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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 1st 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)

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Pentaquarks Θ+(1540)

B=1, S=+1 hypercharge Y=2

We find no Isospin-partners (Θ0Θ++) 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

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Pentaquarks(cont’d) The width matters!

Skyrme soliton modelconst.-quark models;diquarks, chiral quark…

Model calculationsQCD-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. WorkmanPhys.Rev.C68:042201,2003, Erratum-ibid.C69:019901,2004J. Haidenbauer, G. Krein Phys.Rev.C68:052201,2003

The width of Θ+ should be less than 1 MeV

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Present lattice QCD studies of the pentaquarks

Rather controversial

F.Csikor, Z.Fodor, F.Katz, T.Kovacs JHEP 0311 (03) 070S.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, 0404007Lowest lying Pentaquark is in positive parity channelN.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 group4x4 correlation matrix ?

Titech group (Analysis using Hybrid boundary conditions)NO Pentaquark near KN threshold in both parity channel

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1. Proper separation of the possible resonance and continuum →prepare two or more operators , construct the correlation matri

x to be diagonalized c.f. It’s also true with QCD sum rules.２． Close examination determining whether a resonance or continuu

m →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

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Separation of the 1st 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

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Separation of 1st 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 1st 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. SuganumaGluonic excitations in static three quark systemPhys.Rev.Lett.90:182001,2003

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Separation of 1st excited state from gr. st.(cont’d):

Many excited states Only the ground-state

Energy of the 1st E.S.

Energy of the G.S.

Ground-state

1st excited-state

After the diagonalization…. due to the reduction of therank of the correl. Matrix.

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NK scattering state vs Pentaquark state

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

We find volume dependences. NK scattering state????

If it is the scattering statewith the relative momentum p=2π/L,we can estimate the energy as

Here, we assume the absence of the interactionand that nucleon and kaon are point-particles.

Lattice is finite box

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NK scattering state vs Pentaquark state

We can incorporate the interaction between N and K.

Outer wave func.

PeriodicSolution 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 …..

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Simulation conditions

β=5.7 (lattice spacing : 0.2fm) quenchedWilson gauge action and Wilson quark action

83 x 24 [(1.6 fm)3 x 4.8fm] 3000 gauge configurations103 x 24 [(2.0 fm)3 x 4.8fm] 2900 gauge configurations123 x 24 [(2.4 fm)3 x 4.8fm] 1950 gauge configurations163 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

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Interpolating operators

Nucleaon Kaon

N+K like operator

Pentaquark like operator

Same as Csikor et al.Spinor structure : sameColor structure : different

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Ground state in (I, JP)=(0, 1/2-) channel

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L dependence of the mass of the gr. st.

Small L Large L Small L Large L

Small L Large L 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

MN+MK

Mas

s (G

eV)

1fm 2.4fm 4fm

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

coincides with MN+MK

We find almost no volume dependence.

It is expected to be the scattering stateof Nucleon and Kaon,with the relative momentum p=0.

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1st Excited state in (I, JP)=(0, 1/2-) channel

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NK scattering state vs Pentaquark state

Small L Large L Small L Large L

Small L Large L 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

MN+MK

NKscattering

Mas

s (G

eV)

1fm 2.4fm 4fm

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1st excited state

The volume dependence of the 1st excited-stateseem to be rather different from that of NK scattering.

Current quark mass

lightCurrent quark mass

heavy

Above thetheoretical curve

Below thetheoretical curve

Large volume dependence

SmallVolume dependence

Existence of possible resonance statesin I=0, ½- channel ?

Spatial volume (1.6fm)3 ～ (3.2fm)3

Current quark mass 100 MeV ～ 240MeV

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NK scattering state vs Pentaquark state

Small L Large L Small L Large L

Small L Large L 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

MN+MK

NKscattering

Mas

s (G

eV)

1fm 2.4fm 4fm

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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

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Spectral weight

Ground-stateConj. ： NK scattering with p=0 1/V dependence

1st excited-stateConj. ： some resonance no volume dependence

NEGATIVE parity channel

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(I, JP)=(0, 1/2+) channel

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Small L Large L Small L Large L

Small L Large L 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

Mas

s (G

eV)

1fm 2.4fm 4fm

N*+K

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Spectral weight POSITIVE parity channel

behave as 1/V a simple scattering state, i.e., a spread (non-localized) wave function

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Summary

We have studied the ground-state and the 1st excited-state in (I, JP)=(0, 1/2) channel using lattice QCD.

We prepared two operators and constructed 2x2 correlation matricesand diagonalized them. As a result, we have successfullyobtained the ground-state and 1st excited-state energies.

JP=1/2 － Ground-state NK scattering state with the relative momentum p=0

JP=1/2 － 1st excited-state resonance state

JP=1/2 ＋ Ground-state N*K scattering state with the relative momentum p=0

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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 onthe 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.

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Remarks:If the 1st excited-state is Θ+(1540),it is broad object and it’s safe to use larger lattice than (2.5fm)3.

The 1st 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 studyFull QCD calculations Improvement of quark actionLarge β(finer lattice) Higher statisticsDetailed study of the quark mass dependenceAnalysis 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)