Ward identity for loop level soft photon theorem for massless QED coupled to gravity

TL;DR

This work demonstrates that the loop-level, subleading soft photon theorem for massless scalar QED in the presence of dynamical gravity is equivalent to an asymptotic conservation law. By constructing a 1-loop asymptotic charge, the authors show that long-range dressings of both scalar and gauge fields—including gravity-induced dressings of photons—generate the necessary contributions to this charge. They compute the classical and quantum parts of the relevant dressings, establish a Ward identity for the 1-loop charge, and prove that it reproduces the Sahoo–Sen soft theorem. The analysis highlights the role of logarithmic soft modes and the Fadeev–Kulish-like dressing in linking infrared structure to asymptotic symmetries in four dimensions. This provides a deeper understanding of how loop corrections and gravitational interactions shape soft theorems and their symmetry origins.

Abstract

Strominger and his collaborators pioneered the study of equivalence between soft theorems and asymptotic conservation laws. We study this equivalence in the context of loop level subleading soft photon theorem for massless scalar QED in presence of dynamical gravity. Motivated by Campiglia and Laddha \cite{1903.09133}, we show that the Sahoo-Sen soft photon theorem \cite{1808.03288} for loop amplitudes is equivalent to an asymptotic conservation law. This asymptotic charge is directly related to the dressing of fields due to long range forces exclusively present in four spacetime dimensions. In presence of gravity, the new feature is that soft photons also acquire a dressing due to long range gravitational force and this dressing contributes to the asymptotic charge.