Approximate calculation of the eigenvalue function in massless quantum electrodynamics

Abstract

We solve a vertex equation in massless quantum electrodynamics and use the results to calculate an approximation to the eigenvalue function, ${F}_{1}$, in the Johnson-Baker-Willey model. This approximation consists of a summation of the contributions to ${F}_{1}$ of all one-electron-loop diagrams in which no internal photon lines intersect (if all such lines are drawn within the electron loop). Our result reproduces the known low-order terms, ${F}_{1}=\frac{2}{3}+\frac{\ensuremath{\alpha}}{2\ensuremath{\pi}}\ensuremath{-}\frac{1}{4}{(\frac{\ensuremath{\alpha}}{2\ensuremath{\pi}})}^{2}$ exactly. In addition we find branch-point singularities and zeros at points corresponding to values for the fine-structure constant of order unity. Nonperturbative solutions of the vertex equation and ${F}_{1}$ are also shown to exist.