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

256 P. van Nieuwenhuizen, Supergravit’yseems unlikely. Secondly, other N = 8 formulations might exist, notably formulations involvingantisymmetric tensor fields. However, it is known that different field representations for a given spin dolead to the same S-matrices (although they lead to different anomalies [560]).It should, however, not beoverlooked, that using antisymmetric tensor fields, the trace anomalies cancel “miraculously” in theN = 8 model [609].Something is going on here.Extra symmetries usually help to eliminate possible invariants. For N = 8, the maximal on-shellsymmetry group is U(8) (see subsection 3.3), and W,Jk, is manifestly SU(8) covariant, while a U(1)invariance would violate the self duality property. Thus there seem to be no on-shell extra symmetriesleft. (The chiral-dual symmetries first discovered in “miraculous” cancellations were later shown to bepart of the U(N) invariance. In addition, the N = 8 model has a self-duality, see section 6.)In the presence of central charges, the integration measure is no longer d320 since in that case thehighest component of a multiplet varies into a total derivative plus something more. However, centralcharges usually vanish on-shell (section 6).Yet another approach is to pin one’s hope on the spontaneously broken version of the N = 8 model(section 6). However, mimicking in superspace the dimensional reduction from the massless d = 5,N =8 theory to the massive d =4, N =8, one would expect to end up with a dangerous invariant £2? ind = 4 dimensions. In fact, the reverse might happen: it might be that massive d = 4, N = 8 is not evenone-loop finite. This is presently unknown and under study.Of course, miracles do sometimes happen, but they would have to happen at every ioop order. Itseems better to accept a fundamental property of gravitons: that no nearby massless theory exist (thevan Dam—Veltman theorem) and to use nonperturbative methods. Perhaps this is the way quantumsupergravity should go in the future, but it is easier to say that one should do nonperturbativecalculations than to do them.2.10. Explicit calculations2.10.1. Green’s functions are not finiteAs a first application of the Feynman rules derived in the last section we calculate the gravitonenergy in simple supergravity at the one-loop level. This yields at once an important result of quantumsupergravity: Green’s functions are not finite in general [510].Later we shall see that S-matrix elementsare sometimes finite.The general amputated two point function receives contributions from a spin 2 loop, a generalcoordinate (spin 1) ghost, a spin 3/2 loop and a supersymmetry (spin ~) ghost, see fig. 1. The divergent partsare local and the general form is (using dimensional regularization)D~,.,,,0.= —~-—~ [Ap~p,.p~p,, + (2B6~~6,.,, + ~ + (2Dp~p,.6,.,~+ 4Ep~pp6,.c,.)p 2]where one should still symmetrize in (j~ii),in (po) and under (~~v) ~ (,po). As regularization scheme we~graviton spin 1 ghost gravitino spin ~ghostFig. 1. Graviton self energy graphs.