The mixing of the f 0 ( 1370 ) , f 0 ( 1500 ) and f 0 ( 1710 ) and the search for the scalar glueball

For the first time a complete data set of the two-body decays of the f0(1370), f0(1500) and f0(1710) into all pseudoscalar mesons is available. The implications of these data for the flavour content for these three f0 states is studied. We find that they are in accord with the hypothesis that the scalar glueball of lattice QCD mixes with the qq nonet that also exists in its immediate vicinity. We show that this solution also is compatible with the relative production strengths of the f0(1370), f0(1500) and f0(1710) in pp central production, pp̄ annihilations and J/ψ radiative decays. e-mail: [email protected] e-mail: [email protected] The best estimate for the masses of glueballs comes from lattice gauge theory calculations which in the quenched approximation show [1] that the lightest glueball has J = 0 and that its mass should be in the range 1.45− 1.75 GeV. While the lattice remains immature for predicting glueball decays, Amsler and Close [2] have noted that in lattice inspired models, such as the flux tube [3], glueballs will mix strongly with nearby qq states with the same J [4]. This will lead to three isoscalar states of the same J with predictable patterns of decay branching ratios [2, 5, 6]. Such mixing ideas have been applied recently to the three states in the glueball mass region the f0(1370), f0(1500) and f0(1710) [2, 6, 7, 8, 9]. While these papers at first sight differ in detail, nonetheless their conclusions share some common robust features. The flavour content of the states have the nn and ss in phase (SU(3) singlet tendency) for the f0(1370) and f0(1710), and out of phase (octet tendency) for the f0(1500) (that such a pattern is natural is discussed in ref. [10]). In general these mixings will negate the naive folklore that glueball decay branching ratios would be independent of the quark composition of the final state mesons after taking into account phase space effects, so called “flavour blind decays”. Recently ref.[8] has used some of the observed branching ratios as input to constrain the mixing pattern. The results here too agree with the generic structure found in refs. [2, 6, 7, 10]. Most recently, and subsequent to the above work, the WA102 collaboration has now published [11], for the first time in a single experiment, a complete data set for the decay branching ratios of the f0(1370), f0(1500) and f0(1710) to all pseudoscalar meson pairs. These data will be our point of departure. Using methods similar to those proposed in ref. [8] we investigate the implications of the WA102 data for the glueball-quarkonia content of the f0(1370), f0(1500) and f0(1710). This moves the debate forwards in the following ways: • It highlights the sensitivity of the mixings to the input data. • It exposes some assumptions, both explicit and implicit, in the analysis of ref. [8] that can be improved upon. • It allows for the direct decay of glueballs into η and η, which were not manifested in leading order in ref.[8] even though there are reasons to suspect that they could be important [2, 12]. In order to unfold the production kinematics we use the invariant decay couplings (γij) for the observed decays, namely we express the partial width (Γij) as [2] Γij = γ 2 ij|Fij(~q)|Sp(~q) (1) where Sp(~q) denotes the phase space and Fij(~q) are form factors appropriate to exclusive two body decays. Here we have followed ref. [2] and have chosen |Fij(~q)| = qexp(−q/8β) (2) where l is the angular momentum of the final state with daughter momenta q and we have used β = 0.5 GeV/c [2]. The f0(1500) lies very near to threshold in the ηη ′ decay mode, therefore we have used an average value of q (190 MeV/c) derived from a fit to the ηη mass spectrum. 2 The branching ratios measured by the WA102 experiment for the f0(1370), f0(1500) and f0(1710) are given in table 1. The invariant decay couplings (γij) are related to the relevant decay amplitudes Mij by γ ij = cij |Mij | (3) where cij is a weighting factor arising from the sum over the various charge combinations, namely 4 for KK, 3 for ππ, 2 for ηη and 1 for ηη for isoscalar decays. If in the decay of some state the ratios of the decay amplitudes squared (|Mij|) for the ηη/ππ and ηη/KK are simultaneously greater than unity, then this state cannot be a quarkonium decay (see fig. 3 of ref. [2] and also ref. [5]). Table 2 gives these ratios for the f0(1370), f0(1500) and f0(1710), as abstracted using eqs.(1), (2) and (3). As can be seen the ratios for the f0(1710) argue either for non-qq content in this meson or for some further dynamical suppression of the ππ mode, say, as may occur for special values of parameters in some specific models [13]. In the |G〉 = |gg〉, |S〉 = |ss̄〉, |N〉 = |uū + dd̄〉/ √ 2 basis, the mass matrix describing the mixing of a glueball and quarkonia can be written as follows [7]: