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

1. Introduction 2. Formalism 3. Results 4. Summary

•

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

•The overlap action with quark mass “

m

” is defined as

D overlap

(

m

) 

D



m

 1 1 2

a D

•This operator respects the chiral symmetry on the lattice. γ 5

D



D

γ 5 

a D

γ 5

D

　 (GW relation) where

D

 1   1 

a A A

†

A

 

A



aD wilson

(

M

0 )

a



a

/

M

0  

i



t a

 5  •Disadvantages - Numerically costly - So, limited volume On the lattice  

i



t a

 5 ( 1  1 2

a D

) 

Lüscher’s formula

• Two particles with momentum “

k

” confined in a large volume box

k

cot  (

k

)  1 

L

S

 

k

2

L

2 4  2   •

L

: Length of the box where

S

  2

k L

2 4  2    lim   

n

 

n

1

n

2  4  2  4  

n

  • For s-wave, M

.

Lüscher(1986,1991

)

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

L

.

E

 2    

E

2

m

   4 

a

0

m



L

3 1 

c

1

a

0

L



c

2

a

0

L

2   

a

0 : scattering length

Setup

• • Machine – BlueGene/L @ KEK We used gauge configurations generated by JLQCD (Matsufuru) – Lattice spacing = 0.1184(12) [fm] , from

r

0 –

m q



m s

/ 6 ～

m s

~ 0.49 [fm] – pick configs every 100 traj to avoid autocorrelations.

– with Low mode averaging Lattice Size Number of flavors Topological Charge Gauge Action Fermion Action Source Type Gauge Fixing 16 3 32 2 0 Iwasaki Action Overlap Fermion Wall Source Coulomb Gauge

ma

0.015

0.025

0.035

0.050

0.070

0.100

No.Conf

99 96 93 92 92 92

Periodic boundary (1)

Our lattice is periodic in temporal direction Contaminations to the correlation functions exist 1. Two point correlation (1-Pion) 1-Pion Line (not quark line)

C

 (

t

) 

e



m



t



e



m

 (

T



t

) 2. Four point correlation (2-Pion) T : size of the temporal direction

C

2  (

t

) 

e



E

2 

t



e



E

2  (

T



t

)   Extra   2

e

( term

E

2  / 2 )  Constant

t

 (

E

2  / 2 )(

T



t

) 

Periodic boundary (2)

C

2  (

t

) 

A

cosh   

E

2 

T

2    

B

(

A

,

B R effect

2  (

t

) 

C

2 

C

2  (

t

(

t

)   1 ) 

C

2 

C

2 (

t

  (

t

1 ) ) 

E effect

2  (

t

)

R



effect

(

t

) 

C

 (

t

 1 )

C

2  (

t

) 

E



effect

(

t

) : independent of A & B Ex) m=0.050

: Constant )

E

2

effect

 Good plateau is identified from this function.

2 

E



effect excited state

Finite size effects (FSE)

• Lattice volume is not sufficiently large. (L  1.9[fm]) • 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)) • 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 talk ref) • • R. Brower, S.Chandrasekharan , J.W.Negele, and U.-J. Wiese (2003) S. Aoki, H. Fukaya, S. Hashimoto, and T. Onogi (2007)

Result (1): energy spectrum



E

[ GeV ]  

E

2   2

m

  energy spectra for two- and four point correlators

+

Lüscher’s formula Scattering Length • We performed chiral extrapolation using NLO ChPT

a

0

I

 2

m

   8  1

F

0 2    1  2 8 

m

 2

F

2  3 .

5  log  

m

2   2   

l

 (

I

 2 ) (  )       where

F

and

F 0

are pion decay constants in the chiral limit.

• Because this quantity can be described only by decay constant in the chiral limit, we used

F 0

from 2pt function in the chiral limit (Noaki’ talk).

m q

Result (2): correction

a I

0  2 /

m

 [ GeV  2 ]

m

 2  GeV  exp after FSE correction (blue) before FSE correction (yellow) • 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.

From the decay constant

a

0

m

   16  1

F

0 2

a I

0  2 /

m

 [ GeV  2 ] exp

Result (3): chiral fit

m

 2  GeV  ( fit w/o

F

 2 / dof 

F

0  3.5) • Data show curvature toward chiral limit .

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

( fit with

F

 2 / dof 

F

0  70)

a

0

m

   16  1

F

0 2    1 

m

 16  2 2

F

2  3 .

5  log  

m

 2  2   

l

 (

I

 2 ) (  )       • Data at smallest mass (ma=0.015) is away from the fit curve. Possibly FSE?

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.