1. scadron70 page 1 Pattern of Light Scalar Mesons a 0 (1450) and K 0 *(1430) on the Lattice Tetraquark Mesonium – Sigma (600) on the Lattice Pattern of Scalar Mesons and Glueball QCD Collaboration: 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 SCADRON 70 Workshop on "Scalar Mesons and Related Topics" February 11-16, 2008, at IST in Lisbon, Portugal

2. 0 ¯ ¯ (1) 1 ¯ + (1) 0 ++ (0)0 + ¯ (1) 1 + ¯ (1) π (137) 0 + (1/2) ρ (770) σ (600) f 0 (980) f 0 (1370) f 0 (1500) a 0 (980) a 0 (1450) a 1 (1230) K 0 * (1430) J PG (I)) M (MeV) a 2 (1320) 2 + ¯ (1) f 0 (1710) K 0 * (800)

3. scadron70 page 3 Is a 0 (980) is a state? The corresponding K 0 * would be ~ 1100 MeV which is 300 MeV away from both and. The corresponding K 0 * would be ~ 1100 MeV which is 300 MeV away from both and. Hard to explain why a 0 (980) and f 0 (980) are narrow while σ(600) and κ(800) are broad. Hard to explain why a 0 (980) and f 0 (980) are narrow while σ(600) and κ(800) are broad. Large indicates Large indicates in f 0 (980), but cannot be in I=1 a 0 (980). How to explain the mass degeneracy then? in f 0 (980), but cannot be in I=1 a 0 (980). How to explain the mass degeneracy then?

4. scadron70 page 4 Is a 0 (1450) the state? Why is it higher than a 1 (1230) and Why is it higher than a 1 (1230) and a 2 (1320)? a 2 (1320)? Why is it degenerate with K 0 *(1430)? Why is it degenerate with K 0 *(1430)? Why is it higher than a 0 (980) Why is it higher than a 0 (980) ?

5. σ(600) f 0 (980) f 0 (1370) f 0 (1500) f 0 (1710)? Julian Alps, Slovenia 2007

6. scadron70, page 6 Masses of N, ρ, and π in Quenched Lattice Calculation 16 3 x 28 quenched lattice, Iwasaki action with a = 0.200(3) fm 16 3 x 28 quenched lattice, Iwasaki action with a = 0.200(3) fm Overlap fermion Overlap fermion Critical slowing down is gentle Critical slowing down is gentle Smallest m π ~ 180 MeV Smallest m π ~ 180 MeV m π L > 3 m π L > 3

7. Our results shows scalar mass around 1400-1500 MeV, suggesting Our results shows scalar mass around 1400-1500 MeV, suggesting a 0 (1450) is a two quark state. a 0 (1450) is a two quark state. msmsmsms

8. scadron70 page 8 What is the nature of σ (600)? σ (500): Johnson and Teller Two-pion exchange potential: Chembto, Durso, Riska; Stony Brook, Paris, … σ enhancement of Δ I = ½ rule

9. The σ in D + → π ¯ π + π + The σ in D + → π ¯ π + π + σ Without a σ pole With a σ pole M σ = 478 ± 24 23 ± 17MeV Γ σ = 324 ± 42 40 ± 21 MeV E.M. Aitala et. al. Phys. Rev. Lett. 86, 770, (2001)

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

11. 0 0.2 0.4 -0.4 -0.2 0 0.20.40.60.81.0 Re s (GeV ) 2 Im s (GeV ) 2  : I = 0, J = 0 complex s-plane  E791 BES CERN-Munich ZQZXZW Zhou, Qin, Zhang, Xiao, Zheng & Wu CCL Caprini, Colangelo, & Leutwyler M. Pennington Charm 2006

12. ππ four quark operator (I=0) ππ four quark operator (I=0)

13. chiral07 page 13 E |T| 2 in continuum E W on lattice E L E L ?

14. chiral07 page 14 K. Rummukainen and S. Gottlieb, NP B450, 397 (1995)

15. chiral07 page 15 Lüscher formula

16. Tokyo 2005, page 16

17. scadron70 page 17 Further study is needed to check the volume dependence of the observed states. Scattering states Scattering states (Negative scattering length) length) Scattering states Scattering states Possible BOUND state σ(600)? σ(600)?

18. Two Pion Energy Shift

19. scadron70 page 19 Scattering state and its volume dependence 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

20. scadron70 page 20 Scattering state and its volume dependence 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

21. Volume dependence of spectral weights Volume independence suggests the observed state is an one particle state Volume independence suggests the observed state is an one particle state W0W0W0W0 W1W1W1W1

22. 0 ¯ ¯ (1) 1 ¯ + (1) 0 ++ (0)0 + ¯ (1) 1 + ¯ (1) π (137) 0 + (1/2) ρ (770) σ (600) f 0 (980) f 0 (1370) f 0 (1500) a 0 (980) a 0 (1450) a 1 (1230) K 0 * (1430) J PG (I)) M (MeV) a 2 (1320) 2 + ¯ (1) f 0 (1710) K 0 * (800) Kπ Mesonium ππ Mesonium

23. scadron70 page 23 Mixing of First order approximation: exact SU(3) x is annihilation diagram

24. scadron70 page 24 Mixing of with Glueball First order approximation: exact SU(3)

25. SU(3) Breaking and f 0 (1370), f 0 (1500), f 0 (1710) mixing For SU(3) octet f 0 (1500),  = -2  R 1 =0.21 vs. 0.246  0.026 (expt) R 2 =0 vs. 0.145  0.027 (expt) LQCD [Lee, Weingarten]  y= 43  31 MeV, y/y s =1.198  0.072 y and x are of the same order of magnitude ! Need SU(3) breaking in mass matrix to lift degeneracy of a 0 (1450) and f 0 (1500) Need SU(3) breaking in decay amplitudes to accommodate observed strong decays SU(3) breaking effect is weak and can be treated perturbatively H.Y. Cheng, C.K. Chua, and K.F. Liu, PR D74, 094005 (2006) hep-ph/0607206

26. Consider two different cases of chiral suppression in G→PP: (i) (ii) In absence of chiral suppression (i.e. g  =g KK =g  ), the predicted f 0 (1710) width is too small (< 1 MeV)  importance of chiral suppression in G  PP decay

27. M S -M U  25 MeV is consistent with LQCD result  near degeneracy of a 0 (1450), K 0 * (1430), f 0 (1500)  (J/  f 0 (1710)) = 4.1  ( J/   f 0 (1710)) versus 6.6  2.7(expt) no large doubly OZI is needed  (J/   f 0 (1710)) >>  (J/  f 0 (1500)) : primarily a glueball : tend to be an SU(3) octet : SU(3) singlet + glueball content (  13%) M U =1474 MeV, M S =1498 MeV, M G =1666 MeV

28. scadron70, page 28 Scalar Mesons and Glueball glueball arXiv:0706.1262

29. scadron70, page 29Summary Plenty of tetraquark mesonium candidates Plenty of tetraquark mesonium candidates σ(600) is very likely to be a tetraquark mesonium. σ(600) is very likely to be a tetraquark mesonium. f 0 (1710) could be a fairly pure glueball. f 0 (1710) could be a fairly pure glueball. Pattern of light scalar mesons may be repeated in the heavy-light sectors (?) Pattern of light scalar mesons may be repeated in the heavy-light sectors (?)

30. scadron70 page 30 Azimuthal anisotropy in Au + Au collisions with = 200 GeV (STAR collaboration)