Multi-Soft gluon limits and extended current algebras at null-infinity

TL;DR

This paper explores the algebraic structure of multi-soft gluon limits in four-dimensional scattering amplitudes, revealing connections to current algebras, conformal field theory, and extended symmetries at null-infinity.

Contribution

It introduces a novel framework linking multi-soft gluon limits to extended current algebras and conformal structures, including the Sugawara construction and KZ equations.

Findings

Establishes the Sugawara energy-momentum tensor from double-soft limits.

Derives KZ equations for positive helicity gluons in MHV amplitudes.

Defines a family of currents for sub-leading soft behavior and explores their algebra.

Abstract

In this note we consider aspects of the current algebra interpretation of multi-soft limits of tree-level gluon scattering amplitudes in four dimensions. Building on the relation between a positive helicity gluon soft-limit and the Ward identity for a level-zero Kac-Moody current, we use the double-soft limit to define the Sugawara energy-momentum tensor and, by using the triple- and quadruple-soft limits, show that it satisfies the correct OPEs for a CFT. We study the resulting Knizhnik-Zamolodchikov equations and show that they hold for positive helicity gluons in MHV amplitudes. Turning to the sub-leading soft-terms we define a one-parameter family of currents whose Ward identities correspond to the universal tree-level sub-leading soft-behaviour. We compute the algebra of these currents formed with the leading currents and amongst themselves. Finally, by parameterising the ambiguity in the double-soft limit for mixed helicities, we introduce a non-trivial OPE between the holomorphic and anti-holomorphic currents and study some of its implications.