Classical and quantum results on logarithmic terms in the soft theorem in four dimensions

Biswajit Sahoo,Ashoke Sen

doi:10.1007/jhep02(2019)086

Abstract

We explore the logarithmic terms in the soft theorem in four dimensions by analyzing classical scattering with generic incoming and outgoing states and one loop quantum scattering amplitudes. The classical and quantum results are consistent with each other. Although most of our analysis in quantum theory is carried out for one loop amplitudes in a theory of (charged) scalars interacting via gravitational and electromagnetic interactions, we expect the results to be valid more generally.

Highlights

Terms in soft theorem to appropriate terms in the radiative part of the electromagnetic and gravitational fields in classical scattering in generic space-time dimensions
We explore the logarithmic terms in the soft theorem in four dimensions by analyzing classical scattering with generic incoming and outgoing states and one loop quantum scattering amplitudes
The subleading terms in the soft theorem contain a factor of angular momentum jμν of the individual particles involved in the scattering, with the orbital contribution to the angular momentum given by xμpν − xνpμ, where xμ(τ ) and pμ(τ ) label the asymptotic coordinates and momenta of the particle as a function of the proper time

Summary

Introduction

Terms in soft theorem to appropriate terms in the radiative part of the electromagnetic and gravitational fields in classical scattering in generic space-time dimensions. In four space-time dimensions the long range gravitational and/or electromagnetic forces acting on the particles produce an additional term of the form bμ ln τ in the expression for xμ This gives a logarithmically divergent term of the form (bμpν − bνpμ) ln τ in the expression for jμν, making the subleading soft factor divergent. The natural guess is that the soft factor at the subleading order is given by replacing the factors of ln τ in the naive expression by ln ω−1 This has been tested in [89] by considering several examples of classical scattering in four space-time dimensions.. The contribution from this region cannot be derived using the usual soft theorem, and need to be computed explicitly

Objectives

Discussion

Conclusion