Towards ?vacuum simulation in lattice QCD

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Towards ?vacuum simulation in lattice QCD

• Hidenori Fukaya
• YITP, Kyoto Univ.
• Collaboration with
• S.Hashimoto (KEK), T.Hirohashi (Kyoto Univ.),
• K.Ogawa(Sokendai), T.Onogi(YITP)

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Contents

1. Introduction
2. ? vacuum simulation of 2D QED
3. How to fix topology in 4D QCD
4. Tempering for QCD ? vacuum
5. Conclusion

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1. Introduction

• Chiral symmetry and topology are very
• important and related each other...
• Chiral symmetry breaking
• Anomaly
• Construction of chiral gauge theory
• 4D N1 SUSY (Chiral super fields)
• Our aim is to keep chiral symmetry and
• topological properties in 4D lattice QCD.
• ? ? vacuum simulation

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1. Introduction

• Ginsparg-Wilson relation
• ? exact chiral symmetry at classical level.
• Luschers admissibility condition
• ? locality of Dirac operator
• ? Stability of the index (topological charge)

Phys.Rev.D25,2649(82)
Phys.Lett.B428,342(98),Nucl.Phys.B549,295(99)
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1. Introduction

• Our goal is to calculate path integrals with
  ?term

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2. ?vacuum simulation of 2D QED

• The 2-flavor massive Schwinger model
• We have a geometrical definition of the
  topological charge

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2. ?vacuum simulation of 2D QED

• We also developed a method to calculate
• Classical solution ? Constant field
  strength.
• Moduli (constant potential which affects Polyakov
  loops ) integral of the determinants
• ? Householder and QL method.
• Integral of
• ? fitting with polynomiyals.

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2. ?vacuum simulation of 2D QED

• Thus, we could evaluate
• by Luschers gauge action and our reweighting
  method and the results were consistent with the
  continuum theory.

..
Details are shown in HF,T.Onogi,Phys.Rev.D68,0
74503(2003).
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2. ?vacuum simulation of 2D QED

• For example, we calculated the pseudoscalar
• condensates,
• which can be obtained from the anomaly equation,

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2. ?vacuum simulation of 2D QED

• We evaluate the total expectation value in
• ?vacuum

Details are in HF,T.Onogi, Phys.Rev.D70,054508(200
4).
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3. How to fix topology in 4D QCD

• Also in 4D QCD, we expect that can
  be obtained with the action
• but
• No geometrical (and practical ) definition of Q.
• The index of D ? a lot of computational costs.

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3. How to fix topology in 4D QCD

• (New) Cooling method
• We cool the configuration smoothly by
  performing the hybrid Monre Carlo steps with
• decreasing g2 (admissibility is always
  satisfied.).
• ? We obtain a cooled configuration
• close to the classical background at
• very weak g20.0001 then,
• gives an integer with 10 accuracy.

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3. How to fix topology in 4D QCD

• How can e be determined ?
• The stability of Q is proved only when
• elt emax 1/20.49 .
• ? configuration space is too narrow
• on 104204 lattice.

Let us search the parameters ß, e, V, ?t which
can fix Q. Anothor group is also studying this
action. S.Shcheredin, W.Bietenholz, K.Jansen,
K.I.Nagai, S.Necco and L.Scorzato,
hep-lat/0409073
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3. How to fix topology in 4D QCD

• Action Luscher action (quenched)
• Algorithm The hybrid Monte Carlo method.
• Gauge coupling ß (6/g2) 1.0 2.8
• Lattice size 104,144,164
• Admissibility condition 1/e 2/3 1.0
• Topological charge Q - 3 3
• 10 40 molecular dynamics steps with the step
  
• size ?t 0.001-0.02 in one trajectory.

• Numerical simulations

..
The simulations were done on the Alpha work
station at YITP and SX-5 at RCNP.
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3. How to fix topology in 4D QCD

• 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
• topological charge Q.

A.Gonzalez-Arroyo,hep-th/9807108,
M.Hamanaka,H.Kajiura,Phys.Lett.B551,360(03)
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3. How to fix topology in 4D QCD

• Results

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3. How to fix topology in 4D QCD

• Physical scale of the lattice

Both 1/e1.0, ß?1.4 and 1/e2/3, ß?2.6 are
corresponding ß?6.0 (1/a 2.0 GeV) for plaquette
action. ? Q can be fixed on lattices finer than
(2.0GeV)-1.
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4. Tempering for QCD ?vacuum
Our method to calculate the partition function
used in 2D QED is not applicable to 4D QCD
due to its large numerical costs.
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4. Tempering for QCD ?vacuum

• Tempering method
• Let us extend the configuration space

Q is not conserved.
? N0,
? N1,
? N-1, samples,
If we obtain
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4. Tempering for QCD ?vacuum

• There are several tempering methods...
• Simulated tempering G.Parisi et
  al,Europhys.Lett.19,451(92)
• Parallel tempering E.Marianari et
  al,cond-mat/9612010,
• K.Hukushima et al, cond-mat/9512035
• If the tempering is applicable for Luscher
• action, we may able to treat ?vacuum,

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5. Conclusion

• We find at V104 , 144 ,
• Well- controlled topological charge can be
  measured
• by our cooling method .
• Q can be fixed for more than 2000 trajectory
• length (200 uncorrelated samples .) if the
  lattice is
• fine enough (1/a 2.0GeV) even when we set
• 1/e2/3.
• Next we will try
• more quantitative studies
• large volume
• The index of GW fermion
• tempering for ?vacuum
• full QCD

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6. Outlook

• Luschers admissibility condition will be
• important for
• Locality of Dirac operators
• e-regime
• Finite temperatures and Chiral transition
• SUSY
• Majorana fermions?
• Non commutative lattice ??
• Matrix model ???