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Lattice QCD, Random Matrix Theory aLattice QCD, Random Matrix Theory and chiral condensatesnd chiral condensates

Lattice QCD, Random Matrix Theory aLattice QCD, Random Matrix Theory and chiral condensatesnd chiral condensates

JLQCD and TWQCD collaboration, Phys.Rev.Lett.98,172001(2007) (hep-lat/07JLQCD and TWQCD collaboration, Phys.Rev.Lett.98,172001(2007) (hep-lat/0702003), arXiv:0705.3322 [hep-lat] to appear in Phys.Rev.D.02003), arXiv:0705.3322 [hep-lat] to appear in Phys.Rev.D.

[Asahi shimbun, 25Apr07, Nikkei shimbun,30Apr07, Yomiuri shimbun 28May07 ][Asahi shimbun, 25Apr07, Nikkei shimbun,30Apr07, Yomiuri shimbun 28May07 ] [Nature 47, 118 (10 May 2007), CERN COURIER, Vol47, number 5][Nature 47, 118 (10 May 2007), CERN COURIER, Vol47, number 5]

Hidenori Fukaya (RIKEN Wako)Hidenori Fukaya (RIKEN Wako)with S.Aoki, T.W.Chiu, S.Hashimoto, T.Kaneko, H.Matsufuru, J.Nwith S.Aoki, T.W.Chiu, S.Hashimoto, T.Kaneko, H.Matsufuru, J.N

oaki, K.Ogawa,M.Okamoto,T.Onogi and N.Yamadaoaki, K.Ogawa,M.Okamoto,T.Onogi and N.Yamada[JLQCD and TWQCD collaboration][JLQCD and TWQCD collaboration]

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1. Introduction

Chiral symmetryand its spontaneous breaking are important !– Mass gap between pion and the other hadrons

pion as (pseudo) Nambu-Goldstone bosonwhile the other hadrons acquire the mass ~ΛQCD.

– Soft pion theorem– Chiral phase transition at finite temperature…

Banks-Casher relation [Banks & Casher, 1980]

Chiral SSB is caused by Dirac zero-modes.

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1. Introduction Chiral Random Matrix Theory (ChRMT)

is an equivalent description of the moduli integrals of chiral perturbation theory. The spectrum of ChRMT is expected to match with the QCD Dirac spectrum; ChRMT predicts

– Distribution of individual Dirac eigenvalues– Spectral density– Spectral correlation functions as functions of m, Σ and V,

which is helpful to analyze lattice data. [Shuryak & Verbaarschot,1993,

Verbaarschot & Zahed, 1993,Damgaard & Nishigaki, 2001…]

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1. Introduction Chiral Random Matrix Theory (ChRMT)

For example, ChRMT “knows” finite V and m corrections in Banks-Casher relation;– In m->0 limit with V=∞

ρ(λ) is flat and the height gives Σ (Banks-Casher).

– Finite m and V correctionsare analytically known.

Note: the same Σ with finite V and m !

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1. Introduction Lattice QCD is the most promising approach to confirm chiral SSB from 1-st principle calculation of QCD. But…

– Chiral symmetry is difficult. [Nielsen & Ninomiya 1981]– m → 0 is difficult 　 (large numerical cost).– V → ∞ is difficult (large numerical cost).

Therefore, the previous works were limited, with • Dirac operator which breaks chiral symmetry. (Wilson or s

taggered fermions, *Domain-wall fermion is better but still has the breaking effects ~5MeV. )

• Heavier u-d quark masses ~20-50MeV than real value ~ a few MeV.

　　→　 needs unwanted operator mixing with opposite chirality, and m->0 extrapolations.

　　→　 large systematic errors.

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1. Introduction

This work1. We achieved 2-flavor lattice QCD simulations with exact

chiral symmetry. • The Ginsparg-Wilson relation -> exact chiral symmetry.• Luescher’s admissibility condition -> smooth gauge fields.• On (~1.8fm)4 lattice, achieved m~3MeV !

2. Finite V effects evaluated by ChRMT.• m, V, Q dependences of QCD Dirac spectrum are calculated.

3. A good agreement of Dirac spectrum with ChRMT.– Strong evidence of chiral SSB from 1st principle.– obtained

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Contents

1. Introduction2. QCD Dirac spectrum & ChRMT3. Lattice QCD with exact chiral symmet

ry4. Numerical results5. Conclusion

　

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Σ

low density

2. QCD Dirac spectrum & ChRMT　 Banks-Casher relation• In the free theory,

ρ(λ) is given by the surface of S3 with the radius λ:

• With the strong coupling The eigenvalues feel the repulsive fo

rce from each other→becoming non-degenerate→ flowing to the low-density region around zero→ results in the chiral condensate.

[Banks &Casher 1980]

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　 Chiral Random Matrix Theory (ChRMT) Consider the QCD partition function at a fixed topology Q,

• High modes (λ >> ΛQCD) -> weak coupling

• Low modes (λ<< 　 ΛQCD) -> strong coupling

⇒ Let us make an assumption: For low-lying modes,

with an unknown action V(λ) ⇒ ChRMT.

2. QCD Dirac spectrum & ChRMT

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2. QCD Dirac spectrum & ChRMT Chiral Random Matrix Theory (ChRMT) Namely, we consider the partition function (for low-modes)

• Universality of RMT [Akemann et al. 1997] :IF V(λ) 　 is in a certain universality class, in large n limit (n : size of matrices) the low-mode spectrum is proven to be equivalent, or independent of the details of V(λ) (up to a scale factor) !

• From the symmetry, QCD should be in the same universality class with the chiral unitary gaussian ensemble,

and share the same spectrum, up to a overall

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2. QCD Dirac spectrum & ChRMT Chiral Random Matrix Theory (ChRMT) In fact, one can show that the ChRMT is equivalent to the moduli

integrals of chiral perturbation theory.

The second term in the exponential is written aswhere

Let us introduce Nf x Nf real matrix σ1 and σ ２ as

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2. QCD Dirac spectrum & ChRMT Chiral Random Matrix Theory (ChRMT) Then the partition function becomes

where 　　 is a NfxNf complex matrix.With large n, the integrals around the suddle point, which satisfies

leaves the integrals over U(Nf) as

equivalent to the ChPT moduli’s integral in the εregime.

⇒

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Eigenvalue distribution of ChRMTDamgaard & Nishigaki [2001] analytically derived the distribution of each eigenvalue of ChRMT.For example, in Nf=2 and Q=0 case, it is

where and

where

-> spectral density or correlation can be calculated, too.

2. QCD Dirac spectrum & ChRMT

Nf=2, m=0 and Q=0.

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Summary of QCD Dirac spectrum IF QCD dynamically breaks the chiral symmetry, the Dirac spectrum in finite V should look like

2. QCD Dirac spectrum & ChRMT

Banks-Casher

Σ

ρ

λ

Low modes are described by ChRMT.

• the distribution of each eigenvalue is known.

• finite m and V effects controlled by the same Σ.

Higher modes are like free theory ~λ3

ChPT moduli

Note: Analytic　 solution is not known

-> let’s study lattice QCD!

Note: We made a stronger assumption

QCD -> ChRMT = ChPT

than usual,

QCD -> ChPT

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3. Lattice QCD with exact chiral symmetry

The overlap Dirac operator We use Neuberger’s overlap Dirac operator [Neuberger 1998]

(we take m0a=1.6) satisfies the Ginsparg-Wilson [1982] relation:

realizes ‘modified’ exact chiral symmetry on the lattice;the action is invariant under [Luescher 1998]

However, Hw→0 (= topology boundary ) is dangerous.

1. D is theoretically ill-defined. [Hernandez et al. 1998]

2. Numerical cost is suddenly enhanced. [Fodor et al. 2004]

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3. Lattice QCD with exact chiral symmetry

Luescher’s admissibility condition [Luescher 1999]

In order to achieve |Hw| > 0 [Luescher’s admissibility condition], we add “topology stabilizing” term [Vranas 2006, HF et al(JLQCD), 2006]

with μ=0.2. Note: Stop →∞ 　 when Hw→0 and Stop→0 when a→0.

( Note

is extra Wilson fermion and twisted mass bosonic spinor with a cut-off scale mass. )

• With Stop, topological charge , or the index of D, is fixed along

the hybrid Monte Carlo simulations -> ChRMT at fixed Q.

• Ergodicity in a fixed topological sector ? -> O.K.

(Local fluctuation of topology is consistent with ChPT.)

[T.W.Chiu et al, in preparation]

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3. Lattice QCD with exact chiral symmetry

Sexton-Weingarten method [Sexton & Weingarten 1992, Hasenbusch, 2001]

We divide the overlap fermion determinant as

with heavy m’ and performed finer (coarser) hybrid Monte Carlo step for the former (latter) determinant -> factor 4-5 faster.

Other algorithmic efforts1. Zolotarev expansion of D -> 10 -(7-8) accuracy.2. Relaxed conjugate gradient algorithm to invert D.3. Multishift –conjugate gradient for the 1/Hw2.4. Low-mode projections of Hw.

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3. Lattice QCD with exact chiral symmetry

Numerical costSimulation 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 ex

tra factor ~10 numerical cost. ⇒ 　 The cost of D_ov ~ 1000 times of D_wilson’s .However,

– Stop can cut the latter numerical cost ~10 times faster

– Stop can reduce the degree of polynomial ~ 2-3 times– New supercomputer at KEK ~60TFLOPS ~50 times– Many algorithmic improvements ~ 5-10 times

We can overcome this difficulty !

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3. Lattice QCD with exact chiral symmetry

Simulation summary On a 163 32 lattice with a ~ 1.6-1.9GeV (L ~ 1.8-2fm), we achieved 2-flavor QCD simulations with the overlap quarks with the quark mass down to ~3MeV. [ε-regime]

Note m >50MeV with Wilson fermions in previous JLQCD works.

– Iwasaki (beta=2.3,2.35) + Stop(μ=0.2) gauge action– 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.)– The lattice spacings a is calculated from quark potential(consi

stent with rho meson mass input).– Eigenvalues are calculated by Lanzcos algorithm.

(and projected to imaginary axis.)

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4. Numerical resultsIn the following, we mainly focus on the data with m=3MeV.

Bulk spectrum Almost consistent with the Banks-Casher’s

scenario !– Low-modes’

accumulation.– The height

suggests Σ ~ (240MeV)3.

– gap from 0.⇒　 need ChRMT analysis

for the precise measurement of Σ !

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4. Numerical results

Low-mode spectrum　 Lowest eigenvalues qualitatively agree with ChRMT.

k=1 data 　→　 Σ= [240(6)(11) MeV]3

statistical choice of k

12.58(28)14.014

9.88(21)10.833

7.25(13)7.622

[4.30]4.301

LatticeRMT

[] is used as an input.~5-10% lower -> Probably NLO 1/V

effects.

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4. Numerical results

Low-mode spectrum Cumulative histogram is useful to compare the shape of the distribution.

The width agrees with RMT within ~2σ.

1.54(10)1.4144

1.587(97)1.3733

1.453(83)1.3162

1.215(48)1.2341

latticeRMT

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4. Numerical results Heavier quark masses For heavier quark masses, [30~160MeV], the good agreement with RMT is not expected, due to finite m effects of non-zero modes. But our data of the ratio of the eigenvalues still show a qualitative agreement.

NOTE• massless Nf=2 Q=0 gives the same spectrum with Nf=0, Q=2. (flavor-topology duality)• m -> large limit is consistent with QChRMT.

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4. Numerical results

Heavier quark massesHowever, the value of Σ, determined by the lowest-eigenvalue, significantly depend on the quark mass.But, the chiral limit is still consistent with the data with 3MeV.

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4. Numerical results

Renormalization Since Σ=[240(2)(6)]3 is the lattice bare value, it should be renormalized. We calculated 1. the renormalization factor in a non-perturbative RI/MOM scheme

on the lattice,

2. match with MS bar scheme, with the perturbation theory,3. and obtained

(tree)(non-perturbative)

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4. Numerical results Systematic errors• finite m -> small.

As seen in the chiral extrapolation of Σ, m~3MeV is very close to the chiral limit.

• finite lattice spacing a -> O(a2) -> (probably) small.the observables with overlap Dirac operator are automatically free from O(a) error,

• NLO finite V effects -> ~ 5-10%.1. Higher eigenvalue feel pressure from bulk modes.

higher k data are smaller than RMT. (5-10%) 2. 1-loop ChPT calculation also suggests ~ 10% .

statistical systematic

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5. Conclusion

• We achieved lattice QCD simulations with exactly chiral symmetric Dirac operator,

• On (~2fm)4 lattice, simulated Nf=2 dynamical quarks with m~3MeV,

• found a good consistency with Banks-Casher’s scenario,

• compared with ChRMT where finite V and m effects are taken into account,

• found a good agreement with ChRMT,– Strong evidence of chiral SSB from 1st principle.– obtained

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5. Conclusion

Future works– Reduce the NLO V effects (or 1/N effects) of Σ.

• Larger lattices (prepared).• NLO calculations of meson correlators in (partially quenched) Ch

PT. analytic part is done. [P.H.Damgaard & HF, arXiv:0707.3740]

– Hadron spectrum – Test of ChPT (chiral log)– Pion form factor– π±π ０ difference– BK

– Topological susceptibility – 2+1 flavor simulations– Finer and larger lattices…

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6. NLO V effects (preliminary)

Meson correlators compared with ChPTWith a direct comparison of meson correlators with (partially quenched) ChPT, we obtain[P.H.Damgaard & HF, arXiv:0707.3740]

where NLO V correction is taken into account.[JLQCD, in preparation.]

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6. NLO V effects (preliminary)

Meson correlators compared with ChPT