Low-Energy Behavior of Gluons and Gravitons from Gauge Invariance

TL;DR

The paper shows that tree-level soft theorems for gluons and gravitons are fully determined by on-shell gauge invariance in $D$ dimensions, extending Low's photon theorem to nonabelian gauge theories and gravity. It derives explicit universal structures: the first subleading soft-gluon term and the first two subleading soft-graviton terms, expressed in terms of amplitudes without the soft leg and angular-momentum operators. It clarifies how loop corrections enter, distinguishing infrared-induced effects from genuine factorizing contributions, and argues that soft Ward identities related to extended BMS symmetry remain non-anomalous at least to the first subleading order in gravity when IR effects are ignored. The results unify the soft behavior across theories and connect to spinor-helicity formalisms and potential symmetries, providing a robust framework for analyzing soft limits in higher-dimensional contexts.

Abstract

We show that at tree level, on-shell gauge invariance can be used to fully determine the first subleading soft-gluon behavior and the first two subleading soft-graviton behaviors. Our proofs of the behaviors for n-gluon and n-graviton tree amplitudes are valid in D dimensions and are similar to Low's proof of universality of the first subleading behavior of photons. In contrast to photons coupling to massive particles, in four dimensions the soft behaviors of gluons and gravitons are corrected by loop effects. We comment on how such corrections arise from this perspective. We also show that loop corrections in graviton amplitudes arising from scalar loops appear only at the second soft subleading order. This case is particularly transparent because it is not entangled with graviton infrared singularities. Our result suggests that if we set aside the issue of infrared singularities, soft-graviton Ward identities of extended BMS symmetry are not anomalous through the first subleading order.

Figures (5)

• Figure 1: Diagrams of the form (a) give universal leading soft behavior. The subleading behavior comes from both diagrams types (a) and (b).
• Figure 2: Diagrams (a) and (b) give leading universal soft-gluon behavior. The first subleading behavior of the amplitude contained in the non-pole diagram (c) can be determined via on-shell gauge invariance.
• Figure 3: The potential factorizing contributions to the one-loop corrections to the leading soft function which then cancel. Leg $n$ is the soft leg which carries momentum $q$. At subleading order there are additional contributions.
• Figure 4: The diagrams with potential factorizing contributions to the one-loop soft function. At subleading order there are additional contributions.
• Figure 5: The diagrams with potential factorizing contributions to the one-loop soft behavior in gravity with a scalar in the loop. This captures all such potential leading and first subleading contributions, but it is incomplete at second subleading order.