1. Mass spectra of the low-lying nonet scalar mesons in the lattice QCD Motoo Sekiguchi Kokushikan University Scalar collaboration; T. Kunihiro, S. Muroya, A. Nakamura, C. Nonaka, H. Wada, M. S.

2. Plan of the talk Objective and Motivation The sigma meson in lattice simulation with dynamical quarks The kappa meson in quenched lattice simulation Current status of our new simulation for the scalar mesons Summary

3. Objective of Scalar Collaboration Confidence level of the sigma meson and kappa meson has been increasing, and its physical significance in hadron physics and QCD is apparent. Using Lattice QCD, we have been addressing the following Question about the scalar mesons: the sigma and the low-lying scalar mesons are resonances in QCD or something else?

4. The nature of the low-lying nonet scalar mesons is not understood yet. Experimentally well established scalar resonances below 1 GeV are a 0 (980) and f 0 (980).

5. The significance of sigma meson (I=0, J PC =0 ++ ) in low energy hadorn physics There is a strong experimental evidence for a light sigma meson, whose pole was extracted with a small uncertainty from modern analyses in pi-pi scattering (Igi, Hikasa PR D (1999), I. Caprini, G. Colangero and H. Leutwyler PRL (2006)). The significant contributions of the sigma pole were identified in the D meson decay; D + →π － π ＋ π ＋ Fermilab E791; E.M. Aitala et al, Phys Rev. Lett. (86), 770 (2001). Responsible for the intermediate range attraction in the nuclear force. Accounts for ΔI=1/2 enhancement in K0 →2π compared with K+→π+π-. T. Morozumi, C.S. Lim and I. Sanda, PRL (1990). Particle Data Group, Physics Letters B667, 1 (2008) Mass_sigma=(400-1200) MeV, Full width=(600-1000) MeV.

6. Issues with the low-mass Sigma meson in QCD In the quark model; J PC =0 ++ mesons→ 3 P 0 ⇒ the mass in the 1.2 -1.6 GeV region. Some mechanism need to down the mass with ~ 800 MeV Color magnetic interaction between the di-quarks (Jaffe1977) with the bag-model wave functions. ⇒ All the low-lying scalars are tetraquarks! The sigma is a superposition of qq-bar states. The sigma is indentified as the chiral partner of the π meson in Dynamical Chiral symmery Breaking in QCD. The π-π molecule as suggested in π-π scattering. A mixed state with scalar guleball state.

7. The Kappa meson scalar meson with the Strangeness (I=1/2) Recent experimental candidates: –Fermilab E791: hep- ex/0204018, (PRL89(2002)12801). – BES:hep-ex/0304001. Both observed a candidate near 800MeV Even 660MeV! Eur. Phys. J C48(2006)543 (hep- ph/0607133) 0 +

8. The sigma in lattice simulations with dynamical quarks A first work on the sigma in lattice QCD with dynamical quarks. (Phys. Rev. D70, 034504(2004).) The full QCD simulation is necessary to properly describe the sigma with possible contents, ie., the qq-bar, the glueball, tetra quarks and so on.

9. Previous Lattice QCD simulations of the sigma mesons W. Lee and D. Weingarten, Phys.ReV.D61(1999)012015 Quenched simulation Mixing matrix between the Guleball and qq-bar Mass above 1 GeV Alford and Jaffe Nucl. Phys.B578(2000)367. Quenched simulation Tetraquarks type interpolating operator Disconnected diagrams are omitted.

10. I=0, scalar interpolating operator for sigma There is experimental evidence that the sigma consist of only uu-bar and dd-bar compnets. c = 1,2,3 ･･･ color α=1,2,3,4 ･･･ Dirac spin

11. The bound state properties are obtained by calculating expectation values, where O i is the interpolating operator. The path integral is regulated by the introduction of a space-time lattice. The integral is computed in Euclidean space using Monte Carlo techniques on the computer. ground state: m 0 excited state: m 1 The results is a table of numbers. We average and fit exponentials to get masses.

12. Propagator Connected diagram Disconnected diagram - Vacuum contribution Inverse of Fermion Matrix, i.e., Quark Propagators

13. Details of our Calculation (1) Full QCD Update by Hybrid Monte Carlo (SX5 at RCNP) Wilson Fermions (2 flavors) Plaquette Gauge Action Phys.Rev. D70 (2004) 034504 (hep-ph/0310312) Disconnected Part by Z 2 Noise Method (SR8000 at KEK)

14. Details of our Calculation (2) - Simulation parameters Lattice size : ８ ３ × １６ β = 4.8 κ = 0.1846, 0.1874, 0.1891 well established by CP-PACS, a = 0.197(2) fm, κ c = 0.19286(14), Lattice size: ( CP - PACS, Phys. Rev. D60(1999)114508 ) Number of the Z2 noise = 1000 Wilson Fermions & Plaquette gauge action Very small ! Very strong coupling ! Very large !

15. Details of our Calculation (3) Separation between configurations are 10 trajectories [κ = 0.1846] 1110 configurations [κ = 0.1874] 860 configurations [κ = 0.1891] 730 configurations

16. Details of our Calculation (4) κm_π/m_ρ (Our Results) m_π/m_ρ (CP-PACS) 0.18460.825±0.0010.8291±0.0012 0.18740.760±0.0020.7715±0.0017 0.18910.692±0.0050.7026±0.0032 Using the same values of the hoping parameters except for the lattice size. Our results and CP-PACS are nearly equal. The small errors indicate the high precision of our simulation

17. m_π^2, m_ρ and m_sigma as a function of the inverse hopping parameter - Chiral Extrapolation - 5.1410±0.0747 κc = 0.1945±0.0029 ( CP-PACS κ c = 0.19286(14) ) 0.8093 a = 1.05×10 -3 (MeV) -1 = 0.207fm (1=197MeV fm) CP-PACS a = 0.197(2) fm 0.270 m σ =257MeV

18.  We conclude the sigma shows a pole behavior and Here the disconnected diagram plays essential role.

19. Other Lattice QCD simulations of the sigma mesons The Kentucky lattice group (hep-ph/0607110) claimed to get a result for mass of Sigma from quenched lattice QCD with pion masses as low as 180 MeV. Using tetraquarks type operator. However, the Kentucky lattice group reported the full QCD results obscured the tetraquark (arXiv/0810.5512, Lattice2008). UK-QCD (Phys. Rev. D74 (2006)114504.) claimed to get a result for Sigma using the glueball and qq-bar interpolating operators in the dynamical quark simulation. The mass of the glueball is below 1 GeV.

20. The Kappa meson in quenched lattice simulation We perform quenched simulations on kappa meson so as to clarify the structure of the scalar meson rather than to reproduce the experimental value of the mass; a quenched-level simulation should give a rather clear perspective on whether the system can fit with the simple quark model picture or not.

21. Kappa propagator

22. Simulation parameter Quech approximation Lattice size =20X20X20X24 Wilson Fermions Plaquette Gauge Action Laticce spacing a=0.1038 fm, β=5.9 Hopping parameters; h_u,d=0.1589, 0.1583 and 0.1574 h_s=0.1557 and 0_1566 We heve checked that the mass of the π, ρ, K and K* mesons obtained in our simulation are good agreement with those on a large lattice (CP-PACS, 32^3X56).

23. The mass ratios m_K /m_K* and m_kappa/m_K* at chiral limit, and m_φ/m_K* for s quark hopping parameters h_s=0.1566 and 0.1557.

24. m_kappa ~ 1.7 GeV (K*0(1430)?) Large than twice the experimental mass. Our quenched lattice calculation suggests the kappa can not a normal qq-bar state.

25. Current Status of our new full QCD simulation for Scalar mesons We use gauge configurations from International Lattice Data Grid (ILDG). We employ the all-to-all propagator method with the dilution techniques (Trinlat Collab. Compt. Phys. Commun. 172 (2005) 145.). Smearing (Jacobi, Gaussian, Derivative quark) source and sink for errors reduction (Graz group, PR D78 (2008) 034501).

26. Variational Method It is to use several different interpolating operator O i.

27. Setting of Calculation Old Interpolating operator O i New Interpolating operator O i (Additional) Variational method with multiple interpolating operators an explicit inclusion of tetraquark operator

29. We are preparing for the simulation of sigma and the other scalar mesons. We will start the simulation for the kappa meson at October.

30. Summary The sigma meson and other low-lying scalar mesons are still a source of debates. A full QCD lattice simulation suggests the existence of a low-lying sigma as a pole in QCD; the physical content is obscure: the disconnected diagram gives the dominate contribution. A quenched lattice calculation suggests that the kappa can not be a normal qq-bar state. We will present the new results of sigma and kappa meson in the near future.

35. Propagator

36. The flavored scalar mesons are not light as obseved m_kappa ~ 1.8 GeV (Exp. 0.8 GeV) m_a0 ~ 1.9 GeV > (Exp. 0.98GeV)

37. Details of our Calculation Quench Full QCD Cold Start κ=0.1846 1500 trajectory 500 trajectories On every 10 trajectry, we calculate propagators. κ=0.1891 κ=0.1874 ca. 10,000 trajectories

38. Simulation parameter Quech approximation β=5.9 Lattice size =20X20X20X24

39. Alford-Jaffe, Nucl. Phys. B578, 367 (2000), (hep-lat/0001023,hep-lat/0306037) Quench Calculation They consider these Diagrams Deviation from Lueshcer Scattering formula. Bound state ? h１h１ h2

40. π, ρ, σmesons (κ=0.1891 )

41. σmeson propagators Connected and Disconnected Parts ( κ=0.1891 )

43. Summary kappa (m*a = 0.8843)1.55 [GeV] axial victor(m*a =0.9448) 1.65 [GeV]

44. Setting of Calculation Configurations: We use gauge configurations from CP-PACS (Lattice QCD Archive). β Lattice Size kappaconfiguration s 1.8012 3 X 24 0.1409 595 1.8012 3 X 24 0.1430 472 1.8012 3 X 24 0.1445 322 1.8012 3 X 24 0.1464 148 1.9516 3 X 32 0.1375 599 1.9516 3 X 32 0.1390 682 1.9516 3 X 32 0.1400 294 1.9516 3 X 32 0.1410 223

45. Setting of Calculation Moreover, we use also 2+1 flavor full QCD configurations by CP-PACS+JLQCD

46. Project Start We will start new project for the scalar nonet mesons at October. Now we write simulation program for SX-9 at RCNP Osaka University. To do list Program improvred. Speed up (time over) MPI