→ μ μ̄ in Models with Minimal Flavour Violation

The predictions for the Bs,d − B̄s,d mixing mass differences ∆Ms,d and the branching ratios Br(Bs,d → μμ̄) within the Standard Model (SM) and its extensions suffer from considerable hadronic uncertainties present in theBs,d-meson decay constants FBs,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(Bq → μμ̄)/∆Mq (q = s, d) do not depend on FBq and the CKM matrix elements. They involve in addition to the short distance functions and B meson lifetimes only the non-perturbative parameters B̂s,d. The latter are under much better control than FBs,d . Consequently in these models the predictions for Br(Bq → μμ̄) have only small hadronic uncertainties once ∆Mq are experimentally known. Of particular interest is also the relation Br(Bs → μμ̄) Br(Bd → μμ̄) = B̂d B̂s τ(Bs) τ(Bd) ∆Ms ∆Md that is practically free of theoretical uncertainties as B̂s/B̂d = 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(Bs → μμ̄) = (3.4±0.5)·10 −9 and Br(Bd → μμ̄) = (1.00±0.14)·10 −10 were in the first case we assumed as an example ∆Ms = (18.0 ± 0.5)/ps. 1. Among the possible extentions of the Standard Model (SM), of particular interest are the models with minimal flavour violation (MFV), where the only source for flavour mixing is still given by the CKM matrix (see, for instance [1, 2, 3]). In the restricted class of these models [1], in which only the SM low energy operators are relevant, it is possible to derive relations between various observables that are independent of the parameters specific to a given MFV model [1, 4]. Violation of these relations would indicate the relevance of new low energy operators and/or the presence of new sources of flavour violation encountered for instance in general supersymmetric models [5, 6, 7, 8]. In this letter we would like to point out the existence of simple relations between the Bs,d − B̄s,d mixing mass differences ∆Ms,d and the branching ratios for the rare decays Bs,d → μμ̄ that are valid in models with minimal flavour violation (MFV) as defined in [1]. These relations should be of interest for the Run II at Tevatron and later for the LHC and BTeV experiments where ∆Ms and Br(Bs,d → μμ̄) should be measured. Moreover, they allow one to make much more accurate predictions for Br(Bs,d → μμ̄) once ∆Ms,d are precisely known. To our knowledge the relations in question have not been discussed so far in the literature except for a short comment made by us in [9]. 2. Within the MFV models ∆Ms,d and Br(Bs,d → μμ̄) are given as follows (q = d, s) [10] ∆Mq = G2F 6π2 ηBmBq(B̂qF 2 Bq )M 2 W|V ∗ tbVtq| S(xt, xnew), (1) Br(Bq → μμ̄) = τ(Bq) G2F π η Y ( α 4π sin θW )2 F 2 Bqm 2 μmBq |V ∗ tbVtq| Y (xt, x̄new), (2) where FBq is the Bq-meson decay constant and B̂q the renormalization group invariant parameter related to the hadronic matrix element of the operator Q(∆B = 2). See [10] for details. ηB = 0.55 ± 0.01 [11, 12] and ηY = 1.012 [13] are the short distance QCD corrections evaluated using mt ≡ mt(mt). In writing (2) we have neglected the terms O(m 2 μ/m 2 Bq ) in the phase space factor. The short distance functions S(xt, xnew) and Y (xt, x̄new) result from the relevant box and penguin diagrams specific to a given MFV model. They depend on the top quark mass (xt = m 2 t/M 2 W ) and new parameters like the masses of new particles that we denoted collectively by xnew and x̄new. Explicit expressions for these functions in the MSSM at low tan β and in the ACD model [14] in five dimensions can be found in [15] and [16], respectively. The main theoretical uncertainties in (1) and (2) originate in the values of B̂qF 2 Bq and F 2 Bq for which the most recent published values obtained by lattice simulations read [17]