Aspects of HQET on the lattice

Transcription

1 Japan Wochen an Deutschen Hochschulen, May 2011, Bergische Universität, Wuppertal

2 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

3 Introduction: the successfull Standard Model QED + Salam Weinberg + QCD very constrained: 3 coupling constants enormous predictivity

4 Introduction: the successfull Standard Model QED + Salam Weinberg + QCD very constrained: 3 coupling constants enormous predictivity top mass from loops = top mass from Tevatron

5 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

6 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)

7 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

8 _ 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

9 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.

10 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)

11 b to u transitions V ub puzzle

12 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 )

13 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 )

14 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 ) More precise & reliable lattice calculations are needed to check whether such puzzles are for real or others are there.

15 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 ) More precise & reliable lattice calculations are needed to check whether such puzzles are for real or others are there. HQET on the lattice

16 The challenge of B-physics on the lattice multiple scale problem always difficult for a numerical treatment

17 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

18 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

19 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: effective theory, Λ QCD /m b expansion

20 Non-perturbative Heavy Quark Effective Theory Systematic expansion in Λ/m b 1/10 Non-perturbative implementation including 1st order corrections: NIC group [Heitger & S., 2003 ]... [B. Blossier, M. Della Morte, N. Garron, G. von Hippel, T. Mendes, H. Simma, R. S., 2010 ] QCD HQET L1 L1 L2 L2 L ω ω a match SSF S1 S2 S3 S4 S5

21 B physics B s µ + µ at LHCb: sensitive to SUSY contributions NP matrix element: F Bs First lattice computation of 1/m b correction in HQET [ LPHA ACollaboration 2010 ] r 0 3/2 φ RGI r 0 3/2 FBs m Bs 1/2 / CBs r 0 3/2 FPS m PS 1/2 / CPS quenched 1/(r 0 m PS ) 1/m b

22 Beyond the classical theory: Renormalization and Matching at leading order in 1/m a matrix element of A 0 : QCD Z A f A 0 (x) i QCD Φ QCD (m) HQET in static approx. ZA stat (µ) f Astat 0 (x) i stat Φ(µ) m: mass of heavy quark (b) in some definition (all other masses zero for simplicity) µ: arbitrary renormalization scale matching (equivalence): Φ QCD (m) = e C match (m, µ) Φ(µ) + O(1/m) ec match (m, µ) = 1 + c 1 (m/µ)ḡ 2 (µ) +... Physical observables, such as F Bs, are independent of renormalization scheme, scale. switch to Renormalization Group Invariants

23 Better: change to RGI s see e.g. [R.S., arxiv: ] Φ RGI = exp ( Z ) ḡ(µ) dx γ(x) Φ(µ) β(x) β : beta-fct ˆ ( Z 2b 0 ḡ(µ) 2 γ ḡ(µ)» 0 /2b 0 γ(x) exp dx 0 β(x) γ ) 0 Φ(µ) b 0 x Φ QCD = C PS (M/Λ) Φ RGI ( Z g (M/Λ) C PS (M/Λ) = exp dx γ ) match(x) β(x) with ( Λ M = exp Z g (M/Λ) dx 1 τ(x) ) β(x) γ : AD in HQET Λ : Lambda-para, g (M/Λ) M : RGI quark mass γ match : describes the mass dependence

24 Better: change to RGI s see e.g. [R.S., arxiv: ] Φ RGI = exp ( Z ) ḡ(µ) dx γ(x) Φ(µ) = Z RGI (g 0 ) β(x) {z } known, ˆ ( Z 2b 0 ḡ(µ) 2 γ ḡ(µ) 0 /2b 0 exp dx 0 LPHA ACollaboration Φ(g 0 ) {z } bare ME» γ(x) β(x) γ 0 b 0 x ) Φ(µ) β : beta-fct Φ QCD = C PS (M/Λ) Φ RGI ( Z g (M/Λ) C PS (M/Λ) = exp dx γ ) match(x) β(x) with ( Λ M = exp Z g (M/Λ) dx 1 τ(x) ) β(x) γ : AD in HQET Λ : Lambda-para, g (M/Λ) M : RGI quark mass γ match : describes the mass dependence

25 Matching and RGI s M Φ Φ M = Λ M C PS C PS M = γ match(g ) Λ 1 τ(g ), g = g (M/Λ). and with γ match (g ) g 0 γ 0 g 2 γ1 match g , β(ḡ) ḡ 0 b 0 ḡ we can give the leading large mass behaviour C PS M (2b0 g 2 ) γ0/2b0 [log(m/λ)] γ0/2b0

26 The present knowledge For γ match at l loops need γ MS = γ : l loops; C match (g ): l 1 loops γ 0 [Shifmann& Voloshin; Politzer& Wise ]... γ MS,2 [Chetyrkin & Grozin, 2003 ] C match (g ) to 3 loops [Bekavac, S. et al, 2009 ]

27 The present knowledge For γ match at l loops need γ MS = γ : l loops; C match (g ): l 1 loops γ 0 [Shifmann& Voloshin; Politzer& Wise ]... γ MS,2 [Chetyrkin & Grozin, 2003 ] C match (g ) to 3 loops [Bekavac, S. et al, 2009 ]

28 An application [ LPHA ACollaboration ] r 0 3/2 φ RGI r 0 3/2 FPS m PS 1/2 / CPS /(r 0 m PS ) this looks good; one may interpolate to the physical point...

29 An application [ LPHA ACollaboration ] r 0 3/2 φ RGI r 0 3/2 FPS m PS 1/2 / CPS /(r 0 m PS ) this looks good; one may interpolate to the physical point... but

30 What is the accuracy of perturbation theory? C match (g ) γ match to 3 loops [Bekavac, S. et al, 2009 ] also for various bilinears Γ notation O Γ = ψ l (x)γψ h (x) γ 0 γ 5 A 0 P γ 5 γ k γ kl V k T ( Z ) g Φ QCD Γ = Cmatch Γ (g ) Φ(µ) = C γ(x) match(g ) exp dx Φ Γ RGI β(x) ( Z g γmatch Γ (x) ) exp dx Φ RGI β(x) γ Γ match : 3-loops γγ match γγ match : 4-loops [chiral symmetry of light quarks (N light > 1): γ Γγ 5 match = γγ match ]

31 Compare different orders We actually show C Γ/Γ = Cmatch Γ Γ (m, µ)/cmatch (m, µ) B-physics: Λ MS /M b / log(λ MS /M b ) 0.3

32 Compare different orders We actually show C Γ/Γ = Cmatch Γ Γ (m, µ)/cmatch (m, µ) B-physics: Λ MS /M b / log(λ MS /M b ) 0.3 Perturbation theory is badly behaved for charm quarks very badly 1/ log(λ MS /M c) 0.5

33 Different orders of PT the normal behavior for one-scale quantities is O = o 0 + o 1 α + o 2 α α = ḡ 2 /(4π) o i < 1 (o 0 suitably normalized) examples: µ ḡ µ = ḡ { b 0 α + b 1 α } (N f = 3) µ m m µ Φ µ = d 0 α d 1 α Φ µ = γ = γ 0α γ 1 α MS b i d i γ i

34 Different orders of PT the normal behavior for one-scale quantities is O = o 0 + o 1 α + o 2 α α = ḡ 2 /(4π) o i < 1 (o 0 suitably normalized) examples: µ ḡ µ = ḡ { b 0 α + b 1 α } (N f = 3) µ m m µ Φ µ = d 0 α d 1 α Φ µ = γ = γ 0α γ 1 α MS b i d i γ i Perturbation theory is well behaved

35 Different orders of PT (N f = 3, well behaved MS RG functions) MS b i d i γ i but mass-dependence (matching anomalous dimensions): γ match (g ) = m Φ QCD Φ QCD m = γ 0 α γ 1 α (N f = 3) A 0, γ i V 0, γ i A 0 /V, γ i

36 Different orders of PT (N f = 3, well behaved MS RG functions) MS b i d i γ i but mass-dependence (matching anomalous dimensions): γ match (g ) = m Φ QCD Φ QCD m = γ 0 α γ 1 α (N f = 3) A 0, γ i V 0, γ i A 0 /V, γ i Perturbation theory is ill behaved (applicable at very small α)

37 Changing the scale we had Φ QCD (m) = C e match (m, µ) Φ(µ) Chose a convenient scale: µ = m = m(m ), g = ḡ(m ) may set more generally µ = s 1 m = m(m ), g = ḡ(m ) note: in the effective theory one does NOT INTEGRATE OUT DOFs ABOVE µ = m one matches the physics BELOW expect s > 1 is better

38 Changing the scale we had Φ QCD (m) = e C match (m, µ) Φ(µ) Chose a convenient scale: µ = m = m(m ), g = ḡ(m ) may set more generally note: in the effective theory µ = s 1 m = m(m ), g = ḡ(m ) one does NOT INTEGRATE OUT DOFs ABOVE µ = m one matches the physics BELOW expect s > 1 is better the result is simply (ĝ = ḡ(s 1 m ))) γ match (g ) = ˆγ match (ĝ) = ˆγ 0 α ˆγ 1 α ˆγ 0 = γ 0, ˆγ 1 = γ 1 + 2b 0 γ 0... ( Z ĝ C PS (M/Λ) = exp dx ˆγ ) match(x). β(x)

39 Changing the scale γ match (g ) = ˆγ match (ĝ) = ˆγ 0 α ˆγ 1 α ˆγ 0 = γ 0, ˆγ 1 = γ 1 + 2b 0 γ 0... jz ĝ ff C PS (M/Λ) = exp dx ˆγ(x) β(x) s A 0, ˆγ i V 0, ˆγ i A 0 /V, ˆγ i [Very similar for C match (m Q, µ) with µ = s 1 m Q expanded in α(µ)]

40 Changing the scale γ match (g ) = ˆγ match (ĝ) = ˆγ 0 α ˆγ 1 α ˆγ 0 = γ 0, ˆγ 1 = γ 1 + 2b 0 γ 0... jz ĝ ff C PS (M/Λ) = exp dx ˆγ(x) β(x) s A 0, ˆγ i V 0, ˆγ i A 0 /V, ˆγ i The behavior can be improved significantly [Very similar for C match (m Q, µ) with µ = s 1 m Q expanded in α(µ)] but s > 4 is required α(m b /4) is not small!

41 Conversion functions with and without scale optimization The ratio C PS /C V, evaluated in the first column as described here. In columns two and three the expansion in g is generalized to an expansion in ḡ(m /s). The last column contains the conventionally used Ĉmatch PS (m Q, m Q, m Q )/Ĉ match V (m Q, m Q, m Q ). For B-physics we have Λ MS/Mb 0.04 and 1/ ln(λ MS/Mb. The loop order changes from one-loop (long-dashes) up to 4-loop (full line) anomalous ) 0.3 dimension.

42 A conclusion on perturbative matching

43 A conclusion on perturbative matching is not easily drawn the effective scale seems well below µ = m

44 A conclusion on perturbative matching is not easily drawn the effective scale seems well below µ = m seems reliable only for masses beyond m b, where it is of limited use for us

45 A conclusion on perturbative matching is not easily drawn the effective scale seems well below µ = m seems reliable only for masses beyond m b, where it is of limited use for us a similar statement is found in [Bekavac, S. et al, 2009 ]

46 A conclusion on perturbative matching is not easily drawn the effective scale seems well below µ = m seems reliable only for masses beyond m b, where it is of limited use for us a similar statement is found in [Bekavac, S. et al, 2009 ] other ideas?

47 A conclusion on perturbative matching is not easily drawn the effective scale seems well below µ = m seems reliable only for masses beyond m b, where it is of limited use for us a similar statement is found in [Bekavac, S. et al, 2009 ] other ideas? in any case perturbative matching is only theoretically consistent at leading order in 1/m b [ α k (m) 1 2b 0 log(m/λ QCD ) ] k m ΛQCD 1/m

48 With N f = 2 dynamical quarks and NP matching LPHA ACollaboration lattices simulated on JUROPA, JUGENE F B = F B m 2 π =0 ( g 2 ) B B π 4 16π 2 Fπ 2 mπ 2 log(mπ/f 2 π) 2 + b mπ 2 Low energy coupling g B B π is needed (for static quarks).

49 Low energy coupling ĝ = g B B π with unprecedented precision ĝ = const. B k  k (0) B 0, A k = dγ k γ 5 u g ALPHA a=0.08fm ALPHA a=0.05fm ALPHA continuum (prelim.) ALPHA a=0.07fm Becirevic et al. a=0.08fm (GeV ) m π precision due to improved techniques

50 Low energy coupling g B B π with unprecedented precision [Bulava, Donnellan, S. ] quenched STAT g Hyp1 Hyp a 2

51 Determination of ĝ: plateaux N = 3 basis of different size Gaussian wavefunctions E eff (t) Energy level t/a M eff (t) Matrix element ĝ * Hyp 1 B B π (52) t/a ĝ eff (t) = ĝ + O(t N+1,1 e t N+1,1 ) N+1,1 6/fm

52 Determination of ĝ: standard ratio ĝ eff (t, t x ) C 3(t, t x ) C 2 (t) C 3 (t, t x ) = C 3 (t, t x ) = 0 ˆB k n e (t tx )En n  k (0) m e tx Em m ˆB 0 n m = const. e tm B B k Âk(0) B 0 (1 + O(e (t tx ) 2,1, e tx 2,1 ) and ĝ eff (t, t x ) = ĝ + O(e (t tx ) 2,1, e tx 2,1 ) 2,1 = E 2,B E 1,B = E 2,B m B 2/fm

53 Determination of ĝ, comparison previous[becirevic et al, 2009 ] New matrix element ĝ 1 R(t x ) Set 3 Set 2 Set 1 κ sea = κ t x M eff (t) * Hyp 1 B B π L (52) t/a

54 Determination of ĝ, comparison previous[becirevic et al, 2009 ] New matrix element ĝ R(t x ) Set 3 Set 2 Set 1 κ sea = κ t x M eff (t) * Hyp 1 B B π L (52) t/a old method with improved statistical accuracy (all-to-all) M eff (t) t = t x

55 Improved techniques: the GEVP matrix of correlation functions on an infinite time lattice C ij (t) = O i (0)O j (t) = e Ent ψ ni ψ nj, n=1 i, j = 1,..., N ψ ni (ψ n ) i = n Ô i 0 = ψ ni E n E n+1 the GEVP is C(t) v n (t, t 0 ) = λ n (t, t 0 ) C(t 0 ) v n (t, t 0 ), n = 1,..., N t > t 0, Lüscher & Wolff [1990 ] showed that E eff n = 1 a log λ n (t, t 0 ) λ n (t + a, t 0 ) = E n + ε n (t, t 0 ) ε n (t, t 0 ) = O(e En (t t0) ), E n = min m n E m E n.

56 The GEVP method [1990 ] correction term ε n (t, t 0 ) = O(e En (t t0) ), E n = min m n E m E n. [ 2000 ] F.Niedermayer, P.Weisz: private notes on GEVP, including perturbation theory in n > N levels [2009 ] we could prove that [B. Blossier, M. Della Morte, G. von Hippel, T. Mendes, R.S. ] ε n (t, t 0 ) = e N+1,n t if t 0 t/2, N+1,n = E N+1 E n [2009 ] similar formula for a decay constant: excited states as well [2011 ] and now also a formula [J. Bulava, M. Donnellan, R.S. ] (t 0 t/2) f O i = M eff (t, t 0 ) + O(t N+1,n e t N+1,n )

57 Demonstration in a toy model 5 states matrix elements between 1/5 and 1 matrix element 3 h w 3 M eff (t) unsummed 1.4 summed t

58 Demonstration in HQET: energies a = 0.07 fm ae eff,stat 1 (t, t 0 ) curve: E 1 + α N e N+1,1 t E 1 stat (t,t0) (lattice units) E 1 stat from GEVP for β = 6.3 E 1 stat (t,4), from 2x2 E 1 stat (t,5), from 2x2 (shifted) fit E 1 stat (t,4), from 3x3 fit E 1 stat (t,4), from 4x4 fit E 1 stat (t,4), from 5x5 plateau at N+1,1 agree with plateaux of E eff,stat N+1 (t, t 0 ) E eff,stat 1 (t, t 0 ) for large N and t. t (fm)

59 Demonstration in HQET: ĝ GEVP M eff (t) Energies E eff (t) * Hyp 1 B B π (52) t/a t/a 15 20

60 Demonstration in HQET: ĝ GEVP M eff (t) Energies E eff (t) * Hyp 1 B B π (52) t/a t/a best wavefunction (one may be lucky) M eff (t) E eff (t) t = t/a t x

61 f b with N f = 2 dynamical quarks LPHA ACollaboration lattices simulated on JUROPA, JUGENE ( F B = F B m 2 π = g 2 ) B B π 4 16π 2 Fπ 2 mπ 2 log(mπ/f 2 π) 2 + b mπ 2 To be done: Chiral extrapolation with known g B B π continuum extrapolation (from a = 0.08fm fm)

62 Summary B-deacys are an important piece in the validation of the SM of particle physics the search for new physics On the lattice an effective theory is needed NP HQET is well on its way for N f > 0 Precision chiral extrapolations require a determination of ĝ High precision determination of ĝ is done using new methods which are applicable more generally With our preliminary number for B τν (f B ), the V ub puzzle remains. A precise number will come soon (N f = 2). B s µ + µ immediately after (f Bs, LHCb). B πlν is the next step.