MHV Gluon Scattering Amplitudes from Celestial Current Algebras

TL;DR

This work establishes that tree-level n-point MHV gluon amplitudes in pure Yang-Mills, when Mellin-transformed to celestial space, obey a set of (n-2) coupled first-order PDEs that resemble Knizhnik-Zamolodchikov equations but receive a correction from subleading soft gluon symmetry. By translating these equations to momentum space and analyzing the leading and subleading soft limits, the authors derive leading celestial OPEs for gluon primaries and provide recursion relations for subleading OPE coefficients using the subleading current algebra generated by J^a and K^a. The results reproduce known OPE coefficients and reveal how subleading symmetry data constrain celestial amplitudes, suggesting a richer symmetry structure beyond a simple Sugawara tensor in celestial CFT. The analysis also clarifies how non-closure of subleading generators can nonetheless yield tractable, predictive relations for OPE coefficients, with implications for the celestial holographic description of Yang-Mills theory. Overall, the paper connects CCFT symmetries to concrete OPE data in the MHV sector and highlights the role of subleading soft theorems in shaping the celestial operator algebra.

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.