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Status of Lattice QCD

Status of Lattice QCD. Richard Kenway. Parameters of QCD. lattice QCD. a. U  (x ). q(x). U  (x ). g 2 and m f are fundamental parameters of the Standard Model computable in a complete theory … a test of BSM theories but quarks are confined … emergent complexity

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Status of Lattice QCD

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  1. Status of Lattice QCD Richard Kenway

  2. Parameters of QCD Status of Lattice QCD

  3. lattice QCD a U(x) q(x) U(x) • g2 and mf are fundamental parameters of the Standard Model • computable in a complete theory … a test of BSM theories • but quarks are confined … emergent complexity • Euclidean space-time lattice regularisation • lattice spacing a, lattice size L • Monte Carlo approximation to path integral • N gauge configurations Status of Lattice QCD Lattice QCD

  4. lattice QCD Lattice QCD L a • lattice spacing must be extrapolated to zero keeping box large enough • by approaching a critical point • think of the computer as a ‘black box’ quark masses + gauge coupling properties of hadrons Status of Lattice QCD

  5. simulations use dimensionless variables (lattice spacing = 1) quark masses, mf, and gauge coupling, g2, are varied hadronic scheme at each value of g2, fix quark masses mf by matching Nf hadron mass ratios to experiment one dimensionful quantity fixes the lattice spacing in physical units QCD scale Status of Lattice QCD • dimensionless ratios become independent of g2 if a is small enough (scaling)

  6. impose mass-independent renormalisation conditions at p2 = 2 renormalisation • convert matrix elements to a perturbative scheme (matching) • eg to combine with Wilson coefficients in an OPE • or use step scaling • let 1 = L, the linear box size • consider a sequence of intermediate renormalisations at box sizes Ln = 2n L0 Status of Lattice QCD

  7. β function • continuum limit: a→ 0 with L constant and large enough • tune β = 6/g2 → ∞ holding low-energy physics constant • non-perturbative β function Status of Lattice QCD

  8. continuum limit of the quantum theory • symmetries of the lattice theory define the universality class • Lorentz invariance is an “accidental” symmetry as a → 0 • there are no relevant operators to break it • confinement • gauge invariance is preserved at the sites of the lattice • there is no phase transition into an unconfined phase as mf, g are tuned to the critical line (a → 0) • chiral symmetry can be realised correctly • Ginsparg-Wilson formulations realise the full chiral symmetry at a ≠ 0 • flavour symmetry can be realised in full, but is broken by some formulations • Osterwalder-Schrader conditions (reflection positivity) • sufficient for a Lorentz invariant QFT • generally not proven, especially for improved actions Status of Lattice QCD

  9. non-perturbative running • compute α(μ) at mq(a) = 0 for a sequence of box sizes 1 = L in the limit a → 0 • match with perturbation theory at a high scale • PCAC quark masses Status of Lattice QCD

  10. strong coupling • lattice QCD provides a precise determination Status of Lattice QCD

  11. quark masses • high precision is being achieved for light quarks • but there are systematic differences between lattice formulations Status of Lattice QCD staggered Wilson

  12. fermion doubling • the covariant derivative as a difference operator • (naïve) free fermion Dirac operator in momentum space Status of Lattice QCD • 16 (= 2d) degenerate fermion species • couple to axial current with alternating signs so U(1) axial anomaly cancels • giving a fully regularised theory with chiral symmetry • potential disaster for lattice QCD! • different lattice fermion actions to deal with this are the main reason for different systematic errors in lattice calculations

  13. ‘good’ lattice fermions • explicitly break chiral symmetry by adding a dimension 5 operator • change the chiral transformation on the lattice • where D satisfies the Ginsparg-Wilson relation • this is a symmetry of the action • there are several local solutions for D with smooth enough gauge fields • eg Ls  limit of 5-dimensional domain wall fermions (Wilson) • gives doublers masses  cut-off, leaving p = 0 pole unchanged • mixes operators of different chirality, complicating renormalisation • requires fine tuning to get mud = 0 (at Mπ = 0) Status of Lattice QCD

  14. ‘ugly’ lattice fermions • staggered fermions • lattice action may be diagonalised in spinor space • keep only one spinor component 1 =  • 2d/2 continuum species = ‘tastes’ (4 tastes in d = 4) • U(1) remnant of chiral symmetry prevents additive mass renormalisation • QCD with N degenerate quarks • N is a parameter in simulation algorithms • rooted staggered quarks • use one staggered fermion per flavour and take the fourth root of the determinant • cannot be described by a local theory … lose universality • non-locality/non-unitarity is a lattice artefact which vanishes as a→ 0, provided the quark mass is not taken to zero first • remains in the same universality class as QCD Status of Lattice QCD

  15. computational cost • big algorithmic improvements over the past two years • chiral regime and/or physically quark masses now seem reachable Status of Lattice QCD

  16. QCDOC Status of Lattice QCD > 6 teraflops sustained (BNL) > 3 teraflops (Edinburgh)

  17. quenched QCD is wrong • quenched QCD sets N = 0 • an early calculation expedient – avoids the costly determinant • omits virtual quark-antiquark pairs in the vacuum • provides a good phenomenological model, often good to 10% level • dynamical quark effects enter through renormalised quantities MN = 900 (100) MeV Hamber & Parisi 1982 Status of Lattice QCD

  18. … and dynamical sea quark effects are seen • string breaking Status of Lattice QCD

  19. QCD vacuum • isosurfaces of positive (red) and negative (green) topological charge density using • topological susceptibility • χPT with experimental fπ Status of Lattice QCD

  20. vacuum angle θ • QCD allows a gauge invariant CP odd term • CKM phase contributes < 10−30 e cm to dn • lattice relates dn to θ • simulations must sample topology well and contain light dynamical quarks with correct chiral symmetry • in quenched QCD dn is singular in the chiral limit • handle complex action for θ small by • experimental measurements • 2 flavour DWF, a−1 = 1.7 GeV • our 2+1 flavour simulations sample topology much better Status of Lattice QCD

  21. effective theories mQ>> QCD lattice QCD a, mq, mQ, L QCD scale QCD L >> QCD-1 mq <<QCD • simulations at physical parameter values are too expensive • use effective field theories to extrapolate simulation results from parameter regimes where systematic errors can be controlled to the physical regime HQET/NRQCD Lüscher finite volume effective theory Symanzik effective field theory a << QCD-1 Status of Lattice QCD chiral perturbation theory

  22. Status Status of Lattice QCD

  23. as a theoretical tool • the lattice is well established as a rigorous non-perturbative regularisation scheme for QCD • correctly realises all internal symmetries • has the correct continuum limit • may be applied to other QFTs … chiral gauge theories, SUSY, BSM • non-perturbative renormalisation • running couplings and matching to MS • matching to effective theories defines QCD at all parameter values • all sources of uncertainty can be systematically controlled • simulations are computationally tractable • dramatic recent progress in developing faster algorithms • renewed confidence that physically light quarks are within reach • visualisation may yet yield insight • explore topological structures and dominant fermionic modes Status of Lattice QCD

  24. Parameters of the Standard Model Status of Lattice QCD

  25. easily computed quantities • 2-point functions determines matrix elements such as decays asymptotically with energy of lightest state created by O • 3-point functions Status of Lattice QCD at large time separations, 2 >> 1 >> 0, can isolate matrix elements such as • but there is no general method for multi-hadron final states eg K 

  26. finite size effects • ‘rule of thumb’ = keep lattices big enough • χPT gives the correct functional dependence on volume for the pseudoscalar meson mass • but underestimates FSE by an order of magnitude (Wilson Nf = 2) • L ~ 2.5 fm needed for FSE below few % for 300 MeV pions Status of Lattice QCD

  27. quenched hadron spectrum • ‘tour de force’ demonstration of the power of lattice QCD • glueballs • nucleon excited states Status of Lattice QCD • mixing with flavour-singlet mesons is a major challenge for 2+1 flavours • requires flavour symmetry and spatially extended operators

  28. QCD hadron spectrum • prediction of the Bc mass • 2+1 flavours + relativistic effective action (c) + NRQCD (b) • inputs to quark mass and scale setting • Edinburgh plot • 2+1 flavours DWF Status of Lattice QCD

  29. flavour physics, CKM and lattice QCD • 3 generations • search for new physics by over-constraining the unitarity triangle • vastly improved experimental accuracy • lattice uncertainties dominate Status of Lattice QCD

  30. leptonic decays ℓ qx qx ℓ • elegant example of lattice ↔ experiment interaction • access to Vxy • cross-check of fX prediction Status of Lattice QCD

  31. π and Kleptonic decays • 2+1 flavours staggered (MILC) • full χPT analysis Status of Lattice QCD

  32. D leptonic decays • CLEO-c (2005) measured D→ μν • BaBar and CLEO-c (2006) measured Ds→ μν Vcd, τD from PDG 04 Status of Lattice QCD • experimental and lattice uncertainties are similar ~ 10% • sea quark effects are not significant

  33. B leptonic decays n b n b H- W t u u t • sensitive to charged Higgs • first direct measurement of fB (Belle 2006) Vub, τB from PDG 04 • lattice cut-off is too small to simulate both b and ud quarks directly • simulate relativistic b in small volumes … step scaling to large volume • use an effective heavy quark action … continuum limit non-trivial • sea-quark effects increase fBs by 10-15% Status of Lattice QCD

  34. neutral K mixing and K • CP violation in K • indirect CP violation Status of Lattice QCD

  35. indirect CP violation • quenched QCD • 2+1 flavour • a ~ 0.125 fm (RBC-UKQCD, preliminary) stat quench Status of Lattice QCD • next year should see the first realistic determinations of BK

  36. direct CP violation • P(1/2) is dominated by • in quenched QCD this mixes with unphysical operators, requiring additional low-energy constants • the resulting ambiguity means we cannot calculate ε'/ε reliably • the resolution is to use 2+1 flavours in the sea quenched QCD CP-PACS: -7.7  2.0 RBC: - 4.0  2.3 Status of Lattice QCD

  37. Bd andBs mixing • neutral Bq meson mass difference • BSM physics could enter loops Status of Lattice QCD • measurement of ΔMBs allows a theoretically well-controlled estimate using

  38. semileptonic decays Vxy qx qy qx qx  e+ W+ • access to Vxy • form factor embeds q2 dependence • more elaborate example of lattice ↔ experiment interaction • CKM-independent checks of lattice QCD from studying Status of Lattice QCD

  39. Kl3 decays • 2+1 flavour • a ~ 0.125 fm (UKQCD-RBC, preliminary) • f+(0) from lattice QCD should allow a precise determination of |Vus| Status of Lattice QCD

  40. semileptonic D  / Kℓdecays leptons W c s, d • model-independent form factors from lattice QCD • hadron momenta must be small to avoid large discretisation errors • maximum recoil ~ 1 GeV, so lattice data span full kinematic range • |Vcs| is well measured • precision test of lattice form factors against CLEO-c data • 2+1 flavours Status of Lattice QCD D→Ke+, with Vcs = 0.9745 D → e+, with Vcd = 0.2238 lattice lattice CLEO-c CLEO-c

  41. semileptonic B ℓ decays and |Vub| • no symmetry: only lattice QCD can fix the normalisation • lattice kinematic range is restricted to near zero recoil, high q2 experiment (2005): Status of Lattice QCD

  42. … and beyond? Status of Lattice QCD

  43. impact of lattice QCD on flavour physics • lattice QCD needs greater precision to be phenomenologically relevant • ICHEP 06: new physics has not shown up • Bs oscillations fully consistent with SM • flavour physics, including CP violation is governed by CKM (at least predominantly) Status of Lattice QCD

  44. muon g-2 • promising place to look for new physics • must compute SM contributions very accurately • leading order hadronic contribution ignored (small) Status of Lattice QCD • staggered χPT, a = 0.09 fm • lattice uncertainty ~ 3 × experimental

  45. rare and forbidden decays  - b b s s   t W • constraints can come from rare decays + + ... • and forbidden decays Status of Lattice QCD • but both involve QCD matrix elements

  46. B  K*  - b b s s + ...   + ... t W • occurs at 1 loop in SM • contribution from virtual sparticles • neglected in recent lattice QCD studies Status of Lattice QCD • must extrapolate to q2 = 0 where c(3) = 0 and T1(0) = T2(0)

  47. GUTs & SUSY • proton lifetime • SuperKamiokande (~100 kt y) • 1 kt = 1033 protons • colour triplet Higgsino exchange (dim 5)  antisymmetric in flavour Status of Lattice QCD • when dressed by sparticles gives proton decay  dominant decay mode is to strange mesons

  48. proton decay • dimension 6 baryon-number violating operators constrained by SM symmetries • matrix elements from lattice QCD provide model-independent input to SUSY-GUT lifetime estimates • related by chiral perturbation theory to • large uncertainty from lattice scale Status of Lattice QCD

  49. Status Status of Lattice QCD

  50. as a phenomenological / discovery tool • the theoretical control that has been established in principle must be turned into higher precision in practice • the determination of some CKM parameters is now limited by the precision of lattice QCD • operator mixing need be no worse than in the continuum, extending the range of matrix elements that can be computed reliably • some constraints on BSM physics are possible at existing levels of precision • by computing all SM matrix elements, eg for proton decay, B→K*γ • by bounding hadronic uncertainties in well-known parameters, eg muon g-2 • all the theoretical and computing technology required for this exists • there is greater confidence than for many years • beyond lattice QCD? • different representations/gauge groups, scalar fields (Higgs), massless fermions (SUSY) … • no obstacles in principle Status of Lattice QCD

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