1. Pattern of Light Scalar Mesons • a0 (1450) and K0*(1430) on the Lattice • Tetraquark Mesonium – Sigma (600) on the Lattice • Pattern of Scalar Mesons and Glueball χQCDCollaboration: A. Alexandru, Y. Chen, S.J. Dong, T. Draper, I. Horvath, B. Joo, F .X. Lee, K.F. Liu, N. Mathur, T. Streuer, S. Tamhankar, H.Thacker, J.B. Zhang Chiral07, Osaka, Nov. 15, 2007

2. q1 q2 Tetraquark Mesoniums QCD allows a state with more than three quarks Four quarks : Two quarks + two anti-quarks Like molecular state? Like di-quark anti-diquark state?

3. f0(1710) f0(1500) a0(1450) K0*(1430) f0(1370) a2(1320) a1(1230) a0(980) f0(980) M (MeV) ρ(770) K0*(800) σ(600) π(137) JPG(I)) 1+ ¯(1) 0¯ ¯(1) 2+ ¯(1) 0+¯(1) 0++(0) 0+(1/2) 1¯+(1)

4. Why a0(980) is not a state? • The corresponding K0* would be ~ 1100 MeV which is 300 MeV away from both and . • Cannot explain why a0(980) and f0(980) are narrow while σ(600) and κ(800) are broad. • γ γ width of a0(980) and f0(980) are much smaller than expected of states. • Large indicates in f0(980), but cannot be in I=1 a0(980). How to explain the mass degeneracy then?

5. Isa0(1450) the state? • Why is it higher than a1 (1230) and a2(1320)? • Why is it almost degenerate with K0*(1430)? • Why is it higher than a0(980) ?

6. f0(1710)? f0(1370) σ(600) f0(980) f0(1500) Julian Alps, Slovenia 2007

7. Masses of N, ρ, and π inQuenched Lattice Calculation • 163 x 28 quenched lattice, Iwasaki action with a = 0.200(3) fm • Overlap fermion • Critical slowing down is gentle • Smallest mπ~ 180 MeV • mπL > 3

8. Is a0 (1450) (0++) a two quark state? Correlation function for Scalar channel Ground state : πηghost state. First excited state : a0

10. ms Our results shows scalar mass around 1400-1500 MeV, suggesting a0(1450)is a two quark state.

11. Two-pion exchange potential: Chembto, Durso, Riska; Stony Brook, Paris, … σ (500): Johnson and Teller σ enhancement of Δ I = ½ rule What is the nature of σ (600)?

12. Without a σ pole σ With a σ pole The σ in D+→π¯π+π+ Mσ= 478 ± 2423± 17MeV Γσ= 324 ± 4240 ± 21 MeV E.M. Aitala et. al. Phys. Rev. Lett. 86, 770, (2001)

13. J/ψ—> ωπ+π- M. Ablikim et al. (BES), Phys. Lett. B598, 149 (2004) Mσ= 541 ± 39 MeV, Γσ= 504 ± 84 MeV

14. ZQZXZW ZQZXZW CCL Caprini, Colangelo, & Leutwyler Zhou, Qin, Zhang, Xiao, Zheng & Wu M. Pennington Charm 2006  : I = 0, J = 0 0.4 complex s-plane 0.2 CERN-Munich 2 0 Im s (GeV )  E791 -0.2 BES -0.4 0 0.2 0.4 0.6 0.8 1.0 -0.2 2 Re s (GeV )

15. ππfour quark operator (I=0)

16. |T|2 in continuum W on lattice E E ? L L E E

17. K. Rummukainen andS. Gottlieb, NP B450, 397 (1995)

18. Lüscher formula

19. Scattering states Possible BOUND state σ(600)? Scattering states (Negative scattering length) Further study is needed to check the volume dependence of the observed states.

20. Two Pion Energy Shift

21. Scattering state and its volume dependence Normalization condition requires : Two point function : Lattice For one particle bound state spectral weight (W) will NOT be explicitly dependent on lattice volume

22. Scattering state and its volume dependence Normalization condition requires : Two point function : Lattice For two particle scattering state spectral weight (W) WILL be explicitly dependent on lattice volume

23. Volume dependence of spectral weights W0 W1 Volume independence suggests the observed state is an one particle state

24. f0(1710) f0(1500) a0(1450) K0*(1430) f0(1370) a2(1320) a1(1230) a0(980) f0(980) M (MeV) ρ(770) K0*(800) σ(600) Kπ Mesonium ππMesonium π(137) JPG(I)) 1+ ¯(1) 0¯ ¯(1) 2+ ¯(1) 0+¯(1) 0++(0) 0+(1/2) 1¯+(1)

25. Mixing of First order approximation: exact SU(3) x is annihilation diagram

26. Mixing of with Glueball First order approximation: exact SU(3)

27. SU(3) Breaking and f0(1370), f0(1500), f0 (1710) mixing H.Y. Cheng, C.K. Chua, and K.F. Liu, PR D74, 094005 (2006) hep-ph/0607206 • Need SU(3) breaking in mass matrix to lift degeneracy of a0(1450) and f0(1500) • Need SU(3) breaking in decay amplitudes to accommodate observed strong decays For SU(3) octet f0(1500),  = -2  R1=0.21 vs. 0.2460.026 (expt) R2=0 vs. 0.1450.027 (expt) LQCD [Lee, Weingarten]  y= 4331 MeV, y/ys=1.1980.072 y and x are of the same order of magnitude ! SU(3) breaking effect is weak and can be treated perturbatively

28. Consider two different cases of chiral suppression in G→PP: (i) (ii) In absence of chiral suppression (i.e. g=gKK=g), the predicted f0(1710) width is too small (< 1 MeV)  importance of chiral suppression in GPP decay

29. : primarily a glueball : tend to be an SU(3) octet : SU(3) singlet + glueball content ( 13%) MU=1474 MeV, MS=1498 MeV, MG=1666 MeV • MS-MU 25 MeV is consistent with LQCD result  near degeneracy of a0(1450), K0*(1430), f0(1500) • (J/f0(1710)) = 4.1 ( J/ f0(1710)) versus 6.62.7(expt) no large doubly OZI is needed • (J/ f0(1710)) >> (J/f0(1500))

30. Scalar Mesons and Glueball glueball

31. Life is full of splendid details besides Fujiyama. Happy Birthday, Hiroshi

33. Summary • Plenty of tetraquark mesonium candidates • σ(600) is very likely to be a tetraquark mesonium. • f0(1710) could be a fairly pure glueball. • Pattern of light scalar mesons may be repeated in the heavy-light sectors (?)

34. Azimuthal anisotropy in Au + Au collisions with = 200 GeV (STAR collaboration)