Non-Compact Pure Gauge QED in 3D is Free

Abstract

For all Poincaré invariant Lagrangians of the form $\mathcal{L}\equiv f(F_{\mu \nu })$, in three Euclidean dimensions, where $f$ is any invariant function of a non-compact $U(1)$ field strength $F_{\mu \nu }$, we find that the only continuum limit (described by just such a gauge field) is that of free field theory: First we approximate a gauge invariant version of Wilson's renormalization group by neglecting all higher derivative terms $\sim {\partial }^{n}F$ in $\mathcal{L}$, but allowing for a general non-vanishing anomalous dimension. Then we prove analytically that the resulting flow equation has only one acceptable fixed point: the Gaussian fixed point. The possible relevance to high-$T_c$ superconductivity is briefly discussed.

Comments: 11 pages, plain tex, uses harvmac. Minor additions - version to be published in Physics Letters B