1. International School of Subnuclear Physics Erice 2006 Status of Lattice QCD Richard Kenway

2. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 2 Parameters of QCD

3. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 3 lattice QCD  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  Euclidean space-time lattice regularisation –lattice spacing a, lattice size L  Monte Carlo approximation to path integral –N gauge configurations Lattice QCD q(x) U  (x) U (x) a

4. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 4 lattice QCD  lattice spacing must be extrapolated to zero keeping box large enough –by approaching a critical point a L quark masses + gauge coupling Lattice QCD properties of hadrons  think of the computer as a ‘black box’

5. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 5 QCD scale  simulations use dimensionless variables (lattice spacing = 1) –quark masses, m f, and gauge coupling, g 2, are varied  hadronic scheme –at each value of g 2, fix quark masses m f by matching N f hadron mass ratios to experiment –one dimensionful quantity fixes the lattice spacing in physical units –dimensionless ratios become independent of g 2 if a is small enough (scaling)

6. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 6 renormalisation  convert matrix elements to a perturbative scheme (matching) –eg to combine with Wilson coefficients in an OPE  impose mass-independent renormalisation conditions at p 2 =  2  or use step scaling –let   1 = L, the linear box size –consider a sequence of intermediate renormalisations at box sizes L n = 2  n L 0

7. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 7 β function  continuum limit: a → 0 with L constant and large enough –tune β = 6/g 2 → ∞ holding low-energy physics constant  non-perturbative β function

8. International School of Subnuclear Physics Erice 2006 Richard Kenway 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 m f, 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

9. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 9 non-perturbative running  compute α(μ) at m q (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

10. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 10 strong coupling  lattice QCD provides a precise determination

11. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 11 quark masses  high precision is being achieved for light quarks –but there are systematic differences between lattice formulations staggered Wilson

12. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 12 fermion doubling  the covariant derivative as a difference operator  (naïve) free fermion Dirac operator in momentum space  16 (= 2 d ) 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. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 13 ‘good’ lattice fermions  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 L s   limit of 5-dimensional domain wall fermions  explicitly break chiral symmetry by adding a dimension 5 operator –gives doublers masses  cut-off, leaving p = 0 pole unchanged –mixes operators of different chirality, complicating renormalisation –requires fine tuning to get m ud = 0 (at M π = 0) (Wilson)

14. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 14 ‘ugly’ lattice fermions  staggered fermions –lattice action may be diagonalised in spinor space –keep only one spinor component  1 =  –2 d/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

15. International School of Subnuclear Physics Erice 2006 Richard Kenway 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

16. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 16 QCDOC > 6 teraflops sustained (BNL) > 3 teraflops (Edinburgh)

17. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 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 M N = 900 (100) MeV Hamber & Parisi 1982

18. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 18 … and dynamical sea quark effects are seen  string breaking

19. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 19  topological susceptibility –χPT with experimental f π QCD vacuum  isosurfaces of positive (red) and negative (green) topological charge density using

20. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 20  lattice relates d n to θ –simulations must sample topology well and contain light dynamical quarks with correct chiral symmetry –in quenched QCD d n 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 vacuum angle θ  QCD allows a gauge invariant CP odd term –CKM phase contributes < 10 −30 e cm to d n

21. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 21 effective theories Lüscher finite volume effective theory a <<  QCD -1 Symanzik effective field theory HQET/NRQCD chiral perturbation theory lattice QCD a, m q, m Q, L QCD scale  QCD lattice QCD a, m q, m Q, L QCD scale  QCD L >>  QCD -1 m Q >>  QCD m q <<  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

22. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 22 Status

23. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 23  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 as a theoretical tool

24. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 24 Parameters of the Standard Model

25. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 25  2-point functions easily computed quantities decays asymptotically with energy of lightest state created by O determines matrix elements such as  3-point functions 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. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 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 N f = 2)  L ~ 2.5 fm needed for FSE below few % for 300 MeV pions

27. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 27 quenched hadron spectrum  ‘tour de force’ demonstration of the power of lattice QCD –glueballs –nucleon excited states –mixing with flavour-singlet mesons is a major challenge for 2+1 flavours –requires flavour symmetry and spatially extended operators

28. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 28 QCD hadron spectrum  prediction of the B c mass –2+1 flavours + relativistic effective action (c) + NRQCD (b)  inputs to quark mass and scale setting  Edinburgh plot –2+1 flavours DWF

29. International School of Subnuclear Physics Erice 2006 Richard Kenway 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

30. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 30 leptonic decays  elegant example of lattice ↔ experiment interaction  access to V xy  cross-check of f X prediction qxqx qxqx ℓ ℓ

31. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 31 π and K leptonic decays  2+1 flavours staggered (MILC) –full χPT analysis

32. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 32 D leptonic decays  CLEO-c (2005) measured D → μν V cd, τ D from PDG 04  BaBar and CLEO-c (2006) measured D s → μν  experimental and lattice uncertainties are similar ~ 10%  sea quark effects are not significant

33. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 33 B leptonic decays  sensitive to charged Higgs b u  W H-H- b u   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 f Bs by 10-15%  first direct measurement of f B (Belle 2006) V ub, τ B from PDG 04

34. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 34 neutral K mixing and K   CP violation in K   indirect CP violation

35. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 35 indirect CP violation  quenched QCD  2+1 flavour –a ~ 0.125 fm (RBC-UKQCD, preliminary)  next year should see the first realistic determinations of B K statquench

36. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 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

37. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 37 B d and B s mixing  measurement of ΔM Bs allows a theoretically well-controlled estimate using  neutral B q meson mass difference –BSM physics could enter loops

38. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 38 semileptonic decays  access to V xy  form factor embeds q 2 dependence –more elaborate example of lattice ↔ experiment interaction  CKM-independent checks of lattice QCD from studying e+e+ W+W+ qxqx qxqx qxqx qyqy V xy

39. International School of Subnuclear Physics Erice 2006 Richard Kenway 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 |V us |

40. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 40 semileptonic D   / Kℓ decays  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 –|V cs | is well measured –precision test of lattice form factors against CLEO-c data  2+1 flavours cs, d W leptons D →  e +, with V cd = 0.2238 D→Ke +, with V cs = 0.9745 lattice CLEO-c lattice CLEO-c

41. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 41 semileptonic B  ℓ decays and |V ub |  no symmetry: only lattice QCD can fix the normalisation –lattice kinematic range is restricted to near zero recoil, high q 2 experiment (2005):

42. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 42 … and beyond?

43. International School of Subnuclear Physics Erice 2006 Richard Kenway 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 –B s oscillations fully consistent with SM –flavour physics, including CP violation is governed by CKM (at least predominantly)

44. International School of Subnuclear Physics Erice 2006 Richard Kenway 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  staggered χPT, a = 0.09 fm –lattice uncertainty ~ 3 × experimental ignored (small)

45. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 45 rare and forbidden decays  constraints can come from rare decays  but both involve QCD matrix elements  and forbidden decays bs   - - bs  t W ++...

46. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 46 B  K*   occurs at 1 loop in SM  contribution from virtual sparticles  neglected in recent lattice QCD studies –must extrapolate to q 2 = 0 where c (3) = 0 and T 1 (0) = T 2 (0) bs   - - +... bs  t W

47. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 47 GUTs & SUSY  proton lifetime –SuperKamiokande (~100 kt y) –1 kt = 10 33 protons  colour triplet Higgsino exchange (dim 5)  when dressed by sparticles gives proton decay  antisymmetric in flavour  dominant decay mode is to strange mesons

48. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 48  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 proton decay

49. International School of Subnuclear Physics Erice 2006 Richard Kenway Status of Lattice QCD 49 Status

50. International School of Subnuclear Physics Erice 2006 Richard Kenway 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