Michael Luke Department of Physics University …luke/talks/durhamckm.pdfApril 8, 2003 CKM Workshop,...

CKM Workshop, IPPP DurhamApril 8, 2003 1

Michael LukeDepartment of PhysicsUniversity of Toronto

Applications of the heavy quark

expansion:

Vub and spectral moments

2CKM Workshop, IPPP DurhamApril 8, 2003

1. Introduction

2. Vub

• Exclusive decays (brief!)

• Inclusive decays - a guide to phase space and cuts

• Uncertainties: perturbative, nonperturbative, higher twist

3. Spectral moments

• variables, constraints and uncertainties

Outline:


Vcb, sin 2β, |Vtd/Vts|Vub, α, γ

Ô : “easy” (theory and experiment both tractable)Ô : HARD - our ability to test CKM depends on the precision with which these can be measured

The unitarity triangle provides a simple way to visualize SM relations:

VudV∗

ub + VcdV∗

cb + VtdV∗

tb = 0

mixing (Tevatron)Bs − B̄s

CPV in B0 → ψKs


B → π"ν̄, B → ρ"ν̄

〈π(pπ)|V µ|B(pB)〉 = f+(E)

[pµ

B + pµπ − m2

B − m2π

q2qµ

]+ f0(E)

m2B − m2

π

q2qµ

nonperturbative - need to model (QCD sum rules: see P. Ball’s talk) or calculate on lattice (see Onogi’s talk)

vanishes for m!=0

Vub:

(i) Exclusive Decays:

f+(0) = 0.26 ± 0.06 ± 0.05Sum rules:

model dependence - hard to improve on

Lattice: current theoretical error is DVub ≈15-18% + quenching error

- goal for future is unquenched, error of ~ few percent

(from A. Kronfeld, hep-ph/0010074)

10 15 20 25 30q2 (GeV2)

0

0.1

0.2

0.3

0.4

0.5

|Vub

|2 d

G/d

q2 (ps1

GeV

2)

JLQCD '00UKQCD '00FNAL '00

(El-Khadra, et. al., 2001)

(Ball and Zwicky)


“Most” of the time, details of b quark wavefunction are unimportant - only averaged properties (i.e. ) matter

b

“Fermi motion”kµ ∼ ΛQCD

dΓ

d(P.S.)∼ parton model +

∑n

Cn

(ΛQCD

mb

)n

Γ(B̄ → Xu!ν̄!) =G2

F |Vub|2m5b

192π3

(1 − 2.41

αs

π− 21.3

(αs

π

)2

+λ1 − 9λ2

2m2b

+ O

(α3

s,Λ3

QCD

m3b

))⟨k2〉

b

u

l

ν

+b

u

l

ν

+...

︸︷︷︸Inclusive decays are in principle model independent ...

(ii) Inclusive Decays: B̄ → Xu!ν̄!


but ... near perturbative singularities, life gets more complicated:

1 2 3 4 50

5

10

15

20

25

q2 (GeV2)

2 mX (GeV2)nonperturbative(no rate at parton level)

real gluon emission

perturbative singularity(real+virtual gluons)

allowed (huge background)B → Xc!ν̄!

B → Xu!ν̄! Phase Space

(Bigi, Shifman, Vainshtein, Uraltsev; Neubert)


Hadronic invariant mass spectrum:

dmXG_1 dG__

1 2 3 4 5 6

0.2

0.4

0.6

0.8

1

mX (GeV )22

2

(GeV-2)

1 2 3 4 50

5

10

15

20

25

q2 (GeV2)

2 mX (GeV2)

(Falk, Ligeti, Wise; Dikeman, Uraltsev)


dmXΓ_1 dΓ__

1 2 3 4 5 6

0.2

0.4

0.6

0.8

1

mX (GeV )22

2

(GeV-2)

parton model

including fermi motion (model)

kinematic limit of b→c

Hadronic invariant mass spectrum:1 2 3 4 5

0

5

10

15

20

25

q2 (GeV2)

2 mX (GeV2)

m2c ∼ ΛQCDmb

∴ integrated rate below charm threshold is sensitive to details of Fermi motion, so model dependent

(Falk, Ligeti, Wise; Dikeman, Uraltsev)

0.5 1 1.5 2

5

10

15

20

25

Ee (GeV)

q2 (GeV2)

0.5 1 1.5 2

0.2

0.4

0.6

0.8

2.5El (GeV)

ΓdΓ

dEl

_1 __

(GeV-1)


parton model



The same thing happens near the endpoint of the lepton energy spectrum:

q2 (GeV )2

ΓdΓdq2

_1 __

5 10 15 20 25

0.02

0.04

0.06

0.08

(GeV-2)


parton model



1 2 3 4 50

5

10

15

20

25

q2 (GeV2)

2 mX (GeV2)

but not always ... i.e. leptonic q2 spectrum: (Bauer, Ligeti, ML)


cut% of rate

good bad

0.5 1 1.5 2

5

10

15

20

25

Ee (GeV)

q2 (GeV2)

E! >m2

B − m2D

2mB

~10% don’t need neutrino

1 2 3 4 50

5

10

15

20

25

q2 (GeV2)

2 mX (GeV2)

sH < m2D

~80% lots of rate

1 2 3 4 50

5

10

15

20

25

q2 (GeV2)

2 mX (GeV2)

q2 > (mB − mD)2 ~20% insensitive to f(k+)

1 2 3 4 50

5

10

15

20

25

q2 (GeV2)

2 mX (GeV2)

“Optimized cut”

~45%

- insensitive to f(k+)- lots of rate

- can move cuts away from kinematic

limits and still get small uncertainties


• Weak Annihilation (WA)

• Fermi motion - at leading and subleading order

• - rate is proportional to - 100 MeV error is a ~5% error in Vub. But restricting phase space increases this sensitivity - with q2 cut, scale as ~

• perturbative corrections - known (in most cases) to - generally under control. When Fermi motion is important, leading and subleading Sudakov logarithms have been resummed.

Theoretical Issues:

O(α2sβ0)

m5bmb

m10b (Neubert)

(Leibovich, Low, Rothstein)


b

u

soft

B

• Weak annihilation

~3% (?? guess!) contribution to rate at q2=mb2

- an issue for all inclusive determinations- relative size of effect gets worse the more severe the cut- no reliable estimate of size - can test by comparing charged and neutral B’s

(Bigi & Uraltsev, Voloshin, Ligeti, Wise and Leibovich)

Theoretical Issues:

O

(16π2 × Λ3

QCD

m3b

×)

∼ 0.03

(fB

0.2 GeV

) (B2 − B1

0.1

)factorization violation


• f(k+) - for cuts

f(ω) ∼ 〈B|b̄ δ(ω − iD̂ · n)b|B〉︸︷︷︸universal distribution function (applicable to all decays)

b

Options:

(i) model

Ex:

f(k+) = N(1 − x)ae(1+a)x

(de Fazio and Neubert.)

O(ΛQCD)

- 1 - 0.5 0 0.5 10

0.25

0.5

0.75

1

1.25

1.5

f(k+)

k+ (GeV)

(model)

a, N determined by (gets first two moments right .. but the uncertainty in f(k+) is not simply given by the uncertainties in )

sH, E!

Λ̄, λ1

Λ̄, λ1

sH < m2D, E! > (m2

B − m2D)/2mB

Theoretical Issues:

... sensitivity to functional form gets stronger as cut is moved away from kinematic boundary


∣∣∣∣ Vub

VtbV ∗ts

∣∣∣∣2 =3α

π|Ceff

7 |2Γu(Ec)

Γs(Ec)+ O(αs) + O

(ΛQCD

mB

)

Γu(Ec)≡∫ mB/2

Ec

dE!

dΓu

dE!

Γs(Ec)≡ 2

mb

∫ mB/2

Ec

dEγ(Eγ − Ec)dΓs

dEγ

.

(ii) determine from experiment: the SAME function determines the photon spectrum in - can related integrated rates without assuming a functional form for f(k+):

|Vub|2|V ∗

tsVtb|2=3 α C7(mb)2

π

∫ 1

xcB

dxB

dΓ

dxB

×{∫ 1

xcB

dxB

∫ 1

xB

duB u2B

dΓγ

duB

K

[xB;

4

3πβ0log(1 − αsβ0 lxB/uB)

]}−1

,

Including resummation of subleading Sudakov logs:

(Leibovich, Low, Rothstein)

B → Xsγ


f(ω) ∼ 〈B|b̄ δ(ω − iD̂ · n)b|B〉

ΛQCD/mb

k⊥sensitive to

breaks spin symmetry (distinguishes semileptonic from radiative decays)

sensitive to soft gluons

(NB this is just DIS at subleading twist all over again)

h1(ω) ∼ 〈B|b̄ [iDµ, δ(ω + in · D̂)] γλγ5b|B〉 εµλ⊥

hλ2(ω1, ω2) ∼ 〈B|b̄ δ(ω2 + in · D̂)Gµν δ(ω1 + in · D̂)γλγ5b|B〉 εµν

⊥

g2(ω1, ω2) ∼ 〈B|b̄ δ(ω2 + in · D̂)(iD⊥)2 δ(ω1 + in · D̂)b|B〉

T (ω) ∼∫

e−iωt〈B|T (b̄(0)b(t), O1/m(y))|B〉

... at , there is more structure:O(ΛQCD/mb)

b

Also note that this only holds at leading order in ...

(Bauer, ML, Mannell)


leading “twist” terms … sum to f(k+)

subleading “twist” terms … sum to new distribution functions

The effect of these subleading “shape functions” can be surprisingly large ….

corresponding coefficient in BfiXsg is 3

dΓ

dy∼ 2θ(1 − y) − λ1

3m2b

δ′(1 − y) − ρ1

9m3b

δ′′(1 − y) + . . .

− λ1

3m2b

δ(1 − y) − 11λ2

m2b

δ(1 − y) + . . .

2 different models for subleading shape functions...

... and the corresponding effect on the determination of |Vub|

(Leibovich, Ligeti, Wise; Bauer, ML, Mannell)

(Ec)

Ec (GeV)-0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-2

-1

0

1

2

3

2 2.1 2.2 2.3 2.4

-0.2

-0.1

0

0.1

0.2h1, H2

h1

h1, H2

h1, H2

`

`


cut % of rate good bad

0.5 1 1.5 2

5

10

15

20

25

Ee (GeV)

q2 (GeV2) ~10% don’t need neutrino

- depends on f(k+) (and subleading corrections)

- WA corrections may be substantial

- reduced phase space - duality issues?

1 2 3 4 50

5

10

15

20

25

q2 (GeV2)

2 mX (GeV2)

~80% lots of ratedepends on f(k+) (and subleading

corrections)

1 2 3 4 50

5

10

15

20

25

q2 (GeV2)

2 mX (GeV2)

~20% insensitive to f(k+)

- very sensitive to mb

- WA corrections may be substantial

- effective expansion parameter is 1/mc

1 2 3 4 50

5

10

15

20

25

q2 (GeV2)

2 mX (GeV2)

~45%

- insensitive to f(k+)- lots of rate

- can move cuts away from kinematic limits and still get small uncertainties

- sensitive to mb (need +/- 30 MeV

for 5% error)


(a) better determination of mb (moments of B decay distributions - next)

(b) test size of WA (weak annihilation) effects - compare D0 & DS S.L. widths,

extract |Vub| from B± and B0 separately

(c) improve measurement of B→Xsγ photon spectrum - get f(k+) - lowering cut reduces effects of subleading corrections, as well as sensitivity to details of f(k+)

(d) (most important) measure |Vub| in as many CLEAN ways as possible - different techniques have different sources of uncertainty (c.f. inclusive and exclusive determinations of |Vcb|)

Experimental measurements can help beat down the theoretical errors:


Moments of B Decay Spectra:

Constrain different linear combinations of L, l1

Ô fit mb

- like rate, moments of spectra can be calculated as a power series in : αs(mb), ΛQCD/mb

1

m2B

〈sH − m̄2D〉E!>1.5 GeV = 0.21

Λ̄

m̄B+ 0.26

Λ̄2 + 3.8λ1 − 1.2λ2

m̄2B

+ . . .

〈Eγ〉 =mB − Λ̄

2+ . . .

1850801-0052000

1500

1000

500

0-4 -2 0 42 6 8 10

M 2 (GeV2)X

Even

ts / 0

.5 G

eV2

~

1850801-007

40

0Wei

ghts

/ 100

MeV

1.5 2.5 3.5 4.5E (GeV)

(CLEO ‘01)

0.1

0

0.1

0.60.40.2 0.80.5

0.4

0.3

0.2

1.00

ExperimentalTotal

II

II

I

I

< E >

1

1850701-004

< MX -M

D >2 _

2

mB = mb + Λ̄ − λ1 + 3λ2

2m2b

+ . . .


Λ (GeV)

λ 1 (G

eV2 )

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Many moments have now been measured, (i) allowing precision extractions of HQET matrix elements (and mb), and (ii) testing validity of the whole approach:

(Battaglia et. al., PLB556:41, 2003, using DELPHI data)


• lepton energy and hadronic invariant mass moments , photon energy spectrum moments

• measured with varying cutoffs by DELPHI, CLEO and BaBar

S1(E0) = 〈m2X − m̄2

D〉∣∣∣E!>E0

, S2(E0) =⟨(m2

X − 〈m2X〉)2

⟩∣∣∣E!>E0

|Vcb| = (40.8 ± 0.9) × 10−3

m1Sb = 4.74 ± 0.10 GeV

R0(E0, E1) =

∫E1

dΓ

dE!

dE!∫E0

dΓ

dE!

dE!

, Rn(E0) =

∫E0

En!

dΓ

dE!

dE!∫E0

dΓ

dE!

dE!

, n = 1, 2

T1(E0) = 〈Eγ〉∣∣∣Eγ>E0

, T2(E0) =⟨(Eγ − 〈Eγ〉)2

⟩∣∣∣Eγ>E0︸︷︷︸

(B̄ → Xc!ν̄)

(B̄ → Xsγ)

exclusive Vcb extraction, b mass from bb sum rules

Hoang

Beneke ︸︷︷︸

_

(Bauer, Ligeti, ML and Manohar, PRD67:054012, 2003 - BaBar sH spectra not included in fit)

|Vcb| = (41.9 ± 1.1) × 10−3

mb(1 GeV)=4.59 ± 0.08 GeV ⇒ m1Sb = 4.69 GeV

mc(1 GeV)=1.13 ± 0.13 GeV(Battaglia et. al., PLB556:41, 2003, using DELPHI data)

Global fits (summer ‘02):(fit including 1/m3 effects)


(C. Bauer and M. Trott)

TABLE III: Fit predict ions for fract ional moments of the elect ron spect rum. The upper/lowerblocks are fits excluding/including the BABAR dat a.

mX [GeV] R3a R3b R4a R4b D3 D4

0.5 0.302 ± 0.003 2.261 ± 0.013 2.128 ± 0.013 0.684 ± 0.002 0.520 ± 0.002 0.604 ± 0.0021.0 0.302 ± 0.002 2.261 ± 0.011 2.128 ± 0.011 0.684 ± 0.002 0.519 ± 0.002 0.604 ± 0.001

0.5 0.302 ± 0.002 2.261 ± 0.013 2.128 ± 0.012 0.684 ± 0.002 0.520 ± 0.001 0.604 ± 0.0011.0 0.302 ± 0.002 2.262 ± 0.012 2.129 ± 0.012 0.683 ± 0.002 0.519 ± 0.002 0.604 ± 0.001

(some fractional moments of lepton spectrum are very insensitive to O(1/mb

3) effects, and so can be predicted very accurately)

The fit also allows us to make precise predictions of other moments as a cross-check:

... and just for fun, setting all experimental errors to zero we find

δ(|Vcb|) × 103 = ±0.35, δ(mb) = ±35 MeV


data

theory

- for most values of the lepton cut, measured sH is significantly higher than predicted

A problem? - BABAR ‘02:


(Falk, ML and Savage PRD53, 2491 (1996), Gremm and Kapustin PRD55 , 6924 (1997))

〈sH − m̄2D〉 = m̄2

B

(0.051

αs

π+ 0.23

Λ̄

m̄B+ . . .

)≥ ΓD∗∗(m∗∗2

D − m̄2D) + ΓD∗(m∗2

D − m̄2D) + ΓD(m2

D − m̄2D)

obtain a lower limit on first moment by assuming excited state production saturated by D**

Combining this with the measured D/D* ratio:

gives the upper bound

Qu: Problem with theory or with modelling of hadronic states? (i.e. low mass nonresonant states)

… will have to see how this evolves...

... this is related to an older issue:

ΓD = 0.31(1 − ΓD∗∗) ΓD∗ = 0.69(1 − ΓD∗∗)

(expt: ~ 35%)ΓD∗∗ < 0.22

26CKM Workshop, IPPP DurhamApril 8, 2003

Summary:

• 1/mQ expansion allows precise theoretical predictions for inclusive decays - uncertainties are at the 1/m3 level

• measuring |Vub| requires probing restricted regions of phase space - some (but not all!) regions are sensitive to nonperturbative structure function

• a number of regions of phase space may be used to determine |Vub|, with different sources of uncertainty

• spectral moments are proving useful to constrain parameters (get mb, |Vcb|), test validity of the expansion

Date post:	02-Dec-2018
Category:	Documents
Upload:	dangtram
View:	215 times
Download:	0 times

Michael Luke Department of Physics University …luke/talks/durhamckm.pdfApril 8, 2003 CKM Workshop,...

Documents