Purely leptonic decays of the ground charged vector mesons

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,

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,

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,

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,

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 .

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.

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 ,

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 ,

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

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.

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.

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,

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 ,

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.

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.

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.

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.

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.