Loop corrected soft photon theorem as a Ward identity

Abstract

Recently Sahoo and Sen obtained a series of remarkable results concerning subleading soft photon and graviton theorems in four dimensions. Even though the S-matrix is infrared divergent, they have shown that the subleading 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 subleading 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.