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

\subsection*{Predictions from New Physics  models}

In the following, we discuss the size of $\tau$ LFV effects on decays and correlations
that are expected in supersymmetric extensions of the Standard Model and, in particular,
in the so-called constrained MSSM,
The flavor-conserving phenomenology of this framework is
characterized by five parameters: $M_{1/2}$, $M_0$, $A_0$, $\tan\beta$, $\text{sgn}\ \mu$.
We will discuss a subset of the ``Snowmass Points and Slopes'' (SPS)~\cite{Allanach:2002nj}, listed
in Table~\ref{tab:SPS:def:15}, in this
five-dimensional parameter space  to illustrate the main
distinctive features of the model as they relate to lepton flavor violation.

Specifying one such point
is sufficient to determine the phenomenology of the model
relevant for the LHC, but it is not sufficient to unambiguously
compute LFV rates. The amount of
flavor-violation is controlled by other parameters,
which play no role in high-$p_T$ physics.
Nonetheless, specifying the flavor-conserving parameters
allows us to simplify the description of LFV decays
and, in particular, to establish clear correlations
among different processes.


\begin{table}[!htb]
\caption{\label{tab:SPS:def:15}
Values of $M_{1/2}$, $M_0$, $A_0$, $\tan \beta$,
and sign of $\mu$ for the SPS points considered in the analysis.}
\begin{tabular}{cccccc}
\hline\hline
SPS & $M_{1/2}$ (GeV) & $M_0$ (GeV) & $A_0$ (GeV) & $\tan \beta$ &
 $\mu$ \\\hline
 1\,a & 250 & 100 & -100 & 10 &  $>\,0 $ \\
 1\,b & 400 & 200 & 0 & 30 &   $>\,0 $ \\
 2 &  300 & 1450 & 0 & 10 &  $>\,0 $ \\
 3 &  400 & 90 & 0 & 10 &    $>\,0 $\\
 4 &  300 & 400 & 0 & 50 &   $>\,0 $ \\
 5 &  300 & 150 & -1000 & 5 &   $>\,0 $\\\hline\hline
\end{tabular}
\end{table}


At all the SPS points, LFV decays are dominated by the contribution
of dipole-type effective operators of the form ($\bar l_i \sigma_{\mu\nu} l_j F^{\mu\nu}$).
Defining $\mathcal{R}^{(a)}_{(b)}=\BR(\tau \to a)/\BR(\tau \to b)$,
The dipole dominance allows us to establish the
following relations,
\begin{eqnarray}
\mathcal{R}^{(\mu ee)}_{(\mu \gamma)}\approx
1.0 \times 10^{-2} & \to & \BR(\tau \to \mu e^+ e^-) <  5 \times 10^{-10} \nonumber \\
\mathcal{R}^{(\mu \rho^0)}_{(\mu\gamma)}
\approx  2.5 \times 10^{-3} & \to &  \BR(\tau \to \mu \rho^0) < 10^{-10} \nonumber \\
\mathcal{R}^{(3\mu)}_{(\mu \gamma)}
\approx  2.2 \times 10^{-3}  & \to &  \BR(\tau \to 3\mu) < 10^{-10} \nonumber \\
\mathcal{R}^{(\mu \eta)}_{(\mu\gamma)} <  10^{-3} & \to &  \BR(\tau \to \mu \eta)
< 5 \times 10^{-11}, \nonumber
\end{eqnarray}
where the bounds correspond to the present limit
\hbox{$\BR(\tau\to\mu\gamma)< 4.5 \times 10^{-8}$.}  Similar relations hold for
\hbox{$\tau \to e$} transitions. As a result, in such a framework only
$\tau \to \mu \gamma$ and  $\tau \to e \gamma$ decays
are within experimental reach.


To estimate the overall scale of  $\tau \to (\mu,e) \gamma$ rates,
we must specify the value of the LFV couplings, since they are not
determined by the SPS conditions.
In the mass-insertion and leading-log approximation, assuming
that the leading LFV couplings appear in the left-handed
slepton sector, we can write
\begin{equation}
\frac{{\rm \BR}(l_j\to l_i\gamma)}{{\rm \BR}(l_j\to l_i \bar{\nu}_i\nu_{j})}
\approx
\frac{\alpha^3}{G_{F}^2}
\frac{\left|\left(m^2_{\widetilde L} \right)_{ji}\right|^2}{M_{S}^8}
\tan^2\beta \nonumber, \label{eq:min}
\end{equation}
where, to a good approximation,
$M_{S}^8\simeq 0.5 M_0^2 M_{1/2}^2 \times (M_0^2 + 0.6 M_{1/2}^2 )^2$.
In a Grand Unified Theory (GUT) with heavy right-handed neutrinos,
the off-diagonal entries of the slepton mass matrix $m^2_{\widetilde L}$
are likely to be dominated by the flavor mixing in
the (s)neutrino sector.
These terms can be expressed as
\begin{equation}
\label{eq:leadinglog}
\left(m^2_{\widetilde {L}}\right)_{ji} \approx
-\frac{6M_0^2+2A_0^2}{16\pi^2}\ \delta_{ij},
\end{equation}
where $\delta_{ij} = \left(Y^\dagger_\nu Y_\nu\right)_{ji}
\log(M_{GUT}/M_{R})$ in terms of the neutrino
Yukawa couplings ($Y_\nu$), the average
heavy right-handed neutrino mass ($M_R$)
and the GUT scale ($M_{GUT} \sim 10^{15}$--$10^{16}$~GeV).
Given the large phenomenological value
of the 2--3 mixing in the neutrino sector
(and the corresponding suppression of the 1--3 mixing)
we expect $|\delta_{32}| \gg |\delta_{31}|$
hence $\BR(\tau \to \mu \gamma) \gg \BR(\tau \to e \gamma)$.
For sufficiently heavy right-handed neutrinos, the
normalization of $Y_\nu$ is such that
$\BR(\tau \to \mu \gamma)$ can reach
values in the $10^{-9}$ range.
In particular, $\BR(\tau \to \mu \gamma) \gsim 10^{-9}$
if at least one heavy right-handed neutrino has a mass around or above
$10^{13}$~GeV (in SPS 4) or $10^{14}$~GeV (in SPS 1a,1b,2,3,5).

A key issue that must be addressed is the role of
$\BR(\mu \to e \gamma)$ in
constraining  the LFV couplings and,
more generally, the correlations between
$\BR(\tau \to (\mu,e) \gamma)$ and $\BR(\mu \to e \gamma)$
in this framework. An extensive analysis of such questions
has been presented in Ref.~\cite{Antusch:2006vw,Arganda:2005ji},
under the hypothesis of a hierarchical spectrum
for the heavy right-handed neutrinos.

The overall structure of the
$\BR(\tau \to \mu \gamma)$ vs.~$\BR(\mu \to e \gamma)$
correlation in SPS~1a is shown in~Fig.~\ref{fig:herrero1}.
As anticipated, $\BR(\tau \to \mu \gamma)\sim 10^{-9}$
requires a heavy right-handed neutrino around or
above  $10^{14}$~GeV. This possibility is not excluded by
$\BR(\mu \to e \gamma)$ only if the 1--3 mixing in the
lepton sector (the $\theta_{13}$ angle of the neutrino
mixing matrix) is sufficiently small. This is a general
feature, valid at all SPS points, as illustrated in
Fig.~\ref{fig:SPS:t13:ad}.
In Table \ref{tab:SPS:BR} we show the predictions for $\BR(\tau\to\mu\,\gamma)$
and $\BR(\tau\to 3\,\mu)$ corresponding to the neutrino
mass parameters chosen in Fig.~\ref{fig:SPS:t13:ad}
(in particular $M_{N_3} = 10^{14}$~GeV), for the various SPS points.
Note that this case contains points that
are within the \superb sensitivity range,
yet are not excluded by $\BR(\mu \to e \gamma)$
(as illustrated in Fig.~\ref{fig:SPS:t13:ad}).


\begin{figure}[!htb]
\includegraphics[width=\linewidth,angle=0]{herrero1.eps}
\caption{\label{fig:herrero1}
$\BR(\tau \to \mu \gamma)$ vs.~$\BR(\mu \to e \gamma)$
in SPS~1a,
for three reference values of the heavy right-handed neutrino
mass and several values of $\theta_{13}$.
The horizontal dashed (dotted) line denotes the present experimental
    bound (future sensitivity) on $\BR(\mu \to e \gamma)$.
    All other relevant parameters are set to the values specified in
    Ref.~\cite{Antusch:2006vw}.
     }
\end{figure}


\begin{figure}[!htb]
\includegraphics[angle=-90,width=\linewidth]{muegSPS.eps}
\caption{\label{fig:SPS:t13:ad}
$\BR(\mu \to e\, \gamma$) as a function of $\theta_{13}$ (in degrees)
for various SPS points. The dashed (dotted) horizontal line denotes the present experimental
    bound (future sensitivity). All other relevant parameters are
    set to the values specified in
    Ref.~\cite{Antusch:2006vw}.
           }
\end{figure}


\begin{table}[!htb]
\caption{\label{tab:SPS:BR}Predictions for  $\BR(\tau \to \mu\, \gamma$) and \hbox{$\BR(\tau\to 3\,\mu$)}
         corresponding to the SPS points. The values of $m_{N_i}$ and $m_{\nu_1}$
         are as specified in Fig.~\ref{fig:SPS:t13:ad}~\cite{Antusch:2006vw}.}

\begin{tabular}{ccccccc}
\hline\hline
SPS               &  1\,a   &  1\,b   &  2  &  3  &  4  &  5   \\ \hline
$\BR(\tau\to\mu\gamma)\times 10^{-9}$ & 4.2 & 7.9 & 0.18 & 0.26 & 97 & 0.019 \\
$\BR(\tau\to 3\mu)\times 10^{-12}$ & 9.4 & 18 & 0.41 & 0.59 & 220 & 0.043 \\ \hline\hline
\end{tabular}
\end{table}

\subsubsection*{LFV in the NUHM scenario}

At large $\tan\beta$ and not too heavy Higgs masses,
another class of LFV interactions is relevant,
the effective coupling between a  $\mu$--$\tau$ pair
and the heavy (scalar and pseudoscalar) Higgs bosons.
This coupling can
overcome the constraints on $\BR(\tau\to \mu\mu\mu)$
and $\BR(\tau\to\mu\eta)$ dictated by $\BR(\tau\to\mu\gamma)$
in the dipole-dominance scenario.
Such a configuration cannot be realized in the CMSSM, but
it could be realized in the so-called NUHM SUSY scenario,
which is also theoretically well-motivated and rather general.
In such a framework, there are specific regions of the
parameter space in which $\tau\to\mu\eta$ could have
a branching ratio in the $10^{-9}$--$10^{-10}$ range,
comparable or even slightly
larger than $\BR(\tau\to\mu\gamma)$~\cite{Paradisi:2005tk}.


Finally, in more exotic New Physics frameworks, such as SUSY
without R parity, Little Higgs Models with T parity (LHT)
or $Z^{'}$ models with non-vanishing LFV couplings ($Z^{'}\ell_{i}\ell_{j}$),
the $\tau\to \mu\mu\mu$ rate could be as large as,
or even larger than $\tau\to\mu\gamma$ (see {\it e.g.},~\cite{Raidal:2008jk}).
In this respect, an improvement of $\BR(\tau\to \mu\mu\mu)$
at the $10^{-10}$ level would be interesting even
with $\BR(\tau\to\mu\gamma)\lsim 10^{-9}$.
