Master formulas for the dressed scalar propagator in a constant field

Aftab Ahmad,Naser Ahmadiniaz,Olindo Corradini,Sang Pyo Kim,Christian Schubert

doi:10.1016/j.nuclphysb.2017.03.007

Abstract

The worldline formalism has previously been used for deriving compact master formulas for the one-loop N-photon amplitudes in both scalar and spinor QED, and in the vacuum as well as in a constant external field. For scalar QED, there is also an analogous master formula for the propagator dressed with N photons in the vacuum. Here, we extend this master formula to include a constant field. The two-photon case is worked out explicitly, yielding an integral representation for the Compton scattering cross section in the field suitable for numerical integration in the full range of electric and magnetic field strengths.

Highlights

The one-loop effective action in scalar QED has the well-known “worldline” or “Feynman– Schwinger” representation [1], ∞ [A] = − dT e−m2T Dx(τ ) e− T 0 d τ [ 1 4 x 2 +i exμ Aμ (x(τ ))] (1)
This leads to the following path integral representation of the constant-field propagator dressed with N photons:
The x-space version of this formula generalizes the one obtained by McKeon and Sherry for the purely magnetic case [33]; the p-space version generalizes the vacuum master formula of Daikouji et al [25] on one hand, the closed-loop master formula of Shaisultanov [9] on the other

Summary

Introduction

The one-loop effective action in scalar QED has the well-known “worldline” or “Feynman– Schwinger” representation [1],. The same master formula (2) was derived by Bern and Kosower [3,4] from string theory as a generating expression from which to construct the one-loop on-shell N gluon amplitudes by way of a certain set of rules It contains the information on the N -photon amplitudes. As a warm-up, in section 2 we use the path integral representation to rederive the well-known scalar propagator in a constant field, in configuration as well as in momentum space. In Appendix A we give our conventions, while in Appendix B we collect some information on the constant field worldline Green’s functions

Configuration space

Momentum space

Compton scattering in a constant field

Summary and outlook

General properties of the Green’s function GB

Matrix decomposition of the Green’s function GB