On BCFW shifts of integrands and integrals

TL;DR

This paper explores extending BCFW on-shell recursion relations from tree-level amplitudes to loop integrands and integrals, revealing structural similarities and potential for reconstructing integrands from single cuts in gauge theories.

Contribution

It demonstrates that the large BCFW shift limit of integrands mirrors tree-level amplitudes, enabling their reconstruction from single cuts, and investigates the relation between shifts of integrands and integrals at one loop.

Findings

Integrands have the same large shift structure as tree amplitudes in Yang-Mills theories.

Integrands can be reconstructed from a subset of single cuts.

Discrepancies between integrals and integrands are linked to UV and IR divergences.

Abstract

In this article a first step is made towards the extension of Britto-Cachazo-Feng-Witten (BCFW) tree level on-shell recursion relations to integrands and integrals of scattering amplitudes to arbitrary loop order. Surprisingly, it is shown that the large BCFW shift limit of the integrands has the same structure as the corresponding tree level amplitude in any minimally coupled Yang-Mills theory in four or more dimensions. This implies that these integrands can be reconstructed from a subset of their `single cuts'. The main tool is powercounting Feynman graphs in a special lightcone gauge choice employed earlier at tree level by Arkani-Hamed and Kaplan. The relation between shifts of integrands and shifts of its integrals is investigated explicitly at one loop. Two particular sources of discrepancy between the integral and integrand are identified related to UV and IR divergences. This is cross-checked with known results for helicity equal amplitudes at one loop. The nature of the on-shell residue at each of the single-cut singularities of the integrand is commented upon. Several natural conjectures and opportunities for further research present themselves.