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Advances in High Energy Physics

Volume 2013 (2013), Article ID 217858, 14 pages

http://dx.doi.org/10.1155/2013/217858

## and : Which Can Be Assigned as a Hybrid State?

^{1}School of Physics and Electrical Engineering, Anyang Normal University, Anyang 455000, China^{2}Department of Physics, Shanghai University, Shanghai 200444, China

Received 11 July 2013; Revised 15 September 2013; Accepted 23 September 2013

Academic Editor: Gongnan Xie

Copyright © 2013 Bing Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The mass spectrum and strong decays of the and are analyzed. Our results indicate that and are the two different resonances. The narrower seems likely a good hybrid candidate. We support the as the quarkonium. We suggest to search the isospin partner of in the channels of and in the future. The latter channel is very important for testing the hybrid scenario.

#### 1. Introduction

An isoscalar resonant structure of was observed by the BESIII Collaboration with a statistical significance of 7.2 in the processes recently [1]. Its mass and width were given as Here the first errors are statistical and the second ones are systematic. The product branching fraction of was also presented [1]. But the quantum numbers of are still unknown, then the partial wave analysis is required in future.

The mass of is consistent with the , but the width is much narrower than the . In the tables of the Particle Data Group (PDG) [2], the available mass and width of are

The has been observed in reactions [3, 4], annihilation [5–8], and radiative decays [9]. It should be stressed that radiative decay channels (Figure 1(a)) and annihilation processes are the ideal glueball hunting grounds. But the glueball production is suppressed in reaction. By contrast, the hadronic decay are considered “hybrid rich” (Figure 1(b)).

Furthermore, the branching ratio reported by the WA102 Collaboration indicates that the decay channel of is tiny for [7]. This has been confirmed by an extensive reanalysis of the Crystal Barrel data [10]. Differently, the analysis of BESIII Collaboration indicates that the primarily decays via the channel [1]. Then the present measurements of the decay widths, productions, and decay properties suggest that and are two different isoscalar mesons.

If the production process is mainly hadronic, the quantum numbers of should be , , or . One notices that the predicted masses for the light , and hybrids overlap 1.8 GeV in the Bag model [11, 12], the flux tube model [13, 14], and the constituent gluon model [15]. In addition, the decay width of isoscalar hybrid is expected to be narrow [16]. Therefore, becomes a possible hybrid candidate.

In addition, the predicted masses of and glueball are much higher than 1.8 GeV by lattice gauge theory [17–19]. Therefore, is not likely to be a glueball state. Moreover, the molecule and four-quark states are not expected in this region [16]. Then the unclear structure looks more like a good hybrid candidate. But the actual situation is much complicated because the nature of is still ambiguous.(i)Since no lines of evidence have been found in the decay mode of , the disfavors the quarkonium assignment. The mass of seems much smaller for the () state in the Godfrey-Isgur (GI) quark model [20]. Therefore the has been assigned as the hybrid state [10, 21–23]. (ii)However, Li and Wang pointed out that the mass, production, total decay width, and decay pattern of the do not appear to contradict with the picture of it as being the conventional state [24].

Therefore, systematical study of the mass spectrum and strong decay properties is urgently required for and . Some valuable suggestions for the experiments in future are also needed.

The paper is organized as follows. In Section 2, the masses of and will be explored in the GI relativized quark model and the Regge trajectories (RTs) framework. In Section 3, the decay processes that an isoscalar meson decays into light scalar (below 1 GeV) and pseudoscalar mesons will be discussed. The two-body strong decays of and will be calculated within the model and the flux-tube model. Finally, our discussions and conclusions will be presented in Section 4.

#### 2. Mass Spectrum

In the Godfrey-Isgur relativized potential model [20], the Hamiltonian consists of a central potential and the kinetic term in a “relativized” form

The funnel-shaped potentials which include a color coulomb term at short distances and a linear scalar confining term at large distances are usually incorporated as the zeroth-order potential. The typical funnel-shaped potential was proposed by the Cornell group (Cornell potential) with the form [29]

The strong coupling constant , the string tension , and the constant are the model parameters which can be fixed by the well-established experimental states. The remaining spin-dependent terms for mass shifts are usually treated as the leading-order perturbations which include the spin-spin contact hyperfine interaction, spin-orbit and tensor interactions, and a longer-ranged inverted spin-orbit term. They arise from one gluon exchange (OGE) forces and the assumed Lorentz scalar confinement. The expressions for these terms may be found in [20].

It should be pointed out that the nonperturbative contribution may dominate for the hyperfine splitting of light mesons, which is not like the heavy quarkonium [30]. For example, the hyperfine shift of the meson with respect to the center gravity of the mesons is much small: MeV [31]. However, for the light isovector mesons , , , and , the hyperfine shift is MeV. Here the masses of , , , and are taken from PDG [2]. For the complexities of nonperturbative interactions, we are not going to calculate the hyperfine splitting.

Now, the spin-averaged mass, , of multiplet can be obtained by solving the spinless Salpeter equation:

Here we employ a variational approach described in [32] to solve (5). This variational approach has been applied well in solving the Salpeter equation for [33], and [34] mass spectrum.

In the calculations, the basic simple harmonic oscillator (SHO) functions are taken as the trial wave functions. It is given by in the position space. Here the SHO function scale is the variational parameter.

By the Fourier transform, the SHO radial wave function in the momentum is

The wave functions of and meet the normalization conditions:

In the variational approach, the corresponding are given by minimizing the expectation value of where

When all the parameters of the potential model are known, the values of the harmonic oscillator parameter can be fixed directly. With the values of , all the spin-averaged masses will be obtained easily. obtained in this way trend to be better for the higher-excited states [35].

It is unreasonable to treat the spin-spin contact hyperfine interaction as a perturbation for the ground states, because the mass splitting between pseudoscalar mesons and vector mesons are much large. Then we consider the contributions of for the mesons. The following Gaussian-smeared contact hyperfine interaction [36] is taken for convenience:

In this work, we choose the model parameters as follows: GeV, GeV, , GeV^{2}, GeV, and GeV. We take the smaller value of here rather than the value in [20]. The smaller was obtained by the relation between the slope of the Regge trajectory for the Salpeter equation and the slope in the string picture [30]. The Gaussian smearing parameter seems a little smaller than that in [20]. However, the is usually fitted by the hyperfine splitting of low-excited states in the works of literature with a certain arbitrariness.

The values of and for the states , , , , , , , , , , , , and are listed in Table 1. The experimental masses for the relative mesons are taken from PDG [2].

Obviously, the spin-averaged masses of the , , and , and mesons are consistent with the experimental data. Indeed, the predicted masses of higher excited states here are also reasonable, for example, and are very possible the -wave isovector and isoscalar mesons with the masses of MeV and MeV, respectively [2]. The predicted spin-averaged mass of is not incompatible with experiments. Our results are also overall in good agreement with the expectations from [37]. The trend that a higher excited state corresponds to a smaller coincides with [38–40]. For considering the spin-spin contact hyperfine interaction, there are two s for the mesons. The larger one corresponds to the state and the smaller one the state.

As shown in [37, 41], the confinement potential is determinant for the properties of higher excited states. In [37], the masses for higher excited states with GeV^{2} and are closer to experimental data than the results given in [20]. Then we ignored the Coulomb interaction for , , , , and states. In this way, for these states increase about 100 MeV.

The masses of , , , and are usually within 1.8~2.1 GeV in various quark potential models [20, 25–27] (see in Table 2). The predicted spin-averaged masses of , , , and are also within this mass regions (bold ones in Table 1). Due to the uncertainty of the potential models, absolute deviation from experimental data are usually about 100~150 MeV for the higher excited states. Compared with these predicted masses, disfavors the assignment for its low mass. But the possibilities of , , and still exist. Here we do not consider the possibility of as the state because looks more like a good candidate [42–44].

Regge trajectories (RTs) are another useful tool for studying the mass spectrum of the light flavor mesons. In [45], the authors fitted the RTs for all light-quark meson states listed in the PDG tables. A global description was constructed as

Here, and mean the the radial and angular-momentum quantum number. Recently, the authors of [45] repeated their fits with the subset mesons of the paper [46]. They found a little smaller averaged slopes of GeV^{2} and GeV^{2} to be compared with GeV^{2} and GeV^{2} in (12). Here the and are the weighed averaged slopes for radial and angular-momentum RTs [45, 47].

Now , , and have been established as the , , and states in PDG [2]. With the differences between the mass squared of and these states (Table 3), could be assigned for the and . The mass of is too large for the state in the RTs. has been assigned as the meson [2]. Since GeV^{2} which is much smaller than GeV^{2}, looks unlike the state for its low mass. However, the difference of GeV^{2} matches the slopes GeV^{2} well. Then the RTs can not exclude the possibility of as the state.

As mentioned in Section 1, is also a good hybrid candidate since its mass overlaps the predictions given by different models. The predicted masses for , , and states by these models are collected in Table 4.

In this section, the mass of has been studied in the GI quark potential model and the RTs framework. In the GI quark potential model, can be interpreted as the , , or state with a reasonable uncertainty. In the RTs, favors the and assignments. But the assignment can not be excluded thoroughly. is also a good hybrid state candidate. Since the masses of and are nearly equal, the possible assignments of also suit . The investigations of the strong decay properties will be more helpful to distinguish the and .

#### 3. The Strong Decay

##### 3.1. The Final Mesons Include the Scalar Mesons below 1 GeV

Despite many theoretical efforts, the scalar nonet of mesons has never established well. The lowest-lying scalar mesons including (or ), , , and are difficult to be described as states; for example, is associated with nonstrange quarks in the scheme. If this is true, its high mass and decay properties are difficult to be understood simultaneously. So interpretations as exotic states were triggered, For examples, two clusters of two quarks and two antiquarks [48], particular quasi-molecular states [49–51], and uncorrelated four-quark states [52–54] have been proposed.

Though the structures of these scalar mesons below 1 GeV are still in dispute, the viewpoint that these scalar mesons can constitute a complete nonet states has been reached in most works of the literature (as illustrated in Figure 2). In the following, we will denote this nonet as “” multiplet for convenience.

Due to the unclear nature of the mesons, it seems much difficult to study the decay processes when the final mesons include a member. As an approximation, , , and were treated as mesons in [44, 55]. In [24, 43], this kind of decay channel was ignored. However, this kind of decay mode may be predominant for some mesons. For example, the observations indicate that , , and primarily decay via the channel [1].

In what follows, we will extract some useful information about this kind of decay mode by the flavor symmetry. We will show that , , and are the main decay channels for the isoscalar and the mesons when they decay primarily through “” mesons, where the sign “” denotes a light pseudoscalar meson. This will explain why has been first observed in the channel.

We noticed that the nonet could be interpreted like the nonet in the diquark-antidiquark scenario. In Wilczek and Jaffe’s terminology [56, 57], the mesons consist of a “good” diquark and a “good” antidiquark. When , quarks form a “good” diquark, it means that the two light quarks, and , could be treated as a quasiparticle in color , flavor , and the spin singlet. The “good” , diquark is usually denoted as .

In the diquark-antidiquark limit, the parity of a tetraquark is determined by [58] where the refers to the relative angular momentum between two clusters. Thus the mesons are the lightest tetraquark states in the diquark-antidiquark model with . The nonet in the full set of flavor representations is

Because the flavor symmetry is not exact, the two physical isoscalar mesons, and , are usually the mixing states of the and states [48], When the mixing angle equals the so-called ideal mixing angle, that is, , the composition of the and is

It seems that the deviation from the ideal mixing angle of the and is small [48]. In the following calculations, we will treat them in the ideal mixing scheme.

Under the flavor assumption, all the members of the octet have the same basic coupling constant in one type of reaction, while the singlet member has a different coupling constant. Particularly, when a quarkonium decays into and mesons, there are five independent coupling constants, that is, , , , , and , corresponding to five different channels

In order to determine the relations between these coupling constants, we will assume the process that the or meson decays into a and another mesons obey the OZI (Okubo-Zweig-Iizuka) rule; that is, the two quarks in the mother meson go into two daughter mesons, respectively. Therefore, there are four forbidden processes: , , and . With the help of the Clebsch-Gordan coefficients [59], the ratios between the five coupling constants are extracted as

It is well known that the physical states, and , are the mixture of the flavor octet and singlet. They can be written in terms of a mixing angle, , as follows:

The mixing angle has been measured by various means. However, there is still uncertainty for . An excellent fit to the tensor meson decay widths was performed under the symmetry, and was obtained [23]. In our calculation, is taken as . The excited mixtures of and are denoted as

In this scheme, the ideal mixing occurs with the choice of . When and decay into a and pseudoscalar mesons, the relations of decay amplitudes are governed by the coefficients which are model-independent in the limitation of symmetry. With the coupling constants in hand, the coefficients of and versus the mixing angle are shown in the Figures 3 and 4. When and occur in the ideal mixing, the values of are presented in Table 5. In the factorization framework, the decay difference of a hybrid and excited mesons comes from the spatial contraction [60]. Then the coefficients for hybrid states are the same as these of quarkonia.

Here the mixing of and has been considered. It is sure that the are zero for the processes , , and , since they are OZI-forbidden. of has not been considered in Table 5 since lies below the threshold of .

As illustrated in Figures 3 and 4, the primary decay channels of a or predominant excitation are and . If the deviation of from the ideal mixing angle is not large, should be a or predominant state since primarily decays via the channel. At present, only the ground and the isoscalar mesons deviate from the ideal mixing distinctly. In addition, if the is produced via a diagram of Figure 1(b), it should also be or predominant state.

Of course, the symmetry breaking will affect the ratios of these channels listed in Table 5, because the three-momentum of the these products are different. However, the coefficients have presented the valuable information for these specific decay channels. When occupies the state, becomes a good candidate. In the following subsection, we will explore the two-body strong decays of within the model and the flux-tube model. Of course, the analysis of also suits for their nearly equal masses.

##### 3.2. The Strong Decays of and

In [24], the model [61–63] and the flux-tube model [64] were employed to study the two-body strong decays of . There, the pair production (creation) strength and the simple harmonic oscillator (SHO) wave function scale parameter, s, were taken as constants.

However, a series of studies indicate that the strength may depend on both the flavor and the relative momentum of the produced quarks [28, 65]. may also depend on the reduced mass of quark-antiquark pair of the decaying meson [66]. Firstly, the relations of the model to “microscopic” QCD decay mechanisms have been studied in [65]. There, the authors found that the constant corresponds approximately to the dimensionless combination, , where is the mass of produced quark, means the meson wave function scale, and is the string tension. Secondly, the momentum dependent manner of has been studied in [28]. It was found that is dependent on the relative momentum of the created pair, and the form of with was suggested. Thirdly, Segovia et al. proposed that is a function of the reduced mass of quark-antiquark pair of the decaying meson [66]. Based on the first and third points above, will depend on the flavors of both the decaying meson and produced pairs. In our calculations, we will treat the as a free parameter and fix it by the well-measured partial decay widths.

In addition, the amplitudes given by the model and the flux-tube model often contain the nodal-type Gaussian form factors which can lead to a dynamic suppression for some channels. Then the values of are important to extract the decay width for the higher excited mesons in these two strong decay models.

In the following, the two-body strong decay of will be investigated in the model where the strength will be extracted by fitting the experimental data. The SHO wave function scale parameter, s, will be borrowed from Table 1 which are extracted by the GI relativized potential model. We will also check the possibility of as a possible hybrid state by the flux-tube model.

In the nonrelativistic limit, the transition operator of the model is depicted as where the and are the color and flavor wave functions of the pair created from vacuum. Thus, , are color and flavor singlets. The pair is also assumed to carry the quantum numbers of , suggesting that they are in a state. Then represents the pair production in a spin triplet state. The solid harmonic polynomial reflects the momentum-space distribution of the .

The helicity amplitude of is given by where represents the momentum of the outgoing meson in the rest frame of the meson . When the mock state [67] is adopted to describe the spatial wave function of a meson, the helicity amplitude can be constructed in the basis easily [62, 63]. The mock state for meson is

To obtain the analytical amplitudes, the SHO wave functions are usually employed for . For comparison with experiments, one obtains the partial decay width via the Jacob-Wick formula [68]:

Finally, the decay width is derived analytically in terms of the partial wave amplitudes

More technical details of the model can be found in [63]. The inherent uncertainties of the decay model itself have been discussed in [28, 69, 70].

The dimensionless parameter will be fixed by the 8 well-measured partial decay widths which are listed in Table 6. The amplitudes of these decay channels are presented explicitly in Appendix A.

As mentioned before, may depend on the flavors of both the decaying meson and produced pairs. Then we divide the 8 decay channels into two groups: one is , and the other includes and . The values of here are a little different from these given in [28] where an potential (for details of potential, see [35]) was selected to determine the meson wave functions. Of course, the meson wave function given by different potentials will influence the values of .

It is clear in Table 6 that decrease with increase. In addition, our calculation indicates that depends on flavors of both the decaying meson and the produced quark pairs. For example, values of fixed by and are roughly equal.

In the following calculations, we assume that the values of corresponding to the processes of and are determined by one function. Similarly, we take the function, , for the creation vertex. *The function of the creation vertex here is different from the one used in *[28]. With the four decay channels listed in the fifth column of Table 6, we fix the function as . For the processes of (the first column of Table 6), we fix the creation vertex function as . The dependence of on the momentum is plotted in Figure 5. Obviously the functions can describe the dependence of and well. The functions of creation vertex given here need further test.

Since we neglected the mass splitting within the isospin multiplet, the partial width into the specific charge channel should be multiplied by the flavor multiplicity factor (Table 7). This factor also incorporates the statistical factor 1/2 if the final state mesons and are identical (as illustrated in Figure 6). More details of can be found in the Appendix of [71].

The partial decay widths of are shown in Table 8 except the channels of mesons. and are large channels for the state in our work and [24], which are consistent with the experimental observations of the . The partial widths of , , and are narrower in our work than the expectations from [24]. has been observed by the BES Collaboration in the radiative decay channel of [24]. However, no apparent signals were detected in the channels of [72] and [73, 74]. Therefore, improved experimental measurements of the radiative decay channels are needed for the in future.

Next, we will evaluate the partial widths of “” channels which have not been listed in Table 8. *The scheme is proposed as follows.* As illustrated in Figure 7, we assume that decays into via a virtual intermediate meson. We notice that the also dominantly decays into the [1]. Its three-body decay can occur via three intermediate processes: [2]. With the ratio and MeV, the partial width of decaying into is estimated no more than 45 MeV. By the model, the ratio of can be reached easily. If the uncertainty of the coupling vertex of (see in Figure 7) is assumed to be canceled in the ratio of , the value of can be extracted roughly. Although the assumption above seems a little rough, we just need to evaluate magnitudes of these decay channels.

is proposed to be the first radial excited state of . Then the total decay widths of are evaluated no more than 12.6 MeV and MeV. The BESIII Collaboration claimed that primarily decay via [1]. The small partial width of also indicates that the can not be interpreted as the state.

In addition, our results do not support as the state since its observed decay width is much smaller than the theoretical estimate. The is the largest decay channel in our numerical results and in [24] for the state (Table 8). If the partial width of channel is as large as , the predicted width of will be much larger than the observed value.

We adopt the flux tube model to check the possibility of as a hybrid meson. The partial widths are also listed in Table 8 for the comparison. Details of the flux model are collected in Appendix B.

Two groups of the partial widths predicted in [16] are quoted in the Table 8. The left column was given by the flux tube decay model of Isgur, Kokoski, and Paton (IKP) with the “standard parameters” [75]. The right column was by the developed flux tube decay model of Swanson-Szczepaniak (SS). In [16], the masses are taken as 1.8 GeV for the , , and hybrid states.

For a hybrid meson, seems most possible to be the state because the total widths which exclude the channels of are much narrow in our work and in [16]. It is consistent with the narrow width of .

As shown in Table 8, is impossible to be the hybrid state. The predicted width in both our work and in [16] is broader. In addition, is a visible channel for both . A week signal was found in the region of 12001400 MeV in the analysis of (Figure of [1]), which contradicts the large channel of the state. We can exclude the possibility of as the hybrid state preliminarily.

The assignment for as the hybrid seems impossible since the theoretical width of is rather broad in our results and in the IKP model. If the partial width of channel is as large as , the total widths of will be much broader than the experimental value. But the width given by the flux tube decay model for the hybrid is much small. So the possibility of as a hybrid cannot be excluded. We suggest to detect the decay channel of because this channel is forbidden for the state in the IKP flux tube decay model and very small in the flux tube decay model (see Table 8). Then the channel of can discriminate the states and for .

Finally, if is the state, its decay width is predicted about 100 MeV which is much smaller than the experiments. However, the difference can be explained by the remedy of mixing effect. If and have the same quantum numbers, , they should mix with each other with a visible mixing angle. Then the interference enhancement will enlarge the width of . The broad decay width of could be explained naturally. On the other hand, has been observed in the channel of . However, this channel seems much small if is a pure meson. The mixing effect will also enlarge this partial width. Here, we do not plan to discuss the mixing of and further for the complex mechanism.

#### 4. Discussions and Conclusions

A isoscalar resonant structure of was observed by BESIII in the channels recently. Although the mass of is consistent with the , the production, decay width, and decay properties are much different. In this paper, the mass spectrum and strong decays of the and are analyzed.

Firstly, the mass spectrum is studied in the GI potential model and the RTs framework. In the GI potential model, both and could be the , , and states. In RTs, the possible assignments are the , and states. For the mass spectrum, they are also good hybrid candidates since the masses overlap the predictions given by different models (see Table 4).

Secondly, the processes of a quarkonium or a hybrid meson decaying into the “” mesons are studied under the symmetry and the diquark-antidiquark description of the mesons. We assumed that the processes obey the OZI rule. We find that the channels of , , and are the dominant when a quarkonium or a hybrid meson decays primarily through this kind of processes. This result can explain why has been first observed in the channel.

Thirdly, the two-body strong decay of is computed in the model. As the quarkonium, the predicted width of looks much larger than the observations. The broad resonance, , can be a natural candidate for the meson. There, we fix the creation strength, , in two kinds of processes: (1) ; (2) and . The functions of creation vertex are determined as and , respectively. Meanwhile, the SHO wave function scale, s, is obtained by the GI potential model.

We have evaluated the magnitude of the partial widths of “” channels by the ratio, , under a rather crude assumption that through a virtual intermediate meson (see Figure 7). Then the uncertainties of the coupling vertex for are assumed to be canceled in the ratio. The total widths of “” are evaluated no more than 12.6 MeV and MeV. Since primarily decays via , it also indicated that the cannot be interpreted as the state.

We also study the as a hybrid state in the flux tube model. Our results agree well with most of predictions given by [16]. looks most like the state for the narrow predicted width, which is consistent with the experiments. But we cannot exclude the possibility of . A precise measurement of is suggested to pin down this uncertainty.

Finally, some important arguments and useful suggestions are given as follows. If is the state, the broad should be isovector partner of . has been interpreted as the conventional meson in [76]. Indeed, the decay channel of is large enough for [77]. This observation disfavors the as a hybrid candidate for the selection rule that a hybrid meson decaying into two -wave mesons is strongly suppressed [78]. If is a hybrid meson, we suggest to search its isospin partner in the decay channels of and , which are accessible at BESIII, Belle, and BABAR Collaborations. The decay channel of is forbidden for the quarkonium due to the “spin selection rule” [60, 79]. We also suggest to search the in the decay channels of and since these channels are forbidden for the hybrid production.

#### Appendices

#### A. The Expressions of Amplitudes

We have omitted an exponential factor in the following decay amplitudes for compactness: where we defined For , For , For , .

For , For , For , For , For , and .

For For , For , For

and are the masses of quarks in the decaying meson . is the mass of the created quark from the vacuum. For calculating the decay widths, the masses of quarks are taken as GeV, GeV, which are the same as those in Section 2. The above amplitudes, , can be reduced further in the approximation of and . The reduced are consistent with those given by [71] except for an unimportant factor, , since this factor can be absorbed into the coefficient .

#### B. Hybrid Decay in the Flux Tube Model

The flux tube model was motivated by the strong coupling expansion of the lattice QCD. In this model, decay occurs when the flux-tube breaks at any point along its length, with a pair production in a relative state. It is similar to the decay model but with an essential difference. The flux tube model extends the nonrelativistic constituent quark model to include gluonic degrees of freedom in a very simple and intuitive way, where the gluonic field is regarded as tubes of color flux. Then it can be extended to the hybrid research. When the hybrid mesons are assumed to be narrow, and the threshold effects are not taken into account, the partial decay width is given by the flux model as [79] where , , are the “mock-meson” masses of , , [64]. When a hybrid meson decays into -wave and pseudoscalar mesons, the partial wave amplitude (with ) is given as the following form:

The flavor matrix element has been discussed before. are listed in Table 9 for the states of , , and .

Here the , , , and are defined as , , , , , and . The analytical expressions of and are given as where are the confluent hypergeometr