On-shell constructibility of tree amplitudes in general field theories

Abstract

We study "on-shell constructibility" of tree amplitudes from recursion relations in general 4-dimensional local field theories with any type of particles, both massless and massive. Our analysis applies to renormalizable as well as non-renormalizable interactions, with or without supersymmetry. We focus on recursion relations that arise from complex deformations of all external momenta. Under certain conditions, these "all-line shift recursion relations" imply the MHV vertex expansion. We derive a simple sufficient criterion for the validity of the all-line shift recursion relations. It depends only on the mass dimensions of the coupling constants and on the sum of helicities of the external particles. Our proof is strikingly simple since it just relies on dimensional analysis and little-group transformation properties. In particular, the results demonstrate that all tree amplitudes with n>4 external states are constructible in any power-counting renormalizable theory. Aspects of all-line shift constructibility are illustrated in numerous examples, ranging from pure scalar theory and the massless Wess-Zumino model to theories with higher-derivative interactions, gluon-Higgs fusion, and Z-boson scattering. We propose a sharp physical interpretation of our constructibility criterion: the all-line shift fails precisely for those classes of n-point amplitudes that can receive local contributions from independent gauge-invariant n-field operators.