On-shell superamplitudes in $ \mathcal{N} < 4 $ SYM

Abstract

We present an on-shell formalism for superamplitudes of pure N<4 super Yang-Mills theory. Two superfields, Phi and Phi^+, are required to describe the two CPT conjugate supermultiplets. Simple truncation prescriptions allow us to derive explicit tree-level MHV and NMHV superamplitudes with N-fold SUSY. Any N=0,1,2 tree superamplitudes have large-z falloffs under super-BCFW shifts, except under [Phi,Phi^+>-shifts. We show that this `bad' shift is responsible for the bubble contributions to 1-loop amplitudes in N=0,1,2 SYM. We evaluate the MHV bubble coefficients in a manifestly supersymmetric form and demonstrate for the case of four external particles that the sum of bubble coefficients is equal to minus the tree superamplitude times the 1-loop beta-function coefficient. The connection to the beta-function is expected since only bubble integrals capture UV divergences; we discuss briefly how the minus sign arises from UV and IR divergences in dimensional regularization. Other applications of the on-shell formalism include a solution to the N^{K}MHV N=1 SUSY Ward identities and a clear description of the connection between 6d superamplitudes and the 4d ones for both N=4 and N=2 SYM. We outline extensions to N<8 supergravity.