Theoretical issues with staggered fermion simulations Stephan Dürr Uni Bern, ITP based on work together with Christian Hoelbling (Wuppertal) Urs Wenger (DESY Zeuthen) Lattice 05, Dublin, July 25-30, 2005 1.5 st W ov 1 0.5 0 −0.5 −1 162, β=3.2, nsmear=0 0 • • • • S. Dürr, ITP Bern 0.5 1 1.5 2 2.5 3 3.5 −1.5 4 D̂ = iγµpµ + O(p2) Dst, Dov have no additive mass renormalization, but DW has † † γ5DW,ov γ5 = DW,ov and η5Dstη5 = Dst =⇒ det(D) ≥ 0 spec(Dna) = spec(Dst) ⊗ I2 (would be I4 in 4D) Lattice 05, Dublin, July 25-30, 2005 1/32 Controversy in a nutshell • In 4D the staggered action yields 4 “tastes” in the continuum, and 3 of these must be excised from ones physical predictions. • The way how this is done is (in general) different for valence and sea quarks. • Naively, that difference should leave no trace in the continuum limit. Question: Is QCD with Nf = 2+1 staggered quarks fundamentally correct ? [Is it physics from first principles or just a (phenomenologically successful) model of QCD ?] – In the fundamental theory taste splitting may be suppressed through various tricks (improved glue, filtering, RG blocking, ...). – In the effective theory taste splitting effects may be parametrized and thus “taken away” (approximately) for a variety of observables. From a conceptual viewpoint either of these improvements is immaterial; if the staggered approach is correct, it yields the right continuum limit for arbitrary observables without any of these. ⊕: Prove that QCD with Nf = 2+1 staggered quarks is in the right universality class. ª: Find a single observable where the staggered answer, after continuum extrapolation, is wrong. S. Dürr, ITP Bern Lattice 05, Dublin, July 25-30, 2005 2/32 • Review: staggered action and taste representation • Problem: rooting versus locality • Free case: four constructions • Interacting theory: spec(Dst) in 4D • Interacting theory: χsca , χtop in 2D • Correlation of det1/Nt (Dst,m) and det(Dov,m) • Low-energy unitarity and SXPT • Summary S. Dürr, ITP Bern Lattice 05, Dublin, July 25-30, 2005 3/32 Review: staggered action Sna a3 X † = ψ̄(x) γµ [Uµ(x)ψ(x+ µ̂) − Uµ(x− µ̂)ψ(x− µ̂)] 2 x,µ −→ Naive action describes 4 (2D) or 16 (4D) fermions (in general: 2 d) in the continuum. ψ(x) = γ(x)χ(x) Sna/st ψ̄(x) = χ̄(x)γ(x) † γ(x) = x x x x γ1 1 γ2 2 γ3 3 γ4 4 ηµ(x) = P (−1)µ<ν xν a3 X † = χ̄(x) ηµ(x) [Uµ(x)χ(x+ µ̂) − Uµ(x− µ̂)χ(x− µ̂)] 2 x,µ −→ The 2 (2D) or 4 (4D) components (in general: 2d/2) decouple. −→ Downgrade to one component, i.e. to 2 (2D) or 4 (4D) “tastes” (in general: N t = 2d/2). =⇒ KS procedure “thins out d.o.f.”, but distributes/intertwines spinor and taste. χ(x) → eiθA η5(x)χ(x) χ̄(x) → e−iθA η5(x)χ̄(x) (m = 0) −→ Remainder of SU (2d/2)A in taste space is sufficient to forbid additive mass renormalization. =⇒ Further (exact) “thinning” impossible, since resulting spectrum is non-degenerate. S. Dürr, ITP Bern Lattice 05, Dublin, July 25-30, 2005 4/32 Review: taste representation Nf staggered fields χ = u,d,s,... with 4 tastes each; hypercubic decomposition χ(x, x+a1̂, ..., x+a1̂+a2̂+a3̂+a4̂) → q(X) collects 2d/2 tastes with 2d/2 components each in one “blocked node”. ~ With {X} = {N }b and b = 2a the free action takes the form Sst = b 4 X X,µ ²¯ ±° b q̄(X) [∇µ(γµ ⊗I) − 4µ(γ5 ⊗τµτ5)] q(X) 2 with (spinor⊗taste), τµ = γµ∗ ,τ5 = γ5 and the blocked first/second derivative (∇µq)(X) = q(X+bµ̂) − q(X−bµ̂) 2b (4µq)(X) = q(X+bµ̂) − 2q(X) + q(X−bµ̂) b2 ~ ¡ ª ? ¡ ²¯ z ~¾ ±° p=0 ²¯ ¡ ±° ²¯ ~ ±° p=π/a . =⇒ In taste basis the taste interactions stem from a dim=5 Wilson-like term. −→ Do they go away with a → 0 without any trace ? (order of limits?) p=0 p=-π/a Interacting theory: x-space: different tastes (and individual components of each) see slightly different local gauge field. p-space: gluons with p ∼ π/a may kick field from one taste to another (flavor exact with Nf fields). −→ Identification staggered=physical flavor may work only, if taste interactions minimized/eliminated. S. Dürr, ITP Bern Lattice 05, Dublin, July 25-30, 2005 5/32 |q|=0 |q|=1 |q|=4 see text 0.5 staggered overlap 0 -0.5 0 1 3 0 1 3 0 1 # smearing steps 3 0 3 (Schwinger model, β = 4.0, 20 2 ) SD, C.Hoelbling, PRD 69, 034503 (2004) [hep-lat/0311002] S. Dürr, ITP Bern 1 Lattice 05, Dublin, July 25-30, 2005 6/32 Problem: rooting versus locality Marinari,Parisi,Rebbi (1981): “On the lattice the action SG−14 tr log(Dst) will produce a violation of fundamental axioms, but we expect the violation to disappear in the continuum limit and then recover the theory with a single fermion.” • With any undoubled Dirac operator, e.g. D = DW , one has Z Z PR ψ̄Dψ −S [U ]− N −S [U ] Z|Nf = D[U,ψ̄,ψ] e G = D[U ] det f (D) e G . • With a 4-flavor Dirac operator like Dst, one may formally define Z N /4 −S [U ] Z|Nf = D[U ] det f (Dst) e G • but it is not clear whether there is a 1-taste operator Dca such that Z R 1/4 − ψ̄Dca ψ det (Dst) = D[ψ̄,ψ] e . (A) Quenched QCD: quark loops neglected (B) Full QCD To guarantee locality/causality of arbitrary Green’s functions (thus to discuss renormalizability and universality) a “candidate” operator Dca should exist with K.Jansen, NPPS 129, 3 (2004) [hep-lat/0311039] (1) (2) S. Dürr, ITP Bern a↓0 det(Dca ) −→ const·det1/4(Dst) ||Dca(x, y)|| < C e−ν|x−y|/a with C, ν independent of U. Lattice 05, Dublin, July 25-30, 2005 7/32 • naive rooting of Dst P † † † Dst is a normal operator ( [Dst, Dst ] = 0 ), hence Dst = λ λ ψ(x)ψ (y) = U Λ U with U unitary (R/L-eigenvectors in columns) and Λ diagonal, thus f (D st) = U f (Λ) U †. 2 Dst D1/2 st D1/4 1 st Im(λ) (D+ D )1/4 st st ee 0 −1 −2 −2 1/2 −1 0 Re(λ) 1 2 3 1/4 =⇒ Dst ,Dst (in 2D, 4D) unacceptable, since D̂ = iγµpµ +O(p2) violated (and non-analytic in p). S. Dürr, ITP Bern Lattice 05, Dublin, July 25-30, 2005 8/32 † 1/2 • explicit non-locality of (Dst Dst)ee ¯ Measure of localization: f (r) = sup{ ||ψ(x)||2 ¯ ||x−y||1 = r } where P ψ(x) = Dca(x, y)η(y) with η normalized random vector at y . −→ For local Dca one has f (r) ∝ e−r/rloc , ideally with rloc ∝ a (in any case ξphys /rloc → ∞). B.Bunk, M.DellaMorte, K.Jansen, F.Knechtli, NPB 697, 343 (2004) [hep-lat/0403022] 1 β=6.0 β=6.2 β=6.5 6 10 0.9 0 0.8 4 0.7 10 loc 0 0.6 r /r <f(r)>/<f(rref)> r/r =1.86 0 r/r =3.72 2 10 0.5 0.4 0 0.3 10 0.2 −2 0.1 10 0 2 4 r/r 6 8 10 0 0 0.05 0 0.1 a/r0 0.15 0.2 † 1/2 2 Rooted operator Dca = (Dst,m Dst,m )ee = (−Dst +m2)1/2 ee > 0 as a first test. −→ f (r) and rloc (r) are finite quantities; they scale even though (for a local Dca) they should not. S. Dürr, ITP Bern Lattice 05, Dublin, July 25-30, 2005 9/32 −→ Bunk et al. show: rloc = fcn(r/r0)/Mπ in the interacting theory (numerically). −→ Bunk et al. show: rloc = const/m in the free theory (analytically). † −→ In physical units rloc constant under β → ∞, thus (Dst Dst)1/2 is non-local. ee A.Hart, E.Müller, PRD 70, 057502 (2004) [hep-lat/0406030] 2 2 ONE-LINK FAT7TAD ASQTAD β=5.8, 12 3 β=6.0, 12 32 β=6.2, 24 1.5 rloc/r0 1 eff eff rloc/r0 1.5 0.5 0 4 4 1 0.5 0 2 4 r/r0 6 8 0 0 2 4 r/r0 6 8 −→ Same conclusion (of course) with improved/filtered staggered quarks. =⇒ Question: Is there one local Dca with det(Dca ) = const·det1/4(Dst) or modulo cut-off effects ? S. Dürr, ITP Bern Lattice 05, Dublin, July 25-30, 2005 10/32 Free case: four candidates In the free case spec(Dst) highly degenerate, thus “thinning” of d.o.f. much easier. • D.Adams, hep-lat/0411030 In the taste basis Dst,m = ∇µ(γµ ⊗I4) − 2b 4µ(γ5 ⊗τµτ5) + m(I4 ⊗I4) on the blocked lattice may be used to build an operator which is simultaneously diagonal in spinor⊗taste † Dst,m Dst,m b2 2 2 = [−∇ + 4 + m ](I4 ⊗I4) . 4 2 On the blocked lattice a free generalized Wilson operator Dca,m = ∇µγµ + 2b W + m yields b † 2 2 Dca,m Dca,m = [−∇ + ( W + m) ](I4 ⊗1) . 2 † † With det(Dst,m ) = det(Dst,m ) and det(DW,m ) = det(DW,m ) it follows that b b2 2 2 2 4 + m = ( W + m) 4 2 =⇒ −→ q 2 Dca,m = ∇µγµ + b4 42 + m2 p a↓0 rloc = 8a/m −→ 0 S. Dürr, ITP Bern =⇒ det 1/4 (Dst,m ) = det(Dca,m ) . (in the free theory, on the blocked lattice) (i.e. local, but requires m > 0) Lattice 05, Dublin, July 25-30, 2005 11/32 • F.Maresca, M.Peardon, hep-lat/0411029 On the blocked lattice a free 1-taste operator Dca = iγµ Pµ +Q with local (i.e. not ultra-local) Pµ, Q † † yields det(Dca) = det1/4(Dst), if Dca Dca ⊗I4taste = Dst Dst. With Pµ = Pµ†, Q = Q† trivial in spinor (!) † space and [Pµ, Q] = 0 one thus requires that Dca Dca = Pσ Pσ + Q2 = [−4 + have range r . Solution 1: Without further constraints, optimizing locality of Dca yields spectrum and fall r off pattern of ωp,µ , ωqr shown on the left. 0 -0.5 [Sol. 1 for m > 0 even better localized.] S. Dürr, ITP Bern 0 1 0.5 1.5 0 Re λ 1 0.5 1.5 Re λ 0 10 10 -2 pµ, on-axis 10 -4 pµ, diagonal -6 q, on-axis q, diagonal 10 10 -2 10 -4 -8 10 |ωR(x,y)| |ωp,q(x,y)| 0 -0.5 0 Solution 2: Ditto, but restriction to Q = −4R with local R yields spectrum and (slower) fall off pattern shown on the right and {Dca , γ5} = Dca 2Rγ5 Dca. ] ⊗I4spinor . 0.5 Im λ where r≥0 |d|≤r r ωp,µ , ωqr 0.5 Im λ Ansatz: P P r Pµ = ωp,µ (x, y) r≥0 |d|≤r P P r ωq (x, y) Q = b2 2 44 -10 10 -12 10 10 -6 10 -14 10 -8 -16 10 10 -18 10 -20 10 0 1 2 3 4 5 6 7 Field separation, |x-y| Lattice 05, Dublin, July 25-30, 2005 8 9 10 -10 10 0 1 2 3 4 5 6 7 8 9 10 Field separation, |x-y| 12/32 • Y.Shamir, PRD 71, 034509 (2005), hep-lat/0412014 Main idea: Improve taste symmetry through RG blocking. Infinitely many blocking steps would achieve Dn → D∞ ⊗I4, while Dca after n steps satisfies det1/4(Dst) = det(Dca) det1/4(T ) where T contains only cut-off excitations and should maintain Symanzik class, i.e. det(T ) = const·(1+O(a 2)). Note: If one is satisfied with a = 0.4 fm for Dca (optimistic view), then original lattice with a ∼ 0.1 fm allows for 2 steps; for 5 steps original lattice must have a ∼ 0.01 fm (cf. talk by F. Maresca). The massless staggered action on the original lattice satisfies {D0, (γ5 ⊗ τ5)} = 0 or equivalently (γ5 ⊗τ5)D0(γ5 ⊗τ5) = D0†. After n ≥ 1 RG steps (parameter αn) one has the generalized GW relation −1 {Dn , (γ5 ⊗τ5)} = const·δx,y or {Dn, (γ5 ⊗τ5)} = Dn 2 (γ5 ⊗τ5) Dn . αn If one could establish (γ5⊗τ5)-hermiticity of Dn, one would easily obtain Dn +Dn† = Dn α2n Dn† , i.e. spectrum on a circle. P With Dn = j λj uj vj† and vi†uk = δi,k , upon sandwiching vi†(GW)uk , one obtains λivi†(.⊗.)uk +vi†(.⊗.)uk λk = λivi† α2n (.⊗.)uk λk and thus for arbitrary pair (i,k) of L,R-eigenmodes vi†,uk that vi†(γ5 ⊗τ5)uk = 0 or λi +λk − α2n λiλk = 0 . In particular for i = k it follows that vj†(γ5 ⊗τ5)uj 6= 0 implies 2λj − α2n λ2j = 0 or λj ∈ {0, αn}. κf Re λ S. Dürr, ITP Bern H.Neuberger, PRD 70, 097504 (2004), hep-lat/0409144 • J.Giedt, hep-lat/0507002 • Re λ κf Im λ Im λ Similar concepts – exploratory discussion of interacting case. Issue cast into local field-theoretical framework in 6D. Lattice 05, Dublin, July 25-30, 2005 13/32 Interacting theory: spec(Dst) in 4D • concept of filtering − Replace covariant derivative, e.g. Uµ(x)ψ(x+ µ̂) − ψ(x) → UµHYP (x)ψ(x+ µ̂) − ψ(x). − Redefines (diminishes) cut-off effects without changing Symanzik class, i.e. O(a 2) → O(a2). − Designed to render Dst “immune” against p ∼ π/a gluons, impact on taste “symmetry” ? • chirality of low-lying eigenmodes Continuum: Dψ = λψ ⇐⇒ Dγ5ψ = −λγ5ψ and ψ †γ5ψ = 0 (for λ 6= 0). Dζ = 0 and ζ †γ5ζ = ±1 characteristic signature of zero-modes. A.Hasenfratz, http://www.rccp.tsukuba.ac.jp/lat03/Dat/OHP/a.hasenfratz.ps S. Dürr, ITP Bern Lattice 05, Dublin, July 25-30, 2005 14/32 • near-degeneracy of staggered quartets Wilson glue, a ' 0.1 fm 0.8 CHIRALITY 0.7 0.6 0.5 HISQ HYP ASQTAD ONE-LINK 0.6 0.4 0.3 0.2 0.2 0.1 0.1 0 0 -0.1 -0.1 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 10 12 14 16 EIGENVALUES 0.08 0.07 0.01 0.06 0.008 0.05 0.006 0.04 0.004 0.03 0.02 0.002 0 0 0.09 EIGENVALUES 0.012 HISQ ASQTAD ONE-LINK 0.5 0.3 2 CHIRALITY 0.7 0.4 0 Symanzik glue, a ' 0.1 fm 0.8 0.01 0 2 4 6 8 10 12 14 16 0 0 2 4 6 8 E.Follana, A.Hart, C.T.H.Davies, PRL 93, 241601 (2004) [hep-lat/0406010] E.Follana, NPPS 140, 141 (2005) [hep-lat/0409062] −→ Fine lattice plus filtering makes separation into near-zero modes and non-zero modes visible. −→ Near-zero modes have chirality |ζ †γ5ζ| ' 1, non-zero modes have ψ †γ5ψ ' 0. =⇒ Approximate index theorem for staggered fermions (once taste “symmetry” visible). S. Dürr, ITP Bern Lattice 05, Dublin, July 25-30, 2005 15/32 • comparing with overlap spectrum on individual configurations CU 0001: staggered Nt = 2 simulation (β = 5.7, m = 0.01 → a ' 0.1 fm) 0.08 CU_0001, conf_000, ρ=1.0 0.08 overlap staggered 0.06 0.06 0.04 0.04 0.02 0.02 0 0 −0.02 −0.02 −0.04 −0.04 −0.06 −0.06 −0.08 orig 1 APE 3 APE 1 HYP 3 HYP −0.08 CU_0001, conf_006, ρ=1.0 overlap staggered orig 1 APE 3 APE 1 HYP 3 HYP SD, C.Hoelbling, U.Wenger, PRD 70, 094502 (2004) [hep-lat/0406027] −→ Filtering pushes λst out and pulls λ̂ov in (in general: drives ZS → 1). −→ Manifest staggered quartet to single overlap mode correspondence (modulo different Z S factor). =⇒ In particular: 4|q| staggered near-zero modes on (typical) configuration with overlap charge q . S. Dürr, ITP Bern Lattice 05, Dublin, July 25-30, 2005 16/32 • cut-off dependence of taste-splitting Matched lattices (β = 5.66,5.79,6.00,6.18, L4 = 64,84,124,164) with V = (1.12 fm)4 = 1.57 fm4. 4 4 8 , β=5.79, conf_000, ρ=1.0 16 , β=6.18, conf_000, ρ=1.0 0.25 0.2 0.1 0.15 0.1 1 HYP over 1 HYP bare stag 1 HYP ren. stag 0.05 0 0 2 4 6 8 10 12 14 16 0.05 0 1 HYP over 1 HYP bare stag 1 HYP ren. stag 0 2 4 6 8 10 12 14 16 0.25 0.2 0.1 0.15 0.1 3 HYP over 3 HYP bare stag 3 HYP ren. stag 0.05 0 0 2 4 6 8 10 12 14 16 0.05 0 3 HYP over 3 HYP bare stag 3 HYP ren. stag 0 2 4 6 8 10 12 14 16 SD, C.Hoelbling, U.Wenger, PRD 70, 094502 (2004) [hep-lat/0406027] −→ On matched lattices taste “symmetry” (quartet near-degeneracy) improves with a → 0. −→ Rescaling with ZSov /ZSst: quantitative agreement of staggered quartet with single overlap mode. =⇒ Explicit tests passed by Dov (index theorem, RMT agreement, Banks-Casher, ...) transfer to Dst. S. Dürr, ITP Bern Lattice 05, Dublin, July 25-30, 2005 17/32 • explicit check that hλii/hλj i agrees with RMT prediction 7 Q=0 |Q|=1 |Q|=2 6 HISQ HYP ASQTAD ONE-LINK 5 4 3 2 1 2/1 4/1 4/2 3/1 3/2 4/3 2/1 4/1 4/2 3/1 3/2 4/3 2/1 4/1 4/2 3/1 3/2 4/3 E.Follana, A.Hart, C.T.H.Davies, PRL 93, 241601 (2004) [hep-lat/0406010] −→ Sectoral hλii/hλj i agrees with RMT prediction (up to small finite volume effects ?). S. Dürr, ITP Bern Lattice 05, Dublin, July 25-30, 2005 18/32 • explicit check that CED(λmin ) agrees with RMT prediction 1 Q=1 0.6 0.4 Q=2 0.2 UFat7 Asq 0 0 2 4 6 8 0.4 Q=2 0.2 0 10 2 4 ζ 6 8 Q=2 0 2 4 6 8 10 a = 0.077 fm 1 Q=3 0.8 0.6 0.4 0.2 0 Overlap q = 0, quartets 1,2,3,4 Pc(ζ) Pc(ζ) Q=1 Q=2 ζ a = 0.077 fm Q=0 0 10 mode 1, q = 0, 1, 2, 3 0.8 0.4 ζ K.Y.Wong, R.M.Woloshyn, PRD 71, 094508 (2005) [hep-lat/0412001] 1 Q=1 0.6 0.2 HYP 0 Q=0 0.8 Q=1 0.6 none Dov 1 Q=0 0.8 Pcumm (ζ) Pcumm (ζ) 1 Q=0 0.8 HYP Dst Pcumm (ζ) UFat7×Asq Dst k=1 k=2 k=3 k=4 0.6 0.4 0.2 0 2 4 ζ 6 8 10 0 0 2 4 6 8 ζ 10 12 14 16 E.Follana, A.Hart, C.T.H.Davies, Q.Mason, hep-lat/0507011 =⇒ On fine enough lattices (and with filtering) agreement with RMT for individual topological sectors. S. Dürr, ITP Bern Lattice 05, Dublin, July 25-30, 2005 19/32 • evidence that taste-splitting is O(a2) effect ∆ [fuzziness of first would−be zero mode] 0 0.35 orig 1 HYP 3 HYP 0.3 ∆ [fuzziness of first positive mode] 0.5 0.2 0.4 0.15 0.3 0.1 0.2 0.05 0.1 0.005 0.01 0.015 orig 1 HYP 3 HYP 0.6 0.25 0 0 1 0.7 0 0 0.02 0.005 0.01 0.015 0.02 SD, C.Hoelbling, U.Wenger, PRD 70, 094502 (2004) [hep-lat/0406027] |Q|=1 |Q|=2 20 |Q|=1 Q=0 30 ONE-LINK ASQTAD FAT7xASQ |Q|=2 ONE-LINK ASQTAD FAT7xASQ 15 <δΛ1>Q / MeV <Λ0>Q / MeV 20 10 10 5 0 0 0.01 0 0.01 2 (Lattice spacing, a / fm) 0 0 0.01 0 0.01 0 2 (Lattice spacing, a / fm) 0.01 E.Follana, A.Hart, C.T.H.Davies, Q.Mason, hep-lat/0507011 =⇒ Minimal a for taste breaking to possibly be O(a2) effect seems enlarged through filtering. S. Dürr, ITP Bern Lattice 05, Dublin, July 25-30, 2005 20/32 Interacting theory: χsca , χtop in 2D • In 2D rooting issue exists for Nf = 1, 3, ..., since Dst yields 2 fermions in the continuum. • In 2D scale may be set through fundamental coupling: [g] = [e] = 1 (no UV running), β = 1/(ag) 2. µ ¶ anomaly induced • Analytic Nf = 1 result: limm=0hψ̄ψi/g = eγ/(2π 3/2) = 0.1599... J.Schwinger (1962) overlap fermions (ρ = 1): χov sca g = N ov ) P 1 i √ hdet f (Dm β λ̂+m N V ov )i hdet f (Dm ov det(Dm )= λ̂ = staggered fermions: (bare) Q ((1− m 2 )λ + m) 1 (1/λ−1/2) χst sca g = √ hdetNf /2 (D st ) P 1 i m β λ+m 2V Nf /2 st hdet (Dm )i st det(Dm )= (=stereogr. proj.) Q (bare) (λ + m) − Sample quenched, compute all λ [LAPACK] and build observables (variable m) in analysis program. − In plots below theory is unitary (at least with Dov ); we use msea = mval throughout. − Details in S. Dürr, ITP Bern SD, C.Hoelbling, PRD 69, 034503 (2004) [hep-lat/0311002]& PRD 71, 054501 (2005) [hep-lat/0411022]. Lattice 05, Dublin, July 25-30, 2005 21/32 • χsca with Dst and Dov at β = 7.2 Continuum: g/m χsca /g ∝ const m/g (Nf = 0) (Nf = 1) (Nf = 2) β=7.2, 24^2, nsmear=0 ("thin link") 0.3 staggered, Nf=0 staggered, Nf=1 staggered, Nf=2 overlap, Nf=0 overlap, Nf=1 overlap, Nf=2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 Overlap: 0.05 0.1 0.15 m/g 0.2 0.25 staggered, Nf=0 staggered, Nf=1 staggered, Nf=2 overlap, Nf=0 overlap, Nf=1 overlap, Nf=2 0.25 χ’/g 0.2 χ’/g β=7.2, 24^2, nsmear=1 ("thick link") 0.3 0.25 0 (in 2D not an order parameter of SSB) 0.3 0 0 0.05 0.1 0.15 m/g 0.2 0.3 0.25 N =1 f qualitatively correct behavior ∀Nf , and lim χsca /g consistent with 0.1599... for β ≥ 4. m→0 Staggered: qualitatively wrong behavior in chiral limit for Nf = 0,1, since lim χsca /g = 0 for any β , m→0 but filtering shifts point where staggered answer fails more chiral (rel. to taste splitting ?). S. Dürr, ITP Bern Lattice 05, Dublin, July 25-30, 2005 22/32 Question: can one obtain χsca /e = 0.1599... with staggered fermions, if one first extrapolates to the continuum, taking m → 0 afterwards ? • χsca with Dst and Dov in the continuum overlap (nsmear=1) 0.2 L=12 β=1.8 L=16 β=3.2 L=20 β=5 L=24 β=7.2 0.19 0.18 0.17 0.18 0.17 0.16 χ’/e χ’/e L=12 β=1.8 L=16 β=3.2 L=20 β=5 L=24 β=7.2 0.19 0.16 0.15 0.15 0.14 0.14 0.13 0.13 0.12 0.12 0.11 0.11 0.1 staggered (nsmear=1) 0.2 0 0.1 m/e 0.2 0.3 0.1 0 0.1 m/e 0.2 - small cut-off effects - large cut-off effects - for m/e ≤ 0.3 already β ≥ 1.8 sufficient to enter scaling window - for small m/e ever larger β needed to enter scaling window S. Dürr, ITP Bern Lattice 05, Dublin, July 25-30, 2005 0.3 23/32 0.165 exact result staggered overlap 0.16 χ’subt/e 0.155 0.15 0.145 0.14 0.135 0.13 0 0.05 m/e • overlap (“universal behavior”): 0 ov (m/e,a2 ) χsca lima→0 limm→0 e 0 ov χsca (m/e,a2 ) limm→0 lima→0 e = = eγ 2π 3/2 eγ 2π 3/2 0.1 0.15 • staggered (“non-commutativity”): st (m/e,a2 ) χ0sca e st (m/e,a2 ) χ0sca limm→0 lima→0 e lima→0 limm→0 =0 = eγ 2π 3/2 =⇒ Staggered fermions see chiral anomaly, but only if lima→0 is taken first and limm→0 thereafter. S. Dürr, ITP Bern Lattice 05, Dublin, July 25-30, 2005 24/32 • χsca with Dst and Dov in hybrid mode β=7.2, 24^2, nsmear=0 β=7.2, 24^2, nsmear=1 0.3 0.3 stag-val, Nf=0 stag-val, Nf=1 ov-sea stag-val, Nf=2 ov-sea ov-val, Nf=0 ov-val, Nf=1 stag-sea ov-val, Nf=2 stag-sea 0.25 0.25 0.2 χ’/g χ’/g 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 0 0.05 0.1 0.15 m/g 0.2 0.25 stag-val, Nf=0 stag-val, Nf=1 ov-sea stag-val, Nf=2 ov-sea ov-val, Nf=0 ov-val, Nf=1 stag-sea ov-val, Nf=2 stag-sea 0.3 0 0 0.05 0.1 0.15 m/g 0.2 0.25 0.3 −→ Warning for all “hybrid action” studies: −→ Deficiencies of Dst in sea/valence sector may overwhelm good properties of Dov in other sector. =⇒ Failure of χscal in staggered Nt = 1 case cannot be attributed to sea or valence sector alone. S. Dürr, ITP Bern Lattice 05, Dublin, July 25-30, 2005 25/32 • χsca with Dst and Dov in the quenched case χsca /e ∝ Reminder: ½ e/m m/e Chiral condensate for Nf=0 (Nf = 0) (Nf = 2) Chiral condensate for Nf=2 β=7.2, N=24, nsmear=1, coeff. quenched divergence β=7.2, N=24, nsmear=1, coeff. linear piece 0.04 2 overlap staggered 0.03 0.02 χ/m mχ overlap staggered 1 0.01 0 0 0.1 m/e 0.2 0 0 0.1 m/e 0.2 −→ Even in the quenched theory a staggered non-commutativity (in an artificial observable) found. Earlier paper: J.Smit, J.C.Vink, NPB 286, 485 (1987). =⇒ Non-commutativity not genuinely tied to rooted determinant, more likely due to mismatch in sea and valence sector; compare discussion in C.Bernard, PRD 71, 094020 (2005) [hep-lat/0412030]. S. Dürr, ITP Bern Lattice 05, Dublin, July 25-30, 2005 26/32 • χtop with Dst and Dov at β = 4 ov χtop β = V ov )q 2i hdetNf (Dm ov )i hdetNf (Dm st χtop β = V st )q 2i hdetNf /2(Dm st )i hdetNf /2(Dm nsmear=0 (“thin link”) 0.04 χtop/g 2 0.03 2 χtop/g ind = − 12 tr(γ5Dov ) R 1 F12 d2x 2π nsmear=1 (“thick link”) 0.04 0.03 q= ( 0.02 staggered, Nf=1 staggered, Nf=2 overlap, Nf=1 overlap, Nf=2 0.01 0 0 Overlap: 0.1 0.2 0.3 m/g 0.4 0.5 0.6 0.02 staggered, Nf=1 staggered, Nf=2 overlap, Nf=1 overlap, Nf=2 0.01 0.7 0 0 0.1 0.2 0.3 m/g 0.4 0.5 0.6 0.7 small O(a2) effects, fairly insensitive to filtering (overlap yields good IR↔UV separation). Staggered: large O(a2) effects, increase towards chiral limit, substantially reduced through filtering. S. Dürr, ITP Bern Lattice 05, Dublin, July 25-30, 2005 27/32 • χtop with Dst and Dov in the continuum ¯ Matched lattices: β = 1.8, 3.2, 5.0, 7.2, 9.8 with L = 12, 16, 20, 24, 28 yields LM η0 ¯ N =1,m=0 ' 5. f 0.026 0.024 staggered 0.014 nsmear=0 nsmear=1 nsmear=3 overlap Nf=1 χtop/e2 at m/e=0.1 staggered Nf=1 χtop/e2 at m/e=0.1 0.028 0.022 0.02 0.018 0.016 0.014 0.012 0.01 0.008 0 0.1 0.2 0.3 1/β 0.4 0.5 0.6 0.0135 overlap nsmear=0 nsmear=1 nsmear=3 0.013 0.0125 0.012 0.0115 0.011 0.0105 0.01 0.0095 0 0.1 0.2 0.3 1/β 0.4 0.5 0.6 −→ Staggered extrapolation much steeper than with overlap (different scales). −→ Combined fit with several filtering levels yields cost-effective continuum extrapolation. N =1 f =⇒ Results for χtop suggest universal continuum limit, in spite of det1/2(Dst). −→ Similar agreement in other continuum extrapolated quantities, e.g. for F HQ. S. Dürr, ITP Bern Lattice 05, Dublin, July 25-30, 2005 28/32 Correlation of 12 log det(Dst,m) and log det(Dov,m) in 2D Determinant ratio: λ1 λ2 ... ¯¯ λ01 λ02 ... ov ' β=9.8 β=5.0 β=1.8 st0/ov0 γ1 γ2 ... ¯¯ 0 γ 0 ... st γ1 2 ? (γk = st1/ov1 p λ2k−1 λ2k geometric staggered mean) ov1/ov0 st1/st0 4 2 0 -2 -4 4 2 0 -2 -4 4 2 0 -2 -4 -6 -4 -2 0 2 4 -6 -4 -2 0 2 4 -6 -4 -2 0 2 4 -6 -4 -2 0 2 4 1APE −→ At fixed quark mass det1/2(Dst,m ) in 2D generates ensemble that is closer to the one from 1APE none det(Dov,m ) than the latter would be to det(Dov,m ), and the agreement improves with β . =⇒ Maybe, Dov,m is a “candidate” operator with det(Dca,m ) = const·det1/2(Dst,m )(1+O(a2)) . S. Dürr, ITP Bern Lattice 05, Dublin, July 25-30, 2005 29/32 Low-energy unitarity and SXPT • continuum chiral perturbation theory (XPT) Observation: In QCD with p ¿ 1 GeV chiral symmetry constrains interactions of low-energy degrees of freedom with each other and with heavier particles (e.g. nuclei). Consider ππ forward scattering (s = 0,I = 0) at low momentum: Fππ (ν) s=0 I=0 I=0 = T [0, 2Mπ (Mπ +ν), 2Mπ (Mπ −ν)] = Mπ2 4 − 2 + O(p ) Fπ Mπ ,p↓0 −−−→ 0 J.Gasser, H.Leutwyler, Ann.Phys. 158, 142 (1984), NPB, 250, 465 (1985) Taste splitting makes ¯ 5 ⊗T )u most d(γ combinations become non-Goldstone bosons: (Mπ r0)2 • staggered chiral perturbation theory (SXPT) 6 p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p pp p pp p p p p p p p p p pp p p p p p p p pp pp pp pp p p p p p p p p p p p p p p p p pu p P pup p A pup p T pup p V pup p S (a/r0) - 2 W.J.Lee, S.R.Sharpe, PRD 60, 114503 (1999) [hep-lat/9905023] S. Dürr, ITP Bern Lattice 05, Dublin, July 25-30, 2005 30/32 Assumption: With Nf flavors of (4-taste) quark fields the pattern of SSB is SU (4Nf )L ×SU (4Nf )R → SU (4Nf )V leading to 16Nf 2 −1 pseudo-Goldstone bosons, collected in the 12×12 matrix (Nf = 3) U =e iΦ/f 9,16 Φu π + K + X λa b ab − 0 Φ=π Φ T Φd K = 2 a,b=1 K − K̄ 0 Φs m u I4 0 0 M = 0 0 m d I4 0 0 m s I4 that transforms as U → VLU VR† under chiral rotations with unitary VL,VR . With f ' 122 MeV and Σ ' (270 MeV)3 the LO-Lagrangian (counting scheme p2 ∼ m ∼ a2) reads f2 Σ 2m20 † † 2 L= tr(∂µU ∂µU ) − tr(M U+M U ) + (Φu,TS +Φd,TS +Φs,TS ) + a VTB 8 2 3 C.Aubin, C.Bernard, PRD 68, 034014 (2003) [hep-lat/0304014] & PRD 68, 074011 (2003) [hep-lat/0306026] and allows for systematic treatment of quantities covered by XPT, e.g. M π , fπ , MK , fK . ⊕ SXPT analysis includes taste breaking effects. ⊕ ⊕ ª ª Overall fits with horrific covariance matrices (“fitting herds of elephants”) yield acceptable χ 2/d.o.f. Some tests [Nt = 1.28(12) per flavor, SXPT logs] successful, more [e.g. Sharpe, van de Water] to come. What about physical observables not covered by (S)XPT ? Unphysical tastes excised from predictions, but differently in valence and sea sector (unitarity?). S. Dürr, ITP Bern Lattice 05, Dublin, July 25-30, 2005 31/32 Summary • Full QCD with Nf = 2+1 staggered fermions is controversial, since the Boltzmann weight det 1/4(Dst) assumes a taste symmetry which is only approximate. • Formally, the taste symmetry breaking is due to a dimension 5 Wilson-type term in the taste basis and should thus go away in the continuum limit. • Weak coupling, filtering, RG blocking reduce the taste splitting and give staggered quarks more appealing features, but there is no guarantee that no trace is left1 in the continuum. • One legal 1-flavor Dca with det(Dca ) = const·det1/4(Dst)(1+O(a2)) is sufficient to re-interpret existing MILC ensembles as being generated with a local action. • The problem of (exact) unitarity in the fundamental theory remains, unless same D ca is used in valence sector, too. Otherwise “partially quenched” situation with unitarity restored with a → 0 ? 1 Caveat: at least in some theories the cases m = 0 and m > 0 might be different in this respect. S. Dürr, ITP Bern Lattice 05, Dublin, July 25-30, 2005 32/32