Scattering Amplitude Recursion Relations in BV Quantisable 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 MHV amplitudes. We show that the origin of this recursion relation becomes clear in the BV formalism, which encodes a field theory in an $L_{\infty }$-algebra. The recursion relation is obtained in the transition to a smallest representative in the quasi-isomorphism class of that $L_{\infty }$-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 quantisable theory, which, in turn, produces recursion relations for its tree-level scattering amplitudes.

v3: 33 pages, corrections to section 3.1, improvements of presentation, typos fixed