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\begin{document}
\vspace*{4cm}
\title{REVIEW OF $V_{cb}$ AND $V_{ub}$ MEASUREMENTS}
\author{ F. BIANCHI }
\address{Universit\'a di Torino, Dipartimento di Fisica Sperimentale,
\\ v. Pietro Giuria 1, Torino, 10125, Italy}
\maketitle\abstracts{
Flavour mixing is described within the Standard Model by the
Cabibbo-Kobayashi-Maskawa matrix elements.
With the high statistics collected by the experiments at the b-factories,
the matrix elements $|V_{cb}|$ and $|V_{ub}|$ are measured with improved
precision, allowing for more stringent tests of the Standard Model.
In this paper, a review of the current status of their measurements is
presented.}
\section{Introduction}
The Standard Model (SM) accounts for flavor changing quark transition through
the coupling of the V-A charged current operator to a W boson:
\begin{equation}
\mathcal{L}_W=-\sqrt{\frac{1}{2}}g\overline{u_{Li}}\gamma^\mu\overline{V_{ij}}
d_{Lj}W^+_\mu+{\rm h.c.}
\label{chargedcurrent}
\end{equation}
where $V_{ij}$ are the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix
elements.
By convention, the mixing is expressed in terms of the $V$ matrix
operating on the charge -$e/3$ quark mass eigenstates ($d$, $s$ and $b$):
\begin{equation}
\left( \begin{array}{c}
d^{~\prime} \\
s^{~\prime} \\
b^{~\prime} \end{array} \right)
=
\left( \begin{array}{ccc}
V_{ud} & V_{us} & V_{ub} \\
V_{cd} & V_{cs} & V_{cb} \\
V_{td} & V_{ts} & V_{tb}
\end{array} \right)
\left( \begin{array}{c}
d \\
s \\
b \end{array} \right)
\label{CKMmatrix}
\end{equation}
Generation changing transitions between quarks are possible because the
off-diagonal elements are not zero. The values of the CKM matrix elements are
fundamental parameters of the SM and cannot be predicted. In the following, a
review of the current values of $|V_{cb}|$ and $|V_{ub}|$ measured by the
BaBar, Belle and Cleo experiments is presented.
The averages of the Heavy Flavour
Averaging Group (HFAG) \cite{HFAG} will also be quoted.
\section{$|V_{cb}|$ Measurements}
The CKM matrix element $|V_{cb}|$ is measured from the semileptonic inclusive
and exclusive $b \ra c l \nu$ decays.
At the parton level, this decay rate can be calculated accurately;
it is proportional to $|V_{cb}|^2$
and depends on the quark masses, $m_b$ and $m_c$. To relate
measurements of the semileptonic B-meson decay rate to $|V_{cb}|$,
the parton-level calculations have to be corrected
for effects of strong interactions.
\subsection{Inclusive Measurements}
In the kinetic-mass scheme the Heavy Quark Expansion (HQE) to
$\mathcal{O}(1/m^3_b)$
for the rate $\Gamma_{SL}$ of semileptonic decays $B \ra X_c l^- \nu$ can be
expressed as \cite{bbmu}:
\begin{eqnarray}
\Gamma_{SL} & = & \frac{G_F^2 m_b^5}{192\pi^3} |V_{cb}|^2
(1+A_{\mathit{ew}})
A_{\mathit{pert}}(r,\mu) \nonumber\\
& \times & \Bigg [ z_0(r) \Bigg ( 1 - \frac{\mu_{\pi}^2 - \mu_G^2 +
\frac{\rho_D^4 + \rho_{LS}^3}{c^2 m_b}}{2 c^4 m_b^2} \Bigg ) \\
& - & 2(1-r)^4\frac{\mu_G^2 + \frac{\rho_D^3 + \rho_{LS}^3}{c^2 m_b}}
{c^4 m_b^2}
+ d(r)\frac{\rho_D^3}{c^6 m_b^3} \nonumber \\
& + & \mathcal{O}(1/m_b^4)\Bigg]. \nonumber
\label{eq:vcb_gammaslkinetic}
\end{eqnarray}
This expansion contains six parameters: the running kinetic
masses of the b- and c-quarks, $m_b(\mu)$ and $m_c(\mu)$,
and four non-perturbative parameters $\mu_{\pi}$, $\mu_G$, $\rho_D$,
and $\rho_{LS}$: . The parameter
$\mu$ denotes the Wilson normalization scale that separates
effects from long- and short-distance dynamics.
The ratio $r = m_c^2/m_b^2$ enters in the tree level phase space factor
$z_0(r) = 1 - 8r + 8r^3 - r^4 - 12r^2 \ln r$ and in the function
$d(r) = 8 \ln r + 34/3 - 32r/3 - 8r^2 + 32r^3/3 - 10r^4 /3$.
The factor $1 + A_{\mathit{ew}}$ accounts for electroweak corrections. It
is estimated to be $1 + A_{ew} \cong ( 1 + \alpha/\pi \ln M_Z/m_b )^2 = 1.014$,
where $\alpha$ is the electromagnetic coupling constant.
The quantity $A_{\mathit{pert}}$ accounts for perturbative contributions and
is estimated to be $A_{pert}(r,\mu) \approx 0.908$.
The moments of the hadronic-mass and electron-energy distributions
in semileptonic B decays $B \ra X_c l^- \nu$ and the moments of
the photon-energy spectrum in $B \ra X_s \gamma$ decays depend on
the same set of parameters.
BaBar \cite{momentsBBR} and Belle \cite{momentsBLL} have performed a combined
fit to all these moments to
extract values for $|V_{cb}|$, the quark masses $m_b$ and $m_c$, the total
semileptonic branching fraction $\mathcal{B}(B \ra X_c l^- \nu)$, and the
non-perturbative HQE parameters. The fitted value of $|V_{cb}|$,
using expressions in the kinetic scheme, is
$|V_{cb}| = (41.67 \pm 0.43 \pm 0.08 \pm 0.58) \times 10^{-3}$,
where the errors are due to the global fit, the B lifetime and theory,
respectively. However it should be noted that that a fit just to the
$B \ra X_c l^- \nu$ moments tends to give a value of $m_b$
about $1\sigma$ higher than the one from $B \ra X_c l^- \nu$ and
$B \ra X_s \gamma$ moments combined as shown in Fig.~\ref{fig:moments}.
This incertitude impacts
the $|V_{ub}|$ extraction where $m_b$ is used as input in the fitting
procedure.
In addition the $\chi^2/NDF$ of the fit is 29.7/57, a quite small value that
can possibly come from an improper treatment of correlations between the
different moments.
The most recent result from Belle \cite{momentsBLL} does not
shown the dependence of the value of $m_b$ on the set of moment used.
\begin{figure}
\vskip 2.5cm
\psfig{figure=vcb-mb-moments-hfag.eps,height=3.0in}
\psfig{figure=mbmu2pi-moments-hfag.eps,height=3.0in}
\caption{$\Delta_{\chi^2} = 1$ contours for the fit to all
moments and the fit to the $B \ra X_c l^- \nu$ data only. $|V_{cb}|$ vs $m_b$
(left) and $\mu^2_{\pi}$ vs $m_b$ (right).
\label{fig:moments}}
\end{figure}
\subsection{Exclusive Measurements}
The determination of
$|V_{cb}|$ from exclusive $b \ra c l \nu$ decays is based on the
$B \ra D^{(*)} l \nu$ decays, for which, in the assumption of infinite b and
c quark masses, the form factors describing the $B \ra D^{(*)}$ transitions
depend only on the product, w, of the initial,
v, and final, v', state hadron four-velocities, $w = v \times v'$, and relies
on a parametrization of the form factors using the Heavy Quark Symmetry (HQS)
\cite{HQS} and a non-perturbative calculation of the form factor
normalization at w = 1, which corresponds to the maximum momentum transfer to
the leptons.
The form factors for $B \ra D l \nu$ and for $B \ra D^* l \nu$
decays are G(w) and F(w), respectively.
BaBar and Belle adopt the form factor parametrization from Caprini et
al. \cite{caprini}, and lattice QCD to correct the normalization of the form
factor at w = 1, due to the finite quark masses.
Experimentally, the w spectrum is measured and $F(1)|V_{cb}|$ and
$G(1)|V_{cb}|$ are obtained from an extrapolation of the measured w
spectrum to 1.
Several analyses from BaBar \cite{bbrff} and Belle \cite{bllff},
which adopt different experimental techniques,
were recently presented.
The bi-dimensional plots of the form factor at w = 1 times $|V_{cb}|$ versus
the slope parameter for the form factors $\rho^2$ is shown in
Fig.~\ref{fig:ffvcb}.
The fitted values are
$G(1)|V_{cb}| = (42.4 \pm 1.6) \times 10^{-3}$
and
$F(1)|V_{cb}| = (35.4 \pm 0.5) \times 10^{-3}$
rispectively.
Assuming:
$G(1) = 1.074 \pm 0.018 \pm 0.016$ \cite{okamoto} and
$F(1) = 0.924 \pm 0.012 \pm 0.019$ \cite{bernard},
where the errors are statistical and
systematical, respectively, and appling a 1.07 QCD correction factor,
the values
$|V_{cb}| = (39.7 \pm 1.4 \pm 0.9) \times 10^{-3}$ and
$|V_{cb}| = (38.1 \pm 0.6 \pm 0.9) \times 10^{-3}$ are obtained,
for $B \ra D l \nu$ and for $B \ra D^* l \nu$ decays, respectively.
The two results are completely consistent.
The 2$\sigma$ discrepancy between the value of $|V_{cb}|$ extracted from
the moment analysis and the one coming from $B \ra D^{(*)} l \nu$ decays
using the Lattice QCD form factor calculations, is still an open question.
\begin{figure}
\vskip 2.5cm
\psfig{figure=vcbf1-vs-rho2.eps,height=3.0in}
\psfig{figure=vcbg1-vs-rho2.eps,height=3.0in}
\caption{$G(1)|V_{cb}|$ (left) and $F(1)|V_{cb}|$ (right) versus
the form factor slope parameter $\rho^2$.
\label{fig:ffvcb}}
\end{figure}
\section{$|V_{ub}|$ Measurements}
Semileptonic inclusive and exclusive $b \ra u l \nu$ decays are used to
measure the CKM matrix element $|V_{ub}|$.
Different experimental and theoretical approaches are involved,
thus providing complementary ways to extract $|V_{ub}|$.
\subsection{Inclusive Measurements}
The measurement of the inclusive decays rate for $B \ra X_u l^- \nu$
decays is affected by a large background of the
order $|V_{ub}/V_{cb}|^2 = 1/50$, due to $B \ra X_c l^- \nu$ decays.
Stringent kinematic cuts are applied to select regions of the phase space in
which the $B \ra X_c l^- \nu$ background can be kept under control.
Thus, only a partial branching fraction, limited to the particular kinematic
region selected, is measured and needs to be estrapolated to the full
phase space.
Whilst the total branching fraction can be computed using HQE
and QCD perturbation theory, the partial rate
needs further theoretical tools, which have been the subject of intense
theoretical effort, expecially in the last years. Different approches have
been used: BLNP \cite{BLNP} (a shape function approach, where the shape
function represents the momentum distribution function of the b quark
in the B meson), DGE \cite{DGE} (a resummation based approach),
GGOU \cite{GGOU} (an HQE based structure function parametrization approach)
and ADFR \cite{ADFR} (a soft gluon resummation and analytic time-like
QCD coupling approach).
Concerning BLNP, recent NNLO corrections \cite{BLNP-nnlo} were presented.
The models depend strongly on the b quark mass, except for ADFR, so it is
very important to use a precise determination of this quantity.
BaBar \cite{bbrincvub} and Belle \cite{bllincvub} have applied kinematic
cuts using the following variables:
the lepton energy ($E_l$), the invariant mass of the hadron
final state ($M_X$),
the light-cone distribution ($P^+ = E_X - |p_X|$, $E_X$ and
$|p_X|$ being the energy and the magnitude of
the 3-momentum of the hadronic system)
and a two dimensional distribution
in the electron energy and $s^{max}$, the
maximal $M_X^2$ at fixed $q^2$ and $E_l$.
The results obtained by these methods and the corresponding averages are
shown in Fig.~\ref{fig:inclvub}.
\begin{figure}
%\rule{5cm}{0.2mm}\hfill\rule{5cm}{0.2mm}
\vskip 2.5cm
%\rule{5cm}{0.2mm}\hfill\rule{5cm}{0.2mm}
\psfig{figure=vub_adfr.eps,height=3.4in}
\psfig{figure=vub_blnp.eps,height=3.4in}
\psfig{figure=vub_dge.eps,height=3.4in}
\psfig{figure=vub_ggou.eps,height=3.4in}
\caption{Inclusive $|V_{ub}|$ measurements.
\label{fig:inclvub}}
\end{figure}
The values of $|V_{ub}|$ obtained using diffent kinematical cuts and
exctracted using the same theoretical approch are consistent.
On the contrary, different theoretical approches give $|V_{ub}|$ values
that are somehow different.
Very recently, a preliminary result from Belle using an innovative
multivariate analysis \cite{bllmultivar}, in which 90\% of the total rate
is measured, has been presented.
This experimental measurement is extremely interesting as it will help in
a further understanding of $|V_{ub}|$ from inclusive decays.
\subsection{Exclusive Measurements}
$|V_{ub}|$ can be extracted from exclusive charmless semileptonic decays,
$B \ra \pi , \rho, \eta, \eta', \omega l \nu$, where the corresponding rate is
related to $|V_{ub}|$ by the form factor(s) $f(q^2)$, where $q^2$ is the
momentum transfer squared to the lepton pair.
Non perturbative methods for the calculation of the form factors include
unquenched lattice QCD, like the HPQCD \cite{HPQCD} and Fermilab/MILC
\cite{FermiLab} calculations, and QCD light cone sum rules \cite{LCSR}.
BaBar \cite{bbrpieta}, Belle \cite{bllpirho}, and Cleo \cite{cleopi}
have perfomed measurements of $B \ra \pi l \nu$ decays
exploiting different analysis techniques that fall into two broad classes:
untagged and tagged, depending on whether the B in the event that does not
decay into the $\pi l \nu$ final state is tagged or not. The untagged method
has higher statistic and higher background, while the B tagging reduces
significantly the background at a price of a reduced statistics.
The results are presented for the full $q^2$, $q^2 > 16$ $GeV^2$ and
$q^2 < 16$ $GeV^2$ ranges. The last two phase space
regions correspond to regions where the lattice and QCD light cone sum rule
calculations of the form factors are
restricted to, respectively. The corresponding measurements of the total
branching ratio for all the collaborations and their average is shown in
Fig.~\ref{fig:exclvub}(left plot). From the average, and
using both lattice QCD and QCD light cone sum rules, the value of $|V_{ub}|$
are extracted (Fig.~\ref{fig:exclvub}, right plot). The $|V_{ub}|$
results coming from different theoretical calculations are consistent among
themselves. However the uncertainties from form factors calculation are
the dominant systematic in the $|V_{ub}|$ extraction.
In a recent paper by Bailey at al.\cite{bailey}, the $B \ra \pi l \nu$
12 bin $q^2$ spectrum measured by BaBar \cite{bbrpiq2} has been used
to extract $|V_{ub}| = (3.38 \pm 0.36) \times 10^{-3}$.
\begin{figure}
\vskip 2.5cm
\psfig{figure=Bpilnu.eps,height=3.2in}
\psfig{figure=vubexcl.eps,height=3.2in}
\caption{ $B \ra \pi l \nu$ branching fractio measurements (left) and
$|V_{ub}|$ values extracted using different theoretical calculations (right).
\label{fig:exclvub}}
\end{figure}
Moreover, experimental measurements of the
$B \ra \pi , \rho, \eta, \eta', \omega l \nu$ branching
ratio have been performed by BaBar\cite{bbrometa},
Belle\cite{bllpirho} and Cleo\cite{cleoeta} and will provide a test of
the $|V_{ub}|$ extraction from $B \ra \pi l \nu$
decays, once the corresponding form factors will be computed.
\section{Summary}
A significant progress has been made in the past years thanks to the
b-factory measurements of $|V_{cb}|$ and $|V_{ub}|$ and to a remarkable
theoretical effort. However the dominant systematics are the one coming from
the theoretical calculation used to extract the CKM matrix elements from
the experimental observables. More data and theoretical progress will improve
our knowledge of $|V_{cb}|$ and $|V_{ub}|$.
\section*{Acknowledgments}
We are grateful to the experimental collegues from BaBar, Belle, and Cleo
for having shared their latest results when needed.
Also, we would like to thank the theory collegues
for several discussions regarding the $|V_{cb}|$ and $|V_{ub}|$ calculations.
Finally, we would like to thank the MoriondEW09
organizers for the flawless organization and the pleasant and stimulating
atmosphere of the conference.
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\end{document}
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