The scalar and tensor glueball in production and decay

Evidence for the scalar and the tensor glueball is reported. The evidence stems from an analysis of BESIII data on radiative J/ψ data into π0π0, KSKS, ηη, and φω [1]. The coupled-channel analysis is contrained by a large number of further data. The scalar intensity is described by ten scalar isoscalar mesons, covering the range from f0(500) to f0(2330). Five resonances are interpreted as mainly-singlet states in SU(3), five as mainly-octet states. The mainly-singlet resonances are produced over the full mass range, the production of octet states is limited to the 1500 to 2100 MeV mass range and shows a large peak. The peak is interpreted as scalar glueball. Its mass, width and yield are determined to Mglueball = (1865 ± 25)MeV, Γglueball = (370 ± 50+30 −20)MeV, YJ/ψ→γG0 = (5.8± 1.0) · 10 −3. The study of the decays of the scalar mesons identifies significant glueball fractions [2]. The tensor wave shows the f2(1270) and f ′ 2(1525) and a small enhancement at M = 2210 ± 40 MeV, Γ = (355+60 −30)MeV [3]. An interpretation of these data is suggested. Copyright E. Klempt. This work is licensed under the Creative Commons Attribution 4.0 International License. Published by the SciPost Foundation. Received 04-03-2022 Accepted 06-04-2022 Published 31-05-2022 Check for updates doi:10.21468/SciPostPhysProc.6.001


Introduction
Nearly 50 years ago, Fritzsch and Gell-Mann proposed a new theory of strong interactions: Quantum Chromo Dynamics (QCD) was born [4,5]. The new theory predicted not only qq mesons and qqq baryons but also allowed for the existence of quark-less particles called glueballs. Their existence is a direct consequence of the nonabelian nature of QCD and of confinement. First quantitative estimates of glueball masses were given in a bag model [6]. More reliable are calculations on a lattice where the scalar glueball is predicted to have a mass in the 1500 to 1800 MeV range [7][8][9][10]. Analytic approximations to QCD predict the scalar glueball at 1850 to 1980 MeV [11][12][13]. The tensor glueball is expected to have higher mass, with a mass gap of about 600 MeV. QCD sum rules predict a scalar glueball at about 1780 MeV and a tensor glueball 100 MeV higher [14]. We thus expect the mass of the scalar glueball to be between 1500 and 2000 MeV and a tensor glueball mass in the 1900 to 2600 MeV range. The mass of the pseudoscalar glueball is expected slightly above the tensor glueball.
Glueballs are embedded into the spectrum of isoscalar mesons. The scalar and tensor glueball have isospin I = 0, positive G-parity (decaying into an even number of pions), their parity P and their C-parity are positive, and their total spin J is 0 or 2: (I G )J P C = (0 + )0 ++ or (0 + )2 ++ . Glueballs have the same quantum numbers and may mix with them. Most claims for the scalar glueball are based on the observation of three scalar isoscalar resonances, f 0 (1370), f 0 (1500), and f 0 (1710). In this mass range, two isoscalar tensor mesons are known, f 2 (1270) and f 2 (1525) where f 2 (1270) consists mainly of light quarks (nn) and f 2 (1525) of strange quarks (ss). Amsler and Close [15,16] interpreted these three scalar mesons as mixed states of an nn, ss and the scalar glueball (g g). Several authors suggested similar mixing schemes all based on the three resonances f 0 (1370), f 0 (1500), and f 0 (1710) (see [17] and refs. therein).
In this contribution, I present the results on a coupled-channel analysis of BESIII data on radiative J/ψ decays into π 0 π 0 [18], K s K s [19], ηη [20], and ωφ [21]. The results on J/ψ → γ2π + 2π − [22,23] and J/ψ → γωω [24] were included in the interpretation of the results. The analysis was constrained by a large number of further data: from the GAMS collaboration on the charge-exchange reactions π − p → π 0 π 0 n, ηη n and ηη n at 100 GeV/c in a mass range up to 3 GeV, BNL data on π − p → K S K S n, the CERN-Munich data on ππ → ππ elastic scattering, the low-mass ππ interactions from the K e4 of charged Kaons, and by 15 Dalitz plots onpN annihilation. The references to these data can be found elsewhere [1].
The fit to the data -shown in Fig. 1 -requires five pairs of close-by isoscalar resonances.

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Their masses and widths are given in Table 1. Most resonances have been reported before: the five lower-mass resonances are included in the Meson Summary Table of the Review of Particle Physics [27], four states are not considered to be established, one is "new". The agreement between our values and those reported earlier is rather good.
We now assume that the upper states in Table 1 are singlet states, the lower ones octet states. In Fig. 2 (right) we plot the squared meson masses as a function of a consecutive number. A linear relation is found with a slope of 1.1 GeV −2 . The separation is equal to the η − η mass square separation but reversed: the mainly singlet states are lower in mass than the mainly octet states. This pattern is expected for instant-induced interactions [30]. These states could have a glueball component; then they certainly have at least a singlet component. We define high-mass states (H) as resonances that have a mainly-octet qq configuration but that may additionally have a glueball component. The low-mass states (L) are mainly-singlet states. Table 2 lists the yields of scalar mesons in radiative J/ψ decays in units of 10 −5 . RPP numbers are also given for comparison but with two digits only, statistical and systematic uncertainties are added quadratically. The CERN-Munich data on elastic ππ scattering extend up to 1.9 GeV only; the missing intensity can hence be given only up to this mass.

The scalar glueball
The missing intensity is compared with the ρρ and ωω yield in radiative J/ψ decays. The J/ψ yields for f 0 (1750) reported in the RPP should be compared to our sum for the yields of f 0 (1710) and f 0 (1770). The RPP presents yields for f 0 (2100) and f 0 (2200); they should be compared to the yields of our three high-mass states. The J/ψ → γ4π yield [22,23] is distributed among these three states. Figure 3 (left) presents the total yield of H and L scalar mesons in radiative J/ψ decays. Both distributions show a significant yield at about 1900 MeV. The production of mainly-octet scalar mesons is surprising. The production is strong, it could be due to a singlet qq component but this hypothesis does not explain the peak structure. We assign the production of high-mass 001.3 Table 2: 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 [22,23].
The RPP values and those from Refs. [22,23] 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 [19] 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).  To quantify the glueball fractions in the wave functions, we write the wave function of scalar states in the form ϕ s n is the scalar mixing angle, φ G nH and φ G nL are the meson-glueball mixing angles of the highmass state H and of the low-mass state L in the nth nonet. The fractional glueball content of a meson is given by sin 2 φ G nH or sin 2 φ G nL . The qq component of a scalar meson couples to the final states with the SU(3) structure constant γ α and with a decay coupling constant c n . The structure constants γ α are shown in Fig. 4 as functions of the scalar mixing angle. The SU(3) structure constants γ α of a qq singlet and of a glueball are, of course, identical. There is one coupling constant c G for the glueball contents of all scalar mesons.
The coupling of a meson in nonet n to the final state α can be written as The coupling constants were fit to the values derived from the PWA of the BESIII data. Thus, the fractional contributions were determined. The probability that the glueball mixes into one of these resonances is The glueball is distributed, the sum of the fractional contribution is (78±18)%. A small further contribution (of about 10%) can be expected from the two higher mass states f 0 (2200) and f 0 (2330). Figure 3 shows the fractional contribution of the scalar mesons to the glueball. The solid curve is a Breit-Wigner function with mass and width M = 1865 MeV, Γ = 370 MeV, the area is normalized to one. Obviously, one full glueball is observed.
Further evidence for the glueball nature of the peak in Fig. 3 can be derived from a comparison of J/ψ radiative decays with the decayB s → J/ψ f 0 . Figure 5 shows the form factor [31] from production of scalar mesons in J/ψ → γ f 0 andB s → J/ψ f 0 decays [32,33]. The squared form factors are proportional to the yield.
The LHCb data demonstrate that the production of high-mass scalar states is strongly suppressed. The f 0 (980) is produced abundantly, there is some f 0 (1500) intensity but little production of scalar mesons above this mass. The ss → f 0 yield dies out rapidly with increasing mass. In contrast, two gluons couple strongly to high-mass scalar mesons. The difference is particularly large for the f 0 (1710)/ f 0 (1770) resonances in their KK decay. These two resonances decay strongly into KK but are not produced with ss in the initial state, only via two gluons.

The tensor glueball
With a scalar glueball at 1865 MeV and its large yield in radiative J/ψ decays we must expect the tensor glueball with an even larger yield. The experimental mass distributions in the Dwave show large peaks due to f 2 (1270) and f 2 (1525). In addition, there is a small but wide enhancement at M = 2210 ± 40 MeV, Γ = (355 +60 −30 ) MeV. This could be the desired tensor glueball. To have the large expected yield, the resonance should have large unobserved decay modes. Certainly, significant more work is required to decide if this is the tensor glueball.  Table 1 is shown. A phase difference between the ππ and KK decay modes of 180 • is required to reproduce the phase difference. One state is singlet in SU (3), the other one octet. Right: Squared masses of mainly-octet and mainly-singlet scalar isoscalar mesons as functions of a consecutive number.

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Figure 5: The BESIII data on J/ψ → γπ 0 π 0 and K s K s and pion and kaon form factor derived from LHCb data onB s → J/ψπ + π − and K + K − .

Conclusion
In radiative J/ψ decays mainly-octet and mainly-singlet scalar mesons are produced abundantly. The yield of scalar mesons shows a peak structure; mainly-octet mesons are produced with no background, mainly-singlet mesons above a smooth background. The peak is fit with a Breit-Wigner shape with a pole at M = (1865 ± 25) − i(185 ± 25 +15 −10 ) M eV . The yield is determined to Y J/ψ→γG 0 = (5.8 ± 1.0) · 10 −3 . The peak is interpreted as scalar glueball because of the following reasons: 1. Its mass is consistent with QCD predictions.
2. It is produced abundantly in radiative J/ψ decays where glueballs are expected.
3. The yield in radiative J/ψ decays is consistent with QCD predictions. 4. The decay modes of scalar mesons contributing to the glueball yield require a glueball contribution.
5. The glueball fractions of the observed scalar mesons contributing to the glueball add up to (78±18)%. About 10% are expected from higher-mass states. Hence the full glueball is is identified in the decays of scalar mesons.
6. In the reactionB s → J/ψ → f 0 under similar kinematic conditions, scalar mesons of higher mass are only weakly produced. There is little overlap of these scalar mesons with ss in the initial state. In radiative J/ψ with two gluons in the initial state, the yield of high-mass scalar mesons is siginicantly larger: the overlap of these scalar mesons with two gluons is larger.
The search for the tensor glueball in radiative J/ψ decays revealed a several 100 MeV wide peak of little intensity. This could be the tensor glueball but further studies are certainly required to establish its nature.