Pion Correlators in the ε- regime Hidenori Fukaya (YITP) collaboration with S. Hashimoto (KEK) and...

16
Pion Correlators in the ε- regime Hidenori Fukaya (YITP) collaboration with S. Hashimoto (KEK) and K.Ogawa (Sokendai)

description

1. Introduction 1-1. Our Goals  Lattice QCD - 1 st principle and non-perturbative calculation.  Chiral perturbation theory (ChPT) - Low energy effective theory of QCD (pion theory). - Free parameters F π and Σ. It is important to determine F π and Σ from 1-st principle calculation but simulations at m~0 (m 2fm) are difficult... ⇒ Consider fm universe (ε-regime).

Transcript of Pion Correlators in the ε- regime Hidenori Fukaya (YITP) collaboration with S. Hashimoto (KEK) and...

Page 1: Pion Correlators in the ε- regime Hidenori Fukaya (YITP) collaboration with S. Hashimoto (KEK) and K.Ogawa (Sokendai)

Pion Correlators in the ε- regime

Hidenori Fukaya (YITP) collaboration withS. Hashimoto (KEK)

and K.Ogawa (Sokendai)

Page 2: Pion Correlators in the ε- regime Hidenori Fukaya (YITP) collaboration with S. Hashimoto (KEK) and K.Ogawa (Sokendai)

0. Contents

1. Introduction2. Lattice Simulations3. Results ( quenched)4. Conclusion

Page 3: Pion Correlators in the ε- regime Hidenori Fukaya (YITP) collaboration with S. Hashimoto (KEK) and K.Ogawa (Sokendai)

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 atm~0 (m<30MeV) and large V (V>2fm) are difficult... ⇒ ⇒ Consider fm universe (ε-regime).Consider fm universe (ε-regime).

Page 4: Pion Correlators in the ε- regime Hidenori Fukaya (YITP) collaboration with S. Hashimoto (KEK) and K.Ogawa (Sokendai)

1. Introduction1-1. 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)…

Page 5: Pion Correlators in the ε- regime Hidenori Fukaya (YITP) collaboration with S. Hashimoto (KEK) and K.Ogawa (Sokendai)

1-2. Setup To simulate m ~ 0 region, ’Exact’ chiral symmet

ry 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)

Page 6: Pion Correlators in the ε- regime Hidenori Fukaya (YITP) collaboration with S. Hashimoto (KEK) and K.Ogawa (Sokendai)

P.H.Damgaard et al. (02)

1-3. Pion correlators in the ε-regime

Quenched ChPT in small VPion 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.

Page 7: Pion Correlators in the ε- regime Hidenori Fukaya (YITP) collaboration with S. Hashimoto (KEK) and K.Ogawa (Sokendai)

2. Lattice Simulations

2-1. Calculation of D -1Overlap at m~0 Large numerical costs !⇒

Low mode preconditioningWe 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)

Page 8: Pion Correlators in the ε- regime Hidenori Fukaya (YITP) collaboration with S. Hashimoto (KEK) and K.Ogawa (Sokendai)

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 .≦ ≦

Page 9: Pion Correlators in the ε- regime Hidenori Fukaya (YITP) collaboration with S. Hashimoto (KEK) and K.Ogawa (Sokendai)

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 !

Page 10: Pion Correlators in the ε- regime Hidenori Fukaya (YITP) collaboration with S. Hashimoto (KEK) and K.Ogawa (Sokendai)

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

Page 11: Pion Correlators in the ε- regime Hidenori Fukaya (YITP) collaboration with S. Hashimoto (KEK) and K.Ogawa (Sokendai)

3. Results (quenched QCD)

3-1. Pion correlators m = 5 MeV

Q =1

Q =2

Q =3

m = 8 MeV

Q =1Q =2

Q =3

m = 12.8 MeV

Our data show remarkable Q and m dependences.

preliminarypreliminary

Page 12: Pion Correlators in the ε- regime Hidenori Fukaya (YITP) collaboration with S. Hashimoto (KEK) and K.Ogawa (Sokendai)

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

Page 13: Pion Correlators in the ε- regime Hidenori Fukaya (YITP) collaboration with S. Hashimoto (KEK) and K.Ogawa (Sokendai)

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。

Page 14: Pion Correlators in the ε- regime Hidenori Fukaya (YITP) collaboration with S. Hashimoto (KEK) and K.Ogawa (Sokendai)

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.

Page 15: Pion Correlators in the ε- regime Hidenori Fukaya (YITP) collaboration with S. Hashimoto (KEK) and K.Ogawa (Sokendai)

A. Full QCD Lowest 100 eigenvalues

Page 16: Pion Correlators in the ε- regime Hidenori Fukaya (YITP) collaboration with S. Hashimoto (KEK) and K.Ogawa (Sokendai)

A. Full QCD Truncated determinant The truncated determinant is equivalent toadding 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.