Approaching the chiral limit in lattice QCD

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Approaching the chiral limit Approaching the chiral limit in lattice QCDin 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.

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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

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But the lattice regularization spoils a lot of symmetries…

Translational symmetry Lorentz invariance Chiral symmetry and topology Supersymmetry… 　

1. Introduction

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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.

1. Introduction

Nielsen and Ninomiya, Nucl.Phys.B185,20(‘81)

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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 mu, md ~ O(1) MeV.

1. Introduction

Ginsparg & WilsonPhys.Rev.D25,2649(‘82)

Neuberger, Phys.Lett.B417,141(‘98)

M.Luescher,Nucl.Phys.B568,162 (‘00)

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Plan of my talk

1. Introduction 2. Chiral symmetry and topology3. JLQCD’s overlap fermion project4. Finite volume and fixed topology5. Summary and discussion

　

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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)

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02/a 4/a 6/a

Eigenvalue distribution of Dirac operator


continuum

(massive)

m

1/a

-1/a

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Wilson fermion



1/a

-1/a

Naïve lattice fermion

16 lines

02/a 4/a 6/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 heavydense

sparse but nonzero density

until a→0.

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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)

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1/a

-1/a

02/a 4/a 6/a

The overlap fermion

• Doublers are massive.

•　 D is smooth except for

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1/a

-1/a

02/a 4/a 6/a

The overlap fermion(massive)

m

• m is well-defined.

• index is well-defined.

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1/a

-1/a

02/a 4/a 6/a

The overlap fermion

• Theoretically ill-defined.

• Large simulation cost.

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The topology (index) changes


1/a

-1/a

02/a 4/a 6/a

The complex modes make pairs

The real modes are chiral eigenstates.

Hw=Dw-1=0

⇒ Topology boundary

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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.

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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.



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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 Stop

[ Luescher’s admissibility condition]

2. Approach mu, md ~ O(1) MeV.


Ginsparg & WilsonPhys.Rev.D25,2649(‘82)

Neuberger, Phys.Lett.B417,141(‘98)


　

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Numerical costSimulation 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 e

xtra 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

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The details of the simulationAs 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


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Numerical data of test run (Preliminary)

Both data confirm the exact chiral symmetry.


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Systematic error from finite V and fixed QOur 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

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ChPT and ChRMT with finite V and fixed QHowever, 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.


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Dirac spectrum and ChRMT (Preliminary)Nf=2 Nf=0

Lowest eigenvalue (Nf=2) ⇒ Σ=(233.9(2.6)MeV)3


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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 atm~2MeV. we don’t need chiral extrapolation.


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The chiral limit is within our reach now! Exact chiral symmetry at quantum level can be achieved i

n 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

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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…

5. Summary and discussion

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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

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Initial configurationFor 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)

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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

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Fpi

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Mv

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Mps2/m

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Approaching the chiral limit in lattice QCD