1. September 3, 2005 Heraeus Summer School 1 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 lν decay spectra

2. September 3, 2005 Heraeus Summer School 2 Soft-Collinear Factorization Kinematics in heavy-to-light processes, Soft and collinear modes, Effective field theory

3. September 3, 2005Heraeus Summer School3 Motivation Separation of scales (“factorization”) is crucial to many applications of QCD 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 Q 2, …) Wilsonian OPE: integrate out heavy particles or large virtualities (Fermi theory, HQET, correlators at large Q 2, …) Expansion in (Λ QCD /Q) 2n and α s (Q) Expansion in (Λ QCD /Q) 2n and α s (Q) Q 2 » Λ QCD 2

4. September 3, 2005Heraeus Summer School4 Complication Jet-light physics: large energies and momenta, but small virtualities Jet-light physics: large energies and momenta, but small virtualities e + e - →jets, B→light particles, … e + e - →jets, B→light particles, … Light-cone kinematics Light-cone kinematics How to integrate out short-distance physics in a situation where p μ is large, but p 2 small?

5. September 3, 2005Heraeus Summer School5 B-factory physics Much interest in B→light processes: Much interest in B→light processes: |V ub | determinations |V ub | determinations Angles of the unitarity triangle Angles of the unitarity triangle Rare decays, searches for New Physics Rare decays, searches for New Physics Large-recoil processes (fast light particles) Large-recoil processes (fast light particles)

6. September 3, 2005Heraeus Summer School6 Challenge Construct short-distance expansions for processes involving both soft and energetic light partons Construct short-distance expansions for processes involving both soft and energetic light partons Soft: p soft ~ Λ QCD Soft: p soft ~ Λ QCD Collinear: p col 2 « E col 2 Collinear: p col 2 « E col 2  p softp col ~ E col Λ semi-hard scale Technology: effective field theory, OPE Technology: effective field theory, OPE lνlν b B jet

7. September 3, 2005Heraeus Summer School7 Soft-collinear effective theory Systematic power counting in λ=Λ QCD /E Systematic power counting in λ=Λ QCD /E Effective Lagrangians for strong and weak interactions expanded in powers of λ Effective Lagrangians for strong and weak interactions expanded in powers of λ More complicated than previous heavy- quark expansions More complicated than previous heavy- quark expansions Expansion in non-local string operators integrated over light-like field separation Expansion in non-local string operators integrated over light-like field separation Many degrees of freedom Many degrees of freedom [Bauer, Pirjol, Stewart & Fleming, Luke]

8. September 3, 2005Heraeus Summer School8 Different versions of SCET SCET-1: hard-collinear & soft SCET-1: hard-collinear & soft E.g.: inclusive B→X s γ and B→X u l ν decays, jet physics E.g.: inclusive B→X s γ and B→X u l ν decays, jet physics SCET-2: collinear & soft & soft-collinear SCET-2: collinear & soft & soft-collinear E.g.: exclusive B→ππ, B→K * γ decays, B→light form factors E.g.: exclusive B→ππ, B→K * γ decays, B→light form factors Often 2-step matching: Often 2-step matching: [Bauer, Pirjol, Stewart; Beneke, Feldmann et al.; Chay, Kim] [Becher, Hill, MN] QCD → SCET-1 → HQET / SCET-2

9. September 3, 2005 Heraeus Summer School 9 Factorization in B→X s γ Partially inclusive decay rate B XsXs FCNC γ

10. September 3, 2005Heraeus Summer School10 Different scales Consider partial rate integrated over E γ > E 0 Consider partial rate integrated over E γ > E 0 Cut on photon energy (E 0 ≈ 1.8 GeV) introduces new scale Δ = m b - 2E 0 ≈ 1 GeV Cut on photon energy (E 0 ≈ 1.8 GeV) introduces new scale Δ = m b - 2E 0 ≈ 1 GeV Important to disentangle short-distance physics at scale m b from soft physics at scale Δ Important to disentangle short-distance physics at scale m b from soft physics at scale Δ Belle 04

11. September 3, 2005Heraeus Summer School11 Relevant modes Hard: p μ ~ m b Hard: p μ ~ m b Hard-collinear: p - ~ m b, p + ~ , p ┴ ~ m b Δ Hard-collinear: p - ~ m b, p + ~ , p ┴ ~ m b Δ (p 2 ~ m b Δ ~ inv. hadr. mass 2 ) (p 2 ~ m b Δ ~ inv. hadr. mass 2 ) Soft: p μ ~ Δ Soft: p μ ~ Δ 2-step matching: 2-step matching: QCD → SCET-1 → HQET mbmb ΔmbΔΔmbΔ Δ

12. September 3, 2005Heraeus Summer School12 Soft-collinear (QCD) factorization Systematic separation of short- and long- distance physics Systematic separation of short- and long- distance physics order by order in 1/m b : [Korchemsky, Sterman] Soft functions (~  Hard functions (~m b ) Jet functions (~ m b   [Lee, Stewart] [Bosch, MN, Paz]

13. September 3, 2005Heraeus Summer School13 Different kinematical regions Δ ~ Λ QCD : shape-function region Δ ~ Λ QCD : shape-function region  Need for nonperturbative structure functions (matrix elements of light-cone string ops.) m b » Δ » Λ QCD : multi-scale OPE region m b » Δ » Λ QCD : multi-scale OPE region  Model-independent predictions in terms of heavy-quark parameters m b ~ Δ: conventional OPE region m b ~ Δ: conventional OPE region

14. September 3, 2005Heraeus Summer School14 Different kinematical regions E 0 [GeV] Scales mbmb mbΔmbΔ Δ Shape function region OPE region Multi-scale OPE region Nonperturbative !

15. September 3, 2005Heraeus Summer School15 Scale separation (MSOPE) Master formula for the rate: Master formula for the rate: Γ ~ H ( μ h ) * U(μ h,μ i ) * J(μ i ) * U(μ i,μ 0 ) * M(μ 0 ) QCD → SCET → (RG evolution) → HQET → (RG evolution) → local OPE Perturbation theory Nonperturbative physics

16. September 3, 2005Heraeus Summer School16 Partial B→X s γ branching ratio Theoretical calculation with a cut at E 0 = 1.8GeV: Theoretical calculation with a cut at E 0 = 1.8GeV: Experiment (Belle 2004): Experiment (Belle 2004): Br(1.8GeV) = (3.30 ± 0.33[pert] ± 0.17[pars]) 10 -4 Br(1.8GeV) = (3.38 ± 0.30[stat] ± 0.28[syst]) 10 -4 [MN]

17. September 3, 2005Heraeus Summer School17 Implications for New Physics Larger theory errors, and better agreement between theory and experiment, weaken constraints on parameter space of New Physics models! 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: E.g., type-II two-Higgs doublet model: m(H+) > 200 GeV (95% CL) (compared with previous bound of 500 GeV) m(H+) > 200 GeV (95% CL) (compared with previous bound of 500 GeV)

18. September 3, 2005 Heraeus Summer School 18 Factorization in B→X s γ Determination of m b from moments of the photon spectrum B XsXs FCNC γ

19. September 3, 2005Heraeus Summer School19 Moments of photon spectrum Marvelous QCD laboratory Marvelous QCD laboratory Extraction of heavy-quark parameters (m b,μ π 2 ) with exquisite precision Extraction of heavy-quark parameters (m b,μ π 2 ) with exquisite precision Calculations achieved: Calculations achieved: Full two-loop corrections (+ 3-loop running) Full two-loop corrections (+ 3-loop running) Second NNLO calculation in B physics Second NNLO calculation in B physics Same accuracy for leading power corrections ~(Λ QCD /Δ) 2 ; fixed-order results for 1/m b terms Same accuracy for leading power corrections ~(Λ QCD /Δ) 2 ; fixed-order results for 1/m b terms

20. September 3, 2005Heraeus Summer School20 Scale separation (MSOPE) A wonderful formula (exact): A wonderful formula (exact): [MN] with: Scales: with: Scales: μ h ~ m b μ i ~ m b Δ μ 0 ~ Δ μ h ~ m b μ i ~ m b Δ μ 0 ~ Δ Jet functionSoft function Dependence on E 0

21. September 3, 2005Heraeus Summer School21 Perturbation theory Hard, jet, and soft matching coefficients computed at O(α s ) Hard, jet, and soft matching coefficients computed at O(α s ) [Bauer, Manohar; Bosch et al.; MN] Momentum-dependent corrections to jet and soft functions known to 2 loops Momentum-dependent corrections to jet and soft functions known to 2 loops [MN] Cusp anomalous dimension computed to 3 loops Cusp anomalous dimension computed to 3 loops [Moch, Vermaseren, Vogt] Shape-function anomalous dimension computed at 2 loops Shape-function anomalous dimension computed at 2 loops [Korchemsky, Marchesini; Gardi; MN] Jet-function anomalous dimension derived at 2 loops Jet-function anomalous dimension derived at 2 loops [MN]

22. September 3, 2005Heraeus Summer School22 Predictions for moments O(1) O(1/m b ) O(1/m b 2 ) Perturbation Theory Complete resummation at NNLO αs2αs2αs2αs2 αs2αs2αs2αs2 Hadronic Parameters m b, μ π 2 μ π 2 ρ D 3, ρ LS 3 ρ D 3, ρ LS 3

23. September 3, 2005Heraeus Summer School23 Fit to Belle data (E 0 = 1.8 GeV) Fit results: Fit results: Combined results (B→X s γ and B→X c l ν ): Combined results (B→X s γ and B→X c l ν ): Theory uncertainty B→X c l ν moments m b = (4.62±0.10 exp ±0.03 th ) GeV μ π 2 = (0.11±0.13 exp ±0.08 th ) GeV 2 m b = (4.61±0.06) GeV μ π 2 = (0.14±0.06) GeV 2 ! 68% CL 90% CL [MN]

24. September 3, 2005 Heraeus Summer School 24 |V ub | from B→X u l ν Decay Factorization for inclusive decay spectra B XuXu SM l ν

25. September 3, 2005Heraeus Summer School25 Scale separation Master formula for inclusive decay spectra: Master formula for inclusive decay spectra: Γ ~ H ( μ h ) * U(μ h,μ i ) * J(μ i ) * U(μ i,μ 0 ) * S(μ 0 ) QCD → SCET → (RG evolution) → HQET → (RG evolution) → Shape Function Perturbation theory Nonperturbative physics

26. September 3, 2005Heraeus Summer School26 Example: B→X s γ decay Photon spectrum: Photon spectrum: Different components in this formula are obtained from matching calculations Different components in this formula are obtained from matching calculations

27. September 3, 2005Heraeus Summer School27 Matching 1: QCD → SCET QCD graphs: SCET graphs: determines hard function H

28. September 3, 2005Heraeus Summer School28 Matching 1: QCD → SCET Hard function: Hard function:

29. September 3, 2005Heraeus Summer School29 Matching 2: SCET → HQET SCET graphs: HQET graphs: determines jet function J

30. September 3, 2005Heraeus Summer School30 Nonperturbative input Shape function of B meson (parton distribution function) can be measured with good precision in B→X s γ decay Shape function of B meson (parton distribution function) can be measured with good precision in B→X s γ decay Use result to predict aritrary B→X u l ν decay spectra, with arbitrary experimental cuts Use result to predict aritrary B→X u l ν decay spectra, with arbitrary experimental cuts Implemented in a generator (“InclusiveBeauty”) Implemented in a generator (“InclusiveBeauty”) Extraction of |V ub | from a fit to data Extraction of |V ub | from a fit to data  Many different strategies  Many cross checks  Conistent results [Lange, MN, Paz]

31. September 3, 2005Heraeus Summer School31 Inclusive semileptonic decays Factorization theorem analogous to B→X s γ Factorization theorem analogous to B→X s γ Hadronic phase space most transparent in the variables P = E X ± P X Hadronic phase space most transparent in the variables P = E X ± P X In practice, Δ = P + - Λ is always of order Λ QCD for cuts eliminating the charm background In practice, Δ = P + - Λ is always of order Λ QCD for cuts eliminating the charm background Shape-function region OPE region Charm background ±

32. September 3, 2005Heraeus Summer School32 Strategy Exploit universality of shape function Exploit universality of shape function Extract shape function in B→X s γ (fit to photon spectrum), then predict arbitrary distributions in B→X u l ν decay Extract shape function in B→X s γ (fit to photon spectrum), then predict arbitrary distributions in B→X u l ν decay Functional form of fitting function is constrained by model-independent moment relations Functional form of fitting function is constrained by model-independent moment relations  Knowledge of m b and μ π 2 helps! Variant: construct “shape-function independent relations” between spectra (equivalent) Variant: construct “shape-function independent relations” between spectra (equivalent) [Lange, MN, Paz]

33. September 3, 2005Heraeus Summer School33 Results for various cuts 7.0% 9.9% 15.0% 6.6% 18.9% Eff = 86% Eff = 76% 36% Eff = 18% Eff = 66% Eff = 12% Theory Error [Lange, MN, Paz] Rate Γ ~ (m b ) a

34. September 3, 2005Heraeus Summer School34 Facit Combined theory error on |V ub | is 5-10% for several different cuts (10% is now conservative – seemed unrealistic only a few years ago) Combined theory error on |V ub | is 5-10% for several different cuts (10% is now conservative – seemed unrealistic only a few years ago) Average of different extractions will give |V ub | with a total error of less than 10% Average of different extractions will give |V ub | with a total error of less than 10% Needed to match the precision of sin2β Needed to match the precision of sin2β

35. September 3, 2005Heraeus Summer School35 Impact of precise |V ub | Realistic: δ|V ub |: ±7% Realistic: δ|V ub |: ±7%