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]

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.

3 1. Introduction JLQCD collaboration  achieved 2-flavor QCD simulations with the dynamical overlap quarks on a (~1.7-2fm) lattice with a~ fm 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

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.

5 1. Introduction Partially quenched ChPT in the epsilon-regime [P.H.Damgaard & HF, arXiv: , Bernardoni & Hernandez, arXiv: ]  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.

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.

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).

8 2. Lattice simulations Lattice size = (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 = (~3MeV). m v a=0.0005, 0.001,0.002, [1-4MeV]. topological sector is limited to Q= confs from 5000 trj. Details -> Matsufuru’s plenary talk.

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 !

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

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

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.)

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. )

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)

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.

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)

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.

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.

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…