Classical Sub-subleading Soft Photon and Soft Graviton Theorems in Four Spacetime Dimensions

TL;DR

This paper extends classical soft theorems to include sub-subleading terms in four dimensions, deriving new waveforms and analyzing loop amplitudes to reveal detailed late and early time behaviors of electromagnetic and gravitational waves.

Contribution

It introduces the derivation of sub-subleading electromagnetic and gravitational waveforms and analyzes two-loop scalar QED amplitudes to extend soft photon theorems.

Findings

Derived sub-subleading waveforms with $u^{-2}(\

Analyzed two-loop scalar QED amplitudes for soft photon theorems.

Identified non-analytic contributions behaving like $u^{-n}(\

Abstract

Classical soft photon and soft graviton theorems determine long wavelength electromagnetic and gravitational waveforms for a general classical scattering process in terms of the electric charges and asymptotic momenta of the ingoing and outgoing macroscopic objects. Performing Fourier transformation of the electromagnetic and gravitational waveforms in the frequency variable one finds electromagnetic and gravitational waveforms at late and early retarded time. Here extending the formalism developed in \cite{1912.06413}, we derive sub-subleading electromagnetic and gravitational waveforms which behave like $u^{-2}(\ln u)$ at early and late retarded time $u$ in four spacetime dimensions. We also have derived the sub-subleading soft photon theorem analyzing two loop amplitudes in scalar QED. Finally, we give the structure of leading non-analytic contribution to (sub)$^{n}$-leading classical soft photon and graviton theorems which behave like $u^{-n}(\ln u)^{n-1}$ for early and late retarded time $u$.