#  $B^{*}_{c}$ meson parameters and radiative decay width within the covariant
confined quark model

Aidos Issadykov<EMAIL_ADDRESS>Sayabek K. Sakhiyev  The Institute of
Nuclear Physics,
Ministry of Energy of the Republic of Kazakhstan, 050032 Almaty, KAZAKHSTAN

###### Abstract

In this work we tried to predict the parameters of $B^{*}_{c}$ meson. Simple
assumptions gave us following parametres $m_{B_{c}^{*}}=6329\pm 10$ MeV and
$f_{B_{c}^{*}}=535.5\pm 57.8$ MeV (for $\Lambda_{B_{c}^{*}}=2.26\pm 0.14$ GeV
in covariant confined quark model). We calculated widths of radiative decays
of $B^{*}_{q}$ mesons, where $q=u/d,s,c$ and compared them with other
theoretical works. It was shown that the width of the $B_{c}^{*}$ meson very
sensitive to the mass $m_{B_{c}^{*}}$ as expected and less to the size
parameter $\Lambda_{B_{c}^{*}}$.

###### pacs:

12.39.Ki, 13.30.Ce, 14.40.Nd

## I Introduction

The decay mode $B_{c}\to J/\psi\ell\nu$ of $B_{c}$ meson have about 2 standard
deviations disagreement between experimental data and theoretical predictions
Aaij:2017tyk . Meanwhile, its vector partner $B_{c}^{*}$ is still not found.
It is expected that the mass difference is not large to decay strongly to
$B_{c}$ meson and light meson. Thus, $B_{c}^{*}$ mesons cannot decay strongly
but can decay only weakly and electromagnetically. As a result, the partial
widths of electromagnetic decay channels, especially single-photon decay
channels, are dominant. Since the $B^{*}_{c}$ meson was not observed yet,
there are some theoretical predictions of it’s mass and leptonic decay
constants in the relativistic quark modelEbert:2002pp , Lattice
QCDDowdall:2012ab ; Colquhoun:2015oha , QCD Sum RulesWang:2012kw  and
Nonrelativistic renormalization groupPenin:2004xi . Properties of $B^{*}_{c}$
meson in the relativistic quark modelEbert:2002pp  as follows:

$\displaystyle m_{B_{c}^{*}}=6332\quad~{}\text{MeV},\qquad
f_{B_{c}^{*}}=503\quad~{}\text{MeV}.$ (1)

Mass and leptonic decay constant of $B^{*}_{c}$ meson in Lattice
QCDDowdall:2012ab ; Colquhoun:2015oha  looks like:

$\displaystyle m_{B_{c}^{*}}=6332\pm 9\quad~{}\text{MeV},\qquad
f_{B_{c}^{*}}=422\pm 13\quad~{}\text{MeV}.$ (2)

Mass and leptonic decay constant of $B^{*}_{c}$ meson from QCD Sum
RulesWang:2012kw :

$\displaystyle m_{B_{c}^{*}}=6337\quad~{}\text{MeV},\qquad
f_{B_{c}^{*}}=384\quad~{}\text{MeV}.$ (3)

The Nonrelativistic renormalization group Penin:2004xi  gave their prediction
on mass differences of $B^{*}_{c}$ and $B_{c}$ mesons

$\Delta m_{({B_{c}^{*}-B_{c}})}=50\pm 17^{+15}_{-12}\quad~{}\text{MeV}.$ (4)

Radiative decay of $B_{c}^{*}$ meson was calculated in Chang:2020xvu ;
Simonis:2018rld ; Jena:2002is ; Priyadarsini:2016tiu ; Patnaik:2017cbl ;
Ebert:2002xz ; Ebert:2002pp ; Lahde:1999ih ; Lahde:2002wj ; Choi:2007se ;
Choi:2009ai ; Eichten:1994gt ; Kiselev:1994rc ; Fulcher:1998ka ; Nobes:2000pm
; Monteiro:2016rzi ; AbdElHady:2005bv  and have partial widths less than 1 keV
which makes the branching ratios of their weak decay modes may be within the
detection ability of current experiments. There are several works dedicated to
investigate the semileptonic decays of $B_{c}^{*}$ Wang:2012hu ; Dai:2018vzz ;
Wang:2018ryc ; Chang:2020xvu . The purpose of this paper is to extend our
model and predict a model parameters of unobserved $B_{c}^{*}$. We studied
$b\to c$, $b\to s$ and $b\to d(u)$ transitions in the framework of covariant
confined quark model(CCQM) in our previous worksSoni:2021fky ; Soni:2020bvu ;
Issadykov:2018myx ; Dubnicka:2016nyy ; Issadykov:2015iba .

## II Model

The covariant confined quark model Efimov:1988yd ; Efimov:1993ei ;
Branz:2009cd  is an effective quantum field approach to hadronic interactions
based on an interaction Lagrangian of hadrons interacting with their
constituent quarks.

The effective Lagrangian describing the transition of a meson
$M(q_{1}\bar{q}_{2})$ to its constituent quarks $q_{1}$ and $\bar{q}_{2}$

$\displaystyle{\mathcal{L}}_{\rm int}(x)$ $\displaystyle=$ $\displaystyle
g_{M}M(x)\cdot J_{M}(x)+{\rm h.c.},$ $\displaystyle J_{M}(x)$ $\displaystyle=$
$\displaystyle\int\\!\\!dx_{1}\\!\\!\int\\!\\!dx_{2}F_{M}(x,x_{1},x_{2})\bar{q}_{2}(x_{2})\Gamma_{M}q_{1}(x_{1})$
(5)

with $\Gamma_{M}$ a Dirac matrix which projects onto the spin quantum number
of the meson field $M(x)$. The vertex function $F_{M}$ characterizes the
finite size of the meson. Translational invariance requires the function
$F_{M}$ to fulfill the identity
$F_{M}(x+a,x_{1}+a,x_{2}+a)=F_{M}(x,x_{1},x_{2})$ for any four-vector $a$. A
specific form for the vertex function is adopted

$F_{M}(x,x_{1},x_{2})=\delta(x-w_{1}x_{1}-w_{2}x_{2})\Phi_{M}((x_{1}-x_{2})^{2}),$
(6)

where $\Phi_{M}$ is the correlation function of the two constituent quarks
with masses $m_{q_{1}}$ and $m_{q_{2}}$. The ratios of the quark masses
$w_{i}$ are defined as

$w_{q_{1}}=\frac{m_{q_{1}}}{m_{q_{1}}+m_{q_{2}}},\quad
w_{q_{2}}=\frac{m_{q_{2}}}{m_{q_{1}}+m_{q_{2}}},\quad w_{1}+w_{2}=1.$ (7)

A simple Gaussian form of the vertex function $\bar{\Phi}_{M}(-\,k^{2})$ is
selected

$\bar{\Phi}_{M}(-\,k^{2})=\exp\left(k^{2}/\Lambda_{M}^{2}\right)$ (8)

with the parameter $\Lambda_{M}$ linked to the size of the meson. The minus
sign in the argument is chosen to indicate that we are working in the
Minkowski space. Since $k^{2}$ turns into $-\,k_{E}^{2}$ in the Euclidean
space, the form (8) has the appropriate fall-off behavior in the Euclidean
region. Any choice for $\Phi_{M}$ is appropriate as long as it falls off
sufficiently fast in the ultraviolet region of the Euclidean space to render
the corresponding Feynman diagrams ultraviolet finite. We choose a Gaussian
form for calculational convenience.

The coupling constant $g_{M}$ in Eq. (5) is determined by the so-called
compositeness condition. The compositeness condition requires that the
renormalization constant $Z_{B}$ of the elementary meson field $B(x)$ is set
to zero, i.e.

$Z_{B}=1-\widetilde{\Pi}^{\prime}_{B}(p^{2})=0,\qquad(p^{2}=m^{2}_{B})$ (9)

where $\Pi^{\prime}_{B}(p^{2})$ is the derivative of the mass function.

$S$-matrix elements are described by the quark-loop diagrams which are the
convolution of the vertex functions and quark propagators. In the evaluation
of the quark-loop diagrams we use the local Dirac propagator

$S_{q}(k)=\frac{1}{m_{q}-\not\\!k-i\epsilon}=\frac{m_{q}+\not\\!k}{m^{2}_{q}-k^{2}-i\epsilon}$
(10)

with an effective constituent quark mass $m_{q}$.

The meson functions in the case of the pseudoscalar and vector meson are
written as

$\displaystyle\widetilde{\Pi}_{P}(p^{2})$ $\displaystyle=$ $\displaystyle
N_{c}g_{P}^{2}\int\frac{d^{4}k}{(2\pi)^{4}i}\widetilde{\Phi}^{2}_{P}(-k^{2})\mbox{\rm{tr}}\Big{(}\gamma^{5}S_{1}(k+w_{1}p)\gamma^{5}S_{2}(k-w_{2}p)\Big{)},$
(11) $\displaystyle\widetilde{\Pi}^{\mu\nu}_{V}(p^{2})$ $\displaystyle=$
$\displaystyle
N_{c}g_{V}^{2}\int\frac{d^{4}k}{(2\pi)^{4}i}\widetilde{\Phi}^{2}_{V}(-k^{2})\mbox{\rm{tr}}\Big{(}\gamma^{\mu}S_{1}(k+w_{1}p)\gamma^{\nu}S_{2}(k-w_{2}p)\Big{)}$
(12) $\displaystyle=$ $\displaystyle
g^{\mu\nu}\widetilde{\Pi}_{V}(p^{2})+p^{\mu}p^{\nu}\widetilde{\Pi}^{\parallel}_{V}(p^{2}).$

Here $N_{c}=3$ is the number of colors. Since the vector meson is on its mass-
shell $\epsilon_{V}\cdot p=0$ we need to keep the part
$\widetilde{\Pi}_{V}(p^{2})$. Substituting the derivative of the mass
functions into Eq. (9) one can determine the coupling constant $g_{B}$ as a
function of other model parameters. The loop integrations in Eqs. (11) and
(12) proceed by using the Fock-Schwinger representation of quark propagators

$S_{q}(k+wp)=\frac{1}{m_{q}-\not\\!k-w\not\\!p}=(m_{q}+\not\\!k+w\not\\!p)\int\limits_{0}^{\infty}\\!\\!d\alpha\,e^{-\alpha[m_{q}^{2}-(k+wp)^{2}]}.$
(13)

In the obtained integrals over the Fock-Schwinger parameters
$0\leq\alpha_{i}<\infty$ we introduce an additional integration over the
proper time which converts the set of Fock-Schwinger parameters into a
simplex. In general case one has

$\prod\limits_{i=1}^{n}\int\limits_{0}^{\infty}\\!\\!d\alpha_{i}f(\alpha_{1},\ldots,\alpha_{n})=\int\limits_{0}^{\infty}\\!\\!dtt^{n-1}\prod\limits_{i=1}^{n}\int\\!\\!d\alpha_{i}\delta\left(1-\sum\limits_{i=1}^{n}\alpha_{i}\right)f(t\alpha_{1},\ldots,t\alpha_{n}).$
(14)

Finally, we cut the integration over the proper time at the upper limit by
introducing an infrared cutoff $\lambda$. One has

$\int\limits_{0}^{\infty}dt(\ldots)\to\int\limits_{0}^{1/\lambda^{2}}dt(\ldots).$
(15)

This procedure allows us to remove all possible thresholds present in the
initial quark diagram. Thus 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 infrared cutoff parameter $\lambda$ is
taken to be universal for all physical processes.

The model parameters are determined by fitting calculated quantities of basic
processes to available experimental data or lattice simulations (for details,
see Ref. Branz:2009cd ).

## III Matrix elements and one-photon radiative decay width

The free Lagrangian of quarks is gauged in the standard manner by using
minimal substitution which gives

$\mathcal{L}^{\rm em}_{\rm int}(x)=e\,A_{\mu}(x)\,J^{\mu}_{\rm em}(x),\qquad
J^{\mu}_{\rm
em}(x)=e_{b}\,\bar{b}(x)\gamma^{\mu}b(x)+e_{q}\,\bar{q}(x)\gamma^{\mu}q(x)$
(16)

where $e_{b}$ and $e_{q}$ are the quark charges in units of the positron
charge. The radiative decays of a vector mesons into a pseudoscalar meson and
photon $X_{1}\to X_{2}\gamma$ are described by the Feynman diagrams shown in
Fig. 1.

Figure 1:  Feynman diagrams contributing in leading order to the dominant one-
photon radiative transitions $X_{1}(p)\to\gamma(q_{2})+X_{2}(q_{1})$
Ganbold:2021nvj .

The invariant matrix element for the one-photon radiative transition
$X_{1}\to\gamma X_{2}$ reads

$\displaystyle{\cal
M}_{{X_{1}}\to\gamma{X_{2}}}(p;p^{\prime},q)=eg_{X_{1}}g_{X_{2}}\epsilon^{V}_{\nu}(p)\epsilon^{\gamma}_{\mu}(q)\int\\!\\!dx\\!\\!\int\\!\\!dy\\!\\!\int\\!\\!dz\,e^{-ipx+ip^{\prime}y+iqz}\langle\,T\\{\bar{J}_{X_{1}}^{\nu}(x)J^{\mu}_{\rm
em}(z)J_{X_{2}}(y)\\}\rangle_{0}.,$ (17)

One has to note that there is an additional piece in the Lagrangian related to
the gauging nonlocal interactions of hadrons with their constituents
Branz:2009cd . This piece gives the additional contributions to the
electromagnetic processes. However, they are identically zero for the process
$X_{1}\to X_{2}\gamma$ due to its anomalous nature.

Using the Fourier transforms of the quark currents, we come to the final
result

$\displaystyle{\cal M}_{{X_{1}}\to\gamma{X_{2}}}(p;p^{\prime},q)$
$\displaystyle=$
$\displaystyle(2\pi)^{4}i\,\delta(p-p^{\prime}-q)M(p,p^{\prime}),$
$\displaystyle M(p,p^{\prime})$ $\displaystyle=$
$\displaystyle(-3i)eg_{X_{1}}g_{X_{2}}\epsilon^{V}_{\nu}(p)\epsilon^{\gamma}_{\mu}(q)\,\left(e_{b}M^{\mu\nu}_{b}+e_{q}M^{\mu\nu}_{q}\right)$
$\displaystyle M^{\mu\nu}_{b}$ $\displaystyle=$
$\displaystyle\int\\!\\!\frac{dk}{(2\pi)^{4}i}\widetilde{\Phi}_{X_{1}}(-\ell_{1}^{2})\widetilde{\Phi}_{X_{2}}(-\ell_{2}^{2})\mbox{\rm{tr}}\left[S_{q}(k)\gamma^{\nu}S_{b}(k-p)\gamma^{\mu}S_{b}(k-p^{\prime})\gamma^{5}\right]$
$\displaystyle M^{\mu\nu}_{q}$ $\displaystyle=$
$\displaystyle\int\\!\\!\frac{dk}{(2\pi)^{4}i}\widetilde{\Phi}_{X_{1}}(-\ell_{3}^{2})\widetilde{\Phi}_{X_{2}}(-\ell_{4}^{2})\mbox{\rm{tr}}\left[S_{q}(k+p^{\prime})\gamma^{\mu}S_{q}(k+p)\gamma^{\nu}S_{b}(k)\gamma^{5}\right]$
(18)

where $\ell_{1}=k-w_{2}\,p$, $\ell_{2}=k-w_{2}\,p^{\prime}$ and
$\ell_{3}=k+w_{1}\,p$, $\ell_{2}=k+w_{1}\,p^{\prime}$. The ratios of quark
masses are defined by Eq. (7). Now one has $m_{q_{1}}=m_{b}$ and
$m_{q_{2}}=m_{q}$ with $q=u,d,s$. By using the technique of calculations and
taking into account the transversality conditions
$\epsilon^{\gamma}_{\mu}(q)q^{\mu}=0$ and $\epsilon^{V}_{\nu}(p)p^{\nu}=0$ one
can arrives at the standard form of matrix element

$M(p,p^{\prime})=e\,g_{X_{1}X_{2}\gamma}\,\varepsilon^{pq\mu\nu}\epsilon^{\gamma}_{\mu}(q)\epsilon^{V}_{\nu}(p),$
(19)

where
$g_{X_{1}X_{2}\gamma}=e_{b}I_{b}(m^{2}_{X_{1}},m^{2}_{X_{2}})+e_{q}I_{q}(m^{2}_{X_{1}},m^{2}_{X_{2}})$
is radiative decay constant. The quantities $I_{b,q}$ are defined by the two-
fold integrals which are calculated numerically. The electromagnetic decay
width is written as

$\Gamma(X_{1}\to
X_{2}+\gamma)=\frac{\alpha}{24}m_{X_{1}}^{3}\left(1-\frac{m_{X_{2}}^{2}}{m_{X_{1}}^{2}}\right)^{3}g_{X_{1}X_{2}\gamma}^{2}\,.$
(20)

where $\alpha=e^{2}/4\pi=1/137.036$ is the fine-structure constant.

## IV Numerical results

The obvious model parameters include constituent quark masses and meson size
parameters that are fixed by fitting with the basic processes such as leptonic
decay widths with the experimental data or lattice simulations and the
differences are considered to be the absolute uncertainty in the respective
parameter. These parameters are determined by minimizing the functional
$\chi^{2}=\sum\limits_{i}\frac{(y_{i}^{\rm expt}-y_{i}^{\rm
theor})^{2}}{\sigma^{2}_{i}}$ where $\sigma_{i}$ is the experimental
uncertainty. If $\sigma$ is too small then we take its value of 10$\%$.
Besides, we have observed that the errors of the fitted parameters are of the
order of 10$\%$. Thus, the theoretical error of the CCQM is estimated to be of
the order of 10$\%$ at the level of matrix elements and the order of
15$-$20$\%$ at the level of widths. For present computations, we use the model
parameters obtained using the updated least square fit method performed in the
Ref. Ivanov:2015tru ; Ganbold:2014pua ; Dubnicka:2016nyy .

Table 1: Input values for some basic electromagnetic decay widths and our least-squares fit values (in keV). Process | Fit Values | Data ParticleDataGroup:2020ssz
---|---|---
$\rho^{\pm}\to\pi^{\pm}\gamma$ | 75.7$\pm$ 15.1 | 67 $\pm$ 7.5
$\omega\to\pi^{0}\gamma$ | 679$\pm$ 135.8 | 713 $\pm$ 26
$K^{\ast\pm}\to K^{\pm}\gamma$ | 55.8$\pm$ 11.2 | 46.8 $\pm$ 4.7
$K^{\ast 0}\to K^{0}\gamma$ | 132$\pm$ 26.4 | 116 $\pm$ 10
$D^{\ast\pm}\to D^{\pm}\gamma$ | 0.75$\pm$ 0.15 | 1.33 $\pm$ 0.37
$J/\psi\to\eta_{c}\gamma$ | 1.77$\pm$ 0.35 | 1.58 $\pm$ 0.37

The results of the least-squares fit used in the present study can be found in
Table 1. The agreement between the fit and experimental data is quite
satisfactory. The result for $J/\psi\to\eta_{c}\gamma$ agrees with the one
given in Ganbold:2021nvj (please look Table II there).

We think that there are strong relation between pseudoscalar $B_{q}$ and
vector $B^{*}_{q}$ mesons. In Table 2 given the leptonic decay constants and
masses of $B_{q}^{(*)}$ mesons from PDG ParticleDataGroup:2020ssz  and
corresponding fitted size parameters from previous works in CCQM
Issadykov:2015iba ; Dubnicka:2016nyy ; Dubnicka:2017job ; Issadykov:2017wlb ;
Issadykov:2018myx .

The leptonic decay constants in CCQM are defined by Eq.10 in Issadykov:2017wlb
.

Table 2:  The values of the leptonic decay constants and meson masses(in MeV) except the $B^{*}_{c}$ meson parameters from PDG ParticleDataGroup:2020ssz  and corresponding our model parameter $\Lambda$(in GeV)from our previous works Issadykov:2015iba ; Dubnicka:2016nyy ; Dubnicka:2017job ; Issadykov:2017wlb ; Issadykov:2018myx .  | $B_{c}$ | $B^{*}_{s}$ | $B_{s}$ | $B^{*0}$ | $B^{0}$ | $B^{+}$
---|---|---|---|---|---|---
$m$ | $6274.47\pm 0.32$ | $5415.4^{+1.8}_{-1.5}$ | $5366.88\pm 0.14$ | $5324.70\pm 0.21$ | $5279.65\pm 0.12$ | $5279.34\pm 0.12$
$f$ | 489 | 229 | 238.7 | 196 | 193 | 193
$\Lambda$ | 2.73 | 1.79 | 2.05 | 1.80 | 1.96 | 1.96

From Table 2 one can find next mass differences between pseudoscalar and
vector mesons

$\displaystyle\Delta m_{({B_{s}^{*}-B_{s}})}=49\quad~{}\text{MeV},$ (21)
$\displaystyle\Delta m_{({B^{*0}-B^{0}})}=45\quad~{}\text{MeV},$ (22)

so that the mass for $B_{c}^{*}$ meson assumed as:

$\displaystyle\Delta m_{({B_{c}^{*}-B_{c}})}=55\pm
10\quad~{}\text{MeV,}\quad~{}\text{then}\quad m_{B_{c}^{*}}=6329\pm
10\quad~{}\text{MeV,}$ (23)

which is within the predictions of other modelsEbert:2002pp ; Dowdall:2012ab ;
Colquhoun:2015oha ; Wang:2012kw ; Penin:2004xi .

The ratio between size parameters of $B_{q}^{(*)}$ mesons from our previous
works Issadykov:2015iba ; Dubnicka:2016nyy ; Dubnicka:2017job ;
Issadykov:2017wlb ; Issadykov:2018myx  as follows

$\displaystyle\Delta\Lambda_{({B_{s}^{*}/B_{s}})}=0.876,$ (24)
$\displaystyle\Delta\Lambda_{({B^{*0}/B^{0}})}=0.921,$ (25)

so that the size parameter $\Lambda_{B_{c}^{*}}$ assumed as:

$\displaystyle\Delta\Lambda_{({B_{c}^{*}/B_{c}})}=0.83\pm
0.05,\quad~{}\text{then}\quad\Lambda_{B_{c}^{*}}=2.26\pm
0.14\quad~{}\text{GeV.}$ (26)

Taking into account these two parameters we calculated the width of radiative
decay $\Gamma(B^{*+}_{c}\to B^{+}_{c}\gamma)$ and $f_{B^{*}_{c}}$ leptonic
decay constant in In Table 3. We calculated the widths of radiative decay in
dependence from mass(6319$-$6339 MeV) and $\Lambda$(2.12$-$2.40 GeV)
parameters of $B^{*}_{c}$ meson.

Table 3:  The widths of radiative decay of $B^{*}_{c}$ meson in dependence from mass and $\Lambda$ parameters.  $m_{B_{c}^{*}}=6319$ MeV | $\Gamma(B^{*+}_{c}\to B^{+}_{c}\gamma),~{}\text{(keV)}$ | $f_{B_{c}^{*}},~{}\text{(MeV)}$
---|---|---
$\Lambda=2.12$ | 0.023 | 481
$\Lambda=2.19$ | 0.024 | 508.5
$\Lambda=2.26$ | 0.025 | 536.4
$\Lambda=2.33$ | 0.026 | 564.6
$\Lambda=2.40$ | 0.027 | 593.3
$m_{B_{c}^{*}}=6324$ MeV | $\Gamma(B^{*+}_{c}\to B^{+}_{c}\gamma),~{}\text{(keV)}$ | $f_{B_{c}^{*}},~{}\text{(MeV)}$
$\Lambda=2.12$ | 0.032 | 479.9
$\Lambda=2.19$ | 0.033 | 507.3
$\Lambda=2.26$ | 0.034 | 535
$\Lambda=2.33$ | 0.035 | 563.1
$\Lambda=2.40$ | 0.036 | 591.6
$m_{B_{c}^{*}}=6329$ MeV | $\Gamma(B^{*+}_{c}\to B^{+}_{c}\gamma),~{}\text{(keV)}$ | $f_{B_{c}^{*}},~{}\text{(MeV)}$
$\Lambda=2.12$ | 0.042 | 478.8
$\Lambda=2.19$ | 0.044 | 506
$\Lambda=2.26$ | 0.045 | 533.6
$\Lambda=2.33$ | 0.047 | 561.6
$\Lambda=2.40$ | 0.048 | 589.9
$m_{B_{c}^{*}}=6339$ MeV | $\Gamma(B^{*+}_{c}\to B^{+}_{c}\gamma),~{}\text{(keV)}$ | $f_{B_{c}^{*}},~{}\text{(MeV)}$
$\Lambda=2.12$ | 0.069 | 476.5
$\Lambda=2.19$ | 0.072 | 503.5
$\Lambda=2.26$ | 0.074 | 530.8
$\Lambda=2.33$ | 0.077 | 558.5
$\Lambda=2.40$ | 0.079 | 586.5

The width of $\Gamma(B^{*+}_{c}\to B^{+}_{c}\gamma)$ decay strongly depends on
the choice of $B^{*}_{c}$ meson’s mass than on the choice of
$\Lambda_{B^{*}_{c}}$ in our calculations as expected, and shown on the Figure
2. While $f_{B^{*}_{c}}$ leptonic decay constant depends on the choice of
$\Lambda_{B^{*}_{c}}$.

Figure 2:  The width $\Gamma(B^{*+}_{c}\to B^{+}_{c}\gamma)$ in dependence on
the choice of the $B^{*}_{c}$ meson mass and the size parameter
$\Lambda_{B^{*}_{c}}$.

We compared the results of widths of radiative decays of $B^{*}_{q}$ mesons
within the covariant confined quark model with those from other theoretical
predictions in Table 4. For $\Gamma(B^{*+}_{c}\to B^{+}_{c}\gamma)$ we used
central values of assumed parameters($m_{B_{c}^{*}}=6329$ MeV and
$\Lambda_{B_{c}^{*}}=2.26$ GeV).

Table 4:  The widths of radiative decays of $B^{*}_{q}$ mesons in units of keV. | $\Gamma(B^{*0}\to B^{0}\gamma)$ | $\Gamma(B^{*+}\to B^{+}\gamma)$ | $\Gamma(B^{*0}_{s}\to B^{0}_{s}\gamma)$ | $\Gamma(B^{*+}_{c}\to B^{+}_{c}\gamma)$
---|---|---|---|---
This work | $0.117\pm 0.022.$ | $0.362\pm 0.072$ | $0.094\pm 0.018$ | $0.045\pm 0.009$
Ebert:2002xz ; Ebert:2002pp  | 0.070 | 0.19 | 0.054 | 0.033
Simonis:2018rld  | 0.165 | 0.520 | 0.115 | 0.039
Jena:2002is  | 0.14 | 0.52 | 0.06 | 0.030
Chang:2020xvu  | $0.116\pm 0.006$ | $0.349\pm 0.018$ | $0.084^{+11}_{-9}$ | $0.049^{+28}_{-21}$
Priyadarsini:2016tiu ; Patnaik:2017cbl  | 0.181 | 0.577 | 0.119 | 0.023
Lahde:1999ih ; Lahde:2002wj  | 0.0096 | 0.0674 | 0.148 | 0.034
Choi:2007se ; Choi:2009ai  | 0.13 | 0.4 | 0.068 | 0.022
Eichten:1994gt  |  |  |  | 0.135
Kiselev:1994rc  |  |  |  | 0.060
Fulcher:1998ka  |  |  |  | 0.059
Nobes:2000pm  |  |  |  | 0.050
Monteiro:2016rzi  |  |  |  | 0.019
AbdElHady:2005bv  |  |  |  | 0.019

## V CONCLUSION

In this work we made naive assumptions for the $B_{c}^{*}$ meson mass and size
parameter $\Lambda_{B_{c}^{*}}$ as $m_{B_{c}^{*}}=6329\pm 10$ MeV and
$\Lambda_{B_{c}^{*}}=2.26\pm 0.14$ GeV . Further, using this numbers We
calculated leptonic decay constants for the $B_{c}^{*}$ meson, and widths of
radiative decays of $B^{*}_{q}$ mesons, where $q=u/d,s,c$. In Table 3 and Fig.
2 were shown that the width $\Gamma(B^{*+}_{c}\to B^{+}_{c}\gamma)$ very
sensitive to the mass $m_{B_{c}^{*}}$ as expected, and less to the size
parameter $\Lambda_{B_{c}^{*}}$. While the $f_{B^{*}_{c}}$ leptonic decay
constant strongly depends on the choice of $\Lambda_{B^{*}_{c}}$. There is a
significant scatter in the values for the decay widths in Table 4. Therefore,
their experimental measurement will significantly correct the framework of the
existing theoretical approaches to the description of these processes.

## VI ACKNOWLEDGEMENTS

We would like to thank Prof. Mikhail A. Ivanov for useful discussions of some
aspects of this work. This research has been funded by the Science Committee
of the Ministry of Education and Science of the Republic of Kazakhstan (Grant
No. AP09057862).

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