Inclusive |Vub| unlimitedZoltan Ligeti

SCET Workshop, MIT, March 24–26, 2009

Outline

• Introduction

• Complete description of B → Xsγ [ZL, Stewart, Tackmann, arXiv:0807.1926]

• Complete description of B → Xu`ν̄ [ZL, Stewart, Tackmann, to appear]

• A glimpse at SIMBA [Bernlochner, Lacker, ZL, Stewart, Tackmann, Tackmann, to appear]

• Conclusions

p.1

The importance of |Vub|

• |Vub| determined by tree-level decaysCrucial for comparing tree-dominatedand loop-mediated processes

• |Vub|π`ν̄−LQCD = (3.5± 0.5)× 10−3

|Vub|incl−BLNP = (4.32± 0.35)× 10−3

|Vub|τν = (5.2± 0.5± 0.4fB)× 10−3

SM CKM fit, sin 2φ1, favors small value

• Fluctuation, bad theory, new physics?

• The level of agreement between themeasurements often misinterpreted

3φ

3φ

2φ

2φ

dm∆Kε

Kε

sm∆ & dm∆

SLubV

ν τubV

1φsin 2

(excl. at CL > 0.95)

< 01

φsol. w/ cos 2

excluded at CL > 0.95

2φ

1φ

3φ

ρ−1.0 −0.5 0.0 0.5 1.0 1.5 2.0

η

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5excluded area has CL > 0.95

Moriond 09

CKMf i t t e r

• 10–20% non-SM contributions to most loop-mediated transitions are still possible

p.2

|Vub| matters. . .

• Including B → τ ν̄, the SM is “disfavored” at >2σ

dh0.0 0.2 0.4 0.6 0.8 1.0

dσ

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.01-CL

Moriond 09

CKMf i t t e r

ντ →w/o B

dh0.0 0.2 0.4 0.6 0.8 1.0

dσ0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.01-CL

Moriond 09

CKMf i t t e r

Parameterize NP in B0–B0 mixing: M12 = MSM12 (1 + hd e

2iσd)

• 10–20% non-SM contributions to most loop-mediated transitions are still possible

p.3

The challenge of inclusive |Vub| measurements

• Total rate known with ∼ 4% accuracy, similar to B(B → Xc`ν̄) [Hoang, ZL, Manohar]

• To remove the huge charm background(|Vcb/Vub|2 ∼ 100), need phase space cuts

Phase space cuts can enhance perturbativeand nonperturbative corrections drastically

dΓ(b→c)/dEe

10 dΓ(b→u)/dEe

Ee (GeV)

dΓ/d

Ee

∆E

0

20

40

60

80

0 1 2

Nonperturbative effects shift endpoint 12 mb → 1

2 mB and determine shape↗

• Endpoint region determined by b quark PDF in B; want to extract it from data tomake predictions — at lowest order ∝ B → Xsγ photon spectrum

[Bigi, Shifman, Uraltsev, Vainshtein; Neubert]

p.4

|Vub|: lepton endpoint vs. B → Xsγ

b quark decayspectrum

with a model forb quark PDF

0 0.5 1 1.5 2 2.5El

dΓdEl

p.5

|Vub|: lepton endpoint vs. B → Xsγ

b quark decayspectrum

with a model forb quark PDF

0 0.5 1 1.5 2 2.5El

dΓdEl

ddEl−

p.5

|Vub|: lepton endpoint vs. B → Xsγ

b quark decayspectrum

with a model forb quark PDF

0 0.5 1 1.5 2 2.5El

dΓdEl

ddEl−

difference:

2 3 4

p.5

|Vub|: lepton endpoint vs. B → Xsγ

b quark decayspectrum

with a model forb quark PDF

0 0.5 1 1.5 2 2.5El

dΓdEl

ddEl−

difference:

2 3 4

1850801-007

40

0

DataSpectator Model

Wei

ghts

/ 10

0 M

eV

1.5 2.5 3.5 4.5E (GeV)

[CLEO, 2001]

p.5

|Vub|: lepton endpoint vs. B → Xsγ

b quark decayspectrum

with a model forb quark PDF

0 0.5 1 1.5 2 2.5El

dΓdEl

ddEl−

difference:

2 3 4

1850801-007

40

0

DataSpectator Model

Wei

ghts

/ 10

0 M

eV

1.5 2.5 3.5 4.5E (GeV)

[CLEO, 2001]

• Both of these spectra are determined at lowest order by the b quark PDF in the B

• Lots of work toward extending beyond leading order; many open issues...

p.5

Past efforts: BLNP (best so far)

• Treated factorization & resummationin shape function region correctly

• Use fixed functional forms to modelshape function (similar to PDF’s)

• Analysis tied to shape functionscheme for mb, λ1

(One scheme for each approach)

• Need for “tail gluing” to get correctperturbative tail (other approachesdon’t even do this [DGE, GGOU, ADFR])

⇒ Hard to assess uncertainties

[Bosch, Lange, Neubert, Paz]

p.5

Start with B → Xsγ

[ZL, Stewart, Tackmann, PRD 78 (2008) 114014, arXiv:0807.1926]

Regions of phase space

• B → Xsγ gives one of the strongest bounds on NP

• p+X ≡ mB − 2Eγ <∼ 2 GeV, and <1 GeV at the peak

Three cases: 1) Λ ∼ mB − 2Eγ � mB

Three cases: 2) Λ� mB − 2Eγ � mB

Three cases: 3) Λ� mB − 2Eγ ∼ mB

[Sometimes called 1) SCET and 2) MSOPE]

• Neither 1) nor 2) is fully appropriate [GeV]γ

c.m.sE1.5 2 2.5 3 3.5 4

Ph

oto

ns

/ 50

MeV

-4000

-2000

0

2000

4000

6000

[Belle, 0804.1580]

• Include all known results in regions 1) – 2)

LL: 1–loop Γcusp, tree-level matching

NLL: 2–loop Γcusp, 1–loop matching, 1–loop γx

NNLL: 3–loop Γcusp, 2–loop matching, 2–loop γx

We can combine 1) – 2) withoutexpanding shape function in Λ/p+

X

p.6

The shape function (b quark PDF in B)

• The shape function S(ω, µ) contains nonperturbative physics and obeys a RGE

If S(ω, µΛ) has exponentially small tail, any smallrunning gives a long tail and divergent moments

[Balzereit, Mannel, Kilian]

S(ω, µi) =

Zdω′US(ω − ω′, µi, µΛ)S(ω

′, µΛ)

Constraint: moments (OPE) + B → Xsγ shapeConstraint: How to combine these?

– Consistent setup at any order, in any scheme

– Stable results for varying µΛ

– (SF modeling scale, must be part of uncert.)

– Similar to how all matrix elements are defined– e.g., BK(µ) = bBK [αs(µ)]2/9(1 + . . .)

Derive: [ZL, Stewart, Tackmann, 0807.1926]

S(ω, µΛ) =

Zdk C0(ω−k, µΛ)F (k)

0

0

0

0

1

1

1

1 22

0.5

0.5

0.5

0.5

1.5

1.5

1.5

1.5 2.52.5

−0.5−0.5

ω [GeV]ω [GeV]

S(ω

,2.5

GeV

)[G

eV−

1]

S(ω

,2.5

GeV

)[G

eV−

1] µΛ = 2.5 GeV

µΛ = 1.8 GeVµΛ = 1.3 GeVµΛ = 1.0 GeV

nonpert. perturbative

ModelS (dash)F (solid)

run to 2.5GeV

Can consistently impose moment constraints on F (k), but not on S(ω, µΛ)

p.7

The shape function (b quark PDF in B)

• The shape function S(ω, µ) contains nonperturbative physics and obeys a RGE

If S(ω, µΛ) has exponentially small tail, any smallrunning gives a long tail and divergent moments

[Balzereit, Mannel, Kilian]

S(ω, µi) =

Zdω′US(ω − ω′, µi, µΛ)S(ω

′, µΛ)

Constraint: moments (OPE) + B → Xsγ shapeConstraint: How to combine these?

– Consistent setup at any order, in any scheme

– Stable results for varying µΛ

– (SF modeling scale, must be part of uncert.)

– Similar to how all matrix elements are defined– e.g., BK(µ) = bBK [αs(µ)]2/9(1 + . . .)

Derive: [ZL, Stewart, Tackmann, 0807.1926]

S(ω, µΛ) =

Zdk C0(ω−k, µΛ)F (k)

0

0

0

0

1

1

1

1 22

0.5

0.5

0.5

0.5

1.5

1.5

1.5

1.5 2.52.5

−0.5−0.5

ω [GeV]ω [GeV]

S(ω

,2.5

GeV

)[G

eV−

1]

S(ω

,2.5

GeV

)[G

eV−

1] µΛ = 2.5 GeV

µΛ = 1.8 GeVµΛ = 1.3 GeVµΛ = 1.0 GeV

nonpert. perturbative

ModelS (dash)F (solid)

run to 2.5GeV

• Consistent to impose moment constraints on F (k), but not on S(ω, µΛ) w/o cutoff

p.7

Derivation of the magic formula (1)

• The shape function is the matrix element of a nonlocal operator:

S(ω, µ) = 〈B| b̄v δ(iD+ − δ + ω) bv| {z }O0(ω,µ)

|B〉, δ = mB −mb

Integrated over a large enough region, 0 ≤ ω ≤ Λ, one can expand O0 as

O0(ω, µ) =X

Cn(ω, µ) b̄v (iD+ − δ)n bv| {z }Qn

+ . . . =X

Cn(ω − δ, µ) b̄v (iD+)nbv| {z }eQn

+ . . .

The Cn are the same for Qn and Q̃n (since O0 only depends on ω − δ)

• Matching: 〈bv|O0(ω+δ, µ)|bv〉 =X

Cn(ω, µ) 〈bv| eQn|bv〉 = C0(ω, µ), 〈bv| eQn|bv〉 = δ0n

〈bv(k+)|O0(ω + δ, µ)|bv(k+)〉 = C0(ω + k+, µ) =X kn+

n!

dnC0(ω, µ)

dωn

〈bv(k+)|O0(ω + δ, µ)|bv(k+)〉 =X

Cn(ω, µ)〈bv| eQn|bv〉 =X

Cn(ω, µ) kn+

• Comparing last two lines: Cn(ω, µ) =1n!

dnC0(ω, µ)dωn

[Bauer & Manohar]

p.8

Derivation of the magic formula (2)

• Define the nonperturbative function F (k) by: [ZL, Stewart, Tackmann; Lee, ZL, Stewart, Tackmann]

S(ω, µΛ) =∫

dk C0(ω − k, µΛ)F (k), C0(ω, µ) = 〈bv|O0(ω + δ, µ)|bv〉

uniquely defines F (k): F̃ (y) = S̃(y, µ)/C̃0(y, µ)

• Expand in k: S(ω, µ) =∑n

1n!

dnC0(ω, µ)dωn

∫dk (−k)nF (k)

Compare with previous page ⇒∫

dk knF (k) = (−1)n 〈B|Qn|B〉

〈B|Q0|B〉 = 1 , 〈B|Q1|B〉 = −δ , 〈B|Q2|B〉 = −λ1

3+ δ

2

More complicated situation for higher moments, so stop here

• This treatment is fully consistent with the OPE

p.9

Changing schemes: mb

• Converting results to a short distance mass scheme removes dip at small ω:

• Want to define short distance(hatted) quantities such that:

S(ω) =

Zdk C0(ω − k)F (k)

S(ω) =

Zdk bC0(ω − k) bF (k)

Switch from pole to short dis-tance scheme:

mb = bmb + δmb

λ1 = bλ1 + δλ1

0

0

0

0

1

1

1

1 22

0.5

0.5

0.5

0.5

1.5

1.5

1.5

1.5 2.52.5−0.5−0.5

ω [GeV]ω [GeV]

S(ω

,2.5

GeV

)[G

eV−

1]

S(ω

,2.5

GeV

)[G

eV−

1] mpole

b

m1Sb

mkinb

dashed:

solid:

NLL

NNLL

(ω ∼ mB − 2Eγ)

bC0(ω) = C0(ω + δmb)−δλ1

6

d2

dω2C0(ω) =

»1 + δmb

d

dω+

„(δmb)

2

2−δλ1

6

«d2

dω2

–C0(ω)

• Can use any short distance mass scheme (1S, kinetic, PS, shape function, . . . )

p.10

Changing schemes: λ1

• We find that kinetic scheme, µ2π ≡ −λkin

1 , oversubtracts; similar to mb(mb) issues

• Introduce “invisible” scheme: λi1 = λ1−0αs−R2 α

2s(µ)

π2

CFCA

4

„π2

3−1

«(R = 1 GeV)

• Yet to be seen if shape func-tion scheme for λ1 gives goodbehavior

0

0

0

0

1

1

1

1 22

0.20.2

0.40.4

0.50.5

0.60.6

0.80.8

1.21.2

1.51.5 2.52.5

−0.2−0.2

ω [GeV]ω [GeV]

S(ω

,2.5

GeV

)[G

eV−

1]

S(ω

,2.5

GeV

)[G

eV−

1] λ1 NLL

λ1 NNLL

λkin1 NLL

λkin1 NNLL

(All curves use mkinb )

p.11

Scale (in)dependence of B → Xsγ spectrum

• Dependence on 3 scales in the problem:

µΛ = 1.2, 1.5, 1.9 GeV µi = 2.0, 2.5, 3.0 GeV µb = 2.35, 4.7, 9.4 GeV

00

00

1

1

1

1

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0.8

1.2

1.2

1.2

1.2

1.4

1.4

1.4

1.4 1.61.6p+X [GeV]p+X [GeV]

(dΓ

s/dp

+ X)/(Γ

0s|C

incl

7|2 )

[GeV−

1]

(dΓ

s/dp

+ X)/(Γ

0s|C

incl

7|2 )

[GeV−

1]

λ1 NNLL

λi1 NNLL

LL

NLL

00

00

1

1

1

1

0.2

0.2

0.2

0.2

0.4

0.4

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0.8

1.2

1.2

1.2

1.2

1.4

1.4

1.4

1.4 1.61.6p+X [GeV]p+X [GeV]

(dΓ

s/dp

+ X)/(Γ

0s|C

incl

7|2 )

[GeV−

1]

(dΓ

s/dp

+ X)/(Γ

0s|C

incl

7|2 )

[GeV−

1]

LL

NLL

NNLL

00

00

1

1

1

1

0.2

0.2

0.2

0.2

0.4

0.4

0.4

0.4

0.6

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0.8

1.2

1.2

1.2

1.2

1.4

1.4

1.4

1.4 1.61.6p+X [GeV]p+X [GeV]

(dΓ

s/dp

+ X)/(Γ

0s|C

incl

7|2 )

[GeV−

1]

(dΓ

s/dp

+ X)/(Γ

0s|C

incl

7|2 )

[GeV−

1]

LL

NLL

NNLL

dΓs

dp+X

= Γ0sHs(p+X, µb)UH(mb, µb, µi)

Zdk bP (mb, k, µi) bF (p

+X − k)

“p+X

= mB − 2Eγ”

P̂ , F̂ indicate use of short distance schemes: m1Sb and λi

1

• In other approaches, using models for S(ω, µΛ) run up to µi, dependence on µΛ

ignored so far, but it must be considered an uncertainty ⇒ This is how to solve it

p.12

Designer orthonormal functions

• Devise suitable orthonormal basis functions(earlier: fit parameters of model functions to data)

F̂ (λx) = 1λ

[∑cnfn(x)

]2, n th moment ∼ΛnQCD

fn(x) ∼ Pn[y(x)] ← Legendre polynomials

• Approximating a model shape function

Better to add a new term in an orthonormalbasis than a new parameter to a model:– less parameter correlations– errors easier to quantify

“With four parameters I can fit an elephant, and with fiveI can make him wiggle his trunk.” (John von Neumann)

0

0

0

0

1

1

1

1 22 33 44

−1−1

0.5

0.5

0.5

0.5 1.51.5 2.52.5

−0.5−0.5

3.53.5xx

f0(x)f1(x)f2(x)f3(x)f4(x)

00

00

1

1

1

1

22

0.20.2 0.40.4

0.50.5

0.60.6 0.80.8 1.21.2 1.41.4

1.51.5

1.61.6k [GeV]k [GeV]

F̂(k

)[G

eV−

1]

F̂(k

)[G

eV−

1]

F̂ (k)

F̂ (0)(k)

F̂ (1)(k)

F̂ (2)(k)

F̂ (3)(k)

F̂ (4)(k)

p.13

Back to the B → Xsγ spectrum

• 9 models: same 0th, 1st, 2nd moments

Including all NNLL contributions, find:

– Shape in peak region not determined– at all by first few moments

– Smaller shape function uncertainty for– Eγ <∼ 2.1 GeV than earlier studies 00

11

2

2

2

2

0.50.5

1.51.5

1.91.9 2.12.1 2.22.2 2.32.3 2.42.4

2.5

2.5

2.5

2.5 2.62.6Eγ [GeV]Eγ [GeV]

(dΓ

s/dE

γ)/(Γ

0s|C

incl

7|2 )

[GeV−

1]

(dΓ

s/dE

γ)/(Γ

0s|C

incl

7|2 )

[GeV−

1]

c3=c4=0

c3=±0.15, c4=0

c3=0, c4=±0.15c3=±0.1, c4=±0.1

• Not shown in this plot: subleading shape functionsNot shown in this plot: subleading corrections not in C incl

7 (0)Not shown in this plot: kinematic power correctionsNot shown in this plot: boost to Υ(4S) frame

• Same analysis can also be used for: B → Xs`+`−, |Vub|

p.14

More complicated: B → Xu`ν̄

[ZL, Stewart, Tackmann, to appear]

Regions of phase space (again)

• “Natural” kinematic variables: p±X = EX∓|~pX|— “jettyness” of hadronic final state

B → Xsγ: p+X = mB − 2Eγ & p−X ≡ mB, but independent variables in B → Xu`ν̄

• Three cases: 1) Λ ∼ p+X � p−X

Three cases: 2) Λ� p+X � p−X

Three cases: 3) Λ� p+X ∼ p

−X

Make no assumptions how p−X compares to mB

• B → Xu`ν̄: 3-body final state, appreciable ratein region 3), where hadronic final state not jet-like

E.g., m2X < m2

D does not imply p+X � p−X 0

000

1

1

1

1

2

2

2

2

3

3

3

3

4

4

4

4

5

5

5

5

p−X [GeV]p−X [GeV]p

+ X[G

eV

]p

+ X[G

eV

]

mX ≤ mD

1) 2)

3)

• Can combine 1) – 2) just as we did in B → Xsγ, by not expanding in Λ/p+X

p.15

Charmless B → Xu`ν̄ made charming

• To combine all 3 regions: do not expand in Λ/p+X nor in p+

X/p−X (nontrivial!)

• Want to have: result accurate to NNLL and ΛQCD/mb in regions 1) – 2)and to order α2

sβ0 and Λ2QCD/m

2b when phase space limits are in region 3)

Start with triple differential rate (involves a delta-fn at the parton level at O(α0s),

which is smeared by the shape function)

The p+X/p

−X terms, which are not suppressed in local OPE region, start at O(αs)

Recently completed O(α2s) matching computations [Ben’s & Guido’s talks]

[Bonciani & Ferroglia, 0809.4687; Asatrian, Greub, Pecjak, 0810.0987; Beneke, Huber, Li, 0810.1230; Bell, 0810.5695]

p.16

SIMBA

[Bernlochner, Lacker, ZL, Stewart, Tackmann, Tackmann, to appear]

Fitting charmless inclusive decay spectra

• Fit strategy: F̂ (k) enters the spectra linearly ⇒ can calculate independently thecontribution of fm fn in the expansion of F̂ (k):

dΓ =∑

cm cn︸ ︷︷ ︸fit

dΓmn︸ ︷︷ ︸compute

dΓmn = Γ0H(p−

)

Z p+X

0

dkbP (p−, k)

λfm

„p+X − kλ

«fn

„p+X − kλ

«| {z }

basis functions

Fit the ci coefficients from the measured (binned) spectra

• What we hope to achieve:– Correlation and error propagation of SF uncertainties– Simultaneous fit using all available information– Realistic estimate of model uncertainties (fit parameters c0,...,N constrained by– data; vary N , the number of orthonormal basis functions in fit)

p.17

A preliminary B → Xsγ fit

• Belle B → Xsγ spectrum in Υ(4S) restframe

For demonstration purposes only — thereare very strong correlations

Fit with 4 basis functions in the expansion ofthe shape function

Shows that fit works (not as trivial as this plotmight indicate); still issues to resolve

[Belle, 0804.1580 + Preliminary; thanks to Antonio Limosani]

[GeV]γE2 2.5

γ/d

EΓd

00.05

0.10.150.2

0.250.3

0.350.4

• Next step: include various B → Xu`ν̄ measurements

p.18

Conclusions

• Improving accuracy of |Vub| will remain important to constrain new physics(Current situation unsettled, PDG in 2008 inflated error for the first time)

• Qualitatively better inclusive analyses possible than those that exist so far– Modeling F (k) instead of S(ω, µ)– Designer orthonormal functions– Fully consistent combination of 3 phase space regions– Decouple SF shape variation from mb variation

• Developments will allow combining all pieces of data with tractable uncertainties– Consistently combine B → Xu`ν̄, B → Xsγ, B → Xc`ν̄ data to constrain SF’s– Reduce role of shape function modeling

• To draw conclusions about new physics comparing sides and angles, we’ll want≥2 extractions of |Vub| with different uncertainties (inclusive, exclusive, leptonic)

p.19