1. 1 Approaching the chiral limit in lattice QCD Hidenori Fukaya (RIKEN Wako) for JLQCD collaboration Ph.D. thesis [hep-lat/0603008], JLQCD collaboration,Phys.Rev.D74:094505(2006)[hep- lat/0607020], hep-lat/0607093, hep-lat/0610011, hep- lat/0610024 and hep-lat/0610026.

2. 2 Lattice gauge theory gives a non-perturbative definition of the quantum field theory. finite degrees of freedom. ⇒ Monte Carlo simulations ⇒ very powerful tool to study QCD; Hadron spectrum Non-perturbative renormalization Chiral transition Quark gluon plasma 1. Introduction

3. 3 But the lattice regularization spoils a lot of symmetries … Translational symmetry Lorentz invariance Chiral symmetry and topology Supersymmetry … 1. Introduction

4. 4 The chiral limit (m → 0) is difficult.  Losing chiral symmetry to avoid fermion doubling.  Large computational cost for m → 0. Wilson Dirac operator (used in JLQCD’s previous works) breaks chiral symmetry and requires  additive renormalization of quark mass.  unwanted operator mixing with opposite chirality  symmetry breaking terms in chiral perturbation theory.  Complitcated extrapolation from m u, m d > 50MeV. ⇒ Large systematic uncertainties in m~ a few MeV results. 1. Introduction Nielsen and Ninomiya, Nucl.Phys.B185,20(‘81)

5. 5 Our strategy in new JLQCD project 1.Achieve the chiral symmetry at quantum level on the lattice by overlap fermion action [ Ginsparg-Wilson relation] and topology conserving action [ Luescher’s admissibility condition] 2.Approach m u, m d ~ O(1) MeV. 1. Introduction Ginsparg & WilsonPhys.Rev.D25,2649(‘82) Neuberger, Phys.Lett.B417,141(‘98) M.Luescher,Nucl.Phys.B568,162 (‘00)

6. 6 Plan of my talk 1.Introduction 2.Chiral symmetry and topology 3.JLQCD ’ s overlap fermion project 4.Finite volume and fixed topology 5.Summary and discussion

7. 7 Nielsen-Ninomiya theorem: Any local Dirac operator satisfying has unphysical poles (doublers). Example - free fermion – Continuum has no doubler. Lattice has unphysical poles at. Wilson Dirac operator (Wilson fermion) Doublers are decoupled but spoils chiral symmetry. 2. Chiral symmetry and topology Nielsen and Ninomiya, Nucl.Phys.B185,20(1981)

8. 8 0 2/a4/a6/a  Eigenvalue distribution of Dirac operator 2. Chiral symmetry and topology continuum (massive) m 1/a -1/a

9. 9 Wilson fermion  Eigenvalue distribution of Dirac operator 2. Chiral symmetry and topology 1/a -1/a Naïve lattice fermion 16 lines 0 2/a4/a6/a (massive) m Doublers are massive. m is not well-defined. The index is not well-defined. 1 physical 4 heavy 6 heavy 4 heavy 1 heavy dense sparse but nonzero density until a → 0.

10. 10  The overlap fermion action The Neuberger ’ s overlap operator: satisfying the Ginsparg-Wilson relation: realizes ‘ modified ’ exact chiral symmetry on the lattice; the action is invariant under NOTE Expansion in Wilson Dirac operator ⇒ No doubler. Fermion measure is not invariant; ⇒ chiral anomaly, index theorem Phys.Rev.D25,2649(‘82) Phys.Lett.B417,141(‘98) M.Luescher,Phys.Lett.B428,342(1998) (Talk by Kikukawa)

11. 11  Eigenvalue distribution of Dirac operator 2. Chiral symmetry and topology 1/a -1/a 0 2/a4/a6/a The overlap fermion Doublers are massive. D is smooth except for

12. 12  Eigenvalue distribution of Dirac operator 2. Chiral symmetry and topology 1/a -1/a 0 2/a4/a6/a The overlap fermion (massive) m m is well-defined. index is well-defined.

13. 13  Eigenvalue distribution of Dirac operator 2. Chiral symmetry and topology 1/a -1/a 0 2/a4/a6/a The overlap fermion Theoretically ill-defined. Large simulation cost.

14. 14  The topology (index) changes 2. Chiral symmetry and topology 1/a -1/a 0 2/a4/a6/a The complex modes make pairsThe real modes are chiral eigenstates. Hw=Dw-1=0 ⇒ Topology boundary

15. 15  The overlap Dirac operator becomes ill-defined when Hw=0 forms topology boundaries. These zero-modes are lattice artifacts(excluded in a → ∞limit.) In the polynomial expansion of D, The discontinuity of the determinant requires reflection/refraction (Fodor et al. JHEP0408:003,2004) ~ V 2 algorithm.

16. 16  Topology conserving gauge action To achieve |Hw| > 0 [Luescher ’ s “ admissibility ” condition], we modify the lattice gauge action. We found that adding with small μ, is the best and easiest way in the numerical simulations ( See JLQCD collaboration, Phys.Rev.D74:09505,2006 ) Note: S top → ∞ when Hw → 0 and S top → 0 when a → 0. 2. Chiral symmetry and topology M.Luescher,Nucl.Phys.B568,162 (‘00)

17. 17 Our strategy in new JLQCD project 1.Achieve the chiral symmetry at quantum level on the lattice by overlap fermion action [ Ginsparg-Wilson relation] and topology conserving action S top [ Luescher’s admissibility condition] 2.Approach m u, m d ~ O(1) MeV. 2. Chiral symmetry and topology Ginsparg & WilsonPhys.Rev.D25,2649(‘82) Neuberger, Phys.Lett.B417,141(‘98) M.Luescher,Nucl.Phys.B568,162 (‘00)

18. 18  Numerical cost Simulation of overlap fermion was thought to be impossible; D_ov is a O(100) degree polynomial of D_wilson. The non-smooth determinant on topology boundaries requires extra factor ~10 numerical cost. ⇒ The cost of D_ov ~ 1000 times of D_wilson’s. However, S top can cut the latter numerical cost ~10 times faster New supercomputer at KEK ~60TFLOPS ~50 times Many algorithmic improvements ~ 5-10 times we can overcome this difficulty ! 3. JLQCD’s overlap fermion project

19. 19  The details of the simulation As a test run on a 16 3 32 lattice with a ~ 1.6-1.8GeV (L ~ 2fm), we have achieved 2-flavor QCD simulations with overlap quarks with the quark mass down to ~2MeV. NOTE m >50MeV with non-chiral fermion in previous JLQCD works. Iwasaki (beta=2.3) + S top (μ=0.2) gauge action Overlap operator in Zolotarev expression Quark masses ： ma=0.002(2MeV) – 0.1. 1 samples per 10 trj of Hybrid Monte Carlo algorithm. 2000-5000 trj for each m are performed. Q=0 topological sector 3. JLQCD’s overlap fermion project

20. 20  Numerical data of test run (Preliminary) Both data confirm the exact chiral symmetry. 3. JLQCD’s overlap fermion project

21. 21  Systematic error from finite V and fixed Q Our test run on (~2fm) 4 lattice is limited to a fixed topological sector (Q=0). Any observable is different from θ=0 results; where χ is topological susceptibility and f is an unknown function of Q. ⇒ needs careful treatment of finite V and fixed Q. Q=2, 4 runs are started. 24 3 48 (~3fm) 4 lattice or larger are planned. 4. Finite volume and fixed topology

22. 22  ChPT and ChRMT with finite V and fixed Q However, even on a small lattice, V and Q effects can be evaluated by the effective theory: chiral perturbation theory (ChPT) or chiral random matrix theory (ChRMT). They are valid, in particular, when m π L<1 (ε-regime). ⇒ m~2MeV, L~2fm is good. Finite V effects on ChRMT : discrete Dirac spectrum ⇒ chiral condensate Σ. Finite V effects on ChPT : pion correlator is not exponential but quadratic. ⇒ pion decay const. F π. 4. Finite volume and fixed topology

23. 23  Dirac spectrum and ChRMT (Preliminary) Nf=2 Nf=0 Lowest eigenvalue (Nf=2) ⇒ Σ=(233.9(2.6)MeV) 3 4. Finite volume and fixed topology

24. 24  Pion correlator and ChPT (Preliminary) The quadratic fit (fit range=[10,22],β1=0) worked well. [χ2 /dof ~0.25.] Fπ = 86(7)MeV is obtained [preliminary]. NOTE: Our data are at m~2MeV. we don’t need chiral extrapolation. 4. Finite volume and fixed topology

25. 25 The chiral limit is within our reach now!  Exact chiral symmetry at quantum level can be achieved in lattice QCD simulations with Overlap fermion action Topology conserving gauge action  Our test run on (~2fm) 4 lattice, we’ve simulated Nf=2 dynamical overlap quarks with m~2MeV.  Finite V and Q dependences are important. ChRMT in finite V ⇒ Σ~2.193E-03. ChPT in finite V ⇒ F π ~86MeV. 5. Summary and discussion

26. 26 To do  Precise measurement of hadron spectrum, started.  2+1 flavor, started.  Different Q, started.  Larger lattices, prepared.  B K, started.  Non-perturbative renormalization, prepared. Future works  θ-vacuum  ρ → ππ decay  Finite temperature… 5. Summary and discussion

27. 27  How to sum up the different topological sectors Formally, With an assumption, The ratio can be given by the topological susceptibility, if it has small Q and V’ dependences. Parallel tempering + Fodor method may also be useful. V’V’ Z.Fodor et al. hep-lat/0510117

28. 28  Initial configuration For topologically non-trivial initial configuration, we use a discretized version of instanton solution on 4D torus; which gives constant field strength with arbitrary Q. A.Gonzalez-Arroyo,hep-th/9807108, M.Hamanaka,H.Kajiura,Phys.Lett.B551,360(‘03)

29. 29  Topology dependence If, any observable at a fixed topology in general theory (with θvacuum) can be written as Brower et al, Phys.Lett.B560(2003)64 In QCD, ⇒ Unless, （ like NEDM ） Q-dependence is negligible. Shintani et al,Phys.Rev.D72:014504,2005

30. 30 Fpi

31. 31 Mv

32. 32 Mps 2 /m