Conformally Soft Theorem in Gauge Theory

TL;DR

The paper investigates whether a soft-theorem analog exists for celestial amplitudes, where external states are labeled by conformal weights rather than energies. By connecting the familiar energetically soft gluon relations to the celestial Mellin basis under a mild UV-softness assumption, the authors derive a conformally soft theorem at tree level in nonabelian gauge theories. They verify this Ward identity across explicit 4-point Yang-Mills and string-theory amplitudes (Type I and Heterotic) and extend the result to leg permutations and general $n$-point MHV amplitudes, highlighting a current-algebra–like structure in the celestial framework. A conjecture is proposed linking the existence of celestial amplitudes to the possibility of perturbative quantum deformations of the 4D theory, suggesting deep connections between 4D scattering and 2D conformal constraints.

Abstract

Asymptotic particle states in four-dimensional celestial scattering amplitudes are labelled by their $SL(2,\mathbb{C})$ Lorentz/conformal weights $(h,\bar{h})$ rather than the usual energy-momentum four-vector. These boost eigenstates involve a superposition of all energies. As such, celestial gluon (or photon) scattering cannot obey the usual (energetically) soft theorems. In this paper we show that tree-level celestial gluon scattering, in theories with sufficiently soft UV behavior, instead obeys conformally soft theorems involving $h \to 0$ or $\bar{h} \to 0$. Unlike the energetically soft theorem, the conformally soft theorem cannot be derived from low-energy effective field theory.