Anomalous gauge theories revisited

TL;DR

This paper re-examines the formulation of anomalous chiral gauge theories on the lattice, showing that many cannot be consistently formulated within the Ginsparg-Wilson framework, especially in four dimensions.

Contribution

It demonstrates the limitations of lattice formulations for anomalous non-abelian gauge theories and discusses conditions under which abelian theories can be formulated.

Findings

Most anomalous non-abelian theories cannot be formulated on the lattice.

Compact U(1) formulations fail for anomalous abelian theories, but non-compact ones may succeed.

The results do not apply to effective low-energy theories derived from anomaly-free models.

Abstract

A possible formulation of chiral gauge theories with an anomalous fermion content is re-examined in light of the lattice framework based on the Ginsparg-Wilson relation. It is shown that the fermion sector of a wide class of anomalous non-abelian theories cannot consistently be formulated within this lattice framework. In particular, in 4 dimension, {\it all} anomalous non-abelian theories are included in this class. Anomalous abelian chiral gauge theories cannot be formulated with compact $\U(1)$ link variables, while a non-compact formulation is possible at least for the vacuum sector in the space of lattice gauge fields. Our conclusion is not applied to effective low-energy theories with an anomalous fermion content which are obtained from an underlying anomaly-free theory by sending the mass of some of fermions to infinity. For theories with an anomalous fermion content in which the anomaly is cancelled by the Green-Schwarz mechanism, a possibility of a consistent lattice formulation is not clear.