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\begin{document}

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%
%\begin{flushright}
%{\tt hep-ph/}\\
%{CERN-TH/2002-148}\\
%{FTUAM-02-18}\\
%{IFT-UAM/CSIC-02-26} \\
%{FTUV-02-0704}\\
%{IFIC/02-27}
%\end{flushright}
\begin{center}
\title{Superbeams plus Neutrino Factories} 
%\vspace*{0.5cm}
\author{O. Mena\dag}
\address{\dag\ Dept. de F\'{\i}sica Te\'orica, Univ. Aut\'onoma de
Madrid, 28049 Spain}

%\ead{jordi.burguet-castell@cern.ch}
%\ead{gavela@delta.ft.uam.es}
%\ead{gomez@hal.ific.uv.es}
%\ead{pilar.hernandez@cern.ch}
%\ead{mena@delta.ft.uam.es}
%


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\begin{abstract}
\noindent
 We show that the combination of the expected  physics results from a
 SuperBeam facility (SB) and those from a Neutrino Factory
 (NF) resolves fake degenerate solutions which appear in the simultaneous measurement of
 $\theta_{13}$ and leptonic CP violation. Intrinsic, sign($\Delta m_{13}^2$)
 and $\theta_{23}$ degeneracies are largely eliminated because of the
 different average neutrino energy and  baseline of both facilities. A short NF baseline, $L=732$ km, becomes, after such a combination, a very interesting distance.
\end{abstract}
%\submitto{\JPG}
%\pacs{14.60Pq}
%\small{\textit{Keywords:} NUFACT02, nufactory, superbeams, degeneracies.}
%\maketitle
\end{center}
%


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%\pagestyle{plain} 
\setcounter{page}{1}

\section{Introduction}
The development of a neutrino factory from muon storage rings
\cite{reviewnf} requires, by design, the essentials of a superbeam facility
\cite{firstsb,mezzetosb,sb} as an intermediate step. In this perspective, the
correct analysis strategy is to contemplate the combination of their expected
physics results. In this talk, based on the work \cite{nuevo}, we show the
important impact of such a combination, in particular on the disappearance of fake solutions in the simultaneous measurement of $\tetaot$ and $\delta$.

\section{Degenerate solutions}

It was shown in ref.~\cite{burguet} that there exists generically, at a given
(anti)neutrino energy and fixed baseline, a second value of the set ($\tetaotp,\deltap$) that gives the same 
oscillation probabilities for neutrinos and antineutrinos as
the true value chosen by nature. In what follows, these fake solutions 
will be dubbed {\it intrinsic degeneracies}. More recently, it has been pointed out \cite{minakata,bargerdeg} that other fake solutions might appear from 
unresolved degeneracies in two other oscillation parameters: the sign of
$\Delta m_{13}^2$, and the exchange $\theta_{23} \leftrightarrow \pi/2 -
\theta_{23}$ ambiguity for $\theta_{23} \ne \pi/4$ \cite{fogli}. 


Strategies to eliminate some of the fake solutions have previously been discussed, advocating the combination of different baselines \cite{burguet}, an improved experimental technique allowing the measurement of the neutrino 
energy with good precision\cite{lindner}, the supplementary detection of $\nu_e \rightarrow \nu_\tau$ channels \cite{andrea} and a cluster of detectors at
a superbeam facility located at different off-axis angles, so as to have 
different $\langle E\rangle$ \cite{bargernew}, or different superbeam experiments \cite{whisn}. 

We consider the measurement of the subleading probabilities $P_{\nu_ e\nu_\mu
  ( \bar \nu_e \bar \nu_\mu )}$ in the NF complex( for more details see
\cite{golden}), through the detection of ``wrong-sign'' muons, and
$P_{\nu_\mu\nu_e ( \bar \nu_\mu \bar \nu_e )}$ in the SB facility, through the
detection of electrons/positrons.
 
We have performed both analytical and numerical studies of the three types of
degenerate solutions. From the analytical perspective, the vacuum approximation is excellent as regards the SB scenario with a baseline of hundreds of kilometers, 
while in practice it also gives a good indication for the results at the short ($732$ km) and intermediate ($2000$--$3000$km) baselines 
of a NF. 
Of course, in our numerical treatment, all the fits include the exact formulae for the probabilities, including matter effects.

For the NF we consider the experimental setup presented in
\cite{burguet,lmd,golden}. As the SB facility we consider the design proposed
for the CERN SPL \cite{sb2}. 
Whenever it is not specified otherwise, we take the following central values
for the oscillation parameters: $\sin 2 \theta_{12} \;\cdot \Delta m^2_{12} = 10^{-4}$eV$^2$ with $\Delta m^2_{12}>0$, 
$|\Delta m^2_{13}| = 3\times 10^{-3}$ eV$^2$ and $\sin 2 \theta_{23} = 1$.
The small expected uncertainties in these parameters have not been included yet, but in a previous work \cite{burguet}, we did not find dramatic effects
when one considers them.


\begin{figure}[t]
\vglue -0.3cm
\begin{center}
\begin{tabular}{ll}
\hskip -0.5cm
\epsfig{file=2810+sb_06_burguetn.eps, width=5cm} &
\hskip -0.5cm
\epsfig{file=732+sb_8_burguetn.eps, width=5cm} \\
\end{tabular}
\end{center}
\vskip -0.6cm
\caption{\it{Fits  
combining the results from the SPL-SB facility and a NF 
baseline at $L= 2810$ km and $\tetaot=0.6^\circ$ (left) or $L=732$ km and
$\tetaot=8^\circ$ (right).
The true values illustrated correspond to 
 $\delta=- 90^\circ, 0^\circ, 90^\circ, 180^\circ$ and $\tetaot=8^\circ$ (top)
 or $\tetaot=0.6^\circ$ (bottom).
Notice that the fake intrinsic solutions have completely 
disappeared in the combinations.}} 
\label{2810sb}
\end{figure}

\subsection{Intrinsic degeneracies}

In the vacuum approximation, apart from the true solution, $\deltap=\delta$
and $\tetaotp=\tetaot$, there is a fake one. Our results for the NF can be found in ref.~\cite{burguet}, where it was shown
that these intrinsic degeneracies were eliminated when two NF baselines
(intermediate and long)  were combined. The degenerate solutions for the
SPL-SB are presented in ref. \cite{nuevo}. A comparison of the NF and SPL-SB
fits shows that the displacement of the fake solution with respect to the true
one is opposite for the two facilities. In fact, the position (measured in
$\tetaotp-\tetaot$ or $\delta '$) of the degenerate solution  depends strongly
on the baseline and the neutrino energy through the ratio $L/E$, so it is very
different in the NF, the SPL-SB and also JHF setups, as can be clearly seen in
Figs.2 of ref. \cite{nuevo} . As a result, we expect
that any combination of the results of two of these three facilities could in principle exclude the fake solutions. 
 As it is shown in Fig.~\ref{2810sb} (left), the combination of the optimal
 NF baseline $L=2810$ km results with those obtained from the SPL-SB facility is sufficient to get rid of all
 the fake solutions, even for very small values of $\tetaot$! ({\it solar
   regime} \cite{burguet}). 
It is specially important to remark what happens when one considers the
combination of the SPL-SB results and those from the NF baseline of $L=732$ km (see Fig.~\ref{2810sb} (right)) . As explained
 in refs. \cite{burguet,golden}, when this NF baseline is considered alone
 there exists a continuum of almost degenerate solutions. After the combination of the results from this NF baseline with those from
 SPL-SB not only the fake solutions corresponding to the intrinsic 
degeneracies in the superbeam disappear, but the accuracy in the 
determination of the true solution becomes competitive with  
that obtained in the combination with the optimal baseline for large
values of $\tetaot$ ({\it atmospheric regime} \cite{burguet}). At small values
 of $\tetaot$ the latter still gives better results, as expected. 


\subsection{ ${\rm Sign}(\Delta m^2_{13})$ degeneracy} 
 Assume that nature has chosen a given sign for $\Delta m^{2}_{13}$, while the
 data analysis is performed assuming the opposite sign. It is very easy to find the fake sign solutions in the vacuum approximation, 
as they mirror the two solutions (true and fake) obtained in the analysis of
the intrinsic degeneracies \cite{nuevo}, upon the substitution $\delta' \rightarrow \pi - \delta'$. 

\begin{figure}[t]
\vglue -0.3cm
\begin{center}
\begin{tabular}{ll}
\hskip -0.5cm
\epsfig{file=sb_8_sign.eps, width=5cm} &
\hskip -0.5cm
\epsfig{file=732_8_sign.eps, width=5cm} \\
\end{tabular}
\end{center}
\vskip -0.6cm
\caption{\it{Fits for central 
values $\tetaot=8^\circ$ and  
 $\delta=- 90^\circ, 0^\circ, 90^\circ, 180^\circ$ for the SPL-SB (left) and 
NF at $L=732$ km (right). Nature's
sign for $\Delta m_{23}^2$ is assumed to be positive, while the fits have been performed
 with the opposite sign.   
All fake solutions disappear when the two sets of data are combined.}} 
\label{732sbatmsgn}
\end{figure}

One of them mirrors the true (nature) solution and will be called below
solution I. Since it is approximately $E$- and $L$-independent, it will be
hard to overcome this degenerate solution. Fortunately, this fake solution
does not interfere significantly with the determination of $\tetaot$ or
CP-violation (i.e. $\sin \delta$) \footnote{ $\sin \deltap=\sin \delta$.}.
The second fake sign solution, which we will call solution II, is strongly
$L$- and $E$-dependent. Both solutions I and II can be nicely seen in the numerical analysis for the SPL-SB in Fig.~\ref{732sbatmsgn} (left), for $\tetaot=8^\circ$ and positive sign$(\Delta m_{13}^2)$.
Of course matter effects are very important in resolving fake sign
solutions\cite{sign}: the task should thus be  easier at large $ \tetaot$ and
large enough  NF baselines, where matter effects are largest.

At  $L=732$ km some fake sign solutions remain close to the present experimental limit for $\tetaot$, as shown in Figs.~\ref{732sbatmsgn} (right). It should be noticed that, again, in the combination of these latter data with those from the SPL-SB facility, all fake sign solutions disappear for large $\tetaot \geq 4^\circ$, and the sign of $\Delta m_{13}^2$ could thus be determined from it.

The solutions that survive in the combinations for small  $\theta_{13}$ are 
indeed of type I, as shown in ref. \cite{nuevo}, which do not interfere significantly with the determination of  $\tetaot$ and $\delta$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{ $\tetatt\rightarrow\pi/2 - \tetatt$  degeneracy}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The present atmospheric data indicate that $ \tetatt$ is close to maximal,
although not necessarily $45^\circ$. Superkamiokande results give
$90\%$CL-allowed parameter regions for $\sin^{2} 2 \theta_{23}>0.92$ \cite{kajita}, translating into the allowed range $35^\circ<\theta_{23}<55^\circ$. 

 As in the ${\rm sign}( \Delta m^2_{13})$ degeneracy case, there exist one
 solution I which mirrors nature's choice and a second solution, solution II,
 that mirrors the intrinsic degeneracy, as illustrated by the analytical
 formulae, see Ref.~\cite{nuevo}. Because of this parenthood, solution I is {\it a priori} 
expected to present generically less $ L$ and $E$ dependence than solution II,
and be thus more difficult to eliminate in the combination.

\begin{figure}[t]
\vglue -0.3cm
\begin{center}
\begin{tabular}{ll}
\hskip -0.5cm
\epsfig{file=sb_4_th23.eps, width=5cm} &
\hskip -0.5cm
\epsfig{file=2810_4_th23.eps, width=5cm} \\
\end{tabular}
\end{center}
\vskip -0.6cm
\caption{\it{ Fake solutions due to $ \tetatt$ degeneracies for SPL-SB results (left) and  a $L=2810$~km NF baseline (right),   
for $ \tetatt=35^\circ$, $\theta_{13}=4^\circ$ and  $\delta=- 90^\circ,
0^\circ, 90^\circ, 180^\circ$. The combination of the results from both experiments resolves the degeneracies.}}
\label{2810atmosth}
\end{figure}

We have performed fits with the wrong choice of octant for $ \tetatt$ and 
central values of $ \tetatt$ at the limit of the currently allowed domains. 
The results confirm the expectations above and indicate a situation close to that for the fake sign degeneracies, albeit slightly 
more difficult. For instance, at the $L=2810$ km baseline of the NF alone, 
still some fake $\tetatt$ solutions remain down to $\tetaot > 2^\circ$, but again they all disappear when combined with the SPL-SB data.  
As an illustration, in Figs.~\ref{2810atmosth} we show the results for $ \tetatt= 35^\circ$ and $\tetaot=4^\circ$,
 at the SPL-SB facility (left) and the $ L=2810$ km NF baseline (right).
 The same exercise, but for an $ L=732$ km baseline of the NF, results in
 the elimination of the $ \tetatt$ degeneracies only for $\tetaot\geq 8^\circ$.

In the solar regime both fake $\tetatt$ solutions have a sizable  $ L,E$
dependence, when $ \theta_{23}$ is far from maximal.  

In ref. \cite{nuevo} the fits for $ \tetaot=0.6^\circ$, for a NF at
$L=2810$ km as well as the same combined with the results from the SPL-SB facility are shown: only one annoying fake solution remains in 
the latter, which results from the merging of  solution I for SB and solution
II for the NF, owing to the finite resolution. 

In general, our conclusions are rather parallel to those for the fake
sign($\Delta m_{13}^2$) solutions, with the caveat that for the $\tetatt$
ambiguities, solution I, which is harder to resolve, is not that close to
satisfying $\sin \delta^{'} =\sin \delta$, and it is thus potentially more harmful to the measurement of CP violation. 

\section{The silver channels}

One possibility that can help very much to remove degeneracies further is to measure also the $\nu_e \rightarrow \nu_\tau$ and  $\bar{\nu}_e \rightarrow \bar{\nu}_\tau$ transition probabilities. 
The relevance of these {\it silver} channels in reducing 
intrinsic degeneracies was studied in ref.~\cite{andrea}, in the atmospheric
regime: the sign of  $\tetaotp-\tetaot$ will be opposite to that 
for the golden $\nu_e \leftrightarrow\nu_\mu$ ($\bar \nu_e \leftrightarrow \bar \nu_\mu$) channel.
In the solar regime, the intrinsic solutions in these silver channels will be identical upon exchanging $\delta'=0 $ and $\pi$, and the combination of the golden and silver channels remains a promising option.  

 Let us now turn to the fake $ \tetatt$ solutions.
When considering only $ \nu_e\rightarrow \nu_\tau$ and $\bar{\nu}_e\rightarrow
\bar{\nu}_\tau$ oscillations, the fake solutions related to the $\theta_{23}$
ambiguity get different displacements to the ones for the golden channels.
A detailed analysis for a realistic experimental 
setup will be done elsewhere \cite{prep}, but we expect that the combination of the two appearance measurements: $ \nu_e\rightarrow \nu_\mu$ and $\nu_e\rightarrow \nu_\tau$ for both polarities can help to 
resolve the dangerous solution I associated with the $\theta_{23}$ ambiguity, for $\tetaot$ in the atmospheric regime.

Finally, we recall that the disappearance measurements (e.g. $\nu_\mu \rightarrow \nu_\mu$)  should also be helpful in reducing these ambiguities for large  $\tetatt$. 
 Obviously, if the angle $\theta_{23}$ will turn out to be close to maximal (as the best-fit point now indicates), the 
$\tetatt$ degeneracies will be of very little relevance.

\section{Conclusions}
 Superbeams and Neutrino Factory are two successive steps in the same path
 towards the discovery of leptonic CP violation: the combination of the data
 from these two experiments is particularly useful to eliminate the degenerate solutions.
 The position of the fake solution  depends strongly on the baseline and the neutrino energy, and it is very different for the NF and the
 SPL-SB setups. We have studied all the degeneracies both analytically and through simulations, including realistic background 
errors and efficiencies for a magnetized iron detector at a neutrino factory, and a water Cerenkov one at the proposed 
SPL superbeam  facility.
  The intrinsic degeneracies disappear down to the sensitivity limit even for
 the neutrino factory with the detector located at short baselines!.
 The combination of two different super beams with different $ L/E$ should
 also get rid of them.

  We have also shown that the fake solutions associated with the sign and $\tetatt$ ambiguities can be grouped in two sets: 
those closer to nature's values (solutions of type I) and those related to the
intrinsic fake solution (solutions of type II).
Generically solutions I show a milder $ L/E$ dependence  and are thus more difficult to eliminate through this strategy.
 For $\tetaot$ near its present limit, the combination of the SPL-SB data and those from a short NF 
baseline, i.e. $L=732$ km, is sufficient to resolve all of them and deliver a clean measurement of 
$\tetaot$ and leptonic CP violation.
 With lowering $\tetaot$ but still in the atmospheric regime, although the same setup often produces interesting results,
it is necessary to consider an intermediate NF baseline, i.e. $L=2810$ km, together with the superbeam. In particular the 
sign of $\Delta m^2_{13}$ can be measured from the combination of their data down to $\tetaot > 1^\circ$. 

For values of $0.5^\circ< \tetaot\ < 1^\circ$ most degeneracies still
disappear in the combined data from the SPL-SB facility and the $L=2810$ km NF
baseline, but some fake solutions remain, mainly of type I. While those
associated to the sign$(\atmos)$ ambiguity  bias only slightly the extraction
of the true $\tetaot$ and $\delta$ values, those related to $\tetatt$ would
remain a problem, if $\tetatt$ were far from maximal. But, as it has been
pointed out \cite{andrea}, a supplementary  measurement of the {\it silver} channels, i.e. $ \nu_e\leftrightarrow\nu_\tau$ ($\bar{\nu}_e\leftrightarrow\bar{\nu}_\tau$), could help in removing the
intrinsic degeneracy. 
 
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\end{thebibliography}

\end{document}
