Part II: Heavy Quark Expansion

Transcription

1 Part II: Heavy Quark Expansion Thomas Mannel CERN-PH-TH and Theoretische Physik I, Siegen University KITPC, June 24th, 2008

2 Contents 1 Introduction to HQE Set-up: OPE Spectra of Inclusive Decays 2 Theory Status Experimental Input 3 Calculation of high orders in 1/m b Intrinsic charm 4

3 Introduction Set-up: OPE Spectra of Inclusive Decays Inclusive semileptonic Decays: B Xl ν l with X = D, D, D + nπ, π, ρ, 2π, Λ c p,... Sometimes split into X c and X u (Theory point of view) Inclusive Radiative Decays B Xγ with X = all Sometimes split into X s and X d, cut on E γ Lifetimes of heavy hadrons B d X, B u X, Λ b X Ratios of lifetimes, lifetime differences in B d -B d, B s -B s, and D 0 -D 0 -systems HQE: Very precise theoretical tool!

4 Set-up: OPE Spectra of Inclusive Decays Inclusive Decays: Heavy Quark Expansion Operator Product Expansion = Heavy Quark Expansion (Chay, Georgi, Bigi, Shifman, Uraltsev, Vainstain, Manohar. Wise, Neubert, M,...) Γ (2π) 4 δ 4 (P B P X ) X H eff B(v) 2 X = d 4 x B(v) H eff (x)h eff (0) B(v) = 2 Im d 4 x B(v) T {H eff (x)h eff (0)} B(v) = 2 Im d 4 x e imbv x B(v) T { H eff (x) H eff (0)} B(v) Last step: p b = m b v + k, Expansion in the residual momentum k

5 Set-up: OPE Spectra of Inclusive Decays Perform an OPE: m b is much larger than any scale appearing in the matrix element d 4 xe im bvx T { H eff (x) H eff (0)} = ( ) n 1 C n+3(µ)o n+3 n=0 2m Q The rate for B X c l ν l can be written as Γ = Γ Γ Γ m Q mq Γ mq The Γ i are power series in α s (m Q ): Perturbation theory!

6 Set-up: OPE Spectra of Inclusive Decays Γ 0 is the decay of a free quark ( Parton Model ) Γ 1 vanishes due to Heavy Quark Symmetries Γ 2 is expressed in terms of two parameters 2M H µ 2 π = H(v) Q v (id) 2 Q v H(v) 2M H µ 2 G = H(v) Q v σ µν (id µ )(id ν )Q v H(v) µ π : Kinetic energy and µ G : Chromomagnetic moment Γ 3 two more parameters 2M H ρ 3 D = H(v) Q v (id µ )(ivd)(id µ )Q v H(v) 2M H ρ 3 LS = H(v) Q v σ µν (id µ )(ivd)(id ν )Q v H(v) ρ D : Darwin Term and ρ LS : Chromomagnetic moment

7 Set-up: OPE Spectra of Inclusive Decays New: 1/m 4 b Contribution Γ 4 (Dassinger, Turczyk, M.) Five new parameters: E 2 : Chromoelectric Field squared B 2 : Chromomagnetic Field squared ( p 2 ) 2 : Fourth power of the residual b quark momentum ( p 2 )( σ B) : Mixed Chromomag. Mom. and res. Mom. sq. ( p B)( σ p) : Mixed Chromomag. field and res. helicity Some of these can be estimated in naive factorization

8 Spectra of Inclusive Decays Set-up: OPE Spectra of Inclusive Decays _ dγ Γ dy b y Endpoint region: ρ = mc/m 2 b 2, y = 2E l/m b [ ( λ 1 ρ 2 + (m b (1 y)) 2 1 y dγ dy Θ(1 y ρ) ) 2 { 3 4 ρ 1 y Reliable calculation in HQE possible for the moments of the spectrum } ]

9 Set-up: OPE Spectra of Inclusive Decays Shape- or Light-Cone Distribution Functions Resummation into a shape function or light cone distribution function (Bigi, Shifman, Uraltsev, Neubert, M.,...) such that 2M B f (ω) = B(v) b v δ(ω + i(n D)) B(v) dγ dy = G2 F V ub 2 mb 5 96π 3 dω Θ(m b (1 y) ω)f (ω) Moment Expansion of f in terms of HQE parameters: f (ω) = δ(ω) + µ2 π δ (ω) 6mb 2 ρ3 D δ (ω) + 18mb 3

10 Theory Status Experimental Input Determination of V cb : B X c l ν l Γ = V cb 2ˆΓ0 mb(µ)(1 5 + A ew )A pert (r, µ) [ ( ) µ 2 z 0 (r) + z 2 (r) π, µ2 G mb 2 mb 2 ( ρ 3 + z 3 (r) D, ρ3 LS mb 2 mb 2 State of the art: 1/m b Expansion at tree level up to 1/mb 3 (New 1/mb 4 still too new ) Complete O(α s ) corrections for the partonic rate (1/mb 0) and partial O(αs) 2 O(α s ) for 1/mb 2 terms under consideration Radiative Corrections: Scheme Dependence ) ] +...

11 Scheme Dependence Theory Status Experimental Input Pole mass introduces large radiative corrections use a suitably defined short distance mass Two Schemes are commonly used: Kinetic Scheme Bigi, Uraltsev, Shifman... m kin (µ) defined from a sum rule for the kinetic energy of the heavy quark 1S Scheme: Manohar, Hoang, Bauer, Ligeti... m 1S defiend from a (perturbative) calculation of the Υ(1S) mass Both Schemes yield comparable and small uncertainties.

12 Theory Status Experimental Input Heavy to Heavy: B X c l ν l Determine the HQE parameters from Charged lepton energy spectrum Hadronic invariant mass spectrum From the theoretical side: Calculation of moments of the spectra MX n = 1 Γ El n = 1 Γ dm X MX n d 2 Γ de l E cut dm x de l dm X de l E E n d 2 Γ l cut dm x de l

13 Theory Status Experimental Input Hadronic Invariant Mass Moments (Buchmüller, Flächer)

14 Theory Status Experimental Input Lepton Energy Moments I (Buchmüller, Flächer)

15 Theory Status Experimental Input Lepton Energy Moments II (Buchmüller, Flächer)

16 Theory Status Experimental Input V cb,incl = (41.6 ± 0.6) 10 3

17 Perspectives Theory Status Experimental Input Currently: δv cb V cb (2%) theo Main sources of uncertainties: Mass of the b quark: δm b 50 MeV Higher order QED and QCD radiative corrections Higher Order of the 1/m b expansion Mass ratio r = m 2 c/m 2 b Extraction of the HQE Parameters Parton Hadron Duality (?)

18 General Method: OPE Calculation of high orders in 1/m b Intrinsic charm Standard approach: Perform an OPE for the correlator (j µ = b L γ µ c L ) T µν (q, p B ) = d 4 x e iqx B(p B ) T (j µ (x)j ν(0) B(p B ) At tree level: Look at the Feynman diagramm q p b = m b v + k Set p b = m b v + k (with v = p B /M B ) and expand in the residual momentum k

19 Calculation of high orders in 1/m b Intrinsic charm Match the expression to the operators: [(...): symmetrization] k µ bid µ b k µ k ν bid (µ id ν) b k µ k ν k ρ bid (µ id ν id ρ) b... To get the antisymmetric pieces: Calculate (one, two, three...) Gluon matrix elements Allows to identify the order of covariant derivatives

20 Calculation of high orders in 1/m b Intrinsic charm Nonperturbative Input: Hadronic Parameters Forward matrix elements: hadronic input parameters B(p B ) bb B(p B ) = 1 + O(1/mb) 2 B(p B ) bid µ b B(p B ) = 0 + O(1/m b ) B(p B ) b(id) 2 b B(p B ) = 2M B µ 2 π + O(1/mb) 2 B(p B ) b( iσ µν )id µ id ν b B(p B ) = 2M B µ 2 G + O(1/m b ) B(p B ) bid µ (ivd)id µ b B(p B ) = 2M B ρ 3 D + O(1/m b ) B(p B ) b( iσ µν )id µ (ivd)id ν b B(p B ) = 2M B ρ 3 LS + O(1/m b )

21 Calculation of high orders in 1/m b Intrinsic charm Some remarks for experts b is still the full QCD field B(p B ) is still the full state The hadronic parameters still depend on m b : µ π for B mesons differs from µ π for D mesons Tree level only, no discussion of renorm. issues here Advantage 1: No non-local, non-perturbative matrix elements in the expansion. Advantage 2: Simple and straightforward generalization to higher orders in the 1/m b expansion. Disadvantage 1: Non-perturbative Parameters are not universal for all heavy mesons.

22 External Field Method Calculation of high orders in 1/m b Intrinsic charm Keep track of the order of the covariant derivatives Use p b = m b v + k m b v + id Q + id Write the charm Propagator as S BGF = i /Q + i /D m c Charm quark in the gluonic background of B meson Expand as a geometric series ( i)s BGF = + 1 /Q m c 1 (i /D) /Q m c 1 /Q m c (i /D) 1 /Q m c (i /D) 1 /Q m c 1 + /Q m c

23 Calculation of high orders in 1/m b Intrinsic charm T = Insert this into the forward matrix element + + [ Γ [ Γ [ Γ ] i Γ /Q m c αβ b α b β i i γ µ Γ /Q m c /Q m c ] αβ b α id µ b β i i i γ µ γ ν Γ /Q m c /Q m c /Q m c ] αβ b α id µ id ν b β + Keeps track of the order of the covariant derivatives automatically (works only at tree level) Can be generalized to higher order in 1/m b

24 Calculation of high orders in 1/m b Intrinsic charm Hadonic Matrix elements can be expressed in terms of the basic parameters Iterative proceedure, staring from the highest dimension to be considered Trace formulae B(p) b β b α B(p) = ( /v + 1 2M B + 1 ) (ˆµ 2 4 8mb 2 G ˆµ 2 π ) + O(1/mb) 5 B(p) b β (id ρ )b α B(p) = ( 2M B 1 (/v + 1)v ρ (ˆµ 2 G ˆµ 2 π ) 8m b m b (γ ρ v ρ /v)(ˆµ G 2 ˆµ π 2 ) + O(1/m 2 b) αβ ) αβ

25 Calculation of high orders in 1/m b Intrinsic charm Basic Dimension Six Matrix Elements Step 1: Identify the basic dim-7 Matrix elements Spin-independent basic parameters of dimension 7 2M B s 1 = B(p) b v id ρ (iv D) 2 id ρ b v B(p) 2M B s 2 = B(p) b v id ρ (id) 2 id ρ b v B(p) 2M B s 3 = B(p) b v ((id) 2 ) 2 b v B(p) Spin-dependent basic parameters of dimension 7 2M B s 4 = B(p) b v id µ (id) 2 id ν ( iσ µν )b v B(p) 2M B s 5 = B(p) b v id ρ id µ id ν id ρ ( iσ µν )b v B(p)

26 Calculation of high orders in 1/m b Intrinsic charm Physical Interpretation of the s i Spin-independent 2M B s 1 = g 2 E 2 2M B s 2 = g 2 ( E 2 B 2 ) + ( ( p) 2) 2 2M B s 3 = ( ( p) 2) 2 Spin-dependent 2M B s 4 = 3g ( S B)( p) 2 + 2g ( p B)( S p) 2M B s 5 = g ( S B)( p) 2

27 Intrinsic charm Calculation of high orders in 1/m b Intrinsic charm In the naive OPE: 2M B W IC µν = (2π) 4 δ 4 (m b v q) B(p) b v γ µ 1 2 (1 γ 5)c cγ ν 1 2 (1 γ 5)b v B(p) Charm Content of the B Meson... However, this depends on the point of view...

28 Calculation of high orders in 1/m b Intrinsic charm m b m c Λ QCD Bottom and charm integrated out at the same scale ρ = mc/m 2 b 2 is O(1) 1 B(p) b v γ µ (1 γ 1 2 5)c cγ ν (1 γ 2 5)b v B(p) = 0 at and below this scale There is a contribution to the Darwin Term ρ D ( ) dγ (3) dy = G2 F m5 b 24π V cb 2 ρ3 D 1 Θ(1 y ρ) + 3 mb 3 1 y where y = 2E l /m b Integration yields a Log Γ (3) = G2 F m5 b 24π 3 V cb 2 ln ( ) m 2 c ρ 3 D m 2 b m 3 b +

29 Calculation of high orders in 1/m b Intrinsic charm At m b < µ < m c : m b m c Λ QCD B(p) b v γ µ 1 2 (1 γ 5)c cγ ν 1 2 (1 γ 5)b v B(p) 0 Contribution of intrinsic charm : dγ IC dy = 2G2 F m5 b π V cb 2 bc cb δ(1 y) mb 3 Infrared-singular contribution in the Darwin term: dγ (3) dy ( ) = G2 F m5 b 24π V cb 2 ρ3 D 1 Θ(1 y) + 3 mb 3 1 y

30 Calculation of high orders in 1/m b Intrinsic charm Regularize the IR Singularity: [ ] ( ) θ(1 y) θ(1 y) µ 2 δ(1 y) ln 1 y 1 y + m 2 b RG Mixing of intrinsic charm into the Darwin term: µ ( bc cb (µ) 1 ) 1 µ 3 (4π) 2 ρ3 D ln m2 b = 0 µ 2 Generates ln(m 2 b /m2 c) through RG running! At µ = m c ɛ: bc cb = 0 and hence bc cb (m c + ɛ) = (4π) 2 ρ3 D ln m2 b m 2 c

31 Calculation of high orders in 1/m b Intrinsic charm m b m c Λ QCD m c is considered non-perturbative: Corresponds to the b u case Intrinsic charm = weak annihilation RG flow bc cb ρ D remains the same Both ρ D and bc cb are non-perturbative parameters This is not realistic for the b c case

32 Calculation of high orders in 1/m b Intrinsic charm For the realistic case m c Λ QCD : m c is a perturbative scale Intrinsic charm contributions vanish at scales µ m c At the current leve: No additional uncertainty from intrinsic charm matrix elements In higher orders of the 1/m Expansion: At Order 1/m m+n may be terms of order 1/m n b 1/mm c Leading term at tree level: 1/m 3 b 1/m2 c 1/m 5 At order α s (m c ): α s 1/m 3 b 1/m c α s /m 4

33 Recent Developments: Soft Collinear Effective Theory Problem: How to deal with energetic light degrees of freedom = Endpoint regions of the spectra? More than two scales involved! Inclusive Rates in the Endpoint become (Korchemski, Sterman) dγ = H J S with * = Convolution H: Hard Coefficient Function, Scales O(m b ) J: Jet Function, Scales O( m b Λ QCD ) S: Shape function, Scales O(Λ QCD )

34 Basics of Soft Collinear Effective Theory Heavy-to-light decays: Kinematic Situations with energetic light quarks hadronizing into jets or energetic light mesons p fin : Momentum of a light final state meson p 2 fin O(Λ QCD m b ) v p fin O(m b ) Use light-cone vectors n 2 = n 2 = 0, n n = 2: p fin = 1 2 (n p fin) n and v = 1 2 (n + n) Momentum of a light quark in such a meson: p light = 1 2 [(n p light) n + ( n p light )n] + p light

35 SCET Power Counting Define the parameter λ = Λ QCD /m b The light quark invariant mass (or virtuality) is assumed to be p 2 light = (n p light )( n p light ) + (p light) 2 λ 2 m 2 b The components of the quark momentum have to scale as (n p light ) m b ( n p light ) λ 2 m b p light λm b

36 A brief look at SCET (Bauer, Stewart, Pirjol, Beneke, Feldmann...) QCD quark field q is split into a collinear component ξ and a soft one with ξ = 1 4 /n /n + q The Lagrangian L QCD = q(i /D)q is rewritten in terms of the collinear field L = 1 2 ξ/n + (in D)ξ ξi 1 /n + /D in + D + iɛ 2 i /D ξ Expansion according to the above power couning: in + D = in + + gn + A c + gn + A us = in + D c + gn + A us Leading L becomes non-local: Wilson lines

37 B ππ, B K π in QCD Factorization (Beneke, Buchalla, Neubert, Sachrajda) Partial Calculations of the hard scattering kernels to NNLO available Small strong phases due to power suppression / perturbation theory Data indicate sizable power corrections At subleading level: Too many nonperturbative parameters Results agree quantitatively with the ones from QCD LC Sum rules (Khodjamirian, Melic, Melcher, M.)

38 ο 2B (π K + + )/B (π ο K ) 2.5 B (π + K + ο ο )/2B (π K ) 2 B (π + π )/B (π + K ) < 80 ο γ (deg) γ (deg) γ (deg) ο 2.5 τ + /τ ο B (π K + B B )/B (π K ) τ + B /τ ο B B (π + π )/2B (π π ) τ + /τ ο ο ο 2 B B 2B (K π )/B (Κ + π ) γ (deg) (Beneke, Buchalla, Neubert, Sachrajda, 2001 ) > 58 ο γ (deg) γ (deg)

39 Recent NNLO calculation (Beneke Jaeger) Variation of the inverse moment λ B : Is a small value of λ B realistic?

40 Update on the BR s by Neubert (CKM 2005, San Diego)

41 Update on the CP Asymmetries by Neubert (CKM 2005)

42 Conclusion HQET and HQE offer unique possibilities to extract fundamental parameters Possibility of precision calculations Semileptonic decays are in a mature state, some details need to be clarified Non-leptonics (despite of SCET / QCDF / PQCD) remain difficult...