Canonical interpretation of the $X(4140)$ state within the $^3P_0$ model

Recently, LHCb collaboration has confirmed the state $X(4140)$, with a much larger width $\Gamma= 83\pm 21^{+21}_{-14}$ MeV than the previous experimental measurements, which has confused the understanding of its nature. We have investigated the possibility of the $\chi_{c1}(3P)$ interpretation for the $X(4140)$, considering the mass and the strong decay properties.


I. INTRODUCTION
Since the X(3872) was discovered in 2003 by Belle collaboration [1], a lot of unexpected state (charmoniumlike states or XYZ states) have been reported experimentally [2]. Most of them have strange properties, and are difficult to be interpreted as the charmonium states, which make them more like exotic states [3][4][5][6].
In 2009, a new-threshold X(4140) state was first reported in the B + → J/ψφK + process by CDF collaboration [7], with a statistical significance of the signal 3.8σ. This state was confirmed in the same process by CMS [8] and D0 collaborations [9,10], and also in the reanalyzed the B ± → J/ψφK ± process with a larger data sample by CDF collaboration [11]. However, the Belle, LHCb, and Babar collaborations have not found the signal of this state [12][13][14]. Since the X(4140) in only seen in the J/ψφ channel, which is OZI suppressed for the charmonium assignment, there are a lot of theoretical interest about its properties, such as charmonium state, molecular state, tetraquark state, hybrid state, or a rescattering effect (more information can be found in the reviews [4,6]).
In 2017, the LHCb collaboration has also confirmed this state with high statistic data [15,16], with a mass 4146.5 ± 4.5 MeV and a width 83 ± 21 MeV, much larger than the previous experimental measurements (see the  Table I), and the quantum numbers of this state were determined to be J P C = 1 ++ . Thus, the D * sD * s molecular explanation, which prefers the quantum numbers J P C = 0 ++ or 2 ++ , is ruled out [17][18][19][20][21][22][23][24].
However, the X(4140) is still the subject of much theoretical work, and there are many different suggestions about its structure [25][26][27][28], because of the large discrepancy of the width. For instance, the X(4140) state with assignment of the χ c1 (3P ) state is predicted to have a small width in Ref. [26]. In Ref. [27], the partial width of the decay mode X(4140) → J/ψφ is predicted to be 86.9±22.6 MeV, with the axial-vector tetraquark picture * Electronic address: wangen@zzu.edu.cn for the X(4140).
Indeed, it is natural and necessary to exhaust the possible qq description of the observed states before restoring to the more exotic assignments. While the ground states of the P -wave charmonium states, χ cJ (1P ), have been well established, and the first radial excitation, χ cJ (2P ), are predicted to be have the mass around 3900 MeV [2,[29][30][31][32][33], the X(4140), with the quantum numbers of J P C = 1 ++ , could be the second radial excitation χ c1 (3P ), with the predicted mass of 4100 ∼ 4200 MeV in the quark model [31,34].
In this work, taking the meson wave functions by solving the relativistic/non-relativistic Schrödinger equation, we will investigate the decay properties of the X(4140) as the assignment of charmonium state in the 3 P 0 model, and provide more information about the decay modes, and the observation of the X(4140) in other channels could be useful to extract the more precision width. This paper is organized as follows. In Sec. II, we will present a brief review of the 3 P 0 decay model, and in Sec. III, we will give two kinds of wave functions for the mesons. The results and the discussions are shown in Sec. IV. Finally, the summary are given in Sec. V.

II. THE 3 P0 DECAY MODEL
In this section, we will present the 3 P 0 model, which is used to evaluate the Okubo-Zweig-Iizuka (OZI) allowed open charm decays of the χ cJ (3P ). The 3 P 0 model, also known as the quark-pair creation model, was originally introduced by Micu [35] and further developed by Le Yaouanc et al. [36][37][38]. The 3 P 0 model has been widely applied to study strong decays of hadrons with considerable success [39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56]. In this model, the strong decay of hadron occurs through a quark-antiquark pair creating from the vacuum with the vacuum quantum number J P C = 0 ++ , then the new quark-antiquark pair, together with the qq within the initial meson, regroups into two outgoing mesons in all possible quark rearrangement ways, as shown in Fig. 1. The transition operator T of the decay A → BC in the 3 P 0 model can be written, where γ is a dimensionless parameter corresponding the strength of quark-antiquark q 3q4 pair producing from the vacuum, and p 3 and p 4 are the momenta of the created quark q 3 andq 4 , respectively. χ 34 1,−m , φ 34 0 , and ω 34 0 are the spin, flavor, color wave functions of the q 3q4 , respectively.
The partial wave amplitude M LS (P ) of the decay A → BC can be given by [57], where M MJ A MJ B MJ C (P ) is the helicity amplitude and defined as, The |A , |B , and |C denote the mock meson states defined by Ref. [58]. Due to different choices of the pair-production vertex, phase space convention, employed meson space wave function, various 3 P 0 models exist in literature. In this work, we employ the simplest vertex as introduced originally by Micu which assumes a spatially constant pairproduction strength γ [35], the relativistic phase space. We will take into account the two choices of the wave functions for mesons, which will be presented in next section. Finally, the decay width Γ(A → BC) can be expressed in terms of the partial wave amplitude, where , and M A , M B , and M C are the masses of the meson A, B, and C, respectively. The explicit expressions for M LS (P ) can be found in Refs. [47][48][49].

III. WAVE FUNCTIONS
In this section, we will present the two choices of wave functions for the charmonium states, charm and charmed strange mesons, which will be used to calculation the χ cj (3P ) strong decay widths.

A. Non-relativistic quark model
For the wave functions of the open charm mesons in the final states, we use the non-relativistic quark model (NRQM), proposed by Lakhina and Swanson [59]. The non-relativistic quark model has been successfully used to describe the mass spectrum of charm and charmedstrange mesons [53,59], bottom mesons [56].
For the open charm mesons, the total Hamiltonian can be written as [31] where H 0 is the zeroth-order Hamiltonian, H sd is the spin-dependent Hamiltonian, and C qq is a constant. The H 0 can be compressed as where p is the center-of-mass momentum, r is the qq separation, M r = 2m q mq/(m q + mq), m q and mq are the masses of quark q and anti-quarkq, respectively, b is the linear potential slope and α c is the coefficient of Coulomb potential. The explicit expression of the H sd and the corresponding parameters are given in Refs. [53,59]. We have tabulated the spectra of charm and charmed-strange mesons in Table II and Table III, respectively, which are same as those of Ref. [53].
For the wave functions of the charmonium states, we will use the modified non-relativistic quark model (MN-RQM) by taking into account the screening effect, as discussed in Ref. [31]. When screening effect is considered, the modification can be accomplished by the transformation where µ is the characteristic scale for color screening. The mass spectra of the charmonium states are shown in Table IV, which are same as those of Ref. [31].

B. Modified Godfrey-Isgur model
In addition to the non-relativistic quark model, the Godfrey-Isgur (GI) relativistic quark model [60] is one of the most successful models to predict mass spectrum of mesons. Because coupled-channel effect becomes more important for higher radial and orbital excitations, the modified relativistic quark model was proposed [61,62] and widely used to calculate mass spectrum of charm meson [61], charmed-strange meson [62], charmonium [34] and bottomonium [63]. In the relativistic quark model, the Hamiltonian of a meson system is [60] whereH conf qq is spin-independent potential,H hyp qq is colorhyperfine interaction,H so qq is spin-orbit interaction. The explicit expression ofH conf qq ,H hyp qq , andH so qq are given in Refs. [62]. The spin-independent potential contains a constant term, a linear confining potential, and a onegluon exchange potential, Although the GI model has achieved great success in describing the meson spectrum, there still exists a discrepancy between the predictions and the recent experimental observation, as discussed in Ref. [62]. When screening effect is considered, the modification can be accomplished by the transformation With the modified Godfrey-Isgur model, we calculated the mass spectra of charm mesons, charmed-strange mesons, and charmonium in Table II, III, and IV, respectively, which are same as those of Refs. [34,61,62].

IV. RESULTS AND DISCUSSIONS
The mass spectra of the charmonium states predicted by the modified non-relativistic quark model (MNRQM) and modified Godfrey-Isgur model (MGI) are shown in Table IV. According to the PDG [2], the mass of the X(4140) [I G (J P C ) = 0 + (1 ++ )] is 4146.8 ± 2.4 MeV, which is consistent with the predicted mass of χ c1 (3P ) in both models within the uncertainties of the models. Next, we will calculate the strong decay widths of the X(4140) state as the χ c1 (3P ) assignment.
In our calculation, we take two kinds of the wave functions, by solving the Schrödinger equation in the (modified) non-relativistic quark model as discussed in Subsec. III A (Case A), and in the modified Godfrey-Isgur quark model as discussed in Subsec. III B (Case B) for the charm mesons, charmed-strange mesons, and the charmonium states. In the 3 P 0 model, we take the same constituent quark masses as those in Eq. With the above parameters, we have calculated the partial decay widths and total decay width, as shown in Table V  MeV of PDG, considering the uncertainties of the 3 P 0 model. It should be pointed out that the decay modes DD * and D * D * have large decay widths, which are also consistent with the conclusions of Refs. [26,33]. We suggest to search for this state in those two channels, and to measure the width precisely, which can be shed light on its nature.
We also show the dependence of the χ c1 (3P ) decay width on the masses with the wave functions of Case A and Case B, respectively in Fig. 2 and Fig. 3. The decay width of the χ c1 (3P ) state is 13 ± 3 MeV for Case A, and 31 ± 3 MeV for Case B, where the uncertainties corresponds to the mass error of the X(4140) [2]. Thus, if the small width of the X(4140) is confirmed in future high-precision measurements, the X(4140) could be explained as the charmonium state χ c1 (3P ). Indeed, the B + → J/ψφK + decay was investigated in Ref. [67], where the X(4140), with the small width Γ = 19 MeV, and the molecular state X(4160) were taken into account, and it was found that the low J/ψφ invariant mass distributions were better described compared with the analysis in Refs. [15,16] where only the X(4140) resonance was considered. Thus, the high-precision measurement about the X(4140) width is necessary to shed light on its possible nature.
Studying the strong decay properties of the χ c0 (3P ) and χ c2 (3P ) states is also useful to search for those states, and understand the family of the charmonium states. The decay widths of the χ c0 (3P ) and χ c2 (3P ) are tabulated in Table V, and the mass dependence of the total widths are also shown in Fig. 2 and Fig. 3, respectively corresponding to the results of Case A and Case B. The total decay width of χ c0 (3P ) is about 25 ± 3 MeV for Case A with the predicted mass 4131 ± 30 MeV, and about 35 ± 5 MeV for Case B with the predicted mass 4177 ± 30 MeV. For the χ c2 (3P ), the total decay width is predicted to be about 35 ± 5 MeV for Case A with the predicted mass 4208 ± 30 MeV, and about 43 ± 5 MeV for Case B with the predicted mass 4213 ± 30 MeV. In the energies region of 4100 ∼ 4250 [2], there is one state X(4160), with M = 4156 +29 −25 MeV and I G (J P C =? ? (? ?? ), but with Γ = 139 +110 −60 MeV, which is much larger than the predicted total widths of the χ cJ (3P ). Indeed, among the different interpretations of the X(4160), the D * sD * s molecular nature has been widely studied in Refs. [67][68][69][70].

V. SUMMARY
We have investigated the strong decay properties of the X(4140) with the assignment of the χ c1 (3P ) states in the 3 P 0 model, where the modified non-relativistic quark   The total decay width of the χ c1 (3P ) is predicted to be 13 ± 3 MeV for Case A, and 31 ± 3 MeV for Case B, both of which are in agreement with the PDG average width of X(4140). Thus, we conclude that, the X(4140), with a small width, could be explained as the charmonium state χ c1 (3P ), and the high-precision measurement about the X(4140) could shed light on the nature of the X(4140).
We also show the strong decay properties of χ c0 (3P ) and χ c2 (3P ), and the total widths of the χ c0 (3P ) and χ c2 (3P ) are predicted be about 20 ∼ 40 MeV and 30 ∼ 50 MeV, respectively. By comparing with the width of the X(4160), we find it is difficult to interpretation the X(4160) as the charmonium states χ cJ (3P ).