30 parameter version of the MSSM in light of the $B_s\rightarrow\mu^+\mu^-$ observation

The R-parity conserving MSSM in light of the decay $B_s\rightarrow\mu^+\mu^-$ with near-SM branching ratio is an interesting platform for studying the complementarity between direct and indirect searches for beyond the SM physics. Based on this, we have analysed the possible impact of the $B_s\rightarrow\mu^+\mu^-$ observation on the posterior sample from the global fit of a 30-parameter MSSM (MSSM-30), and the related Wilson Coefficients. The MSSM-30 is a systematically constructed, symmetry-guided, MSSM parametrization, as opposed to the traditional frames (e.g. pMSSM) with crude treatment of flavor violation parameters. This paper illustrates why phenomenological frames like the MSSM-30 should be preferred to study flavor physics. For the current and future B-physics experimental precision, such a consideration is crucial for suitably assessing supersymmetric contributions to flavor observables.


Contents 1 Introduction
The decay B s → µ + µ − has been used traditionally as an indicator of how contributions from extended Higgs sectors, with respect to the Standard Model (SM), can give sizeable contributions to leptonic decays. This happens because its branching fraction undergoes a helicity suppression by m 2 µ /M 2 Bs , where m µ is the mass of the muon and M Bs is the mass of the B s meson. This helicity suppression can be lifted in models with extra Higgs doublets where chirality-changing quark flavor violation is present and it is particularly strong for large values of tan β [1,2]. The chirality-changing quark flavor violation contributions to B s → µ + µ − are proportional to m 2 Bs /m 2 b instead. Hence, these decays provide a good opportunity to look for physics beyond the Standard Model (BSM). For the particular BSM case of the R-parity conserving minimal supersymmetric standard model (MSSM), with diagonal soft-squared mass matrices and trilinear terms, analytical approximations indicate that the B s → µ + µ − decay amplitude can be proportional to tan 3 β [3,4,5,6,7]. Here tan β is the ratio of the MSSM Higgs fields vacuum expectation values: H 2 / H 1 that can take values between 2 and 60. However, full-fledged (numerical) analyses including global fits of models to experimental data have shown that BSM contributions to B s → µ + µ − behave in a multi-dimensional manner.
Experimental constraints mostly suppress, or require opposite signs with similar magnitudes, the various supersymmetric contributions with respect to the SM one. Examples showing the manifestation of this suppression were shown within the phenomenological MSSM (pMSSM) framework [8,9], where the BR(B s → µ + µ − ) posterior distribution lies around the SM prediction despite the moderately high values of tan β, and in the more recent work of [10], as required by experiments which indicate the absence of large deviations of BR(B s → µ + µ − ) from the SM prediction [11,12,13,14,15,16,17,18].
Given the amazing consistency of the predictions of the SM with flavor observables one may wonder if in supersymmetry there is a mechanism, that just as it happens in the SM, effectively forbids flavor changing neutral current (FCNC) processes and controls CP violation. Specific models with a MSSM spectrum which avoid FCNC and CP violating processes can be constructed successfully [30,31,32]. However, without a specific model for generation of flavor within the MSSM, it is necessary to work within a phenomenological framework with a systematically constructed parametrization of flavor violation. Following this rationale, we consider a MSSM framework with 30 parameters [20,21] which goes beyond the constrained MSSM (see [22,23,24,25,26,27,28] for related works within the constrained MSSM set ups) and the pMSSM [29,33,8] where flavor violation in the SUSY breaking mass terms is manipulated by hand albeit with reasonable motivations.
In the pMSSM there is no information about how flavor violation in the soft-squared masses and trilinear terms generated by radiative corrections will impact flavor observables. We know that once full 3 × 3 Yukawa matrices are considered, flavor violation is automatically generated through radiative effects. In the MSSM-30, a counting rule keeps track of the hierarchical structure of the Yukawa matrices, which are expanded in terms of the Cabibbo angle. Then trilinear terms and soft squared masses can be expanded in that basis. The off-diagonal parameters generated in that way can be thought of as the effective off-diagonal parameters generated through radiative corrections. This rationale discards dangerous terms for FCNC and CP violation. In this sense, this work builds further on the project for MSSM explorations within systematically built frames, in this case a specific frame for flavor violation, deriving inference from experimental data [33,8,34,35,36,37,38,39,21,40] with fewer theoretical or traditional prejudices compared to other MSSM phenomenology frameworks.
The structure of flavor violation 1 and the flat distribution chosen for this work make our results obviously model dependent but one of our points is to exemplify how a realistic treatment for flavor effects in the MSSM can influence the allowed parameter space regions of supersymmetric parameters. The flavor structure that we have chosen is a realisation of minimal flavour violation (MFV), and within this framework the posterior sample from the MSSM-30 fit in [21] indicates results which are more restrictive than those from the LHC searches for gluinos and squarks. The MFV parametrization favours heavy gluinos and squarks [21], in order to satisfy flavor and electric dipole moment constraints.
This work is organised as follows. In Sec. 2, we present a brief review of the MSSM-30 construction and the global fit of its parameters to data from indirect searches for BSM physics. The effect of the B s → µ + µ − measurement, and the possible future accuracy of the measurement, on the MSSM-30 parameters is analysed there. In Sec. 3, we present respectively the numerical anatomy and analyses of the Wilson Coefficients, in terms of contributions to B s → µ + µ − classified according to kind of diagrams and kind of particles. For some interesting points, we give comparisons and contrast the MSSM-30 results to the pMSSM case. The summary and conclusions are presented in Sec. 4.
where M 1,2,3 are as for MSSM-30, mf the sfermion mass parameters were allowed in the range 100 GeV to 4 TeV. The trilinear couplings A t,b,τ,µ=e ∈ [− 8,8] TeV. The Higgs doublet masses m 2 H 1 , m 2 H 2 were allowed according to m 2 ∈ sign(m) [−4, 4] 2 TeV 2 . Here sign(µ) is the sign of the Higgs doublets mixing parameter (allowed to be randomly ±1). The SM parameters were fixed at their experimentally determined central values for the MSSM-30 but varied in a Gaussian manner for the pMSSM.

The MSSM-30 global fit to data
The posterior distribution used for our analysis came from a Bayesian fit of the MSSM-30 to data [21]. In order to make this paper a self-contained exposition of the statistical details of our work, in what follows we describe the fitting procedure.
The Bayesian fit was performed within a context, H, where the MSSM-30 neutralino lightest supersymmetric particle is assumed to be a least part of the cold dark matter (CDM) relic. The thirty parameters, detailed in Eq. (1), were varied according to a flat prior probability density, p(θ|H). The SM parameters fixed were: the mass of the Z-boson, m Z = 91.2GeV, the top quark mass, m t = 165.4GeV, the bottom quark mass, m b = 4.2GeV, the electromagnetic coupling, α −1 em = 127.9, and the strong interaction coupling, α s = 0.119. The data set, d, used for fitting the MSSM-30 are summarised in Tab. 1. It is composed of the experimental central values, µ i , and errors, σ i , for the Higgs boson mass, the electroweak physics, B-physics, dipole moment of leptons and the CDM relic density observables set where the index i runs over the list of observables O, the variable x represents the predicted value of neutralino CDM relic density at an MSSM-30 parameter space point and Here y = 0.11 is the CDM relic density central value and s = 0.02 the corresponding inflated (to allow for theoretical uncertainties) error. By passing parameters to SPHENO [62,44] via the SLHA2 [43] interface, the corresponding MSSM-30 predictions for the branching ratios BR(B s → µ + µ − ), BR(B → sγ), R BR(Bu→τ ν) , BR(B d → µ + µ − ), ∆M Bs , ∆M B d and d e,µ,τ were obtained. Similarly, using the SLHA1 [42] interface, the neutralino CDM relic density was computed using mi-crOMEGAs [41], while susyPOPE [63,64] was used for computing precision observables that include the W -boson mass m W , the effective leptonic mixing angle variable sin 2 θ lep ef f , the total Z-boson decay width, Γ Z , and the other electroweak observables whose experimentally determined central values and associated errors are summarised in Tab. 1.
Using MultiNest [65,66] which implements the Nested Sampling algorithm [67], Bayes' theorem then gives the MSSM-30 posterior probability distribution 2.3 BR(B s → µ + µ − ) prediction and measurement The tagged average branching fraction of the rare decay BR(B s → µ + µ − ) is given by where the operators that we use above are related to those of [46] by for X = π/( √ 2G F αV * ts V tb ), and the Wilson operators and coefficients are defined through the Hamiltonian as where O V are vector and O S scalar operators respectively and P X are the chirality projectors. The contributions proportional to |C S − C ′ S | 2 and |C P − C ′ P | 2 are not any longer proportional to m 2 µ /m 2 Bs and hence lift the helicity suppression exhibited in the SM. We use SUSY FLAVOR [45] to obtain the contributions from the different particles and kinds of diagrams 2 , Higgs and Z penguins and box diagrams.
It is well established that in the SM, C 10 gets its larger contribution from the Z penguin with a top loop, about 75% and its second largest contribution from the W box, 24% and we have The experimental measured quantity (denoted here with an overline) is the untagged branching fraction which is related to Eq. (8), the theoretical (tagged) expression, as where y s = ∆Γ s /2Γ s , ∆Γ s being the decay width difference between the B s mass eigenstates and Γ s = τ −1 Bs is the average B s decay width, using the LHCb measurement (y s = 0.087 ± 0.014 [69]), and that in the SM A µµ ∆Γ =1, we obtain 3 on the other hand, the experimental value measured by the LHCb collaboration is [18,19], including Run 1 and Run 2 data, while the CMS value is [70], As we can see, both values in agreement with the SM. In the MSSM, A µµ ∆Γ = (|P | 2 cos(2ϕ P )− |S| 2 cos(2ϕ S ))/(|P | 2 + |S| 2 ), [71], where ϕ S = arg(S), ϕ P = arg(P ), and S and P are related to our notation for the Wilson Coefficients as follows In the upper panels of Fig. 1 we compare the planes tan β vs the tagged value of BR(B s → µ + µ − ) (top-left), as produced by the official 2 53 version of SUSY F LAV OR, and tan β vs the untagged value of BR(B s → µ + µ − ) (top-right) using a modified version of it. This comparison shows the importance of appropriately comparing the measurement of the BR(B s → µ + µ − ) with the theoretical value. Although the contributions from the pre-factors in Eq. (12) do not differ greatly from point to point (due to the smallness of the supersymmetric contributions), they have a significant impact in pushing up the values of BR(B s → µ + µ − ). In the lower part of the figure we present just the tagged distribution for the linear prior of the pMSSM, which is in agreement with that of [9]. In the plane tan β vs BR(B s → µ + µ − ) is clear that for the MSSM-30, contrary to the pMSSM, values of tan β < 10 are not excluded. This shows that allowing a richer structure in the soft-squared terms, opens up regions of parameter space in comparison to the pMSSM. One of the main results of this work is that we found that the MSSM-30 predictions for BR(B s → µ + µ − ) with tan β ∈ (10, 20) are within the experimental limit, while for the pMSSM in that range are not. For the pMSSM instead the preferred values for tan β are above 25.
From the second plot of Fig. 1, we can see that supersymmetric contributions add up little to the SM contribution, except for values between tan β ∼ (10,20), where there could be both enhancing or suppressing effects. In Sec. 3, we present the numerical anatomy of the BR(B s → µ + µ − ) based on the MSSM-30 frame for which there are sources for CP violation beyond the CKM. We shall comment on the interplay of the contributions coming from the neutral Higgs, H 0 and Z penguin diagrams after introducing the theory of supersymmetric contributions to C S , C P and C 10 . The box diagram contribution to any of the Wilson Coefficients entering into BR(B s → µ + µ − ) is small in comparison to the SM [46]. Although this is strictly true in the case where the CKM matrix is the only source of CP and flavor violation, in our case the contributions from the extra sources of CP violation are generally negligible. We comment very briefly about the box contribution in Sec. 3.1.
For this work, the relevant posterior probability distribution, of the form of Eq. (7), which is a marginalised over the 2D (m A , tan β) plane is shown in Fig. 2. A 3D scatter plot showing the variations of BR(B s → µ + µ − ) on the same place is also shown. One of the aims of this article is to analyse the different contributions to the BR(B s → µ + µ − ) within the MSSM-30 posterior and to assess the impact of the recent BR(B s → µ + µ − ) measurement on the MSSM-30 parameters posterior. We shall address the latter case in what follows and the former in Sec. 3. The posterior distribution in Fig. 2 (left) shows that high tan β values are disfavoured. The result of the MSSM-30 global fit shows that tan β lies in the range 4.5 to 26.9 at 95% Bayesian probability interval. This feature is new compared to the pMSSM fits in [33,8]. The reason why the two distributions have different shape is mainly due to the inclusion of new CP-violating parameters in the MSSM-30 and the leptonic electric dipole moment constraints which tend to be proportional to tan β [68]. In order to limit the over production of the dipole moments, relatively lower, in comparison to the pMSSM, tan β values are needed. In Fig. 2 (right), the weight-free scatter plot shows the correlations of BR(B s → µ + µ − ) along the m A or tan β directions within the global fit posterior. It can be seen that independently of m A above some few 100s of GeV and for tan β ∼ 5, the value of BR(B s → µ + µ − ) ∼ 3.5 × 10 −9 is constant. This indicates a possible tension between the MSSM-30 global fit posterior described here with the BR(B s → µ + µ − ) measurement [18] given that, for instance, assuming a future BR(B s → µ + µ − ) precision of 15% relative to the  3.1 Diagram-by-diagram and particle-by-particle contributions From here on, we refer to the Box, Higgs penguin and Z penguin diagrams as kind of diagrams for which some examples are shown in Fig. 3. As mentioned in Sec. 2.3, C 10 in the SM gets its larger contribution from the Z penguin with a top loop, about 75% and its second largest contribution from the W box, 24%. Higgs penguin contributions in the SM are highly suppressed. On the other hand, the largest contribution in the pMSSM comes from the second diagram of Fig. 3 since the degeneracy of scalar massesQ is broken by radiative effects induced by Yukawa couplings. This produces and effective flavor offdiagonal piece which does not go away when rotating to the mass eigenstates basis [6]. In the MSSM-30 off-diagonal elements are present and compete with the contribution coming from the afore mentioned radiative effects. In the case of MSSM-30, contributions from the Z penguin diagrams are in general suppressed, except for low value of tan β ( 10). Figure 3: Higgs penguin and Z penguin diagrams contributing to BR(B s → µ + µ − ). The first diagram represents the general Higgs penguin diagram. The most important contribution in the pMSSM comes from the second diagram of the first line, depicted in flavor basis. Even in the pMSSM this is the most important contribution since the degeneracy of scalar massesQ is broken by radiative effects induced by Yukawa couplings and hence induces and effective flavor off-diagonal piece which does not go away when rotating to mass eigenstates basis [6]. The last Z penguin gives the leading SM contribution.
In Fig. 4 and Fig. 5 we compare the pMSSM and the MSSM-30 in terms of their contributions to BR(B s → µ + µ − ) coming from different diagrams and particles respectively. Here by "particles", we refer to "gluino", "chargino", "neutralino" and "W + charged Higgs boson", understanding that these particles can only come in their respective loops together with squarks type down, squarks type up, squarks type down and quarks type up respectively. The contributions for the pMSSM (solid black lines) are compared to the MSSM-30 case (red dashed lines). We can see that the distributions for the Z penguin and Box dia-grams become narrower in the MSSM-30, in comparison to those of the pMSSM, but overall these contributions shift BR(B s → µ + µ − ) to higher values than in the pMSSM case. For the MSSM-30, the contributions to the Higgs-Penguin diagrams become a bit suppressed, because of the preferred bigger masses for m A and mt and lower values for tan β.
3.2 Contributions to C P and C S H 0 -penguin As mentioned above, the most important contribution in the pMSSM comes from the Higgs penguin diagram depicted in the second diagram of Fig. 3. This happens because the degeneracy of scalar massesQ is broken by radiative effects induced by Yukawa couplings and hence this induces and effective flavor off-diagonal piece which does not go away when rotating to mass eigenstate basis [6]. The large tan β region of this contribution can play a very significant role. This can be understood by writing the simplified contribution at LO as [1] which is quite sensitive to m A and mt and therefore drops noticeably with the increase of their values. From this expression, we can also understand that the lower the value of tan β, the lower the contribution to BR(B s → µ + µ − ) from this diagram. Since for our fits, the preferred values of mt are O(1) TeV for the pMSSM, the suppression of this contribution becomes considerable.
In the MSSM-30, the off-diagonal parameters in the soft-squared masses and trilinear terms, Eq. (1), add up to the contributions given by the broken degeneracy of the of scalar massesQ. In this case, the contributions to C H 0 S,P (χ ± ) cannot be written in the form of Eq. (18), because non-zero off-diagonal terms are present even before the breaking of the degeneracy of the diagonal soft-squared masses. However, we find that the differences between the contributions of C H 0 S,P (χ ± ) in the pMSSM and in the MSSM-30 is only at the percent level. In this respect, in the MSSM-30, there could be cancellations among these two contributions for equally heavy/high magnitude parameters (e.g. µ and A t ). These cancellations could make the BR(B s → µ + µ − ) independent of m A as shown in Fig. 1 (top-right), and make the contribution to the Wilson Coefficients not so different from the pMSSM.

Z-penguin and Box diagrams
Supersymmetric particles propagating in the loop cannot generate a contributions to C S,P , C ′ S,P due to the vector coupling to Z 0 . Diagrams with charginos propagating in the box also give rise to non zero values of C χ S and C χ P but leave C Using SUSY FLAVOR we find that the total contribution to C 10 is , O(10 −2 )), adding up little to the total of C 10 , including the SM contributions. For our set of experimental values, we obtain that C SM 10 = −4.13 ± 0.05. In the SM the current accuracy for this coefficient is better than 0.1% level when allowing only the top-quark mass and and the strong coupling constant to deviate from their corresponding central value [72]. In principle, then the contributions from supersymmetric particles could be disentangled from the SM uncertainty.
For illustration of our discussion, in Fig. 6   represented by the orange-light-dashed line). On the other hand, contributions to both C Z 10T (H ± ) and C H 0 S (χ ± ) for values of tan β around 10 are typically less than a third of the corresponding values when tan β > 20. Since for the pMSSM sample, values for tan β above 20 dominate the sample, it is clear that the most important contributions come from C H 0 S (χ ± ). For the MSSM-30 sample however, smaller values than 10 for tan β are an important part of the sample and in this case they can give the largest supersymmetric contribution to the Wilson Coefficients via C Z 10T (H ± ). This is appreciated in the plot of the left in Fig. 6 where C Z 10T (H ± ) for tan β = 5 and m H − > 700 GeV is the dominant of the supersymmetric contributions (solid blue-dark line).

Interplay of Wilson Coefficients
In order to understand the different contributions to Eq. (8), it is customary to compare the relative size of the Wilson Coefficients C S and C P to C 10 . This is useful because C S and C P can only have supersymmetric contributions and supersymmetric contributions to C 10 are highly suppressed.
In order to do this comparison, we compare the value of C 10 to the Wilson Coefficients C S and C P but weighted in the same way that C 10 contributes to BR(B s → µ + µ − ), Eq. (8). This allows a direct comparison among C S , C P and C 10 . The weighted Wilson Coefficients C S and C P are respectively denoted byĈ S andĈ P : In Fig. 7, we plot our results in planes C 10 vsĈ S (we do not present C 10 vsĈ P sincê C s ≈ −Ĉ P ), comparing the Wilson Coefficients for the two samples. For the pMSSM sample we present only the results when fixing the top mass value, since for our MSSM-30 sample the top mass was kept fixed. In this sense, even if the sample is not complete for the pMSSM, we give a fair comparison to the MSSM-30 sample and we can be sure that the increase (in absolute value) of the Wilson Coefficient C 10 arises due to the supersymmetric contributions. While in the pMSSM, the center value of C 10 is -4.61 in the MSSM-30 is -4.64. Overall the contour plots for C 10 vs C S and C 10 vs C P cover a bigger area in the pMSSM (see also e.g. [9]) than in the MSSM-30 but the one and two sigma regions of this last sample are shifted to the left, Fig. 7. As mentioned in Section 3.2, C SM 10 = −4.13 ± 0.05 and the current accuracy is of the order 0.1%. Hence supersymmetric contributions could be also disentangled from the SM error. 10T (H ± ), as mentioned before (Fig. 6). We recall the reader that the present work builds further on the project for MSSM explorations within systematically built frameworks, in this case a specific frame for flavor violation and a Bayesian approach for deriving inference from experimental data. The specific flavor violation determined by the MSSM-30 is a realistic one, owning to the fact that it can be understood as taking into account the running of off-diagonal elements of softsquared masses and trilinears. In contrast, the pMSSM sets these elements to zero to start with. Since there are many supersymmetric contributions in a less constrained MSSM, it is natural to expect that these contributions will have a constructive interference instead of a destructive one, resulting in the increase of the value of B s → µ + µ − . This is effectively what is happening in the MSSM-30 in comparison to the pMSSM: owning to the fact that the MSSM-30 is less constrained, it does increase the value of B s → µ + µ − in comparison to the pMSSM. It is out of the scope of this work to identify a region where actually cancellations could take place, but it is definitely a project that it should be peformed.

Interplay of off-diagonal soft-squared elements
It is customary to assess the impact of flavor violation in terms of the flavor violating parameters which measure the amount of off-diagonal allowed contributions constrained by all relevant flavor observables (see Table 1 of [21]). Here we present only the relevant parameters for BR(B s → µ + µ − ). In order to make manifest how the Z penguin contributions dominate the supersymmetric contributions of the coefficient C 10 for most part of the parameter space. We present in Fig. 8 (top row) the individual contributions to BR(B s → µ + µ − ) from Higgs penguin, Z penguin and Box diagrams, as a function of (δ QLR ) 23 . In this figure, we can clearly appreciate how Z penguin contributions are in general the most dominant for practically all values of (δ QLR ) 23 and as emphasized in [46], for small values of tan β (≤ 20), Higgs penguin contributions are small. Box contributions are also small, bu they tend to be even bigger than the Higgs penguin contributions. In the second row of Fig. 8 we represent the individual contributions to BR(B s → µ + µ − ) from Higgs penguin, Z penguin and Box diagrams, as a function of (δ QLL ) 23 . As it is usual with LR flavor violation, it tends to be more constrained, as in this case it is allowed to be only O(10 −3 ), while LL flavor violation it can be of O(10 −2 ).

Conclusions
We have continued with our studies in order to explore features of the MSSM by using Bayesian statistical techniques on systematically constructed, symmetry-guided, MSSM frameworks beyond the traditional constructions. Here, the phenomenological framework considered is the 30-parameter-MSSM, called MSSM-30, and the observable of interest is  the B s → µ + µ − decay. The measured branching ratio BR(B s → µ + µ − ) is compatible with the SM prediction but it still has a large (order 20%) uncertainty. Future precision measurements of this observable will be excellent for assessing the MSSM as New Physics beyond the SM. Within the MSSM-30 a posterior sample was considered, for which a 15% uncertainty on the measured BR(B s → µ + µ − ) would favour a sub-TeV pseudo-scalar Higgs boson.
Knowing that the decay B s → µ + µ − is a good indicator for assessing models with extended Higgs sectors, we compared the MSSM-30 to the pMSSM to see if there are any physics or features in the parameters of the former which are not accessible in the latter. It turned out to be the case since the MSSM-30 sample prefers lower values of tan β ∼ (10,20) in comparison to the pMSSM which prefers tan β ∼ (20,40). It is then possible to find higher values for BR(B s → µ + µ − ) in the MSSM-30 mainly due to bigger contributions coming from diagrams involving charginos and Z-penguin diagrams.
We found that the best way to analyse the contributions to the branching ratio BR(B s → µ + µ − ) was by comparing kind of diagrams: Z penguin, box and Higgs penguin diagrams, instead of comparing contributions of supersymmetric particles (i.e. gluinos, neutralinos or charginos). The reason is that in the pMSSM the supersymmetric contributions become quite suppressed due to the large values of m A and mt (well into the multi-TeV region) and that in the SM the Z penguin and box contributions are dominant ( ∼ 75% and 24% respectively). This last fact then in principle helps to look for contributions coming from supersymmetry. In the pMSSM it is well established that the major BSM contributions to BR(B s → µ + µ − ) come from a Higgs penguin and a chargino in the loop. Hence, this is a specific example of how analysis by kind of diagrams becomes relevant.
The MSSM-30 has by construction non-zero off-diagonal soft-squared mass terms, contrary to the pMSSM where they are set to zero by hand. When analysing both samples using a Bayesian fit, both samples are constrained by the same observables. Therefore, if no cancellations appear, the value of the allowed effective off-diagonal soft-squared mass terms should be of the same order in both samples. The best place to look for the difference between both samples is in the contribution coming from the charginos, mainly from Higgspenguins, (Fig. 5) where we can see that the distributions from pMSSM and MSSM-30 are different but mostly indistinguishable towards higher values of BR(B s → µ + µ − ).
In order to assess the impact of BSM contributions to B observables, it is customary to compare the size of the Wilson Coefficients which receive the majority of the BSM contributions with the Wilson Coefficients present only in the SM case. For the decay B s → µ + µ − , the relevant coefficients are C S and C P , which are scalar operators sensitive to the chirality of BSM contributions. In the SM the only contribution comes from the vector operator C SM 10 , for which we find a value of −4.13 ± 0.05. We have compared the coefficients C S and C P to C 10 , using C 10 vs C S and C 10 vs C P planes and found that in the MSSM-30, C S and C P represent typically only a O(1%) contribution to the branching ratio B s → µ + µ − . Within the pMSSM C 10 = −4.58 ± 0.06 ranging from −4.68 to −4.50 at 95% Bayesian probability region. For the MSSM-30, C 10 = −4.64 ± 0.01 ranging from −4.66 to −4.63 at 95% Bayesian probability region. The MSSM-30 is more severely away from the SM value compared to the pMSSM. The current SM accuracy in determining the value of C 10 is of the order 0.1% and therefore supersymmetric contributions can be disentangled from the SM uncertainty.
Finally, as an outlook we highlight that future improvements of the measurement of BR(B s → µ + µ − ), along with the measurement of other observables, like A µµ ∆Γ and BR(B d → µ + µ − ) which are correlated with BR(B s → µ + µ − ), will play a crucial role in shaping the parameter space of the MSSM-30. On the other hand, systematically constructed frameworks which can capture the flavor structure of the MSSM, like the one presented here, should be favored over simplified scenarios which cannot capture the rich flavor structure of the MSSM.