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

258 P. van Nieuwenhuizen, Supergravitvpgravitino—graviton loopFig. 2. Gravitino self energy corrections.spin i—spin 1/2 ghost loopfirst consider matter coupling theories (i.e., not the extended supergravities) and we will later come backto the pure (simple and extended) supergravities.Only an explicit calculation can demonstrate that matter couplings diverge, as a moment of thoughtwill convince most readers. As an example of such a calculation, consider photon—photon scattering atthe one-loop in supersymmetric Maxwell—Einstein theory. The spin content of this system is (2, 3/2) +(1, 1/2), and the action reads= 2(gauge) + ~2’(kinetic) + ~ eç~u. Fy~LA+ (4,4, 4i2A2 A 4)terms.The process photon—photon scattering is considered since it yields the simplest diagrams; these aredepicted in fig. 3. The diagram “ME” stands for all diagrams which are already present in the ordinaryEinstein—Maxwell system and which contain only gravitons and photons. These one-loop divergenceswere evaluated with the background field method and with normal field theory. As expected, the sameresult was obtained. This result was rather amusing: (137/60) times (n — 4)_i times the energymomentumtensorof thephoton squared. The other diagrams in fig. 3 are what supergravityadds, and threeconclusions are drawn [510]:(i) The supersymmetric Maxwell—Einstein system is invariant under combined ~uality-chiralitytransformationsup to order K2 at least6A —aiy5A.The duality invariance of the ordinary Maxwell—Einstein system and the chirality invariance of themassless Dirac system is well-known. However, the Noether coupling is invariant under 2, sincecombinedF,,,.4 andduality-chirality transformations. This explains why all divergences are of the form T,,,.60 120 402Fig. 3.