MHV Gluon Scattering Amplitudes from Celestial Current Algebras

TL;DR

This paper derives differential equations governing celestial MHV gluon amplitudes, revealing their structure akin to KZ equations with additional corrections, and uses these to compute operator product expansions and subleading terms.

Contribution

It introduces a system of differential equations for celestial MHV gluon amplitudes, incorporating subleading soft gluon effects, and proposes a method to handle non-closure of subleading current algebra generators.

Findings

Derived differential equations for celestial amplitudes.

Connected equations to KZ equations with corrections.

Matched subleading OPE terms with explicit Mellin amplitude calculations.

Abstract

We show that the Mellin transform of an $n$-point tree level MHV gluon scattering amplitude, also known as the celestial amplitude in pure Yang-Mills theory, satisfies a system of $(n-2)$ linear first order partial differential equations corresponding to $(n-2)$ positive helicity gluons. Although these equations closely resemble Knizhnik-Zamolodchikov equations for $SU(N)$ current algebra there is also an additional "correction" term coming from the subleading soft gluon current algebra. These equations can be used to compute the leading term in the gluon-gluon OPE on the celestial sphere. Similar equations can also be written down for the momentum space tree level MHV scattering amplitudes. We also propose a way to deal with the non closure of subleading current algebra generators under commutation. This is then used to compute some subleading terms in the mixed helicity gluon OPE and our results match with those obtained from an explicit calculation using the Mellin MHV amplitude.