Vanishing β-functions in extended supergravities

Given that extended supergravity theories can be written in N = 1 superfield form, the non-renormalization of chiral superspace integrals implies a vanishing β-function for the SO(N) gauged Poincare supergravity theories for N >= 5 at one loop and probably to all orders. In particular, this explains the results of existing one-loop calculations of the β-function. (orig.)