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 {\it the increase} of the rates of the decays $B^0_{s,d}\to \mu^+\mu^-$ and {\it 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 $6\cdot 10^{-7}\ge BR(B^0_s\ra \mu^+\mu^-)^{\rm max} \ge 4\cdot 10^{-8}$ and $1.4\cdot 10^{-8}\ge BR(B^0_d\ra \mu^+\mu^-)^{\rm max}\ge 1\cdot 10^{-9}$. 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 $(\Delta M_s)^{\rm exp}>15.0/$ps a conservative analysis of $(\Delta M_s)^{\rm SM}$ gives $BR(B^0_s\ra \mu^+\mu^-)\simlt1.2\cdot10^{-6}$ and $BR(B^0_d\ra \mu^+\mu^-)\simlt3\cdot10^{-8}$. 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 $B^0_{s,d}\to \mu^+\mu^-$ 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.