Low-energy behavior of gluons and gravitons from gauge invariance

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 Bondi, van der Burg, Metzner and Sachs symmetry are not anomalous through the first subleading order.

Figure 1

Diagrams of the form (a) give universal leading soft behavior. The subleading behavior comes from both diagram 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 nonpole diagram (c) can be determined via on-shell gauge invariance.

Figure 3

The potential factorizing contributions to the one-loop corrections to the soft operators. Leg $n$ is the soft leg which carries momentum $q$.

Figure 4

The diagrams contributing to the factorizing contribution to the one-loop soft function.

Figure 5

The diagrams with potential factorizing contributions to the one-loop soft operator in gravity with a scalar in the loop.