Physics results from dynamical overlap fermion simulations. Shoji Hashimoto Lattice 2008 at Williamsburg, Virginia, July 16, 2008.

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1 Physics results from dynamical overlap fermion simulations Shoji Hashimoto Lattice 008 at Williamsburg, Virginia, July 16, 008.

2 JLQCD+TWQCD collaborations JLQCD SH, T. Kaneko, H. Matsufuru, J. Noaki, E. Shintani, N. Yamada (KEK) H. Ohki, T. Onogi (Kyoto) S. Aoki, N. Ishizuka, K. Kanaya, Y. Kuramashi, Y. Taniguchi, A. Ukawa, T. Yoshie (Tsukuba) K. Ishikawa, M. Okawa (Hiroshima) H. Fukaya (Niels Bohr Inst) TWQCD T.W. Chiu, T.H. Hsieh, K. Ogawa (National Taiwan Univ) Machines at KEK (since 006) SR11000 (.15 Tflops) BlueGene/L (10 racks, 57.3 Tflops)

3 Project: dynamical overlap fermions Theoretically clean = Respect the symmetry Slow to develop 3

4 Project: dynamical overlap fermions First large scale simulation with exact chiral symmetry Theoretical interest Dirac operator spectrum: Banks-Casher relation, chiral RMT Chiral symmetry breaking: chiral condensate and related Topology: θ-vacuum, topological susceptibility Phenomenological interest Controlled chiral extrapolation with the continuum ChPT Physics applications: B K, form factors, etc. Sum rules, OPE Flavor-singlet physics 4

5 Publications from the project Not including conference proceedings 1. Fukaya et al. Lattice gauge action suppressing near-zero modes of H W, Phys. Rev. D, (006).. Fukaya et al. Two-flavor QCD simulation in the ε-regime, Phys. Rev. Lett 98, (007). 3. Fukaya et al. Two-flavor lattice QCD in the ε-regime, Phys. Rev. D76, (007). 4. Aoki, Fukaya, SH, Onogi, Finite volume QCD at fixed topological charge, Phys. Rev. D76, (007). 5. Aoki et al., Topological susceptibility in two-flavor QCD, Phys. Lett. B665, 94 (008). 6. Fukaya et al., Lattice study of meson correlators in the ε-regime, Phys. Rev. D77, (008). 7. Aoki et al. B K with two flavors of dynamical overlap fermions, Phys. Rev. D77, (008). 8. Aoki et al. Two-flavor QCD simulation with exact chiral symmetry, arxiv: [hep-lat], to appear in PRD. 9. Noaki et al. Convergence of the chiral expansion, arxiv: [hep-lat]. 10. Shintani et al. S-parameter and pseudo NG boson mass, arxiv: [hep-lat]. 11. Ohki et al., Nucleon sigma term and strange quark content, arxiv: [hep-lat]. 1. Shintani et al., Lattice calculation of strong coupling constant, arxiv: [hep-lat]. 5

6 Plan 1. Simulation status Overlap implementation, runs,. Topology issues Physics from fixed topology Topological susceptibility 3. Physics applications Chiral condensate Convergence of the chiral expansion (m π, f π ) Pion form factor, B K VV-AA, αs Nucleon sigma-term, strange content 6

7 1. Simulation status See also, Matsufuru, poster session

8 Overlap fermion Neuberger-Narayanan (1998) 1 X D = 1 +, X = ad 1 W a X X 1 = [ 1 + γ5 sgn( ahw ) ], ahw = γ5 ( adw 1) a Exact chiral symmetry through the Ginsparg-Wilson relation. Dγ 5 + γ 5D = adγ 5D Continuum-like Ward-Takahashi identities hold Index theorem (relation to topology) satisfied 8

9 Sign function Rational approximation (Zolotarev) ε[ x] = x p + N pole 0 l= 1 x pl + q l Problem of near-zero modes of H W. Their density is non-zero at any finite β (Edwards, Heller, Narayanan, 1998) Need to subtract before approximate = potentially O(V ) 9

10 Near-zero mode suppression Near-zero modes are unphysical (associated with a local lump or dislocation = lattice artifact) lattice action to suppress them Introduce unphysical (heavy negative mass) Wilson fermions (Vranas, JLQCD, 006) Plaquette gauge, β=5.83, μ=0; β=5.70, μ=0. det H HW ( m ( m ) W 0 0 ) + μ Completely wash-out the near-zero modes. 10

11 Topological freezing Suppress the dislocations, e.g. by adding extra Wilson fermions det H HW ( m ( m ) W 0 0 ) + μ If the MD-type algorithm is used, the global topology never changes. Provided that the step size is small enough. Zero probability to have an exact zero-mode No chance to tunnel between different topological sectors. Property of the continuum QCD: common for all lattice formulations as the continuum limit is approached. 11

12 Dynamical overlap Recent attempts: Fodor-Katz-Szabo (003) Reflection/refraction trick Cundy et al. (004) Many algorithmic improvements DeGrand-Schaefer (005) Fat-link Some physics results Our project: Aoki et al., arxiv: [hep-lat] Fixed topology Large scale simulation with L fm, m q ~m s /6. -flavor and +1-flavor runs Broad physics program: Pion/kaon physics ε-regime Nucleon sigma term etc. 1

13 Parameters N f = runs many physics analysis completed/on-going. β=.30 (Iwasaki), a=0.1 fm, 16 3 x3 6 sea quark masses covering m s /6~m s 10,000 HMC traj. Q=0 sector only, except Q=, 4 runs at m q =0.050 N f = +1 runs some physics analysis begun. β=.30 (Iwasaki), a=0.11 fm, 16 3 x48 5 ud quark masses, covering m s /6~m s x s quark masses,500 HMC traj. Using 5D solver Q=0 sector only 13

14 Lattice spacing N f = N f =+1 β fixed (=.30) with varying m q Overlap fermion: close to the mass independent renormalization = no O(am q ) term. Sommer scale r 0, from the static quark potential N f = a = 0.118() fm N f = +1 a = 0.108() fm 14

15 SELENE (KAGUYA). Topology issues Instanton behind the moon?

16 Topology is fixed. Any problem? Yes = the real QCD vacuum is the θ-vacuum, a superposition of different topological sectors. A serious problem for everyone Topological tunneling occurs through rough gauge configs. If you observe frequent topology change, you are far apart from the continuum. A solution: accept it and reconstruct the θ-vacuum physics Aoki, Fukaya, SH, Onogi, PRD76, (007) Finite volume effect of O(1/V) Topological susceptibility calculable on a fixed topology configs. 16

17 Cluster decomposition In QCD, the real vacuum has a certain distribution of the topological charge = the θ vacuum. Required to satisfy the cluster decomposition property: topology distribution must satisfy f ( Q + Q ) = f( Q ) f( Q ) Ω 1 Ω 1 1 f ( Q) = i Q e θ Can one reproduce the physics of the θ vacuum from the fixed topology simulations? Sum-up the topology! Or, not? 17

18 Sum-up the topology! Partition function of the vacuum [ VE θ ] Z( θ ) = exp ( ) Vacuum energy density E(θ) Partition function for a fixed Q χ c 1 t 4 4 ( θ) = θ + θ i Q 1 Z = + π + π θ Q dθ Z( θ) e dθexp VE( θ) iθq π = π π + π Using a saddle point expansion around (θ c =iq/v), one can evaluate the θ integral to obtain Z Q Then, the original partition function can be recovered, if one i Q knows χ t, c 4, etc., as 18 E [ ] 1 Q c 4 1 Q = exp 1 + O, πχ tv 8V tv χ χ t ( χtv) ( χtv) Z( θ ) = ZQe θ Q

19 Or, not? Fixing topology = Finite volume effect Brower et al., PLB560, 64 (003); Aoki, Fukaya, SH, Onogi, PRD76, (007) When the volume is large enough, the global topology is irrelevant. Topological charge fluctuate locally, according to χ t, topological susceptibility. Physics of the θ-vacuum can be recovered by a similar saddle-point analysis, e.g. Some Green s function: CPeven () 1 Q c 4 (4) 1 GQ = G(0) + G (0) 1 (0)... + G + χtv χtv χtv 8χtV 19

20 Topological susceptibility χ t = Q /V Applying the same formula for the flavor-singlet PS density, χ t can be extracted. 1 Q = χt + + V V m x η ' lim mp( x) mp(0)... O( e ) x N f = example Q Talk by Chiu, Tue :50, Chesapeake B arxiv: [hep-lat] Look at a (negative) constant correlation of the local topological charges. Found a clear plateau. Results from other topological sectors are consistent. 0

21 Sea quark mass dependence Talk by Chiu, Tue :50, Chesapeake B N f = N f = done; +1 on-going Disconnected loops constructed from low modes (saturation confirmed) N f =+1 Leutwyler-Smilga (199) χ = mσ/ N or t = f, Σ χt mu + md + ms Clear evidence of the sea quark effects: and +1. Fit with ChPT expectation N f =: Σ = [4(5)(10) MeV]3 N f =+1: Σ = [40(5)() MeV]3 1

22 3. Physics applications

23 Chiral condensate 3

24 Chiral condensate Thanks to the exact chiral symmetry, additive renormalization is prohibited. 1 ( ψψ ) = Z ( ψψ ) + Z (1) a cont lat lat S mix 3 Many ways to extract Banks-Casher relation Low-lying eigenmodes (ChRMT) ε-regime correlator Topological susceptibility GMOR relation Σ mπ = ( mu + md) f [ ] Banks-Casher relation (N f =) 4

25 ε-regime Matching the low-lying eigenvalue distribution with Chiral Random Matrix Theory (ChRMT) Matching the PS and A correlators with ε-regime ChPT Σ = [ 51(7)(11) MeV] 3 F = 87(6)(8) MeV Σ = [ 40(4)(7) MeV] 3 5

26 Two-flavor results Σ 1/3 (GeV) Phys. Rev. Lett 98, (007) Phys. Rev. D76, (007) Phys. Rev. D77, (008) arxiv: [hep-lat] arxiv: [hep-lat] In good agreement 6

27 m π and f π 7

28 Convergence of chiral expansion Chiral expansion The region of convergence is not known a priori. Test with lattice QCD; conceptually clear with exact chiral symmetry. Expand in either 8 m m π q = B + x x+ c x+ O x 1 ln 3 ( ), fπ = f x x+ c x+ O x 1 ln 4 ( ). m m m x (4 π f ) (4 π f) (4 π f ) π π, xˆ, ξ π ξ extends the region significantly. Talk by Noaki, Tue 6:0, Auditorium arxiv: [hep-lat]

29 Two-loop analysis Talk by Noaki, Tue 6:0, Auditorium Analysis including NNLO With the ξ-expansion For reliable extraction of the low energy constants, the NNLO terms are mandatory. 9

30 +1 flavors Similar analysis including NNLO is on-going for +1- flavor data. Preliminary results. m m ud s (GeV)=3.76(45) MeV, ( GeV)=116(1) MeV, fk = 1.01(30). f π Talk by Noaki, Tue 6:0, Auditorium 30

31 Pion form factors 31

32 Pion form factors Another testing ground of ChPT Vector and scalar π π ( p') Vμ π( p) = i( pμ + pμ ') FV ( q ), ( p') S π( p) = FS ( q ), qμ pμ ' pμ Charge and scalar radius 1 π 4 FV ( q ) = 1 + r q + O( q ), 6 V 1 F q F r q O q 6 π 4 S( ) = S(0) ( ), S Calculation using the all-to-all technique. Talk by Kaneko, Thu 9:10, Chesapeake A Vector form factor q dependence well described by a vector meson pole + corrections. F 1 ( q ) = q / m cq π + + V 3

33 All-to-all Talk by Kaneko, Thu 9:10, Chesapeake A Disconnected diagram Relevant for the scalar form factor. Calculated using the all-to-all technique. Lowmodes are averaged over space-time. Disconnected contribution is visible. 33

34 Chiral extrapolation Fit with NNLO ChPT Talk by Kaneko, Thu 9:10, Chesapeake A Data do not show clear evidence of the chiral log. But, it is expected to show up even smaller pion masses. NNLO contribution is significant; necessary to reproduce the phenomenological values. r π V r π S 34

35 Vacuum polarization functions 35

36 Vacuum polarization functions Vector and axial correlators in the momentum space. ( ) J J = g q q q Π ( Q ) q q Π ( Q ) (1) (0) μ ν μν μ ν J μ ν J ds (1) (0) = ( g s s s ) Im J ( s) s s Im J ( s) μν μ ν Π μ ν Π s q + iε 0 Directly calculable on the lattice for space-like momenta Weinberg sum rules: f = lim Q Π ( Q ) Π ( Q ), (1+ 0) (1+ 0) π V A Q 0 Talk by Yamada, Fri 3:30, Tidewater A (1+ 0) (1+ 0) S = lim Q Π ( ) ( ) or V Q ΠA Q L Q 0 Q Another probe of the chiral symmetry breaking. S is relevant for the precision EW test of new strong dynamics

37 Pion electromagnetic mass splitting Das-Guralnik-Mathur-Low-Young sum rule (1967) 3α EM (1+ 0) (1+ 0) π V A 4π f π 0 Δ m = dq Q Π ( Q ) Π ( Q ) Valid in the chiral limit (soft pion theorem) Gives dominant contribution to the π ± -π 0 splitting. Related to the pseudo-ng boson mass in the context of new strong dynamics. Exact chiral symmetry is essential. Talk by Yamada, Fri 3:30, Tidewater A The quantity of interest is obtained after huge cancellation between V and A. 37

38 Lattice artifact Lorentz violation + currents not conserving J = V or A. Only B (0) J and C (1,1) J are physical. Thanks to the exact chiral symmetry, B J and C J are common (up to m q ) between V and A, thus cancel in V-A. Talk by Yamada, Fri 3:30, Tidewater A 4 ( ) (, ) 1 1 ( ) iq x 0 ( ) ( ) 0 n n m n m n Jμν q d xe T J x J y μ ν BJ qμ CJ qμ qν n= 0 m, n= 1 Π = = + 38

39 Lattice results Can be fitted with ChPT in the low q region (1) fπ r Π V A( q ) = 8 L 10( μ) q 1 m 1 π ln H ( x) + 4π μ 3 L 10 is extracted. + + L ( m ρ ) = 5.()( )( ) 10 r Talk by Yamada, Fri 3:30, Tidewater A arxiv: [hep-lat]. OPE in the high q region. In the massless limit, 1/Q 6 is the leading. Summing up the two regions, Δm π is obtained. Δ = + 0 m π 993(1)( 135 )(149) MeV 39

40 Strong coupling constant Matching of Π J (0+1) (Q ) with its perturbative expansion Adler function dπ J ( Q ) DJ ( Q ) Q dq is finite, renormalization scheme independent. Lattice artifacts are nonperturbatively subtracted. Talk by Shintani, Fri 3:10, Chesapeake A 40

41 Strong coupling constant High Q region described by OPE. J (0+ 1) Cm ( Q ) J ( Q ) c C0( Q, μ ) Π = + + α s mqq π 4 GG 4 GG J + Cqq ( Q ) Q + C ( Q ) Q +... Q Talk by Shintani, Fri 3:10, Chesapeake A Perturbative expansions known to α s. Chiral condensate is an input. Λ = 34(9)( ) MeV () + 16 MS 0 cf.) ALPHA: 50(16)(16) MeV QCDSF: 49(16)(5) MeV 41

42 Nucleon structure 4

43 Nucleon sigma term Finite quark mass effect on the nucleon mass σ = m N uu + dd N π N ud Contains both connected and disconected contrib. Strange quark content y N ss N Nuu+ ddn Disconnect contrib only. Talk by Ohki, Thu 9:50, Auditorium Feynman-Hellman theorem: Relates them to derivatives of nucleon mass in terms of m val and m sea. M m M m N val N sea = Nuu+ ddn = Nuu+ ddn ( = N ss N ) conn disc Analysis with partially quenched data set (m val m sea ) Use PQChPT, 43

44 Fit with HBChPT Nucleon mass Talk by Ohki, Thu 9:50, Auditorium Valence and Sea derivatives M m M m N val N sea = Nuu+ ddn = Nuu+ ddn conn disc, finite volume corrected Finite volume effect significant. Downward shift observed (non-analytic + ~g 0 + A m5 = 5()( )( ) π3 MeV ) σ π N 0 Disconnected contribution relatively small. y = 0.030(16)( )( )

45 So what? Previous lattice results: Fukugita et al. (1995) y = 0.66(15), quenched Dong-Lagae-Liu (1996) y = 0.36(3), quenched SESAM (1999) y = 0.59(13), N f = UKQCD (001) y = 0.8(33), N f = JLQCD (008) y = 0.030(18), N f = Talk by Ohki, Thu 9:50, Auditorium Problem and solution Was difficult to calculate due to m sea dependence of m cr.= easily spoils the physical effect. Problem persists in the quenched calculation. If subtracted, too large error. Exact chiral symmetry is the key. 45

46 Conclusion = clean approach producing interesting physics. Feasible with O(10 Tflops) machines 16 3 x x48: test runs started Frozen topology = the property of continuum QCD New strategy successful, e.g. topological susceptibility Physics applications (so far) Chiral condensates Test of continuum ChPT Sum rules, OPE Nucleon structure, flavor-singlet physics More to come 46

47 Backup slides 47

48 Locality Lattice Dirac operator must be local in order that a local theory is obtained in the continuum limit. Locality is not obvious for the overlap operator due to 1/. 1 X D= 1 +, X = ad 1 W a X X Locality in the sense that D <exp(-μx), with μ a number of order 1/a, may be satisfied. Proof is known for smooth enough gauge canfigurations (Hernandez, Jansen, Luscher (1999)). No mathematical proof in more realistic situations where there is non-zero density of the near-zero modes. Okay if near-zero modes are always localized. 48

49 Locality Maybe analyzed by looking at individual eigenmodes of H W. Near-zero modes are more localized. Higher modes are extended. There is a critical value above which the modes are extended = mobility edge (Golterman, Shamir (003)). An important lessen: do not use the overlap fermion in the Aoki phase (where the near-zero modes are extended). 49

50 FAQ on topology fixing 1. Extra Wilson fermions: Don t they spoil the continuum limit or the O(a ) scaling? No. They are heavy: m~1/a. Low-lying modes of H W are local and irrelevant in the continuum limit.. Ergordicity: Is the ergordicity maintained? No. HMC visits only the fixed topological sector. It restricts the path integral to a given Q. But the same physics can be obtained as explained above. Probably yes, within a given Q. A fixed Q manifold is connected in the continuum theory. No proof on the lattice; no counter example, either. Lattice aims at approaching the continuum, anyway. 50

51 Measurement techniques Measurements at every 0 traj 500 conf / m sea Improved measurements 50 pairs of low modes calculated and stored. Used for low mode preconditioning (deflation) (multi-mass) solver is then x8 faster Low mode averaging (and all-to-all) u ( x) u ( y) D ( xy, ) D ( xy, ) N 1 k k ( h) 1 m = + m k = 1 λk + m C( x, y) = C + C ll hl ( x, y) + C ( x, y) + C hh lh ( x, y) ( x, y) 51

52 All-to-all To improve the signal Usually, the quark propagator is calculated with a fixed initial point (one-to-all) Average over initial point (or momentum config) will improve statistics; possible with all-to-all 1 D ( x, y) u ( x) u ( y) D ( x) ( y) Nev Nd 1 ( k) ( k) 1 ( d) ( d) = + ( k ) highη η k= 1 λ d= 1 Low mode contribution High mode propagation From the random noice Random noise 5

53 An example: two-point func Dramatic improvement of the signal, thanks to the averaging over source points Similar to the low mode averaging; but all-to-all can be used for any n- point func. PP correlator is dominated by the low-modes 53

54 B K First (unquenched) lattice calculation with exact chiral symmetry: O LL JLQCD collab, arxiv: [hep-lat]. No problem of operator mixing; otherwise, mixes with O LR, for instance. Enhanced by its wrong chiral behavior. Another test of chiral log. Here the data follows the NLO ChPT K O ( μ) K = B ( μ) f m LL K K K χ 6mP m 4 P BP = BP 1 ln + bm ( ) P + O mp (4 π f ) μ K A μ A μ B (GeV) = 0.534(5)(30)