On All-loop Integrands of Scattering Amplitudes in Planar N=4 SYM

TL;DR

This paper explores the connection between momentum twistor MHV vertex expansion and all-loop BCFW recursion in planar N=4 SYM, demonstrating their compatibility and introducing a cyclically invariant reformulation.

Contribution

It establishes that MHV vertex expansions satisfy all-loop recursion relations and introduces a new cyclically invariant sum over non-crossing partitions.

Findings

MHV vertex expansions satisfy recursion relations at all loops

Rewriting in terms of non-crossing partitions maintains cyclic invariance

Explicit examples confirm the theoretical framework

Abstract

We study the relationship between the momentum twistor MHV vertex expansion of planar amplitudes in N=4 super-Yang-Mills and the all-loop generalization of the BCFW recursion relations. We demonstrate explicitly in several examples that the MHV vertex expressions for tree-level amplitudes and loop integrands satisfy the recursion relations. Furthermore, we introduce a rewriting of the MHV expansion in terms of sums over non-crossing partitions and show that this cyclically invariant formula satisfies the recursion relations for all numbers of legs and all loop orders.