Pion Correlators in the ε- regime

Hidenori Fukaya (YITP)

collaboration with

S. Hashimoto (KEK)

and K.Ogawa (Sokendai)

0. Contents

1. Introduction

2. Lattice Simulations

3. Results　（ quenched）4. Conclusion

1. Introduction

1-1. Our Goals 　 Lattice QCD

- 1st principle and non-perturbative calculation. Chiral perturbation theory (ChPT)

- Low energy effective theory of QCD (pion theory).

- Free parameters Fπ and Σ.

It is important to determine Fπ and Σ from

1-st principle calculation but simulations at

m~0 (m<30MeV) and large V (V>2fm) are difficult...

⇒ ⇒ Consider fm universe (ε-regime).Consider fm universe (ε-regime).

1. Introduction

1-1. Our Goals 　In the ε- regime ( mπL < 1 , LΛQCD>>1), we have ChPT with finite V correction. Quenched QCD simulation

⇒ low energy constants (Σ, Fπ, α…) of

quenched ChPT (in small V). Full QCD simulation⇒ 　 those of ChPT (in small V).In particular, dependence on topological charge Qand X ≡ mΣV is important .

J.Gasser,H.Leutwyler(‘87),F.C.Hansen(‘90),

H.Leutwyler,A.Smilga(92)…

S.R.Sharpe(‘01)P.H.Damgaard et al.(‘02)…

1-2. Setup

To simulate m ～ 0 region, ’Exact’ chiral symmetry is required.

⇒ 　 Overlap operator (Chebychev

polynomial (of order ～ 150 )) which

satisfies Ginsparg-Wilson relation. Fitting pion correlators in small V at different Q

and m with ChPT in the ε-regime 　⇒　 extract Σ, Fπ, α, m0

P.H.Ginsparg,K.G.Wilson(‘82), H.Neuberger(‘98)

P.H.Damgaard et al. (02)

1-3. Pion correlators in the ε-regime

Quenched ChPT in small V

Pion correlators are not exponential but

ChPT in small V (Nf=2)

where

and

Fitting the coefficient of H1(t) and H2(t) with

lattice data at various Q and m, we extract

Σ, Fπ, α, m0.

2. Lattice Simulations

2-1. Calculation of D -1

Overlap at m~0 Large numerical costs !⇒ Low mode preconditioning

We calculate lowest 100 eigenvalues and eigen functions so that we deform D as

⇒ 　 costs for at m=0 ~ costs for at m=100MeV !

L.Giusti et al.(03)

2-2. Low-mode contribution in pion correlators

Is the low-mode contribution dominant ?As m→0 　⇒　 low-modes must be important.

We find the contribution from is negligible

( ~ only 0.5 %.) for m<0.008 (12.8MeV)

and Q ≠ 0 at large t , so we can approximate

for large |x-y|.The difference <　0.5% for 3 t 7 .≦ ≦

2-2. Low-mode contribution in pion correlators

Pion source averaging over space-timeNow we know at all x. we know ⇒

at any x and y. Averaging over x0 and t0;

reduces the noise almost 10 times !

2-3. Numerical Simulations Size :β=5.85, 1/a = 1.6GeV, V=104 (1.23fm)4

Gauge fields: updated by plaquette action (quenched).

Fermion mass: m=0.016,0.032,0.048,0.064,0.008 ( 2.6MeV m 12.8 MeV !!)≦ ≦

100 eigenmodes are calculated by ARPACK. Q is evaluated from # of zero modes. Source pion is averaged over x=odd sites for

Q ≠ 0.

|Q| 0 1 2 3

# of conf.

50 76 57 19

3. Results (quenched QCD)

3-1. Pion correlators m = 5 MeV

Q =1

Q =2

Q =3

m = 8 MeV

Q =1

Q =2

Q =3

m = 12.8 MeV

Our data show remarkable Q and m dependences.

preliminarypreliminary

Using

we simultaneously fit all of our data

(15 correlators ) with the function;

←　 Ogawa’s talkP.H.Damgaard (02)

3-2. Low energy parameters

m=2.6MeVm=5 MeVm=10.2MeV

We obtain

Σ = (307±23 　 MeV)3, Fπ= 111.1±5.2MeV,

α = 0.07±0.65, m0 = 958±44 MeV, χ2/dof=1.5.

preliminarypreliminary

4. ConclusionIn quenched QCD in the ε-regime, using Overlap operator ‘exact’ ⇒ chiral symmetry, 2.6 MeV m ≦ ≦ 12.8 MeV , lowest 100 eigenmodes (dominance~99.5%), Pion source averaging over space-time,

( equivalent to 100 times statistics )

we compare the pion correlators with ChPT .

⇒ 　 The correlators show remarkable Q and

m dependences.

⇒ Σ=(307±23 MeV)3, Fπ=111.1±5.2 MeV,

　　 α=0.07±0.65, m0=958±44 MeV.

まとめ

（実質）１００倍の統計をためると

できなかったことができた。

4. ConclusionAs future works, a → 0 limit and renormalization, isosinglet meson correlators, full QCD 　 ( → Ogawa’s talk), consistency check with p-regime results,

will be important.

A. Full QCD

Lowest 100 eigenvalues

A. Full QCD Truncated determinant

The truncated determinant is equivalent to

adding a Pauli-Villars regulator as

where, for example,

γ→0 limit usual Pauli-Villars (gauge inv,local).⇒ Λ→0 limit quench QCD (good overlap config. ?)⇒ If Λa is fixed as a→0, unitarity is also restored.