SUPERGRAVITY P. van NIEUWENHUIZEN To Joel Scherk 0370 ...

254 P. van Nieuwenhuizen. Supergravitvof the quantum field equations (now also of the ghosts) as derived from the full quantum action. Forexample, two BRST transformations on the antighost C*a yield a result proportional to ~5(F 0y’3”) =F0,,R’7C” which is indeed proportional to 6.2~(qu)/6C*a.(Nilpotency is important in renormalizabletheories to analyze the higher order quantum correction.) It is not known whether such theories with 6and higher ghost-couplings exist, and, if they exist, whether one can eliminate these ghost couplings byintroducing auxiliary fields.2.9. Finiteness of quantum supergravity?Quantum supergravity is more finite as far as ultraviolet divergences are concerned than ordinarygeneral relativity. Since this subject has been discussed in detail at many conferences, we refer fordetails to these sources as well as the original literature. For a review, see ref. [286].The first breakthrough [512]occurred in the N = 2 model, where an explicit computation of theone-loop correction to photon—photon scattering revealed that all divergences in the process cancel. Atheoretical explanation was given [512], based on the observation that(i) graviton—graviton 2 and on-shellscattering Rh,. = 0. is finite because possible counter terms are of the form £2? =aR~,.+ (ii) Using /3R (global) supersymmetry of the S-matrix, one can “rotate” gravitons away and replace bygravitinos, etc. This then shows that all four-point processes are finite. (Higher point processes have alsobeen shown to be finite in this way.)Of course one must show that starting from graviton—graviton scattering one can indeed reach allother processes. This has been done.An alternative proof, valid for all n-point functions, was given [123]by showing that no £2? existson-shell which is invariant under the on-shell transformations of supergravity. In particular it was shownthat no £2? can start with gravitino terms but must start with R~,.2and R2. Not all details of this analysishave been published.At the two-loop level, again both proofs of finiteness have been used. Since the only bosonic £2?which does not vanish on-shell is given by [503]£2? = a~2R,,0R~,.”R~’3 (1)and describes helicity-ifipping of graviton—graviton scattering, whereas supersymmetry (global as well aslocal) conserves helicity, a =0. Hence, as before, graviton—graviton scattering is two-loop finite insupergravity, and rotating gravitons away and replacing them by other particles, one shows that also atthe two-loop level supergravity is finite [283].The counter term analysis has matched also this result byclaiming that one cannot extend £2? in (1) to a full on-shell supersymmetnc invariant [123].For N = 1 supergravity, the tensor calculus [189],or, equivalently, superspace methods [191]havesuperseded these proofs and prove one- and two-loop finiteness. In particular, the counter termanalysis was performed only to order 4,2, whereas the tensor calculus and superspace give completeresults with considerable less algebra. For N = 2 supergravity similar results have recently beenobtained.Possible problems (but no more than that!) arise at the three-loop level. Namely, it seems that in allN-extended pure supergravities on-shell a supersymmetric invariant three-loop counter term exists[1291.Whether it actually will have nonvanishing coefficient is the big question. This £2? has in itsbosonic sector the Bel—Robinson tensor.