Approaching the chiral limit in lattice QCD

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.

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

1. Introduction 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 mu, md > 50MeV . ⇒Large systematic uncertainties in m~ a few MeV results. Nielsen and Ninomiya, Nucl.Phys.B185,20(‘81)

1. Introduction Our strategy in new JLQCD project • 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] • Approach mu, md ~ O(1) MeV. Neuberger, Phys.Lett.B417,141(‘98) Ginsparg & WilsonPhys.Rev.D25,2649(‘82) M.Luescher,Nucl.Phys.B568,162 (‘00)

Plan of my talk • Introduction • Chiral symmetry and topology • JLQCD’s overlap fermion project • Finite volume and fixed topology • Summary and discussion

2. Chiral symmetry and topology 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. Nielsen and Ninomiya, Nucl.Phys.B185,20(1981)

continuum (massive) 0 2/a 4/a 6/a m 2. Chiral symmetry and topology • Eigenvalue distribution of Dirac operator 1/a -1/a

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

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.Lett.B417,141(‘98) Phys.Rev.D25,2649(‘82) M.Luescher,Phys.Lett.B428,342(1998) (Talk by Kikukawa)

The overlap fermion 0 2/a 4/a 6/a • Doublers are massive. • D is smooth except for 2. Chiral symmetry and topology • Eigenvalue distribution of Dirac operator 1/a -1/a

The overlap fermion (massive) 0 2/a 4/a 6/a m 2. Chiral symmetry and topology • Eigenvalue distribution of Dirac operator 1/a -1/a • m is well-defined. • index is well-defined.

The overlap fermion 0 2/a 4/a 6/a 2. Chiral symmetry and topology • Eigenvalue distribution of Dirac operator 1/a -1/a • Theoretically ill-defined. • Large simulation cost.

The complex modes make pairs 0 2/a 4/a The real modes are chiral eigenstates. 6/a 2. Chiral symmetry and topology • The topology (index) changes Hw=Dw-1=0 ⇒ Topology boundary 1/a -1/a

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) ~ V2 algorithm.

2. Chiral symmetry and topology • 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: Stop →∞when Hw→0 and Stop→0 when a→0. M.Luescher,Nucl.Phys.B568,162 (‘00)

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

3. JLQCD’s overlap fermion project • 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, • Stop 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 • The details of the simulation As a test run on a 163 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) + Stop(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 • Numerical data of test run (Preliminary) Both data confirm the exact chiral symmetry.

4. Finite volume and fixed topology • 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. • 24348 (~3fm)4 lattice or larger are planned.

4. Finite volume and fixed topology • 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 mπL<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. Fπ.

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

4. Finite volume and fixed topology • 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.

5. Summary and discussion The chiral limit is within our reach now! • Exact chiral symmetry at quantum levelcan 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 ⇒ Fπ~86MeV.

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

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’ Z.Fodor et al. hep-lat/0510117

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)

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

Fpi

Mps2/m

Approaching the chiral limit in lattice QCD