September 3, 2005 Heraeus Summer School 1 Lecture 2 Factorization in Inclusive B Decays...

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September 3, 2005 September 3, 2005 Heraeus Summer School Heraeus Summer School 1 Lecture 2 Lecture 2 Factorization in Inclusive B Factorization in Inclusive B Decays 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

Transcript of September 3, 2005 Heraeus Summer School 1 Lecture 2 Factorization in Inclusive B Decays...

Page 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.

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

Page 2: 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.

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

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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

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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?

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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)

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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

b

B

jet

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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]

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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

Page 9: 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.

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

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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

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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

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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]

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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

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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 !

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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

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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]

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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)

Page 18: 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.

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

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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

Page 20: 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.

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

Page 21: 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.

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]

Page 22: 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.

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

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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]

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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

Page 25: 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.

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

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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

Page 27: 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.

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

Page 28: 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.

September 3, 2005September 3, 2005 Heraeus Summer SchoolHeraeus Summer School 2828

Matching 1: QCD → SCETMatching 1: QCD → SCET

Hard function:Hard function:

Page 29: 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.

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

Page 30: 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.

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]

Page 31: 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.

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

±

Page 32: 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.

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]

Page 33: 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.

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

Page 34: 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.

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

Page 35: 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.

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%