Structure of the vertex function in finite quantum electrodynamics

Abstract

We study the structure of the renormalized electromagnetic current vertex, ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Gamma}}}_{\ensuremath{\mu}}(p, p+q, q)$, in finite quantum electrodynamics. Using conformal invariance we find that ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Gamma}}}_{\ensuremath{\mu}}(p, p, 0)$ takes the simple form of ${Z}_{1}{\ensuremath{\gamma}}_{\ensuremath{\mu}}$ when the external fermions are far off the mass shell. We interpret this result as an old theorem on the structure of the vertex function due to Gell-Mann and Zachariasen. We give the general structure of the vertex for arbitrary momentum transfer parametrically, and discuss how the Bethe-Salpeter equation and the Federbush-Johnson theorem are satisfied. We contrast the meaning of pointlike in a finite field theory with the meaning understood in the parton model. We discuss to what extent the condition ${Z}_{1}=0$, which may hold in conformal theories other than finite quantum electrodynamics, may be interpreted as a bootstrap condition. We show that the vanishing of ${Z}_{1}$ prevents there being bound states in the Migdal-Polyakov bootstrap.