# Purely leptonic decays of the ground charged vector mesons

Yueling Yang  Institute of Particle and Nuclear Physics, Henan Normal
University, Xinxiang 453007, China  Zhenglin Li  Institute of Particle and
Nuclear Physics, Henan Normal University, Xinxiang 453007, China  Kang Li
Institute of Particle and Nuclear Physics, Henan Normal University, Xinxiang
453007, China  Jinshu Huang  School of Physics and Electronic Engineering,
Nanyang Normal University, Nanyang 473061, China  Junfeng Sun  Institute of
Particle and Nuclear Physics, Henan Normal University, Xinxiang 453007, China

###### Abstract

The study of the purely leptonic decays of the ground charged vector mesons is
very interesting and significant in determining the CKM matrix elements,
obtaining the decay constant of vector mesons, examining the lepton flavor
universality, and searching for new physics beyond the standard model. These
purely leptonic decays of the ground charged vector mesons are induced by the
weak interactions within the standard model, and usually have very small
branching ratios, ${\cal B}({\rho}^{-}{\to}{\ell}^{-}{\nu}_{\ell})$ ${\sim}$
${\cal O}(10^{-13})$, ${\cal B}(K^{{\ast}-}{\to}{\ell}^{-}{\nu}_{\ell})$
${\sim}$ ${\cal O}(10^{-13})$, ${\cal
B}(D_{d}^{{\ast}-}{\to}{\ell}^{-}{\nu}_{\ell})$ ${\sim}$ ${\cal O}(10^{-10})$,
${\cal B}(B_{u}^{{\ast}-}{\to}{\ell}^{-}{\nu}_{\ell})$ ${\sim}$ ${\cal
O}(10^{-10})$, ${\cal B}(D_{s}^{{\ast}-}{\to}{\ell}^{-}{\nu}_{\ell})$ ${\sim}$
${\cal O}(10^{-6})$ and ${\cal B}(B_{c}^{{\ast}-}{\to}{\ell}^{-}{\nu}_{\ell})$
${\sim}$ ${\cal O}(10^{-6})$. Inspired by the potential prospects of LHCb,
Belle-II, STCF, CEPC and FCC-ee experiments, we discussed the probabilities of
experimental investigation on these purely leptonic decays. It is found that
the measurements of these decays might be possible and feasible with the
improvement of data statistics, analytical technique, and measurement
precision in the future. (1) With the hadron-hadron collisions, the purely
leptonic decays of ${\rho}^{-}$, $K^{{\ast}-}$, $D_{d,s}^{{\ast}-}$ and
$B_{u,c}^{{\ast}-}$ mesons might be accessible at LHC experiments. (2) With
the $e^{+}e^{-}$ collisions, the purely leptonic decays of $D_{d,s}^{{\ast}-}$
and $B_{u,c}^{{\ast}-}$ mesons might be measurable with over $10^{12}$ $Z^{0}$
bosons available at CEPC and FCC-ee experiments. In addition, the
$D_{d,s}^{{\ast}-}$ ${\to}$ ${\ell}^{-}{\nu}_{\ell}$ decays could also be
studied at Belle-II and SCTF experiments.

Published : Eur. Phys. J. C 81, 1110 (2021).

##  I Introduction

In the quark model pdg2020 ; pl.8.214 ; zweig , mesons are generally regarded
as bound states of the valence quark $q$ and antiquark $\bar{q}^{\prime}$. The
classifications of mesons are usually based on the spin-parity quantum number
$J^{P}$ of the $q\bar{q}^{\prime}$ system. The spin $J$ of meson is given by
the relation ${|}L-S{|}$ ${\leq}$ $J$ ${\leq}$ ${|}L+S{|}$. The orbital
angular momentum and total spin of the $q\bar{q}^{\prime}$ system are
respectively $L$ and $S$, where $S$ $=$ $0$ for antiparallel quark spins, and
$S$ $=$ $1$ for parallel quark spins. By convention, quarks have a positive
parity and antiquarks have a negative parity. Hence, the parity of meson is
$P$ $=$ $(-1)^{L+1}$. The $L$ $=$ $0$ states are the ground-state
pseudoscalars with $J^{P}$ $=$ $0^{-}$ and vectors with $J^{P}$ $=$ $1^{-}$.
Both quarks and leptons are fermions with spin $S$ $=$ $1/2$. Mesons are
composed of a pair of fermions — quark and antiquark, therefore, they could in
principle decay into a pair of fermions, for example, lepton and antilepton.
The experimental observation of the two-body purely leptonic decays of mesons
could be a clear and characteristic manifestation of the quark model. These
leptonic decays provide us with valuable opportunities to fully investigate
the microstructure and properties of mesons. The study of two-body purely
leptonic decays of mesons is very interesting and significant.

The valence quarks of the electrically charged mesons must have different
flavors. Within the standard model (SM) of elementary particles, the purely
leptonic decays of the charged mesons (PLDCM) are typically induced by the
tree-level exchange of the gauge bosons $W$, the quanta of the weak
interaction fields. Up to today, the masses of all the experimentally observed
mesons are much less than those of $W$ bosons. Consequently, the massive $W$
bosons are virtual propagators rather than physical particles in the true
picture of PLDCM. Phenomenologically, by integrating out the contributions
from heavy dynamical degrees of freedom such as the $W$ fields, PLDCM can be
properly described by the low-energy effective theory in analogy with the
Fermi theory for ${\beta}$ decays. Considering the fact that leptons are free
from the strong interactions, the corresponding effective Hamiltonian
RevModPhys.68.1125  for PLDCM could be written as the product of quark
currents and leptonic currents,

${\cal H}_{\rm
eff}\,=\,\frac{G_{F}}{\sqrt{2}}\,V_{q_{1}q_{2}}\,\big{[}\bar{q}_{1}{\gamma}_{\mu}(1-{\gamma}_{5})q_{2}\big{]}\,\big{[}\bar{\ell}\,{\gamma}^{\mu}(1-{\gamma}_{5}){\nu}_{\ell}\big{]}+{\rm
h.c.},$ (1)

where the contributions of the $W$ bosons are embodied in the Fermi coupling
constant $G_{F}$ ${\simeq}$ $1.166{\times}10^{-5}\,{\rm GeV}^{-2}$ pdg2020 ,
and $V_{q_{1}q_{2}}$ is the Cabibbo-Kobayashi-Maskawa (CKM) Cabibbo ;
Kobayashi  matrix element between the quarks in the charged mesons. The decay
amplitudes can be written as,

${\langle}{\ell}\bar{\nu}_{\ell}{|}{\cal H}_{\rm
eff}{|}M{\rangle}\,=\,\frac{G_{F}}{\sqrt{2}}\,V_{q_{1}q_{2}}\,{\langle}{\ell}\bar{\nu}_{\ell}{|}\bar{\ell}\,{\gamma}^{\mu}(1-{\gamma}_{5}){\nu}_{\ell}{|}0{\rangle}\,{\langle}0{|}\bar{q}_{1}{\gamma}_{\mu}(1-{\gamma}_{5})q_{2}{|}M{\rangle}.$
(2)

The leptonic part of amplitudes can be calculated reliably with the
perturbative theory. The hadronic matrix elements (HMEs) interpolating the
diquark currents between the mesons concerned and the vacuum can be
parameterized by the decay constants.

With the conventions of Refs. JHEP.0605.004 ; JHEP.0703.069 , the HMEs of
diquark currents are defined as,

$\displaystyle{\langle}0{|}\bar{q}_{1}(0)\,{\gamma}_{\mu}\,q_{2}(0){|}P(k){\rangle}$
$\displaystyle=$ $\displaystyle 0,$ (3)
$\displaystyle{\langle}0{|}\bar{q}_{1}(0)\,{\gamma}_{\mu}\,{\gamma}_{5}\,q_{2}(0){|}P(k){\rangle}$
$\displaystyle=$ $\displaystyle i\,f_{P}\,k_{\mu},$ (4)
$\displaystyle{\langle}0{|}\bar{q}_{1}(0)\,{\gamma}_{\mu}\,q_{2}(0){|}V(k,{\epsilon}){\rangle}$
$\displaystyle=$ $\displaystyle f_{V}\,m_{V}\,{\epsilon}_{\mu},$ (5)
$\displaystyle{\langle}0{|}\bar{q}_{1}(0)\,{\gamma}_{\mu}\,{\gamma}_{5}\,q_{2}(0){|}V(k,{\epsilon}){\rangle}$
$\displaystyle=$ $\displaystyle 0,$ (6)

where the nonperturbative parameters of $f_{P}$ and $f_{V}$ are the decay
constants of pseudoscalar $P$ and vector $V$ mesons, respectively; and $m_{V}$
and ${\epsilon}_{\mu}$ are the mass and polarization vector, respectively. To
the lowest order, the decay widths are written as,

$\displaystyle{\Gamma}(P{\to}{\ell}\,\bar{\nu}_{\ell})$ $\displaystyle=$
$\displaystyle\frac{G_{F}^{2}}{8{\pi}}\,{|}V_{q_{1}q_{2}}{|}^{2}\,f_{P}^{2}\,m_{P}\,m_{\ell}^{2}\,\Big{(}1-\frac{m_{\ell}^{2}}{m_{P}^{2}}\Big{)}^{2},$
(7) $\displaystyle{\Gamma}(V{\to}{\ell}\,\bar{\nu}_{\ell})$ $\displaystyle=$
$\displaystyle\frac{G_{F}^{2}}{12{\pi}}\,{|}V_{q_{1}q_{2}}{|}^{2}\,f_{V}^{2}\,m_{V}^{3}\,\Big{(}1-\frac{m_{\ell}^{2}}{m_{V}^{2}}\Big{)}^{2}\,\Big{(}1+\frac{m_{\ell}^{2}}{2\,m_{V}^{2}}\Big{)},$
(8)

where $m_{P}$ and $m_{\ell}$ are the masses of the charged pseudoscalar meson
and lepton, respectively.

It is clearly seen from the above formula that the highly precise measurements
of PLDCM will allow the relatively accurate determinations of the product of
the decay constants and CKM elements, ${|}V_{q_{1}q_{2}}{|}f_{P,V}$.
Theoretically, the decay constants are nonperturbative parameters, and they
are closely related with the $\bar{q}_{1}q_{2}$ wave functions at the origin
which cannot be computed from first principles. There still exist some
discrepancies among theoretical results of the decay constants with different
methods, such as the potential model, QCD sum rules, lattice QCD, and so on.
If the magnitudes of CKM element ${|}V_{q_{1}q_{2}}{|}$ are fixed to the
values of Ref. pdg2020 , the decay constants $f_{P,V}$ will be experimentally
measured, and be used to seriously examine the different calculations on the
decay constants with various theoretical models. Likewise, if the decay
constants $f_{P,V}$ are well known to sufficient precision, the magnitudes of
the corresponding CKM element will be experimentally determined, and provide
complementary information to those from other processes. Within SM, the $P$
${\to}$ ${\ell}\bar{\nu}_{\ell}$ and $V$ ${\to}$ ${\ell}\bar{\nu}_{\ell}$
decays are induecd by the axial-vector current of Eq.(4) and vector current of
Eq.(5), respectively; and the electroweak interactions assign the vector-
minus-axial-vector ($V-A$) currents to the $W$ bosons. The CKM elements
determined from two different and complementary parts of the electroweak
interactions, charged vector and axial-vector currents, could be independently
examined. The latest CKM elements determined by PLDCM, such as ${|}V_{us}{|}$,
${|}V_{cd}{|}$ and ${|}V_{cs}{|}$, differ somewhat from those by exclusive and
inclusive semileptonic meson decays pdg2020 . The CKM elements extracted from
various processes can be combined to test the electroweak characteristic
charged-current $V-A$ interactions.

Within SM, the lepton-gauge-boson electroweak gauge couplings are generally
believed to be universal and process independent, which is called lepton
flavor universality (LFU). However, there are some hints of LFU discrepancies
between SM predictions and experimental measurements, such as the ratios of
branching fractions of semileptonic $B$ decays $R(D^{(\ast)})$ ${\equiv}$
${\cal B}(\bar{B}{\to}D^{(\ast)}{\tau}\bar{\nu}_{\tau})/{\cal
B}(\bar{B}{\to}D^{(\ast)}{\ell}\bar{\nu}_{\ell})$ with ${\ell}$ $=$ $e/{\mu}$
pdg2020 . The LFU validity can be carefully investigated through the PLDCM
processes. Beyond SM, some possible new heavy particles accompanied with novel
interactions, such as the charged higgs bosons, would affect PLDCM and LFU,
and might lead to detectable effects. So PLDCM provide good arenas to search
for the smoking gun of new physics (NP) beyond SM.

By considering the angular momentum conservation and the final states
including a left-handed neutrino or right-handed antineutrino, the purely
leptonic decay width of charged pseudoscalar meson, Eq.(7), is proportional to
the square of the lepton mass. This is called helicity suppression. While
there is no helicity suppression for the purely leptonic decay of charged
vector meson (PLDCV). From the analytical expressions of Eq.(7) and Eq.(8),
the decay width of pseudoscalar meson is suppressed by the factor
$m_{\ell}^{2}/m_{P}^{2}$ compared with that of vector meson. What’s more, both
the masses and the decay constants of vector mesons are relatively larger than
those of corresponding pseudoscalar mesons, which would result in an
enhancement of the decay widths for vector mesons. Of course, the vector
mesons decay dominantly through the strong and/or electromagnetic
interactions. The branching ratios for the PLDCV weak decays are usually very
small, sometimes even close to the accessible limits of the existing and the
coming experiments.

Inspired by the potential prospects of the future high-intensity and high-
energy frontiers, along with the noticeable increase of experimental data
statistics, the remarkable improvement of analytical technique and the
continuous enhancement of measurement precision, the carefully experimental
study of PLDCV might be possible and feasible. In this paper, we will focus on
the PLDCV within SM to just provide a ready reference. The review of the
purely leptonic decays of charged pseudoscalar mesons can be found in Ref.
pdg2020 .

##  II ${\rho}^{-}$ ${\to}$ ${\ell}^{-}\bar{\nu}_{\ell}$ decays

The mass of the ${\rho}^{\pm}$ meson, $m_{\rho}$ $=$ $775.11(34)$ MeV pdg2020
, is much larger than that of two-pion pair. The rate of the ${\rho}$ meson
decay into two pions via the strong interactions is almost 100%, which results
in the very short lifetime ${\tau}_{\rho}$ ${\sim}$ $4.4{\times}10^{-24}$ s
pdg2020 . The direct measurements of the electroweak properties of the
${\rho}$ meson would definitely be very challenging. It is evident from Eq.(8)
that the parameter of ${|}V_{ud}{|}\,f_{\rho}$ could be experimentally
determined from the observations of decay widths for the ${\rho}^{-}$ ${\to}$
${\ell}^{-}\bar{\nu}_{\ell}$ decays (if it is not specified, the corresponding
charge-conjugation processes are included in this paper), with the coupling
constant $G_{F}$, the masses of lepton $m_{\ell}$ and meson $m_{\rho}$.

The precise values of the CKM element ${|}V_{ud}{|}$ in ascending order of
measurement accuracy mainly come from ${\beta}$ transitions between the super-
allowed nuclear analog states with quantum number of both $J^{P}$ $=$ $0^{+}$
and isospin $I$ $=$ $1$, between mirror nuclei with $I$ $=$ $1/2$, between
neutron and proton, between charged and neutral pions CKM2016-vud . These four
results for ${|}V_{ud}{|}$ are basically consistent with one another. The
result of the super-allowed $0^{+}$ ${\to}$ $0^{+}$ nuclear ${\beta}$
transitions has an uncertainty a factor of about 10 smaller than the other
results, and thus dominates the weighted average value CKM2016-vud . The best
value from super-allowed nuclear ${\beta}$ transitions is ${|}V_{ud}{|}$ $=$
$0.97370(14)$ pdg2020 , which is smaller compared with the 2018 value
${|}V_{ud}{|}$ $=$ $0.97420(21)$ pdg2018 , as illustrated Fig. 1. This
reduction of the value of ${|}V_{ud}{|}$ leads to a slight deviation from the
first row unitarity requirement ${|}V_{ud}{|}^{2}$ $+$ ${|}V_{us}{|}^{2}$ $+$
${|}V_{ub}{|}^{2}$ $=$ $1$. The current precision of the CKM element
${|}V_{ud}{|}$ is about 0.01%. The latest value from the global fit in SM,
${|}V_{ud}{|}$ $=$ $0.97401(11)$ pdg2020 , will be used in our calculation.

Figure 1: The values of the CKM element ${|}V_{ud}{|}$ from Particle Data Group (PDG). Table 1: Decay constant $f_{\rho}$ (in the unit of MeV) obtained from the diffractive vector meson production using deep inelastic scattering (DVP), QCD sum rules (SR), relativistic quark model (RQM), light-front quark model (LFQM), lattice QCD (LQCD) and so on. DVP |  $143$ PhysRevD.69.094013  111With light-cone wave functions and parameters of Ref. epjc.22.655 . |  $147$ PhysRevD.69.094013  222With light-cone wave functions and parameters of Ref. PhysRevD.60.114023 . |  $153$ PhysRevD.69.094013  333With Gaussian wave functions and parameters of Ref. epjc.22.655 . |  $161$ PhysRevD.69.094013  444With Gaussian wave functions and parameters of Ref. PhysRevD.60.114023 . |  $211$ PhysRevD.94.074018
---|---|---|---|---|---
SR |  $201{\pm}5$ plb.436.351 555With nonlocal condensates like functions. |  $205{\pm}10$ plb.436.351  666With Ball-Braun wave functions PhysRevD.54.2182 . |  $194$ plb.436.351  777With Chernyak-Zhitnitsky wave functions PhysRept.112.173 . |  $198{\pm}7$ PhysRevD.58.094016  |  $206{\pm}7$ JHEP.0604.046
RQM |  $168.3$ epja.24.411  888With Gaussian spatial wave functions and adjusted parameters of Ref. epja.24.411 . |  $151.3$ epja.24.411  999With Gaussian spatial wave functions and parameters of Ref. plb.602.212 . |  $190.1$ epja.24.411  101010With rational spatial wave functions and adjusted parameters of Ref. epja.24.411 . |  $175.4$ epja.24.411  111111With rational spatial wave functions and parameters of Ref. plb.602.212 . |  $219$ plb.635.93
LFQM |  $246$ PhysRevD.75.034019  121212With Coulomb plus linear potential model. |  $215$ PhysRevD.75.034019  131313With Coulomb plus harmonic oscillator potential model. |  $205$ PhysRevC.92.055203  121212With Coulomb plus linear potential model. |  $166^{+2}_{-4}$ cpc.42.073102  141414With a dilation parameter ${\kappa}$ $=$ $0.54$ GeV. |  $210{\pm}6$ cpc.42.073102  151515With a dilation parameter ${\kappa}$ $=$ $0.68$ GeV.
LFQM |  $215{\pm}5$ jpg.39.025005  |  $211{\pm}1$ PhysRevD.98.114018  |  $242^{+23}_{-24}$ PhysRevD.100.014026  |  |
LQCD |  $239.4{\pm}7.3$ PhysRevD.65.054505  |  $210{\pm}15$ PTP.119.599  |  $239{\pm}18$ PhysRevD.80.054510  |  $199{\pm}4$ JHEP.1704.082  |  $208.5{\pm}5.6$ cpc.42.063102
other |  $490$ plb.635.93  |  $207$ PhysRevC.60.055214  |  $254$ pan.79.444  |  |

The decay constant $f_{\rho}$ is an very important characteristics of the
${\rho}$ meson. Compared with the CKM element ${|}V_{ud}{|}$, the present
precision of decay constant $f_{\rho}$ is still not very high and needs to be
improved. Theoretically, the estimations from different methods are more or
less different from each other and even calculations with the same method
sometimes give the diverse results. Some theoretical estimations on the decay
constant $f_{\rho}$ are presented in Table 1. Experimentally, the decay
constant $f_{\rho}$ can be obtained from the 1-prong hadronic ${\tau}^{\pm}$
${\to}$ ${\rho}^{\pm}{\nu}_{\tau}$ decay. The partial width for the ${\tau}$
${\to}$ $V{\nu}_{\tau}$ decay is given by Ref. JHEP.1504.101 ,

${\Gamma}({\tau}{\to}V{\nu})\,=\,{\cal
S}\,\frac{G_{F}^{2}\,m_{\tau}^{3}}{16\,{\pi}}\,{|}V_{q_{1}q_{2}}{|}^{2}\,f_{V}^{2}\,\Big{(}1-\frac{m_{V}^{2}}{m_{\tau}^{2}}\Big{)}^{2}\Big{(}1+\frac{2\,m_{V}^{2}}{m_{\tau}^{2}}\Big{)},$
(9)

where the factor ${\cal S}$ $=$ $1.0154$ includes the electroweak corrections
JHEP.1504.101 ; PhysRevLett.61.1815 ; PhysRevD.42.3888 . With the mass
$m_{\tau}$ $=$ $1776.86(12)$ MeV and lifetime ${\tau}_{\tau}$ $=$ $290.3(5)$
fs pdg2020 , and branching ratio ${\cal B}({\tau}{\to}{\rho}{\nu})$ $=$
$25.19(33)$ % pdg2020 , one can easily extract the decay constant
$f_{\rho}^{\rm exp}$ $=$ $207.7{\pm}1.6$ MeV, which agrees well with the
latest numerical simulation result from lattice QCD $f_{\rho}$ $=$
$208.5{\pm}5.5{\pm}0.9$ MeV cpc.42.063102 . The more accurate decay constant
$f_{\rho}^{\rm exp}$ will be used in our calculation.

For the ${\rho}^{-}$ ${\to}$ ${\ell}^{-}\bar{\nu}_{\ell}$ decays, one can
obtain the PLDCV partial decay widths with Eq.(8) and the corresponding
branching ratios with the full width ${\Gamma}_{\rho}$ $=$ $149.1{\pm}0.8$ MeV
pdg2020 ,

$\displaystyle{\Gamma}({\rho}^{-}{\to}e^{-}\bar{\nu}_{e})$ $\displaystyle=$
$\displaystyle 68.8{\pm}1.2\,{\mu}\,{\rm eV},$ (10)
$\displaystyle{\Gamma}({\rho}^{-}{\to}{\mu}^{-}\bar{\nu}_{\mu})$
$\displaystyle=$ $\displaystyle 66.9{\pm}1.1\,{\mu}\,{\rm eV},$ (11)
$\displaystyle{\cal B}({\rho}^{-}{\to}e^{-}\bar{\nu}_{e})$ $\displaystyle=$
$\displaystyle(4.6{\pm}0.1){\times}10^{-13},$ (12) $\displaystyle{\cal
B}({\rho}^{-}{\to}{\mu}^{-}\bar{\nu}_{\mu})$ $\displaystyle=$
$\displaystyle(4.5{\pm}0.1){\times}10^{-13},$ (13)

where the uncertainties come from the uncertainties of mass $m_{\rho}$, decay
constant $f_{\rho}$ and CKM element ${|}V_{ud}{|}$, and additional decay width
${\Gamma}_{\rho}$ for branching ratios. Clearly, the branching ratios are very
small. Given the identification efficiency and pollution from background, the
${\rho}^{-}$ ${\to}$ ${\ell}^{-}\bar{\nu}_{\ell}$ decays might be measured
only with more than $10^{14}$ ${\rho}^{\pm}$ events available.

There are at least three possible ways to experimentally produce the charged
${\rho}$ mesons in the electron-position collisions, (a) the prompt pair
production $e^{+}e^{-}$ ${\to}$ ${\rho}^{+}{\rho}^{-}$, (b) the pair
production via $V$ decay $1^{--}$ ${\to}$ ${\rho}^{+}{\rho}^{-}$, and (c) the
single production via $V$ decay $1^{--}$ ${\to}$ ${\rho}^{\pm}h^{\mp}$. The
cross section ${\sigma}(e^{+}e^{-}{\to}{\rho}^{+}{\rho}^{-})$ has been
determined by the BaBar group to be $19.5{\pm}1.6{\pm}3.2$ fb near the center-
of-mass energy $\sqrt{s}$ $=$ $10.58$ GeV PhysRevD.78.071103 . Assuming the
production cross section ${\sigma}$ ${\propto}$ $1/s$ PhysRevC.75.065202 ;
plb.763.87 , it could be speculated that
${\sigma}(e^{+}e^{-}{\to}{\rho}^{+}{\rho}^{-})$ ${\sim}$ $230$ fb near
$\sqrt{s}$ $=$ $3.1$ GeV. There would be only about $10^{6}$
${\rho}^{+}{\rho}^{-}$ pairs with a data sample of $50$ ${\rm ab}^{-1}$
PTEP.2019.123C01  near $\sqrt{s}$ ${\approx}$ $m_{\Upsilon(4S)}$ at the Belle-
II detector or a data sample of $10$ ${\rm ab}^{-1}$ epjconf.212.01010  near
$\sqrt{s}$ ${\approx}$ $m_{J/{\psi}}$ with the future super-tau-charm factory
like STCF or SCTF PhysRevLett.127.012003 ; STCF ; SCTF . The charge ${\rho}$
mesons can in principle be produced from the ${\Upsilon}(4S)$, $J/{\psi}$ and
${\phi}$ decays. The branching ratios are

$\displaystyle{\cal B}({\Upsilon}(4S){\to}{\rho}^{+}{\rho}^{-})$
$\displaystyle<$ $\displaystyle 5.7{\times}10^{-6}\
\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{PhysRevD.78.071103}{\@@citephrase{(}}{\@@citephrase{)}}}},$
(14) $\displaystyle{\cal B}(J/{\psi}{\to}{\rho}^{+}{\rho}^{-})$
$\displaystyle{\sim}$ $\displaystyle 10^{-3}\
\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{PhysRevD.32.2961,pdg2020}{\@@citephrase{(}}{\@@citephrase{)}}}},$
(15) $\displaystyle{\cal B}(J/{\psi}{\to}{\rho}^{\pm}{\pi}^{\mp})$
$\displaystyle{\sim}$ $\displaystyle 10^{-2}\
\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{pdg2020}{\@@citephrase{(}}{\@@citephrase{)}}}},$ (16)
$\displaystyle{\cal B}({\phi}{\to}{\rho}^{\pm}{\pi}^{\mp})$
$\displaystyle{\sim}$ $\displaystyle 10^{-1}\
\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{pdg2020}{\@@citephrase{(}}{\@@citephrase{)}}}},$ (17)

where the brancing ratio ${\cal B}(J/{\psi}{\to}{\rho}^{+}{\rho}^{-})$ is
assumed to be the same order of magnitude as ${\cal
B}(J/{\psi}{\to}K^{{\ast}+}K^{{\ast}-})$ ${\sim}$ $10^{-3}$ pdg2020  from the
phenomenological analysis based on the flavor-$SU(3)$ symmetry
PhysRevD.32.2961 . Now, there are $7.7{\times}10^{8}$ ${\Upsilon}(4S)$ events
at Belle epjc74.3026 , $10^{10}$ $J/{\psi}$ events PhysRevLett.127.012003  at
BES-III, and $2.4{\times}10^{10}$ ${\phi}$ events at KLOE/KLOE-2 KLOE2.2020
available. It is expected that only about $5{\times}10^{10}$ ${\Upsilon}(4S)$
events PTEP.2019.123C01  and $10^{13}$ $J/{\psi}$ events at SCTF or STCF
PhysRevLett.127.012003  could be accumulated. It is clearly seen that unless a
very significant enhancement to branching ratios from some NP, the
experimental data on the ${\rho}^{\pm}$ meson are too scarce to search for the
${\rho}^{-}$ ${\to}$ ${\ell}^{-}\bar{\nu}_{\ell}$ decays at the electron-
position collisions in the near future, which result in the natural
difficulties to understand the ${\rho}$ meson.

The production cross sections of prompt $J/{\psi}$ and $J/{\psi}$-from-b
mesons in proton–proton collisions at $\sqrt{s}$ $=$ $13$ TeV are measured by
LHCb to be $15.0{\pm}0.6{\pm}0.7$ ${\mu}$b and $2.25{\pm}0.09{\pm}0.10$
${\mu}$b, respectively, JHEP.2015.10.172 . It is expected that some $10^{12}$
$J/{\psi}$ events could be accumulated at $\sqrt{s}$ $=$ $13$ TeV with an
integrated luminosity of $300$ ${\rm fb}^{-1}$ at LHCb 1808.08865 . There are
only about $10^{10}$ ${\rho}^{\pm}$ mesons from $J/{\psi}$ decays available
for prying into the ${\rho}^{\pm}$ PLDCV decays. At the same time, the
inclusive cross-sections for prompt charm production at LHCb at $\sqrt{s}$ $=$
$13$ TeV are measured to be ${\cal O}(1\,{\rm mb})$ JHEP.2016.03.159 .
Analogically assuming the inclusive cross section of prompt ${\rho}^{\pm}$
meson production at LHCb at $\sqrt{s}$ $=$ $13$ TeV is ${\cal O}(10\,{\rm
mb})$, some $3{\times}10^{15}$ ${\rho}^{\pm}$ events would be accumulated with
an integrated luminosity of $300$ ${\rm fb}^{-1}$ at LHCb 1808.08865 .
Optimistically assuming the reconstruction efficiency is about $10\,\%$, there
would be about ${\cal O}(10^{2})$ events of the ${\rho}^{-}$ ${\to}$
${\ell}^{-}\bar{\nu}_{\ell}$ decays at LHCb, and more events with the enhanced
branching ratios from NP contributions. Even through it will be very
challenging for experimental analysis due to the complex background in hadron-
hadron collisions, there is still a strong presumption that the ${\rho}^{-}$
${\to}$ ${\ell}^{-}\bar{\nu}_{\ell}$ decays could be explored and studied at
LHC in the future. In addition, it is expected that an integrated luminosity
exceeding $10$ ${\rm ab}^{-1}$ would be reached at the future HE-LHC
experiments epjst.228.1109 . More experimental data at HE-LHC would make the
study of the ${\rho}^{-}$ ${\to}$ ${\ell}^{-}\bar{\nu}_{\ell}$ decays indeed
feasible in hadron-hadron collisions.

##  III $K^{{\ast}-}$ ${\to}$ ${\ell}^{-}\bar{\nu}_{\ell}$ decays

The parameter product ${|}V_{us}{|}\,f_{K^{\ast}}$ could be experimentally
determined from the $K^{{\ast}-}$ ${\to}$ ${\ell}^{-}\bar{\nu}_{\ell}$ decays
using Eq.(8). Like the ${\rho}^{\pm}$ meson, the mass of the $K^{{\ast}{\pm}}$
meson, $m_{K^{{\ast}{\pm}}}$ $=$ $895.5(8)$ MeV, is above the threshold of
$K{\pi}$ pair, and the partial branching ratio of the $K^{\ast}$ meson decay
into $K{\pi}$ pair via the strong interactions is almost 100% pdg2020 . It is
not hard to imagine that the very short lifetime ${\tau}_{K^{\ast}}$ ${\sim}$
$1.4{\times}10^{-23}$ s would enable the measurements of the electroweak
properties of the $K^{\ast}$ meson to be very challenging or nearly
impossible.

The CKM element ${|}V_{us}{|}$ ${\simeq}$ ${\lambda}$ up to the order of
${\cal O}({\lambda}^{6})$, where ${\lambda}$ is a Wolfenstein parameter. The
current precision of the CKM element ${|}V_{us}{|}$ from purely leptonic and
semileptonic $K$ meson decays and hadronic ${\tau}$ decays are 0.2%, 0.3% and
0.6%, respectively. It is seen from Fig. 2 that these three results for
${|}V_{us}{|}$ are not very consistent with one another. So if the
$K^{{\ast}-}$ ${\to}$ ${\ell}^{-}\bar{\nu}_{\ell}$ decays could be measured,
they would provide another determination and constraint to ${|}V_{us}{|}$.
Probably due to the reduction of the value of ${|}V_{ud}{|}$, the latest value
from the global fit in SM, ${|}V_{us}{|}$ $=$ $0.22650(48)$ pdg2020 , is
slightly larger than the 2018 value, to satisfy the first row unitarity
requirement.

Figure 2: The values of the CKM element ${|}V_{us}{|}$ from PDG. Table 2: The theoretical values of decay constant $f_{K^{\ast}}$ (in the unit of MeV), where the legends are the same as those in Table 1. RQM |  $170.5$ epja.24.411  111With Gaussian spatial wave functions and adjusted parameters of Ref. epja.24.411 . |  $153.1$ epja.24.411  222With Gaussian spatial wave functions and parameters of Ref. plb.602.212 . |  $192.6$ epja.24.411  333With rational spatial wave functions and adjusted parameters of Ref. epja.24.411 . |  $177.6$ epja.24.411  444With rational spatial wave functions and parameters of Ref. plb.602.212 . |  $236$ plb.635.93
---|---|---|---|---|---
LFQM |  $256$ PhysRevD.75.034019  555With Coulomb plus linear potential model. |  $223$ PhysRevD.75.034019  666With Coulomb plus harmonic oscillator potential model. |  $224$ PhysRevC.92.055203  555With Coulomb plus linear potential model. |  $186^{+2}_{-3}$ cpc.42.073102  777With a dilation parameter ${\kappa}$ $=$ $0.54$ GeV. |  $204^{+7}_{-9}$ cpc.42.073102  888With a dilation parameter ${\kappa}$ $=$ $0.68$ GeV.
LFQM |  $217{\pm}5$ jpg.39.025005  |  $223{\pm}1$ PhysRevD.98.114018  |  $253{\pm}25$ PhysRevD.100.014026  |  |
LQCD |  $255.5{\pm}6.5$ PhysRevD.65.054505  999The $K$ meson mass is used as input. |  $257.7{\pm}6.4$ PhysRevD.65.054505  101010The ${\phi}$ meson mass is used as input. |  $240{\pm}18$ PhysRevD.84.014505  |  |
other |  $226{\pm}28$ PhysRevD.58.094016  |  $508$ plb.635.93  |  $241$ PhysRevC.60.055214  |  |

Some theoretical results on the decay constant $f_{K^{\ast}}$ are presented in
Table 2. Like the case of the decay constant $f_{\rho}$, the model dependence
of theoretical estimations on the decay constant $f_{K^{\ast}}$ is also
obvious. Experimentally, the decay constant $f_{K^{\ast}}$ can be obtained
from the hadronic ${\tau}^{\pm}$ ${\to}$ $K^{{\ast}{\pm}}{\nu}_{\tau}$ decays.
Using Eq.(9) and experimental data on branching ratio ${\cal
B}({\tau}{\to}K^{\ast}{\nu})$ $=$ $1.20(7)$ % pdg2020 , one can obtain the
decay constant $f_{K^{\ast}}^{\rm exp}$ $=$ $202.5^{+6.5}_{-6.7}$ MeV. The
value of $f_{K^{\ast}}^{\rm exp}$ is much less than that of LQCD results, and
will be used in our calculation.

For the $K^{{\ast}-}$ ${\to}$ ${\ell}^{-}\bar{\nu}_{\ell}$ decays, the SM
expectations on the partial decay widths and branching ratios are,

$\displaystyle{\Gamma}(K^{{\ast}-}{\to}e^{-}\bar{\nu}_{e})$ $\displaystyle=$
$\displaystyle 5.5{\pm}0.4\,{\mu}\,{\rm eV},$ (18)
$\displaystyle{\Gamma}(K^{{\ast}-}{\to}{\mu}^{-}\bar{\nu}_{\mu})$
$\displaystyle=$ $\displaystyle 5.3{\pm}0.4\,{\mu}\,{\rm eV},$ (19)
$\displaystyle{\cal B}(K^{{\ast}-}{\to}e^{-}\bar{\nu}_{e})$ $\displaystyle=$
$\displaystyle(1.2{\pm}0.1){\times}10^{-13},$ (20) $\displaystyle{\cal
B}(K^{{\ast}-}{\to}{\mu}^{-}\bar{\nu}_{\mu})$ $\displaystyle=$
$\displaystyle(1.2{\pm}0.1){\times}10^{-13}.$ (21)

The decay width ${\Gamma}_{K^{\ast}}$ $=$ $46.2{\pm}1.3$ MeV pdg2020  is used
in our calculation. It is apparent that more than $10^{14}$ $K^{{\ast}{\pm}}$
events are the minimum requirement for experimentally studying the
$K^{{\ast}-}$ ${\to}$ ${\ell}^{-}\bar{\nu}_{\ell}$ decays.

Based on the $U$-spin symmetry, the production mechanism of the
$K^{{\ast}{\pm}}$ mesons in electron-position collisions is similar to that of
the ${\rho}^{\pm}$ mesons. An educated guess is that the cross section
${\sigma}(e^{+}e^{-}{\to}K^{{\ast}+}K^{{\ast}-})$ ${\sim}$ $20$ fb and $230$
fb near $\sqrt{s}$ ${\sim}$ $m_{{\Upsilon}(4S)}$ and $m_{J/{\psi}}$,
respectively. The branching ratios of $J/{\psi}$ decays are pdg2020 ,

$\displaystyle{\cal B}(J/{\psi}{\to}K^{{\ast}+}K^{{\ast}-})$ $\displaystyle=$
$\displaystyle(1.00^{+0.22}_{-0.40}){\times}10^{-3},$ (22) $\displaystyle{\cal
B}(J/{\psi}{\to}K^{{\ast}{\pm}}K^{\mp})$ $\displaystyle=$
$\displaystyle(6.0^{+0.8}_{-1.0}){\times}10^{-3},$ (23) $\displaystyle{\cal
B}(J/{\psi}{\to}K^{{\ast}{\pm}}K^{\mp}{\pi}^{0})$ $\displaystyle=$
$\displaystyle(4.1{\pm}1.3){\times}10^{-3},$ (24) $\displaystyle{\cal
B}(J/{\psi}{\to}K^{{\ast}{\pm}}K_{S}^{0}{\pi}^{\mp})$ $\displaystyle=$
$\displaystyle(2.0{\pm}0.5){\times}10^{-3},$ (25) $\displaystyle{\cal
B}(J/{\psi}{\to}K^{{\ast}{\pm}}K_{2}^{\ast}(1430)^{\mp})$ $\displaystyle=$
$\displaystyle(3.4{\pm}2.9){\times}10^{-3},$ (26) $\displaystyle{\cal
B}(J/{\psi}{\to}K^{{\ast}{\pm}}K^{\ast}(700)^{\mp})$ $\displaystyle=$
$\displaystyle(1.0^{+1.0}_{-0.6}){\times}10^{-3}.$ (27)

It is approximately estimated that ${\cal
B}(J/{\psi}{\to}K^{{\ast}{\pm}}X^{\mp})$ ${\sim}$ $1.8$ %. Hence, the
experiemtal data on the $K^{{\ast}{\pm}}$ mesons at the $e^{+}e^{-}$
collisions, which would be available by either the prompt
$K^{{\ast}+}K^{{\ast}-}$ pair production at SuperKEKB and SCTF experiments or
the production via $10^{13}$ $J/{\psi}$ decay at SCTF, are far from sufficient
for investigating the $K^{{\ast}-}$ ${\to}$ ${\ell}^{-}\bar{\nu}_{\ell}$
decays. If we assume that the inclusive cross section of prompt
$K^{{\ast}{\pm}}$ meson production in $pp$ collisions at the center-of-mass
energy of $13$ TeV is similar to that of ${\rho}^{\pm}$ mesons, about ${\cal
O}(10\,{\rm mb})$, there would be some $3{\times}10^{15}$ $K^{{\ast}{\pm}}$
events to be available with an integrated luminosity of $300$ ${\rm fb}^{-1}$
at LHCb, which correspond to about ${\cal O}(10^{2})$ events of the
$K^{{\ast}-}$ ${\to}$ ${\ell}^{-}\bar{\nu}_{\ell}$ decays. It should be some
glimmer of hope for observation and scrutinies of the $K^{{\ast}-}$ ${\to}$
${\ell}^{-}\bar{\nu}_{\ell}$ decays at hadron-hadron collisions in the future,
particularly at the planning HE-HLC.

##  IV $D_{d}^{{\ast}-}$ ${\to}$ ${\ell}^{-}\bar{\nu}_{\ell}$ decays

The mass of $D_{d}^{\ast}$ mesons, $m_{D_{d}^{\ast}}$ $=$ $2010.26(5)$ MeV,
are just above the threshold of $D{\pi}$ pair. The $D_{d}^{\ast}$ meson decays
via the strong interactions are dominant, and the ratio of branching ratios
pdg2020 , ${\cal B}(D_{d}^{{\ast}{\pm}}{\to}D_{d}^{\pm}{\pi}^{0})/{\cal
B}(D_{d}^{{\ast}{\pm}}{\to}D_{u}{\pi}^{\pm})$ $=$ $30.7(5)\,\%/67.7(5)\,\%$
${\sim}$ $1/2$, basically agrees with the relations of isospin symmetry. It
should be pointed out that the $D_{d}^{\ast}$ strong decays are highly
suppressed by the compact phase spaces because of $m_{D_{d}^{\ast}}$ $-$
$m_{D}$ $-$ $m_{\pi}$ $<$ 6 MeV. The branching ratio of the magnetic dipole
transition is small, ${\cal B}(D_{d}^{\ast}{\to}D_{d}{\gamma})$ $=$
$1.6(4)\,\%$ pdg2020 . Hence, the decay width of $D_{d}^{\ast}$ mesons is
narrow, ${\Gamma}_{D_{d}^{\ast}}$ $=$ $83.4{\pm}1.8$ keV pdg2020 . From the
$D_{d}^{{\ast}-}$ ${\to}$ ${\ell}^{-}\bar{\nu}_{\ell}$ decays, the parameter
${|}V_{cd}{|}\,f_{D_{d}^{\ast}}$ is expected to be experimentally determined.

Figure 3: The values of the CKM element ${|}V_{cd}{|}$ from PDG. Table 3: The theoretical values of decay constant $f_{D_{d}^{\ast}}$ (in the unit of MeV), where the legends are the same as those in Table 1, NRQM is an abbreviation for nonrelativistic quark model. NRQM |  $223^{+23}_{-19}$ PhysRevD.71.113006  |  $307$ epl.115.21002  111Without QCD radiative corrections. |  $253$ epl.115.21002  222With QCD radiative corrections. |  $353.8$ epjp.132.80  111Without QCD radiative corrections. |  $290.3$ epjp.132.80  222With QCD radiative corrections.
---|---|---|---|---|---
NRQM |  $391$ plb.635.93  |  $290$ ahep.2018.7032041  111Without QCD radiative corrections. |  $210$ ahep.2018.7032041  222With QCD radiative corrections. |  $332$ mpla.17.803  |
RQM |  $310$ plb.635.93  |  $315$ mpla.17.803  |  $327{\pm}13$ PhysRevD.55.6944  333With constituent quark masses for the light quarks $m_{s}$ and $m_{d}$. |  $252{\pm}10$ PhysRevD.55.6944  444With current quark masses for the light quarks $m_{s}$ and $m_{d}$. |
LFQM |  $254$ PhysRevD.75.073016  555With Coulomb plus linear potential model. |  $228$ PhysRevD.75.073016  666With Coulomb plus harmonic oscillator potential model. |  $259.6{\pm}14.6$ PhysRevD.81.114024  777With the Gaussian type wave functions. |  $306.3^{+18.2}_{-17.7}$ PhysRevD.81.114024  888With the power-law type wave functions. |  $253{\pm}7$ PhysRevD.98.114018
LFQM |  $230$ PhysRevC.92.055203  999With Coulomb plus linear potential model. |  $226.6^{+~{}5.9}_{-10.2}$ cpc.42.073102  101010With a dilation parameter ${\kappa}$ $=$ $0.54$ GeV. |  $230.1{\pm}6.2$ cpc.42.073102  111111With a dilation parameter ${\kappa}$ $=$ $0.68$ GeV. |  $245^{+35}_{-34}$ jpg.39.025005  |  $230{\pm}29$ PhysRevD.100.014026
LFQM |  $252.0^{+13.8}_{-11.6}$ epjc.76.313  777With the Gaussian type wave functions. |  $264.9^{+10.2}_{-~{}9.5}$ epjc.76.313  888With the power-law type wave functions. |  $272$ epjp.133.134  121212With Martin potential model plb.93.338 . |  $260$ epjp.133.134  131313With Cornell potential model zpc.33.135 . |  $269$ epjp.133.134  141414With logarithmic potential model plb.71.153 .
LQCD |  $245{\pm}20$ PhysRevD.60.074501  |  $234{\pm}26$ npb.619.507  151515With $m_{D^{\ast}}/f_{D^{\ast}}$ $=$ $8.6{\pm}0.3^{+0.5}_{-0.9}$ npb.619.507  and $m_{D_{d}^{\ast}}$ $=$ $2010.26(5)$ MeV pdg2020 . |  $278{\pm}16$ jhep.1202.042  |  $223.5{\pm}8.7$ PhysRevD.96.034524  |  $234(6)$ cpc.45.023109
SR |  $242^{+20}_{-12}$ PhysRevD.88.014015  |  $263{\pm}21$ epjc.75.427  |  $252.2{\pm}22.7$ plb.735.12  |  $250{\pm}11$ ijmpa.30.1550116  |
other |  $186$ epjp.133.134  161616With harmonic plus Yukawa potential model epjp.132.80 . |  $273{\pm}13$ PhysRevD.75.116001  |  $341{\pm}23$ plb.633.492  |  $237$ PhysRevD.58.014007  |

Currently, the precise values of the CKM element ${|}V_{cd}{|}$ comes mainly
from the leptonic and semileptonic $D$ meson decays pdg2020 , as illustrated
in Fig. 3. Because of the decay width of Eq.(7) being proportional to
$m_{\ell}^{2}$, the $D^{-}$ ${\to}$ $e^{-}\bar{\nu}_{e}$ decay is helicity
suppressed. And the $D^{-}$ ${\to}$ ${\tau}^{-}\bar{\nu}_{\tau}$ decay suffers
from the complications caused by the additional neutrino in ${\tau}$ decays.
The $D^{-}$ ${\to}$ ${\mu}^{-}\bar{\nu}_{\mu}$ decay is the most favorable
mode for experimental measurement. For the values of ${|}V_{cd}{|}$ from the
purely leptonic decay $D^{-}$ ${\to}$ ${\mu}^{-}\bar{\nu}$, the experimentally
statistical uncertainties are dominant uncertainties. For the values of
${|}V_{cd}{|}$ from the semileptonic $D$ meson decays, the theoretical
uncertainties from the form factor controlled by nonperturbative dynamics are
dominant uncertainties. It is clearly seen from Fig. 3 that the experimental
uncertainties have not decreased significantly recently. Besides,
${|}V_{cd}{|}$ can also be determined from the neutrino-induced charm
production data pdg2020 , but the relevant experimental data have not been
updated after the measurements given by the CHARM-II Collaboration in 1999
epjc.11.19 . According to the Wolfenstein parameterization of the CKM matrix,
there is an approximate relation between its elements ${|}V_{cd}{|}$ $=$
${|}V_{us}{|}$ $=$ ${\lambda}$ up to ${\cal O}({\lambda}^{4})$. However, the
measurement precision of the CKM element ${|}V_{cd}{|}$ from both leptonic and
semileptonic $D$ meson decays is generally about an order of magnitude smaller
than that of ${|}V_{us}{|}$ from leptonic and semileptonic $K$ meson decays
for the moment. The most precise values are from the global fit in SM,
${|}V_{cd}{|}$ $=$ $0.22636(48)$ pdg2020  with uncertainties ${\sim}$
$0.2\,\%$.

The information about the decay constant $f_{D_{d}^{\ast}}$ has not yet been
obtained experimentally by now. Some theoretical results on $f_{D_{d}^{\ast}}$
are listed in Table 3. The theoretical discrepancies among various methods are
obvious. In our calculation, as a conservative estimate, we will take the
recent value $f_{D_{d}^{\ast}}$ $=$ $230{\pm}29$ MeV PhysRevD.100.014026  from
the light front quark model, which agrees basically with the values
$f_{D_{d}^{\ast}}$ $=$ $234{\pm}6$ MeV cpc.45.023109  from the recent lattice
QCD simulation.

After some simple computation with Eq.(8), we obtain the partial decay widths
and branching ratios for the $D_{d}^{{\ast}-}$ ${\to}$
${\ell}^{-}\bar{\nu}_{\ell}$ decays as follows.

$\displaystyle{\Gamma}(D_{d}^{{\ast}-}{\to}{\ell}^{-}\bar{\nu}_{\ell})$
$\displaystyle=$ $\displaystyle 79^{+22}_{-19}\,{\mu}\,{\rm
eV},\quad\text{for}\ {\ell}\,=\,e,\,{\mu},$ (28)
$\displaystyle{\Gamma}(D_{d}^{{\ast}-}{\to}{\tau}^{-}\bar{\nu}_{\tau})$
$\displaystyle=$ $\displaystyle 5{\pm}1\,{\mu}\,{\rm eV},$ (29)
$\displaystyle{\cal B}(D_{d}^{{\ast}-}{\to}{\ell}^{-}\bar{\nu}_{\ell})$
$\displaystyle=$
$\displaystyle(9.5^{+2.9}_{-2.4}){\times}10^{-10},\quad\text{for}\
{\ell}\,=\,e,\,{\mu},$ (30) $\displaystyle{\cal
B}(D_{d}^{{\ast}-}{\to}{\tau}^{-}\bar{\nu}_{\tau})$ $\displaystyle=$
$\displaystyle(0.6{\pm}0.2){\times}10^{-10}.$ (31)

These branching ratios are consistent with those of Ref. ctp.67.655  if the
different values of decay constants $f_{D_{d}^{\ast}}$ are considered. The
relatively large uncertainties of branching ratios come from the uncertainties
of mass $m_{D_{d}^{\ast}}$, width ${\Gamma}_{D_{d}^{\ast}}$, decay constant
$f_{D_{d}^{\ast}}$ and the CKM element ${|}V_{cd}{|}$. To experimentally study
the $D_{d}^{{\ast}-}$ ${\to}$ ${\ell}^{-}\bar{\nu}_{\ell}$ decays, more than
$10^{11}$ $D_{d}^{\ast}$ events are needed. Due to the short lifetime of
lepton ${\tau}^{\pm}$ and the lepton number conservation in ${\tau}^{\pm}$
decays, additional neutrinos will make the measurement of the
$D_{d}^{{\ast}-}$ ${\to}$ ${\tau}^{-}\bar{\nu}_{\tau}$ decay to have a poor
reconstruction efficiency and to be very challenging. Perhaps some $10^{12}$
or more $D_{d}^{\ast}$ events are necessarily required to study the
$D_{d}^{{\ast}-}$ ${\to}$ ${\tau}^{-}\bar{\nu}_{\tau}$ decay.

Figure 4: The exclusive cross sections (in the unit of nb) as functions of
$\sqrt{s}$ (in the unit of GeV) for $e^{+}e^{-}$ ${\to}$ $D^{+}D^{{\ast}-}$ in
(a) and $e^{+}e^{-}$ ${\to}$ $D^{{\ast}+}D^{{\ast}-}$ in (b). The Belle, BaBar
and CLEO-c data are from Ref. PhysRevD.97.012002 , Ref. PhysRevD.79.092001
and Ref. cpc.42.043002 , respectively.

Above the open charm production threshold, there are several charmonium
resonances and charmonium-like structures decaying predominantly into pairs of
charmed meson final states. The studies of Belle PhysRevD.97.012002 , BaBar
PhysRevD.79.092001  and CLEO-c PhysRevD.80.072001 ; cpc.42.043002
collaborations have shown that there is a sharply peaked
$D_{d}^{+}D_{d}^{{\ast}-}$ structure and a broad
$D_{d}^{{\ast}+}D_{d}^{{\ast}-}$ plateau just above threshold, as illustrated
in Fig. 4. Assuming the exclusive cross sections near threshold
${\sigma}(e^{+}e^{-}{\to}D_{d}^{+}D_{d}^{{\ast}-})$ ${\sim}$ $4\,{\rm nb}$ and
${\sigma}(e^{+}e^{-}{\to}D_{d}^{{\ast}+}D_{d}^{{\ast}-})$ ${\sim}$ $3\,{\rm
nb}$, there will be about $10^{11}$ $D_{d}^{{\ast}{\pm}}$ events corresponding
to the total integrated luminosity of $10$ ${\rm ab}^{-1}$ at future STCF, and
about $5{\times}10^{11}$ $D_{d}^{{\ast}{\pm}}$ events corresponding to a data
sample of $50$ ${\rm ab}^{-1}$ at SuperKEKB. In addition, about $10^{12}$ $Z$
bosons will be produced on the on the schedule of the large international
scientific project of Circular Electron Positron Collider (CEPC) cepc  and
$10^{13}$ $Z$ bosons at Future Circular $e^{+}e^{-}$ Collider (FCC-ee) fcc .
Considering the branching ratio ${\cal B}(Z{\to}D^{{\ast}{\pm}}X)$ $=$
$(11.4{\pm}1.3)\,\%$ pdg2020 , the $Z$ boson decays will yield more than
$10^{11}$ $D_{d}^{{\ast}{\pm}}$ events at the tera-Z factories. So the
$D_{d}^{{\ast}-}$ ${\to}$ ${\ell}^{-}\bar{\nu}_{\ell}$ decays could be
investigated at Belle-II, SCTF, CEPC and FCC-ee experiments.

In hadron-hadron collisions, the inclusive cross sections for the $c\bar{c}$
pair and $D^{{\ast}+}$ meson production are measured to be
${\sigma}(pp{\to}c\bar{c}X)$ $=$ $2369{\pm}3{\pm}152{\pm}118$ ${\mu}$b and
${\sigma}(pp{\to}D^{{\ast}+}X)$ $=$ $784{\pm}4{\pm}87{\pm}118$ ${\mu}$b at the
center-of-mass energy of $\sqrt{s}$ $=$ $13$ TeV by the LHCb group, with the
transverse momentum $p_{T}$ within the range of $1\,{\rm GeV}$ $<$ $p_{T}$ $<$
$8\,{\rm GeV}$ JHEP.2016.03.159 . Some $2{\times}10^{14}$
$D_{d}^{{\ast}{\pm}}$ events could be accumulated with the integrated
luminosity $300\,{\rm fb}^{-1}$ at LHCb. The total cross sections of charm and
$D^{{\ast}+}$ production measured at $\sqrt{s}$ $=$ $7$ TeV by the ALICE group
are ${\sigma}_{c\bar{c}}^{\rm tot}$ ${\simeq}$ $8.5$ mb and
${\sigma}_{D^{{\ast}+}}^{\rm tot}$ ${\simeq}$ $2.11$ mb, respectively
JHEP.2012.07.191 . The total cross sections of charm production measured at
$\sqrt{s}$ $=$ $7$ TeV by the ATLAS group are ${\sigma}_{c\bar{c}}^{\rm tot}$
${\simeq}$ $8.6$ mb npb.907.717 . The $D^{{\ast}+}$ production cross section
at ATLAS should be very close to that at ALICE based on an educated guess. In
addition, the $D^{\ast}$ meson can also produced from $b$ decays with the
fragmentation fraction about $f(b{\to}D_{d}^{\ast})$ ${\simeq}$ $23\,\%$
epjc.75.19 . The $b$-quark production cross sections at $\sqrt{s}$ $=$ 13 TeV
determined by LHCb and ALICE are about ${\sigma}(pp{\to}b\bar{b}X)$ ${\simeq}$
$495$ ${\mu}$b JHEP.2015.10.172  and $541$ ${\mu}$b 2108.02523 , respectively.
So more than $10^{13}$ $D_{d}^{{\ast}{\pm}}$ events from $b$ decays could be
accumulated with the integrated luminosity $300\,{\rm fb}^{-1}$ at LHCb. All
in all, the large cross section of $D^{\ast}$ meson plus the high luminosity
at hadron-hadron collisions result in the abundant $D_{d}^{{\ast}{\pm}}$
events, and make the carefully experimental study of the $D_{d}^{{\ast}-}$
${\to}$ $e^{-}\bar{\nu}_{e}$, ${\mu}^{-}\bar{\nu}_{\mu}$ decays, even the
$D_{d}^{{\ast}-}$ ${\to}$ ${\tau}^{-}\bar{\nu}_{\tau}$ decay, to be possible
and practicable.

##  V $D_{s}^{{\ast}-}$ ${\to}$ ${\ell}^{-}\bar{\nu}_{\ell}$ decays

The $D_{s}^{{\ast}{\pm}}$ mesons have explicitly nonzero quantum number of
electric charges, charm and strange, $Q$ $=$ $C$ $=$ $S$ $=$ ${\pm}1$.
Considering the conservation of the charm and strange quantum number in the
strong and electromagnetic interactions, and the mass of $D_{s}^{\ast}$
mesons, $m_{D_{s}^{\ast}}$ $=$ $2112.2(4)$ MeV pdg2020 , being just above the
threshold of $D_{s}{\pi}$ pair but below the threshold of $DK$ pair, the
$D_{s}^{\ast}$ ${\to}$ $D_{s}{\pi}$ decays are the only allowable hadronic
decay modes. However, the $D_{s}^{\ast}$ ${\to}$ $D_{s}{\pi}$ decays are
highly suppressed due to four factors: (1) from the dynamical view, the
$D_{s}^{\ast}$ ${\to}$ $D_{s}{\pi}$ decays are induced by the the
electromagnetic interactions rather than the strong interactions because of
the isospin non-conservation between the initial and final states, (2) from
the perspective of the conservation of angular momentum, the orbital angular
momentum of final states should be $L$ $=$ $1$, so the $D_{s}^{\ast}$ ${\to}$
$D_{s}{\pi}$ decays are induced by the contributions of the $P$-wave
amplitudes, (3) from the phenomenological view, the $D_{s}^{\ast}$ ${\to}$
$D_{s}{\pi}$ decays are suppressed by the the Okubo-Zweig-Iizuka rules zweig ;
ozi-o ; ozi-i  because the quark lines of pion disconnect from those of the
$D_{s}^{\ast}D_{s}$ system, (4) from the kinematic view, the phase spaces of
final states are very compact because of $m_{D_{s}^{\ast}}$ $-$ $m_{D_{s}}$
$-$ $m_{{\pi}^{0}}$ ${\sim}$ 9 MeV. Hence, the branching ratio for the
hadronic decay is very small ${\cal B}(D_{s}^{\ast}{\to}D_{s}{\pi})$ $=$
$5.8(7)\,\%$ pdg2020 . And the branching ratio of the electromagnetic
radiative decay is dominant, ${\cal B}(D_{s}^{\ast}{\to}D_{s}{\gamma})$ $=$
$93.5(7)\,\%$ pdg2020 . Except for the $D_{s}{\pi}$, $D_{s}{\gamma}$ and
$D_{s}e^{+}e^{-}$ final states, other decay modes of the $D_{s}^{\ast}$ mesons
have not yet been observed pdg2020 . The $D_{s}^{{\ast}-}$ ${\to}$
${\ell}^{-}\bar{\nu}_{\ell}$ weak decays are favored by the CKM element
${|}V_{cs}{|}$. The information about the ${|}V_{cs}{|}f_{D_{s}^{\ast}}$ can
be experimentally obtained from the $D_{s}^{{\ast}-}$ ${\to}$
${\ell}^{-}\bar{\nu}_{\ell}$ decays.

Figure 5: The values of the CKM element ${|}V_{cs}{|}$ from PDG. Table 4: The theoretical values of decay constant $f_{D_{s}^{\ast}}$ (in the unit of MeV), where the legends including the footnotes are the same as those in 3. NRQM |  $326^{+21}_{-17}$ PhysRevD.71.113006  |  $344$ epl.115.21002  111Without QCD radiative corrections. |  $275$ epl.115.21002  222With QCD radiative corrections. |  $382.1$ epjp.132.80  111Without QCD radiative corrections. |  $303.5$ epjp.132.80  222With QCD radiative corrections.
---|---|---|---|---|---
NRQM |  $447$ plb.635.93  |  $310$ ahep.2018.7032041  111Without QCD radiative corrections. |  $212$ ahep.2018.7032041  222With QCD radiative corrections. |  $384$ mpla.17.803  |
RQM |  $315$ plb.635.93  |  $335$ mpla.17.803  |  $362{\pm}15$ PhysRevD.55.6944  333With constituent quark masses for the light quarks $m_{s}$ and $m_{d}$. |  $288{\pm}11$ PhysRevD.55.6944  444With current quark masses for the light quarks $m_{s}$ and $m_{d}$. |  $272$ PhysRevD.96.016017
LFQM |  $290$ PhysRevD.75.073016  555With Coulomb plus linear potential model. |  $268$ PhysRevD.75.073016  666With Coulomb plus harmonic oscillator potential model. |  $338.7{\pm}29.7$ PhysRevD.81.114024  777With the Gaussian type wave functions. |  $391.0{\pm}28.9$ PhysRevD.81.114024  888With the power-law type wave functions. |  $314{\pm}6$ PhysRevD.98.114018
LFQM |  $260$ PhysRevC.92.055203  999With Coulomb plus linear potential model. |  $254.7^{+6.3}_{-6.7}$ cpc.42.073102  101010With a dilation parameter ${\kappa}$ $=$ $0.54$ GeV. |  $289.7^{+6.3}_{-4.5}$ cpc.42.073102  111111With a dilation parameter ${\kappa}$ $=$ $0.68$ GeV. |  $272^{+39}_{-38}$ jpg.39.025005  |  $253{\pm}32$ PhysRevD.100.014026
LFQM |  $318.3^{+15.3}_{-12.6}$ epjc.76.313  777With the Gaussian type wave functions. |  $330.9^{+9.9}_{-9.0}$ epjc.76.313  888With the power-law type wave functions. |  $303$ epjp.133.134  121212With Martin potential model plb.93.338 . |  $291$ epjp.133.134  131313With Cornell potential model zpc.33.135 . |  $302$ epjp.133.134  141414With logarithmic potential model plb.71.153 .
LQCD |  $272{\pm}16^{+~{}3}_{-20}$ PhysRevD.60.074501  |  $254{\pm}17$ npb.619.507  151515With $m_{D_{s}^{\ast}}/f_{D_{s}^{\ast}}$ $=$ $8.3{\pm}0.2{\pm}0.5$ npb.619.507  and $m_{D_{s}^{\ast}}$ $=$ $2112.2(4)$ MeV pdg2020 . |  $311{\pm}9$ jhep.1202.042  |  |
LQCD |  $268.8{\pm}6.6$ PhysRevD.96.034524  |  $274{\pm}6$ PhysRevLett.112.212002  |  $274{\pm}7$ cpc.45.023109  |  |
SR |  $293^{+19}_{-14}$ PhysRevD.88.014015  |  $308{\pm}21$ epjc.75.427  |  $305.5{\pm}27.3$ plb.735.12  |  $270{\pm}19$ ijmpa.30.1550116  |
other |  $240$ epjp.133.134  161616With harmonic plus Yukawa potential model epjp.132.80 . |  $307{\pm}18$ PhysRevD.75.116001  |  $375{\pm}24$ plb.633.492  |  $242$ PhysRevD.58.014007  |

The direct determinations of the CKM element ${|}V_{cs}{|}$ come mainly from
leptonic $D_{s}$ decays and semileptonic $D$ decays, as shown in Fig. 5. The
uncertainties of ${|}V_{cs}{|}$ from the $D_{s}$ leptonic decays, about
$1\,\%$, are dominated by the experimental uncertainties. The uncertainties of
${|}V_{cs}{|}$ from the $D$ semileptonic decays, about $4\,\%$, are dominated
by the theoretical calculations of the form factors. It is wroth noting that
the recent CKM element ${|}V_{cs}{|}$ determined by the BES-III group from the
$D_{s}^{+}$ ${\to}$ ${\mu}^{+}{\nu}_{\mu}$ and $D_{s}^{+}$ ${\to}$
${\tau}^{+}{\nu}_{\tau}$ decays based on available $6.32$ ${\rm fb}^{-1}$ data
is ${|}V_{cs}{|}$ $=$ $0.978{\pm}0.009{\pm}0.014$ PhysRevD.104.052009 , where
the systematic (second) uncertainties has outweighed the statistical (first)
one. This value is very close to the precise result from the global fit,
${|}V_{cs}{|}$ $=$ $0.97320(11)$ pdg2020  that will be used in this paper.

By now, a relatively little information about the properties of the
$D_{s}^{\ast}$ mesons is available. For example, the quantum number of
$J^{P}$, the decay constant $f_{D_{s}^{\ast}}$, and the width
${\Gamma}_{D_{s}^{\ast}}$ have not yet been determined or confirmed explicitly
by experiments. It is generally thought that the $J^{P}$ of the $D_{s}^{\ast}$
mesons is consistent with $1^{-}$ from decay modes PhysRevLett.75.3232 . Some
theoretical results on the decay constant $f_{D_{s}^{\ast}}$ are listed in
Table 4. It can be seen that the theoretical results are various. The recent
LQCD results on the decay constant from ETM PhysRevD.96.034524 , HPQCD
PhysRevLett.112.212002  and ${\chi}$QCD cpc.45.023109  groups are in
reasonable agreement with each other within an error range. The latest decay
constant $f_{D_{s}^{\ast}}$ $=$ $274{\pm}7$ MeV from LQCD calculation
cpc.45.023109  will be used for an estimation for PLDCV of the $D_{s}^{\ast}$
mesons in this paper. The experimental upper limit of the decay width is
${\Gamma}_{D_{s}^{\ast}}$ $<$ $1.9$ MeV at the 90 % confidence level set by
the CLEO collaboration in 1995 PhysRevLett.75.3232 . An approximate relation
for the decay width, ${\Gamma}_{D_{s}^{\ast}}$ ${\simeq}$
${\Gamma}(D_{s}^{\ast}{\to}{\gamma}D_{s})$, is often used in theoretical
calculation. The radiative transition process, $D_{s}^{\ast}$ ${\to}$
${\gamma}D_{s}$, is a parity conserving decay. The parity and angular momentum
conservation implies that the orbital angular momentum of final states $L$ $=$
$1$. There are many theoretical calculation on the decay width
${\Gamma}_{D_{s}^{\ast}}$, for example, Refs. PhysRevLett.112.212002 ;
PhysRevD.18.2537 ; PhysRevD.21.203 ; PhysRevD.31.1081 ; PhysRevD.37.2564 ;
plb.284.421 ; plb.334.169 ; PhysRevD.47.1030 ; plb.316.555 ; plb.334.175 ;
plb.336.113 ; PhysRevD.49.299 ; zpc.67.633 ; PhysRevD.52.6383 ; mpla.12.3027 ;
npa.658.249 ; npa.671.380 ; jpg.27.1519 ; PhysRevD.64.094007 ; epja.13.363 ;
plb.537.241 ; PhysRevD.68.054024 ; PhysRevD.72.094004 ; ijmpa.25.2063 ;
epjc.75.243 ; epja.52.90 ; ijmpa.31.1650109 ; PhysRevD.94.113011 ;
PhysRevD.100.016019 ; PhysRevD.101.054019 ; jhep.2020.04.023 ; 2106.13617 .
The partial decay width for the magnetic dipole transition is generally
written as fayyazuddin ,

${\Gamma}(V{\to}P{\gamma})\,=\,\frac{4}{3}\,{\alpha}_{\rm
em}\,k_{\gamma}^{3}\,{\mu}_{VP}^{2},$ (32)

with the definition of the magnetic dipole moment ${\mu}_{VP}$ and the
momentum of photon $k_{\gamma}$ in the rest frame of the vector meson,

${\mu}_{VP}\,=\,{\langle}P{|}\hat{\mu}_{z}{|}V{\rangle}\,=\,{\langle}P{|}\sum\limits_{i}\frac{Q_{i}}{2\,m_{i}}\,\hat{\sigma}_{z}{|}V{\rangle},$
(33) $k_{\gamma}\,=\,\frac{m_{V}^{2}-m_{P}^{2}}{2\,m_{V}},$ (34)

where $Q_{i}$ and $m_{i}$ are the electric charge in the unit of ${|}e{|}$ and
mass of the constituent quark, respectively. With $m_{d}$ ${\approx}$ $336$
MeV, $m_{s}$ ${\approx}$ $490$ MeV, $m_{c}$ ${\approx}$ $1500$ MeV and the

${\mu}_{D_{d}^{\ast}D_{d}}\,=\,\frac{1}{6}\Big{(}\frac{2}{m_{c}}-\frac{1}{m_{d}}\Big{)},$
(35)
${\mu}_{D_{s}^{\ast}D_{s}}\,=\,\frac{1}{6}\Big{(}\frac{2}{m_{c}}-\frac{1}{m_{s}}\Big{)},$
(36)

one can obtain ${\Gamma}(D_{d}^{\ast}{\to}{\gamma}D_{d})$ ${\approx}$ $1.8$
keV and ${\Gamma}(D_{s}^{\ast}{\to}{\gamma}D_{s})$ ${\approx}$ $0.36$ keV
fayyazuddin . The theoretical value of partial decay width
${\Gamma}(D_{d}^{\ast}{\to}{\gamma}D_{d})$ is roughly consistent with the
corresponding experimental data ${\Gamma}(D_{d}^{\ast}{\to}{\gamma}D_{d})$ $=$
${\Gamma}_{D_{d}^{\ast}}{\times}{\cal B}(D_{d}^{\ast}{\to}{\gamma}D_{d})$ $=$
$1.33{\pm}0.33$ keV within $2\,{\sigma}$ regions pdg2020 . For the moment, we
will use ${\Gamma}_{D_{s}^{\ast}}$ $=$ $0.36$ keV in the calculation to give
an estimate of branching ratios for the $D_{s}^{{\ast}-}$ ${\to}$
${\ell}^{-}\bar{\nu}_{\ell}$ decays.

$\displaystyle{\Gamma}(D_{s}^{{\ast}-}{\to}{\ell}^{-}\bar{\nu}_{\ell})$
$\displaystyle=$ $\displaystyle 2.4{\pm}0.1\,{\rm meV},\quad\text{for}\
{\ell}\,=\,e,\,{\mu},$ (37)
$\displaystyle{\Gamma}(D_{s}^{{\ast}-}{\to}{\tau}^{-}\bar{\nu}_{\tau})$
$\displaystyle=$ $\displaystyle 0.28{\pm}0.02\,{\rm meV},$ (38)
$\displaystyle{\cal B}(D_{s}^{{\ast}-}{\to}{\ell}^{-}\bar{\nu}_{\ell})$
$\displaystyle=$ $\displaystyle(6.7{\pm}0.4){\times}10^{-6},\quad\text{for}\
{\ell}\,=\,e,\,{\mu},$ (39) $\displaystyle{\cal
B}(D_{s}^{{\ast}-}{\to}{\tau}^{-}\bar{\nu}_{\tau})$ $\displaystyle=$
$\displaystyle(7.8{\pm}0.4){\times}10^{-7}.$ (40)

If considering the experimental measurement efficiency, there are at least
more than $10^{7}$ $D_{s}^{\ast}$ events to experimentally study the
$D_{d}^{{\ast}-}$ ${\to}$ ${\ell}^{-}\bar{\nu}_{\ell}$ decays. And more than
$10^{8}$ $D_{s}^{\ast}$ events might be needed to explore the
$D_{s}^{{\ast}-}$ ${\to}$ ${\tau}^{-}\bar{\nu}_{\tau}$ decay.

Figure 6: The exclusive cross sections (in the unit of nb) as functions of
$\sqrt{s}$ (in the unit of GeV) for $e^{+}e^{-}$ ${\to}$
$D_{s}^{-}D_{s}^{{\ast}+}$ in (a) and $e^{+}e^{-}$ ${\to}$
$D_{s}^{{\ast}-}D_{s}^{{\ast}+}$ in (b). The Belle, BaBar and CLEO-c data are
from Ref. PhysRevD.83.011101 , Ref. PhysRevD.82.052004  and Ref. cpc.42.043002
, respectively.

In the electron-positron collisions, the cross sections of
$D_{s}^{+}D_{s}^{{\ast}-}$ and $D_{s}^{{\ast}+}D_{s}^{{\ast}-}$ production
have been experimentally studied by the Belle PhysRevD.83.011101 , BaBar
PhysRevD.82.052004  and CLEO-c PhysRevD.80.072001 ; cpc.42.043002  groups, as
illustrated in Fig. 6. Assuming the exclusive cross sections near threshold
${\sigma}(e^{+}e^{-}{\to}D_{s}^{+}D_{s}^{{\ast}-})$ ${\sim}$ $1.0\,{\rm nb}$
and ${\sigma}(e^{+}e^{-}{\to}D_{s}^{{\ast}+}D_{s}^{{\ast}-})$ ${\sim}$
$0.2\,{\rm nb}$, there will be about $10^{10}$ $D_{s}^{{\ast}{\pm}}$ events
corresponding to a data sample of $10$ ${\rm ab}^{-1}$ at STCF, and about
$5{\times}10^{10}$ $D_{s}^{{\ast}{\pm}}$ events corresponding to a data sample
of $50$ ${\rm ab}^{-1}$ at SuperKEKB. In addition, considering the branching
ratio ${\cal B}(Z{\to}c\bar{c})$ $=$ $(12.03{\pm}0.21)\,\%$ pdg2020  and the
fragmentation fraction $f(c{\to}D_{s}^{\ast})$ ${\simeq}$ $5.5\,\%$
epjc.76.397 , there will be more than $6{\times}10^{9}$ (and
$6{\times}10^{10}$) $D_{s}^{{\ast}{\pm}}$ events corresponding to $10^{12}$
cepc  (and $10^{13}$ fcc ) $Z$ bosons at the future CEPC (and FCC-ee). So the
$D_{s}^{{\ast}-}$ ${\to}$ ${\ell}^{-}\bar{\nu}_{\ell}$ decays (with ${\ell}$
$=$ $e$, ${\mu}$ and ${\tau}$) could be measured at Belle-II, SCTF, CEPC and
FCC-ee experiments.

In hadron-hadron collisions, the inclusive cross sections for the $c\bar{c}$
pair production are ${\sigma}(pp{\to}c\bar{c}X)$ ${\simeq}$ $2.4$ mb at the
center-of-mass energy of $\sqrt{s}$ $=$ $13$ TeV at LHCb JHEP.2016.03.159 ,
${\sigma}_{c\bar{c}}^{\rm tot}$ ${\simeq}$ $8.5$ mb and $8.6$ mb at $\sqrt{s}$
$=$ $7$ TeV at ALICE JHEP.2012.07.191  and ATLAS npb.907.717 , respectively.
With the fragmentation fraction $f(c{\to}D_{s}^{\ast})$ ${\simeq}$ $5.5\,\%$
epjc.76.397 , there will be about $4{\times}10^{13}$ $D_{s}^{{\ast}{\pm}}$
events corresponding a data sample of $300\,{\rm fb}^{-1}$ at LHCb, and more
$D_{s}^{{\ast}{\pm}}$ events available at ALICE and ATLAS. So the
$D_{s}^{{\ast}-}$ ${\to}$ $e^{-}\bar{\nu}_{e}$, ${\mu}^{-}\bar{\nu}_{\mu}$,
${\tau}^{-}\bar{\nu}_{\tau}$ decays could be measured precisely at LHCb, ALICE
and ATLAS experiments.

##  VI $B_{u}^{{\ast}-}$ ${\to}$ ${\ell}^{-}\bar{\nu}_{\ell}$ decays

The experimental information about the $B_{u}^{\ast}$ mesons are very scarce.
The already known information about the $B_{u}^{\ast}$ mesons are their quark
composition $b\bar{u}$ with the quark model assignment, the isospin $I$ $=$
$1/2$, the spin-parity quantum number $J^{P}$ $=$ $1^{-}$ and the mass
$m_{B_{u}^{\ast}}$ $=$ $5324.70(21)$ MeV pdg2020 . Due to the mass difference
$m_{B_{u}^{\ast}}$ $-$ $m_{B_{u}}$ $=$ $45$ MeV $<$ $m_{\pi}$, the
electromagnetic radiative transition $B_{u}^{\ast}$ ${\to}$ $B_{u}{\gamma}$
certainly will be the important and dominant decay mode. The photon in the
$B_{u}^{\ast}$ ${\to}$ $B_{u}{\gamma}$ decay is very soft, with the momentum
$k_{\gamma}$ ${\sim}$ $45$ MeV in the center-of-mass of the $B_{u}^{\ast}$
mesons. No signal event of the $B_{u}^{\ast}$ ${\to}$ $B_{u}{\gamma}$ decay
has yet been found. The $B_{u}^{{\ast}-}$ ${\to}$ ${\ell}^{-}\bar{\nu}_{\ell}$
decays offer a complementary decay modes of the $B_{u}^{\ast}$ meson. It can
be seen from Eq.(8) that the information about
${|}V_{ub}{|}\,f_{B_{u}^{\ast}}$ could be obtained, however, the partial width
for the purely leptonic decays $B_{u}^{{\ast}-}$ ${\to}$
${\ell}^{-}\bar{\nu}_{\ell}$ are highly suppressed by the CKM element of
${|}V_{ub}{|}^{2}$ ${\sim}$ ${\cal O}({\lambda}^{6})$.

The precise determinations of the CKM element $V_{ub}$ $=$
${|}V_{ub}{|}\,e^{-i\,{\gamma}}$ are very central and important to verify the
CKM picture of SM, where ${\gamma}$ is the angle of the unitarity triangle of
$V_{ud}V_{ub}^{\ast}$ $+$ $V_{cd}V_{cb}^{\ast}$ $+$ $V_{td}V_{tb}^{\ast}$ $=$
$0$. The experimental determination of ${|}V_{ub}{|}$ from the inclusive $B$
${\to}$ $X_{u}{\ell}\bar{\nu}_{\ell}$ decay is complicated mainly by the large
backgrounds from the CKM-favored $B$ ${\to}$ $X_{c}{\ell}\bar{\nu}_{\ell}$
decay. The experimental extraction of ${|}V_{ub}{|}$ from exclusive $B$
${\to}$ ${\pi}{\ell}\bar{\nu}_{\ell}$ decay is subject to the form factors
calculated with the lattice QCD or QCD sum rules. The latest values obtained
from inclusive and exclusive determinations are pdg2020

${|}V_{ub}{|}\,{\times}10^{3}\,=\,4.25{\pm}0.12_{\rm\,exp}{}^{+0.15}_{-0.14}{}_{\rm\,theo}{\pm}0.23_{\,{\Delta}BF}\,=\,4.25^{+0.30}_{-0.29}\quad\text{(inclusive)},$
(41)
${|}V_{ub}{|}\,{\times}10^{3}\,=\,3.70{\pm}0.10_{\rm\,exp}{\pm}0.12_{\rm\,theo}\,=\,3.70{\pm}0.16\quad\text{(exclusive)}.$
(42)

It is clearly seen that (1) the difference between inclusive and exclusive
determinations of ${|}V_{ub}{|}$ is obvious. (2) The best determinations of
${|}V_{ub}{|}$ are from exclusive semileptonic decays, with a precision of
about $4\,\%$. The experimental errors for the exclusive semileptonic decays
are expected to decrease from the current $2.7\,\%$ to $1.2\,\%$ based on a
dataset of $50$ ${\rm ab}^{-1}$ at Belle-II experiments PTEP.2019.123C01 . (3)
The theoretical uncertainties are larger than experimental ones. Moreover,
${|}V_{ub}{|}$ can also be experimentally determined from the leptonic decay
$B_{u}$ ${\to}$ ${\tau}\bar{\nu}_{\tau}$ and the semileptonic hyperon decay
${\Lambda}_{b}$ ${\to}$ $p{\mu}\bar{\nu}_{\mu}$. The constraint from the
global fit gives ${|}V_{ub}{|}$ $=$ $(3.61^{+0.11}_{-0.09}){\times}10^{-3}$
pdg2020 , which will be used in our calculation.

Table 5: The theoretical values of decay constant $f_{B_{u}^{\ast}}$ (in the unit of MeV), where the legends are the same as those in Table 3. NRQM |  $280$ plb.635.93  |  $151^{+15}_{-13}$ PhysRevD.71.113006  |  $234.7$ epjp.132.80  111Without QCD radiative corrections. |  $225.1$ epjp.132.80  222With QCD radiative corrections. |
---|---|---|---|---|---
NRQM |  $196$ ahep.2018.7032041  111Without QCD radiative corrections. |  $182$ ahep.2018.7032041  222With QCD radiative corrections. |  $213$ mpla.17.803  |  $242.37$ epl.116.31004  111Without QCD radiative corrections. |  $232.47$ epl.116.31004  222With QCD radiative corrections.
RQM |  $219$ plb.635.93  |  $195$ mpla.17.803  |  $252{\pm}10$ PhysRevD.55.6944  333With $f_{B_{u}^{\ast}}$ $=$ $f_{B_{d}^{\ast}}$, and the constituent quark masses for the light quarks $s$ and $d$. |  $193{\pm}8$ PhysRevD.55.6944  444With $f_{B_{u}^{\ast}}$ $=$ $f_{B_{d}^{\ast}}$, and the current quark masses for the light quarks $s$ and $d$. |
LFQM |  $188$ PhysRevC.92.055203  555With Coulomb plus linear potential model. |  $198.7^{+~{}4.9}_{-11.3}$ cpc.42.073102  666With a dilation parameter ${\kappa}$ $=$ $0.54$ GeV. |  $193.1^{+4.3}_{-4.6}$ cpc.42.073102  777With a dilation parameter ${\kappa}$ $=$ $0.68$ GeV. |  $196^{+28}_{-27}$ jpg.39.025005  |  $172^{+24}_{-23}{\pm}6$ PhysRevD.100.014026
LFQM |  $204$ PhysRevD.75.073016  888With Coulomb plus linear potential model. |  $193$ PhysRevD.75.073016  999With Coulomb plus harmonic oscillator potential model. |  $225{\pm}38$ PhysRevD.81.114024  101010With the Gaussian type wave functions. |  $249^{+44}_{-42}$ PhysRevD.81.114024  111111With the power-law type wave functions. |  $205{\pm}5$ PhysRevD.98.114018
LFQM |  $201.9^{+43.2}_{-41.4}$ epjc.76.313  101010With the Gaussian type wave functions. |  $220.2^{+49.1}_{-46.2}$ epjc.76.313  111111With the power-law type wave functions. |  $227$ epjp.133.134  121212With Martin potential model plb.93.338 . |  $211$ epjp.133.134  131313With Cornell potential model zpc.33.135 . |  $228$ epjp.133.134  141414With logarithmic potential model plb.71.153 .
LQCD |  $196{\pm}24^{+39}_{-~{}2}$ PhysRevD.60.074501  |  $190{\pm}28$ npb.619.507  151515With $m_{B^{\ast}}/f_{B^{\ast}}$ $=$ $28{\pm}1^{+3}_{-4}$ npb.619.507  and $m_{B^{\ast}}$ $=$ $5324.70(21)$ MeV pdg2020 . |  $185.9{\pm}7.2$ PhysRevD.96.034524  |  $175{\pm}6$ PhysRevD.91.114509  |
SR |  $210^{+10}_{-12}$ PhysRevD.88.014015  |  $213{\pm}18$ epjc.75.427  |  $209{\pm}8$ ijmpa.30.1550116  |  $209^{+23}_{-22}$ 2106.13617  |  $181.8{\pm}13.7$ PhysRevD.91.116009
other |  $138$ epjp.133.134  161616With harmonic plus Yukawa potential model epjp.132.80 . |  $200{\pm}10$ PhysRevD.75.116001  |  $238{\pm}18$ plb.633.492  |  $164$ PhysRevD.58.014007  |

Some theoretical results on the decay constant $f_{B_{u}^{\ast}}$ are
collected in Table 5. The recent result $f_{B_{u}^{\ast}}$ $=$ $185.9{\pm}7.2$
MeV from LQCD calculation PhysRevD.96.034524  will be used in this paper.
There are many theoretical calculation on the decay width
${\Gamma}_{B_{u}^{\ast}}$, for example, Refs. PhysRevD.47.1030 ; plb.316.555 ;
plb.334.175 ; PhysRevD.49.299 ; zpc.67.633 ; mpla.12.3027 ; npa.658.249 ;
npa.671.380 ; jpg.27.1519 ; PhysRevD.64.094007 ; epja.13.363 ; plb.537.241 ;
PhysRevD.68.054024 ; ijmpa.25.2063 ; epja.52.90 ; PhysRevD.94.113011 ;
PhysRevD.100.016019 ; jhep.2020.04.023 ; 2106.13617 . With the formula of
Eq.(32), the quark mass $m_{u}$ $=$ $336$ MeV fayyazuddin  and $m_{b}$ $=$
$4.78$ GeV pdg2020 , and the magnetic dipole momentum

${\mu}_{B_{u}^{\ast}B_{u}}\,=\,\frac{1}{6}\Big{(}\frac{2}{m_{u}}-\frac{1}{m_{b}}\Big{)},$
(43)

one can obtain ${\Gamma}_{B_{u}^{\ast}}$ ${\simeq}$
${\Gamma}(B_{u}^{\ast}{\to}{\gamma}B_{u})$ ${\simeq}$ $820$ eV. The partial
decay width and branching ratios for the $B_{u}^{{\ast}-}$ ${\to}$
${\ell}^{-}\bar{\nu}_{\ell}$ decays are

$\displaystyle{\Gamma}(B_{u}^{{\ast}-}{\to}{\ell}^{-}\bar{\nu}_{\ell})$
$\displaystyle=$ $\displaystyle 0.25^{+0.04}_{-0.03}\,{\mu}\,{\rm
eV},\quad\text{for}\ {\ell}\,=\,e,\,{\mu},$ (44)
$\displaystyle{\Gamma}(B_{u}^{{\ast}-}{\to}{\tau}^{-}\bar{\nu}_{\tau})$
$\displaystyle=$ $\displaystyle 0.20^{+0.03}_{-0.02}\,{\mu}\,{\rm eV},$ (45)
$\displaystyle{\cal B}(B_{u}^{{\ast}-}{\to}{\ell}^{-}\bar{\nu}_{\ell})$
$\displaystyle=$ $\displaystyle(3.0{\pm}0.4){\times}10^{-10},\quad\text{for}\
{\ell}\,=\,e,\,{\mu},$ (46) $\displaystyle{\cal
B}(B_{u}^{{\ast}-}{\to}{\tau}^{-}\bar{\nu}_{\tau})$ $\displaystyle=$
$\displaystyle(2.5{\pm}0.4){\times}10^{-10}.$ (47)

It is expected that there should be at least more than $10^{11}$
$B_{u}^{\ast}$ events available for experimental study of the
$B_{u}^{{\ast}-}$ ${\to}$ ${\ell}^{-}\bar{\nu}_{\ell}$ decays.

The experimental study has shown that the exclusive cross sections for the
final states of $B\overline{B}^{\ast}$, $B^{\ast}\overline{B}^{\ast}$ and
$B\overline{B}^{\ast}{\pi}$ will have a large share of the total $b\bar{b}$
cross sections above the open bottom threshold, for example pdg2020 ,

$\displaystyle{\cal
B}(Z_{b}(10610){\to}B^{+}\overline{B}^{{\ast}0}+B^{{\ast}+}\overline{B}^{0})\,=\,85.6^{+2.1}_{-2.9}\,\%,$
(48) $\displaystyle{\cal
B}(Z_{b}(10650){\to}B^{{\ast}+}\overline{B}^{{\ast}0})\,=\,74^{+4}_{-6}\,\%,$
(49) $\displaystyle{\cal
B}({\Upsilon}(5S){\to}B\overline{B}^{\ast}+c.c.)\,=\,13.7{\pm}1.6\,\%,$ (50)
$\displaystyle{\cal
B}({\Upsilon}(5S){\to}B^{\ast}\overline{B}^{\ast})\,=\,38.1{\pm}3.4\,\%,$ (51)
$\displaystyle{\cal
B}({\Upsilon}(5S){\to}B\overline{B}^{\ast}{\pi}+B^{\ast}\overline{B}{\pi})\,=\,7.3{\pm}2.3\,\%,$
(52) $\displaystyle{\cal
B}({\Upsilon}(5S){\to}B^{\ast}\overline{B}^{\ast}{\pi})\,=\,1.0{\pm}1.4\,\%.$
(53)

There are about $36{\times}10^{6}$ ${\Upsilon}(5S)$ events corresponding to
the dataset of $121$ ${\rm fb}^{-1}$ at Belle experiments at the disposal
PTEP.2019.123C01 . About $1.5{\times}10^{10}$ ${\Upsilon}(5S)$ events with a
dataset $50$ ${\rm ab}^{-1}$ at Belle-II are an outside estimate. Assuming the
inclusive branching ratio ${\cal B}({\Upsilon}(5S){\to}B_{u}^{\ast}X)$
${\simeq}$ $30\,\%$, there will be some $4.5{\times}10^{9}$ $B_{u}^{\ast}$
events at most at Belle-II. And it is more important that the vast majority of
the data will be taken at ${\Upsilon}(4S)$ resonance rather than
${\Upsilon}(5S)$ mesons at Belle-II experiments, and ${\Upsilon}(4S)$ lies
below the $B\overline{B}^{\ast}$ threshold. So the probability of direct
observation of the $B_{u}^{{\ast}-}$ ${\to}$ ${\ell}^{-}\bar{\nu}_{\ell}$
decays at Belle-II experiments should be very tiny. Considering about
$10^{13}$ $Z$ bosons at FCC-ee fcc  and branching ratio ${\cal
B}(Z{\to}b\bar{b})$ $=$ $12.03{\pm}0.21\,\%$ pdg2020 , and assuming the
fragmentation fraction $f(b{\to}B_{u}^{\ast})$ ${\sim}$ $20\,\%$ plb.576.29 ,
there will be more than $4{\times}10^{11}$ $B_{u}^{\ast}$ events to search for
the $B_{u}^{{\ast}-}$ ${\to}$ ${\ell}^{-}\bar{\nu}_{\ell}$ decays. The
$b$-quark production cross sections at the center-of-mass energy $\sqrt{s}$
$=$ 13 TeV is about ${\sigma}(pp{\to}b\bar{b}X)$ ${\simeq}$ $495$ ${\mu}$b at
LHCb JHEP.2015.10.172 . There will be more than $5{\times}10^{13}$
$B_{u}^{\ast}$ events with a dateset of $300\,{\rm fb}^{-1}$ at LHCb and
fragmentation fraction $f(b{\to}B_{u}^{\ast})$ ${\sim}$ $20\,\%$. Hence, the
$B_{u}^{{\ast}-}$ ${\to}$ $e^{-}\bar{\nu}_{e}$, ${\mu}^{-}\bar{\nu}_{\mu}$,
${\tau}^{-}\bar{\nu}_{\tau}$ decays could be investigated at FCC-ee and LHCb
experiments in the future.

##  VII $B_{c}^{{\ast}-}$ ${\to}$ ${\ell}^{-}\bar{\nu}_{\ell}$ decays

According to the conventional quark-model assignments, the $B_{c}^{\ast}$
mesons consist of two heavy quarks with different flavor numbers $B$ $=$ $C$
$=$ $-Q$ $=$ ${\pm}1$. Up to today, the experimental information of the
$B_{c}^{\ast}$ meson is still very limited. For example, the potential
candidate of the $B_{c}^{\ast}$ meson has not yet been determined. It is
generally believed that the mass of the $B_{c}^{\ast}$ meson should be in the
region between $m_{B_{c}}$ $=$ $6274.47{\pm}0.27{\pm}0.17$ MeV recently
measured by LHCb JHEP.2020.07.123  and $m_{B_{c}(2S)}$ $=$
$6872.1{\pm}1.3{\pm}0.1{\pm}0.8$ MeV obtained by LHCb PhysRevLett.122.232001
(or $6871.0{\pm}1.2{\pm}0.8{\pm}0.8$ MeV given by CMS PhysRevLett.122.132001
), where the $B_{c}$, $B_{c}^{\ast}$ and $B_{c}(2S)$ particles correspond to
the sibling isoscalar states with quantum numbers of $n^{2S+1}L_{J}$ $=$
$1^{1}S_{0}$, $1^{3}S_{1}$ and $2^{1}S_{0}$, respectively. So the branching
ratios for the strong decays $B_{c}^{\ast}$ ${\to}$ $BD$ are zero, because
$B_{c}^{\ast}$ meson is below the $BD$ pair threshold. The experimental
particle physicists are earnestly looking for and identifying the
$B_{c}^{\ast}$ meson, a long-expected charming beauty. For the moment, almost
all of the information available about the properties of $B_{c}^{\ast}$ meson
(such as the mass, decay constant, lifetime, decay modes and so on) come from
theoretical estimates. There are too many estimations on the $B_{c}^{\ast}$
meson mass with various theoretical models, for example, in Refs. zpc.3.165 ;
PhysRevD.21.3180 ; PhysRevD.23.2724 ; PhysRevD.24.132 ; zpc.12.63 ;
PhysRevD.32.189 ; ijmpa.6.2309 ; PhysRevD.44.212 ; PhysRevD.46.1165 ;
PhysRevD.49.5845 ; zpc.64.57 ; Phys.Usp.38.1 ; PhysRevD.51.1248 ;
PhysRevD.51.3613 ; PhysRevD.52.5229 ; PhysRevD.53.312 ; plb.382.131 ;
epjc.4.107 ; PhysRevD.59.094001 ; PhysRevD.60.074006 ; PhysRevD.62.114024 ;
PhysRevD.70.054017 ; ijmpa.19.1771 ; epjc.37.323 ; PhysRevD.71.034006 ;
Pramana.66.953 ; plb.651.171 ; PhysRevC.78.055202 ; jpg.36.035003 ;
npa.848.299 ; PhysRevD.81.076005 ; PhysRevLett.104.022001 ; PhysRevD.81.071502
; epjc.71.1825 ; pan.74.631 ; PhysRevD.86.094510 ; epja.49.131 ; epja.50.154 ;
ijmpa.32.1750021 ; PhysRevLett.121.202002 ; epjc.78.592 ; PhysRevD.99.054025 ;
PhysRevD.99.096020 ; PhysRevD.100.096002 ; PhysRevD.100.114032 ; epja.55.82 ;
npb.947.114726 ; epjc.80.223 ; plb.807.135522 ; PhysRevD.101.056002 ;
PhysRevD.102.034002 ; pan.83.634 ; plb.816.136277 ; fewbody.62.39 ;
epjc.81.327 . The recent result from lattice QCD calculation,
$m_{B_{c}^{\ast}}$ $=$ $6331{\pm}7$ MeV PhysRevLett.121.202002 , which are
basically consistent with other estimations, will be used in this paper.
Clearly, it is foreseeable that the isospin violating decay $B_{c}^{\ast}$
${\to}$ $B_{c}{\pi}$ is explicitly forbidden by the law of energy
conservation, because of $m_{B_{c}^{\ast}}$ $-$ $m_{B_{c}}$ ${\simeq}$ $57$
MeV $<$ $m_{\pi}$. Hence, the electromagnetic radiative transition
$B_{c}^{\ast}$ ${\to}$ $B_{c}{\gamma}$ should be the dominant decay mode. In
addition, the photon in the magnetic dipole transition $B_{c}^{\ast}$ ${\to}$
$B_{c}{\gamma}$ is very soft in the rest frame of the $B_{c}^{\ast}$ meson.
This might be one main reason why the unambiguously experimental
identification of the $B_{c}^{\ast}$ meson is very challenging. As an
important complementary decay modes, the $B_{c}^{\ast}$ meson has very rich
weak decay channels, which could be approximately classified into three
classes: (1) the valence $b$ quark weak decay accompanied by the spectator $c$
quark, (2) the valence $c$ quark weak decay accompanied by the spectator $b$
quark, and (3) the $b$ and $c$ quarks annihilation into a virtual $W$ boson.
The purely leptonic decays $B_{c}^{{\ast}-}$ ${\to}$
${\ell}^{-}\bar{\nu}_{\ell}$ belong to the third case, which are favored by
the CKM element ${|}V_{cb}{|}$. And the information of
${|}V_{cb}{|}\,f_{B_{c}^{\ast}}$ could be obtained from the $B_{c}^{{\ast}-}$
${\to}$ ${\ell}^{-}\bar{\nu}_{\ell}$ decays.

Table 6: The theoretical values of decay constant $f_{B_{c}^{\ast}}$ (in the unit of MeV), where the legends are the same as those in Table 3. NRQM |  $562$ mpla.17.803  |  $434.64$ epjc.78.592  |  $544.3$ pan.83.634  |  |
---|---|---|---|---|---
RQM |  $503$ mpla.17.803  |  $510{\pm}80$ PhysRevD.51.3613  111With Martin potential model plb.93.338 . |  $456{\pm}70$ PhysRevD.51.3613  222With Coulomb plus linear potential model. |  $460{\pm}60$ PhysRevD.51.3613  333Obtained from the scaling relation. |
LFQM |  $391^{+4}_{-5}$ PhysRevC.92.055203  |  $440^{+51}_{-52}$ jpg.39.025005  |  $387$ PhysRevD.81.114024  444With the Gaussian type wave functions. |  $423$ PhysRevD.81.114024  555With the power-law type wave functions. |  $465{\pm}7$ PhysRevD.98.114018
LFQM |  $473.4{\pm}18.2$ epjc.76.313  444With the Gaussian type wave functions. |  $487.6{\pm}19.2$ epjc.76.313  555With the power-law type wave functions. |  $398$ PhysRevD.80.054016  666With Coulomb plus linear potential model. |  $551$ PhysRevD.80.054016  777With Coulomb plus harmonic oscillator potential model. |  $474{\pm}42$ PhysRevD.98.114038
LQCD |  $422{\pm}13$ PhysRevD.91.114509  |  $387{\pm}12$ lattice2018.273  |  |  |
SR |  $384{\pm}32$ epja.49.131  888With the current quark mass. |  $415{\pm}31$ epja.49.131  999With the pole quark mass. |  $300{\pm}30$ npb.947.114726  |  $442{\pm}44$ plb.807.135522  101010With inputs from the inverse Laplace-type model. |  $387{\pm}15$ plb.807.135522  111111With inputs from Heavy Quark Symmetry.
other |  $453{\pm}20$ PhysRevD.75.116001  |  $418{\pm}24$ plb.633.492  |  $471$ PhysRevD.99.054025  |  |

The current values of the CKM element ${|}V_{cb}{|}$ come mainly from
inclusive and exclusive semileptonic decays of $B$ meson to charm pdg2020 .
The average values obtained from inclusive $b$ ${\to}$
$c{\ell}\bar{\nu}_{\ell}$ decays and exclusieve $B$ ${\to}$
$D^{(\ast)}{\ell}\bar{\nu}_{\ell}$ decays are ${|}V_{cd}{|}{\times}10^{3}$ $=$
$42.2(8)$ and $39.5(9)$, respectively pdg2020 . The lepton flavor non-
universality in the ratio $R(D^{({\ast})})$ complicate the determination of
${|}V_{cb}{|}$. In addition, ${|}V_{cb}{|}$ can also be obtained from the
PLDCM $B_{c}^{-}$ ${\to}$ ${\ell}\bar{\nu}_{\ell}$ decays, although none of
the measurements has reached a competitive level of precision due to either
the serious helicity suppression for $B_{c}^{-}$ ${\to}$ $e\bar{\nu}_{e}$,
${\mu}\bar{\nu}_{\mu}$ decays or other additional neutrinos from ${\tau}$
decay for $B_{c}^{-}$ ${\to}$ ${\tau}\bar{\nu}_{\tau}$ decay. The global SM
fit value is ${|}V_{cb}{|}$ $=$ $40.53^{+0.83}_{-0.61}{\times}10^{-3}$ pdg2020
, which will be used in this paper.

Both valence quarks of the $B_{c}^{({\ast})}$ mesons are regarded as heavy
quarks. Their Compton wave lengths ${\sim}$ $1/m_{b,c}$ are much shorter than
a typical hadron size. The spin-flavor symmetry in the heavy quark limit would
lead to an approximation between decay constants $f_{B_{c}}$ ${\approx}$
$f_{B_{c}^{\ast}}$. Some theoretical results on the decay constant
$f_{B_{c}^{\ast}}$ are collected in Table 6. The recent lattice QCD
calculation $f_{B_{c}^{\ast}}$ $=$ $387{\pm}12$ MeV lattice2018.273  will be
used in this paper. As it is well known that the magnetic momentum of both $b$
and $c$ quarks are inversely proportional to their mass. The magnetic dipole
momentum

${\mu}_{B_{c}^{\ast}B_{c}}\,=\,\frac{1}{6}\Big{(}\frac{2}{m_{c}}-\frac{1}{m_{b}}\Big{)},$
(54)

should be very small. With the quark mass $m_{c}$ $=$ $1.5$ GeV and $m_{b}$
$=$ $4.78$ GeV, one can obtain ${\Gamma}_{B_{c}^{\ast}}$ ${\simeq}$
${\Gamma}(B_{c}^{\ast}{\to}{\gamma}B_{c})$ ${\simeq}$ $60$ eV using the
formula of Eq.(32). The partial decay width and branching ratios for the
$B_{c}^{{\ast}-}$ ${\to}$ ${\ell}^{-}\bar{\nu}_{\ell}$ decays are estimated to
be,

$\displaystyle{\Gamma}(B_{c}^{{\ast}-}{\to}{\ell}^{-}\bar{\nu}_{\ell})$
$\displaystyle=$ $\displaystyle 225^{+25}_{-21}\,{\mu}\,{\rm
eV},\quad\text{for}\ {\ell}\,=\,e,\,{\mu},$ (55)
$\displaystyle{\Gamma}(B_{c}^{{\ast}-}{\to}{\tau}^{-}\bar{\nu}_{\tau})$
$\displaystyle=$ $\displaystyle 198^{+22}_{-18}\,{\mu}\,{\rm eV},$ (56)
$\displaystyle{\cal B}(B_{c}^{{\ast}-}{\to}{\ell}^{-}\bar{\nu}_{\ell})$
$\displaystyle=$
$\displaystyle(3.8^{+0.4}_{-0.3}){\times}10^{-6},\quad\text{for}\
{\ell}\,=\,e,\,{\mu},$ (57) $\displaystyle{\cal
B}(B_{c}^{{\ast}-}{\to}{\tau}^{-}\bar{\nu}_{\tau})$ $\displaystyle=$
$\displaystyle(3.3^{+0.4}_{-0.3}){\times}10^{-6}.$ (58)

To experimentally investigate the $B_{c}^{{\ast}-}$ ${\to}$
${\ell}^{-}\bar{\nu}_{\ell}$ decays, there should be at least more than
$10^{7}$ $B_{c}^{\ast}$ events available.

More than $10^{12}$ $Z$ bosons are expected at the future $e^{+}e^{-}$
colliders of CEPC cepc  and FCC-ee fcc . Considering the branching ratio
${\cal B}(Z{\to}b\bar{b})$ $=$ $12.03{\pm}0.21\,\%$ pdg2020  and fragmentation
fraction $f(b{\to}B_{c}^{\ast})$ ${\sim}$ $6{\times}10^{-4}$ npa.953.21 ;
cpc.43.083101 ; PhysRevD.100.034004 , there will be more than $10^{8}$
$B_{c}^{\ast}$ events to search for the $B_{c}^{{\ast}-}$ ${\to}$
$e^{-}\bar{\nu}_{e}$, ${\mu}^{-}\bar{\nu}_{\mu}$, ${\tau}^{-}\bar{\nu}_{\tau}$
decays. In addition, the $B_{c}^{\ast}$ production cross sections at LHC are
estimated to be about $100$ nb for $pp$ collisions at $\sqrt{s}$ $=$ $13$ TeV,
about $8$ mb for $p$-Pb collisions at $\sqrt{s}$ $=$ $8.16$ TeV and some $920$
mb for Pb-Pb collisions at $\sqrt{s}$ $=$ $5.02$ TeV, respectively
PhysRevD.97.114022 . There will be more than $3{\times}10^{10}$ $B_{c}^{\ast}$
events corresponding to a dataset of $300\,{\rm fb}^{-1}$ at LHCb for $pp$
collisions. Hence, the $B_{c}^{{\ast}-}$ ${\to}$ $e^{-}\bar{\nu}_{e}$,
${\mu}^{-}\bar{\nu}_{\mu}$, ${\tau}^{-}\bar{\nu}_{\tau}$ decays are expected
to be carefully measured at LHCb experiments in the future.

##  VIII summary

The mass of the charged vector mesons are generally larger than that of the
corresponding ground pseudoscalar mesons. The vector mesons decay mainly
through the strong or/and electromagnetic interactions. These facts will
inevitably result in that the branching ratios of the vector meson weak decays
are often very tiny. Inspired by the potential prospects of existing and
coming high-luminosity experiments, more and more experimental data will be
accumulated, and higher measurement precision level will be reached. The
probabilities of experimental investigation on the purely leptonic decays of
charged vector mesons are discussed in this paper. We found that (1) for both
${\rho}^{\pm}$ and $K^{{\ast}{\pm}}$ mesons, their widths are large due to the
dominance of strong decay. Their PLDCV branching ratios are estimated at the
order of ${\cal O}(10^{-13})$. Although extremely complicated and difficult,
the PLDCV decays ${\rho}^{\pm}$, $K^{{\ast}{\pm}}$ ${\to}$
$e^{-}\bar{\nu}_{e}$, ${\mu}^{-}\bar{\nu}_{\mu}$ might be measurable due to
the huge data of the ${\rho}^{\pm}$ and $K^{{\ast}{\pm}}$ mesons at LHCb. (2)
The PLDCV $D_{s}^{\ast}$ decays are favored by the CKM element ${|}V_{cs}{|}$.
Their branching ratios are about ${\cal O}(10^{-6})$. The PLDCV decays
$D_{d,s}^{\ast}$ ${\to}$ $e^{-}\bar{\nu}_{e}$, ${\mu}^{-}\bar{\nu}_{\mu}$,
${\tau}^{-}\bar{\nu}_{\tau}$ could be carefully studied at the Belle-II, SCTF
or STCF, CEPC, FCC-ee, LHCb experiments. (3) For the $B_{u}^{\ast}$ mesons
below the $B{\pi}$ thresholds and the $B_{c}^{\ast}$ mesons below both $BD$
and $B_{c}{\pi}$ thresholds, they decay predominantly through the magnetic
dipole transitions. The branching ratios of the PLDCV $B_{c}^{\ast}$ decays
favored by the CKM element ${|}V_{cb}{|}$ could reach up to ${\cal
O}(10^{-6})$. The PLDCV decays $B_{u,c}^{\ast}$ ${\to}$ $e^{-}\bar{\nu}_{e}$,
${\mu}^{-}\bar{\nu}_{\mu}$, ${\tau}^{-}\bar{\nu}_{\tau}$ might be searched for
at the CEPC, FCC-ee, LHCb experiments. Our rough estimations and findings are
summed in Table 7. We wish that our investigation could provoke physicists’
researching interest in PLDCV and offer a ready reference for the future
experimental analysis.

Table 7: The probabilities of experimetal investigation on PLDCV, where ${\cal B}$ denotes the branching ratio, and the symbol ${\star}$ denotes that the PLDCV process might be experimentally accessible in the future. decay modes | ${\cal B}$ | Belle-II | SCTF/STCF | CEPC | FCC-ee | LHCb
---|---|---|---|---|---|---
${\rho}^{-}$ ${\to}$ $e^{-}\bar{\nu}_{e}$, ${\mu}^{-}\bar{\nu}_{\mu}$ | ${\cal O}(10^{-13})$ |  |  |  |  | ${\star}$
$K^{{\ast}-}$ ${\to}$ $e^{-}\bar{\nu}_{e}$, ${\mu}^{-}\bar{\nu}_{\mu}$ | ${\cal O}(10^{-13})$ |  |  |  |  | ${\star}$
$D_{d}^{{\ast}-}$ ${\to}$ $e^{-}\bar{\nu}_{e}$, ${\mu}^{-}\bar{\nu}_{\mu}$, ${\tau}^{-}\bar{\nu}_{\tau}$ | ${\cal O}(10^{-10})$ | ${\star}$ | ${\star}$ | ${\star}$ | ${\star}$ | ${\star}$
$D_{s}^{{\ast}-}$ ${\to}$ $e^{-}\bar{\nu}_{e}$, ${\mu}^{-}\bar{\nu}_{\mu}$, ${\tau}^{-}\bar{\nu}_{\tau}$ | ${\cal O}(10^{-6})$ | ${\star}$ | ${\star}$ | ${\star}$ | ${\star}$ | ${\star}$
$B_{u}^{{\ast}-}$ ${\to}$ $e^{-}\bar{\nu}_{e}$, ${\mu}^{-}\bar{\nu}_{\mu}$, ${\tau}^{-}\bar{\nu}_{\tau}$ | ${\cal O}(10^{-10})$ |  |  | ${\star}$ | ${\star}$ | ${\star}$
$B_{c}^{{\ast}-}$ ${\to}$ $e^{-}\bar{\nu}_{e}$, ${\mu}^{-}\bar{\nu}_{\mu}$, ${\tau}^{-}\bar{\nu}_{\tau}$ | ${\cal O}(10^{-6})$ |  |  | ${\star}$ | ${\star}$ | ${\star}$

## Acknowledgments

The work is supported by the National Natural Science Foundation of China
(Grant Nos. 11705047, 11981240403, U1632109, 11547014), the Chinese Academy of
Sciences Large-Scale Scientific Facility Program (1G2017IHEPKFYJ01) and the
Program for Innovative Research Team in University of Henan Province
(19IRTSTHN018). We thank Prof. Haibo Li (IHEP@CAS), Prof. Shuangshi Fang
(IHEP@CAS), Prof. Frank Porter (Caltech), Prof. Antimo Palano (INFN), Prof.
Chengping Shen (Fudan University), Dr. Xiao Han (Fudan University), Prof.
Xiaolin Kang (China University of Geosciences), Ms. Qingping Ji (Henan Normal
University), Ms. Huijing Li (Henan Normal University) for their kindly help
and valuable discussion.

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