Self-dual Yang--Mills Theory and One-Loop Maximally Helicity Violating Multi-Gluon Amplitudes

TL;DR

This work shows that one can reproduce all-positive-helicity (MHV) QCD amplitudes at one loop from a scalar action whose SDYM equations are realized in a light-cone framework, establishing a direct link between SDYM structure and QCD helicity amplitudes. It develops a Bethe Ansatz construction that connects SDYM solutions to tree-level MHV amplitudes (which vanish) and demonstrates that one-loop MHV amplitudes arise from scalar loops in the SDYM background, reproducing known QCD results for arbitrary multiplicity. The analysis highlights a deep role for SDYM symmetries, including an affine current algebra, in governing the vanishing of tree amplitudes and the simplicity of one-loop amplitudes, and discusses connections to the Donaldson–Nair–Schiff action and possible anomaly-driven mechanisms. Together, these results suggest integrability and symmetry principles underlying the all-plus helicity sector of gauge theory, with open questions about higher-loop extensions and the quantum realization of these symmetries.

Abstract

A scalar cubic action that classically reproduces the self-dual Yang--Mills equations is shown to generate one-loop QCD amplitudes for external gluon all with the same helicity. This result is related to the symmetries of the self-dual Yang--Mills equations.