%% -*- mode: latex; TeX-master: "tau-all.tex" -*-
%%
%% LFV section, experiment
%%

\subsection*{\texorpdfstring{\superb experimental reach}{SuperB experimental reach}}
%%{\superb experimental reach}

A sensitive search for lepton flavor-violating $\tau$ decays at \superb
requires signal to be selected with as high an efficiency as possible, while allowing
minimal, and preferably zero, background.
A  candidate $e^+e^- \to \tau^+\tau^-$ events obtained from an initial screening selection
is divided into hemispheres in the center-of-mass frame,
each containing the decay products of one $\tau$ lepton.
Unlike Standard Model $\tau$~decays, which contain at least one neutrino,
the decay products from a LFV decay have a combined energy in the center-of-mass frame
equal to $\sqrt{s}/2$ and a mass equal to that of the $\tau$.
A requirement on the two dimensional signal region in the $E_{\ell X}$--$M_{\ell X}$ plane
therefore provides a powerful tool to reject backgrounds,
which arise from well-understood Standard Model $\tau$ decays.
Consequently, residual background rates and distributions
are reliably estimated from Monte Carlo simulations and validated
using quantitative comparisons with data as various selection requirements are applied.
Global event properties and an explicit identification of the non-signal
$\tau$ decay can be applied to suppress non-$\tau$ backgrounds with only marginal
loss of efficiency.

The considerable experience developed in searching for these decays
in the ${\sim}0.5$~ab$^{-1}$ data set at \babar enables
 us confidently to estimate background levels to be expected with 75~ab$^{-1}$
 for selection strategies similar to those of the existing experiments.
These lead us to classify the LFV decay modes into two categories for the
purposes of estimating the experimental $\tau$ LFV discovery reach of
\superb: (i) modes having ``irreducible backgrounds'' and (ii)
modes that do not have irreducible backgrounds.
For luminosities of $10^{36}\,\mathrm{cm^{-2}s^{-1}}$,
\taulg decays fall into category (i), whereas $\tau\ra\ell\ell\ell$ and \taulh generally fall
 into category (ii),
where $\ell$ is either a muon or electron and h$^0$ is a hadronic system.
The hadronic system may be identified as a pseudoscalar or vector meson
 ($\pi^0$, $\eta$, $\eta'$, $K^0_S$, $\omega$,$\phi$,$K^{*}$ {\it etc.}) or
a non-resonant system of two pions, two kaons or a pion and kaon.

The category (ii) decay modes have the property that with perfect particle identification
no known process or combination of processes can mimic the signal at rates
relevant to \superb.
The challenge in searching for these decays
is thus to remove all non-$\tau$ backgrounds and to provide as powerful
a particle identification as possible.
For category (i) modes, however, even with perfect particle identification, there
exist backgrounds that limit the discovery sensitivity.
In fact, there are no \taulg Standard Model processes expected
at these luminosities, but there are combinations
of processes that can mimic this signal, even with perfect measurements.
In the case of $\taumg\!\!,$ for example, the irreducible background arises from
events having a $\tau\ra\mu\nu\bar{\nu}$ decay
and a $\gamma$ from initial state radiation (ISR)
in which the photon combines with the muon to form a candidate that accidentally
falls into the signal region in the $E_{\ell X}$--$M_{\ell X}$ plane.
At sufficiently high rates, $\tau\ra\ell\pi^0$ and  $\tau\ra\ell\eta \, (\eta\ra\gamma\gamma)$
searches will suffer the same problems when two hard ISR
photons accidentally reconstruct to a $\pi^0$ or $\eta$ mass,
 but the rate for two hard-photon ISR emission will be roughly 100 times lower
 than  the rate for a signal hard photon emission and lower still
 when requiring a $\gamma\gamma$ mass to match that of a $\pi^0$ or $\eta$.
 Consequently, this  is not expected to  be an issue at \superb luminosities.
Similarly, $\tau\ra ee^+e^-$ and $\tau\ra \mu e^+e^-$ can, in principle,
suffer a background from $\tau\ra\ell\nu\bar{\nu}e^+e^-$ events where
the ISR photon undergoes internal pair production. Such background events are expected
 to start to just become measurable for luminosities roughly 100 times higher than
current experiments, and so might just begin to impact the experimental
bounds placed on those modes at \superb.


The experimental reach is expressed here in terms of
``the expected $90\%$ CL upper limit'' assuming no signal,
as well as in terms of a 4$\sigma$ discovery branching fraction
 in the presence of projected backgrounds.
In the absence of signal,
for large numbers of background events $N_{\rm bkd}$,
the $90\%$ CL upper limit for the number of signal events can be given as
$N^{UL}_{90} \sim 1.64 \sqrt{N_{\rm bkg}}$,
whereas for small $N_{\rm bkg}$ a value for $N^{UL}_{90}$
is obtained using the method described in~\cite{Cousins:1991qz},
which gives, for  $N_{\rm bkg} \sim 0$, $N^{UL}_{90} \sim 2.4$.
If a signal is determined from counting events within a signal region
rather than from a fit, the $90\%$ CL branching ratio upper limit is:
\begin{equation}
  \B^{UL}_{90} =
  \frac{N^{UL}_{90}}{2 N_{\tau\tau} \epsilon} =
  \frac{N^{UL}_{90}}{2 \cal{L}\sigma_{\tau\tau} \epsilon}\,,
\end{equation}
where  $N_{\tau\tau}=\cal{L}\sigma_{\tau\tau}$
is the number of $\tau$-pairs produced in $e^+e^-$ collisions;
$\cal{L}$ is the integrated luminosity,
$\sigma_{\tau\tau}$=0.919~nb~\cite{Banerjee:2007is}
 is the $\tau$-pair production cross section, and
$\epsilon$ is the signal efficiency.


The \taumg projected sensitivity is based on the published
\babar\ analysis~\cite{Aubert:2005ye},
but incorporating changes designed for a very high luminosity
data set and using the improved muon particle identification efficiencies
that became available with a hardware upgrade to the \babar\ muon system.
The published analysis explicitly identifies  the non-signal $\tau$ decays
as specific Standard Model decay modes. In the published analysis, this
 set of tag modes includes $\tau\ra\mu\nu\bar{\nu}$, which has a disproportionate amount
 of $\mu$-pair background compared to the other tag modes. For \superb\ luminosities
it would appear that a more optimal analysis would not include this mode.
The consequence is that the efficiency for a 2$\sigma$ signal ellipse region suffers
a decrease from dropping the $\mu$-tag, but increases from the other improvements
 to both the analysis and the hardware, so that the net efficiency is 7.4\%.
The background levels for 75~ab$^{-1}$ are projected from the Monte Carlo
to be $200\pm50$ events from the $\tau\ra\mu\nu\bar{\nu}(\gamma)$ irreducible background.
This leads to an expected 90\%CL upper limit of $2.3\times 10^{-9}$ and 4$\sigma$ discovery
reach of $5.6\times 10^{-9}$. It is important to note that further improvements can be obtained using the \superb\ polarized electron beam. For a 100\% polarized electron beam,
the polar angles of the signal decay products provide additional background suppression,
as is evident from Figure~\ref{fig:tau-lfv-exp-1}. The ``irreducible background'' would be cut
by 70\% for a 39\% loss in signal efficiency. This would result in approximately a
10\% improvement in the sensitivity:
an expected upper limit of $2.1\times 10^{-9}$ and 4$\sigma$ discovery
level of $5.0\times 10^{-9}$. However, by far the most important aspect
of having the polarization is the possibility to determine the helicity
structure of the LFV coupling from the final state momenta
distributions (see for instance Ref.\cite{Matsuzaki:2007hh} for the
$\tau\to\mu\mu\mu$ process).
Note that for a data sample of  15~ab$^{-1}$ using a
 machine with no polarization, the same analysis and detector
can be expected to yield an expected upper limit of
$5.2\times 10^{-9}$ with a discovery potential of $1.3\times 10^{-8}$.
Similar analyses can be expected to yield comparable sensitivities for the
\taueg LFV decay mode, based on the published \babar\ analysis~\cite{Aubert:2005wa}.

The situation for the other LFV decays,
$\tau\to\ell_1\ell_2\ell_3$\ and $\tau\to\ell h$, is different,
as these modes do not suffer the problem of accidental photons
 with which the \taulg searches must contend.
In these cases, one can project sensitivities assuming $N_{\rm bkg}$
comparable to backgrounds in existing analyses
for approximately the same efficiencies.
For illustrative purposes, we demonstrate how this is accomplished
for the \taummm based on modifications to the published \babar\ analysis~\cite{:2007pw}.
The published analysis managed to suppress the backgrounds for the
data set without explicitly identifying the  Standard Model $\tau$ decays for
 the non-signal $\tau$ and using the loosest muon identification
algorithms.




\begin{figure}[!htb]
  \begin{center}
    \begin{overpic}[width=\linewidth,trim=0 20 0 0,clip]{muonCosThetaqmu}
      \put(0,-4){\makebox[\linewidth][r]{\larger\textbf{$\text{sign}(q_\mu)\cdot\cos(\theta_\mu)$}}}
    \end{overpic}
  \end{center}
  \vspace{1ex}
  \caption{\label{fig:tau-lfv-exp-1}%
    Distribution of the cosine of the signal-side muon multiplied by the
    muon charge for signal and background events with and without electron
    beam polarization in the \taumg search analysis at \superb.}
\end{figure}

Table~\ref{tab:LFVExptSensitivities} summarizes the sensitivities
for various LFV decays.

\begin{table}[!htb]
  \caption{
    \label{tab:LFVExptSensitivities}
    Expected $90\%$ CL upper limits and 4$\sigma$ discovery reach
    on \taumg and \taummm LFV decays with $75 \ {\rm ab}^{-1}$
    with  a polarized electron beam.
  }
  \begin{center}
    \begin{tabular}{lll}
      \hline \hline
      Process &  Expected 90\%CL & 4$\sigma$ Discovery \\
              &  upper limited   &  Reach  \\
      \hline
      $\BR(\tau \to \mu\,\gamma)$          &  $2 \times 10^{-9}$ &  $5 \times 10^{-9}$  \\
%      $\BR(\tau \to e\,\gamma)$            &  $2 \times 10^{-9}$     \\
      $\BR(\tau \to \mu\, \mu\, \mu)$      &  $2 \times 10^{-10}$ & $8.8 \times 10^{-10}$  \\
%      $\BR(\tau \to e e e )$               &  $2 \times 10^{-10}$  \\
%      % $\B(\tau \to \ell_1 \ell_2 \ell_3)$ &  $1-3  \times 10^{-10}$  \\
%      $\BR(\tau \to \mu \eta)$             &  $4 \times 10^{-10}$    \\
%      $\BR(\tau \to e \eta)$               &  $6 \times 10^{-10}$    \\
%      $\BR(\tau \to \ell \KS)$           &  $2 \times 10^{-10}$    \\
      \hline\hline
    \end{tabular}
  \end{center}
\end{table}
