Pentaquark in Anisotropic Lattice QCD --- A possibility of a new 5Q resonance around 2.1 GeV N. Ishii (TITECH, Japan) T. Doi (RIKEN BNL) H. Iida (TITECH, Japan) Y. Nemoto (Nagoya Univ.) M. Oka (TITECH, Japan) F. Okiharu (Nihon Univ., Japan) H. Suganuma (TITECH, Japan) Plan of the talk: 1. Introduction 2. General Formalism 3. Numerical Result on J P =1/2(±) 4. A Further Investigation of the Negative parity state a. New method with Hybrid Boundary Condition(HBC) b. Numerical Result II 5.First Lattice QCD result on J P =3/2( - ) --- A possibility of a new 5Q resonance around m=2.1 GeV. 6. Summary/Discussion preliminary

Since the first discovery of a manifestly exotic baryon by LEPS group at SPring-8, enormous efforts have been devoted to the studies of penta quarks. 1. Introduction ★ The parity of Θ + (1540) is one of the most important topics. 1.Experimental determination of the parity of Θ + (1540) is difficult. 2.Theoretical opinions are divided into two pieces. i.Positive parity is supported by Soliton models, Jaffe-Wilczek diquark model,... ii.Negative parity is supported by Naive quark models, QCD sum rule, …

Lattice QCD studies of the penta quarks There are a number of lattice QCD studies of penta quarks. (1) F.Scikor et al., JHEP11(2003)070. (2) S.Sasaki, PRL93 (2004) (3) T.-W.Chiu et al., hep-ph/ (4) N.Mathur et al., PRD70(2004) (5) N.Ishii et al., PRD71(2005) (6) C.Alexandrou et al., hep-lat/ ; hep-lat/ (7) T.T.Takahashi et al., hep-lat/ ; hep-lat/ (8) D.Sigaev et al., MIT group. (9) B.G.Lasscock et al., hep-lat/ (10) F.Scikor et al., hep-lat/ However, these studies have not reached the consensus yet. The aim of this talk is (1) to provide a accurate data using anisotropic lattice QCD. (2) to provide a further studies of negative parity state using a new method with the Hybrid boundary condition(HBC). (3) to provide the 1st lattice QCD result on J P =3/2( - ) channel. preliminary

2.General Formalism (Part I: J P =1/2(±)) A non-NK type operator: (I=0, J=1/2) To reduce the overlap with NK scattering states Interpolating field for Θ + Temporal correlator T Positive parity states dominate. Negative parity contribution cannot become negligible. (“upper component ”) Positive parity contribution cannot become negligible. As adopted in (1) J.Sugiyama et al., PLB581,167(2004). (2) S.Sasaki, PRL93, (2004). T (“lower component ”) Negative parity states dominate.

Lattice Parameter 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 f.The gauge configurations are separated by 500 pseudo heat-bath sweeps, after skipping thermalization sweeps. 2.O(a) improved Wilson quark (clover) action. 3.Smeared source to reduce higher spectral contributions 3. Numerical Result I (2)784(1)893(1)1005(1) 1011(5)1085(4)1162(3)1240(3) These values covers 2.2 fm Finer lattice spacing along the temporal direction time

Negative parity channel (J P =1/2( - )) NK threshold(s-wave) By neglecting the interaction between N and K: Higher spectral contribution is gradually reduced. Plateau Single-state saturation is achieved. Effective Mass: If then Existence of the plateau indicates the single-state saturation of the correlator G(t). negligible ! Effective mass Correlator best fit in the plateau

L Positive parity channel J P =1/2(+) Higher spectral contribution is gradually reduced. Plateau NK threshold (p-wave) The quantized spatial momenta are due to the finiteness of the box. Effective mass Correlator best fit in the plateau L L Single-state saturation is achieved.

Chiral extrapolation At physical point (1) Positive parity: 2.25(11) GeV (2) Negative parity: 1.75(3) GeV NK threshold (p-wave) NK threshold (s-wave) 1.Our data does not support the low-lying positive parity. 2.For negative parity channel, m=1.75 GeV is rather close to the empirical value 1.54 GeV. However, it should be clarified whether this state is a compact 5Q resonance or not. (We will perform a further study in this direction from the next slide)

L L L 4. Further study of the negative parity state. (a) NEW METHOD with Hybrid BC(HBC) quark contentsspatial BCminimum momentum Nanti-periodic BC K periodic BC Hybrid Boundary Condition(HBC) The spatial BOX Spatial momentum is quantized due to finite volume effect: 1. periodic BC: 2. anti-periodic BC: NK scattering states Hybrid BC: u quarkspatially anti-periodic BC d quarkspatially anti-periodic BC s quarkspatially periodic BC Cosequence on hadrons Standar BC: Expected consequences on the spectra 1.NK threshold is raised due to finite volume effect. 2.Compact 5Q resonance states are expected to be less sensitive to the change of boundary condition. HBC helps us detecting existence of compact 5Q resonance in the region as:

An example Response of a compact resonance state to the change of boundary condition. A localized resonance is less sensitive to the change of boundary condition ! spatially periodic BC For this purpose, nucleon is not appropriate, because nucleon is sujbect to the anti-periodic BC.

Numerical result II Hybrid BCStandard BC The hopping parameter leads to m N =1.74 GeV, m K =0.79 GeV Expected shift of the NK threshold for L=2.15 fm is The plateau is shifted above by the expected amount. (1) No compact 5Q resonance exists in the region as (2) The state observed in the negative parity channel turns out to be an NK scattering state.

Combining the results from the other quark masses data points The best fit value on the plateau. solid lines NK(s-wave) threshold We have not found a compact 5Q resonance in J P =1/2( - ) in our calculation.

1.Spin of Θ + is also not yet determined experimentally. 2.J P =3/2( - ) possibility can solve the puzzle of the narrow decay width. (proposed by A.Hosaka et al., hep-ph/ ) Advantage: (a) It allows the configuration of (0s) 5. (b) It decays into a d-wave KN state. Suppressed overlap to d-wave KN state The decay width is expected to be significantly narrow. Disadvantage: (a) The color-magnetic interaction makes it massive. If some contribution can cancel the color-magnetic interaction to make its mass around 1540, we will obtain a penta-quark with a significantly narrow width. 3.There have been no lattice QCD calculations for J P =3/2 penta-quark yet. Part II First lattice QCD result on J P =3/2(-) channel

Interpolating field (J P =3/2(-)) NK * -type interpolating field (I=0, Rarita-Schwinger formalism) Temporal correlator spin 3/2 projection matrix: spin 1/2 contributions + higher spectral contributions Negative parity contribution cannot become negligible. T (“lower component ”) Positive parity states dominate. T Negative parity states dominate. Positive parity contribution cannot become negligible. (“upper component ”)

effective mass plot (J P =3/2(-)) NK* threshold (s-wave): NK threshold (d-wave): Best-fit mass in the plateau: The best-fit mass is located above the NK* threshold and NK threshold ! plateau Excited state’s contribution is gradually reduced. best-fit NK(d-wave) Single-state saturation is expected to be achieved. NK*(s-wave)

Standard BC v.s. Hybrid BC (J P =3/2(-)) plateau best-fit NK(d-wave) NK * (s-wave) best-fit NK(d-wave) NK * (s-wave) StandardBC Hybrid BC 200MeV up 40MeV up 70MeV down twist After twisting the boundary condition to HBC: 1.The location of the best fit mass is almost unchanged. 2.It appears below NK* threshold by 70 MeV. 70 MeV This state may be a compact resonance state. preliminary

Chiral extrapolation (J P =3/2( - )) Physical region: m 5Q = 2.14(5) GeV 1.m 5Q = 2.14(5) GeV would be too massive to be identified as Θ + (1540). 2.This may be a new compact 5Q resonance around 2.1 GeV. (J P =3/2( - ), I=0, S=+1) Several comments on this state: (1) Quenched QCD results should be understood to contain about ±10% error. (The mass is better understood to be located in the region 1.9 GeV – 2.3 GeV.) (2) The decay width could be less narrow. --- The state appears above NK* threshold. --- Quenched QCD tends to underestimate the decay width. (K* does not decay) (3) Still, it would be interesting to investigate this state into detail. J P =3/2( - ) NK* threshold (s-wave) preliminary

1.We have studied Θ + (1540) 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 2.J P =1/2(±) i.Non-NK type interpolating field: ii.Positive parity: m 5Q = 2.25(11) GeV --- too massive to be identified as Θ + (1540) iii.Negative parity: m 5Q = 1.75(4) GeV --- rather close to the observed value. iv.We have proposed a new method (Hybrid BC [HBC]). HBC analysis shows the state(1.75 GeV) is not a compact 5Q state but an NK scattering state. 3.J P =3/2(-) [1 st lattice QCD result] i.NK*-type interpolating field: ii.HBC analysis indicates there is a state, which may be a compact 5Q resonance. iii.Chiral extrapolation leads to m 5Q = 2.14(5) GeV --- too massive to be identified as Θ + (1540). A possibility of a new 5Q resonance. (J P =3/2(-), I=0, S=+1) 4.Following possibilies would be interesting for Θ + (1540): (a) small quark mass effects (and/or more elaborate chiral extrapolation), (b) large spatial volume, (c) dynamical quark(including πKN hepta-quark picture), (d) elaborate interpolating fields to fit the diquark picture. 6. Summary/discussion preliminary Analysis of the other operators are currently going on. (Operator dependences are seen in our results)