Study of the $\omega$ meson family and newly observed $\omega$-like state $X(2240)$

Since the present $\omega$ meson family has not been established, in this work, we carry out an investigation of the mass spectrum and Okubo-Zweig-Iizuka a allowed two-body strong decay of the $S$-wave and the $D$-wave $\omega$ mesons, and make the comparison with the experimental data of these reported $\omega$ states and the $\omega$-like $X(2240)$ state observed by BESIII. By this study, we not only suggest the possible assignments to these observed $\omega$ states under the framework of the $\omega$ meson family, but also predict three $\omega$ mesons ($\omega(5S)$, $\omega(2D)$ , and $\omega(4D)$) which are still missing in experiment. The present study may provide valuable information to further construct the $\omega$ meson family. Considering the present running status of BESIII, we also suggest that BESIII should pay more attention to the issue of $\omega$ meson with accumulating more data.


I. INTRODUCTION
There is abundant information on ω states collected in Particle Data Group(PDG) [1], which provides that their spinparity J PC could be 1 −− and all of them are isospin scalar. In In the other hand, very recently, the BESIII Collaboration observed a resonant structure in the line shape when they analyzed the cross section of the e + e − → K + K − process at center-of-mass energies varying from 2.00 to 3.08 GeV, which has the mass of 2239.2± 7.1± 11.3 MeV and the width of 139.8±12.3±20.6 MeV (we name it X(2240)) [2]. Reference [3] treats it as the candidate of a ssss. Given its production process, the quantum number of this resonant structure can be assigned as J PC = 1 −− . In addition, BESIII Collaboration observed a structure in partial-wave analysis of J/Ψ → K + K − π 0 , whose resonance parameters are M=2039.2±8 +36 −18 MeV and Γ =193±23 +25 −27 MeV with J PC = 1 −− (we name it X(2040)) [4].
In view of these ω states and the two new observed states, a systemical study for the ω meson family becomes very necessary and urgently.
Both ω and ρ family have the quantum number J PC = 1 −− and the same quark flavor(u and d quark). The only difference between them is that ω is iso-scalar while ρ is iso-vector, so the mass spectrum and decay information of ω and ρ family are very similar. In order to study ω meson family, we compare ω states with ρ states which were studied by Ref. [8]. We present the mass information of ω and ρ states in Table I.
In this paper, we will study the excited states of ω meson. By using modified Godfrey-Isgur model(MGI) and quark pair creation(QPC) model, the mass spectrum and strong decay behavior of excited states of ω meson are analyzed, which indicates that X(2040) is the candidate of ω(2D) meson with I(J P ) = 0(1 − ). Our research does not support X(2240) as the ω(4S ) or ρ(4S ) assignment. At the same time, the mass and the widths of ω(5S ) and ω(4D) are predicted, respectively.
The spectrum of the ω meson family are studied by using Regge trajectory and MGI model. Regge trajectory is very useful for the light meson mass spectrum analyzing [9][10][11]. MGI model contains the screening effect and works well for describing the higher excited states of ω meson [12][13][14][15][16][17][18][19]. Then, for further study the properties of ω mesons, their Okubo-Zweig-Iizuka (OZI)-allowed two-body strong decays are studied by taking input with the spatial wave functions obtaining in mass spectrum by numerically calculation. Their partial and total decay widths are calculated by using the QPC model which was proposed by Micu [20] and extensively applied to studies of strong decay of other hadrons [8,14,19,. The effort will be helpful to uncover the structure of X(2040) and X(2240), and establish ω meson family.
This paper is organized as follows. We first analyze the mass spectrum and decay behavior phenomenologically of ω mesons in Sec. II and Sec. III. In Sec. IV, we discuss our results and summarize the main conclusions. In 1985, Godfrey and Isgur propound GI model to describe relativistic meson spectra with great success, exactly in lowlying mesons [12]. As for the excited states, the screening potential should be taken into account for its coupled-channel effect [15,16,19,46].
The mesons' internal interaction is depicted by the Hamiltonian of potential model, which can be written as where m 1 and m 2 denote the mass of quark and antiquark respectively, the relation betweenṼ eff (p, r) and V eff (p, r) will be illustrated later, and the effective potential has a familiar for- mat in the non-relativistic limit [12,47], where S 1 /S 2 indicates the spin of quark/antiquark and L is the orbital momentum. F 1 and F 2 are related to the Gell-Mann matrices in color space . For a meson, F 1 · F 2 = −4/3, and the running coupling constant α s (r) has following form, where k is varying from 1 to 3 and the corresponding α k and γ k are constant, α 1,2,3 = 0.25, 0.15, 0.2 and γ 1,2,3 = 1 2 , √ 10 2 , √ 1000 2 [12]. H conf consists two pieces, the spin-independent linear confinement piece S (r) and Coulomb-like potential G(r), H hyp also includes two parts, tensor and contact terms, which is the color-hyperfine interaction. H SO denotes the spin-orbit interaction with colour magnetic term causing of one-gluonexchange and the Thomas precession term which can be written as In light meson system, we must consider the relativistic effects introduced by GI model.
On one hand, GI model introduces a smearing function for a qq meson which is the nonlocal interactions and new r dependence, i.e. ρ r − r ′ = σ 3 π 3/2 e −σ 2 (r−r ′ ) 2 , (2.9) the S (r) and G(r) become smeared potentialsS (r) andG(r) by the following procedurẽ where the values of σ 0 and s are defined in Table II.  On the other hand, for the sake of making up the losing of relativistic effects in the non-relativistic limit, a general potential relying on the the center-of-mass of interacting quarks and momentum are applied as whereṼ i (r) delegates the contact, tensor, vector spin-orbit and scalar spin-orbit terms, and ǫ i is the relevant modification parameter as shown in Table II. After the above revises in two points,Ṽ eff (p, r) is replaced by V eff (p, r). Via solving the Hamiltonian in Eq. (2.1) by exploiting harmonic oscillator (HO) basis, we obtain the mass spectrum and wave functions. In configuration and momentum space, HO wave functions have explicit forms respectively, where Y LM L (Ω) is the spherical harmonic function, and L L+1/2 n−1 (x) is the associated Laguerre polynomial, and β = 0.4 GeV for the calculation.
With diagonalizing the Hamiltonian matrix, the mass and wave function of meson which are available to the following strong decay process can be obtained.
In addition, we study the mass of ω family using Regge trajectory which is an effective approach to quantitatively study meson mass spectrum [48,49]. In general, there exists an expression [9,50] where M 0 is the mass of ground state and µ 2 denotes the trajectory slope and n is the radial quantum number of the corresponding meson with mass M. The relation expressed by Eq.    2000), ω(2330) and ρ(2270). At the same time. we found that X(2040) can be a good candidate of ω(2D). As the extension of Regge trajectory of ω, ω(5S ) and ω(4D) are predicted with the mass of 2.63 GeV and 2.64 GeV, respectively.
The mass spectrum of ω family can be obtained by applying the MGI model and the parameters are shown in Table II. Otherwise, besides the GI model can be used to calculate the mass spectrum of mesons with J P = 1 − , Reference [10] also gave a spectrum for ω/ρ meson. The mass spectrum for these ω states obtained by the MGI model is listed in Table II. We compare the numerical results of GI model [12], Ref. [10], MGI model, and experimental value in Table II. The results indicate that the mass of ω obtained in this work is more approaching the experimental values than the value of GI [12] and Ebert [10] for 2S , 1D, 2D and 3D states. More important is that the mass of MGI result and Regge trajectories for the ω(ρ) family are very close. For example, both MGI and Regge trajectories give the mass of ω(ρ)(4S) state is 2.258 GeV.
From Regge trajectories' and MGI's results, we make a summary as follows: 1. X(2240) may be the same state with ω(2205) as the candidate of 4 3 S 1 assignment. However, the widths of the two states are different. We will give a deep discussion for the two states in next section.
3. Result of MGI model shows that the mass of ω(5S ) and ω(4D) is 2.55 GeV and 2.62 GeV, respectively. Regge trajectories predict the mass of ω(5S ) and ω(4D) will be 2.63 GeV and 2.64 GeV, respectively. The physical mass of ω(5S ) and ω(4D) may be between the two theoretical value.
The above summary is just obtained by analyzing the mass spectrum. In the following section, we will give a deep discussion by the means of their two-body strong decay.
For the process A → B + C, 1m; 1 − m|00 dp 3 dp 4 δ 3 (p 3 + p 4 ) where the quark and antiquark are denoted by indices 3 and 4, respectively, γ depicts the strength of the creation of qq from vacuum Y ℓm (p) = |p| ℓ Y ℓm (p) are the solid harmonics. χ, φ, and ω denote the spin, flavor, and color wave functions respectively, which can be treated separately. Subindices i and j denote the color of a qq pair. The decay width reads where m A is the mass of an initial state A, and the two decay amplitudes can related by the Jacob-Wick formula [60] as In the calculation, the spatial wave functions of the discussed mesons can be numerically obtained by the MGI model.

B. numerical result of the strong decay of ω meson family
The study of ω and ρ meson families can be borrowed from each other, so we can estimate the mass of the missing states in these meson families. Similarly, we can study the strong decay of ω meson family comparing with that of ρ meson family. The γ value in Eq. 3.2 is taken by the following method. When fitting the ρ meson experimental width value (with error) using the theoretical total width, the range of γ value can be fixed. Then we use this γ value to calculate the width of ω meson. By analyzing the above mass results of ω, we know that ω(1420), ω(1960), ω(2205) and ω(2553) are the radial excitations of ω(782). We will discuss their two-body strong decay behaviour.

S-wave ω mesons
We can see from TABLE IV and educe the width of ω(1420). ω(1420) dominantly decay into ρπ which is consist with experiment [1] and Ref. [61]. We can also find that ηρ, KK, b 1 π, KK * are the main decay modes in which b 1 π was observed in experiment [1]. The total width of ω(1420) in our calculation has a overlap with the experiment value [5]. ω(1420) is a good candidate of ω(2S ) which agree with Ref. [41].
ω(1960) is observed in the pp → ωη, ωππ process [7]. As shown in TABLE V, ω(1960) dominantly decay into ρπ under ω(3S ) assignment. The decay modes, b 1 π, KK 1 , KK ′ , contribute much to the total width, and b 1 π can decay to ωππ which is the final channel observed in experiment [7]. According to our calculation, the total width of ω(1960) is 143 ± 34 MeV and this is consistent with the experimental data [7]. In addition, ωη is sizable final channel and has been observed in experiment [7]. Other detailed information is demonstrated in TABLE V. Our calculation indicates that ω(1960) can be assigned as ω(3S ) state.
X(2240) is observed in the e + e − → K + K − process by the BESIII Collaboration, which has the mass of 2239.2± 7.1± 11.3 MeV and the width of 139.8±12.3±20.6 MeV [2]. The quantum number of this resonant structure can be assigned as J PC = 1 −− . Assuming that it is a isospin scalar state, it may be a ω(4S ) candidate from our previous mass analysis. Meanwhile, it should be noted that the mass of X(2240) is larger than the mass of ω(2205), which means if we treat X(2240) as ω(4S ), the total width of X(2240) in theoretical will be large than that of ω(2205). However, the experimental width of X(2240) is just 193 ± 35.5 MeV which doesn't support it's ω(4S ) assignment. The new state X(2240) needs more theoretical and experimental research to recover its structure. ω(5S ) has not been observed in experiment. Regge trajectory and MGI model predict that ω(5S ) may have the mass of 2550-2620 MeV as discussed in previous section. Here, we obtained the two-body strong decay of ω(5S ) by QPC model with the mass of 2553 MeV in TABLE VII. As predicted in TABLE VII, ω(2550)'s main decay modes are πρ(1450), a 2 ρ, b 1 π, a 1 ρ, ρ 3 π, KK * (1680), K * K(1630), f 2 ω are its important decay channels. Other channels, like KK * (1410), KK * 3 (1780), KK, ηω, f 2 (1525)ω, ηω 3 , K * K * 0 (1425), are very narrow. As we cannot sure for the parameter γ in QPC model, we select the value 11.6 in Ref. [46] and obtain its width about 330 MeV. We hope our predication can be helpful for searching for this state in experiment.

D-wave ω mesons
In this section, we will give a analysis for the D-wave ω mesons, ω(1650), X(2040), and ω(2290), by the means of their two-body strong decay behaviour. ω(1650) was observed in many final channels, such as ρπ, ωππ, and ηω [1,7]. In theory, ω(1650) is well established as a 1D state in ω family [1,41]. Here, we compared ω(1650) with ρ(1700) and educed the width of ω(1650) by the width of ρ(1700) as shown in TABLE VIII. As the same time, we can conclude that b 1 π, ρπ, KK, ηω, and KK * are the main decay modes of ω(1600) in which b 1 π can decay into the three body final state ωππ. The total width of ω(1650) is 284±183 MeV which agrees with the experimental value 315 ± 35MeV [1]. Our theoretical result shows that ω(1650) is a good candidate of ω(1D). X(2040) was observed in partial wave analysis of J/ψ → K + K − π 0 by BESIII Collaboration [4]. It's J PC = 1 −− and the and width are 2039 ± 19.7 MeV and 193 ± 35.5 MeV, respectively. By our previous mass analysis, it may be a ω(2D) state if it is a iso-scalar. Examining its strong decay information in theory, we can obtain that the total width of X(2040) with the range of 126 MeV to 178 MeV, which has a overlap with the result of experiment [4]. The main decay mode is b 1 π. πρ(1450), πρ, ρa 1 , and ηh 1 are the important decay channels too as shown in TABLE IX. The other decay modes are extraordinary narrow such as KK(1460), KK * (1410), ηω(1425), ηω and so on. As mentioned previously, b 1 π can decay to ωππ, so the channel of ωππ will be a main final channel for X(2040)(2D) state. Because Kπ is a mail decay channel of K * (1410), KKπ will be a important final state too. This is consistent with the experimental result [4]. If it is a iso-vector, it may be the ρ(2D) state as shown in Tab. III, which is the same state as ρ(2000). If we arrange X(2040) as a ρ(2D) state, it will have the same decay behaviour with ρ(2000), i.e. it mainly decay into a 1 (1640)π and ρρ. ππ 2 , ππ, and a 1 π are very important final states too. In experiment, ππ will be an ideal final channel to distinguish whether X(2040) is a ρ(2D) state. With above analysis, X(2040) is a favor ω(2D) state if it only has a uū(dd) component. ω(2330) was observed in the process γp → ρ ± ρπ∓ by OMEG Collaboration [63] and ω(2290) was found in the partial wave analysis of the data pp → ΛΛ [62]. The mass of ω(2290) and ω(2330) are very close but their widths are very different. We need to analyze their decay behaviours to distinguish which of them is a good candidate of ω(3D). TABLE X presents the decay behaviours of ω(2290) and ρ(2270). We can educe that the width of ω(2290) with the range of 134 MeV to 224 MeV. The primary decay modes is b 1 π. π 1 ρ(1450), KK(1630), a 2 ρ, and ρπ are the main decay channels of ω(2290). The decay modes like KK, KK 1 , ηω and K * K 1 have the sizeable contribution to the total width of ω(2290) as shown in TABLE X. The decay behaviour of ω(2330) is very similar to ω(2290) besides that a 0 (1450)ρ and a 2 ρ have the lager widths and KK(1630) has little width value.
Our calculation shows that both the total widths of ω(2290) and ω(2330) have a little overlap with experimental widths [62] and they are good candidates of ω(3D) state. We hope there are more experiments to research the two ω states in order to identify which one of them is the ω(3D) state and whether they are the same state.
ω(4D) has not been observed in the experiment. Regge trajectory and MGI model predict that the mass of ω(4D) may be 2630-2640 MeV (we name it ω(2630)) which is similar with ω(5S ). In TABLE VII, we give the two-body strong decay of ω(4D) with the mass of 2663 MeV. According to TABLE VII, one can note that ω(2550) mainly decay to a 0 (1450)ρ and πb 1 . ρa 2 , ρ(1465)π, f 2 ω are the important decay modes of ω(2630)(4D). KK(1630), a 1 b 1 , ρ 3 π, π 2 ρ, ρπ, h 2 η and other channels as shown in TABLE VII are smaller. Since we cannot determine the parameter γ in QPC model, we select the value 11.6 in Ref. [46] and obtain its width is about 300 MeV. We hope our predication can be helpful to search for this state in experiment.

IV. SUMMARY
In this paper, we systematically study the mass spectrum and the two-body strong decay of the ω meson family and the newly observed X(2040) and X(2240) with (J P ) = (1 − ) by the BESIII Collaboration. We discussed the mass spectrum and strong decay behaviors of ω meson excited states, and whether or not X(2040) and X(2240) are the candidates of ω(2D) and ω(3D) state. In addition, we predict ω(4D) and ω(5S ) states and the abundant information of their two-body strong decays of ω family. This will be helpful to further study of these ω states.
3. If X(2040) is a uū(dd)-component meson, it is a good candidate of ω(2D) state, and not rule out the possible of ρ(2D) arrangement before experiment determine its iso-spin. Our research does not support X(2240) as the ω(4S ) or ρ(4S ) assignment, if X(2240) only has uū(dd) component.
Also, we give more mass and decay information of ρ excitations which will be helpful for the establish the ω meson family.
The purpose of this study is crucial not only to establish the ω meson family and future search for the higher excita-tions, but also to help us reveal the structure information of the newly observed X(2040) and X(2240) states. Thus, more experimental measurements of the resonance parameters should be conducted by the BESIII and other experiments. This can help us to identify the nature of X(2040) and X(2240) and establish the ω meson family in the future. We hope this work can assist in exploring the ω states in both experiment and theory.