\subsection{The physics case for advanced neutrino experiments}
\label{subsec:physicscase}

The unambiguous observation of neutrino oscillations is arguably the
most significant development in particle physics of the past two
decades \cite{GonzalezGarcia:2007ib}.  
It revealed, beyond reasonable doubt, that neutrinos have mass and
that leptons mix.  
Neutrino masses are not accounted for in the Standard Model of
particle physics.  
It is not hard to modify the Standard Model Lagrangian to accommodate
neutrino masses \cite{Mohapatra:2005wg}.
Indeed, one can write down several different ``New Standard Model''
Lagrangians that render neutrinos massive that are consistent with the
available data.
The candidates for the New Standard Model introduce various new
degrees of freedom at a variety of energy scales (from below the
electron-Volt scale all the way to the Grand Unification scale).  
Part of the mission of particle-physics experiments in the next few
decades will be to elucidate the mechanism by which neutrino mass is
generated. 
The next generation of neutrino experiments will represent a moderate
extension of existing techniques.
This makes them technologically attractive, since the technical risks
are relatively low, but also limits their physics sensitivity. 
To exploit to the full the discoveries made to date in neutrino
physics, advanced neutrino experiments, based on novel technologies
are required.
The Neutrino Factory is the ultimate advanced neutrino oscillation
facility.
In this section, the scientific case for advanced neutrino oscillation
experiments in general and of the Neutrino Factory in particular is
reviewed.

While the mechanism behind neutrino-mass generation remains unknown, a 
very successful phenomenological description exists.  
It considers the existence of three massive neutrinos 
$\nu_1$, $\nu_2$, and $\nu_3$ with masses $m_1$, $m_2$, and $m_3$,
respectively.
These mass eigenstates are linear combinations of the ``active''
neutrino flavours, $\nu_e$, $\nu_{\mu}$, and $\nu_{\tau}$, labelled
according to the way they interact with the $W$-boson and the
charged-leptons, $e$, $\mu$, and $\tau$.
One can pick a weak-basis in which neutrinos with a well-defined
flavour are related to neutrinos with a well-defined mass via the
unitary lepton mixing matrix $U$: $\nu_{\alpha}=U_{\alpha i}\nu_i$
($\alpha=e,\mu,\tau$, $i=1,2,3$).  
It is customary to define the
neutrino masses as follows: $m_1<m_2$ while $m_3$ is either the
heaviest or the lightest neutrino.  
To identify $m_3$, $|m_3^2-m^2_1|$ and $|m_3^2-m^2_2|$ are evaluated
and the smaller combination is chosen such that it is larger than
$m_2^2-m_1^2$. 
With the masses labelled in this way, $m_3>m_2$ if $m_3^2-m_1^2$ is
positive and $m_3<m_1$ if $m_3^2-m_1^2$ is negative.  
Note that
$\Delta m^2_{21}=m_2^2-m_1^2$ is a positive-definite quantity, while
$\Delta m^2_{31}=m_3^2-m_1^2$ is allowed to have either sign.  
It is
also customary to refer to the spectrum $m_3>m_2>m_1$ as ``normal'',
in which case $\Delta m^2_{31}>0$, while the $m_2>m_1>m_3$ spectrum is
referred to as ``inverted,'' in which case $\Delta m^2_{31}<0$.  
Once the neutrino mass eigenstates are properly defined, it is also
customary to parametrise $U$ as prescribed in the Particle Data Book
\cite{Nakamura:2010zzi}.
The neutrino mixing angles ($\theta_{ij}$) are related to the
mixing-matrix elements by:
\begin{equation}
  \frac{|U_{e2}|^2}{|U_{e1}|^2}\equiv \tan^2\theta_{12};
  ~~~~\frac{|U_{\mu3}|^2}{|U_{\tau3}|^2}\equiv \tan^2\theta_{23};
  ~~~~U_{e3}\equiv\sin\theta_{13}e^{-i\delta};
\end{equation}
where $\delta$ is the CP-odd, ``Dirac'', phase.
Two other CP-odd phases might be required to complete the
parametrisation of $U$ if the neutrinos are Majorana fermions. 
However, these ``Majorana'' phases can not be observed in
neutrino oscillations and will henceforth be ignored.  
Neutrino-oscillation data are sensitive to the neutrino mass-squared
differences, the values of the mixing angles $\theta_{12}$,
$\theta_{13}$ , $\theta_{23}$, and $\delta$.  
Two of the three mixing angles are large, in stark contrast to the
quark-mixing matrix, in which all mixing parameters are either small 
($\sin\theta_c\sim 0.23$, where $\theta_c$ is the Cabibbo angle) or
very small ($|V_{ub}|\sim 0.004$).  
In the neutrino sector, three parameters remain unknown or only poorly
constrained: the sign of $\Delta m^2_{31}$; $\delta$; and the mixing
angle $\theta_{13}$.  
The remaining parameters are known at the $\sim 5$--$10$\% level
\cite{GonzalezGarcia:2010er,Mezzetto:2010zi}.  

If $\theta_{13}$ is ``large'', the current generation of experiments
(which includes Double Chooz \cite{Ardellier:2004ui}, RENO
\cite{Ahn:2010vy}, Daya Bay \cite{Guo:2007ug}, T2K \cite{Itow:2001ee}
and No$\nu$a \cite{Ayres:2004js}) may determine that $\theta_{13}$ is
not zero and No$\nu$a may provide a non-trivial hint regarding the
sign of $\Delta m^2_{31}$. 
It is also expected that the precision with which $|\Delta m^2_{31}|$
and $\sin^22\theta_{23}$ are known will improve significantly.  
Even under these conditions, it is widely anticipated that by the end
of the current decade we shall not know the sign of 
$\Delta m^2_{31}$ and will have no information regarding whether
CP-invariance is violated in neutrino interactions 
($\delta\neq 0,\pi$).   
On the other hand, if $\theta_{13}$ is ``small'', not withstanding a
measurement of $|\Delta m^2_{31}|$ with significantly improved
precision, a more precise measurement of $\sin^22\theta_{23}$, and a
more stringent bound on $\theta_{13}$, our understanding of the
neutrino-oscillation parameters might not be too different from what
it is today.

To complete our knowledge of the parameters that describe neutrino
oscillations, the goals of advanced experiments must be:
\begin{itemize}
  \item 
    To search for CP-invariance violation in neutrino oscillations by
    seeking to measure the CP-odd phase $\delta$;
  \item
    To establish whether the neutrino-mass spectrum is normal or
    inverted by determining the sign of $\Delta m^2_{31}$;
  \item
    To determine $\theta_{13}$ with good precision (or constrain it to
    be very small indeed); and
  \item
    To measure as precisely as possible all the parameters the oscillation
    parameters.
\end{itemize}
The search for CP-invariance violation in neutrino oscillations
provides a unique opportunity to further our understanding of
CP-invariance.  
Experiments have demonstrated that the CP-odd phase of the quark
mixing matrix, $\delta_{\rm CKM}$, controls all CP-invariance
violation in the quark sector.
The past generation of neutrino-oscillation experiments has revealed
that there is at least one more CP-violating parameter in the lepton
mixing matrix, $\delta$. 
In practise, $\delta$ can only be observed in neutrino oscillation
experiments and an advanced neutrino experiment is required to explore
this new window into CP-invariance violation.

To determine how precisely the oscillation parameters must be
measured, it is important to assess the precision at which our
understanding would change qualitatively.
Attempts to understand the underlying physics behind the pattern of
lepton mixing provide some guidance 
\cite{Altarelli:2004za,Mohapatra:2005wg,Albright:2006cw}).  
Such attempts often make predictions regarding the specific values for
the different parameters or, perhaps more generically, make predictions
regarding relations among different oscillation parameters. 
One guiding principle for precision measurements of neutrino
oscillations should be the ability to provide unambiguous tests of
these relations. 
An example is given by the tri-bimaximal mixing pattern
($\sin^2\theta_{23}=1/2$, $\sin^2\theta_{12}=1/3$, and
$\sin^2\theta_{13}=0$) \cite{Harrison:2002er}; a popular, zeroth-order
ansatz that guides theoretical research in neutrino flavour. 
Different models make different predictions regarding deviations from
this ansatz, which are usually small and proportional to some other
small parameters in flavour physics. 
Neutrino physics offers one known small (but non-zero) parameter: the
ratio of the two mass-squared differences, 
$\Delta m^2_{12}/|\Delta m^2_{31}|\equiv\epsilon\sim 0.03$
\cite{deGouvea:2004gr}. 
Current data constrain
$\sin^2\theta_{13}\lesssim\epsilon$, $\sin^2\theta_{12}-1/3\lesssim
\epsilon$, while $\sin^2\theta_{23}-1/2\lesssim \sqrt{\epsilon}$.  
An important next-to-next-generation goal would be to test whether
$\sin^2\theta_{13}\lesssim\epsilon^2$, $\sin^2\theta_{12}-1/3\lesssim
\epsilon^2$, and $\sin^2\theta_{23}-1/2\lesssim\epsilon$ (or better).

Another well-known example of the importance of high-precision
neutrino oscillation measurements is the very ambitious goal of
testing whether the origin of neutrino masses is also the origin of
the baryon asymmetry of the Universe, through leptogenesis
\cite{Fukugita:1986hr,Davidson:2008bu}.  
Several indirect generic predictions of leptogenesis can be verified
with neutrino experiments: neutrinos are expected to be Majorana
fermions, and it is expected, quite generically, that neutrino
oscillations violate CP-invariance, even if no model independent
relation between ``high-energy'' and ``low-energy'' CP-invariance
violation exists.   
Quantitative tests are all model dependent and will rely on very
precise measurements in neutrino oscillations and elsewhere.  
For example, precise measurements of oscillation parameters could
validate a specific flavour model, which allows one to relate
``high-energy'' and ``low-energy'' parameters.  
Furthermore, discoveries at collider experiments and searches for
charged-lepton flavour violation may provide other hints that render
leptogenesis either ``very likely'' or ``most improbable''
\cite{Buckley:2006nv}.
We are very far from testing leptogenesis conclusively, but precise
measurements of all neutrino oscillation parameters -- far beyond
where we are now -- are a {\it conditio sine qua non}.

Precision measurements the oscillation parameters are also required to
confirm or refute with confidence the current three-active-neutrino
formalism.
Ultimately, one aims not only at constraining the mixing-parameter
space, but also at {\it over-constraining} it.  
Several important questions need to be addressed: 
\begin{itemize}
  \item
    Are there really only three light neutrinos and is $U$ a unitary
    matrix? 
  \item
    Are there other neutrino interactions?  And, 
  \item
    Is there only one source of CP-invariance violation in the
    neutrino sector?  
\end{itemize}
While our understanding of the neutrino has increased tremendously
over the past decade, we are far from providing a satisfactory answer
to any of these questions.   
In this respect, our understanding of the lepton-flavour sector is far
behind our understanding of the quark-flavour sector.

Some manifestations of new physics are best investigated with neutrino
oscillation experiments.  
For example, the search for light sterile neutrinos, which may be
related to the origin of neutrino masses.
Sterile neutrinos can be detected via the observation of new
oscillation frequencies and mixing angles. 
Tests of the unitarity of the lepton-mixing matrix may also point to
new ``neutrino'' degrees of freedom that are too heavy to be seen in
oscillation experiments. 
Neither of these phenomena can be studied outside of neutrino
oscillation experiments.
Other new physics ideas to which measurements of neutrino oscillation
are uniquely sensitive are related to physics at the electroweak
scale, including new four-fermion neutrino interactions (of the
current--current type, 
$\propto(\bar{\nu}_{\alpha}\Gamma\nu_{\beta})(\bar{f}\Gamma' f)$,
where $f$ is a charged fermion).  
Some of these possibilities will be discussed in more detail later in
this section. 
Indeed, non-standard neutrino interactions are often used as proxies
for the discussion of the sensitivity of neutrino-oscillation
experiments to new physics. 

In many candidate new-physics scenarios, a combination of different
experimental probes will be required in order to piece together a more
fundamental description of how nature works at the smallest distance
scales.  
In addition to studies of neutrino flavour-change, these include
collider experiments (for example, the LHC and a next-generation
lepton collider), searches for charged-lepton flavour violation in the
muon and tau sectors, searches for the permanent electric dipole
moments of fundamental particles, including the electron and the muon,
searches for lepton-number violation, especially searches for
neutrinoless double-beta decay, and direct and indirect searches for
dark matter.  
In many new-physics scenarios, in particular when it
comes to identifying the physics responsible for neutrino masses,
advanced neutrino oscillation experiments are guaranteed to play a
leading role.

Neutrino experiments have proved, over the past few
decades, that our ability to predict what will be detected, and to
identify what are the important questions, is limited at best.  
It is
safe to state, however, that a Neutrino Factory, combined with a
multi-kTon detector at an underground facility, offers a unique and 
powerful tool for the study of fundamental physics.  
In order to prepare for the unexpected, it is vital that advanced
set-ups be versatile and multifaceted.  
The Neutrino Factory fits the bill.
In addition to providing the neutrino beams required for the
definitive, precision neutrino-oscillation programme, the Neutrino
Factory also provides an ideal environment in which to study a variety
of other phenomena.   
The well-characterised neutrino beam from the muon storage ring allows
a programme of extremely precise neutrino scattering measurements to
be carried ot at a near detector, including studies of neutrino
flavour-change at very short baselines and precision measurements of
neutrino scattering on nucleons \cite{Mangano:2001mj} and electrons
\cite{deGouvea:2006cb}.
The latter allow for uniquely sensitive tests of the electroweak
theory.

Activities not directly related to neutrino physics can also be
addressed at a Neutrino Factory Complex.  
The availability
of a large number of muons allows one to consider new set-ups for
searching for rare muon processes \cite{Aysto:2001zs}, especially
$\mu\to e$ conversion in nuclei, and for measurements of the
electroweak properties of the muon, including the muon electric and
magnetic dipole moments.  
The availability of a large number of protons -- used
to make the muons for the storage ring -- allows one to consider a
suite of hadronic experiments including, for example, those required
to study very rare kaon phenomena 
($K\to\pi\bar{\nu}\nu$, $K\to\pi \mu^{\pm} e^{\mp}$, $K^+\to\pi^- \ell^+\ell'^+$, 
etc).   
On the opposite end of the neutrino beam, the very large detector
complex also serves many purposes.  
Depending on the location and composition, very large detectors can be
used to study naturally occurring neutrino--especially the
atmospheric neutrinos and, perhaps, neutrinos from Supernova
explosions. 
Finally, the large
instrumented volumes can be used for searching for proton decay.
Indeed, the Kamiokande and IMB experiments were originally constructed
to look for proton decay, and stumbled upon atmospheric neutrino
oscillations along the way.
 
