New physics in \(b\rightarrow s\) transitions after LHC run 1
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Abstract
We present results of global fits of all relevant experimental data on rare \(b \rightarrow s\) decays. We observe significant tensions between the Standard Model predictions and the data. After critically reviewing the possible sources of theoretical uncertainties, we find that within the Standard Model, the tensions could be explained if there are unaccounted hadronic effects much larger than our estimates. Assuming hadronic uncertainties are estimated in a sufficiently conservative way, we discuss the implications of the experimental results on new physics, both model independently as well as in the context of the minimal supersymmetric standard model and models with flavourchanging \(Z'\) bosons. We discuss in detail the violation of leptonflavour universality as hinted by the current data and make predictions for additional leptonflavouruniversality tests that can be performed in the future. We find that the ratio of the forward–backward asymmetries in \(B \rightarrow K^* \mu ^+\mu ^\) and \(B \rightarrow K^* e^+e^\) at low dilepton invariant mass is a particularly sensitive probe of lepton flavour universality and allows to distinguish between different new physics scenarios that give the best description of the current data.
1 Introduction
Rare decays based on the flavourchanging neutral current \(b\rightarrow s\) transition are sensitive probes of physics beyond the Standard Model (SM). In recent years, a plethora of observables, including branching ratios, CP and angular asymmetries in inclusive and exclusive B decay modes, has been measured at the B factories and at LHC experiments. This wealth of data allows to investigate the helicity structure of flavourchanging interactions as well as possible new sources of CP violation.
In 2013, the observation by LHCb of a tension with the SM in \(B\rightarrow K^*\mu ^+\mu ^\) angular observables [1] has received considerable attention from theorists and it was shown that the tension could be softened by assuming the presence of new physics (NP) [2, 3, 4, 5]. In 2014, another tension with the SM has been observed by LHCb, namely a suppression of the ratio \(R_K\) of \(B\rightarrow K\mu ^+\mu ^\) and \(B\rightarrow Ke^+e^\) branching ratios at low dilepton invariant mass [6]. Assuming new physics in \(B\rightarrow K\mu ^+\mu ^\) only, a consistent description of these anomalies seems possible [7, 8, 9, 10]. In addition, also branching ratio measurements of \(B\rightarrow K^*\mu ^+\mu ^\) and \(B_s\rightarrow \phi \mu ^+\mu ^\) decays published recently [11, 12] seem to be too low compared to the SM predictions when using stateoftheart form factors from lattice QCD or lightcone sum rules (LCSR) [13, 14, 15, 16]. Finally, in the latest update of the LHCb \(B\rightarrow K^*\mu ^+\mu ^\) analysis from 2015 [17], the tensions in angular observables persist.
 1.
Is there a significant tension with SM expectations in the current data on \(b\rightarrow s\) transitions?
 2.
Assuming the absence of NP, which QCD effects could have been underestimated and how large would they have to be to bring the data into agreement with predictions, assuming they are wholly responsible for an apparent tension?
 3.
Assuming the QCD uncertainties to be estimated sufficiently conservatively, what do the observations imply for NP, both model independently and in specific NP models?

In our global \(\chi ^2\) fits, we take into account all the correlations of theoretical uncertainties between different observables and between different bins. This has become crucial to assess the global significance of any tension, as the experimental data are performed in more and more observables in finer and finer bins.

We assess the impact of different choices for the estimates of theoretical uncertainties on the preferred values for the Wilson coefficients.

We model the subleading hadronic uncertainties in exclusive semileptonic decays in a different way, motivated by discussions of these effects in the recent literature (see e.g. [16, 19, 20, 22, 23, 24, 25]); see Sect. 2 for details.

We use the information on \(B\rightarrow K^*\) and \(B_s\rightarrow \phi \) form factors from the most precise LCSR calculation [13, 16], taking into account all the correlations between the uncertainties of different form factors and at different \(q^2\) values. This is particularly important to estimate the uncertainties in angular observables that involve ratios of form factors.

We include in our analysis the branching ratio of \(B_s\rightarrow \phi \mu ^+\mu ^\), showing that there exists a significant tension between the recent LHCb measurements and our SM predictions.
2 Observables and uncertainties
In this section, we specify the effective Hamiltonian encoding potential new physics contributions and we discuss the most important observables entering our analysis. The calculation of the observables included in our previous analyses [3, 33, 34] (see also [16, 35]) have been discussed in detail there and in references therein; here we only focus on the novel aspects of the present analyses – like the \(B_s\rightarrow \phi \mu ^+\mu ^\) decay – and on our refined treatment of theoretical uncertainties.
2.1 Effective Hamiltonian

Fourquark operators (including currentcurrent, QCD penguin, and electroweak penguin operators). These operators only contribute to the observables considered in this analysis through mixing into the operators listed above and through higher order corrections. Moreover, at low energies they are typically dominated by SM contributions. Consequently, we expect the impact of NP contributions to these operators on the observables of interested to be negligible.^{1}

Chromomagnetic dipole operators. In the radiative and semileptonic decays we consider, their Wilson coefficients enter at leading order only through mixing with the electromagnetic dipoles and thus enter in a fixed linear combination, making their discussion redundant.

Tensor operators. Our rationale for not considering these operators is that they do not appear in the dimension6 operator product expansion of the Standard Model [37, 38, 39]. Consequently, they are expected to receive only small NP contributions unless the scale of new physics is very close to the electroweak scale, which is in tension with the absence of new light particles at the LHC.

Scalar operators of the form \((\bar{s} P_{A} b)( \bar{\ell } P_{B} \ell )\). The operators with \(AB=LL\) or RR do not appear in the dimension6 operator product expansion of the Standard Model either. While the ones with \(AB=LR\) and RL do appear at dimension 6, their effects in semileptonic decays are completely negligible once constraints from \(B_s\rightarrow \mu ^+\mu ^\) are imposed [39]. The constraints from \(B_s\rightarrow \mu ^+\mu ^\) can only be avoided for a new physics scale close to the electroweak scale such that scalar LL and RR operators can have nonnegligible impact.
2.2 \(B\rightarrow K\mu ^+\mu ^\)
2.2.1 Observables
2.2.2 Theoretical uncertainties
For the form factors, we perform a combined fit of the recent lattice computation by the HPQCD collaboration [41], valid at large \(q^2\), and form factor values at \(q^2=0\) obtained from LCSR [42, 43], to a simplified series expansion. Details of the fit are discussed in “Appendix A”. The results are 3parameter (4parameter) fit expressions for the form factors \(f_{+,T}\) (\(f_0\)) as well as the full \(10\times 10\) covariance matrix. We retain the correlations among these uncertainties throughout our numerical analysis.

Virtual corrections to the matrix elements of the fourquark operators \(O_1\) and \(O_2\). We include them to NNLL accuracy using the results of Ref. [44].

Contributions from weak annihilation and hard spectator scattering. These have been estimated in QCD factorisation to be below a percent [43] and we neglect them.

Soft gluon corrections to the virtual charm quark loop at low \(q^2\). This effect was computed recently in LCSR with B meson distribution amplitudes in Ref. [22] and was found to be “unimportant at least up to \(q^2\sim 5\)–\(6~\text {GeV}^2\)” (see also [24]).

Violation of quark–hadron duality at high \(q^2\), above the open charm threshold, due to the presence of broad charmonium resonances. Employing an OPE in inverse powers of the dilepton invariant mass, this effect has been found to be under control at a few percent in Ref. [23].
2.3 \(B\rightarrow K^*\mu ^+\mu ^\) and \(B\rightarrow K^*\gamma \)
2.3.1 Observables

the CPaveraged differential branching ratio \(\mathrm{d}\text {BR}/\mathrm{d}q^2\),

the CPaveraged \(K^*\) longitudinal polarisation fraction \(F_L\) and forward–backward asymmetry \(A_\text {FB}\),

the CPaveraged angular observables \(S_{3,4,5}\),

the Todd CP asymmetries \(A_{7,8,9}\).
In the case of \(B\rightarrow K^*\gamma \), we consider the following observables: the branching ratio of \(B^\pm \rightarrow K^{*\pm }\gamma \), the branching ratio of \(B^0\rightarrow K^{*0}\gamma \), the direct CP asymmetry \(A_\text {CP}\) and the mixinginduced CP asymmetry \(S_{K^*\gamma }\) in \(B^0\rightarrow K^{*0}\gamma \). Since we take all known correlations between the observables into account in our numerical analysis, including the branching ratios of the charged and neutral B decays is to a very good approximation equivalent to including one of these branching ratios and the isospin asymmetry.
2.3.2 Theoretical uncertainties
Similarly to the \(B\rightarrow K\mu ^+\mu ^\) decay, the main challenges of \(B\rightarrow K^*\mu ^+\mu ^\) are the form factors and the contributions of the hadronic weak Hamiltonian.
For the form factors, we use the preliminary results of a combined fit [16] to a LCSR calculation of the full set of seven form factors [13] with correlated uncertainties as well as lattice results for these form factors [14]. This leads to strongly reduced uncertainties in angular observables.

The NNLL contributions to the matrix elements of \(O_{1,2}\) as in the case of \(B\rightarrow K\mu ^+\mu ^\).

At low \(q^2\), hard spectator scattering at \(O(\alpha _s)\) from QCD factorisation [45] including the subleading doubly Cabibbosuppressed contributions [46].

At low \(q^2\), weak annihilation beyond the heavy quark limit as obtained from LCSR [47].

At low \(q^2\), contributions from the matrix element of the chromomagnetic operator as obtained from LCSR [48].
At high \(q^2\), as in the case of \(B\rightarrow K\mu ^+\mu ^\), we do not have to consider a \(q^2\)dependent correction as we are only considering observables integrated over the full high \(q^2\) region. Analogous to \(B\rightarrow K\mu ^+\mu ^\), we parametrise the subleading uncertainties by a relative correction to \(C_9\). To be conservative, we allow it to be up to \(7.5\,\%\) in magnitude, independently for the three helicity amplitudes, with an arbitrary strong phase.
2.3.3 Direct CP asymmetry in \(B\rightarrow K^*\gamma \)
The problem with using this observable as a constraint on NP is that it is proportional to a strong phase that appears only at subleading order and is afflicted with a considerable uncertainty. With our error treatment described above, taking the subleading contributions from Ref. [48], we find an overall relative uncertainty of 20 % in the presence of a large imaginary \(C_7\). However, to be conservative, we will not include \(A_\text {CP}(B^0\rightarrow K^{*0}\gamma )\) in our global fits, but we will discuss the impact of including it separately in Sect. 3.3.
2.4 \(B_s\rightarrow \phi \mu ^+\mu ^\)

The form factors are of course different; we use the combined fit of lattice and LCSR results obtained in [16] including the correlated uncertainties.

The subleading nonfactorisable corrections are parametrised as in the case of \(B\rightarrow K^*\mu ^+\mu ^\), and the coefficients \(a_\lambda \), \(b_\lambda \) and \(c_\lambda \) are varied in the same ranges. We assume the uncertainty in these coefficients to be \(90\,\%\) correlated between \(B_s\rightarrow \phi \mu ^+\mu ^\) and \(B\rightarrow K^*\mu ^+\mu ^\) since we do not see a physical reason why they should be drastically different.^{4}
 In contrast to \(B\rightarrow K^*\mu ^+\mu ^\), the \(B_s\rightarrow \phi \mu ^+\mu ^\) decay is not selftagging. Therefore, the only observables among the ones mentioned at the beginning of Sect. 2.3.1 that are experimentally accessible in a straightforward way at a hadron collider are [50]:

the differential branching ratio \(\mathrm{d}\text {BR}/\mathrm{d}q^2\),

the CPaveraged angular observables \(F_L\) and \(S_4\),

the angular CP asymmetry \(A_9\).


An additional novelty is the impact of the sizable \({ B}_{ s}\) width difference. As shown in [16] (see also [51]), this effect is small in the SM and we have checked that it is also negligible in the presence of NP at the current level of experimental precision, unless the Wilson coefficients assume extreme values that are already excluded by other constraints. Therefore, we have neglected the effect in our numerical analysis.
3 Global numerical analysis
3.1 Fit methodology
We determine \(C_\text {th}\) by evaluating all observables of interest for a large set of the parameters parametrising the theory uncertainties, randomly distributed following normal distributions according to the uncertainties and correlations described above. In this way, we retain not only correlated uncertainties between different observables, but also between different bins of the same observable. We find these correlations to have a large impact on our numerical results. Concerning \(C_\text {exp}\), we symmetrise the experimental error bars and include the experimental error correlations provided by the latest LHCb update of the \(B \rightarrow K^* \mu ^+\mu ^\) analysis [17]. For branching ratio measurements, where no error correlations are available, we include a rough guess of the correlations by assuming the statistical uncertainties to be uncorrelated and the systematic uncertainties to be fully correlated for measurements of the same observable by a single experiment. We have checked that this treatment has only a small impact on the overall fit at the current level of experimental and theoretical uncertainties on branching ratios.

\(B \rightarrow K^* \mu ^+\mu ^\) branching ratios and angular observables from LHCb [1, 11, 17, 52], CMS [53], ATLAS [54], and CDF [55, 56, 57];

\(B \rightarrow K \mu ^+\mu ^\) branching ratios and angular observables from LHCb [11, 58] and CDF [55, 56, 57];

\(B_s \rightarrow \phi \mu ^+\mu ^\) branching ratios and angular observables from LHCb [12] and CDF [55, 57];

branching ratios for \(B \rightarrow K^* \gamma \) and \(B \rightarrow X_s \gamma \) and the mixinginduced CP asymmetry in \(B \rightarrow K^* \gamma \) from HFAG [59];

the combined result of the \(B_s \rightarrow \mu ^+\mu ^\) branching ratio from LHCb and CMS [60, 61, 62];

the \(B \rightarrow X_s \mu ^+\mu ^\) branching ratio measurement from BaBar [63].
We would like to stress that for none of the observables, we use low \(q^2\) bins that extend into the region above the perturbative charm threshold \(q^2 > 6\) GeV, where hadronic uncertainties cannot be estimated reliably. This applies in particular to the bin [4.3, 8.68] GeV\(^2\) that has been used in several fits in the past [2, 5, 9] as well as the bin [6, 8] GeV\(^2\) in the recent \(B\rightarrow K^*\mu ^+\mu ^\) angular analysis by LHCb [17].
For the \(B^0 \rightarrow K^{*0} \mu ^+\mu ^\) observables at low \(q^2\), we choose the smallest available bins satisfying this constraint, since they are most sensitive to the nontrivial \(q^2\) dependence of the angular observables. For \(B_s\rightarrow \phi \mu ^+\mu ^\), we use the [1, 6] GeV\(^2\) bin, since the branching ratio does not vary strongly with \(q^2\) and since the statistics is limited. In the high \(q^2\) region, we always consider the largest \(q^2\) bins available that extend to values close to the kinematical end point. All the experimental measurements used in our global fits are listed in “Appendix B” along with their theory predictions. All theory predictions are based on our own work and on [16], except the \(B_s \rightarrow \mu ^+\mu ^\), \(B \rightarrow X_s \gamma \) and \(B \rightarrow X_s \ell ^+\ell ^\) branching ratios that we take from [68, 69, 70],^{5} respectively. In the case of the SM prediction for BR\((B_s \rightarrow \mu ^+\mu ^)\) we rescale the central value and uncertainty obtained in [68], to reflect our choice of \(V_{cb}\) (see Sect. 3.2.2 below).
3.2 Compatibility of the data with the SM
Observables where a single measurement deviates from the SM by \(1.8\sigma \) or more. The full list of observables is given in “Appendix B”. Differential branching ratios are given in units of GeV\(^{2}\)
Decay  Obs.  \(q^2\) bin  SM pred.  Measurement  Pull  

\(\bar{B}^0\rightarrow \bar{K}^{*0}\mu ^+\mu ^\)  \(10^{7}~\frac{\mathrm{d}\text {BR}}{\mathrm{d}q^2}\)  [2, 4.3]  \(0.44 \pm 0.07\)  \(0.29 \pm 0.05\)  LHCb  \(+1.8\) 
\(\bar{B}^0\rightarrow \bar{K}^{*0}\mu ^+\mu ^\)  \(10^{7}~\frac{\mathrm{d}\text {BR}}{\mathrm{d}q^2}\)  [16, 19.25]  \(0.47 \pm 0.06\)  \(0.31 \pm 0.07\)  CDF  \(+1.8\) 
\(\bar{B}^0\rightarrow \bar{K}^{*0}\mu ^+\mu ^\)  \(F_L\)  [2, 4.3]  \(0.81 \pm 0.02\)  \(0.26 \pm 0.19\)  ATLAS  \(+2.9\) 
\(\bar{B}^0\rightarrow \bar{K}^{*0}\mu ^+\mu ^\)  \(F_L\)  [4, 6]  \(0.74 \pm 0.04\)  \(0.61 \pm 0.06\)  LHCb  \(+1.9\) 
\(\bar{B}^0\rightarrow \bar{K}^{*0}\mu ^+\mu ^\)  \(S_5\)  [4, 6]  \(0.33 \pm 0.03\)  \(0.15 \pm 0.08\)  LHCb  \(2.2\) 
\(B^\rightarrow K^{*}\mu ^+\mu ^\)  \(10^{7}~\frac{\mathrm{d}\text {BR}}{\mathrm{d}q^2}\)  [4, 6]  \(0.54 \pm 0.08\)  \(0.26 \pm 0.10\)  LHCb  \(+2.1\) 
\(\bar{B}^0\rightarrow \bar{K}^{0}\mu ^+\mu ^\)  \(10^{8}~\frac{\mathrm{d}\text {BR}}{\mathrm{d}q^2}\)  [0.1, 2]  \(2.71 \pm 0.50\)  \(1.26 \pm 0.56\)  LHCb  \(+1.9\) 
\(\bar{B}^0\rightarrow \bar{K}^{0}\mu ^+\mu ^\)  \(10^{8}~\frac{\mathrm{d}\text {BR}}{\mathrm{d}q^2}\)  [16, 23]  \(0.93 \pm 0.12\)  \(0.37 \pm 0.22\)  CDF  \(+2.2\) 
\(B_s\rightarrow \phi \mu ^+\mu ^\)  \(10^{7}~\frac{\mathrm{d}\text {BR}}{\mathrm{d}q^2}\)  [1, 6]  \(0.48 \pm 0.06\)  \(0.23 \pm 0.05\)  LHCb  \(+3.1\) 
\(B\rightarrow X_se^+e^\)  \(10^{6}~\text {BR}\)  [14.2, 25]  \(0.21 \pm 0.07\)  \(0.57 \pm 0.19\)  BaBar  \(1.8\) 
3.2.1 Underestimated hadronic effects?
We will see in Sect. 3.3 that the agreement of the theory predictions with the experimental data is improved considerably assuming nonstandard values for the Wilson coefficient \(C_9\). Since this coefficient corresponds to a lefthanded quark current and a leptonic vector current, it is conceivable that a NP effect in \(C_9\) is mimicked by a hadronic SM effect that couples to the lepton current via a virtual photon, e.g. the charm loop effects at low \(q^2\) and the resonance effects at high \(q^2\) as discussed in Sect. 2 (see e.g. [19]). In our numerical analysis, in addition to the known nonfactorisable contributions taken into account as described in Sect. 2, subleading effects of this type are parametrised by the parameters \(a_i, b_i, c_i\) in (10), (11), and analogously for \(B_s\rightarrow \phi \mu ^+\mu ^\). Since they parametrise unknown subleading uncertainties, the central values of these parameters are 0 in our SM predictions.
Any underestimation of a nonperturbative QCD effect (not related to form factors) should then manifest itself as a drastic reduction of the \(\chi ^2\) for a sizable value of one of the parameters, when treating them as completely free. To investigate this question, we have constructed a \(\chi ^2\) function analogous to (15), but writing the central values \(\vec {O}_\text {th}\) as functions of the parameters \(a_i, b_i, c_i\) instead of the Wilson coefficients.
 The \(\chi ^2\) can be reduced by up to 4 when pushing the parameter \(b_K\), parametrising subleading corrections in \(B\rightarrow K\mu ^+\mu ^\) at low \(q^2\), to the border of our estimated uncertainty. The fit does not improve significantly when changing the parameter \(c_K\) from 0, i.e. when assuming large violations of quark–hadron duality in the global (integrated) high \(q^2\) observables in \(B\rightarrow K\mu ^+\mu ^\), unless \(b_K\) is shifted at the same time.

A simultaneous positive shift in the subleading corrections to the \(\lambda =\) and 0 helicity amplitudes in \(B\rightarrow K^*\mu ^+\mu ^\) can significantly reduce the \(\chi ^2\) as well. \(\Delta \chi ^2=9\) requires a shift in both parameters that is four times larger than our error estimate.

Corrections to quark–hadron duality in the global high \(q^2\) observables in \(B\rightarrow K^*\mu ^+\mu ^\) do not lead to any significant reduction of the \(\chi ^2\).
3.2.2 Underestimated parametric uncertainties?
To see whether this has an impact on the significance of the tensions, we multiply all branching ratios by a scale factor \(\eta _\text {BR}\) and fit this scale factor to the data. We find \(\eta _\text {BR}=0.79\pm 0.08\), i.e. a \(21\,\%\) reduction of the branching ratios with respect to our central values is preferred. The \(\chi ^2\) is improved by 7.0 with respect to the SM. The obtained central value for \(\eta _\text {BR}\) would correspond to \(V_{cb} \simeq 3.6 \times 10^{2}\), which is in tension with both the inclusive and exclusive determinations.
We conclude that underestimated parametric uncertainties are unlikely to be responsible for the observed tensions in the branching ratio measurements. Needless to say, the angular observables and \(R_K\) would be unaffected by a shift in \(V_{cb}\) anyway.
3.2.3 Underestimated form factor uncertainties?
The tensions between data and SM predictions could also be due to underestimated uncertainties in the form factor predictions from LCSR, lattice, or both. A first relevant observation in this respect is that the tensions in Table 1 include observables in decays involving \(B\rightarrow K\), \(B\rightarrow K^*\) and \(B_s\rightarrow \phi \) transitions, both at low \(q^2\) (where LCSR calculations are valid) and at high \(q^2\) (where the lattice predictions are valid). Explaining all of them would imply underestimated uncertainties in several completely independent theoretical form factor determinations.
In the case of \(B\rightarrow K\mu ^+\mu ^\) and \(B_s\rightarrow \phi \mu ^+\mu ^\), tensions are present only in branching ratios, which seem to be systematically below the SM predictions. This could be straightforwardly explained if the form factor predictions were systematically too high. Note that the largest tensions in the \(B \rightarrow K \mu ^+\mu ^\) branching ratios appear in the neutral mode. The branching ratio of the charged mode, \(B^+ \rightarrow K^+ \mu ^+\mu ^\), is measured with considerably smaller statistical uncertainty and agrees better with the SM predictions (see “Appendix B”). Nevertheless, also the charged mode seems to be systematically below the SM prediction and would profit from a reduction of the form factors.
Finally, an important observation in the case of \(B\rightarrow K^*\mu ^+\mu ^\) angular observables is that the tensions are only present at low \(q^2\), where the seven form factors can be expressed in terms of two independent “soft” form factors up to power corrections of naive order \(\Lambda _\text {QCD}/m_b\). It is then possible to construct angular observables that do not depend on the soft form factors, but only on the power corrections [32]. The tensions can then be seen by estimating the power corrections by dimensional analysis [20]. This shows that an explanation of the tensions by underestimated form factor uncertainties would imply that the values of the power corrections are very different from what LCSR calculations predict for them.
3.3 New physics in a single Wilson coefficient
We now investigate whether new physics could account for the tension of the data with the SM predictions. We start by discussing the preferred ranges for individual Wilson coefficients assuming our nominal size of hadronic uncertainties. We determine the \(1\sigma \) (\(2\sigma \)) ranges by computing \(\Delta \chi ^2=1~(4)\), while fixing all the other coefficients to their SM values. We also set the imaginary part of the respective coefficient to 0. In addition to the Wilson coefficients \(C_{7,9,10}^{(\prime )}\), we also consider the case where the NP contributions to \(C_9^{(\prime )}\) and \(C_{10}^{(\prime )}\) are equal up to a sign, since this pattern of effects is generated by \(\mathrm{SU}(2)_L\)invariant four fermion operators in the dimension6 SM effective theory.
 A negative NP contribution to \(C_9\), approximately \(25\,\%\) of \(C_9^\text {SM}\), leads to a sizable decrease in the \(\chi ^2\). The bestfit point corresponds to a p value of \(11.3\,\%\), compared to \(2.1\%\) for the SM. This was already found in fits of low\(q^2\) angular observables only [2] and in global fits not including data released this year [3, 4, 5, 20], as well as in a recent fit to a subset of the available data [9]. We find that the significance of this solution has increased substantially. This is due in part to the reduced theory uncertainties, in particular the form factors, as well as due to the new measurements by LHCb.Table 2
Constraints on individual Wilson coefficients, assuming them to be real. The pull in the last column is defined as \(\sqrt{\chi ^2_\text {SM}  \chi ^2_\text {b.f.}}\)
Coeff.
Best fit
\(1\sigma \)
\(2\sigma \)
\(\chi ^2_\text {SM}\chi ^2_\text {b.f.}\)
Pull
\(C_7^\text {NP}\)
\(0.04\)
\([0.07,0.01]\)
\([0.10,0.02]\)
2.0
1.4
\(C_7'\)
0.01
\([0.04,0.07]\)
\([0.10,0.12]\)
0.1
0.2
\(C_9^\text {NP}\)
\(1.07\)
\([1.32,0.81]\)
\([1.54,0.53]\)
13.7
3.7
\(C_9'\)
0.21
\([0.04,0.46]\)
\([0.29,0.70]\)
0.7
0.8
\(C_{10}^\text {NP}\)
0.50
[0.24, 0.78]
\([0.01,1.08]\)
3.9
2.0
\(C_{10}'\)
\(0.16\)
\([0.34,0.02]\)
\([0.52,0.21]\)
0.8
0.9
\(C_9^\text {NP}=C_{10}^\text {NP}\)
\(0.22\)
\([0.44,0.03]\)
\([0.64,0.33]\)
0.8
0.9
\(C_9^\text {NP}=C_{10}^\text {NP}\)
\(0.53\)
\([0.71,0.35]\)
\([0.91,0.18]\)
9.8
3.1
\(C_9'=C_{10}'\)
\(0.10\)
\([0.36,0.17]\)
\([0.64,0.43]\)
0.1
0.4
\(C_9'=C_{10}'\)
0.11
\([0.01,0.22]\)
\([0.12,0.33]\)
0.9
0.9

A significant improvement is also obtained in the \(\mathrm{SU}(2)_L\) invariant direction \(C_9^\text {NP}=C_{10}^\text {NP}\), corresponding to an operator with lefthanded muons.

A positive NP contribution to \(C_{10}\) alone can also improve the fit, although to a lesser extent.

NP contributions to individual righthanded Wilson coefficients hardly lead to improvements of the fit.
The global constraints in the complex planes of all Wilson coefficients are shown in Fig. 11 of “Appendix C”.
3.4 Constraints on pairs of Wilson coefficients
3.5 Minimal flavour violation
The expression (23) can be used to easily impose the combined fit constraints in phenomenological analyses of models satisfying CMFV. For scenarios with nonstandard CP violation or righthanded currents, it can be understood from Figs. 11 and 12 that at present the constraints are not stringent enough to allow a quadratic expansion of the \(\chi ^2\) and we cannot provide a comparably simple expression in general.
3.6 Testing leptonflavour universality
So far, in our numerical analysis we have only considered the muonic \(b \rightarrow s \mu ^+\mu ^\) modes and the leptonflavourindependent radiative \(b \rightarrow s \gamma \) modes to probe the Wilson coefficients \(C_7^{(\prime )}\), \(C_9^{(') \mu }\) and \(C_{10}^{(\prime ) \mu }\), where the superscript \(\mu \) indicates that in the semileptonic operators (3) and (4) only muons are considered. In this section we will extend our analysis and include also semileptonic operators that contain electrons. In particular, we will allow new physics in the Wilson coefficients \(C_9^e\) and \(C_{10}^e\) and confront them with the available data on \(B \rightarrow K e^+e^\) from LHCb [6] and \(B \rightarrow X_s e^+e^\) from BaBar [63].
 new physics only in \(C_9^\mu \);Table 3
Predictions for ratios of observables with muons vs. electrons for four different scenarios with NP only in one or two Wilson coefficients with muons. Ratios deviating from the SM prediction 1.00 by more than 30 % are highlighted in boldface. Differential branching ratios are given in units of GeV\(^{2}\)
Observable
Ratio of muon vs. electron mode
\(C_9^\text {NP}=1.07\)
\(1.10\)
\(0.53\)
\(1.06\)
\(C_9'=0\)
0.45
0
0
\(C_{10}^\text {NP}=0\)
0
0.53
0.16
\(10^{7}~\frac{\mathrm{d}\text {BR}}{\mathrm{d}q^2}(\bar{B}^0\rightarrow \bar{K}^{*0}\ell ^+\ell ^)_{[1,6]}\)
0.83
0.77
0.77
0.79
\(10^{7}~\frac{\mathrm{d}\text {BR}}{\mathrm{d}q^2}(\bar{B}^0\rightarrow \bar{K}^{*0}\ell ^+\ell ^)_{[15,19]}\)
0.78
0.72
0.75
0.74
\(F_L(\bar{B}^0\rightarrow \bar{K}^{*0}\ell ^+\ell ^)_{[1,6]}\)
0.93
0.90
0.98
0.93
\(F_L(\bar{B}^0\rightarrow \bar{K}^{*0}\ell ^+\ell ^)_{[15,19]}\)
1.00
0.97
1.00
1.00
\(A_\text {FB}(\bar{B}^0\rightarrow \bar{K}^{*0}\ell ^+\ell ^)_{[4,6]}\)
0.33
0.33
0.74
0.35
\(A_\text {FB}(\bar{B}^0\rightarrow \bar{K}^{*0}\ell ^+\ell ^)_{[15,19]}\)
0.90
0.96
0.99
0.92
\(S_5(\bar{B}^0\rightarrow \bar{K}^{*0}\ell ^+\ell ^)_{[4,6]}\)
0.73
0.77
0.93
0.74
\(S_5(\bar{B}^0\rightarrow \bar{K}^{*0}\ell ^+\ell ^)_{[15,19]}\)
0.91
0.97
0.99
0.92
\(10^{8}~\frac{\mathrm{d}\text {BR}}{\mathrm{d}q^2}(B^+\rightarrow K^+\ell ^+\ell ^)_{[1,6]}\)
0.77
0.85
0.76
0.74
\(10^{8}~\frac{\mathrm{d}\text {BR}}{\mathrm{d}q^2}(B^+\rightarrow K^+\ell ^+\ell ^)_{[15,22]}\)
0.78
0.86
0.76
0.74
\(10^{6}~\text {BR}(B\rightarrow X_s\ell ^+\ell ^)_{[1,6]}\)
0.83
0.83
0.77
0.79
\(10^{6}~\text {BR}(B\rightarrow X_s\ell ^+\ell ^)_{[14.2,25]}\)
0.78
0.78
0.75
0.74

new physics in \(C_9^\mu \) and \(C_9^{\prime ~\mu }\);

new physics along the \(\mathrm{SU}(2)_L\) invariant direction \(C_9^\mu =  C_{10}^\mu \);

new physics independently in \(C_9^\mu \) and \(C_{10}^\mu \).
4 Constraints on new physics models
The results from the modelindependent fit of the Wilson coefficients in the effective Hamiltonian can be interpreted in the context of new physics models. Here we discuss implications for the minimal supersymmetric standard model (MSSM) and models that contain massive \(Z^\prime \) gauge bosons with flavourchanging couplings.
4.1 SUSY models with generic flavour violation
Recently, the \(B \rightarrow K^* \mu ^+\mu ^\) decay has been studied in MSSM scenarios that do not contain sources of flavour violation beyond the CKM matrix [79]. We do not find sizable SUSY contributions to \(C_9\) and \(C_{10}\) in such scenarios. In the following, we will therefore allow for generic flavour violation.
Experimental data on flavourchanging neutral current processes lead to strong constraints on new sources of flavour violation that can be present in the MSSM [80, 81]. In particular, the experimental information on rare \(b \rightarrow s \mu ^+\mu ^\) decays can be used to put constraints on flavourviolating trilinear couplings in the upsquark sector, which are only poorly constrained otherwise [82, 83, 84, 85, 86]. In principle, the general MSSM also allows for leptonflavour nonuniversality effects and we will comment to which extent the \(R_K\) measurement can be accommodated.
4.1.1 Bounds on flavourchanging trilinear couplings
 contributions to \(C_{7,8}^\prime \), are suppressed by \(m_s/m_b\) with respect to contributions to \(C_{7,8}\);

contributions to \(C_{9,10}^\prime \) are suppressed by \(m_s m_b / m_t^2\) with respect to contributions to \(C_{9,10}\);

contributions proportional to \(A_{tc}\) are suppressed by \(m_c / m_t\) compared to contributions proportional to \(A_{ct}\).
The contributions to \(C_7\) and \(C_8\) from \(A_{ct}\) arise first at the dimension8 level, i.e. they are suppressed by \(m_\text {EW}^4/m_\text {SUSY}^4\). The last terms in (29a) and (29b) are the leading irreducible MFV contributions to \(C_7\) and \(C_8\) from Higgsino stop loops. They arise already at dimension 6 and are typically much larger than the contributions proportional to \(A_{ct}\).
This suggests that there are regions of MSSM parameter space, where a contribution to \(C_{10}^Z\) of O(1) is indeed possible. MSSM contributions to \(C_9^Z\) on the other hand are suppressed by the accidentally small vector coupling of the Z boson to leptons, \((4s_W^2 1) \sim 0.08\), and therefore negligible.
Recalling the modelindependent results from Sect. 3, a positive new physics contribution to the Wilson coefficient \(C_{10}^\text {NP} \simeq O(1)\), can improve the agreement with the current experimental \(b \rightarrow s \mu ^+\mu ^\) data significantly (albeit to a lesser extent than NP in \(C_9\)). Negative NP contributions to \(C_{10}\) on the other hand are strongly disfavoured with the current data. We use these results to probe regions of MSSM parameter space with sizable flavourchanging trilinear couplings.
In principle, additional constraints on \(A_{ct}\) can be obtained from the experimental bounds on electric dipole moments (EDMs). In particular, if \(A_{ct}\) and \(A_t\) contain a relative phase, a strange quark EDM and chromo EDM will be induced analogous to the new physics contributions to \(C_7\) and \(C_8\). However, predicting an experimentally accessible EDM of a hadronic system, like the neutron, given a strange quark EDM or chromo EDM involves large theoretical uncertainties [94, 95]. Due to these uncertainties, existing EDM bounds do not give appreciable constraints in our setup. Note also that bounds on the charm quark chromo EDM [96] do not constrain the parameter space of our scenario. A sizable charm quark chromo EDM would be generated in the presence of both \(A_{ct}\) and \(A_{tc}\) couplings, but here we only consider a nonzero \(A_{ct}\).
We now describe the SUSY spectrum that we chose to illustrate the bounds on the trilinear couplings from the \(b\rightarrow s \mu ^+\mu ^\) data. The soft masses for the lefthanded stop and charm squark are set to a common value \(m_{\tilde{t}_L} = m_{\tilde{c}_L} = 1\) TeV. The soft mass of the righthanded stop is set to \(m_{\tilde{t}_R} = 500\) GeV. All other squarks and sleptons as well as the gluino are assumed to be heavy, with masses of 2 TeV. Concerning the trilinear couplings, we only consider nonzero \(A_t\) and \(A_{ct}\). Due to these trilinear couplings, the lightest upsquark mass eigenstate can have a mass \(m_{\tilde{t}_1} < 500\) GeV and is potentially subject to strong bounds from direct stop searches. Higgsinos, Winos and Binos are assumed to have mass parameters \(m_{\tilde{B}} = 250\) GeV, \(m_{\tilde{W}} = 300\) GeV, \(\mu = 350\) GeV. In that way the mass of the lightest neutralino is given by \(m_{\tilde{\chi }_1^0} \simeq 225\) GeV and the mass of the lightest chargino is \(m_{\tilde{\chi }_1^\pm } \simeq 250\) GeV. Such a chargino–neutralino spectrum is heavy enough to avoid the bounds from the direct stop searches [97, 98, 99, 100]^{12} as well as bounds from electroweakino searches [103, 104]. Finally, we set \(\tan \beta = 3\) to minimise contributions to the dipole Wilson coefficients.
In Fig. 8 we show bounds on the trilinear couplings that can be derived from the \(b\rightarrow s \mu ^+\mu ^\) data in the described scenario. We evaluate all MSSM 1loop contributions to the Wilson coefficients \(C_{7,8,9,10}^{(\prime )}\) and compute the \(\chi ^2\) as defined in (15) as a function of the trilinear couplings. For the numerical evaluation of the Wilson coefficients in the MSSM, we use an adapted version of the SUSY_FLAVOR code [105, 106, 107]. The plot on the left hand side of Fig. 8 shows constraints in the \(A_t\)–\(A_{ct}\) plane, assuming real trilinears. The plot on the right hand side shows constraints in the Re\((A_{ct})\)–Im\((A_{ct})\) plane, for a fixed \(A_t = 1.5\) TeV.^{13} The red region is excluded by the \(b\rightarrow s \mu ^+\mu ^\) data by more than \(2\sigma \) with respect to the SM (\(\chi ^2 > \chi ^2_\text {SM} + 6\)). In the blue region the agreement between the theory predictions and the experimental \(b\rightarrow s \mu ^+\mu ^\) data is improved by more than \(1\sigma \) with respect to the SM (\(\chi ^2 < \chi ^2_\text {SM}  2.3\)). In the bestfit point in the left plot of Fig. 8, the \(\chi ^2\) is reduced by 4.2 compared to the SM. This improvement is rather moderate compared to the results of the modelindependent fits and also compared to the \(Z^\prime \) scenarios discussed below. In the black corners, the lightest upsquark mass eigenstate is lighter than the lightest neutralino. Outside the dashed contours there exist charge and color breaking minima in the MSSM scalar potential that are deeper than the electroweak minimum. Inside the contours, the NP effects in the Wilson coefficients are rather moderate. In particular, we find that in this region of parameter space the SUSY contribution to \(C_{10}\) does not exceed 0.3; the SUSY contribution to \(C_9\) is smaller by approximately one order of magnitude, as expected.
Note that the regions outside of the vacuum stability contours are not necessarily excluded. Even though a deep charge and color breaking minimum exists in these regions, the electroweak vacuum might be metastable with a live time longer than the age of the universe. Studies show that requiring only metastability, relaxes the stability bounds on the trilinear couplings slightly [108, 109, 110, 111, 112]. A detailed analysis of vacuum metastability is beyond the scope of the present work.
4.1.2 Leptonflavour nonuniversality in the MSSM
The Z penguin effects discussed above are leptonflavour universal, i.e. they lead to the same effects in \(b \rightarrow s e^+e^\) and \(b \rightarrow s \mu ^+\mu ^\) decays. Breaking of e\(\mu \) universality as hinted by the \(R_K\) measurement can only come from box contributions as they involve sleptons of different flavours. If there are large mass splittings between the first and second generations of sleptons, or more precisely, if the selectrons are decoupled but smuons are kept light, Wino box diagrams (and to a lesser extent also Bino box diagrams) can contribute to \(C_9^\mu \) and \(C_{10}^\mu \) but not to \(C_9^e\) and \(C_{10}^e\).
Box contributions are, however, typically rather modest in size. As discussed above, boxes that are induced by flavourchanging trilinears arise only at the dimension8 level and are completely negligible. Nonnegligible box contributions (at the dimension 6 level) are only possible in the presence of flavour violation in the squark soft masses. However, even allowing for maximal mixing of lefthanded bottom and strange squarks, it was found in [3] that Winos and smuons close to the LEP bound of \(\sim \)100 GeV as well as bottom and strange squarks with masses of few hundred GeV would be required to obtain contributions to \(C_9^\mu \) and \(C_{10}^\mu \) of \(\gtrsim \)0.5, which could give \(R_K \sim 0.75\). A careful collider analysis would be required to ascertain if there are holes in the LHC searches for stops [97, 98, 99, 100], sbottoms [113, 114, 115], sleptons [103, 116, 117] and electroweakinos [103, 104] that would allow such an extremely light spectrum. We also note that a sizable splitting between the lefthanded smuon and selectron masses required to break e\(\mu \) universality is only possible if the slepton mass matrix is exactly diagonal in the same basis as the charged lepton mass matrix, since even a tiny misalignment would lead to an excessive \(\mu \rightarrow e \gamma \) decay rate.
4.2 Flavourchanging \(Z'\) bosons
4.2.1 \(Z'\) with coupling to lefthanded muons
We conclude that, in order to lead to visible effects in \(b\rightarrow s\mu ^+\mu ^\) transitions, a heavy \(Z'\) with \(M_{Z'} \gtrsim 3~\text {TeV}\) can have weakinteraction strength couplings to firstgeneration quarks without being in conflict with the bounds from contact interactions. Such a heavy \(Z'\) must have strong couplings to muons (\(\Delta _L^{\mu \mu } \gtrsim 1\)). A lighter \(Z'\) can be weakly coupled to muons but requires a suppression of the coupling to firstgeneration quarks by roughly two orders of magnitude to avoid the bounds from direct searches.
4.2.2 \(Z'\) with vectorlike coupling to muons
4.2.3 \(Z'\) with universal coupling to leptons
Concerning collider searches, the new feature of the leptonuniversal case is that there is an absolute lower bound on the \(Z'\) mass from LEP2, \(M_{Z'}>209\) GeV. LHC bounds on the coupling to firstgeneration quarks, on the other hand, are qualitatively similar to the nonuniversal case discussed above.
5 Summary and conclusions
Several recent results on rare B decays by the LHCb collaboration show tensions with Standard Model predictions. Those include discrepancies in angular observables in the \(B \rightarrow K^* \mu ^+\mu ^\) decay, a suppression in the branching ratios of \(B \rightarrow K^* \mu ^+\mu ^\) and \(B_s \rightarrow \phi \mu ^+\mu ^\), as well as a hint for the violation of leptonflavour universality in the form of a \(B \rightarrow K \mu ^+\mu ^\) branching ratio that is suppressed not only with respect to the SM prediction but also with respect to \(B \rightarrow K e^+e^\). In this paper we performed global fits of the experimental data within the SM and in the context of new physics.
For our SM predictions we use stateoftheart \(B \rightarrow K\), \(B \rightarrow K^*\) and \(B_s \rightarrow \phi \) form factors taking into account results from lattice and lightcone sum rule calculations. All relevant nonfactorisable corrections to the \(B \rightarrow K \mu ^+\mu ^\), \(B \rightarrow K^* \mu ^+\mu ^\) and \(B_s \rightarrow \phi \mu ^+\mu ^\) amplitudes that are known are included in our analysis. Additional unknown contributions are parametrised in a conservative manner, such that existing estimates of their size are within the \(1\sigma \) range of our parametrisation. We take into account all the correlations of theoretical uncertainties between different observables and between different bins of dilepton invariant mass. As experimental data is available for more and more observables in finer and finer bins, the theory error correlations have a strong impact on the result of the fits.^{15}
Making use of all relevant experimental data on radiative, leptonic and semileptonic \(b \rightarrow s\) decays we find that there is on overall tension between the SM predictions and the experimental results. Assuming the absence of new physics, we investigated to which extent nonperturbative QCD effects can be responsible for the apparent disagreement. We find that large nonfactorisable corrections, a factor of 4 above our error estimate, could improve the agreement for the \(B \rightarrow K^* \mu ^+\mu ^\) angular observables and the branching ratios considerably. Alternatively, the branching ratio predictions could also be brought into better agreement with the experimental data, if the involved form factors were all systematically below the theoretical determinations from the lattice and from LCSR. On the other hand, we find that nonstandard values of the form factors could at most lead to a modest improvement of \(B \rightarrow K^* \mu ^+\mu ^\) angular observables. In both cases, however, the hint for violation of leptonflavour universality cannot be explained.
Assuming that in our global fits the hadronic uncertainties are estimated in a sufficiently conservative way, we discussed the implications of the experimental results on new physics. Effects from new physics at short distances can be described model independently by an effective Hamiltonian and the experimental data can be used to obtain allowed regions for the new physics contributions to the Wilson coefficients. We find that the by far largest decrease in the \(\chi ^2\) can be obtained either by a negative new physics contribution to \(C_9\) (with \(C_9^\text {NP} \sim 25\,\% \times C_9^\text {SM}\)), or by new physics in the \(\mathrm{SU}(2)_L\) invariant direction \(C_9^\text {NP}=C_{10}^\text {NP}\), (with \(C_9^\text {NP} \sim 12\,\% \times C_9^\text {SM}\)). A positive NP contribution to \(C_{10}\) alone would also improve the fit, although to a lesser extent.
Concerning the hint for violation of leptonflavour universality, we observe that new physics exclusively in the muonic decay modes leads to an excellent description of the data. We do not find any preference for new physics in the electron modes. We provide predictions for other leptonflavouruniversality tests. We find that the ratio \(R_{A_\text {FB}}\) of the forward–backward asymmetries in \(B \rightarrow K^* \mu ^+\mu ^\) and \(B \rightarrow K^* e^+e^\) at low dilepton invariant mass is a particularly sensitive probe of new physics in \(C_9^\mu \). A precise measurement of \(R_{A_\text {FB}}\) would allow to distinguish the new physics scenarios that give the best description of the current data.
Finally we also discussed the implications of the modelindependent fits for the minimal supersymmetric standard model and models that contain \(Z^\prime \) gauge bosons with flavourchanging couplings. In the MSSM, large flavourchanging trilinear couplings in the upsquark sector can give sizable contributions to the Wilson coefficient \(C_{10}\) and we identified regions of MSSM parameter space that are favoured or disfavoured by the current experimental data. Heavy \(Z'\) bosons can have the required properties to explain the discrepancies observed in the \(b \rightarrow s \ell \ell \) data. If the \(Z'\) couples to muons but not to electrons (as preferred by the data), it is only weakly constrained by indirect probes. On the other hand, if the \(Z'\) couplings to leptons are flavour universal, LEP constraints on four lepton contact interactions imply that an explanation of the \(b \rightarrow s \ell \ell \) discrepancies results in new physics effects in \(B_s\) mixing of at least \(\sim \)10 %. In all scenarios, the couplings of the \(Z'\) to firstgeneration quarks are strongly constrained by ATLAS and CMS measurements of dilepton production.
We look forward to the updated experimental results using the full LHCb data set, which will be crucial in helping to establish or to refute the exciting possibility of new physics in \(b \rightarrow s\) transitions.
Footnotes
 1.
Note that the situation is different when also nonleptonic decays are considered; see e.g. [36].
 2.
We take the nonzero lepton mass into account in our numerics; the zeromass limit is taken here just for illustration.
 3.
Here, we gloss over the fact that the B factories actually measure the direct CP asymmetry in an admixture between charged and neutral B decays. However, the isospin difference between the CP asymmetries generated by an imaginary \(C_7\) or \(C_7'\) turns out to be negligibly small, so this is not relevant for our purposes.
 4.
In the case of \(B\rightarrow K^*\mu ^+\mu ^\), all known spectatordependent nonfactorisable effects are very small (see e.g. [47]), while e.g. the sizable effect discussed in Ref. [22] does not depend on the flavour of the spectator quark and we therefore expect it to be very similar between \(B_s\rightarrow \phi \mu ^+\mu ^\) and \(B\rightarrow K^*\mu ^+\mu ^\). We also stress that this guess for the correlation has a small impact on the numerical results as the uncertainty of BR(\(B_s\rightarrow \phi \mu ^+\mu ^\)) is by far dominated by form factor uncertainties [16], which we assume to be uncorrelated between \(B\rightarrow K^*\) and \(B_s\rightarrow \phi \) to be conservative.
 5.
Note also the recent update [71] which appeared after our analyses had been completed. We expect the changes to be much smaller than the experimental uncertainty.
 6.
By “parametric” here we refer to uncertainties that are not due to the form factors or other nonperturbative QCD effects.
 7.
For our global numerical analysis, we use a threeparameter z expansion as in [16]. The twoparameter expansion is only used in this case for simplicity. Note that two of the 14 parameters are redundant due to two exact kinematical relations at \(q^2=0\).
 8.
Here we use the transversity basis of form factors, cf. [14].
 9.
 10.
We do not quote uncertainties in Table 3 since any significant deviation from 1 would constitute a clear sign of NP. However, it should be noted that for a fixed value of the NP contributions to the Wilson coefficients, there are nonzero uncertainties in the observables.
 11.
 12.
Note that the most important bounds from [97, 98, 99, 100] assume 100 % branching ratio to either \(\tilde{t}_1 \rightarrow t \tilde{\chi }_1^0\) or \(\tilde{t}_1 \rightarrow b \tilde{\chi }_1^\pm \). In our scenario, both decay modes will compete with each other, weakening the bounds slightly. In addition, in our scenario there is significant second–third generation mixing and the lightest stop can also have a sizable branching ratio \(\tilde{t}_1 \rightarrow c \tilde{\chi }_1\). Thus the actual bounds from direct searches are further loosened; see e.g. [101, 102].
 13.In the MSSM not all the parameter space shown in Fig. 8 would be compatible with a lightest Higgs mass of 125 GeV. However, there exist various extensions of the MSSM Higgs sector that allow one to treat the Higgs mass independently from the stop sector. As the considered SUSY effects in \(b \rightarrow s \ell \ell \) do not depend on the details of the Higgs sector, we do not consider the Higgs mass constraint in the plots of Fig. 8.
 14.
The only exception relevant in the context of NP in \(b\rightarrow s\mu ^+\mu ^\) is a very low mass window between 10 GeV \(\lesssim M_{Z'} \lesssim \) 50 GeV, where the \(Z \rightarrow 4\mu \) branching ratio measured at the LHC gives a constraint that is slightly stronger than the one obtained from neutrino tridents [131].
 15.
To quantify this statement: when all correlations of theory uncertainties are set to zero, the \(\Delta \chi ^2\) of the fit with NP in \(C_9\) only increases from 13.7 to 38.9. This huge overestimate of the significance is easy to understand, as tensions in the same direction in adjacent bins are less significant if one knows that they are highly correlated.
Notes
Acknowledgments
We thank Martin Beneke, Aoife Bharucha, Christoph Bobeth, Gerhard Buchalla, Danny van Dyk, Thorsten Feldmann, Christoph Niehoff, Yuming Wang, Roman Zwicky, and all the participants of the “Workshop on \(b \rightarrow s ll\) processes” at Imperial College in April 2014 for useful discussions. We also thank the National Science Foundation for partial support (under Grant No. PHYS1066293), the Aspen Center for Physics for hospitality and the German national football team for moral support during the workshop “Connecting Flavor Physics with Naturalness: from Theory to Experiment” in July 2014. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development and Innovation. The research of D.S. was supported by the DFG cluster of excellence “Origin and Structure of the Universe”.
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