inclusive jv jat superb - agenda.infn.it · overview reducing theoretical uncertainties reducing...

Overview Reducing Theoretical Uncertainties Reducing Parametric Uncertainties

Inclusive |Vub| at SuperB

Frank Tackmann

Massachusetts Institute of Technology

Workshop on New Physics with SuperBUniversity of Warwick, April 14-17, 2009

[Ligeti, Stewart, FT: PRD 78 (2008) 114014 [arXiv:0807.1926]]

[Ligeti, Stewart, FT: work in progress]

[Bernlochner, Lacker, Ligeti, Stewart, FT, Tackmann: work in progress]

Frank Tackmann (MIT) Inclusive |Vub| at SuperB Warwick, 2009-04-14 0 / 16


Outline

1 Overview

2 Reducing Theoretical Uncertainties

3 Reducing Parametric Uncertainties



|Vub| is Crucial as Standard Model Reference

)α(γ

ubV

α

βγ

ρ−0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0

η

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

excl

uded

are

a ha

s C

L >

0.9

5

Moriond 09

CKMf i t t e r γ

α

α

dm∆

Kε

Kε

sm∆ & dm∆

ubV

βsin 2(excl. at CL > 0.95)

< 0βsol. w/ cos 2

α

βγ

ρ−0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0

η

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

excl

uded

are

a ha

s C

L >

0.9

5

Moriond 09

CKMf i t t e r

|Vub| determined from tree-level decays⇒ crucial for SM reference UTsin 2β favors small |Vub| ⇒ > 2σ tension (amplified by B → τν)Tension between different methods

leptonic: |Vub|B→τν = 5.2 ± 0.5[exp] ± 0.4[fB]

inclusive OPE: |Vub|BLL = 4.87 ± 0.24[exp] ± 0.38[theory]

inclusive SCET: |Vub|BLNP = 4.32 ± 0.16[exp]+0.32−0.27[theory]

exclusive: |Vub|B→π`ν = 3.5 ± 0.2[exp] ± 0.5[lattice]

⇒ PDG inflated error for first time in 2008Frank Tackmann (MIT) Inclusive |Vub| at SuperB Warwick, 2009-04-14 1 / 16


|Vub| from Inclusive B → Xu`ν

Removing huge charm background requiresstringent phase space cuts

B(B → Xc`ν)/B(B → Xu`ν) ' 50

Cuts can drastically enhance perturbative andnonperturbative corrections

Rates become sensitive to b-quark PDFs in B mesonDetermine shape of spectraLeading order: Universal shape function (SF)[Bigi et al., Neubert]

O(ΛQCD/mb): Several more subleading shapefunctions [Bauer, Luke, Mannel]

Need to be extracted from data (like any PDF)

/ GeVlE0 0.5 1 1.5 2 2.5

/ GeVlE0 0.5 1 1.5 2 2.5

/ GeVXm0 0.5 1 1.5 2 2.5 3 3.5

/ GeVXm0 0.5 1 1.5 2 2.5 3 3.5

50dΓ(b→u)

dE`

dΓ(b→c)

dE`

50dΓ(b→u)

dmX

dΓ(b→c)

dmX



Summary of Non-Experimental Uncertainties

Theoretical uncertaintiesOptimal theory description depends on phase space region

I Higher order αs correctionsI Higher order nonperturbative correctionsI Combining descriptions for different phase space regions

Scheme changes (e.g. m1Sb ↔ mSF

b ↔ mkinb )

Weak annihilation

Uncertainties from input parametersmb, λ1, ...Leading and subleading shape functionsWhose responsibility?

I Not purely a theoretical uncertaintyI Not purely an experimental uncertaintyI Requires combined effort from theory and experiment



Strategy Towards Precision |Vub|Ultimate precision on (inclusive) |Vub| will depend on

How well we know mb and leading SFAbility to (consistently) combine many different measurements

I Different kinematic cuts: mX , p+X , E`

I Different analysis techniques: hadronic tag, leptonic tag, untagged

Can we push inclusive |Vub| precision to 3%? [arXiv:0810.1312]

At SuperB we will have roughly as much data on B → Xu`ν as we havenow on B → Xc`ν

But B → Xu`ν has larger backgrounds and more complicated theory

To repeat success of inclusive |Vcb| (∼ 2% precision), repeat strategy:Perform global fit to all available dataSimultaneously determine |Vub| and inputs (mb, leading SF, ...)



What to Do About the Shape Function?

Try to avoid itPush cuts deep into B → Xc`ν background [e.g. BABAR PRL 96 (2006) 221801]

I Trade off between mb, SF and systematic uncertaintyI Shape function uncertainty reappears through MC signal model

“Shape-function independent” relationsI Only avoids modeling of the shape function (same underlying theory)I Still dependence on subleading shape functions

⇒ Hard to combine in a global fit, no way to include additional constraints

Better: Just deal with it! Combine all known constraints on SFPerturbative constraints (perturbative tail and RGE)Moment constraints from B → Xc`ν

Shape information from B → Xsγ and B → Xu`ν spectra



Outline

1 Overview





Regions of Phase Space

Kinematic variables: p±X = EX ∓ |~pX |

(1) Nonpert. SF: ΛQCD ∼ p+X � p−X

(2) SF OPE: ΛQCD � p+X � p−X

(3) Local OPE: ΛQCD � p+X ∼ p−X

00

00

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

[GeV

]p

+ X[G

eV

]

mX < mD (1)

(2)

(3)

(1) + (2) SCET region (SF region): p+X � p−X

(2) Perturbative information on SF (perturbative SF tail)(1) and (2) are not really distinct in practice⇒ need to combine them

(3) Local OPE region: p+X ∼ p−X

Formally no shape functions (large q2, small E`)mX < mD does not imply p+

X � p−X ⇒ receives contribution from (3)



Limitations of Previous Approaches

Current and very likely future optimal experimental cuts depend on multiplephase space regions

Previous approaches [BLNP, DGE, GGOU, ADFR] based on theory for singleregion, extrapolated/modeled into other regions

For example: BLNP (best so far)Correct factorization and resummationbased on region (1)Fixed functional forms to model SFTied to shape function scheme for mb, λ1

Awkward “tail gluing” needed to getcorrect shape function tail 0 0.5 1 1.5 2 2.5

-0.5

0

0.5

1

1.5

PSfrag repla ements ! [GeV℄^ S(^!;�)[GeV�1 ℄ �0 = 1GeV�i = 1:5GeV

Intrinsic model uncertainties are hard to assess (often neglected)Current uncertainty in inclusive |Vub| is underestimatedExisting approaches do not scale to accuracies . 10%



New and Improved Approach to Shape Function[Ligeti, Stewart, FT, 0807.1926]

Difference in (1) and (2) only due to SF

(1) Nonpert. peak region ΛQCD ∼ ω(2) Perturbative tail ΛQCD � ω

⇒ RGE running always yields non-exponential tail [Balzereit, Mannel, Kilian]

S(ω, µ) run up to µ = 2.5 GeV

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

peak

tail

dashed: model S(ω, µΛ)

solid: model F (k)

Can derive factorized form of SF

S(ω, µΛ) =∫

dk C0(ω − k, µΛ) F (k)

Perturbative C0(ω, µΛ) (partonic SF) gives tail consistent with RGE

Peak determined by purely nonperturbative function F (k)Moments of F (k) are given by HQE parameters mb, λ1, ...

⇒ Consistently combines (1) and (2)



New and Improved Approach to Shape Function[Ligeti, Stewart, FT, 0807.1926]

Difference in (1) and (2) only due to SF

(1) Nonpert. peak region ΛQCD ∼ ω(2) Perturbative tail ΛQCD � ω

⇒ RGE running always yields non-exponential tail [Balzereit, Mannel, Kilian]

S(ω, µ) run up to µ = 2.5 GeV

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

peak

tail

dashed: model S(ω, µΛ)

solid: model F (k)

Can derive factorized form of SF

S(ω, µΛ) =∫

dk C0(ω − k, µΛ) F (k)

Perturbative C0(ω, µΛ) (partonic SF) gives tail consistent with RGE

Peak determined by purely nonperturbative function F (k)Moments of F (k) are given by HQE parameters mb, λ1, ...

⇒ Consistently combines (1) and (2)Frank Tackmann (MIT) Inclusive |Vub| at SuperB Warwick, 2009-04-14 8 / 16


Perturbative Shape Function UncertaintiesCan use any short distance scheme

S(ω) =∫

dkCpole0 (ω − k)F pole(k)

=∫

dkC1S0 (ω − k)F 1S(k)

=∫

dkCkin0 (ω − k)F kin(k)

= . . .

⇒ No need to translate mb schemes!

Can assess perturbative uncertainty inS(ω, µΛ) from µΛ dependence

Previously part of neglected modeluncertaintyC0(ω) known to α2

s [Becher, Neubert]

and NNLL [Korchemsky et al., Moch et al., Gardi]

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

00

00

1

1

1

1

0.2

0.2

0.2

0.2

0.4

0.4

0.4

0.4

0.6

0.6

0.6

0.6

0.8

0.8

0.8

0.8

1.2

1.2

1.2

1.2

1.4

1.4

1.4

1.4 1.61.6ω [GeV]ω [GeV]

S(ω

,2.5

GeV

)[G

eV−

1]

S(ω

,2.5

GeV

)[G

eV−

1] LL

NLL

NNLL



Combining SCET Region and Local OPE Region

[Ligeti, Stewart, FT: in preparation]

Left with two regions (mX < mD is sensitive to both)

(1+2) SCET region: ΛQCD, p+X � p−X

(3) Local OPE: ΛQCD � p+X ∼ p−X

Next step: Consistently combine (1+2) and (3) 00

00

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

[GeV

]p

+ X[G

eV

]

mX < mD (1)

(2)

(3)

dΓ =∫

dk dΓpert(E`, p+X , p

−X ; k) F (k)

Nontrivial due to different structure of perturbation series for (1+2) and (3),but we know how to do itWant combined result for dΓ accurate to

I (1+2): NNLO/NNLL and ΛQCD/mb (missing α2s recently completed

[Bonciani, Ferroglia; Asatrian et al.; Beneke et al.; Bell])I (3): α2

sβ0 [Aquila, Gambino, Ridolfi, Uraltsev] (α2s not known) and Λ2

QCD/m2b



Outline

1 Overview





Designer Orthonormal Basis FunctionsDesign suitable orthonormal basis

F (λx) =1

λ

[ ∞∑n=0

cnfn(x)]2

fn(x) ∼ Pn[y(x)] (Legendre pol.)

Formally model independentWill facilitate fitting F (k) from data

In practice, series is truncated at n ≤ nmax

Better to add new term in orthonormalbasis than new parameter to a modelResidual model dependence can beestimated by size of last expansion termand varying nmax

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]

F(k

)[G

eV−

1]

F(k

)[G

eV−

1]

F (k)

F (0)(k)

F (1)(k)

F (2)(k)

F (3)(k)

F (4)(k)

⇒ More systematic and quantifiable uncertainties than using few modelfunctions and fitting model parameters (currently done)



Incorporating Moment Constraintsmb, λ1, ... dependence of dΓ enters via moments of F (k)∫

dk k F 1Si(k) = mB−m1Sb

∫dk k2 F 1Si(k) = −λ

i1

3+(mB−m1S

b )2

Can consistently include previous knowledge of m1Sb , λi

1, ...(from B → Xc`ν, sum rules, etc.) via constraints on coefficients cnCleanly separates mb and SF dependence

F 1Si(k) with fixed m1Sb , λi

1

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]

F(k

)[G

eV−

1]

F(k

)[G

eV−

1]

c3 =c4 =0

c3 =±0.15, c4 =0

c3 =0, c4 =±0.15

c3 =±0.1, c4 =±0.1 ⇒

dΓs/dEγ for fixed m1Sb , λi

1

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



SIMBA

[Bernlochner, Lacker, Ligeti, Stewart, FT, Tackmann: work in progress]



SIMBA: Towards a Global |Vub| Fit

Use basis expansion for F (k) to get

dΓ =∑n,m

cn cm︸︷︷︸∫

dk dΓpert(E`, p±X ; k) fn(k) fm(k)︸︷︷︸

fit compute

Mi(mb, λ1) =∑n,m

︷︸︸︷cn cm

︷︸︸︷∫dk ki fn(k) fm(k)

Perform global fit which combines all available informationSimultaneously fit cn from all available measured (binned) spectra dΓ

All known perturbative information included through dΓpert

Consistently combines prior knowledge of mb, λ1 with information fromB → Xu`ν and B → Xsγ⇒ improved determination of mb, λ1

⇒ SF and mb uncertainties are determined by data, including correlations



SIMBA: Some Preliminary Results

Fit to Belle B → Xsγ spectrum in Υ(4S)[Belle, 0804.1580 + preliminary, thanks to A. Limosani]

Fit with 4 basis functionsFit works but nontrivial (still issuesdue to large correlations in data)Fitting several spectra and momentconstraints works as well

[GeV]γE2 2.5

γ/d

EΓd

00.05

0.10.150.2

0.250.3

0.350.4

Demonstration only

DataFit

To be able to use measurements, one needsDetector response matrix or unfolded spectra(usually not included in publications, please do!)Correlation matrix for spectra (please include in publications!)Spectra in B frame (hadronic tag) are nicer, but do not correct spectrameasured in Υ(4S) frame to B frame (boost correction depends on SF)



Constraining Subleading Shape Functions

Decompose triple differential rate

d3Γ

dp+X dp−X dz

=3

8

[(1 + z2)HT (p±X) + 2zHA(p±X) + 2(1− z2)HL(p±X)

]z = cos θ = 2

E` − Eνp−X − p+

X

θ is angle between neutrino and B meson in W rest framez dependence is exactStructure functions depend on different combinations of subleading SFs

Can use increased statistics to constrain subleading shape functionsSeparately measure HT,A,L(mX) and/or HT,A,L(p+

X)

Separate B+ and B0 (constrains weak annihilation)

Theory hits a wall at O(αsΛQCD/mb) and O(Λ2QCD/m

2b)



Conclusions

Accuracy of |Vub| is crucial to overconstrain CKM and constrain new physicsCurrent inclusive determinations do not scale to accuracies . 10%

Need to combine all available information into global fit

Consistently combine optimal theory descriptions fordifferent phase space regionsUtilize increased statistics to constrain nonperturbativeinputs (SF) and uncertainties by dataδmb ∼ 10− 20 MeV seems feasible

⇒ |Vub| at SuperB at 3% is ambitious but might be possible

⇒ Precise B → Xu`ν important to look for new physics in B → Xs`+`−

I focused on inclusive |Vub|. Ultimately, it will be important to have ≥ 2independent determinations (inclusive, exclusive, leptonic).


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