The light hadron spectrum from lattice QCD

Transcription

1 Ch. Hoelbling with S. Durr Z. Fodor J. Frison S. Katz S. Krieg T. Kurth L. Lellouch Th. Lippert K. Szabo G. Vulvert The Budapest-Marseille-Wuppertal collaboration Rab 2008

2 Outline 1 Motivation 2 Setup 3 Simulation details 4 Analysis 5 Treatment of systematic errors 6 Combining all errors 7 Final Result

3 QCD AS THE SIGNAL Asymptotic freedom: good agreement between theory and experiment ρ mass[gev] Λ φ K * N Σ Ξ Σ Ξ Ω - input φ - input Ω Good evidence that QCD describes the strong interaction in the non-perturbative domain (e.g. CP-PACS 07, N f =2 + 1, 210MeV M π 730MeV, a fm, L 2.8 fm, M πl 2.9) However, systematic errors not yet under control

4 QCD AS BACKGROUND At the quark level As seen in experiment V ub V ub π ūγ µb B 0 V ub from experiment must evaluate non-perturbative strong interaction corrections Must be done in QCD to test quark-flavor mixing and CP violation and possibly reveal new physics Must match accuracy of (BaBar, BELLE, CDF, D0, ALEPH, DELPHI, KLOE, NA48, KTEV, LHC-b, etc.) High-precision Lattice QCD

5 WHY THE LIGHT HADRON SPECTRUM? Goal: Firmly establish (or invalidate?) QCD as the theory of strong interaction in the low energy region Method: Post-diction of light hadron spectrum Octet baryons Decuplet baryons Vector mesons Challenge: Minimize and control all systematics 2+1 dynamical fermion flavors Physical quark masses Continuum Infinite volume (treatment of resonant states)

6 LATTICE QCD Lattice gauge theory mathematically sound definition of NP QCD: UV (and IR) cutoffs and a well defined path integral in Euclidean spacetime: Z O = DUD ψdψ e S G R ψd[m]ψ O[U, ψ, ψ] Z = DU e S G det(d[m]) O[U, ψ, ψ] Wick T Ψ(x) U 0 (x) e S G det(d[m]) 0 and finite # of dof s evaluate numerically using stochastic methods L NOT A MODEL: Lattice QCD is QCD when a 0, V and stats In practice, limitations...

7 STATISTICAL AND SYSTEMATIC ERROR SOURCES Limited computer resources a, L and m q are compromises and statistics finite Associated errors: Statistical: 1/ N conf ; eliminate with N conf Discretization: aλ QCD, am q, a p, with a GeV 1/m b < a < 1/m c b quark cannot be simulated directly rely on effective theories (large m Q expansions of QCD) Eliminate with continuum extrapolation a 0: need at least three a s Chiral extrapolation: m q m u, m d Use χpt to give functional form chiral logs M 2 π ln(m 2 π/λ χ) Requires a number of M π 500 MeV Finite volume: for simple quantities e MπL and M πl 4 usually safe Eliminate with L (χpt gives functional form) Renormalization: LQCD gives bare quantities must renormalize: can be done in PT, best done non-perturbatively

8 THE BERLIN WALL CA Unquenched calculations very demanding: # of d.o.f. O(10 9 ) and large overhead for computing det(d[m]) ( matrix) as m q m u,d Tflops x year/500 configs L = 2.5 fm, T = 8.6 fm, a = 0.09 fm Physical point Staggered (R-algorithm) Wilson M π /M ρ Staggered and Wilson with traditional unquenched algorithms ( 2004) cost N confv 5/4 mq a 7 (Gottlieb 02, Ukawa 02) Both formulations have a cost wall Wall appears for lighter quarks w/ staggered MILC got a head start w/ staggered fermions: N f = simulations with M π 250 MeV Impressive effort: many quantities studied Detailed study of chiral/continuum extrapolation with staggered χpt

9 fπ Devil s advocate! potential problems: fk 3MΞ MN 2MBs MΥ ψ(1p 1S) Υ(1D 1S) Υ(2P 1S) Υ(3S 1S) Υ(1P 1S) LQCD/Exp t (nf = 0) LQCD/Exp t (nf = 3) det(d[m]) Nf =1 det(d[m] stagg) 1/4 to eliminate spurious tastes corresponds to non-local theory (Durr, C.H ; Shamir, Bernard, Golterman, Sharpe, ) more difficult to argue that a 0 is QCD at current a, significant lattice artefacts complicated chiral extrapolations w/ SχPT (Davies et al 04) it is important that approaches on firmer theoretical ground also be used Wilson fermions strike back: Schwarz-preconditioned Hybrid Monte Carlo (SAP) (Lüscher 03-04) HMC algorithm with multiple time-scale integration and mass preconditioning (Sexton et al 92, Hasenbusch 01, Urbach et al 06) Crucial insight: seperate scales (even better: also remove UV junk )

10 N f =2+1 WILSON FERMIONS À LA BMW (Dürr et al (BMW Coll.) arxiv: ) 3 essential components: Separation of scales in HMC evolution: Mass preconditioning (w/ multiple time scales, mixed precision inverters, Omelyan integrator) Effective supression of irrelevant UV modes: Stout link smearing (6-step, ρ = 0.11) Action improvement: Tree level O(a) improved Wilson fermion action, tree level O(a 2 ) improved gauge action Why not go beyond tree level? Keeping it simple (parameter fine tuning), no real improvement This is a crucial advantage of our approach Last two ingredients were shown in the quenched case to lead to excellent improvement (Capitani, Durr, C.H., 2006) Better chiral behavior renormalization constants, improvement coefficients closer to tree level

11 LOCALITY PROPERTIES smearing locality in position space: D(x, y) < const e λ x y with λ=o(a 1 ) for all couplings. Our case: D(x, y)=0 as soon as x y >1 (despite 6 smearings). locality of gauge field coupling: δd(x, y)/δa(z) < const e λ (x+y)/2 z with λ=o(a 1 ) for all couplings.

12 GAUGE FIELD COUPLING LOCALITY D(x,y)/ U m (x+z) a~0.125 fm a~0.085 fm a~0.065 fm z /a

13 SCALING OF OUR ACTION ( Dürr et al (BMW Coll.) arxiv: ) scaling study: N f = 3 w/ action described above, 5 lattice spacings, M πl > 4 fixed and q M π/m ρ = 2(M ph K )2 (Mπ ph ) 2 /M ph φ 0.67 i.e. m q m s fm 0.10 fm 0.15 fm 0.18 fm 0.20 fm 1700 MeV Excellent scaling up to a 0.2fm M MN a 2 /fm 2

14 FERMIONIC FORCE HISTORY 15 pseudofermion 1 pseudofermion 2 pseudofermion 3 s-quark (RHMC) gauge field trajectory

15 INVERSE ITERATION COUNT DISTRIBUTION 0.25 action, residue 10 9 force, residue /n CG

16 λ 1 min DISTRIBUTION 0.6 β =2.8 β =3.23 β =3.4 β =3.59 β = λ min/m PCAC Simulations

17 THERMAL CYCLE P cycle down cycle up am π

18 TUNING THE STRANGE QUARK MASS We use a N f = 3 simulation to set the strange quark mass For each beta, search for 0.08 the m q where q m ps 2mK 2 = 0.06 m2 π m v m Φ We determined the β 0.04 dependency in the range (β = ) Note: this is a rough papameter tuning; we will properly interpolate to the physical strange quark mass point later! 0.1

19 SIMULATION POINTS β am ud M π [GeV] am s L 3 T # traj , , , 24 3, ,2000, , , , , , , , ,

20 OUR LANDSCAPE (M K 2 -Mπ 2 /2) 1/ β=3.7 β=3.57 β=3.3 physical point M π

21 NUCLEON AUTOCORR. (M π = 550 MeV, β = 3.3)

22 PION AUTOCORR. (M π = 190 MeV, β = 3.57)

23 SOURCES Point-Point Gauss-Gauss Point-Point Gauss-Gauss 2 2 am π eff 1.5 am N eff t/a Gaussian sources r = 0.32 fm Coulomb gauge Gauss-Gauss less contaminated by excited states t/a

24 EFFECTIVE MASSES AND CORRELATED FITS a M t/a N K

25 SETTING THE LATTICE SPACING VIA HADRON MASS The particle selected should have a mass 1 that is experimentally well known 2 that is independent of light quark mass large strange content 3 that can be simulated with small statistical errors octet better suited than decuplet All points cannot be fulfilled simultaneously, but Ξ: largest strange content of the octet, but still dependent on ud mass Ω: member of the decuplet, but largest strange content of particles included in analysis

26 QUARK MASS DEPENDENCE Goal: Method: Extra-/Interpolate M X (baryon/vector meson mass) to physical point (M π, M K ) Use M Ξ or M Ω to set the scale Variables to parametrize Mπ 2 and MK 2 dependence of M X : Use bare masses am y, y {X, π, K } and a (bootstrapped) Use dimensionless ratios r y := My M Ξ/Ω (cancellations) We use both procedures systematic error

27 QUARK MASS DEPENDENCE (ctd.) Method (ctd.): Parametrization: M X = M (0) x + αmπ 2 + βmk 2 + higher orders Leading order sufficinet for MK 2 dependence We include higher order term in Mπ 2 Next order χpt (around Mπ 2 = 0): Mπ 3 Taylor expansion (around Mπ 2 0): Mπ 4 Both procedures fine systematic error No sensitivity to any order beyond these Vector mesons: higher orders not significant Baryons: higher orders significant Restrict fit range to further estimate systematics: full range, M π < 550/450MeV We use all 3 ranges systematic error

28 CHIRAL FIT M [GeV] physical M p O N a~0.125 fm a~0.085 fm a~0.065 fm M p 2 [GeV 2 ]

29 CHIRAL FIT USING RATIOS M/M X physical M p O N a~0.125 fm a~0.085 fm a~0.065 fm (M p /M X ) 2

30 CONTINUUM EXTRAPOLATION Goal: Method: Eliminate discretization effects Formally in our action: O(α s a) and O(a 2 ) Discretization effects are tiny Not possible to distinguish between O(a) and O(a 2 ) include both in systematic error

31 FINITE VOLUME EFFECTS FROM VIRTUAL PIONS Goal: Eliminate virtual pion finite V effects Method: Best practice: use large V We use M π L 4 (and one point to study finite V ) Effects are tiny and well described by M X (L) M X M X = cmπ 1/2 L 3/2 e MπL (Colangelo et. al., 2005) am p M p L=4 Pion c 1 + c 2 e -M L p L -3/2 fit Colangelo et. al am N M p L=4 Nucleon c 1 + c 2 e -M L p L -3/2 fit Colangelo et. al L/a L/a

32 FINITE VOLUME EFFECTS IN RESONANCES Goal: Eliminate spectrum distortions from resonances mixing with scattering states Method: Stay in region where resonance is ground state Otherwise no sensitivity to resonance mass in ground state Systematic treatment (Lüscher, ) Conceptually satisfactory basis to study resonances Coupling as parameter (related to width) Fit for coupling (assumed constant, related to width) No sensitivity on width (compatible within large error) Small but dominant FV correction for light resonances

33 RESONANCES CTD

34 SYSTEMATIC UNCERTAINTIES Goal: Method: Accurately estimate total systematic error We account for all the above mentioned effects When there are a number of sensible ways to proceed, we take them: Complete analysis for each of 18 fit range combinations ratio/nonratio fits (r X resp. M X ) O(a) and O(a 2 ) discretization terms NLO χpt M 3 π and Taylor M 4 π chiral fit 3 χ fit ranges for baryons: M π < 650/550/450 MeV resulting in 432 (144) predictions for each baryon (vector meson) mass with each 2000 bootstrap samples for each Ξ and Ω scale setting

35 SYSTEMATIC UNCERTAINTIES II Method (ctd.): Weigh each of the 432 (144) central values by fit quality Q Median of this distribution final result Central 68% systematic error Statistical error from bootstrap of the medians median Nucleon 0.3 Omega median M N [MeV] M O [MeV]

36 THE LIGHT HADRON SPECTRUM 2000 Budapest-Marseille-Wuppertal collaboration M[MeV] r K * N L S X D O X* S* p K experiment width input QCD

37 Mass predictions in GeV Exp. Ξ scale Ω scale ρ (29)(13) 0.778(30)(33) K (14)(4) 0.907(15)(8) N (25)(22) 0.953(29)(19) Λ (15)(5) 1.103(23)(10) Σ (18)(15) 1.157(25)(15) Ξ (16)(13) (97)(61) 1.234(82)(81) Σ (46)(35) 1.404(38)(27) Ξ (26)(15) 1.561(15)(15) Ω (20)(15)