Charmonia in an unquenched quark model

Qian Deng¹, Ru-Hui Ni¹ ¹¹1Ru-Hui Ni and Qian Deng contributed equally to this work. ²²2E-mail: [email protected], Qi Li², and Xian-Hui Zhong^1,3 ³³3E-mail: [email protected] 1) Department of Physics, Hunan Normal University, and Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Changsha 410081, China 2) School of Science, Tianjin Chengjian University, Tianjin 300000, China 3) Synergetic Innovation Center for Quantum Effects and Applications (SICQEA), Hunan Normal University, Changsha 410081, China

Abstract

In this work, we study the charmonium spectrum within an unquenched quark model including coupled-channel effects. In couple-channel calculations, we include all of the opened charmed meson channels with the once-subtracted method, meanwhile adopt a suppressed factor to soften the hard vertices given by the ${}^{3}P_{0}$ model in the high momentum region. We obtain a good description of both the masses and widths for the well-established states in the charmonium spectrum. Furthermore, we give predictions for the higher $S$ -, $P$ - and $D$ -wave charmonium states up to mass region of $\sim 5.0$ GeV. The magnitude of mass shifts due to the coupled-channel effects is estimated to be about $10s$ MeV. Although many decay channels are opened for the higher charmonium states, they are relatively narrow states. Their widths scatter in the range of $\sim 10s-100$ MeV. Many charmonium-like states, such as $\chi_{c1}(3872)$ , $\chi_{c1}(4274)$ , $\chi_{c0}(3915)$ , $\chi_{c0}(4500)$ , $\chi_{c0}(4700)$ , $X(4160)$ , $X(4350)$ , $Y(4500)$ , and $\psi(4660)$ / $Y(4710)$ , can be accommodated by the charmonium spectrum when the unquenched coupled-channel effects are carefully considered.

I Introduction

Since the $X(3872)$ was first observed at Belle in 2003 Belle:2003nnu , many charmonium-like states have been observed in the past two decades at experiments such as BABAR, Belle, LHCb, and BESIII, etc.. Sixteen states $\chi_{c1}(3872)$ (known as $X(3872)$ ), $\chi_{c0}(3860)$ , $\chi_{c0}(3915)$ , $X(3940)$ , $X(3960)$ , $\chi_{c1}(4140)$ , $X(4160)$ , $\psi(4230)$ , $\chi_{c1}(4274)$ , $X(4350)$ , $\psi(4360)$ , $\chi_{c0}(4500)$ , $X(4630)$ , $\psi(4660)$ , $\chi_{c1}(4685)$ and $\chi_{c0}(4700)$ , are listed in the Review of Particle Physics 2023 (RPP 2023) by the Particle Data Group (PDG) ParticleDataGroup:2022pth . Recently, the center-of-mass energies of BESIII experiments have been extended to 4.95 GeV, which bring new opportunities for searching for the higher charmonium states. Lately, in the higher mass region three new vector states $Y(4500)$ BESIII:2022joj ; BESIII:2023cmv , $Y(4710)$ BESIII:2022kcv and $Y(4790)$ BESIII:2023wsc were observed at BESIII. Since these charmonium-like states contain a $c\bar{c}$ pair, they may be candidates of charmonium states. However, as a $c\bar{c}$ assignment the observed properties, such as mass, for some of them are out of the conventional quark model expectations. They may be also candidates of exotic states, such as multi-quark state, hadronic molecule, or hybrid. About the nature of the charmonium-like states, there are many debates in the literature. To know about the experimental and theoretical status of the charmonium-like states, one can refer to the review works Dong:2021bvy ; Ali:2017jda ; Esposito:2016noz ; Olsen:2017bmm ; Lebed:2016hpi ; Liu:2019zoy ; Chen:2016qju ; Brambilla:2019esw ; Chen:2022asf ; Guo:2017jvc .

As we know that the exotic states, such as tetraquark states, with normal quantum numbers can hardly be distinguished from the normal ones, in order to search for exotic states, one need have a good knowledge of the normal hadron spectrum. In the charmonium spectrum, the low-lying states can be very well described by various quark models with linear confinement potentials, such as the famous nonrelativistic Cornell model Eichten:1974af ; Eichten:1978tg , relativized Godfrey-Isgur model Godfrey:1985xj ; Barnes:2005pb , relativistic quark model Ebert:2011jc , and so on. However, for the description of higher charmonium states, the conventional quark model will be questionable. The predictions often deviate from the observations. These discrepancies may be caused by the quantum fluctuation, i.e., the creation and annihilation of $q\bar{q}$ pairs in vacuum, which will bring an important effect. That is the so-called “coupled-channel effect” through virtual charm meson loops. This effect would become essential for some higher excited states.

The unquenched coupled-channel effects for charmonium states were evaluated within various coupled-channel models based on different strategies in the literature, such as Refs. Ferretti:2021xjl ; Ferretti:2020civ ; Duan:2021alw ; Chen:2023wfp ; Heikkila:1983wd ; Pennington:2007xr ; Eichten:1978tg ; Ortega:2019tby ; Fu:2018yxq ; Ferretti:2018tco ; Li:2009ad ; Barnes:2007xu ; Kalashnikova:2005ui ; Anwar:2021dmg ; Kanwal:2022ani ; Zhou:2013ada ; Ferretti:2013faa ; Bruschini:2021cty ; Lu:2017yhl . It is found that the masses of the higher excited states are often lowered by the coupled-channel effects. Thus, some screened potential models, which roughly includes such unquenched effects, were adopted to deal with the mass spectrum in the literature Dong:1996ci ; Ding:1993uy ; Li:2009zu ; Li:2009ad ; Deng:2016stx ; Wang:2019mhs . The screened potential model results show that the mass suppression tends to be strengthened from lower levels to higher ones, compared to the quenched potential model. Both the screened potential model and coupled-channel model seem to have the similar global features in describing the charmonium spectrum Li:2009ad . In theory, to carry out unquenched calculations of charmonium spectrum, the best way is Lattice QCD. There are some attempts in the literature Wilson:2023anv ; Wilson:2023hzu ; Dudek:2007wv ; HadronSpectrum:2012gic ; DeTar:2011nn ; Bali:2011rd ; Lang:2015sba .

For the aspect of phenomenology, one should face several challenges Ni:HL to seriously carry out a study of the charmonium spectrum within the unquenched framework. i) The first challenge is how to select the coupled channels. There are many coupled channels for a charmonium state. However, only the low-lying channels $D^{(*)}D^{(*)}$ and $D_{s}^{(*)}D_{s}^{(*)}$ are included in most of the calculations. Can the contributions of the other higher channels be neglected for the high charmonium states? ii) The second challenge is how to evaluate the coupled channel effects in the high momentum region. Where the nonphysical contributions may be involved due to the hard vertices given by the phenomenological models, such as the ${}^{3}P_{0}$ model Micu:1968mk ; LeYaouanc:1972vsx ; LeYaouanc:1973ldf . There may be too large mass correction of the coupled channel effects with a bare vertex. iii) The third challenge is how to obtain a unified description of both the masses and widths for the whole charmonium spectrum within the coupled-channel framework. Usually, most of the works only care about mass corrections due to the coupled-channel effects, a comprehensive description of the strong decays together with the mass spectrum is very scarce.

In this work, we study both the masses and widths for the high charmonium spectrum within a unified framework, where the unquenched coupled-channel effects are included by combining the semirelativistic linear potential model with the ${}^{3}P_{0}$ model Micu:1968mk ; LeYaouanc:1972vsx ; LeYaouanc:1973ldf . Two strategies are adopted to overcome the challenges existing in the unquenched framework as mentioned above. (i) First, in the coupled-channel calculations, we include all possible Okubo-Zweig-Iizuka (OZI) allowed charmed meson channels with mass thresholds below the bare $c\bar{c}$ states together with the nearby virtual channels. The contributions from the other virtual channels (whose mass thresholds are significantly above the bare $c\bar{c}$ states) are subtracted from the dispersion relation by redefining the bare mass with the once-subtracted method suggested in Ref. Pennington:2007xr . In other words, the mass shift of a bare state is entirely given by the opened and nearby virtual channels. (ii) Second, a factor is adopted to suppress the unphysical contributions of the coupled-channel effects in the high momentum region as done in the literature Morel:2002vk ; Silvestre-Brac:1991qqx ; Zhong:2022cjx ; Ortega:2016mms ; Ortega:2016pgg ; Yang:2022vdb ; Yang:2021tvc . As we know the vertices of the ${}^{3}P_{0}$ model are only effective in the non-perturbative region, which reflect the ability of $q\bar{q}$ creation in the vacuum. This ability will be suppressed in the high momentum region due to the weak interactions between the valence quarks. Thus, one need introduce a suppressed factor when extending the vertices of ${}^{3}P_{0}$ model to the high momentum region.

Our main purposes of this work are as follows. (i) To obtain a more comprehensive understanding of the charmonium spectrum within the unquenched quark model. Here, we focus not only the explanation of the masses but also the decay widths. (ii) To systematically explore the magnitude of mass shifts of the higher charmonium states up to the mass region of $\sim 5.0$ GeV due to the coupled-channel effects. By this study, we expect to clarify whether the mass shifts have an obvious increasing trend from lower levels to higher ones as that predicted with screened potentials. (iii) To know whether the charmonium-like states could be accommodated by the charmonium spectrum when including the unquenched coupled-channel effects. It is also crucial for seeking out the genuine exotic states from the charmonium-like states. (iv) To predict the masses and decay properties of the missing higher charmonium states within a unquenched quark model. We expect our predictions can provide useful references for the future experimental observations.

The paper is organized as follows. In Sec. II, we give an introduction of the framework of the unquenched quark model including coupled-channel effects. In Sec. III, the masses and strong decay properties of the $S$ -, $P$ - and $D$ -wave charmonium states up to mass region of $\sim 5.0$ GeV are given. Then, we further give some discussions based on the results. Finally, a summary is given in Sec. IV.

II Framework

II.1 Quark potential model

The bare $c\bar{c}$ states are described within a quenched semirelativistic quark potential model. In the model, the effective Hamiltonian is described by

H_{0}=\sqrt{\mathbf{p}_{1}^{2}+m_{q}^{2}}+\sqrt{\mathbf{p}_{2}^{2}+m_{\bar{q}}^{2}}+V_{0}(r)+V_{sd}(r),

(1)

where the first two terms stand for the kinetic energies for the quark and antiquark, respectively. Their masses are labeled with $m_{q}$ and $m_{\bar{q}}$ , respectively. While their three momenta are labeled with $\mathbf{p}_{1}$ and $\mathbf{p}_{2}$ , respectively. $r$ is the distance between the quark and antiquark. $V_{0}(r)$ is the well-known Cornell potential Eichten:1978tg

\displaystyle V_{0}(r)=-\frac{4}{3}\frac{\alpha_{s}}{r}+br+C_{0},

(2)

which includes the color Coulomb interaction and linear confinement, and zero point energy $C_{0}$ . The parameters $b$ and $\alpha_{s}$ denote the strength of the confinement and strong coupling of the one-gluon-exchange potential, respectively. The spin-dependent potential, $V_{sd}(r)$ , is adopted the widely used form Godfrey:1985xj ; Barnes:2005pb

\displaystyle\begin{aligned} V_{sd}(r)&=\frac{32\pi\alpha_{s}\cdot\sigma^{3}e^{-\sigma^{2}r^{2}}}{9\sqrt{\pi^{3}}m_{q}m_{\bar{q}}}\mathbf{S_{q}}\cdot\mathbf{S_{\bar{q}}}\\ &+\frac{4}{3}\frac{\alpha_{s}}{m_{q}m_{\bar{q}}}\frac{1}{r^{3}}\left(\frac{3\mathbf{S}_{q}\cdot\mathbf{r}\mathbf{S}_{{\bar{q}}}\cdot\mathbf{r}}{r^{2}}-\mathbf{S}_{q}\cdot\mathbf{S}_{{\bar{q}}}\right)+H_{LS}.\end{aligned}

(3)

In the above equation, the first and second terms are the spin-spin contact hyperfine potential and tensor potential, respectively. The third term is the spin-orbit interaction, which can be further decomposed into symmetric and antisymmetric terms Godfrey:2004ya ; Li:2019tbn , i.e.,

\displaystyle H_{LS}=H_{sym}+H_{anti},

(4)

with

\displaystyle H_{sym}=\frac{\mathbf{S_{+}\cdot L}}{2}\left[\left(\frac{1}{2m_{q}^{2}}+\frac{1}{2m_{\bar{q}}^{2}}\right)\left(\frac{4\alpha_{s}}{3r^{3}}-\frac{b}{r}\right)+\frac{8\alpha_{s}}{3m_{q}m_{\bar{q}}r^{3}}\right],

(5)

\displaystyle H_{anti}=\frac{\mathbf{S_{-}\cdot L}}{2}\left(\frac{1}{2m_{q}^{2}}-\frac{1}{2m_{\bar{q}}^{2}}\right)\left(\frac{4\alpha_{s}}{3r^{3}}-\frac{b}{r}\right).

(6)

In these equations, $\mathbf{L}$ is the relative orbital angular momentum of the $q\bar{q}$ system; $\mathbf{S}_{q}$ and $\mathbf{S}_{\bar{q}}$ are the spins of the quark $q$ and antiquark $\bar{q}$ , respectively, and $\mathbf{S}_{\pm}\equiv\mathbf{S}_{q}\pm\mathbf{S}_{\bar{q}}$ . It should be mentioned that, for a $c\bar{c}$ state, there are no contributions from the the antisymmetric term $H_{anti}$ due to the equal masses for the quark and antiquark. The parameter set $\{m_{q},m_{\bar{q}},b,\alpha_{s},\sigma,C_{0}$ in the above potentials is determined by fitting the mass spectrum. To solve the radial Schrodinger equation, the Gaussian expansion method Hiyama:2003cu is adopted in this work.

Figure 1: Coupled-channel effects via the

BC

hadronic loops for a bare meson state

|A\rangle

shown at the hadronic level (a), and the quark level (b), respectively.

II.2 Coupled-channel effects

There are creation and annihilation of $q\bar{q}$ pairs in vacuum for a physical hadron state. Thus, a bare meson state $|A\rangle$ described within the quenched quark model can further couple to two-hadron $BC$ continuum via hadronic loops, as shown in Fig. 1. Including such unquenched coupled-channel effects, in the simple coupled-channel model Kalashnikova:2005ui , the physical state is described by

|\Psi\rangle=c_{A}|A\rangle+\sum_{BC}\int c_{BC}(\mathbf{p})d^{3}\mathbf{p}|BC,\mathbf{p}\rangle,

(7)

where $\mathbf{p}=\mathbf{p}_{B}=-\mathbf{p}_{C}$ is the final two-hadron relative momentum in the initial hadron static system, $c_{A}$ and $c_{BC}(\mathbf{p})$ denote the probability amplitudes of the bare state $|A\rangle$ and $|BC,\mathbf{p}\rangle$ continuum, respectively.

The effective Hamiltonian of the physical state $|\Psi\rangle$ is given by

H=H_{0}+H_{c}+H_{I},

(8)

where $H_{0}$ is the Hamiltonian for describing the bare state $|A\rangle$ , which has been given by Eq. (1). $H_{c}$ is a Hamiltonian for describing the continuum state $|BC,\mathbf{p}\rangle$ . Neglecting the interactions between the $B$ and $C$ hadrons, the eigenenergy of $|BC,\mathbf{p}\rangle$ is given by

E_{BC}=\sqrt{m_{B}^{2}+\mathbf{p}^{2}}+\sqrt{m_{C}^{2}+\mathbf{p}^{2}}.

(9)

While $H_{I}$ is an effective Hamiltonian for describing the coupling of the bare state $|A\rangle$ with the $|BC,\mathbf{p}\rangle$ continuum. In the present work, this coupling is adopted from the widely used ${}^{3}P_{0}$ model Micu:1968mk ; LeYaouanc:1972vsx ; LeYaouanc:1973ldf , which is expressed as

	$\displaystyle H_{I}$	$\displaystyle=$	$\displaystyle-3\gamma\sqrt{96\pi}\sum_{m}\langle 1,m;1,-m\mid 0,0\rangle\int d\mathbf{p}_{3}d\mathbf{p}_{4}\delta^{3}\left(\mathbf{p}_{3}+\mathbf{p}_{4}\right)$		(10)
			$\displaystyle\times\mathcal{Y}_{1}^{m}\left(\frac{\mathbf{p}_{3}-\mathbf{p}_{4}}{2}\right)\chi_{1-m}^{34}\phi_{0}^{34}\omega_{0}^{34}b_{3i}^{\dagger}\left(\mathbf{p}_{3}\right)d_{4j}^{\dagger}\left(\mathbf{p}_{4}\right),$		(10)

where $\gamma$ is a dimensionless constant that denotes the strength of the quark-antiquark pair creation with momentum $\mathbf{p}_{3}$ and $\mathbf{p}_{4}$ from vacuum; $b_{3i}^{\dagger}\left(\mathbf{p}_{3}\right)$ and $d_{4j}^{\dagger}\left(\mathbf{p}_{4}\right)$ are the creation operators for the quark and antiquark, respectively; the subscripts, $i$ and $j$ , are the SU(3)-color indices of the created quark and antiquark; $\phi_{0}^{34}=\left(u\bar{u}+d\bar{d}+s\bar{s}\right)/\sqrt{3}$ and $\omega_{0}^{34}=\delta_{ij}/\sqrt{3}$ correspond to flavor and color singlets, respectively; $\chi_{1-m}^{34}$ is a spin triplet state; $\mathcal{Y}_{\ell m}(\mathbf{k})\equiv|\mathbf{k}|^{\ell}Y_{\ell m}\left(\theta_{\mathbf{k}},\phi_{\mathbf{k}}\right)$ is the $\ell$ -th solid harmonic polynomial.

The Schrödinger equation including coupled-channel effects can be expressed as

		$\displaystyle\left(\begin{matrix}H_{0}~{}~{}~{}~{}~{}~{}H_{I}\\ H_{I}~{}~{}~{}~{}~{}~{}H_{c}\end{matrix}\right)~{}\left(\begin{matrix}c_{A}\|A\rangle\\ \sum_{BC}\int c_{BC}(\mathbf{p})d^{3}\mathbf{p}\|BC,\mathbf{p}\rangle\end{matrix}\right)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$		(11)
		$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=M\left(\begin{matrix}c_{A}\|A\rangle\\ \sum_{BC}\int c_{BC}(\mathbf{p})d^{3}\mathbf{p}\|BC,\mathbf{p}\rangle\end{matrix}\right),$		(11)

where $M$ is the mass of the physical state $|\Psi\rangle$ . It can be determined by the coupled-channel equation

M=M_{A}+\Delta M(M).

(12)

$M_{A}$ is the bare mass of $|A\rangle$ obtained from the potential model. While $\Delta M(M)$ is the mass shift due to the coupled-channel effects, which is given by

	$\displaystyle\Delta M(M)$	$\displaystyle=$	$\displaystyle\mathrm{Re}\sum_{BC}\int_{0}^{\infty}\frac{\|\langle BC,\mathbf{p}\|H_{I}\|A\rangle\|^{2}}{(M-E_{BC})}p^{2}dpd\Omega_{p}$		(13)
		$\displaystyle=$	$\displaystyle\operatorname{Re}\sum_{BC}\int_{0}^{\infty}\frac{\overline{\left\|\mathcal{M}_{A\rightarrow BC}(\mathbf{p})\right\|^{2}}}{\left(M-E_{BC}\right)}p^{2}dp.$		(13)

In principle, all $|BC\rangle$ hadronic loops should contribute a mass shift to the bare state $|A\rangle$ . However, it is unfeasible to calculate the self-energy function including an unlimited number of loops. In our calculations, we included all OZI-allowed two-body hadronic channels with mass thresholds below the bare $|A\rangle$ states. Additionally, we account for channels with thresholds slightly (about $50$ MeV) higher than the bare mass. The contributions from the other far away virtual channels (whose mass thresholds are significant above the bare $|A\rangle$ states) are subtracted from the dispersion relation by redefining the bare mass with the once-subtracted method suggested in Ref. Pennington:2007xr . In this approach, the mass shift can be evaluated with

$\displaystyle\Delta M(M)$	$\displaystyle=\operatorname{Re}\sum_{BC}\int_{0}^{\infty}\frac{\overline{\left\|\mathcal{M}_{A\rightarrow BC}(\mathbf{p})\right\|^{2}}}{\left(M-E_{BC}\right)}p^{2}dp$	(14)
	$\displaystyle-\operatorname{Re}\sum_{BC}\int_{0}^{\infty}\frac{\overline{\left\|\mathcal{M}_{A\rightarrow BC}(\mathbf{p})\right\|^{2}}}{\left(M_{0}-E_{BC}\right)}p^{2}dp$
	$\displaystyle=\operatorname{Re}\sum_{BC}\int_{0}^{\infty}\frac{\left(M_{0}-M\right)\overline{\left\|\mathcal{M}_{A\rightarrow BC}(\mathbf{p})\right\|^{2}}}{\left(M-E_{BC}\right)\left(M_{0}-E_{BC}\right)}p^{2}dp,$

where $M_{0}$ represents the subtracted zero-point for the $c\bar{c}$ states. It is chosen as $M_{0}=3097$ MeV, which is the mass of $J/\psi$ . This method has been successfully applied to study the coupled-channel effects on heavy-light and charmonium states in the literature Ni:HL ; Zhou:2011sp ; Duan:2021alw .

The vertices given by the ${}^{3}P_{0}$ model are too hard at high momenta. To suppress the unphysical contributions to the mass shift from the higher momentum region, as done in the literature Silvestre-Brac:1991qqx ; Zhong:2022cjx ; Ortega:2016mms ; Ortega:2016pgg ; Yang:2022vdb ; Yang:2021tvc , a suppressed factor $e^{-\mathbf{p}^{2}/(2\Lambda^{2})}$ is introduced in the strong transition amplitudes, i.e.,

\displaystyle\mathcal{M}_{A\rightarrow BC}(\mathbf{p})\to\mathcal{M}_{A\rightarrow BC}(\mathbf{p})e^{-\frac{\mathbf{p}^{2}}{2\Lambda^{2}}},

(15)

where, $\Lambda$ is a cut-off parameter. In this study, we adopt $\Lambda=780$ MeV to consist with our recent study of the heavy-light meson spectrum including unquenched coupled-channel effects Ni:HL .

By combining Eq. (12) and Eq. (14), one can determine the physical mass $M$ together with the mass shift $\Delta M$ . If the mass of the initial state $A$ is above the mass threshold of final hadron states $B$ and $C$ , a strong decay process $A\to BC$ will happen. The partial decay width for the opened $BC$ channel is given by

\displaystyle\Gamma=2\pi\frac{|\mathbf{p}|E_{B}E_{C}}{M_{A}}\overline{\left|\mathcal{M}_{A\rightarrow BC}(\mathbf{p})\right|^{2}},

(16)

which is equal to two times of the imaginary part of the self-energy of the $BC$ loop as shown in Fig. 1.

Table 1: The measured widths (in MeV) of the four well-established states,

\psi(3770)

\psi(4160)

\psi(4040)

, and

\chi_{c2}(3930)

, compared with the fitted theoretical results. The creation quark pair strength

\gamma=0.422

is obtained by fitting the data with

\chi^{2}/\mathrm{d.o.f}=5.4/(4-1)=1.8

. It should be noted that the errors of some data are properly adjusted to obtain an overall successful fitting.

$n^{2S+1}L_{J}$	Observed State	$\Gamma_{\mathrm{exp}}$ ParticleDataGroup:2022pth	$\Gamma_{\mathrm{Error}}$	$\Gamma_{\mathrm{th}}$
$1^{3}D_{1}$	$\psi(3770)$	$27.2\pm 1.0$	$5.0$	$35.1$
$2^{3}D_{1}$	$\psi(4160)$	$70.0\pm 10.0$	$10.0$	$61.8$
$3^{3}S_{1}$	$\psi(4040)$	$80.0\pm 10.0$	$10.0$	$77.9$
$2^{3}P_{2}$	$\chi_{c2}(3930)$	$35.2\pm 2.2$	$10.0$	$21.0$

II.3 Model parameters

To consist with our previous work for the study of the charmed and charmed-strange mesons in the potential model Ni:2021pce , the constituent masses for charmed, strange, and up/down quarks are taken as $m_{c}=1.70$ GeV, $m_{s}=0.50$ GeV, and $m_{u,d}=0.40$ GeV, respectively. The slope parameter $b$ for the linear confining potential is taken as $b=0.18~{}\mathrm{GeV}^{2}$ . The other three parameters $\{\alpha_{s},\sigma,C_{0}\}$ are taken as $\alpha_{s}=0.445$ , $\sigma=1.20$ GeV, and $C_{0}=-495$ MeV, which are determined by fitting the masses of the low-lying well established $1S$ , $2S$ , $1P$ and $1D$ states. To overcome the singular behavior of $1/r^{3}$ in the spin-dependent potentials, following the method of our previous works li:2021hss ; Deng:2016ktl ; Deng:2016stx ; Li:2019tbn ; Li:2020xzs ; Ni:2021pce , we introduce a cutoff distance $r_{c}$ in the calculations, i.e., let $1/r^{3}=1/r_{c}^{3}$ within a small range $r\in(0,r_{c})$ . The cutoff distance $r_{c}$ is taken as $r_{c}=0.166$ fm, which is determined by fitting the mass of $\chi_{c0}(1P)$ , since its mass is sensitive to the cutoff distance $r_{c}$ .

When calculating the strong transition amplitudes, one also need the masses and wave functions of the initial and final hadron states, and the strength parameter for the quark pair creation of the ${}^{3}P_{0}$ model. The masses of the well-established hadrons are taken from PDG ParticleDataGroup:2022pth . For the unestablished charmonium states, their masses are taken from our predictions in the present work; while for the unestablished $D$ - and $D_{s}$ -meson states, their masses are taken from our previous predictions in Ref. Ni:2021pce . The wave functions of the charmonium states and $D_{(s)}$ -meson states are adopted the numerical forms predicted in the present work and our previous work Ni:2021pce , respectively. The strength parameter for the quark pair creation is taken as $\gamma=0.422$ , which is determined by fitting the widths of the well-established charmonium states $\psi(3770)$ , $\psi(4040)$ , $\psi(4160)$ and $\chi_{c2}(2P)$ with $\chi^{2}/\mathrm{d.o.f}=1.8$ , as shown in Table 1.

Figure 2: Charmonium mass spectrum compared with the observations. The well established states in the charmonium mass spectrum are labeled with blue color. The results obtained within the unquenched and quenched quark models are labeled by a red cross and a solid green circle, respectively.

Table 2: Charmonium mass spectrum described within both the quenched

(Q)

and unquenched

(UQ)

quark models. The mass shifts (

\Delta M

) of the bare states due to the coupled-channel effects are given in the fourth column. The unit of mass is MeV. For a comparison, the experimental data and some results from other works are also listed in the table. The notation “✗” stands for the case of no mass shift for a

c\bar{c}

state.

$n^{2S+1}L_{J}$	$J^{PC}$	$Q$	$\Delta M$	$UQ$	Exp ParticleDataGroup:2022pth	LC Li:2009zu	EFG Ebert:2011jc	BGS Barnes:2005pb	WCLM Wang:2019mhs	DLGZ Deng:2016stx	GI Godfrey:1985xj
$1^{1}S_{0}$	$0^{-+}$	$2984$	✗	$2984$	$2984$	$2979$	$2981$	$2982$	$2981$	$2983$	$2970$
$1^{3}S_{1}$	$1^{--}$	$3097$	✗	$3097$	$3097$	$3097$	$3096$	$3090$	$3096$	$3097$	$3100$
$2^{1}S_{0}$	$0^{-+}$	$3634$	✗	$3634$	$3639$	$3623$	$3635$	$3630$	$3642$	$3635$	$3620$
$2^{3}S_{1}$	$1^{--}$	$3685$	$-9$	$3676$	$3686$	$3673$	$3685$	$3672$	$3683$	$3679$	$3680$
$3^{1}S_{0}$	$0^{-+}$	$4057$	$-35$	$4022$	$...$	$3991$	$3989$	$4043$	$4013$	$4048$	$4060$
$3^{3}S_{1}$	$1^{--}$	$4094$	$-50$	$4044$	$4039$	$4022$	$4039$	$4072$	$4035$	$4078$	$4100$
$4^{1}S_{0}$	$0^{-+}$	$4404$	$-54$	$4350$	$...$	$4250$	$4401$	$4384$	$4260$	$4388$	$...$
$4^{3}S_{1}$	$1^{--}$	$4433$	$-52$	$4381$	$4421$	$4273$	$4427$	$4406$	$4274$	$4412$	$4450$
$5^{1}S_{0}$	$0^{-+}$	$4708$	$-46$	$4662$	$...$	$4446$	$4811$	$...$	$4433$	$4690$	$...$
$5^{3}S_{1}$	$1^{--}$	$4733$	$-51$	$4682$	$4652$ Belle:2014wyt	$4463$	$4837$	$...$	$4443$	$4711$	$...$
$6^{1}S_{0}$	$0^{-+}$	$4983$	$-81$	$4902$	$...$	$4595$	$5155$	$...$	$...$	$...$	$...$
$6^{3}S_{1}$	$1^{--}$	$5005$	$-64$	$4941$	$...$	$4608$	$5167$	$...$	$...$	$...$	$...$
$1^{3}P_{0}$	$0^{++}$	$3415$	✗	$3415$	$3415$	$3433$	$3413$	$3424$	$3464$	$3415$	$3440$
$1^{1}P_{1}$	$1^{+-}$	$3544$	✗	$3544$	$3525$	$3519$	$3525$	$3516$	$3538$	$3522$	$3520$
$1^{3}P_{1}$	$1^{++}$	$3539$	✗	$3539$	$3511$	$3510$	$3511$	$3505$	$3530$	$3516$	$3510$
$1^{3}P_{2}$	$2^{++}$	$3575$	✗	$3575$	$3556$	$3554$	$3555$	$3556$	$3571$	$3552$	$3550$
$2^{3}P_{0}$	$0^{++}$	$3865$	$+40$	$3905$	$3922$	$3842$	$3870$	$3852$	$3896$	$3869$	$3920$
$2^{1}P_{1}$	$1^{+-}$	$3968$	$-7$	$3961$	$...$	$3908$	$3926$	$3934$	$3933$	$3940$	$3960$
$2^{3}P_{1}$	$1^{++}$	$3965$	$-94/+25$	$3871/3990$	$3872/...$	$3901$	$3906$	$3925$	$3929$	$3937$	$3950$
$2^{3}P_{2}$	$2^{++}$	$4002$	$-55$	$3947$	$3923$	$3937$	$3949$	$3972$	$3952$	$3967$	$3980$
$3^{3}P_{0}$	$0^{++}$	$4232$	$+2$	$4234$	$...$	$4131$	$4301$	$4202$	$4177$	$4230$	$...$
$3^{1}P_{1}$	$1^{+-}$	$4318$	$-11$	$4307$	$...$	$4184$	$4337$	$4279$	$4200$	$4285$	$...$
$3^{3}P_{1}$	$1^{++}$	$4316$	$-29$	$4287$	$4286$	$4178$	$4319$	$4271$	$4197$	$4284$	$...$
$3^{3}P_{2}$	$2^{++}$	$4352$	$-15$	$4337$	$4351$ Belle:2009rkh	$4208$	$4354$	$4317$	$4213$	$4310$	$...$
$4^{3}P_{0}$	$0^{++}$	$4551$	$-62$	$4489$	$4474$	$...$	$4698$	$...$	$4374$	$...$	$...$
$4^{1}P_{1}$	$1^{+-}$	$4625$	$-52$	$4573$	$...$	$...$	$4744$	$...$	$4389$	$...$	$...$
$4^{3}P_{1}$	$1^{++}$	$4625$	$-63$	$4562$	$...$	$...$	$4728$	$...$	$4387$	$...$	$...$
$4^{3}P_{2}$	$2^{++}$	$4660$	$-89$	$4571$	$...$	$...$	$4763$	$...$	$4398$	$...$	$...$
$5^{3}P_{0}$	$0^{++}$	$4839$	$-59$	$4780$	$...$	$...$	$...$	$...$	$...$	$...$	$...$
$5^{1}P_{1}$	$1^{+-}$	$4905$	$-30$	$4875$	$...$	$...$	$...$	$...$	$...$	$...$	$...$
$5^{3}P_{1}$	$1^{++}$	$4905$	$-49$	$4856$	$...$	$...$	$...$	$...$	$...$	$...$	$...$
$5^{3}P_{2}$	$2^{++}$	$4939$	$-57$	$4882$	$...$	$...$	$...$	$...$	$...$	$...$	$...$
$1^{3}D_{1}$	$1^{--}$	$3823$	$-35$	$3788$	$3774$	$3787$	$3783$	$3785$	$3830$	$3787$	$3820$
$1^{1}D_{2}$	$2^{-+}$	$3842$	$-28$	$3814$	$...$	$3796$	$3807$	$3799$	$3848$	$3806$	$3840$
$1^{3}D_{2}$	$2^{--}$	$3844$	$-35$	$3809$	$3824$	$3798$	$3795$	$3800$	$3848$	$3807$	$3840$
$1^{3}D_{3}$	$3^{--}$	$3847$	$-4$	$3843$	$3843$	$3799$	$3813$	$3806$	$3859$	$3811$	$3850$
$2^{3}D_{1}$	$1^{--}$	$4176$	$-20$	$4156$	$4191$	$4089$	$4150$	$4142$	$4125$	$4144$	$4190$
$2^{1}D_{2}$	$2^{-+}$	$4206$	$-14$	$4192$	$4146$ LHCb:2021uow	$4099$	$4196$	$4158$	$4137$	$4164$	$4210$
$2^{3}D_{2}$	$2^{--}$	$4207$	$-15$	$4192$	$...$	$4100$	$4190$	$4158$	$4137$	$4165$	$4210$
$2^{3}D_{3}$	$3^{--}$	$4216$	$-3$	$4213$	$...$	$4103$	$4220$	$4167$	$4144$	$4172$	$4220$
$3^{3}D_{1}$	$1^{--}$	$4486$	$-23$	$4463$	$4485$ BESIII:2022joj	$4317$	$4507$	$...$	$4334$	$4456$	$4520$
$3^{1}D_{2}$	$2^{-+}$	$4522$	$-61$	$4461$	$...$	$4326$	$4549$	$...$	$4343$	$4478$	$...$
$3^{3}D_{2}$	$2^{--}$	$4523$	$-55$	$4468$	$...$	$4327$	$4544$	$...$	$4343$	$4478$	$...$
$3^{3}D_{3}$	$3^{--}$	$4535$	$-74$	$4461$	$...$	$4331$	$4574$	$...$	$4348$	$4486$	$...$
$4^{3}D_{1}$	$1^{--}$	$4767$	$-30$	$4737$	$\sim 4710$ BESIII:2023wqy ; BESIII:2022kcv	$...$	$4857$	$...$	$4484$	$...$	$...$
$4^{1}D_{2}$	$2^{-+}$	$4808$	$-32$	$4776$	$...$	$...$	$4898$	$...$	$4490$	$...$	$...$
$4^{3}D_{2}$	$2^{--}$	$4808$	$-31$	$4777$	$...$	$...$	$4896$	$...$	$4490$	$...$	$...$
$4^{3}D_{3}$	$3^{--}$	$4823$	$-32$	$4791$	$...$	$...$	$4920$	$...$	$4494$	$...$	$...$
$5^{3}D_{1}$	$1^{--}$	$5029$	$-41$	$4988$	$...$	$...$	$...$	$...$	$...$	$...$	$...$
$5^{1}D_{2}$	$2^{-+}$	$5073$	$-29$	$5044$	$...$	$...$	$...$	$...$	$...$	$...$	$...$
$5^{3}D_{2}$	$2^{--}$	$5074$	$-44$	$5030$	$...$	$...$	$...$	$...$	$...$	$...$	$...$
$5^{3}D_{3}$	$3^{--}$	$5091$	$-30$	$5061$	$...$	$...$	$...$	$...$	$...$	$...$	$...$

Table 3: The mass shifts and strong decay widths for the

3S

- and

4S

-wave charmonium states (in MeV). The “✗” is labeled the strong decay channels which are not open or forbidden. These channels are considered to have no contributions to the mass shifts and decay widths. The “

...

” stands for the predicted values are negligibly small.

	$3^{1}S_{0}$		$3^{3}S_{1}$
	$\Delta M$	$\Gamma$	$\Delta M$	$\Gamma$
Channel	$[4057]$	$[4022]$	$[4094]$	$as~{}\psi(4040)$
$DD$	✗	✗	$-1.24$	$0.96$
$D_{s}D_{s}$	✗	✗	$+0.07$	$2.10$
$DD^{*}$	$+17.06$	$52.75$	$+9.26$	$17.00$
$D_{s}D_{s}^{*}$	$-6.15$	✗	$-3.89$	✗
$D^{}D^{}$	$-45.56$	$9.56$	$-54.39$	$57.88$
$D_{s}^{}D_{s}^{}$	✗	✗	✗	✗
Total	$-34.65$	$62.31$	$-50.19$	$77.94$
$M_{th},\Gamma_{th}$	$\bm{4022,62.31}$		$\bm{4044,77.94}$
$M_{exp},\Gamma_{exp}$	$\bm{-,-}$		$\bm{4039,80\pm 10}$
	$4^{1}S_{0}$		$4^{3}S_{1}$
	$\Delta M$	$\Gamma$	$\Delta M$	$\Gamma$
Channel	$[4404]$	$[4350]$	$[4433]$	$as~{}\psi(4415)$
$DD$	✗	✗	$-0.31$	$0.70$
$D_{s}D_{s}$	✗	✗	$-0.05$	$...$
$DD^{*}$	$+1.40$	$1.11$	$-0.07$	$0.13$
$D_{s}D_{s}^{*}$	$-0.68$	$2.26$	$-0.12$	$1.15$
$D^{}D^{}$	$+1.08$	$11.69$	$+2.74$	$5.77$
$D_{s}^{}D_{s}^{}$	$+0.24$	$0.43$	$-0.89$	$1.27$
$DD_{0}(2550)$	✗	✗	$-3.50$	$7.46$
$DD_{0}^{*}(2300)$	$+2.62$	$34.05$	✗	✗
$D_{s}D_{s0}^{*}(2317)$	$-6.29$	$3.36$	✗	✗
$D^{}D_{0}^{}(2300)$	✗	✗	$-4.41$	$20.99$
$D_{s}^{}D_{s0}^{}(2317)$	✗	✗	$-1.33$	✗
$DD_{2}^{*}(2460)$	$-25.00$	$7.65$	$-9.37$	$18.40$
$DD_{1}(2430)$	✗	✗	$-9.89$	$17.48$
$D_{s}D_{s1}(2460)$	✗	✗	$-3.86$	✗
$DD_{1}(2420)$	✗	✗	$-2.98$	$15.39$
$D^{*}D_{1}(2430)$	$-12.14$	✗	$-7.57$	✗
$D^{*}D_{1}(2420)$	$-15.28$	✗	$-10.58$	✗
Total	$-54.05$	$60.55$	$-52.19$	$88.74$
$M_{th},\Gamma_{th}$	$\bm{4350,60.55}$		$\bm{4381,88.74}$
$M_{exp},\Gamma_{exp}$	$\bm{-,-}$		$\bm{4421\pm 4,62\pm 20}$

III Result and Discussion

The masses for the $S$ -, $P$ - and $D$ -wave charmonium states up to mass region of $\sim 5.0$ GeV obtained within the unquenched quark models are listed in Table 2. For a comparison, the results of the quenched scenario, the experimental data, and some predictions from other works are also presented in the same table. For clarity, our theoretical mass spectra compared with the data are plot Fig. 2 as well. Furthermore, the mass shift and partial strong decay width contributed by each channels for the charmonium states are given in Tables 3-15.

III.1 $S$ -wave states

The mass spectrum up to $6S$ -wave states is given in Table 2 and also shown in Fig. 2. The strong decay properties for the higher $S$ -wave states are given in Tables 3-6. The masses for the low-lying $S$ -wave $c\bar{c}$ states $\eta_{c}$ , $\eta_{c}(2S)$ , $J/\psi(1S)$ and $\psi(2S)$ can be well described within the quark model. For the high-lying $3,4,5,6S$ -wave states, from Table 2 one can see that the unquenched coupled-channel effects of intermediate hadron loops have sizeable corrections to the masses of the bare states, the mass shifts are predicted to be in the range of $\sim(-80,-30)$ MeV.

III.1.1 $\psi(4040)$

The well established vector state $\psi(4040)$ , as the assignment of $\psi(3S)$ , from Table 3, one can see that both its mass and width are consistent with the predictions. The predicted partial width ratio between $DD$ and $DD^{*}$ channels,

\displaystyle\frac{\Gamma(DD)}{\Gamma(DD^{*})}\simeq 0.06,

(17)

is close to the lower limit of the data $0.24\pm 0.05\pm 0.12$ measured by the BABAR collaboration BaBar:2009elc . However, the predicted ratio

\displaystyle\frac{\Gamma(D^{*}D^{*})}{\Gamma(DD^{*})}\simeq 3.4,

(18)

is inconsistent with the measured value $0.18\pm 0.14\pm 0.03$ by BABAR. It should be mentioned that the $D^{*}D^{*}$ as the dominant channel of $\psi(4040)$ is supported by the measurements of Belle Belle:2006hvs .

III.1.2 $\psi(4415)$

Considering $\psi(4415)$ as the $\psi(4S)$ assignment, from Table 3, one can see that both the mass and width are consistent with the quark model expectations. The predicted partial width ratios,

\displaystyle\frac{\Gamma(DD)}{\Gamma(D^{*}D^{*})}\simeq 0.12,\ \ \frac{\Gamma(DD^{*})}{\Gamma(D^{*}D^{*})}\simeq 0.02,

(19)

are consistent with the data $0.14\pm 0.12\pm 0.03$ and $0.17\pm 0.25\pm 0.03$ measured by the BABAR collaboration BaBar:2009elc , respectively, within uncertainties. Recently, the $\psi(4415)$ was observed in the $D_{s}^{*+}D_{s}^{*-}$ final state by the BESIII collaboration BESIII:2023wsc . As the $\psi(4S)$ assignment, we find that the branching fraction of $\psi(4415)$ into $D_{s}^{*+}D_{s}^{*-}$ is predicted to be $\sim 1.5\%$ (see Table 3), which is similar to that into $\bar{D}_{s}D_{s}^{*}+c.c.$ . Thus, $\psi(4415)$ should be observed in the $\bar{D}_{s}D_{s}^{*}+c.c.$ final states as well.

It should be mentioned that the quark model classification for $\psi(4415)$ still bears some controversies. In the literature Li:2009zu ; Wang:2023zxj ; Wang:2022jxj , $\psi(4415)$ is suggested to be a $5S$ state or a $5S$ - $4D$ mixing state based on the screening potential model. The screening effect is considered to be partly equivalent to the coupled-channel effect. The magnitude of mass shifts due to the coupled-channel effects estimated within the screening potential model reaches up to a fairly large value $\sim 200-300$ MeV, which is about a factor of $\sim 5-10$ larger than our estimations with a more comprehensive consideration of the coupled-channel effects (see Table 3). In Ref. Gui:2018rvv , the strong decays of $\psi(4415)$ as the $\psi(5S)$ assignment based on the screening potential model have been studied by our group, it is found that the decay width $\Gamma\simeq 8.4$ MeV is notably smaller than the experimental value $\Gamma_{exp}\simeq 62\pm 20$ MeV. Thus, $\psi(4415)$ as the $\psi(5S)$ assignment should be excluded according to our study within the unquenched quark model.

III.1.3 $\psi(5S)$ and $\psi(6S)$

The center-of-mass energies of BESIII experiments have been extended to 4.95 GeV, which provide good opportunities for establishing high vector charmonium states $\psi(5S)$ and $\psi(6S)$ . Our predictions about their masses and decay properties have been listed in Tables 5 and 6.

The $\psi(5S)$ may favor the $\psi(4660)$ assignment listed in RPP ParticleDataGroup:2022pth . The predicted mass and width of $\psi(5S)$ , $M=4682$ MeV and $\Gamma=76$ MeV, are in good agreement with the data $M_{exp}=4652\pm 21$ MeV and $\Gamma_{exp}=68\pm 16$ MeV measured at Belle Belle:2014wyt . Recently, a new vector resonance $Y(4710)$ was observed in $e^{+}e^{-}\to K\bar{K}J/\psi$ at BESIII BESIII:2023wqy ; BESIII:2022kcv , the measured mass and width are close to those of $\psi(4660)$ . The $Y(4710)$ and $\psi(4660)$ may correspond to the same state. Further measurements of the main decay channels listed in Table 5, such as $D^{*}D^{*}_{1}(2600)$ and $DD_{1}(2430)$ , may be useful to establish the $\psi(5S)$ state.

Finally, it is should be mentioned that $\psi(4660)$ is also suggested to be assigned as the $\psi(5S)$ state in Refs. Ding:2007rg ; Gui:2018rvv ; Segovia:2008zz ; Zhao:2023hxc based on some quenched quark model studies.

III.2 $P$ -wave states

The mass spectrum up to $5P$ -wave states is given in Table 2 and also shown in Fig. 2. The strong decay properties for the $P$ -wave states are given in Tables 7-10. All of the four low-lying $1P$ -wave states, $h_{c}(1P)$ and $\chi_{c0,1,2}(1P)$ , have been well established. However, the situation become complicated and confusing when toward establishing the higher $P$ -wave states. The unquenched coupled-channel effects play crucial roles for understanding the nature of $\chi_{cJ}(2P)$ states. The coupled-channel effects on the $3P$ -wave states are small. For the $4P$ - and $5P$ -wave states, the unquenched coupled-channel effects systematically lower the mass spectrum, the mass shifts are predicted to be in the range of $\sim(-90,-30)$ MeV.

Figure 3: The mass shift

\Delta M(M)

(upper panel) and spectral density function

\omega(M)

(lower panel) for the

\chi_{c1}(2P)

including the coupled-channel effect of

DD^{*}

. Three solutions are found by solving the coupled-channel equation (12), i.e., the intersection points of the two lines of

\Delta M(M)

and

M-M_{A}

. The first and third solution with masses of 3871 MeV and 3990 MeV correspond to the narrow and broad structures shown in the spectral density function, respectively. The middle solution with a mass of 3888 MeV is unphysical.

Table 4: The components of the two

1^{++}

resonances obtained by the

\chi_{c1}(2P)

state coupling to the

DD^{*}

channel.

Physical Mass	$c\bar{c}$ core	$DD^{*}$
$3871$ MeV	$9.68\%$	$90.32\%$
$3990$ MeV	$61.11\%$	$38.89\%$

III.2.1 $2P$ -wave states

For the $2P$ -wave states, there are several candidates, $\chi_{c1}(3872)$ , $\chi_{c0}(3860)$ , $\chi_{c0}(3930)$ , $X(3915)$ , and $\chi_{c2}(3930)$ , from experiments ParticleDataGroup:2022pth . However, about their assignments one will face several problems: (i) Can $\chi_{c1}(3872)$ be assigned to $\chi_{c1}(2P)$ indeed? (ii) Is the mass gap between $\chi_{c0}(2P)$ and $\chi_{c2}(2P)$ small or large? (iii) Is the width of $\chi_{c0}(2P)$ broad or narrow? The unquenched quark model including couple-channel effects may shed light on these puzzles.

First, let’s focus on the $\chi_{c0}(2P)$ state. Within the unquenched quark model, the mass of $\chi_{c0}(2P)$ is predicted to be $M=3905$ MeV. The coupled-channel effects due to the $DD$ -loop have a significant correction to bare mass of $\chi_{c0}(2P)$ . By solving the coupled-channel equation, it is interesting to find that there is a positive mass shift $\Delta M\simeq+40$ MeV. The positive mass shift was also found by the Lanzhou Group Duan:2020tsx . The $\chi_{c0}(2P)$ should be a narrow state with a width of $\Gamma\simeq 16.8$ MeV, which is nearly saturated by the $DD$ channel. Our prediction is consistent with that in Refs. Duan:2020tsx ; Liu:2009fe . Recently, in the $D^{+}D^{-}$ final state the LHCb collaboration observed a new $0^{++}$ charmonium resonance $\chi_{c0}(3930)$ with mass $M_{exp}=3923.8\pm 1.9$ MeV and width $\Gamma_{exp}=17.4\pm 5.9$ MeV LHCb:2020pxc . Comparing the observations with our predictions, we find that $\chi_{c0}(3930)$ perfectly favors the $\chi_{c0}(2P)$ assignment. The $\chi_{c0}(3930)$ resonance may correspond to $X(3915)$ observed in the $\omega J/\psi$ final state by Belle Belle:2004lle ; Belle:2009and and BABAR BaBar:2012nxg ; BaBar:2007vxr .

In Ref. Guo:2012tv , the authors analyzed the Belle and BABAR data of $\gamma\gamma\to D\bar{D}$ . From the data, a broad resonance with mass $3837.6\pm 11.5$ MeV and width $221\pm 19$ MeV was extracted. They claimed that this broad resonance should correspond to $\chi_{c0}(2P)$ , rather than the narrow $X(3915)$ resonance. Soon after the work was published, the Belle collaboration reported the observation of a $\chi_{c0}(3860)$ state with a mass of $M_{exp}=3862^{+66}_{-45}$ MeV and a width of $\Gamma_{exp}=201^{+242}_{-149}$ MeV in the process $e^{+}e^{-}\to J/\psi D\bar{D}$ Belle:2017egg . However, based our present study, the $\chi_{c0}(3860)$ resonance disfavors the $\chi_{c0}(2P)$ assignment. The broad width is out of theoretical expectation, although the mass seems to be consistent with the unquenched quark model predictions. Our conclusion is consistent with that in Refs. Duan:2020tsx ; Gui:2018rvv . It should be pointed out that the $\chi_{c0}(3860)$ is not seen in a recent observation of $B\to D^{+}D^{-}K^{+}$ at LHCb LHCb:2020pxc , where such a state might be expected to play a significant role. The broad $\chi_{c0}(3860)$ resonance may be contributed by several states or nonresonant backgrounds.

Then, we focus on the $\chi_{c1}(2P)$ state. This state has a strong $S$ -wave coupling to the $DD^{*}$ channel. When solving the coupled-channel Eq. (12), it is seen that there are three solutions with masses $3871$ MeV, 3888 MeV and 3990 MeV, respectively, as shown in the upper panel of Fig. 3. The first solution with a mass of $3871$ MeV lies just below the $D^{0}D^{*0}$ threshold. The second solution with a mass of $3888$ MeV is just above the $D^{\pm}D^{*\mp}$ threshold. While the third solution with a mass of 3990 MeV is heavier than the bare state. To uncover the nature of the solutions obtained from the coupled-channel model, we further analyze the spectral density function,

\displaystyle\omega(M)=\frac{1}{2\pi}\frac{\Gamma}{[M-(M_{A}+\Delta M(M))]^{2}+\Gamma^{2}/4},

(20)

as adopted in the literature Baru:2003qq ; Kalashnikova:2005ui ; Kalashnikova:2009gt ; Wang:2023snv . The line shape of the spectral density function is shown in the lower panel of Fig. 3. It is found that the first solution with a mass of $3871$ MeV corresponds to a very narrow state with a width of about several MeV shown in the spectral density function. The third solution with a mass of $3990$ MeV corresponds to a broader resonance with a width of $\Gamma\simeq 60$ MeV. However, the second solution with a mass of $3888$ MeV does not exhibit any resonance structures in the spectral density function, thus, this solution is unphysical. Similar line shape of the spectral density function was also found in the previous studies Kalashnikova:2005ui ; Giacosa:2019zxw . Two similar solutions are also found by the other coupled-channel analysis of the $\chi_{c1}(2P)$ Zhou:2017dwj ; Ortega:2009hj ; Kalashnikova:2005ui ; Giacosa:2019zxw . Considering the integral intervals as shown in the lower panel of Fig. 3, from the spectral density function, one can estimate the $c\bar{c}$ core components for the resonance structures. Our results are given in Table 4. The narrow state with a mass of $3871$ MeV is dominated by the $DD^{*}$ component, while the $c\bar{c}$ core component is only $\sim 10\%$ . The broad resonance with a mass of $3990$ MeV is a $c\bar{c}$ dominant state, the $c\bar{c}$ core component is estimated to be $~{}60\%$ . The narrow state favors the famous $\chi_{c1}(3872)$ resonance. The nature of $\chi_{c1}(3872)$ originating from the $\chi_{c1}(2P)$ is also suggested in the literature Zhou:2017dwj ; Zhou:2017txt ; Duan:2020tsx ; Pennington:2007xr ; Li:2009ad ; Meng:2007cx ; Ferretti:2014xqa ; Ferretti:2013faa ; Meng:2014ota ; Coito:2012vf ; Wang:2023ovj . To confirm the nature of $\chi_{c1}(3872)$ , it is crucial to look for the broad state with a mass of about $3990$ MeV in the $DD^{*}$ final state.

Finally, we focus on the $\chi_{c2}(2P)$ state. The unquenched coupled-channel effects play a significant role as well. The $D^{*}D^{*}$ and $DD^{*}$ -loops lower the bare mass of the $\chi_{c2}(2P)$ , 4002 MeV, to the physical point $\sim 3947$ MeV. The mass shift reaches up to a fairly large value $\Delta M\simeq-55$ MeV. Considering the $\chi_{c2}(3930)$ resonance as $\chi_{c2}(2P)$ , the measured mass is well described within the unquenched framework, which can be clearly seen from Fig. 2. Within the unquenched picture, the mass gap between $\chi_{c2}(2P)$ and $\chi_{c0}(2P)$ is predicted to be $\sim 40$ MeV, which is much smaller than $\sim 60-120$ MeV predicted in the quenched quark model (see Table 2). Our predictions are consistent with that in Ref. Duan:2020tsx . Taking $\chi_{c2}(3930)$ as the $\chi_{c2}(2P)$ state, one find the decay width $\Gamma\simeq 21$ MeV predicted in theory is also compatible with the data $\Gamma_{exp}=35.2\pm 2.2$ MeV ParticleDataGroup:2022pth . The $\chi_{c2}(2P)$ state dominantly decays into both $DD$ and $DD^{*}$ channels, the partial width ratio between them is predicted to be

\displaystyle\frac{\Gamma(DD^{*})}{\Gamma(DD)}\simeq 0.74.

(21)

The $DD$ decay mode has been observed by the Belle and BABAR experiments ParticleDataGroup:2022pth .

In summary, the unquenched coupled-channel effects are crucial for uncovering the puzzles in the $\chi_{cJ}(2P)$ states. Two resonance structures are found when considering coupled-channel effects for the bare $\chi_{c1}(2P)$ state. The $\chi_{c1}(3872)$ may correspond to the low-mass resonance dominated by the $DD^{*}$ component. The $\chi_{c0}(3930)$ and $\chi_{c2}(3930)$ resonances can be well explained with the assignments $\chi_{c0}(2P)$ and $\chi_{c2}(2P)$ , respectively, when the unquenched coupled-channel effects are properly included. The broad structure $\chi_{c0}(3860)$ cannot be explained as $\chi_{c0}(2P)$ , which may be contributed by several states or nonresonant backgrounds.

III.2.2 $3P$ -wave states

For the $3P$ -wave states, from Table 2 one can see that the mass corrections due to the coupled-channel effects are not significant. The magnitude of the mass shifts is estimated to be within 30 MeV. Except the $\chi_{c0}(3P)$ has a relatively low mass $4234$ MeV, the masses for the other three states $\chi_{c1,2}(3P)$ and $h_{c}(3P)$ are predicted to be around $\sim 4.3$ GeV. Our results are consistent with the those predicted with linear potentials Barnes:2005pb ; Deng:2016stx ; Ebert:2011jc , however, are about $100$ MeV larger than those predicted with screened potentials Wang:2019mhs ; Li:2009zu .

The strong decay properties of the $3P$ -wave states are also studied, the results have been given in Table 7. The $\chi_{c1}(3P)$ and $h_{c}(3P)$ are predicted to be narrow states with a comparable width of $\sim 30$ MeV. While the $\chi_{c0}(3P)$ and $\chi_{c2}(3P)$ are predicted to be moderate width states with a comparable width of $\sim 60$ MeV. The decay properties are roughly consistent with the predictions with a linear potential model in previous work of our group Gui:2018rvv . The $\chi_{c0}(3P)$ may have good potentials to be observed in the $D^{*}D^{*}$ and $D_{s}D_{s}$ channels, the branching fractions for these two channels are predicted to be $68\%$ and $12\%$ , respectively. The $\chi_{c2}(3P)$ may have a potential to be observed in the $DD^{*}$ and $D_{s}^{*}D_{s}^{*}$ channels, the predicted branching fractions may reach up to $\sim 10\%$ . The $\chi_{c1}(3P)$ may have a good potential to be observed in the $D_{s}D_{s}^{*}$ and $D^{*}_{s}D^{*}_{s}$ channels, the branching fractions may reach up to $\sim 20\%$ . The $h_{c}(3P)$ mainly decays into the $DD^{*}$ , $D_{s}D^{*}_{s}$ , $D^{*}D^{*}$ , and $D^{*}_{s}D^{*}_{s}$ channels with a comparable branching fraction $\sim 20-30\%$ .

The $\chi_{c1}(4274)$ (known as $X(4274)$ ) observed in the $J/\psi\phi$ channel in the decay $B^{+}\to J/\psi\phi K^{+}$ at CDF CDF:2011pep and LHCb LHCb:2016axx ; LHCb:2016nsl ; LHCb:2021uow may be a good candidate of $\chi_{c1}(3P)$ . The averaged mass and width of $\chi_{c1}(4274)$ , $M_{exp}=4286^{+8}_{-9}$ MeV and $\Gamma_{exp}=51\pm 7$ MeV ParticleDataGroup:2022pth , are consistent with the our predictions, $M=4287$ MeV and $\Gamma\simeq 34$ MeV. The $\chi_{c1}(4274)$ is also suggested to be assigned as $\chi_{c1}(3P)$ in Refs. Lu:2016cwr ; Duan:2021alw ; Gui:2018rvv ; Wang:2022dfd ; Ferretti:2020civ . The $\chi_{c1}(4274)$ , as the $\chi_{c1}(3P)$ assignment, should be observed in its main decay channels $D_{s}D_{s}^{*}$ and $D^{*}_{s}D^{*}_{s}$ via the $B^{+}\to D^{(*)}_{s}D^{(*)}_{s}K^{+}$ decays.

The $X(4350)$ structure observed in the $\gamma\gamma\to J/\psi\phi$ process at Belle Belle:2009rkh may be a good candidate of $\chi_{c2}(3P)$ . The measured mass and width are $M_{exp}=4350.6^{+4.6}_{-5.1}\pm 0.8$ MeV and $\Gamma_{exp}=13^{+18}_{-9}\pm 4$ MeV, respectively. Assigning $X(4350)$ to $\chi_{c2}(3P)$ , the predicted mass, $M\simeq 4337$ MeV is in good agreement with the data, while the predicted width $\Gamma\simeq 56$ MeV is slightly broader than the data. In Refs. Gui:2018rvv ; Liu:2009fe ; Wang:2022dfd , such a possible assignment was also considered. To confirm this assignment future experimental search for its decays into $DD^{*}$ and $D_{s}^{*}D_{s}^{*}$ are strongly recommended.

Finally, it should be mentioned that recently the LHCb collaboration carried out observations of the $B^{+}\to D^{+}_{s}D^{-}_{s}K^{+}$ decay LHCb:2022dvn ; LHCb:2022aki , there seems to be a bump structure around $4.3$ GeV in the $D^{+}_{s}D^{-}_{s}$ invariant-mass spectrum, which may be contributed by the $\chi_{c0}(3P)$ state. With more statistics, this state is most likely to be established in the $D_{s}D_{s}$ channel by using the $B^{+}\to D^{+}_{s}D^{-}_{s}K^{+}$ decay in forthcoming experiments.

As a whole, it’s time to establish the $3P$ -wave states, which have a relatively narrow width of $\sim 20-60$ MeV. The $\chi_{c1}(4274)$ and $X(4350)$ are good candidates of the $\chi_{c1}(3P)$ and $\chi_{c2}(3P)$ , respectively. Some weak signals of $\chi_{c0}(3P)$ may have been found in the recent LHCb experiments. The $D^{(*)}_{s}D^{(*)}_{s}$ may be good channel to establish the $3P$ -wave states.

III.2.3 $4P$ -wave states

For the $4P$ -wave states, from Table 2 one can see that the mass corrections due to the coupled-channel effects are significant. The magnitude of the mass shifts is estimated to be $\sim 50-90$ MeV. Except the $\chi_{c0}(4P)$ has a relatively low mass of $\sim 4.5$ GeV, the masses for the other three states $\chi_{c1,2}(4P)$ and $h_{c}(4P)$ are predicted to be around $4.6$ GeV. There are some studies of the higher $4P$ -wave states within some quenched Gui:2018rvv ; Sultan:2014oua ; Cao:2012du ; Ebert:2011jc ; Soni:2017wvy ; Chaturvedi:2019usm ; Mansour:2021rru ; Fang:2022bft and unquenched Duan:2021alw ; Ferretti:2021xjl quark models. Our results are close to the predictions with a linear potential in Refs. Gui:2018rvv ; Sultan:2014oua ; Cao:2012du , however, is about $100-200$ MeV smaller than the predictions in Refs. Ebert:2011jc ; Soni:2017wvy ; Chaturvedi:2019usm , and about $200$ MeV higher than the predictions with screened potentials Wang:2019mhs and other modified confinement potentials Mansour:2021rru ; Fang:2022bft .

The strong decay properties of the $4P$ -wave states are also studied, the results have been given in Table 8. The $\chi_{c0}(4P)$ and $\chi_{c2}(4P)$ are predicted to be moderate width states with a comparable width of $\sim 70-90$ MeV. While the $\chi_{c1}(4P)$ and $h_{c}(4P)$ have a slightly broader width of $\sim 100$ MeV. The decay properties are roughly consistent with the predictions with a linear potential model in the previous work of our group Gui:2018rvv . The $4P$ -wave states mainly decay into the $1P$ -wave and/or $2S$ -wave $D$ -meson excitations by emitting a $D$ or $D^{*}$ meson. The rates decaying into the OZI-allowed $D^{(*)}D^{(*)}$ , $D^{(*)}_{s}D^{(*)}_{s}$ channels are often small.

The charmonium-like resonance $\chi_{c0}(4500)$ (known as $X(4500)$ ) listed in RPP is a good candidate of $\chi_{c0}(4P)$ . This resonance is found by LHCb in the $J/\psi\phi$ final state via the $B^{+}\to J/\psi\phi K^{+}$ decay LHCb:2016axx ; LHCb:2021uow ; LHCb:2016nsl . The newly measured mass and width of $\chi_{c0}(4500)$ are $M_{exp}=4474\pm 6$ MeV and $\Gamma_{exp}=77\pm 6^{+10}_{-8}$ MeV, respectively LHCb:2021uow , which are consistent with theoretical predictions $M\simeq 4486$ MeV and $\Gamma\simeq 106$ MeV. If $\chi_{c0}(4500)$ corresponds to $\chi_{c0}(4P)$ indeed, the decay rate into $D_{s}^{+}D_{s}^{-}$ channel is sizeable, the branching fraction is estimated to be $\sim 1\%$ . Thus, the $\chi_{c0}(4500)$ resonance should be established in the $D_{s}^{+}D_{s}^{-}$ final state by using the $B^{+}\to D_{s}^{+}D_{s}^{-}K^{+}$ decay. Recently, this process has been observed by LHCb LHCb:2022dvn ; LHCb:2022aki . There seems to be a vague bump structure around $4.5$ GeV in the $D^{+}_{s}D^{-}_{s}$ invariant mass spectrum. With more statistics, the $\chi_{c0}(4500)$ is most likely to be established in $B^{+}\to D_{s}^{+}D_{s}^{-}K^{+}$ .

III.2.4 $5P$ -wave states

For the higher $5P$ -wave states, from Table 2 one can see that the mass corrections due to the coupled-channel effects are significant. The magnitude of the mass shifts is estimated to be $\sim 30-60$ MeV. Except the $\chi_{c0}(5P)$ has a relatively low mass $4780$ MeV, the masses for the other three states $\chi_{c1,2}(5P)$ and $h_{c}(5P)$ are predicted to be around $4.9$ GeV. There are some studies of the higher $5P$ -wave states within some quenched Gui:2018rvv ; Sultan:2014oua ; Soni:2017wvy ; Mansour:2021rru and unquenched Duan:2021alw ; Ferretti:2021xjl quark models. Strong model dependencies exist in the predictions. Our results are close to those predicted with a linear potential in Refs. Ferretti:2021xjl ; Gui:2018rvv ; Sultan:2014oua , however, is about $100-200$ MeV smaller than those predicted in Refs. Ebert:2011jc ; Soni:2017wvy , and about $300-400$ MeV higher than the those predicted with a unquenched quark model Duan:2021alw and other modified confinement potentials Mansour:2021rru .

The strong decay properties of the $5P$ -wave states are also studied, the results have been given in Tables 9 and 10. It is found that the $\chi_{c0}(5P)$ has a width of $\Gamma\simeq 70$ MeV, while the other three states $\chi_{c1,2}(5P)$ and $h_{c}(5P)$ have a slightly broader width of $\sim 100$ MeV. The strong decay properties are roughly consistent with those predicted with a linear potential model in the previous work of our group Gui:2018rvv . The $5P$ -wave states mainly decay into the $1P$ / $2P$ -wave and/or $2S$ -wave $D$ -meson excitations by emitting a light $D$ or $D^{*}$ meson. The rates decaying into the OZI-allowed $D^{(*)}D^{(*)}$ , $D^{(*)}_{s}D^{(*)}_{s}$ channels are negligibly small.

The charmonium-like resonance $\chi_{c0}(4700)$ (known as $X(4700)$ ) listed in RPP may be a good candidate of $\chi_{c0}(5P)$ . The $\chi_{c0}(4700)$ was first observed by LHCb in the $J/\psi\phi$ invariant mass spectrum via the $B^{+}\to J/\psi\phi K^{+}$ decays in 2016 LHCb:2016nsl ; LHCb:2016axx , and confirmed in the same process with more statistics in 2021 LHCb:2021uow . The lately measured mass and width are $M_{exp}=4694\pm 6^{+16}_{-3}$ MeV and $\Gamma_{exp}=87\pm 8\pm^{+16}_{-6}$ MeV, respectively LHCb:2021uow . By using the $B^{0}_{s}\to J/\psi\phi\pi^{+}\pi^{-}$ decays, the LHCb collaboration also observed a similar structure in the $J/\psi\phi$ channel with a mass of $M_{exp}=4741\pm 6\pm 6$ MeV and $\Gamma_{exp}=53\pm 15\pm 11$ MeV, respectively LHCb:2020coc . The measured mass and width of $\chi_{c0}(4700)$ are consistent with our predictions, $M\simeq 4780$ MeV and $\Gamma\simeq 71$ MeV, with the $\chi_{c0}(5P)$ assignment. The coupled-channel effects are crucial for understanding the mass of $\chi_{c0}(4700)$ . These unquenched effects can lower the bare mass of $\chi_{c0}(5P)$ with a value of $\sim 60$ MeV. If $\chi_{c0}(4700)$ corresponds to $\chi_{c0}(5P)$ indeed, it is most likely to be observed in the $DD_{1}(2420)$ channel via the $B^{+}\to\chi_{c0}(4700)K^{+}$ process. The branching fraction for $\chi_{c0}(4700)\to DD_{1}(2420)$ may reach up to $\sim 7\%$ .

III.3 $D$ -wave states

The mass spectrum up to $5D$ -wave states is given in Table 2 and also shown in Fig. 2. The strong decay properties for the $D$ -wave states are given in Tables 11-15. The mass shifts due to the unquenched coupled-channel effects on the are predicted to be in the range of $\sim(-70,-20)$ MeV. The high $D$ -wave states have a relatively narrow width of $\sim 10s-100$ MeV, although many decay channels are fully opened.

III.3.1 $1D$ -wave states

For the low-lying $1D$ states, except the spin singlet $\eta_{c2}(1D)$ , all the spin triplets, $\psi(1D)$ , $\psi_{2}(1D)$ , and $\psi_{3}(1D)$ , have been well established. Assigning the $\psi(3770)$ , $\psi_{2}(3823)$ and $\psi_{3}(3842)$ as the spin triplets of the $1D$ -wave states, from Fig. 2 it is seen that their masses are consistent with the quark model expectations.

The decays of both $\psi(3770)$ and $\psi_{3}(3842)$ are governed by the $DD$ channel, their decay widths are consistent with the quark model expectations as well (see Table 11). For $\psi_{2}(3823)$ , the OZI-allowed two-body strong decays are kinematic forbidden, thus, its decays is dominated by the electromagnetic transitions. The measured electromagnetic decay properties are also consistent with the theoretical predictions Deng:2016stx ; BESIII:2021qmo .

How to find the missing spin singlet $\eta_{c2}(1D)$ is still a challenge in experiments. It may be produced via the $B\to\eta_{c2}(1D)K$ decay as suggested in Refs. Eichten:2002qv ; Xu:2016kbn ; Fan:2009cj , and established by the the two-photon cascade decay process $\eta_{c2}(1D)\to h_{c}(1P)\gamma\to\eta_{c}\gamma\gamma$ Deng:2016stx .

III.3.2 $2D$ -wave states

In the $2D$ -wave states, only the $\psi(2D)$ with $J^{PC}=1^{--}$ has been established. The $\psi(4160)$ resonance is usually assigned as the $\psi(2D)$ state. With this assignment, the measured width $\Gamma_{exp}=70\pm 10$ MeV are consistent with the theoretical prediction $\Gamma\simeq 62$ MeV. However, the mass $M_{exp}=4191\pm 5$ MeV extracted from experiments is about $40$ MeV higher than most of the predictions in theory (See Table 2). This state dominantly decays into $D^{*}D^{*}$ channel with a branching fraction of $\sim 70\%$ , while it also has sizeable decay rates into $DD$ and $D_{s}D_{s}^{*}$ channels with a comparable branching fraction of $\sim 10\%$ (see Table 11). The branching fraction ratios predicted in the present work and other works Gui:2018rvv ; Eichten:2005ga ; Segovia:2013kg are very different from the old observations at BABAR BaBar:2009elc . In these $P$ -wave decay channels there exist obvious interfering effects between $\psi(4040)$ and $\psi(4160)$ Gui:2018rvv . A coherent partial wave analysis combining all these exclusive channels is suggested to be carried out for extracting the resonance parameters and branching fractions for these two states.

The other three states $\psi_{3}(2D)$ , $\eta_{c2}(2D)$ and $\psi_{2}(2D)$ have a similar mass of $\sim 4.2$ GeV. The corrections to the bare masses due to the coupled-channel effects are not significant, the magnitude of the mass shifts is within $20$ MeV. Our predicted masses for these $2D$ -wave states are in agreement with most of the predictions in the literature, such as Refs. Ebert:2011jc ; Godfrey:1985xj ; Deng:2016stx ; Barnes:2005pb . The theoretical strong decay properties have been given in Table 11. It is found that $\psi_{3}(2D)$ , as the narrowest state in the $2D$ -wave states, has a width of $\Gamma\simeq 38$ MeV, while mainly decays into the $DD^{*}$ and $D^{*}D^{*}$ channels with branching fractions $\sim 34\%$ and $\sim 47\%$ , respectively. Both the $J^{P}=2^{-}$ states, $\psi_{2}(2D)$ and $\eta_{c2}(2D)$ , have a comparable width of $\Gamma\simeq 60$ MeV. The $\psi_{2}(2D)$ ( $\eta_{c2}(2D)$ ) mainly decays into the $DD^{*}$ , $D_{s}D^{*}_{s}$ , and $D^{*}D^{*}$ channels with branching fractions $\sim 30\%$ ( $\sim 35\%$ ), $\sim 17\%$ ( $\sim 11\%$ ) and $\sim 52\%$ ( $\sim 43\%$ ), respectively.

Some signals of $\eta_{c2}(2D)$ may have been observed in experiments. In 2021, the LHCb collaboration observed a new resonance $X(4160)$ with a significance of $4.8$ $\sigma$ in the $J/\psi\phi$ final state by using an amplitude analysis of the $B^{+}\to J/\psi\phi K^{+}$ decays LHCb:2021uow . The $J^{PC}=2^{-+}$ assignment is favored over other assignments with a significance of more than $4.8$ $\sigma$ . The observed mass and width are $M_{exp}=4146\pm 18\pm 33$ MeV and $\Gamma_{exp}=135\pm 28^{+25}_{-30}$ MeV, respectively, which are consistent with the state observed in $e^{+}e^{-}\to J/\psi X$ with $X\to D^{*}D^{*}$ by Belle Belle:2007woe . Considering $X(4160)$ as the $J^{PC}=2^{-+}$ state $\eta_{c2}(1D)$ , both the observed mass and width are consistent with the theoretical predictions, $M\simeq 4192$ MeV and $\Gamma\simeq 60$ MeV. To confirm the nature of $X(4160)$ , the other two dominant decay modes, $DD^{*}$ and $D_{s}D^{*}_{s}$ , are expected to be searched for in future experiments.

III.3.3 $3D$ -wave states

The $3D$ -wave states, $\psi(3D)$ , $\psi_{3}(3D)$ , $\eta_{c2}(3D)$ and $\psi_{2}(3D)$ , may highly degenerate with each other in a very narrow mass range of $\sim 4460-4470$ MeV. From Table 2, one can see that the coupled-channel effects have a significant correction to the bare masses of the $3D$ -wave states with $J^{P}=2^{-}$ and $3^{-}$ . The magnitude of the mass shifts due to these unquenched effects can reach up to about $60-70$ MeV. The masses of the $3D$ -wave states predicted in the present work are consistent with the those predicted with linear potentials Deng:2016stx ; Sultan:2014oua ; Kher:2018wtv and a Martin-like potential Shah:2012js , however, are about $130$ MeV larger than those predicted with screened potentials Deng:2016stx ; Wang:2019mhs ; Li:2009zu , while about $100$ MeV smaller than those predicted with other potential models Ebert:2011jc ; Soni:2017wvy .

The strong decay properties of the $3D$ -wave states are given in Table 12. From the table, one can see that the widths for the $3D$ -wave states are not broad. In these states, the $\psi(3D)$ is the broadest state with a width of $\Gamma\simeq 119$ MeV. This state dominantly decays into $DD_{0}(2550)$ , $DD_{2}^{*}(2460)$ , $DD_{1}(2420)$ , $D^{*}D_{1}(2420)$ channels with branching fractions $\sim 24\%$ , $\sim 9\%$ , $\sim 27\%$ , and $\sim 18\%$ , respectively. The $\psi_{3}(3D)$ has a narrow width of $\Gamma\simeq 38$ MeV, and mainly decays into $DD_{2}^{*}(2460)$ , $DD_{1}(2420)$ , $D^{*}D_{1}(2430)$ channels with branching fractions $\sim 17\%$ , $\sim 15\%$ , and $\sim 24\%$ , respectively. The $\psi_{2}(3D)$ has a moderate width of $\Gamma\simeq 65$ MeV, and mainly decays into $DD_{2}^{*}(2460)$ and $D^{*}D_{1}(2420)$ channels with a comparable branching fraction of $\sim 30\%$ . The $\eta_{c2}(3D)$ is the narrowest state with a width of $\Gamma\simeq 30$ MeV. This state has large decay rates into the $DD_{2}^{*}(2460)$ , $D^{*}D_{1}(2430)$ , and $D^{*}D_{1}(2420)$ channels with branching fractions of $\sim 22\%$ , $\sim 33\%$ , and $\sim 13\%$ , respectively.

Recently, the BESIII collaboration observed a vector resonance $Y(4500)$ in the line shape of the cross sections of the $e^{+}e^{-}\to K^{+}K^{-}J/\psi$ process BESIII:2022joj . The mass and width are measured to be $M_{exp}=4484.7\pm 13.3\pm 24.1$ MeV and $\Gamma_{exp}=111.1\pm 30.1\pm 15.2$ MeV, respectively. This state is confirmed in $e^{+}e^{-}\to D^{*0}D^{*-}\pi^{+}$ by BESIII BESIII:2023cmv . The observed mass, width and quantum numbers of $Y(4500)$ can be well understood in theory with the $\psi(3D)$ assignment. If $Y(4500)$ corresponds to $\psi(3D)$ indeed, it is most likely to be observed in the dominant decay channels, such as $DD_{2}^{*}(2460)$ , $DD_{1}(2420)$ , $D^{*}D_{1}(2420)$ , in future experiments. The $D^{*0}D^{*-}\pi^{+}$ decay mode of $Y(4500)$ observed by BESIII BESIII:2023cmv may be mainly contributed by $D^{*}D_{1}(2420)\to D^{*}D^{*}\pi$ .

III.3.4 $4D$ -wave states

The $4D$ -wave states are predicted to lie in the mass range of $\sim 4730-4800$ MeV (see Table 2). The mass corrections due to the unquenched coupled-channel effects are not large, the values are about $30$ MeV. The masses of the $4D$ -wave states predicted in the present work are consistent with the those predicted with linear potentials Deng:2016stx ; Sultan:2014oua , however, are about $130$ MeV larger than those predicted with a screened potential Wang:2019mhs and a Martin-like potential Shah:2012js , while about $100$ MeV smaller than those predicted with other potential models Chaturvedi:2019usm ; Ebert:2011jc ; Soni:2017wvy .

The strong decay properties of the $4D$ -wave states are given in Table 13. In these states, the $\psi(4D)$ and $\psi_{3}(4D)$ have a relatively broader width of $\Gamma\sim 110$ MeV, while the two $J^{P}=2^{-}$ states $\eta_{c2}(4D)$ and $\psi_{2}(4D)$ have a moderate width of $\Gamma\sim 80$ MeV. The decay rates of the $4D$ -wave states into the OZI-allowed $D^{(*)}D^{(*)}$ and $D_{s}^{(*)}D_{s}^{(*)}$ channels are tiny. The branching fractions for these channels are predicted to be less than $1\%$ . The $1^{--}$ state $\psi(4D)$ has large decay rates into the $D^{*}D_{1}^{*}(2600)$ , $D^{*}D_{1}(2420)$ , and $DD(1D_{2}^{\prime})$ channels with branching fractions $\sim 37\%$ , $\sim 8\%$ , and $\sim 16\%$ , respectively. The $3^{--}$ state $\psi_{3}(4D)$ has large decay rates into the $D^{*}D_{1}^{*}(2600)$ , $D^{*}D_{2}^{*}(2460)$ , and $D^{*}D_{3}^{*}(2750)$ channels with branching fractions $\sim 17\%$ , $\sim 12\%$ , and $\sim 37\%$ , respectively. The $2^{--}$ state $\psi_{2}(4D)$ has large decay rates into the $D^{*}D_{1}^{*}(2600)$ , $D^{*}D_{2}^{*}(2460)$ , $D^{*}D_{1}(2420)$ , and $DD_{3}^{*}(2750)$ channels with branching fractions $\sim 28\%$ , $\sim 8\%$ , $\sim 7\%$ , and $\sim 10\%$ , respectively. The $2^{-+}$ state $\eta_{c2}(4D)$ has large decay rates into the $D^{*}D_{1}^{*}(2600)$ , $D^{*}D_{2}^{*}(2460)$ , and $DD_{3}^{*}(2750)$ channels with branching fractions $\sim 27\%$ , $\sim 11\%$ , and $\sim 16\%$ , respectively.

Some signals of the $1^{--}$ state $\psi(4D)$ may have been seen in experiments. Recently, a new vector resonance $Y(4710)$ was observed in $e^{+}e^{-}\to K\bar{K}J/\psi$ at BESIII BESIII:2023wqy ; BESIII:2022kcv , the measured mass and width are $\sim 4710$ GeV and $\sim 100$ MeV, respectively. The predicted mass and width of $\psi(4D)$ , $M\simeq 4737$ MeV and $\Gamma\simeq 107$ MeV are consistent with those of $Y(4710)$ . Lately, in the Born cross sections of $e^{+}e^{-}\to D_{s}^{*+}D_{s}^{*-}$ , a structure ( $Y(4790)$ ) with a mass of $\sim 4.7-4.8$ GeV and a width of $\sim 27-60$ MeV was observed by BESIII BESIII:2023wsc . This might be the same state observed in $e^{+}e^{-}\to K\bar{K}J/\psi$ . It should be mentioned that the $\psi(4D)$ may highly overlap with $\psi(5S)$ in the mass range of $\sim 4.7$ GeV. Only a small mass gap ( $\sim 55$ MeV) between them is predicted in theory. Both $\psi(4D)$ and $\psi(5S)$ have a comparable decay rate, $\mathcal{O}(10^{-3})$ , into the $D_{s}^{*+}D_{s}^{*-}$ channel. Thus, the structure observed at around $4.7-4.8$ GeV may be contributed by $\psi(4D)$ and/or $\psi(5S)$ . Further observations with more data samples may be useful to uncover the nature of the $Y(4710)$ structure.

III.3.5 $5D$ -wave states

The $5D$ -wave states are predicted to lie in the mass range of $\sim 4988-5060$ MeV (see Table 2). The mass corrections due to the unquenched coupled-channel effects are about $30-40$ MeV. The masses of the $5D$ -wave states predicted in the present work are consistent with the those predicted within a linear potential model Sultan:2014oua .

The strong decay properties of the $5D$ -wave states are given in Tables 14 and 15. It is found that these high $5D$ -wave states have a relatively narrow width of $\sim 30-60$ MeV. The decay rates of the $5D$ -wave states into the OZI-allowed $D^{(*)}D^{(*)}$ and $D_{s}^{(*)}D_{s}^{(*)}$ channels are tiny. The $1^{--}$ state $\psi(5D)$ has relatively large decay rates into the $D^{*}D_{1}^{*}(2600)$ and $D_{1}(2420)D_{1}(2420)$ channels with branching fractions $\sim 6\%$ and $\sim 8\%$ , respectively. The $3^{--}$ state $\psi_{3}(5D)$ has relatively large decay rates into the $D^{*}D_{3}^{*}(2750)$ and $D_{2}^{*}(2460)D_{2}^{*}(2460)$ channels with branching fractions $\sim 14\%$ and $\sim 9\%$ , respectively. The $2^{--}$ state $\psi_{2}(5D)$ has relatively large decay rates into the $D_{s0}^{*}(2317)D_{s1}(2460)$ and $D_{2}^{*}(2460)D_{2}^{*}(2460)$ channels with branching fractions $\sim 17\%$ and $\sim 4\%$ , respectively. The $2^{-+}$ state $\eta_{c2}(5D)$ has relatively large decay rates into the $D_{2}^{*}(2460)D_{2}^{*}(2460)$ and $D_{2}^{*}(2460)D_{1}(2420)$ channels with branching fractions $\sim 7\%$ and $\sim 5\%$ , respectively.

IV Summary

In this work, the mass spectrum and strong decay properties of the $S$ -, $P$ -, and $D$ -wave charmonium states up to mass region of $\sim 5.0$ GeV are systematically studied within the unquenched quark model including coupled-channel effects from all of the OZI-allowed opened charmed meson channels. We can obtain a good description of both the masses and widths for the well-established states in the charmonium spectrum. Although many decay channels are fully opened for the higher charmonium states, they are relatively narrow states. Their widths scatter in the range of $\sim 10s-100$ MeV. We expect our study can provide a useful reference for establishing an abundant charmonium spectrum. Some key results from this study are emphasized as follows.

•

The magnitude of mass shifts of the bare states due to the coupled-channel effects are estimated to be within $10s$ MeV. The mass shifts do not show an obvious increasing trend from lower levels to higher ones as that predicted in the screened potential models.
•

Two resonance structures are found when considering coupled-channel effects for the bare $\chi_{c1}(2P)$ state. The $\chi_{c1}(3872)$ favors the low-mass resonance dominated by the $DD^{*}$ component.
•

The $\chi_{c0}(3915)$ resonance can be well understood with the dressed $\chi_{c0}(2^{3}P_{0})$ states in the coupled-channel model.
•

The vector resonances $\psi(4415)$ and $\psi(4660)$ favor the $\psi(4S)$ and $\psi(5S)$ assignments, respectively.
•

The newly observed vector states $Y(4500)$ and $Y(4710)$ at BESIII may favor the $\psi(3D)$ and $\psi(4D)$ assignments, respectively. However, $Y(4710)$ as the $\psi(5S)$ assignment cannot be excluded.
•

The $\chi_{c0}(4500)$ and $\chi_{c0}(4700)$ resonances observed by LHCb favor the $\chi_{c0}(4P)$ and $\chi_{c0}(5P)$ assignments, respectively.
•

The $\chi_{c1}(4274)$ resonance observed at CDF and LHCb and $X(4350)$ observed at Belle may be good candidates of $\chi_{c1}(3P)$ and $\chi_{c2}(3P)$ , respectively.
•

The $X(4160)$ resonance observed at LHCb and Belle favor the $\eta_{c2}(1D)$ assignment
•

The $\chi_{c0}(3860)$ , $\psi(4230)$ , $Y(4360)$ , $\chi_{c1}(4140)$ , and $\chi_{c1}(4685)$ resonances listed in RPP cannot be accommodated by the charmonium spectrum.

Table 5: The mass shifts and strong decay widths for the

5^{1}S_{0}

and

6^{1}S_{0}

charmonium states (in MeV). The “✗” is labeled the strong decay channels which are not open or forbidden. These channels are considered to have no contributions to the mass shifts and decay widths. The “

...

” stands for the predicted values are negligibly small.

Channel	$\Delta M$ [4708]	$\Gamma$ [4662]	$\Delta M$ [4983]	$\Gamma$ [4902]	Continue	$\Delta M$ [4708]	$\Gamma$ [4662]	$\Delta M$ [4983]	$\Gamma$ [4902]
	$5^{1}S_{0}$		$6^{1}S_{0}$			$5^{1}S_{0}$		$6^{1}S_{0}$
$DD^{*}$	$-0.06$	$...$	$-0.03$	$...$	$D_{s}^{*}D_{s1}(2536)$	$-2.90$	$0.52$	$-0.01$	$0.29$
$D_{s}D_{s}^{*}$	$-0.01$	$0.27$	$-0.01$	$0.03$	$D_{1}^{*}(2600)D_{1}(2430)$	✗	✗	$-3.00$	✗
$D^{}D^{}$	$+0.53$	$1.12$	$+0.09$	$0.20$	$D_{1}^{*}(2600)D_{1}(2420)$	✗	✗	$-3.86$	✗
$D_{s}^{}D_{s}^{}$	$-0.23$	$0.36$	$-0.03$	$0.06$	$DD(1^{3}D_{1})$	$-0.29$	$2.85$	$-0.50$	$0.48$
$D_{0}(2550)D^{*}$	$-4.37$	$5.12$	$-0.09$	$2.93$	$D_{s}D_{s}(1^{3}D_{1})$	✗	✗	$-0.01$	$...$
$D(3^{1}S_{0})D^{*}$	✗	✗	$-2.38$	✗	$DD(2^{3}D_{1})$	✗	✗	$-0.80$	✗
$DD_{1}^{*}(2600)$	$-3.75$	$9.79$	$+0.38$	$2.52$	$DD(1^{3}D_{3})$	$-11.61$	$1.69$	$-1.68$	$3.45$
$D^{}D_{1}^{}(2600)$	✗	✗	$-2.88$	$2.33$	$D_{s}D_{s}(1^{3}D_{3})$	✗	✗	$-0.77$	$0.24$
$DD^{*}(3^{3}S_{1})$	✗	✗	$-3.37$	✗	$D^{*}D(1^{3}D_{1})$	✗	✗	$-0.08$	$0.19$
$D_{s}(2^{1}S_{0})D_{s}^{*}$	$-1.58$	✗	$-0.01$	$...$	$D_{s}^{*}D_{s}(1^{3}D_{1})$	✗	✗	$-0.05$	✗
$D_{s}D_{s}^{*}(2700)$	$-1.74$	✗	$-0.10$	$0.05$	$D^{*}D(1^{3}D_{3})$	✗	✗	$+2.04$	$1.06$
$D_{s}^{}D_{s}^{}(2700)$	✗	✗	$-0.15$	$0.20$	$D_{s}^{*}D_{s}(1^{3}D_{3})$	✗	✗	$-0.89$	✗
$DD^{*}(2300)$	$+3.37$	$6.70$	$+1.43$	$1.91$	$D^{*}D(1D_{2})$	✗	✗	$-0.49$	$2.33$
$D_{0}(2550)D^{*}(2300)$	✗	✗	$-2.40$	$1.18$	$D_{s}^{*}D_{s}(1D_{2})$	✗	✗	$-1.37$	✗
$DD(2^{3}P_{0})$	✗	✗	$-2.33$	$5.22$	$D^{*}D(1D^{\prime}_{2})$	✗	✗	$-4.67$	$2.61$
$D_{s}D_{s0}^{*}(2317)$	$+0.36$	$5.16$	$+0.49$	$1.56$	$D_{s}^{*}D_{s}(1D^{\prime}_{2})$	✗	✗	$-0.70$	✗
$D_{s}(2^{1}S_{0})D_{s0}^{*}(2317)$	✗	✗	$-1.44$	✗	$DD(1^{3}F_{2})$	✗	✗	$-0.21$	✗
$D_{s}D_{s}(2^{3}P_{0})$	✗	✗	$-1.02$	✗	$DD(1^{3}F_{4})$	✗	✗	$-3.07$	✗
$DD_{2}^{*}(2460)$	$+0.65$	$7.73$	$+0.36$	$1.40$	$D^{*}(2300)D_{1}(2430)$	$-0.72$	✗	$-0.35$	$...$
$D_{0}(2550)D_{2}^{*}(2460)$	✗	✗	$-3.41$	✗	$D_{s0}^{*}(2317)D_{s1}(2460)$	✗	✗	$-0.07$	$0.03$
$DD(2^{3}P_{2})$	✗	✗	$+3.10$	$...$	$D^{*}(2300)D_{1}(2420)$	$-1.79$	✗	$-0.12$	$0.06$
$D_{s}D_{s}(2^{3}P_{2})$	✗	✗	$-1.04$	✗	$D_{s0}^{*}(2317)D_{s1}(2536)$	✗	✗	$+0.06$	$0.12$
$D_{s}D_{s2}^{*}(2573)$	$+0.29$	$0.41$	$-0.18$	$0.09$	$D_{2}^{*}(2460)D_{1}(2430)$	✗	✗	$-9.89$	$0.32$
$D^{}D_{2}^{}(2460)$	$-3.88$	$2.34$	$-1.50$	$2.96$	$D_{2}^{*}(2460)D_{1}(2420)$	✗	✗	$-5.24$	$8.53$
$D^{*}D(2^{3}P_{2})$	✗	✗	$-7.85$	✗	$D_{1}(2430)D_{1}(2430)$	✗	✗	$-0.06$	$0.21$
$D_{s}^{}D_{s2}^{}(2573)$	$-2.48$	✗	$-0.23$	$0.71$	$D_{s1}(2460)D_{s1}(2460)$	✗	✗	$-0.08$	✗
$D^{*}D_{1}(2430)$	$-9.60$	$7.53$	$-1.62$	$7.13$	$D_{1}(2430)D_{1}(2420)$	✗	✗	$+0.01$	$0.04$
$D^{*}D(2P_{1})$	✗	✗	$-11.10$	✗	$D_{s1}(2460)D_{s1}(2536)$	✗	✗	$-0.06$	✗
$D_{s}^{*}D_{s1}(2460)$	$-1.01$	$0.29$	$-0.73$	$1.05$	$D_{1}(2420)D_{1}(2420)$	✗	✗	$+0.03$	$...$
$D^{*}D_{1}(2420)$	$-4.90$	$7.55$	$-0.78$	$2.89$	Total	$-45.72$	$59.43$	$-81.24$	$54.38$
$D^{*}D(2P^{\prime}_{1})$	✗	✗	$-6.52$	✗	$M_{th},\Gamma_{th}$	$\bm{4662,59.43}$		$\bm{4902,54.38}$

Table 6: The mass shifts and strong decay widths for the

5^{3}S_{1}

and

6^{3}S_{1}

...

” stands for the predicted values are negligibly small.

Channel	$\Delta M$ [4733]	$\Gamma$ [4682]	$\Delta M$ [5005]	$\Gamma$ [4941]	Continue	$\Delta M$ [4733]	$\Gamma$ [4682]	$\Delta M$ [5005]	$\Gamma$ [4941]
	$5^{3}S_{1}$		$6^{3}S_{1}$			$5^{3}S_{1}$		$6^{3}S_{1}$
$DD$	$-0.07$	$0.17$	$-0.02$	$0.06$	$D^{*}D(2P^{\prime}_{1})$	✗	✗	$-6.71$	$1.37$
$D_{s}D_{s}$	$-0.03$	$0.01$	$-0.01$	$0.01$	$D^{*}D_{1}(2420)$	$-2.40$	$6.28$	$-0.01$	$0.99$
$DD^{*}$	$-0.20$	$0.08$	$-0.07$	$0.10$	$D^{*}(2600)D_{1}(2420)$	✗	✗	$-2.36$	✗
$D_{s}D_{s}^{*}$	$...$	$0.10$	$-0.01$	$...$	$DD(1^{3}D_{1})$	$+0.01$	$0.21$	$-0.05$	$0.22$
$D^{}D^{}$	$+0.45$	$0.56$	$+0.05$	$...$	$DD(2^{3}D_{1})$	✗	✗	$-0.15$	✗
$D_{s}^{}D_{s}^{}$	$-0.20$	$0.50$	$-0.02$	$0.08$	$DD(1^{3}D_{3})$	$-5.19$	$1.94$	$-0.14$	$0.53$
$DD_{0}(2550)$	$+0.59$	$0.32$	$-0.02$	$0.19$	$D_{s}D_{s}(1^{3}D_{1})$	✗	✗	$-0.01$	$...$
$D_{s}D_{s}(2^{1}S_{0})$	$-0.30$	$0.29$	$-0.01$	$0.08$	$D_{s}D_{s}(1^{3}D_{3})$	✗	✗	$-0.02$	$0.06$
$DD(3^{1}S_{0})$	✗	✗	$+0.92$	$0.97$	$DD(1D_{2})$	$...$	$2.21$	$-0.46$	$1.73$
$D_{0}(2550)D^{*}$	$-2.54$	$5.08$	$+0.46$	$0.36$	$DD(2D_{2})$	✗	✗	$-1.01$	✗
$DD_{1}^{*}(2600)$	$-0.99$	$6.14$	$+0.52$	$0.12$	$D_{s}D_{s}(1D_{2})$	✗	✗	$-0.08$	$0.06$
$D(3^{1}S_{0})D^{*}$	✗	✗	$-1.76$	✗	$DD(1D^{\prime}_{2})$	$-2.62$	✗	$-0.71$	$1.19$
$DD^{*}(3^{3}S_{1})$	✗	✗	$-3.27$	$0.48$	$D_{s}D_{s}(1D^{\prime}_{2})$	✗	✗	$-0.32$	$0.02$
$D_{s}(2^{1}S_{0})D_{s}^{*}$	$-1.13$	✗	$-0.14$	$0.48$	$D^{*}D(1^{3}D_{1})$	$-0.76$	✗	$-0.11$	$0.07$
$D_{s}D_{s}^{*}(2700)$	$-1.51$	✗	$-0.19$	$0.54$	$D_{s}^{*}D_{s}(1^{3}D_{1})$	✗	✗	$-0.14$	$0.22$
$D^{}D_{1}^{}(2600)$	$+3.94$	$14.95$	$-2.91$	$7.40$	$D^{*}D(1^{3}D_{3})$	$-10.54$	✗	$+0.65$	$5.70$
$D_{s}^{}D_{s}^{}(2700)$	✗	✗	$-0.89$	$0.12$	$D_{s}^{*}D_{s}(1^{3}D_{3})$	✗	✗	$-1.35$	$0.31$
$D^{}D^{}(2300)$	$+1.54$	$4.35$	$+0.77$	$1.33$	$D^{*}D(1D_{2})$	$-3.15$	✗	$+0.10$	$1.92$
$D^{}D^{}(2^{3}P_{0})$	✗	✗	$-0.24$	$0.57$	$D_{s}^{*}D_{s}(1D_{2})$	✗	✗	$-0.47$	$0.10$
$D^{}(2600)D^{}(2300)$	✗	✗	$-0.48$	$0.47$	$D^{*}D(1D^{\prime}_{2})$	$-5.88$	✗	$+0.76$	$0.64$
$D_{s}^{}D_{s}^{}(2317)$	$-0.75$	$1.89$	$+0.33$	$1.07$	$D_{s}^{*}D_{s}(1D^{\prime}_{2})$	✗	✗	$-0.70$	✗
$D_{s}^{}(2700)D_{s}^{}(2317)$	✗	✗	$-0.48$	✗	$DD(1^{3}F_{2})$	✗	✗	$-0.06$	$0.10$
$D_{s}^{}D_{s}^{}(2^{3}P_{0})$	✗	✗	$-0.86$	✗	$DD(1^{3}F_{4})$	✗	✗	$-1.71$	$0.63$
$DD_{2}^{*}(2460)$	$+0.71$	$3.27$	$+0.19$	$0.02$	$DD(1F_{3})$	✗	✗	$-0.26$	$0.04$
$DD(2^{3}P_{2})$	✗	✗	$+0.19$	$2.83$	$DD(1F^{\prime}_{3})$	✗	✗	$-1.01$	$...$
$D_{0}(2550)D_{2}^{*}(2460)$	✗	✗	$-2.04$	$2.79$	$D^{*}D(1F_{3})$	✗	✗	$-0.45$	✗
$D_{s}D_{s2}^{*}(2573)$	$-0.15$	$...$	$-0.11$	$0.26$	$D^{*}D((1^{3}F_{4})$	✗	✗	$-4.29$	✗
$D_{s}D_{s}(2^{3}P_{2})$	✗	✗	$-0.55$	✗	$D_{1}(2430)D_{1}(2420)$	✗	✗	$+0.04$	$0.25$
$D^{}D_{2}^{}(2460)$	$-7.25$	$6.70$	$-0.90$	$4.38$	$D^{}(2300)D^{}(2300)$	$-0.09$	$0.29$	$+0.02$	$0.12$
$D^{}D^{}(2^{3}P_{2})$	✗	✗	$-13.17$	$4.48$	$D_{s}^{}(2317)D_{s}^{}(2317)$	$-0.02$	$0.08$	$-0.02$	$0.06$
$D_{s}^{}D_{s2}^{}(2573)$	$-2.30$	$0.21$	$+0.04$	$0.02$	$D^{}(2300)D_{2}^{}(2460)$	$-1.25$	✗	$-0.08$	$0.41$
$DD_{1}(2430)$	$+0.49$	$8.36$	$+0.78$	$1.84$	$D^{*}(2300)D_{1}(2430)$	$-0.56$	✗	$-0.25$	$0.50$
$D_{0}(2550)D_{1}(2430)$	✗	✗	$+0.15$	$0.23$	$D_{1}(2420)D_{1}(2420)$	✗	✗	$-0.21$	$1.21$
$DD(2P_{1})$	✗	✗	$-2.09$	$6.08$	$D^{*}(2300)D_{1}(2420)$	$-0.45$	✗	$-0.07$	$0.21$
$D_{s}D_{s1}(2460)$	$-1.10$	$2.30$	$...$	$1.19$	$D_{s}^{*}(2317)D_{s1}(2536)$	✗	✗	$-0.01$	$0.05$
$D_{s}(2^{1}S_{0})D_{s1}(2460)$	✗	✗	$-0.30$	✗	$D_{s1}(2460)D_{s1}(2536)$	✗	✗	$-0.52$	$0.05$
$D_{s}D_{s}(2P_{1})$	✗	✗	$-0.62$	$...$	$D_{2}^{}(2460)D_{2}^{}(2460)$	✗	✗	$-7.28$	$7.43$
$DD_{1}(2420)$	$+0.48$	$0.78$	$+0.04$	$0.03$	$D_{2}^{*}(2460)D_{1}(2430)$	✗	✗	$-2.31$	$2.21$
$D_{0}(2550)D_{1}(2420)$	✗	✗	$-2.32$	$2.07$	$D_{s}^{}(2317)D_{s2}^{}(2573)$	✗	✗	$+0.06$	$0.36$
$DD(2P^{\prime}_{1})$	✗	✗	$-1.14$	$3.57$	$D_{s}^{*}(2317)D_{s1}(2460)$	✗	✗	$-0.05$	$...$
$D_{s}D_{s1}(2536)$	$-0.22$	$0.08$	$-0.07$	$0.16$	$D_{2}^{*}(2460)D_{1}(2420)$	✗	✗	$+0.15$	$2.58$
$D_{s}D_{s}(2P^{\prime}_{1})$	✗	✗	$-0.49$	$0.05$	$D_{s1}(2460)D_{s2}^{*}(2573)$	✗	✗	$-0.65$	✗
$D^{*}D_{1}(2430)$	$-5.37$	$6.73$	$-0.28$	$5.33$	$D_{1}(2430)D_{1}(2430)$	✗	✗	$-0.25$	$1.12$
$D^{*}(2600)D_{1}(2430)$	✗	✗	$-1.80$	✗	$D_{s1}(2460)D_{s1}(2460)$	✗	✗	$-0.18$	$0.06$
$D^{*}D(2P_{1})$	✗	✗	$+2.54$	$0.46$
$D_{s}^{*}D_{s1}(2460)$	$-0.88$	$0.03$	$-0.53$	$1.53$	Total	$-51.43$	$75.05$	$-63.50$	$84.38$
$D_{s}^{*}D_{s1}(2536)$	$-1.74$	$1.14$	$-0.01$	$0.11$	$M_{th},\Gamma_{th}$	$\bm{4682,75.05}$		$\bm{4941,84.38}$

Table 7: The mass shifts and strong decay widths for the

2P

- and

3P

...

” stands for the predicted values are negligibly small.

	$2^{3}P_{0}$		$2^{3}P_{2}$		$2^{1}P_{1}$		$2^{3}P_{1}$
	$\Delta M$	$\Gamma$	$\Delta M$	$\Gamma$	$\Delta M$	$\Gamma$	$\Delta M$	$\Gamma$
Channel	$[3865]$	$as~{}\chi_{c0}(3915)$	$[4002]$	$as~{}\chi_{c2}(3930)$	$[3968]$	$[3961]$	$[3965]$	$as~{}\chi_{c1}(3872)/[3990]$
$DD$	$+39.86$	$15.82$	$-0.23$	$12.10$	✗	✗	✗	✗
$D_{s}D_{s}$	✗	✗	$-1.24$	✗	✗	✗	✗	✗
$DD^{*}$	✗	✗	$-19.57$	$8.86$	$-6.82$	$66.15$	$-93.92/+24.79$	✗ $\bm{/59.56}$
$D_{s}D_{s}^{*}$	✗	✗	✗	✗	✗	✗	✗	✗
$D^{}D^{}$	✗	✗	$-33.60$	✗	✗	✗	✗	✗
Total	$+39.86$	$15.82$	$-54.64$	$20.96$	$-6.82$	$66.15$	$-93.92/+24.79$	✗ $\bm{/59.56}$
$M_{th},\Gamma_{th}$	$\bm{3905,15.82}$		$\bm{3947,20.96}$		$\bm{3961,66.15}$		$\bm{3871/3990,-/59.56}$
$M_{exp},\Gamma_{exp}$	$\bm{3922,18.8\pm 3.5}$		$\bm{3923,35.2\pm 2.2}$		$\bm{-,-}$		$\bm{3872,1.19\pm 0.21}$
	$3^{3}P_{0}$		$3^{3}P_{2}$		$3^{1}P_{1}$		$3^{3}P_{1}$
	$\Delta M$	$\Gamma$	$\Delta M$	$\Gamma$	$\Delta M$	$\Gamma$	$\Delta M$	$\Gamma$
Channel	$[4232]$	$[4234]$	$[4352]$	$[4337]$	$[4318]$	$[4307]$	$[4316]$	$as~{}\chi_{c1}(4274)$
$DD$	$+4.62$	$1.16$	$+0.28$	$1.69$	✗	✗	✗	✗
$D_{s}D_{s}$	$+1.82$	$7.89$	$-0.23$	$0.37$	✗	✗	✗	✗
$DD^{*}$	✗	✗	$-1.43$	$6.06$	$-2.03$	$4.95$	$+0.97$	$2.25$
$D_{s}D_{s}^{*}$	✗	✗	$-0.11$	$0.05$	$+0.59$	$3.94$	$+0.10$	$7.60$
$D^{}D^{}$	$+11.31$	$48.27$	$+2.34$	$3.32$	$+4.02$	$7.19$	$+2.74$	$2.00$
$D_{s}^{}D_{s}^{}$	$-8.44$	$4.47$	$-2.81$	$6.09$	$-3.38$	$4.65$	$-4.08$	$4.98$
$DD_{0}^{*}(2300)$	✗	✗	✗	✗	$+2.31$	$2.03$	$-0.06$	$0.03$
$D_{s}D_{s0}^{*}(2317)$	✗	✗	✗	✗	$-2.27$	$2.62$	$-0.02$	✗
$D^{}D_{0}^{}(2300)$	$-6.94$	✗	$+0.53$	$14.34$	$-0.04$	$0.05$	$-7.83$	$13.37$
$DD_{2}^{*}(2460)$	✗	✗	$-4.01$	✗	$-10.02$	✗	$-7.59$	✗
$DD_{1}(2430)$	✗	✗	$-1.27$	$21.41$	$-0.06$	$0.05$	$-8.00$	$3.95$
$DD_{1}(2420)$	✗	✗	$-8.18$	$2.67$	$-0.04$	$0.01$	$-5.41$	✗
Total	$+2.37$	$61.79$	$-14.89$	$56.00$	$-10.92$	$25.49$	$-29.18$	$34.18$
$M_{th},\Gamma_{th}$	$\bm{4234,61.79}$		$\bm{4337,56.00}$		$\bm{4307,25.49}$		$\bm{4287,34.18}$
$M_{exp},\Gamma_{exp}$	$\bm{-,-}$		$\bm{-,-}$		$\bm{-,-}$		$\bm{4286^{+8}_{-9},51\pm 7?}$

Table 8: The mass shifts and strong decay widths for the

4P

...

” stands for the predicted values are negligibly small.

Channel	$\Delta M$ [4551]	$\Gamma$ [4489]	$\Delta M$ [4660]	$\Gamma$ [4571]	$\Delta M$ [4625]	$\Gamma$ [4573]	$\Delta M$ [4625]	$\Gamma$ [4562]
	$4^{3}P_{0}$		$4^{3}P_{2}$		$4^{1}P_{1}$		$4^{3}P_{1}$
$DD$	$+1.10$	$0.34$	$...$	$0.40$	✗	✗	✗	✗
$D_{s}D_{s}$	$+0.46$	$1.29$	$-0.05$	$0.06$	✗	✗	✗	✗
$DD^{*}$	✗	✗	$-0.42$	$1.02$	$-0.54$	$1.01$	$-0.02$	$0.43$
$D_{s}D_{s}^{*}$	✗	✗	$-0.06$	$0.01$	$+0.07$	$0.41$	$+0.23$	$0.93$
$D^{}D^{}$	$+1.68$	$7.75$	$+0.26$	$0.39$	$+0.52$	$0.76$	$+0.03$	$0.05$
$D_{s}^{}D_{s}^{}$	$-1.06$	$0.67$	$-0.24$	$1.25$	$-0.30$	$1.38$	$-0.22$	$1.03$
$DD_{0}(2550)$	$-15.04$	$32.77$	$-0.10$	$0.01$	✗	✗	✗	✗
$D_{s}D_{s}(2^{1}S_{0})$	$-2.75$	✗	$-0.82$	$0.12$	✗	✗	✗	✗
$D_{0}(2550)D^{*}$	✗	✗	$-5.54$	$10.36$	$-7.45$	$14.21$	$-0.98$	$10.79$
$DD_{1}^{*}(2600)$	✗	✗	$-1.25$	$12.31$	$-5.50$	$21.15$	$-8.32$	$12.30$
$D_{s}D_{s}^{*}(2700)$	✗	✗	$-0.63$	✗	$-1.40$	✗	$-1.46$	✗
$D^{}D_{1}^{}(2600)$	✗	✗	$-12.81$	✗	$-12.31$	✗	$-10.60$	✗
$DD_{0}^{*}(2300)$	✗	✗	✗	✗	$+0.59$	$0.14$	$-0.01$	$0.03$
$D_{s}D_{s0}^{*}(2317)$	✗	✗	✗	✗	$+0.18$	$1.05$	$-0.01$	$...$
$D^{}D_{0}^{}(2300)$	$-2.10$	$7.14$	$+0.58$	$1.22$	$-0.01$	$...$	$+0.95$	$3.93$
$D_{s}^{}D_{s0}^{}(2317)$	$-0.25$	$1.08$	$-0.29$	$0.24$	$...$	$...$	$-0.60$	$0.31$
$DD_{2}^{*}(2460)$	✗	✗	$+0.58$	$1.90$	$-3.25$	$7.74$	$-1.41$	$4.20$
$D^{}D_{2}^{}(2460)$	$-27.58$	$5.25$	$-4.44$	$12.81$	$-6.28$	$20.14$	$-14.86$	$31.90$
$D_{s}D_{s2}^{*}(2573)$	✗	✗	$-0.34$	✗	$-1.00$	✗	$-0.77$	✗
$D_{s}^{}D_{s2}^{}(2573)$	✗	✗	$-2.87$	✗	$-2.40$	✗	$-2.38$	✗
$DD_{1}(2430)$	$-3.35$	$5.86$	$+0.70$	$1.86$	$-0.01$	$...$	$+0.47$	$5.11$
$D_{s}D_{s1}(2460)$	$-0.55$	$0.83$	$-0.35$	$0.28$	$...$	$...$	$-0.71$	$0.32$
$DD_{1}(2420)$	$-6.93$	$15.01$	$-0.26$	$0.05$	$-0.01$	$...$	$+0.16$	$5.28$
$D_{s}D_{s1}(2536)$	$-2.90$	✗	$-0.69$	$0.68$	$...$	$...$	$+0.04$	$1.05$
$D^{*}D_{1}(2430)$	$+0.34$	$8.42$	$-31.28$	$49.17$	$-0.01$	$0.01$	$-3.75$	$2.36$
$D_{s}^{*}D_{s1}(2460)$	$-1.25$	✗	$-14.04$	$...$	$-0.01$	$...$	$-1.40$	✗
$D^{*}D_{1}(2420)$	$+0.66$	$22.29$	$-4.24$	$2.46$	$...$	$...$	$-4.48$	$8.77$
$D_{s}^{*}D_{s1}(2536)$	✗	✗	$-3.25$	✗	$...$	✗	$-1.83$	✗
$DD(1^{3}D_{1})$	✗	✗	$-0.15$	$0.42$	$-2.08$	$2.44$	$-0.73$	$0.20$
$DD(1^{3}D_{3})$	✗	✗	$-2.71$	✗	$-8.63$	✗	$-5.78$	✗
$DD(1D_{2})$	✗	✗	$-1.38$	✗	$-0.01$	✗	$-0.36$	✗
$DD(1D^{\prime}_{2})$	✗	✗	$-2.05$	✗	$-0.01$	✗	$-2.81$	✗
$D^{}(2300)D^{}(2300)$	$-2.26$	✗	$-0.07$	$0.20$	✗	✗	✗	✗
$D_{s}^{}(2317)D_{s}^{}(2317)$	✗	✗	$-0.02$	✗	✗	✗	✗	✗
$D^{*}(2300)D_{1}(2430)$	✗	✗	$-0.17$	✗	$-0.56$	✗	$-1.25$	✗
$D^{*}(2300)D_{1}(2420)$	✗	✗	$-0.11$	✗	$-1.27$	✗	$-0.48$	✗
Total	$-61.78$	$108.70$	$-88.51$	$97.22$	$-51.68$	$70.44$	$-63.34$	$88.99$
$M_{th},\Gamma_{th}$	$\bm{4489,108.70}$		$\bm{4571,97.22}$		$\bm{4573,70.44}$		$\bm{4562,88.99}$
$M_{exp},\Gamma_{exp}$	$\bm{4474\pm 4,77^{+12}_{-10}?}$		$\bm{-,-}$		$\bm{-,-}$		$\bm{-,-}$

Table 9: The mass shifts and strong decay widths for the

5^{3}P_{0}

and

5^{3}P_{2}

...

” stands for the predicted values are negligibly small.

Channel	$\Delta M$ [4839]	$\Gamma$ [4780]	$\Delta M$ [4939]	$\Gamma$ [4882]	Continue	$\Delta M$ [4839]	$\Gamma$ [4780]	$\Delta M$ [4939]	$\Gamma$ [4882]
	$5^{3}P_{0}$		$5^{3}P_{2}$			$5^{3}P_{0}$		$5^{3}P_{2}$
$DD$	$+0.28$	$0.04$	$+0.01$	$0.10$	$D^{*}D(2^{3}P_{2})$	✗	✗	$-12.12$	✗
$D_{s}D_{s}$	$+0.15$	$0.20$	$-0.01$	$0.02$	$D^{*}D_{1}(2430)$	$-2.11$	$2.79$	$+0.42$	$21.36$
$DD^{*}$	✗	✗	$-0.07$	$0.32$	$D^{*}D(2P_{1})$	✗	✗	$-3.77$	✗
$D_{s}D_{s}^{*}$	✗	✗	$-0.03$	$0.01$	$D_{s}^{*}D_{s1}(2460)$	$-0.31$	$0.13$	$-0.27$	$4.63$
$D^{}D^{}$	$+0.45$	$1.21$	$-0.11$	$0.11$	$D^{*}D_{1}(2420)$	$-1.13$	$1.24$	$+0.29$	$9.16$
$D_{s}^{}D_{s}^{}$	$-0.20$	$0.43$	$+0.03$	$0.17$	$D^{*}D(2P^{\prime}_{1})$	✗	✗	$-3.77$	✗
$DD_{0}(2550)$	$+3.32$	$8.66$	$-0.29$	$0.97$	$D_{s}^{*}D_{s1}(2536)$	✗	✗	$-0.23$	$0.08$
$DD(3^{1}S_{0})$	$-4.12$	✗	$-1.75$	✗	$DD(1^{3}D_{1})$	✗	✗	$+0.01$	$0.08$
$D_{s}D_{s}(2^{1}S_{0})$	$-1.71$	$2.15$	$+0.01$	$0.08$	$D_{s}D_{s}(1^{3}D_{1})$	✗	✗	$-0.01$	$0.04$
$D_{0}(2550)D^{*}$	✗	✗	$+0.15$	$0.07$	$DD(1^{3}D_{3})$	✗	✗	$+0.18$	$1.27$
$DD_{1}^{*}(2600)$	✗	✗	$-0.19$	$0.02$	$D_{s}D_{s}(1^{3}D_{3})$	✗	✗	$-0.38$	$0.52$
$DD^{*}(3^{3}S_{1})$	✗	✗	$-1.84$	✗	$D^{*}D(1^{3}D_{1})$	$-1.66$	$8.23$	$...$	$...$
$D_{s}(2^{1}S_{0})D_{s}^{*}$	✗	✗	$-0.27$	$0.31$	$D^{*}D(1^{3}D_{3})$	$-11.69$	$0.07$	$...$	$...$
$D_{s}D_{s}^{*}(2700)$	✗	✗	$-0.23$	$0.43$	$DD(1D_{2})$	$-0.41$	$...$	$-0.18$	$1.63$
$D^{}D_{1}^{}(2600)$	$+0.44$	$0.62$	$-1.34$	$13.16$	$DD(1D^{\prime}_{2})$	$+0.82$	$9.53$	$+0.38$	$2.24$
$D_{s}^{}D_{s}^{}(2700)$	$-1.38$	✗	$-1.14$	$3.11$	$D_{s}D_{s}(1D_{2})$	$-0.40$	✗	$-0.10$	$0.25$
$DD_{2}^{*}(2460)$	✗	✗	$-0.23$	$0.15$	$D_{s}D_{s}(1D^{\prime}_{2})$	✗	✗	$-0.32$	$0.02$
$DD(2^{3}P_{2})$	✗	✗	$-2.97$	$6.61$	$D^{*}D(1D_{2})$	$-0.89$	$0.31$	$+0.11$	$1.04$
$D_{0}(2550)D_{2}^{*}(2460)$	✗	✗	$-1.60$	✗	$D^{*}D(1D^{\prime}_{2})$	$-4.93$	✗	$-2.75$	$3.43$
$D_{0}(2550)D(2^{3}P_{2})$	✗	✗	$-1.01$	✗	$D^{}(2300)D^{}(2300)$	$-0.77$	$3.58$	$-0.01$	$...$
$D_{s}D_{s2}^{*}(2573)$	✗	✗	$-0.07$	$0.29$	$D_{s}^{}(2317)D_{s}^{}(2317)$	$-0.54$	$0.51$	$...$	$0.01$
$D^{}D^{}(2300)$	$+0.23$	$2.68$	$+0.10$	$0.02$	$D^{*}(2300)D_{1}(2430)$	✗	✗	$-0.05$	$0.24$
$D^{*}D(2^{3}P_{0})$	$-1.51$	✗	$-0.86$	$6.44$	$D_{s0}^{*}(2317)D_{s1}(2460)$	✗	✗	$+0.01$	$0.03$
$D_{1}^{}(2600)D^{}(2300)$	✗	✗	$-1.87$	$0.32$	$D^{*}(2300)D_{1}(2420)$	✗	✗	$-0.05$	$0.12$
$D_{s}^{}D_{s0}^{}(2317)$	$-0.38$	$0.55$	$+0.03$	$0.20$	$D_{s0}^{*}(2317)D_{s1}(2536)$	✗	✗	$-0.04$	$0.05$
$DD_{1}(2430)$	$-0.24$	$3.40$	$+0.13$	$0.03$	$D^{}(2300)D_{2}^{}(2460)$	$-0.45$	$4.17$	$...$	$...$
$DD(2P_{1})$	$-3.77$	$5.23$	$-1.07$	$4.01$	$D_{s}^{}(2317)D_{s2}^{}(2573)$	✗	✗	$...$	✗
$D_{0}(2550)D_{1}(2430)$	✗	✗	$-0.94$	✗	$D_{2}^{*}(2460)D_{1}(2430)$	✗	✗	$-4.51$	$4.44$
$D_{s}D_{s1}(2460)$	$-0.53$	$0.67$	$+0.03$	$0.23$	$D_{2}^{*}(2460)D_{1}(2420)$	✗	✗	$-4.01$	✗
$D_{s}D_{s}(2P_{1})$	✗	✗	$-0.45$	✗	$D_{2}^{}(2460)D_{2}^{}(2460)$	✗	✗	$...$	✗
$DD_{1}(2420)$	$+0.38$	$4.69$	$-0.34$	$0.44$	$D_{1}(2430)D_{1}(2430)$	$-1.43$	✗	$-0.18$	$1.81$
$DD(2P^{\prime}_{1})$	$-10.37$	✗	$-2.03$	$6.74$	$D_{s1}(2460)D_{s1}(2460)$	✗	✗	$-0.15$	✗
$D_{0}(2550)D_{1}(2420)$	✗	✗	$-1.58$	✗	$D_{1}(2420)D_{1}(2420)$	$-1.70$	✗	$-1.08$	$1.28$
$D_{s}D_{s1}(2536)$	$-0.26$	$...$	$...$	$0.17$	$D_{1}(2430)D_{1}(2420)$	$-6.85$	✗	$-3.38$	$1.22$
$D^{}D_{2}^{}(2460)$	$-4.88$	$10.14$	$+0.87$	$7.61$	Total	$-58.55$	$71.23$	$-56.97$	$107.74$
$D_{s}^{}D_{s2}^{}(2573)$	✗	✗	$-0.08$	$0.65$	$M_{th},\Gamma_{th}$	$\bm{4780,71.23}$		$\bm{4882,107.74}$

Table 10: The mass shifts and strong decay widths for the

5^{1}P_{1}

and

5^{3}P_{1}

...

” stands for the predicted values are negligibly small.

Channel	$\Delta M$ [4905]	$\Gamma$ [4875]	$\Delta M$ [4905]	$\Gamma$ [4856]	Continue	$\Delta M$ [4905]	$\Gamma$ [4875]	$\Delta M$ [4905]	$\Gamma$ [4856]
	$5^{1}P_{1}$		$5^{3}P_{1}$			$5^{1}P_{1}$		$5^{3}P_{1}$
$DD^{*}$	$-0.15$	$0.37$	$-0.05$	$0.16$	$D^{*}D(2P_{1})$	$-6.84$	✗	$-5.60$	✗
$D_{s}D_{s}^{*}$	$-0.01$	$0.04$	$+0.05$	$0.10$	$D_{s}^{*}D_{s1}(2460)$	$-0.76$	$0.63$	$-0.54$	$0.28$
$D^{}D^{}$	$-0.01$	$0.09$	$-0.13$	$0.02$	$D^{*}D_{1}(2420)$	$+0.83$	$2.95$	$+0.23$	$4.24$
$D_{s}^{}D_{s}^{}$	$+0.02$	$0.24$	$...$	$0.14$	$D_{s}^{*}D_{s1}(2536)$	$-0.26$	$0.17$	$-0.05$	$0.37$
$D_{0}(2550)D^{*}$	$+1.28$	$2.20$	$+1.42$	$4.77$	$DD(1^{3}D_{1})$	$+0.41$	$1.76$	$+0.10$	$0.64$
$DD_{1}^{*}(2600)$	$+1.01$	$1.33$	$+1.73$	$3.76$	$DD(1^{3}D_{3})$	$+0.57$	$6.10$	$-0.17$	$4.95$
$D_{s}(2^{1}S_{0})D_{s}^{*}$	$-0.55$	$0.28$	$-0.38$	$0.02$	$D_{s}D_{s}(1^{3}D_{1})$	$-0.06$	$0.27$	$...$	$0.16$
$D_{s}D_{s}^{*}(2700)$	$-0.69$	$0.69$	$-0.75$	$0.28$	$D_{s}D_{s}(1^{3}D_{3})$	$-1.31$	$1.87$	$-1.08$	$0.81$
$D^{}D_{1}^{}(2600)$	$-2.38$	$13.55$	$-2.70$	$11.63$	$D^{*}D(1^{3}D_{1})$	$-0.34$	$0.09$	$-1.39$	$0.43$
$D_{s}^{}D_{s}^{}(2700)$	$-1.24$	$3.78$	$-2.78$	$3.46$	$D^{*}D(1^{3}D_{3})$	$-2.19$	$11.68$	$-8.27$	$12.34$
$DD^{*}(2300)$	$+0.19$	$0.01$	$...$	$0.01$	$DD(1D_{2})$	$...$	$...$	$-0.20$	$0.36$
$DD(2^{3}P_{0})$	$-0.49$	$3.93$	$-0.01$	$...$	$DD(1D^{\prime}_{2})$	$...$	$...$	$-1.07$	$0.50$
$D_{0}(2550)D^{*}(2300)$	$-1.13$	$0.86$	$...$	$...$	$D^{*}D(1D_{2})$	$+0.06$	$1.80$	$+0.53$	$3.02$
$D_{s}D_{s0}^{*}(2317)$	$+0.13$	$0.17$	$...$	$...$	$D^{*}D(1D^{\prime}_{2})$	$-0.26$	$3.76$	$-3.61$	$4.85$
$D_{s}D_{s}(2^{3}P_{0})$	$-0.39$	✗	$...$	✗	$D_{s}D_{s}(1D_{2})$	$...$	$...$	$-0.09$	$0.09$
$D_{s}(2^{1}S_{0})D_{s0}^{*}(2317)$	$-0.33$	✗	$...$	✗	$D_{s}D_{s}(1D^{\prime}_{2})$	$...$	✗	$-0.46$	✗
$D^{}D^{}(2300)$	$...$	$0.01$	$+0.48$	$0.63$	$D^{}(2300)D_{2}^{}(2460)$	$...$	$...$	$-0.08$	$...$
$D^{*}D(2^{3}P_{0})$	$-0.01$	$0.02$	$-3.31$	✗	$D_{s}^{}(2317)D_{s2}^{}(2573)$	$...$	✗	$-0.18$	✗
$D_{1}^{}(2600)D^{}(2300)$	$-0.01$	✗	$-1.22$	✗	$D^{*}(2300)D_{1}(2430)$	$-0.48$	$0.92$	$-1.02$	$1.15$
$D_{s}^{}D_{s0}^{}(2317)$	$...$	$...$	$-0.03$	$0.63$	$D_{s0}^{*}(2317)D_{s1}(2460)$	$-0.06$	$0.07$	$-0.17$	$0.30$
$DD_{2}^{*}(2460)$	$+0.10$	$0.42$	$+0.40$	$0.77$	$D^{*}(2300)D_{1}(2420)$	$-1.03$	$1.27$	$-0.36$	$0.33$
$DD(2^{3}P_{2})$	$-4.38$	$16.78$	$-4.03$	$20.07$	$D_{s0}^{*}(2317)D_{s1}(2536)$	$-0.18$	$0.73$	$-0.27$	$0.50$
$D_{s}D_{s2}^{*}(2573)$	$-0.32$	$0.75$	$-0.31$	$0.42$	$D_{2}^{*}(2460)D_{1}(2430)$	$-4.23$	$0.87$	$-5.65$	✗
$D^{}D_{2}^{}(2460)$	$+0.51$	$6.51$	$+0.82$	$7.00$	$D_{2}^{*}(2460)D_{1}(2420)$	$-4.60$	✗	$-2.72$	✗
$D_{s}^{}D_{s2}^{}(2573)$	$-0.03$	$0.67$	$+0.10$	$0.64$	$D_{2}^{}(2460)D_{2}^{}(2460)$	$...$	✗	$...$	✗
$DD_{1}(2430)$	$-0.01$	$0.01$	$+0.52$	$1.11$	$D_{1}(2430)D_{1}(2430)$	$-0.20$	$1.02$	$+0.22$	$1.03$
$DD(2P_{1})$	$...$	$...$	$-0.30$	$0.01$	$D_{s1}(2460)D_{s1}(2460)$	$-0.17$	✗	$-0.20$	✗
$D_{s}D_{s1}(2460)$	$...$	$...$	$-0.08$	$0.77$	$D_{1}(2420)D_{1}(2420)$	$-0.06$	$0.32$	$-2.11$	$2.71$
$DD_{1}(2420)$	$...$	$...$	$+0.02$	$0.05$	$D_{1}(2430)D_{1}(2420)$	$-0.47$	$1.34$	$-6.03$	$3.99$
$DD(2P^{\prime}_{1})$	$...$	$...$	$+2.31$	$6.43$
$D_{s}D_{s1}(2536)$	$...$	$...$	$-0.09$	$0.13$	Total	$-30.23$	$96.46$	$-48.71$	$110.37$
$D^{*}D_{1}(2430)$	$+0.29$	$6.13$	$-0.15$	$4.31$	$M_{th},\Gamma_{th}$	$\bm{4875,96.46}$		$\bm{4856,110.37}$

Table 11: The mass shifts and strong decay widths for the

1D

- and

2D

...

” stands for the predicted values are negligibly small.

Channel	$[3842]$	$[3814]$	$[3844]$	$[3809]$
	$1^{3}D_{1}$		$1^{3}D_{3}$
	$\Delta M$	$\Gamma$	$\Delta M$	$\Gamma$
Channel	$[3823]$	$as~{}\psi(3770)$	$[3847]$	$as~{}\psi_{3}(3842)$
$DD$	$-35.12$	$35.11$	$-4.07$	$1.24$
Total	$-35.12$	$35.11$	$-4.07$	$1.24$
$M_{th},\Gamma_{th}$	$\bm{3788,35.11}$		$\bm{3843,1.24}$
$M_{exp},\Gamma_{exp}$	$\bm{3774,27.2\pm 1.0}$		$\bm{3843,2.8\pm 0.6}$
	$1^{1}D_{2}$		$1^{3}D_{2}$
	$\Delta M$	$\Gamma$	$\Delta M$	$\Gamma$
$DD^{*}$	$-27.65$	✗	$-34.76$	✗
Total	$-27.65$	✗	$-34.76$	✗
$M_{th},\Gamma_{th}$	$\bm{3814,-}$		$\bm{3809,-}$
$M_{exp},\Gamma_{exp}$	$\bm{-,-}$		$\bm{3824,<2.5}$
	$2^{3}D_{1}$		$2^{3}D_{3}$
	$\Delta M$	$\Gamma$	$\Delta M$	$\Gamma$
Channel	$[4176]$	$as~{}\psi(4160)$	$[4216]$	$[4213]$
$DD$	$-1.65$	$7.25$	$+0.96$	$1.07$
$D_{s}D_{s}$	$+0.37$	$...$	$-0.14$	$2.70$
$DD^{*}$	$+1.30$	$0.65$	$+2.49$	$13.44$
$D_{s}D_{s}^{*}$	$-1.12$	$7.65$	$-2.22$	$2.62$
$D^{}D^{}$	$-19.03$	$46.27$	$+2.75$	$18.24$
$D_{s}^{}D_{s}^{}$	✗	✗	$-7.09$	✗
Total	$-20.13$	$61.82$	$-3.25$	$38.07$
$M_{th},\Gamma_{th}$	$\bm{4156,61.82}$		$\bm{4213,38.07}$
$M_{exp},\Gamma_{exp}$	$\bm{4191\pm 5,70\pm 10}$		$\bm{-,-}$
	$2^{1}D_{2}$		$2^{3}D_{2}$
	$\Delta M$	$\Gamma$	$\Delta M$	$\Gamma$
Channel	$[4206]$	$[4192]$	$[4207]$	$[4192]$
$DD^{*}$	$-1.08$	$20.64$	$-2.64$	$16.68$
$D_{s}D_{s}^{*}$	$-1.75$	$7.34$	$-0.53$	$9.81$
$D^{}D^{}$	$-1.84$	$26.14$	$-7.74$	$29.49$
$D_{s}^{}D_{s}^{}$	$-4.29$	✗	$-3.71$	✗
$DD_{0}^{*}(2300)$	$-5.04$	$5.44$	$-0.23$	$0.27$
Total	$-14.00$	$59.56$	$-14.85$	$56.25$
$M_{th},\Gamma_{th}$	$\bm{4192,59.56}$		$\bm{4192,56.25}$

Table 12: The mass shifts and strong decay widths for the

3D

...

” stands for the predicted values are negligibly small.

Channel	$\Delta M$ [4486]	$\Gamma$ [4463]	$\Delta M$ [4535]	$\Gamma$ [4461]	$\Delta M$ [4522]	$\Gamma$ [4461]	$\Delta M$ [4523]	$\Gamma$ [4468]
	$3^{3}D_{1}$		$3^{3}D_{3}$		$3^{1}D_{2}$		$3^{3}D_{2}$
$DD$	$-0.23$	$0.86$	$+0.15$	$0.46$	✗	✗	✗	✗
$D_{s}D_{s}$	$+0.04$	$0.01$	$-0.11$	$0.34$	✗	✗	✗	✗
$DD^{*}$	$+0.28$	$0.01$	$-0.15$	$2.46$	$-0.81$	$3.55$	$-0.88$	$2.96$
$D_{s}D_{s}^{*}$	$+0.30$	$0.94$	$-0.22$	$0.07$	$-0.12$	$0.36$	$+0.04$	$0.53$
$D^{}D^{}$	$-1.64$	$2.43$	$-0.22$	$1.20$	$-0.41$	$1.79$	$-1.18$	$2.23$
$D_{s}^{}D_{s}^{}$	$+0.10$	$3.21$	$-0.38$	$2.31$	$-0.20$	$2.02$	$+0.02$	$1.66$
$DD_{0}(2550)$	$+6.20$	$29.03$	$-4.30$	$4.33$	✗	✗	✗	✗
$D_{0}(2550)D^{*}$	✗	✗	$-2.98$	✗	$-6.70$	✗	$-7.69$	✗
$DD_{1}^{*}(2600)$	$-4.50$	✗	$-3.08$	✗	$-7.71$	✗	$-9.43$	✗
$DD_{0}^{*}(2300)$	✗	✗	✗	✗	$-0.73$	$1.04$	$-0.03$	$0.07$
$D_{s}D_{s0}^{*}(2317)$	✗	✗	✗	✗	$+0.12$	$0.41$	$-0.01$	$...$
$D^{}D_{0}^{}(2300)$	$+2.00$	$2.81$	$+0.49$	$0.67$	$-0.03$	$...$	$+1.03$	$1.02$
$D_{s}^{}D_{s0}^{}(2317)$	$-1.29$	$0.88$	$-0.56$	$0.33$	$-0.02$	$0.01$	$-0.85$	$0.74$
$DD_{2}^{*}(2460)$	$+1.39$	$10.59$	$-1.83$	$6.71$	$+0.96$	$6.78$	$-5.05$	$22.25$
$D^{}D_{2}^{}(2460)$	$-5.59$	✗	$-41.50$	✗	$-22.85$	✗	$-19.35$	✗
$D_{s}D_{s2}^{*}(2573)$	✗	✗	$-0.73$	✗	$-0.40$	✗	$-1.87$	✗
$DD_{1}(2430)$	$+1.22$	$4.19$	$+1.06$	$1.66$	$-0.02$	$...$	$+1.23$	$2.07$
$D_{s}D_{s1}(2460)$	$-0.92$	$0.55$	$-0.98$	$0.61$	$-0.02$	$0.01$	$-0.83$	$0.70$
$DD_{1}(2420)$	$+1.47$	$33.28$	$-2.86$	$6.00$	$-0.01$	$...$	$+1.05$	$2.25$
$D_{s}D_{s1}(2536)$	$-3.16$	✗	$-0.58$	✗	$-0.01$	✗	$-0.63$	✗
$D^{*}D_{1}(2430)$	$-7.70$	$8.00$	$-8.76$	$9.40$	$-9.38$	$10.11$	$-6.38$	$8.56$
$D^{*}D_{1}(2420)$	$-10.63$	$22.14$	$-6.86$	$1.56$	$-12.27$	$4.11$	$-4.00$	$20.02$
Total	$-22.66$	$118.93$	$-74.4$	$38.11$	$-60.61$	$30.19$	$-54.81$	$65.06$
$M_{th},\Gamma_{th}$	$\bm{4463,118.93}$		$\bm{4461,38.11}$		$\bm{4461,30.19}$		$\bm{4468,65.06}$

Table 13: The mass shifts and strong decay widths for the

4D

...

” stands for the predicted values are negligibly small.

Channel	$\Delta M$ [4767]	$\Gamma$ [4737]	$\Delta M$ [4823]	$\Gamma$ [4791]	$\Delta M$ [4808]	$\Gamma$ [4776]	$\Delta M$ [4808]	$\Gamma$ [4777]
	$4^{3}D_{1}$		$4^{3}D_{3}$		$4^{1}D_{2}$		$4^{3}D_{2}$
$DD$	$-0.03$	$0.13$	$+0.04$	$...$	✗	✗	✗	✗
$D_{s}D_{s}$	$+0.02$	$...$	$-0.01$	$0.07$	✗	✗	✗	✗
$DD^{*}$	$+0.11$	$...$	$+0.02$	$0.14$	$-0.16$	$1.00$	$-0.19$	$0.87$
$D_{s}D_{s}^{*}$	$+0.08$	$0.19$	$-0.06$	$0.28$	$-0.08$	$0.07$	$-0.04$	$0.06$
$D^{}D^{}$	$-0.39$	$0.42$	$-0.33$	$0.61$	$-0.24$	$0.55$	$-0.35$	$0.74$
$D_{s}^{}D_{s}^{}$	$...$	$0.52$	$+0.03$	$0.21$	$+0.02$	$0.25$	$-0.01$	$0.16$
$DD_{0}(2550)$	$+0.80$	$1.16$	$...$	$2.28$	✗	✗	✗	✗
$D_{s}D_{s}(2^{1}S_{0})$	$-0.48$	$0.40$	$-0.18$	$2.28$	✗	✗	✗	✗
$D_{0}(2550)D^{*}$	$+0.09$	$4.50$	$-0.89$	$0.02$	$-0.15$	$3.69$	$+0.46$	$5.36$
$DD_{1}^{*}(2600)$	$+0.74$	$4.55$	$-0.99$	$1.00$	$-0.65$	$1.47$	$+0.05$	$1.94$
$D_{s}(2^{1}S_{0})D_{s}^{*}$	$-0.21$	✗	$-0.26$	$0.16$	$-0.55$	$1.27$	$-0.52$	$2.02$
$D_{s}D_{s}^{*}(2700)$	$-0.24$	$0.94$	$-0.16$	$2.10$	$-0.64$	$1.55$	$-0.80$	$1.06$
$D_{s}^{}D_{s}^{}(2700)$	✗	✗	$-0.63$	✗	$-0.50$	✗	$-0.47$	✗
$D^{}D_{1}^{}(2600)$	$-5.63$	$39.75$	$-2.85$	$21.30$	$-2.54$	$20.40$	$-1.23$	$22.71$
$DD_{0}^{*}(2300)$	✗	✗	✗	✗	$-0.15$	$0.21$	$...$	$0.02$
$D_{s}D_{s0}^{*}(2317)$	✗	✗	✗	✗	$-0.03$	$...$	$-0.02$	$0.01$
$D^{}D_{0}^{}(2300)$	$+0.53$	$1.37$	$-0.08$	$0.03$	$-0.01$	$...$	$...$	$0.24$
$D_{s}^{}D_{s0}^{}(2317)$	$-0.07$	$0.72$	$+0.03$	$0.13$	$...$	$...$	$+0.06$	$0.33$
$DD_{2}^{*}(2460)$	$+0.38$	$0.85$	$-0.53$	$1.06$	$+0.54$	$0.42$	$+0.57$	$1.86$
$D^{}D_{2}^{}(2460)$	$-0.68$	$2.19$	$+0.25$	$14.86$	$-0.60$	$8.69$	$-0.61$	$6.40$
$D_{s}D_{s2}^{*}(2573)$	$-0.24$	$0.10$	$-0.09$	$0.41$	$-0.22$	$0.42$	$-0.23$	$0.94$
$D_{s}^{}D_{s2}^{}(2573)$	$-0.11$	$0.36$	$-0.69$	$3.69$	$-0.82$	$2.73$	$-0.83$	$2.83$
$DD_{1}(2430)$	$+0.54$	$1.27$	$-0.14$	$0.12$	$-0.01$	$0.02$	$+0.26$	$0.05$
$DD(2P_{1})$	$-1.39$	✗	$-2.12$	$3.70$	$...$	$...$	$-0.52$	$0.03$
$DD(2^{3}P_{2})$	✗	✗	$-0.99$	✗	$-1.13$	✗	$-1.15$	✗
$DD(2^{3}P_{0})$	✗	✗	✗	✗	$-1.14$	✗	$-0.02$	✗
$DD(2P^{\prime}_{1})$	$-4.14$	✗	$-0.81$	✗	$-0.02$	✗	$-1.82$	✗
$D_{s}D_{s1}(2460)$	$-0.21$	$0.77$	$+0.05$	$0.19$	$-0.01$	$...$	$+0.05$	$0.40$
$DD_{1}(2420)$	$+1.52$	$3.25$	$-0.28$	$1.68$	$-0.01$	$0.01$	$-0.38$	$0.17$
$D_{s}D_{s1}(2536)$	$-1.00$	$0.93$	$-0.09$	$0.10$	$...$	$...$	$-0.07$	$0.32$
$D^{*}D_{1}(2430)$	$-0.60$	$5.35$	$+1.08$	$2.20$	$+1.07$	$4.21$	$+0.71$	$2.97$
$D_{s}^{*}D_{s1}(2460)$	$-0.29$	$...$	$-0.54$	$0.78$	$-0.66$	$0.57$	$-0.47$	$0.41$
$D^{*}D_{1}(2420)$	$-1.18$	$8.66$	$-0.65$	$0.76$	$-0.31$	$1.85$	$-0.04$	$5.88$
$D_{s}^{*}D_{s1}(2536)$	$-0.35$	$1.99$	$-0.31$	$1.04$	$-0.46$	$1.23$	$-0.47$	$1.33$
$DD(1^{3}D_{1})$	$-0.22$	$0.08$	$+0.02$	$0.10$	$-0.36$	$1.20$	$-0.16$	$0.36$
$D_{s}D_{s}(1^{3}D_{1})$	✗	✗	$-0.01$	✗	$-0.15$	✗	$-0.05$	✗
$DD(1^{3}D_{3})$	$-0.72$	$5.67$	$-0.63$	$3.95$	$-2.79$	$12.05$	$-2.67$	$7.86$
$D_{s}D_{s}(1^{3}D_{3})$	✗	✗	$-0.25$	✗	$-0.79$	✗	$-0.59$	✗
$DD(1D_{2})$	$-1.08$	$2.69$	$-0.62$	$2.52$	$-0.01$	$0.01$	$-0.34$	$0.59$
$D_{s}D_{s}(1D_{2})$	✗	✗	$-0.26$	✗	$...$	✗	$-0.12$	✗
$DD(1D^{\prime}_{2})$	$-1.24$	$17.78$	$-0.90$	$1.22$	$-0.01$	$0.02$	$-1.23$	$4.31$
$D_{s}D_{s}(1D^{\prime}_{2})$	✗	✗	$-0.17$	✗	$...$	✗	$-0.26$	✗
$D^{*}D(1^{3}D_{1})$	$-2.11$	✗	$-0.84$	$0.43$	$-0.57$	$0.49$	$-1.71$	$1.75$
$D^{*}D(1^{3}D_{3})$	$-5.06$	✗	$-11.92$	$45.01$	$-7.73$	$4.37$	$-6.16$	$1.84$
$D^{*}D(1D_{2})$	$-1.34$	✗	$-1.99$	$5.31$	$-4.78$	$4.30$	$-3.62$	$3.61$
$D^{*}D(1D^{\prime}_{2})$	$-3.56$	✗	$-1.61$	✗	$-3.85$	✗	$-3.50$	✗
$D^{*}(2300)D_{1}(2430)$	$-0.02$	✗	$-0.06$	$0.04$	$-0.29$	$0.62$	$-0.52$	$0.94$
$D^{}(2300)D^{}(2300)$	$-0.08$	$0.21$	$-0.01$	$0.07$	✗	✗	✗	✗
$D^{}(2300)D^{}(2460)$	$-2.00$	✗	$+0.05$	✗	$-0.01$	✗	$-1.14$	✗
$D_{s}^{}(2317)D_{s}^{}(2317)$	$-0.16$	$0.06$	$...$	$0.02$	✗	✗	✗	✗
$D_{s}^{*}(2317)D_{s1}(2460)$	$-0.14$	✗	$-0.09$	$...$	$-0.24$	✗	$-0.50$	$...$
$D_{s}^{*}(2317)D_{s1}(2536)$	✗	✗	$-0.03$	✗	$-0.24$	✗	$-0.12$	✗
$D^{*}(2300)D_{1}(2420)$	$+0.03$	✗	$-0.09$	$0.01$	$-0.54$	$0.32$	$-0.25$	$0.12$
Total	$-30.13$	$106.86$	$-31.59$	$119.88$	$-31.82$	$73.99$	$-31.02$	$80.49$
$M_{th},\Gamma_{th}$	$\bm{4737,106.86}$		$\bm{4791,119.88}$		$\bm{4776,73.99}$		$\bm{4777,80.49}$

Table 14: The mass shifts and strong decay widths for the

5^{3}D_{1}

and

5^{3}D_{3}

...

” stands for the predicted values are negligibly small.

	$5^{3}D_{1}$		$5^{3}D_{3}$			$5^{3}D_{1}$		$5^{3}D_{3}$
	$\Delta M$ [5029]	$\Gamma$ [4988]	$\Delta M$ [5091]	$\Gamma$ [5061]	Continue	$\Delta M$ [5029]	$\Gamma$ [4988]	$\Delta M$ [5091]	$\Gamma$ [5061]
$DD$	$+0.01$	$0.02$	$+0.01$	$0.02$	$DD(1D_{2})$	$+0.08$	$1.51$	$+0.16$	$0.46$
$D_{s}D_{s}$	$+0.02$	$0.01$	$...$	$0.02$	$D_{s}D_{s}(1D_{2})$	$-0.08$	$0.04$	$-0.10$	$0.07$
$DD^{*}$	$+0.04$	$...$	$+0.01$	$0.14$	$DD(1D^{\prime}_{2})$	$-0.72$	$1.61$	$-0.30$	$0.35$
$D_{s}D_{s}^{*}$	$+0.03$	$0.05$	$-0.01$	$0.02$	$D_{s}D_{s}(1D^{\prime}_{2})$	$-0.18$	$0.27$	$-0.05$	$0.04$
$DD_{0}(2550)$	$+0.20$	$...$	$+0.09$	$0.41$	$D^{*}D(1^{3}D_{1})$	$-0.99$	$0.16$	$-0.07$	$0.53$
$DD(3^{1}S_{0})$	$-1.19$	$3.00$	$-0.32$	$0.24$	$D_{s}^{*}D_{s}(1^{3}D_{1})$	$-0.28$	$0.31$	$-0.03$	$0.08$
$D_{s}D_{s}(2^{1}S_{0})$	$+0.14$	$0.58$	$-0.06$	$0.09$	$D^{*}D(1^{3}D_{3})$	$-0.81$	$3.13$	$-1.62$	$6.05$
$D_{s}D_{s}(3^{1}S_{0})$	✗	✗	$-0.05$	✗	$D_{s}^{*}D_{s}(1^{3}D_{3})$	$-0.51$	$0.03$	$-0.73$	$0.72$
$D_{0}(2550)D^{*}$	$+0.32$	$0.64$	$-0.21$	$0.62$	$D^{*}D(1D_{2})$	$-0.50$	$0.17$	$-0.47$	$1.28$
$D(3^{1}S_{0})D^{*}$	$-0.26$	✗	$-0.25$	$0.04$	$D_{s}^{*}D_{s}(1D_{2})$	$-0.17$	$0.08$	$-0.16$	$0.20$
$DD_{1}^{*}(2600)$	$+0.42$	$0.60$	$-0.15$	$0.74$	$D^{*}D(1D^{\prime}_{2})$	$-0.58$	$1.14$	$-0.41$	$0.68$
$DD^{*}(3^{3}S_{1})$	$-0.74$	$0.10$	$-0.28$	$...$	$D_{s}^{*}D_{s}(1D^{\prime}_{2})$	$-0.35$	✗	$-0.16$	$0.03$
$D_{s}(2^{1}S_{0})D_{s}^{*}$	$-0.16$	$0.25$	$-0.02$	$0.02$	$DD(2^{3}D_{1})$	$-0.14$	✗	$-0.06$	$0.03$
$D_{s}D_{s}^{*}(2700)$	$-0.16$	$0.43$	$-0.05$	$...$	$DD(2^{3}D_{3})$	$-0.30$	✗	$-0.40$	✗
$D^{}D^{}$	$-0.08$	$0.08$	$-0.09$	$0.19$	$DD(2D_{2})$	$-0.71$	✗	$-1.35$	$0.04$
$D^{}D_{1}^{}(2600)$	$+0.41$	$4.06$	$+0.24$	$0.93$	$DD(2D^{\prime}_{2})$	$-0.99$	✗	$-0.32$	✗
$D^{}D^{}(3^{3}S_{1})$	✗	✗	$-1.17$	✗	$DD(1^{3}F_{2})$	$-0.06$	$0.03$	$...$	$...$
$D_{s}^{}D_{s}^{}$	$...$	$0.10$	$...$	$0.02$	$DD(1^{3}F_{4})$	$-0.27$	$0.25$	$...$	$...$
$D_{s}^{}D_{s}^{}(2700)$	$-0.27$	$0.03$	$-0.26$	$0.27$	$D_{s}D_{s}(1^{3}F_{4})$	✗	✗	$...$	✗
$D^{}D^{}(2300)$	$+0.18$	$0.15$	$-0.04$	$0.04$	$DD(1F_{3})$	$-5.28$	$1.50$	$-0.61$	$0.81$
$D_{1}^{}(2600)D^{}(2300)$	$-0.50$	$...$	$-0.49$	$1.40$	$D_{s}D_{s}(1F_{3})$	✗	✗	$-0.12$	✗
$D^{*}D(2^{3}P_{0})$	$-0.80$	$0.40$	$-0.13$	$1.00$	$DD(1F^{\prime}_{3})$	$-5.54$	✗	$-0.23$	$0.03$
$D_{s}^{}D_{s0}^{}(2317)$	$+0.02$	$0.21$	$+0.01$	$0.01$	$D^{*}D(1^{3}F_{2})$	✗	✗	$...$	✗
$D_{s}^{}(2700)D_{s0}^{}(2317)$	$-0.17$	✗	$-0.03$	$0.14$	$D^{*}D(1^{3}F_{4})$	✗	✗	$...$	$...$
$DD_{2}^{*}(2460)$	$+0.09$	$0.10$	$-0.13$	$0.39$	$D^{*}D(1F_{3})$	✗	✗	$-0.72$	$0.39$
$D_{0}(2550)D_{2}^{*}(2460)$	$-1.00$	$0.20$	$-0.58$	$0.82$	$D^{*}D(1F^{\prime}_{3})$	✗	✗	$-0.45$	✗
$DD(2^{3}P_{2})$	$-0.59$	$1.48$	$-0.42$	$0.28$	$D^{}(2300)D^{}(2300)$	$+0.12$	$0.53$	$-0.01$	$0.01$
$D_{s}D_{s2}^{*}(2573)$	$-0.05$	$0.13$	$-0.02$	$0.02$	$D_{s}^{}(2317)D_{s}^{}(2317)$	$-0.05$	$0.24$	$...$	$...$
$D_{s}D_{s}(2^{3}P_{2})$	$-0.11$	✗	$-0.32$	$0.01$	$D^{*}(2300)D_{1}(2430)$	$-0.24$	$0.47$	$+0.03$	$0.07$
$D^{}D_{2}^{}(2460)$	$-0.03$	$0.62$	$+0.38$	$1.16$	$D_{s0}^{*}(2317)D_{s1}(2460)$	$-0.07$	$...$	$-0.01$	$0.05$
$D^{*}D(2^{3}P_{2})$	$-0.43$	$0.23$	$-2.85$	$0.72$	$D^{*}(2300)D_{1}(2420)$	$-0.11$	$0.21$	$+0.02$	$0.05$
$D_{s}^{}D_{s2}^{}(2573)$	$-0.01$	$...$	$-0.43$	$0.63$	$D_{s0}^{*}(2317)D_{s1}(2536)$	$-0.02$	$0.01$	$-0.01$	$0.01$
$DD_{1}(2430)$	$+0.21$	$0.60$	$-0.06$	$0.08$	$D^{}(2300)D_{2}^{}(2460)$	$-0.47$	$1.09$	$-0.13$	$1.22$
$D_{0}(2550)D_{1}(2430)$	$-0.31$	$2.16$	$-0.58$	$0.29$	$D_{s}^{}(2317)D_{s2}^{}(2573)$	$-0.06$	$0.31$	$-0.03$	$0.09$
$D_{s}D_{s1}(2460)$	$-0.03$	$0.34$	$...$	$0.01$	$D_{2}^{}(2460)D_{2}^{}(2460)$	$-2.65$	$1.02$	$-0.36$	$3.91$
$DD_{1}(2420)$	$+0.39$	$0.45$	$-0.03$	$0.43$	$D_{2}^{*}(2460)D_{1}(2430)$	$-0.50$	$3.68$	$-0.75$	$1.36$
$D_{0}(2550)D_{1}(2420)$	$-1.65$	✗	$-0.39$	$1.22$	$D_{s}1(2460)D_{s2}^{*}(2573)$	$-0.52$	✗	$-0.35$	$0.23$
$D_{s}D_{s1}(2536)$	$-0.13$	$0.58$	$-0.04$	$0.01$	$D_{2}^{*}(2460)D_{1}(2420)$	$-0.45$	$1.96$	$-0.23$	$0.77$
$D^{*}D_{1}(2430)$	$+0.07$	$1.90$	$+0.25$	$0.20$	$D_{s}1(2536)D_{s2}^{*}(2573)$	✗	✗	$-0.23$	✗
$D_{s}^{*}D_{s1}(2460)$	$-0.24$	$0.29$	$-0.01$	$0.37$	$D_{1}(2430)D_{1}(2430)$	$-0.30$	$0.60$	$-0.26$	$0.50$
$D^{*}D_{1}(2420)$	$+0.14$	$1.74$	$-0.26$	$0.33$	$D_{s1}(2460)D_{s1}(2460)$	$-0.01$	$0.11$	$-0.02$	$0.03$
$D_{s}^{*}D_{s1}(2536)$	$-0.19$	$0.03$	$-0.03$	$0.17$	$D_{1}(2430)D_{1}(2420)$	$-0.64$	$5.26$	$-0.23$	$0.68$
$DD(2P_{1})$	$-0.32$	$3.00$	$-0.18$	$0.03$	$D_{s1}(2460)D_{s1}(2536)$	$-1.01$	✗	$-0.22$	$0.06$
$D_{s}D_{s}(2P_{1})$	$-0.20$	$0.02$	$-0.20$	$0.04$	$D_{1}(2420)D_{1}(2420)$	$-0.53$	$1.16$	$-0.33$	$0.33$
$DD(2P^{\prime}_{1})$	$-2.77$	$7.00$	$-0.58$	$0.51$	$D_{s1}(2536)D_{s1}(2536)$	$-0.21$	✗	$-0.08$	✗
$D_{s}D_{s}(2P^{\prime}_{1})$	$-0.53$	✗	$-0.11$	$0.02$	$D_{s}(2^{1}S_{0})D_{s1}(2460)$	$-0.17$	✗	$-0.24$	$0.05$
$D^{*}D(2P_{1})$	$-0.65$	$0.51$	$-1.74$	$2.91$	$D_{1}^{}(2600)D_{2}^{}(2460)$	✗	✗	$-3.94$	✗
$D^{*}D(2P^{\prime}_{1})$	$-1.42$	$1.35$	$-0.70$	$0.80$	$D_{1}^{*}(2600)D_{1}(2430)$	$-1.20$	✗	$-2.57$	$2.41$
$DD(1^{3}D_{1})$	$-0.08$	$0.26$	$...$	$...$	$D_{1}^{*}(2600)D_{1}(2420)$	$-1.49$	✗	$+1.21$	$0.96$
$D_{s}D_{s}(1^{3}D_{1})$	$-0.03$	$...$	$-0.01$	$0.01$
$DD(1^{3}D_{3})$	$+0.10$	$0.49$	$-0.24$	$0.38$	Total	$-41.35$	$61.07$	$-29.55$	$42.88$
$D_{s}D_{s}(1^{3}D_{3})$	$-0.08$	$...$	$-0.08$	$0.10$	$M_{th},\Gamma_{th}$	$\bm{4988,61.07}$		$\bm{5061,42.88}$

Table 15: The mass shifts and strong decay widths for the

5^{1}D_{2}

and

5^{3}D_{2}

...

” stands for the predicted values are negligibly small.

	$5^{1}D_{2}$		$5^{3}D_{2}$			$5^{1}D_{2}$		$5^{3}D_{2}$
	$\Delta M$ [5073]	$\Gamma$ [5044]	$\Delta M$ [5074]	$\Gamma$ [5030]	Continue	$\Delta M$ [5073]	$\Gamma$ [5044]	$\Delta M$ [5074]	$\Gamma$ [5030]
$DD^{*}$	$-0.04$	$0.25$	$-0.05$	$0.20$	$DD(1^{3}D_{1})$	$+0.08$	$0.32$	$+0.02$	$0.14$
$D_{s}D_{s}^{*}$	$-0.02$	$0.02$	$-0.01$	$0.01$	$D_{s}D_{s}(1^{3}D_{1})$	$-0.06$	$0.03$	$-0.02$	$...$
$D^{}D^{}$	$-0.07$	$0.15$	$-0.09$	$0.18$	$DD(1^{3}D_{3})$	$-0.26$	$1.15$	$-0.01$	$1.20$
$D_{s}^{}D_{s}^{}$	$...$	$0.03$	$-0.01$	$0.02$	$D_{s}D_{s}(1^{3}D_{3})$	$-0.29$	$0.20$	$-0.19$	$0.05$
$D_{0}(2550)D^{*}$	$-0.29$	$0.49$	$-0.12$	$0.27$	$DD(1D_{2})$	$...$	$...$	$+0.01$	$0.46$
$D(3^{1}S_{0})D^{*}$	$-0.61$	$0.09$	$-0.63$	✗	$D_{s}D_{s}(1D_{2})$	$...$	$...$	$-0.06$	$0.02$
$DD_{1}^{*}(2600)$	$-0.31$	$0.71$	$-0.18$	$0.40$	$DD(1D^{\prime}_{2})$	$...$	$...$	$+0.02$	$0.61$
$DD^{*}(3^{3}S_{1})$	$-0.81$	$0.90$	$-1.10$	$1.16$	$D_{s}D_{s}(1D^{\prime}_{2})$	$...$	$...$	$-0.10$	$0.03$
$D_{s}(2^{1}S_{0})D_{s}^{*}$	$-0.02$	$0.34$	$-0.06$	$0.48$	$D^{*}D(1^{3}D_{1})$	$-0.13$	$0.26$	$-0.44$	$0.59$
$D_{s}D_{s}^{*}(2700)$	$-0.01$	$0.29$	$-0.01$	$0.48$	$D_{s}^{*}D_{s}(1^{3}D_{1})$	$-0.03$	$0.02$	$-0.07$	$0.04$
$D^{}D_{1}^{}(2600)$	$+0.31$	$1.36$	$+0.11$	$0.94$	$D^{*}D(1^{3}D_{3})$	$-1.18$	$2.60$	$-0.97$	$2.31$
$D^{}D^{}(3^{3}S_{1})$	$-0.85$	✗	$-0.72$	✗	$D_{s}^{*}D_{s}(1^{3}D_{3})$	$-0.47$	$0.24$	$-0.46$	$0.18$
$D_{s}^{}D_{s}^{}(2700)$	$-0.24$	$0.21$	$-0.22$	$0.17$	$D^{*}D(1D_{2})$	$-1.07$	$2.03$	$-0.84$	$1.42$
$DD^{*}(2300)$	$-0.03$	$0.12$	$...$	$0.01$	$D_{s}^{*}D_{s}(1D_{2})$	$-0.25$	$0.21$	$-0.18$	$0.14$
$D_{0}(2550)D^{*}(2300)$	$+0.24$	$0.80$	$-0.01$	$...$	$D^{*}D(1D^{\prime}_{2})$	$-1.20$	$1.13$	$-0.87$	$0.58$
$DD(2^{3}P_{0})$	$-0.31$	$0.01$	$-0.01$	$0.01$	$D_{s}^{*}D_{s}(1D^{\prime}_{2})$	$-0.37$	$0.14$	$-0.35$	$0.09$
$D_{s}D_{s0}^{*}(2317)$	$-0.01$	$...$	$...$	$...$	$DD(2^{3}D_{1})$	$-0.97$	$0.08$	$-0.27$	$0.01$
$D_{s}(2^{1}S_{0})D_{s0}^{*}(2317)$	$-0.20$	$0.10$	$...$	$...$	$DD(2^{3}D_{3})$	$-1.38$	✗	$-1.04$	✗
$D^{}D^{}(2300)$	$...$	$0.01$	$+0.03$	$...$	$DD(2D_{2})$	$...$	$...$	$-0.47$	✗
$D_{1}^{}(2600)D^{}(2300)$	$-0.01$	$...$	$-0.83$	$0.96$	$DD(2D^{\prime}_{2})$	$...$	✗	$-0.54$	✗
$D^{*}D(2^{3}P_{0})$	$-0.01$	$...$	$-0.45$	$1.22$	$DD(1^{3}F_{2})$	$...$	$...$	$...$	$...$
$D_{s}^{}D_{s0}^{}(2317)$	$...$	$...$	$+0.03$	$0.06$	$DD(1^{3}F_{4})$	$...$	$...$	$...$	$...$
$D_{s}^{}(2700)D_{s0}^{}(2317)$	$...$	$...$	$-0.17$	✗	$D_{s}D_{s}(1^{3}F_{4})$	$...$	$...$	$...$	✗
$DD_{2}^{*}(2460)$	$+0.15$	$0.08$	$-0.05$	$0.31$	$DD(1F_{3})$	$...$	$...$	$-0.19$	$0.16$
$D_{0}(2550)D_{2}^{*}(2460)$	$+0.07$	$1.96$	$-1.66$	$2.01$	$D_{s}D_{s}(1F_{3})$	$...$	$...$	$-0.04$	✗
$DD(2^{3}P_{2})$	$+0.09$	$2.05$	$-0.74$	$3.65$	$DD(1F^{\prime}_{3})$	$...$	$...$	$-0.14$	$0.04$
$D_{s}D_{s2}^{*}(2573)$	$...$	$0.20$	$-0.03$	$0.29$	$D^{*}D(1^{3}F_{2})$	$...$	✗	$...$	✗
$D_{s}D_{s}(2^{3}P_{2})$	$-0.17$	$0.02$	$-0.33$	$...$	$D^{*}D(1^{3}F_{4})$	$...$	$...$	$...$	✗
$D^{}D_{2}^{}(2460)$	$+0.12$	$1.03$	$-0.06$	$0.63$	$D^{*}D(1F_{3})$	$-0.81$	$0.70$	$-0.91$	$...$
$D_{1}^{}(2600)D_{2}^{}(2460)$	$-2.39$	✗	$-2.27$	✗	$D^{*}D(1F^{\prime}_{3})$	$-1.08$	✗	$-0.92$	✗
$D^{*}D(2^{3}P_{2})$	$-1.72$	$1.07$	$-1.81$	$1.06$	$D^{*}(2300)D_{1}(2430)$	$+0.01$	$0.43$	$-0.02$	$0.73$
$D_{s}^{}D_{s2}^{}(2573)$	$-0.28$	$0.41$	$-0.27$	$0.47$	$D_{s0}^{*}(2317)D_{s1}(2460)$	$-0.06$	$0.06$	$-9.16$	$8.53$
$D_{s}D_{s1}(2460)$	$...$	$...$	$+0.03$	$0.10$	$D^{*}(2300)D_{1}(2420)$	$-0.04$	$0.93$	$-0.05$	$0.38$
$DD_{1}(2430)$	$...$	$0.01$	$+0.08$	$0.02$	$D_{s0}^{*}(2317)D_{s1}(2536)$	$-0.07$	$0.01$	$-2.88$	$0.68$
$D_{0}(2550)D_{1}(2430)$	$-0.01$	$...$	$-0.10$	$0.14$	$D^{}(2300)D_{2}^{}(2460)$	$...$	$...$	$-0.13$	$1.04$
$DD(2P_{1})$	$-0.01$	$0.01$	$+0.26$	$0.97$	$D_{s}^{}(2317)D_{s2}^{}(2573)$	$...$	$...$	$-0.04$	$0.06$
$D_{s}D_{s}(2P_{1})$	$...$	$...$	$-0.17$	$...$	$D_{2}^{}(2460)D_{2}^{}(2460)$	$-0.90$	$2.59$	$-1.18$	$2.22$
$DD_{1}(2420)$	$...$	$0.01$	$-0.12$	$0.11$	$D_{2}^{*}(2460)D_{1}(2430)$	$-1.32$	$2.74$	$-0.55$	$2.96$
$D_{0}(2550)D_{1}(2420)$	$...$	$...$	$-0.15$	$0.16$	$D_{s1}(2460)D_{s2}^{*}(2573)$	$-0.87$	$0.54$	$-0.80$	$0.03$
$DD(2P^{\prime}_{1})$	$-0.01$	$...$	$-0.03$	$0.45$	$D_{2}^{*}(2460)D_{1}(2420)$	$-0.33$	$2.01$	$-0.22$	$1.93$
$D_{s}D_{s1}(2536)$	$...$	$...$	$-0.01$	$0.04$	$D_{s1}(2536)D_{s2}^{*}(2573)$	$-0.41$	✗	$-0.28$	✗
$D_{s}D_{s}(2P^{\prime}_{1})$	$...$	$...$	$-0.10$	$0.02$	$D_{1}(2430)D_{1}(2430)$	$-0.21$	$0.12$	$-0.18$	$0.05$
$D^{*}D_{1}(2430)$	$+0.49$	$0.82$	$+0.32$	$0.73$	$D_{s1}(2460)D_{s1}(2460)$	$-0.03$	$0.12$	$-0.07$	$0.13$
$D_{1}^{*}(2600)D_{1}(2430)$	$-2.49$	$0.53$	$-1.31$	✗	$D_{1}(2430)D_{1}(2420)$	$-0.13$	$0.06$	$-0.54$	$0.33$
$D^{*}D(2P_{1})$	$-2.26$	$2.28$	$-1.42$	$0.63$	$D_{s1}(2460)D_{s1}(2536)$	$-0.04$	$0.04$	$-0.17$	$0.11$
$D_{s}^{*}D_{s1}(2460)$	$-0.09$	$0.57$	$-0.09$	$0.37$	$D_{1}(2420)D_{1}(2420)$	$-0.03$	$0.02$	$-0.11$	$0.16$
$D^{*}D_{1}(2420)$	$-0.23$	$0.23$	$+0.10$	$0.71$	$D_{s1}(2536)D_{s1}(2536)$	$-0.02$	✗	$-0.07$	✗
$D_{1}^{*}(2600)D_{1}(2420)$	$-1.56$	✗	$-2.02$	✗
$D^{*}D(2P^{\prime}_{1})$	$-1.24$	$0.86$	$-1.40$	$0.74$	Total	$-28.84$	$37.09$	$-43.54$	$47.90$
$D_{s}^{*}D_{s1}(2536)$	$-0.09$	$0.29$	$-0.21$	$0.31$	$M_{th},\Gamma_{th}$	$\bm{5044,37.09}$		$\bm{5030,47.90}$

Acknowledgements

We thank useful discussions from Long-Cheng Gui, Qi-Fang Lü, Xiang Liu, Zhi-Yong Zhou, Ying Chen, and Qiang Zhao. This work is supported by the National Natural Science Foundation of China under Grants Nos. 12235018, 12175065, 12205216.

References

[1] S. K. Choi et al. [Belle], Observation of a narrow charmonium-like state in exclusive $B^{\pm}\to K^{\pm}\pi^{+}\pi^{-}J/\psi$ decays, Phys. Rev. Lett. 91, 262001 (2003).
[2] R. L. Workman et al. [Particle Data Group], Review of Particle Physics, PTEP 2022, 083C01 (2022).
[3] M. Ablikim et al. [(BESIII), and BESIII], Observation of the Y(4230) and a new structure in $e^{+}e^{-}\to K^{+}K^{-}J/\psi$ , Chin. Phys. C 46, 111002 (2022).
[4] M. Ablikim et al. [BESIII], Observation of Three Charmoniumlike States with $J^{PC}=1^{--}$ in $e^{+}e^{-}\to D^{*0}D^{*-}\pi^{+}$ , Phys. Rev. Lett. 130, 121901 (2023).
[5] M. Ablikim et al. [BESIII], Observation of the Y(4230) and evidence for a new vector charmoniumlike state Y(4710) in $e^{+}e^{-}\to K^{0}_{S}K^{0}_{S}J/\psi$ , Phys. Rev. D 107, no.9, 092005 (2023).
[6] M. Ablikim et al. [BESIII], Precise Measurement of the $e^{+}e^{-}\to D_{s}^{*+}D_{s}^{*-}$ Cross Sections at Center-of-Mass Energies from Threshold to 4.95 GeV, Phys. Rev. Lett. 131, 151903 (2023).
[7] F. K. Guo, C. Hanhart, U. G. Meißner, Q. Wang, Q. Zhao and B. S. Zou, Hadronic molecules, Rev. Mod. Phys. 90, 015004 (2018) [erratum: Rev. Mod. Phys. 94, no.2, 029901 (2022)]
[8] N. Brambilla, S. Eidelman, C. Hanhart, A. Nefediev, C. P. Shen, C. E. Thomas, A. Vairo and C. Z. Yuan, The $XYZ$ states: experimental and theoretical status and perspectives, Phys. Rept. 873, 1-154 (2020).
[9] H. X. Chen, W. Chen, X. Liu, Y. R. Liu and S. L. Zhu, An updated review of the new hadron states, Rept. Prog. Phys. 86, 026201 (2023).
[10] H. X. Chen, W. Chen, X. Liu and S. L. Zhu, The hidden-charm pentaquark and tetraquark states, Phys. Rept. 639, 1-121 (2016).
[11] Y. R. Liu, H. X. Chen, W. Chen, X. Liu and S. L. Zhu, Pentaquark and Tetraquark states, Prog. Part. Nucl. Phys. 107, 237-320 (2019).
[12] S. L. Olsen, T. Skwarnicki and D. Zieminska, Nonstandard heavy mesons and baryons: Experimental evidence, Rev. Mod. Phys. 90, 015003 (2018).
[13] R. F. Lebed, R. E. Mitchell and E. S. Swanson, Heavy-Quark QCD Exotica, Prog. Part. Nucl. Phys. 93, 143-194 (2017).
[14] A. Esposito, A. Pilloni and A. D. Polosa, Multiquark Resonances, Phys. Rept. 668, 1-97 (2017).
[15] A. Ali, J. S. Lange and S. Stone, Exotics: Heavy Pentaquarks and Tetraquarks, Prog. Part. Nucl. Phys. 97, 123-198 (2017).
[16] X. K. Dong, F. K. Guo and B. S. Zou, A survey of heavy–heavy hadronic molecules, Commun. Theor. Phys. 73, no.12, 125201 (2021).
[17] E. Eichten, K. Gottfried, T. Kinoshita, J. B. Kogut, K. D. Lane and T. M. Yan, The Spectrum of Charmonium, Phys. Rev. Lett. 34, 369-372 (1975) [erratum: Phys. Rev. Lett. 36, 1276 (1976)].
[18] E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane and T. M. Yan, Charmonium: The Model, Phys. Rev. D 17, 3090 (1978) [erratum: Phys. Rev. D 21, 313 (1980)].
[19] S. Godfrey and N. Isgur, Mesons in a Relativized Quark Model with Chromodynamics, Phys. Rev. D 32, 189-231 (1985).
[20] T. Barnes, S. Godfrey and E. S. Swanson, Higher charmonia, Phys. Rev. D 72, 054026 (2005).
[21] D. Ebert, R. N. Faustov and V. O. Galkin, Spectroscopy and Regge trajectories of heavy quarkonia and $B_{c}$ mesons, Eur. Phys. J. C 71, 1825 (2011).
[22] K. Heikkila, S. Ono and N. A. Tornqvist, Heavy $c\bar{c}$ and $b\bar{b}$ quarkonium states and unitarity effects, Phys. Rev. D 29, 110 (1984) [erratum: Phys. Rev. D 29, 2136 (1984)].
[23] Y. S. Kalashnikova, Coupled-channel model for charmonium levels and an option for $X(3872)$ , Phys. Rev. D 72, 034010 (2005).
[24] T. Barnes and E. S. Swanson, Hadron loops: General theorems and application to charmonium, Phys. Rev. C 77, 055206 (2008) [arXiv:0711.2080 [hep-ph]].
[25] M. R. Pennington and D. J. Wilson, Decay channels and charmonium mass-shifts, Phys. Rev. D 76, 077502 (2007).
[26] B. Q. Li, C. Meng and K. T. Chao, Coupled-Channel and Screening Effects in Charmonium Spectrum, Phys. Rev. D 80, 014012 (2009) [arXiv:0904.4068 [hep-ph]].
[27] J. Ferretti, G. Galatà and E. Santopinto, Interpretation of the X(3872) as a charmonium state plus an extra component due to the coupling to the meson-meson continuum, Phys. Rev. C 88, 015207 (2013) [arXiv:1302.6857 [hep-ph]].
[28] Z. Y. Zhou and Z. Xiao, Comprehending heavy charmonia and their decays by hadron loop effects, Eur. Phys. J. A 50, 165 (2014).
[29] Y. Lu, M. N. Anwar and B. S. Zou, $X(4260)$ Revisited: A Coupled Channel Perspective, Phys. Rev. D 96, 114022 (2017).
[30] J. Ferretti and E. Santopinto, Threshold corrections of $\chi_{c}(2P)$ and $\chi_{b}(3P)$ states and $J/\psi\rho$ and $J/\psi\omega$ transitions of the $\chi(3872)$ in a coupled-channel model, Phys. Lett. B 789, 550-555 (2019).
[31] H. F. Fu and L. Jiang, Coupled-channel-induced $S{-}D$ mixing of Charmonia and testing possible assignments for $Y$ (4260) and $Y$ (4360), Eur. Phys. J. C 79, 460 (2019).
[32] P. G. Ortega, D. R. Entem and F. Fernández, Unquenching the Quark Model in a Nonperturbative Scheme, Adv. High Energy Phys. 2019, 3465159 (2019).
[33] J. Ferretti, E. Santopinto, M. N. Anwar and Y. Lu, Quark structure of the $\chi_{\mathrm{c}}(3P)$ and $X(4274)$ resonances and their strong and radiative decays, Eur. Phys. J. C 80, 464 (2020).
[34] J. Ferretti and E. Santopinto, Quark Structure of the $X(4500)$ , $X(4700)$ and $\chi_{c}(4P,5P)$ States, Front. in Phys. 9, 76 (2021).
[35] M. N. Anwar and Y. Lu, Heavy quark spin partners of the Y(4260) in coupled-channel formalism, Phys. Rev. D 104, 094006 (2021).
[36] M. X. Duan and X. Liu, Where are 3P and higher P-wave states in the charmonium family?, Phys. Rev. D 104, 074010 (2021).
[37] R. Bruschini and P. González, Diabatic description of charmoniumlike mesons II: mass corrections and strong decay widths, Phys. Rev. D 103, 074009 (2021).
[38] S. Kanwal, F. Akram, B. Masud and E. S. Swanson, Charmonium spectrum in an unquenched quark model, Eur. Phys. J. A 58, 219 (2022).
[39] X. Chen, Y. Tan and Y. Yang, Charmonium mass shifts in an unquenched quark model, Eur. Phys. J. Plus 138, 653 (2023).
[40] Y. B. Ding, K. T. Chao and D. H. Qin, Screened $Q\bar{Q}$ potential and spectrum of heavy quarkonium, Chin. Phys. Lett. 10, 460-463 (1993).
[41] Y. B. Dong, Color screening potential and dissolution of heavy quarkonium, Int. J. Mod. Phys. E 5, 643-648 (1996).
[42] B. Q. Li and K. T. Chao, Higher Charmonia and X,Y,Z states with Screened Potential, Phys. Rev. D 79, 094004 (2009).
[43] W. J. Deng, H. Liu, L. C. Gui and X. H. Zhong, Charmonium spectrum and their electromagnetic transitions with higher multipole contributions, Phys. Rev. D 95, 034026 (2017).
[44] J. Z. Wang, D. Y. Chen, X. Liu and T. Matsuki, Constructing $J/\psi$ family with updated data of charmoniumlike $Y$ states, Phys. Rev. D 99, 114003 (2019).
[45] J. J. Dudek, R. G. Edwards, N. Mathur and D. G. Richards, Charmonium excited state spectrum in lattice QCD, Phys. Rev. D 77, 034501 (2008).
[46] L. Liu et al. [Hadron Spectrum], Excited and exotic charmonium spectroscopy from lattice QCD, JHEP 07, 126 (2012).
[47] C. DeTar, Charmonium Spectroscopy from Lattice QCD, Int. J. Mod. Phys. Conf. Ser. 02, 31-35 (2011).
[48] G. S. Bali, S. Collins and C. Ehmann, Charmonium spectroscopy and mixing with light quark and open charm states from $n_{F}$ =2 lattice QCD, Phys. Rev. D 84, 094506 (2011).
[49] C. B. Lang, L. Leskovec, D. Mohler and S. Prelovsek, Vector and scalar charmonium resonances with lattice QCD, JHEP 09, 089 (2015).
[50] D. J. Wilson, C. E. Thomas, J. J. Dudek and R. G. Edwards, Scalar and tensor charmonium resonances in coupled-channel scattering from QCD, [arXiv:2309.14070 [hep-lat]].
[51] D. J. Wilson, C. E. Thomas, J. J. Dudek and R. G. Edwards, Charmonium $\chi_{c0}$ and $\chi_{c2}$ resonances in coupled-channel scattering from lattice QCD, [arXiv:2309.14071 [hep-lat]].
[52] R. H. Ni, J. J. Wu and X. H. Zhong, Unified unquenched quark model for heavy-light mesons with chiral dynamics, [arXiv:2312.04765 [hep-ph]].
[53] L. Micu, Decay rates of meson resonances in a quark model, Nucl. Phys. B 10, 521-526 (1969).
[54] A. Le Yaouanc, L. Oliver, O. Pene and J. C. Raynal, Naive quark pair creation model of strong interaction vertices, Phys. Rev. D 8, 2223-2234 (1973).
[55] A. Le Yaouanc, L. Oliver, O. Pene and J. C. Raynal, Naive quark pair creation model and baryon decays, Phys. Rev. D 9, 1415-1419 (1974).
[56] D. Morel and S. Capstick, Baryon meson loop effects on the spectrum of nonstrange baryons, [arXiv:nucl-th/0204014 [nucl-th]].
[57] B. Silvestre- Brac and C. Gignoux, Unitary effects in spin orbit splitting of P wave baryons, Phys. Rev. D 43, 3699-3708 (1991).
[58] P. G. Ortega, J. Segovia, D. R. Entem and F. Fernandez, Molecular components in P-wave charmed-strange mesons, Phys. Rev. D 94, no.7, 074037 (2016).
[59] P. G. Ortega, J. Segovia, D. R. Entem and F. Fernández, Threshold effects in P-wave bottom-strange mesons, Phys. Rev. D 95, no.3, 034010 (2017).
[60] Z. Yang, G. J. Wang, J. J. Wu, M. Oka and S. L. Zhu, Novel Coupled Channel Framework Connecting the Quark Model and Lattice QCD for the Near-threshold Ds States, Phys. Rev. Lett. 128, 112001 (2022).
[61] Z. Yang, G. J. Wang, J. J. Wu, M. Oka and S. L. Zhu, The investigations of the P-wave B_s states combining quark model and lattice QCD in the coupled channel framework, JHEP 01, 058 (2023).
[62] H. H. Zhong, R. H. Ni, M. Y. Chen, X. H. Zhong and J. J. Xie, Further study of $\Omega^{*}|1P_{3/2^{-}}\rangle$ within a chiral quark model, Chin. Phys. C 47, 063104 (2023).
[63] S. Godfrey, Spectroscopy of $B_{c}$ mesons in the relativized quark model, Phys. Rev. D 70, 054017 (2004).
[64] Q. Li, M. S. Liu, L. S. Lu, Q. F. Lü, L. C. Gui and X. H. Zhong, Excited bottom-charmed mesons in a nonrelativistic quark model, Phys. Rev. D 99, 096020 (2019).
[65] E. Hiyama, Y. Kino and M. Kamimura, Gaussian expansion method for few-body systems, Prog. Part. Nucl. Phys. 51, 223-307 (2003).
[66] Z. Y. Zhou and Z. Xiao, Hadron loops effect on mass shifts of the charmed and charmed-strange spectra, Phys. Rev. D 84, 034023 (2011).
[67] R. H. Ni, Q. Li and X. H. Zhong, Mass spectra and strong decays of charmed and charmed-strange mesons, Phys. Rev. D 105, 056006 (2022).
[68] Q. li, R. H. Ni and X. H. Zhong, Towards establishing an abundant $B$ and $B_{s}$ spectrum up to the second orbital excitations, Phys. Rev. D 103, 116010 (2021).
[69] W. J. Deng, H. Liu, L. C. Gui and X. H. Zhong, Spectrum and electromagnetic transitions of bottomonium, Phys. Rev. D 95, 074002 (2017).
[70] Q. Li, L. C. Gui, M. S. Liu, Q. F. Lü and X. H. Zhong, Mass spectrum and strong decays of strangeonium in a constituent quark model, Chin. Phys. C 45, 023116 (2021).
[71] B. Aubert et al. [BaBar], Exclusive Initial-State-Radiation Production of the $D\bar{D}$ , $D^{*}\bar{D}$ , and $D^{*}\bar{D}^{*}$ Systems, Phys. Rev. D 79, 092001 (2009).
[72] K. Abe et al. [Belle], Measurement of the near-threshold $e^{+}e^{-}\to D^{(*)\pm}D^{(*)\mp}$ cross section using initial-state radiation, Phys. Rev. Lett. 98, 092001 (2007).
[73] J. Z. Wang and X. Liu, Identifying a characterized energy level structure of higher charmonium well matched to the peak structures in $e^{+}e^{-}\to\pi^{+}D^{0}D^{*-}$ , [arXiv:2306.14695 [hep-ph]].
[74] J. Z. Wang and X. Liu, Confirming the existence of a new higher charmonium $\psi(4500)$ by the newly released data of $e^{+}e^{-}\to K^{+}K^{-}J/\psi$ , Phys. Rev. D 107, 054016 (2023).
[75] L. C. Gui, L. S. Lu, Q. F. Lü, X. H. Zhong and Q. Zhao, Strong decays of higher charmonium states into open-charm meson pairs, Phys. Rev. D 98, 016010 (2018).
[76] X. L. Wang et al. [Belle], Measurement of $e^{+}e^{-}\to\pi^{+}\pi^{-}\psi(2S)$ via Initial State Radiation at Belle, Phys. Rev. D 91, 112007 (2015).
[77] M. Ablikim et al. [BESIII], Observation of a vector charmoniumlike state at 4.7 ${\rm GeV}/c^{2}$ and search for $Z_{cs}$ in $e^{+}e^{-}\to K^{+}K^{-}J/\psi$ , Phys. Rev. Lett. 131, 211902 (2023).
[78] G. J. Ding, J. J. Zhu and M. L. Yan, Canonical Charmonium Interpretation for Y(4360) and Y(4660), Phys. Rev. D 77, 014033 (2008).
[79] J. Segovia, A. M. Yasser, D. R. Entem and F. Fernandez, $J^{PC}=1^{--}$ hidden charm resonances, Phys. Rev. D 78, 114033 (2008).
[80] Z. Zhao, K. Xu, A. Limphirat, W. Sreethawong, N. Tagsinsit, A. Kaewsnod, X. Liu, K. Khosonthongkee, S. Cheedket and Y. Yan, Mass spectrum of $1^{--}$ heavy quarkonium, [arXiv:2304.06243 [hep-ph]].
[81] M. X. Duan, S. Q. Luo, X. Liu and T. Matsuki, Possibility of charmoniumlike state $X(3915)$ as $\chi_{c0}(2P)$ state, Phys. Rev. D 101, 054029 (2020).
[82] X. Liu, Z. G. Luo and Z. F. Sun, X(3915) and X(4350) as new members in P-wave charmonium family, Phys. Rev. Lett. 104, 122001 (2010).
[83] R. Aaij et al. [LHCb], Amplitude analysis of the $B^{+}\to D^{+}D^{-}K^{+}$ decay, Phys. Rev. D 102, 112003 (2020).
[84] S. Uehara et al. [Belle], Observation of a charmonium-like enhancement in the $\gamma\gamma to\omega J/\psi$ process, Phys. Rev. Lett. 104, 092001 (2010).
[85] K. Abe et al. [Belle], Observation of a near-threshold $\omega J/\psi$ mass enhancement in exclusive $B\to K\omega J/\psi$ decays, Phys. Rev. Lett. 94, 182002 (2005).
[86] J. P. Lees et al. [BaBar], Study of $X(3915)\to J/\psi\omega$ in two-photon collisions, Phys. Rev. D 86, 072002 (2012).
[87] B. Aubert et al. [BaBar], Observation of $Y(3940)$ $\to J/\psi\omega$ in $B\to J/\psi\omega K$ at BABAR, Phys. Rev. Lett. 101, 082001 (2008).
[88] F. K. Guo and U. G. Meissner, Where is the $\chi_{c0}(2P)$ ?, Phys. Rev. D 86, 091501 (2012).
[89] K. Chilikin et al. [Belle], Observation of an alternative $\chi_{c0}(2P)$ candidate in $e^{+}e^{-}\rightarrow J/\psi D\bar{D}$ , Phys. Rev. D 95, 112003 (2017).
[90] Y. F. Wang, U. G. Meißner, D. Rönchen and C. W. Shen, On the nature of the $N^{*}$ and $\Delta$ resonances via coupled-channel dynamics, [arXiv:2307.06799 [nucl-th]].
[91] V. Baru, J. Haidenbauer, C. Hanhart, Y. Kalashnikova and A. E. Kudryavtsev, Evidence that the $a_{0}(980)$ and $f_{0}(980)$ are not elementary particles, Phys. Lett. B 586, 53-61 (2004).
[92] Y. S. Kalashnikova and A. V. Nefediev, Nature of X(3872) from data, Phys. Rev. D 80, 074004 (2009).
[93] F. Giacosa, M. Piotrowska and S. Coito, $X(3872)$ as virtual companion pole of the charm–anticharm state $\chi_{c1}(2P)$ , Int. J. Mod. Phys. A 34, 1950173 (2019).
[94] P. G. Ortega, J. Segovia, D. R. Entem and F. Fernandez, Coupled channel approach to the structure of the X(3872), Phys. Rev. D 81, 054023 (2010).
[95] Z. Y. Zhou and Z. Xiao, Understanding $X(3862)$ , $X(3872)$ , and $X(3930)$ in a Friedrichs-model-like scheme, Phys. Rev. D 96, 054031 (2017) [erratum: Phys. Rev. D 96, 099905 (2017)].
[96] S. Coito, G. Rupp and E. van Beveren, $X(3872)$ is not a true molecule, Eur. Phys. J. C 73, 2351 (2013).
[97] C. Meng and K. T. Chao, Decays of the $X(3872)$ and $\chi_{c1}(2P)$ charmonium, Phys. Rev. D 75, 114002 (2007).
[98] C. Meng, J. J. Sanz-Cillero, M. Shi, D. L. Yao and H. Q. Zheng, Refined analysis on the X(3872) resonance, Phys. Rev. D 92, no.3, 034020 (2015).
[99] J. Ferretti, G. Galatà and E. Santopinto, Quark structure of the $X(3872)$ and $\chi_{b}(3P)$ resonances, Phys. Rev. D 90, no.5, 054010 (2014).
[100] Z. Y. Zhou and Z. Xiao, Comprehending Isospin breaking effects of $X(3872)$ in a Friedrichs-model-like scheme, Phys. Rev. D 97, 034011 (2018).
[101] G. J. Wang, Z. Yang, J. J. Wu, M. Oka and S. L. Zhu, New insight into the exotic states strongly coupled with the $D\bar{D}^{*}$ from the $T^{+}_{cc}$ , [arXiv:2306.12406 [hep-ph]].
[102] T. Aaltonen et al. [CDF], Observation of the $Y(4140)$ Structure in the $J/\psi\phi$ Mass Spectrum in $B^{\pm}\to J/\psi\phi K^{\pm}$ Decays, Mod. Phys. Lett. A 32, 1750139 (2017).
[103] R. Aaij et al. [LHCb], Observation of New Resonances Decaying to $J/\psi K^{+}$ + and $J/\psi\phi$ , Phys. Rev. Lett. 127, 082001 (2021).
[104] R. Aaij et al. [LHCb], Observation of $J/\psi\phi$ structures consistent with exotic states from amplitude analysis of $B^{+}\to J/\psi\phi K^{+}$ decays, Phys. Rev. Lett. 118, 022003 (2017).
[105] R. Aaij et al. [LHCb], Amplitude analysis of $B^{+}\to J/\psi\phi K^{+}$ decays, Phys. Rev. D 95, 012002 (2017).
[106] Z. H. Wang and G. L. Wang, Two-body strong decays of the 2P and 3P charmonium states, Phys. Rev. D 106, no.5, 054037 (2022).
[107] Q. F. Lü and Y. B. Dong, X(4140) , X(4274) , X(4500) , and X(4700) in the relativized quark model, Phys. Rev. D 94, no.7, 074007 (2016).
[108] C. P. Shen et al. [Belle], Evidence for a new resonance and search for the $Y(4140)$ in the $\gamma\gamma\to\phi J/\psi$ process, Phys. Rev. Lett. 104, 112004 (2010).
[109] R. Aaij et al. [LHCb], First observation of the $B^{+}\to D_{s}^{+}D_{s}^{-}K^{+}$ decay, Phys. Rev. D 108, 034012 (2023).
[110] R. Aaij et al. [LHCb], Observation of a Resonant Structure near the Ds+Ds- Threshold in the $B^{+}\to D_{s}^{+}D_{s}^{-}K^{+}$ Decay, Phys. Rev. Lett. 131, 071901 (2023).
[111] N. R. Soni, B. R. Joshi, R. P. Shah, H. R. Chauhan and J. N. Pandya, $Q\bar{Q}$ ( $Q\in\{b,c\}$ ) spectroscopy using the Cornell potential, Eur. Phys. J. C 78, 592 (2018).
[112] R. Chaturvedi and A. K. Rai, Charmonium spectroscopy motivated by general features of pNRQCD, Int. J. Theor. Phys. 59, 3508-3532 (2020).
[113] M. A. Sultan, N. Akbar, B. Masud and F. Akram, Higher Hybrid Charmonia in an Extended Potential Model, Phys. Rev. D 90, 054001 (2014).
[114] H. Mansour and A. Gamal, Meson spectra using Nikiforov-Uvarov method, Results Phys. 33, 105203 (2022).
[115] Z. Y. Fang, Y. R. Wang and C. Q. Pang, $Q\bar{Q}(Q\in\{b,c\})$ Spectroscopy Using the Modified Rovibrational Model, Phys. Part. Nucl. Lett. 20, 589-597 (2023).
[116] L. Cao, Y. C. Yang and H. Chen, Charmonium states in QCD-inspired quark potential model using Gaussian expansion method, Few Body Syst. 53, 327-342 (2012).
[117] R. Aaij et al. [LHCb], Study of $B^{0}_{s}\to J\psi\pi^{+}\pi^{-}K^{+}K^{-}$ decays, JHEP 02, 024 (2021) [erratum: JHEP 04, 170 (2021)].
[118] M. Ablikim et al. [BESIII], Search for new decay modes of the $\psi_{2}(3823)$ and the process $e^{+}e^{-}\rightarrow\pi^{0}\pi^{0}\psi_{2}(3823)$ , Phys. Rev. D 103, L091102 (2021).
[119] E. J. Eichten, K. Lane and C. Quigg, B Meson Gateways to Missing Charmonium Levels, Phys. Rev. Lett. 89, 162002 (2002).
[120] H. Xu, X. Liu and T. Matsuki, Understanding $B^{-}\rightarrow X(3823)K^{-}$ via rescattering mechanism and predicting $B^{-}\to\eta_{c2}(^{1}D_{2})/\psi_{3}(^{3}D_{3})K^{-}$ , Phys. Rev. D 94, no.3, 034005 (2016).
[121] Y. Fan, Z. G. He, Y. Q. Ma and K. T. Chao, Predictions of Light Hadronic Decays of Heavy Quarkonium $1D_{2}$ States in NRQCD, Phys. Rev. D 80, 014001 (2009).
[122] E. J. Eichten, K. Lane and C. Quigg, New states above charm threshold, Phys. Rev. D 73, 014014 (2006) [erratum: Phys. Rev. D 73, 079903 (2006)].
[123] J. Segovia, D. R. Entem and F. Fernandez, Strong charmonium decays in a microscopic model, Nucl. Phys. A 915, 125-141 (2013).
[124] P. Pakhlov et al. [Belle], Production of New Charmoniumlike States in $e^{+}e^{-}\to J/\psi D^{(*)}\bar{D}^{(*)}$ at $\sqrt{s}\approx 10.6$ GeV, Phys. Rev. Lett. 100, 202001 (2008).
[125] V. Kher and A. K. Rai, Spectroscopy and decay properties of charmonium, Chin. Phys. C 42, 083101 (2018).
[126] M. Shah, A. Parmar and P. C. Vinodkumar, Leptonic and Digamma decay Properties of S-wave quarkonia states, Phys. Rev. D 86, 034015 (2012).

$\displaystyle\Delta M(M)$	$\displaystyle=\operatorname{Re}\sum_{BC}\int_{0}^{\infty}\frac{\overline{\left\|\mathcal{M}_{A\rightarrow BC}(\mathbf{p})\right\|^{2}}}{\left(M-E_{BC}\right)}p^{2}dp$	(14)
	$\displaystyle-\operatorname{Re}\sum_{BC}\int_{0}^{\infty}\frac{\overline{\left\|\mathcal{M}_{A\rightarrow BC}(\mathbf{p})\right\|^{2}}}{\left(M_{0}-E_{BC}\right)}p^{2}dp$
	$\displaystyle=\operatorname{Re}\sum_{BC}\int_{0}^{\infty}\frac{\left(M_{0}-M\right)\overline{\left\|\mathcal{M}_{A\rightarrow BC}(\mathbf{p})\right\|^{2}}}{\left(M-E_{BC}\right)\left(M_{0}-E_{BC}\right)}p^{2}dp,$

Charmonia in an unquenched quark model

Abstract

I Introduction

II Framework

II.1 Quark potential model

II.2 Coupled-channel effects

II.3 Model parameters

III Result and Discussion

III.1 SS-wave states

III.1.1 ψ​(4040)\psi(4040)

III.1.2 ψ​(4415)\psi(4415)

III.1.3 ψ​(5​S)\psi(5S) and ψ​(6​S)\psi(6S)

III.2 PP-wave states

III.2.1 2​P2P-wave states

III.2.2 3​P3P-wave states

III.2.3 4​P4P-wave states

III.2.4 5​P5P-wave states

III.3 DD-wave states

III.3.1 1​D1D-wave states

III.3.2 2​D2D-wave states

III.3.3 3​D3D-wave states

III.3.4 4​D4D-wave states

III.3.5 5​D5D-wave states

IV Summary

Acknowledgements

References

III.1 $S$ -wave states

III.1.1 $\psi(4040)$

III.1.2 $\psi(4415)$

III.1.3 $\psi(5S)$ and $\psi(6S)$

III.2 $P$ -wave states

III.2.1 $2P$ -wave states

III.2.2 $3P$ -wave states

III.2.3 $4P$ -wave states

III.2.4 $5P$ -wave states

III.3 $D$ -wave states

III.3.1 $1D$ -wave states

III.3.2 $2D$ -wave states

III.3.3 $3D$ -wave states

III.3.4 $4D$ -wave states

III.3.5 $5D$ -wave states