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

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

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

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  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  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  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  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  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  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  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 → Chiral symmetry and topology M.Luescher,Nucl.Phys.B568,162 (‘00)

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  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  The details of the simulation As a test run on a lattice with a ~ GeV (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) – samples per 10 trj of Hybrid Monte Carlo algorithm trj for each m are performed. Q=0 topological sector 3. JLQCD’s overlap fermion project

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

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 (~3fm) 4 lattice or larger are planned. 4. Finite volume and fixed topology

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  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  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 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 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  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/

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/ , M.Hamanaka,H.Kajiura,Phys.Lett.B551,360(‘03)

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 Fpi

31 Mv

32 Mps 2 /m