$b \rightarrow s \gamma$ and $\epsilon_b$ Constraints on Two Higgs Doublet Model

We perform a combined analysis of two stringent constraints on the 2 Higgs doublet model, one coming from the recently announced CLEO II bound on $B(b \rightarrow s \gamma)$ and the other from the recent LEP data on $\epsilon_b$. We have included one-loop vertex corrections to $Z \rightarrow b \overline b$ through $\epsilon_b$ in the model. We find that the new $\epsilon_b$ constraint excludes most of the less appealing window $\tan\beta\lsim 1$ at $90\%$C.~L. for $m_t=150\GeV$. We also find that although $b \rightarrow s \gamma$ constraint is stronger for $\tan\beta>1$, $\epsilon_b$ constraint is stronger for $\tan\beta\lsim 1$, and therefore these two are the strongest and complimentary constraints present in the charged Higgs sector of the model.