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Lecture 2 Factorization in Inclusive B Decays

Lecture 2 Factorization in Inclusive B Decays. Soft-collinear factorization Factorization in B→X s γ decay m b from B→X s γ moments |V ub | from B→X u l ν decay spectra. Soft-Collinear Factorization.

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Lecture 2 Factorization in Inclusive B Decays

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  1. Lecture 2Factorization in Inclusive B Decays • Soft-collinear factorization • Factorization in B→Xsγ decay • mb from B→Xsγmoments • |Vub| from B→Xulν decay spectra Heraeus Summer School

  2. Soft-Collinear Factorization Kinematics in heavy-to-light processes, Soft and collinear modes, Effective field theory Heraeus Summer School

  3. Motivation • Separation of scales (“factorization”) is crucial to many applications of QCD • Wilsonian OPE: integrate out heavy particles or large virtualities (Fermi theory, HQET, correlators at large Q2, …) • Expansion in (ΛQCD/Q)2n and αs(Q) Q2» ΛQCD2 Heraeus Summer School

  4. Complication • Jet-light physics: large energies and momenta, but small virtualities • e+e-→jets, B→light particles, … • Light-cone kinematics How to integrate out short-distance physics in a situation where pμ is large, but p2 small? Heraeus Summer School

  5. B-factory physics • Much interest in B→light processes: • |Vub| determinations • Angles of the unitarity triangle • Rare decays, searches for New Physics • Large-recoil processes (fast light particles) Heraeus Summer School

  6. lν jet Challenge • Construct short-distance expansions for processes involving both soft and energetic light partons • Soft: psoft ~ ΛQCD • Collinear: pcol2 « Ecol2 • psoft•pcol ~ EcolΛ semi-hard scale • Technology: effective field theory, OPE B b Heraeus Summer School

  7. Soft-collinear effective theory [Bauer, Pirjol, Stewart & Fleming, Luke] • Systematic power counting in λ=ΛQCD/E • Effective Lagrangians for strong and weak interactions expanded in powers of λ • More complicated than previous heavy-quark expansions • Expansion in non-local string operators integrated over light-like field separation • Many degrees of freedom Heraeus Summer School

  8. Different versions of SCET [Bauer, Pirjol, Stewart; Beneke, Feldmann et al.; Chay, Kim] • SCET-1: hard-collinear & soft • E.g.: inclusive B→Xsγ and B→Xulν decays, jet physics • SCET-2: collinear & soft & soft-collinear • E.g.: exclusive B→ππ, B→K*γ decays, B→light form factors • Often 2-step matching: [Becher, Hill, MN] QCD → SCET-1 → HQET / SCET-2 Heraeus Summer School

  9. Factorization in B→Xsγ Partially inclusive decay rate γ B Xs FCNC Heraeus Summer School

  10. Different scales • Consider partial rate integrated over Eγ> E0 • Cut on photon energy (E0 ≈1.8 GeV) introduces new scaleΔ = mb - 2E0 ≈ 1 GeV • Important to disentangle short-distance physics at scale mb from soft physics at scale Δ Belle 04 Heraeus Summer School

  11. Relevant modes • Hard: pμ ~ mb • Hard-collinear: p- ~ mb, p+ ~ , p┴ ~ mbΔ (p2 ~ mbΔ ~ inv. hadr. mass2) • Soft: pμ ~ Δ • 2-step matching: mb mbΔ QCD → SCET-1 → HQET Δ Heraeus Summer School

  12. Soft-collinear (QCD) factorization • Systematic separation of short- and long-distance physics order by order in 1/mb: [Korchemsky, Sterman] [Lee, Stewart] [Bosch, MN, Paz] Soft functions (~) Hard functions (~mb) Jet functions (~ mb) Heraeus Summer School

  13. Different kinematical regions • Δ ~ ΛQCD:shape-function region • Need for nonperturbative structure functions (matrix elements of light-cone string ops.) • mb » Δ » ΛQCD:multi-scale OPE region • Model-independent predictions in terms of heavy-quark parameters • mb ~ Δ:conventional OPE region Heraeus Summer School

  14. Multi-scale OPE region Shape function region OPE region Different kinematical regions mb mbΔ Scales Δ Nonperturbative ! E0 [GeV] Heraeus Summer School

  15. Scale separation (MSOPE) • Master formula for the rate: Γ ~ H(μh) * U(μh,μi) * J(μi) * U(μi,μ0) * M(μ0) QCD → SCET → (RG evolution) → HQET → (RG evolution) → local OPE Perturbation theory Nonperturbative physics Heraeus Summer School

  16. Partial B→Xsγ branching ratio • Theoretical calculation with a cut at E0 = 1.8GeV: • Experiment (Belle 2004): Br(1.8GeV) = (3.30 ± 0.33[pert] ± 0.17[pars]) • 10-4 [MN] Br(1.8GeV) = (3.38 ± 0.30[stat] ± 0.28[syst]) • 10-4 Heraeus Summer School

  17. Implications for New Physics • Larger theory errors, and better agreement between theory and experiment, weaken constraints on parameter space of New Physics models! • E.g., type-II two-Higgs doublet model: • m(H+) > 200 GeV (95% CL) (compared with previous bound of 500 GeV) Heraeus Summer School

  18. Factorization in B→Xsγ Determination of mb from moments of the photon spectrum γ B Xs FCNC Heraeus Summer School

  19. Moments of photon spectrum • Marvelous QCD laboratory • Extraction of heavy-quark parameters (mb,μπ2) with exquisite precision • Calculations achieved: • Full two-loop corrections (+ 3-loop running) • Second NNLO calculation in B physics • Same accuracy for leading power corrections ~(ΛQCD/Δ)2; fixed-order results for 1/mb terms Heraeus Summer School

  20. Jet function Soft function Dependence on E0 Scale separation (MSOPE) • A wonderful formula (exact): [MN] with: Scales: μh ~ mb μi ~ mbΔ μ0 ~ Δ Heraeus Summer School

  21. Perturbation theory • Hard, jet, and soft matching coefficients computed at O(αs) [Bauer, Manohar; Bosch et al.; MN] • Momentum-dependent corrections to jet and soft functions known to 2 loops [MN] • Cusp anomalous dimension computed to 3 loops [Moch, Vermaseren, Vogt] • Shape-function anomalous dimension computed at 2 loops [Korchemsky, Marchesini; Gardi; MN] • Jet-function anomalous dimension derived at 2 loops [MN] Heraeus Summer School

  22. Predictions for moments Heraeus Summer School

  23. Fit to Belle data (E0= 1.8 GeV) ! • Fit results: • Combined results (B→Xsγ and B→Xclν): mb = (4.62±0.10exp±0.03th) GeV μπ2 = (0.11±0.13exp±0.08th) GeV2 Theory uncertainty B→Xclν moments 68% CL 90% CL mb = (4.61±0.06) GeV μπ2 = (0.14±0.06) GeV2 [MN] Heraeus Summer School

  24. |Vub| from B→Xulν Decay Factorization for inclusive decay spectra l ν B Xu SM Heraeus Summer School

  25. Scale separation • Master formula for inclusive decay spectra: Γ ~ H(μh) * U(μh,μi) * J(μi) * U(μi,μ0) * S(μ0) QCD → SCET → (RG evolution) → HQET → (RG evolution) → Shape Function Perturbation theory Nonperturbative physics Heraeus Summer School

  26. Example: B→Xsγ decay • Photon spectrum: • Different components in this formula are obtained from matching calculations Heraeus Summer School

  27. QCD graphs: SCET graphs: Matching 1: QCD → SCET determines hard function H Heraeus Summer School

  28. Matching 1: QCD → SCET • Hard function: Heraeus Summer School

  29. SCET graphs: HQET graphs: Matching 2: SCET → HQET determines jet function J Heraeus Summer School

  30. Nonperturbative input • Shape function of B meson (parton distribution function) can be measured with good precision in B→Xsγ decay • Use result to predict aritrary B→Xulν decay spectra, with arbitrary experimental cuts • Implemented in a generator (“InclusiveBeauty”) • Extraction of |Vub| from a fit to data • Many different strategies • Many cross checks • Conistent results [Lange, MN, Paz] Heraeus Summer School

  31. Inclusive semileptonic decays • Factorization theorem analogous to B→Xsγ • Hadronic phase space most transparent in the variables P= EX ± PX • In practice, Δ = P+ - Λ is always of order ΛQCD for cuts eliminating the charm background Charm background ± OPE region Shape-function region Heraeus Summer School

  32. Strategy • Exploit universality of shape function • Extract shape function in B→Xsγ(fit to photon spectrum), then predict arbitrary distributions in B→Xulνdecay • Functional form of fitting function is constrained by model-independent moment relations • Knowledge of mb and μπ2 helps! • Variant: construct “shape-function independent relations” between spectra (equivalent) [Lange, MN, Paz] Heraeus Summer School

  33. Results for various cuts Theory Error 7.0% Eff = 86% 7.0% Eff = 76% 9.9% 36% 15.0% Eff = 18% 6.6% Eff = 66% 18.9% Eff = 12% Rate Γ ~ (mb)a [Lange, MN, Paz] Heraeus Summer School

  34. Facit • Combined theory error on |Vub| is 5-10% for several different cuts (10% is now conservative – seemed unrealistic only a few years ago) • Average of different extractions will give |Vub| with a total error of less than 10% • Needed to match the precision of sin2β Heraeus Summer School

  35. Impact of precise |Vub| • Realistic: δ|Vub|: ±7% Heraeus Summer School

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