Hadronic Effects in B -Decays

Transcription

1 Hadronic Effects in B -Decays (from the b-quark to the B-meson) Thorsten Feldmann Neckarzimmern, March 2007 Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

2 Outline 1 b cdū decays b cdū decays at Born level Quantum-loop contributions to b cdū decay From b cdū to B Dπ 2 b s(d) q q decays Penguin operators Charmless non-leptonic decays: B ππ and B πk 3 b s(d)γ decays Inclusive B X s γ decays Sensitivity to the B-meson shape function Exclusive B K γ decay B K l + l Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

3 ... Some introductory remarks... Physical processes involve different typical energy/length scales Different physical phenomena are described in terms of different degrees of freedom and different input parameters. In particle physics, the description of low-energy phenomena can be formulated as an effective theory, where the physics at small scales (high energies) is irrelevant, or can be absorbed into appropriate effective quantities. In our case, the relevant scales are: The scale of possible new physics : M X > 100 GeV The electroweak scale : M W 80 GeV The heavy quark masses : m b 4.6 GeV m c 1.4 GeV The strong interaction scale : Λ QCD 0.3 GeV Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

4 ... Some introductory remarks... Physical processes involve different typical energy/length scales Different physical phenomena are described in terms of different degrees of freedom and different input parameters. In particle physics, the description of low-energy phenomena can be formulated as an effective theory, where the physics at small scales (high energies) is irrelevant, or can be absorbed into appropriate effective quantities. In our case, the relevant scales are: The scale of possible new physics : M X > 100 GeV The electroweak scale : M W 80 GeV The heavy quark masses : m b 4.6 GeV m c 1.4 GeV The strong interaction scale : Λ QCD 0.3 GeV Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

5 Central Notions to be explained The dynamics of strong interactions in B-decays is very complex and has many faces. I will not be able to cover everything, but I hope that some theoretical and phenomenological concepts become clearer... Factorization separation of scales in perturbation theory simplification of exclusive hadronic matrix elements Operators in the weak effective Hamiltonian (current-current, strong penguins, electroweak penguins) Naive factorization and its improvement (BBNS) Form factors, light-cone distribution amplitudes,... Isospin and SU(3) F Inclusive decays and shape functions Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

6 First Example: b cdū decays Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

7 b cdū decay at Born level Full theory (SM) Fermi model ( ) 2 g αβ + qα q β g 2 (b c) M 2 J W 2 α q 2 MW 2 J (d u) β q M W 2 G F J α (b c) αβ (d u) g J β Energy/Momentum transfer limited by mass of decaying b-quark. b-quark mass much smaller than W -boson mass. q m b m W Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

8 Effective Theory: Analogously to muon decay, transition described in terms of current-current interaction, with left-handed charged currents J α (b c) = V cb [ c γ α (1 γ 5 ) b], J(d u) β = Vud [ d γβ (1 γ 5 ) u ] Effective operators only contains light fields ("light" quarks, electron, neutrinos, gluons, photons). Effect of large scale M W in effective Fermi coupling constant: g 2 8MW 2 G F GeV 2 Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

9 Quantum-loop contributions to b cdū decay Momentum q of the W -boson is an internal loop momentum that is integrated over and can take values between and +. We cannot simply expand in q /M W! Need a method to separate the cases q M W and q M W. Factorization Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

10 For illustration 1-dimensional toy integral Full theory integral with two distinct scales : m M I 0 dk M ln[m/m] = M (k + M)(k + m) M m ln [ m M In this simple example, the calculation in the full theory is easy. Why to switch to effective theories anyway? Integrals with different mass scales become more difficult to calculate in realistic cases, in particular at higher-loop order. The product of coupling constants and large logarithms may be too large for fixed-order perturbation theory to work, g 2 ln m M O(1) ] The physics at the small scale m might involve non-perturbative phenomena, for instance, if m represents the light quark masses in QCD. Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

11 For illustration 1-dimensional toy integral Full theory integral with two distinct scales : m M I 0 dk M ln[m/m] = M (k + M)(k + m) M m ln [ m M In this simple example, the calculation in the full theory is easy. Why to switch to effective theories anyway? Integrals with different mass scales become more difficult to calculate in realistic cases, in particular at higher-loop order. The product of coupling constants and large logarithms may be too large for fixed-order perturbation theory to work, g 2 ln m M O(1) ] The physics at the small scale m might involve non-perturbative phenomena, for instance, if m represents the light quark masses in QCD. Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

12 Hard and soft regions of an integral via cut-off procedure Intuitive procedure: introduce a cut-off m < µ < M: I cut s (µ) = µ 0 dk h (µ) = dk I cut µ M k µ M (k + M)(k + m) M k µ m (k + M)(k + m) µ 0 µ dk dk 1 [ µ ] k + m ln m M [ µ ] (k + M)(k) ln M Interpretation: The soft (low-energy) part does not depend on ln M. Can be calculated within the low-energy effective theory. The hard (short-distance) does not depend on ln m. Take into account by re-adjusting the (Born-level) effective coupling constants. The cut-off dependence cancels in the combination of soft and hard pieces. Sub-leading terms correspond to power-suppressed operators. Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

13 Hard and soft regions of an integral via cut-off procedure Intuitive procedure: introduce a cut-off m < µ < M: I cut s (µ) = µ 0 dk h (µ) = dk I cut µ M k µ M (k + M)(k + m) M k µ m (k + M)(k + m) µ 0 µ dk dk 1 [ µ ] k + m ln m M [ µ ] (k + M)(k) ln M Interpretation: The soft (low-energy) part does not depend on ln M. Can be calculated within the low-energy effective theory. The hard (short-distance) does not depend on ln m. Take into account by re-adjusting the (Born-level) effective coupling constants. The cut-off dependence cancels in the combination of soft and hard pieces. Sub-leading terms correspond to power-suppressed operators. Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

14 Comment: For technical reasons, one often uses dimensional regularization instead of momentum cut-offs, to separate hard and soft momentum regions... Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

15 Back to the real case: Hard gluon corrections to the current-current interaction modify the effective coupling constants. Colour matrices attached to the quark-gluon vertex, second current-current operator with another colour structure. H eff = V cb V ud G F 2 i=1,2 C i (µ) O i (b cdū) The so-called Wilson coefficients C i (µ) contain all the information about short-distance physics above the scale µ (SM and NP) [see Buchalla/Buras/Lautenbacher 1996] Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

16 What did we gain? At 1-loop, Wilson coefficients have generic form: C i (µ) = { } 1 + α ( s(µ) a (1) 0 i ln µ ) + δ (1) i + O(α 4π M s) 2 W Wilson coefficients depend on the scale µ Matching For µ M W the logarithmic term is small, and C i (M W ) can be calculated in fixed-order perturbation theory, since α s (M W )/π 1. { } 1 C i (M W ) = + δ (1) α s (M W ) 0 i π Here M W is called the matching scale. Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

17 Evolution of effective couplings to small scales Remember: soft loop integrals in effective theory involve ln µ/m. In order not to obtain large logarithmic coefficients, we would like to perform calculations in the low-energy effective theory in terms of C i (µ m b ) rather than C i (M W ). In perturbation theory, we can calculate the scale dependence: ( ) ln µ C αs (µ) i(µ) γ ji (µ) C j (µ) = 4π γ(1) ji +... C j (µ) γ ij (µ) is called anomalous dimension matrix Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

18 Comment: RG evolution at leading-log accuracy Formal solution of differential equation: (separation of variables) [ ln µ ] C(µ) = C(M) exp d ln µ γ(µ ) Perturbative expansion of anomalous dimension and β-function ln M γ(µ) = αs 4π γ(1) β(µ) d d ln µ αs(µ) = 2β 0 4π α2 s(µ) +... change variables, d ln µ = dα s /2β, to obtain ( ) γ αs (µ) (1) /2β 0 C(µ) C(M W ) (LeadingLogApprox) α s (M W ) Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

19 Numerical values for C 1,2 operator: O 1 = ( s L γ µ u L )( c L γ µ b L ) O 2 = ( s L γ µ T a u L )( c L γ µ T a b L ) C i (m b ): (LL) (LL) (NLL) (NLL) depends on M W, M Z, sin θ W, m t, m b, α s (SM) to be modified in NP scenarios Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

20 Summary: Effective Theory for b-quark decays Full theory all modes propagate Parameters: M W, m q, g, α s... µ > M W ( ) C i (M W ) = C tree i 1 + δ (1) α s(m W ) i 4π +... matching: µ M W Eff. theory low-energy modes propagate. High-energy modes are integrated out. Parameters: m b, α s, C i (µ)... µ < M W ln µ C i(µ) = γ ji (µ) C j (µ) anomalous dimensions Expectation values of operators O i at µ = m b. All dependence on M W absorbed into C i (m b ) resummation of logs Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

21 From b cdū to B 0 D + π In experiment, we cannot see the quark transition directly. Rather, we observe exclusive hadronic transitions, described by hadronic matrix elements, like e.g. D + π H SM B 0 d = V cbv ud r i (µ) = D + π O i B 0 d µ G F 2 i C i (µ) r i (µ) The hadronic matrix elements r i contain QCD (and also QED) dynamics below the scale µ m b. Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

22 "Naive" Factorization of hadronic matrix elements r i = D + J (b c) i B d 0 }{{} π J (d u) i 0 }{{} Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

23 "Naive" Factorization of hadronic matrix elements r i = D + J (b c) i B d 0 }{{} π J (d u) i 0 }{{} form factor decay constant Part of the gluon effects encoded in simple/universal had. quantities Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

24 "Naive" Factorization of hadronic matrix elements r i (µ) = D + J (b c) i B d 0 }{{} π J (d u) i 0 }{{} + corrections(µ) form factor decay constant Gluon cross-talk between π and B D QCD corrections Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

25 QCD factorization [Beneke/Buchalla/Neubert/Sachrajda 2000] light quarks in π have large energy (in B rest frame) gluons from the B D transition see small colour-dipole corrections to naive factorization dominated by gluon exchange at short distances 1/m b New feature: Light-cone distribution amplitudes φ π (u) momenta/energies of light quarks in π cannot be neglected compared to reference scale µ m b Short-distance corrections to naive factorization given as convolution r i (µ) j F (B D) j (m 2 π) 1 0 du ( 1 + α ) sc F 4π t ij(u, µ) +... f π φ π (u, µ) φ π (u) : distribution of momentum fraction u of a quark in the pion. t ij (u, µ) : perturbative coefficient function (depends on u) Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

26 QCD factorization [Beneke/Buchalla/Neubert/Sachrajda 2000] light quarks in π have large energy (in B rest frame) gluons from the B D transition see small colour-dipole corrections to naive factorization dominated by gluon exchange at short distances 1/m b New feature: Light-cone distribution amplitudes φ π (u) momenta/energies of light quarks in π cannot be neglected compared to reference scale µ m b Short-distance corrections to naive factorization given as convolution r i (µ) j F (B D) j (m 2 π) 1 0 du ( 1 + α ) sc F 4π t ij(u, µ) +... f π φ π (u, µ) φ π (u) : distribution of momentum fraction u of a quark in the pion. t ij (u, µ) : perturbative coefficient function (depends on u) Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

27 Light-cone distribution amplitude for the pion ΦΠ u 1.4 (for illustration only) u Exclusive analogue of parton distribution function: PDF: probability density (all Fock states) LCDA: probability amplitude (one Fock state, e.g. q q) Phenomenologically relevant u 1 π 3.3 ± 0.3 [from sum rules, lattice, exp.] Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

28 Complication: Annihilation in B d D + π Second topology for hadronic matrix element possible: Tree (class-i) Annihilation (class-iii) annihilation is power-suppressed by Λ/m b numerically difficult to estimate Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

29 Still more complicated: B D 0 π Second topology with spectator quark going into light meson: Tree (class-i) Tree (class-ii) class-ii amplitude does not factorize into simpler objects (colour-transparency argument does not apply) again, it is power-suppressed compared to class-i topology Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

30 Non-factorizable: B 0 D 0 π 0 In this channel, class-i topology is absent: Tree (class-ii) Annihilation (class-iii) The whole decay amplitude is power-suppressed! Naive factorization is not even an approximation! Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

31 QCDF - Generic statements: QCD corrections to hadronic matrix elements match µ dependence of Wilson coefficients Strong phases are suppressed by α s or Λ QCD /m b to be tested in experiment Discussion of hadronic input B D form factors rather well known (heavy-quark limit, exp. data on B Dlν) pion decay constant pion distribution amplitude: expanded into set of polynomials first few coefficients known from lattice/sum rules power corrections of order Λ QCD /m b contain genuinely non-perturbative ( non-factorizable ) correlations between B, D and π. (?) Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

32 QCDF - Generic statements: QCD corrections to hadronic matrix elements match µ dependence of Wilson coefficients Strong phases are suppressed by α s or Λ QCD /m b to be tested in experiment Discussion of hadronic input B D form factors rather well known (heavy-quark limit, exp. data on B Dlν) pion decay constant pion distribution amplitude: expanded into set of polynomials first few coefficients known from lattice/sum rules power corrections of order Λ QCD /m b contain genuinely non-perturbative ( non-factorizable ) correlations between B, D and π. (?) Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

33 Alternative: Isospin analysis for B Dπ Employ isospin symmetry between (u, d) of strong interactions. Final-state with pion (I = 1) and D-meson (I = 1/2). The three possible decay modes (+ CP conjugates) described by only two isospin amplitudes: 1 2 A( B d D + π ) = 3 A 3/2 + 3 A 1/2, A( Bd D 0 π 0 ) = 3 A 3/2 3 A 1/2, A(B D 0 π ) = 3 A 3/2, QCDF: A 1/2 /A 3/2 = 2 + corrections Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

34 Isospin amplitudes from experimental data [BaBar hep-ph/ ] A 1/2 2 A3/2 = , cos θ = corrections to naive factorization of order 35% relative strong phases from FSI of order 30 Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

35 Next Example: b s(d) q q decays Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

36 b s(d) q q decays current-current operators Now, there are two possible flavour structures: V ub V us(d) (ū Lγ µ b L )( d(s) L γ µ u L ) V ub V us(d) Ou 1, V cb V cs(d) ( c Lγ µ b L )( d(s) L γ µ c L ) V cb V cs(d) Oc 1, Again, α s corrections induce independent colour structure O u,c 2. Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

37 b s(d) q q decays strong penguin operators New feature: Penguin diagrams induce additional operator structures V tb V ts(d) O 3 6 Wilson coeff. C 3 6 numerically suppressed by loop factor. Potential sensitivity to new-physics contribution. Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

38 Comments on operator basis for b s(d) q q Also electroweak penguin diagrams with γ/z instead of gluon. electroweak penguin operators O 7 10 Also electromagnetic and chromomagnetic operators O γ 7 and Og 8. Mixing between operators under renormalization. Use unitarity of CKM matrix, V tb V ts(d) = V ubv us(d) V cbv cs(d) yields two sets of operators with different weak phase. Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

39 Charmless non-leptonic decays: B ππ and B πk Naive factorization: Both final-state mesons are light and energetic. Colour-transparency argument applies for class-i and class-ii topologies. B π(k ) form factors fairly well known (QCD sum rules) Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

40 Corrections to naive factorization 4 kinds of partonic momentum configurations (in B rest frame) heavy b quark: p b m b (1, 0, 0) soft spectator in B-meson: p s (0, 0, 0) collinear quarks in meson 1 : p c1 m b /2 (1, 0, +1) collinear quarks in meson 2 : p c2 m b /2 (1, 0, 1) Internal interactions: heavy soft coll 1 coll 2 heavy heavy hard hard soft heavy soft hard-coll 1 hard-coll 2 coll 1 hard hard-coll 1 coll 1 hard coll 2 hard hard-coll 2 hard coll 2 where hard modes have virtualities of order m b hard-collinear modes have invariant mass Λm b Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

41 Corrections to naive factorization 4 kinds of partonic momentum configurations (in B rest frame) heavy b quark: p b m b (1, 0, 0) soft spectator in B-meson: p s (0, 0, 0) collinear quarks in meson 1 : p c1 m b /2 (1, 0, +1) collinear quarks in meson 2 : p c2 m b /2 (1, 0, 1) Internal interactions: heavy soft coll 1 coll 2 heavy heavy hard hard soft heavy soft hard-coll 1 hard-coll 2 coll 1 hard hard-coll 1 coll 1 hard coll 2 hard hard-coll 2 hard coll 2 where hard modes have virtualities of order m b hard-collinear modes have invariant mass Λm b Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

42 QCDF for B ππ and B πk decays (BBNS 1999) Factorization formula has to be extended: "Hard" corrections are treated as in B Dπ Take into account new penguin diagrams! ( Fig.) "Hard-collinear" corrections involve spectator quark in B-meson ( Fig.) Sensitive to the distribution of the spectator momentum ω light-cone distribution amplitude φ B (ω) Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

43 Additional diagrams for hard corrections in QCDF (example) additional contributions to the hard coefficient functions t ij (u, µ) r i (µ) hard j F (B π) 1 j (mk 2 ) du 0 ( 1 + α ) s 4π t ij(u, µ) +... f K φ K (u, µ) Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

44 Additional diagrams for hard corrections in QCDF (example) additional contributions to the hard coefficient functions t ij (u, µ) r i (µ) hard j F (B π) 1 j (mk 2 ) du 0 ( 1 + α ) s 4π t ij(u, µ) +... f K φ K (u, µ) Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

45 Spectator corrections with hard-collinear gluons in QCDF additive correction to naive factorization ( αs ) r i (µ) = du dv dω spect. 4π h i(u, v, ω, µ) +... f K φ K (u, µ) f π φ π (v, µ) f B φ B (ω, µ) Distribution amplitudes for all three mesons involved! Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

46 New ingredient: LCDA for the B-meson ΦB Ω Ω GeV Phenomenologically relevant: ω 1 B (1.9 ± 0.2) GeV 1 (at µ = m b Λ 1.5 GeV) (from QCD sum rules [Braun/Ivanov/Korchemsky]) (from HQET parameters [Lee/Neubert]) Large logarithms with ratio of hard and hard-collinear scale appear! Can be resummed using Soft-Collinear Effective Theory Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

47 Complications for QCDF in B ππ, πk etc. Annihilation topologies are numerically important. BBNS use conservative model estimates. Some power-corrections are numerically enhanced by chiral factor µ π = m2 π f π 2f π m q Many decay topologies interfere with each other. Many hadronic parameters to vary. Hadronic uncertainties sometimes quite large. Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

48 Phenomenology of B ππ decays Phenomenological parametrization: (using isospin; neglect EW peng.) A( B 0 d π+ π ) = V ub V ud T ππ + V cb V cd P ππ, 2 A(B π π 0 ) V ub V ud (T ππ + C ππ ), 2 A( B0 d π 0 π 0 ) = V ub V ud C ππ V cb V cd P ππ Recent predictions from QCDF Ratio Value/Range P ππ /T ππ ( ) i C ππ /T ππ ( ) i Remark: P/T ratio important to extract the CKM angle α from CP asymmetries in B ππ. (incl. part of NNLO corrections) [Beneke/Jäger, hep-ph/ ] Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

49 Phenomenology of B ππ decays Phenomenological parametrization: (using isospin; neglect EW peng.) A( B 0 d π+ π ) = V ub V ud T ππ + V cb V cd P ππ, 2 A(B π π 0 ) V ub V ud (T ππ + C ππ ), 2 A( B0 d π 0 π 0 ) = V ub V ud C ππ V cb V cd P ππ Recent predictions from QCDF Ratio Value/Range P ππ /T ππ ( ) i C ππ /T ππ ( ) i Remark: P/T ratio important to extract the CKM angle α from CP asymmetries in B ππ. (incl. part of NNLO corrections) [Beneke/Jäger, hep-ph/ ] Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

50 Phenomenology of B ππ decays Phenomenological parametrization: (using isospin; neglect EW peng.) A( B 0 d π+ π ) = V ub V ud T ππ + V cb V cd P ππ, 2 A(B π π 0 ) V ub V ud (T ππ + C ππ ), 2 A( B0 d π 0 π 0 ) = V ub V ud C ππ V cb V cd P ππ Recent predictions from QCDF Ratio Value/Range P ππ /T ππ ( ) i C ππ /T ππ ( ) i Remark: P/T ratio important to extract the CKM angle α from CP asymmetries in B ππ. (incl. part of NNLO corrections) [Beneke/Jäger, hep-ph/ ] Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

51 Phenomenology of B πk decays Important: Tree amplitudes are CKM suppressed. Electroweak penguins are non-negligible. Phenomenological parametrization: (4th amplitude from isospin) A(B π K 0 ) = P ( 1 + ɛ a e iφa e iγ), 2 A(B π 0 K ) = P ( 1 + ɛ a e iφa e iγ ɛ 3/2 e iφ ( e iγ qe iω)), A( B d 0 π+ K ) = P ( 1 + ɛ a e iφa e iγ ɛ T e iφ ( T e iγ q C e iω )) C with ɛ 3/2,T,a V ub Vus V cb Vcs ɛ KM = O(λ 2 ) q, q C = O(1/λ 2 ) makes a priori 11 independent hadronic parameters! Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

52 Qualitative Results for B πk parameters: SU(3) F symmetry predicts: q e iω 3 2ɛ KM C 9 + C 10 C 1 + C ɛ a e iφa is negligible in QCDF. Consequence: Direct CP asymmetry in B π K 0 is tiny. q C e iω C is of minor numerical importance. ɛ T and ɛ 3/2 are of the order 30%. Related strong phases are of the order of 10. Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

53 Exploiting the approximate SU(3) F symmetry Relate e.g. B πk and B ππ amplitudes Explicitely realized in BBNS approach (small SU(3) F violation from f π f K, F B π F B K etc.) To control SU(3) F violations in a model-independent way, one needs the whole set of B u,d,s (π, K, η) 2 decays [see, e.g., Buras/Fleischer/Recksiegel/Schwab] Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

54 Other 2-body charmless B-decays Similar considerations for B PV and B VV decays. [ Beneke/Neubert 2004] Different interference pattern between various decay topologies. New issues with flavour-singlet mesons (η, η,...) Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

55 Last Example: b s(d)γ decays Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

56 Operator basis for b s(d)γ decays Same operator basis as for b s(d) q q decays: current-current operators: O u,c 1,2 strong penguins: O 3 6 EW penguins: O 7 10 chromomagnetic: O g 8 electromagnetic: O γ 7 Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

57 Inclusive B X s γ decays Decay signature allows for an almost inclusive measurement of the b sγ transition: Sum over all final states with one (energetic) photon, and hadronic system with open strangeness. To first approximation, the inclusive decay rate equals the partonic rate for b sγ, since the probability to find a b-quark with (almost) the same momentum as the B-meson is nearly one. Radiative corrections to the partonic rate can be calculated in terms of α s (m b ). Corrections to the partonic rate arise from the binding effects of the b-quark in the B-meson, and can be systematically expanded in Λ/m b. Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

58 Leading partonic contribution Γ O 7 B X sγ = αg2 F m5 b 32π 4 V tb V ts 2 [ C γ 7 (m b) ] 2 ( 1 + δ m 2 b +... ) δ = λ 1 2 9λ 2 2 arises from the kinetic and chromomagnetic term in HQET. related uncertainty can be reduced by normalizing to B X c lν rate. Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

59 Example: Correction term involving O2 c ( Γ Oc 2 B X = sγ ΓLO B X sγ 1 9 C 2 C γ 7 λ 2 m 2 c ) +... Only suppressed by 1/m 2 c. Gives an order 3% correction. Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

60 Status of the theoretical calculation: [Misiak et al, hep-ph/ ] B( B X s γ) th(1) = (3.15 ± 0.23) 10 4 for photon-energy cut E γ > 1.6 GeV. includes corrections of order α 2 s 5% uncertainty from non-perturbative parameters 3% uncertainty from perturbative input parameters 3% from higher-order effects 3% from m c-dependence of loop integrals (not exactly treated) Comparison with exp. value (HFAG): B( B X s γ) exp = (3.55 ± ± 0.03) 10 4 (E γ > 1.6) Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

61 Sensitivity to the B-meson shape function Presence of the photon-energy cut induces a second scale = m b 2E γ 1.4 GeV, m b Perturbative coefficients enhanced by ln [ /m b ] 1 What scale to choose for α s? α s (m b ) 0.2, vs. α s ( ) Need method to separate the scales and m b (At least, if we aim for precision tests of the SM) Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

62 Decay kinematics: p µ b := m b (1, 0) + k µ, k Λ p µ γ := E γ (1, 0, 1), γ m b 2E γ < cut ps 2 = (p b p γ ) 2 = mb 2 + 2m b k 0 + k 2 2E γ (m b + k 0 k z ) = m b ( γ + k 0 + k z ) + γ (k 0 k z ) + k 2 m b ( γ + k 0 + k z ) For small γ invariant mass of hadronic system small Jet Dynamics at the intermediate scale ps 2 mb 2 sensitive to residual momentum of the b-quark in the B-meson! Shape function: S(k + = k 0 + k z ) = Parton Distribution Function Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

63 Momentum configuration photon b-quark }{{} low-energy quarks and gluons } hadronic jet from energetic quarks and gluons B-Meson rest frame: m b 2E γ = γ m b long-distance dynamics: heavy b-quark: p µ b = m bv + k µ s soft quarks and gluons: p s µ Λ m b soft matrix elements in HQET (shape function) short-distance dynamics: hard modes: p 2 O(mb) 2 perturbative coefficients from QCD ( collinear ) jet modes: pjet 2 m b mb 2 perturbative jet function from SCET Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

64 Momentum configuration photon b-quark }{{} low-energy quarks and gluons } hadronic jet from energetic quarks and gluons B-Meson rest frame: m b 2E γ = γ m b long-distance dynamics: heavy b-quark: p µ b = m bv + k µ s soft quarks and gluons: p s µ Λ m b soft matrix elements in HQET (shape function) short-distance dynamics: hard modes: p 2 O(mb) 2 perturbative coefficients from QCD ( collinear ) jet modes: pjet 2 m b mb 2 perturbative jet function from SCET Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

65 Momentum configuration photon b-quark }{{} low-energy quarks and gluons } hadronic jet from energetic quarks and gluons B-Meson rest frame: m b 2E γ = γ m b long-distance dynamics: heavy b-quark: p µ b = m bv + k µ s soft quarks and gluons: p s µ Λ m b soft matrix elements in HQET (shape function) short-distance dynamics: hard modes: p 2 O(mb) 2 perturbative coefficients from QCD ( collinear ) jet modes: pjet 2 m b mb 2 perturbative jet function from SCET Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

66 Ingredients of the factorization theorem Hard Function: describes the dynamics on the scale µ = m b perturbatively calculable in QCD in terms of α s (m b ) Jet Function: describes the dynamics on the intermediate scale µ i = m b perturbatively calculable in SCET in terms of α s (µ i ) large logarithms resummed via renormalization in SCET Shape Function: describes the non-perturbative dynamics of the b-quark if is not too small it can be expanded in terms of moments, which are related to the usual HQET parameters. Numerical estimate: [Neubert/Becher] B( B X s γ) th(2) = ( ) 10 4 (5% smaller value than fixed order result, more conservative(?) error estimate, 19% smaller than experiment (1.4 σ)) Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

67 Alternative view: E γ spectrum measures the shape function: Shape-function independent relations Shape functions are universal (depend only on B-meson bound state) Only one SF at leading power in 1/m b (= leading twist in DIS) Decay spectra for B X u lν and B X s γ only differ by perturbatively calculable short-distance functions Shape-function independent relations (re-weighting the B X sγ spectrum to predict B X ulν) Rather precise determination of V ub (2-loop + estimate of power corr.) [Lange/Neubert/Paz, hep-ph/ ] Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

68 Exclusive B K γ decay Naive factorization B K form factor fairly well known (QCD sum rules) The photon does not couple to gluons,...,... should there be any corrections at all?! Actually, the photon can split into (q q) just like a meson! Factorization of QCD effects, similar to B ππ. Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

69 Exclusive B K γ decay Naive factorization B K form factor fairly well known (QCD sum rules) The photon does not couple to gluons,...,... should there be any corrections at all?! Actually, the photon can split into (q q) just like a meson! Factorization of QCD effects, similar to B ππ. Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

70 Example: Spectator scattering correction to B K γ LCDAs for B- and K -meson enter Photon still behaves point-like, to first approximation Power corrections also require LCDA for the photon! Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

71 Application: V td /V ts from B ργ vs. B K γ Constraint on Wolfenstein parameters ( ρ, η) compared to global CKM fit [Plot from UTfit Collaboration] Ratio of decay widths (B-factories): Γ[B 0 ρ 0 γ] Γ[B 0 K 0 γ] V td 2 F B ρ V ts F B K (1 + ) form factor ratio = 1 in SU(3) F symmetry limit deviations can be estimated from non-perturbative methods correction term estimated from QCD factorization [for more details, see Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

72 Generalization to B K l + l Two new operators O ll 9 = ( s Lγ µ b L )( lγ µ l), O ll 10 = ( s Lγ µ b L )( lγ µ γ 5 l) Form factors for B K ll fulfill approximate symmetry relations. QCD corrections perturbatively calculable for m 2 ρ < q 2 < 4m 2 c. dγ[b K l + l ] dq 2 d cos θ»(1 + cos 2 θ) 2q2 m 2 B h i 2 f (q 2 ) C 9, (q 2 ) 2 + C10 2 h i 2 +(1 cos 2 θ) f (q 2 ) C 9, (q 2 ) 2 + C10 2 h i cos θ 8q2 2 h f mb 2 (q 2 ) Re C 9, (q 2 ) ic 10 Functions C 9,, (q 2 ) contain the non-trivial QCD corrections! Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

73 Important application: FB asymmetry zero Leading hadronic uncertainties from form factors drop out: 0 =! Re [ C 9, (q0) 2 ] [ = Re C9 ll + 2m ] bm B q0 2 C γ 7 + Y (q2 0) + QCDF corr. + power corr Theoretical uncertainties incl.: hadronic input parameters electro-weak SM parameters variation of factorization scale [from Beneke/TF/Seidel 01] Asymmetry zero in SM: q 2 0 = (4.2 ± 0.6) GeV2 (incl. estimated 10% uncertainty from undetermined power corrections) Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60

74 Summary When looking for new physics, do not forget about the complexity of the old physics! Th. Feldmann (Uni Siegen) Hadronic Effects in B -Decays March / 60