%
% ---------------------------------------------------------------------------
\subsection{Non-standard neutrino interactions}
\label{sec:nsi}
% --

Since the Neutrino Factory is the most ambitious concept available for
advancing our knowledge about lepton flavour physics, the search for new and
exotic phenomena in the lepton sector will be an important part of its physics
program.  In this section, we will outline the results that can be obtained on
non-standard neutrino interactions, non-unitarity in the lepton mixing matrix,
and sterile-neutrino scenarios.


%-------------------------------------------------------------------------

\subsubsection{Effective theory of new physics in the neutrino sector}

Without any knowledge of possible high-energy extensions of the Standard
Model, an effective theory approach is well-suited to parameterising the
effects of such extensions on neutrino-oscillation experiments.  Non-standard
interactions (NSI) mediated by effective higher-dimensional operators can lead
to additional, possibly flavour-violating, contributions to the MSW potential
that neutrinos experience when travelling through matter, and they can affect
the neutrino production and detection processes~\cite{Wolfenstein:1977ue,
Grossman:1995wx}.  The neutrino oscillation probability in the presence of NSI
can be written:
\begin{align}
  P(\nu^s_\alpha \rightarrow \nu^d_\beta)
    &= \Big| \big[ \big( 1 + \eps^d \big)^T \,\, e^{-i H_{\rm NSI} L} \,\,
                   \big( 1 + \eps^s \big)^T \big]_{\beta\alpha} \Big|^2 \,,
  \label{eq:P-NSI}
\end{align}
with the Hamiltonian:
\begin{align}
  H_{\rm NSI} &= U \begin{pmatrix}
    0 &                      & \\
    & \Delta m_{21}^2 / 2E & \\
    & & \Delta m_{31}^2 / 2E
                   \end{pmatrix} U^\dag
  + V_{\rm MSW} (\mathbb{1} + \eps^m) \,.
\end{align}
Here, $U$ is the lepton-mixing matrix, $V_{\rm MSW}$ is the
Mikheyev-Smirnov-Wolfenstein (MSW) potential corresponding to the matter
traversed by the neutrinos, and $\eps^d$, $\eps^s$, and $\eps^m$ are complex $3
\times 3$ matrices parameterising the relative strength of the non-standard
effects compared to Standard Model weak interactions. If $\eps^s_{\alpha\beta}$
is non-zero, neutrinos produced in interactions with charged leptons of flavour
$\alpha$ will have a non-standard component of flavour $\beta$. If
$\eps^d_{\beta\alpha} \neq 0$, neutrinos of flavour $\beta$ can produce a
charged lepton of flavour $\alpha$ in the detector. The Hermitian matrix
$\eps^m$ is responsible for non-standard matter effects.  Note that in most
concrete models, the elements of $\eps^d$, $\eps^s$, and $\eps^m$ are not all
independent.

Model-independent experimental constraints on the $\eps^{s,d,m}_{\alpha\beta}$
range from $10^{-4}$ to $\mathcal{O}(1)$~\cite{Fornengo:2001pm,
Davidson:2003ha, GonzalezGarcia:2007ib, Biggio:2009kv, Biggio:2009nt}, while
model-dependent limits are often much tighter~\cite{Antusch:2008tz,
Gavela:2008ra} since, in concrete models, NSI in the neutrino sector are
usually accompanied by non-standard effects in the charged-lepton sector that
are constrained, for example, by rare decay searches or measurements of the
weak mixing angle.

If NSI are induced by dimension-6 operators that arise when heavy mediator fields
are integrated out, we expect $\eps^{s,d,m}_{\alpha\beta} \sim g^2 M_W^2 /
g_{\rm NSI}^2 M_{\rm NSI}^2$, where $g$ is the weak gauge coupling, $g_{\rm
NSI}$ is the coupling constant of the heavy mediators, and $M_{\rm NSI}$ is
their mass.  If $g_{\rm NSI} \sim g$ and $M_{\rm NSI} \sim \text{few} \times
100$~GeV, we naively expect $\eps^{s,d,m}_{\alpha\beta} \sim 0.01$. Most
dimension-6 operators of this magnitude are, however, already ruled out by
several orders of magnitude by constraints from the charged-lepton sector. 

Such constraints are avoided if NSI are generated by dimension-8
operators such as $(\bar{L}_\alpha H) \gamma^\rho (\bar{H}^\dag
L_\beta) (\bar{E}_\gamma \gamma_\rho E_\delta)$, where $L$, $H$, and
$E$ are a left-handed lepton field, the Standard Model Higgs field,
and a right-handed charged lepton field, respectively, and $\alpha$,
$\beta$, $\gamma$, $\delta$ are flavour indices.  When $H$ acquires a
non-zero vacuum expectation value $v$, this operator leads to
$SU(2)$-violating four-fermion contact interactions, so that NSI in
the neutrino sector can be generated without inducing
phenomenologically-problematic four-charged-lepton couplings or other
harmful effects. On the other hand, this procedure leads to an
additional suppression of order $v2/M2$ and requires some cancellation
conditions at tree level~\cite{Gavela:2008ra}. Larger non-standard
effects could be generated in theories involving \emph{light}
mediators ($\ll M_W$) with very small couplings. Models containing
light new particles have recently received a lot of interest in the
context of Dark Matter searches (see for example references
\cite{Fitzpatrick:2010em, Hooper:2010uy, Arina:2010rb, Lavalle:2010yw,
  Cumberbatch:2010hh, Kuflik:2010ah}), but possible effects in the
neutrino sector are less well explored in the literature.  For very
light mediators, processes with small or no momentum transfer, such as
coherent forward scattering leading to non-standard neutrino matter
effects, will be enhanced compared to processes with large momentum
transfer from which most NSI constraints are derived.

Finally, there are more exotic scenarios of new interactions in the neutrino
sector that could be tested at the Neutrino Factory. One example for such a
scenario is long-range leptonic forces~\cite{Joshipura:2003jh, Grifols:2003gy,
GonzalezGarcia:2006vp, Bandyopadhyay:2006uh, Samanta:2010zh, Heeck:2010pg},
which are phenomenologically equivalent to NSI even though the theoretical
motivation is very different.


\subsubsection{Neutrino Factory sensitivities to non-standard interactions}

The far detectors of the Neutrino Factory will offer a unique opportunity to
constrain or measure the elements of $\eps^m$, corresponding to non-standard
matter effects. This is shown in figure \ref{fig:NSI-discreach} under the
assumption that only one of the $\eps^m_{\alpha\beta}$ is non-zero at a time.
We see that existing model-independent constraints can be significantly
improved, and that the Neutrino Factory would begin to probe the parameter
region $\eps^m_{\alpha\beta} \lesssim 0.01$ that is most interesting from the
model-building point of view. We also compare the sensitivity
to standard oscillation observables ($\theta_{13}$, CP violation, MH)
in a conventional fit to the sensitivities that may be achieved if one
NSI parameter ($\epsilon^m_{e\tau}$) is allowed to be non-zero. 
It turns out that allowing for the possibility of non-standard effects
reduces the sensitivity to standard oscillation observables.
\begin{figure}
  \begin{center}
    \includegraphics[width=13cm]{01-PPEG/Figures/NSI-discreach.eps}
  \end{center}
  \vspace{-0.5cm}
  \caption{Summary of the Neutrino Factory performance with and
    without the presence of non-standard interactions. The IDS-NF
    setup with two 50~kTon magnetised iron detectors (MIND) at
    baselines of 4\,000~km and 7\,500~km was used, and the ``true''
    parameter values $\sin^2 2\theta_{13} = 0.001$ and 
    $\delta_{\rm CP} = 3 \pi/2$ were assumed (the detector mass used
    differs from the present IDS-NF baseline).
    The plot shows that sensitivities are poor at
    $E_\mu = 5$~GeV (light bars) due to the high energy threshold of
    the MIND detector, but increase dramatically at $E_\mu = 25$~GeV
    (medium light bars). The benefit from increasing $E_\mu$ further
    to 50~GeV (dark bars) is only marginal, as is the benefit from
    including a silver channel $\nu_\tau$ detector at 4\,000~km
    (``Silver*''). Current model-independent limits on NSI are taken
    from~\cite{Davidson:2003ha, Biggio:2009nt, GonzalezGarcia:2007ib}.
    No strong limits exist for $\epsilon^m_{ee}$ and
    $\epsilon^m_{e\tau}$. Figure taken from reference~\cite{Kopp:2008ds}.}
  \label{fig:NSI-discreach}
\end{figure}

For NSI affecting the neutrino production and detection mechanisms,
even tighter constraints can often be obtained from the Neutrino
Factory \emph{near} detector.  To demonstrate this, we consider
non-standard contributions to the charged-current neutrino production
and detection processes~\cite{Grossman:1995wx}. New contributions to
$\nu_e$ and $\nu_\mu$ production or detection are difficult to observe
because they are degenerate with systematic biases in the neutrino
cross section calculations or in the normalisation of the beam flux. A
near detector with charge identification capabilities, however, would
be sensitive to flavour-violating processes such as $\mu \to e \nu_e
\nu_e$, $\mu \to e \nu_\mu \nu_\mu$, $\nu_e + N \to \mu + X$, $\nu_\mu
+ N \to e + X$~\cite{Tang:2009na}.  Another interesting possibility is
the construction of a detector sensitive to $\nu_\tau$ to look for
processes such as $\mu \to e \nu_{\alpha} \nu_\tau$ or $\nu_\alpha + N
\to \tau + X$ for $\alpha = e, \mu$. This possibility has been
considered in references
\cite{Tang:2009na,Antusch:2009pm,Meloni:2009cg,MINSIS2009Madrid}.  In
table~\ref{tab:cc-nsi-bounds}, we summarise some expected bounds on CC
NSI operators for the Neutrino Factory with and without a near
detector capable of $\nu_\tau$ detection.  We see that some bounds can
be improved by almost two orders of magnitude compared to their
current values, and that the Neutrino Factory would probe the
parameter region expected if NSI are induced by
$\mathcal{O}(\text{TeV})$ mediators.
\begin{table}
  \caption{Expected sensitivities and current bounds for the coupling constants
    (relative to Standard Model weak interactions) of some NSI operators
    relevant to the Neutrino Factory experiment.  Numbers in
    parenthesis are for a model with charged-singlet
    scalar mediation, while numbers without parenthesis are model-independent.}
  \begin{center}
    \begin{tabular}{cccc}
    \hline \hline
      & Sensitivity w/o $\nu_\tau$ ND & Sensitivity w/ $\nu_\tau$ ND
      & Current bound~\cite{Tang:2009na,Antusch:2008tz} \\
      \hline
      $\epsilon^{s}_{e\tau}$ ($\mu \to e \nu_\mu \nu_\tau$)
          & $ 4 \cdot 10^{-3}$
          & $ 7 \cdot 10^{-4}$
          & $ 3.2 \cdot 10^{-2}$ ($ 1.8 \cdot 10^{-3}$) \\
      $\epsilon^{s}_{\mu\tau}$ ($\mu \to e \nu_e \nu_\tau$)
          & $ 4 \cdot 10^{-1}$ ($3\cdot 10^{-3}$)
          & $ 6 \cdot 10^{-4}$ ($6 \cdot 10^{-4}$)
          & $ 3.2 \cdot 10^{-2}$ ($ 1.9 \cdot 10^{-3}$) \\
      \hline \hline
    \end{tabular}
  \end{center}
  \label{tab:cc-nsi-bounds}
\end{table}


\subsubsection{Non-standard interactions in a low-energy Neutrino Factory}

Let us now comment on the NSI sensitivity of a low-energy Neutrino Factory
(LENF).  We have analysed the LENF setup defined in
\cite{FernandezMartinez:2010zza}, using the MonteCUBES (Monte Carlo
Utility Based Experiment Simulator) software package
\cite{Blennow:2009pk,MCBhome}. 
The LENF has leading order sensitivity to 
$\varepsilon^m_{e\mu}$ and $\varepsilon^m_{e\tau}$ from the golden and platinum
channels. We have found that the sensitivity is limited by the correlations
between the standard oscillation and non-standard parameters, and thus
that increasing statistics does \emph{not} increase the sensitivity of the
setup, in direct contrast to standard oscillation measurements. Instead, it is
necessary to include information from a second baseline, or from a
complementary channel such as the platinum channel. With the performance of the
platinum channel as assumed in \cite{FernandezMartinez:2010zza} ($47\%$
efficiency, $10^{-2}$ background), the platinum channel has the greatest impact
for \emph{large} values of $\theta_{13}$. However, we have found that a
platinum channel with hypothetically perfect performance ($\sim100\%$
efficiency, negligible background) has an effect even for small values of
$\theta_{13}$, indicating that there is a critical signal-to-background
threshold that must be overcome in order for it to become effective.

In figure \ref{fig:th13d} we show how the platinum channel helps to maintain
sensitivity to the standard oscillation parameters, when marginalisation over
all NSI parameters, as well as the oscillation parameters is performed: we show
the $68\%$, $90\%$ and $95\%$ regions obtained in the $\theta_{13}-\delta$
plane when only the golden channel and $\nu_{\mu} / \bar{\nu}_{\mu}$
disappearance channels are used (`Scenario 1' - dotted blue lines), and when
combined with the platinum channel (`Scenario 2' - solid red lines). The
left-hand figure shows the results when \emph{only} the standard oscillation
parameters are marginalised over, and the right-hand figure the results when
all oscillation \emph{and} NSI parameters are taken into account. This shows
that the sensitivity of Scenario 1 to $\theta_{13}$ and $\delta$ is diminished
by taking into account the possibility of non-zero NSIs, but that Scenario 2
is almost unaffected---the addition of the platinum channel
makes the setup more robust.
\begin{figure}
\includegraphics[width=7.5cm,height=6cm]{01-PPEG/Figures/./so.eps}
\includegraphics[width=7.5cm,height=6cm]{01-PPEG/Figures/./nsi.eps}
\caption{$68\%$, $90\%$ and $95\%$ regions in the $\theta_{13}-\delta$ plane
  when marginalising over standard oscillation parameters only (left) and when
  marginalising over all standard oscillation and NSI parameters (right), for
  true values of $\theta_{13}=5^{\circ}$, $\delta=0$ and
  $\varepsilon^m_{e\mu}=\varepsilon^m_{e\tau}=0$. Scenario 1 is a LENF setup without
  platinum channel; Scenario 2 is the LENF with platinum channel. Figure taken from reference~\cite{FernandezMartinez:2010zza}.}
  \label{fig:th13d}
\end{figure}

For measuring the NSI parameters themselves, the LENF running for 5 years per
polarity (i.e., 10 years running are assumed, the same assumption that
has been made in evaluating the performance of the other long-baseline
experiments) has sensitivity to $\varepsilon^m_{e\mu}$ and
$\varepsilon^m_{e\tau}$ down to $\sim10^{-2}$ (figure \ref{fig:eps}). 
If the running time is doubled, no significant improvement
is gained, indicating that the limiting factor is not statistics. 
These bounds are a significant improvement upon the current bounds 
\cite{Biggio:2009kv,Biggio:2009nt}, but not as strong as those which
the Neutrino Factory can obtain.
\begin{figure}
  \includegraphics[width=7.5cm,height=6cm]{01-PPEG/Figures/./emu.eps}
  \includegraphics[width=7.5cm,height=6cm]{01-PPEG/Figures/./et.eps}
  \caption{$68\%$, $90\%$ and $95\%$ regions in the
    $\theta_{13}-\varepsilon^m_{e\mu}$ (left) and
    $\theta_{13}-\varepsilon^m_{e\tau}$ (right) plane for
    $\theta_{13}=5^{\circ}$, $\delta=0$, and
    $\varepsilon^m_{e\mu}=\varepsilon^m_{e\tau}=0$.  Figure taken from
    reference~\cite{FernandezMartinez:2010zza}.}
  \label{fig:eps}
\end{figure}

% ---------------------------------------------------------------------------
\subsubsection{Non-unitarity of the leptonic mixing matrix}
% ---------------------------------------------------------------------------

A non-unitary mixing matrix in the charged current-coupling of
neutrinos and charged leptons is a general prediction of extensions 
of the Standard Model (SM) that include new fermion degrees of freedom that can
mix with the SM leptons \cite{Langacker:1988ur}. Indeed, the full mixing matrix
including the extra degrees of freedom would thus be larger than the standard
three-by-three matrix. 
While the full mixing matrix should be unitary, such a
constraint does not apply to the $3 \times 3$ sub-matrix accessible at low
energies. In particular, many extensions of the SM that try to accommodate the
evidence for neutrino masses and mixings from neutrino-oscillation experiments
introduce such extra leptonic degrees of freedom. This is the case for example
of the type-I and type-III seesaw models. The study of deviations from
unitarity of the leptonic mixing matrix can therefore provide a tool to explore
the origin of neutrino masses beyond the SM~\cite{Broncano:2002rw,Broncano:2003fq,
Antusch:2009gn,Gavela:2009cd,Antusch:2009gn,Ohlsson:2010ca}.

In a completely general way, the non-unitary leptonic mixing matrix
$N$ can be parametrised as the product of a Hermitian matrix,
$1 + \eta$, times a unitary matrix $U$
\cite{FernandezMartinez:2007ms}: 
\begin{equation}
N=(1+\eta)U\;.
\label{eq:param}
\end{equation}
Such a modification of the SM charged-current interactions would affect lepton
universality tests, the measurement of $G_F$ from muon decay compared to other
measurements, rare lepton decays and the invisible width of the $Z$,
and strong constraints can be derived on the allowed size of
$\eta$~\cite{Langacker:1988ur,Nardi:1994iv,Tommasini:1995ii,Antusch:2006vwa,Antusch:2008tz}:
$\eta_{ee}<2.0 \times 10^{-3}$, $\eta_{e \mu}<5.9 \times 10^{-5}$, $\eta_{e
\tau}<1.6 \times 10^{-3}$, $\eta_{\mu \mu}<8.2 \times 10^{-4}$, $\eta_{\mu
\tau}<1.0 \times 10^{-3}$ and $\eta_{\tau \tau}<2.6 \times 10^{-3}$, at the
$90~\%$ CL Up to order $\eta$ effects, the matrix $U$ can be identified with
the PMNS matrix as extracted from experimental data without taking
non-unitarity into account. Since $\eta$ is much smaller than the experimental
uncertainties on the PMNS matrix elements, the corrections are negligible.

The effects of a non-unitary mixing at the Neutrino Factory have been
studied in references \cite{FernandezMartinez:2007ms, Holeczek:2007kk,
Goswami:2008mi, Antusch:2009pm, Meloni:2009cg}.  In particular,
in~\cite{Antusch:2009pm} this study was performed in detail for a setup very
similar to the IDS-NF baseline Neutrino Factory (muon energy 25~GeV,
two 50~kt magnetised iron detectors at baselines of 4\,000~km and
7\,500~km) and considering simultaneously the effect of all extra
non-unitarity parameters in the fit.  
To 
perform a fit in this high-dimensional parameter space, the Markov Chain Monte
Carlo tool MonteCUBES, a plug-in for the widely used GLoBES
software~\cite{Huber:2004ka,Huber:2007ji}, has been used.  It was found that,
given the very stringent present limits on the allowed deviations from
unitarity, the Neutrino Factory could only improve constraints on the 
$\eta_{e \tau}$ and $\eta_{\mu \tau}$ elements of the matrix $\eta$.
Indeed, it is
easier to search for off diagonal elements since they imply lepton flavour
violation and the present bounds on $\eta_{e \mu}$ from $\mu \to e \gamma$ are
too strong to improve through measurements of neutrino oscillations. 
The sensitivity of the Neutrino Factory to these parameters could be
significantly improved if a near detector capable of $\tau$ detection
is included, since the zero-distance effects caused by the unitarity
deviations could be tested.  
\begin{figure}
 \begin{center}
  \includegraphics[width=0.49\textwidth]{01-PPEG/Figures/tm-sens.eps}
  \includegraphics[width=0.49\textwidth]{01-PPEG/Figures/te-sens.eps}
 \end{center}
 \caption{ The 90~\% confidence level sensitivity of the IDS Neutrino
   Factory to the unitarity violating parameters $\eta_{\tau\mu} =
   |\varepsilon_{\tau\mu}|e^{i\phi_{\tau\mu}}$ (left) and $\eta_{\tau
     e} = |\varepsilon_{\tau e}|e^{i\phi_{\tau e}}$ (right). The
   different curves correspond to different sizes of the near $\tau$
   detector, from left to right, 10~kTon, 1~kTon, 100~ton, no near
   detector.  Figure taken from reference~\cite{Antusch:2009pm}.}
 \label{fig:sens}
\end{figure}

In figure \ref{fig:sens} the sensitivity limits on the parameters 
$\eta_{\tau\mu} = |\varepsilon_{\tau\mu}|e^{i\phi_{\tau\mu}}$ and 
$\eta_{\tau e} = |\varepsilon_{\tau e}|e^{i\phi_{\tau e}}$ are
depicted. 
The different curves correspond to different sizes of the $\tau$
detector close to the muon storage ring; from left to right, 
10~kTon, 1~kTon, 100~ton, no near detector. 
As can be seen, the sensitivity to
the real part of $\eta_{\mu \tau}$ is remarkable, down to $\sim 5 \times
10^{-4}$ without the inclusion of any near detector. This stems from the fact
that matter effects in the $\nu_\mu$ disappearance channel depend linearly on
this parameter. The inclusion of near detectors able to detect $\nu_\tau$
appearance with increasing size improves also the sensitivity to the imaginary
part of $\eta_{\mu \tau}$, since the zero-distance effect only depends on the
modulus, but the sensitivity is not as good, reaching $\sim 10^{-3}$ for a
1~kTon detector, since the effect is second order. 
A larger detector would thus
be required to improve over present limits. 
Regarding $\eta_{e \tau}$, the
sensitivity of the Neutrino Factory in the absence of a near
detector sensitive to $\tau$ production is much poorer, down to 
$\sim 2 \times 10^{-3}$ for some values of the phase. 
This mainly stems
from second-order terms in the $\nu_\mu$ appearance channel. The inclusion of
near detectors to search for $\tau$ appearance again improves the sensitivity.
However, as for the $\eta_{\mu \tau}$ element, detectors larger than 1~kTon
would be needed to improve over present bounds.  

\subsubsection{Discovery reach for sterile neutrinos}

Sterile neutrinos (Standard Model singlet fermions) are a very common
prediction in many models of new physics and, if they are light and mix with
the three standard neutrinos, it is possible to search for them at neutrino
oscillation experiments. Sterile neutrinos have received a lot of attention in
the context of seemingly anomalous results from the LSND and MiniBooNE
experiments~\cite{Aguilar:2001ty, AguilarArevalo:2010wv}.
%, see e.g.\
%\cite{Maltoni:2007zf, Meloni:2010zr, Dighe:2007uf, Giunti:2010wz, Akhmedov:2010vy}
Here, we will discuss
the discovery reach for sterile neutrinos at the Neutrino Factory.  We consider
the simplest scenario in which only one sterile neutrino is added to the three
active neutrinos of the Standard Model in the so-called 3+1 scheme, which
recovers the standard picture in the case of small active-sterile
mixings.  
For the sake of simplicity, we focus only on the scheme where the
fourth state is the heaviest and the normal hierarchy is assumed in
the standard sector. 

\paragraph{Theoretical formalism/notation}
\label{sec:ana}
The numerical results on the discovery reach can be understood from the
analytical expressions of the transition probabilities.  Instead of using a
parametrisation-independent approach, where all the probabilities are directly
expressed in terms of matrix elements of the $4 \times 4$ unitary mixing matrix
$U$, we rely on a particular  parametrisation which allows the
standard PMNS matrix to be recovered for small ``new'' mixing angles: 
\begin{equation}
    \label{equ:3+1param1}
    U =
    R_{34}(\theta_{34} ,\, 0) \; R_{24}(\theta_{24} ,\, 0) \;
    R_{14}(\theta_{14} ,\, 0) \;
    R_{23}(\theta_{23} ,\, \delta_3) \;
    R_{13}(\theta_{13} ,\, \delta_2) \; 
    R_{12}(\theta_{12} ,\, \delta_1) \,,
\end{equation}
where $R_{ij}(\theta_{ij},\ \delta_l)$ are the complex rotation matrices in the
$ij$-plane.  In the short-baseline limit $|\Delta_{41}|= \Delta m^2_{41}L/4 E
\sim \mathcal{O}(1) \gg |\Delta_{31}|$,  the matter effects can be safely
ignored, and the relevant probabilities read:
\begin{align}
&\mathcal{P}_{e\mu}= \mathcal{P}_{\mu e} = 4 c_{14}^2 s_{14}^2 s_{24}^2 \sin ^2 \Delta_{41} \label{equ:pem2}\,;  \\
&\mathcal{P}_{ee}=1-\sin ^2\left(2 \theta _{14}\right)\sin ^2 \Delta_{41}  \label{equ:pee2}\,;\\
&\mathcal{P}_{\mu \tau}= 4 c_{14}^4 c_{24}^2 s_{24}^2 s_{34}^2 \sin ^2
  \Delta_{41}\,; {\rm and} \\
&\mathcal{P}_{\mu\mu}=1-c_{14}^2 s_{24}^2  \left[3+2 c_{14}^2 \cos \left(2 \theta _{24}\right)-\cos \left(2 \theta _{14}\right)\right]\sin ^2 \Delta_{41}  \label{equ:pmm2}\,;
\end{align}
where we used the short-hand notation $\Delta_{ij}= \Delta m^2_{ij} L/4 E$.
From these probabilities we can see that $\theta_{24}$ can be measured by
$\mathcal{P}_{\mu \mu}$  and  $\theta_{14}$  by   $\mathcal{P}_{e e}$. On the
other hand, $\theta_{34}$ only shows up in combination with the other small
mixing angles.  For long baselines,  some of the relevant features of the
probability transitions can be well understood using simple perturbative
expansions: for $\Delta_{31} = \mathcal{O}(1) \ll \Delta_{41}$ and up to the
second order in
$
s_{13},  s_{14},  s_{24},  s_{34}, \hat s_{23}=\sin\theta_{23}-\frac{1}{\sqrt{2}}
$,
and considering
$
\Delta_{21}
$
as small as $s_{ij}^2$,
we obtain:
\begin{eqnarray}
\mathcal P_{\mu\mu} &=& \cos^2 (\Delta_{31}) (1 - 2 s_{24}^2) + 8 \hat s_{23}^2\sin^2(\Delta_{31}) + c_{12}^2 \Delta_{12}   \sin(2\Delta_{31}) +  \nonumber \\
&&   2 s_{24} s_{34} \cos \delta_3 \Delta_n  \sin(2\Delta_{31}) - \label{pmumulo}\\ 
&& 2 s_{13}^2 \Delta_{31}\cos( \Delta_{31})\,\frac{(\Delta_{31}-\Delta_e) \Delta_e \sin (\Delta_{31})-\Delta_{31}\sin\left(\Delta_{31}-\Delta_e \right)\sin (\Delta_e )}{(\Delta_{31}-\Delta_e)^2} \, ;
 and \nonumber \\
\mathcal P_{\mu\tau} &=&\sin^2 (\Delta_{31})(1-8 \hat s_{23}^2-s_{24}^2-s_{34}^2)-c_{12}^2 \Delta_{12} \sin (2\Delta_{31})-\nonumber \\
&& s_{24}s_{34}\sin (2\Delta_{31})\left[2 \Delta_n \cos \delta_3-\sin\delta_3\right] - \label{pmutaulo}  \\ 
&& s_{13}^2 \Delta_{31}\sin \Delta_{31} \,\frac{\Delta_{31}
\left\{\sin (\Delta_{31}-\Delta_e) +\sin (\Delta_e )\right\}-2 (\Delta_{31}-\Delta_e) 
\Delta_e \cos (\Delta_{31})}{(\Delta_{31}-\Delta_e)^2} \, ;\nonumber 
\end{eqnarray}
from which we learn that with long baselines, $\theta_{24}$ is best
accessed via $\mathcal{P}_{\mu \mu}$ with the first term proportional to
$\cos^2 (\Delta_{13})$.  The leading sensitivity to $\theta_{34}$ can be
expected from $\mathcal{P}_{\mu \tau}$  (the {\it discovery channel} as introduced
in \cite{Donini:2008wz}).  Notice also the dependence on the phase $\delta_3$
in both probabilities, which makes them useful to check whether other sources of
CP violation beside the standard one ($\delta_2$ in our parametrisation) can
be tested at the Neutrino Factory.


\paragraph{Results for IDS-NF baseline}

We first discuss general constraints on the new mixing angles $\theta_{14}$,
$\theta_{24}$, and $\theta_{34}$,  and on the additional mass squared
difference $\Delta m_{41}^2$ without any additional assumptions
\cite{Meloni:2010zr}.  
The oscillation channels considered, simulated with a
modified version of the GLoBES software, are electron-to-muon neutrino
oscillations (appearance channels) and muon-to-muon neutrino
oscillations (disappearance channels) (see \cite{Meloni:2010zr} for
all details of the simulations).  
In addition to the IDS-NF baseline setup, we have also considered
near detectors with a fiducial mass of $32$\,Ton and a distance of
$d=2$~km from the end of the decay straight, which corresponds to an
effective baseline of 2.28~km. 
We show the 
resulting exclusion limits in the $\theta_{i4}$---$\Delta m_{41}^2$ planes in
figure \ref{fig:theta-mass}.
\begin{figure}
\includegraphics[width=0.32\textwidth]{01-PPEG/Figures/th14.eps}
\includegraphics[width=0.32\textwidth]{01-PPEG/Figures/th24.eps}
\includegraphics[width=0.32\textwidth]{01-PPEG/Figures/th34.eps}
\caption{
  The exclusion limit at 90\% CL (2 d.o.f) for
  $\sin^22\theta_{i4}$--$\Delta m_{41}^2 (i=1,2,3)$ (region on r.h.s.\ of
  curves excluded). We have assumed a Neutrino Factory with two 50\,kTon
  magnetised iron far detectors (FD) at  $4\,000$~km and $7\,500$~km, and with
  32\,Ton near detectors (ND) at $\sim 2$~km. The thick solid black curve is for
  the full ND + FD setup, while the dashed blue curve shows the impact of the
  near detectors separately. The thin red dotted curve illustrates how the
  sensitivity would improve if parameter correlations and degeneracies were
  absent, so that only statistical and systematical uncertainties would
  contribute. Figure taken from reference~\cite{Meloni:2010zr}.}
\label{fig:theta-mass}
\end{figure}

The main sensitivity is obtained at about $\Delta m_{41}^2 \simeq 10 \,
\mathrm{eV}^2$, which comes from the distance chosen for the near detectors.
Since the efficiencies for muon-neutrino detection are typically better than
those for $\nu_e$, the sensitivity to $\theta_{24}$ is slightly better than
that to $\theta_{14}$ for our assumptions. As expected, there is no sensitivity
to $\theta_{34}$ coming from the near detectors, because the $\nu_\tau$
disappearance channel does not exist.  For the effect at long baselines, it is
useful to consider first the thin dashed curves in figure \ref{fig:theta-mass},
which correspond to a simulation including only statistical and systematic
errors, but no parameter correlations. In all three panels, the sensitivity
changes as a function of $\Delta m_{41}^2$ in the region where $\Delta m_{41}^2
\sim \Delta m_{31}^2$. This is due to the fact that the Neutrino Factory is
sensitive to the atmospheric oscillation frequency, whereas for $\Delta
m_{41}^2 \sim \Delta m_{21}^2$, no particular additional effects from the solar
frequency can be found. As expected (see equation (\ref{pmumulo})), the main
sensitivity is found for $\theta_{24}$.  However, there is also some
sensitivity to $\theta_{14}$, which vanishes after the marginalisation, and
some sensitivity to $\theta_{34}$, which is even present for $\Delta
m_{41}^2=0$ for systematics only. After we include parameter correlations by
marginalising over all free oscillation parameters (thick solid curves), only
the sensitivities to $\theta_{24}$ and $\theta_{34}$ remain in the $\Delta
m_{41}^2$ regions close to the atmospheric $\Delta m_{31}^2$ and above, where
the effects of $\Delta m_{41}^2$ average out. Very interestingly, note that
mixing-angle correlations destroy the sensitivities for $\Delta m_{41}^2=\Delta
m_{31}^2$, where $m_4=m_3$ and no additional $\Delta m_{41}^2$ is observable,
leading to small gaps (see horizontal lines).  We have tested that the
sensitivity to $\theta_{34}$ is a matter-potential-driven, statistics-limited,
higher-order effect present in the muon-neutrino disappearance channels.  We
see from the three panels of figure \ref{fig:theta-mass} that it is not easy to
disentangle the parameters for arbitrarily-massive sterile neutrinos.  Parameter
correlations lead to a pollution of the exclusion limit of a particular mixing
angle with $\Delta m_{41}^2$. In addition, there is a competition between
$\Delta m_{41}^2$ and $\Delta m_{31}^2$ at the long baseline. Near detectors,
on the other hand, have very good sensitivities to $\theta_{14}$ and
$\theta_{24}$ but cannot measure $\theta_{34}$. Nevertheless, the absolute
values of the sensitivities are quite impressive. Especially, $\theta_{24}$ can
be very well constrained close to the atmospheric mass squared difference
range. This indicates that sterile neutrino bounds in that range should be also
obtainable from current atmospheric neutrino oscillation experiments.  Note
that while one may expect some effect from $\mathcal P_{\mu \tau}$, we have
tested the impact of $\nu_\tau$ detectors at all baselines, and we have not
found any improvement of the sensitivities.  The reason is that the
$\theta_{34}$ effect at the long baseline comes with the same energy dependence
as the $\theta_{24}$ effect, which means that one cannot disentangle these, as
discussed in section \ref{sec:ana}.

\paragraph{Optimisation issues and sensitivity of alternative setups}

Although a similar analysis for CP-violating signals has not been carried out,
the authors of reference \cite{Donini:2008wz} have presented a first
analysis of the precision achievable in the simultaneous measurement
of mixing angles and CP-violating phases (but see
\cite{Donini:1999jc,Kalliomaki:1999ii,Donini:2001xp,Donini:2001xy,Dighe:2007uf,Donini:2008wz,Goswami:2008mi,Giunti:2009en,Yasuda:2010rj}
for preliminary analyses). The experimental setup 
adopted there is a Neutrino Factory with 50 GeV stored muons, with two
detectors of the hybrid-MIND type (a magnetised emulsion cloud chamber
followed by a magnetised iron tracking calorimeter), located at 
$L = 3\,000$~km and $7\,500$~km. 
The $\nu_\tau$ 
identification turns out to be of primary importance for the study of CP
violation. As an example, for the simultaneous measurement of $\theta_{34}$ and
$\delta_3$ it has been shown that,  using the $\nu_\mu$ disappearance channel only,
we are able to measure a non-vanishing $\delta_3$ for values of $\theta_{34}$
above $\sin^2 2 \theta_{34} \geq 0.4\ (\theta_{34} \geq 18^\circ)$,  and that
the detector at $L = 3\,000$~km has no $\delta_3$-sensitivity
whatsoever.  
Adding 
the $\nu_\mu \to \nu_\tau$ discovery channel data, the $L=3000$ km detector is
no longer useless for the measurement of $\delta_3$: spikes of
$\delta_3$-sensitivity for particular values of $\delta_3$ can be
observed, in some cases outperforming the far-detector results. 
However, it is in the combination of the two
baselines that a dramatic improvement in the $\delta_3$-discovery potential
is achieved.  When the $\nu_\mu \to \nu_\tau$ data are included, a
non-vanishing $\delta_3$ can be measured for values of $\theta_{34}$ as small
as $\sin^2 2 \theta_{34} = 0.06\ (\theta_{34} = 7^\circ)$ for $\theta_{24} =
5^\circ$ and $\sin^2 2 \theta_{34} = 0.10\ (\theta_{34} = 9^\circ)$ for
$\theta_{24} = 3^\circ$.

% ---------------------------------------------------------------------------
\subsubsection{New physics summary}
% ---------------------------------------------------------------------------

We have shown that, in addition to its unique capability of clarifying the
flavour structure of neutrinos in the standard framework, the Neutrino Factory
is also able to probe possible modifications of the oscillations pattern due to
new physics. For example, it is very sensitive to non-standard matter effects,
non-standard charged current interactions, non-unitarity in the leptonic mixing
matrix, and sterile neutrinos. Especially for non-standard effects modifying
the neutrino-production and -detection processes and for short-baseline
oscillations into sterile neutrinos, a crucial role is played by the near
detector.  A very important aspect of new physics searches at a Neutrino
Factory is the possibility to identify new sources of CP violation by studying
their effect on the oscillation pattern. Many other electroweak precision
experiments do not rely on interference phenomena and are therefore insensitive
to CP-violating phase factors.

Quantitatively, we have shown that the constraints the Neutrino Factory can set
on non-unitarity in the mixing matrix and on active-sterile neutrino mixing are
comparable to or slightly better than the existing ones, while current
model-independent bounds on non-standard neutrino interactions can be
significantly improved. The Neutrino Factory could probe interactions
100 times weaker than Standard Model weak interactions, and the low energy
option, though less sensitive, would still allow for improvements compared to
existing bounds.

The NSI sensitivity of the Neutrino Factory naively translates into
sensitivity to new physics scales of several 100~GeV, implying that
the Neutrino Factory could provide measurements complementary to those
expected from the LHC.  On the other hand, in many concrete models,
non-standard neutrino interactions large enough to be seen at the
Neutrino Factory are already ruled out by other experiments probing
correlated effects in the charged-lepton sector.  Therefore, from a
model-builder's point of view the new physics discovery reach of the
Neutrino Factory is limited.  However, as neutrinos have proved
theoretical prejudices to be wrong in the past, the versatility and
precision of a Neutrino Factory will still make new physics searches a
worthwhile and integral part of its physics program.
