Proving the dimension-shift conjecture

TL;DR

This paper proves a conjecture linking one-loop N=4 MHV amplitudes and all-plus helicity amplitudes in pure Yang-Mills theory through a dimension-shifting relationship, using advanced unitarity techniques.

Contribution

It provides a rigorous proof of the dimension-shift conjecture for all orders in dimensional regularisation, employing a novel combination of massive spinor-helicity and Coulomb-branch supersymmetry.

Findings

Established a dimension-shifting relationship between different amplitude types.

Captured full amplitude structure using D-dimensional unitarity with pentagon and box integrals.

Demonstrated the proof applies to all multiplicities in the theory.

Abstract

We prove the conjecture made by Bern, Dixon, Dunbar, and Kosower that describes a simple dimension shifting relationship between the one-loop structure of N = 4 MHV amplitudes and all-plus helicity amplitudes in pure Yang-Mills theory. The proof captures all orders in dimensional regularisation using unitarity cuts, by combining massive spinor-helicity with Coulomb-branch supersymmetry. The form of these amplitudes can be given in terms of pentagon and box integrals using a generalised D-dimensional unitarity technique which captures the full amplitude to all multiplicities.