Loops on the Celestial Sphere

TL;DR

This work develops a framework for celestial amplitudes that exposes how loop corrections impact the conformal structure at null infinity. By expressing loop effects as differential operators acting on the celestial tree-level amplitudes, the authors derive an all-loop operator formalism and demonstrate a celestial analogue of the BDS exponentiation. The approach is applied to finite one-loop YM amplitudes and infrared-divergent planar N=4 SYM, with explicit constructions showing how IR divergences shift conformal dimensions and can be resummed into a celestial BDS formula. The results suggest a universal operator mechanism linking loop corrections to celestial conformal data, with potential extensions to other theories and higher-point amplitudes.

Abstract

We study the effect of loop corrections to conformal correlators on the celestial sphere at null infinity. We first analyze finite one-loop celestial amplitudes in pure Yang-Mills theory and Einstein gravity. We then turn to our main focus: infrared divergent loop amplitudes in planar $\mathcal{N}=4$ super Yang-Mills theory. We compute the celestial one-loop amplitude in dimensional regularization and show that it can be recast as an operator acting on the celestial tree-level amplitude. This extends to any loop order and the re-summation of all planar loops enables us to write down an expression for the all-loop celestial amplitude. Finally, we show that the exponentiated all-loop expression given by the BDS formula gets promoted on the celestial sphere to an operator acting on the tree-level conformal correlation function, thus yielding, the celestial BDS formula.