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

P. van Nieuwenhuizen. Supergravity 259are not duality invariant. The combined chirality-duality invariance first found in this system (asa result of “miraculous regularities” in an explicit calculation!) has been found also in the extendedsupergravities as a complete invariance to all orders in K (see section 6).(ii) Mattercouplings are not finite. The sum of the coefficients in fig. 3 is 25/12 and hence the S-matrixdiverges. Also all other matter couplings have been found to be divergent. These include(a) in the (2, ~)+ (~,0~,0—) system photon—photon and scalar—scalar scattering is not finite [516].(b) in N = 4 extended supergravity with spin content (2, 3/2~,16, 1/2~,02) coupled to m matter 0(4)multiplets with spin content (1, 1/2~,06) one finds that gauge—photon gauge—photon scattering isone-loop finite (although virtual matter contributes!) [223]. This can be explained by helicityarguments [512, 515]. Anyprocess whose externalfields are related to the graviton by symmetries is oneand two-loop finite, even though virtual matter fields contribute. However, matter-photon matterphotonscattering is not finite [223].(iii) It is incorrect to state that unitarity prevents cancellation of all divergences (as the finitenessof extended supergravities shows). In a background field calculation, the coefficient of R,L,.2, R2, F,,,.’5 etc.are indeed positive, but on-shell G,,,. = T,,,. and for example R,LIF~”’F~~ can become negative. (For adiscussion about the meaning of G,L,,,. = T,~,.for diagrams, see ref. [607].)Thus we conclude that it seems likely that no matter coupled system exists with a finite S-matrix atthe one-loop level.2.10.3. Calculations of finite processesFor pure N = 1 supergravity, the simplest and at the same time only rigorous proof on one- andtwo-loop finiteness is by means of the tensor calculus. For N> 1, no rigorous proofs exist, but it israther probable that also here one- and two-loop finiteness exists. Historically, though, it was onlythrough explicit calculations of certain processes that one-loop finiteness was discovered. Since thesecalculations have been done up to N = 8, it is very probable that all extended supergravities areone-loop finite. About two-loop finiteness no rigorous results beyond order 1(2 are known because hereno tensor calculus is yet available. Also two-loop computations in supergravity are hard. Hence here atensor calculus might bring relief.N =2. The first successful result in quantum supergravity was a calculation of photon—photonscattering in N =2 extended supergravity [512].The only terms in the action which contribute inaddition to the N =2 gauge action is the Noether coupling= ~ I/F,L(F’~”+ ~ (4,,,. and ç,. are the two gravitons). (1)The set of diagrams is given in fig. 4.137 (23\~ (-i)Fig. 4.