LHCb anomaly in B→ K ∗ μ + μ − optimised observables and potential of Z ′ Model
LLHCb anomaly in B → K ∗ µ + µ − optimised observables and potential of Z (cid:48) Model
Ishtiaq Ahmed ∗ , Abdur Rehman † National Centre for Physics, Quaid-i-Azam University Campus, Islamabad, 45320 Pakistan (Dated: April 10, 2018)Over the last few years LHCb with present energies found some discrepancies in b → s(cid:96) + (cid:96) − FCNCtransitions including anomalies in the angular observables of B → K ∗ µ + µ − , particularly in P (cid:48) , inlow 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 arebeing interpreted as indirect hint of new physics. In this context, we study the implication of nonuniversal family of Z (cid:48) model to the angular observables P , , , P (cid:48) , , and newly proposed leptonflavor universality violation observables, Q , , in B → K ∗ ( → Kπ ) µ + µ − decay channel in the lowdimuon mass region. To see variation in the values of these observables from their standard modelvalues, we have chosen the different scenarios of the Z (cid:48) model. It is found that these angularobservables are sensitive to the values of the parameters of Z (cid:48) model. We have also found thatwith the present parametric space of Z (cid:48) model, the P (cid:48) -anomaly could be accommodated. However,more statistics on the anomalies in the angular observables are helpful to reveal the status of theconsidered model and, in general, the nature of new physics. I. INTRODUCTION
In flavor physics, the study of rare B meson decaysprovide us a powerful tool, not only to test the standardmodel (SM) at loop level but also to search the possi-ble new physics (NP). searching of NP in rare decays ofB-meson demands to focus on those observables whichcontain minimum hadronic uncertainties such that theycan be predicted precisely in the SM and are availableat current colliders. In exclusive rare B meson decays,the main source of hadronic uncertainties come from theform factors which are non-perturbative quantities andare difficult to compute. In addition, these uncertaintiesmay preclude the signature of any possible NP. From thispoint of view, among all rare decays, the four body decaychannel, B → K ∗ ( → Kπ ) µ + µ − , have a special interestin literature due to the fact that it gives a large varietyof angular observables, namely, P i ( i = 1 , ,
3) and P (cid:48) i ( i = 4 , ,
6) [1] which are free from hadronic uncertainties[2]. The comparison between the theoretical predictionsof these kind of observables in the SM with the experi-mental data could be helpful to clear some smog on thephysics beyond the SM.From experimental point of view, few years back,LHCb measured the values of these angular observablesfor the decay channel B → K ∗ ( → Kπ ) µ + µ − . Thesemeasurements found a 3.7 σ deviation in the value of P (cid:48) ,with 1 fb − luminosity in the s ∈ [4 . , .
68] GeV bin[3]. Recently, this discrepancy again seen at LHCb witha 3 σ deviation with 3 fb − luminosity in comparativelytwo shorter adjacent bins s ∈ [4 ,
6] GeV [4] and s ∈ [6 , which is also confirmed by Belle in the larger bin s ∈ [4 ,
8] GeV [6, 7]. The very recent results from AT-LAS [8] and CMS [9, 10] collaborations, presented in ∗ [email protected] † [email protected] Moriond 2017, are also confirmed this discrepancy. Fur-thermore, LHCb also found 2.6 σ deviation in the valueof R K = Br( B → Kµ + µ − ) / Br( B → Ke + e − ) [12], and > ∼ σ in the Br( B s → φµ + µ − ) [13]. Interestingly, allthese deviations belong to the flavor changing neutralcurrent (FCNC) transitions, b → s(cid:96) + (cid:96) − , where (cid:96) − de-notes the final state leptons.The anomalies, mentioned above, are slowly piled upand received a considerable attention in the literature(see for instance [11, 14]). It is also important to men-tion here that even the angular observables are form fac-tor independent (FFI) but for precise theoretical predic-tions, one needs to incorporate the factorizable and non-factorizable QCD corrections. The factorizable correc-tions absorb in the hadronic form factors while the non-factorizable corrections arise from hard scattering of theprocess and do not belong to the form factors. In this re-spect, there are some studies which focus to the questionwhether these anomalies emerge from unknown factor-izable power corrections or from NP [15, 16]. However,global fit analysis with present data, strongly pointed outthat the interpretation of mentioned anomalies throughthe NP is a valid option [11]. In the present study, todetermine the values of angular observables, we have in-cluded both type of corrections up to next-to-leading or-der (NLO) and their expressions are given in AppendixB.From NP point of view, several extensions of the SMhave been put forwarded [17–25]. Among these, the Z (cid:48) model is economical due to the fact that besides theSM gauge group, it requires only one extra U (1) (cid:48) gaugesymmetry associated with a neutral gauge boson, called Z (cid:48) . The nature of couplings of the Z (cid:48) boson with thequarks and leptons leads the FCNC transitions to thetree level. In this model, the NP effects comes onlythrough the short distance Wilson coefficients which areencapsulated in the new coefficients C tot9 = C SM9 + C Z (cid:48) , C tot10 = C SM10 + C Z (cid:48) , while operator basis remained un-change. a r X i v : . [ h e p - ph ] A p r Several previous studies shown a possible interpreta-tion to alleviate the mismatch between the experimentaldata of different observables for the decay B → K ∗ µ + µ − and their SM predictions in terms of Z (cid:48) model [26–31]without any conflict. Therefore, it is natural to askwhether the Z (cid:48) model could explain the recently observedanomalies in the angular observables of the decay chan-nel B → K ∗ ( → Kπ ) µ + µ − . With this motivation, in thecurrent study, we have analyzed the optimal observables P , , and P (cid:48) , , , in the low dimuon mass region, for the B → K ∗ ( → Kπ ) µ + µ − in the SM and in the Z (cid:48) model.Besides these observables, we have also calculated theviolation of lepton flavor universality (LFU) observablesnamely, Q = P (cid:48) µ − P (cid:48) e [32]. For numerical cal-culations of these observables, we have used the LCSRvalues of the hadronic form factors [33] and for Z (cid:48) pa-rameters, we have used the Utfit collaboration values,called as S , S and another different scenario, called S which numerical values are listed in Tab. (V).We would like to mention here that the consideredscenarios labeled as S , S and S have same couplingstructure of the Z (cid:48) boson with the quarks and the lep-tons. However, the underlying difference between thesescenarios is related to the different fit values of parame-ters such as new weak phase and couplings of Z (cid:48) model,for considered decay process, available in the literature.For example, by using the all available experimental dataon B s − ¯ B s mixing, Utfit collaboration has found two so-lutions of new weak phase, φ sb , that arises due to themeasurement ambiguities in the data and referred as S and S . Similarly, another possible constraint on param-eters of Z (cid:48) model is discussed in [49] that, hereafter, welabel as S .This paper is organized as follows: Section II A, con-tains the effective Hamiltonian for the b → s(cid:96) + (cid:96) − tran-sition in the SM and in the Z (cid:48) model. The B → K ∗ matrix elements in terms of form factors and the expres-sion of differential decay distributions are also given inthis section. Formulae for the angular observables in sec-tion II B. In section III, we have plotted the angular ob-servables and their average values against dimuon mass s and we have given phenomenological analysis of these ob-servables. In the last section we conclude our work. Ap-pendix A contains the analytical expressions of the angu-lar observables and the values of input parameters. Thecontributions of factorizable and non-factorizable correc-tions at NLO are summarized in Appendix B. II. FORMULATION FOR THE ANALYSISA. Matrix Elements and Form Factors
In the standard model, FCNC transition, b → s(cid:96) + (cid:96) − ,occurs at loop level which amplitude can be written as, M SM ( b → s(cid:96) + (cid:96) − ) = − α G F √ π V tb V ∗ ts × (cid:26) (cid:104) K ∗ ( p K ∗ , (cid:15) ) | ¯ sγ µ Lb | B ( p B ) (cid:105) ( C eff9 ¯ (cid:96)γ µ (cid:96) + C SM10 ¯ (cid:96)γ µ γ (cid:96) ) − m b C eff7 (cid:104) K ∗ ( p K ∗ , (cid:15) ) | ¯ siσ µν q ν q Rb | B ( p B ) (cid:105) ¯ (cid:96)γ µ (cid:96) (cid:27) , (1)where L, R = (1 ∓ γ ), p K ∗ and (cid:15) are momentum andpolarization of K ∗ meson, respectively, while p B is themomentum of B meson.In the presence of Z (cid:48) the FCNC transitions could occurat tree level and the Hamiltion can be written in thefollowing form (see detail in the refs. [34–37] ) H Z (cid:48) eff = − G F √ V tb V ∗ ts (cid:104) Λ sb C Z (cid:48) O + Λ sb C Z (cid:48) O (cid:105) , (2)where, Λ sb = 4 πe − iφ sb α em V tb V ∗ ts , C Z (cid:48) = |B sb | S LR(cid:96)(cid:96) , and C Z (cid:48) = |B sb | D LR(cid:96)(cid:96) with, S LR(cid:96)(cid:96) = B L(cid:96)(cid:96) + B R(cid:96)(cid:96) , D LR(cid:96)(cid:96) = B L(cid:96)(cid:96) − B
R(cid:96)(cid:96) . (3)The B sb is the coupling of Z (cid:48) with quarks and B L(cid:96)(cid:96) , B R(cid:96)(cid:96) are left and right-handed couplings fo Z (cid:48) with leptons.One can notice from Eq. (3) that in the Z (cid:48) model, oper-ator basis remains the same as in the SM while Wilsoncoefficients, C and C , get modified. The total ampli-tude for the decay B → K ∗ (cid:96) + (cid:96) − is the sum of SM and Z (cid:48) contributions, and can be written as follows, M tot ( B → K ∗ (cid:96) + (cid:96) − ) = − α G F √ π V tb V ∗ ts × (cid:26) (cid:104) K ∗ ( p K ∗ , (cid:15) ) | ¯ sγ µ Lb | B ( p B ) (cid:105) ( C tot9 ¯ (cid:96)γ µ (cid:96) + C tot10 ¯ (cid:96)γ µ γ (cid:96) ) − m b C eff7 (cid:104) K ∗ ( p K ∗ , (cid:15) ) | ¯ siσ µν q ν q Rb | B ( p B ) (cid:105) ¯ (cid:96)γ µ (cid:96) (cid:27) , (4)where C tot9 = C eff9 + Λ sb C Z (cid:48) and C tot10 = C SM10 + Λ sb C Z (cid:48) .The matrix elements for B → K ∗ transition, appearsin Eq. (4), can be written in terms of form factors asfollows (cid:104) K ∗ ( p K ∗ , (cid:15) ) | ¯ sγ µ Lb | B ( p B ) (cid:105) = − iq µ m K ∗ s (cid:15) ∗ · q × (cid:20) A ( s ) − A ( s ) (cid:21) − (cid:15) µνλσ (cid:15) ∗ ν p λK ∗ q σ V ( s )( m B + m K ∗ )+ i(cid:15) ∗ µ ( m B + m K ∗ ) A ( s ) ∓ i ( p B + p K ∗ ) µ (cid:15) ∗ · q A ( s )( m B + m K ∗ ) , (cid:104) K ∗ ( p K ∗ , (cid:15) ) | ¯ siσ µν q ν Rb | B ( p B ) (cid:105) = 2 (cid:15) µνλσ (cid:15) ∗ ν p λK ∗ q σ T ( s )+ i(cid:15) ∗ · q (cid:26) q µ − ( p B + p K ∗ ) µ s ( m B − m K ∗ ) (cid:27) T ( s )+ i (cid:26) (cid:15) ∗ µ ( m B − m K ∗ ) − ( p B + p K ∗ ) µ (cid:15) ∗ · q (cid:27) T ( s ) , (5)where, A ( s ) = m B + m K ∗ m K ∗ A ( s ) − m B − m K ∗ m K ∗ A ( s ) . (6)Here A , , ( s ), V ( s ), T , , ( s ) are the form factors andcontain hadronic uncertainties. At leading order by usingthe heavy quark limit, the QCD form factors follow thesymmetry relations and can be expressed in terms of twouniversal form factors ξ ⊥ and ξ (cid:107) [38, 39]. ξ ⊥ = m B m B + m K ∗ V,ξ (cid:107) = m B + m K ∗ E K ∗ A − m B − m K ∗ m B A . (7)It is also important to mention here that the angular ob-servables are soft form factor independent at LO in α s (i.e., not totally dependent of FF). There is residual de-pendence has been discussed, computed systematicallyand included in the predictions of the main papers ofthe field and even if, as expeted, does not play an im-portant role induce certain mild dependence on FF. Inaddition, for the s dependence of the universal form fac-tors there are different parametrization [5], however, wehave analyzed that the choice of parametrization is notso important at low s . In the current study, we use thefollowing parametrization of LCSR approach [33]. V ( s ) = r − s/m R + r − s/m fit , A ( s ) = r − s/m fit ,A ( s ) = r − s/m fit + r (1 − s/m fit ) , (8)where the parameters r , , m R and m fit are listed in Tab.(I). The uncertainty in the universal form factors ξ ⊥ and ξ (cid:107) arises from the uncertainty in the different parametersusing in LCSR approach which is about 11% and 14%,respectively, as discussed in [38]. At NLO, the relations TABLE I:
The values of the fit parameters involved in the cal-culations of the form factors given in Eq. (8) [33]. r r m R (GeV ) m fit (GeV ) V ( s ) 0.923 -0.511 28.30 49.40 A ( s ) 0.290 40.38 A ( s ) -0.084 0.342 52.00 between the T i ( s ) where ( i = 1 , ,
3) and the invariantamplitudes T ⊥ , (cid:107) ( s ), where T ⊥ , (cid:107) = T −⊥ , (cid:107) , read as [40]. T ( s ) = T ⊥ , T ( s ) = 2 E K ∗ m B T ⊥ , T ( s ) = T ⊥ + T (cid:107) , (9) where E K ∗ = ( m B + m K ∗ − s ) / m B is the energy of kaonin the rest frame of B -meson and T ⊥ , (cid:107) ( s ) are defined inEq. (B4) of Appendix B.The four-fold differential decay distribution for the cas-cade decay B → K ∗ ( → Kπ ) (cid:96) + (cid:96) − is completely describedby the four independent kinematical variables: the threeangles; θ K ∗ is the angle between the K and B mesonsin the rest frame of K ∗ , θ (cid:96) is the angle between leptonand B meson in the dilepton rest frame while φ is theazimuthal angle between the dilepton rest frame and K ∗ rest frame and the fourth variable is dilepton invariantsquared mass s . The explicit dependence of differentialdecay distribution on these kinematical variables can beexpressed as follows d Γ ds d cos θ (cid:96) d cos θ K ∗ dφ = 932 π (cid:101) Γ ( s, θ (cid:96) , θ K ∗ , φ ) , (10)where (cid:101) Γ ( s, θ (cid:96) , θ K ∗ , φ ) = J s sin θ K ∗ + J c cos θ K ∗ + (cid:0) J s sin θ K ∗ + J c cos θ K ∗ (cid:1) cos 2 θ (cid:96) + J sin θ K ∗ sin θ (cid:96) cos 2 φ + J sin 2 θ K ∗ sin 2 θ (cid:96) cos φ + J sin 2 θ K ∗ sin θ (cid:96) cos φ + ( J s sin θ K ∗ + J c cos θ K ∗ ) cos θ (cid:96) + J sin 2 θ K ∗ sin θ (cid:96) sin φ + J sin 2 θ K ∗ sin 2 θ (cid:96) sin φ + J sin θ K ∗ sin θ (cid:96) sin 2 φ . (11) The full physical region phase space of kinematical vari-ables is given by4 m (cid:96) (cid:54) s (cid:54) ( m B − m K ∗ ) , (cid:54) θ (cid:96) (cid:54) π, (cid:54) θ K ∗ (cid:54) π, (cid:54) φ (cid:54) π, (12)where m B , m K ∗ , m (cid:96) are the masses of B meson, K ∗ meson and lepton, respectively.The expressions of coefficients J ( a ) i = J ( a ) i ( s ) for i =1 , ...., a = s, c as a function of the dilepton mass s ,are given in Appendix A in Eq. (A1). As we do not takethe scalar contribution in this study, therefore, J c = 0. B. Expressions of the Angular Observables
The definitions of FFI angular observables (optimalobservables) are given in ref. [14], P ( s ) = J J s , P ( s ) = β (cid:96) J s J s , P ( s ) = − J J s ,P ( s ) = √ J (cid:112) − J c (2 J s − J ) , P ( s ) = β (cid:96) J (cid:112) − J c (2 J s + J ) ,P ( s ) = − β (cid:96) J (cid:112) − J c (2 J s − J ) . (13)The primed observables (related to the P i ( i = 4 , , (cid:45) (cid:45) (cid:45) (cid:45) (cid:72) GeV (cid:76) P (cid:45) (cid:45) (cid:45) (cid:72) GeV (cid:76) (cid:60) P (cid:62) (cid:45) (cid:45) (cid:72) GeV (cid:76) P (cid:45) (cid:45)
101 s (cid:72)
GeV (cid:76) (cid:60) P (cid:62) (cid:45) (cid:45) (cid:72) GeV (cid:76) P (cid:72) GeV (cid:76) (cid:60) P (cid:62) FIG. 1:
The dependence of the optimal observables, P , , and (cid:104) P , , (cid:105) for the decay B → K ∗ ( → Kπ ) l + l − on s . The black dashedline correspond to the SM while green, blue and red bands correspond to the S , S and S scenarios of the Z (cid:48) model, respectively. are defined as, P (cid:48) ≡ P (cid:112) − P = J (cid:112) − J c J s ,P (cid:48) ≡ P (cid:112) P = J (cid:112) − J c J s ,P (cid:48) ≡ P (cid:112) − P = − J (cid:112) − J c J s . (14) III. RESULTS AND DISCUSSION
In this section, we will present the numerical analysis ofthe angular observables. The authors would like to men-tion here that all of the numerical results are taken fromthe self-written
Mathematica code. Before the analysis,we would like to write the different definitions of angularobservables that are opted by LHCb [4] and theoreticallyused in the literature, P exp2 = − P , P exp3 = − P , P (cid:48) exp4 = − P (cid:48) ,P (cid:48) exp6 = − P (cid:48) , P exp1 = P , P (cid:48) exp5 = P (cid:48) . (15) (cid:45) (cid:72) GeV (cid:76) P (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:72) GeV (cid:76) (cid:60) P (cid:62) (cid:45) (cid:45) (cid:72) GeV (cid:76) P (cid:45) (cid:45) (cid:72) GeV (cid:76) (cid:60) P (cid:62) (cid:45) (cid:45) (cid:72) GeV (cid:76) P (cid:45) (cid:45) (cid:72) GeV (cid:76) (cid:60) P (cid:62) FIG. 2:
The dependence of the optimal observables, P (cid:48) , , and (cid:104) P (cid:48) , , (cid:105) for the decay B → K ∗ ( → Kπ ) l + l − on s , the legends aresame as in Fig. (1). For the numerical analysis the values of LCSR formfactors, and relevant fit parameters are listed in Tab. (I).The values of Wilson coefficients and other input param-eters are listed in Appendix A in Tabs. (IV) and (VI), re-spectively. Regarding the coupling parameters of Z (cid:48) withquarks and leptons, there are some severe constraintscome from different inclusive and exclusive B − mesonchannels [42]. Particularly, coming from the two differ-ent fitting values for B s − ¯ B s mixing data by the UTFitcollaboration[43]. In this study, we call these two fittingvalues as S and S and their numerical values are listedin Tab.(V). We have considered another scenario whichdenoted by S in the present study that are obtained fromthe analysis of B → X s µ + µ − [45], B → K ∗ µ + µ − [46, 47] and B → µ + µ − [48]. The numerical values of scenario S Z (cid:48) parameters could accomodate theanomalies in the angular observables, particularly, in P (cid:48) . A. P -observables in different bin size The numerical values of angular observables in differ-ent low s bins in SM and in S , S and S are given inTab. (II). For comparison with experimental measure-ments, the maximum likelihood fit results of LHCb [4]are also given in the table. The ranges in the values of - - B ℓℓ L B ℓℓ R FIG. 3: Gray dots represent the left (right) couplings, ( B L(cid:96)(cid:96) , B R(cid:96)(cid:96) ), of Z (cid:48) with leptons in S while the red dots show thevalues of these couplings after accommodating the P (cid:48) anomalyin the s ∈ [4 . , .
0] GeV angular observables in S , S and S is found by set-ting the upper and lower values of parametric space ofthese scenarios. These results are also shown graphicallyin Figs. (1) and (2) where black crosses are the datapoints taken from the last column of Tab. (II) and blackdashed line correspond to the SM while green, red andblue bands correspond to the S , S and S scenarios ofthe Z (cid:48) model, respectively. The upper curve of the bandcorresponds the upper values of parametric space whilethe lower curve of the band corresponds the lower val-ues of parametric space of the scenario. In our differentbin sized analysis, we have not included the preliminaryresults from Belle [6, 7], ATLAS [8] and CMS [9, 10] be-cause their bin intervals are different from LHCb [4] thatwe have discussed in this section. In Fig. (1), the grayshaded region corresponds to the uncertainty in the SMvalues due to the uncertainty in different input parame-ters. One can see from the left panel of Figs. (1) and (2)that the uncertainty band in SM not preclude the effectsof Z (cid:48) model. Therefore, we have not provided the SMuncertainty in Tab. (II) and hence in the right panel ofFigs. (1) and (2).The plots in first and third rows of Fig. (1), representthe variation in the values of P , and their average values (cid:104) P , (cid:105) as a function of s in the SM and in the differentscenarios of Z (cid:48) model. From these graphs one can seethat the values of these observables are quite small inthe SM and not much enhanced when we incorporate the Z (cid:48) effects. One can also see from Fig. (1) that the SMvalues of (cid:104) P (cid:105) lie inside the measured values. As the errorin the measurement is huge, therefore, no potent resultcan be drawn from this observable with the current data.On the other hand the values of (cid:104) P (cid:105) in last two bins arewithin the measured values while in first two bins theSM values are out of the measured bars. However, to say see figure 6 of [11] for the recent analysis with these new results. something about any discrepancy in these observables,reduction in the experimental uncertainties are required.Plots in second row of Fig. (1), show the variationin the values of P and its average (cid:104) P (cid:105) against dilep-ton mass s . It could be seen from these figures that thevalues of these observables are significantly influenced inthe presence of Z (cid:48) effects. The right plot in the secondrow of Fig. (1) shows that the SM values of (cid:104) P (cid:105) in thebins s ∈ [1 . , .
5] and s ∈ [2 . , .
0] lie within the mea-surements and also in the bin s ∈ [4 . , ,
0] when the the-oretical uncertainties of the input parameters are takeninto account. However, in the first bin s ∈ [0 . , . (cid:104) P (cid:105) looks mismatch from the experimentalvalue. But it is worthy to mention here that the measure-ment performed by LHCb in this bin is without includingthe m (cid:96) − suppressed terms which are important at verylow s region and it was found in [41], that the impactof these terms is about 23% reduction in the value of (cid:104) P (cid:105) . Regarding this, it is mentioned in [15] that in thefirst bin, LHCb actually measured (cid:104) ˆ P (cid:105) instead of (cid:104) P (cid:105) .Therefore, in principle, one could say that, up-till now,there is no mismatch between the SM predicted values of (cid:104) P (cid:105) with the experimental values.In the first row of Fig. (2), we have displayed P (cid:48) andit’s average value, (cid:104) P (cid:48) (cid:105) , in the SM and in the differentscenarios of Z (cid:48) model as a function of s . One can seefrom these plots that the Z (cid:48) effects are quite significantin the P (cid:48) values at low s region but mild at larger valuesof s . However, the SM values of (cid:104)P (cid:48) (cid:105) in all four bins lieinside the measured values.The results of P (cid:48) and it’s average value (cid:104) P (cid:48) (cid:105) in the SMand in the Z (cid:48) models are presented in the second row ofFigs. (2). The values are significantly changed from theSM values when we incorporate the Z (cid:48) effects. It canbe noticed in the bin s = 4 to 6 GeV , the SM averagevalue (cid:104) P (cid:48) (cid:105) mismatch with the experimental values andas mentioned in the introduction that LHCb found 3 σ deviation in this bin. It could be seen from the figurethat this discrepancy can be alleviated by S (red band)of Z (cid:48) model. On the other hand for the Utfit scenarios,namely, S and S it can be noticed that when we takethe upper and lower limit values of the current paramet-ric space of these scenarios (green and blue bands), the P (cid:48) anomaly in the bin s ∈ [4 , can not be accom-modated. However, if the values of different parametersare chosen randomly within the allowed range then onecould accommodate the P (cid:48) anomaly in this bin by S butnot with S . Therefore, it looks that the S of Utfit isnot consistent with the present data while the parametricspace of S , the left (right) couplings, ( B L(cid:96)(cid:96) , B R(cid:96)(cid:96) ), of Z (cid:48) with leptons is severely constraint as shown in Fig. (3).In the third row of Fig. (2), we have shown the varia-tion of P (cid:48) and (cid:104) P (cid:48) (cid:105) as a function of s . Similar to P , , theSM value of this observable is also suppressed. As seenfrom the graph that SM value of P (cid:48) consistent with thedata with large error bars, however there is 2 σ deviationin one bin s ∈ [1 . , .
5] which probably will be disappearwhen data will increase. One can also notice that in con- (cid:45) (cid:45) (cid:72)
GeV (cid:76) (cid:60) P (cid:62) (cid:45) (cid:45) (cid:72) GeV (cid:76) (cid:60) P (cid:62) (cid:72) GeV (cid:76) (cid:60) P (cid:62) (cid:45) (cid:45) (cid:45) (cid:72) GeV (cid:76) (cid:60) P (cid:62) (cid:45) (cid:45) (cid:72) GeV (cid:76) (cid:60) P (cid:62) (cid:45) (cid:45) (cid:45) (cid:72) GeV (cid:76) (cid:60) P (cid:62) FIG. 4:
Optimal observables for s ∈ [1 . , . GeV where, magenta [6] and yellow [7] error bars correspond to Belle measurementsavailable for some of these observables. The empty red box in (cid:104) P (cid:105) and (cid:104) P (cid:48) (cid:105) represents the S when we choose φ sb = − ± given in Tab. V of Appendix B. Other legends are same as in Figs. (1) and (2). trast to the P , , the value of P (cid:48) significantly enhancedin the Z (cid:48) model. It is also noticed that in the Z (cid:48) modelthe value of P (cid:48) is positive in scenarios S and S whilebecomes negative in S . As for the present analysis in S , we set the value of φ sb = 150 ±
10, in contrast to this,if we choose φ sb = − ±
10 which is also allowed (seeTab. (V)), then this negative value becomes positive. B. P -observables in s ∈ [1 . , . GeV Besides the analysis of angular observables in shorterbins at low s region (discussed in previous section), wehave also analyzed these observables in the full s ∈ [1 . , .
0] GeV region. The results for P -observables in s ∈ [1 . , .
0] GeV are summarized in Tab. (III) andcorresponding plots are shown in Fig. (4). In this figure,black error bar corresponds to LHCb result [4] while, ma-genta and yellow error bars correspond to Belle measure-ments for some of these observables [6, 7]. However, it is TABLE II:
Results for (cid:104) P (cid:105) -observables and their comparison with maximum likelihood fit results of ref. [4] in different bin size.Obs. SM Prediction S S S Measurement [4]0 . < s < . (cid:104) P (cid:105) − . − . ↔ − . − . ↔ − . − . ↔ − . − . +0 . − . ± . (cid:104) P (cid:105) − . − . ↔ − . − . ↔ − .
102 0 . ↔ − . − . +0 . − . ± . (cid:104) P (cid:105) − . . ↔ − . − . ↔ − . − . ↔ − . . +0 . − . ± . (cid:104) P (cid:48) (cid:105) .
267 0 . ↔ .
155 0 . ↔ .
171 0 . ↔ .
380 0 . +0 . − . ± . (cid:104) P (cid:48) (cid:105) .
740 0 . ↔ .
473 0 . ↔ . − . ↔ .
424 0 . +0 . − . ± . (cid:104) P (cid:48) (cid:105) − . − . ↔ − . − . ↔ − .
566 0 . ↔ .
400 0 . +0 . − . ± . . < s < . (cid:104) P (cid:105) − . − . ↔ − . − . ↔ − . − . ↔ − . − . +0 . − . ± . (cid:104) P (cid:105) − . − . ↔ − . − . ↔ − .
187 0 . ↔ − . − . +0 . − . ± . (cid:104) P (cid:105) . − . ↔ .
001 0 . ↔ .
001 0 . ↔ .
001 0 . +0 . − . ± . (cid:104) P (cid:48) (cid:105) . − . ↔ − . − . ↔ − .
13 0 . ↔ . − . +0 . − . ± . (cid:104) P (cid:48) (cid:105) .
225 0 . ↔ − .
208 0 . ↔ − .
141 0 . ↔ .
160 0 . +0 . − . ± . (cid:104) P (cid:48) (cid:105) − . − . ↔ − . − . ↔ − .
536 0 . ↔ . − . +0 . − . ± . . < s < . (cid:104) P (cid:105) -0.023 − . ↔ − . − . ↔ − . − . ↔ − .
024 0 . +2 . − . ± . (cid:104) P (cid:105) − . − . ↔ . − . ↔ . − . ↔ − . − . +0 . − . ± . (cid:104) P (cid:105) − . ↔ .
002 0 . ↔ .
002 0 . ↔ .
004 0 . +2 . − . ± . (cid:104) P (cid:48) (cid:105) − . − . ↔ − . − . ↔ − . − . ↔ − . − . +0 . − . ± . (cid:104) P (cid:48) (cid:105) − . − . ↔ − . − . ↔ − .
628 0 . ↔ − . − . +0 . − . ± . (cid:104) P (cid:48) (cid:105) − . − . ↔ − . − . ↔ − .
372 0 . ↔ .
508 0 . +0 . − . ± . . < s < . (cid:104) P (cid:105) -0.055 − . ↔ − . − . ↔ − . − . ↔ − .
062 0 . +0 . − . ± . . (cid:104) P (cid:105) .
206 0 . ↔ .
357 0 . ↔ . − . ↔ .
146 0 . +0 . − . ± . (cid:104) P (cid:105) . − . ↔ .
003 0 . ↔ .
003 0 . ↔ .
004 0 . +0 . − . ± . (cid:104) P (cid:48) (cid:105) − . − . ↔ − . − . ↔ − . − . ↔ − . − . +0 . − . ± . (cid:104) P (cid:48) (cid:105) − . − . ↔ − . − . ↔ − .
829 0 . ↔ − . − . +0 . − . ± . (cid:104) P (cid:48) (cid:105) − . − . ↔ − . − . ↔ − .
214 0 . ↔ . − . +0 . − . ± . good to mention here that LHCb results are in the bin s ∈ [1 . , .
0] GeV while Belle measurements [6, 7] arein the s ∈ [1 . , .
0] GeV . In addition, recently, the AT-LAS collaboration announced its results for s ∈ [0 . , . [8] which is not included in the current analysis.The empty red boxes in the plots of (cid:104) P (cid:105) and (cid:104) P (cid:48) (cid:105) rep-resent the S scenario when we choose φ sb = − ± (cid:104) P (cid:105) and (cid:104) P (cid:48) (cid:105) in the SM and in all the threescenarios of Z (cid:48) lie within the current measurements, how-ever, the error bars are huge. Therefore, to extract anyinformation about the NP requires the precise measure-ment of these observables. It is also noticed that thevalues of (cid:104) P (cid:105) in the SM and in the Z (cid:48) scenarios are veryclose, consequently, this observable even after the reduc-tion of error bars not a good candidate to constrained the Z (cid:48) parametric space. On the other hand (cid:104) P (cid:48) (cid:105) could behelpful to constraint the Z (cid:48) parametric space, if any mis-match will appear in future in the bin [1,6] GeV . TheSM value of (cid:104) P (cid:105) is small and not enhanced in Z (cid:48) model.However, the measured value is well above the SM pre- diction with huge error bars and need precision to drawany conclusion from this observable as well. From graphof (cid:104) P (cid:105) in Fig. (4), one can deduced that the SM valueof (cid:104) P (cid:105) not lie within the measured value of LHCb. How-ever, the values of (cid:104) P (cid:105) in S and S are within the mea-surements while in S , the value is out side the measurederror bars. For (cid:104) P (cid:48) (cid:105) , we have two different measurementsas shown in the plot and contrast to the (cid:104) P (cid:105) , the valueof (cid:104) P (cid:48) (cid:105) lie within these measurements. However, similarto (cid:104) P (cid:105) the values of (cid:104) P (cid:48) (cid:105) in S and S lie within themeasurements while the value in S lies outside the mea-sured values (see red bands in both plots). Regarding S ,it is interesting to check whether the values of (cid:104) P (cid:105) and (cid:104) P (cid:48) (cid:105) could be reduced to current measurements. For thispurpose, we choose the weak phase with opposite signi.e., φ sb = − ±
10 (see Tab. V in Appendix A) andrepresent them in plots by empty red boxes. In Fig. 4,by looking the empty red box in (cid:104) P (cid:105) plot, the value isreduced but still well above the current measurement. Incontrast, the value of (cid:104) P (cid:48) (cid:105) reduce to the Belle measure-ments [6]. However, more statistics on the observables (cid:104) P (cid:105) and (cid:104) P (cid:48) (cid:105) are helpful to constrained the Z (cid:48) param-eters, particularly, the sign and the magnitude of newweak phase φ sb .For (cid:104) P (cid:48) (cid:105) plots of Fig. (III), the values in the SM andin S , S lie out side the error bars of experimental datapoints while the values in the S well inside the all datapoints shown in figure. In general, from the plots of Fig.(III), one concludes that the considered model do havepotential to remove mismatch between theory and ex-periment but it is not so conclusive at present. We hopemore precise measurements will clear the situation. C. Q , for s ∈ [1 . , . GeV In Fig. (5), we have plotted the lepton flavor uni-versality violation (LFUV) observables (cid:104) Q (cid:105) against s . The values are quite small in the SM approximately (cid:104) Q (cid:105) = 8 . ± . × − (7 . ± . × − ) in the bin s ∈ [1 , . We have also found that the effects of Z (cid:48) are negligible. This fact is trivial, since Eq. (3) implies C Z (cid:48) ,µ , = C Z (cid:48) ,e , . However, error bars are quite large andneed more experimental data to find the accurate valuesof these observables. IV. CONCLUSION
In the present study, we have calculated the angularobservables P i and their average values (cid:104) P i (cid:105) in the SMand in the noun-universal family of Z (cid:48) model for the de-cay channel B → K ∗ ( → Kπ ) (cid:96) + (cid:96) − . The expressions ofthe angular observables are given in the form of coeffi-cient J i ( s ) which are written in terms of auxiliary func-tions g i ( h i ) in Eq. (A1). As in the literature, these J i ( s ) coefficients, in general, expressed via transversityamplitudes, A ⊥ , A (cid:107) and A , so the relations of thesetransversity amplitudes with auxiliary function g i ( h i ) arealso given in Eq. (A4). To see the Z (cid:48) effects on theseobservables, we have used the UtFit collaboration con-straints for the Z (cid:48) parameters, called as scenarios S and S . Besides, we also consider another scenario, called S ,in the present study. From the present analysis, in allthree scenarios of Z (cid:48) for small values of s i.e the largerecoil region, the values of angular observables are sig-nificantly changed from their SM values. The currentanalysis shown that except the sceanario S , the sce-narios S and S of Z (cid:48) model has potential to accom-modate the mismatch between the recent experimentalmeasurements and the SM values of some of the angularobservables in some bins of s . For instance, there is adiscrepancy between experimentally measured value andSM value of P (cid:48) in the region s ∈ [4 ,
6] GeV and in thecurrent study it is found that scenario S of Z (cid:48) could beadjusted this mismatch value with the measured value inthis bin. On the other hand, this mismatch can not beaccommodated on taking the maximum and minimumvalues of different parameters of scenarios S and S of UtFit collaborations. However, when we choose the ran-dom values of different parameters in the allowed regionof these scenarios, one can accommodate the P (cid:48) anomalywith scenario S but not with scenario S . It is also no-ticed that the P (cid:48) anomaly further constraint on the al-lowed parametric space of S . Furthermore, we have alsocalculated the angular observables (cid:104) P i (cid:105) and the LFUVobservables (cid:104) Q , (cid:105) in the large bin s ∈ [1 ,
6] and plottedwith the measured data, however, the error bar is quitelarge in this bin and more static is needed to draw results.Here, we would like to comment that CMS and ATLAScollaborations recently announced preliminary results onangular observables in Moriond 2017 which still show thetension between experimental measurements and the SMpredictions. Therefore, in general, one can say, as datawill be enlarge and the statistical error will be reducedthen these observables are quite promising to say some-thing about the constraints on coupling of Z (cid:48) boson withthe quarks and leptons and consequently about the sta-tus of Z (cid:48) model. Appendix A
The expressions of J i appeared in Eqs. (13) and (14)are as follows, J s = 3 sβ (cid:96) (cid:20) p K ∗ s (cid:0) | g | + | h | (cid:1) + | g | + | h | (cid:21) + 8 m (cid:96) s ( p K ∗ s | h | + | h | ) ,J c = 2 m K ∗ (cid:20)
32 a C tot m K ∗ m (cid:96) p K ∗ + β (cid:96) s | E K ∗ g + 2 √ sp K ∗ g | + (cid:0) − β (cid:96) (cid:1) s | E K ∗ h + 2 √ sp K ∗ h | (cid:21) ,J s = 12 sβ (cid:96) (cid:20) P K ∗ s (cid:0) | g | + | h | (cid:1) + | g | + | h | (cid:21) ,J c = − β (cid:96) sm K ∗ (cid:20) | E K ∗ g + 2 √ sp K ∗ g | + | E K ∗ h +2 √ sp K ∗ h | (cid:21) , (cid:45) (cid:72) GeV (cid:76) (cid:60) Q (cid:62) (cid:45) (cid:72) GeV (cid:76) (cid:60) Q (cid:62) FIG. 5:
Optimal observables Q , Q for s ∈ [1 . , . GeV where, yellow error bar corresponds to recent Belle measurements [7].Other legends are same as in previous figures. TABLE III:
Results for (cid:104) P (cid:105) -observables for s ∈ [1 . , . GeV and their comparison with LHCb maximum likelihood fit results ofref. [4] in different bin size, Belle results [6, 7].Obs. SM Prediction S S S Measurement (cid:104) P (cid:105) -0.033 ± − . ↔ − . − . ↔ − . − . ↔ − .
036 0 . +0 . − . ± .
044 [4] (cid:104) P (cid:105) . ± .
033 0 . ↔ − .
162 0 . ↔ − .
135 0 . ↔ . − . +0 . − . ± .
010 [4] (cid:104) P (cid:105) . ± . − . ↔ .
002 0 . ↔ .
002 0 . ↔ .
003 0 . +0 . − . ± .
017 [4] (cid:104) P (cid:48) (cid:105) − . ± . − . ↔ − . − . ↔ − . − . ↔ − . − . +0 . − . ± .
12 [4] − . +0 . − . ± .
174 [6] − . +0 . − . ± .
15 [7] (cid:104) P (cid:48) (cid:105) − . ± . − . ↔ − . − . ↔ − .
583 0 . ↔ − . − . +0 . − . ± .
014 [4]0 . +0 . − . ± .
099 [6]0 . +0 . − . ± .
10 [7] (cid:104) P (cid:48) (cid:105) − . ± − . − . ↔ − . − . ↔ − .
345 0 . ↔ . − . +0 . − . ± .
021 [4] − . +0 . − . ± .
172 [6] J = sβ (cid:96) (cid:20) p K ∗ s (cid:0) | g | + | h | (cid:1) − | g | − | h | (cid:21) ,J = √ sβ (cid:96) m K ∗ (cid:20) E K ∗ (cid:0) | g | + | h | (cid:1) + p K ∗ ( s ) / R e ( g g ∗ + h h ∗ ) (cid:21) ,J = − √ p K ∗ ( s ) / β (cid:96) m K ∗ (cid:20) E K ∗ R e ( g h ∗ + g h ∗ )+2 p K ∗ s / R e ( g h ∗ + g h ∗ ) (cid:21) ,J s = − p K ∗ ( s ) / β (cid:96) (cid:20) R e ( g h ∗ + g h ∗ ) (cid:21) ,J = √ p K ∗ ( s ) / β (cid:96) m K ∗ (cid:20) I m ( g h ∗ + g ∗ h ) (cid:21) , J = √ p K ∗ ( s ) / β (cid:96) m K ∗ (cid:20) E K ∗ I m ( g ∗ g + h ∗ h )+2 p K ∗ s / I m ( g ∗ g + h ∗ h ) (cid:21) ,J = 2 p K ∗ ( s ) / β (cid:96) (cid:20) I m ( g g ∗ + h h ∗ ) (cid:21) , (A1)1where g i ( h i ), i = 1 , · · · , h = 4 m b s T ⊥ + 2 M B + m K ∗ C tot9 V ( s ) ,g = 2 M B + m K ∗ C tot10 V ( s ) ,h = − ( M B + m K ∗ ) C tot9 A ( s ) − m b (cid:0) m B − m K ∗ (cid:1) s E K ∗ M B T ⊥ ,g = − ( M B + m K ∗ ) A ( s ) C tot10 ,h = A M B + m K ∗ C tot9 + 2 m b s (cid:20) s ( T ⊥ + T (cid:107) ) m B − m K ∗ + 2 E K ∗ M B T ⊥ (cid:21) ,g = A M B + m K ∗ C tot10 , (A2) E K ∗ = m B − m K ∗ − s √ s , p K ∗ = (cid:113) E K ∗ − m K ∗ ,β (cid:96) = (cid:114) − m (cid:96) s , (A3)and a = E K ∗ m K ∗ ξ (cid:107) ∆ (cid:107) where ∆ (cid:107) is given in Appendix B inEq. (B1).Traditionally, the J ’s are given in terms of transversityamplitudes but we have written in terms of g i ( h i ) func-tions given in Eq. (A2) A , (cid:107) , ⊥ . The A , (cid:107) , ⊥ are relatedwith g i ( h i ) as follows A L,R = N m K ∗ (cid:20) E k ∗ ( h ∓ g ) + 2 p k ∗ √ s ( h ∓ g ) (cid:21) ,A L,R (cid:107) = √ N (cid:2) h ∓ g (cid:3) , A L,R ⊥ = √ s N p k ∗ (cid:2) h ∓ g (cid:3) , (A4)where N = α G F | V tb V ∗ ts | (cid:113) s β (cid:96) p K ∗ · π m B . We would like tomention here that our expressions of J ’s are consistentwith the literature for example given in refs. [14, 50].The values of Wilson coefficients at NNLO, Z (cid:48) param-eters and other input parameters are listed in Tabs. (IV),(V) and (VI), respectively. Appendix B
The expression of ∆ (cid:107) , appear in the definition of a below Eq. (A3), written as follows∆ (cid:107) ( s ) = 1 + α s C F π [(2 L − − sE K ∗ π f B f K ∗ (cid:107) λ − B + N c m B ( E K ∗ /m K ∗ ) ξ (cid:107) ( s ) (cid:90) du ¯ u Φ ¯ K ∗ , (cid:107) (cid:35) , (B1) and contributes only for massive leptons. The light-conedistribution amplitude (LCDA) Φ ¯ K ∗ ,a for transversely( a = ⊥ ) and longitudinally ( a = (cid:107) ) polarized K ∗ can bewritten as [40, 51]Φ ¯ K ∗ ,a = 6 u (1 − u ) { a (cid:0) ¯ K ∗ (cid:1) a C (3 / (2 u − a (cid:0) ¯ K ∗ (cid:1) a C (3 / (2 u − } , (B2)where L = − ( m b − s ) /s ln (cid:0) − s/m b (cid:1) and a i (cid:0) ¯ K ∗ (cid:1) a arethe Gegenbauer coefficients. The moments are λ − B, + = (cid:90) ∞ dω Φ B, + ( ω ) ω ,λ − B, − = (cid:90) ∞ dω Φ B, − ( ω ) ω − s/m B − i(cid:15) where Φ B, ± are the two B -meson light-cone distributionamplitudes [40]. The λ − B, − ( s ) can be expressed as: λ − B, − ( s ) = e − s/ ( m B ω ) ω [ − Ei ( s/m B ω ) + iπ ] , where ω = 2( m B − m b ). The ξ a are the universal formfactors, ξ ⊥ = m B m B + m K ∗ Vξ (cid:107) = m B + m K ∗ E K ∗ A − m B − m K ∗ m B A . (B3)The B → K ∗ matrix elements in heavy quark limitdepend on four independent functions T ± a ( a = ⊥ , (cid:107) ). Inthe low s , (1 . < s < . ), the invariant amplitudes T ⊥ , (cid:107) at NLO within QCDf are given in [38, 40, 50], T a = ξ a C a + π N c f B f K ∗ ,a m B Ξ a (cid:88) ± (cid:90) dωω × Φ B, ± ( ω ) (cid:90) du Φ K ∗ ,a ( u ) T a, ± ( u, ω ) , (B4)where Ξ ⊥ ≡
1, Ξ (cid:107) ≡ m K ∗ /E K ∗ and the factorizationscale µ f = (cid:112) m b Λ QCD . The coefficient functions C a andhard scattering functions T a, ± are written as C a = C (0) a + α s ( µ b ) C F π C (1) a T a, ± = T (0) a, ± ( u, ω ) + α s ( µ f ) C F π T (1) a, ± ( u, ω ) . (B5)The form factor terms C (0) a at LO are C (0) ⊥ = C eff7 + s m b m B Y ( s ) , and C (0) (cid:107) = − C eff7 − m B m b Y ( s ) . TABLE IV:
Values of Wilson coefficients at µ b = 4 · . C ( µ b ) C ( µ b ) C ( µ b ) C ( µ b ) C ( µ b ) C ( µ b ) C eff7 ( µ b ) C eff8 ( µ b ) C ( µ b ) C ( µ b )-0.2632 1.0111 -0.0055 -0.0806 0.0004 0.0009 -0.2923 -0.1663 4.0749 -4.3085TABLE V: The numerical values of the Z (cid:48) parameters [42, 43]. |B sb | × − φ sb (Degree) S LR(cid:96)(cid:96) × − D LR(cid:96)(cid:96) × − S . ± . − ± − . ± . − . ± . S . ± . − ± − . ± . − . ± . S . ± . ±
10 or ( − ±
10) 0 . − . Y ( s ) = h ( s, m c ) (cid:18) C + C + 6 C + 60 C (cid:19) − h ( s, m pole b ) (cid:18) C + 43 C + 76 C + 643 C (cid:19) − h ( s, (cid:18) C + 43 C + 16 C + 643 C (cid:19) + 43 C + 649 C + 6427 C , where h ( s, m q ) is well-known fermionic loop function.The coefficients C (1) a at NLO is divided into a factorizableand a non-factorizable part as C (1) a = C ( f ) a + C ( nf ) a . (B6)At NLO the factorizable correction reads [40, 52] C ( f ) ⊥ = C eff7 (cid:18) ln m b µ − L + ∆ M (cid:19) C ( f ) (cid:107) = − C eff7 (cid:18) ln m b µ + 2 L + ∆ M (cid:19) . The non-factorizable corrections are, C F C ( nf ) ⊥ = − ¯ C F (7)2 − C eff8 F (7)8 − s m b m B (cid:20) ¯ C F (9)2 + 2 ¯ C (cid:18) F (9)1 + 16 F (9)2 (cid:19) + C eff8 F (9)8 (cid:21) ,C F C ( nf ) (cid:107) = ¯ C F (7)2 + C eff8 F (7)8 + m B m b (cid:20) ¯ C F (9)2 + 2 ¯ C (cid:18) F (9)1 + 16 F (9)2 (cid:19) + C eff8 F (9)8 (cid:21) , where ∆ M depends on the mass renormalization conven-tion for m b . These corrections are obtained from the ma-trix elements of four-quark and chromomagnetic dipoleoperators [40] that are embedded in F (7 , , and F (7 , [53, 54].At LO the hard-spectator scattering term T (0) a, ± ( u, ω )from weak annihilation diagram is [40] T (0) ⊥ , + ( u, ω ) = T (0) ⊥ , − ( u, ω ) = T (0) (cid:107) , + ( u, ω ) = 0 ,T (0) (cid:107) , − ( u, ω ) = − e q m B ωm B ω − s − i(cid:15) m B m b (cid:0) ¯ C + 3 ¯ C (cid:1) . The contributions to T (1) a at NLO also contain a factor-izable as well as non-factorizable part T (1) a = T ( f ) a + T ( nf ) a . (B7)Including O ( α s ) corrections the factorizable term to T (1) a, ± are given by [40, 52] T ( f ) ⊥ , + ( u, ω ) = C eff7 m B ¯ uE K ∗ , T ( f ) (cid:107) , + ( u, ω ) = C eff7 m B ¯ uE K ∗ T ( f ) ⊥ , − ( u, ω ) = T ( f ) (cid:107) , − ( u, ω ) = 0 , where ¯ u = 1 − u . The non-factorizable correction comesthrough the matrix elements of four-quark operators andthe chromomagnetic dipole operator T ( nf ) ⊥ , + ( u, ω ) = − e d C eff8 u + ¯ us/m B + m B m b [ e u t ⊥ ( u, m c ) (cid:0) ¯ C + ¯ C − ¯ C (cid:1) + e d t ⊥ ( u, m b ) (cid:0) ¯ C + ¯ C − ¯ C − m b /m B ¯ C (cid:1) + e d t ⊥ ( u,
0) ¯ C ] ,T ( nf ) ⊥ , − ( u, ω ) = 0 ,T ( nf ) (cid:107) , + ( u, ω ) = m B m b [ e u t (cid:107) ( u, m c ) (cid:0) ¯ C + ¯ C − ¯ C (cid:1) + e d t (cid:107) ( u, m b ) (cid:0) ¯ C + ¯ C − ¯ C (cid:1) + e d t (cid:107) ( u,
0) ¯ C ] ,T ( nf ) (cid:107) , − ( u, ω ) = e q m B ωm B ω − s − i(cid:15) (cid:34) C eff8 ¯ u + us/m B + 6 m B m b (cid:32) h (cid:0) ¯ um B + us, m c (cid:1) (cid:0) ¯ C + ¯ C + ¯ C (cid:1) + h (cid:16) ¯ um B + us, m poleb (cid:17) (cid:0) ¯ C + ¯ C + ¯ C (cid:1) + h (cid:0) ¯ um B + us, (cid:1) (cid:0) ¯ C + 3 ¯ C + 3 ¯ C (cid:1) − (cid:0) ¯ C − ¯ C −
15 ¯ C (cid:1) (cid:33)(cid:35) . TABLE VI:
Values of input parameters. α em ( M Z ) = 1 / .
940 [55] α s ( M Z ) = 0 . ± . m e = 0 . × − GeV [56] m µ = 0 . m B = 5 . m K ∗ = 0 . m Sb = 4 . ± .
03 GeV [57] m s = 0 . ± .
005 GeV [56] m MSc ( m c ) = 1 . ± .
09 GeV [56] | V tb | = 0 . ± . | V ts | = (40 . ± . · − [56] f B = 194 ±
10 MeV [60] λ B = 460 ±
110 MeV [59] f K ∗ , || = 220 ± f K ∗ , ⊥ = 185 ± a , || = 0 . ± .
03 [33] a , || = 0 . ± .
06 [33] a , ⊥ = 0 . ± .
03 [33] a , ⊥ = 0 . ± .
06 [33]
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