Propagators, BCFW Recursion and New Scattering Equations at One Loop

TL;DR

This paper explores the derivation of one-loop propagators from tree-level amplitudes using BCFW recursion and scattering equations, introducing new worldsheet formulas and connecting different modern amplitude approaches.

Contribution

It presents a novel connection between BCFW recursion and scattering equations at one loop, including new worldsheet formulas based on one-loop scattering equations.

Findings

Explicit one-loop integrand examples with and without supersymmetry

Derivation of standard Feynman propagators from new scattering equations

Proof of a worldsheet formula for all multiplicities

Abstract

We investigate how loop-level propagators arise from tree level via a forward-limit procedure in two modern approaches to scattering amplitudes, namely the BCFW recursion relations and the scattering equations formalism. In the first part of the paper, we revisit the BCFW construction of one-loop integrands in momentum space, using a convenient parametrisation of the D-dimensional loop momentum. We work out explicit examples with and without supersymmetry, and discuss the non-planar case in both gauge theory and gravity. In the second part of the paper, we study an alternative approach to one-loop integrands, where these are written as worldsheet formulas based on new one-loop scattering equations. These equations, which are inspired by BCFW, lead to standard Feynman-type propagators, instead of the `linear'-type loop-level propagators that first arose from the formalism of ambitwistor strings. We exploit the analogies between the two approaches, and present a proof of an all-multiplicity worldsheet formula using the BCFW recursion.