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Asymptotic behaviour of the QED perturbation series. (English) Zbl 1404.81298
Summary: 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.

##### MSC:
 81V10 Electromagnetic interaction; quantum electrodynamics 81T15 Perturbative methods of renormalization applied to problems in quantum field theory
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