Scalar isoscalar mesons and the scalar glueball from radiative $J/\psi$ decays

A coupled-channel analysis of BESIII data on radiative $J/\psi$ decays into $\pi\pi$, $K\bar K$, $\eta\eta$ and $\omega\phi$ has been performed. The partial-wave amplitude is constrained by a large number of further data. The analysis finds ten isoscalar scalar mesons. Their masses, widths and decay modes are determined. The scalar mesons are interpreted as mainly SU(3)-singlet and mainly octet states. Octet isoscalar scalar states are observed with significant yields only in the 1500-2100\,MeV mass region. Singlet scalar mesons are produced over a wide mass range but their yield peaks in the same mass region. The peak is interpreted as scalar glueball. Its mass and width are determined to $M=1865$\er25$^{+10}_{-30}$ {\rm MeV} and $\Gamma= 370$\er$50^{+30}_{-20}$ {\rm MeV}, its yield in radiative $J/\psi$ decays to ($5.8\pm 1.0)\,10^{-3}$.


Introduction
Scalar mesons -mesons with the quantum numbers of the vacuum -are most fascinating objects in the field of strong interactions. The lowest-mass scalar meson f 0 (500), traditionally often called σ, reflects the symmetry breaking of strong interactions and plays the role of the Higgs particle in quantum chromodynamics (QCD) [1,2]. The f 0 (500) is accompanied by further low-mass scalar mesons filling a nonet of particles with spin J = 0 and parity P = +1: The three charge states a 0 (980), the four K * 0 (700), and the two isoscalar mesons f 0 (980), f 0 (500) are supposed to be dynamically generated from meson-meson interactions [3]. Alternatively -or complementary -these mesons are interpreted as four-quark or tetraquark states [4].
The continued quest for scalar isoscalar mesons at higher masses is driven by a prediction -phrased for the first time nearly 50 years ago [5,6] -that QCD allows for the existence of quark-less particles called glueballs. Their existence is a direct consequence of the nonabelian nature of QCD and of confinement. However, the strength of the strong interaction in the confinement region forbids analytical solutions of full QCD. First quantitative estimates of glueball masses were given in a bag model [7]. Closer to QCD are calculations on a lattice. In quenched approximation, i.e. when qq loops are neglected, the lowest-mass glueball is predicted to have scalar quantum numbers, and to have a mass in the 1500 to 1800 MeV range [8,9,10]; unquenching lattice QCD predicts a scalar glueball at (1795 ± 60) MeV [11]. Exploiting a QCD Hamiltonian in Coulomb gauge generating an instantaneous interaction, Szczepaniak and Swanson [12] calculate the low-lying glueball masses with no free parameters. The scalar glueball of lowest mass is found at 1980 MeV. Huber, Fischer and Sanchis-Alepuz [13] calculate the glueball spectrum using a parameterfree fully self-contained truncation of Dyson-Schwinger and Bethe-Salpeter equations and determine the lowest-mass scalar glueball to (1850 ± 130) MeV. In gravitational (string) theories -an analytic approach to QCD -glueballs are predicted as well [14] at 1920 MeV. Glueballs are predicted consistently within a variety of approaches to QCD. They seem to be a safe prediction.
Scalar glueballs are embedded into the spectrum of scalar isoscalar mesons. These have isospin I = 0, positive G-parity (decaying into an even number of pions), their total spin J vanishes, their parity P and their C-parity are positive: (I G )J PC = (0 + )0 ++ . Scalar glueballs have the same quantum numbers as scalar isoscalar mesons and may mix with them. In quark models, mesons are described as bound states of a quark and an antiquark. Their quantum numbers are often defined in spectroscopic notation by the orbital angular momentum of the quark and the antiquark L, the total quark spin S , and the total angular momentum J. Scalar mesons have 2S +1 L J = 3 P 0 .
Experimentally, the scalar glueball was searched for intensively but no generally accepted view has emerged. The most promising reaction to search for glueballs are radiative decays of J/ψ. In this process, the dominant contribution to direct photon production is expected to come from the process J/ψ → γ plus two gluons, where the final-state hadrons are produced by the hadronization of the two gluons. QCD predicts the two gluons to interact forming glueballs -if they exist. Lattice gauge calculations predict a branching ratio for radiative J/ψ decays to produce the scalar glueball of (3.8 ± 0.9)10 −3 [15]. This is a significant fraction of all radiative J/ψ decays, (8.8±1.1)%. There was hence great excitement when a broad bump in the radiatively produced ηη mass spectrum [16] was discovered by the Crystal Ball collaboration at the Stanford Linear Accelerator (even though with tensor quantum numbers). However, a resonance with the reported properties was not reproduced by any other experiment. The DM2 collaboration reported a strong peak at 1710 MeV in the KK invariant mass distribution [17], a peak that is now known as f 0 (1710). Data from the CLEO experiment on radiative J/ψ decays into pairs of pseudoscalar mesons were studied in a search for glueballs [18] but no definite conclusions were obtained.
The data with the highest statistics nowadays stem from BE-SIII in Bejing. The partial wave amplitudes for J/ψ radiative decays into π 0 π 0 [19] and K S K S [20] were determined in fits to the data in slices in the invariant mass of the two outgoing mesons. Data on J/ψ → γηη [21] and J/ψ → γφω [22] were presented including an interpretation within a partial wave analysis. In the reactions J/ψ → γ2π + 2π − [23,24] and J/ψ → γωω [25], the 2π + 2π − and into ωω branching ratios of contributing resonances were deduced from a smaller data sample.
A new understanding of the spectrum of light-quark scalar mesons emerged from the results obtained with the Crystal Barrel experiment at the Low-Energy Antiproton Ring at CERN. Inpp annihilation at rest, annihilation into 3π 0 [26], π 0 ηη [27], π 0 ηη [28], and π 0 K L K L [29] was studied. These data established the existence of the f 0 (1500) resonance; the existence of the f 0 (1370) had been proposed in 1966 [30] but its existence was accepted only after its rediscovery at LEAR inpp [31] and pn annihilation [32,33].
Central production in hadron-hadron collisions is mostly interpreted as collision of two Pomerons, and this process is supposed to be gluon-rich. Data on this reaction were taken at CERN by the WA102 collaboration that reported results on π + π − and K S K S [34], ηη [35], ηη and η η [36], and into four pions [37]. The GAMS collaboration reported a study of the π 0 π 0 system in the charge-exchange reactions π − p → π 0 π 0 n, ηη n and ηη n at 100 GeV/c [38] in a mass range up to 3 GeV. The charge exchange reaction π − p → K S K S n was studied at the Brookhaven National Laboratory [39]. An energydependent partial-wave analysis based on a slightly increased data set was reported in Ref. [40]. A reference for any analysis in light-meson spectroscopy are the amplitudes for ππ → ππ elastic scattering [41]. The low-mass ππ interactions are known precisely from the K e4 of charged kaons [42]. In these experiments, a series of scalar isoscalar mesons was found. The Review of Particle Properties (RPP) [43] lists nine states; only the five states at lower mass are considered to be established. None of these states sticks out and identifies itself as the scalar glueball of lowest mass.

Our data base
It seems obvious that the scalar glueball can be identified reliably only once the spectrum of scalar mesons is understood into which the glueball is embedded. Decisive for the interpretation are the data on radiative J/ψ decays. But many experiments contribute to our knowledge on scalar isoscalar mesons and provide additional constraints. In this coupled-channel analysis we fit meson-pairs in S -wave from radiative J/ψ decays and include the S -wave contributions to ππ elastic scattering [38] and ππ → K S K S [39,40], the CERN-Munich [41] data and the K e4 [42] data. Further, we use 15 Dalitz plots for different reactions frompN annihilation at rest [26,27,29], [57]- [63,64,65].
The real and imaginary parts of the mass-dependent S -wave amplitudes were derived for J/ψ → γπ 0 π 0 in Ref. [19] and J/ψ → γK S K S in Ref. [20]. Assuming dominance of resonances with spin J = 0 and J = 2, the partial-wave analysis returned -for each mass bin -two possible solutions, called black (b) and red (r). In some mass regions, the two amplitudes practically coincide. We assume continuity between regions in which the two amplitudes are similar, and divide the full mass range into five regions: in three regions, the two amplitudes are identical, in two regions, the red and black amplitudes are different. Thus there are four sets of amplitudes, (r, r); (r, b); (b, r); (b, b). For the data on J/ψ → γK S K S , we again define five mass regions and four sets of amplitudes. The amplitudes (r, r) give the best χ 2 for ππ, and (b, b) for K S K S . Figure 1a,b shows the π 0 π 0 [19] and K S K S [20] invariant mass distributions from radiative J/ψ decays for the best set of amplitudes. The "data" are represented by triangles with error bars, the solid curve represents our fit. (i) The π 0 π 0 invariant mass distribution shows rich structures. So far, no attempt has been reported to understand the data within an energy-dependent partial-wave analysis. The mass distribution starts with a wide enhancement at about 500 MeV and a narrow peak at 975 MeV: with f 0 (500) and f 0 (980). A strong enhancement follows peaking at 1450 MeV, and a second minimum at 1535 MeV. Three more maxima at 1710 MeV, 2000 MeV, and 2400 MeV are separated by a deep minimum at about 1850 MeV and a weak one at 2285 MeV. (ii) The K S K S invariant mass distribution exhibits a small peak at 1440 MeV immediately followed by a sharp dip at 1500 MeV. The subsequent enhancement -with a peak position of 1730 MeV -has a significant low-mass shoulder. There is a wide minimum at 1940 MeV followed by a further structure peaking at 2160 MeV. It is followed by a valley at 2360 MeV, a small shoulder above 2400 MeV, and a smooth continuation. (iii) Data on J/ψ → γηη [21] and J/ψ → γφω [22] were published including an interpretation within a partial wave analysis. We extracted the S -wave contributions (Fig. 1c,d) (for [22] only the contribution of the dominant threshold resonance), and included them in the fits.  The ηη mass distribution (Fig. 1d) resembles the one observed at SLAC [16], but now greatly improved statistics. The BE-SIII data show an asymmetry that reveals the presence of at least two resonances, f 0 (1500) and f 0 (1710). The sharp dropoff above 1.75 GeV indicates destructive interference between two resonances. (iv) The φω mass distribution exhibits a strong threshold enhancement. It was assigned to a scalar resonance at (1795±7) MeV [22] but the data can also be described by a resonance at 1770 MeV that was suggested earlier [66] and that is required in our fit. The two figures 1a,b look very different. Obviously, interference between neighboring states plays a decisive role. Figures 2a,b simulate the observed pattern: in Fig. 2a the lowmass part of f 0 (1500) interferes constructively with f 0 (1370) and leads to a sharp drop-off at its high-mass part. In Fig. 2b, the f 0 (1500) is responsible for the dip. The phase difference between the amplitudes for f 0 (1370) and f 0 (1500) in figures 2a,b changes by 180 • when going from ππ to KK. This is an important observation: these two states do not behave like a 1 √ 2 (uū + dd) and a ss state but rather like a singlet and an octet state, like 1 √ 3 (uū + dd + ss) and 1 √ 6 (uū + dd − 2ss). This change in the sign of the coupling constant in ππ and KK decays for f 0 (1500) with respect to the f 0 (1370) "background" has first been noticed in Ref. [67].
The two resonances f 0 (1710) and f 0 (1770) form the large enhancement in figure 1b while their contribution to figure 1a is much smaller. This again is due to interference: the two resonances interfere destructively in the ππ channel and constructively in the KK channel. Again, the f 0 (1710) and f 0 (1770) wave functions must contain significant uū + dd and ss contri-butions of opposite signs: there must one singlet-like and one octet-like state.
In J/ψ radiative decays, the final-state mesons are produced by two gluons in the initial state. It is illuminating to compare the ππ and KK mass distributions shown in Figs. 1 with the ones produced when an ss pair forms the initial state. Ropertz, Hanhart and Kubis [68] analyzed the ππ and KK systems produced in the reaction B 0 s → J/ψπ + π − [69] and B 0 s → J/ψK + K − [70]. Here, the ππ and KK systems stem from an ss pair recoiling against the J/ψ. The pion and the kaon form factors are dominated by f 0 (980), followed by a bump-drop-off (ππ) or peak (KK) structure at 1500 MeV and a small enhancement just below 2000 MeV. The form factors (as well as the mass spectra) are decisively different from the spectra shown in Fig. 1 that originate from two interacting gluons and that are dominated by a large intensity in the 1700 to 2100 MeV mass range.

The PWA
The different sets of partial-wave amplitudes were fitted with a modified K-matrix approach [71] that takes into account dispersive corrections and the Adler zero. We fit data in which resonances are produced and data in which they are formed in scattering processes. The scattering amplitude between the channels a and b is described as while the production of a resonance is given by a P-vector amplitude: The Λ α are production couplings of the resonances, the F j represent non-resonant transitions from the initial to the final states, and the g R(α) a and g L(α) b are right-hand (R) and left-hand (L) coupling constants of the state α into channels a and b. Here the "state" represents either the bare resonant state or a nonresonant contribution. For resonances the vectors of right-hand and left-hand vertices are identical (but transposed), for nonresonant contributions, the vertices can be different, even the sign can differ. The d αα are elements of the diagonal matrix of the propagatorsd: where the first N elements describe propagators of resonances and R α are propagators of non-resonant contributions. Here, these are constants and or a pole well below the ππ threshold. The block D αβ describes the transition between the bare state α and the bare state β (with the propagator of the β state included). For this block one can write the equation (with summation over the double indices γ, η) The elements B j γη describe the loop diagrams of the channel j between states γ and η: In matrix form, Eqn. (4) can be written aŝ where the elements of the B αβ matrix are equal to the sum of the loop diagrams between states α and β: If only the imaginary part of the integral from Eqn. (5) is taken into account, Eqn. (1) corresponds to the standard K-matrix amplitude. In this case, the energy dependent non-resonant terms can be described by left-hand vertices only, the right-hand vertices can be set to 1.
Here, the real part of the loop diagrams is taken into account, and we parameterize the non-resonant contributions either as constants or as a pole below all relevant thresholds. The latter parameterization reproduces well the projection of the t and uchannel exchange amplitudes into particular partial waves.
The elements of the B j αβ are calculated using one subtraction taken at the channel threshold M j = (m 1 j + m 2 j ): Our parameterization of the non-resonant contributions allows us to rewrite theB-matrix as where the parameters b j depend on decay channels only: In this form the D-matrix approach would be equivalent to the K-matrix approach when the substitution is made. The Adler zero (set to s A = m 2 π /2) is introduced by a modification of the phase volume, with s A 0 =0.5 GeV 2 : Branching ratios of a resonance into the final state α were determined by defining a Breit-Wigner amplitude in the form The Breit-Wigner mass M 0 and the parameter f are fitted to reproduce the T -matrix pole position. For all states the factor f was between 0.95 and 1.10; the Breit-Wigner mass exceeded the pole mass by 10 − 20 MeV. The decay couplings g α and production couplings g J/ψ were calculated as residues at the pole position. Then we use the definition (Eqn. (49.16) in [43]) Branching ratios into final states with little phase space only were determined by integration over the (distorted) Breit-Wigner function. This procedure is used in publications of the Bonn-Gatchina PWA group and was compared to other definitions in Ref. [72]. The K-matrix had couplings to ππ, KK, ηη, ηη , φω, to ωω, and to the four-pion phase-space representing unseen multibody final states. The ωω and the four-pion intensities are treated as missing intensity. Fits with K-matrix poles above 1900 MeV were found to be unstable: the CERN-Munich data on elastic scattering and the GAMS data on ππ → π 0 π 0 , ηη and ηη stop at about 1900 MeV. For resonances above, only the product of the coupling constants for production and decay can be determined. Therefore we used K-matrix poles for resonances below and Breit-Wigner amplitudes above 1900 MeV. The latter amplitudes had the form The total amplitude was thus written as the sum of the P-vector amplitude (Eqn. 2) and a summation over for Breit-Wigner amplitudes (Eqn. 15). Table 1 gives the χ 2 of our best fit for the various data sets. This fit requires contributions from ten resonances. Their masses and widths are given in Table 2, their decay properties in Table 3. The errors stem from the spread of results from different sets of S -wave amplitudes [19,20], and cover the spread of results when the background amplitude was altered (constant transition amplitudes or left-hand pole in the K-matrix), when the number of high-mass resonances was changed (between 7 and 11), and part of the data were excluded from the fit. The resulting values are compared to values listed in the RPP [43]. The overall agreement is rather good. Only the five low-mass resonances are classified as established in the RPP, the other states needed confirmation. We emphasize that the solution presented here was developed step by step, independent from the RPP results. The comparison was made only when the manuscript was drafted.
The sum of Breit-Wigner amplitudes is not manifestly unitary. However, radiative J/ψ decays and the two-body decays of high-mass resonances are far from the unitarity limit. To check possible systematic errors due to the use of Breit-Wigner amplitudes, we replace the four resonances f 0 (1370), f 0 (1500), f 0 (1710), f 0 (1770) by Breit-Wigner amplitudes (imposing mass and width of f 0 (1370)). The fit returns properties of these resonances within the errors quoted in Table 2 and 3.
The four lower-mass resonances, one scalar state at about 1750 MeV and the f 0 (2100), are mandatory for the fit: if one of them is excluded, no acceptable description of the data is obtained. Only one state, f 0 (1770), is "new". Based on different peak positions, Bugg [66] had suggested that f 0 (1710) should have a close-by state called f 0 (1770). When f 0 (1710) and f 0 (1770) are replaced by one resonance, the χ 2 /N data increases by 58/167 for J/ψ → γπ 0 π 0 , 8/121 for J/ψ → γK S K S , 50/21 for J/ψ → γηη, or by (58,8,50) in short. When f 0 (2020), f 0 (2200), or f 0 (2330) are removed, the χ 2 increases by (48,6,5); (30,6,1); (23,5,0). In addition, there is a very significant deterioration of the fit to the Dalitz plots forpp annihilation when only one scalar resonance in the 1700 to 1800 MeV range is admitted. When high-mass poles are removed, a small change in χ 2 is observed also in the data onpp annihilation due to a change of the interference between neighboring poles. All ten states contribute to the reactions studied here.
In view of these arguments and the interference pattern discussed above, we make a very simple assumption: we assume that the upper states in Table 2 all have large SU(3)-singlet components, while the lower states have large octet components. For the two lowest-mass mesons, f 0 (500) and f 0 (980), Oller [73] determined the mixing angle to be small, (19±5) • : f 0 (500) is dominantly SU(3) singlet, f 0 (980) mainly octet.
We choose f 0 (1500) as reference state and plot a (M 2 , n) trajectory with M 2 n = 1.483 2 + n a GeV 2 , n = −1, 0, 1, · · · , where a = 1.08 is the slope of the trajectory. States close to this trajectory are assumed to be mainly SU(3) octet states. In instantoninduced interactions, the separation in mass square of scalar singlet and octet mesons is the same as the one for pseudoscalar mesons, but reversed [74]. Hence we calculate a second trajectory m 2 n = 1.483 2 + m 2 η − m 2 η + n a GeV 2 , n = −1, 0, 1, · · · . The low-mass singlet mesons are considerably wider than their octet partners. With increasing mass, the width of singlet mesons become smaller (except for f 0 (2020)), those of octet mesons increase (except for f 0 (2330)). Figure 3 shows (M 2 , n) trajectories for "mainly-octet" and "mainly-singlet" resonances. The agreement is not perfect but astonishing for a two-parameter prediction. The interpretation neglects singlet-octet mixing; there could be tetraquark, mesonmeson or glueball components that can depend on n; final-state interactions are neglected; close-by states with the same decay modes repel each other. There is certainly a sufficient number of reasons that may distort the scalar-meson mass spectrum. In spite of this, the mass of none of the observed states is incompatible with the linear trajectory by more than its half-width. Now we comment on the φω decay mode. The prominent peak in Fig. 1d is ascribed to f 0 (1770) → φω decays. The BE-SIII collaboration interpreted the reaction as doubly OZI suppressed decay [22]. We assume that all scalar mesons have a tetraquark component as suggested by Jaffe [4] for the light scalar meson-nonet: the price in energy to excite a qq pair to orbital angular momentum L = 1 (to 3 P 0 ) is similar to the energy required to create a new qq pair with all four quarks in the S -state. Thus, a tetraquark component in scalar mesons should not be surprising. The tetraquark component may de-    Table 3: J/ψ radiative decay rates in 10 −5 units. Small numbers represent the RPP values, except the 4π decay modes that gives our estimates derived from [23,24].
The RPP values and those from Refs. [23,24] are given with small numbers and with two digits only; statistical and systematic errors are added quadratically. The missing intensities in parentheses are our estimates. Ratios for KK are calculated from K S K S by multiplication with a factor 4. Under f 0 (1750) we quote results listed in RPP as decays of f 0 (1710), f 0 (1750) and f 0 (1800). The RPP values should be compared to the sum of our yields for f 0 (1710) and f 0 (1770). BES [20] uses two scalar resonances, f 0 (1710) and f 0 (1790) and assigns most of the KK intensity to f 0 (1710). Likewise, the yield of three states at higher mass should be compared to the RPP values for f 0 (2100) or f 0 (2200 145±32 11.0 +6.5

Multiparticle decays
Scalar mesons may also decay into multi-meson final states. This fraction is determined here as missing intensity in the mass range where data on ππ elastic scattering are available. The results are also given in Table 3 and compared to earlier determinations. The reaction J/ψ → γπ + π − π + π − has been studied in Refs. [23,24]. The partial wave analyses determined σσ as main decay mode of the scalar mesons. Then, the yields seen in 2π + 2π − need to be multiplied by 9/4 to get the full four-pion yield. These estimated yields for J/ψ → γ4π are given in Table 3 by small numbers. Assuming different decay modes (like those reported in Table III in [33]) leads to small changes only in the four-pion yields. The ωω yield, determined in J/ψ → γωω [25], is unexpectedly large and inconsistent with the small ρ 0 ρ 0 yield.
The missing intensity of f 0 (1370) reported here is not inconsistent with the branching ratio found in radiative decays J/ψ → γπ + π − π + π − but contradicts the findings frompN annihilation into five pions and from central production of four pions. This discrepancy can only be resolved by analyzing data on J/ψ → γ4π andpN annihilation in a coupled-channel analysis. The inclusion of both data sets seems to be of particular importance.
First analyses of the reactions J/ψ → γρρ and J/ψ → γωω [75,76] revealed only small scalar contributions. A few scalar resonances found here were identified in J/ψ → γ4π [23,24] and J/ψ → γωω [25]. We compare our missing intensities for f 0 (1710) and f 0 (1770) with the measured 4π and the ωω decay modes assigned to f 0 (1750). Our missing intensities are mostly well compatible with the measured 4π and 2ω yields. For the high-mass states we distribute the measured 4π intensity equally among the three resonances with masses close to the f 0 (2100). Our estimated intensities are given in Table 3 in parentheses. Changing how the 4π intensity is distributed has little effect on the properties of the peak shown in Fig. 4. Figure 4 shows the yields of scalar SU(3)-singlet and octet resonances as functions of their mass, for two-body decays in Fig. 4a and for all decay modes (except 6π) in Fig. 4b. SU(3)octet mesons are produced in a limited mass range only. In this mass range, a clear peak shows up. SU(3)-singlet mesons are produced over the full mass range but at about 1900 MeV, their yield is enhanced. Obviously, the two gluons from radiative J/ψ decays couple to SU(3) singlet mesons in the full mass range while octet mesons are formed only in a very limited mass range. But both, octet and singlet scalar isoscalar mesons are formed preferentially in the 1700 to 2100 MeV mass range.

The integrated yield
The peak structure is unlikely to be explained as kinematics effect. Billoire et al. [77] have calculated the mass spectrum of two gluons produced in radiative J/ψ decay. For scalar quantum numbers, the distribution has a maximum at about 2100 MeV and goes down smoothly in both directions. Körner et al. [78] calculated the (squared) amplitude to produce scalar mesons in radiative J/ψ decays. The smooth amplitude does not show any peak structure, neither.

The scalar glueball
We suggest to interpret this enhancement as the scalar glueball of lowest mass. The scalar isoscalar mesons that we assigned to the SU(3) octet seem to be produced only via their mixing with the glueball. Indeed, J/ψ → γ f 8 0 decays are expected to be suppressed: two gluons cannot couple to one SU(3) octet meson. Mesons interpreted as singlet scalar isoscalar mesons are produced over the full mass range. This finding supports strongly the interpretation of the scalar mesons as belonging to SU(3) singlet and octet. Figure 4a and 4b are fitted with a Breit-Wigner amplitude. We determined the yield of the scalar glueball as sum of the yield of "octet" scalar mesons plus the yield of "singlet" scalar mesons above a suitably chosen phenomenological background. Different background shapes were assumed. In the shown one, a background of the form x · exp {−αM 2 } (x = 149, α = 0.73/GeV 2 ) was used. For tetraquarks components belong to the octet). We suggest their wave functions could contain a small qq component (a qq seed), a small tetraquark component as discussed above, and a large glueball component. At the present statistical level, there seem to be no direct decays of the glueball to mesons; the two gluons forming a glueball are seen only since the glueball mixes with scalar mesons. We observe no "extra" state.

Summary
Summarizing, we have performed a first coupled-channel analysis of the S -wave partial-wave amplitudes for J/ψ radiative decays into ππ, K S K S , ηη, and φω decays. The fits were constrained by a large number of further data. The observed pattern of peaks and valleys in the ππ and KK invariant mass distributions depends critically on the interference between neighboring states. We are convinced that only a coupled-channel analysis has the sensitivity to identify reliably the position of resonances.
Scalar mesons seem to show up as mainly-singlet and mainly-octet states in SU(3). The masses of both, of singlet and octet states, are compatible with a linear (M 2 , n) behavior. Only the f 0 (500), mostly interpreted as dynamically generated ππ molecule, does not fall onto the trajectory. The ωφ decay mode of some scalar resonances suggests that these may have a tetraquark component as it was suggested for the lowest-mass scalar-meson nonet by Jaffe 45 years ago. Thus, a simple picture of the scalar-meson mass spectrum has emerged. The yield of scalar mesons in radiative J/ψ decays shows a significant structure that we propose to interpret as scalar glueball.
The BESIII collaboration has recorded data with significantly improved quality and statistics. It seems very important to repeat this analysis with the full statistics and including all final states into which scalar mesons can decay.