SUSY Ward identities, Superamplitudes, and Counterterms

Ward identities of SUSY and R-symmetry relate n-point amplitudes in supersymmetric theories. We review recent work in which these Ward identities are solved in N=4 SYM and N=8 supergravity. The solution, valid at both tree and loop level, expresses any (Next-to)^K MHV superamplitude in terms of a basis of ordinary amplitudes. Basis amplitudes are classified by semi-standard tableaux of rectangular N-by-K Young diagrams. The SUSY Ward identities also impose constraints on the matrix elements of candidate ultraviolet counterterms in N=8 supergravity, and they can be studied using superamplitude basis expansions. This leads to a novel and quite comprehensive matrix element approach to counterterms, which we also review. This article is an invited review for a special issue of Journal of Physics A devoted to "Scattering Amplitudes in Gauge Theories".