Remarks on Asymptotic Symmetries and the Sub-leading Soft Photon Theorem

TL;DR

The paper challenges the notion that each order in soft theorems requires a distinct asymptotic symmetry by showing that the sub-leading soft photon theorem can arise from the same residual large gauge transformations that yield the leading term, via a $1/r$ expansion of the associated charge. By solving the Maxwell equations in four dimensions and expanding the symmetry charge in $r$, it identifies $Q^{(0)}$ with the leading soft factor and $Q^{(1)}$ with the sub-leading factor, both obtained from Ward identities. The approach demonstrates that no infinite tower of new soft theorems emerges and argues for the broader applicability of this mechanism to other gauge theories and gravity. It also clarifies that Newman–Penrose-type charges do not straightforwardly generalize to produce the sub-leading soft photon term, shaping future explorations of asymptotic symmetries in flat space holography and infrared structure.

Abstract

A deep connection has been recently established between soft theorems and symmetries at null infinity in gravity and gauge theories, recasting the former as Ward identities of the latter. In particular, different orders (in the frequency of the soft particle) in the soft theorems are believed to be controlled by different asymptotic symmetries. In this paper we argue that this needs not be the case by focusing on the soft photon theorem. We argue that the sub-leading soft factor follows from the same symmetry responsible for the leading one, namely certain residual (large) gauge transformations of the gauge theory. In particular, expanding the associated charge in inverse powers of the radial coordinate, the (sub-)leading charge yields the (sub-)leading soft factor.