Continuum strong-coupling expansion for quantum electrodynamics

Abstract

We derive from the path integral a continuum strong-coupling expansion for QED in $d$-dimensional Euclidean space-time. It is a double expansion in the fermion and boson kinetic energy (inverse free propagators), which leads to a double power series for the Green's functions of the cutoff theory in terms of $\frac{1}{{e}^{2}}$ and $\frac{{\ensuremath{\Lambda}}^{2}}{{M}^{2}}$. $\ensuremath{\Lambda}$ is a smooth cutoff in Euclidean momentum space, and $M$ is an infrared regulator mass for the photons needed to define the local part of the path integral. We demonstrate how dimensional continuation is necessary to control the broken gauge invariance of the cutoff theory. Restricting to $d=2$ (the Schwinger model) we show how to remove the cutoff using Pad\'e approximants. We find some evidence that as $\frac{{\ensuremath{\Lambda}}^{2}}{{M}^{2}}\ensuremath{\rightarrow}\ensuremath{\infty}$ gauge invariance is restored and we calculate the vector-meson mass, keeping the first three terms in the expansion in powers of the bare photon inverse propagator.