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Pion Correlators in the ε- regime

Pion Correlators in the ε- regime. Hidenori Fukaya (YITP) collaboration with S. Hashimoto (KEK) and K.Ogawa (Sokendai). 0. Contents. Introduction Lattice Simulations Results 　（ quenched ） Conclusion. 1. Introduction. 1-1. Our Goals Lattice QCD

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Pion Correlators in the ε- regime

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

  2. 0. Contents • Introduction • Lattice Simulations • Results　（quenched） • Conclusion

  3. 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).

  4. 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 ofChPT (in small V). In particular, dependence on topological charge Q and X ≡ mΣV is important . S.R.Sharpe(‘01)P.H.Damgaard et al.(‘02)… J.Gasser,H.Leutwyler(‘87),F.C.Hansen(‘90), H.Leutwyler,A.Smilga(92)…

  5. 1-2. Setup • To simulatem～0region,’Exact’ chiral symmetry is required. ⇒Overlap operator(Chebychev polynomial (of order～150 )) which satisfiesGinsparg-Wilson relation. • Fittingpion correlatorsin smallVat differentQandmwith ChPT in the ε-regime　⇒　extractΣ, Fπ, α, m0 P.H.Ginsparg,K.G.Wilson(‘82), H.Neuberger(‘98)

  6. 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. 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) P.H.Damgaard et al. (02)

  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. The difference <0.5% for 3 ≦ t ≦ 7 . 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|.

  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 x0 and t0; reduces the noise almost 10 times !

  10. 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.

  11. m = 12.8 MeV m = 5 MeV m = 8 MeV Q =3 Q =3 Q =2 Q =2 Q =1 Q =1 3. Results (quenched QCD) preliminary 3-1. Pion correlators Our data show remarkable Q and m dependences.

  12. m=10.2MeV m=2.6MeV m=5 MeV 3-2. Low energy parameters preliminary Using we simultaneously fit all of our data (15 correlators ) with the function; ←Ogawa’s talk P.H.Damgaard (02) We obtain Σ = (307±23MeV)3, Fπ= 111.1±5.2MeV, α = 0.07±0.65, m0 = 958±44 MeV,χ2/dof=1.5.

  13. 4. Conclusion 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。 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, Fπ=111.1±5.2 MeV, α=0.07±0.65, m0=958±44 MeV.

  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.

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