# Two-Body Strong Decays of the $2P$ and $3P$ Charmonium states

Zhi-Hui<EMAIL_ADDRESS>Guo-Li Wang[4],[5]222
<EMAIL_ADDRESS>
1 Key Laboratory of Physics and Photoelectric Information Functional
Materials, North Minzu University, Yinchuan, 750021, China,
2School of Electrical and Information Engineering, North Minzu University,
Yinchuan, 750021, China
3School of Physics and Center of High Energy Physics, Peking University,
Beijing 100871, China
4Department of Physics, Hebei University, Baoding, 071002, China,
5Hebei Key Laboratory of High-precision Computation and Application of Quantum
Field Theory, Baoding, 071002, China

###### Abstract

Two-body open charm strong decays of the $2P$ and $3P$ charmonium states are
studied by the Bethe-Salpeter(BS) method combined with the ${}^{3}P_{0}$
model. The wave functions and mass spectra of the $2P$ and $3P$ charmonium
states are obtained by solving the BS equation with the relativistic
correction. The strong decay widths and relative ratios of the $2P$ and $3P$
charmonium states are calculated. Comparing our results with the experimental
data, we obtain some interesting results. Considering the $X^{*}(3860)$ as the
$\chi_{c0}(2P)$, the total strong decay width is smaller than the experimental
data. But the strong decay width depends on the parameter $\gamma$ in the
${}^{3}P_{0}$ model, and the mass and width of the $X^{*}(3860)$ have large
errors, we cannot rule out the possibility that the $X^{*}(3860)$ is the
$\chi_{c0}(2P)$. The $X(4160)$ is a good candidate for the $\chi_{c0}(3P)$,
not only the strong decay width of the $\chi_{c0}(3P)$ is same as the
experimental data, but the relative ratios $\frac{\Gamma(\chi_{c0}(3P)\to
D\bar{D})}{\Gamma(\chi_{c0}(3P)\to D^{*}\bar{D}^{*})}\approx 0.0019<0.09$, and
$\frac{\Gamma({\chi_{c0}(3P)\to D\bar{D}^{*}})}{\Gamma({\chi_{c0}(3P)\to
D^{*}\bar{D}^{*}})}=0<0.22$ are consistent with the experimental results of
the $X(4160)$. Taking the $X(4274)$ as the $\chi_{c1}(3P)$, the strong decay
width is consistent with the experimental data, so the $X(4274)$ is a good
candidate for the $\chi_{c1}(3P)$. Assigning the $X(4350)$ as the
$\chi_{c2}(3P)$, the corresponding strong decay width is slightly larger than
the experimental data. To identify if the $X(4350)$ is $\chi_{c2}(3P)$, many
more investigations are needed. All of the strong decay widths and relative
ratios of the $2P$ and $3P$ charmonium states can provide the useful
information to discover and confirm these particles in the future.

##  I Introduction

Since the Belle Collaboration reported the first observation of the $X(3872)$
3872 , many more charmonium-like states have been observed experimentally.
Belle observed the $X(4160)$ from the process $e^{+}e^{-}\to J/\psi
D^{*}\bar{D}^{*}$, which has the mass and width $M=(4156^{+25}_{-20}\pm 15)$
MeV and $\Gamma=(139^{+111}_{-61}\pm 21)$ MeV, respectively 39404160 . They
also gave the upper limits of relative ratios:
$\frac{B_{D\bar{D}}(X(4160))}{B_{D^{*}\bar{D}^{*}}(X(4160))}<0.09,$
$\frac{B_{D^{*}\bar{D}}(X(4160))}{B_{D^{*}\bar{D}^{*}}(X(4160))}<0.22$. The
$X(4140)$ was first observed by the CDF Collaboration in the exclusive decay
$B\to J/\psi\phi K$ 4140 , then another charmonium-like states the $X(4274)$
also was observed in the same decay channel 4274 . These two charmonium-like
states also were observed by LHCb Collaboration lhcb1 ; lhcb2 , the mass and
natural width of $X(4140)$ and $X(4274)$ were $M=(4146.8\pm 2.4)$ MeV,
$\Gamma=(22^{+8}_{-7})$ MeV, and $M=(4274^{+8}_{-6})$ MeV, $\Gamma=(49\pm 12)$
MeV PDG , respectively. In 2010, BABAR observed the $Z(3930)$ in the
$\gamma\gamma$ production of $D\bar{D}$ system Z3930-BABAR . Now the Particle
Data Group(PDG) gives the mass and width of the $Z(3930)$ as $M=(3927.2\pm
2.6)$ MeV and $\Gamma=(24\pm 6)$ MeV PDG . And the properties of $Z(3930)$ are
consistent with the expectations for the $\chi_{c2}(2P)$ state chic2 ; slzhu1
. Belle also explored a charmonium-like state $X(4350)$ in the process
$J/\psi\phi$ in 2010, the extracted mass and width were
$(4350.6^{+4.6}_{-5.1})$ MeV and $(13^{+18}_{-9}\pm 4)$ MeV 4350 . The
$X^{*}(3860)$ was observed in the process $e^{+}e^{-}\to J/\psi D\bar{D}$ by
Belle in 2017, the corresponding mass and width are
$M=(3862^{+26}_{-32}$${}^{+40}_{-13})$ MeV and
$\Gamma=(201^{+154}_{-67}$${}^{+88}_{-82})$ MeV 3860 , respectively.

The properties of these charmonium-like states have inspired great interest in
both theoretical and experimental research fields of hadronic physics. Many
theoretical approaches have studied the properties of these charmonium-like
states th1 ; th2 ; th201 ; th3 ; th4 ; th7 ; liux1 ; th9 ; th10 ; th12 ; th13
; zhaoqiang ; th6 ; Godfrey ; liuxiang2021 ; Swanson . The Ref. th201
investigated that the $X(4140)$ and $X(4274)$ can be both interpreted as the
$S$-wave $cs\bar{c}\bar{s}$ tetraquark states of $J^{P}=1^{+}$. The Ref. th3
computed the open-charm strong decay widths of the $\chi_{c}(3P)$ states and
their radiative transitions, and they suggested the $X(4274)$ could be
interpreted as the $\chi_{c1}(3P)$ state. Taking the $Z(3930)$ as
$\chi_{c2}^{\prime}(2P)$, the Ref. liux1  investigated the decay $Z(3930)$
into $J/\psi\omega$. Considering the $X(4350)$ as the
$\chi_{c2}^{\prime\prime}$, the Ref. th13  analyzed the mass and calculated
the open charm strong decay of the $X(4350)$, which were consistent with the
existing experimental data. The results of Ref. th6  preferred the
$J^{PC}=0^{++}$ assignment for the $X^{*}(3860)$ over the $2^{++}$ assignment,
which was also in agreement with the experiment. Calculating the observable
quantity (such as the spectrum or the strong decay width) of these charmonium-
like states, then comparing with the experimental data, may help us to better
understand the quark structure of the charmonium-like states.

In addition to the mass spectrum and strong decay width, the electromagnetic
decay also can help us to determine the structure these charmonium-like
states. According to E1 transition widths for the $\chi_{c1}(2P)\to\gamma
J/\psi$ and $\chi_{c1}(2P)\to\gamma\psi(2S)$ and other results, the Ref. th8
argued that the $X(3872)$ may be a $\chi_{c1}(2P)$ dominated charmonium state
with some admixture of the $D^{0}\bar{D}^{*0}$ component. The Ref.
electromagnetic1  calculated the one- and two-photon decay widths of
$Y(3940)$, $Z(3930)$, $X(3915)$ and $X(4160)$ mesons. Considering $X(4660)$,
$X(3872)$, $X(3900)$, $X(3915)$ and $X(4274)$ as $5^{3}S_{1}$, $2^{3}P_{1}$,
$2^{1}P_{1}$, $2^{3}P_{0}$ and $3^{3}P_{1}$, respectively, the Ref.
electromagnetic2  studied the E1 and M1 transition width, and annihilation
decays of these charmonium states. But the electromagnetic decay widths of
these charmonium states are about the order of keV, which are smaller than the
results of Okubo-Zweig-Iizuka (OZI)-allowed strong decay. These
electromagnetic decays can only be detected experimentally when large amounts
of data are available in the future. For now, the strong decay widths and the
relative ratios of these charmonium states are good ways to determine their
properties.

In this paper we will focus on the strong decay widths of the $2P$ and $3P$
charmonium states. The relativistic correction of the $2P$ and $3P$ charmonium
states are larger than that of the corresponding $1P$ states, therefore, we
need a relativistic model. The BS method is a relativistic framework that
describes the bound state with definite quantum number, the corresponding
relativistic form of wave functions are the solutions of the full Salpeter
equations. Using the BS method, we have discussed the properties of some
radial excited states in previous work, such as the semileptonic and
nonleptonic $B_{c}$ decays to the $Z(3930)$ and $X(4160)$ as the
$\chi_{c2}(2P)$ and $\chi_{c2}(3P)$ bcweak1 , the strong decays of the
$X(3940)$ and $X(4160)$ as radial high excited states the $\eta_{c}(3S)$ and
$\eta_{c}(4S)$ 4S4160 , the radiative E1 decay of the $X(3872)$ thwang1 ;
thwang2 , two-body strong decay of the $Z(3930)$ which was the $\chi_{c2}(2P)$
state thwang . All the theoretical results are consistent with the
experimental data or other theoretical results. So the BS method is a good way
to describe the properties and decays of the radially higher excited states.
In this paper, we will study the strong decays of the $2P$ and $3P$ charmonium
states by the BS method with the ${}^{3}P_{0}$ model.

For the $2P$ and $3P$ charmonium states, the dominant decay is the Okubo-
Zweig-Iizuka (OZI)-allowed two-body open charm strong decay. We will adopt the
${}^{3}P_{0}$ model to calculate the two-body open charm strong decay. The
${}^{3}P_{0}$ model was used to calculate the decay rates of the meson
resonances in Ref. 3p01 , which assumed that the $q\bar{q}$ pair is produced
from vacuum with quantum number $J^{PC}=0^{++}(^{3}P_{0})$, was applied to
calculate the strong decay of heavy-light mesons th13 ; 3p0hl1  and heavy
quarkonia Godfrey ; 3p05 . We also studied the strong decays of some heavy
quarkonia by the ${}^{3}P_{0}$ model combine with the BS method in Refs.
4S4160 ; thwang ; fu , the results were in accordance with the experimental
data or other theoretical results. So we take the same model to calculate the
two-body open charm decay of the $2P$ and $3P$ charmonium states.

The paper is organized as follows. We give the formulation of two-body strong
decay of charmonium state in Section II; In Sec. III, we show the numerical
results and discussions; We give the corresponding conclusions in Sec. IV.
Finally, we present the instantaneous BS equation and the relativistic wave
functions of $P$-wave charmonium states in the Appendix.

##  II two-Body Strong Decay of charmonium state

To calculate the two-body open charm strong decays of the $2P$ and $3P$
charmonium states by the relativistic BS method, we extend the ${}^{3}P_{0}$
model to relativistic form: $H=-ig\int d^{4}x\bar{\psi}\psi$ thwang ; fu .
Here $\psi$ is the dirac quark field, $g=2\gamma m_{q}$, $m_{q}$ is the quark
mass of the light quark-pairs, $\gamma$ is a dimensionless constant that
describes the pair-production strength. In this paper, we take $\gamma=0.35$,
which is the best-fit value for the usual ${}^{3}P_{0}$ model Swanson .
Combining the ${}^{3}P_{0}$ model with the BS wave functions of the initial
and final mesons, the corresponding amplitude in Fig. 1 can be written as

$\mathcal{M}=\langle BC|H|A\rangle=-ig\int\frac{{\rm d}^{4}q}{(2\pi)^{4}}{\rm
Tr}[\chi_{P}(q)S_{2}^{-1}(p_{2})\bar{\chi}_{P_{f2}}(q_{f2})\bar{\chi}_{P_{f1}}(q_{f1})S_{1}^{-1}(p_{1})]$
(1)

Figure 1: The Feynman diagram of two-body open charm strong decay.

Integrating out the momentum $q_{0}$ with instantaneous approximation, and
neglecting all the negative energy contributions which have very small
influence on the amplitude fu , then the leading order amplitude Eq.(1) is the
overlap integration of the positive BS wave functions for the initial and
final states,

$\mathcal{M}=\langle BC|H|A\rangle=g\int\frac{{\rm
d}^{3}\vec{q}}{(2\pi)^{3}}{\rm
Tr}\left[\frac{\not\\!{P}}{M}\varphi_{p}^{++}(\vec{q})\frac{\not\\!{P}}{M}\bar{\varphi}_{p_{f2}}^{++}(\vec{q}_{f2})\bar{\varphi}_{p_{f1}}^{++}(\vec{q}_{f1})\right]\left(1-\frac{M-\omega_{1}-\omega_{2}}{2\omega_{12}}\right),$
(2)

where $\varphi^{++}_{P}(\vec{q})$, $\varphi^{++}_{P_{f1}}(\vec{q}_{f1})$ and
$\varphi^{++}_{P_{f2}}(\vec{q}_{f2})$ are the positive BS wave functions of
the initial meson $A$, finial meson $B$ and $C$, respectively.
$\bar{\varphi}=\gamma^{0}\varphi^{\dagger}\gamma^{0}$. $\vec{q}$,
$\vec{q}_{f1}$, $\vec{q}_{f2}$ are the three dimensions relative momentum
between the quark and anti-quark of the initial meson $A$, finial meson $B$
and $C$, respectively.
$\vec{q}_{f1}=\vec{q}-\frac{m_{c}}{m_{c}+m_{u,d,s}}\vec{P}_{f1}$,
$\vec{q}_{f2}=\vec{q}+\frac{m_{c}}{m_{c}+m_{u,d,s}}\vec{P}_{f2}$.
$\vec{P}_{f1}$ and $\vec{P}_{f2}$ are the three momentum of finial mesons $B$
and $C$. $w_{12}=\sqrt{m_{u,d,s}^{2}+\vec{q}^{2}_{f1}}$.

Finally, the two-body open charm strong decay width of the $2P$ and $3P$
charmonium states can be expressed as

$\displaystyle\Gamma=\frac{|\vec{P}_{f1}|}{8\pi
M^{2}(2J+1)}\sum_{\lambda}|\mathcal{M}|^{2},$ (3)

where
$|\vec{P}_{f1}|=\sqrt{[M^{2}-(M_{f1}-M_{f2})^{2}][M^{2}-(M_{f1}+M_{f2})^{2}]}/(2M)$
which is the three momentum of the final mesons.

##  III Numerical results and discussions

In order to fix the parameters in Cornell potential in Eq.(11) and masses of
quarks, we take $a=e=2.7183,\lambda=0.21$ GeV2, ${\Lambda}_{QCD}=0.27$ GeV,
$\alpha=0.06$ GeV, $m_{c}=1.62$ GeV, $m_{u}=0.305$ GeV, $m_{d}=0.311$ GeV,
$m_{s}=0.500$ GeV, $etc$ w1 ; mass1 , which give the best to fit the mass
spectra of the ground charmonium states and other heavy meson states. The
corresponding mass spectra of the $P$-wave charmonium states are shown in
Table 1 which are obtained by solving the coupled Salpeter equations Eq.(10).

Table 1: The Mass spectra of $P$-wave charmonia (unit in MeV) mass1 . ‘Ex.’ means the experimental data from PDG PDG . $(n^{2S+1}L_{J})J^{PC}$ | Our results | screened potential modelth8  | GI(NR) modelGodfrey  | Ex
---|---|---|---|---
$(1^{3}P_{0})0^{++}$ | 3414.7(input) | 3433 | 3445(3424) | $3414.71\pm 0.30$
$(2^{3}P_{0})0^{++}$ | 3836.8 | 3842 | 3916(3852) | –
$(3^{3}P_{0})0^{++}$ | 4140.1 | 4131 | 4292(4202) | –
$(1^{3}P_{1})1^{++}$ | 3510.3(input) | 3510 | 3510(3505) | $3510.67\pm 0.05$
$(2^{3}P_{1})1^{++}$ | 3928.7 | 3901 | 3953(3925) | –
$(3^{3}P_{1})1^{++}$ | 4228.8 | 4178 | 4317(4271) | –
$(1^{3}P_{2})2^{++}$ | 3556.1(input) | 3554 | 3550(3556) | $3556.17\pm 0.07$
$(2^{3}P_{2})2^{++}$ | 3972.4 | 3937 | 3979(3972) | –
$(3^{3}P_{2})2^{++}$ | 4271.0 | 4208 | 4337(4317) | –
$(1^{1}P_{1})1^{+-}$ | 3526.0(input) | 3519 | 3517(3516) | $3525.38\pm 0.11$
$(2^{1}P_{1})1^{+-}$ | 3943.0 | 3908 | 3956(3934) | –
$(3^{1}P_{1})1^{+-}$ | 4242.4 | 4184 | 4318(4279) |

###  III.1 $\chi_{c0}(2P)$ and $\chi_{c0}(3P)$

Table 2: The strong decay type and decay widths of the $\chi_{c0}(2P)$ and $\chi_{c0}(3P)$ (unit in MeV). The results in the parentheses are calculated with $M_{\chi_{c0}(2P)}=3862.0$ MeV and $M_{\chi_{c0}(3P)}=4156.0$ MeV. Meson | State | Mode | Our Result | LP(SP)zhaoqiang  | Godfrey  | liuxiang2021  | Swanson
---|---|---|---|---|---|---|---
$\chi_{c0}$ | $2^{3}P_{0}$ | $D\bar{D}$ | 21.0(16.4) | 22(28) | 30 | – | 23
| $3^{3}P_{0}$ | $D\bar{D}$ | 0.17(0.13) | 0.04(0.08) | 0.5 | 2 | –
|  | $D^{*}\bar{D}^{*}$ | 79.0(69.4) | 21(30) | 43 | 67 | –
|  | $D_{s}D_{s}$ | 1.7(1.8) | 8.9(9) | 6.8 | 3 | –
|  | $D_{s}^{*}D_{s}^{*}$ | – | 2.7(–) | – | – | –
|  | total | 81(71) | 33(39) | 51 | 72 | –

First, we study the higher charmonium states with $J^{PC}=0^{++}$,
$\chi_{c0}(2P)$ and $\chi_{c0}(3P)$, and give the two-body open charm strong
decay results in Table 2. The mass $M_{\chi_{c0}(2P)}=3836.8$ MeV is close to
the result of the screened potential model (3842 MeV) th8  and the
nonrelativistic potential model (3852 MeV) Godfrey . Limited by phase space ,
there is one decay mode $0^{++}\to 0^{-}0^{-}$ for the $\chi_{c0}(2P)$, the
corresponding decay channel only includes $D\bar{D}$ within the kinematic
ranges. And we also get the total strong decay widths of the $\chi_{c0}(2P)$:
$\Gamma=21.0$ MeV, which is in accordance with the result of the linear
potential quark model (22 MeV) zhaoqiang  and the usual ${}^{3}P_{0}$ model
(23 MeV) Swanson .

Belle reported a charmonium-like state $X^{*}(3860)$, and they claimed the
$X^{*}(3860)$ seems to be a candidate of the $\chi_{c0}(2P)$ state 3860 . The
Ref. zhaoqiang  studied the strong decays of the $X^{*}(3860)$ as
$\chi_{c0}(2P)$ by LP and SP models, and they gave a similar value
$\Gamma\approx 22\sim 28$ MeV. The analysis of Ref. th5  showed that the
$X^{*}(3860)$ was an indication of the $\chi_{c0}(2P)$ state. Assuming the
$X^{*}(3860)$ as the $\chi_{c0}(2P)$, the Ref. x3860zgwang  calculated the
strong decay of the $\chi_{c0}(2P)$, the total decay of the $X^{*}(3860)$
state ranged from 110 to 180 MeV with $R=2.3\sim 2.5$ GeV-1, the corresponding
decay mode and total decay width were consistent with the experimental data.
Taking the $X^{*}(3860)$ as the $\chi_{c0}(2P)$, we get the strong decay width
$\Gamma=16.4$ MeV, which is smaller than the center value of the
$X^{*}(3860)$: $(201^{+154}_{-67}$${}^{+88}_{-82})$ MeV. But the strong decay
width is related to the parameter $\gamma$ in the ${}^{3}P_{0}$ model, the
result increases with the parameter $\gamma$. In addition, considering the
large uncertainties of the mass and decay width for the $X^{*}(3860)$, we
can’t exclude that the $X^{*}(3860)$ is $\chi_{c0}(2P)$. And more
investigations are needed to confirm the property of the $X^{*}(3860)$ in the
future.

By solving the Eq. (10), we get the mass of $\chi_{c0}(3P)$ as:
$M_{\chi_{c0}(3P)}=4140.1$ MeV, which is close to the result of the screened
potential model (4131 MeV) th8 . The dominant strong decay of the is OZI-
allowed two-body open charm strong decay. And there are two decay types:
$0^{++}\to 0^{-}0^{-}$ and $0^{++}\to 1^{-}1^{-}$, while decay mode $0^{++}\to
0^{-}1^{-}$ is forbidden. Therefore, the final mesons include $D\bar{D}$,
$D_{s}D_{s}$ and $D^{*}\bar{D}^{*}$ within the kinematic ranges, and there is
no $D\bar{D}^{*}$. The decay channel $\chi_{c0}(3P)\to D\bar{D}$ has the
largest phase space, but due to the node structure of $\chi_{c0}(3P)$’s wave
functions, the integrand (which consists of the overlapped wave functions)
oscillates accordingly in the amplitude. The positive contribution of the
integrand almost cancels the negative contribution in $\chi_{c0}(3P)\to
D\bar{D}$, leading to a smallest value of $\chi_{c0}(3P)\to D\bar{D}$ in Table
2. So the dominant contribution comes from $\chi_{c0}(3P)\to
D^{*}\bar{D}^{*}$, which is consistent with the result of the screened
potential model th8 . Then we calculate all of the two-body open charm strong
decays, the total strong decay width $\Gamma_{\chi_{c0}(3P)}=81$ MeV is close
to the result of the unquenched quark model (71 MeV) liuxiang2021 .

The $X(4160)$ was observed by Belle from the process $e^{+}e^{-}\to J/\psi
D^{*}\bar{D}^{*}$ 39404160 . The Ref. th8  and zhao1  discussed possible
interpretations for the $X(4160)$ based on the NRQCD calculations and the
potential model, two likely assignments for the $X(4160)$ were $\chi_{c0}(3P)$
and $\eta_{c}(4S)$. The Ref. x41601  calculated the strong decays of the
$\eta_{c}(nS)$, they found that the explanation of the $X(3940)$ as the
$\eta_{c}(3S)$ is possible and the assignment of the $X(4160)$ as the
$\eta_{c}(4S)$ can not be excluded. The Ref. 416011  calculated the strong
decay of the $X(4160)$ which was assumed as the $\chi_{c0}(3P)$,
$\chi_{c1}(3P)$, $\eta_{c2}(2D)$ or $\eta_{c}(4S)$ by the ${}^{3}P_{0}$ model,
they thought that the excited charmonium state $\eta_{c}(4S)$ cannot be ruled
out as an assignment for the $X(4160)$. Considering the $X(4160)$ as the
$\eta_{c}(4S)$ state, we also calculated the strong decay of the
$\eta_{c}(4S)$ in Ref. 4S4160 , the ratio of the decay width
$\frac{\Gamma(D\bar{D}^{*})}{\Gamma(D^{*}\bar{D}^{*})}$ of $\eta_{c}(4S)$ was
larger than the experimental data of the $X(4160)$, thus, the $X(4160)$ was
not the candidate of the $\eta_{c}(4S)$. In this work, the mass of
$\chi_{c0}(3P)$ is close to the $X(4160)$, assigning the $X(4160)$ as the
$\chi_{c0}(3P)$, the total strong decay width $\Gamma_{\chi_{c0}(3P)}=71$ MeV
is rough consistent with the result of experimental results for $X(4160)$:
$(139^{+111}_{-61}\pm 21)$ MeV. Then we calculate relative ratios
$\frac{\Gamma(\chi_{c0}(3P)\to D\bar{D})}{\Gamma(\chi_{c0}(3P)\to
D^{*}\bar{D}^{*})}\approx 0.0019<0.09$, and $\frac{\Gamma({\chi_{c0}(3P)\to
D\bar{D}^{*}})}{\Gamma({\chi_{c0}(3P)\to D^{*}\bar{D}^{*}})}=0<0.22$, both of
which agree with the experimental results of $X(4160)$ by Belle collaboration
39404160 . Therefore, $X(4160)$ is a good candidate for $\chi_{c0}(3P)$.

###  III.2 $\chi_{c1}(2P)$ and $\chi_{c1}(3P)$

Using the BS method, we get the masses and wave functions of the
$\chi_{c1}(2P)$ and $\chi_{c1}(3P)$. The $M_{\chi_{c1}(2P)}=3928.7$ MeV is
close to the result of the nonrelativistic potential model (3925 MeV) Godfrey
. The two-body open charm strong decay results have been shown in Table 3. For
the $\chi_{c1}(2P)$, there is only one decay mode $1^{++}\to 0^{-}1^{-}$, so
the final state include $DD^{*}$ state. The corresponding decay width
$\Gamma_{\chi_{c1}(2P)}=103$ MeV is the same as the results of linear
potential quark model (102 MeV) zhaoqiang , but smaller than the results of
other methods.

The mass $M_{\chi_{c1}(3P)}=4228.8$ MeV is larger than the result of the
screened potential model (4178 MeV) th8 , but smaller the than results of the
relativized Godfrey-Isgur model (4317 MeV) and nonrelativistic potential
model(4271) MeV Godfrey . The two-body open charm strong decay of
$\chi_{c1}(3P)$ have three decay modes: $1^{++}\to
0^{-}1^{-},\;1^{-}1^{-},\;0^{-}0^{+}$ and include five final states: $DD^{*}$,
$D^{*}\bar{D}^{*}$, $D_{s}D_{s}^{*}$, $D_{s}^{*}D_{s}^{*}$, $DD_{0}$. The
dominant strong decay channels are $DD^{*}$, $D^{*}\bar{D}^{*}$ and
$D_{s}D_{s}^{*}$. The ratios between different partial width are independent
of the strength parameter $\gamma$, they are $\frac{\Gamma(\chi_{c1}(3P)\to
D^{*}\bar{D}^{*})}{\Gamma(\chi_{c1}(3P)\to DD^{*})}\approx 0.63$ and
$\frac{\Gamma(\chi_{c1}(3P)\to D_{s}D_{s}^{*})}{\Gamma(\chi_{c1}(3P)\to
DD^{*})}\approx 0.42$, which can be explored in the future experiment. The
total strong decay width $\Gamma_{\chi_{c1}(3P)}=29.7$ MeV, which is similar
to the result of the linear potential quark model (23 MeV) and the relativized
Godfrey-Isgur model (39 MeV) Godfrey , but smaller than the result of the
unquenched quark model (48 MeV) liuxiang2021 .

Table 3: The strong decay type and decay widths of the $\chi_{c1}(2P)$ and $\chi_{c1}(3P)$ (unit in MeV). The results in the parentheses are calculated with $M_{\chi_{c1}(3P)}=4274.0$ MeV. Meson | State | Mode | Our Result | th3  | LP(SP)zhaoqiang  | Godfrey  | liuxiang2021  | Swanson
---|---|---|---|---|---|---|---|---
$\chi_{c1}$ | $2^{3}P_{1}$ | $DD^{*}$ | 103 | – | 102(127) | 165 | – | 127
| $3^{3}P_{1}$ | $DD^{*}$ | 14.3(24.9) | 6.6 | 7.1(5.3) | 6.8 | 20 | –
|  | $D^{*}\bar{D}^{*}$ | 9.0(14.7) | 28 | 0.2(1.1) | 19 | 26 | –
|  | $D_{s}D_{s}^{*}$ | 6.0(10.2) | 6.3 | 11(8.0) | 9.7 | – | –
|  | $D_{s}^{*}D_{s}^{*}$ | 0.4(2.9) | 2.5 | 5.5(–) | 2.7 | 2 | –
|  | $DD_{0}$ | 0.01(0.02) | 0.2 | 0.001(–) | 0.1 | – | –
|  | total | 29.7(52.7) | 43.6 | 23(14) | 39 | 48 | –

Some charmonium-like states with $J^{PC}=1^{++}$: $X(3872)$, $X(4274)$, have
been discovered in experiments PDG . They may be the good candidates for the
$\chi_{c1}(nP)$. The Ref. 3872-1  calculated the E1 radiative and strong
decays of the $X(3872)$ as all possible $1D$ and $2P$ $c\bar{c}$ states. The
Ref. 3872-2  explored the $1^{3}D_{2}$, $1^{3}D_{3}$ and $2^{1}P_{1}$
charmonium candidates for $X(3872)$, and the $1^{3}D_{2}$, $1^{3}D_{3}$ were
favored candidates for the $X(3872)$, both have prominent radiative decays.
The $X(3872)$ was examined by the molecule model and the charmonium model in
Ref. 3872-3 , the author thought that the $X(3872)$ may fit more likely to the
excited ${}^{3}P_{1}$ charmonium than to the molecule. The quantum number of
the $X(3872)$ is the same as $\chi_{c1}(2P)$, but its mass is about 50 MeV
lighter than the result of our prediction. Considering the $X(3872)$ as the
$\chi_{c1}(2P)$, we have calculated the radiative $E1$ decay widths of the
$X(3872)$ through the BS method in Ref. thwang1 , the result was in agreement
with the experimental data. However, there is no two-body open charm strong
decay for $X(3872)$ which has a narrow width $\Gamma_{X(3872)}<1.2$ MeV PDG .
Because the mass of $X(3872)$ happens to lie around the $D\bar{D}^{*}$
threshold, many authors believed that it’s an ideal candidate for the
$D\bar{D}^{*}$ exotic hadrons. Various scenarios have been discussed in the
literature Ref. Olsen ; fkguo ; swanson1 ; Ferretti ; slzhu2 , but until now,
the nature of the $X(3872)$ still remains unclear, many more investigations
are very essential to understand the property of the $X(3872)$ in the future.

The Ref. th3  calculated the strong decay of $X(4274)$ as $\chi_{c1}(3P)$ with
$M=4317$ MeV, and they got the width $\Gamma=43.6$ MeV. The Ref. liuxiang2021
gave the decay width of $\chi_{c1}(3P)$: $\Gamma=48$ MeV, which was consistent
with experimental data, and it was possible to assign $X(4274)$ as the
$\chi_{c1}(3P)$ state. Considering the $X(4274)$ as the $\chi_{c1}(3P)$ state,
we calculate its strong decays by the BS method. The total strong decay width
$\Gamma=52.7$ MeV is larger than 29.7 MeV with $M_{\chi_{c1}(3P)}=4228.8$ MeV,
so the strong decay width is sensitive to the mass of the $\chi_{c1}(3P)$. The
total strong decay width $\Gamma=52.7$ MeV is agreement with the world average
data $\Gamma_{X(4274)}=(49\pm 12)$ MeV PDG . Therefore, the $X(4274)$ is a
good candidate for the $\chi_{c1}(3P)$ state. Assigning the $X(4274)$ as the
$\chi_{c1}(3P)$, the relative ratios are $\frac{\Gamma(\chi_{c1}(3P)\to
D^{*}\bar{D}^{*})}{\Gamma(\chi_{c1}(3P)\to DD^{*})}\approx 0.59$ and
$\frac{\Gamma(\chi_{c1}(3P)\to D_{s}D_{s}^{*})}{\Gamma(\chi_{c1}(3P)\to
DD^{*})}\approx 0.41$, which can provide more useful information to observe
the $X(4274)$ in the future experiment.

Table 4: The strong decay type and decay widths of the $\chi_{c2}(2P)$ and $\chi_{c2}(3P)$ (unit in MeV). The results in the parentheses are calculated with $M_{\chi_{c2}(2P)}=3930.0$ MeV and $M_{\chi_{c2}(3P)}=4350.0$ MeV. Meson | State | Mode | Our Result | LP(SP)zhaoqiang  | Godfrey  | liuxiang2021  | Swanson
---|---|---|---|---|---|---|---
$\chi_{c2}$ | $2^{3}P_{2}$ | $D\bar{D}$ | 24.6(20.4) | – | 42 | – | 32
|  | $DD^{*}$ | 21.1(6.9) | – | 37 | – | 28
|  | $D_{s}D_{s}$ | 0.6(–) | – | 0.7 | – | 0.5
|  | total | 46(27.3) | – | 80 | – | 60.5
| $3^{3}P_{2}$ | $D\bar{D}$ | 5.7(8.7) | 8.1(7.3) | 8.0 | 7 | –
|  | $DD^{*}$ | 2.2(2.7) | 17(13) | 2.4 | 3 | –
|  | $D^{*}\bar{D}^{*}$ | 24.3(28.6) | 4.2(7.1) | 24 | 39 | –
|  | $D_{s}D_{s}$ | 0.4(0.9) | 1.0(0.6) | 0.8 | 0 | –
|  | $D_{s}D_{s}^{*}$ | 3.0(4.2) | 0.3(1.4) | 11 | – | –
|  | $D_{s}^{*}D_{s}^{*}$ | 0.6(1.8) | 4.8(–) | 7.2 | 1 | –
|  | $DD_{1}$ | –(0.9) | 1.0(0) | 1.1 | – | –
|  | $DD_{1}^{\prime}$ | –(10.9) | 7.3(0) | 12 | – | –
|  | $DD_{2}$ | –(1.4) | – | – | – | –
|  | $D^{*}D_{0}$ | –(0.3) | – | – | – | –
|  | total | 36(60) | 43(30) | 66 | 50 | –

###  III.3 $\chi_{c2}(2P)$ and $\chi_{c2}(3P)$

By solving the Eq.(10), we also obtain the mass of $\chi_{c2}(2P)$ and
$\chi_{c2}(3P)$, the $M_{\chi_{c2}(2P)}=3972.4$ MeV is in accordance with the
relativized Godfrey-Isgur model (3979 MeV) and the nonrelativistic potential
model (3972 MeV) Godfrey . The $M_{\chi_{c2}(3P)}=4271.0$ MeV is smaller than
the mass in the relativized Godfrey-Isgur model (4337 MeV) and the
nonrelativistic potential model (4317 MeV) Godfrey . The two-body open charm
strong decays of the $\chi_{c2}(2P)$ and $\chi_{c2}(3P)$, are shown in Table
4. The $\chi_{c2}(2P)$ state has two decay modes: $2^{++}\to
0^{-}0^{-},\;0^{-}1^{-}$, and the corresponding final states include
$D\bar{D}$, $DD^{*}$ and $D_{s}D_{s}$. We find that the dominant channels are
$D\bar{D}$ and $DD^{*}$, the $\chi_{c2}(2P)\to D_{s}D_{s}$ is very small with
the small phase space. Then the total strong decay width
$\Gamma_{\chi_{c2}(2P)}=46$ MeV, which is smaller than the results of the
nonrelativistic potential model (80 MeV) Godfrey  and the usual ${}^{3}P_{0}$
model (60.5 MeV) Swanson .

There are three decay modes for the $\chi_{c2}(3P)$: $2^{++}\to
0^{-}0^{-},\;0^{-}1^{-},\;1^{-}1^{-}$, and six strong decay channels
$D\bar{D}$, $DD^{*}$, $D^{*}\bar{D}^{*}$,$D_{s}D_{s}$, $D_{s}D_{s}^{*}$ and
$D_{s}^{*}D_{s}^{*}$. The total strong decay width $\Gamma=36$ MeV is
consistent with the results of linear potential quark model (43 MeV) and
screened potential quark model (30 MeV) zhaoqiang . $\chi_{c2}(3P)\to
D^{*}\bar{D}^{*}$ is the dominant decay channel, which contribute about $68\%$
of the total strong decay width. We also predict the relative ratios
$\frac{\Gamma(\chi_{c2}(3P)\to D\bar{D})}{\Gamma(\chi_{c2}(3P)\to
D^{*}\bar{D}^{*})}\approx 0.23$ and $\frac{\Gamma(\chi_{c2}(3P)\to
DD^{*})}{\Gamma(\chi_{c2}(3P)\to D^{*}\bar{D}^{*})}\approx 0.091$ with
$M_{\chi_{c2}(3P)}=4271.0$ MeV.

The Ref. th11  studied the strong decays of the $\chi_{cJ}(2P)$ and
$\chi_{cJ}(3P)$, the mass of $\chi_{c2}(2P)$ was very close to the
experimental data of the $Z(3930)$, but the decay width $\Gamma\approx 68$
MeV, which was 3 times that of the experimental value. The $Z(3930)$ was
assigned to the $\chi_{c2}(2P)$, and the strong decay width was $\Gamma=19.0$
MeV in the Ref. 39301 . Assigning the $Z(3930)$ as the $\chi_{c2}(2P)$, we
have taken two methods to calculate the OZI-allowed two-body strong decay
processes of the $\chi_{c2}(2P)$ state in detail: the BS method and the
extended ${}^{3}P_{0}$ model in Ref. thwang . The total decay width is
consistent with the experimental data, which means the $Z(3930)$ is a good
candidate for the $\chi_{c2}(2P)$, so we only list the total strong decay
width of the $\chi_{c2}(2P)$ in this paper.

The $X(4350)$ was observed by Belle in the $\phi J/\psi$ mass spectrum, which
is a candidate for the $\chi_{c2}(3P)$ 4350 . The open-charm decay of
$\chi_{c2}(3P)$ with $R=1.9\approx 2.3$ GeV-1 was well consistent with
experimental data of the $X(4350)$, which showed that the $X(4350)$ as a good
candidate of $\chi_{c2}(3P)$ in Ref. th13 . Assigning the $X(4350)$ as the
$\chi_{c2}(3P)$ state, the Ref. zhaoqiang  studied its the strong decay, and
got the decay width which was about 90 MeV. Considering the $X(4350)$ as the
$\chi_{c2}(3P)$, some new decay channels are allowed with the increasing mass,
such as $DD_{1}$, $DD_{1}^{\prime}$, $DD_{2}$ and $D^{*}D_{0}$. So the total
decay width increases to $\Gamma=60$ MeV, which is consistent with the results
of the nonrelativistic potential model (66 MeV) Godfrey , but larger than the
result of experimental dada $\Gamma_{X(4350)}=(13^{+18}_{-9}\pm 4)$ MeV. Thus,
if one takes the $\chi_{c2}(3P)$ as an assignment of the $X(4350)$, the
precision measurements are needed in further experiments. The relative ratios
$\frac{\Gamma(\chi_{c2}(3P)\to D\bar{D})}{\Gamma(\chi_{c2}(3P)\to
D^{*}\bar{D}^{*})}\approx 0.30$ and $\frac{\Gamma(\chi_{c2}(3P)\to
DD^{*})}{\Gamma(\chi_{c2}(3P)\to D^{*}\bar{D}^{*})}\approx 0.094$ with
$M_{\chi_{c2}(3P)}=4350.0$ MeV, also can provide evidence to discover the
$X(4350)$ for the future experiment.

Table 5: The strong decay type and decay widths of the $h_{c}(2P)$ and $h_{c}(3P)$ (unit in MeV). Meson | State | Mode | Our Result | LP(SP)zhaoqiang  | Godfrey  | Swanson
---|---|---|---|---|---|---
$h_{c}$ | $2^{1}P_{1}$ | $DD^{*}$ | 48 | 64(68) | 87 | 67
| $3^{1}P_{1}$ | $DD^{*}$ | 16.0 | 14(11) | 3.0 | –
|  | $D^{*}\bar{D}^{*}$ | 3.6 | 4.8(7.5) | 22 | –
|  | $D_{s}D_{s}^{*}$ | 6.4 | 6.5(6.3) | 15 | –
|  | $D_{s}^{*}D_{s}^{*}$ | 0.2 | 3.6(–) | 7.5 | –
|  | $DD_{0}$ | 4.6 | 15(5.0) | 28 | –
|  | total | 31 | 44(30) | 75 | –

###  III.4 $h_{c}(2P)$ and $h_{c}(3P)$

Finally, we study the higher charmonium states with $J^{PC}=1^{+-}$,
$h_{c}(2P)$ and $h_{c}(3P)$. The $M_{h_{c}(2P)}=3943.0$ MeV is consistent with
the mass of the relativized Godfrey-Isgur model (3956 MeV) and the
nonrelativistic potential model (3934 MeV) Godfrey . The
$M_{h_{c}(3P)}=4242.4$ MeV is close to the results of the nonrelativistic
potential model (4279 MeV) Godfrey . And the two-body open charm strong decay
results are shown in Table 5. The $h_{c}(2P)$ has one decay mode $1^{+-}\to
0^{-}1^{-}$, and only can decay into $DD^{*}$. The total strong decay width
$\Gamma_{h_{c}(2P)}=48$ MeV, which is smaller than the results of other
theoretical models.

The main decay modes of the $h_{c}(3P)$ include $1^{+-}\to
0^{-}1^{-},\;1^{-}1^{-},\;0^{-}0^{+}$. The final states $DD^{*}$,
$D^{*}\bar{D}^{*}$, $D_{s}D_{s}^{*}$, $D_{s}^{*}D_{s}^{*}$, $DD_{0}$ are
allowed. The total strong decay width $\Gamma_{h_{c}(3P)}=31$ MeV is in
accordance with the result of the screened potential quark model (30 MeV)
zhaoqiang . The main decay channel $h_{c}(3P)\to DD^{*}$ contributes about
$52\%$ of the total strong decay width. The corresponding relative ratios:
$\frac{\Gamma(h_{c}(3P)\to D^{*}\bar{D}^{*})}{\Gamma(h_{c}(3P)\to
DD^{*})}\approx 0.23$ and $\frac{\Gamma(h_{c}(3P)\to
D_{s}D_{s}^{*})}{\Gamma(h_{c}(3P)\to DD^{*})}\approx 0.40$ can provide
theoretical assistance to confirm the $h_{c}(3P)$ in future experiments.

##  IV summary

In conclusion, we have studied the two-body open charm strong decays of the
$2P$ and $3P$ charmonium states by the BS method combined with the
${}^{3}P_{0}$ model. The wave functions and mass spectra of the initial $2P$
and $3P$ charmonium states are obtained by solving the BS equation with the
relativistic correction. Considering the relativistic correction, the masses
of some $3P$ charmonium states in our model will have a difference with the
results of the nonrelativistic potential model. Then we get the two-body open
charm strong decay widths of the $2P$ and $3P$ charmonium states.

Considering the $X^{*}(3860)$ as $\chi_{c0}(2P)$, the narrow strong decay
width is smaller than the experimental data, because the strong decay width
depend on the parameter, and there are large errors in the mass and the width
of the $X^{*}(3860)$, we cannot rule out that the $X^{*}(3860)$ is
$\chi_{c0}(2P)$. Taking the $X(4160)$ as the $\chi_{c0}(3P)$, the total strong
decay width is in accordance with the experimental result of the $X(4160)$
with the uncertainty. In addition, because of the node structure of the
$\chi_{c0}(3P)$’s wave functions, the integrand oscillates accordingly in the
amplitude, the decay $X(4160)\to D\bar{D}$ is strong suppressed. Then the
relative ratios $\frac{\Gamma(\chi_{c0}(3P)\to
D\bar{D})}{\Gamma(\chi_{c0}(3P)\to D^{*}\bar{D}^{*})}\approx 0.0019<0.09$, and
$\frac{\Gamma({\chi_{c0}(3P)\to D\bar{D}^{*}})}{\Gamma({\chi_{c0}(3P)\to
D^{*}\bar{D}^{*}})}=0<0.22$ are consistent with the experimental results.
Therefore, the $X(4160)$ is a good candidate for the $\chi_{c0}(3P)$.

Assigning the $X(4274)$ as the $\chi_{c1}(3P)$, we find that the total strong
decay width is in agreement with the data of the $X(4274)$, so the $X(4274)$
resonance is a good candidate for the $\chi_{c1}(3P)$. The relative ratios
$\frac{\Gamma(\chi_{c1}(3P)\to D^{*}\bar{D}^{*})}{\Gamma(\chi_{c1}(3P)\to
DD^{*})}\approx 0.59$ and $\frac{\Gamma(\chi_{c1}(3P)\to
D_{s}D_{s}^{*})}{\Gamma(\chi_{c1}(3P)\to DD^{*})}\approx 0.41$ can provide
useful information in experiments.

The $Z(3930)$ has been confirmed as the $\chi_{c2}(2P)$ state in our previous
work, so we only show the strong decay width in this work. Considering the
$X(4350)$ as $\chi_{c2}(3P)$, we find that the total strong decay width of the
$\chi_{c2}(3P)$ is larger than the result of the $X(4350)$, thus, if we take
the $\chi_{c2}(3P)$ as an assignment of the $X(4350)$, we need many more
investigations in the future.

Finally, we also calculate the two-body open charm strong decay of the
$h_{c}(2P)$ and $h_{c}(3P)$, and give the total strong decay widths of the
$h_{c}(2P)$ and $h_{c}(3P)$. The corresponding relative ratios
$\frac{\Gamma(h_{c}(3P)\to D^{*}\bar{D}^{*})}{\Gamma(h_{c}(3P)\to
DD^{*})}\approx 0.23$ and $\frac{\Gamma(h_{c}(3P)\to
D_{s}D_{s}^{*})}{\Gamma(h_{c}(3P)\to DD^{*})}\approx 0.40$ can give the
theoretical assistance in future experiments.

Acknowledgements We would like to thank Shi-Lin Zhu for many valuable
discussions and assistance during this work. This work was supported by the
National Natural Science Foundation of China (NSFC) under Grant No. 11865001
and No. 12075073, and the CAS ”Light of West China” Program.

##  Appendix A Instantaneous Bethe-Salpeter Equation

The BS equation which is used to describe the heavy mesons can be written as
BS :

$(\not\\!{p_{1}}-m_{1})\chi(q)(\not\\!{p_{2}}+m_{2})=i\int\frac{d^{4}k}{(2\pi)^{4}}V(P,k,q)\chi(k)\;,$
(4)

where $\chi(q)$ and $P$ are the wave function and the momentum of the bound
state, respectively. $q$ is the relative momentum between quark and anti-quark
in meson, $p_{1}=\frac{m_{1}}{m_{1}+m_{2}}P+q$,
$p_{2}=\frac{m_{2}}{m_{1}+m_{2}}P-q$ and $m_{1}$, $m_{2}$ are the momentum and
the mass of the quark and anti-quark, respectively. The $V(P,k,q)$ is the
interaction kernel between the quark and antiquark.

In order to solve the Eq. (4), the instantaneous approximation is adopted in
the interaction kernel $V(P,k,q)$ Salp :

$V(P,k,q)\Rightarrow V(|\vec{k}-\vec{q}|)\;.$

For convenience, the relative momentum $q$ is decomposed into two parts
$q_{\parallel}$ and $q_{\perp}$,

$q^{\mu}=q^{\mu}_{\parallel}+q^{\mu}_{\perp}\;,$
$q^{\mu}_{\parallel}\equiv(q_{P}/M)P^{\mu}\;,\;\;\;q^{\mu}_{\perp}\equiv
q^{\mu}-q^{\mu}_{\parallel}\;,\;\;\;q_{P}=P\cdot q/M.$

Then the Eq. (4) can be expressed as:

$\chi(q_{\parallel},q_{\perp})=S_{1}(p_{1})\eta(q_{\perp})S_{2}(p_{2})\;.$ (5)

$\eta(q^{\mu}_{\perp})$ is related to three dimensional BS wave function
$\varphi_{p}(q^{\mu}_{\perp})$ as follows:

$\varphi_{P}(q^{\mu}_{\perp})\equiv i\int\frac{{\rm
d}q_{p}}{2\pi}\chi(q^{\mu}_{\parallel},q^{\mu}_{\perp})\;,$
$\displaystyle\eta(q^{\mu}_{\perp})\equiv\int\frac{{\rm
d}k_{\perp}}{(2\pi)^{3}}V(k_{\perp},q_{\perp})\varphi_{p}(k^{\mu}_{\perp})\;,$
(6)

$S_{1}(p_{1})$ and $S_{2}(p_{2})$ are the propagators of the quark and anti-
quark which can be decomposed as:

$S_{i}(p_{i})=\frac{\Lambda^{+}_{ip}(q_{\perp})}{J(i)q_{p}+\alpha_{i}M-\omega_{i}+i\epsilon}+\frac{\Lambda^{-}_{ip}(q_{\perp})}{J(i)q_{p}+\alpha_{i}M+\omega_{i}-i\epsilon}\;,$
(7)

with

$\omega_{i}=\sqrt{m_{i}^{2}+q^{2}_{{}_{T}}}\;,\;\;\;\Lambda^{\pm}_{ip}(q_{\perp})=\frac{1}{2\omega_{ip}}\left[\frac{\not\\!{P}}{M}\omega_{i}\pm
J(i)(m_{i}+{\not\\!q}_{\perp})\right]\;,$ (8)

where $i=1,2$ for quark and anti-quark, respectively, and $J(i)=(-1)^{i+1}$.

The positive and negative energy projected wave functions
$\varphi^{\pm\pm}_{p}(q_{\perp})$ are defined as:

$\varphi^{\pm\pm}_{p}(q_{\perp})\equiv\Lambda^{\pm}_{1p}(q_{\perp})\frac{\not\\!{P}}{M}\varphi_{p}(q_{\perp})\frac{\not\\!{P}}{M}\Lambda^{{\pm}}_{2p}(q_{\perp})\;.$
(9)

Then under instantaneous approximation, with contour integration over $q_{p}$
on both sides of Eq. (5), we obtain the full Salpeter equation:

$(M-\omega_{1}-\omega_{2})\varphi^{++}_{p}(q_{\perp})=\Lambda^{+}_{1p}(q_{\perp})\eta_{p}(q_{\perp})\Lambda^{+}_{2p}(q_{\perp})\;,$
$(M+\omega_{1}+\omega_{2})\varphi^{--}_{p}(q_{\perp})=-\Lambda^{-}_{1p}(q_{\perp})\eta_{p}(q_{\perp})\Lambda^{-}_{2p}(q_{\perp})\;,$
$\varphi^{+-}_{p}(q_{\perp})=\varphi^{-+}_{p}(q_{\perp})=0\;.$ (10)

The wave functions are different for the bound states with different quantum
$J^{PC}$ (or $J^{P}$). First, we give the original BS wave functions for the
different bound state, then reduce the wave functions through the last
equation of Eq. (10). Finally the numerical result of the wave functions and
mass spectrum are obtained by solving the first and second equations in Eq.
(10). And the detailed solution of the Salpeter equation also has been
discussed in Ref. w1 ; BS1 ; glwang ; mass1 .

To solve the Eq. (10), we take the Cornell potential as the instantaneous
interaction kernel $V$, which include a linear scalar interaction and a vector
interaction. In the momentum space and the C.M.S of the bound state, the
interaction potential is read as:

$V(\vec{q})=V_{s}(\vec{q})+\gamma_{{}_{0}}\otimes\gamma^{0}V_{v}(\vec{q})~{},$
$V_{s}(\vec{q})=-(\frac{\lambda}{\alpha}+V_{0})\delta^{3}(\vec{q})+\frac{\lambda}{\pi^{2}}\frac{1}{{(\vec{q}}^{2}+{\alpha}^{2})^{2}}~{},~{}~{}V_{v}(\vec{q})=-\frac{2}{3{\pi}^{2}}\frac{\alpha_{s}(\vec{q})}{{(\vec{q}}^{2}+{\alpha}^{2})}~{},$
(11)

where $\lambda$ is the string constant and
$\alpha_{s}(\vec{q})=\frac{12\pi}{33-2N_{f}}\frac{1}{\log(a+{\vec{q}}^{2}/\Lambda^{2}_{QCD})}$
is the running coupling constant. In order to fit the data of heavy quarkonia,
a constant $V_{0}$ is often added to confining potential. We also introduce a
small parameter $a$ to avoid the divergence in the denominator. The constants
$\lambda$, $\alpha$, $V_{0}$ and $\Lambda_{QCD}$ are the parameters that
characterize the potential.

##  Appendix B The relativistic wave functions

In this paper, we focus on the OZI-allowed two-body open charm strong decay of
the $2P$ and $3P$ charmonium states. The detailed wave functions have been
obtained in Refs. w1 ; mass1 ; BS1 ; glwang . We mainly introduce the
relativistic BS wave functions of the $\chi_{c0}$, $\chi_{c1}$, $\chi_{c2}$
and $h_{c}$ in this section.

###  B.1 The relativistic BS wave function of the $\chi_{c0}$ with
$J^{PC}=0^{++}$

The original BS wave functions of the $\chi_{c0}$ with $J^{PC}=0^{++}$ can be
written as

$\varphi_{0^{++}}(q_{\perp})=M\left[\frac{\not\\!{q}_{\perp}}{M}f_{1}(q_{\perp})+\frac{\not\\!{P}\not\\!{q}_{\perp}}{M^{2}}f_{2}(q_{\perp})+f_{3}(q_{\perp})+\frac{\not\\!{P}}{M}f_{4}(q_{\perp})\right]$
(12)

where $M$ is the mass of bound state $\chi_{c0}$, $f_{i}(q_{\perp})$ is the
original radial wave functions that are related to $|\vec{q}|^{2}$. Taking Eq.
(12) to Eq. (10), the relativistic BS wave functions and the mass spectrum can
be obtained by solving the Salpeter equations Eq. (10). Then the relativistic
positive BS wave function is shown as,

$\varphi^{++}_{0^{++}}(q_{\perp})=A_{1}(q_{\perp})+\frac{\not\\!{P}}{M}A_{2}(q_{\perp})+\frac{\not\\!{q}_{\perp}}{M}A_{3}(q_{\perp})+\frac{\not\\!{P}\not\\!{q}_{\perp}}{M^{2}}A_{4}(q_{\perp}).$
(13)

The corresponding coefficients are

$A_{1}=\frac{(\omega_{1}+\omega_{2})q_{\perp}^{2}}{2(m_{1}\omega_{2}+m_{2}\omega_{1})}\left[f_{1}+\frac{m_{1}+m_{2}}{\omega_{1}+\omega_{2}}f_{2}\right],\;\;\;A_{2}=\frac{(m_{1}-m_{2})q_{\perp}^{2}}{2(m_{1}\omega_{2}+m_{2}\omega_{1})}\left[f_{1}+\frac{m_{1}+m_{2}}{\omega_{1}+\omega_{2}}f_{2}\right],\;\;$
$A_{3}=\frac{M}{2}\left[f_{1}+\frac{m_{1}+m_{2}}{\omega_{1}+\omega_{2}}f_{2}\right],\;\;\;A_{4}=\frac{M}{2}\left[\frac{\omega_{1}+\omega_{2}}{m_{1}+m_{2}}f_{1}+f_{2}\right].$

where $m_{1},m_{2}$ and
$\omega_{1}=\sqrt{m_{1}^{2}+\vec{q}^{2}},\omega_{2}=\sqrt{m_{2}^{2}+\vec{q}^{2}}$
are the masses and the energies of the quark and anti-quark in the $\chi_{c0}$
state.

###  B.2 The relativistic BS wave function of the $\chi_{c1}$ with
$J^{PC}=1^{++}$

The original BS wave functions of the $\chi_{c1}$ with $J^{PC}=1^{++}$ is
constructed by $P$, ${q}_{\perp}$ and the polarization vector $\epsilon$,

$\varphi_{1^{++}}(q_{\perp})=i\varepsilon_{\mu\nu\alpha\beta}\frac{P^{\nu}}{M}q_{\perp}^{\alpha}\epsilon^{\beta}\left[g_{1}\gamma^{\mu}+g_{2}\frac{\not\\!{P}}{M}\gamma^{\mu}+g_{3}\frac{\not\\!{q}_{\perp}}{M}\gamma^{\mu}+g_{4}\frac{\not\\!{P}\gamma^{\mu}\not\\!{q}_{\perp}}{M^{2}}\right]$
(14)

where $\epsilon$ is the polarization vector of the axial vector meson. The
corresponding relativistic positive BS wave function is obtained in Eq. (15)
by solving the Eq. (10),

$\varphi^{++}_{1^{++}}(q_{\perp})=i\varepsilon_{\mu\nu\alpha\beta}\frac{P^{\nu}}{M}q_{\perp}^{\alpha}\epsilon^{\beta}\gamma^{\mu}\left[B_{1}+B_{2}\frac{\not\\!{P}}{M}+B_{3}\frac{\not\\!{q}_{\perp}}{M}+B_{4}\frac{\not\\!{P}\not\\!{q}_{\perp}}{M^{2}}\right]$
(15)
$B_{1}=\frac{1}{2}\left[g_{1}+\frac{\omega_{1}+\omega_{2}}{m_{1}+m_{2}}g_{2}\right],\;\;\;B_{2}=-\frac{1}{2}\left[\frac{m_{1}+m_{2}}{\omega_{1}+\omega_{2}}g_{1}+g_{2}\right],$
$B_{3}=\frac{M(\omega_{1}-\omega_{2})}{m_{1}\omega_{2}+m_{2}\omega_{1}}B_{1},\;\;\;B_{4}=-\frac{M(m_{1}+m_{2})}{m_{1}\omega_{2}+m_{2}\omega_{1}}B_{1}.$

###  B.3 The Relativistic BS Wave function of the $\chi_{c2}$ with
$J^{PC}=2^{++}$

The original BS wave function of the $\chi_{c2}$ is constructed by $P$,
${q}_{\perp}$, the polarization tensor $\epsilon_{\mu\nu}$ and the gamma
matrices,

$\displaystyle\varphi_{2^{++}}(q_{\perp})$ $\displaystyle=$
$\displaystyle\epsilon_{\mu\nu}q_{\perp}^{\mu}q_{\perp}^{\nu}\left[h_{1}(q_{\perp})+\frac{\not\\!{P}}{M}h_{2}(q_{\perp})+\frac{\not\\!{q}_{\perp}}{M}h_{3}(q_{\perp})+\frac{\not\\!{P}\not\\!{q}_{\perp}}{M^{2}}h_{4}(q_{\perp})\right]$
$\displaystyle+$ $\displaystyle
M\epsilon_{\mu\nu}\gamma^{\mu}q_{\perp}^{\nu}\left[h_{5}(q_{\perp})+\frac{\not\\!{P}}{M}h_{6}(q_{\perp})+\frac{\not\\!{q}_{\perp}}{M}h_{7}(q_{\perp})+\frac{\not\\!{P}\not\\!{q}_{\perp}}{M^{2}}h_{8}(q_{\perp})\right]$

where $\epsilon_{\mu\nu}$ is the polarization tensor of the $\chi_{c2}$ with
$J^{PC}=2^{++}$. According to the solve the Eq. (10), we get the relativistic
positive BS wave function,

$\displaystyle\varphi_{2^{+}}(q_{\perp})$ $\displaystyle=$
$\displaystyle\epsilon_{\mu\nu}q_{\perp}^{\mu}q_{\perp}^{\nu}\left[C_{1}(q_{\perp})+\frac{\not\\!{P}}{M}C_{2}(q_{\perp})+\frac{\not\\!{q}_{\perp}}{M}C_{3}(q_{\perp})+\frac{\not\\!{P}\not\\!{q}_{\perp}}{M^{2}}C_{4}(q_{\perp})\right]$
$\displaystyle+$ $\displaystyle
M\epsilon_{\mu\nu}\gamma^{\mu}q_{\perp}^{\nu}\left[C_{5}(q_{\perp})+\frac{\not\\!{P}}{M}C_{6}(q_{\perp})+\frac{\not\\!{q}_{\perp}}{M}C_{7}(q_{\perp})+\frac{\not\\!{P}\not\\!{q}_{\perp}}{M^{2}}C_{8}(q_{\perp})\right]$

where the coefficients are

$C_{1}=\frac{1}{2M(m_{1}\omega_{2}+m_{2}\omega_{1})}\left[(\omega_{1}+\omega_{2})q_{\perp}^{2}h_{3}+(m_{1}+m_{2})q_{\perp}^{2}h_{4}+2M^{2}\omega_{2}h_{5}-2M^{2}m_{2}h_{6}\right],$
$C_{2}=\frac{1}{2M(m_{1}\omega_{2}+m_{2}\omega_{1})}\left[(m_{1}-m_{2})q_{\perp}^{2}h_{3}+(\omega_{1}-\omega_{2})q_{\perp}^{2}h_{4}-2M^{2}m_{2}h_{5}+2M^{2}\omega_{2}f_{6}\right],$
$C_{3}=\frac{1}{2}\left[h_{3}+\frac{m_{1}+m_{2}}{\omega_{1}+\omega_{2}}h_{4}-\frac{2M^{2}}{m_{1}\omega_{2}+m_{2}\omega_{1}}h_{6}\right],C_{4}=\frac{1}{2}\left[\frac{\omega_{1}+\omega_{2}}{m_{1}+m_{2}}h_{3}+h_{4}-\frac{2M^{2}}{m_{1}\omega_{2}+m_{2}\omega_{1}}h_{5}\right],$
$C_{5}=\frac{1}{2}\left[h_{5}-\frac{\omega_{1}+\omega_{2}}{m_{1}+m_{2}}h_{6}\right],~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}C_{6}=\frac{1}{2}\left[-\frac{m_{1}+m_{2}}{\omega_{1}+\omega_{2}}h_{5}+h_{6}\right],$
$C_{7}=\frac{M}{2}\frac{\omega_{1}-\omega_{2}}{m_{1}\omega_{2}+m_{2}\omega_{1}}\left[h_{5}-\frac{\omega_{1}+\omega_{2}}{m_{1}+m_{2}}h_{6}\right],C_{8}=\frac{M}{2}\frac{m_{1}+m_{2}}{m_{1}\omega_{2}+m_{2}\omega_{1}}\left[-h_{5}+\frac{\omega_{1}+\omega_{2}}{m_{1}+m_{2}}h_{6}\right]$

###  B.4 The Relativistic BS Wave function of the $h_{c}$ with $J^{PC}=1^{+-}$

The original BS wave functions of the $h_{c}$ with $J^{PC}=1^{+-}$ also is
constructed by $P$, ${q}_{\perp}$ and the polarization vector $\epsilon$,

$\displaystyle\varphi_{1^{+-}}(q_{\perp})=q_{\perp}\cdot\epsilon\left[t_{1}(q_{\perp})+\frac{\not\\!{P}}{M}t_{2}(q_{\perp})+\frac{\not\\!{q}_{\perp}}{M}t_{3}(q_{\perp})+\frac{\not\\!{P}\not\\!{q}_{\perp}}{M^{2}}t_{4}(q_{\perp})\right]\gamma_{5}\
$ (18)

where $\epsilon$ is the polarization vector of the $h_{c}$. The corresponding
relativistic positive BS wave function is obtained in Eq. (15) by solving the
Eq. (10),

$\displaystyle\varphi^{++}_{1^{+-}}(q_{\perp})=q_{\perp}\cdot\epsilon\left[K_{1}(q_{\perp})+\frac{\not\\!{P}}{M}K_{2}(q_{\perp})+\frac{\not\\!{q}_{\perp}}{M}K_{3}(q_{\perp})+\frac{\not\\!{P}\not\\!{q}_{\perp}}{M^{2}}K_{4}(q_{\perp})\right]\gamma_{5}$
(19)

where the coefficients are

$K_{1}=\frac{1}{2}\left[t_{1}+\frac{\omega_{1}+\omega_{2}}{m_{1}+m_{2}}t_{2}\right],\;\;\;K_{2}=\frac{1}{2}\left[\frac{m_{1}+m_{2}}{\omega_{1}+\omega_{2}}t_{1}+t_{2}\right],$
$K_{3}=-\frac{M(\omega_{1}-\omega_{2})}{m_{1}\omega_{2}+m_{2}\omega_{1}}K_{1},\;\;\;K_{4}=-\frac{M(m_{1}+m_{2})}{m_{1}\omega_{2}+m_{2}\omega_{1}}K_{1}.$

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