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Scalar isoscalar and the scalar from radiative J/ψ 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/ψ 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/ψ decays to (5.8 ± 1.0) 10 .

1. Introduction glueball to (1850 ± 130) MeV. In gravitational (string) theo- ries – an analytic approach to QCD – 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 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 in quan- isoscalar mesons. These have isospin I = 0, positive G- tum chromodynamics (QCD) [1, 2]. The f0(500) is accom- (decaying into an even number of ), 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 = 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 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 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/ψ. 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/ψ → γ tion in the confinement region forbids analytical solutions of plus two , where the final-state 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/ψ 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/ψ 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/ψ 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/ψ 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/ψ → γηη [21] and J/ψ → γφω [22] were 2. Our data base presented including an interpretation within a partial wave anal- ysis. In the reactions J/ψ → γ2π+2π− [23, 24] and J/ψ → It seems obvious that the scalar glueball can be identified reli- γωω [25], the 2π+2π− and into ωω 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/ψ 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 Ring at CERN. sis we fit meson-pairs in S -wave from radiative J/ψ 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/ψ → γπ0π0 in Ref. [19] and Central production in -hadron collisions is mostly in- J/ψ → γKS KS in Ref. [20]. Assuming dominance of reso- terpreted as collision of two , and this process is sup- nances with spin J = 0 and J = 2, the partial-wave analysis posed to be -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/ψ → 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]. The low-mass ππ interactions are known mass distributions from radiative J/ψ decays for the best set precisely from the Ke4 of charged [42]. In these experi- of amplitudes. The “data” are represented by triangles with ments, a series of scalar isoscalar mesons was found. The Re- error bars, the solid curve represents our fit. (i) The π0π0 view of Particle Properties (RPP) [43] lists nine states; only the invariant mass distribution shows rich structures. So far, no five states at lower mass are considered to be established. None attempt has been reported to understand the data within an of these states sticks out and identifies itself as the scalar glue- energy-dependent partial-wave analysis. The mass distribution ball of lowest mass. starts with a wide enhancement at about 500 MeV and a narrow Amsler and Close [44, 45] suggested that the three scalar res- peak at 975 MeV: with f0(500) and f0(980). A strong enhance- onances f0(1370), f0(1500) and f0(1710) could originate from ment follows peaking at 1450 MeV, and a second minimum at a mixing of the scalar glueball with a scalar √1 (uu¯ + dd¯) and 2 1535 MeV. Three more maxima at 1710 MeV, 2000 MeV, and a scalar ss¯ state. A large number of follow-up papers sug- 2400 MeV are separated by a deep minimum at about 1850 gested different mixing schemes based on these three mesons. MeV and a weak one at 2285 MeV. (ii) The KS KS invari- We mention here one recent paper [46] that takes into account ant mass distribution exhibits a small peak at 1440 MeV im- the production characteristics of these three mesons in radia- mediately followed by a sharp dip at 1500 MeV. The subse- tive J/ψ decays. However, doubts arose whether a compara- quent enhancement - with a peak position of 1730 MeV - has tively narrow f0(1370) exists [47, 48]. Klempt and Zaitsev [47] a significant low-mass shoulder. There is a wide minimum at suggested that the resonances f0(1500), f0(1710), f0(2100) 1940 MeV followed by a further structure peaking at 2160 MeV. could be SU(3) octet states while the singlet states merge to It is followed by a valley at 2360 MeV, a small shoulder above a continuous scalar background. This continuous scalar back- 2400 MeV, and a smooth continuation. (iii) Data on J/ψ → γηη ground was interpreted as scalar glueball by Minkowski and [21] and J/ψ → γφω [22] were published including an interpre- Ochs (and called red dragon) [49]. In Refs. [50, 51], a fourth tation within a partial wave analysis. We extracted the S -wave + 90 isoscalar scalar meson was identified at 1530−250 MeV with contributions (Fig. 1c,d) (for [22] only the contribution of the 560 ± 140 MeV width. This broad state was interpreted as dominant threshold resonance), and included them in the fits. 2

N MeV/bin a (events) 15 J/ψ→γπ0π0

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30000 15 MeV/bin b ψ→γΚ Κ J/ s s

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20000 40 MeV/bin c J/ψ→γηη

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7500 40 MeV/bin d J/ψ→γωΦ 5000

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0 0 0.5 1 1.5 2 2.5 GeV

0 0 Figure 1: Number of events in the S -wave as functions of the two-meson invariant mass from the reactions J/ψ → γ π π (a), KS KS (b), ηη (c), φω (d). (a) and (b) are based on the analysis of 1.3 · 109 J/ψ decays, (c) and (d) on 0.225 · 109 J/ψ decays.

3 butions of opposite signs: there must one singlet-like and one 500 a octet-like state. o Φ=55 In J/ψ radiative decays, the final-state mesons are produced 400 by two gluons in the initial state. It is illuminating to compare 300 the ππ and KK¯ mass distributions shown in Figs. 1 with the ones produced when an ss¯ pair forms the initial state. Ropertz, 200 Hanhart and Kubis [68] analyzed the ππ and KK¯ systems pro- 0 + − 0 + − 100 duced in the reaction Bs → J/ψπ π [69] and Bs → J/ψK K [70]. Here, the ππ and KK¯ systems stem from an ss¯ pair re- coiling against the J/ψ. The and the form factors are dominated by f (980), followed by a bump-drop-off (ππ) or 100 o b 0 Φ=235 peak (KK¯ ) structure at 1500 MeV and a small enhancement just 75 below 2000 MeV. The form factors (as well as the mass spec- tra) are decisively different from the spectra shown in Fig. 1 that 50 originate from two interacting gluons and that are dominated by a large intensity in the 1700 to 2100 MeV mass range. 25

0 3. The PWA 0 0.5 1 1.5 2 2.5 3 Minv (GeV) The different sets of partial-wave amplitudes were fitted with a modified K-matrix approach [71] that takes into account dis- Figure 2: Interference of two K-matrix poles for f0(1370) and f0(1500) with a persive corrections and the Adler zero. We fit data in which ◦ ◦ relative phase of 55 (a) and 235 (b). Shown is the intensity in arbitrary units resonances are produced and data in which they are formed as a function of the two-meson invariant mass. in scattering processes. The scattering amplitude between the channels a and b is described as X The ηη mass distribution (Fig. 1d) resembles the one observed R(α) L(β) Aab = ga dααDαβgb , (1) at SLAC [16], but now greatly improved statistics. The BE- α,β SIII data show an asymmetry that reveals the presence of at while the production of a resonance is given by a P-vector am- least two resonances, f (1500) and f (1710). The sharp drop- 0 0 plitude: off above 1.75 GeV indicates destructive interference between X two resonances. (iv) The φω mass distribution exhibits a strong ˜ (α) L(β) ˜ Ab = P dααDαβgb P=(Λ1,... Λn, F1,...) . (2) threshold enhancement. It was assigned to a scalar resonance α,β 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 The Λα are production couplings of the resonances, the F j is required in our fit. represent non-resonant transitions from the initial to the final R(α) L(α) The two figures 1a,b look very different. Obviously, inter- states, and the ga and gb are right-hand (R) and left-hand ference between neighboring states plays a decisive role. Fig- (L) coupling constants of the state α into channels a and b. Here ures 2a,b simulate the observed pattern: in Fig. 2a the low- the “state” represents either the bare resonant state or a non- resonant contribution. For resonances the vectors of right-hand mass part of f0(1500) interferes constructively with f0(1370) and leads to a sharp drop-off at its high-mass part. In Fig. 2b, and left-hand vertices are identical (but transposed), for non- resonant contributions, the vertices can be different, even the the f0(1500) is responsible for the dip. The phase difference between the amplitudes for f (1370) and f (1500) in figures sign can differ. The dαα are elements of the diagonal matrix of 0 0 ˆ 2a,b changes by 180◦ when going from ππ to KK¯ . This is an the propagators d: important observation: these two states do not behave like a   1  1 1  √ (uu¯ + dd¯) and a ss¯ state but rather like a singlet and an octet dˆ = diag  ,..., , R1, R2 ... , (3) 2  2 2  M1 − s MN − s state, like √1 (uu¯ + dd¯+ ss¯) and √1 (uu¯ + dd¯− 2ss¯). This change 3 6 in the sign of the coupling constant in ππ and KK¯ decays for where the first N elements describe propagators of resonances and Rα are propagators of non-resonant contributions. Here, f0(1500) with respect to the f0(1370) “background” has first been noticed in Ref. [67]. these are constants and or a pole well below the ππ threshold. The block D describes the transition between the bare state The two resonances f (1710) and f (1770) form the large en- αβ 0 0 α and the bare state β (with the propagator of the β state in- hancement in figure 1b while their contribution to figure 1a is cluded). For this block one can write the equation (with sum- much smaller. This again is due to interference: the two reso- mation over the double indices γ, η) nances interfere destructively in the ππ channel and construc- ¯ X j tively in the KK channel. Again, the f0(1710) and f0(1770) D = D B d + d . (4) ¯ αβ αγ γη ηβ αβ wave functions must contain significant uu¯ + dd and ss¯ contri- j 4 j Table 1: χ2/N contribution from radiative J/ψ decays, charge exchange re- The elements Bγη describe the loop diagrams of the channel j data actions, Ke4 decays, and Dalitz plots frompN ¯ annihilation at rest. Dalitz plots between states γ and η: are given by the number of events in cells, N is the number of cells.

Z 0 R(α) 0 L(β) ds g j ρ j(s , m1 j, m2 j)g j B j = . (5) J/Ψ→ χ2/NN Ref. pp¯ (liq) → χ2/NN Ref. αβ π s0 − s − i0 γπ0π0 1.28 167 [19] π0π0π0 1.40 7110 [26] 0 In matrix form, Eqn. (4) can be written as γKS KS 1.21 121 [20] π ηη 1.28 3595 [27] 0 0 Dˆ = Dˆ Bˆdˆ + dˆ Dˆ = dˆ(I − Bˆdˆ)−1, (6) γηη 0.80 21 [21] π π η 1.23 3475 [59] γφω 0.2 17 [22] π+π0π− 1.24 1334 [60] − + 0 where the elements of the Bαβ matrix are equal to the sum of π π → KLKLπ 1.08 394 [29] the loop diagrams between states α and β: π0π0 0.89 110 [38] K+K−π0 0.97 521 [61] X ± ∓ Bˆ = B j . (7) ηη 0.67 15 [38] KS K π 2.13 771 [62] αβ αβ 0 ± ∓ j ηη 0.23 9 [38] KLK π 0.76 737 [63] K+K− 1.06 35 [40] pn¯ (liq) → If only the imaginary part of the integral from Eqn. (5) is taken π+π− 1.32 845 [41] π+π−π− 1.39 823 [58] into account, Eqn. (1) corresponds to the standard K-matrix am- δ(elastic) 0.91 17 [42] π0π0π− 1.57 825 [60] plitude. In this case, the energy dependent non-resonant terms − 0 can be described by left-hand vertices only, the right-hand ver- pp¯ (gas) → KS K π 1.33 378 [62] 0 0 0 − tices can be set to 1. π π π 1.36 4891 [57] KS KS π 1.62 396 [62] Here, the real part of the loop diagrams is taken into account, π0ηη 1.32 1182 [57] and we parameterize the non-resonant contributions either as π0π0η 1.24 3631 [58] constants or as a pole below all relevant thresholds. The latter parameterization reproduces well the projection of the t and u- The Breit-Wigner mass M0 and the parameter f are fitted to channel exchange amplitudes into particular partial waves. reproduce the T-matrix pole position. For all states the factor j The elements of the Bαβ are calculated using one subtraction f was between 0.95 and 1.10; the Breit-Wigner mass exceeded α taken at the channel threshold M j = (m1 j + m2 j): the pole mass by 10−20 MeV. The decay couplings g and pro- duction couplings gJ/ψ were calculated as residues at the pole j j 2 2 Bαβ(s) = Bαβ(M j ) + (s − M j ) position. Then we use the definition (Eqn. (49.16) in [43]) ∞ Z 0 R(α) 0 L(β) α α 2 α 2 α α ds g j ρ j(s , m1 j, m2 j)g j M Γ = (g ) ρ (M ); BR = Γ /Γ . (14) × . (8) 0 0 BW π 0 − − 0 − 2 (s s i0)(s M j ) Branching ratios into final states with little phase space only M2 j were determined by integration over the (distorted) Breit- Our parameterization of the non-resonant contributions allows Wigner function. This procedure is used in publications of the us to rewrite the Bˆ-matrix as Bonn-Gatchina PWA group and was compared to other defini-

j R(α)  j 2  L(β) tions in Ref. [72]. B s g b s − M b s g , 0 αβ( ) = a 0 + ( j ) j( ) b (9) The K-matrix had couplings to ππ, KK¯ , ηη, ηη , φω, to ωω, and to the four-pion phase-space representing unseen multibody where the parameters b j depend on decay channels only: final states. The ωω and the four-pion intensities are treated as ∞ Z 0 0 missing intensity. Fits with K-matrix poles above 1900 MeV ds ρ j(s , m1a, m2a) b (s) = . (10) were found to be unstable: the CERN-Munich data on elastic j π 0 − − 0 − 2 (s s i0)(s M j ) → 0 0 0 M2 scattering and the GAMS data on ππ π π , ηη and ηη stop j at about 1900 MeV. For resonances above, only the product of In this form the D-matrix approach would be equivalent to the the coupling constants for production and decay can be deter- K-matrix approach when the substitution mined. Therefore we used K-matrix poles for resonances below and Breit-Wigner amplitudes above 1900 MeV. The latter am- iρ (s) → b j + (s − M2)b (s) (11) j 0 j j plitudes had the form 2 g is made. The Adler zero (set to sA = mπ/2) is introduced by a A = BW . (15) 2 BW 2 − − modification of the phase volume, with sA0 =0.5 GeV : MBW s i MBW ΓBW r 2 The total amplitude was thus written as the sum of the P-vector s − sA s − 4mπ ρ1(s, mπ, mπ) = . (12) amplitude (Eqn. 2) and a summation over for Breit-Wigner am- s + s s A0 plitudes (Eqn. 15). Branching ratios of a resonance into the final state α were de- Table 1 gives the χ2 of our best fit for the various data termined by defining a Breit-Wigner amplitude in the form sets. This fit requires contributions from ten resonances. Their masses and widths are given in Table 2, their decay properties in f gJ/ψ gα A = (13) Table 3. The errors stem from the spread of results from differ- α 2 P 2 α M0 − s − i f gα ρ (s) α ent sets of S -wave amplitudes [19, 20], and cover the spread 5 of results when the background amplitude was altered (con- In view of these arguments and the interference pattern dis- stant transition amplitudes or left-hand pole in the K-matrix), cussed above, we make a very simple assumption: we assume when the number of high-mass resonances was changed (be- that the upper states in Table 2 all have large SU(3)-singlet tween 7 and 11), and part of the data were excluded from the components, while the lower states have large octet compo- fit. The resulting values are compared to values listed in the nents. For the two lowest-mass mesons, f0(500) and f0(980), RPP [43]. The overall agreement is rather good. Only the five Oller [73] determined the mixing angle to be small, (19±5)◦: low-mass resonances are classified as established in the RPP, f0(500) is dominantly SU(3) singlet, f0(980) mainly octet. the other states needed confirmation. We emphasize that the so- 2 lution presented here was developed step by step, independent We choose f0(1500) as reference state and plot a (M , n) tra- 2 2 2 from the RPP results. The comparison was made only when the jectory with Mn = 1.483 + n a GeV , n = −1, 0, 1, ··· , where manuscript was drafted. a = 1.08 is the slope of the trajectory. States close to this trajec- The sum of Breit-Wigner amplitudes is not manifestly uni- tory are assumed to be mainly SU(3) octet states. In instanton- tary. However, radiative J/ψ decays and the two-body decays induced interactions, the separation in mass square of scalar of high-mass resonances are far from the unitarity limit. To singlet and octet mesons is the same as the one for pseudoscalar check possible systematic errors due to the use of Breit-Wigner mesons, but reversed [74]. Hence we calculate a second trajec- 2 2 2 − 2 2 − ··· amplitudes, we replace the four resonances f0(1370), f0(1500), tory mn = 1.483 + mη mη0 + n a GeV , n = 1, 0, 1, . The f0(1710), f0(1770) by Breit-Wigner amplitudes (imposing mass low-mass singlet mesons are considerably wider than their octet and width of f0(1370)). The fit returns properties of these reso- partners. With increasing mass, the width of singlet mesons be- nances within the errors quoted in Table 2 and 3. come smaller (except for f0(2020)), those of octet mesons in- The four lower-mass resonances, one scalar state at about crease (except for f0(2330)). 1750 MeV and the f (2100), are mandatory for the fit: if one 0 2 of them is excluded, no acceptable description of the data is Figure 3 shows (M , n) trajectories for “mainly-octet” and “mainly-singlet” resonances. The agreement is not perfect but obtained. Only one state, f0(1770), is “new”. Based on dif- astonishing for a two-parameter prediction. The interpretation ferent peak positions, Bugg [66] had suggested that f0(1710) neglects singlet-octet mixing; there could be tetraquark, meson- should have a close-by state called f0(1770). When f0(1710) 2 meson or glueball components that can depend on n; final-state and f0(1770) are replaced by one resonance, the χ /Ndata in- 0 0 interactions are neglected; close-by states with the same decay creases by 58/167 for J/ψ → γπ π , 8/121 for J/ψ → γKS KS , modes repel each other. There is certainly a sufficient number 50/21 for J/ψ → γηη, or by (58, 8, 50) in short. When f0(2020), 2 of reasons that may distort the scalar-meson mass spectrum. In f0(2200), or f0(2330) are removed, the χ increases by (48, 6, 5); (30,6,1); (23, 5, 0). In addition, there is a very significant de- spite of this, the mass of none of the observed states is incom- terioration of the fit to the Dalitz plots forpp ¯ annihilation when patible with the linear trajectory by more than its half-width. only one scalar resonance in the 1700 to 1800 MeV range is Now we comment on the φω decay mode. The prominent admitted. When high-mass poles are removed, a small change peak in Fig. 1d is ascribed to f (1770) → φω decays. The BE- in χ2 is observed also in the data onpp ¯ annihilation due to a 0 SIII collaboration interpreted the reaction as doubly OZI sup- change of the interference between neighboring poles. All ten pressed decay [22]. We assume that all scalar mesons have a states contribute to the reactions studied here. tetraquark component as suggested by Jaffe [4] for the light scalar meson-nonet: the price in energy to excite a qq¯ pair to 3 4. The flavor wave functions orbital angular momentum L = 1 (to P0) is similar to the en- The interference pattern in Fig. 1a,b suggests that the two ergy required to create a new qq¯ pair with all four in the S -state. Thus, a tetraquark component in scalar mesons pairs of resonances, f0(1370)- f0(1500) and f0(1710) - f0(1770), have wave flavor functions in which the uu¯ + dd¯ and ss¯ should not be surprising. The tetraquark component may de- components have opposite signs. Vector (ω, φ) and tensor cay to two mesons by rearrangement of color; thus a small 0 tetraquark component could have a significant impact on the ( f2(1270), f (1525)) mesons show ideal mixing, with approx- √2 decays. The φ and ω are orthogonal in SU(3), thus f (1770) imately 1/ 2(uu¯ + dd¯) and ss¯ configurations. (We neglect the 0 must have a large octet component. The φω decay is assigned small mixing angles in this discussion.) The mass difference is to the √1 (2uud¯ d¯ − uus¯ s¯ − dds¯ s¯) component that can easily dis- 200 to 250 MeV, and the ss¯ mesons decay preferably into KK¯ . 6 This is not the case for the scalar mesons: The mass difference integrate into φω. Scalar SU(3) singlet mesons might have a √1 (uud¯ d¯ + uus¯ s¯ + dds¯ s¯) component but their coupling to varies, and the KK¯ /ππ decay ratio is often less than 1 and never 3 0 φω is small since these two mesons cannot come from a pure ∼ 100 like for f2(1525). The decay pattern rules out the pos- sibility that the scalar resonances are states with a dominant ss¯ SU(3) singlet state. The φ and ω are orthogonal in SU(3), thus component. f0(1770) must have a large octet component. The φω decay is assigned to the √1 (2uud¯ d¯ − uus¯ s¯ − dds¯ s¯) component that can Pseudoscalar mesons are different: the isoscalar mesons are 6 better approximated by SU(3) singlet√ and octet configurations. easily disintegrate into φω. Scalar SU(3) singlet mesons might ¯ have a √1 (uud¯ d¯ + uus¯ s¯ + dds¯ s¯) component but their coupling Pseudoscalar mesons have both, 1/ 2(uu¯ + dd) and ss¯ com- 3 ponents as evidenced by the comparable rates for J/ψ → to φω is small since these two mesons cannot come from a pure ηω, ηφ, η0ω, η0φ. SU(3) singlet state. 6 Table 2: Pole masses and widths (in MeV) of scalar mesons. The RPP values are listed as small numbers for comparison.

Name f0(500) f0(1370) f0(1710) f0(2020) f0(2200) M [MeV] 410±20 1370±40 1700±18 1925±25 2200±25 400→550 1200→1500 1704±12 1992±16 2187±14 Γ [GeV] 480±30 390±40 255±25 320±35 150±30 400→700 100→500 123±18 442±60 ∼ 200

Name f0(980) f0(1500) f0(1770) f0(2100) f0(2330) M [GeV] 1014±8 1483 ± 15 1765±15 2075±20 2340±20 +20 990±20 1506 ± 6 2086−24 ∼2330 Γ [MeV] 71±10 116±12 180±20 260±25 165±25 +60 10→100 112±9 284−32 250±20

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 KS KS by multiplication with a factor 4. Under f0(1750) we quote results listed in RPP as decays of f0(1710), f0(1750) and f0(1800). The RPP values should be compared to the sum of our yields for f0(1710) and f0(1770). BES [20] uses two scalar resonances, f0(1710) and f0(1790) and assigns most of the KK¯ intensity to f0(1710). Likewise, the yield of three states at higher mass should be compared to the RPP values for f0(2100) or f0(2200). ¯ 0 BRJ/ψ→γ f0→ γππ γKK γηη γηη γωφ missing total unit γ4π γωω

−5 f0(500) 105±20 5±5 4±3 ∼0 ∼0 ∼0 114±21 ·10 −5 f0(980) 1.3±0.2 0.8±0.3 ∼0 ∼0 ∼0 ∼0 2.1±0.4 ·10 f (1370) 38±10 13±4 3.5±1 0.9±0.3 ∼0 14±5 69±12 0 −5 42±15 27±9 ·10 f (1500) 9.0±1.7 3±1 1.1±0.4 1.2±0.5 ∼0 33±8 47±9 0 · −5 +0.6 +1.0 10 10.9±2.4 2.9±1.2 1.7−1.4 6.4−2.2 36±9

f0(1710) 6±2 23±8 12±4 6.5±2.5 1±1 7±3 56±10 −5 f0(1770) 24±8 60±20 7±1 2.5±1.1 22±4 65±15 181±26 ·10 ± +10 +12 ± ± ± f0(1750) 38 5 99 −6 24 −7 25 6 97 18 31 10

f0(2020) 42±10 55±25 10±10 (38±13) 145±32

f0(2100) 20±8 32±20 18±15 (38±13) 108±25 ·10−5 f0(2200) 5±2 5±5 0.7±0.4 (38±13) 49±17 + 8 +6.5 f0(2100)/ f0(2200) 62±10 109−19 11.0−3.0 115±41

f0(2330) 4±2 2.5±0.5 1.5±0.4 8±3 ·10−5 20±3

7 6. The integrated yield

) 6 2 Figure 4 shows the yields of scalar SU(3)-singlet and octet 5

(GeV resonances as functions of their mass, for two-body decays in 2

M 4 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 3 mass range, a clear peak shows up. SU(3)-singlet mesons are 2 produced over the full mass range but at about 1900 MeV, their yield is enhanced. Obviously, the two gluons from radiative 1 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 0 -2 -1 0 1 2 3 4 range. But both, octet and singlet scalar isoscalar mesons are n formed preferentially in the 1700 to 2100 MeV mass range. The peak structure is unlikely to be explained as kinematics Figure 3: Squared masses of mainly-octet and mainly-singlet scalar isoscalar effect. Billoire et al. [77] have calculated the mass spectrum of mesons as functions of a consecutive number. 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. Korner¨ et al. [78] 5. Multiparticle decays calculated the (squared) amplitude to produce scalar mesons in radiative J/ψ decays. The smooth amplitude does not show any peak structure, neither.

Scalar mesons may also decay into multi-meson final states. 7. The scalar glueball This fraction is determined here as missing intensity in the mass range where data on ππ elastic scattering are available. The re- We suggest to interpret this enhancement as the scalar glue- sults are also given in Table 3 and compared to earlier deter- ball of lowest mass. The scalar isoscalar mesons that we as- minations. The reaction J/ψ → γπ+π−π+π− has been studied signed to the SU(3) octet seem to be produced only via their 8 in Refs. [23, 24]. The partial wave analyses determined σσ mixing with the glueball. Indeed, J/ψ → γ f0 decays are ex- as main decay mode of the scalar mesons. Then, the yields pected to be suppressed: two gluons cannot couple to one SU(3) seen in 2π+2π− need to be multiplied by 9/4 to get the full octet meson. Mesons interpreted as singlet scalar isoscalar four-pion yield. These estimated yields for J/ψ → γ4π are mesons are produced over the full mass range. This finding given in Table 3 by small numbers. Assuming different decay supports strongly the interpretation of the scalar mesons as be- modes (like those reported in Table III in [33]) leads to small longing to SU(3) singlet and octet. changes only in the four-pion yields. The ωω yield, determined Figure 4a and 4b are fitted with a Breit-Wigner amplitude. in J/ψ → γωω [25], is unexpectedly large and inconsistent with We determined the yield of the scalar glueball as sum of the small ρ0ρ0 yield. the yield of “octet” scalar mesons plus the yield of “singlet” scalar mesons above a suitably chosen phenomenological back- The missing intensity of f0(1370) reported here is not in- consistent with the branching ratio found in radiative decays ground. Different background shapes were assumed. In the x · {−αM2} x J/ψ → γπ+π−π+π− but contradicts the findings frompN ¯ an- shown one, a background of the form exp ( = , α . 2 M, nihilation into five pions and from central production of four 149 = 0 73/GeV ) was used. For Fig. 4a, we find ( Γ) = , M, , pions. This discrepancy can only be resolved by analyzing data (1872 332) MeV, for Fig. 4b ( Γ) = (1856 396) MeV. The on J/ψ → γ4π andpN ¯ annihilation in a coupled-channel anal- results depend on the background chosen. From the spread of ysis. The inclusion of both data sets seems to be of particular results when the background function is changed, we estimate importance. the uncertainty. Our best estimate for the scalar glueball mass and width is given as First analyses of the reactions J/ψ → γρρ and J/ψ → γωω +10 +30 [75, 76] revealed only small scalar contributions. A few scalar MG = (1865 ± 25 −30) MeV ΓG = (370 ± 50 −20) MeV. resonances found here were identified in J/ψ → γ4π [23, 24] and J/ψ → γωω [25]. We compare our missing intensities for The integrated yields depend on the not well-known 4π and ωω f (1710) and f (1770) with the measured 4π and the ωω de- (and unknown 6π) contributions. The (observed) yield of octet 0 0 scalar isoscalar mesons plus the yield of singlet mesons above cay modes assigned to f0(1750). Our missing intensities are mostly well compatible with the measured 4π and 2ω yields. the background is determined to For the high-mass states we distribute the measured 4π inten- −3 YJ/ψ→γG = (5.8 ± 1.0) 10 . sity equally among the three resonances with masses close to the f0(2100). Our estimated intensities are given in Table 3 in The two states with the largest glueball component are f0(1770) parentheses. Changing how the 4π intensity is distributed has and f0(2020). The “mainly octet” f0(1770) acquires a glueball little effect on the properties of the peak shown in Fig. 4. component (and is no longer “mainly octet”, only the qq¯ and 8 250 250 -5 a -5 b 200 200

150 150 decay rate 10

Ψ 100 100 J/ decay rate 10 J/ Ψ 50 50

0 0 0 500 1000 1500 2000 2500 3000 0 500 1000 1500 2000 2500 3000 Mres (MeV) Mres (MeV)

Figure 4: Yield of radiatively produced scalar isoscalar octet mesons (open circles) and singlet (full squares) mesons. a) Yield for ππ, KK¯ , ηη, ηη0, and φω decays. b) Yield when 4π decays and ωω are included.

components belong to the octet). We suggest their Acknowledgement wave functions could contain a small qq¯ component (a qq¯ seed), a small tetraquark component as discussed above, and a large Funded by the NSFC and the Deutsche Forschungsgemein- glueball component. At the present statistical level, there seem schaft (DFG, German Research Foundation) through the funds to be no direct decays of the glueball to mesons; the two gluons provided to the Sino-German Collaborative Research Center forming a glueball are seen only since the glueball mixes with TRR110 “Symmetries and the Emergence of Structure in QCD” scalar mesons. We observe no “extra” state. (NSFC Grant No. 12070131001, DFG Project-ID 196253076 - TRR 110) and the Russian Science Foundation (RSF 16-12- 10267).

8. Summary References

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