Low-energy limit of N-photon amplitudes in a constant field: Part II

TL;DR

This paper uses the worldline formalism to derive a series representation of low-energy N-photon amplitudes in constant background fields, providing compact expressions and explicit helicity component formulas involving Bernoulli numbers.

Contribution

It introduces a new series representation for N-photon amplitudes in constant fields and simplifies calculations for specific cases like crossed fields.

Findings

Series representation terminates for crossed fields

Explicit formulas for helicity amplitudes in terms of Bernoulli numbers

Compact integral expressions for low-energy N-photon amplitudes

Abstract

We employ the worldline formalism to derive a series representation of the low-energy limit of the N -photon amplitude in a constant background field for both scalar and spinor QED. The amplitudes are then written in terms of a single proper-time integral. The above-mentioned series representation terminates when considering a constant crossed field. This allows us to obtain even more compact expressions for these particular amplitudes for which the result of the proper-time integral, for fixed parameters, takes the form of a factorial function. In addition, we derive all helicity components of these amplitudes and express them explicitly in terms of Bernoulli numbers and spinor products.