1. Introduction2. Formalism3. Results4. Summary

I=2 pi-pi scattering length with dynamical overlap fermionI=2 pi-pi scattering length

with dynamical overlap fermion

•Takuya Yagi (Univ.Tokyo,KEK)•Munehisa Ohtani (Univ.Regensburg)•Osamu Morimatsu (KEK)•Shoji Hashimoto (KEK)

July 31 @Lattice07

Introduction

• Study of hadron interactions from QCD– nuclear force– interesting physics when strangeness is involved

• I=2 Pion-Pion: simplest among other hadron interactions– well controlled by ChPT

– do not have annihilation or rectangular topology

• Overlap fermion – exact chiral symmetry application of ChPT straightforward

(If not overlap, mixed action ChPT with domain-wall was worked out by NPLQCD(2006))

Overlap fermion

DamDmDoverlap 2

11)(

DamDmDoverlap 2

11)(

•The overlap action with quark mass “m” is defined as

AA

A

aD

†1

1

)( 0MaDA wilson

0/Maa

where

•This operator respects the chiral symmetry on the lattice.

　γγγ DDaDD 555 　γγγ DDaDD 555 (GW relation)

5ati 5ati )

2

11(5 Dati a )

2

11(5 Dati a

On the lattice

•Disadvantages- Numerically costly

- So, limited volume

Lüscher’s formula

2

22

4

1)(cot

Lk

Lkk S

2

22

4

1)(cot

Lk

Lkk S

• Two particles with momentum “k” confined in a large volume box

2 2

2

2 2

2 2

4

1lim 4

4 nn

nk L

n

k L

n

S

��������������

•L : Length of the box

• : Phase Shift

M. Lüscher(1986,1991)

where

• For s-wave,

this formula can be expanded as a function of scattering length divided by L.

2

02

013

02 1

42

L

ac

L

ac

Lm

amE

2

02

013

02 1

42

L

ac

L

ac

Lm

amE

a0 : scattering length

E

Setup

• Machine

– BlueGene/L @ KEK

Lattice Size

Number of flavors

Topological Charge

16332

2

0

Gauge Action

Fermion Action

Iwasaki Action

Overlap Fermion

Source Type

Gauge Fixing

Wall Source

Coulomb Gauge

• We used gauge configurations generated by JLQCD (Matsufuru)

– Lattice spacing = 0.1184(12) [fm] , from r0 ~ 0.49 [fm]

–

– pick configs every 100 traj to avoid autocorrelations.

– with Low mode averaging

ma No.Conf

0.015

0.025

0.035

0.050

0.070

0.100

99

96

93

92

92

92

ssq mmm ～6/

Periodic boundary (1)

Our lattice is periodic in temporal direction

Contaminations to the correlation functions exist

1. Two point correlation (1-Pion)

2. Four point correlation (2-Pion)

)()( tTmtm eetC

)(2

22)( tTEtE eetC

1-Pion Line (not quark line)1-Pion Line (not quark line)

T : T : size of the temporal direction

Constant

2))(2/()2/( 22

tTEtEe

termExtra

Periodic boundary (2)

)Constant:,(2

cosh)( 22 BABT

tEAtC

Ex) m=0.050

excited state

: independent of A & B

Good plateau is identified Good plateau is identified from this function.from this function.

Good plateau is identified Good plateau is identified from this function.from this function.

)1()(

)()1()(

22

222

tCtC

tCtCtReffect

)(

)1()(

2 tC

tCtReffect

effectE 2

effectE2

)(2 tE effect

)(tE effect

Finite size effects (FSE)

• Lattice volume is not sufficiently large.

• Still, FSE can be estimated using ChPT a) Corrections to the pion mass and Decay constants (Recent study: Colangelo et al)

b) Subleading terms to the Lüscher’s formula (Badeque and Sato (2006),(2007))

ref) • R. Brower, S.Chandrasekharan , J.W.Negele, and U.-J. Wiese (2003)

• S. Aoki, H. Fukaya, S. Hashimoto, and T. Onogi (2007)

1.9[fm])(L

• Additional effect due to fixed topology-Correction term to the n-point Green function

-I=2 Pi-Pi scattering length has no correction

at LO, but pion mass has correction at LO

-In our analysis, only pion mass is corrected From Noaki’s talkFrom Noaki’s talk

Result (1): energy spectrum

• We performed chiral extrapolation using NLO ChPT

where F and F0 are pion decay constants in the chiral limit.

• Because this quantity can be described only by decay constant in the chiral limit, we used F0 from 2pt function in the chiral limit (Noaki’ talk).

)(log5.38

18

1 )2(2

2

22

2

20

20

II

lm

F

m

Fm

a

energy spectra for two- and four-point correlators

qm

]GeV[E

Lüscher’s formulaLüscher’s formula+

Scattering Length Scattering Length

mE 22

Result (2): correction

GeV2m]GeV[/ 22

0

ma I

exp

after FSE correction (blue)

before FSE correction (yellow)

From the decay constant

• Energy difference is converted to the scattering length.

• Massless limit is from the decay constant data.

• FSE correction is made: relevant only near the chiral limit.

• Energy difference is converted to the scattering length.

• Massless limit is from the decay constant data.

• FSE correction is made: relevant only near the chiral limit.

20

0

16

1

Fm

a

Result (3): chiral fit GeV2

m]GeV[/ 220

ma I

exp

• Data show curvature toward chiral limit .

• One- loop ChPT has too strong curvature to fit the data.

• Data at smallest mass (ma=0.015) is away from the fit curve. Possibly FSE?

• Data show curvature toward chiral limit .

• One- loop ChPT has too strong curvature to fit the data.

• Data at smallest mass (ma=0.015) is away from the fit curve. Possibly FSE?

0

2

fit w/o

( / dof 3.5)

F F

0

2

fit with

( / dof 70)

F F

)(log5.3

161

16

1 )2(2

2

22

2

20

0

Ilm

F

m

Fm

a

Summary

• We calculated I=2 scattering length with two-flavors of dynamical overlap fermions.– Exact chiral symmetry allows us to use the standard ChPT

formulas.

• Lattice volume is not large enough– Finite size effect is corrected using known analytic results.

• Fit with one-loop ChPT is attempted.– Not well fitted over the whole mass region.– Two-loop analysis to be done.