1. Sasa PrelovsekScadron70, February 20081 Simulations of light scalar mesons on the lattice and related difficulties Scadron 70, IST Lisbon, Portugal (February 2008) Sasa Prelovsek Sasa Prelovsek (sasa.prelovsek@ijs.si) University of Ljubljana and Jozef Stefan Institute some of the results presented obtained in my collaboration with RBC, MILC and BGR lattice collaborations (C. Dawson, K. Orginos, T. Izubuchi, A. Soni, C. Bernard, C. Detar, Z. Fu, C. Lang, C. Gattringer, L. Glozman, D. Mohler,..)

2. Sasa PrelovsekScadron70, February 20082 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. Sasa PrelovsekScadron70, February 20083 Determination of sigma pole from experiment using Roy equations and theoretical input on pi-pi scattering

4. Sasa PrelovsekScadron70, February 20084 a0: dynamical quenched formulations with various unphysical approx. - tetraquark interpolators One of the main difficulties to determine scalar masses: bound states are accompanied by scattering states in lattice correlators Outline Extracting scalar meson masses from lattice simulations:

5. Sasa PrelovsekScadron70, February 20085 J P =0 + I=1 a0 C(t)= point-point correlator J P =0 + I=1 t=0t m a0 from scalar correlator

6. Sasa PrelovsekScadron70, February 20086 m a0 from scalar correlator J P =0 + I=1 : J P =0 +, I=1 : dominates if

7. Sasa PrelovsekScadron70, February 20087 extracting hadron masses from lattice correlator

8. Sasa PrelovsekScadron70, February 20088 Lattice simulations of a0 with u,d,s dynamical quarks dynamical staggered u,d,s quarksdynamical staggered u,d,s quarks [MILC] u,d,s close to physical masses MILC 2001 [PRD64] similar observation UKCD Irving et al. [Pos LAT05:027] ? Resolving this puzzle: S.P. [PRD 73, 2006]: Prediction for with Staggered ChPT Taste breaking effects of staggered fermions allow at finite lattice spacing

9. Sasa PrelovsekScadron70, February 20089 Simulation of a0 and f0 correlators with dynamical staggered u,d,s quarks Simulation of a0 and f0 correlators with dynamical staggered u,d,s quarks C. Bernard, C. Detar, S.P., Z. Fu [PRD76] Staggered ChPT describes I=0,1 point-point scalar correlators well (with no free parameters) Extracted point-point scalar correlators uncertain due to large unphysical scattering contribution Staggered ChPT

10. Sasa PrelovsekScadron70, February 200810 a0 with u,d dynamical quarks 3 flavor QCD:  KK  ’ 2 flavor QCD  ’ )   ’ 2 flavor QCD with m 0 =inf. : none I=1 t=0t

11. Sasa PrelovsekScadron70, February 200811 a0 with dynamical u,d : a0(980) or a0(1450) ? group m m a0 comment SCALAR Coll, 2003 S.P. et al [RBC] 2004 1.58 +/- 0.34 GeV scattering states accounted for; fit of point-point correlators at low t due to noise; value high possibly due to excited state contribution Hashimoto, Izubuchi [RBC] 2008, not published yet 1.111 +/- 0.081 GeV variational method ; scattering state seems slightly higher UKQDC 2003 1.0 +/- 0.2 GeV variational method UKQDC 2006 1.01 +/- 0.04 GeV deduced from [m(b1)-m(a0)] lat and [m(b1)] exp ; scattering state seems slightly higher BGR 2007 ~ 1 GeV variational method ; scattering state seems slightly higher

12. Sasa PrelovsekScadron70, February 200812 quenched a0: a0(980) or a0(1450) ? group m m a0 Bardeen et al. 2001 1.34 +/- 0.09 GeV BGR 2006, 2007~ 1.4 GeV Mathur, Liu et al., 2006 1.42 +/- 0.13 GeV u d scattering states: S.P. et al [RBC] 2003 Quenched ChPT prediction Bardeen et al. [2001]

13. Sasa PrelovsekScadron70, February 200813 scalar mesons : Important scattering state in all lattice formulations (even quenched) Scattering state subtracted (to my knowledge) only in quenched simulation of K.F.Liu, N.Mathur et al. [hep-ph/0607110], PRD: m=1.41 +/- 0.12 GeV s

14. Sasa PrelovsekScadron70, February 200814 scattering states vs. bound states  prediction for scattering state to point-point a0 correlator from corresponding ChPT:  volume dependence: scattering state: resonant state: small volume dependence  various boundary conditions on quark fields  variational method with a number of interpolators  Quenched: no sea quarks, Bardeen et al [2001]  Staggered: staggered sea and valence quarks, S.P. [2006], PRD  Partially quenched: S.P. et al. [2004], PRD  Mixed: staggered sea and chiral valence qcuarks; S.P. et al. [2006], PRD  Twisted mass: Abdel-Rehim et al., ETM coll.

15. Sasa PrelovsekScadron70, February 200815 I=0 scalar mesons   disconnected quark contributions  mixing with glueballs  sizable dependence on lattice spacingmcomment SCALAR Coll, 2003 large scale simulation with dynamical u,d m s slightly bellow 2m p Wilson fermions Lee, Weingarten 99 f0(1710) 74% glueball mixing of various f0, sigma and glue quenched McNeile, Michael, et al. [UKQCD] 2006 m~ 1 GeV for lightest isosinglet state mixing of various f0, sigma and glue dynamical u,d, quarks

16. Sasa PrelovsekScadron70, February 200816 Tetraquark simulations of light scalars: all quenched  K.-F.Liu, N. Mathur et al. (Kentucky), hep-ph/0607110, PRD, 2006 - - small u,d quark masses (good chiral properties) - quenched - volume dependence of spectral weight to distinguish between resonance (W~L 0 ) and pp scattering (W~1/L 3 ):

17. Sasa PrelovsekScadron70, February 200817 Two Pion Energy Shift

18. Sasa PrelovsekScadron70, February 200818 Tetraquark simulations of light scalars  Alford & Jaffe, 2000 - exotic channel, I=2 of SU(2) flavor - interpreted as repulsive pp scattering relation for scattering at finite volume (Luscher 1986) - non-exotic channel, I=0 of SU(2) - interpreted as repulsive pp scattering + possible tetraquark resonance

19. Sasa PrelovsekScadron70, February 200819 Tetraquark simulations of light scalars  Suganuma et al., 2007 - quenched - hybrid boundary coditions - result: The lowest state found is pp scattering, No indication of tetraquark found for

20. Sasa PrelovsekScadron70, February 200820 Recent reviews: Craig McNeile: Lattice approach to light scalars, 0710.2470 [hep-lat] Hard Hadron spectroscopy, Lattice 2007, 0710.0985 [hep-lat]  determining scalar meson spectra on the lattice is not a simple problem final concusions not ready yet  effect of the scattering states has to be disentangeled !  “vague” conclusions for the time beeing: a0 - a0: simulations with dynamical u,d seem to indicate that the lightest state that couples to is perhaps closer to a0(980) than to a0(1450) kappa - kappa: dynamical simulation which dissentangles p K stattering state missing sigma - sigma: two indications for light sigma, which take into account pp scattering: * K.F. Liu et al., tetraquark, quenched: m sigma ~600 MeV * SCALAR Coll,, dynamical: m sigma ~m rho Summary