Amplitudes in YM and GR as a Minimal Model and Recursive Characterization

Abstract

We study the tree scattering amplitudes of Yang–Mills and General Relativity as functions of complex momenta, using a homological and geometrical approach. This approach uses differential graded Lie algebras, one for YM and one for GR, whose Maurer Cartan equations are the classical field equations. The tree amplitudes are obtained as the L_infty minimal model brackets, given by a trivalent Feynman tree expansion. We show that they are sections of a sheaf on the complex variety of momenta, and that their residues factor in a characteristic way. This requires classifying the irreducible codimension one subvarieties where poles occur; constructing dedicated gauges that make the factorization manifest; and proving a flexible version of gauge independence to be able to work with different gauges. The residue factorization yields a simple recursive characterization of the tree amplitudes of YM and GR, by exploiting Hartogs’ phenomenon for singular varieties. This is similar to and inspired by Britto–Cachazo–Feng–Witten recursion, but avoids BCFW’s trick of shifting momenta, hence avoids conditions at infinity under such shifts.