New physics effects in purely leptonic $B^*_s$ decays

Recently several measurements in neutral current sector $b\rightarrow s \mu^+\mu^-$ as well as in the charged current sector $b \rightarrow c \tau \bar{\nu}$ show significant deviation from the Standard Model predictions. Two different new physics operators which can explain all measurements in $b\rightarrow s \mu^+\mu^-$ sector, have been identified previously. In order to discriminate between them, we study the longitudinal polarization asymmetry of final state muons in $B^*_s\rightarrow \mu^+\,\mu^-$ decay. We find that the muon longitudinal polarization asymmetry can be a good discriminant between the two new physics solutions if it can be measured to a precision of $10\%$. We also investigate the potential impact of $b \rightarrow c \tau \bar{\nu}$ anomalies on $b \rightarrow s \tau^+ \tau^-$ transitions. We consider a model where the NP contributions to these two transitions are strongly correlated and compute the branching ratio of $B^*_s\rightarrow \tau^+\,\tau^-$ and $\tau$ longitudinal polarization asymmetry in $B^*_s\rightarrow \tau^+\,\tau^-$. We find that the branching ratio can be boosted by three orders of magnitude. If the NP amplitude of $b \rightarrow s \tau^+ \tau^-$ is comparable to its Standard Model amplitude the $\tau$ longitudinal polarization asymmetry becomes a powerful tool to distinguish it.

Recently several measurements in neutral current sector b → sµ + µ − as well as in the charged current sector b → cτν show significant deviation from the Standard Model predictions. Different new physics operators which can explain all measurements in b → sµ + µ − sector, are identified previously. In order to discriminate between these new physics operators, we study the longitudinal polarization asymmetry of final state muons in B * s → µ + µ − decay. We find the muon longitudinal polarization asymmetry can be a good discriminant of new physics solution C N P 9 < 0 where C N P 9 is the Wilson coefficient corresponding to the operator (sγ µ PLb)(µγµµ). We also investigate the impact of b → cτν anomalies on the branching ratio of B * s → τ + τ − and τ longitudinal polarization asymmetry in B * s → τ + τ − . We find that these anomalies imply three orders of magnitude enhancement in the branching ratio of B * s → τ + τ − whereas a suppression in τ longitudinal polarization asymmetry as compared to their SM predictions.

I. INTRODUCTION
There are several measurements in B meson sector which do not agree with the predictions of the Standard Model (SM). The decays related to these measurements are induced by quark level transitions b → sµ + µ − and b → c lν.
The R K ( * ) anomaly can be due to new physics (NP) in either electron or muon sector or both whereas other anomalies are only related to the muon sector. It was shown in [13] that all anomalies in b → sµ + µ − sector can be accounted by considering NP only in the muon sector.
2. The measured value of R J/ψ = B(B→J/ψ τν) B(B→J/ψµν) by LHCb collaboration, is 1.7σ away from its SM prediction [15]. All these measurements are indication of LFU violation and hence towards presence of NP in b → c τν transition.
After the measurement of R K * , several groups [13,[16][17][18][19][20][21][22][23] performed global fits to identify the type of NP which can explain anomalies in the b → sµ + µ − sector. It was observed that there are several NP solutions. However, these solutions are all in the form of vector (V) and axial-vector (A) operators. Therefore one needs to look for new observables in the b → sµ + µ − sector in order to discriminate between several NP solutions. These observables can be related to the observed decay modes or can be associated with the decay modes yet to be observed, see for example [24].
The decay of B * s meson into di-muon is such a decay mode [25]. In [26] a model independent analyses of B * s → µ + µ − was performed to identify NP operators which can enhance the branching ratio of B * s → µ + µ − and to disentangle various NP solutions. But it was found that neither the branching ratio of B * s → µ + µ − can be enhanced nor any distinction between various NP solutions can be made. As B * s → µ + µ − decay mode is theoretically very clean, it would be desirable to construct a new observable related to this decay mode to see whether it has the potential to discriminate between the existing NP solutions in b → sµ + µ − sector.
In this work we introduce a new observable, the longitudinal polarization asymmetry of muon in B * s → µ + µ − decay, A LP (µ). This observable is even more theoretically clean as compared to the branching ratio of B * s → µ + µ − as it is independent of the total decay width of B * s meson which is not well determined either theoretically or experimentally. We first obtained the SM prediction of A LP (µ) and then study its sensitivity to the existing NP solutions in the b → sµ + µ − sector.
As far as b → cτν transition is concerned, ref. [27] identified NP solutions for R D ( * ) anomaly. In ref. [28] it was shown that there are only four independent NP solutions which can explain present data in the b → cτν sector. Methods to discriminate between these NP solutions were suggested in ref. [29]. An impact of R D ( * ) and R J/ψ measurements on several decays induced by b → s τ + τ − quark level transition was studied in ref. [30]. In this work we study the correlation between R D ( * ) and R J/ψ data and the branching ratio of B * s → τ + τ − and τ polarization asymmetry A LP (τ ).
This paper is organized as follows. In section II we obtain theoretical expressions for the longitudinal polarization asymmetry of final state leptons in B * s → l + l − decays, where l = e, µ or τ , in the SM as well as in the presence of NP V and A operators. In section III we obtain predictions of A LP (µ) in the SM as well as for various NP scenarios which provide a good fit to all b → sµ + µ − data. In the same section we study correlations between R D ( * ) and R J/ψ and branching ratio of B * s → τ + τ − and A LP (τ ). Finally in the section IV, we present our conclusions.

A. Longitudinal Polarization Asymmetry in the SM
The pure leptonic decay B * s → l + l − is induced by the quark level transition b → sl + l − . In the SM the effective Hamiltonian for b → s l + l − transition is given as where G F is the Fermi constant, V ts and V tb are the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements and . The B * s → l + l − amplitude can be parameterized in terms of the following form factors [25] where µ is the polarization vector of the B * s meson and f B * s and f T B * s are the decay constant of B * s meson. In the heavy quark limit they are related to f Bs , the decay constant of B s meson, with 0|sγ µ γ 5 b|B s (p Bs ) = −if Bs p µ Bs .
The SM amplitude for B * s → l + l − decay is given by This leads to the following decay rate We now define longitudinal polarization asymmetry for the final state leptons in B * s → l + l − decay. The unit longitudinal polarization four-vector in the rest frame of l + is defined as After implementation of Lorentz boost to the unit polarization vectors in the di-lepton rest frame (which is also the rest frame of B * s meson), we get where α is 4-vector Lorentz index and E l , − → p l and m l are the energy, momentum and mass of the l + respectively. The We compute each decay rates with different polarization combination of the final state leptons and those are given below.
where N = and ε 1234 is the Levi-Civita symbol. Using these, we get (11)

B. Longitudinal polarization asymmetry in presence of NP
We now investigate the lepton polarization asymmetry in the presence of NP. As the NP solutions to the b → sl + l − anomalies are in the form of V and A operators, we consider the addition of V and A NP operators to the SM effective Hamiltonian of b → sl + l − . Including the NP terms, the effective Hamiltonian is given by where H V A is Here C N P 9(10) and C N P 9(10) are the NP Wilson coefficients. Within this NP framework, the A LP for final state leptons in B * s → l + l − decay, using the definition of eq. (8), is obtained as We present our results in the next section.

III. RESULTS AND DISCUSSION
A. Impact of ALP (µ) in B * s → µ + µ − decay In this section first we will calculate A LP (µ) for B * s → µ + µ − decay. The SM prediction of muon longitudinal polarization asymmetry is A +  We now study A LP (µ) in the presence of NP operators. Several observables in the b → sµ + µ − sector are in disagreement with the SM. Assuming the NP is indeed present in b → sµ + µ − transition, ref. [13] identified three scenarios: (I) C N P 9 < 0, (II) C N P 9 = −C N P 10 < 0 and (III) C N P 9 = −C N P 9 < 0, as a possible explanation of all observed anomalies in this sector. In ref. [22], it was shown that though scenario (III) can explain all b → sµ + µ − data but it predicts R K = 1 which disagrees with measurement. Therefore one can exclude this third possibility. The predictions of A ± LP (µ) for remaining two scenarios are given in table II.  From this table it is obvious that the predictions of A LP (µ) for scenario (I) deviates from the SM at the level of 3.4 σ where as A LP (µ) for the scenario (II) is consistent with the SM. Hence any large deviation in A LP (µ) can only be due to NP scenario (I).

Scenario NP WCs
The prediction for the branching ratio of B * s → τ + τ − in the SM is whereas the SM prediction of τ longitudinal polarization asymmetry is obtained to be We now study the impact of NP on these two observables. We consider NP contribution to the Wilson coefficients of V and A operators only. Assuming that NP only affects left handed quarks and leptons, the Wilson coefficients of b → s τ + τ − transition can be written as [30] C τ τ 9 = C SM 9 − ∆, where ∆ = 2π is the error weighted average of current experimental values of R D , R D * and R J/ψ without taking the correlation into account, R X /R SM X 1.22 ± 0.06. The value of R X /R SM X with 1σ and 2σ ranges is indicated in fig. 1. We calculate B(B * s → τ + τ − ) and A LP (τ ) as a function of R X /R SM X . The plot of B(B * s → τ + τ − ) vs. R X /R SM X is shown in left panel of fig. 1. From this plot it is obvious that B(B * s → τ + τ − ) can be enhanced up to 10 −9 which is about three orders of magnitude above its SM prediction. Thus the present R D ( * ) ,J/ψ data imply large enhancement in the branching ratio of B * s → τ + τ − . We now consider A LP (τ ) as a function of R X /R SM X . The plot of A LP (τ ) w.r.t. R X /R SM X is shown in the right panel of fig. 1. It can be seen from the plot that A LP (τ ) is suppressed in comparison to its SM value. Hence the present anomalies in R D ( * ) ,J/ψ imply suppression in A LP (τ ).

IV. CONCLUSION
There are several measurements in the b → sµ + µ − and b → c lν sectors which do not agree with the SM. These discrepancies can be considered as signatures of NP. Many groups performed global fits to identify the Lorentz structure of the possible NP in b → sµ + µ − . However, there is no unique solution. Three scenarios (I) C N P 9 < 0, (II) C N P 9 = −C N P 10 < 0 and (III) C N P 9 = −C N P 9 < 0 provide a good fit to all b → sµ + µ − data. As scenario (III) predicts R K = 1, one can exclude this scenario. Therefore there are basically two solutions. Hence one needs to find new observables in order to distinguish between these two solutions. In this work we define a new observable, the muon longitudinal polarization asymmetry in B * s → µ + µ − decay. This observable is theoretically very clean. However its experimental measurement would be a challenging task. We show that any large deviation in the di-muon longitudinal polarization asymmetry, as compared to the SM, can only be due to NP scenario (I). s → τ + τ − ) and ALP (τ ) respectively. In both panels the yellow band represents 1σ range of these observables. In both panels, the 1σ and 2σ ranges of RX /R SM X are indicated by blue and pink bands respectively.
Further, we study the impact of R D ( * ) ,J/ψ measurements on the branching ratio of B * s → τ + τ − and A LP (τ ). We find that the present data in R D ( * ) ,J/ψ sector imply about three orders of magnitude enhancement in the branching ratio of B * s → τ + τ − whereas a suppression in A LP (τ ) as compared to their SM predictions. To measure A LP (µ) or A LP (τ ) in experiments, the final state leptons have to decay into secondary particles. But for muon it is quite difficult to measure as muon does not decay within the detector. In case of A LP (τ ), it may be possible for LHCb to reconstruct τ where τ decays into multiple hadrons. This technique is used to identify τ in B → D * τν decay. Therefore a precise reconstruction from the decay products of τ is necessary to measure τ longitudinal polarization asymmetry in B * s → τ + τ − .