The structure of IR divergences in celestial gluon amplitudes

TL;DR

This work establishes that the all-loop infrared (IR) factorization of gauge-theory amplitudes persists in the celestial (2D conformal) basis. The universal IR-divergent sector is encoded as a correlator of Wilson lines on the celestial sphere, governed by the cusp anomalous dimension, while the finite (hard) piece maps to dressed celestial operators with infinite conformal-dimension shifts tied to the same cusp data. In the planar $\mathcal{N}=4$ SYM setting, the finite part exponentiates and, for 4- and 5-gluon amplitudes, the Mellin-transformed celestial amplitude converges due to the positivity of the cusp anomalous dimension; large-$N$ reduces the soft sector to a Coulomb-gas-like description of color primaries. The authors connect the Wilson-line renormalization to a clean 2D CFT interpretation, defining dressed operators $\hat{\mathcal{O}}_{\Delta,\ell}$ whose correlators yield IR-safe celestial amplitudes, and provide explicit 4- and 5-point results. They also discuss four-loop corrections to the soft sector, potential extensions to non-supersymmetric theories and fermions, and the broader implications for dual conformal structures and memory effects in celestial holography.

Abstract

The all-loop resummation of SU$(N)$ gauge theory amplitudes is known to factorize into an IR-divergent (soft and collinear) factor and a finite (hard) piece. The divergent factor is universal, whereas the hard function is a process-dependent quantity. We prove that this factorization persists for the corresponding celestial amplitudes. Moreover, the soft/collinear factor becomes a scalar correlator of the product of renormalized Wilson lines defined in terms of celestial data. Their effect on the hard amplitude is a shift in the scaling dimensions by an infinite amount, proportional to the cusp anomalous dimension. This leads us to conclude that the celestial-IR-safe gluon amplitude corresponds to a expectation value of operators dressed with Wilson line primaries. These results hold for finite $N$. In the large $N$ limit, we show that the soft/collinear correlator can be described in terms of vertex operators in a Coulomb gas of colored scalar primaries with nearest neighbor interactions. In the particular cases of four and five gluons in planar $\mathcal{N}=4$ SYM theory, where the hard factor is known to exponentiate, we establish that the Mellin transform converges in the UV thanks to the fact that the cusp anomalous dimension is a positive quantity. In other words, the very existence of the full celestial amplitude is owed to the positivity of the cusp anomalous dimension.