1. 1 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

2. 2 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 KEK BlueGene (10 racks, 57.3 TFlops) TWQCD Collaboration National Taiwan U. T.W.Chiu, K. Ogawa,

3. 3 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 ~  QCD. –Soft pion theorem –Chiral phase transition at finite temperature … But QCD is highly non-perturbative.

4. 4 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.)

5. 5 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

6. 6 Contents 1.Introduction 2.QCD Dirac spectrum & ChRMT 3.Lattice QCD with exact chiral symmetry 4.Numerical results 5.NLO effects 6.Conclusion

7. 7 2. QCD Dirac spectrum & ChRMT Banks-Casher relation [Banks &Casher 1980]

8. 8 Σ low density 2. QCD Dirac spectrum & ChRMT Banks-Casher relation In the free theory,  is given by the surface of S 3 with the radius : 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. [Banks &Casher 1980]

9. 9 Chiral Random Matrix Theory (ChRMT) Consider the QCD partition function at a fixed topology Q, High modes ( >>  QCD ) -> weak coupling Low modes (  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 … ]

10. 10 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

11. 11 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  1 and  ２ as

12. 12 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. ⇒

13. 13 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 whereand where -> spectral density or correlation can be calculated, too. 2. QCD Dirac spectrum & ChRMT Nf=2, m=0 and Q=0.  V

14. 14 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 !

15. 15 3. Lattice QCD with exact chiral symmetry The overlap Dirac operator We use Neuberger’s overlap Dirac operator [Neuberger 1998] (we take m 0 a=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]

16. 16 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: S top -> ∞ when Hw->0 and S top -> 0 when a->0. ( Note is extra Wilson fermion and twisted mass bosonic spinor with a cut-off scale mass. ) With S top, 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]

17. 17 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 1.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/Hw 2. 5.Low-mode projections of Hw.

18. 18 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...]

19. 19 3. Lattice QCD with exact chiral symmetry Simulation summary On a 16 3 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 (Sommer scale r0). –Eigenvalues are calculated by Lanzcos algorithm. (and projected to imaginary axis.)

20. Runs Run 1 (epsilon-regime) Nf=2: 16 3 x32, a=0.11fm  -regime ( m sea ~ 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 : 16 3 x48, a=0.11fm (in progress)‏  2 strange quark masses around physical m s  5 ud quark masses covering (1/6~1)m s  Trajectory length = 1  About 2 hours/traj on BG/L 1024 nodes Run 2 (p-regime) Nf=2: 16 3 x32, a=0.12fm 6 quark masses covering (1/6~1) m s –10,000 trajectories with length 0.5 –20-60 min/traj on BG/L 1024 nodes –Q=0, Q=−2,−4 ( m sea ~ m s /2) ‏

21. 21 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  ~ (240MeV) 3. –gap from 0. ⇒ need ChRMT analysis for the precise measurement of  !

22. 22 4. Numerical results Low-mode spectrum Lowest eigenvalues qualitatively agree with ChRMT. k=1 data ->  = [240(6)(11) MeV] 3 statisticalNLO 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.

23. 23 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…]

24. 24 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.

25. 25 4. Numerical results Heavier quark masses However, 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.

26. 26 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)

27. 27 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(a 2 ) -> (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%) 2.1-loop ChPT calculation also suggests ~ 10%. statistical systematic

28. 28 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]

29. 29 5. NLO V effects Meson correlators compared with ChPT But how about NNLO ? O(a 2 ) ? -> need larger lattices.

30. 30 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 1 st principle. –obtained

31. 31 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] –B K [arXiv:0710.0462] –Topological susceptibility [arXiv:0710.1130] –2+1 flavor simulations [arXiv:0710.2730] –…

32. 32 6. Conclusion The future works –Large volume (L~3fm) –Finer lattice (a ~ 0.08fm) We need 24 3 48 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.