The Precision Determination of |V | and |Vub

The Precision Determination of |Vcb| and |Vub|

Phillip UrquijoExperimental Particle Physics GroupThe University of Melbourne

EPFL LPHE Seminar Oct 2007

EPFL LPHE Seminar Oct 2007 Phillip Urquijo 2

Why 3 sets (= generations) of particles? How do they differ? How do they interact with each other?

strong E&M weak

g γW ±

Z 0

µ− cνµ sτ − tντ b

leptons quarkse− uνe d

1st generation

2nd generation

3rd generation

It turns out there are two “extra”

copies of particles

Simplified Standard Model

EPFL LPHE Seminar Oct 2007 Phillip Urquijo

Strong force is transmitted by the gluon

Electromagnetic force by the photon

Weak force by the W± and Z0 bosons

3


uu

g

dd

g

uu

γ

dd

γ

e−

γ

e−

uu

Z0

dd

Z0 Z0

e−e−

Z0

νeνe

du

W+

νee−

W−


Strong force is transmitted by the gluon

Electromagnetic force by the photon

Weak force by the W± and Z0 bosons

3


uu

g

dd

g

uu

γ

dd

γ

e−

γ

e−

uu

Z0

dd

Z0 Z0

e−e−

Z0

νeνe

du

W+

νee−

W−Note W± can “convert”

u ↔ d, e ↔ ν


Mass and the Generations

Fermions They differ only by the masses The Standard Model has no explanation

for the mass spectrum The masses come from the interaction

with the Higgs field ... whose nature is unknown

Par

ticle

mas

s (e

V/c

2 )

Q = −1 0 +2/3 −1/3

The origin of mass is one of the most urgent questions in particle physics today


Masses uniquely define the u, c, t, and d, s, b states We don’t know what creates masses

We don’t know how the weak eigenstates are chosen

couples quark charged currents to W± Mixes (qj=d,s,b) quark mass eigenstates to give weak eigenstates The Standard Model does not predict V

5

Weak decays

6

Cabibbo-Kobayashi-Maskawa matrix

matrix elements are fundamental parameters of the Standard Model, must be determined experimentally

quark mixing in the SM

d′

s′

b′

=

Vud Vus Vub

Vcd Vcs Vcb

Vtd Vts Vtb

dsb

CKM Workshop Nagoya 2006C K M

増川小林

6


d′

s′

b′

=

Vud Vus Vub

Vcd Vcs Vcb

Vtd Vts Vtb

dsb

€

1− λ2

2λ Aλ3(ρ− iη)

−λ 1− λ2

2Aλ2

Aλ3(1−ρ− iη) −Aλ2 1

+ O(λ4)Vij=

Wolfenstein ParameterisationCKM Workshop Nagoya 2006

One irreducible complex phase → CP violationThe only source of CP violation in the minimal Standard Model

C K M増川小林

6


d′

s′

b′

=

Vud Vus Vub

Vcd Vcs Vcb

Vtd Vts Vtb

dsb

€

1− λ2

2λ Aλ3(ρ− iη)

−λ 1− λ2

2Aλ2

Aλ3(1−ρ− iη) −Aλ2 1

+ O(λ4)Vij=

Wolfenstein ParameterisationCKM Workshop Nagoya 2006

One irreducible complex phase → CP violationThe only source of CP violation in the minimal Standard Model

C K M増川小林

Kobayashi and Maskawa winners of EPS High Energy and Particle Physics Prize, 2007


CP violation and New Physics

Are there additional (non-CKM) sources of CP violation?


CP violation and New Physics

The CKM mechanism fails to explain the amount of matter-antimatter imbalance in the Universe ... by several orders of magnitude

New Physics beyond the SM is expected at 1-10 TeV scale e.g. to keep the Higgs mass < 1 TeV/c2

Almost all theories of New Physics introduce new sources of CP violation

New physics effects in quarkflavour mixing can only appear as small corrections to the leading CKM mechanism

New sources of CP violation almost certainly exist

Are there additional (non-CKM) sources of CP violation?


The Unitarity Triangle

V†V = 1 gives us

Measurements of angles and sides constrain the apex (ρ, η)

This one has the 3 terms in the same order of magnitude

A triangle on the complex plane


The Unitarity Triangle

V†V = 1 gives us

Measurements of angles and sides constrain the apex (ρ, η)

This one has the 3 terms in the same order of magnitude

A triangle on the complex plane

Φ1

Φ2

Φ3


What can we measure in B decays?

φ1φ3

φ2

CPV in SM ∝ Area

€

VudVub*

€

VtdVtb*

€

VcdVcb*

€

VtbVts* +VcbVcs

* +VubVus* = 0

CP asymmetries

φ1φ3

φ2

€

(ρ,η)

€

(0,0)

€

(1,0)Measurements often plotted in (ρ,η) plane€

≈Vub

*

λVcb*

€

≈Vtd

λVcb*

€

φ3 ≈ argVub*,φ1 ≈ −argVtd ,φ2 ≈ π −φ1 −φ3


The sides

To measure the lengths of thetwo sides, we must measure|Vub| ≈ 0.004 and |Vtd| ≈ 0.008 The smallest elements – not easy!

Main difficulty: Controlling theoretical errors due to hadronic physics

γ

α

β

Vtd

Vub

Vcb


€

s =10.58GeV,Ee+ = 3.5GeV,Ee− = 8.0GeV

Υ(4S)→B/B(96%)

Data to date: >700 million B/B

Asymmetric e+e- collider running at the Υ(4S) resonance

KEK-B


€

s =10.58GeV,Ee+ = 3.5GeV,Ee− = 8.0GeV

Υ(4S)→B/B(96%)

Data to date: >700 million B/B

Asymmetric e+e- collider running at the Υ(4S) resonance

KEK-B

Crab cavity commissioning

• Current precision

• sin2φ1 ~ 4%

• |Vcb| ~ 1-2%

• |Vub| ~ 8%


Significance of |Vub| and |Vcb| Measurement

∝|Vub|/|Vcb|

• The precise determination of |Vub|/|Vcb| provides a benchmark for testing new physics in other processes

• Precision of sin2β outstrips the other measurements

• We must make the green ring thinner!

• Large new physics contributions to penguins would have already been seen.

• New physics contributions to decays such as B → τν is still open (e.g. minimum flavour violation)


• sin2φ1 ~ 4%

• |Vcb| ~ 1-2%

• |Vub| ~ 8%



∝|Vub|/|Vcb|






!-1 -0.5 0 0.5 1

"

-1

-0.5

0

0.5

1#

cbVubV

!-1 -0.5 0 0.5 1

"

-1

-0.5

0

0.5

1

tree level constraints


• sin2φ1 ~ 4%

• |Vcb| ~ 1-2%

• |Vub| ~ 8%



∝|Vub|/|Vcb|






!-1 -0.5 0 0.5 1

"

-1

-0.5

0

0.5

1#

cbVubV

!-1 -0.5 0 0.5 1

"

-1

-0.5

0

0.5

1

tree level constraints

4% we’re aiming for

the task is quite challenging!


b

decay properties depend directly on |Vcb| & |Vub| and mb

perturbative regime (αsn)

tree level, short distance:

Semileptonic B decays

13

c

e

ν

b c e ν

W

b


b

decay properties depend directly on |Vcb| & |Vub| and mb

perturbative regime (αsn)

tree level, short distance:

Semileptonic B decays

13

But quarks are bound by softgluons: non-perturbative

long distance interactions of b quark with light quark

+ long distance:

B D e ν

W

e

ν

]Dc

B[ b

14

B-decay prediction

c

u

d

b → c d u

b

W

14

B-decay prediction

c

u

d

b → c d u

b

WW

b c

u

d]π]DB[

B → D π

Very difficult

14

B-decay predictionb → c e ν

W

e

ν

cbc

u

d

b → c d u

b

WW

b c

u

d]π]DB[

B → D π

Very difficult

14


W

e

ν

cbc

u

d

b → c d u

b

WW

b c

u

d]π]DB[

B → D π

Very difficult

B →D e ν

W

]DB[Still hadronic

14


W

e

ν

cbc

u

d

b → c d u

b

WW

b c

u

d]π]DB[

B → D π

Very difficult

B →D e ν

W

]DB[Still hadronic

B →τ ν

W τ

νB[b

simple!

u

14


W

e

ν

cbc

u

d

b → c d u

b

WW

b c

u

d]π]DB[

B → D π

Very difficult

B →D e ν

W

]DB[Still hadronic

B →τ ν

W τ

νB[b

simple!

u

..but helicity suppressed


b

15

Semileptonic Decays: Inclusive

One hadronic current.

Inclusive decays b→q l ν:

Weak quark decay + QCD corrections.

Γsl described by Heavy Quark Expansion in (1/mb)n and αs

k

Non perturbative parameters at each order in mb

need to be derived from data, i.e. from inclusive spectral moments of the semileptonic decay products.

Γ(B → Xc!ν) =G2

Fm5b

192π3|Vcb|

2[[1 + Aew]AnonpertApert]

mb is a renormalisation scheme dependent quantity


b

15

Semileptonic Decays: Inclusive

One hadronic current.

Inclusive decays b→q l ν:

Weak quark decay + QCD corrections.

Γsl described by Heavy Quark Expansion in (1/mb)n and αs

k

Non perturbative parameters at each order in mb

need to be derived from data, i.e. from inclusive spectral moments of the semileptonic decay products.

Γ(B → Xc!ν) =G2

Fm5b

192π3|Vcb|

2[[1 + Aew]AnonpertApert]

mb is a renormalisation scheme dependent quantity

Measurement of Vxbb x

Vxb

Wl

ν

! Measurement is very straightforward, use a relation

Γ(b → x!−ν) =G2F192π2

|Vxb|2m5b(

1 +)

! Only need to count the number of b→ x!−ν events, however in reality


Inclusive semileptonic decays Vcb Vs. Vub

Vub: @8% precision, dominated by theoretical uncertainties

Vcb: At the precision of the cabibbo angle, @~1%, with the inclusion of radiative decay measurements


B meson - “Beam”

tag - charge - momentum

B+ and B0 decays studied

separately

Fully reconstruct the tag-side B meson by searching the decay modes

B→D(*)π, B→D(*)ρ, and B→D(*)a1

Bsig→Xlν

π

π

Kγ

γ

l-

ν

B B00

Υ(4S)

Btag→DX

17

Low efficiency (~0.1%), but plenty of data at Belle!

Elepton: in rest frame of signal B

MX: all remaining particles (X),

pX = pbeam-pBtag-pl-pν


Particle identification in BelledE/dx from DriftchamberE/p from E.M.Calorimeter Shower Shape

ee eπ

ππ

1 < p < 2 GeV/c 0.8 < p < 1.2 GeV/cE/p > 0.5

0.8 < p < 1.2 GeV/c

Magnet Coil

Vertex Detector

π π

π

Light yield in Cerenkov Detector

Track and cluster matching

Range and transverse scattering in Muon/hadron

detector

@KEK-B

(GeV/c)e*Bp

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

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trie

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er

0.1

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V/c

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+B

(GeV/c)e*Bp

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

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trie

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er

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V/c

0

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Belle data

0B

! e c X"B

! e u X"B

Secondaries

Combinatorial

Continuum

FIG. 2: Measured electron momentum spectra from B+ and B0 decays before background subtrac-

tion, overlaid with the various backgrounds and the MC signal. The errors shown are statistical

only.

TABLE I: Electron yields for p∗Be ≥ 0.4 GeV/c. The errors are statistical only.

B candidate B+ B0

On Resonance Data 6423 ± 80 5403 ± 74

Scaled Off Resonance 249 ± 48 209 ± 39

Combinatorial Background 1244 ± 20 696 ± 13

Secondary 558 ± 11 1847 ± 22

B → Xueν 74 ± 5 57 ± 6

Background Subtracted 4297 ± 96 2593 ± 87

THE ELECTRON ENERGY SPECTRUM

Unfolding

To measure the moments of the electron energy spectrum, we need to determine the true

electron energy spectrum in the B meson rest frame, E∗Be . In this analysis we assume the

10

• beam background.

The photon spectra for ON and scaled OFF data samples along with the results ofsubsequent background subtractions are plotted in Fig. 1(a). The B → Xsγ photon energyspectrum that has been corrected for efficiency is shown in Fig. 1(b). The analysis measuredthe branching fraction,

B(B → Xsγ) = (3.55 ± 0.32+0.30+0.11−0.31−0.07) × 10−4, (1)

where the errors are statistical, systematic and theoretical, respectively. This result agreedwith the latest theoretical calculations [15, 16], as well as with previous measurements madeby CLEO [17] and Belle [18].

102

103

104

105

1.5 2 2.5 3 3.5 4 4.5 5

E*! [GeV]

Ev

en

ts/1

00

Me

V

On-resonance

Scaled off-resonance

On-Off Difference

Background subtracted

E*! [GeV]

Ev

en

ts/1

00

Me

V

-5000

0

5000

10000

15000

20000

25000

1.5 2 2.5 3 3.5 4

(a) (b)

FIG. 1: From [12]. (a) Photon energy spectra in the Υ(4S) frame. (b) Efficiency-corrected photon

energy spectrum. The two error bars show the statistical and total errors.

MOMENT MEASUREMENTS

We follow the procedure used in the published analysis [12], with some slight variations.No attempt is made to correct for the part of the spectrum that is not measured witha satisfactory precision i.e from energy below 1.8 GeV. We apply lower energy thresholdcuts as measured in the Υ(4S) rest frame (E∗

cut) to the efficiency corrected spectrum, fromwhich we obtain truncated first and second moments. Corrections are applied to recover themoments such that the lower energy thresholds correspond to quantities measured in theB-meson rest frame (Ecut).

A simple procedure is used to unfold the effects of detector resolution, the small B-mesonboost in the Υ(4S) frame, and that of the 100 MeV wide bins. We define the first momentas 〈Eγ〉 (mean) and the second moment as ∆E2

γ ≡⟨

E2γ

⟩

−〈Eγ〉2 (variance). The correctionsare as follows:

5


b

19

Inclusive decays- Big Picture

rate

shape

shape

|Vcb|

mb,mc

μ2G, μ2

πInclusive Mx spectrum

Semileptonic B decay

|Vub|

2|| ubV∝

€

Shape(B→ Xs γ)

(GeV/c)e*Bp

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

En

trie

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0.1

GeV

/c

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400

500

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200

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+B

(GeV/c)e*Bp

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

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trie

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er

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GeV

/c

0

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200

300

400

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600

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100

200

300

400

500

600

Belle data

0B

! e c X"B

! e u X"B

Secondaries

Combinatorial

Continuum



only.


B candidate B+ B0




Secondary 558 ± 11 1847 ± 22

B → Xueν 74 ± 5 57 ± 6



Unfolding



10

Inclusive El spectrum

Belle0

200

400

600

800

1000

1200

1400

1600

0 5 10 15

M2X (GeV

2/c

4)

entr

ies

/ 0.3

33 G

eV2/c

4

Belle

E*l ! 0.7 GeV

data

fake/secondary

combinatorial

B ! Xul"

B ! Xcl"

0

200

400

600

800

1000

1200

1400

1600

0 5 10 15

M2X (GeV

2/c

4)

entr

ies

/ 0.3

33 G

eV2/c

4

Belle

E*l ! 1.1 GeV

0

200

400

600

800

1000

0 5 10 15

M2X (GeV

2/c

4)

entr

ies

/ 0.3

33 G

eV2/c

4

Belle

E*l ! 1.5 GeV

0

50

100

150

200

250

300

350

0 5 10 15

M2X (GeV

2/c

4)

entr

ies

/ 0.3

33 G

eV2/c

4

Belle

E*l ! 1.9 GeV

FIG. 1: Measured hadronic mass spectrum for different lepton energy thresholds. The pointswith error bars are the experimental data after subtraction of the continuum background. Thehistograms show the B → Xc!ν signal and the different background components, explained in moredetail in the text.

and B0 muon). From the unfolded spectrum, we calculate the first moment and its statisticaluncertainty squared,

〈M2X〉 =

∑i(M

2X)ix′

i∑i x

′i

, σ2(〈M2X〉) =

∑i,j(M

2X)iXij(M2

X)j

(∑

i x′i)2

. (3)

Here, x′ is the unfolded spectrum corrected for slightly different bin-to-bin efficiencies and Xis its covariance matrix, also determined by the SVD algorithm. (M2

X)i is the central valueof the i-th bin of the unfolded spectrum. The second central and non-central moments,

8

Experimental Challenge: go from

the measured shapes to the true shapes.


Moments of inclusive spectra

El>Ecut

Theory calculations predict these moments as a function of the minimum lepton energy in the B rest frame, Ecut :

higher moments are sensitive to 1/mb3 terms → reduce theory error on

| Vcb| and Heavy Quark parameters

comparison of results from different measurements and schemes provides a test of the consistency of predictions and of underlying assumptions


Electron energy spectrum

Boost into B rest frame

Add bremsstrahlung photons to the electron

Subtract background

Unfold detector effects and QED radiative effects

Sample preparations:

(GeV/c)e*Bp

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

En

trie

s p

er

0.1

GeV

/c

0

100

200

300

400

500

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40

100

200

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500Belle data

+B

(GeV/c)e*Bp

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

En

trie

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er

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GeV

/c

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100

200

300

400

500

600

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100

200

300

400

500

600

Belle data

0B

! e c X"B

! e u X"B

Secondaries

Combinatorial

Continuum



only.


B candidate B+ B0




Secondary 558 ± 11 1847 ± 22

B → Xueν 74 ± 5 57 ± 6



Unfolding



10

(GeV/c)e*Bp

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

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trie

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/c

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200

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+B

(GeV/c)e*Bp

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trie

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GeV

/c

0

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200

300

400

500

600

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40

100

200

300

400

500

600

Belle data

0B

! e c X"B

! e u X"B

Secondaries

Combinatorial

Continuum



only.


B candidate B+ B0




Secondary 558 ± 11 1847 ± 22

B → Xueν 74 ± 5 57 ± 6



Unfolding



10

pe>0.4GeV/cP.Urquijo et al. Phys. Rev. D 75, 032001 (2007)


Electron energy spectrum

Boost into B rest frame

Add bremsstrahlung photons to the electron

Subtract background

Unfold detector effects and QED radiative effects

Sample preparations:

(GeV/c)e*Bp

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

En

trie

s p

er

0.1

GeV

/c

0

100

200

300

400

500

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40

100

200

300

400

500Belle data

+B

(GeV/c)e*Bp

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

En

trie

s p

er

0.1

GeV

/c

0

100

200

300

400

500

600

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40

100

200

300

400

500

600

Belle data

0B

! e c X"B

! e u X"B

Secondaries

Combinatorial

Continuum



only.


B candidate B+ B0




Secondary 558 ± 11 1847 ± 22

B → Xueν 74 ± 5 57 ± 6



Unfolding



10

(GeV/c)e*Bp

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

En

trie

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GeV

/c

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500

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+B

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0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

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trie

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/c

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100

200

300

400

500

600

Belle data

0B

! e c X"B

! e u X"B

Secondaries

Combinatorial

Continuum



only.


B candidate B+ B0




Secondary 558 ± 11 1847 ± 22

B → Xueν 74 ± 5 57 ± 6



Unfolding



10

pe>0.4GeV/cP.Urquijo et al. Phys. Rev. D 75, 032001 (2007)

Measure up to 4th moment!

0.4 GeV electron energy threshold

electron to be massless, imposing E∗Be = p∗Be . The measured electron energy spectrum is

distorted by various detector effects. Hence, the true electron energy spectrum is extracted

by performing an unfolding procedure based on the Singular Value Decomposition (SVD)

algorithm [34]. The reliability of the unfolding procedure is dependent on the agreement

between data and MC simulation, both for the physics models and the detector response.

The unfolded spectrum is corrected for QED radiative effects using the PHOTOS algo-

rithm [33], as the OPE does not have O(α) QED corrections. The unfolded electron energy

spectrum and the bin to bin statistical covariance matrix calculated with the unfolding al-

gorithm are shown in Figure 3 (for illustrative purposes only, as the full error analysis is

performed on a moment measurement basis).

(GeV/c)e*BE

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4E

ntr

ies p

er

0.1

GeV

/c0

200

400

600

800

1000

Belle

Generated (GeV)e*BE

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

R

eco

nstr

ucte

d (

GeV

)e*B

E

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

FIG. 3: Unfolded electron energy distribution in the B meson rest frame(left), combining con-

tributions from B0 and B+ decays, and corrected for QED radiative effects. The errors shown

are statistical. On the right is the corresponding unfolded electron energy distribution covariance

matrix, where the filled boxes represent negative elements. (These plots are shown for illustrative

purposes only.)

Moments and Branching Fractions

We measure the first four central moments with nine electron energy threshold values

(Ecut = 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0 GeV) combining the spectra from B+ and B0

11

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

Ge

V)

-3 (

10

1M

1400

1500

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1700

1800

1900

2000

2100

Belle

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

)2

Ge

V-3

(1

02

M

0

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120

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180

Belle

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

)3

Ge

V-3

(1

03

M

-25

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-15

-10

-5

0

5

Belle

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

)4

Ge

V-3

(1

04

M

0

10

20

30

40

50

60

70

Belle

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

-2

10

!B

R

0

2

4

6

8

10 Belle

FIG. 4: First, second, third and fourth electron energy moments and branching fraction (M1,

M2, M3, M4, B), as a function of the electron threshold energy Ecut . The errors shown are the

statistical and systematic errors added in quadrature.

24

Electron energy moments and partial BR

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

Ge

V)

-3 (

10

1M

1400

1500

1600

1700

1800

1900

2000

2100

Belle

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

)2

Ge

V-3

(1

02

M

0

20

40

60

80

100

120

140

160

180

Belle

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

)3

Ge

V-3

(1

03

M

-25

-20

-15

-10

-5

0

5

Belle

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

)4

Ge

V-3

(1

04

M

0

10

20

30

40

50

60

70

Belle

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

-2

10

!B

R

0

2

4

6

8

10 Belle




24

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

Ge

V)

-3 (

10

1M

1400

1500

1600

1700

1800

1900

2000

2100

Belle

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

)2

Ge

V-3

(1

02

M

0

20

40

60

80

100

120

140

160

180

Belle

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

)3

Ge

V-3

(1

03

M

-25

-20

-15

-10

-5

0

5

Belle

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

)4

Ge

V-3

(1

04

M

0

10

20

30

40

50

60

70

Belle

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

-2

10

!B

R

0

2

4

6

8

10 Belle




24

Systematics: b→c model, background, electron detection

BR(B+)0.4GeV=(10.79±0.25±0.27)BR(B0)0.4GeV=(10.08±0.30±0.22)

x10-2

P.Urquijo et al. Phys. Rev. D 75, 032001 (2007)

22

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

Ge

V)

-3 (

10

1M

1400

1500

1600

1700

1800

1900

2000

2100

Belle

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

)2

Ge

V-3

(1

02

M

0

20

40

60

80

100

120

140

160

180

Belle

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

)3

Ge

V-3

(1

03

M

-25

-20

-15

-10

-5

0

5

Belle

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

)4

Ge

V-3

(1

04

M

0

10

20

30

40

50

60

70

Belle

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

-2

10

!B

R

0

2

4

6

8

10 Belle




24

Electron energy moments and partial BR

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

Ge

V)

-3 (

10

1M

1400

1500

1600

1700

1800

1900

2000

2100

Belle

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

)2

Ge

V-3

(1

02

M

0

20

40

60

80

100

120

140

160

180

Belle

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

)3

Ge

V-3

(1

03

M

-25

-20

-15

-10

-5

0

5

Belle

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

)4

Ge

V-3

(1

04

M

0

10

20

30

40

50

60

70

Belle

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

-2

10

!B

R

0

2

4

6

8

10 Belle




24

Decrease of truncated BR

Incr

ease

of t

he m

ean Decrease of the width

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

Ge

V)

-3 (

10

1M

1400

1500

1600

1700

1800

1900

2000

2100

Belle

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

)2

Ge

V-3

(1

02

M

0

20

40

60

80

100

120

140

160

180

Belle

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

)3

Ge

V-3

(1

03

M

-25

-20

-15

-10

-5

0

5

Belle

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

)4

Ge

V-3

(1

04

M

0

10

20

30

40

50

60

70

Belle

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

-2

10

!B

R

0

2

4

6

8

10 Belle




24

Systematics: b→c model, background, electron detection

BR(B+)0.4GeV=(10.79±0.25±0.27)BR(B0)0.4GeV=(10.08±0.30±0.22)

x10-2

P.Urquijo et al. Phys. Rev. D 75, 032001 (2007)

22

The hadronic XThe hadronic Xcc systemsystem

!"#

$!#

! !%#

??

&

&'

&''

HQET:

GroundGround

statesstatesBroadBroad

statesstatesNarrowNarrow

statesstates

for L=1 " jq=1/2, jq=3/2

A. OyangurenEPS ‘05 - Lisboa 4

21%

BR(B→Xcl ν)~10.5%

54%

~25%

The hadronic XThe hadronic Xcc systemsystem

!"#

$!#

! !%#

??

&

&'

&''

HQET:

GroundGround

statesstatesBroadBroad

statesstatesNarrowNarrow

statesstates

for L=1 " jq=1/2, jq=3/2

A. OyangurenEPS ‘05 - Lisboa 4

Grounds states Broad states Narrow states

Hadronic Xc system

23

Systematics: b→c model, background subtraction, unfolding

3.9

4

4.1

4.2

4.3

4.4

4.5

1 1.5 2

Emin

* (GeV)

<M

2 X>

(G

eV

2/c

4)

Belle

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

1 1.5 2

Emin

* (GeV)

<(M

2 X-<

M2 X>

)2>

(G

eV

4/c

8)

Belle

16

17

18

19

20

21

22

1 1.5 2

Emin

* (GeV)

<M

4 X>

(G

eV

4/c

8)

Belle

FIG. 2: Graphical representation of the results in Table II. The error bars indicate the statistical

and total experimental errors.

for B → D∗!ν (B → D!ν) [32]. For B → D∗!ν, we also vary the form factor ratios R1 and

R2 [33].

The LLSW model [18] predicts the relative abundance and the form factor shape of the

different components in B → D∗∗!ν only. To obtain the absolute branching fractions of

the B → D∗∗!ν components and of B → (D(∗)π)non−res.!ν, we use B(B+ → D01!

+ν) =

(5.6 ± 1.6) × 10−3 [31], the recent Belle measurement of B(B → D(∗)π!ν) [34] and the total

semileptonic branching fraction [31]. The uncertainty assigned to the B → D∗∗!ν branching

10

3.9

4

4.1

4.2

4.3

4.4

4.5

1 1.5 2

Emin

* (GeV)

<M

2 X>

(G

eV

2/c

4)

Belle

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

1 1.5 2

Emin

* (GeV)

<(M

2 X-<

M2 X>

)2>

(G

eV

4/c

8)

Belle

16

17

18

19

20

21

22

1 1.5 2

Emin

* (GeV)

<M

4 X>

(G

eV

4/c

8)

Belle




R2 [33].




+ν) =



10


Hadron invariant mass moments

Moments of unfolded spectra.

decrease in higher mass final states

Measured Mx2

spectra for varying El

*cut

C.Schwanda, P.Urquijo et al., Phys. Rev. D 75, 032005 (2007)

0

200

400

600

800

1000

1200

1400

1600

0 5 10 15

M2X (GeV

2/c

4)

entr

ies

/ 0.3

33 G

eV2/c

4

Belle

E*l ! 0.7 GeV

data

fake/secondary

combinatorial

B ! Xul"

B ! Xcl"

0

200

400

600

800

1000

1200

1400

1600

0 5 10 15

M2X (GeV

2/c

4)

entr

ies

/ 0.3

33 G

eV2/c

4

Belle

E*l ! 1.1 GeV

0

200

400

600

800

1000

0 5 10 15

M2X (GeV

2/c

4)

entr

ies

/ 0.3

33 G

eV2/c

4

Belle

E*l ! 1.5 GeV

0

50

100

150

200

250

300

350

0 5 10 15

M2X (GeV

2/c

4)

entr

ies

/ 0.3

33 G

eV2/c

4

Belle

E*l ! 1.9 GeV

0

200

400

600

800

1000

1200

1400

1600

0 5 10 15

M2X (GeV

2/c

4)

entr

ies

/ 0.3

33 G

eV2/c

4

Belle

E*l ! 0.7 GeV

data

fake/secondary

combinatorial

B ! Xul"

B ! Xcl"

0

200

400

600

800

1000

1200

1400

1600

0 5 10 15

M2X (GeV

2/c

4)

entr

ies

/ 0.3

33 G

eV2/c

4

Belle

E*l ! 1.1 GeV

0

200

400

600

800

1000

0 5 10 15

M2X (GeV

2/c

4)

entr

ies

/ 0.3

33 G

eV2/c

4

Belle

E*l ! 1.5 GeV

0

50

100

150

200

250

300

350

0 5 10 15

M2X (GeV

2/c

4)

entr

ies

/ 0.3

33 G

eV2/c

4

Belle

E*l ! 1.9 GeV

Systematics: b→c model, background subtraction, unfolding

3.9

4

4.1

4.2

4.3

4.4

4.5

1 1.5 2

Emin

* (GeV)

<M

2 X>

(G

eV

2/c

4)

Belle

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

1 1.5 2

Emin

* (GeV)

<(M

2 X-<

M2 X>

)2>

(G

eV

4/c

8)

Belle

16

17

18

19

20

21

22

1 1.5 2

Emin

* (GeV)

<M

4 X>

(G

eV

4/c

8)

Belle




R2 [33].




+ν) =



10

3.9

4

4.1

4.2

4.3

4.4

4.5

1 1.5 2

Emin

* (GeV)

<M

2 X>

(G

eV

2/c

4)

Belle

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

1 1.5 2

Emin

* (GeV)

<(M

2 X-<

M2 X>

)2>

(G

eV

4/c

8)

Belle

16

17

18

19

20

21

22

1 1.5 2

Emin

* (GeV)

<M

4 X>

(G

eV

4/c

8)

Belle




R2 [33].




+ν) =



10


Hadron invariant mass moments

Moments of unfolded spectra.

decrease in higher mass final states

Measured Mx2

spectra for varying El

*cut

C.Schwanda, P.Urquijo et al., Phys. Rev. D 75, 032005 (2007)

|Vcb| DeterminationMulti-parameter Global fits of HQE to inclusive B decay spectra

INPUT Belle measured moments:

Ee moments: Phys. Rev. D 75, 032001 (2007)

Mx moments: Phys. Rev. D 75, 032005 (2007)

Eγ moments: hep-ex/0508005

P. Urquijo et al., Belle Conf 0669, hep-ex/0611047

25


Heavy quark parameters

−

Decay rate in terms of Operator Product Expansion up to 1/mb3

Expansions in terms of:

mkinb ,mkin

c (m1Sb) - mass of b and c quarks

ΛQCD2/mb2μπ2(λ1) - kinetic energy of b quark,μG

2(λ 2) - chromomagnetic coupling

ΛQCD3/mb3 ρD, ρLS (ρ1,τ1-3)

Two approaches:

... and |Vcb|2 dependence on partial branching fractions

P. Urquijo et. al. , Belle Conf 0669, hep-ex/0611047

Pole mass not used anymore: not well behaved: irreducible error on mb

1S mass (C.Bauer, Z.Ligeti, M.Luke,

A.Manohar, M.Trott PRD 70 094017)

Kinetic running mass(P. Gambino and N.Uraltsev, Eur.

Phys. J. C 34, 181 (2004))

26

Theory calculation for B→Xs γ is at 1-loop.


Experimental data incl. errors and correlaions

Theory (OPE)

|Vcb| and HQ parameters (kinetic scheme)

|Vcb| and HQ parameters (1S scheme)

HQ parameters (SF scheme)

Fit Strategyfit to set of equations

extract

convert

Without truncation of perturbation theory, any path to a given scheme would lead to the same result, e.g.:

[Fit in kinetic scheme] = [Fit in 1S scheme] ⊕ [Translation: 1S → kinetic]

In practice, results differ at finite order in αs.

(GeV)bm4.55 4.6 4.65 4.7 4.75 4.8 4.85 4.9

)2

(G

eV

1!

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

Belle1S scheme

(GeV)bm4.55 4.6 4.65 4.7 4.75 4.8 4.85 4.9

)2

(G

eV

1!

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

Belle1S scheme


Fit to Belle data: |Vcb| and HQ parameters Preliminary

χ2/d.o.f. =17.8/24

mb = 4.564 ± 0.076 GeV

mc = 1.105 ± 0.116 GeV

|Vcb| = ( 41.93 ± 0.65fit ± 0.07αs ± 0.63th ) × 10-3

P. Urquijo et. al. , Belle Conf 0669, hep-ex/0611047

|Vcb| = (41.55 ± 0.68(fit) ± 0.08(τB) ) × 10-3

mb1S = (4.723 ± 0.055) GeVλ1 = (-0.30 ± 0.05) GeV2 χ2/d.o.f. =6/17

(GeV)bm4.55 4.6 4.65 4.7 4.75 4.8 4.85 4.9

)2 (G

eV1!

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

Belle1S scheme1S scheme

kinetic scheme

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

4.2 4.4 4.6 4.8 5

M2X only

El only

E! only

combined

mb (GeV)

!2 " (

GeV

2)

-0.05

0

0.05

0.1

0.15

0.2

0.25

4.2 4.4 4.6 4.8 5

M2X only

El only

combined

mb (GeV)

#3 D (

GeV

3)

FIG. 3: Stability of the fit. The fit is repeated using lepton energy moments only, hadron mass

moments only and photon energy moments only. The ellipses are ∆χ2 = 1.

IV. SUMMARY

We have performed a fit to the Belle measured moments of the lepton energy and hadronic

mass spectrum in B → Xc!ν decays and the photon energy spectrum in B → Xsγ decays

using HQE expressions derived in the kinetic scheme. We have extracted |Vcb|, mb and other

heavy quark parameters.

It is expected that improved calculations of the perturbative corrections to the hadronic

mass moments could improve the quality of the fit and thus reduce the fit error.

[1] P. Urquijo, Moments of the Electron Energy Spectrum in B → Xc"ν decays at Belle, Belle

note #904.

[2] C. Schwanda, Moments of the hadronic invariant mass spectrum in B → Xc"ν decays at Belle,

Belle note #920.

[3] K. Abe et al. [Belle Collaboration], “Moments of the photon energy spectrum from B –¿ X/s

gamma decays measured arXiv:hep-ex/0508005.

[4] P. Gambino and N. Uraltsev, Eur. Phys. J. C 34, 181 (2004) [arXiv:hep-ph/0401063].

8

Kinetic


contours Δχ2=1

1S

Global fit with all available results (except the latest BaBar moments)

Kinetic Vcb (10-3) mb (GeV)

41.91±0.19±0.28±0.59

4.613±0.022+0.027

no b→sγ 41.68±0.39±0.58 4.677±0.053

1S Vcb (10-3) mb (GeV)

41.78±0.30±0. 08 4.70±0.03

no b→sγ 41.56±0.39±0.08 4.751±0.058

Vcb and HQ parameters (world fit)

c.f. mb1S=4.71±0.03 GeV (Hoang PRD 61 034005)

BaBar, Belle, CDF, CLEO & DELPHI measured moments

using upsilon sum rules29


Problem with using B→Xsγmoment data

“ B→XSγ should not be used to extract heavy quark parameters ” (M. Neubert Lepton-Photon 2007)

No model-independent treatment is possible ( Neubert & Becher, Phys. Rev. Lett. 98, 022003 (2007 )

Present analyses either use OPE directly (not fully justified), or use a model to apply a “bias correction” to the data before using the OPE

Tension in data, which lowers mb, and raises |Vcb| and |Vub|

contours Δχ2=1

1S

|Vub| DeterminationLimiting factor in CKM precision tests; known much less well than |Vcb|

CKM suppressed Vub~0.1xVcb- therefore harder to measure

The problem: b → clv decay

Tree level

How can we suppress 50× larger background?How can we reach 4% precision on |Vub|?


Cut away b → clv Lose a part of the b → ulv signal

We measure

Fraction of the signal that pass the cut

Requires the knowledge of the b quark’s motion inside the B meson

Figuring out what we see

Total b → ulv rate

Cut-dependentconstant predicted

by theory

Must be corrected for QCD

Main uncertainty (±5%) from mb5 → ±2.5% on |Vub|, correlated

between all measurements/experiments!

But we need the reasonable fraction (e.g., Eℓ > 2 GeV) of the rate.

32


Detecting b → ulν• Inclusive: Use mu << mc difference in kinematics

• Maximum lepton energy 2.64 vs. 2.31 GeV• First observations (1990) used this technique• Only 6% of signal accessible

• How accurately do we know this fraction?

2.64

2.31

b→c background

Other background

ΔBR x 10-4 % total phase space

Belle 27fb-1:Ee>1.9 GeV 8.47 ± 0.37stat ± 1.53sys ~24%

(a)

Ele

ctr

on

Can

did

ate

s / (

50 M

eV

/c)

Ele

ctr

on

Can

did

ate

s / (

50 M

eV

/c)

(b)

Momentum (GeV/c)

102

103

104

0

2500

5000

7500

2 2.2 2.4 2.6 2.8 3

Fig. 2. The electron momentum spectrum in the Υ(4S) rest frame: (a) ON data(filled circles), scaled OFF data (open circles), sum of scaled OFF data and esti-mated BB backgrounds (histogram). (b) ON data after subtraction of backgroundsand correction for efficiency (filled circles) and model spectrum of B → Xueνe

decays with final state radiation (histogram, normalised to the data yield in the1.9 − 2.6 GeV/c momentum range).

the size of the effect. The correction factors for each HI region are given inTable 2.

The values of fu for the different momentum intervals are determined withthe DeFazio-Neubert prescription, using three different forms of the shapefunction with the parameters, ΛSF and λSF

1 , determined from fits to the Bellemeasured photon energy spectrum in B → Xsγ decays [21,24]. The resultantvalues of fu are given in Table 2, they range from 3 − 32% as the lowermomentum limit is decreased. The statistical uncertainty, averaged over eachshape function form, is determined from the half-difference of maximum and

15

(a)

Ele

ctr

on

Ca

nd

ida

tes

/ (

50

Me

V/c

)E

lec

tro

n C

an

did

ate

s /

(5

0 M

eV

/c)

(b)

Momentum (GeV/c)

102

103

104

0

2500

5000

7500

2 2.2 2.4 2.6 2.8 3






15

(a)

Ele

ctr

on

Can

did

ate

s / (

50 M

eV

/c)

Ele

ctr

on

Can

did

ate

s / (

50 M

eV

/c)

(b)

Momentum (GeV/c)

102

103

104

0

2500

5000

7500

2 2.2 2.4 2.6 2.8 3






15

(a)

Ele

ctr

on

Can

did

ate

s / (

50 M

eV

/c)

Ele

ctr

on

Can

did

ate

s / (

50 M

eV

/c)

(b)

Momentum (GeV/c)

102

103

104

0

2500

5000

7500

2 2.2 2.4 2.6 2.8 3






15

signalGeV

b → u

b → c

Phys.Lett.B621:28-40 (2005)

ELEP

(GeV)

A.U

.

0

0.025

0.05

0.075

0.1

0 1 2 3

mX

(GeV)

A.U

.

b ! u

b ! c

0

0.2

0.4

0 1 2 3

q2 (GeV

2)

A.U

.

b ! u

b ! c

0

0.02

0.04

0.06

0 5 10 15 20

P+ (GeV)

A.U

.

0

0.02

0.04

0.06

0.08

0 0.5 1 1.5 2


ELEP

(GeV)

A.U

.

0

0.025

0.05

0.075

0.1

0 1 2 3

mX

(GeV)

A.U

.

b ! u

b ! c

0

0.2

0.4

0 1 2 3

q2 (GeV

2)

A.U

.

b ! u

b ! c

0

0.02

0.04

0.06

0 5 10 15 20

P+ (GeV)

A.U

.

0

0.02

0.04

0.06

0.08

0 0.5 1 1.5 2E

LEP (GeV)

A.U

.

0

0.025

0.05

0.075

0.1

0 1 2 3

mX

(GeV)

A.U

.

b ! u

b ! c

0

0.2

0.4

0 1 2 3

q2 (GeV

2)

A.U

.

b ! u

b ! c

0

0.02

0.04

0.06

0 5 10 15 20

P+ (GeV)

A.U

.

0

0.02

0.04

0.06

0.08

0 0.5 1 1.5 2

34

Accessing a little more phase space

q2 = lepton-neutrino mass squared

mX = hadron system mass

El = lepton energy

b→u

b→c

b→u

b→c

b→ub→c

Not

to

scal

e!

u-quark turns into one or more hadrons

Best access to kinematics with B meson “beam” 41


Hadronic Mass

The background subtracted hadron

mass spectrum

Same recoil technique as for B->Xclν

ΔBR x 10-4 % total phase space

Belle 253 fb-1: mX < 1.7 12.4 ± 1.1stat ± 1.0sys ~47%

Belle 253 fb-1: mX < 1.7, q2 > 8 8.4 ± 0.8stat ± 1.0sys ~24%

Phys.Rev.Lett.95:241801,2005


Shape Function

Shape Function needs to be determined from experimental data

Theory doesn’t work everywhere in the phase space →resumming turns non-perturbative terms into a Shape Function F(K+)

0

F(K+)

Λ = MB -mb

K+

We can determine mean and width of shape function.

Shape not well constrained.

Two ways to solve the problem:

1. Cutoff-based “shape Function” approach2. Use something different : Dressed Gluon Exponentiation

Theory calculation for SF is at 2-loop.


Predicting b → ulν Spectra

Fit the b → sγ spectrum to constrain the shape function

Additional information from b → clv decays b-quark: mass and kinetic energy

Plug the SF into the b → ulv spectrum calculations

Ready to extract |Vub|

€

dxxicut

∫ dΓ(B→Xulυ )dxi

= dEγ∫ W(xi,cutEγ )dΓ(B→Xsγ )

dEγ


How Things Mesh Together

Inclusiveb → ulv

q2

b→sγ

ShapeFunction

Eγ

mb

Inclusive b → clv

mXEl

HQE Fit

mX

El

WA

duality

|Vub|

SSFs

|Vcb|

AKA: The HQE plumbing diagram

]-3 10×| [ub|V2 4 6

]-3 10×| [ub|V2 4 6

) eCLEO (E 0.44± 0.46 ±3.91

) 2, qXBELLE sim. ann. (m 0.36± 0.45 ±4.23

) eBELLE (E 0.38± 0.43 ±4.67

) eBABAR (E 0.39± 0.24 ±4.23

) hmax, seBABAR (E

0.49± 0.29 ±4.37

) XBELLE (m 0.32± 0.26 ±3.92

) XBABAR (m 0.39± 0.20 ±4.09

Average +/- exp +/- (mb,theory) 0.35± 0.17 ±4.31

HFAGLP 2007

OPE-HQET-SCET (BLNP)Phys.Rev.D72:073006,2005

moments! s " and b# c l " input from bbm

/dof = 6.1/ 6 (CL = 41 %)2$


Inclusive |Vub|: BLNP framework

Statistical ±2.0%

Expt. syst. ±3.3%

b → clν model ±1.8%

b → ulν model ±1.1%

HQ params. ±6.9%

Theory ±3.8%

WA ±1.7%

|Vub| world average as of Summer 07 |Vub| determined to ± 8.9%

CKM Workshop Nagoya 2006

LNP


Theory Errors1.7% Weak annihilation

Expected to be <2% of the total rate

3.8% Higher order non-perturbative corrections

Cannot be constrained with b → sγ6.9% HQ parameters

mainly mb

b

u

soft

B

From my |Vcb| fit!

only limits determined

All of these uncertainties can be reduced significantly if we increase the measured phase space.

The power dependence of mb and related theoretical uncertainties scale inversely with phase space coverage.

ubV0.003 0.0035 0.004 0.0045 0.005

)ub

(V!

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5-310×

0

1

2

3

4

5

6 !


The side versus the angleMost probable value of |Vub| from measurements of other CKM parameters in the Standard Model

|Vub| inclusive

!"#$%&'%()#*++, -.#-/(001#2%(3%(4 56

!"##$%&

! 7(890:8#48;8(<0&%;0/&#/=#>!"#>#9/<?@8<8&;:#:0&*! ;/#;8:;#;A8

B0&C9/<?@8;8&8:: /=#;A8#D;%&4%(4#-/48@

! "".,E#%99'(%9)#%9A08384#:/#=%(" FE#?/::0G@8H

! I@/:8#9/@@%G/(%;0/&#G8;J88&#;A8/()#%&4#8K?8(0<8&;#0:#9('90%@

! L%?04#?(/M(8::#0&#0&9@':038#>!"#>#0&#;A8#@%:;#*#)8%(:

! N<?(/38<8&;#0& $! #!$ =/(<#=%9;/(#0:#&88484

!

>!"# >

φ1


[GeV]0E2 2.1 2.2 2.3 2.4

]-3| [

10ub

|V

0

2

4

6

8

10

12 Calculation

LLRLNP

Graph

A novel approach for |Vub| determinationar

Xiv

:hep

-ph/0

702072v1 7 F

eb 2

007

SLAC-PUB-12336

Extraction of |Vub| with Reduced Dependenceon Shape Functions.

Vladimir B. Golubev, Yuri I. SkovpenBudker Institute for Nuclear Physics, Novosibirsk 630090, Russia.

Vera G. LuthStanford Linear Accelerator Center, Stanford, CA 94309, USA.

February 5, 2007

Abstract

Using BABAR measurements of the inclusive electron spectrum in B → Xueν decays andthe inclusive photon spectrum in B → Xsγ decays, we extract the magnitude of the CKMmatrix element Vub. The extraction is based on several theoretical calculations designed toreduce the theoretical uncertainties by exploiting the assumption that the leading shapefunctions are the same for all b → q transitions (q is a light quark). The current resultsagree well with the previous analysis, have indeed smaller theoretical errors, but arepresently limited by the knowledge of the photon spectrum and the experimental errorson the lepton spectrum.

1

Babar data

Novel method for |Vub| extraction theoretical calculations designed toreduce theoretical uncertainties by exploiting the assumption that the leading shape functions are the same for all b→q transitions.

Uncertainties due to non-perturbative power corrections increase rapidly near the kinematic end point (not included in LLR).

Note:

The results agree with analyses that rely on an assumed shape function form.

42


Rectangular cuts? A linear boundary? A nonlinear one?

• Exploit the many non-linear correlations between kinematic variables that separate b→u and b→c.

• Use it to reduce theoretical errors traditionally large due to limited phase space.

• No need to place stringent hard cuts that result in zero efficiency.

Insight into the futureMy latest approach for “hunting down” b→u

43


Almost fully inclusive: Shape Function not needed!

Insight into the future

|Vub| measurement (using BLNP)

Phys.Rev.Lett. 96 (2006) 221801 BaBar: Mx<2.5GeV

Phys.Rev.Lett. 96 (2006) 221801 Belle: Mx<1.7GeV

MVA “guestimate”

fu uncertainties (%)stat sys the tot

~90% 18.2 7.8 2.6 ~20

~47% 5.0 4.8 5.9 ~9

~90% ~5 ~4 ~2.6 ~6.9

|Vub| theory errors are highly correlated. Measurements of the full phase space are crucial to reducing the error on

|Vub| in the world average!

• Work is underway on a novel- non-linear approach to selecting signal events, allowing for ~90% phase space coverage without a loss in statistical power! Coming soon from Belle.

44

HQE param from S.L. and

radiativeResults for various cuts

3.64±0.18±0.3513.64.09±0.20±0.390.41BaBar, MX>1.55 GeV

3.91±0.42±0.339.34.23±0.45±0.360.24Belle, MX>1.7 GeV,

q2>8 GeV2

3.58±0.24±0.2910.53.92±0.26±0.320.47Belle, MX<1.7 GeV

3.88±0.26±0.4413.84.37±0.29±0.490.13BaBar, Ee>2.0 GeV,

shmax<3.5 GeV2

3.84±0.22±0.3511.44.23±0.24±0.390.19BaBar, Ee>2.0 GeV

4.31±0.40±0.339.54.67±0.43±0.360.24Belle, Ee>1.9 GeV

3.77±0.15±0.31

4.31±0.17±0.35

4.18±0.16±0.34

Average (HFAG)

My average

3.46±0.41±0.3914.43.91±0.46±0.440.13CLEO, Ee>2.1 GeV

|Vub| [10-3](mb=4.71±0.06 GeV)

a|Vub| [10-3]

(mb=4.63±0.06 GeV)fu


Results for various cuts

45

M.Neubert (LP07)

<

<

fraction of total phase space HQE param from S.L. only

HQE param from S.L. and

radiativeResults for various cuts

3.64±0.18±0.3513.64.09±0.20±0.390.41BaBar, MX>1.55 GeV

3.91±0.42±0.339.34.23±0.45±0.360.24Belle, MX>1.7 GeV,

q2>8 GeV2

3.58±0.24±0.2910.53.92±0.26±0.320.47Belle, MX<1.7 GeV

3.88±0.26±0.4413.84.37±0.29±0.490.13BaBar, Ee>2.0 GeV,

shmax<3.5 GeV2

3.84±0.22±0.3511.44.23±0.24±0.390.19BaBar, Ee>2.0 GeV

4.31±0.40±0.339.54.67±0.43±0.360.24Belle, Ee>1.9 GeV

3.77±0.15±0.31

4.31±0.17±0.35

4.18±0.16±0.34

Average (HFAG)

My average

3.46±0.41±0.3914.43.91±0.46±0.440.13CLEO, Ee>2.1 GeV

|Vub| [10-3](mb=4.71±0.06 GeV)

a|Vub| [10-3]

(mb=4.63±0.06 GeV)fu


Results for various cuts

45

M.Neubert (LP07)

<

<

fraction of total phase space

ubV0.003 0.0035 0.004 0.0045 0.005

)ub

(V!

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5-310×

0

1

2

3

4

5

6 !

HQE param from S.L. only


Conclusionsb→ clν

|Vcb| with 1% error in the 1S scheme (2% Belle only), mb @1% (kinetic and 1S schemes), mc @5% (kinetic scheme).

A precise mb is an essential ingredient for reducing the error on inclusive Vub.

b→ ulν Inclusive Vub ~8% error shared between theoretical and experimental. Went up in the past year!

|Vub| @ 4~5% possible? Intelligent techniques to hunt down b→ ulν will help, and work on the theoretical error is ongoing.

|Vub| Vs. sin2φ1 puzzle resolved when only model-independent information on mb is used.mb extracted from semileptonic moments yields |Vub|↓10% lower than if photon moments are included.

mb↑ |Vub|↓ |Vcb|↓

!"#$%&'%()#*++, -.#-/(001#2%(3%(4 56

!"##$%&

! 7(890:8#48;8(<0&%;0/&#/=#>!"#>#9/<?@8<8&;:#:0&*! ;/#;8:;#;A8

B0&C9/<?@8;8&8:: /=#;A8#D;%&4%(4#-/48@

! "".,E#%99'(%9)#%9A08384#:/#=%(" FE#?/::0G@8H

! I@/:8#9/@@%G/(%;0/&#G8;J88&#;A8/()#%&4#8K?8(0<8&;#0:#9('90%@

! L%?04#?(/M(8::#0&#0&9@':038#>!"#>#0&#;A8#@%:;#*#)8%(:

! N<?(/38<8&;#0& $! #!$ =/(<#=%9;/(#0:#&88484

!

>!"# >

φ1New physics hiding skillfully! (behind theoretical uncertainties)

|Vub| & |Vcb| @ Belle

Backup slides


Inclusive |Vub|: DGE framework

Statistical ±2.0%

Expt. syst. ±3.2%

b → clν model ±1.9%

b → ulν model ±1.0%

DGE calc. ±3.1%

WA ±1.9%

Theory ±4.4%

|Vub| determined to ± 6.8%|Vub| world average as of Summer 07

DGE calc.: matching scheme method and scale Theory: mb(MS): ± 1.3%

αs: ± 1.0%

total SL width: ± 3.0 %

Here, 〈X〉meas.i are the measured moments. 〈X〉kin

i are the corresponding kinetic scheme

predictions that depend on these free parameters. The covariance matrix is the sum of the

experimental and theoretical error matrices.

We determine the CKM element |Vcb| by treating it as an eighth free parameter in the

fit. |Vcb| is related to the semileptonic width Γ(B → Xc!ν) by [5]

|Vcb|0.0417

=

(Γ(B → Xc!ν)

1.55 ps

0.105

)1/2

× (1− 0.0018)× (1 + 0.30(αs − 0.22)) (10)

×(1− 0.66(mb − 4.6 GeV) + 0.39(mc − 1.15 GeV)

+0.013(µ2π − 0.4 GeV2) + 0.09(ρ3

D − 0.1 GeV3)

+0.05(µ2G − 0.35 GeV2)− 0.01(ρ3

LS + 0.15 GeV3) .

Using this expression, we calculate Γ(B → Xc!ν) from |Vcb| and add the following term to

the χ2 function,

χ′2 = χ2 + (BXc"ν

Γ(B → Xc!ν)− τB)2/σ2

τB. (11)

As µ2G and ρ3

LS are determined from B∗ − B mass splitting and heavy quark sum rules

and because the expressions depend only weakly on these parameters, we fix µ2G and ρ3

LS to

0.35± 0.07 GeV2 and −0.15± 0.1 GeV3, respectively, by adding the following terms to the

χ2 function,

χ′′2 = χ′2 + (µ2G − 0.35 GeV2)2/(0.07 GeV2)2 + (ρ3

LS + 0.15 GeV3)2/(0.1 GeV3)2 . (12)

The minimization of the χ′′2 is performed using MINUIT [15].

C. Fit results and discussion

The result of the kinetic scheme analysis is shown in Table IV and in Figs. 4 and 5. The

value of the χ2 function at the minimum is 17.76, compared to (31− 7) degrees of freedom.

All results are preliminary.

To assess the stability of the fit, we have repeated the analysis using lepton energy

moments only, hadron mass moments only and photon energy moments only. The result

is shown in Fig. 6. In general, changes are well covered by the fit uncertainty though the

B → Xsγ data seems to prefer lower values of mb.

Finally, the result for |Vcb| reads

|Vcb| = (41.93± 0.65(fit)± 0.48(αs)± 0.63(th))× 10−3 .

11







|Vcb|0.0417

=

(Γ(B → Xc!ν)

1.55 ps

0.105

)1/2

× (1− 0.0018)× (1 + 0.30(αs − 0.22)) (10)

×(1− 0.66(mb − 4.6 GeV) + 0.39(mc − 1.15 GeV)

+0.013(µ2π − 0.4 GeV2) + 0.09(ρ3

D − 0.1 GeV3)

+0.05(µ2G − 0.35 GeV2)− 0.01(ρ3

LS + 0.15 GeV3) .


the χ2 function,

χ′2 = χ2 + (BXc"ν


τB. (11)

As µ2G and ρ3



LS to


χ2 function,

χ′′2 = χ′2 + (µ2G − 0.35 GeV2)2/(0.07 GeV2)2 + (ρ3

LS + 0.15 GeV3)2/(0.1 GeV3)2 . (12)











|Vcb| = (41.93± 0.65(fit)± 0.07(αs)± 0.63(th))× 10−3 .

14







|Vcb|0.0417

=

(Γ(B → Xc!ν)

1.55 ps

0.105

)1/2

× (1− 0.0018)× (1 + 0.30(αs − 0.22)) (10)

×(1− 0.66(mb − 4.6 GeV) + 0.39(mc − 1.15 GeV)

+0.013(µ2π − 0.4 GeV2) + 0.09(ρ3

D − 0.1 GeV3)

+0.05(µ2G − 0.35 GeV2)− 0.01(ρ3

LS + 0.15 GeV3) .


the χ2 function,

χ′2 = χ2 + (BXc"ν


τB. (11)

As µ2G and ρ3



LS to


χ2 function,

χ′′2 = χ′2 + (µ2G − 0.35 GeV2)2/(0.07 GeV2)2 + (ρ3

LS + 0.15 GeV3)2/(0.1 GeV3)2 . (12)











|Vcb| = (41.93± 0.65(fit)± 0.07(αs)± 0.63(th))× 10−3 .

14







|Vcb|0.0417

=

(Γ(B → Xc!ν)

1.55 ps

0.105

)1/2

× (1− 0.0018)× (1 + 0.30(αs − 0.22)) (10)

×(1− 0.66(mb − 4.6 GeV) + 0.39(mc − 1.15 GeV)

+0.013(µ2π − 0.4 GeV2) + 0.09(ρ3

D − 0.1 GeV3)

+0.05(µ2G − 0.35 GeV2)− 0.01(ρ3

LS + 0.15 GeV3) .


the χ2 function,

χ′2 = χ2 + (BXc"ν


τB. (11)

As µ2G and ρ3



LS to


χ2 function,

χ′′2 = χ′2 + (µ2G − 0.35 GeV2)2/(0.07 GeV2)2 + (ρ3

LS + 0.15 GeV3)2/(0.1 GeV3)2 . (12)











|Vcb| = (41.93± 0.65(fit)± 0.07(αs)± 0.63(th))× 10−3 .

14







|Vcb|0.0417

=

(Γ(B → Xc!ν)

1.55 ps

0.105

)1/2

× (1− 0.0018)× (1 + 0.30(αs − 0.22)) (10)

×(1− 0.66(mb − 4.6 GeV) + 0.39(mc − 1.15 GeV)

+0.013(µ2π − 0.4 GeV2) + 0.09(ρ3

D − 0.1 GeV3)

+0.05(µ2G − 0.35 GeV2)− 0.01(ρ3

LS + 0.15 GeV3) .


the χ2 function,

χ′2 = χ2 + (BXc"ν


τB. (11)

As µ2G and ρ3



LS to


χ2 function,

χ′′2 = χ′2 + (µ2G − 0.35 GeV2)2/(0.07 GeV2)2 + (ρ3

LS + 0.15 GeV3)2/(0.1 GeV3)2 . (12)











|Vcb| = (41.93± 0.65(fit)± 0.07(αs)± 0.63(th))× 10−3 .

14


Kinetic Scheme fit procedure

49

Belle Conf 0669, hep-ex/0611047

B-B* mass splitting constrained

Non-Perturbative Perturbative (vary αs) Bias correction ΓSL

vary 1/mb2 parameters by 20%

vary 1/mb3 parameters by 30%

lepton/photon: 0.220 +/- 0.040

hadron: 0.3 +/- 0.1

|Vcb|: 0.220 +/- 0.008

vary bias correction by 30% (uncorrelated

between Emin)

1.5% uncertainty due to the limited accuracy of the

ΓSL expression.

Theoretical uncertainties:

Constraints:=

=

BXc!ν (%) mb (GeV) mc (GeV) µ2π (GeV2) ρ3

D (GeV3) µ2G (GeV2) ρ3

LS (GeV3)

value 10.590 4.564 1.105 0.557 0.162 0.358 -0.174

σ(fit) 0.164 0.076 0.116 0.091 0.053 0.060 0.098

σ(αs) 0.006 0.003 0.005 0.013 0.008 0.003 0.003

BXc!ν 1.000 −0.023 0.003 0.229 0.192 −0.147 −0.024

mb 1.000 0.983 −0.729 −0.623 −0.024 −0.111

mc 1.000 −0.716 −0.633 −0.124 −0.033

µ2π 1.000 0.851 0.005 0.052

ρ3D 1.000 −0.046 −0.156

µ2G 1.000 −0.071

ρ3LS 1.000

TABLE IV: Results of the kinetic scheme fit. The error from the fit contains the uncertainties

related to the experiment, the non-perturbative corrections and the bias correction. σ(αs) is the

uncertainty related to the perturbative corrections. In the lower part of the table, the correlation

matrix of the parameters is shown.

The first error is due to all uncertainties taken into account in the fit (experimental error

in the moment measurements, non-perturbative corrections and bias correction to the mo-

ments). The second error is obtained by varying αs in the expressions for the moments and

in Eq. 10. In Eq. 10, we vary αs by ±0.04 around the nominal central value of 0.22. The

last error is a 1.5% uncertainty due to the limited accuracy of the theoretical expression for

the semileptonic width, assessed in Ref. [5].

V. SUMMARY

We have performed a fit to the Belle measured spectral moments of the lepton energy

and hadronic mass spectrum in charmed semileptonic B decays, and the photon energy

spectrum of inclusive radiative B decays using expressions for the moments in terms of

HQE parameters in the 1S mass and kinetic mass schemes. The fits produce values of |Vcb|

12


Preliminary

Kinetic SchemeBelle Conf 0669, hep-ex/0611047

|Vcb| = ( 41.93 ± 0.65fit ± 0.07αs ± 0.63th ) × 10-3

Fit parameter correlations

mb & mc highly correlated

introducing extra terms in the χ2 function of the fit,

χ2param(mχ, Mχ) =

0 |O| ≤ m3χ ,

(|〈O〉|−m3

χ

)2/M6

χ |O| > m3χ ,

(7)

where (mχ,Mχ) are both quantities of order ΛQCD, and 〈O〉 represents any O(1/m3b) param-

eters. In the fit, we take Mχ = mχ = 500 MeV after Ref [2]. The parameter mχ may have

a value anywhere between 500 MeV and 1 GeV.

The overall form of the χ2 function used in the 1S fit is

χ2 =∑

i,j

(〈X〉meas.i − 〈X〉1Si )cov−1

ij (〈X〉meas.j − 〈X〉1Sj ) +

2∑

i=1

χ2ρi

+3∑

i=1

χ2τi

, (8)

where 〈X〉meas.i are the measured moments and 〈X〉1S

i are the corresponding 1S scheme

predictions.


Minimizing the χ2 function in Eq. 8 using MINUIT [15], we find the following results for

the fit parameters,

|Vcb| = (41.49± 0.52fit ± 0.20τB)× 10−3 ,

m1Sb = (4.729± 0.048) GeV/c2 , and

λ1 = (−0.30± 0.04) GeV2 .

The first error is the uncertainty from the fit including experimental and theory errors, and

the second error (on |Vcb| only) is due to the uncertainty on the average B lifetime. The

correlations between these fit parameters are provided in Table II. Using the measurement of

the partial branching fraction at Emin = 0.6 GeV, we obtain for the semileptonic branching

fraction (over the full lepton energy range),

B(B → Xc#ν) = (10.62± 0.25)% .

The measured moments compared to the 1S scheme predictions are shown in Figs. 1 and 2.

We assess the stability of the fit in two ways (Table III): by repeating the fit only to

B → Xc#ν data (21 measurements), (a) to (c) ; and by releasing the mχ constraint on

6

In this analysis, we determine a total of seven parameters: |Vcb|, Λ, λ1, τ1, τ2, τ3 and ρ1.

One of the higher order parameters, τ4 is set to zero , and from available constraints, e.g.

B∗−B mass splitting, the remaining parameters in Eq. 1 are set to: λ2 = 0.1227−0.0145λ1

and ρ2 = 0.1361 + τ2, following advice from Ref. [2]. The parameter Λ is the difference

between the b-quark mass and the reference value about which it is expanded, i.e., Λ =

mΥ(1S)/2−m1Sb . We will present our results in terms of m1S

b in place of Λ.

B. The χ2 function

The fit takes into account both experimental and theoretical uncertainties. Following the

approach in Ref. [2], an element of the combined experimental and theoretical error matrix

is given by

σ2ij = σiσjcij, (3)

where i and j denote the observables and cij is the experimental correlation matrix element.

The total error on the observable i is defined as

σi =√

(σexpi )2 + (Afnm2n

B )2 + (Bi/2)2 for the nth hadron moment , (4)

σi =√

(σexpi )2 + (Afn(mB/2)n)2 + (Bi/2)2 for the nth lepton moment , (5)

σi =√

(σexpi )2 + (Afn(mB/2)n)2 + (Bi/2)2 for the nth photon moment , (6)

and f0 = f1 = 1, f2 = 1/4 and f3 = 1/(6√

3). Here, σexpi are the experimental errors, Bi =

X(16) are the coefficients of the last computed terms in the perturbation series (providing the

error on the uncalculated higher order perturbative terms). The dimensionless parameter A

contains various theoretical errors (uncalculated power corrections, uncalculated effects of

order (αs/4π)Λ2QCD/m2

b , and effects not included in the OPE, i.e., duality violation), and is

multiplied by dimensionful quantities. We fix A = 0.001 as in Ref. [2]. For B → Xsγ, the

accessible phase space is limited, and the theoretical extraction of mb is affected by shape

function effects. So, A is multiplied by the ratio of the difference from the end point relative

to Emin = 1.8 GeV.

As the fit does not provide strong constraints on the O(1/m3b) parameters, it is necessary

to provide constraints to ensure their convergence to sensible values. We achieve this by

8

1S scheme kinetic scheme

n = 0 Emin = 0.6, 1.0, 1.4 n = 0 Emin = 0.4, 0.8

Lepton moments 〈En! 〉Emin n = 1 Emin = 0.6, 0.8, 1.0, 1.2, 1.4 n = 1 Emin = 0.4, 0.8, 1.0, 1.2 1.4

n = 2 Emin = 0.6, 1.0, 1.4 n = 2 Emin = 0.4, 0.8, 1.0, 1.2 1.4

n = 3 Emin = 0.8, 1.2 n = 3 Emin = 0.4, 0.8, 1.0, 1.2 1.4

Hadron moments 〈M2nX 〉Emin n = 1 Emin = 0.7, 1.1, 1.3, 1.5 n = 1 Emin = 0.7, 0.9, 1.1, 1.3

n = 2 Emin = 0.7, 0.9, 1.3 n = 2 Emin = 0.7, 0.9, 1.1, 1.3

Photon moments 〈Enγ 〉Emin n = 1 Emin = 1.8, 2.0 n = 1 Emin = 1.8, 1.9, 2.0

n = 2 Emin = 1.8, 2.0 n = 2 Emin = 1.8, 1.9, 2.0

TABLE I: Experimental input used in the 1S and kinetic scheme analyses. The values of Emin are

given in GeV. The 1S (kinetic) scheme analysis uses a total of 24 (31) measurements.

III. 1S SCHEME ANALYSIS

A. Theoretical input

The inclusive spectral moments of B → Xc!ν decays have been derived in the 1S scheme

up to O(1/m3b) [2]. The theoretical expressions for the truncated moments have the following

form (where 〈X〉Emin represents any of the experimental observables):

〈X〉Emin = X(1) + X(2)Λ + X(3)Λ2 + X(4)Λ3 + X(5)λ1 + X(6)Λλ1 + X(7)λ2 + X(8)Λλ2 + X(9)ρ1

+X(10)ρ2 + X(11)τ1 + X(12)τ2 + X(13)τ3 + X(14)τ4 + X(15)ε + X(16)ε2BLM + X(17)εΛ .

(1)

The coefficients X(k), determined by theory, are functions of Emin. The non-perturbative

corrections are parametrized by Λ (O(mb)), λ1 and λ2 (O(1/m2b)), and τ1, τ2, τ3, τ4, ρ1 and

ρ2 (O(1/m3b)).

Predictions for the partial branching fractions are obtained using the following expression,

B(B → Xc!ν)Emin = 〈X〉B,Emin

ηQED|Vcb|2G2F m5

192π3τB, (2)

where 〈X〉B,Emin is an expression of the form of Eq. 1, m is the 1S reference mass, m =

mΥ(1S)/2, ηQED = 1.007, and G2F m5/(192π3) = 5.4× 10−11.

7

EPFL LPHE Seminar Oct 2007 Phillip Urquijo51

1S Scheme procedureBelle Conf 0669, hep-ex/0611047

Theoretical uncertainties:

Constraints:

mχ~500 MeV







b in place of Λ.

B. The χ2 function



is given by




σi =√



σi =√


σi =√


and f0 = f1 = 1, f2 = 1/4 and f3 = 1/(6√










to Emin = 1.8 GeV.



8







b in place of Λ.

B. The χ2 function



is given by




σi =√



σi =√


σi =√


and f0 = f1 = 1, f2 = 1/4 and f3 = 1/(6√










to Emin = 1.8 GeV.



8

Various Perturbative Shape function effects.

A contains: Uncalculated power corrections, uncalculated effects of order and

effects not included in the OPE (i.e. duality) (A=0.001)

Last term in pert. expansion gives error on higher

order pert. terms.

A is scaled up as Emin approaches the B→sγ end

point

The Precision Determination of |V | and |Vub

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