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\begin{document}
\title{Neutrino Oscillations, Rare Decays and Warped Geometry}

\author{Stephan J.~Huber\dag
\footnote[3]{stephan.huber@desy.de}
and 
Qaisar Shafi\ddag  
\footnote[4]{shafi@bartol.udel.edu}
}

\address{\dag\ Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany}

\address{\ddag\ Bartol Research Institute, University of Delaware, Newark, USA}


\begin{abstract}
We discuss how the solar and atmospheric neutrino anomalies
can be satisfactorily resolved via neutrino oscillations in a five
dimensional framework with warped geometry. In one particularly significant
example, no new fields such as singlet (right handed) neutrinos have to be 
introduced. We also consider rare processes including proton and
neutrinoless double beta decays. The Kaluza-Klein excitations of the standard
model particles are predicted to possess masses in the 10 TeV range. 
\end{abstract}

\maketitle

\section{Introduction}  
The subject of extra dimensions has created much recent excitement for a 
variety of reasons. Before discussing them, let us recall some earlier motivations:
\\[.2cm]
$\bullet$ Extra dimensions may lead to 'deeper' unification than grand unification. For instance, 
\\
\hspace*{.35cm}through inclusion of gravity, and unification of  'matter', 
             gauge and 'Higgs' sectors.
\\[.2cm]
$\bullet$ Superstring theories for their consistency require 10 space-time dimensions.
\\[.2cm]
$\bullet$ A particularly promising approach is based on the presumed existence of M-theory,
\newline
\hspace*{.35cm}whose low energy limit , it is speculated, is 11D supergravity.
\\[.2cm]
Alas, the predictive power of these higher dimensional theories is quite limited. 
Indeed, it is far from clear how this top-down approach leads to the SM (or MSSM) 
at low energies. 
It may greatly help matters if a bridge can be 
developed  between the low energy world of the SM and MSSM and physics 
at the highest energies, such that the presence of the extra dimensions is 
somehow clearly exploited. 

A number of recent attempts try to show, for instance, how the 
presence of  one or more extra dimension  can be exploited to yield 
possible resolution of the gauge hierarchy problem, the 
doublet-triplet problem, the apparent stability of the proton, and so on. 
Three of these approaches will be briefly discussed:

  1) Large extra dimension(s). It is assumed that the fundamental 
(quantum gravity)  scale $M_f$ is more or less comparable to the electroweak
scale, and this can be made consistent within a $4+N$ 
  dimensional setting through a relation  of the type
$$ M_P^2 \sim M_f ^ {2+N}\cdot V_N,$$
where $V_N$ denotes the characteristic volume of the N dimensional 
compact space, and $M_P$, as usual, is the reduced Planck scale. 
Clearly, $V_N$ must be orders of magnitude larger than $M_f ^{-N}$, 
for $M_P\gg M_f$. For $M_f \sim$ TeV, say, $R \sim 10 ^ {30 / N - 17}$ cm. 
Since the gravitational force is currently being tested in the sub-millimeter 
region, the large extra dimension scenario is still plausible 
for $N=2$ from this point of view.  

With $M_f$ close to the electroweak scale, it was hoped that the gauge hierarchy
problem would simply go away. There are at least two difficulties with this idea:
\\[.2cm]
$\bullet$
A new hierarchy has been introduced,  $M_f^{-N} / V_N$$\ll$$1$. How does this arise ?
\\[.2cm]
$\bullet$ Even though laboratory constraints on $M_f$  are still very mild 
($\gsim$1 to 0.6 TeV) as $N$ varies from  2 to 5, far more stringent lower 
bounds on $M_f$  arise from astrophysical considerations based on 
supernova and neutron stars. Thus, for N=2 (3), $M_f$  may be at least 
as high as 1700 ( 60) TeV  \cite{HR}. This makes 
  the approach look somewhat less appealing .

 2) SUSY GUTS and  (flat) extra dimension(s).
  The presence of a fifth dimension with the topology of $S^1 / Z_2 \times Z_2$ 
has been extensively utilized to construct supersymmetric models that:
Achieve doublet-triplet splitting (without many tears!);
eliminate dimension five nucleon decay;
incorporate fermion mass hierarchies and fermion mixings without 
introducing small Yukawa couplings;
implement  the idea of 'neutrino' democracy which yields large 
neutrino mixings (without making them bimaximal) \cite{ST};
realize inflation  and leptogenesis in realistic SUSY GUTS \cite{KS}.

As far as fermion masses and mixings are concerned, the five 
dimensional approach is not 
necessarily more predictive, but it does provide a new and geometric 
way to think about the 
hierarchies. The fermion wave functions carry a non-trivial dependence 
on the fifth dimension, which 
can be exploited to yield the effective four dimensional Yukawa 
(determined by their overlap with the Higgs doublets).
More interestingly, perhaps, a fifth dimension plays an essential role in 
realizing the democratic neutrino ansatz in SUSY GUTS \cite{ST2}.

3) Warped geometry. The idea that one or more of the extra dimensions 
may be warped can be motivated as follows:
-- It provides possible resolution of the gauge hierarchy problem, namely 
how $M_W \ll M_P$ \cite{RS}. Note that this can be achieved without invoking SUSY;
-- The wave functions of bulk fields are far more constrained compared 
to the case of 'flat' geometry. In principle, this could lead to more predictive power;
-- The observed atmospheric and solar neutrino oscillations can be 
accommodated within the SM 
  framework without invoking additional fields such as singlet (right handed) neutrinos;
-- The approach may be consistent with grand unification, which is not the 
case say when dealing with large extra dimensions;
-- There exists the intriguing possibility that 'warping' could be exploited to 
understand the 'tiny' vacuum energy density  $\rho_V \sim 10^{-120} M_P^4$ \cite{HSCC};
-- There are experimental predictions which can be experimentally tested. 
For instance, the
   KK excitations of bulk SM fields are expected to have masses in the 
several TeV range which,
  hopefully, can be tested at the LHC. Since there are no large extra dimensions, 
it is warping
  which makes these excitations experimentally accessible (without 
warping the masses would
  be of order $M_P$ and therefore inaccessible for ever!).
 


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\section{Fermions in a warped background}
By the Kaluza-Klein (KK) procedure the 5D fields are decomposed
into an infinite tower of 4D fields. The wave functions encode 
information on where the KK states are localized in the extra dimension.
Together with the KK masses they are obtained by solving the
5D equations of motion. In five dimensions fermions are vector-like,
and we can associate with them a 5D Dirac mass term, parameterized 
by $m_{\Psi}=c\cdot k ~{\rm sgn}(y)$, where $y$ is the 5th coordinate. 
Depending on the $Z_2$ orbifold transformation property of the fermion, 
$\Psi(-y)_{\pm}=\pm \gamma_5 \Psi(y)_{\pm}$,
the left-handed (right-handed) zero mode, $f_0\sim e^{(2-c)k|y|}$, 
of the KK decomposition is projected out by the boundary conditions \cite{GN,GP,HS2}.
The KK excited states come in left- and right-handed pairs, which 
are degenerate in mass. Note that the 5D Dirac mass parameter regulates 
whether the zero mode is localized towards the Planck-brane $(c>1/2)$
or the TeV-brane $(c<1/2)$.


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\section{The Dirac case}
It was
suggested in ref.~\cite{GN} that in warped models small Dirac neutrino 
masses can arise from a coupling to sterile (right-handed) bulk neutrinos. 
In order to generate masses in the sub-eV range, the sterile 
neutrinos have to be localized close to the Planck-brane, while 
the SM model neutrinos were confined to the 
TeV-brane. We generalized this scenario to incorporate 
bulk SM neutrinos \cite{HS3}.  

After electroweak symmetry breaking, the Dirac neutrino masses  
are generated from 
the Yukawa-type coupling between the SM neutrinos, contained in the
SU(2) doublets $L_i$, and the three sterile neutrinos $\psi_i$
\begin{equation}
{\cal L}=h^{(5)}_{ij}\bar{L}_i\psi_jH+{\rm h.c.~} +~\dots
\end{equation}
where $h^{(5)}_{ij}$ are the 5D Yukawa couplings. The 
4D Dirac masses are given by
\begin{equation} \label{3.2}
m_{ij}=\int_{-\pi R}^{\pi R}\frac{dy}{2\pi R}h^{(5)}_{ij}
e^{-4\sigma}H(y) f_{0iL}(y)f_{0j\psi}(y),
\end{equation}
where the Higgs field is confined to the TeV-brane, 
i.e.~$H(y)=H_0\delta(y-\pi R)$, to maintain the solution
of the hierarchy problem.

The fermion wave functions and 
consequently the induced  4D Dirac masses crucially depend 
on the 5D mass parameters of the
left- and right-handed fermions, $c_L$ and $c_{\psi}$ respectively, 
which enter eq.~(\ref{3.2}). As the 5D Dirac mass, i.e.~$c$ 
parameter of the fermion increases, it gets localized closer  
towards the Planck-brane. Its overlap with the Higgs profile 
at the TeV-brane is consequently reduced, which is reflected in 
a smaller 4D fermion mass from electroweak symmetry breaking.
In ref.~\cite{HS2} we have shown that this geometrical picture 
nicely generates the charged 
lepton and quark mass hierarchies, as well as quark mixings, 
by employing $c$-parameters and 5D Yukawa couplings 
of order unity. 

From the KK reduction of the left-handed neutrino field $\nu_L$
we obtain a left-handed zero mode $\nu_L^{(0)}$, corresponding to
the SM neutrino,  and an infinite
tower of left- and right-handed KK excited states $\nu_L^{(i)}$ and 
 $\nu_R^{(i)}$, where we omit flavor indices. The sterile neutrinos
decompose into the right-handed zero mode $\psi_R^{(0)}$ and the
KK excited states $\psi_L^{(i)}$ and $\psi_R^{(i)}$. After electroweak
symmetry breaking the mass matrix takes the form
\begin{equation} \label{nu_mass}
M_{\nu}= (\nu_L^{(0)},\nu_L^{(1)}, \psi_L^{(1)},\dots) \left(\begin{array}{cccc} 
m^{(0,0)} & 0 &m^{(0,1)} & \cdots \\[.1cm] 
m^{(1,0)} & m_{L,1} & m^{(1,1)}& \cdots \\[.1cm]  
0 & 0 & m_{\psi,1} & \cdots \\
\vdots & \vdots & \vdots & \ddots
\end{array}\right)\left(\begin{array}{c}\psi_R^{(0)} \\[.1cm]   \nu_R^{(1)} \\[.1cm]   
\psi_R^{(1)} \\ \vdots \end{array}\right) 
\end{equation} 
where we again suppress flavor indices, i.e.~every entry represents
a $3\times 3$ matrix in flavor space. The masses $m^{(i,j)}$ 
are obtained by inserting the relevant wave functions
into eq.~(\ref{3.2}), and $m_{L,i}$ and $m_{\psi,i}$ denote the KK masses
of the excited neutrino states. The zeros in (\ref{nu_mass}) follow 
from the $Z_2$ orbifold properties of the wave functions.

The squares of the physical neutrino masses are the eigenvalues of the 
hermitian mass matrix $M_{\nu}M_{\nu}^{\dagger}$. The unitary matrix $U_{\nu}$,
such that $U^{\dagger}_{\nu}M_{\nu}M_{\nu}^{\dagger}U_{\nu}$ is diagonal,
relates the left-handed mass eigenstates $N_L^{\rm phys}$ to the 
interaction eigenstates $N_L=(\nu_L^{(0)},\nu_L^{(i)},\psi_L^{(i)})$ via
$N_L=U_{\nu}N_L^{\rm phys}$. The observed neutrinos $\nu_L^{\rm phys}$ 
correspond to the three lightest states in $N_L^{\rm phys}$. The 
right-handed mass eigenstates are obtained from a unitary matrix
$V_{\psi}$ which diagonalizes  $M_{\nu}^{\dagger}M_{\nu}$.
The physical neutrino mixings
\begin{equation}
U=U_l^{\dagger}U_{\nu}
\end{equation}
also depend on the rotations of the left-handed charged 
leptons $U_l$. If the Yukawa couplings of the charged leptons are diagonal, the
mixing angles of the physical neutrinos can be directly read off 
from $U$. 
Like in the CKM matrix, there are three mixing angles, and a single complex 
phase $\delta$ which induces CP violation in the lepton sector.  

The atmospheric neutrino data imply \cite{SK}
\begin{equation} \label{atm}
1\cdot10^{-3}{\rm eV}^2 <\Delta m^2_{\rm atm}<5\cdot10^{-3}{\rm eV}^2,
\quad \sin^22\theta_{\rm atm}>0.85.
\end{equation}
There are several solutions to the solar neutrino anomaly \cite{BGP}.
Most favored is the LMA solution
\begin{equation} \label{sol}
2.5\cdot10^{-5}{\rm eV}^2 <\Delta m^2_{\rm sol}<4\cdot10^{-4}{\rm eV}^2,
\quad 0.3 <\sin^22\theta_{\rm sol}< 0.93.
\end{equation}
The SMA solution gives only a poor fit to the data. The CHOOZ reactor
experiment constrains $|U_{e3}|^2\equiv s_2^2$ to be at most a few 
percent \cite{CHOOZ}. Nothing is known about the CP violating phase $\delta$. 

Various constraints on the scenario with bulk gauge and fermion fields
have been discussed in the literature \cite{HS,HLS,HPR}.
With bulk gauge fields for instance, the SM relationship between the gauge couplings 
and masses of the Z and W bosons gets modified. The electroweak 
precision data then requires the lowest KK excitation of the gauge bosons 
to be heavier than about 10 TeV \cite{HS,HLS}.
This bound becomes especially important if the fermions are localized
towards the Planck-brane $(c>1/2)$. As discussed in ref.~\cite{HS2}, 
this applies to all SM fermions, with the exception of the top-quark, if the 
fermion mass hierarchy arises from their different locations in the
extra dimension. In this case the bounds induced by the contribution
of KK  excitations of the SM gauge bosons to  the electroweak 
precision observables are weak. They only require the KK masses
to be above about 1 TeV \cite{GP}. 
In the following we will therefore assume that
the mass of the first KK gauge boson is $m_1^{(G)}=10$ TeV.
The corresponding masses of the KK fermions are then in the range
10 to 16 TeV, for $0<c<1.5$. The mass of the lightest KK graviton
is 16 TeV.

In order to induce large mixings, the neutrinos have to be located at 
similar positions in the extra dimension. Therefore, we take 
$c_L^{(1)}=c_L^{(2)}=c_L^{(3)}\equiv c_L$. The charged lepton
masses are, for instance,  reproduced by $c_L=0.567$,
$c_E^{(1)}=0.787$, $c_E^{(2)}=0.614$ and $c_E^{(3)}=0.50$,
where for simplicity we have assumed diagonal Yukawa couplings 
for the charged leptons. The neutrino mixings are then solely governed by
the neutrino mass matrix $M_{\nu}$. We take $k=M_5=M_{\rm Pl}$. 
From $m_1^{(G)}=10$ TeV we determine the brane separation 
$kR=10.83$.  A parameter set which implements the 
large angle MSW solution is
\begin{eqnarray} \label{LMSW}
c_{\psi}^{(1)}=1.43,~ c_{\psi}^{(2)}=1.36, ~c_{\psi}^{(3)}=1.30, 
\nonumber\\[.2cm]
\frac{h_{ij}^{(5)}}{g_2^{(5)}}=\left(\begin{array}{ccc} 
-2.0 & 1.5 & -0.5 \\
-1.8 & -1.1 & 1.9 \\
0.5 & 1.9 & 1.7
\end{array}\right).
\end{eqnarray}
We obtain the light neutrino masses $m_{\nu1}=1.0\cdot 10^{-3}$ eV,
$m_{\nu2}=1.0\cdot 10^{-2}$ eV and $m_{\nu3}=7.1\cdot 10^{-2}$ eV.
For the neutrino oscillation parameters we find $\Delta m^2_{\rm atm}=
4.9\cdot10^{-3}$ eV$^2$, $\sin^22\theta_{\rm atm}=0.99$,
$\Delta m^2_{\rm sol}=1.0\cdot10^{-4}$ eV$^2$ and 
$\sin^22\theta_{\rm sol}=0.90$. Also, $|U_{e3}|^2=0.036$ 
is close to the experimental bound which is typical for the 
solutions we find. 


The SM neutrinos mix not only with each other but also
with the left-handed KK states of the sterile neutrinos 
$\psi_L^{(i)}$ \cite{GN}. This effect diminishes the effective weak
charge of the light neutrinos. Thus, the effective number
of neutrinos contributing to the width of the Z boson is reduced
to $n_{\rm eff}=3-\delta n$, where $\delta n$ is obtained from 
summing the relevant squared entries of $U_{\nu}$. Measurements
of the Z width induce the constraint $\delta n \lsim 0.005$. 
For the parameter set (\ref{LMSW}) we find the result $\delta n = 2\cdot 10^{-5}$, 
well below the experimental sensitivity.


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\section{The Majorana case}
In the warped SM the suppression scale for non-renormalizable
operators can be anywhere between a few TeV and the Planck scale,
depending on the localization of the fermions \cite{GP,HS2}.
Majorana masses for the left-handed neutrinos are generated by the 
dimension-five operator \cite{GP,HS2,HS4}, 
\begin{equation} \label{nu3}
\int d^4x\int dy \sqrt{-g}\frac{l_{ij}}{M_5^2}H^2\Psi_{iL}C\Psi_{jL}
\equiv \int d^4x ~M^{(\nu)}_{ ij}\Psi_{iL}^{(0)}C\Psi_{jL}^{(0)},
\end{equation}
where $l_{ij}$ are dimensionless couplings constants and 
$C$ is the charge conjugation operator. The neutrino mass matrix
reads
\begin{equation} \label{nu}
M^{(\nu)}_{ ij}=\int_{-\pi R}^{\pi R}\frac{dy}{2\pi R}\frac{l_{ij}}{M_5^2}e^{-4\sigma(y)}
H^2(y)f_{0iL}(y) f_{0jL}(y).
\end{equation}
By localizing $f_{0L}$ close to the Planck-brane 
($c_L\gg1/2$), $M^{(\nu)}$ can be made very small. However,
at the same time, the masses of the charged leptons, residing
in SU(2) multiplets with the neutrinos, can become very small 
as well \cite{GP,HS2}. Since the tau is the heaviest lepton,
it has to reside closest to the TeV-brane, and induces the
most stringent constraint on $l$. 
In ref.~\cite{HS4} we
considered the somewhat favorable case $\lambda_{\tau}^{(5)}=10g^{(5)}_2$ 
and $k/M_5=0.01$, obtaining  $m_{\nu,\tau}=l\cdot 25$ eV. A neutrino
mass of 50 meV can then be generated by $l=0.002$, which
is only a moderately small number. 


Following the spirit of ``anarchy'', we parametrize the coefficients 
in (\ref{nu}) by
\begin{equation}
l_{ij}\equiv \Lambda\cdot \tilde l_{ij}.
\end{equation}
The supposedly 
fundamental theory responsible for the effective interaction 
(\ref{nu3}) is represented through the order unity coefficients $\tilde l_{ij}$.
In order to generate large mixings we locate the left-handed
doublets at the same position in the extra dimension,
\begin{equation}
c_{e,L}=c_{\mu,L}=c_{\tau,L}=0.72.
\end{equation}
To accommodate the tau, muon and electron masses we are led to
$c_{\tau,R}=1/2$, $c_{\mu,R}=0.64$ and $c_{e,R}=0.85$. 
Taking the order unity coefficients to be homogeneously 
distributed $1/2<|\tilde l_{ij}|<2$ with random phases from 0 to $2\pi$, we find
the most favorable value for the overall scale to be 
$\Lambda=1.9\cdot10^{-3}$. We randomly generate 
parameter sets for $\tilde l_{ij}$, calculate the neutrino
mass matrix from (\ref{nu}) and compute the neutrino
masses and mixings. The left-handed charged lepton rotations 
are similar to the neutrino rotations because both fields sit
together  in the extra dimension. We include
them by generating random Yukawa couplings for the charged leptons
as well. 



Focusing on the LMA solution of the solar neutrino anomaly,
which turns out to be clearly favored, we find the following 
picture \cite{HS4}. From our neutrino parameter sets about 70 \% reproduce
$\Delta m^2_{\rm atm}$ (\ref{atm}). Imposing in addition the
constraint from $\Delta m^2_{\rm sol}$ (\ref{sol}), we are
left with about 28 \% of the parameter sets. The solar
and atmospheric mixings angles bring this number down
to about 6 \%, which is still a considerable fraction 
given the number of constraints. The most stringent
constraint turns out to come from the CHOOZ experiment.
Adding the requirement $|U_{e3}|^2<0.05$, the fraction
of viable parameter sets shrinks to about 0.7 \%.
This result is closely related to an average $\langle |U_{e3}|^2\rangle =0.22$,
which is considerably above the experimental bound.
Averaging only over the sets which pass all experimental constraints,
we find $\langle |U_{e3}|^2\rangle =0.022$.
This means that  $|U_{e3}|^2$ is probably close to the experimental bound
and can likely be tested at future neutrino experiments, 
such as MINOS, ICARUS and OPERA which are
 sensitive down to $|U_{e3}|^2\sim0.01$ \cite{LONGB}.
Since we typically start with large phases in the neutrino and
charged lepton mass matrices, the complex phase $\delta$ in the mixing 
matrix $U$ is found to be most likely of order unity.
\begin{table}[t] \centering
\begin{tabular}{|c||c|c|c|} \hline
&  $\Delta m^2_{\rm atm, sol}$ &$+\sin^22\theta_{\rm atm, sol}$ & $+|U_{e3}|^2<0.05$  
\\ \hline 
LMA & 44.4 (28.1) & 5.8 (5.9) & 5.0 (0.7)   \\ \hline
SMA & 1.3 (0.04) & 0.3 ($<0.001$) & 0.3 ($<0.001$)  \\ \hline
LOW & 0.008 ($<0.001$) & 0.002 ($<0.001$)  &  0.002 ($<0.001$)  \\ \hline 
\end{tabular} 
\caption{Probability in percent that a randomly generated set of coefficients
$1/2<|\tilde l_{ij}|<2$ satisfies the constraints from $\Delta m^2_{\rm atm, sol}$
(first column) and $\sin^22\theta_{\rm atm, sol}$ (second column) and
 $|U_{e3}|^2<0.05$  (third column). The results are given for the case
$c_{e,L}>c_{\mu,L}=c_{\tau,L}$ ($c_{e,L}=c_{\mu,L}=c_{\tau,L}$). 
 }
\label{t_L} 
\end{table}
The SMA, LOW and VAC solutions to the solar neutrino anomaly 
can be realized only with severe fine-tuning. 
Our results  are summarized in table \ref{t_L}.


So far the constraint from the CHOOZ experiment is the most 
stringent one. Its inclusion reduces the probability that a parameter 
set realizes the LMA solution from about 6 \% to
less than one percent. With all entries in the neutrino mass matrix being
of similar magnitude, the ensuing mixing angles are typically
large. The fit to the neutrino data  improves considerably if the 
electron neutrino is somewhat separated from the other two
neutrino species. Shifting the electron neutrino closer to the 
Planck-brane induces small elements in the neutrino mass
matrix. As a result, small values of  $\Delta m^2_{\rm sol}$
and $|U_{e3}|^2$ become more probable. The neutrino mass 
matrix (\ref{nu}) 
acquires the following structure
\begin{equation} \label{eps}
M^{(\nu)}\sim\left(\begin{array}{ccc} 
\epsilon^2 & \epsilon & \epsilon  \\[.1cm] 
\epsilon & 1 & 1 \\[.1cm]  
\epsilon & 1 & 1  
\end{array}\right),
\end{equation}  
where $\epsilon\approx f_{0,e}^{(\nu)}(\pi R)/f_{0,\mu}^{(\nu)}(\pi R)
\approx \exp(-(c_{e,L}-c_{\mu,L})\pi kR)$. 

We keep the muon and tau neutrinos at the previous locations 
$c_{\mu,L}=c_{\tau,L}=0.72$. A separation of these two
would make the fit to the atmospheric neutrino data
more difficult since the corresponding mixing is reduced.
For the electron neutrino
the most favorable choice turns out to be $c_{e,L}=0.79$ leading to
$\epsilon=0.15$.   
The positions
of the right-handed leptons are found to be $c_{e,R}=0.79$,
$c_{\mu,R}=0.62$ and $c_{\tau,R}=0.49$.

Let us again focus on the LMA solution.
Assuming $1/2<|\tilde l_{ij}|<2$ for the order unity coefficients,
we find the best value  $\Lambda=2.6\cdot10^{-3}$. The correct values
of the $\Delta m^2$'s are fitted by 44 \% of the parameter sets.
Taking into account also the constraints from the solar and
atmospheric mixing angles, this fraction shrinks to 5.8 \%.  
This reduction is mostly due to the solar mixing angle which is
suppressed if $\epsilon$ is small. 
Most important, however, the CHOOZ constraint $|U_{e3}|^2<0.05$
is now satisfied almost automatically, and we are finally left with 5.0 \%
of the parameter sets. Compared to the case of $\epsilon=1$, the
probability for a parameter set to satisfy all observational
constraints is enhanced by almost an order of magnitude.
The distribution of $|U_{e3}|^2$ is now peaked at
small values. Averaging only over parameter sets
which satisfy all constraints, we find a moderately small value 
$\langle |U_{e3}|^2\rangle=0.010$. As a consequence,
the next generation neutrino experiments still has a chance
to detect a non-vanishing $|U_{e3}|^2$ \cite{LONGB}. 


Taking even smaller values of $\epsilon$, it is possible to
implement the SMA solution to the solar neutrino problem. 
The LOW and VAC solutions are very difficult to accommodate 
because of their small solar  $\Delta m^2$. Thus, the LMA solution 
is by far the most favored scenario. A collection of our results 
is given table \ref{t_L}.


\section{Rare processes}

The most direct evidence
for a Majorana mass of neutrinos would be the discovery of
neutrinoless double beta decay ($0\nu\beta\beta$). 
The non-observation of this process in the Heidelberg-Moscow 
experiment implies an upper bound on 
$|M^{(\nu)}_{11}|\equiv m_{ee}<0.35$ eV \cite{HMC}.  There are
plans to bring this limit down to about 0.01 eV. 
If all neutrinos are at the same position in the extra dimension,
we expect $m_{ee}\sim0.02$ eV, which is far below the current
experimental sensitivity but within reach of future experiments. 
Once the electron neutrino is localized closer towards the Planck-brane
$0\nu\beta\beta$ becomes drastically suppressed proportional to
$\epsilon^2$. For the clearly favored LMA scenario we find 
$m_{ee}\sim0.001$ eV, which is too small to be detected in the
near future. For the SMA, LOW and VAC solutions $m_{ee}$ is
even smaller. It is therefore questionable if our model induces
$0\nu\beta\beta$ at a detectable level. In any case the Majorana
masses we find are far below the recently claimed evidence
of about 0.4 eV \cite{KDHK}. 

The low cut-off scale dramatically amplifies the impact of
non-renormalizable operators on rare processes, such as
proton decay and flavor violation. With bulk fields localized
towards the Planck-brane the corresponding suppression 
scales can be significantly enhanced \cite{GP,HS2}.
However, there are limits because the SM fermions need
to have sufficient overlap with the Higgs field at the TeV-brane
to acquire their observed masses. In the case of proton
decay, typical dimension-six operators still have to be suppressed
by small couplings of order $10^{-8}$ to be compatible with
observations. One may be tempted to completely eliminate 
these small numbers by imposing additional symmetries.
In ref.~\cite{HS3} we used lepton number conservation
to make the proton stable and to forbid the potentially
large Majorana neutrino masses of eq.~(\ref{nu3}).
In the case of Majorana neutrinos proton stability
can be ensured by lepton parity. In both cases baryon
number violation may still be present, and processes
like  neutron anti-neutron oscillations and double 
nucleon decay could be close to their
experimental bounds.

In the scenario of Dirac neutrino masses the rate of 
$\mu\rightarrow e\gamma$ transitions is considerably 
enhanced by the presence of heavy sterile neutrino states. 
If the SM neutrinos are confined to the TeV-brane,
its large branching ratio pushes the KK scale up to 25 TeV
and thus imposes the most stringent constraint on the model 
\cite{K00}.  
However, the rate for $\mu\rightarrow e\gamma$ is very
sensitive to the mixing between light and heavy neutrino states.
With bulk neutrinos the mixing with heavy states is considerably
reduced. In the case of the large angle MSW solution (\ref{LMSW})
we obtain Br$(\mu\rightarrow e\gamma)\approx 10^{-15}$ \cite{HS3}.



\section{Conclusions}
We have discussed how the atmospheric and solar neutrino 
oscillations, especially the bimaximal mixing scenario, are 
incorporated in models with a small but warped extra dimension. 
In one scenario Dirac neutrino masses arise from a coupling to
right-handed neutrinos in the bulk. Alternatively, dimension five
interactions may induce a Majorana mass for the neutrinos. In 
both cases the mixing angle $U_{e3}$ is expected be close to 
its present bound. While the direct detection of KK states is still
waiting, rare processes, such as proton decay, 
neutrinoless double beta decay and flavor violation can provide 
important hints to the higher dimensional theory. 



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