\chi_{c0,2} decay into light meson pairs and its implication of the scalar meson structures

In light of the recent data from BES collaboration for $\chi_{c0}\to VV$, $PP$ and $SS$, and from CLEO-c for $\eta\eta$, $\eta^\prime\eta^\prime$ and $\eta\eta^\prime$, we present a detailed analysis of the decays of heavy quarkonia into light meson pairs such as $\chi_{c0,2}\to VV$, $PP$ and $SS$ in a recently proposed parametrization scheme. An overall agreement with the data is achieved in $\chi_{c0,2}\to VV$ and $PP$, while in $\chi_{c0}\to SS$ we find that a possible existence of glueball-$q\bar{q}$ mixings is correlated with the OZI-rule violations, which can be further examined at CLEO-c and BESIII in $\chi_{c0}\to SS$ measurement.

II. PARAMETRIZATION FOR χc0,2 → M M In Ref. [26] the decay of χ c0,2 → V V , P P and SS was investigated in a parametrization scheme where the production of the final state hadrons were described by a set of transition amplitudes for either SOZI or DOZI processes. Such a parametrization as a leading order approximation is useful for identifying the roles played by different transition mechanisms and will avoid difficulties arising from our poor knowledge about the nonperturbative dynamics. Associated with the up-to-date experimental data, we can constrain the model parameters and make predictions which can be tested in future measurements.
The detailed definition of the parametrization was given in Ref. [26], we only summarize the main ingredients here with slightly rephrased expressions: i) The basic transition amplitude is defined to be the cc annihilation into two gluons which then couple to two non-strange quark pairs to form final state mesons: where V 0 is the interaction potential, and q(q) is non-strange quark (antiquark) with g 14 = g 23 = g 0 . Basically, such a coupling will depend on the quantum numbers of the initial quarkonium. We separate the partial decay information by introducing a conventional form factor in the calculation, i.e., F (|p|) ≡ |p| 2l exp(−|p| 2 /8β 2 ) with β = 0.5 GeV, for the relative l-wave two-body decay. ii) To include the SU(3) flavour symmetry breaking effects, we introduce which implies the occurrence of the SU(3) flavour symmetry breaking at each vertex where a pair of ss is produced, and R = 1 is in the SU(3) flavour symmetry limit. For the production of two ss pairs via the SOZI potential, the recognition of the SU(3) flavor symmetry breaking in the transition is accordingly iii) The DOZI process is parametrized by introducing parameter r accounting for its relative strength to the SOZI amplitude: where V 1 denotes the interaction potential. iv) Scalar glueball state can be produced in company with an isoscalar qq or in pair in the final state. We parametrize their amplitudes by introducing an additional quantity t for the relative strength of the process of glueball production recoiling a qq to the basic amplitude g 2 0 : A reasonable assumption for the glueball coupling is that the glueball does not pay a price to couple to gg, namely, the so-called "flavor-blind assumption" following the gluon counting rule. Under such a condition, parameter t has a value of unity, and the glueball production amplitude is of the same strength as the basic amplitude g 2 0 . Similarly, the production of a glueball pair can be expressed as Considering a general expression for isoscalar meson pair production with qq and glueball components, e.g. M 1,2 = x 1,2 |G + y 1,2 |ss + z 1,2 |nn , we can write the transition amplitude for χ c → M 1 M 2 as For meson pair production with isospin I = 1/2 and 1, the transitions only occur via potential V 0 , and they can be expressed as The modification of the above parametrization rule compared to Ref. [26] is on the glueball production. Here, parameters r and t are explicitly separated out. Parameter r describes the property of the qq-gg couplings in the DOZI processes. Apparent contributions from the DOZI processes generally demonstrate the importance of the OZIrule violations due to long-range interactions [27]. In contrast, parameter t distinguishes the G-gg coupling from the qq-gg, and will allow us to investigate the role played by glueball productions. In the present scheme the underlying physics denoted by the parameters can be more clearly identified.

III. DECAY OF χc0,2 → M M
In this Section we revisit χ c0,2 → V V , P P and SS taking into account the new data from both BES and CLEO-c.
A. χc0,2 → V V For χ c0,2 → V V , three channels, i.e. φφ, ωω and K * 0K * 0 , have been measured by BES collaboration [1,2,3]. Since we neglect glueball component in ω and φ, and assume that ω is pure nn and φ is pure ss due to ideal mixing, we can determine parameters g 0 , r, and R. Predictions for χ c0,2 → ρρ and ωφ can then be made.
In Table I, the parameters are presented. In Table II, we list the fitting results for χ c0,2 → V V in comparison with the experimental data [1,2,3]. Also, the result by fitting the PDG average values for χ c0,2 → φφ, ωω and K * 0K * 0 are included.
One apparent feature is that the OZI-rule violation and SU(3) flavor symmetry breaking are much obvious in χ c0 → V V than in χ c2 → V V . Parameter r is found to be about 20% for χ c0 , while its central values are about 1% for χ c2 though the uncertainties are about 10%. The consequence of small DOZI process contributions is that the production branching ratios for χ c0,2 → ωφ become rather small. For instance, predictions for the branching ratio of χ c0 → ωφ are at least one order of magnitude smaller than φφ channel, and the PDG averaged values for the experimental data lead to a negligibly small branching ratio for χ c2 → ωφ. Further experimental measurement confirmation of this prediction will be extremely interesting.
The ρρ branching ratio turns to be sensitive to the experimental uncertainties carried by those available data. Different from other decay channels, which are determined by parameters r, R and g 0 in a correlated way, it only depends on parameter g 0 . Therefore, the ρρ channel is ideal for testing this parametrization scheme, and can put further constraint on the parameters. B. χc0,2 → P P Decay channels of χ c0,2 → ηη, K + K − , K 0 s K 0 s and ππ have been measured at BES [1,4,5,6]. However, as studied in Ref. [26], the relatively large uncertainties with χ c0 → ηη brought significant errors to parameter r, and the role played by the DOZI processes cannot be clarified. It was shown in Ref. [26] that within the uncertainties of BR χc0→ηη = (2.1 ± 1.1) × 10 −3 [6], the relative branching ratios of χ c0,2 → ηη, ηη ′ and η ′ η ′ were very sensitive to the OZI-rule violation effects, and the branching ratio fractions can vary drastically. The world averaged data for χ c0 → K + K − , K 0 s K 0 s , and ππ [28] do not deviated significantly from the BES data [1,4,5,6] except that BR χc0→ηη = (1.9 ± 0.5) × 10 −3 has much smaller errors. Recently, CLEO-c publishes their results for χ c0,2 → ηη, Adopting the world-average data from PDG [28] and including the new data from CLEO-c [7], we can now make a constraint on the model parameters for χ c0 → P P . We also make a fit for χ c2 → P P in a similar way with the experimental bound limits. The fitted parameters and branching ratios are listed in Table III and IV, respectively.
It shows that the decay of χ c0 → P P can be described consistently with small χ 2 . A prominent feature is that the SU(3) flavor symmetry breaking effects turn out to be small, i.e. R = 1.035 ± 0.067 does not deviate significantly from unity. Meanwhile, parameter r = −0.120 ± 0.044 suggests that contributions from the DOZI processes are not important. The production of ηη ′ is thus strongly suppressed which is consistent with CLEO-c results [7]. These features indicate that pQCD transitions play a dominant role in P P decay channels.
In χ c2 → P P , by fitting the PDG data and adopting the CLEO-c bound limits for ηη, η ′ η ′ and ηη ′ , we obtain results with large χ 2 . Contrary to χ c0 → P P , the fitted parameter R = 0.778 ± 0.067 indicates significant SU(3) flavor symmetry breakings. The OZI-rule violation parameter r = −0.216 ± 0.102 also suggests that the DOZI processes are relatively more influential than in χ c0 . However, this could be due to the poor status of the data. Notice that BR χc2→K + K − = (0.77 ± 0.14) × 10 −3 and BR χc2→K 0 s K 0 s = (0.67 ± 0.11) × 10 −3 have violated the isospin relation drastically. It needs further experiment to check whether this is due to datum inconsistency or unknown mechanisms.

0K
* 0 pair has a branching ratio of (1.05 +0.39 −0.30 ) × 10 −3 in its decay into π + π − K + K − and a set of f i 0 f j 0 pairs are measured, where i, j = 1, 2, 3 denotes f 0 (1710), f 0 (1500) and f 0 (1370), respectively. The interesting feature is that the f 0 (1370)f 0 (1710) pair production is found to have the largest branching ratio in comparison with other f 0 pairs. Theoretical interpretation for such an observation is needed and in Ref. [26], a parametrization for the SOZI and DOZI processes suggests that glueball-qq mixings can lead to an enhanced f 0 (1370)f 0 (1710) branching ratio in χ c0 decays. However, due to the unavailability of the data for other scalar meson pair decays, estimate of the absolute branching ratios were not possible. Here, incorporated by the data for K * 0 (1430)K * 0 (1430), we expect to have more quantitative estimates of the χ c0,2 → SS branching ratios.
To proceed, several issues have to be addressed: i) The scalars, f 0 (1370), f 0 (1500) and f 0 (1710), are assumed to be mixing states between scalar qq and glueball G. On the flavor singlet basis, the state mixing can be expressed as where x i , y i and z i are the mixing matrix elements determined by the perturbation transitions [33,34,35]. We adopt the mixing matrix U from Ref. [35]: In order to examine the sensitivities of the branching ratios to the scalar meson structures in the numerical calculations, we will also apply several other mixing schemes [36,37,38] which are different from Ref. [35]. ii) In χ c0,2 → V V and P P the SU(3) flavor symmetry breaking turns to be at a magnitude of 10∼ 20%. Namely, the deviation of the SU(3) flavor symmetry parameter R from unity is small. Due to lack of data we assume that a similar order of magnitude of the SU(3) flavor symmetry breaking appears in χ c0 → SS, and it is natural to assume R = 1 as a leading order estimate.
We can thus determine the basic transition strength g 0 via where p is the three-vector momentum of the final state K * 0 in the χ c0 -rest frame, and F (|p|) is the form factor for the relative l-wave two-body decay. The partial decay width Γ(χ c0 → K * 0K * 0 ) has been measured by BES [6]: with BR(K * 0 → K + π − ) = BR(K * 0 → K − π + ) = 0.465 [28]. iii) Since there is no constraint on the parameter t, we apply the flavor-blind assumption, t = 1, as a leading order approximation. iv) In order to accommodate the BES data [6], we adopt the same branching ratios for f 0 → P P as used in Ref. [35]: It should be noted that the final predictions for χ c0 → f i 0 f j 0 → π + π − K + K − are sensitive to the above branching ratios. For the charged decay channel, factor 1/2 and 2/3 will be included in the branching ratio of f 0 → K + K − and π + π − , respectively. Detailed analysis of the f 0 states can be found in Ref. [39] and references therein. Now, we are left with only one undetermined parameter r. By taking the measured branching ratio [6]: we determine r = 1.31 ± 0.19. Consequently, predictions for other SS decay channels can be made and the results are listed in Table V. A remarkable feature arising from the prediction is that BR(χ c0 → f 0 (1370)f 0 (1710)) turns out to be the largest one in all the f 0 pair productions with the constraint from K * 0 (1430)K * 0 (1430). As listed in Table V branching ratios of f 0 (1370)f 0 (1370) and f 0 (1370)f 0 (1500) are at order of 1%. Their signals in π + π − K + K − are suppressed due to their small branching ratios to π + π − and K + K − [40,41,42]. As a comparison decay channels with f 0 (1710) → K + K − are less suppressed. Apart from the dominant channel f 0 (1370)f 0 (1710), our calculation shows that χ c0 has also large branching ratios into π + π − K + K − via f 0 (1500)f 0 (1710). It shows that our results for χ c0 → f i 0 f j 0 → π + π − K + K − provide a consistent interpretation for the BES data [6] though some of the predictions strongly depend on the estimates of the branching ratios of f 0 → π + π − and K + K − .
The value of r = 1.31 ± 0.19 suggests an important contribution from the DOZI processes in χ c0 → f i 0 f j 0 , which is very different from the results in V V and P P channels. This certainly depends on the mixing matrix for the scalars, and also correlated with parameters R and t. At this moment, we still lack sufficient experimental information to constrain these parameters simultaneously. But it is worth noting that large contributions from the DOZI processes are also found in the interpretation [35] of the data for J/ψ → ωf 0 (1710), φf 0 (1710), ωf 0 (1370) and φf 0 (1370) [41,42]. The branching ratio for f 0 (1710) recoiled by ω in the J/ψ decays is found to be larger than it being recoiled by φ, while branching ratio for φf 0 (1370) is larger than ωf 0 (1370). Since f 0 (1710) is coupled to KK strongly and f 0 (1370) prefers to couple to ππ than KK, a simple assumption for these two states is that f 0 (1710) and f 0 (1370) are dominated by ss and nn, respectively. Due to this, one would expect that their production via SOZI processes should be dominant, i.e. BR(J/ψ → φf 0 (1710)) > BR(J/ψ → ωf 0 (1710)) and BR(J/ψ → ωf 0 (1370)) > BR(J/ψ → φf 0 (1370)). Surprisingly, the data do not favor such a prescription. In Ref. [35], we find that a glueball-qq mixing can explain the scalar meson decay pattern with a strong contribution from the DOZI processes. In fact, this should not be out of expectation if glueball-qq mixing occurs in the scalar sector.
We compute two additional decay channels for χ c0 → f i 0 f j 0 , i.e. χ c0 → f i 0 f j 0 → π + π − π + π − and K + K − K + K − , which can be examined in experiment. The results are listed in the last two columns of Table V. It shows that the largest decay in the 4π channel is via f 0 (1370)f 0 (1500), and the smallest channel is via f 0 (1500)f 0 (1710). Branching ratios are at order of 10 −4 , the same as the dominant f 0 (1370)f 0 (1500) channel. This means that an improved measurement will allow access to most of those intermediate states if the prescription is correct. In contrast, decays into four kaons are dominantly via f 0 (1500)f 0 (1710) and f 0 (1370)f 0 (1710) at order of 10 −5 , while all the others are significantly suppressed. The branching ratio pattern can, in principle, be examined by future experiment, e.g. at BESIII with much increased statistics. Nonetheless, uncertainties arising from the f 0 → P P decays can be reduced.
It should be noted that our treatment for the SU(3) flavor symmetry breaking in order to reduce the number of free parameters can be checked by measuring χ c0 → a 0 (1450)a 0 (1450). In the SU(3) symmetry limit, we predict BR χc0→a0(1450)a0(1450) = 5.60 × 10 −3 , which is not independent of K * 0 (1430)K * 0 (1430). Experimental information about this channel will be extremely valuable for clarifying the role played by the DOZI processes.
In order to examine how this model depends on the scalar mixings, and learn more about the scalar meson structures, we apply another two mixing schemes from different approaches and compute the branching ratios for χ c0 → f i 0 f j 0 → π + π − K + K − , π + π − π + π − and K + K − K + K − . The first one is from Ref.
With the data from Eqs. (13) and (18), we determine r = 0.90 ± 0.21. Predictions for other decay channels are given in Table VI. In Model-GGLF, four mixing solutions were provided. We apply the first two as an illustration of the effects from the mixing schemes. The Solution-I gives and Solution-II reads We then determine r = 1.93 ± 0.29 and r = −2.07 ± 0.79 for Solution-I and II, respectively. The predictions for the branching ratios are listed in Tables VII and VIII. Among all these outputs the most predominant feature is that large DOZI contributions are needed to explain the available data for χ c0 → f 0 (1370)f 0 (1710) and χ c0 → K * 0 (1430)K * 0 (1430). This also leads to the result that χ c0 → f 0 (1370)f 0 (1710) → π + π − K + K − is a dominant decay channel. Thinking that all these scalar mixing schemes have quite different mixing matrix elements, the dominance of f 0 (1370)f 0 (1710) gives an impression that the SS branching ratios are not sensitive to the scalar wavefunctions. However, this is not the case, we note that the data cannot be explained if f 0 (1710) is nearly pure glueball while f 0 (1500) a pure ss, namely, a mixing such as shown by the fourth solution of Ref. [37].
It turns more practical to extract information about the scalar structures in an overall study of the SS branching ratio pattern arising from χ c0 → SS → π + π − K + K − , 4π and 4K. For instance, in the χ c0 → SS → 4K, the dominant channels are predicted to be via f 0 (1370)f 0 (1710) and f 0 (1500)f 0 (1710) in the mixing of Eq. (11), while in the other models the f 0 (1500)f 0 (1710) channel turns out to be small. In contrast, the f 0 (1370)f 0 (1370) channel is dominant in 4π channel as predicted by Solution-II of Model-GGLF, while it is compatible with other channels in other solutions. Systematic analysis of these decay channels should be helpful for pinning down the glueball-qq mixings.

IV. SUMMARY
A systematic investigation of χ c0,2 → V V , P P and SS in a general parametrization scheme is presented in line with the new data from BES and CLEO-c. It shows that the exclusive hadronic decays of the χ c0,2 are rich of information about the roles played by the OZI-rule violations and SU(3) flavour breakings in the decay transitions. For χ c0,2 → V V and P P , we obtain an overall self-contained description of the experimental data. Contributions from the DOZI processes turn out to be suppressed. For the channels with better experimental measurement, i.e. χ c0,2 → V V , and χ c0 → P P , the SU(3) flavor symmetry is also better respected. Significant SU(3) breaking turns up in χ c2 → P P which is likely due to the poor status of the experimental data and future measurement at BESIII and CLEO-c will be crucial to disentangle this.
The BES data for χ c0 → SS allows us to make a quantitative analysis of the branching ratios in the scalar meson decay channel. In particular, it allows a test of the scalar f 0 mixings motivated by the scalar glueball-qq mixing scenario. Including the new data for χ c0 → K * 0K * 0 from BES Collaboration, we find that the decay of χ c0 → f i 0 f j 0 favors strong contributions from the DOZI processes. This phenomenon is consistent with what observed in J/ψ → φf i 0 and ωf i 0 [41,42], where large contributions from the DOZI processes are also favored [35]. The SS decay branching ratio pattern turns out to be sensitive to the scalar mixing schemes. An overall study of χ c0 → SS → π + π − K + K − , 4π and 4K may be useful for us to gain some insights into the scalar meson structures and extract more information about the glueball signals in its production channel.     The branching ratios obtained for χc0,2 → P P by fitting the world-average data from PDG (quoted in the round bracket) [28] together with the new data from CLEO-c (quoted in the square bracket) [7].  The branching ratios obtained for BRχ c0 →SS . B0 ≡ BR(χc0 → SS) · BR(S → π + π − ) · BR(S → K + K − ) are branching ratios to be compared with the BES data [6]. B1 and B2 are branching ratios of χc0 → SS → π + π − π + π − and χc0 → SS → K + K − K + K − , respectively.