# Loop Corrected Soft Photon Theorem as a Ward Identity

## TL;DR

This paper demonstrates that loop-corrected sub-leading soft photon theorems in four-dimensional QED are equivalent to Ward identities of asymptotic charges, extending the connection between soft theorems and symmetries beyond tree level.

## Contribution

It establishes that the loop-corrected sub-leading soft photon theorem in scalar QED corresponds to an infinite set of conservation laws and Ward identities, generalizing known symmetry relations.

## Key findings

- Loop corrections introduce ln(ω) terms in soft theorems.
- Sub-leading soft photon theorems are linked to asymptotic symmetries.
- Infinite conservation laws correspond to Ward identities.

## 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 {\omega} (where {\omega} 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.