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Semileptonic $B_{(s)}$ meson decays to $D_{0}^{\ast}(2300),D_{s0}^{\ast}(2317),D_{s1}(2460),D_{s1}(2536),D_{1}(2420)$ and $D_{1}(2430)$ within the covariant light-front approach

You-Ya Yang,¹ Zhi-Qing Zhang,^1,¹¹1 zhangzhiqing@haut.edu.cn (corresponding author) and Hao Yang¹ ¹ School of Physics, Henan University of Technology, Zhengzhou, Henan 450001, China

Abstract

In this work, we investigate the semileptonic decays of $B_{(s)}$ meson to $D_{0}^{\ast}(2300)$ , $D_{s0}^{\ast}(2317)$ , $D_{s1}(2460)$ , $D_{s1}(2536)$ , $D_{1}(2420)$ and $D_{1}(2430)$ in the covariant light-front quark model (CLFQM). We combine the helicity amplitudes via the corresponding form factors to obtain the branching ratios of the semileptonic decays $B_{(s)}\to D^{**}_{(s)}\ell\nu_{\ell}$ with $D^{**}_{(s)}$ referring to a P-wave exicted charmed meson $D_{0}^{\ast}(2300)$ , $D_{s0}^{\ast}(2317)$ , $D_{s1}(2460)$ , $D_{s1}(2536)$ , $D_{1}(2420)$ or $D_{1}(2430)$ and $\ell=e,\mu,\tau$ . Furthermore, we also take into account another two physical observables, namely the longitudinal polarization fraction $f_{L}$ and the forward-backward asymmetry $A_{FB}$ . Most of our predictions are comparable to the results given by other theoretical approaches and the present available data. The branching ratios of the semileptonic decay channels $B_{s}\to D_{s1}(2460)\ell\nu_{\ell}$ and $B\to D_{1}(2420)\ell\nu_{\ell}$ are larger than those of the semileptonic decays $B_{s}\to D_{s1}(2536)\ell\nu_{\ell}$ and $B\to D_{1}(2430)\ell\nu_{\ell}$ , respectively. We find that the long-standing ’ $1/2$ vs $3/2$ puzzle’ in the decays $B^{+}\to\bar{D}_{1}^{(\prime)0}\ell^{\prime+}\nu_{\ell^{\prime}}$ $(\ell^{\prime}=e,\mu)$ can be solved by taking some negative mixing angle $\theta_{s}$ values within a range from $-30.3^{\circ}$ to $-24.9^{\circ}$ , corresponding to $\theta$ of about $5^{\circ}\sim 10.4^{\circ}$ . While Belle collaboration updated their measurements for the decays $B^{0}\to D^{*-}_{0}\ell^{\prime+}\nu_{\ell^{\prime}}$ with only a small upper limit $Br(B^{0}\to D^{*-}_{0}\ell^{\prime+}\nu_{\ell^{\prime}})<0.44\times 10^{-3}$ obtained, which is much larger than most theoretical predictions and causes a new puzzle.

pacs:

13.25.Hw, 12.38.Bx, 14.40.Nd

I Introduction

In the conventional quark model, these P-wave orbitally excited charmed mesons $D_{0}^{\ast}(2300)$ , $D_{s0}^{\ast}(2317)$ , $D_{s1}(2460)$ , $D_{s1}(2536)$ , $D_{1}(2420)$ and $D_{1}(2430)$ can be view as constituent quark-antiquark pairs. They are usually classified according to the quantum numbers ${}^{(2S+1)}L_{J}$ : the scalar mesons $D_{0}^{\ast}(2300)$ and $D_{s0}^{\ast}(2317)$ correspond to ${}^{3}P_{0}$ . While there exist two different kinds of axial-vector mesons, namely ${}^{1}P_{1}$ and ${}^{3}P_{1}$ , which can undergo mixing when the two constituent quarks are different. In the heavy quark limit, the heavy quark spin and the total angular momentum of the light quark are good quantum numbers, it is more convenient to use the $L^{j}_{J}$ configurations ²²2 $j(L)$ being the total (orbital) angular momentum of the light quark. to classify them: the scalar mesons $D_{0}^{\ast}(2300)$ and $D_{s0}^{\ast}(2317)$ belong to $P^{1/2}_{0}$ , $D_{1}(2430)(D_{s1}(2536))$ and $D_{1}(2420)(D_{s1}(2460))$ correspond to $P^{3/2}_{1}$ and $P^{1/2}_{1}$ , respectively. However, beyond the heavy quark limit, there is a mixing between $P^{3/2}_{1}$ and $P^{1/2}_{1}$ , denoted by $D^{3/2}_{1}$ and $D^{1/2}_{1}$ , respectively³³3Here we take the physical states $D_{1}(2430)$ and $D_{1}(2420)$ as an example to explain, it is similar to the states $D_{s1}(2536)$ and $D_{s1}(2460)$ ., that is

	$\displaystyle\|D_{1}(2420)\rangle$	$\displaystyle=$	$\displaystyle\|D_{1}^{1/2}\rangle\sin\theta_{s}+\|D_{1}^{3/2}\rangle\cos\theta_{s},$
	$\displaystyle\|D_{1}(2430)\rangle$	$\displaystyle=$	$\displaystyle-\|D_{1}^{3/2}\rangle\sin\theta_{s}+\|D_{1}^{1/2}\rangle\cos\theta_{s}.$		(1)

While the states $D_{1}^{1/2}$ and $D_{1}^{3/2}$ are expected to be a mixture of states ${}^{1}P_{1}$ and ${}^{3}P_{1}$ denoted by ${}^{1}D_{1}$ and ${}^{3}D_{1}$ , respectively,

	$\displaystyle\|D^{3/2}_{1}\rangle$	$\displaystyle=$	$\displaystyle\sqrt{\frac{2}{3}}\|^{1}D_{1}\rangle+\sqrt{\frac{1}{3}}\|^{3}D_{1}\rangle,$
	$\displaystyle\|D^{1/2}_{1}\rangle$	$\displaystyle=$	$\displaystyle-\sqrt{\frac{1}{3}}\|^{1}D_{1}\rangle+\sqrt{\frac{2}{3}}\|^{3}D_{1}\rangle.$		(2)

Combining Eq. (1) and Eq. (2), one can find that the physical states $D_{1}(2420)$ and $D_{1}(2430)$ can be written as

	$\displaystyle\|D_{1}(2420)\rangle$	$\displaystyle=$	$\displaystyle\|^{1}D_{1}\rangle\cos\theta+\|^{3}D_{1}\rangle\sin\theta,$
	$\displaystyle\|D_{1}(2430)\rangle$	$\displaystyle=$	$\displaystyle-\|^{1}D_{1}\rangle\sin\theta+\|^{3}D_{1}\rangle\cos\theta.$		(3)

where $\theta_{s}=7^{\circ}$ and $\theta=\theta_{s}+35.3^{\circ}$ zhw . There exist many puzzles in these several P-wave excited states, such as the low mass puzzle for the states $D^{*}_{0}(2300),D^{*}_{s0}(2317)$ and $D_{s1}(2460)$ belle ; quark ; babar2 ; godfrey2 , the SU(3) mass hierarchy puzzle between $D^{*}_{0}(2300)$ and $D^{*}_{s0}(2317)$ , large width difference between them Gubernari , especially, the long-standing ’ $1/2$ vs $3/2$ puzzle’ morenas ; bigi ; scora ; colangelo , that is the theoretical predictions for the branching ratios of semileptonic B decays into $D^{1/2+}$ are much smaller than those into $D^{3/2+}$ , which conflicts with the experimental measurements belle ; babar2 ; Belle:2022yzd . These unexpected disparities between theory and experiment have sparked many explanations about their inner structures, such as the molecular states, the compact tetraquark states, the states of $\bar{c}s$ mixed with four-quark states, and so on guo ; cleven ; close ; guofk ; lutz ; lutz2 ; ylma ; maiani ; wangzg ; hycheng2 ; yqchen ; kim ; bardeen ; nowak ; browder ; vijande .

In this paper we investigate the semileptonic $B_{(s)}$ meson decays to $D_{0}^{\ast}(2300)$ , $D_{s0}^{\ast}(2317)$ , $D_{s1}(2460)$ , $D_{s1}(2536)$ , $D_{1}(2420)$ and $D_{1}(2430)$ by using the covariant light-front quark model (CLFQM). For the semileptonic decays, the hadronic transition matrix element between the initial and final mesons is most crucial for theoretical calculations, which can be characterized by several form factors. The form factors can be extracted from data or relied on some non-perturbative methods. The $B_{(s)}\to D^{**}_{(s)}$ transition form factors were initially calculated in the improved version of the Isgur-Scora-Ginstein-Wise (ISGW) quark model, the so-called ISGW2 hycheng . Some of them were calculated using the CLFQM Cheng , QCD sum rules (QCDSR) Y.B. ; M. Q. ; Zuo:2023ksq and light-cone sum rules (LCSRs) Gubernari . Additionally, with the available experimental data as inputs, the form factors of the $B_{(s)}$ to these excited charmed meson transitions and the corresponding semileptonic decays were also investigated based on the heavy quark effective theory (HQET), including the next-to-leading order corrections of heavy quark expansion and new physics (NP) effects A.K.1 ; A.K.2 ; F.U. ; F.U.1 . As one of the popular non-perturbative methods, the CLFQM has been used successfully to study the form factors Cheng ; Cheng1 ; Hwang ; Lu ; Wang . Based on the form factors and helicity formalisms, we also calculate another two physical observables: the forward-backward asymmetry $A_{FB}$ and the longitudinal polarization fraction $f_{L}$ , respectively.

The arrangement of this paper is as follows: In Section II, the formalism of the CLFQM, the hadronic matrix elements and the helicity amplitudes combined via form factors are presented. The numerical results for the $B_{(s)}$ meson to $D_{0}^{\ast}(2300)$ , $D_{s0}^{\ast}(2317)$ , $D_{s1}(2460)$ , $D_{s1}(2536)$ , $D_{1}(2420)$ and $D_{1}(2430)$ transition form factors, the branching ratios, the forward-backward asymmetries $A_{FB}$ and the longitudinal polarization fractions $f_{L}$ for the corresponding decays are presented in Section III. In addition, the detailed numerical analysis and discussion, including comparisons with the data and other model calculations, are carried out. The conclusions are presented in the final part.

II Formalism

II.1 The covariant light-front quark model

Under the covariant light-front quark model, the light-front coordinates of a momentum $p$ are defined as $p=(p^{-},p^{+},p_{\perp})$ with $p^{\pm}=p^{0}\pm p_{z}$ and $p^{2}=p^{+}p^{-}-p^{2}_{\perp}$ . If the momenta of the quark and antiquark with masses $m_{1}^{\prime(\prime\prime)}$ and $m_{2}$ in the incoming (outgoing) meson are denoted as $p_{1}^{\prime(\prime\prime)}$ and $p_{2}$ , respectively, the momentum of the incoming (outgoing) meson with mass $M^{\prime}(M^{\prime\prime})$ can be written as $P^{\prime}=p_{1}^{\prime}+p_{2}(P^{\prime\prime}=p_{1}^{\prime\prime}+p_{2})$ . Here, we use the same notation as those in Refs. Jaus ; Cheng and $M^{\prime}$ refers to $m_{B}$ for $B$ meson decays. These momenta can be related each other through the internal variables $(x_{i},p{\prime}_{\perp})$

\displaystyle p_{1,2}^{\prime+}=x_{1,2}P^{\prime+},\quad p_{1,2\perp}^{\prime}=x_{1,2}P_{\perp}^{\prime}\pm p_{\perp}^{\prime},

(4)

with $x_{1}+x_{2}=1$ . Using these internal variables, we can define some quantities for the incoming meson which will be used in the following calculations

	$\displaystyle M_{0}^{\prime 2}$	$\displaystyle=$	$\displaystyle\left(e_{1}^{\prime}+e_{2}\right)^{2}=\frac{p_{\perp}^{\prime 2}+m_{1}^{\prime 2}}{x_{1}}+\frac{p_{\perp}^{2}+m_{2}^{2}}{x_{2}},\quad\widetilde{M}_{0}^{\prime}=\sqrt{M_{0}^{\prime 2}-\left(m_{1}^{\prime}-m_{2}\right)^{2}},$
	$\displaystyle e_{i}^{(\prime)}$	$\displaystyle=$	$\displaystyle\sqrt{m_{i}^{(\prime)2}+p_{\perp}^{\prime 2}+p_{z}^{\prime 2}},\quad\quad p_{z}^{\prime}=\frac{x_{2}M_{0}^{\prime}}{2}-\frac{m_{2}^{2}+p_{\perp}^{\prime 2}}{2x_{2}M_{0}^{\prime}},$		(5)

where the kinetic invariant mass of the incoming meson $M^{\prime}_{0}$ can be expressed as the energies of the quark and the antiquark $e^{(\prime)}_{i}$ . It is similar to the case of the outgoing meson.

Refer to caption — Figure 1: Feynman diagrams for $B_{(s)}$ decay (left) and transition (right) amplitudes, where $P^{\prime(\prime\prime)}$ is the incoming (outgoing) meson momentum, $p^{\prime(\prime\prime)}_{1}$ is the quark momentum and $p_{2}$ is the anti-quark momentum. The X in the diagrams denotes the vector or axial-vector transition vertex.

To calculate the amplitudes for the transition form factors, we need the Feynman rules for the meson-quark-antiquark vertices $i\Gamma^{\prime}_{M}$ ⁴⁴4In the following we take the transitions $B\to D^{*}_{0}$ and $B\to^{3}D_{1},^{1}D_{1}$ as examples. It is similar for the transitions $B_{s}\to D^{*}_{s0}$ and $B_{s}\to^{3}D_{s1},^{1}D_{s1}$ . From now on, we will use $D^{*}_{0},D^{*}_{s0},D_{s1},D_{s1}^{\prime},D_{1}$ and $D_{1}^{\prime}$ to represent $D^{*}_{0}(2300),D^{*}_{s0}(2317),D_{s1}(2460),D_{s1}(2536),D_{1}(2420)$ and $D_{1}(2430)$ , respectively, for simplicity. , which are listed as

$\displaystyle i\Gamma_{D^{*}_{0}}^{\prime}$	$\displaystyle=$	$\displaystyle-iH_{D^{*}_{0}}^{\prime},$	(6)
$\displaystyle i\Gamma_{\;{}^{3}D_{1}}^{\prime}$	$\displaystyle=$	$\displaystyle-iH_{\;{}^{3}D_{1}}^{\prime}\left[\gamma_{\mu}+\frac{1}{W_{\;{}^{3}D_{1}}^{\prime}}\left(p_{1}^{\prime}-p_{2}\right)_{\mu}\right]\gamma_{5},$	(7)
$\displaystyle i\Gamma_{\;{}^{1}D_{1}}^{\prime}$	$\displaystyle=$	$\displaystyle-iH_{\;{}^{1}D_{1}}^{\prime}\left[\frac{1}{W_{\;{}^{1}D_{1}}^{\prime}}\left(p_{1}^{\prime}-p_{2}\right)_{\mu}\right]\gamma_{5}.$	(8)

The form factors of the transitions $B\to D^{*}_{0}$ and $B\to\ ^{i}D_{1}(i=1,3)$ induced by the vector and aixal-vector currents are defined as

$\displaystyle\left\langle D^{*}_{0}\left(P^{\prime\prime}\right)\left\|A_{\mu}\right\|B\left(P^{\prime}\right)\right\rangle$	$\displaystyle=$	$\displaystyle i\left[u_{+}(q^{2})P_{\mu}+u_{-}(q^{2})q_{\mu}\right],$	(9)
$\displaystyle\left\langle\;{}^{i}D_{1}\left(P^{\prime\prime},\varepsilon\right)\left\|A_{\mu}\right\|B\left(P^{\prime}\right)\right\rangle$	$\displaystyle=-$	$\displaystyle q(q^{2})\epsilon_{\mu\nu\alpha\beta}\varepsilon^{*\nu}P^{\alpha}q^{\beta},$	(10)
$\displaystyle\left\langle\;{}^{i}D_{1}\left(P^{\prime\prime},\varepsilon\right)\left\|V_{\mu}\right\|B\left(P^{\prime}\right)\right\rangle$	$\displaystyle=$	$\displaystyle i\left\{l(q^{2})\varepsilon_{\mu}^{}+\varepsilon^{}\cdot P\left[P_{\mu}c_{+}(q^{2})+q_{\mu}c_{-}(q^{2})\right]\right\}.$	(11)

In calculations, the Bauer-Stech-Wirbel (BSW) bsw transition form factors are more frequently used and defined by

$\displaystyle\left\langle D^{*}_{0}\left(P^{\prime\prime}\right)\left\|A_{\mu}\right\|B\left(P^{\prime}\right)\right\rangle$	$\displaystyle=$	$\displaystyle\left(P_{\mu}-\frac{m_{B}^{2}-m_{D^{}_{0}}^{2}}{q^{2}}q_{\mu}\right)F_{1}^{BD^{}_{0}}\left(q^{2}\right)+\frac{m_{B}^{2}-m_{D^{}_{0}}^{2}}{q^{2}}q_{\mu}F_{0}^{BD^{}_{0}}\left(q^{2}\right),\;\;\;\;\;\;\;\;$	(12)
$\displaystyle\left\langle{}^{i}D_{1}\left(P^{\prime\prime},\varepsilon^{\mu*}\right)\left\|V_{\mu}\right\|B\left(P^{\prime}\right)\right\rangle$	$\displaystyle=$	$\displaystyle-i\left\{\left(m_{B}-m_{{}^{i}D_{1}}\right)\varepsilon_{\mu}^{}V_{1}^{B\ ^{i}D_{1}}\left(q^{2}\right)-\frac{\varepsilon^{}\cdot P}{m_{B}-m_{{}^{i}D_{1}}}P_{\mu}V_{2}^{B\ ^{i}D_{1}}\left(q^{2}\right)\right.$	(13)
		$\displaystyle\left.-2m_{{}^{i}D_{1}}\frac{\varepsilon^{*}\cdot P}{q^{2}}q_{\mu}\left[V_{3}^{B\ ^{i}D_{1}}\left(q^{2}\right)-V_{0}^{B\ ^{i}D_{1}}\left(q^{2}\right)\right]\right\},$	(13)
$\displaystyle\left\langle{}^{i}D_{1}\left(P^{\prime\prime},\varepsilon^{\mu*}\right)\left\|A_{\mu}\right\|B\left(P^{\prime}\right)\right\rangle$	$\displaystyle=$	$\displaystyle-\frac{1}{m_{B}-m_{{}^{i}D_{1}}}\epsilon_{\mu\nu\alpha\beta}\varepsilon^{*\nu}P^{\alpha}q^{\beta}A^{B\ ^{i}D_{1}}\left(q^{2}\right),$	(14)

where $P=P^{\prime}+P^{\prime\prime},q=P^{\prime}-P^{\prime\prime}$ and the convention $\epsilon_{0123}=1$ is adopted.

To smear the singularity at $q^{2}=0$ in Eq. (13), the following relations are required

	$\displaystyle V^{B\ ^{i}D_{1}}_{3}(0)$	$\displaystyle=$	$\displaystyle V^{B\ ^{i}D_{1}}_{0}(0),$		(15)
	$\displaystyle V^{B\ ^{i}D_{1}}_{3}(q^{2})$	$\displaystyle=$	$\displaystyle\frac{m_{B}-m_{{}^{i}D_{1}}}{2m_{{}^{i}D_{1}}}V^{B\ ^{i}D_{1}}_{1}(q^{2})-\frac{m_{B}+m_{{}^{i}D_{1}}}{2m_{{}^{i}D_{1}}}V^{B\ {{}^{i}D_{1}}}_{2}(q^{2}).$		(16)

These two kinds of form factors are related to each other via

$\displaystyle F^{BD^{*}_{0}}_{1}(q^{2})$	$\displaystyle=$	$\displaystyle-u_{+}(q^{2}),F^{BD^{*}_{0}}_{0}(q^{2})=-u_{+}(q^{2})-\frac{q^{2}}{q\cdot P}u_{-}(q^{2}),$	(17)
$\displaystyle A^{B\ ^{i}D_{1}}(q^{2})$	$\displaystyle=$	$\displaystyle-(m_{B}-m_{{}^{i}D_{1}})q(q^{2}),V^{B\ ^{i}D_{1}}_{1}(q^{2})=-\frac{l(q^{2})}{m_{B}-m_{{}^{i}D_{1}}},$	(18)
$\displaystyle V^{B\ ^{i}D_{1}}_{2}(q^{2})$	$\displaystyle=$	$\displaystyle(m_{B}-m_{{}^{i}D_{1}})c_{+}(q^{2}),V^{B\ ^{i}D_{1}}_{3}(q^{2})-V^{B\ ^{i}D_{s1}}_{0}(q^{2})=\frac{q^{2}}{2m_{{}^{i}D_{1}}}c_{-}(q^{2}).$	(19)

For the general $B\rightarrow M$ transitions with $M$ being a scalar or axial-vector meson, the decay amplitude at the lowest order is Cheng:2003sm

\displaystyle\mathcal{M}^{BM}=-i^{3}\frac{N_{c}}{(2\pi)^{4}}\int d^{4}p_{1}^{\prime}\frac{H_{B}^{\prime}\left(H_{M}^{\prime\prime}\right)}{N_{1}^{\prime}N_{1}^{\prime\prime}N_{2}}S_{\mu}^{BM},

(20)

where $N_{1}^{\prime(\prime\prime)}=p_{1}^{\prime(\prime\prime)2}-m_{1}^{\prime(\prime\prime)2}$ and $N_{2}=p_{2}^{2}-m_{2}^{2}$ arise from the quark propagators. For our considered transitions $B\rightarrow D^{*}_{0}$ and $B\to\ ^{1}D_{1},\ ^{3}D_{1}$ , the traces $S_{\mu}^{BD^{*}_{0}},S_{\mu\nu}^{B\;^{1}D_{1}}$ and $S_{\mu\nu}^{B\;^{3}D_{1}}$ can be directly written out by using the Lorentz contraction as follows

$\displaystyle S_{\mu}^{BD^{*}_{0}}$	$\displaystyle=$	$\displaystyle Tr\left[\left(\not p_{1}^{\prime\prime}+m_{1}^{\prime\prime}\right)\gamma_{\mu}\gamma_{5}\left(\not p_{1}^{\prime}+m_{1}^{\prime}\right)\gamma_{5}\left(-\not p_{2}+m_{2}\right)\right],$	(21)
$\displaystyle S_{\mu\nu}^{B\;^{1}D_{1}}$	$\displaystyle=$	$\displaystyle\left(S_{V}^{B\;^{1}D_{1}}-S_{A}^{B\;^{1}D_{1}}\right)_{\mu\nu}$	(22)
	$\displaystyle=$	$\displaystyle\operatorname{Tr}\left[\left(-\frac{1}{W_{\;{}^{1}D_{1}}^{\prime\prime}}\left(p_{1}^{\prime\prime}-p_{2}\right)_{\nu}\right)\gamma_{5}\left(\not p_{1}^{\prime\prime}+m_{1}^{\prime\prime}\right)\left(\gamma_{\mu}-\gamma_{\mu}\gamma_{5}\right)\left(\not p_{1}^{\prime}+m_{1}^{\prime}\right)\gamma_{5}\left(-\not p_{2}+m_{2}\right)\right],\;\;\;$	(22)
$\displaystyle S_{\mu\nu}^{B\;^{3}D_{1}}$	$\displaystyle=$	$\displaystyle\left(S_{V}^{B\;^{3}D_{1}}-S_{A}^{B\;^{3}D_{1}}\right)_{\mu\nu}$	(23)
	$\displaystyle=$	$\displaystyle\operatorname{Tr}\left[\left(\gamma_{\nu}-\frac{1}{W_{\;{}^{3}D_{1}}^{\prime\prime}}\left(p_{1}^{\prime\prime}-p_{2}\right)_{\nu}\right)\gamma_{5}\left(\not p_{1}^{\prime\prime}+m_{1}^{\prime\prime}\right)\left(\gamma_{\mu}-\gamma_{\mu}\gamma_{5}\right)\left(\not p_{1}^{\prime}+m_{1}^{\prime}\right)\gamma_{5}\left(-\not p_{2}+m_{2}\right)\right].\;\;\;$	(23)

The form factors can be obtained by matching the coefficients listed in Eqs. (9)-(11) with the corresponding amplitudes given Eq. (20). The specific expressions for these transition form factors are collected in Appendix B. It is noticed that the form factors of the transitions $B\to D_{1}$ and $B\to D_{1}^{\prime}$ can be obtained from those of the transitions $B\to\ ^{1}D_{1}$ and $B\to\ ^{3}D_{1}$ through Eq. (3).

II.2 Wave functions and decay constants

The light-front wave functions (LFWFs) are needed in the form factor calculations. Although the LFWFs can be derived from solving the relativistic Schr $\ddot{o}$ dinger equation theoretically, it is difficult to obtain their exact solutions in many cases. Consequently, we will use the phenomenological Gaussian-type wave functions in this work,

	$\displaystyle\varphi^{\prime}$	$\displaystyle=$	$\displaystyle\varphi^{\prime}\left(x_{2},p_{\perp}^{\prime}\right)=4\left(\frac{\pi}{\beta^{\prime 2}}\right)^{\frac{3}{4}}\sqrt{\frac{dp_{z}^{\prime}}{dx_{2}}}\exp\left(-\frac{p_{z}^{\prime 2}+p_{\perp}^{\prime 2}}{2\beta^{\prime 2}}\right),$
	$\displaystyle\varphi_{p}^{\prime}$	$\displaystyle=$	$\displaystyle\varphi_{p}^{\prime}\left(x_{2},p_{\perp}^{\prime}\right)=\sqrt{\frac{2}{\beta^{\prime 2}}}\varphi^{\prime},\quad\frac{dp_{z}^{\prime}}{dx_{2}}=\frac{e_{1}^{\prime}e_{2}}{x_{1}x_{2}M_{0}^{\prime}},$		(24)

where the parameter $\beta^{\prime}$ describes the momentum distribution and is approximately of order $\Lambda_{QCD}$ . It can be usually determined by the decay constants through the following analytic expressions Jaus ; Cheng ,

$\displaystyle f_{D^{*}_{0}}$	$\displaystyle=$	$\displaystyle\frac{N_{c}}{16\pi^{3}}\int dx_{2}d^{2}p_{\perp}^{\prime}\frac{h_{D^{*}_{0}}^{\prime}}{x_{1}x_{2}\left(M^{\prime 2}-M_{0}^{\prime 2}\right)}4\left(m_{1}^{\prime}x_{2}-m_{2}x_{1}\right),$	(25)
$\displaystyle f_{\;{}^{3}D_{1}}$	$\displaystyle=$	$\displaystyle-\frac{N_{c}}{4\pi^{3}M^{\prime}}\int dx_{2}d^{2}p_{\perp}^{\prime}\frac{h_{{}^{3}D_{1}}^{\prime}}{x_{1}x_{2}\left(M^{\prime 2}-M_{0}^{\prime 2}\right)}$	(26)
		$\displaystyle\times\left[x_{1}M_{0}^{\prime 2}-m_{1}^{\prime}\left(m_{1}^{\prime}+m_{2}\right)-p_{\perp}^{\prime 2}-\frac{m_{1}^{\prime}-m_{2}}{w_{\;{}^{3}D_{1}}^{\prime}}p_{\perp}^{\prime 2}\right],$	(26)
$\displaystyle f_{\;{}^{1}D_{1}}$	$\displaystyle=$	$\displaystyle\frac{N_{c}}{4\pi^{3}M^{\prime}}\int dx_{2}d^{2}p_{\perp}^{\prime}\frac{h_{\;{}^{1}D_{1}}^{\prime}}{x_{1}x_{2}\left(M^{\prime 2}-M_{0}^{\prime 2}\right)}\left(\frac{m_{1}^{\prime}-m_{2}}{w_{\;{}^{1}D_{1}}^{\prime}}p_{\perp}^{\prime 2}\right),$	(27)

where $m_{1}^{\prime}$ and $m_{2}$ represent the constituent quarks of the states $D^{*}_{0},\ ^{3}D_{1}$ and ${}^{1}D_{1}$ . The decay constants can be obtained through experimental measurements for the purely leptonic decays or theoretical calculations. The explicit forms of $h^{\prime}_{M}$ are given by Cheng:2003sm

	$\displaystyle h_{D^{*}_{0}}^{\prime}$	$\displaystyle=$	$\displaystyle\sqrt{\frac{2}{3}}h_{\;{}^{3}D_{1}}^{\prime}=\left(M^{\prime 2}-M_{0}^{\prime 2}\right)\sqrt{\frac{x_{1}x_{2}}{N_{c}}}\frac{1}{\sqrt{2}\widetilde{M}_{0}^{\prime}}\frac{\widetilde{M}_{0}^{\prime 2}}{2\sqrt{3}M_{0}^{\prime}}\varphi_{p}^{\prime},$		(28)
	$\displaystyle h_{\;{}^{1}D_{1}}^{\prime}$	$\displaystyle=$	$\displaystyle\left(M^{\prime 2}-M_{0}^{\prime 2}\right)\sqrt{\frac{x_{1}x_{2}}{N_{c}}}\frac{1}{\sqrt{2}\widetilde{M}_{0}^{\prime}}\varphi_{p}^{\prime}.$		(29)

II.3 Helicity amplitudes and observables

Since the form factors involving the fitted parameters for most of the transitions $B_{(s)}\to D_{(s)}^{**}$ have been investigated in our recent work Zhang:2023ypl , so it is convenient to obtain the differential decay widths of these semileptontic $B$ decays by the combination of the helicity amplitudes via form factors, which are listed as following

	$\displaystyle\frac{d\Gamma(B\to D^{*}_{0}\ell\nu_{\ell})}{dq^{2}}$	$\displaystyle=$	$\displaystyle(\frac{q^{2}-m_{\ell}^{2}}{q^{2}})^{2}\frac{{\sqrt{\lambda(m_{B}^{2},m_{D^{*}_{0}}^{2},q^{2})}}G_{F}^{2}\|V_{cb}\|^{2}}{384m_{B}^{3}\pi^{3}}\times\frac{1}{q^{2}}$		(30)
			$\displaystyle\;\;\;\times\left\{(m_{\ell}^{2}+2q^{2})\lambda(m_{B}^{2},m_{D^{}_{0}}^{2},q^{2})F_{1}^{2}(q^{2})+3m_{\ell}^{2}(m_{B}^{2}-m_{D^{}_{0}}^{2})^{2}F_{0}^{2}(q^{2})\right\},$		(30)

$\displaystyle\frac{d\Gamma_{L}(B\to D^{(\prime)}_{1}\ell\nu_{\ell})}{dq^{2}}$	$\displaystyle=$	$\displaystyle(\frac{q^{2}-m_{\ell}^{2}}{q^{2}})^{2}\frac{{\sqrt{\lambda(m_{B}^{2},m_{D^{(\prime)}_{1}}^{2},q^{2})}}G_{F}^{2}\|V_{cb}\|^{2}}{384m_{B}^{3}\pi^{3}}\times\frac{1}{q^{2}}\left\{3m_{\ell}^{2}\lambda(m_{B}^{2},m_{D^{(\prime)}_{1}}^{2},q^{2})V_{0}^{2}(q^{2})+(m_{\ell}^{2}+2q^{2})\right.$	(31)
		$\displaystyle\times\left.\left\|\frac{1}{2m_{D^{(\prime)}_{1}}}\left[(m_{B}^{2}-m_{D^{(\prime)}_{1}}^{2}-q^{2})(m_{B}-m_{D^{(\prime)}_{1}})V_{1}(q^{2})-\frac{\lambda(m_{B}^{2},m_{D^{(\prime)}_{1}}^{2},q^{2})}{m_{B}-m_{D^{(\prime)}_{1}}}V_{2}(q^{2})\right]\right\|^{2}\right\},$	(31)
$\displaystyle\frac{d\Gamma_{\pm}(B\to{D^{(\prime)}_{1}}\ell\nu_{\ell})}{dq^{2}}$	$\displaystyle=$	$\displaystyle(\frac{q^{2}-m_{\ell}^{2}}{q^{2}})^{2}\frac{{\sqrt{\lambda(m_{B}^{2},m_{D^{(\prime)}_{1}}^{2},q^{2})}}G_{F}^{2}\|V_{cb}\|^{2}}{384m_{B}^{3}\pi^{3}}$	(32)
		$\displaystyle\;\;\times\left\{(m_{\ell}^{2}+2q^{2})\lambda(m_{B}^{2},m_{D^{(\prime)}_{1}}^{2},q^{2})\left\|\frac{A(q^{2})}{m_{B}-m_{D^{(\prime)}_{1}}}\mp\frac{(m_{B}-m_{D^{(\prime)}_{1}})V_{1}(q^{2})}{\sqrt{\lambda(m_{B}^{2},m_{D^{(\prime)}_{1}}^{2},q^{2})}}\right\|^{2}\right\},$	(32)

where $\lambda(a,b,c)=(a+b-c)^{2}-4ab$ and $m_{\ell}$ is the mass of the lepton $\ell$ with $\ell=e,\mu,\tau$ ⁵⁵5For now on, we will use $\ell$ to represent $e,\mu,\tau$ and use $\ell^{\prime}$ to represent $e,\mu$ for simplicity. . The helicity amplitudes for the decays $B_{s}\to D^{*}_{s0}\ell\nu_{\ell}$ and $B_{s}\to D^{(\prime)}_{s1}\ell\nu_{\ell}$ can be obtained from Eqs. (30), (31) and (32), respectively, with simple replacement. The combined transverse and total differential decay widths are defined as

\displaystyle\frac{d\Gamma_{T}}{dq^{2}}=\frac{d\Gamma_{+}}{dq^{2}}+\frac{d\Gamma_{-}}{dq^{2}},\quad\frac{d\Gamma}{dq^{2}}=\frac{d\Gamma_{L}}{dq^{2}}+\frac{d\Gamma_{T}}{dq^{2}}.

(33)

For the decays with $D^{(\prime)}_{1}$ and $D^{(\prime)}_{s1}$ involved, it is meaningful to define the polarization fraction due to the existence of different polarizations

\displaystyle f_{L}=\frac{\Gamma_{L}}{\Gamma_{L}+\Gamma_{+}+\Gamma_{-}}.

(34)

As to the forward-backward asymmetry, the analytical expression is defined as ptau3 ,

\displaystyle A_{FB}=\frac{\int^{1}_{0}{d\Gamma\over dcos\theta}dcos\theta-\int^{0}_{-1}{d\Gamma\over dcos\theta}dcos\theta}{\int^{1}_{-1}{d\Gamma\over dcos\theta}dcos\theta}=\frac{\int b_{\theta}(q^{2})dq^{2}}{\Gamma_{B\to D^{**}\ell\nu_{\ell}}},

(35)

where $\theta$ is the angle between the 3-momenta of the lepton $\ell$ and the initial $B$ meson in the $\ell\nu$ rest frame. The the angular coefficient $b_{\theta}(q^{2})$ for the decays $B\to D^{*}_{0}\ell\nu_{\ell}$ is given as ptau3

\displaystyle b_{\theta}(q^{2})={G_{F}^{2}|V_{cb}|^{2}\over 128\pi^{3}m_{B}^{3}}q^{2}\sqrt{\lambda(q^{2})}\left(1-{m_{\ell}^{2}\over q^{2}}\right)^{2}{m_{\ell}^{2}\over q^{2}}(H^{s}_{V,0}H^{s}_{V,t}),

(36)

with the helicity amplitudes

\displaystyle H^{s}_{V,0}\left(q^{2}\right)=\sqrt{\frac{\lambda\left(q^{2}\right)}{q^{2}}}F_{1}\left(q^{2}\right),H^{s}_{V,t}\left(q^{2}\right)=\frac{m_{B}^{2}-m_{{D^{*}_{0}}}^{2}}{\sqrt{q^{2}}}F_{0}\left(q^{2}\right).

(37)

Here $\lambda(q^{2})=((m_{B}-m_{{D^{*}_{0}}})^{2}-q^{2})((m_{B}+m_{{D^{*}_{0}}})^{2}-q^{2})$ . While for the decays $B\to D^{(\prime)}_{1}\ell\nu_{\ell}$ , the function $b_{\theta}(q^{2})$ is written as

\displaystyle b_{\theta}(q^{2})={G_{F}^{2}|V_{cb}|^{2}\over 128\pi^{3}m_{B}^{3}}q^{2}\sqrt{\lambda(q^{2})}\left(1-{m_{\ell}^{2}\over q^{2}}\right)^{2}\left[{1\over 2}(H_{V,+}^{2}-H_{V,-}^{2})+{m_{\ell}^{2}\over q^{2}}(H_{V,0}H_{V,t})\right],

(38)

where the corresponding helicity amplitudes are listed as

$\displaystyle H_{V,\pm}\left(q^{2}\right)$	$\displaystyle=$	$\displaystyle\left(m_{B_{s}}-{m_{D_{1}}}\right)V_{1}\left(q^{2}\right)\mp\frac{\sqrt{\lambda\left(q^{2}\right)}}{m_{B}-m_{D_{1}}}A\left(q^{2}\right),$
$\displaystyle H_{V,0}\left(q^{2}\right)$	$\displaystyle=$	$\displaystyle\frac{m_{B}-m_{D_{1}}}{2m_{D_{1}}\sqrt{q^{2}}}\left[-\left(m_{B}^{2}-m_{D_{1}}^{2}-q^{2}\right)V_{1}\left(q^{2}\right)+\frac{\lambda\left(q^{2}\right)V_{2}\left(q^{2}\right)}{\left(m_{B}-m_{D_{1}}\right)^{2}}\right],$
$\displaystyle H_{V,t}\left(q^{2}\right)$	$\displaystyle=$	$\displaystyle-\sqrt{\frac{\lambda\left(q^{2}\right)}{q^{2}}}V_{0}\left(q^{2}\right),$	(39)

It is noticed that the subscript $V$ in each helicity amplitude refers to the $\gamma_{\mu}(1-\gamma_{5})$ current.

III Numerical results and discussions

Table 1: The values of the input parameters BN ; pdg22 ; Cheng:2003id ; Becirevic:1998ua ; Li:2009wq .

Mass(GeV)	$m_{b}=4.8$	$m_{c}=1.4$	$m_{u}=0.25$	$m_{s}=0.37$
	$m_{e}=0.000511$	$m_{\mu}=0.106$	$m_{\tau}=1.78$	$m_{B_{s}}=5.367$
	$m_{B}=5.279$	$m_{D_{0}^{\ast}}=2.343$	$m_{D_{s0}^{\ast}}=2.317$	$m_{D_{s1}}=2.460$
	$m_{D_{s1}^{\prime}}=2.536$	$m_{D_{1}}=2.422$	$m_{D^{\prime}_{1}}=2.412$

CKM	$V_{cb}=(40.8\pm 1.4)\times 10^{-3}$

shape parameters(GeV)	$\beta^{\prime}_{B}=0.555^{+0.048}_{-0.048}$	$\beta^{\prime}_{B_{s}}=0.628^{+0.035}_{-0.034}$	$\beta^{\prime}_{D_{0}^{\ast}}=0.373^{+0.063}_{-0.059}$
	$\beta^{\prime}_{D_{s0}^{\ast}}=0.325^{+0.043}_{-0.043}$	$\beta^{\prime}_{{}^{3}D_{s1}}=0.342^{+0.030}_{-0.034}$	$\beta^{\prime}_{{}^{1}D_{s1}^{\prime}}=0.342^{+0.039}_{-0.039}$
	$\beta^{\prime}_{{}^{3}D_{1}}=0.332^{+0.031}_{-0.034}$	$\beta^{\prime}_{{}^{1}D_{1}^{\prime}}=0.329^{+0.038}_{-0.040}$

Lifetimes(s)	$\tau_{B_{s}}=(1.520\pm 0.005)\times 10^{-12}$	$\tau_{B^{\pm}}=(1.638\pm 0.004)\times 10^{-12}$
	$\tau_{B^{0}}=(1.519\pm 0.004)\times 10^{-12}$

The adopted input parameters pdg22 , such as the constituent quark masses, the hadron and lepton masses, the $B$ meson lifetime and the Cabibbo-Kobayashi-Maskawa (CKM) matrix element $V_{cb}$ , in our numerical calculations are listed in Table 1. In the calculations of the helicity amplitudes, the transition form factors are the most important inputs, some of which have been calculated in our previous work Zhang:2023ypl . The parameterized form factors are extrapolated from the space-like region to the time-like region by using following expression,

\displaystyle F\left(q^{2}\right)=\frac{F(0)}{1-aq^{2}/m^{2}+bq^{4}/m^{4}},

(40)

where $m$ represents the initial meson mass and $F(q^{2})$ denotes the different form factors. The values of $a$ and $b$ can be obtained by performing a 3-parameter fit to the form factors in the range $-15\text{GeV}^{2}\leq q^{2}\leq 0$ , which are collected in Table 2. The uncertainties arise from the decay constants of the initial $B_{(s)}$ meson and the final state mesons. Certainly, in order to compare with the results given in other works, we also give the form factors of the transitions $B_{(s)}\to D^{3/2}_{(s)1},D^{1/2}_{(s)1}$ under the heavy quark limit, which are shown in Table 3. Obviously, our results are consistent with the previous CLFQM Cheng:2003sm and ISGW2 Verma:2011yw calculations. The signs of the form factors of the transitions $B_{(s)}\to D^{1/2}_{(s)1}$ between ours and the other two theoretical predictions Verma:2011yw ; Cheng:2003sm are opposite because the definations for the $D^{1/2}_{(s)1}$ mixing formula shown in Eq. (2) are different.

Table 2: The form factors of the transtions

B_{(s)}\to D^{*}_{0},D^{*}_{s0},D_{s1},D^{\prime}_{s1},D_{1}

and

D_{1}^{\prime}

in the CLFQM. The uncertainties are from the decay constants of

B_{(s)}

and the final state mesons.

	$F_{i}(q^{2}=0)$	$F_{i}(q^{2}_{max})$	a	b
$F_{1}^{BD^{\ast}_{0}}$	$0.25^{+0.03+0.05}_{-0.02-0.05}$	$0.30^{+0.03+0.06}_{-0.03-0.07}$	$0.70^{+0.04+0.03}_{-0.05-0.11}$	$0.65^{+0.08+0.03}_{-0.07-0.07}$
$F_{0}^{BD^{\ast}_{0}}$	$0.25^{+0.03+0.05}_{-0.02-0.05}$	$0.22^{+0.02+0.04}_{-0.01-0.04}$	$-0.38^{+0.04+0.05}_{-0.04-0.02}$	$0.21^{+0.07+0.08}_{-0.07-0.08}$
$F_{1}^{B_{s}D^{*}_{s0}}$	$0.21^{+0.02+0.04}_{-0.01-0.04}$	$0.24^{+0.02+0.05}_{-0.01-0.05}$	$0.63^{+0.05+0.07}_{-0.06-0.12}$	$0.78^{+0.08+0.01}_{-0.09-0.04}$
$F_{0}^{B_{s}D^{*}_{s0}}$	$0.21^{+0.02+0.04}_{-0.01-0.04}$	$0.18^{+0.02+0.03}_{-0.01-0.03}$	$-0.43^{+0.01+0.01}_{-0.00-0.02}$	$0.28^{+0.03+0.01}_{-0.06-0.04}$
$A^{B_{s}D_{s1}}$	$0.20^{+0.01+0.02}_{-0.01-0.02}$	$0.18^{+0.02+0.03}_{-0.01-0.02}$	$-0.27^{+0.06+0.8}_{-0.07-0.09}$	$0.11^{+0.02+0.02}_{-0.02-0.03}$
$V^{B_{s}D_{s1}}_{0}$	$0.40^{+0.02+0.04}_{-0.02-0.04}$	$0.42^{+0.02+0.05}_{-0.02-0.05}$	$-0.17^{+0.02+0.04}_{-0.04-0.06}$	$-0.02^{+0.01+0.00}_{-0.00-0.01}$
$V^{B_{s}D_{s1}}_{1}$	$0.58^{+0.01+0.03}_{-0.02-0.04}$	$0.57^{+0.01+0.03}_{-0.02-0.04}$	$-0.05^{+0.01+0.01}_{-0.01-0.01}$	$0.02^{+0.00+0.01}_{-0.00-0.00}$
$V^{B_{s}D_{s1}}_{2}$	$-0.05^{+0.01+0.02}_{-0.00-0.01}$	$-0.05^{+0.01+0.03}_{-0.00-0.02}$	$0.56^{+0.06+0.22}_{-0.06-0.25}$	$2.50^{+0.25+1.67}_{-0.20-1.30}$
$A^{B_{s}D^{\prime}_{s1}}$	$0.08^{+0.01+0.02}_{-0.01-0.02}$	$0.03^{+0.01+0.01}_{-0.02-0.03}$	$2.05^{+0.13+0.35}_{-0.10-0.35}$	$5.57^{+0.25+0.50}_{-0.20-0.41}$
$V^{B_{s}D^{\prime}_{s1}}_{0}$	$-0.08^{+0.01+0.04}_{-0.01-0.04}$	$-0.05^{+0.02+0.05}_{-0.01-0.04}$	$1.24^{+0.05+0.23}_{-0.06-0.25}$	$0.74^{+0.02+0.21}_{-0.02-0.17}$
$V^{B_{s}D^{\prime}_{s1}}_{1}$	$0.17^{+0.02+0.04}_{-0.03-0.03}$	$0.15^{+0.01+0.02}_{-0.03-0.03}$	$-0.52^{+0.06+0.06}_{-0.05-0.06}$	$0.36^{+0.01+0.03}_{-0.00-0.08}$
$V^{B_{s}D^{\prime}_{s1}}_{2}$	$0.11^{+0.01+0.01}_{-0.02-0.02}$	$0.10^{+0.01+0.01}_{-0.02-0.02}$	$0.25^{+0.06+0.06}_{-0.07-0.07}$	$-0.07^{+0.03+0.01}_{-0.04-0.03}$
$A^{BD_{1}}$	$0.21^{+0.02+0.02}_{-0.01-0.01}$	$0.20^{+0.02+0.02}_{-0.02-0.02}$	$-0.23^{+0.10+0.09}_{-0.09-0.09}$	$0.09^{+0.02+0.02}_{-0.02-0.02}$
$V^{BD_{1}}_{0}$	$0.42^{+0.03+0.04}_{-0.02-0.04}$	$0.44^{+0.04+0.05}_{-0.02-0.05}$	$0.16^{+0.02+0.03}_{-0.02-0.05}$	$-0.02^{+0.00+0.01}_{-0.00-0.01}$
$V^{BD_{1}}_{1}$	$0.57^{+0.03+0.03}_{-0.01-0.04}$	$0.56^{+0.03+0.03}_{-0.02-0.04}$	$-0.07^{+0.00+0.01}_{-0.01-0.00}$	$0.02^{+0.00+0.00}_{-0.00-0.00}$
$V^{BD_{1}}_{2}$	$-0.06^{+0.01+0.02}_{-0.01-0.02}$	$-0.06^{+0.01+0.04}_{-0.01-0.03}$	$0.57^{+0.06+0.17}_{-0.07-0.20}$	$2.13^{+0.21+1.30}_{-0.18-1.17}$
$A^{BD^{\prime}_{1}}$	$0.07^{+0.02+0.02}_{-0.02-0.02}$	$0.02^{+0.03+0.01}_{-0.02-0.02}$	$-1.90^{+0.09+0.12}_{-0.10-0.14}$	$5.64^{+0.22+1.59}_{-0.18-1.29}$
$V^{BD^{\prime}_{1}}_{0}$	$-0.08^{+0.02+0.04}_{-0.02-0.03}$	$-0.06^{+0.02+0.06}_{-0.02-0.03}$	$-0.11^{+0.01+0.06}_{-0.04-0.11}$	$3.67^{+0.05+0.51}_{-0.05-0.51}$
$V^{BD^{\prime}_{1}}_{1}$	$0.14^{+0.04+0.03}_{-0.04-0.03}$	$0.13^{+0.03+0.03}_{-0.04-0.02}$	$-0.36^{+0.04+0.04}_{-0.06-0.06}$	$0.09^{+0.01+0.01}_{-0.01-0.01}$
$V^{BD^{\prime}_{1}}_{2}$	$0.10^{+0.03+0.02}_{-0.02-0.02}$	$0.11^{+0.03+0.02}_{-0.04-0.04}$	$0.19^{+0.06+0.06}_{-0.09-0.09}$	$0.00^{+0.01+0.01}_{-0.00-0.01}$

The branching ratios of the decays $B_{(s)}\to D^{*}_{(s)0}\ell{\nu}_{\ell}$ are shown in Table 4. For comparison, the values given by other theoretical approaches and the current experiments are also listed. One can find that the branching ratios of the decays $B^{+}\to\bar{D}^{*0}_{0}\ell^{+}{\nu}_{\ell}$ are larger than the results from the QCD sum rules (QCDSRs) Zuo:2023ksq and the heavy quark effective field theory (HQEFT) W.Y. . Obviously, our predictions are consistent with the previous CLFQM calculations Kang:2018jzg and the differences are mainly due to the different values of the decay constant $f_{D^{*}_{0}}$ . In Figure 2, the changing trends of the form factors $F^{BD^{*}_{0}}_{1}(q^{2}=0)(F^{BD^{*}_{0}}_{0}(q^{2}=0))$ and the branching ratios of the decays $B^{+}\to\bar{D}^{*0}_{0}\ell^{+}\nu_{\ell}$ with $f_{D^{*}_{0}}$ are plotted, respectively. Both of them increase with $f_{D^{*}_{0}}$ . The branching ratios of the decays $B^{0}\to D^{*-}_{0}\ell^{+}{\nu}_{\ell}$ can agree with QCD LCSRs under scenario 2 (S2), where the broad resonance $D^{*}_{0}$ was considered as consisting of the two resonances $D^{*}_{0}(2105)$ and $D^{*}_{0}(2451)$ Gubernari:2023rfu . While it seems to be too large in scenario 1 (S1), where $D^{*}_{0}$ was considered as a single resonance Gubernari:2023rfu . Certainly, there still exist large errors in both S1 and S2. Our predictions are much smaller than those given in the LCSRs approach Shen:2012mm , where a very large form factor $F_{0}^{BD^{*}_{0}}(q^{2}=0)=0.94$ was used in the calculations. There exists a similar situation for the decays $B_{s}^{0}\to D^{*-}_{s0}\ell^{+}{\nu}_{\ell}$ . It is surprising that a large value $(2.2\pm 0.86)\times 10^{-2}$ was measured by Belle Belle:2007uwr in 2008, latter a more large one $(4.4\pm 1.0)\times 10^{-2}$ was given by BaBar BaBar:2008ozy , while Belle updated their measurement with only a small upper limit $<0.44\times 10^{-3}$ obtained. In theory, the branching ratios of the decays $B^{+}\to\bar{D}^{*0}_{0}\ell^{\prime+}\nu_{\ell^{\prime}}$ and $B^{0}\to D^{*-}_{0}\ell^{\prime+}\nu_{\ell^{\prime}}$ should be not much difference. In order to clarifying this puzzle, we urge our experimental colleagues to perform further more precise measurements. In Ref. A.L. , the authors calculated the branching ratios based on the general HQET expansion, the so called the Leibovich-Ligeti-Stewart-Wise (LLSW) scheme, combining other theoretical results and the constrains from experimental measurements, which are smaller than nearly all the present avaiable predictions.

Table 3: Form factors of the transitions

B_{(s)}\to D^{3/2}_{s1},D^{1/2}_{s1},D^{3/2}_{1}

and

D^{1/2}_{1}

q^{2}=0

together with other theoretical results.

Transitions	References	$A(0)$	$V_{0}(0)$	$V_{1}(0)$	$V_{2}(0)$
$B_{s}\to D^{3/2}_{s1}$	This work	$0.19$	$0.41$	$0.55$	$-0.07$
	CLFQM^a Verma:2011yw	$0.24$	$0.49$	$0.57$	$-0.09$
$B_{s}\to D^{1/2}_{s1}$	This work	$0.10$	$-0.03$	$0.24$	$0.11$
	CLFQM^a Verma:2011yw	$-0.17$	$0.13$	$-0.25$	$-0.17$
$B\to D^{3/2}_{1}$	This work	$0.20$	$0.43$	$0.55$	$-0.07$
	CLFQM Cheng:2003sm	$0.23$	$0.47$	$0.55$	$-0.09$
	ISGW2 Cheng:2003sm	$0.16$	$0.43$	$0.40$	$-0.12$
	CLFQM Verma:2011yw	$0.25$	$0.52$	$0.58$	$-0.10$
$B\to D^{1/2}_{1}$	This work	$0.09$	$-0.02$	$0.21$	$0.09$
	CLFQM^a Cheng:2003sm	$-0.12$	$0.08$	$-0.19$	$-0.12$
	ISGW2^aCheng:2003sm	$-0.16$	$0.18$	$-0.19$	$-0.18$
	CLFQM^a Verma:2011yw	$-0.13$	$0.11$	$-0.19$	$-0.14$

^a Due to the signs of the $D^{1/2}_{(s)1}$ mixing formula Eq. (2) between this work and Refs. Cheng:2003sm ; Verma:2011yw are opposite, the corresponding results are just contrary with our predictions.

Table 4: Branching ratios (

10^{-3}

) of the semileptonic decays

B_{(s)}\to D^{*}_{(s)0}\ell\nu_{\ell}

References	$B^{+}\to\bar{D}^{*0}_{0}e^{+}\nu_{e}$	$B^{+}\to\bar{D}^{*0}_{0}\mu^{+}\nu_{\mu}$	$B^{+}\to\bar{D}^{*0}_{0}\tau^{+}\nu_{\tau}$
This work	$1.66^{+0.43+0.74}_{-0.27-0.62}$	$1.65^{+0.43+0.74}_{-0.26-0.62}$	$0.21^{+0.06+0.10}_{-0.03-0.08}$
QCDSRs Zuo:2023ksq	$0.61^{+0.28+0.04}_{-0.21-0.05}$	$0.61^{+0.28+0.04}_{-0.21-0.05}$	$0.04^{+0.02+0.00}_{-0.01-0.00}$
CLFQM Kang:2018jzg	$2.31\pm 0.25$	$2.31\pm 0.25$	$0.30\pm 0.03$
HQEFT W.Y.	$0.50\pm 0.16$	$0.50\pm 0.16$	$-$
PDG pdg22	$1.35\pm 0.75$	$1.35\pm 0.75$	$-$
References	$B^{0}\to D^{*-}_{0}e^{+}\nu_{e}$	$B^{0}\to D^{*-}_{0}\mu^{+}\nu_{\mu}$	$B^{0}\to D^{*-}_{0}\tau^{+}\nu_{\tau}$
This work	$1.54^{+0.40+0.69}_{-0.25-0.57}$	$1.53^{+0.40+0.69}_{-0.25-0.57}$	$0.19^{+0.05+0.09}_{-0.03-0.07}$
QCD LCSRs Gubernari:2023rfu ^a	$3.6^{+5.1}_{-3.0}$	$3.6^{+5.1}_{-3.0}$	$0.39^{+0.51}_{-0.31}$
QCD LCSRs Gubernari:2023rfu ^b	$1.6^{+3.2}_{-1.4}$	$1.6^{+3.2}_{-1.4}$	$0.24^{+0.47}_{-0.21}$
LCSRs Shen:2012mm	$8.7^{+5.1}_{-2.8}$	$8.7^{+5.1}_{-2.8}$	$1.1^{+0.6}_{-0.3}$
LLSW A.L.	$0.51\pm 0.12$	$0.51\pm 0.12$	$0.050\pm 0.013$
BaBar BaBar:2008ozy	$4.4\pm 1.0$	$4.4\pm 1.0$	$-$
Belle Belle:2007uwr	$2.0\pm 0.86$	$2.0\pm 0.86$	$-$
Belle Belle2023	$<0.44$	$<0.44$	$-$
References	$B_{s}^{0}\to D^{*-}_{s0}e^{+}\nu_{e}$	$B_{s}^{0}\to D^{*-}_{s0}\mu^{+}\nu_{\mu}$	$B_{s}^{0}\to D^{*-}_{s0}\tau^{+}\nu_{\tau}$
This work	$1.26^{+0.26+0.55}_{-0.13-0.45}$	$1.25^{+0.26+0.55}_{-0.13-0.45}$	$0.18^{+0.04+0.08}_{-0.02-0.07}$
QCDSRs Zuo:2023ksq	$0.72^{+0.30+0.04}_{-0.26-0.06}$	$0.71^{+0.33+0.04}_{-0.25-0.05}$	$0.06^{+0.03+0.00}_{-0.02-0.00}$
CUM Navarra:2015iea	$1.3$	$1.3$	$-$
QCD LCSRs Gubernari:2023rfu	$1.9^{+3.8}_{-1.7}$	$1.9^{+3.8}_{-1.7}$	$0.26^{+0.49}_{-0.22}$
QCDSRs M. Q.	$0.9\sim 2.0$	$0.9\sim 2.0$	$-$
QCDSRs T. M.	$1.0$	$1.0$	$0.1$
LCSRs Li:2009wq	$2.3^{+1.2}_{-1.0}$	$2.3^{+1.2}_{-1.0}$	$0.57^{+0.28}_{-0.23}$
CQM Zhao:2006at	$4.90\sim 5.71$	$4.9\sim 5.71$	$-$
RQM Faustov:2012mt	$3.6\pm 0.4$	$3.6\pm 0.4$	$0.19\pm 0.02$
LSCRs Shen:2012mm	$6.0\pm 1.9$	$6.0\pm 1.9$	$0.82^{+0.18}_{-0.20}$

^a Results obtained in scenario 1 (S1), where $D^{*}_{0}$ was considered as a single broad resonance with mass being $(2343\pm 10)$ MeV and width $(229\pm 16)$ MeV.
^b Results obtained in scenario 2 (S2), where $D^{*}_{0}$ was assumed to consist of two scalar resonances $D^{*}_{0}(2105)$ and $D^{*}_{0}(2451)$ .

Then we compare our predictions for the branching ratios of the decays $B_{s}^{0}\to D^{*-}_{s0}\ell^{+}{\nu}_{\ell}$ with the results obtained from other approaches. One can find that our results are consistent with those given in the chiral unitary approach (CUA) Navarra:2015iea , the QCDSRs Zuo:2023ksq , the QCD LCSRs Gubernari:2023rfu and the LCSRs Li:2009wq within errors. While they are much smaller than the constituent quark meson (CQM) Zhao:2006at and the LCSRs Shen:2012mm calculations. Although the LCSRs was used in both Ref. Li:2009wq and Ref. Shen:2012mm , their results are very different. It is because of the different correlation function, which is taken between the vacuum and $D^{*}_{s0}(B)$ with the $B(D^{*}_{s0})$ meson being interploated by a local current for the former (the latter). The form factor of the transition $B_{s}\to D^{*}_{s0}$ obtained in the latter (the so called B-meson LCSRs) is about $0.80$ , which is larger than $0.53$ calculated by the former (the so called the conventional light meson LCSRs). Further experimental and theoretical researches are needed to clarify these divergences and puzzles.

Table 5: Branching ratios (

10^{-3}

) of the semileptonic decays

B\to D^{(\prime)}_{1}\ell\nu_{\ell}

and

B_{s}\to D^{(\prime)}_{s1}\ell\nu_{\ell}

References	$B^{+}\to\bar{D}_{1}^{0}e^{+}\nu_{e}$	$B^{+}\to\bar{D}_{1}^{0}\mu^{+}\nu_{\mu}$	$B^{+}\to\bar{D}_{1}^{0}\tau^{+}\nu_{\tau}$
This work	$6.21^{+0.94+0.43+1.00}_{-0.67-0.28-0.90}$	$6.16^{+0.93+0.42+0.99}_{-0.66-0.27-0.89}$	$0.57^{+0.08+0.03+0.07}_{-0.05-0.04-0.07}$
QCDSRs Zuo:2023ksq	$7.26^{+3.60+0.51}_{-2.87-0.49}$	$7.19^{+3.56+0.50}_{-2.84-0.48}$	$0.46^{+0.22+0.03}_{-0.17-0.03}$
LLSW A.L.	$6.40\pm 0.44$	$6.40\pm 0.44$	$0.63\pm 0.06$
HQEFT W.Y.2	$5.9\pm 1.6$	$5.9\pm 1.6$	$-$
PDG pdg22	$4.26\pm 0.26$	$4.26\pm 0.26$	$-$
References	$B^{+}\to\bar{D}_{1}^{\prime 0}e^{+}\nu_{e}$	$B^{+}\to\bar{D}_{1}^{\prime 0}\mu^{+}\nu_{\mu}$	$B^{+}\to\bar{D}_{1}^{\prime 0}\tau^{+}\nu_{\tau}$
This work	$0.15^{+0.15+0.07+0.08}_{-0.09-0.03-0.06}$	$0.15^{+0.15+0.07+0.08}_{-0.09-0.03-0.06}$	$0.02^{+0.01+0.00+0.00}_{-0.01-0.02-0.00}$
QCDSRs Zuo:2023ksq	$0.68^{+0.31+0.04}_{-0.24-0.05}$	$0.67^{+0.31+0.04}_{-0.23-0.05}$	$0.05^{+0.05+0.00}_{-0.02-0.00}$
LLSW A.L.	$0.46\pm 0.37$	$0.46\pm 0.37$	$0.03\pm 0.03$
HQEFT W.Y.	$0.48\pm 0.16$	$0.48\pm 0.16$	$-$
PDG pdg22	$2.55\pm 0.90$	$2.55\pm 0.90$	$-$
References	$B_{s}^{0}\to D_{s1}^{-}e^{+}\nu_{e}$	$B_{s}^{0}\to D_{s1}^{-}\mu^{+}\nu_{\mu}$	$B_{s}^{0}\to D_{s1}^{-}\tau^{+}\nu_{\tau}$
This work	$6.17^{+0.94+0.21+0.42}_{-0.91-0.62-0.52}$	$6.13^{+0.91+0.21+0.42}_{-0.94-0.61-0.52}$	$0.63^{+0.03+0.04+0.00}_{-0.05-0.06-0.00}$
QCDSRs Zuo:2023ksq	$6.31^{+3.07+0.44}_{-2.44-0.43}$	$6.25^{+3.03+0.44}_{-2.42-0.42}$	$0.38^{+0.18+0.03}_{-0.14-0.03}$
CQM Zhao:2006at	$7.52\sim 8.69$	$7.52\sim 8.69$	$-$
QCDSRs T.M.2	$4.90$	$4.90$	$-$
RQM Faustov:2012mt	$8.40\pm 0.90$	$8.40\pm 0.90$	$0.49\pm 0.05$
References	$B_{s}^{0}\to D_{s1}^{\prime-}e^{+}\nu_{e}$	$B_{s}^{0}\to D_{s1}^{\prime-}\mu^{+}\nu_{\mu}$	$B_{s}^{0}\to D_{s1}^{\prime-}\tau^{+}\nu_{\tau}$
This work	$0.18^{+0.06+0.07+0.07}_{-0.07-0.07-0.09}$	$0.18^{+0.06+0.07+0.07}_{-0.07-0.07-0.09}$	$0.02^{+0.01+0.01+0.00}_{-0.00-0.01-0.00}$
QCDSRs Zuo:2023ksq	$0.65^{+0.30+0.04}_{-0.23-0.05}$	$0.64^{+0.30+0.04}_{-0.23-0.05}$	$0.05^{+0.03+0.00}_{-0.02-0.00}$
RQM Faustov:2012mt	$1.90\pm 0.02$	$1.90\pm 0.02$	$0.15\pm 0.02$

We calculate the branching ratios of the decays $B^{0}_{s}\to D^{(\prime)-}_{s1}\ell^{+}\nu_{\ell}$ and $B^{+}\to\bar{D}^{(\prime)0}_{1}\ell^{+}\nu_{\ell}$ , which are listed in Table 5 with other theoretical predictions and data for comparison. All the theoretical predictions show that the branching ratios of the decays $B^{0}_{s}\to D^{-}_{s1}\ell^{+}{\nu}_{\ell}$ are (much) lager than those of the decays $B^{0}_{s}\to D^{\prime-}_{s1}\ell^{+}\nu_{\ell}$ . This is because that the related form factors of the transition $B_{s}\to D_{s1}$ are much larger than those of the transition $B_{s}\to D^{\prime}_{s1}$ . There exists a similar situation between the decays $B^{+}\to\bar{D}^{0}_{1}\ell^{+}\nu_{\ell}$ and $B^{+}\to\bar{D}^{\prime 0}_{1}\ell^{+}\nu_{\ell}$ . One can find that the branching ratios of the decays $B_{(s)}\to D_{(s)1}\ell\nu_{\ell}$ are comparable with the results given by most theoretical calculations, such as the QCDSRs Zuo:2023ksq ; T.M.2 , the CQM Zhao:2006at , the relativistic quark model (RQM) Faustov:2012mt , the LLSW A.L. , the HQEFT W.Y. , and so on. Certainly, they are also consistent well with the present avalable data pdg22 . While for the decays $B^{+}\to\bar{D}^{\prime 0}_{1}\ell^{+}\nu_{\ell}$ , their branching ratios given by all the theoretical predictions are smaller than the data measured by Belle Belle:2022yzd . Therefore we urge our experimental colleagues to accurately measure these decays. It is very helpful to probe the inner structures of the resonant states $D_{(s)1}$ and $D^{\prime}_{(s)1}$ by clarifying the tension between theory and experiment, which is the so called ‘1/2 vs 3/2 puzzle’. In order to explain this puzzle, we calculate the dependences of the branching ratios of the decays $B\to D^{(\prime)}_{1}\ell^{\prime}\nu_{\ell^{\prime}}$ on the mixing angle $\theta_{s}$ , which are shown in Figure 3, where the branching ratios for the decays $B^{+}\to D^{(\prime)0}_{1}\ell^{\prime}\nu_{\ell^{\prime}}$ increase (decrease) with the mixing angle $\theta_{s}$ . The upper (lower) shadow band and its horizontal image center line refers to the experimentally achievable range and the center value for the branching ratios of the decays $B^{+}\to D^{0}_{1}\ell^{\prime}\nu_{\ell^{\prime}}(B^{+}\to\bar{D}^{\prime 0}_{1}\ell^{\prime}\nu_{\ell^{\prime}})$ , respectively. One can find that taking some negative mixing angle $\theta_{s}$ values within a range from $-30.3^{\circ}$ to $-24.9^{\circ}$ can explain the data, which correspond to $\theta$ within the range $5^{\circ}\sim 10.4^{\circ}$ . It is similar for the decays $B^{0}_{s}\to{D}^{(\prime)-}_{s1}\ell^{+}\nu_{\ell}$ and $B^{+}\to\bar{D}^{(\prime)0}_{1}\tau^{+}\nu_{\tau},B^{0}_{s}\to{D}^{(\prime)-}_{s1}\tau^{+}\nu_{\tau}$ , the dependencies of their branching ratios on the mixing angle $\theta_{s}$ are shown in Figure 4. All of these decays shown that the branching ratios of the decays with $D_{(s)1}$ involved increase with the mixing angle $\theta_{s}$ , while it is contrary for those of the decays with $D^{\prime}_{(s)1}$ involved.

In Table 6, we also calculate the lepton flavor universality ratios, which are defined as

\displaystyle R(D^{**}_{(s)})=\frac{\Gamma\left(B\rightarrow D^{**}_{(s)}\tau\nu_{\tau}\right)}{\Gamma\left(B\rightarrow D^{**}_{(s)}\ell^{\prime}\nu_{\ell^{\prime}}\right)},

(41)

where a large part of the theoretical and experimental uncertainties, especially the errors from the form factors, can be canceled. One can find that most of our predictions are comparable with other theoretical results. Compared to the S1 and S2 values given in the QCD LCSRs Gubernari:2023rfu , our prediction for $R(D^{*}_{0})$ gives a moderate value.

III.1 Physical observables

In our study of semileptonic decays, we define two additional physical observables, namely the longitudinal polarization fraction $f_{L}$ and the forward-backward asymmetry $A_{FB}$ , to account for the impact of lepton mass and provide a more detailed physical picture. The results of these two physical observables are listed in Tables 7 and 8, respectively. In Table 7, we can clearly find that the longitudinal polarization fractions $f_{L}$ between the decays $B_{(s)}\to{D}^{(\prime)}_{(s)1}e^{+}\nu_{e}$ and $B_{(s)}\to{D}^{(\prime)}_{(s)1}\mu^{+}\nu_{\mu}$ are very close to each other, which reflect the lepton flavor universality (LFU). In order to investigate the dependences of the polarizations on the different $q^{2}$ , we divide the full energy region into two segments for each decay and calculate the longitudinal polarization fractions accordingly. Region 1 is defined as $m_{\ell}^{2}<q^{2}<\frac{(m_{B_{(s)}}-m_{D^{(\prime)}_{(s1)}})^{2}+m_{\ell}^{2}}{2}$ and Region 2 is $\frac{(m_{B_{(s)}}-m_{D_{(s)1}^{(\prime)}})^{2}+m_{\ell}^{2}}{2}<q^{2}<(m_{B_{(s)}}-m_{D_{(s)1}^{(\prime)}})^{2}$ . Obviously, the longitudinal polarization fraction in Region 1 is larger than that in Region 2 for each decay. Furthermore, for the decays with $D_{(s)1}$ involved in the final states the longitudinal polarization is dominant, while it is contrary for the decays with $D^{\prime}_{(s)1}$ involved. These results can be validated by the future high-luminosity experiments.

Table 6: The lepton flavor universality ratios of the transitions

B_{(s)}\to D^{*}_{0}

D^{*}_{s0}

D^{(\prime)}_{s1}

D^{(\prime)}_{1}

Decays	Ratios	Predicted values	Decays	Ratios	Predicted values
$B^{+}\to\bar{D}_{0}^{*0}\ell^{+}\nu_{\ell}$	$R(\bar{D}^{*}_{0})$	$0.127^{+0.003+0.003}_{-0.001-0.000}$	$B_{s}^{0}\to D_{s1}^{-}\ell^{+}\nu_{\ell}$	$R(D_{s1})$	$0.102^{+0.007+0.003}_{-0.009-0.001}$
		$0.063^{+0.001}_{-0.002}$ Zuo:2023ksq			$0.061\pm 0.003$ Zuo:2023ksq
		$0.08\pm 0.03$ F.U.			$0.09\pm 0.02$ F.U.1
$B^{0}\to D_{0}^{*-}\ell^{+}\nu_{\ell}$	$R(D^{*}_{0})$	$0.127^{+0.001+0.002}_{-0.001-0.002}$	$B_{s}^{0}\to D_{s1}^{\prime-}\ell^{+}\nu_{\ell}$	$R(D^{\prime}_{s1})$	$0.111^{+0.014+0.009}_{-0.029-0.020}$
		$0.099\pm 0.015$ A.L.			$0.084^{+0.001}_{-0.002}$ Zuo:2023ksq
		$0.11^{+0.03}_{-0.01}\textnormal{(S1)},0.16^{+0.04}_{-0.02}\textnormal{(S2)}^{1}$ Gubernari:2023rfu			$0.07\pm 0.03$ F.U.1
$B^{0}_{s}\to D_{s0}^{*}\ell^{+}\nu_{\ell}$	$R(D^{*}_{s0})$	$0.147^{+0.003+0.001}_{-0.004-0.003}$	$B^{+}\to\bar{D}_{1}^{0}\ell^{+}\nu_{\ell}$	$R(D_{1})$	$0.091^{+0.001+0.003}_{-0.002-0.004}$
		$0.080^{+0.001}_{-0.002}$ Zuo:2023ksq			$0.064^{+0.003}_{-0.003}$ Zuo:2023ksq
		$0.09\pm 0.04$ F.U.1			$0.10\pm 0.02$ F.U.
		$0.14^{+0.07}_{-0.02}$ Gubernari:2023rfu			$0.098\pm 0.007$ A.L.
			$B^{+}\to\bar{D}_{1}^{\prime 0}\ell^{+}\nu_{\ell}$	$R(D^{\prime}_{1})$	$0.110^{+0.030+0.051}_{-0.010-0.025}$
					$0.076^{+0.001}_{-0.002}$ Zuo:2023ksq
					$0.05\pm 0.02$ F.U.
					$0.074\pm 0.012$ A.L.

¹ The definitions of S1 and S2 are given in Table 4.

Table 7: The longitudinal polarization fractions

f_{L}

for the decays

B_{s}\to D^{(\prime)-}_{s1}\ell^{+}\nu_{\ell}

and

B^{+}\to\bar{D}^{(\prime)0}_{1}\ell^{+}\nu_{\ell}

in Region 1 and Region 2.

Observables	Region 1	Region 2	Total	Observables	Region 1	Region 2	Total
$f_{L}(B_{s}^{0}\to D_{s1}^{-}e^{+}\nu_{e})$	$0.81$	$0.50$	$0.71$	$f_{L}(B_{s}^{0}\to D_{s1}^{\prime-}e^{+}\nu_{e})$	$0.42$	$0.12$	$0.31$
$f_{L}(B_{s}^{0}\to D_{s1}^{-}\mu^{+}\nu_{\mu})$	$0.81$	$0.50$	$0.71$	$f_{L}(B_{s}^{0}\to D_{s1}^{\prime-}\mu^{+}\nu_{\mu})$	$0.42$	$0.12$	$0.31$
$f_{L}(B_{s}^{0}\to D_{s1}^{-}\tau^{+}\nu_{\tau})$	$0.66$	$0.48$	$0.56$	$f_{L}(B_{s}^{0}\to D_{s1}^{\prime-}\tau^{+}\nu_{\tau})$	$0.23$	$0.24$	$0.24$
$f_{L}(B^{+}\to\bar{D}_{1}^{0}e^{+}\nu_{e})$	$0.81$	$0.50$	$0.72$	$f_{L}(B^{+}\to\bar{D}_{1}^{\prime 0}e^{+}\nu_{e})$	$0.54$	$0.11$	$0.39$
$f_{L}(B^{+}\to\bar{D}_{1}^{0}\mu^{+}\nu_{\mu})$	$0.81$	$0.50$	$0.72$	$f_{L}(B^{+}\to\bar{D}_{1}^{\prime 0}\mu^{+}\nu_{\mu})$	$0.54$	$0.11$	$0.39$
$f_{L}(B^{+}\to\bar{D}_{1}^{0}\tau^{+}\nu_{\tau})$	$0.67$	$0.49$	$0.56$	$f_{L}(B^{+}\to\bar{D}_{1}^{\prime 0}\tau^{+}\nu_{\tau})$	$0.32$	$0.25$	$0.28$

In Figure 5, we plot the $q^{2}$ dependencies of the differential decay rates for the channels $B_{(s)}\to D^{*}_{(s)0}\ell{\nu}_{\ell}$ and $B_{(s)}\to{D}^{(\prime)}_{(s)1}\ell\nu_{\ell}$ . One can find that the line shapes of these differential distributions are constrained by the phase space, in other words, the lepton mass. The polarization needs to be considered in the decays $B_{s}\to{D}^{(\prime)}_{s1}\ell\nu_{\ell}$ and $B\to D^{(\prime)}_{1}\ell\nu_{\ell}$ , which is shown in Figures 5(c)-5(h). It is obvious that for the decays with $D_{(s)1}$ involved the longitudinal polarization is dominant in small $q^{2}$ region and comparable with the transverse ones in large $q^{2}$ region. While for the decays with $D^{\prime}_{(s)1}$ involved the transverse polarizations are dominant, especially in large $q^{2}$ region. It is interesting that taking some special $q^{2}$ values for the decays $B\to{D}^{\prime}_{1}\ell^{\prime}\nu_{\ell^{\prime}}$ and $B_{s}\to{D}^{\prime}_{s1}\ell^{\prime}\nu_{\ell^{\prime}}$ , we can find that the contribution from the longitudinal polarization almost disappears with only the transverse polarizations left, which is shown in Figures 5(d) and 5(f). Maybe such a phenomenon can be checked in the future LHC and Super KEKB experiments to test the present mixing mechanism.

Table 8: The forward-backward asymmetries

A_{FB}

for the decays

B_{(s)}\to D^{*}_{(s)0}\ell{\nu}_{\ell}

and

B_{(s)}\to{D}^{(\prime)}_{(s)1}\ell\nu_{\ell}

Channels		$B^{+}\to\bar{D}^{*}_{0}e^{+}\nu_{e}$	$B^{+}\to\bar{D}^{*}_{0}\mu^{+}\nu_{\mu}$	$B^{+}\to\bar{D}^{*}_{0}\tau^{+}\nu_{\tau}$
$A_{FB}$	This work	$(5.820^{+1.494+2.582}_{-0.904-2.108})\times 10^{-7}$	$0.020^{+0.005+0.009}_{-0.003-0.007}$	$0.383^{+0.102+0.177}_{-0.063-0.143}$
Channels		$B_{s}^{0}\to D^{*-}_{s0}e^{+}\nu_{e}$	$B_{s}^{0}\to D^{*-}_{s0}\mu^{+}\nu_{\mu}$	$B_{s}^{0}\to D^{*-}_{s0}\tau^{+}\nu_{\tau}$
$A_{FB}$	This work	$(5.465^{+1.100+2.298}_{-0.515-1.898})\times 10^{-7}$	$0.019^{+0.004+0.010}_{-0.002-0.007}$	$0.381^{+0.079+0.168}_{-0.038-0.138}$
	Albertus:2014bfa	$8.22\times 10^{-7}$	$0.016$	$0.39$
Channel		$B_{s}^{0}\to D_{s1}^{-}e^{+}\nu_{e}$	$B_{s}^{0}\to D_{s1}^{-}\mu^{+}\nu_{\mu}$	$B_{s}^{0}\to D_{s1}^{-}\tau^{+}\nu_{\tau}$
$A_{FB}$	This work	$-0.175^{+0.016+0.027+0.011}_{-0.014-0.016-0.017}$	$-0.174^{+0.016+0.026+0.010}_{-0.014-0.017-0.017}$	$-0.129^{+0.010+0.021+0.003}_{-0.015-0.017-0.016}$
	Albertus:2014bfa	$-0.19$	$-0.18$	$0.10$
Channels		$B_{s}^{0}\to D_{s1}^{\prime-}e^{+}\nu_{e}$	$B_{s}^{0}\to D_{s1}^{\prime-}\mu^{+}\nu_{\mu}$	$B_{s}^{0}\to D_{s1}^{\prime-}\tau^{+}\nu_{\tau}$
$A_{FB}$	This work	$-0.407^{+0.104+0.129+0.020}_{-0.101-0.185-0.029}$	$-0.405^{+0.104+0.129+0.020}_{-0.102-0.186-0.029}$	$-0.248^{+0.026+0.080+0.048}_{-0.064-0.027-0.007}$
	Albertus:2014bfa	$-0.41$	$-0.40$	$-0.20$
Channels		$B^{+}\to\bar{D}_{1}^{0}e^{+}\nu_{e}$	$B^{+}\to\bar{D}_{1}^{0}\mu^{+}\nu_{\mu}$	$B^{+}\to\bar{D}_{1}^{0}\tau^{+}\nu_{\tau}$
$A_{FB}$	This work	$-0.178^{+0.014+0.018+0.008}_{-0.030-0.025-0.017}$	$-0.177^{+0.012+0.016+0.006}_{-0.032-0.028-0.019}$	$-0.131^{+0.024+0.037+0.014}_{-0.034-0.036-0.025}$
Channels		$B^{+}\to\bar{D}_{1}^{\prime 0}e^{+}\nu_{e}$	$B^{+}\to\bar{D}_{1}^{\prime 0}\mu^{+}\nu_{\mu}$	$B^{+}\to\bar{D}_{1}^{\prime 0}\tau^{+}\nu_{\tau}$
$A_{FB}$	This work	$-0.316^{+0.115+0.085+0.083}_{-0.190-0.090-0.048}$	$-0.314^{+0.111+0.081+0.079}_{-0.096-0.095-0.043}$	$-0.186^{-0.013+0.041+0.024}_{-0.032-0.013-0.017}$

From Table 8, we find that the ratios of the forward-backward asymmetries $A^{\mu}_{FB}$ / $A^{e}_{FB}$ between the semileptonic decays $B_{(s)}\to{D}^{*}_{(s)0}\mu^{+}\nu_{\mu}$ and $B_{(s)}\to{D}^{*}_{(s)0}e^{+}\nu_{e}$ are about $3.5\times 10^{4}$ . The reason is that the forward-backward asymmetries $A_{FB}$ for the decays $B_{(s)}\to{D}^{*}_{(s)0}\ell^{+}\nu_{\ell}$ are proportional to the square of the lepton mass. Undoubtedly, the effect of lepton mass can be well checked in such decay mode with a scalar meson involved in the final states. While for the decays $B_{(s)}\to{D}^{(\prime)}_{(s)1}\ell^{\prime}\nu_{\ell^{\prime}}$ , the values of the forward-backward asymmetries $A^{\mu}_{FB}$ and $A^{e}_{FB}$ are almost equal to each other. The magnitudes of the $A_{FB}$ for the decays $B_{(s)}\to{D}^{\prime}_{(s)1}\ell\nu_{\ell}$ are larger than those for the decays $B_{(s)}\to{D}_{(s)1}\ell\nu_{\ell}$ . It is worth mentioning that our results are consistent well with those calculated in the nonrelativistic constituent quark models Albertus:2014bfa , which are shown in Table 8. In Figure 6, we also display the $q^{2}$ -dependencies of the forward-backward asymmetries $A_{FB}$ for the decays $B_{(s)}\to D^{*}_{(s)0}\ell{\nu}_{\ell}$ and $B_{(s)}\to{D}^{(\prime)}_{(s)1}\ell\nu_{\ell}$ . Obviously, the signs of the $A_{FB}$ for the decays $B_{(s)}\to D^{*}_{(s)0}\ell{\nu}_{\ell}$ are contrary with those for the decays $B_{(s)}\to{D}^{(\prime)}_{(s)1}\ell\nu_{\ell}$ . The lepton mass effects can be easily observed in these figures.

IV Summary

In this work, we used the covariant light-front quark method to comprehensively investigate the semileptonic $B_{(s)}$ decays to $D_{0}^{\ast}$ , $D_{s0}^{\ast}$ , $D^{(\prime)}_{s1}$ and $D^{(\prime)}_{1}$ , which can provide an important reference for future experiments. We calculated the branching ratios, the longitudinal polarization fractions $f_{L}$ , and the forward-backward asymmetries $A_{FB}$ for these semileptonic $B_{(s)}$ decays using the helicity amplitudes combined with form factors. We found the following points:

1.

The small form factors of the transitions $B_{(s)}\to D_{0}^{\ast},D_{s0}^{\ast}$ are related to the small decay constants $f_{D_{0}^{\ast}}$ and $f_{D_{s0}^{\ast}}$ . Unfortunately, there are large uncertainties in these two decay constants. Combined with the data, our predictions for the branching ratios of the semileptonic $B_{(s)}$ meson decays with $D_{0}^{\ast}$ and $D_{s0}^{\ast}$ involved are helpful in probing the inner structures of these two resonances. Recently, Belle updated their measurement for the decays $B^{0}\to D^{*-}_{0}\ell^{\prime+}\nu_{\ell^{\prime}}$ with only a small upper limit $Br(B^{0}\to D^{*-}_{0}\ell^{\prime+}\nu_{\ell^{\prime}})<0.44\times 10^{-3}$ obtained, which is much larger than most theoretical predictions. We urge our experimental colleagues to perform further more precise measurements to clarify this new puzzle.
2.

In our considered decays, the branching ratios of the channels $B^{0}_{s}\to D^{-}_{s1}\ell^{+}{\nu}_{\ell}$ are (much) lager than those of the decays $B^{0}_{s}\to D^{\prime-}_{s1}\ell^{+}\nu_{\ell}$ . This is because the related form factors of the transition $B_{s}\to D_{s1}$ are much larger than those of the transition $B_{s}\to D^{\prime}_{s1}$ . There exists a similar situation between the decays $B^{+}\to\bar{D}^{0}_{1}\ell^{+}\nu_{\ell}$ and $B^{+}\to\bar{D}^{\prime 0}_{1}\ell^{+}\nu_{\ell}$ . In addition, we calculated the dependencies of the branching ratios of the decays $B\to D^{(\prime)}_{1}\ell\nu_{\ell}$ and $B_{s}\to D^{(\prime)}_{s1}\ell\nu_{\ell}$ on the mixing angle $\theta_{s}$ . One can find that taking some negative mixing angle $\theta_{s}$ values within a range from $-30.3^{\circ}$ to $-24.9^{\circ}$ can explain the data, which correspond to $\theta$ within the range $5^{\circ}\sim 10.4^{\circ}$ .
3.

In these semileptonic decays $B_{(s)}\to D^{(\prime)}_{(s)1}\ell\nu_{\ell}$ , the longitudinal polarization fractions in small $q^{2}$ region are always larger than those in large $q^{2}$ region. Furthermore, the longitudinal polarization for the decays with $D_{(s)1}$ involved in the final states is dominant, while it is contrary for the decays with $D^{\prime}_{(s)1}$ involved. It is interesting that taking some special $q^{2}$ values for the decays $B\to D^{\prime}_{1}\ell^{\prime}\nu_{\ell^{\prime}}$ and $B_{s}\to{D}^{\prime}_{s1}\ell^{\prime}\nu_{\ell^{\prime}}$ , we find that the contribution from the longitudinal polarization almost disappears with only the transverse polarizations left. Maybe such a phenomenon can be searched for in the future LHC and Super KEKB experiments to test the present mixing mechanism.

Acknowledgment

We thank Prof. Guo-Li Wang for helpful discussions. This work is partly supported by the National Natural Science Foundation of China under Grant No. 11347030, the Program of Science and Technology Innovation Talents in Universities of Henan Province 14HASTIT037, as well as the Natural Science Foundation of Henan Province under Grant No. 232300420116.

Appendix A Some specific rules under the $p^{-}$ intergration

When preforming the integraion, we need to include the zero-mode contributions. It amounts to performing the integration in a proper way in the CLFQM. Specificlly we use the following rules given in Refs. Cheng:2003sm ; Jaus

$\displaystyle\hat{p}_{1\mu}^{\prime}$	$\displaystyle\doteq$	$\displaystyle P_{\mu}A_{1}^{(1)}+q_{\mu}A_{2}^{(1)},$	(42)
$\displaystyle\hat{p}_{1\mu}^{\prime}\hat{p}_{1\nu}^{\prime}$	$\displaystyle\doteq$	$\displaystyle g_{\mu\nu}A_{1}^{(2)}+P_{\mu}P_{\nu}A_{2}^{(2)}+\left(P_{\mu}q_{\nu}+q_{\mu}P_{\nu}\right)A_{3}^{(2)}+q_{\mu}q_{\nu}A_{4}^{(2)},$	(43)
$\displaystyle Z_{2}$	$\displaystyle=$	$\displaystyle\hat{N}_{1}^{\prime}+m_{1}^{\prime 2}-m_{2}^{2}+\left(1-2x_{1}\right)M^{\prime 2}+\left(q^{2}+q\cdot P\right)\frac{p_{\perp}^{\prime}\cdot q_{\perp}}{q^{2}},$	(44)
$\displaystyle A_{1}^{(1)}$	$\displaystyle=$	$\displaystyle\frac{x_{1}}{2},\quad A_{2}^{(1)}=A_{1}^{(1)}-\frac{p_{\perp}^{\prime}\cdot q_{\perp}}{q^{2}},\quad A_{3}^{(2)}=A_{1}^{(1)}A_{2}^{(1)},$	(45)
$\displaystyle A_{4}^{(2)}$	$\displaystyle=$	$\displaystyle\left(A_{2}^{(1)}\right)^{2}-\frac{1}{q^{2}}A_{1}^{(2)},\quad A_{1}^{(2)}=-p_{\perp}^{\prime 2}-\frac{\left(p_{\perp}^{\prime}\cdot q_{\perp}\right)^{2}}{q^{2}},\quad A_{2}^{(2)}=\left(A_{1}^{(1)}\right)^{2}.$	(46)

Appendix B EXPRESSIONS OF $B\rightarrow D^{*}_{0},\;^{i}D_{1}$ FORM FACTORS

$\displaystyle F_{1}^{BD^{*}_{0}}\left(q^{2}\right)$	$\displaystyle=$	$\displaystyle\frac{N_{c}}{16\pi^{3}}\int dx_{2}d^{2}p_{\perp}^{\prime}\frac{h_{B}^{\prime}h_{D^{*}_{0}}^{\prime\prime}}{x_{2}\hat{N}_{1}^{\prime}\hat{N}_{1}^{\prime\prime}}\left[x_{1}\left(M_{0}^{\prime 2}+M_{0}^{\prime\prime 2}\right)+x_{2}q^{2}\right.$	(47)
		$\displaystyle\left.-x_{2}\left(m_{1}^{\prime}+m_{1}^{\prime\prime}\right)^{2}-x_{1}\left(m_{1}^{\prime}-m_{2}\right)^{2}-x_{1}\left(m_{1}^{\prime\prime}+m_{2}\right)^{2}\right],$	(47)
$\displaystyle F_{0}^{BD^{*}_{0}}\left(q^{2}\right)$	$\displaystyle=$	$\displaystyle F_{1}^{BD^{}_{0}}\left(q^{2}\right)+\frac{q^{2}}{q\cdot P}\frac{N_{c}}{16\pi^{3}}\int dx_{2}d^{2}p_{\perp}^{\prime}\frac{2h_{B}^{\prime}h_{D^{}_{0}}^{\prime\prime}}{x_{2}\hat{N}_{1}^{\prime}\hat{N}_{1}^{\prime\prime}}\left\{-x_{1}x_{2}M^{\prime 2}-p_{\perp}^{\prime 2}-m_{1}^{\prime}m_{2}\right.$	(48)
		$\displaystyle\left.-\left(m_{1}^{\prime\prime}+m_{2}\right)\left(x_{2}m_{1}^{\prime}+x_{1}m_{2}\right)+2\frac{q\cdot P}{q^{2}}\left(p_{\perp}^{\prime 2}+2\frac{\left(p_{\perp}^{\prime}\cdot q_{\perp}\right)^{2}}{q^{2}}\right)+2\frac{\left(p_{\perp}^{\prime}\cdot q_{\perp}\right)^{2}}{q^{2}}\right.$
		$\displaystyle\left.-\frac{p_{\perp}^{\prime}\cdot q_{\perp}}{q^{2}}\left[M^{\prime\prime 2}-x_{2}\left(q^{2}+q\cdot P\right)-\left(x_{2}-x_{1}\right)M^{\prime 2}+2x_{1}M_{0}^{\prime 2}\right.\right.$
		$\displaystyle\left.\left.-2\left(m_{1}^{\prime}-m_{2}\right)\left(m_{1}^{\prime}-m_{1}^{\prime\prime}\right)\right]\right\},$
$\displaystyle A^{B\;^{i}D_{1}}(q^{2})$	$\displaystyle=$	$\displaystyle(M^{\prime}-M^{\prime\prime})\frac{N_{c}}{16\pi^{3}}\int dx_{2}d^{2}p_{\perp}^{\prime}\frac{2h_{B}^{\prime}h_{\;{}^{i}D_{1}}^{\prime\prime}}{x_{2}\hat{N}_{1}^{\prime}\hat{N}_{1}^{\prime\prime}}\left\{x_{2}m_{1}^{\prime}+x_{1}m_{2}+\left(m_{1}^{\prime}+m_{1}^{\prime\prime}\right)\frac{p_{\perp}^{\prime}\cdot q_{\perp}}{q^{2}}\right.$	(49)
		$\displaystyle\left.+\frac{2}{w_{{}^{i}D_{s1}}^{\prime\prime}}\left[p_{\perp}^{\prime 2}+\frac{\left(p_{\perp}^{\prime}\cdot q_{\perp}\right)^{2}}{q^{2}}\right]\right\},$	(49)
$\displaystyle V_{1}^{B\;^{i}D_{1}}(q^{2})$	$\displaystyle=$	$\displaystyle-\frac{1}{M^{\prime}-M^{\prime\prime}}\frac{N_{c}}{16\pi^{3}}\int dx_{2}d^{2}p_{\perp}^{\prime}\frac{h_{B}^{\prime}h_{\;{}^{i}D_{1}}^{\prime\prime}}{x_{2}\hat{N}_{1}^{\prime}\hat{N}_{1}^{\prime\prime}}\{2x_{1}\left(m_{2}-m_{1}^{\prime}\right)\left(M_{0}^{\prime 2}+M_{0}^{\prime\prime 2}\right)+4x_{1}m_{1}^{\prime\prime}M_{0}^{\prime 2}$	(50)
		$\displaystyle+2x_{2}m_{1}^{\prime}q\cdot P\left.+2m_{2}q^{2}-2x_{1}m_{2}\left(M^{\prime 2}+M^{\prime\prime 2}\right)+2\left(m_{1}^{\prime}-m_{2}\right)\left(m_{1}^{\prime}-m_{1}^{\prime\prime}\right)^{2}+8\left(m_{1}^{\prime}-m_{2}\right)\right.$
		$\displaystyle\left.\times\left[p_{\perp}^{\prime 2}+\frac{\left(p_{\perp}^{\prime}\cdot q_{\perp}\right)^{2}}{q^{2}}\right]+2\left(m_{1}^{\prime}-m_{1}^{\prime\prime}\right)\left(q^{2}+q\cdot P\right)\frac{p_{\perp}^{\prime}\cdot q_{\perp}}{q^{2}}-4\frac{q^{2}p_{\perp}^{\prime 2}+\left(p_{\perp}^{\prime}\cdot q_{\perp}\right)^{2}}{q^{2}w_{{}^{i}D_{s1}}^{\prime\prime}}\right.$
		$\displaystyle\left.\times\left[2x_{1}\left(M^{\prime 2}+M_{0}^{\prime 2}\right)-q^{2}-q\cdot P-2\left(q^{2}+q\cdot P\right)\frac{p_{\perp}^{\prime}\cdot q_{\perp}}{q^{2}}-2\left(m_{1}^{\prime}+m_{1}^{\prime\prime}\right)\left(m_{1}^{\prime}-m_{2}\right)\right]\right\},\;\;\;\;\;\;\;$
$\displaystyle V_{2}^{B\;^{i}D_{1}}(q^{2})$	$\displaystyle=$	$\displaystyle(M^{\prime}-M^{\prime\prime})\frac{N_{c}}{16\pi^{3}}\int dx_{2}d^{2}p_{\perp}^{\prime}\frac{2h_{B}^{\prime}h_{\;{}^{i}D_{1}}^{\prime\prime}}{x_{2}\hat{N}_{1}^{\prime}\hat{N}_{1}^{\prime\prime}}\left\{(x_{1}-x_{2}\right)\left(x_{2}m_{1}^{\prime}+x_{1}m_{2}\right)-[2x_{1}m_{2}-m_{1}^{\prime\prime}$	(51)
		$\displaystyle+\left(x_{2}-x_{1}\right)m_{1}^{\prime}]\times\frac{p_{\perp}^{\prime}\cdot q_{\perp}}{q^{2}}-2\frac{x_{2}q^{2}+p_{\perp}^{\prime}\cdot q_{\perp}}{x_{2}q^{2}w_{{}^{i}D_{1}}^{\prime\prime}}[p_{\perp}^{\prime}\cdot p_{\perp}^{\prime\prime}+\left(x_{1}m_{2}+x_{2}m_{1}^{\prime}\right)$
		$\displaystyle\times\left(x_{1}m_{2}+x_{2}m_{1}^{\prime\prime}\right)]\},$

$\displaystyle V_{0}^{B\;^{i}D_{1}}(q^{2})$	$\displaystyle=$	$\displaystyle\frac{M^{\prime}-M^{\prime\prime}}{2M^{\prime\prime}}V_{1}^{B\;^{i}D_{1}}(q^{2})-\frac{M^{\prime}+M^{\prime\prime}}{2M^{\prime\prime}}V_{2}^{B\;^{i}D_{1}}(q^{2})-\frac{q^{2}}{2M^{\prime\prime}}\frac{N_{c}}{16\pi^{3}}\int dx_{2}d^{2}p_{\perp}^{\prime}\frac{h_{B}^{\prime}h_{\;{}^{i}D_{1}}^{\prime\prime}}{x_{2}\hat{N}_{1}^{\prime}\hat{N}_{1}^{\prime\prime}}$	(52)
		$\displaystyle\times\{2\left(2x_{1}-3\right)\left(x_{2}m_{1}^{\prime}+x_{1}m_{2}\right)-8\left(m_{1}^{\prime}-m_{2}\right)\left[\frac{p_{\perp}^{\prime 2}}{q^{2}}+2\frac{\left(p_{\perp}^{\prime}\cdot q_{\perp}\right)^{2}}{q^{4}}\right]-[\left(14-12x_{1}\right)m_{1}^{\prime}$
		$\displaystyle+2m_{1}^{\prime\prime}-\left(8-12x_{1}\right)m_{2}]\frac{p_{\perp}^{\prime}\cdot q_{\perp}}{q^{2}}+\frac{4}{w_{{}^{i}D_{1}}^{\prime\prime}}(\left[M^{\prime 2}+M^{\prime\prime 2}-q^{2}+2\left(m_{1}^{\prime}-m_{2}\right)\left(-m_{1}^{\prime\prime}+m_{2}\right)\right]$
		$\displaystyle\times\left(A_{3}^{(2)}+A_{4}^{(2)}-A_{2}^{(1)}\right)+Z_{2}\left(3A_{2}^{(1)}-2A_{4}^{(2)}-1\right)+\frac{1}{2}[x_{1}\left(q^{2}+q\cdot P\right)-2M^{\prime 2}-2p_{\perp}^{\prime}\cdot q_{\perp}$
		$\displaystyle-2m_{1}^{\prime}\left(-m_{1}^{\prime\prime}+m_{2}\right)\left.-2m_{2}\left(m_{1}^{\prime}-m_{2}\right)\right]\left(A_{1}^{(1)}+A_{2}^{(1)}-1\right)$
		$\displaystyle\left.\left.\times q\cdot P\left[\frac{p_{\perp}^{\prime 2}}{q^{2}}+\frac{\left(p_{\perp}^{\prime}\cdot q_{\perp}\right)^{2}}{q^{4}}\right]\left(4A_{2}^{(1)}-3\right)\right)\right\},\;\;\;$

with $i=1,3$ .

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	$\displaystyle\|D_{1}(2420)\rangle$	$\displaystyle=$	$\displaystyle\|D_{1}^{1/2}\rangle\sin\theta_{s}+\|D_{1}^{3/2}\rangle\cos\theta_{s},$
	$\displaystyle\|D_{1}(2430)\rangle$	$\displaystyle=$	$\displaystyle-\|D_{1}^{3/2}\rangle\sin\theta_{s}+\|D_{1}^{1/2}\rangle\cos\theta_{s}.$		(1)

	$\displaystyle\|D^{3/2}_{1}\rangle$	$\displaystyle=$	$\displaystyle\sqrt{\frac{2}{3}}\|^{1}D_{1}\rangle+\sqrt{\frac{1}{3}}\|^{3}D_{1}\rangle,$
	$\displaystyle\|D^{1/2}_{1}\rangle$	$\displaystyle=$	$\displaystyle-\sqrt{\frac{1}{3}}\|^{1}D_{1}\rangle+\sqrt{\frac{2}{3}}\|^{3}D_{1}\rangle.$		(2)

	$\displaystyle\|D_{1}(2420)\rangle$	$\displaystyle=$	$\displaystyle\|^{1}D_{1}\rangle\cos\theta+\|^{3}D_{1}\rangle\sin\theta,$
	$\displaystyle\|D_{1}(2430)\rangle$	$\displaystyle=$	$\displaystyle-\|^{1}D_{1}\rangle\sin\theta+\|^{3}D_{1}\rangle\cos\theta.$		(3)

Semileptonic B(s)B_{(s)} meson decays to D0∗​(2300),Ds​0∗​(2317),Ds​1​(2460),Ds​1​(2536),D1​(2420)D_{0}^{\ast}(2300),D_{s0}^{\ast}(2317),D_{s1}(2460),D_{s1}(2536),D_{1}(2420) and D1​(2430)D_{1}(2430) within the covariant light-front approach