Pion Correlators in the e- regime

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Pion Correlators in the e- regime

• Hidenori Fukaya (YITP)
• collaboration with
• S. Hashimoto (KEK)
• and K.Ogawa (Sokendai)

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0. Contents

1. Introduction
2. Lattice Simulations
3. Results (quenched)
4. Conclusion

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1. Introduction

• 1-1. Our Goals
• Lattice QCD
• - 1st principle and non-perturbative calculation.
• Chiral perturbation theory (ChPT)
• - Low energy effective theory of QCD (pion
  theory).
• - Free parameters Fp and S.
• It is important to determine Fp and S from
• 1-st principle calculation but simulations at
• m0 (mlt30MeV) and large V (Vgt2fm) are
  difficult...
• ? Consider fm universe
  (e-regime).

4
1. Introduction

• 1-1. Our Goals
• In the e- regime ( mpL lt 1 , L?QCDgtgt1), we have
• ChPT with finite V correction.
• Quenched QCD simulation
• ? low energy constants (S, Fp, a) of
• quenched ChPT (in small V).
• Full QCD simulation
• ? those of ChPT (in small V).
• In particular, dependence on topological charge Q
• and X mSV is important .

S.R.Sharpe(01)P.H.Damgaard et al.(02)
J.Gasser,H.Leutwyler(87),F.C.Hansen(90),
H.Leutwyler,A.Smilga(92)
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1-2. Setup

• To simulate m0 region, Exact chiral symmetry
  is required.
• ? Overlap operator (Chebychev
• polynomial (of order150 )) which
• satisfies Ginsparg-Wilson relation.
• Fitting pion correlators in small V at different
  Q and m with ChPT in the e-regime ? extract S,
  Fp, a, m0

P.H.Ginsparg,K.G.Wilson(82), H.Neuberger(98)
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1-3. Pion correlators in the e-regime

• Quenched ChPT in small V
• Pion correlators are not exponential but
• ChPT in small V (Nf2)

P.H.Damgaard et al. (02)
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2. Lattice Simulations

• 2-1. Calculation of D -1
• Overlap at m0 ? Large numerical costs !
• Low mode preconditioning
• We calculate lowest 100 eigenvalues and eigen
• functions so that we deform D as
• ? costs for at m0 costs for
• at m100MeV !

L.Giusti et al.(03)
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2-2. Low-mode contribution in pion correlators

• Is the low-mode contribution dominant ?
• As m?0 ? low-modes must be important.
• We find the contribution from is negligible
• ( only 0.5 .) for mlt0.008 (12.8MeV)
• and Q ? 0 at large t , so we can approximate
• for large x-y.

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2-2. Low-mode contribution in pion correlators

• Pion source averaging over space-time
• Now we know at all x. ? we know
• at any x and y. Averaging over x0 and t0
• reduces the noise almost 10 times !

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2-3. Numerical Simulations

• Size ß5.85, 1/a 1.6GeV, V104 (1.23fm)4
• Gauge fields updated by plaquette action
• (quenched).
• Fermion mass m0.016,0.032,0.048,0.064,0.008
  ( 2.6MeV ? m ?12.8 MeV !!)
• 100 eigenmodes are calculated by ARPACK.
• Q is evaluated from of zero modes.
• Source pion is averaged over xodd sites for
• Q ? 0.

Q 0 1 2 3
of conf. 50 76 57 19
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3. Results (quenched QCD)
preliminary

• 3-1. Pion correlators

Our data show remarkable Q and m dependences.
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3-2. Low energy parameters
preliminary

• Using
• we simultaneously fit all of our data
• (15 correlators ) with the function

? Ogawas talk
P.H.Damgaard (02)
We obtain S (30723 MeV)3, Fp 111.15.2MeV,
a 0.070.65, m0 95844 MeV, ?2/dof1.5.
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4. Conclusion
??? (??)100????????? ?????????????

• In quenched QCD in the e-regime, using
• Overlap operator ? exact chiral symmetry,
• 2.6 MeV ? m ? 12.8 MeV ,
• lowest 100 eigenmodes (dominance99.5),
• Pion source averaging over space-time,
• ( equivalent to 100 times statistics )
• we compare the pion correlators with ChPT .

? The correlators show remarkable Q and m
dependences.
? S(30723 MeV)3, Fp111.15.2 MeV,
a0.070.65, m095844 MeV.
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4. Conclusion

• As future works,
• a ? 0 limit and renormalization,
• isosinglet meson correlators,
• full QCD ( ? Ogawas talk),
• consistency check with p-regime results,
• will be important.

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A. Full QCD

• Lowest 100 eigenvalues

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A. Full QCD

• Truncated determinant
• The truncated determinant is equivalent to
• adding a Pauli-Villars regulator as
• where, for example,
• ??0 limit ? usual Pauli-Villars (gauge
  inv,local).
• ??0 limit ? quench QCD (good overlap config. ?)
• If ?a is fixed as a?0, unitarity is also
  restored.