# Systematic analysis of the form factors of $B_{c}\rightarrow\eta_{c}$,
$J/\psi$ and corresponding weak decays

Bin Wu1 Guo-Liang Yu1,2<EMAIL_ADDRESS>Jie Lu1,2 Zhi-Gang Wang1
<EMAIL_ADDRESS>1 Department of Mathematics and Physics, North China
Electric Power University, Baoding 071003, People’s Republic of China
2 Hebei Key Laboratory of Physics and Energy Technology, North China Electric
Power University, Baoding 071000, China

###### Abstract

The form factors of $B_{c}\rightarrow\eta_{c}$ and $B_{c}\rightarrow J/\psi$
are analyzed in the framework of three-point QCD sum rules. In these analyses,
the contributions of the vacuum condensate terms $\langle g_{s}^{2}GG\rangle$
and $\langle g_{s}^{3}GGGf\rangle$ are considered. In addition, the decay
widths and branching ratios of several decay channels are obtained by using
the calculated form factors. These decay processes include the nonleptonic
decays of $B_{c}^{-}\to\eta_{c}\pi^{-}$, $\eta_{c}K^{-}$, $\eta_{c}\rho^{-}$,
$\eta_{c}K^{*-}$, $B_{c}^{-}\to J/\psi\pi^{-}$, $J/\psi K^{-}$,
$J/\psi\rho^{-}$, $J/\psi K^{*-}$, and the semileptonic decays of
$B_{c}^{-}\to\eta_{c}\mathcal{l}\bar{\nu}$, $B_{c}^{-}\to
J/\psi\mathcal{l}\bar{\nu}$. These results about the form factors and decay
properties of $B_{c}$ meson provide useful information for us to study the
heavy-quark dynamics and find new physics(NP) beyond Standard Model(SM).

###### pacs:

13.25.Ft; 14.40.Lb

##  I Introduction

Since the pesudoscalar meson $B_{c}$ was firstly observed in 1998CDF:1998axz ,
physicists have focused attention on it for its special properties. Because of
the instability of top quark, $B_{c}$ meson is the only quarknoium with mixed
heavy flavor quarks. Moreover, $B_{c}$ meson lies below the threshold of
$B\bar{D}$, thus can only decay through weak interaction. Correspondingly, it
is relatively more stable than other heavy quarkonium states consisting of
only one flavor quark such as $\bar{c}c$ and $\bar{b}b$. Naturally, $B_{c}$
meson is an excellent laboratory to study the heavy-quark dynamics and the
weak decay process about heavy quark.

The decay process of $B_{c}$ to charmonium is especially interesting and
valuable. At quark level, the decay process that $B_{c}$ decays to charmonium
is accompanied by $b\to c$ transition, which is useful to extract the Cabibbo-
Kobayashi-Maskawa (CKM) matrix element $V_{cb}$. Furthermore, the decay
process $B_{c}\to J/\psi\mathcal{l}\nu$ is similar to $B\to
D^{(*)}\mathcal{l}\nu$. Both of these processes are relevant to the transition
of $b\to c\mathcal{l}\nu$. As for the decay ratio
$R(D^{(*)})=\Gamma(\bar{B}\to D^{(*)}\tau^{-}\bar{\nu})/\Gamma(\bar{B}\to
D^{(*)}\mu^{-}\bar{\nu})$, the experimental results and predictions of the
Standard Model (SM) were not agree with each other for a long time. Though
Belle and LHCb carried out repeated measurements and theorists also updated
their results for several times basing on SM predictions, there is still
$3\sigma\sim 4\sigma$ discrepancy between them. The decay process $B_{c}\to
J/\psi\mathcal{l}\nu$ which is similar to that of $B\to
D^{(*)}\mathcal{l}\nu$, is expected to find the new physics (NP) beyond the
Standard Model (SM). Recently, the LHCb collaboration measured the decay
ratioLHCb:2017vlu ,

$\displaystyle R(J/\psi)=\frac{{\Gamma(B_{c}^{-}\to
J/\psi{\tau^{-}}\bar{\nu})}}{{\Gamma(B_{c}^{-}\to
J/\psi{\mu^{-}}\bar{\nu})}}=0.71\pm 0.17(\mathrm{stat})\pm
0.18(\mathrm{syst})$ (1)

which waits for further confirmation by theorist.

Both experimental and theoretical investigations about $B_{c}$ decaying to
charmonium have made great progress. The $B_{c}$ meson was firstly found
through the decay process $B_{c}\to J/\psi\mathcal{l}\nu$ in $p\bar{p}$
collisions at the Fermilab Tevatron by the CDF collaboration. Up to now, many
decay modes have been observed in experiments, such as $B_{c}^{+}\to
J/\psi\pi^{+}$LHCb:2012ihf , $B_{c}^{+}\to
J/\psi\pi^{+}\pi^{-}\pi^{+}$LHCb:2012ag , $B_{c}^{+}\to J/\psi
K^{+}$LHCb:2013hwj , $B_{c}^{+}\to J/\psi D^{(*)+}_{s}$LHCb:2013kwl ,
$B_{c}^{+}\to J/\psi K^{+}K^{-}\pi^{+}$LHCb:2013rud . Thanks to the
development of accelerator and detector techniques, it is now easy to
reconstruct the decay process of $B_{c}\to J/\psi$. Ref.Gao:2010zzc  estimated
that the inclusive production cross section of $B_{c}$ meson (including
excited states) at the LHC is at a level of $1ub$ for $\sqrt{14}$ TeV. This
means that $O(10^{9})$ $B_{c}^{+}$ meson can be anticipated with 1 fb-1, so
more precise experimental results and more decay channels of $B_{c}$ are
expected. As for the decay process $B_{c}\to\eta_{c}$, it has not been
observed in experiments. In theory, the processes that $B_{c}$ decays to
charmnoium have been studied by many theoretical methods, such as the lattice
QCDHarrison:2020gvo ; Colquhoun:2016osw , the relativistic quark model
(RQM)Nobes:2000pm ; Faustov:2022ybm , the covariant confined quarkmodel
(CCQM)Tran:2018kuv ,the non-relativistic QCD (NRQCD)Qiao:2012hp ; Tang:2022nqm
, the light-front quark model (LFQM)Zhang:2023ypl ; Shi:2016gqt , the QCD sum
rules (QCDSR)Kiselev:2000pp ; Azizi:2013zta ; Azizi:2019aaf ; Colangelo:1992cx
, the light-cone QCD sum rules (LCSR)Wang:2007fs ; Huang:2007kb ;
Leljak:2019eyw ; Bordone:2022drp  and the perturbative QCD (pQCD)Wang:2012lrc
; Liu:2023kxr ; Liu:2020upy ; Liu:2018kuo . Nevertheless, these results are
not consistent well with the experimental data or with each other, thus it is
necessary for us to perform further confirmations about this weak transition
by different theoretical methods.

The decay process of $B_{c}$ to charmonium is realized according to the
spectator mechanism that $b$ quark decays and $c$ quark plays as a spectator.
This process is related to the weak interaction and its effective Hamiltonian
at quark level has been well formulated. The last step in calculating the
decay width of $B_{c}$ decaying to charmonium is properly evaluating the
hadron transition matrix element, which is commonly factorized as form
factors. Generally, the form factors are non-perturbative in QCD, which can be
obtained by non-perturbative methods. The QCDSR is a powerful non-perturbative
approach which was proposed in Refs. Shifman:1978bx ; Shifman:1978by . Besides
of the evaluation about hadron mass spectra and decay constantsWei:2006wa ;
Wang:2012kw ; Aliev:2012iv ; Aliev:2012tt , QCDSR has been also used to
analyze the electroweak and electromagnetic form factorBelyaev:1993wp ;
Dai:1996xv ; Yang:2005bv ; Shi:2019hbf ; Zhao:2020mod ; Zhang:2023nxl ;
Lu:2024tgy , coupling constant of the strong interactionBracco:2011pg ;
Yu:2015xwa ; Lu:2023gmd ; Lu:2023pcg ; Lu:2023lvu ; Colangelo:2000dp , etc. In
this work, we will calculate the form factors of $B_{c}$ decaying to
$\eta_{c}$, $J/\psi$ in the framework of QCDSR with unified input parameters.
In addition, these form factors will be used to obtain the decay widths of
different decay processes including nonleptonic decays of
$B_{c}^{-}\to\eta_{c}\pi^{-}$, $\eta_{c}K^{-}$, $\eta_{c}\rho^{-}$,
$\eta_{c}K^{*-}$, $B_{c}^{-}\to J/\psi\pi^{-}$, $J/\psi K^{-}$,
$J/\psi\rho^{-}$, $J/\psi K^{*-}$, and semileptonic decays of
$B_{c}^{-}\to\eta_{c}\mathcal{l}\bar{\nu}$, $B_{c}^{-}\to
J/\psi\mathcal{l}\bar{\nu}$.

The article is organized as follows. After introduction in Sec. I, the basic
procedure of three-point QCDSR are discussed in detail in Sec. II. The
numerical results of form factors are organized in Sec. III. The calculation
and discussion about decay width are performed in Sec. IV. In the last section
a brief conclusion is presented.

##  II The three-point QCDSR

In the framework of QCDSR, the analysis about form factors starts from the
three-point correlation function,

$\displaystyle\Pi(p,p^{\prime})$
$\displaystyle={i^{2}}\int{{d^{4}}x{d^{4}}y{e^{ip^{\prime}\cdot
x}}{e^{i(p-p^{\prime})\cdot y}}}$ $\displaystyle\times\left\langle
0\right|T\\{{J_{C}}(x)J(y)J_{{B_{c}}}^{+}(0)\\}\left|0\right\rangle$ (2)

where $T$ is the time ordering operation, $p$ and $p^{\prime}$ are the
momentum of $B_{c}$ meson and $C$ meson with $C=\eta_{c}$, $J/\psi$. $J_{C}$
and $J_{B_{c}}$ are interpolating currents which have the same quantum numbers
with corresponding mesons. In this work, the scalar, vector and tensor form
factors of $B_{c}\to\eta_{c}$, and vector, axial vector and tensor form
factors of $B_{c}\to J/\psi$ are evaluated. Different form factors are
distinguished from each other by the current $J(y)$. These currents in Eq.
(II) can be written as,

$\displaystyle{J_{{B_{c}}}}(0)=\bar{c}(0)i{\gamma_{5}}b(0)$ $\displaystyle
J(y)=\bar{c}(y)\Gamma b(y)$
$\displaystyle{J_{{\eta_{c}}}}(x)=\bar{c}(x)i{\gamma_{5}}c(x)$
$\displaystyle{J^{J/\psi}_{\mu}}(x)=\bar{c}(x){\gamma_{\mu}}c(x)$ (3)

where $\Gamma$ is the Dirac matrix,
$\Gamma=I,{\gamma_{\mu}},{\gamma_{\mu}}{\gamma_{5}},{\sigma_{\mu\nu}(\gamma_{5})}$
for scalar, vector, axial vector and tensor form factors, respectively.

The correlation function will be calculated in two sides which are named as
phenomenological side and QCD side. The former is realized according to
phenomenological method where the form factors are extracted, whereas the
latter is obtained by using the operations product expansion (OPE) in deep
Euclidean region. Combining these two sides by the quark-hadron duality, the
sum rules of form factors will be derived.

###  II.1 The Phenomenological Side

In phenomenological side, the correlation function is inserted with complete
sets of hadronic states with the same quantum numbers as $\eta_{c}$, $J/\psi$
and $B_{c}$. Performing integrations over $x$ and $y$, and using the
dispersion relation, we can represent the correlation function of Eq. (II) as
fallowsColangelo:2000dp ,

$\displaystyle\Pi^{\mathrm{had}1}(p,p^{\prime})$
$\displaystyle=\frac{{\left\langle
0\right|{J_{{\eta_{c}}}}(0)\left|{{\eta_{c}}(p^{\prime})}\right\rangle\left\langle{{B_{c}}(p)}\right|J_{{B_{c}}}^{+}(0)\left|0\right\rangle}}{{(m_{{B_{c}}}^{2}-{p^{2}})(m_{{\eta_{c}}}^{2}-{{p^{\prime}}^{2}})}}$
$\displaystyle\quad\times\left\langle{{\eta_{c}}(p^{\prime})}\right|\bar{c}(0)b(0)\left|{{B_{c}}(p)}\right\rangle+h.r.$
$\displaystyle\Pi^{\mathrm{had}2}_{\mu}(p,p^{\prime})$
$\displaystyle=\frac{{\left\langle
0\right|{J_{{\eta_{c}}}}(0)\left|{{\eta_{c}}(p^{\prime})}\right\rangle\left\langle{{B_{c}}(p)}\right|J_{{B_{c}}}^{+}(0)\left|0\right\rangle}}{{(m_{{B_{c}}}^{2}-{p^{2}})(m_{{\eta_{c}}}^{2}-{{p^{\prime}}^{2}})}}$
$\displaystyle\quad\times\left\langle{{\eta_{c}}(p^{\prime})}\right|\bar{c}(0){\gamma_{\mu}}\left|{{B_{c}}(p)}\right\rangle+h.r.$
$\displaystyle\Pi^{\mathrm{had}3}_{\mu\nu}(p,p^{\prime})$
$\displaystyle=\frac{{\left\langle
0\right|{J_{{\eta_{c}}}}(0)\left|{{\eta_{c}}(p^{\prime})}\right\rangle\left\langle{{B_{c}}(p)}\right|J_{{B_{c}}}^{+}(0)\left|0\right\rangle}}{{(m_{{B_{c}}}^{2}-{p^{2}})(m_{{\eta_{c}}}^{2}-{{p^{\prime}}^{2}})}}$
$\displaystyle\quad\times\left\langle{{\eta_{c}}(p^{\prime})}\right|\bar{c}(0){\sigma_{\mu\nu}}{\gamma_{5}}b(0)\left|{{B_{c}}(p)}\right\rangle+h.r.$
$\displaystyle\Pi^{\mathrm{had}4}_{\lambda\mu}(p,p^{\prime})$
$\displaystyle=\frac{{\left\langle
0\right|{J^{{J/\psi}}_{\lambda}}(0)\left|{{J/\psi}(p^{\prime})}\right\rangle\left\langle{{B_{c}}(p)}\right|J_{{B_{c}}}^{+}(0)\left|0\right\rangle}}{{(m_{{B_{c}}}^{2}-{p^{2}})(m_{{J/\psi}}^{2}-{{p^{\prime}}^{2}})}}$
$\displaystyle\quad\times\left\langle{{J/\psi}(p^{\prime})}\right|\bar{c}(0){\gamma_{\mu}}b(0)\left|{{B_{c}}(p)}\right\rangle+h.r.$
$\displaystyle\Pi^{\mathrm{had}5}_{\lambda\mu}(p,p^{\prime})$
$\displaystyle=\frac{{\left\langle
0\right|{J^{{J/\psi}}_{\lambda}}(0)\left|{{J/\psi}(p^{\prime})}\right\rangle\left\langle{{B_{c}}(p)}\right|J_{{B_{c}}}^{+}(0)\left|0\right\rangle}}{{(m_{{B_{c}}}^{2}-{p^{2}})(m_{{J/\psi}}^{2}-{{p^{\prime}}^{2}})}}$
$\displaystyle\quad\times\left\langle{{J/\psi}(p^{\prime})}\right|\bar{c}(0){\gamma_{\mu}}{\gamma_{5}}b(0)\left|{{B_{c}}(p)}\right\rangle+h.r.$
$\displaystyle\Pi^{\mathrm{had}6}_{\lambda\mu\nu}(p,p^{\prime})$
$\displaystyle=\frac{{\left\langle
0\right|{J^{{J/\psi}}_{\lambda}}(0)\left|{{J/\psi}(p^{\prime})}\right\rangle\left\langle{{B_{c}}(p)}\right|J_{{B_{c}}}^{+}(0)\left|0\right\rangle}}{{(m_{{B_{c}}}^{2}-{p^{2}})(m_{{J/\psi}}^{2}-{{p^{\prime}}^{2}})}}$
$\displaystyle\quad\times\left\langle{{J/\psi}(p^{\prime})}\right|\bar{c}(0){\sigma_{\mu\nu}}b(0)\left|{{B_{c}}(p)}\right\rangle+h.r.$
(4)

where $h.r.$ denotes the contributions of excited and continuum states.
$\Pi^{\mathrm{had}1}$, $\Pi^{\mathrm{had}2}_{\mu}$ and
$\Pi^{\mathrm{had}3}_{\mu\nu}$ are used to analyze the scalar, vector, and
tensor form factors of $B_{c}\to\eta_{c}$, and
$\Pi^{\mathrm{had}4}_{\lambda\mu}$, $\Pi^{\mathrm{had}5}_{\lambda\mu}$ and
$\Pi^{\mathrm{had}6}_{\lambda\mu\nu}$ are used for the analysis of vector,
axial vector and tensor form factors of $B_{c}\to J/\psi$, respectively. The
meson vacuum matrix elements in these above correlation functions can be
factorized as,

$\displaystyle\left\langle{\rm{0}}\right|{J_{{\eta_{c}}}}(0)\left|{{\eta_{c}}(p^{\prime})}\right\rangle=\frac{{{f_{{\eta_{c}}}}m_{{\eta_{c}}}^{2}}}{{2{m_{c}}}}$
$\displaystyle\left\langle{\rm{0}}\right|{J^{J/\psi}_{\lambda}}(0)\left|{J/\psi(p^{\prime})}\right\rangle={{f_{J/\psi}}m_{J/\psi}\xi_{\lambda}}$
$\displaystyle\left\langle{{B_{c}}(p)}\right|J_{{B_{c}}}^{+}(0)\left|0\right\rangle=\frac{{{f_{{B_{c}}}}m_{{B_{c}}}^{2}}}{{{m_{c}}+{m_{b}}}}$
(5)

where $f_{\eta_{c}}$, $f_{J/\psi}$ and $f_{B_{c}}$ are decay constants of
corresponding mesons. $\xi_{\lambda}$ is the polarization vector of $J/\psi$
meson. The other transition matrix elements in Eq. (II.1) can be expressed in
terms of form factorsColangelo:2022lpy ,

$\displaystyle\left\langle{{\eta_{c}}(p^{\prime})}\right|\bar{c}(0)b(0)\left|{{B_{c}}(p)}\right\rangle$
$\displaystyle={f_{S}}({q^{2}})$
$\displaystyle\left\langle{{\eta_{c}}(p^{\prime})}\right|\bar{c}(0){\gamma_{\mu}}b(0)\left|{{B_{c}}(p)}\right\rangle$
$\displaystyle={f_{+}}({q^{2}})\left({{p_{\mu}}+{{p^{\prime}}_{\mu}}-\frac{{m_{{B_{c}}}^{2}-m_{{\eta_{c}}}^{2}}}{{{q^{2}}}}{q_{\mu}}}\right)+{f_{0}}({q^{2}})\frac{{m_{{B_{c}}}^{2}-m_{{\eta_{c}}}^{2}}}{{{q^{2}}}}{q_{\mu}}$
$\displaystyle\left\langle{{\eta_{c}}(p^{\prime})}\right|\bar{c}(0){\sigma_{\mu\nu}}{\gamma_{5}}b(0)\left|{{B_{c}}(p)}\right\rangle$
$\displaystyle=-\frac{{2{f_{T}}({q^{2}})}}{{{m_{{B_{c}}}}+{m_{{\eta_{c}}}}}}{\varepsilon_{\mu\nu\alpha\beta}}{p^{\alpha}}{{p^{\prime}}^{\beta}}$
$\displaystyle\left\langle{J/\psi(p^{\prime},\xi)}\right|\bar{c}(0){\gamma_{\mu}}b(0)\left|{{B_{c}}(p)}\right\rangle$
$\displaystyle=\frac{{2V({q^{2}})}}{{{m_{{B_{c}}}}+{m_{J/\psi}}}}{\varepsilon_{\mu\nu\alpha\beta}}{\xi^{*\nu}}{p^{\alpha}}{{p^{\prime}}^{\beta}}$
$\displaystyle\left\langle{J/\psi(p^{\prime},\xi)}\right|\bar{c}(0){\gamma_{\mu}}{\gamma_{5}}b(0)\left|{{B_{c}}(p)}\right\rangle$
$\displaystyle=i({m_{{B_{c}}}}+{m_{J/\psi}})\left({\xi_{\mu}^{*}-\frac{{({\xi^{*}}\cdot
q)}}{{{q^{2}}}}{q_{\mu}}}\right){A_{1}}({q^{2}})+i({\xi^{*}}\cdot
q)\frac{{2{m_{J/\psi}}}}{{{q^{2}}}}{q_{\mu}}{A_{0}}({q^{2}})$
$\displaystyle\quad-i\frac{{({\xi^{*}}\cdot
q)}}{{{m_{{B_{c}}}}+{m_{J/\psi}}}}\left({{{(p{\rm{+}}p^{\prime})}_{\mu}}-\frac{{m_{{B_{c}}}^{2}-m_{J/\psi}^{2}}}{{{q^{2}}}}{q_{\mu}}}\right){A_{2}}({q^{2}})$
$\displaystyle\left\langle{J/\psi(p^{\prime},\xi)}\right|\bar{c}(0){\sigma_{\mu\nu}}b(0)\left|{{B_{c}}(p)}\right\rangle$
$\displaystyle={T_{0}}({q^{2}})\frac{{{\xi^{*}}\cdot
q}}{{{{({m_{{B_{c}}}}+{m_{J/\psi}})}^{2}}}}{\varepsilon_{\mu\nu\alpha\beta}}{p^{\alpha}}{{p^{\prime}}^{\beta}}+{T_{1}}({q^{2}}){\varepsilon_{\mu\nu\alpha\beta}}{p^{\alpha}}{\xi^{*\beta}}+{T_{2}}({q^{2}}){\varepsilon_{\mu\nu\alpha\beta}}{{p^{\prime}}^{\alpha}}{\xi^{*\beta}}$
(6)

where $f_{S}$, $f_{+}/f_{0}$ and $f_{T}$ represent the scalar, vector and
tensor form factors of $B_{c}\to\eta_{c}$, and $V$, $A_{0}/A_{1}/A_{2}$ and
$T_{0}/T_{1}/T_{2}$ denote the vector, axial vector and tensor form factors of
$B_{c}\to J/\psi$, respectively. These form factors are the function of four-
momentum squared of $q$ with $q=p-p^{\prime}$. With Eqs. (II.1)$\sim$(II.1)
and the following relation of polarization vector,

$\displaystyle{\xi_{\mu}}\xi_{\nu}^{*}=-{g_{\mu\nu}}+\frac{{{{p^{\prime}}_{\mu}}{{p^{\prime}}_{\nu}}}}{{{{p^{\prime}}^{2}}}}$
(7)

we can obtain the expressions of the correlation functions at phenomenological
side. Take the vector form factor of $B_{c}\to J/\psi$ as an example, it is
written as,

$\displaystyle{\Pi^{\mathrm{had}4}_{\lambda\mu}}(p,p^{\prime})=$
$\displaystyle
2\frac{{{f_{{B_{c}}}}{f_{J/\psi}}m_{{B_{c}}}^{2}{m_{J/\psi}}}}{{({m_{{B_{c}}}}+{m_{J/\psi}})({m_{c}}+{m_{b}})}}V(q^{2})$
$\displaystyle\times\frac{{{\varepsilon_{\lambda\mu\alpha\beta}}{p^{\alpha}}{{p^{\prime}}^{\beta}}}}{{(m_{J/\psi}^{2}-{{p^{\prime}}^{2}})(m_{{B_{c}}}^{2}-{p^{2}})}}+h.r.$
(8)

Finally, we take the change of variables $p^{2}\rightarrow-P^{2}$, $p^{\prime
2}\rightarrow-P^{\prime 2}$ and $q^{2}\rightarrow-Q^{2}$, and perform the
double Borel transformations for $P^{2}$ and $P^{\prime 2}$. The correlation
function is written as,

$\displaystyle{\mathscr{B}\mathscr{B}}[{\Pi^{\mathrm{had4}}_{\lambda\mu}}(P,P^{\prime})]=$
$\displaystyle
2\frac{{{f_{{B_{c}}}}{f_{J/\psi}}m_{{B_{c}}}^{2}{m_{J/\psi}}{\varepsilon_{\lambda\mu\alpha\beta}}{p^{\alpha}}{{p^{\prime}}^{\beta}}}}{{({m_{{B_{c}}}}+{m_{J/\psi}})({m_{c}}+{m_{b}})}}{V}(Q^{2})$
$\displaystyle\times\exp(-\frac{{m_{{B_{c}}}^{2}}}{{{M^{2}}}}-\frac{{m_{J/\psi}^{2}}}{{k{M^{2}}}})+{\mathscr{B}\mathscr{B}}(h.r.)$
(9)

where $M$ is the Borel parameter. Theoretically there are two parameters
introduced by the double Borel transformations, we can relate one to the other
with factor $k=m_{\eta_{c}}^{2}/m_{B_{c}}^{2}$Bracco:2011pg .

###  II.2 The QCD Side

The correlation functions which have been written as Eq. (II.1) at
phenomenological side, now can be represented as follows according to the
Wick‘s theorem at the QCD side,

$\displaystyle\Pi^{\mathrm{QCD}1}(p,p^{\prime})$
$\displaystyle=\int{{d^{4}}x{d^{4}}y{e^{ip^{\prime}x}}{e^{i(p-p^{\prime})y}}}$
$\displaystyle\quad\times
Tr\\{{C}(-x){\gamma_{5}}{C}(x-y){B}(y){\gamma_{5}}\\}$
$\displaystyle\Pi^{\mathrm{QCD}2}_{\nu}(p,p^{\prime})$
$\displaystyle=\int{{d^{4}}x{d^{4}}y{e^{ip^{\prime}x}}{e^{i(p-p^{\prime})y}}}$
$\displaystyle\quad\times
Tr\\{{C}(-x){\gamma_{5}}{C}(x-y)\gamma_{\nu}{B}(y){\gamma_{5}}\\}$
$\displaystyle\Pi^{\mathrm{QCD}3}_{\mu\nu}(p,p^{\prime})$
$\displaystyle=\int{{d^{4}}x{d^{4}}y{e^{ip^{\prime}x}}{e^{i(p-p^{\prime})y}}}$
$\displaystyle\quad\times
Tr\\{{C}(-x){\gamma_{5}}{C}(x-y){\sigma_{\mu\nu}}{\gamma_{5}}{B}(y){\gamma_{5}}\\}$
$\displaystyle\Pi^{\mathrm{QCD}4}_{\lambda\mu}(p,p^{\prime})$
$\displaystyle=\int{{d^{4}}x{d^{4}}y{e^{ip^{\prime}x}}{e^{i(p-p^{\prime})y}}}$
$\displaystyle\quad\times
Tr\\{{C}(-x){\gamma_{\lambda}}{C}(x-y)\gamma_{\mu}(y){\gamma_{5}}\\}$
$\displaystyle\Pi^{\mathrm{QCD}5}_{\lambda\mu}(p,p^{\prime})$
$\displaystyle=\int{{d^{4}}x{d^{4}}y{e^{ip^{\prime}x}}{e^{i(p-p^{\prime})y}}}$
$\displaystyle\quad\times
Tr\\{{C}(-x){\gamma_{\lambda}}{C}(x-y)\gamma_{\mu}{\gamma_{5}}{B}(y){\gamma_{5}}\\}$
$\displaystyle\Pi^{\mathrm{QCD}6}_{\lambda\mu\nu}(p,p^{\prime})$
$\displaystyle=\int{{d^{4}}x{d^{4}}y{e^{ip^{\prime}x}}{e^{i(p-p^{\prime})y}}}$
$\displaystyle\quad\times
Tr\\{{C}(-x){\gamma_{\lambda}}{C}(x-y){\sigma_{\mu\nu}}{B}(y){\gamma_{5}}\\}$
(10)

where $C(x)$ and $B(x)$ are the full propagator of $c$ and $b$ quarks, which
have the following expressionReinders:1984sr ,

$\displaystyle Q_{ij}(x)$
$\displaystyle=\frac{i}{{{{(2\pi)}^{4}}}}\int{{d^{4}}k{e^{-ik\cdot
x}}}\bigg{\\{}\frac{{{\delta_{ij}}}}{{\not{k}-{m_{Q}}}}-\frac{{{g_{s}}G_{\alpha\beta}^{n}t_{ij}^{n}}}{4}\frac{{{\sigma^{\alpha\beta}}(\not{k}+{m_{Q}})+(\not{k}+{m_{Q}}){\sigma^{\alpha\beta}}}}{{{{({k^{2}}-m_{Q}^{2})}^{2}}}}+\frac{{{g_{s}}{D_{\alpha}}G_{\beta\lambda}^{n}t_{ij}^{n}({f^{\lambda\beta
a}}+{f^{\lambda\alpha\beta}})}}{{3{{({k^{2}}-m_{Q}^{2})}^{4}}}}$
$\displaystyle\quad-\frac{{g_{s}^{2}{{({t^{a}}{t^{b}})}_{ij}}G_{\alpha\beta}^{a}G_{\mu\nu}^{b}({f^{\alpha\beta\mu\nu}}+{f^{\alpha\mu\beta\nu}}+{f^{\alpha\mu\nu\beta}})}}{{4{{({k^{2}}-m_{Q}^{2})}^{5}}}}+\frac{{i{\delta_{ij}}\left\langle{f{G^{3}}}\right\rangle}}{{48}}\frac{{(\not{k}+{m_{Q}})[\not{k}({k^{2}}-3m_{Q}^{2})+2{m_{Q}}(2{k^{2}}-m_{Q}^{2})](\not{k}+{m_{Q}})}}{{{{({k^{2}}-m_{Q}^{2})}^{6}}}}+...\bigg{\\}}$
(11)

Here the subscript $i$ and $j$ are color indices, and
${\sigma_{\alpha\beta}}=i[{\gamma_{\alpha}},{\gamma_{\beta}}]/2$.
${D_{\alpha}}={\partial_{\alpha}}-i{g_{s}}G_{\alpha}^{n}{t^{n}}$,
$G_{\alpha}^{n}$ is the gluon field, ${t^{n}}={\lambda^{n}}/2$, and
${\lambda^{n}}$ is the Gell-Mann matrix. ${f^{\lambda\beta\alpha}}$ and
${f^{\alpha\beta\mu\nu}}$ are defined as

$\displaystyle{f^{\lambda\alpha\beta}}=$
$\displaystyle(\not{k}+{m_{c(b)}}){\gamma^{\lambda}}(\not{k}+{m_{c(b)}}){\gamma^{a}}(\not{k}+{m_{c(b)}})$
$\displaystyle\times{\gamma^{\beta}}(\not{k}+{m_{c(b)}})$
$\displaystyle{f^{\alpha\beta\mu\nu}}=$
$\displaystyle(\not{k}+{m_{c(b)}}){\gamma^{\alpha}}(\not{k}+{m_{c(b)}}){\gamma^{\beta}}(\not{k}+{m_{c(b)}})$
$\displaystyle\times{\gamma^{\mu}}(\not{k}+{m_{c(b)}}){\gamma^{\nu}}(\not{k}+{m_{c(b)}})$
(12)

Substituting with the full propagator in Eq. (II.2), the correlation functions
can be decomposed into different tensor structures except for
$\Pi^{\mathrm{QCD}1}$,

$\displaystyle\Pi_{\nu}^{\mathrm{QCD2}}(p,p^{\prime})={\Pi^{\mathrm{QCD2}}_{1}}(p,p^{\prime}){p_{\nu}}+{\Pi^{\mathrm{QCD2}}_{2}}(p,p^{\prime}){{p^{\prime}}_{\nu}}$
$\displaystyle\Pi_{\mu\nu}^{\mathrm{QCD3}}(p,p^{\prime})={\Pi^{\mathrm{QCD3}}_{1}}(p,p^{\prime}){\varepsilon_{\mu\nu\alpha\beta}}{p^{\alpha}}{{p^{\prime}}^{\beta}}$
$\displaystyle\Pi_{\lambda\mu}^{\mathrm{QCD4}}(p,p^{\prime})={\Pi^{\mathrm{QCD4}}_{1}}(p,p^{\prime}){\varepsilon_{\lambda\mu\alpha\beta}}{p^{\alpha}}{{p^{\prime}}^{\beta}}$
$\displaystyle\Pi_{\lambda\mu}^{\mathrm{QCD5}}(p,p^{\prime})={\Pi^{\mathrm{QCD5}}_{1}}(p,p^{\prime}){g_{\lambda\mu}}$
$\displaystyle\quad+{\Pi^{\mathrm{QCD5}}_{2}}(p,p^{\prime}){p_{\lambda}}{p_{\mu}}+{\Pi^{\mathrm{QCD5}}_{3}}(p,p^{\prime}){{p^{\prime}}_{\lambda}}{{p^{\prime}}_{\mu}}$
$\displaystyle\quad+{\Pi^{\mathrm{QCD5}}_{4}}(p,p^{\prime}){p_{\lambda}}{{p^{\prime}}_{\mu}}+{\Pi^{\mathrm{QCD5}}_{5}}(p,p^{\prime}){{p^{\prime}}_{\lambda}}{p_{\mu}}$
$\displaystyle\Pi_{\lambda\mu\nu}^{\mathrm{QCD6}}(p,p^{\prime})$
$\displaystyle\quad={\Pi^{\mathrm{QCD6}}_{1}}(p,p^{\prime}){\varepsilon_{\lambda\mu\nu\alpha}}{p^{\alpha}}+{\Pi^{\mathrm{QCD6}}_{2}}(p,p^{\prime}){\varepsilon_{\lambda\mu\nu\alpha}}{{p^{\prime}}^{\alpha}}$
$\displaystyle\quad+{\Pi^{\mathrm{QCD6}}_{3}}(p,p^{\prime}){\varepsilon_{\mu\nu\alpha\beta}}{p^{\alpha}}{{p^{\prime}}^{\beta}}{p_{\lambda}}+{\Pi^{\mathrm{QCD6}}_{4}}(p,p^{\prime}){\varepsilon_{\mu\nu\alpha\beta}}{p^{\alpha}}{{p^{\prime}}^{\beta}}{{p^{\prime}}_{\lambda}}$
$\displaystyle\quad+{\Pi^{\mathrm{QCD6}}_{5}}(p,p^{\prime}){\varepsilon_{\lambda\nu\alpha\beta}}{p^{\alpha}}{{p^{\prime}}^{\beta}}{{p}_{\mu}}+{\Pi^{\mathrm{QCD6}}_{6}}(p,p^{\prime}){\varepsilon_{\lambda\nu\alpha\beta}}{p^{\alpha}}{{p^{\prime}}^{\beta}}{{p^{\prime}}_{\mu}}$
$\displaystyle\quad+{\Pi^{\mathrm{QCD6}}_{7}}(p,p^{\prime}){\varepsilon_{\lambda\mu\alpha\beta}}{p^{\alpha}}{{p^{\prime}}^{\beta}}{p_{\nu}}+{\Pi^{\mathrm{QCD6}}_{8}}(p,p^{\prime}){\varepsilon_{\lambda\mu\alpha\beta}}{p^{\alpha}}{{p^{\prime}}^{\beta}}{{p^{\prime}}_{\nu}}$
(13)

In these above equations, the expression $\Pi^{\mathrm{QCD}j}_{i}$ in the
right side of the equations without lorentz index is called the invariant
amplitude. Theoretically, each structure can be used to analyze the form
factors. No matter which structure is used, it is feasible as long as two
conditions are satisfied for the QCDSR. These two conditions will be discussed
in detail in Sec. III. In Eq. (II.2), each invariant amplitude can be divided
into two parts,

$\displaystyle\Pi_{i}(p,p^{\prime})=\Pi^{\mathrm{pert}}(p,p^{\prime})+\Pi^{\mathrm{{non-
pert}}}(p,p^{\prime})$ (14)

where the non-perturbative term includes the two gluons condensate
$\left\langle{g_{s}^{2}{G^{2}}}\right\rangle$ and three gluons condensate
$\left\langle{f{G^{3}}}\right\rangle$. The contributions of higher dimensions
than $\left\langle{f{G^{3}}}\right\rangle$ are neglected because these
contributions are small and further suppressed by $O(\alpha_{s}^{2})$.

The next procedure is to obtain the expressions of spectral densities. Still
taking the vector form factor $B_{c}\to J/\psi$ as an example, the
perturbative part of the correlation function
$\Pi_{\lambda\mu}^{\mathrm{QCD4}}$ in Eq. (II.2) can be written as,

$\displaystyle\Pi_{\lambda\mu}^{\mathrm{QCD4pert}}(p,p^{\prime})$
$\displaystyle=\frac{{3{i^{3}}}}{{{{(2\pi)}^{4}}}}\int{{d^{4}}k}\frac{1}{{[{{(k-p^{\prime})}^{2}}-m_{c}^{2}]({k^{2}}-m_{c}^{2})[{{(k+p-p^{\prime})}^{2}}-m_{b}^{2}]}}$
$\displaystyle\times
Tr\\{[(\not{k}-\not{p}^{\prime})+{m_{c}}]{\gamma_{\lambda}}(\not{k}+{m_{c}}){\gamma_{\mu}}[(\not{k}+\not{p}-\not{p}^{\prime})+{m_{b}}]{\gamma_{5}}{\rm{\\}}}$
(15)

The spectral density can be derived according to the Cutkoskys’s
rulesCutkosky:1960sp  which is illustrated in Fig. 1. In this process, each
denominator of quark propagator is substituted by a $\delta$ function,

Figure 1: The feynman diagram for the perturbative part. The dashed lines
denote the Cutkosky cuts. Figure 2: Feynman diagrams for the two-gluon (a-h)
and three-gluon (i-p) condensate terms.

$\displaystyle\frac{1}{{{k^{2}}-{m^{2}}+i\varepsilon}}\to-2\pi
i\delta({k^{2}}-{m^{2}})$ (16)

Thus, there will be an integration for three delta functions, which can be
derived as,

$\displaystyle\int{{d^{4}}k\delta[{{(k-p^{\prime})}^{2}}-m_{c}^{2}]\delta({k^{2}}-m_{b}^{2})\delta[{{(k+p-p^{\prime})}^{2}}-m_{c}^{2}]}$
$\displaystyle=-\frac{\pi}{{2\sqrt{\lambda(s,u,{q^{2}})}}}$ (17)

with

$\displaystyle\lambda(s,u,{q^{2}})={s^{2}}+{u^{2}}+{q^{4}}-2(su+u{q^{2}}+s{q^{2}})$
(18)

and $s=p^{2}$, $u=p^{\prime 2}$. During this integral process, the following
condition should be satisfied,

$\displaystyle-1\leq\frac{{2su-(m_{b}^{2}-m_{c}^{2}+u-{q^{2}})(s+u-{q^{2}})}}{{\sqrt{\left[{{{(m_{b}^{2}-m_{c}^{2}+u-{q^{2}})}^{2}}-4sm_{c}^{2}}\right]\lambda(s,u,{q^{2}})}}}\leq
1$ (19)

Finally, we obtain the spectral density of perturbative part,

$\displaystyle\rho^{\mathrm{{QCD4}}}_{pert}(s,u,{q^{2}})=\frac{3}{{4{\pi^{2}}{{[{{(s+u-{q^{2}})}^{2}}-4su]}^{3/2}}}}$
$\displaystyle\quad\times\\{{m_{c}}[-2m_{b}^{2}u+{q^{4}}-{q^{2}}(2s+u)+{s^{2}}-su]$
$\displaystyle\quad+{m_{b}}u(2m_{b}^{2}-{q^{2}}-s+u)-2{m_{b}}m_{c}^{2}u+2m_{c}^{3}u\\}$
(20)

As for the contributions of condensate terms, which are illustrated in Fig. 2,
the calculating process is similar to that of the perturbative part. In this
process, the following integration will be encountered,

$\displaystyle{I_{abc}}=\int{{d^{4}}k\frac{1}{{{{[{{(k-p^{\prime})}^{2}}-m_{1}^{2}]}^{a}}{{({k^{2}}-m_{2}^{2})}^{b}}{{[{{(k+p-p^{\prime})}^{2}}-m_{3}^{2}]}^{c}}}}}$
(21)

By reducing the power of denominators, we can also perform the integration
with the Cutkosky’s rules,

$\displaystyle{I_{abc}}=\frac{1}{{(a-1)!(b-1)!(c-1)!}}\frac{{{\partial^{a-1}}}}{{\partial
m_{1}^{a-1}}}\frac{{{\partial^{b-1}}}}{{\partial
m_{2}^{b-1}}}\frac{{{\partial^{c-1}}}}{{\partial m_{3}^{c-1}}}$
$\displaystyle\times\int{{d^{4}}k\frac{1}{{[{{(k-p^{\prime})}^{2}}-m_{1}^{2}]({k^{2}}-m_{2}^{2})[{{(k+p-p^{\prime})}^{2}}-m_{3}^{2}]}}}$
$\displaystyle\to\frac{{{{(-2\pi i)}^{3}}}}{{{{(2\pi
i)}^{2}}}}\frac{1}{{(a-1)!(b-1)!(c-1)!}}\frac{{{\partial^{a-1}}}}{{\partial
m_{1}^{a-1}}}\frac{{{\partial^{b-1}}}}{{\partial
m_{2}^{b-1}}}\frac{{{\partial^{c-1}}}}{{\partial m_{3}^{c-1}}}$
$\displaystyle\times\frac{{-\pi}}{{2\sqrt{\lambda(s,u,{q^{2}})}}}$ (22)

After all of the spectral densities are obtained, the correlation function can
be represented by spectral densities according to the dispersion relation,

$\displaystyle\Pi_{\lambda\mu}^{\mathrm{QCD4}}(p,p^{\prime})=\int\limits_{s_{min}}^{\infty}{\int\limits_{u_{min}}^{\infty}dsdu{\frac{{{\rho}^{\mathrm{QCD4}}(s,u,{q^{2}})}}{{(s-{p^{2}})(u-p{{}^{\prime
2}})}}{\varepsilon_{\lambda\mu\alpha\beta}}{p^{\alpha}}{{p^{\prime}}^{\beta}}}}$

where $s_{min}=(m_{b}+m_{c})^{2}$ and $u_{min}=4m_{c}^{2}$. The Borel
transformation can enhance the contribution of ground state and suppress those
of excited and continuum states. After performing the double Borel
transformations, we finally obtain the expression of correlation function at
QCD side,

$\displaystyle{\mathscr{B}\mathscr{B}}\Pi^{\mathrm{QCD4}}_{\lambda\mu}(M^{2},Q^{2})=\int\limits_{{s_{min}}}^{s_{0}}{\int\limits_{{u_{min}}}^{u_{0}}{dsdu}{\rho}^{\mathrm{QCD4}}(s,u,{Q^{2}})}$
$\displaystyle\times\exp(-\frac{s}{{{M^{2}}}}-\frac{u}{{k{M^{2}}}}){\varepsilon_{\lambda\mu\alpha\beta}}{p^{\alpha}}{{p^{\prime}}^{\beta}}$
(23)

where $s_{0}$ and $u_{0}$ are the threshold parameters. These two parameters
are introduced to subtract the contributions of excited and continuum states.

After matching the phenomenological and QCD sides by using quark-hadron
duality, the sum rule for the vector form factor of $B_{c}\to J/\psi$ is
obtained,

$\displaystyle{V}({Q^{2}})=\int\limits_{{s_{min}}}^{{s_{0}}}{\int\limits_{{u_{min}}}^{{u_{0}}}{dsdu}{{{\rho}^{\mathrm{QCD4}}}(s,u,{Q^{2}})\exp(-\frac{s}{{{M^{2}}}}-\frac{u}{{k{M^{2}}}})}}$
$\displaystyle\times\frac{{({m_{{B_{c}}}}+{m_{J/\psi}})({m_{c}}+{m_{b}})}}{{{\rm{2}}{f_{{B_{c}}}}m_{{B_{c}}}^{2}{f_{J/\psi}}{m_{J/\psi}}}}\exp(\frac{{m_{{B_{c}}}^{2}}}{{{M^{2}}}}{\rm{+}}\frac{{m_{J/\psi}^{2}}}{{k{M^{2}}}})$
(24)

As for the other form factors, they can also be obtained by similar process as
that of ${V}({Q^{2}})$. All of the sum rules for other form factors are
explicitly shown in Appendix A.

Figure 3: The ladder Feynman diagram for the Coulomb-like interaction.

The Coulomb-like $\alpha_{s}/v$ correction is essential for heavy quarkonium
$B_{c}$ meson and charmonium, where the relative velocity of quark movement is
small. In this physical sketch, the Coulomb-like correction can be illustrated
by the ladder diagram which is illustrated in Fig. 3. In the nonrelativistic
approximation, this correction will lead to a renormalization coefficient for
the spectral density of perturbative partKiselev:1993ea ; Kiselev:1999sc ,

$\displaystyle\rho^{pert}_{c}=\bm{C}\rho^{pert}$ (25)

$\bm{C}$ is the correct coefficient with assumption
$\alpha_{s}^{C}=\alpha_{s}(\mu)$,

$\displaystyle\bm{C}=$
$\displaystyle\sqrt{\frac{{4\pi\alpha_{s}^{C}}}{{3{v_{1}}}}{{\left[{1-\exp\left({-\frac{{4\pi\alpha_{s}^{C}}}{{3{v_{1}}}}}\right)}\right]}^{-1}}}$
$\displaystyle\times$
$\displaystyle\sqrt{\frac{{4\pi\alpha_{s}^{C}}}{{3{v_{2}}}}{{\left[{1-\exp\left({-\frac{{4\pi\alpha_{s}^{C}}}{{3{v_{2}}}}}\right)}\right]}^{-1}}}$
(26)

where $v_{1}$ and $v_{2}$ are the relative velocities of quarks in the $B_{c}$
meson and charmonium with

$\displaystyle{v_{1}}$
$\displaystyle=\sqrt{1-\frac{{4{m_{b}}{m_{c}}}}{{{s_{0}}-{{({m_{b}}-{m_{c}})}^{2}}}}}$
$\displaystyle{v_{2}}$ $\displaystyle=\sqrt{1-\frac{{4m_{c}^{2}}}{{{u_{0}}}}}$
(27)

##  III Numerical Results of form factors

The masses of quarks are energy-scale dependent, which can be expressed as the
renormalization equation,

$\displaystyle{m_{Q}}(\mu)$
$\displaystyle={m_{Q}}({m_{Q}}){\Big{[}\frac{{{\alpha_{s}}(\mu)}}{{{\alpha_{s}}({m_{Q}})}}\Big{]}^{\frac{{12}}{{33-2{n_{f}}}}}}$
(28) $\displaystyle{\alpha_{s}}(\mu)$
$\displaystyle=\frac{1}{{{b_{0}}t}}\Big{[}1-\frac{{{b_{1}}}}{{b_{0}^{2}}}\frac{{\log
t}}{t}+\frac{{b_{1}^{2}({{\log}^{2}}t-\log
t-1)+{b_{0}}{b_{2}}}}{{b_{0}^{4}{t^{2}}}}\Big{]}$ (29)

where $t=\log\frac{{{\mu^{2}}}}{{\Lambda_{QCD}^{2}}}$,
${b_{0}}=\frac{{33-2{n_{f}}}}{{12\pi}}$,
${b_{1}}=\frac{{153-19{n_{f}}}}{{24{\pi^{2}}}}$,
${b_{2}}=\frac{{2857-\frac{{5033}}{9}{n_{f}}+\frac{{325}}{{27}}n_{f}^{2}}}{{128{\pi^{3}}}}$.
$\Lambda_{QCD}=213$ MeV for the flavors $n_{f}=5$ in this work. The
$\overline{\mathrm{MS}}$ masses are taken from the Particle Date Group
(PDG)ParticleDataGroup:2022pth , where $m_{c}(m_{c})=1.275\pm 0.025$ GeV and
$m_{b}(m_{b})=4.18\pm 0.03$ GeV. In our previous studiesWang:2013iia ;
Wang:2013cha , energy scale $\mu=2$ GeV worked well in analyzing the
properties of $B_{c}$ meson, so this value is still employed in the present
work. The values of gluons condensate are taken as the standard values from
Refs. Narison:2010cg ; Narison:2011xe ; Narison:2011rn . Threshold parameters
$s_{0}$ and $u_{0}$ are used to eliminate the contributions of excited and
continuum states, which are commonly expressed as,

$\displaystyle{s_{0}}$ $\displaystyle={({m_{{B_{c}}}}+\Delta)^{2}}$
$\displaystyle{u_{0}}$ $\displaystyle={({m_{C}}+\Delta)^{2}}$ (30)

Here the values of $s_{0}$ and $u_{0}$ should be larger than the ground state
and lower than the first excited state. Commonly, the value of $\Delta$ with
$0.4\sim 0.6$ GeV can satisfy this condition. In present work, $\Delta=$0.5
GeV is used to obtain the central values of the final results, and 0.4, 0.6
GeV is for the lower and upper bounds, respectivelyBracco:2011pg . All the
values of parameters used in this work are listed in Tab. 1.

Table 1: Input parameters used in calculation of form factors, the values with no reference have been mentioned in text.  Parameters | Values | Parameters | Values
---|---|---|---
$m_{B_{c}}$ | 6.27 GeVParticleDataGroup:2022pth  | $f_{B_{c}}$ | 0.371 GeVZGW:2024Bc ; Narison:2020guz
$m_{\eta_{c}}$ | 2.98 GeVParticleDataGroup:2022pth  | $f_{\eta_{c}}$ | 0.387 GeVBecirevic:2013bsa
$m_{J/\psi}$ | 3.10 GeVParticleDataGroup:2022pth  | $f_{J/\psi}$ | 0.418 GeVBecirevic:2013bsa
$m_{c}(2\mathrm{GeV})$ | 1.16 GeV | $\left\langle{g_{s}^{2}{G^{2}}}\right\rangle$ |  $(0.88\pm 0.15)$ GeV4
$m_{b}(2\mathrm{GeV})$ | 4.76 GeV | $\left\langle{g^{3}_{s}{GGG}f}\right\rangle$ |  $(8.8\pm 5.5)$ GeV${}^{2}\left\langle{g_{s}^{2}{GG}}\right\rangle$

To obtain stable and reliable results, two conditions should be satisfied for
the QCDSR, which are the pole dominance and convergence of OPE. The pole
contribution is defined as,

$\displaystyle\mathrm{Pole}=\frac{{\Pi_{\mathrm{pole}}({M^{2}})}}{{\Pi_{\mathrm{total}}({M^{2}})}}$
(31)

with

$\displaystyle{\Pi_{\mathrm{pole}}}({M^{2}})$
$\displaystyle=\int\limits_{{s_{min}}}^{{s_{0}}}{\int\limits_{{u_{min}}}^{{u_{0}}}{{\rho^{QCD}}(s,u,{Q^{2}})\exp(-\frac{s}{{{M^{2}}}}-\frac{u}{{k{M^{2}}}})dsdu}}$
$\displaystyle{\Pi_{\mathrm{total}}}({M^{2}})$
$\displaystyle=\int\limits_{{s_{min}}}^{\infty}{\int\limits_{{u_{min}}}^{\infty}{{\rho^{QCD}}(s,u,{Q^{2}})\exp(-\frac{s}{{{M^{2}}}}-\frac{u}{{k{M^{2}}}})dsdu}}$
(32)

That is to say, we should find an appropriate region of the Borel parameters
$M^{2}$, where the results have good stability and at the same time the
condition of pole dominance($\geq 40\%$) is also satisfied. This region is
called the ’Borel platform’. Fixed $Q^{2}=1$ GeV2, the Borel parameters are
determined and Borel platform is chosen according to repeated trial and error.
The variations of pole contribution, perturbative part and condensate terms
with the Borel parameters are explicitly shown in Figs. 10 and 11 in Appendix
B. From these figures, we can see that the condition of pole dominance is
satisfied. At the same time, it is obvious that the contribution of
perturbative term is dominant while those of gluon condensate terms are very
small, which indicates the convergence of OPE. The Borel platform and the pole
contribution are listed in Tab. 2.

Table 2: The Borel platform and pole contribution for different form factors Modes | Form factors | Borel platforms | Pole contributions($\%$)
---|---|---|---
$B_{c}\to\eta_{c}$ | $f_{s}$ | 14 $\sim$ 18 | 41.65
$f_{+}$ | 18 $\sim$ 22 | 40.91
$f_{0}$ | 18 $\sim$ 22 | 41.22
$f_{T}$ | 19 $\sim$ 23 | 40.24
$B_{c}\to J/\psi$ | $V$ | 20 $\sim$ 24 | 40.14
$A_{0}$ | 16 $\sim$ 20 | 41.79
$A_{1}$ | 15 $\sim$ 19 | 41.89
$A_{2}$ | 13 $\sim$ 17 | 41.26
$T_{0}$ | 23 $\sim$ 27 | 40.98
$T_{1}$ | 19 $\sim$ 23 | 40.21
$T_{2}$ | 12 $\sim$ 16 | 40.74
Figure 4: Fitting results of form factors for $B_{c}\to\eta_{c}$. The legends
with diamond denote form factors with Coulomb-like correction. Figure 5:
Fitting results of form factors for $B_{c}\to J/\psi$. The legends with
diamond denote form factors with Coulomb-like correction.

By changing the value of $Q^{2}$, we obtain the form factors in space-like
region ($1\leq Q^{2}\leq 9$), where the results are shown in Figs. 4 and 5.
The values of form factors at $Q^{2}=0$ can be obtained by fitting the results
in space-like region ($Q^{2}>0$) with appropriate analytical functions and
then extrapolating these numerical results into the time-like
region($Q^{2}<0$). The polar function, exponential function and the $z$-series
parameterization approach are commonly used to fit the form factorWang:2007ys
; Wang:2008xt ; Wang:2012vna ; Biswas:2023bqz . The $z$-series
parameterization approach, which satisfies the unitarity, analyticity and
perturbative QCD scaling, is adopted in this work. There are different forms
of this approach used in different studiesBoyd:1994tt ; Bourrely:2008za ;
Wang:2015vgv . Among them the method proposed in Ref. Bourrely:2008za
considered the asymptotic behaviors of the vector form factors near threshold.
In this way form factors are expanded as,

$\displaystyle F({Q^{2}})$
$\displaystyle=\frac{1}{{1+{Q^{2}}/m_{R}^{2}}}\sum\limits_{k=0}^{N-1}{{b_{k}}[z{{({Q^{2}},{t_{0}})}^{k}}-{{(-1)}^{k-N}}\frac{k}{N}z{{({Q^{2}},{t_{0}})}^{N}}]}$
(33)

where $t_{+}\equiv(m_{B_{c}}+m_{C})^{2}$. $m_{R}$ denotes the masses of the
low-lying $B_{c}$ resonances. In present work, $m_{R}$ is taken in accordance
with Ref. Leljak:2019eyw . $b_{k}$ are the real coefficients and
$z(Q^{2},t_{0})$ is the function,

$\displaystyle z({Q^{2}},{t_{0}})$
$\displaystyle=\frac{{\sqrt{{t_{+}}+{Q^{2}}}-\sqrt{{t_{+}}-{t_{0}}}}}{{\sqrt{{t_{+}}+{Q^{2}}}+\sqrt{{t_{+}}-{t_{0}}}}}$
(34)

which maps the complex $Q^{2}$-plane onto the unit disk $\lvert
z(Q^{2},t_{0})\rvert\leq 1$ in the $z$ complex plane. The free parameter
$t_{0}$ corresponds to the value of $Q^{2}$ mapping onto the origin in the
$z$-plane. Aimed to maximally reduce the interval of $z$ obtained after
mapping of the semileptonic domain $Q^{2}\in[-(m_{B_{c}}-m_{C})^{2},0]$, the
auxiliary parameter $t_{0}$ are taken asCui:2022zwm ,

$\displaystyle{t_{0}}$ $\displaystyle={t_{+}}-\sqrt{{t_{+}}({t_{+}}-{t_{-}})}$
$\displaystyle t_{-}$ $\displaystyle=(m_{B_{c}}-m_{C})^{2}$ (35)

The $z$-series are truncated at $N=3$ in practical calculation. The vector and
tensor form factors are expanded as Eq. (33), while the scalar form factor can
be well parameterized with a more simple functionBourrely:2008za ; Cui:2022zwm
,

$\displaystyle
f_{S}({Q^{2}})=\frac{1}{{1+{Q^{2}}/m_{R}^{2}}}\sum\limits_{k=0}^{N-1}{{b_{k}}[z{{({Q^{2}},{t_{0}})}^{k}}]}$
(36)

Table 3: Fitting parameters of the $z$-series parameterized approach. $b_{i}^{*}$ denote the fitting parameters with Coulomb-like correction. Modes | Form factors | $b_{0}$ | $b_{1}$ | $b_{2}$ | $b_{0}^{*}$ | $b_{1}^{*}$ | $b_{2}^{*}$
---|---|---|---|---|---|---|---
$B_{c}\to\eta_{c}$ | $f_{s}$ | 2.1 | -17 | 37 | 6.5 | -67 | 22
$f_{+}$ | 0.25 | -3.3 | 15 | 0.78 | -11 | 55
$f_{0}$ | 0.22 | -1.5 | 1.4 | 0.69 | -5.6 | 7.7
$f_{T}$ | 0.38 | -4.9 | 22 | 1.2 | -17 | 81
$B_{c}\to J/\psi$ | $V$ | 0.52 | -6.6 | 30 | 1.6 | -22 | 100
$A_{0}$ | 0.34 | -4.9 | 24 | 1.1 | -17 | 90
$A_{1}$ | 0.30 | -2.2 | 3.8 | 0.94 | -8.5 | 22
$A_{2}$ | 0.31 | -3.0 | 7.9 | 0.97 | -11 | 40
$T_{0}$ | 1.1 | -20 | 140 | 3.5 | -61 | 380
$T_{1}$ | 0.37 | -2.8 | 5.0 | 1.1 | -9.3 | 13
$T_{2}$ | 0.12 | -0.83 | -0.95 | 0.38 | -4.7 | 21
Table 4: Form factors at $Q^{2}=0$, ’This work∗’ denotes the results with Coulomb-like correction. Modes | Form factors | This work | This work∗ | QCDSRColangelo:1992cx  | Lattice QCDHarrison:2020gvo  | LCSRLeljak:2019eyw  | LFQMZhang:2023ypl  | pQCDWang:2012lrc  | RQMNobes:2000pm
---|---|---|---|---|---|---|---|---|---
$B_{c}\to\eta_{c}$ | $f_{S}$ | $1.78^{+0.25}_{-0.26}$ | $5.44^{+0.65}_{-0.70}$ | - |  | - | - | - | -
$f_{+}$ | $0.20^{+0.03}_{-0.03}$ | $0.60^{+0.07}_{-0.09}$ | $0.20^{+0.02}_{-0.02}$ | - | $0.62^{+0.05}_{-0.05}$ | 0.60 | $0.48^{+0.07}_{-0.07}$ | 0.54
$f_{0}$ | $0.20^{+0.03}_{-0.03}$ | $0.60^{+0.07}_{-0.09}$ | - | - | - | 0.60 | $0.48^{+0.07}_{-0.07}$ | -
$f_{T}$ | $0.30^{+0.04}_{-0.05}$ | $0.92^{+0.11}_{-0.12}$ | - | - | $0.93^{+0.07}_{-0.07}$ | - | - | -
$B_{c}\to J/\psi$ | $V$ | $0.43^{+0.06}_{-0.06}$ | $1.29^{+0.15}_{-0.16}$ | $0.19^{+0.05}_{-0.05}$ | 0.725 | $0.73^{+0.06}_{-0.06}$ | 0.76 | $0.42^{+0.02}_{-0.02}$ | 0.73
$A_{0}$ | $0.27^{+0.04}_{-0.04}$ | $0.82^{+0.1}_{-0.1}$ | - | 0.477 | $0.54^{+0.04}_{-0.04}$ | 0.55 | $0.59^{+0.03}_{-0.03}$ | 0.53
$A_{1}$ | $0.27^{+0.04}_{-0.04}$ | $0.81^{+0.09}_{-0.1}$ | $0.27^{+0.03}_{-0.03}$ | 0.457 | $0.55^{+0.04}_{-0.04}$ | 0.53 | $0.46^{+0.03}_{-0.03}$ | 0.52
$A_{2}$ | $0.27^{+0.03}_{-0.04}$ | $0.81^{+0.09}_{-0.1}$ | $0.28^{+0.09}_{-0.09}$ | 0.417 | $0.35^{+0.03}_{-0.03}$ | 0.49 | $0.64^{+0.03}_{-0.03}$ | 0.51
$T_{0}$ | $0.86^{+0.12}_{-0.12}$ | $2.63^{+0.3}_{-0.32}$ | - | - | - | - | - | -
$T_{1}$ | $0.33^{+0.05}_{-0.05}$ | $0.98^{+0.11}_{-0.12}$ | - | - | - | - | - | -
$T_{2}$ | $0.10^{+0.01}_{-0.01}$ | $0.31^{+0.04}_{-0.04}$ | - | - | - | - | - | -

All the fitting parameters $b_{k}$ are listed in Tab. 3. The fitting diagrams
about the form factors of $B_{c}\to\eta_{c}$ and $B_{c}\to J/\psi$ are
explicitly shown in Figs. 4 and 5. From these figures, we can see that the
numerical results are all perfectly fitted by the analytical functions, which
indicates the reliability of the final results. In addition, these figures
also show that the values of form factors decrease evidently with increase of
four-momentum squared $Q^{2}$. This $Q^{2}$ dependence is consistent well with
the results of other collaborationsZhang:2023ypl ; Colangelo:1992cx ;
Leljak:2019eyw ; Harrison:2020gvo . The physical transition processes of
$B_{c}\rightarrow J/\psi$ and $B_{c}\rightarrow\eta_{c}$ are relevant to the
time-like form factors which can be obtained by setting $Q^{2}\leq 0$ in
analytical functions. These time-like form factors will be used to analyze the
nonleptonic and semileptonic decay processes of $B_{c}$ meson in Sec. IV. As a
contrast, the values of the form factors at $Q^{2}=0$ are all displayed in
Tab. 4. It is indicated that the values of bare form factors calculated in
this work are consistent well with the results of Ref.Colangelo:1992cx , but
some results are relatively small than those of other collaborations. After
considering the Coulomb-like $\alpha_{s}/v$ correction, the numerical results
are about three times of those without correction, which indicates the
Coulomb-like correction has a significant impact on the results.

##  IV Weak decays of $B_{c}$ to charmonium

###  IV.1 Nonleptonic decay processes

With the results of form factors obtained in the above section, the decay
widths and branching ratios of several weak decay processes about
$B_{c}\to\eta_{c}$ and $B_{c}\to J/\psi$ are calculated with the factorization
approachMohammadi:2014rpa ; Mohammadi:2023itb . The effective Hamiltonian of
several nonleptonic decay processes of $B_{c}$ to charmonium are expressed as
follows,

$\displaystyle{H^{B_{c}^{-}\to
C\pi(\rho)^{-}}_{eff}}=\frac{{{G_{F}}}}{{\sqrt{2}}}{V_{cb}}V_{ud}^{*}{a_{1}}\bar{c}{\gamma_{\mu}}(1-{\gamma_{5}})b\bar{d}{\gamma^{\mu}}(1-{\gamma_{5}})u$
$\displaystyle{H^{B_{c}^{-}\to
CK^{(*)-}}_{eff}}=\frac{{{G_{F}}}}{{\sqrt{2}}}{V_{cb}}V_{us}^{*}{a_{1}}\bar{c}{\gamma_{\mu}}(1-{\gamma_{5}})b\bar{u}{\gamma^{\mu}}(1-{\gamma_{5}})s$
(37)

where $V_{cb}$, $V_{ud}$ and $V_{us}$ are CKM matrix elements, $C=\eta_{c}$,
$J/\psi$, and $a_{1}$ is the Wilson coefficients. All of the parameters used
in this section are listed in Tab. 5

Table 5: Input parameters used to analyze the decay processes of $B_{c}$. Parameters | Values | Parameters | Values
---|---|---|---
$m_{\pi}$ | 0.14 GeVParticleDataGroup:2022pth  | $f_{\pi}$ | 0.131 GeVParticleDataGroup:2020ssz
$m_{K}$ | 0.494 GeVParticleDataGroup:2022pth  | $f_{K}$ | 0.160 GeVParticleDataGroup:2020ssz
$m_{\rho}$ | 0.775 GeVParticleDataGroup:2022pth  | $f_{\rho}$ | 0.216 GeVParticleDataGroup:2010dbb
$m_{K^{*}}$ | 0.892 GeVParticleDataGroup:2022pth  | $f_{K^{*}}$ | 0.217 GeVParticleDataGroup:2020ssz
$V_{cb}$ | 0.041ParticleDataGroup:2022pth  | $V_{ud}$ | 0.974ParticleDataGroup:2022pth
$V_{us}$ | 0.224ParticleDataGroup:2022pth  | $a_{1}$ | 1.07Buchalla:1995vs

The decay amplitude can be explicitly expressed as,

$\displaystyle\Gamma=\frac{1}{{2J+1}}\sum\limits_{J}{\frac{1}{{2{m_{{B_{c}}}}}}\int{d\Phi(p\to
p^{\prime},q)}|T{|^{2}}}$ (38)

where $J$ is the angle momentum of $B_{c}$ meson. $d\Phi(p\to p^{\prime},q)$
is the two-body phase factor,

$\displaystyle d\Phi(p\to
q,p^{\prime})={(2\pi)^{4}}{\delta^{4}}(p-q-p^{\prime})\frac{{{d^{3}}\vec{q}}}{{{{(2\pi)}^{3}}2{q_{0}}}}\frac{{{d^{3}}\vec{p}^{\prime}}}{{{{(2\pi)}^{3}}2{p^{\prime}_{0}}}}$
(39)

where $p$, $p^{\prime}$ and $q$ are the four-momentum of $B_{c}$ meson,
charmoniums($\eta_{c}$, $J/\psi$) and the other final state mesons($\pi^{-}$,
$K^{-}$, $\rho$, $K^{*}$). After performing integration of the two-body phase
factor, we can obtain the following expression,

$\displaystyle\int{d\Phi(p\to p^{\prime},q)=}\frac{1}{{8\pi
m_{{B_{c}}}^{2}}}\sqrt{\lambda(m_{{B_{c}}}^{2},m_{C}^{2},{m^{2}})}$ (40)

where $m_{C}$ and $m$ are the masses of charmonium $m_{\eta_{c}}$,
$m_{J/\psi}$, and the other final state mesons $m_{\pi^{-}}$, $m_{K^{-}}$
$m_{\rho}$, $m_{K^{*}}$. $\lambda$ function can be expressed as,

$\displaystyle\lambda(a,b,c)={a^{2}}+{b^{2}}+{c^{2}}-2(ab+bc+ac)$ (41)

Figure 6: Feynman diagram for the decay processes
$B_{c}^{-}\to\eta_{c}\pi^{-},\eta_{c}K^{-},\eta_{c}\rho^{-},\eta_{c}K^{*-}$,
and $B_{c}^{-}\to J/\psi\pi^{-},J/\psi K^{-},J/\psi\rho^{-},J/\psi K^{*-}$

In Eq. (38), $T$ is the transition matrix element. Under the factorization
approach, this process can be illustrated as Fig. 6. Transition matrix element
$T$ can be expressed as a product of two matrix elements,

$\displaystyle\left\langle{{C}(p^{\prime})\pi^{-}(q)}\right|{H_{eff}}\left|{{B_{c}^{-}}(p)}\right\rangle$
$\displaystyle=\frac{{{G_{F}}}}{{\sqrt{2}}}{V_{cb}}V_{ud}^{*}{a_{1}}\left\langle{{C}(p^{\prime})}\right|\bar{c}{\gamma_{\mu}}(1-{\gamma_{5}})b\left|{{B^{-}_{c}}(p)}\right\rangle$
$\displaystyle\quad\times\left\langle{\pi^{-}(q)}\right|\bar{d}{\gamma^{\mu}}(1-{\gamma_{5}})u\left|0\right\rangle$
$\displaystyle\left\langle{{C}(p^{\prime})K^{-}(q)}\right|{H_{eff}}\left|{{B_{c}^{-}}(p)}\right\rangle$
$\displaystyle=\frac{{{G_{F}}}}{{\sqrt{2}}}{V_{cb}}V_{us}^{*}{a_{1}}\left\langle{{C}(p^{\prime})}\right|\bar{c}{\gamma_{\mu}}(1-{\gamma_{5}})b\left|{{B^{-}_{c}}(p)}\right\rangle$
$\displaystyle\quad\times\left\langle{K^{-}(q)}\right|\bar{u}{\gamma^{\mu}}(1-{\gamma_{5}})s\left|0\right\rangle$
$\displaystyle\left\langle{{C}(p^{\prime})\rho^{-}(q)}\right|{H_{eff}}\left|{{B_{c}^{-}}(p)}\right\rangle$
$\displaystyle=\frac{{{G_{F}}}}{{\sqrt{2}}}{V_{cb}}V_{ud}^{*}{a_{1}}\left\langle{{C}(p^{\prime})}\right|\bar{c}{\gamma_{\mu}}(1-{\gamma_{5}})b\left|{{B^{-}_{c}}(p)}\right\rangle$
$\displaystyle\quad\times\left\langle{\rho^{-}(q)}\right|\bar{c}{\gamma^{\mu}}(1-{\gamma_{5}})s\left|0\right\rangle$
$\displaystyle\left\langle{{C}(p^{\prime})K^{*-}(q)}\right|{H_{eff}}\left|{{B_{c}^{-}}(p)}\right\rangle$
$\displaystyle=\frac{{{G_{F}}}}{{\sqrt{2}}}{V_{cb}}V_{us}^{*}{a_{1}}\left\langle{{C}(p^{\prime})}\right|\bar{c}{\gamma_{\mu}}(1-{\gamma_{5}})b\left|{{B^{-}_{c}}(p)}\right\rangle$
$\displaystyle\quad\times\left\langle{K^{*-}(q)}\right|\bar{c}{\gamma^{\mu}}(1-{\gamma_{5}})s\left|0\right\rangle$
(42)

where $C=\eta_{c}$, $J/\psi$. In these above equations (Eq. (IV.1)), each
matrix element of $B_{c}\rightarrow\eta_{c}$ or $B_{c}\rightarrow J/\psi$ can
be factorized as the form factor which has been listed in Eq. (II.1). The
other meson vacuum matrix elements can be factorized as decay constant. As for
pseudoscalar meson $\pi^{-}$ and $K^{-}$, and axisvector meson $\rho^{-}$ and
$K^{*-}$, the decay constants are respectively defined as,

$\displaystyle\left\langle{P(q)}\right|\bar{q}{\gamma_{\mu}}(1-{\gamma_{5}})q\left|0\right\rangle$
$\displaystyle=-i{f_{P}}{q_{\mu}}$
$\displaystyle\left\langle{A(q)}\right|\bar{q}{\gamma_{\mu}}(1-{\gamma_{5}})q\left|0\right\rangle$
$\displaystyle=-i{f_{A}}{m_{A}}{\xi_{\mu}}$ (43)

where $P=\pi^{-}$, $K^{-}$, and $A=\rho^{-}$, $K^{*-}$. Substituting these
matrix elements in Eq. (38), the expression of decay width can be derived. All
the expressions of decay widths are explicitly shown in Appendix C. Finally,
the decay widths and branching ratios of the nonleptonic decays are calculated
and the corresponding numerical results are explicitly shown in Tab. 6.

Table 6: Decay widths (in $10^{-7}$eV) and Branching ratios (in ‰) of $B_{c}$ decaying to charmonium. Branching ratios are calculated at $\tau_{B_{c}}$ = 0.51 psLHCb:2014ilr . The superscript star denotes the results obtained by considering Coulomb-like correction. Decay channels | Decay widths | Branching ratios
---|---|---
| This work | This work∗ | This work | This work∗ | QCDSRKiselev:2002vz  | LFQMZhang:2023ypl  | pQCDWang:2012lrc  | RQMEbert:2003cn  | LCSRLeljak:2019eyw
$B_{c}^{-}\to\eta_{c}\pi^{-}$ | $1.8^{+0.6}_{-0.6}$ | $17^{+5}_{-5}$ | $0.15^{+0.05}_{-0.05}$ | $1.4^{+0.4}_{-0.4}$ | 2 | $2.36^{+0.1}_{-0.1}$ | - | 0.85 | -
$B_{c}^{-}\to\eta_{c}K^{-}$ | $0.15^{+0.05}_{-0.05}$ | $1.4^{+0.4}_{-0.4}$ | $0.012^{+0.003}_{-0.003}$ | $0.11^{+0.03}_{-0.03}$ | 0.13 | $0.19^{+0}_{-0.01}$ | - | 0.07 | -
$B_{c}^{-}\to\eta_{c}\rho^{-}$ | $4.9^{+1.5}_{-1.5}$ | $46^{+12}_{-13}$ | $0.41^{+0.12}_{-0.12}$ | $3.8^{+0.1}_{-0.1}$ | 4.2 | $3.8^{+1.0}_{-1.0}$ | - | 2.1 | -
$B_{c}^{-}\to\eta_{c}K^{*-}$ | $0.26^{+0.08}_{-0.08}$ | $2.5^{+0.6}_{-0.7}$ | $0.022^{+0.007}_{-0.007}$ | $0.21^{+0.05}_{-0.06}$ | 0.20 | $0.34^{+0.02}_{-0.02}$ | - | 0.11 | -
$B_{c}^{-}\to\eta_{c}e\bar{\nu}_{e}$ | $11^{+3}_{-3}$ | $105^{+23}_{-26}$ | $0.9^{+0.2}_{-0.3}$ | $8.7^{+1.9}_{-2.2}$ | 7.5 | - | 4.41 | 4.2 | 8.2
$B_{c}^{-}\to\eta_{c}\mu\bar{\nu}_{\mu}$ | $11^{+3}_{-3}$ | $105^{+23}_{-26}$ | $0.9^{+0.2}_{-0.3}$ | $8.7^{+1.9}_{-2.2}$ | - | - | 4.41 | - | 8.2
$B_{c}^{-}\to\eta_{c}\tau\bar{\nu}_{\tau}$ | $3.3^{+0.8}_{-0.9}$ | $33^{+7}_{-8}$ | $0.28^{+0.07}_{-0.07}$ | $2.8^{+0.5}_{-0.6}$ | 2.3 | - | 1.37 | - | 2.6
$B_{c}^{-}\to J/\psi\pi^{-}$ | $3.3^{+1}_{-0.9}$ | $30^{+8}_{-7}$ | $0.27^{+0.08}_{-0.07}$ | $2.5^{+0.6}_{-0.6}$ | 1.3 | $1.97^{+0.28}_{-0.29}$ | - | 0.61 | -
$B_{c}^{-}\to J/\psi K^{-}$ | $0.26^{+0.07}_{-0.07}$ | $2.3^{+0.6}_{-0.5}$ | $0.022^{+0.006}_{-0.006}$ | $0.19^{+0.05}_{-0.04}$ | 0.11 | $0.16^{+0.02}_{-0.02}$ | - | 0.05 | -
$B_{c}^{-}\to J/\psi\rho^{-}$ | $11^{+2.8}_{-3.0}$ | $96^{+22}_{-23}$ | $0.88^{+0.23}_{-0.25}$ | $8.0^{+1.9}_{-2.0}$ | 4.0 | $5.34^{+0.16}_{-0.16}$ | - | 1.6 | -
$B_{c}^{-}\to J/\psi K^{*-}$ | $0.60^{+0.16}_{-0.17}$ | $5.4^{+1.2}_{-1.3}$ | $0.049^{+0.013}_{-0.014}$ | $0.45^{+0.10}_{-0.11}$ | 0.22 | $0.31^{+0.14}_{-0.14}$ | - | 0.10 | -
$B_{c}^{-}\to J/\psi e\bar{\nu}_{e}$ | $61^{+15}_{-14}$ | $554^{+150}_{-91}$ | $5.1^{+1.2}_{-1.2}$ | $46^{+12}_{-08}$ | 19 | - | 10.03 | 12.3 | 22.4
$B_{c}^{-}\to J/\psi\mu\bar{\nu}_{\mu}$ | $61^{+15}_{-14}$ | $552^{+149}_{-90}$ | $5.1^{+1.2}_{-1.2}$ | $46^{+12}_{-8}$ | - | - | 10.03 | - | 22.4
$B_{c}^{-}\to J/\psi\tau\bar{\nu}_{\tau}$ | $15^{+3.5}_{-3.4}$ | $144^{+35}_{-23}$ | $1.3^{+0.3}_{-0.3}$ | $12^{+3}_{-2}$ | 4.8 | - | 2.92 | - | 5.3

###  IV.2 Semileptonic decay processes

As for the semileptonic decay of $B_{c}$ meson, the effective Hamilton of the
decay modes $B_{c}^{-}\to\eta_{c}\mathcal{l}^{-}\bar{\nu}_{\mathcal{l}}$ and
$B_{c}^{-}\to J/\psi\mathcal{l}^{-}\bar{\nu}_{\mathcal{l}}$ can be expressed
as,

$\displaystyle
H_{eff}=\frac{{{G_{F}}}}{{\sqrt{2}}}{V_{cb}}\bar{c}{\gamma_{\mu}}(1-{\gamma_{5}})b{\bar{\nu}_{l}}{\gamma_{\mu}}(1-{\gamma_{5}})l$
(44)

The decay amplitude is the same as Eq. (38). The three-body phase factor can
be decomposed as a product of double two-body phase factors,

$\displaystyle d\Phi(p\to p^{\prime},l,\nu)$
$\displaystyle={(2\pi)^{4}}{\delta^{4}}(p-p^{\prime}-l-\nu)$
$\displaystyle\quad\times\frac{{{d^{3}}\vec{p}^{\prime}}}{{{{(2\pi)}^{3}}2{{p^{\prime}}_{0}}}}\frac{{{d^{3}}\vec{l}}}{{{{(2\pi)}^{3}}2{l_{0}}}}\frac{{{d^{3}}\vec{\nu}}}{{{{(2\pi)}^{3}}2{\nu_{0}}}}$
$\displaystyle=\frac{{d{q^{2}}}}{{2\pi}}d\Phi(p\to p^{\prime},q)d\Phi(q\to
l,\nu)$ (45)

Here $q$ is the four-momentum of $W^{-}$ boson, which is off shell. This decay
process can be illustrated as the feynman diagram in Fig.7, where the
transition matrix element is decomposed into a product of
$\left\langle{C}\right|\bar{c}{\gamma_{\mu}}(1-{\gamma_{5}})b\left|{{B_{c}}}\right\rangle$
and
$\left\langle{\ell\bar{\nu}}\right|{{\bar{\nu}}_{l}}{\gamma_{\mu}}(1-{\gamma_{5}})l\left|0\right\rangle$
with $C=\eta_{c}$, $J/\psi$,

$\displaystyle\left\langle{C(p^{\prime}),{l}(l),\bar{\nu_{l}}(\nu)}\right|{H_{eff}}\left|{{B^{-}_{c}(p)}}\right\rangle$
$\displaystyle=\frac{{{G_{F}}}}{{\sqrt{2}}}{V_{cb}}\left\langle{\ell\bar{\nu}}\right|{{\bar{\nu}}_{l}}{\gamma_{\mu}}(1-{\gamma_{5}})l\left|0\right\rangle$
$\displaystyle\quad\times\left\langle{C(p^{\prime})}\right|\bar{c}{\gamma_{\mu}}(1-{\gamma_{5}})b\left|{{B^{-}_{c}(p)}}\right\rangle$
(46)

Figure 7: Feynman diagram for the decay processes $B_{c}^{-}\to
J/\psi\mathcal{l}\bar{\nu}_{\mathcal{l}}$ and
$B_{c}^{-}\to\eta_{c}\mathcal{l}\bar{\nu}_{\mathcal{l}}$.

It is similar as the two-body nonleptonic decay process, we can also derive
the width of this semileptonic decay. The expressions of differential decay
widths about these decay processes are listed in Appendix D. The differential
decay widths with variations of $q^{2}$ are plotted in Figs. 8 and 9. The
predictions for partial decay widths and branching ratios are also shown in
Tab. 6.

Figure 8: The differential decay widths $d\Gamma/dq^{2}$ with variations of
$q^{2}$ for the processes $B_{c}^{-}\to C\mathcal{l}\bar{\nu}_{\mathcal{l}}$.
These results are obtained with the bare form factors which are calculated
without considering the Coulomb-like corrected. Figure 9: It is the same as
Fig. 8, but these results are obtained with the form factors which are
calculated by considering the Coulomb-like corrected.

In most of the relevant references, only the branching ratios about these weak
decay processes are shown explicitly. In this work, the differential decay
widths, partial widths and the branching ratios are all listed explicitly.
Comparing with other collaborations, the values of branching ratios without
Coulomb-like $\alpha_{s}/v$ correction are lower than those of others. After
considering the Coulomb-like $\alpha_{s}/v$ correction, the results are
compatible with others, which indicates the importance of the correction.
Using the values of form factors with or without Coulomb-like $\alpha_{s}/v$
correction, the ratio $\frac{{\Gamma(B_{c}^{-}\to
J/\psi{\tau^{-}}\bar{\nu}_{\tau})}}{{\Gamma(B_{c}^{-}\to
J/\psi{\mu^{-}}\bar{\nu}_{\mu})}}$ is the same with each other,

$\displaystyle R(J/\psi)=\frac{{\Gamma(B_{c}^{-}\to
J/\psi{\tau^{-}}\bar{\nu}_{\tau})}}{{\Gamma(B_{c}^{-}\to
J/\psi{\mu^{-}}\bar{\nu}_{\mu})}}=0.25$ (47)

This result is consistent well with that of other collaborations, where their
results are in the range of $0.25\sim 0.28$. However, the experimental data
about this ratio is $0.71\pm 0.17\pm 0.18$, which is higher than the
theoretical predictions. There are two possible interpretations about this
deviation. First, the theoretical results have a certain model dependency.
This deviation may be caused by the uncertainties of the experimental data and
theoretical predictions. Besides, it is indicated by other researches that the
similar problem also exist to the branching fraction $\frac{{\Gamma(B\to
D^{(*)}{\tau^{-}}\bar{\nu}_{\tau})}}{{\Gamma(B\to
D^{(*)}{\mu^{-}}\bar{\nu}_{\mu})}}$. Many theorists speculated that this kind
of discrepancy may be the signs of new physics beyond the SM.

##  V Conclusions

In this work, the form factors of $B_{c}\rightarrow\eta_{c}$ and
$B_{c}\rightarrow J/\psi$ are calculated in the framework of three-point
QCDSR. The Coulomb-like correction is considered, which indicates about triple
the form factor at $Q^{2}=0$. All the form factors are obtained with the
condition of pole dominance and convergence of OPE being well satisfied. We
find that the values of form factors without Coulomb-like correction are in
agreement well with the results of Ref.Colangelo:1992cx , but lower than those
of other theoretical methods. After considering the Coulomb-like correction,
the results are compatible with those of other theoretical methods. Besides
the form factors at $Q^{2}=0$, we also obtain their values of $Q^{2}>0$ by
using the fitting function which is obtained with $z$-series parametrization
approach. Using these values of form factors and basing on factorization
approach, we also systematically analyze the decay widths and branching ratios
of different decay channels. Our predictions are comparative with the results
obtained by other theoretical approaches. However, the ratio $R(J/\psi)$ is
somewhat lower than the experimental data, which indicates existence of the
discrepancy between the prediction based on SM and experimental data. In
summary, the results of form factors, decay widths and branching ratios
obtained in this work provide useful information for further studying the
heavy-quark dynamics and finding new physics beyond Standard Model.

## Acknowledgements

This project is supported by National Natural Science Foundation, Grant Number
12175068 and Natural Science Foundation of HeBei Province, Grant Number
A2018502124.

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##  Appendix A The expressions of form factors

$\displaystyle{f_{S}}({Q^{2}})=\frac{{{m_{b}}+{m_{c}}}}{{{f_{{B_{c}}}}m_{{B_{c}}}^{2}}}\frac{{2{m_{c}}}}{{{f_{{\eta_{c}}}}m_{{\eta_{c}}}^{2}}}\int\limits_{{s_{\min}}}^{{s_{0}}}{\int\limits_{{u_{\min}}}^{{u_{0}}}{dsdu{\rho^{\mathrm{QCD1}}}(s,u,{Q^{2}})}}\exp(-\frac{s}{{{M^{2}}}}-\frac{u}{{k{M^{2}}}})\exp(\frac{{m_{{B_{c}}}^{2}}}{{{M^{2}}}}+\frac{{m_{{\eta_{c}}}^{2}}}{{k{M^{2}}}})$
(48) $\displaystyle\left({\begin{array}[]{*{20}{c}}{{f_{+}}({Q^{2}})}\\\
{{f_{0}}({Q^{2}})}\end{array}}\right)$
$\displaystyle=\int\limits_{{s_{\min}}}^{{s_{0}}}{\int\limits_{{u_{\min}}}^{{u_{0}}}{dsdu}}\left({\begin{array}[]{*{20}{c}}{\frac{{{m_{c}}({m_{c}}+{m_{b}})}}{{m_{{B_{c}}}^{2}m_{{\eta_{c}}}^{2}{f_{{B_{c}}}}{f_{{\eta_{c}}}}}}}&{\frac{{{m_{c}}({m_{c}}+{m_{b}})}}{{m_{{B_{c}}}^{2}m_{{\eta_{c}}}^{2}{f_{{B_{c}}}}{f_{{\eta_{c}}}}}}}\\\
{\frac{{{Q^{2}}}}{{m_{{\eta_{c}}}^{2}-m_{{B_{c}}}^{2}}}+\frac{1}{2}}&{\frac{{{Q^{2}}}}{{m_{{B_{c}}}^{2}-m_{{\eta_{c}}}^{2}}}+\frac{1}{2}}\end{array}}\right)\left({\begin{array}[]{*{20}{c}}{\rho_{1}^{\mathrm{QCD2}}(s,u,{Q^{2}})}\\\
{\rho_{2}^{\mathrm{QCD2}}(s,u,{Q^{2}})}\end{array}}\right)$ (55)
$\displaystyle\quad\times\exp(-\frac{s}{{{M^{2}}}}-\frac{u}{{k{M^{2}}}})\exp(\frac{{m_{{B_{c}}}^{2}}}{{{M^{2}}}}+\frac{{m_{{\eta_{c}}}^{2}}}{{k{M^{2}}}})$
(56)
$\displaystyle{f_{T}}({Q^{2}})=-\frac{{{m_{c}}({m_{b}}+{m_{c}})({m_{{B_{c}}}}+{m_{{\eta_{c}}}})}}{{{f_{{\eta_{c}}}}m_{{\eta_{c}}}^{2}{f_{{B_{c}}}}m_{{B_{c}}}^{2}}}\int\limits_{{s_{\min}}}^{{s_{0}}}{\int\limits_{{u_{\min}}}^{{u_{0}}}{dsdu{\rho^{\mathrm{QCD3}}}(s,u,{Q^{2}})\exp(-\frac{s}{{{M^{2}}}}-\frac{u}{{k{M^{2}}}})}}\exp(\frac{{m_{{B_{c}}}^{2}}}{{{M^{2}}}}+\frac{{m_{{\eta_{c}}}^{2}}}{{k{M^{2}}}})$
(57)
$\displaystyle{V}({Q^{2}})=\int\limits_{{s_{min}}}^{{s_{0}}}{\int\limits_{{u_{min}}}^{{u_{0}}}{dsdu}{{{\rho}^{\mathrm{QCD4}}}(s,u,{Q^{2}})\exp(-\frac{s}{{{M^{2}}}}-\frac{u}{{k{M^{2}}}})}}\frac{{({m_{{B_{c}}}}+{m_{J/\psi}})({m_{c}}+{m_{b}})}}{{{\rm{2}}{f_{{B_{c}}}}m_{{B_{c}}}^{2}{f_{J/\psi}}{m_{J/\psi}}}}\exp(\frac{{m_{{B_{c}}}^{2}}}{{{M^{2}}}}{\rm{+}}\frac{{m_{J/\psi}^{2}}}{{k{M^{2}}}})$
(58) $\displaystyle\left({\begin{array}[]{*{20}{c}}{{A_{0}}({Q^{2}})}\\\
{{A_{1}}({Q^{2}})}\\\ {{A_{2}}({Q^{2}})}\end{array}}\right)$
$\displaystyle=\int\limits_{{s_{\min}}}^{{s_{0}}}{\int\limits_{{u_{\min}}}^{{u_{0}}}{dsdu}}\left({\begin{array}[]{*{20}{c}}{-\frac{{{m_{J/\psi}}}}{{m_{{B_{c}}}^{2}+m_{J/\psi}^{2}+{Q^{2}}}}}&{\frac{{-m_{{B_{c}}}^{2}+m_{J/\psi}^{2}+{Q^{2}}}}{{4{m_{J/\psi}}}}}&{\frac{{{m_{J/\psi}}(m_{{B_{c}}}^{2}-m_{J/\psi}^{2}+{Q^{2}})}}{{2(m_{{B_{c}}}^{2}+m_{J/\psi}^{2}+{Q^{2}})}}}\\\
{-\frac{1}{{{m_{{B_{c}}}}+{m_{J/\psi}}}}}&0&0\\\
{-\frac{{{m_{{B_{c}}}}+{m_{J/\psi}}}}{{m_{{B_{c}}}^{2}+m_{J/\psi}^{2}+{Q^{2}}}}}&{\frac{{{m_{{B_{c}}}}+{m_{J/\psi}}}}{2}}&{-\frac{{m_{J/\psi}^{2}({m_{{B_{c}}}}+{m_{J/\psi}})}}{{m_{{B_{c}}}^{2}+m_{J/\psi}^{2}+{Q^{2}}}}}\end{array}}\right)\left({\begin{array}[]{*{20}{c}}{\rho_{1}^{\mathrm{QCD5}}(s,u,{Q^{2}})}\\\
{\rho_{2}^{\mathrm{QCD5}}(s,u,{Q^{2}})}\\\
{\rho_{3}^{\mathrm{QCD5}}(s,u,{Q^{2}})}\end{array}}\right)$ (68)
$\displaystyle\quad\times\frac{{{m_{b}}+{m_{c}}}}{{{m_{J/\psi}}m_{{B_{c}}}^{2}{f_{J/\psi}}{f_{{B_{c}}}}}}\exp(-\frac{s}{{{M^{2}}}}-\frac{u}{{k{M^{2}}}})\exp(\frac{{m_{{B_{c}}}^{2}}}{{{M^{2}}}}+\frac{{m_{{\eta_{c}}}^{2}}}{{k{M^{2}}}})$
(69)
$\displaystyle{T_{0}}({Q^{2}})=-{({m_{{B_{c}}}}+{m_{J/\psi}})^{2}}\int\limits_{{s_{\min}}}^{{s_{0}}}{\int\limits_{{u_{\min}}}^{{u_{0}}}{dsdu\rho_{1}^{\mathrm{QCD6}}(s,u,{Q^{2}})\exp(-\frac{s}{{{M^{2}}}}-\frac{u}{{k{M^{2}}}})}}\exp(\frac{{m_{{B_{c}}}^{2}}}{{{M^{2}}}}+\frac{{m_{{\eta_{c}}}^{2}}}{{k{M^{2}}}})$
$\displaystyle{T_{1}}({Q^{2}})=\int\limits_{{s_{\min}}}^{{s_{0}}}{\int\limits_{{u_{\min}}}^{{u_{0}}}{dsdu\rho_{2}^{\mathrm{QCD6}}(s,u,{Q^{2}})\exp(-\frac{s}{{{M^{2}}}}-\frac{u}{{k{M^{2}}}})}}\exp(\frac{{m_{{B_{c}}}^{2}}}{{{M^{2}}}}+\frac{{m_{{\eta_{c}}}^{2}}}{{k{M^{2}}}})$
$\displaystyle{T_{2}}({Q^{2}})=\int\limits_{{s_{\min}}}^{{s_{0}}}{\int\limits_{{u_{\min}}}^{{u_{0}}}{dsdu\rho_{3}^{\mathrm{QCD6}}(s,u,{Q^{2}})\exp(-\frac{s}{{{M^{2}}}}-\frac{u}{{k{M^{2}}}})}}\exp(\frac{{m_{{B_{c}}}^{2}}}{{{M^{2}}}}+\frac{{m_{{\eta_{c}}}^{2}}}{{k{M^{2}}}})$
(70)

##  Appendix B The graphs of pole contributions and the Borel platform

Figure 10: The pole and continuum contributions with variation of the Borel
parameters. (a-d) are for the form factors $f_{S},f_{+},f_{0}$ and $f_{T}$ of
decay process $B_{c}\to\eta_{c}$, (e-k) are for
$V,A_{0},A_{1},A_{2},T_{0},T_{1}$ and $T_{2}$ of decay process $B_{c}\to
J/\psi$. Figure 11: The contributions of perturbative part and different
vacuum condensate terms with variation of the Borel parameter. The legend with
star denotes the results obtained with Coulomb-like correction.

##  Appendix C The decay widths of nonleptonic decay

$\displaystyle\Gamma_{B^{-}_{c}\to\eta_{c}\pi^{-}}=\frac{{\sqrt{\lambda(m_{{B_{c}}}^{2},m_{{\eta_{c}}}^{2},m_{\pi}^{2})}}}{{{\rm{32}}\pi
m_{{B_{c}}}^{\rm{3}}}}{(m_{{B_{c}}}^{2}-m_{{\eta_{c}}}^{2})^{2}}G_{F}^{2}|{V_{cb}}{|^{2}}|{V_{ud}}{|^{2}}{a_{1}^{2}}f_{\pi}^{2}f_{0}^{2}({q^{2}})\left|{{}_{{q^{2}}=m_{\pi}^{2}}}\right.$
(71)
$\displaystyle\Gamma_{B^{-}_{c}\to\eta_{c}K^{-}}=\frac{{\sqrt{\lambda(m_{{B_{c}}}^{2},m_{{\eta_{c}}}^{2},m_{K}^{2})}}}{{32\pi
m_{{B_{c}}}^{3}}}{(m_{{B_{c}}}^{2}-m_{{\eta_{c}}}^{2})^{2}}G_{F}^{2}|{V_{cb}}{|^{2}}|{V_{us}}{|^{2}}{a_{1}^{2}}f_{K}^{2}{f_{0}}{({q^{2}})^{2}}\left|{{}_{{q^{2}}=m_{K}^{2}}}\right.$
(72)
$\displaystyle\Gamma_{B^{-}_{c}\to\eta_{c}\rho^{-}}=\frac{{\sqrt{\lambda(m_{{B_{c}}}^{2},m_{{\eta_{c}}}^{2},m_{\rho}^{2})}}}{{32\pi
m_{{B_{c}}}^{3}}}[m_{{B_{c}}}^{2}-{({m_{{\eta_{c}}}}+{m_{\rho}})^{2}}][m_{{B_{c}}}^{2}-{({m_{{\eta_{c}}}}-{m_{\rho}})^{2}}]G_{F}^{2}|{V_{cb}}{|^{2}}|{V_{ud}}{|^{2}}{a_{1}^{2}}f_{\rho}^{2}{f_{+}}{({q^{2}})^{2}}\left|{{}_{{q^{2}}=m_{\rho}^{2}}}\right.$
(73)
$\displaystyle\Gamma_{B^{-}_{c}\to\eta_{c}K^{*-}}=\frac{{\sqrt{\lambda(m_{{B_{c}}}^{2},m_{{\eta_{c}}}^{2},m_{K^{*}}^{2})}}}{{32\pi
m_{{B_{c}}}^{3}}}[m_{{B_{c}}}^{2}-{({m_{{\eta_{c}}}}+{m_{K^{*}}})^{2}}][m_{{B_{c}}}^{2}-{({m_{{\eta_{c}}}}-{m_{K^{*}}})^{2}}]G_{F}^{2}|{V_{cb}}{|^{2}}|{V_{us}}{|^{2}}{a_{1}^{2}}f_{K^{*}}^{2}{f_{+}}{({q^{2}})^{2}}\left|{{}_{{q^{2}}=m_{K^{*}}^{2}}}\right.$
(74) $\displaystyle\Gamma_{B^{-}_{c}\to
J/\psi\pi^{-}}=\frac{{\sqrt{\lambda(m_{{B_{c}}}^{2},m_{J/\psi}^{2},m_{\pi}^{2})}}}{{8\pi
m_{{B_{c}}}^{3}}}[\frac{{{{(m_{{B_{c}}}^{2}-m_{J/\psi}^{2}-m_{\pi}^{2})}^{2}}}}{{4m_{J/\psi}^{2}}}-m_{\pi}^{2}]m_{J/\psi}^{2}G_{F}^{2}|{V_{cb}}{|^{2}}|{V_{ud}}{|^{2}}{a_{1}^{2}}f_{\pi}^{2}{A_{0}}{({q^{2}})^{2}}\left|{{}_{{q^{2}}=m_{\pi}^{2}}}\right.$
(75) $\displaystyle\Gamma_{B^{-}_{c}\to J/\psi
K^{-}}=\frac{{\sqrt{\lambda(m_{{B_{c}}}^{2},m_{J/\psi}^{2},m_{K}^{2})}}}{{8\pi
m_{{B_{c}}}^{3}}}[\frac{{{{(m_{{B_{c}}}^{2}-m_{J/\psi}^{2}-m_{K}^{2})}^{2}}}}{{4m_{J/\psi}^{2}}}-m_{K}^{2}]m_{J/\psi}^{2}G_{F}^{2}|{V_{cb}}{|^{2}}|{V_{us}}{|^{2}}{a_{1}^{2}}f_{K}^{2}{A_{0}}{({q^{2}})^{2}}\left|{{}_{{q^{2}}=m_{K}^{2}}}\right.$
(76) $\displaystyle\Gamma_{B^{-}_{c}\to J/\psi\rho^{-}}=$
$\displaystyle\frac{{\sqrt{\lambda(m_{{B_{c}}}^{2},m_{J/\psi}^{2},m_{\rho}^{2})}}}{{16\pi
m_{{B_{c}}}^{2}}}G_{F}^{2}|{V_{cb}}{|^{2}}|{V_{cs}}{|^{2}}{a_{1}^{2}}f_{\rho}^{2}m_{\rho}^{2}$
$\displaystyle\times\\{-\\{\frac{{8V{{({q^{2}})}^{2}}}}{{{{({m_{{B_{c}}}}+{m_{J/\psi}})}^{2}}}}[m_{{B_{c}}}^{2}m_{J/\psi}^{2}-{(\frac{{m_{{B_{c}}}^{2}+m_{J/\psi}^{2}-m_{\rho}^{2}}}{2})^{2}}]$
$\displaystyle-3{({m_{{B_{c}}}}+{m_{J/\psi}})^{2}}{A_{1}}{({q^{2}})^{2}}$
$\displaystyle-2({m_{{B_{c}}}}+{m_{J/\psi}}){A_{1}}({q^{2}})B({q^{2}})[-\frac{{m_{{B_{c}}}^{2}-m_{J/\psi}^{2}+m_{\rho}^{2}}}{2}+\frac{{(m_{{B_{c}}}^{2}-m_{J/\psi}^{2}-m_{\rho}^{2})(m_{{B_{c}}}^{2}+m_{J/\psi}^{2}-m_{\rho}^{2})}}{{4m_{J/\psi}^{2}}}]$
$\displaystyle+[-m_{\rho}^{2}+\frac{{{{(m_{{B_{c}}}^{2}-m_{J/\psi}^{2}-m_{\rho}^{2})}^{2}}}}{{4m_{J/\psi}^{2}}}][B({q^{2}})m_{{B_{c}}}^{2}-B({q^{2}})C({q^{2}})(m_{{B_{c}}}^{2}+m_{J/\psi}^{2}-m_{\rho}^{2})+C{({q^{2}})^{2}}m_{J/\psi}^{2}]\\}$
$\displaystyle+\frac{1}{{m_{\rho}^{2}}}{A_{0}}{({q^{2}})^{2}}[{(m_{{B_{c}}}^{2}-m_{J/\psi}^{2}-m_{\rho}^{2})^{2}}-4m_{\rho}^{2}m_{J/\psi}^{2}]\\}$
(77) $\displaystyle\Gamma_{B^{-}_{c}\to J/\psi K^{*-}}=$
$\displaystyle\frac{{\sqrt{\lambda(m_{{B_{c}}}^{2},m_{J/\psi}^{2},m_{K^{*}}^{2})}}}{{16\pi
m_{{B_{c}}}^{2}}}G_{F}^{2}|{V_{cb}}{|^{2}}|{V_{ud}}{|^{2}}{a_{1}^{2}}f_{K^{*}}^{2}m_{K^{*}}^{2}$
$\displaystyle\times\\{-\\{\frac{{8V{{({q^{2}})}^{2}}}}{{{{({m_{{B_{c}}}}+{m_{J/\psi}})}^{2}}}}[m_{{B_{c}}}^{2}m_{J/\psi}^{2}-{(\frac{{m_{{B_{c}}}^{2}+m_{J/\psi}^{2}-m_{K^{*}}^{2}}}{2})^{2}}]$
$\displaystyle-3{({m_{{B_{c}}}}+{m_{J/\psi}})^{2}}{A_{1}}{({q^{2}})^{2}}$
$\displaystyle-2({m_{{B_{c}}}}+{m_{J/\psi}}){A_{1}}({q^{2}})B({q^{2}})[-\frac{{m_{{B_{c}}}^{2}-m_{J/\psi}^{2}+m_{K^{*}}^{2}}}{2}+\frac{{(m_{{B_{c}}}^{2}-m_{J/\psi}^{2}-m_{K^{*}}^{2})(m_{{B_{c}}}^{2}+m_{J/\psi}^{2}-m_{K^{*}}^{2})}}{{4m_{J/\psi}^{2}}}]$
$\displaystyle+[-m_{K^{*}}^{2}+\frac{{{{(m_{{B_{c}}}^{2}-m_{J/\psi}^{2}-m_{K^{*}}^{2})}^{2}}}}{{4m_{J/\psi}^{2}}}][B({q^{2}})m_{{B_{c}}}^{2}-B({q^{2}})C({q^{2}})(m_{{B_{c}}}^{2}+m_{J/\psi}^{2}-m_{K^{*}}^{2})+C{({q^{2}})^{2}}m_{J/\psi}^{2}]\\}$
$\displaystyle+\frac{1}{{m_{K^{*}}^{2}}}{A_{0}}{({q^{2}})^{2}}[{(m_{{B_{c}}}^{2}-m_{J/\psi}^{2}-m_{K^{*}}^{2})^{2}}-4m_{K^{*}}^{2}m_{J/\psi}^{2}]\\}$
(78) $\displaystyle\mathrm{where}$ $\displaystyle
B({q^{2}})=\frac{{{m_{{B_{c}}}}+{m_{J/\psi}}}}{{{q^{2}}}}{A_{1}}({q^{2}})+\frac{{{A_{2}}({q^{2}})}}{{{m_{{B_{c}}}}+{m_{J/\psi}}}}-\frac{{({m_{{B_{c}}}}-{m_{J/\psi}}){A_{2}}({q^{2}})}}{{{q^{2}}}}-\frac{{2{m_{J/\psi}}{A_{0}}({q^{2}})}}{{{q^{2}}}}$
$\displaystyle
C({q^{2}})=\frac{{{m_{{B_{c}}}}+{m_{J/\psi}}}}{{{q^{2}}}}{A_{1}}({q^{2}})-\frac{{{A_{2}}({q^{2}})}}{{{m_{{B_{c}}}}+{m_{J/\psi}}}}-\frac{{({m_{{B_{c}}}}-{m_{J/\psi}}){A_{2}}({q^{2}})}}{{{q^{2}}}}-\frac{{2{m_{J/\psi}}{A_{0}}({q^{2}})}}{{{q^{2}}}}$
(79)

##  Appendix D The differential decay widths of semileptonic decay

$\displaystyle\frac{{d\Gamma_{B^{-}_{c}\to\eta_{c}\mathrm{l}\bar{\nu}}}}{{d{q^{2}}}}=$
$\displaystyle-\frac{{G_{F}^{2}|{V_{cb}}{|^{2}}\sqrt{\lambda(m_{{B_{c}}}^{2},m_{{\eta_{c}}}^{2},{q^{2}})}\sqrt{\lambda({q^{2}},m_{l}^{2},0)}}}{{384{\pi^{3}}m_{{B_{c}}}^{3}{{({q^{2}})}^{4}}}}({q^{2}}-m_{l}^{2})$
$\displaystyle\times\\{(2{q^{2}}+m_{l}^{2})[{D(q^{2})^{2}}m_{{B_{c}}}^{2}+D(q^{2})E(q^{2})(m_{{B_{c}}}^{2}-{q^{2}}+m_{{\eta_{c}}}^{2})+{E(q^{2})^{2}}m_{{\eta_{c}}}^{2}]$
$\displaystyle-2{q^{2}}({q^{2}}+2m_{l}^{2}){f_{0}}{({q^{2}})^{2}}{(m_{{B_{c}}}^{2}-m_{{\eta_{c}}}^{2})^{2}}\\}$
(80)

$\displaystyle\frac{{d\Gamma_{B^{-}_{c}\to
J/\psi\mathrm{l}\bar{\nu}}}}{{d{q^{2}}}}=$
$\displaystyle-\frac{{G_{F}^{2}|{V_{cb}}{|^{2}}\sqrt{\lambda(m_{{B_{c}}}^{2},m_{J/\psi}^{2},{q^{2}})}\sqrt{\lambda({q^{2}},m_{l}^{2},0)}}}{{192{\pi^{3}}m_{{B_{c}}}^{3}{{({q^{2}})}^{2}}}}({q^{2}}-m_{l}^{2})$
$\displaystyle\times\\{(2{q^{2}}+m_{l}^{2})\\{\frac{{8V{{({q^{2}})}^{2}}}}{{{{({m_{{B_{c}}}}+{m_{J/\psi}})}^{2}}}}[m_{{B_{c}}}^{2}m_{J/\psi}^{2}-{(\frac{{m_{{B_{c}}}^{2}+m_{J/\psi}^{2}-{q^{2}}}}{2})^{2}}]$
$\displaystyle-3{({m_{{B_{c}}}}+{m_{J/\psi}})^{2}}{A_{1}}{({q^{2}})^{2}}$
$\displaystyle-2({m_{{B_{c}}}}+{m_{J/\psi}}){A_{1}}({q^{2}})B({q^{2}})[-\frac{{m_{{B_{c}}}^{2}-m_{J/\psi}^{2}+{q^{2}}}}{2}+\frac{{(m_{{B_{c}}}^{2}-m_{J/\psi}^{2}-{q^{2}})(m_{{B_{c}}}^{2}+m_{J/\psi}^{2}-{q^{2}})}}{{4m_{J/\psi}^{2}}}]$
$\displaystyle+[-{q^{2}}+\frac{{{{(m_{{B_{c}}}^{2}-m_{J/\psi}^{2}-{q^{2}})}^{2}}}}{{4m_{J/\psi}^{2}}}][B({q^{2}})m_{{B_{c}}}^{2}-B({q^{2}})C({q^{2}})(m_{{B_{c}}}^{2}+m_{J/\psi}^{2}-{q^{2}})+C{({q^{2}})^{2}}m_{J/\psi}^{2}]\\}$
$\displaystyle-\frac{{({q^{2}}+2m_{l}^{2})}}{{2{q^{2}}}}{A_{0}}{({q^{2}})^{2}}[{(m_{{B_{c}}}^{2}-m_{J/\psi}^{2}-{q^{2}})^{2}}-4{q^{2}}m_{J/\psi}^{2}]$
(81) $\displaystyle\mathrm{where}$ $\displaystyle
B({q^{2}})=\frac{{{m_{{B_{c}}}}+{m_{J/\psi}}}}{{{q^{2}}}}{A_{1}}({q^{2}})+\frac{{{A_{2}}({q^{2}})}}{{{m_{{B_{c}}}}+{m_{J/\psi}}}}-\frac{{({m_{{B_{c}}}}-{m_{J/\psi}}){A_{2}}({q^{2}})}}{{{q^{2}}}}-\frac{{2{m_{J/\psi}}{A_{0}}({q^{2}})}}{{{q^{2}}}}$
$\displaystyle
C({q^{2}})=\frac{{{m_{{B_{c}}}}+{m_{J/\psi}}}}{{{q^{2}}}}{A_{1}}({q^{2}})-\frac{{{A_{2}}({q^{2}})}}{{{m_{{B_{c}}}}+{m_{J/\psi}}}}-\frac{{({m_{{B_{c}}}}-{m_{J/\psi}}){A_{2}}({q^{2}})}}{{{q^{2}}}}-\frac{{2{m_{J/\psi}}{A_{0}}({q^{2}})}}{{{q^{2}}}}$
$\displaystyle
D({q^{2}})=f_{+}^{{B_{c}}\to{\eta_{c}}}({q^{2}}){q^{2}}+(f_{0}^{{B_{c}}\to{\eta_{c}}}({q^{2}})-f_{+}^{{B_{c}}\to{\eta_{c}}}({q^{2}}))(m_{{B_{c}}}^{2}-m_{{\eta_{c}}}^{2})$
$\displaystyle
E({q^{2}})=f_{+}^{{B_{c}}\to{\eta_{c}}}({q^{2}}){q^{2}}-(f_{0}^{{B_{c}}\to{\eta_{c}}}({q^{2}})-f_{+}^{{B_{c}}\to{\eta_{c}}}({q^{2}}))(m_{{B_{c}}}^{2}-m_{{\eta_{c}}}^{2})$
(82)