# CP violation in non-leptonic $B_{c}$ decays to excited final states

Tian<EMAIL_ADDRESS>Tianhong<EMAIL_ADDRESS>(Corresponding author), Hui-Feng<EMAIL_ADDRESS>Zhi-Hui
<EMAIL_ADDRESS>Lei<EMAIL_ADDRESS>
Guo-Li<EMAIL_ADDRESS>
1School of Physics, Harbin Institute of Technology, Harbin, 150001, China
2Center for Theoretical Physics, College of Physics, Jilin University,
Changchun 130012, China
3School of Electrical and Information Engineering, North Minzu University,
Yinchuan 750021, China

###### Abstract

We study the CP violation in two-body nonleptonic decays of $B_{c}$ meson. We
concentrate on the decay channels which contain at least one excited heavy
meson in the final states. Specifically, the following channels are
considered: $B_{c}\to c\bar{c}(2S,2P)+\bar{c}q(1S,1P)$, $B_{c}\to
c\bar{c}(1S)+\bar{c}q(2S,2P)$, $B_{c}\to c\bar{c}(1P)+\bar{c}q(2S)$, $B_{c}\to
c\bar{c}(1D)+\bar{c}q(1S,1P)$, and $B_{c}\to c\bar{c}(3S)+\bar{c}q(1S)$. The
improved Bethe-Salpeter method is applied to calculate the hadronic transition
matrix element. Our results show that some decay modes have large branching
ratios, which is of the order of $10^{-3}$. The CP violation effect in
$B_{c}\rightarrow\eta_{c}(1S)+D(2S)$,
$B_{c}\rightarrow\eta_{c}(1S)+D_{0}^{*}(2P)$, and $B_{c}\rightarrow
J/\psi+D^{*}(2S)$ are most likely to be found. If the detection precision of
the CP asymmetry in such channels can reach the $3\sigma$ level, at least
$10^{7}$ $B_{c}$ events are needed.

##  I Introduction

The $B_{c}$ meson was discovered by CDF more than two decades ago Abe  _et
al._ (1998). Since then, there have been a lot of studies both theoretical and
experimental on this particle. The reason why it’s so interesting is that the
$B_{c}$ meson is the lowest bound state which consists of two heavy quarks
with different flavors. It cannot decay through strong interaction or
electromagnetic interaction, and only the weak decay channels are allowed.
This makes the $B_{c}$ meson an ideal platform to study the properties of
heavy quarks and test accurately the Standard Model (SM) predictions.
Experimentally, about $5\times 10^{10}$ $B_{c}$ mesons per year could be
produced at the LHC Gouz  _et al._ (2004), which provides us the opportunity
to get more information of this particle, such as the CP violation effect in
two-body nonleptonic decay modes.

The CP asymmetry in the non-leptonic $B$ decays has been extensively studied
by using different methods, such as the QCD factorization approach Beneke  _et
al._ (1999); Du  _et al._ (2002), the soft-collinear effective theory (SCET)
Bauer  _et al._ (2004, 2004), and the perturbative QCD (pQCD) method Keum  _et
al._ (2001); L$\ddot{\text{u}}$  _et al._ (2001); Li (2003). For $B_{c}$
meson, the CP violation effects have also attracted some attentions. Based on
the pQCD method, Ref. Zhang and Yu (2009); Zou  _et al._ (2018); Rui  _et al._
(2012); Xiao and Liu (2014) studied the direct CP asymmetry parameter of two-
body decays of $B_{c}$ meson with one light final state. In Ref. Ivanov  _et
al._ (2003), a relativistic quark model was applied to explore the decay
channels of $B_{c}^{+}\to D_{s}^{+}\bar{D}^{0}$ and $B_{c}^{+}\to
D_{s}^{+}D^{0}$. In Ref. Kar  _et al._ (2013), with a relativistic independent
quark model the authors predicted that there were significant CP violation in
the $B_{c}\to D^{\ast}D^{\ast},~{}D^{\ast}D_{s}^{\ast}$ channels. In Refs. Dai
and Du (1999); Kiselev (2004); Fleischer and Wyler (2000); Giri  _et al._
(2002); Bhattacharya and Petrov (2017), the model-independent method was used
to estimate the Cabibbo-Kobayashi-Maskawa (CKM) angle $\gamma$. In Ref. Dai
and Du (1999), the authors pointed out that to observe the CP violation, about
$10^{8}$ $B_{c}$ events were needed.

In Refs. Hui Feng  _et al._ (2011); Chen  _et al._ (2012), the Bethe-Salpeter
(BS) method was used to study the CP violation of $B_{c}$ and $B_{s}$ mesons.
In this formalism, by solving the instantaneous BS equations, we get the wave
functions of heavy mesons, which are used to calculate the hadronic transition
matrix elements (Ref. Liu and Chao (1997) applied a similar formalism to study
the CP violation of $B_{c}$ with $S$-wave final states). In this paper, we
consider the direct CP violation in two-body nonleptonic decays of $B_{c}$
meson, where a radial excited state is included in the final states. There are
three reasons why we make such study. First, it is still lack of theoretical
calculations of these decays. However, as more and more excited $D$ and
$D_{s}$ states are discovered, such as $D_{s1}^{*}(2710)^{\pm}$,
$D(2550)^{0}$, $D^{*}(2640)^{\pm}$, and $D_{1}^{*}(2680)^{0}$ Zyla  _et al._
(2020), which can be identified as $D_{s}^{*}(2S)$, $D(2S)^{0}$,
$D^{*}(2S)^{\pm}$, and $D^{*}(2S)^{0}$, respectively, those calculations will
become necessary. Second, these decay channels can be used to test the
validity of the potential models. According to Ref. Geng  _et al._ (2019), the
relativistic effects are important when the excited states are involved. So
the improved BS method Zhou  _et al._ (2020), which writes the amplitude in a
more covariant form, will be applied. Third, some interesting information can
be obtained by studying the CP violation effects. For example, in the decays
to the $1^{+(\prime)}$ states, the mixing angle can affect the CP violation
severely in some region.

As the direct CP violation comes from the interference of different diagrams,
we will consider the color-favored tree diagram, color-suppressed tree
diagram, and the time-like penguin diagram. The annihilation diagram and the
space-like penguin diagram will be helicity suppressed if the masses of the
final mesons are small compared with that of the inital meson (this
suppression is not very large at this case as the final mesons are also
heavy). Besides, the creation of the $c\bar{c}$ pair also leads to the form
factor suppression. So we will not consider these two kinds of diagrams in
this work. The soft strong phase arising from the rescattering effects of
final states may also bring considerable contribution. For example, in Ref.
Liu and Li (2008) the CP asymmetry of the $B_{c}^{+}\to D^{0}\pi^{+}$ channel
is changed about $22\%$ by such effects. To estimate the uncertainties brought
by the final state interaction in the channels with two heavy final states is
the project of our ongoing work.

The paper is organized as follows. In Section II, we present the theoretical
formalism of the CP asymmetry in nonleptonic decays of $B_{c}$ meson. In
Section III, we use the improved BS method to calculate the hadronic
transition matrix elements which are expressed as the overlap integral of the
wave functions of heavy mesons. In Section IV, we give the numerical values of
the CP asymmetry and make discussions of the results.

##  II CP violation in non-leptonic decays

The NLO effective Hamiltonian for $|\Delta B|=1$ nonleptonic decays can be
written as Buchalla  _et al._ (1996)

$\displaystyle\mathcal{H}_{\mathrm{eff}}(|\Delta
B|=1)=\frac{G_{F}}{\sqrt{2}}{\sum_{q=c\atop
q^{\prime}=d,s}}V_{qq^{\prime}}V_{qb}^{*}\left\\{C_{1}(\mu)Q_{1}+C_{2}(\mu)Q_{2}+\sum_{k=3}^{10}C_{k}(\mu)Q_{k}\right\\}+\mathrm{h.c.},$
(1)

where $C_{i}(\mu)(i=1,2,\cdots,10)$ are the Wilson coefficients and $\mu$ is
the renormalization scale. $Q_{1}$ and $Q_{2}$ are the tree diagram operators.
$Q_{3}$, $Q_{4}$, $Q_{5}$, and $Q_{6}$ are the QCD penguin diagram operators.
$Q_{7}$, $Q_{8}$, $Q_{9}$, and $Q_{10}$ are the electroweak penguin diagram
operators. Specifically, these operators have the following forms Dai and Du
(1999)

$\displaystyle
Q_{1}=\left(\bar{q}^{\prime}_{\alpha}q_{\beta}\right)_{V-A}\left(\bar{q}_{\beta}b_{\alpha}\right)_{V-A},$
(2) $\displaystyle
Q_{2}=\left(\bar{q}^{\prime}q\right)_{V-A}(\bar{q}b)_{V-A},$ $\displaystyle
Q_{3(5)}=\left(\bar{q}^{\prime}b\right)_{V-A}\sum_{q^{\prime\prime}}\left(\bar{q}^{\prime\prime}q^{\prime\prime}\right)_{V-A(V+A)},$
$\displaystyle
Q_{4(6)}=\left(\bar{q}_{\alpha}^{\prime}b_{\beta}\right)_{V-A}\sum_{q^{\prime\prime}}\left(\bar{q}^{\prime\prime}_{\beta}q_{\alpha}^{\prime\prime}\right)_{V-A(V+A)},$
$\displaystyle
Q_{7(9)}=\frac{3}{2}(\overline{q^{\prime}}b)_{V-A}\sum_{q^{\prime\prime}}e_{q^{\prime\prime}}\left(\bar{q}^{\prime\prime}q^{\prime\prime}\right)_{V+A(V-A)},$
$\displaystyle
Q_{8(10)}=\frac{3}{2}\left(\overline{q^{\prime}}_{\alpha}b_{\beta}\right)_{V-A}\sum_{q^{\prime\prime}}e_{q^{\prime\prime}}\left(\bar{q}^{\prime\prime}_{\beta}q_{\alpha}^{\prime\prime}\right)_{V+A(V-A)},$

where $e_{q^{\prime\prime}}$ is the electric charge of the quark
$q^{\prime\prime}$ which can be $u$, $d$, $s$, $c$, or $b$; the subscripts
$\alpha$ and $\beta$ are color indices; $(\bar{q}_{1\alpha}q_{2\beta})_{V\pm
A}\equiv\bar{q}_{1\alpha}\gamma^{\mu}(1\pm\gamma_{5})q_{2\beta}$. As Ref.
Buras  _et al._ (1992) did, we will use the factorization approximation, under
which the amplitude can be factorized into the product of hadronic transition
matrix element and decay constants. This is expected to be true when the final
mesons are light, so the strong interaction effects can be neglected and the
color transparency is valid. If the final mesons are heavy and in the region
close to zero recoil, such as the case considered in this work, the
factorization approximation may not hold up well, and the final state
interactions may give considerable contribution. In Ref. Chen  _et al._ (2012)
both the factorization approximation and the pQCD method are used to study the
two-body nonleptonic decays of $B_{s}$ meson, and the results of two methods
are close to each other. This gives us some confidence that the factorization
approximation can be applied in these cases, at least as a preliminary
estimate.

(a) Color-favored tree diagram $\mathcal{A}_{1}$ (b) Color-suppressed tree
diagram $\mathcal{A}_{2}$

(c) Time-like penguin diagram $\mathcal{A}_{3}$ (d) Annihilation diagram

(e) Space-like penguin diagram

Figure 1:  The Feynman diagrams for $B_{c}^{-}$ decaying into two mesons
$X^{0}$ and $X^{-}$ in the spectator approximation, where $q,~{}q_{v}=u,~{}c$
and $q^{\prime},~{}q_{v}^{\prime}=d,~{}s$. The subscript $v$ denotes ‘vacuum’.

The amplitudes corresponding to the first three Feynman diagrams in Fig. 1 are
Dai and Du (1999)

$\displaystyle\mathcal{A}_{1}$
$\displaystyle=a_{1}A,~{}~{}~{}~{}~{}~{}\mathcal{A}_{2}=a_{2}B,$ (3)
$\displaystyle\mathcal{A}_{3}$
$\displaystyle=\left[\left(a_{3}+a_{9}\right)+\xi_{f}\left(a_{5}+a_{7}\right)+\left(1+\xi_{f}\right)b^{q}G\left(m_{q},k^{2}\right)\right]A,$

where A and B are the factorized hadronic matrix elements for
$B_{c}^{-}\rightarrow X^{-}X^{0}$, which have the forms

$\displaystyle A=\frac{G_{F}}{\sqrt{2}}\langle
X^{-}(q_{1}\bar{q}_{2})\left|\left(\bar{q}_{1}q_{2}\right)_{V-A}\right|0\rangle\langle
X^{0}(q_{3}\bar{c})\left|\left(\bar{q}_{3}b\right)_{V-A}\right|B_{c}^{-}\rangle,$
(4) $\displaystyle B=\frac{G_{F}}{\sqrt{2}}\langle
X^{0}(q_{1}\bar{q}_{2})\left|\left(\bar{q}_{1}q_{2}\right)_{V-A}\right|0\rangle\langle
X^{-}(q_{3}\bar{c})\left|\left(\bar{q}_{3}b\right)_{V-A}\right|B_{c}^{-}\rangle.$

The $a_{k}$ in Eq. (3) are defined by Wilson coefficients at the
renormalization scale $\mu\sim m_{b}$

$a_{2i-1}\equiv\frac{C_{2i-1}}{N_{c}}+C_{2i},\quad
a_{2i}\equiv\frac{C_{2i}}{N_{c}}+C_{2i-1},\quad(i=1,2,3,4,5)$ (5)

where $N_{c}$ is the number of colors. In the actual calculation, one often
uses the effective number of colors Leitner  _et al._ (2002)

$\frac{1}{\left(N_{c}^{eff}\right)_{i}}=\frac{1}{3}+\delta_{i},\text{ with
}i=1,\cdots,10$ (6)

where $\delta_{i}$ stands for the nonfactorizable part. In this paper we will
choose $N_{c}=3$ for all operators to get the results, and we will also
discuss the effect of different $N_{c}^{eff}$.

The parameter $\xi_{f}$ in the Eq. (3) comes from the Fiertz rearrangement,
which transforms the $(V\mp A)(V\pm A)$ currents into the $(S\pm P)(S\mp P)$
currents. For decay channels with different final states, this parameter takes
different values Chen  _et al._ (2012)

$\xi_{f}=\left\\{\begin{aligned}
+\frac{2M_{B_{c}}^{2}}{\left(m_{b}-m_{q_{v}}\right)\left(m_{q^{\prime}}+m_{q_{v}}\right)}&~{}~{}~{}~{}~{}~{}~{}~{}X_{1}\left(0^{-}\right)X_{2}\left(0^{-}\right)~{}\text{or}~{}X_{1}\left(1^{+}\right)X_{2}\left(0^{-}\right),\\\
-\frac{2M_{B_{c}}^{2}}{\left(m_{b}-m_{q_{v}}\right)\left(m_{q^{\prime}}-m_{q_{v}}\right)}&~{}~{}~{}~{}~{}~{}~{}~{}X_{1}\left(0^{-}\right)X_{2}\left(0^{+}\right)~{}\text{or}~{}X_{1}\left(1^{+}\right)X_{2}\left(0^{+}\right),\\\
-\frac{2M_{B_{c}}^{2}}{\left(m_{b}+m_{q_{v}}\right)\left(m_{q^{\prime}}+m_{q_{v}}\right)}&~{}~{}~{}~{}~{}~{}~{}~{}X_{1}\left(1^{-}\right)X_{2}\left(0^{-}\right)~{}\text{or}~{}X_{1}\left(0^{+}\right)X_{2}\left(0^{-}\right),\\\
+\frac{2M_{B_{c}}^{2}}{\left(m_{b}+m_{q_{v}}\right)\left(m_{q^{\prime}}-m_{q_{v}}\right)}&~{}~{}~{}~{}~{}~{}~{}~{}X_{1}\left(1^{-}\right)X_{2}\left(0^{+}\right)~{}\text{or}~{}X_{1}\left(0^{+}\right)X_{2}\left(0^{+}\right),\\\
\end{aligned}\right.$ (7)

where $X_{1}$ is the final meson which contains the spectator $\bar{c}$ and
$X_{2}$ is the other one. Their $J^{P}$ numbers are presented in the
parentheses.

The $b^{q}$ in Eq. (3) have the form

$\displaystyle
b^{q}=\left\\{\frac{\alpha_{s}}{8\pi}\left(1-\frac{1}{3N_{c}}\right)C_{1}+\frac{\alpha_{e}}{3\pi}\frac{1}{N_{c}}\left(\frac{C_{1}}{N_{c}}+C_{2}\right)\right\\}\left[\frac{10}{9}-G\left(m_{q},k^{2}\right)\right],$
(8)

where we take
$\alpha_{s}(m_{Z})=0.1176,~{}\alpha_{e}(m_{Z})=1/128,~{}m_{W}=91.1876$ GeV,
and $\Lambda_{\mathrm{QCD}}^{(f=5)}=220.9$ MeV. The function $G(m_{q},k^{2})$
is defined as Dai and Du (1999)

$G\left(m_{q},k^{2}\right)=\frac{3}{2}\left[\frac{10}{9}-F_{q}\left(k^{2}\right)\right],$
(9)

where $F_{q}(k^{2})$ is the penguin loop-integral function of the squared
momentum $k^{2}$ carried by the virtual gluon at the renormalization scale
$\mu\sim m_{b}$,

$F_{q}\left(k^{2}\right)=-4\int_{0}^{1}\mathrm{d}xx(1-x)\ln\left[\frac{m_{q}^{2}-x(1-x)k^{2}}{m_{b}^{2}}\right].$
(10)

According to Ref. Gerard and Hou (1991), we can rewrite the function
$G(m_{q},k^{2})$ as

$\displaystyle
G\left(m_{q},k^{2}\right)=\ln\frac{m_{q}^{2}}{m_{b}^{2}}-r_{q}+\left(1+\frac{r_{q}}{2}\right)\sqrt{1-r_{q}}\ln\frac{1+\sqrt{1-r_{q}}}{1-\sqrt{1-r_{q}}}+i\pi\left(1+\frac{r_{q}}{2}\right)\sqrt{1-r_{q}},$
(11)

where $r_{q}=4m_{q}^{2}/k^{2}$. As Ref. Hui Feng  _et al._ (2011) did, we will
take $k^{2}$ to be its average value $\bar{k}^{2}$, which is defined as

$\frac{\bar{k}^{2}}{m_{b}^{2}}=\frac{1}{2}\left(1+\left(m_{\bar{q}_{v}}^{2}-m_{q}^{2}\right)\left(1-\frac{m_{\bar{q}_{v}}^{2}}{m_{b}^{2}}\right)/m_{X^{-}}^{2}+\left(m_{q}^{2}+2m_{\bar{q}_{v}}^{2}-m_{X^{-}}^{2}\right)/m_{b}^{2}\right).$
(12)

In the calculation of the QCD penguin diagram, we will take the current quark
masses Zyla  _et al._ (2020)

$\displaystyle m_{u}=0.0022~{}\text{GeV},\quad m_{d}=0.0047~{}\text{GeV},\quad
m_{s}=0.095~{}\text{GeV},$ (13) $\displaystyle m_{c}=1.275~{}\text{GeV},\quad
m_{b}=4.18~{}\text{GeV},\quad m_{t}=173~{}\text{GeV}.$

Finally the amplitude of two-body nonleptonic decays of $B_{c}$ meson can be
written as Dai and Du (1999)

$\mathcal{M}=\frac{G_{F}}{\sqrt{2}}\left\\{\left\\{\lambda_{c}a_{1}+\sum_{q=u,c}\lambda_{q}\left[a_{3}+a_{9}+\xi_{f}\left(a_{5}+a_{7}\right)+(1+\xi_{f})b^{q}\right]\right\\}A+\lambda_{c}a_{2}B\right\\},$
(14)

where $\lambda_{q}=V_{qb}V_{qd}^{*}$, red which is accepted for the cause
$B_{c}\to c\bar{c}+cd$ and it should be changed to
$\lambda_{q}=V_{qb}V_{qs}^{*}$ for the channel $B_{c}\to c\bar{c}+cs$, and the
$q$ is the quark inside the QCD penguin. By combining terms with the same
$\lambda_{q}$, we can write $\mathcal{M}$ as (14)

$\mathcal{M}=V_{cb}V_{cd}^{*}T_{1}+V_{ub}V_{ud}^{*}T_{2}.$ (15)

The CP asymmetry of two-body decays of $B_{c}$ meson is defined as

$\mathcal{A}_{cp}=\frac{\Gamma\left(B_{c}^{+}\rightarrow\bar{f}\right)-\Gamma\left(B_{c}^{-}\rightarrow
f\right)}{\Gamma\left(B_{c}^{+}\rightarrow\bar{f}\right)+\Gamma\left(B_{c}^{-}\rightarrow
f\right)}.$ (16)

By calculating $|\mathcal{M}(B_{c}^{+}\rightarrow\bar{f})|^{2}$ and
$|\mathcal{M}(B_{c}^{+}\rightarrow\bar{f})|^{2}$ and inserting them into Eq.
(16), we get

$\mathcal{A}_{cp}=\frac{-2{\rm Im}\left(T_{1}T_{2}^{*}\right){\rm
Im}\left(\frac{V_{ub}V_{ud}^{*}}{V_{cb}V_{cd}^{*}}\right)}{\left|T_{1}\right|^{2}+\left|\frac{V_{ub}V_{ud}^{*}}{V_{cb}V_{cd}^{*}}\right|^{2}\left|T_{2}\right|^{2}+2{\rm
Re}\left(T_{1}T_{2}^{*}\right){\rm
Re}\left(\frac{V_{ub}V_{ud}^{*}}{V_{cb}V_{cd}^{*}}\right)}.$ (17)

According to Ref. Wolfenstein (1983), the CKM matrix can be parameterized as

$\displaystyle V=\left(\begin{array}[]{ccc}V_{ud}&V_{us}&V_{ub}\\\
V_{cd}&V_{cs}&V_{cb}\\\
V_{tb}&V_{ts}&V_{tb}\end{array}\right)=\left(\begin{array}[]{ccc}1-\frac{\lambda^{2}}{2}&\lambda&A\lambda^{3}(\rho-i\eta)\\\
-\lambda&1-\frac{\lambda^{2}}{2}&A\lambda^{2}\\\
A\lambda^{3}(1-\rho-i\eta)&-A\lambda^{2}&1\end{array}\right)+O\left(\lambda^{4}\right).$
(18)

Inserting Eq. (18) into Eq. (17), we get

$\displaystyle\mathcal{A}_{cp}$ $\displaystyle=\frac{-2{\rm
Im}\left(T_{1}T_{2}^{*}\right)\sin\gamma}{\left|T_{1}\right|^{2}/\Delta+\Delta\left|T_{2}\right|^{2}-2{\rm
Re}\left(T_{1}T_{2}^{*}\right)\cos\gamma}$ (19) $\displaystyle\equiv
D_{1}\frac{\sin\gamma}{1+D_{2}\cos\gamma},$

where we have defined
$\gamma\equiv\arg\left(-\frac{V_{ub}^{*}V_{ud}}{V_{cb}^{*}V_{cd}}\right)\simeq\arg\left(\frac{V_{ub}^{*}V_{us}}{V_{cb}^{*}V_{cs}}\right)$;
$\Delta\equiv\left|\frac{V_{ub}V_{ud}^{*}}{V_{cb}V_{cd}^{*}}\right|$.

##  III The improved BS method

In this section, we give a brief introduction to the improved BS method, which
is used to calculate the hadronic transition matrix element $\left\langle
X^{0(-)}\left|J^{V-A}_{\mu}\right|B_{c}\right\rangle$. More details about this
method can be found in our previous work Zhou  _et al._ (2020).

According to the Mandelstam formalism, the hadronic matrix can be written as

$\left\langle
X\left(P_{f}\right)\left|\left(\bar{c}\Gamma^{\mu}b\right)\right|B_{c}(P)\right\rangle=\int\frac{\mathrm{d}^{4}q}{(2\pi)^{4}}\frac{\mathrm{d}^{4}q_{f}}{(2\pi)^{4}}\operatorname{Tr}\left[\bar{\chi}_{P_{f}}\left(q_{f}\right)\Gamma^{\mu}\chi_{P}(q)\mathrm{i}S_{2}^{-1}\left(p_{2}\right)\right](2\pi)^{4}\delta^{(4)}\left(p_{2}-p_{2f}\right),$
(20)

where $\Gamma^{\mu}=\gamma^{\mu}\left(1-\gamma^{5}\right)$; $\chi$ is the BS
wave function; $p_{i}$ ($q$) and $p_{fi}$ ($q_{f}$) are the (relative) momenta
of quark and antiquark within initial and final mesons, respectively; $S_{i}$
is the quark propagator. By applying the instantaneous approximation, Eq. (20)
can be reduced into the three-dimensional form,

$\displaystyle\langle
D_{q}(P_{f})|(\overline{c}\Gamma^{\mu}b)|B_{q}(P)\rangle=-i\int\frac{d\vec{q}}{(2\pi)^{3}}{\rm
Tr}\left[\frac{\not{P}_{f}}{M_{f}}\overline{\varphi}^{++}(P_{f},q_{f\perp\\!\\!\\!\perp})\frac{\not{P}_{f}}{M_{f}}L_{r}\Gamma^{\mu}\varphi^{++}(P,q_{\perp})\right],$
(21)

where

$\displaystyle\varphi^{++}(P,q_{\perp})=\Lambda_{1}^{+}(p_{1\perp})\frac{\not{P}}{M}\varphi(P,q_{\perp})\frac{\not{P}}{M}\Lambda_{2}^{+}(p_{2\perp}),$
(22)
$\displaystyle\varphi^{++}(P_{f},q_{f\perp\\!\\!\\!\perp})=\widetilde{\Lambda}_{1}^{+}(p_{1f\perp\\!\\!\\!\perp})\frac{\not{P}_{f}}{M_{f}}\varphi(P_{f},q_{f\perp\\!\\!\\!\perp})\frac{\not{P}_{f}}{M_{f}}\widetilde{\Lambda}_{2}^{+}(p_{2f\perp\\!\\!\\!\perp})$

are the positive energy projection of wave functions; the symbol $L_{r}$ is
expressed as

$L_{r}=\frac{\left(M_{f}-\widetilde{\omega}_{1f}-\widetilde{\omega}_{2f}\right)}{\left(P_{f_{P}}-\omega_{1f}-\omega_{2}\right)}\Lambda_{1}^{+}\left(p_{1f\perp}\right).$
(23)

In the above equations, we have definited the projection operator as follows

$\displaystyle\Lambda^{\pm}_{i}(p_{i\perp})=\frac{1}{2\omega_{i}}\left[\frac{\not{P}}{M}\omega_{i}\pm(Jm_{i}+\not{p}_{i\perp})\right],$
(24)
$\displaystyle\widetilde{\Lambda}^{\pm}_{i}(p_{if\perp\\!\\!\\!\perp})=\frac{1}{2\widetilde{\omega}_{if}}\left[\frac{\not{P}_{f}}{M_{f}}\widetilde{\omega}_{if}\pm(Jm_{if}+\not{p}_{if\perp\\!\\!\\!\perp})\right],$

where $J=1$ and $-1$ for quark and antiquark, respectively; $m_{i}$ and
$m_{if}$ are quark masses. The quark energies $\omega_{i}$, $\omega_{if}$, and
$\widetilde{\omega}_{if}$ have the forms

$\displaystyle\omega_{i}=\sqrt{m_{i}^{2}-q_{\perp}^{2}},~{}~{}~{}~{}\omega_{if}=\sqrt{m_{if}^{2}-q_{f\perp}^{2}},~{}~{}~{}\widetilde{\omega}_{if}=\sqrt{m_{if}^{2}-q_{f\perp\\!\\!\\!\perp}^{2}},$
(25)

where

$q_{f\perp\\!\\!\\!\perp}=q_{f\perp}-\frac{q_{f\perp}\cdot
P_{f\perp}}{M_{f}^{2}}P_{f}+s_{r}\left(\frac{1}{M}P-\frac{P_{f}\cdot
P}{MM_{f}^{2}}P_{f}\right),$ (26)

with $s_{r}=\frac{m_{2}}{m_{1}+m_{2}}\frac{P_{f}\cdot P}{M}-\omega_{2f}$. The
symbols $\perp$ and $\perp\\!\\!\\!\perp$ mean projecting onto the momenta of
the initial and final mesons, respectively.

The instantaneous BS wave functions of heavy mesons with different quantum
numbers ($0^{-}$, $0^{+}$, $1^{-}$, $1^{+-}$, and $1^{++}$) have the following
forms

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\varphi_{0^{-}}(q_{\perp})=M\left[\frac{\not{P}}{M}f_{1}(q_{\perp})+f_{2}(q_{\perp})+\frac{\not{q}_{\perp}}{M}f_{3}(q_{\perp})+\frac{\not{P}\not{q}_{\perp}}{M^{2}}f_{4}(q_{\perp})\right]\gamma_{5},$
(27)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\varphi_{0^{+}}(q_{\perp})=M\left[\frac{\not{q}_{\perp}}{M}g_{1}(q_{\perp})+\frac{\not{P}\not{q}_{\perp}}{M^{2}}g_{2}(q_{\perp})+g_{3}(q_{\perp})+\frac{\not{P}}{M}g_{4}(q_{\perp})\right],$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\varphi_{1^{-}}(q_{\perp})=(q_{\perp}\cdot\epsilon)\left[h_{1}(q_{\perp})+\frac{\not{P}}{M}h_{2}(q_{\perp})+\frac{\not{q}_{\perp}}{M}h_{3}(q_{\perp})+\frac{\not{P}\not{q}_{\perp}}{M^{2}}h_{4}(q_{\perp})\right]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+M\not{\epsilon}\left[h_{5}(q_{\perp})+\frac{\not{P}}{M}h_{6}(q_{\perp})+\frac{\not{q}_{\perp}}{M}h_{7}(q_{\perp})+\frac{\not{P}\not{q}_{\perp}}{M^{2}}h_{8}(q_{\perp})\right],$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\varphi_{1^{++}}(q_{\perp})=i\varepsilon_{\mu\nu\alpha\beta}\frac{P^{\nu}}{M}q_{\perp}^{\alpha}\epsilon^{\beta}\left[r_{1}(q_{\perp})+\frac{\not{P}}{M}r_{2}(q_{\perp})+\frac{\not{q}_{\perp}}{M}r_{3}(q_{\perp})-\frac{\not{P}\not{q}_{\perp}}{M^{2}}r_{4}(q_{\perp})\right]\gamma^{\mu},$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\varphi_{1^{+-}}(q_{\perp})=q_{\perp}\cdot\epsilon\left[s_{1}(q_{\perp})+\frac{\not{P}}{M}s_{2}(q_{\perp})+\frac{\not{q}_{\perp}}{M}s_{3}(q_{\perp})+\frac{\not{P}\not{q}_{\perp}}{M^{2}}s_{4}(q_{\perp})\right]\gamma_{5},$

where $f_{i}$, $g_{i}$, $h_{i}$, $r_{i}$, and $s_{i}$ are the radial wave
functions; $\epsilon^{\mu}$ is the polarization vector of the vector or axial
vector mesons. The $1^{+}(P_{1}^{1/2})$ and $1^{+\prime}(P_{1}^{3/2})$ states
are the mixing of ${}^{1}P_{1}$ and ${}^{3}P_{1}$ states, which can be written
as Godfrey and Moats (2016)

$\displaystyle
P_{1}^{1/2}={{}^{1}P}_{1}\cos\theta_{nP}+{}^{3}P_{1}\sin\theta_{nP},$ (28)
$\displaystyle
P_{1}^{3/2}=-{{}^{1}P}_{1}\sin\theta_{nP}+{}^{3}P_{1}\cos\theta_{nP}.$

According to Ref. Godfrey and Moats (2016), the mixing angle for different
states takes the values:

$\displaystyle\theta_{1P}(c\bar{d})=-25.68^{\circ},$
$\displaystyle\theta_{1P}(c\bar{s})=-37.48^{\circ},$ (29)
$\displaystyle\theta_{2P}(c\bar{d})=-29.39^{\circ},$
$\displaystyle\theta_{2P}(c\bar{s})=-30.40^{\circ}.$

##  IV Numerical Results and Discussions

The numerical results of the wave functions are achieved by solving the
corresponding instantaneous BS equations. The Cornell-like potential is
adopted, whose detailed forms can be found in Ref. Zhou  _et al._ (2020). The
values of the related parameters are fixed by fitting the masses of the ground
states. The constituent quark masses used in this work, which are different
from those of the current quarks when calculating the QCD penguin diagrams,
have the values:

$\displaystyle
m_{u}=0.305~{}\mathrm{GeV},~{}m_{d}=0.311~{}\mathrm{GeV},~{}m_{s}=0.500~{}\mathrm{GeV},$
(30) $\displaystyle m_{c}=1.62~{}\mathrm{GeV},~{}m_{b}=4.96~{}\mathrm{GeV}.$

The PDG values of the CKM matrix elements are Zyla  _et al._ (2020):

$\displaystyle\left|V_{ud}\right|=0.97425,~{}\left|V_{us}\right|=0.2252,~{}\left|V_{ub}\right|=3.89\times
10^{-3},$ (31)
$\displaystyle\left|V_{cd}\right|=0.230,~{}\left|V_{cb}\right|=0.0406,~{}\left|V_{cs}\right|=0.97345.$

For the Wilson coefficients, we use the results in Ref. Sun  _et al._ (2008):

$\displaystyle C_{1}=-0.1902,\quad C_{2}=1.0849,\quad C_{3}=0.0148,\quad
C_{4}=-0.0362,\quad C_{5}=0.0088,$ (32) $\displaystyle
C_{6}=-0.0422,\quad\frac{C_{7}}{\alpha_{e}}=-0.0007,\quad\frac{C_{8}}{\alpha_{e}}=0.0565,\quad\frac{C_{9}}{\alpha_{e}}=-1.3039,\quad\frac{C_{10}}{\alpha_{e}}=0.2700.$

The masses and decay constants of mesons in the ground and excited states are
presented in Table 1. Here we have used the method in Refs. Cvetic  _et al._
(2004); Wang (2006, 2007) to calculate the decay constants except those of
$D^{\pm}(1S)$ and $D_{s}^{\pm}(1S)$ for which the experimental data are
adopted.

Table 1:  The masses and decay constants of mesons. particle |  Mass(MeV) Zyla  _et al._ (2020) |  $f_{P}$(MeV) | particle |  Mass(MeV) Zyla  _et al._ (2020) |  $f_{P}$(MeV)
---|---|---|---|---|---
$D^{\pm}(1S)$ | $1869.65\pm 0.05$ |  $203.2\pm 5.3\pm 1.8$ Ablikim  _et al._ (2014) | $D_{s1}^{\pm}(1P)$ | $2459.5\pm 0.6$ | 258
$D_{s}^{\pm}(1S)$ | $1968.34\pm 0.07$ |  $241.0\pm 16.3\pm 6.5$ Ablikim  _et al._ (2016) | $D^{\prime\pm}_{s1}(1P)$ | $2535.11\pm 0.06$ | 83.5
$\eta_{c}(1S)$ | $2983.9\pm 0.5$ | 420 | $D_{1}^{\pm}(1P)$ | $2423.2\pm 1.6$ | 283
$D^{*\pm}(1S)$ | $2010.26\pm 0.05$ | 419 | $D^{\prime\pm}_{1}(1P)$ | $2427\pm 26\pm 25$ | 70.2
$D_{s}^{*\pm}(1S)$ | $2112.2\pm 0.4$ | 469 | $D_{0}^{*\pm}(1P)$ | $2350.6\pm 5.9$ | 138
$J/\psi$ | $3096.900\pm 0.006$ | 560 | $D_{s0}^{*\pm}(1P)$ | $2317.8\pm 0.5$ | 109
$B_{c}(1S)$ | $6274.9\pm 0.8$ |  | $\chi_{c1}(1P)$ | $3510.67\pm 0.05$ | 239
particle | Masses (MeV) |  $f_{p}$(MeV) | particle | Masses (MeV) |  $f_{p}$(MeV)
$D^{\pm}(2S)$ |  2581 Godfrey and Moats (2016) | 178 | $D_{s1}^{\pm}(2P)$ |  3018 Godfrey and Moats (2016) | 250
$D_{s}^{\pm}(2S)$ |  2673 Godfrey and Moats (2016) | 204 | $D_{s1}^{\prime\pm}(2P)$ |  3038 Godfrey and Moats (2016) | 58.0
$\eta_{c}(2S)$ |  $3637.5\pm 1.1$ Zyla  _et al._ (2020) | 284 | $D_{1}^{\pm}(2P)$ |  2924 Godfrey and Moats (2016) | 217
$D^{*\pm}(2S)$ |  $2637\pm 2\pm 6$ Abreu  _et al._ (1998) | 325 | $D_{1}^{\prime\pm}(2P)$ |  2961 Godfrey and Moats (2016) | 53.6
$D_{s}^{*\pm}(2S)$ |  $2732\pm 4.3\pm 5.8$ Aaij  _et al._ (2016) | 372 | $D_{0}^{*\pm}(2P)$ |  2931 Godfrey and Moats (2016) | 79.0
$\psi(2S)$ |  $3686.10\pm 0.06$ Zyla  _et al._ (2020) | 434 | $D_{s0}^{*\pm}(2P)$ |  3005 Godfrey and Moats (2016) | 78.6
$\psi(1D)$ |  $3773\pm 0.4$ Zyla  _et al._ (2020) | 336 | $\chi_{c1}(2P)$ |  $3871.69\pm 0.17$ Zyla  _et al._ (2020) | 229
$\psi(3S)$ |  $4039\pm 1$ Zyla  _et al._ (2020) | 375 |  |  |

By using the numerical results of wave functions and the parameter values
mentioned above, we get the form factors of different channels. As an example,
we present the form factors $F_{0}^{B_{c}\rightarrow\eta_{c}}$ and
$A_{0}^{B_{c}\rightarrow J/\psi}$ at $q^{2}=(P-P_{f})^{2}=0$ in Table 2. Their
expressions can be found in our previous work Zhou  _et al._ (2020). For
comparison, we also present the results of other models, which are close to
ours.

Table 2:  The form factors $F_{0}^{B_{c}\rightarrow\eta_{c}}$ and $A_{0}^{B_{c}\rightarrow J/\psi}$ at $q^{2}=0$. | Ours |  pQCD Rui and Zou (2014) |  SDY Sun  _et al._ (2009) |  Kiselev Kiselev  _et al._ (2000) |  IKP Ivanov  _et al._ (2001) |  WSL Wang  _et al._ (2009) |  HZ Huang and Zuo (2007) |  DSV Dhir  _et al._ (2008) |  EFG Ebert  _et al._ (2003)
---|---|---|---|---|---|---|---|---|---
$F_{0}^{B_{c}\rightarrow\eta_{c}}$ | 0.577 | 0.72 | 0.66 | 0.66 | 0.79 | 0.61 | 0.87 | 0.58 | 0.47
$A_{0}^{B_{c}\rightarrow J/\psi}$ | 0.515 | 0.64 | 0.65 | 0.60 | 0.69 | 0.53 | 0.27 | 0.58 | 0.40

The branching ratios ($Br$) and CP asymmetry ($\mathcal{A}_{cp}$) of different
decay channels are presented in Table 3, Table 4, and Table 5. The
uncertainties come from varying the parameter values simultaneously by $\pm
5\%$ when solving the instantaneous BS equations. The values of $D_{1,2}$
defined in Eq. (19) are also given. Following Refs. Hui Feng  _et al._ (2011);
Dai and Du (1999), we also give an estimation of how many $B_{c}$ events are
needed to observe the CP violation effect. If it is observed at three standard
deviation ($3\sigma$) level, the $B_{c}^{\pm}$ events needed are
$\epsilon_{f}N\sim\frac{9}{Br\mathcal{A}_{cp}^{2}}$, where $\epsilon_{f}$ is
the detecting efficiency of the final state Hui Feng  _et al._ (2011). For the
cases when the radial excited state is the charmonium (Table 3 and 5), one can
see that the $\psi(2S)D_{s}^{*}$ channel has the largest branching ratio but a
small $\mathcal{A}_{cp}$; the $\psi(1D)D_{0}^{*}$ channel has the largest
$\mathcal{A}_{cp}$ but a small branching ratio; the $\epsilon_{f}N$ for
$\eta_{c}(2S)D(1S)$ and $\psi(2S)D^{*}(1S)$ channels are of the order of
$10^{8}$, which is possible to be observed by the current experiments. For the
cases when the radial excited state is the heavy-light meson (Table 4), there
are five channels whose $\epsilon_{f}N$ is of the order of $10^{7}\sim
10^{8}$: $B_{c}\rightarrow\eta_{c}(1S)D(2S)$,
$B_{c}\rightarrow\eta_{c}(1S)D_{0}^{*}(2P)$,
$B_{c}\rightarrow\eta_{c}(1S)D^{*}(2S)$,
$B_{c}\rightarrow\eta_{c}(1S)D_{s}(2S)$, $B_{c}\rightarrow J/\psi D^{*}(2S)$.
One also notices that the decay channels $J/\psi D_{s}^{*}(2S)$ and
$\eta_{c}D(2S)$ have the largest branching ratio and $\mathcal{A}_{cp}$,
respectively.

The CP asymmetry is related to the weak CP phase $\gamma$ by Eq. (19).
Experimentally, $\gamma$ is constrained by the nonleptonic decays of $B$
meson, the latest results of which in PDG2020 is
$(72.1^{+4.1}_{-4.5})^{\circ}$ Zyla  _et al._ (2020). In Fig. 2 we draw
$|\mathcal{A}_{cp}|$ as a function of $\gamma$. The eight decay channels,
whose $|\mathcal{A}_{cp}|$ are of the order of $10^{7}\sim 10^{8}$, have been
plotted. The maxima of the curves are achieved when $\gamma$ takes about
$100^{\circ}$. The gray band indicates the experimental upper and lower limits
of $\gamma$. So when we calculate $\mathcal{A}_{cp}$ of these channels,
$\gamma$ will bring errors to the results. We can see that for the
$B_{c}\rightarrow\eta_{c}(2S)D(1S)$ and $B_{c}\rightarrow\psi(2S)D(1S)$
channels, $\mathcal{A}_{cp}$ changes about $7\%$, while for the other
channels, the changes of $\mathcal{A}_{cp}$ are very small.

Next we study how the results are affected by the effective color number
$N_{c}^{eff}$ and the squared momentum $k^{2}$ carried by the virtual gluon.
According to Refs. Leitner  _et al._ (2002); Gerard and Hou (1989), $k^{2}$
varies in the ranges $[m_{b}^{2}/4,~{}m_{b}^{2}/2]$ or $[0,~{}m_{b}^{2}]$. So
here we will let $k^{2}$ take four different values: $0.35m_{b}^{2}$,
$0.5m_{b}^{2}$, $0.65m_{b}^{2}$, and $0.8m_{b}^{2}$. Besides, the
$N_{c}^{eff}$ will vary from 2 to 10, which accounts for the errors brought by
the non-factorization effect. The results are shown in Figs. 4 $\sim$ 9. Here
we only consider the decay channels with $\epsilon_{f}N\sim 10^{7}$ or
$10^{8}$. One can see that as $N_{c}^{eff}$ changes, $\mathcal{A}_{cp}$ and
$Br$ can change at most several times. The changes of $k^{2}$ have even little
effect on these quantities. For example, the branching ratio of
$\eta_{c}(2S)D_{0}^{*}(1S)$ channel (see Fig. 7) changes less than $50\%$ when
$k^{2}$ changes from $0.35m_{b}^{2}$ to $0.80m_{b}^{2}$.

At last, we study how the $P$-wave mixing angle affects the CP asymmetry. The
results are shown in Fig. 10 and Fig. 11. One can see that except $B_{c}\to
J/\psi D_{s1}^{(\prime)}(2P)$, all the other channels with a $1^{+(\prime)}$
final state have a critical angle around which $\mathcal{A}_{cp}$ changes
severely.

In conclusion, we have calculated the CP violation in two-body nonleptonic
decays of $B_{c}$, where excited states are included in the final states. Some
decay modes have large branching ratios, which is of the order of $10^{-3}$.
We studied in detail seven decay channels whose CP asymmetry could be detected
by the current experiments. Among these channels,
$B_{c}\rightarrow\eta_{c}(1S)D(2S)$,
$B_{c}\rightarrow\eta_{c}(1S)D_{0}^{*}(2P)$, and $B_{c}\rightarrow J/\psi
D^{*}(2S)$ are the most promising ones, for which, about $10^{7}$ $B_{c}$
events are needed, if the CP violation is observed at $3\sigma$ level.

Table 3:  The results for $B_{c}\rightarrow c\bar{c}(2S,2P)+\bar{c}q(1S,1P)$.  Channel | $D_{1}$ | $D_{2}$ | $Br(B_{c}^{+})$ | $Br(B_{c}^{-})$ |  $\mathcal{A}_{cp}$% | $\epsilon_{f}N$
---|---|---|---|---|---|---
$\eta_{c}(2S)D$ | 0.0734 | 0.133 | $3.64^{+0.54}_{-0.36}\times 10^{-6}$ | $3.19^{+0.54}_{-0.36}\times 10^{-6}$ | $6.53^{+0.09}_{-0.08}$ | $(5.19\sim 7.08)\times 10^{8}$
$\eta_{c}(2S)D_{0}^{*}$ | 0.0349 | 0.0723 | $6.22^{+0.21}_{-0.27}\times 10^{-6}$ | $5.84^{+0.21}_{-0.27}\times 10^{-6}$ | $3.17^{+0.11}_{-0.15}$ | $(1.33\sim 1.71)\times 10^{9}$
$\eta_{c}(2S)D^{*}$ | 0.0123 | 0.0235 | $2.92^{+0.18}_{-0.14}\times 10^{-5}$ | $2.85^{+0.18}_{-0.14}\times 10^{-5}$ | $1.13^{+0.07}_{-0.06}$ | $(2.04\sim 2.85)\times 10^{9}$
$\eta_{c}(2S)D_{1}$ | 0.00148 | 0.00299 | $8.58^{+0.01}_{-0.02}\times 10^{-9}$ | $8.56^{+0.01}_{-0.02}\times 10^{-9}$ | $0.138^{+0.001}_{-0.001}$ | $(5.42\sim 5.60)\times 10^{12}$
$\eta_{c}(2S)D_{1}^{\prime}$ | 0.0213 | 0.0412 | $9.39^{+0.05}_{-0.09}\times 10^{-7}$ | $9.04^{+0.05}_{-0.09}\times 10^{-7}$ | $1.96^{+0.02}_{-0.04}$ | $(2.39\sim 2.57)\times 10^{10}$
$\eta_{c}(2S)D_{s}$ | -0.00156 | -0.00316 | $4.46^{+0.01}_{-0.02}\times 10^{-4}$ | $4.47^{+0.01}_{-0.02}\times 10^{-4}$ | $-0.145^{+0.003}_{-0.005}$ | $(9.17\sim 10.33)\times 10^{9}$
$\eta_{c}(2S)D_{s0}^{*}$ | -0.0171 | -0.0228 | $2.49^{+0.08}_{-0.09}\times 10^{-6}$ | $2.57^{+0.08}_{-0.09}\times 10^{-6}$ | $-1.61^{+0.06}_{-0.05}$ | $(1.33\sim 1.43)\times 10^{10}$
$\eta_{c}(2S)D_{s}^{*}$ | -0.000634 | -0.00122 | $3.56^{+0.02}_{-0.02}\times 10^{-4}$ | $3.56^{+0.02}_{-0.02}\times 10^{-4}$ | $-0.0591^{+0.0003}_{-0.0003}$ | $(7.17\sim 7.32)\times 10^{10}$
$\eta_{c}(2S)D_{s1}$ | -0.000186 | -0.000375 | $1.76^{+0.07}_{-0.06}\times 10^{-5}$ | $1.76^{+0.07}_{-0.06}\times 10^{-5}$ | $-0.0173^{+0.0012}_{-0.0011}$ | $(1.44\sim 2.02)\times 10^{13}$
$\eta_{c}(2S)D_{s1}^{\prime}$ | -0.00135 | -0.00261 | $3.46^{+0.18}_{-0.07}\times 10^{-6}$ | $3.47^{+0.18}_{-0.07}\times 10^{-6}$ | $-0.126^{+0.004}_{-0.005}$ | $(1.55\sim 1.74)\times 10^{12}$
$\psi(2S)D$ | 0.00431 | 0.000567 | $1.56^{+0.01}_{-0.05}\times 10^{-6}$ | $1.55^{+0.01}_{-0.05}\times 10^{-6}$ | $0.402^{+0.003}_{-0.001}$ | $(3.31\sim 4.07)\times 10^{11}$
$\psi(2S)D_{0}^{*}$ | -0.00199 | -0.00712 | $2.52^{+0.08}_{-0.09}\times 10^{-6}$ | $2.53^{+0.08}_{-0.09}\times 10^{-6}$ | $-0.186^{+0.024}_{-0.034}$ | $(0.82\sim 1.12)\times 10^{12}$
$\psi(2S)D^{*}$ | 0.00981 | 0.0189 | $1.29^{+0.12}_{-0.23}\times 10^{-4}$ | $1.26^{+0.12}_{-0.23}\times 10^{-4}$ | $0.908^{+0.071}_{-0.058}$ | $(6.88\sim 10.44)\times 10^{8}$
$\psi(2S)D_{1}$ | 0.00177 | 0.00357 | $3.70^{+0.02}_{-0.01}\times 10^{-6}$ | $3.69^{+0.02}_{-0.01}\times 10^{-6}$ | $0.164^{+0.001}_{-0.001}$ | $(0.98\sim 1.02)\times 10^{10}$
$\psi(2S)D_{1}^{\prime}$ | 0.0204 | 0.0394 | $6.42^{+0.08}_{-0.04}\times 10^{-6}$ | $6.18^{+0.08}_{-0.04}\times 10^{-6}$ | $1.88^{+0.01}_{-0.02}$ | $(3.83\sim 4.03)\times 10^{9}$
$\psi(2S)D_{s}$ | -0.0000893 | -0.0000238 | $2.69^{+0.03}_{-0.03}\times 10^{-4}$ | $2.69^{+0.03}_{-0.03}\times 10^{-4}$ | $-0.00832^{+0.00020}_{-0.00020}$ | $(3.13\sim 8.37)\times 10^{12}$
$\psi(2S)D_{s0}^{*}$ | -0.00252 | -0.00682 | $2.82^{+0.08}_{-0.06}\times 10^{-7}$ | $2.83^{+0.08}_{-0.06}\times 10^{-7}$ | $-0.236^{+0.005}_{-0.004}$ | $(5.65\sim 5.80)\times 10^{12}$
$\psi(2S)D_{s}^{*}$ | -0.000515 | -0.00100 | $2.72^{+0.05}_{-0.07}\times 10^{-3}$ | $2.73^{+0.05}_{-0.07}\times 10^{-3}$ | $-0.0481^{+0.0007}_{-0.0005}$ | $(1.21\sim 1.60)\times 10^{10}$
$\psi(2S)D_{s1}$ | -0.000314 | -0.000631 | $1.15^{+0.13}_{-0.10}\times 10^{-5}$ | $1.15^{+0.13}_{-0.10}\times 10^{-5}$ | $-0.0293^{+0.0031}_{-0.0029}$ | $(6.69\sim 12.3)\times 10^{11}$
$\psi(2S)D_{s1}^{\prime}$ | -0.00133 | -0.00256 | $3.43^{+0.93}_{-0.60}\times 10^{-5}$ | $3.44^{+0.93}_{-0.60}\times 10^{-5}$ | $-0.124^{+0.031}_{-0.031}$ | $(1.41\sim 1.96)\times 10^{11}$
$\chi_{c1}(2P)D$ | -0.0356 | -0.0878 | $8.23^{+4.23}_{-2.38}\times 10^{-8}$ | $8.81^{+4.23}_{-2.38}\times 10^{-8}$ | $-3.43^{+1.32}_{-1.13}$ | $(3.13\sim 27.7)\times 10^{10}$
$\chi_{c1}(2P)D_{0}^{*}$ | -0.0164 | -0.0376 | $4.27^{+0.29}_{-0.80}\times 10^{-8}$ | $4.19^{+0.29}_{-0.80}\times 10^{-8}$ | $1.00^{+0.22}_{-0.21}$ | $(0.42\sim 13.4)\times 10^{12}$
$\chi_{c1}(2P)D^{*}$ | 0.0246 | 0.0428 | $2.49^{+0.12}_{-0.08}\times 10^{-5}$ | $2.45^{+0.12}_{-0.08}\times 10^{-5}$ | $0.781^{+0.042}_{-0.030}$ | $(5.16\sim 6.69)\times 10^{9}$
$\chi_{c1}(2P)D_{s}$ | -0.000734 | -0.00154 | $1.68^{+0.11}_{-0.06}\times 10^{-5}$ | $1.69^{+0.11}_{-0.06}\times 10^{-5}$ | $-0.0685^{+0.0122}_{-0.0189}$ | $(0.72\sim 1.58)\times 10^{12}$
$\chi_{c1}(2P)D_{s0}^{*}$ | 0.000632 | 0.00149 | $8.06^{+1.81}_{-0.38}\times 10^{-7}$ | $8.05^{+1.81}_{-0.38}\times 10^{-7}$ | $0.0589^{+0.0114}_{-0.0031}$ | $(1.84\sim 3.76)\times 10^{13}$
$\chi_{c1}(2P)D_{s}^{*}$ | -0.000381 | -0.000744 | $3.76^{+0.03}_{-0.05}\times 10^{-4}$ | $3.77^{+0.03}_{-0.05}\times 10^{-4}$ | $-0.0355^{+0.0021}_{-0.0011}$ | $(1.67\sim 2.05)\times 10^{11}$
Table 4:  The results for $B_{c}\rightarrow c\bar{c}(1S)+\bar{c}q(2S,2P)$ and $B_{c}\rightarrow c\bar{c}(1P)+\bar{c}q(2S)$. Channel | $D_{1}$ | $D_{2}$ | $Br(B_{c}^{+})$ | $Br(B_{c}^{-})$ |  $\mathcal{A}_{cp}$% | $\epsilon_{f}N$
---|---|---|---|---|---|---
$\eta_{c}D(2S)$ | 0.0990 | 0.184 | $4.01^{+0.06}_{-0.06}\times 10^{-5}$ | $3.37^{+0.06}_{-0.06}\times 10^{-5}$ | $8.65^{+0.13}_{-0.14}$ | $(3.11\sim 3.42)\times 10^{7}$
$\eta_{c}D_{0}^{*}(2P)$ | 0.0982 | 0.190 | $3.44^{+0.06}_{-0.04}\times 10^{-5}$ | $2.90^{+0.06}_{-0.04}\times 10^{-5}$ | $8.56^{+0.15}_{-0.10}$ | $(3.59\sim 3.93)\times 10^{7}$
$\eta_{c}D^{*}(2S)$ | 0.0136 | 0.0270 | $1.83^{+0.11}_{-0.34}\times 10^{-4}$ | $1.78^{+0.11}_{-0.34}\times 10^{-4}$ | $1.25^{+0.08}_{-0.32}$ | $(2.65\sim 7.10)\times 10^{8}$
$\eta_{c}D_{1}(2P)$ | 0.00636 | 0.0132 | $1.38^{+0.56}_{-0.09}\times 10^{-5}$ | $1.36^{+0.56}_{-0.09}\times 10^{-5}$ | $0.590^{+0.211}_{-0.051}$ | $(0.72\sim 2.42)\times 10^{10}$
$\eta_{c}D_{1}^{\prime}(2P)$ | 0.0182 | 0.0361 | $3.05^{+0.06}_{-0.04}\times 10^{-5}$ | $2.95^{+0.06}_{-0.04}\times 10^{-5}$ | $1.67^{+0.02}_{-0.03}$ | $(1.02\sim 1.13)\times 10^{9}$
$\eta_{c}D_{s}(2S)$ | -0.00337 | -0.00677 | $2.65^{+0.06}_{-0.06}\times 10^{-3}$ | $2.66^{+0.06}_{-0.06}\times 10^{-3}$ | $-0.315^{+0.004}_{-0.004}$ | $(3.40\sim 3.43)\times 10^{8}$
$\eta_{c}D_{s0}^{*}(2P)$ | -0.00841 | -0.0147 | $9.23^{+0.26}_{-0.36}\times 10^{-5}$ | $9.38^{+0.26}_{-0.36}\times 10^{-5}$ | $-0.788^{+0.032}_{-0.022}$ | $(1.53\sim 1.64)\times 10^{9}$
$\eta_{c}D_{s}^{*}(2S)$ | -0.000738 | -0.00147 | $2.31^{+0.25}_{-0.26}\times 10^{-3}$ | $2.31^{+0.25}_{-0.26}\times 10^{-3}$ | $-0.0688^{+0.0027}_{-0.0021}$ | $(6.87\sim 10.4)\times 10^{9}$
$\eta_{c}D_{s1}(2P)$ | -0.000704 | -0.00143 | $6.24^{+0.05}_{-0.05}\times 10^{-5}$ | $6.25^{+0.05}_{-0.05}\times 10^{-5}$ | $-0.0656^{+0.0042}_{-0.0052}$ | $(2.76\sim 4.15)\times 10^{11}$
$\eta_{c}D_{s1}^{\prime}(2P)$ | -0.000861 | -0.00174 | $3.61^{+0.52}_{-0.41}\times 10^{-4}$ | $3.62^{+0.52}_{-0.41}\times 10^{-4}$ | $-0.0803^{+0.0002}_{-0.0002}$ | $(3.82\sim 3.89)\times 10^{10}$
$J/\psi D(2S)$ | -0.0105 | -0.0324 | $2.43^{+0.05}_{-0.07}\times 10^{-5}$ | $2.48^{+0.05}_{-0.07}\times 10^{-5}$ | $-0.989^{+0.003}_{-0.002}$ | $(3.69\sim 3.84)\times 10^{9}$
$J/\psi D_{0}^{*}(2P)$ | -0.0163 | -0.0475 | $4.56^{+0.19}_{-0.26}\times 10^{-6}$ | $4.70^{+0.19}_{-0.26}\times 10^{-6}$ | $-1.55^{+0.09}_{-0.06}$ | $(7.95\sim 8.76)\times 10^{9}$
$J/\psi D^{*}(2S)$ | 0.0168 | 0.0332 | $4.93^{+1.37}_{-0.58}\times 10^{-4}$ | $4.78^{+1.37}_{-0.58}\times 10^{-4}$ | $1.55^{+0.34}_{-0.21}$ | $(0.40\sim 1.17)\times 10^{7}$
$J/\psi D_{1}(2P)$ | 0.00912 | 0.0185 | $5.45^{+0.27}_{-0.18}\times 10^{-5}$ | $5.36^{+0.27}_{-0.18}\times 10^{-5}$ | $0.845^{+0.028}_{-0.039}$ | $(2.08\sim 2.65)\times 10^{9}$
$J/\psi D_{1}^{\prime}(2P)$ | 0.0164 | 0.0326 | $3.07^{+0.17}_{-0.27}\times 10^{-4}$ | $2.98^{+0.17}_{-0.27}\times 10^{-4}$ | $1.51^{+0.15}_{-0.08}$ | $(9.01\sim 10.19)\times 10^{9}$
$J/\psi D_{s}(2S)$ | 0.000347 | 0.00109 | $2.54^{+0.02}_{-0.01}\times 10^{-3}$ | $2.53^{+0.02}_{-0.01}\times 10^{-3}$ | $0.0347^{+0.0002}_{-0.0001}$ | $(2.89\sim 2.97)\times 10^{10}$
$J/\psi D_{s0}^{*}(2P)$ | 0.00126 | 0.00372 | $2.62^{+0.04}_{-0.03}\times 10^{-5}$ | $2.61^{+0.04}_{-0.03}\times 10^{-5}$ | $0.118^{+0.003}_{-0.002}$ | $(2.31\sim 2.58)\times 10^{11}$
$J/\psi D_{s}^{*}(2S)$ | -0.000738 | -0.00147 | $1.75^{+0.13}_{-0.02}\times 10^{-2}$ | $1.75^{+0.13}_{-0.02}\times 10^{-2}$ | $-0.0688^{+0.0027}_{-0.0021}$ | $(0.94\sim 1.17)\times 10^{9}$
$J/\psi D_{s1}(2P)$ | -0.000309 | -0.000629 | $1.90^{+0.03}_{-0.02}\times 10^{-3}$ | $1.90^{+0.03}_{-0.02}\times 10^{-3}$ | $-0.0288^{+0.0022}_{-0.0042}$ | $(4.61\sim 7.90)\times 10^{10}$
$J/\psi D_{s1}^{\prime}(2P)$ | -0.000805 | -0.00163 | $6.32^{+0.09}_{-0.18}\times 10^{-3}$ | $6.33^{+0.09}_{-0.18}\times 10^{-3}$ | $-0.0751^{+0.0023}_{-0.0011}$ | $(2.34\sim 2.67)\times 10^{9}$
$\chi_{c1}D(2S)$ | 0.337 | 0.408 | $11.50^{+0.45}_{-0.38}\times 10^{-9}$ | $6.55^{+0.45}_{-0.38}\times 10^{-9}$ | $27.4^{+1.3}_{-1.2}$ | $(0.57\sim 4.44)\times 10^{10}$
$\chi_{c1}D^{*}(2S)$ | 0.0127 | 0.0253 | $2.43^{+0.43}_{-0.22}\times 10^{-5}$ | $2.37^{+0.43}_{-0.23}\times 10^{-5}$ | $1.17^{+0.19}_{-0.13}$ | $(1.71\sim 3.83)\times 10^{9}$
$\chi_{c1}D_{s}(2S)$ | -0.00225 | -0.00473 | $2.53^{+0.71}_{-0.92}\times 10^{-6}$ | $2.54^{+0.71}_{-0.92}\times 10^{-6}$ | $-0.210^{+0.112}_{-0.043}$ | $(2.67\sim 19.98)\times 10^{11}$
$\chi_{c1}D_{s}^{*}(2S)$ | -0.000986 | -0.00194 | $1.29^{+0.45}_{-0.28}\times 10^{-4}$ | $1.29^{+0.45}_{-0.28}\times 10^{-4}$ | $-0.0920^{+0.0221}_{-0.0201}$ | $(7.08\sim 10.58)\times 10^{10}$
Table 5:  The results for $B_{c}\rightarrow c\bar{c}(1D)+\bar{c}q(1S,1P)$ and $B_{c}\rightarrow c\bar{c}(3S)+\bar{c}q(1S)$. Channel | $D_{1}$ | $D_{2}$ | $Br(B_{c}^{+})$ | $Br(B_{c}^{-})$ |  $\mathcal{A}_{cp}$% | $\epsilon_{f}N$
---|---|---|---|---|---|---
$\psi(1D)D$ | 0.00160 | 0.000216 | $8.25^{+0.14}_{-0.32}\times 10^{-6}$ | $8.23^{+0.14}_{-0.32}\times 10^{-6}$ | $0.150^{+0.002}_{-0.005}$ | $(4.65\sim 5.40)\times 10^{11}$
$\psi(1D)D_{0}^{*}$ | 0.0890 | 0.236 | $9.21^{+0.05}_{-0.03}\times 10^{-10}$ | $7.90^{+0.05}_{-0.03}\times 10^{-10}$ | $7.64^{+0.04}_{-0.02}$ | $(1.77\sim 1.81)\times 10^{12}$
$\psi(1D)D^{*}$ | 0.00864 | 0.0159 | $9.47^{+0.24}_{-0.03}\times 10^{-6}$ | $9.32^{+0.24}_{-0.03}\times 10^{-6}$ | $0.801^{+0.020}_{-0.030}$ | $(1.38\sim 1.61)\times 10^{10}$
$\psi(1D)D_{s}$ | -0.000229 | -0.0000605 | $2.96^{+0.03}_{-0.04}\times 10^{-4}$ | $2.96^{+0.03}_{-0.04}\times 10^{-4}$ | $-0.0213^{+0.0002}_{-0.0002}$ | $(6.51\sim 6.98)\times 10^{12}$
$\psi(1D)D_{s0}^{*}$ | -0.00146 | -0.00418 | $2.54^{+0.05}_{-0.06}\times 10^{-7}$ | $2.55^{+0.05}_{-0.06}\times 10^{-7}$ | $-0.136^{+0.005}_{-0.004}$ | $(1.74\sim 2.07)\times 10^{12}$
$\psi(1D)D_{s}^{*}$ | -0.000291 | -0.000529 | $1.70^{+0.03}_{-0.02}\times 10^{-4}$ | $1.71^{+0.03}_{-0.02}\times 10^{-4}$ | $-0.0271^{+0.0005}_{-0.0003}$ | $(6.81\sim 7.44)\times 10^{11}$
$\psi(1D)D_{s1}$ | 0.000141 | 0.000290 | $2.48^{+0.17}_{-0.19}\times 10^{-4}$ | $2.48^{+0.17}_{-0.19}\times 10^{-4}$ | $-0.0131^{+0.0014}_{-0.0016}$ | $(1.61\sim 2.98)\times 10^{13}$
$\psi(3S)D$ | -0.00783 | -0.00114 | $3.62^{+0.04}_{-0.02}\times 10^{-8}$ | $3.67^{+0.04}_{-0.02}\times 10^{-8}$ | $-0.731^{+0.003}_{-0.008}$ | $(4.53\sim 4.75)\times 10^{12}$
$\psi(3S)D^{*}$ | 0.00559 | 0.0109 | $3.38^{+0.12}_{-0.09}\times 10^{-5}$ | $3.34^{+0.12}_{-0.09}\times 10^{-5}$ | $0.519^{+0.021}_{-0.018}$ | $(8.86\sim 10.97)\times 10^{9}$
$\psi(3S)D_{s}$ | -0.0000571 | -0.0000153 | $3.76^{+0.49}_{-0.66}\times 10^{-5}$ | $3.76^{+0.49}_{-0.66}\times 10^{-5}$ | $-0.00533^{+0.00020}_{-0.00020}$ | $(7.43\sim 10.116)\times 10^{13}$
$\psi(3S)D_{s}^{*}$ | -0.000277 | -0.000545 | $5.89^{+0.21}_{-0.87}\times 10^{-4}$ | $5.90^{+0.20}_{-0.38}\times 10^{-4}$ | $-0.0258^{+0.0012}_{-0.0023}$ | $(1.51\sim 5.63)\times 10^{11}$
Figure 2:  The dependence of $|\mathcal{A}_{CP}|$ on the weak phase $\gamma$.
The gray band is the constrained range of $\gamma$ measured in $B$ decay,
which is about $67.6^{\circ}\sim 76.2^{\circ}$ Zyla  _et al._ (2020).

Figure 3:  $\mathcal{A}_{cp}$ and averaged branching ratio of
$B_{c}\rightarrow\psi(1S)D(2S)$ change wiht $k^{2}$ and $N_{c}^{eff}$.

Figure 4:  $\mathcal{A}_{cp}$ and averaged branching ratio of
$B_{c}\rightarrow\eta_{c}(2S)D(1S)$ change with $k^{2}$ and $N_{c}^{eff}$.

Figure 5:  $\mathcal{A}_{cp}$ and averaged branching ratio of
$B_{c}\rightarrow\psi(2S)D^{*}(1S)$ change with $k^{2}$ and $N_{c}^{eff}$.

Figure 6:  $\mathcal{A}_{cp}$ and averaged branching ratio of
$B_{c}\rightarrow\eta_{c}(1S)D(2S)$ change with $k^{2}$ and $N_{c}^{eff}$.

Figure 7:  $\mathcal{A}_{cp}$ and averaged branching ratio of
$B_{c}\rightarrow\eta_{c}(2S)D_{0}^{*}(1S)$ change with $k^{2}$ and
$N_{c}^{eff}$.

Figure 8:  $\mathcal{A}_{cp}$ and averaged branching ratio of
$B_{c}\rightarrow\eta_{c}(1S)D^{*}(2S)$ change with $k^{2}$ and $N_{c}^{eff}$.

Figure 9:  $\mathcal{A}_{cp}$ and averaged branching ratio of
$B_{c}\rightarrow J/\psi D^{*}(2S)$ change with $k^{2}$ and $N_{c}^{eff}$.

Figure 10: $A_{cp}$ varies along with the mixing angle for the $B_{c}\to
c\bar{c}(2S)+D_{(s)1}(1P)$.

Figure 11: $A_{cp}$ varies along with the mixing angle for the $B_{c}\to
c\bar{c}(1S)+D_{(s)1}(2P)$.

##  V Acknowledgments

This work was supported in part by the National Natural Science Foundation of
China (NSFC) under Grant No. 12075073, 12047569, and 11865001. We also thank
the HEPC Studio at Physics School of Harbin Institute of Technology for access
to computing resources through<EMAIL_ADDRESS>
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