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.
Highlights
Celestial amplitudes reveal conformal properties of fourdimensional scattering amplitudes of massless particles as the standard plane wave basis is replaced by a basis of boost eigenstates
We study the effect of loop corrections to conformal correlators on the celestial sphere at null infinity
We show that the exponentiated all-loop expression given by the Bern-Dixon-Smirnov (BDS) formula gets promoted on the celestial sphere to an operator acting on the tree-level conformal correlation function, yielding, the celestial BDS formula
Summary
Celestial amplitudes reveal conformal properties of fourdimensional scattering amplitudes of massless particles as the standard plane wave basis is replaced by a basis of boost eigenstates. We will focus on the case of maximal helicity violation (MHV) four-point amplitudes in planar N 1⁄4 4 super–Yang-Mills theory This has the benefit that the four-gluon amplitude is known to all orders in the loop expansion. The four-gluon MHV amplitude in planar N 1⁄4 4 super–Yang-Mills can be conveniently written in factorized form in terms of the tree-level amplitude containing the helicity structure and an infrared divergent piece containing the information about the loop order. The celestial one-loop amplitude in planar N 1⁄4 4 super–Yang-Mills can be recast as an operator acting on the celestial tree-level amplitude.3 This structure is shown to persist at any loop order. We move on to the main focus of this paper in Sec. III B and analyze how divergent loop corrections in planar N 1⁄4 4 super–Yang-Mills theory correct the corresponding tree-level celestial correlators. Notice that no mention to perturbation theory has been made, and as long as the amplitude is of the form (2.11), the statements made so far apply to the exact four-point S-matrix element
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