Semileptonic and nonleptonic decays of B_u,d,s,c^* in the covariant light-front approach

Both on the theoretical and experimental sides, the b-flavored pseudoscalar mesons, namely $ B_{u,d,s} $ and $ B_{c} $, have been extensively studied about their properties through the weak decays, which are important to precisely test the Standard Model (SM) and search for new physics (NP). However, the vector partners $ B^{*}_{u,d,s} $ and $ B^{*}_{c} $ still lack sufficient experimental measurements, due to their low production rates and detection efficiencies compared to the pseudoscalar $ B_{u,d,s} $ and $ B_c $ mesons. With the increasing amount of data collected by LHCb adn Belle II, discovering $ B^*_c $ and precisely detecting the properties of $ B^{*}_{u,d,s} $ become possible. We can expect that these b-flavored vector mesons will provide another platform to study the heavy flavor meson weak decays in the future.

The vector mesons $ B^* $ and $ B^*_s $ have been found in experiments with masses $ m_{B^*}=5324.71 $ MeV and $ m_{B^*_s}=5415.4 $ MeV, respectively. Because of the small mass difference between $ m_{B^*_{(s)}} $ and $ m_{B_{(s)}} $, $ B^{*}_{(s)} $ can not decay to the corresponding pseudoscalar partner and a light meson, such as $ \pi, K $. That is to say, the strong decays are forbidden and the radiative decays $ B^*_{(s)}\to B_{(s)}\gamma $ become the dominant channels. Thanks to the rapid development of the heavy flavor experiments [1−3], even though the $ B^{*}_{(s)} $ decays are dominated by the electromagnetic processes $ B^{*}_{(s)}\to B_{(s)}\gamma $, Belle-II still has hope to observe the branching ratios with the order of 10^-8 [4, 5] in the future. In addition, due to the high production rates of $ b\bar b $ pair on pp collisions [6], LHCb may also provide a lot of experimental information for the $ B^{*} $ decays [7]. Theoretically, some researches about the $ B^{*}_{(s)} $ weak decays in the SM [5, 7−10] and the NP scenarios [11] have been performed. The $ B_{c}^{*} $ meson consists of two different heavy quarks $ \bar b $ and c, which can decay individually. So one can expect that there should exist rich weak decay channels for the $ B_{c}^{*} $ meson. While it has not been found in experiments up to now and the corresponding theoretical investigations are very few. It is similar to the $ B^*_{(s)} $ mesons, the mass difference between $ B_{c}^{*} $ and $ B_c $ is much smaller ($ \simeq 50 $ MeV) than the pion mass. Therefore, the $ B_{c}^{*} $ meson decays only through electromagnetic and weak interactions. With the promotion of the collider energy and the luminosity, it is possible to find the $ B^*_c $ on LHC.

Compared with other quark-model approaches, the covariant light-front quark model (CLFQM) has some unique advantages. First, the light-front wave functions used in this approach are independent of the hadron momentum and thus are manifestly Lorentz invariant. Moreover, the hadron spin is correctly constructed using the so-called Melosh rotation. Second, this model provides a relativistic treatment of the hadron. Since the final state meson at the maximum recoil point ($ q^2=0 $) or in heavy-to-light transitions can be highly relativistic, it is important to consider the relativistic effects. Then it is expected that the non-relativistic quark model may not work well. Last, in the CLFQM, the spurious contribution, which is dependent on the orientation of the light-front, is just canceled by the zero-mode contribution, thereby the covariance of the current matrix elements lost in the previous standard light-fron quark model is restored. The CLFQM has been successfully extended to investigate the transition form factors and hadronic weak decays [12−15]. This paper is organized as follows. The formalism of the CLFQM, the hadronic matrix elements and the helicity amplitudes combined via form factors are listed in Sec. II. In addition to the numerical results for the form factors of the transitions $ B^{*}\to D, \pi $, $ B_{s}^{*}\to D_{s}, K $ and $ B_{c}^{*}\to D, B, B_{s}, \eta_{c}, \eta_{c}(2S) $, the branching ratios, the forward-backward asymmetries $ A_{FB} $ and the polarization fractions $ f_{L, {\|}} $ for the corresponding decays are presented in Sec. III. Detailed comparisons with other theoretical values and relevant discussions are also included. The summary is presented in Sec. IV. Some specific rules when performing the $ p^- $ integration and the expression for each form factor are collected in the Appendix A and B, respectively.

The Bauer-Stech-Wirble (BSW) form factors for the transitions $ B^* \to P $ are defined as follows,

where $ p=p'+p'', q=p'-p'' $ and the convention $ \epsilon_{0123}=1 $ is adopted. In above equations, $ V_{\mu} $ and $ A_{\mu} $ are the corresponding vector and axial-vector currents, which are dominant contributions in the weak decays. Here $ p'(p'') $ is the four-momentum of the initial (final) meson.

Following the convention and calculation rules for the form factors of the transition $ J/\Psi\to D_{(s)} $ given in our previous work [13], one can write out the decay amplitude in the lowest order for the transitions $ B^* \to P $, whose Feynman diagram is shown in Fig. 1,

Figure 1.  Feynman diagram for the transition $ B^*\to P $ with P referring a pseudoscalar meson, i.e., $ \pi, K, \eta_c(1S,2S), D_{(s)}, B_{(s)} $, where $ P^{\prime(\prime\prime)} $ is the $ B^* $ (P) meson momentum, $ p^{\prime(\prime\prime)}_1 $ is the quark momentum, $ p_2 $ is the antiquark momentum and X denotes the vector or axial vector transition vertex.

where the quark propagators $ N^{\prime(\prime\prime)}, N_2 $ and the trace $ S_{\mu \nu}^{B^*P} $ can be obtained from those given in Ref. [13] by the replacements $ J/\Psi\to B^*, D_{(s)}\to P $. The covariant vertex function $ h^\prime_{B^*} $ is defined as

where $ M'_{0B^*} $ is the kinetic invariant mass of the initial $ B^* $ meson and can be expressed as the energies $ e^{(\prime)}_i $ of the constituent quark and anti-quark with masses (momentum fractions) being $ m^\prime_1(x_1) $ and $ m_2(x_2) $, respectively. Their definitions including the denominator $ \widetilde{M}_{0}^{\prime} $ are given as follows

where the $ p_{z}^{\prime}=\dfrac{x_{2} M_{0}^{\prime}}{2}-\dfrac{m_{2}^{2}+p_{\perp}^{\prime 2}}{2 x_{2} M_{0}^{\prime}} $. In principle, the light-front wave functions (LFWFs) can be derived from solving the relativistic Schr$ \ddot{o} $dinger equation, while it is difficult to obtain their exact solutions in many cases. Here the harmonic oscillator wave function with an exponential term is used and its expression for the S-wave mesons is written as

where $ \beta^\prime $ is a phenomenological parameter describing the momentum distribution and can be fixed by fitting the corresponding decay constant. For example, the shape parameter $ \beta^\prime_{B^*} $ can be obtained through

where $ w_{B^*}^{\prime}=M_{0B^*}^{\prime}+m_{b}^{\prime}+m_{q} $, $ M^{\prime}_{B^*} $ and $ m_b^\prime (m_q) $ are the $ B^* $ meson and $ b, (q=u,d) $ quark masses, respectively. The expressions of the vertex functions $ h^{\prime\prime}_P $ for our considered pseudoscalar mesons are similar. After expanding the trace $ S_{\mu \nu}^{B^*P} $ using the Lortentz contraction, then one can get the form factors $ V(q^2), A_0(q^2), A_1(q^2) $ and $ A_0(q^2) $ by matching their coefficients given in Eq. (1). Their specific expressions are listed in Appendix B.

The differential decay widths of the semileptonic $ B^* $ decays can be obtained by the combinations of the helicity amplitudes via the form factors, which are listed as follows

where $ \lambda(q^2)=\lambda(m^{2}_{B^*},m^{2}_{P},q^{2})=(m^{2}_{B^*}+m^{2}_{P}-q^{2})^{2}-4m^{2}_{B^*}m^{2}_{P} $ and $ m_{\ell} $ is the mass of the lepton $ \ell $ with $ \ell=e,\mu, \tau $ ¹. It is noted that although the electron and nuon are very light, we do not ignore their masses in our calculations in order to check the mass effects. The combined transverse and total differential decay widths are defined as

For the $ B^* $ decays, it is meaningful to define the polarization fraction due to the existence of different polarizations

As to the forward-backward asymmetry, the analytical expression is defined as [16]

where $ \theta $ is the angle between the 3-momenta of the lepton $ \ell $ and the initial meson in the $ \ell\nu_\ell $ rest frame. The function $ b_{\theta}(q^2) $ represents the angular coefficient, which can be written as [16]

where the helicity amplitudes

with the subscript V in each helicity amplitude referring to the $ \gamma_\mu(1-\gamma_5) $ current.

Based on the effective Hamiltonian, the amplitudes for the decays $ B^{*} \rightarrow P M_{1} $ with $ M_{1}= \pi, K, D_{(s)} $ can be expressed as

It is noticed that the second equality is an approximation. Here we only consider the dominant contributions from the tree-emission diagrams. In some cases, W-exchange and W-annihilation diagrams might give sizeable deviations to the results. For the decays of $ B^{*} \to \pi D_{(s)} $, the amplitudes are given as

where $ \epsilon $ is the polarization four vector of the $ B^* $ meson. The combinations of the Wilson coefficients $ a_1=C_2+C_1/3 $ and $ a_2=C_1+C_2/3 $. Similarly, the amplitudes for the decays $ B^{*0}_{(s)} \to D_{(s)} K $ are listed as

The amplitudes of the decays $ B_{c}^{*+} \to B_{(s)}^{0}\pi(K), \eta_{c}(1S,2S)\pi(K) $ are written as

For the decays $ B^{*}\rightarrow P M_{2} $ with $ M_2 $ being $ \rho, K^* $, the hadronic matrix elements can be expressed as

where $ \lambda=0, \pm $ denotes the helicity of vector meson, q stands for $ s,d $ and b quarks corresponding to the transitions $ B^*_c\to B, B^*_c\to B_s $ and $ B_c\to \eta_c(1S,2S), B^*_{(s)}\to D_{(s)} $, respectively, $ {\cal{H}}_{\lambda}=\langle M_{2}\left|J^{\mu}\right| 0\rangle\langle P \left|J_{\mu}\right| B^{*}\rangle $ is given as follows

Along with the branching fraction, the polarization fractions are defined as

where $ H_{\parallel} $ and $ H_{\perp} $ are parallel and perpendicular amplitudes, respectively, and can be obtained through $ H_{\parallel,\perp}=\dfrac{(H_{-}\pm H_{+})}{\sqrt{2}} $.

The input parameters, including the masses of the initial and the final mesons, the CKM matrix elements, the shape parameters fitted by the decay constants and so on are listed in Table 1.

Mass(GeV)	$ m_{b}=4.8 $	$ m_{c}=1.4 $	$ m_{s}=0.37 $	$ m_{u,d}=0.25 $	$ m_{e}=0.000511 $
	$m_{\pi}=0.140$	$m_{K}=0.494$	$m_{\rho}=0.775$	$m_{K^*}=0.892$	$m_{\mu}=0.106$
	$m_{\eta_c}=2.9839$	$m_{\eta_c(2S)}=3.6377 $	$ m_{D}=1.86966$	$m_{D_{s}}=1.96835 $	$m_{\tau}=1.78 $
	$m_{B}=5.27965 $	$m_{B_{s}}=5.36688 $	$m_{B^*}=5.32470 $	$m_{B^*_{s}}=5.4154 $	$m_{B^*_{c}}=6.332 $
CKM	$V_{ub}=(0.00382\pm0.0002)$		$V_{cb}=(0.0408\pm0.0014)$
	$V_{cd}=(0.221\pm0.004)$		$V_{us}=(0.2243\pm0.0008)$
	$V_{ud}=(0.97373\pm0.00031)$		$V_{cs}=(0.975\pm0.006)$
Decay constants(GeV)	$f_{\pi}=0.130\pm0.002$		$f_{K}=0.16\pm0.004$	$f_{\rho}=0.209\pm0.002$
	$f_{K^*}=0.217\pm0.005$		$f_{\eta_c}=0.335\pm0.075$	$f_{\eta_c(2S)}=0.243^{+0.079}_{-0.111}$
	$f_{D}=0.235\pm0.016$		$f_{D_s}=0.290\pm0.046$	$f_{B^*}=0.210\pm0.020$
	$f_{B^*_{s}}=0.260\pm0.020$		$f_{B^*_{c}}=0.5355\pm0.0578$	$f_{B}=0.190\pm0.020$
	$f_{B_{s}}=0.230\pm0.030$
Shape parameters(GeV)	$\beta^{'}_{\eta_{c}}=0.814^{+0.092}_{-0.086}$		$\beta^{'}_{\eta_{c}(2S)}=0.488^{+0.140}_{-0.187}$	$\beta^{'}_{B^*_{c}}=0.954^{+0.065}_{-0.069}$
	$\beta^{'}_{B^*}=0.528^{+0.033}_{-0.034}$		$\beta^{'}_{B^*_{s}}=0.599^{+0.033}_{-0.032}$	$\beta^{'}_{B}=0.555^{+0.048}_{-0.048}$
	$\beta^{'}_{D}=0.464^{+0.011}_{-0.014}$		$\beta^{'}_{D_{s}}=0.497^{+0.032}_{-0.028}$

Table 1.  The values of the input parameters [12, 17, 18, 20−22].

Although the $ B^{*} $ and $ B_{s}^{*} $ mesons have been experimentally discovered [17], while there are still lack of available experimental and theoretical information on their decay widths at present. In order to evaluate the branching ratio, the total decay width is essential. In view of the prohibition of the strong decays in phase space, the total decay widths of these vector b-flavored mesons can be estimated by the partial widths of the single-photon decays. In this work we will adopt the result of Refs. [23, 24] for the decay width $ \Gamma(B^{*+}\to B^+\gamma) $ and take the most recent results [25] for the $ \Gamma(B^{*0}_{s}\to B^0_s\gamma) $ and $ \Gamma(B^{*0}\to B^0\gamma) $ to evaluate the branching fractions,

One can find that there exists obvious difference between the charged and neutral $ B^* $ mesons, whose weak decays will be calculated later. The difference will induce large discrepancy in the branching ratios between their decay channels. Certainly, there are many other theoretical predictions on the single-photon decay widths, which can be found in Table 2.

References	[25]	[18, 19]	[27]	[28]	[29]	[23, 24]	[30, 31]	[32, 33]	[34]	[35]	[36]
$ \Gamma(B^{*+}\to B^{+}\gamma) $	$ 0.362\pm0.072 $	0.19	0.520	0.52	$ 0.349\pm0.018 $	0.577	0.0674	0.4	−	−	−
$ \Gamma(B^{*0}\to B^{0}\gamma) $	$ 0.117\pm0.022 $	0.070	0.165	0.14	$ 0.116\pm0.006 $	0.181	0.0096	0.13	−	−	−
$ \Gamma(B^{*0}_{s}\to B^{0}_{s}\gamma) $	$ 0.094\pm0.018 $	0.054	0.115	0.06	$ 0.084^{+0.011}_{-0.009} $	0.119	0.148	0.068	−	−	−
$ \Gamma(B^{*+}_{c}\to B^{+}_{c}\gamma) $	$ 0.045\pm0.009 $	0.033	0.039	0.030	$ 0.049^{+0.028}_{-0.021} $	0.023	0.034	0.022	0.135	0.060	0.050

Table 2.  The widths of radiative decays of $ B^{*}_{u,d,s,c} $ mesons in units of keV.

For the $ B^{*}_{c} $ meson has not been experimentally discovered, here we take the theoretical value $ m_{B^{*+}_{c}}=6.332 $ GeV predicted by the relativistic quark model [18]. The predictions on its single-photon decay width have been given in many different theoretical models, which are also listed in Table 2. In this paper, we will take an intermediate value [26],

The form factor as a nonperturbative hadronic parameter is essential in evaluating the hadronic matrix element. we adopt the CLFQM [37−40] to calculate the form factor values. All the computations are carried out within the $ q^+=0 $ reference frame, where the form factors can only be obtained at spacelike momentum transfers $ q^2=-q^2_{\bot}\leq0 $. It is need to know the form factors in the timelike region for the physical decay processes. Here we use the following double-pole approximation to parametrize the form factors obtained in the spacelike region and then extend to the timelike region,

where m represents the initial meson mass and $ F(q^{2}) $ denotes the different form factors, such as $ V,A_{0},A_{1} $ and $ A_{2} $. The values of a and b can be obtained by performing a 3-parameter fit to the form factors in the range $ -(m_{B^*}-m_P)^2\; {\rm{GeV}}^2\leq q^2\leq0 $ with the subscript P representing the final states, such as $ \pi $, K, $ B_{(s)} $, $ D_{(s)} $ and $ \eta_{c}(1S,2S) $. The results are collected in Tables 3-4, where the uncertainties arise from the decay constants of the initial and final mesons.

F	F(0)	$ F(q^2_{\rm {max}}) $	a	b
$ V^{B^{*} D} $	$ 0.75^{+0.00+0.00}_{-0.00-0.01} $	$ 0.79^{+0.08+0.16}_{-0.07-0.14} $	$ 0.70^{+0.30+0.25}_{-0.27-0.25} $	$ 1.39^{+0.07+0.07}_{-0.06-0.07} $
$ A_0^{B^{*} D} $	$ 0.63^{+0.00+0.00}_{-0.00-0.00} $	$ 0.69^{+0.06+0.09}_{-0.05-0.08} $	$ 0.43^{+0.30+0.27}_{-0.27-0.27} $	$ 0.59^{+0.06+0.08}_{-0.07-0.06} $
$ A_1^{B^{*} D} $	$ 0.66^{+0.00+0.00}_{-0.01-0.00} $	$ 0.72^{+0.02+0.05}_{-0.01-0.04} $	$ 0.35^{+0.28+0.24}_{-0.25-0.23} $	$ 0.43^{+0.06+0.06}_{-0.04-0.05} $
$ A_2^{B^{*} D} $	$ 0.57^{+0.00+0.00}_{-0.00-0.00} $	$ 0.61^{+0.01+0.07}_{-0.01-0.06} $	$ 0.62^{+0.26+0.23}_{-0.24-0.23} $	$ 1.16^{+0.04+0.06}_{-0.04-0.05} $
$ V^{B^{*} \pi} $	$ 0.29^{+0.00+0.00}_{-0.00-0.00} $	$ 0.29^{+0.07+0.16}_{-0.07-0.14} $	$ 1.21^{+0.23+0.21}_{-0.21-0.21} $	$ 1.27^{+0.04+0.06}_{-0.04-0.06} $
$ A_0^{B^{*} \pi} $	$ 0.25^{+0.00+0.00}_{-0.00-0.00} $	$ 0.43^{+0.06+0.08}_{-0.05-0.08} $	$ 0.78^{+0.23+0.22}_{-0.21-0.23} $	$ 0.36^{+0.06+0.06}_{-0.04-0.06} $
$ A_1^{B^{*} \pi} $	$ 0.30^{+0.00+0.01}_{-0.00-0.00} $	$ 0.46^{+0.01+0.04}_{-0.01-0.04} $	$ 0.58^{+0.21+0.19}_{-0.19-0.19} $	$ 0.22^{+0.03+0.04}_{-0.03-0.04} $
$ A_2^{B^{*} \pi} $	$ 0.21^{+0.00+0.00}_{-0.00-0.00} $	$ 0.23^{+0.01+0.06}_{-0.02-0.06} $	$ 1.04^{+0.19+0.18}_{-0.18-0.18} $	$ 0.97^{+0.03+0.04}_{-0.04-0.04} $
$ V^{B^{*}_{s} D_{s}} $	$ 0.76^{+0.01+0.01}_{-0.01-0.01} $	$ 0.78^{+0.03+0.09}_{-0.04-0.08} $	$ 0.74^{+0.13+0.10}_{-0.13-0.10} $	$ 1.62^{+0.02+0.03}_{-0.03-0.03} $
$ A_0^{B^{*}_{s} D_{s}} $	$ 0.63^{+0.00+0.01}_{-0.00-0.01} $	$ 0.67^{+0.02+0.07}_{-0.02-0.07} $	$ 0.47^{+0.13+0.11}_{-0.13-0.11} $	$ 0.72^{+0.02+0.02}_{-0.03-0.02} $
$ A_1^{B^{*}_{s} D_{s}} $	$ 0.66^{+0.00+0.01}_{-0.00-0.01} $	$ 0.70^{+0.00+0.05}_{-0.01-0.05} $	$ 0.39^{+0.12+0.12}_{-0.12-0.12} $	$ 0.56^{+0.01+0.02}_{-0.02-0.02} $
$ A_2^{B^{*}_{s} D_{s}} $	$ 0.56^{+0.00+0.01}_{-0.00-0.00} $	$ 0.59^{+0.02+0.05}_{-0.03-0.04} $	$ 0.66^{+0.11+0.11}_{-0.10-0.11} $	$ 1.36^{+0.01+0.02}_{-0.01-0.02} $
$ V^{B^{*}_{s} K} $	$ 0.34^{+0.00+0.01}_{-0.00-0.01} $	$ 0.36^{+0.03+0.09}_{-0.04-0.08} $	$ 1.23^{+0.13+0.10}_{-0.13-0.10} $	$ 1.43^{+0.02+0.03}_{-0.03-0.03} $
$ A_0^{B^{*}_{s} K} $	$ 0.27^{+0.00+0.01}_{-0.00-0.01} $	$ 0.42^{+0.02+0.07}_{-0.02-0.07} $	$ 0.77^{+0.13+0.11}_{-0.13-0.11} $	$ 0.42^{+0.02+0.02}_{-0.03-0.02} $
$ A_1^{B^{*}_{s} K} $	$ 0.31^{+0.00+0.01}_{-0.00-0.01} $	$ 0.45^{+0.00+0.05}_{-0.01-0.05} $	$ 0.60^{+0.12+0.12}_{-0.12-0.12} $	$ 0.28^{+0.01+0.02}_{-0.02-0.02} $
$ A_2^{B^{*}_{s} K} $	$ 0.22^{+0.00+0.00}_{-0.00-0.01} $	$ 0.26^{+0.02+0.05}_{-0.03-0.04} $	$ 1.02^{+0.11+0.11}_{-0.10-0.11} $	$ 1.02^{+0.01+0.02}_{-0.01-0.02} $

Table 3.  The form factors of the transitions $ B^{*}\to D,\pi, B_{s}^{*}\to D_{s},K $ in the CLFQM. The uncertainties are from the decay constants of the initial and final mesons, respectively.

F	F(0)	$ F(q^2_{\rm {max}}) $	a	b
$ V^{B_{c}^{*} D} $	$ 0.26^{+0.01+0.01}_{-0.01-0.02} $	$ 0.42^{+0.08+0.16}_{-0.07-0.14} $	$ 1.58^{+0.30+0.25}_{-0.27-0.25} $	$ 1.65^{+0.07+0.07}_{-0.06-0.07} $
$ A_0^{B_{c}^{*} D} $	$ 0.15^{+0.01+0.01}_{-0.01-0.01} $	$ 0.24^{+0.06+0.09}_{-0.05-0.08} $	$ 1.18^{+0.30+0.27}_{-0.27-0.27} $	$ 0.82^{+0.06+0.08}_{-0.07-0.06} $
$ A_1^{B_{c}^{*} D} $	$ 0.16^{+0.01+0.01}_{-0.01-0.01} $	$ 0.26^{+0.02+0.05}_{-0.01-0.04} $	$ 1.10^{+0.28+0.24}_{-0.25-0.23} $	$ 0.70^{+0.06+0.06}_{-0.04-0.05} $
$ A_2^{B_{c}^{*} D} $	$ 0.13^{+0.01+0.00}_{-0.01-0.01} $	$ 0.20^{+0.01+0.07}_{-0.01-0.06} $	$ 1.33^{+0.26+0.23}_{-0.24-0.23} $	$ 1.21^{+0.04+0.06}_{-0.04-0.05} $
$ V^{B_{c}^{*} B} $	$ 3.31^{+0.10+0.18}_{-0.10-0.22} $	$ 3.91^{+0.07+0.16}_{-0.07-0.14} $	$ 6.22^{+0.23+0.21}_{-0.21-0.21} $	$ 24.49^{+0.04+0.06}_{-0.04-0.06} $
$ A_0^{B_{c}^{*} B} $	$ 0.43^{+0.01+0.01}_{-0.02-0.00} $	$ 0.48^{+0.06+0.08}_{-0.05-0.08} $	$ 3.93^{+0.23+0.22}_{-0.21-0.23} $	$ 10.86^{+0.06+0.06}_{-0.04-0.06} $
$ A_1^{B_{c}^{*} B} $	$ 0.44^{+0.00+0.01}_{-0.00-0.01} $	$ 0.49^{+0.01+0.04}_{-0.01-0.04} $	$ 4.32^{+0.21+0.19}_{-0.19-0.19} $	$ 11.26^{+0.03+0.04}_{-0.03-0.04} $
$ A_2^{B_{c}^{*} B} $	$ 0.28^{+0.15+0.15}_{-0.18-0.16} $	$ 0.27^{+0.01+0.06}_{-0.02-0.06} $	$ -0.65^{+0.19+0.18}_{-0.18-0.18} $	$ 9.85^{+0.03+0.04}_{-0.04-0.04} $
$ V^{B_{c}^{*} B_{s}} $	$ 3.61^{+0.09+0.11}_{-0.10-0.14} $	$ 4.12^{+0.03+0.09}_{-0.04-0.08} $	$ 5.78^{+0.13+0.10}_{-0.13-0.10} $	$ 17.20^{+0.02+0.03}_{-0.03-0.03} $
$ A_0^{B_{c}^{*} B_{s}} $	$ 0.50^{+0.01+0.01}_{-0.02-0.01} $	$ 0.54^{+0.02+0.07}_{-0.02-0.07} $	$ 3.70^{+0.13+0.11}_{-0.13-0.11} $	$ 7.75^{+0.02+0.02}_{-0.03-0.02} $
$ A_1^{B_{c}^{*} B_{s}} $	$ 0.52^{+0.00+0.00}_{-0.00-0.01} $	$ 0.57^{+0.00+0.05}_{-0.01-0.05} $	$ 4.01^{+0.12+0.12}_{-0.12-0.12} $	$ 7.98^{+0.01+0.02}_{-0.02-0.02} $
$ A_2^{B_{c}^{*} B_{s}} $	$ 0.27^{+0.17+0.16}_{-0.21-0.16} $	$ 0.27^{+0.02+0.05}_{-0.03-0.04} $	$ -0.90^{+0.11+0.11}_{-0.10-0.11} $	$ 7.57^{+0.01+0.02}_{-0.01-0.02} $
$ V^{B_{c}^{*} \eta_{c}} $	$ 0.88^{+0.01+0.04}_{-0.01-0.06} $	$ 0.87^{+0.03+0.09}_{-0.04-0.08} $	$ 1.10^{+0.13+0.10}_{-0.13-0.10} $	$ 4.09^{+0.02+0.03}_{-0.03-0.03} $
$ A_0^{B_{c}^{*} \eta_{c}} $	$ 0.56^{+0.01+0.01}_{-0.02-0.02} $	$ 0.59^{+0.02+0.07}_{-0.02-0.07} $	$ 0.77^{+0.13+0.11}_{-0.13-0.11} $	$ 2.15^{+0.02+0.02}_{-0.03-0.02} $
$ A_1^{B_{c}^{*} \eta_{c}} $	$ 0.59^{+0.01+0.02}_{-0.02-0.03} $	$ 0.62^{+0.00+0.05}_{-0.01-0.05} $	$ 0.72^{+0.12+0.12}_{-0.12-0.12} $	$ 1.94^{+0.01+0.02}_{-0.02-0.02} $
$ A_2^{B_{c}^{*} \eta_{c}} $	$ 0.45^{+0.02+0.01}_{-0.02-0.02} $	$ 0.46^{+0.02+0.05}_{-0.03-0.04} $	$ 0.93^{+0.11+0.11}_{-0.10-0.11} $	$ 3.11^{+0.01+0.02}_{-0.01-0.02} $
$ V^{B_{c}^{*} \eta_{c}(2S)} $	$ 0.56^{+0.01+0.01}_{-0.01-0.03} $	$ 0.58^{+0.03+0.09}_{-0.04-0.08} $	$ 0.77^{+0.13+0.10}_{-0.13-0.10} $	$ 3.00^{+0.02+0.03}_{-0.03-0.03} $
$ A_0^{B_{c}^{*} \eta_{c}(2S)} $	$ 0.31^{+0.00+0.05}_{-0.00-0.07} $	$ 0.30^{+0.02+0.07}_{-0.02-0.07} $	$ 0.18^{+0.13+0.11}_{-0.13-0.11} $	$ 1.51^{+0.02+0.02}_{-0.03-0.02} $
$ A_1^{B_{c}^{*} \eta_{c}(2S)} $	$ 0.34^{+0.00+0.02}_{-0.00-0.10} $	$ 0.34^{+0.00+0.05}_{-0.01-0.05} $	$ 0.27^{+0.12+0.12}_{-0.12-0.12} $	$ 1.42^{+0.01+0.02}_{-0.02-0.02} $
$ A_2^{B_{c}^{*} \eta_{c}(2S)} $	$ 0.22^{+0.01+0.04}_{-0.01-0.15} $	$ 0.20^{+0.02+0.05}_{-0.03-0.04} $	$ -0.32^{+0.11+0.11}_{-0.10-0.11} $	$ 2.08^{+0.01+0.02}_{-0.01-0.02} $

Table 4.  The form factors of the transitions $ B^{*}_c\to D, B_{(s)},\eta_c(1S, 2S) $ in the CLFQM. The uncertainties are the same with those listed in Table 3.

In Tables 5-6, we compare the form factor values at maximum recoil ($ q^{2}=0 $) with those obtained within other different models. The form factors of the transitions $ B^{*}_{c}\to B_{(s)}, D, \eta_{c} $ and $ B^{*}_{s}\to D_{s}, K $ were investigated with the WSB model in Refs. [7, 9, 41]. Recently, the authors in Ref. [42] investigated the form factors of the transitions $ B^{*}_{c}\to B_{(s)}, \eta_{c}(1S,2S) $ within the light-front quark model. Certainly, the form factors of the $ B^{*} \to D $ and $ B_{s}^{*} \to D_{s}, K $ were analysed within the model independent approach [43] as well. For the form factors related to the $ B^*_{(s)} $ transitions, one can find that even in the same WSB model, the valuses of $ A_2 $ for the transitions $ B^*_s\to D_s, K $ given by different authors [7, 41] are opposite in signs. Besides the serval sign differences among these predictions, most of the results given by different approaches are consistent well with each other. As to the form factors related to the $ B^*_c $ transitions, there exist large differences predicted by the BSW approach with different values for $ \omega $, which refers to the average transverse quark momentum. Obviously, for the form factors of the transitions $ B^{*}_{c}\to B_{(s)}, \eta_{c}(1S,2S) $, our predictions are comparable with the previous light-front quark model calculations [42] within errors except for the $ A_2 $ values. These differences can be clarified by the future experiments.

Transitions	References	V(0)	$ A_{0}(0) $	$ A_{1}(0) $	$ A_{2}(0) $
$ B^{*}\rightarrow D $	This work	0.75	0.63	0.66	0.57
	[43]	0.76	0.71	0.75	0.62
	[7]	0.70	0.63	0.66	0.56
$ B^{*}\rightarrow \pi $	This work	0.29	0.25	0.30	0.21
	[7]	0.34	0.34	0.38	0.29
$ B^{*}_{s}\rightarrow D_{s} $	This work	0.76	0.63	0.66	0.56
	[41]1	0.68	0.61	0.63	-0.55
	[41]2	0.77	0.67	0.72	-0.56
	[43]	0.72	0.66	0.69	0.59
	[7]	0.67	0.59	0.61	0.54
$ B^{*}_{s}\rightarrow K $	This work	0.34	0.27	0.31	0.22
	[41]1	0.30	0.27	0.28	-0.25
	[41]2	0.27	0.24	0.26	-0.21
	[43]	0.30	0.28	0.29	0.26
	[7]	0.32	0.29	0.31	0.28
1 The form factors are computed with the parameter $\omega=0.4$ GeV in the WSB model. 2 The form factors are computed with flavor dependent parameter $\omega$ in the WSB model.

Table 5.  The form factors of the transitions $ B^{*}\to D,\pi, $$ B_{s}^{*}\to D_{s},K $ at $ q^{2}= 0 $ in the CLFQM, together with other theoretical results.

Transition	References	V(0)	$ A_{0}(0) $	$ A_{1}(0) $	$ A_{2}(0) $
$ B^{*}_{c}\rightarrow D $	This work	0.26	0.15	0.16	0.13
	[41]1	0.020	0.013	0.012	-0.016
	[41]2	0.084	0.045	0.052	-0.033
	[41]3	0.60	0.29	0.37	-0.15
$ B^{*}_{c}\rightarrow B $	This work	3.31	0.43	0.44	0.28
	[41]1	2.62	0.31	0.35	0.13
	[41]2	5.09	0.43	0.67	2.19
	[41]3	6.93	0.90	0.91	-0.83
	[44]1	1.95	0.54	0.29	3.29
	[44]4	2.88	0.78	0.43	4.70
	[44]3	3.61	0.94	0.54	5.41
	[42]	3.08	0.60	0.65	0.91
$ B^{*}_{c}\rightarrow B_{s} $	This work	3.61	0.50	0.52	0.27
	[41]1	2.89	0.37	0.43	0.43
	[41]2	5.56	0.58	0.83	2.45
	[41]3	6.52	0.92	0.98	-0.24
	[44]1	1.87	0.61	0.36	3.62
	[44]4	2.55	0.82	0.49	4.79
	[44]3	3.10	0.95	0.60	5.27
	[42]	3.40	0.69	0.75	0.96
$ B^{*}_{c}\rightarrow \eta_{c} $	This work	0.88	0.56	0.59	0.45
	[41]1	0.21	0.15	0.15	-0.15
	[41]2	0.79	0.53	0.57	-0.40
	[41]3	1.21	0.73	0.87	-0.33
	[42]	0.91	0.66	0.69	0.59
$ B^{*}_{c}\rightarrow \eta_{c}(2S) $	This work	0.56	0.31	0.34	0.22
	[42]	0.59	0.43	0.41	0.51
1 The form factors are computed with the parameter $\omega=0.4$ GeV in the WSB model. 2 The form factors are computed with flavor dependent parameter $\omega$ in the WSB model. 3 The form factors are computed with the QCD inspired parameter $\omega = m_{B^*_c}\alpha_{s}$ in the WSB model. 4 The form factors are computed with flavor dependent parameter $\omega=0.6$ GeV in the WSB model.

Table 6.  The form factors of the transitions $ B^{*}_c\to D, B_{(s)}, $$ \eta_c(1S, 2S) $ at $ q^{2}= 0 $, together with other theoretical results.

We plot the $ q^2 $-dependencies of the form factors of the transitions $ B^*\to D, \pi $, $ B^*_s\to D_{s}, K $ and $ B^*_c\to B_{(s)}, D, \eta_c(1S,2S) $ shown in Figure 2. Compared to the form factors of the transitions $ B^*_{(s)}\to D_{(s)} $, those of the transitions $ B^*\to \pi $ and $ B^*_{s}\to K $ are more sensitive to the change of $ q^2 $. Compared with the single pole expression, the double-pole approximation makes the predictions less model dependent. Nonetheless, when the mass difference between the initial and final sates is large, for example, the final state is $ \pi $ or K, there still exists the model dependent, that is the non-monotonous $ q^2 $ behaviours. This is the reason why the transition form factors predicted by the CLFQM in small $ q^2 $ region are more reliable than those given in large $ q^2 $ region. It is just contrary to the case of lattice QCD predictions. If the transitions, such as $ B^*_c\to B_{(s)} $, have small phase space, their corresponding form factors can indeed display a monotonous $ q^2 $ behaviours. Among the form factors of the transitions $ B^*_c\to B_{(s)} $, the values of $ V(q^2) $ are much larger than those of $ A_{0,1,2}(q^2) $.

Figure 2.  (color online) Form factors $ V(q^2) $ and $ A_{0,1,2}(q^2) $ for the transitions $ B^*\to D, \pi $, $ B^*_s\to D_{s}, K $ and $ B^*_c\to B_{(s)}, D, \eta_c(1S,2S) $.

As the flavor changing processes in the Standard Model, the semileptonic heavy flavor meson decays are important to extract the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements. The form factors involving the dynamical information play an essential role in these semileptonic decays. Based on the form factors and the helicity amplitudes provided in the previous section, the branching ratios of the semileptonic $ B^{*+} $, $ B_{s}^{*0} $ and $ B_{c}^{*+} $ decays are presented in Table 7. In order to study the impact of lepton mass and provide a more detailed physical picture for the semileptonic decays, we also define other two physical observables, that is the forward-backward asymmetry $ A_{FB} $ and the longitudinal polarization fraction $ f_{L} $. The corresponding results are listed in Tables 8 and 9, respectively. The theoretical uncertainties for our calculations are caused by the decay widths of the initial mesons, the decay constants of the initial and final state mesons, respectively. The following are some discussions:

Channels	This work	BS [45]	WSB [7]
$ B^{*+} \to \bar{D}^{0}e^{+}{\nu}_{e} $	$ (5.19^{+1.36+0.17+0.19}_{-0.89-0.02-0.04})\times10^{-8} $	$ 3.29\times10^{-8} $	$ 2.29\times10^{-8} $
$ B^{*+} \to \bar{D}^{0}\mu^{+}{\nu}_{\mu} $	$ (5.16^{+1.35+0.17+0.18}_{-0.89-0.02-0.04})\times10^{-8} $	$ 3.27\times10^{-8} $	$ 2.29\times10^{-8} $
$ B^{*+} \to \bar{D}^{0}\tau^{+}{\nu}_{\tau} $	$ (1.30^{+0.34+0.04+0.04}_{-0.22-0.01-0.00})\times10^{-8} $	$ 0.82\times10^{-8} $	$ 0.68\times10^{-8} $
$ B_{s}^{*0} \to D_{s}^{-}e^{+}{\nu}_{e} $	$ (3.25^{+0.77+0.01+0.11}_{-0.52-0.11-0.01})\times10^{-7} $	$ 2.04\times10^{-7} $	$ 1.39\times10^{-7} $
$ B_{s}^{*0} \to D_{s}^{-}\mu^{+}{\nu}_{\mu} $	$ (3.23^{+0.77+0.01+0.11}_{-0.52-0.11-0.01})\times10^{-7} $	$ 2.03\times10^{-7} $	$ 1.39\times10^{-7} $
$ B_{s}^{*0} \to D_{s}^{-}\tau^{+}{\nu}_{\tau} $	$ (8.09^{+1.92+0.03+0.21}_{-1.30-0.24-0.03})\times10^{-8} $	$ 5.35\times10^{-8} $	$ 4.08\times10^{-8} $
$ B_{s}^{*0} \to K^{-}e^{+}{\nu}_{e} $	$ (6.06^{+1.43+0.56+0.31}_{-0.98-0.44-0.17})\times10^{-9} $	−	$ 1.67\times10^{-9} $
$ B_{s}^{*0} \to K^{-}\mu^{+}{\nu}_{\mu} $	$ (6.03^{+1.43+0.56+0.30}_{-0.97-0.44-0.17})\times10^{-9} $	−	$ 1.67\times10^{-9} $
$ B_{s}^{*0} \to K^{-}\tau^{+}{\nu}_{\tau} $	$ (2.56^{+0.61+0.18+0.45}_{-0.41-0.13-0.08})\times10^{-9} $	−	$ 1.07\times10^{-9} $
$ B_{c}^{*+} \to D^{0} e^{+}{\nu}_{e} $	$ (1.36^{+0.35+0.13+0.20}_{-0.23-0.13-0.15})\times10^{-9} $	$ 1.60\times10^{-9} $
$ B_{c}^{*+} \to D^{0} \mu^{+}{\nu}_{\mu} $	$ (1.36^{+0.35+0.13+0.20}_{-0.23-0.13-0.15})\times10^{-9} $	$ 1.60\times10^{-9} $	−
$ B_{c}^{*+} \to D^{0} \tau^{+}{\nu}_{\tau} $	$ (0.65^{+0.17+0.07+0.08}_{-0.11-0.06-0.07})\times10^{-9} $	$ 1.08\times10^{-9} $	−
$ B_{c}^{*+} \to B^{0}_{s} e^{+}{\nu}_{e} $	$ (7.51^{+1.92+0.17+0.10}_{-1.27-0.08-0.37})\times10^{-7} $	$ 9.43\times10^{-7} $	−
$ B_{c}^{*+} \to B^{0}_{s} \mu^{+}{\nu}_{\mu} $	$ (7.06^{+1.80+0.14+0.06}_{-1.19-0.10-0.32})\times10^{-7} $	$ 8.96\times10^{-7} $	−
$ B_{c}^{*+} \to B^{0} e^{+}{\nu}_{e} $	$ (4.37^{+1.11+0.14+0.27}_{-0.74-0.12-0.27})\times10^{-8} $	$ 5.78\times10^{-8} $	−
$ B_{c}^{*+} \to B^{0} \mu^{+}{\nu}_{\mu} $	$ (4.15^{+1.06+0.11+0.24}_{-0.70-0.10-0.24})\times10^{-8} $	$ 5.57\times10^{-8} $	−
$ B_{c}^{*+} \to \eta_{c} e^{+}{\nu}_{e} $	$ (4.48^{+1.14+0.28+0.40}_{-0.76-0.10-0.50})\times10^{-7} $	$ 4.20\times10^{-7} $	−
$ B_{c}^{*+} \to \eta_{c} \mu^{+}{\nu}_{\mu} $	$ (4.45^{+1.14+0.27+0.39}_{-0.75-0.10-0.49})\times10^{-7} $	$ 4.19\times10^{-7} $	−
$ B_{c}^{*+} \to \eta_{c} \tau^{+}{\nu}_{\tau} $	$ (1.03^{+0.26+0.07+0.07}_{-0.17-0.04-0.10})\times10^{-7} $	$ 1.26\times10^{-7} $	−
$ B_{c}^{*+} \to \eta_{c}(2S) e^{+}{\nu}_{e} $	$ (5.65^{+1.44+0.02+0.59}_{-0.96-0.03-0.29})\times10^{-8} $	−	−
$ B_{c}^{*+} \to \eta_{c}(2S) \mu^{+}{\nu}_{\mu} $	$ (5.60^{+1.43+0.02+0.60}_{-0.95-0.03-0.29})\times10^{-8} $	−	−
$ B_{c}^{*+} \to \eta_{c}(2S) \tau^{+}{\nu}_{\tau} $	$ (4.92^{+1.26+0.11+0.12}_{-0.83-0.11-0.23})\times10^{-9} $	−	−

Table 7.  Branching ratios of the semileptonic decays $ B^{*}_{u,s,c}\to P \ell^{+}{\nu}_{\ell} $. For each entry, the first error is from the decay width of the initial meson, the second and third errors are caused by the decay constants of the initial and final mesons, respectively.

Channels	$ B^{*+} \to \bar{D}^{0}e^{+}\nu_{e} $	$ B^{*+} \to \bar{D}^{0}\mu^{+}\nu_{\mu} $	$ B^{*+} \to \bar{D}^{0}\tau^{+}\nu_{\tau} $
$ A_{FB} $	$ -0.19^{+0.00+0.04+0.04}_{-0.01-0.04-0.04} $	$ -0.19^{+0.00+0.04+0.04}_{-0.01-0.04-0.04} $	$ -0.14^{+0.02+0.00+0.00}_{-0.04-0.00-0.00} $
Channel	$ B_{s}^{*0} \to D^{-}_{s}e^{+}\nu_{e} $	$ B_{s}^{*0} \to D^{-}_{s}\mu^{+}\nu_{\mu} $	$ B_{s}^{*0} \to D^{-}_{s}\tau^{+}\nu_{\tau} $
$ A_{FB} $	$ -0.19^{+0.03+0.00+0.00}_{-0.04-0.00-0.00} $	$ -0.18^{+0.03+0.01+0.00}_{-0.04-0.01-0.00} $	$ -0.13^{+0.02+0.00+0.00}_{-0.03-0.00-0.00} $
Channels	$ B_{s}^{*0} \to K^{-}e^{+}\nu_{e} $	$ B_{s}^{*0} \to K^{-}\mu^{+}\nu_{\mu} $	$ B_{s}^{*0} \to K^{+}\tau^{+}\nu_{\tau} $
$ A_{FB} $	$ -0.10^{+0.02+0.00+0.01}_{-0.02-0.00-0.01} $	$ -0.10^{+0.02+0.00+0.01}_{-0.02-0.00-0.01} $	$ -0.09^{+0.02+0.00+0.01}_{-0.02-0.00-0.01} $
Channels	$ B_{c}^{*+} \to D^{0}e^{+}\nu_{e} $	$ B_{c}^{*+} \to D^{0}\mu^{+}\nu_{\mu} $	$ B_{c}^{*+} \to D^{0}\tau^{+}\nu_{\tau} $
$ A_{FB} $	$ -0.28^{+0.05+0.03+0.04}_{-0.07-0.02-0.03} $	$ -0.28^{+0.05+0.03+0.04}_{-0.07-0.02-0.03} $	$ -0.24^{+0.04+0.02+0.03}_{-0.06-0.02-0.02} $
Channels	$ B_{c}^{*+} \to \eta_{c}e^{+}\nu_{e} $	$ B_{c}^{*+} \to \eta_{c}\mu^{+}\nu_{\mu} $	$ B_{c}^{*+} \to \eta_{c}\tau^{+}\nu_{\tau} $
$ A_{FB} $	$ -0.19^{+0.03+0.01+0.02}_{-0.05-0.01-0.01} $	$ -0.19^{+0.03+0.01+0.02}_{-0.05-0.01-0.01} $	$ -0.14^{+0.02+0.00+0.02}_{-0.04-0.01-0.01} $
Channels	$ B_{c}^{*+} \to \eta_{c}(2S)e^{+}\nu_{e} $	$ B_{c}^{*+} \to \eta_{c}(2S)\mu^{+}\nu_{\mu} $	$ B_{c}^{*+} \to \eta_{c}(2S)\tau^{+}\nu_{\tau} $
$ A_{FB} $	$ -0.16^{+0.03+0.00+0.03}_{-0.04-0.00-0.02} $	$ -0.16^{+0.03+0.00+0.03}_{-0.04-0.00-0.02} $	$ -0.11^{+0.02+0.00+0.02}_{-0.03-0.00-0.05} $
Channels	$ B_{c}^{*+} \to B^{0}_{s}e^{+}\nu_{e} $	$ B_{c}^{*+} \to B^{0}_{s}\mu^{+}\nu_{\mu} $	−
$ A_{FB} $	$ -0.22^{+0.04+0.00+0.01}_{-0.06-0.01-0.01} $	$ -0.22^{+0.04+0.00+0.01}_{-0.06-0.01-0.01} $	−
Channels	$ B_{c}^{*+} \to B^{0}e^{+}\nu_{e} $	$ B_{c}^{*+} \to B^{0}\mu^{+}\nu_{\mu} $	−
$ A_{FB} $	$ -0.26^{+0.04+0.01+0.02}_{-0.07-0.01-0.02} $	$ -0.26^{+0.04+0.01+0.02}_{-0.07-0.01-0.02} $	−

Table 8.  The forward-backward asymmetries $ A_{FB} $ for the decays $ B^*_{u,s,c} \to P\ell^{+}\nu_{\ell} $.

Observables	Region 1	Region 2	Total		Observables	Region 1	Region 2	Total
$ f_{L}(B^{*+} \to \bar{D}^{0}l^{'+}\nu_{l^{'}}) $	0.69	0.43	$ 0.57^{+0.15+0.02+0.02}_{-0.10-0.00-0.01} $		$ f_{L}(B_{s}^{*0} \to D^{-}_{s}l^{'+}\nu_{l^{'}}) $	0.69	0.43	$ 0.57^{+0.14+0.00+0.02}_{-0.09-0.02-0.00} $
$ f_{L}(B^{*+} \to \bar{D}^{0}\tau^{+}\nu_{\tau}) $	0.56	0.42	$ 0.47^{+0.12+0.02+0.02}_{-0.08-0.00-0.00} $		$ f_{L}(B_{s}^{*0} \to D^{-}_{s}\tau^{+}\nu_{\tau}) $	0.56	0.42	$ 0.47^{+0.11+0.00+0.00}_{-0.08-0.02-0.01} $
Observables	Region 1	Region 2	Total		Observables	Region 1	Region 2	Total
$ f_{L}(B_{s}^{*0} \to K^{-}l^{'+}\nu_{l^{'}}) $	0.88	0.70	$ 0.82^{+0.18+0.06+0.18}_{-0.14-0.08-0.03} $		$ f_{L}(B_{c}^{*+} \to D^{0}l^{'+}\nu_{l^{'}}) $	0.65	0.42	$ 0.53^{+0.14+0.05+0.09}_{-0.09-0.05-0.05} $
$ f_{L}(B_{s}^{*0} \to K^{-}\tau^{+}\nu_{\tau}) $	0.83	0.68	$ 0.76^{+0.17+0.05+0.13}_{-0.13-0.04-0.03} $		$ f_{L}(B_{c}^{*+} \to D^{0}\tau^{+}\nu_{\tau}) $	0.57	0.42	$ 0.48^{+0.12+0.05+0.07}_{-0.08-0.04-0.05} $
Observables	Region 1	Region 2	Total		Observables	Region 1	Region 2	Total
$ f_{L}(B_{c}^{*+} \to \eta_{c}l^{'+}\nu_{l^{'}}) $	0.68	0.42	$ 0.57^{+0.14+0.03+0.06}_{-0.10-0.10-0.07} $		$ f_{L}(B_{c}^{*+} \to \eta_{c}(2S)l^{'+}\nu_{l^{'}}) $	0.70	0.43	$ 0.58^{+0.15+0.00+0.02}_{-0.10-0.00-0.32} $
$ f_{L}(B_{c}^{*+} \to \eta_{c}\tau^{+}\nu_{\tau}) $	0.55	0.41	$ 0.46^{+0.12+0.03+0.03}_{-0.08-0.01-0.05} $		$ f_{L}(B_{c}^{*+} \to \eta_{c}(2S)\tau^{+}\nu_{\tau}) $	0.50	0.39	$ 0.43^{+0.11+0.01+0.09}_{-0.07-0.01-0.20} $
Observables	Region 1	Region 2	Total		Observables	Region 1	Region 2	Total
$ f_{L}(B_{c}^{*+} \to B^{0}_{s}e^{+}\nu_{e}) $	0.65	0.40	$ 0.53^{+0.13+0.00+0.01}_{-0.09-0.02-0.03} $		$ f_{L}(B_{c}^{*+} \to B^{0}e^{+}\nu_{e}) $	0.63	0.39	$ 0.51^{+0.13+0.03+0.04}_{-0.19-0.02-0.04} $
$ f_{L}(B_{c}^{*+} \to B^{0}_{s}\mu^{+}\nu_{\mu}) $	0.64	0.40	$ 0.52^{+0.13+0.00+0.01}_{-0.09-0.02-0.03} $		$ f_{L}(B_{c}^{*+} \to B^{0}\mu^{+}\nu_{\mu}) $	0.62	0.39	$ 0.51^{+0.13+0.03+0.04}_{-0.19-0.02-0.04} $

Table 9.  The longitudinal polarization fractions $ f_{L} $ for the decays $ B^*_{u,s,c} \to P\ell^{+}\nu_{\ell} $ in Region 1, Region 2 and total physical region.

1. From Table 7, one can find that the branching ratios of the decays $ B^{*}_{s}\to K\ell{\nu}_{\ell} $ are significantly $ 1\sim2 $ orders smaller compared to those of the decays $ B^{*}_{(s)}\to D_{(s)}\ell{\nu}_{\ell} $. This is mainly caused by the smaller CKM matrix element $ V_{ub}=0.00382 $ compared to $ V_{cb}=0.0408 $. Furthermore, the form factor $ A_0^{B^{*}_{s}K} $ is only about 0.43 times of $ A_0^{B^{*}_{(s)}D_{(s)}} $. In addition, $ {\cal{B}}r(B_{s}^{*0} \to D_{s}^{-}\ell^{+}{\nu}_{\ell}) $ is about six times of $ {\cal{B}}r(B^{*+} \to \bar{D}^{0}\ell^{+}{\nu}_{\ell}) $, it is attributed to the small total decay width $ \Gamma_{tot}(B_{s}^{*0}) $, which is only about one sixth of $ \Gamma_{tot}(B^{*+}) $. Among these $ B^*_{(s)} $ decays, the channels $ B_{s}^{*0} \to D_{s}^{-}\ell^{\prime+}{\nu}_{\ell^\prime} $ with $ \ell^{\prime}=e, \mu $ ² have the largest branching fractions with 10^-7 order, which make them a top priority for observation. The branching ratios of the decays $ B^{*+}\to \bar{D}^{0}\ell^{+}{\nu}_{\ell} $ and $ B^{*0}_{s}\to D^{-}_s\tau^{+}{\nu}_{\tau} $ are at the order of 10^-8, which can be in the measurement scopes of LHC and SuperKEKB in the future. It is may be difficult to observe the decays $ B^{*0}_{s}\to K^{-}\ell^{+}{\nu}_{\ell} $ in experiments, because of their smallness branching fractions with 10^-9 order.

2. For the branching ratios of the $ B^{*+} $ and $ B_{s}^{*0} $ decays, our results are two times or even more larger than those given in the WSB model [7]. Although the form factors calculated using the CLFQM and the WSB are consistent with each other at $ q^{2}=0 $, the dependencies of them on $ q^2 $ between these two approaches are very different, which induce to the difference of the branching ratios. While the Bethe-Salpeter (BS) equation approach [45] gave the intermediate values, which were obtained by solving the instantaneous BS equation with a Cornell-like potential.

3. The branching ratios of the $ B^{*}_{c} $ decays induced by the $ \bar b\to \bar c(\bar u) $ and $ c\to d(s) $ transitions are calculated, which are also shown in Table 7. Our predictions are consistent with the results obtained in the BS method [45]. The decays $ B_{c}^{*+} \to B^{0}_{s} \ell^{\prime+}{\nu}_{\ell^\prime} $ and $ B_{c}^{*+} \to \eta_c \ell^{+}{\nu}_{\ell} $, which possess larger branching ratios compared to other semileptonic $ B^{*}_{c} $ decays, can be served as the golden channels to seek for the $ B^*_c $ meson in experiments. The branching ratios of the decays $ B_{c}^{*+} \to D^{0}\ell^{+}{\nu}_{\ell} $ are much smaller than those of other channels, the reason is that the CKM matrix element $ V_{ub}=3.82\times10^{-3} $ involved in these decays is very small. Comparing the branching ratios of the semileptonic decays $ B_{c}^{*+} \to \eta_{c}\ell^{+}{\nu}_{\ell} $ and $ B_{c}^{*+} \to \eta_{c}(2S)\ell^{+}{\nu}_{\ell} $, we can find that the later are about one order smaller than the former, which is because of the smaller form factor of the transition $ B_{c}^{*} \to \eta_{c}(2S) $ and the smaller phase space for the decays $ B_{c}^{*+} \to \eta_{c}(2S)\ell^{+}{\nu}_{\ell} $.

4. Some ratios of the branching fractions about the $ B_{(s)} $ and $ B_{c} $ meson decays have garnered significant attention due to their deviations between theoretical predictions and experimental data, such as $ R_{J/\Psi} $ anomaly [46], which may indicate possible new physics beyond the SM. It is meaningful to check whether the similar cases also exist in their vector partner decays. In view of this purpose, we define the following quantities

where P represents a pseudoscalar meson, such as $ K, D_{(s)}, \eta_c(1S,2S) $ and so on.

The corresponding values are listed as

From above ratios, we can find that our predictions are consistent with the results calculated by using the BS method in Ref. [45] within errors, where $ {\cal{R}}_{\bar D_{0}} $=0.249, $ {\cal{R}}_{D_{s}} $=0.262, $ {\cal{R}}_{{D}_{0}} $=0.675, $ {\cal{R}}_{\eta_{c}} $=0.300. Certainly, our results are also comparable with those obtained by the BSW model in Ref. [7], where $ {\cal{R}}_{\bar D_{0}} $=0.297, $ {\cal{R}}_{D_{s}} $=0.294, $ {\cal{R}}_{K} $=0.641. Since the numerator and denominator involving the same initial and final hadrons for each ratio share the same CKM matrix element and the form factor, part of uncertainties can be cancelled in the calculations. Therefore, these ratios are less model-dependent and can be checked by the future experiments.

5. In the calculations we can clearly find that the longitudinal polarization fractions $ f_{L} $ between the decays $ B^*_{u,s,c} \to P e\nu_{e} $ and $ B^*_{u,s,c} \to P\mu\nu_{\mu} $ are very close to each other, while $ f_L(B^*_{u,s,c} \to P \tau\bar{\nu}_{\tau}) $ is smaller by a few to $ 15 $ percent shown in Table 9, that is

which reflects the lepton flavor universality (LFU). In order to investigate the dependence of the polarization on $ q^2 $, we calculate the longitudinal polarization fractions by dividing the full energy region into two regions for each decay. Region 1 is defined as $ m_{\ell}^{2}<q^{2}<\dfrac{(m_{B^*}-m_{P})^{2}+m_{\ell}^{2}}{2} $ and Region 2 is $ \dfrac{(m_{B^*}-m_{P})^{2}+m_{\ell}^{2}}{2} <q^{2}<(m_{B^*}-m_{P})^{2} $. Interestingly, these decays $ B^*_{u,s,c} \to P \ell^{+}{\nu}_{\ell} $ except for the channels $ B_{s}^{*0} \to K^{-}\ell^{+}\nu_{\ell} $ are dominated by the transverse polarization in Region 2. Furthermore, among these decays $ B^*_{u, s, c}\to P \tau^{+}{\nu}_{\tau} $, only the channel $ B_{s}^{*0} \to K^{-}\tau^{+}\nu_{\tau} $ is dominated by the longitudinal polarization in the entire physical region. In a word, the special polarization for the decay $ B_{s}^{*0} \to K^{-}\ell^{+}\nu_{\ell} $ can be tested by the future high-luminosity experiments.

6. In Figures 3 and 4, we also display the $ q^{2} $-dependencies of the differential decay rates $ d\Gamma_{(L)}/dq^{2} $ and the forward-backward asymmetries $ A_{FB} $, respectively. In Figure 3, one can find that the $ q^2 $ dependencies of the differential decay rates $ d\Gamma/dq^2 $ ($ d\Gamma^{L}/dq^2 $) for the decays $ B^*\to P e\nu_e $ and $ B^*\to P \mu\nu_\mu $ almost coincide with each other. The longitudinal polarizations are dominant in the decays $ B_{s}^{*0} \to K^{-}\ell^{+}\nu_{\ell} $, which are obviously different from those of other decays. As to the forward-backward asymmetries $ A_{FB} $ for the decays $ B_{s}^{*0} \to K^{-}\ell^{\prime+}\nu_{\ell^\prime} $ and $ B_{s}^{*0} \to K^{-}\tau^{+}\nu_{\tau} $, their extreme values are more close to each other, which are shown in Figure 4.

Figure 3.  (color online) The $ q^2 $ dependencies of the differential decay rates $ d\Gamma/dq^2 $ (the solid lines) and $ d\Gamma^{L}/dq^2 $ (the dashed lines refer to the decays with $ e^+\nu_e $ involved, the dotted lines represent for the decays with $ \mu^+\nu_\mu $ involved, and the dash-dotted lines are for the decays with $ \tau^+\nu_\tau $ involved) for the decays $ B^*_{u,s,c} \to P\ell^{+}\nu_{\ell} $.

Figure 4.  (color online) The $ q^{2} $-dependencies of the forward-backward asymmetries $ A_{FB} $ for the decays $ B^*_{u,s,c} \to P\ell^{+}\nu_{\ell} $.

The decay rates of the nonleptonic $ B^{*}_{u,d,s,c} $ decays can be written as

where $ p_{cm} $ represents the three-momentum of the final states and is defined as

Using the input parameters given in Table 1, we can calculate the branching ratios of the nonleptonic decays $ B^{*}_{u,d,s,c}\to M_{1} M_{2} $, which are listed in Tables 12, 11 and 10 with the values obtained from Refs. [41, 42, 44, 45, 47, 48] for comparison. The uncertainties of our results arise from the decay widths of the vector mesons $ B^{*}_{u,d,s,c} $, the decay constants of the initial and final state mesons, respectively. The decay modes considered here are dominated by the color-favored factorizable contributions and insensitive to the nonfactorizable contributions. Numerically, we adopt the Wilson coefficients $ a_{1}=1.2 $, $ a_{2}=-0.5 $. The following are some comments:

Channel	This work	[5]	[10]	Unit
$ B^{*+} \to \pi^{+} {D}^{0} $	$ 3.14^{+0.82+0.06+0.12}_{-0.54-0.08-0.10} $	$ [0.5,3.2] $	−	10−14
$ B^{*+} \to \pi^{+} \bar D^{0} $	$ 0.76^{+0.20+0.03+0.08}_{-0.13-0.09-0.09} $	$ [0.6,3.9] $	−	10−9
$ B^{*0} \to \pi^{-} D^{+} $	$ 0.86^{+0.20+0.02+0.03}_{-0.14-0.02-0.03} $	$ [0.4,3.0] $	−	10−12
$ B^{*0} \to \pi^{-} D_{s}^{+} $	$ 2.46^{+0.57+0.05+0.10}_{-0.39-0.07-0.08} $	[1.3,8.4]	−	10−11
$ B^{*0} \to \pi^{+} D^{-} $	$ 6.36^{+1.47+0.05+0.03}_{-1.00-0.09-0.05} $	[2,13]	−	10−9
$ B^{*0} \to K^{+} D^{-} $	$ 4.81^{+1.11+0.04+0.02}_{-0.76-0.07-0.04} $	[1.5,9.8]	−	10−10
$ B^{*0} \to \rho^{+} D^{-} $	$ 1.77^{+0.41+0.01+0.01}_{-0.28-0.03-0.01} $	−	1.51	10−8
$ B^{*0} \to K^{*+} D^{-} $	$ 1.05^{+0.24+0.02+0.01}_{-0.17-0.01-0.01} $	−	0.87	10−9
$ B_{s}^{*0} \to K^{-} D^{+} $	$ 12.58^{+2.98+0.34+0.14}_{-2.02-0.38-0.14} $	[3,21]	−	10−12
$ B_{s}^{*0} \to \pi^{+} D_{s}^{-} $	$ 7.85^{+1.86+0.06+0.14}_{-1.26-0.11-0.19} $	−	−	10−9
$ B_{s}^{*0} \to K^{+} D_{s}^{-} $	$ 5.92^{+1.40+0.05+0.11}_{-0.95-0.08-0.15} $	[1.4,8.7]	−	10−10
$ B_{s}^{*0} \to K^{-} D_{s}^{+} $	$ 3.62^{+0.86+0.09+0.04}_{-0.58-0.11-0.04} $	[0.9,5.9]	−	10−11
$ B_{s}^{*0} \to \rho^{+} D_{s}^{-} $	$ 2.18^{+0.52+0.02+0.05}_{-0.35-0.03-0.06} $	−	2.89	10−8
$ B_{s}^{*0} \to K^{*-} D_{s}^{-} $	$ 1.30^{+0.31+0.01+0.03}_{-0.21-0.02-0.04} $	−	1.66	10−9

Table 12.  Branching ratios of the decays $ B^{*}_{(s)} \to \pi(K) D_{(s)}, \rho(K^*)D_{(s)} $.

Decay modes	This work	Ref. [42]	Ref. [44]			Unit
			$ \omega=0.4 $GeV	$ \omega=0.6 $ GeV	$ \omega=m\alpha_{s} $
$ B_{c}^{*+} \to B \pi^{+} $	$ 1.12^{+0.29+0.07+0.03}_{-0.19-0.09-0.04} $	2.1	2.14	4.50	6.53	10−8
$ B_{c}^{*+} \to B K^{+} $	$ 6.28^{+1.60+0.38+0.02}_{-1.06-0.52-0.21} $	$ 11 $	11.5	24.1	35.0	10−10
$ B_{c}^{*+} \to B \rho^{+} $	$ 4.76^{+1.22+0.03+0.13}_{-0.80-0.09-0.26} $	9.7	3.5	7.5	11.5	10−8
$ B_{c}^{*+} \to B K^{*+} $	$ 2.47^{+0.63+0.01+0.08}_{-0.42-0.04-0.14} $	4.4	1.45	3.13	4.84	10−9
$ B_{c}^{*+} \to B_{s} \pi^{+} $	$ 2.34^{+0.60+0.13+0.08}_{-0.39-0.18-0.01} $	4.4	4.00	7.26	9.82	10−7
$ B_{c}^{*+} \to B_{s} K^{+} $	$ 1.21^{+0.31+0.07+0.04}_{-0.20-0.09-0.01} $	2.2	1.98	3.60	4.87	10−8
$ B_{c}^{*+} \to B_{s} \rho^{+} $	$ 9.36^{+2.39+0.04+0.01}_{-1.58-0.18-0.23} $	$ 19 $	6.3	11.7	16.8	10−7
$ B_{c}^{*+} \to B_{s} K^{*+} $	$ 3.99^{+1.02+0.01+0.01}_{-0.67-0.06-0.11} $	7.2	2.21	4.12	6.01	10−8

Table 11.  Branching ratios of the decays $ B^{*}_{c} \to B_{(s)}\pi(K) $ and $ B^{*}_{c} \to B_{(s)}\rho(K^*) $.

Decay modes	This work	NF [42]	PQCD [48]	Ref. [47]		BSW [41]	Unit
				BSW	LFQM
$ {\cal{B}}r(B_{c}^{*+} \to \eta_{c} \pi^{+}) $	$ 1.14^{+0.29+0.06+0.03}_{-0.19-0.08-0.07} $	1.5	2.2	−	−	0.77	10−8
$ {\cal{B}}r(B_{c}^{*+} \to \eta_{c} K^{+}) $	$ 8.69^{+2.21+0.46+0.25}_{-1.47-0.59-0.57} $	$ 11 $	$ 17 $	−	−	−	10−10
$ {\cal{B}}r(B_{c}^{*+} \to \eta_{c} \rho^{+}) $	$ 3.21^{+0.82+0.16+0.12}_{-0.54-0.21-0.23} $	4.3	−	3.02	2.45	7.72	10−8
$ {\cal{B}}r(B_{c}^{*+} \to \eta_{c} K^{*+}) $	$ 1.93^{+0.49+0.10+0.07}_{-0.33-0.12-0.14} $	2.3	−	1.7	1.4	4.18	10−9
$ {\cal{B}}r(B_{c}^{*+} \to \eta_{c}(2S) \pi^{+}) $	$ 2.32^{+0.59+0.04+0.67}_{-0.39-0.05-0.89} $	4.1	2.4	−	−	−	10−9
$ {\cal{B}}r(B_{c}^{*+} \to \eta_{c}(2S) K^{+}) $	$ 1.75^{+0.45+0.03+0.52}_{-0.29-0.04-0.67} $	2.9	3.4	−	−	−	10−10
$ {\cal{B}}r(B_{c}^{*+} \to \eta_{c}(2S) \rho^{+}) $	$ 6.91^{+1.76+0.09+0.17}_{-1.17-0.13-0.29} $	$ 12 $	−	−	−	−	10−9
$ {\cal{B}}r(B_{c}^{*+} \to \eta_{c}(2S) K^{*+}) $	$ 3.89^{+0.99+0.05+0.14}_{-0.66-0.07-0.26} $	6.1	−	−	−	−	10−10

Table 10.  Branching ratios of the decays $ B^{*+}_{c}\to \eta_c(1S,2S)P $ with P referring to $ \pi, K, \rho, K^* $.

1. For the hierarchical relations between the CKM matrix elements $ |V_{ud}|>|V_{us}| $, the branching ratios of the decays $ B^{*+}_{c}\to\eta_{c}\pi^{+}(\rho^{+}) $ are at least an order of magnitude larger than those of the decays $ B^{*+}_{c}\to\eta_{c}K^{+} (K^{*+}) $. It is the similar to the cases with $ \eta_c $ replaced by $ \eta_c(2S) $ in these decays. Due to the smaller decay constant $ f_{\eta_{c}(2S)} $ compared with $ f_{\eta_c} $ and the compacter phase space for the final states $ \eta_{c}(2S)P(V) $ compared with the $ \eta_c P(V) $, the branching ratios of the decays $ B^*_c\to \eta_c(2S)P(V) $ are smaller than those of the corresponding decays $ B^*_c\to \eta_cP(V) $ by about a factor of 5. In addition, the CKM-suppressed decays $ B_{c}^{*}\to \eta_{c}(1S,2S)K^{(*)} $ have relatively small branching ratios, which only reach up to 10^-10～10^-9 order. Our predictions are comparable with the naive factorization (NF) approach [42], the perturbative QCD (PQCD) approach [48] and the BSW model [41, 47]. Certainly, our results for the decays $ B_{c}^{*+} \to \eta_{c}\rho^{+}(K^{*+}) $ are consistent with the previous LFQM calculations [47], and the difference is caused by the different input values.

2. Due to the large CKM factors $ V_{ud}V_{cs}=0.95 $, the Cabibbo-favored decays $ B_{c}^{*+}\to B_{s}\pi^+(\rho^+) $ have the largest branching ratios, which can reach up to 10^-7 order, especially for the decay $ B_{c}^{*+}\to B_{s} \rho^+ $, its branching ratio is very close to 10^-6 order. These two decays might be observable at the future LHCb and Super KEKB experiments and can be served as the golden channels to seek for the $ B^*_c $ meson. While the decays $ B_{c}^{*}\to B\pi(\rho), B_{s}K^{(*)} $ have relatively smaller branching ratios which are only at the order of 10⁻⁸ due to the smaller CKM factors $ V_{ud}V_{cd}\approx V_{us}V_{cs}\approx0.22 $. The Cabibbo-double-suppressed decays $ B_{c}^{*}\to BK^{(*)} $ have the smallest branching ratios within the range of 10^-10～10^-9 for the smallest CKM factors $ V_{us}V_{cd}\approx0.050 $. A clear hierarchical relationship among their branching ratios can be obtained as following

Furthermore, there exist three partial polarization amplitudes for the decays $ B^{*}_c\to B_{(s)}\rho(K^*) $, where the transverse polarizations are dominant and will promote obviously the branching ratios, while only the p-wave amplitude is involved for the decays $ B^{*}_c\to B_{(s)}\pi(K) $, so the branching ratios of the decays $ B^{*}_c\to B_{(s)}\rho(K^*) $ are about 3~4 times larger than those of the corresponding decays $ B^{*}_c\to B_{(s)}\pi(K) $. It is similar to the cases with $ B_{(s)} $ replaced by $ \eta_c(1S,2S) $ in these decays: $ {\cal{B}}r(B_{c}^{*} \to \eta_{c}(1S,2S)\rho(K^*)) $ are about 2~3 times larger than $ {\cal{B}}r(B_{c}^{*} \to \eta_{c}(1S,2S)\pi(K)) $.

3. We give the following SU(3) symmetry breaking relations $ R^{K^*/\rho}_M $ and $ R^{K/\pi}_M $ with M being $ B_{(s)},\eta_c(1S,2S) $.

which are consistent with the estimations $ R^{K^*/\rho}_M=\left|\dfrac{V_{us}f_{K^{*}}}{V_{ud}f_{\rho}}\right|^{2}\approx0.057 $ and $ R^{K/\pi}_M=\left|\dfrac{V_{us}f_{K}}{V_{ud}f_{\pi}}\right|^{2}\approx0.078 $ obtained from the factorization assumption within errors. Furthermore, these ratios $ R^{K^*/\rho}_{B_{(s)}} $ and $ R^{K/\pi}_{B_{(s)}} $ also agree with the results given in the naive factorization approach [42], the WSB model [44, 47] and the PQCD approach [48].

4. In Ref. [5], a range of the total decay width for the initial meson $ B^{*}_{(s)} $ was used in the numerical calculations, so the ranges of the branching ratios were obtained, which are shown in Table 12. One can find that all of our predictions fall within the ranges given in the BSW model [5]. In addition, for the branching ratios of the decays $ B_{(s)}^{*} \to \rho D_{(s)}, K^{*} D_{(s)} $, our results are consistent well with the QCDF calculations [10] within errors. In these decays, the channels $ B_{(s)}^{*} \to \rho^+ D_{(s)}^- $ have the largest branching ratios, which can reach up to 10^-8 order and may be observed in the future high-luminosity LHC experiments.

5. We present the results of the polarization fractions in Table 13. It can be clearly observed that the decays $ B_{c}^{*+} \to \eta_{c}(1S,2S) \rho^{+}, \eta_{c}(1S,2S) K^{*+} $ and $ B^{*0}_{(s)} \to \rho^{+} D^{-}_{(s)}, K^{*+} D^{-}_{(s)} $ are dominated by the longitudinal polarization, while the channels $ B_{c}^{*+} \to B_{(s)} \rho^{+}, B_{(s)} K^{*+} $ are dominated by the transverse polarizations.

Channels	$ B_{c}^{*+} \to \eta_{c} \rho^{+} $	$ B_{c}^{*+} \to \eta_{c} K^{*+} $	$ B_{c}^{*+} \to \eta_{c}(2S) \rho^{+} $	$ B_{c}^{*+} \to \eta_{c}(2S) K^{*+} $
$ f_{L}$[%]	86.60	83.07	81.64	77.19
BSW [47]	88.0	84.6	-	-
LFQM [47]	86.6	83.1	-	-
$ f_{\|}$[%]	10.58	13.42	15.52	19.37
BSW [47]	10.0	12.7	-	-
LFQM [47]	10.2	13.0	-	-
Channel	$ B_{c}^{*+} \to B^0 \rho^{+} $	$ B_{c}^{*+} \to B^0 K^{*+} $	$ B_{c}^{*+} \to B^0_{s} \rho^{+} $	$ B_{c}^{*+} \to B^0_{s} K^{*+} $
$ f_{L}$[%]	42.49	37.69	40.30	35.56
$ f_{\|}$[%]	47.43	55.12	53.43	61.49
Channels	$ B^{*0} \to D^{+}\rho^{-} $	$ B^{*0} \to D^{-}K^{*+} $	$ B_{s}^{*0} \to D_{s}^{+}\rho^{-} $	$ B_{s}^{*0} \to D_{s}^{-}K^{*+} $
$ f_{L}$[%]	87.65	84.33	87.62	82.29
$ f_{\|}$[%]	9.67	12.33	9.73	12.40

Table 13.  Polarization fractions $ f_L, f_{\|} $ for the decays $ B^*\to P\rho(K^*) $ with P referring to $ \eta_c(1S,2S), B_{(s)}, D_{(s)} $.

In this work, we use the covariant light-front method to study the semileptonic and nonleptonic rare weak decays of the vector heavy mesons $ B^{*}_{u,d,s,c} $ in detail, with the aim of providing an important reference for the future experiments. Here the transition form factors $ B^*\to D,\pi $, $ B^*_s\to D_s, K $ and $ B^*_c\to B_{(s)},D,\eta_c(1S,2S) $ were calculated, which play a crucial role in our considered decays. It is noted that the total decay widths of these b-flavored vector mesons were estimated by the partial widths of their corresponding single-photon decay channels. The following are some points

1. In these semileptonic decays, the channels $ B_{s}^{*0}\to D_{s}^{-}\ell^{\prime+}{\nu}_{\ell^\prime} $ and $ B_{c}^{*+}\to B_{s}^{0}\ell^{\prime+}{\nu}_{\ell^\prime}, \eta_{c}\ell^{\prime+}{\nu}_{\ell^\prime} $ have the largest branching ratios, which can reach up to 10^-7 order. So they are very possible to be observed by the future LHCb experiments or can be served as golden channels to seek for the $ B^*_c $ meson in experiments.

2. The ratio of the branching fractions is an interesting quantity in experiments. We present some ratios of the branching fractions for these vector meson decays

which are agree well with other theoretical predictions.

3. In the nonleptonic decays, the Cabibbo-favored channels $ B_{c}^{*+}\to B^0_{s} \pi^{+}, B^0_{s} \rho^{+} $ have the largest branching ratios, which can reach up to 10^-7 order. Obviously, these two decays are also golden channels to search for the $ B_c^* $ meson in experiments.

4. Using the obtained branching ratios, we can consider the SU(3) symmetry breaking parameters $ R^{V}_M $ and $ R^{K/\pi}_M $ with M being $ B_{(s)} $ and $ \eta_c(1S,2S) $,

which are consistent with the estimations $ R^{K^*/\rho}_M=\left|\dfrac{V_{us}f_{K^{*}}}{V_{ud}f_{\rho}}\right|^{2}\approx0.057 $ and $ R^{K/\pi}_M=\left|\dfrac{V_{us}f_{K}}{V_{ud}f_{\pi}}\right|^{2}\approx0.078 $.

In the future, the self-consistency about the CLFQM is needed to systematically improve. A known example is the vector meson decay constant, that is $ f_V $ inconsistency puzzle [49, 50]. It is resolved by replacing the physical mass M with the invariant mass $ M_0 $. In the past, authors usually investigated the distribution amplitudes (DAs) up to the leading-twist, which describes the longitudinal momentum distribution of valence quarks inside the hadrons. The self-consistency is needed to further check carefully when the higher-twist DAs are involved [51]. It is meaningful to recognize the higher-twist effects to the hadron structures. Furthermore, it is also worth examining such effects in detail when the radially and orbitally excited states are involved.

When preforming the integration, we need to include the zero-mode contribution. It amounts to performing the integration in a proper way in the CLFQM. Specifically we use the following rules given in Refs. [37−40]

The following are the analytical expressions of the form factors of the transitions $ B^* \to P $ in the CLFQM,