# Purely leptonic decay of the charged vector mesons ($D_{(s)}^{*+}$)

Kalpalata Dash1111email<EMAIL_ADDRESS>P. C. Dash1, R. N.
Panda1, Susmita Kar2, N. Barik3 1 Department of Physics, Siksha
${}^{\prime}O^{\prime}$ Anusandhan Deemed to be University,
Bhubaneswar-751030, India.
2 Department of Physics, Maharaja Sriram Chandra Bhanja Deo University,
Baripada-757003, India.
3 Department of Physics, Utkal University, Bhubaneswar-751004, India.

###### Abstract

In the wake of the first-ever experimental study of the decay: $D_{s}^{*+}\to
e^{+}\nu_{e}$ with $e^{+}e^{-}$ collision data taken from the BESIII detector
at the BEPCII collider, we study the purely leptonic decay of the charged
vector mesons ($D^{*}_{(s)}$) in the framework of the relativistic independent
quark (RIQ) model based on an average flavor-independent confining potential
in equally mixed scalar-vector harmonic form. We calculate the decay
constants: $f_{D^{*}}$ and $f_{D_{s}^{*}}$ representing the decay amplitudes
using the meson wave function derivable in the RIQ model. Our results for the
decay constants: $f_{D^{*}}=197\pm 14$ MeV, $f_{D_{s}^{*}}=236\pm 19$ MeV,
decay widths: $\Gamma(D^{*}\to e\nu_{e})=(57.523\pm 8.183)\ \mu eV$,
$\Gamma(D_{s}^{*}\to e\nu_{e})=(1.802\pm 0.275)\ meV$ and corresponding
branching fractions (BFs): ${\cal B}(D^{*}\to e\nu_{e})=(6.897\pm 0.992)\times
10^{-10}$ and ${\cal B}(D_{s}^{*}\to e\nu_{e})=(2.574\pm 1.102)\times 10^{-5}$
are in good agreement with available experimental data and other model
predictions including those from the lattice QCD calculation. Our predictions:
${\cal R}_{D^{*+}}=0.0665$ and ${\cal R}_{D_{s}^{*+}}=0.1157$, which
correspond to the ratios of the BFs for $D_{(s)}^{*+}$ decays in their
$\tau^{-}$ mode to the corresponding values in $e^{-}$ mode, are in good
agreement with the results of the LQCD and LFQM calculations.

##  I Introduction

Recently, the BESIII Collaboration [1] reported the first experimental study
of purely leptonic decay: $D_{s}^{*+}\to e^{+}\nu_{e}$, which yields its BFs
of $(2.1^{+1.2}_{-0.9_{stat.}}{\pm 0.2}_{syst.})\times 10^{-5}$, corresponding
to an upper limit of $4.0\times 10^{-5}$ at the $90\%$ confidence level.
Considering the total decay width: $\Gamma_{D_{s}^{*+}}$ = $(0.07\pm 0.028)$
keV from the lattice chromodynamics as the input, its decay constant
$f_{D_{s}^{*+}}$ is found to be $(213.6^{+61.0}_{-45.8_{stat.}}{\pm
43.9}_{syst.})$ MeV, corresponding to an upper limit of 353.8 MeV at the
$90\%$ confidence level. Intense experimental investigations of the ground
pseudoscalar meson decays: $D_{(s)}^{+}\to e^{+}\nu_{e}$ over the last two
decades, have allowed precision tests of the Standard Model (SM). The
measurement of purely leptonic $D_{(s)}^{*+}$ decays was not available until
very recent data from BESIII Collaboration are obtained in Oct. 2023 for
$D_{s}^{*+}\to e^{+}\nu_{e}$ only. The $D^{*+}\to e^{+}\nu_{e}$ decay is yet
be measured. It is expected that the data samples from BESIII experiment and
its future upgrades would allow applications of double-tag methods to fully
reconstruct events even when invisible particles such as the neutrinos are
present in the final states. This would provide unique opportunities to obtain
absolute BFs ${\cal B}(D_{(s)}^{*+}\to l^{+}\nu_{l})$ in the charm and
charmed-strange meson sector. The weak decays of $D_{(s)}^{*+}$ thus remain an
exciting field for both experimental and theoretical investigations.

The purely leptonic $D_{(s)}^{*+}$ decays with final lepton-neutrino pair or
lepton-lepton pair, in general, are considered as rare decays, which have
relatively simpler physics than hadron decays. The study of these decays is
important as their decay rates are connected with the Cobibbo-Kobayashi-
Maskawa (CKM) matrix elements [2, 3] that parameterize the strength of $c\to
s$ and $c\to d$ weak transition, respectively. These decays also determine the
strong-interaction phases in charm meson ($D^{*}$) decays, that are essential
inputs to the measurements to the $CP$-violating phase $\gamma$ of the CKM-
matrix element $V_{ub}$ in $B$-meson decays. Thus the study of purely leptonic
decay of charged vector mesons (PLDCVMs): $D_{(s)}^{*+}\to l^{+}\nu_{l}$,
besides giving unique opportunities to fully investigate the microstructure
and properties of the decaying meson-bound-states, are also significant to
determine the leptonic decay constants, ($f_{D_{(s)}^{*+}}$) that enter into
the expression of the experimentally measurable physical quantity: the decay
cross-section $\Gamma(D_{(s)}^{*+}\to l^{+}\nu_{l})$.

Within the SM, the PLDCVMs are typically induced by the tree-level exchange of
the gauge boson $W$. The description of these decays requires the calculation
of corresponding heavy-to-light decay constants $f_{D_{(s)}^{*+}}$, which are
closely related to the constituent quark ($q_{1}q_{2}$) wave functions at the
origin and hence they cannot be computed from the first principle QCD
applications. Alternate routes via different phenomenological and theoretical
approaches have been extended to predict these decays. Some of these
approaches include the nonrelativistic quark model (NRQM) [4, 5, 6, 7, 8, 9,
10], relativistic quark model (RQM) [4, 9, 10, 11, 12, 13], light front quark
model (LFQM) [14, 15, 16, 17, 18, 19, 20, 21, 22], lattice QCD [23, 24, 25,
26, 27, 28], sum rule (SR) [29, 30, 31, 32, 33, 34] and the Bathe-Salpeter
method [35, 36]. The prediction of the decay constant in different approaches
are obtained in the wide range of 186 [22] to 391 [4] MeV for $f_{D^{*+}}$ and
212 [8] to 447 [4] MeV for $f_{D_{s}^{*+}}$.

The purely leptonic decays of charged pseudoscalar mesons (PLDCPMs) in the
light and heavy flavor sector have been predicted in the RIQ model [37] in
reasonable agreement with the available experimental data and other model
predictions. The recent BESIII data on $D_{s}^{*+}\to e^{+}\nu_{e}$ [1]
provide us necessary motivation to investigate the PLDCVMs: (${D_{(s)}^{*+}}$)
to predict the decay constants $f_{D_{(s)}^{*+}}$ as well as BFs ${\cal
B}(D_{(s)}^{*+}\to l^{+}\nu_{l})$ in the framework of the RIQ model. The
obtained results could be used to constrain the unknown total decay widths of
$D_{(s)}^{*+}$, that can distinguish the difference in different
model/theoretical predictions. We would also like to predict the BFs and their
ratios ${\cal R}_{D_{(s)}^{*+}}=\frac{{\cal
B}(D_{(s)}^{*+}\to\tau^{+}\nu_{\tau})}{{\cal B}(D_{(s)}^{*+}\to
l^{+}\nu_{l})}$ with $l=e,\mu$, in comparison with other SM predictions.

The assumptions involved in our calculation are mainly as follows. Here, we
consider the quark-antiquark pair inside the decaying meson-bound-state that
annihilates to a single massive virtual $W$-boson, which subsequently decays a
lepton pair ($l\bar{\nu}_{l}$). Although in principle, the decay can take
place at any arbitrary momentum $\vec{k}$ of the parent meson, for the sake of
simplicity we consider the decay in the meson rest frame. Thus we are led to
believe that a strong correlation exists between the quark-antiquark momenta
so as to have their total momentum identically zero in the meson rest frame.
With such consideration, the ground state of the decaying vector meson can be
suitably represented with an appropriate momentum distribution of the bound
quark-antiquark pair in the corresponding $SU(6)$-spin-flavor configuration.
Then the transition probability amplitude for the weak leptonic decay,
calculated from the appropriate Feynman diagram, can be expressed effectively
as the free quark-antiquark pair annihilation amplitude integrated over the
model momentum distribution function. There is, of course, an obvious
difficulty relating to the energy conservation at the hadron-boson vertex,
since the sum total of the kinetic energy of the annihilating quark-antiquark
pair in the process is not equal to the rest mass energy of the decaying
meson. In the absence of a fully rigorous field-theoretic formulation of the
bound quark-antiquark annihilation inside the meson, such difficulty arises as
a common feature with all the phenomenological models based on leading-order
calculation. We therefore content ourselves with the usual assumption that the
differential amount of energy is somehow made available to the $W$-boson, when
quark-antiquark annihilation occurs with the disappearance of the meson-bound
state.

The paper is organised as follows. In Sec. II, we present the theoretical
framework that includes the description of the invariant transition matrix
element ${\cal M}_{fi}$ leading to the general expression of the decay widths
$\Gamma(D_{(s)}^{*+}\to l^{+}\nu_{l})$ and a brief account of the RIQ model
conventions, meson-bound-state and extraction of decay constants
$f_{D_{(s)}^{*+}}$ in terms of model quantities. Sec. III is devoted to the
numerical results and discussion. Finally, Sec. IV encompasses our summary and
conclusion.

##  II Theoretical framework

###  II.1 Invariant transition matrix amplitude and decay width:
$\Gamma(D_{(s)}^{*+}\to l^{+}\nu_{l})$

Since leptons are free from the strong interaction, the effective Hamiltonian
[38, 39] for PLDCVMs ($D_{(s)}^{*+}$) could be written as the product of quark
current and leptonic current,

${\cal
H}_{eff}=\frac{G_{F}}{\sqrt{2}}V_{cq^{{}^{\prime}}}[\bar{c}\gamma_{\mu}(1-\gamma_{5})q^{{}^{\prime}}][\bar{l}\gamma^{\mu}(1-\gamma_{5})\nu_{l}]+h.c.,$
(1)

where the contributions of the $W$ bosons are embodied in the Fermi coupling
constant $G_{F}$ and $V_{cq^{{}^{\prime}}}$ is the CKM matrix element between
the constituent quark $c$ and $q^{{}^{\prime}}(s,d)$ in the decaying charged
vector mesons ($D_{(s)}^{*+}$). The decay amplitude can therefore be written
as

${\cal M}_{fi}=\langle\bar{l}\nu_{l}|{\cal
H}_{eff}|D_{(s)}^{*+}\rangle=\frac{G_{F}}{\sqrt{2}}V_{cq^{{}^{\prime}}}\langle\bar{l}\nu_{l}|\bar{l}\gamma^{\mu}(1-\gamma_{5})\nu_{l}|0\rangle\langle
0|\bar{c}\gamma_{\mu}(1-\gamma_{5})q^{{}^{\prime}}|D_{(s)}^{*+}(\vec{k},\epsilon)\rangle.$
(2)

The leptonic part of amplitude can be calculated reliably with perturbative
theory. The hadronic matrix element (HME) interpolating the diquark current
between the vector meson $D_{(s)}^{*+}(\vec{k},\epsilon)$ and the vacuum
states, can be expressed in terms of a non-perturbative parameter called the
decay constant $f_{D_{(s)}^{*+}}$ as

$\displaystyle\langle
0|\bar{c}(0)\gamma_{\mu}q^{{}^{\prime}}(0)|D_{(s)}^{*+}(\vec{k},\epsilon)\rangle$
$\displaystyle=$ $\displaystyle
f_{D_{(s)}^{*+}}m_{D_{(s)}^{*+}}\epsilon_{\mu},$ $\displaystyle\langle
0|\bar{c}(0)\gamma_{\mu}\gamma_{5}q^{{}^{\prime}}(0)|D_{(s)}^{*+}(\vec{k},\epsilon)\rangle$
$\displaystyle=$ $\displaystyle 0.$ (3)

Here $m_{D_{(s)}^{*+}}$, $\vec{k}$ and $\epsilon_{\mu}$ are the mass, three
momentum and polarization vector of the decaying vector meson $D_{(s)}^{*+}$,
respectively.
In the present model calulation, the timelike component of HME vanishes.
Therefore, with the nonvanishing spacelike component of HME, the invariant
transition amplitude squared: $|{\cal M}_{fi}|^{2}$ is expressed in terms of
hadronic and leptonic tensor. The hadronic tensor $H_{ij}$ from Eq. (3) and
leptonic tensor $L^{ij}$ in our normalization are obtained in the form

$\displaystyle H_{ij}$ $\displaystyle=$
$\displaystyle{f^{2}_{D_{(s)}^{*+}}}{m^{2}_{D_{(s)}^{*+}}}\epsilon_{i}\epsilon_{j}^{\dagger},$
$\displaystyle L^{ij}$ $\displaystyle=$ $\displaystyle
8(k_{l}^{i}k_{\nu}^{j}-k_{l\alpha}k_{\nu}^{\alpha}g^{ij}+k_{l}^{j}k_{\nu}^{i}+i\epsilon^{i\alpha
j\beta}k_{l\alpha}k_{\nu\beta}),$ (4)

respectively. The decay width $\Gamma$ is then calculated from the generic
expression:

$\Gamma=\frac{1}{(2\pi)^{2}}\int\frac{d\vec{k}_{l}d\vec{k}_{\nu}}{2m_{D_{(s)}^{*+}}2E_{k_{l}}2E_{k_{\nu}}}\delta^{(4)}(k_{l}+k_{\nu}-{\hat{O}m_{D_{(s)}^{*+}}})\sum|{{\cal
M}_{fi}|^{2}}.$ (5)

We use two frames of reference to evaluate the contribution of Lorentz
invariant pieces ($L^{ij}$ and $H_{ij}$). For the sake of simplicity, we
evaluate the contribution of $L^{ij}$ in the lepton rest frame and that of
$H_{ij}$ in the parent meson rest frame to finally obtain the decay width in
the form

$\Gamma=\frac{G_{F}^{2}}{12\pi}|V_{cq^{{}^{\prime}}}|^{2}{m^{3}_{D_{(s)}^{*+}}}{f^{2}_{D_{(s)}^{*+}}}\left(1-\frac{m_{l}^{2}}{m^{2}_{D_{(s)}^{*+}}}\right)^{2}\left(1+\frac{m_{l}^{2}}{2m^{2}_{D_{(s)}^{*+}}}\right).$
(6)

###  II.2 Quark orbitals, meson-bound-state and decay constant in the RIQ
model

In the RIQ model, a meson is picturized as a color-singlet assembly of a quark
and an antiquark independently confined by an effective and average flavor-
independent potential in the form:
$U(r)=\frac{1}{2}(1+\gamma^{0})(ar^{2}+V_{0})$, where ($a$, $V_{0}$) are the
potential parameters. It is believed that the zeroth-order quark dynamics
generated by the phenomenological confining potential $U(r)$, taken in equally
mixed scalar-vector harmonic form, can provide an adequate tree-level
description of the decay process. With the interaction potential $U(r)$ put
into the zeroth-order quark lagrangian density, the ensuing Dirac equation
admits a static solution of positive and negative energy as:

$\displaystyle\psi^{(+)}_{\xi}(\vec{r})\;$ $\displaystyle=$
$\displaystyle\;\left(\begin{array}[]{c}\frac{ig_{\xi}(r)}{r}\\\
\frac{{\vec{\sigma}}.{\hat{r}}f_{\xi}(r)}{r}\end{array}\;\right){{\chi}_{l,j,m_{j}}}(\hat{r}),$
(9) $\displaystyle\psi^{(-)}_{\xi}(\vec{r})\;$ $\displaystyle=$
$\displaystyle\;\left(\begin{array}[]{c}\frac{i({\vec{\sigma}}.{\hat{r}})f_{\xi}(r)}{r}\\\
\frac{g_{\xi}(r)}{r}\end{array}\;\right)\
{\tilde{\chi}}_{l,j,m_{j}}(\hat{r}),$ (12)

where $\xi=(nlj)$ represents a set of Dirac quantum numbers specifying the
eigenmodes. $\chi_{ljm_{j}}(\hat{r})$ and ${\tilde{\chi}}_{ljm_{j}}(\hat{r})$
are the spin angular parts given by,

$\displaystyle\chi_{ljm_{j}}(\hat{r})$ $\displaystyle=$
$\displaystyle\sum_{m_{l},m_{s}}<lm_{l}\;{1\over{2}}m_{s}|jm_{j}>Y_{l}^{m_{l}}(\hat{r})\chi^{m_{s}}_{\frac{1}{2}},$
$\displaystyle{\tilde{\chi}}_{ljm_{j}}(\hat{r})$ $\displaystyle=$
$\displaystyle(-1)^{j+m_{j}-l}\chi_{lj-m_{j}}(\hat{r}).$ (13)

With the quark binding energy parameter $E_{q}$ and quark mass parameter
$m_{q}$ written in the form $E_{q}^{\prime}=(E_{q}-V_{0}/2)$,
$m_{q}^{\prime}=(m_{q}+V_{0}/2)$ and
$\omega_{q}=E_{q}^{\prime}+m_{q}^{\prime}$, one can obtain solutions to the
radial equation for $g_{\xi}(r)$ and $f_{\xi}(r)$ in the form:

$\displaystyle g_{nl}$ $\displaystyle=$ $\displaystyle
N_{nl}\Big{(}\frac{r}{r_{nl}}\Big{)}^{l+1}\exp(-r^{2}/2r^{2}_{nl})L_{n-1}^{l+1/2}(r^{2}/r^{2}_{nl}),$
$\displaystyle f_{nl}$ $\displaystyle=$
$\displaystyle\frac{N_{nl}}{r_{nl}\omega_{q}}\Big{(}\frac{r}{r_{nl}}\Big{)}^{l}\exp(-r^{2}/2r^{2}_{nl})$
(14) $\displaystyle\times$
$\displaystyle\left[\Big{(}n+l-\frac{1}{2}\Big{)}L_{n-1}^{l-1/2}(r^{2}/r^{2}_{nl})+nL_{n}^{l-1/2}(r^{2}/r^{2}_{nl})\right],$

where $r_{nl}=(a\omega_{q})^{-1/4}$ is a state independent length parameter,
$N_{nl}$ is an overall normalization constant given by

$N^{2}_{nl}=\frac{4\Gamma(n)}{\Gamma(n+l+1/2)}\frac{(\omega_{nl}/r_{nl})}{(3E_{q}^{\prime}+m_{q}^{\prime})},$
(15)

and $L_{n-1}^{l+1/2}(r^{2}/r_{nl}^{2})$ are associated Laguerre polynomials.
The radial solutions yield an independent quark bound-state condition in the
form of a cubic equation:

$\sqrt{(\omega_{q}/a)}(E_{q}^{\prime}-m_{q}^{\prime})=(4n+2l-1).$ (16)

In the relativistic independent particle picture of this model, the
relativistic constituent quark and antiquark are thought to move independently
inside the meson bound-state $|D_{(s)}^{*+}(\vec{k},S_{D_{(s)}^{*+}})\rangle$
with their momentum $\vec{p}_{c}$ and ${\vec{p}}_{q^{{}^{\prime}}}$,
respectively. For the study of the decay process that takes place in the
momentum eigenstates of participating mesons, we take the momentum space
projection of the quark orbitals and obtain the momentum probability amplitude
of the constituent quark and antiquark: ($G_{c}(\vec{p}_{c})$ and
${\tilde{G}}_{q^{{}^{\prime}}}(\vec{p}_{q^{{}^{\prime}}})$) in the following
form:
For the ground state mesons:($n=1$,$l=0$),

$\displaystyle G_{c}(\vec{p}_{c})$ $\displaystyle={{i\pi{\cal
N}_{c}}\over{2\alpha_{c}\omega_{c}}}\sqrt{{(E_{p_{c}}+m_{c})}\over{E_{p_{c}}}}(E_{p_{c}}+E_{c})\times\exp{\Big{(}-{{\vec{p_{c}}}^{2}\over{4\alpha_{c}}}\Big{)}},$
$\displaystyle{\tilde{G}}_{q^{{}^{\prime}}}(\vec{p}_{q^{{}^{\prime}}})$
$\displaystyle=-{{i\pi{\cal
N}_{q^{{}^{\prime}}}}\over{2\alpha_{q^{{}^{\prime}}}\omega_{q^{{}^{\prime}}}}}\sqrt{{(E_{p_{q^{{}^{\prime}}}}+m_{q^{{}^{\prime}}})}\over{E_{p_{q^{{}^{\prime}}}}}}(E_{p_{q^{{}^{\prime}}}}+E_{q^{{}^{\prime}}})\times\exp{\Big{(}-{{\vec{p}}_{q^{{}^{\prime}}}^{2}\over{4\alpha_{q^{{}^{\prime}}}}}\Big{)}}.$
(17)

In this model an effective momentum distribution function for the meson state
is taken in the form ${\cal G}(\vec{p}_{c},\
\vec{p}_{q^{{}^{\prime}}})=\sqrt{G_{c}(\vec{p}_{c})G_{q^{{}^{\prime}}}(\vec{c}_{q^{{}^{\prime}}})}$
in the light of the ansatz of Margolis and Mendel in their bag model
description of the meson bound state [40]. With the momentum distribution
function ${\cal G}(\vec{p}_{c},\ \vec{p}_{q^{{}^{\prime}}})$ that embodies the
bound-state characteristics of
$|D_{(s)}^{*+}(\vec{k},S_{D_{(s)}^{*+}})\rangle$, the meson-bound-state at
definite momentum $\vec{k}$ and spin projection $S_{D_{(s)}^{*+}}$, is
constructed in the present model in the form:

$\displaystyle|D_{(s)}^{*+}(\vec{k},S_{D_{(s)}^{*+}})\rangle$
$\displaystyle={\hat{\Lambda}}(\vec{k},S_{D_{(s)}^{*+}})|(\vec{p}_{c},\lambda_{c});(\vec{p}_{q^{{}^{\prime}}},\lambda_{q^{{}^{\prime}}})\rangle$
(18)
$\displaystyle={\hat{\Lambda}}(\vec{k},S_{D_{(s)}^{*+}})\hat{b}^{\dagger}_{c}(\vec{p}_{c},\lambda_{c})\hat{\tilde{b}}^{\dagger}_{q^{{}^{\prime}}}(\vec{p}_{q^{{}^{\prime}}},\lambda_{q^{{}^{\prime}}})|0\rangle,$

where
$|(\vec{p}_{c},\lambda_{c});(\vec{p}_{q^{{}^{\prime}}},\lambda_{q^{{}^{\prime}}})\rangle$
is the Fock-space representation of unbound quark and antiquark in a color-
singlet configuration with respective momentum and spin:
$(\vec{p}_{c},\lambda_{c})$ and
$(\vec{p}_{q^{{}^{\prime}}},\lambda_{q^{{}^{\prime}}})$.
$\hat{b}^{\dagger}_{c}(\vec{p}_{c},\lambda_{c})$ and
$\hat{\tilde{b}}^{\dagger}_{q^{{}^{\prime}}}(\vec{p}_{q^{{}^{\prime}}},\lambda_{q^{{}^{\prime}}})$
are the quark and antiquark creation operators and
${\hat{\Lambda}}(\vec{k},S_{D_{(s)}^{*+}})$ is a bag-like integral operator
taken in the form:

$\displaystyle{\hat{\Lambda}}(\vec{k},S_{D_{(s)}^{*+}})=\frac{\sqrt{3}}{\sqrt{N_{{D_{(s)}^{*+}}}(\vec{k})}}\sum_{\lambda_{c},\lambda_{q^{{}^{\prime}}}}\zeta_{c,{q^{{}^{\prime}}}}^{{D_{(s)}^{*+}}}\int
d^{3}\vec{p}_{c}\
d^{3}\vec{p}_{q^{{}^{\prime}}}\delta^{(3)}(\vec{p}_{c}+\vec{p}_{q^{{}^{\prime}}}-\vec{k})\
{\cal G}_{{D_{(s)}^{*+}}}(\vec{p}_{b},\vec{p}_{q^{{}^{\prime}}}).$ (19)

Here, $\sqrt{3}$ is the effective color factor,
$\zeta_{c,q^{{}^{\prime}}}^{{D_{(s)}^{*+}}}$ is the $SU(6)$ spin-flavor
coefficients for the meson state
$|{D_{(s)}^{*+}}(\vec{k},S_{D_{(s)}^{*+}})\rangle$ and
$N_{{D_{(s)}^{*+}}}(\vec{k})$ is the meson state normalization which is
obtained via the normalization condition
$\langle{D_{(s)}^{*+}}(\vec{k})|{D_{(s)}^{*+}}(\vec{k}^{{}^{\prime}})\rangle=(2\pi)^{3}{2E_{k}}\delta^{(3)}{(\vec{k}-\vec{k}^{{}^{\prime}})}$
in an integral form:

$N_{{D_{(s)}^{*+}}}(\vec{k})=\frac{1}{(2\pi)^{3}{2E_{k}}}\int
d^{3}\vec{p}_{c}\ |{\cal G}_{{D_{(s)}^{*+}}}(\vec{p}_{c},\
\vec{p}_{q^{{}^{\prime}}})|^{2}=\frac{{\bar{N}_{{D_{(s)}^{*+}}}(\vec{k})}}{(2\pi)^{3}{2E_{k}}}.$
(20)

For the sake of convenience, we consider the PLDCVMs: ($D_{(s)}^{*+}$) in the
parent meson rest frame. Accordingly, we put $\vec{k}=0$ in the relevant terms
althrough and then calculate the decay constants $f_{D_{(s)}^{*+}}$ by
substituting Eqs. (13-15) in (3) to find

$\displaystyle f_{D_{(s)}^{*+}}=$
$\displaystyle\frac{2\sqrt{3}}{\sqrt{(2\pi)^{3}{\bar{N}_{{D_{(s)}^{*+}}}(0)}m_{{D_{(s)}^{*+}}}}}{\int\frac{d\vec{p}_{c}}{\sqrt{2E_{p_{c}}2E_{-{p_{c}}}}}}{{\cal
G}_{{D_{(s)}^{*+}}}(\vec{p}_{c},-\vec{p}_{c}})$ (21)
$\displaystyle\times\sqrt{(E_{p_{c}}+m_{c})(E_{-{p_{c}}}+m_{q^{{}^{\prime}}})}{\left(1+\frac{|\vec{p}_{c}|^{2}}{3(E_{p_{c}}+m_{c})(E_{-{p_{c}}}+m_{q^{{}^{\prime}}})}\right)}.$

##  III Numerical results and discussion

In this section, we evaluate the decay constants: $f_{D_{(s)}^{*+}}$, partial
decay widths: $\Gamma({D_{(s)}^{*+}}\to l^{+}\nu_{l})$ and corresponding BFs:
${\cal B}({D_{(s)}^{*+}}\to l^{+}\nu_{l})$ in the RIQ model framework. For
numerical analysis, we use the following model parameters ($a,V_{0}$), and
quark mass $\&$ quark binding energy parameters ($m_{q}$, $E_{q}$).

$\displaystyle(a,V_{0})$ $\displaystyle=$ $\displaystyle(0.017166\
GeV^{3},-0.1375\ GeV)$ $\displaystyle(m_{c},m_{s},m_{d})$ $\displaystyle=$
$\displaystyle(1.49276,0.31575,0.07875)\ GeV$
$\displaystyle(E_{c},E_{s},E_{d})$ $\displaystyle=$
$\displaystyle(1.57951,0.591,0.47125)\ GeV$ (22)

The input parameters (17), already fixed in this model from hadron
spectroscopy by fitting the data of light and heavy-flavored mesons [37, 41,
42], have been used in the description of radiative weak and radiative
leptonic decays of charm and charmed strange mesons [43, 44] as well as
radiative, rare radiative, radiative leptonic, electromagnetic, semileptonic
and nonleptonic decays of $b$-flavored mesons [45, 46, 47, 48, 49, 50, 51, 52,
53, 54, 55]. The meson masses ($m_{D_{(s)}^{*+}}$), considered as
phenomenological inputs and other relevant numerical inputs, used in the
present calculation, are shown in Table I.

Table 1: Numerical Inputs $G_{F}$ | $1.166\times 10^{-5}$ | GeV-2 | [56]
---|---|---|---
$|V_{cd}|$ | $0.22486\pm 0.00067$ | …… | [56]
$|V_{cs}|$ | $0.97349\pm 0.00016$ | …… | [56]
$m_{D^{*+}}$ | $2.0102\pm 0.00005$ | GeV | [56]
$m_{D_{s}^{*+}}$ | $2.1122\pm 0.0004$ | GeV | [56]

Figure 1: The predicted decay constants in comparison with the observed value
from BESIII Collaboration and theoretical predictions from LQCD, LFQM and NRQM
approaches.

Using the input parameters (17), we evaluate the decay constants:
$f_{D_{(s)}^{*+}}$ from the expression in Eq. (16) and our predictions with
corresponding uncertainties are:

$\displaystyle f_{D^{*+}}$ $\displaystyle=$ $\displaystyle 197\pm 14\ MeV,$
$\displaystyle f_{D_{s}^{*+}}$ $\displaystyle=$ $\displaystyle 236\pm 19\
MeV.$ (23)

The uncertainties in our predictions (18) are obtained by varying all the
input parameters simultaneously within $\pm 10\%$ of the central values and
taking the largest variation of the decay constant. Our predicted decay
constant values compare well with the latest observed data for
$f_{D_{s}^{*+}}$ from the BESIII measurement [1] and other theoretical
predictions based on LQCD [23, 24, 26], LFQM [15, 16, 17, 18, 22] and NRQM [5,
8] calculations as shown in Table II. For the sake of illustration we also
provide in Fig.1 a comparison of our predicted decay constants with that from
the BESIII measurement [1] and LQCD [24, 26], LFQM [15, 17, 18] and NRQM [5]
calculations.

Table 2: Predicted Decay Constants (in the unit of MeV) in comparison with available experimental data and other theoretical predictions. $f_{D^{*+}}$ | Reference | $f_{D_{s}^{*+}}$ | Reference
---|---|---|---
$197\pm 14$ | This work | $236\pm 19$ | This work
$234\pm 26$ | LQCD [24] | $(213.6^{+61.0}_{-45.8_{stat.}}{\pm 43.9}_{syst.})$ | BESIII Expt. [1]
$223.5\pm 8.7$ | LQCD [26] | $272\pm 16^{+3}_{-20}$ | LQCD [23]
$226.6^{+5.9}_{-10.2}$ | LFQM [17] | $254\pm 17$ | LQCD [24]
$230\pm 29$ | LFQM [18] | $268.8\pm 6.6$ | LQCD [26]
186 | LFQM [22] | $254.7^{+6.3}_{-6.7}$ | LFQM [15]
$223^{+23}_{-19}$ | NRQM [5] | $272^{+39}_{-38}$ | LFQM [16]
210 | NRQM [8] | $253\pm 32$ | LFQM [18]
|  | 240 | LFQM [22]
|  | 212 | NRQM [8]

The ratios of vector to pseudoscalar decay constants, which are sensitive to
the difference between the vector and pseudoscalar wave functions, are also
estimated in the present calculation. Our estimates:
$\frac{f_{D^{*+}}}{f_{D^{+}}}=1.22(11)$ and
$\frac{f_{D_{s}^{*+}}}{f_{D_{s}^{+}}}=1.15(11)$ are in reasonable agreement
with corresponding results: 1.078(36) and 1.087(20) of the lattice calculation
by the ETM Collaboration [26].

The CKM matrix elements $|V_{cd}|$ and $|V_{cs}|$, that enter into the
calculation of partial decay widths $\Gamma(D^{*+}\to l^{+}\nu_{l})$ and
$\Gamma(D_{s}^{*+}\to l^{+}\nu_{l})$, come mainly from their leptonic and
semileptonic decays. For the CKM matrix elements from purely leptonic decays,
experimentally the statistical uncertainties are dominant and those for the
semileptonic decays, the theoretical uncertainties from the form factors,
controlled by nonperturbative dynamics, are dominant uncertainties. The
experimental uncertainties have not decreased significantly recently.
Therefore, in the present calculation, we consider the most precise values of
CKM matrix elements from the global fit in SM i.e. $|V_{cd}|=0.22486(67)$ and
$|V_{cs}|=0.97349(16)$ [56]. With these CKM matrix elements and our predicted
decay constants: $f_{D_{(s)}^{*+}}$ (18), we calculate the partial decay
widths: $\Gamma(D_{(s)}^{*+}\to l^{+}\nu_{l})$ and our predictions are:

$\displaystyle\Gamma(D^{*+}\to e^{+}\nu_{e})$ $\displaystyle=$ $\displaystyle
57.523\pm 8.183\ \mu eV,$ $\displaystyle\Gamma(D^{*+}\to\mu^{+}\nu_{\mu})$
$\displaystyle=$ $\displaystyle 57.284\pm 8.149\ \mu eV,$
$\displaystyle\Gamma(D^{*+}\to\tau^{+}\nu_{\tau})$ $\displaystyle=$
$\displaystyle 3.827\pm 0.544\ \mu eV,$ (24)

and

$\displaystyle\Gamma(D_{s}^{*+}\to e^{+}\nu_{e})$ $\displaystyle=$
$\displaystyle 1.802\pm 0.275\ meV,$
$\displaystyle\Gamma(D_{s}^{*+}\to\mu^{+}\nu_{\mu})$ $\displaystyle=$
$\displaystyle 1.795\pm 0.274\ meV,$
$\displaystyle\Gamma(D_{s}^{*+}\to\tau^{+}\nu_{\tau})$ $\displaystyle=$
$\displaystyle 0.208\pm 0.032\ meV.$ (25)

The partial decay widths for the PLDCVMs (19, 20) are expected to be larger
than those for the PLDCPMs. This is because the partial decay widths for
PLDCVMs do not suffer from the helicity suppression as in the case of PLDCPMs.
In addition, the relatively larger mass and decay constant of the vector
mesons, compared to those of corresponding pseudoscalar mesons also contribute
to the enhancement of their partial decay widths. In the present analysis, our
predicted partial decay widths: $\Gamma(D^{*+}\to l^{+}\nu_{l})$ and
$\Gamma(D_{s}^{*+}\to l^{+}\nu_{l})$ are found $\sim 2$ and $\sim 3$ orders of
magnitude higher than $\Gamma(D^{+}\to l^{+}\nu_{l})$ and $\Gamma(D_{s}^{+}\to
l^{+}\nu_{l})$, respectively, as expected.

For evaluation of BFs for PLDCVMs, we take the observed value of the total
decay width $\Gamma^{total}_{D^{*+}}=83.4\pm 1.8$ keV from PDG [56]. Since
$\Gamma^{total}_{D_{s}^{*+}}$ has not yet been observed, we consider its
value: $0.070\pm 0.028$ keV [28] from LQCD calculation. Our predicted BFs for
different lepton pairs are cited in Table III in good comparison with the
observed values from the BESIII measurement [1] and other theoretical
predictions [57].

Table 3: The predicted BFs ${\cal B}(D_{(s)}^{*+}\to l^{+}\nu_{l})$ in
comparison with the observed value and other theoretical predictions.

${\cal B}(V\to l^{+}\nu_{l})$ |  0000 This work |  000000 [39] |  000000 [57] |  00000Experiment [1]
---|---|---|---|---
${\cal B}(D^{*+}\to e^{+}\nu_{e})$ | $(6.897\pm 0.992)\times 10^{-10}$ | $({9.5}^{+2.9}_{-2.4})\times 10^{-10}$ | $(11\pm 1)\times 10^{-10}$ |  ——
${\cal B}(D^{*+}\to\mu^{+}\nu_{\mu})$ | $(6.869\pm 0.988)\times 10^{-10}$ | $({9.5}^{+2.9}_{-2.4})\times 10^{-10}$ | $(11\pm 1)\times 10^{-10}$ |  ——
${\cal B}(D^{*+}\to\tau^{+}\nu_{\tau})$ | $(0.459\pm 0.067)\times 10^{-10}$ | $(0.6\pm 0.2)\times 10^{-10}$ | $(0.72\pm 0.08)\times 10^{-10}$ |  ——
${\cal B}(D_{s}^{*+}\to e^{+}\nu_{e})$ | $(2.574\pm 1.102)\times 10^{-5}$ | $(6.7\pm 0.4)\times 10^{-6}$ | $(3.1\pm 0.4)\times 10^{-6}$ | $(2.1^{+1.2}_{{-0.9}_{stat.}}\pm{0.2}_{syst.})\times 10^{-5}$
${\cal B}(D_{s}^{*+}\to\mu^{+}\nu_{\mu})$ | $(2.565\pm 1.098)\times 10^{-5}$ | $(6.7\pm 0.4)\times 10^{-6}$ | $(3.1\pm 0.4)\times 10^{-6}$ |  ——
${\cal B}(D_{s}^{*+}\to\tau^{+}\nu_{\tau})$ | $(0.298\pm 0.127)\times 10^{-5}$ | $(0.78\pm 0.04)\times 10^{-6}$ | $(0.36\pm 0.04)\times 10^{-6}$ |  ——

In the present analysis, we evaluate the ratios of BFs: ${\cal
R}_{D_{(s)}^{*+}}=\frac{{\cal B}(D_{(s)}^{*+}\to\tau^{+}\nu_{\tau})}{{\cal
B}(D_{(s)}^{*+}\to l^{+}\nu_{l})}$ with $l=e,\mu$. Our predictions: ${\cal
R}_{D^{*+}}=0.0665$ and ${\cal R}_{D_{s}^{*+}}=0.1157$ are in good agreement
with the SM predictions including those of the LQCD [27] and LFQM [18]
calculations.

##  IV Summary and Conclusion

In the present work, we study the purely leptonic decays of charged vector
mesons: $D_{(s)}^{*+}$ in the framework of the RIQ model based on a flavor
independent average interaction potential in equally mixed scalar-vector
harmonic form. The relevant decay constants: $f_{D_{(s)}^{*+}}$ representing
the decay amplitudes are calculated from the overlapping integral between the
decaying meson and vacuum states, using the constituent quark ($q_{1}q_{2}$)
wavefunctions derivable from the RIQ model dynamics. Our predictions:
$f_{D^{*+}}=197\pm 14\ $MeV and $f_{D_{s}^{*+}}=236\pm 19\ $ MeV compare well
with the available experimental data and other model/theoretical predictions
based on LQCD, LFQM and NRQM approaches. Our predicted ratios of vector to
pseudoscalar decay constants: $\frac{f_{D^{*+}}}{f_{D^{+}}}=1.22(11)$ and
$\frac{f_{D_{s}^{*+}}}{f_{D_{s}^{+}}}=1.15(11)$ are found in reasonable
agreement with the calculated values of the ETM Collaborations based on their
lattice calculation. We have also predicted the partial decay widths for the
PLDCVMs and that of the PLDCPMs. Our predicted partial decay widths
$\Gamma(D^{*+}\to l^{+}\nu_{l})$ and $\Gamma(D_{s}^{*+}\to l^{+}\nu_{l})$ are
found larger about $\sim 2$ and $\sim 3$ order of magnitude higher than
$\Gamma(D^{+}\to l^{+}\nu_{l})$ and $\Gamma(D_{s}^{+}\to l^{+}\nu_{l})$,
respectively, as expected.

Using the total decay width $\Gamma^{total}_{D^{*+}}=83.4\pm 1.8$ keV from PDG
(2022) and $\Gamma^{total}_{D_{s}^{*+}}=0.070\pm 0.028$ keV from the LQCD
calculation, we predict the BFs: ${\cal B}(D_{(s)}^{*+}\to l^{+}\nu_{l})$
separately for different lepton pairs. Our results are obtained in good
comparison with the recently observed value from the BESIII measurement and
other theoretical predictions based on LQCD and LFQM calculations. Our
predicted ratios of branching fractions ($R_{D_{(s)}^{*+}}$) corresponding to
the decays in their $\tau^{-}$ mode to that in $e^{-}$ mode, which are found
to be $R_{D^{*+}}=0.0665$ and $R_{D_{s}^{*+}}=0.1157$, also agree well with
those from LQCD and LFQM calculation. The potential prospects of precision
measurements, increase of data statistics and improved analytical techniques
in the high energy experimental frontier, are likely to yield BFs for PLDCVMs
including $D^{*+}\to l^{+}\nu_{l}$, which has not yet been observed, even
close to the accessible limits of the existing and upcoming experiments. The
PLDCVMs, thus would continue to provide a fascinating area of experimental and
theoretical research in the charm flavor sector.

## Acknowledgments

The library and computational facilities provided by authorities of Siksha ’O’
Anusandhan Deemed to be University, Bhubaneswar, 751030, India are duly
acknowledged.

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