Study of b → c induced B̄ ∗ → V ` ν̄ ` decays

In this paper, we investigate the tree-dominated B̄∗ u,d,s,c → V `ν̄` (V = D∗ u,d , D∗ s , J/ψ and ` = e , μ , τ) decays in the Standard Model with the relevant form factors obtained in the light-front quark model. These decays involve much more helicity states relative to the corresponding B̄∗ → P`ν̄` and B̄ → V `ν̄` decays, and moreover, the contribution of longitudinal polarization mode (V meson) is relatively small, ∼ 30%, compared with the corresponding B meson decays. We have also computed the branching fraction, lepton spin asymmetry, forward-backward asymmetry and ratio R ∗(L) V ≡ B(B̄∗→V τ−ν̄τ ) B(B̄∗→V `ν̄`′ ) (`′ = e , μ). Numerically, the branching fractions of B̄∗ → V `ν̄`′ decays are at the level of O(10−7), and are hopeful to be observed by LHC and Belle-II experiments. The ratios R ∗(L) D∗ ,D∗ s ,J/ψ have relatively small theoretical uncertainties and are close to each other, R ∗(L) D∗ ' R ∗(L) D∗ s ' R J/ψ ' [0.26, 0.27] ([0.27, 0.29]), which are a bit different from the predictions in some previous works. The future measurements are expected to make tests on these predictions. 1 ar X iv :2 00 3. 08 60 0v 1 [ he pph ] 1 9 M ar 2 02 0

The spin-triplet vector B * q meson with quantum number of n 2s+1 L J = 1 3 S 1 and J P = 1 − [50][51][52][53] has the same flavor components as the spin-singlet pseudoscalar B q (q = u, d, s and c) meson, and can also decay through the b → c ν transition at quark-level, therefore its b → c induced semileptonic decays can play a similar role as B meson decays for testing the SM and probing possible hints of NP.
The B * q meson is unstable particle, it cannot decay via strong interaction due to that m B * q −m Bq 50 MeV<m π [54]; B * q meson decay is dominated by the radiative process [54], B * q → B q γ; the weak decay modes via the bottom-changing transition (for instance, the b → c induced semileptonic B * q decays considered in this work) are generally very rare, and their branching fractions are expected to be very small within the SM. Until now, there is no experimental information and few theoretical works concentrating on the B * q weak decays. Fortunately, thanks to the high luminosity and large production cross section at the running LHC and SuperKEKB/Belle-II experiments, a huge amount of the B * q meson data samples would be accumulated. At Belle-II experiment, the B * and B * s mesons are produced mainly via Υ(5S) decays. With the target annual integrated luminosity, ∼ 13 ab −1 [55], and the cross section of Υ(5S) production in e + e − collisions, σ(e + e − → Υ(5S)) = (0.301 ± 0.002 ± 0.039) nb [56], it is expected that about 4 × 10 9 Υ(5S) samples could be produced per year by Belle-II. Further considering that Υ(5S) meson mainly decays to final states with a pair of B ( * ) (s) mesons and using the branching fractions of Υ(5S) decays given by PDG [54], it can be estimated that about N (B * +B * )/year ∼ 4 × 10 9 and N (B * s +B * s )/year ∼ 2 × 10 9 samples can be accumulated by Belle-II per year. Unfortunately, the B * c meson and its decays are out of the scope of Belle-II experiment. In addition, a lot of B * q samples can also be produced via pp collision and be accumulated in the future by LHC with high collision energy, high luminosity and rather large production cross section [57][58][59], and some B * q weak decays are hopeful to be observed, such as the leptonic B * s → + − decay with branching fraction ∼ O(10 −11 ) [60]. Encouraged by the abundant B * q data samples at future heavy-flavor experiments, some interesting theoretical studies for the B * q weak decays have been made within the SM, for instance, the pure leptonicB * s → + − andB * u,c → −ν decays [60], the impact ofB * s,d → µ + µ − onB s,d → µ + µ − decays [61], the studies of the semileptonic B * c decays within the QCD sum rules [62][63][64], the semileptonic B * u,d,c,s → (P, V ) −ν with P = D, D s , η c , V = D * , D * s , J/ψ decays within the Bethe-Salpeter (BS) method [65] and a approach under the assumption of heavy quark symmetry (HQS) [66],B * → P −ν with P = D, D s , π, K [67] and the nonleptonic , B * →DD [73] and B * c → ψ(1S, 2S)P, η c (1S, 2S)P [74] decays. Moreover, the NP effects on the semileptonicB * → P −ν with P = D, D s , π, K decays have been investigated in a model-independent scheme [75] and the vector leptoquark model [76]. In this paper, we pay our attention to the CKM-favored and tree-dominated semileptonicB * u,d,s,c → V ν (V = D * u,d , D * s , J/ψ) weak decays, which are generally much more complicated than the corresponding B decay modes because they involve much more allowed helicity states.
Our paper is organized as follows. In section 2, the helicity amplitudes and observables of B * → V ν decays are calculated. Section 3 is devoted to the numerical results and discussions, and theB * → V transition form factors obtained within the covariant light-front quark mode are used in the computation. Finally, we give our summary in section 4.

Effective Lagrangian and amplitude
In the SM,B * u,d,s,c → V ν (V = D * u,d , D * s , J/ψ) decays are induced by b → c ν transition at quark level via W-exchange, and can be described by the effective Lagrangian and hadronic (H µν ) tensors built from the respective products of the leptonic and hadronic currents, the square amplitude can be expressed as Inserting the completeness relation of the polarization vector of virtual W * boson, the product of L µν and H µν can be rewritten as where L(m, n) ≡ L µν¯ µ (m)¯ * ν (n) and H(m, n) ≡ H µν¯ * µ (m)¯ ν (n) are Lorentz invariant and therefore can be evaluated in different reference frames. In our following evaluation, H(m, n) and L(m, n) will be calculated in the B * -meson rest frame and the −ν center-of-mass frame, respectively.
Turning to the −ν center-of-mass frame, the four-momenta of lepton and antineutrino are given as where E = (q 2 + m 2 )/2 q 2 , | p | = (q 2 − m 2 )/2 q 2 , and θ is the angle between V and three-momenta. In this frame, the polarization vectors¯ µ (λ W * ) have the form

Hadronic helicity amplitudes
For hadronic part, one has to calculate the hadronic helicity amplitudes which describes the decay of three helicity states of B * meson into the three helicity states of daughter V meson and the four helicity states of virtual W * . For the B * → V transition, the can be factorized in terms of ten form factors V 1,2,3,4,5,6 (q 2 ) and A 1,2,3,4 (q 2 ) as [79,80] with the sign convention 0123 = −1.
Then, by contracting these hadronic matrix elements with the polarization vector of virtual W * boson, we can finally obtain the non-vanishing hadronic helicity amplitudes, given as Obviously, only the amplitudes with λ B * = λ V − λ W * survive due to the helicity conservation.

Helicity amplitudes and observables
For the leptonic part, the leptonic tensor could be expanded in terms of a complete set of Wigner's d J -functions, which has been widely used in the study of hadron semileptonic [77,81,82]. As a result, L µν H µν can be reduced to a very compact form where J and J run over 1 and 0, λ ( ) W * and λ run over their components. For the standard expression of d J function, we take their value from PDG [54]. The leptonic helicity amplitude h λ ,λν in Eq. (24) defined as Taking the exact forms of spinors and W * polarization vectors given in Eq. (10), we obtain which are the same as the results obtained in semileptonic B and hyperon decays [81,82].
Using the amplitudes obtained above, we can then further evaluate the observables ofB * → V −ν decays. The double differential decay rate is written as where the factor 1/3 is caused by averaging over the spins of initialB * meson. The double differential decay rate with a given helicity state of lepton (λ = ± 1 2 ) is written as Integrating over cos θ and summing over the lepton helicity, we can obtain the differential decay rate written as in which, the three non-diagonal interference terms in Eq. (29) vanish. In addition, paying attention to the polarization states of V meson, one can obtain the longitudinal differential decay width dΓ L /dq 2 by picking out H 2 t00 , H 2 +−0 , H 2 −+0 and H 2 000 terms in Eq. (30). Using Eqs. (28) and (29) given above, we can also construct some useful observables as follows. The q 2 dependent ratios is defined as where, denotes the light leptons µ and e (in the following calculations, we take m e,µ = 0).
The lepton spin asymmetry and forward-backward asymmetry are defined as and respectively. These observables are independent of the CKM matrix elements, and the hadronic uncertainties canceled to a large extent, therefore, they can be predicted with a rather high accuracy.

Numerical results and discussions
In our numerical calculation, for the well-known Fermi coupling constant G F and the masses of mesons and τ , we take their central values given by PDG [54]. For the CKM element, we take |V cb | = 41.80 +0.28 −0.60 × 10 −3 given by CKMFitter Group [83]. In order to evaluate the branching fractions, the total decay widths (or lifetimes), Γ tot (B * u,d,s,c ), are also essential inputs. However, there is no available experimental or theoretical information until now. While, due to the fact that the electromagnetic processes B * → Bγ dominates B * decays, we can take the approximation Γ tot (B * ) Γ(B * → Bγ). In the light-front quark model (LFQM), the decay width of B * → Bγ decay is given by [84] Γ is the kinematically allowed energy of the outgoing photon. The radial wavefunction (WF) ψ(x, k ⊥ ) of bound-state is responsible for describing the momentum distribution of the constituent quarks. In this paper, we shall use the Gaussian-type WF where k z is the relative momentum in z-direction and has the form k z = (x− 1 2 )M 0 + m 2 2 −m 2 1 2M 0 . One can refer to Ref. [84] for more details. Using the constituent quark masses and the Gaussian parameter β given in Table 1, we obtain the numerical results for Γ(B * → Bγ) as follows, These theoretical predictions are generally in agreement with the ones obtained in the previous work based on different theoretical models [84,[87][88][89][90][91][92]. obtained by fitting to the data of decay constants [85,86], where q = u , d.
where F denotes A 1−4 and V 1−6 . Using the inputs given in Table 1, we then present our theoretical prediction for the form factors ofB * → D * ,B * s → D * s andB * c → J/ψ transitions in Table 2. Their q 2 -dependences are shown in Fig. 1.
Using the formulas given in the last section and inputs given above, we then present our numerical results for the q 2 -integrated observables ofB * → V −ν decays in Tables 3 and   4. For the branching fractions, the three errors in Table 3 are caused by the uncertainties of form factors, V cb and Γ tot (B * ), respectively. For the other observables listed in Table 4, the theoretical uncertainties are caused only by the form factors. Besides, the q 2 -dependence of differential decay rates dΓ (L) /q 2 and A * V λ ,θ , R * (L) V are shown in Figs. 2 and 3. The following are    (1) From Table 3      In Table 3, the previous predictions based on the Bethe-Salpeter (BS) method [65] and the assumption of heavy quark symmetry (HQS) [66] are also listed for comparison. It can be found that the results based on the BS method and the assumption of HQS are a little bit smaller and larger respectively than our results, but they are also in agreement Table 4: Predictions for q 2 -integrated observables A * V λ ,θ ( = τ ) , R * (L) V and F * V L . (2) Deviations from the SM predictions inB → D ( * ) ν decay modes have been observed by the BaBar [11,12], Belle [13][14][15] and LHCb [16,17]

Obs. Prediction Obs. Prediction Obs. Prediction
which show tensions of about 2 and 4σ, respectively, with the SM predictions [97]. Very recent measurement of R D * by Belle [98] results in values more compatible with the SM and yield a downward shift in the average. However, even though such measurement is included in the global average, the deviation is still larger than 3σ [97]. If this "R D * anomaly" is the truth, it possibly exists also in the b → c inducedB * → V ν decays, which therefore can provide another useful test on the lepton flavor universality and the various method based on the SM and NP for resolving "R D * anomaly". Our numerical results for R * (L) V are summarized in Table 4, and the q 2 -spectra of R * (L) V are shown in Fig. 3. It can be found that within theoretical uncertainties. Moreover, their q 2 -spectra almost overlap with each other as shown in Figs. 3 (a) and (b). Using the results summarized in Table 3, we can also obtain the predictions based on the BS method and HQS, It can be found that these results are different from our predictions more or less because different models and parameterizations are used for evaluating form factors, which has been observed in the case of R D * [18]. Future measurement will make a judgement on these results.
(3) Besides, the lepton spin asymmetry and the forward-backward asymmetry are also important observables for testing the SM and NP scenarios, for instance, two-Higgs-doublet models, R-parity violating supersymmetry models and so on [42][43][44][45][46][47], because their theoretical uncertainties can be well controlled and the zero-crossing points of their q 2 -spectra are sensitive to the NP effects [42]. Our numerical results for q 2 -integrated A * V λ and A * V θ are collected in Table 4, and the q 2 dependences of A * V λ (q 2 ) and A * V θ are shown by (4) The D * longitudinal polarization fraction in semileptonic B 0 → D * − τ + ν τ decay, defined as F D * L =Γ λ D * =0 (B 0 → D * − τ + ν τ )/Γ(B 0 → D * − τ + ν τ ), has been measured by Belle experiment with F D * L = 0.60 ± 0.08(stat.) ± 0.04(syst.) [99], which deviates from the SM prediction (F D * L ) SM = 0.457±0.010 [100] by 1.6σ. Similarly, we can define the longitudinal polarization fraction forB * → V τ −ν τ decay modes. From the numerical results given in the last row of Table 4, one can easily find that which implies thatB * → V τ −ν τ decay is dominated by the transverse polarization. It is obviously different from the correspondingB → V τ −ν τ decay mode, which is dominated by the longitudinal polarization state.

Summary
In this paper, motivated by abundant B * data samples at high-luminosity heavy-flavor experi- B(B * c → J/ψ −ν ) ∼ 5 × 10 −7 is the largest one, and are hopeful to be observed at running LHC and SuperKEKB/Belle-II experiments; in addition, for theB * → V τ −ν τ decay, the longitudinal polarization state of V meson presents only about 30% contribution to the integrated decay width, which is obviously different from the correspondingB → V τ −ν τ decay. All of results and findings in this paper are waiting for the experimental test in the future.

Data Availability Statement
This manuscript has no associated data or the data will not be deposited. This is a theoretical research work, no additional data are associated with this work.