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 Introduction2 General Formalisms3 Numerical Results4 Summary/Discussion(5 Tetra-quarks(4Q))

See Phys.Rev.D71,034001(2005); D72,074503(2005) for detail.

START

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 ⇒ JP=1/2(+) and 3/2(+)

c. πKN hepta-quark picture ⇒ JP=1/2(+)

d. The string picture

e. JP=3/2(-) assignment ⇒ JP=3/2(-) In this talk, we are mainly interested in JP=3/2(±) possibilities:

1. We first present our numerical results on JP=1/2(±) penta-quarks brieflyemplyoing a diquak-type interpolating fieldusing a flavor dependent boundary condition(HBC)

2. We then present our numerical results on JP=3/2(±) penta-quarksemploying three Rarita-Schwinger interpolating fieldsusing 1000 gauge field configurations for high statistics

Lattice QCD Setup:

1. Gauge Config by standard Wilson gauge action: 　a. Lattice size : 123×96 　　　　 [(2.2fm)3×4.4fm in physical unit]

b. β= 5.75

c. Lattice spacing: from Sommer parameter r0.

d. Anisotropic latticeRenormalized anisotropy: as/at=4for accurate measurements of correlators and masses

e. #(gauge config) = 504 for JP=1/2(±) = 1000 for JP=3/2(±)

2. O(a) improved Wilson quark (clover) action.The quark mass covers the region ms < mq < 2 m s

3. Smeared source to reduce higher spectral contributions

2.General Formalism

[GeV]4.41 a

0.1240 0.1230 0.1220 0.1210

656(2) 784(1) 893(1) 1005(1)

1011(5) 1085(4) 1162(3) 1240(3)[MeV]m

[MeV]m

2.2 fm

Finer lattice spacing along the temporal direction

tim

e

The interpolating fields

ddcbTaabcddcb

Taabc usddCudsudCu 55NK*-type

cddbTaabccddb

Taabc usddCudsudCu 55color-twisted NK*-type

geTdb

Tacfgdefabc sCdCudCu 555 diquark-type

★ Three Rarita-Schwinger interplating fields for JP=3/2(±) states:

★ A diquark-type interplating fields for JP=1/2(±) states: T

cgTfe

Tdbfgadeabc sCCdudCu 5

We consider the following iso-scalar interpolating fields:

(scalar) (pseudo scalar)

(scalar) (vector)

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 contents

spatial BC minimum momentum

N anti-periodic BC

K,K*

anti-periodic BC

periodic BC

udduud ,sdsu ,suudd

LLLp ,,min

0,0,0min p

LLLp ,,min

Lp /3min

0min p

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:

L

np ii

2

L

np ii

12

u quark spatially anti-periodic BC

d quark spatially anti-periodic BC

s quark spatially periodic BC

Lp /3min

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).

0min pWit

h HBC

3.Numerical Results: JP=1/2(±) states (effective mass plots)

“Effective mass” is defined as

which can be considered as an “weighted average” of massesat each time-slice t.

)1(

)(log)(

tG

tGtmeff

plateau

JP=1/2(-)

plateau

JP=1/2(+)

KNth mmE NK-threshold (s-wave)

2min

22min

2 pmpmE KNth

NK-threshold (p-wave)

1. JP=1/2(-) state:A state appears slightly above the NK threshold (mN+mK).

2. JP=1/2(+) state:A state appears above the raised NK threshold (due to the finite box).⇒ rather massive !

Excite

d st

ate

cont

ribu

tion

s

are

redu

cing

A single state dominate the correlator G(t) in this region.

Chiral extrapolation (JP=1/2(±))

At physical point

(1) JP=1/2(+): 2.24(11) GeV

(2) JP=1/2(-): 1.75(3) GeV

NK threshold (p-

wave)

NK threshold (s-

wave)

1. Our data does not support a low-lying JP=1/2(+) penta-quark.

2. For JP=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

HBC analysis (JP=1/2(-) state)

PBC HBC

KNth mmE NK-threshold (PBC)

2min

22min

2 pmpmE KNth

NK-threshold (HBC)

1. NK(s-wave) threshold is raised up by 210 MeV.

2. The best fit mass m5Q is raised up by a similar amount.

★ No compact 5Q resonance exists in the region:

★ The state observed in JP=1/2(-) is an NK scattering state.

2min

22min

2 pmpmmmm KNKN

Numerical Results: JP=3/2(-) state (effective mass plot)

This correlator is too noisy !Fit is not performed.

The plateaus appearabove the NK*-threshold and above the raised NK threshold.

plateau

plateau

twisted

×

“Effective mass” is defined as

which can be considered as an “weighted average” of massesat each time-slice t.

)1(

)(log)(

tG

tGtmeff

Chiral extrapolation (JP=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: m5Q= 2.17(4) GeV

(2) Color-twisted NK*-type: m5Q= 2.11(4) GeV

No evidence for a low-lying 5Q state

HBC analysis suggeststhese states are NK*(s-wave) scattering states

Due to the limited

time, we cannot

show HBC

analysis.

JP=3/2(+) state (effective mass plot)

The plateaus appearabove the raised NK*-threshold and above the raised NK threshold.

plateau

twisted

plateau

plateau

Chiral extrapolation (JP=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: m5Q= 2.64(7) GeV

(2) Color-twisted NK*-type: m5Q= 2.48(10) GeV

(3) Diquark-type: m5Q=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.

1. We have studied spin=1/2 and 3/2 penta-quarks by using the anisotropic lattice QCD. For acuracy,(a) renormalized anisotropy as /at = 4(b) O(a) improved Wilson (clover) action for quarks(c) smeared source(d) large number of gauge configurations: Ncf=1000 for JP=3/2(±)

2. JP=1/2(±) [with a diquark-type interpolating field]

i. JP=1/2(-) state: JP=1/2(+) state:

ii. HBC analysis shows that the state at 1.75 GeV is an NK scattering state.

3. JP=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:

JP=3/2(-) state: JP=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

GeV17.2,11.25 Qm GeV42.2,48.2,64.25 Qm

GeV25.25 QmGeV75.15 Qm

Too heavy

to

be identifi

ed

as Θ

+ (1540)

See for detail:

Phys. Rev. D71,034001

(2005)

Phys. Rev. D72,074503

(2005)