The gauge dependence of the conformal anomaly for spin 3/2 and spin 2 fields
in non-conformal supergravities has been a long standing puzzle. In this Letter
we argue that the `correct' gauge choice is the one that follows from requiring
all terms that would imply a violation of the Wess-Zumino consistency condition
to be absent in the counterterm, because otherwise the usual link between the
anomaly and the one-loop divergence becomes invalid. Remarkably, the `good'
choice of gauge is the one that confirms our previous result that a complete
cancellation of conformal anomalies in D=4 can only be achieved for N-extended
(Poincar\'e) supergravities with $N\geq 5$

Meissner, Krzysztof A.

Nicolai, Hermann

English

arXiv

The gauge dependence of the conformal anomaly for spin 3/2 and spin 2 fields in non-conformal supergravities has been a long standing puzzle. In this Letter we argue that the `correct' gauge choice is the one that follows from requiring all terms that would imply a violation of the Wess-Zumino consistency condition to be absent in the counterterm, because otherwise the usual link between the anomaly and the one-loop divergence becomes invalid. Remarkably, the `good' choice of gauge is the one that confirms our previous result that a complete cancellation of conformal anomalies in D=4 can only be achieved for N-extended (Poincar\'e) supergravities with $N\geq 5$

Meissner, K.

Nicolai, H.

MPG.PuRe

arXiv:1705.02685v1  [hep-th]  7 May 2017Conformal Anomaly and Off-Shell Extensions of GravityKrzysztof A. Meissner1 and Hermann Nicolai21Faculty of Physics, University of Warsawul. Pasteura 5, 02-093 Warsaw, Poland2Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut)Mühlenberg 1, D-14476 Potsdam, GermanyThe gauge dependence of the conformal anomaly for spin- 32and spin-2 fields in non-conformalsupergravities has been a long standing puzzle. In this Letter we argue that the ‘correct’ gaugechoice is the one that follows from requiring all terms that would imply a violation of the Wess-Zumino consistency condition to be absent in the counterterm, because otherwise the usual linkbetween the anomaly and the one-loop divergence becomes invalid. Remarkably, the ‘good’ choiceof gauge is the one that confirms our previous result [1] that a complete cancellation of conformalanomalies in D = 4 can only be achieved for N-extended (Poincaré) supergravities with N > 5.PACS numbers: 04.62+v, 04.65+e1. Introduction. Conformal anomalies have been asubject of investigation for a long time [2–11]. Whiletheir relevance is obvious for theories that are manifestlyclassically conformal, they are equally relevant for non-conformal theories, in the same way that gauge anoma-lies are relevant even if gauge invariance is spontaneouslybroken. While the anomaly coefficients are unambiguousfor spin s 6 1 fields, it has long been known that in thecontext of (non-conformal) Poincaré or AdS supergravi-ties the anomaly coefficients for spin- 32and spin-2 fieldssuffer from an apparent ‘gauge-dependence’, in the sensethat they depend on how the theory is taken off the massshell (defined in lowest order by Rµν = 0) [5, 6]. This isa puzzling feature, because the anomaly, being a physicalquantity, should not depend on the choice of gauge, un-like the divergent counterterms that must be introducedto render the results finite. This is in marked contrast togauge anomalies, where such a gauge dependence is notobserved, a feature that is known to be absolutely cru-cial for the consistency of the Standard Model of particlephysics. In this Letter we propose a simple prescriptionto resolve this puzzle and to arrive at a unique and phys-ical answer also for conformal anomalies.We refer to the papers cited above for reviews andbasic results on conformal anomalies. We assume thatthere exists a (non-local) effective action functional Γeff ,depending on the metric gµν and various matter fields,such that the conformal anomaly A can be representedas the variational derivativeT µµ(x) ≡ −1√−g(x)δΓeffδσ(x)= A(x) + · · · (1)where σ(x) ≡√−g(x) is the conformal factor. The dotson the r.h.s. stand for non-gravitational contributionsdue to matter interactions, or due to explicit or sponta-neous breaking of conformal invariance (such as explicitmass terms ∝ m2ϕ2) that are not relevant to our argu-ment. We emphasize that (1) remains a perfectly validdefinition of the trace of the quantized energy momen-tum tensor also for non-conformal theories. As a directconsequence of (1) the trace, and thus the anomaly, mustsatisfy the integrability (alias Wess-Zumino) consistencyconditionδ(√−gA(x))δσ(y)=δ(√−gA(y))δσ(x)(2)independently of whether we are dealing with a conformalor a non-conformal theory.As is well known (see for instance [8]) the gravitationalpart of the conformal anomaly in four dimensional space-time takes the formA(x) = 1180 · 16π2(aE4(x) + cC2(x))(3)whereC2 ≡ CµνρσCµνρσ = RµνρσRµνρσ − 2RµνRµν +13R2E4 = RµνρσRµνρσ − 4RµνRµν +R2 (4)and the coefficients a and c depend on the type of fieldcoupling to gravity that induces the anomaly. E4 is theGauss-Bonnet density, which is a total derivative. Inwriting (3) we omit a possible further term ∝ R whichcan be removed by a local counterterm (∝ R2). It isstraightforward to check that the two terms displayed in(3) do satisfy the consistency condition (2), whereas thesquare of the scalar curvature R2 by itself does not. Im-portantly, possible extra terms indicated by dots in (1),and more specifically any terms arising for non-conformaltheories, must also satisfy (2).Existing calculations of the conformal anomaly rely onthe relation between the conformal anomaly and the one-loop divergence. This was in particular explained in [3]in the framework of dimensional regularization, where,however, spin s > 1 fields were not considered. The di-vergence can be generally represented in the formΓ∞ =116π2∫d4x√−gL∞(x) (5)2where [6]L∞ = b0L4 + b2L2 + b4 ln(L2/µ2) + . . . (6)with the UV cutoff L, and where the dots stand for finite(non-local) contributions. The terms b0 and b2 are re-moved by appropriate counterterms, while the coefficientof the logarithmic divergence contains the physically rel-evant information. More precisely, using the notationfrom [6] which we follow throughout,b4 = β1E4 +β22(C2 − E4) +β33R2 + · · · (7)where the ellipses denote further terms e.g. involvinga cosmological constant, as well as possible matter fieldcontributions. In the context of supergravity the coeffi-cients βi were computed in [6], building on earlier workin [5]. It was also pointed out there that the countertermcoefficients β2 and β3 depend on the gauge choice, whileβ1 does not.The link between L∞ and the anomaly A is encapsu-lated in the relationA(x) ∝ limL→∞[LddL(√−gL∞(x))]F.P.(8)where “F.P.” denotes the finite part (after subtraction ofquartic and quadratic infinities). The formula (8) cap-tures the anomalous scale dependence of the theory inpresence of a (scale breaking) cutoff L (and incidentallyalso explains the appearance of β-functions on the r.h.s.of (1) in the presence of matter interactions). Howeverit is clear that the relation b4 = A cannot hold if β3 6= 0because β3 multiplies the term R2 which does not satisfy(2). Because β3 is gauge dependent our proposal is there-fore to identify the ‘good’ gauge as the one where β3 = 0,because only then do we obtain a consistent anomaly.Let us now examine existing results in view of the po-tential discrepancy for β3 6= 0. For low spins s 6 1 theanomaly coefficients have been known for a long time[5, 6, 12–16], and there is no issue (and hence no gaugedependence) here, as the relevant actions are conformalalso for non-conformal supergravities. For all of these onefinds β3 = 0. However, for spins s >32there are two dif-ferent possible actions. One is conformal (super-)gravity,with kinetic terms of higher order (cubic for spin- 32andfourth order for spin-2), whereas the non-conformal su-pergravity actions are of first and second order, respec-tively, just like the actions for spin s 6 1. Higher deriva-tive actions for higher spin fields have been investigatedrecently in [11], and for them (and thus for conformalsupergravities) one gets again β3 = 0, whence the rela-tion (8) is preserved. For those theories it is found that acomplete cancellation of conformal anomalies is possibleonly for the maximal N = 4 conformal supergravity cou-pled to an N = 4 Yang-Mills gauge theory with gaugegroup SU(2)× U(1) or U(1)4 [7].For non-conformal (Poincaré or AdS) (super)gravitiesthe two explicit calculations with different off-shell for-mulations available in the literature give different resultsfor spins s > 32, including ones with non-vanishing β3[6]. In the Table we show the relevant coefficients to-gether with the values for β3 for the two different gauges,namely the Feynman gauge [6] and the harmonic gauge[11], respectively. As already mentioned, there is no am-biguity for spins s 6 1, and β3 = 0. By contrast, fors > 32the numbers do differ. In the first three columnsof Table 1 we give the coefficients as extracted from [11],cf. appendix of [1]; for these β3 vanishes for all spins.The coefficients in the remaining columns of the Tableare based on Table 1 of [6], yielding β3 = − 13 for spin-32,and β0 =34for spin-2. We thus see that only the a andc coefficients from the first two columns are fully consis-tent with (2), whereas the ones from the fourth and fifthcolumns are not.[11] [6]cs as β3 cs as β30 32− 120 32− 1201292− 1140 92− 11401 18 −31 0 18 −31 032− 411258940 − 23122294− 132 783 −571 0 63 149 34Table 1. Anomaly related coefficients for two gauges.Remarkably, the cancellations exhibited in [1], to wit,c2 + 5c 32+ 10c1 + 11c 12+ 10c0 = 0c2 + 6c 32+ 16c1 + 26c 12+ 30c0 = 0 ,c2 + 8c 32+ 28c1 + 56c 12+ 70c0 = 0 , (9)work only with the numbers from [11], which are the onesyielding a consistent anomaly. Inspecting the differencesin the coefficients for s > 32, viz.∆a3/2 = −∆c3/2 = −90 , ∆a2 = −∆c2 = 720 (10)(which are in accord with the fact that only the sum(a + c) is gauge invariant [4, 6]) we see that the cancel-lation persists for N = 8 supergravity (as already notedin [6]), but fails for lower N . From the present point ofview, however, this is just an accident and the cancella-tions might not work for yet different ways of taking thetheory off shell. Further clarification of the ambiguitiescould come from a Feynman diagram computation of athree-point correlator of energy-momentum tensors (asopposed to the determination of the conformal anomalyfrom the one-loop divergence, as in previous work), anal-ogous to the textbook derivation of the axial anomaly.To the best of our knowledge such a calculation, whichcould exhibit the dependence of β3 on a continuous gaugeparameter, is currently not available.3The phenomenon that we describe is similar in spirit tothe one discovered in [17]. There is a continuous familyof α′ actions of the string gravity (gravity + dilaton +antisymmetric tensor) that can be changed by the lowerorder equations of motion but only one action exhibitsO(d, d) symmetry [18] and this action was proposed in[17] as the ’correct’ one.We consider the cancellations in (9) as indicative of ahidden conformal structure of unknown type underlyingthese theories (and possibly M theory). In [1] wefurthermore suggested a link between these cancellationsand the cancellation of composite anomalies in extendedsupergravities [19], as well as the so far unexplainedfiniteness of N > 5 supergravities [20]; further evidencefor such a link has been exhibited in very recent work[21, 22].Acknowledgments: K.A.M. thanks the AEI for hos-pitality and support during this work; his researchwas also supported by the Polish NCN grant DEC-2013/11/B/ST2/04046. We are most grateful to ArkadyTseytlin for correspondence and clarifications of his ear-lier work. H.N. would also like to thank GianguidoDall’Agata and the Padova group for an enjoyable visitand discussions related to this work.[1] K.A. Meissner and H. Nicolai, arXiv:1607.07312[gr-qc].[2] S. Deser, M.J. Duff and C. Isham, Nucl. Phys. B111(1976) 45.[3] M.J. Duff, Nucl. Phys. B125(1977) 334[4] S.M. Christensen and M.J. Duff, Phys. Lett. 76B (1978)571.[5] S. M. Christensen and M. J. Duff, Nucl. Phys. B154(1979) 301.[6] E. 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Conformal Anomaly and Off-Shell Extensions of Gravity

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