The ${\Bbb Z}_2$ anomaly in some chiral gauge theories

We revisit the simplest Bars-Yankielowicz (BY) model (the $\psi\eta$ model), starting from a model with an additional Dirac pair of fermions in the fundamental representation, together with a complex color-singlet scalar $\phi$ coupled to them through a Yukawa interaction. This model possesses a color-flavor-locked 1-form ${\Bbb Z}_N$ symmetry, due to the intersection of the color $SU(N)$ and two nonanomalous $U(1)$ groups. In the bulk, the model reduces to the $\psi\eta$ model studied earlier when $\phi$ acquires a nonzero vacuum expectation value and the extra fermions pair up, get massive and decouple (thus we will call our extended theory as the ``X-ray model"), while it provides a regularization of the $\Bbb Z_2$ fluxes needed to study the $\Bbb Z_2$ anomaly. The anomalies involving the 1-form ${\Bbb Z}_N$ symmetry reduce, for $N$ even, exactly to the mixed ${\Bbb Z}_2$ anomaly found earlier in the $\psi\eta$ model. The present work is a first significant step to clarify the meaning of the mixed ${\Bbb Z}_2-[{\Bbb Z}_N^{(1)}]^2$ anomaly found in the $\psi\eta$ and in other BY and Georgi-Glashow type $SU(N)$ models with even $N$.