Generalized Bohr-Sommerfeld rules for anomalies with applications to symmetry breakdown and decoupling

In the presence of anomalies, the requirement that a classical symmetry group $G$ has a proper action on the fermion measure or in the effective Lagrangian description imposes Bohr-Sommerfeld conditions on the anomalies, and often implies that $G$ is broken to a subgroup $H$ as well. We show these results in this paper and apply them to QCD and SU(5). In particular, constraints on the QCD order parameter are derived, and an argument is presented which suggests that the breakdown of the chiral flavor symmetry and the emergence of some sort of generation structure in QCD may be natural.