MHV Diagrams from an All-Line Recursion Relation

TL;DR

This work develops an all-line shift-based recursion for loop integrands in planar $\mathcal{N}=4$ SYM, formulated in both region-momentum space and momentum-twistor space. It proves that the recursion has the correct large-$z$ falloff and yields a complete, diagrammatic MHV expansion at all loop orders when the same reference twistor is used throughout. The authors solve the recursion in representative one- and two-loop examples, showing the forward- and factorisation-terms structure generates the full MHV diagram content and reproduces standard box expansions at one loop. The result provides a rigorous proof that the MHV diagram formalism applies to all loop amplitudes in planar $\mathcal{N}=4$ SYM and suggests extensions to other theories and Wilson-loop formulations.

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.

Figures (28)

• Figure 1: Examples to illustrate evaluation of MHV diagrams in momentum twistor space.
• Figure 2: Chiral integrals with unit leading singularities used to illustrate large $z$ behavior.
• Figure 3: A graphical representation of the all-line recursion relation for generic integrands.
• Figure 4: The standard factorisation terms in the all-line recursion relation.
• Figure 5: The geometry of poles in the loop integrand in momentum twistor space.