Relations between \Delta M_{s,d} and B_{s,d}\to \mu\bar\mu in Models with Minimal Flavour Violation

The predictions for the B_{s,d}-\bar B_{s,d} mixing mass differences \Delta M_{s,d} and Br(B_{s,d}\to\mu\bar\mu) within the Standard Model (SM) and its extensions suffer from considerable hadronic uncertainties present in the B_{s,d}-meson decay constants F_{B_{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}\to\mu\bar\mu)/\Delta M_q (q=s,d) do not depend on F_{B_{q}} and the CKM matrix elements. They involve in addition to the short distance functions and B-meson lifetimes only the non-perturbative parameters \hat B_{s,d}. The latter are under much better control than F_{B_{s,d}}. Consequently in these models the predictions for Br(B_{q}\to\mu\bar\mu) have only small hadronic uncertainties once \Delta M_q are experimentally known. Of particular interest is also the relation Br(B_{s}\to\mu\bar\mu)/Br(B_{d}\to\mu\bar\mu)=\hat B_{d}/\hat B_{s} \tau(B_{s})/\tau(B_{d}) \Delta M_{s}/\Delta M_{d} that is practically free of theoretical uncertainties as \hat B_{s}/\hat B_{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}\to\mu\bar\mu)=(3.4\pm 0.5)\cdot 10^{-9} and Br(B_{d}\to\mu\bar\mu)=(1.00\pm 0.14)\cdot 10^{-10} were in the first case we assumed as an example \Delta M_s=(18.0\pm 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 B s,d − B s,d mixing mass differences ∆M s,d and the branching ratios for the rare decays B s,d → µμ that are valid in models with minimal flavour violation (MFV) as defined in [1].
where F Bq is the B q -meson decay constant andB q the renormalization group invariant parameter related to the hadronic matrix element of the operator Q(∆B = 2). See [10] for details. η B = 0.55 ± 0.01 [11,12] and η Y = 1.012 [13] are the short distance QCD corrections evaluated using m t ≡ m t (m t ). 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(x t , x new ) and Y (x t ,x new ) result from the relevant box and penguin diagrams specific to a given MFV model. They depend on the top quark mass and new parameters like the masses of new particles that we denoted collectively by x new andx new . 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) Similar results are obtained by means of QCD sum rules [18]. Consequently the hadronic uncertainties in ∆M s,d and Br(B s,d → µμ) are in the ballpark of ±30% which is clearly disturbing.
The uncertainties in the B q -meson lifetimes are substantially smaller [19]: As noticed by many authors in the past, the uncertainties in ∆M s,d and Br(B s,d → µμ) can be considerably reduced by considering the ratios that can be used to determine |V td /V ts | without the pollution from new physics [1]. In particular the relation (6) will offer after the measurement of ∆M s a powerful determination of the length of one side of the unitarity triangle, denoted usually by R t . As [17] (see also ξ = 1.22 ± 0.07 [20]) the uncertainties in the relations (6) and (7) are in the ballpark of ±15% and thus by roughly a factor of two smaller than in (1) and (2).

3.
Here we would like to point out three useful relations that do not involve the decay constants F B d and consequently contain substantially smaller hadronic uncertainties than the formulae considered so far. These relations follow directly from (1) and (2) and read with where we have used α = 1/129, sin 2 θ W = 0.23 and M W = 80.423 GeV [21].
The simple relation between ∆M s /∆M d and Br(B s → µμ)/Br(B d → µμ) in (9) [22,17,20], the chiral extrapolation in the case ofB q is well controlled and very little variation is observed between quenched and N f = 2 results. Consequently the error in B s /B d = 1 is very small and also the separate values forB s andB d given in (12) are rather accurate [22,17,20,23]. These results should be further improved in the future. Consequently (9) is one of the cleanest relations in B physics but also (10) is rather clean theoretically.
We note that once ∆M s /∆M d has been precisely measured, the relation (9) Using m t (m t ) = (167 ± 5) GeV, the lifetimes in (5) These results are substantially more accurate than the ones found in the literature (see for instance [2,3,25,26]) where the errors are in the ballpark of ±(30 − 50)%.
In calculating the errors in (15) we have added first the experimental errors in τ (B q ), m t (m t ) and ∆M q in quadrature to find ±6.8% and ±4.9% for (13) and (14), respectively. We have then added linearly the error of ±9% fromB Bq . Consequently the total uncertainties in Br(B s → µμ) and Br(B d → µμ) are found to be ±15.8% and ±13.9%, respectively. If all errors are added in quadrature we find ±11.3% and ±10.2%, respectively. As the errors in τ (B s ), m t (m t ) and ∆M s will be decreased considerably in the coming years, the only significant errors in (13) and (14) will be then due to the uncertainties inB Bq . Future lattice calculations should be able to reduce these errors as well, so that predictions for Br(B s,d → µμ) in the SM will become very accurate and in other MFV their accuracy will mainly depend on the knowledge of the short distance functions S and Y . (10) is given entirely by the ratio Y 2 /S. As hadronic uncertainties in (10) are substantially smaller than in (1) and (2), the differences between various MFV models can be easier seen. For instance in the ACD model with five dimensions [14] the ratio Y 2 /S with m t (m t ) = 167 GeV equals 0.58, 0.53, 0.49 and 0.46 for the compactifications scales 1/R = 200, 250, 300, 400 GeV, respectively [16]. In the SM one has Y 2 /S = 0.40 and the effects of the Kaluza-Klein modes could in principle be seen when (10) is used, whereas it is very difficult by means of (2).

The dependence on new physics in
The relation (9) (9) is violated. Finally, we expect that it is generally violated in models with nonminimal flavour violation [5,6,7,8].

5.
In summary we have presented stringent relations between Br(B s,d → µμ) and ∆M s,d that are valid in the MFV models as defined in [1]. The virtue of these relations is their theoretical cleanness that allows to obtain improved predictions for Br(B s,d → µμ) as demonstrated above.
Other useful relations in the MFV models can be found in [4]. It will be interesting to follow the developments at Tevatron, LHC, BTeV, BaBar, Belle and K physics dedicated experiments to see whether these relations are satisfied. While the present experimental upper bounds are still rather weak Br(B s → µμ) < 2.6 × 10 −6 (95% C.L. [29]), Br(B d → µμ) < 2.0 × 10 −7 (95% C.L. [30]), considerable progress is expected in the coming years.
Needless to say the improvement on the accuracy of F Bq is very important as F Bs /F B d is crucial for the determination of the CKM element |V td | as seen in (6) and (7). Moreover the measurements of ∆M s and Br(B s → µμ) in conjunction with accurate values of B s F Bs and F Bs will determine the function S and Y , respectively. This information will allow one to distinguish between various MFV models.
We would like to thank A. Dedes and F. Krüger for useful remarks on the manuscript.