Infinite Soft Theorems from Gauge Symmetry
Zhi-Zhong Li, Hung-Hwa Lin, Shun-Qing Zhang
TL;DR
The paper shows that ordinary on-shell gauge invariance suffices to fix photon and graviton soft behavior to infinite order in a suitably projected amplitude, reproducing the infinite soft theorem associated with large gauge transformations for photons and extending it for gravitons. By carefully applying Ward identities to relate pole and non-pole contributions and projecting away homogeneous ambiguities, the authors derive a general, infinite-order soft expansion and discuss how higher-dimensional operators modify these relations. The results illuminate the role of gauge symmetry in constraining scattering amplitudes beyond leading orders and clarify the impact of EFT corrections on soft theorems, with implications for how asymptotic symmetries may manifest in observable quantities.
Abstract
In this letter we show that the soft behaviour of photons and graviton amplitudes, after projection, can be determined to infinite order in soft expansion via ordinary on-shell gauge invariance. In particular, as one of the particle's momenta becomes soft, gauge invariance relates the non-singular diagrams of an n-point amplitude to that of the singular ones up to possible homogeneous terms. We demonstrate that with a particular projection of the soft-limit, the homogeneous terms do not contribute, and one arrives at an infinite soft theorem. This reproduces the result recently derived from the Ward identity of large gauge transformations. We also discuss the modification of these soft theorems due to the presence of higher-dimensional operators.