1. 1 PENTA-Quark Baryons on the Lattice T. T.Takahashi, T. Umeda, T. Onogi and T.K. (Yukawa Institute for Theoretical Physics, Kyoto) ・ Pentaquarks ・ Present lattice QCD studies of the pentaquarks ・ Ground state and 1 st excited state in I=0,J=1/2 channel ・ NK scattering state vs Pentaquark state ・ Summary KIAS-HANYANG joint workshop KIAS, Oct. 25-27 (’04) (hep-lat/0410025 and in preparation)

2. 2 Pentaquarks Θ + (1540) B=1, S=+1  hypercharge Y=2 We find no Isospin-partners (Θ 0 Θ ++ ) in experiments Θ + (1540) should be a member of 10 * representation Minimal quark contents is 5 quarks Isospin of the pentaquark is I=0 I=0, Y=2 Spin and Parity remain undetermined!! Has a very narrow width

3. 3 Pentaquarks(cont’d) The width matters! Skyrme soliton model const.-quark models; diquarks, chiral quark… Model calculations QCD-based (direct) QCD sum rules Lattice QCD study ---- Theoretical studies of pentaquark baryons ---- Reliable nonperturbative method based on QCD Can pentaquark baryons exist………………? R.A. Arndt, I.I. Strakovsky, R.L. Workman Phys.Rev.C68:042201,2003, Erratum-ibid.C69:019901,2004 J. Haidenbauer, G. Krein Phys.Rev.C68:052201,2003 The width of Θ+ should be less than 1 MeV

4. 4 Present lattice QCD studies of the pentaquarks Rather controversial F.Csikor, Z.Fodor, F.Katz, T.Kovacs JHEP 0311 (03) 070 S.Sasaki, Phys. Rev. Lett. In press (hep-lat/0310014) Pentaquark state near KN threshold in negative parity channel Ting-Wai Chiu, Tung-Han Hsieh hep-lat/0403020, 0404007 Lowest lying Pentaquark is in positive parity channel N.Mathur et al. (Kentucky group)hep-ph/0406196 NO Pentaquark near KN threshold in both parity channel We investigate the I=0, J=1/2, state using lattice QCD MIT group 4x4 correlation matrix ? Titech group (Analysis using Hybrid boundary conditions) NO Pentaquark near KN threshold in both parity channel

5. 5 1. Proper separation of the possible resonance and continuum →prepare two or more operators, construct the correlation matrix to be diagonalized c.f. It’s also true with QCD sum rules. ２． Close examination determining whether a resonance or continuum →the volume dependence of the energy obtained → and that of the overlaps with the operators Something should lack in the previous investigations to be conclusive! In a finite lattice, momenta are discretized and volume-dependent → so are the relative momenta of Ｎ and Ｋ →a simple scattering state has a large and specific momentum dependence

6. 6 Separation of the 1 st excited state from the gr. st.: Difficult in the lattice QCD calculation We suffer from contaminations from the NK scattering states. Energy of the ground state signal Correlation between operators

7. 7 Separation of 1 st excited state from gr. st.(cont’d): How to separate the states?  We prepare two independent operators and make the correlation matrix and then diagonalize it to obtain the G.S and 1 st E.S.  In effect, we are constructing the operator which selectively couples to the n-th excited-state from a linear combination of the independent op’s. : independent operators which couple to the same quantum number T. T. Takahashi and H. Suganuma Gluonic excitations in static three quark system Phys.Rev.Lett.90:182001,2003

8. 8 Separation of 1 st excited state from gr. st.(cont’d): Many excited statesOnly the ground-state Energy of the 1 st E.S. Energy of the G.S. Ground-state 1 st excited-state After the diagonalization…. due to the reduction of the rank of the correl. Matrix.

9. 9 NK scattering state vs Pentaquark state 1 st excited state in (I,J,P)=(0,1/2,-) channel We find volume dependences.  NK scattering state???? If it is the scattering state with the relative momentum p=2π/L, we can estimate the energy as Here, we assume the absence of the interaction and that nucleon and kaon are point-particles. Lattice is finite box

10. 10 NK scattering state vs Pentaquark state We can incorporate the interaction between N and K. Outer wave func.  Periodic  Solution of Helmholtz Eq. Inner wave func.  ordinary scattering wave M.Luesher Nucl.Phys.B354(1991)531 The Eigenstate in the finite-volume lattice should be connected smoothly. However, we obtain almost thesame behavior as 2π/L  The weakness of NK interaction in this channel I = 2 PI PI SCATTERING PHASE SHIFT WITH TWO FLAVORS OF O(A) IMPROVED DYNAMICAL QUARKS. By CP-PACS Collaboration (T. Yamazaki et al.) hep-lat/0402025 For example, this was applied to …..

11. 11 Simulation conditions β=5.7 (lattice spacing : 0.2fm) quenched Wilson gauge action and Wilson quark action 8 3 x 24 [(1.6 fm) 3 x 4.8fm] 3000 gauge configurations 10 3 x 24 [(2.0 fm) 3 x 4.8fm] 2900 gauge configurations 12 3 x 24 [(2.4 fm) 3 x 4.8fm] 1950 gauge configurations 16 3 x 24 [(3.2 fm) 3 x 4.8fm] 950 gauge configurations Current quark mass : (u, d, s) ～ (240MeV, 240MeV, 240MeV) (100MeV, 100MeV, 240MeV) (240MeV, 240MeV, 100MeV) (170MeV, 170MeV, 100MeV) (100MeV, 100MeV, 100MeV) We adopt Dirichlet boundary condition on time-direction m_pi/m_rho=.7 to.85; Csikor et al ;.5 to.7; S.Sasaki;.68 to.90

12. 12 Interpolating operators NucleaonKaon N+K like operator Pentaquark like operator Same as Csikor et al. Spinor structure : same Color structure : different

13. 13 Ground state in (I, J P )=(0, 1/2 - ) channel

14. 14 L dependence of the mass of the gr. st.  Small L Large L  (u,d,s)=(100,100,100)MeV(u,d,s)=(100,100,240)MeV (u,d,s)=(240,240,100)MeV (u,d,s)=(240,240,240)MeV M N +M K Mass (GeV) 1fm 2.4fm 4fm

15. 15 Summary on Ground state in (I,J,P)=(0,1/2, - ) channel  coincides with M N +M K  We find almost no volume dependence. It is expected to be the scattering state of Nucleon and Kaon, with the relative momentum p=0.

16. 16 1 st Excited state in (I, J P )=(0, 1/2 - ) channel

17. 17 NK scattering state vs Pentaquark state  Small L Large L  (u,d,s)=(100,100,100)MeV (u,d,s)=(100,100,240)MeV (u,d,s)=(240,240,100)MeV(u,d,s)=(240,240,240)MeV M N +M K NK scattering Mass (GeV) 1fm 2.4fm 4fm

18. 18 1 st excited state The volume dependence of the 1st excited-state seem to be rather different from that of NK scattering. Current quark mass light Current quark mass heavy Above the theoretical curve Below the theoretical curve Large volume dependence Small Volume dependence Existence of possible resonance states in I=0, ½- channel ? Spatial volume (1.6fm) 3 ～ (3.2fm) 3 Current quark mass 100 MeV ～ 240MeV

19. 19 NK scattering state vs Pentaquark state  Small L Large L  (u,d,s)=(100,100,100)MeV (u,d,s)=(100,100,240)MeV (u,d,s)=(240,240,100)MeV(u,d,s)=(240,240,240)MeV M N +M K NK scattering Mass (GeV) 1fm 2.4fm 4fm

20. 20 Need of further examination of the volume dependence Spectral weight  The overlaps of the operators and the each state Eg. the relative w.f. of N and K The volume of lattice V Owing to the normalization of each state, a localized w.f. hardly gives volume dependence of say, the value of the w.f. at the origin, in comparison with continuum states. Resonance states  small V-dependence of overlaps

21. 21 Spectral weight Ground-state Conj. ： NK scattering with p=0  1/V dependence 1 st excited-state Conj. ： some resonance  no volume dependence NEGATIVE parity channel

22. 22 (I, J P )=(0, 1/2 + ) channel

23. 23  Small L Large L  (u,d,s)=(100,100,100)MeV (u,d,s)=(100,100,240)MeV (u,d,s)=(240,240,100)MeV (u,d,s)=(240,240,240)MeV Mass (GeV) 1fm 2.4fm 4fm N*+K

24. 24 Spectral weight POSITIVE parity channel behave as 1/V  a simple scattering state, i.e., a spread (non-localized) wave function

25. 25 Summary We have studied the ground-state and the 1 st excited-state in (I, J P )=(0, 1/2) channel using lattice QCD. We prepared two operators and constructed 2x2 correlation matrices and diagonalized them. As a result, we have successfully obtained the ground-state and 1 st excited-state energies. J P =1/2 － Ground-state  NK scattering state with the relative momentum p=0 J P =1/2 － 1st excited-state  resonance state J P =1/2 ＋ Ground-state  N*K scattering state with the relative momentum p=0

26. 26 Caveats The quark masses used are rather heavy!  pentaquark resonance states tend to exisit for heavier quark systems, involving the strangenes, charm, b, t(?). Conversely speaking, the pentaquark with lighter quark has a rather spread wave function. The states created by the lattice cal. can strongly depend on the interpolating fields; （ eg. Only single positive parity state was excited by the present operator for the positive parity states)  The operator used may not be general enough to excite all the physical states.

27. 27 Remarks: If the 1 st excited-state is Θ+(1540), it is broad object and it’s safe to use larger lattice than (2.5fm) 3. The 1 st excited-state seems to be rather different from NK scattering state. If it were NK scattering state, (the case Θ+(1540) does not exist……..?) Our result shows the volume dependence of the scattering state is nontrivial, which implies that we need careful treatment of the scattering state, for example, in the study of the phase shift using lattice QCD. Further study Full QCD calculations Improvement of quark action Large β(finer lattice) Higher statistics Detailed study of the quark mass dependence Analysis of the wave function ….. c.f. S.Sasaki: L=2.2 fm; F.Csikor,Fodor, Katz: L=1.3 2.7 fm For Roper and S(1535), L=3 fm is needed. (S.Sasaki, nucl-th/0305014)