Non-Compact Pure Gauge QED in 3D is Free

TL;DR

This paper proves that all Poincaré invariant pure gauge theories in 3D with non-compact U(1) fields flow to a free theory in the continuum limit, indicating no interacting fixed points exist.

Contribution

It analytically demonstrates that such 3D gauge theories have only the Gaussian fixed point, extending understanding of their renormalization group behavior.

Findings

Only the free theory fixed point exists for these models.

Higher derivative terms do not alter the fixed point structure.

Implications for high-Tc superconductivity are discussed.

Abstract

For all Poincar\'e invariant Lagrangians of the form ${\cal L}\equiv f(F_{\mu\nu})$, 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 $\sim \partial^nF$ in ${\cal L}$, 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.