The ℤ₂ anomaly in some chiral gauge theories

We revisit the simplest Bars-Yankielowicz (BY) model (the ψη model), starting from a model with an additional Dirac pair of fermions in the fundamental representation, together with a complex color-singlet scalar ϕ coupled to them through a Yukawa interaction. This model possesses a color-flavor-locked 1-form 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 ψη model studied earlier when ϕ 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 ℤ2 fluxes needed to study the ℤ2 anomaly. The anomalies involving the 1-form ℤ N symmetry reduce, for N even, exactly to the mixed ℤ2 anomaly found earlier in the ψη model. The present work is a first significant step to clarify the meaning of the mixed ℤ2 − [ℤN(1)]2 anomaly found in the ψη and in other BY and Georgi-Glashow type SU(N) models with even N. $\psi\eta$ and in other BY and Georgi-Glashow type $SU(N)$ models with even $N$.