LHCb anomaly in $\boldsymbol{B\to K^*\mu^+ \mu^-}$ optimised observables and potential of $\boldsymbol{Z^\prime}$ Model

Over the last few years LHCb with present energies found some discrepancies in $b\to s\ell^+\ell^-$ FCNC transitions including anomalies in the angular observables of $B\to K^*\mu^+\mu^-$, particularly in $P_5^\prime$, in low dimuon mass region. Recently, these anomalies are confirmed by Belle, CMS and ATLAS. As the direct evidence of physics beyond-the-SM is absent so far, therefore, these anomalies are being interpreted as indirect hint of new physics. In this context, we study the implication of non universal family of $Z^\prime$ model to the angular observables $P_{1,2,3}$, $P^{\prime}_{4,5,6}$ and newly proposed lepton flavor universality violation observables, $Q_{4,5}$, in $B\to K^*(\to K\pi)\mu^+ \mu^-$ decay channel in the low dimuon mass region. To see variation in the values of these observables from their standard model values, we have chosen the different scenarios of the $Z^\prime$ model. It is found that these angular observables are sensitive to the values of the parameters of $Z^\prime$ model. We have also found that with the present parametric space of $Z^\prime$ model, the $P_5^\prime$-anomaly could be accommodated. However, more statistics on the anomalies in the angular observables are helpful to reveal the status of the considered model and, in general, the nature of new physics.


Introduction
In flavor physics, study of rare B meson decays provide us a powerful tool to test standard model (SM) at loop level as well as to search possible new physics (NP). To search NP in rare decays of B-meson demands to focus on these observables which contain minimum hadronic uncertainties such that they can be predicted precisely in the SM and are available at current colliders. As in the exclusive rare B meson decays, the long distance physics encapsulated in the transition matrix elements that can be parameterized in terms of form factors. The main source of hadronic uncertainties comes from these form factors which are non-perturbative quantities and are difficult to compute and may preclude the signature of any possible NP. From this point of view, among all rare decays, the four body decay channel, B → K * (→ Kπ)µ + µ − , have a special interest in literature due to the fact that it gives a large variety of angular observables which are free from hadronic uncertainties and are available experimentally, for instance, optimal observables that are denoted in the literature as P i (i = 1, 2, 3) and P i (i = 4, 5, 6). The comparison between the theoretical predictions of these kind of observables in and beyond the SM with the experimental data could be helpful to clear some smog on beyond-SM physics.
The primed observables, introduced in ref. [1] and are free of form-factor uncertainties [2]. From experimental point of view, recently, LHCb analysis on these angular observables for B → K * (→ Kπ)µ + µ − indicates notably deviation from SM expectations, mainly 3.7σ discrepancy with data set of 1 fb −1 in the P 5 observable, at large recoil, in the s ∈ [4. 30,8.68] [3], where s is the dimuon invariant mass squared, q 2 . This discrepancy again seen at LHCb with a 3σ deviation with 3 fb −1 luminosity, comparatively, in two shorter adjacent bins s ∈ [4,6] [4] and s ∈ [6,8] which is also confirmed by Belle in the larger bin s ∈ [4,8] [6,7]. The very recent results from ATLAS [8] and CMS [9,10] collaborations are presented in Moriond 2017. With these results a state of the art global analysis, performed by the authors of ref. [11], show that there is a sizable discrepancy between data and SM. In addition, LHCb also found 2.6σ deviation from the SM prediction in the ratio R K = Br(B → Kµ + µ − )/Br(B → Ke + e − ) [12], and > ∼ 2σ in the Br(B s → φµ + µ − ) [13]. Interestingly, all these deviations belong to the rare B meson decays which is flavor changing neutral current (FCNC) transitions and at quark level it is shown as, b → s + − , where − denotes final state leptons. It is important to mention here that even angular observables are form factor independent (FFI) but for precise theoretical predictions of their values, one needs to incorporate the QCD corrections. There are two classes of QCD corrections contributing to B → meson transitions, namely, factorisable and non-factorisable. The factorisable means those contributions which comes through soft interaction of the spectator quarks and can be absorbed in hadronic form factors while the non-factorisable contributions arise from hard scattering of the process and do not belong to the form factors. In the present study, to determine the values of angular observables, we have incorporated both type of corrections in our numerical analysis available up to next-to-leading order (NLO) and the expressions of these contributions are given in Appendix B.
As these anomalies are slowly piled up, they received a considerable attention in the literature (see for instance [11,14]). Regarding these anomalies, another interesting debate recently appeared in the literature, whether they emerge from unknown factorisable power corrections or from NP. As the authors of Ref. [15] claim that these anomalies cannot be accommodate by large hadronic power corrections while the authors of Ref. [16] indicated that one cannot eliminate the choice that the current anomalies are partly outcome of underrated uncertainties in the form factor calculations which arise from power corrections. However, by keeping both of these arguments, global fit analysis [11] with present data, strongly pointed out that instead of this inconclusive situation, the interpretation of anomalies through the NP is a valid option.
To alleviate these anomalies, several NP models have been put forward [17][18][19][20][21][22]. From NP point of view, an extension in the SM consider to be good and simple which is economical and alleviate most of the existing anomalies simultaneously through a common mechanism in a consistent way. In this regard, the Z model is economical due to the fact that it requires only one extra U (1) gauge symmetry associate with the neutral gauge boson, called Z with the SM gauge group. The nature of couplings of the Z boson with the quarks and leptons leads the FCNC transitions to the tree level. This feature of the model arises that NP effects comes only through the short distance Wilson coefficients which are encapsulated in the new coefficients C tot 9 = C SM 9 + C Z 9 , C tot 10 = C SM 10 + C Z 10 , while operator basis remained unchange.
In addition, several previous studies on different observables of decay B → K * µ + µ − in which SM values deviate from their experimental measurements have already been discussed and shown a possible interpretation of these mismatch in terms of Z model [23][24][25][26][27][28].
It is natural to ask whether the Z model could explain the recently observed anomalies in the angular observables of the decay channel B → K * (→ Kπ)µ + µ − . With this motivation, in the current study, we have analyzed the optimal observables P 1,2,3 and P 4,5,6 , in the low dimuon mass region, for the B → K * (→ Kπ)µ + µ − in the SM and in the Z models. Besides this we also calculated the violation of lepton flavor universality (LFU) observables namely, Q 4(5) = P µ 4(5) − P e 4(5) [29]. For numerical calculations of these observables, we have used the light cone sum rules (LCSR) values of the hadronic form factors [30] and the numerical values of Z parameters are given in Tab. (5).
This paper is organized as follows: in section 2.1, we write the effective Hamiltonian for the b → s + − transition in the SM and the modified Hamiltonian after incorporating contributions that come through the Z boson. The B → K * matrix elements in terms of form factors and the expression of four differential decay distributions are also given in this section. Formulae for the angular observables and their analytical expressions are given in section 2.2. In section 3, we have plotted the angular observables and their average values against dimuon mass s and we have given phenomenological analysis of these observables. In the last section we conclude our work. Appendix A contains the values of input parameters needed for our numerical calculations. The factorisable and non-factorisable contributions at NLO are summarized in Appendix B.

Matrix Elements and Form Factors
In the standard model, FCNC processes occur at loop level and the amplitude of b → s + − can be written as following, where L, R = (1±γ 5 ), p K * and are momentum and polarization of K * meson, respectively, while p B is the momentum of B meson.
In the presence of Z the FCNC transitions could occur at tree level and the Hamiltion is written in the following form (see detail in the refs. [31][32][33][34] ) where, The B sb is the coupling of Z with quarks and B L , B R are left and right-handed couplings fo Z with leptons. One can notice from Eq. (2.3) that in the Z-prime model, operator basis remains the same as in the SM and C 9 and C 10 get modifications while C eff 7 remains unchanged. The total amplitude for the decay B → K * + − is the sum of SM and Z contributions, and can be written as follows, where C tot 9 = C eff 9 + Λ sb C Z 9 and C tot 10 = C SM 10 + Λ sb C Z 10 . The matrix elements of the quark operators for the decay amplitude for B → K * + − appears in Eq. (2.4) can be written in terms of form factors as follows where, Here A 0,1,2 (s), V (s), T 1,2,3 (s) are the form factors and contain hadronic uncertainties. At leading order by using the heavy quark limit, the QCD form factors follow the symmetry relations and can be expressed in terms of two universal form factors ξ ⊥ and ξ [35,36].
For the s dependence of the universal form factors, we use the following parameterization of light cone sum rule (LCSR) approach [30].
where the parameters r 1,2 , m 2 R and m 2 f it are listed in Tab. (1). The uncertainty in the universal form factors ξ ⊥ and ξ arises from the uncertainty in the different parameters using in LCSR approach which is about 11% and 14%, respectively, as discussed in [35]. At NLO, the relations between the T i (s) where (i = 1, 2, 3) and the invariant amplitudes T ⊥, (s), where T ⊥, = T − ⊥, , read as [37]. The four-fold differential decay distribution for the cascade decay B → K * (→ Kπ) + − is completely described by the four independent kinematical variables: the three angles; θ K * is the angle between the K and B mesons in the rest frame of K * , θ is the angle between lepton and B meson in the dilepton rest frame while φ is the azimuthal angle between the dilepton rest frame and K * rest frame. The fourth variable is dilepton invariant squared mass s. The explicit dependence of differential decay distribution on these kinematical variables can be expressed as follows (2.11) The full physical region phase space of kinematical variables is given by where m B , m K * , m are the masses of B meson, K * and lepton, respectively. The expressions of coefficients J i (s) for i = 1, ...., 9 and a = s, c as a function of the dilepton mass s, are given in Eq. (2.15). As we do not take the scalar contribution in this study, therefore, J c 6 = 0.

Expressions of the Angular Observables
The definitions of form factors independent optimal observables are given in ref. [14], .
(2.13) Table 1. The values of the fit parameters involved in the calculations of the form factors given in Eq.
The primed observables (related to the P i (i = 4, 5, 6)) which are simpler and more efficient to fit experimentally are defined as, (2.14) We get the following expressions of J i as follows, where g i (h i ), i = 1, · · · , 3 are the auxiliary functions and given as follows, where ∆ is given in Appendix B in Eq. (B.1). Traditionally, the J's are given in terms of transversity amplitudes but we have written in terms of g i (h i ) functions given in Eq. (2.16) A 0, ,⊥ . The A 0, ,⊥ are related with g i (h i ) as follows

Results and Discussion
In this section, we will present the numerical analysis of the angular observables. Before start the analysis, in the following we would like to write the different definitions of angular observables that are opted by LHCb [4] and theoretically used in the literature,

P -observables in different bin size
Our results for P -observables in the SM, in different Z scenarios and their comparison with maximum likelihood fit results of ref. [4] by LHCb in different bin size are summarized in Tab. (2) and graphically shown in Figs. (1) and (2) where black crosses are the data points taken from the last column of Tab. (2) and black dashed line correspond to the SM while green, red and blue bands correspond to the S 1 , S 2 and S 3 scenarios of the Z model, respectively. In our different bin sized analysis, we have not included the preliminary results from Belle [6,7], ATLAS [8] and CMS 1 [9, 10] because their bin intervals are different from LHCb [4] that we have discussed in this section. The continuous plots over s are presented with SM uncertainty band in light gray color. This uncertainty is mainly coming through parametric uncertainty including form factors, though these observables are FFI. Let us also explicitly mention that while performing these calculations using different form factors (e.g, [5]), it is indeed the case that these observables are form factors independent. One can see from the left panel of Figs. (1) and (2) that the uncertainty band in SM not preclude the effects of Z model. Keeping in mind this we have not provided the SM uncertainty bands for different bins in Tab. (2) and hence in the right panel of Figs. (1) and (2). However, for s ∈ [1, 6]GeV 2 , we do have listed in Tab. (3) and in Fig. (3). The plots in first and third rows of Fig. (1), represent the variation in the values of P 1,3 and their average values P 1,3 as a function of s in the SM and in the different scenarios of Z model. From these graphs one can see that the values of these observables are quite small in the SM and not much enhanced when we incorporate the Z effects. One can also see from Fig. (1) that the SM values of P 1 lie inside the measured values, however with huge error bars, so no potent result can be drawn from this observable with current data. On the other hand the values of P 3 in last two bins are within the measured values while in first two bins the SM values are out of the measured bars. However, to say something about any discrepancy in these observables, reduction in the experimental uncertainties are required.
Plots in second row of Fig. (1), show the variation in the values of P 2 and its average P 2 against dilepton mass s. It could be seen from these figures that the values of these observables are significantly influenced in the presence of Z effects. The right plot in the second row of Fig. (1) shows that the SM values of P 2 in the bins s ∈ [1.1, 2.5] and s ∈ [2.5, 4.0] lie within the measurements and also in the bin s ∈ [4.0, 6, 0] when the theoretical uncertainties of the input parameters are taken into account. However, in the first bin s ∈ [0.1, 0.98], the SM value of P 2 looks mismatch from the experimental value but it is worthy to mention here that the measurement performed by LHCb in this bin is without including the m − suppressed terms which are important at very low s region and it was found in [38], that the impact of these terms results is about 23% reductionn in the value of P 2 . Regarding this, it is mentioned in [15] that in the first bin, LHCb actually measured P 2 instead of P 2 . Therefore, in principle, one could say that, up-till now, there is no mismatch between the SM predicted values of P 2 with the experimental values.  Figure 1. The dependence of the optimal observables, P 1,2,3 and P 1,2,3 for the decay B → K * (→ Kπ)l + l − on s. The black dashed line correspond to the SM while green, blue and red bands correspond to the S 1 , S 2 and S 3 scenarios of the Z model, respectively.
In the first row plots of Fig. (2), we have displayed P 4 and it's average value P 4 in the SM and in the different scenarios of Z model as a function of s. One can see from these plots that the Z effects are quite significant in the P 4 values at low s region but mild at larger values of s. However, the SM values of P 4 in all four bins lie inside the experimentally measured values.
The results of P 5 and it's average value P 5 in the SM and in the Z models are presented in the second row plots of Figs. (2). The values are significantly changed from the SM values when we incorporate the Z effects. It can be noticed in the bin s = 4 to 6 GeV 2 , the SM average value P 5 mismatch with the experimental values and as mentioned in the introduction that LHCb found 2.6σ deviation in this bin. Regarding this discrepancy, our analysis shows that the average value P 5 in this bin could be adjusted with the measured value by using S 3 parametric space of Z model, as shown in right plot of second row of Fig. (2) by red band.
In the third row plots of Fig. (2), we have shown the variation of P 6 and P 6 as a function of s. Similar to P 1,3 , the SM value of this observable is also suppressed. As seen from the graph that SM value of P 6 consistent with the data with large error bars, however there is 2σ deviation in one bin s ∈ [1.1, 2.5] which probably will be disappear when data will increase. On the NP point of view, in contrast to the P 1,3 , the value of P 6 significantly enhanced in the Z model. It is also noticed that in the Z model the value of P 6 is positive in scenarios S 1 and S 2 while becomes negative in S 3 .

P -observables in
The results for P -observables in s ∈ [1.0, 6.0] GeV 2 are summarized in Tab. (3) and corresponding plots are shown in Fig. (3) where, magenta [6] and yellow [7] error bars correspond to available Belle measurements for some of these observables. It is worth to mention that LHCb results [4] are for s ∈ [1.1, 6.0] GeV 2 whereas, both Belle measurements [6,7] are for s ∈ [1.0, 6.0] GeV 2 . Moreover, recently the ATLAS collaboration [8] announced its results for s ∈ [0.04, 6.0] GeV 2 that we have not include in our analysis. The empty red box in P 2 and P 6 represents the S 3 scenario when we choose φ sb = −150 ± 10 given in Appendix A, while, other legends are same as in Figs. (1) and (2). From Fig. (3), one can notice immediately that the values of P 1 and P 4 in the SM and in all the three scenarios of Z lie within the current measurements. It is also noticed that the values of P 1 in the SM and in the Z scenarios are very close, consequently, this observable even after the reduction of error bars not a good candidate to constrained the Z parametric space. On the other hand P 4 could be helpful to constrained the Z parametric space, if any mismatch will appear in future in the bin [1,6] GeV 2 . The P 3 has small values in the SM and in all the three scenarios of considered model while measured value is well above the prediction, however, more data is required to cast any result from this observable. In the plots of P 2 and P 6 , one can deduced that the SM value of P 2 not lie within the measured error bars while the SM value of P 6 lie within the measurements, however for both observables, values in S 1 and S 2 lie also within the measurements while the values in S 3 lie outside the measured values (see red bands in both figures). Regarding S 3 , it is interesting to check whether the value in this scenario could be  Figure 2. The dependence of the optimal observables, P 4,5,6 and P 4,5,6 for the decay B → K * (→ Kπ)l + l − on s, the legends are same as in Fig. (1).
reduced to accommodate with current measurement, for this purpose, we choose S 3 with the opposite sign of φ sb angle i.e, given in the parenthesis of Tab. 5 in Appendix A. This is indeed the case as one can see from the empty red boxes in these two plots. In P 2 the empty red box is close to measurements as compare to red band. Similarly, in P 6 red box is within the Belle measurements [6]. Therefore, more statistics on the observables P 2 and P 6 are helpful to constrained the Z parameters, particularly, the sign and the magnitude of new weak phase φ sb .
For P 5 plots of Fig. (3), the values in the SM and in S 1 , S 2 lie out side the error basrs of experimental data points while the values in the S 3 well inside the all data points shown in figure. In general, from the plots of Fig. (3), one concludes that the considered model do have potential to remove mismatch between theory and experiment but it is not so conclusive at present. We hope more precise measurements will clear the situation. Table 3. Results for P -observables for s ∈ [1.0, 6.0] GeV 2 and their comparison with LHCb maximum likelihood fit results of ref. [4] in different bin size, Belle results [6,7]. 3.3 Q 4,5 for s ∈ [1.0, 6.0] GeV 2 In Fig. (4), we have plotted the lepton flavor universality violation (LFUV) observables Q 4(5) against s. The values are quite small in the SM approximately Q 4(5) = 8.8 ± 2.1 × 10 −3 (7.5 ± 3.6 × 10 −3 ) in the bin s ∈ [1, 6]GeV 2 . We have also found that the effects of Z are very mild, consequently, there is no enhancement in the values of these observable. However, error bars are quite large and need more experimental data to find the accurate values of these observables.

Conclusion
In the present study, we have calculated the FFI observables P i and their average values P i in the SM and in the Z model for B → K * (→ Kπ) + − . The expressions of these observables are given in the form of J i (s) which are written in terms of auxiliary functions g i (h i ) in Eq. (2.15). These coefficients, in general, expressed via transversity amplitudes that are A ⊥ , A and A 0 , the relations of these transversity amplitudes with g i (h i ) are given in Eq. (2.18). To see the Z effects, we use the UtFit collaboration constraints for the Z parameters. It is found that the values of these angular observables are significantly changed from their SM values in the presence of Z , particularly, at small values of s, i.e the large recoil region. We have also found that the SM values of some of the observables under consideration, in some bins of s, are mismatch with the recent experimental result and the Z parametric space has potential to accommodate these mismatch. For instance, there is a discrepancy between experimentally measured value and SM value of P 5 in the region s ∈ [4, 6] GeV 2 and in the current study it is found that scenario S 3 of Z could be adjust this mismatch value with the measured value in this bin. Furthermore, we have also calculated the angular observables P i and the LFUV observables Q 4,5 in the large bin s ∈ [1,6] and plotted with the measured data, however, the error bar is quite large in this bin and more static is needed to draw results. Here, we would like to comment that CMS and ATLAS collaborations recently announced preliminary results on angular observables  Figure 3. Optimal observables for s ∈ [1.0, 6.0] GeV 2 where, magenta [6] and yellow [7] error bars correspond to Belle measurements available for some of these observables. The empty red box in P 2 and P 6 represents the S 3 when we choose φ sb = −150 ± 10 given in Tab in Moriond 2017 which still show the tension between experimental measurements and the SM predictions. Therefore, in general, one can say, as data will be enlarge and the statistical error will be reduced then these observables are quite promising to say something about the constraints on coupling of Z boson with the quarks and leptons and consequently about the status of Z model.  Figure 4. Optimal observables Q 4 , Q 5 for s ∈ [1.00, 6.00] GeV 2 where, yellow error bar corresponds to recent Belle measurements [7]. Other legends are same as in previous figures.

A Input parameters
The values of Wilson coefficients at NNLO, Z parameters and other input parameters are listed in Tabs. (4), (5) and (6), respectively.

B NLO contributions in the low dilepton mass limit
The expression of ∆ , appear in the definition of a 0 below Eq. (2.17), written as follows and contributes only for massive leptons. The light-cone distribution amplitude (LCDA) ΦK * ,a for transversely (a =⊥) and longitudinally (a = ) polarized K * can be written as [37,43] ΦK * ,a = 6u where L = −(m 2 b − s)/s ln 1 − s/m 2 b and a i K * a are the Gegenbauer coefficients. The moments are where Φ B,± are the two B-meson light-cone distribution amplitudes [37]. The λ −1 B,− (s) can be expressed as: where ω 0 = 2(m B − m b ). The ξ a are the universal form factors, 3) The B → K * matrix elements in heavy quark limit depend on four independent functions T ± a (a =⊥, ). In the low s, (1.0 < s < 6.0 GeV 2 ), the invariant amplitudes T ⊥, at NLO within QCDf are given in [35,37,42], where Ξ ⊥ ≡ 1, Ξ ≡ m K * /E K * and the factorization scale µ f = m b Λ QCD . The coefficient functions C a and hard scattering functions T a,± are written as The form factor terms C  where h(s, m q ) is well-known fermionic loop function. The coefficients C a at NLO is divided into a factorizable and a non-factorizable part as The non-factorizable corrections are, 2 − C eff 8 F 2 + 2C 1 F , 2 + C eff 8 F 2 + 2C 1 F , where ∆M depends on the mass renormalization convention for m b . These corrections are obtained from the matrix elements of four-quark and chromomagnetic dipole operators [37] that are embedded in F (7,9) 1,2 and F (7,9) 8 [45,46].
At LO the hard-spectator scattering term T (0) a,± (u, ω) from weak annihilation diagram is [37] T The t a (u, m q ) functions are given by where B 0 and I 1 are