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

P. van Nieuwenhuizen, Supergrav0y 247Let us now turn to supergravity. There are 16 real fields ,l/a so one expects after gauge fixing thatthere are 8 gravitino modes propagating. Indeed, there are 8 modes in the propagator as an explicitdecomposition shows [522].However, the complex spin 1/2 supersymmetry ghost is expected to subtractonly 4 modes. (Fermions have always half as many modes as bosons.) Where do the remaining 2 modesgo?The solution of this puzzle can be found in the proof of unitarity of ref. [522].To see what happens,we begin by stating that the unitary gravitino propagator is given by~&a(yXy 0)~~~k2. (6)The Ward identities now become (considering a gravitino loop)6BRST(C*Y. çfj) = 0 = ( ~ y,ø)(-y. i/i)) + (C*JC), cf (2) (7)hence (yx)i~fy)~y) = 0 on-shell, since there are extra factors I in (C*/C), and= 0 = (y• i/i(x) i/i~(y))+(C*a(x) O~C”(y)), cf. (3). (8)Notice now that y’4(4i,.,4,,.) = 2k,.k2 so that we may use conservation to replace in one of the two cutpropagators the unitary propagator in (6) by the renormalizable propagator ~y,.X’y~k2. (We also usethat a term k~k,,in (6) vanishes on-shell.) In the other propagator one has leftk2(F~(unit)— F~L~(ren)) = —~(k,.XXy~+ y,,k’R’k,. — 2k,Yk~)(k kr’. (9)The second term can be written as k~times a factor (—1~,.)(k . k)’ times the gravitino propagator~ Cycling as before with k,.,, once on the left, and once on the right, both times clockwise, oneends up with k,, times —t (k . ~ hence with precisely one ghost loop. Notice that contracting agravitino with its own momentum on one side of a blob first of all ffips this fermion line into a ghostline, but, as is clear from (8), it also obliterates the numerator of the gravitino propagator at the otherend, replacing it by the numerator of the ghost propagator.Thus the first two terms in (9) are just equal to the ghost contributions. That leaves us with the lastterm. It represents indeed two modes in the propagator (since R’ can be written as ~ + wóY withspinors w~,w~orthogonal to the physical spinors u~and u); but they decouplefrom the S-matrix. Thisfollows easily if one cycles once with the factor k,. on the left and then uses (7).The above sketch of the unitarity proofhasbeen extendedto general N-particle cuts, taking into accountall ghosts and antighosts. The results are unchanged. One can choose different commutation propertiesbetween ghosts without violating unitarity [553].2.7. Supersymmetric regularizationAll preceding manipulations only make sense if one would use a regularization scheme to givemeaning to the divergent loops. Preferably, but not necessarily, a scheme should be used whichmaintains manifest supersymmetry at all stages. Such a scheme exists. It is an extension of dimensionalregularization [417,425], but there are also some differences. It is based on the idea of dimensionalreduction.