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Correlation between Delta M_s and B^0_{s,d} --> mu^+ mu^- in Supersymmetry at Large tan beta

Considering the MSSM with the CKM matrix as the only source of flavour violation and heavy supersymmetric particles at large $\tanβ$, we analyze the correlation between {\it the increase} of the rates of the decays $B^0_{s,d}\to μ^+μ^-$ and {\it the suppression} of $ΔM_s$, that are caused by the enhanced flavour changing neutral Higgs couplings to down-type quarks. We give analytic formulae for the neutral and charged Higgs couplings to quarks including large $\tanβ$ resummed corrections in the $SU(2)\times U(1)$ limit and comment briefly on the accuracy of this approximation. For $0.8\le (ΔM_s)^{\rm exp}/(ΔM_s)^{\rm SM}\le 0.95$ we find $6\cdot 10^{-7}\ge BR(B^0_s\ra μ^+μ^-)^{\rm max} \ge 4\cdot 10^{-8}$ and $1.4\cdot 10^{-8}\ge BR(B^0_d\ra μ^+μ^-)^{\rm max}\ge 1\cdot 10^{-9}$. For $(ΔM_s)^{\rm exp} \ge (ΔM_s)^{\rm SM}$ substantial enhancements of $B^0_{s,d}\raμ^+μ^-$ relative to the expectations based on the Standard Model are excluded. With $(ΔM_s)^{\rm exp}>15.0/$ps a conservative analysis of $(ΔM_s)^{\rm SM}$ gives $BR(B^0_s\ra μ^+μ^-)\simlt1.2\cdot10^{-6}$ and $BR(B^0_d\ra μ^+μ^-)\simlt3\cdot10^{-8}$. However, we point out that in the less likely scenario in which the squark mixing is so large that the neutral Higgs contributions dominate $ΔM_s$, the rates for $B^0_{s,d}\to μ^+μ^-$ increase with increasing $ΔM_s$ and the bounds in question are weaker. Violation of all these correlations and bounds would indicate new sources of flavour violation.