Lattice QCD, Random Matrix Theory and chiral condensates

Lattice QCD, Random Matrix Theory and 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) for JLQCD collaboration

JLQCD Collaboration KEK S. Hashimoto, T. Kaneko, H. Matsufuru, J. Noaki,M. Okamoto, E. Shintani, N. Yamada RIKEN -> Niels Bohr H. Fukaya Tsukuba S. Aoki, T. Kanaya, Y. Kuramashi, N. Ishizuka, Y. Taniguchi,A. Ukawa, T. Yoshie Hiroshima K.-I. Ishikawa, M. Okawa YITP H. Ohki, T. Onogi TWQCD Collaboration National Taiwan U. T.W.Chiu, K. Ogawa, KEK BlueGene (10 racks, 57.3 TFlops)

1. Introduction Chiral symmetry and its spontaneous breaking are important. • Mass gap between pion and the other hadrons pion as (pseudo) Nambu-Goldstone boson while the other hadrons acquire the mass ~LQCD. • Soft pion theorem • Chiral phase transition at finite temperature… But QCD is highly non-perturbative.

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.)

1. Introduction This work • 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 ! • 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

Contents • Introduction • QCD Dirac spectrum & ChRMT • Lattice QCD with exact chiral symmetry • Numerical results • NLO effects • Conclusion

2. QCD Dirac spectrum & ChRMT [Banks &Casher 1980] Banks-Casher relation

low density Σ 2. QCD Dirac spectrum & ChRMT [Banks &Casher 1980] Banks-Casher relation • In the free theory, r(l) is given by the surface of S3 with the radius l: • With the strong coupling The eigenvalues feel the repulsive force from each other→becoming non-degenerate→ flowing to the low-density region around zero→ results in the chiral condensate.

2. QCD Dirac spectrum & ChRMT Chiral Random Matrix Theory (ChRMT) Consider the QCD partition function at a fixed topology Q, • High modes ( l >> LQCD ) -> weak coupling • Low modes ( l << LQCD ) -> strong coupling ⇒ Let us make an assumption: For low-lying modes, with an unknown action V(l) ⇒ ChRMT. [Shuryak & Verbaarschot,1993, Verbaarschot & Zahed, 1993,Nishigaki et al, 1998, Damgaard & Nishigaki, 2001…]

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(l)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(l) (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

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 [Osborn et al, 1999]; The second term in the exponential is written as where Let us introduce Nf x Nf real matrix s1 and s２as

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 e regime. ⇒

V Nf=2, m=0 and Q=0. 2. QCD Dirac spectrum & ChRMT Eigenvalue distribution of ChRMT Damgaard & 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.

Higher modes are like free theory ~l3 ChPT moduli • Low modes are described by ChRMT. • the distribution of each eigenvalue is known. • finite m and V effects controlled by the same S. 2. QCD Dirac spectrum & ChRMT Summary of QCD Dirac spectrum IF QCD dynamically breaks the chiral symmetry, the Dirac spectrum in finite V should look like r Banks-Casher S Analytic solution not known -> Let us compare with lattice QCD ! l

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. • D is theoretically ill-defined. [Hernandez et al. 1998] • Numerical cost is suddenly enhanced. [Fodor et al. 2004]

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 m=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]

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 efforts • Zolotarev expansion of D -> 10 -(7-8) accuracy. • Relaxed conjugate gradient algorithm to invert D. • 5D solver. • Multishift –conjugate gradient for the 1/Hw2. • Low-mode projections of Hw.

3. Lattice QCD with exact chiral symmetry Numerical cost Simulation 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 extra 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 ...]

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. [e-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 (Sommer scale r0). • Eigenvalues are calculated by Lanzcos algorithm. (and projected to imaginary axis.)

Runs • Run 1 (epsilon-regime) Nf=2: 163x32, a=0.11fm e-regime (msea ~ 3MeV)‏ • 1,100 trajectories with length 0.5 • 20-60 min/traj on BG/L 1024 nodes • Q=0 • 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)‏ • 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

4. Numerical results In 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 S ~ (240MeV)3. • gap from 0. ⇒need ChRMT analysis for the precise measurement of S !

RMT Lattice 1 4.30 [4.30] 2 7.62 7.25(13) 3 10.83 9.88(21) 4 14.01 12.58(28) [] is used as an input. ~5-10% lower -> Probably NLO 1/V effects. k=1 data ->S = [240(6)(11) MeV]3 statistical NLO effect 4. Numerical results Low-mode spectrum Lowest eigenvalues qualitatively agree with ChRMT.

RMT lattice 1 1.234 1.215(48) 2 1.316 1.453(83) 3 1.373 1.587(97) 4 1.414 1.54(10) 4. Numerical results Low-mode spectrum Cumulative histogram is useful to compare the shape of the distribution. The width agrees with RMT within ~2s. [Related works: DeGrand et al.2006, Lang et al, 2006, Hasenfratz et al, 2007…]

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.

4. Numerical results Heavier quark masses However, the value of S, determined by the lowest-eigenvalue, significantly depends on the quark mass. But, the chiral limit is still consistent with the data with 3MeV.

4. Numerical results Renormalization Since S=[240(2)(6)]3 is the lattice bare value, it should be renormalized. We calculated • the renormalization factor in a non-perturbative RI/MOM scheme on the lattice, • match with MS bar scheme, with the perturbation theory, • and obtained (non-perturbative) (tree)

4. Numerical results Systematic errors • finite m -> small. As seen in the chiral extrapolation of S , 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%. • Higher eigenvalue feel pressure from bulk modes. higher k data are smaller than RMT. (5-10%) • 1-loop ChPT calculation also suggests ~ 10% . systematic statistical

5. NLO V effects Meson correlators compared with ChPT With 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]

5. NLO V effects Meson correlators compared with ChPT But how about NNLO ? O(a2) ? -> need larger lattices.

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

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] • …

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.

Lattice QCD, Random Matrix Theory and chiral condensates