Logarithmic Terms in the Soft Expansion in Four Dimensions

TL;DR

The paper investigates soft theorems in four spacetime dimensions, where infrared divergences complicate the quantum S-matrix. By relating soft factors to the classical low-frequency radiation in scattering, it shows that subleading soft factors acquire logarithmic corrections, $\ln \omega^{-1}$, due to long-range forces. It derives explicit forms for both electromagnetic and gravitational soft factors in various scattering setups, including contributions from phase factors arising from gravitational drag. The results illuminate the structure of soft theorems in 4D and provide a framework for obtaining finite classical quantities despite IR divergences in the quantum theory.

Abstract

It has been shown that in larger than four space-time dimensions, soft factors that relate the amplitudes with a soft photon or graviton to amplitudes without the soft particle also determine the low frequency radiative part of the electromagnetic and gravitational fields during classical scattering. In four dimensions the S-matrix becomes infrared divergent making the usual definition of the soft factor ambiguous beyond the leading order. However the radiative parts of the electromagnetic and gravitational fields provide an unambiguous definition of soft factor in the classical limit up to the usual gauge ambiguity. We show that the soft factor defined this way develops terms involving logarithm of the energy of the soft particle at the subleading order in the soft expansion.