is a non-trivial cancellation which would require 6-100~ calculations to verify by direct means.From
170
S. J. HATHRELL
now on we will confine attention to the leading terms to demonstrate this cancellation. The RGE relating the poles of r,,, is obtained by applying p-’ Dp^” to Eq. (7.38), and using the RGEs for [#*I and {04} given by Eqs. (3.2), (4.8) and (7.19). Thus pr44
n-4 ’ + (p2)2pn-4 --J( 1 l@-4)+Dlh
= (finite) + 0
A2
a.
(n _ 4)1 (n _ 4)2 ,... . 1 (
The contribution from the 3-point functions is removed from the leading order because of the O(A*) factor of d’ in Eq. (7.19). The significance of this is that the leading terms in the poles of L, are shown to depend only on the leading poles of I- 44, known from previous work, and have been deduced without the necessity of calculating any Feynman diagrams for r444. Multiplying (7.38) by p’ gives p3r444(p2, 0,O) + (p2)*pnP4(n
- 4)3 LT = (finite) + O(A’j){poles},
so the single pole of b3r444 is given by T,(A) + O@“). r,(A) substituting for r44 in Eq. (5.9) ,+4)+D]L,=P’T;
is obtained
by
(7.43)
LY + 0
(n-4)
(7.42)
and then integrating iteratively. The higher poles of L, are given in terms of the lowest pole Y,(A) using the RGE for L, derived earlier in Eq. (5.13). It is not necessary to know the actual values of the coefficients yet, just their order in A, so if we put
P(l) =p*lz* + O(k’), Yl(A) = cd* + O(P)
(7.44)
then (5.13) gives
Substitution
Y,(l)
= ap,A’ + O(A4),
Y’(A)
=
O(P).
(7.45)
of these into Eq. (7.43), with the known fact that
T,(A) = O(P)
(7.46a)
then gives T*(d) = gfp,~3 + O(A4), T’(A)
= fc&A4 + O(A5),
T,(n)
=
0(/l”).
(7.46b)
TRACE
ANOMALIES
AND
14”
171
THEORY
Conclusion: the leading pole of /?3T444(pz, 0,O) is not O(d’) after all, but O(L6). This is quite a striking prediction, which as mentioned above would require 6-100~ calculations to verify directly. There is, however, another prediction of this analysis which can be put to the test quite easily: the double-pole residue r,(L) is given by Eqs. (7.44) and (7.46b) solely in terms of the l-loop p-function and the leading pole of r,,. Substituting into the original equation for the renormalisation of l-444, Eq. (7.38), gives the leading order (4-100~) result L(P2*
2P* 03 0) + (n _ 4) L(P2) + (P*)*P”-~
I
5%
O(l)
cn _ 4j2 + cn _ 4j
i
= (finite)
+ WI2
(7.47)
a non-trivial prediction about the subtraction of both the leading, O(1) double pole and the ln( p’/p’) terms in the single pole of r,,,. Using the values listed in Appendix El9 and the value of r444(p2, 0,O) given by Eq. (7.3) this is indeed seento be correct. Finally, Eq. (7.40) shows that the leading contribution to b2(IZ) is after all given only by the leading contribution to cl(l), i.e., b,(l) = 2c,(A) + O(P).
(7.48)
(The mixing of pole residuescausedby the various factors of (n - 2), (3~ - IO), etc., only contributes at O(A6).) 6,(L) and p6 can then be recovered through Eqs. (7.16) and (4.11). c1(A) is determined by the analysis of Section 5 and rests on an explicit calculation of the leading pole in I-,,. The values of all these terms are listed in Appendix B. In view of the length of the argument in this section, the result in Eq. (7.48) is rather conspicuous in its simplicity. However, it only appears to be a feature of the leading order of the theory. In higher orders all the terms listed in (7.38) become relevant, and so do other terms on the right-hand side of (7.40), and so does the mixing of different pole residuescaused by functions of n. These considerations are mainly bound up with the mixing of a’[#‘] in the renormalisation of $i, and all have first effect in c, and b, at O(L6). It therefore seemsunlikely that the simplicity of the relationship between b, and c, remains beyond O@‘). One of the key points of the analysis was that it gave the leading behaviour of the double-pole residue b,, at 0(;1j). The single-pole residue and the P-function therefore have the form {constant + O(,J4)}, with the constant term remaining undetermined. We conjecture that this constant term could be found within the general flat-space approach of this paper by considering instead a l-loop, 3-point Green’s function analogous tO r, but using untraced metric derivatives -26/fi6g,, with the action 9 Calculation of rd4 is straightforward using only (A.]). The required expansion may, however, be read off from Table I as well. This calculation also provides the value of T,(L) as (-1 l/3456) (L3/(4n)8), which would be needed for the O(1”) term in 8,.
S. J. HATHRELL
172
principle instead of s/&2. It would involve functions like i2(8”B,,P,), and would avoid the (n - 4) factor multiplying b, in view of its absence from the untraced Eq. (6.5). Coupling constants could be ignored completely. This would be a rather clumsy approach, however, and it is better to take advantage of the fact that it is a lloop result and use a fully covariant position-space determination of the l-loop effective action, for example, either in the “proper-time” representation of Schwinger and Dewitt [4], or on a topologically non-trivial background [9]. The result has been obtained elsewhere [4,9, lo], and is incorporated in the values quoted at the end of the paper. This concludes the general analysis of the trace anomalies and their relationships to flat-space functions of normal products. The remainder of the paper concentrates on using the methods explained to calculate the actual values of the parameters.
8. ABSORPTIVE
PARTS AND THE CALCULATION
OFP~,P,
It was argued earlier in the paper that PA and /I, were both O(1) in leading order, and given by l-loop calculations. These leading-order, I-independent terms have been calculated by Brown and Collins [lo] in the case of /?,,, and by several authors in the case of /I,. lo In this section we determine the effect of introducing the scalar quartic self-interaction by calculating both these p-functions to the 3-100~ level and O(L’). Some of the Feynman integrals appearing at this level can not easily be evaluated directly, and for these we use a technique based on a calculation of the absorptive part of the integral [ 14, 151. The full Feynman amplitudes can be reconstructed from their absorptive parts, and this approach enables all the 3-100~ amplitudes to be obtained explicitly in closed form as a function of the space-time dimension n. In particular we find that the “difficult” amplitudes can be expressedrather simply in terms of generalised hypergeometric functions, a form which compares favourably with the infinite double sums appearing in calculations based on the use of Gegenbauer polynomials [ 191. As explained in Section 5, p,, is obtained from L, and the divergences of the 2point function Zz2. The relevant diagrams up to the 3-100~ level are shown in Fig. 7, and their values denoted Zi”‘-I?‘. Th ese include an overall factor of i, and symmetry ’ Generic operator insertions are denoted by factors, respectively, of 4, 7, ’ 8, ’ 5, i 7. double lines. Exactly the same set of diagrams appears in the calculation of /I,, but with the scalar operator insertion $[@‘I replaced by the tensor operator @“’ (cf. (6.19)); and their values denoted I,(a)-I, @. In both cases I,-I, are easily evaluated using only the result (A.l). I, is more difficult, and for this the absorptive-part approach is used. We consider Z,, by way of illustration; the argument for i(@‘“guL,) is identical. The argument is clearer if in fact all the contributions I,-I, are at first taken together. It follows from the form of Eq. (5.4) that the absorptive (imaginary) part of ” See Duff.
Ref. 17 1, for a summary.
TRACE
FIG.
ANOMALIES
7.
Diagrams
AND
Ad4
for /3, and
THEORY
173
8,.
Z,, is finite, and it also follows from the unitarity of the theory expressed in terms of a sum over intermediate states. Thus
that it can be
Im Z2*( p) = (finite)
where CN represents a complete sum over both phase-space and all intermediate particle states. The right-hand side corresponds to the various ways of cutting the diagrams in Fig. 7, replacing the cut propagators on the (zero) mass-shell, and complex-conjugating the amplitudes to the right of the cut. The result really only depends on the structure of the Feynman integrals, while the various symmetry factors or identical-particle factors match exactly, and it is actually true diagram by diagram, using the symmetry factors appropriate to that diagram. However, the tricky point concerns the ir divergences and the normalisation of the intermediate states IN). These states correspond to the cutting of the bare propagators (#0$0), and do not include factors of Z, for wave-function renormalisation. Also all diagrams containing self-energy insertions on external on-shell legs are regularised to zero. Although [d’] IS a finite operator, the states (0 I+[#‘][ N) are therefore not finite just to the extent that these factors of Z, are absent. But the integration over phase-space incorporated in C,,, contains ir divergences, and Eq. (8.1) shows that these must exactly balance the uv divergences of the unrenormalised matrix elements to give an overall finite answer. In the present case there is an ir divergence for the phase-space integral in the cut version of I, in Fig. 7, and this balances the uv divergences of the 2-particle states in some of the cuts of I,. (There is in principle a second category of ir divergence, though not present in the current examples, which may arise from putting the intermediate states on shell. For such divergences the cancellation comes instead between different cuts of the same diagram. Both categories do occur for massless QED, and the cancellation of divergences is illustrated in detail in Ref. [ 141.) For the purposes of diagram-by-diagram use of the absorptive-part analysis, however, it is sufficient just to calculate using the unrenormalised states.
174
S. J. HATHKELL
Furthermore it is only necessary to evaluate I, this way-the “direct” method suffices for the other diagrams. The full amplitudes can be reconstructed from their absorptive parts by means of a very simple dispersion relation. On dimensional grounds a k-loop amplitude Z(p) appearing as part of T,,(p) must have the form
I(p) = F(n, A)(-p’ - i&)k’“‘* - 2) +‘(,,,
A)
for some function F(n, A), so that Im I(p) Im I(p) = -F(n,
(8.2)
i-k(n-4)(~2)k(n/2-2),
= 0 for p* ( 0. Therefore
A) sin
P2 > 0,
(8.3)
and Z(p) is reconstructed from Im I(p) by the simple expedient of multiplying by ( -iCk”/sin(n/2 - 2) kn}. Th’ is clearly introduces one extra pole in each component. The correctness of the whole procedure can easily be verified for a wide variety of simple examples. As a preliminary, 2, and Z, are needed to O(L*). Z, is obtained as usual from the finiteness of the renormalised propagator (#), and
;1* ‘1
= ’ +
(4n)4
y(i.)=/?$lnZ,
1 24(n _ 4) + o(A3)y A2 =-I--. 12 (47r)
(8.5)
Z, is obtained from the finiteness of the insertion of f[#‘] into a flat-space 2-point function, for which the diagrams Vi-V, are exhibited in Fig. 8. The simple vertex V, and V, and V, are obtained using formula (A.1). When the scalar legs is just Z;‘, are on shell, qy = qi = 0, V, is easily obtained from (A.l) and (AS):
v, =z;’
f$
(p’)“-”
/I;
q2 - n/2) qn/2 - 1)’ f(4 - n) qn - 3) . 2(n - 3) r(3(n/2) - 4)
V* FIG.
v, 8.
Diagrams
for i(f[$‘]
44) to O(l.‘).
V4
(8.6)
TRACE
ANOMALIES
AND
Ad4
175
THEORY
Then Z, is fixed by incorporating the coupling constant (Appendix B), and requiring that Zf Ci Vi is finite. Hence
4 5 A2 (4x)2+-- 12 (471)
1+
1 (n-4)2
2L2 (4n)4
renormalisation
[ 1
+ O(i3).
(8.7)
and 5 /I2 A @I = (471)2 - 6(4n)4
+ W3).
There is once again a slightly tricky point involving ir divergences. The usual theory is that an off-shell Green’s function is renormalised either by including selfenergy insertions on external legs and multiplying each leg by Z;‘, or by excluding insertions ‘on external legs and multiplying each by Z,. But only the latter version remains finite if the external legs are put on the zero mass-shell. This is because the extra factor of Z;’ for each leg in the former case subtracts the extra poles in the self-energy insertions and gives finite terms in ln(q2/p2); but these logarithms diverge when q2 + 0. (An alternative view is that all the self-energy insertions are regularised to zero when q2 = 0 because they involve multiples of (q2)n’2-2, and so no extra factors of ,Z;’ are needed to subtract the poles.) The argument above gives the right values of ,Z;’ in (8.7) for all the ln(p2/p2) poles to be correctly cancelled from T2,(p2) at the 3-100~ level. Since the absence of ln(p2/p2) poles in Zz2(p2) is essential to the general argument, we will demonstrate it explicitly to O(L2). It is convenient to introduce the quantity w defined by n’2-2 Z(3 - n/2)Z-(n/2 - 1)2 = 1 + (n - 4)w. r(n - 2)
(8.9)
fr om the diagrams in Fig. 7 are easily evaluated using the The integrals II*‘-Zy’ result (A.l). Combining these with the value of Z;’ obtained from (8.7) and the renormalisation (321 of 1, from Appendix B gives
1 + (n _ 4)
[
/I2 A20 1 - 8(4n)4 + ---co (47c)4
2A2 (4r)z,
2 + O(1) + W3). I I
I, is calculated through its absorptive part. The cut diagrams Fig. 9. I,. , and I, . 2 are complex conjugates, and I,., and I,.,, each other. The 2-particle on-shell amplitude to the left of the V,, Eq. (8.6), and the 2-particle phase-space integrals [33,34]
(8.10)
I,.,-I,,, are shown in are real and equal to cut in Ii!,) is given by are given by formula
176
S. J. HATHRELL
-:-cp+=k5@+* I 5.1
I 5.3
I 5.2
FIG.
(A.6), with an additional Hence
9.
Cut diagrams
I 54
for I,.
symmetry factor of 4. (V, already included a factor of f.) Z(2 - n/2) Z(3 - n) T(n/2 - 1)” cos n?r 8(n - 3) Z(3(n/2) - 4) *
(8.11)
The 4-particle phase-space integrals in I:“‘: and Z::,j are obtained from the general result (A.lO), with an extra factor of a: T(n/2 - 1)4 B(n - 3) Z(3(n/2) - 3) Z(2n - 5) 1, (n - 2), (3w2) (3(n/2)-3),(2n-5)
The full amplitude Ii”’ ‘:^)
- 4) ; l
(8.12)
1*
can then be reconstructed from the argument of (8.2), (8.3):
= ‘;’
W-‘”
(&~3’“,2’
(P2 >3(n/2-6
w2
-
r(5
-
3(n/2))
2(n - 3)
X IT (2-+(3-n)cosnn-
(3n2;(;;;c;nL)5)
1, (n - 2), (3w2) - 4) ; 1 (3(n/2)3), (2n-5)
Expanding this in poles”
‘I3
II
(8.13)
*
and in terms of the function o introduced above gives
w’] + (finite)/
+ O(L’).
(8.14)
” The first four terms in the expansion (i.e., including the finite term) agree with the result quoted in Ref. 1191 based on the use of Gegenbauer polynomials. The first three terms only are required here and are easily extracted, but the fourth term is more easily obtained if the ,F, is first transformed to make it “Saalschutzian.” See Refs. [ 16-181.
TRACE
ANOMALIES
AND
A#”
177
THEORY
Adding this to the other terms in Eq. (8.10) shows that the poles in w do indeed disappear, and comparing with Eq. (5.4) gives 1 1 1 1 A2 1 1 L (n 4)2 -5(4n)4+-6 qi 4) 2 (4n)2 48 (47r)6 I LA = __ [ -+ I +
5 /I2 18 (4n) I (8.15)
and hence (8.16)
The higher poles in L, are consistent with the RGE (4.13). The absence of an O(L) termin /3, agrees with the prediction in Section 7. The calculation of L, and p, is essentially the same as that above, but with f[$‘] replaced b:y @” and therefore with the additional complication of numerator factors in the Feynman intregrals. The insertions into 2-point functions, epicted in Fig. 8, are now represented by the tensor functions Vy”-VgV. The simple vertex Vg’ is derived from Eq. (6.20) and is vyyq,
3q2) = ;2 vl’q,
.
92
- s:s”; - 93’;
+ (n - 2) ( p”p” - + p2f” 1 . 2(n- 1)
(8.17)
This has the properties v,“,
= 0
(8.18)
+d)-qY9:-@?:~
(8.19)
and
which vanishes when q:, q: are put on shell. It follows from the structure of the diagram V;" and Vy" in Fig. 8 that they must be functions of p” only, and so Eqs. (8.18) and (8.19) immediately imply that VP” 2
=
VU” 3
=
0
.
(8.20)
178
S. J. HATHRELL
Consequently
the corresponding
diagrams in Fig. 7 give vanishing integrals also: z(a) = ,y
= 0
2
The on-shell function
Vft” is obtained using formula (AS): 4;
C”(919
(8.21)
q2)
=
-(dryI x
ir2yp2y4
V’;“(Sl, 42)
Z-(4 - n/2) Z(n/2 - l)? Z(3 - n) Z(n - 3) 2Z-(3(n/2) - 2) I
(8.22)
Notice that these functions Vy” satisfy x4=, Zy Vy” = (finite), which is necessary because @‘” is a finite operator. Evaluation of the integrals Zy’ and Zy’ again uses only formula (A.l). Some details concerning the numerator factors in the integrand are given in Appendix C. The results are .-n z:a)
=
(417)n,2 (P
n,2 r(2
2
-
1
42)
=-my&i
I-&
(p2)3’“‘2’-41;
=-(P’)’=j T4n,2 1 + (n-4)
-
1)
WG)
-2) (8.23)
[+1+0(l)/.
.-3n
I$’ = (4;)3(“,2)
W/2
4(n - l)Z(n
Z-@I/~)~ Z(n/2 - 1) Z-@/2 - 2) Z-(4 - 3(n/2)) n(n - l)(n - 3) Z(2n - 4)
l [&&‘J
cn _ 412
-7
lb2
[ 216(4x)4+36
1
/I* -w] (47r)4
+ (finite)!
+ O(L”).
(8.24)
The evaluation of Zy) is again through its absorptive part and the cut diagrams in Fig. 9. The on-shell 2-particle state to the left of the cut in Zyi is given by Eq. (8.22), and the 2- and 4-particle phase-space integrals and the appropriate reduction of the integrands are obtained from Appendices A and C. The results for the absorptive part are 1:“; +I$=-
~;(p2)3’n/2’-4 3(n/2)-1
(471)
cos n?rZ(4 - n/2) Z-@/2 - 1)” Z(3 - n) , (8.25) 48(n - l)(n - 3) Z-(3(n/2) - 3)
TRACE
ANOMALIES
AND
@”
179
THEORY
z-@/2) - 1)4 48(n - l)(n - 3) r(3(n/2) - 3) r(2n - 4) 1on3 - 39n2 + 25n + 18 n
I
X
- + (n - 2)(2n - 5) 3FZ [ :;&;t
-
2(n
-
l>Cn
-
3, 3F2
[
n(n - l)(n - 3) (3n - 4)
;i;;)154) 3
; I]
I, cn - 2), (3w) - 3) (3tn/2) _ 2, (zn _ 4)
;’I
1, (n - 2), (3(n/2) - 2) (3(n/2)-1),(2n-3) ” Ii ’
(8.26)
(The relative complexity of this expression is a consequence of the complexity of the numerator factors in the integrand, rather than of the basic loop integral.) The full amplitude Zy) is regenerated as before by adding these two expressions together and multiplying by {-i-3”/sin 3z(n/2 - 2)}; and the expansion in poles is 1-m 5
(p*)*
p-4
=
(41r)Z 2 A2 ~04-~(4rr)4~
1
A2
1
tWte)+W’)
The doublle pole, and the single pole in w, cancel between Zip’ and I?‘. Eqs. (4.11) and (6.19) we deduce12 La=
1
120(47Q2
1 ~(n - 4)
!
[1--7
5 A2 108 (47~)
pa=-l 1-L A2 120(4x)*
1+W)i3
I .
(8.27) Recalling
(8.28) (8.29)
36 4(47~) + O(A3)
Unlike L,, L, contains no double or treble poles below O(A3). The difference is a consequence of the 26(A) term in the RGE for L,, and its absence from the otherwise identical R.GE for L,. The absence of these low order double and treble poles in L, means that the function of n premultiplying L, in Eq. (6.19) does not mix pole residues below O(A’), so there is no possible ambiguity in the value of 8, up to O(A2). In view of the fact that the double-pole residue u*(A) is O(A’) and depends only on the O@‘) term in a,(A), it might be supposed that a,(A) could be directly obtained by an argument analogous to that used for b,(A) in Section 7. This is indeed the case following a consideration of the 3-point function (B@‘“$,,), whose renormalisation ” A calculation result.
based on the use of the Gegenbauer
polynomial
formulae
of Ref. [ 191 gives the same
180
S.J. HATHRELL
involves the pole-terms in (n - 4) L,. However, there is no advantage to be gained this way because the leading, non-vanishing contribution to ((04} &‘#,,) is O(n) and dependson just those 3-100~ diagrams appearing in Fig. 7. It seemslikely that whatever route is adopted, at least in flat space, the determination of /I, ultimately rests on a calculation of these diagrams-the chief reason being that there are no factors of p^ which can be extracted from the problem using normal-product and renormalisation-group arguments.
9. A
~-LOOP
CALCULATION,AND
THE OTHER PARAMETERS
One of the most significant consequencesof the analysis of Brown and Collins [lo] was that the HZ term in the anomaly only appeared at O@‘) and not O(J”)neglecting, for the moment, the dependenceon q. The O(L”) term is obtained from Eq. (5.14), and it only depends on the 2-100~ p-function and the leading, 3-100~ divergence of r,,. A further consequenceof this, following from Eq. (4.33) is that e(L) is 0(14) and not O(L3). Since these results depend in no small part on the specific details of the renormalization-group analysis and the nature of the divergences of functions of normal products, and since the consistency of field theory in curved space is something of an open question, it is perhaps of interest that the results can be verified by direct calculation. To this end we calculated the set of amplitudes exhibited in Figs. 1Oaand b, J,-J,. Apart from showing the usefulnessof the absorptive-part approach in certain contexts, this 5-100~ calculation serves altogether three purposes. (i) With operator insertions of (n - 4) &$:/4! at each end it gives the qindependent poles of i(&?), and hence L,, to O@“). The term r10a2@iin 8 does not contribute below O(L6) becauseL, is O(L4). (ii) With an operator insertion of (n - 4)&#:/$4!) at one end and (n - 4)&#$‘4! at the other it gives the O(L4) contribution to e(A), the parameter which describes the q-independent part of the mixing of aZH in 4:. This is because the diagrams are just those appearing in the function (d/&2)@4P”[$4]/4!) in flat
FIG. 10a.
The 3- and 4-100~
integrals.
TRACE
ANOMALIES
AND
Ad4
181
THEORY
+=@= +=a++=e J,
J,
e==a= J,
“J
5
J,
b FIG.
lob.
The
space, and the finiteness of this function be precise, we have
5-100~
integrals.
provided the original definition
= (linite) + O(L”). Again the various terms in a’#’
of e(L). To
(9.1)
do not contribute
to the leading order.
(iii) With operator insertions of (n - 4) &#:/@4!) at both ends it gives the value of the function r,, up to O@*). Used in collaboration with the earlier renormalis,ation-group analysis and a 3-100~ calculation of j? from elsewhere [32], it provides the values of PC to O(L6), and e to O(L5). (In fact r,, is only needed up to O(n), the 41-100~ calculation involving only J, and J,.) To summarise, if we define the amplitudes Ji as for (i) with operator insertions of (n - 4) &$:/4! at both ends, we have 4- J.I + (p’)’
pLlnP48L, = (finite) + O@‘),
(9.2)
(e + L,) = (finite) t O(L4),
(9.3)
i=l
L
+ Ji + (p’)’
P
l
,P-4 +
iTi
6
p p,
J,
'
+
(p2)2$--4
^ (
n-4 P
2 1
L, = (finite) t O(L3).
(9.4)
182
S. J. HATHRELL
These are obtained from Eqs. (5.10), (9.1) and (5.9). The value of 8-r and the renormalisation of ,I, are given in Appendix B. The basic Sloop calculation of cs Ji therefore gives a direct determination of L,, (e + L,) and L, to O(d4) from their definitions, without assuming any relationships between them. Computation of the 3-100~ amplitude Jr, the 4-100~ amplitude J2, and the first five of the 5-100~ amplitudes J,J, is entirely straightforward, and uses only the standard TABLE Values
Coeff.of:
J,
1 ~ (n-4) 1
I
of J,-.I,
1
I4
16 (47r)’
1
(n-4)?
(n - 4)
J3
1
-72(4n)6
.k2
43204 I3 + (4n)6
35 i 864
I 24 wj
+O(IJ) J*
-3
A4
16 (47~)’
1
I’
I3
6(4n)
48(4n)6
-49
1
(576+12w
!
O(P)
+ O@“)
J,
1
/IJ
I”
40
(47?)8
(4n)8
J4
1
A4
60(4x)8
J,
0
J,
0
J,
0
-21
1
160+TW
1
W”)
m”)
W”)
O(l.“,
W”)
o(14)
W”)
W’)
W4)
O(i”)
/I4
216008 1
,I4
1080(4K)8 0
J*
1 /I4 60(4x)8
W”)
O@“)
J,
1 A4 rs(4n)8
O(A4)
OCR”,
Note.
A factor
of (-(P’)~
p”-‘/(4x)‘}
has been removed
from
each term.
TRACE
ANOMALIES
AND
liJ
THEORY
183
a FIG.
1 la.
The 4-100~
subdiagram
of J,.
integral (A.l). J, is also easily obtained using the result (8.13) for the 3-100~ subdiagram in it which has the form of I, (Fig. 7)-the remaining integrations depend onl:y on the momentum factor (p2)‘3’“‘2’P6’ in (8.13), and are again of type (A.l). The expansion of these amplitudes in poles is given in Table I. The term w appearing in the expansions is the same as that defined in the previous section, Eq. (8.9), and it contains all the “unwanted” dependence on ln( p’/p 2), Euler’s number and ln(4z). (Each amplitude includes an overall factor of i, and the symmetry factors for ,the nine diagrams are, respectively 3 L24 3 L8 9 J-16 5 r32 9 L36 3 L36 7 L36 7 18 and f.) J, is obtained through an absorptive-part calculation of the 4-100~ subdiagram in Fig. 1 la. The corresponding cut subdiagrams J,.,-J,,, are shown in Fig. 1 lb. The absorptive-part argument for this 4-100~ subdiagram works perfectly well even though the cut diagrams do not actually appear in their own right as “physical” on-shell amplitudes in the cuts of J, itself. It only matters that the Feynman propagators have the correct L-prescription, and that the various other parts of the diagram have the correct factors of i for unitarity. The symmetry factors are those appropriate to the diagram J, and not to the N-particle intermediate states of J,,,-J,.,. The loops of J,., are evaluated as usual with (A.l), and the remaining 3-particle
FIG.
595/139/l-13
1 lb.
Cut subdiagrams.
184
S. J. HATHRELL
phase-space integral factors the result is
is of type (A.8). Neglecting
A:
J,.,=
(4n)2”-, (P*)*“-’
for the moment
the symmetry
q3 - n/2)2 qn/2 - 1)’ (n - 3)* (n - 4)2 r(n - 2) z-(5(n/2) - 7) ’
(9.5)
J,., and J,., are complex conjugates. The on-shell amplitude to the left of the cut in J,,, (see Fig. 1 lc) is effectively a 3-point function, W, with only one leg on shell; i.e., kf = 0, k,, s k, + k,, and it is given in terms of ordinary hypergeometric functions
from (A.2):
4
2 ” ‘, 2r(3 - n)T(n/2 - 1)3 [if+,;;; ) - (n-4)Z-(3(n/2)-4) 2F1
w= ---2”(p (47c)”
(1 --$)I.
(9.6)
The 3-particle phase-space integrals can then be evaluated using (A.9), so that J,,,
+
Jg
3 =
2cosn~rr(3 -n)T(n/21)” (n - 4)(3n - 8)T(n - 2) r(3(n/2) - 4)*
‘itp2)2n-7
(47+-l
1, (4 -n>, (n - 2) ’ 3F2 (3 - n/2), (3(n/2) - 3) ’ ’ I
(9.7)
Note that this 3F2 has the simple expansion 1 - $(n - 4) + O(n - 4)2. Next, examination of the 5-particle phase-space integral J,., shows that there are no ir divergences, and as there are no closed loops either in the on-shell amplitudes, the whole integral must therefore be finite:
Jw = O(l).
k,
(9.8)
The value could be obtained in leading order in the expansion in (n - 4) by calculating in four rather than in n dimensions, but since it is only relevant to the third, single-pole term in the expansion of J, (J,.,-J,.3 each contain two more poles than J9.& we have left it undetermined. It would be necessary for a calculation of /I, to O(3L’) by the “indirect” route.
-c P
FIG.
1 Ic.
An on-shell
3-point
k2
function
in J,,,.
TRACE
ANOMALIES
AND
@”
185
THEORY
Finally, ,J9 is regenerated according to the arguments of the last section by adding the cut dliagrams together, multiplying the sum by {---““/sin 4n(n/2 - 2)}, performing the last trivial loop integration using (A.l), and including the overall symmetry factor of f. The appropriate expansion in poles is given in Table I along with the expansions fo_r the other diagrams. Adding all these together and multiplying by inverse powers of /3 give the results required for Eqs. (9.2)-(9.4):
;7 J,= _ (PZYP4 -
1
i=l
(W
+ (n - 4)3 O(A’) + O(L”)
i
+
P
Ji
=
-
1 +
, I
)I (9.9)
(lo’)’ Pn-4 (4n)4
iG
-1081 1 12,960 +16O
&j’ol+[&&+&
432(4a)2+
I 1
A2 (47r)4
(9.10)
1 I
+ (finite) + O(L3)
I
.
(9.11)
As required by the arguments of Sections 4 and 5, the O(L4) pole has vanished from (9.9), the O(L2) single pole and the O(L”) double pole have vanished from (9.10), and all the poles in w have vanished from all three expressions. Comparison with Eqs. (9.2)-(9.4) gives L, = O(#q,
(9.12)
186
S. J. HATHRELL
17 L4 e(A) = ----1. 12,960 (4n)
(9.13)
+ W%
L, = O(P),
A2
Yl(L) = J-
432 (4n)6
(9.14) 7 A3 - -7 + 0(/l”). 1728 (47~)
(9.15)
In addition, the pole residues in (9.11) satisfy the RGE (5.13). More than predicting (9.12), however, the arguments of Section 5 showed that L, and e were given in terms of Y, and the standard P-function. According to Eqs. (5.11) and (5.14), L, is given to O(L”) by the 3-100~ p-function [ 321, by Y, to O(L3), and by the leading values of d, X, and X,; then Eq. (4.33) gives e to O(A5) using also the leading values of /?,, p, and f: The only “new” information provided by the calculation in this section is the 4-100~ result contained in Eq. (9.15), and the 5-100~ calculation contained in Eq. (9.13) serves only to show consistency with thefirst term in the expansion in (9.15). Clearly the arguments [lo] of Section 5 provide a much more efficient route to the values of the terms in the anomaly than does direct calculation. The leading-order values of the terms d, Xi, X2, ,!?,, /?, and f are readily determined from the definitions and relations earlier in the section, the calculations requiring only the use of (A.l) and (A.5). In brief, d is determined by the requirement that the divergences of the graphs in Fig. 1 should cancel, yielding the result
dd-
A3
6 + O(A”). 36 (471)
Hence Eqs. (4.32) and (8.8) give
+ O(A’).
(9.17)
From the RGE (4.9) Ll=-L= From Eqs. (5.6), (see Fig. 3b),
(5.8)
1 Q-4)
1 yq$p
and a calculation
L4
1
+O(A5).
of the function
(9.18) r24 in leading order
f-=x&- 36 m ,I2 +O@“). So from Eq. (4.34), the RGE (4.14) for L,, and the results of Section 8 for p,, , L,: (9.20)
P, = W”), LK=
l-(n-4)
[O@“)]+ (n-4)2l
A4 [ T&p
1+O(k5).
(9.21)
TRACE ANOMALIES
AND&h'THEoRy
187
Thus Eq. (5.15) gives
(9.22) The fact that the double pole of L, is O(A”), and determined in leading order, whereas the single pole and j3, vanish to O(A3), suggeststhat the cancellation is “accidental,” unlike the other cancellations found so far. The values of Pbr /?, and e may now be found to high orders of A. First, using the 3-100~p-function and the values of the various other parameters now established,L, and hence /?, can be deducedI from Eqs. (5.11) and (5.14): c,@) =
17 A5 A6 - 14 103,680 m (471)
-17 A5
A6
(4n)
5191
4,354,560 5191
622,080
+
+ O(k’),
+1 C(3)) + O(A’>. 288
(9.23) (9.24)
From Eq. (4.33) we next obtain 17 II” e(l)=-y-7 12,960 (4n)
/I5
(9.25)
(471)
The first term of this is confirmed by the results of the direct calculation given in Eq. (9.13). Last of all, L, and Pb are found by combining the A-independent result from elsewhere 14, 7. 9, lo] with the results of Section 7. Eq. (7.48), and the RGE (4.11): 17 /I4 1 +288m+0’L5)
L,= + 2(n - 4)2 [&$]
P/7=360(4~)~ ’
+O(n”)j.
1 (9.26) (9.27)
I
10. SUMMARY
AND DISCUSSION
We have shown how to extend the analysis of Brown and Collins to include the determination of the coefficients of the F and G terms in the trace anomaly. With this extension all the terms can be found by relating them to flat-space functions of renorI’ In agreement
with
Ref. [lo],
but recall
footnote
2.
188
S. J. HATHRELL
malised composite operators, or “normal products,” in the massless theory. In particular the coefficient of the F term, a,, depends especially on the traceless symmetric part of the energy-momentum tensor of the scalar field, &I”“, and can be found in perturbation theory from the amplitude i(&“$,,,). The coefficient 6, of the topological G term can be found from a consideration of 3-point functions of normal products. Moreover the leading L-dependent terms in b, depend only on the leading poles in c,,, the coefficient of the Hz (i.e., the R*) term, and can be found without the necessity of calculating any 3-point functions. This is a consequence of the renormalisation-group structure of the theory, which ensures a cancellation in the O@“) poles of the 3-point function i*(&W), in much the same way as the 0(14) poles of the 2-point function i(&9) are cancelled. In order to calculate the actual values of a number of the coefficients in the anomaly, as well as to verify the consistency of the general arguments based on the renormalisation-group analysis, an absorptive-part argument was used and extended to provide analytic, closed-form expressions for certain classes of multiloop Feynman diagrams. These expressions involve only gamma functions and generalised hypergeometric functions of the space-time dimension n, and provide a convenient alternative to the double series appearing in evaluations based on the use of Gegenbauer polynomials. Combining these multiloop integrals with the general renormalisation arguments enabled calculation of higher order, l-dependent corrections to /I,, Pb, p,, p,, and e(A). A direct 5-100~ calculation of the leading term in e(L) agreed with the value obtained through the more efficient but indirect arguments, and the same calculation also directly confirmed the vanishing of the O(L”) term in the HZ anomaly. Higher order corrections have not been calculated for d(jL) or f(L), nor has the leading O@“) term in /I, been found. Although there seems to be no obvious reason for /I, to vanish, it would be interesting if this could be confirmedparticularly as in the massive theory it is relevant to the renormalisation of the Einstein gravitational action [lo]. The value appears to depend unavoidably on the O(L4) term in d(L), and the next-to-leading terms in the 2-point function r24. All three terms d, f and /I, are closely related to each other and to the mixing of the operator 8’(4’] in the renormalisation of 4:. We have found, then, that the trace anomaly is given in terms of manifestly finite quantities by the expression
e= -/P 4-“M41 4r. + I +,P-~{~,F+&G+
(1-~-r)[~l+(tl-d)~‘[~‘]~ &+@,
+v2Pn)H2
+/@,+@)+H[#‘]I + (4c-e-rlf)a*H).
(10.1)
The definitions of all these terms are given in Section 2 and 3, and the values of the coefficients are listed in Appendix B. The whole expression is independent of the unit of mass ,u by construction, and it is in a form suitable for taking the limit as n -+ 4. In this sense it is a regularisation-independent result, although the expression of the normal products in terms of the bare operators of the theory depends upon some
TRACE
ANOMALIES
AND
189
Ad4 THEORY
regularisation and renormalisation prescription. i4 Through the renormalisation-group equations the finite “anomalous” coefficients provide complete information about the renormalisation of the parameters describing the whole scalar theory on a general curved background. The expression in the first curly bracket is the remainder when the curvature is set to zero, and is then the ordinary flat-space trace anomaly for @” theory [35, 361. It is interesting to investigate the parameter q a little more closely. It was introduced in Section 3 as an independent finite parameter, but obeying the inhomogeneous RGE (10.2) CD - @7= P,(A). If we split 7 into two parts, II = W)
(10.3)
+ $9
where f obeys the corresponding homogeneous RGE, then we can solve (10.2) in the n = 4 limit. Using the values given in Appendix B we have -1 A3 - -+ O@“>, ’ = 288 (47r)6 t=
kP(1
+ O(l)),
(10.5)
where k is now a truly independent, arbitrary constant. The value of k must be determined in principle by experiment, but the non-perturbative aspect of 7 is clearly shown by (10.5) and it would be interesting to examine the consequences of this term for non-zero k in more depth. Finally, we can compare these results for the renormalisation of the R#’ term with the related results of Birrell [37], and Bunch and Panangaden [ 3 11. They obtained
si,= rR- ~[(5,-&$ (n_ 4) l + (n - 4)’ so that even if (corresponding to disagree with f= 0) appears
[2
+G-(4)&J
~R-&$]+o(i~)~
(10.6)
the renormalised coupling CR is chosen as b, the bare coupling & to (< + qO) here) contains a pole at O(L’). At first sight this appears the arguments in this section suggesting that the first pole in no (with at 0(14), but the disagreement is illusory: if we write
(n- 2) c-+I;lo=5R+ (4(np1)-5X+% 1 1 I4 Also the value of the 3-100~ p-function, and hence the values it, depends upon the renormalisation prescription. See Ref. [321.
of the other
parameters
derived
from
190
S.J.HATHRELL
then the poles appearing in Z,( (n - 2)/4(n - 1) - & + rO} to 0(3,‘)-which multiplies the finite operator R [$‘I-are just the poles appearing in (10.6). The O(I’) pole in C$for & = i therefore appears as a consequenceof regularising < to d rather than (n - 2)/4(n - l), and really only depends on Z,. There seems to be no advantage in breaking the conformal invariance of the L = 0 theory this way when it can be maintained using the regularisation (n - 2)/4(n - l), and then the introduction of L only forces this symmetry to be broken by a pole at O(1”). The “true” effect on the trace anomaly is a finite O(L”) contribution multiplying a’[$‘].
APPENDIX
A: FEYNMAN
INTEGRALS
All integrals or subintegrals involving only two invariant momenta in the integrand can be evaluated using the well-known result
i$
(k2)’((k+ s>‘>” - s- 42) e/2 + f-1w2 + $1 , (A.1) (42>n,2+r+s q--r I-(--r) z-(-s) II z-(n+r+s) ! I
where the usual is-prescription in Minkowski space is implied. The factor i’-” is particularly important to the absorptive-part analysis, and there is one such factor for each closed loop. The terms in curly brackets exhibit, respectively, the ultra-violet and the infra-red behaviour of the integral. Notice that any symmetry factors, vertex factors or factors of i for propagators have to be included separately as appropriate, both here and for subsequentformulae. The 3-point functions with one external momentum on shell can be evaluated using
ig
(k2)’ ((k +p>‘)” ((k + d2)’
(A-2) where p = k, + q and k: = 0. (A. 1) is a special caseof this when k, = 0. The function ,F, is a standard hypergeometric function with seriesexpansion 2
F1
(A.31
TRACE
Here (a), is Pochammer’s (a>, =
ANOMALIES
AND
Ad4
191
THEORY
symbol: r(u + m) qu) (A.4)
=a@+
I)***
(u+m-
1)
for
m>l.
The ,F, in (A.2) can be summed in a number of cases (including when s = t = -1). A special case of particular interest is when a second leg is put on shell, i.e., p=k,+k:t, k2-1- kg = 0. Then
Is
[(k - k2)2]’
[(k + k,)2]”
(k’)’
= -$+
(P~)~“+~+‘+’
x
l--r I
x
The 2-particle phase-space
- s - t - n/2) I-(-r) l--s) I
T(n/2+r+t)T(n/2+s+t) r(n + r + s + t) I
integrals can be performed
I-
using
(P2)“‘2-2 T(n/2 - 1) + (271)n J dp(2) = (4x)n/2- 1 4r(n _ 2) ) where the :Lorentz-invariant
(A.51
(A-6)
phase-space for N particles is given by [33, 341 (‘4.7)
and kf = c~f - kf = 0. The phase-space vanishes for p2 < 0, so the integrals carry an implicit step-function. The 3-particle phase-space integrals encountered here can be evaluated using either + (2~)”
I’ W)Kp - kA21’ [(p - kJ21S
qn/2+r-l)qn/2+s-l)qn/2-1) = (p2)r+s+n-3 4r(n - 2) T(r + s + 3(n/2) (4?r)“- ’
- 3)
’
(A.8)
or
= (P’Y’ (4x)“-’
T(n/2 - 1)’ * 4r(n - 2)I-(3(n/2)
(n - 2)nl - 3) * (3(n/2) - 3), *
64.9)
192
S.J.HATHRELL
A general result for 4-particle phase-space integrations, needed here, is + (2+ =
J dp(4)[(k,
+ k4)T
(p2)rtst1t3w2)-4
[(P - k)T
[(P -
which includes all those
k,)*i’
T(n/2-1)3z-(r+n/2-1)~(r+s+n-2) 4Z+r-2)T(r+3@/2)-3)T(r+s+2n-4)
3(x/2)--1
(47r)
3F2
-t,(n - 2),(r+s + 3(n/2)- 3) (A.lO) ' [ (r+s + 2n -4),(r+ 3(n/2)- 3) " I ' The function 3F2 is a generalised hypergeometric function with definition analogous to (A.3): 3F2
a, b, c = 5 (4, W, (4, zm [ 1 d,e
-. ( m!
;z
,zo
(4,
(4,
(A.1 1)
1
There are a number of standard results known [ 16-181 about this function, especially when z = 1. When either or both of s, t are non-negative integers the 3F2 in (A.lO) can be reduced to a product or (terminating) sum of products of gamma functions only. This corresponds to the case when the original Feynman integrals can be expressed in terms of integrals and subintegrals of type (A.l) only, and then the result (A.lO) combined with the rest of the absorptive-part argument reproduces the correct answers. For example, the value of J, in Fig. 10a can be obtained from (A.lO) when r = s = t = 0. By construction the right-hand side of (A.lO) must be symmetric in s and t, but the direct proof of this seems to be non-trivial and we have only been able to confirm it for some particular cases. APPENDIX
B: VALUES OF THE PARAMETERS
The new results obtained in this paper are 1 1 PA = - T .- (4n)*
1 A2 -+ I+ s (47r)4
O@‘)
I
,
(B.1) P.2) (B.3) 03.4) (B-5)
TRACE
ANOMALIES
AND
A#”
193
THEORY
Previous results gave only the first term of each expansion. For convenience we also list the values of the other finite parameters, and some of the pole series. A2 17 1’ + O(O, q + 121(3)) & p = 3 (47# -3(4x)4+ ( A
5 k2 6 (4~)
J=pj--4
+ O(L”>,
,I3 + O(J4), dL-36 (471)6
p = (n - 4)L + P(A), & = -(n - 4)a + P,(A), 6, = -(n - 4)b + B&L PC = -(n - 4) + P,(A), q, a, b, c are independent finite parameters.
1 1
d3*7+-T (4x)
+(n [ A1 _
zl=l+i
z;l
4)3
9
(4n)4
+
17 I 3 (4n)
1
W’),
,&$,+o@3,,
= l+
1 -I. (n - 4) [ (47q
5 + 12
A2 -]+&)[2&]
(47r)4
+w3)?
194
La=--
S.J.HATHRELL
1 1 1 ~ 120 (47c)2 ! (n-4)
5 I2 + W’) l--108 (47~) I [ + O(P)
A2 m
7 A3 -4(4n)6
I
7
1
+ O(A4) + O(A”> . I !
Other pole series can be constructed from the finite parameters above and the equations given earlier.
APPENDIX
C: SOME DETAILS
OF THE CALCULATIONS
The simplification of the numerator factors in the integrands of Ip’-Zy’ and appearing in i(P”B,,) can be derived from the following general case (see Figs. 7 and 9):
It&I:(f:
= 4;n--:) KP=)= + @:I=+ @:I=+ ((P- k,)=)’+ ((P- kJ=)= I - 2p=(k: + k: + (P -k,)’
+ (P - k,)*)l
+ (q*)=+p*q= - q=[k: + k: + (p -k,)* + (P - kA*] +q
[k;k;+k;(p-k,)=+k;(p-k,)=
+ 2(nn 1>[k:(p - k)= + k:(p - kJ=l 13
+(P+)=
(P--J’]
cc.11
TRACE ANOMALIES
195
AND,@'THEORY
where q”=p”--y-k@
(C.2)
23
and q“ is the momentum flowing “down the middle.” (Cl) is arranged so that the right-hand side contains only the invariant momenta which appear in the denominators of the Feynman integrals. The evaluation of Z:(2)and Zy’ requires (C. 1) with qw = 0, kz = (p - k,)“. The absorptive-part calculations in Z:‘fi and Ziyi also use (C.1) with qL1= 0, and in addition kf = k: = 0, leaving only the single term ((n - 2)/(4(n - l))(~~)~. The absorptive-part cuts in I,., and Z5,4use (C.1) and (C.2) again with the on-shell condition k: = kz = 0; and with q’ = k: + k:, k: = k: = 0. The evaluation of I,(‘) based instead upon the use of Gegenbauer polynomials requires all the terms in (C.l) as they stand.
APPENDIX
D: DEFINITIONS,
CONVENTIONS
AND SOME USEFUL RESULTS
The curvature tensor is defined by
so that
V,i:,,r- V,,:,, = RK,tw V, for a covariant vector V,. The metric tensor g,,,, has signature (+---),
and
rxr” = WY g&l,” + &?A”,@ - &“.il)’ The Ricci tensor and scalar are defined by R,” = R”,,,,, R = g”“R,“, and in the weak-field limit with g,,(x) = q,, + h,,(x) these curvature terms have the expansion
Functional derivatives with respect to g,, are most conveniently obtained by using Riemann normal coordinates, in which TX,,,, = 0, since while ZKLlcdoes not transform like a tensor, i3ZKctUdoes. Hence
196
S.J. HATHRELL
Also
In particular,
Two further standard results which prove useful here are -1 =-&.,(, sin(n/2 - 2) kn used in conjunction given by
(+)T(k(+)
with the absorptive-part
lnQl+c)=---YE+
g
+l),
analysis; and the expansion of r(l + E)
(-)m*Emm, C(m)
m=2
where y is Euler’s constant (zO.58), and
ACKNOWLEDGMENTS I am grateful to I. T. Drummond for many conversations and useful suggestions, and to the U. K. Science Research Council for supporting part of this work.
REFERENCES 1. D. M. CAPPER AND M. J. DUFF, Nuouo Cimenfo A 23 (1974), 173. 2. D. M. CAPPER, M. J. DUFF, AND L. HALPERN, Phys. Rev. D 10 (1974), 3. S. DESER, M. J. DUFF, AND C. J. ISHAM, Nucl. Phys. B 111 (1976), 45. 4. L. S. BROWN, Phys. Rev. D 15 (1977), 1469. 5. A. DUNCAN, Phys. Left. B 66(2) (1977), 170. 6. H. S. TSAO, Phys. Lett. B 68 (1977), 79. 7. M. J. DUFF, Nucl. Phys. B 125 (1977), 334. 8. N. D. BIRRELL
AND
P. C. W.
DAVIES,
Phys. Rev.
D 18
(1978), 4408.
461.
TRACE
9. I. T. DRUMMOND
AND
G. M.
ANOMALIES
Phys.
SHORE,
AND J. C. COLLINS, AND M. VELTMAN,
Princeton Nucl. Phys.
AND
,+$”
197
THEORY
Rev. D 19 (1979),
1134.
10. 11.
L. S. BROWN G. ‘THOOFT
12. 13.
C. G. BNDLLINI G. M. C’ICUTA
14. 15.
I. T. DRUMMOND T. CURTRIGHT.
16. 17. 18. 19.
Cambridge Univ. Press, Cambridge, L. J. SL.4TER, “Generalised Hypergeometric Functions,” Vol. 3, McGraw-Hill, New York, A. ERDELYI et al., “The Bateman Manuscript Project,” W. N. BAILEY, “Generalised Hypergeometric Series,” Cambridge Univ. Press, Cambridge. K. G. CHETYRKIN AND F. V. TKACHOV, preprint INR, II-125 (Moscow, 1979).
20. 21.
J. C. COLLINS. Phys. Rev. D lO(4) (1974), 1213. P. BREITENLOHNER AND D. MAISON, Commun. Math.
22. 23.
J. C. COLLINS, L. S. BROWN,
24. 25.
J. C. COLLINS. W. ZIMMERMAN,
26. 27.
Grisaru, and H. Pendleton, Eds.), Vol. I, MIT Press, Cambridge, J. H. LOWENSTEIN, Commun. Math. Phys. 24 (I 97 1 ), 1. J. C. COLLINS, PhJw. Rev. D 14(8) (1976), 1965.
28. 29. 30.
G. ‘T HOOFT, J. C. COLLINS, T. S. B~INCH,
1980. B 44 (1972), 189.
Phys. Lett. B 40 (1972), 566. Left. Nuovo Cimento 4 (1972), 329. AND S. J. HATHRELL. Phys. Rev. D 21(4) (1980), 958. Phys. Rev. D 21(6) (1980), 1543.
AND AND
J. J. GIAMBIAGI, E. MONTALDI,
Phys.
52 (1977).
11, 39, and
1966. 1955. 1935.
55.
Phys. B 80 (1974), 341. Ann. Phys. (N.Y.) 126(l) (1980), 135. Nucl. Phys. B 92(3) (1975), 477. Nucl.
“Lecture
on
Elementary
Particles
Nucl. Phvs. B 61 (1973), 455. Nuovo Cimenta A 25(l) (1975), P. PANANGADEN. AND L. PARKER,
3 1. T. S. BUNCH AND 32. A. A. VLADIMIROV,
P. PANANGADEN,
Theor.
Math.
J. Phys.
Phys.
33. 34.
R. GAslrM.&xs AND R. MEULDERMANS, W. J. MARCIANO AND A. SIRLIN. Nucl.
35. 36.
J. H. LOWENSTEIN, B. SCHROER. Lea.
37.
N.
D. BIRRELL.
preprint,
and
Quantum
47. J. Phys. A 13 (1980),
A 13 (1980),
919.
36 (1978), 732. Phys. B 63 (1973), Phys. B 88 (1975). 86.
Nurl.
Phys. Rev. D 4 (1971). 2281. Nuovo Cimento 2 (1971). 867. J. Phys. A 13 (1980). 569.
Field Mass.,
271.
Theory” 1970.
901.
(S.
Deser,
M.