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%% section 3 Extraction of $V_{ub}$ from the Hadron Energy Spectrum [slacpub7245003 in slacpub7245003: slacpub7245004]
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\section{\usemenu{slacpub7245::context::slacpub7245003}{Extraction of $V_{ub}$ from the Hadron Energy Spectrum}}\label{section::slacpub7245003}
As discussed in the Introduction,
theoretical predictions for the hadron energy spectrum
are expected to be rather reliable in the relevant kinematical region
below the $D$meson mass, once a sufficiently high lowercut in the
hadron energy is made at the same time.
Choosing the lower cut at 1 GeV, the invariant hadronic mass
ranges from $m_\pi$ up to $E_{had} \ge 1 \, \mbox{GeV}$ for all values
of the hadronic energy in the window
$1 \, \mbox{GeV} \le E_{had} \le m_D$.
In this case a large number of different
hadronic final states contributes
to the spectrum; hence, quarkhadron duality should hold quite well.
We therefore propose to measure experimentally the
kinematical branching ratio defined by
\be
\label{brkin}
BR^{kin} ( B \to X_u \ell \nu) =
\int_{1 \, GeV}^{m_D} \, \frac{dBR(B \to X_u \ell \nu)}{
dE_{had}} \, dE_{had} \quad .
\ee
A comparison with the corresponding theoretical quantity then allows
to extract the CKM ratio $V_{ub}/V_{cb}$. While leaving aside
the experimental question how accurately the kinematical branching
ratio can be measured now or in the future, we would like to point out
in the following that the theoretical uncertainties in this
observable are small enough to reduce substantially the present
theoretical error on $V_{ub}$.
As it turns out that the theoretical error of the kinematical
branching ratio is dominated by the uncertainty of the
effective $b$ quark mass (i.e., $m_b$ in the HQET approach and
$\langle m_b \rangle$ in the ACCMM approach),
we take into account only this effect in the following analysis
\footnote{The dependence of the kinematical branching ratio
on the numerical value
of $\a_s$ is very small. In our numerical evaluations we have taken
$\a_s=0.205$.}.
A reasonable range for the value of $m_b$ can be inferred from the
measurement of the lepton energy spectrum in the inclusive decays
$B \to X_{c,u} \ell \nu$. Fitting to the ACCMM model,
CLEO extracted $p_F$ to be $264 \pm 16$ MeV (using a value of
$m_{sp}=150$ MeV for the mass of the spectator quark)
\cite{2,21}. In a more recent
analysis using lepton tags \cite{22}, they extracted the value
$p_F=347 \pm 68$ MeV. We thus choose to vary $p_F$ in the somewhat
larger range $200 \, \mbox{MeV} \le p_F \le 435 \, \mbox{MeV}$
which covers both measurements. Using
Eq.~(\docLink{slacpub7245002.tcx}[mbeff]{22}), this $p_F$range translates into the
$\langle m_b \rangle$range
\be
\label{mbrange}
4.75 \, \mbox{GeV} \le \langle m_b \rangle \le 5 \, \mbox{GeV}.
\ee
We note that this range is also compatible with the best value
($\langle m_b \rangle$=4.77 GeV)
fitted from the photon energy spectrum in $B \to X_s \gamma$
\cite{23}.
So far, we have not discussed the question,
if the cascade decay $b \to c X \to s \ell \nu X$ , where
the symbol $X$ denotes light quarks, can fake a $b \to u \ell \nu$
transition. If the $c$quark is offshell, this process can
kinematically fake a $b \to u \ell \nu$ transition, but
in this case the process is higher order in the weak interaction,
hence
negligible. On the other hand, if the $c$ quark is onshell,
the energy of the hadronic system $X$ is larger than $m_D$.
Therefore, this cascade process is not a backgound to the
$b \to u \ell \nu$ transition;
this is in contrast to the endpoint analysis of the lepton
energy spectrum, where the cascade process has to be subtracted out.
To get an idea what fraction $R$ of the total semileptonic
$b \to u$ events will be captured in the energy window
$1 \, \mbox{GeV} \le E_{had} \le m_D$, we
have used the present central
value for $V_{ub}/V_{cb}=0.08$ \cite{24}.
The $m_b$ dependence of the kinematical branching ratio and of the
fraction $R$ are shown in Table 1 for both the ACCMM and the HQET
approaches.
\begin{table}[htb]
\label{tabelle}
\begin{center}
\begin{tabular}{ c  c  c  c  c  }
\hline
$m_b$ (GeV) & $BR^{kin}_{ACCMM}*10^4$ & $R_{ACCMM}$ (\%)
& $BR^{kin}_{HQET}*10^4$
& $R_{HQET} (\%)$\\
\hline \hline
4.60 & $2.41$ & $23$ & $2.17$ & $21$ \\
4.65 & $2.58$ & $24$ & $2.40$ & $22$ \\
4.70 & $2.76$ & $25$ & $2.64$ & $24$ \\
4.75 & $2.94$ & $26$ & $2.89$ & $25$ \\
4.80 & $3.15$ & $27$ & $3.14$ & $27$ \\
4.85 & $3.37$ & $28$ & $3.41$ & $28$ \\
4.90 & $3.59$ & $29$ & $3.68$ & $29$ \\
4.95 & $3.83$ & $30$ & $3.95$ & $31$ \\
5.00 & $4.07$ & $31$ & $4.23$ & $32$ \\
\hline
\end{tabular}
\end{center}
\caption[]{The kinematical branching ratio $BR^{kin}$ defined
in Eq.~(\docLink{slacpub7245003.tcx}[brkin]{34}) and the corresponding fraction $R$
of the semileptonic $b \to u$ events lying in the energy
window $1 \, \mbox{GeV} \le E_{had} \le m_D$ are given as a function
of $m_b$ for the ACCMM and the HQET approach.
$V_{ub}/V_{cb}=0.08$ is assumed.
\label{table1}}
\end{table}
Varying $m_b$ in the range specified in Eq. (\docLink{slacpub7245003.tcx}[mbrange]{35})
and using the results
for $BR^{kin}_{HQET}$ in Table 1 (where the variation is
somewhat larger than in the ACCMM model), we obtain
\be
\label{predtheory}
BR^{kin} = \frac{V_{ub}^2}{V_{cb}^2} \, \times
(4.51  6.61) \, \times 10^{2} \quad .
\ee
Denoting the measured kinematical branching ratio by $BR_{exp}^{kin}$,
one can extract $V_{ub}/V_{cb}$ to be
\be
\label{vubres}
\frac{V_{ub}}{V_{cb}} = \sqrt{BR^{kin}_{exp}} \, \cdot
\, (4.30 \pm 0.41) \quad .
\ee
This implies that the theoretical error of the ratio
$V_{ub}/V_{cb}$ is approximately $\pm 10\%$.
Taking the somewhat larger range $m_b=(4.80 \pm 0.15)$ GeV
adopted in~\cite{1,25},
we get from Table 1
\be
\label{vubres1}
\frac{V_{ub}}{V_{cb}} = \sqrt{BR^{kin}_{exp}} \, \cdot
\, (4.60 \pm 0.56) \quad ,
\ee
which implies a theoretical error of about 12 \%.
To illustrate that our proposal has the potential to
lead to a more precise
determination of $V_{ub}$, it is instructive to briefly review the
present situation. The traditional
inclusive lepton endpoint spectrum analysis done at
CLEO \cite{2} and at ARGUS \cite{3}, leads to
the result
$V_{ub}/V_{cb}=0.08 \pm 0.03$,
where the error is dominated by theory \cite{26}.
A recent new input to this quantity is provided by the measurements
of the exclusive semileptonic decays
$ B \to (\pi,\rho,\omega) \, \ell \nu$ \cite{4,5}.
The value for $V_{ub}$
quoted in the most recent analysis \cite{5}
is $V_{ub}=(3.3 \pm 0.2^{+0.3}_{0.4} \pm 0.7) \times 10^{3}$, where
the errors are statistical, systematic and estimated model dependence,
after excluding models which are unable to predict the correct
$\pi/\rho$ ratio.
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