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arXiv:hep-lat/9811005v2 2 Dec 1998 uain[] na16 cal- a a by on given now [1], is culation scalar lightest the a anyit ttscnann an containing states into mainly cay ∗ uroim rbe ihti suggestion, this with that problem is however, A quarkonium. l lentv to alternative ble glue- scalar lightest the [1]. ball of primarily composed otnu ii fSaa assadMxn Energies Mixing Lee and W. Masses Scalar of Limit Continuum ur 7.Rf 8 hsinterprets thus [8] Ref. [7]. B eerh ..Bx28 okonHihs Y158 USA 10598, NY Heights, Yorktown 218, Box P.O. Research, IBM uroim ute rbe o n fthese of any of for problem identifications further A quarkonium. clrqaknu-lealmxn nryfrarneo diff of of range identification proposed a the for by raised energy questions mixing quarkonium-glueball scalar clrgubl aste onsto points then mass glueball mass the scalar underestimate The will approximation that valence experiment. [5] the expectation in the glue- easily with combined scalar seen prediction lightest be the to for total ball a enough yields small states re- width multibody the to and width volume, maining lattice finite finite of effect spacing, the lattice for guesses reasonable any combined with result width (quenched) decay valence The four the approximation. All in done mass. were glueball of calculations scalar lightest continuum [4] the volume, of prediction infinite limit the combined as a MeV calcu- 1632(49) with independent pseu- [2,3] three possible by lations all and to light- pairs, decay the to doscalar for glueball width scalar the est as MeV 108(29) yielding rps that propose n leal ota hsclsae hudbe to mix- lead extreme, should could the ing In states both. physical of combinations that linear so quarkonium couples and QCD full of Hamiltonian the ihetsaa lealand glueball scalar lightest rsn drs:T8 AL o lms M87545 NM Alamos, Los LANL, T-8, address: present vdnethat Evidence mn salse eoacsteol plausi- only the resonances established Among eeaut h otnu ii ftevlneapproximatio valence the of limit continuum the evaluate We ∗ n .Weingarten D. and f 0 10)cnit anyof mainly consists (1500) f f f 0 f f 0 0 10)aprnl osntde- not does apparently (1500) 0 0 11)i opsdmil of mainly composed is (1710) 11)and (1710) 11)is (1710) 10)and (1500) 3 × 4ltieat lattice 24 f f 0 11)as (1710) 0 f 10) es [1,6] Refs. (1500). 0 f 10)ec half each (1500) 0 f 11)i that is (1710) 0 10)a the as (1500) f s 0 β 11)as (1710) n an and s s s f5.70, of s scalar scalar f 0 s 11)a opsdpiaiyo h ihetsaa gluebal scalar lightest the of primarily composed as (1710) opsdo 3895%gubl and glueball 73.8(9.5)% of composed ossigo 8414%qaknu,mainly quarkonium, 98.4(1.4)% of consisting iumlmto h aso h lightest the of mass the con- have of the we limit to tinuum approximation mass, valence the quark found of now values different several rvainw dp o ( for adopt we breviation otasto hsclmxdsae with states mixed physical of set sup- a then port states, the scalar among isosinglet only discrete lightest mixing sim- the considering with of combined plification energy mixing for values lealadhl quarkonium. half and glueball iumlmto h aso h lightest the of mass the of limit tinuum With glueball. scalar L lightest the of largely posed alsilfrhr hsi per ou htthe that us to appears it of Thus identification further. still fall f dition, asdb h dnicto of identification the by raised reported are work this of [6,5,9]. Refs. stages in Earlier period fm. lattice for 2.3 also lat- results have largest we two four spacings the tice For at spacings. calculations lattice predic- from different Continuum extrapolated are glueball. tions scalar and lightest states these the between energy mixing the of and ean mainly remains stkn h aso h lightest limit the volume of infinite mass the the as taken, that show is fm with calculations 2.3 Our of mass. of glueball limit scalar continuum volume, the infinite approxi- the valence to the mation below significantly is find we osams nieyt h state the to entirely almost goes h lealapiuewihlasfrom leaks which amplitude glueball The 0 f16f,tevlneapoiaint h con- the to approximation valence the fm, 1.6 of 11)a mainly as (1710) o xdltieperiod lattice fixed a For u eut rvd nwr otequestions the to answers provide results Our rn ur ass u eut nwrseveral answer results Our masses. quark erent otems fsaa uroimadt the to and quarkonium scalar of mass the to n f 0 10)aqie an acquires (1500) f n 0 n 10)a anygubl with glueball mainly as (1500) omlatnra,teab- the normal-antinormal, , s s sqieipoal.Our improbable. quite is u L u faot16f and fm 1.6 about of n + n f d 0 f mltd with amplitude d 11)a com- as (1710) 0 ) 19) which (1390), s / s √ clrwill scalar .I ad- In 2. q s f f f q s 0 0 0 (1710) (1500) (1710) states scalar s l. s L 1 . 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 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 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 . 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. Finally, the state vector for f0(1500) we approximation glueball and quarkonium states find has a relative negative sign between the ss which arises in full QCD from quark-antiquark and nn components leading, by interference, to a annihilation. The Hamiltonian coupling together suppression of the partial width for this state to the scalar glueball, the scalar ss and the scalar decay to KK by a factor of 0.39(16) in compari- nn isosinglet is son to the KK rate for an unmixed ss state. This suppression is consistent with the experimentally mg E(ms) √2rE(ms) observed suppression. E(ms) mss 0

√2rE(ms) 0 mnn. REFERENCES where r is E(mn)/E(ms), and mg, mss and mnn are, respectively, the glueball mass, the ss 1. J. Sexton, A. Vaccarino and D. Weingarten, quarkonium mass and the nn quarkonium mass Phys. Rev. Lett. 75, 4563 (1995). before mixing. 2. H. Chen, J. Sexton, A. Vaccarino and D. The three unmixed mass parameters and Weingarten, Nucl. Phys. B (Proc. Suppl.) 34, E(ms), for which our measured value has a large 357 (1994); fractional error bar, we determine from four ob- 3. G. Bali, et al., Phys. Lett. B 309, 378 (1993). served masses. To leading order in the valence C. Morningstar and M. Peardon, Phys. Rev. approximation, with valence quark-antiquark an- D56, 4043 (1997). nihilation turned off, corresponding isotriplet and 4. A. Vaccarino and D. Weingarten, to appear. isosinglet states composed of u and d will 5. D. Weingarten, Nucl. Phys. B (Proc. Suppl.) be degenerate. For mnn we thus take the observed 53, 232 (1997). isovector value of 1470(25) MeV [7]. The three 6. W. Lee and D. Weingarten, Nucl. Phys. B remaining unknowns we tune to give the mixing (Proc. Suppl.) 53, 236 (1997). Hamiltonian eigenvalues of 1697(4) MeV, 1505(9) 7. C. Amsler, et al., Phys. Lett. B355, 425 MeV and 1404(24) MeV, respectively the (1995). Data Group’s masses for f0(1710) and f0(1500), 8. C. Amsler and F. Close, Phys. Rev. D53, 295 and the weighted average of Refs. [7,13] masses (1996). for f0(1390). 9. W. Lee and D. Weingarten, Nucl. Phys. B We find mg becomes 1622(29) MeV, mss be- (Proc. Suppl.) 63, 198 (1998). comes 1514(11) MeV, and E(ms) becomes 64(13) 10. M. Boglione and M. Pennington, Phys. Rev. MeV, with error bars including the uncertain- Lett. 79, 1998 (1997). ties in the four input physical masses. The un- 11. W. Lee and D. Weingarten, hep-lat/9811024. mixed mg is consistent with the world average va- 12. F. Butler, et al., Nucl. Phys. B 430, 179 lence approximation glueball mass 1632(49) MeV, (1994). E(ms) is consistent with our measured value of 13. SLAC-PUB-5669, 1991; SLAC-PUB-7163; 43(31) MeV, and mss is about 13% above the va- W. Dunwoodie, private communication.