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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. Ishizuka, 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 symmetry

and 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 extr

a 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)14.0149.88(21)10.8337.25(13)7.622[4.30]4.301LatticeRMT

[] 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)1.41441.587(97)1.37331.453(83)1.31621.215(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.