\section{Phenomenology}
\label{sec:B:pheno}

\subsection*{Golden processes}
\label{sec:B:pheno:gold}

\begin{table*}[!htb]
 \caption{
    \label{tab:golden}
     Golden modes in different New Physics scenarios. A ``X'' indicates the golden channel of a given scenario.
     An ``O'' marks modes which are not the ``golden'' one of a given scenario but can still display a measurable deviation from the Standard Model.
     The label $CKM$ denotes golden modes which require the high-precision determination of the CKM parameters achievable at \superb. }
   \begin{tabular}{lcccccc}%%%%\begin{tabular}{|c|cccccc|}
    \hline\hline
                                      &      $H^+$      &   ~Minimal~  &   ~Non-Minimal~   &  ~Non-Minimal~    &     NP            & ~Right-Handed~   \\
                                      & high tan$\beta$ &     FV       &     FV (1-3)      &   FV (2-3)        & ~Z-penguins~      &   currents       \\
    \hline
    \BR$(B \to X_s \gamma)$            &                 &     X        &                   &       O           &                   &      O           \\
    $A_{\CP}(B \to X_s \gamma)$        &                 &              &                   &       X           &                   &      O           \\
    \BR$(B \to \tau\nu)$               &      X-$CKM$    &              &                   &                   &                   &                  \\
    \BR$(B \to X_s l^+l^-)$            &                 &              &                   &       O           &      O            &      O           \\
    \BR$(B \to K \nu \overline{\nu})$  &                 &              &                   &       O           &      X            &                  \\
    $S(K_S \pi^0 \gamma)$              &                 &              &                   &                   &                   &      X           \\
    $\beta$                           &                 &              &      X-$CKM$      &                   &                   &      O           \\
    \hline\hline
  \end{tabular}
\end{table*}


At \superb, a golden channel is any channel that is very well known in the Standard Model.  This
includes ``null tests'' (observables that are zero, at least approximately, in
the Standard Model) but also other channels predicted with small errors.  This places more emphasis on
inclusive modes than on exclusive decays.  While there are probably
specific channels that can be selected in charm and in $\tau$ physics, in $B$
physics there are so many golden channels that selecting one or two risks
missing the point. In addition processes that are golden (\ie\ display a
measurable deviation from Standard Model) for given New Physics scenario could be uninteresting
in a different scenario.
The rationale for building \superb\ based on the New Physics-sensitivity of any individual
channel can certainly be challenged -- the motivation is the large range of golden
channels for which \superb\ has unsurpassed sensitivity.
We will nonetheless, in response to the IRC, select some specific channels for which \superb\ has unique potential. However, the argument given above makes it clear that golden modes are defined only in the context of a limited
and non-orthogonal set of New Physics scenarios.
We thus want to stress once more that one of the most sensitive searches for
New Physics will be the 1$\%$ determination of CKM parameters; the possibility
of performing such a precise determination in the presence of New Physics is a unique
feature of \superb. The precision measurements required to achieve this goal are $|V_{ub}|$ and
the CKM angles. In the spirit of indicating the golden modes, we select
$|V_{ub}|$ and $\alpha$, being $\beta$ and $\gamma$ precisely measured at LHCb.
In the following, we denote by $CKM$ those places in which the improvements
on the CKM parameters achieved by \superb\ are crucial to the corresponding New Physics
searches. We do not include rare kaon decays in which a precise CKM measurement is also extremely
important. Notice that whenever a high precision CKM determination is required,
progress on Lattice QCD calculations, as discussed in the Appendix of the CDR,
is needed.
In Table \ref{tab:golden} we show the result of our selection of golden modes in
different New Physics scenarios. For each scenario, ``X'' marks the golden channel while ``O'' marks
those modes which can display measurable deviation from the Standard Model.

A few comments are in order on this selection. Notice first that ${\cal B}(B \to X_s \gamma)$ is important in several scenarios,  in
particular in the MFV scenarios, and therefore we put it in the list, even though at \superb\ it is limited by theoretical errors,
unless a major breakthrough in non-perturbative calculations of power suppressed corrections is achieved. In some
of the scenarios considered, of course, this list is far from complete; many other measurements are expected to show deviations from their Standard Model
values. For example, in the case of non-minimal flavor violation in the transitions between third and second generations, the entire cohort
of $b \to s$ penguins-dominated non-leptonic modes could show a deviation in the measured value of time-dependent \CP\  asymmetries
compared to those measured in $b \to c \bar{c} s$ transitions.


\noindent
\subsection*{Benchmarks}
\label{sec:B:pheno:benchmarks}

The problem of defining proper benchmarks for \superb\ has not been addressed yet. In fact benchmarks
for flavor physics clearly require the specification of the New Physics flavor structure, which is not needed
(at least at first approximation) for high-$p_T$ physics. Nonetheless, stimulated by the IRC, we estimate
the relevant flavor observable measured at \superb\ within the mSUGRA models at the SPS$1a$, SPS$4$ and
SPS5 benchmark points defined for the LHC in ~\cite{Allanach:2002nj} .
The purpose of this exercise is to evaluate the deviation from the Standard Model of flavor observables in a
MFV scenario where LHC can reconstruct a large part of the SUSY spectrum. We consider a set of measurements
which are likely to be affected in the MFV model under consideration.

In terms of the fundamental parameters at the high scale, the SPS considered points are
defined as:
\begin{eqnarray}
{\rm SPS}1a: &&\quad m_0 = 100 {\rm GeV}, \quad m_{1/2} = 250 {\rm GeV}, \\ \nonumber
             &&\quad A_0 = -100 {\rm GeV}, \quad \tan\beta = 10, \quad \mu > 0 \\ \nonumber
{\rm SPS}4:  &&\quad m_0 = 400 {\rm GeV}, \quad m_{1/2} = 300 {\rm GeV}, \\ \nonumber
             &&\quad A_0 = 0, \quad \tan\beta = 50, \quad \mu > 0 , \\ \nonumber
{\rm SPS}5:  &&\quad m_0 = 150 {\rm GeV}, \quad m_{1/2} = 300 {\rm GeV}, \\ \nonumber
             &&\quad A_0 = -1000,\quad \tan\beta = 5, \quad \mu > 0 . \nonumber
\end{eqnarray}

Note that SPS$1a$, a ``typical'' mSUGRA scenario with intermediate tan$\beta$, is extremely good for LHC and indeed the most studied -
the pattern of sparticle masses allows them all to be measured with very good accuracy~\cite{Weiglein:2004hn}.
By contrast, the relatively high squark masses and the low value of $\tan\beta$ suppress effects on flavor observables.
SPS4 is an mSUGRA scenario with large $\tan\beta$. Unfortunately, no detailed studies are available at
LHC for this point. Nevertheless we roughly estimated the LHC performance by studying the decay chain starting
from the computed SUSY spectrum. We found a single study at SPS5~\cite{Borjanovic:2005tv}, a parameter configuration with relatively
light stop quark and low $\tan\beta$. Here again the LHC performance in measuring the SUSY spectrum is rather good.

Based on these studies, and using the tools developed at the recent CERN-Workshop ``Flavour in the LHC Era''
\cite{Buchmueller:2007zk} we produced the predictions presented in Table \ref{tab:spspoints}.

\begin{table*}[!htb]
 \caption{
    \label{tab:spspoints}
    Predictions of flavor observables based on expected measurements
    from LHC in mSUGRA at SPS1a, SPS4, SPS5 benchmark points. Quantities denoted ${\cal R}$ are the ratios of
    the branching fractions to their Standard Model values. Quoted uncertainties (when available) come from the
    errors on the measurement of the New Physics parameters at LHC. Uncertainties on the Standard Model predictions of flavor
    observables are not included. For the SPS4 benchmark point the sensitivity study at LHC are not available.
  }
  \bigskip
   \begin{tabular}{lccc|c}     %%%%%%% \begin{tabular}{|l|c|c|c|}
    \hline\hline %
                                                       &        SPS1a             &       SPS4            &     SPS5           \\ \hline
    ${\cal R}(B \to X_s\gamma)$                          &     0.919  $\pm$ 0.038   &  0.248  & 0.848  $\pm$ 0.081 \\
    ${\cal R}(B \to \tau\nu)$                          &     0.968  $\pm$ 0.007   &  0.436  & 0.997  $\pm$ 0.003 \\
    ${\cal R}(B \to X_s l^+l^-)$                       &     0.916  $\pm$ 0.004   &  0.917  & 0.995  $\pm$ 0.002 \\
    ${\cal R}(B \to K \nu \overline{\nu})$             &     0.967  $\pm$ 0.001   &  0.972  & 0.994  $\pm$ 0.001 \\
    ${\cal B}(B_d \to \mu^+\mu^-)/10^{-10}$            &     1.631  $\pm$ 0.038   &  16.9   & 1.979  $\pm$ 0.012 \\
    ${\cal R}(\Delta m_s)$                             &     1.050  $\pm$ 0.001   &  1.029  & 1.029  $\pm$ 0.001 \\
    ${\cal B}(B_s \to \mu^+\mu^-)/10^{-9}$             &     2.824  $\pm$ 0.063  &   29.3   & 3.427  $\pm$ 0.018 \\
    ${\cal R}(K \to \pi^0 \nu \overline{\nu})$         &     0.973  $\pm$  0.001  &  0.977  & 0.994  $\pm$ 0.001 \\  \hline\hline %
  \end{tabular}
\bigskip
\end{table*}


The most striking feature of this result is that SPS4 is already ruled out by the present measurement
of ${\cal B}(B \to s\gamma)$ with high significance, showing the impact of flavor observables on the SUSY parameter space even in a MFV case. Indeed, from Eqs.~(\ref{eq:btosgamma:HFAG}) and (\ref{eq:btosgamma:NNLL}) one obtains R$^\mathrm{exp}(B\to X_s\gamma)=1.13\pm 0.12$. In the absence of a detailed analysis, we have not  attempted an estimate of the errors associated with the predictions of Table~\ref{tab:spspoints} at SPS4.
Nevertheless, even assuming an error of 50\%, much larger than the other points, R($B\to X_s\gamma$)=0.25 at SPS4  is more than 5$\sigma$ away from the present experimental value.
SPS5 is marginally compatible with present measurement of ${\cal B}(B \to s\gamma)$. Clearly this point will produced a measurable effect on ${\cal B}(B \to s\gamma)$ at \superb. Considering these results, it is not surprising that the recent MSSM analysis in \cite{Buchmueller:2007zk} found that the best fit to present data, using ${\cal B}(B \to s\gamma)$ among the constraints, resembles SPS$1a$.

SPS$1a$ is clearly the least favorable point from the flavor point of view. However, even here
\superb\ could see a definite pattern of 1-2 $\sigma$ deviations from the Standard Model in
${\cal R}(B \to \tau\nu)$, ${\cal R}(b \to s\gamma)$ and ${\cal R}(B \to X_s l^+l^-)$,
although this does depend, to some extent, on improvements in theory.
In any case, \superb\ flavor measurements are required to establish that
the New Physics flavor couplings are small as predicted by mSUGRA, since
LHC alone cannot establish which model is behind the measured SUSY spectrum.

One of the lessons of this exercise is that the benchmarks for flavor physics,
if needed, should mainly address the problem of defining a ``typical'' non-minimal flavor
structure with an economical number of parameters. A possible way to further investigate is using
models of SUSY-breaking as discussed below.


\noindent
\subsection*{Update on the $B \to \ell \nu$ predictions }
\label{sec:B:pheno:btaunu}

We update in this section the analysis of the decay $B \to \ell \nu$ in the
2HDM. The case of SUSY, discussed in the CDR, is very similar.

Figure~\ref{fig:btaunu} shows a comparison of the exclusion plot in the
$m(H^+)$--$\tan\beta$ plane coming from a measurement of ${\cal B}(B \to \tau
\nu)$ with different data samples, 2 ab$^{-1}$, 10 ab$^{-1}$, 75 ab$^{-1}$ and
200 ab$^{-1}$, assuming that the result is consistent with the Standard Model.

Note that moving from 10 ab$^{-1}$ to 75 ab$^{-1}$ the channel $B \to \mu \nu$
begins to give a significant contribution to the average, and the scale is then
larger than the naive statistical gain. With further increases in the integrated
luminosity beyond 75 ab$^{-1}$, $B \to \tau \nu$ become systematics-dominated
but $B \to \mu \nu$ still scales with statistics.


\begin{figure*}[!htb]
  \begin{center}
    \includegraphics[width=4.cm,angle=-90]{btaunu_bfact.eps}
    \includegraphics[width=4.cm,angle=-90]{btaunu_kek.eps} \\

    \includegraphics[width=4.cm,angle=-90]{btaunu_superb.eps}
    \includegraphics[width=4.cm,angle=-90]{btaunu-200ab.eps}
    \caption{\label{fig:btaunu}
      Exclusion regions in the $m(H^+)$--$\tan\beta$ plane arising from the
      combinations of the measurement of ${\cal B}(B \to \tau \nu)$ and
      ${\cal B}(B \to \mu \nu)$ using 2 ab$^{-1}$ (top left), 10 ab$^{-1}$ (top right)
      75 ab$^{-1}$ (bottom left) and 200 ab$^{-1}$ (bottom right). We assume that
      the result is consistent with the Standard Model.
    }
  \end{center}
  
\end{figure*}

To give an example of a positive signal as seen at \superb,
Figure~\ref{fig:btaunu-sig}
shows the deviation of ${\cal B}(B \to \ell \nu)$ with
respect to its Standard Model value computed in the 2HDM for $m(H^+)$=500GeV and
$\tan\beta$=30 as it would be measured with a sample of 75 ab$^-1$. It's clear that the deviation is established with very high significance.

\begin{figure*}[!htb]
  \begin{center}
    \includegraphics[width=4.cm,angle=-90]{btaunu-signal.eps}
    \caption{\label{fig:btaunu-sig}
      Distribution of $R= {\cal B}(B \to \ell \nu)/{\cal B}_{SM}(B \to\ell \nu)$
      in 2HDM, using $m(H^+)$=500GeV and $\tan\beta$=30 as it would be measured
      in 5 years at \superb.
    }
  \end{center}

\end{figure*}
\vfill
\clearpage
\newpage

\subsection*{SUSY-breaking models}
\label{sec:B:pheno:SUSY-b}

Within supersymmetric extensions of the Standard Model, the flavor structure is directly
linked to the crucial question of the supersymmetry-breaking mechanism.
Indeed,
the bulk of soft SUSY-breaking terms is given by the sfermion bilinear and
trilinear couplings, which are matrices in flavor space.
Thus, once some SUSY particles have been found, the measurement of the flavor
sector can provide important information for distinguishing among models of
supersymmetry. This is a manifestation of the complementary nature of flavor
physics and collider physics. At the LHC direct searches for supersymmetric
particles are essential in establishing the existence of new physics. On the
other hand, there are a variety of possibilities for the origin of SUSY
breaking
and of flavor structures within supersymmetry. Flavour physics provides an
unique tool with which fundamental questions, such as how supersymmetry is
broken, can be addressed.

\begin{figure}[!htb]
  \begin{center}
    \hspace{-0.1cm}\includegraphics[width=0.45\textwidth]{sbsg2.eps}
    \hspace{-0.5cm}\includegraphics[width=0.45\textwidth]{dsphikz2.eps}
    \caption{
      Time-dependent \CP\ asymmetry of $B \rightarrow  K_s \pi^0 \gamma$
      and the difference between the time-dependent asymmetries of
      $B \to \phi K_S$ and $B \to \jpsi\,K_S$ modes
      for three SUSY breaking scenarios:
      mSUGRA(left), SU(5) SUSY GUT with right-handed neutrinos
      in non-degenerate case (middle),
      and MSSM with U(2) flavor symmetry (right).
      The \superb\ sensitivities are also shown.
    }
    \label{Okada1}
  \end{center}
\end{figure}


A comprehensive analysis of the flavor patterns generated in
SUSY models with different SUSY-breaking sector has been
recently presented in Ref.~\cite{Goto:2007ee}. The models under study are
mSUGRA, MSSM with U(2) flavor symmetry, MSSM with right-handed neutrinos,
and SU(5) SUSY-GUT with right-handed neutrinos. Different scenarios for
the neutrino mass spectrum and Yukawa couplings have also been considered.
For our purpose, it is sufficient to consider a few examples. We refer the
reader
to the original publication for all the details.

Figs.~\ref{Okada1}--\ref{Okada2} are examples of the power of \superb\ in
discriminating
different SUSY-breaking scenarios.  Additional information can certainly be
obtained
from a systematic study of correlations among flavor observables. It is
interesting to
notice that the plot in Fig.~\ref{Okada2} calls for a determination of
$\gamma$ ($\phi_3$)
with a sub-degree precision, which could be obtained at \superb\ with 100
$ab^{-1}$.

\begin{figure}[!htb]
  \begin{center}
%     \includegraphics[width=0.25\textwidth]{msgr-t30-DmsDmd-phi3.eps}
%     \includegraphics[width=0.25\textwidth]{su5nr-nd2-4e14-t30-DmsDmd-phi3.eps}
%     \includegraphics[width=0.25\textwidth]{u2-t30-DmsDmd-phi3.eps}
     \includegraphics[width=0.5\textwidth]{shindou3.eps}
    \caption{
      Correlation of $\Delta m_s/\Delta m_d$ and $\gamma$ ($\phi_3$)
      for three SUSY breaking scenarios:
      mSUGRA(left), SU(5) SUSY GUT with right-handed neutrinos
      in non-degenerate case (middle),
      and MSSM with U(2) flavor symmetry (right).
%      The \superb\ sensitivities are also shown.
    }
    \label{Okada2}
  \end{center}
\end{figure}


%\subsection*{SUSY-breaking models}
%\label{sec:B:pheno:SUSY-b}
%%
%
%{\bf MARCO WILL WORK ON IT...... !!!!!!!!!!!!}\\%
%
%see ref \cite{Goto:2007ee}%
%
%\begin{figure}[!htb]
%  \begin{center}
%    \hspace{-0.7cm}\includegraphics[width=0.45\textwidth]{sbsg2.eps}
%    \hspace{-0.5cm}\includegraphics[width=0.45\textwidth]{dsphikz2.eps}
%    \caption{
%      Time-dependent asymmetry of $B \rightarrow  K_s \pi^0 \gamma$
%      and the difference between the time-dependent asymmetries of
%      $B \to \phi K_s$ and $B \to J/\psi\,K_s$ modes
%      for three SUSY breaking scenarios:
%      mSUGRA(left), SU(5) SUSY GUT with right-handed neutrinos
%      in non-degenerate case (middle),
%      and MSStandard Model with U(2) flavor symmetry (right).
%      The expected SFF sensitivities are also shown.
%    }
%    \label{Okada1}
%  \end{center}
%\end{figure}
