Building the full fermion-photon vertex of QED by imposing multiplicative renormalizability of the Schwinger-Dyson equations for the fermion and photon propagators

In principle, calculation of a full Green’s function in any field theory requires knowledge of the infinite set of multipoint Green’s functions, unless one can find some way of truncating the corresponding Schwinger-Dyson equations. For the fermion and boson propagators in QED this requires an ansatz for the full 3-point vertex. Here we illustrate how the properties of gauge invariance, gauge covariance and multiplicative renormalizability impose severe constraints on this fermion-boson interaction, allowing a consistent truncation of the propagator equations. We demonstrate how these conditions imply that the 3-point vertex in the propagator equations is largely determined by the behavior of the fermion propagator itself and not by knowledge of the many higher-point functions. We give an explicit form for the fermion-photon vertex, which in the fermion and photon propagator fulfills these constraints to all orders in leading logarithms for massless QED, and accords with the weak coupling limit in perturbation theory at $\mathcal{O}(\alpha )$. This provides the first attempt to deduce nonperturbative Feynman rules for strong physics calculations of propagators in massless QED that ensure a more consistent truncation of the 2-point Schwinger-Dyson equations. The generalization to next-to-leading order and masses will be described in a longer publication.

Figure 1

Flow diagram of the Schwinger-Dyson calculation presented here.

Figure 2

Fermion-boson vertex.

Figure 3

Unquenched Schwinger-Dyson equation for fermion propagator.

Figure 4

Unquenched Schwinger-Dyson equation for photon propagator.