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September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 11
Lecture 2Lecture 2Factorization in Inclusive B DecaysFactorization in Inclusive B Decays
• Soft-collinear factorization• Factorization in B→Xsγ decay• mb from B→Xsγ moments• |Vub| from B→Xulν decay spectra
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 22
Soft-Collinear FactorizationSoft-Collinear Factorization
Kinematics in heavy-to-light Kinematics in heavy-to-light processes, Soft and collinear processes, Soft and collinear modes, Effective field theorymodes, Effective field theory
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 33
MotivationMotivation
Separation of scales (“factorization”) is Separation of scales (“factorization”) is crucial to many applications of QCDcrucial to many applications of QCD
Wilsonian OPE: integrate out heavy Wilsonian OPE: integrate out heavy particles or large virtualities (Fermi theory, particles or large virtualities (Fermi theory, HQET, correlators at large QHQET, correlators at large Q22, …), …)
Expansion in (Expansion in (ΛΛQCDQCD/Q)/Q)2n2n and and ααss(Q)(Q)
Q2 » ΛQCD2
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 44
ComplicationComplication
Jet-light physics: large energies and Jet-light physics: large energies and momenta, but small virtualitiesmomenta, but small virtualities ee++ee--→jets, B→light particles, …→jets, B→light particles, …
Light-cone kinematicsLight-cone kinematics
How to integrate out short-distance physics in a situation where pμ is large, but p2 small?
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 55
B-factory physicsB-factory physics
Much interest in B→light processes:Much interest in B→light processes: |V|Vubub| determinations| determinationsAngles of the unitarity Angles of the unitarity
triangletriangleRare decays, searches Rare decays, searches
for New Physicsfor New PhysicsLarge-recoil processes Large-recoil processes
(fast light particles)(fast light particles)
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 66
ChallengeChallenge
Construct short-distance expansions for Construct short-distance expansions for processes involving both soft and processes involving both soft and energetic light partonsenergetic light partonsSoft: pSoft: psoft soft ~ Λ~ ΛQCDQCD
Collinear: pCollinear: pcolcol2 2 «« E Ecolcol
22
ppsoftsoft••ppcol col ~ E~ EcolcolΛΛ
semi-hard scalesemi-hard scaleTechnology: effective field theory, OPETechnology: effective field theory, OPE
lν
b
B
jet
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 77
Soft-collinear effective theorySoft-collinear effective theory
Systematic power counting in Systematic power counting in λλ==ΛΛQCDQCD/E/EEffective Lagrangians for strong and weak Effective Lagrangians for strong and weak
interactions expanded in powers of λinteractions expanded in powers of λMore complicated than previous heavy-More complicated than previous heavy-
quark expansionsquark expansionsExpansion in non-local string operators Expansion in non-local string operators
integrated over light-like field separationintegrated over light-like field separationMany degrees of freedomMany degrees of freedom
[Bauer, Pirjol, Stewart & Fleming, Luke]
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 88
Different versions of SCETDifferent versions of SCET
SCET-1: SCET-1: hard-collinear & softhard-collinear & softE.g.: inclusive B→XE.g.: inclusive B→Xssγγ and B→X and B→Xuullνν decays, decays,
jet physicsjet physicsSCET-2: SCET-2: collinear & soft & soft-collinearcollinear & soft & soft-collinear
E.g.: exclusive B→ππ, B→KE.g.: exclusive B→ππ, B→K**γγ decays, decays, B→light form factorsB→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-2QCD → SCET-1 → HQET / SCET-2
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 99
Factorization in B→XFactorization in B→Xssγγ
Partially inclusive decay rate Partially inclusive decay rate
BXsFCNC
γ
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 1010
Different scalesDifferent scales
Consider partial rate integrated over EConsider partial rate integrated over Eγγ > E> E00
Cut on photon energy (ECut on photon energy (E0 0 ≈≈ 1.8 GeV) 1.8 GeV)
introduces new scaleintroduces new scale ΔΔ = m = mb b - 2E- 2E00 ≈ 1 GeV ≈ 1 GeV Important to disentangle Important to disentangle
short-distance physics short-distance physics at scale mat scale mbb from soft from soft
physics at scale physics at scale ΔΔ
Belle 04
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 1111
Relevant modesRelevant modes
Hard: Hard: ppμμ ~ m ~ mbb
Hard-collinear: Hard-collinear: pp-- ~ m ~ mbb, p, p++ ~ ~ , p, p┴┴ ~ m ~ mbbΔΔ
(p(p22 ~ m ~ mbbΔΔ ~ inv. hadr. mass ~ inv. hadr. mass22))
Soft: Soft: ppμμ ~ ~ ΔΔ2-step matching:2-step matching:
QCD → SCET-1 → HQETQCD → SCET-1 → HQET
mb
mbΔΔ
ΔΔ
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 1212
Soft-collinear Soft-collinear (QCD)(QCD) factorization factorization
Systematic separation of short- and long-Systematic separation of short- and long-distance physics distance physics order by order in 1/mb:
[Korchemsky, Sterman]
Soft functions(~
Hard functions(~mb)
Jet functions(~ mb
[Lee, Stewart]
[Bosch, MN, Paz]
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 1313
Different kinematical regionsDifferent kinematical regions
ΔΔ ~ ~ ΛΛQCDQCD:: shape-function regionshape-function region Need for nonperturbative structure functions Need for nonperturbative structure functions
(matrix elements of light-cone string ops.)(matrix elements of light-cone string ops.)
mmb b » » ΔΔ » » ΛΛQCDQCD:: multi-scale OPE regionmulti-scale OPE region Model-independent predictions in terms of Model-independent predictions in terms of
heavy-quark parametersheavy-quark parameters
mmb b ~ ~ ΔΔ:: conventional OPE regionconventional OPE region
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 1414
Different kinematical regionsDifferent kinematical regions
E0 [GeV]
Scal
es
mb
mbΔ
Δ
Shape function region
OPE region
Multi-scale OPE region
Nonperturbative !
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 1515
Scale separation (MSOPE)Scale separation (MSOPE)
Master formula for the rate: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 theoryNonperturbative physics
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 1616
Partial B→XPartial B→Xssγ γ branching ratiobranching ratio
Theoretical calculation with a cut at Theoretical calculation with a cut at EE0 0 = 1.8GeV:= 1.8GeV:
Experiment (Belle 2004):Experiment (Belle 2004):
Br(1.8GeV) = (3.30 ± 0.33[pert] ± 0.17[pars]) • 10-4 Br(1.8GeV) = (3.30 ± 0.33[pert] ± 0.17[pars]) • 10-4
Br(1.8GeV) = (3.38 ± 0.30[stat] ± 0.28[syst]) • 10-4 Br(1.8GeV) = (3.38 ± 0.30[stat] ± 0.28[syst]) • 10-4
[MN]
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 1717
Implications for New PhysicsImplications for New Physics
Larger theory errors, and better agreement Larger theory errors, and better agreement between theory and experiment, weaken between theory and experiment, weaken constraints on parameter space of New constraints on parameter space of New Physics models!Physics models!
E.g., type-II two-Higgs doublet model:E.g., type-II two-Higgs doublet model:m(H+) > 200 GeVm(H+) > 200 GeV (95% CL) (95% CL)
(compared with previous bound of (compared with previous bound of 500 GeV)500 GeV)
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 1818
Factorization in B→XFactorization in B→Xssγγ
Determination of mDetermination of mbb from from
moments of the photon spectrummoments of the photon spectrum
BXsFCNC
γ
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 1919
Moments of photon spectrumMoments of photon spectrum
Marvelous QCD laboratoryMarvelous QCD laboratoryExtraction of heavy-quark parameters Extraction of heavy-quark parameters
(m(mbb,μ,μππ22)) with exquisite precision 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 physicsSecond NNLO calculation in B physicsSame accuracy for leading power corrections Same accuracy for leading power corrections
~(~(ΛΛQCDQCD//ΔΔ))22; fixed-order results for 1/m; fixed-order results for 1/mbb terms terms
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 2020
Scale separation (MSOPE)Scale separation (MSOPE)
A wonderful formula (exact): A wonderful formula (exact): [MN]
with: with: Scales:Scales:
μh ~ mb
μi ~ mbΔ μ0 ~ Δ
μh ~ mb
μi ~ mbΔ μ0 ~ Δ
Jet function Soft function Dependence on E0
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 2121
Perturbation theoryPerturbation theory
Hard, jet, and soft matching coefficients Hard, jet, and soft matching coefficients computed at O(computed at O(ααss) ) [Bauer, Manohar; Bosch et al.; MN]
Momentum-dependent corrections to jet and soft Momentum-dependent corrections to jet and soft functions known to 2 loops functions known to 2 loops [MN]
Cusp anomalous dimension computed to Cusp anomalous dimension computed to 3 loops 3 loops [Moch, Vermaseren, Vogt]
Shape-function anomalous dimension computed Shape-function anomalous dimension computed at 2 loops at 2 loops [Korchemsky, Marchesini; Gardi; MN]
Jet-function anomalous dimension derived at Jet-function anomalous dimension derived at 2 loops 2 loops [MN]
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 2222
Predictions for momentsPredictions for moments
O(1)O(1) O(1/mO(1/mbb)) O(1/mO(1/mbb22))
Perturbation Perturbation TheoryTheory
Complete Complete resummation resummation
at NNLOat NNLOααss
22 ααss22
Hadronic Hadronic ParametersParameters
mmbb, μ, μππ22
μμππ22 ρ ρDD
33, ,
ρρLSLS33 ρρDD
33, ρ, ρLSLS33
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 2323
Fit to Belle data (EFit to Belle data (E00 = 1.8 GeV)= 1.8 GeV)
Fit results:Fit results:
Combined results Combined results (B(B→X→Xssγγ and B and B→→XXccllνν):):
Theory uncertainty
B→Xclν moments
mb = (4.62±0.10exp±0.03th) GeV
μπ2 = (0.11±0.13exp±0.08th) GeV2
mb = (4.61±0.06) GeV
μπ2 = (0.14±0.06) GeV2
!
68% CL90% CL
[MN]
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 2424
|V|Vubub| from B→X| from B→Xuullνν Decay Decay
Factorization for inclusive decay Factorization for inclusive decay spectra spectra
BXuSM
l ν
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 2525
Scale separationScale separation
Master formula for inclusive decay spectra: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 theoryNonperturbative physics
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 2626
Example: B→XExample: B→Xssγγ decay decay
Photon spectrum:Photon spectrum:
Different components in this formula are Different components in this formula are obtained from matching calculationsobtained from matching calculations
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 2727
Matching 1: QCD → SCETMatching 1: QCD → SCET
QCD graphs: SCET graphs:
determines hard function determines hard function HH
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 2828
Matching 1: QCD → SCETMatching 1: QCD → SCET
Hard function:Hard function:
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 2929
Matching 2: SCET → HQETMatching 2: SCET → HQET
SCET graphs: HQET graphs:
determines jet function determines jet function JJ
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 3030
Nonperturbative inputNonperturbative input
Shape function of B meson (parton distribution Shape function of B meson (parton distribution function) can be measured with good precision function) can be measured with good precision in B→Xin B→Xssγγ decay decay
Use result to predict aritrary Use result to predict aritrary B→XB→Xuullνν decay decay spectra, with arbitrary experimental cutsspectra, with arbitrary experimental cuts
Implemented in a generator (“InclusiveBeauty”)Implemented in a generator (“InclusiveBeauty”) Extraction of |VExtraction of |Vubub| from a fit to data| from a fit to data
Many different strategiesMany different strategies Many cross checksMany cross checks Conistent resultsConistent results
[Lange, MN, Paz]
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 3131
Inclusive semileptonic decaysInclusive semileptonic decays
Factorization theorem Factorization theorem analogous to Banalogous to B→X→Xssγγ
Hadronic phase space Hadronic phase space most transparent in the most transparent in the variables Pvariables P = E= EXX ± P ± PXX
In practice, In practice, ΔΔ = P = P++ - - ΛΛ
is always of order is always of order ΛΛQCDQCD
for cuts eliminating the for cuts eliminating the charm backgroundcharm background
Shape-function region
OPE regionOPE region
Charm background
±
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 3232
StrategyStrategy
Exploit universality of shape functionExploit universality of shape function Extract shape function in BExtract shape function in B→X→Xssγγ (fit to photon (fit to photon
spectrum), then predict arbitrary distributions in spectrum), then predict arbitrary distributions in BB→X→Xuullνν decaydecay
Functional form of fitting function is constrained Functional form of fitting function is constrained by model-independent moment relationsby model-independent moment relations Knowledge of Knowledge of mmbb and and μμππ
22 helps! helps!
Variant: construct “shape-function independent Variant: construct “shape-function independent relations” between spectra (equivalent)relations” between spectra (equivalent) [Lange, MN, Paz]
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 3333
Results for various cutsResults for various cuts
7.0%
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 ~ (mb)a
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 3434
FacitFacit
Combined theory error on |VCombined theory error on |Vubub| is 5-10% | is 5-10%
for several different cuts (10% is now for several different cuts (10% is now conservative – seemed unrealistic only a conservative – seemed unrealistic only a few years ago)few years ago)
Average of different extractions will give |Average of different extractions will give |VVubub| with a total error of less than 10%| with a total error of less than 10%
Needed to match the precision of sin2Needed to match the precision of sin2ββ
September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 3535
Impact of precise |VImpact of precise |Vubub||
Realistic: Realistic: δδ|V|Vubub|: ±7%|: ±7%