Abstract
Extending work by Gies and Karbstein on the Euler–Heisenberg Lagrangian, it has recently been shown that the one-loop propagator of a charged scalar particle in a constant electromagnetic field has a one-particle reducible contribution in addition to the well-studied irreducible one. Here we further generalize this result to the spinor case, and find the same relation between the reducible term, the tree-level propagator and the one-loop Euler–Heisenberg Lagrangian as in the scalar case. Our demonstration uses a novel worldline path integral representation of the photon-dressed spinor propagator in a constant electromagnetic field background.
Highlights
The QED one-loop one-photon amplitude vanishes in vacuum by Furry’s theorem
In the presence of a constant external field this theorem does not imply that the one-photon diagram (Fig. 1) vanishes
That diagram is still usually discarded, since it formally vanishes by momentum conservation
Summary
The QED one-loop one-photon amplitude vanishes in vacuum by Furry’s theorem. In the presence of a constant external field this theorem does not imply that the one-photon diagram (Fig. 1) vanishes. Gies and Karbstein [2] discovered that this diagram can cause non-vanishing contributions when appearing as part of a larger diagram, due to the infrared singularity of the photon propagator connecting it to the rest of the diagram With the ingredients ((11))[k , ε ; F ] (the one-loop one-photon amplitude in the constant field) and S(x1)x[k, ε; F ] (the x-space spinor propagator in the field with one photon attached)2 The former contains a delta function δD(k) and one factor of momentum (see (7) below), so that by itself it vanishes.
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