One-loop pseudoscalar mass in a 2HDM with a $Z_3$ symmetry

A two-Higgs doublet model with a discrete $Z_3$ symmetry acquires, in its scalar and gauge sectors, an accidental continuous $U(1)$ symmetry. One therefore finds, after spontaneous symmetry breaking of those symmetries, that a massless pseudoscalar arises, as expected by Goldstone's theorem. In the fermion sector, however, it is possible to obtain Yukawa matrix textures which distinguish between the $Z_3$ and $U(1)$ symmetries, so one expects loop corrections to originate a non-zero pseudoscalar mass for the $Z_3$ case. We perform an explicit calculation that shows that at one-loop the pseudoscalar remains massless for all but one tested $Z_3$ Yukawa matrices. The pseudoscalar mass thus found is highly suppressed by the hierarchy of fermion masses.