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

TL;DR

The paper extends classical soft photon and soft graviton theorems in four dimensions to sub-subleading order, deriving universal $u^{-2}\ln u$ tails for both EM and gravity and establishing a precise connection between long-wavelength waveforms and asymptotic scattering data. It provides a classical derivation of sub-subleading waveforms, a two-loop derivation of the sub-subleading soft photon theorem in scalar QED, and a conjectured structure for the leading non-analytic contributions at arbitrary order, expressed through universal coefficients $K_{em}^{cl}$ and $K_{gr}^{cl}$ and undetermined gauge-invariant functions $\mathcal{B}^{(n)}$. The results separate universal, theory-independent parts from non-universal, scattering-detail-dependent pieces, elucidating memory effects and highlighting how spinning internal structure affects higher-order tails. The work connects classical waveform memory to quantum loop analyses, offers a framework for combined EM and gravitational long-range interactions, and discusses implications for gravitational tail memory in spinning or non-spinning scattering. Overall, it advances understanding of how soft theorems govern long-wavelength radiation and memory across EM and gravitational interactions, with concrete universal formulas and guiding conjectures for future higher-order analyses.

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 $u$.

Figures (3)

• Figure 1: A scattering process describing $M$ number of particles are coming into region $\mathcal{R}$, going through unspecified interaction inside the region $\mathcal{R}$ and disperse to $N$ number of particles. Outside the region $\mathcal{R}$ only long range electromagnetic and/or gravitational interaction is present.
• Figure 2: Sets of two loop Feynman diagrams with two virtual photons connected to three different scalar lines which potentially can contribute to order $\mathcal{O}(\omega(\ln\omega)^{2})$. The thick lines represent massive complex scalars and the thin lines represent photons.
• Figure 3: Sets of two loop Feynman diagrams with two virtual photons connected to two different scalar lines which potentially can contribute to order $\mathcal{O}(\omega(\ln\omega)^{2})$. The thick lines represent massive complex scalars and the thin lines represent photons.