Approaching the chiral limit in lattice QCD

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.D74094505(2006)hep-
  lat/0607020, hep-lat/0607093, hep-lat/0610011,
  hep-lat/0610024 and hep-lat/0610026.

2
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

3
1. Introduction

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

4
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 JLQCDs 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 gt 50MeV .
• ? Large systematic uncertainties in m a few MeV
  results.

Nielsen and Ninomiya, Nucl.Phys.B185,20(81)
5
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
• Lueschers 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)
6
Plan of my talk

• Introduction
• Chiral symmetry and topology
• JLQCDs overlap fermion project
• Finite volume and fixed topology
• Summary and discussion

7
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)
8
2. Chiral symmetry and topology

• Eigenvalue distribution of Dirac operator

1/a
-1/a
9
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
10

• The overlap fermion action
• The Neubergers 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)
11
2. Chiral symmetry and topology

• Eigenvalue distribution of Dirac operator

1/a
-1/a
12
2. Chiral symmetry and topology

• Eigenvalue distribution of Dirac operator

1/a
-1/a

• m is well-defined.
• index is well-defined.

13
2. Chiral symmetry and topology

• Eigenvalue distribution of Dirac operator

1/a
-1/a

• Theoretically ill-defined.
• Large simulation cost.

14
2. Chiral symmetry and topology

• The topology (index) changes

HwDw-10 ? Topology boundary
1/a
-1/a
15

• The overlap Dirac operator
• becomes ill-defined when
• Hw0 forms topology boundaries.
• These zero-modes are lattice artifacts(excluded
  in a?8limit.)
• In the polynomial expansion of D,
• The discontinuity of the determinant requires
• reflection/refraction (Fodor et al.
  JHEP0408003,2004)
• V2 algorithm.

16
2. Chiral symmetry and topology

• Topology conserving gauge action
• To achieve Hw gt 0 Lueschers 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.D7409505,2006)
• Note Stop ?8 when Hw?0 and Stop?0
  when a?0.

M.Luescher,Nucl.Phys.B568,162 (00)
17
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
• Lueschers 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)
18
3. JLQCDs 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_wilsons .
• 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 !

19
3. JLQCDs 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 gt50MeV with non-chiral fermion in
  previous JLQCD works.
• Iwasaki (beta2.3) Stop(µ0.2) gauge action
• Overlap operator in Zolotarev expression
• Quark masses ma0.002(2MeV) 0.1.
• 1 samples per 10 trj of Hybrid Monte Carlo
  algorithm.
• 2000-5000 trj for each m are performed.
• Q0 topological sector

20
3. JLQCDs overlap fermion project

• Numerical data of test run (Preliminary)
• Both data confirm the exact chiral symmetry.

21
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 (Q0). Any observable is
  different from ?0 results
• where ? is topological susceptibility and f is
  an unknown function of Q.
• ? needs careful treatment of finite V and fixed
  Q .
• Q2, 4 runs are started.
• 24348 (3fm)4 lattice or larger are planned.

22
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 mpLlt1
  (e-regime) .
• ? m2MeV, L2fm is good.
• Finite V effects on ChRMT
• discrete Dirac spectrum ? chiral condensate S.
  
• Finite V effects on ChPT
• pion correlator is not exponential but
  quadratic.
• ? pion decay const. Fp.

23
4. Finite volume and fixed topology

• Dirac spectrum and ChRMT (Preliminary)
• Nf2 Nf0
• Lowest eigenvalue (Nf2) ? S(233.9(2.6)MeV)3

24
4. Finite volume and fixed topology

• Pion correlator and ChPT (Preliminary)
• The quadratic fit (fit range10,22,ß10) worked
  well.
• ?2 /dof 0.25.
• Fp 86(7)MeV
• is obtained preliminary.
• NOTE Our data are at
• m2MeV. we dont need
• chiral extrapolation.

25
5. Summary and discussion

• 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, weve simulated
  Nf2 dynamical overlap quarks with m2MeV.
• Finite V and Q dependences are important.
• ChRMT in finite V ? S2.193E-03.
• ChPT in finite V ? Fp86MeV.

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5. Summary and discussion

• To do
• Precise measurement of hadron spectrum,
  started.
• 21 flavor, started.
• Different Q, started.
• Larger lattices, prepared.
• BK , started.
• Non-perturbative renormalization, prepared.
• Future works
• ?-vacuum
• ??pp decay
• Finite temperature

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
Z.Fodor et al. hep-lat/0510117
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

• Topology dependence
• If , any observable at a fixed topology
  in general theory (with ?vacuum) 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.D72014504,2005
30
Fpi
31
Mv
32
Mps2/m