Volume 113B, number 1 PHYSICS LETTERS 3 June 1982 VANISHING p-FUNCTIONS IN EXTENDED SUPERGRAVITIES K.S. STELLE and P.K. TOWNSEND Laboratoire de Physique Th~orique de l'Eeole Normale Supdrieure 1, 75231 Paris Cedex 05, France Received 22 February 1982 Given that-extended supergravity theories can be written in N = I superfield form, the non-renormalization of chiral superspace integrals implies a vanishing #-function for the SO(N) gauged Poinear~ supergravity theories for N 1> 5 at one loop and probably to all orders. In particular, this explains the results of existing one-loop calculations of the #-function. 1. Introduction: N = 4 Y-M theory. Calculations in perturbation theory have shown that unexpected cancellations of ultraviolet divergences occur in ex- tended supersymmetric field theories. The N = 4 super Yang-Mills (Y-M) theory, has vanishing/3-function up to three loops [1-3] while N~> 5 gauged supergravity theories have vanishing one-loop B-function [4,5]. Re- cently it has also been shown that N = 4 conformal supergravity has vanishing one-100p/5-function [6], subject to verification of an assumption concerning the scalar interactions of the, as yet unknown, full ac- tion. These cancellations are unexpected in the sense that they are not explainable solely as a consequence ofsupersymmetry invariance since the counterterm(s), that would be needed if/3 were not zero, do exist. The cancellations ofN = 4 super Y-M theory can be explained in two ways. The first explanation [7] relies on an analysis of the possible N = 1 anomaly supermultiplets consistent with SU(4) or SO(4) inter- nal symmetry, and will not concern us here. The sec- ond explanation, which was mentioned in ref. [2], and which has recently been discussed and contrasted with the previous one in ref. [8], is more directly linked with the manner in which the calculations in perturbation theory are most easily performed, name- ly by means ofN = 1 superfield perturbation theory. Extended supersymmetric Y-M theories can be writ- 1 Laboratoire Propre du Centre National de la Recherche Scientifique, associ6 ~ l'Ecole Normale Sup~rieure et l'Universit6 de Paris-Sud, France. ten as a sum ofN = 1 super-invariants consisting of in- tegrals over the full N = 1 superspace or over the chiral N = 1 subspace. While various internal symme- tries and the remaining N- 1 supersymmetries will not be manifest in this formulation, the techniques of N = 1 superfield perturbation theory may be applied. In particular, there is the remarkable property that terms in the action that are written as chiral integrals, and that can only be written locally in this way, are never renormalized. The corresponding cancellations were noticed in particular models long ago [9,10], while the general theorem was proved in ref. [11]. In practice it is only integrals of chiral potentials, f d20 f(~.) for chiral fields Z, that cannot be rewritten locally as f d40 integrals: Hence mass and interaction terms for chiral fields are never renormalized. This theorem is remarkable in that it applies equally to non-renormalizabie as to renormalizable theories, be- ing a result of 0-counting rather than power counting. Note, however, that the theorem depends on the local structure of the counterterms and so should be ap- plied off-shell. This is because f d20 Z = f d40(D2/ ff]) Z and it may be posaible to remove the non-locality of the d40 expression by use of a field equation. The N = 4 Y-M theory has the following action in N= 1 superfields [12,11]: I= tr[64~ f d4x d20 WaW + f d4x d40 e,g Vcbi egVcj 0 031-9163/82/0000=0000/$02.75 © 1982 North-Holland 25