\subsection{Comparison with the physics performance of alternative experiments}
\label{sec:competition}

A Neutrino Factory is not the only facility that has been proposed for
the study of neutrino oscillations with great accuracy.  Several other
approaches, in particular beta-beams and super-beams, are currently
being examined.  Given the existence of several options, each of which
comes with its own advantages and disadvantages, a critical comparison
of the various facilities is called for. However, this report is
about the feasibility and physics reach of the Neutrino Factory and,
therefore, we feel that we have neither the space, the expertise, nor
a mandate to provide an in-depth critical comparison.  At the same
time, it would be a disservice to the reader to shun the comparison
altogether and thus, we will present a comparison of the physics
sensitivities only, which in turn are based on the outcome of studies
of the facilities that are presently under way.  All results in this
section have been computed using the GLoBES
software~\cite{Huber:2004ka,Huber:2007ji}.

\subsubsection{Super-Beams}

There are two large super-beam studies under way, one is for the SPL at
CERN as part of EUROnu and the other is in the context of LBNE in the
US.  Both experiments will employ a horn-focused pion beam as the
neutrino source and use baselines of $130$~km (SPL) and $1\,300$~km
(LBNE).  In both experiments the primary proton-beam power will be at
the MW level, however the proton energies will be very different,
4~GeV for the SPL and 120~GeV for LBNE.  The SPL will use a water
Cherenkov detector, with $440$~kTon fiducial volume \cite{wp2}; the
detector performance is taken from
references~\cite{Campagne:2004wt,Campagne:2006yx}.  Note, that the SPL
beam has been studied in quite some detail as part of EUROnu and there
it was found that the neutrino fluxes can be optimised by changing the
target and horn configuration~\cite{wp2}, which improves the
anticipated performance compared with that obtained in earlier
studies.

LBNE will use either a water Cherenkov detector or a liquid-Argon
detector with a 6-times smaller mass.  
It has been shown \cite{LBNE-physics,Huber:2010dx} that
physics-wise the two detector configurations are equivalent and
therefore we will consider only the water-Cherenkov option.  
The default fiducial volume for the
water Cherenkov detector in LBNE is $200$~kTon and the baseline is from
Fermilab to DUSEL, which is $1\,300$~km.  The implementation of both
the beam spectrum and detector performance follows
reference~\cite{LBNE-physics}. We also consider a possible luminosity
upgrade as provided by Project X.

\subsubsection{Beta-beams}

Beta-beams have been proposed in reference~\cite{Zucchelli:2002sa} and
the idea is to exploit the $\beta$-decay of relatively short-lived
radio-isotopes with half-lives of around $1$~s which are
ionised, accelerated to relativistic speeds and put into a storage
ring, where they eventually decay. 
As a result of the high Lorenz
$\gamma\simeq 50-500$, the (anti-)neutrino emission will be directed
in the forward direction with an opening angle of $1/\gamma$.
Neutrino beams can be produced using $\beta^+$ emitters.  
The maximum neutrino energy that is available, $E_\text{max}$, is
given by the end-point of the $\beta$-spectrum $Q_\beta$ and the
Lorenz boost, $E_\text{max}=\gamma Q_\beta$.  
Given the fact that $Q_\beta$ lies in
the range of $1-10\,\mathrm{MeV}$, it follows that neutrino beam
energies in the range from $0.1-5\,\mathrm{GeV}$ are possible.
Currently, four isotopes are being studied: $^{18}$Ne and $^8$B for
neutrino production and $^6$He and $^8$Li for anti-neutrino
production; their properties are listed in table~\ref{tab:ions}.
Clearly, the ions with a large $Q_\beta$ will yield higher neutrino
energies for a given Lorenz $\gamma$, but for a given energy they will
yield less flux since the flux is proportional to $\gamma^{-2}$. 
All
of these isotopes are too short-lived to have any natural abundance and
thus they need to be produced artificially. 
It turns out that the
achievable production rates ultimately limit the luminosity of these
kinds of neutrino beams. 
The intensities considered to be the goals are
$1.1 \times 10^{18}$ ions per year for $^{18}$Ne and $2.9 \times 10^{18}$
ions per year for $^{6}$He~\cite{EURISOL}. 
For some time it was not quite clear whether these ion intensities
are feasible, but recent developments within EUROnu indicate that, for
$^{18}$Ne, the required intensity can be achievede \cite{Wildner:2010zz}.
\begin{table}
\caption{\label{tab:ions} $A/Z$, half-life and end-point energies for
  two $\beta^+$-emitters ($^{18}$Ne and $^8$B) and two
  $\beta^-$-emitters ($^6$He and $^8$Li).Table adapted
  from~\cite{Bernabeu:2010rz}.}
\begin{center}
\begin{tabular}{|c|c|c|c|c|} \hline \hline
   Element  & $A/Z$ & $T_{1/2}$ (s) & $Q_\beta$  [MeV] & Decay Fraction \\ 
\hline
  $^{18}$Ne &   1.8 &     1.67      &        3.41         &      92.1\%    \\
           
  $^{8}$B   &   1.6 &     0.77      &       13.92         &       100\%    \\
\hline
 $^{6}$He   &   3.0 &     0.81      &        3.51         &       100\%    \\ 
 $^{8}$Li   &   2.7 &     0.83      &       12.96         &       100\%    \\ 
\hline
\hline
\end{tabular}
\end{center}
\end{table}

In the phenomenological literature it has been pointed out that large
Lorenz boosts, around 350, possibly combining data from all four ion
species, can lead to physics sensitivities which are on par with that
of the Neutrino Factory
\cite{Burguet-Castell:2003vv,Burguet-Castell:2005pa,Agarwalla:2005we,%
  Huber:2005jk,Donini:2006tt,Agarwalla:2006vf,Agarwalla:2008ti,%
  Agarwalla:2008gf,Coloma:2010wa}.  However, there are currently no
attempts to perform a feasibility study of any of these so called
``high-$\gamma$'' beta-beams.  As a consequence, it it is not possible
to assess the cost and technological risks associated with the
high-$\gamma$ options.  Therefore, we will not consider them here and
instead will focus on those options which are currently studied, at
a machine level, within the EUROnu Beta-Beam work package; i.e., only
the $\gamma=100$ option will be considered.  
For this reason, we also do
not consider $^8$Li or $^8$B because they require higher values of
$\gamma$ to be useful, see, for example, \cite{Coloma:2010wa}.

\subsubsection{Comparison}

It is possible that a next-generation super-beam (or combined
super-beam/beta-beam) experiment could be constructed on the same
timescale as the Neutrino Factory, or might even start earlier. 
Possible
candidates are the LBNE and SPL experiments
shown in figure \ref{fig:euronu2009} with their $\theta_{13}$, CP
violation, and mass hierarchy discovery potentials.
\begin{figure}[tp]
  \begin{center}
    \includegraphics[width=0.49\textwidth]%
      {01-PPEG/Figures/idr-comparison-cpv.eps} %
    \includegraphics[width=0.49\textwidth]%
      {01-PPEG/Figures/idr-comparison-sgn.eps}
    \includegraphics[width=0.49\textwidth]%
      {01-PPEG/Figures/idr-comparison-the.eps} %
  \end{center}
  \caption{ Comparison of the physics reach of different future
    facilities for the discovery of CP violation (top left panel), the
    mass hierarchy (top right panel), and $\sin^2 2 \theta_{13}$
    bottom panel.  The sensitivities of the SPL super-beam are taken
    from \cite{Campagne:2006yx}.  The beta-beam curves are also taken
    from \cite{Campagne:2006yx}, however with the ion intensities
    reduced to the EURISOL values \cite{EURISOL}.  Curves for LBNE are
    taken from \cite{Huber:2010dx} and correspond to the results in
    \cite{LBNE-physics}.  The $\theta_{13}$ sensitivities expected
    from current experiments are shown as vertical lines
    \cite{Huber:2009cw}.  MIND LE is a single-baseline Neutrino
    Factory optimised for large $\stheta>0.01$, see also
    section~\ref{sec:opti1}.  }
  \label{fig:euronu2009}
\end{figure}

If $\stheta \gtrsim 0.01$ is discovered before the RDR is prepared, CP
violation can be measured at the $3 \sigma$ CL, depending on the
option and exact value of $\stheta$, for between 60\% and 75\% of all
values of $\delta$.  For the current hint $\stheta=0.06$, a discovery
can be made for up to 70\% of all possible values of $\delta$.  The
mass hierarchy discovery is, in this case, guaranteed by the LBNE
option; for the alternatives the discovery reach strongly depends on
$\stheta$ in this range.  For comparison, in figure
\ref{fig:euronu2009} the $3 \sigma$ curves from the existing
experiments in 2025 are also shown (curves ``2025''). As we have
demonstrated, the Neutrino Factory can do significantly better for the
CP violation discovery, and, of course, also measure the mass
hierarchy in that range. Very importantly, the CP fraction in figure
\ref{fig:euronu2009} can be related to the precision measurement of
$\delta$ for the CP conserving values. Therefore, the higher the CP
fraction, the higher the precision. In either case, a Neutrino Factory
will provide a more precise measurement than any other experiment,
especially if a single baseline configuration, optimised for the, by
then known, value of $\stheta$ were employed.

If the data indicate that $\stheta \lesssim 0.01$ at the time the RDR
is in preparation, alternatives to the Neutrino Factory have a
discovery potential for $\stheta$ significantly exceeding the existing
experiments, in some cases by up to an order of magnitude.  
Depending on $\stheta$, CP violation may be discovered for up to 65\%
of possible values of $\delta$ (SPL \& LBNE).  
The mass hierarchy can only by accessed by LBNE in a small fraction
of the parameter space.  The CP violation plot demonstrates that these
experiments have limited potential for $\stheta \lesssim 0.01$, since
the small data samples that can be expected will cut off the
sensitivity at some value of $\sin^2 2 \theta_{13}$.  Figure
\ref{fig:euronu2009} shows that the Neutrino Factory can do
significantly better.  It is also interesting from figure
\ref{fig:euronu2009} that the alternatives are either optimised for
the CP violation (SPL/BB100) or the mass hierarchy discovery (LBNE).
No option other than the Neutrino Factory can do all these
measurements equally well.

In summary, even if $\theta_{13} > 0$ is discovered by the generation
of experiments currently under construction, it is likely that the
discovery of CP violation and precision measurement of the CP phase
require data from advanced experiments, like the Neutrino Factory.
If $\theta_{13}$ is not discovered by the generation of experiments
presently under construction, there will be no further information on
CP violation and mass hierarchy from these experiments, and an advanced
experiment is mandatory to make further progress.  In this case, the
super-beam and beta-beam based alternatives will be strongly limited
by the size of the data sample that can be collected and only the
Neutrino Factory provides a robust sensitivity to all observables
throughout the accessible parameter space.
