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Continuum Limit of Scalar Masses and Mixing Energies

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Continuum Limit of Scalar Masses and Mixing Energies 1 Continuum Limit of Scalar Masses and Mixing Energies W. Lee ∗ and D. Weingarten IBM Research, P.O. Box 218, Yorktown Heights, NY 10598, USA We evaluate the continuum limit of the valence approximation to the mass of scalar quarkonium and to the scalar quarkonium-glueball mixing energy for a range of different quark masses. Our results answer several questions raised by the proposed identification of f0(1710) as composed primarily of the lightest scalar glueball. Evidence that f0(1710) is composed mainly of glueball and half quarkonium. the lightest scalar glueball is now given by a cal- For a fixed lattice period L of about 1.6 fm and culation [1], on a 163 24 lattice at β of 5.70, several different values of quark mass, we have yielding 108(29) MeV as× the width for the light- now found the valence approximation to the con- est scalar glueball to decay to all possible pseu- tinuum limit of the mass of the lightest qq states doscalar pairs, and by three independent calcu- and of the mixing energy between these states and lations [2,3] with a combined prediction [4] of the lightest scalar glueball. Continuum predic- 1632(49) MeV as the infinite volume, continuum tions are extrapolated from calculations at four limit of the lightest scalar glueball mass. All four different lattice spacings. For the two largest lat- calculations were done in the valence (quenched) tice spacings we have results also for lattice period approximation. The decay width result combined 2.3 fm. Earlier stages of this work are reported with any reasonable guesses for the effect of finite in Refs. [6,5,9]. lattice spacing, finite lattice volume, and the re- Our results provide answers to the questions maining width to multibody states yields a total raised by the identification of f0(1710) as com- width small enough for the lightest scalar glue- posed largely of the lightest scalar glueball. With ball to be seen easily in experiment. The mass L of 1.6 fm, the valence approximation to the con- prediction combined with the expectation [5] that tinuum limit of the mass of the lightest ss scalar the valence approximation will underestimate the we find is significantly below the valence approxi- scalar glueball mass then points to f0(1710) as mation to the infinite volume, continuum limit of composed primarily of the lightest scalar glue- the scalar glueball mass. Our calculations with L ball [1]. of 2.3 fm show that as the infinite volume limit Among established resonances the only plausi- is taken, the mass of the lightest ss scalar will arXiv:hep-lat/9811005v2 2 Dec 1998 ble alternative to f0(1710) is f0(1500). Refs. [1,6] fall still further. Thus it appears to us that the propose that f0(1500) consists mainly of ss scalar identification of f0(1500) as mainly glueball with quarkonium. A problem with this suggestion, f0(1710) as mainly ss is quite improbable. Our however, is that f0(1500) apparently does not de- values for mixing energy combined with the sim- cay mainly into states containing an s and an s plification of considering mixing only among the quark [7]. Ref. [8] thus interprets f0(1500) as the lightest discrete isosinglet scalar states, then sup- lightest scalar glueball and f0(1710) as ss scalar port a set of physical mixed states with f0(1710) quarkonium. A further problem for any of these composed of 73.8(9.5)% glueball and f0(1500) identifications of f0(1500) and f0(1710) is that consisting of 98.4(1.4)% quarkonium, mainly ss. the Hamiltonian of full QCD couples quarkonium The glueball amplitude which leaks from f0(1710) and glueballs so that physical states should be goes almost entirely to the state f0(1390), which linear combinations of both. In the extreme, mix- remains mainly nn, normal-antinormal, the ab- ing could lead to f0(1710) and f0(1500) each half breviation we adopt for (uu + dd)/√2. In ad- dition, f0(1500) acquires an nn amplitude with ∗present address: T-8, LANL, Los Alamos, NM 87545 2 1.4 ) 8.0 s 1.2 )/E(m f (1710) n L~2.3 fm 7.0 0 E(m L~1.6 fm 1.0 MS (0) 2.0 f0(1500) Λ L~2.3 fm m/ 6.0 MS glueball L~1.6 fm (0) ss, L ∼ 1.6 fm Λ 1.0 )/ ss, L ∼ 2.3 fm s 5.0 0.0 0.1 0.2 E(m 0.0 (0) 0.0 0.1 0.2 Λ a (0) MS Λ a Figure 1. Continuum limit of the scalar ss mass, MS Figure 2. Continuum limit of the mixing energy the scalar glueball mass, and one sigma upper and E(m ) and of the ratio E(m )/E(m ). lower bounds on observed masses. s n s sign opposite to its ss component. Interference mass, scalar glueball mass and quarkonium- glueball mixing energy E. These calculations between the nn and ss components of f0(1500) suppresses the state’s decay to KK final states were done for a range of quark masses starting by a factor consistent, within uncertainties, with a bit below the strange quark mass and running the experimentally observed suppression. to a bit above twice the strange quark mass. An alternative calculation of mixing between For L near 1.6 fm, Figure 1 shows the ss scalar mass in units of Λ(0) as a function of lattice spac- valence approximation quarkonium and glueball MS states is given in Ref. [10]. We show in Ref. [11], ing in units of 1/Λ(0) . A linear extrapolation of MS however, that the calculation of Ref. [10] is not the mass to zero lattice spacing gives 1322(42) correct. MeV, far below our valence approximation infi- Our calculations, using Wilson fermions and nite volume continuum glueball mass of 1632(49) the plaquette action, were done with ensembles of MeV. Figure 1 shows also values of the ss scalar 2 2749 configurations on a lattice 12 10 24 with mass at β of 5.70 and 5.93 with L of 2.24(7) and ×3 × β of 5.70, 1972 configurations on 16 24 with β 2.31(6) fm, respectively. The ss mass with L near 2 × of 5.70, 2328 configurations on 16 14 20 with 2.3 fm lies below the 1.6 fm result for both values ×4 × β of 5.93, 1733 configurations on 24 at β of 5.93, of lattice spacing. Thus the infinite volume con- 2 1000 configurations on 24 20 32 with β of tinuum ss mass should lie below 1322(42) MeV. 6.17, and 1003 configurations× on× 322 28 40 × × For comparison with our data, Figure 1 shows the with β of 6.40. The smaller lattices with β of valence approximation value for the infinite vol- 5.70 and 5.93 and the lattices with β of 6.17 ume continuum limit of the scalar glueball mass and 6.40 have periods in the two (or three) equal and the observed value of the mass of f0(1500) space directions of 1.68(5) fm, 1.54(4) fm, 1.74(5) and of the mass of f0(1710). fm, 1.66(5) fm, respectively, permitting extrapo- Figure 2 shows a linear extrapolation to zero lations to zero lattice spacing with nearly con- lattice spacing of quarkonium-glueball mixing en- stant physical volume. Conversions from lattice ergy at the strange quark mass E(ms). The ex- to physical units in this paper are made [2,4] us- trapolation uses the points with L near 1.6 fm. ing the exact solution to the two-loop zero-flavor The points with larger lattice period suggest that Callan-Symanzik equation for Λ(0) a with Λ(0) of MS MS E(ms) rises a bit with lattice volume, but the 234.9(6.2) MeV [12]. trend is not statistically significant. As a func- From each ensemble of configurations, we eval- tion of quark mass with lattice spacing fixed, we uated, following Ref. [2,6,9], the lightest pseu- found the mixing energy to be extremely close to doscalar quarkonium mass, scalar quarkonium linear. We were thereby able to extrapolate our 3 data reliably down to the normal quark mass mn. lence approximation value 1322(42) MeV for lat- Figure 2 shows also a linear extrapolation to zero tice period 1.6 fm. This 13% gap is comparable lattice spacing of the ratio E(mn)/E(ms). The fit to the largest disagreement, about 10%, found is to the set of points with L near 1.6 fm but is also between the valence approximation and experi- consistent with the points for larger L. Thus the mental values for the masses of light hadrons. In continuum limit we obtain for E(mn)/E(ms) is addition, the physical mixed f0(1710) has a glue- also the infinite volume limit. The limiting value ball content of 73.8(9.5)%, the mixed f0(1500) of E(ms) is 43(31) MeV and of E(mn)/E(ms) is has a glueball content of 1.6(1.4)% and the mixed 1.198(72). f0(1390) has a glueball content of 24.5(10.7)%. The infinite volume continuum value for These predictions are supported by a recent re- E(mn)/E(ms) we now take as an input to a sim- analysis of Mark III data [13] for J/Ψ radiative plified treatment [5] of the mixing among valence decays.
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