MHV Diagrams from an All-Line Recursion Relation

TL;DR

This paper derives a recursion relation for loop integrands in planar N=4 SYM using all-line shifts, proving the MHV diagram formalism for all loop amplitudes.

Contribution

It introduces a new all-line shift recursion relation for loop integrands and demonstrates it generates the MHV vertex expansion in planar N=4 SYM.

Findings

Valid recursion relation for all cases examined.

Recursion relation formulated in region momentum and twistor space.

Proof that the MHV diagram formalism applies to all loop amplitudes.

Abstract

We consider the recursion relation for loop integrands in planar N = 4 SYM generated by an all-line shift of momentum twistors. We examine the behaviour of the rational loop integrands when the shift parameter becomes large, and find that a valid recursion relation may be obtained in all cases. The recursion relation is then formulated both in region momentum space and in momentum twistor space, and solved in detail for some one and two-loop examples. Finally, we show that the general iterative solution of the recursion relation generates the MHV vertex expansion for all loop integrands, providing a proof of the MHV diagram formalism for all loop amplitudes in planar N = 4 SYM.