Scattering Amplitude Recursion Relations in BV Quantisable Theories

TL;DR

This paper explains the origin of recursion relations for tree-level scattering amplitudes in Yang-Mills theory using the BV formalism and $L_$-algebras, providing a systematic way to derive these relations in BV quantisable theories.

Contribution

It demonstrates how the BV formalism and minimal models of $L_$-algebras naturally produce recursion relations for scattering amplitudes, clarifying their origin.

Findings

Recursion relations arise from minimal models in BV formalism.

The approach applies to any BV quantisable theory.

Provides a systematic method to derive scattering amplitude recursions.

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.