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Pion Correlators in the ε- regime Hidenori Fukaya (YITP) collaboration with S. Hashimoto (KEK) and K.Ogawa (Sokendai)
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0. Contents 1.Introduction 2.Lattice Simulations 3.Results ( quenched ) 4.Conclusion
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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).
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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)…
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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)
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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.
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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)
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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.
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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 !
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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
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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
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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
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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。
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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.
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A. Full QCD Lowest 100 eigenvalues
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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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