# A comprehensive study on the semileptonic decay of heavy flavor mesons

Lu Zhang1, Xian-Wei Kang1,2, Xin-Heng Guo1, Ling-Yun Dai3, Tao Luo4, Chao
Wang5 1 Key Laboratory of Beam Technology of Ministry of Education, College of
Nuclear Science and Technology, Beijing Normal University, Beijing 100875,
People’s Republic of China
2 Beijing Radiation Center, Beijing 100875, China
3 School of Physics and Electronics, Hunan University, Changsha 410082, China
4 Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and
Institute of Modern Physics, Fudan University, Shanghai 200443, China
5 School of Ecology and Environment, Northwestern Polytechnical University,
Xi’an 710072, China

###### Abstract

The semileptonic decay of heavy flavor mesons offers a clean environment for
extraction of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, which
describes the CP-violating and flavor changing process in the Standard Model.
The involved form factors where the dynamical information is encoded play an
essential role in achieving any conclusive statement. That is, the knowledge
of the form factors should be under good control, requiring one to examine
more observables in addition to the branching fraction. In this paper, we
provide the mean value and the $q^{2}$-dependent shape for further observables
[differential decay distribution ($d\Gamma/dq^{2}$), forward-backward
asymmetry ($\mathcal{A}_{FB}^{l}$), longitudinal ($P_{L}^{l}$) and transverse
($P_{T}^{l}$) polarization of a charged lepton, longitudinal polarization of a
vector meson in the final state ($F_{L}^{l}(V)$), leptonic convexity parameter
($C_{F}^{l}$), and trigonometric moments ($W_{i}^{l}$) in the decay of
$D_{(s)}$ and $B_{(s)}$ to $P/Vl^{+}\nu_{l}$ ($l=e$, $\mu$ or $\tau$)], based
on the predictions of the relevant form factors from the covariant light-front
quark model. $P$ and $V$ denote the pseudoscalar and vector meson,
respectively. As a comparison, we find a good agreement with the results from
the covariant confining quark model and the relativistic quark model in the
literature. As it has been observed that the $P_{L}^{l}$ and $F_{L}^{l}(V)$
are crucial quantities to discriminate various New Physics models, the
reexamination of these observables from a different method is also essential
and necessary.

##  I INTRODUCTION

In charged-current transition, the semileptonic decay of heavy flavor mesons
offers a clean environment (compared to the pure hadronic decay) to determine
the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, which describes the quark
flavor mixing Cabibbo (1963); Kobayashi and Maskawa (1973) in the Standard
Model (SM). The precise determination of the matrix elements is one of the
main tasks for both theorists and experimentalists. Any deviation from the
unitarity relation encoded in the CKM matrix will certainly be an inspiring
signal of New Physics (NP). As a result, the semileptonic decays are
comprehensively studied to test the SM, or instead, to constrain NP
parameters. In fact, there exist tensions for both $|V_{ub}|$ and $|V_{cb}|$
from their respective determinations by inclusive and exclusive modes Kang et
al. (2014a); Tanabashi et al. (2018); Amhis et al. (2019). For a most recent
review, see Ref. Gambino et al. (2020). Another aspect of hunting for NP via
the semileptonic channel concerns the lepton-flavor universality violation;
see the most recent review in Bifani et al. (2019). To gain a deeper
understanding of these issues, re-examination of the semileptonic channel is
not obsolete but rather is worth doing again.

The hadronic and leptonic currents are involved in the semileptonic decay, and
the former is parameterized by Lorentz-invariant form factors containing the
dynamical information. The theoretical precision of this channel relies on our
good control of the knowledge of form factors. In this paper, we exploit the
results from the covariant light-front quark model (CLFQM) (Ref. Cheng et al.
(2004) and updated in Ref. Verma (2012)). The light-front form was initiated
by Dirac in a formulation of Hamiltonian dynamics Dirac (1949) and later
developed and applied to the hadron transition in Refs. Jaus (1990, 1991);
Choi and Ji (1999). However, owing to the absence of a zero-mode effect, the
conventional light-front quark model is noncovariant. To compensate for this
deficiency, the CLFQM was proposed Jaus (1999) (some recent applications can
be found, e.g., Refs. Chang et al. (2019, 2020); Zhu et al. (2019); Ke et al.
(2019); Shi et al. (2016)), where all four-momentum of each vertex is
conserved, and the constituent quarks are off-shell. In Ref. Cheng et al.
(2004), the authors fixed the free parameters $\beta$ in the Gaussian-type
wave function by the input value of the decay constant from other theory
predictions, lattice and/or experiment results and then predicted the decay
form factors. In Refs. Cheng and Kang (2017); Kang et al. (2018), the authors
calculated the corresponding branching ratios of some relevant decay channels,
forming a direct bridge between phenomenological calculation and experiment.
The result shows that the form factors given in Ref. Cheng et al. (2004);
Verma (2012) nicely reproduce the experimental values.

We stress that the branching fraction is not the whole landscape and is only a
number. More observables are needed to distinguish various model predictions.
The correlations among a series of observables are crucial to test a model
calculation. We elucidate this point in two examples. One example concerns the
$K\pi$ form factor in $\tau$ decay. Currently, the most sophisticated
description of the form factors corresponds to the dispersive representation
with imposed chiral symmetry and short-distance QCD behavior, e.g., in Refs.
Bernard (2014); Jamin et al. (2006). However, other options are available,
e.g., in Ref. Kimura et al. (2013) and Belle paper Epifanov et al. (2007),
where in the former reference the form factors are calculated up to one-loop
using the Lagrangian, including vector mesons. The shape in Ref. Kimura et al.
(2013) differs dramatically from that in Ref. Bernard (2014), which may be
attributed to the perturbative versus nonperturbative treatment. In the
experimental study of Ref. Epifanov et al. (2007), the superposition of Breit-
Wigner functions is used in a pragmatic way, which violates the Watson final-
state-interaction theorem and also the unitarity. However, all these variants
of the form factor describe equally well the event distribution data, clearly
indicating that one may need deeper theoretical founds or test more observable
predictions. The other example is related to antinucleon-nucleon interaction.
We will demonstrate this example by the Jülich potential Mull and Holinde
(1995), where one sees that the cross section is well reproduced by various
models there (models A(Box), C, D), but the predictions on the polarizations
and spin observables differ dramatically. This situation also holds for the
interaction constructed in chiral effective field theory (ChEFT) Kang et al.
(2014b); Dai et al. (2017). Such a potential was applied to various hadron
processes, e.g., in $e^{+}e^{-}\to p\bar{p}$ Haidenbauer et al. (2014) and
$J/\psi\to\gamma p\bar{p}$ Kang et al. (2015); Dai et al. (2018a). There, and
especially in Ref. Kang et al. (2015), both the model A(OBE) and that for
ChEFT successfully reproduce the phase shift (on-shell behavior) in the
partial wave of ${}^{1}S_{0}$; however, they predict very different decay
amplitudes in the distorted-wave Born approximation that involves the off-
shell $T$-matrix elements. In a word, probing other observables apart from the
branching fraction (integrated rate) is crucial to examine the theoretical
models, which drives us to access the differential decay rate and the
polarization.

In this paper, we will present such observables mentioned above in the
semileptonic decays of heavy flavor mesons ($D,\,D_{s},\,B,\,B_{s}$), namely,
$D_{(s)}$, $B_{(s)}$ decaying to $Pl^{+}\nu_{l}$ and $Vl^{+}\nu_{l}$ ($l=e$,
$\mu$, or $\tau$), with $P$ and $V$ denoting the pseudoscalar and vector
meson, respectively. For the ease of the reader, and to define our notation
consistently, we closely follow Ref. Ivanov et al. (2019) and write out the
relevant equations111We perceive that the helicity formalism may be elegant.
One example is that the decay rate for the semileptonic decay of a baryon,
where the helicity formalism is analyzed in Ref. Gutsche et al. (2015), and
the direct calculation is given in Ref. Garcia and Huerta (1992). The former
is mathematically more straightforward than the latter.. In such a framework,
one can evaluate the hadronic tensor in the mother-particle (e.g., $D$) rest
frame but the leptonic tensor in the rest frame of a $W$ boson, avoiding the
somewhat complicated boost. Then, the observables are expressed by the
combination of form factors via helicity amplitudes, including the
differential decay rate, forward-backward asymmetry (denoted by $A_{FB}^{l}$),
longitudinal ($P_{L}^{l}$) and transverse ($P_{T}^{l}$) polarization of the
charged lepton ($P_{L}^{l}$), longitudinal polarization of vector in the final
state ($F_{L}^{l}(V)$), and trigonometric moments ($W_{i}^{l}$).

Additionally, we notice that the observables of $A_{FB}^{l}$, $P_{L}^{l}$ have
already been calculated in the covariant confining quark model (CCQM) Soni et
al. (2018); Ivanov et al. (2015) and the relativistic quark model (RQM)
Faustov et al. (2020). From a theoretical viewpoint, we need to complete the
corresponding calculation in light of the CLFQM as an indispensable complement
and comparison. In particular, some of the observables have been measured in
the experiment, which will be used to rectify our theoretical predictions.
These constituents are another motivation for the current work, continuing the
series of our previously published papers Cheng and Kang (2017); Kang et al.
(2018); Faustov et al. (2020); Cheng and Kang (2018).

In this work, we will consider the prediction of various observables, as
mentioned above, based on the form factors calculated in the CLFQM. A full
comparison will be made with the ones from CCQM and RQM, wherever available.
Here we add some comments on the similarity and difference between these three
models. They are all a kind of quark model and thus share some common
features, e.g., containing the wave function (bound state property) of initial
and final state mesons, the quark masses as parameters. Some differences are
in the treatments of wave functions. The CCQM Ivanov et al. (2015) is based on
the effective Lagrangian describing the coupling of hadron to its constituent
quarks. The coupling constant is fixed by saturating the Weinberg
compositeness to guarantee the meson as a pure bound state of a quark and
antiquark. For the vertex function, a simple Gaussian function is adopted with
cutoff $\Lambda$ characterizing the finite size of meson. The CLFQM Cheng et
al. (2004) is based on Dirac’s light-front Hamiltonian dynamics and the
relevant variables are expressed in the light-front system. The light-front
vertex function derived from the Melosh transformation encodes the bound state
nature. In the RQM Faustov et al. (2020), the wave function can be obtained by
solving the quasipotential equation numerically, not necessary to assume a
Gaussian function. In addition, the form factors in the CCQM and RQM can be
calculated in the whole $q^{2}$ region while CLFQM usually predicts the form
factor in the spacelike region and then one analytically continue it to the
(physical) timelike region.

This paper is organized as follows. In Sec. II, we introduce the form factors
and helicity amplitudes of the transitions $D(B)_{(s)}\rightarrow
Pl^{+}\nu_{l}$ and $D(B)_{(s)}\rightarrow Vl^{+}\nu_{l}$. In Sec. III, we
provide the differential decay distribution, expressed by the helicity
structure functions, while in Sec. IV the definitions of various observables,
as mentioned above, are presented. In Sec. V, we give the numerical results of
these physical observables in their mean value and their $q^{2}$ spectrum. We
also compare the results with those from the constituent confining quark model
Ivanov et al. (2019) and the relativistic quark model Faustov et al. (2020),
and excellent agreement is obtained. We conclude in Sec. VI.

##  II FORM FACTORS AND HELICITY AMPLITUDES

In this paper, we take the decay process of $D$ as an example to demonstrate
our derivation for the relevant equations. For the $D_{s}$, $B$, and $B_{s}$
meson, just replacing some relevant physical parameters, the calculation is
completely similar.

In the SM, the invariant matrix element for semileptonic decay of $D$ meson to
a pseudoscalar ($P$) or a vector ($V$) meson can be written as

$\displaystyle\mathcal{M}\left(D\rightarrow P(V)l^{+}\nu_{l}\right)$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}V_{cq}\left\langle
P(V)\mid\bar{q}O^{\mu}c\mid D_{(s)}\right\rangle l^{+}O_{\mu}\nu_{l}$ (1)
$\displaystyle=\frac{G_{F}}{\sqrt{2}}V_{cq}H^{\mu}L_{\mu},$

where $O^{\mu}=\gamma^{\mu}(1-\gamma_{5})$, $q=d,s$, and $G_{F}=1.166\times
10^{-5}\,\text{GeV}^{-2}$ is the fermion coupling constant.

The hadronic part can be parameterized by the invariant form factors, which
are a function of the momentum transfer squared $(q^{2})$, between the initial
and final meson. For the form factors, we stick to the commonly used Bauer-
Stech-Wirbel (BSW) form Wirbel et al. (1985).

For the transition of $D\rightarrow Pl^{+}\nu_{l}$, one has

$\left\langle P(p_{2})|V_{\mu}|D(p_{1})\right\rangle=T_{\mu},$ (2)

and the form factors are given by

$T_{\mu}=\left(P_{\mu}-\frac{m_{1}^{2}-m_{2}^{2}}{q^{2}}q_{\mu}\right)F_{1}(q^{2})+\frac{m_{1}^{2}-m_{2}^{2}}{q^{2}}q_{\mu}F_{0}(q^{2}).$
(3)

For the transition of $D\rightarrow Vl^{+}\nu_{l}$, one has

$\left\langle
V(p_{2})|V_{\mu}-A_{\mu}|D(p_{1})\right\rangle=\epsilon_{2}^{\dagger\nu}T_{\mu\nu},$
(4)

where

$\begin{split}T_{\mu\nu}&=-\left(m_{1}+m_{2}\right)\left[g_{\mu\nu}-\frac{P_{\nu}}{q^{2}}q_{\mu}\right]A_{1}(q^{2})+\frac{P_{\nu}}{m_{1}+m_{2}}\left[P_{\mu}-\frac{m_{1}^{2}-m_{2}^{2}}{q^{2}}q_{\mu}\right]A_{2}(q^{2})\\\
&-2m_{2}\frac{P_{\nu}}{q^{2}}q_{\mu}A_{0}(q^{2})+\frac{i}{m_{1}+m_{2}}\varepsilon_{\mu\nu\alpha\beta}P^{\alpha}q^{\beta}V(q^{2}).\end{split}$
(5)

In the above equations, $p_{1}$ ($m_{1}$) and $p_{2}$ ($m_{2}$) are the four-
momentum (mass) of the initial and final meson, respectively. The momenta
$P_{\mu}$ and $q_{\mu}$ are defined as $P_{\mu}={(p_{1}+p_{2})}_{\mu}$ and
$q_{\mu}={(p_{1}-p_{2})}_{\mu}$, respectively. $\epsilon_{2}$ is the
polarization vector of the final state meson $V$ and satisfies
$\epsilon_{2}^{\dagger}\cdot p_{2}=0$.

In the covariant contraction of hadronic and leptonic tensors,
$H^{\mu\nu}L_{\mu\nu}$, the hadronic tensor can be written as

$\begin{split}H^{\mu\nu}&=\sum\left\langle
X|V^{\mu}-A^{\mu}|D\right\rangle\left\langle
X|V^{\nu}-A^{\nu}|D\right\rangle^{\dagger}\\\
&=\begin{cases}T_{\mu}T_{\nu}^{\dagger},\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
D\rightarrow P,\\\
T_{\mu\alpha}T_{\nu\beta}^{\dagger}\left(-g^{\alpha\beta}+\frac{p_{2}^{\alpha}p_{2}^{\beta}}{m_{2}^{2}}\right),\quad\quad
D\rightarrow V.\end{cases}\end{split}$ (6)

We can define the explicit representations of polarization four-vectors
$\epsilon^{\mu}(\lambda_{W})$, where $\lambda_{W}$ denotes the polarization
index of $W_{\text{off-shell}}$, and $\epsilon^{\mu}(\lambda_{W})q_{\mu}=0$
for $\lambda_{W}=\pm,0$. These four-vectors $\epsilon^{\mu}(\lambda_{W})$ have
the orthonormality property

$\epsilon_{\mu}^{\dagger}(\lambda_{W})\epsilon^{\mu}({\lambda^{\prime}_{W}})=g_{\lambda_{W}{\lambda^{\prime}_{W}}},\quad(\lambda_{W},{\lambda^{\prime}_{W}}=t,\pm,0)$
(7)

and satisfy the completeness relation

$\epsilon_{\mu}(\lambda_{W})\epsilon_{\nu}^{\dagger}({\lambda^{\prime}_{W}})g_{\lambda_{W}{\lambda^{\prime}_{W}}}=g_{\mu\nu}.$
(8)

We can rewrite the contraction of leptonic and hadronic tensors by using the
orthonormality and completeness relations as

$\begin{split}L^{\mu\nu}H_{\mu\nu}&=L_{\mu^{\prime}\nu^{\prime}}g^{\mu^{\prime}\mu}g^{\nu^{\prime}\nu}H_{\mu\nu}\\\
&=L_{\mu^{\prime}\nu^{\prime}}\epsilon^{\mu^{\prime}}(\lambda_{W})\epsilon^{\dagger\mu}(\lambda_{W}^{\prime\prime})g_{\lambda_{W}\lambda_{W}^{\prime\prime}}\epsilon^{\dagger\nu^{\prime}}(\lambda_{W}^{\prime})\epsilon^{\nu}(\lambda_{W}^{\prime\prime\prime})g_{\lambda_{W}^{\prime}\lambda_{W}^{\prime\prime\prime}}H_{\mu\nu}\\\
&=L\left(\lambda_{W},\lambda_{W}^{\prime}\right)g_{\lambda_{W}\lambda_{W}^{\prime\prime}}g_{\lambda_{W}^{\prime}\lambda_{W}^{\prime\prime\prime}}H\left(\lambda_{W}^{\prime\prime}\lambda_{W}^{\prime\prime\prime}\right),\end{split}$
(9)

where $L(\lambda_{W},\lambda_{W}^{\prime})$ and
$H(\lambda_{W},\lambda_{W}^{\prime})$ are the leptonic and hadronic tensors in
the helicity-component space:

$L\left(\lambda_{W},\lambda_{W}^{\prime}\right)=\epsilon^{\mu}(\lambda_{W})\epsilon^{\dagger\nu}(\lambda_{W}^{\prime})L_{\mu\nu},\quad
H\left(\lambda_{W},\lambda_{W}^{\prime}\right)=\epsilon^{\dagger\mu}(\lambda_{W})\epsilon^{\nu}(\lambda_{W}^{\prime})H_{\mu\nu}.$
(10)

It is necessary to stress that the leptonic and hadronic tensor can be
evaluated in two different Lorentz frames within this framework: the leptonic
tensor is evaluated in the $W$ rest frame, and the hadronic tensor is
evaluated in the $D$ rest frame.

###  II.1 The helicity amplitudes of $D\rightarrow Pl^{+}\nu_{l}$

For the transition $D\rightarrow Pl^{+}\nu_{l}$ in the helicity-component
space, the hadronic tensor is given by

$\begin{split}H\left(\lambda_{W},\lambda_{W}^{\prime}\right)&=\epsilon^{\dagger\mu}(\lambda_{W})\epsilon^{\nu}(\lambda_{W}^{\prime})H_{\mu\nu}\\\
&=[\epsilon^{\dagger\mu}(\lambda_{W})T_{\mu}][\epsilon^{\dagger\nu}(\lambda_{W}^{\prime})T_{\nu}]^{\dagger}=H_{\lambda_{W}}H_{\lambda_{W}^{\prime}}^{\dagger},\end{split}$
(11)

with $H_{\lambda_{W}}\equiv\epsilon^{\mu\dagger}(\lambda_{W})T_{\mu}$. In the
rest frame of the initial meson $D$, one has the explicit representations
Ivanov et al. (2015) of the polarization four-vectors
$\epsilon^{\mu}(\lambda_{W})$:

$\epsilon^{\mu}(t)=\frac{1}{\sqrt{q^{2}}}\left(q_{0},0,0,|\vec{p}_{2}|\right),\quad\epsilon^{\mu}(\pm)=\frac{1}{\sqrt{2}}\left(0,\mp
1,-i,0\right),\quad\epsilon^{\mu}(0)=\frac{1}{\sqrt{q^{2}}}\left(\left|\vec{p}_{2}\right|,0,0,q_{0}\right).$
(12)

Among these representations, $q_{0}$ is the energy and $|\vec{p}_{2}|$ is the
momentum of the $W_{\text{off-shell}}$ in the $D$ rest frame. Additionally, in
this frame (see Fig. 2),

$\displaystyle p_{1}^{\mu}=\left(m_{1},0,0,0\right),$ (13) $\displaystyle
p_{2}^{\mu}=\left(E_{2},0,0,-|\vec{p}_{2}|\right),$ $\displaystyle
q^{\mu}=\left(q_{0},0,0,|\vec{p}_{2}|\right),$

where $E_{2}$ is the energy of the final meson, and $q_{0}$ and
$|\vec{p}_{2}|$ are given by

$2m_{1}q_{0}=m_{1}^{2}-m_{2}^{2}+q^{2},\quad\left|\vec{p}_{2}\right|^{2}=E_{2}^{2}-m_{2}^{2},\quad
2m_{1}E_{2}=m_{1}^{2}+m_{2}^{2}-q^{2}.\quad$ (14)

To calculate the relations between form factors and helicity amplitudes, we
project out the relevant helicity amplitudes using the polarization four-
vectors $\epsilon^{\mu}(\lambda_{W})$. Combining Eq. (3) and Eq. (12), for the
$D\rightarrow P$ transition, we obtain

$\displaystyle H_{t}=\frac{1}{\sqrt{q^{2}}}(m_{1}^{2}-m_{2}^{2})F_{0}(q^{2}),$
(15) $\displaystyle H_{\pm}=0,$ $\displaystyle
H_{0}=\frac{2m_{1}|\vec{p}_{2}|}{\sqrt{q^{2}}}F_{1}(q^{2}).$

The invariant form factors are only a function of $q^{2}$. From this equation,
we find that $H_{t}=H_{0}$ at the maximum recoil $(q^{2}=0)$, and $H_{0}=0$ at
the zero recoil $(q^{2}=q^{2}_{\text{max}})$.

###  II.2 The helicity amplitudes of $D\rightarrow Vl^{+}\nu_{l}$

For the transition $D\rightarrow Vl^{+}\nu_{l}$, we also need the polarization
four-vectors $\epsilon_{2}^{\nu}(\lambda_{V})$ to obtain the expressions of
the helicity amplitudes. The index $\lambda_{V}$ denotes the polarization
index of vector meson $V$. The hadronic tensor of the $D\rightarrow V$
transition can be rewritten as

$\begin{split}H\left(\lambda_{W},\lambda_{W}^{\prime}\right)&=\epsilon^{\dagger\mu}(\lambda_{W})\epsilon^{\nu}(\lambda_{W}^{\prime})H_{\mu\nu}\\\
&=\epsilon^{\dagger\mu}(\lambda_{W})\epsilon^{\nu}(\lambda_{W}^{\prime})\epsilon_{2}^{\dagger\alpha}(\lambda_{V})T_{\mu\alpha}\left[\epsilon_{2}^{\dagger\beta}(\lambda_{V}^{\prime})T_{\nu\beta}\right]^{\dagger}\delta_{\lambda_{V}\lambda_{V}^{\prime}}\\\
&=\epsilon^{\dagger\mu}(\lambda_{W})\epsilon_{2}^{\dagger\alpha}(\lambda_{V})T_{\mu\alpha}\left[\epsilon^{\dagger\nu}(\lambda_{W}^{\prime})\epsilon_{2}^{\dagger\beta}(\lambda_{V}^{\prime})T_{\nu\beta}\right]^{\dagger}\delta_{\lambda_{V}\lambda_{V}^{\prime}}\\\
&=H_{\lambda_{W}\lambda_{V}}H_{\lambda_{W}^{\prime}\lambda_{V}}^{\dagger},\end{split}$
(16)

with

$H_{\lambda_{W}\lambda_{V}}\equiv\epsilon^{\dagger\mu}(\lambda_{W})\epsilon_{2}^{\dagger\alpha}(\lambda_{V})T_{\mu\alpha}.$
(17)

According to the angular momentum conservation,
$\lambda_{V}=\lambda_{W},\lambda_{V}^{\prime}=\lambda_{W}^{\prime}$ for
$\lambda_{W},\lambda_{W}^{\prime}=\pm,0$; and
$\lambda_{V},\lambda_{V}^{\prime}=0$ for $\lambda_{W},\lambda_{W}^{\prime}=t$.

In the $D$ rest frame, the explicit representations of the polarization four-
vectors for the final vector meson $\epsilon_{2}^{\nu}(\lambda_{V})$ can be
written as

$\epsilon_{2}^{\nu}(\pm)=\frac{1}{\sqrt{2}}\left(0,\pm
1,-i,0\right),\quad\epsilon_{2}^{\nu}(0)=\frac{1}{m_{2}}\left(|\vec{p}_{2}|,0,0,-E_{2}\right).$
(18)

They satisfy the orthonormality condition

$\epsilon_{2\nu}^{\dagger}(\lambda_{V})\epsilon_{2}^{\nu}({\lambda^{\prime}_{V}})=-\delta_{\lambda_{V}{\lambda^{\prime}_{V}}},$
(19)

and the completeness relation

$\sum\epsilon_{2\mu}(\lambda_{V})\epsilon_{2\nu}^{\dagger}({\lambda^{\prime}_{V}})\delta_{\lambda_{V}{\lambda^{\prime}_{V}}}=-g_{\mu\nu}+\frac{p_{2\mu}p_{2\nu}}{m_{2}^{2}}.$
(20)

In terms of the Eq. (17), we use a similar method to obtain the nonvanishing
helicity amplitudes of the transition $D\rightarrow Vl^{+}\nu_{l}$:

$\displaystyle H_{t}$
$\displaystyle\equiv\epsilon^{\dagger\mu}(t)\epsilon_{2}^{\dagger\nu}(0)T_{\mu\nu}=-\frac{2m_{1}|\vec{p}_{2}|}{\sqrt{q^{2}}}A_{0}(q^{2}),$
(21) $\displaystyle H_{\pm}$
$\displaystyle\equiv\epsilon^{\dagger\mu}(\pm)\epsilon_{2}^{\dagger\nu}(\pm)T_{\mu\nu}=-(m_{1}+m_{2})A_{1}(q^{2})\pm\frac{2m_{1}|\vec{p}_{2}|}{m_{1}+m_{2}}V(q^{2}),$
$\displaystyle H_{0}$
$\displaystyle\equiv\epsilon^{\dagger\mu}(0)\epsilon_{2}^{\dagger\nu}(0)T_{\mu\nu}=-\frac{m_{1}+m_{2}}{2m_{2}\sqrt{q^{2}}}\left(m_{1}^{2}-m_{2}^{2}-q^{2}\right)A_{1}(q^{2})+\frac{1}{m_{1}+m_{2}}\frac{2m_{1}^{2}|\vec{p}_{2}|^{2}}{m_{2}\sqrt{q^{2}}}A_{2}(q^{2}).$

At the maximum recoil ($q^{2}=0$), we have $H_{t}(0)=H_{0}(0)$, and $H_{t}$
and $H_{0}$ dominate, while at the zero recoil ($q^{2}=q^{2}_{\text{max}}$),
$H_{t}=0$ and $H_{\pm}=H_{0}$.

###  II.3 The hadronic tensor including the cascade decay $V\rightarrow
P_{1}P_{2}$

Taking into account the finite width effect of the vector meson, we need to
consider the cascade decay $V\rightarrow P_{1}P_{2}$ with $P$ denoting a
pseudoscalar. For a more intuitive description, we give an example of cascade
decay in Fig. 1.

Figure 1: Illustration of the semileptonic decay process $D\rightarrow
K^{*}(\rightarrow K\pi)l\nu_{l}$.

The kinematics for four-body decay is elaborated in Refs. Kang et al. (2014a);
Cheng and Kang (2018); Stoffer (2014) and earlier papers Cabibbo and
Maksymowicz (1965); Pais and Treiman (1968). As shown in Fig. 2, we define
$\theta$ as the polar angle of the lepton in the $l^{+}\nu_{l}$ center-of-mass
frame with respect to the $W_{\text{off-shell}}$ line of flight in the $D$
rest frame. Additionally, $\theta^{*}$ is defined as the polar angle of
$P_{1}$ in the $V$ rest frame with respect to the $V$ meson line of flight in
the $D$ rest frame. $\chi$ is the dihedral angle between the $l^{+}\nu_{l}$
and the $P_{1}P_{2}$ planes. The $z$-axis is along the momentum direction of
$W_{\text{off-shell}}$ in the $D$ rest frame, and the momenta of $P_{1}$ and
$P_{2}$ lie in the $xz$ plane.

Figure 2: The definitions of $\theta$, $\theta^{*}$ and the azimuthal angle
$\chi$.

In the rest frame of final meson $V$, the corresponding momenta are

$\displaystyle p_{2}^{\mu}$ $\displaystyle=\left(m_{2},0,0,0\right),$ (22)
$\displaystyle p_{3}^{\mu}$
$\displaystyle=\left(E_{3},|\vec{p}_{3}|\sin\theta^{*},0,-|\vec{p}_{3}|\cos\theta^{*}\right),$
$\displaystyle p_{4}^{\mu}$
$\displaystyle=\left(E_{4},-|\vec{p}_{3}|\sin\theta^{*},0,|\vec{p}_{3}|\cos\theta^{*}\right),$

and the polarization vectors of $V$ meson are

$\epsilon_{2}^{\mu}(+)=\frac{1}{\sqrt{2}}\left(0,+1,-i,0\right),\;\epsilon_{2}^{\mu}(-)=\frac{1}{\sqrt{2}}\left(0,-1,-i,0\right),\;\epsilon_{2}^{\mu}(0)=\left(0,0,0,-1\right).$
(23)

In the decay process of $D\to V(\to P_{1}P_{2})l\nu_{l}$, the hadronic tensor
is given by

$\displaystyle H\left(\lambda_{W},\lambda_{W}^{\prime}\right)$
$\displaystyle=\epsilon^{\dagger\mu}(\lambda_{W})\epsilon^{\nu}(\lambda_{W}^{\prime})H_{\mu\nu}$
(24)
$\displaystyle=\epsilon^{\dagger\mu}(\lambda_{W})\epsilon^{\nu}(\lambda_{W}^{\prime})\epsilon_{2}^{\dagger\alpha}(\lambda_{V})\epsilon_{2}^{\beta}(\lambda_{V}^{\prime})T_{\mu\alpha}(T_{\nu\beta})^{\dagger}g^{2}_{VPP}p_{3\alpha^{\prime}}p_{3\beta^{\prime}}\epsilon_{2}^{\alpha^{\prime}}(\lambda_{V})\epsilon_{2}^{\dagger\beta^{\prime}}(\lambda_{V}^{\prime})$
$\displaystyle=g^{2}_{VPP}p_{3\alpha^{\prime}}p_{3\beta^{\prime}}\epsilon_{2}^{\alpha^{\prime}}(\lambda_{V})\epsilon_{2}^{\dagger\beta^{\prime}}(\lambda_{V}^{\prime})\times
H_{\lambda_{W},\lambda_{V}}H^{\dagger}_{\lambda_{W}^{\prime},\lambda_{V}^{\prime}},$

where $g_{VPP}\cdot\epsilon_{2}^{\rho}(\lambda_{V})\cdot p_{3\rho}$ describes
the amplitude $\mathcal{A}(V\to P_{1}P_{2})$, and the effective coupling
constant $g_{VPP}$ is related to the branching ratio $Br(V\to P_{1}P_{2})$ by

$g^{2}_{VPP}=\frac{24\pi m_{2}^{2}\Gamma_{V}Br}{|\vec{p}_{3}|^{3}},$ (25)

with $\Gamma_{V}$ being the finite width of vector meson. Notice that
$S^{\alpha\alpha^{\prime}}(p_{2})=\displaystyle\sum_{\lambda_{V}=\pm
1,0}\epsilon_{2}^{\alpha}(\lambda_{V})\epsilon_{2}^{\dagger\alpha^{\prime}}(\lambda_{V})$
is just the numerator of the propagator of intermediate meson $V$.
Additionally, Eq. (24) agrees with Eq. (37) in Ref. Ivanov et al. (2019),
except for some extra factors that will be compensated in the final result.

##  III THE TWOFOLD AND FOURFOLD DIFFERENTIAL DECAY DISTRIBUTIONS

In this section, we consider the differential decay distribution. For the
$D\rightarrow P(V)l^{+}\nu_{l}$ transition, we have

$\begin{split}\frac{d^{2}\Gamma(D\rightarrow
P(V)l^{+}\nu_{l})}{dq^{2}d\cos\theta}&=\frac{|\vec{p}_{2}|}{(2\pi)^{3}32m_{1}^{2}}\left(1-\frac{m_{l}^{2}}{q^{2}}\right)\sum\limits_{\text{spins}}\left|\mathcal{M}\right|^{2}\\\
&=\frac{G_{F}^{2}|V_{cq}|^{2}}{(2\pi)^{3}}\frac{|\vec{p}_{2}|}{64m_{1}^{2}}\left(1-\frac{m_{l}^{2}}{q^{2}}\right)L\left(\lambda_{W},\lambda_{W}^{\prime}\right)g_{\lambda_{W}\lambda_{W}^{\prime\prime}}g_{\lambda_{W}^{\prime}\lambda_{W}^{\prime\prime\prime}}H\left(\lambda_{W}^{\prime\prime},\lambda_{W}^{\prime\prime\prime}\right),\end{split}$
(26)

where $m_{l}$ is the lepton mass, and the factor in front of
$|\mathcal{M}|^{2}$ corresponds to the three-body phase factor. In terms of
the helicity amplitudes, Eqs. (11) and (16), the hadronic tensor is

$H\left(\lambda_{W},\lambda_{W}^{\prime}\right)=\begin{cases}H_{\lambda_{W}}H_{\lambda_{W}^{\prime}}^{\dagger},\quad\quad\quad\quad
D\rightarrow P,\\\
H_{\lambda_{W}\lambda_{V}}H_{\lambda_{W}^{\prime}\lambda_{V}}^{\dagger},\quad\quad
D\rightarrow V.\end{cases}$ (27)

We then evaluate the leptonic tensor
$L(\lambda_{W},\lambda_{W}^{\prime})=\epsilon^{\mu}(\lambda_{W})\epsilon^{\dagger\nu}(\lambda_{W}^{\prime})L_{\mu\nu}$.
The leptonic tensor can be given by

$\begin{split}L_{\mu\nu}&=\begin{cases}\text{tr}\left[\left(\not{k}_{1}+m_{l}\right)O_{\mu}\not{k}_{2}O_{\nu}\right],\quad\quad
W^{-}\rightarrow l^{-}\bar{\nu}_{l},\\\
\text{tr}\left[\left(\not{k}_{1}-m_{l}\right)O_{\nu}\not{k}_{2}O_{\mu}\right],\quad\quad
W^{+}\rightarrow l^{+}\nu_{l},\end{cases}\\\
&=8\left(k_{1\mu}k_{2\nu}+k_{1\nu}k_{2\mu}-k_{1}\cdot k_{2}g_{\mu\nu}\pm
i\varepsilon_{\mu\nu\alpha\beta}k_{1}^{\alpha}k_{2}^{\beta}\right),\end{split}$
(28)

where $k_{1},k_{2}$ denote the momentum of a charged lepton and neutrino,
respectively. The evaluation of the leptonic tensor is performed in the
$W_{\text{off-shell}}$ rest frame, and the explicit expressions of the
momentum can be written as

$\begin{split}&q^{\mu}=\left(\sqrt{q^{2}},0,0,0\right),\\\
&k_{1}^{\mu}=\left(E_{1},|\vec{k}_{1}|\sin\theta\cos\chi,|\vec{k}_{1}|\sin\theta\sin\chi,|\vec{k}_{1}|\cos\theta\right),\\\
&k_{2}^{\mu}=\left(|\vec{k}_{1}|,-|\vec{k}_{1}|\sin\theta\cos\chi,-|\vec{k}_{1}|\sin\theta\sin\chi,-|\vec{k}_{1}|\cos\theta\right),\end{split}$
(29)

where $E_{1}=\left(q^{2}+m_{l}^{2}\right)/2\sqrt{q^{2}}$ is the energy and
$|\vec{k_{1}}|=\left(q^{2}-m_{l}^{2}\right)/2\sqrt{q^{2}}$ is the three-
momentum of the lepton. The polarization vectors of $W$ in this rest frame are
given by

$\epsilon^{\mu}(0)=\left(0,0,0,1\right),\quad\epsilon^{\mu}(\pm)=\frac{1}{\sqrt{2}}\left(0,\mp
1,-i,0\right),\quad\epsilon^{\mu}(t)=\left(1,0,0,0\right).$ (30)

Combining the momentum and polarization vectors, Eqs. (28)-(30), we obtain the
expression for $L(\lambda_{W},\lambda_{W}^{\prime})$, where the matrix columns
and rows are ordered in the sequence $(t,+,0,-)$ (In Ref. Ivanov et al.
(2019), the element in the first row and second column of the matrix after
$\delta$ contains a typo, $\chi$ should be replaced by $\chi$):

$\displaystyle(2q^{2}v)^{-1}L(\lambda_{W},\lambda_{W}^{\prime})(\theta,\chi)$
$\displaystyle=\left(\begin{array}[]{cccc}0&0&0&0\\\
0&(1\mp\cos\theta)^{2}&\mp\frac{2}{\sqrt{2}}(1\mp\cos\theta)\sin\theta
e^{i\chi}&\sin^{2}\theta e^{2i\chi}\\\
0&\mp\frac{2}{\sqrt{2}}(1\mp\cos\theta)\sin\theta
e^{-i\chi}&2\sin^{2}\theta&\mp\frac{2}{\sqrt{2}}(1\pm\cos\theta)\sin\theta
e^{i\chi}\\\ 0&\sin^{2}\theta
e^{-2i\chi}&\mp\frac{2}{\sqrt{2}}(1\pm\cos\theta)\sin\theta
e^{-i\chi}&(1\pm\cos\theta)^{2}\end{array}\right)$ (31)
$\displaystyle+\delta_{l}\left(\begin{array}[]{cccc}4&-\frac{4}{\sqrt{2}}\sin\theta
e^{-i\chi}&4\cos\theta&\frac{4}{\sqrt{2}}\sin\theta e^{i\chi}\\\
-\frac{4}{\sqrt{2}}\sin\theta
e^{i\chi}&2\sin^{2}\theta&-\frac{2}{\sqrt{2}}\sin 2\theta
e^{i\chi}&-2\sin^{2}\theta e^{2i\chi}\\\ 4\cos\theta&-\frac{2}{\sqrt{2}}\sin
2\theta e^{-i\chi}&4\cos^{2}\theta&\frac{2}{\sqrt{2}}\sin 2\theta e^{i\chi}\\\
\frac{4}{\sqrt{2}}\sin\theta e^{-i\chi}&-2\sin^{2}\theta
e^{-2i\chi}&\frac{2}{\sqrt{2}}\sin 2\theta
e^{-i\chi}&2\sin^{2}\theta\end{array}\right).$

where the upper/lower sign refers to the $(l^{-}\bar{\nu}_{l})/(l^{+}\nu_{l})$
configuration. To obtain the distribution on the polar angular $\theta$, we
integrate over the azimuthal angel $\chi$, and then, the matrix becomes

$(2q^{2}v)^{-1}L(\lambda_{W},\lambda_{W}^{\prime})(\theta)=\left(\begin{array}[]{cccc}0&0&0&0\\\
0&(1\mp\cos\theta)^{2}&0&0\\\ 0&0&2\sin^{2}\theta&0\\\
0&0&0&(1\pm\cos\theta)^{2}\end{array}\right)+\delta_{l}\left(\begin{array}[]{cccc}4&0&4\cos\theta&0\\\
0&2\sin^{2}\theta&0&0\\\ 4\cos\theta&0&4\cos^{2}\theta&0\\\
0&0&0&2\sin^{2}\theta\end{array}\right),$ (32)

where $v=1-m_{l}^{2}/q^{2}$ is the velocity-type parameter and
$\delta_{l}=m_{l}^{2}/2q^{2}$ is the helicity-flip factor.

We consider now the circumstance of $W^{+}\rightarrow l^{+}\nu_{l}$, and the
$W^{-}\rightarrow l^{-}\bar{\nu}_{l}$ case will be discussed later. Combing
the Eq. (32) and Eq. (27), one has the contraction
$H^{\mu\nu}L_{\mu\nu}(\theta)$ in helicity amplitude space as follows:

$\begin{split}H^{\mu\nu}L_{\mu\nu}(\theta)=&L\left(\lambda_{W},\lambda_{W}^{\prime}\right)g_{\lambda_{W}\lambda_{W}^{\prime\prime}}g_{\lambda_{W}^{\prime}\lambda_{W}^{\prime\prime\prime}}H\left(\lambda_{W}^{\prime\prime},\lambda_{W}^{\prime\prime\prime}\right)\\\
=&\left(2q^{2}v\right)\times\left\\{\left[\left(1+\cos^{2}\theta\right)+2\delta_{l}\sin^{2}\theta\right]\left(\left|H_{+}\right|^{2}+\left|H_{-}\right|^{2}\right)+\left(2\sin^{2}\theta+4\delta_{l}\cos^{2}\theta\right)\left|H_{0}\right|^{2}\right.\\\
&\left.+2\cos\theta\left(\left|H_{+}\right|^{2}-\left|H_{-}\right|^{2}\right)+4\delta_{l}\left|H_{t}\right|^{2}-8\delta_{l}\cos\theta\,\text{Re}\left(H_{0}H_{t}^{\dagger}\right)\right\\}\\\
=&\left(2q^{2}v\right)\times\left[\left(1+\cos^{2}\theta\right)\mathcal{H}_{U}+2\sin^{2}\theta\mathcal{H}_{L}+2\cos\theta\mathcal{H}_{P}\right.\\\
&\left.+2\delta_{l}\left(\sin^{2}\theta\mathcal{H}_{U}+2\cos^{2}\theta\mathcal{H}_{L}+2\mathcal{H}_{S}-4\cos\theta\mathcal{H}_{SL}\right)\right].\end{split}$
(33)

The above equation contains relevant bilinear combinations $\mathcal{H}_{i}$,
called by the helicity structure functions Ivanov et al. (2015), whose
definitions are listed in Table 1.

Table 1: Definitions of helicity structure functions Parity-conserving | Parity-violating
---|---
$\mathcal{H}_{U}=|H_{+}|^{2}+|H_{-}|^{2}$ | $\mathcal{H}_{P}=|H_{+}|^{2}-|H_{-}|^{2}$
$\mathcal{H}_{L}=|H_{0}|^{2}$ | $\mathcal{H}_{A}=\frac{1}{2}\text{Re}(H_{+}H_{0}^{\dagger}-H_{-}H_{0}^{\dagger})$
$\mathcal{H}_{T}=\text{Re}(H_{+}H_{-}^{\dagger})$ | $\mathcal{H}_{IA}=\frac{1}{2}\text{Im}(H_{+}H_{0}^{\dagger}-H_{-}H_{0}^{\dagger})$
$\mathcal{H}_{IT}=\text{Im}(H_{+}H_{-}^{\dagger})$ | $\mathcal{H}_{SA}=\frac{1}{2}\text{Re}(H_{+}H_{t}^{\dagger}-H_{-}H_{t}^{\dagger})$
$\mathcal{H}_{I}=\frac{1}{2}\text{Re}(H_{+}H_{0}^{\dagger}+H_{-}H_{0}^{\dagger})$ | $\mathcal{H}_{ISA}=\frac{1}{2}\text{Im}(H_{+}H_{t}^{\dagger}-H_{-}H_{t}^{\dagger})$
$\mathcal{H}_{II}=\frac{1}{2}\text{Im}(H_{+}H_{0}^{\dagger}+H_{-}H_{0}^{\dagger})$ |
$\mathcal{H}_{S}=|H_{t}|^{2}$ |
$\mathcal{H}_{ST}=\frac{1}{2}\text{Re}(H_{+}H_{t}^{\dagger}+H_{-}H_{t}^{\dagger})$ |
$\mathcal{H}_{IST}=\frac{1}{2}\text{Im}(H_{+}H_{t}^{\dagger}+H_{-}H_{t}^{\dagger})$ |
$\mathcal{H}_{SL}=\text{Re}(H_{0}H_{t}^{\dagger})$ |
$\mathcal{H}_{ISL}=\text{Im}(H_{0}H_{t}^{\dagger})$ |
$\mathcal{H}_{tot}=\mathcal{H}_{U}+\mathcal{H}_{L}+\delta_{l}\left(\mathcal{H}_{U}+\mathcal{H}_{L}+3\mathcal{H}_{S}\right)$ |

Then, we obtain the twofold differential decay distribution on $q^{2}$ and
$\cos\theta$:

$\begin{split}\frac{d\Gamma\left(D\rightarrow
P(V)l^{+}\nu_{l}\right)}{dq^{2}d\cos\theta}=&\frac{G_{F}^{2}|V_{cq}|^{2}|\vec{p}_{2}|q^{2}v^{2}}{32(2\pi)^{3}m_{1}^{2}}\times\left[\left(1+\cos^{2}\theta\right)\mathcal{H}_{U}+2\sin^{2}\theta\mathcal{H}_{L}+2\cos\theta\mathcal{H}_{P}\right.\\\
&\left.+2\delta_{l}\left(\sin^{2}\theta\mathcal{H}_{U}+2\cos^{2}\theta\mathcal{H}_{L}+2\mathcal{H}_{S}-4\cos\theta\mathcal{H}_{SL}\right)\right].\end{split}$
(34)

Further integrating over $\cos\theta$, the differential $q^{2}$ distribution
will be

$\displaystyle\frac{d\Gamma\left(D\rightarrow
P(V)l^{+}\nu_{l}\right)}{dq^{2}}=$
$\displaystyle\frac{G_{F}^{2}|V_{cq}|^{2}|\vec{p}_{2}|q^{2}v^{2}}{12(2\pi)^{3}m_{1}^{2}}\times\mathcal{H}_{tot},$
(35)

with
$\mathcal{H}_{tot}=\mathcal{H}_{U}+\mathcal{H}_{L}+\delta_{l}\left(\mathcal{H}_{U}+\mathcal{H}_{L}+3\mathcal{H}_{S}\right)$.

For the cascade decays, we need to consider the fourfold distribution.
Equation (24) shows the hadronic tensor, including the cascade decay.
Combining the momenta of $P_{1}$ Eq. (22) and the polarization vectors of $V$
meson Eq. (23), the hadronic tensor in the helicity space can be written as

$\displaystyle\frac{1}{g^{2}_{VPP}|\vec{p}_{3}|^{2}}H\left(\lambda_{W},\lambda_{W}^{\prime}\right)=$
$\displaystyle\left(\begin{array}[]{cccc}\cos^{2}\theta^{*}|H_{t}|^{2}&\frac{1}{2\sqrt{2}}\sin
2\theta^{*}H_{t}H_{+}^{\dagger}&\cos^{2}\theta^{*}H_{t}H_{0}^{\dagger}&-\frac{1}{2\sqrt{2}}\sin
2\theta^{*}H_{t}H_{-}^{\dagger}\\\ \frac{1}{2\sqrt{2}}\sin
2\theta^{*}H_{+}H_{t}^{\dagger}&\frac{1}{2}\sin^{2}\theta^{*}|H_{+}|^{2}&\frac{1}{2\sqrt{2}}\sin
2\theta^{*}H_{+}H_{0}^{\dagger}&-\frac{1}{2}\sin^{2}\theta^{*}H_{+}H_{-}^{\dagger}\\\
\cos^{2}\theta^{*}H_{0}H_{t}^{\dagger}&\frac{1}{2\sqrt{2}}\sin
2\theta^{*}H_{0}H_{+}^{\dagger}&\cos^{2}\theta^{*}|H_{0}|^{2}&-\frac{1}{2\sqrt{2}}\sin
2\theta^{*}H_{0}H_{-}^{\dagger}\\\ -\frac{1}{2\sqrt{2}}\sin
2\theta^{*}H_{-}H_{t}^{\dagger}&-\frac{1}{2}\sin^{2}\theta^{*}H_{-}H_{+}^{\dagger}&-\frac{1}{2\sqrt{2}}\sin
2\theta^{*}H_{-}H_{0}^{\dagger}&\frac{1}{2}\sin^{2}\theta^{*}|H_{-}|^{2}\end{array}\right),$
(36)

with the matrix orders $\lambda_{W},\lambda_{W}^{\prime}=t,+,0,-$.

The four-body phase space for $D\to V(\to P_{1}P_{2})l\nu_{l}$ depends on five
variables: $P_{1}P_{2}$ invariant mass squared $s$, $l\nu_{l}$ invariant mass
squared $q^{2}$, the angles $\theta$, $\theta^{*}$ and $\chi$. See the
definitions in Fig. 2. Considering the narrow width approximation

$\frac{1}{(s-m_{2}^{2})^{2}+m_{2}^{2}\Gamma_{V}^{2}}\xrightarrow[]{\Gamma_{V}\to
0}\frac{\pi}{m_{2}\Gamma_{V}}\delta(s-m_{2}^{2})$ (37)

in the four-body phase space integral, one arrives at

$\frac{d\Gamma\left(D\rightarrow V(\rightarrow
P_{1}P_{2})l^{+}\nu_{l}\right)}{dq^{2}d\cos\theta d\chi
d\cos\theta^{*}}=\frac{3G_{F}^{2}|V_{cq}|^{2}v|\vec{p}_{2}|}{(2\pi)^{4}128m_{1}^{2}}Br(V\to
P_{1}P_{2})\frac{1}{|\vec{p}_{3}|^{2}}H(\lambda_{W},\lambda_{W}^{\prime})L(\lambda_{W},\lambda_{W}^{\prime}),$
(38)

which is then generally expressed as

$\frac{d\Gamma\left(D\rightarrow V(\rightarrow
P_{1}P_{2})l^{+}\nu_{l}\right)}{dq^{2}d\cos\theta
d\frac{\chi}{2\pi}d\cos\theta^{*}}=\frac{G_{F}^{2}|V_{cq}|^{2}|\vec{p}_{2}|q^{2}v^{2}}{12(2\pi)^{3}m_{1}^{2}}Br(V\rightarrow
P_{1}P_{2})W(\theta,\theta^{*},\chi),$ (39)

and the angular distribution $W(\theta,\theta^{*},\chi)$ is given by

$\displaystyle W\left(\theta,\theta^{*},\chi\right)=$
$\displaystyle\frac{9}{32}\left(1+\cos^{2}\theta\right)\sin^{2}\theta^{*}\mathcal{H}_{U}+\frac{9}{8}\sin^{2}\theta\cos^{2}\theta^{*}\mathcal{H}_{L}+\frac{9}{16}\cos\theta\sin^{2}\theta^{*}\mathcal{H}_{P}$
(40) $\displaystyle-\frac{9}{16}\sin^{2}\theta\sin^{2}\theta^{*}\cos
2\chi\mathcal{H}_{T}+\frac{9}{8}\sin\theta\sin
2\theta^{*}\cos\chi\mathcal{H}_{A}$ $\displaystyle+\frac{9}{16}\sin
2\theta\sin 2\theta^{*}\cos\chi\mathcal{H}_{I}-\frac{9}{8}\sin\theta\sin
2\theta^{*}\sin\chi\mathcal{H}_{II}$ $\displaystyle-\frac{9}{16}\sin
2\theta\sin
2\theta^{*}\sin\chi\mathcal{H}_{IA}+\frac{9}{16}\sin^{2}\theta\sin^{2}\theta^{*}\sin
2\chi\mathcal{H}_{IT}$ $\displaystyle+$
$\displaystyle\delta_{l}\left[\frac{9}{4}\cos^{2}\theta^{*}\mathcal{H}_{S}-\frac{9}{2}\cos\theta\cos^{2}\theta^{*}\mathcal{H}_{SL}+\frac{9}{4}\cos^{2}\theta\cos^{2}\theta^{*}\mathcal{H}_{L}\right.$
$\displaystyle+\frac{9}{16}\sin^{2}\theta\sin^{2}\theta^{*}\mathcal{H}_{U}+\frac{9}{8}\sin^{2}\theta\sin^{2}\theta^{*}\cos
2\chi\mathcal{H}_{T}$ $\displaystyle+\frac{9}{4}\sin\theta\sin
2\theta^{*}\cos\chi\mathcal{H}_{ST}-\frac{9}{8}\sin 2\theta\sin
2\theta^{*}\cos\chi\mathcal{H}_{I}$ $\displaystyle-\frac{9}{4}\sin\theta\sin
2\theta^{*}\sin\chi\mathcal{H}_{ISA}+\frac{9}{8}\sin 2\theta\sin
2\theta^{*}\sin\chi\mathcal{H}_{IA}$
$\displaystyle\left.-\frac{9}{8}\sin^{2}\theta\sin^{2}\theta^{*}\sin
2\chi\mathcal{H}_{IT}\right].$

Note that the observables such as differential decay rate, forward-backward
asymmetry, polarization of $\tau$ and vector mesons, can be constructed from
three-body decays directly, without the help of Eq. (40), which is derived
assuming the narrow width limit. The uncertainty due to the narrow width
approximation influence only the following $W_{i}$ measurements. Even so, the
finite-width effect is generally very small for vector menson, less than a few
percent, as explored in Ref. Cheng et al. (2020).

##  IV THE PHYSICAL OBSERVABLES

To study the effect of the lepton mass and provide a more detailed physical
picture in semileptonic decays beyond the branching fraction, we can also
define other physical observables that can be measured experimentally, such as
forward-backward asymmetry $\left(\mathcal{A}_{FB}^{l}\right)$, longitudinal
$\left(P_{L}^{l}\right)$ and transverse $\left(P_{T}^{l}\right)$ polarization
of the charged lepton, longitudinal polarization $\left(F_{L}^{l}(V)\right)$
of the final vector meson, leptonic convexity parameter
$\left(C_{F}^{l}\right)$, and trigonometric momentum $\left(W_{i}\right)$ in
the angular distribution. These observables are expressed again by the above
helicity structure functions. The hadronic convexity parameter
$\left(C_{F}^{h}\right)$ is simply related to the longitudinal polarization of
vector, and we will not repetitively calculate $C_{F}^{h}$.

First, we consider the forward-backward asymmetry. The “forward” region
requires the $\theta\in\left[0,\pi/2\right]$ and the “backward” region
$\theta\in\left[\pi/2,\pi\right]$. Then, $\mathcal{A}_{FB}^{l}$ is defined as

$\displaystyle\mathcal{A}_{FB}^{l}(q^{2})$
$\displaystyle=\frac{\int_{0}^{1}d\cos\theta\frac{d\Gamma}{dq^{2}d\cos\theta}-\int_{-1}^{0}d\cos\theta\frac{d\Gamma}{dq^{2}d\cos\theta}}{\int_{0}^{1}d\cos\theta\frac{d\Gamma}{dq^{2}d\cos\theta}+\int_{-1}^{0}d\cos\theta\frac{d\Gamma}{dq^{2}d\cos\theta}}$
(41) $\displaystyle=\frac{3}{4}\frac{H_{P}-4\delta_{l}H_{SL}}{H_{tot}}.$

Next, we consider the polarization observables, which can be derived from the
three-body decay. We then define a system where the leptons lie in the $xz$
plane, leading to
$k_{1}=\left(E_{1},\;|\vec{k_{1}}|\sin\theta,\;0,\;|\vec{k_{1}}|\cos\theta\right)$.
That is, the observables have no $\chi$ dependence and integration over $\chi$
trivially gives $2\pi$ factor. Generally, the expression of the spin four-
vector $s$ is Greiner and Muller (1993)

$s^{\mu}=\left(\frac{\vec{k}_{1}\cdot\hat{\vec{s}}}{m_{l}},\;\hat{\vec{s}}+\frac{\vec{k}_{1}\left(\vec{k}_{1}\cdot\hat{\vec{s}}\right)}{m_{l}\left(k_{1}^{0}+m_{l}\right)}\right),$
(42)

which can be obtained by imposing a Lorentz transformation to $(0,\vec{s})$,
and $\vec{s}$ is a unit vector defining the direction of spin in the rest
frame. For the longitudinal polarization of the charged lepton,
$\vec{k}_{1}\cdot\hat{\vec{s}}=|\vec{k}_{1}|$, we obtain the longitudinal
polarization vector:

$s_{L}^{\mu}=\frac{1}{m_{l}}\left(|\vec{k}_{1}|,\;E_{1}\sin\theta,\;0,\;E_{1}\cos\theta\right),$
(43)

which satisfies $s_{L\mu}s_{L}^{\mu}=-1$, $s_{L\mu}k_{1}^{\mu}=0$.

For the leptonic tensor $L_{\mu\nu}$ in Eq.(28), one sums over the spins in
the the product of Dirac spinors,
$\sum\limits_{s}u(p,s)\bar{u}(p,s)=\left(\not{p}+m\right)$ and
$\sum\limits_{s}v(p,s)\bar{v}(p,s)=\left(\not{p}-m\right)$. While the lepton
is polarized, one has
$u(p,s_{L})\bar{u}(p,s_{L})=\frac{1}{2}\left[\left(\not{p}+m\right)\left(1+\gamma_{5}\not{s}\right)\right]$
and
$v(p,s_{L})\bar{v}(p,s_{L})=\frac{1}{2}\left[\left(\not{p}-m\right)\left(1+\gamma_{5}\not{s}\right)\right]$.
The expression of the leptonic tensor for the longitudinally polarized lepton
is

$\displaystyle L_{\mu\nu}(s_{L})=4$
$\displaystyle\left(k_{1\mu}k_{2\nu}+k_{1\nu}k_{2\mu}-k_{1}\cdot
k_{2}g_{\mu\nu}\pm
i\varepsilon_{\mu\nu\alpha\beta}k_{1}^{\alpha}k_{2}^{\beta}\right.$ (44)
$\displaystyle\left.\mp m_{l}s_{L\mu}k_{2\nu}\mp m_{l}s_{L\nu}k_{2\mu}\pm
m_{l}s_{L}\cdot
k_{2}g_{\mu\nu}-im_{l}\varepsilon_{\mu\nu\alpha\beta}s_{L}^{\alpha}k_{2}^{\beta}\right)$

where the upper/lower sign refers to the $(l^{-}\bar{\nu}_{l})/(l^{+}\nu_{l})$
configuration. In Ref. Gutsche et al. (2015) the leptonic tensor is written as

$L_{\mu\nu}(s_{L})=\mp
8m_{l}\left(s_{L\mu}k_{2\nu}+s_{L\nu}k_{2\mu}-s_{L}\cdot k_{2}g_{\mu\nu}\pm
i\varepsilon_{\mu\nu\alpha\beta}s_{L}^{\alpha}k_{2}^{\beta}\right),$ (45)

i.e., making the replacement $k_{1}^{\mu}\rightarrow\mp m_{l}s_{L}^{\mu}$ from
the unpolarized case. Equations (44) and (45) are equivalent in the sense of
$L(\lambda_{W},\lambda_{W}^{\prime})$, cf. Eq. (10).

With Eq. (45) we obtain the polarized differential decay distribution:

$\displaystyle\frac{d\Gamma(s_{L})}{dq^{2}}$
$\displaystyle=\frac{G_{F}^{2}|V_{cq}|^{2}|\vec{p}_{2}|q^{2}v^{2}}{12(2\pi)^{3}m_{1}^{2}}\left[-3\delta_{l}\left|H_{t}\right|^{2}+\left(1-\delta_{l}\right)\left(|H_{+}|^{2}+|H_{-}|^{2}+|H_{0}|^{2}\right)\right]$
(46)
$\displaystyle=\frac{G_{F}^{2}|V_{cq}|^{2}|\vec{p}_{2}|q^{2}v^{2}}{12(2\pi)^{3}m_{1}^{2}}\left[\mathcal{H}_{U}+\mathcal{H}_{L}-\delta_{l}\left(\mathcal{H}_{U}+\mathcal{H}_{L}+3\mathcal{H}_{S}\right)\right].$

The longitudinal polarization of the lepton is then defined as the ratio of
polarized decay distribution Eq. (46) to the unpolarized decay distribution
Eq. (35):

$\displaystyle P_{L}^{l}(q^{2})$
$\displaystyle=\frac{\mathcal{H}_{U}+\mathcal{H}_{L}-\delta_{l}\left(\mathcal{H}_{U}+\mathcal{H}_{L}+3\mathcal{H}_{S}\right)}{\mathcal{H}_{tot}}$
(47)

Similarly, one can obtain the definition of the leptonic transverse
polarization $P_{T}^{l}(q^{2})$, where the leptonic polarization direction is
perpendicular to its momentum direction:

$s_{T}^{\mu}=\left(0,\;\hat{\vec{s}}_{T}\right)=\left(0,\;\cos\theta,\;0,\;-\sin\theta\right).$
(48)

$\hat{\vec{s}}_{T}$ is obtained from rotating
$\hat{\vec{s}}_{L}=(\sin\theta,0,\cos\theta)$ by $\pi/2$ in the
counterclockwise direction. Making the substitution $k_{1}^{\mu}\rightarrow\mp
m_{l}s_{T}^{\mu}$, one obtains the transversely polarized leptonic tensor, and
the resulting differential decay distribution is

$\displaystyle\frac{d\Gamma(s_{T})}{dq^{2}}$
$\displaystyle=\frac{G_{F}^{2}|V_{cq}|^{2}|\vec{p}_{2}|q^{2}v^{2}}{12(2\pi)^{3}m_{1}^{2}}\frac{3\pi\sqrt{\delta_{l}}}{4\sqrt{2}}\left[-\left|H_{+}\right|^{2}+\left|H_{-}\right|^{2}-2\text{Re}(H_{0}H_{t}^{\dagger})\right]$
(49)
$\displaystyle=\frac{G_{F}^{2}|V_{cq}|^{2}|\vec{p}_{2}|q^{2}v^{2}}{12(2\pi)^{3}m_{1}^{2}}\frac{3\pi\sqrt{\delta_{l}}}{4\sqrt{2}}\left[-\mathcal{H}_{P}-2\mathcal{H}_{SL}\right].$

The transverse polarization of lepton will be

$\displaystyle P_{T}^{l}(q^{2})$
$\displaystyle=-\frac{3\pi\sqrt{\delta_{l}}}{4\sqrt{2}}\frac{\mathcal{H}_{P}+2\mathcal{H}_{SL}}{\mathcal{H}_{tot}}.$
(50)

In addition, the fourfold decay distribution allows us to define more physical
quantities. The angular distribution can be normalized as

$\widetilde{W}\left(\theta^{*},\theta,\chi\right)=\frac{W\left(\theta^{*},\theta,\chi\right)}{\mathcal{H}_{tot}}.$
(51)

Integrating $W\left(\theta^{*},\theta,\chi\right)$ over $\cos\theta^{*}$ and
$\chi$, one can recover the angular distribution of $\theta$ appearing in the
$D\to Vl\nu_{l}$ transition (Eq. (34)):

$\displaystyle W\left(\theta\right)=$
$\displaystyle\frac{3}{8}\times\left[\left(1+2\delta_{l}\right)\mathcal{H}_{U}+2\mathcal{H}_{L}+4\delta_{l}\mathcal{H}_{S}+\left(2\mathcal{H}_{P}-8\delta_{l}\mathcal{H}_{SL}\right)\cos\theta\right.$
(52)
$\displaystyle\left.+((1-2\delta_{l})\mathcal{H}_{U}-2(1-2\delta_{l})\mathcal{H}_{L})\cos^{2}\theta\right].$

The normalized angular distribution of $\theta$ is then

$\widetilde{W}(\theta)=\frac{W(\theta)}{\mathcal{H}_{tot}}=\frac{a+b\cos\theta+c\cos^{2}\theta}{2\left(a+c/3\right)},$
(53)

where

$\displaystyle a$
$\displaystyle=3/8\times\left[\left(1+2\delta_{l}\right)\mathcal{H}_{U}+2\mathcal{H}_{L}+4\delta_{l}\mathcal{H}_{S}\right],$
(54) $\displaystyle b$
$\displaystyle=3/8\times\left(2\mathcal{H}_{P}-8\delta_{l}\mathcal{H}_{SL}\right),$
$\displaystyle c$
$\displaystyle=3/8\times\left[\left(1-2\delta_{l}\right)\mathcal{H}_{U}-2\left(1-2\delta_{l}\right)\mathcal{H}_{L}\right].$

By the above coefficient definition, it is clear that the linear coefficient
indicates the forward-backward asymmetry

$\mathcal{A}_{FB}(q^{2})=\frac{b}{2\left(a+c/3\right)}=\frac{3}{4}\frac{\mathcal{H}_{P}-4\delta_{l}\mathcal{H}_{SL}}{\mathcal{H}_{tot}}.$
(55)

The leptonic convexity parameter is defined as

$\displaystyle C_{F}^{l}(q^{2})$
$\displaystyle=\frac{d^{2}\widetilde{W}(\theta)}{d(\cos\theta)^{2}}=\frac{c}{a+c/3}=\frac{3}{4}\left(1-2\delta_{l}\right)\frac{\mathcal{H}_{U}-2\mathcal{H}_{L}}{\mathcal{H}_{tot}}.$
(56)

In the same manner, we can obtain the $\theta^{*}$ angular distribution by
integrating the $W(\theta,\theta^{*},\chi)$ over $\theta$ and $\chi$:

$\displaystyle
W(\theta^{*})=\frac{3}{4}\times\left[\left(1+\delta_{l}\right)\mathcal{H}_{U}+\left(-\left(1+\delta_{l}\right)\mathcal{H}_{U}+2\left(1+\delta_{l}\right)\mathcal{H}_{L}+6\delta_{l}\mathcal{H}_{S}\right)\cos^{2}\theta^{*}\right].$
(57)

The normalized $\theta^{*}$ angular distribution is described by

$\widetilde{W}(\theta^{*})=\frac{W(\theta^{*})}{\mathcal{H}_{tot}}=\frac{a^{\prime}+c^{\prime}\cos^{2}\theta^{*}}{2a^{\prime}+2/3c^{\prime}},$
(58)

where

$\displaystyle a^{\prime}$
$\displaystyle=\left(1+\delta_{l}\right)\mathcal{H}_{U}$ (59) $\displaystyle
c^{\prime}$
$\displaystyle=-\left(1+\delta_{l}\right)\mathcal{H}_{U}+2\left(1+\delta_{l}\right)\mathcal{H}_{L}+6\delta_{l}\mathcal{H}_{S}.$

The hadronic convexity parameter can be extracted by taking the second
derivative of $\widetilde{W}(\theta^{*})$ as

$\displaystyle C^{h}_{F}(q^{2})$
$\displaystyle=\frac{d^{2}\widetilde{W}(\theta^{*})}{d(\cos\theta^{*})^{2}}=\frac{c^{\prime}}{a^{\prime}+c^{\prime}/3}=-\frac{3}{2}\frac{\mathcal{H}_{U}-2\mathcal{H}_{L}+\delta_{l}\left(\mathcal{H}_{U}-2\mathcal{H}_{L}-6\mathcal{H}_{S}\right)}{\mathcal{H}_{tot}}.$
(60)

The longitudinal polarization fraction of the final vector meson is given by

$\displaystyle F_{L}^{l}(q^{2})$
$\displaystyle=\frac{d\Gamma(\lambda_{V}=0)/dq^{2}}{d\Gamma/dq^{2}}=\frac{(1+\delta_{l})\mathcal{H}_{L}+3\delta_{l}\mathcal{H}_{S}}{\mathcal{H}_{tot}},$
(61)

and the transverse polarization fraction is
$F_{T}^{l}(q^{2})=1-F_{L}^{l}(q^{2})$. The hadronic convexity parameter is
related to the longitudinal polarization of vector meson by

$C_{F}^{h}(q^{2})=\frac{3}{2}(3F_{L}(q^{2})-1).$ (62)

Moreover, to examine more helicity structure functions in Eq. (40), we can
define the trigonometric moments through the normalized angular decay
distribution $\widetilde{W}(\theta^{*},\theta,\chi)$

$W_{i}=\int d\cos\theta
d\cos\theta^{*}d\left(\chi/2\pi\right)M_{i}\left(\theta,\theta^{*},\chi\right)\widetilde{W}\left(\theta,\theta^{*},\chi\right)=\left\langle
M_{i}\left(\theta,\theta^{*},\chi\right)\right\rangle.$ (63)

Some explicit examples include

$\displaystyle W_{T}(q^{2})$ $\displaystyle=\left\langle\cos
2\chi\right\rangle=-\frac{1}{2}\left(1-2\delta_{l}\right)\frac{\mathcal{H}_{T}}{\mathcal{H}_{tot}},$
(64) $\displaystyle W_{I}(q^{2})$
$\displaystyle=\left\langle\cos\theta\cos\theta^{*}\cos\chi\right\rangle=\frac{9\pi^{2}\left(1-2\delta_{l}\right)}{512}\frac{\mathcal{H}_{I}}{\mathcal{H}_{tot}},$
$\displaystyle W_{A}(q^{2})$
$\displaystyle=\left\langle\sin\theta\sin\theta^{*}\cos\chi\right\rangle=\frac{3\pi}{16}\frac{\mathcal{H}_{A}+2\delta_{l}\mathcal{H}_{ST}}{\mathcal{H}_{tot}}.$

For final mesons being pseudoscalars, the helicity amplitudes $H_{\pm}=0$,
which leads to vanishing trigonometric moments. Therefore, Eq. (64) is only
applied to $D\rightarrow V$ transitions.

In practice, due to the limited statistics in the experiment, we show the
average values of all mentioned physical observables. In such a case, we
reinstate the phase factor
$C(q^{2})=\left|\vec{p_{2}}\right|(q^{2}-m_{l}^{2})^{2}/q^{2}$ in both
numerator and denominator and integrate them separately. More explicitly,

$\langle\mathcal{A}_{FB}^{l}\rangle=\frac{3}{4}\frac{{\int}dq^{2}C(q^{2})\left(\mathcal{H}_{P}-4\delta_{l}\mathcal{H}_{SL}\right)}{{\int}dq^{2}C(q^{2})\left[\mathcal{H}_{U}+\mathcal{H}_{L}+\delta_{l}\left(\mathcal{H}_{U}+\mathcal{H}_{L}+3\mathcal{H}_{S}\right)\right]}.$
(65)

A similar operation holds for all others. We also present the full momentum
dependence of $\mathcal{A}_{FB}^{l}$, $P_{L}^{l}$ and $P_{T}^{l}$.

For the case in which the charge of the lepton is negative, i.e.,
$l^{-}\bar{\nu}_{l}$, the relative sign in Eq. (28) and Eq. (45) will
influence the expressions of some observables. To better compare with
experimental and other theoretical results, we also give the definitions of
forward-backward asymmetry, leptonic longitudinal and transverse polarization
in such a case:

$\displaystyle\mathcal{A}_{FB}(q^{2})=-\frac{3}{4}\frac{\mathcal{H}_{P}+4\delta_{l}\mathcal{H}_{SL}}{\mathcal{H}_{tot}},$
(66) $\displaystyle
P_{L}^{l}(q^{2})=-\frac{\mathcal{H}_{U}+\mathcal{H}_{L}-\delta_{l}\left(\mathcal{H}_{U}+\mathcal{H}_{L}+3\mathcal{H}_{S}\right)}{\mathcal{H}_{tot}},$
$\displaystyle
P_{T}^{l}(q^{2})=-\frac{3\pi\sqrt{\delta_{l}}}{4\sqrt{2}}\frac{\mathcal{H}_{P}-2\mathcal{H}_{SL}}{\mathcal{H}_{tot}}.$

The expressions for other observables do not change.

##  V NUMERICAL RESULTS AND DISCUSSION

In this section, we will calculate the branching fractions of $D_{(s)}$ and
$B_{(s)}$ to $(P,V)l^{+}\nu_{l}$ and the average values of
$\mathcal{A}_{FB}^{l}$, $P_{L}^{l}$, $P_{T}^{l}$, $C_{F}^{l}$, and $W_{i}$, on
the basis of related form factors and the abovementioned helicity amplitudes.
The average values for these observables and the $q^{2}$ dependence are
presented in the tables and figures below.

Concerning the related form factors in these physical quantities equations, we
stick to the predictions in the covariant light-front quark model (CLFQM)
Cheng et al. (2004); Verma (2012). The momentum dependence of form factors can
be parameterized as 222The shape of form factor is generally smooth, and such
a parametrization works very accurately. The difference between it and the
exact numerical value is less than 3%. However, there are indeed exceptions.
See discussions between Eqs. (33) and (34) in Ref. Cheng et al. (2004). For
the $B(D)\to{}^{1}P_{1}$ decays, a new parametrisation is proposed, but it is
not relevant to our current study.

$F(q^{2})=\frac{F(0)}{1-a\left(q^{2}/m_{D}^{2}\right)+b\left(q^{2}/m_{D}^{2}\right)^{2}}.$
(67)

The values of relevant parameters that appear in form factors are listed in
Table 2.

For the final state of the $D_{(s)}$ transitions, the pseudoscalar mesons $P$
contain $\eta,\eta^{\prime},\pi^{0},K^{0}$, $\bar{K^{0}}$ and the vector
mesons $V$ contain $\rho,\omega,\phi,K^{*}$,$\bar{K^{*}}$; for the $B_{(s)}$
transitions, the pseudoscalar mesons $P$ contain
$\eta,\eta^{\prime},\pi^{0},\bar{D}^{0},D_{s}^{-}$ and the vector mesons $V$
contain $\rho,\omega,K^{*-},\bar{D}^{*0}$, $D_{s}^{*-}$. For $\eta$ and
$\eta^{\prime}$, we need to consider the quark mixing scheme Feldmann et al.
(1998); Feldmann (2000)

$\left(\begin{array}[]{c}\eta\\\
\eta^{\prime}\end{array}\right)=\left(\begin{array}[]{cc}\cos\phi&-\sin\phi\\\
\sin\phi&\cos\phi\end{array}\right)\left(\begin{array}[]{c}\eta_{q}\\\
\eta_{s}\end{array}\right),$ (68)

where $\eta_{q}=\frac{1}{\sqrt{2}}(\mu\bar{\mu}+d\bar{d})$,
$\eta_{s}=s\bar{s}$, and the mixing angle $\phi=39.3^{\circ}$ from Ref. Zyla
et al. (2020); Dai et al. (2018b).

Table 2: The form factor parameters predicted by CLFQM in $D_{(s)}/B_{(s)}\rightarrow P(V)$ transitions. The values are taken from Ref. Verma (2012). $F$ | $F(0)$ | $a$ | $b$ | $F$ | $F(0)$ | $a$ | $b$ | $F$ | $F(0)$ | $a$ | $b$ | $F$ | $F(0)$ | $a$ | $b$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$F_{1}^{D\pi}$ | 0.66 | 1.19 | 0.35 | $F_{0}^{D\pi}$ | 0.66 | 0.51 | 0.00 | $F_{1}^{B\pi}$ | 0.25 | 1.70 | 0.90 | $F_{0}^{B\pi}$ | 0.25 | 0.82 | 0.09
$F_{1}^{D\eta_{q}}$ | 0.71 | 1.13 | 0.27 | $F_{0}^{D\eta_{q}}$ | 0.71 | 0.43 | $-0.01$ | $F_{1}^{B\eta_{q}}$ | 0.29 | 1.63 | 0.74 | $F_{0}^{B\eta_{q}}$ | 0.29 | 0.75 | 0.04
$F_{1}^{DK}$ | 0.79 | 1.05 | 0.25 | $F_{0}^{DK}$ | 0.79 | 0.47 | $-0.00$ | $F_{1}^{BD}$ | 0.67 | 1.22 | 0.36 | $F_{0}^{BD}$ | 0.67 | 0.63 | $-0.01$
$F_{1}^{D_{s}K}$ | 0.66 | 1.11 | 0.48 | $F_{0}^{D_{s}K}$ | 0.66 | 0.56 | 0.04 | $F_{1}^{B_{s}K}$ | 0.23 | 1.88 | 1.58 | $F_{0}^{B_{s}K}$ | 0.23 | 1.05 | 0.35
$F_{1}^{D_{s}\eta_{s}}$ | 0.76 | 1.02 | 0.40 | $F_{0}^{D_{s}\eta_{s}}$ | 0.76 | 0.60 | 0.04 | $F_{1}^{B_{s}D_{s}}$ | 0.67 | 1.28 | 0.52 | $F_{0}^{B_{s}D_{s}}$ | 0.67 | 0.69 | $0.07$
$V^{D\rho}$ | 0.88 | 1.23 | 0.40 | $A_{0}^{D\rho}$ | 0.69 | 1.08 | 0.45 | $A_{1}^{D\rho}$ | 0.60 | 0.46 | 0.01 | $A_{2}^{D\rho}$ | 0.47 | 0.89 | 0.23
$V^{D\omega}$ | 0.85 | 1.24 | 0.45 | $A_{0}^{D\omega}$ | 0.64 | 1.08 | 1.50 | $A_{1}^{D\omega}$ | 0.58 | 0.49 | 0.02 | $A_{2}^{D\omega}$ | 0.49 | 0.95 | 0.28
$V^{DK^{*}}$ | 0.98 | 1.10 | 0.32 | $A_{0}^{DK^{*}}$ | 0.78 | 1.01 | 0.34 | $A_{1}^{DK^{*}}$ | 0.72 | 0.45 | 0.01 | $A_{2}^{DK^{*}}$ | 0.60 | 0.89 | 0.21
$V^{D_{s}K^{*}}$ | 0.87 | 1.13 | 0.69 | $A_{0}^{D_{s}K^{*}}$ | 0.61 | 0.90 | 0.62 | $A_{1}^{D_{s}K^{*}}$ | 0.56 | 0.59 | 0.08 | $A_{2}^{D_{s}K^{*}}$ | 0.46 | 0.90 | 0.43
$V^{D_{s}\phi}$ | 0.98 | 1.04 | 0.54 | $A_{0}^{D_{s}\phi}$ | 0.72 | 0.92 | 0.62 | $A_{1}^{D_{s}\phi}$ | 0.69 | 0.56 | 0.07 | $A_{2}^{D_{s}\phi}$ | 0.59 | 0.90 | 0.38
$V^{B\rho}$ | 0.29 | 1.77 | 1.06 | $A_{0}^{B\rho}$ | 0.32 | 1.67 | 1.01 | $A_{1}^{B\rho}$ | 0.24 | 0.86 | 0.15 | $A_{2}^{B\rho}$ | 0.22 | 1.56 | 0.85
$V^{B\omega}$ | 0.27 | 1.81 | 1.18 | $A_{0}^{B\omega}$ | 0.28 | 1.62 | 1.22 | $A_{1}^{B\omega}$ | 0.23 | 0.91 | 0.18 | $A_{2}^{B\omega}$ | 0.21 | 1.62 | 0.97
$V^{BD^{*}}$ | 0.77 | 1.25 | 0.38 | $A_{0}^{BD^{*}}$ | 0.68 | 1.21 | 0.36 | $A_{1}^{BD^{*}}$ | 0.65 | 0.60 | 0.00 | $A_{2}^{BD^{*}}$ | 0.61 | 1.12 | 0.31
$V^{B_{s}K^{*}}$ | 0.23 | 2.03 | 2.27 | $A_{0}^{B_{s}K^{*}}$ | 0.25 | 1.95 | 2.20 | $A_{1}^{B_{s}K^{*}}$ | 0.19 | 1.24 | 0.62 | $A_{2}^{B_{s}K^{*}}$ | 0.16 | 1.83 | 1.85
$V^{B_{s}D_{s}^{*}}$ | 0.75 | 1.37 | 0.67 | $A_{0}^{B_{s}D_{s}^{*}}$ | 0.66 | 1.33 | 0.63 | $A_{1}^{B_{s}D_{s}^{*}}$ | 0.62 | 0.76 | 0.13 | $A_{2}^{B_{s}D_{s}^{*}}$ | 0.57 | 1.25 | 0.56
|  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Table 3: Branching fraction for the semileptonic decays of $D^{+}$ and
$D_{s}^{+}$, compared to the PDG Zyla et al. (2020) and BES${\mathrm{III}}$
results. All values are in unit of $10^{-3}$

$e$ mode $\mu$ mode  CLFQM PDG BES${\mathrm{III}}$ CLFQM PDG
BES${\mathrm{III}}$ $D^{+}\rightarrow P$ $D^{+}\rightarrow\eta l^{+}\nu_{l}$
$1.20$ $1.11\pm 0.07$ $1.16$ $1.04\pm 0.15$ Ablikim (2020a)
$D^{+}\rightarrow\eta^{\prime}l^{+}\nu_{l}$ $0.179$ $0.20\pm 0.04$ $0.169$
$D^{+}\rightarrow\pi^{0}l^{+}\nu_{l}$ $4.09$ $3.72\pm 0.17$ $3.63\pm 0.13$
Ablikim et al. (2017) $4.04$ $3.50\pm 0.15$
$D^{+}\rightarrow\bar{K}^{0}l^{+}\nu_{l}$ $103.2$ $87.3\pm 1.0$ $86.0\pm 2.1$
Ablikim et al. (2017) $100.7$ $87.6\pm 1.9$ $D^{+}\rightarrow V$
$D^{+}\rightarrow\rho l^{+}\nu_{l}$ $2.32$ $2.18^{+0.17}_{-0.25}$ $2.22$
$2.4\pm 0.4$ $D^{+}\rightarrow\omega l^{+}\nu_{l}$ $2.07$ $1.69\pm 0.11$
$1.69\pm 0.11$ Ablikim (2020b) $1.98$ $1.77\pm 0.29$ Ablikim (2020b)
$D^{+}\rightarrow\bar{K}^{*0}l^{+}\nu_{l}$ $73.2$ $54.0\pm 1.0$ $69.3$
$52.7\pm 1.5$ $D_{s}^{+}\rightarrow P$ $D_{s}^{+}\rightarrow\eta l^{+}\nu_{l}$
$21.9$ $23.2\pm 0.8$ $23.0\pm 3.9$ Ablikim et al. (2018) $21.5$ $24\pm 5.0$
$24.2\pm 5.7$ Ablikim et al. (2018)
$D_{s}^{+}\rightarrow\eta^{\prime}l^{+}\nu_{l}$ $8.82$ $8.0\pm 0.7$ $9.3\pm
3.5$ Ablikim et al. (2018) $8.41$ $11\pm 5.0$ $10.6\pm 6.1$ Ablikim et al.
(2018) $D_{s}^{+}\rightarrow K^{0}l^{+}\nu_{l}$ $2.54$ $3.4\pm 0.4$ $2.49$
$D_{s}^{+}\rightarrow V$ $D_{s}^{+}\rightarrow\phi l^{+}\nu_{l}$ $30.7$
$23.9\pm 1.6$ $22.6\pm 5.4$ Ablikim et al. (2018) $28.9$ $19.0\pm 5.0$
$19.4\pm 6.2$ Ablikim et al. (2018) $D_{s}^{+}\rightarrow K^{*0}l^{+}\nu_{l}$
$1.90$ $2.15\pm 0.28$ $1.82$

Table 4: Branching fraction for the semileptonic decays of $B^{+}$ and $B_{s}$, compared to the PDG Tanabashi et al. (2018) results. |  |  $e$ mode | PDG ($l^{+}\nu_{l}$) |  $\tau$ mode | PDG ($\tau^{+}\nu_{\tau}$)
---|---|---|---|---|---
$B^{+}\rightarrow P$ | $B^{+}\rightarrow\eta l^{+}\nu_{l}$ | $4.96\times 10^{-5}$ | $(3.9\pm 0.5)\times 10^{-5}$ | $3.03\times 10^{-5}$ |
| $B^{+}\rightarrow\eta^{\prime}l^{+}\nu_{l}$ | $2.41\times 10^{-5}$ | $(2.3\pm 0.8)\times 10^{-5}$ | $1.28\times 10^{-5}$ |
| $B^{+}\rightarrow\pi^{0}l^{+}\nu_{l}$ | $7.20\times 10^{-5}$ | $(7.8\pm 0.27)\times 10^{-5}$ | $4.89\times 10^{-5}$ |
| $B^{+}\rightarrow\bar{D}^{0}l^{+}\nu_{l}$ | $2.59\times 10^{-2}$ | $(2.35\pm 0.09)\times 10^{-2}$ | $0.78\times 10^{-2}$ |  $(0.77\pm 0.25)\times 10^{-2}$
$B^{+}\rightarrow V$ | $B^{+}\rightarrow\rho l^{+}\nu_{l}$ | $2.00\times 10^{-4}$ | $(1.58\pm 0.11)\times 10^{-4}$ | $1.09\times 10^{-4}$ |
| $B^{+}\rightarrow\omega l^{+}\nu_{l}$ | $1.89\times 10^{-4}$ | $(1.19\pm 0.09)\times 10^{-4}$ | $1.00\times 10^{-4}$ |
| $B^{+}\rightarrow\bar{D}^{*0}l^{+}\nu_{l}$ | $6.67\times 10^{-2}$ | $(5.66\pm 0.22)\times 10^{-2}$ | $1.66\times 10^{-2}$ |  $(1.88\pm 0.20)\times 10^{-2}$
$B_{s}\rightarrow P$ | $B_{s}\rightarrow K^{-}l^{+}\nu_{l}$ | $9.23\times 10^{-5}$ |  | $6.18\times 10^{-5}$ |
| $B_{s}\rightarrow D_{s}^{-}l^{+}\nu_{l}$ | $2.41\times 10^{-2}$ |  | $0.72\times 10^{-2}$ |
$B_{s}\rightarrow V$ | $B_{s}\rightarrow K^{*-}l^{+}\nu_{l}$ | $3.01\times 10^{-4}$ |  | $1.56\times 10^{-4}$ |
| $B_{s}\rightarrow D_{s}^{*-}l^{+}\nu_{l}$ | $5.91\times 10^{-2}$ |  | $1.46\times 10^{-2}$ |
|  |  |  |  |

For the leptons involved, we consider the electron and muon mode for the
$D_{(s)}$ transition, separately. In addition, the corresponding tau mode is
prohibited by the phase space. By contrast, for the $B_{(s)}$ transition, the
electron and tau mode are included, where the results of the electron and muon
mode are almost identical, as expected, due to the large mass of $B_{(s)}$,
and the difference between them is beyond the current experimental
uncertainties.

Table 5: Forward-backward asymmetry, lepton polarization, leptonic convexity parameter, and longitudinal polarization of the vector meson for semileptonic decays of $D^{+}$ and $D_{s}^{+}$, compared with other predictions from the CCQM Ivanov et al. (2019) and RQM Faustov et al. (2020). The dash indicates that the value does not exist. In Ref. Faustov et al. (2020), given the specific scale, some numbers of $\langle A_{FB}^{e}\rangle$ and $\langle P_{T}^{e}\rangle$ (with orders of $10^{-6}$ and $10^{-3}$, respectively) are shown as 0. |  |  | $\langle\mathcal{A}_{FB}^{e}\rangle$ | $\langle\mathcal{A}_{FB}^{\mu}\rangle$ | $\langle P_{L}^{e}\rangle$ | $\langle P_{L}^{\mu}\rangle$ | $\langle P_{T}^{e}\rangle(10^{-2})$ | $\langle P_{T}^{\mu}\rangle$ | $\langle C_{F}^{e}\rangle$ | $\langle C_{F}^{\mu}\rangle$ | $\langle F_{L}^{e}\rangle$ |  $\langle F_{L}^{\mu}\rangle$
---|---|---|---|---|---|---|---|---|---|---|---|---
$D^{+}\rightarrow P$ | $D^{+}\rightarrow\eta l^{+}\nu_{l}$ | CLFQM | $-6.0\times 10^{-6}$ | $-0.05$ | $1.00$ | $0.84$ | $-0.27$ | $-0.43$ | $-1.50$ | $-1.36$ | - | \-
|  | CCQM | $-6.4\times 10^{-6}$ | $-0.06$ | $1.00$ | $0.83$ | $-0.28$ | $-0.44$ | $-1.50$ | $-1.32$ | - | \-
|  | RQM |  | $-0.052$ | 1.00 | 0.85 |  | $-0.40$ | $-1.50$ | $-1.34$ | - | \-
| $D^{+}\rightarrow\eta^{\prime}l^{+}\nu_{l}$ | CLFQM | $-13.2\times 10^{-6}$ | $-0.10$ | $1.00$ | $0.71$ | $-0.41$ | $-0.57$ | $-1.50$ | $-1.27$ | - | \-
|  | CCQM | $-13.0\times 10^{-6}$ | $-0.10$ | $1.00$ | $0.70$ | $-0.42$ | $-0.59$ | $-1.50$ | $-1.19$ | - | \-
|  | RQM |  | $-0.097$ | 1.00 | 0.72 |  | $-0.56$ | $-1.50$ | $-1.20$ | - | \-
| $D^{+}\rightarrow\pi^{0}l^{+}\nu_{l}$ | CLFQM | $-3.4\times 10^{-6}$ | $-0.04$ | $1.00$ | $0.90$ | $-0.20$ | $-0.34$ | $-1.50$ | $-1.40$ | - | \-
|  | CCQM | $-4.1\times 10^{-6}$ | $-0.04$ | 1.00 | 0.88 | $-0.22$ | $-0.36$ | $-1.50$ | $-1.37$ | - | \-
|  | RQM |  | $-0.040$ | 1.00 | 0.89 |  | $-0.36$ | $-1.50$ | $-1.38$ | - | \-
| $D^{+}\rightarrow\bar{K}^{0}l^{+}\nu_{l}$ | CLFQM | $-5.8\times 10^{-6}$ | $-0.05$ | $1.00$ | $0.84$ | $-0.27$ | $-0.42$ | $-1.50$ | $-1.36$ | - | \-
|  | CCQM | $-6.4\times 10^{-6}$ | $-0.06$ | 1.00 | 0.83 | $-0.28$ | $-0.43$ | $-1.50$ | $-1.32$ | - | \-
|  | RQM |  | $-0.053$ | 1.00 | 0.85 |  | $-0.42$ | $-1.50$ | $-1.34$ | - | \-
$D^{+}\rightarrow V$ | $D^{+}\rightarrow\rho l^{+}\nu_{l}$ | CLFQM | $-0.24$ | $-0.26$ | $1.00$ | $0.92$ | $-0.10$ | $-0.13$ | $-0.48$ | $-0.40$ | 0.55 | 0.54
|  | CCQM | $-0.21$ | $-0.24$ | $1.00$ | $0.92$ | $-0.09$ | $-0.13$ | $-0.44$ | $-0.36$ | 0.53 | 0.51
|  | RQM | $-0.26$ | $-0.28$ | $1.00$ | $0.92$ |  | $-0.12$ | $-0.42$ | $-0.34$ | 0.52 | 0.52
| $D^{+}\rightarrow\omega l^{+}\nu_{l}$ | CLFQM | $-0.24$ | $-0.26$ | $1.00$ | $0.92$ | $-0.09$ | $-0.12$ | $-0.45$ | $-0.37$ | 0.53 | 0.53
|  | CCQM | $-0.21$ | $-0.24$ | $1.00$ | $0.92$ | $-0.09$ | $-0.12$ | $-0.43$ | $-0.35$ | 0.52 | 0.50
|  | RQM | $-0.25$ | $-0.27$ | $1.00$ | $0.93$ |  | $-0.11$ | $-0.39$ | $-0.32$ | 0.51 | 0.50
| $D^{+}\rightarrow\bar{K}^{*0}l^{+}\nu_{l}$ | CLFQM | $-0.19$ | $-0.22$ | $1.00$ | $0.90$ | $-0.11$ | $-0.15$ | $-0.48$ | $-0.39$ | 0.55 | 0.54
|  | CCQM | $-0.18$ | $-0.21$ | 1.00 | 0.91 | $-0.11$ | $-0.15$ | $-0.47$ | $-0.37$ | 0.54 | 0.52
|  | RQM | $-0.22$ | $-0.25$ | $1.00$ | $0.90$ |  | $-0.15$ | $-0.47$ | $-0.37$ | 0.54 | 0.54
$D_{s}^{+}\rightarrow P$ | $D_{s}^{+}\rightarrow\eta l^{+}\nu_{l}$ | CLFQM | $-5.6\times 10^{-6}$ | $-0.05$ | $1.00$ | $0.84$ | $-0.27$ | $-0.43$ | $-1.50$ | $-1.33$ | - | \-
|  | CCQM | $-6.0\times 10^{-6}$ | $-0.06$ | $1.00$ | $0.84$ | $-0.27$ | $-0.42$ | $-1.50$ | $-1.33$ | - | \-
|  | RQM |  | $-0.043$ |  | 0.88 |  | $-0.35$ |  | $-1.37$ | - | \-
| $D_{s}^{+}\rightarrow\eta^{\prime}l^{+}\nu_{l}$ | CLFQM | $-11.1\times 10^{-6}$ | $-0.09$ | $1.00$ | $0.74$ | $-0.38$ | $-0.55$ | $-1.50$ | $-1.23$ | - | \-
|  | CCQM | $-11.2\times 10^{-6}$ | $-0.09$ | $1.00$ | $0.75$ | $-0.38$ | $-0.54$ | $-1.50$ | $-1.23$ | - | \-
|  | RQM |  | $-0.080$ |  | 0.77 |  | $-0.51$ |  | $-1.26$ | - | \-
| $D_{s}^{+}\rightarrow K^{0}l^{+}\nu_{l}$ | CLFQM | $-5.1\times 10^{-6}$ | $-0.05$ | $1.00$ | $0.86$ | $-0.25$ | $-0.41$ | $-1.50$ | $-1.35$ | - | \-
|  | CCQM | $-5.0\times 10^{-6}$ | $-0.05$ | $1.00$ | $0.86$ | $-0.24$ | $-0.39$ | $-1.50$ | $-1.35$ | - | \-
|  | RQM |  | $-0.038$ |  | 0.89 |  | $-0.34$ |  | $-1.38$ | - | \-
$D_{s}^{+}\rightarrow V$ | $D_{s}^{+}\rightarrow\phi l^{+}\nu_{l}$ | CLFQM | $-0.18$ | $-0.21$ | $1.00$ | $0.91$ | $-0.11$ | $-0.14$ | $-0.48$ | $-0.38$ | 0.54 | 0.53
|  | CCQM | $-0.18$ | $-0.21$ | $1.00$ | $0.91$ | $-0.11$ | $-0.14$ | $-0.43$ | $-0.34$ | 0.53 | 0.50
|  | RQM | $-0.21$ | $-0.24$ | $1.00$ | $0.90$ |  | $-0.15$ | $-0.49$ | $-0.35$ | 0.54 | 0.54
| $D_{s}^{+}\rightarrow K^{*0}l^{+}\nu_{l}$ | CLFQM | $-0.22$ | $-0.25$ | $1.00$ | $0.92$ | $-0.09$ | $-0.12$ | $-0.47$ | $-0.38$ | 0.54 | 0.54
|  | CCQM | $-0.22$ | $-0.25$ | $1.00$ | $0.92$ | $-0.09$ | $-0.11$ | $-0.40$ | $-0.33$ | 0.51 | 0.49
|  | RQM | $-0.26$ | $-0.29$ | $1.00$ | $0.92$ |  | $-0.11$ | $-0.41$ | $-0.33$ | 0.52 | 0.51
|  |  |  |  |  |  |  |  |  |  |  |

In Tables 3 and 4, we list the numerical results of the branching fractions
for $D_{(s)}$ and $B_{(s)}$ decays, respectively, calculated through the
helicity formalism in Eq. (35). The results are in perfect agreement with the
values of Ref. Cheng and Kang (2017), which are calculated directly via form
factors. These two ways are cross-checked well. We also compare our
predictions with the results from the PDG Tanabashi et al. (2018) and
BES${\mathrm{III}}$ collaboration Ablikim (2020a, b); Ablikim et al. (2018,
2017), and we find a good agreement among them within the uncertainty. But
they differ a little larger for $D\to Kl\nu_{l}$, and especially for $D\to
K^{*}l\nu_{l}$. Assuming 15% uncertainty for the decay rate, our result
$\mathcal{B}(D^{+}\to\bar{K}^{*0}e^{+}\nu_{e})=73.2\times 10^{-3}$ differs
from the experimental one $54.0\times 10^{-3}$ by $1.7\sigma$, while
$2.5\sigma$ for assuming 10% uncertainty. This also happens for CCQM, see
Table V in Faustov et al. (2020), where the RQM result agrees better with PDG.
It shows again that investigation in a different model is necessary. For the
values of the $B^{+}\rightarrow P(V)l^{+}\nu_{l}$ decay from PDG in Table 4,
the $l$ indicates an electron or a muon, not a sum over these two modes. We
also note that in Ref. Ivanov et al. (2019), the branching fractions for
$D^{+}\to D^{0}e^{+}\nu_{e}$ and $D_{s}^{+}\to D^{0}e^{+}\nu_{e}$ are
calculated to be on the order of $10^{-13}$ and $10^{-8}$, respectively. These
numbers are far beyond the current and even future scope of the experiment. We
will refrain from considering these two channels in our work.

Table 6: Forward-backward asymmetry, lepton polarization, and leptonic convexity parameter and the longitudinal polarization of vector meson for semileptonic decays of $B^{+}$ and $B_{s}$. The dash indicates that the value does not exist.  |  | $\langle\mathcal{A}_{FB}^{e}\rangle$ | $\langle\mathcal{A}_{FB}^{\tau}\rangle$ | $\langle P_{L}^{e}\rangle$ | $\langle P_{L}^{\tau}\rangle$ | $\langle P_{T}^{e}\rangle$ | $\langle P_{T}^{\tau}\rangle$ | $\langle C_{F}^{e}\rangle$ | $\langle C_{F}^{\tau}\rangle$ | $\langle F_{L}^{e}\rangle$ |  $\langle F_{L}^{\tau}\rangle$
---|---|---|---|---|---|---|---|---|---|---|---
$B^{+}\rightarrow P$ | $B^{+}\rightarrow\eta l^{+}\nu_{l}$ | $-0.39\times 10^{-6}$ | $-0.29$ | $1.00$ | $0.11$ | $-0.64\times 10^{-3}$ | $-0.86$ | $-1.50$ | $-0.60$ | - | \-
| $B^{+}\rightarrow\eta^{\prime}l^{+}\nu_{l}$ | $-0.49\times 10^{-6}$ | $-0.31$ | $1.00$ | $0.026$ | $-0.72\times 10^{-3}$ | $-0.87$ | $-1.50$ | $-0.52$ | - | \-
| $B^{+}\rightarrow\pi^{0}l^{+}\nu_{l}$ | $-0.35\times 10^{-6}$ | $-0.28$ | $1.00$ | $0.087$ | $-0.62\times 10^{-3}$ | $-0.85$ | $-1.50$ | $-0.59$ | - | \-
| $B^{+}\rightarrow\bar{D}^{0}l^{+}\nu_{l}$ | $-1.04\times 10^{-6}$ | $-0.36$ | $1.00$ | $-0.32$ | $-1.07\times 10^{-3}$ | $-0.84$ | $-1.50$ | $-0.27$ | - | \-
$B^{+}\rightarrow V$ | $B^{+}\rightarrow\rho l^{+}\nu_{l}$ | $-0.32$ | $-0.39$ | $1.00$ | $0.60$ | $-0.18\times 10^{-3}$ | $-0.10$ | $-0.39$ | $-0.12$ | 0.51 | 0.49
| $B^{+}\rightarrow\omega l^{+}\nu_{l}$ | $-0.30$ | $-0.36$ | $1.00$ | $0.65$ | $-0.15\times 10^{-3}$ | $-0.06$ | $-0.42$ | $-0.15$ | 0.51 | 0.49
| $B^{+}\rightarrow\bar{D}^{*0}l^{+}\nu_{l}$ | $-0.22$ | $-0.30$ | $1.00$ | $0.51$ | $-0.29\times 10^{-3}$ | $-0.10$ | $-0.42$ | $-0.056$ | 0.52 | 0.45
$B_{s}\rightarrow P$ | $B_{s}\rightarrow K^{-}l^{+}\nu_{l}$ | $-0.43\times 10^{-6}$ | $-0.29$ | $1.00$ | $-0.10$ | $-0.72\times 10^{-3}$ | $-0.86$ | $-1.50$ | $-0.46$ | - | \-
| $B_{s}\rightarrow D_{s}^{-}l^{+}\nu_{l}$ | $-1.05\times 10^{-6}$ | $-0.36$ | $1.00$ | $-0.33$ | $-1.07\times 10^{-3}$ | $-0.84$ | $-1.50$ | $-0.26$ | - | \-
$B_{s}\rightarrow V$ | $B_{s}\rightarrow K^{*-}l^{+}\nu_{l}$ | $-0.21$ | $-0.28$ | $1.00$ | $0.65$ | $-0.17\times 10^{-3}$ | $-0.13$ | $-0.59$ | $-0.26$ | 0.59 | 0.56
| $B_{s}\rightarrow D_{s}^{*-}l^{+}\nu_{l}$ | $-0.22$ | $-0.29$ | $1.00$ | $0.51$ | $-0.29\times 10^{-3}$ | $-0.10$ |  $-0.4$3 | $-0.058$ | 0.52 | 0.45
|  |  |  |  |  |  |  |  |  |  |

In Tables 5 and 6, we list the average values of other observables for the
$D_{(s)}$ and $B_{(s)}$ transitions, including the forward-backward asymmetry
$\langle\mathcal{A}_{FB}^{l}\rangle$, leptonic longitudinal and transverse
polarization $\langle P_{L}^{l}\rangle$, $\langle P_{T}^{l}\rangle$, leptonic
convexity parameter $\langle C_{F}^{l}\rangle$ and longitudinal polarization
$\langle F_{L}^{l}\rangle$ of the final vector meson. In Table 5, we compare
our results with other theoretical predictions from Ref. Ivanov et al. (2019)
and Ref. Faustov et al. (2020). In these references, the form factors are
obtained based on the covariant confining quark model (CCQM) and the
relativistic quark model (RQM), separately, and the magnitude of these
numerical results agree well. From these two tables, we find that the average
values of the forward-backward asymmetry $A_{FB}^{\mu(\tau)}$ are similar to
those for $\mathcal{A}_{FB}^{e}$ for both the $D_{s}\rightarrow V$ and
$B_{s}\rightarrow V$ transition, while for the $D_{s}\rightarrow P$ and
$B_{s}\rightarrow P$ transitions, we obtain
$\langle\mathcal{A}_{FB}^{\mu}\rangle/\langle\mathcal{A}_{FB}^{e}\rangle\sim
10^{4}$ and
$\langle\mathcal{A}_{FB}^{\tau}\rangle/\langle\mathcal{A}_{FB}^{e}\rangle\sim
10^{7}$. Therefore, the lepton mass effect is more apparent for the mode of
the pseudoscalar in final state. In fact, it is a natural result of Eq. (41),
where for the pseudoscalar case $H_{\pm}=0$, then $\mathcal{H}_{P}=0$ and
$\mathcal{A}_{FB}$ is proportional to the lepton mass squared; the above ratio
just corresponds to $m_{\mu}^{2}/m_{e}^{2}$ and $m_{\tau}^{2}/m_{e}^{2}$,
respectively. Concerning the average values of the leptonic transverse
polarization, the absolute values of $\langle P_{T}^{\mu}\rangle$ are much
larger than $\langle P_{T}^{e}\rangle$ for $D_{(s)}$ decay and $|\langle
P_{T}^{\tau}\rangle|\gg|\langle P_{T}^{e}\rangle|$ for $B_{(s)}$ decays. More
precisely, one has $\langle P_{T}^{\mu}\rangle/\langle P_{T}^{e}\rangle\sim
10^{2}$ for $D_{(s)}$ decay and $\langle P_{T}^{\mu}\rangle/\langle
P_{T}^{e}\rangle\sim 10^{3}$ for $B_{(s)}$ decay. Again, these ratios
correspond to $m_{\mu}/m_{e}$ and $m_{\tau}/m_{e}$, cf. Eq. (50). $\langle
P_{L}^{e}\rangle$ equals 1 for all the cases since in Eq. (47), the term
proportional to $\delta_{l}$ almost vanishes for a numerical calculation.
Clearly, in the zero lepton mass limit, $\langle P_{T}^{l}\rangle=0$ and
$\langle P_{L}^{e}\rangle=1$ for all channels, and $\langle
C_{F}^{e}\rangle=-1.5$ for the $D_{(s)}/B_{(s)}\rightarrow P$ case. In Table
6, the results from other models are still lacking. We note that some of those
channels have been explored e.g., in the RQM Ebert et al. (2007, 2010);
Faustov and Galkin (2012, 2014); Faustov and Galkin (2013a, b), in the QCD sum
rule Azizi (2008); Azizi and Bayar (2008), in the light-cone sum rule Wang and
Shen (2015); Wang et al. (2017); Lü et al. (2019); Gao et al. (2020); Azizi et
al. (2011), however, the relevant observables are not given.

Through the longitudinal polarization of the final vector meson, we can obtain
the ratios of the partial decay rates $\Gamma_{L}/\Gamma_{T}=\langle
F_{L}\rangle/(1-\langle F_{L}\rangle)$. Experimental measurements give the
results regarding these ratios $\Gamma_{L}/\Gamma_{T}$ for
$D^{+}\rightarrow\bar{K}^{*0}l^{+}\nu_{l}$, and $D_{s}^{+}\rightarrow\phi
l^{+}\nu_{l}$ decays, and we give the comparison in Table 7. Our predictions
are in good agreement with the available experimental values and other
theoretical results from the CCQM Ivanov et al. (2019) and RQM Faustov et al.
(2020). The PDG average for $\Gamma_{L}/\Gamma_{T}$ in
$D^{+}\rightarrow\bar{K}^{*0}l^{+}\nu_{l}$ is $1.13\pm 0.08$, which is fully
taken from the measurements for $D^{+}\rightarrow\bar{K}^{*0}\mu^{+}\nu_{\mu}$
with the exclusion of the data from Ref. Anjos et al. (1990) for the electron
mode. The PDG average for $\Gamma_{L}/\Gamma_{T}$ in $D_{s}^{+}\rightarrow\phi
l^{+}\nu_{l}$ is $0.72\pm 0.18$, which is based on the value from Ref. Avery
et al. (1994) as well as from Refs. Frabetti et al. (1994) and Kodama et al.
(1993). The last two treat $\Gamma_{L}/\Gamma_{T}$ for a lepton mass of zero.

Table 7: The ratios of the partial decay rates $\Gamma_{L}/\Gamma_{T}$ for $D^{+}\rightarrow\bar{K}^{*0}l^{+}\nu_{l}$ and $D_{s}\rightarrow\phi l^{+}\nu_{l}$ decays. The PDG averages are discussed in the text. |  CLFQM |  CCQM Ivanov et al. (2019) |  RQM Faustov et al. (2020) | Experimental
---|---|---|---|---
$D^{+}\rightarrow\bar{K}^{*0}e^{+}\nu_{e}$ |  $1.21$ |  $1.17$ |  $1.17$ |
$D^{+}\rightarrow\bar{K}^{*0}\mu^{+}\nu_{\mu}$ |  1.17 |  1.08 |  1.17 |  $1.13\pm 0.08$ Zyla et al. (2020)
$D_{s}\rightarrow\phi e^{+}\nu_{e}$ |  $1.17$ |  $1.12$ |  $1.17$ |  $1.0\pm 0.3\pm 0.2$ Avery et al. (1994)
$D_{s}\rightarrow\phi\mu^{+}\nu_{\mu}$ |  $1.12$ |  $1$ |  $1.17$ |
|  |  |  |

Considering the $l^{-}\bar{\nu}_{l}$ mode, we recalculate the relevant
physical observables for the $\bar{B}^{0}\rightarrow D^{+}l^{-}\bar{\nu}_{l}$
and $\bar{B}^{0}\rightarrow D^{*+}l^{-}\bar{\nu}_{l}$ decay channels and
compare our results with another prediction from the CCQM Ivanov et al. (2015)
in Table 8. Again there is a good agreement.

Table 8: Forward-backward asymmetry, lepton polarization, and convexity parameters for semileptonic decays of $\bar{B}^{0}\rightarrow D^{+}l^{-}\bar{\nu}_{l}$ and $\bar{B}^{0}\rightarrow D^{*+}l^{-}\bar{\nu}_{l}$, compared with the prediction from the CCQM Ivanov et al. (2015), where $\langle P_{T}^{e}\rangle$ is shown as 0 since it is on the order of $10^{-3}$.  |  | $\langle\mathcal{A}_{FB}^{e}\rangle$ | $\langle\mathcal{A}_{FB}^{\tau}\rangle$ | $\langle P_{L}^{e}\rangle$ | $\langle P_{L}^{\tau}\rangle$ | $\langle P_{T}^{e}\rangle$ | $\langle P_{T}^{\tau}\rangle$ | $\langle C_{F}^{e}\rangle$ |  $\langle C_{F}^{\tau}\rangle$
---|---|---|---|---|---|---|---|---|---
$\bar{B}^{0}\rightarrow D^{+}l^{-}\bar{\nu}_{l}$ | CLFQM | $-1.04\times 10^{-6}$ | $-0.36$ | $-1$ | $0.32$ | $1.06\times 10^{-3}$ | $0.84$ | $-1.5$ |  $-0.27$
| CCQM | $-1.17\times 10^{-6}$ | $-0.36$ | $-1$ | $0.33$ |  | $0.84$ | $-1.5$ |  $-0.26$
$\bar{B}^{0}\rightarrow D^{*+}l^{-}\bar{\nu}_{l}$ | CLFQM | $0.22$ | $0.054$ | $-1$ | $-0.51$ | $0.46\times 10^{-3}$ | $0.47$ | $-0.42$ |  $-0.056$
| CCQM | $0.19$ | $0.027$ | $-1$ | $-0.50$ |  | $0.46$ | $-0.47$ |  $-0.062$
|  |  |  |  |  |  |  |  |
Table 9: The average values for trigonometric moments (see Eq.(64)), compared with the prediction from the CCQM Ivanov et al. (2019). |  |  $\langle W_{T}^{e}\rangle$ |  $\langle W_{T}^{\mu}\rangle$ |  $\langle W_{I}^{e}\rangle$ |  $\langle W_{I}^{\mu}\rangle$ |  $\langle W_{A}^{e}\rangle$ |  $\langle W_{A}^{\mu}\rangle$
---|---|---|---|---|---|---|---
$D^{+}\rightarrow\rho l^{+}\nu_{l}$ | CLFQM |  $-0.078$ |  $-0.077$ |  $0.052$ |  $0.049$ |  $-0.078$ |  $-0.072$
| CCQM |  $-0.091$ |  $-0.089$ |  $0.054$ |  $0.051$ |  $-0.067$ |  $-0.061$
$D^{+}\rightarrow\omega l^{+}\nu_{l}$ | CLFQM |  $-0.081$ |  $-0.080$ |  $0.052$ |  $0.049$ |  $-0.077$ |  $-0.071$
| CCQM |  $-0.093$ |  $-0.091$ |  $0.054$ |  $0.051$ |  $-0.066$ |  $-0.060$
$D^{+}\rightarrow\bar{K^{*}}l^{+}\nu_{l}$ | CLFQM |  $-0.091$ |  $-0.089$ |  $0.054$ |  $0.051$ |  $-0.062$ |  $-0.055$
| CCQM |  $-0.097$ |  $-0.094$ |  0.055 |  0.051 |  $-0.056$ |  $-0.049$
$D_{s}^{+}\rightarrow\phi l^{+}\nu_{l}$ | CLFQM |  $-0.095$ |  $-0.093$ |  $0.054$ |  $0.051$ |  $-0.057$ |  $-0.050$
| CCQM |  $-0.101$ |  $-0.098$ |  $0.055$ |  $0.052$ |  $-0.055$ |  $-0.048$
$D_{s}^{+}\rightarrow K^{*}l^{+}\nu_{l}$ | CLFQM |  $-0.085$ |  $-0.083$ |  $0.053$ |  $0.050$ |  $-0.072$ |  $-0.066$
| CCQM |  $-0.094$ |  $-0.092$ |  $0.054$ |  $0.051$ |  $-0.068$ |  $-0.062$
|  |  $\langle W_{T}^{e}\rangle$ |  $\langle W_{T}^{\tau}\rangle$ |  $\langle W_{I}^{e}\rangle$ |  $\langle W_{I}^{\tau}\rangle$ |  $\langle W_{A}^{e}\rangle$ |  $\langle W_{A}^{\tau}\rangle$
$B^{+}\rightarrow\rho l^{+}\nu_{l}$ | CLFQM |  $-0.050$ |  $-0.042$ |  $0.048$ |  $0.031$ |  $-0.109$ |  $-0.056$
$B^{+}\rightarrow\omega l^{+}\nu_{l}$ | CLFQM |  $-0.056$ |  $-0.048$ |  $0.049$ |  $0.032$ |  $-0.103$ |  $-0.058$
$B^{+}\rightarrow\bar{D^{*}}l^{+}\nu_{l}$ | CLFQM |  $-0.091$ |  $-0.056$ |  $0.054$ |  $0.025$ |  $-0.069$ |  $-0.005$
$B_{s}\rightarrow K^{*-}l^{+}\nu_{l}$ | CLFQM |  $-0.063$ |  $-0.053$ |  $0.051$ |  $0.034$ |  $-0.085$ |  $-0.042$
$B_{s}\rightarrow D_{s}^{*-}l^{+}\nu_{l}$ | CLFQM |  $-0.092$ |  $-0.056$ |  $0.054$ |  $0.025$ |  $-0.067$ |  $-0.043$
|  |  |  |  |  |  |

In Table 9, we present our predictions for the trigonometric moments of
$D_{(s)}$ and $B_{(s)}$ decay. Except for the electron mode, for $D_{(s)}$
decay, we have also calculated the muon mode, while for $B_{(s)}$, the tau
mode is also illustrated. For the transitions of meson $D_{(s)}$, we give the
comparison with the predictions of the CCQM Soni et al. (2018), and we find
that our predictions are in good agreement with them.

As a drawback of the model, it is hard to quantify the theoretical
uncertainty. Comparing the central values between Refs. Verma (2012) and Cheng
et al. (2004), one finds that a general 7%-10% uncertainly attached to form
factor is reasonable. Reference Verma (2012) is just an update of Cheng et al.
(2004) with utilizing the latest experimental information, wherever available,
or the lattice results to fix the parameter $\beta$ in the wave function of
meson involved. Considering also the uncertainty from CKM matrix elements (the
uncertainty for $|V_{ub}|$ is larger due to the difference between the
extractions from the exclusive and inclusive mode), a 10%-15% uncertainty will
be assigned to the observable.

Table 10: Our predictions for $F_{L}^{\tau}(D_{(s)}^{*})$ and $P_{L}^{\tau}(D_{(s)}^{(*)})$, compared with other models as well as experimental values. In parenthesis, we also include the value of $F_{L}^{e}(D^{*})$ for $\bar{B}\rightarrow D^{*}\tau^{-}\bar{\nu}_{\tau}$. Observables | Approach | $\bar{B}\rightarrow D\tau^{-}\bar{\nu}_{\tau}$ | $\bar{B}\rightarrow D^{*}\tau^{-}\bar{\nu}_{\tau}\,(e^{-}\bar{\nu}_{e})$ | $B_{s}\rightarrow D_{s}\tau^{-}\bar{\nu}_{\tau}$ |  $B_{s}\rightarrow D_{s}^{*}\tau^{-}\bar{\nu}_{\tau}$
---|---|---|---|---|---
$F_{L}^{\tau}(D_{(s)}^{*})$ | CLFQM | - | 0.451 (0.521) | - | 0.453
| SM1 Alok et al. (2017) | - | $0.46\pm 0.04$ | - | \-
| SM2 Das and Dutta (2020) | - | 0.455 | - | 0.433
| PQCD Hu et al. (2020) | - | 0.43 | - | 0.43
| Belle Abdesselam et al. (2019) | - |  $0.60\pm 0.08\pm 0.04$ ($0.56\pm 0.02$) | - | \-
$P_{L}^{\tau}(D_{(s)}^{(*)})$ | CLFQM | $0.32$ | $-0.51$ | $0.33$ |  $-0.51$
| SM1 |  $0.325\pm 0.09$ Tanaka and Watanabe (2010) |  $-0.497\pm 0.013$ Tanaka and Watanabe (2013) | - | \-
| SM2 Das and Dutta (2020) | 0.352 | $-0.501$ | - |  $-0.520$
| PQCD Hu et al. (2020) | 0.30 | $-0.53$ | 0.30 |  $-0.53$
| Belle Hirose et al. (2017) | - | $-0.38\pm 0.51^{+0.21}_{-0.16}$ | - | \-
|  |  |  |  |

The Belle collaboration has measured the longitudinal polarization of the
final vector meson as $F_{L}^{\tau}(D^{*})=0.60\pm 0.08\pm 0.04$ Abdesselam et
al. (2019) and the longitudinal polarization of the $\tau$ lepton333Notice
that the Belle collaboration measures the longitudinal polarization of lepton
with $\tau$ decays $\tau\rightarrow\pi^{-}\nu_{\tau}$ and
$\tau\rightarrow\rho^{-}\nu_{\tau}$ in the rest frame of $\tau$. In fact, it
could also be measured in laboratory frame of $\tau$ or in the rest frame of
virtual W Hagiwara et al. (1990). Nevertheless, in the rest frame of $\tau$,
the formula becomes much simpler. as $P_{L}^{\tau}(D^{*})=-0.38\pm
0.51^{+0.21}_{-0.16}$ Hirose et al. (2017) for the decay $\bar{B}\rightarrow
D^{*}\tau^{-}\nu_{\tau}$. Our results from the CLFQM, as well as other
predictions, are compared to those from the experiment in Table 10. The
results agree well within uncertainties; however, the uncertainty for
$P_{L}^{\tau}$ of the Belle measurement is very large. Further precise
measurement is important and desirable. As discussed in many works Hirose et
al. (2017); Blanke et al. (2019); Shi et al. (2019); Ivanov et al. (2020);
Alok et al. (2018, 2020), the polarization observables $F_{L}^{\tau}(D^{*})$,
$P_{L}^{\tau}(D^{*})$, and the yet unmeasured $P_{L}^{\tau}(D)$ could
potentially discriminate the effects of new operator structure beyond the SM.
Their correlation with other NP models—the two Higgs doublet model and a
leptoquark model—are discussed in Refs. Tanaka and Watanabe (2010, 2013). We
also notice that the values of $P^{\tau}$ and $\mathcal{A}_{FB}$ in Table
${\mathrm{VI}}$ in Ref. Hu et al. (2020) should correspond to the
$\tau^{-}\bar{\nu}_{\tau}$ mode, not the $\tau^{+}\nu_{\tau}$ mode. As
mentioned, the two modes will lead to different results with not only an
overall sign difference, cf. Eq. (66) and Eqs. (41), (47), and (50).

As a matter of fact, there are very different landscapes for polarizations of
$\tau$ and lighter ones $e$ and $\mu$. The polarisation of $\tau$ can be
accessed by analysing its decay product, as has been done by Belle
collaboration, but it is not possible for $e$ and $\mu$. In a complete angular
analysis, cf. Eq. (40), the helicity structure functions
$\mathcal{H}_{U,L,S,P,SL}$ could be extracted, and from them the polarisation
observables are defined.

Figure 3: The differential decay rates of the decays $D_{s}^{+}\rightarrow
K^{(*)}l^{+}\nu_{l}$, $B^{+}\rightarrow\bar{D}^{(*)}l^{+}\nu_{l}$ and
$B_{s}\rightarrow D_{s}^{(*)-}l^{+}\nu_{l}$. The lepton mode is indicated by
the corresponding legend. The solid and dashed lines denote the central values
and the band demonstrates our estimated uncertainty.

We also show the differential decay distribution $d\Gamma/dq^{2}$ within the
full range of the momentum transfer squared in Fig. 3 for some selected
channels. These are direct observables in the experiment and will be tested in
the future.

Figure 4: The differential decay rate for the decays $D\rightarrow
Ke^{+}\nu_{e}$ and $D\rightarrow\pi e^{+}\nu_{e}$. The solid line indicates
our central values and the band indicates the estimated uncertainty. We have
used the experimental data from BES ${\mathrm{III}}$ for neutral $D^{0}$
Ablikim et al. (2015) (red dots with error bars) and charged $D^{+}$ Ablikim
et al. (2017) (green dots with error bars), BaBar Lees et al. (2015); Aubert
et al. (2007) (blue dots and error bars) and CLEO Besson et al. (2009) for
neutral $D^{0}$ (orange dots and error bars) and charged $D^{+}$ (brown dots
with error bars).

Moreover, we compare our results with the experimental results of the
differential decay rate for $D\rightarrow Ke^{+}\nu_{e}$ and $D\rightarrow\pi
e^{+}\nu_{e}$ in Fig. 4. The experimental data are from the
BES${\mathrm{III}}$ Ablikim et al. (2015, 2017), BABAR Lees et al. (2015);
Aubert et al. (2007) and CLEO Besson et al. (2009) collaboration. The blue
band is obtained by assigning the central values a $10\%$ uncertainty,
demonstrating to some extent our theoretical uncertainty. Our results for
$D\rightarrow\pi e^{+}\nu_{e}$ agree very well with the experimental findings,
while for the $D\rightarrow Ke^{+}\nu_{e}$ case, they only agree within their
uncertainties. As already noted in Ref. Faustov et al. (2020), our result for
$D\rightarrow Ke^{+}\nu_{e}$ is larger than theirs and the experiment.

Figure 5: The forward-backward asymmetries of the decays $D_{(s)}\rightarrow
P(V)l^{+}\nu_{l}$ and $B_{(s)}\rightarrow P(V)l^{+}\nu_{l}$.

Figure 6: The longitudinal polarization of a charged lepton of the decays
$D_{(s)}\rightarrow P(V)l^{+}\nu_{l}$ and $B_{(s)}\rightarrow
P(V)l^{+}\nu_{l}$.

Figure 7: The transverse polarization of a charged lepton of the decays
$D_{(s)}\rightarrow P(V)l^{+}\nu_{l}$ and $B_{(s)}\rightarrow
P(V)l^{+}\nu_{l}$.

In Figs. 5-7, we represent the $q^{2}$ dependence of the forward-backward
asymmetry and the leptonic longitudinal and transverse polarization,
respectively. For the decays $D_{(s)}\rightarrow Vl^{+}\nu_{l}$ and
$B_{(s)}\rightarrow Vl^{+}\nu_{l}$, we can find that the values of
$\mathcal{A}_{FB}$ and $P_{T}$ coincide with 0 at the zero recoil point
($q^{2}=q_{\text{max}}^{2}$) since $\mathcal{H}_{P}=0$, $\mathcal{H}_{SL}=0$,
cf. Eq. (41) and Eq. (50). Owing to the same reason, the leptonic longitudinal
polarization Eq. (47) for the mode of the vector meson in final states is
reduced to $(1-\delta_{l})/(1+\delta_{l})$ at the zero recoil point; then, the
values of $P_{L}^{e(\mu)}$ approach 1, as shown in Fig. 6. In all these
figures, one finds that the longitudinal and transverse polarization values
for $\mu$ and $\tau$ are larger than the electron, which illustrates the
lepton mass effect as expected.

##  VI SUMMARY

Based on the form factors computed from the covariant light front quark model
Cheng et al. (2004); Verma (2012), we perform a comprehensive analysis of the
semileptonic decays $D_{(s)},B_{(s)}\rightarrow P(V)l\nu_{l}$ in the helicity
formalism, where all the observables are expressed by form factors via
helicity amplitude. We provide the detailed derivation for the differential
decay rate $(d\Gamma/dq^{2})$, forward-backward asymmetry
$(\mathcal{A}_{FB}^{l})$, longitudinal $(P_{L}^{l})$ and transverse
polarization $(P_{T}^{l})$ of a lepton, longitudinal polarization of a vector
meson $(F_{L}^{l}(V))$, leptonic convexity parameter $(C_{F}^{l})$, and
trigonometric moments $(W_{L}^{l},W_{I}^{l},W_{A}^{l})$. The numerical results
are shown in Sec. V.

The values of the branching fractions are in good agreement with the
experiment for $D_{(s)}$ and $B$ decay; however, the experimental information
is lacking for $B_{s}$ decay. For the polarization observables, the only
measurements are $F_{L}^{\tau}(D^{*})$ and $P_{L}^{\tau}(D^{*})$ for the decay
$\bar{B}\rightarrow D^{*}\tau^{-}\nu_{\tau}$ from the Belle collaboration. The
current Standard Model prediction agrees with the experimental value for the
polarization of $D^{*}$ ($F_{L}^{\tau}(D^{*})$), within 1.6 standard
deviations of the mean, while the experimental values for the polarization of
$\tau$ ($P_{L}^{\tau}(D^{*})$) suffer from large uncertainty. As recognized by
our community, these polarization observables are crucial inputs for testing
and investigating New Physics. Thus, further examination from both the
theoretical and experimental sides is of great importance.

These observables could be measured in BES${\mathrm{III}}$, Belle, and LHCb.
From a practical perspective, the forward-backward asymmetry could be measured
as a first step. In the near future, Belle${\mathrm{II}}$ will accumulate data
samples 50 times as large as those for Belle, which will provide a major
opportunity for such further measurements and the determination of variously
unmeasured observables discussed in the current work. Concerning the charm
part, the super $\tau$-charm factory is planned to be built in China.

###### Acknowledgements.

The author XWK acknowledges very useful discussions with Profs. M. A. Ivanov
and V. Galkin. The author XWK is supported by the National Natural Science
Foundation of China (NSFC) under Project No. 11805012 and the Fundamental
Research Funds for the Central Universities (BNU). XHG acknowledges NSFC under
Grant No. 11775024. LYD acknowledges NSFC with Grant No. 11805059 and Joint
Large Scale Scientific Facility Funds of the NSFC and Chinese Academy of
Sciences (CAS) under Contract No. U1932110, and Fundamental Research Funds for
the Central Universities (HNU). CW acknowledges NSFC No. 11805153. TL
acknowledges NSFC with Project Nos. 11805037 and U1832121 and Shanghai Pujiang
Program under Grant No. 18PJ1401000 and Open Research Program of Large
Research Infrastructures (2017), CAS.

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