\subsection{The baseline configuration}
\label{sec:henf}

Here we describe the optimisation and performance of the Neutrino
Factory baseline setup.  Two different optimisation strategies are
possible depending on the available information on $\stheta$: if
$\stheta$ is not discovered prior to the start of the Neutrino Factory
project, then the prime goal of a Neutrino Factory would be to
determine $\stheta$ \emph{and} to be able to address the mass
hierarchy and CP violation for the largest range of $\stheta$
feasible.  We will refer to this as the ``small $\stheta$'' case.  On
the other hand, if $\stheta$ has been measured prior to the start of
the Neutrino Factory project, then the goal is to obtain the most
definitive measurements of the CP phase and the atmospheric
parameters.  We will refer to this as the ``large $\stheta$'' case.
Note that, for large $\stheta$, the mass hierarchy would be measured
by NO$\nu$A at less than $3\,\sigma$ for the vast majority of CP
phases and that the precision on $\stheta$ obtained by Daya Bay would
not be likely to be improved by any accelerator-based experiment.  For
practical purposes, the limiting value of $\stheta$ that separates the
large and small $\stheta$ regimes is given by the sensitivity of Daya
Bay which will eventually reach $\stheta\simeq0.01$ at the $3\,\sigma$
level~\cite{Huber:2009cw}.  The Long Baseline Neutrino Experiment
(LBNE)~\cite{LBNE-physics} at FNAL has the potential to improve upon
the sensitivity of Daya Bay by a factor of between 1--5.  However,
this limit would be approached only towards the end of the 10 years of
LBNE running and thus would be too late to inform the Neutrino Factory
optimisation process.  For a more detailed discussion of these
possible optimisation strategies see reference \cite{Tang:2009wp}.
The optimisation of the Neutrino Factory has been studied in great
detail, for instance, in references
\cite{Barger:1999fs,Cervera:2000kp,Burguet-Castell:2001ez,Freund:2001ui,Huber:2003ak,Donini:2005db,Huber:2006wb,Gandhi:2006gu,Kopp:2008ds,Agarwalla:2010xx}.
Here we will give a brief summary of the considerations which led to
the IDS-NF baseline setup. All results in this section have been
computed using the General Long Baseline Experiment Simulator (GLoBES)
software package~\cite{Huber:2004ka,Huber:2007ji}.

\subsubsection{Optimisation of a single baseline Neutrino Factory}
\label{sec:opti1}

In figure~\ref{fig:elopt} we show the sensitivity reach in $\stheta$
and $\delta$ as a function of the baseline, $L$, and stored-muon
energy, $E_\mu$.  The left panel shows the $\stheta$ sensitivity
(exclusion limit) including correlations and degeneracies while the
right panel shows the CP violation discovery reach for one particular
value of maximal CP violation.  
The results shown in figure \ref{fig:elopt} have been used to optimise
the facility for the largest possible reach in $\stheta$, in other
words, for the small $\stheta$ case.  
The optimum discovery reach in $\stheta$ is for the $L$ and $E_\mu$
combinations in the white regions of the plots; for details, see
reference \cite{Huber:2006wb}.  The excellent sensitivity to $\stheta$
at about 7500~km can be easily understood in terms of the magic
baseline \cite{Huber:2003ak,Smirnov:2006sm}.  At this baseline, which
only depends on the matter-density profile, the effect of the CP phase
vanishes, which results in a clean signal for $\stheta$ and the mass
hierarchy.  The optimum baseline for the measurement of CP-violation
is found for $L$, $E_\mu$ combinations in the white region of the
right panel.  It turns out that, in terms of the $\stheta$ reach, a
baseline between 3000~km and 5000~km is close to optimal.  Therefore,
the IDS-NF baseline setup specifies two long-baseline detectors, one
between 3\,000~km to 5\,000~km and the at between 7\,000~km
and 8\,000~km.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.48\textwidth]{01-PPEG/Figures/theta13sensel} %
\includegraphics[width=0.48\textwidth]{01-PPEG/Figures/cpviolel}
\end{center}
\caption{\label{fig:elopt} Optimisation of a single baseline Neutrino
  Factory as a function of baseline $L$ and muon energy $E_\mu$. The
  left panel shows the $\stheta$ sensitivity (exclusion limit) with
  the optimum sensitivity $\stheta \simeq 5.0 \times 10^{-4}$ (diamond).
  The contours show the regions within a factor of 0.5, 1, 2, 5, and
  10 of the optimal sensitivity. The right panels shows the
  discovery reach for maximal CP violation, $\delta=\pi/2$, in terms
  of $\stheta$. Figure taken from reference \cite{Huber:2006wb}.  }
\end{figure}

As far as the muon energy dependence is concerned, there are two
competing factors.  The oscillation maximum is at relatively low
energies in the Neutrino Factory spectrum, therefore a low energy
threshold, has been identified as the main priority for the
optimisation of the MIND \cite{Huber:2006wb}.  
The low energy threshold provided by the MIND (see section
\ref{Sect:DetWG}) allows for a reduction of $E_\mu$ to values as low
as about $\sim 20$~GeV without significant loss of sensitivity
\cite{Huber:2006wb}. 
Therefore, the IDS-NF baseline setup with a MIND detector uses
$E_\mu=25$~GeV.  On the other hand, matter effects play an important
role, such as in the determination of the mass hierarchy.  Assuming an
averaged matter density $\rho=3.4 \, \mathrm{g/cm^3}$, the
matter-resonance energy for a baseline of 4\,000~km is at 9.3~GeV.
One can already see from the left panel in figure \ref{fig:elopt} that
this leads to the requirement that the peak of the neutrino flux
exceeds this energy, which is equivalent to $E_\mu \gtrsim 15$~GeV for
optimal performance at the very long baseline.  Therefore, for the
baseline Neutrino Factory, $E_\mu$ between about 15~GeV and 25~GeV can
be considered optimal, where the lower end of this range is determined
by Earth matter effects, and the upper end by detector performance and
cost.
These results, which are valid for the small $\stheta$ case, are fully
confirmed by a recent analysis \cite{Agarwalla:2010xx} based on the
updated MIND performance as described in section~\ref{sec:MIND}.

On the one hand, the results on the optimisation of a single baseline
serve as a starting point for the optimisation of the two baseline
setup.  As we will discuss later, two baselines are essential to
ensure uniformly robust performance throughout the parameter space.
On the other hand, a single baseline Neutrino Factory may be feasible,
or even appropriate, under certain conditions, for example large
$\theta_{13}$.

A Neutrino Factory with a single baseline of $\sim 1000\,\mathrm{km}$
was first proposed in references \cite{lenf1,lenf2} 
and was termed the ``low-energy Neutrino Factory'' (LENF), since the
associated muon beam energies were much lower than the usual $25$~GeV.
At the LENF baseline, the oscillation spectrum at energies below
$5$~GeV is very rich, potentially allowing for very precise
measurements of the unknown oscillation parameters $\theta_{13}$,
$\delta$ and the mass hierarchy.  Since the initial studies
\cite{lenf1,lenf2}, developments of the accelerator and detector
designs have enabled the experimental simulations to be refined and
more detailed optimisation studies to be performed
\cite{LENF,mythesis}.  The key finding of these studies is that,
given sufficiently high statistics and a detector with excellent
detection efficiency at low neutrino energy ($\lesssim 1$\,GeV), an
optimised LENF can have excellent sensitivity to the standard
oscillation parameters, competitive with the usual two baseline, high
energy setup for $\sin^{2}2\theta_{13}\gtrsim 10^{-2}$.

In order to exploit the rich signatures at low energies, alternative
detector types with a lower neutrino energy threshold and a better
energy resolution compared to the MIND are necessary.  Two possible
alternative detector technologies have been considered: either a
20\,kTon totally active scintillating detector (TASD) \cite{ISS} or a
100\,kTon liquid argon (LAr) detector \cite{LAr}, both of which would
need to be magnetised; a challenging requirement.  
These detectors would be capable of detecting and
identifying the charges of both electrons and muons, providing access
to multiple oscillation channels: the $\nu_{\mu}$ ($\bar{\nu}_{\mu}$)
disappearance channels, as well as the golden
($\nu_{e}\rightarrow\nu_{\mu}$ and
$\bar{\nu}_{e}\rightarrow\bar{\nu}_{\mu}$)~\cite{Cervera:2000kp} and platinum
channels ($\nu_{\mu}\rightarrow\nu_{e}$ and
$\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$).  A detailed description of
the assumptions used to simulate the detectors is given in reference
\cite{LENF}.  The main features of the TASD that were assumed were an
energy threshold of 0.5 GeV with a detection efficiency of $94\%$
above 1~GeV and $74\%$ below 1~GeV for $\nu_{\mu}$ and
$\bar{\nu}_{\mu}$ with a background of $10^{-3}$.  For $\nu_{e}$ and
$\bar{\nu}_{e}$ an efficiency of $47\%$ above 1~GeV and $37\%$ below
1~GeV with a background of $10^{-2}$ were assumed.  The main sources of
background were assumed to arise from charge mis-identification and
neutral current events.  The energy resolution was assumed to be
$10\%$ for all channels.  The possibility of using non-magnetised TASD
or LAr detectors was explored in reference \cite{nonmagnetic}; however
the reference LENF setup assumes that magnetisation will be possible.
Recent interest in the LENF has also prompted the authors of reference
\cite{LENF_LENA} to study a setup with a 50~kTon liquid-scintillator
detector.  Each of the alternative detector technologies considered in
the LENF studies is essentially fully active throughout its volume
and therefore offer, for the first time, the possibility of observing
the \emph{platinum} channels in addition to the golden channels.
Indeed, there were early indications that such detector technologies
might be capable of detecting and distinguishing between $e^{-}$ and
$e^{+}$ as well as $\mu^{-}$ and $\mu^{+}$.  It is found that the
complementarity between the platinum and golden channels can be of
great benefit if statistics are limited, improving the sensitivity to
the standard oscillation parameters.  The inclusion of the platinum
channels also turns out to be crucial for resolving degeneracies
between the oscillation parameters and non-standard
effects~\cite{mynsi,mythesis} in the absence of a second baseline.
\begin{figure}[tp!]
  \centering \subfigure[
  $\sin^22\theta_{13}=10^{-1}$]{\includegraphics[width=.45\textwidth,
    viewport=-25 -25 440 440, clip,
    angle=270]{01-PPEG/Figures/Cons-1.eps}} \subfigure[
  $\sin^22\theta_{13}=10^{-2}$]{\includegraphics[width=.45\textwidth,
    viewport=-25 -25 440 440, clip,
    angle=270]{01-PPEG/Figures/Cons-2.eps}}

 \subfigure[
  $\sin^22\theta_{13}=10^{-3}$]{\includegraphics[width=.45\textwidth,
    viewport=-25 -25 440 440, clip,
    angle=270]{01-PPEG/Figures/Cons-3.eps}} \subfigure[
  $\sin^22\theta_{13}=10^{-4}$]{\includegraphics[width=.45\textwidth,
    viewport=-25 -25 440 440, clip,
    angle=270]{01-PPEG/Figures/Cons-4.eps}}
  \caption{CP fraction as a function of baseline, $L$, and stored muon
    energy, $E_\mu$.  Results are shown for a $20\,\mathrm{kTon}$ TASD
    using the default muon intensity of $5\times10^{21}$ $\mu^+$ and
    $5\times10^{21}$ $\mu^-$ for four different values of
    $\theta_{13}$: 
    (a) $\sin^2 2 \theta_{13} = 10^{-1}$;
    (b) $\sin^2 2 \theta_{13} = 10^{-2}$;
    (c) $\sin^2 2 \theta_{13} = 10^{-3}$; and
    (d) $\sin^2 2 \theta_{13} = 10^{-4}$.
  }
\label{fig:plots1}
\end{figure}

In previous work \cite{Pascoli-Li2010, Pascoli2008}, the LENF was
configured with a stored muon energy of between $4$~GeV and $5$~GeV
and a baseline of around $1300$ to $1500\,$km, with $1480\,$km
corresponding to the configuration Fermilab to Henderson mine and
$1300\,$km to the combination Fermilab to Homestake
mine\footnote{Homestake is the host site the proposed Deep Underground
  Science and Engineering Laboratory (DUSEL).}.  
Here we 
focus on optimising a LENF for the purpose of measuring CP-violation.
To this end, we compute the ability, under a range of experimental
scenarios, of ruling out all CP-conserving values of $\delta$. For
each detector, a CP-violation discovery-potential-plot was first
produced for each baseline and muon energy combination.  
Using a Poissonian $\chi^2$ function and marginalising over all
parameters except $\delta$, plots showing the range of values of
``true'' $\theta_{13}$ and $\delta$ for which CP-violation could be
detected at the $3\sigma$ significance were produced.
The simulations assumed a normal mass hierarchy and the hierarchy
degeneracy was taken into account during the marginalisation.  
The simulated oscillation parameters were chosen 
to be $\sin^22\theta_{12}=0.3$, $\theta_{23}=\pi/4$, $\Delta
m^2_{12}=8.0\times10^{-5}\,\mathrm{eV}^2$ and $|\Delta
m^2_{13}|=2.5\times10^{-3}\,\mathrm{eV}^2$ with an uncertainty of $4$
and $10\,$\% on the solar and atmospheric parameters respectively. For
a few illustrative values of true $\theta_{13}$, the CP fraction was
then calculated for each choice of energy and baseline.  The CP
fraction is defined for each true $\theta_{13}$ as the fraction of
true $\delta$ values which fall inside the CP discovery potential
curve. Using these values, a contour plot was then produced of how the
CP fraction varies as a function of baseline and muon energy for a
given true $\theta_{13}$, as shown in figure \ref{fig:plots1}.

>From figure~\ref{fig:plots1}, we see that for large values of
$\theta_{13}$, around $\sin^22\theta_{13}\gtrsim10^{-2}$, CP fractions
of $70$ to $90\,$\% for most energies and baselines can be obtained.
The only deviation from this is in the region of the lowest energies
but longest baselines where we see a worse performance at around $40$
to $80\,$\%.  This poor performance may partially be explained by
considering the uncertainty in $\theta_{13}$: as this parameter is
being marginalised over, the ability to measure the CP phase is
diminished if $\theta_{13}$ cannot be constrained.  The best
constraints on $\theta_{13}$ angle are found near the first
oscillation maximum; in the poor-performance region, the majority of
neutrinos have baseline-energy ratios not matching this condition.
This region is also affected by the decrease of the neutrino flux at
large baselines which compounds the poor performance.

Looking at smaller values of $\theta_{13}$, around
$\sin^22\theta_{13}\approx10^{-3}$ we see a general decrease in CP
fraction to a range of $50$ to $60\,$\% and the region of poor
performance which was mentioned earlier worsens to $0$ to $40\,$\%.
We also see another region in which the performance is significantly
reduced; the region of highest-energies but shortest-baselines.  This
drop in performance is attributed to a decrease in statistics: this
region has the smallest baseline-energy ratio which means it is the
furthest away from the oscillation maxima.  When these effects are
combined with the drop in statistics that arises from the lowering of
$\theta_{13}$, experiments in this region start to be severely
compromised by the number of events.  The same behaviour is seen for
the updated MIND detector \cite{Agarwalla:2010xx} and in figure
\ref{fig:LvsE}.

As the true value of $\theta_{13}$ drops to
$\sin^22\theta_{13}\lesssim10^{-4}$, the performance decreases rapidly
and we generally see CP fractions in the range of $0$ to $50\,$\%.
The two poorly performing regions continue to worsen and are mostly in
the range $0$ to $30\,$\%.  Eventually, only the regions of
$E\gtrsim10$ and $L\gtrsim2000$ have any appreciable CP fractions.

The high-energy regions of our CP fraction plots overlap with part of
a previous optimisation study that considered the baseline Neutrino
Factory \cite{Huber:2006wb,Agarwalla:2010xx}.  In
figure~\ref{fig:LvsE} we show the same type of result as in
figure~\ref{fig:plots1}, but now for a $50\,\mathrm{kTon}$ MIND detector
corresponding to the one described in~\ref{sec:MIND}.  Qualitatively
the results in figures~\ref{fig:plots1} and~\ref{fig:LvsE} are very
similar and demonstrate that the decision between shorter baseline and
lower energy versus longer baseline and higher energy is more driven
by the magnitude of $\stheta$ than by the detector technology alone.
Although the two experimental simulations have many differences, we
see similar generic features: higher energies require longer baselines
and as $\theta_{13}$ decreases the low energy, short baseline setups
become increasingly disfavoured. 
Also, not surprisingly, when using a MIND
somewhat larger energies at large $\stheta$ are favoured in order to
avoid the effects from the energy threshold of the detector. For large
$\theta_{13}$, the baseline Neutrino Factory has a CP fraction around
$80\,$\%~(see figure~\ref{fig:heperf}) and the LENF has CP fractions
of $70$ to $80\,$\% over a comparable region.  But also, a MIND
detector with 10~GeV muon energy has a similar performance of 80\%,
see figure~\ref{fig:LvsE}.  Thus, low energy setups are preferable at
large $\stheta>0.01$ for \emph{both} MIND and TASD detectors. Any
perceived performance advantage of one detector type with respect to
the other strongly depends on the relative detector sizes and assumed
beam luminosities.
%
\begin{figure}[tp]
 \centering
 \includegraphics[width=0.8\textwidth]{01-PPEG/Figures/LvsE}
 % spectra-log.pdf: 600x600 pixel, 72dpi, 21.17x21.17 cm, bb=0 0 600 600
 \caption{ \label{fig:LvsE} Fraction of $\delta$ for which CPV will
   be discovered ($3 \sigma$ CL) as a function of $L$ and $E_\mu$ for
   the single baseline Neutrino Factory.  The different panels
   correspond to different true values of $\stheta$, as given there.
   Here SF=1 ($2.5\times10^{20}$ muons per year and polarity) is used
   with a 50 kt detector. The optimal performance is marked by a dot:
   (2200,10.00), (2288,13.62), (3390,20.00) and (4345,22.08). Figure
   and caption taken from reference~\cite{Agarwalla:2010xx}.}
\end{figure}

\paragraph*{The platinum channel}

The most significant addition relative to the initial LENF studies
would be the inclusion of the platinum channels,
$\nu_{\mu}\rightarrow\nu_{e}$ and
$\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$. 
We find that, if backgrounds
are at a negligibly low level, the platinum channels always give a
significant improvement to the setup. 
However, once realistic
background levels of at least $\sim10^{-2}$ on the platinum channels
are included, much of this benefit is lost. 
There is still some improvement if statistics are limited to below the
$5.0\times10^{20}$ decays per year per polarity assumed in references
\cite{lenf1,lenf2}, but virtually nothing is gained in the case of
higher statistics.
This is illustrated in figure \ref{fig:plat} where the $68\%$, $90\%$
and $95\%$ contours in the $\theta_{13}-\delta$ plane are shown for a
true simulated value of $\theta_{13}=1^{\circ}$ and $\delta=0$,
$\pm90^{\circ}$ and $\pm180^{\circ}$, comparing the results from a
setup without the platinum channel (`Scenario 1' - blue dashed lines)
and a setup which does include the platinum channel with a background
of $10^{-2}$ (`Scenario 2' - solid red lines). 
Figure \ref{fig:plat}a shows the
results in the low statistics case and figure \ref{fig:plat}b the high
statistics case.
\begin{figure}[tp]
   \centering
   \subfigure[~$5.0\times10^{20}$ decays per year.]{
         \includegraphics[width=7.5cm,height=6cm]{01-PPEG/Figures/./plat_lo_2.eps}}
   \subfigure[~$1.4\times10^{21}$ decays per year.]{
         \includegraphics[width=7.5cm,height=6cm]{01-PPEG/Figures/./plat_hi_2.eps}}
       \caption{~Sensitivity to $\theta_{13}$ and $\delta$ without the
         platinum channel (Scenario 1 - dashed blue lines) and with
         the platinum channel with a $10^{-2}$ background (Scenario 2
         - solid red lines), in the case of a) low statistics
         ($5.0\times10^{20}$ decays per year and b) high statistics
         ($1.4\times10^{21}$ decays per year, significantly larger
         than the present IDS-NF baseline), for
         $\theta_{13}=1^{\circ}$. From reference \cite{LENF}.}
\label{fig:plat}
\end{figure}

\subsubsection{Optimisation of a two baseline Neutrino Factory}
\label{sec:opti2}

The simultaneous optimisation of two Neutrino Factory baselines has
been studied in reference \cite{Kopp:2008ds}, see figure \ref{fig:allopt}
(shaded regions). 
One easily recovers the behaviour discussed above: $\stheta$
(upper left panel) and mass hierarchy (upper right panel) sensitivity
prefer a long baseline, but the combination with a shorter baseline
is not significantly worse.
For CP violation (lower left panel), however, one
baseline has to be considerably shorter. 
If all performance indicators
are combined (lower right panel), the optimum at 4\,000~km plus
7\,500~km is obtained. 
A very interesting study is performed in the dashed
curves: these curves illustrate the effect of a potential non-standard
interaction $\epsilon_{e \tau}$, which is known to harm especially the
appearance channels. 
While the absolute performance for all
performance indicators decreases, the position of the optimum in two
baseline space remains qualitatively unchanged. 
Therefore, the two
baseline setup is not only optimal for standard physics, but also
robust with respect to unknown effects (see also
reference \cite{Ribeiro:2007ud}).
\begin{figure}[tp]
  \begin{center}
    \includegraphics[width=\textwidth]{01-PPEG/Figures/BLPlot.eps}
  \end{center}
  \caption{ \label{fig:allopt} Two-baseline optimisation of the
    Neutrino Factory for the standard oscillation performance
    indicators. The upper left panel shows the (shaded) region where
    the sensitivity to $\stheta$ better than $10^{-3.4}$ ($5 \sigma$),
    the upper right panel shows the (shaded) region where the
    sensitivity to the normal mass hierarchy (MH) is given for all
    $\stheta \ge 10^{-3.8}$ ($5 \sigma$, $\delta_{\rm CP}^{\rm true} =
    3\pi/2$), the lower left panel shows the (shaded) region where the
    sensitivity to maximal CP violation (CPV) is given for all
    $\stheta \ge 10^{-3.8}$ ($5 \sigma$, $\delta_{\rm CP}^{\rm true} =
    3\pi/2$), and the lower right panel shows the intersection of the
    three regions as the shaded region.  The dotted curves have been
    obtained from a fit including a non-standard interaction parameter
    $\epsilon_{e \tau}$ marginalised (for the $\stheta$ ranges given
    in the plots).  The diamonds show the setups with optimal
    sensitivities (coloured/grey for the shaded contours, black for the
    dotted contours), whereas the circles correspond to the {\sf
      IDS-NF} standard choices $L_1 = 4 \, 000$~km and $L_2 = 7 \,
    500$~km. Figure taken from reference \cite{Kopp:2008ds}.}
\end{figure}

All this optimisation is based on the assumption that the experiment
is optimised in the $\stheta$ direction. On the other hand, it has
been demonstrated in the literature that intermediate values of
$\stheta$ require the magic baseline as a degeneracy resolver (see
reference \cite{Barger:2001yr} for degeneracy issues). Therefore, the
optimisation of the Neutrino Factory does change qualitatively.
However, note that for larger values of $\stheta$, which
may be found in a first low energy phase of the Neutrino Factory,
somewhat shorter baselines than 3000~km can be used; cf., figure 8 in
reference \cite{Tang:2009wp}.

\paragraph*{Robustness of performance}

Given the excellent performance of single baseline configurations over
a wide range of $\stheta$ values, the question arises whether a
second baseline is strictly required.  The answer is a resounding yes,
as the results shown in figure~\ref{fig:robustness} decisively
demonstrate.  In this figure, we show the sensitivity reach for CP
violation as a function of $\stheta$ in the left hand panel for three
different configurations.  ``IDS-NF 2010/2.0'' refers to a the
reference setup defined in section~\ref{sec:performance}, it employs
two baselines of $4\,000$~km with a $100\,\mathrm{kTon}$ MIND and
$7\,500$\,km with a $50\,\mathrm{kTon}$ MIND.  The muon energy is
$25$~GeV.  ``MIND LE'' uses a single baseline of $2\,000$~km with a
$100$~kTon MIND and a muon energy of $10$~GeV.  ``TASD LE'' uses a
single baseline of $1\,300$~km with a $20$~kTon TASD and a muon energy 
of $5$~GeV.  The MIND performance is described in
section~\ref{sec:MIND} and the TASD performance is taken from
reference \cite{LENF}.  All setups are assumed to run for 10 years
with $10^{21}$ useful muon decays per year summed over both
polarities.  For a two baseline setup, this means that each detector
receives only one half of this flux.  The systematics for all setups
is the same: 1\% on signal and 20\% on background.  The results are
shown at the $5\,\sigma$ level, which helps to emphasise the crucial
difference.  For values of $\stheta>0.01$ (thin vertical line) all
three setups show a qualitatively very similar behaviour, but
nonetheless there a distinct differences in performance; the 
single-baseline setups perform better, at least for very large values
of $\stheta$.  
For values of $\stheta<0.01$ the picture changes
dramatically and the single-baseline setups suffer from a phenomenon
called $\pi$-transit resulting in large regions of parameter space
where sensitivity is lost.  The reason for the occurrence of the
$\pi$-transit was described in reference \cite{Huber:2002mx}: for
certain combinations of the true values of the CP phase and $\stheta$,
a CP violating true solution can be fitted with the wrong mass
hierarchy and $\delta=\pi$.  The location of the $\pi$-transit depends
sensitively on the baseline and energy, and therefore MIND~LE and
TASD~LE show this effect for slightly different values of $\stheta$,
whereas IDS-NF~2010/2.0, the two baseline setup, is immune against
this effect due the ability of the magic baseline to identify
unambiguously the mass hierarchy.  This is a rather general
conclusion: all single-baseline setups we have studied suffer from the
$\pi$-transit problem due to their relatively weak ability to
determine the mass hierarchy.
%
\begin{figure}
  \begin{center}
    \includegraphics[width=1\textwidth]{01-PPEG/Figures/robustness}
  \end{center}
  \caption{ 
    CP violation discovery reach (left hand panel) and CP fraction
    (right hand panel) comparing possible single-baseline setups with
    the two-baseline default configuration (IDS-NF 2010/2.0)
    TASD LE stands for a $20\,\mathrm{kTon}$ TASD detector at
    $1\,300$~km baseline with $E_\mu=5$~GeV and MIND LE stands for a
    $100$~kTon MIND at $2\,000$~km with $E_\mu=10$~GeV.  
    Note, that the results are shown at the $5\,\sigma$ confidence
    level.
  }
  \label{fig:robustness}
\end{figure}

We conclude that single-baseline setups, whether they use a MIND or
TASD, all suffer from problems with degeneracies, also, in the context
of new physics as will be discussed later in this section, for values
of $\stheta\lesssim 0.01$. The actual extent of the problem and the
precise boundary in $\stheta$ depend crucially on the detector
performance and available statistics. However, once these factors are
known it is straightforward to derive the limiting values of $\stheta$
above which a given single baseline setup is safe from degeneracies.
In those cases, single baseline setups typically achieve a higher
precision for the CP phase than our standard two baseline
configuration. Hence, for sufficiently large values of $\stheta$, a
single baseline configuration offers a cost-effective way to improve
the ability to measure the leptonic CP phase. Fortunately, the typical
magnitude of the delimiting value of $\stheta$ is within the
sensitivity reach of Daya Bay and therefore, sufficient data for a
informed decision will be available.

\subsubsection{Performance of the Neutrino Factory}
\label{sec:performance}

% Standard discovery reaches

Here we discuss the performance of the high energy Neutrino Factory
for standard oscillation physics.  We show in figure \ref{fig:heperf}
the performance of the high energy Neutrino Factory (100~kt at 4000~km
plus 50~kt at 7500~km) in terms of the $\theta_{13}$, CP violation,
and mass hierarchy discovery reach at $3\sigma$ CL using the updated
MIND performance.  The figure shows that for all performance
indicators discovery reaches for $\stheta$ between $10^{-5}$ and
$10^{-4}$ can be achieved, which corresponds to oscillation amplitudes
of about four orders of magnitude smaller than the current bounds.
Non-zero $\theta_{13}$ and the mass hierarchy are discovered almost
independently of the true value of $\delta$.  The CP-violation
discovery reach is very stable over about two orders of magnitude
$10^{-3.5} \lesssim \stheta \lesssim 10^{-1.5}$.  Only for large
values of $\stheta$, the fraction of $\delta$ (leftmost panel)
decreases.  This effect comes from correlations among matter density
uncertainty (see references~\cite{Ohlsson:2003ip}), oscillation
parameters (see, e.g., reference \cite{Huber:2002mx}), and other
systematics.  In addition, the explicit treatment of near detectors
affects this range~\cite{Tang:2009na}.  To illustrate the effect of
beam and detector systematics we show the thin black line, which is
computed for zero beam and detectors systematics.  Even with the
updated MIND the baseline setup is still found to be optimal for
$\stheta<0.01$ and the impact of tau contamination on the standard
performance indicators is found to be
negligible~\cite{Agarwalla:2010xx}.  This issue is discussed in detail
in section~\ref{sec:tau}.
\begin{figure}
  \begin{center}
    \includegraphics[width=0.3\textwidth]{01-PPEG/Figures/cpv-2010-20}%
    \includegraphics[width=0.3\textwidth]{01-PPEG/Figures/sgn-2010-20}%
    \includegraphics[width=0.3\textwidth]{01-PPEG/Figures/the-2010-20}
  \end{center}
  \caption{
    Performance of the baseline Neutrino Factory (100~kt at 4\,000~km
    plus 50~kt at 7\,500~km, updated MIND) in terms of the
    $\theta_{13}$, CP violation, and mass hierarchy discovery reach at
    $3\sigma$ CL. 
    The thick black line assumes default systematics, whereas the thin
    black line is for the case with no beam or detector systematics. 
  }
  \label{fig:heperf} 
\end{figure}

Apart from the discovery reaches, for which canonical performance
indicators representing the whole parameter space exist, the precision
measurements of $\theta_{13}$ or $\delta$ strongly depend on the true
values of $\theta_{13}$ and $\delta$ themselves.  This is illustrated
in figure \ref{fig:cppattern} for two performance indicators.  In the
left panel, the precision of $\theta_{13}$ is shown as a function of
the true $\stheta$ for a single baseline Neutrino Factory
``NuFact-II''.  The bands reflect the unknown true value of $\delta$.
One can easily see from figure \ref{fig:cppattern} that different
experiments can be compared in this representation.  In the right
panel, the precision of $\delta$ (``CP coverage'') is shown as a
function of the true $\delta$.  Note that a CP coverage of 360$^\circ$
means that all values of $\delta$ fit the chosen true value, \ie, no
information on $\delta$ can be obtained (for details, see reference
\cite{Huber:2004gg}).  The maximum in the figure of 180$^\circ$ means
that about 50\% of all values of $\delta$ can be excluded.  Comparing
the two solid curves, one can easily see that the magic baseline
combination acts as a risk minimiser: The curve becomes relatively
flat with a $3\sigma$ (full width) error of 30$^\circ$ to 60$^\circ$,
corresponding to a $1\sigma$ (half width) error of 5$^\circ$ to
10$^\circ$.  This corresponds to a precision similar to the quark
sector.  Note that the right panel is shown for a particular true
value of $\stheta$.  
Another performance indicator that is often used is obtained by
fitting for $\theta_{13}$ and $\delta$ simultaneously at several
selected sets of true $\{ \theta_{13}, \delta \}$.  
While this result corresponds most closely with what would be expected
from the Neutrino Factory, the analysis has yet to be repeated for the
revised Neutrino Factory baseline presented here.
Figure \ref{fig:cppattern} is therefore included to provide examples
of the precision with which $\theta_{13}$ and $\delta$ would be
measured at represent facilities. 
\begin{figure}
  \begin{center}
  \begin{tabular}{cc}
\raisebox{6.3cm}{\includegraphics[angle=-90,width=0.55\textwidth]{01-PPEG/Figures/th13accuracy}} & 
\includegraphics[width=0.40\textwidth]{01-PPEG/Figures/cppattern} 
\end{tabular}
\end{center}
\caption{\label{fig:cppattern} Performance indicators for the
  precision of $\theta_{13}$ (left panel) and $\delta$ (right panel)
  measurement at the Neutrino Factory. The precision of $\theta_{13}$
  is shown as a function of the true $\stheta$ for a single baseline
  Neutrino Factory ($90\%$\,CL); taken from reference \cite{Huber:2002mx}.
  The bands reflect the unknown true value of $\delta$. The
  precision of $\delta$ (``CP coverage'') is shown as a function of
  the true $\delta$ for a true $\stheta=0.001$ ($3 \sigma$ CL);
  taken from reference \cite{Huber:2004gg}. The different curves correspond
  to a single baseline version with all muons in one storage ring
  (dark/blue solid curve) and a two baseline version (light/yellow
  solid curve). The dashed curve corresponds to not taking 
  degeneracies into account.}
\end{figure}

There are other measurements in standard oscillation physics which can
be performed at the Neutrino Factory and which are not related to the
above performance indicators. For example, we show in figure \ref{fig:cid} the
combined precision to $\ldm$ and $\theta_{23}$ for maximal atmospheric
mixing. The different shaded and dashed regions correspond to the
normal and inverted hierarchy fit (for the true normal hierarchy),
which are separated by $2 \cdot \Delta m_{21}^2 \cdot \cos^2
\theta_{12}$ (see, e.g., reference \cite{deGouvea:2005mi}). Obviously, one
can measure $\ldm$ with a precision better than $\sdm$.
The splitting of the measured region is a relic of choosing 
$\Delta m_{31}^2$ instead of an appropriate intermediate value between
$\Delta m_{31}^2$ and $\Delta m_{32}^2$, which results in a different
mass-squared splitting for normal and inverted hierarchy for fixed
$|\Delta m_{31}^2|$. 
It does not mean that one can measure the mass hierarchy using this
effect \cite{deGouvea:2005mi}.
By comparing the left and right panels, it can be seen that it is
important to use a data sample without charge identification (CID) for
the analysis of the disappearance channels, because the overall
efficiency and threshold are much better and the CID background is
comparatively small \cite{Huber:2006wb}. 
In reference \cite{Huber:2006wb} it also has been demonstrated that
the very long baseline clearly helps to constrain $\theta_{23}$ even
further.
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{01-PPEG/Figures/cid}
\end{center}
\caption{\label{fig:cid} 
  Comparison of $\ldm$-$\theta_{23}$-precision in an analysis with
  charge identification (CID, left panel) and without CID (right
  panel) in the disappearance channel including all correlations
  ($1\sigma$, $2 \sigma$, $3\sigma$, 2 d.o.f., $\stheta=0$). 
  The appearance
  information is added as usual with CID. Dashed curves correspond to
  the inverted hierarchy solution. Figure taken from reference
  \cite{Huber:2006wb}.} 
\end{figure}


Other potential measurements within the standard oscillation framework
at the Neutrino Factory include deviations from maximal
mixings~\cite{Antusch:2004yx}, the high-confidence check of the MSW
effect in Earth matter, even for $\stheta=0$, by the solar term of the
appearance channel~\cite{Winter:2004mt}, the percent-level
determination of the average matter density along the
baseline~\cite{Winter:2005we,Minakata:2006am,Gandhi:2006gu}, and the
mass hierarchy determination if
$\stheta=0$~\cite{deGouvea:2005hk,deGouvea:2005mi}. For these
measurements, a very long baseline is typically the key ingredient.

\subsubsection{The problem of $\tau$-contamination}
\label{sec:tau}

The problem of $\tau$-contamination was first studied in reference
\cite{Indumathi:2009hg} in the context of precision measurements of
the atmospheric parameters $(\Delta m_{32}^2,\theta_{23})$ using the
$\nu_\mu \to \nu_\mu$ channel at a Neutrino Factory.  It was shown in
section \ref{sec:performance}, figure \ref{fig:cid}, that it is useful
to add all muons in the final state without CID.  The improved
efficiency in the low-energy part of the neutrino spectrum, however,
has the drawback that a previously irrelevant background now becomes
potentially harmful.  Oscillations into $\nu_\tau$, unless suppressed
by low efficiency at low energy, enhance both the right- and
wrong-sign muon samples.  In previous analyses the poor efficiency of
the detector below 10~GeV allowed $\tau$-contamination to be neglected
\cite{Cervera:2000vy,Cervera:2000kp}.  In the case of the detection of
the platinum channel, $\tau$-contamination also affects the electron
sample.  However, detectors capable of measuring the platinum channel
are usually assumed to be capable of distinguishing the electron from
tau decay from the true platinum-electron sample.  Oscillations of
$\nu_e, \nu_\mu \to \nu_\tau$ will produce $\tau$s through $\nu_\tau
N$ CC interactions within the detector that will, eventually, decay
into muons (approximately 17\% of them). These muons from taus will,
therefore, ``contaminate'' the ``direct'' muon samples (coming from
$\nu_e,\nu_\mu \to \nu_\mu$ oscillations). Notice that muons from taus
are not background but as good a signal as the direct muons.  Muons
from $\tau$-decay accumulate in low-energy muon bins, since the
associated neutrinos produced in $\tau$-decay result in a
``secondary'' muon which has, on average, 1/3 of the $\tau$
energy \cite{Indumathi:2009hg}.

It is very hard to remove the muons from tau decay using kinematic
cuts in a MIND-like detector.  Any cuts that attempt to do so
drastically reduce the direct muon events as well and hence worsen the
sensitivity to the oscillation parameters.  They escape essentially
all filters designed to kill the dominant backgrounds and directly add
to the direct muon sample, see reference \cite{Indumathi:2009hg}.  On
the other hand, neglect of the tau contribution will lead to an
incorrect conclusion about the precision achievable at the Neutrino
Factory on a given observable.  The ``$\tau$-contamination'' must be
added to the signal and it must be studied together with it.

\paragraph{Impact on the atmospheric parameters measurement}
\label{sec:rightsigntaus}

The fit of the atmospheric parameters for a typical sample set of
inputs 
($\Delta m^2 = \Delta m^2_{31} = 2.4\times 10^{-3}$ eV$^2$, 
$\theta_{23} = 41.9^\circ; \theta_{13} = 1^\circ; \delta = 0$) is
shown in figure \ref{fig:contt13fix}\,(left), from reference
\cite{Indumathi:2009hg}. 
The contours represent the allowed region in $\Delta
m^2$--$\theta_{23}$ parameter space at 99\% CL ($\Delta \chi^2 =
9.21$).  The solid line corresponds to direct muon events alone and
the dashed line to the total right-sign muon sample, including those
from tau decay.  It can be seen that the allowed region is much more
constrained with direct muons only than when including muons from
taus.  In particular, the inclusion of the tau contribution worsens
the precision with which $\theta_{23}$ and its deviation from
maximality can be measured: a spread of $\sim 2^\circ$ if tau events
are removed turns into $\sim 4.5^\circ$ when they are included.  The
reason is the following: since the $\theta_{23}$-dependent terms come
with opposite sign in $P_{\mu\mu}$ and $P_{\mu\tau}$, and the
statistics of $\tau$-events is significant, the combination of muons
from direct production and from tau decays marginally decreases the
sensitivity of the event rates to this angle.  On the other hand, the
$\tau$-contamination does not affect the determination of (the modulus
of) $\Delta m^2$.  The largest true value of $\theta_{23}$ that can be
discriminated at 99\% CL from maximal $\theta_{23}$ is shown in figure
\ref{fig:contt13fix}\,(right) as a function of $\Delta m^2$.  While
there is a distinct but mild dependence on $\Delta m^2$, it is seen
that $\tau$-contamination worsens the ability to discriminate
$\theta_{23}$ from maximality, thus making this measurement harder
than originally expected.
\begin{figure}
  \begin{tabular}{cc}
    \includegraphics[width=0.50\textwidth]%
      {01-PPEG/Figures/contt13fix} & 
    \includegraphics[width=0.50\textwidth]%
      {01-PPEG/Figures/disct13fix} 
  \end{tabular}
  \caption{ Left: 99\% CL contours in $\Delta m^2$--$\theta_{23}$ from
    CC events directly producing muons (solid line) and with the
    inclusion of muons from tau decay as well (dashed line). The input
    parameters are: $\Delta m^2 = \Delta m^2_{31} = 2.4\times 10^{-3}$ eV$^2$,
    $\theta_{23} = 41.9^\circ$.  From reference
    \cite{Indumathi:2009hg}.  Right: The largest (smallest) true value
    of $\theta_{23}$ in the first (second) octant that can be
    discriminated at 99\% CL from $\theta_{23} = \pi/4$ as a function
    of $\Delta m^2$ when only Direct ($D$) and total ($D + \tau$)
    events are considered.  In both panels, $\theta_{13}$ has been
    fixed to $1^\circ$ and $\delta = 0$.  From
    reference~\cite{Indumathi:2009hg}.  }
  \label{fig:contt13fix}
\end{figure}

\paragraph{Impact on the $\theta_{13}$ and $\delta$ measurement}
\label{sec:wrongsigntaus}

The problem of $\tau$-contamination of the $\nu_e \to \nu_\mu$ channel
was studied in detail in reference \cite{Donini:2010xk}.  
A good signal-to-background ratio is crucial to determine
simultaneously, and with good precision,
$\theta_{13}$ and $\delta$, since in this channel we have a sample
of tens of events at most.  
In order to separate high-energy charged
currents from the low-energy dominant neutral current background good
energy reconstruction is crucial. 
For this reason, in the standard
MIND analysis at the Neutrino Factory, the neutrino energy is
reconstructed by adding the energy of the muon and that of the
hadronic jet.  
This operation, however, yields a biased result when the
muon comes from a tau decay and it is detected in an iron calorimeter
such as MIND, since no additional information regarding the missing
energy arising from the neutrinos in the $\tau \to \nu_\tau \bar
\nu_\mu \mu^-$ decay can be provided.
This would not be the case in an emulsion cloud chamber or
liquid-argon detector capable of separating the $\nu_e \to \nu_\tau$
signal from $\nu_e \to \nu_\mu$ and of measuring precisely the
kinematics of the process under study.
In this case the neutrino energy could also be reconstructed 
for $\nu_e \to \nu_\tau \to \tau \to \mu$ transitions. 
The sample of wrong-sign muons from the decay of wrong-sign taus will
be distributed erroneously in neutrino energy bins, thus contaminating
the wrong-sign muon sample by events in which the parent neutrino
energy reconstruction is biased.

Consider a $\nu_\tau$ of energy $E_{\nu_\tau}$, interacting in MIND
and producing a wrong-sign $\tau$ of energy $E_\tau$ together with a
hadronic jet of energy $E_h$. After $\tau$-decay, $E_{\nu_\tau} =
E_\tau + E_h = (E_\mu + E_{miss}) + E_h$, where $E_{miss}$~is the
missing energy carried away by the two neutrinos in the $\tau$-decay.
Experimentally, we observe the secondary muon and a hadronic jet, a
signal essentially indistinguishable from that of a wrong-sign muon
from CC $\nu_\mu$ interactions. However, in the latter case, the
addition of the (primary) muon energy $E_\mu$ and of the hadronic jet
energy $E_h$ results in the correct parent $\nu_\mu$ energy,
$E_{\nu_\mu} = E_\mu + E_h$. On the other hand, in the former case the
addition of the (secondary) muon energy $E_\mu$ and of the hadronic
jet energy $E_h$ results in the wrongly reconstructed fake neutrino
energy $E_{fake} = E_\mu + E_h = E_{\nu_\tau} - E_{miss}$. If we
divide the $\tau$ three-body decay energy distribution in discrete
fake neutrino energy bins, we find that for a monochromatic $\nu_\tau$
beam of energy $E_{\nu_\tau}$, the final muon will be assigned to a
given fake neutrino energy bin of energy $E_{\mu,j}$ with probability
$V_j(E_{\nu_\tau})$, where $j = 1, \dots, N^\mu_{bin}$.  We can
compute the distribution of $\nu_\tau$ of a given energy
$E_{\nu_\tau}$ and divide them into $\nu_\tau$ energy bins of energy
$E_{\tau,i}$, where $i = 1, \dots, N^\tau_{bin}$.  The ensemble of the
probability vectors $V_j (E_{\tau.i})$, for $i$ and $j$ running over all
the $\nu_\mu$ and $\nu_\tau$ energy bins, is represented by the
migration matrix $M_{ij}$. After having computed $M_{ij}$, the number
of total wrong-sign muons in a given neutrino energy bin is given by:
\begin{equation}
\label{eq:ntot}
N_i (\theta_{13},\delta) = \sum_{i = 1, N_{bin}} \left [ N^\mu_i (\theta_{13},\delta) + \sum_{j = 1, N_{bin}} M_{ij} N^\tau_j (\theta_{13},\delta) \right ] \, .
\end{equation}
%
The resulting fraction of wrong sign muons which stem from
$\tau$-decay is found to be as large as 60\% in the energy range from
$5-10\,\mathrm{GeV}$ but only a few percent for higher energies.
\begin{figure}
  \begin{tabular}{cc}
    \includegraphics[width=0.45\textwidth]%
      {01-PPEG/Figures/Chi2ContourIB_th13_large_exa_HS.eps} & 
    \includegraphics[width=0.45\textwidth]%
      {01-PPEG/Figures/CPfractionComparisonIB.eps} 
  \end{tabular}
  \caption{ 
    Left: Test of the hypothesis that a
    simulation of the data that includes the effect of the
    $\tau$-contamination at $L = 4\,000$~km can be fitted with the
    golden muon theoretical distribution.  
    In the regions to the right of the
    contour lines, the hypothesis can be rejected at 1, 2 or 3$\sigma$
    (from left to right), assuming the goodness-of-fit statistics
    follows the $\chi^2$ distribution with $n =8$ dof. From
    reference \cite{Donini:2010xk}.  
    Right: Comparison of the CP-fraction
    achievable at the IDS-NF baseline Neutrino Factory when
    the $\tau$-contamination is correctly taken into account (dashed
    red line) with respect to the ideal one in which no muons from
    taus are considered (solid blue line).  
  }
  \label{fig:taucontapp}
\end{figure}

If not properly treated, the $\tau$-contamination introduces an
intolerable systematic error, in particular for large $\theta_{13}$.
Figure \ref{fig:taucontapp}~(left), from reference \cite{Donini:2010xk}, shows
that the test of the hypothesis that simulated data including the
$\tau$-contamination, $N_i (\theta_{13},\delta)$, can be fitted using
the direct wrong-sign muon distribution, $N^\mu_i
(\theta_{13},\delta)$, fails at more than 3$\sigma$ for $\theta_{13}
\geq 5^\circ$.  For $\theta_{13} \in [1^\circ,5^\circ]$, even if
$N^\mu_i (\theta_{13},\delta)$ can fit the $\tau$-contaminated data
(albeit with a relatively poor $\chi^2$), the error in the joint
measurement of $\theta_{13}$ and $\delta$ can be so large that it
could actually obviate the use of the Neutrino Factory as a precision
facility (see, again, reference \cite{Donini:2010xk} for a detailed
analysis of the errors introduced by a wrong treatment of the
$\tau$-contamination). On the other hand, once $M_{ij}$ has been
statistically computed, experimental data distributed in reconstructed
neutrino energy bins, can be fitted using the complete wrong-sign
muons distribution $N_i (\theta_{13},\delta)$, properly taking into
account the $\tau$-contamination of the golden-muon sample. Using this
procedure, the systematic error introduced by the muons from taus is
completely removed. The remaining error is the statistical error of
the migration matrix elements, which is under control.

It is worth noting that the $\tau$-contamination of the wrong-sign
muon sample, once properly treated, does not worsen the measurement of
$\theta_{13}$ and $\delta$, as was the case for the measurement of the
atmospheric parameters.  This is illustrated in figure
\ref{fig:taucontapp}~(right) where the CP fraction achieved using the
golden-muon sample only and the total wrong-sign muon sample are
compared.  It can be seen that the only difference between the two
lines is a slight displacement of the wiggles at $\sin^2 2 \theta_{13}
\sim 10^{-3}$ ($\theta_{13} \sim 1^\circ$).  The wiggles are a
consequence of the loss of sensitivity to CP violation introduced by
the so-called ``sign clones'' for negative $\delta$ (a phenomenon
known as $\pi$-transit, \cite{Huber:2002mx}).  Since the location of
the clones in the two samples differs, a small difference in the
location of the wiggles is found when the two lines are compared.  We
can see, however, that once the $\tau$-contamination is properly
treated, no (significant) loss in the CP-fraction is found anywhere
else.  This result differs from that shown in section
\ref{sec:rightsigntaus}.  The main difference between the two channels
is the statistical weight of the two signals.  In the case of the
$\theta_{23}$-measurement through $\nu_\mu \to \nu_\mu$ disappearance,
the right-sign muon direct sample is represented by tens of thousands
of events, and the corresponding $\tau$-contamination by $\sim$ 10\%
of the signal.  This is still a huge number of events, and it affects
the precision with which we are able to measure $\theta_{23}$ and its
deviation from maximality.  On the other hand, when dealing with the
golden channel measurement of $\theta_{13}$ and $\delta$, the signal
is represented by tens of events.  Once the problem of the wrong
assignment of muons from taus into reconstructed energy bin is solved
by means of the migration matrix approach, the residual statistical
impact of the $\tau$-contamination for $\theta_{13} \leq 10^\circ$ is
small.

