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Trace anomalies and QED in curved space
ANNALS
OF PHYSICS
142, 34-63 (1982)
Trace Anomalies
and QED in Curved
Space
S. J. HATHRELL Department University
of Applied Mathematics and Theoretical Physics, of Cambridge, Cambridge CB3-9E W, England
Received September 9, 1981; revised February 1, 1982
It is shown, for massless QED in a weakly curved background, how to obtain full information about the trace anomaly in perturbation theory, including the “topological” term in the gravitational part of the anomaly. The arguments used follow as straightforward adaptations of those presented recently for A)” theory, and rely on a renormalisation-group analysis combined with strong connections between renormalisability of the curved space theory, finiteness of the energy-momentum tensor, and the use of normal products. The first non-zero a-dependent terms appear at O(a’) for the topological part, and at O(a’) for the non-conformal R* part of the anomaly. Both values can be deduced via the renormalisationgroup arguments from simple flat Feynman diagrams. A direct 3-100~ calculation confirms the vanishing of the O(a’) term in the R* anomaly. The analysis follows the scalar theory closely, while being simpler in many places.
1. INTRODUCTORY
REMARKS
Having established a framework for the investigation of the trace anomalies for a self-interacting scalar theory [ 11, we wish to see whether the same ideas can be applied to theories of a more directly practical relevance. The obvious first choice is quantum electrodynamics, not only becausethe theory is experimentally well verified in many circumstances, but also because as a gauge theory it is the simplest prototype of a wide range of such theories. The question of compatibility of QED and gravitation has not of course been fully answered yet, but it seemsreasonable to investigate QED on a curved background as a possible approximation, and it seems equally reasonable to choose the simplest coordinate-invariant extension available in the absenceof arguments to the contrary. There are two obvious points of contrast with the scalar theory. First, we now have a theory containing not one type of field but two different mutually interacting fields. Second, QED possessesa local gauge symmetry with respect to the background manifold. It turns out that the former does not significantly complicate the investigation, while the gauge symmetry actually constrains the theory sufficiently to make it substantially simpler in places than its scalar counterpart. There are, however, minor complications of detail for QED which have to be considered in the process, but which do not affect the genera1line of argument. They are principally the necessity for vierbein fields, and the necessity of a gauge-fixing term to define the 34 0003-49 16/82/090034-30$05.@0/0 Copyright All rights
0 1982 by Academic Press, Inc. of reproduction in any form reserved.
TRACE ANOMALIES
35
perturbation theory. These are considered in Section 2, where the theory is defined in more detail. In Section 3 the renormalisation of the parameters in the theory is determined, and explicit forms are found for the renormalised composite operators-“normal products.” This enables the trace of the energy-momentum tensor to be given as a manifestly finite expression. In Section 4 we introduce twopoint functions of normal products, and show how to derive the renormalisation of the F part of the anomaly to O(e’) and the R2 part to O(e”). The latter involves a cancellation of the O(e”) term specifically predicted by the renormalisation-group structure of the theory, closely analogous to the behaviour of the scalar theory in Ref. [ 11. In Section 5 we extend the analysis to three-point functions of normal products, and show how to derive the renormalisation of the topological, G term to O(e”) from flat-space calculations. Again the renormalisation-group structure of the theory entails a particular cancellation of the leading divergence of a certain threepoint function. The next sections are concerned with more specific details of the calculations of the parameters describing the anomaly. In Section 6 we set up the Feynman rules and incorporate the results of an earlier investigation’ [2] to calculate the F term to O(e’). The cancellation of infra-red divergences was displayed explicitly in this paper. In Section 7 we calculate the R* term to O(e”) using the indirect methods, and then verify the cancellation of the O(e”) term by a direct 3-100~ calculation. Using the arguments of Section 5 this result also gives the G term to O(e4). Finally in Section 8 we discuss the results, with particular emphasis on the status of the #R term in the anomaly.
2. DESCRIPTION
OF THE THEORY
The action S is given by (2*1)
where the curvature terms in the Lagrangian are ~g=ua,F+b,G+c,H2,
and the matter Lagrangian
(2.2)
is given by Pm = -$F~~,F;u + ftji,iSy,.
(2.3)
The terms F, G and H* are as defined in Eqs. (2.9), (2.10) and (2.7) of Ref. [ 11, and ’ The conjecture about the nature of the RZ term in this earlier paper was erroneous, but the calculation of the F term presented remains valid.
36
S. J. HATHRELL
a,, b, and c,, are bare coupling constants whose renormalisation tromagnetic charge e. The covariant derivative 3 is given by B = yaeo”(x)(iiw
- l5J
depends on the elec(2.4)
and e,l(x) is a vierbein field relating the coordinates of the manifold to an orthonormal tetrad for the Dirac gamma-matrices. Its relationship to the metric tensor g,, is described by
e,,@,= g,,.(x),
(2.5)
The connections in the covariant derivative for the vierbein and spinor labels can be determined sequentially by requiring that the covariant derivatives of eou and then of y” should vanish, giving
where 0 rob
=
eaA{ebA,M
+
rApL&,l’}T
u ab = $ [y”, yb].
(2.7) (2.8)
P I1v is the usual Levi-Civita connection for the manifold, and A,, can be any vector quantity which commutes with y”--in this case of course it is the potential for the electromagnetic field. e, is the bare charge. As for the scalar theory there is no mass term, nor any Einstein or cosmological term, because their contributions to the trace of the energy-momentum tensor are not “anomalous” and they can consistently be taken to vanish with dimensional renormalisation. When combined with the gauge-invariance of the action this means that the number of possible terms in rP, and Yg with the correct symmetries is actually smaller than for the scalar case. Notably there is no analogue of the Rqh2term which was responsible for so many of the complications. To define the perturbation theory, however, we must break the gauge-invariance and introduce a gauge-fixing term. It actually proves quite convenient to break the coordinate-invariance at the same time and use the coordinate-dependent density p(A) given by S,.,. =$l
\/-g~(Ah
(2.9)
0
P(A) = &
W‘l’%,Aoo)2.
(2.10)
The useful features of this choice become apparent if we briefly summarise the usual,
TRACE
31
ANOMALIES
formal, path-integral arguments leading to this term. No claims of rigour are made for the argument. The vacuum generating functional Z[ g,,,] is multiplied by the quantity
J. (do) d(A)
Gdf(A,) - B) = 1,
(2.11)
= ?‘“a,&>,
(2.12)
where f(A”)
inserted into the integral over the gauge potential, and the integral in (2.11) is over the U(1) gauge group. d(A) is the functional determinant (2.13) The gauge-fixing parameter C;, is then introduced in a second multiplicative
factor (2.14)
Combining
(2.11) and (2.14) we get a multiplicative [ (do) d(A) eis”,f.
factor (2.15)
which is inserted into the integration over the gauge potential A. The functional determinant can then be expressed as an integral over ghost fields, and the integration over the gauge group is factored out of the connected amplitudes. From these arguments we see the following: (i) The gauge-fixing term S,.r, and ghost contribution in d(A) are both independent of the metric g,,L,. Therefore functional derivatives of the action with respect to the metric generate insertions of the canonical gauge-invariant and coordinate-covariant energy-momentum tensor. (ii) The ghosts are decoupled from both the gauge-field and the background metric, and are completely factored out of the connected amplitudes. They need not be considered any further. (iii) The gauge-fixing term gives the usual Feynman rules for calculation in the flat-space limit. (iv) The multiplicative nature of the gauge-fixing process leading to (2.15). emphasized in the argument, leads to orthodox treatment for the Ward identities for connected amplitudes. In particular, connected Green’s functions derived from Z[ g,,] by functional derivatives with respect to gUL, must be gauge-invariant and independent of <,, .
38
S.J.HATHRELL
We make the usual assumption that action (2.1) generates finite Green’s functions, and that the renormalisation of Green’s functions containing external legs is determined by the flat-space limit. That is, in the minimal subtraction scheme of dimensional renormalisation, and in accordance with the usual flat-space theory, we supply a factor of Z;‘j2 for each external gauge-field leg and a factor of Z;‘12 for each external fermion leg. From the Ward identities we get the usual charge renormalisation e,=P 2-nIZez; l/2,
(2.16)
where p is the ‘t Hooft unit of mass. The gauge-fixing parameter 5, is renormalised by
To= z, 6
(2.17)
Z, has the usual form { 1 + C (poles)} and the residues are functions of e2 only, but Z, depends on < as well. All the bare quantities in the action are defined to be independent of ,u. Functional derivatives with respect to the metric are generalised, where necessary, to include the vierbein tield by (2.18) and the contraction of these with g,, gives the scaling derivative S/&II in the notation of Ref. [ 11. The trace of the energy-momentum tensor is obtained in bare form by applying 6/&2 to action (2.1)-(2.3)-the gauge-fixing term does not contribute, as argued above-and ee*,y
=$=
{d(n-4)E;,,,F;“-$(n-
+ {-(n - 4)(a,F+
l>P,i&,)
b,G + coH2) + 4c, a2H}.
(2.19)
This is the expression for the anomaly in bare form, and to express it in a form suitable for taking the limit n + 4 we need to find it in terms of finite operators. Just as for the scalar theory 19must be finite as a whole (given the initial assumptions), since its insertions into finite Green’s functions are generated through the renormalised action principle with a finite functional derivative. We must therefore turn to the normal products in the theory.
3. RENORMALISATION
AND NORMAL
PRODUCTS
The connected amplitudes containing only insertions of 0’” are independent of <, so the renormalised action principle with <~?/a< on such amplitudes produces zero.
TRACE ANOMALIES
39
From this we deduce
(3.2)
The former is a Ward identity for an insertion of the gauge-fixing term, and the latter shows that the renormalisation of the parameters a,, b, and c0 depends only on e*. We will actually find it convenient to work directly with the fine-structure constant* a = e2/4z We therefore have (Z)
=P
(ZJ.
(3.3)
where a, b and c are independent finite parameters, while L,, L, and L, are Laurent series of poles whose residues are functions of a only, (3.4)
and similar expressions for L,, L,. The notation is the natural adaptation of the notation in Ref. [ 11, and will be used subsequently in an entirely systematic way for other series of poles appearing in the analysis. Although the energy-momentum trace 0 contains only gauge-invariant operators, the route to the determination of the normal products involves considering the action principle applied to general off-shell Green’s functions containing external gauge-field legs, site the normal products are just those operators which are finite when inserted into such Green’s functions. Off-shell Green’s functions with external gauge-field legs are not gauge-invariant, so the derivation involves some gauge-dependent operators as well. These must disappear from the final expression for the gauge-invariant normal products, however. We consider first therefore the dimension 4 “equation-of-motion” operator [E,] for the electromagnetic field, which is not gauge-invariant. Only the integrated version is needed:
(3.5)
* Strictly, this is only an intermediate-renormalised parameter. The physical constant corresponds to on-shell renormalisation in the massive theory in the low-energy limit.
40
S. .I. HATHRELL
This is the only operator obtainable directly through the renormalised action principle which contains a term involving Fo,,Fou’“. It is finite and produces a multiplicative factor of iNY when inserted into Green’s functions containing ‘N, external A-fields, for reasons analogous to those in the scalar theory, Eq. (4.1) of Ref. [ 11: it is the curved-space version of the photon counting identity [3,4]. The notation [ ] is again used here and subsequently to denote normal products. The other term in expression (2.19) for 8 which explicitly involves the matter fields is tjT,,i&,,, and this is given by the other equation-of-motion operator. We define for convenience the notation
Then we have (3.7) [E,] is (formally) shown to be finite in Green’s functions using once again a functional integration by parts as in Eq. (4.1) of Ref. [ 11. When integrated over all space it is just the curved-space version of the fermion counting identity [3,4]: )‘fi[E,]=iN,.
(3.8)
That is, when inserted into a Green’s function of elementary fields it multiplies the function by i times the number of external fermion lines. It is apparent from (3.5) that we must also consider the normal product for the gauge-fixing term.3 This is found by applying -it a/a< to an elementary Green’s function containing NY external photons and NO external fermions, which produces an insertion of {@B/C?<) + (i/2) N,&~/cF?~) In Z,}. This must be finite, and SO
defining [p(A)], as the right-hand side has the correct form for this normal product in the minimal subtraction scheme. Next we consider the effect of the finite derivative a a/&, as this produces insertions involving e, Iv&,, v/,,, and by substitution with (3.5) leads to an expression for [F,,F”“]. This derivative is defined holding the renormalised parameters finite, so r The analysis in this section has several features in common with Ref. 141, in which the finiteness of the flat-space trace anomaly is proved. The significant differences are, however, (i) that here we have the full curved-space theory, and (ii) finiteness of the energy-momentum tensor is immediate following the assumption of renormalisability of the curved-space theory, and is used instead to generate other results.
41
TRACE ANOMALIES
we need to know how the bare parameters vary with a. We must therefore turn to the renormalisation-group equations [5] (RGEs) for this information. As in Ref. [l] the RGEs are obtained by applying the differential operator D, (3.10)
where the bare parameters e,, <,,, a,, b,, c,, and the bare fields are defined to be independent ofp. From (2.16) we find that the ~-function4 is given by
where D In 2, = /?(a).
(3.12)
The anomalous dimension ~*(a, ?J of Z, is given by DlnZ,=y,.
(3.13)
Also Eq. (2.17) gives <-’ D< = --P(a).
(3.14)
From (3.3) we find (61
(3.15)
with
(3.16)
Notice that while a, b and c are independent finite parameters, they obey inhomogeneous RGEs. We can now deduce how the bare parameters vary with a when the other finite parameters ,u, C, a, b, c are held constant. We get
a 1 a-e,=-e-eo=-e 3a 2 4 The function /?(a) is the same as y, in Ref. 151.
a
(n - 4)
3e
2p
O’
(3.17)
42
S. J. HATHRELL
&I I I La+
a
(n-4) BP
aaa
“-
(3.18)
(n-4)
Pb
4 Lbt(n-4)
(3.19)
*
P
Lc+(n
Also
a
a
a-lnZ,=-r aa
B (
Y2+/I<-lnZz
,
at
(3.20)
1
(3.21) Acting on the Green’s function (nj”=y, (A,) n;?li=, (p or v)J, produces an insertion of
a/as therefore
[EY] +$ (y2+13t$lnZ2) [E,l
iq(J --
-ia
--@$e0P0A0(ir0+$? 0
(n - 4) pn-4 B
L,+(n-4)
PO
K
1
F+
L,+(n-4) (
and this must be finite. We now eliminate using (3.9), which gives
Pb
1
G+
(
L’+(“prq)
eotJo~o~o using (34,
1
H2
)I ’ (3.22)
and then p(A)/<,
(3.23)
As for the scalar theory, the distinction between p and /? is of fundamental importance. In particular 8-l in perturbation theory becomes an ascending series of poles (see (3.11)): P=p4)
-I 1+$-q [ I *
(3.24)
TRACE
43
ANOMALIES
We see therefore that the second of terms in (3.23) is gauge-dependent but finite; whereas all the terms in the first set are gauge-invariant and coordinate-invariant, and the whole set has the general form 1 + x (poles)
(3.25)
fFOeCVFOLLV + c (poles){other operators},
which is exactly the form the normal product f [F,,F“““] must take [7]. Furthermore [F,,F”“] must be composed only of gauge-invariant and coordinate-invariant terms with the correct charge conjugation symmetry as well [4]-in fact all those terms appearing in the gauge-invariant Lagrangian, plus any which vanish when integrated over all space. So from (3.23) we see that the poles have disappeared from the gaugedependent terms as required for consistency, and that [F,,F’“]=~F,,,F,Y”+$
(y*--(n-4)<$lnZ,)
- 4(n - 4&-4 B
P Ii La+(n
P + (Lc+(n
HZ
)
[E,]
P* F+ Lb+@--4) ) c
_ 4@ + 47) (n-4)
3ZH
i.
1G (3.26)
The only “new” term is the last one, involving a2H, as this is the only possible term with the correct symmetries which vanishes at zero momentum. There is no term in a,,(yl~~“w) as this has the wrong charge conjugation. The new coefficient a(a) is a finite function of a only, and L, is a pole series of the usual type (3.4) with residue functions of a only. The particular form of the coefficient of a2H is chosen for convenience, but as a whole it is just a pole series in a. u and L, do not depend on < because they are determined completely by the poles of the function (d/&2)( [F,,,,F""]) in flat space, (A
(]F,,P‘“]))
flat
= i(e[F,,l,F”“])
+ 32(~~)~ pnP4 F=
(finite), (3.27)
and (OIF,,Fuu]) is easily seen to be independent of <. We can now substitute for normal products in the earlier, bare expression for 8, Eq. (2.19), and obtain the manifestly finite version 8= @[F/I']-
{(n-
1) + 7) #,]
+p"-4{/?,F++bG+~cH2+4(c--)~2H},
(3.28)
together with the consistency conditions L, = L,,
(3.29)
44
S.J. HATHRELL
y2G-G0 - (n - 4)< & In 2, = T(a) (3.30)
= I$@, 0).
Note that /?a = -(n - 4)~ + /?,(a), with similar expressions for B* and DC, and a, b, c are independent finite parameters; but Q = o(a) is a finite function of a only. Result (3.30) follows because 8 is finite and independent of r, and it is the standard flatspace result for QED showing that the only dependence on r in In Z, is a linear term in the single pole [8]. In fact (3.3 1) jr is just the anomalous dimension of Z, in the Landau gauge, and is O(a’). More information is available from the RGE for [F,,F”“]. We apply B-‘D Eq. (3.28), using the fact that 6’ is independent of ,u, and find f~-lD{@,,P’]}
to
= (finite)
=+aj7[Eo] --pnd4a(~~F+~~G+PLH’)-4~(au)a’H I
, (3.32) I
together with
p, + up= 0.
(3.33)
This last equation is a strong relation between finite functions of a only, valid to all orders of perturbation theory. As for the scalar theory, it was important to consider j?-‘D rather than just D, otherwise this result would have been lost. The role played by u(a) and L, here is analogous to that played by e(A) and L, for the scalar theory ill* We now turn to functions of more than one normal product to find economical ways of calculating the functions of a describing the curved-space theory. 4. TWO-POINT FUNCTIONS OF NORMAL PRODUCTS The procedure for calculating /?,, /16, /3, and u is based on the obvious application of the ideas presented in Ref. [ 11. p,, /3, and u are found from the renormalisation of flat-space 2-point functions of the energy-momentum tensor. The remaining function pb, the coefficient of the topological G term in the anomaly, is found from the renormalisation of flat-space 3-point functions, and is dealt with in the next section. With the benefit of hindsight from Ref. [ 1] we start straight away with the 2-point
TRACE
45
ANOMALIES
functions obtained by two derivatives of the vacuum functional with respect to the metric, followed by taking the flat-space limit. As before this gives Par”
+ a,A”‘“”
+ cOPeD = (finite),
(4.1)
= qyq?““),
(4.2)
where r+(p)
and A”“” and Cc”‘” are the 2-point vertices corresponding to F and HZ in Pg’,, as given in Ref. [l] in Eqs. (6.6), (6.7). The flat-space energy-momentum tensor is 141 p*=-
-1 qG
6s 6S e nat a’ de,, + eOp dear --I I
The covariant derivative reduces to $”
= $(p -a”‘)
t ieoAo’.
(4.4)
The information necessary for p, and j3, is obtained from two different contractions of Eq. (4.1). First, contraction with qKll vIV gives ~‘“yL,(P2) t (P’)’ Pn-4
4(n - 3)(n + 1) L, t & 1
LcI = (finite),
(4.5 1
where the expansions of a, and cO from (3.3) have been inserted. We could take this slightly further by expressing 13’” in terms of its irreducible components, (4.6 1 and then work with i(~wv~u,,) instead of i(P”8,,,) as in Ref. [ 11. But in practice there is little to be gained, and we choose here to remain with the full 8”“. It has the advantage that the absorptive-part analysis used in the evaluation of i(tP”~,,) then involves the somewhat more physical on-shell matrix elements of the complete energy-momentum tensor, and the calculations are then as presented in Ref. [2]. The important point for the present is that the expression of 8“’ in terms of normal products does not lead to any overall factors of p^ appearing in the expression for PvMv, and so the renormalisation-group analysis gives no advantageous route to its computation. The second contraction of Eq. (4.1) is with qKnq,,,: i(kW) + (p’)’ ,une48L, = (finite).
(4.7)
46
S. J. HATHRELL
The flat-space trace 0 is given by I9 = f(n - 4) F,,“FO;,“” - gn - 1) l&i&0 = &F,“F’“]
(4.8)
- ;((n - 1) + y?(E,]
(4.9)
which can be read off from the corresponding expressions (2.19) and (3.28) in curved-space. We can now use the fact that [E,] makes no contributions in 2-point functions: if P(y) is a polynomial composite in the fields t+?(y) and w(y), not necessarily finite, then
i([4(x)lP(y))= j(~)j(d~dyl)P(+&f =- -WY)=. in flat space. (6x(x) i
(4.10)
This vanishes in flat space because (i) I (dw)(J/Jw) gives zero, and likewise for p; (ii) terms like (~/SW(X)) w(x ) contain d(“)(O) and are dimensionally regularised to zero for a massless theory. These arguments are not rigorous, but their validity can easily be checked in a number of examples-in terms of Feynman diagrams they are a simple consequence of the Feynman rules for the insertion of an [E,] vertex. Using this result, and anticipating the requirements of the next section along the lines argued for the scalar theory, we define another operator
{AZ} = f [F,“F’“] - $ [E,],
(4.11)
= (n-4) FowvJ?“” ( 4B 1 0 * The two versions are, respectively, the normal operator. We then define the 2-point function
product
(4.12) and bare forms of the (4.13)
Ld~‘>=i(~~%~*1).
Although {A*} is not actually a finite operator in general, its only contributions to r,, are from the finite term +[F,,P”] because of (4.10), and so r,, has the renormalisation n-4 * r,4.4 +(P2)2P”-4 -p( 1 L,
= (finite),
defining L, as a pole series in a. The key feature is that r,, contains no poles in ln(p2/p2). Multiplying (4.14) by 8’ and comparing with (4.7) then gives 8L, - (n - 4)2 L, = (finite),
(4.15)
47
TRACE ANOMALIES
so that in particular (4.16)
C,(a) = $X,(4;
i.e., the single pole of L, is completely determined by the triple pole of L,. The triple pole of L, can then be found from its single pole by using the appropriate RGE derived from (4.14). For this we need the RGE for {A * }. From (4.12) we have simply fi- ‘D(B{A * }) = 0,
(4.17)
another particularly simple property of the operator (A ‘}. Therefore from applying to (4.14) we find that the RGE for L, is
,!-‘Ofi’
[(n - 4) + D] L, = (finite) = -$
(ax,).
(4.18)
The right-hand side follows because it is the only finite term on the left. This can then be multiplied through by @/(n - 4))*, expanded and integrated iteratively to yield (4.19) and hence X,(a). The leading, O(a*) terms in X, have cancelled, just as the O(A4) terms cancelled for the corresponding result in the scalar theory, and the leading term is in fact O(a”). The reason is essentially that X3 depends on the derivative of (/I/a). This implies [9] through (4.16) and (3.15) that the coefficient of the H* anomaly, /I,, is also O(a3) and not O(a*) as might superficially have been expected from power counting alone. X,(a) is found from the residue of the single-pole divergence of the function r,,-its original definition in (4.14). In leading order it is 0( 1) and determined by the one-loop, a-independent contribution shown in Fig. 1. If we write P(a)=&a
+@*a2 + O(a3)
(4.20)
we can then determine the leading behaviour to be c,(a)=-
(4.21)
(81~6(‘)a3+O(a4),
P,(a) = - -& (ac,) = (‘l !:“I
) a3 + O(a4).
FIG. 1. Leading contribution to r,, . S95/ 14211.4
(4.22)
48
S.J.HATHRELL
From this and Eq. (3.33) we can also derive I: &Xl 24
u(a) = -
t
a* + O(a3).
(4.23)
1
It would take three- and four-loop calculations to verify these results for u(a) and However, a tractable three-loop calculation suffices to confirm the vanishing of the O(a*) term’ in p,, and this is done in Section 7.
PC(a) directly.
5. THREE-POINT FUNCTIONS OF NORMAL PRODUCTS
Once again the pattern for QED follows the scalar theory quite closely, but with simplifications consequent from the absence of an R$* term. The G term in 9 contributes to the flat-space 3-point functions which are obtained from (a/&2)! applied to the vacuum functional. Thus i2(exeyez)+i
%)
B,-$+8,++8, (
z
+ ( ~~~~Oz)
= (finite).
(5.1)
x
The labelling is as in Section 7 of Ref. [I], suppressing the distinction between position and momentum space. The insertions of [E,] into 3-point functions do not in general vanish, but its contribution to Eq. (5.1) can be removed by defining e, = 8, + f(n - l>[E,(x)]
(5.2)
and substituting into the above. The 3-point functions containing [Eclr] can be translated into 2-point functions using the analogue of Eq. (7.4) of [ 11: if P(y) and Q(Z) are polynomial composites in the fields p and w, then in flat space we have i’( [E,(x)]
PYQz) = -i
+
4,~a’,
(5.3)
from a functional integration by parts; cf. (4.10). (It is straightforward to verify this result in a number of examples-it is a simple consequence of the Feynman rules.) We also need
Substitution of these results into (5.1), using the fact that [E,] makes no contribution in l- or 2-point functions, then yields the modified equation
’ Work done in collaboration with I. T. Drummond.
TRACE
49
ANOMALIES
The terms in this equation are (5.6)
= (n - 4) F,,y,@“G(y
- z) = 4&A*} 6(y -z)
(5.7)
and (e,,,) = b,,B(p:,
P:, P:) + C&P:,
P:, P:).
(5.8)
This last term contains only the contributions from G and HZ in Yg, which give the 3-point vertices B and C. B and C are of course exactly the same as they were for the scalar theory, Eqs. (7.13), (7.14) of [ 11, and we are interested in the particular cases B(p*, p*, 0) = 0, C(P*, P*, 0) = 8(n - 4)(p*)*
(5.9)
and B(p*, 0,O) = 2(n - 2)(n - 3)(n - 4)(p*)*,
(5.10)
C(p2, 0, 0) = 4(n + 2)(P*)*. If we define the 3-point function r,,,
by
as the obvious extension of the definition of r,, , we can obtain two special cases of Eq. (5.5) from the momentum configurations T(p’, p2, 0). Note that these on-shell configurations do not introduce any infra-red divergences. We get (9
83rAaa(~2; P’, 0) + @*~,AP*)
+ 8(n + 4)(p2)* ,u”-~L, = (finite), (ii)
~r,&~*,
(5.12)
0,O) + 4~2~A,(P2)
+ (p’)’
pnm4{ 2(n - 2)(n - 3)(n - 4) L, + 4(n + 2) L,} = (finite).
(5.13)
The first of these equations is at p, = 0 and does not contain any information about L,--it is a consistency check on the analysis using the action principle to provide insertions at zero momentum. The second equation leads to a relation between L,, L, and the poles of the functions r,,, and I’,, . We can further simplify both equations by substituting for B’r,, using Eq. (4.7). (Recall (5.2), (5.6) and the fact that [E,] makes no contribution to 2-point functions.) Thus
50
S. J. HATHRELL
(i) (ii)
B3raAA(p2, p*, 0) + 8(n - 4)(p*)*
,unP4Lc = (finite),
B3raAA(p2, 0,O) + (p2)2p”-4{2(n
- 2)(n - 3)(n - 4) L, + 4(n - 6) L,}
= (finite).
(5.14)
(5.15)
It remains to determine the renormalisation of the 3-point function r,,,. The first term in this is the one-loop, a-independent amplitude shown in Fig. 2. (The momentum factors at the vertices make this ultra-violet divergent, and are also responsible for removing infra-red divergences at zero-momentum and on-shell configurations.) Superficially, therefore, the first poles in j?“r,,, can be expected at O(a”). L, is also O(a3) by the arguments of the last section, so from (5.15) we see that b,, the double pole residue of L,, is O(a3). Hence the single pole residue b, , and the p-fucntion pb, are derived from the RGE for L, and must be of the form {constant + O(a*)}, where the constant term is not determined by this analysis. Information about the renormalisation of r,.,,, is obtained by using the action principle with the equations for the renormalisation of r,,, , i.e., Eqs. (4.13), (4.14) through application of the finite derivative a i?/aa. Using (3.5), (3.17), (3.18) and (4.12) we have (in flat space) (5.16) a$-{A*/=-
f
(A*].
The contribution arising from the insertion of the gauge-fixing term vanishes because of (3.1)-the Ward identity-and the contribution from the photon equation-of-motion operator [E,] can be translated into 2-point functions by the usual functional integration by parts. Using
J. d”z A “(z)
(5.18)
we have i*({A:}{A;}
FIG.
2.
1’d”z[E,(z)])
Leading
contribution
= -4I’,,
to r,,,
.
.
(5.19)
TRACE
Combining
51
ANOMALIES
all these in the derivative of Eq. (4.14) we arrive at
= (finite),
L,
! where we have also added twice Eq. (4.14) to turn the coefficient of r,, into a simple expansion in poles starting at O(a’)/(n - 4). If we multiply through by /?” and use the relationships between r,, , L, and L, given in (4.7), (4.14), (4.15), and the RGE (3.15) for L,, we find that this is indeed consistent with Eq. (5.14). The new information comes on comparing Eq. (5.20) with the general form of the equation for the renormalisation of T,,,(p:, p:, p:), just as for the corresponding result in the scalar theory. Although (A ‘} is actually a divergent operator in elementary Green’s functions, the extra subtractions needed when it appears in the composite function FAAA are of the same type as the subtractions for the corresponding finite operator a [F,,F”“] in its place (see (4.11)). This is because all the contributions in r,,, involving [E,] can be translated by means of Eq. (5.3) into poles multiplying 2-point functions. These latter 2-point functions are just the functions appearing in the subtractions required for i2 < {d [F,,,P“] )“). We therefore deduce that the renormalisation of r,,, takes the general form
r,,,(p:, P$pL2) + 5’ (poles){terms inr,,(pT)i Ii +c4 I-v
(poles)
1{terms
in pfpi } = (finite).
(5.21)
The pole-terms involving L’,,(pf) subtract the poles in ln(pi/p*), and the third term subtracts the remaining divergences. (We should emphasize that this result is only stated here-the proof would require a detailed application of renormalisation theory [lo, 111.) We can now compare this with Eq. (5.20) and deduce the nature of the polecoefficients multiplying r,, . Then going to the momentum configuration pz = pi = 0 we have
raAa(p*,O,O)+ I$:
(4)
/T,,(p2)+(p2)*pne4
(y)3L,=(linite). (5.22)
The pure-pole term L, cannot be deduced from (5.20) because there are too many possible combinations of pfpj in (5.21). L, is therefore defined by Eq. (5.22), as a pole series like (3.4), and the factor of (n - 4)3 B-’ has been extracted from its definition for convenience. It is a “new” quantity, reflecting the appearance of the “new” quantity L, in Eq. (5.15).
52
S. J. HATHRELL
We relate L, to L, by multiplying Eq. (5.22) by j?’ and comparing with Eq. (5.15). Substituting for B’r,, = i(f%) with (4.7) we obtain 2(n - 2)(n - 3)(n - 4) L, + 4L,
(n - 6) + 2a2 &
f
I
c
- (n - 4)3 L, = (finite). Ii
(5.23)
The double pole of L, is therefore determined by the single pole of L, and the quadruple pole of L,. The latter can be determined from its single pole through the appropriate RGE. Once again there is a close analogy with the scalar theory because it transpires that the renormalisation-group structure of Eq. (5.22) entails a cancellation of the leading term in the quadruple pole of L,, and so the leading term in the single pole of /?“r,,, . We now confine attention to the leading terms to demonstrate this cancellation. The RGE for L, is obtained by applying B-’ D/?’ to Eq. (5.22)and using the RGE (4.17) for {A’}. Thus [(n - 4) + D] L,, = (finite).
(5.24)
The single pole residue of L,, Y,, is determined in leading order by the divergence of the graph in Fig. 2, and is clearly O(1). The first term in (5.24) therefore does not contribute to the leading terms in L,, and we deduce [(n-4)+D]L,=$(aY,)+O
(a’, (n ‘*
3 (n a-4j2 ’
,...
This can now be multiplied through by /?‘/(n - 4)3 and integrated iteratively for the pole residues of L,, using the expansion /3(a) = /I, a + O(a’) as before, and gives Y2 = 4/l, Y, a + O(a*), Y3 = $P: Y, a2 t O(a3),
(5.26)
Y4 = O(a4).
Returning now to Eq. (5.23), this shows that the leading double pole of L, is determined only by the leading single pole of L,. The functions of n do not affect the leading behaviour, as argued in Ref. [ 11, and so (5.23) reduces to the simple result b, = 2c, t O(a”).
(5.27)
This is exactly the same relationship as was found for the scalar theory. Substituting the results at the end of Section 4 and using the RGE (3.15) for L, leads to
53
TRACE ANOMALIES
b, = const+pg
a* + O(a3), (5.28)
/lb = const - ‘+
cf2+ O(a”).
The constant term is not determined because b, only depends on b, through ab,/h, but it is known from elsewhere [ 12, 131 and only depends on the non-interacting theory. Presumably it could in principle be found in the general flat-space approach here from a consideration of the one-loop, 3-point function involving i2(8”A0A,,tl”‘,), but other techniques ar less cumbersome in this case. We summarise the argument in this section, as it is quite involved. The coefftcient L, of the topological G term in the action is found through its double pole residue from its contributions to the 3-point function obtained by applying (~5/&f2)~ to the vacuum functional. The contributions from the fermion equation-of-motion operator [E,] can be absorbed by two manoeuvres: part is translated into 2-point functions, and the rest is incorporated into a new operator (A*}. Although not a finite operator, (A’} has simple renormalisation and renormalisation-group properties. Then L, is determined by the divergences of 2- and 3-point functions of (A *}, i.e., r,, and r,,, . Furthermore the renormalisation-group structure of r,,, implies a cancellation of the leading pole of /“r,,, , so that it does not contribute to the O(a3) double pole of L,. This cancellation is much like the cancellation of the O(a’) pole of j?‘r,, shown in the last section, and the whole analysis mirrors the scalar theory rather closely. These results for the topological part of the anomaly do not agree with the results of Shore [ 141, who finds a non-vanishing O(a) contribution by direct calculation in a spherical spacetime. We remark in this context that the vanishing of the O(a) term found here does not depend on the vanishing of the O(a3) pole in j?“r,,, , which is relevant only to the O(a’) term in Pb, but it is sensitive to the vanishing of the O(a’) pole in /?rAA, and hence the O(a*) term in the R* part of the anomaly. The latter cancellation is confirmed here by explicit calculation in Section 7. This concludes the general analysis of massless QED in curved space with particular emphasis on the renormalisation of the trace anomaly. The remainder of this paper is concerned with applying the arguments presented so far to the actual calculation of the coefficients. /3, is found to O(a) by a 2-100~ calculation [2] described in the next section. /I, is found to O(a3) by a simple one-loop calculation described in Section 7, from which Pb and u are determined to O(a’). A related 3. loop calculation directly confirms the vanishing of the O(a’) term in /?, as predicted by the more powerful renormalisation-group arguments. 6. ABSORPTIVE
PARTS,~NFRA-RED AND THE CALCULATION
DIVERGENCES OF/Z,
It was shown earlier in Section 4 that i(80) and L, had their first poles at O(a3). The RGE for L, shows that the double and higher poles depend on pa(&.z,/k?a) and
S. J. HATHRELL
54
FIG. 3. l-loop contributions to i(B““B,,,.).
must be O(a2) at least. So from Eq. (4.5) we deduce that the F part of the anomaly, i.e., the coeffkient p,, is determined to O(a) by a,(a) i(@‘“@,,) + 20(p2)2 PnP4 n-4 = (finite) + O(a2), I I
(6.1)
and therefore by the divergences of the I- and 2-100~ amplitudes i(fP”19,~,), evaluated in flat space. The relevant diagrams are shown in Figs. 3 and 4. This analysis predicts the overall cancellation of the double poles in the 2-100~ amplitudes. The Feynman rules are the standard ones for massless QED, together with the vertices for the insertion of P‘“(p) which are derived from the expressions in Eq. (4.3). These vertices are depicted in Fig. 5 and have the following values: (i)
For P”e+e-: VL;“(kI, k,) = {[y”(k,
(ii)
+ k2y + y”(k, + k,)’ - W”‘(K~ + KJ].
(6.2)
For P”yy: V;“““(k,,
k2) = q’“(k, . k, ya4 - k;kf) -k,
. k,(t,+=q”” + tf’“rfD)
- qu4(k:k; + kfjk;) + (k:k;tyv4 f kt’k;y’”
(iii)
+ k;kfq””
+ k;kfq”“).
(6.3)
For e”“e+e-y: e, V$J”, = - ~eo(yP~“a + y”tj+” - 2q”“y”).
FIG. 4. 2-100~ contributions to i(S““S,,.).
(6.4)
TRACE
FIG.
55
ANOMALIES
5. Basic vertices for the insertion of 8”“.
Notice that Vyua5 satisfies V;ua4k,a = V;“a4k,,, = 0
in accordance with the Ward identity expression
(6.5)
(3.1). We remark also for later use that the
Pab(k,, k,) E k, . k,ge4 - k’;‘k:
(6.6)
appearing in (6.3) is a gauge-invariant projection tensor. The calculation of the l-loop amplitudes in Fig. 3 and the 2-100~ amplitudes in Fig. 4 by an absorptive-part argument was described in some detail in an earlier paper [2]. These details are not central to the rest of the analysis here, so it will be sufficient to quote the result: i(e’“e,,> = -
Er’-4(P2)2 I& t4nj2
(3 +$
(+)I
+ O(a’)[
+ (finite).
(6.7)
All the infra-red divergences cancel when the diagrams are summed, and in addition we see that the double ultra-violet pole is absent to O(a) as predicted in the preceding arguments. Comparing with (6.1) we deduce (6.8)
giving the coefficient of the F term in the anomaly to O(a). It should be noticed that while the 0( 1) terms in the coefficient of F in the bare and the normal product forms of the anomaly are the same, i.e., the O(1) terms in -(n - 4) a, and ,u”“-“ba, respectively (from (2.19) and (3.28)), the O(a) terms differ by a factor of 2. This is because the finite operator [F,,P”] contains a subtraction involving F at O(a); cf. (3.26). It is the finite, normal product version of the anomaly with the a-dependence given by P,(a) which is appropriate when taking the n + 4 limit. Similar considerations apply to the other curvature terms in the anomaly.
56
S.J.HATHRELL
7. CALCULATION
OF THE REMAINING
COEFFICIENTS
It was shown in Section 4 that the coefficient of the HZ term in the anomaly, j?,, was determined to O(a3) by the l-loop divergence of the 2-point function r,, . This in turn provides the [F,,F”“] - c?*H mixing coefficient a(a) to O(a’), and through the arguments of Section 5 it also provides the coeffkient of the remaining topological G term pb to O(a’). Furthermore the renormalisation-group structure of r,, implied [9] a cancellation of the O(a’) pole of i(M), and therefore the O(a) term in a(a). In order both to evaluate the parameters Pb, /?, and u in leading order, and also to verify the cancellation of the O(a’) pole in i(M), we therefore examine the set of amplitudes up to the 3-100~ level shown in Fig. 6, labelled K,-K,. These are defined for an operator insertion of $(n - 4)FOU”F0PI’ =jqA2)
(7.1)
at each end, and include an overall factor of i. In complete analogy with the scalar theory [ 11, we then have qT Ki $ (p’)” ,Pe48L, iC1
’ + Ki + (p*)*$-’ 7 ,r,
= (finite) t O(a’),
(7.2)
‘(’ i ‘O) = (finite) t O(a”),
(7.3)
b
FIG. 6. (a) I-loop, (b) 2-loop, and (c) 34oop amplitudes K,-K,.
TRACE
57
ANOMALIES
l 4 Ki+ (p*)*p Bn-4 2 L, F iY1 ( 1 Equations (7.2)-(7.4) follow, respectively, from only compute the leading terms in the (n - 4) and this information provides, respectively, the of u(a) to O(a), and the single pole of L, to amount to two- and one-loop calculations only,
= (finite) + O(a3).
(4.7), (3.27) and (4.13). However, we expansion of the amplitudes K,-K,, single pole of L, to O(a’), the value O(1). In effect these last two results and reduce to (finite) + O(a2),
;(K,+K,)+(p’)‘p’-4&=
1 (n - 4)2
K, + (P’)’ pnV4 (n
J-1
= (finite) + O(a).
We emphasize two points. First, Eqs. (7.2)-(7.4) amount to the definitions of the terms L,, (a + L,), L, and do not depend on any relationships between them. Second, although (7.6) is the simplest of these results to evaluate, depending only on the single-loop diagram in Fig. 6(a), through the powerful renormalisation-group arguments it also provides more information than the direct 2- and 3-100~ calculations: using only the value of the b-function it gives c, to O(a-‘) and u to O(a’), according to Eqs. (4.20~(4.23). The calculations leading to (7.2) and (7.5) serve only to verify the vanishing of the low-order terms in c, and rr. The Feynman rules are as usual [ 181 for massless QED, together with the vertex VP0 for the insertion of (7.1), obtained from (6.3) by contraction (see Fig. 7), V(k,
3k2) = (n - 4) Pyk,,
k,),
(7.7)
with Paa the gauge-invariant projection tensor defined earlier in (6.6). Calculation of K,-K, is straightforward. In the case of K, the photon self-energy 2-100~ subdiagram can be obtained through the absorptive-part approach. In fact this calculation gives the well-known 2-100~ b-function [4, 19,201 as a by-product, since the photon renormalisation to O(a’) appears as part of the diagrams K,-K,:
JJ FIG.
I.
Vy4, vertex
for the insertion
P kz
of f(n - 4) F,,,.F,U“.
S. J. HATHRELL
=,=l+(n-4)
-q”(L)+4
3
47r
P(a) = Ba $lnZ,=$(~)+8
(+J2]+o(a~)’
(7.8)
(+-)2+O(cf3).
(7.9)
Working to O(a’), the results for these amplitudes are
K,=_ (P”)’r4 3 w* I K2 = _ (P’)’ e4 (47r)*
(n - 4) + O(n - 4)2 1)
l&j
[--F
(C)‘]
(7.10)
+ [4
+ O(n - 4) * O(a) , i K,+K,+K,=-
“;~~~-” 7l
K,+K,=-(p~~~~~~4
(2)
+O(a2)] (7.11) (7.12)
(0(1).0(a2)},
]j---+[~(-f-)*]+O(l)-O(az)~. n ?t
(7.13)
K, includes the charge renormalisation to O(a). The absence of a pole in (7.12) below O(a”) reflects the fact that the double poles in the one-particle irreducible
photon self-energy insertions cancel between each other at the 2-100~ level, which in turn reflects the absence of the double pole in Z, below O(a3). Explicit calculation of K, and K, is more difficult. However, it can be seen that the poles arising from the fermion loops cancel between the two diagrams for the same reason that the ordinary 4-photon interaction in QED is actually finite, although superficially it is logarithmically divergent. (The symmetry factors for K, and K, are 1 and $, and match the permutations of the 4-photon Green’s function.) The remaining loop integrations can only contribute a double pole, which is removed by the (n - 4) factors appearing in the vertices for the operator insertions. Hence K, + K, = - (‘;,
Combining
I:-’ ?t
{O(l) . O(a*)}.
(7.14)
all these amplitudes we find i i=
Ki = (finite) + O(a3), I
f (K, + K,) = (finite) + O(a*),
(7.16)
TRACE
1
K, = _ (P212F4
(n - 4)2 Comparison
59
ANOMALIES
1
(47Q2
I
&-q+o(l)/.
(7.17)
with (7.2) and (7.5), (7.6) then gives L, = O(d) = L,,
(7.18)
u = O(a2),
X,(a)=-L a2 2
(7.19) 1
(4x)
+ O(a).
We may now insert the value of /I(a) into the earlier results (5.28) and (4.22), (4.23) to deduce (7.21) (7.22) 1 a2 ~7=-~(4n)4+
O(d),
(7.23)
where we have now incorporated the constant term in & obtained from Refs. [ 12, 131. Equation (7.21) does not agree with the O(a) result of Shore [ 141, obtained on a spherical background. As a final check on the consistency of the renormalisation-group analysis leading to the value of /Ib, we consider the 3-point function TAAA(p2, 0,O) to O(a). The diagrams are shown in Fig. 8, and two of the three operator insertions are at p2 = 0. The general arguments of Section 5 led to Eq. (5.22) for the renormalisation of this function, and the form of (5.22) was essential to the renormalisation-group analysis involved in the evaluation of fib to O(a’)--though it was not essential to the vanishing of the O(a) term. A particular prediction of Eq. (5.22) which can be verified quite easily is that there are no poles multiplying ln(p2/,u2) in TAaa(p2, 0,O) below O(a’), whereas superficial power-counting suggests the appearance of such
FIG.
8. Diagrams for r,,,
to O(a).
60
S.J. HATHRELL
poles at O(a). This feature is rather in contrast with the scalar theory, in which the ln(p2/p2) poles appeared in leading order; cf. (7.38) of [I]. The Feynman rules for the insertion of {A’} are just as in (7.7), but with an additional factor of p^-‘. Evaluation of the integrals is straightforward using formulae (A.l) and (AS) of [l], and leads to
(7.24) The O(a) poles in ln(p2/p2) have indeed cancelled between the diagrams. Comparing with (5.22), we also have
($) + O(a’)f, W4 = --!c4K12 I+-+Y,(a)
= (4;)’ + Oh’>. 2
(7.26)
These values are consistent with (5.26). Although the actual value of Y, was not relevant to the calculation of the O(a") term in Pbr higher order corrections to Ph will depend on it. In fact if we combined (7.25) with the calculation of the diagrams K, and K, in Fig. (6), a fuller treatment of the RGE for r,,,, , and the value of ,@a) to the 3-100~ level, this would provide suffkient to find Pb to O(a"), u to O(a") and /3, to
O(a"). 8. DISCUSSION AND CONCLUSIONS The final expression for the trace anomaly is 8 = $[I;,“F’“]
- {(n - 1) + 7) f[&l
+~““-4{~~i,++~G+~~~2+4(c-~)~2~),
(8.1)
and the values obtained for the parameters are
&-(n-4)aA
20(&2
/3++(-J-)+O(aq' IA--+(--J-)'+O(a')[, 360
(8.2)
(8.3) (8.4)
TRACE
61
ANOMALIES
P-5)
In addition, y(a) is the usual anomalous dimension of the fermion fields, evaluated in the Landau gauge; and &(n-4)+
If
(-f-)+8
(+)‘+W3)/
(8.6)
a, b and c are independent finite parameters obeying inhomogeneous RGEs, [(n - 4) + D]c = PC@>,
(8.7)
and similar equations for a and 6. The normal products [E,] and (F,,,P”‘] are given by Eqs. (3.7) and (3.26). Equation (8.1) is in a suitable form for taking the limit as n -+ 4. Then the dependence on the unspecified parameters p, a and b disappears, but c remains with a finite effect through the coefficient (c - a(a)) of the curvature term a*H--which is perhaps more familiar as the “ambiguous” i3*R in the anomaly. This needs a rather more careful discussion. The independent finite parameter c obeys the inhomogeneous RGE (8.7), so it cannot consistently be set to zero under a scaling of the other renormalised variables. It is independent by virtue of the independent solutions of the corresponding homogeneous equation. To examine this more closely, split c into two parts, c = F(a) + c’, where c’ satisfies the homogeneous equation. In the limit n --) 4 we have
Nab $ F= P&l, p d c’ = 0. & We can solve (8.8) perturbatively
(8.9)
in a and combine with (8.5) to find 2
4(F - a) = *
+ O(a”).
(8.10)
In fact F and u are closely related by
aaaac=-*
(8.11)
from comparison with (3.26). Notice however that the n-dimensional solution of (8.7) expanded as a power series in (n - 4) is not perturbative in a. Then F contains terms in In a and powers of a-’ starting at O(n - 4)‘. The remaining constant c’ is truly
62
S. J. HATHRELL
independent, and can be chosen to vanish-or in physical terms, it must be determined by experiment. It is precisely the arbitrariness of this finite parameter c’ which provides the “ambiguity” in the cT*R part of the anomaly. There are two further points concerning the ambiguity which should be made. First, the c?*R term as a whole plays a significant role in the analysis: in fact it is the only term which contributes to the (p’)’ pne48L, term in Eq. (4.7), and is essential to the cancellation of the poles in the flat-space function i(88). It is also essential as a term in the subtractions required to make the normal product [F,,,P”] finite at nonzero momentum. Second, the choice of the term (F + $‘R) for the anomaly, which is quite common in the literature, is no more than an artefact of the definition of the curvature term F. If F is taken to be the square of the n-dimensional Weyl tensor in the regularised theory, as here and in [I], then its behaviour under conformal scaling is given by Eq. (2.19) and there is no i?*R term from this source. But if F is defined as the 4-dimensional Weyl tensor instead, so that the coefficients (see (2.9) of ] 11) take their n = 4 values even in the regularised theory, then in the right-hand side of Eq. (2.19), F is replaced by (F + $3’R). This amounts to taking slightly different linear combinations of the three curvature terms F, G and HZ as a basis for q. In conclusion, the results of this paper and Ref. [ I] show that the trace anomaly contains all the essential information about the curved-space theory, not surprisingly in view of the close relation between the energy-momentum tensor and the coordinate-invariance of the theory; and that the various parameters in the description can be determined in perturbation theory from flat-space calculations alone. In so far as the assumptions about renormalisability and the use of the renormalised action principle have not been proved here, one might take the view that the results found here require conventional proof of, for example, the finiteness of the energy-momentum tensor in the flat-space theory. But it seems that this approach does not give sufficient weight to the fundamentally important role of the energymomentum tensor in the curved-space theory, and we believe a more profitable attitude is to emphasize the importance of renormalisability in curved space as a starting-point for generating results-if indeed the theories are renormalisable, although we have found no evidence to the contrary. It should be possible to extend the ideas to a discussion of non-abelian gauge theories, and in this context the absorptive-part approach to the calculation of Feynman amplitudes, based on unitarity considerations, should be useful in simplifying the contributions from ghost loops. Further extension to supersymmetric theories would be interesting in view of the connections between the axial and trace anomalies [21,22], although the simple dimensional regularisation employed here would not then suffice.
ACKNOWLEDGMENTS I am grateful to I. T. Drummond for many conversations, useful suggestions, and assistance with some of the calculations, and to the U. K. Science Research Council for supporting part of this work.
TRACE
ANOMALIES
63
REFERENCES 1. S. J. HATHRELL, Ann. Phys. (N.Y.) 139, No. 1 (1982), 136. See also S. J. HATHRELL, University of Cambridge Ph.D. thesis, 1980. 2. I. T. DRUMMOND AND S. J. HATHRELL. Phys. Rev. D 21, No. 4 (1980). 958. 3. J. H. LOWENSTEXN, Commun. Math. Phys. 24 (1971), 1. 4. S. L. ADLER, J. C. COLLINS, AND A. DUNCAN, Phys. Rev. D 15. No. 6 (1977), 1712. 5. J. C. COLLINS AND A. J. MACFARLANE, Phys. RLv. D 10 (1974), 1201. 6. J. C. COLLINS, private communication. 7. L. S. BROWN, Ann. Phys. (N.Y.) 126. No. 1 (1980). 135. 8. B. LAUTRUP, Nucl. Phys. i3 105 (1976), 23. 9. L. S. BROWN AND J. C. COLLINS, Ann. Phys. (N.Y.) 130, No. 1 (1980), 215. 10. W. ZIMMERMANN, in “Lectures on Elementary Particles and Quantum Field Theory” (S. Deser, M. Grisaru, and H. Pendleton, Eds.), Vol. 1, MIT Press, Cambridge, Mass., 1970. 11. P. BREITENLOHNER AND D. MAISON, Commun. Math. Phys. 52 (1977), 11. 39, 55. 12. M. J. DUFF, Nucl. Phys. B 125 (1977). 334. 13. G. M. SHORE, Ann. Phys. (N.Y.) 117 (1979), 121. 14. G. M. SHORE, Phys. Rev. D 21, No. 8 (1980), 2226. 15. R. GASTMANS, J. VERWAEST, AND R. MEULDERMANS, Nucl. Phys. B 105 (1976). 454. 16. W. J. MARCIANO, Phys. Rev, D 12 (1975), 3861. 17. D. A. AKYEAMPONG AND R. DELBOURGO, Nuovo Cimento A 17 (1973), 578. 18. G. ‘T HOOFT AND M. VELTMAN, “Diagrammar,” CERN Publication 73-9, Geneva, 1973. 19. M. S. CHANOWITZ AND J. ELLIS, Phys. Rev. D 7 (1973), 2490. 20. J. D. BJORKEN AND S. D. DRELL. “Relativistic Quantum Fields,” McGraw-Hill, New York, 1965. (Eq. (19.160)) 21. J. LUKIERSKI. Phys. Let?. B 70 (1977). 183. 22. T. CURTRIGHT, Phys. Lett. B 71 (1977). 185.
595/142/I-5
OF PHYSICS
142, 34-63 (1982)
Trace Anomalies
and QED in Curved
Space
S. J. HATHRELL Department University
of Applied Mathematics and Theoretical Physics, of Cambridge, Cambridge CB3-9E W, England
Received September 9, 1981; revised February 1, 1982
It is shown, for massless QED in a weakly curved background, how to obtain full information about the trace anomaly in perturbation theory, including the “topological” term in the gravitational part of the anomaly. The arguments used follow as straightforward adaptations of those presented recently for A)” theory, and rely on a renormalisation-group analysis combined with strong connections between renormalisability of the curved space theory, finiteness of the energy-momentum tensor, and the use of normal products. The first non-zero a-dependent terms appear at O(a’) for the topological part, and at O(a’) for the non-conformal R* part of the anomaly. Both values can be deduced via the renormalisationgroup arguments from simple flat Feynman diagrams. A direct 3-100~ calculation confirms the vanishing of the O(a’) term in the R* anomaly. The analysis follows the scalar theory closely, while being simpler in many places.
1. INTRODUCTORY
REMARKS
Having established a framework for the investigation of the trace anomalies for a self-interacting scalar theory [ 11, we wish to see whether the same ideas can be applied to theories of a more directly practical relevance. The obvious first choice is quantum electrodynamics, not only becausethe theory is experimentally well verified in many circumstances, but also because as a gauge theory it is the simplest prototype of a wide range of such theories. The question of compatibility of QED and gravitation has not of course been fully answered yet, but it seemsreasonable to investigate QED on a curved background as a possible approximation, and it seems equally reasonable to choose the simplest coordinate-invariant extension available in the absenceof arguments to the contrary. There are two obvious points of contrast with the scalar theory. First, we now have a theory containing not one type of field but two different mutually interacting fields. Second, QED possessesa local gauge symmetry with respect to the background manifold. It turns out that the former does not significantly complicate the investigation, while the gauge symmetry actually constrains the theory sufficiently to make it substantially simpler in places than its scalar counterpart. There are, however, minor complications of detail for QED which have to be considered in the process, but which do not affect the genera1line of argument. They are principally the necessity for vierbein fields, and the necessity of a gauge-fixing term to define the 34 0003-49 16/82/090034-30$05.@0/0 Copyright All rights
0 1982 by Academic Press, Inc. of reproduction in any form reserved.
TRACE ANOMALIES
35
perturbation theory. These are considered in Section 2, where the theory is defined in more detail. In Section 3 the renormalisation of the parameters in the theory is determined, and explicit forms are found for the renormalised composite operators-“normal products.” This enables the trace of the energy-momentum tensor to be given as a manifestly finite expression. In Section 4 we introduce twopoint functions of normal products, and show how to derive the renormalisation of the F part of the anomaly to O(e’) and the R2 part to O(e”). The latter involves a cancellation of the O(e”) term specifically predicted by the renormalisation-group structure of the theory, closely analogous to the behaviour of the scalar theory in Ref. [ 11. In Section 5 we extend the analysis to three-point functions of normal products, and show how to derive the renormalisation of the topological, G term to O(e”) from flat-space calculations. Again the renormalisation-group structure of the theory entails a particular cancellation of the leading divergence of a certain threepoint function. The next sections are concerned with more specific details of the calculations of the parameters describing the anomaly. In Section 6 we set up the Feynman rules and incorporate the results of an earlier investigation’ [2] to calculate the F term to O(e’). The cancellation of infra-red divergences was displayed explicitly in this paper. In Section 7 we calculate the R* term to O(e”) using the indirect methods, and then verify the cancellation of the O(e”) term by a direct 3-100~ calculation. Using the arguments of Section 5 this result also gives the G term to O(e4). Finally in Section 8 we discuss the results, with particular emphasis on the status of the #R term in the anomaly.
2. DESCRIPTION
OF THE THEORY
The action S is given by (2*1)
where the curvature terms in the Lagrangian are ~g=ua,F+b,G+c,H2,
and the matter Lagrangian
(2.2)
is given by Pm = -$F~~,F;u + ftji,iSy,.
(2.3)
The terms F, G and H* are as defined in Eqs. (2.9), (2.10) and (2.7) of Ref. [ 11, and ’ The conjecture about the nature of the RZ term in this earlier paper was erroneous, but the calculation of the F term presented remains valid.
36
S. J. HATHRELL
a,, b, and c,, are bare coupling constants whose renormalisation tromagnetic charge e. The covariant derivative 3 is given by B = yaeo”(x)(iiw
- l5J
depends on the elec(2.4)
and e,l(x) is a vierbein field relating the coordinates of the manifold to an orthonormal tetrad for the Dirac gamma-matrices. Its relationship to the metric tensor g,, is described by
e,,@,= g,,.(x),
(2.5)
The connections in the covariant derivative for the vierbein and spinor labels can be determined sequentially by requiring that the covariant derivatives of eou and then of y” should vanish, giving
where 0 rob
=
eaA{ebA,M
+
rApL&,l’}T
u ab = $ [y”, yb].
(2.7) (2.8)
P I1v is the usual Levi-Civita connection for the manifold, and A,, can be any vector quantity which commutes with y”--in this case of course it is the potential for the electromagnetic field. e, is the bare charge. As for the scalar theory there is no mass term, nor any Einstein or cosmological term, because their contributions to the trace of the energy-momentum tensor are not “anomalous” and they can consistently be taken to vanish with dimensional renormalisation. When combined with the gauge-invariance of the action this means that the number of possible terms in rP, and Yg with the correct symmetries is actually smaller than for the scalar case. Notably there is no analogue of the Rqh2term which was responsible for so many of the complications. To define the perturbation theory, however, we must break the gauge-invariance and introduce a gauge-fixing term. It actually proves quite convenient to break the coordinate-invariance at the same time and use the coordinate-dependent density p(A) given by S,.,. =$l
\/-g~(Ah
(2.9)
0
P(A) = &
W‘l’%,Aoo)2.
(2.10)
The useful features of this choice become apparent if we briefly summarise the usual,
TRACE
31
ANOMALIES
formal, path-integral arguments leading to this term. No claims of rigour are made for the argument. The vacuum generating functional Z[ g,,,] is multiplied by the quantity
J. (do) d(A)
Gdf(A,) - B) = 1,
(2.11)
= ?‘“a,&>,
(2.12)
where f(A”)
inserted into the integral over the gauge potential, and the integral in (2.11) is over the U(1) gauge group. d(A) is the functional determinant (2.13) The gauge-fixing parameter C;, is then introduced in a second multiplicative
factor (2.14)
Combining
(2.11) and (2.14) we get a multiplicative [ (do) d(A) eis”,f.
factor (2.15)
which is inserted into the integration over the gauge potential A. The functional determinant can then be expressed as an integral over ghost fields, and the integration over the gauge group is factored out of the connected amplitudes. From these arguments we see the following: (i) The gauge-fixing term S,.r, and ghost contribution in d(A) are both independent of the metric g,,L,. Therefore functional derivatives of the action with respect to the metric generate insertions of the canonical gauge-invariant and coordinate-covariant energy-momentum tensor. (ii) The ghosts are decoupled from both the gauge-field and the background metric, and are completely factored out of the connected amplitudes. They need not be considered any further. (iii) The gauge-fixing term gives the usual Feynman rules for calculation in the flat-space limit. (iv) The multiplicative nature of the gauge-fixing process leading to (2.15). emphasized in the argument, leads to orthodox treatment for the Ward identities for connected amplitudes. In particular, connected Green’s functions derived from Z[ g,,] by functional derivatives with respect to gUL, must be gauge-invariant and independent of <,, .
38
S.J.HATHRELL
We make the usual assumption that action (2.1) generates finite Green’s functions, and that the renormalisation of Green’s functions containing external legs is determined by the flat-space limit. That is, in the minimal subtraction scheme of dimensional renormalisation, and in accordance with the usual flat-space theory, we supply a factor of Z;‘j2 for each external gauge-field leg and a factor of Z;‘12 for each external fermion leg. From the Ward identities we get the usual charge renormalisation e,=P 2-nIZez; l/2,
(2.16)
where p is the ‘t Hooft unit of mass. The gauge-fixing parameter 5, is renormalised by
To= z, 6
(2.17)
Z, has the usual form { 1 + C (poles)} and the residues are functions of e2 only, but Z, depends on < as well. All the bare quantities in the action are defined to be independent of ,u. Functional derivatives with respect to the metric are generalised, where necessary, to include the vierbein tield by (2.18) and the contraction of these with g,, gives the scaling derivative S/&II in the notation of Ref. [ 11. The trace of the energy-momentum tensor is obtained in bare form by applying 6/&2 to action (2.1)-(2.3)-the gauge-fixing term does not contribute, as argued above-and ee*,y
=$=
{d(n-4)E;,,,F;“-$(n-
+ {-(n - 4)(a,F+
l>P,i&,)
b,G + coH2) + 4c, a2H}.
(2.19)
This is the expression for the anomaly in bare form, and to express it in a form suitable for taking the limit n + 4 we need to find it in terms of finite operators. Just as for the scalar theory 19must be finite as a whole (given the initial assumptions), since its insertions into finite Green’s functions are generated through the renormalised action principle with a finite functional derivative. We must therefore turn to the normal products in the theory.
3. RENORMALISATION
AND NORMAL
PRODUCTS
The connected amplitudes containing only insertions of 0’” are independent of <, so the renormalised action principle with <~?/a< on such amplitudes produces zero.
TRACE ANOMALIES
39
From this we deduce
(3.2)
The former is a Ward identity for an insertion of the gauge-fixing term, and the latter shows that the renormalisation of the parameters a,, b, and c0 depends only on e*. We will actually find it convenient to work directly with the fine-structure constant* a = e2/4z We therefore have (Z)
=P
(ZJ.
(3.3)
where a, b and c are independent finite parameters, while L,, L, and L, are Laurent series of poles whose residues are functions of a only, (3.4)
and similar expressions for L,, L,. The notation is the natural adaptation of the notation in Ref. [ 11, and will be used subsequently in an entirely systematic way for other series of poles appearing in the analysis. Although the energy-momentum trace 0 contains only gauge-invariant operators, the route to the determination of the normal products involves considering the action principle applied to general off-shell Green’s functions containing external gauge-field legs, site the normal products are just those operators which are finite when inserted into such Green’s functions. Off-shell Green’s functions with external gauge-field legs are not gauge-invariant, so the derivation involves some gauge-dependent operators as well. These must disappear from the final expression for the gauge-invariant normal products, however. We consider first therefore the dimension 4 “equation-of-motion” operator [E,] for the electromagnetic field, which is not gauge-invariant. Only the integrated version is needed:
(3.5)
* Strictly, this is only an intermediate-renormalised parameter. The physical constant corresponds to on-shell renormalisation in the massive theory in the low-energy limit.
40
S. .I. HATHRELL
This is the only operator obtainable directly through the renormalised action principle which contains a term involving Fo,,Fou’“. It is finite and produces a multiplicative factor of iNY when inserted into Green’s functions containing ‘N, external A-fields, for reasons analogous to those in the scalar theory, Eq. (4.1) of Ref. [ 11: it is the curved-space version of the photon counting identity [3,4]. The notation [ ] is again used here and subsequently to denote normal products. The other term in expression (2.19) for 8 which explicitly involves the matter fields is tjT,,i&,,, and this is given by the other equation-of-motion operator. We define for convenience the notation
Then we have (3.7) [E,] is (formally) shown to be finite in Green’s functions using once again a functional integration by parts as in Eq. (4.1) of Ref. [ 11. When integrated over all space it is just the curved-space version of the fermion counting identity [3,4]: )‘fi[E,]=iN,.
(3.8)
That is, when inserted into a Green’s function of elementary fields it multiplies the function by i times the number of external fermion lines. It is apparent from (3.5) that we must also consider the normal product for the gauge-fixing term.3 This is found by applying -it a/a< to an elementary Green’s function containing NY external photons and NO external fermions, which produces an insertion of {@B/C?<) + (i/2) N,&~/cF?~) In Z,}. This must be finite, and SO
defining [p(A)], as the right-hand side has the correct form for this normal product in the minimal subtraction scheme. Next we consider the effect of the finite derivative a a/&, as this produces insertions involving e, Iv&,, v/,,, and by substitution with (3.5) leads to an expression for [F,,F”“]. This derivative is defined holding the renormalised parameters finite, so r The analysis in this section has several features in common with Ref. 141, in which the finiteness of the flat-space trace anomaly is proved. The significant differences are, however, (i) that here we have the full curved-space theory, and (ii) finiteness of the energy-momentum tensor is immediate following the assumption of renormalisability of the curved-space theory, and is used instead to generate other results.
41
TRACE ANOMALIES
we need to know how the bare parameters vary with a. We must therefore turn to the renormalisation-group equations [5] (RGEs) for this information. As in Ref. [l] the RGEs are obtained by applying the differential operator D, (3.10)
where the bare parameters e,, <,,, a,, b,, c,, and the bare fields are defined to be independent ofp. From (2.16) we find that the ~-function4 is given by
where D In 2, = /?(a).
(3.12)
The anomalous dimension ~*(a, ?J of Z, is given by DlnZ,=y,.
(3.13)
Also Eq. (2.17) gives <-’ D< = --P(a).
(3.14)
From (3.3) we find (61
(3.15)
with
(3.16)
Notice that while a, b and c are independent finite parameters, they obey inhomogeneous RGEs. We can now deduce how the bare parameters vary with a when the other finite parameters ,u, C, a, b, c are held constant. We get
a 1 a-e,=-e-eo=-e 3a 2 4 The function /?(a) is the same as y, in Ref. 151.
a
(n - 4)
3e
2p
O’
(3.17)
42
S. J. HATHRELL
&I I I La+
a
(n-4) BP
aaa
“-
(3.18)
(n-4)
Pb
4 Lbt(n-4)
(3.19)
*
P
Lc+(n
Also
a
a
a-lnZ,=-r aa
B (
Y2+/I<-lnZz
,
at
(3.20)
1
(3.21) Acting on the Green’s function (nj”=y, (A,) n;?li=, (p or v)J, produces an insertion of
a/as therefore
[EY] +$ (y2+13t$lnZ2) [E,l
iq(J --
-ia
--@$e0P0A0(ir0+$? 0
(n - 4) pn-4 B
L,+(n-4)
PO
K
1
F+
L,+(n-4) (
and this must be finite. We now eliminate using (3.9), which gives
Pb
1
G+
(
L’+(“prq)
eotJo~o~o using (34,
1
H2
)I ’ (3.22)
and then p(A)/<,
(3.23)
As for the scalar theory, the distinction between p and /? is of fundamental importance. In particular 8-l in perturbation theory becomes an ascending series of poles (see (3.11)): P=p4)
-I 1+$-q [ I *
(3.24)
TRACE
43
ANOMALIES
We see therefore that the second of terms in (3.23) is gauge-dependent but finite; whereas all the terms in the first set are gauge-invariant and coordinate-invariant, and the whole set has the general form 1 + x (poles)
(3.25)
fFOeCVFOLLV + c (poles){other operators},
which is exactly the form the normal product f [F,,F“““] must take [7]. Furthermore [F,,F”“] must be composed only of gauge-invariant and coordinate-invariant terms with the correct charge conjugation symmetry as well [4]-in fact all those terms appearing in the gauge-invariant Lagrangian, plus any which vanish when integrated over all space. So from (3.23) we see that the poles have disappeared from the gaugedependent terms as required for consistency, and that [F,,F’“]=~F,,,F,Y”+$
(y*--(n-4)<$lnZ,)
- 4(n - 4&-4 B
P Ii La+(n
P + (Lc+(n
HZ
)
[E,]
P* F+ Lb+@--4) ) c
_ 4@ + 47) (n-4)
3ZH
i.
1G (3.26)
The only “new” term is the last one, involving a2H, as this is the only possible term with the correct symmetries which vanishes at zero momentum. There is no term in a,,(yl~~“w) as this has the wrong charge conjugation. The new coefficient a(a) is a finite function of a only, and L, is a pole series of the usual type (3.4) with residue functions of a only. The particular form of the coefficient of a2H is chosen for convenience, but as a whole it is just a pole series in a. u and L, do not depend on < because they are determined completely by the poles of the function (d/&2)( [F,,,,F""]) in flat space, (A
(]F,,P‘“]))
flat
= i(e[F,,l,F”“])
+ 32(~~)~ pnP4 F=
(finite), (3.27)
and (OIF,,Fuu]) is easily seen to be independent of <. We can now substitute for normal products in the earlier, bare expression for 8, Eq. (2.19), and obtain the manifestly finite version 8= @[F/I']-
{(n-
1) + 7) #,]
+p"-4{/?,F++bG+~cH2+4(c--)~2H},
(3.28)
together with the consistency conditions L, = L,,
(3.29)
44
S.J. HATHRELL
y2G-G0 - (n - 4)< & In 2, = T(a) (3.30)
= I$@, 0).
Note that /?a = -(n - 4)~ + /?,(a), with similar expressions for B* and DC, and a, b, c are independent finite parameters; but Q = o(a) is a finite function of a only. Result (3.30) follows because 8 is finite and independent of r, and it is the standard flatspace result for QED showing that the only dependence on r in In Z, is a linear term in the single pole [8]. In fact (3.3 1) jr is just the anomalous dimension of Z, in the Landau gauge, and is O(a’). More information is available from the RGE for [F,,F”“]. We apply B-‘D Eq. (3.28), using the fact that 6’ is independent of ,u, and find f~-lD{@,,P’]}
to
= (finite)
=+aj7[Eo] --pnd4a(~~F+~~G+PLH’)-4~(au)a’H I
, (3.32) I
together with
p, + up= 0.
(3.33)
This last equation is a strong relation between finite functions of a only, valid to all orders of perturbation theory. As for the scalar theory, it was important to consider j?-‘D rather than just D, otherwise this result would have been lost. The role played by u(a) and L, here is analogous to that played by e(A) and L, for the scalar theory ill* We now turn to functions of more than one normal product to find economical ways of calculating the functions of a describing the curved-space theory. 4. TWO-POINT FUNCTIONS OF NORMAL PRODUCTS The procedure for calculating /?,, /16, /3, and u is based on the obvious application of the ideas presented in Ref. [ 11. p,, /3, and u are found from the renormalisation of flat-space 2-point functions of the energy-momentum tensor. The remaining function pb, the coefficient of the topological G term in the anomaly, is found from the renormalisation of flat-space 3-point functions, and is dealt with in the next section. With the benefit of hindsight from Ref. [ 1] we start straight away with the 2-point
TRACE
45
ANOMALIES
functions obtained by two derivatives of the vacuum functional with respect to the metric, followed by taking the flat-space limit. As before this gives Par”
+ a,A”‘“”
+ cOPeD = (finite),
(4.1)
= qyq?““),
(4.2)
where r+(p)
and A”“” and Cc”‘” are the 2-point vertices corresponding to F and HZ in Pg’,, as given in Ref. [l] in Eqs. (6.6), (6.7). The flat-space energy-momentum tensor is 141 p*=-
-1 qG
6s 6S e nat a’ de,, + eOp dear --I I
The covariant derivative reduces to $”
= $(p -a”‘)
t ieoAo’.
(4.4)
The information necessary for p, and j3, is obtained from two different contractions of Eq. (4.1). First, contraction with qKll vIV gives ~‘“yL,(P2) t (P’)’ Pn-4
4(n - 3)(n + 1) L, t & 1
LcI = (finite),
(4.5 1
where the expansions of a, and cO from (3.3) have been inserted. We could take this slightly further by expressing 13’” in terms of its irreducible components, (4.6 1 and then work with i(~wv~u,,) instead of i(P”8,,,) as in Ref. [ 11. But in practice there is little to be gained, and we choose here to remain with the full 8”“. It has the advantage that the absorptive-part analysis used in the evaluation of i(tP”~,,) then involves the somewhat more physical on-shell matrix elements of the complete energy-momentum tensor, and the calculations are then as presented in Ref. [2]. The important point for the present is that the expression of 8“’ in terms of normal products does not lead to any overall factors of p^ appearing in the expression for PvMv, and so the renormalisation-group analysis gives no advantageous route to its computation. The second contraction of Eq. (4.1) is with qKnq,,,: i(kW) + (p’)’ ,une48L, = (finite).
(4.7)
46
S. J. HATHRELL
The flat-space trace 0 is given by I9 = f(n - 4) F,,“FO;,“” - gn - 1) l&i&0 = &F,“F’“]
(4.8)
- ;((n - 1) + y?(E,]
(4.9)
which can be read off from the corresponding expressions (2.19) and (3.28) in curved-space. We can now use the fact that [E,] makes no contributions in 2-point functions: if P(y) is a polynomial composite in the fields t+?(y) and w(y), not necessarily finite, then
i([4(x)lP(y))= j(~)j(d~dyl)P(+&f =- -WY)=. in flat space. (6x(x) i
(4.10)
This vanishes in flat space because (i) I (dw)(J/Jw) gives zero, and likewise for p; (ii) terms like (~/SW(X)) w(x ) contain d(“)(O) and are dimensionally regularised to zero for a massless theory. These arguments are not rigorous, but their validity can easily be checked in a number of examples-in terms of Feynman diagrams they are a simple consequence of the Feynman rules for the insertion of an [E,] vertex. Using this result, and anticipating the requirements of the next section along the lines argued for the scalar theory, we define another operator
{AZ} = f [F,“F’“] - $ [E,],
(4.11)
= (n-4) FowvJ?“” ( 4B 1 0 * The two versions are, respectively, the normal operator. We then define the 2-point function
product
(4.12) and bare forms of the (4.13)
Ld~‘>=i(~~%~*1).
Although {A*} is not actually a finite operator in general, its only contributions to r,, are from the finite term +[F,,P”] because of (4.10), and so r,, has the renormalisation n-4 * r,4.4 +(P2)2P”-4 -p( 1 L,
= (finite),
defining L, as a pole series in a. The key feature is that r,, contains no poles in ln(p2/p2). Multiplying (4.14) by 8’ and comparing with (4.7) then gives 8L, - (n - 4)2 L, = (finite),
(4.15)
47
TRACE ANOMALIES
so that in particular (4.16)
C,(a) = $X,(4;
i.e., the single pole of L, is completely determined by the triple pole of L,. The triple pole of L, can then be found from its single pole by using the appropriate RGE derived from (4.14). For this we need the RGE for {A * }. From (4.12) we have simply fi- ‘D(B{A * }) = 0,
(4.17)
another particularly simple property of the operator (A ‘}. Therefore from applying to (4.14) we find that the RGE for L, is
,!-‘Ofi’
[(n - 4) + D] L, = (finite) = -$
(ax,).
(4.18)
The right-hand side follows because it is the only finite term on the left. This can then be multiplied through by @/(n - 4))*, expanded and integrated iteratively to yield (4.19) and hence X,(a). The leading, O(a*) terms in X, have cancelled, just as the O(A4) terms cancelled for the corresponding result in the scalar theory, and the leading term is in fact O(a”). The reason is essentially that X3 depends on the derivative of (/I/a). This implies [9] through (4.16) and (3.15) that the coefficient of the H* anomaly, /I,, is also O(a3) and not O(a*) as might superficially have been expected from power counting alone. X,(a) is found from the residue of the single-pole divergence of the function r,,-its original definition in (4.14). In leading order it is 0( 1) and determined by the one-loop, a-independent contribution shown in Fig. 1. If we write P(a)=&a
+@*a2 + O(a3)
(4.20)
we can then determine the leading behaviour to be c,(a)=-
(4.21)
(81~6(‘)a3+O(a4),
P,(a) = - -& (ac,) = (‘l !:“I
) a3 + O(a4).
FIG. 1. Leading contribution to r,, . S95/ 14211.4
(4.22)
48
S.J.HATHRELL
From this and Eq. (3.33) we can also derive I: &Xl 24
u(a) = -
t
a* + O(a3).
(4.23)
1
It would take three- and four-loop calculations to verify these results for u(a) and However, a tractable three-loop calculation suffices to confirm the vanishing of the O(a*) term’ in p,, and this is done in Section 7.
PC(a) directly.
5. THREE-POINT FUNCTIONS OF NORMAL PRODUCTS
Once again the pattern for QED follows the scalar theory quite closely, but with simplifications consequent from the absence of an R$* term. The G term in 9 contributes to the flat-space 3-point functions which are obtained from (a/&2)! applied to the vacuum functional. Thus i2(exeyez)+i
%)
B,-$+8,++8, (
z
+ ( ~~~~Oz)
= (finite).
(5.1)
x
The labelling is as in Section 7 of Ref. [I], suppressing the distinction between position and momentum space. The insertions of [E,] into 3-point functions do not in general vanish, but its contribution to Eq. (5.1) can be removed by defining e, = 8, + f(n - l>[E,(x)]
(5.2)
and substituting into the above. The 3-point functions containing [Eclr] can be translated into 2-point functions using the analogue of Eq. (7.4) of [ 11: if P(y) and Q(Z) are polynomial composites in the fields p and w, then in flat space we have i’( [E,(x)]
PYQz) = -i
+
4,~a’,
(5.3)
from a functional integration by parts; cf. (4.10). (It is straightforward to verify this result in a number of examples-it is a simple consequence of the Feynman rules.) We also need
Substitution of these results into (5.1), using the fact that [E,] makes no contribution in l- or 2-point functions, then yields the modified equation
’ Work done in collaboration with I. T. Drummond.
TRACE
49
ANOMALIES
The terms in this equation are (5.6)
= (n - 4) F,,y,@“G(y
- z) = 4&A*} 6(y -z)
(5.7)
and (e,,,) = b,,B(p:,
P:, P:) + C&P:,
P:, P:).
(5.8)
This last term contains only the contributions from G and HZ in Yg, which give the 3-point vertices B and C. B and C are of course exactly the same as they were for the scalar theory, Eqs. (7.13), (7.14) of [ 11, and we are interested in the particular cases B(p*, p*, 0) = 0, C(P*, P*, 0) = 8(n - 4)(p*)*
(5.9)
and B(p*, 0,O) = 2(n - 2)(n - 3)(n - 4)(p*)*,
(5.10)
C(p2, 0, 0) = 4(n + 2)(P*)*. If we define the 3-point function r,,,
by
as the obvious extension of the definition of r,, , we can obtain two special cases of Eq. (5.5) from the momentum configurations T(p’, p2, 0). Note that these on-shell configurations do not introduce any infra-red divergences. We get (9
83rAaa(~2; P’, 0) + @*~,AP*)
+ 8(n + 4)(p2)* ,u”-~L, = (finite), (ii)
~r,&~*,
(5.12)
0,O) + 4~2~A,(P2)
+ (p’)’
pnm4{ 2(n - 2)(n - 3)(n - 4) L, + 4(n + 2) L,} = (finite).
(5.13)
The first of these equations is at p, = 0 and does not contain any information about L,--it is a consistency check on the analysis using the action principle to provide insertions at zero momentum. The second equation leads to a relation between L,, L, and the poles of the functions r,,, and I’,, . We can further simplify both equations by substituting for B’r,, using Eq. (4.7). (Recall (5.2), (5.6) and the fact that [E,] makes no contribution to 2-point functions.) Thus
50
S. J. HATHRELL
(i) (ii)
B3raAA(p2, p*, 0) + 8(n - 4)(p*)*
,unP4Lc = (finite),
B3raAA(p2, 0,O) + (p2)2p”-4{2(n
- 2)(n - 3)(n - 4) L, + 4(n - 6) L,}
= (finite).
(5.14)
(5.15)
It remains to determine the renormalisation of the 3-point function r,,,. The first term in this is the one-loop, a-independent amplitude shown in Fig. 2. (The momentum factors at the vertices make this ultra-violet divergent, and are also responsible for removing infra-red divergences at zero-momentum and on-shell configurations.) Superficially, therefore, the first poles in j?“r,,, can be expected at O(a”). L, is also O(a3) by the arguments of the last section, so from (5.15) we see that b,, the double pole residue of L,, is O(a3). Hence the single pole residue b, , and the p-fucntion pb, are derived from the RGE for L, and must be of the form {constant + O(a*)}, where the constant term is not determined by this analysis. Information about the renormalisation of r,.,,, is obtained by using the action principle with the equations for the renormalisation of r,,, , i.e., Eqs. (4.13), (4.14) through application of the finite derivative a i?/aa. Using (3.5), (3.17), (3.18) and (4.12) we have (in flat space) (5.16) a$-{A*/=-
f
(A*].
The contribution arising from the insertion of the gauge-fixing term vanishes because of (3.1)-the Ward identity-and the contribution from the photon equation-of-motion operator [E,] can be translated into 2-point functions by the usual functional integration by parts. Using
J. d”z A “(z)
(5.18)
we have i*({A:}{A;}
FIG.
2.
1’d”z[E,(z)])
Leading
contribution
= -4I’,,
to r,,,
.
.
(5.19)
TRACE
Combining
51
ANOMALIES
all these in the derivative of Eq. (4.14) we arrive at
= (finite),
L,
! where we have also added twice Eq. (4.14) to turn the coefficient of r,, into a simple expansion in poles starting at O(a’)/(n - 4). If we multiply through by /?” and use the relationships between r,, , L, and L, given in (4.7), (4.14), (4.15), and the RGE (3.15) for L,, we find that this is indeed consistent with Eq. (5.14). The new information comes on comparing Eq. (5.20) with the general form of the equation for the renormalisation of T,,,(p:, p:, p:), just as for the corresponding result in the scalar theory. Although (A ‘} is actually a divergent operator in elementary Green’s functions, the extra subtractions needed when it appears in the composite function FAAA are of the same type as the subtractions for the corresponding finite operator a [F,,F”“] in its place (see (4.11)). This is because all the contributions in r,,, involving [E,] can be translated by means of Eq. (5.3) into poles multiplying 2-point functions. These latter 2-point functions are just the functions appearing in the subtractions required for i2 < {d [F,,,P“] )“). We therefore deduce that the renormalisation of r,,, takes the general form
r,,,(p:, P$pL2) + 5’ (poles){terms inr,,(pT)i Ii +c4 I-v
(poles)
1{terms
in pfpi } = (finite).
(5.21)
The pole-terms involving L’,,(pf) subtract the poles in ln(pi/p*), and the third term subtracts the remaining divergences. (We should emphasize that this result is only stated here-the proof would require a detailed application of renormalisation theory [lo, 111.) We can now compare this with Eq. (5.20) and deduce the nature of the polecoefficients multiplying r,, . Then going to the momentum configuration pz = pi = 0 we have
raAa(p*,O,O)+ I$:
(4)
/T,,(p2)+(p2)*pne4
(y)3L,=(linite). (5.22)
The pure-pole term L, cannot be deduced from (5.20) because there are too many possible combinations of pfpj in (5.21). L, is therefore defined by Eq. (5.22), as a pole series like (3.4), and the factor of (n - 4)3 B-’ has been extracted from its definition for convenience. It is a “new” quantity, reflecting the appearance of the “new” quantity L, in Eq. (5.15).
52
S. J. HATHRELL
We relate L, to L, by multiplying Eq. (5.22) by j?’ and comparing with Eq. (5.15). Substituting for B’r,, = i(f%) with (4.7) we obtain 2(n - 2)(n - 3)(n - 4) L, + 4L,
(n - 6) + 2a2 &
f
I
c
- (n - 4)3 L, = (finite). Ii
(5.23)
The double pole of L, is therefore determined by the single pole of L, and the quadruple pole of L,. The latter can be determined from its single pole through the appropriate RGE. Once again there is a close analogy with the scalar theory because it transpires that the renormalisation-group structure of Eq. (5.22) entails a cancellation of the leading term in the quadruple pole of L,, and so the leading term in the single pole of /?“r,,, . We now confine attention to the leading terms to demonstrate this cancellation. The RGE for L, is obtained by applying B-’ D/?’ to Eq. (5.22)and using the RGE (4.17) for {A’}. Thus [(n - 4) + D] L,, = (finite).
(5.24)
The single pole residue of L,, Y,, is determined in leading order by the divergence of the graph in Fig. 2, and is clearly O(1). The first term in (5.24) therefore does not contribute to the leading terms in L,, and we deduce [(n-4)+D]L,=$(aY,)+O
(a’, (n ‘*
3 (n a-4j2 ’
,...
This can now be multiplied through by /?‘/(n - 4)3 and integrated iteratively for the pole residues of L,, using the expansion /3(a) = /I, a + O(a’) as before, and gives Y2 = 4/l, Y, a + O(a*), Y3 = $P: Y, a2 t O(a3),
(5.26)
Y4 = O(a4).
Returning now to Eq. (5.23), this shows that the leading double pole of L, is determined only by the leading single pole of L,. The functions of n do not affect the leading behaviour, as argued in Ref. [ 11, and so (5.23) reduces to the simple result b, = 2c, t O(a”).
(5.27)
This is exactly the same relationship as was found for the scalar theory. Substituting the results at the end of Section 4 and using the RGE (3.15) for L, leads to
53
TRACE ANOMALIES
b, = const+pg
a* + O(a3), (5.28)
/lb = const - ‘+
cf2+ O(a”).
The constant term is not determined because b, only depends on b, through ab,/h, but it is known from elsewhere [ 12, 131 and only depends on the non-interacting theory. Presumably it could in principle be found in the general flat-space approach here from a consideration of the one-loop, 3-point function involving i2(8”A0A,,tl”‘,), but other techniques ar less cumbersome in this case. We summarise the argument in this section, as it is quite involved. The coefftcient L, of the topological G term in the action is found through its double pole residue from its contributions to the 3-point function obtained by applying (~5/&f2)~ to the vacuum functional. The contributions from the fermion equation-of-motion operator [E,] can be absorbed by two manoeuvres: part is translated into 2-point functions, and the rest is incorporated into a new operator (A*}. Although not a finite operator, (A’} has simple renormalisation and renormalisation-group properties. Then L, is determined by the divergences of 2- and 3-point functions of (A *}, i.e., r,, and r,,, . Furthermore the renormalisation-group structure of r,,, implies a cancellation of the leading pole of /“r,,, , so that it does not contribute to the O(a3) double pole of L,. This cancellation is much like the cancellation of the O(a’) pole of j?‘r,, shown in the last section, and the whole analysis mirrors the scalar theory rather closely. These results for the topological part of the anomaly do not agree with the results of Shore [ 141, who finds a non-vanishing O(a) contribution by direct calculation in a spherical spacetime. We remark in this context that the vanishing of the O(a) term found here does not depend on the vanishing of the O(a3) pole in j?“r,,, , which is relevant only to the O(a’) term in Pb, but it is sensitive to the vanishing of the O(a’) pole in /?rAA, and hence the O(a*) term in the R* part of the anomaly. The latter cancellation is confirmed here by explicit calculation in Section 7. This concludes the general analysis of massless QED in curved space with particular emphasis on the renormalisation of the trace anomaly. The remainder of this paper is concerned with applying the arguments presented so far to the actual calculation of the coefficients. /3, is found to O(a) by a 2-100~ calculation [2] described in the next section. /I, is found to O(a3) by a simple one-loop calculation described in Section 7, from which Pb and u are determined to O(a’). A related 3. loop calculation directly confirms the vanishing of the O(a’) term in /?, as predicted by the more powerful renormalisation-group arguments. 6. ABSORPTIVE
PARTS,~NFRA-RED AND THE CALCULATION
DIVERGENCES OF/Z,
It was shown earlier in Section 4 that i(80) and L, had their first poles at O(a3). The RGE for L, shows that the double and higher poles depend on pa(&.z,/k?a) and
S. J. HATHRELL
54
FIG. 3. l-loop contributions to i(B““B,,,.).
must be O(a2) at least. So from Eq. (4.5) we deduce that the F part of the anomaly, i.e., the coeffkient p,, is determined to O(a) by a,(a) i(@‘“@,,) + 20(p2)2 PnP4 n-4 = (finite) + O(a2), I I
(6.1)
and therefore by the divergences of the I- and 2-100~ amplitudes i(fP”19,~,), evaluated in flat space. The relevant diagrams are shown in Figs. 3 and 4. This analysis predicts the overall cancellation of the double poles in the 2-100~ amplitudes. The Feynman rules are the standard ones for massless QED, together with the vertices for the insertion of P‘“(p) which are derived from the expressions in Eq. (4.3). These vertices are depicted in Fig. 5 and have the following values: (i)
For P”e+e-: VL;“(kI, k,) = {[y”(k,
(ii)
+ k2y + y”(k, + k,)’ - W”‘(K~ + KJ].
(6.2)
For P”yy: V;“““(k,,
k2) = q’“(k, . k, ya4 - k;kf) -k,
. k,(t,+=q”” + tf’“rfD)
- qu4(k:k; + kfjk;) + (k:k;tyv4 f kt’k;y’”
(iii)
+ k;kfq””
+ k;kfq”“).
(6.3)
For e”“e+e-y: e, V$J”, = - ~eo(yP~“a + y”tj+” - 2q”“y”).
FIG. 4. 2-100~ contributions to i(S““S,,.).
(6.4)
TRACE
FIG.
55
ANOMALIES
5. Basic vertices for the insertion of 8”“.
Notice that Vyua5 satisfies V;ua4k,a = V;“a4k,,, = 0
in accordance with the Ward identity expression
(6.5)
(3.1). We remark also for later use that the
Pab(k,, k,) E k, . k,ge4 - k’;‘k:
(6.6)
appearing in (6.3) is a gauge-invariant projection tensor. The calculation of the l-loop amplitudes in Fig. 3 and the 2-100~ amplitudes in Fig. 4 by an absorptive-part argument was described in some detail in an earlier paper [2]. These details are not central to the rest of the analysis here, so it will be sufficient to quote the result: i(e’“e,,> = -
Er’-4(P2)2 I& t4nj2
(3 +$
(+)I
+ O(a’)[
+ (finite).
(6.7)
All the infra-red divergences cancel when the diagrams are summed, and in addition we see that the double ultra-violet pole is absent to O(a) as predicted in the preceding arguments. Comparing with (6.1) we deduce (6.8)
giving the coefficient of the F term in the anomaly to O(a). It should be noticed that while the 0( 1) terms in the coefficient of F in the bare and the normal product forms of the anomaly are the same, i.e., the O(1) terms in -(n - 4) a, and ,u”“-“ba, respectively (from (2.19) and (3.28)), the O(a) terms differ by a factor of 2. This is because the finite operator [F,,P”] contains a subtraction involving F at O(a); cf. (3.26). It is the finite, normal product version of the anomaly with the a-dependence given by P,(a) which is appropriate when taking the n + 4 limit. Similar considerations apply to the other curvature terms in the anomaly.
56
S.J.HATHRELL
7. CALCULATION
OF THE REMAINING
COEFFICIENTS
It was shown in Section 4 that the coefficient of the HZ term in the anomaly, j?,, was determined to O(a3) by the l-loop divergence of the 2-point function r,, . This in turn provides the [F,,F”“] - c?*H mixing coefficient a(a) to O(a’), and through the arguments of Section 5 it also provides the coeffkient of the remaining topological G term pb to O(a’). Furthermore the renormalisation-group structure of r,, implied [9] a cancellation of the O(a’) pole of i(M), and therefore the O(a) term in a(a). In order both to evaluate the parameters Pb, /?, and u in leading order, and also to verify the cancellation of the O(a’) pole in i(M), we therefore examine the set of amplitudes up to the 3-100~ level shown in Fig. 6, labelled K,-K,. These are defined for an operator insertion of $(n - 4)FOU”F0PI’ =jqA2)
(7.1)
at each end, and include an overall factor of i. In complete analogy with the scalar theory [ 11, we then have qT Ki $ (p’)” ,Pe48L, iC1
’ + Ki + (p*)*$-’ 7 ,r,
= (finite) t O(a’),
(7.2)
‘(’ i ‘O) = (finite) t O(a”),
(7.3)
b
FIG. 6. (a) I-loop, (b) 2-loop, and (c) 34oop amplitudes K,-K,.
TRACE
57
ANOMALIES
l 4 Ki+ (p*)*p Bn-4 2 L, F iY1 ( 1 Equations (7.2)-(7.4) follow, respectively, from only compute the leading terms in the (n - 4) and this information provides, respectively, the of u(a) to O(a), and the single pole of L, to amount to two- and one-loop calculations only,
= (finite) + O(a3).
(4.7), (3.27) and (4.13). However, we expansion of the amplitudes K,-K,, single pole of L, to O(a’), the value O(1). In effect these last two results and reduce to (finite) + O(a2),
;(K,+K,)+(p’)‘p’-4&=
1 (n - 4)2
K, + (P’)’ pnV4 (n
J-1
= (finite) + O(a).
We emphasize two points. First, Eqs. (7.2)-(7.4) amount to the definitions of the terms L,, (a + L,), L, and do not depend on any relationships between them. Second, although (7.6) is the simplest of these results to evaluate, depending only on the single-loop diagram in Fig. 6(a), through the powerful renormalisation-group arguments it also provides more information than the direct 2- and 3-100~ calculations: using only the value of the b-function it gives c, to O(a-‘) and u to O(a’), according to Eqs. (4.20~(4.23). The calculations leading to (7.2) and (7.5) serve only to verify the vanishing of the low-order terms in c, and rr. The Feynman rules are as usual [ 181 for massless QED, together with the vertex VP0 for the insertion of (7.1), obtained from (6.3) by contraction (see Fig. 7), V(k,
3k2) = (n - 4) Pyk,,
k,),
(7.7)
with Paa the gauge-invariant projection tensor defined earlier in (6.6). Calculation of K,-K, is straightforward. In the case of K, the photon self-energy 2-100~ subdiagram can be obtained through the absorptive-part approach. In fact this calculation gives the well-known 2-100~ b-function [4, 19,201 as a by-product, since the photon renormalisation to O(a’) appears as part of the diagrams K,-K,:
JJ FIG.
I.
Vy4, vertex
for the insertion
P kz
of f(n - 4) F,,,.F,U“.
S. J. HATHRELL
=,=l+(n-4)
-q”(L)+4
3
47r
P(a) = Ba $lnZ,=$(~)+8
(+J2]+o(a~)’
(7.8)
(+-)2+O(cf3).
(7.9)
Working to O(a’), the results for these amplitudes are
K,=_ (P”)’r4 3 w* I K2 = _ (P’)’ e4 (47r)*
(n - 4) + O(n - 4)2 1)
l&j
[--F
(C)‘]
(7.10)
+ [4
+ O(n - 4) * O(a) , i K,+K,+K,=-
“;~~~-” 7l
K,+K,=-(p~~~~~~4
(2)
+O(a2)] (7.11) (7.12)
(0(1).0(a2)},
]j---+[~(-f-)*]+O(l)-O(az)~. n ?t
(7.13)
K, includes the charge renormalisation to O(a). The absence of a pole in (7.12) below O(a”) reflects the fact that the double poles in the one-particle irreducible
photon self-energy insertions cancel between each other at the 2-100~ level, which in turn reflects the absence of the double pole in Z, below O(a3). Explicit calculation of K, and K, is more difficult. However, it can be seen that the poles arising from the fermion loops cancel between the two diagrams for the same reason that the ordinary 4-photon interaction in QED is actually finite, although superficially it is logarithmically divergent. (The symmetry factors for K, and K, are 1 and $, and match the permutations of the 4-photon Green’s function.) The remaining loop integrations can only contribute a double pole, which is removed by the (n - 4) factors appearing in the vertices for the operator insertions. Hence K, + K, = - (‘;,
Combining
I:-’ ?t
{O(l) . O(a*)}.
(7.14)
all these amplitudes we find i i=
Ki = (finite) + O(a3), I
f (K, + K,) = (finite) + O(a*),
(7.16)
TRACE
1
K, = _ (P212F4
(n - 4)2 Comparison
59
ANOMALIES
1
(47Q2
I
&-q+o(l)/.
(7.17)
with (7.2) and (7.5), (7.6) then gives L, = O(d) = L,,
(7.18)
u = O(a2),
X,(a)=-L a2 2
(7.19) 1
(4x)
+ O(a).
We may now insert the value of /I(a) into the earlier results (5.28) and (4.22), (4.23) to deduce (7.21) (7.22) 1 a2 ~7=-~(4n)4+
O(d),
(7.23)
where we have now incorporated the constant term in & obtained from Refs. [ 12, 131. Equation (7.21) does not agree with the O(a) result of Shore [ 141, obtained on a spherical background. As a final check on the consistency of the renormalisation-group analysis leading to the value of /Ib, we consider the 3-point function TAAA(p2, 0,O) to O(a). The diagrams are shown in Fig. 8, and two of the three operator insertions are at p2 = 0. The general arguments of Section 5 led to Eq. (5.22) for the renormalisation of this function, and the form of (5.22) was essential to the renormalisation-group analysis involved in the evaluation of fib to O(a’)--though it was not essential to the vanishing of the O(a) term. A particular prediction of Eq. (5.22) which can be verified quite easily is that there are no poles multiplying ln(p2/,u2) in TAaa(p2, 0,O) below O(a’), whereas superficial power-counting suggests the appearance of such
FIG.
8. Diagrams for r,,,
to O(a).
60
S.J. HATHRELL
poles at O(a). This feature is rather in contrast with the scalar theory, in which the ln(p2/p2) poles appeared in leading order; cf. (7.38) of [I]. The Feynman rules for the insertion of {A’} are just as in (7.7), but with an additional factor of p^-‘. Evaluation of the integrals is straightforward using formulae (A.l) and (AS) of [l], and leads to
(7.24) The O(a) poles in ln(p2/p2) have indeed cancelled between the diagrams. Comparing with (5.22), we also have
($) + O(a’)f, W4 = --!c4K12 I+-+Y,(a)
= (4;)’ + Oh’>. 2
(7.26)
These values are consistent with (5.26). Although the actual value of Y, was not relevant to the calculation of the O(a") term in Pbr higher order corrections to Ph will depend on it. In fact if we combined (7.25) with the calculation of the diagrams K, and K, in Fig. (6), a fuller treatment of the RGE for r,,,, , and the value of ,@a) to the 3-100~ level, this would provide suffkient to find Pb to O(a"), u to O(a") and /3, to
O(a"). 8. DISCUSSION AND CONCLUSIONS The final expression for the trace anomaly is 8 = $[I;,“F’“]
- {(n - 1) + 7) f[&l
+~““-4{~~i,++~G+~~~2+4(c-~)~2~),
(8.1)
and the values obtained for the parameters are
&-(n-4)aA
20(&2
/3++(-J-)+O(aq' IA--+(--J-)'+O(a')[, 360
(8.2)
(8.3) (8.4)
TRACE
61
ANOMALIES
P-5)
In addition, y(a) is the usual anomalous dimension of the fermion fields, evaluated in the Landau gauge; and &(n-4)+
If
(-f-)+8
(+)‘+W3)/
(8.6)
a, b and c are independent finite parameters obeying inhomogeneous RGEs, [(n - 4) + D]c = PC@>,
(8.7)
and similar equations for a and 6. The normal products [E,] and (F,,,P”‘] are given by Eqs. (3.7) and (3.26). Equation (8.1) is in a suitable form for taking the limit as n -+ 4. Then the dependence on the unspecified parameters p, a and b disappears, but c remains with a finite effect through the coefficient (c - a(a)) of the curvature term a*H--which is perhaps more familiar as the “ambiguous” i3*R in the anomaly. This needs a rather more careful discussion. The independent finite parameter c obeys the inhomogeneous RGE (8.7), so it cannot consistently be set to zero under a scaling of the other renormalised variables. It is independent by virtue of the independent solutions of the corresponding homogeneous equation. To examine this more closely, split c into two parts, c = F(a) + c’, where c’ satisfies the homogeneous equation. In the limit n --) 4 we have
Nab $ F= P&l, p d c’ = 0. & We can solve (8.8) perturbatively
(8.9)
in a and combine with (8.5) to find 2
4(F - a) = *
+ O(a”).
(8.10)
In fact F and u are closely related by
aaaac=-*
(8.11)
from comparison with (3.26). Notice however that the n-dimensional solution of (8.7) expanded as a power series in (n - 4) is not perturbative in a. Then F contains terms in In a and powers of a-’ starting at O(n - 4)‘. The remaining constant c’ is truly
62
S. J. HATHRELL
independent, and can be chosen to vanish-or in physical terms, it must be determined by experiment. It is precisely the arbitrariness of this finite parameter c’ which provides the “ambiguity” in the cT*R part of the anomaly. There are two further points concerning the ambiguity which should be made. First, the c?*R term as a whole plays a significant role in the analysis: in fact it is the only term which contributes to the (p’)’ pne48L, term in Eq. (4.7), and is essential to the cancellation of the poles in the flat-space function i(88). It is also essential as a term in the subtractions required to make the normal product [F,,,P”] finite at nonzero momentum. Second, the choice of the term (F + $‘R) for the anomaly, which is quite common in the literature, is no more than an artefact of the definition of the curvature term F. If F is taken to be the square of the n-dimensional Weyl tensor in the regularised theory, as here and in [I], then its behaviour under conformal scaling is given by Eq. (2.19) and there is no i?*R term from this source. But if F is defined as the 4-dimensional Weyl tensor instead, so that the coefficients (see (2.9) of ] 11) take their n = 4 values even in the regularised theory, then in the right-hand side of Eq. (2.19), F is replaced by (F + $3’R). This amounts to taking slightly different linear combinations of the three curvature terms F, G and HZ as a basis for q. In conclusion, the results of this paper and Ref. [ I] show that the trace anomaly contains all the essential information about the curved-space theory, not surprisingly in view of the close relation between the energy-momentum tensor and the coordinate-invariance of the theory; and that the various parameters in the description can be determined in perturbation theory from flat-space calculations alone. In so far as the assumptions about renormalisability and the use of the renormalised action principle have not been proved here, one might take the view that the results found here require conventional proof of, for example, the finiteness of the energy-momentum tensor in the flat-space theory. But it seems that this approach does not give sufficient weight to the fundamentally important role of the energymomentum tensor in the curved-space theory, and we believe a more profitable attitude is to emphasize the importance of renormalisability in curved space as a starting-point for generating results-if indeed the theories are renormalisable, although we have found no evidence to the contrary. It should be possible to extend the ideas to a discussion of non-abelian gauge theories, and in this context the absorptive-part approach to the calculation of Feynman amplitudes, based on unitarity considerations, should be useful in simplifying the contributions from ghost loops. Further extension to supersymmetric theories would be interesting in view of the connections between the axial and trace anomalies [21,22], although the simple dimensional regularisation employed here would not then suffice.
ACKNOWLEDGMENTS I am grateful to I. T. Drummond for many conversations, useful suggestions, and assistance with some of the calculations, and to the U. K. Science Research Council for supporting part of this work.
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