Realistic simulations w/ exact chiral symmetry T. Kaneko for the JLQCD/TWQCD collaborations 1 High Energy Accelerator Research Organization (KEK) 2 Graduate University for Advanced Studies Workshop @ Atami, Feb. 27, 2009 T.Kaneko Realistic simulations w/ exact chiral symmetry
introduction introduction 1. introduction: JLQCD/TWQCD collaborations JLQCD collaboration study lattice QCD on super computer system at KEK KEK: S.Hashimoto, H.Ikeda, TK, M.Matsufuru, J.Noaki, N.Yamada YITP: H.Ohki, T.Onogi, E.Shintani, T.Umeda Tsukuba: S.Aoki, K.Takeda, T.Yamazaki Nagoya: H.Fukaya Hiroshima: K-I.Ishikawa, M.Okawa – FY2000: w/ Fujitsu VPP500 (128GFLOPS) N f =0 : KS or Wilson/clover: B K , f B,D , B B , m s , ... – FY2005: w/ Hitachi SR8000 (1.2TFLOPS) N f =2 : clover : M msn , M brn , f π,K , B B N f =2 + 1 : clover : only M msn , M brn , f π,K (w/ CP-PACS Collab.) chiral symmetry breaking due to lattice action ⇒ severely limits applications of gauge ensembles! T.Kaneko Realistic simulations w/ exact chiral symmetry
introduction introduction 1. introduction: chiral symmetry on the lattice chiral symmetry characterize low-energy dynamics of QCD through SSB M 2 msn ∝ m q , ... price of chiral symmetry breaking operator mixing in renormalization : B K modified ChPT : ex. Wilson ChPT (Sharpe-Singleton, 1998, ...) a 2 ln[ m q ] singularities additional LECs, OK for M msn , f P S at NLO; maybe not for others / NNLO Nielsen-Ninomiya no-go theorem, 1981 Ginsparg-Wilson fermions, 1982 GW relation : The following properties can NOT hold { γ 5 , D } = aDγ 5 D/m 0 ⇔ simultaneously: modified chiral symmetry locality δ ¯ q = ¯ qγ 5 , δq = γ 5 (1 − aD/ 2) q translational invariance correct anomaly, no mixing no doublers unphys.species, anomaly examples of D ............? chiral sym. : { γ 5 , D } = 0 T.Kaneko Realistic simulations w/ exact chiral symmetry
introduction introduction 1. introduction: dynamical overlap simulations overlap fermions (Neuberger, 1998) X S q = q D ( m ) q, ¯ m 0 + m m 0 − m “ ” “ ” γ 5 sgn [ H w ( − m 0 )] D ( m ) = + 2 2 ( m = quark mass, m 0 = a tunable parameter) satisfies GW relation exactly! computationally demanding... JLQCD/TWQCD’s project precise determination of MEs ⇒ test of Standard Model, ... current supercomputer system @ KEK (FY2006– FY2010) IBM Blue Gene/L + Hitachi SR11K : 59 TFLOPS in total ⇒ embark large-scale unquenched simulations with overlap fermion in collaboration w/ T.-W.Chiu and T.H.Sheh in Taiwan (TWQCD) T.Kaneko Realistic simulations w/ exact chiral symmetry
introduction introduction Outline outline Introduction simulation method implementation; parameters fixed topology physics results for N f =2 : on-going N f =3 runs � J µ ( x ) J ν (0) � ⇒ talk by E. Shintani (Kyoto) summary T.Kaneko Realistic simulations w/ exact chiral symmetry
implementation simulations parameters measurement 2. simulation method T.Kaneko Realistic simulations w/ exact chiral symmetry
implementation simulations parameters measurement 2.1 implementation: overlap action overlap-Dirac operator ∋ sgn [ H W ] m 0 + m m 0 − m “ ” “ ” γ 5 sgn [ H w ( − m 0 )] , D ( m ) = + m 0 = 1 . 6 2 2 sgn [ H W ] ∼ polynomial / rational approx. Zolotarev (min-max) approx: sgn [ H W ] = H W { p 0 + P l =1 p l / ( H 2 W + q l ) } nested inversion of Dirac operators : very time consuming D ( m ) − 1 ∋ Dx ∋ H − 1 W discontinuity in S q ⇒ modified HMC (Fodor-Katz-Szabo, 2004) : very time consuming locality ⇐ OK, if σ [ H W ] has a positive lower bound ⇒ suppress (near-)zero modes of H W by auxiliary Boltzman weight (JLQCD, 2006) det[ H W ( − m 0 ) 2 ] ∆ W = det[ H W ( − m 0 ) 2 + µ 2 ] , µ = 0 . 2 T.Kaneko Realistic simulations w/ exact chiral symmetry
implementation simulations parameters measurement 2.1 implementation: auxiliary determinant ∆ W : modification of gauge action ⇒ modification of a dependence ∆ W : suppress (near)zero modes of H W ⇒ locality, reduced cost w/ extra-Wilson w/o extra-Wilson 0.04 0.04 0.03 0.03 | λ | 0.02 | λ | 0.02 0.01 0.01 0 0 0 100 200 300 0 300 600 0 100 200 0 50 100 HMC trajectory hitogram hitogram HMC trajectory ∆ W : fix global topology during HMC ⇒ effects should be studied T.Kaneko Realistic simulations w/ exact chiral symmetry
implementation simulations parameters measurement 2.1 implementation: other techniques other techniques multiplications of D : accuracy of sgn [ H W ] = accuracy of chiral sym. σ [ H W ] ⇒ [ λ min , λ thrs ] ∪ [ λ thrs , λ max ] low modes : Lanczos ⇒ low mode projection high modes : Zolotarev approx. (min-max) ⇒ �∇ 4 A 4 P † � / � PP † �| m =0 � 0 . 1(0 . 1) MeV D solver = 5D solver (Edwards et al.,2005) : Schur decomposition ( D ⇒ M 5 ) + even/odd precond. ⇒ CPU cost × 1 / 2 HMC algorithm mass preconditioning +multiple time scale MD ⇒ CPU cost × 1 / 5 Hasenbusch, 2001; Sexton-Weingargen, 1992 no reflection/refraction steps : ⇒ CPU cost × 1 / 4 assembler code for H W multiplications : ⇒ CPU cost × 1 / 3 T.Kaneko Realistic simulations w/ exact chiral symmetry
implementation simulations parameters measurement 2.2 parameters N f =2 runs : completed Iwasaki-gauge + overlap + ∆ W β =2 . 30 ⇒ a ≈ 0 . 118(2) fm ( r 0 =0 . 49 fm) 16 3 × 32 lattice ⇒ L ≃ 2 fm 6 values of m ud ∈ [ m s, phys / 6 , m s, phys ] for Q =0 10000 HMC trajectories at each m ud Q = − 2 , − 4 at m ud ∼ m s , phys / 2 N f =2 + 1 runs : on-going β =2 . 30 ⇒ a ≈ 0 . 107(1) fm ( r 0 =0 . 49 fm) 16 3 × 48 lattice ⇒ L ≃ 1 . 7 fm 2 m s ’s = m s, phys , 1 . 3 × m s, phys ; 5 m ud ’s ∈ [ m s, phys / 6 , m s ] ; 2500 traj. 24 3 × 48 lattice ⇒ L ≃ 2 . 6 fm, M π L � 3 . 7 m s ∼ m s, phys ; m ud ∼ m s, phys / 6 , m s, phys / 4 ; 2500 traj. simulations with Q � =0 are on-going T.Kaneko Realistic simulations w/ exact chiral symmetry
implementation simulations parameters measurement 2.3 measurements: low-mode averaging low-mode averaging (LMA) PS correlator 100 low-modes ⇐ Lanczos 0.2 D u ( k ) λ ( k ) u ( k ) = m π ,eff 1 X u k u † ( D − 1 ) low = k λ k k 0.2 conventional Dn ( D − 1 ) low + ( D − 1 ) high o C Γ = Γ LMA 0 5 10 15 t n o E ( D − 1 ) low + ( D − 1 ) high × Γ vector correlator = C Γ ,LL + C Γ ,HL + C Γ ,LH + C Γ ,HH 1.0 exact low-mode contrib. 0.8 m ρ ,eff ⇒ reduce stat. error remarkably 0.6 preconditioning for D − 1 solver 0.4 conventional 100 modes ⇒ × 8 speed up LMA 0 5 10 15 t ρ ( λ ) , χ t , spectrum, B K , ... T.Kaneko Realistic simulations w/ exact chiral symmetry
implementation simulations parameters measurement 2.3 measurements: all-to-all propagator all-to-all quark propagator (TrinLat, 2005) � π | V 4 | π � 6 high mode ⇐ noise method m = 0.025, |p| = sqrt(2), |p ′ |=0, ∆ t = 7, ∆ t ′ = 7 average over x average over (x, t ) and p D x ( r ) (1 − P low ) η ( r ) = 4 (conn) ( ∆ t , ∆ t ′ ;p,p ′ ) 1 ( D − 1 ) high X x ( r ) η ( r ) † = N r 2 r can construct any correlators C π V 4 π 0 w/o small additional CPU costs smearing function -2 0 20 40 60 80 100 jackknife sample meson momenta ⇒ form factors, hadron decays needs large N r for M HL , M N , ... ⇔ O (100) TB disk pion form factors T.Kaneko Realistic simulations w/ exact chiral symmetry
fixed topology fixed topology topological susceptibility 3. fixed topology T.Kaneko Realistic simulations w/ exact chiral symmetry
fixed topology fixed topology topological susceptibility 3.1 fixed topology fixed topology simulations auxiliary determinant ∆ W ⇒ fix global topology during HMC : problematic ? do NOT suppress local topological fluctuations ⇒ χ t is calculable from fixed Q simulations (next slide) suppressed by 1 /V : small effects ( � 1 %) in MEs effects can be corrected systematically (Aoki et al., 2007) G θ =0 − G Q = G (2) 1 − Q 2 „ c 4 « + O ( V − 3 ) θ =0 χ t V − 2 χ 2 2 χ t V t V topological tunneling will be suppressed at a → 0 ⇒ fixing Q is inevitable for any lattice regularization (w/o modifications of HMC) T.Kaneko Realistic simulations w/ exact chiral symmetry
fixed topology fixed topology topological susceptibility 3.2 topological susceptibility: determination determination from η ′ correlator ( N f =2 ) η ′ correlator ⇒ a constant term ⇒ χ t 1 − Q 2 − χ t „ c 4 « m 2 q � η ′ ( t + ∆ t ) η ′ ( t ) � Q + O ( e − M η ∆ t ) + O ( V − 3 ) = χ t V + 2 χ 2 V t V η ′ correlator for Q =0 Q =0 : 2nd term vanishes 0.004 disconnected connected 4pt function ⇒ c 4 / (2 χ 2 V ) : small η ' 0.002 A + cosh η ′ contamination rapidly damps C(t) A ⇒ clear plateau ( M ′ η > M π ) 0.000 3 sectors Q =0 , − 2 , − 4 at m ud =0 . 050 -0.002 ⇒ consistent results for χ t 5 10 15 20 25 t T.Kaneko Realistic simulations w/ exact chiral symmetry
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