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:
CKM Workshop, IPPP DurhamApril 8, 2003 3
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
CKM Workshop, IPPP DurhamApril 8, 2003 4
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)
CKM Workshop, IPPP DurhamApril 8, 2003 5
“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!
CKM Workshop, IPPP DurhamApril 8, 2003 6
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)
CKM Workshop, IPPP DurhamApril 8, 2003 7
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)
CKM Workshop, IPPP DurhamApril 8, 2003 8
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)
CKM Workshop, IPPP DurhamApril 8, 2003 9
parton model
including fermi motion (model)
kinematic limit of b→c
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)
CKM Workshop, IPPP DurhamApril 8, 2003 10
parton model
including fermi motion (model)
kinematic limit of b→c
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)
CKM Workshop, IPPP DurhamApril 8, 2003 11
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
CKM Workshop, IPPP DurhamApril 8, 2003 12
• 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)
CKM Workshop, IPPP DurhamApril 8, 2003 13
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
CKM Workshop, IPPP DurhamApril 8, 2003 14
• 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
CKM Workshop, IPPP DurhamApril 8, 2003 15
∣∣∣∣ 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
CKM Workshop, IPPP DurhamApril 8, 2003 16
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)
CKM Workshop, IPPP DurhamApril 8, 2003 17
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
`
`
CKM Workshop, IPPP DurhamApril 8, 2003 18
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)
CKM Workshop, IPPP DurhamApril 8, 2003 19
(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:
CKM Workshop, IPPP DurhamApril 8, 2003 20
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
+ . . .
CKM Workshop, IPPP DurhamApril 8, 2003 21
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)
CKM Workshop, IPPP DurhamApril 8, 2003 22
• 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)
CKM Workshop, IPPP DurhamApril 8, 2003 23
(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
CKM Workshop, IPPP DurhamApril 8, 2003 24
data
theory
- for most values of the lepton cut, measured sH is significantly higher than predicted
A problem? - BABAR ‘02:
CKM Workshop, IPPP DurhamApril 8, 2003 25
(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