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σ (600) and Pattern of Scalar Mesons from Lattice QCD

σ (600) and Pattern of Scalar Mesons from Lattice QCD. a 0 (1450) on the Lattice Tetraquark Mesonium – Sigma (600) on the Lattice Pattern of Scalar Mesons and Glueball. χ QCD Collaboration : A. Alexandru, Y. Chen, S.J. Dong, T. Draper, I. Horvath, B. Joo,

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σ (600) and Pattern of Scalar Mesons from Lattice QCD

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  1. σ(600) and Pattern of Scalar Mesons from Lattice QCD • a0 (1450) 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 Tokyo U., Nov. 17, 2006

  2. Introduction to Lattice Gauge Theory Path Integral Formulation in Discrete Euclidean Space-Time • Path integral of the partition function of continuum QCD in Minkowski space • Imaginary time with Wick rotation and then • Integrating Grassmann numbers and gives Euclidean partition function • Note the Grassmann number integration • The Green’s function

  3. a Lattice QCD Why Lattice? • Regularization • Lattice spacing a • Hard cutoff, p ≤ π/a • Scale introduced (dimensional transmutation) • Renormalization • Perturbative • Non-perturbative Regularization independent Scheme Schroedinger functional Current algebra relations • Numerical Simulation • Quantum field theory classical statistical mechanics • Monte Carlo simulation (importance sampling)

  4. Lattice QCD Correspondence between Euclidean field theory and classical statistical mechanics

  5. Hadron Mass and Decay Constant The two-point Green’s function decays exponentially at large separation of time Mass M= Ep(p=0), decay constant ~ Φ

  6. QCD Vacuum

  7. ρ π Ξ φ Λ σ Σ N Δ ω * N K S11 Creation Operator QCD Vacuum

  8. Θ+ ?? ?? Tetraquark Pentaquark Creation Operator QCD Vacuum

  9. Le Taureau of Pablo Picasso (1945) 5thstage 11th stage Dynamical chiral fermion Quenched approximation with Chiral symmetry, and light quark masses

  10. Masses of N, ρ, and π • 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

  11. η η π π Quenched Artifacts • Chiral log in mπ2 x

  12. Evidence of η’N GHOST State in S11 (1535) Channel η η - - - - W > 0 W<0

  13. 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?

  14. 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)

  15. 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. • Large indicates in f0(980), but cannot be in I=1 a0(980). How to explain the mass degeneracy then?

  16. M. Pennington Charm 2006 Scalar mesons

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

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

  19. 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)?

  20. 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)

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

  22. 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 )

  23. ππfour quark operator (I=0)

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

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

  26. Lüscher formula

  27. Scattering Length and energy shift • ππ energies : • Threshold energy shift on the finite lattice:

  28. 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.

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

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

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

  32. 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)

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

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

  35. SU(3) Breaking and f0(1370), f0(1500), f0 (1710) mixing H.Y. Cheng, C.K. Chua, and K.F. Liu, 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

  36. 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

  37. : 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))

  38. Scalar Mesons and Glueball glueball

  39. Constituent Quark Scaling Anisotropy in Au + Au at = 200 GeV (STAR) Meson n=2 and Baryon n=3 grouping Some deviation due to internal hadron structure

  40. Summary • Plenty of tetraquark mesonium candidates • σ(600) is very likely to be a tetraquark mesonium. • Pattern of light scalar mesons may be repeated in the heavy-light sectors (?)

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

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