Relations between ΔM_s,d and B_s,d→μμ̄ in models with minimal flavour violation

The predictions for the B_s,d-B̄_s,d mixing mass differences ΔM_s,d and the branching ratios Br(B_s,d→μμ̄) within the Standard Model (SM) and its extensions suffer from considerable hadronic uncertainties present in the B_s,d-meson decay constants FB_s,d that enter these quantities quadratically. We point out that in the restricted class of models with minimal flavour violation (MFV) in which only the SM low energy operators are relevant, the ratios Br(B_q → μμ̄)/ΔM_q (q = s,d) do not depend on F_Bq and the CKM matrix elements. They involve in addition to the short distance functions and B-meson lifetimes only the non-perturbative parameters B̂_s,d. The latter are under much better control than F_Bs,d. Consequently in these models the predictions for Br(B_q → μμ̄ have only small hadronic uncertainties once ΔM_q are experimentally known. Of particular interest is also the relation Br(B_s→μμ̄)/Br(B_d→μμ̄) = B̂_d τ(B_s) ΔM_s/B̂_s τ(B_d) ΔM_d that is practically free of theoretical uncertainties as B̂_s/B̂_d = 1 up to small SU(3) breaking corrections. Using these ideas within the SM we find much more accurate predictions than those found in the literature: Br(B_s → μμ̄) = (3.4 ± 0.5) × 10^-9 and Br(B_d → μμ̄) = (1.00 ± 0.14) × 10^-10 were in the first case we assumed as an example ΔM_s = (18.0 ± 0.5)/ps.