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\begin{document}\sloppy
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\title{
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\vspace{-3.0cm}
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TTP13-006, SFB/CPP-13-09
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On the ${\cal O}(\alpha_s^2)$ corrections to
$b \to X_u e \bar \nu$
inclusive decays }
\author{Mathias Brucherseifer}
\address{ Institut f\"ur Theoretische Teilchenphysik,
Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany}
\author{Fabrizio Caola and Kirill Melnikov}
\address{
Department of Physics and Astronomy,
Johns Hopkins University,
Baltimore, MD, USA}
\begin{abstract}
\noindent We present
${\cal O}(\alpha_s^2)$ QCD corrections to
the fully-differential decay rate of a $b$-quark
into inclusive semileptonic charmless final states.
Our calculation provides genuine two-loop QCD corrections,
beyond the Brodsky-Lepage-Mackenzie (BLM) approximation,
to {\it any } infra-red safe partonic observable that can be probed
in $b \to X_u e \bar \nu$ decays. Kinematic cuts
that closely match those
used in experiments can be fully accounted for. To illustrate these points,
we compute the non-BLM corrections to moments of the hadronic invariant
mass and the hadronic energy with cuts on the lepton
energy and the hadronic invariant mass.
Our results remove one of the sources of theoretical uncertainty
that affect
the extraction of the CKM matrix element $|V_{ub}|$ from charmless
inclusive $B$-decays.
\end{abstract}
\maketitle
Studies of CP violation in $B$-mesons performed by BELLE and BABAR, firmly
established the correctness of the Cabbibo-Kobayashi-Maskawa paradigm at
the few percent level. These studies will be continued, when the super-$B$
factory in Japan will come on line.
A powerful tool to test the CKM picture is the unitarity of
the CKM-matrix. A combined fit to all available data gives
$|V_{ub}| = 3.58(13) \times 10^{-3}$ \cite{ufit}. This number should
be compared with the value $|V_{ub}| = 3.38(36) \times 10^{-3}$ extracted
from exclusive $B \to h e \bar \nu$
decays, where the hadron $h$ is either a pion or a $\rho$-meson,
and with $|V_{ub}| = 4.27(38) \times 10^{-3}$ which is obtained from
{\it inclusive} measurements of $B \to X_u e \bar \nu$ decays
\cite{pdg}. Although exclusive and inclusive results
are not in serious disagreement, they are clearly different and further
scrutiny of both exclusive
and inclusive determinations of $|V_{ub}|$ is certainly warranted.
The most complicated theoretical issue for both exclusive and inclusive methods
is the control of non-perturbative effects. This is hard to do for exclusive
decays and important input in this case is provided by {\it ab initio}
lattice
QCD calculations of exclusive $B \to \pi,\rho,..$ transitions. In contrast,
in case of
{\it inclusive} semileptonic decays of $B$-mesons, non-perturbative
difficulties
can be largely circumvented by the application of local operator
product expansion (OPE) \cite{ope1,ope2,ope3}.
The OPE allows to compute
sufficiently inclusive observables related to
semileptonic decays of $B$-mesons, such as the total rate and moments
of various kinematic distributions, by correcting
distributions and rates of semileptonic decays of $b$-quarks with a
limited number
of universal non-perturbative parameters.
These non-perturbative parameters
can be determined from fits to semileptonic decays of $B$-mesons
to charmed final states $B \to X_c e \bar \nu$
\cite{fit1,fit1a,fit2,babar2009,Gambino:2004qm,Gambino:2011cq}
and then used in the description of
$B \to X_ue \bar \nu$ transitions,
facilitating the extraction of the CKM matrix element $|V_{ub}|$
from observables in the latter.
While this procedure is well-defined theoretically, it was not used
in the determination of $|V_{ub}|$ right away because
$B \to X_ue \bar \nu$ transitions suffer from a much larger
$B \to X_ce \bar \nu$ background.
One can place severe cuts on the kinematics
of final state particles to suppress it; for example,
requiring that the hadronic invariant mass is smaller
than the mass of the $D$-meson, $m_D \sim 1.87~{\rm GeV}$, clearly
eliminates the charm background.
However, it was realized early on that such cuts lead
to problems with the convergence of the operator product
expansion and infinitely many terms in the OPE
need to be summed up to obtain reliable results. Such a resummation is
usually
expressed through the so-called shape function \cite{neub,bigi} which
parametrizes the residual motion
of a heavy quark inside a heavy meson. A recent discussion
of $B \to X_u e \bar \nu$ decay in the shape-function
region, that includes next-to-next-to-leading order (NNLO)
QCD effects, can be found in Ref.~\cite{gnp}.
Unfortunately, current uncertainties
in the functional form of both leading and sub-leading shape-functions
are significant and affect a precise determination of $|V_{ub}|$.
In parallel to the studies of the shape function region,
it was suggested that
a combination of cuts
on hadronic and leptonic invariant masses \cite{bauer} allows
one to extend the phase-space coverage in $B \to X_u e \bar \nu$
decays and make the impact of the shape functions
smaller. Measurements that use
these selection criteria
were performed by the BELLE collaboration \cite{belle34}.
Further advances in experimental techniques allowed to achieve
an almost complete phase-space coverage in $B \to X_u e \bar \nu$
decays. Indeed, in recent experimental measurements it was possible
to fully reconstruct
the $B \bar B$ kinematics from their decay products, thereby
allowing to extend selection cuts for the $b \to u$ process
into the charm-rich regions and yet, successfully reject
the $b \to X_c e\bar \nu$ background.
For example, two recent measurements by
BELLE \cite{belle1} and BABAR \cite{bab1}
present partial decay
rates and a variety of kinematic distributions for $b \to u$ transitions
with the cut on the electron energy as low as $E_{l} > 1~{\rm GeV}$.
These cuts are inclusive enough so that the local OPE expansion
can be used with confidence to describe $B \to X_u e \bar \nu$ decays.
We summarize now the status of the theoretical
description of $B \to X_u e \bar \nu $
decays, under the assumption that the local OPE is applicable.
The OPE expansion in the inverse $b$-quark
mass
$m_b$ is well-established for moments of the hadronic invariant mass
and the hadronic energy \cite{ope1,ope2,ope3}.
The leading order term
in the OPE
expansion is given by the partonic $b \to u$ transition.
The total decay rate for $b \to u$ is known in perturbative QCD through
${\cal O}(\alpha_s^2)$ \cite{timo}
and a large number of kinematic distributions and
their moments are known through ${\cal O}(\alpha_s)$
\cite{ural1,jk,cz,li}. Also, the so-called BLM
${\cal O}(\beta_0 \alpha_s^2)$ corrections \cite{Brodsky:1982gc},
that can be derived by considering the
contribution of a massless $q \bar q$ pair
to the $b \to u$ transition,
are known
for the decay rate and main kinematic distributions
\cite{Luke:1994du,hoang,Gambino:2006wk}.
The only kinematic distribution in $b \to u$ decays
that is known beyond the BLM approximation is the
electron-neutrino
invariant mass
distribution,
computed in Ref.~\cite{cz1}.
While the BLM-approximation is known to account for a significant
fraction of the complete ${\cal O}(\alpha_s^2)$ correction, the precision
of current and, especially, forthcoming measurements of $|V_{ub}|$,
the relatively large value of $\alpha_s(m_b)$
and a large variety of kinematic cuts employed in experimental analyses
make it very desirable to compute NNLO QCD corrections
to the fully-differential $b \to u $ decay rate
beyond the BLM approximation. The goal of this paper is to provide
such a computation.
The calculation of NNLO QCD corrections to the $b \to u e \bar \nu $
decay requires three ingredients: {\it i}) two-loop amplitudes for the
$b \to u e \bar \nu$ transition; {\it ii}) one-loop amplitudes for
$b \to ug e\bar \nu$; {\it iii}) tree amplitudes for
$b \to u gge \bar \nu$ and $b \to u q \bar q e \bar \nu$.
The two-loop amplitudes were computed
by several authors in recent years \cite{bon,bell,astr,ben}.
The one-loop amplitudes for $b \to uge\bar \nu$ can be extracted from
the computation reported in Ref.~\cite{Campbell:2005bb}. Finally,
the tree amplitudes for
$b \to u gg e \bar \nu$ and $b \to u q \bar q e \bar \nu$ are straightforward
to calculate and compact results can be obtained by using the spinor-helicity
formalism. These amplitudes can be found in Ref.~\cite{bck}.
The well-known challenge for fully-differential NNLO QCD computations
is to put these different contributions together in a consistent way.
This is not easy to do since individual contributions
exhibit infra-red and collinear
divergences and correspond to processes with different final-state
multiplicities. For the computation reported in this paper, we use
a method proposed in Refs.~\cite{Czakon:2010td,Czakon:2011ve}
(see also \cite{Boughezal:2011jf}) which combines the idea of
sector decomposition \cite{binothheinrich1,binothheinrich2,an1}
with the phase-space partitioning \cite{Frixione:1995ms} in such a way
that singularities are extracted from matrix elements in
a process-independent way. This framework leads to a parton level
integrator which can be used to compute an arbitrary number of kinematic
distributions in a single run of the program.
We have recently given a detailed description
of the relevant computational techniques in a paper ~\cite{bck}
that describes a calculation of NNLO QCD corrections to a related
process $t \to be^+\nu$ and so we do not repeat it here.
Instead, we focus on the illustration of
phenomenological capabilities of the program that are relevant for the
description of $b \to u$ transitions.
\begin{figure}[t]
\centering
\includegraphics[scale=0.55]{q2fullnocuts.eps}
\caption{
The coefficient of the second order correction to the lepton
invariant mass distribution. The solid curve is the analytic result
of Ref.~\cite{cz1}.
}\label{fig1}
\end{figure}
Numerical results reported below are obtained within the standard framework
for perturbative QCD computations. We
employ the on-shell renormalization
for the $b$-quark field and the $b$-quark mass.
The strong coupling constant is
renormalized in the ${\overline {\rm MS}}$-scheme.
We note that we do not include the charm mass dependence when we compute
contributions of additional $q \bar q$ pairs to the decay rate.
As was explicitly shown in Ref.~\cite{bell}, contributions of virtual
charm loops for physical value of $m_c$ can be obtained, with a good
accuracy, from the bottom quark loops by equating charm and
bottom masses. We will use this recipe in what follows.
We write the differential decay
rate for $b \to X_u e \bar \nu $ through NNLO in perturbative QCD as
\be
{\rm d} \Gamma = {\rm d} \Gamma^{(0)}
+ a_s {\rm d} \Gamma^{(1)}
+ a_s^2 {\rm d} \Gamma^{(2)}+
{\cal O}(\alpha_s^3),
\ee
where $a_s = \alpha_s/\pi$ and
$\alpha_s$ is the ${\overline {\rm MS}}$
strong coupling constant at the scale $\mu = m_b$.
By integrating the fully differential decay rate over all the available
phase-space for final state particles, we obtain a prediction for the
${\cal O}(\alpha_s^2)$ correction to the total decay rate. We write
the result of our numerical integration in the following way
\be
\begin{split}
\Gamma^{(2)} = \Gamma^{(0)} & \left (
-29.98(8) + 2.143(7) N_f
\right. \\
& \left. - 0.0243 N_h \right ),
\end{split}
\label{eq1}
\ee
where $N_f=3$ denotes the number of massless quarks in the theory
and $N_h=2$ denotes the number of quarks whose mass coincides
with the $b$-quark mass.
Also, $\Gamma_b^{(0)} = G_F^2 |V_{ub}|^2 m_b^5/(192 \pi^3)$ is the total
decay rate for $b \to u e \bar \nu$ at leading order in perturbative
QCD. Comparing our computation to the analytic results presented
in Ref.~\cite{timo}, we find agreement for each term shown in Eq.(\ref{eq1})
to better than five per mille.
Having reproduced the known result for the NNLO QCD
corrections to the total rate, we can now proceed to the discussion of
kinematic distributions. Our numerical program is set up in such a way
that it can compute various kinematic distributions, both conventional
and cumulative, in a single run. To illustrate this, we show
in Fig.~\ref{fig1} ${\rm d} \Gamma^{(2)}/{\rm d} q^2$, where $q^2$ is
the invariant mass of the lepton pair. The solid curve is the
result of the analytic calculation from Ref.~\cite{cz1}. The numerical
and analytical results perfectly agree for all values of $q^2$ except
in the region $q^2\sim 0$ where some discrepancy is observed. This
discrepancy is not surprising since the
analytic results of Ref.~\cite{cz1} were obtained as
an expansion around $q^2=m_b^2$ so that deviations at small $q^2$ reflect
convergence problems of the analytic computation in that
region.
\begin{figure}[t]
\centering
\includegraphics[scale=0.77]{l00noBLMnocuts.eps}
\caption{
The cumulative histogram that shows $L^{(2)}_{00}$
as a function of the cut on the charged lepton energy
$E_l > E_{\rm cut}$. No cut on the hadronic invariant
mass is applied.
}\label{fig2}
\end{figure}
To further discuss kinematic distributions, we
follow Ref.~\cite{ural1} and define
moments of the partonic invariant mass $M_X^2=(p_b-p_e-p_\nu)^2$
and energy $E_X=E_b-E_e-E_\nu$, in dependence of the lower cut on the
electron energy $E_{\rm cut}$
and the upper cut on the partonic invariant mass $M_{\rm cut}$. More
specifically, we write
\be
L_{ij} = \langle M_X^{2 i} E_X^{j} \theta(E_{e} - E_{\rm cut})
\theta(M_{\rm cut} - M_{X} )
\rangle\label{defL}
\ee
where $\langle ... \rangle$ denotes the normalized
phase-space average for final-state particles in
$b \to X_u e \bar \nu$
\be
\langle \mathcal F \rangle \equiv \frac{1}{\Gamma^{(0)}} \int \mathrm{d} \Gamma \mathcal F.
\ee
We note~\cite{ural1} that one can use $L_{ij}$'s
defined in Eq.(\ref{defL}) to obtain
moments of the lepton invariant mass $q^2$.
We write the moments in Eq.(\ref{defL})
as an expansion in the strong coupling
constant and explicitly separate the BLM corrections
\be
L_{ij} =
L_{ij}^{(0)} + a_s L_{ij}^{(1)}
+ a_s^2 \left (
L_{ij}^{(2), \rm BLM}
+ L_{ij}^{(2)} \right ).
\ee
The BLM correction to these moments is
obtained by computing the contributions of a massless $q \bar q$ pair
$L_{ij}^{(2),n_f}$
and then by rescaling it by the full leading order QCD $\beta$-function
for three massless flavors
\be
L_{ij}^{(2), \rm BLM} = -27/2 \;L_{ij}^{(2),n_f=1}.
\ee
For the numerical calculation of the moments reported below,
we use $m_b = 4.6~{\rm GeV}$ as the value of the $b$-quark pole mass.
To specify partonic
cuts, we introduce the physical hadronic invariant mass
\be
M_H^2 = {\bar \Lambda}^2 + 2 m_b \bar \Lambda E_X + m_b^2 M_X^2,
\label{eq5}
\ee
where ${\bar \Lambda} = m_{B^\pm} - m_b = 0.6769~{\rm GeV}$, for our choice
of the $b$-quark mass. We impose a cut on $M_H$ and translate it
into a cut on $M_X$ and $E_X$ using Eq.(\ref{eq5}). Throughout
the paper, we use $M_H < 2.5~{\rm GeV}$ as the cut on the hadronic
invariant mass. Also, as we already mentioned,
the BLM corrections are well-known. They were
discussed previously in the literature (see e.g. Ref.~\cite{ural1})
and, for this reason, we focus on non-BLM corrections in the remainder
of this paper.
\begin{figure}[t]
\centering
\includegraphics[scale=0.77]{l03noBLMnocuts.eps}
\caption{
The cumulative histogram that shows $L^{(2)}_{03}$
as a function of the cut on the charged lepton energy
$E_l > E_{\rm cut}$. No cut on the hadronic invariant
mass is applied.
}\label{fig3}
\end{figure}
Our results for the
moments are presented
in Figs.~\ref{fig2},\ref{fig3} and in
Table~\ref{table1}.
In Figs.~\ref{fig2},\ref{fig3} we show cumulative
histograms for the non-BLM contributions to two moments
$L_{0j}^{(2)}$ for $j=0$ and $j=3$,
with no cut on hadronic invariant mass. In both cases, the
$x$-axis shows the applied cut on the the charged lepton energy.
These figures illustrate that our numerical program
works as a parton level Monte Carlo integrator and that it can
reliably compute large number of infra-red safe
observables for the $b \to X_ue \bar \nu$
decay with various cuts in a {\it single
run}. This should be useful for further studies of
charmless decays of $B$-mesons given, in particular, a large
number of kinematic cuts employed in experimental
analyses.\footnote{We also note that our numerical program is rather
fast. For example, all numerical results reported in this paper,
including distributions shown in Figs.~\ref{fig1},\ref{fig2},\ref{fig3} and in
Table~\ref{table1} were obtained in an overnight run on a modest-size
computer cluster.}
To illustrate the dependence of the non-BLM corrections on the
applied cuts, we show the
ratio of non-BLM contributions $L^{(2)}$ to NLO ones $L^{(1)}$,
as a function of the electron energy cut
in lower panes of Figs.~\ref{fig2},\ref{fig3}.
We note that this ratio is renormalization-scale independent.
We are interested in this ratio because, if
it is independent of $E_{\rm cut}$, we could have found
the corrections to the moments without fully-differential NNLO computations.
However, it is apparent from Figs.~\ref{fig2},\ref{fig3}
that this is not possible and that
non-BLM corrections have a different functional dependence on
$E_{\rm cut}$ as compared to the NLO ones.
In addition, the cut-dependence is
strongly moment-dependent and it is more pronounced
for higher-$j$ moments.
We will now take a closer look at the numerical values of the computed
corrections. To facilitate this, we show in
Table~\ref{table1} our results for leading, next-to-leading and
next-to-next-to-leading order {\it partonic} moments $L_{ij}$ computed with the
lepton energy cut of $1~{\rm GeV}$ and the hadronic energy cut
$M_H < 2.5~{\rm GeV}$. This set of cuts was previously studied in
Ref.~\cite{ural1}.
\begin{tiny}
\begin{table*}[t]
\vspace{0.1cm}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline\hline
$i$ & j & $L_{ij}^{(0)}$ & $L_{ij}^{(1)}$ & $L_{ij}^{(2,{\rm BLM})}$ &
$L_{ij}^{(2)}$ \\ \hline\hline
$0$ & $0$ & 0.87135 & -2.261(4) & -27.7(1) & 5.1(1) \\ \hline
$0$ & $1$ & 0.29306 & -0.738(2) & -8.13(1) & 1.38(2) \\ \hline
$0$ & $2$ & 0.10789 & -0.2558(8) & -2.55(1) & 0.38(1) \\ \hline
$0$ & $3$ & 0.04210 & -0.0920(4) & -0.815(2) & 0.091(6) \\ \hline
$1$ & $0$ & 0.0 & 0.13110(7) & 2.231(1) & -0.638(3) \\ \hline
$1$ & $1$ & 0.0 & 0.05265(3) & 0.882(1) & -0.256(1) \\ \hline
$1$ & $2$ & 0.0 & 0.02207(2) & 0.365(1) & -0.106(1)\\ \hline
$2$ & $0$ & 0.0 & 4.973(4) $\cdot 10^{-3}$ & 6.83(1) $\cdot 10^{-2}$ & -9.8(1) $\cdot 10^{-3}$ \\ \hline
$2$ & $1$ & 0.0 & 2.144(2) $\cdot 10^{-3}$ & 2.93(1) $\cdot 10^{-2}$ & -4.3(1) $\cdot 10^{-3}$ \\ \hline
$3$ & $0$ & 0.0 & 3.452(7) $\cdot 10^{-4}$ & 4.41(1) $\cdot 10^{-3}$ & -4.9(1) $\cdot 10^{-4}$ \\ \hline \hline
\end{tabular}
\caption{\label{table1} Moments of the partonic invariant mass $M_X^2$ and
the partonic energy $E_X$ with the hadronic invariant mass cut
$M_{H} < 2.5~{\rm GeV}$ and the charged lepton energy
cut $E_l < 1~{\rm GeV}$. See text for details.
}
\vspace{-0.1cm}
\end{center}
\end{table*}
\end{tiny}
It follows from Table~\ref{table1} that similar to the total rate BLM and
non-BLM corrections have opposite size, so that the total result for NNLO
corrections is {\it smaller} than the BLM corrections taken alone.
The non-BLM corrections seem to be more important for lower-moments than
for higher moments. Indeed, the ratio $L_{ij}^{(2)}/L_{ij}^{2,\rm BLM}$
decreases
monotonically by about a factor of $1.6$,
from $0.1842$ to $0.112$, for $i=0$ and $j$ changing from $j=0$ to $j=3$.
This trend is also visible in the absolute magnitude of the corrections.
Taking $\alpha_s(m_b) = 0.24$, we find that for $i=0,j=0$,
the non-BLM corrections increase the moment by about three percent while
for $i=0,j=3$, they become as small as one percent.
While these corrections look small compared to the current
${\cal O}(10\%)$ uncertainty in the
$|V_{ub}|$ determined from inclusive
decays, we note that Ref.~\cite{ural2} estimates
the total theoretical uncertainty
on $|V_{ub}|$ that can be achieved with
various kinematic cuts on $M_H,E_e$, and $q^2$ to be
close to six percent.
The uncertainty in perturbative corrections, which mainly refers
to non-BLM ${\cal O}(\alpha_s^2)$ effects that we discuss
in this paper, is believed \cite{ural2} to be
responsible for $30$ to $50\%$ of the
full theory uncertainty. Our calculation allows to remove this part
of the theory
uncertainty by providing explicit results for non-BLM corrections.
For example, one of the scenarios considered in Ref.~\cite{ural2}
is a high-cut on the lepton energy $E_l > 2~{\rm GeV}$; it corresponds
to the measurement by the BABAR collaboration reported in Ref.~\cite{babar2}.
We find the non-BLM correction to $L_{00}(E_l > 2~{\rm GeV})$
using the cumulative histogram in
Fig.~\ref{fig2} and observe that
it changes the leading order moment
$L_{00}^{(0)}(E_l > 2~{\rm GeV}) = 0.257$
by $6\%$.\footnote{For comparison,
we note that the corresponding BLM corrections
to $L_{00}^{(0)}$ is $-30\%$.}
Since the experimental measurement corresponds
to $|V_{ub}|^2 L_{00}$, a $6\%$ shift in $L_{00}$
due to non-BLM corrections
translates into a $-3 \%$ shift in $V_{ub}$. We stress that
the above number is given to illustrate the magnitude of the
expected effect; a precise statement about the impact of non-BLM
corrections requires a dedicated analysis along the lines
of Ref.~\cite{ural2}. However, it is clear that
our computation should help in removing a
significant fraction of the full theory error
in $|V_{ub}|$ as estimated in \cite{ural2} for the
$E_l > 2~{\rm GeV}$ cut.
We also note that it is customary to consider
{\it normalized} moments, which are defined as
$C_{ij} = L_{ij}/L_{00}$. Since both the numerator and the denominator
in the definition of $C_{ij}$ receive perturbative corrections,
we need to consistently expand $C_{ij}$ in a series in $\alpha_s$
to establish how stable it is against radiative corrections.
We find that in case of $C_{0j}$, the non-BLM corrections
are close to one-fifth of the BLM corrections for all values
of $j$ and they change the normalized
moment by $-0.5\%$ for $j=1$ and by $-1.45\%$ for $j=3$.
The situation changes
dramatically for partonic invariant mass moments
$L_{ij}$, with
$i \ne 0$. In this case the leading order partonic moments vanish since
in the $b \to u e^- \bar \nu$ process the partonic invariant mass is zero.
As the result, for these moments our NNLO calculation is, essentially,
next-to-leading order and the significance of non-BLM corrections increases.
As follows from Table~\ref{table1} for
$L_{ij}$ moments with $i \ne 0$ and $j \ne 0$, the non-BLM
corrections can be as large as $30 \%$.
To conclude, we presented a computation of
${\cal O}(\alpha_s^2)$ corrections to the fully-differential decay rate
of charmless semileptonic $b$ decay,
$b \to X_u e \bar \nu$. Our calculation provides a NNLO QCD
description of
arbitrary infra-red safe observables and
allows arbitrary kinematic cuts
including those that closely match the ones employed in experimental
analyses. We constructed a parton-level
Monte-Carlo integrator which can be used to compute
large number of relevant observables
and kinematic distributions in a single run of the program.
This calculation, together
with earlier results on NNLO QCD
corrections to fully-differential
$b \to c l \bar \nu$ transition \cite{Melnikov:2008qs,Biswas:2009rb},
makes all inclusive semileptonic
decays of $b$-quarks upgraded to that accuracy.
We hope that these results will contribute to the reduction of the
theoretical error on $|V_{ub}|$ and $|V_{cb}|$ that will be achieved
in the forthcoming $B$-physics experiments.
\vspace*{0.2cm}
{\bf Acknowledgments}
We are grateful to P.~Gambino and N.~Uraltsev for comments
on the manuscript.
The research of F.C. and K.M. is supported by US NSF under grant PHY-1214000.
The research of K.M. and M.B.
is partially
supported by Karlsruhe Institute of Technology through a grant provided
by its Distinguished Researcher Fellowship program.
M.B. is partially supported by the DFG through
the SFB/TR~9 ``Computational particle physics''.
Calculations described in this paper were performed at the Homewood
High Performance Computer Cluster at Johns Hopkins University.
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\end{document}