Model-independent determination of |V_{ub}|

5 Pages·0.1 MB·English

by  Changhao Jin

hep-ph/0001324
Model-independent determination of |V |
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Changhao Jin
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3 School of Physics, University of Melbourne

Victoria 3010, Australia
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v E-mail: [email protected]
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0 Abstract
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/ The decay distribution of the kinematic variable ξu in inclusive charmless semilep-
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tonic decays of B mesons is unique. The novel method for a model-independent
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determination of |V | is described.
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To appear in Proceedings of the 3rd International Conference on B Physics
and CP Violation
Taipei, Taiwan, December 3-7, 1999
MODEL-INDEPENDENT DETERMINATION OF |V |
ub
CHANGHAOJIN
School of Physics, University of Melbourne, Victoria 3010, Australia
E-mail: [email protected]
Thedecaydistributionofthekinematicvariableξu ininclusivecharmlesssemilep-
tonic decays of B mesons is unique. The novel method for a model-independent
determinationof|Vub|isdescribed.
1 Problems and Solutions
The fundamental Cabibbo-Kobayashi-Maskawa(CKM) matrix element |V |
ub
has been determined from inclusive charmless semileptonic decays of B
mesons, B → X ℓν. However, there are experimental and theoretical prob-
u
lems that obstruct a precise determination of |V |. Experimentally, it is
ub
very difficult to separate signals from the rare B → X ℓν decay from large
u
B →X ℓνbackground. Theoretically,QCDuncertaintiesariseincalculations
c
that relate the measured quantity to |V |. The potential theoretical uncer-
ub
tainties from perturbative and nonperturbative QCD can be comparable.
The solutions for the problems are provided by a novelmethod.1,2 It has
been proposed to use the kinematic cut on the variable ξ = (q0 +|q|)/M
u B
(q is the momentumtransferto the leptonpair)abovethe kinematic limitfor
B →X ℓν, ξ >1−M /M , to separate B →X ℓν signal from B →X ℓν
c u D B u c
background. Most of B → X ℓν events pass the above cut. This kinematic
u
requirement provides a very efficient way for background suppression. |V |
ub
canthenbeextractedfromtheweightedintegralofthemeasuredξ spectrum
u
via the sum rule for inclusive charmless semileptonic decays of B mesons
with little theoretical uncertainty. The sum rule is derived from the light-
cone expansion and beauty quantum number conservation. Thus a model-
independent determination of |V | can be achieved, minimizing the overall
ub
(experimental and theoretical) error.
2 Sum Rule
Because of the large B meson mass, the light-cone expansion is applicable
to inclusive B decays that are dominated by light-cone singularities. For
inclusivecharmlesssemileptonicdecaysofB mesons,thelight-coneexpansion
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and beauty quantum number conservation lead to the sum rule 1
1 1 dΓ G2M5
S ≡ dξ (B →X ℓν)=|V |2 F B . (1)
Z u ξ5 dξ u ub 192π3
0 u u
This sum rule has the following advantages:
• Independent of phenomenological models
• No perturbative QCD uncertainty
• Dominant hadronic uncertainty avoided
The sum rule (1) establishes a relationship between |V | and the observable
ub
quantity S in the leading twist approximation of QCD. The only remaining
theoretical uncertainty in the relation comes from higer-twist corrections to
the sum rule, which are suppressed by a power of Λ2 /M2.
QCD B
3 The ξ Spectrum
u
Nowlet meexplainwhy the decaydistributionofthe kinematicvariableξ is
u
unique andwhythe kinematic cutonξ isveryefficientinthe discrimination
u
between B →X ℓν signal and B →X ℓν background.
u c
Without QCD corrections, the tree-level ξ spectrum in the free quark
u
decay b → uℓν in the b-quark rest frame is a discrete line at ξ = m /M .
u b B
This is simply a consequence of kinematics that fixes ξ to the single value
u
m /M , no other values of ξ are kinematically allowed in b → uℓν decays.
b B u
This discrete line at ξ = m /M ≈ 0.9 lies well above the charm threshold,
u b B
ξ >1−M /M =0.65.
u D B
The O(α ) perturbative QCD correction to the ξ spectrum has been
s u
calculated.2 The ξ spectrum remains a discrete line at ξ = m /M , even
u u b B
if virtualgluon emission occurs. Gluon bremsstrahlung generates a small tail
below the parton-level endpoint ξ =m /M .
u b B
To calculate the real physical decay distribution in B → X ℓν, we must
u
also account for hadronic bound-state effects. In the framework of the light-
cone expansion, the leading nonperturbative QCD effect is incorporated in
the b-quark distribution function 3
1 d(y·P) 0
f(ξ)= 4π Z y·P eiξy·PhB|¯b(0)y/Pexp[igsZ dzµAµ(z)]b(y)|Bi|y2=0,
y
(2)
where P denotes path ordering. Although several important properties of
it are known3 in QCD, the form of the distribution function has not been
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completely determined. The distribution function f(ξ) has a simple physical
interpretation: It is the probability of finding a b-quark with momentum ξP
inside the B meson with momentum P. The real physical spectrum is then
obtained from a convolution of the hard perturbative spectrum with the soft
nonperturbative distribution function:
dΓ 1 dΓ
(B →X ℓν)= dξf(ξ) (b→uℓν,p =ξP), (3)
dξ u Z dξ b
u ξu u
where the b-quark momentum p in the perturbative spectrum is replaced by
b
ξP. The interplay between nonperturbative and perturbative QCD effects
has been accounted for.
Bound-state effects lead to the extension of phase space from the parton
levelto the hadronlevel, also stretchthe spectrum downwardbelow m /M ,
b B
and are solely responsible for populating the spectrum upward in the gap
betweentheparton-levelendpointξ =m /M andthehadron-levelendpoint
u b B
ξ =1. TheinterplaybetweennonperturbativeandperturbativeQCDeffects
u
eliminates the singularity at the endpoint of the perturbative spectrum, so
that the physical spectrum shows a smooth behaviour over the entire range
of ξ , 0≤ξ ≤1.
u u
Although the monochromatic ξ spectrum at tree level is smeared by
u
gluonbresstrahlungandbound-stateeffects aroundξ =m /M ,about80%
u b B
of B → X ℓν events remain above the charm threshold. The uniqueness of
u
the decaydistributionofthe kinematicvariableξ implies thatthe kinematic
u
cut on ξ is very efficient in disentangling B →X ℓν signal from B →X ℓν
u u c
background.
4 How Do You Measure S?
To measure the observable S defined in Eq. (1), one needs to measure the
weighted ξ spectrum ξ−5dΓ(B → X ℓν)/dξ , using the kinematic cut ξ >
u u u u u
1−M /M againstB →X ℓν background. S canthen be obtainedfroman
D B c
extrapolation of the weighted spectrum measured above the charm threshold
to low ξ .
u
While the normalizationof the weighted spectrum given by the sum rule
(1) does not depend on the b-quark distribution function f(ξ), thus being
model-independent, the shape of the weighted spectrum does. The detailed
analysis is presented in Ref. 2. Gluon bremsstrahlung and hadronic bound-
state effects strongly affect the shape of the weighted ξ spectrum. However,
u
theshapeofthe weightedξ spectrumisinsensitivetothevalueofthe strong
u
coupling α , varied in a reasonable range. The overall picture appears to be
s
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that the weighted ξ spectrum is peaked towards larger values of ξ with a
u u
narrowwidth. Thecontributionbelowξ =0.65issmallandrelativelyinsen-
u
sitive to forms of the distribution function. This suggests that extrapolating
the weighted ξ spectrum downto low ξ would not introduce a considerable
u u
uncertainty in the value of S.
5 Summary
The kinematic cut on ξ , ξ > 1−M /M , and the semileptonic B decay
u u D B
sumrule,Eq.(1),makeanoutstandingopportunityfortheprecisedetermina-
tion of |V | from the observable S. This method is both exceptionally clean
ub
theoretically and very efficient experimentally in background suppression.
Thereremaintwokindsoftheoreticalerrorinthe model-independentde-
termination of |V |. First, higher-twist (or power suppressed) corrections to
ub
thesumrulecauseanerroroftheorderO(Λ2 /M2)∼1%in|V |. Second,
QCD B ub
theextrapolationoftheweightedξ spectrumtolowξ givesrisetoasystem-
u u
aticerrorinthemeasurementofS. Thesizeofthiserrordependsonhowwell
the weightedspectrum canbe measured, since the measured spectrum would
directlydeterminetheformofthedistributionfunction. Inaddition,theform
of the universal distribution function can also be determined directly by the
measurement of the B → X γ photon energy spectrum.3 The experimental
s
determinationofthedistributionfunctionwouldprovideamodel-independent
way to make the extrapolation, allowing an error reduction.
Eventually, the error in |V | determined by this method would mainly
ub
depend on how well the observable S can be measured. To measure S ex-
perimentally one needs to be able to reconstruct the neutrino. This poses a
challenge to experiment. The unique potential of determining |V | warrants
ub
a feasibility study for the experiment.
Acknowledgments
I would like to thank Hai-Yang Cheng and Wei-Shu Hou for the very stimu-
lating and enjoyable conference. This work was supported by the Australian
Research Council.
References
1. C.H. Jin, Mod. Phys. Lett. A 14, 1163 (1999).
2. C.H. Jin, hep-ph/9911396.
3. C.H. Jin, Eur. Phys. J. C 11, 335 (1999).
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