Asymptotic Behaviour of the QED Perturbation Series

Abstract EN

I will summarize the present state of a long-term effort to obtain information on the large-order asymptotic behaviour of the QED perturbation series through the effective action.

Starting with the constant-field case, I will discuss the Euler-Heisenberg Lagrangian in various dimensions and up to the three-loop level.

This Lagrangian holds the information on the N-photon amplitudes in the low-energy limit, and combining it with Spinor helicity methods explicit all-N results can be obtained at the one-loop and, for the “all +” amplitudes, also at the two-loop level.

For the imaginary part of the Euler-Heisenberg Lagrangian, an all-loop formula has been conjectured independently by Affleck, Alvarez, and Manton for Scalar QED and by Lebedev and Ritus for Spinor QED.

This formula can be related through a Borel dispersion relation to the leading large-N behaviour of the N-photon amplitudes.

It is analytic in the fine structure constant, which is puzzling and suggests a diagrammatic investigation of the large-N limit in perturbation theory.

Preliminary results of such a study for the 1+1 dimensional case throw doubt on the validity of the conjecture.