The Glueball among the Light Scalar Mesons

The lightest gluonic meson is expected with J = 0, calculations in full QCD point towards a mass of around 1 GeV. The interpretation of the scalar meson spectrum is hindered as some states are rather broad. In a largely modelindependent analysis of ππ → ππ, ππ scattering in the region 600-1800 MeV a unique solution for the isoscalar S-wave is obtained. The resonances f0(980), f0(1500) and the broad f0(600) or “σ” are clearly identified whereas f0(1370) is not seen at the level B(f0(1370) → ππ) & 10%. Arguments for the broad state to be a glueball are recalled. We see no contradiction with the reported large B(σ → γγ) and propose some further experimental tests. 1 QCD predictions for the lightest glueball The existence of gluonic mesons belongs to the early predictions of QCD and first scenarios have been developed back in 1975 1). Today, quantitative results are available from 1. Lattice QCD: In full QCD both glue and qq̄ states couple to the flavour singlet 0 states and first “unquenched” results for the lightest gluonic state point towards a mass of around 1 GeV 2). This is a considerably lower mass value than what is obtained in the pure Yang Mills theory for gluons (quenched approximation) where the lightest glueball is found at masses around 1700 MeV (recent review 3)). Further studies concerning the dependence on lattice spacing and the quark mass appear important. 2. QCD sum rules: Results on the scalar glueball and various decays are obtained in 4). The lightest gluonic state is found in the mass range (7501000) MeV with a decay width of (300-1000) MeV into ππ and the width into γγ of (0.2-0.3) keV. Other analyses find similar or slightly higher masses (1250± 200) MeV for the lightest glueball 5). 2 The scalar meson spectrum and its interpretation In the search for glueballs one attempts to group the scalar mesons into flavour multiplets (either qq̄ or tetraquarks) and to identify supernumerous states. The existence of such states could be a hint for glueballs either pure or mixed with qq̄ isoscalars. In other experimental activities one looks for states which are enhanced in “gluon rich” processes and are suppressed in γγ processes. The lightest isoscalar states listed in the particle data group 6) are f0(600)(or σ), f0(980), f0(1370)(?), f0(1500), f0(1710), f0(2080), (1) where the question mark behind f0(1370) will be explained below. There are different routes to group these states into multiplets together with a0 and K ∗ 0 states. In a popular approach the two lightest isoscalars in (1) are combined with κ(800) and a0(980) to form the lightest nonet, either of qq̄ or of qq − q̄q̄ type. Then the next higher multiplet from qq̄ would include a0(1450), K ∗ 0 (1430); near these masses three isoscalars are found in the list (1) at 1370, 1500 and 1710 MeV and this suggests to consider these three mesons as mixtures of the two members of the qq̄ nonet and one glueball (for an early reference, see 7)). A potential problem in this scheme for the glueball is the very existence of f0(1370), otherwise there is no supernumerous state in this mass range. Some problems with this state will be discussed below, see also the review 8). The low mass multiplet depends on the existence of κ which we consider as not beyond any doubt: its observed phase motion is rather weak and it is markedly different from the one of “σ”, see below. There are other approaches for the classification of the scalar mesons where f0(980) is the lightest qq̄ scalar. In the scheme we prefer 9) the lightest qq̄ nonet contains f0(980), f0(1500) together with a0(1450), K ∗ 0 (1430). The supernumerous state f0(600), called previously f0(400− 1200), corresponds to a very broad object which extends from ππ threshold up to about 2 GeV and is interpreted as largely gluonic. No separate f0(1370) is introduced, nor κ(800). Our classification is consistent with various findings on production and decay processes including D,Ds, B and J/ψ decays 9, 10, 11). Related schemes are the Bonn model 12) with a similar mixing scheme for the isoscalars and the K-matrix model 13) which finds a similar classification (but with f0(1370) included) and a broad glueball, centered at the higher masses around 1500 MeV. 3 Study of ππ scattering from 600 to 1800 MeV 3.1 Selection of the physical solution for mππ > 1000 MeV We are interested here in particular in the problem of f0(1370) and also in the behaviour of the broad “background” which is related to f0(600) or “σ”, alias f0(400−1200) and describe the results from an ongoing analysis (see also 14)). Information on ππ scattering can be obtained from production experiments like πp → ππn by isolating the contribution of the one-pion-exchange process. In an unpolarised target experiment these amplitudes can be extracted by using dynamical assumptions, such as “spin and phase coherence”, which have been tested by experiments with polarised target. At the level of the process ππ → ππ in different charge states one measures the distribution in scattering angle, z = cos θ, or their moments 〈Y L M 〉, in a sequence of mass intervals. The ππ partial wave amplitudes S, P,D, F, . . . can be obtained in each bin from the measured moments up to the overall phase and a discrete ambiguity (characterised by the “Barrelet Zeros”). The overall phase can be fixed by fitting a Breit Wigner amplitude for the leading resonances ρ, f2(1270) and ρ3(1690) to the experimental moments 〈Y 2 0 〉, 〈Y 4 0 〉 and 〈Y 6 0 〉 respectively. Phase shift analyses of this type for ππ scattering have been performed by the CERN-Munich group: an analysis guided by a global resonance fit (CM-I 15)) and a fully energy-independent analysis by CM-II 17) and by Estabrooks and Martin 16); the latter two analyses found 4 different solutions above 1 GeV in mass. Up to 1400 MeV a unique solution has been found 20) using results from polarised target and unitarity. Two solutions remain above 1400 MeV, classified according to Barrelet zeros in 17) as (− − −) and (− + −). corresponding to sols. A,C in 16). A new result has been added recently 14) by the construction of the isoscalar S wave S0 from the π π → ππ data (GAMS collaboration 19)) and the I = 2 scattering data. This S0 wave shows a qualitatively similar behaviour to S0 obtained from π π → ππ scattering above, namely a resonance circle in the complex plane (Argand diagram) related to f0(1500) above a slowly moving circular background amplitude. This has lead us to select the solution (−+−) as unique solution. We relate the differences in the two results to systematic errors introduced through the overall phase and the S2 wave, but these are only slowly varying effects as function of mass. 3.2 Resonance fit to the isoscalar S wave The resulting amplitude S0(−+ −) = (η 0 exp(2iδ 0)− 1)/2i is shown in Fig. 1 using the CM-II data after correction for the more recent I = 2 amplitudes. The curves refer to a fit of the data (CM-II for Mππ > 1 GeV, CM-I for Mππ < 1 GeV) to an S-matrix in the space of 3 reaction channels (ππ,KK̄, 4π) as product of individual S-matrices for resonances SR = 1 + 2iTR S = Sf0(980)Sf0(1500)Sbroad (2) TR = [M 2 0 −M ππ − i(ρ1g 1 + ρ2g 2 + ρ3g 3)] × ρ 1 2T (gigj)ρ 1 2 (3) where ρi = 2ki/ √ s. As can be seen in Fig. 1 the fit including 3 resonances gives a reasonable description of the data. For f0(1500) the fit parametersM0 = 1510 MeV, Γtot = 88 MeV, B(f0 → ππ) = 38% are obtained in remarkable agreement to the PDG numbers, despite the different approaches involved. 3.3 Note on f0(600), κ(800) and f0(1370) The broad object is also described by a resonance form with mass parameter M0 ∼ 1100 MeV and width Γ ∼ 1450 MeV. The elastic width is about 85% 0 20 40 60 80 100 120 140 160 180 0.6 0.8 1 1.2 1.4 1.6 1.8 2 phase shifts resonance fit ,