Are Soft Theorems Renormalized?

TL;DR

The paper addresses whether soft theorems in gravity survive loop corrections when regulators are present, by treating soft theorems as distributions and enforcing a priority order where the soft-limit expansion is performed before regulator expansions. It develops a precise, integrand-level framework and tests it on a five-point one-loop amplitude in $\mathcal{N}=8$ supergravity, using an all-orders-in-$\epsilon$ representation for the five-point amplitude. The main result is that the leading and subleading soft theorems remain unrenormalized up to $\mathcal{O}(\epsilon^0)$ when the soft-limit order is respected, with the analysis highlighting crucial cancellations between box and pentagon contributions across dimensions. The work clarifies why a naive box-only expansion fails and argues for a distributional, integrand-first approach to soft limits, with implications for gauge theories and pure gravity alike.

Abstract

We show that the distributional nature of soft theorems requires the soft limit expansion to take priority over the regulator expansion of Feynman loop integrals. We start the study of soft graviton theorems at loop level from this perspective by considering a five-particle one-loop amplitude in ${\cal N}=8$ supergravity. Surprisingly, we find that a soft theorem recently introduced by one of the authors and Strominger is not renormalized in this case. Computations are done in $4-2ε$ dimensions and for terms of order $ε^{-2}$, $ε^{-1}$ and $ε^{0}$.

Figures (2)

• Figure 1: Pentagon and box scalar integrals involved in the soft limit expansion.
• Figure 2: Examples of the scalar integrals needed in the check. The $a$'s denote the weights of the corresponding propagators. The label on each external legs denotes the momentum flowing out of that leg.