ΔM _d,s , B ⁰ _d,s →μ ⁺ μ ^- and B→X _s γ in supersymmetry at large tan β

We present an effective Lagrangian formalism for the calculation of flavour changing neutral and charged scalar currents in weak decays including SU(2)×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 tanβ 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 tanβ resummed corrections in the SU(2)×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 tanβ 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 →μ ⁺ μ ^- and the B ⁰ _d,s -B ⁰ _d,s mass differences ΔM _d,s that include simultaneously the resummed large tanβ corrections, SU(2)×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 →μ ⁺ μ ^- and the suppression of ΔM _s , that for large tanβ are caused by the enhanced flavour changing neutral Higgs couplings to down quarks. We take into account the constraint from B→X _s γ clarifying some points in the calculation of the large tanβ enhanced corrections to this decay.