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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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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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1. Introduction JLQCD collaboration

achieved 2-flavor QCD simulations with the dynamical overlap quarks on a 16332(~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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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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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. A0+V0 calculated by the latter authors.

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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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Contents1. Introduction2. Lattice simulations3. Results4. 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), “BK” by N.Yamada (weak).

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2. Lattice simulations Lattice size = 16332 (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). mv 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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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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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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Axial vector correlator (mv=msea=3MeV) We use the ultra local definition of A0 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, chi2/d.o.f. ~ 0.01)are obtained.

Note: A0A0 is not very sensitive to .

3. Results

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3. Results Pseudoscalar correlators (mv=msea=3MeV)

With as an input, 1-parameter fit of PP correlator works well and condensate is obtained.

(fit range: t=12-20, chi2/d.o.f.=0.07.) PP correlator is sensitive to . A0A0 is sensitive to .　　 -> 　 With the simultaneous 2-parameter fit with P

P and A0A0 correlator, we obtain to

in lattice unit. (fit range : t=12-20, chi2/d.o.f.=0.02.)

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3. Results

SS V0V0

Consistency with SS and V0V0 (mv=msea=3MeV)

are consistent with SS and V0V0 channels !

(No free parameter left. )

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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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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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3. Results Non-perturbative renormalization

Since is the lattice bare value,

it should be renormalized. We calculated 1. the renormalization factor in a non-perturbative RI/

MOM scheme on the lattice,

2. match with MS bar scheme, with the perturbation theory,

3. and obtained

(tree)(non-perturbative)

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3. Results Systematic errors

1. Different channels, PP, A0A0, SS, V0V0, their partially quenched correlators, and the Dirac spectrum are all consistent.

2. Fit range : from tmin~10(1.1fm) to 15 (1.7fm), both are stable (within 1%) with similar error-bars.

3. Finite V : 　 taken into account in the analysis.

4. Finite a : overlap fermion is automatically free from O(a).

5. 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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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 A0A0 (sensitive to ) channels, compared with ChPT,

to accuracy, are obtained (preliminary). They are consistent with SS and V0V0 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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4. Conclusion Future works

Larger volumes Smaller lattice spacings Partially quenched analysis for A0A0 and V0V0 channel

s. 2+1 flavors…