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K. Kleinknecht and U. Uwer
5. |V cb |. This matrix element is determined from semileptonic B decays to D
or D ∗ mesons. Two experimentally and theoretically different approaches are
used. The inclusive approach measures the inclusive semileptonic decay rate to
any charmed final state together with the moments of the leptonic energy and
the hadronic invariant mass spectra. An operator product expansion within the
Heavy Quark Effective Theory (HQET) allows the calculation of decay rates
and the energy and mass spectra in dependence of expansion parameters α s and
the inverse of the heavy quark mass. The simultaneous measurement of several
distributions over-constrains the physical parameters and allows a determination
of |V cb| . An analysis and an averaging of existing measurements is performed in
[145] and leads to |V cb | = (42.2 ± 0.8) × 10 3 with uncertainties arising mainly
from higher-order perturbative and non-perturbative corrections.
In the exclusive approach semileptonic B decays to exclusive channels containing
D and D ∗ mesons are studied. In the infinite quark mass limit with m b , m c
Λ QCD , heavy quark symmetry predicts that all form-factors are given by a
single function which depends on the product of the four-velocities v of the
B and v of the D (∗) (mesons) and are normalized at the point of maximum
momentum transfer to the lepton system (v · v = 1). The matrix element
|V cb | is obtained from an extrapolation to this point. The precise determination
of normalization and the shape of the form factor function requires additional
corrections calculated using HQET. Reference [145] quotes |V cb | = (41.9 ±
2.0) × 10 3 as the average exclusive value of |V cb |.
The inclusive and the exclusive determination of |V cb | are in agreement and [145]
performs the average of both values:
|V cb | = (42.2 ± 0.8) × 10
3 .
6. |V ub |. Similar to |V cb | also |V ub | is determined analysing inclusive and exclusive
semileptonic B decays with a b → uu¯ ν transition. The determination of |V ub |
from inclusive decays however suffers experimental and theoretical difficulties.
The total inclusive decay rate is hard to measure due to the large background
from CKM-favored b → cc¯ ν transitions. Therefore, strong kinematic cuts
are introduced to suppress these background contributions. The restriction to
tight kinematic regions however complicates the theoretical description. The
calculation of partial decay rates in the various kinematic regions requires
the introduction of non-perturbative distribution functions—the so called shape
functions—to describe the effect of hadronic physics. At leading order there is
only a single shape function which can be determined using inclusive ¯
B → X s γ
decays. Subleading effects are considered using different theoretical models
based on Heavy Quark Expansion (HQE). A recent summary of the |V ub | values
extracted within different models from measurements by BABAR, BELLE and
CLEO can be found in [145]. All calculations give similar values for |V ub | and
similar error estimates. As average [145] quotes
|V ub | = (4.49 ± 0.15
+0.16
−0.17 ) × 10
−3 .
K. Kleinknecht and U. Uwer
5. |V cb |. This matrix element is determined from semileptonic B decays to D
or D ∗ mesons. Two experimentally and theoretically different approaches are
used. The inclusive approach measures the inclusive semileptonic decay rate to
any charmed final state together with the moments of the leptonic energy and
the hadronic invariant mass spectra. An operator product expansion within the
Heavy Quark Effective Theory (HQET) allows the calculation of decay rates
and the energy and mass spectra in dependence of expansion parameters α s and
the inverse of the heavy quark mass. The simultaneous measurement of several
distributions over-constrains the physical parameters and allows a determination
of |V cb| . An analysis and an averaging of existing measurements is performed in
[145] and leads to |V cb | = (42.2 ± 0.8) × 10 3 with uncertainties arising mainly
from higher-order perturbative and non-perturbative corrections.
In the exclusive approach semileptonic B decays to exclusive channels containing
D and D ∗ mesons are studied. In the infinite quark mass limit with m b , m c
Λ QCD , heavy quark symmetry predicts that all form-factors are given by a
single function which depends on the product of the four-velocities v of the
B and v of the D (∗) (mesons) and are normalized at the point of maximum
momentum transfer to the lepton system (v · v = 1). The matrix element
|V cb | is obtained from an extrapolation to this point. The precise determination
of normalization and the shape of the form factor function requires additional
corrections calculated using HQET. Reference [145] quotes |V cb | = (41.9 ±
2.0) × 10 3 as the average exclusive value of |V cb |.
The inclusive and the exclusive determination of |V cb | are in agreement and [145]
performs the average of both values:
|V cb | = (42.2 ± 0.8) × 10
3 .
6. |V ub |. Similar to |V cb | also |V ub | is determined analysing inclusive and exclusive
semileptonic B decays with a b → uu¯ ν transition. The determination of |V ub |
from inclusive decays however suffers experimental and theoretical difficulties.
The total inclusive decay rate is hard to measure due to the large background
from CKM-favored b → cc¯ ν transitions. Therefore, strong kinematic cuts
are introduced to suppress these background contributions. The restriction to
tight kinematic regions however complicates the theoretical description. The
calculation of partial decay rates in the various kinematic regions requires
the introduction of non-perturbative distribution functions—the so called shape
functions—to describe the effect of hadronic physics. At leading order there is
only a single shape function which can be determined using inclusive ¯
B → X s γ
decays. Subleading effects are considered using different theoretical models
based on Heavy Quark Expansion (HQE). A recent summary of the |V ub | values
extracted within different models from measurements by BABAR, BELLE and
CLEO can be found in [145]. All calculations give similar values for |V ub | and
similar error estimates. As average [145] quotes
|V ub | = (4.49 ± 0.15
+0.16
−0.17 ) × 10
−3 .