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1 Meson correlators of two-flavor QCD in the epsilon-regime Hidenori Fukaya (RIKEN) with S.Aoki, S.Hashimoto, T.Kaneko, H.Matsufuru, J.Noaki, K.Ogawa, T.Onogi and N.Yamada [JLQCD collaboration]

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2 1. Introduction The chiral limit is difficult. The standard way requires before. Lattice QCD in ( ) [Necco (plenary), Akemann, DeGrand, Shindler (poster), Cecile, Hierl (chiral), Hernandez (weak)…] Finite effects can be estimated within ChPT ( ). is not very expensive. -> the chiral symmetry is essential. -> the dynamical overlap fermions.

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3 1. Introduction JLQCD collaboration achieved 2-flavor QCD simulations with the dynamical overlap quarks on a 16 3 32(~1.7-2fm) lattice with a~0.11- 0.13fm at Q=0 sector. the quark mass down to ~3MeV ! (enough to reach the epsilon-regime.) The Dirac spectrum [JLQCD, Phys.Rev.Lett.98,172001(2007)] shows a good agreement with Banks-Casher relation. with finite V correction via Random Matrix Theory (RMT), we obtained the chiral condensate, statistical systematic

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4 1. Introduction ChPT in the epsilon-regime [Gasser & Leutwyler, 1987] RMT does not know. Direct comparison with ChPT at -> more accurate (condensate). -> pion decay constant Meson correlators in the epsilon-regime [Hansen, 1990, 1991, Damgaard et al, 2002] are quadratic function of t; where A and B are expressed by the “finite volume” condensate, which is sensitive to m and topological charge Q.

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5 1. Introduction Partially quenched ChPT in the epsilon-regime [P.H.Damgaard & HF, arXiv:0707.3740, Bernardoni & Hernandez, arXiv:0707.3887] The previous known results are limited to degenerate cases. We extend ChPT to the partially quenched theory. Pseudoscalar and scalar channels are done; the correlators are expressed by of the “partially quenched finite volume” condensate, with which we can use the different valence quark masses to extract and. Axial vector and vector channels are in preparation. A 0 +V 0 calculated by the latter authors.

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6 1. Introduction The goal of this work On a (1.8fm) 4 lattice with a~0.11fm, 2-flavor QCD simulation with m~3MeV is achieved. The Dirac spectrum shows a qualitative agreement with RMT prediction, however, has ~10% error of effects. Therefore, our goal is to determine to by comparing meson correlators with (partially quenched) ChPT.

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7 Contents Introduction Lattice simulations Results Conclusion Related talks and posters Plenary talk by H.Matsufuru, “meson spectrum” by J.Noaki (chiral), “2+1 flavor simulations” by S.Hashimoto (hadron spectroscopy), “topology” by T.W.Chiu and T.Onogi (chiral), “pion form factor” by T.Kaneko (hadron structure), “pi±pi0 difference” by E.Shintani (hadron spectroscopy), “B K ” by N.Yamada (weak).

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8 2. Lattice simulations Lattice size = 16 3 32 (L~1.8fm.). a~0.11 fm. (determined by Sommer scale r0=0.49fm.) Iwasaki gauge action with. Extra topology fixing determinant. 2-flavor dynamical overlap quarks. ma = 0.002 (~3MeV). m v a=0.0005, 0.001,0.002, 0.003 [1-4MeV]. topological sector is limited to Q=0. 460 confs from 5000 trj. Details -> Matsufuru’s plenary talk.

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9 2. Lattice simulations Numerical cost Finite volume helps us to simulate very light quarks since the lowest eigenvalue of the Dirac operator are uplifted by an amount of 1/V. m~3MeV is possible with L~1.8fm !

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10 2. Lattice simulations Low-mode averaging [DeGrand & Schaefer, 2004,, Giusti,Hernandez,Laine,Weisz & Wittig,2004.] We calculate PS, S, V0, A0 correlation functions with a technique called low-mode averaging (LMA) with the lowest 100 Dirac-eigenmodes. PS, S -> the fluctuation is drastically suppressed. V0, A0 -> the improvement is marginal. PS-PS A0-A0

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11 Axial vector correlator (m v =m sea =3MeV) We use the ultra local definition of A 0 which is not a conserved current -> need renormalization. We calculate From 2-parameter fit with ChPT, chiral condensate, pion decay const, （ Fit range : t=12-20, chi 2 /d.o.f. ~ 0.01) are obtained. Note: A 0 A 0 is not very sensitive to. 3. Results

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12 3. Results Pseudoscalar correlators (m v =m sea =3MeV) With as an input, 1-parameter fit of PP correlator works well and condensate is obtained. (fit range: t=12-20, chi 2 /d.o.f.=0.07.) PP correlator is sensitive to. A0A0 is sensitive to. -> With the simultaneous 2-parameter fit with PP and A 0 A 0 correlator, we obtain to in lattice unit. (fit range : t=12-20, chi 2 /d.o.f.=0.02.)

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13 3. Results SS V 0 V 0 Consistency with SS and V 0 V 0 (m v =m sea =3MeV) are consistent with SS and V 0 V 0 channels ! (No free parameter left. )

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14 3. Results Consistency with Partially quenched ChPT are also consistent with partially quenched ChPT but the valence quark mass dependence is weak. (No free parameter left)

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15 3. Results Consistency with Dirac spectrum If non-zero modes of ChPT are integrated out, there remains the zero-mode integral with “effective” chiral condensate, In fact, this value agree well with the value via Dirac spectrum compared with RMT, -> support our estimate of correction.

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16 3. Results Non-perturbative renormalization Since is the lattice bare value, it should be renormalized. We calculated the renormalization factor in a non-perturbative RI/MOM scheme on the lattice, match with MS bar scheme, with the perturbation theory, and obtained (tree) (non-perturbative)

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17 3. Results Systematic errors Different channels, PP, A 0 A 0, SS, V 0 V 0, their partially quenched correlators, and the Dirac spectrum are all consistent. Fit range : from t min ~10(1.1fm) to 15 (1.7fm), both are stable (within 1%) with similar error-bars. Finite V : taken into account in the analysis. Finite a : overlap fermion is automatically free from O(a). Finite m : m~3MeV is already very close to the chiral limit. But =87.3(5.5)MeV slightly different from the value [~78(3)(1)MeV] (Noaki’s talk) in the p-regime.

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18 4. Conclusion On a (1.8fm) 4 lattice with a~0.11fm, 2-flavor QCD simulation with m~3MeV is achieved, which is in the epsilon-regime. We calculate the various meson correlators with low-mode averaging (LMA). From PP (sensitive to ) and A 0 A 0 (sensitive to ) channels, compared with ChPT, to accuracy, are obtained (preliminary). They are consistent with SS and V 0 V 0 channels. Also consistent with partially quenched ChPT. Also consistent with result from Dirac spectrum. But slightly deviate from p-regime results.

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19 4. Conclusion Future works Larger volumes Smaller lattice spacings Partially quenched analysis for A 0 A 0 and V 0 V 0 channels. 2+1 flavors…

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