The supersymmetric Higgs sector and $B-\overline{B}$ mixing for large tan $\beta$

We match the Higgs sector of the most general flavour breaking and CP violating minimal supersymmetric standard model (MSSM) onto a generic two-Higgs-doublet model, paying special attention to the definition of tan beta in the effective theory. In particular no tan beta-enhanced loop corrections appear in the relation to tan beta defined in the DRbar scheme in the MSSM. The corrections to the Higgs-mediated flavour-changing amplitudes which result from this matching are especially relevant for the B_d and B_s mass differences dM_s,d for minimal flavour violation, where the superficially leading contribution vanishes. We give a symmetry argument to explain this cancellation and perform a systematic study of all Higgs-mediated effects, including Higgs loops. The corrections to dM_s are at most 7% for mu>0 and M_A < 600 if constraints from other observables are taken into account. For mu<0 they can be larger, but are always less than about 20%. Contrary to recent claims we do not find numerically large contributions here, nor do we find any tan beta-enhanced contributions from loop corrections to the Higgs potential in B^+ -> tau^+ nu or B -> X_s gamma. We further update supersymmetric loop corrections to the Yukawa couplings, where we include all possible CP-violating phases and correct errors in the literature. The possible presence of CP-violating phases generated by Higgs exchange diagrams is briefly discussed as well. Finally we provide improved values for the bag factors P^VLL_1, P^LR_2, and P^SLL_1 at the electroweak scale.