Two--loop Contributions of Flavor Changing Neutral Higgs Boson to $\mu \to e\gamma$

The two--loop mechanism of Bjorken and Weinberg is used to constrain flavor changing neutral Higgs bosons. We calculate the complete set of two--loop diagrams for the rare decay $\mu \rightarrow e + \gamma$ induced by such neutral Higgs bosons, for arbitrary Higgs and top masses. The analytic result is used to set limits on Higgs masses for some recent models with specific ansatz about the flavor changing couplings. For example, in the Cheng--Sher scenario of multi-Higgs doublet models, all neutral Higgs bosons possess flavor changing ($f_i \leftrightarrow f_j)$ couplings proportional to $\sqrt{m_i m_j}$. We find that the present limit on $\mu\to e\gamma$ implies that, in such scheme, these neutral Higgs bosons should be heavier than $200$ GeV.