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Nf=2 lattice QCD & Random Matrix TheoNf=2 lattice QCD & Random Matrix Theory in the ε-regimery in the ε-regime

Hidenori Fukaya (Riken Wako)for JLQCD collaboration

HF et al, [JLQCD collaboration], hep-lat/0702003 (accepted by Phisical Review Letters )

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JLQCD’s overlap fermion project (->Noaki’s 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 >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(3MeV) – 0.1. 1 samples per 10 trj of Hybrid Monte Carlo algorithm. 5000 trj for each m are performed. Q=0 topological sector (No topology change.)

1. Introduction

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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;

Brower et al, Phys.Lett.B560(2003)64

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. Check of ergodicity in fixed topological sector.

1. Introduction

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Effective theory with finite V and fixed QDue to the large mass gap between mπ 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 Σ, Fpi is important.

1. Introduction

Gasser & Leutsyler, 1987, Hansen, 1990, 1991, Damgaard et al, 2002, ……

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Dirac spectrum and ChRMTIn particular, in the ε-regime, when m~0, 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 Σ, Fπ and V effects…

1. Introduction

Shuryak & Verbaarschot, 1993, Damgaard & Nishigaki, 2001, Akemann, Damgaard, Osborn, Splitorff, 2006, etc.

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Consider the QCD partition function at a fixed topology Q,

Weak coupling (λ >> ΛQCD)

Strong coupling (λ<<　 ΛQCD)

⇒ An assumption:

for the low-modes with an unknown function V ⇒ ChRMT.

2. QCD → RMT → ChPT

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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) -> SU(Nf) SSB. Randomness -> kinetic term neglected. RMT predicts Dirac low-modes -> pion zero-mode !

2. QCD → RMT → ChPT

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Dirac spectrum and analytic prediction of ChRMTNf=2 　（ m=3MeV) results

Lowest eigenvalue ⇒Σ=(251(7)(11)MeV)3

3. Numerical results

• Direct evidence of chiral SSB of QCD !!

• Σ obtained without “chiral extrapolation”

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Dirac spectrum with imaginary isospin chemical potential (preliminary)

2-point correlation function The eigenvalues of

is predicted by Ch2-RMT.

Fπ ~ 70 MeV.

3. Numerical results

See Akemann, Damgaard, Osborn, Splitorff, hep-th/0609059 for the details.

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The chiral limit is within our reach now! On (~2fm)4 lattice, JLQCD have simulated Nf=2 dynamical

overlap quarks with m~3MeV. Finite V and Q dependences are important. ChPT and ChRMT are helpful to estimate finite V and Q ef

fects. Comparing QCD in the ε-regime with RMT,

Direct evidence of chiral SSB from 1st principle. ChRMT in the ε-regime ⇒ 　 Σ~(250 MeV)3. Ch2-RMT in the ε-regime ⇒ Fπ~ 70MeV.

4. 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, almost done.

Future works θ-vacuum ρ→ππ decay Finite temperature…

4. Summary and discussion

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Numerical result (Preliminary)

Both data confirm the exact chiral symmetry.

3. JLQCD’s overlap fermion projectM2/mFpi

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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 effects = V effects.

Shintani et al,Phys.Rev.D72:014504,2005

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Fπ chiral log ?

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Mv

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Mps2/m