Towards Precision Flavour Physics

Transcription

1 : The Status of Semi-Leptonic B Decays Thomas Mannel CERN PH-TH / Universität Siegen CERN-TH Seminar, May 2008

2 Contents 1 Introduction 2 External Field Method Results to order 1/m 4 b 3 Towards the O(α s /m 2 b ) corrections Reparametrization Invariance Results 4 m b m c versus m b m c 5 Effective Theory Parametrization of new physics Calculation and Results

3 Why is this interesting? 1.5 Important Ingredient for the Unitarity Traingle excluded area has CL > 0.95 η sin 2β ε K V ub /V cb CKM f i t t e r FPCP 06 α γ γ γ α excluded at CL > 0.95 sol. w/ cos 2β < 0 (excl. at CL > 0.95) β α m d m s & m d ρ ε K Standard Fit of for the Unitarity Traingle Unitarity Clock : V ub /V cb Buras Relation between Kaon CP violation and the Unitarity Trangle

4 How to compute Inclusive Decays Heavy Quark Expansion = Operator Product 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 Perform an OPE: m b is much larger than any scale appearing in the matrix element d 4 xe imbvx T { H eff (x) H eff (0)} ( ) n 1 = C n+3(µ)o n+3 2m Q n=0 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 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 Present state of the Calculations What has been achieved in semileptonic decays? Good theoretical control over b c inclusive decays using OPE and HQE b c Exclusive transitions: control channels Lattice Results become very precise future V cb? Reasonable theoretical control over inclusive b u decays Phase space cuts twist expansion, shape functions b u exclusive channels are harder: Lattice QCD and QCD Sum rules Is there anything else that can be done with this data?

8 State of the art in b c semileptonics: Tree level terms up to and including 1/mb 4 known O(α s ) and full O(αs) 2 for the partonic rate known O(α s ) for the µ 2 π/mb 2 is known Mass logarithms In the pipeline: Complete α s /mb 2, including the µ G terms Phase space effects Enormous (and still incresing) ammont of data The upshot: V cb = (41.2 ± 1.1) 10 3 V ub = (3.95 ± 0.35) 10 3

9 In this talk: Inclusive b c transitions Pushing the Standard Model Calculation further: 1/m 4 Contributions (at tree level) B. Dassinger, S. Turczyk, TM: JHEP 0703:087 (2007) α s /m 2 b corrections T. Becher, H. Boos and E. Lunghi: JHEP 0712, 062 (2007) Is there intrinsic charm in the B meson? C. Breidenbach, T. Feldmann, TM, S. Turzcyk, CERN-PH-TH/ Use of semileptonic data to search for new physics in charged current decays B. Dassinger, R. Feger, TM: Phys.Rev.D75:095007(2007) B. Dassinger, R. Feger, TM: arxiv... (2008)

10 External Field Method Results to order 1/m 4 b Part 1: 1/m 4 b at tree level

11 General Method: OPE External Field Method Results to order 1/m 4 b 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

12 External Field Method Results to order 1/m 4 b 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

13 External Field Method Results to order 1/m 4 b 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 )

14 External Field Method Results to order 1/m 4 b 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.

15 External Field Method External Field Method Results to order 1/m 4 b 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 i S BGF = /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

16 External Field Method Results to order 1/m 4 b 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

17 External Field Method Results to order 1/m 4 b 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

18 External Field Method Results to order 1/m 4 b 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)

19 External Field Method Results to order 1/m 4 b 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

20 External Field Method Results to order 1/m 4 b Step 2: Derive Trace formulae for the matrix elements: Start at dim-7: b(id µ )(id ν )(id ρ )(id σ )b : static limit lengthy, but systematically calculable Compute dim-6, including the 1/m b corrections: b(id µ )(id ν )(id ρ )b... Compute dim-3, including corrections up to 1/m 4 b Step 3: Compute the trace with the result from the geometric series

21 Quantitative Results Guestimates of the new parameters s 1 = g 2 E 2 (g p E ) 2 s 2 ρ6 D µ 2 π p 2 External Field Method Results to order 1/m 4 b ρ6 D, etc. µ 2 π µ 4 G + µ 4 π, s 3 µ 4 π, s 4 s 5 µ 2 Gµ 2 π Our guess for basic parameters s i (µ 2 π, µ2 G, ρ3 D, ρ3 LS from Buchmüller and Flächer) s 1 / GeV 4 s 2 / GeV 4 s 3 / GeV 4 s 4 / GeV 4 s 5 / GeV ± ± ± ± ± 0.04 More sophisticated estimates possible: Bigi, Zwicky, Uraltsev: hep-ph/ , similar numbers

22 External Field Method Results to order 1/m 4 b Effect has been studied in detail on the moments El n = dm X de l E E n d 2 Γ l cut dm x de l MX n = dm X MX n d 2 Γ de l E cut dm x de l small effects of expected size! δ (4) Γ Effect on the total rate: 0.25% Γ Impact on V cb : Slight improvement of the uncertainly related to the application of the HQE Total improvement small, O(0.25%)

23 Reparametrization Invariance Results Part 2: α s /m 2 b Corrections

24 O(α s µ 2 π/m 2 b ) corrections Reparametrization Invariance Results One-Loop α s corrections known since a long time Corrections to the leading (partonic) rate Make use of Reparametrization invariance: v v = v + k m b Relates different orders of the 1/m b expansion Valid to all orders in α s Compute O(α s )-Correction with p b = m b v + k and expand in k k µ k ν (g µν v µ v ν ) µ2 π 3

25 Reparametrization Invariance Results

26 Reparametrization Invariance Results Typical size of the corrections: Will shift the extracted value for µ π substantially ( Global fit) Only small impact of the V cb determination

27 Reparametrization Invariance Results For the complete α s /m 2 b also the O(α sµ 2 G /m2 b ) Corrections need to be computed Significantly more complicated Needs the one gluon matrix elements at one loop Doable, is in the pipeline

28 m b m c versus m b m c Part 3: Intrinsic charm

29 Intrinsic charm m b m c versus m b m c 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...

30 m b m c versus m b m c 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 +

31 m b m c versus m b m c 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

32 m b m c versus m b m c 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

33 m b m c versus m b m c 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

34 m b m c versus m b m c For the realistic case m c Λ QCD : m c is a perturbative scale Intrinsic charm contributions vanish at scales µ m c No additional uncertainty from intrinsic charm matrix elements Similar statements hold for higher dimensional operators

35 Effective Theory Parametrization of new physics Calculation and Results Part 4: New Physics in s.l. Decays

36 Effective Theory Parametrization of new physics Calculation and Results : General Remarks General believe: Charged Currents are tree level no sensitivity to new physics Test of the left handedness: Michel Parameter Analysis of e.g. µ e ν e ν µ Can a precise Michel Paramater analysis be done for hadronic currents?... in particular for the b c current, given the large ammount of data? How do we parametrize new physics in this case?

37 Effective Theory Parametrization of new physics Calculation and Results Effective Theory Parametrization We write quark and lepton fields Q L = Q R = L L = L R = ( ul d L ( ur d R ( νe,l e L ( νe,r e R ), ), ), ), ( cl s L ( cr s R ( νµ,l µ L ( νµ,r ), ), ), ), µ R ( tl b L ( tr ) left handed quarks ) right handed quarks ) left handed leptons ) right handed leptons b R ( ντ,l τ L ( ντ,r τ R

38 Effective Theory Parametrization of new physics Calculation and Results and the Higgs field H = 1/ ( φ0 + iχ 2 0 2φ+ 2φ φ 0 iχ ) Write down all operators which are compatible with SU(3) c SU(2) L U(1) Y containing a b cl ν transition

39 Effective Theory Parametrization of new physics Calculation and Results At the weak scale we can have two classes of dim-6 operators (relevent to the question) Two Quark Operators: LL O (1) LL = Q L /L Q L O (2) LL = Q L /L 3 Q L with L µ = H (id µ H) +(id µ H) H L µ 3 = Hτ 3 (id µ H) +(id µ H) τ 3 H with Two Quark Operators: RR O (1) RR = Q R /R Q R O (2) RR = Q R {τ 3, /R} Q R O (3) RR = i Q R [τ 3, /R] Q R O (4) RR = Q R τ 3 /Rτ 3 Q R R µ = H (id µ H) + (id µ H) H

40 Effective Theory Parametrization of new physics Calculation and Results Two Quark Operators: LR O (1) LR = Q L (σ µν B µν ) H Q R + h.c. O (2) LR = Q L (σ µν W µν ) H Q R + h.c. O (3) LR = Q L (id µ H) id µ Q R + h.c. We may omit possible two-lepton operators (see below)

41 Effective Theory Parametrization of new physics Calculation and Results Two Quark-Two Lepton Operators: SU(2) L SU(2) R invariant: O (i) LL,LL = ( Q L Γ i Q L )(L L Γ i L L ) P (i) LL,LL = ( Q L τ a Γ i Q L )(L L τ a Γ i L L ) O (i) LL,RR = ( Q L Γ i Q L )(L R Γ i L R ) O (i) RR,LL = ( Q R Γ i Q R )(L L Γ i L L ) O (i) RR,RR = ( Q R Γ i Q R )(L R Γ i L R ) P (i) RR,RR = ( Q R τ a Γ i Q R )(L R τ a Γ i L R ) Two Quark-Two Lepton Operators: SU(2) L invariant: R (i) LL,RR = ( Q L Γ i Q L )(L R Γ i τ 3 L R ) R (i) RR,LL = ( Q R Γ i τ 3 Q R )(L L Γ i L L ) R (i) RR,RR = ( Q R Γ i Q R )(L R Γ i τ 3 L R ) S (i) RR,RR = ( Q R τ a Γ i Q R )(L R τ a τ 3 Γ i L R ) T (i) RR,RR = ( Q R τ a τ 3 Γ i Q R )(L R τ a τ 3 Γ i L R )

42 Effective Theory Parametrization of new physics Calculation and Results Helicity structure: Γ i Γ i = 1 1, γ µ γ µ, γ µ γ 5 γ 5 γ µ, σ µν σ µν These operators are not independent! Operators with (LR)(LR) combinations cannot appear at dim 6 We assume all ν R to be very heavy (see-saw) We look at charged currents: The leptonic side is a l ν transition Only the left handed (charged) leptonic current is relevant! (Massless Leptons) Only P (i) LL,LL (b Lγ µ c L )( ν L γ µ l L ) contributes.

43 Effective Theory Parametrization of new physics Calculation and Results Spontaneous symmetry breaking: Two Quark Operators O LL and O RR yield anomalous gauge boson couplings of the order v 2 /Λ 2 NP Two Quark Operators O RL yield anomalous couplings of the order vm q /Λ 2 NP: Additional supression likely! At the bottom scale: Integrate out the W boson Yields a factor g 2 /M 2 W = 1/v 2 At µ = m b : Only two quark-two lepton operators left with couplings O(1/Λ 2 NP) for LL and RR or O((m b /v)1/λ 2 NP) for LR. Interaction can be written as a current-current structure

44 Effective Theory Parametrization of new physics Calculation and Results Leptonic current remains as in the Standard Model At the scale m b we get the effective interaction H eff = 4G FV cb 2 J q,µ J µ l, Generalized b c current (P ± = 1 2 (1 ± γ 5)) J h,µ = c L cγ µ P b + c R cγ µ P + b +g L c idqcd µ m P b + g c idqcd µ R P + b b m b i µ i µ +d L ( ciσ µν P b) + d R ( ciσ µν P + b) m b m b Orders of magnitude: c L, c R v 2 Λ 2 g L, g R, d L, d R vm b Λ 2

45 Effective Theory Parametrization of new physics Calculation and Results Re-compute the inclusive decay with the new effective interaction Only the interference term with the Standard-Model contribution is relevant... This is why the leptonic current has to be as in the Standard Model (for massless leptons) Study the influence of the new couplings on the observables

46 Calculation Introduction Effective Theory Parametrization of new physics Calculation and Results Naive Starting Point: Parton Model: Lepton energy ½º ½º¾ ½ ¼º ¼º ¼º Γ y ¼º¾ ¼ ¹¼º¾ ¹¼º ¹¼º ¹¼º ¹½ ¹½º¾ ¹½º ¹½º c L c L c L c R c L d L c L d R c L g L c L g R ¼ ¼º½ ¼º¾ ¼º ¼º ¼º ¼º ¼º ¼º ¼º y

47 Effective Theory Parametrization of new physics Calculation and Results Expect a significant influence on the shape of the spectra Changes in the moments relative to the Standard Model Corrections matter: 1/mb 2 corrections: Too small to matter B. Dassinger, R. Feger, Diploma Thesis QCD Radiative corrections: Known to be relevant Calculate the QCD corrections to the various currents in the parton model B. Dassinger, R. Feger, T. M.: hep-ph/ , B. Dassinger, R. Feger, T. M.: SI-HEP , in preparation

48 Effective Theory Parametrization of new physics Calculation and Results Compute Feynman Diagrams Virtual corrections ν e ν e ν e W c g g b b Real Corrections e ν e W e W c g b ν e e c ν e W e W e W c c g b b g b g e c

49 Simple Case: V + A admixture Effective Theory Parametrization of new physics Calculation and Results In case of V + A: No Anomalous dimension Infrared sensitivity cancells between real and virtual corrections Leptonic Moments L n = 1 Γ 0 ZE cut de l E n l dγ mit Γ 0 = G2 F V cb 2 m 5 h b de l 192π 3 1 8ρ 12ρ 2 ln ρ+8ρ 3 ρ 4i Hadronic Moments H ij = 1 Γ 0 ZE cut de l Z de had dmhad 2 (M2 had m2 c )i E j d 3 Γ had de l de had dmhad 2

50 Numerical Results Effective Theory Parametrization of new physics Calculation and Results L n = L (0) n H ij = H (0) ij + α s π mn b l (1) n + + α s π m2i+j b h (1) ij + α s /π-coefficients l (1) n (left) and h (1) ij (right) n cl 2 c l c r i j cl 2 c l c r

51 Numerical Results Effective Theory Parametrization of new physics Calculation and Results L n = L (0) n H ij = H (0) ij + α s π mn b l (1) n + + α s π m2i+j b h (1) ij + α s /π-coefficients l (1) n (left) and h (1) ij (right) n cl 2 c l c r i j cl 2 c l c r

52 Effective Theory Parametrization of new physics Calculation and Results General Case: Scalar and Tensor Currents Scalar and Tensor Currents have anomalous dimensions RG Mixing into left handed currents However, for the hadronic moments with i > 0 only the real contribution matters RG discussion affects only the leptonic moments and the hadronic ones with i = 0

53 Effective Theory Parametrization of new physics Calculation and Results α s /π-coefficients h (1) ij i j cl 2 c l c r c l d l c l d r c l g l c l g r Also the virtual contributions have been calculated One loop RG mixing One loop finite terms Needed for the correct scale setting in the finite terms: Two loop mixing: Results quoted for µ = m b Results for arbitrary lepton-energy cut available

54 What is in the pipeline? Effective Theory Parametrization of new physics Calculation and Results Use the data on the moments in B X c l ν to perform a combined fit to Masses: m b, m c HQE Parameters: µ π, µ G, ρ D,... V cb New Physics contributions c L c R, c L g R, c L g L,... needs full access to the experimental data, BaBar, BELLE

55 Conclusions Introduction Effective Theory Parametrization of new physics Calculation and Results Large amount of data + good theoretical control Precise determination of SM parameters Search for deviations from the Standard Model Good job need full access to the data of BaBar / Belle Further improvements of inclusive s.l. decays: Partonic αs n with n 3: technical challenge BLM αs/m 2 b can probably be done 1/mb 5 can probably be done, but there is no good reason (yet?) Inclusive V cb has almost reached its theory limits