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TRANSCRIPT
Inclusive |Vub| unlimitedZoltan Ligeti
SCET Workshop, MIT, March 24–26, 2009
Outline
• Introduction
• Complete description of B → Xsγ [ZL, Stewart, Tackmann, arXiv:0807.1926]
• Complete description of B → Xu`ν̄ [ZL, Stewart, Tackmann, to appear]
• A glimpse at SIMBA [Bernlochner, Lacker, ZL, Stewart, Tackmann, Tackmann, to appear]
• Conclusions
p.1
The importance of |Vub|
• |Vub| determined by tree-level decaysCrucial for comparing tree-dominatedand loop-mediated processes
• |Vub|π`ν̄−LQCD = (3.5± 0.5)× 10−3
|Vub|incl−BLNP = (4.32± 0.35)× 10−3
|Vub|τν = (5.2± 0.5± 0.4fB)× 10−3
SM CKM fit, sin 2φ1, favors small value
• Fluctuation, bad theory, new physics?
• The level of agreement between themeasurements often misinterpreted
3φ
3φ
2φ
2φ
dm∆Kε
Kε
sm∆ & dm∆
SLubV
ν τubV
1φsin 2
(excl. at CL > 0.95)
< 01
φsol. w/ cos 2
excluded at CL > 0.95
2φ
1φ
3φ
ρ−1.0 −0.5 0.0 0.5 1.0 1.5 2.0
η
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5excluded area has CL > 0.95
Moriond 09
CKMf i t t e r
• 10–20% non-SM contributions to most loop-mediated transitions are still possible
p.2
|Vub| matters. . .
• Including B → τ ν̄, the SM is “disfavored” at >2σ
dh0.0 0.2 0.4 0.6 0.8 1.0
dσ
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.01-CL
Moriond 09
CKMf i t t e r
ντ →w/o B
dh0.0 0.2 0.4 0.6 0.8 1.0
dσ0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.01-CL
Moriond 09
CKMf i t t e r
Parameterize NP in B0–B0 mixing: M12 = MSM12 (1 + hd e
2iσd)
• 10–20% non-SM contributions to most loop-mediated transitions are still possible
p.3
The challenge of inclusive |Vub| measurements
• Total rate known with ∼ 4% accuracy, similar to B(B → Xc`ν̄) [Hoang, ZL, Manohar]
• To remove the huge charm background(|Vcb/Vub|2 ∼ 100), need phase space cuts
Phase space cuts can enhance perturbativeand nonperturbative corrections drastically
dΓ(b→c)/dEe
10 dΓ(b→u)/dEe
Ee (GeV)
dΓ/d
Ee
∆E
0
20
40
60
80
0 1 2
Nonperturbative effects shift endpoint 12 mb → 1
2 mB and determine shape↗
• Endpoint region determined by b quark PDF in B; want to extract it from data tomake predictions — at lowest order ∝ B → Xsγ photon spectrum
[Bigi, Shifman, Uraltsev, Vainshtein; Neubert]
p.4
|Vub|: lepton endpoint vs. B → Xsγ
b quark decayspectrum
with a model forb quark PDF
0 0.5 1 1.5 2 2.5El
dΓdEl
p.5
|Vub|: lepton endpoint vs. B → Xsγ
b quark decayspectrum
with a model forb quark PDF
0 0.5 1 1.5 2 2.5El
dΓdEl
ddEl−
p.5
|Vub|: lepton endpoint vs. B → Xsγ
b quark decayspectrum
with a model forb quark PDF
0 0.5 1 1.5 2 2.5El
dΓdEl
ddEl−
difference:
2 3 4
p.5
|Vub|: lepton endpoint vs. B → Xsγ
b quark decayspectrum
with a model forb quark PDF
0 0.5 1 1.5 2 2.5El
dΓdEl
ddEl−
difference:
2 3 4
1850801-007
40
0
DataSpectator Model
Wei
ghts
/ 10
0 M
eV
1.5 2.5 3.5 4.5E (GeV)
[CLEO, 2001]
p.5
|Vub|: lepton endpoint vs. B → Xsγ
b quark decayspectrum
with a model forb quark PDF
0 0.5 1 1.5 2 2.5El
dΓdEl
ddEl−
difference:
2 3 4
1850801-007
40
0
DataSpectator Model
Wei
ghts
/ 10
0 M
eV
1.5 2.5 3.5 4.5E (GeV)
[CLEO, 2001]
• Both of these spectra are determined at lowest order by the b quark PDF in the B
• Lots of work toward extending beyond leading order; many open issues...
p.5
Past efforts: BLNP (best so far)
• Treated factorization & resummationin shape function region correctly
• Use fixed functional forms to modelshape function (similar to PDF’s)
• Analysis tied to shape functionscheme for mb, λ1
(One scheme for each approach)
• Need for “tail gluing” to get correctperturbative tail (other approachesdon’t even do this [DGE, GGOU, ADFR])
⇒ Hard to assess uncertainties
[Bosch, Lange, Neubert, Paz]
p.5
Start with B → Xsγ
[ZL, Stewart, Tackmann, PRD 78 (2008) 114014, arXiv:0807.1926]
Regions of phase space
• B → Xsγ gives one of the strongest bounds on NP
• p+X ≡ mB − 2Eγ <∼ 2 GeV, and <1 GeV at the peak
Three cases: 1) Λ ∼ mB − 2Eγ � mB
Three cases: 2) Λ� mB − 2Eγ � mB
Three cases: 3) Λ� mB − 2Eγ ∼ mB
[Sometimes called 1) SCET and 2) MSOPE]
• Neither 1) nor 2) is fully appropriate [GeV]γ
c.m.sE1.5 2 2.5 3 3.5 4
Ph
oto
ns
/ 50
MeV
-4000
-2000
0
2000
4000
6000
[Belle, 0804.1580]
• Include all known results in regions 1) – 2)
LL: 1–loop Γcusp, tree-level matching
NLL: 2–loop Γcusp, 1–loop matching, 1–loop γx
NNLL: 3–loop Γcusp, 2–loop matching, 2–loop γx
We can combine 1) – 2) withoutexpanding shape function in Λ/p+
X
p.6
The shape function (b quark PDF in B)
• The shape function S(ω, µ) contains nonperturbative physics and obeys a RGE
If S(ω, µΛ) has exponentially small tail, any smallrunning gives a long tail and divergent moments
[Balzereit, Mannel, Kilian]
S(ω, µi) =
Zdω′US(ω − ω′, µi, µΛ)S(ω
′, µΛ)
Constraint: moments (OPE) + B → Xsγ shapeConstraint: How to combine these?
– Consistent setup at any order, in any scheme
– Stable results for varying µΛ
– (SF modeling scale, must be part of uncert.)
– Similar to how all matrix elements are defined– e.g., BK(µ) = bBK [αs(µ)]2/9(1 + . . .)
Derive: [ZL, Stewart, Tackmann, 0807.1926]
S(ω, µΛ) =
Zdk C0(ω−k, µΛ)F (k)
0
0
0
0
1
1
1
1 22
0.5
0.5
0.5
0.5
1.5
1.5
1.5
1.5 2.52.5
−0.5−0.5
ω [GeV]ω [GeV]
S(ω
,2.5
GeV
)[G
eV−
1]
S(ω
,2.5
GeV
)[G
eV−
1] µΛ = 2.5 GeV
µΛ = 1.8 GeVµΛ = 1.3 GeVµΛ = 1.0 GeV
nonpert. perturbative
ModelS (dash)F (solid)
run to 2.5GeV
Can consistently impose moment constraints on F (k), but not on S(ω, µΛ)
p.7
The shape function (b quark PDF in B)
• The shape function S(ω, µ) contains nonperturbative physics and obeys a RGE
If S(ω, µΛ) has exponentially small tail, any smallrunning gives a long tail and divergent moments
[Balzereit, Mannel, Kilian]
S(ω, µi) =
Zdω′US(ω − ω′, µi, µΛ)S(ω
′, µΛ)
Constraint: moments (OPE) + B → Xsγ shapeConstraint: How to combine these?
– Consistent setup at any order, in any scheme
– Stable results for varying µΛ
– (SF modeling scale, must be part of uncert.)
– Similar to how all matrix elements are defined– e.g., BK(µ) = bBK [αs(µ)]2/9(1 + . . .)
Derive: [ZL, Stewart, Tackmann, 0807.1926]
S(ω, µΛ) =
Zdk C0(ω−k, µΛ)F (k)
0
0
0
0
1
1
1
1 22
0.5
0.5
0.5
0.5
1.5
1.5
1.5
1.5 2.52.5
−0.5−0.5
ω [GeV]ω [GeV]
S(ω
,2.5
GeV
)[G
eV−
1]
S(ω
,2.5
GeV
)[G
eV−
1] µΛ = 2.5 GeV
µΛ = 1.8 GeVµΛ = 1.3 GeVµΛ = 1.0 GeV
nonpert. perturbative
ModelS (dash)F (solid)
run to 2.5GeV
• Consistent to impose moment constraints on F (k), but not on S(ω, µΛ) w/o cutoff
p.7
Derivation of the magic formula (1)
• The shape function is the matrix element of a nonlocal operator:
S(ω, µ) = 〈B| b̄v δ(iD+ − δ + ω) bv| {z }O0(ω,µ)
|B〉, δ = mB −mb
Integrated over a large enough region, 0 ≤ ω ≤ Λ, one can expand O0 as
O0(ω, µ) =X
Cn(ω, µ) b̄v (iD+ − δ)n bv| {z }Qn
+ . . . =X
Cn(ω − δ, µ) b̄v (iD+)nbv| {z }eQn
+ . . .
The Cn are the same for Qn and Q̃n (since O0 only depends on ω − δ)
• Matching: 〈bv|O0(ω+δ, µ)|bv〉 =X
Cn(ω, µ) 〈bv| eQn|bv〉 = C0(ω, µ), 〈bv| eQn|bv〉 = δ0n
〈bv(k+)|O0(ω + δ, µ)|bv(k+)〉 = C0(ω + k+, µ) =X kn+
n!
dnC0(ω, µ)
dωn
〈bv(k+)|O0(ω + δ, µ)|bv(k+)〉 =X
Cn(ω, µ)〈bv| eQn|bv〉 =X
Cn(ω, µ) kn+
• Comparing last two lines: Cn(ω, µ) =1n!
dnC0(ω, µ)dωn
[Bauer & Manohar]
p.8
Derivation of the magic formula (2)
• Define the nonperturbative function F (k) by: [ZL, Stewart, Tackmann; Lee, ZL, Stewart, Tackmann]
S(ω, µΛ) =∫
dk C0(ω − k, µΛ)F (k), C0(ω, µ) = 〈bv|O0(ω + δ, µ)|bv〉
uniquely defines F (k): F̃ (y) = S̃(y, µ)/C̃0(y, µ)
• Expand in k: S(ω, µ) =∑n
1n!
dnC0(ω, µ)dωn
∫dk (−k)nF (k)
Compare with previous page ⇒∫
dk knF (k) = (−1)n 〈B|Qn|B〉
〈B|Q0|B〉 = 1 , 〈B|Q1|B〉 = −δ , 〈B|Q2|B〉 = −λ1
3+ δ
2
More complicated situation for higher moments, so stop here
• This treatment is fully consistent with the OPE
p.9
Changing schemes: mb
• Converting results to a short distance mass scheme removes dip at small ω:
• Want to define short distance(hatted) quantities such that:
S(ω) =
Zdk C0(ω − k)F (k)
S(ω) =
Zdk bC0(ω − k) bF (k)
Switch from pole to short dis-tance scheme:
mb = bmb + δmb
λ1 = bλ1 + δλ1
0
0
0
0
1
1
1
1 22
0.5
0.5
0.5
0.5
1.5
1.5
1.5
1.5 2.52.5−0.5−0.5
ω [GeV]ω [GeV]
S(ω
,2.5
GeV
)[G
eV−
1]
S(ω
,2.5
GeV
)[G
eV−
1] mpole
b
m1Sb
mkinb
dashed:
solid:
NLL
NNLL
(ω ∼ mB − 2Eγ)
bC0(ω) = C0(ω + δmb)−δλ1
6
d2
dω2C0(ω) =
»1 + δmb
d
dω+
„(δmb)
2
2−δλ1
6
«d2
dω2
–C0(ω)
• Can use any short distance mass scheme (1S, kinetic, PS, shape function, . . . )
p.10
Changing schemes: λ1
• We find that kinetic scheme, µ2π ≡ −λkin
1 , oversubtracts; similar to mb(mb) issues
• Introduce “invisible” scheme: λi1 = λ1−0αs−R2 α
2s(µ)
π2
CFCA
4
„π2
3−1
«(R = 1 GeV)
• Yet to be seen if shape func-tion scheme for λ1 gives goodbehavior
0
0
0
0
1
1
1
1 22
0.20.2
0.40.4
0.50.5
0.60.6
0.80.8
1.21.2
1.51.5 2.52.5
−0.2−0.2
ω [GeV]ω [GeV]
S(ω
,2.5
GeV
)[G
eV−
1]
S(ω
,2.5
GeV
)[G
eV−
1] λ1 NLL
λ1 NNLL
λkin1 NLL
λkin1 NNLL
(All curves use mkinb )
p.11
Scale (in)dependence of B → Xsγ spectrum
• Dependence on 3 scales in the problem:
µΛ = 1.2, 1.5, 1.9 GeV µi = 2.0, 2.5, 3.0 GeV µb = 2.35, 4.7, 9.4 GeV
00
00
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1.2
1.2
1.2
1.4
1.4
1.4
1.4 1.61.6p+X [GeV]p+X [GeV]
(dΓ
s/dp
+ X)/(Γ
0s|C
incl
7|2 )
[GeV−
1]
(dΓ
s/dp
+ X)/(Γ
0s|C
incl
7|2 )
[GeV−
1]
λ1 NNLL
λi1 NNLL
LL
NLL
00
00
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1.2
1.4
1.4
1.4
1.4 1.61.6p+X [GeV]p+X [GeV]
(dΓ
s/dp
+ X)/(Γ
0s|C
incl
7|2 )
[GeV−
1]
(dΓ
s/dp
+ X)/(Γ
0s|C
incl
7|2 )
[GeV−
1]
LL
NLL
NNLL
00
00
1
1
1
1
0.2
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1.2
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1.2
1.4
1.4
1.4
1.4 1.61.6p+X [GeV]p+X [GeV]
(dΓ
s/dp
+ X)/(Γ
0s|C
incl
7|2 )
[GeV−
1]
(dΓ
s/dp
+ X)/(Γ
0s|C
incl
7|2 )
[GeV−
1]
LL
NLL
NNLL
dΓs
dp+X
= Γ0sHs(p+X, µb)UH(mb, µb, µi)
Zdk bP (mb, k, µi) bF (p
+X − k)
“p+X
= mB − 2Eγ”
P̂ , F̂ indicate use of short distance schemes: m1Sb and λi
1
• In other approaches, using models for S(ω, µΛ) run up to µi, dependence on µΛ
ignored so far, but it must be considered an uncertainty ⇒ This is how to solve it
p.12
Designer orthonormal functions
• Devise suitable orthonormal basis functions(earlier: fit parameters of model functions to data)
F̂ (λx) = 1λ
[∑cnfn(x)
]2, n th moment ∼ΛnQCD
fn(x) ∼ Pn[y(x)] ← Legendre polynomials
• Approximating a model shape function
Better to add a new term in an orthonormalbasis than a new parameter to a model:– less parameter correlations– errors easier to quantify
“With four parameters I can fit an elephant, and with fiveI can make him wiggle his trunk.” (John von Neumann)
0
0
0
0
1
1
1
1 22 33 44
−1−1
0.5
0.5
0.5
0.5 1.51.5 2.52.5
−0.5−0.5
3.53.5xx
f0(x)f1(x)f2(x)f3(x)f4(x)
00
00
1
1
1
1
22
0.20.2 0.40.4
0.50.5
0.60.6 0.80.8 1.21.2 1.41.4
1.51.5
1.61.6k [GeV]k [GeV]
F̂(k
)[G
eV−
1]
F̂(k
)[G
eV−
1]
F̂ (k)
F̂ (0)(k)
F̂ (1)(k)
F̂ (2)(k)
F̂ (3)(k)
F̂ (4)(k)
p.13
Back to the B → Xsγ spectrum
• 9 models: same 0th, 1st, 2nd moments
Including all NNLL contributions, find:
– Shape in peak region not determined– at all by first few moments
– Smaller shape function uncertainty for– Eγ <∼ 2.1 GeV than earlier studies 00
11
2
2
2
2
0.50.5
1.51.5
1.91.9 2.12.1 2.22.2 2.32.3 2.42.4
2.5
2.5
2.5
2.5 2.62.6Eγ [GeV]Eγ [GeV]
(dΓ
s/dE
γ)/(Γ
0s|C
incl
7|2 )
[GeV−
1]
(dΓ
s/dE
γ)/(Γ
0s|C
incl
7|2 )
[GeV−
1]
c3=c4=0
c3=±0.15, c4=0
c3=0, c4=±0.15c3=±0.1, c4=±0.1
• Not shown in this plot: subleading shape functionsNot shown in this plot: subleading corrections not in C incl
7 (0)Not shown in this plot: kinematic power correctionsNot shown in this plot: boost to Υ(4S) frame
• Same analysis can also be used for: B → Xs`+`−, |Vub|
p.14
More complicated: B → Xu`ν̄
[ZL, Stewart, Tackmann, to appear]
Regions of phase space (again)
• “Natural” kinematic variables: p±X = EX∓|~pX|— “jettyness” of hadronic final state
B → Xsγ: p+X = mB − 2Eγ & p−X ≡ mB, but independent variables in B → Xu`ν̄
• Three cases: 1) Λ ∼ p+X � p−X
Three cases: 2) Λ� p+X � p−X
Three cases: 3) Λ� p+X ∼ p
−X
Make no assumptions how p−X compares to mB
• B → Xu`ν̄: 3-body final state, appreciable ratein region 3), where hadronic final state not jet-like
E.g., m2X < m2
D does not imply p+X � p−X 0
000
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
p−X [GeV]p−X [GeV]p
+ X[G
eV
]p
+ X[G
eV
]
mX ≤ mD
1) 2)
3)
• Can combine 1) – 2) just as we did in B → Xsγ, by not expanding in Λ/p+X
p.15
Charmless B → Xu`ν̄ made charming
• To combine all 3 regions: do not expand in Λ/p+X nor in p+
X/p−X (nontrivial!)
• Want to have: result accurate to NNLL and ΛQCD/mb in regions 1) – 2)and to order α2
sβ0 and Λ2QCD/m
2b when phase space limits are in region 3)
Start with triple differential rate (involves a delta-fn at the parton level at O(α0s),
which is smeared by the shape function)
The p+X/p
−X terms, which are not suppressed in local OPE region, start at O(αs)
Recently completed O(α2s) matching computations [Ben’s & Guido’s talks]
[Bonciani & Ferroglia, 0809.4687; Asatrian, Greub, Pecjak, 0810.0987; Beneke, Huber, Li, 0810.1230; Bell, 0810.5695]
p.16
SIMBA
[Bernlochner, Lacker, ZL, Stewart, Tackmann, Tackmann, to appear]
Fitting charmless inclusive decay spectra
• Fit strategy: F̂ (k) enters the spectra linearly ⇒ can calculate independently thecontribution of fm fn in the expansion of F̂ (k):
dΓ =∑
cm cn︸ ︷︷ ︸fit
dΓmn︸ ︷︷ ︸compute
dΓmn = Γ0H(p−
)
Z p+X
0
dkbP (p−, k)
λfm
„p+X − kλ
«fn
„p+X − kλ
«| {z }
basis functions
Fit the ci coefficients from the measured (binned) spectra
• What we hope to achieve:– Correlation and error propagation of SF uncertainties– Simultaneous fit using all available information– Realistic estimate of model uncertainties (fit parameters c0,...,N constrained by– data; vary N , the number of orthonormal basis functions in fit)
p.17
A preliminary B → Xsγ fit
• Belle B → Xsγ spectrum in Υ(4S) restframe
For demonstration purposes only — thereare very strong correlations
Fit with 4 basis functions in the expansion ofthe shape function
Shows that fit works (not as trivial as this plotmight indicate); still issues to resolve
[Belle, 0804.1580 + Preliminary; thanks to Antonio Limosani]
[GeV]γE2 2.5
γ/d
EΓd
00.05
0.10.150.2
0.250.3
0.350.4
• Next step: include various B → Xu`ν̄ measurements
p.18
Conclusions
• Improving accuracy of |Vub| will remain important to constrain new physics(Current situation unsettled, PDG in 2008 inflated error for the first time)
• Qualitatively better inclusive analyses possible than those that exist so far– Modeling F (k) instead of S(ω, µ)– Designer orthonormal functions– Fully consistent combination of 3 phase space regions– Decouple SF shape variation from mb variation
• Developments will allow combining all pieces of data with tractable uncertainties– Consistently combine B → Xu`ν̄, B → Xsγ, B → Xc`ν̄ data to constrain SF’s– Reduce role of shape function modeling
• To draw conclusions about new physics comparing sides and angles, we’ll want≥2 extractions of |Vub| with different uncertainties (inclusive, exclusive, leptonic)
p.19