Multiloop Information from the QED Effective Lagrangian

TL;DR

This paper derives high-order photon amplitude information in QED from the effective Lagrangian, proposing a conjecture on series convergence and extending known results to self-dual fields using Borel analysis.

Contribution

It introduces a conjectured all-order formula for the imaginary part of the effective Lagrangian in a self-dual field and analyzes the asymptotic growth of QED photon amplitudes.

Findings

Proposes a conjecture for the all-order imaginary part of the Lagrangian in self-dual fields.

Derives the asymptotic growth rate of multi-photon amplitudes at large N.

Suggests the convergence of the perturbation series for on-shell QED N-photon amplitudes in the quenched approximation.

Abstract

We obtain information on the QED photon amplitudes at high orders in perturbation theory starting from known results on the QED effective Lagrangian in a constant electric field. A closed-form all-order result for the weak field limit of the imaginary part of this Lagrangian has been given years ago by Affleck, Alvarez and Manton (for scalar QED) and by Lebedev and Ritus (for spinor QED). We discuss the evidence for its correctness, and conjecture an analogous formula for the case of a self-dual field. From this extension we then obtain, using Borel analysis, the leading asymptotic growth for large N of the maximally helicity violating component of the L - loop N - photon amplitude in the low energy limit. The result leads us to conjecture that the perturbation series converges for the on-shell renormalized QED N - photon amplitudes in the quenched approximation.