1. Charm 2006, page 1 Pentaquark Baryons and Tetraquark Mesoniums from Lattice QCD Hadron Calculation with overlap fermion Pentaquark Baryons and Tetraquark Mesoniums on the Lattice χ QCD Collaboration: A. Alexandru, Y. Chen, S.J. Dong, T. Draper, I. Horvath, B. Joo, F.X. Lee, K.F. Liu, N. Mathur, S. Tamhankar, H.Thacker, J.B. Zhang Charm 2006, June 6, 2006

2. ITP, 2005, page 2 Hadron Mass and Decay Constant The two-point Green’s function decays exponentially at large separation of time Mass M= E p (p=0), decay constant ~ Φ

3. Charm 2006, page 3 QCD Vacuum QCD Vacuum

4. π ρ σ N Δ φ Σ Λ Ξ ω K S 11 N * CreationOperator QCD Vacuum

5. CreationOperator Θ+Θ+Θ+Θ+ ?? Pentaquark Tetraquark ??

6. Charm 2006, page 6 Le Taureau of Pablo Picasso (1945) Dynamical chiral fermion Quenched approximation with chiral symmetry, and light quark masses 5 th stage11 th stage

7. Charm 2006, page 7 Masses of N, ρ, and π 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

8. Charm 2006, page 8 Evidence of η’N GHOST State in S 11 (1535) Channel Evidence of η’N GHOST State in S 11 (1535) Channel - - - - ηη W > 0 W<0

9. Charm 2006, page 9 Possible Pentaquark candidate Possible Pentaquark candidate Possible candidate Signal reported to be “observed” recently θ + (1540) u d u d

10. u d u d u d u d NK NK u u d d u u d d ٭ Not sufficient to study the masses/energies of the states, particularly in the 1/2 - channel. ٭ Not sufficient to study the masses/energies of the states, particularly in the 1/2 - channel. ٭ Volume study of spectral weight W is much more sensitive. ٭ Volume study of spectral weight W is much more sensitive. m K + m N m K + m N ~ 1432 MeV ~ 1432 MeV m(Θ + ) ~1540MeV 1) Need to discern the nature of the states

11. Lattice Pentaquark Volume Variational Light enough Ghost Group signal claim Test Calculation Quark Mass State Lattice Pentaquark Volume Variational Light enough Ghost Group signal claim Test Calculation Quark Mass State Csikor Yes No Yes (2) No No Csikor Yes No Yes (2) No No et.al No No Yes (9/6) No No et.al No No Yes (9/6) No No Sasaki Yes No No No No Sasaki Yes No No No No Chiu et.al Yes No Yes No No Chiu et.al Yes No Yes No No Mathur et.al No Yes No Yes Yes Mathur et.al No Yes No Yes Yes Ishii et.al No Yes No No No Ishii et.al No Yes No No No Alexandrou Yes Yes No No No Alexandrou Yes Yes No No No Takahashi Yes Yes Yes(2) No No Takahashi Yes Yes Yes(2) No No Negele et.al ? Yes Yes(19) Negele et.al ? Yes Yes(19) Lasscock et. al No No No No No Lasscock et. al No No No No No Holland et. al No No Yes No No Holland et. al No No Yes No No Lattice Results on Θ + Pentaquark

12. Charm 2006, page 12 Status of Lattice Pentaquark Calculation KFL and N. Mathur (hep-lat/0510036) No convincing evidence for the claims of observing pentaquarks on the lattice. No convincing evidence for the claims of observing pentaquarks on the lattice. Absence of evidence is not evidence of absence. Absence of evidence is not evidence of absence. Definite calculation should involve dynamic chiral fermion with light quarks, variational approach, volume test, and removal of ghosts. Definite calculation should involve dynamic chiral fermion with light quarks, variational approach, volume test, and removal of ghosts.

13. Charm 2006, page 13 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

14. 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)

15. Charm 2006, page 15 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

16. 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

17. Charm 2006, page 17 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

18. 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)

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

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

21. Charm 2006, page 21 Scattering Length and energy shift Scattering Length and energy shift ππ energies : ππ energies : Threshold energy shift on the finite lattice :

22. Charm 2006, page 22 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)?

23. Charm 2006, page 23 Scattering state and its volume dependence Scattering state and its volume dependence Normalization condition requires : Two point function : Lattice Continuum For one particle bound state spectral weight (W) will NOT be explicitly dependent on lattice volume

24. Charm 2006, page 24 Scattering state and its volume dependence Scattering state and its volume dependence Normalization condition requires : Two point function : Lattice Continuum For two particle scattering state spectral weight (W) WILL be explicitly dependent on lattice volume

25. 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

26. 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

27. Charm 2006, page 27 Mixing of First order approximation: exact SU(3) x is annihilation diagram

28. Charm 2006, page 28 Mixing of with Glueball First order approximation: exact SU(3)

29. Charm 2006, page 29 Scalar Mesons and Glueball glueball

30. Charm 2006, page 30 Summary No credible evidence of pentaquarks in lattice calculations No credible evidence of pentaquarks in lattice calculations 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. 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 (?)

31. Charm 2006, page 31 Azimuthal anisotropy in Au + Au collisions with = 200 GeV (STAR collaboration)