\vspace {-3mm}

\section{Studies of selected $B$ decay channels}
\label{sec:B:channels}

In this section we present new studies on a selected set of
$B$ meson decay channels,
updating the determination of the following processes: \\
\par\noindent
{{\it The CKM matrix element $|V_{ub}|$.} This measurement, crucial to the
model-independent determination of the CKM matrix,
can only be done at an \epem  machine. We update the calculation of the \superb\ reach, as
suggested by the International Review Committee.\\
\par\noindent
%{{\it The branching fractions  \hbox{${\cal B}(B \to X_s \gamma)$ and ${\cal B}(B \to X_s \ell^+ \ell^-)$.}} }
{{\it The branching fractions  \hbox{${\cal B}(B \to X_s \gamma)$, ${\cal B}(B \to X_s \ell^+ \ell^-)$.}} }
These channels were not thoroughly studied in the CDR, as they are limited by experimental
and theoretical systematic uncertainties. In the CDR we concentrated on
other observables, such as the photon polarization and $\CP$ and isospin asymmetries.
However ${\cal B}(B \to X_s \gamma)$, at present  one of the most powerful New Physics probes,
remains a powerful constraint, even in the Minimal Flavour Violation case. We have therefore reassessed
the experimental and theoretical sensitivities for these modes at \superb. We have also
done a preliminary sensitivity study for $B \to K^{(*)} \tau^+ \tau^-$.\\
\par\noindent
{{\it  The branching fraction ${\cal B}(B \to X_s \nu \overline{\nu})$.}} A new detailed
study has been performed on this mode, evaluating the possibility of measuring the
branching fraction with the full \superb\  data sample. This information complements
the measurements of $B \to X_s \gamma$ and $B \to X_s \ell^+ \ell^-$ in accessing
New Physics that can contribute to $\Delta B=1$ box, photon penguin, and $Z^0$ penguin diagrams.\\
\par\noindent
{{\it  Leptonic decay modes.}} The precise measurement of ${\cal B}(B \to \ell \nu$)
is particularly interesting in New Physics scenarios with a charged
Higgs at high tan$\beta$. Following the suggestion of the IRC, we discuss
possible improvements in signal efficiency and systematic uncertainties at \superb.
We also present a new study of radiative leptonic decays and some discuss considerations
relevant to LFV modes.\\

\noindent
\subsection*{Precise determination of the CKM element $|V_{ub}|$}
\label{sec:B:channels:Vub}

The precise measurement of $|V_{ub}|$ is a crucial ingredient in
the determination of the CKM parameters $\bar{\rho}$ and $\bar{\eta}$
in the presence of New Physics. At the time \superb\ commences operation,
LHCb will have already provided precise measurements of $\sin2\beta$ and
$\gamma$. This will allow for an improved determination of CKM parameters within the
Standard Model.  However, in the presence of generic New Physics contributions,
this information alone is not sufficient to obtain the same precision. As precise information
on CKM parameters is essential for any New Physics flavor analysis
in the $K$ and $B$ sectors, an improved determination of $|V_{ub}|$ turns out to be quite important
in New Physics searches.

%All the available experimental data exclude the possibility of sizable
%contributions to tree-level Standard Model processes, so that extensions of the Standard Model
%in which New Physics enters low-energy processes at the tree level are strongly
%disfavoured.
%Many of the measurements used for the Standard Model determination of
%$\bar{\rho}$--$\bar{\eta}$ can be affected by the presence of New Physics at the
%loop level. Unambiguous New Physics searches require a determination
%of $\bar{\rho}$ and $\bar{\eta}$ in the presence of arbitrary New Physics
%contributions. It is straightforward to generalize the Standard Model
%analysis including generic New Physics effects in $\Delta F = 2$ processes.
%
%The processes which are mediated only by tree-level $B$ decays
%plays a special role, since they strongly contribute to determine
%$\bar \rho$ and $\bar \eta$ parameters independently of New Physics contributions.
%There are two powerful constraints: the measurement of the CKM elements
%$V_{ub}$ and $V_{cb}$ determined from semileptonic $B$ decays,
%and the angle $\gamma$ obtained by measuring the phase of $V_{ub}$ appearing
%in the interference between $b \to c$ and $b \to u$ transitions to $DK$ final
%states.
%% \footnote{
%%   We neglect possible New Physics contributions to $D^0$--$\bar D^0$ mixing,
%%   since their contribution is expected to be well below the present
%%   experimental accuracy~\cite{ddbar1}. In the future, it might become
%%   necessary to take them into account following Ref.~\cite{ddbar2}.
%% }

The precise study of both inclusive and exclusive $B$
semileptonic branching fractions is a unique feature of \superb. \\
%The question we address here is: ``Could $|V_{ub}|$ be determined with a
%relative precision of $3\%$ or better ?''
\par\noindent
{{\it Inclusive decays}}\\
\vskip -9pt
\noindent
The current $5$--$10\%$ theoretical error on the inclusive determination
of $|V_{ub}|$ is due mainly to uncertainties in the $b$ quark mass,
in weak annihilation (WA) contributions, in missing higher order perturbative
corrections, and in the modeling of the shape functions.

At the time \superb\ takes data, new
calculations should decrease the perturbative error,
and  lattice calculations, together with
improved analyses of $e^+e^- \to \ {\rm hadrons}$ and
measurements of the moments of semileptonic and radiative $B$ decay spectra should
provide better determinations of $m_b$;
a precision of $20 \ {\rm MeV}$ on $m_b$ is possible.
Weak annihilation contributions are relevant only at high $q^2$, and
can be efficiently constrained by studying the $q^2$ spectrum.
The shape functions can also be better-constrained by studies of the
$B \to X_u \ell \nu$ spectra, but their importance will
decrease as the measurements become increasingly more inclusive.
A pioneering analysis in this area has
recently been published by \babar~\cite{Aubert:2006qi}.
In this analysis the $M_X$ cut is raised to values for which
the shape function sensitivity becomes negligible.
Such measurements are not competitive now,
but the situation will be quite different at \superb.

As a result, we expect the theoretical uncertainty on the inclusive
determination of $|V_{ub}|$ to eventually be dominated by the uncertainty in the $b$ quark mass.
In this respect, it should be stressed that in current analyses, $|V_{ub}|$
depends quite strongly on the precise value of $m_b$.
Typically, for a cut of $M_X < 1.7 \ {\rm GeV}$,
the relative error on $V_{ub}$ scales as  $4(\delta m_b)/m_b$.
Currently, with $\delta m_b = 40 \ {\rm MeV}$, the error induced on $V_{ub}$ is about $3.5\%$.
If the error on $m_b$ were halved, $|V_{ub}|$ extracted in this way would
have a parametric uncertainty below $2\%$.
However, the presence of the $M_X$ cut increases the sensitivity to $m_b$,
because the distribution functions also strongly depend on $m_b$.
Increasing the $M_X$ cut (as mentioned above) reduces the sensitivity to $m_b$.
Indeed, the total rate is proportional to $m_b^5$,
and for a totally inclusive measurement one has
$\delta V_{ub}/V_{ub} \simeq 2.5 (\delta m_b)/m_b$.
Therefore, if one could measure the total $B \to X_u \ell \nu$ rate,
the uncertainty induced by $\delta m_b = 20 \ {\rm MeV}$ on $|V_{ub}|$
would be only $1\%$.

A promising way to deal with the large $B \to X_c \ell \nu$ backgrounds with no cut
on the inclusive $B \to X_u \ell \nu$ decays phase space is to reconstruct the semileptonic decays in the recoil
against the other  $B$ fully reconstructed in a hadronic final state in $e^+e^- \to \FourS \to B\bar{B}$
events (the so-called ``hadronic tag technique'').
This technique provides full knowledge of the event, including the flavor of the $B$,
and allows the precise reconstruction of the neutrino four-momentum,
significantly improving background rejection against, for example.
events with several neutrinos or with one or more $K_L$ mesons.
At present, these measurements
% using $m_X$ to extract the signal
are limited by low signal efficiency, and have large statistical uncertainty.
% and to the dilution due to a S/B of the order of 0.1.
At \superb\, however, the statistical uncertainty will be less than $\sim1\%$.
The leading systematic errors will also be reduced: those due to detector effects
could reach 2$\%$ using the large data control samples available.
The current analyses have uncertainties due to $B \to X_c \ell \nu$ background
(branching fractions and form factors) as low as $4\%$ --
it is possible to reduce this by a factor of two.
Indeed, higher statistics and improvements in the detector and analysis will yield better measurements of these quantities.
Moreover, the enhanced hermeticity and superior vertexing capability of the \superb\ detector
will further improve background rejection through more precise neutrino
reconstruction and the detection of the displaced $D$ meson vertex.
A total experimental uncertainty on $|V_{ub}|$ of approximately $2$--$3\%$
can thus be achieved with this method.

Combined with the theoretical uncertainty discussed above,
an overall precision of $3\%$ on the determination of $|V_{ub}|$
using inclusive $B \to X_u \ell \nu$ decays at \superb\ will be possible.\\
\par\noindent
{{\it Exclusive decays}}\\
\vskip -9 pt
\noindent
The measurement of $|V_{ub}|$ using exclusive decays
% (primarily $B \to \pi \ell \nu$)
is presently limited by theoretical
uncertainties on the form factors (about $12\%$).
Lattice calculations are expected to improve significantly
in the next five years, mainly due to an increase of available computing power.
Results from these calculations will decrease the uncertainty to approximately
2--3\% in the case of the most promising decay $B \to \pi \ell \nu$ (see
the Appendix of the CDR~\cite{Bona:2007qt}).

Using the hadronic tag approach (as in Refs.~\cite{Aubert:2006ry,Abe:2006gb}),
the statistical uncertainty on $|V_{ub}|$ will be below $1\%$.
This measurement being almost background-free, the systematic uncertainties
are dominated by detector effects, and should be of the order of $2\%$.
A total experimental uncertainty of 2--3\% on $|V_{ub}|$ can thus be achieved,
leading to an overall precision as good as 3--4\%.

These figures basically confirm the sensitivities presented in the CDR
for the measurement of $|V_{ub}|$.

\noindent
\subsection*{Rare radiative decays}
\label{sec:B:channels:rare}

%The crucial problem in the new physics search within flavor physics is
%the optimal separation of new physics effects and hadronic uncertainties.
%This can be successfully solved only for a selected number of
%{\it golden} observables in flavor physics,
%where hadronic physics can be disentangled to a large extent
%and clean tests of the Standard Model  are possible.
%The golden channels include inclusive decay modes which are
%dominated by the partonic perturbative contributions.
%Certain ratios of exclusive decay modes such as asymmetries
%also have relatively small theoretical errors,
%as large parts of the hadronic uncertainties cancel out.

%Among the flavor-changing neutral current processes, the inclusive
%$b \to s \gamma$ and $b \to s \ell^+ \ell^-$ modes remain the most prominent.
%Precise studies of $b \to s \nu \bar{\nu}$ decays are also increasingly important.
%The stringent bounds on various non-standard scenarios that can be obtained
%from these  modes provide a clear example of the importance of clean FCNC
%observables in discriminating New Physics models, as discussed further in the next section. \\


\subsubsection*{The branching fraction ${\cal B}(B \to X_s \gamma)$}

% {\it experimental considerations}.\\

%The radiative penguin decay, $B\to X_s \gamma$, has played
%an important role in constraining New Physics models.  In the Standard Model, this flavor
%changing neutral current decay occurs only at the loop level, which
%makes it a good hunting ground for effects coming from New Physics:  new heavy
%particles may contribute in the virtual loop, modifying the Standard Model decay
%rate or, possibly, the CP asymmetry. Inclusive measurements are the
%most powerful in searching for New Physics, since the theoretical prediction
%for the inclusive rate is much more precise than for rates of
%exclusive decays.

The inclusive branching fraction ${\cal B}(B\to X_s \gamma)$ has been
measured at the $B$ factories
~\cite{Chen:2001fja,Abe:2001hk,Koppenburg:2004fz,Aubert:2005cu,Aubert:2006gg};
the current experimental world average is~\cite{Barberio:2006bi}:
\[
%\begin{equation}
  \label{eq:btosgamma:HFAG}
    {\cal B}(B\rightarrow X_s \gamma)|_{E_\gamma >1.6 \ \rm{GeV}} =
  (3.55 \pm 0.26) \times 10^{-4}\,.
%\end{equation}
\]
The $7\%$ error on the branching fraction is a mixture of statistical,
systematic and theoretical contributions, where the latter comes
primarily from extrapolating the partial branching fraction,
typically measured for photon energies above $1.9 \ {\rm GeV}$, down to
the value of $1.6 \ {\rm GeV}$ used for the theoretical prediction.

Several different experimental approaches have been pursued to make a
measurement of the inclusive $B\rightarrow X_s \gamma$ branching fraction.
The approach that yields the most precise measurement depends on
the available statistics.
Untagged inclusive analyses, in which only the high-momentum photon is
reconstructed, have been carried out at $B$ factories, but are limited by
systematic errors that will make them uncompetitive in the \superb\ era.
Similarly, the semi-inclusive approach, which attempts to reconstruct as
many exclusive modes as possible, and then applies a correction
due to the missing rate, is already limited by the $X_s$ fragmentation
properties, {\it i.e.}, by uncertainty in the estimate of the fraction of the
total rate that is not reconstructed.
This systematic uncertainty amounts to about $15\%$
on the branching ratio~\cite{Aubert:2005cu}. More detailed studies are
needed to evaluate how much this systematic could be reduced with the
statistics available at \superb.
%and appears unlikely to be
%significantly reduced in the near future.

The most promising approaches for \superb\ are those that make use of
recoil analysis, in which the ``other $B$'' in the $B\bar B$ event is
tagged in either a semileptonic or hadronic decay.
This allows backgrounds to be reduced to acceptable levels
without putting constraints on the $X_s$ system.
The most recent semileptonic tag analysis~\cite{Aubert:2006gg} currently has
comparable statistical and systematic uncertainties (about $8\%$ each), but a
sizable portion of the systematic uncertainty is actually statistical in
nature, since it depends on the size of control samples derived from the data.
The current systematic uncertainty of the hadronic tagged
analysis~\cite{Aubert:2007my} is larger, but it seems probable that
refinements to this relatively new technique will be able significantly to reduce the
systematic error.

With the data sample of \superb, all approaches will
be systematics-limited.
We estimate that the hadronic and semileptonic tagged analyses
will be able to reduce systematic uncertainties to about $4$--$5\%$.
Since the systematics are mostly uncorrelated, the combined branching fraction
can be expected to have a systematic error of around $3\%$.\\

The Standard Model prediction of ${\cal B}(B\to X_s \gamma$) for $E_{\gamma} > 1.6 \ {\rm GeV}$ is
%\begin{equation}
\[
  \label{eq:btosgamma:NNLL}
  {\cal B}( B \to X_s\gamma)|_{E_\gamma >1.6 \ \rm{GeV}} =
  \left\{
    \begin{array}{ll}
%      (3.15  \pm 0.23) \times 10^{-4}\, & \cite{Misiak:2006zs} \\
      (3.15  \pm 0.23) \times 10^{-4}   & \cite{Misiak:2006zs} \\
      (2.98  \pm 0.26) \times 10^{-4}   & \cite{Becher:2006pu}.
    \end{array}
  \right.
\]
%\end{equation}
The two predictions differ in their use of resummation of log-enhanced terms which are included
in the result of \cite{Becher:2006pu}. There is no consensus on the consistency of the resummed result~\cite{misiaktalk}. We therefore quote both predictions pending clarification.
For both results, the overall uncertainty consists of non-perturbative ($5\%$),
parametric ($3\%$), higher-order ($3\%$) and $m_c$-interpolation ($3\%$), which
have been added in quadrature.

There are other perturbative NNLL corrections that are not yet included
in the present NNLL estimate, but are expected to be smaller than the current
uncertainty, producing a shift of the central value of about 1.6$\%$.
% \cite{ref:misiaktalk}.

While the uncertainties due to the input parameters and due to the
$m_c$~interpolation could be further reduced, the perturbative error
of $3\%$ will remain until a new major effort to compute the NNNLO
is carried out.
However, the theoretical prediction has now reached the non-perturbative
boundaries. The largest uncertainty is presently due to nonperturbative
corrections that scale with $\alpha_s \Lambda_{\rm QCD}/m_b$.
A local expansion is not possible for these contributions;
it is not clear if the corresponding uncertainty of $5\%$
(based on a simple dimensional estimate) can be reduced.
Recently, a specific piece of these additional nonperturbative corrections
has been estimated~\cite{Lee:2006wn}, and found to be consistent with the
dimensional estimate. It is also included in the prediction of Ref.~\cite{Becher:2006pu}.\\

Two explicit examples should demonstrate the stringent constraints that
can, with these uncertainties, be derived from the measurement of the $B \rightarrow X_s \gamma$
branching fractions.

Fig.~\ref{fig:MHc} shows the dependence of ${\cal B}(B \to X_s \gamma)$ on the
charged Higgs mass in the 2-Higgs-doublet model (2HDM-II) ~\cite{Misiak:2006zs}.
\begin{figure}[!htb]
  \begin{center}
    \includegraphics[width=8cm,angle=0]{MHc.ps}
  \end{center}
  \caption{
    ${\cal B}(B \to X_s \gamma) \times 10^{-4}$ as a function of the charged
    Higgs boson mass $M_{H^{+}}$ (GeV) in the 2HDM~II for $\tan\beta=2$ (solid lines).
    Dashed and dotted lines show the Standard Model and experimental results,
    respectively.
    \label{fig:MHc}
  }
\end{figure}
The bound on $M_{H^{+}}$ = 295 GeV at 95$\%$~CL, shown in Fig.~\ref{fig:MHc}, is currently
the strongest available lower limit on the charged Higgs mass.

Similarly, the bound on the inverse compactification radius of the minimal
universal extra dimension model (mACD) derived from ${\cal B}(B \rightarrow X_s \gamma)$~\cite{Haisch}
is $1/R > 600$ GeV at $95\%$ confidence level, as shown in Fig.~\ref{fig:ACDLO}.

\begin{figure}[!htb]
\begin{center}
\vspace{2mm}
\makebox{
\begin{psfrags}
\newcommand{\psfragtextscale}{1}
% -- content from the Prologue option --
% The `auto-ignore' makes arXiv.org ignore this file:
%auto-ignore
% This file replaces labels in an EPS graphic by LaTeX macros,
% i.e. it should be accompanied by an EPS file.
\providecommand{\psfragtextscale}{1}
\providecommand{\psfragmathscale}{\psfragtextscale}
\providecommand{\psfragnumericscale}{\psfragtextscale}
\providecommand{\psfragtextstyle}{}
\providecommand{\psfragmathstyle}{}
\providecommand{\psfragnumericstyle}{}
% -- end of Prologue option --

% psfrag replacement commands
\psfrag{x}[cc][cc][1.1][0]{$1/R~[{\rm TeV}]$}
\psfrag{y}[bc][bc][1.1][0]{${\cal B} (\bar{B} \to X_s \gamma)~[10^{-4}]$}
\makebox{\hspace{-1.1cm} \includegraphics[width=3.5in,height=2.25in]{ACD_LO.eps}}
\end{psfrags}
}
\end{center}
\vspace{-4mm}
\caption{Branching fraction for $E_0 = 1.6 \, GeV$ as a function of $1/R$.
  The red (dark gray) band corresponds to the LO mUED result.
  The $68 \%$ CL range and central value of the
  experimental/Standard Model result is indicated by the yellow/green
  (light/medium gray) band underlying the straight solid line.}
\label{fig:ACDLO}
\end{figure}

\subsubsection*{$B \rightarrow X_s \ell^+ \ell^-$ decay modes}

% {\it theoretical considerations}.\\

The decay $B \rightarrow X_s \ell^+\ell^-$ is particularly important to the
\superb\ physics programme, due to the sensitivity to New Physics effects on
kinematic observables, such as the dilepton invariant  mass spectrum and the
forward--backward asymmetry $A_{\rm FB}$.
%The complete analysis of these decays provides potential to detect
%subtle patterns and to distinguish between various scenarios beyond the Standard Model.
% In the indirect search for new physics, it is also mandatory to go beyond the
% analysis of branching ratios and to measure more complex kinematical
% distributions such as  CP, forward-backward, and isospin asymmetries to detect
% subtle patterns and to distinguish between various scenarios beyond the Standard Model.

In the $B \rightarrow X_s \ell^+\ell^-$ system, one has to remove contributions from $c \bar c$
resonances that appear as large peaks in the dilepton invariant mass spectrum, by appropriate  kinematic cuts.
%the $B \rightarrow X_s \ell^+ \ell^-$ observables
%are dominated by perturbative contributions.
It is conventional to define ``perturbative windows'' with $s=q^2/m_b^2$ away from charmonium resonances,
namely the low dilepton-mass region $ 1 \ {\rm GeV} < q^2 < 6 \ {\rm GeV}$
and the high dilepton-mass region with $q^2 > 14.4 \ {\rm GeV}$.
In these windows theoretical predictions for the invariant mass spectrum
are dominated by the perturbative contributions;
a theoretical precision of order $10\%$ is, in principle, possible.

%Recent calculations of the NNLL QCD contributions~\cite{Asatryan:2001zw,Ghinculov:2002pe,Ghinculov:2003qd,Asatrian:2002va,Bobeth:1999mk,Bobeth:2003at}
%and also of subleading contributions  like  electromagnetic
%corrections~\cite{Bobeth:2003at,Huber:2005ig,Huber:2007vv}
%and non-perturbative corrections scaling with $1/m_b^2$ or
%$1/m_c^2$~\cite{Falk:1993dh,Ali:1996bm,Buchalla:1997ky,Chen:1997dj,Buchalla:1998mt,Bauer:1999kf,Ligeti:2007sn}
%have significantly improved the sensitivity of the inclusive
%$B \rightarrow X_s \ell^+ \ell^-$ decay in  testing extensions of the Standard Model.
%In particular
In the following, we collect the most accurate predictions for
observables in $B \rightarrow X_s \mu^+ \mu^-$ decay. Formulae for
the electron case should be modified to take into account
the experimental resolution for collinear photons.

The value of the dilepton invariant mass $q^2_0$,
for which the differential asymmetry $A_{\rm  FB}$ vanishes,
is one of the most precise predictions in flavor physics,
with a theoretical uncertainty of order $5\%$~\cite{Huber:2007vv}:
\begin{eqnarray}
(q_0^2)_{\mu\mu} & = &
\Big[ 3.50 \pm 0.10_{\rm scale} \pm 0.002_{m_t} \pm 0.04_{m_c,C}\nnb \\
&& \hspace*{-15pt}
\pm 0.05_{m_b} \pm 0.03_{\alpha_s(M_Z)} \pm 0.001_{\lambda_1} \pm 0.01_{\lambda_2}
\Big] \, \gev^2 \nnb \\
& = & ( 3.50 \pm 0.12) \, \gev^2\,. \label{muonzero}
\end{eqnarray}
This accuracy cannot be reached with the analogous exclusive
observable in $B \rightarrow K^* \ell^+\ell^-$, due to the unknown
$\Lambda_{\rm QCD}/m_b$ corrections.

The latest update of the dilepton mass spectrum, integrated
over the low and the high dilepton invariant mass region in the muonic case,
leads respectively to~\cite{Huber:2005ig}:
\bea
{\cal B}_{\mu\mu}^{\rm low} & = &
\Big[ 1.59 \pm 0.08_{\rm scale} \pm 0.06_{m_t} \pm 0.024_{ C,m_c } \nnb \\
&& \hspace*{-40pt}
\pm 0.015_{m_b} \pm 0.02_{\alpha_s(M_Z)} \pm 0.015_{\rm CKM} \pm 0.026_{{\rm BR}_{sl}}
\Big] \times 10^{-6} \nnb \\
& = & (  1.59  \pm 0.11 ) \times 10^{-6} \, \label{muonBRlow}
\eea

and
%In the high-${s}$ region, one encounters the breakdown of the heavy-mass
%expansion at the endpoint.
%However, for the integrated quantity an effective expansion in inverse powers of
%$m_b^{\rm eff} = m_b \times (1 - \sqrt{s_{\rm min}})$
%(depending on the lower dilepton-mass cut $s_{\rm min}$
%exists, albeit with large uncertainties due to poorly known
%${\cal O}(1/m_b^3)$ power corrections~\cite{Huber:2007vv}:

\bea
{\cal B}_{\mu\mu}^{\rm high} & = &
2.40 \times 10^{-7} \Big(
1 + \left[ {}^{+0.01}_{-0.02} \right]_{\mu_0} + \left[ {}^{+0.14}_{-0.06} \right]_{\mu_b}
\pm 0.02_{m_t} \nnb \\
&& \hspace*{-25pt}
+ \left[ {}^{+0.006}_{-0.003} \right]_{{C,m_c}} \pm 0.05_{m_b} + \left[ {}^{+0.0002}_{-0.001} \right]_{\alpha_s} \nnb \pm 0.002_{\rm CKM} \\
&& \hspace*{-25pt}
\pm 0.02_{{\rm BR}_{sl}} \pm 0.05_{\lambda_2} \pm 0.19_{\rho_1} \pm 0.14_{f_s} \pm 0.02_{f_u} \Big)  \nnb \\
&=& 2.40 \times 10^{-7} \; (1\,^{+0.29}_{-0.26} ) \,. \label{muonBR}
\eea

In the high ${s}$ region, the uncertainties are larger, due to the breakdown of
the heavy-mass expansion at the endpoint. However the uncertainties can be
significantly reduced by considering quantities normalized to the semileptonic
$b\to u \ell \nu$ rate integrated over the same $s$ interval~\cite{Ligeti:2007sn}:

\begin{equation}
  {\cal R}(s_{\rm min}) = \frac{
    \int_{s_{\rm min}}^1 {\rm d}s \,
    \frac{{\rm d}\Gamma(B\to X_s \ell^+\ell^-)}{{\rm d}s}
  }{
    \int_{s_{\rm min}}^1 {\rm d}s \,
    \frac{{\rm d}\Gamma(B\to X_u \ell \nu)}{{\rm d}s}
  } \, .
\end{equation}

The numerical analysis shows that the uncertainties due ${\cal O}(1/m_b)$
power corrections which correspond to the parameters $\lambda_2$, $\rho_1$,
$f_u^0+f_s$ and $f_u^0-f_s$ are now under control~\cite{Huber:2007vv}:
\bea
{\cal R}(s_{\rm min})_{\mu\mu}^{\rm high} & = &
2.29 \times 10^{-3} \Big(
1 \pm 0.04_{\rm scale} \pm 0.02_{m_t} \nnb \\
&& \hspace*{-25pt}
\pm 0.01_{{C,m_c}} \pm 0.006_{m_b} \pm 0.005_{\alpha_s} \pm 0.09_{\rm CKM}
\nnb \\
&& \hspace*{-25pt}
\pm 0.003_{\lambda_2} \pm 0.05_{\rho_1} \pm 0.03_{f_u^0+f_s} \pm 0.05_{f_u^0-f_s} \Big) \nnb \\
&=& 2.29 \times 10^{-3} ( 1 \pm 0.13 ) \,. \label{muonR}
\eea
The largest remaining source of error is now $|V_{ub}|$, which will be further reduced
with the precise CKM determination at \superb.
As in the $B \rightarrow X_s \gamma$ case, additional uncertainties, such as the still
unknown non-perturbative corrections that scale with $\alpha_s\Lambda_{\rm QCD}/m_b$,
are about $5\%$.
The cuts in the hadronic invariant mass spectrum lead to
additional uncertainties of order $5\%$, which correspond to the effects of
subleading shape functions~\cite{Lee:2005pk,Lee:2005pw}. \\

% {\it experimental considerations}.\\
Published analyses
for $B \to X_s l^+ l^-$~\cite{Aubert:2004it,Iwasaki:2005sy}
have used a semi-inclusive approach ($X_s = 1K + n\pi, n \le 3$).
This technique is affected by large systematics arising from uncertainties
on the ratio used to extrapolate from the semi-inclusive to the inclusive
branching ratio. This type of analysis is expected to be systematics
dominated, with statistics around $1 \ {\rm ab}^{-1}$.

With larger statistics, a fully inclusive analysis using
semileptonic or hadronic tags is likely to be more sensitive.
Feasibility studies for such an analysis show that about 40 signal events per
${\rm ab}^{-1}$ can be expected with a signal-to-background ratio of $\sim 1.5$.
At \superb, a few percent statistical error on the inclusive branching ratio
can be achieved, well below the present theoretical error
(see Eqs.~\ref{muonBRlow} and~\ref{muonBR}).
No detailed studies are available for the systematic uncertainties,
but they are likely to become dominant over
experimental statistical uncertainties at this level of precision. \\

\subsubsection*{$B \to K^{(*)} \tau \tau$ decay modes}

The branching ratio of
$B \to X_s \tau^+ \tau^-$ is smaller by a factor of about 20,
with respect to $B \to X_s \ell^+ \ell^-$ ($\ell = e,\mu$),
in the low $q^2$ region,
but is expected to be about $2$--$3 \times 10^{-7}$,
comparable to $B \to X_s \ell^+ \ell^-$ (see Eq.~\ref{muonBR}),
in the high $q^2$ region.

An inclusive experimental determination is essentially impossible,
but an analysis of the exclusive decays $B \to K^{(*)} \tau \tau$ might be possible.
These decays are predicted to make up $50$--$60\%$ of the total inclusive
rate~\cite{Du:1993sh}.
Preliminary simulation studies using the hadronic tag technique indicate
that the Standard Model branching fractions could be measurable with the full \superb\
integrated luminosity. Other interesting measurements such as the polarization
asymmetry~\cite{Hewett:1995dk} are under study. \\

\subsubsection*{$B \to K^{(*)} \nu \overline \nu$ decay modes}

The rare decay $B \to K^{(*)} \nu \overline \nu$ is an
interesting probe for New Physics in $Z^0$ penguins~\cite{Buchalla:2000sk}, such as
chargino-up-squark contributions in a generic supersymmetric theory.
Moreover, since only the $b \to s$ +
missing energy process can be detected, the measured rate can be affected by
exotic sources of missing energy, such as light dark matter~\cite{Bird:2004ts}
or ``unparticle physics''~\cite{Georgi:2007ek,Aliev:2007gr}. Notice also that
New Physics effects can modify the kinematics of the decay, which implies that any
selection applied on kinematical variables has an impact on the theoretical
interpretation of the measured branching ratio. The best upper limit among the
exclusive decay channels is
${\cal B}(B^+ \to K^+\nu{\overline\nu}) < 14 \times 10^{-6}$~\cite{Chen:2007zk},
still far above the Standard Model branching fraction of $4 \times 10^{-6}$~\cite{Buchalla:2000sk}.

Due to the undetected neutrinos, it is not possible to reject
background by means of the usual kinematical constraints, so the search for
these decays must be performed using a recoil analysis.
%one of
%the two $B$ mesons produced in the $\Upsilon(4{\rm S})$ decay
%(the $B_{\rm tag}$) is reconstructed in a semileptonic or hadronic decay,
%then the signal is searched for in the rest of the event (the signal side).
%The $B_{\rm tag}$ reconstruction allows a strong reduction of the background
%contamination, but the corresponding efficiency is of the order of few per mille.

In the $B^+ \to K^+ \nu \overline \nu$ analysis, only one track is required on
the signal side. A selection on the kaon momentum is usually applied.
A final selection is applied on the extra energy $E_{\rm extra}$, defined as
the sum of the energies of the neutral electromagnetic calorimeter clusters that are not associated with
the $B_{\rm tag}$ or the signal side. Current analyses employ a counting
technique, but a maximum likelihood (ML) fit to the $E_{\rm extra}$ distribution can be
used to improve performance. To be conservative, we assume the current
analysis technique. From toy MC simulations, combining the results from the
semileptonic and the hadronic recoil, the observation of the decay is expected with
between 10 and $20 \ {\rm ab}^{-1}$ with an expected error of $18\%$, in the most
conservative scenario, at $50 \ {\rm ab}^{-1}$.
The improvement in the precision as a function of luminosity is shown in
Fig.~\ref{fig:snunu}.

\begin{figure}[!htb]
  \begin{center}
    \includegraphics[width=\columnwidth]{knunu_superb.eps}
    \includegraphics[width=\columnwidth]{kstpnunu_superb.eps}
    \caption{
      Expected precision of the measurements of the branching fractions of
      (top) $B^+ \to K^+ \nu \bar{\nu}$ and
      (bottom) $B^+ \to K^{*+} \nu \bar{\nu} \ (K^{*+} \to K_S \pi^+)$
      evaluated as a function of the integrated luminosity,
      assuming efficiencies and backgrounds as in the current \babar\ analyses.
      The bands indicate the range of the Standard Model predictions.
      \label{fig:snunu}
    }
    \end{center}
\end{figure}


In the $B^0 \to K^{*0} \nu \overline \nu$ analysis, the $K^{*0}$ is
reconstructed in the $K^{*0} \to K^+ \pi^-$ channel, with no cut on the
kinematical variables. A maximum likelihood fit is used to extract the signal yield from the
$E_{\rm extra}$ distribution. Observation of this decay is expected between 10
and $20 \ {\rm ab}^{-1}$ with an expected error of $20\%$, in the most conservative
scenario, at $50 \ {\rm ab}^{-1}$.

The same approach is adopted in the $B^\pm \to K^{*\pm} \nu \overline \nu$
analysis, where $K^{*\pm} \to K_S^0 \pi^\pm$ or $K^{*\pm} \to K^\pm \pi^0$.
The observation is expected around $40 \ {\rm ab}^{-1}$ with an expected error of  $25\%$,
in the most conservative scenario, at $50 \ {\rm ab}^{-1}$
(see Fig.~\ref{fig:snunu}).

An irreducible background contribution from $B \to \tau \nu$ decays
is expected in the $B \to K^{(*)\pm} \nu \overline \nu$ analyses.
However, the effect of this background can be controlled with
improvements in the analyses (such as using a maximum likelihood fit).
Moreover, the performance of the recoil technique will be improved by
the improved hermeticity of the \superb\ detector, making the cuts usually applied on
the track multiplicity of the signal side more effective. Preliminary studies have shown
that a $30\%$ reduction in the background contamination with the baseline
\superb\ design is possible.
For background dominated channels such as $B \to K^{(*)\pm} \nu \overline \nu$,
a reduction in background of $30\%$ can be shown to be roughly equivalent to
an increase in statistics of $1/0.7$, {\it i.e.} about $40\%$.
Therefore, such an improvement has a significant effect on the sensitivity.

If the background can be reduced sufficiently, it will be possible to do higher
multiplicity studies of $b \to s\nu\bar{\nu}$ decays such as
 $B^+ \to K_1^+ \nu\bar{\nu} \,, \ K_1^+ \to K^+ \pi^+\pi^-$.
This information could be used to make a semi-inclusive measurement of
${\cal B}(b \to s \nu \bar{\nu})$.
Further background rejection can come from an improved vertex detector,
that allows to apply vertexing requirements (poorly used now)
and secondary vertex information.
The semi-inclusive approximation may provide the best possible
analysis of $B \to X_s \nu \overline \nu$ decay.
% At present, there is no feasibility study available for the inclusive
% $B \to X_s \nu \overline \nu$ analysis.
Owing to the complete absence of any powerful constraint to be applied
on the signal side, the fully inclusive analysis appears to be difficult
in the face of large backgrounds.
If a fully inclusive analysis could be performed at \superb, it may be possible
to make a test of the theoretically clean Standard Model prediction~\cite{Buras:2000dm}
\begin{equation}
\frac{
  {\cal B}(B \to X_d \nu \bar{\nu})
}{
  {\cal B}(B \to X_s \nu \bar{\nu})
} = \left| \frac{V_{td}}{V_{ts}} \right|^2 \, .
\end{equation}
Studies of corresponding ratios using exclusive modes are less theoretically
clean, however the prospects for measuring $B \to \pi \nu \bar{\nu}$ at
\superb\ look good~\cite{Bona:2007qt}.


% To assess the impact of these improvements, we have simulated three different scenarios:
% \begin{enumerate}
%   \setlength{\itemsep}{-2mm}
%   \vspace{-2mm}
% \item BaBar-like scenario
%   (efficiency and background from the current BaBar analysis);
% \item $30\%$ background reduction;
% \item $50\%$ background reduction.
% \end{enumerate}



\noindent
\subsection*{Leptonic $B$ decays}
\label{sec:B:channels:leptonic}

\noindent
{\underline{The branching fraction of $B \to \ell \nu$}}

The decays \btolnu\ can be used to constrain the Standard Model mechanism of
quark mixing.  New Physics contributions can enhance the branching fractions
of \btolnu, as described in the \superb\ CDR~\cite{Bona:2007qt}.
% and elsewhere~\cite{Hou:1992sy,Isidori:2006pk}.
Precision measurements of the branching fraction
of \btolnu\ where $\ell = e, \mu, \tau$ can be used to constrain New Physics.

Recent measurements have provided evidence for
$B \to \tau \nu$~\cite{Ikado:2006un,Aubert:2007bx,Aubert:2007xj}
These measurements rely on recoil analyses in which fully (partially)
reconstructed $B$ meson decays to hadronic (semileptonic) final states
of the non-signal $B$ in the event ($B_{\rm tag}$) are used to help reduce
background for the partially reconstructed signal.
This approach is required for the $B \to \tau \nu$ analysis,
in which there are at least two missing neutrinos in the final state.
For $B \to \mu \nu$ and $B \to e \nu$~\cite{Satoyama:2006xn,Aubert:2004yu,Aubert:2006at},
on the other hand, the high momentum lepton alone provides a characteristic
signature.
Nevertheless, recoil analyses appear preferable also for these channels, due
to the additional kinematic constraints and the reduction in background.

A number of possible improvements to the \btolnu\ analyses
% \superb\ will have to investigate.
are being explored.
These include
\begin{enumerate}
\item All existing \babar\ measurements rely on reconstructing $B_{\rm tag}$
  modes with a $D$ or $D^*$ in the final state.  However it is also possible
  to increase the signal efficiency by including charmonium decay modes
  with a \jpsi\ in the $B_{\rm tag}$ final state.
\item The $B_{\rm tag}$ categories all have different purities.
%  , and different energy  flow in the $B_{\rm tag}$ and in the signal $B$,
%  $B_{\rm sig}$, (in analogy with flavor tagging in a time-dependent $CP$ analysis).
  It is a natural extension of the existing analyses to investigate the gain
  in precision that one can obtain by subdividing the data according to the
  $B_{\rm tag}$ purity in a multi-dimensional maximum likelihood fit, and if
  necessary, exclude any $B_{\rm tag}$ category in which systematic uncertainties
  are not under control.
  (Similar strategies have been successfully employed in time-dependent \CP\
  asymmetry measurements at the $B$ factories.)%
\item In the case of \btotaunu, $B_{\rm sig}$ has contributions from several
  reconstructed $\tau$ decay channels that have different purities; so one
  should subdivide the data according to $B_{\rm tag}$ and $B_{\rm sig}$ purity.%
\item Existing analyses rely heavily on a variable constructed from the sum of
  electromagnetic energy unassociated wither with the $B_{\rm sig}$ or
  $B_{\rm tag}$ to isolate signal ($E_{\rm extra}$).  In order to do this reliably,
  one has to understand, and accurately simulate, noise in the calorimeter as
  well as the geometric acceptance of the detector to backgrounds in which final
  state particles escape down the beam pipe, or into uninstrumented regions of
  the detector.  Not only does this rely on accurate accounting of material in
  the inner parts of the detector, but also in the calorimeter itself, and
 a finely-tuned understanding of the production mechanisms for all types of $B$
  and non-$B$ backgrounds.  It is not clear if the continual use of such a
  variable would facilitate a precision measurement of \btolnu\ branching
  fractions.  It would be possible to improve control of systematic
  uncertainties by limiting the analysis to high purity $B_{\rm tag}$ samples
  and/or  to $B_{\rm sig}$ channels only.  During the detector R\&D stage, one
  should also consider the effects of non-active material in the
  calorimeter, and material in front of the calorimeter, as it is critical that %
  this is correctly accounted for in GEANT simulations of \superb.%
\item The current analyses that extract the yield from a fit of the %
  $E_{\rm extra}$ distribution determine the shape of the signal PDF %
  using a control sample of semileptonic $B \to D^{(*)}\ell\nu$ decays %
  on the recoil of $B_{\rm tag}$.
  With \superb\ statistics it would be possible to use hadronic $B$
  decays for the control sample, which could lead to a reduction of systematic
  uncertainty.  This approach has been used as a systematic cross-check in one
  search for \btotaunu~\cite{Aubert:2006at}, and has also been employed by
  CLEO in the measurement of $f_{D_s}$ using
  $D_s^+ \to \tau^+ \nu_\tau$~\cite{Ecklund:2007zm}.%
\item There are alternatives to the $E_{\rm extra}$ variable that do not rely
  so critically on our understanding of the detector material, acceptance,
  response and details of the background kinematics.  Examples of such
  variables include the highest energy cluster unassociated with $B_{\rm sig}$ or
  $B_{\rm tag}$.%
\item Improvements in the detector hermeticity would, as well as increasing
  the signal efficiency, lead to smaller backgrounds due to particles that
  travel down the beam pipe.  Similarly, improvements in the efficiency with
  which $K_L$ mesons are detected would help to reduce the background.%
\end{enumerate}

The emphasis in these improvements is on increasing the signal efficiency, and
on better control of systematic uncertainties associated with measuring
\btolnu\ branching fractions. It must be emphasized that, while the Standard Model
expectation for the branching fraction of \btomunu\ is significantly lower
than that of \btotaunu, the experimental signature, a high momentum muon
with missing energy, is much cleaner than that of a $\tau$ lepton.
Therefore, at very high luminosities, \btomunu\ is expected to provide
a more precise branching fraction measurement, as it will not be systematics
limited.
Measurements of \btomunu\ and \btotaunu\ are central to the New Physics
search capability of \superb.
The phenomenological impact of these measurements is discussed in
Section~\ref{sec:B:pheno}. \\

\subsubsection*{Radiative leptonic decays}

Radiative leptonic decays, namely
$B_{u}\to \ell \nu\gamma$, $B_{d(s)}\to \ell\ell\gamma$ and $B_{d(s)}\to \gamma\gamma$,
do not contain any hadrons except the $B$ meson.
This simple observation drastically reduces theoretical uncertainties
originating from the strong interaction, such as final state interactions.
\superb\ may be able to observe these extremely rare processes,
due to its good efficiency for reconstruction of the radiative photon.

It has been shown that, in the Standard Model, the strong interaction factorizes at the
large $m_b$ limit, making it possible to describe these three processes in
terms of an universal non-perturbative
form-factor~\cite{DescotesGenon:2002ja}.
Rough estimates of the branching ratios yield
${\cal B}(B_{u}\to \ell \nu\gamma)�\sim {\cal O}(10^{-6})$,
${\cal B}(B_{d(s)}\to \ell\ell\gamma) \sim {\cal O}(10^{-10(-9)})$ and
${\cal B}(B_{d(s)}\to \gamma\gamma) \sim {\cal O}(10^{-8(-6)})$.
It should be emphasized that the helicity suppression,
which diminishes the branching ratio of the
pure-leptonic processes corresponding to the first two channels,
$B_u\to \ell \nu$ and $B_{d (s)} \to \ell\ell$,
does not occur here, due to the additional photon.
As a result, one can take advantage of all three final states
with $\ell = e, \mu, \tau$, which have similar decay rates.

A strategy to search for New Physics with these channels would be to first determine
the form factor through
the tree level $B^+ \to \ell \nu\gamma$ process~\cite{Korchemsky:1999qb}
and then use it to extract New Physics effects from the loop level
$B_{d(s)}\to \ell\ell\gamma$ and $B_{d(s)}\to \gamma\gamma$ processes.
In the former, the most recent experimental results~\cite{Aubert:2007yh} are
already close to the Standard Model expectation.
\superb\ can make a precise measurement of this decay; theoretical
uncertainties due to the restricted phase space used in the analysis
(necessary to reduce backgrounds from final state radiation photons) may then
become a limiting factor.
The current experimental upper limits on $B_d\to \ell\ell\gamma$ are
at the $10^{-7}$ level~\cite{Aubert:2006ft}; since these are not background-limited,
\superb\ can improve the limits to close to the Standard Model level.
Once observed, kinematical distributions in these processes provide additional New Physics sensitivity.
New Physics effects on the branching ratio and the forward-backward asymmetry
of the $B_{d(s)}\to \ell\ell\gamma$ process have been investigated,
e.g. in~\cite{Yilmaz:2004tr,Choudhury:2005rz}.
For example, those effects could come from an anomalous $bd(s)Z$ coupling, that
could be also seen the in $B\to K^{(*)} \ell\ell$ and $B_{d(s)}\to \ell\ell$
processes.

On the other hand, the new physics effect to $B_{d(s)}\to \gamma\gamma$ process
could come from two kinds of short-distance contributions: anomalous  $bs\gamma$ coupling
and the $bs\gamma\gamma$ coupling. In particular, the later contribution has not been
explored yet and \superb\ sensitivity will reveal these couplings for the first time.
It should be noted that this contribution can be also studied in
$B\to K\gamma\gamma$~\cite{Hiller:2004wc}.
Detailed investigations of the supersymmetric contributions to
$B_s\to\gamma\gamma$ and $B\to X_s\gamma\gamma$
have been performed~\cite{Bertolini:1998hp}.
As discussed in the CDR~\cite{Bona:2007qt}, $B_s\to\gamma\gamma$ could be
observed at \superb\ after accumulating about $1 \ {\rm ab}^{-1}$ at the
\FiveS.
Extrapolating from existing upper limits on the $B_d \to\gamma\gamma$
decay~\cite{Aubert:2001fm,Abe:2005bs}, \superb\ could probe down to the Standard Model
level of this New Physics-sensitive decay.


%\subsubsection*{Lepton Flavour Violating Modes.}
%
%{\BF WE REMOVE THIS PARAGRAPH SINCE EVEN IN THE MOST FAVOURABLE SCENARIO
%TANBETA=60 ma=100 GEV  --> BR(B-->mutau) =2 10-8 }
%
%A peculiarity of \superb\ is the possibility to study lepton flavor violating
%(LFV) decays including tau leptons.
%These are expected to be sensitive to New Physics in the third generation couplings.
%In addition to studying LFV tau decays, it is also possible to search for LFV
%effects in $B$ (and charm) decays, using channels such as $B \to \tau \mu$,
%$B \to \tau e$ and $B \to K \tau \mu$.
%The current upper limits are
%${\cal B}(B \to \mu\tau) <22 \times 10^{-6}$~\cite{Aubert:2008cu},
%${\cal B}(B \to \mu\tau) <28 \times 10^{-6}$~\cite{Aubert:2008cu} and
%${\cal B}(B \to  K \mu\tau) <28 \times 10^{-6}$~\cite{Aubert:2007rn},
%all at $90\%$ confidence level.
%
%With present analysis techniques the background levels are approximately
%1 event per ${\rm ab}^{-1}$, and we expect the limits to scale roughly as
%$1/\sqrt{\rm luminosity}$ so that limits of ${\cal O}(10^{-6})$ can be
%achieved at \superb\. However, there is also significant room for improvements
%in the analyses which could improve the sensitivity.





