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Dynamical overlap fermions in the ε-regime. Hidenori Fukaya (RIKEN), T.Kaneko, S.Hashimoto, K.Kanaya,H.Matsufuru, K.Ogawa,M.Okamoto, T.Onogi, N.Yamada,for JLQCD collaboration. Contents. Introduction Numerical cost Static quark potential with almost massless sea quarks

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## Dynamical overlap fermions in the ε-regime

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**Dynamical overlap fermions in the ε-regime**Hidenori Fukaya (RIKEN), T.Kaneko, S.Hashimoto, K.Kanaya,H.Matsufuru, K.Ogawa,M.Okamoto, T.Onogi, N.Yamada,forJLQCD collaboration**Contents**• Introduction • Numerical cost • Static quark potential with almost massless sea quarks • Low eigenmodes • Pion correlators in the ε-regime • Summary and discussion**Introduction**• Standard lattice QCD simulations require • V→∞, m→0, a→0 limits. • In the ε-regime ( 1/ΛQCD < L < 1/mπ), • Finite V effects are calculated on ChPT side. • m→0 can be reached in small V. • Low-energy constants of ChPT can be extracted without chiral extrapolation. • m→0 ⇒ Chiral symmetry is crucial ⇒ The overlap fermions Neuberger,Phys.Lett.B417(1998)141,Phys.Lett.B427(1998)353**Introduction**• JLQCD’s dynamical overlap fermion project • Iwasaki gauge action + Nf=2 overlap fermions • Q=0 topological sector only. • β~2.3-2.4 (1/a~1.69 -1.9GeV) . • Lattice size = 163×32 (L~1.8fm with β=2.35). • Hasenbusch acceleration. [Hasenbusch,Phys.Lett.B519(2001)177] • Sea quark mass is 2MeV – 120MeV (with an assumption Zm ~ Zmquenched=1.8 ). • Details →Matsufuru’s talk (Algorithm part2 15:40 today). • m~2.0MeV on L~1.8fm is in the ε-regime.**Introduction – references -**• ε-regime in ChPT • Gasser & Leutwyler, Phys.Lett.B188(1967)477, Nucl.Phys.B307(1988)763 • Hansen, Nucl.Phys.B345(1990)685 • Hansen & Leutwyler, Nucl.Phys.B350(1991)201 • Leutwyler & Smilga, Phys.Rev.D46(1992)5607 • Sharpe, Phys.Rev.D46(1992)3146 • Damgaard, Diamantini,Hernandez & Jansen, Nucl.Phys.B629(2002)445, • Damgaard,Hernandez,Jansen,Laine & Lellouch,Nucl.Phys.B656(2003)226 • Hernandez & Laine, JHEP301(2003)063, hep-lat/0607027**Introduction – references -**• Quenched lattice • Prelovsek & Orginos,Nucl.Phys.Proc.Suppl.119(2003)822 • Giusti,Hoelbling,Luscher & Wittig,Compt.Phys.Commun.153(2003)31 • Giusti,Luscher,Weisz & Wittig,JHEP0311(2003)023 • Bietenholz,Chiarappa,Jansen,Nagai & Shcheredin, JHEP0402(2004)023 • Giusti,Hernandez,Laine,Weisz & Wittig,JHEP0401(2004)003 • Giusti,Hernandez,Laine,Weisz & Wittig,JHEP0404(2004)013 • Ogawa & Hashimoto,Prog.Theor.Phys.114(2005)609 • HF,Hashimoto & Ogawa,Prog.Theor.Phys.114(2005)451 • Mehen & Tiburzi,Phys.Rev.D72,014501(2005) • Bietenholz & Shcheredin,PoSLAT2005,138(2006) • Giusti & Necco,PoSLAT2005,132(2006) • Damgaard,Heller,Splittorff & Svetitsky,Phys.Rev.D72,091501(2005) • Bietenholz & Shcheredin, hep-lat/0605013**Introduction – references -**• Nf=2 overlap fermion (m~35-100MeV) • DeGrand & Schaefer, Phys.Rev.D72,054503 • Nf=2 staggered fermion with isospin chemical potential • Damgaard,Heller,Splittorff,Svetitsky, & Toublan,Phys.Rev.D73,074023(2006) →Plenary talk by Splittorff. • Related talks –overlap fermions in JLQCD- • Kaneko -> overview of JLQCD overlap fermion project. • Yamada-> Locality of dynamical overlap quarks. • Hashimoto-> dynamical overlap at a fixed topology. • Okamoto-> pion mass, decay constant. • Matsufuru-> algorithm**Numerical cost**• CG iteration ∝|λmax + m|2/ |λmin+ m|2. • In large V regime,λmin ~ 1/(ΣV) ~ 0 →CG iteration ∝1/m2. • In the ε-regime, λlow feels repulsive force from 0; Chiral Random Matrix Theory (RMT) indicates 〈 λmin 〉ΣV ~ 4.34 >> mΣV [Nf=2,Q=0] →CG iteration ∝ 1/ λ2minRMT (independent of m). • But auto-correlation time seems longer. • On BlueGene/L 512node （2.8Tflops）, 1trj / 1h , autocorrelation ~ 100 trj • We’ve generated 1400trj (100 confs) in Q=0 sector. (400trj was discarded for thermalization.)**Static quark potential with almost massless quarks**• String breaking at m~2MeV? → No indication. • Too small V for two static-light mesons? • Wilson loop has small overlap with static-light meson operators? • Anyway, a ~ 0.11fm [Sommer scale r0~0.49fm].**Nf=2,Q=0 (Nf=0,Q=2)**Low eigenmodes - compared with ChRMT - • λ’ ≡ Im λ/ (1- Re λ/2) (λ is eigenvalue of Dov ) is compared with ChRMT. [λ was calculated by Lanczos algorithm with Chebyshev acceleration] • The lowest mode should feel stronger repulsive force from 0 (Nf=2, m=0); 〈λmin〉ΣV ~ 4.34. than quenched case; 〈λmin〉ΣV ~ 1.77. • The ratio 〈λi〉/〈λj〉 (λi: i-th eigenvalue) shows a good agreement in both quenched and Nf=2 cases. • m dependence is also interesting… • From λmin, Σ1/3= 228.9(3.6) MeV [bare lattice value] is obtained.**Pion correlators in the ε-regime**• Pion correlator in the ε-regime ChPT is not exponential but quadratic; • Low-mode averaging DeGrand & Schaefer,Comp.Phys.Commun. 159 (2004)185, Giusti,Hernandez,Laine,Weisz & Wittig,JHEP0404(2004)013 is very effective when m→0 since ~90% contribution is from 100 lowest-modes, which are obtained with Lanczos algorithm. Higher mode contribution is calculated with a point source, using low-mode preconditioning.**Pion correlators in the ε-regime**• With an input; Σ= 228.9(3.6) MeV , from lowest eigenvalue, quadratic fit (fit range=[10,22],β1=0) worked well. [χ2 /dof ~0.25.] • Fπ = 86.2(6.9)MeV is obtained [preliminary]. • Error from Σ is not taken into account. • Consistency with the other correlators (axial, scalar etc.) would be important. • Fitting range is appropriate? • O(a2) effects? • Higher order ChPT contribution？**Summary and discussion**• Nf=2 dynamical overlap fermion at fixed topology can reach m~2MeV on a 16332 lattice (a~0.11fm, L~1.8fm). CG iteration ≠1/m2 but 1/(λRMT+m)2 • Wilson loops shows no indication of string breaking. • The eigenvalue distribution is consistent with ChRMT in the ε-regime⇒ Σ • Sea quark mass dependence of low-modes was also observed. • Low-mode averaging is effective to evaluate the meson correlators. ⇒ Fπ**Summary and discussion**For future works, • The other connected meson correlators and condensates with higher accuracy. • Q dependence. In fact, Q≠0 sector would be easier. ( λmin is higher after projecting out 0-modes.) • Disconnected correlators with low-mode averaging. • Baryon correlators. • Including chemical potentials to valence quark to extract Fπ. • String breaking ?? • ρ→ππ decay ??? • …

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