The decay Bs ->mu+ mu-: updated SUSY constraints and prospects

We perform a study of the impact of the recently released limits on BR(Bs ->mu+ mu-) by LHCb and CMS on several SUSY models. We show that the obtained constraints can be superior to those which are derived from direct searches for SUSY particles in some scenarios, and the use of a double ratio of purely leptonic decays involving Bs ->mu+ mu- can further strengthen such constraints. We also discuss the experimental sensitivity and prospects for observation of Bs ->mu+ mu- during the sqrt(s)=7 TeV run of the LHC, and its potential implications.


Bs
(1) where the coefficients C Q 1 , C Q 2 , and C 10 parametrize different contributions. Within the SM, C Q 1 and C Q 2 are negligibly small, whereas the main contribution entering through C 10 is helicity suppressed. In SUSY, both C Q 1 and C Q 2 can receive large contributions from scalar exchange, which was first pointed out (in the context of a different decay, b → sl + l − ) in [28]. The explicit expressions for the different coefficients can be found in e.g. [17]. The B s decay constant, f Bs , constitutes the main source of uncertainty in BR(B s → µ + µ − ). As of the year 2009 there were two unquenched lattice QCD calculations of f Bs , by the HPQCD collaboration [29] and FNAL/MILC [30] respectively, which when averaged gave the value f Bs = 238.8 ± 9.5 MeV [31]. The calculation of [30] was updated in [32], which gave rise to a higher world average of f Bs = 250 ± 12 MeV in the year 2010. Recently, the ETM collaboration announced its result of f Bs = 232 ± 10 MeV [33]. At the Lattice 2011 conference [34], new results by FNAL/MILC (f Bs = 242 ± 9 MeV) and the HPQCD collaboration (f Bs = 226 ± 10 MeV [35] and f Bs = 225 ± 4 MeV [36]) suggest that an updated world average would be lower than that of the year 2009. In our numerical analysis we will use f Bs = 238.8 ± 9.5 MeV [31].
To study the constraints on the parameter spaces of SUSY scenarios, we use the newly released combined limit from LHCb and CMS at 95% C.L. [15]: More details are given in section 5. In order to take into account the theoretical uncertainties, in our numerical analysis we will use the following limit BR(B s → µ + µ − ) < 1.26 × 10 −8 (3) to constrain the parameter spaces of the SUSY models under consideration.

The double ratios of purely leptonic decays
The main uncertainty in the theoretical prediction of B s → µ + µ − is f Bs . As described in Section 2, f Bs is now being evaluated in the unquenched approximation by various lattice collaborations. The error (which is currently around 5% or less) has been reduced over time, and the central values of f Bs from the various collaborations are in reasonable agreement. The prospects for a further reduction of the error are good. However, despite the continuing improvement in the calculations of f Bs our view is that it is instructive to consider other observables which involve B s → µ + µ − but do not depend on the decay constants, and to compare the constraints on the SUSY parameter space with those which are obtained from BR(B s → µ + µ − ) alone. One such observable which involves B s → µ + µ − , but has essentially no dependence on the absolute values of the decay constants, is a double ratio involving the leptonic decays B u → τ ν, B s → µ + µ − , D → µν and D s → µν/τ ν [19,20]. One such double ratio is defined by: The quantity (f B /f Bs )/(f D /f Ds ) deviates from unity by small corrections of the form m s /m b and m s /m c . The double ratio would be equal to one in the heavy quark limit of a very large mass for the b and c quarks, and in the limit of exact SU(3) flavour symmetry (m s → 0). A calculation in [19] gives (f B /f Bs )/(f D /f Ds ) = 0.967, and subsequent works [37] also give values very close to 1, with a very small error. Unquenched lattice calculations of the ratios f Ds /f D and f Bs /f B have a precision of the order of 1% (e.g. [32]), from which it can be inferred that the numerical value of the double ratio is very close to 1. In our numerical analysis we will take (f B /f Bs )/(f D /f Ds ) = 1. Importantly, the absolute values of the decay constants do not determine the value of the double ratio, in contrast to the case of BR(B s → µ + µ − ) alone in Eq. (1). Instead, |V ub | replaces f Bs as the only major source of uncertainty, as can be seen from Eq. (4). Information on |V ub | is available from direct measurements of semileptonic decays of B mesons, both inclusive (B → X u ν) and exclusive (B → π ν). Moroever, global fits [38] in the context of the SM give additional experimental information on |V ub |. Due to its different theoretical uncertainties, the double ratio is an alternative observable which includes B s → µ + µ − , and can provide competitive constraints on SUSY parameters. A comparison of the constraints on specific SUSY models from the double ratio and from BR(B s → µ + µ − ) alone is of much interest because the theoretical input parameters |V ub | and f Bs for these two observables are independent. Such a comparative study was performed for the first time in [21], and it was shown that the double ratio can provide stronger constraints. In particular, the constraints from the double ratio are maximised (minimised) for smaller (larger) |V ub |, while the constraints from BR(B s → µ + µ − ) are maximised (minimised) for larger (smaller) f Bs .
We will perform an updated study of these two observables using the recently improved bounds on BR(B s → µ + µ − ) from the LHCb [13] and CMS [14] collaborations. In the previous study of the double ratio [21] the value |V ub | = (3.92 ± 0.45 ± 0.09) × 10 −3 [39] was used, which is an average of the exclusive and inclusive determinations of |V ub |. We note that this world average does not include three recent measurements of |V ub |, of which two are from the exclusive channel [40,41] and one is from the inclusive channel [42]. The inclusion of these measurements would only have a small effect on the world average, and so for simplicity we will use |V ub | = (3.92 ± 0.45 ± 0.09) × 10 −3 , as done in [21].
We note that the exclusive determination of |V ub | suggests values of |V ub | which are below the central value of the world average. The exclusive determination of |V ub | requires a theoretical calculation of one hadronic form factor f + (q 2 ) (where q is the momentum of ). For q 2 > 16 GeV 2 one can use lattice QCD to calculate f + (q 2 ), while for q 2 < 16 GeV 2 non-lattice techniques must be used. In both regions of q 2 the extracted value of |V ub | is below the central value of the world average. The inclusive determination of |V ub |, which does not have a dependence on lattice QCD, suggests values which are above the central value of the world average. Prospects for precise measurements of |V ub | in the inclusive channel are very good at highluminosity B factories. In particular, the method used in [42,43] is a very promising approach because the theoretical errors are greatly reduced by employing a low cut on the momentum of the (p > 1 GeV), which keeps 90% of the phase space of B → X u ν. This anticipated experimental improvement in the measurement of |V ub | bodes well for the double ratio as an alternative observable with which to constrain SUSY. It is important to emphasise that f Bs is currently known with greater precision than |V ub |, and this may also be the case in the era of a high-luminosity B factory. However, we note that the central values of these unrelated input parameters plays a major role in determining which observable gives the stronger constraints, as will be discussed in our numerical analysis. The double ratio also has the attractive feature of using ongoing measurements of BR(D s → µν/τ ν) and BR(D → µν). Such decays are not usually discussed when constraining SUSY parameters (although see [44] for a discussion of D s → µν/τ ν in this regard), but increased precision in their measurements would enhance the capability of the double ratio to probe the SUSY parameter space. The decay B u → τ ν alone is very sensitive to the presence of a charged Higgs boson (H ± ) and provides a strong constraint on tan β and the mass of H ± in SUSY models [45][46][47][48]. The experimental prospects for precise measurements of all the decays in the double ratio are very promising. The precision in the measurements of BR(D s → µν) and BR(D s → τ ν) will be improved at the ongoing BES-III experiment [49], and at high-luminosity B factories operating at a centre-of-mass energy of √ s ∼ 10.6 GeV (and also possibly at energies in the charm threshold region). Similar comments apply to the prospects for significantly improved measurements of BR(D → µν) and BR(B u → τ ν). For more details about the calculation of these decays we refer the reader to [21].
In this analysis we use where The theoretical evaluation of η SM gives (2.47 ± 0.58) × 10 −7 where the main uncertainty comes from V ub . To determine the experimental limit on the ratio R, we combine the limits on the individual branching fractions, namely BR(B s → µ + µ − ), BR(B u → τ ν), BR(D s → τ ν) and BR(D → µν). To compute the p.d.f of R, we use a Gaussian distribution for the measured decays, and a "truncated" Gaussian p.d.f for the upper limit in (2). We consider two different approaches. The first approach consists in building first the p.d.f for BR(B s → µ + µ − ) which reproduces the 90% and 95% C.L. experimental limits, and to combine it with the Gaussian p.d.f of the other involved decays. The second approach determines the p.d.f from the derivative For B s → µ + µ − the p.d.f was obtained based on the C.L. from [15]. For the other three decays, their measurements are modelled as Gaussians.
of the CL s+b with BR(B s → µ + µ − ), which is extracted from the derivative of CL s shown in Ref. [15] and the almost constant behaviour of CL b . Fig. 1 shows the R p.d.f . Both approaches agree and provide the upper limit for R, at 95% C.L.: in which the uncertainty from V ub is taken into account. In our numerical analysis we use (7) to constrain the supersymmetric parameter space in various scenarios in the MSSM and NMSSM.

Constraints on SUSY Models
We consider five distinct SUSY models in order to illustrate the impact of the new limits on BR(B s → µ + µ − ) on the SUSY parameter spaces. All previous studies have been carried out before the LHCb [13] and CMS [14] limits were released 1 . Some very recent works [51] study the impact of the latest CDF result [12] only, and address the case of the excess of events being a genuine signal. Moreover, none of the previous studies have considered the double ratio, apart from our earlier work in [21] in which two of the five SUSY scenarios were discussed. For each scenario we also check the constraints from direct searches for Higgs bosons and delimit the regions where the lightest supersymmetric particle (LSP) is charged. All the flavour observables are calculated with the SuperIso v3.2 program [16][17][18]. The spectrum of the MSSM points is generated with SOFTSUSY-3.1.7 [52] and we used NMSPEC program from the NMSSMTools 3.0.0 package [53] for the NMSSM points. For every generated MSSM point we check if it fulfills the constraints from the Higgs searches using HiggsBounds-3.2.0 [54,55]. The value of m t = 173.3 GeV [56] is used throughout.

CMSSM
The first model we consider is the constrained MSSM (CMSSM) [57], which is characterized by the set of parameters {m 0 , m 1/2 , A 0 , tan β, sgn(µ)}. The CMSSM model invokes unification boundary conditions at a very high scale m GU T where the universal mass parameters are specified.
The results are displayed in Fig. 2, where the four-dimensional space is projected into a plane. When interpreting these results it is therefore important to remember that each point in the figures corresponds to a multi-dimensional parameter space in the variables which are not displayed on the x-axis and the y-axis.
In order to show the viable parameter space of the SUSY scenario under investigation, in all the figures we introduce a colour coding which is applied sequentially. Areas which are disallowed theoretically are in white. Next, the points which are disallowed phenomenologically are plotted, which are those with a charged LSP (in violet) and those which are excluded by the direct searches for Higgs bosons (in black). In this way, these points lie in the background. On top of them, the points excluded by the double ratio R (in orange) are displayed, superseded by the points excluded by BR(B s → µ + µ − ) (in yellow). Finally the allowed points (in green) are shown in the foreground.
These indirect constraints on the CMSSM parameter space from BR(B s → µ + µ − ) are competitive with the direct constraints from searches for squarks and gluinos by ATLAS and CMS [58]. As expected, one can see strong constraints on small m A and large tan β values. At large tan β ( 30), these constraints are stronger than those obtained from BR(B → X s γ) [3].
In order to better quantify the impact of BR(B s → µ + µ − ) and R, we show in Fig. 3 the constraints for fixed values of tan β (=30, 40 and 50) and A 0 = 0. One striking result here is that the double ratio, being a combination of four different flavour observables, extends impressively the constraints obtained by BR(B s → µ + µ − ) alone, as was pointed out in [21]. Also, for tan β = 50, the constraints from the flavour observables go far beyond the direct search limits by the ATLAS and CMS collaborations for the same scenario.
The SUSY contributions to B u → τ ν gives rise to a scale factor which multiplies BR(B u → τ ν). When we manually set this scale factor to be equal to 1 (as in the SM), the excluded region of the plane [m 0 , m 1/2 ] does not change much. Therefore we conclude that the points i) and ii) above are the main reasons why the double ratio gives the superior constraints.

NUHM
The second model we consider involves non-universal Higgs masses (NUHM) [59]. This model generalizes the CMSSM, allowing for the GUT scale mass parameters of the Higgs doublets to have values different from m 0 , i.e. m H 1 = m H 2 = m 0 . These two additional parameters with dimension of mass can be traded for two other parameters at a lower scale, which can be conveniently chosen as the µ parameter and the mass m A of the CP-odd Higgs boson.     Fig. 4. Again the constraints are very important, and restrict strongly the region of large tan β / small m A .
In Fig. 5 we show two examples in the two-dimensional parameter planes (µ, m A ) and (m H + , tan β) with the rest of parameters being fixed. As can be seen from the figures, a large part of the parameter space is restricted by BR(B s → µ + µ − ) and R observables, whereas in the same plane one would not get any constraints from BR(B → X s γ) for µ > 0 [3].

AMSB
We can now focus on another supersymmetry breaking scenario, namely the Anomaly Mediated Supersymmetry Breaking (AMSB) [60]. This is a special case of gravity mediation in which there is no direct tree-level coupling that transmits the SUSY breaking in the hidden sector to the visible one. The breaking is communicated through the conformal anomaly. The free parameters of the minimal model consist of {m 0 , m 3/2 , tan β, sgn(µ)}.

GMSB
The last MSSM scenario that we consider is the Gauge Mediated Supersymmetry Breaking (GMSB) scenario [62], which consists of the SUSY breaking sector and the messenger sector. The latter can be taken as a 5 +5 of the SU(5) which contains the Standard Model group, and therefore the gauge coupling unification is not affected. The minimal model is characterized by the set of parameters {Λ, M mess , N 5 , c grav , tan β, sgn(µ)}. For our study, we consider N 5 = 1, c grav = 1 and generate about 300,000 random points in the ranges Λ ∈ [10, 500] TeV, M mess ∈ [10 2 , 10 14 ] TeV and tan β ∈ [1, 60] with Λ < M mess .
In Fig. 9 we show the results in the parameter planes (Λ, tan β) and (M mess , tan β). Again, the region of large tan β is the most restricted by the flavour observables. To see better the regions in the parameter space which are excluded by BR(B s → µ + µ − ) and the double ratio R, we show in Fig. 10 the results in the plane (M mess , tan β) for a fixed value of Λ = 100 TeV for both µ > 0 and µ < 0. It is remarkable to see that tan β 40 is excluded regardless of the value of M mess , while the same plane is probed by the well-known BR(B → X s γ) constraints only for a very large messenger scale (M mess 10 10 TeV) [3].

CNMSSM
The last scenario that we consider is a constrained version of the NMSSM (CNMSSM) with semi-universal parameters defined at the GUT scale [63]. The choice of a semi-universal scenario instead of the case of strict universality facilitates the obtention of valid NMSSM points [64]. In this scenario, κ, λ and m 2 S are computed from the minimization equations and the free parameters are {m 0 , m 1/2 , A 0 , A κ , λ, tan β, sgn(µ)}. Previous studies were performed in [65,66]. The results are displayed in Fig. 12 in the parameter planes (m H + , tan β) and (λ, tan β). The constraints are more severe for large tan β, small m H + and large λ. In Fig. 13 we fix two of the parameters, namely λ = 0.01 and tan β = 50. This allows us to see in a clearer way the effect of the constraints on the other parameters. In Fig. 14 the same results are shown for λ = 0.1. As mentioned before, the constraints are more pronounced for larger λ.
As a final example we fix all the parameters except two, to see the results in a twodimensional plane. This is done in Fig. 15 for A 0 = 1000 GeV, A κ = −60 GeV, tan β = 50 and λ = 0.1. As can be seen, a large part of this parameter plane is excluded by BR(B s → µ + µ − ) and the double ratio R.

Discussion
In the above subsections, we investigated the constraining power of BR(B s → µ + µ − ) and the double ratio R for different SUSY scenarios. As explained in sections 2 and 3, the main input parameter for BR(B s → µ + µ − ) is f Bs , while |V ub | is the most important input for R. To examine how the choice of these input parameters can affect our results, we consider here two scenarios, namely the "least constraining" (with high |V ub | and low f Bs ) and "most constraining" (with low |V ub | and high f Bs ) cases for BR(B s → µ + µ − ) and the double ratio. For the least constraining scenario we consider the inclusive determination of |V ub | with the central value being 4.34 × 10 −3 [39], and f Bs = 232 MeV [33]. For the most constraining case we take the exclusive value |V ub | = 3.42 × 10 −3 [39] and f Bs = 250 MeV [32]. To compare these two cases we take an example in the CMSSM scenario with tan β = 40 and A 0 = 0. The results are presented in Fig. 16. As can be seen, in the most constraining case, the exclusion limits are greatly increased while in the least constraining case the results are only slightly changed. This shows that the analysis in the previous subsections does not correspond to a particularly optimistic choice of the input parameters.
The next point we discuss here is the effect of B u → τ ν in the double ratio. The constraints from B u → τ ν alone on the parameter space of [m 0 , m 1/2 ] have been presented in several works (e.g. [17]) and the excluded region differs from that obtained from BR(B s → µ + µ − ) alone, as can be seen in Fig. 17. In most of the parameter space, BR(B u → τ ν) is reduced with respect to the SM value, leading to the large blue excluded strip in Fig. 17. On the other hand, in the small strip, BR(B u → τ ν) is larger than in the SM. In the narrow region in between, a cancellation happens since the charged Higgs contribution is roughly twice that of the SM contribution and so B u → τ ν cannot exclude this parameter space. As can be seen from the figure, BR(B s → µ + µ − ) probes larger values of m 1/2 than B u → τ ν, although B u → τ ν can exclude part of the region 1300 GeV < m 0 < 1600 GeV and m 1/2 < 200 GeV which cannot be excluded from BR(B s → µ + µ − ) alone and the double ratio.    The reason why the double ratio is more constraining than BR(B s → µ + µ − ) alone is mainly due to two reasons: i) |V ub | is used as an input parameter in the double ratio, instead of f Bs . Although these two parameters have comparable errors, their current central values give rise to stronger constraints from the double ratio, as discussed in the preceding paragraph. This could not have been expected, and a value of f Bs much larger than that preferred by lattice QCD would have ensured that BR(B s → µ + µ − ) alone had the stronger constraints; ii) The experimental value of BR(B u → τ ν), which enters the derivation of η in Eq. (6), is larger than the SM expectation, and so reduces R in Eq. (7), leading to a stronger constraint on the SUSY parameter space. The SUSY contributions to B u → τ ν gives rise to a scale factor which multiplies BR(B u → τ ν). When we manually set this scale factor to be equal to 1 (as in the SM), the excluded region of the plane [m 0 , m 1/2 ] does not change much. Therefore we conclude that the points i) and ii) above are the main reasons why the double ratio gives the superior constraints.
Finally we discuss the effect of a hypothetical measurement of BR(B s → µ + µ − ) at the SM value (3.5 ± 0.3) × 10 −9 . Fig. 18 shows the obtained impact in the CMSSM plane (mt 1 , tan β) with all the parameters being varied in the intervals given in section 4.1. For comparison, the same parameter plane with the current experimental limits is also provided. As can be seen, almost no scenario with tan β 45 remains viable regardless of the other parameters in the case of a SM-like discovery, and the parameter space of the CMSSM becomes very restricted.

Experimental prospects
At present, the best upper limit for BR(B s → µ + µ − ) measured in a single experiment comes from LHCb [13]: at 95% C.L. This upper limit is followed closely by the result from CMS [14]: at 95% C.L. These two results were officially combined for EPS conference in Ref. [15], giving the upper limit of which we will use to constrain the parameter space of SUSY models. The CDF collaboration obtains a 95% C.L. upper limit [12]: together with a one sigma interval coming from an observed excess over the expected background which corresponds to a p−value of 0.27%. Finally, the D0 collaboration obtains the 95% C.L. upper limit [11]: The preliminary result on BR(B s → µ + µ − ) [15] from the combination of the limits from LHCb and CMS shows an excess of more than one sigma (CL b ≈ 0.92 for values of the BR around the SM value) with respect to the background-only hypothesis. This excess can be accounted for by a BR(B s → µ + µ − ) ≈ (3.7 +3.7 −2.7 ) × 10 −9 . However, the signal significance is not enough to claim evidence. In this section we study the experimental sensitivity to B s → µ + µ − and the prospects for its measurement in the period of operation of the LHC at √ s = 7 TeV.

Combination LHC-CDF
The CDF experiment at the Tevatron has reported a p−value of 0.27% for the background only hypothesis [12]. In order to evaluate whether a combination of results from CMS, LHCb and CDF could lead to evidence for a signal, we perform an approximate combination of the results of the three experiments, based on the signal and background expectations and the observed pattern of events. We use mc limit [67] to combine the results of the different experiments and to extract the confidence levels. We have also scaled f d /f s to the value measured at LHCb [68] in order to be consistent with the value used in the LHC combined result. According to this study, a hypothetical combination of the LHCb and CMS results with that of CDF would increase CL b to ∼ 0.994 (for values of the BR close to the most probable value), which is close to a 3σ deviation. Note that this is approximately the same signal significance that CDF obtains alone. This approximate study leads to the following averaged branching ratio: However, at the time of writing this paper, this kind of combination has not been performed officially.

Sensitivity to BR(B s → µ + µ − ) at the LHC
We perform a toy MC study in order to determine how much luminosity is needed to obtain evidence for B s → µ + µ − at the LHC. For this, we scale the signal and background expectations accordingly with the increase of luminosity. Fig. 19 shows the integrated luminosity that is needed in order to obtain a 3(5) σ evidence (discovery) of a given BR(B s → µ + µ − ) in either LHCb or CMS.
Assuming that the ratio of luminosities between CMS and LHCb remains at the value of the current analysis (i.e. CMS takes approximately four times more data than LHCb over the same period of time), we show in Fig. 20 the integrated luminosity scale factor (with respect to the amount of data used in [15]) that would be needed for the discovery of a given BR(B s → µ + µ − ) in the case of a CMS+LHCb combination. The width of the bands reflects possible scenarios for the evolution of the systematic uncertainties, where the lower side assumes negligible systematics and the upper side assumes that the systematics do not get reduced with time. It can be seen that with 6-8 times more luminosity than that used in Ref. [15] a CMS+LHCb combination could provide evidence at the 3σ level for BR(B s → µ + µ − ) of the SM. This corresponds to between 2 and 3 fb −1 for LHCb and between 7 and 10 fb −1 for CMS. As the sensitivity of CMS is equivalent to that of LHCb for four times more luminosity, a scenario in which CMS takes up to 14 fb −1 and LHCb takes 2 fb −1 would afford equal sensitivity as a combination of CMS with 10 fb −1 and LHCb with 3 fb −1 . From this toy MC  Figure 19: Required luminosity in order to provide a 3σ evidence (orange) or a 5σ discovery (green) of a given BR(B s → µ + µ − ) on the left for LHCb and on the right for CMS. study we conclude that the SM prediction for BR(B s → µ + µ − ) is likely to be probed during the operation of the LHC at √ s = 7 TeV (i.e. before the end of the year 2012). If ATLAS can manage to obtain sensitivity to BR(B s → µ + µ − ) which is comparable to that of CMS, then even a 5σ discovery for a SM-like BR(B s → µ + µ − ) would be possible during the run at √ s = 7 TeV. However, from pre-LHC MC studies in Ref. [69] the sensitivity of ATLAS was found to be inferior to that of CMS. If experimental evidence of B s → µ + µ − is achieved at the LHC, the double ratio in Eq. (4) would be measured for the first time. Moreover, limits on the ratio BR(B d → µ + µ − )/BR(B s → µ + µ − ) (which is a very interesting test of Minimal Flavour Violation) would also be set. If BR(B s → µ + µ − ) is much smaller than the SM prediction (as can happen for example in the MSSM [70] and NMSSM), values down to O(5 × 10 −10 ) can still be discovered with an upgrade of the LHCb.

NP discovery with
In section 5.2 we discussed the luminosity needed for discovery of B s → µ + µ − . However, a measurement of B s → µ + µ − with a branching ratio larger than the SM prediction does not necessarily mean a New Physics (NP) discovery. In such a case, the compatibility with the SM prediction has to be computed. Fig. 21 is the equivalent of Fig. 20 but with the SM rate for BR(B s → µ + µ − ) being considered as a background, and the signal corresponds to the NP contribution to BR(B s → µ + µ − ). We can see that for the same luminosity needed for a 3σ evidence of a SM-like signal, the LHC could alternatively claim NP at 3σ if the NP contribution is of the order of 4 − 5 × 10 −9 , i.e, if the actual BR(B s → µ + µ − ) is O(8×10 −9 ). Finally, with the current uncertainties in f d /f s (7.9%) and in the SM prediction (8%), only values of BR(B s → µ + µ − ) that are at least 33%(55)% larger than the SM prediction can allow exclusion of a SM-like rate at 3(5)σ.  Figure 20: Required luminosity in order to provide a 3σ evidence (orange) or a 5σ discovery (green) of a given BR(B s → µ + µ − ) for LHCb and CMS combined. The luminosity is expressed in terms of the luminosity used in [15], (0.34 fb −1 for LHCb and 1.14 fb −1 for CMS).  Figure 21: Required luminosity in order to provide a 3σ evidence (orange) or a 5σ discovery (green) of a given NP contribution to BR(B s → µ + µ − ) for LHCb and CMS combined. The luminosity is expressed in terms of the luminosity used for Ref. [15], (0.34 fb −1 for LHCb and 1.14 fb −1 for CMS).

Conclusions
The decay B s → µ + µ − is known to be a very effective probe of SUSY models with large (> 30) tan β, and its importance has been emphasised in numerous studies over the past decade. Due to its distinct signature, this decay can be searched for by three LHC collaborations: LHCb, CMS and ATLAS. Recently, searches by LHCb and CMS have been released, and have improved the upper limit on its branching ratio to BR(B s → µ + µ − ) < 1.1 × 10 −8 . Using this new bound, we performed a study of the constraints on the parameter space of five distinct SUSY models.
We emphasised that such indirect constraints can be stronger than those which are obtained from the ongoing direct searches for SUSY particles at the LHC. For instance, in the CMSSM for tan β ∼ 50, the SUSY particles have to be very heavy and in particular squarks cannot be lighter than ∼ 1.2 − 2 TeV in order to be compatible with the upper limit on BR(B s → µ + µ − ). Nevertheless, in the scenarios we investigated here, in spite of the severe constraints we obtained, there is still room for SUSY contributions in large parts of the parameter space, especially for small tan β.
In addition, we considered an alternative observable which includes BR(B s → µ + µ − ), namely a double ratio formed from the decays B s → µ + µ − , B u → τ ν, D → µν and D s → µν/τ ν. The magnitude of the double ratio depends on the CKM matrix element |V ub |, a parameter for which there is already considerable experimental information, and the prospects for further precision in measurements of |V ub | are promising. In contrast, the magnitude of BR(B s → µ + µ − ) depends on the absolute value of the decay constant f Bs , and thus a comparative study of the constraints obtained from these two observables is instructive. We showed that the double ratio can provide stronger constraints on the SUSY parameter space, and we advocate its use when discussing the impact of BR(B s → µ + µ − ) alone on SUSY models.
The final integrated luminosity of the operation of the LHC at √ s = 7 TeV is likely to be significantly larger than the amount that was anticipated at the start of the run. Both CMS and LHCb will have a chance to obtain a significant signal by the end of the run, even if BR(B s → µ + µ − ) is as small as the prediction in the SM. Throughout the run at √ s = 7 TeV, the ongoing searches for B s → µ + µ − will continue to compete with the direct searches for SUSY particles as a probe of the parameter space of SUSY models.