Study of Unflavored Light Mesons with $J^{PC}=2^{--}$

The unflavored light meson families, namely $\omega_2$, $\rho_2$, and $\phi_2$, are studied systematically by investigating the spectrum and the two-body strong decays allowed by Okubo-Zweig-Iizuka rule. Including the four experimentally observed states and other predicted states, phenomenological analysis of the partial decay widths can verify the corresponding assignments of these states into the families. Moreover, we provide typical branching ratios of the dominant decay channels, especially for missing ground states, which is helpful to search for or confirm them and explore more properties of these families at experiment.


I. INTRODUCTION
Since Quark model had been proposed in 1964 [1,2], big progresses on the study of light hadron spectrum have been made by joint effort from both theorists and experimentalists [3]. Among the light hadron families, light mesons are an indispensable part and still are unclear, which is the main reason why investigating and exploring light mesons becomes an interesting physical aim of the present running experiments like BESIII and COMPASS, and forthcoming experiments like GlueX and PANDA.
When further checking the experimental status of light mesons [3], we notice an interesting phenomenon, i.e., the ground states of unflavored light meson families with J PC = 2 −− are still absent, their higher excited states are not well established yet, and they are only collected into Further States in Particle Date Group (PDG) [3]. This phenomenon stimulates our interests in exploring unflavored light mesons with J PC = 2 −− thoroughly.
According to the isospin, unflavored light mesons with J PC = 2 −− can be categorized into three groups, which are isovector ρ 2 meson family, and isoscalar ω 2 and φ 2 meson families. Though experimental information of 2 −− unflavored light ground meson is totally barren, their counterparts 1 −− and 3 −− meson are properly recognized. By comparing the similarity of them, the mass spectrum of ω 2 , ρ 2 and φ 2 ground * Electronic address: guod13@lzu.edu.cn † Electronic address: xuehua45@163.com ‡ Electronic address: liuzhanwei@lzu.edu.cn § Electronic address: xiangliu@lzu.edu.cn states is successfully constructed. In addition, we predict the behaviors of two-body strong decays allowed by Okuba-Zweig-Iizuka (OZI) rule, which are the key information of explaining the disappearance reason for them. Here, potential decay channels and typical branching ratios which are valuable to explore in future experiment are also provided. On the other hand, combining with the theoretical investigation of K * 2 meson [17], we can found an integrated nonet, and then systematically compare with 1 −− and 3 −− light mesons.
In PDG [3], we can find ρ 2 (1940), ω 2 (1975), ω 2 (2195), and ρ 2 (2225) as further states in pp reaction. Based on the experimental information of these four 2 −− states, we continue to discuss their possible assignments into ρ 2 and ω 2 meson families with the mass spectrum analysis and two-body OZI-allowed strong decay calculation. Besides, we predict the properties of the corresponding undiscovered φ 2 mesons. Additionally, masses and decay behaviors of third radial excitations are also calculated based on previous results. We hope that our effort will be helpful to establish ρ 2 , ω 2 and φ 2 meson families.
Comparing with other theoretical calculations [18][19][20], not only a more detailed analysis of the missing ground states but also a theoretical study of newly observed four resonances are included in this work. Moreover, based on the consistence of experimental information, we predict the decay properties of the third excitation states.
We organize this paper as follows. In Sec. II, the mass spectrum of unflavored light meson is studied, where one can fix the mass range of ground state in unflavored light meson families with J PC = 2 −− and give the possible assignments of radial excitation states. In Sec. III, we first introduce the Quark-Pair-Creation (QPC) model, and then we discuss the strong decay behaviors of these discussed states with fixed mass spectrum. In the end, Sec. V is devoted into the summary and discussion.

II. ANALYSIS OF MASS SPECTRUM
Although the light isoscalar and isovector states with 2S +1 L J = 3 D 1 or 3 D 3 are relatively well established and categorized because of many discovered candidates [6,8], there are only four unflavored light 3 D 2 mesons discovered at experiment. By using the Crystal Barrel detector's data, four 2 −− resonances are reported for pp collision in 2002 [21,22], whose masses and widths are listed According to our previous works [6,8] and the review Quark Model in PDG [3], the accompanying isoscalar ω(1650) and isovector ρ(1700) are generally considered as 3 D 1 ground states, and isoscalar ω 3 (1670) and isovector ρ 3 (1690) as 3 D 3 ground states. Those are consistent with the rough estimations as follows. Because of same spin angular momentum and orbital angular momentum, the masses of ω 2 and ρ 2 ground states should be around 1.7 GeV by comparing masses of the 3 D 1 state and 3 D 3 state. Similarly, the mass of φ 2 ground state is about 1.9 GeV by comparing that of φ 3 (1850).
In most situation, the Godfrey-Isgur (GI) model [23] is used for mass spectrum analysis and has achieved great success since proposed. Here, we employ this model to obtain masses of ground states.
The Regge trajectory is another effective approach to quantitatively study the mass spectrum of the radial excited light mesons [24,25]. As to higher excitations, masses of GI model calculation generally are larger than experimental results because of the coupled channel effect and the relativity effect. Thus, we use the Regge trajectory to obtain the masses of the radical excitation states. A general expression for the Regge trajectory is where M 0 is the mass of ground state, M is the mass of state with radical excitation number n, and µ 2 denotes the trajectory slope.
A general mass assignment of unflavored light mesons with 3 D 2 is shown in Fig. 1, and the four experimental states are well-arranged as n = 2 or 3 state. Here, we need to specify that the masses of ρ 2 (1D), ω 2 (1D) and φ 2 (1D) are taken from the GI model calculation [23]. By combining these theoretical inputs and experimental data, we can construct three Regge trajectories just shown in Fig. 1, by which can further predict the masses of other missing states. As shown in Fig. 1, linear typical Regge trajectories of ω 2 and ρ 2 family are observed, and we can obtain the slope µ 2 = 1.0 GeV 2 , which is gotten by fitting the experimental and the GI ground states data. Moreover, when we take the mass of the ground state φ 2 (1D) from GI model, the masses of φ 2 family can also be obtained with the extrapolation from the Regge trajectory using the same slope µ 2 = 1.0 GeV 2 .
Besides these ω 2 , ρ 2 , φ 2 mesons, there exist the corresponding K 2 partners. In our previous work [16], the resonance parameter and partial decay width of these K * 2 mesons were calculated. An integrated nonet is established as shown in Fig. 2 and thus we can have a complete comprehension of 2 −− light mesons. The masses of unflavored mesons with different radical number n are summarized in Table I, and these masses are employed to the following study for the decay widths. On the other hand, analysis of decay behaviors will examine the reasonability of our assignments.  Presently, in addition to the study of ground state masses from Godfrey and Isgur [23], and Ebert et al. [20], there are also another two works [18,19] indicating ground state masses of ω 2 and ρ 2 as ∼1.7 GeV comparable with ours. Furthermore, Ebert et al. also predict the masses of the first and second excitations, and the masses of ω 2 and ρ 2 families are generally consistent with ours, but the masses of φ 2 excitations is higher than ours. In the next section, we give a complete analysis of strong decay behaviors based on the wellestablished mass assignments.

III. STRONG DECAY BEHAVIORS
In this section, we firstly introduce the phenomenological model applied to obtain the information for the OZI-allowed hadronic strong decays. We then illustrate strong decay behaviors of ω 2 , ρ 2 and φ 2 families in details, respectively.

A. A Brief Introduction to the QPC Model
The QPC model is proposed by Micu [26] firstly in 1968 and developed by Orsay group [27][28][29][30][31]. The QPC model assumes a pair of quark-antiquark qq is created from vacuum with J PC = 0 ++ and then rearranged with the initial hadron to form two daughter hadrons. For instance, the transition operator T of a meson decay progress A → B + C can be expressed as is the solid spherical harmonic polynomial, and p 3 and p 4 depict momenta of quark and antiquark created from vacuum. b † 3 (d † 4 ) denotes quark (antiquark) creation operator. χ 34 1,−m , φ 34 0 , and ω 34 0 denote spin triplet, flavor singlet, and color singlet wavefunctions, respectively. The dimensionless parameter γ describes the quark pair creation strength, which can be fitted by the experimental width data. From our fitted results of 2 −− light states with minimal χ 2 = 17.8 (χ 2 = i (Theo. − Exp.) 2 /Error 2 ), γ = 7.1 for the uū (dd) pair creation, and the ss quark pair creation strength sets as 7.1/ √ 3 [31]. Then the transition matrix of decay process in the the rest frame of particle A reads as To sum up, the general decay width writes as where S ≡ 1/(1 + δ BC ) denotes the statistic factor which is responsible for a situation if B and C are identical particles, and m A is the mass of initial particle. Furthermore, the meson wavefunction is defined as mock state, i.e.
and here the spacial wavefunction Ψ nLM L (p) of meson adopts the simple harmonic oscillator (SHO) wavefunction which has explicit form where Y LM L (Ω) is spherical harmonic function, and L L+1/2 n−1 (x) is the associated Laguerre polynomial. The parameter β = 1/R, and R is obtained by reproducing the realistic root mean square radius via solving the Schrödinger equation (see more details in Ref. [33], though we proceed some improvements of the method and results).
In the following subsections, we perform phenomenological analysis of total widths and partial widths calculated by QPC model by comparing our results with experimental data. This analysis is helpful to explain why there does not exist any information of 2 −− unflavored light ground states and reveal the underlying properties of these ground states and their radical excitations for more future experimental measurements, especially in BESIII detector, CMD-3, SND and KEDR detector.

B. ω 2 Family
As shown in Fig. 3, the calculated total widths of ω 2 family states are illustrated by contour lines which depend on R and γ. At present, experiment only observed ω 2 (1975), ω 2 (2195), ρ 2 (1940), ρ 2 (2225), where their widths were measured. Thus, we may take these experimental data to fix the γ value to be 7.1. Considering the uncertainty, we set γ = 7.1 ± 0.3 in our concrete calculation. Generally, the total widths of ω 2 (2D) and ω 2 (3D) in our results agree well with those of ω 2 (1975) and ω 2 (2195) at experiment. It demonstrates the reasonability of setting the two states as ω 2 (2D) and ω 2 (3D) respectively, and also predicts there exists an undiscovered ground state.
By assuming the ground state ω 2 (1D) with mass equalling to 1696 MeV, the total decay width of missing ω 2 (1D) reach up to 220 MeV when choosing fitted γ = 7.1 and recommended R value [33] shown in the top left of Fig. 3. We present the branching ratios of ω 2 (1D) at the top of Fig. 4. Corresponding partial decay width can be obtained by multiplying the total width and branching ratio. The largest branching ratio comes from the πρ channel and is about 71%. The πb 1 (1235), ωη, and KK * channels are also important. Besides, all branching ratios are not strongly dependent on R, and those of the πb 1 (1235) and ωη channels are close to each other.
The information of stable partial width ratios will be helpful for experimental searches, and we list them for ω 2 (1D), 18. In Ref. [18,19], the similar branching ratios of πρ are obtained, where the branching ratio of this πρ decay channel for ω 2 (1D) is 74% and 60% given by Ref. [18,19] respectively, and other branching ratios are also analogous. As mentioned in Ref. [19], π 2 (1670) is with the mass and total width similar to ω 2 (1D), and also decays to πρ, so that ω 2 (1D) is possibly masked by π 2 (1670) in the πρ channel. Therefore, ωη and KK * are the ideal channels suggested to search for ω 2 (1D). The process ω 2 (1D) → b 1 (1235)π → ωππ are also valuable since ω 2 (1975) and ω 2 (2195) are discovered in the ωππ channel. The structure ω 2 (1975) and ω 2 (2195) are well determined by both pp → ωη and pp → ωππ processes with ω decaying to π + π − π 0 [22]. The two resonances are classified as furthur states in PDG, which means these states are not confirmed well in experimental and more measurements are needed. Thus, a detailed analysis of their categorization in nonet and decay properties are valuable to confirm them.
According to the analysis of Regge trajectory in previous section, ω 2 (1975) is assigned as n = 2, whose total width and branching ratios of calculated two-body decays are presented in the top right of Fig. 3 and at the bottom of Fig. 4, respectively. The experimental total width of ω 2 (1975) is 175±25 MeV, which is well reproduced in our results.
Since no experimental signal of ω 2 (4D), we predict its mass as 2424 MeV, and plot total width and branching ratios in the bottom-right corner of Fig. 3  As the isovector counterpart of ω 2 family, ρ 2 family takes the same R ranges. The contour lines of total widths from n=1 to 4 are illustrated in Fig. 6. Corresponding branching ratios are illustrated in Figs. 7 and 8. We can roughly conclude that the higher radial excitations rely more intensively on R variation, and take larger R values.
The mass of φ 2 (1D) is suggested as 1904 MeV, near its partner φ 3 (1850). If taking a typical value of γ = 7.1, its total width is predicted about 255 MeV with mild R dependence shown in the top left of Fig. 9, and it is comparable with the calculation in Ref. [19]. Two-body decay properties of φ 2 (1D) are presented at the top of Fig. 10, where KK * decay mode is dominant with the fraction up to 58.5-66.2%, and K * K * , KK 1 (1270), and ηφ channels also hold considerable proportions. Similar branching ratio results appear in Ref. [19], except that of φη is slightly larger than our prediction. Besides, all channel ratios exhibit linearly dependence with changing R, which probably results from the smooth wavefunction of the ground state. Thus, studying partial width ratios at experiment can appropriately test our prediction. These are The branching ratios of φ 2 (3D) and φ 2 (4D) with R dependence. Decay channels with branching ratios less than 1% are neglected.

IV. SUMMARY AND DISCUSSION
Since the quark model proposed, it dose achieve great success for explaining and predicting numerous hadrons. However, there also remain some unexplained exotic states beyond the quark model, such as glueballs, hybrids. To distinguish between conventional and exotic mesons, it is crucial that we comprehend conventional-meson spectroscopy very well. Currently, the experimental information of J PC = 2 −− unflavored light mesons is still scarce except the results of Crystal Barrel detector in 2002, and thus they are not well established. Therefore, a systematical and complete study of 2 −− unflavored light mesons is necessary to stimulate mounting experimental efforts to this sector.
In this work, an investigation of mass spectrum is firstly performed by combining the analysis of Regge trajectory and GI model in Sec. II, which indicates that the four experimental resonances are assigned as the first and second excitation of ω 2 and ρ 2 families, respectively. In addition, this categorization is also supported by the corresponding features of twobody decays allowed by OZI rule. More importantly, with predicted masses ∼1.7 GeV for ω 2 and ρ 2 ground states, total decay widths are suggested as 220 MeV and 390 MeV. Furthermore, since ω 2 (1D) is masked by π 2 (1670) and ρ 2 (1D) is difficult to reconstruct with broad width and 4π final state, we can explain why they are blind at experiment. Detailed decay properties of the missing ground resonances are given by QPC model in Sec. III. In addition, for the absolutely missing φ 2 family, their partial decay behaviors are also discussed by adopting advisable masses from n = 1 to 4. The branching ratios of main and subordinate channels are illustrated in Sec. III, respectively. Certain partial decay width is obtained by multiplying the corresponding branching ratio with total width. Definitely, the main decay widths of ground states are listed in Table II.   TABLE II: The partial decay width of ground states in units of MeV. Corresponding R value ranges are 3.7-4.3 for ω 2 and ρ 2 ground states, and 3.1-3.7 for φ 2 ground state.
The strong decay information is vital to search for missing ground states and higher excitations. We hope our study can inspire more experimental interests to find them and establish 2 −− light meson family at experiment eventually. BE-SIII, CMD-3, SND, KEDR, COMPASS, and E852 are suitable platforms to explore these issues. Especially, BESIII is the promising facility with the highest luminosity running in 2.0-4.6 GeV, and its main focus still is exploring light hadrons in the following years. Besides, it is hopeful that many states can be experimentally better understood in relatively near future at the Jefferson Lab. Potential channels, ωη for ω 2 , ρη for ρ 2 , ηφ for φ 2 , are suggested to explore these ground states. KK * is an ideal channel which contributes to all 2 −− light unflavored states, and contributes to sizable branching ratios.

V. ACKNOWLEDGEMENT
This project is supported by the China National Funds for Distinguished Young Scientists under Grant No. 11825503.