Loop Corrected Soft Photon Theorem as a Ward Identity

TL;DR

This work shows that in four-dimensional QED with massive charges, the loop-corrected subleading soft photon theorem, which features a logarithmic term in frequency, is exactly captured by Ward identities of an infinite family of asymptotic charges $Q[V]$. The charges decompose into soft and hard parts, with the hard piece splitting into jet and homogeneous contributions tied to both current dynamics and a logarithmic radiative mode, and they connect to Kulish–Faddeev dressing of charged states. The results extend the tree-level soft-theorem–Ward-identity correspondence to the loop-corrected regime and illuminate the role of dressing and large-gauge structures in the infrared, suggesting a refined symmetry picture for the S-matrix in QED. Further work is needed to understand the precise action of these charges on radiative data, their algebra, and how to incorporate gravity into this loop-corrected asymptotic-symmetry framework.

Abstract

Recently Sahoo and Sen obtained a series of remarkable results concerning sub-leading soft photon and graviton theorems in four dimensions. Even though the S- matrix is infrared divergent, they have shown that the sub-leading soft theorems are well defined and exact statements in QED and perturbative Quantum Gravity. However unlike the well studied Cachazo-Strominger soft theorems in tree-level amplitudes, the new sub-leading soft expansion is at the order ln ω (where ω is the soft frequency) and the corresponding soft factors structurally show completely different properties then their tree-level counterparts. Whence it is natural to ask if these theorems are associated to asymptotic symmetries of the S-matrix. We consider this question in the context of sub-leading soft photon theorem in scalar QED and show that there are indeed an infinity of conservation laws whose Ward identities are equivalent to the loop-corrected soft photon theorem. This shows that in the case of four dimensional QED, the leading and sub-leading soft photon theorems are equivalent to Ward identities of (asymptotic) charges.