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 tan beta, we analyze the correlation between the increase of the rates of the decays B^0_{s,d} --> mu^+ mu^- and the suppression of Delta M_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 tan beta resummed corrections in the $SU(2)\times U(1)$ limit and comment briefly on the accuracy of this approximation. For $0.8\le (\Delta M_s)^{\rm exp}/(\Delta M_s)^{\rm SM}\le 0.95$ we find $6\cdot 10^{-7}\ge BR(B^0_s\to \mu^+\mu^-)^{\rm max} \ge 3\cdot 10^{-8}$ and $1.2\cdot 10^{-8}\ge BR(B^0_d\to \mu^+\mu^-)^{\rm max}\ge 1\cdot 10^{-9}$. For $(\Delta M_s)^{\rm exp} \ge (\Delta M_s)^{\rm SM}$ substantial enhancements of $B^0_{s,d}\ra\mu^+\mu^-$ relative to the expectations based on the Standard Model are excluded. With the present bound $(\Delta M_s)^{\rm exp}>15.0/$ps a conservative analysis of $(\Delta M_s)^{\rm SM}$ gives $BR(B^0_s\to \mu^+\mu^-)\simlt1.2\cdot10^{-6}$ and $BR(B^0_d\to \mu^+\mu^-) \simlt3\cdot10^{-8}$. However, we point out that in the less likely scenario in which the squark mixing is so large that the neutral Higgs contributions dominate Delta M_s, the rates for B^0_{s,d} --> mu^+ mu^- increase with increasing Delta M_s and the bounds in question are weaker. Violation of all these correlations and bounds would indicate new sources of flavour violation.