Nf=2 lattice QCD

1
Nf2 lattice QCD Random Matrix Theory in the
e-regime

• Hidenori Fukaya (Riken Wako)
• for JLQCD collaboration
• HF et al, JLQCD collaboration, hep-lat/0702003
  (accepted by Phisical Review Letters )

2
1. Introduction

• JLQCDs overlap fermion project (-gtNoakis talk)
• On a 163 32 lattice with a 1.6-1.8GeV (L
  1.8-2fm),
• we have achieved 2-flavor QCD simulations with
• the overlap quarks with the quark mass down to
  3MeV.
• 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(3MeV) 0.1.
• 1 samples per 10 trj of Hybrid Monte Carlo
  algorithm.
• 5000 trj for each m are performed.
• Q0 topological sector (No topology change.)

3
1. Introduction

• 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
• Brower et al, Phys.Lett.B560(2003)64
• 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.
• Check of ergodicity in fixed topological sector.

4
1. Introduction

• Effective theory with finite V and fixed Q
• Due to the large mass gap between mp and the
  other hadron masses, the pion should be most
  responsible for the finite V or Q effects.
• ? finite V and Q effects can be evaluated in
  pion effective theory ( ChPT or ChRMT)
• Examples
• where
• ? precise measurement of S, Fpi is important.

Gasser Leutsyler, 1987, Hansen, 1990, 1991,
Damgaard et al, 2002,
5
1. Introduction

• Dirac spectrum and ChRMT
• In particular, in the e-regime, when m0, s.t.
  
• chiral Random Matrix Theory (ChRMT) is helpful
  to evaluate the finite V scaling of the Dirac
  eigen spectrum
• ChRMT ? low-mode Dirac spectrum
• Controlled by
• Or by
• with chemical potential.
• ? precise measurement of S, Fp and V effects

Shuryak Verbaarschot, 1993, Damgaard
Nishigaki, 2001, Akemann, Damgaard, Osborn,
Splitorff, 2006, etc.
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2. QCD ? RMT ? ChPT

• Consider the QCD partition function at a fixed
  topology Q,
• Weak coupling (? gtgt ?QCD)
• Strong coupling (?ltlt ?QCD)
• ? An assumption
• for the low-modes with an unknown function V ?
  ChRMT.

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2. QCD ? RMT ? ChPT

• From the universality and symmetry of RMT, QCD
  should
• have the same low-mode spectrum with chiral
  unitary
• gaussian ensemble,
• up to overall factor
• In fact,
• SU(Nf)SU(Nf) -gt SU(Nf) SSB.
• Randomness -gt kinetic term neglected.
• RMT predicts Dirac low-modes -gt pion zero-mode
  !

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3. Numerical results

• Dirac spectrum and analytic prediction of ChRMT
• Nf2 (m3MeV) results
• Lowest eigenvalue ?S(251(7)(11)MeV)3

• Direct evidence of chiral SSB of QCD !!
• S obtained without chiral extrapolation

9
3. Numerical results

• Dirac spectrum with imaginary isospin chemical
  potential (preliminary)
• 2-point correlation function
• The eigenvalues of
• is predicted by Ch2-RMT.
• Fp 70 MeV.

See Akemann, Damgaard, Osborn, Splitorff,
hep-th/0609059 for the details.
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4. Summary and discussion

• The chiral limit is within our reach now!
• On (2fm)4 lattice, JLQCD have simulated Nf2
  dynamical overlap quarks with m3MeV.
• Finite V and Q dependences are important.
• ChPT and ChRMT are helpful to estimate finite V
  and Q effects.
• Comparing QCD in the e-regime with RMT,
• Direct evidence of chiral SSB from 1st principle.
• ChRMT in the e-regime ? S(250 MeV)3.
• Ch2-RMT in the e-regime ? Fp 70MeV.

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4. 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, almost
  done.
• Future works
• ?-vacuum
• ??pp decay
• Finite temperature

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3. JLQCDs overlap fermion project

• Numerical result (Preliminary)
• Both data confirm the exact chiral symmetry.

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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 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)
15

• 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 effects V effects.

Shintani et al,Phys.Rev.D72014504,2005
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Fp chiral log ?
17
Mv
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Mps2/m