Asymptotic Behavior of the Vertex Function in Quantum Electrodynamics

Abstract

Using recently developed infinite-momentum techniques, we study the asymptotic behavior of the three-point function in quantum electrodynamics (massive photons) by summing the leading behavior of the perturbation series. We find that the leading diagrams are those in which photons are exchanged in all permutations across the external photon insertion. When the fermion lines are on-shell, we confirm an earlier speculation of Jackiw based on a fourth-order calculation, namely, that the vertex exponentiates in the form $\overline{u}({p}_{f}){\ensuremath{\Lambda}}^{\ensuremath{\mu}}u({p}_{i})=\overline{u}({p}_{f}){\ensuremath{\gamma}}^{\ensuremath{\perp}}u({p}_{i}){e}^{\ensuremath{-}\ensuremath{\Phi}}$, where $\ensuremath{\Phi}=(\frac{{e}^{2}}{16{\ensuremath{\pi}}^{2}}){\mathrm{ln}}^{2}(\frac{{Q}^{2}}{{\ensuremath{\mu}}^{2}})$. In a fashion which is reminiscent of eikonalization in a scattering process, $\ensuremath{\Phi}$ is characteristic of single-photon exchange across the vertex. We also identify and compute an $O({e}^{4})$ contribution to $\ensuremath{\Phi}$ coming from vacuum polarization. We discuss the concept of infinite-momentum pathways, the possibility of exponentiation in scalar theories, and speculate on extensions of our work.