First results for two-flavor QCD with light quarks

Transcription

1 First results for two-flavor QCD with light quarks Leonardo Giusti CERN - Theory Group CERN L. Giusti Nara November 2005 p.1/30

2 Outline Flavor physics with lattice QCD The Berlin wall in full QCD simulations A new algorithm for full QCD simulations: SAP First preliminary results and contact with ChPT Perspective for B-parameters and form factors Conclusions and outlook L. Giusti Nara November 2005 p.2/30

3 Unitarity triangle fit in the Standard Model η sin2β M d M d M s C K M f i t t e r Winter 2004 sin 2β m d excluded area has CL < 0.05 m s & m d ε K η ε K α V ub V cb ρ γ β V ub /V cb ρ Indirect CP/ in K ππ ε K = T[K L (ππ) 0 ] T[K S (ππ) 0 ] CP asymmetry in B J/ψ K S Mass differences of neutral B M Bq = M H q M L q q = d, s Semileptonic B decays a = Γ[B0 d J/ψ K S] Γ[ B 0 d J/ψ K S] Γ[B 0 d J/ψ K S] + Γ[ B 0 d J/ψ K S] Γ[B X u l ν l ] Γ[B X c l ν l ] L. Giusti Nara November 2005 p.3/30

4 Unitarity triangle fit in the Standard Model η sin2β M d M d M s C K M f i t t e r Winter 2004 sin 2β m d excluded area has CL < 0.05 m s & m d ε K η ε K α V ub V cb ρ γ β V ub /V cb ρ ɛ K M Bs M Bs M Bd B πρ lν B DD lν CKM Im[V td ] V ts 2 V ts 2 / V td 2 V ub 2 V cb 2 Matrix Elements ˆB K f 2 B s ˆB Bs f 2 B s ˆBBs f 2 B d ˆBBd π ρ J ub L B 2 D D JL cb B 2 L. Giusti Nara November 2005 p.3/30

5 An impressive improvement thanks also to LQCD M. Bona et al. [UTfit] but quenching systematics still not under control L. Giusti Nara November 2005 p.4/30

6 Why we need LQCD? The K 0 - K 0 mixing case d W s u, c, t u, c, t s W d d W s u, c, t u, c, t s W d OPE d s s d ε K 1 2 e i π 4 ImM 12 M K 2 m K M 12 = K 0 H S=2 eff K 0 In the Standard Model (λ i = V is V id) H S=2 eff = G2 F M 2 W 4π 2 h i λ 2 cη 1 S0 c + λ 2 t η 2 S0 t + 2λ c λ t η 3 S0 ct bo S=2 Notice: Im[λ 2 t ] (Aλ2 ) 4 λ 2 η(1 ρ) The long-distance non-perturbative QCD effects are parameterized as K 0 Ô S=2 K 0 = Ẑ K 0 ( sγ µ P d)( sγ µ P d) K 0 = 4 3 F 2 K m2 K ˆB K L. Giusti Nara November 2005 p.5/30

7 Fermions on the lattice We know how to discretize fermions preserving the relevant symmetries 1. Gauge invariance 2. Flavor symmetry 3. Chiral symmetry (recent! Kaplan 92, Neuberger 97, Hasenfratz 98, Lüscher 98...) Ginsparg Wilson fermions numerically more expensive (typically times): chiral symmetry to be used only when needed! Faster simulations if we renounce to some global symmetry: (Wilson, Staggered, tmqcd fermions), symmetry recovered for a 0 L. Giusti Nara November 2005 p.6/30

8 First-principle results from lattice simulations First-principle results when all systematic uncertainties quantified Main sources of errors: 1. Statistical errors 2. Finite volume: L = fm 3. Continuum limit: a = fm 4. Chiral extrapolation: M π = MeV On the lattice they can be estimated and (eventually) removed without extra free parameters or dynamical assumptions (QFT,V, Alg., CPU) L. Giusti Nara November 2005 p.7/30

9 The Witten Veneziano formula: a recent example of systematics under control The Witten Veneziano formula is F 2 πm 2 η 2N f m=0 N f = χym =0 Nc where Z χ YM = d 4 x Q(x)Q(0) YM By setting the scale F K = 113(1) MeV r 0 4 χl YM χ YM = (191 ± 5 MeV) to be compared with F 2 π 2N f (M 2 η + M 2 η 2M 2 K ) exp (180 MeV) (a/r 0 ) 2 [Del Debbio, L. G., Pica 04] In QCD the bulk of m η is due to the quantum anomaly as conjectured by WV L. Giusti Nara November 2005 p.8/30

10 Quenched approximation Fermion determinant replaced by its average value Z, O = DUe S G [DetD] N f O CP-PACS Coll 02 Quenched light hadron spectrum: 10% discrepancy with experiment m (GeV) K K* φ N Λ Σ Ξ Σ* Ξ* K input φ input experiment Ω 0.4 For some quantities quenching is the only systematics not quantified with confidence L. Giusti Nara November 2005 p.9/30

11 Standard unquenching algorithms: the Berlin wall For Wilson-type fermions [A. Ukawa Lattice Berlin] a=0.080 fm L=2.0 fm C ost N conf m 3 q L 5 a 8 With a O(10) Tflops machine (example) Simulation cost Staggered root trick Berlin 01 L = 2 fm m π m ρ = a 1 = GeV With O(1) Tflop machines some points feasible m π /m ρ L. Giusti Nara November 2005 p.10/30

12 Improved staggered with the root trick Impressive effort: 2+1 flavor simulations with improved staggered fermions Fermilab, HPQCD, MILC, UKQCD Coll. 04 Detailed study of chiral extrapolation f π f K 3M Ξ M N The formulation requires four deg. flavors (tastes) Dynamical simulations with the root trick detd = (detd stagg ) 1/4 2M Bs M Υ ψ(1p 1S) Υ(1D 1S) Υ(2P 1S) Υ(3S 1S) Υ(1P 1S) LQCD/Exp t (n f = 0) LQCD/Exp t (n f = 3) No local action known with this determinant: = no first-principle computation = unitarity violations = unknown systematic errors L. Giusti Nara November 2005 p.11/30

13 Schwarz-preconditioned Hybrid Monte Carlo (SAP) Lüscher Decomposition of the lattice into blocks with Dirichlet b.c. with q π/l > 1 GeV Asymptotic freedom: quarks are weakly interacting in the blocks = QCD easy (cheaper) to simulate Block interactions are weak and are taken into account exactly S(x, y) 1 x y 3 L. Giusti Nara November 2005 p.12/30

14 Block decomposition of the Dirac operator The Wilson Dirac operator D = 1 2 γµ ( µ + µ) µ µ + m0 can be decomposed as D = D Ω + D Ω + D Ω + D Ω where D Ω = X D Λ D Ω = X white Λ black Λ D Λ Ω, Ω are white and black blocks, Ω, Ω are exterior boundaries L. Giusti Nara November 2005 p.13/30

15 Factorization of the determinant The determinant of the Dirac operator written as det D = Y allλ det ˆD Λ det R with the block interaction R = 1 P Ω D 1 Ω D ΩD 1 Ω D Ω For two flavors can be written as integral over scalar fields S φχ = X all Λ ˆD 1 Λ φ Λ 2 + R 1 χ 2 where φ Λ defined on Λ and χ on Ω L. Giusti Nara November 2005 p.14/30

16 Schwarz-preconditioned Hybrid Monte Carlo (SAP) Lüscher In molecular dynamics force naturally split d dt Π(x, µ) = F G(x, µ) F Λ (x, µ) F R (x, µ) 10 F k (x,µ) 1 k=0 k=1 d U(x, µ) = Π(x, µ)u(x, µ) dt 0.1 k=2 Integration step-sizes chosen such that d ɛ G F G ɛ Λ F Λ ɛ R F R i.e. the most expensive force computed less often! Do not give up first-principles: teach Physics to exact algorithms for being smarter (faster)! C ost N conf m 1 q L 5 a 6 L. Giusti Nara November 2005 p.15/30

17 First simulations with SAP Lüscher 04; Del Debbio, L.G., Lüscher, Petronzio, Tantalo 05 a=0.080 fm L=2.0 fm PC cluster with 80 Nodes (160 Xeon procs) 64 used for this project ( 200 Gflops sustained) Simulation cost Berlin 01 SAP m π /m ρ Volume a[fm] m/m s N conf (100) TBD (100) Full statistics for small lattice: nodes All confs CERN First goal: making contact w. ChPT L. Giusti Nara November 2005 p.16/30

18 A new PC cluster at CERN 32 Xeon 2.8 GHz 800 MByte/s bi-directional 160 GFlop/s sustained with SAP L. Giusti Nara November 2005 p.17/30

19 Correlation functions on the finer lattice X O(x, t)o(0, 0) e m O (t)t x 0.5 V=32 3 * 64, β=5.8, K= a m ρ eff 0.2 a m π t/a Algorithm stable over the relevant parameter ranges: 1. Quark mass: m m s /6 2. Lattice spacing: a fm 3. Volume: L 2 fm L. Giusti Nara November 2005 p.18/30

20 First results for pion mass and decay constant Volume a[fm] am am π af π (3) 0.274(2) (8) (2) 0.197(2) (9) (2) 0.155(3) (17) (2) 0.121(4) (23) m π ~ 676 MeV (am π ) af π am am L. Giusti Nara November 2005 p.19/30

21 Chiral behavior of M π At the NLO in SU(2) ChPT [J. Gasser, H. Leutwyler 84] j Mπ 2 = M M 2 ff 32π 2 F 2 log(m 2 /µ 2 π) with M 2 = 2B ˆm 0.08 m π ~ 676 MeV Data below M π 500 MeV are compatible (within errors) with NLO ChPT More lattice spacings needed to confirm the picture (am π ) For comparison: from Nature M 2 π/m 2 const 0.956(8) am in the range M = MeV L. Giusti Nara November 2005 p.20/30

22 Chiral behavior of F π NLO SU(2) ChPT gives [J. Gasser, H. Leutwyler 84] F π = F j1 M 2 ff 16π 2 F 2 log(m 2 /µ 2 F ) 0.08 Fitting points below M π 500 MeV (Preliminary!) F π 80(7)MeV af π with Z A from 1-loop PT 0.02 More lattices needed to remove systematics am Also in this case data are compatible (within errors) with NLO ChPT L. Giusti Nara November 2005 p.21/30

23 Where we are...lüscher 04; Del Debbio, L.G., Lüscher, Petronzio, Tantalo 05 m π 2 [GeV 2 ] 0.3 SPQcdR 04 CP-PACS 01 TχL 00 BNL 04 N F = 2 m~m s /2 0.2 UKQCD 01 qq+q CERN-TOV 05 CP-PACS 04 m~m s /6 Exp a [fm] L. Giusti Nara November 2005 p.22/30

24 B K from quenched QCD From JLQCD (S. Aoki et al. 98) B K MS (2 GeV) KS-PT (JLQCD) KS-PT (KGS) WI-WI (JLQCD) WI-NP (SPQcdR) DW-PT (CP-PACS) DW-NP (RBC) OV-NP (GGHLR) OV-PT (DeGrand) tmqcd (Alpha) a/r 0 B MS K (2GeV) = 0.63 ± 0.04 ˆB K = 0.86 ± 0.06 Systematic uncertainties: - Statistical errors - Finite volume effects - Continuum limit - Renormalization - Degenerate quarks (m s = m d )? - Quenched Approximation? (Full) QCD simulations required at this stage L. Giusti Nara November 2005 p.23/30

25 Error milestones for B K Total non-b K error in ε K is 9.5%, mainly from 2.2% error in Aλ 2 = V cb Lattice ˆB K value used in CKM fits [CKM CERN Workshop 02] ˆB K = 0.86 ± 0.06 [stat&rin.] ± 0.14 [quench&chiral] (18% error) Error milestones for lattice calculations of ˆB K 1. 10%: match the error from other sources 2. 5%: half that of other sources L. Giusti Nara November 2005 p.24/30

26 Status and perspective for f Bd q ˆBBd and f Bs q ˆB Bs M Bs not measured yet! Probably soon at Tevatron Lattice values used in CKM fits [CKM CERN Workshop 02] f Bs q ˆBBs = 276 ± 38 [stat + syst] (14% error) q ˆBB s ξ = f Bs q = 1.24 ± 0.04 [stat] ± 0.06 [syst&chiral] (6% error) f Bd ˆBBd Error milestones for lattice calculations: 1. 5% for f Bs q ˆBBs % for ξ L. Giusti Nara November 2005 p.25/30

27 B πlν l from quenched QCD π ūγ µ b B = f + (q 2 )(p B + p π m2 B m2 π q 2 + f 0 (q 2 ) m2 B m2 π q 2 q µ Systematic uncertainties: - Statistical errors - Finite volume effects - Continuum limit - Renormalization - Quenched Approximation? Full QCD simulations required q 2 can be reduced by imposing θ-boundary conditions [de Devitis et al. 04] 10% as an error milestone in lattice calculations of these form factors L. Giusti Nara November 2005 p.26/30

28 More information from experiments CLEO, Babar, Belle S.B. Athar et al. (CLEO Coll. 03) dependence of the physical rate for B (π, ρ)lν l q 2 First analyses presented also by Babar and Belle Leptonic decays of charged B-mesons B[B τ ν τ ] = G2 F V ub 2 fb 2 8π τ B m B m 2 τ 1 m2 2 τ m 2 B A measurement by the end of the B-factories? Competitive to extract V ub once f B known at 5 10% L. Giusti Nara November 2005 p.27/30

29 Conclusions and Outlook m π 2 [GeV 2 ] 0.3 SPQcdR 04 CP-PACS 01 TχL 00 BNL 04 N F = 2 m~m s /2 0.2 UKQCD 01 qq+q CERN-TOV 05 CP-PACS 04 m~m s /6 Exp a [fm] First results with SAP: a breakthrough in full QCD simulations First goal: contact with ChPT established It will be worth to attack computations of weak MEs and form factors in full QCD with O(1 10) Tflops computer (but do not expect results by tomorrow!) It is getting interesting for flavor physics, stay tuned! L. Giusti Nara November 2005 p.28/30

30 Summarizing error milestones If we are able to run full QCD simulations with machines O(1 10) Tflops L = fm m π m ρ = a = fm Observable ε M Bs M Bs M Bd B ( π ρ )lν B ( D D )lν CKM Im[V td ] V ts 2 V ts 2 / V td 2 V ub 2 V cb 2 Matr. Elem. ˆBK f 2 B s ˆBBs f 2 B s ˆBBs f 2 B d ˆBBd π ρ J ub L B 2 D D JL cb B 2 Err. in Quen. 7% 10% 6% 30% 8% Quen+χ Err. 15% 20% 10% - 3% Proj. error 5% 10% 5% 20% 6%...it is timely to attack these computations in full QCD L. Giusti Nara November 2005 p.29/30

31 UT fit with same observables but extr. exp+theo err. Conservative! More observables may be relevant for the fit in future η sin2β M d M d M s 0.2 ε K V ub V cb C K M f i t t e r Future (2) sin 2β m d excluded area has CL < 0.05 m s & m d ρ η α ε K γ β V ub /V cb ρ L. Giusti Nara November 2005 p.30/30