%% slacpub7245: page file slacpub7245004.tcx.
%% section 4 Summary and Discussion [slacpub7245004 in slacpub7245004: slacpub7245004u1]
%%%% latex2techexplorer block:
%% latex2techexplorer page setup:
\iftechexplorer
\setcounter{section}{3}
\fi
\iftechexplorer
\setcounter{secnumdepth}{2}
\setcounter{tocdepth}{2}
\def\thepart#1{}%
\def\thechapter#1{}%
\newcommand{\partLink}[3]{\docLink{#1.tcx}[part::#2]{#3}\\}
\newcommand{\chapterLink}[3]{\docLink{#1.tcx}[chapter::#2]{#3}\\}
\newcommand{\sectionLink}[3]{\docLink{#1.tcx}[section::#2]{#3}\\}
\newcommand{\subsectionLink}[3]{\docLink{#1.tcx}[subsection::#2]{#3}\\}
\newcommand{\subsubsectionLink}[3]{\docLink{#1.tcx}[subsubsection::#2]{#3}\\}
\newcommand{\paragraphLink}[3]{\docLink{#1.tcx}[paragraph::#2]{#3}\\}
\newcommand{\subparagraphLink}[3]{\docLink{#1.tcx}[subparagraph::#2]{#3}\\}
\newcommand{\partInput}{\partLink}
\newcommand{\chapterInput}{\chapterLink}
\newcommand{\sectionInput}{\sectionLink}
\else
\newcommand{\partInput}[3]{\input{#2.tcx}}
\newcommand{\chapterInput}[3]{\input{#2.tcx}}
\newcommand{\sectionInput}[3]{\input{#2.tcx}}
\fi
\newcommand{\subsectionInput}[3]{\input{#2.tcx}}
\newcommand{\subsubsectionInput}[3]{\input{#2.tcx}}
\newcommand{\paragraphInput}[3]{\input{#2.tcx}}
\newcommand{\subparagraphInput}[3]{\input{#2.tcx}}
\aboveTopic{slacpub7245.tcx}%
\previousTopic{slacpub7245003.tcx}%
\nextTopic{slacpub7245004u1.tcx}%
\bibfile{slacpub7245004u1.tcx}%
\newmenu{slacpub7245::context::slacpub7245004}{
\docLink{slacpub7245.tcx}[::Top]{Top}%
\sectionLink{slacpub7245003}{slacpub7245003}{Previous: 3. Extraction of $V_{ub}$ from the Hadron Energy Spectrum}%
\sectionLink{slacpub7245004u1}{slacpub7245004u1}{Next: Bibliography}%
}
%%%% end of latex2techexplorer block.
%%%% code added by add_nav perl script
\docLink{slacpub7245.tcx}[::Top]{Top of Paper}%

\docLink{pseudo:previousTopic}{Previous Section}%
\bigskip%
%%%% end of code added by add_nav
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%% author definitions added by nc_fix
% \renewcommand{\theequation}{\thesection.\arabic{equation}}
% \renewcommand{\theequation}{\thesection.\arabic{equation}}
\newcommand{\epsl}{\varepsilon \hspace{5pt} / }
\newcommand{\rsl}{r \hspace{5pt} / }
\newcommand{\qsl}{q \hspace{5pt} / }
\newcommand{\pbsl}{p \hspace{5pt} / }
\newcommand{\pssl}{p' \hspace{7pt} / }
\newcommand{\hm}{\hat{m}_c^2}
\newcommand{\BR}{\mbox{BR}}
\newcommand{\ra}{\rightarrow}
\newcommand{\BGAMAXS}{B \ra X _{s} + \gamma}
\def\Vcb{V_{cb}}
\def\Vub{V_{ub}}
\def\Vtd{V_{td}}
\def\Vts{V_{ts}}
\def\Vtb{V_{tb}}
\newcommand{\ba}{\begin{array}}
\newcommand{\ea}{\end{array}}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
\def\qb{\bar{q}}
\def\ub{\bar{u}}
\def\db{\bar{d}}
\def\cb{\bar{c}}
\def\sb{\bar{s}}
\def\bra{\langle}
\def\ket{\rangle}
\def\a{\alpha}
\def\b{\beta}
\def\g{\gamma}
\def\d{\delta}
\def\e{\epsilon}
\def\p{\pi}
\def\ve{\varepsilon}
\def\ep{\varepsilon}
\def\et{\eta}
\def\l{\lambda}
\def\m{\mu}
\def\n{\nu}
\def\G{\Gamma}
\def\D{\Delta}
\def\L{\Lambda}
\def\to{\rightarrow}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%% end of definitions added by nc_fix
\section{\usemenu{slacpub7245::context::slacpub7245004}{Summary and Discussion}}\label{section::slacpub7245004}
In this paper we have proposed to measure the hadron energy spectrum
from inclusive semileptonic $B$ decays
in order to extract the CKM matrix element
$V_{ub}$ with improved precision.
The main advantage of our proposal is that
the energy window, in which
the $b \to c \ell^ {\overline \nu}$ transition is kinematically absent
$(E_{had} < m_D)$, is much wider than the
corresponing window
available in the traditional
lepton energy endpoint spectrum analysis.
Even after imposing a relatively high lowercut at
$E_{had}=1$ GeV (in order to avoid
the region of phase space where the range in the invariant
hadronic mass is too narrow to invoke quarkhadron duality),
a much larger fraction of the $b \to u \ell^ {\overline \nu}$
events is captured in the remaining window $1 \,
\mbox{GeV} \le E_{had} \le m_D$
than in the lepton spectrum endpoint analysis.
After calculating the hadron energy spectrum in QCD improved
perturbation theory, we have implemented the
boundstate effects using both
the ACCMM and the HQET approaches; the two methods gave
essentially the same result, in particular
in the relevant kinematical window.
Qualitatively, the dominant bound state effect
is a uniform shift of the parton level spectrum.
We have pointed out that
the theoretical error in the ratio $V_{ub}/V_{cb}$,
which is dominated
by the present uncertainties in
the $m_b$ quark mass, is about a factor of 2 to 3 smaller than the one
in the
lepton endpoint spectrum analysis and about a factor of 2
smaller than the model uncertainties in the
branching fractions of the exclusive decays $B \to
(\pi, \rho, \omega) \, \ell \nu$.
Therefore, for both statistical and theoretical reasons, the
extraction of $V_{ub}$ from the hadron energy spectrum seems
very attractive.
On the experimental side, our
proposal is most suited for a symmetric $B$factory
running at the $\Upsilon(4S)$ resonance, which is
currently available at the
CLEO experiment.
Tagging the events from the
$\Upsilon(4S)$ decay, in which one $B$meson is decaying
semileptonically and the other one nonleptonically,
the energy of the final state hadrons stemming from the semileptonically
decaying $B$meson
is easily obtained by adding up the energies
of all the hadrons in the final state
and then subtracting $m_{\Upsilon}/2$.
On the other hand, in asymmetric $B$ factories, the hadron
energy spectrum is harder to measure because one first
has to reconstruct the
corresponding distribution in the rest frame of $\Upsilon (4S)$;
to perform the corresponding boost,
one has to measure precisely
both the energy and the momentum
of each final state hadron, which requires accurate particle
identification.
The spectrum of the invariant hadronic mass ($d\G/dm_{X}$)
in inclusive semileptonic decays
$B \to X_{c,u} \ell \nu$
is another source from
which one may try to extract $V_{ub}$.
Requiring $m_X$ to be below $m_D$, the
process $B \to X_c \ell \nu$ can be totally supressed.
Consequently, the integral of the hadron invariant mass distribution
below $m_D$ is another observable proportional to $V_{ub}^2$.
As the invariant mass is integrated over a large range,
which covers
about 95 \%
of all the $b \to u \ell^ \overline{\nu}$ events \cite{27},
this observable is also theoretically viable.
However, we believe that the invariant mass spectrum is more
difficult to measure than the hadron energy spectrum proposed in
this paper, because first,
one has to measure the four momenta of all the final state
hadrons from the $\Upsilon(4S)$ decay and second,
one has to find a subset of final state hadrons with an invariant
mass of $m_B$, corresponing to the $B$meson which decays
hadronically. Only then the invariant mass of the semileptonically
decaying $B$meson can be determined.
Finally, we point out a potentially interesting
possibility of a direct measurement of $\alpha_s (m_b)$
or ${\overline \Lambda}$. We have shown that, once the
real gluon bremsstrahlung correction is taken into account,
the kinematic maximum of the hadron energy shifts from the
treelevel (and virtual gluon correction) endpoint $(m_b^2 + m_q^2)/2m_b$
to $m_b$. For the $b \rightarrow u$ case,
Fig. 1 shows that the bremsstrahlung
tail extends approximately 400 MeV beyond $m_b/2$. We also have shown that
the dominant boundstate effect is to shift the spectrum by
${\overline \Lambda}$ for both ACCMM and HQET approaches. Therefore, once
${\overline \Lambda}$ is known accurately, then $\alpha_s (m_b)$ can be
extracted directly from the bremsstrahlung tail spectrum.
Alternatively, given an accurate value of $\alpha_s (m_b)$,
the nonperturbative parameter $\overline \Lambda$ can be
accurately measured.
Since the bremsstrahlung spectrum of $b \rightarrow c$ extends further out
than that of $b \rightarrow u$, the above proposal should be most suited
for the $b \to c$ transition.
In this case, one has to take into account the finite $c$quark mass
dependence of the perturbative QCD corrections and boundstate effects.
\vskip0.5cm
We thank A. Ali, S.J. Brodsky, B. Grinstein, S.K. Kim,
J.H. K\"uhn, Y.J. Kwon, M. Lu,
P. Minkowski,
R. Poling, E. Thorndike, and R. Wang for helpful discussions.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% code added by add_nav perl script
\bigskip%
\docLink{pseudo:nextTopic}{Next Section}%

\docLink{slacpub7245004u1.tcx}[::Bottom]{Bottom of Paper}%
%%%% end of code added by add_nav