Logarithmic Terms in the Soft Expansion in Four Dimensions

TL;DR

This paper investigates the behavior of soft factors in four-dimensional spacetime, revealing the emergence of logarithmic terms at subleading order due to infrared divergences, and clarifies their classical interpretation.

Contribution

It demonstrates that soft factors in four dimensions include logarithmic terms at subleading order, resolving ambiguities caused by infrared divergences.

Findings

Soft factors in four dimensions contain logarithmic terms at subleading order.

Classical radiative fields provide an unambiguous definition of soft factors.

Infrared divergences affect the soft expansion beyond leading order.

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.