Scattering amplitude recursion relations in Batalin-Vilkovisky–quantizable theories

Abstract

Tree-level scattering amplitudes in Yang-Mills theory satisfy a recursion relation due to Berends and Giele which yields e.g., the famous Parke-Taylor formula for maximally helicity violating amplitudes. We show that the origin of this recursion relation becomes clear in the Batalin-Vilkovisky (BV) formalism, which encodes a field theory in an ${L}_{\ensuremath{\infty}}$-algebra. The recursion relation is obtained in the transition to a smallest representative in the quasi-isomorphism class of that ${L}_{\ensuremath{\infty}}$-algebra, known as a minimal model. In fact, the quasi-isomorphism contains all the information about the scattering theory. As we explain, the computation of such a minimal model is readily performed in any BV quantizable theory, which, in turn, produces recursion relations for its tree-level scattering amplitudes.