chiral07 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 χ 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 Chiral07, Osaka, Nov. 15, 2007

chiral07 page 2 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? q1q1 q2q2

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)

chiral07 page 4 Why a 0 (980) is not 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. Cannot explain why a 0 (980) and f 0 (980) are narrow while σ(600) and κ(800) are broad. Cannot explain why a 0 (980) and f 0 (980) are narrow while σ(600) and κ(800) are broad. γ γ width of a 0 (980) and f 0 (980) are much smaller than expected of states. γ γ width of a 0 (980) and f 0 (980) are much smaller than expected of states. 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?

chiral07 page 5 Is a 0 (1450) the state? Why is it higher than a 1 (1230) and a 2 (1320)? Why is it higher than a 1 (1230) and a 2 (1320)? Why is it almost degenerate with K 0 *(1430)? Why is it almost degenerate with K 0 *(1430)? Why is it higher than a 0 (980) Why is it higher than a 0 (980) ?

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

Tokyo 2006, page 7 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

Tokyo 2006, page 8 Is a 0 (1450) (0 ++ ) a two quark state? Is a 0 (1450) (0 ++ ) a two quark state? Ground state : π η ghost state. Ground state : π η ghost state. First excited state : a 0 First excited state : a 0 CorrelationfunctionforScalarchannel

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

chiral07 page 11 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

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 ± ± 17MeV Γ σ = 324 ± ± 21 MeV E.M. Aitala et. al. Phys. Rev. Lett. 86, 770, (2001)

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

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

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

chiral07 page 16 E |T| 2 in continuum E W on lattice E L E L ?

chiral07 page 17 K. Rummukainen and S. Gottlieb, NP B450, 397 (1995)

chiral07 page 18 Lüscher formula

chiral07 page 19 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)?

Two Pion Energy Shift

chiral07 page 21 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

chiral07 page 22 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

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

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

chiral07 page 25 Mixing of First order approximation: exact SU(3) x is annihilation diagram

chiral07 page 26 Mixing of with Glueball First order approximation: exact SU(3)

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  (expt) R 2 =0 vs  (expt) LQCD [Lee, Weingarten]  y= 43  31 MeV, y/y s =1.198  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, (2006) hep-ph/

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

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

Tokyo 2006, page 30 Scalar Mesons and Glueball glueball

Tokyo 2006, page 31 Life is full of splendid details besides Fujiyama. Happy Birthday, Hiroshi

Tokyo 2006, page 33Summary 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 (?)

chiral07 page 34 Azimuthal anisotropy in Au + Au collisions with = 200 GeV (STAR collaboration)