Low-energy QCD II Status of Lattice Calculations

Transcription

1 Low-energy QCD II Status of Lattice Calculations Hartmut Wittig Institute for Nuclear Physics and Helmholtz Institute Mainz Determination of the Fundamental Parameters of QCD Nanyang Technical University 18 March 2013

2 2 Introduction

3 3 Introduction Low-Energy QCD in the LHC Era Quantitative understanding of hadronic properties: Hadron masses and decays Structural properties Precision tests of the Standard Model at low energies: QCD corrections to weak decay amplitudes Hadronic contributions to the muon (g 2)

4 4 Introduction Wilson s 1989 dictum I still believe that an extraordinary increase in computer power (10 8 is I think not enough) and equally powerful algorithmic advances will be necessary before a full interaction with experiment takes place. Peak performance in GFlops/s Is there interaction with experiment in the Peta-Flops age?

5 5 Outline Outline I. A Lattice QCD Primer II. Successes Hadron Spectroscopy Lattice constraints on CKM physics III. Challenges Axial charge of the nucleon Scalar form factor of the pion Hadronic contributions to g 2 μ IV. Conclusions

6 6 A Lattice QCD Primer

7 7 A Lattice QCD Primer Non-perturbative approach: regularised Euclidean functional integrals Minkowski space-time, continuum Euclidean space-time, discrete lattice spacing: finite volume: a, x μ = n μ a, a 1 Λ UV L 3 T Regularisation respects gauge invariance Perform stochastic evaluation of observables Ω : D lat : Discretisation of the quark action

8 8 A Lattice QCD Primer Simulation algorithm for dynamical fermions: [Duane et al., 1987] Hybrid Monte Carlo Obstacle: strong growth of numerical cost near physical m u, m d Main progress due to algorithmic improvements: Low-mode reweighting Deflation Domain Decomposition Mass Preconditioning Hierarchical Integration Schemes Rational Hybrid Monte Carlo [Lüscher & Palombi 2008, Hasenfratz et al. 2008] [Lüscher 2007; Morgan & Wilcox 2007] [Lüscher ] [Hasenbusch, Jansen 2001, Marinkovic et al. 2011] [Lüscher 2005, Urbach et al. 2005] [Clark & Kennedy 2006] Pion mass, i.e. lightest mass in the pseudoscalar channel: 500 MeV (2001) MeV (2013)

9 9 A Lattice QCD Primer Systematic Effects Lattice artefacts: extrapolate to continuum limit from a fm Finite volume effects: Observables distorted by finite box size L fm and m π L > 4 sufficient for many purposes Finite volume corrections computable in ChPT Unphysical quark masses: Chiral extrapolation to physical values of m u, m d required Use ChPT as a guide Insufficient sampling of SU(3) group manifold: Strong growth of autocorrelation time of topological charge as a 0 Use open boundary conditions in time direction

10 10 Successes

11 11 Hadron Spectroscopy Hadron Spectroscopy Hadron masses encoded in correlation functions: O had x : interpolating operator in a given hadron channel, e.g. projects on all states with the same quantum numbers Ground state dominates at large Euclidean time separations Excited states: sub-leading contributions

12 12 Hadron Spectroscopy Light Hadron Spectrum the Genesis of an Icon Masses of the lowest-lying hadrons: Benchmark of Lattice QCD GF11 UKQCD CP-PACS BMW Collaboration (1992): (2008): (1999): (1998): (2002): Quenched Dynamical approximation u, d, s quarks No Agreement Confirmation Better Excellent significant agreement with of deviation the with with from GF11/CP-PACS experiment experiment at after in continuum the results inclusion within level using of 10 different of limit dynamical 15% discretisation quarks Statistically significant deviations Masses of the lowest-lying hadrons computable from first principles

13 13 Lattice constraints on CKM physics First Row Unitarity and V us Unitarity implies V ud 2 + V us 2 + V ub 2 = 1, V ub = Experimental constraints: V ud 2 = ± Super-allowed nuclear β-decays [Hardy & Towner, 2009] V us f + 0 = , V us f K V ud f π = (5) Branching fractions for K l2 and K l3 decays [Antonelli et al., 2010] Use experimental and lattice data to Obtain precise determinations of V ud and V us Perform precision tests of CKM unitarity

14 14 Lattice constraints on CKM physics K l3 Decays on the Lattice Matrix element and form factors e i x i x p f e i x x i p i φ π x f V μ x φ K (x i ) = π(p π ) V μ 0 K(p K ) x i, x f π(p π ) V μ 0 K(p K ) = f + q 2 p K + p π μ + f q 2 p K p π μ Lattice momenta: p i, p f = 0,0,0, 1,0,0, 1,1,0,. times 2π L Employ twisted boundary conditions: ψ x + Le k = e iθ k ψ(x) 2π p k = n k L + θ k L, q2 2 = E K p i E π p f pi + θ i L p f + θ f L 2

15 15 Lattice constraints on CKM physics Compilation & Comparison: The FLAG Working Group [Colangelo et al., Eur Phys J C71 (2011) 1695] Provide PDG-style global averages Rate lattice results according to quality criteria when the systematic error has been estimated in a satisfactory manner and convincingly shown to be under control when a reasonable attempt at estimating the systematic error has been made, although this could be improved when no or a clearly unsatisfactory attempt at estimating the systematic error has been made Example: Continuum Finite-volume Chiral extrapolation extrapolation effects 3 mor min ππ more L < > lattice or MeV spacings, least 3 volumes at least 2 points below a = 0.1 fm 2 m250 or min π more MeV L > 3 lattice < and m min π at spacings, least < volumes MeV at least 1 point below a = 0.1 fm otherwise m min π > 400 MeV

16 16 Lattice constraints on CKM physics Compilation & Comparison: The FLAG Working Group [Colangelo et al., Eur Phys J C71 (2011) 1695] Provide PDG-style global averages Rate lattice results according to quality criteria when the systematic error has been estimated in a satisfactory manner and convincingly shown to be under control when a reasonable attempt at estimating the systematic error has been made, although this could be improved when no or a clearly unsatisfactory attempt at estimating the systematic error has been made Example: Chiral extrapolation FLAG-2 m min π < 200 MeV 200 MeV < m min π < 300 MeV m min π > 300 MeV

17 17 Lattice constraints on CKM physics FLAG Analysis of K l3 Decays [Colangelo et al., Eur Phys J C71 (2011) 1695] Lattice results consistent with Ademollo-Gatto theorem [Leutwyler & Roos 1984] FLAG global estimates: f + 0 = ( ) 14, N f = , N f = 2

18 18 Lattice constraints on CKM physics V us from K l2 Decays [Marciano, Phys Rev Lett 93 (2004) ] Leptonic decay rate: Γ K μν μ γ Γ K eν e γ V us 2 m K 2 fk V ud f 2 π m π Lattice determination of f K f π yields CKM matrix elements Published FLAG averages: f K f π = , N f = f K f π = (17), N f = 2 [FLAG-2 - preliminary]

19 19 Lattice constraints on CKM physics Testing the Standard Model [Colangelo et al., Eur Phys J C71 (2011) 1695] First row unitarity: V 2 ud + V 2 us + V 2 ub V 2 u = 1 Measured branching fractions: V us f + 0 = (5) V us f K V ud f π = (5) Global estimates N f = : V u 2 = 1.002(15) Combine lattice results with V ud from nuclear β-decay: f + (0)-input: V u 2 = (7) f K f π -input: V u 2 = (6)

20 20 Lattice constraints on CKM physics V ud and V us from Lattice QCD [Colangelo et al., Eur Phys J C71 (2011) 1695] Experimental branching fractions plus unitarity fix three values from the set V ud, V us, f + 0, f K f π Lattice estimate for V us agrees with determination from nuclear β-decay and is equally precise

21 21 Challenges

22 22 Challenges Current lattice simulations do not reproduce experimental results on nucleon structure Average Nucleon momentum axial charge fraction [H-W [Alekhin, Lin, Blümlein, PoS LATTICE2012 Moch, (2012) 013] Phys Rev D86 (2012) ] Systematic effects not fully controlled Lattice artefacts Chiral extrapolation to physical pion mass Finite-volume effects Contamination from excited states Quark-disconnected diagrams ignored

23 23 Challenges Many observables require evaluation of quark-disconnected diagrams: Exponentially increasing noise-to-signal ratio: Nucleon at rest: R NS x 0 e m N 3 2 m π x 0 Pion at p 0: R NS x 0 e m π 2 +p 2 m π x 0 Serious obstacle for accurate determinations of hadron form factors

24 24 Axial Charge of the Nucleon Axial Charge of the Nucleon g A : nucleon matrix element of the axial current: N p, s A μ u d N p, s = 2g A s μ Ideal benchmark quantity for lattice QCD: Simple quark bilinear: A μ = q γ μ γ 5 q No momentum transfer between initial and final state No quark-disconnected diagram Lattice calculations underestimate g A by 10%

25 25 Axial Charge of the Nucleon Standard Method Extract nucleon hadronic matrix elements from ratios of three- and two-point correlation functions, e.g. Statistical fluctuations impose t s 1.3 fm m π = 312 MeV t s = fm

26 26 Nucleon Form Factors and Axial Charge Summed Insertions [Maiani et al. 1987, Güsken et al. 1989, Bulava et al., Capitani et al.] Standard method: R Γ q, t, t s = M Γ (q) + O e Δt, e Δ t s t Summed insertion: t s S Γ t s R Γ q, t, t s = K Γ + t s M Γ (q) + O e Δt s, e Δ t s t=0 Excited state contributions more strongly suppressed Determine M Γ (q) from linear slope of summed ratio (axial charge)

27 27 Axial Charge of the Nucleon Nucleon axial charge in two-flavour QCD [Capitani, Della Morte, von Hippel, Jäger, Jüttner, Knippschild, Meyer, H.W., Phys Rev D86 (2012) ] O a improved Wilson fermions Three lattice spacings: a = 0.05, 0.063, fm Pion masses m min π = MeV, m min π L 4.0

28 28 Axial Charge of the Nucleon Nucleon axial charge in two-flavour QCD [Capitani, Della Morte, von Hippel, Jäger, Jüttner, Knippschild, Meyer, H.W., Phys Rev D86 (2012) ] Standard plateau method and summed insertions Chiral fits: A: α + βm π 2 B: α + βm π 2 γ m π 2 ln m π 2 Λ 2 C: α + βm π 2 γ e m πl D: Heavy Baryon ChPT

29 29 Axial Charge of the Nucleon Nucleon axial charge in two-flavour QCD [Capitani, Della Morte, von Hippel, Jäger, Jüttner, Knippschild, Meyer, H.W., Phys Rev D86 (2012) ] g A = ± stat ( ) syst (Fit A, m π < 540 MeV) Still to come: Smaller pion masses Smaller lattice spacings Better statistics

30 30 Axial Charge of the Nucleon Nucleon axial charge in two-flavour QCD [Capitani, Della Morte, von Hippel, Jäger, Jüttner, Knippschild, Meyer, H.W., Phys Rev D86 (2012) ] g A = ± stat ( ) syst (Fit A, m π < 540 MeV) Still to come: Smaller pion masses Smaller lattice spacings Better statistics

31 31 Axial Charge of the Nucleon Summary: Nucleon Form Factors and Axial Charge Short source-sink separation t s dictated by noisy signal Addressing excited-state contamination indispensable LHP Collaboration [Green et al., arxiv: ] Physical pion mass at a = 0.12 fm Excited-state contribution addressed [H-W Lin, PoS LATTICE2012 (2012) 013]

32 32 Scalar Form Factor of the Pion Scalar form factor of the pion Definition; charge radius: q 2 = p f p i 2 = Q 2 + F s 0 σ π (pion σ-term) Chiral expansion: l 4 = ± r 2 s = ± fm 2 Phenomenological determination from ππ-scattering [Colangelo, Gasser & Leutwyler, Nucl Phys B603 (2001) 125] r 2 s = 0.61 ± 0.04 fm 2

33 33 Scalar Form Factor of the Pion Lattice calculation [Gülpers, von Hippel, H.W., PoS (Lattice2012) 181, and in prep.] Quark-disconnected diagrams: Stochastic evaluation of disconnected contribution: N stochastic sources Hopping parameter expansion of Dirac operator to order k Choose k and N to minimise computational effort at fixed variance loop = Tr D 1 x, x x G

34 34 Scalar Form Factor of the Pion Lattice calculation [Gülpers, von Hippel, H.W., PoS (Lattice2012) 181, and in prep.] Scalar form factor: momentum dependence m π 324 MeV Quark-disconnected contributions: Twisted boundary conditions not applicable Signal lost for q > 2π L

35 35 Scalar Form Factor of the Pion Lattice calculation [Gülpers, von Hippel, H.W., PoS (Lattice2012) 181, and in prep.] Preliminary results at a = fm, m π = MeV Inclusion of quark-disconnected contributions crucial for agreement with phenomenology

36 36 Hadronic contributions to the muon (g 2) Hadronic vacuum polarisation contribution to g 2 μ a μ VP;had = ( ± 4.75) (combined e + e data) Theory prediction uses experimental data as input Ab initio estimate from Lattice QCD: total accuracy of 0. 5% required!

37 37 Hadronic contributions to the muon (g 2) Hadronic vacuum polarisation contribution Lattice approach: evaluate convolution integral over Euclidean momenta Convolution function peaked far below lowest Fourier momentum: Maximum of f q 2 located near 5 2 m μ GeV 2 2π T GeV 2 Subtracted amplitude enters convolution integral: Π q 2 Π q 2 Π(0) determine Π(0) via extrapolation q 2 0 Quark-disconnected diagrams:

38 38 Hadronic contributions to the muon (g 2) Hadronic vacuum polarisation contribution Current lattice estimates not competitive with dispersive approach: Improve statistical accuracy Simulate at the physical pion mass Include quark-disconnected diagrams

39 39 Hadronic contributions to the muon (g 2) Hadronic vacuum polarisation contribution [Della Morte, Jäger, Jüttner, H.W., JHEP 1203 (2012) 055, and to appear] Apply twisted boundary conditions to reach lower momenta New ensembles: m min π 195 MeV, m min π L 4.0, a = fm q 2 -dependence described by [2,3] Padé ansatz [Blum, Golterman & Peris, 2012] Analysis of new data in progress; increase statistics further

40 40 Hadronic contributions to the muon (g 2) Lattice approach to hadronic light-by-light scattering Brute force: Compute 4-point function Integrate over internal momenta Extrapolate: q 2 0 Simultaneous QCD+QED simulations: [Blum, Chowdury et al.] Difference yields desired contribution up to O(α 3 )

41 41 Conclusions Conclusions: Lattice QCD an established and mature method to study QCD at low energies Successful post-diction of the light hadron spectrum Predictions for decay constants, form factors, low-energy constants, Is there full interaction with experiment in the Pflops age? First-row unitarity QCD Thermodynamics: freeze-out conditions in heavy-ion collisions Tackle more ambitious quantities Nucleon form factors and axial charge: 2 nd generation benchmark Excitation spectrum Hadronic contributions to muon g 2