Heavy Quarks on the lattice

Transcription

1 Intro Cont Latt Ren Struct NP HQET Res LPHA ACollaboration NIC, DESY, A Research Centre of the Helmholtz Association Lectures at HISS school, September 2011

2 Intro Cont Latt Ren Struct NP HQET Res Plan Introduction Naive HQ on the lattice Continuum HQET Action Propagator Symmetries Renormalizability Lattice HQET Action Propagator Renormalization and matching Perturbative matching Structure of the 1/M expansion Toy model HQET at order 1/m Non-perturbative HQET Tests Non-perturbative matching Large volume Some results [arxiv: ] and Les Houches school 2009 Oxford University Press look there for more references

3 Intro Cont Latt Ren Struct NP HQET Res Lattice Introduction: Particle Physics Observations (e, µ,... Z,... t, Lorenz invariance... ) + Principles (Unitarity, Causality, Renormalizability) + theory calculations including lattice QCD (spectrum, F π ) Standard Model of Particle Physics local Quantum Field Theory (gauge theory) QED + Salam-Weinberg + QCD + GR

4 Intro Cont Latt Ren Struct NP HQET Res Lattice Introduction: the successfull Standard Model QED + Salam Weinberg + QCD very constrained: 3 coupling constants enormous predictivity

5 Intro Cont Latt Ren Struct NP HQET Res Lattice Introduction: the successfull Standard Model QED + Salam Weinberg + QCD very constrained: 3 coupling constants enormous predictivity top mass from loops = top mass from Tevatron

6 Intro Cont Latt Ren Struct NP HQET Res Lattice Introduction: the successfull Standard Model QED + Salam Weinberg + QCD very constrained: 3 coupling constants + masses of elementary fields + CKM-matrix enormous predictivity top mass from loops = top mass from Tevatron

7 Intro Cont Latt Ren Struct NP HQET Res Lattice Introduction: the successfull Standard Model QED + Salam Weinberg + QCD very constrained: 3 coupling constants + masses of elementary fields + CKM-matrix enormous predictivity top mass from loops = top mass from Tevatron too successfull (all particle physics experiments match)

8 Intro Cont Latt Ren Struct NP HQET Res Lattice Introduction: the incomplete Standard Model But from other sources we know that there are missing pieces dark matter too little CP-violation for the observed matter / antimatter asymmetry There is an intense search for deviations from the Standard Model in particle physics experiments

9 Intro Cont Latt Ren Struct NP HQET Res Lattice _ Crab cavities will be installed and tested with beam in e+ 4.1 A The superconducting cavities will be upgraded to absorb more higher-order mode power up to 50 kw. The beam pipes and all vacuum components will be replaced with higher-current-proof design. e- 9.4 A * -1 The state-of-art ARES coppe cavities will be upgraded with higher energy storage ratio to support higher current. * x y * Two Frontiers to search for missing pieces High Energy Tevatron LHC ) 2 Events/(8 GeV/c Bkg Sub Data (4.3 fb Gaussian WW+WZ (a) ) High Intensity virtual (quantum) effects _ b B W W _ d _ B M jj 2 [GeV/c ] [CDF: arxiv: ] SuperKEKB *y = z = 3 mm d u,c,t b less tested interactions time L = ± 1+ y I ± ±y 2er e will reach cm -2 s -1. [Yutaka Ushiroda, May 2008 ] R L R y

10 Intro Cont Latt Ren Struct NP HQET Res Lattice High Intensity Frontier Less tested interactions: quark-flavour changing interactions L int =... g weak W µ + Ūγ µ(1 γ 5 )D... B-decays 0 D d 1 0 s A = V d 1 s A = V CKM D b b {z } {z } weak int. strong int. 0 V CKM V 1 ud V us V ub V cd V cs V cb A V td V ts V tb Confinement: V ij are not directly measurable. QCD matrix elements (or assumptions/approximations) are needed.

11 Intro Cont Latt Ren Struct NP HQET Res Lattice b to u transitions clean transitions: B = bū W lν 1. inclusive: B X u lν optical theorem + heavy quark expansion perturbatively calculable: (accuracy?) double expansion in α s (m b ) 0.2, Λ QCD /m b semileptonic: B πlν (three-body, form factor) 3. leptonic: B lν (decay constant)

12 Intro Cont Latt Ren Struct NP HQET Res Lattice b to u transitions V ub puzzle

13 Intro Cont Latt Ren Struct NP HQET Res Lattice b to u transitions V ub puzzle G. Isidori Quark flavour mixing with right-handed currents Euroflavour2010, Munich Motivation Exp. side: RH currents provide a natural solution to the V ub puzzle B(B π lν) V ub 2 B(B τν) V ub 2 B(B X u lν) V ub 2 Within SM B πlν B X u lν B τν V ub ε Re(Vub R /Vub L )

14 Intro Cont Latt Ren Struct NP HQET Res Lattice b to u transitions V ub puzzle G. Isidori Quark flavour mixing with right-handed currents Euroflavour2010, Munich Motivation Exp. side: RH currents provide a natural solution to the V ub puzzle B(B π lν) V ub 2 B(B τν) V ub 2 B(B X u lν) V ub 2 Lattice ME ME Within SM B πlν B X u lν B τν V ub ε Re(Vub R /Vub L )

15 Intro Cont Latt Ren Struct NP HQET Res Lattice b to u transitions V ub puzzle G. Isidori Quark flavour mixing with right-handed currents Euroflavour2010, Munich Motivation Exp. side: RH currents provide a natural solution to the V ub puzzle B(B π lν) V ub 2 B(B τν) V ub 2 B(B X u lν) V ub 2 Lattice ME ME Within SM B πlν B X u lν B τν V ub ε Re(Vub R /Vub L ) Precise & reliable lattice calculations are needed to check whether such puzzles are for real or others are there.

16 Intro Cont Latt Ren Struct NP HQET Res Lattice b to u transitions V ub puzzle G. Isidori Quark flavour mixing with right-handed currents Euroflavour2010, Munich Motivation Exp. side: RH currents provide a natural solution to the V ub puzzle B(B π lν) V ub 2 B(B τν) V ub 2 B(B X u lν) V ub 2 Lattice ME ME Within SM B πlν B X u lν B τν V ub ε Re(Vub R /Vub L ) Precise & reliable lattice calculations are needed to check whether such puzzles are for real or others are there. V ub is one example. Others such as B B oscillations....

17 Intro Cont Latt Ren Struct NP HQET Res Lattice The challenge of B-physics on the lattice multiple scale problem always difficult for a numerical treatment

18 Intro Cont Latt Ren Struct NP HQET Res Lattice The challenge of B-physics on the lattice multiple scale problem always difficult for a numerical treatment lattice cutoffs: Λ UV = a 1 Λ IR = L 1

19 Intro Cont Latt Ren Struct NP HQET Res Lattice The challenge of B-physics on the lattice multiple scale problem always difficult for a numerical treatment lattice cutoffs: Λ UV = a 1 Λ IR = L 1 L 1 m π,..., m D, m B a 1 O(e Lmπ ) m D a 1/2 L 4/m π 6 fm a 0.05 fm L/a 120

20 Intro Cont Latt Ren Struct NP HQET Res Lattice The challenge of B-physics on the lattice multiple scale problem always difficult for a numerical treatment lattice cutoffs: Λ UV = a 1 Λ IR = L 1 L 1 m π,..., m D, m B a 1 O(e Lmπ ) m D a 1/2 L 4/m π 6 fm a 0.05 fm L/a 120 beauty not yet accomodated: we ll discuss what to do

21 Intro Cont Latt Ren Struct NP HQET Res Lattice The challenge of B-physics on the lattice The issue is cutoff effects = discretization errors = lattice artefacts

22 Intro Cont Latt Ren Struct NP HQET Res Lattice The challenge of B-physics on the lattice The issue is cutoff effects = discretization errors = lattice artefacts Look at a simple one: Dispersion relation, free fermion, Wilson discretization

23 Intro Cont Latt Ren Struct NP HQET Res Lattice The challenge of B-physics on the lattice The issue is cutoff effects = discretization errors = lattice artefacts Look at a simple one: Dispersion relation, free fermion, Wilson discretization continuum: E 2 = m 2 + p 2

24 Intro Cont Latt Ren Struct NP HQET Res Lattice The challenge of B-physics on the lattice The issue is cutoff effects = discretization errors = lattice artefacts Look at a simple one: Dispersion relation, free fermion, Wilson discretization lattice E 2 = m 2 + p 2 + O(a 2 )

25 Intro Cont Latt Ren Struct NP HQET Res Lattice The challenge of B-physics on the lattice The issue is cutoff effects = discretization errors = lattice artefacts Look at a simple one: Dispersion relation, free fermion, Wilson discretization lattice E 2 = m 2 + p 2 + O(a 2 ) enhanced by m p

26 Intro Cont Latt Ren Struct NP HQET Res Lattice Dispersion relation Wilson fermion action [ µf (x) = 1 a (f (x) f (x aˆµ)) ] S lat = a 4 x = π/a π/a { ψ(x) m 0 + µ + µ γ µ a µ µ 2 2 d 4 p ψ(p) { m 0 + i p µ γ µ + a 2 ˆp2} ψ(p) } ψ(x) p µ = 1 a sin(p µa), ˆp 2 = µ (ˆp µ ) 2, ˆp µ = 2 a sin(ap µ 2 )

27 Intro Cont Latt Ren Struct NP HQET Res Lattice Dispersion relation Wilson fermion action [ µf (x) = 1 a (f (x) f (x aˆµ)) ] S lat = a 4 x = π/a π/a { ψ(x) m 0 + µ + µ γ µ a µ µ 2 2 d 4 p ψ(p) { m 0 + i p µ γ µ + a 2 ˆp2} ψ(p) } ψ(x) p µ = 1 a sin(p µa), ˆp 2 = µ (ˆp µ ) 2, ˆp µ = 2 a sin(ap µ 2 ) propagator G W (p) = (m 0 + a 2 ˆp2 ) i p µ γ µ (m 0 + a 2 ˆp2 ) 2 + p 2

28 Intro Cont Latt Ren Struct NP HQET Res Lattice Dispersion relation Wilson fermion action [ µf (x) = 1 a (f (x) f (x aˆµ)) ] S lat = a 4 x = π/a π/a { ψ(x) m 0 + µ + µ γ µ a µ µ 2 2 d 4 p ψ(p) { m 0 + i p µ γ µ + a 2 ˆp2} ψ(p) } ψ(x) p µ = 1 a sin(p µa), ˆp 2 = µ (ˆp µ ) 2, ˆp µ = 2 a sin(ap µ 2 ) propagator G W (p) = (m 0 + a 2 ˆp2 ) i p µ γ µ (m 0 + a 2 ˆp2 ) 2 + p 2 This determines E(p, m R, a)

29 Intro Cont Latt Ren Struct NP HQET Res Lattice Dispersion relation Transfer matrix representation for euclidean correlation functions,. : generally here (free theory) expect O(x 0 )O(0) x0 Be Ex0 G(x 0, p) = ψ α (x 0, p) ψ β (0, p) = B αβ e Ex0, ψα (x 0, p) = a 3 x ψ α (x)e ipx

30 Intro Cont Latt Ren Struct NP HQET Res Lattice Dispersion relation Transfer matrix representation for euclidean correlation functions,. : generally here (free theory) expect O(x 0 )O(0) x0 Be Ex0 G(x 0, p) = ψ α (x 0, p) ψ β (0, p) = B αβ e Ex0, ψα (x 0, p) = a 3 x ψ α (x)e ipx G W (x 0, p) = π/a π/a dp 0 e ip0x0 G W (p)

31 Intro Cont Latt Ren Struct NP HQET Res Lattice Dispersion relation Transfer matrix representation for euclidean correlation functions,. : generally here (free theory) expect O(x 0 )O(0) x0 Be Ex0 G(x 0, p) = ψ α (x 0, p) ψ β (0, p) = B αβ e Ex0, ψα (x 0, p) = a 3 x ψ α (x)e ipx G W (x 0, p) = π/a π/a dp 0 e ip0x0 G W (p) π/a i p 0 π/a

32 Intro Cont Latt Ren Struct NP HQET Res Lattice Dispersion relation i p 0 G W (x 0, p) = π/a π/a dp 0 e ip0x0 G W (p) E(p) from G W (p) 1 = 0 : Exercise π/a π/a Interpretation: 2 cosh(ea) = 1 + a2 p 2 1. Renormalization (in the free theory!) A + A, A = 1 + am 0 + a2 2 ˆp2 p = 0: e ae + e ae = A + 1 A E = 1 a log(1 + am 0) m R Now expand in a: 2 cosh(ea) = 2 + a 2 E a4 E

33 Intro Cont Latt Ren Struct NP HQET Res Lattice Dispersion relation log(1 + am 0 ) m R and expand in a: 2 cosh(ea) = 2 + a 2 E a4 E find: E 2 = p 2 + mr 2 ( 1 3 p m2 Rp pk) 4 a 2 + O(a 4 ) k

34 Intro Cont Latt Ren Struct NP HQET Res Lattice Dispersion relation log(1 + am 0 ) m R and expand in a: 2 cosh(ea) = 2 + a 2 E a4 E find: E 2 = p 2 + mr 2 ( 1 3 p m2 Rp pk) 4 a 2 + O(a 4 ) k cutoff effects are enhanced by large m R O(a 2 ) (in the free Wilson theory automatically) break O(3) symmetry (not H(3))

35 Intro Cont Latt Ren Struct NP HQET Res Lattice Dispersion relation Numerical example, relevant for B-physics: Wilson discretization p = 0.5GeV a = 1 2GeV GeV m R = 4GeV units: GeV

36 Intro Cont Latt Ren Struct NP HQET Res Lattice Dispersion relation Numerical example, relevant for B-physics: Wilson discretization p = 0.5GeV a = 1 2GeV GeV m R = 4GeV units: GeV asymptotics (a 2 -behavior needs am < 1/2) am = 1/2... 1/4 needed; therefore: charm: yes, beauty: no

37 Intro Cont Latt Ren Struct NP HQET Res Lattice Relativistic heavy quark action [El Khadra, Kronfeld, Mackenzie; Aoki, Kuramashi; Christ, Li, Lin ] S lat = a { 4 ψ(x) x ξ = ξ(am 0 ) p = 0.5GeV a = 1 2GeV GeV m R = 4GeV E 2 = p 2 + mr 2 ( 1 3 p k p4 k ) a2 +O(a 4 ) m γ 0 + ξ k + k γ k a µ µ 2 2 } ψ(x)

38 Intro Cont Latt Ren Struct NP HQET Res Lattice Relativistic heavy quark action [El Khadra, Kronfeld, Mackenzie; Aoki, Kuramashi; Christ, Li, Lin ] S lat = a { 4 ψ(x) x ξ = ξ(am 0 ) p = 0.5GeV a = 1 2GeV GeV m R = 4GeV E 2 = p 2 + mr 2 ( 1 3 p k p4 k ) a2 +O(a 4 ) m γ 0 + ξ k + k γ k a µ µ 2 2 } ψ(x)

39 Intro Cont Latt Ren Struct NP HQET Res Lattice Options to do B-physics on the lattice relativistic heavy quark actions extrapolations in the quark mass effective theories: expansions in Λ/m b Heavy Quark Effective Theory Nonrelativistic QCD

40 Intro Cont Latt Ren Struct NP HQET Res Lattice Options to do B-physics on the lattice relativistic heavy quark actions consistent beyond tree-level? at the non-perturbative level? extrapolations in the quark mass effective theories: expansions in Λ/m b Heavy Quark Effective Theory Nonrelativistic QCD

41 Intro Cont Latt Ren Struct NP HQET Res Lattice Options to do B-physics on the lattice relativistic heavy quark actions consistent beyond tree-level? at the non-perturbative level? extrapolations in the quark mass need continuum limit before the extrapolation effective theories: expansions in Λ/m b Heavy Quark Effective Theory Nonrelativistic QCD

42 Intro Cont Latt Ren Struct NP HQET Res Lattice Plan Introduction Naive HQ on the lattice Continuum HQET Action Propagator Symmetries Renormalizability Lattice HQET Action Propagator Renormalization and matching Perturbative matching Structure of the 1/M expansion Toy model HQET at order 1/m Non-perturbative HQET Tests Non-perturbative matching Large volume Some results [arxiv: ] and Les Houches school 2009 Oxford University Press look there for more references

43 Intro Cont Latt Ren Struct NP HQET Res Propagator On continuum HQET Want to describe hadrons with a single very heavy quark e.g. a B-meson like a hydrogen atom hydrogen atom : heavy proton + light electron B-meson : heavy b-quark + light anti-quark b-baryons : heavy b-quark + two light quarks... m b Rest-frame of B rest-frame of b (quark) antiquarks can t be created D k ψ = 0

44 Intro Cont Latt Ren Struct NP HQET Res Propagator On continuum HQET Dirac Lagrangian: ψ{d µ γ µ + m}ψ D k ψ=0 ψ{d 0 γ 0 + m}ψ = L stat h + L stat h }{{} anti-quark L stat h = ψ h (m + D 0 )ψ h, P + ψ h = ψ h, ψ h P + = ψ h, P ± = 1±γ0 2 Corrections by treating D k γ k perturbatively: Couple quark and anti-quark fields. D k ψ mψ

45 Intro Cont Latt Ren Struct NP HQET Res Propagator Deriving the form of the continuum HQET Lagrangian Decouple by Fouldy Wouthuysen-Tani (FTW) transformations ψ ψ = e S e S ψ, S = 1 2m D kγ k = S, S = 1 4m 2 γ 0γ k F k0 ψ ψ = ψe S e S... rename ψ ψ L = Lh stat + 1 2m L (1) h + L h stat + 1 2m L (1) + O( h 1 m ) 2 L (1) h = (O kin + O spin ), L (1) = (Ō h kin + Ō spin ), O kin (x) = ψ h (x) D 2 ψ h (x), O spin (x) = ψ h (x) σ B(x) ψ h (x), σ k = 1 2 ɛ ijkσ ij, B k = i 1 2 ɛ ijkf ij,

46 Intro Cont Latt Ren Struct NP HQET Res Propagator Comments on the Derivation It is classical: in a path integral D k ψ mψ is not satisfied. All momentum components are integrated over. Have not integrated out any components The derivation is order by order in 1/m We now have the classical effective Lagrangian. Its renormalization could need more terms. to be discussed.

47 Intro Cont Latt Ren Struct NP HQET Res Propagator Comments on the Derivation It is classical: in a path integral D k ψ mψ is not satisfied. All momentum components are integrated over. Have not integrated out any components The derivation is order by order in 1/m We now have the classical effective Lagrangian. Its renormalization could need more terms. to be discussed. Lagrangian for a b-hadron at rest. B lν, B πlν, B B,...: ok

48 Intro Cont Latt Ren Struct NP HQET Res Propagator The static quark propagator The propagator G h (x, y) satisfies Solution ( x0 + A 0 (x) + m)g h (x, y) = δ(x y) P + G h (x, y) = θ(x 0 y 0 ) exp( m (x 0 y 0 )) δ(x y) P + { x0 } P exp dz 0 A 0 (z 0, x) y 0 P: path ordering explicit solution (check it as an exercise) δ(x y): static

49 Intro Cont Latt Ren Struct NP HQET Res Propagator Mass dependence G h (x, y) = θ(x 0 y 0 ) exp( m (x 0 y 0 )) δ(x y) P + j Z x0 ff P exp dz 0 A 0 (z 0, x) y 0 explicit factor exp( m x 0 y 0 ) for any gauge field also after path integration over the gauge fields example: C h (x, y; m) = C h (x, y; 0) exp( m (x 0 y 0 )). C PP h (x, y; m) = ψ l (x)γ 5ψ h (x) ψ h (y)γ 5 ψ l (y), ψ l (x): a light-quark fermion field remove m from Lagrangian all energies are shifted by m L stat h L stat h = ψ h (D 0 + ɛ)ψ h = ψ h ( D 0 + ɛ)ψ h E QCD h/ h = E stat h/ h + m

50 Intro Cont Latt Ren Struct NP HQET Res Propagator Heavy Quark Symmetries 1. Flavor F heavy quarks symmetry ψ h ψ h = (ψ h1,..., ψ hf ) T L stat h = ψ h (D 0 + ɛ)ψ h. ψ h (x) V ψ h (x), ψ h (x) ψ h (x)v, V SU(F ) emerges as (F = 2) m b m c = c Λ QCD, or m b /m c = c, m b c or c fixed when taking m b

51 Intro Cont Latt Ren Struct NP HQET Res Propagator Heavy Quark Symmetries 2. Spin ψ h (x) e iα k σ k ψ h (x), ψ h (x) ψ h (x)e iα k σ k, in Dirac representation where ( ) 1 0 γ 0 =, P = σ k = 1 ( ) 2 ɛ σk 0 ijkσ ij, 0 σ k ( ) 1 0, P 0 0 = ( )

52 Intro Cont Latt Ren Struct NP HQET Res Propagator Heavy Quark Symmetries 3. Local Flavor-number ψ h (x) e iη(x) ψ h (x), ψ h (x) ψ h (x)e iη(x), is a symmetry for any local phase η(x). For every point x there is a corresponding Noether charge Q h (x) = ψ h (x)ψ h (x) [ = ψ h (x)γ 0 ψ h (x) ] 0 Q h (x) = 0 x Q h (x): local (heavy) Flavor number All heavy quark symmetries are broken at order 1/m. But it is essential to have them at the lowest order.

53 Intro Cont Latt Ren Struct NP HQET Res Propagator Summary first lecture B-physics is interesting for searching for deviations from the standard model m b a 1 impossible to realize for a while to come Expansion in Λ/m b 1/10 Lowest order Term m b scale is removed L stat h = ψ h (D 0 + ɛ)ψ h, P + ψ h = ψ h Symmetries SU(2) Flavor (for m b and m c ) spin symmetry local flavor number

54 Intro Cont Latt Ren Struct NP HQET Res Propagator Renormalizability of the static theory L stat h = ψ h (D 0 + ɛ)ψ h local Lagrangian with field O 1 (x) = ψ h (x)d 0 ψ h (x), [O 1 ] = 4 standard wisdom: renormalized by adding all local fields with [O j ] 4 O 2 (x) = ψ h (x)ψ h (x), [O 2 ] = 3 no other fields compatible with the symmetries complete renormalized Lagrangian L stat h = ψ h (D 0 + δm + ɛ)ψ h δm = (e 1 g0 2 + e 2g ) Λcut δm given for massless light quarks 1 O 1 (x) possible by choosing wave function renormalization Energies of any state are E QCD = E stat + δm + m = E h/ h h/ h δm=0 stat + m h/ h bare δm=0

55 Intro Cont Latt Ren Struct NP HQET Res Propagator Renormalizability of the static theory Note: none of this is proven (e.g. to all orders of PT), but worked out in PT so far NP tests (later)

56 Intro Cont Latt Ren Struct NP HQET Res Propagator Predictions (if charm is heavy enough) E QCD h/ h = E stat + δm + m f = E stat + mbare f δm=0 δm=0 h/ h h/ h Considering different levels (e.g. radial excitations or different angular momentum: Also predictions for decays: E b E b = E c E c + O(Λ QCD /m c ) B τ ν τ amplitude 0 ū(x)γ µ γ 5 b(x) B(p) = p µ e ipx F B p = 0 : 0 ū(0)γ 0 γ 5 b(0) B(p) = m B F B

57 Intro Cont Latt Ren Struct NP HQET Res Propagator Normalization of states, scaling of decay constants relativistic normalization of states p p rel = (2π) 3 2E(p) δ(p p ). The factor E(p) introduces a spurious mass-dependence. non-relativistic normalization is p p NR p p = 2 (2π) 3 δ(p p ) p rel = p E(p) p. To lowest order in 1/m b the FTW transformation is trivial: A HQET 0 (x) = A stat 0 (x) + O(1/m b ), A stat 0 (x) = ū(x)γ 0 γ 5 ψ h (x). 0 A stat 0 (0) B (p = 0) = Φ {z stat } mass independent Φ stat = m 1/2 B p 0 F B = m 1/2 B F B = m 1/2 D F D in the limit m b, m c ; rather doubtful for charm. also up to logarithmic corrections (see later)

58 Intro Cont Latt Ren Struct NP HQET Res Static action on the lattice no chiral symmetry for a static quark discretize à la Wilson (with r = 1) D 0 γ {( 0 + 0)γ 0 a 0 0 }, ( µψ(x) = 1 a [ψ(x) ψ(x aˆµ)], µψ(x) = 1 [ψ(x + aˆµ) ψ(x)]) a with P + ψ h = ψ h, P ψ h = ψ h, get lattice identities D 0 ψ h (x) = 0ψ h (x), D 0 ψ h(x) = 0 ψ h(x). convenient normalization factor L h = L h = aδm ψ h(x)[ 0 + δm]ψ h (x), aδm ψ h (x)[ 0 + δm]ψ h(x).

59 Intro Cont Latt Ren Struct NP HQET Res Static action on the lattice The following points are worth noting. Formally, this is just a one-dimensional Wilson fermion replicated for all space points x No doubler modes Positive hermitian transfer matrix for Wilson fermions can be taken over The choice of the backward derivative for the quark and the forward derivative for the anti-quark is selected by the Wilson term. Selects forward/backward propagation; an ɛ-prescription is not needed First written down by Eichten and Hill. Preserves all the continuum heavy quark symmetries

60 Intro Cont Latt Ren Struct NP HQET Res Propagator a δm ( 0 + δm)g h (x, y) = δ(x y)p + a 4 Y µ δ x µ a y µ a P +. Writing G h (x, y) = g(n 0, k 0 ; x)δ(x y)p +, x 0 = an 0, y 0 = ak 0 simple recursion for g(n 0 + 1, k 0 ; x) in terms of g(n 0, k 0 ; x) solution g(n 0, k 0 ; x) = θ(n 0 k 0 )(1 + aδm) (n 0 k 0 ) P(y, x; 0), P(x, x; 0) = 1, P(x, y + aˆ0; 0) = P(x, y; 0)U(y, 0), where θ(n 0 k 0 ) = ( 0 n 0 < k 0 1 n 0 k 0. G h (x, y) = θ(x 0 y 0 ) δ(x y) exp ` c δm (x 0 y 0 ) P(y, x; 0) P +, cδm = 1 ln(1 + aδm). a

61 Intro Cont Latt Ren Struct NP HQET Res Propagator G h (x, y) = θ(x 0 y 0 ) δ(x y) exp ` c δm (x 0 y 0 ) P(y, x; 0) P +, cδm = 1 ln(1 + aδm). a θ(0) = 1 for the lattice θ-function mass counter term δm just yields an energy shift on the lattice: E QCD h/ h = E stat + m h/ h bare, m bare = δm δm=0 c + m. the split between δm and the finite m is convention dependent

62 Intro Cont Latt Ren Struct NP HQET Res Symanzik analysis of cutoff effects I do not derive it here... result: automatic O(a) improvement of the action, energy levels O(a 2 ) cutoff effects O(a) improvement term for currents, eg. A stat 0 = ψ l γ 0 γ 5 ψ h (A stat R ) 0 = ZA stat 0 + aca stat 0) δa stat 0 ) irrespective of the light-quark action other (components) of currents, similarly

63 Intro Cont Latt Ren Struct NP HQET Res Numerical test of the renormalizability fa stat (x 0, θ) = a6 X D E (A stat I ) 0 (x) ζ 2 h (y)γ 5 ζ l (z) y,z : x 0=0 T f stat 1 (θ) = a12 2L 6 X u,v,y,z D E ζ l (u)γ 5 ζ h (v) ζ h (y)γ 5 ζ l (z) : x 0=0 T f hh 1 (x 3, θ) = a8 2L 2 X x 1,x 2,y,z ζ h (x)γ 5 ζ h (0) ζ h (y)γ 5 ζ h(z) : x 0=0 T In a Schrödinger functional. Double lines are static quark propagators. θ angle in spatial BC s. Renormalization according to standard wisdom [ f stat A ]R = Z A stat Z ζh Z ζ fa stat, [ f stat 1 ]R = Z ζ 2 h Zζ 2 f1 stat, [ f hh 1 ]R = Z ζ 4 h f1 hh.

64 Intro Cont Latt Ren Struct NP HQET Res Numerical test of the renormalizability [Della Morte, Shindler, S., 2005 ] ξ A (θ, θ ) = f A stat (T /2, θ) fa stat(t /2, θ ), ξ 1 (θ, θ ) = f 1 stat (θ) f1 stat (θ ), h(d/l, θ) = f 1 hh (d, θ) f1 hh (L/2, θ).

65 Intro Cont Latt Ren Struct NP HQET Res match Beyond the classical theory: Renormalization and Matching a matrix element of A 0 : QCD Z A f A 0 (x) i QCD Φ QCD (m) HQET in static approx. Z stat A (µ) f Astat m: mass of heavy quark (b) in some definition (all other masses zero for simplicity) µ: arbitrary renormalization scale matching (equivalence): 0 (x) i stat Φ(µ) Φ QCD (m) = C match (m, µ) Φ(µ) + O(1/m) C match (m, µ) = 1 + c 1 (m/µ) }{{} γ 0 log(µ/m)+const. ḡ 2 (µ) +... M, Λ, Φ RGI : Renormalization Group Invariants =

66 Intro Cont Latt Ren Struct NP HQET Res match Beyond the classical theory: Renormalization and Matching a matrix element of A 0 : QCD Z A f A 0 (x) i QCD Φ QCD (m) HQET in static approx. Z stat A (µ) f Astat m: mass of heavy quark (b) in some definition (all other masses zero for simplicity) µ: arbitrary renormalization scale matching (equivalence): 0 (x) i stat Φ(µ) Φ QCD (m) = C match (m, µ) Φ(µ) + O(1/m) C match (m, µ) = 1 + c 1 (m/µ) }{{} γ 0 log(µ/m)+const. ḡ 2 (µ) +... = C PS (M/Λ) Φ RGI M, Λ, Φ RGI : Renormalization Group Invariants

67 Intro Cont Latt Ren Struct NP HQET Res Toy model The nature of the expansion QFT divergencies, log(m) Strongly interacting: non-perturbative in α perturbative in 1/M A toy model with the essential features Φ QCD (β) = C(M/Λ) {z } Φ 0 (β) + c 1 M Φ 1(β) + c 2 M 2 Φ 2(β) +... Φ QCD : a (renormalized) observable energy level decay constant (F B mb ) β: Quantum number (e.g. pseudo-scalar vector) M: the mass of the quark (RGI)

68 Intro Cont Latt Ren Struct NP HQET Res Toy model The nature of the expansion QFT divergencies, log(m) Strongly interacting: non-perturbative in α perturbative in 1/M A toy model with the essential features Φ QCD (β) = C(M/Λ) Φ 0 (β) + c 1 {z } M Φ 1(β) + c 2 M 2 Φ 2(β) +... (log(m/λ)) r 1+k[log(M/Λ)] 1 Φ QCD : a (renormalized) observable energy level decay constant (F B mb ) β: Quantum number (e.g. pseudo-scalar vector) M: the mass of the quark (RGI) C(M/Λ) not constant; not a naive expansion! as just discussed: from renormalization of HQET and matching to QCD [Eichten, Hill ] from QCD mass dependence at large masses [Shifmann, Voloshin; Politzer, Wise ]

69 Intro Cont Latt Ren Struct NP HQET Res Toy model The nature of the expansion Φ QCD (β) = (log(m/λ)) r Φ 0 (β) 1 + k[log(m/λ)] 1 +Φ 1 (β) c 1 M + Φ 2(β) c 2 M {z } C(M/Λ) [log(m/λ)] 1 α(m) 2 C(M/Λ) = (log(m/λ)) r 41 + X l 1( k) l (log(m/λ)) l 5 2 = α(m) r 41 + X l 1( k) l α(m) l simplified: summable, finite radius of convergence ( not in real QCD) neglected log-corrections and mixing in M n terms r given by anomalous dimension in HQET (γ 0 )

70 Intro Cont Latt Ren Struct NP HQET Res Toy model The standard approach [log(m/λ)] 1 α(m) 1 matching (computation of the coefficients) can be done in perturbation theory Wilson coefficients can be computed in perturbation theory in our model this means C(M/Λ) = α(m) r "1 + error : # LX ( k) l α(m) l ± [C(M/Λ)] l=1 = (log(m/λ)) r "1 + [C(M/Λ)] C(M/Λ) # LX ( k) l (log(m/λ)) l ± [C(M/Λ)] l=1 = O α(m) L+1 fine for leading term in 1/M if pertubration theory is well behaved but including M 1 -term is theoretically ill defined log(m/λ) L 1 Λ/M for M

71 Intro Cont Latt Ren Struct NP HQET Res Toy model The standard approach [log(m/λ)] 1 α(m) 1 matching (computation of the coefficients) can be done in perturbation theory Wilson coefficients can be computed in perturbation theory in our model this means C(M/Λ) = α(m) r "1 + error : # LX ( k) l α(m) l ± [C(M/Λ)] l=1 = (log(m/λ)) r "1 + [C(M/Λ)] C(M/Λ) # LX ( k) l (log(m/λ)) l ± [C(M/Λ)] l=1 = O α(m) L+1 fine for leading term in 1/M if pertubration theory is well behaved but including M 1 -term is theoretically ill defined log(m/λ) L 1 Λ/M for M Need C(M/Λ), c 1, c 2 non-perturbatively

72 Intro Cont Latt Ren Struct NP HQET Res Toy model The standard approach [log(m/λ)] 1 α(m) 1 matching (computation of the coefficients) can be done in perturbation theory Wilson coefficients can be computed in perturbation theory in our model this means C(M/Λ) = α(m) r "1 + error : # LX ( k) l α(m) l ± [C(M/Λ)] l=1 = (log(m/λ)) r "1 + [C(M/Λ)] C(M/Λ) # LX ( k) l (log(m/λ)) l ± [C(M/Λ)] l=1 = O α(m) L+1 fine for leading term in 1/M if pertubration theory is well behaved but including M 1 -term is theoretically ill defined log(m/λ) L 1 Λ/M for M Need C(M/Λ), c 1, c 2 non-perturbatively Also matching QCD HQET on the lattice

73 Intro Cont Latt Ren Struct NP HQET Res Toy model Including 1/m b corrections ( O(1/mb 2 ) dropped without notice) NRQCD path integral weight: L (1) h (x) = (ω kin O kin (x) + ω spin O spin (x)). W NRQCD exp( a 4 X x [L light (x) + Lh stat (x) + L (1) h (x)]) non-renormalizable ([O kin ] = [O spin ] = 5), no continuum limit the real trouble is not the effective theory, but that at the same time we want non-perturbative results: not a finite number of loops HQET path integral weight: W HQET exp( a 4 X x [L light (x) + Lh stat (x)]) ( 1 a 4 X x L (1) h (x) ) part of the definition of HQET

74 Intro Cont Latt Ren Struct NP HQET Res Toy model Expanding correlation functions. W HQET exp( a X 4 x ( 1 + a 4 X x [L light (x) + Lh stat (x)]) (ω kin O kin (x) + ω spin O spin (x)) ) This yields O = O stat + ω kin a 4 X x OO kin (x) stat + ω spin a 4 X x OO spin (x) stat O stat + ω kin O kin + ω spin O spin, with O stat = 1 Z Z fields O exp( a 4 X x [L light (x) + Lh stat (x)])

75 Intro Cont Latt Ren Struct NP HQET Res Toy model Expanding correlation functions. O = O stat + ω kin O kin + ω spin O spin, one more point: also fields in correlation functions need to be expanded: O QCD = A 0 (x)a 0 (0) A 0 (x) A HQET 0 (x) = Z HQET A [A stat 0 (x) + 2X i=1 c (i) A A(i) 0 (x)], A (1) 0 (x) = ψ l (x) 1 2 γ 5γ i ( s i s i )ψ h (x), A (2) 0 (x) = e i A stat i, c (i) A symmetric derivatives: = O(1/m) [A(i) 0 (x)] = 4 e i = 1 2 ( i + i ), s i = 1 2 ( i + i ), s i = 1 2 ( i + i ).

76 Intro Cont Latt Ren Struct NP HQET Res Toy model Expanding correlation functions. Example its HQET expansion with C QCD AA,R (x 0) = Z 2 A a3 X x DA 0 (x)a 0 (0) E QCD h C QCD AA (x 0) = e m x 0 (Z HQET A ) 2 CAA stat (x 0) + c (1) A C δaa stat (x 0) i + ω kin CAA kin (x 0) + ω spin C spin AA (x 0) h e mx 0 (Z HQET A ) 2 CAA stat (x 0) 1 + c (1) A Rstat δa (x 0) i + ω kin RAA kin (x 0) + ω spin R spin AA (x 0) C stat δaa (x 0) = a 3 X x C kin AA (x 0) = a 3 X x C spin AA (x 0) = a 3 X x A stat 0 (x)(a (1) 0 (0)) stat + a 3 X x A stat 0 (x)(a stat 0 (0)) kin A stat 0 (x)(a stat 0 (0)) spin. A (1) 0 (x)(astat 0 (0)) stat,

77 Intro Cont Latt Ren Struct NP HQET Res Toy model Expanding correlation functions. h C QCD AA (x 0) = e m x 0 (Z HQET A ) 2 CAA stat (x 0) + c (1) A C δaa stat (x 0) i + ω kin CAA kin (x 0) + ω spin C spin AA (x 0) parameters in the effective theory (ω 1,... ω 5 ) = (m bare = m + δm, ln(z HQET A ), c (1) A, ω kin, ω spin ) ω i = ω i (g 0, am b ) bare parameters renormalization all terms needed for the renormalization of the correlation functions with insertions of O kin, O spin are present in the expression ω i are the necessary free coefficients keep M b fixed change g 0 0, a 0: all divergences (logarithmic and power) absorbed in ω i a more detailed explanation in [R.S., arxiv: ] matching finite parts of the ω i by matching to QCD (more precisiely later) Φ HQET i ({ω i }) = Φ QCD i (M b )

78 Intro Cont Latt Ren Struct NP HQET Res Toy model Expansion of energies... m B = lim e 0 ln C QCD x 0 AA (x 0) = m bare lim e x 0 0ˆ ln C stat AA (x 0) + c (1) A Rstat δa (x 0) + + ω kin R kin AA (x 0) + ω spin R spin AA (x 0) δm=0 = m bare + E stat + ω kin E kin + ω spin E spin,

79 Intro Cont Latt Ren Struct NP HQET Res Toy model Expansion of energies... m B = lim e 0 ln C QCD x 0 AA (x 0) = m bare lim e x 0 0ˆ ln C stat AA (x 0) + c (1) A Rstat δa (x 0) + + ω kin R kin AA (x 0) + ω spin R spin AA (x 0) δm=0 = m bare + E stat + ω kin E kin + ω spin E spin, E stat = lim e 0 ln C stat x 0 AA (x 0), δm=0 E kin = lim e 0 R kin x 0 AA (x 0) = 1 2L 3 B a3 X z O kin (0, z) B stat = 1 2 B O kin(0) B stat E spin = lim e 0 R spin x 0 AA (x 0) = 1 2 B O spin(0) B stat, 0 = lim e 0 R stat x 0 δa (x 0).

80 Intro Cont Latt Ren Struct NP HQET Res Toy model... and matrix elements F B mb = lim x 0 2 exp(mb x 0 ) C QCD AA (x 0) 1/2 = Z HQET A Φ stat lim x x 0 ˆωkin E kin + ω spin E spin c(1) A Rstat δa (x 0) ω kinraa kin 0) ω spinr spin AA (x 0), Φ stat = lim 2 exp(e stat x 0 ) C stat x 0 AA (x 0) 1/2.

81 Intro Cont Latt Ren Struct NP HQET Res Toy model... and matrix elements F B mb = lim x 0 2 exp(mb x 0 ) C QCD AA (x 0) 1/2 = Z HQET A Φ stat lim x x 0 ˆωkin E kin + ω spin E spin c(1) A Rstat δa (x 0) ω kinraa kin 0) ω spinr spin AA (x 0), Φ stat = lim 2 exp(e stat x 0 ) C stat x 0 AA (x 0) 1/2. always a strict expansion (c (1) A, ω kin, ω spin ) = O(1/m b ) Z HQET A ω kin ZA stat ω kin (Z stat A + Z (1/m) A ) ω kin

82 Intro Cont Latt Ren Struct NP HQET Res Tests of HQET [Fritzsch, Jüttner, Heitger, S., Wennekers ] Example: SF boundary-to-boundary correlators ( ( R 1 = 1 f1 (θ 1 )k 1 (θ 1 ) 3 )) 4 ln f 1 (θ 2 )k 1 (θ 2 ) 3 z = LM 1 R (θ1, θ2) = (0, 1) (θ1, θ2) = ( 1 2, 1) (θ1, θ2) = (0, 1 2 ) 0.2 spin averaged /z

83 Intro Cont Latt Ren Struct NP HQET Res Tests of HQET [Fritzsch, Jüttner, Heitger, S., Wennekers ] Example: SF boundary-to-boundary correlators R 1 = 3 ( ) 4 ln f1 ω spin k θ =0.0 θ =0.5 θ =1.0 z = LM R 1 spin symmetry violating /z

84 Intro Cont Latt Ren Struct NP HQET Res LPHA Non-perturbative determination of parameters [ ACollaboration, ] static parameters parameters at first order ω stat = ( m stat bare, [ln(z A)] stat ) t, N HQET = 2 ω HQET = ( m bare, ln(z HQET A ), c (1) A, ω kin, ω spin ) t N HQET = 5 ω (1/m) = ω HQET ω stat matching: L fm Why L 1 = 0.5fm? Φ i (L 1, M, a) = Φ QCD i (L 1, M, 0), i = 1... N HQET. a = fm is accessible b-quark can be simulated, continuum limit can be taken

85 Intro Cont Latt Ren Struct NP HQET Res LPHA Non-perturbative determination of parameters [ ACollaboration, ] L 1 = 0.5fm: a = fm: b-quark can be simulated, continuum limit can be taken Φ QCD x 10 3 Φ QCD x 10 3 (a/l 1 ) 2 (a/l 1 ) 2 Φ 1 = L m B (L 1 ) = O(z) Φ 2 = ln(l 3/2 1 [F B mb ](L 1 ) = O(z 0 ) three different z

86 Intro Cont Latt Ren Struct NP HQET Res LPHA Non-perturbative determination of parameters [ ACollaboration, ] L 1 = 0.5fm: a = fm: b-quark can be simulated, continuum limit can be taken Φ QCD x 10 3 (a/l 1 ) 2 R stat (a/l 1 ) 2 Φ 4 = O(1) = R1 stat + ω kin Φ (1/m) 4 three different z three different θ

87 Intro Cont Latt Ren Struct NP HQET Res LPHA Non-perturbative determination of parameters [ ACollaboration, ] natural: Φ i (L 1, M, a) = Φ QCD i (L 1, M, 0), i = 1... N HQET. Φ 1 = LΓ P L e 0 ln( f A (x 0 )) x0 =L/2 L LmB Φ 2 = ln(z A f A f1 ) L ln(l 3/2 F B p mb /2), HQET expansion in general Φ 1 = L [m bare + Γ stat ] + O(1/m b ) Φ 2 = ln(za stat A + O(1/m b ) Φ(L, M, a) = η(l, a) + φ(l, a) ω(m, a) η Γstat ζ A A, φ L A......

88 Intro Cont Latt Ren Struct NP HQET Res Full strategy to determine ω(m b, a), a = 0.05fm fm (in QCD HQET L 1 L 1 L 2 L 2 L ω ω a match SSF S 1 S 2 S 3 S 4 S 5 the realistic implementation Rainer finer resolutions Sommer are Heavy used) Quarks on the lattice

89 Intro Cont Latt Ren Struct NP HQET Res a Full strategy (1-2a) QCD HQET L1 L1 L2 L2 L ω ω match SSF S1 S2 S3 S4 S5 (1) continuum limit in QCD a = fm fm Φ QCD i (L 1, M, 0) = lim a/l 1 0 ΦQCD i (L 1, M, a), (2a) HQET parameters a = 0.05 fm fm ω(m, a) φ 1 (L 1, a) [Φ(L 1, M, 0) η(l 1, a)] 0 L 1 1 Φ 1 1(L 1, M, 0) Γ stat (L 1, a) Φ 2 (L 1, M, 0) ζ A (L 1, a) A... L 1 /a 1, am b irrelevant

90 Intro Cont Latt Ren Struct NP HQET Res a Full strategy QCD HQET L1 L1 L2 L2 L ω ω match SSF S1 S2 S3 S4 S5 (2b) step scaling to L 2 = 2L 1 a = 0.05 fm fm Insert ω into Φ(L 2, M, a) Φ(L 2, M, 0) = lim 2, a) + φ(l 2, a) ω(m, a)} a/l 2 0 = 0 L 2 Γ stat (L 2, a) + L 2 Φ L 1 (L 1, M, 0) L 2 Γ stat 1 (L 1, a) 1 ζ A (L 2, a) + Φ 2 (L 1, M, 0) ζ A (L 1, a) A a/l = 0 L 2[Γ stat (L 2, a) Γ stat 1 0 L (L 1, a)] 2 1 Φ L 1 (L 1, M, 0) ζ A (L 2, a) ζ A (L 1, a) A 1 Φ 2 (L 1, M, 0) A a/l {z } {z }. finite HQET SSF s QCD, mass dependence

91 Intro Cont Latt Ren Struct NP HQET Res a Full strategy QCD HQET L1 L1 L2 L2 L ω ω match SSF S1 S2 S3 S4 S5 (3.) Repeat (2a.) for L 1 L 2 : ω(m, a) φ 1 (L 2, a) [Φ(L 2, M, 0) η(l 2, a)]. With same resolutions L 2 /a = now a = 0.1 fm fm (4.) insert ω into the expansion of large volume observables, e.g. m B = ω 1 + E stat from above: m B = m B (M b ) determine M b

92 Intro Cont Latt Ren Struct NP HQET Res a Full strategy QCD HQET L1 L1 L2 L2 L ω ω match SSF S1 S2 S3 S4 S5 m B = m B (M b ) determine M b explicitly in static approximation m B = lim [E stat Γ stat (L 2, a)] a =0.1fm fm [S 4, S 5 ] a 0 + lim a 0 [Γ stat (L 2, a) Γ stat (L 1, a)] a =0.05fm fm [S 2, S 3 ] + 1 L 1 lim a 0 Φ 1 (L 1, M b, a) a =0.025 fm fm [S 1 ].

93 Intro Cont Latt Ren Struct NP HQET Res Large volume statistical and systematic precision statistical [Della Morte, Shindler, S., 2005; Hasenfratz & Knechtli, 2001 ] self energy: signal/noise becomes worse as a 0 change action: HYP-smearing

94 Intro Cont Latt Ren Struct NP HQET Res Large volume statistical and systematic precision statistical [Della Morte, Shindler, S., 2005; Hasenfratz & Knechtli, 2001 ] self energy: signal/noise becomes worse as a 0 change action: HYP-smearing systematic F B mb = lim 2 exp(mb x 0 ) C QCD x 0 AA (x 0) 1/2 excited state contaminations typically non-negligible at a 0.7fm a common problem, e.g. nucleon structure the GEVP helps: an operator ˆQ eff n can be constructed such that (rigorous [Blossier, Della Morte, von Hippel, Mendes & S., 2009 ]) ) 0 ˆQ eff n e Ĥt ˆPe Ĥt ( ˆQ eff n ) 0 = Q eff n (2t)P(t)(Q eff n (0)) = n ˆP n + O(e [E N+1 E n] t 0 ) e.g. t = t 0 + a

95 Intro Cont Latt Ren Struct NP HQET Res Examples of results: M b [Blossier, Della Morte, Garron, Mendes, Papinutto, Simma, S. ] static approximation m B = 19 lim [E stat Γ stat (L 2, a)] a =0.1fm fm [S 4, S 5 ] a 0 + lim a 0 [Γ stat (L 2, a) Γ stat (L 1, a)] a =0.05fm fm [S 2, S 3 ] + 1 L 1 lim a 0 Φ 1 (L 1, M b, a) a =0.025 fm fm [S 1 ] exp L 2 m B - L2 (E stat stat -Γ (L1 1 )) - σ m 2 Φ 2 (L 1, M) L 1 M b stat z

96 Intro Cont Latt Ren Struct NP HQET Res Examples of results: M b LO (static) NLO (static + O(1/m)) (θ 1, θ 2 ) = (0, 0.5) (θ 1, θ 2 ) = (0.5, 1) (θ 1, θ 2 ) = (0, 1) θ 0 = ± ± ± ± 0.2 θ 0 = ± ± ± ± 0.2 θ 0 = ± ± ± ± 0.3 Table: Dimensionless b-quark mass, r 0 M b, obtained from the B s meson mass, for different values of θ i. small 1/m b corrections weak dependence on matching conditions

97 Intro Cont Latt Ren Struct NP HQET Res Examples of results: quenched F Bs mbs r 3/2 0 Φ HYP1 HYP2 static limit Φ RGI with 1/m b : Φ HQET = F Bs mbs /C PS a 2 [fm 2 ]

98 Intro Cont Latt Ren Struct NP HQET Res Examples of results: quenched F Bs mbs LO (static) NLO (static + O(1/m)) (θ 1, θ 2 ) = (0, 0.5) (θ 1, θ 2 ) = (0.5, 1) (θ 1, θ 2 ) = (0, 1) θ 0 = ± ± ± ± 9 θ 0 = ± ± ± ± 9 θ 0 = ± ± ± ± 8 Table: Pseudo-scalar heavy-light decay constant f Bs values of θ i. in MeV, for different small 1/m b corrections weak dependence on matching conditions

99 Intro Cont Latt Ren Struct NP HQET Res Comparison to relativistic (not so) heavy quarks r 0 3/2 φ RGI r 0 3/2 φbs HQET / CPS r 0 3/2 fps m PS 1/2 / CPS /(r 0 m PS ) surprisingly consistent picture C PS inserted from perturbation theory (unclear theoretical status)

100 Intro Cont Latt Ren Struct NP HQET Res Examples of results: quenched level splittings r0 En, HYP1 HYP a 2 [fm 2 ] 3s 1s splitting static 2s 1s splitting static + 1/m b 2s 1s splitting static Static results for splittings are in agreement with [T. Burch et al. ] Also ratio of ground state / excited state decay constant

101 Intro Cont Latt Ren Struct NP HQET Res Example of a results for N f = 2 (Lattice 2011) [ LPHA ACollaboration ] 0.24 A: a=0.08fm E,F: a=0.07fm N,O: a=0.05fm 0.22 fb/gev A4 E5 F7 O7 A5 F6 N m 2 π/gev 2 F B = F B m 2 π = ! 1 + 3gB 2 B π 16π 2 Fπ 2 mπ 2 log(mπ/f 2 π) 2 + b mπ 2 + ca 2

102 Intro Cont Latt Ren Struct NP HQET Res Example of a results for N f = 2 (Lattice 2011) [ LPHA ACollaboration ] 0.24 A: a=0.08fm E,F: a=0.07fm N,O: a=0.05fm 0.22 fb/gev A4 E5 F7 O7 A5 F6 N m 2 π/gev 2 F B = F B m 2 π = V ub becomes slightly more puzzling! 1 + 3gB 2 B π 16π 2 Fπ 2 mπ 2 log(mπ/f 2 π) 2 + b mπ 2 + ca 2

103 Intro Cont Latt Ren Struct NP HQET Res We are at the beginning of applications Lots of work remaining to be done... and phenomenology waiting to be explored