# Radiative decays $D^{*}_{(s)}\to D_{(s)}\gamma$ in covariant confined quark
model

C. T. Tran<EMAIL_ADDRESS>(corresponding author) Department of
Physics, HCMC University of Technology and Education,
Vo Van Ngan 1, 700000 Ho Chi Minh City, Vietnam  M. A. Ivanov
<EMAIL_ADDRESS>Bogoliubov Laboratory of Theoretical Physics, Joint
Institute for Nuclear Research, 141980 Dubna, Russia  P. Santorelli
<EMAIL_ADDRESS>Dipartimento di Fisica “E. Pancini”, Università
di Napoli Federico II,  Complesso Universitario di Monte S. Angelo, Via
Cintia, Edificio 6, 80126 Napoli, Italy Istituto Nazionale di Fisica Nucleare,
Sezione di Napoli, 80126 Napoli, Italy Q. C. Vo  University of Science, 700000
Ho Chi Minh City, Vietnam

###### Abstract

Radiative decays $D^{*}_{(s)}\to D_{(s)}\gamma$ are revisited in light of new
experimental data from BABAR and BESIII collaborations. The radiative
couplings $g_{D^{*}D\gamma}$ encoding nonperturbative QCD effects are
calculated in the framework of the covariant confined quark model developed by
us. We compare our results with other theoretical studies and experimental
data. The couplings (in $\textrm{GeV}^{-1}$) $|g_{D^{*+}D^{+}\gamma}|=0.45(9)$
and $|g_{D^{*0}D^{0}\gamma}|=1.72(34)$ calculated in our model agree with the
experimental data $|g_{D^{*+}D^{+}\gamma}|=0.47(7)$ and
$|g_{D^{*0}D^{0}\gamma}|=1.77(16)$. The most interesting case is the decay
$D^{*}_{s}\to D_{s}\gamma$, for which the recent prediction based on light-
cone sum rules at next-to-leading order $|g_{D^{*}_{s}D_{s}\gamma}|=0.60(19)$
deviates from the first (and so far the only) lattice QCD result
$|g_{D^{*}_{s}D_{s}\gamma}|=0.11(2)$ at nearly $3\sigma$. Our calculation
yields $|g_{D^{*}_{s}D_{s}\gamma}|=0.29(6)$, which falls somehow between the
two mentioned results, but is larger than those predicted in other studies
using quark models or QCD sum rules.

covariant quark model, radiative decay, charmed meson

###### pacs:

13.20.Fc, 12.39.Ki

## I Introduction

Decays of $D^{*}$ mesons are dominated by the radiative $D^{*}\to D\gamma$
($\gamma$ emission) and strong $D^{*}\to D\pi$ ($\pi$ emission) processes. The
hadronic effects in these decays are encoded in the so-called magnetic moment
coupling $g_{D^{*}D\gamma}$ and the axial coupling $g_{D^{*}D\pi}$,
respectively. Due to experimental difficulties, data on their width is rare.
The first measurement of the width of a vector $D$ meson was done by the CLEO
collaboration for the charged $D^{*+}$ meson. They observed
$\Gamma(D^{*+})=(96\pm 4\pm 22)\,\textrm{keV}$ [1], where the uncertainties
are statistical and systematic, respectively. In 2013, the BABAR collaboration
obtained the width with much better precision: $\Gamma(D^{*+})=(83.3\pm 1.2\pm
1.4)\,\textrm{keV}$ [2]. In 2014, the BESIII collaboration reported their
measurement of the branching fractions of the $D^{*0}$ decays to be
$\mathcal{B}(D^{*0}\to D^{0}\pi^{0})=(65.5\pm 0.8\pm 0.5)\%$ and
$\mathcal{B}(D^{*0}\to D^{0}\gamma)=(34.5\pm 0.8\pm 0.5)\%$, assuming the
dominance of these two channels [3]. This is the most precise measurement of
the branching fractions up to now. Very recently, the BESIII collaboration has
announced the measurement of the branching of $D_{s}^{*+}\to D_{s}^{+}\pi^{0}$
relative to $D_{s}^{*+}\to D_{s}^{+}\gamma$ to be $(6.16\pm 0.43\pm 0.19)\%$
[4]. This measurement confirmed and made more accurate the previous BABAR’s
result [5]. It should be noted that the decay $D_{s}^{*+}\to D_{s}^{+}\pi^{0}$
violates isospin symmetry and is highly suppressed as compared to the
dominating mode $D_{s}^{*+}\to D_{s}^{+}\gamma$. The mechanism of
$D_{s}^{*+}\to D_{s}^{+}\pi^{0}$ is still an open question, but obviously not
the same $\pi$ emission as for $D^{*+}\to(D\pi)^{+}$ and $D^{*0}\to
D^{0}\pi^{0}$. It can be seen that in light of these new experimental data, as
well as data to be appreared in the near future from BESIII, the study of
$D^{*}$-meson radiative decays is fruitful and promising.

Radiative decays of $D^{*}$ meson can shed light on the dynamics of many
hadronic processes. They provide tests of various non-perturbative approaches
in QCD. This explains the large amount of theoretical attempts in the
calculation of these decays, especially prior to the CLEO first measurement of
$\Gamma(D^{*})$. The radiative decays $D^{*}\to D\gamma$ have been studied in
the framework of QCD sum rules (QCDSR) [6, 7, 8, 9], light cone sum rules
(LCSR) [11, 10], and various quark models [12, 13, 15, 14, 17, 16, 18, 19, 20,
21, 22]. Lattice QCD (LQCD) calculation has been carried out for the decays
$D^{*}\to D\gamma$ [23] and $D^{*}_{s}\to D_{s}\gamma$ [24]. However, the
theoretical uncertainties in these calculations are still large. Recently, the
first complete next-to-leading order (NLO) computation of the decays $D^{*}\to
D\gamma$ using LCSR has been provided [25]. In Ref. [25], the authors have
pointed out some tension between the first LQCD prediction [24] of
$g_{D_{s}^{*}D_{s}\gamma}=0.11(2)\,\textrm{GeV}^{-1}$ and the current
experimental value $g_{D^{*}D\gamma}=-0.47(7)\,\textrm{GeV}^{-1}$, assumming
the level of $D$-spin breaking to be no more than 20–30%. Moreover, the LQCD
result [24] predicted the decay width to be $\Gamma(D_{s}^{*}\to
D_{s}\gamma)=0.066(26)$ keV. This value disagrees with the result obtained in
light front quark model $\Gamma(D_{s}^{*}\to D_{s}\gamma)=0.18(1)$ keV [21] by
$4\sigma$, and with the result in QCDSR $\Gamma(D_{s}^{*}\to
D_{s}\gamma)=0.25(8)$ keV [7] by $2\sigma$. The central value of this
prediction is one order of magnitude smaller than most of other predictions
available in the literature. Given that $D^{*}_{s}\to D_{s}\gamma$ dominates
the decay of $D^{*}_{s}$ meson, it is interesting to look more carefully into
this case.

In this paper, we revisit the radiative decays of the $D^{*}$ mesons in the
framework of the covariant confined quark model (CCQM) developed previously by
our group. This is a model based on quantum field theory, in which the
interaction between the constituent quarks and the corresponding hadronic
bound state is described by a Lagrangian in the form of a hadron field coupled
to a quark current. Our model has the advantages of being able to treat
multiquark states in a similar manner and to obtain observables in the full
physical range without any extrapolation. We aim to provide predictions for
both $D^{*}$ and $D_{s}^{*}$ decays, but will focus more on the decay
$D^{*}_{s}\to D_{s}\gamma$, for which theoretical predictions still vary
largely, and experimental data on its width seems to be available soon.

The rest of the paper is organized as follows. In Sec. II we briefly introduce
the covariant confined quark model. The calculation of the radiative decays
$D^{*}_{(s)}\to D\gamma$ in our model is demonstrated in Sec. III. Numerical
results are presented in Sec. IV. We also provide a detailed comparison of our
results with experimental data and other theoretical studies. Finally, a brief
summary is given in Sec. V.

## II The Covariant Confined Quark Model

The covariant confined quark model (CCQM) is a model based on quantum field
theory. In this model, hadronic bound states are described by quantum fields
that interact with their constituent quarks via an interaction Lagrangian. In
the case of a meson $M(q_{1}\bar{q}_{2})$, this Lagrangian has the form

$\mathcal{L}_{\mathrm{int}}(x)=g_{M}M(x)J(x)+\mathrm{H.c.},\qquad J(x)=\int
dx_{1}\int
dx_{2}F_{M}(x;x_{1},x_{2})[\bar{q}_{2}(x_{2})\Gamma_{M}q_{1}(x_{1})],$ (1)

where $\Gamma_{M}$ is the relevant Dirac matrix and $g_{M}$ is the meson-quark
coupling constant. The vertex function $F_{M}(x;x_{1},x_{2})$ effectively
represents the finite size of the meson and the relative quark-hadron
position. The vertex function has to obey the identity
$F_{M}(x+a;x_{1}+a,x_{2}+a)=F_{M}(x;x_{1},x_{2})$ for any given four-vector
$a$ so as to satisfy the translational invariance. It’s form is chosen as
follows

$F_{M}(x;x_{1},x_{2})=\delta^{(4)}(x-\omega_{1}x_{1}-\omega_{2}x_{2})\Phi_{M}[(x_{1}-x_{2})^{2}],$
(2)

where $\omega_{i}=m_{q_{i}}/(m_{q_{1}}+m_{q_{2}})$ with $m_{q_{i}}$ being the
constituent quark mass. One therefore has $\omega_{1}+\omega_{2}=1$. This
chosen form satisfies the translational invariance mentioned above. Besides,
it corresponds to an effective description of a meson. The Dirac delta
function effectively represents the barycenter of the two-quark system. The
function $\Phi_{M}[(x_{1}-x_{2})^{2}]$ depends on the distance between the two
quarks, which can be understood as the effective size of the meson. The
Fourier transform of the function $\Phi_{M}[(x_{1}-x_{2})^{2}]$, which we
denote as $\widetilde{\Phi}_{M}(-k^{2})$, in principle can be calculated from
the solutions of the Bethe-Salpeter equation for meson bound states. In a
series of our previous papers [26, 27, 28], by trying various forms for the
vertex function, we have found that the hadronic observables are insensitive
to the details of the functional form of the quark-hadron vertex function.
This observation is used as a guiding principle, and the function
$\Phi_{M}[(x_{1}-x_{2})^{2}]$ is assumed to be Gaussian for simplicity, and is
written in the momentum representation as

$\widetilde{\Phi}_{M}(-k^{2})=\exp(k^{2}/\Lambda^{2}_{M}).$ (3)

The size parameter $\Lambda_{M}$ of the meson $M$ is one of the free
parameters of the model. We use the minus sign in the argument of the function
$\widetilde{\Phi}_{M}(-k^{2})$ to emphasize that we are working in Minkowski
space. Since $k^{2}$ turns into $-k^{2}$ in Euclidean space, the Gaussian
function $\exp(k^{2}/\Lambda^{2}_{M})$ falls off appropriately in the
Euclidean region. We stress again that any choice for $\widetilde{\Phi}_{M}$
is applicable as long as it has the sufficiently fast fall-off behavior in the
ultraviolet region of Euclidean space in order to render the ultraviolet
finite of Feynman diagrams.

The normalization of particle-quark vertices is provided by the compositeness
condition [29, 30]

$Z_{M}=1-\widetilde{\Pi}^{\prime}_{M}(m^{2}_{M})=0,$ (4)

where $Z_{M}$ is the wave function renormalization constant of the meson $M$
and $\widetilde{\Pi}^{\prime}_{M}$ is the derivative of the meson mass
function taken on the mass-shell $p^{2}=m_{M}^{2}$. It is worth mentioning
that the square root of the renormalization constant $Z_{M}^{1/2}$ can be
interpreted as the matrix element between the physical state and the
corresponding bare state. When $Z_{M}=0$, the physical state does not contain
the bare state, and is therefore described as a bound state. The Lagrangian
given in Eq. (1) describes the interaction of the constituents (quarks) and
the corresponding physical state (meson) which is the bound state of the
constituents. As a result of the interaction, the physical state is dressed,
and its mass and wave function have to be renormalized.

Note that even before the rise of QCD it was well understood that describing
composite particles such as hadrons is not an easy task within QFT. In QFT,
free fields are quantized by imposing (anti)commutator relations between
creation and annihilation operators, which act on vacuum and help construct
the asymptotic in- and out-states. Physical processes are described by the
S-matrix elements which are convolutions of propagators. The Lagrangian
describing free fields and their interactions must be renormalized, or in
other words, bare or unrenormalized quantities such as wave functions,
coupling constants, and mass must be renormalized so that they represent
physical (renormalized) ones. When describing a bound state such as a hadron
in a physical process, say its decays, one has to use physical (renormalized)
quantities such as its mass. The relation between the bare field and the
dressed one can be expressed using the wave function renormalization constant
as $\phi_{0}=Z^{1/2}\phi_{r}$. One sees that the bare field $\phi_{0}$ can be
eliminated from the Lagrangian by setting its wave function renormalization
constant to be zero, i.e. $Z=0$. Jouvet [31] suggested that the equation $Z=0$
can be used as a compositeness condition. In particular, he proved the
equivalence between a four-fermion theory and a Yukawa-type theory if the
renormalization constant of the boson field is set to zero. Then one can
express the renormalized boson mass and Yukawa coupling in terms of the Fermi
constant via the compositeness condition. A more detailed discussion of the
compositeness condition is given in the Appendix.

The compositeness condition effectively excludes the constituent degrees of
freedom from the physical state space and avoids double counting. In other
words, the constituents exist only in virtual states. To illustrate this point
in a more intuitive way, we draw the idea here in Fig. 1.

Figure 1: The compositeness condition: how it works. Figure 2: One-loop self-
energy diagram for a meson.

The meson mass function in Eq. (4) is defined by the Feynman diagram shown in
Fig. 2 and has the following form for a pseudoscalar meson and a vector meson,
respectively:

$\displaystyle\widetilde{\Pi}_{P}(p)$ $\displaystyle=$ $\displaystyle
3g_{P}^{2}\int\\!\\!\frac{dk}{(2\pi)^{4}i}\,\widetilde{\Phi}^{2}_{P}\left(-k^{2}\right)\mbox{\rm{tr}}\left[S_{1}(k+w_{1}p)\gamma^{5}S_{2}(k-w_{2}p)\gamma^{5}\right],$
(5) $\displaystyle\widetilde{\Pi}_{V}(p)$ $\displaystyle=$ $\displaystyle
g_{V}^{2}\left[g^{\mu\nu}-\frac{p^{\mu}p^{\nu}}{p^{2}}\right]\int\\!\\!\frac{dk}{(2\pi)^{4}i}\,\widetilde{\Phi}^{2}_{V}\left(-k^{2}\right)\mbox{\rm{tr}}\left[S_{1}(k+w_{1}p)\gamma_{\mu}S_{2}(k-w_{2}p)\gamma_{\nu}\right],$
(6)

where we use the free quark propagator

$S_{i}(k)=\frac{1}{m_{q_{i}}-\not\\!k-i\epsilon}=\frac{m_{q_{i}}+\not\\!k}{m^{2}_{q_{i}}-k^{2}-i\epsilon}.$
(7)

It is convenient to calculate the derivatives of the meson mass functions by
using the following identities

$\displaystyle\frac{d}{dp^{2}}\widetilde{\Pi}_{M}(p^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2p^{2}}p^{\mu}\frac{d}{dp^{\mu}}\widetilde{\Pi}_{M}(p^{2}),$
$\displaystyle p^{\mu}\frac{d}{dp^{\mu}}S(k+wp)$ $\displaystyle=$
$\displaystyle w\,S(k+wp)\not\\!p\,S(k+wp).$ (8)

The derivatives of the meson mass functions then have the form

$\displaystyle\widetilde{\Pi}^{\prime}_{P}(p^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2p^{2}}\,\frac{3g^{2}_{P}}{4\pi^{2}}\int\\!\\!\frac{dk}{4\pi^{2}i}\widetilde{\Phi}^{2}_{P}\left(-k^{2}\right)$
(9) $\displaystyle\times$
$\displaystyle\Big{\\{}w_{1}\,\mbox{\rm{tr}}\left[S_{1}(k+w_{1}p)\not\\!p\,S_{1}(k+w_{1}p)\gamma^{5}S_{2}(k-w_{2}p)\gamma^{5}\right]$
$\displaystyle-
w_{2}\,\mbox{\rm{tr}}\left[S_{1}(k+w_{1}p)\gamma^{5}S_{2}(k-w_{2}p)\not\\!p\,S_{2}(k-w_{2}p)\gamma^{5}\right]\Big{\\}},$
$\displaystyle\widetilde{\Pi}^{\prime}_{V}(p^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2p^{2}}\,\frac{1}{3}\left(g^{\mu\nu}-\frac{p^{\mu}p^{\nu}}{p^{2}}\right)\frac{3g^{2}_{V}}{4\pi^{2}}\int\\!\\!\frac{dk}{4\pi^{2}i}\widetilde{\Phi}^{2}_{V}\left(-k^{2}\right)$
(10) $\displaystyle\times$
$\displaystyle\Big{\\{}w_{1}\,\mbox{\rm{tr}}\left[S_{1}(k+w_{1}p)\not\\!p\,S_{1}(k+w_{1}p)\gamma_{\mu}S_{2}(k-w_{2}p)\gamma_{\nu}\right]$
$\displaystyle-
w_{2}\,\mbox{\rm{tr}}\left[S_{1}(k+w_{1}p)\gamma_{\mu}S_{2}(k-w_{2}p)\not\\!p\,S_{2}(k-w_{2}p)\gamma_{\nu}\right]\Big{\\}}.$

The Fock-Schwinger representation of quark propagators is then used to
calculate the loop integrations in Eqs. (9) and (10):

$\displaystyle S_{q}(k+wp)$ $\displaystyle=$
$\displaystyle\frac{1}{m_{q}-\not\\!k-w\not\\!p}=\frac{m_{q}+\not\\!k+w\not\\!p}{m^{2}_{q}-(k+wp)^{2}}$
(11) $\displaystyle=$
$\displaystyle(m_{q}+\not\\!k+w\not\\!p)\int\limits_{0}^{\infty}\\!\\!d\alpha\,e^{-\alpha[m_{q}^{2}-(k+wp)^{2}]}.$

As can be seen later, the Fock-Schwinger representation gives an efficient
approach to the calculation of tensor loop integrals since loop momenta can be
converted into derivatives of the exponent function.

Loop integrations are performed in Euclidean space. The transition from
Minkowski space to Euclidean space is done by the Wick rotation

$k_{0}=e^{i\frac{\pi}{2}}k_{4}=ik_{4}$ (12)

so that $k^{2}=k_{0}^{2}-\vec{k}^{2}=-k_{4}^{2}-\vec{k}^{2}=-k_{E}^{2}\leq 0.$
Simultaneously, all external momenta are rotated, i.e. $p_{0}\to ip_{4}$ so
that $p^{2}=-p_{E}^{2}\leq 0$. One sees that the quadratic form in Eq. (11)
now becomes positive-definite,

$m^{2}_{q}-(k+wp)^{2}=m^{2}_{q}+(k_{E}+wp_{E})^{2}>0,$

and the integral over $\alpha$ is absolutely convergent. The Minkowski
notation will be kept to avoid excessive relabeling. One simply keeps in mind
that $k^{2}\leq 0$ and $p^{2}\leq 0$.

Using the representations of the vertex functions and quark propagators given
by Eqs. (3) and (11), respectively, we then calculate the Gaussian integration
in the derivatives of the mass functions in Eqs. (9) and (10). The exponent
has the form $ak^{2}+2kr+z_{0}$, where $r=b\,p$. Next, with the help of the
properties ($k$ is the loop momentum)

$\left.\begin{aligned}
k^{\mu}\,\exp(ak^{2}+2kr+z_{0})&=\frac{1}{2}\frac{\partial}{\partial
r_{\mu}}\exp(ak^{2}+2kr+z_{0})\\\
k^{\mu}k^{\nu}\,\exp(ak^{2}+2kr+z_{0})&=\frac{1}{2}\frac{\partial}{\partial
r_{\mu}}\frac{1}{2}\frac{\partial}{\partial r_{\nu}}\exp(ak^{2}+2kr+z_{0})\\\
\text{etc.}&\end{aligned}\right\\},$ (13)

we replace $\not\\!k$ by
${\not\\!\partial}_{r}=\gamma^{\mu}\frac{\partial}{\partial r_{\mu}}$ which
allows us to exchange the tensor integrations for a differentiation of the
Gaussian exponent. For example, Eq. (6) now has the form

$\widetilde{\Pi}_{P}(p^{2})=\frac{3g^{2}_{P}}{16\pi^{2}}\int\limits_{0}^{\infty}\\!\\!\int\limits_{0}^{\infty}\\!\frac{d\alpha_{1}d\alpha_{2}}{a^{2}}\,\mbox{\rm{tr}}\left[\gamma^{5}(m_{1}+{\not\\!\partial}_{r}+w_{1}\not\\!p)\gamma^{5}(m_{2}+{\not\\!\partial}_{r}-w_{2}\not\\!p)\right]e^{-\frac{r^{2}}{a}+z_{0}}.\,$
(14)

The $r$-dependent Gaussian exponent $e^{-r^{2}/a}$ is moved to the left
through the differential operator $\not\\!\partial_{r}$ by using the following
properties

$\displaystyle\frac{\partial}{\partial r_{\mu}}\,e^{-r^{2}/a}$
$\displaystyle=$ $\displaystyle
e^{-r^{2}/a}\left[-\frac{2r^{\mu}}{a}+\frac{\partial}{\partial
r_{\mu}}\right],$ $\displaystyle\frac{\partial}{\partial
r_{\mu}}\,\frac{\partial}{\partial r_{\nu}}\,e^{-r^{2}/a}$ $\displaystyle=$
$\displaystyle e^{-r^{2}/a}\left[-\frac{2r^{\mu}}{a}+\frac{\partial}{\partial
r_{\mu}}\right]\cdot\left[-\frac{2r^{\nu}}{a}+\frac{\partial}{\partial
r_{\nu}}\right],$ etc. (15)

Finally, the derivatives are moved to the right by using the commutation
relation

$\left[\frac{\partial}{\partial r_{\mu}},r^{\nu}\right]=g^{\mu\nu}.$ (16)

The last step is done by using a form code which works for any numbers of
loops and propagators.

The remaining integrals over the Fock-Schwinger parameters
$0\leq\alpha_{i}<\infty$ are treated by introducing an additional integration
which converts the set of these parameters into a simplex as follows

$\prod\limits_{i=1}^{n}\int\limits_{0}^{\infty}\\!\\!d\alpha_{i}f(\alpha_{1},\ldots,\alpha_{n})=\int\limits_{0}^{\infty}\\!\\!dtt^{n-1}\prod\limits_{i=1}^{n}\int\\!\\!d\alpha_{i}\delta\left(1-\sum\limits_{i=1}^{n}\alpha_{i}\right)f(t\alpha_{1},\ldots,t\alpha_{n}).$
(17)

For illustration, we show below the expression for the derivative of the mass
function of a pseudoscalar meson:

$\displaystyle\widetilde{\Pi}^{\prime}_{P}(p^{2})$ $\displaystyle=$
$\displaystyle\frac{3g^{2}_{P}}{4\pi^{2}}\int\limits_{0}^{\infty}\\!\\!\frac{dt\,t}{a_{P}^{2}}\int\limits_{0}^{1}\\!\\!d\alpha\,e^{-t\,z_{0}+z_{P}}\,f_{P}(t,\alpha),$
(18) $\displaystyle z_{0}$ $\displaystyle=$ $\displaystyle\alpha
m^{2}_{q_{1}}+(1-\alpha)m^{2}_{q_{2}}-\alpha(1-\alpha)p^{2},$ $\displaystyle
z_{P}$ $\displaystyle=$ $\displaystyle\frac{2s_{P}t}{a_{P}}(\alpha-
w_{2})^{2}p^{2},$ $\displaystyle a_{P}$ $\displaystyle=$ $\displaystyle
2s_{P}+t,\qquad s_{P}=1/\Lambda^{2}_{P}.$

The function $f_{P}(t,\alpha)$ arises from the trace evaluation.

At this stage, an infrared cutoff is introduced to avoid any possible
thresholds in the Feynman diagram:

$\int\limits_{0}^{\infty}dt(\ldots)\to\int\limits_{0}^{1/\lambda^{2}}dt(\ldots).$
(19)

The infrared cutoff parameter $\lambda$ effectively guarantees the confinement
of quarks within hadrons. This method is quite general and can be used for
diagrams with an arbitrary number of loops and propagators. In the CCQM the
parameter $\lambda$ is taken to be universal. The numerical evaluation of the
final integral in Eq. (19) is done by FORTRAN code using NAG library.

Let us enumerate the number of free parameters in the CCQM and describe the
fitting process to determine their values. For a given meson $M_{i}$, five
parameters are needed: the size parameter $\Lambda_{M_{i}}$, the meson-quark
coupling $g_{M_{i}}$, two of the four constituent quark masses $m_{q_{j}}$,
($m_{u}=m_{d},m_{c},m_{s},m_{b}$), and the universal cutoff parameter
$\lambda$. If one considers $n_{M}$ mesons one has $2n_{M}+5$ model
parameters. However, the compositeness condition in Eq. (4) provides $n_{M}$
constraints. These constraints are used to exclude the coupling $g_{M_{i}}$
from the parameter set. The remaining $n_{M}+5$ parameters are obtained by a
fit to experimental data. Naturally, we first choose experimental values for
the leptonic decay constants of $n_{M}$ mesons to fit the model parameters.
Whenever experimental data is not available, or available but with large
errors, we decided to use LQCD results instead.

The matrix elements of the leptonic decays are described by the Feynman
diagram shown in Fig. 3.

Figure 3: Quark model diagram for meson leptonic decay.

The leptonic decay constants of the pseudoscalar and vector mesons are defined
in the CCQM as

$\displaystyle
N_{c}\,g_{P}\\!\int\\!\\!\frac{d^{4}k}{(2\pi)^{4}i}\,\widetilde{\Phi}_{P}(-k^{2})\,{\rm
tr}\biggl{[}O^{\,\mu}S_{1}(k+w_{1}p)\gamma^{5}S_{2}(k-w_{2}p)\biggr{]}$
$\displaystyle=$ $\displaystyle f_{P}p^{\mu},$ $\displaystyle
N_{c}\,g_{V}\\!\int\\!\\!\frac{d^{4}k}{(2\pi)^{4}i}\,\widetilde{\Phi}_{V}(-k^{2})\,{\rm
tr}\biggl{[}O^{\,\mu}S_{1}(k+w_{1}p)\not\\!\epsilon_{V}S_{2}(k-w_{2}p)\biggr{]}$
$\displaystyle=$ $\displaystyle m_{V}f_{V}\epsilon_{V}^{\mu},$ (20)

where $N_{c}=3$ is the number of colors, and
$O^{\mu}=\gamma^{\mu}(1-\gamma_{5})$ is the weak Dirac matrix with left
chirality. The mesons are taken on their mass-shells. The matrix elements in
Eq. (20) are calculated in a similar way as for the case of the mass functions
described above.

In addition, we decided to use eight fundamental electromagnetic decays to
give further constraints on the model parameters. These decays are listed in
Table 2. The results of the (overconstrained) least-squares fit to the
leptonic decay constants and the electromagnetic decay widths are given in
Tables 1 and 2. The results of the fit for the values of quark masses, the
size parameters, and the infrared cutoff parameter are given in Tables 3 and
4.

Table 1: Input values for the leptonic decay constants $f_{M}$ (in MeV) and our least-squares fit values [38] | Fit Values | Input | Ref. |  | Fit Values | Input | Ref.
---|---|---|---|---|---|---|---
$f_{\pi}$ | 128.7(6.4) | $130.4\pm 0.2$ | [33, 34] | $f_{\omega}$ | 198.5(9.9) | $198\pm 2$ | [33]
$f_{K}$ | 156.1(7.8) | $156.1\pm 0.8$ | [33, 34] | $f_{\phi}$ | 228.2(11.4) | $227\pm 2$ | [33]
$f_{D}$ | 205.9(10.3) | $206.7\pm 8.9$ | [33, 34] | $f_{J/\psi}$ | 415.0(20.8) | $415\pm 7$ | [33]
$f_{D_{s}}$ | 257.5(12.9) | $257.5\pm 6.1$ | [33, 34] | $f_{K^{*}}$ | 213.7(10.7) | $217\pm 7$ | [33]
$f_{B}$ | 191.1(9.6) | $192.8\pm 9.9$ | [35] | $f_{D^{*}}$ | 243.3(12.2) | $245\pm 20$ | [37]
$f_{B_{s}}$ | 234.9(11.7) | $238.8\pm 9.5$ | [35] | $f_{D_{s}^{*}}$ | 272.0(13.6) | $272\pm 26$ | [37]
$f_{B_{c}}$ | 489.0(24.5) | $489\pm 5$ | [36] | $f_{B^{*}}$ | 196.0(9.8) | $196\pm 44$ | [37]
$f_{\rho}$ | 221.1(11.1) | $221\pm 1$ | [33] | $f_{B_{s}^{*}}$ | 229.0(11.5) | $229\pm 46$ | [37]
Table 2: Input values for some basic electromagnetic decay widths and our least-squares fit values (in keV) [38]. Process | Fit Values | Input [33]
---|---|---
$\pi^{0}\to\gamma\gamma$ | $5.06(25)\times 10^{-3}$ | $(7.7\pm 0.4)\times 10^{-3}$
$\eta_{c}\to\gamma\gamma$ | 1.61(8) | $1.8\pm 0.8$
$\rho^{+}\to\pi^{+}\gamma$ | 76.0(3.8) | $67\pm 7$
$\omega\to\pi^{0}\gamma$ | 672(34) | $703\pm 25$
$K^{*+}\to K^{+}\gamma$ | 55.1(2.8) | $50\pm 5$
$K^{*0}\to K^{0}\gamma$ | 116(6) | $116\pm 10$
$D^{*+}\to D^{+}\gamma$ | 1.22(6) | $1.5\pm 0.5$
$J/\psi\to\eta_{c}\gamma$ | 1.43(7) | $1.58\pm 0.37$

Once the free parameters are fixed, one can employ the CCQM as a frame-
independent tool for hadronic calculation. Note that this fit was done in 2011
(see, Ref. [38]). One of the advantages of the CCQM is the possibility to
calculate hadronic quantities in the full physical range of momentum transfer
without any extrapolation. Another advantage of the model is that it can be
used to treat not only mesons [39, 40, 41, 42], but also baryons [43, 44, 45],
tetraquarks [46, 47], and other multiquark states [48, 49] in a consistent
way.

It is worth mentioning the estimation of theoretical errors in the CCQM. We
use MINUIT to fit, and this fit is based on $\chi^{2}$ minimalizing. When we
fit, we obtained the best-fit values for model parameters, which are given in
Tables 3 and 4. We do not consider the errors of these parameters because it
is complicated to take into account the error propagations into the final
physical predictions. Instead, we choose the easier but less rigorous approach
of error estimation. That is, we only use the best-fit parameters for further
calculation. Then, we observe that in the fit, fitted values deviate from
experimental data at the order of $5\%-10\%$. In our model, hadronic
quantities are calculated in a similar manner as for the leptonic and
electromagnetic decay constants, we therefore estimate the errors of the
hadronic quantities to be about $5\%-10\%$. When one uses these hadronic
quantities for further calculations, for example, the decay widths, the errors
accumulate. We therefore estimate the error for the decay widths to be of
order $10\%-20\%$. In particular, in this paper, we estimated the error for
the couplings $g_{D*D\gamma}$ to be $10\%$ and the error for the decay widths
to be $20\%$.

Table 3: Meson size parameters (in GeV) [38]. $\Lambda_{\pi}$ | $\Lambda_{K}$ | $\Lambda_{D}$ | $\Lambda_{D_{s}}$ | $\Lambda_{B}$ | $\Lambda_{B_{s}}$ | $\Lambda_{B_{c}}$ | $\Lambda_{\rho}$
---|---|---|---|---|---|---|---
0.87(10) | 1.04(26) | 1.47(39) | 1.57(42) | 1.88(26) | 1.95(27) | 2.42(10) | 0.61(7)
$\Lambda_{\omega}$ | $\Lambda_{\phi}$ | $\Lambda_{J/\psi}$ | $\Lambda_{K^{*}}$ | $\Lambda_{D^{*}}$ | $\Lambda_{D^{*}_{s}}$ | $\Lambda_{B^{*}}$ | $\Lambda_{B_{s}^{*}}$
0.47(5) | 0.88(10) | 1.48(17) | 0.72(18) | 1.16(22) | 1.17(22) | 1.72(27) | 1.71(27)
Table 4: Quark masses and infrared cutoff parameter (in GeV) [38]. $m_{u/d}$ | $m_{s}$ | $m_{c}$ | $m_{b}$ | $\lambda$
---|---|---|---|---
0.235(12) | 0.424(19) | 2.160(71) | 5.090(21) | 0.181(9)

## III The radiative decays $D^{*}_{(s)}\to D_{(s)}\gamma$

The radiative decays $D^{*}_{(s)}\to D_{(s)}\gamma$ are described by the
Feynman diagrams in Fig. 4. The coupling between the quarks and the photon is
described by the interaction Lagrangian

$\mathcal{L}_{\mathrm{int}}^{\mathrm{em}}(x)=eA_{\mu}(x)J_{\mathrm{em}}^{\mu}(x),\qquad
J_{\mathrm{em}}^{\mu}(x)=e_{c}\bar{c}(x)\gamma^{\mu}c(x)+e_{q}\bar{q}(x)\gamma^{\mu}q(x),$
(21)

where $e_{c}$ and $e_{q}$ are the quark charges in units of $e$.

|
---|---
Figure 4:  Feynman diagrams for radiative decays $D^{*}_{(s)}\to
D_{(s)}\gamma$.

The transition amplitude is written as

$\displaystyle\langle\vec{p^{\prime}};\vec{q},\epsilon_{\mu}^{\gamma}|S-I|\vec{p},\epsilon_{\nu}^{D^{*}}\rangle$
$\displaystyle=$
$\displaystyle(i)^{3}eg_{D^{*}}g_{D}\epsilon^{D^{*}}_{\nu}(p)\epsilon^{\gamma}_{\mu}(q)$
(22) $\displaystyle\times\iiint
dx\,dy\,dz\,e^{-ipx+ip^{\prime}y+iqz}\langle\mathrm{T}{\bar{J}_{D^{*}}^{\nu}(x)J_{\mathrm{em}}^{\mu}(z)J_{D}(y)}\rangle_{0}$
$\displaystyle=$
$\displaystyle(2\pi)^{4}i\delta(p-p^{\prime}-q)\mathcal{M}(D^{*}\to D\gamma),$

where

$\displaystyle\mathcal{M}(D^{*}\to D\gamma)$ $\displaystyle=$
$\displaystyle(-3i)eg_{D^{*}}g_{D}\epsilon^{D^{*}}_{\nu}(p)\epsilon^{\gamma}_{\mu}(q)(e_{c}\mathcal{M}^{\mu\nu}_{c}+e_{q}\mathcal{M}^{\mu\nu}_{q}),$
(23) $\displaystyle\mathcal{M}^{\mu\nu}_{c}$ $\displaystyle=$
$\displaystyle\int\frac{dk}{(2\pi)^{4}i}\widetilde{\Phi}_{D^{*}}\big{[}-(k-\omega_{2}p)^{2}\big{]}\widetilde{\Phi}_{D}\big{[}-(k-\omega_{2}p^{\prime})^{2}\big{]}$
(24)
$\displaystyle\times\mathrm{tr}\big{[}S_{q}(k)\gamma^{\nu}S_{c}(k-p)\gamma^{\mu}S_{c}(k-p^{\prime})\gamma^{5}\big{]},$
$\displaystyle\mathcal{M}^{\mu\nu}_{q}$ $\displaystyle=$
$\displaystyle\int\frac{dk}{(2\pi)^{4}i}\widetilde{\Phi}_{D^{*}}\big{[}-(k+\omega_{1}p)^{2}\big{]}\widetilde{\Phi}_{D}\big{[}-(k+\omega_{1}p^{\prime})^{2}\big{]}$
(25)
$\displaystyle\times\mathrm{tr}\big{[}S_{q}(k+p^{\prime})\gamma^{\mu}S_{q}(k+p)\gamma^{\nu}S_{c}(k)\gamma^{5}\big{]}.$

The ratios of quark masses now read $\omega_{1}=m_{c}/(m_{c}+m_{q})$ and
$\omega_{2}=m_{q}/(m_{c}+m_{q})$ with $q=u,d,s$.

In order to proceed the calculation of the matrix element
$\mathcal{M}(D^{*}\to D\gamma)$ one substitutes the Gaussian form for the
vertex functions in Eq. (3) and the Fock-Schwinger representation of quark
propagators in Eg. (11), and then one uses the techniques described in Eqs.
(12)–(19).

Next, using the transversality conditions
$\epsilon_{\mu}^{\gamma}(q)q^{\mu}=0$ and
$\epsilon_{\nu}^{D^{*}}(p)p^{\nu}=0$, the matrix element can be written in the
form

$\mathcal{M}(D^{*}\to
D\gamma)=e\,g_{D^{*}D\gamma}\,\varepsilon^{\alpha\beta\mu\nu}\,p_{\alpha}\,q_{\beta}\,\epsilon_{\mu}^{\gamma}(q)\,\epsilon_{\nu}^{D^{*}}(p),$
(26)

where
$g_{D^{*}D\gamma}=e_{c}\,I_{c}(m^{2}_{D^{*}},m^{2}_{D})+e_{q}\,I_{q}(m^{2}_{D^{*}},m^{2}_{D})$
is the radiative decay constant. The quantities
$I_{c(q)}(m^{2}_{D^{*}},m^{2}_{D})$ are threefold integrals which are
calculated numerically. The expression for $I_{c}(m^{2}_{D^{*}},m^{2}_{D})$
reads

$\displaystyle I_{c}(m^{2}_{D^{*}},m^{2}_{D})$ $\displaystyle=$ $\displaystyle
g_{D}g_{D^{*}}\frac{N_{c}}{4\pi^{2}}\int\limits_{0}^{1/{\lambda^{2}}}\\!\\!\frac{dt\,t^{2}}{(s+t)^{2}}\int\limits\\!\\!d\alpha^{3}\delta\big{(}1-\sum\limits_{i=1}^{3}\alpha_{i}\big{)}$
$\displaystyle\times\left[m_{c}\omega_{2}+m_{q}\omega_{1}+\frac{t}{s+t}(m_{c}-m_{q})(\omega_{1}-\alpha_{2})\right]\exp\left(-tz_{0}+\frac{st}{s+t}z_{1}\right),$
$\displaystyle z_{0}$ $\displaystyle=$
$\displaystyle(1-\alpha_{2})m_{c}^{2}+\alpha_{2}m_{q}^{2}-\alpha_{1}\alpha_{2}m_{D^{*}}^{2}-\alpha_{2}\alpha_{3}m_{D}^{2},$
$\displaystyle z_{1}$ $\displaystyle=$ $\displaystyle
m_{D^{*}}^{2}\left(\alpha_{1}-\omega_{2}\frac{s_{D^{*}}}{s}\right)(\omega_{1}-\alpha_{2})+m_{D}^{2}(\alpha_{2}-\omega_{1})\left(\alpha_{1}+\alpha_{2}-\frac{s_{D^{*}}}{s}-\omega_{1}\frac{s_{D}}{s}\right),$
$\displaystyle s$ $\displaystyle=$ $\displaystyle s_{D^{*}}+s_{D},\qquad
s_{D^{(*)}}=1/\Lambda^{2}_{D^{(*)}}.$

The expression for $I_{q}(m^{2}_{D^{*}},m^{2}_{D})$ can be obtained simply by
exchanging $m_{c}\leftrightarrow m_{q}$ and
$\omega_{1}\leftrightarrow\omega_{2}$, i.e.
$I_{q}(m^{2}_{D^{*}},m^{2}_{D},m_{q},m_{c},\omega_{1},\omega_{2})=I_{c}(m^{2}_{D^{*}},m^{2}_{D},m_{c},m_{q},\omega_{2},\omega_{1})$.

Finally, the decay width for $D^{*}\to D\gamma$ is given by

$\Gamma(D^{*}\to
D\gamma)=\frac{\alpha}{24}m_{D^{*}}^{3}\left(1-\frac{m_{D}^{2}}{m_{D^{*}}^{2}}\right)^{3}g_{D^{*}D\gamma}^{2}.$
(28)

## IV Numerical results and discussion

Our results for the radiative decay constants are listed in Table 5 together
with results obtained in other theoretical approaches. It is worth mentioning
that the sign difference in the predictions in Table 5 is simply due to the
choice when one defines the matrix element and does not affect the physical
observables, i.e. the decay width, since the last is proportional to the
square of the matrix element. In other words, the sign of the radiative decay
constants cannot be measured experimentally. The experimental value
$g_{D^{*+}D^{+}\gamma}=-0.47(7)\,\textrm{GeV}^{-1}$ was extracted from the
total width of $D^{*+}$ and the measured branching fraction
$\mathcal{B}(D^{*+}\to D^{+}\gamma)$ [25]. Meanwhile, the total width of
$D^{*0}$ is still unknown experimentally. However, it can be calculated based
on isospin symmetry relating the strong couplings $g_{D^{*+}D\pi}$ and
$g_{D^{*0}D\pi}$ [50, 23], which yields
$\Gamma(D^{*0})=56.5(14.0)\,\textrm{keV}$ [25]. Then the value
$g_{D^{*0}D^{0}\gamma}=1.77(16)\,\textrm{GeV}^{-1}$ was obtained with the help
of the measured $\mathcal{B}(D^{*0}\to D^{0}\gamma)$ [25]. One can see that
our predictions $g_{D^{*+}D^{+}\gamma}=-0.45(9)\,\textrm{GeV}^{-1}$ and
$g_{D^{*0}D^{0}\gamma}=1.72(34)\,\textrm{GeV}^{-1}$ are in full agreement with
the current experimental data. We note here that the decay width of the
channel $D^{*+}\to D^{+}\gamma$ was included in the fit for the model
parameter (see, Ref. [38] and Table 2). However, the channels $D^{*0}\to
D^{0}\gamma$ and $D^{*+}_{s}\to D_{s}^{+}\gamma$ were not included in the fit.

Table 5: Radiative decay constants in the CCQM and other approaches (all in $\textrm{GeV}^{-1}$) Ref. | $g_{D^{*+}D^{+}\gamma}$ | $g_{D^{*0}D^{0}\gamma}$ | $g_{D^{*+}_{s}D^{+}_{s}\gamma}$
---|---|---|---
This work | $-0.45(9)$ | $1.72(34)$ | $-0.29(6)$
LCSR (NLO) [25] | $0.40^{+0.12}_{-0.13}$ | $-2.11^{+0.35}_{-0.34}$ | $0.60^{+0.19}_{-0.18}$
LCSR (NLO)[10] | $-0.15^{+0.11}_{-0.10}$ | $1.48^{+0.29}_{-0.27}$ | $-0.079^{+0.086}_{-0.078}$
HQET$+$CQM [22] | $-0.38^{+0.05}_{-0.04}$ | $1.91^{+0.09}_{-0.09}$ | –
LQCD [23]†, [24]∗ | $-0.2(3)^{\dagger}$ | $2.0(6)^{\dagger}$ | $0.11(2)^{*}$
Bag Model [16] | 0.5 | 1.1 | –
RQM [18] | $-0.30$ | 1.85 | –
RQM [19] | $-0.44(6)$ | $2.15(11)$ | $-0.19(3)$
LCSR [11] | $-0.50(12)$ | $1.52(25)$ | –
QCDSR [7] | $-0.19^{+0.03}_{-0.02}$ | $0.62^{+0.03}_{-0.03}$ | $-0.20^{+0.03}_{-0.03}$
HQET$+$VMD [51] | $-0.29^{+0.19}_{-0.11}$ | $1.60^{+0.35}_{-0.45}$ | $-0.19^{+0.19}_{-0.08}$
HH$\chi$PT [52] | $-0.27(5)$ | $2.19(11)$ | $0.041(56)$
Experiment [53] | $-0.47(7)$ | $1.77(16)$ | –

In the case of $g_{D^{*+}_{s}D^{+}_{s}\gamma}$, theoretical predictions vary
largely, as compared to the case of $g_{D^{*+}D^{+}\gamma}$ and
$g_{D^{*0}D^{0}\gamma}$. The central value of our result
$|g_{D^{*+}_{s}D^{+}_{s}\gamma}|=0.29(6)\,\textrm{GeV}^{-1}$ is larger than
all other theoretical predictions, except for the recent value
$|g_{D^{*+}_{s}D^{+}_{s}\gamma}|=0.60(19)\,\textrm{GeV}^{-1}$ obtained by
using LCSR(NLO) [25]. Our result deviates from other within $2\sigma$ due to
large uncertainties in the predictions. In Ref. [25], a discussion of $D$-spin
symmetry for the decays $D^{*+}\to D^{+}\gamma$ was also provided. Here, the
$D$-spin symmetry is understood as the symmetry under the exchange of down and
strange quarks $d\leftrightarrow s$. As pointed out in Ref. [25], this is a
good approximate symmetry in QED. Since the decay $D_{s}^{*+}\to
D_{s}^{+}\gamma$ can be obtained from the decay $D^{*+}\to D^{+}\gamma$ by the
exchanging $d\leftrightarrow s$, one expects that the order of this symmetry
breaking is not too far from that in QED. It was roughly estimated that the
deviation between $g_{D^{*+}D^{+}\gamma}$ and $g_{D^{*+}_{s}D^{+}_{s}\gamma}$
is about $20$–$30\%$. Our results show a deviation of about $35\%$ which is
quite reasonable. One also notices from Table 5 the difference between the
predictions of Refs. [25] and [10] despite of the similarity in the approach
which is LCSR at NLO. The authors of Ref. [25] explained this issue as the
result of computational differences between the two studies. In particular,
the twist-1 $\mathcal{O}(\alpha_{s})$-corrections were calculated in Ref. [25]
while being neglected in Ref. [10], and the twist-4 corrections were dropped
in Ref. [25] while being considered in Ref. [10].

Table 6: Decay widths $\Gamma(D^{*}_{(s)}\to D_{(s)}\gamma)$ in the CCQM and other approaches (all in keV) Ref. | $\Gamma(D^{*+}\to D^{+}\gamma)$ | $\Gamma(D^{*0}\to D^{0}\gamma)$ | $\Gamma(D^{*+}_{s}\to D^{+}_{s}\gamma)$
---|---|---|---
This work | $1.21(48)$ | $18.4(7.4)$ | $0.55(22)$
LCSR (NLO) [25] | $0.96^{+0.58}_{-0.62}$ | $27.83^{+9.23}_{-9.50}$ | $2.36^{+1.49}_{-1.41}$
LQCD [23]†, [24]∗ | $0.8(7)^{\dagger}$ | $27(14)^{\dagger}$ | $0.066(26)^{*}$
HQET$+$CQM [22] | $0.9\pm 0.3$ | $22.7\pm 2.2$ | –
Bag Model [16] | 1.73 | 7.18 | –
NJL Model [17] | 0.7 | 19.4 | 0.09
RQM [14] | 0.46 | 23.05 | 0.38
RQM [18] | 0.56 | 21.69 | –
RQM [19] | (0.94–1.42) | (26.0–32.0) | (0.2–0.3)
RQM [20] | 1.04 | 11.5 | 0.19
LFQM [21] | $0.90(2)$ | $20.0(3)$ | $0.18(1)$
QCDSR [9] | $0.23(10)$ | $12.9(2.0)$ | $0.13(5)$
LCSR [11] | 1.50 | 14.40 | –
QCDSR [7] | $0.22(6)$ | $2.43(21)$ | $0.25(8)$
HQET$+$VMD [51] | $0.51(57)$ | $16.0(10.8)$ | $0.24(24)$
Experiment [53] | $1.33(36)$ | $19.9(5.5)$ | –

In Table 6 we compare predictions for the width of the $D^{*}_{(s)}$ radiative
decays . Available theoretical predictions for $\Gamma(D^{*+}\to D^{+}\gamma)$
vary in the range $(0.2-1.7)$ keV. Our result $\Gamma(D^{*+}\to
D^{+}\gamma)=1.21(48)$ agrees well with experimental data, suggesting the
width of the decay $D^{*+}\to D^{+}\gamma$ to be at the higher end of the
predicted range. Note that the experimental data $\Gamma(D^{*+}\to
D^{+}\gamma)=1.33(36)\,\textrm{keV}$ was obtained from the world average of
the total width $\Gamma(D^{*+})=83.4(1.8)\,\textrm{keV}$ and the branching
fraction $\mathcal{B}(D^{*+}\to D^{+}\gamma)=0.016(4)$ given by Particle Data
Group (PDG) [53]. We stress again that the decay $D^{*+}\to D^{+}\gamma$ was
included in the fit for our model parameters (see Table 2). However, the fit
was done in 2011 [38], and the input value $\Gamma(D^{*+}\to
D^{+}\gamma)=1.5(5)\,\textrm{keV}$ was taken from PDG at that time [33]. The
slight difference of the central value of $\Gamma(D^{*+}\to D^{+}\gamma)$
given in the Tables 6 and 2, which are 1.21 and 1.22, respectively, is due to
the fact that in this paper we have used the most updated values for meson
masses.

The experimental value $\Gamma(D^{*0}\to D^{0}\gamma)=19.9(5.5)\,\textrm{keV}$
in Table 6 was calculated by using the estimation
$\Gamma(D^{*0})=56.5(14.0)\,\textrm{keV}$ mentioned above and the branching
fraction $\mathcal{B}(D^{*0}\to D^{0}\gamma)=0.353(9)$ fitted by PDG [53]. Our
prediction $\Gamma(D^{*0}\to D^{0}\gamma)=18.4(7.4)\,\textrm{keV}$ also agrees
very well with this experimental value.

At this point, one can check the ratio of decay widths $\Gamma(D^{*+}\to
D^{+}\gamma)/\Gamma(D^{*0}\to D^{0}\gamma)$. It follows from Eq. (28) that

$\displaystyle\frac{\Gamma(D^{*+}\to D^{+}\gamma)}{\Gamma(D^{*0}\to
D^{0}\gamma)}$ $\displaystyle=$
$\displaystyle\frac{m^{3}_{D^{*0}}}{m^{3}_{D^{*+}}}\left(\frac{m^{2}_{D^{*+}}-m^{2}_{D^{+}}}{m^{2}_{D^{*0}}-m^{2}_{D^{0}}}\right)^{3}\left[\frac{e_{c}I_{c}(m^{2}_{D^{*+}},m^{2}_{D^{+}})+e_{d}I_{d}(m^{2}_{D^{*+}},m^{2}_{D^{+}})}{e_{c}I_{c}(m^{2}_{D^{*0}},m^{2}_{D^{0}})+e_{u}I_{u}(m^{2}_{D^{*0}},m^{2}_{D^{0}})}\right]^{2}$
(29) $\displaystyle\approx$
$\displaystyle\frac{1}{4}\left[\frac{2I_{c}(m^{2}_{D^{*+}},m^{2}_{D^{+}})-I_{d}(m^{2}_{D^{*+}},m^{2}_{D^{+}})}{I_{c}(m^{2}_{D^{*+}},m^{2}_{D^{+}})+I_{u}(m^{2}_{D^{*+}},m^{2}_{D^{+}})}\right]^{2}.$

At the final step we have neglected the mass differences between charged and
neutral $D^{(*)}$ mesons. Note also that we have assumed from the beginning
that $m_{d}=m_{u}$. In the heavy quark limit (HQL) $m_{c}\to\infty$ the
integral $I_{c}$ is suppressed as $1/m_{c}$ and the width ratio yields $0.25$.
However, from Table 6 one gets $\Gamma(D^{*+}\to D^{+}\gamma)/\Gamma(D^{*0}\to
D^{0}\gamma)=0.066$. Therefore, the use of HQL is not suitable for the decays
$D^{*}\to D\gamma$. This is opposed to the case of the decays $B^{*}\to
B\gamma$, in which the use of HQL for $b$ quark is reliable, as pointed out in
our previous paper [54].

In the case of the decay $D^{*}_{s}\to D_{s}\gamma$, there is still much
speculation about the width. Predictions for $\Gamma(D^{*}_{s}\to
D_{s}\gamma)$ range from 0.07 keV to 2.4 keV. Our result $\Gamma(D^{*}_{s}\to
D_{s}\gamma)=0.55(22)\,\textrm{keV}$ deviates from the recent prediction based
on LCSR at NLO $\Gamma(D^{*}_{s}\to
D_{s}\gamma)=2.36^{+1.49}_{-1.41}\,\textrm{keV}$ [25] and the first LQCD
result $\Gamma(D^{*}_{s}\to D_{s}\gamma)=0.066(26)\,\textrm{keV}$ [24] by
about $2\sigma$. One should notice that the central value of our result
$\Gamma(D^{*}_{s}\to D_{s}\gamma)=0.55(22)\,\textrm{keV}$ is larger than that
of most predictions, and in particular, exceeds the LQCD central value
$\Gamma(D^{*}_{s}\to D_{s}\gamma)=0.066\,\textrm{keV}$ [24] by roughly an
order of magnitude. On the other hand, the recent prediction based on LCSR at
NLO $\Gamma(D^{*}_{s}\to D_{s}\gamma)=2.36^{+1.49}_{-1.41}\,\textrm{keV}$ [25]
has the central value being about four times as large as our value. Other
theoretical studies seem to agree on the central value $\Gamma(D^{*}_{s}\to
D_{s}\gamma)\approx 0.2\,\textrm{keV}$.

It is instructive to estimate the width of the decay $D^{*}_{s}\to
D_{s}\gamma$ using the very recent measurement of the leptonic decay
$D_{s}^{*+}\to e^{+}\nu_{e}$ performed by BESIII collaboration [55]. This is
the first measurement of the decay branching fraction and the result reads
$\mathcal{B}(D_{s}^{*+}\to
e^{+}\nu_{e})=(2.1^{+1.2}_{-0.9_{\textrm{stat.}}}\pm
0.2_{\textrm{syst.}})\times 10^{-5}$. By taking the ratio of branching
fractions $\mathcal{B}(D_{s}^{*+}\to
e^{+}\nu_{e})/\mathcal{B}(D_{s}^{+}\to\mu^{+}\nu_{\mu})$ one gets

$\Gamma^{\textrm{total}}_{D_{s}^{*+}}=\frac{2.04\times
10^{-3}}{\mathcal{B}(D_{s}^{*+}\to
e^{+}\nu_{e})}\left(\frac{f_{D_{s}^{*+}}}{f_{D^{+}_{s}}}\right)^{2}\,\textrm{eV}.$
(30)

Using the world average $\mathcal{B}(D_{s}^{+}\to\mu^{+}\nu_{\mu})$ and the
average value $\frac{f_{D_{s}^{*+}}}{f_{D^{+}_{s}}}=1.12(1)$ given by LQCD
calculations, BESIII collaboration obtained the total width of $D_{s}^{*+}$ to
be $\Gamma^{\textrm{total}}_{D_{s}^{*+}}=121.9^{+69.6}_{-52.2}\pm
11.8\,\textrm{eV}$ [55], in agreement with the LQCD prediction
$70(28)\,\textrm{eV}$ [24] within $1\sigma$. Now one can use the world average
value $\mathcal{B}(D^{*+}_{s}\to D^{+}_{s}\gamma)=0.381(29)$ [53] to obtain
$\Gamma(D^{*+}_{s}\to D^{+}_{s}\gamma)=46(30)\,\textrm{eV}$ which is much
smaller that most theoretical predictions given in Table 6, including ours. It
should be noted that this estimation is based on the first measured branching
fraction $\mathcal{B}(D_{s}^{*+}\to
e^{+}\nu_{e})=(2.1^{+1.2}_{-0.9_{\textrm{stat.}}}\pm
0.2_{\textrm{syst.}})\times 10^{-5}$ which still suffers from large
uncertainties. A more precise measurement of $\mathcal{B}(D_{s}^{*+}\to
e^{+}\nu_{e})$, very likely in the near future, will help clarify the value of
the decay width $\Gamma(D^{*+}_{s}\to D^{+}_{s}\gamma)$.

Since our predictions for the case of $D^{*+}$ and $D^{*0}$ fully agree with
experiment, we want to check if the discrepancy in the case of $D^{*}_{s}$ is
due to the value of the constituent $s$-quark mass. We vary the mass of
$s$-quark by $\pm 20\%$ about its fitted value $m_{s}=0.424$ GeV while keeping
other parameters unchanged. The dependence of the width $\Gamma(D^{*}_{s}\to
D_{s}\gamma)$ on $m_{s}$ is shown in Fig. 5. One sees that our prediction
cannot be consistent with experiment within the errors. More accurate
experimental data is needed in the case of $D^{*}_{s}$ radiative decay. If
this discrepancy still remains, then we need to refit our model parameters in
the future.

Figure 5: $\Gamma(D^{*}_{s}\to D_{s}\gamma)$ vs. strange-quark mass.

Finally, we compare the contribution from charm quark and the other one (up,
down, and strange quark) into the radiative coupling constants. As mentioned
above, one has

$\displaystyle g_{D^{*}D\gamma}$ $\displaystyle=$ $\displaystyle
e_{c}\,I_{c}+e_{q}\,I_{q},$ $\displaystyle
I_{q}(m^{2}_{D^{*}},m^{2}_{D},m_{q},m_{c},\omega_{1},\omega_{2})$
$\displaystyle=$ $\displaystyle
I_{c}(m^{2}_{D^{*}},m^{2}_{D},m_{c},m_{q},\omega_{2},\omega_{1}),$ (31)

where the expression of $I_{c(q)}$ is given in Eq. (III). The electromagnetic
aspect of the decays is mostly visible in the proportionality of the quark
contribution to its electric charge. Since $e_{c}=e_{u}=+2/3$ and
$e_{s}=e_{d}=-1/3$, there is a large cancelation between the contributions
from the charm and the down quarks in $g_{D^{*+}D^{+}\gamma}$, as well as from
the charm and the strange quarks in $g_{D_{s}^{*+}D_{s}\gamma}$. Meanwhile,
for $g_{D^{*0}D^{0}\gamma}$, the contributions from the charm and the up
quarks add up (see, Table 5). The strong effects in the decays are captured in
the quantities $I_{c(q)}$. In Table 7 we compare $I_{c}$ and $I_{q}$,
$(q=u,d,s)$ by taking the ratios $I_{q}/I_{c}$. We also list the numerical
values for the ratios $e_{q}I_{q}/e_{c}I_{c}$.

Table 7: Comparison of quark contributions to radiative decay constant Channel |  Quark $q$ | $I_{q}/I_{c}$ | $e_{q}I_{q}/e_{c}I_{c}$ |  $g_{D^{*}D\gamma}$ (GeV-1)
---|---|---|---|---
$D^{*+}\to D^{+}\gamma$ | $d$ | 3.74 | $-1.87$ | $-0.45(9)$
$D^{*0}\to D^{0}\gamma$ | $u$ | 3.74 | $+3.74$ | $+1.72(34)$
$D^{*+}_{s}\to D^{+}_{s}\gamma$ | $s$ | 2.97 | $-1.49$ | $-0.29(6)$

## V Summary

Radiative decays of the vector mesons $D^{*+}$, $D^{*0}$, and $D^{*+}_{s}$ are
studied in the framework of the Covariant Confined Quark Model in light of new
experimental data from BABAR and BESIII collaborations. Predictions for the
radiative couplings $g_{D^{*+}D^{+}\gamma}$, $g_{D^{*0}D^{0}\gamma}$, and
$g_{D^{*+}_{s}D^{+}_{s}\gamma}$, as well as the decay widths are given. A full
agreement between the model’s prediction and experimental data is found for
the decays $D^{*+}\to D^{+}\gamma$ and $D^{*0}\to D^{0}\gamma$. In the case of
$D^{*+}_{s}\to D^{+}_{s}\gamma$, our result for the decay width
$0.55(22)\,\textrm{keV}$ is smaller than the recent LCSR prediction
$2.36^{+1.49}_{-1.41}\,\textrm{keV}$ by more than $1\sigma$ but exceeds the
first LQCD prediction $0.066(26)\,\textrm{keV}$ by more than $2\sigma$. Using
the branching fraction of the purely leptonic decay $D_{s}^{*+}\to
e^{+}\nu_{e}$ recently measured by BESIII collaboration, one estimates
$\Gamma(D^{*+}_{s}\to D^{+}_{s}\gamma)$ to be about $0.05(3)\,\textrm{keV}$.
Meanwhile, theoretical predictions for this width range from
$0.07\,\textrm{keV}$ to $2.4\,\textrm{keV}$, and most of them are around
$0.2\,\textrm{keV}$. In particular, this estimated value is smaller than our
prediction by more that $2\sigma$. The decay $D^{*+}_{s}\to D^{+}_{s}\gamma$
is therefore a very interesting case, and further theoretical and experimental
studies are needed to give a final answer.

###### Acknowledgements.

This work is supported by Ho Chi Minh City University of Technology and
Education under Grant №T2022-26. CTT deeply thanks Dr. Anh-Vu Phan-Gia and Dr.
Hai-Cat Tran for their help and fruitful discussion. This paper is dedicated
to the memory of our dear friend and colleague Jürgen Körner.

##  Appendix A The compositeness condition

Let us consider the simplest case of a zero-spin bound state of which
constituents are fermions. The Yukawa-type “bare” Lagrangian describing the
coupling of the boson field $\phi_{0}$ to the fermion field $q$ has the form

$\mathcal{L}_{Y}=\bar{q}(i\not\\!\partial-m_{q})q+\frac{1}{2}\phi_{0}(\Box-
m_{0}^{2})\phi_{0}+g_{0}\phi_{0}(\bar{q}\Gamma
q),\quad\textrm{where}\quad\Box=-\partial^{\mu}\partial_{\mu},\quad\Gamma=I,i\gamma^{5}.$
(32)

The vacuum generating functional of the Yukawa theory is given by

$Z_{Y}=\int\\!\mathcal{D}\phi_{0}\int\\!\mathcal{D}\bar{q}\int\\!\mathcal{D}q\,e^{i\int\\!dx\mathcal{L}_{Y}(x)}.$
(33)

Integrating out quark fields and dropping all irrelevant normalization factors
one gets

$\displaystyle Z_{Y}$
$\displaystyle=\int\\!\mathcal{D}\phi_{0}\exp\Big{\\{}\frac{i}{2}\int\\!dx\,\phi_{0}(x)(\Box-
m_{0}^{2})\phi_{0}(x)$
$\displaystyle-\sum_{n=1}^{\infty}\frac{i^{n}g_{0}^{n}}{n}\int\\!dx_{1}\cdots\int\\!dx_{n}\,\phi_{0}(x_{1})\cdots\phi_{0}(x_{n})\mbox{\rm{tr}}[\Gamma
S_{q}(x_{1}-x_{2})\cdots\Gamma S_{q}(x_{n}-x_{1})]\Big{\\}},$ (34)

where $\displaystyle
S_{q}(x-y)=\int\frac{d^{4}k}{(2\pi)^{4}i}\frac{e^{-ik(x-y)}}{m_{q}-\not\\!k}$
is the quark Green function (propagator).

The terms bi-linear in boson field read

$\displaystyle L_{Y}^{(2)}$
$\displaystyle=\frac{1}{2}\int\\!dx\,\phi_{0}(x)(\Box-m^{2})\phi_{0}(x)$
$\displaystyle-\frac{i}{2}g_{0}^{2}\int\\!dx_{1}\int\\!dx_{2}\,\phi_{0}(x_{1})\phi_{0}(x_{2})\mbox{\rm{tr}}[\Gamma
S_{q}(x_{1}-x_{2})\Gamma S_{q}(x_{2}-x_{1})]$
$\displaystyle=\frac{1}{2}\int\\!dx\,\phi_{0}(x)(\Box-m^{2})\phi_{0}(x)+\frac{1}{2}g_{0}^{2}\int\\!dx_{1}\int\\!dx_{2}\,\phi_{0}(x_{1})\Pi_{S=0}(x_{1}-x_{2})\phi_{0}(x_{2}),$
(35)

where $\displaystyle\Pi_{S=0}(x_{1}-x_{2})=i\left\langle T[(\bar{q}\Gamma
q)_{x}(\bar{q}\Gamma q)_{y}]\right\rangle_{0}=-i\,\mbox{\rm{tr}}[\Gamma
S_{q}(x_{1}-x_{2})\Gamma S_{q}(x_{2}-x_{1})]$ is the mass function of the
spin-0 boson.

Expanding the Fourier transform of the mass function at the physical boson
mass up to second order

$\tilde{\Pi}_{S=0}(p^{2})=\int\\!dxe^{-ipx}\Pi_{S=0}(x)=\tilde{\Pi}_{S=0}(m^{2})+(p^{2}-m^{2})\tilde{\Pi}^{\prime}_{S=0}(m^{2})+\tilde{\Pi}^{\textrm{ren}}_{S=0}(p^{2})$
(36)

one obtains

$\displaystyle L_{Y}^{(2)}$
$\displaystyle=\frac{1}{2}\int\\!dx\,\phi_{0}(x)\big{[}\Box-
m_{0}^{2}+g_{0}^{2}\tilde{\Pi}_{S=0}(m^{2})+(\Box-m^{2})\tilde{\Pi}^{\prime}_{S=0}(m^{2})\big{]}\phi_{0}(x)$
$\displaystyle+\frac{1}{2}g_{0}^{2}\int\\!dx_{1}\int\\!dx_{2}\,\phi_{0}(x_{1})\Pi^{\textrm{ren}}_{S=0}(x_{1}-x_{2})\phi_{0}(x_{2}).$
(37)

The wave function, the boson mass, and the Yukawa coupling are renormalized as
follows:

$\displaystyle\phi_{r}$ $\displaystyle=$ $\displaystyle
Z^{-1/2}\phi_{0},\qquad m^{2}=m_{0}^{2}-g_{0}^{2}\tilde{\Pi}_{S=0}(m^{2}),$
$\displaystyle g_{r}$ $\displaystyle=$ $\displaystyle Z^{1/2}g_{0},\qquad
Z=\frac{1}{1+g_{0}^{2}\tilde{\Pi}^{\prime}_{S=0}(m^{2})}.$ (38)

One sees that the wave function renormalization constant $Z$ is related to the
renormalized coupling constant $g_{r}$ via

$Z=1-g_{r}^{2}\tilde{\Pi}^{\prime}_{S=0}(m^{2}).$ (39)

The renormalized generating functional of the Yukawa theory is finally written
as

$\displaystyle Z_{Y}^{\textrm{ren}}$
$\displaystyle=\int\\!\mathcal{D}\phi_{r}\exp\Big{\\{}\frac{i}{2}\int\\!dx\,\phi_{r}(x)(\Box-m^{2})\phi_{r}(x)$
$\displaystyle+\frac{ig_{r}^{2}}{2}\int\\!dx_{1}\int\\!dx_{2}\,\phi_{r}(x_{1})\Pi_{S=0}^{\textrm{ren}}(x_{1}-x_{2})\phi_{r}(x_{2})$
$\displaystyle-\sum_{n=3}^{\infty}\frac{i^{n}g_{r}^{n}}{n}\int\\!dx_{1}\cdots\int\\!dx_{n}\,\phi_{r}(x_{1})\cdots\phi_{r}(x_{n})\mbox{\rm{tr}}[\Gamma
S_{q}(x_{1}-x_{2})\cdots\Gamma S_{q}(x_{n}-x_{1})]\Big{\\}}.$ (40)

We drop the linear boson term since for pseudoscalar mesons it is absent, and
for scalar mesons it can be removed by a shift of the field.

Next, we move on to the Fermi theory for which the generating functional is
written as

$Z_{F}=\int\\!\mathcal{D}\bar{q}\int\\!\mathcal{D}qe^{i\int\\!dx\mathcal{L}_{F}(x)},$
(41)

where the Fermi-type Lagrangian $\mathcal{L}_{F}(x)$ is given by

$\mathcal{L}_{F}=\bar{q}(i\not\\!\partial-m_{q})q+\frac{G}{2}(\bar{q}\Gamma
q)^{2}.$ (42)

For the exponential of the four-fermion interaction one can use the Gaussian
functional representation as follows:

$e^{i\frac{G}{2}\langle(\bar{q}\Gamma
q)^{2}\rangle}=N^{-1}_{+}\int\\!\mathcal{D}\phi\exp\big{\\{}-\frac{i}{2}\frac{1}{G}\langle\phi^{2}\rangle+i\langle\phi\cdot(\bar{q}\Gamma
q)\rangle\big{\\}},\quad\textrm{where}\quad\langle(\cdots)\rangle=\int\\!dx(\cdots).$
(43)

One then has

$Z_{F}=N^{-1}_{+}\int\\!\mathcal{D}\phi\int\\!\mathcal{D}\bar{q}\int\\!\mathcal{D}q\,\exp\big{\\{}-\frac{i}{2}\frac{1}{G}\langle\phi^{2}\rangle+i\langle\bar{q}[i\not\\!\partial-
m_{q}+\phi\Gamma]q\rangle\big{\\}}.$ (44)

Integrating out quark fields and dropping all irrelevant normalization factors
one obtains

$\displaystyle Z_{F}$
$\displaystyle=\int\\!\mathcal{D}\phi\exp\Big{\\{}-\frac{i}{2}\frac{1}{G}\int\\!dx\phi^{2}(x)$
$\displaystyle-\sum_{n=1}^{\infty}\frac{i^{n}}{n}\int\\!dx_{1}\cdots\int\\!dx_{n}\phi(x_{1})\cdots\phi(x_{n})\mbox{\rm{tr}}[\Gamma
S_{q}(x_{1}-x_{2})\cdots\Gamma S_{q}(x_{n}-x_{1})]\Big{\\}}.$ (45)

One then collects all terms bi-linear in boson field and introduces the
renormalized mass function similar to the Yukawa case as follows:

$\displaystyle L_{F}^{(2)}$
$\displaystyle=\frac{1}{2}\int\\!dx\,\phi(x)\Big{[}-\frac{1}{G}+\tilde{\Pi}_{S=0}(m^{2})+(\Box-m^{2})\tilde{\Pi}^{\prime}_{S=0}(m^{2})\Big{]}\phi(x)$
$\displaystyle+\frac{1}{2}\int\\!dx_{1}\int\\!dx_{2}\,\phi(x_{1})\Pi^{\textrm{ren}}_{S=0}(x_{1}-x_{2})\phi(x_{2}).$
(46)

If one requires the condition $\displaystyle G\tilde{\Pi}_{S=0}(m^{2})=1$ and
rescales the boson field as
$\displaystyle\phi\to\frac{\phi}{\sqrt{\tilde{\Pi}^{\prime}_{S=0}(m^{2})}}$
one obtains the free Lagrangian of the boson field with mass $m$ and the
correct residue of the Green function. The renormalized generating functional
of the Fermi theory finally reads

$\displaystyle Z_{F}^{\textrm{ren}}$
$\displaystyle=\int\\!\mathcal{D}\phi\exp\Big{\\{}\frac{i}{2}\int\\!dx\,\phi(x)(\Box-m^{2})\phi(x)$
$\displaystyle+\frac{i}{2}\frac{1}{\tilde{\Pi}^{\prime}_{S=0}(m^{2})}\int\\!dx_{1}\int\\!dx_{2}\,\phi(x_{1})\Pi_{S=0}^{\textrm{ren}}(x_{1}-x_{2})\phi(x_{2})$
$\displaystyle-\sum_{n=3}^{\infty}\frac{i^{n}}{n}\left[1/\sqrt{\tilde{\Pi}^{\prime}_{S=0}(m^{2})}\right]^{n}\int\\!dx_{1}\cdots\int\\!dx_{n}\,\phi(x_{1})\cdots\phi(x_{n})$
$\displaystyle\times\mbox{\rm{tr}}[\Gamma S_{q}(x_{1}-x_{2})\cdots\Gamma
S_{q}(x_{n}-x_{1})]\Big{\\}}.$ (47)

Comparing the renormalized generating functionals of the Yukawa theory in Eq.
(A) and the Fermi theory in Eq. (A) one sees that the condition for their
equality is

$g_{r}=1/\sqrt{\tilde{\Pi}^{\prime}_{S=0}(m^{2})}=0.$ (48)

Taking into account the relation (39), this condition is equivalent to

$Z=1-g_{r}^{2}\tilde{\Pi}^{\prime}_{S=0}(m^{2})=0.$ (49)

Therefore, the vanishing of the wave function renormalization constant in the
Yukawa theory can be used as the condition for the bare (unrenormalized) field
$\phi_{0}=Z^{1/2}\phi_{r}$ to vanish for a composite boson.

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