New Asymptotic Conservation laws for Electromagnetism

We obtain the subleading tail to the memory term in the late time electromagnetic radiative field generated due to a generic scattering of charged bodies. We show that there exists a new asymptotic conservation law which is related to the subleading tail term. The corresponding charge is made of a mode of the asymptotic electromagnetic field that appears at $\mathcal{O}(e^5)$ and we expect that it is uncorrected at higher orders. This hints that the subleading tail arises from classical limit of a 2-loop soft photon theorem. Building on the $m=1$ \cite{1903.09133, 1912.10229} and $m=2$ cases, we propose that there exists a conservation law for every $m$ such that the respective charge involves an $\mathcal{O}(e^{2m+1})$ mode and is conserved exactly. This would imply a hierarchy of an infinite number of $m$-loop soft theorems. We also predict the structure of $m^{th}$ order tails to the memory term that are tied to the classical limit of these soft theorems.