Soft-Collinear Effective Theory and B-Meson Decays

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1 Soft-Collinear Effective Theory and B-Meson Decays Thorsten Feldmann (TU München) presented at 6th VIENNA CENTRAL EUROPEAN SEMINAR on Particle Physics and Quantum Field Theory, November 27-29, 2009 Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

2 Outline: 1 Motivation 2 Formalism 3 Applications SCET in Inclusive B Decays SCET in Exclusive B Decays 4 Summary Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

3 Factorization in QCD: Short-distance modes: Separate short- and long-distance modes Heavy massive particles (e.g. electroweak gauge bosons, top quark, Higgs) Quantum fluctuations with large virtualities. Dynamics encoded in short-distance coefficients (coefficient functions). Calculable in (RG-improved) Perturbation Theory. Long-distance modes: IR-divergences of short-distance coefficients factorization scale µ UV-divergences of operator matrix elements Particles with small masses/virtualities (light quarks, gluons, photons). Dynamics in (hadronic/partonic) matrix elements of composite operators. Requires non-perturbative methods to deal with masses/virtualities of order Λ Λ QCD. Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

4 Factorization in QCD: Short-distance modes: Separate short- and long-distance modes Heavy massive particles (e.g. electroweak gauge bosons, top quark, Higgs) Quantum fluctuations with large virtualities. Dynamics encoded in short-distance coefficients (coefficient functions). Calculable in (RG-improved) Perturbation Theory. Long-distance modes: IR-divergences of short-distance coefficients factorization scale µ UV-divergences of operator matrix elements Particles with small masses/virtualities (light quarks, gluons, photons). Dynamics in (hadronic/partonic) matrix elements of composite operators. Requires non-perturbative methods to deal with masses/virtualities of order Λ Λ QCD. Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

5 QCD Factorization in B-decays We are interested in fundamental flavour parameters, CKM angles, quark masses,... in the SM or its NP extensions. We have to analyze weak decays of b-quarks, but within a non-perturbative hadronic environment! The Procedure: Integrate out EW gauge bosons (NP particles) at (above) the EW scale: Effective Hamiltonian for semi-leptonic decays ( tree ) non-leptonic decays, FCNCs and meson-mixings ( penguins, boxes ) RG-running of Wilson coefficients from µ M W to µ m b QCD factorization to separate perturbative effects (i.e. virtualities scale with mb Λ QCD ) genuine hadronic properties (i.e. universal for B-meson and its decay products) Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

6 QCD Factorization in B-decays We are interested in fundamental flavour parameters, CKM angles, quark masses,... in the SM or its NP extensions. We have to analyze weak decays of b-quarks, but within a non-perturbative hadronic environment! The result: Precise determination of the CKM triangle η excluded area has CL > 0.95 sin 2β ε K α CKM f i t t e r Moriond 09 γ V ub SL V ub τ ν γ α γ excluded at CL > 0.95 ρ β α m d & m s m d ε K sol. w/ cos 2β < 0 (excl. at CL > 0.95) Hadronic parameters: decay constants transition form factors HQET parameters shape functions light-cone distribution amplitudes... Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

7 Nature of IR divergences: for instance, b X sγ (X s : s-quark jet) Large recoil energy: EX m b /2 Invariant mass of hard-collinear jet: p 2 X Λm b γ γ γ b b b s g s g s etc. b-quark interactions with soft gluons described by HQET interactions between soft and collinear jet modes: Sudakov logarithms in b s form factor (involving ln 2 px 2 /m2 b ) propagation of s-quark in soft background described by jet function non-trivial dependence on (residual) b-quark light-cone momentum: b-quark PDF ( shape function for inclusive decays) B-meson LCDA ( light-cone distribution amplitude for exclusive decays) Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

8 Effective Field Theory Approach: QCD SCET ( HQET) Introduce separate field operators for each type of (relevant) IR-mode: collinear fields (light quarks and gluons/photons) for each jet direction. soft fields (light quarks and gluons/photons) quasi-static heavy-quark fields (with soft residual momentum) Construct interaction terms, performing multipole-expansion of soft-collinear vertices SCET Feynman rules. Perturbative matching to QCD at hard scale µ h m b : short-distance coefficient functions (still depend on jet-energy) RG running in SCET resums large logs between hard and jet scale (incl. Sudakov logarithms) matching onto HQET at jet scale yields non-local operators b-quark PDF (for inclusive B decays) B-meson LCDA (for exclusive B decays) [Bauer/Fleming/Luke/Manohar/Pirjol/Rothstein/Stewart/ldots, ], [Chay/Kim, ], [Beneke/Diehl/Feldmann/Chapovsky/..., ], [Becher/Hill/Lange/Neubert/Paz/..., ], [... ] Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

9 Formalism: (see also Andre Hoang s talk... ) Light-cone kinematics Specify collinear momentum direction by light-like vectors n µ + and n µ, (e.g. in rest frame or c.m.s.: n µ + = (1, 0, 0, 1) and nµ = (1, 0, 0, 1)) Decomposition of any Lorentz vector: p µ = (n +p) nµ 2 + pµ + (n p) nµ + 2 collinear particles: (n+p) p (n p) soft particles: (n+p) p (n p) Physical applications require different variants of SCET SCET I: (inclusive reactions (jets); intermediate step for exclusive reactions) (hard-)collinear particles: p 2 (n p)(n +p) Λ (n +p) SCET II: (exclusive reactions) collinear particles: p 2 (n p)(n +p) Λ 2 Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

10 Large and small collinear spinor components different derivative terms for (massless) collinear quarks scale as " # L q coll. = q (in +D) n/ 2 + id/ + (in D) n/+ q 2 O(n +p) O(p ) O(n p) large and small spinor components ξ(x) = n/ n/+ 4 q(x), η(x) = n/+n/ 4 q(x) solve e.o.m. for small spinor component η(x) " # L ξ = ξ 1 (in D) + id/ (in id/ n/ + +D) 2 ξ (still exact for interactions with collinear gluons) Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

11 Multipole Expansion (SCET I ) For some directions, soft fields have larger wavelengths than collinear ones: soft: (n x, x, n +x) ( Λ 1, Λ 1, Λ 1 ) collinear: (n x, x, n +x) ( (n +p) 1, (n +p Λ) 1/2, Λ 1 ) At soft-collinear vertices, expand all soft fields around x µ = (n+x) nµ 2. Field Redefinitions Leading interactions with soft gluons via i(n D) i(n ) + g(n A c)(x) + g(n A s)(x ) Perform field redefinitions for collinear quarks and gluons, with soft Wilson line ξ c(x) Y s(x ) ξ c(x), A c(x) Y s(x ) A c(x) Y s (x ) R ig dt n A s(x +tn ) Y s(x ) = P e 0, (in + gn A s) Y s = 0. Soft gluons decouple from collinear fields to first approximation. Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

12 Matching of external heavy-to-light currents in SCET I Expansion in 1/m b yields: ψ(x) Γ Q(x) C Γ (µ) ( ξ W c)(x) Γ (Y s h v)(x ) +... Collinear Wilson line W c with (in + + n +A c) W c = 0, from resummation of (unsuppressed) collinear gluon radiation from heavy quark. Wilson coefficient C Γ absorbs short-distance corrections from virtual hard gluons. Soft and collinear divergences Sudakov logarithms: C Γ (µ) = 1 αsc «F 2 ln 2 (n +p) π µ +... (sub-leading currents in the power-expansion can be identified in a similar manner) Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

13 Current renormalization in SCET Resum logarithms between hard scale (n +p) and jet scale µ p Λm b using renormalization group in SCET I dc(µ) d ln µ = Γ cusp(α s) ln (n+p) «+ γ(α s) C(µ) µ with (universal) cusp anomalous dimension Γ cusp = 4 αsc F 4π +... Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

14 Jet Function in inclusive reactions J(p 2, µ) 1 Z ji π Im ff d 4 x e ipx 0 T (W c ξ c)(0) ( ξ cw c)(x) 0 factorize soft and (hard-)collinear corrections to propagation of energetic quark (e.g. in hadronic tensor for b sγ) one-loop result in terms of modified plus distributions: 8 < : (7 π2 ) δ(p 2 ) 3 J(p 2, µ) = δ(p 2 ) + αsc F 4π «1 [µ 2 ] p ln(p 2 /µ 2 ) p 2! [µ 2 ] 9 = ; two-loop result and solution of RG-equation also known [Becher/Neubert 06] Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

15 Applications Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

16 1.) B X u lν determination of V ub Factorization Theorem (leading power) p X = E X p X spectrum in B X ulν: dγ u dp X Z 1 0 dy y 1 2a H u(y; µ h ) {z } U(µ h, µ i ) Z p X 0 d ˆω J ym b (p X ˆω); µ i {z } bs(ˆω; µ i ) {z } hard function (QCD) jet function (SCET) shape fct. Cut p X < M2 D /M B 0.66 GeV, in order to suppress charm background RG-evolution functions U(µ h, µ i ) and a = a(µ h, µ i ) similar factorization theorem for B X sγ at large photon energy factorization at higher orders complicated by resolved photon effects [Paz/Lee/Neubert 09] Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

17 Issues: Perturbative calculation of hard function(s) NLO: [Bosch/Lange/Neubert/Paz 04; Bauer/Manohar 03; Bauer/Fleming/Pirjol/Stewart 00] NNLO: [Asatrian et al. 08; Beneke et al. 08; Bell 08] Perturbative calculation of jet function (massless) NNLO: [Becher/Neubert 05/06] (massive) NLO: [Boos/TF/Mannel/Pecjak 05; Fleming/Hoang/Mantry/Stewart 07] Shape-function evolution 2-loop: [Becher/Neubert 05] Extracting the shape function from B X sγ: shape-function independent relations [Lange/Neubert/Paz, Lange 05; Hoang/Ligeti/Luke 05; Leibovich/Low/Rothstein 00] model parametrizations and theoretical uncertainties [see below ] Sub-leading shape-function effects classification [Lee/Stewart 04; Bosch/Neubert/Paz 04; Beneke et al. 04; Tackmann 05] shape-function independent relations (tree-level) [K. Lee 08] Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

18 Issues: Perturbative calculation of hard function(s) NLO: [Bosch/Lange/Neubert/Paz 04; Bauer/Manohar 03; Bauer/Fleming/Pirjol/Stewart 00] NNLO: [Asatrian et al. 08; Beneke et al. 08; Bell 08] Perturbative calculation of jet function (massless) NNLO: [Becher/Neubert 05/06] (massive) NLO: [Boos/TF/Mannel/Pecjak 05; Fleming/Hoang/Mantry/Stewart 07] Shape-function evolution 2-loop: [Becher/Neubert 05] Extracting the shape function from B X sγ: shape-function independent relations [Lange/Neubert/Paz, Lange 05; Hoang/Ligeti/Luke 05; Leibovich/Low/Rothstein 00] model parametrizations and theoretical uncertainties [see below ] Sub-leading shape-function effects classification [Lee/Stewart 04; Bosch/Neubert/Paz 04; Beneke et al. 04; Tackmann 05] shape-function independent relations (tree-level) [K. Lee 08] Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

19 The B-meson shape function (= b-quark pdf in HQET) Definition and Properties bs(ˆω = Λ ω) = B h v δ(ω in D) h v B, (n v = 1, Λ = M B m b ) support 0 ˆω < depends on renormalization scheme for b-quark mass radiative tail at large ˆω positive moments diverge Phenomenological constraints Moments of B X clν spectra (HQET parameters Λ, µ 2 π SF scheme) Photon spectrum in B X sγ (through factorization formula) Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

20 The B-meson shape function (= b-quark pdf in HQET) Approach 1: Parametrization at low input scales, e.g. S(ω, µ 0 ) = N Λ «b 1 ˆω exp b ˆω «Λ Λ + αs(µ 0) π [Becher/Lange/Neubert/Paz] [radiative tail] adjust to HQET parameters Λ, µ 2 π RG evolution to intermediate scale µ i compare with B X sγ spectrum predict B X ulν spectrum ÈË Ö Ö ÔÐ Ñ ÒØ Ë µ Î ½ ¼ ½ Î ½ Î Î Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

21 The B-meson shape function (= b-quark pdf in HQET) Approach 2: [Ligeti/Stewart/Tackmann, arxiv: ] Calculate partonic matrix element: Spart.(ˆω, b µ 0 ) = δ(ˆω) + αs(µ 0) [...] π Generate model shape function via convolution Z bs(ˆω, µ 0 ) := dk S b part.(ˆω k, µ 0 ) F b (k) F b (k) normalized to HQET parameters can be expanded in terms of suitable basis functions systematic studies of theoretical uncertainties in global fits [... to be done] (dγs/deγ) / (Γ0s C 7 incl 2 ) [GeV 1 ] c3=c4=0 c3=±0.15, c4=0 c3=0, c4=±0.15 c3=±0.1, c4=± Eγ [GeV] Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

22 Determination of V ub [update of BLNP-method, Greub/Neubert/Pecjak 09] Values of V ub determined at NLO and NNLO. In the columns labeled V ub the first error is experimental, the second is the sum of all theoretical and parametric errors except for that from m b, and the third is that from m b. Method B exp [10 4 ] V ub [10 3 ] V ub [10 3 ] NLO NNLO E l > 2.1 GeV 3.3 ± 0.2 ± ± ± CLEO E l > 2.0 GeV 5.7 ± 0.4 ± ± ± BABAR E l > 1.9 GeV 8.5 ± 0.4 ± ± ± BELLE M X < 1.7 GeV 12.3 ± 1.1 ± ± ± BELLE M X < 1.55 GeV 11.7 ± 0.9 ± ± ± BABAR P + < 0.66 GeV 11.0 ± 1.0 ± ± ± BELLE P + < 0.66 GeV 9.4 ± 1.0 ± ± ± BABAR NNLO effects important ( 10% shift in V ub ) Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

23 2.) SCET in Exclusive B Decays Factorization Theorems for Decay Amplitudes [Beneke/Buchalla/Neubert/Sachrajda 99; Korchemsky/Pirjol/Yan 99; Beneke/Feldmann 00/03; Bauer/Pirjol/Stewart 01; Lunghi/Pirjol/Wyler 02; Bosch/Hill/Lange/Neubert 03; Becher/Hill/Neubert 05 ] A i (B γ + lept.) = + T II i (µ) φ B (µ) A i (B M + lept.) = ξ M (µ) T I i (µ) + T II i (µ) φ B (µ) φ M (µ) A i (B MM ) = ξ M (µ) T I i (µ) φ M (µ) + T II i (µ) φ B (µ) φ M (µ) φ M (µ) universal transition form factor ξ M (non-perturbative input) two-particle LCDAs for B-meson and light hadrons M (non-perturbative input) perturbative coefficient functions: Ti I ( non-factorizable ) Ti II = H i J ( factorizable in SCET I SCET II) Λ/m b Power-corrections lead to more factorizable and non-factorizable terms! Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

24 Recent Perturbative Results NNLO vertex corrections in non-leptonic B decays (T I I ): imaginary part [Bell 07] real part [Bell 09; Beneke/Huber/Li 09] NLO Spectator scattering in non-leptonic B decays (T II I ): tree amplitudes [Beneke/Jäger 05; Kivel 06; Pilipp 07] leading penguin amplitudes [Beneke/Jäger 06] O(α 2 s) corrections in B V γ decays: Contributions from O γ 7 and Og 8 [Ali/Greub/Pecjak 07] Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

25 Endpoint divergencies and non-factorizable effects Collinear modes in exclusive final state have the same virtualities (i.e. same transverse momenta) as soft spectators in B-meson. Dimensional regularization not sufficient to render integration over individual soft and collinear momentum regions IR-finite. Soft and collinear dynamics entangled in a non-factorizable manner. Still, in the endpoint region certain symmetry relations hold, similar to those known from the Isgur-Wise function in HQET. Universal non-factorizable matrix elements in SCET I, e.g.» /n+ /n π(p) ( ξ hcw hc) Γ (Y sh v) B(v) ξ π(n +p, µ) tr Γ 1 + /v 4 2 Corrections to symmetry relations are factorizable (!) (or power-suppressed) (similar statement for corrections to naive factorization in non-leptonic B decays) Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

26 A toy integral: [Beneke/TF] Consider (UV-finite) integral in D = 4 2ɛ dimensions Z I = [ dk] f 1 [(k l) 2 ][k 2 m 2 ][(p k) 2 m 2 ], with l 2 = m 2, p 2 = 0, and large momentum transfer p l m 2 Integral decomposes into 3 momentum regions: [see Beneke/Smirnov] hard-collinear: Z I hc = [ dk] f 1 = (n +p)(n l) 1 [k 2 (n +k)(n l)][k 2 ][k 2 (n k)(n +p)] ( 1 ɛ ɛ ln µ 2 (n + 1 µ 2 +p)(n l) 2 ln2 (n π2 +p)(n l) 12 ), well-defined in dim-reg can be reproduced by SCET I Feynman rules. Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

27 A toy integral: [Beneke/TF] Consider (UV-finite) integral in D = 4 2ɛ dimensions Z I = [ dk] f 1 [(k l) 2 ][k 2 m 2 ][(p k) 2 m 2 ], with l 2 = m 2, p 2 = 0, and large momentum transfer p l m 2 Integral decomposes into 3 momentum regions: [see Beneke/Smirnov] collinear: Z I c = [ dk] f [ ν 2 ] δ [ (n +k)(n l)] 1+δ [k 2 m 2 ][k 2 m 2 (n k)(n +p)] 1 = 1 «! (n+p)(n l) 1 µ2 + ln + ln, (n +p)(n l) δ ν 2 ɛ m 2 not defined in dim-reg requires additional IR regulator (here analytic reg.) Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

28 A toy integral: [Beneke/TF] Consider (UV-finite) integral in D = 4 2ɛ dimensions Z I = [ dk] f 1 [(k l) 2 ][k 2 m 2 ][(p k) 2 m 2 ], with l 2 = m 2, p 2 = 0, and large momentum transfer p l m 2 Integral decomposes into 3 momentum regions: [see Beneke/Smirnov] soft: Z I s = [ dk] f [ ν 2 ] δ [(k l) 2 ] 1+δ [k 2 m 2 ][ (n k)(n +p)] " # " # 1 1 m2 1 µ2 = ln + ln 1 (n +p)(n l) δ ν 2 ɛ m 2 ɛ 1 2 ɛ µ2 ln m 1! µ ln2 m + 5π not defined in dim-reg requires additional IR regulator (here analytic reg.) Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

29 A toy integral: [Beneke/TF] Consider (UV-finite) integral in D = 4 2ɛ dimensions Z I = [ dk] f 1 [(k l) 2 ][k 2 m 2 ][(p k) 2 m 2 ], with l 2 = m 2, p 2 = 0, and large momentum transfer p l m 2 Integral decomposes into 3 momentum regions: [see Beneke/Smirnov] additional IR regulator drops out in sum, I = I hc + (I c + I s) {z } +O( m 2 p l ) non-perturbative / non-factorizable Recent ideas to solve the issue by so-called zero-bin subtractions [Manohar/Stewart] have been shown not to work consistently in exclusive decays [Beneke/Vernazza] Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

30 SCET Sum Rules for Heavy-to-light Form Factors [De Fazio/Feldmann/Hurth 05/07] Consider correlation function in SCET I : exclusive final state (e.g. pion) is replaced by (off-shell) interpolating current. Factorization theorem for correlation function (soft hard-collinear) Dispersion relation between (unphysical) region of large (hc) space-like momenta physical spectral function, containing the hadronic state Sum rule for non-factorizable form factor in SCET I: ξ π(q 2, µ) in terms of light-cone distribution amplitudes of B meson correlation function at tree level: J 0 Z Π 0 (n p φ B ) = f B m B dω (ω) ω n p iη 0 m b v ω n+ 2 hc J π (p ) φ B (ω) : distribution amplitude for light-cone momentum of B-meson spectator. Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

31 SCET Sum Rules for Heavy-to-light Form Factors [De Fazio/Feldmann/Hurth 05/07] Consider correlation function in SCET I : exclusive final state (e.g. pion) is replaced by (off-shell) interpolating current. Factorization theorem for correlation function (soft hard-collinear) Dispersion relation between (unphysical) region of large (hc) space-like momenta physical spectral function, containing the hadronic state Sum rule for non-factorizable form factor in SCET I: ξ π(q 2, µ) in terms of light-cone distribution amplitudes of B meson Sum rule after continuum subtraction (ω s) and Borel trafo (ω M ) : m b f π ξ π = 1 π Z ωs 0 dω e ω /ω M Im ˆΠ 0 (ω ) [ω s, ω M : intrinsic sum-rule parameters to be optimized] Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

32 Radiative corrections Corrections involving φ B (ω): J0 J0 J0 s s hc Jπ s s hc Jπ s s hc Jπ J0 J0 J0 s s hc Jπ s s hc L (2) ξq Explicit calculation: Factorization works (on the level of correlator) But O(αs) result for form factor contains (non-resummed) large logarithms involving sum rule parameters. (!) Jπ s s hc L (2) ξq Jπ Corrections involving 3-particle LCWF: (only tree-level available so far [Khodjamirian/Mannel/Offen]) Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

33 Predictions for non-leptonic B decays: Specify hadronic input (lattice, sum rules,... or data). Guesstimate size of power corrections. Calculate corrections to naive factorization to sufficient accuracy (including renormalization running in SCET II ) Compare with experiment. [here: for tree-dom. decays at NNLO, Beneke/Huber/Li 09] Theory I (lattice,sr) Theory II (data) Experiment B π π ( ) ( ) B 0 d π+ π ( ) ( ) 5.16 ± 0.22 B 0 d π0 π ± 0.19 B π ρ ( ) ( ) B π 0 ρ ( ) ( ) B 0 π + ρ ( ) ( ) 15.7 ± 1.8 B 0 π ρ ( ) ( ) 7.3 ± 1.2 B 0 π ± ρ ( ) ( ) 23.0 ± 2.3 B 0 π 0 ρ ± 0.5 B ρ L ρ0 L ( ) ( ) B 0 d ρ+ L ρ L ( ) ( ) B 0 d ρ0 L ρ0 L CP-averaged branching fractions in units of 10 6 of tree-dominated B ππ, πρ and ρ L ρ L decays. The first error on a quantity comes from the CKM parameters, while the second one stems from all other parameters added in quadrature. (,, : additional overall uncertainty from form factors) Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

34 Summary SCET helps: to separate effects associated to different dynamical scales appearing in processes involving soft and energetic particles, to establish the corresponding factorization theorems, to define/identify process-independent non-perturbative input parameters/functions. to resum large logarithms in RG-improved perturbation theory. Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

35 Summary SCET Applications: Inclusive B decays: Factorization theorems ( B-meson shape function) Precise determination of V ub from B X ulν (NNLO, using HQET moments) Not discussed: SM Precision Tests in B X sγ [Lee/Neubert/Paz 09] Not discussed: Shape-function effects in B X sl + l [Lee/Ligeti/Stewart/Tackmann 06] Exclusive B decays: Factorization theorems ( hadronic LCDAs) Endpoint divergencies non-factorizable dynamics ( form factor, power corr.) QCD corrections to non-leptonic B decays ((N)NLO, depending on hadronic input) SCET sum rules for (soft) form factors Collider Physics (QCD + EW): [see A. Hoang s talk] Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

36 Backup Slides: Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

37 Modified plus distributions satisfying Z M «1 [m] du F(u) 0 u Z M «ln(u/m) [m] du F(u) 0 u = = Z M 0 Z M 0 F (u) F (0) du + F (0) ln u 1»ln π Im u «1 1 [m] = m u u 1» π Im ln 2 u 1 ln(u/m) = 2 m u u M m «, F (u) F (0) du ln u u m + F (0) 2 ln2 «[m] π2 3 δ(u), M m «. Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

38 Theoretical accuracy in B X s γ Phenomenological Importance: (large photon energy) Good understanding of dγ/de γ important for V ub extraction (see above) Γ(B X sγ) sensitive to New Physics. Complication: Operators in weak effective Hamiltonian for b s transitions contribute differently to hadronization process: electromagnetic operator O 7 (b sγ) chromomagnetic operator O 8 (b sg) 4-quark operators O 1 6 (b s q q) New effects at sub-leading order in 1/m b expansion: Photon does not couple directly to short-distance b s transition. New Factorization Theorem [Paz/Lee/Neubert 09] Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

39 Structure of New Factorization Formula Features of resolved photon contribution: Involves new jet function J in opposite direction to X s New soft functions from operators that are non-local in 2 light-cone directions Potential mechanism to observe CP Violation in B X sγ Leading mechanism for Isospin Violation in B X sγ Difficult to estimate Vacuum Insertion Approximation 5% Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

40 Light-Cone Distribution Amplitudes B-mesons: [Grozin/Neubert 96] 2-particle LCDAs defined from HQET matrix elements: 0 q(z) β [z, 0] h v(0) α B(v) (with z 2 = 0) 2 independent Dirac structures φ + B (ω), φ B (ω), with light-cone momentum ω of the light quark (after Fourier transform.) Properties: 1/ω moment of φ + B (ω) relevant for leading contribution in factorization theorem. 1-loop evolution equation for φ + B (ω, µ) [Lange/Neubert 03] phenomenological parametrizations: sum rules: ω 1 µ=1 GeV = (2.15 ± 0.5)/GeV [Braun/Ivanov/Korchemsky 03] moment analysis : ω 1 µ=1 GeV = (2.09 ± 0.24)/GeV [Lee/Neubert 05] Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31

41 Non-relativistic Toy-Model for B-meson LCDAs: Light constituent quark mass m (assumption: m Λ) φ ± B (ω, µ m) δ(ω m) Study evolution towards relativistic scales µ m: (WW approx. for φ B (ω)) m φ + B (ω, µ) m φ + B (ω, µ) ω/m ω/m m φ B (ω, µ) m φ B (ω, µ) ω/m ω/m αs (µ)/αs (m) = 1/2 αs (µ)/αs (m) = 1/5 αs (µ)/αs (m) = 1/10 φ + B (ω) ω for ω 0 φ B (ω) const. for ω 0 radiative tail for φ ± B (ω): positive moments do not exist (analogous to SF) [Bell/Feldmann 08, Talk by G. Bell at SCET 08] Th. Feldmann (TUM) SCET and B-decays Vienna, Nov / 31