1 Recent results from lattice QCD Tetsuya Onogi (YITP, Kyoto Univ.) for JLQCD collaboration

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 Outline 1.Introduction 2.Method for dynamical overlap simulation 3.Applications QCD vacuum Chiral symmetry breaking, topological susceptibility Spectrum and Chiral Perturbation Theoy (ChPT) Flavor Physics New directions 4.Summary

4 1. Introduction Many unquenched simulations are performed or starting now. New era for lattice QCD. Lattice QCD with dynamical quark having exact chiral symmetry enables us to attack new problems which was impossible otherwise.

5 Advantage of exact chiral symmetry Exact results from chiral symmetry can be reproduced good for Chiral symmetry breaking Exact chiral anomaly relations good for topological susceptibility Chiral behavior is correct at finite lattice spacing fit with Chiral Perturbation Theory (ChPT) is valid good for chiral extrapolation for Operator mixing with wrong chirality is prohibited. good for No extra divergence due to lack of chiral symmetry appears good for No O(a) error good for scaling and nonperturbative renormalization

6 Neuberger’s overlap fermion Ginsparg-Wilson relation Ginsparg and Wilson, Phys.Rev.D 25(1982) Exact chiral symmetry on the lattice (index theorem) Hasenfratz, Laliena and Niedermayer, Phys.Lett. B427(1998) 125 Luscher, Phys.Lett.B428(1998)342. Overlap fermion

7 [1] Cost grows towards chiral limit as [2] Numerical implementation by rational approximation Nrational  large : huge numerical cost. [3] Sign function discontinuity at Hw=0. Special care is need to make reflection/refraction at the discontinuity Additional numerical cost. Becomes much more serious with larger volume. Problems in dynamical overlap simulation 300 years or more on 10Tflops supercomputer with naïve algorithm. We should reduce it to one year

8 KEK Blue/Gene 57Tflops ( x 5 ) Algorithm –Hasenbusch mass preconditioning ( x 5 ) solves the problem [1] by separating the low and high modes –5 dimensional solver ( x 4 ) solves the problem [2] by more efficient rationalization Gauge action ( x 3 or much more ) solves the problem [3] by prohibiting the topology change. Speed up by factor 5 x 5 x 4 x 3 = 300 !! H. Matsufuru, Plenary talk at lattice Method for dynamical overlap fermion simulation

Suppressing near-zero modes of H W Topology fixing term: extra Wilson fermion/ghost (Vranas, 2000, Fukaya, 2006, JLQCD, 2006)‏ N f = 2, a~0.125fm, m sea ~ m s, with S E without S E avoids zero mode of Hw during MD evolution  No need of reflection/refraction  Cheeper sign function

10 2. Applications

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)‏

12 Mass parameters for Run 2 (p-regime Nf=2) We have 6 sea quark masses and 9 valence quark masses.

13 QCD vacuum

Chiral condensate Banks-Casher relation (Banks & Casher, 1980)‏ –Accumulation of low modes Chiral SSB – : spectral density of D epsilon-regime: at finite V  Low-energy effective theory  Q-dependence is manifest  Random Matrix Theory (RMT)‏ Finite V

Result in the -regime –Nf=2, 16 3 x32, a=0.11fm –m~3MeV Good agreement with RMT –lowest level distrib. –Flavor-topology duality Chiral condensate: –Nonperturbative renorm. effect: correctable by meson correlator H.Fukaya et al. (JLQCD, 2007, JLQCD and TWQCD, 2007)‏ Low-lying spectrum of D(m)‏

Simulation at fixed topology Out of the -regime, fixing topology could be a problem In the infinite V limit, –Fixing topology is irrelevant –Local fluctuation of topology is active In practice, V is finite –Topology fixing finite V effect –Theta vacuum physics can be reconstructed (see below)‏ –Must check local topological fluctuation topological susceptibility –Questions: Ergodicity ? S. Aoki et al.‏

Physics at fixed topology Reconstruct QCD in vacuum from fixed topology (Bowler et al., 2003, Aoki, Fukaya, Hashimoto, & Onogi, 2007)‏ Partition function at fixed topology for, Q distribution is Gaussian Physical observables –Saddle point analysis for –Example: pion mass

Topological susceptibility Topological susceptibility can be extracted from correlation functions (Aoki et al., 2007)‏ ma =0.025 where Numerical result at Nf=2, a=0.12fm m t mP 0

19 Mass Spectrum and Test with Chiral Perturbation Theory

Low-mode averaging Technical improvement: –50 pairs of eigenmodes of D predetermined –Solver with low mode projection (8 times faster)‏ –Low-mode averaging (DeGrand, 2004, Giusti et al., 2004)‏ Averaging over source points only for low mode contrib. Nf=2, a=0.12fm, Q=0 3 smallest m sea =m val. J.Noak, lattice 2007 proceedings

Two-loop ChPT test Chiral extrapolation –Fit parameter: –NLO vs NNLO of ChPT --- NLO tends to fail; NNLO successful –Low energy constants: J.Noaki lattice 2007 proceedings Nf=2, a =0.12fm, Q =0

Finite volume effect Finite volume correction –R: finite size effect from 2-loop ChPT (Colangelo et al, 2005)‏ –T: Fixed topology effect (Aoki et al, 2007)‏ At most 5% effect --- largely cancel between R and T No Q-dependence (consistent with expectation )‏ Nf=2, a =0.12fm, Q =0 Q=-2,-4 ( m sea =0.05)‏ (ampi) 2 afpi

23 Flavor Physics

BKBK Indirect CP violation for K Problems unquenched lattice calculations Wilson fermion : huge operator mixing wrong chirality. Staggered fermion : How to treat doubler contribution ? Domain-wall fermion: Exponentially suppressed but sizable operator mixing Overlap fermion is free from operator mixing problem N.Yamada, lattice 2007 proceedings

Bare Bag parameter is obtained by the simultaneous fit of 2-pt, 3-pt functions Contaminations from other states can be removed using fits with different Fit the mass dependence with Partially Quenched Chiral Perturbation (PQChPT) formula

Renormalization factor is obtained nonperturbatively with RI-MOM scheme (off-shell quark amplitude in Landau gauge) Use continuum perturbation theory to convert to MS bar scheme –Nf=2, a=0.12fm, preliminary result:

Pion form factor Precisely calculated with the all-to-all technique (J.Foley et al., 2001)‏ Pion charge radius Nf=2, a=0.12fm, preliminary result: T.Kaneko lattice 2007 proceedings ‏ This method also applies to Kl3

28 New directions

Vacuum polarizations Das-Guralnik-Mathur-Low-Young sum rule (1967)‏ –Vacuum polarization Low Q 2 : pole dominance High Q 2 : OPE Nf=2, a=0.12fm Cf. exp MeV 2 Exact chiral symmetry plays essential role ! Talk by E.Shintani

30 Nucleon sigma term Definitions of the nucleon sigma term sigma term = scalar form factor of the nucleon at zero recoil related quantities

31 Why sigma term is important? A crucial parameter for the WIMP dark matter detection rate. The interaction with nucleon is mediated by the higgs boson exchange in the t-channel. Also related to the chromo-electric contributions to neutron EDM assuming (1) PCAC + (2) QCD sum rule with saturation by lightest O++ state ex. J. Hisano and Y. Shimizu, Phys.Rev.D70, (2004) heavy quark loopstrange quark K. Griest, Phys.Rev.Lett.62,666(1988) Phys,Rev,D38, 2375(1988)

32 Previous results

33 Basic Methods Nucleon mass spectrum Feynman - Hellman theorem From now on we will denote as for referring for the sake of brevity.

34 Nucleon masses from 2-pt functions Effective mass plot for amq=0.035 Solid lines are the mass from the fit Nice plateau for t >4 We fit the 2-pt function with a single expoential function with fitting range t=5-10

35 Sea and valence quark mass dependences The valence quark mass dependence is very clear, while the sea quark mass dependence is small.

36 Fit of the quark mass dependence Fit for interpolation using polynomial Fit for chiral extrapolation with diagonal (unitary) points Chiral Perturbation Theory (ChPT)

37 valence quark contribution to (polynomial)

38 Sea quark contribution to (polynomial fit)

39 Nucleon mass in the chiral limit is consistent with the experimental value. similar analysis was done by Procula et al.(2004) Sea and valence quark contribution to (ChPT)

40 sigma term ChPT fit Preliminary

41 (polynomial) Preliminary

42 Comparison with other results – ChPT results and previous results are consistent. –Our results with ChPT is consistent –Previous lattice calculation sea/valence. Is larger than 1 –Our lattice calculation sea/valence n is about 0.1 –ChPT predicts –Previous lattice results due to large sea quark contribution –Our results with ChPT gives

43 Comparison with other results – ChPT results and previous results are consistent. –Our results with ChPT is consistent –Previous lattice calculation sea/valence. Is larger than 1 –Our lattice calculation sea/valence n is about 0.1 –ChPT predicts –Previous lattice results due to large sea quark contribution –Our results with ChPT gives

44 Additive mass shift and sigma term Wilson fermion has an additive mass shift through tadpole diagram which mimics mass operator insertion to the valence quark due to the lack of chiral symmetry Normal disconnected contribution (long distance effect) Operator mixing induced connected contribution (short distance effect)

45 Additive mass shift and sigma term Same phenomena can be understood in the mass spectrum approach Fixing the bare valence quark mass and changing the sea quark mass effectively induces the change of the ‘physical’ valence quark mass due to the additive mass shift.

Summary/Outlook We are performing dynamical overlap project at fixed topological charge –Nf=2 on 16 3 x32, a~0.12fm: producing rich physics results –Nf=2+1 on 16 3 x48, a~0.11fm: in progress Understanding chiral dynamics with exact chiral symmetry Application to matrix elements in progress Opens new directions for phenomenological study from lattice QCD Outlook More measurements planned Larger lattices 24 3 x48: need further improved algorithms

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