Delta expansion for local gauge theories. II. Nonperturbative calculation of the anomaly in the Schwinger model.

Abstract

This is the second paper in a series in which we show how to use the principles of the $\ensuremath{\delta}$ expansion to obtain nonperturbative solutions to gauge theories. The approach consists of replacing the usual minimal-coupling term $\overline{\ensuremath{\psi}}(\ensuremath{\partial}\ensuremath{-}eA)\ensuremath{\psi}$ by $\overline{\ensuremath{\psi}}{(i\ensuremath{\partial}\ensuremath{-}eA)}^{\ensuremath{\delta}}\ensuremath{\psi}$ and then expanding the new theory in powers of $\ensuremath{\delta}$. For all values of $\ensuremath{\delta}$ the theory is locally gauge invariant. Thus, local gauge invariance holds order by order in powers of $\ensuremath{\delta}$. In this paper we show how to calculate the photon propagator and thus the anomaly in the Schwinger model (two-dimensional massless quantum electrodynamics) to first order in $\ensuremath{\delta}$. At $\ensuremath{\delta}=1$ the exact value for the anomaly, $\frac{{e}^{2}}{\ensuremath{\pi}}$, is obtained.