1. Anisotropic lattice QCD studies of penta-quarks and tetra-quarks N. Ishii (Univ. of Tokyo) in collaboration with T. Doi (Riken BNL) H. Iida (TITECH) Y. Nemoto (Nagoya Univ.) M. Oka (TITECH) F. Okiharu (Nihon Univ.) H. Suganuma (Kyoto Univ.) K. Tsumura (Kyoto Univ.) Plan of the talk: 1 Introduction 2 General Formalisms 3 Numerical Results 4 Summary/Discussion (5 Tetra-quarks(4Q)) See Phys.Rev.D71,034001(2005); D72,074503(2005) for detail. START

2. 1.Introduction One of the most important issues for Θ + (1540) is to understand its extremely narrow decay width Γ<1 MeV. Several ideas have been proposed as a.I=2 assignment b.Jaffe-Wilczek’s diquark picture ⇒ J P =1/2(+) and 3/2(+) c.πKN hepta-quark picture ⇒ J P =1/2(+) d.The string picture e.J P =3/2(-) assignment ⇒ J P =3/2(-) In this talk, we are mainly interested in J P =3/2(±) possibilities: 1.We first present our numerical results on J P =1/2(±) penta-quarks briefly emplyoing a diquak-type interpolating field using a flavor dependent boundary condition(HBC) 2.We then present our numerical results on J P =3/2(±) penta-quarks employing three Rarita-Schwinger interpolating fields using 1000 gauge field configurations for high statistics

3. 1.Spin of Θ + has not yet been determined experimentally. 2.J P =3/2( - ) assignment can solve the puzzle of the narrow decay width. (proposed by A.Hosaka et al., PRD71,074021(2005).) Advantage: (a) It allows the configuration of (0s) 5. (b) It only decays into a d-wave KN state. Suppressed overlap between these two states The decay width is expected to be significantly narrow. Disadvantage: (a) The constatituent quark picture suggests such a 5Q state is quite heavy. (← due to the color-magnetic interaction) Since it is not apriori clear whether such a conventional framework can be applied to a new exotic 5Q system as Θ + (1540) or not, it is desirable to perform a direct lattice QCD calculation. 3.J P =3/2(+) is also interesting, which is suggested by the diquark picture. J P =3/2(±) possibilities HOWEVER S-diq The total spin(parity) is 1/2(+) or 3/2(+). ⇒ 3/2(+) penta-quark may have a narrow decay width ! p-wave

4. Lattice QCD studies of the penta quarks ★ There are several lattice QCD calculations of penta-quarks available. (published one only) SPIN 1/2 SPIN 3/2 Most of them are devoted to spin 1/2 states except for the recent two.

5. Lattice QCD Setup : 1.Gauge Config by standard Wilson gauge action: a.Lattice size : 12 3 ×96 [(2.2fm) 3 ×4.4fm in physical unit] b.β= 5.75 c.Lattice spacing: from Sommer parameter r 0. d.Anisotropic lattice Renormalized anisotropy: a s /a t =4 for accurate measurements of correlators and masses e.#(gauge config) = 504 for J P =1/2(±) = 1000 for J P =3/2(±) 2.O(a) improved Wilson quark (clover) action. The quark mass covers the region m s < m q < 2 m s 3.Smeared source to reduce higher spectral contributions 2.General Formalism 0.12400.12300.12200.1210 656(2)784(1)893(1)1005(1) 1011(5)1085(4)1162(3)1240(3) 2.2 fm Finer lattice spacing along the temporal direction time

6. The interpolating fields NK * -type color-twisted NK * -type diquark-type ★ Three Rarita-Schwinger interplating fields for J P =3/2(±) states: ★ A diquark-type interplating fields for J P =1/2(±) states: We consider the following iso-scalar interpolating fields: (scalar) (pseudo scalar) (scalar) (vector)

7. Hybrid Boundary Condition(HBC) We utilize a flavor dependent spatial BC (Hybrid BC (HBC)). (We use HBC in addition to the standard periodic BC(PBC)) quark contentsspatial BCminimum momentum Nanti-periodic BC K,K*anti-periodic BC periodic BC Hybrid Boundary Condition(HBC) L L L The spatial BOX Spatial momentum is quantized due to finite volume effect: 1. periodic BC: 2. anti-periodic BC: u quarkspatially anti-periodic BC d quarkspatially anti-periodic BC s quarkspatially periodic BC Cosequence on hadrons ◎ NK and NK* threshold energies(s-wave) are raised due to, ◎ Θ +,if it is a compact resonance, will not be affected so much. HBC can be used to determine whether a state is a compact resonance or not. ※ In the case of p/d wave, HBC serves as another boundary condition(other than PBC). With HBC

8. 3.Numerical Results: J P =1/2(±) states (effective mass plots) “Effective mass” is defined as which can be considered as an “weighted average” of masses at each time-slice t. plateau J P =1/2(-) plateau J P =1/2(+) NK-threshold (s-wave) NK-threshold (p-wave) 1.J P =1/2(-) state: A state appears slightly above the NK threshold (m N +m K ). 2.J P =1/2(+) state: A state appears above the raised NK threshold (due to the finite box). ⇒ rather massive ! Excited state contributions are reducing A single state dominate the correlator G(t) in this region.

9. A plateau in the effective mass plot indicates G(t) is saturated by a single-state contribution. “Effective mass plot” The correlator can be expressed as a sum: The Effective Mass is defined as: negligible ! If G(t) is dominated by a single state: Then we have, This can be considered as “average” of masses at each time-slice t (Constant effective mass)

10. Chiral extrapolation (J P =1/2(±)) At physical point (1) J P =1/2(+): 2.24(11) GeV (2) J P =1/2(-): 1.75(3) GeV NK threshold (p-wave) NK threshold (s-wave) 1.Our data does not support a low-lying J P =1/2(+) penta-quark. 2.For J P =1/2(-) state, the mass(1.75 GeV) is OK ! Still, it is necessary to check whether it is not an NK scattering state but a compact resonance. ⇒ HBC analysis

11. HBC analysis (J P =1/2(-) state) PBCHBC NK-threshold (PBC)NK-threshold (HBC) 1.NK(s-wave) threshold is raised up by 210 MeV. 2.The best fit mass m 5Q is raised up by a similar amount. ★ No compact 5Q resonance exists in the region: ★ The state observed in J P =1/2(-) is an NK scattering state.

12. Combining the results from all the other hopping parameters. data points The best fit value over the plateau. solid lines NK(s-wave) threshold The states observed in are NK scattering states !

13. Numerical Results: J P =3/2(-) state (effective mass plot) This correlator is too noisy ! Fit is not performed. The plateaus appear above the NK*-threshold and above the raised NK threshold. plateau twisted × “Effective mass” is defined as which can be considered as an “weighted average” of masses at each time-slice t.

14. NK*(JP=3/2( - )) 1.NK*(s-wave) threshold is raised up by 179 MeV. 2.NK(d-wave) threshold is lowered down by 66 MeV. 3.Best-fit(m 5Q ) is raised up by 80 MeV. Its value is almost consistent with NK*-threshold(s-wave). This state is an NK* scattering state. A large number of config. Nconf=1000 has played a crucial role.

15. color-twisted NK*(JP=3/2( - )) The situation is similar to the NK*-correlator. NK*(s-wave) threshold is raised up by 179 MeV. NK(d-wave) threshold is lowered down by 66 MeV. Best-fit(m 5Q ) is raised up by 90 MeV. Its value is almost consistent with the NK*-threshold. This state is also an NK* scattering state. twisted

16. Chiral extrapolation (J P =3/2( - )) ○ (circle) from NK*-type correlator □ (box) from color-twisted NK*-type correlator Physical quark mass region In the physical quark mass region (1)NK*-type: m 5Q = 2.17(4) GeV (2)Color-twisted NK*-type: m 5Q = 2.11(4) GeV No evidence for a low-lying 5Q state HBC analysis suggests these states are NK*(s- wave) scattering states Due to the limited time, we cannot show HBC analysis.

17. J P =3/2(+) state (effective mass plot) The plateaus appear above the raised NK*-threshold and above the raised NK threshold. plateau twisted plateau

18. NK*(J P =3/2(+)) Changes in the two-particle spectrum are too small in J P =3/2(+) channel. N*K*(s-wave) threshold is raised up by 170 MeV. NK*(p-wave) threshold is lowered down by 57 MeV. NK(p-wave) threshold is lowered down by 66 MeV. Best-fit(m 5Q ) is raised up by 60 MeV. Its value coincides with N*K*(s-wave) threshold. This state is a N*K*(s-wave) threshold.

19. Color-twisted NK*(J P =3/2(+)) Changes in two-particle spectrum are too small in J P =3/2(+) channel. NK*(p-wave) threshold is lowered down by 57 MeV. NK(p-wave) threshold is lowered down by 66 MeV. Best-fit(m 5Q ) is raised up by 90 MeV. This state is an NK* scattering state. twisted

20. Diquark-type(J P =3/2(+)) Changes in the two-particle spectrum are too small in J P =3/2(+) channel. NK*(p-wave) threshold is lowered down by 57 MeV. NK(p-wave) threshold is lowered down by 66 MeV. Best-fit(m 5Q ) is raised up by 80 MeV. This state is an NK*-scattering state.

21. Chiral extrapolation (J P =3/2(+)) ○ (circle) from NK*-type correlator □ (box) from color-twisted NK*-type correlator △ (triangle) from diquark-type correlator In the physical quark mass region, (1)NK*-type: m 5Q = 2.64(7) GeV (2)Color-twisted NK*-type: m 5Q = 2.48(10) GeV (3)Diquark-type: m 5Q =2.42(6) GeV No evidence for a low-lying 5Q states. Physical quark mass region NK*(p-wave) scattering states N*K*(s-wave) scattering state HBC analysis suggests: Due to the limited time, we cannot show HBC analysis.

22. 1.We have studied spin=1/2 and 3/2 penta-quarks by using the anisotropic lattice QCD. For acuracy, (a) renormalized anisotropy a s /a t = 4 (b) O(a) improved Wilson (clover) action for quarks (c) smeared source (d) large number of gauge configurations: Ncf=1000 for J P =3/2(±) 2.J P =1/2(±) [with a diquark-type interpolating field] i.J P =1/2(-) state: J P =1/2(+) state: ii.HBC analysis shows that the state at 1.75 GeV is an NK scattering state. 3.J P =3/2(±) [A large statistics as Ncf=1000 has played an important role.] i.Three interpolating fields (NK*-type, color-twisted NK*-type, diquark-type) ii.Only massive states after the chiral extrapolation: J P =3/2(-) state: J P =3/2(+) state: iii.HBC analysis suggests that these 5Q states are NK* and N*K* scattering states. 4.Following possibilies would be interesting for Θ + (1540): i.Small quark mass effect(and/or elaborate chiral extrapolation) ii.Large spatial volume iii.Dynamical quarks iv.Elaborate interpolating fields to fit the diquark picture v.πKN hepta-quark picture 4. Summary/discussion Too heavy to be identified as Θ + (1540 ) See for detail: Phys. Rev. D71,034001 (2005) Phys. Rev. D72,074503 (2005)

23. 5.Tetra-quarks(4Q): (work in progress) A lattice QCD calculation using the πKN interpolating field(7-body op.) is difficult. πK subsystem is much easier to study. It is a 4Q system with the quantum number of κ. κ together with f 0 (600), f 0 (980), a 0 (980) forms the scalar nonet, which are candidates of tetra(4Q) quark states. 4Q with “HBC” 4Q with PBC 0 + meson (chiral extrapolation) Our preliminary result suggests an existence of a 4Q resonance. work in progress. ππ threshold(“HBC”) ππ threshold(PBC) PRELIMINARY ππ scattering state Since these tetra-quarks are interesting target in their own right, we are currently performing 4Q lattice QCD calculations from a more general point of view.