Approximate calculation of the eigenvalue function in massless quantum electrodynamics

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, $F_1=\frac{2}{3}+\frac{\alpha }{2\pi }-\frac{1}{4}{\left(\frac{\alpha }{2\pi }\right)}^2$ 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.