Asymptotic Behaviour of the QED Perturbation Series

Abstract

The publication of this article was funded by SCOAP 3 .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 lowenergy 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.

Note on all-order Landau-level structures of the Heisenberg-Euler effective actions for QED and QCD

“…Extensions to finite temperature/density [73][74][75][76][77] and higher-loop diagrams [78][79][80][81] are also left as interesting future works. While there is no interlevel mixing at the one-loop level, we would expect the occurrence of interlevel transitions via interactions with the dynamical gauge fields in the higher-loop diagrams.…”

Thermal Schwinger pair production at arbitrary coupling

“…Further it gives an all-orders correction to the known weak coupling results. This will be of interest to current and future experimental studies of Schwinger pair production (see for example [62] for a discussion), as well as to multi-loop and asymptotic analyses of the QED perturbation series [36][37][38].…”

Electron electromagnetic-mass melting in strong fields

“…(2) Nonlocal corrections to the action are required in order to account for nonlinear mixing between the external fields and the inhomogeneous fields of the particles [26][27][28][29]. (3) Another effect has been pointed out by Gies and Karbstein [30,31]: in addition to the well-studied internal photon line corrections to the EHS action [32,33], reducible connecting photon line corrections to the EHS action have been shown to be nonvanishing. This contribution may also be accounted for in a self-consistent manner.…”