Abstract
The invariants in D=4, N=4 supergravity are discussed up to the three-loop order (where one expects a general R4 structure). Because there is an anomaly in the rigid SL(2,R) symmetry of this theory, the analysis of possible restrictions on three-loop divergences due to duality needs careful treatment. We show that this anomalous symmetry is still strong enough at the three-loop order to require duality invariance of candidate counterterms. Provided one makes the additional assumption that there exists a full 16-supercharge off-shell formulation of the theory, counterterms at L⩾2 loops would also have to be writable as full-superspace integrals. At the three-loop order such a duality-invariant full-superspace integral candidate counterterm exists, but its duality invariance is marginal in the sense that the full-superspace counter-Lagrangian is not itself duality-invariant. We show that, subject to the assumption that a full off-shell quantisation formalism exists, such marginal invariants are not allowable as counterterms.
Highlights
Developments in the evaluation of scattering amplitudes using unitarity methods over the past decade or so have made it possible to push the investigation of the onset of ultra-violet divergences in maximal supergravity theories to higher loop orders than would have been possible using conventional Feynman-diagram techniques
Since these invariants are F-type, i.e. correspond to integrals over fewer than the maximal number of odd superspace coordinates, it might have been thought that they should be protected by superspace non-renormalisation theorems [5], but it is difficult to justify this argument because there are no known off-shell versions of maximal supergravity that realise all of the supersymmetries linearly
Such off-shell versions cannot exist in every dimension because it is known that divergences do occur for F-type counterterms in D = 6 and D = 7 above one loop [6]
Summary
Developments in the evaluation of scattering amplitudes using unitarity methods over the past decade or so have made it possible to push the investigation of the onset of ultra-violet divergences in maximal supergravity theories to higher loop orders than would have been possible using conventional Feynman-diagram techniques. It is known that the volume of superspace vanishes on-shell for any N in D = 4 there is still an N = 8 seven-loop invariant that can be written as a manifestly dualityinvariant harmonic-superspace integral over 28 odd coordinates [19] The situation at this order is somewhat ambiguous, it is unlikely that there is an offshell formulation of the maximal supergravity theory preserving all the supersymmetries linearly which could be used to try to justify the absence of a seven-loop divergence. The borderline F/D problem is difficult to analyse from our field-theoretic point of view, but for higher loops there will certainly be candidate counterterms that are purely D-type and whose integrands are invariant with respect to all known symmetries In this sense, an unambiguous test of “miraculous” ultra-violet cancellations in half-maximal supergravity requires calculations at one loop higher than those that have been carried out to date.
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