Asymptotic conservation laws for loop level soft photon theorems

Abstract

In the thesis, we study new asymptotic conservation laws in electromagnetism that could reproduce the loop effects in the soft expansion of QED amplitudes. We also explore interesting properties of asymptotic expansions of the classical as well as quantum gauge field.

Incorporating the effect of long range electromagnetic force (present in four spacetime dimensions) acting on the scattered particles, we analyse the new modes that arise in the asymptotic radiative field emitted in a generic classical scattering process. We show that there exist new asymptotic conservation laws (Qm;m = 1; 2) that are obeyed by the classical radiative field. Building on the m = 1; 2 cases, we propose that there exists a conservation law for every m. The corresponding charges are made of modes of the asymptotic electromagnetic field that appear at O(e2m+1) and are expected to be conserved exactly.

The asymptotic behaviour of the gauge field is modified in the quantum theory due to use of Feynman boundary condition. We derive the analogue of the first of above asymptotic conservation laws upon imposing Feynman boundary condition on the radiative field. We also discuss purely quantum modes in the Feynman solution which are absent in the classical radiative solution. These modes lead to quantum corrections to the asymptotic charges.

We anticipate that the Qm charges imply existence of m-loop soft theorems for every m. In particular we show that the Ward identity for the Q1 charge is equivalent to the 1-loop exact subleading soft photon theorem for loop level QED amplitudes. We demonstrate this equivalence in the context of massless scalar QED in presence of dynamical gravity. This asymptotic charge is directly related to the dressing of fields due to long range forces. In presence of gravity, the new feature is that the soft photon also acquires a dressing due to long range gravitational force and contributes to the asymptotic charge.

We expect this story to hold beyond 1-loop as well. The Q2 conservation law that we derived is expected to be related to the 2-loop exact soft photon theorem that has been obtained recently in the literature. It would be very interesting to explore this equivalence for m > 1.