Possibility of charmoniumlike state $X(3915)$ as $\chi_{c0}(2P)$ state

As an important group of the whole hadron spectrum, the charmonium family plays a very important role to provide the hint for quantitatively understanding how quarks form different types of hadrons, which has a close relation to non-perturbative behavior of strong interactions. In 1974, the first charmonium state $J/\psi$ was found Aubert:1974js ; Augustin:1974xw . Then, in the subsequent eight years from 1974 to 1982, most of charmonia listed in the present Particle Data Group (PDG) were observed, which becomes the main body of the charmonium family. Here, the typical states include $J/\psi$ Aubert:1974js ; Augustin:1974xw , $\psi(3686)$ Abrams:1974yy , $\psi(4040)$ Goldhaber:1977qn , $\psi(4415)$ Siegrist:1976br , $\psi(3770)$ Rapidis:1977cv , $\psi(4160)$ Brandelik:1978ei , $\chi_{c0}(1P)$ Biddick:1977sv , $\chi_{c1}(1P)$ Tanenbaum:1975ef , $\chi_{c2}(1P)$ Whitaker:1976hb , $\eta_{c}(1S)$ Partridge:1980vk , and $\eta_{c}(2S)$ Edwards:1981mq . With these observations, the Cornel model was proposed by Eichten $et$ $al$. Eichten:1974af in 1975, from which different versions of a potential model Krasemann:1979ir ; Stanley:1980zm ; Godfrey:1985xj ; Radford:2007vd ; Badalian:1999fe ; Barnes:2005pb applied to depict the interaction between quarks were developed by different groups.

However, the present observed charmonium spectrum is not complete in the sense that higher states in the charmonium family are still absent, where the higher states refer to the charmonia with higher radial and orbital quantum numbers. These missing higher states include three $1D$ states accompanied by $\psi(3770)$ and $2P$ states in the charmonium family. In fact, there is a big window without discovery of more new charmonia from 1982 to 2003, except $h_{c}$ reported by the R704 Collaboration Baglin:1986yd in 1986. In Fig. 1, all the observed charmonia and possible candidates are shown for the present status of charmonium family.

This situation has been dramatically changed as a series of charmoniumlike $XYZ$ states have been observed in experiments. $X(3872)$, as the first $XYZ$ states reported by the Belle collaboration Choi:2003ue , stimulated theorists’ interests in exploring $D\bar{D}^{*}$ molecular pictures Swanson:2003tb ; Wong:2003xk ; AlFiky:2005jd , which has continued to date and shed light on the nature of $X(3872)$. For $X(3872)$, the experimental mass and decay width are measured as $M_{X(3872)}=$3.871 GeV and $\Gamma^{exp}_{X(3872)}<1.2$ MeV. The mass and width are far lower than predictions of potential models. By introducing coupled-channel effects, the low mass puzzle of $X(3872)$ can be well understood Barnes:2003vb ; Ortega:2009hj ; Kalashnikova:2005ui . Thus, $X(3872)$ can be explained as a $\chi_{c1}(2P)$ state containing a $D\bar{D}^{*}$ component. And, two candidates of $1D$ states were announced by the Belle and LHCb Collaborations Bhardwaj:2013rmw ; Aaij:2019evc , which are $X(3823)$ from the $X(3823)\to\chi_{c1}\gamma$ decay channel and $X(3842)$ from the $X(3842)\to D\bar{D}$ process. In addition, the Lanzhou group indicated that there exists a narrow $Y$ state around 4.2 GeV, which corresponds to $\psi(4S)$ He:2014xna . Later, BESIII indeed observed this narrow structure in the $e^{+}e^{-}\to\pi^{+}\pi^{-}h_{c}$ and $e^{+}e^{-}\to\omega\chi_{cJ}$ processes Chang-Zheng:2014haa ; Ablikim:2014qwy . Recently, they again published one paper to illustrate how to construct higher vector states of the $J/\psi$ family with updated data of charmoniumlike $Y$ states Wang:2019mhs . From these examples, some of the charmoniumlike $XYZ$ states may be good candidates of missing charmonia. Thus, the above facts tell us a lesson, i.e., before introducing exotic hadronic state assignments to $XYZ$, we should carefully check whether there exists a possibility to group it into the charmonium family. Up to date, such a study has become an interesting research issue Barnes:2003vb ; Kalashnikova:2005ui ; Liu:2009fe ; Chen:2012wy .

In 2009, focusing on $2P$ states, the Lanzhou group carried out the study of the mass spectrum and strong decay behaviors of $2P$ charmonia by combining the experimental data of $X(3872)$, $Z(3930)$, and $X(3915)$. Here, $Z(3930)$ and $X(3915)$ are from $\gamma\gamma\to D\bar{D}$ Uehara:2005qd and $\gamma\gamma\to J/\psi\omega$ processes Uehara:2009tx , respectively. Linking these $XZ$ states to charmonia, they indicated that $Z(3930)$ is the $\chi_{c2}(2P)$ state and decoded $X(3915)$ as the $\chi_{c0}(2P)$ state with definite $J^{PC}=0^{++}$ quantum number Liu:2009fe . Later, the BaBar Collaboration confirmed this quantum number of $X(3915)$ Lees:2012xs . Thus, $X(3915)$ as the $\chi_{c0}(2P)$ state was listed into the 2013 version of PDG Beringer:1900zz .

After three years, this situation was changed by the paper Guo:2012tv with the title “Where is the $\chi_{c0}(2P)$?”. In this work, three questions were raised if treating $X(3915)$ as $\chi_{c0}(2P)$: 1) why $X(3915)\to J/\psi\omega$ has large width, 2) why the main decay mode “$X(3915)\to D\bar{D}$” was not reported in experiment, and 3) why the mass gap between $X(3915)$ and $Z(3930)$ is far smaller than that between $\chi_{b0}(2P)$ and $\chi_{b2}(2P)$. Then, two groups joined the discussion of whether $X(3915)$ can be the $\chi_{c0}(2P)$ state Olsen:2014maa ; Olsen:2019lcx ; Zhou:2015uva . As a consequence, labeling $X(3915)$ as $\chi_{c0}(2P)$ was removed in the 2016 version of PDG Patrignani:2016xqp .

Guo $et$ $al$. claimed that the $\chi_{c0}(2P)$ state should have mass around 3837.6$\pm$11.5 MeV and width about 221$\pm$19 MeV by their analysis to the $D\bar{D}$ invariant mass spectrum of the $\gamma\gamma\to D\bar{D}$ process Guo:2012tv . In 2017, the Belle Collaboration made an analysis with $e^{+}e^{-}\to J/\psi D\bar{D}$ process, and found a broad structure named as $X(3860)$ Chilikin:2017evr . Here, its mass and width are $M=3862$ MeV and $\Gamma=201$ MeV, respectively. Belle indicated that $X(3860)$ favors the $J^{PC}=0^{++}$ assignment. Therefore, Belle assigned the observed $X(3860)$ as $\chi_{c0}(2P)$. In Ref. Ortega:2017qmg , the authors studied charmoniumlike structures around 3.9 GeV in the framework of a constituent quark model. Here, their result favors the hypothesis that $X(3915)$ and $Z(3930)$ resonances arise as different decay mechanisms of the same $J^{PC}=2^{++}$ state, and explained $X(3860)$ to be a $\chi_{c0}(2P)$ state Ortega:2017qmg .

It is obvious that the situation of establishing the $\chi_{c0}(2P)$ candidate gets into a mess, which should be urgently clarified as soon as possible.

In the past years, we have been paying close attention to this problem. In Ref. Chen:2012wy , the Lanzhou group proposed a solution to the second problem mentioned above. The structure corresponding to $Z(3930)$ observed in the $D\bar{D}$ decay channel may contain two $P-$wave higher charmonia $\chi_{c0}(2P)$ and $\chi_{c2}(2P)$, which can be supported by the analysis of the $D\bar{D}$ invariant mass spectrum and $\cos\theta^{*}$ distribution of $\gamma\gamma\to D\bar{D}$ Uehara:2005qd . This means that the second problem raised in Ref. Guo:2012tv can be solved. We suggest Belle II to reanalyze the $\gamma\gamma\to D\bar{D}$ process with more precise data.

We still believe that $X(3915)$ observed in $\gamma\gamma\to J/\psi\omega$ is a good candidate of $\chi_{c0}(2P)$. Thus, we must face the third problem raised in Ref. Guo:2012tv just mentioned above. In a quenched potential model, the mass splitting between $\chi_{c0}(2P)$ and $\chi_{c2}(2P)$ is far larger than that between $X(3915)$ and $Z(3930)$. According to the quenched quark model estimate, this relation $\mid m_{\chi_{c2}(2P)}-m_{\chi_{c0}(2P)}\mid$ $>$ $\mid m_{\chi_{b2}(2P)}-m_{\chi_{b0}(2P)}\mid$ can be naively obtained as claimed in Ref. Guo:2012tv . In fact, we should be careful with this point. $X(3872)$ is a typical example, where there exists the low mass puzzle, i.e., the mass of $X(3872)$ is around 100 MeV lower than the value from the quenched quark model calculation Godfrey:1985xj . This puzzle can be solved by a coupled-channel effect by calculating mass with an unquenched quark model Kalashnikova:2005ui . In fact, for other $2P$ states which are above the threshold of open-charm decay channels, the coupled-channel effect should be seriously considered, which will be the task in this work. We will illustrate why the mass gap of $\chi_{c0}(2P)$ and $\chi_{c2}(2P)$ is far smaller than that of $\chi_{b0}(2P)$ and $\chi_{b2}(2P)$ by an unquenched quark model calculation. In the following sections, we will give a detailed illustration.

Finally, when treating $X(3915)$ as $\chi_{c0}(2P)$, we need to answer the remaining problem whether or not $\chi_{c0}(2P)$ has wide width, which is a crucial point we have to face. In this work, we will explicitly present that $\chi_{c0}(2P)$ should be a narrow state which is due to the node effect. Thus, two $\chi_{c0}(2P)$ candidates like $X(3840)$ in Ref. Guo:2012tv and $X(3860)$ reported by the Belle Collaboration Chilikin:2017evr should be excluded.

This paper is organized as follows. After the Introduction, we will introduce the mass problem of a quenched quark model. Next, we will give a coupled-channel picture for the discussed $\chi_{c0}(2P)$ state in Sec. II. In Sec. III, the numerical result will be presented. Especially, we give an analysis why we can get consistent results with experimental data of $X(3915)$. At last, this paper ends with a summary in Sec. IV.

With the observation of a series of charmonia, the Cornell model for quantitatively depicting the strong interactions between quarks was proposed by Eichten et al. Eichten:1974af . Since then, different versions of a potential model were developed by different groups. Among them, the Godfrey-Isgur (GI) model Godfrey:1985xj was extensively applied to study the hadron spectrum. In this work, we firstly illustrate the mass problem of a quenched quark model by presenting the spectrum of $2P$ charmonium states, where the GI model was adopted.²²2Here, we need to comment on the calculated result of the mass of ${}^{3}P_{0}$ $c\bar{c}$ state by the nonrelativistic quark model. In Ref. Barnes:2005pb , the authors adopted the nonrelativistic quark model to give the mass spectrum of the charmonium family. We may reproduce most of their results by applying a perturbation method, where $H_{0}$ and $H^{\prime}$ are treated as a solvable part and a perturbation term, respectively. However, for $1^{3}P_{0}$ and $2^{3}P_{0}$ states, the calculated masses are not stable and convergent when including higher order perturbation contributions. For example, if adopting the potential suggested in Ref. Barnes:2005pb , mass of $1^{3}P_{0}$ is 3.525, 3.425, 3.351, and 3.266 GeV and mass of $2^{3}P_{0}$ is 3.943, 3.854, 3.781 and 3.701 GeV when zeroth-order, first-order, second-order, and third-order perturbation contributions are considered step by step in calculation. If adopting the potential given in Ref. Badalian:1999fe , there exists the same problem for the calculation of the mass of $1^{3}P_{0}$ and $2^{3}P_{0}$ states. This problem is due to the singularity of $1/r^{3}$-like terms in the potential near $r=0$. However, in the GI model, this singularity is smeared. Thus, such a problem does not exist.

The GI model is a semirelativistic potential model with a Hamiltonian

where $m_{1}$ and $m_{2}$ are masses of quark and antiquark. The potential $\tilde{V}(\textbf{p,r})$ is composed of a short-range $\gamma^{\mu}\otimes\gamma_{\mu}$ interaction of one-gluon exchange and a long-range $1\otimes 1$ linear color confining interaction. When taking the nonrelativistic limit, a familiar nonrelativistic potential can be obtained from $\tilde{V}(\textbf{p,r})$. In the GI model, the relativistic corrections can be considered by smearing transformation and momentum-dependent factors. Here, the smearing function should be introduced, i.e.,

by which the confining potential $S(r)=br+c$ and one-gluon exchange potential $G(r)=-4\alpha_{s}(r)/(3r)$ can be smeared out by

For a general relativistic form of the potential, it should be dependent on momenta of interacting quarks in the center-of-mass system. Thus, we should further modify this smeared potential $\tilde{V}(r)$ by

with $E_{c}=(p^{2}+m_{c}^{2})^{1/2}$ and $E_{\bar{c}}=(p^{2}+m_{\bar{c}}^{2})^{1/2}$, where a parameter $\epsilon_{i}$ corresponds to different types of interactions. The details of the GI model can be found in Ref. Godfrey:1985xj .

In Table 1, we list the parameters of the GI model, which can be obtained by refitting the masses of the low-lying well-established charmonia ($\eta_{c}(1S)$, $J/\psi$, $\psi(3686)$, $\psi(3770)$, $h_{c}(1P)$, $\chi_{c0}(1P)$, $\chi_{c1}(1P)$, $\chi_{c2}(1P)$, $\psi(4040)$, and $\psi(4160)$) Tanabashi:2018oca . The obtained values are slightly different from those given in Ref. Godfrey:1985xj . Here, the obtained masses (in units of GeV) of $1^{1}S_{0}$, $1^{3}S_{1}$, $2^{1}S_{0}$, $2^{3}S_{1}$, $3^{3}S_{1}$, $1^{1}P_{1}$, $1^{3}P_{0}$, $1^{3}P_{1}$, $1^{3}P_{2}$, $1^{3}D_{1}$, $1^{3}D_{2}$, $1^{3}D_{3}$, and $2^{3}D_{1}$ are $2.996$, $3.098$, $3.634$, $3.676$, $4.090$, $3.513$, $3.417$, $3.500$, $3.549$, $3.805$, $3.828$, $3.841$, and $4.172$, respectively. Just shown in above, these low-lying charmonia can be well reproduced.

With the same parameters as input, we may give the masses of $2P$ states and make a comparison with the observed $X(3872)$, $X(3915)$, and $Z(3930)$. There exists the 64 MeV difference between $2^{3}P_{1}$ charmonium and $X(3872)$, which is the famous low mass puzzle of $X(3872)$. In addition, the mass gap (89 MeV) between $2^{3}P_{0}$ and $2^{3}P_{2}$ $c\bar{c}$ states is far larger than that between $X(3915)$ and $Z(3930)$, which is 12 MeV. In Fig. 2, the difference of mass spectrum between the $2P$ states given by the GI model and the observed three charmoniumlike states is explicitly illustrated.

This is the mass problem of the $2P$ charmonium spectrum by the quenched quark model. Hence, we should develop an unquenched picture when facing such a mass problem since the allowed open-charm decay channels are open for these $2P$ states. This will be the crucial task dedicated in this paper.

When checking the masses from a quenched quark model like the GI model, we notice that the discussed $2P$ $c\bar{c}$ states are above the $D\bar{D}$ and $D\bar{D}^{*}$ thresholds. For $\chi_{c1}(2P)$, $S$-wave and $D$-wave interactions occur for the $\chi_{c1}(2P)$ coupling with the $D\bar{D}^{*}$. For $\chi_{c0}(2P)$, it can couple with $D\bar{D}$ via an $S$-wave interaction while $\chi_{c2}(2P)$ may interact with the $D\bar{D}$ and $D\bar{D}^{*}$ via a $D$-wave coupling. Thus, in this section we exam the coupled-channel effect from the $D\bar{D}$ and $D\bar{D}^{*}$ channels to the mass spectrum of $2P$ charmonia. In the following subsection, we first introduce some historical results of $\chi_{c0}(2P)$ presented in some published literatures. After that, the unquenched model adopted in this paper will be introduced.