$\Delta M_{d,s}$, $B^0_{d,s}\to\mu^+\mu^-$ and $B\to X_s\gamma$ in Supersymmetry at Large $\tan\beta$

We present an effective Lagrangian formalism for the calculation of flavour changing neutral and charged scalar currents in weak decays including $SU(2)\times U(1)$ symmetry breaking effects and the effects of the electroweak couplings $g_1$ and $g_2$. We apply this formalism to the MSSM with large $\tan\beta$ with the CKM matrix as the only source of flavour violation, heavy supersymmetric particles and light Higgs bosons. We give analytic formulae for the neutral and charged Higgs boson couplings to quarks including large $\tan\beta$ resummed corrections in the $SU(2) \times U(1)$ limit and demonstrate that these formulae can only be used for a semi-quantitative analysis. In particular they overestimate the effects of large $\tan\beta$ resummed corrections. We give also improved analytic formulae that reproduce the numerical results of the full approach within $5-10%$. We present for the first time the predictions for the branching ratios $B^0_{s,d}\to \mu^+\mu^-$ and the $B^0_{d,s}-\bar B^0_{d,s}$ mass differences $\Delta M_{d,s}$ that include simultaneously the resummed large $\tan\beta$ corrections, $SU(2)\times U(1)$ breaking effects and the effects of the electroweak couplings. We perform an anatomy of the correlation between the increase of the rates of the decays $B^0_{s,d}\to\mu^+\mu^-$ and the suppression of $\Delta M_s$, that for large $\tan\beta$ are caused by the enhanced flavour changing neutral Higgs couplings to down quarks. We take into account the constraint from $B\to X_s \gamma$ clarifying some points in the calculation of the large $\tan\beta$ enhanced corrections to this decay.