Integer solutions to the anomaly equations for a class of chiral gauge theories

We find all the integer charge solutions to the equations for the cancellation of local gauge anomalies in a class of gauge theories which extend the Standard Model (SM) by a gauge group of the form $G \times U(1)$, where $G$ is an arbitrary semisimple compact Lie group. The SM fermions are assumed to be neutral under $G \times U(1)$ gauge interactions, while the new fermions transform in non-trivial representations of both the new and the SM gauge groups. Our analysis is valid also when the latter is embedded in an arbitrary semisimple compact Lie group. Theories with this structure have been recently studied as models of composite axions based on accidental symmetries and can provide a field theory resolution to the axion quality problem. We apply our results to cases of phenomenological interest and prove the existence of charge assignments with Peccei-Quinn symmetry protected up to dimension 18.