Kaon-Nucleon potential from lattice QCD

S. Aoki (Univ. of Tsukuba),S. Aoki (Univ. of Tsukuba),T. Doi (Univ. of Tsukuba),T. Doi (Univ. of Tsukuba),T. Hatsuda (Univ. of Tokyo),T. Hatsuda (Univ. of Tokyo),T. Inoue (Univ. of Tsukuba), T. Inoue (Univ. of Tsukuba), N. Ishii (Univ. of Tokyo), N. Ishii (Univ. of Tokyo), K. Murano (Univ. of Tokyo),K. Murano (Univ. of Tokyo),H. Nemura (Tohoku Univ.),H. Nemura (Tohoku Univ.),K. Sasaki (Univ. of Tsukuba)K. Sasaki (Univ. of Tsukuba)

Kaon-Nucleon potential from lattice QCDKaon-Nucleon potential from lattice QCD

Yoichi Ikeda (Univ. of Tokyo)Yoichi Ikeda (Univ. of Tokyo)

for for

HAL QCD collaborationHAL QCD collaboration

Plan of this talkPlan of this talk

Introduction and our motivationIntroduction and our motivation

FormalismFormalism

Numerical results and discussionsNumerical results and discussions

Summary and future plansSummary and future plans

Experimental study of Experimental study of + + state by LEPS collaborationstate by LEPS collaboration

nKnK++ invariant mass distribution shows a narrow peak around 1.524 GeV invariant mass distribution shows a narrow peak around 1.524 GeV

Statistical significance of the peak is 5.1 Statistical significance of the peak is 5.1

The obtained results support the evidence of the The obtained results support the evidence of the state state

IntroductionIntroduction

Many experiments at high energy show negative results.Many experiments at high energy show negative results.

No narrow peak was observed by CLAS collaboration.No narrow peak was observed by CLAS collaboration.

Open problemsOpen problems

Reaction-dependent production mechanism?Reaction-dependent production mechanism?


LQCD studies for signal of LQCD studies for signal of + + statestate

Quenched LQCD for energy of penta-quark (J=1/2, 3/2)Quenched LQCD for energy of penta-quark (J=1/2, 3/2)(e.g., Csikor, Sasaki, Chiu, Mathur, Ishii, Takahashi, Alexandrou, Lasscock, Holland)(e.g., Csikor, Sasaki, Chiu, Mathur, Ishii, Takahashi, Alexandrou, Lasscock, Holland)

Penta-quark might be allowed, Penta-quark might be allowed,

but most probably NK scattering state.but most probably NK scattering state.

Penta-quark might be allowed, Penta-quark might be allowed,

but most probably NK scattering state.but most probably NK scattering state.

Current status from LQCDCurrent status from LQCD

One possible explanation of One possible explanation of ++ state state

Hadronic molecule (NK) or bound state (NHadronic molecule (NK) or bound state (NK)K)

Need precise information on NK interactionNeed precise information on NK interaction


NK scattering phase shift NK scattering phase shift repulsive repulsive

NK (I=0, 1) scattering phase shiftsNK (I=0, 1) scattering phase shiftsHashimoto, PRC 29.

S01S01

S11S11

I=0I=0

I=1I=1

No attraction in any range were found.No attraction in any range were found.

Potential from Quark modelPotential from Quark model Barnes-Swanson, PRCBarnes-Swanson, PRC4949..

Consistent with meson-exchange model?Consistent with meson-exchange model?

e.g.,) Julich group, NPe.g.,) Julich group, NPA506A506..

LQCD simulations solve the ambiguities of the NK potential.LQCD simulations solve the ambiguities of the NK potential.LQCD simulations solve the ambiguities of the NK potential.LQCD simulations solve the ambiguities of the NK potential.

Our motivationOur motivation

Previous LQCD studies of the Previous LQCD studies of the ++ are all quenched. are all quenched.

FullFull LQCD simulation is necessary. LQCD simulation is necessary.

PACS-CS config. (2PACS-CS config. (2+1 flavors)+1 flavors)

We are almost on the We are almost on the “physical point”.“physical point”.

PRDPRD7979(2009).(2009).

Our motivationOur motivation

Production mechanism of Production mechanism of + + state might be reaction dependent.state might be reaction dependent.

The nKThe nK++ potential derived from potential derived from QQCCDD

powerful tool to analyze the nature of the powerful tool to analyze the nature of the ++..

Previous LQCD studies of the Previous LQCD studies of the ++ are all quenched. are all quenched.

FullFull LQCD simulation is necessary. LQCD simulation is necessary.

This studyThis study

We derive the We derive the nKnK++ potential potential from from Full LQCD Full LQCD simulation. simulation.

Formalism (HAL procedure)Formalism (HAL procedure)

Developed by Ishii, Aoki, and Hatsuda for NN systemDeveloped by Ishii, Aoki, and Hatsuda for NN systemPRLPRL9999, 022001(2007)., 022001(2007).

3) Derive the potential from Schrodinger Eq.3) Derive the potential from Schrodinger Eq.3) Derive the potential from Schrodinger Eq.3) Derive the potential from Schrodinger Eq.

1) Define interpolating operators1) Define interpolating operators1) Define interpolating operators1) Define interpolating operators

Wave func. Wave func. Potential Potential Observable Observable

2) Calculate the equal-time BS amplitude2) Calculate the equal-time BS amplitude2) Calculate the equal-time BS amplitude2) Calculate the equal-time BS amplitude

This workThis work

Formalism (This work)Formalism (This work)

Schrodinger Eq.Schrodinger Eq.

We investigate the s-wave nKWe investigate the s-wave nK++ potential. potential.We investigate the s-wave nKWe investigate the s-wave nK++ potential. potential.

S-wave projected nKS-wave projected nK++ wave function and potential wave function and potentialS-wave projected nKS-wave projected nK++ wave function and potential wave function and potential

Numerical set-upNumerical set-up

2+1 flavor full QCD configuration by CP-PACS/JLQCD2+1 flavor full QCD configuration by CP-PACS/JLQCD

RG improved gauge action &RG improved gauge action &

O(a) improved Wilson-clover quark actionO(a) improved Wilson-clover quark action

Lattice spacing : a=0.1209 [fm]Lattice spacing : a=0.1209 [fm]

Size of Lattice : 16Size of Lattice : 1633×32 ×32 L=1.93 [fm] L=1.93 [fm]

Hopping parameters : ,Hopping parameters : ,

# of conf. = 700# of conf. = 700

Flat wall source to provide NK state.Flat wall source to provide NK state.

Numerical results (Hadron effective masses)Numerical results (Hadron effective masses)

GG22(t) : 2-point function(t) : 2-point function

Wal

l sou

rce

Wal

l sou

rce

Diri

chle

t B

.C.

Diri

chle

t B

.C.

Mass [MeV]Mass [MeV] Fit rangeFit range

PionPion 870.7(1.9)870.7(1.9) 5-105-10

KaonKaon 911.5(1.9)911.5(1.9) 5-105-10

NucleonNucleon 1795.5(6.9)1795.5(6.9) 6-116-11

NK thresholdNK threshold

2707(8) MeV2707(8) MeV

nKnK++ effective mass in J=1/2 effective mass in J=1/2-- channel channel

Wal

l sou

rce

Wal

l sou

rce

Diri

chle

t B

.C.

Diri

chle

t B

.C.PlateauPlateau

NK threshold (s-wave)NK threshold (s-wave)

Single state saturation is achieved.Single state saturation is achieved.

The best fit in the plateau gives MThe best fit in the plateau gives Meffeff=2723(10) MeV.=2723(10) MeV.

GG44(t) : NK temporal correlator(t) : NK temporal correlator

S-wave nKS-wave nK++ BS wave function and potential BS wave function and potential

PotentialPotentialWave functionWave function

Repulsive core at short distance ( r<0.5[fm] )Repulsive core at short distance ( r<0.5[fm] )

Attractive pocket in middle range ( 0.5<r<1.2 [fm] )Attractive pocket in middle range ( 0.5<r<1.2 [fm] )

I=0I=0

I=1I=1

S-wave nKS-wave nK++ scattering phase shift scattering phase shift

S01S01

S11S11

Qualitatively consistent withQualitatively consistent with

experimental dataexperimental data Pion mass is 871 MeV in our set-upPion mass is 871 MeV in our set-up NoteNote

Gaussian coreGaussian core

++

(Yukawa x form factor)(Yukawa x form factor)22


++


The range of Yukawa-type potential = 805 MeV

Fitting functionFitting function

SummarySummary S-wave nKS-wave nK++ potential derived from full LQCD potential derived from full LQCD is studied.is studied.

- We found repulsive core at short distance and weak attractive pocket - We found repulsive core at short distance and weak attractive pocket in the middle range.in the middle range.

We calculateWe calculate the scattering phase shift the scattering phase shift

- Qualitatively consistent with experimental data- Qualitatively consistent with experimental data

We also study the effective mass of the nKWe also study the effective mass of the nK++ state in J=1/2 state in J=1/2-- channel. channel.

- The low-lying state is consistent with nK- The low-lying state is consistent with nK++ threshold. threshold.

Future plansFuture plans We will study We will study the quark mass dependence of the s-wave the quark mass dependence of the s-wave nKnK++ potential. potential.

P and D-wave nKP and D-wave nK++ potentials potentials

The small width of the The small width of the ++ might be explained might be explained

due to the centrifugal barrier.due to the centrifugal barrier.

back upback up

Convergence of nK+ potentialConvergence of nK+ potential

Potential fitPotential fit


++



++


The range of Yukawa-type potential = 805 MeV

Scattering lengthScattering length

rr(r) fit(r) fit

Calculated from LS eq. with fitted potentialCalculated from LS eq. with fitted potential

Very weak interaction at thresholdVery weak interaction at thresholdVery weak interaction at thresholdVery weak interaction at threshold

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Kaon-Nucleon potential from lattice QCD