The possible members of the $5^1S_0$ meson nonet

The strong decays of the $5^1S_0$ $q\bar{q}$ states are evaluated in the $^3P_0$ model with two types of space wave functions. Comparing the model expectations with the experimental data for the $\pi(2360)$, $\eta(2320)$, $X(2370)$, and $X(2500)$, we suggest that the $\pi(2360)$, $\eta(2320)$, and $X(2500)$ can be assigned as the members of the $5^1S_0$ meson nonet, while the $5^1S_0$ assignment for the $X(2370)$ is not favored by its width. The $5^1S_0$ kaon is predicted to have a mass of about 2418 MeV and a width of about 163 MeV or 225 MeV.


I. INTRODUCTION
In the framework of quantum chromodynamics (QCD), apart from the ordinary qq states, other exotic states such as glueballs, hybrids, and tetraquarks are permitted to exist in meson spectra. To identify these exotic states, one needs to distinguish them from the background of ordinary qq states, which requires one to understand well the conventional qq meson spectroscopy both theoretically and experimentally. To be able to understand the nature of a newly observed state, it is natural and necessary to exhaust the possible qq description before restoring to more exotic assignments.

As shown in
With the assignment of the X(2500) as the ss member of the 5 1 S 0 nonet, one can expect that other members of the 5 1 S 0 nonet should be lighter than the X(2500). Along this line, considering that other pseudoscalar states have discussed in our previous works [4,5], we shall focus on the π(2360) and η(2320) shown in Table I, and check whether they can be explained as the 5 1 S 0 qq states or not. Study on the pseudoscalar radial qq excitations in the mass region of 2.3 ∼ 2.6 GeV is especially interesting because the pseudoscalar glueball is predicted to exist in this mass region [7][8][9].
Both the mass and width of a resonance are related to its inner structure. Although the masses of the π(2360), η(2320), and X(2500) are consistent with them belonging to the 5 1 S 0 meson nonet, their decay properties also need to be compared with model expectations in order to identify the possible candidates for the 5 1 S 0 meson nonet. Below, we shall evaluate their strong decays in the framework of the 3 P 0 model. This paper is organized as follows. In Sec. II, we present the 3 P 0 model parameters used in our calculations. The results and discussions are given in Sec. III. Finally, a short summary is given in Sec. IV.
Following the conventions in Ref. [4], the transition operator T of the decay A → BC in the 3 P 0 model is given by where the γ is a dimensionless parameter denoting the production strength of the quark-antiquark pair q 3q4 with quantum number J PC = 0 ++ . p 3 and p 4 are the momenta of the created quark q 3 and antiquarkq 4 , respectively. χ 34 1,−m , φ 34 0 , and ω 34 0 are the spin, flavor, and color wave functions of q 3q4 , respectively. The solid harmonic polynomial Y m 1 (p) ≡ |p| 1 Y m 1 (θ p , φ p ) reflects the momentum-space distribution of the q 3q4 .
The S matrix of the process A → BC is defined by where |A (|B ,|C ) is the mock meson defined by [37] |A(n 2S A +1 Here m 1 and m 2 (p 1 and p 2 ) are the masses (momenta) of the quark q 1 and the antiquarkq 2 , respectively; A , and ψ n A L A M L A (p A ) are the spin, flavor, color, and space wave functions of the meson A composed of q 1q2 with total energy E A , respectively. n A is the radial quantum number of the meson A. S A = s q 1 + sq 2 , J A = L A + S A , s q 1 (sq 2 ) is the spin of q 1 (q 2 ), and L A is the relative orbital angular momentum between q 1 andq 2 .
The transition matrix element BC|T |A can be written as where M M J A M J B M J C (P ) is the helicity amplitude. In the center of mass frame of meson A, the helicity amplitude is where P = P B = −P C , p = p 3 , m 3 is the mass of the created quark q 3 . The partial wave amplitude M LS (P ) can be given by [38], Various 3 P 0 models exist in literature and typically differ in the choices of the pair-production vertex, the phase space conventions, and the meson wave functions employed. In this work, we restrict to the simplest vertex as introduced originally by Micu [39] which assumes a spatially constant pairproduction strength γ, adopt the relativistic phase space, and employ two types of meson space wave functions: the simple harmonic oscillator (SHO) wave functions and the relativized quark model (RQM) wave functions [40].
With the relativistic phase space, the decay width Γ(A → BC) can be expressed in terms of the partial wave amplitude where The parameters used in the 3 P 0 model calculations involve the qq pair production strength γ, the parameters associated with the meson wave functions, and the constituent quark masses. In the SHO wave functions case (case A), we follow the parameters used in Ref. [25], where the SHO wave function scale is β = β A = β B = β C = 0.4 GeV, the constituent quark masses are m u = m d = 330 MeV, m s = 550 MeV, and γ = 8.77 obtained by fitting to 32 well-established decay modes. In the RQM wave functions case (case B), we take m u = m d = 220 MeV, and m s = 419 MeV as used in the relativized quark model of Godfrey and Isgur [40], and γ = 15.28 by fitting to the same decay modes used in Ref. [25] except for three decay modes without the specific branching ratios K * ′ → ρK, K * ′ → K * π, and a 2 → ρπ [2]. The meson flavor wave functions follow the conventions of Refs. [24,40]. We assume that the a 0 (1450), K * 0 (1430), f 0 (1370), and f 0 (1710) are the ground scalar mesons as in Refs. [23,24,41]. Masses of the final state mesons are taken from Ref. [2].

A. π(2360)
The decay widths of the π(2360) as the π(5 1 S 0 ) are listed in Table II. The π(5 1 S 0 ) total width is predicted to be about 281 MeV in case A or 285 MeV in case B, both in agreement with the π(2360) width of Γ = 300 +100 −50 MeV [10,11]. The dependence of the π(5 1 S 0 ) width on the initial mass is shown in Fig. 2. Within the π(2360) mass errors (2360 ± 25 MeV), in both cases, the predicted width of the π(5 1 S 0 ) always overlaps with the π(2360) width. Therefore, the measured width for the π(2360) supports that the π(2360) can be identified as the π(5 1 S 0 ). The flux-tube model calculations in Ref. [42] also favor this assignment.
As shown in Eqs. (6) and (7), the partial width from the 3 P 0 model depends on the overlap integrals of flavor, spin, and space wave functions of initial and final states. For a given decay mode, the overlap integrals of the flavor and spin wave functions of initial and final mesons are identical in both RQM and SHO cases, therefore, the partial width difference between the RQM and SHO cases results from the different choices of meson space wave functions. Generally speaking, the different space wave functions would lead to different decay widths. Especially, if the overlap is near to the nodes of space wave functions, the decay width would strongly depend on the details of wave functions, and the small wave function difference could generate a large discrepancy of the decay width. However, for some modes, the possibility that the different wave functions can give the similar decay widths also exists. To our knowledge, there is no some rules to judge whether the RQM and SHO wave functions can give the similar or different results before the numerical calculations.
The difference between the predictions in case A and those in case B provides a chance to distinguish among different meson space wave functions. At present, we are unable to conclude which type of wave function is more reasonable due to the lack of the branching ratios for the π(2360). However, as suggested by Ref. [45], we should keep in mind that it is essential to treat the wave functions accurately in the 3 P 0 model calculations. B. η(2320) and X(2500) The η(2320) was observed inpp → ηηη process, and its mass and width are 2320 ± 15 MeV and 230 ± 35 MeV [46]. The predicted η(5 1 S 0 ) mass in the relativistic quark model is about 2385 MeV [13], close to the η(2320) mass. In the presence of the X(2500) as the isoscalar member of the 5 1 S 0 meson nonet [6], we shall discuss the possibility of the η(2320) as the isoscalar partner of the X(2500).  [10,11] In a meson nonet, the two physical isoscalar states can mix. The mixing of the two isoscalar states can be parametrized as where nn = (uū+dd)/ √ 2 and ss are the pure 5 1 S 0 nonstrange and strange states, respectively, and the φ is the mixing angle. Accordingly, the partial widths for the η(5 1 S 0 ) and X(2500) can be expressed as Γ(X(2500) → BC) = π P 4M 2 Under the mixing of η(2320) and X(2500), their decays in the case A are listed in Table III and those in the case B are listed in Table IV. The dependence of the η(2320) and X(2500) total widths on the mixing angle φ is displayed in Fig. 3. In order to simultaneously reproduce the measured widths for the η(2320) and X(2500), the mixing angle φ is required to satisfy −0.5 ≤ φ ≤ 0.45 radians in case A or −0.69 ≤ φ ≤ 0.59 radians in case B. Below, we shall estimate the value of φ to check whether it satisfies these constraints based on the mass-squared describing the mixing of two isoscalar mesons.
In above discussions, we focus on the possibility of the pseudoscalar states π(2360), η(2320), and X(2500) as the 5 1 S 0 mesons. Apart from the states listed in Table I, the X(2120) and X(2370) also probably are the J PC = 0 −+ resonances. The X(2120) and X(2370) were observed by the BESIII collaboration in the π + π − η ′ invariant mass spectrum and their spin parities are not determined [1]. Based on the observed decay mode π + π − η ′ , the possible J PC for the X(2120) and X(2370) are 0 −+ , 1 ++ , · · · . The natures of the X(2120) and X(2370) are not clear [42,[49][50][51][52]. Since the X(2370) mass is also close to the quark model expectation for the η(5 1 S 0 ) mass [13], we shall discuss the possibility of the X(2370) as the isoscalar partner of the X(2500).
With the X(2370)-X(2500) mixing, the decay widths for the X(2370) are listed in Table V. The dependence of the X(2370) and X(2500) total widths on the mixing angle is plotted in Fig. 4. Obviously, the X(2370) width can not be reproduced in the whole region of the mixing angle. Therefore, our calculations do not support the 5 1 S 0 assignment for the X(2370). Other calculations from the 3 P 0 model suggest that the X(2370) is unlikely to be the 4 1 S 0 qq state [50,51]. If the X(2370) turns out to have J PC = 0 −+ in future, in order to explain its properties, more complicate scheme such as the qq-glueball mixing may be necessary, since the X(2370) mass is close to the pseudoscalar glueball mass of about 2.3 − 2.6 GeV predicted by the lattice QCD [7][8][9].
C. K(5 1 S 0 ) As mentioned in Sec. III B, with the π(2360), η(2320), and X(2500) as the members of 5 1 S 0 meson nonet, from Eq. (17), the K(5 1 S 0 ) mass is predicted to be about 2418 MeV as shown in Eqs. (19) and (20). At present, no candidate for the I(J P ) = 1/2(0 − ) state around 2418 MeV is reported experimentally. It is noted that with our estimated masses for the K(4 1 S 0 ) and K(5 1 S 0 ), M K(4 1 S 0 ) = 2153 ± 20 MeV [5] and M K(5 1 S 0 ) = 2418 ± 49 MeV, the K, K(1460), K(1830), K(2153), and K(2418) approximately populate a trajectory as shown in Fig. 5, which indicates that the K(2153) and K(2418) could be the good candidates for the 4 1 S 0 and 5 1 S 0 kaons, respectively. The decay widths of the K(2418) as the 5 1 S 0 kaon are listed in Table VI. The total width of the K(5 1 S 0 ) is predicated to be about 163 MeV in case A or 225 MeV in case B. This could be of use in looking for the candidate for the 5 1 S 0 kaon experimentally.

IV. SUMMARY AND CONCLUSION
In this work, we have discussed the possible members of the 5 1 S 0 meson nonet by analysing the masses and calculating the strong decay widths in the 3 P 0 model with the SHO and RQM meson space wave functions. Both the mass and width for the π(2360) are consistent with the quark model expectations for the π(5 1 S 0 ). In the presence of the X(2500) as the 5 1 S 0 isoscalar state, the possibility of the η(2320) and X(2370) as the isoscalar partner of the X(2500) is discussed. The X(2370) width can not be reproduced for any value of the mixing angle φ, thus, the assignment of the X(2370) as the 5 1 S 0 isoscalar state is not favored by its width. Both the η(2320) and X(2500) widths can be reproduced with −0.5 ≤ φ ≤ 0.45 radians for the SHO wave functions or −0.69 ≤ φ ≤ 0.59 for the RQM wave functions. The assignment of the π(2360), η(2320), and X(2500) as the members of the 5 1 S 0 nonet not only gives φ = −0.1 radians, which naturally accounts for the η(2320) and X(2500) widths, but also shows that the 5 1 S 0 kaon has a mass of about 2418 MeV. The K, K(1460), K(1830), K(2153), and K(2418) approximately populate a common trajectory. The K(2418) is predicted to have a width of about 163 MeV for SHO wave functions or 225 MeV for the RQM wave functions. We tend to conclude that the π(2360), η(2320), X(2500), together with the unobserved K(2418), construct the 5 1 S 0 meson nonet.
Our numerical results show that the 3 P 0 model predictions depend on the choice of meson space wave functions. It is essential to treat the wave functions accurately in the 3 P 0 model calculations. The difference between the predictions in SHO case and those in RQM case provides a chance to    2). In our fit, the we don't use the data of K(1460) since the K(1460) mass error is not given experimentally. The masses of the K and K(1830) are taken from Ref. [2]. The masses of the K(2153) and K(2418) are taken to be 2153 ± 20 MeV and 2418 ± 49 MeV, respectively. The K(1460) mass is taken to be 1460 MeV [53]. distinguish among different meson space wave functions. To conclude which type of wave function is preferable, the further experimental study on the decays of π(2360), η(2320), and X(2500) is needed. Also, in our calculations, all the states are assumed to be qq. It is noted that some resonances such as h 1 (1170), h 1 (1380), f 1 (1285), b 1 (1235), a 1 (1260), and K 1 (1270), can also be explained as the dynamically gener-ated resonances [54][55][56], which means they might have large hadron-molecular components in their wave functions. If so, both the SHO and RQM wave functions derived from the simple qq picture, would be not appropriate and could lead to the big discrepancies between the experiments and the 3 P 0 model predictions. To test this point, the further experimental information about the partial widths is also needed.