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

2. 0. Contents 1.Introduction 2.Lattice Simulations 3.Results （ quenched ） 4.Conclusion

3. 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).

4. 1. Introduction 1-1. Our Goals In the ε- regime ( m π L >1), we have ChPT with finite V correction. Quenched QCD simulation ⇒ low energy constants ( Σ, F π, α …) of quenched ChPT (in small V). Full QCD simulation ⇒ those of ChPT (in small V). In particular, dependence on topological charge Q and 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)…

5. 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, F π, α, m 0 P.H.Ginsparg,K.G.Wilson(‘82), H.Neuberger(‘98)

6. 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, m 0.

7. 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)

8. 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.

9. 2-2. Low-mode contribution in pion correlators  Pion source averaging over space-time Now we know at all x. ⇒ we know at any x and y. Averaging over x 0 and t 0 ; reduces the noise almost 10 times !

10. 2-3. Numerical Simulations  Size :β=5.85, 1/a = 1.6GeV, V=10 4 (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|0123 # of conf.50765719

11. 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. preliminary

12. Using we simultaneously fit all of our data (15 correlators ) with the function; ← Ogawa’s talk P.H.Damgaard (02) 3-2. Low energy parameters m=2.6MeV m=5 MeV m=10.2MeV We obtain Σ = (307±23 MeV) 3, F π = 111.1±5.2MeV, α = 0.07±0.65, m 0 = 958±44 MeV, χ 2 /dof=1.5. preliminary

13. 4. Conclusion In 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, F π =111.1±5.2 MeV, α=0.07±0.65, m 0 =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。

14. 4. Conclusion As future works,  a → 0 limit and renormalization,  isosinglet meson correlators,  full QCD ( → Ogawa’s talk),  consistency check with p-regime results, will be important.

15. A. Full QCD  Lowest 100 eigenvalues

16. 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.