The possible assignments of the scalar $K_0^*(1950)$ and $K_0^*(2130)$ within the $^3P_0$ model

We have evaluated the strong decays of the $K_0^*(1950)$ and $K_0^*(2130)$ within the $^3P_0$ model, by employing the meson wave functions from the relativized quark model. By comparing with the experimental measurements, the $K_0^*(2130)$ could be assigned as $K_0^*(3^3P_0)$, while the $K_0^*(1950)$ seems like an exotic state, because its width can not be reasonably reproduced within the $^3P_0$ model. We also predict that the $K_0^*(2^3P_0)$ state has a mass of about $1811$ MeV and a width of about $656$ MeV, while the $K_0^*(4^3P_0)$ state has a mass of about $2404$ MeV and a width of about $180$ MeV.

We have evaluated the strong decays of the K * 0 (1950) and K * 0 (2130) within the 3 P 0 model, by employing the meson wave functions from the relativized quark model. By comparing with the experimental measurements, the K * 0 (2130) could be assigned as K * 0 (3 3 P 0 ), while the K * 0 (1950) seems like an exotic state, because its width can not be reasonably reproduced within the 3 P 0 model. We also predict that the K * 0 (2 3 P 0 ) state has a mass of about 1811 MeV and a width of about 656 MeV, while the K * 0 (4 3 P 0 ) state has a mass of about 2404 MeV and a width of about 180 MeV. PACS numbers:

I. INTRODUCTION
According to the theory of quantum chromodynamics (QCD), in addition to the conventional qq mesons, the so-called exotic states are also permitted, such as tetraquarks, molecules, glueballs, and hybrids [1][2][3]. Although many exotic states have been observed experimentally, such as X(3872), Z c (3900), Z c (4025), P c , T cc , it is still difficult to distinguish between the exotic states with conventional quantum numbers and the ordinary qq mesons.
One puzzle in hadron spectra is the scalar mesons, since there are too many states to be accommodated within the quark model without difficulty [4]. For example, K * 0 (700) state (also known as κ), together with its multiple partners a 0 (980), f 0 (500)(σ), and f 0 (980), does not fit well into the predictions of the quark model, since the observed mass ordering of these lowest scalar states is m σ < m κ < m a 0 , f 0 [5], while in the conventional quark model, by a naive counting of the quark mass, the mass ordering of the scalar qq nonet should be m σ ∼ m a 0 < m κ < m f 0 . These scalar states below 1 GeV are generally believed not to be qq states [2,3,[6][7][8][9].
Within the naive quark model, it is natural to assume that the a 0 (1450), K * 0 (1430), f 0 (1710), and f 0 (1370) are the 1 3 P 0 members of the SU(3) flavor nonet [5]. The isovector scalar mesons a 0 (2020)/a 0 (1950) are suggested to be the good candidates of the a 0 (3 3 P 0 ) in our previous work [10]. However, the assignments of the excited scalar K * 0 states are still unclear. Up to now, above the K * 0 (1430) mass, only two scalar states K * 0 (1950) and K * 0 (2130) are reported. Their masses and widths are listed in Table I . Since K * 0 (1950) was first reported in the Kπ invariant mass distribution of the K − p → K − π + n reaction by LASS in 1988 [12], it is difficult to interpret its properties within the quark model. The K * 0 (1950) mass is close to the K * 0 (2 3 P 0 ) mass of about 1890 MeV predicted by the Godfrey-Isgur (GI) quark model [13]. However, it is expected that the K * 0 (2 3 P 0 ) with a mass of 1850 MeV has a width of about 450 MeV within the 3 P 0 decay model [14], larger than the K * 0 (1950) width. In addition, Ref. [15] recently analyzed the kaon family within the modified GI model involving the color screening effect, and predicted the mass and width of the K * 0 (2 3 P 0 ) to be M = 1829 MeV and Γ = 1000 MeV, respectively, both of which disfavor the assignment of K * 0 (1950) as the candidate the K * 0 (2 3 P 0 ) state. The K * 0 (2130) was recently observed in the η c decays by the BABAR Collaboration [11], and its mass is close to the K * 0 (3 3 P 0 ) mass of 2176 MeV predicted by the modified GI model [15]. In addition, we have estimated that the nn(3 3 P 0 ) mass is about 1.9 ∼ 2.0 GeV [10], thus one can naturally expect the K * 0 (3 3 P 0 ) mass should be about 100 ∼ 200 MeV larger than the a 0 (3 3 P 0 ) mass. Based on its mass information, the K * 0 (2130) seems a good candidate of the K * 0 (3 3 P 0 ). The mass information alone is insufficient to identify the K * 0 (2130) as the K * 0 (3 3 P 0 ) state. We shall discuss the possibility of the K * 0 (2130) as the K * 0 (3 3 P 0 ) state by studying its strong decay properties.
In this work, we will investigate the possible assignment of K * 0 (2130) by analyzing the strong decay behaviors within the 3 P 0 decay model. For completeness, we also check the possibility of the K * 0 (1950) as the ordinary scalar mesons, since it is natural and necessary to exhaust the possible qq descriptions of a newly observed state before restoring to the more exotic assignments. This paper is organized as follows. In Sec. II, we introduce the 3 P 0 strong decay model used in our calculations, and the results and discussions are given in Sec. III. Finally, a summary is given in Sec. IV.

II. MODEL AND PARAMETERS
The 3 P 0 model has been widely used to study the Okubo-Zweig-Iizuka (OZI)-allowed open flavor two-body strong decays, it was originally introduced by Micu [16] and further developed by Le Yaouanc et al. [17][18][19]. In the 3 P 0 model, the meson strong decay takes place by producing a quark-antiquark pair with vacuum quantum number J PC = 0 ++ . The newly produced quark-antiquark pair, together with the qq within the initial meson, regroups into two outgoing mesons in two possible quark rearrangement ways, as shown in Fig. 1. The 3 P 0 model has been widely applied to study strong decays of hadrons with considerable success [10,14,[20][21][22][23][24][25][26][27][28][29][30][31][32][33]. Following the conventions in Refs. [20,21], 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 corresponding to 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 system, 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 ) are the wave functions of the mock mesons defined by Ref. [34]. 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. The partial wave amplitude M LS (p) can be given by [35], 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 [16] which assumes a spatially constant pair-production strength γ, adopt the relativistic phase space, and employ the relativized quark model (RQM) wave functions [13].
With the relativistic phase space, the decay width Γ(A → BC) can be expressed in terms of the partial wave amplitude
We take γ = 0.52 by fitting to the total width of K * 0 (1430) as the 1 3 P 0 state. The decay widths of K * 0 (1430) as the K * 0 (1 3 P 0 ) state are listed in Table II. According to our results, the dominant decay mode of K * 0 (1430) is Kπ, which is consistent with the experimental data [5].

Channel
Mode MeV [5], it is hard to assign the K * 0 (1950) as the ordinary scalar qq meson. Then we will discuss the possible assignment of the K * 0 (2130) state. We have shown the decay widths of K * 0 (2130) as the K * 0 (3 3 P 0 ) and K * 0 (4 3 P 0 ) states in Table IV, and total width is expected to be about 105 MeV and 20 MeV, respectively for the cases of the K * 0 (3 3 P 0 ) and K * 0 (4 3 P 0 ). The dependences of the total widths of K * 0 (3 3 P 0 ) and K * 0 (4 3 P 0 ) on the initial state mass are shown in Fig. 2 and Fig. 3, respectively. Within the uncertainties of the K * 0 (2130) mass, the total width of K * 0 (3 3 P 0 ) is in agreement with the experimental data Γ = 95±42±76 MeV [5], which implies that the K * 0 (2130) should be the good candidate of the K * 0 (3 3 P 0 ) state.
Phenomenologically, it is suggested that the light mesons could be grouped into the following Regge trajectories [29,33,38], where M 0 is the lowest-lying meson mass, n is the radial quantum number, and µ 2 is the slope parameter of the IV: Decay widths of K * 0 (2130) as 3 3 P 0 and 4 3 P 0 states (in MeV). The initial state mass is 2128 MeV.

Channel
Mode  corresponding trajectory. In the presence of K * 0 (1430) and K * 0 (2130) being the K * 0 (1 3 P 0 ) and K * 0 (3 3 P 0 ) states, we can roughly estimate the K * 0 (2 3 P 0 ) mass to be about 1811 MeV and K * 0 (4 3 P 0 ) mass to be about 2404 MeV as shown in Fig.4. It should be pointed out that, in our previous work [10], the mass scale for the nn(2 3 P 0 ) nonets is expected to be 1700 ∼ 1800 MeV. In addition, the mass of K * 0 (2 3 P 0 ) state is predicted  to be 1890 MeV by the GI model [13] and to be 1829 MeV by the modified GI model [15]. The prediction of the K * 0 (2 3 P 0 ) mass from the Regge trajectories is consistent with these previous predictions. The strong decays of K * 0 (2 3 P 0 ) with a mass of 1811 MeV are presented in Table V. The dependence of the total width of K * 0 (2 3 P 0 ) on the initial state mass is shown in Fig. 5. When the initial state mass varies from 1700 MeV to 1900 MeV, the total width of the K * 0 (2 3 P 0 ) varies from about 215 MeV to 1111 MeV. The decay width varies greatly, since some decay modes are open gradually. The total width of K * 0 (2 3 P 0 ) is expected to be about 656 MeV. The dominant decay modes of K * 0 (2 3 P 0 ) are π(1300)K, Kη(1295), and πK(1460). The K * 0 (2 3 P 0 ) is predicted to be a broad state, which could be the reason that the K * 0 (2 3 P 0 ) candidate is not yet observed experimentally.
We show the partial decay widths and the total decay width of the K * 0 (4 3 P 0 ) state with a mass of 2404 MeV in Table VI. The total width of K * 0 (4 3 P 0 ) is expected to be about 180 MeV. The dominant decay modes of K * 0 (4 3 P 0 ) include K * (1410) + ω, K * (1410) + ρ, K * (892)ω(1420). The dependence of the total width of the K * 0 (4 3 P 0 ) state on the initial state mass is shown in Fig. 3. When the initial state mass varies from 2300 MeV to 2500 MeV, the total width of the K * 0 (4 3 P 0 ) varies from about

IV. SUMMARY
In this work, we have discussed the possible assignments of K * 0 (1950) and K * 0 (2130) by calculating the strong decay widths within the 3 P 0 strong decay model. We suggest that the K * 0 (2130) could be assigned as K * 0 (3 3 P 0 ) based on its mass and width. However, the K * 0 (1950) seems like an exotic state, because its width can not be reasonably reproduced within the 3 P 0 model.
With the assignment of the K * 0 (2130) as the K * 0 (3 3 P 0 ) state, we have roughly estimated the masses of K * 0 (2 3 P 0 ) and K * 0 (4 3 P 0 ) to be about 1811 MeV and 2404 MeV, respectively, within the Regge trajectories. The total width of K * 0 (2 3 P 0 ) is predicted to be about 656 MeV, which implies that this state is not easy to be observed experimentally. The total width of K * 0 (4 3 P 0 ) is predicted to be about 180 MeV, which could be helpful to search for the K * 0 (4 3 P 0 ) state in future.