Scalar Quarkonium and the Scalar Glueball

The identification of fJ(1710) as the lightest scalar glueball is supported at present by two different sets of calculations. For the valence approximation to the infinite volume continuum limit of the lightest scalar glueball mass, a calculation on GF11 [1], using 25000 to 30000 gauge configurations, gives 1740 ± 71 MeV. An independent calculation by the UKQCD-Wuppertal [2] collaboration, using 1000 to 3000 gauge configurations, gives 1625±94 MeV when extrapolated to zero lattice spacing according to Ref. [3]. The GF11 and UKQCDWuppertal data combined predicted 1707 ± 64 MeV. The calculation with larger statistics and the combined result both favor fJ(1710) as the lightest scalar glueball. The mass calculations by themselves, however, leave open the possibility that the lightest glueball may have too large a total width to be found in experiment. A valence approximation calculation on GF11 [4] of couplings for glueball decay to all possible pseudoscalar pairs, π + π, K +K, η+η, and η+η, using a 163×24 lattice at β = 5.7, gets 108± 29 MeV for the total two-body width. Based on this number, any reasonable guess for the width to multibody states yields a total width small enough for the lightest scalar glueball to be seen easily in experiment. In fact, the predicted total two-body width agrees with the fJ(1710) width of 99±15 MeV of Ref. [5], as do the partial widths to individual channels. Among established resonances with the quantum numbers to be a scalar glueball, aside from fJ(1710) all are clearly inconsistent with the mass calculation expect f0(1500). The mass of f0(1500) is still more than 3 sigma away both from the prediction with larger statistics or from the combined result. Ref. [4] proposes to interpret f0(1500) as dominantly composed of ss scalar quarkonium. The interpretation of f0(1500) as ss quarkonium, however, encounters three difficulties. First, it appears possible that the gap between 1740 ± 71 MeV and 1500 MeV might simply be an error arising from the valence approximation. Second, f0(1500) does not seem to decay mainly into states containing an s and an s quark [6]. Third, the Hamiltonian of full QCD couples quarkonium and glueballs so that fJ(1710) and f0(1500) could both be linear combinations of quarkonium and a glueball, perhaps even half glueball and half quarkonium each. In the remainder of this article, I show that the pattern of established quarkonium masses, an estimate of the error in valence approximation mass calculations, a calculation of the ss scalar quarkonium mass and a model of quarkoniumglueball mixing help resolve these difficulties and support the interpretation of fJ(1710) as dominantly a glueball and f0(1500) as dominantly ss scalar quarkonium.