07/30/2007 Lattice 2007

Sigma meson contribution Sigma meson contribution in the in the I=1/2 ruleI=1/2 rule

Takumi Doi(Univ. of Kentucky)

In collaboration with

T.Draper (Univ. of Kentucky) K.-F.Liu (Univ. of Kentucky)N.Mathur (JLAB)J.-B. Zhang (Zhejiang Univ.)

07/30/2007 Lattice 2007

Outline The long-standing puzzle in the Scalar mesons

a0(980), a0(1450), etc. The lattice QCD study for the scalar mesons

Study of the spectrum Standard q-qbar meson or tetraquark ?

The volume dependence study to extract the nature of the signal

as a tetraquark mesonium The role of meson in K 2 decay

07/30/2007 Lattice 2007 0¯ ¯(1)1¯+(1)

0++(0)0+ ¯(1)1+ ¯(1)

π(137)

0+ (1/2)

ρ(770)

σ(600)

f0(980)

f0(1370)

f0(1500)

a0(980)

a0(1450)

a1(1230)

K0*(1430)

JPG(I)

M (

MeV

)

a2(1320)

2+ ¯(1)

f0(1710)

K0*(800)

07/30/2007 Lattice 2007

Puzzle in scalar mesons Too many states compared to the other chan

nels Strange meson is lighter than non-stranged mesons ?? Some have very broad width, others have narrow width

(600), (800), f0(980), a0(980) may be tetraquark states R.L.Jaffe (1977)

Diquark model, molecular model

etc.

07/30/2007 Lattice 2007

The controversial state: σ meson

σ (500): Johnson and Teller

Two-pion exchange potential:

Chembto, Durso, Riska; Stony Brook, Paris, …N-N potential

07/30/2007 Lattice 2007

M. Ablikim et al. (BES), Phys. Lett. B598, 149 (2004)

Mσ = 541 ± 39 MeV, Γσ = 504 ± 84 MeV

J/ψ —> ωπ+π-

07/30/2007 Lattice 2007

Simulation Parameters

Overlap fermion with quenched approx. Iwasaki gauge action a = 0.200(3)fm (a-1 = 1GeV) (beta=2.264) V=163X28 L3 = (3.2fm)3

m >= 180 MeV, m L > 3 About 200-300 configurations

Exact chiral symmetry !N.Mathur et al.( QCD collab.), hep-lat/0607110

07/30/2007 Lattice 2007

Our results shows scalar mass around 1400-1500 MeV, suggesting

a0(1450) is a two quark state.

mmss

Eliminate ’ ghost state for a0 study

N.Mathur et al.( QCD collab.), hep-lat/0607110

07/30/2007 Lattice 2007

)(2

1

, 3

1

55

5

5

dduu

ud

du

o

oo

ππππ four quark operator (I=0)four quark operator (I=0)

Disconnected diagrams are found to be small neglected at this moment

)]0(0 )(JI ,[ PCG55

Further study is needed to check the Further study is needed to check the volume dependencevolume dependence of the of the observed states.observed states.

Scattering statesScattering states(Negative scattering(Negative scattering length)length)

)0()0( pEpE

)1()1( pEpE

Scattering statesScattering states

Possible BOUND statePossible BOUND state

σσ (600)?(600)?

07/30/2007 Lattice 2007

Volume dependence to distinguish one/two particle state

Spectral weight on the lattice Normalization for the particle

If the signal is one particle state:

If the signal is two particle state:

Physical meaning: O(1/V) corresponds to the encounter possibility of two particles

07/30/2007 Lattice 2007

Volume dependence of spectral weights

Volume independence suggests the observed state is an one particle state

WW00

WW11

3D-Volume 123 vs. 163

N.Mathur et al.( QCD collab.), hep-lat/0607110

07/30/2007 Lattice 2007

The role of Sigma meson in Physics

K2 decay

I = 1/2 rule Sigma can enhance the I=0 channel only

T.N.Pham, Phys.Rev.D33 (1986) 1499

T.Morozumi, C.S.Lim, A.I.Sanda, PRL65 (1990) 404

Usual lattice formulation K , K vac. + chPT No consideration !

(However, T.Yamazaki at lat06, direct K 2 for I=2)

07/30/2007 Lattice 2007

Strategy

3pt correlator G(t) Almost no energy-

momentum mismatch (m(K)=m()=500MeV)

Hw

K0

For the contribution to physical K2, one needs additional 2, for which one may use experimental information

Coulomb wall

t

07/30/2007 Lattice 2007

Results

The correlator is found to be very noisy Further work is necessary to extract the quantita

tive signal

O2 O6

K

Hw HwWall src

Wall src

K

07/30/2007 Lattice 2007

Summary/Outlook We have investigated the scalar mesons with overlap

fermion at the quenched level with m > 180MeV a0(1450) is two quark state ) is likely to be a tetraquark state

3D-Volume dependence study is essential Dynamical quark effect, disconnected diagram study is desi

rable for future study The existence of meson may have substantial impact:

The enhancement of I=0 for K2 decay, which may solve the I=1/2 rule

Need further calculation to extract the quantitative results