The Soft $\mathcal{S}$-Matrix in Gravity

TL;DR

Infrared divergences in four-dimensional gravity are reformulated as a celestial current algebra on the sphere, generated by the supertranslation current P_z and its Goldstone partner. The soft sector is captured by Wilson-line–like operators built from the Goldstone field C, with explicit OPEs and an off-diagonal current-level structure tied to the cusp anomalous dimension and the IR cutoff. Gravitational memory emerges as an IR-safe observable from the P_z–Goldstone interactions, and the framework extends to massive external states via hyperbolic slicing and generalized Wilson lines. Collectively, the work unifies soft theorems, memory, and IR divergences within a CCFT-based description of the gravitational S-matrix.

Abstract

The gravitational $\mathcal{S}$-matrix defined with an infrared (IR) cutoff factorizes into hard and soft factors. The soft factor is universal and contains all the IR and collinear divergences. Here we show, in a momentum space basis, that the intricate expression for the soft factor is fully reproduced by two boundary currents, which live on the celestial sphere. The first of these is the supertranslation current, which generates spacetime supertranslations. The second is its symplectic partner, the Goldstone current for spontaneously broken supertranslations. The current algebra has an off-diagonal level structure involving the gravitational cusp anomalous dimension and the logarithm of the IR cutoff. It is further shown that the gravitational memory effect is contained as an IR safe observable within the soft $\mathcal{S}$-matrix.