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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 collaboration, Phys.Rev.Lett.98,172001(2007) [hep-lat/0702003], Phys.Rev.D76,054503 (2007) [arXiv:0705.3322] ,arXiv:0711.4965.

Hidenori Fukaya (Niels Bohr Institute)Hidenori Fukaya (Niels Bohr Institute)for JLQCD collaborationfor JLQCD collaboration

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JLQCD Collaboration KEK 　 　　　　　 S. Hashimoto, T. Kaneko, H. Matsufuru, J. Noaki, 　 M. Okamoto, E. Shintani, N. YamadaRIKEN -> Niels Bohr H. FukayaTsukuba 　　　 S. Aoki, T. Kanaya, Y. Kuramashi, N. Ishizuk

a, Y. Taniguchi, 　 A. Ukawa, T. YoshieHiroshima 　 K.-I. Ishikawa, M. OkawaYITP H. Ohki, T. Onogi

KEK BlueGene (10 racks, 57.3 TFlops)

TWQCD Collaboration

National Taiwan U. T.W.Chiu, K. Ogawa,

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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 ~QCD.

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

But QCD is highly non-perturbative.

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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…

1. Chiral symmetry is difficult. [Nielsen & Ninomiya 1981]

Recently chiral symmetry is redefined [Luescher 1998] with a new type of Dirac operator [Hasenfratz 1994, Neuberger 1998] satisfies the Ginsparg-Wilson [1982] relation

but numerical implementation and m->0 require a large computational cost.

2. Large finite V effects when m-> 0. as m->0, the pion becomes massless.

(the pseudo-Nambu-Goldstone boson.)

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1. Introduction This work

1. We achieved lattice QCD simulations with exact chiral symmetry.

• Exact chiral symmetry with the overlap fermion.• With a new supercomputer at KEK ( 57 TFLOPS )• Speed up with new algorithms + topology fixing => On (~1.8fm)4 lattice, achieved m~3MeV !

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

by the Chiral Random Matrix Theory (ChRMT). -> 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. NLO effects6. Conclusion

　

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2. QCD Dirac spectrum & ChRMT　 Banks-Casher relation [Banks &Casher

1980]

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Σ

low density

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

is given by the surface of S3 with the radius :

• 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 ( >> QCD ) -> weak coupling

• Low modes ( << QCD ) -> strong coupling

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

with an unknown action V ⇒ ChRMT.

2. QCD Dirac spectrum & ChRMT

[Shuryak & Verbaarschot,1993, Verbaarschot & Zahed, 1993,Nishigaki et al, 1998, Damgaard & Nishigaki, 2001…]

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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() 　 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() (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 in

tegrals of chiral perturbation theory [Osborn et al, 1999];

The second term in the exponential is written aswhere

Let us introduce Nf x Nf real matrix 1 and 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 regime.

⇒

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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.

V

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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

Low modes are described by ChRMT.

• the distribution of each eigenvalue is known.

• finite m and V effects controlled by the same .

Higher modes are like free theory ~3

ChPT moduli

Analytic solution not known

-> Let us compare with lattice QCD !

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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

Topology fixing In order to achieve |Hw| > 0 [Hernandez et al.1998, Luescher 1998,1999], we add “topology stabilizing” term [Izubuchi et al. 2002, Vranas 2006, JLQCD 2006]

with =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 ? -> (probably) O.K.

(Local fluctuation of topology is consistent with ChPT.)

[JLQCD, arXiv:0710.1130]

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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. 5D solver.4. Multishift –conjugate gradient for the 1/Hw2.5. 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,

– Topology fixing cut the latter cost ~ 10 times faster– New supercomputer at KEK ~60TFLOPS ~ 10 times– Mass preconditioning ~ 5 times– 5D solvor ~ 2 times

10*10*5*2 = 1000 ! [See recent developments: Fodor et al, 2004, DeGrand &

Schaefer, 2004, 2005, 2006 ...]

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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. [regime]

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

– Iwasaki (beta=2.3,2.35) + Q fixing action– Fixed topological sector (No topology change.)– The lattice spacings a is calculated from quark potential (Som

mer scale r0).– Eigenvalues are calculated by Lanzcos algorithm.

(and projected to imaginary axis.)

Runs• Run 1 (epsilon-regime) Nf=2: 163x32, a=0.11fm

-regime (msea ~ 3MeV)– 1,100 trajectories with length 0.5– 20-60 min/traj on BG/L 1024 nodes– Q=0

Run 3 (p-regime) Nf=2+1 : 163x48, a=0.11fm (in progress)

2 strange quark masses around physical ms

5 ud quark masses covering (1/6~1)ms

Trajectory length = 1 About 2 hours/traj on BG/L 1024 nodes

• Run 2 (p-regime) Nf=2: 163x32, a=0.12fm 6 quark masses covering (1/6~1) ms

– 10,000 trajectories with length 0.5– 20-60 min/traj on BG/L 1024 nodes

– Q=0, Q=−2,−4 (msea ~ ms/2)

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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 ~ (240MeV)3.

– gap from 0.⇒　 need ChRMT analysis

for the precise measurement of !

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4. Numerical results

Low-mode spectrum　 Lowest eigenvalues qualitatively agree with ChRMT.

k=1 data ->　 = [240(6)(11) MeV]3

statistical NLO effect

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 ~2.

1.54(10)1.4144

1.587(97)1.3733

1.453(83)1.3162

1.215(48)1.2341

latticeRMT

[Related works: DeGrand et al.2006, Lang et al, 2006, Hasenfratz et al, 2007…]

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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 , determined by the lowest-eigenvalue, significantly depends 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 =[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 , 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 -> ~ 10%.1. Higher eigenvalue feel pressure from bulk modes.

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

statistical systematic

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5. NLO V effects

Meson correlators compared with ChPTWith a comparison of meson correlators with (partially quenched) ChPT, we obtain[P.H.Damgaard & HF, Nucl.Phys.B793(2008)160]

where NLO V correction is taken into account.[JLQCD, arXiv:0711.4965]

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5. NLO V effects

Meson correlators compared with ChPT

But how about NNLO ? O(a2) ? -> need larger lattices.

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6. 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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6. Conclusion

The other works– Hadron spectrum [arXiv:0710.0929]

– Test of ChPT (chiral log) – Pion form factor [arXiv:0710.2390]

– difference [arXiv:0710.0691]

– BK [arXiv:0710.0462]

– Topological susceptibility [arXiv:0710.1130]

– 2+1 flavor simulations [arXiv:0710.2730]

– …

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6. Conclusion

The future works– Large volume (L~3fm)– Finer lattice (a ~ 0.08fm)

We need 24348 lattice (or larger).We plan to start it with a~0.11fm, ma=0.015 (ms/6) [not enough to e-regime] in March 2008.