The asymmetries of the $B \to K^* \mu^+ \mu^-$ decay and the search of new physics beyond the Standard Model

In this paper, we compute the forward-backward asymmetry and the isospin asymmetry of the $B \to K^* \mu^+ \mu^-$ decay. The $B \to K^*$ transition form factors (TFFs) are key components of the decay. To achieve a more accurate QCD prediction, we adopt a chiral correlator for calculating the QCD light-cone sum rules for those TFFs with the purpose of suppressing the uncertain high-twist distribution amplitudes. Our predictions show that the asymmetries under the Standard Model and the Minimal Supersymmetric Standard Model with minimal flavor violation are close in shape for $q^2 \ge 6~{\rm GeV}^2$ and are consistent with the Belle, LHCb and CDF data within errors. When $q^2<2~{\rm GeV}^2$, their predictions behave quite differently. Thus a careful study on the $B \to K^* \mu^+ \mu^-$ decay within small $q^2$-region could be helpful for searching new physics beyond the Standard Model. As a further application, we also apply the $B \to K^*$ TFFs to the branching ratio and longitudinal polarization fraction of the $B\to K^*\nu\bar\nu$ decay within different models.


I. INTRODUCTION
Processes involving flavor changing neutral current (FCNC) provide important platforms for testing the Standard Model (SM) and for searching of new physics beyond the SM. Among them, the B-meson exclusive decays, such as the B → K * µ + µ − with the cascade decay K * → Kπ, is important. This is because the measurements of their four-body final state angular distributions provide abundant information on probing and discriminating different scenarios of new physics.
The B-meson exclusive decay requires a proper factorization of the long-distance and the short-distance physics, which could generally be distinguished by the heavy quark mass m b emerged in the hadronic matrix elements. By further taking the heavy-quark limit, m b → ∞, the hadronic amplitudes arising from the hard gluon exchanges can be factorized into the perturbative scattering kernels and the nonperturbative but universal hadronic quantities. This treatment has been successfully introduced in dealing with the nonleptonic Bmeson decays, the heavy-to-light transition form factors, and the radiative B-meson decays [1][2][3][4].
In the paper, we shall focus on the forward-backward and the isospin asymmetries of the B → K * µ + µ − exclusive decay, which are sensitive to the Wilson coefficients and could be used to test the new physics scenario beyond SM. The new physics part of Wilson coefficients are * Electronic address: fuhb@cqu.edu.cn † Electronic address: wuxg@cqu.edu.cn ‡ Electronic address: chengw@cqu.edc.cn § Electronic address: zhongtao@htu.edu.cn ¶ Electronic address: zhansun@cqu.edu.cn model dependent, which have been dealed with various methods [5][6][7][8][9]. According to the minimal supersymmetric standard model (MSSM) with minimal flavour violation (MFV), all flavour transitions occur only in the charged-current sector and are determined by the known CKM mixing angles. This idea is also adopted by several theoretical schemes in which the communication of the supersymmetry breaking to the observable particles occurs via flavour-independent interactions. In many of those schemes the departure from the MFV hypothesis is rather small [10]. To increase the predictivity of the MSSM with MFV, one may follow several restrictions which state that all supersymmetric particles except for charginos, sneutrinos and charged Higgs fields are about 1 TeV; heavy particles shall be integrated out, resulting in a "low-energy" effective theory in terms of light SUSY and SM particles; the weak effective hamiltonian includes only the SM operators and the down-squark sector including a flavour diagonal mass matrix [11]. In the paper, we shall adopt the Wilson coefficients under the SUSY MFV model as an explanation of how the new physics terms could affect the SM predictions. Furthermore, large uncertainties in predicting the B → K * µ + µ − decay come from the nonperturbative quantities, namely the B → K * transition form factors (TFFs). Those TFFs have been studied within various approaches such as the relativistic quark model [12,13], the lightcone sum rules (LCSR) [14][15][16][17][18][19], the lattice QCD [20][21][22], and etc.. The LCSR predictions are reliable from the hard region around the large recoil point to the soft contribution below m 2 b − 2m b χ (χ ∼ 500 MeV is the typical hadronic scale of the decay), thus it provides an important bridge for connecting the results of various approaches and for comparing with the data.
The LCSR is based on the operator product expansion (OPE), which parameterizes the nonperturbative dynam-ics into light-cone distribution amplitudes (LCDAs). To compare with that of the usually considered pseudoscalar LCDA, the vector meson's LCDA has much complex twist structures. Even though the high-twist LCDAs are generally power suppressed, their contributions are sizable, especially in specific kinematic region. The inaccurateness of high-twist LCDAs then lead to important systematic errors for the LCSR predictions. A practical way to suppress the uncertainties from those uncertain high-twist LCDAs is to take a proper LCSR correlator. For example, the contributions from the high-twist LCDAs can be highly suppressed by using chiral correlators [23][24][25]. As an application of those more accurate TFFs, we shall recalculate the forward-backward and isospin asymmetries of the B → K * µ + µ − decay and also the branching ratio and longitudinal polarization fraction of the B → K * νν decay.
The remaining parts of the paper are organized as follows. In Sec.II, we describe our calculation technology for deriving the forward-backward and isospin asymmetries. In Sec.III, we present numerical results and discussions on the TFFs and the asymmetries of the B → K * µ + µ − decay within the SM and the SUSY with MFV. And we also present the results for the branching ratio and longitudinal polarization fraction of the B → K * νν decay within different models in Sec.III. Sec.IV is reserved for a summary.

II. CALCULATION TECHNOLOGY
Within the SM, the B → K * µ + µ − decay is induced by a set of operators O i appearing in the weak effective hamiltonian [26], where λ q = V * qs V qb , and the Wilson coefficients C i are perturbatively calculable, whose values shall be alternated when new particles beyond the SM are included.
The differential decay width of B → K * µ + µ − over the squared transition momentum (q 2 ) and the angle (θ) takes the form [2], of K ⊥ 1,2 and K 1 up to subleading Λ h /m B expansion can be found in Ref. [2,3], whose effects are small for b q (q 2 ) but are sizable for b ⊥ q (q 2 ). As a cross-check, by taking the limit q 2 → 0, due to fact that the photon pole dominates the C , which rightly equals to the isospin asymmetry of the B → K * γ decay.
The expressions of the TFFs ξ λ (q 2 ) can be related to the usually defined B → K * TFFs A 1,2 (q 2 ) and T 1 (q 2 ) via the following way [27] As mentioned in the Introduction, we adopt the expressions of the TFFs A 1,2 (q 2 ) and T 1 (q 2 ) that have been derived under the LCSR approach by using a right-handed chiral correlator [23,25] to get the final LCSRs for ξ ⊥ (q 2 ) and ξ (q 2 ), which take the form where is the usual step function, Θ(c(̺, s 0 )) and Θ(c(̺, s 0 )) are step functions with surface terms which are defined in Ref. [23].
Those two formulas show that the LCSRs for ξ ⊥ (q 2 ) and ξ (q 2 ) are free of contributions from most of the hightwist LCDAs, and the remaining high-twist ones are generally suppressed by δ 2 ∼ (m * K /m b ) 2 ∼ 0.03 to compare with the leading-twist terms; thus uncertainties from the high-twist LCDAs themselves are effectively suppressed and a more accurate prediction for the TFFs ξ ⊥ (q 2 ) and ξ (q 2 ) can be achieved.
Ref. [15] 0.126(11) 0.333(28) Ref. [30] 0.118(8) 0.266(32) Ref. [27] 0.076 0.245 We adopt the usual criteria to set the LCSR parameters, the Borel window and the continuum threshold s 0 , of the B → K * TFFs: I) The continuum contribution is required to be less than 30% of the total LCSR, and all high-twist DAs' contributions are suppressed to be less than 15% of the total LCSR; II) The derivatives of the LCSRs over (−1/M 2 ) give the LCSRs for m B , and for self-consistency, we require all the predicted B-meson masses to be full-filled in comparing with the experimental one, e.g. |m LCSR 1%. We present the B → K * TFFs at the large recoil point q 2 = 0 GeV 2 in Table I, where the LCSR predictions of Refs. [15,27,30] are presented as a comparison. The LCSRs of Refs. [15,30] are derived by using the usual correlator, in which all twist-2, 3, 4 LCDAs are in the LCSRs. Table I shows that the LCSRs under different choice of correlators are consistent with each other within errors, indicating the LCSRs are independent to the choice of correlators. A detailed discussion of the consistency of the LCSRs under different choice of correlators can be found in Ref. [25]. The differences among different LCSRs are mainly caused by different choice of the dominant leading-twist K * LCDA. For example, the use of AdS/QCD holographic leading-twist LCDA leads to a much smaller ξ (0) [27].
The contribution from the leading-twist LCDA φ ⊥ 2;K * has been amplified by using the chiral correlator, thus the systematic errors from the the φ ⊥ 2;K * parameters shall be amplified. This leads to slightly larger error than those LCSRs for usual correlator. By comparing with the data, this fact can be inversely adopted to achieve a better constraint on φ ⊥ 2;K * . The high-twist terms for the LC-SRs [15,30] follow the δ-power counting rule, which could be large for δ 1 twist-3 terms. By using the chiral correlator, the high-twist LCDAs' contributions are greatly suppressed due to chiral suppression, thus their own uncertainties to the LCSR can be safely neglected and the accuracy of the LCSRs can be greatly improved. For example, we find that the contributions from the twist-3 LCDA Φ 3;K * and the twist-4 LCDA Ψ ⊥ 4;K * provide less than 0.1% of the total LCSRs.
The LCSR approach is applicable in large and intermediate recoil region, 0 ≤ q 2 ≤ 15 GeV 2 . We extrapolate its prediction to the physically allowed q 2 -region by using a simplified series expansion [17,31], which is based on a rapidly converging series over the parameter z(t), i.e. where The form factors are them expanded as where F i stand for the TFFs ξ λ (q 2 ), and the resonance masses m R,i can be found in Ref. [32]. The coefficients a i 0 = F i (0). The parameters a i 1 and a i 2 are determined by requiring the "quality of fit (∆)" to be less than one [15], which is defined as 27 2 , 14]GeV 2 . The extrapolated B → K * TFFs are presented in Fig.1, in which the lattice QCD prediction [22] have also been presented. Fig.1 shows that our present LCSR predictions are consistent with the lattice QCD predictions within errors. In the following, we adopt the extrapolated TFFs to study the forward-backward asymmetry and the isospin asymmetry for the B → K * µ + µ − decay.
B. The forward-backward and the isospin asymmetries of the B → K * µ + µ − decay The Wilson coefficients are scale-dependent, whose values at the lower scales such as the typical momentum FIG. 1: The extrapolated B → K * TFFs ξ λ (q 2 ) based on the present LCSR predictions. The shaded bands stand for the theoretical errors. The lattice QCD [22] prediction has also been presented.
where i = (1, · · · , 10). The expression of C (0) i and C (1) i can be found in Ref. [33] and those of δC (0,1) 7,8 can be found in Ref. [34]. Up to NLO level, the first six Wilson coeffi-cientsC i (µ) can be rewritten as (16) in which T ij is the transformation matrix [2]. The SUSY contributions to C 9 and C 10 have been calculated in Refs. [35,36]. The interactions among the charged Higgs and the up-type quarks, which are from the SUSY model or the SM models with two Higgs doublet, may have sizable contributions, those terms are represented by a subscript H. Because the NLO SUSY contribution to the four-quark penguin operators are also sizable, we treat it as δC  Table II, where as suggested by Ref. [2], we take C eff We adopt MSSM with MFV as a typical SUSY model to probe the possible new physics effect. The basic input for the SUSY parameters is the ratio of the vacuum expectation values of the Higgs doublet, i.e. tan β, and we take tan β ∈ [2, 40] to do the discussion. A larger tan β could lead to a flip of sign for C eff 7,8 [38], which arouses people's great interests. The behaviours of the two ranges as a function of the free parameters are quite different, we call the model for tan β ∈ [2, 10] as MFV-I and the model for tan β = 40 as MFV-II. The mechanisms that enhance the SUSY contribution to C eff 7 at large tan β are not working for C 9,10 [35,36]. For instance, the charged Higgs contribution dominant for C eff 7 at large tan β is suppressed for C 9,10 , and the modifications for the forwardbackward and the isospin asymmetries shall be mainly due to the new physics contributions to C eff 7,8 . In allowable parameter space, the ranges of the new physics part of C eff 7,8 at the scales µ b and µ h are where I corresponds to the MFV-I and II corresponds to the MFV-II, respectively.  [39], the CDF [40], the LHCb [41], and the CMS [42] collaborations have also been presented.
By using the parameters in the SM and the MSSM MFV scenarios allowed from the constraints discussed in above, we give our prediction on the forward-backward and the isospin asymmetries of the B → K * µ + µ − decay in the following paragraphs.
Such a smaller effect to the SM prediction at the large-q 2 region indicates that one can not distinguish those MSSM MFV models with the SM one in the large q 2 -region. In large q 2 -region, the predicted forward-backward asymmetry agrees with the Belle [39] and the CDF [40] measurements within errors. However even by including the MSSM MFV-I or MFV-II terms, we still cannot explain the trends of a smaller forward-backward asymmetry around q 2 > 16 GeV 2 as indicated by the LHCb [41] and the CMS [42] measurements. Thus we need new SUSY models to explain this discrepancy, or we need more data to confirm those measurements.
• Main differences among various models lie in low q 2 -region, e.g. q 2 ≤ 6 GeV 2 , indicating the MSSM effects could be important and sizable. The SM prediction has a cross-over around q 2 ∼ 3.2 GeV 2 , −0.05 ± 0.03 ATLAS [43] 0.07 ± 0.20 ± 0.07 which shifts to a smaller value for MSSM MFV-I. The forward-backward asymmetries of SM and MSSM MFV-I behave closely in shape, both of which are negative for small q 2 -region and are consistent with the measurements. Meanwhile, the forward-backward asymmetries of MSSM MFV-II are always positive in low q 2 -region, which is due to the flip of sign for C eff 7,8 at large tan β and is out of the LHCb and CMS measurements. Thus the present data prefers a smaller tan β, i.e. MSSM MFV-I. Due to different behaviors of the forwardbackward asymmetries under MSSM MFV-I and MFV-II, the more precise measurements in low q 2region shall be helpful for constraining a more reliable range for the key MSSM parameter tan β.
Next, we present the integrated forward-backward asymmetry for q 2 ∈ [1, 6]GeV 2 in Table III. Here the first uncertainty is the SM error which is mainly from the LCSR predictions and the second one is the MSSM MFV-I or MFV-II error which is dominated by the possible choices of C eff 7,8 . In Table III We also present the Belle [39], the CMS [42], and the ATLAS [43] data as a comparison. Table III confirms our above observation that the MSSM MFV-I gives SM-like prediction, both of which are consistent with the measurements within errors; while, the MSSM MFV-II prefers a quite large asymmetry A FB .
Finally, we present the differential distribution for the The isospin asymmetry of the B → K * µ + µ − decay. The SM prediction by using the AdS/QCD LCDA [27] is presented in the first diagram as a comparison. The measurements from the Belle [39] and the LHCb [41] collaborations are also presented.
isospin asymmetry of the B → K * µ + µ − decay in Fig.3. The B → K * νν decay has the virtue that the angular distribution of the K * decay products allows to extract information on the K * polarization, similar to the B → K * µ + µ − decays. The longitudinal and transverse differential distributions versus q 2 , the square of the invariant mass of the νν pair, is given as [44][45][46] with the coefficient N = The hadronic transversity amplitudes H ⊥, ,0 (q 2 ) are The total differential decay width dΓ/q 2 = dΓ L /q 2 + dΓ T /q 2 . To calculate the branching ratios, we use the average value from the B ± lifetime τ B + and the B 0 lifetime τ B 0 for B → K * νν decay. Meanwhile, the K *meson longitudinal and transverse polarization fraction F L,T are defined as which satisfy F L +F T = 1. The TFFs A 1,2 (q 2 ) and V (q 2 ) have also been calculated by using a right-handed chiral correlator under the LCSR approach [23,25]. Principally, the Wilson coefficients C L and C R are complex. One usually defines two real parameters, and the differential decay branching ratio and longitudinal polarization fraction can be expressed as The Wilson coefficient C SM R for the SM is negligibly small, leading to η SM ≃ 0. The Wilson coefficient C SM L for the SM has been calculated at the next-to-leading order QCD corrections [47,48], which gives C SM L = −X(x t )/ sin 2 θ W , where x t = m 2 t /m 2 W and X(x i ) is the corresponding loop function which gives C SM L = −6.38(6) [49]. Different to the above considered case of two leptons in final state which uses MSSM with MFV to deal with the new physics effect, as suggested by Ref. [49], we adopt the MSSM with generic flavour violating (GFV) to deal with the two Wilson coefficients C L and C R for the present case of two neutrinos in final state. In this model, the MSSM contributions to C R turn out to be very small, which implies that η ≃ 0, thus leads to a SM-like prediction on F L (q 2 ), i.e. F L (q 2 ) ≃ F SM L (q 2 ). Thus one cannot use the observable F L (q 2 ) along to probe the MSSM.  [30], LCSR-II result [15] and the QM result [50], respectively. We present our prediction of the K * -meson longitu-dinal polarization fraction F SM L in Fig.4. As a comparison we also present the QM result [50] and the other two LCSR predictions, i.e. LCSR-I [30]and LCSR-II [15] in the figure. Fig.4 shows our results are consistent with LCSR-II and QM results within reasonable errors in whole q 2 -region, while the LCSR-I has a larger F L (q 2 ) in intermediate and large q 2 -region such as 6 GeV 2 < q 2 < (m B − m * K ) 2 . We present a comparison of the SM differential branching ratio of the B → K * νν decay under various approaches in Fig.5. It shows that different TFFs leads to different behaviors, the LCSR-I and QM results agree with our differential branching ratio within errors; while the LCSR-II agrees with our prediction only for small q 2 -region, e.g. 0 < q 2 < 10GeV 2 . Thus a more accurate prediction on the B → K * TFF shall be helpful for a more accurate SM prediction.
The possible visible MSSM effects in C L are generated by chargino contributions through a large (δ RL u ) 32 mass insertion. Because those chargino contributions are not sensitive to the choice of tan β, we choose to work in the low tan β regime, i.e. tan β = 5. The necessary inputs for the MSSM with GFV can be found in the Ref. [49], in which there are two typical sets of parameters, and we call them as MSSM GFV-I and GFV-II, respectively. By using the parameters in the SM and the MSSM with GFV scenarios allowed from the constraints discussed in above, we give our prediction on the differential branching ratio in Fig.6. The SM plus MSSM GFV-I and GFV-II predictions are consistent with the SM prediction in low q 2 -region, but are different at high q 2 -region, e.g. q 2 > 7 GeV 2 . Thus a careful measurement at high q 2region could be helpful to clarify whether there is new physics, and which one, MSSM GFV-I or MSSM GFV-II, is more preferable.
To have a clear look at the differences among different models and approaches, we integrate the momentum transfer in whole physical region 0 < q 2 < (m B −m * K ) 2 to  get the total branching ratio and the q 2 -integrated form of F L , which is define as where We present the results for the branching ratio B and the longitudinal polarization fraction F L of B → K * νν in Table IV. As a comparison, we also present the 2017 Belle Collaboration measurements [51], the results of Ref. [49] (ABSW) and the SM prediction of Ref. [52] (NWA) in Table IV. As mentioned above, the MSSM effect to F L is negligibly small due to η → 0, and the predicted values of F L within the SM are listed in the second series at Table IV, which shows that our prediction is close to NWA one. Table IV shows by including the new physics effect, the branching ratio shall be suppressed by ∼ 22% and increased by ∼ 28% for MSSM GFV-I and GFV-II, respectively. Our SM prediction of the branching ratio B are consistent with the ABSW SM and NWA predictions within errors, all of which agree with the newest upper limit predicted by Belle Collaboration in 2017 (B Belle < 18 × 10 −6 ). Thus we still need more accurate data to draw definite conclusions.

IV. SUMMARY
In the paper, we recalculate the B → K * TFFs ξ ⊥, (q 2 ) by using the LCSR approach, in which a chiral correlator has been adopted to suppress the large uncertainties from the twist-2 and twist-3 structures at the δ 1order. For each LCSR, except the dominate twist-2 contribution which are proportional to φ ⊥ 2;K * , the remaining non-zero twist-3 and twist-4 terms as shown by Eqs. (10,11) shall be at least δ 2 -suppressed, which totally only provide less than 10% contributions to the LCSRs. Thus the resultant LCSRs are more accurate than the previous ones derived in the literature. The extrapolated B → K * TFFs as shown in Fig.1 are consistent with the Lattice QCD predictions within errors. This new achievement helps for probing new physics beyond the SM.
Based on the definitions of the forward-backward and the isospin asymmetries, we calculate their differential distributions over q 2 under three models and present our results in Figs.2 and 3. The SM and the SM+MSSM MFV-I predictions are consistent with each other; while the SM+MSSM MFV-II prediction shows quite different behavior, especially in low q 2 -region. Thus a careful comparing with data could be helpful for judging whether we need new physics scenario for those observables or which new physics scenario is more credible: • For the forward-backward asymmetry A FB , the MSSM MFV-I only slightly changes the SM prediction and does not change its arising trends, both of which agree with the Belle, the CDF and the CMS measurements in low q 2 -region. On the contrary, due to the flip of sign for C eff 7,8 , the MSSM MFV-II give large corrections to the SM prediction in low q 2 -region, leading to a positive A FB in whole q 2 -region. This differences make it possible to draw the conclusion of whether MFV-I or MFV-II is preferable by using more accurate data measured in low q 2 -region. Table III prefers a small tan β for the MSSM MFV model.
• For the forward-backward asymmetry A FB at the large q 2 -region, we have found that the new physics effect shall be suppressed by 1/q 4 to compare with the SM prediction. Fig.2 and Fig.3 show that even by including the MSSM MFV-I or MFV-II terms, we still cannot explain the trends of the smaller forward-backward asymmetry around q 2 > 16 GeV 2 as indicated by the present LHCb [41] and the CMS [42] measurements. Thus we may need new SUSY models to explain this large q 2discrepancy, or we need more measurements to confirm those data in large q 2 -region.
• As shown by Fig.3, the flip of sign for the Wilson coefficients C eff 7,8 also makes the isospin asymmetry of MSSM MFV-II a little different from the SM prediction in low q 2 -region. The LHCb data prefers a positive isospin asymmetry for q 2 → 0 which could be explained by the SM and the MSSM MFV-I models. However the LHCb data is still of large errors, thus at present, we can not draw definite conclusions on which scenario is preferable via using the present isospin asymmetry data.
Thus, we think the forward-backward and the isospin asymmetries of the B → K * µ + µ − decay are interesting observables to probe possible new physics beyond the SM. More accurate data, especially those in low q 2region, at the LHCb or the future super B-factory are important for clarifying this point.