Scalar isoscalar mesons and the scalar glueball from radiative J= decays A.V. Sarantseva,b, I. Denisenkoc, U. Thomaa, and E. Klempta aHelmholtz–Institut f¨urStrahlen– und Kernphysik, Universit¨atBonn, Germany bNRC “Kurchatov Institute”, PNPI, Gatchina 188300, Russia cJoint Institute for Nuclear Research, Joliot-Curie 6, 141980 Dubna, Moscow region, Russia Abstract A coupled-channel analysis of BESIII data on radiative J= decays into ππ, KK¯ , ηη and ωφ 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 +10 +30 −3 determined to M = 1865±25−30 MeV and Γ = 370±50−20 MeV, its yield in radiative J= decays to (5:8 ± 1:0) 10 . 1. Introduction glueball to (1850 ± 130) MeV. In gravitational (string) theo- ries – an analytic approach to QCD – glueballs are predicted Scalar mesons – mesons with the quantum numbers of the as well [14] at 1920 MeV. Glueballs are predicted consistently vacuum – are most fascinating objects in the field of strong in- within a variety of approaches to QCD. They seem to be a safe teractions. The lowest-mass scalar meson f0(500), tradition- prediction. ally often called σ, reflects the symmetry breaking of strong Scalar glueballs are embedded into the spectrum of scalar interactions and plays the role of the Higgs particle in quan- isoscalar mesons. These have isospin I = 0, positive G-parity tum chromodynamics (QCD) [1, 2]. The f0(500) is accom- (decaying into an even number of pions), their total spin J van- panied by further low-mass scalar mesons filling a nonet of ishes, their parity P and their C-parity are positive: (IG)JPC = particles with spin J = 0 and parity P = +1: The three (0+)0++. Scalar glueballs have the same quantum numbers as ∗ charge states a0(980), the four K0(700), and the two isoscalar scalar isoscalar mesons and may mix with them. In quark mod- mesons f0(980), f0(500) are supposed to be dynamically gen- els, mesons are described as bound states of a quark and an erated from meson-meson interactions [3]. Alternatively - or antiquark. Their quantum numbers are often defined in spectro- complementary - these mesons are interpreted as four-quark or scopic notation by the orbital angular momentum of the quark tetraquark states [4]. and the antiquark L, the total quark spin S , and the total angular 2S +1 3 The continued quest for scalar isoscalar mesons at higher momentum J. Scalar mesons have LJ = P0. masses is driven by a prediction – phrased for the first time Experimentally, the scalar glueball was searched for inten- nearly 50 years ago [5, 6] – that QCD allows for the exis- sively but no generally accepted view has emerged. The most tence of quark-less particles called glueballs. Their existence promising reaction to search for glueballs are radiative decays is a direct consequence of the nonabelian nature of QCD and of J= . In this process, the dominant contribution to direct pho- of confinement. However, the strength of the strong interac- ton production is expected to come from the process J= ! γ tion in the confinement region forbids analytical solutions of plus two gluons, where the final-state hadrons are produced by full QCD. First quantitative estimates of glueball masses were the hadronization of the two gluons. QCD predicts the two glu- arXiv:2103.09680v2 [hep-ph] 25 Mar 2021 given in a bag model [7]. Closer to QCD are calculations on ons to interact forming glueballs – if they exist. Lattice gauge a lattice. In quenched approximation, i.e. when qq¯ loops are calculations predict a branching ratio for radiative J= decays neglected, the lowest-mass glueball is predicted to have scalar to produce the scalar glueball of (3:8 ± 0:9)10−3 [15]. This is quantum numbers, and to have a mass in the 1500 to 1800 MeV a significant fraction of all radiative J= decays, (8.8±1.1)%. range [8, 9, 10]; unquenching lattice QCD predicts a scalar There was hence great excitement when a broad bump in the glueball at (1795 ± 60) MeV [11]. Exploiting a QCD Hamilto- radiatively produced ηη mass spectrum [16] was discovered by nian in Coulomb gauge generating an instantaneous interaction, the Crystal Ball collaboration at the Stanford Linear Acceler- Szczepaniak and Swanson [12] calculate the low-lying glueball ator (even though with tensor quantum numbers). However, a masses with no free parameters. The scalar glueball of low- resonance with the reported properties was not reproduced by est mass is found at 1980 MeV. Huber, Fischer and Sanchis- any other experiment. The DM2 collaboration reported a strong Alepuz [13] calculate the glueball spectrum using a parameter- peak at 1710 MeV in the KK¯ invariant mass distribution [17], free fully self-contained truncation of Dyson-Schwinger and a peak that is now known as f0(1710). Data from the CLEO Bethe-Salpeter equations and determine the lowest-mass scalar experiment on radiative J= decays into pairs of pseudoscalar Preprint submitted to Elsevier March 29, 2021 mesons were studied in a search for glueballs [18] but no defi- glueball. Bugg, Peardon and Zou [52] suggested that the four nite conclusions were obtained. known mesons f0(1500), f2(1980), f0(2105), η(2190) should The data with the highest statistics nowadays stem from BE- be interpreted as scalar, tensor, excited scalar and pseudoscalar SIII in Bejing. The partial wave amplitudes for J= radiative glueball. Recent reviews of scalar mesons and the scalar glue- 0 0 decays into π π [19] and KS KS [20] were determined in fits ball can be found elsewhere [47, 53, 54, 55, 56]. to the data in slices in the invariant mass of the two outgoing mesons. Data on J= ! γηη [21] and J= ! γφω [22] were 2. Our data base presented including an interpretation within a partial wave anal- ysis. In the reactions J= ! γ2π+2π− [23, 24] and J= ! It seems obvious that the scalar glueball can be identified reli- γ!! [25], the 2π+2π− and into !! branching ratios of con- ably only once the spectrum of scalar mesons is understood into tributing resonances were deduced from a smaller data sample. which the glueball is embedded. Decisive for the interpretation are the data on radiative J= decays. But many experiments A new understanding of the spectrum of light-quark scalar contribute to our knowledge on scalar isoscalar mesons and mesons emerged from the results obtained with the Crystal Bar- provide additional constraints. In this coupled-channel analy- rel experiment at the Low-Energy Antiproton Ring at CERN. sis we fit meson-pairs in S -wave from radiative J= decays and Inpp ¯ annihilation at rest, annihilation into 3π0 [26], π0ηη [27], include the S -wave contributions to ππ elastic scattering [38] π0ηη0 [28], and π0K K [29] was studied. These data estab- L L and ππ ! K K [39, 40], the CERN-Munich [41] data and the lished the existence of the f (1500) resonance; the existence of S S 0 K [42] data. Further, we use 15 Dalitz plots for different reac- the f (1370) had been proposed in 1966 [30] but its existence e4 0 tions frompN ¯ annihilation at rest [26, 27, 29], [57]-[63, 64, 65]. was accepted only after its rediscovery at LEAR inpp ¯ [31] and The real and imaginary parts of the mass-dependent S -wave pn¯ annihilation [32, 33]. amplitudes were derived for J= ! γπ0π0 in Ref. [19] and Central production in hadron-hadron collisions is mostly in- J= ! γKS KS in Ref. [20]. Assuming dominance of reso- terpreted as collision of two Pomerons, and this process is sup- nances with spin J = 0 and J = 2, the partial-wave analysis posed to be gluon-rich. Data on this reaction were taken at returned - for each mass bin - two possible solutions, called CERN by the WA102 collaboration that reported results on black (b) and red (r). In some mass regions, the two ampli- + − 0 0 0 π π and KS KS [34], ηη [35], ηη and η η [36], and into tudes practically coincide. We assume continuity between re- four pions [37]. The GAMS collaboration reported a study gions in which the two amplitudes are similar, and divide the 0 0 − of the π π system in the charge-exchange reactions π p ! full mass range into five regions: in three regions, the two am- π0π0 n; ηη n and ηη0 n at 100 GeV/c [38] in a mass range up to − plitudes are identical, in two regions, the red and black am- 3 GeV. The charge exchange reaction π p ! KS KS n was stud- plitudes are different. Thus there are four sets of amplitudes, ied at the Brookhaven National Laboratory [39]. An energy- (r; r); (r; b); (b; r); (b; b). For the data on J= ! γKS KS , we dependent partial-wave analysis based on a slightly increased again define five mass regions and four sets of amplitudes. The data set was reported in Ref. [40]. A reference for any analysis 2 amplitudes (r; r) give the best χ for ππ, and (b; b) for KS KS . in light-meson spectroscopy are the amplitudes for ππ ! ππ 0 0 Figure 1a,b shows the π π [19] and KS KS [20] invariant elastic scattering [41].