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Type-Z models, where charged leptons, up type quarks, and down type quarkseach couple to a different scalar, are only possible when there are three ormore Higgs doublets. We consider the Type-Z three Higgs doublet model imposedby a softly broken $Z_3$ symmetry. We take into account all theoretical andexperimental constraints, highlighting the importance of perturbative unitarityand bounded from below conditions that we develop here, and which have a strongimpact on $B \rightarrow X_s \gamma$. Since there can be cancellations betweenthe two charged Higgs in $B \rightarrow X_s \gamma$ (and in $h \rightarrow\gamma \gamma$), the lower bounds obtained on the charged Higgs masses arealleviated. We also discuss in detail the important physical differencesbetween exact alignment and approximate alignment, and present some usefulbenchmark points.

Current bounds on the Type-Z $Z_3$ three Higgs doublet model