Geometric Origin of CP Violation in an Extra-Dimensional Brane World
David Dooling
Montreal
Quebec
H3A 2T8
CANADA
The fermion mass hierarchy and finding a predictive mechanism of the flavor mixing parameters remains one of the least understood puzzles facing particle physics today. In this work, we demonstrate how the realization of the Dirac algebra in the presence of two extra spatial dimensions leads to complex fermion field profiles in the extra dimensions. Dimensionally reducing to four dimensions then leads to complex quark mass matrices in such a fashion that CP violation necessarily follows. We also present the generalization of the Randall-Sundrum scenario to the case of a multi-brane, six-dimensional brane-world and discuss how multi-brane worlds may shed light on the generation index of the SM matter content.
toolj.tex
\documentclass{JHEP3}
\usepackage{psfig}


\title{Geometric Origin of CP Violation in an
Extra-Dimensional Brane World}

\author{
David Dooling \\
McGill University \\
Montreal, Quebec, H3A 2T8, CANADA \\
E-mail:
\email{dooling@hep.physics.mcgill.ca}}
\author{
Damien A. Easson \\
McGill University \\
Montreal, Quebec, H3A 2T8, CANADA\\
E-mail:
\email{easson@hep.physics.mcgill.ca}}
\author{
Kyungsik Kang \\
Brown University \\
Providence, RI 02912, USA \\
E-mail: \email{kang@het.brown.edu}}

\abstract{
The fermion mass hierarchy and finding a predictive mechanism
of the flavor mixing parameters remain two of the least
understood puzzles facing particle physics today.
In this work, we demonstrate how the realization of the Dirac
algebra in the presence of two extra spatial
dimensions leads to complex fermion field profiles in
the extra dimensions.
Dimensionally reducing to four dimensions leads to 
complex quark mass matrices in such a fashion that CP
violation necessarily follows.
We also present the generalization of the Randall-Sundrum
scenario to the case of a multi-brane, six-dimensional
brane-world and discuss how multi-brane worlds may shed light on 
the generation index of the SM matter content.
}
\vskip .2cm
\keywords{CP violation; dimension, 6}
\preprint{MCGILL 02-03 \\ BROWN-HET-1298 \\ {\tt hep-ph/0202206}}
%\bibliographystyle{unsrt}
\begin{document}
\section{Introduction}

A fundamental explanation of both the quark flavor-mixing matrix and the 
fermion masses and their hierarchical structure persist to be two of the
most challenging problems of particle physics today.
A predictive mechanism for fermion mass generation is currently
lacking.
After spontaneous symmetry breaking, the quark mass term in
the langrangian reads:
\begin{equation}
\mathcal{L}_{\mathnormal \mbox{mass} } \mathnormal =
\frac{v}{\sqrt{2}} \left( \overline{u_{L_{i}}} h_{ij}^{(u)} u_{R_{j}} +
 \overline{d_{L_{i}}} h_{ij}^{(d)} d_{R_{j}} \right) + \mbox{h.c.}
\end{equation}
where the $h_{ij}$ are arbitrary $3 \times 3$ complex Yukawa
coupling matrices.
Within the standard model (SM) the fermion masses, the quark 
flavor-mixing angles and the CP violating phase are free parameters and
no relation exists among them.
The SM can accommodate the observed mass spectrum of the fermions but
unfortunately does not predict it.
Thus the calculability of the fermion masses remains an outstanding
theoretical challenge.
Our hope is that some predictive mechanism of fermion
mass generation exists and will place the understanding of fermion mass on a 
par with that of gauge boson mass.

The number of free parameters in the Yukawa sector eliminates any real
predictive power of this sector of the SM.
A first step one may take is to modify the Yukawa sector
in such a way that the four quark flavor-mixing parameters depend
solely on the quark masses themselves.
As an attempt to derive a relationship between the quark masses and
flavor-mixing parameters, mass matrix ans\"{a}tze based on 
flavor democracy with a suitable breaking so as to allow mixing 
between the quarks of nearest kinship
via nearest neighbor interactions was suggested about two decades
ago \cite{Weinberg:1977hb,Wilczek:1977uh,Rothman:1979ft,Kang:1981yg,Fritzsch:1978vd,Fritzsch:1979zq,DeRujula:1977ry,Georgi:1979dq}.
These early attempts are the first examples of ``strict calculability'';
i.e., mass matrices such that all flavor-mixing parameters 
depend solely on, and are determined by, the quark masses.
But the simple symmetric NNI texture leads to the experimentally
violated inequality $M_{top} < 110$ GeV, prompting
consideration of a less restricted form for the mass matrices
so as to retain calculability, yet be consistent
with experiment \cite{Kang:1997uv}.


After implementing this first step towards
gaining a deeper understanding of the Yukawa sector of the SM in the
guise of calculability, one may then attempt to
 confront the fundamental problem of explaining
the fermion mass hierarchy itself.
In this paper, we will address neither of the above issues, but
instead shall retreat even further into a simpler domain of the 
overall problem.
The quark mixing matrix contains four physical 
parameters, the three mixing angles and the single CP violating
phase of the Cabibo-Kobayashi-Maskawa (CKM) matrix.
Here we conjecture that the CP phase
in the quark flavor-mixing matrix may be a result of the existence of 
extra dimensions and the Dirac algebra realized in the presence
of these extra dimensions.
The notion of CP violation arising from the presence of extra
dimensions is not new, but was studied long ago
 \cite{Thirring:1972de}, and more recently in
\cite{Casadio:2001fe,Chang:2001uk,Chang:2001yn,Huang:2001np,Branco:2000rb,Sakamura:2000ik,Sakamura:1999fa}.


Our work is largely inspired by the papers \cite{Hung:2001hw,Huber:2000ie,Arkani-Hamed:1999dc,Mouslopoulos:2001uc,Kaplanet:2001} in which the fermion
mass and mixing hierarchies have been addressed within the context
of large extra dimensions (LED), both in the five- and six-dimensional
cases, as well as within the context of a single extra dimension
with warped geometry.
This paper addresses the same problem using two
extra dimensions with warped geometry, so as to explain the
existence of the CP phase in the quark-flavor mixing matrix.
Six dimensional extensions of the RS scenario have been
studied in \cite{Chodos:1999zt,Gherghettaet:2000,Collins:2001ni,Kogan:2001yr,Burgess:2001bn,Kim:2001rm}.



Another major theoretical problem in physics is that
of the hierarchy between the Planck scale 
(the rest mass of a flea) and particle physics scales,
such as the masses of the $W$ and $Z$ bosons.
String/M-theory seems to be the most promising candidate to form
a tight conceptual connection between gravity and particle
physics, with its attendant extra dimensions.
Some novel ideas concerning solutions to the gauge
hierarchy problem are rooted in the possibility that
the hierarchy is controlled by exotic features of the extra 
dimensions; namely, either they are very large and so the
hierarchy is generated by the volume of the extra
dimensions \cite{Arkani-Hamed:1998rs,Arkani-Hamed:1998nn,
Antoniadis:1998ig} or that the extra dimensions are warped, with the 
hierarchy generated by an exponential damping
\cite{Gogberashvili:1998vx,Randall:1999ee,Randall:1999vf}.
%Without the recent ADD and RS developments, it seems unlikely
%that the study of the fermion mass hierarchy would have
%developed within the context of extra dimensions.
%As is well-known, string theory has little data to constrain
%the ideas of its metatopology.
%Less constraints promote creative ideas.
%In particle physics, and in particular in the quark sector, we
%are blessed with increasingly precise data, which
%serve as tighter constraints on
%theoretical understanding.
Here we explore
the implications of extra dimensions for the fermion
mass and mixing problem.
This investigation sheds light on the source of CP
violation. 
%making this exchange of ideas between these two
%branches of theoretical physics more promising and worthy
%of continued investigation.

The organization of this paper is as follows.
We briefly review the Randall-Sundrum scenario and introduce
its extension to two extra dimensions.
We then introduce fundamental six-dimensional fermions
(eight-component objects), as well as a fundamental
Higgs scalar.
We present the equations for the fermion
zero modes, as well as the boundary conditions.
Previous treatments of the fermion boundary
conditions in six dimensions differ from the ones
presented here.
We solve for the Higgs zero mode and show,
for particular values of its six-dimensional
mass, how its profile is peaked away from the
six dimensional analogue of the hidden brane in the
brane set-up to be described.
The possibility of bulk SM fields within the RS scenario
has been extensively studied in \cite{Chang:1999nh}.
We demonstrate that the presence of two extra dimensions leads to
complex fermion field profiles in the extra dimension and 
show how this leads to CP violation.
We then conclude and briefly discuss some future avenues to be
investigated.


\section{Many-Brane, Six-Dimensional Extension of the 
Randall-Sundrum Solution}

Here we present a simple generalization of the RS scenario to the six
dimensional, multi-brane case.
In what follows, we consider one fundamental cell of the 
brane lattice (to be described below), but for completeness we
write down the entire brane-lattice solution.
Ultimately, one wants to understand not only the fermion mass
hierarchy and the flavor mixing parameters, but also the very
existence of the three fermion generations of the SM.
An interesting observation of Kogan et al \cite{Kogan:2001wp} is that
in multi-brane worlds, there exist ultra-light localized
and strongly coupled bulk fermion KK modes.
This leads to the possibility that for a 
given fundamental bulk 
fermion field with given SM gauge group transformation
properties, the generation index may be associated with KK
mode number, so that ultimately there is only one six-dimensional
species of up-type quarks, for example, and that the generation
structure is just a reflection of the existence of ultra-light
KK modes arising from the brane set-up and geometry of the
extra dimensions.
One stumbling block confronting this mode of 
understanding fermion generation structure is that the very
multi-brane set-up that gives rise to the family
structure also gives rise to the same number of ultralight
KK modes for all other fields, including the graviton.
This approach to the family index is currently under investigation,
though in the applications to follow we will consider only one 
fundamental cell of this brane crystal and hence will not
have any ultralight KK modes.

We now present a solution for the metric corresponding to a 
six dimensional, multi-brane extension of the RS scenario.
A five dimensional multi-brane extension is 
presented in \cite{Hatanaka:1999ac}.


To generalize to six dimensions, we consider a 
N $\times$ M lattice of N parallel 4-branes localized in the $\phi$ dimension 
orthogonal to M parallel 4-branes localized in the $\rho$ dimension.
3-Branes reside at their loci of intersection.
The action describing this set-up is given by the following three
terms:
\normalcolor
\begin{displaymath}
S = S_{gravity} + \sum_{i=1}^{N} S_{i} + \sum_{j=1}^{M} S_{j}
\end{displaymath}
%\Large
where $S_{grav}$ is
\begin{displaymath}
S_{grav} = \int d^{4} x \int_{0}^{2 \pi} d \phi \int_{0}^{2 \pi} d
\rho \sqrt{-g} \left( \frac{1}{ \kappa_{6}^{2} } R - \sum_{i,j} 
\Lambda_{ij} \left[ \Theta \left( \phi - \phi_{i} \right) - \Theta
 \left( \phi - \phi_{i+1} \right) \right] \times \right.
\end{displaymath}
\begin{displaymath}
\left.
 \left[ \Theta \left( \rho
- \rho_{j} \right) - \Theta \left( \rho - \rho_{j+1} \right) \right] \right)
\end{displaymath}
%\large
and the terms in the action representing the 4-branes
are
\begin{displaymath}
S_{i} = - \int d^{4} x \int_{0}^{2 \pi} d \rho \sqrt{g^{\left( \phi = \phi_{i} \right)}} T_{\phi_{i}}
\end{displaymath}
and
\begin{displaymath}
S_{j} = -\int d^{4} x \int_{0}^{2 \pi} d \phi \sqrt{g^{\left( \rho = \rho_{j} \right)}} T_{\rho_{j}},
\end{displaymath}
where the $T_{\phi_{i}}$ are the tensions of the 4-branes
located at
 $\phi_{i}$ and $T_{\rho_{j}}$ are the tensions of the 4-branes
located at $\rho_{j}$. 
%\Large
We are interested in the case of both extra dimensions being compact
and impose the following $S^{1}$ periodicity conditions on the 
extra coordinates:
\begin{displaymath}
\rho_{M+1} = 2 \pi
\end{displaymath}
\begin{displaymath}
\phi_{N+1} = 2 \pi
\end{displaymath}
Furthermore, we consider an orbifold
by imposing a pair of
$Z_{2}$ symmetries on the solution.
As in \cite{Hatanaka:1999ac}, we use the $S^{1}$ symmetry(ies) to 
define the position of the first set of 
 brane sources to be at the origin of the extra dimensions,
\begin{displaymath}
\phi_{1} = 0 = \rho_{1}
\end{displaymath}
In the above expressions for the 4-brane actions, 
the induced metric is
\begin{displaymath}
g_{ab}^{\rho = \rho_{j}} = g_{ab} \left( x^{\mu}, \phi, \rho = \rho_{j} \right)
\end{displaymath}
for the 4-branes localized in the $\phi$ direction and
\begin{displaymath}
g_{\alpha \beta}^{\phi = \phi_{i}} = g_{\alpha \beta} \left( x^{\mu}, \phi = 
\phi_{i}, \rho \right)
\end{displaymath}
for the 4-branes localized in the $\rho$ direction.
%\newpage
The six dimensional Einstein equations are
\begin{displaymath}
R_{N}^{M} - \frac{1}{2} \delta_{N}^{M} R = \frac{\kappa_{6}^{2}}{2} T_{N}^{M}
\end{displaymath}
where
\begin{eqnarray*}
T_{N}^{M} & = & -\sum_{i,j}^{N,M} \Lambda_{ij} \left[ \Theta \left(
\phi - \phi_{i} \right) - \Theta \left( \phi - \phi_{i+1} \right) 
\right] \left[ \Theta \left( \rho - \rho_{j} \right) - \Theta
 \left( \rho - \rho_{j+1} \right) \right] \delta_{N}^{M} \\
& = & -\sum_{i}^{N} \sqrt{\frac{-\mbox{det} g^{\phi = \phi_{i}}}{\mbox{det} g}} T_{\phi_{i}} \delta \left( \phi - \phi_{i} \right) \delta_{a}^{M} \delta^{a}_{N} \\
& = & -\sum_{j=1}^{M} \sqrt{\frac{-\mbox{det} g^{\rho = \rho_{j}}}{\mbox{det} g}} T_{\rho_{j}} \delta \left( \rho - \rho_{j} \right) \delta_{\alpha}^{M} \delta_{N}^{\alpha}
\end{eqnarray*}
%\Huge

We are not addressing any cosmological issues in this work, and for 
simplicity consider the following
static ans\"{a}tze for the metric:
\normalcolor
%\Large
\begin{displaymath}
ds^{2} = A^{2} \left( \phi, \rho \right) \eta_{\mu \nu} dx^{\mu}
dx^{\nu} - B^{2} \left( \phi, \rho \right) d \phi^{2} - C^{2} \left(
\phi, \rho \right) d \rho^{2}
\end{displaymath}


With this ans\"{a}tz, the left-hand side of the Einstein 
equations are given by the following expressions, where dots
denote derivatives with respect to the $\rho$ coordinate and
primes denote derivatives with respect to the $\phi$ coordinate:
%\begin{displaymath}
\begin{eqnarray*}
G_{\nu}^{\mu} & = & \frac{2}{A^{2}} \left[ \left(
\frac{\dot{A}}{A} \right)^{2} + \left( \frac{A^{\prime}}{A} \right)^{2} 
+ 2 \left( \frac{\ddot{A}}{A} + \frac{A^{\prime \prime}}{A} \right)
\right] \delta_{\nu}^{\mu} \\
G_{{\phi}}^{{\phi}} & = & \frac{2}{A^{2}} \left[ 5 \left( \frac{A^{\prime}}{A} \right)^{2} + 2 \frac{\ddot{A}}{A} + \left(
\frac{\dot{A}}{A} \right)^{2} \right] \\
%G_{{\rho}}^{{\rho}} & =  & \frac{2}{A^{2}} \left[ 5 \left(
%\frac{\dot{A}}{A} \right)^{2} + 2 \frac{ A^{\prime \prime}}{A} + 
%\left( \frac{A^{\prime}}{A} \right)^{2} \right] \\
%G_{{\rho}}^{{\phi}} & = & \frac{4}{A^{4}} \left[
%-A\dot{A^{\prime}} + 2 \dot{A} A^{\prime} \right] \\
%G_{{\phi}}^{{\rho}} & = & G_{{\rho}}^{{\phi}}
\end{eqnarray*}
%\newpage
\begin{eqnarray*}
G_{{\rho}}^{{\rho}} & =  & \frac{2}{A^{2}} \left[ 5 \left(
\frac{\dot{A}}{A} \right)^{2} + 2 \frac{ A^{\prime \prime}}{A} + 
\left( \frac{A^{\prime}}{A} \right)^{2} \right] \\
G_{{\rho}}^{{\phi}} & = & \frac{4}{A^{4}} \left[
-A\dot{A^{\prime}} + 2 \dot{A} A^{\prime} \right] \\
G_{{\phi}}^{{\rho}} & = & G_{{\rho}}^{{\phi}}
\end{eqnarray*}
%\end{displaymath}
%\color{green}
Taking the warp factor to be
\normalcolor

\begin{displaymath}
A = \frac{1}{e^{\sigma \left( \phi \right)} +
 e^{\gamma \left( \rho \right)}
 + 1}
\end{displaymath}
and
\begin{displaymath}
B \left( \phi, \rho \right) = A \left( \phi, \rho \right) e^{ \sigma \left( \phi \right)}
\end{displaymath}
\begin{displaymath}
C \left( \phi, \rho \right) = A \left( \phi, \rho \right) e^{ \gamma \left( \rho \right)}
\end{displaymath}
one can easily check that the nondiagonal elements of the Einstein tensor
vanish,
$G_{{\rho}}^{{\phi}} = 0$, and thus the ${\phi} - 
{\rho}$ component of the Einstein equations are trivially 
satisfied.


The remaining equations will be satisfied if the following relations
are fulfilled:
\normalcolor
\begin{displaymath}
10 \left( k_{\phi_{i}}^{2} + k_{\rho_{j}}^{2} \right) =
-\frac{\kappa_{6}^{2}}{2} \sum_{i,j} \Lambda_{ij} \left[ \Theta \left(
\phi - \phi_{i} \right) - \Theta \left( \phi - \phi_{i+1} \right)
\right] \times
\end{displaymath}
\begin{displaymath}
 \left[ \Theta \left( \rho - \rho_{j} \right) - \Theta \left(
\rho - \rho_{j} \right) \right],
\end{displaymath}
\begin{displaymath}
8 \left( k_{\rho_{j}} - k_{\rho_{j-1}} \right) = \frac{\kappa_{6}^{2}}{2}
T_{\rho_{j}},
\end{displaymath}
\begin{displaymath}
8 \left( k_{\phi_{i}} - k_{\phi_{i-1}} \right) = \frac{\kappa_{6}^{2}}{2}
T_{\phi_{i}},
\end{displaymath}
%\newpage
So we have
%\normalcolor
%\begin{displaymath}
%A \left( \phi, \rho \right) = \frac{1}{\left( e^{ \sigma \left( \phi \right) } + e^{\gamma \left( \rho \right)} -1 \right)}
%\end{displaymath}
%\begin{displaymath}
%B \left( \phi, \rho \right) = A \left( \phi, \rho \right) e^{ \sigma \left( \phi \right)}
%\end{displaymath}
%\begin{displaymath}
%C \left( \phi, \rho \right) = A \left( \phi, \rho \right) e^{ \gamma \left( \rho \right)}
%\end{displaymath}
%where
\normalcolor
\begin{displaymath}
\sigma \left( \phi \right) = 
k_{\phi_{1}} | \phi - \phi_{1} | \Theta \left( \phi - \phi_{1} \right) + 
\left( k_{\phi_{2}} - k_{\phi_{1}} \right) | \phi - \phi_{2} | \Theta \left( \phi - \phi_{2} \right) +
\end{displaymath}
\begin{displaymath}
 \left( k_{\phi_{3}} - k_{\phi_{2}} \right) | \phi
- \phi_{3} | \Theta \left( \phi - \phi_{3} \right) + \cdots
 +
\left( k_{\phi_{N}} - k_{\phi_{N-1}} \right) | \phi - \phi_{N}| 
\Theta \left( \phi - \phi_{N} \right)
\end{displaymath}
\begin{displaymath}
\gamma \left( \rho \right) =
k_{\rho_{1}} | \rho - \rho_{1} | \Theta \left( \rho - \rho_{1} \right) +
\left( k_{\rho_{2}} - k_{\rho_{1}} \right) | \rho - \rho_{2} | \Theta
\left( \rho - \rho_{2} \right) +
\end{displaymath}
\begin{displaymath}
 \left( k_{\rho_{3}} - k_{\rho_{2}} \right)
| \rho - \rho_{3} | \Theta \left( \rho - \rho_{3} \right) + \cdots
+ \left( k_{\rho_{M}} - k_{\rho_{M-1}} \right) |\rho - \rho_{M} |
\Theta \left( \rho - \rho_{M} \right)
\end{displaymath}
and
\normalcolor
\begin{displaymath}
k_{i} - k_{i-1} = \frac{\kappa_{6}^{2}}{16} T_{i}
\end{displaymath}
\begin{displaymath}
\kappa_{6}^{2} = \frac{ 16 \pi}{M_{6}^{4}}
\end{displaymath}
We 
work in units where $M_{6}^{-4} = 1$, so that
\normalcolor
\begin{displaymath}
k_{i} - k_{i-1} = \pi T_{i}
\end{displaymath}
and
for simplicity we take all the bulk cosmological constants to be equal
and
the magnitudes of the brane tensions to be the same:
$| T_{\phi_{i}} | = |T_{\rho_{j}}| = T$
Thus we have the relations

\begin{displaymath}
10 \left( 2 \right) k_{1}^{2} = -\frac{ 16 \pi}{\left( 2 \right) M_{6}^{4}} \Lambda
\end{displaymath}
\begin{displaymath}
k_{1} = \sqrt{ -\frac{2 \pi \Lambda}{5}} = k_{\phi_{1}} = k_{\rho_{1}}
\end{displaymath}
%\newpage
%\LARGE
 Our expressions for the functions
$\sigma \left( \phi \right)$ and $\gamma \left( \rho \right)$
then become 
\begin{displaymath}
\sigma \left( \phi \right) = \sqrt{-\frac{2 \pi \Lambda}{5}} \phi \Theta 
\left( \phi \right) + \pi T \left[ -\left( \phi - \phi_{2} \right) 
\Theta \left( \phi - \phi_{2}\right) \right.
\end{displaymath}
\begin{displaymath}
+ \left( \phi - \phi_{3} \right) \Theta \left( \phi - \phi_{3} \right) - 
\left( \phi - \phi_{4} \right) \Theta \left( \phi - \phi_{4} \right)
\end{displaymath}
\begin{displaymath}
+ \left( \phi - \phi_{5} \right) \Theta \left( \phi - \phi_{5} \right) -
\left( \phi - \phi_{6} \right) \Theta \left( \phi - \phi_{6} \right)
\end{displaymath}
\begin{displaymath}
\left. + \ldots
 - \left( \phi - \phi_{N} \right) \Theta \left( \phi - \phi_{N}
\right) \right]
\end{displaymath}
\begin{displaymath}
\sigma \left( \phi_{c} \right) = \sigma \left( 0 \right) = 0 \rightarrow
\end{displaymath}
\begin{displaymath}
\phi_{c} = \frac{ \pi T \left( \phi_{2} - \phi_{3} + \phi_{4} - \phi_{5} +
\phi_{6} - \ldots + \phi_{N} \right)}{\left( \pi T - \sqrt{
-\frac{2 \pi \Lambda}{5}} \right)}
\end{displaymath}
and similarly,
\normalcolor
\begin{displaymath}
\gamma \left( \rho \right) = \sqrt{ -\frac{ 2 \pi \Lambda}{5}} \rho
\Theta \left( \rho \right) + \pi T \left[ -\left( \rho - \rho_{2} \right)
\Theta \left( \rho - \rho_{2} \right) \right.
\end{displaymath}
\begin{displaymath}
\left. + \ldots - \left( \rho - \rho_{M} \right) \Theta
\left( \rho - \rho_{M} \right) \right]
\end{displaymath}
\begin{displaymath}
\gamma \left( \rho_{c} \right) = \gamma \left( 0 \right) = 0 \rightarrow
\end{displaymath}
\begin{displaymath}
\rho_{c} = \frac{ \pi T \left( \rho_{2} - \ldots + \rho_{M} \right)}{
\left( \pi T - \sqrt{-\frac{ 2 \pi \Lambda}{5}} \right)}.
\end{displaymath}

\section{Six Dimensional Dirac Fermions}

In the seminal work of \cite{Arkani-Hamed:1999dc}, a framework is introduced for 
understanding both the fermion mass hierarchy and proton
stability without recourse to flavor symmetries 
in terms of higher dimensional geography.
Additional physics must be assumed in order to localize the fermions in
the flat extra dimension, but once this additional
scalar field is introduced, any coupling between chiral fermions
is exponentially suppressed because the two fields are
separated in space.
A key observation made by Huber and Shafi in \cite{Huber:2000ie} is that one
can get this exponential damping automatically from a 
non-factorizable geometry and that there is no need to assume 
additional physics.

 In both \cite{Huber:2000ie} and 
\cite{Arkani-Hamed:1999dc}, the effective four dimensional masses arise
from integrating over the extra dimensional Yukawa interactions
between the five dimensional Higgs and chiral fermion
fields.
The resulting four dimensional Yukawa coupling matrices
exhibit phenomenologically acceptable spectrums and mixing
(excluding CP violation) because of how these overlap
integrals can be tuned depending on the values of the
five dimensional mass term for the fermions in the action.
In the flat space scenario of \cite{Arkani-Hamed:1999dc}, the five dimensional mass
term serves to translate the gaussian profile of the fermions
along the extra dimension, so that the overlap integral of two
chiral fermions and the flat zero mode of the Higgs is itself
a gaussian, thus generating exponentially small effective
four dimensional Yukawa couplings.
In the warped space scenario of \cite{Huber:2000ie},
 with the $\frac{S^{1}}{Z_{2}}$
geometry of Randall-Sundrum, the right-handed zero modes
are peaked at the origin on the positive tension 3-brane, while
the left-handed zero modes are peaked at the other orbifold
fixed point around the negative tension 3-brane.
In this case, varying the value of the five dimensional mass
does not translate the fermion field profiles along the 
extra dimension; the right-handed and left-handed fermions remain
localized around the positive and negative tension 3-branes,
respectively.
What does change, however, is the width of the profile.
Hence, just as in the flat space scenario, small changes in the 
values of the five dimensional mass parameters lead to 
greatly amplified changes in the resulting four dimensional
Yukawa coupling matrices.

In both the flat and warped space scenarios in five dimensions,
the profiles for all fields in the extra dimension are real
and CP violation cannot be realized in a natural way.
While CP violation is not addressed in
\cite{Huber:2000ie}, a numerical
example is given where nine input parameters, essentially the ratios
of the five dimensional masses for the fermions appearing
in the five dimensional action to the AdS curvature scale, 
result in very good agreement with the quark mass spectrum and
absolute values of the elements of the $V_{CKM}$ matrix.
The agreement is striking, with just enough
disagreement in the mixing matrix to generate the suspicion 
that inclusion of CP violation may somehow bring about
even closer agreement between this type of model's predictions
and experiment.

The action for a fermion in the six dimensional background
considered here is~\cite{Grossman:1999ra}


\begin{equation} 
S = \int d^{4}x \int d \phi d \rho \sqrt{G} \left\{ E^{A}_{\alpha} \left[ \frac{i}{2} \overline{\psi} \gamma^{\alpha} \left( \stackrel{\rightarrow}{\partial_{A}} - \stackrel{\leftarrow}{\partial_{A}} \right) \psi \right] -m \left( \phi, \rho \right) \overline{\psi} \psi \right\}
\end{equation}
\begin{equation}\label{metric}
ds^{2} = A^{2} \left( \phi, \rho \right) \eta_{\mu \nu} dx^{\mu} dx^{\nu} -
B^{2} \left( \phi, \rho \right) d \phi^{2} - C^{2} \left( \phi, \rho \right)^{2}
\end{equation}
\begin{equation}
A = \frac{1}{e^{\sigma} + e^{ \gamma} -1}
\end{equation}
\begin{equation}
B = e^{\sigma} A
\end{equation}
\begin{equation}
C = e^{\gamma} A
\end{equation}
\begin{equation}
\sqrt{g} = \frac{ e^{\sigma} e^{\gamma} }{\left( e^{\sigma} + e^{\gamma} - 1\right)^{6}}
\end{equation}

We now show that the introduction
of another extra dimension leads to a natural explanation of 
CP violation.
In six spacetime dimensions the Dirac algebra is minimally
realized by $8 \times 8$ matrices.
A particularly convenient representation is that of~\cite{Hung:2001hw},
in which the ideas of \cite{Arkani-Hamed:1999dc}  have been extended to the six dimensional case.  This provides a theoretical motivation for
the so-called democratic mass matrices that have served as the 
starting point for many flavor symmetry approaches to the
quark mass hierarchy problem.
This convenient representation is presented below, where the
$\gamma_{5}$ in $\Gamma_{\phi}$ is the same $\gamma_{5}$
constructed from the four dimensional $\gamma$'s.


\begin{equation}
\Gamma_{0} = \left( \begin{array}{cc}
0 & +i \gamma_{0} \\
-i \gamma^{0} & 0  
\end{array} \right)
\end{equation}
\begin{equation}
\Gamma_{1} = \left( \begin{array}{cc}
0 & +i \gamma_{1} \\
-i \gamma^{1} & 0  
\end{array} \right)
\end{equation}
\begin{equation}
\Gamma_{2} = \left( \begin{array}{cc}
0 & +i \gamma_{2} \\
-i \gamma^{2} & 0  
\end{array} \right)
\end{equation}
\begin{equation}
\Gamma_{3} = \left( \begin{array}{cc}
0 & +i \gamma_{3} \\
-i \gamma^{3} & 0  
\end{array} \right)
\end{equation}
\begin{equation}
\Gamma_{\phi} = \left( \begin{array}{cc}
0 &  \gamma_{5} \\
 -\gamma^{5} & 0  
\end{array} \right)
\end{equation}
\begin{equation}
\Gamma_{\rho} = \left( \begin{array}{cc}
0 & +i_{4 \times 4} \\
+i_{4 \times 4} & 0  
\end{array} \right)
\end{equation}
\begin{equation}
\Gamma_{7} = \left( \begin{array}{cc}
1 & 0 \\
0 & -1  
\end{array} \right)
\end{equation}
\begin{equation}
\gamma_{0} = \left( \begin{array}{cccc}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \end{array} \right)
\end{equation}
\begin{equation}
\gamma_{1} = \left( \begin{array}{cccc}
0 & 0 & 0 & -1 \\
0 & 0 & -1 & 0 \\
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \end{array} \right)
\end{equation}
\begin{equation}
\gamma_{2} = \left( \begin{array}{cccc}
0 & 0 & 0 & i \\
0 & 0 & -i & 0 \\
0 & -i & 0 & 0 \\
i & 0 & 0 & 0 \end{array} \right)
\end{equation}
\begin{equation}
\gamma_{3} = \left( \begin{array}{cccc}
0 & 0 & -1 & 0 \\
0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \end{array} \right)
\end{equation}
\begin{equation}
\gamma_{5} = \left( \begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1 \end{array} \right)
\end{equation}


Both the $\Gamma$ 's and the $\gamma$'s have the mostly minus signature,
accounting for the discrepancy between this representation and the
one
presented in \cite{Hung:2001hw}.


Using $\Gamma_{7}$, we can construct  projection operators
in the usual way to
get two four-component objects we call $\psi_{+}$ and $\psi_{-}$.
This procedure is
in perfect analogy with what is done in four dimensions 
(expressing a four dimensional fermion field in terms of its
left and right-handed components).

\begin{equation}
\psi_{+} = \frac{1}{2} \left( 1 - \Gamma_{7} \right) \psi
\end{equation}
\begin{equation}
\psi_{-} = \frac{1}{2} \left( 1 + \Gamma_{7} \right) \psi
\end{equation}
\begin{equation}
\psi = \psi_{+} + \psi_{-}
\end{equation}

Note that in this representation 
the six dimensional lorentz invariant fermion bilinear
$\overline{\psi} \psi$ has a complex coefficient when
expressed in terms of the four component fields $\psi_{+}$ and
$\psi_{-}$ and their four component conjugate fields
$\overline{\psi_{+}}$ and $\overline{\psi_{-}}$.
This leads to a real six dimensional mass
term which is complex when expressed in terms of the 
four component projections $\psi_{+}$ and $\psi_{-}$ (the 
six dimensional analogs of $\psi_{L}$ and $\psi_{R}$).


\begin{equation}
\overline{\psi} = \psi^{\dagger} \Gamma_{0} = \left( \psi_{+}^{\dagger}, \psi_{-}^{\dagger} \right) \left( \begin{array}{cc}
0 & i \gamma_{0} \\
-i \gamma_{0} & 0 \end{array} \right) = \left( -i \overline{\psi_{-}} , i \overline{\psi_{+}} \right)
\end{equation}

As in \cite{Mouslopoulos:2001uc},
we assume that the higher dimensional fermion mass term is the 
result of the coupling of the fermions with a scalar field
which has a nontrivial stable vacuum.
We assume this VEV to have a multi-kink solution, which we can 
express in terms of the functions $\sigma \left( \phi \right)$
and $\gamma \left( \rho \right)$.

After integrating by parts, we can recast the action as:

\begin{eqnarray*}
S & =  & \int \mbox{d}^4{x} \int \mbox{d} \phi \mbox{d} \rho \left(
\frac{i e^{\sigma} e^{\gamma} }{\left( e^{\sigma} + e^{\gamma} - 1 \right)^{5}}
\left[
\overline{\psi_{R+}} \gamma^{\mu} \partial_{\mu} \psi_{R+} + 
\overline{\psi_{L+}} \gamma^{\mu} \partial_{\mu} \psi_{L+} +
\overline{\psi_{R-}} \gamma^{\mu} \partial_{\mu} \psi_{R-} +
\overline{\psi_{L-}} \gamma^{\mu} \partial_{\mu} \psi_{L-} \right] \right. \\
 & & -\frac{1}{2} \overline{\psi_{R-}} \left( 
\frac{e^{\gamma}}{ \left( e^{\sigma} + e^{\gamma} - 1 \right)^{5}} 
\partial_{\phi} + \partial_{\phi} \frac{e^{\gamma}}{ \left( e^{\sigma} +
 e^{\gamma} - 1 \right)^{5}} \right) \psi_{L-} \\
 & &  +\frac{1}{2} \overline{\psi_{L-}} \left( 
\frac{e^{\gamma}}{ \left( e^{\sigma} + e^{\gamma} - 1 \right)^{5}} 
\partial_{\phi} + \partial_{\phi} \frac{e^{\gamma}}{ \left( e^{\sigma} +
 e^{\gamma} - 1 \right)^{5}} \right) \psi_{R-} \\
 & &  -\frac{1}{2} \overline{\psi_{R+}} \left( 
\frac{e^{\gamma}}{ \left( e^{\sigma} + e^{\gamma} - 1 \right)^{5}} 
\partial_{\phi} + \partial_{\phi} \frac{e^{\gamma}}{ \left( e^{\sigma} +
 e^{\gamma} - 1 \right)^{5}} \right) \psi_{L+} \\
 & & +\frac{1}{2} \overline{\psi_{L+}} \left( 
\frac{e^{\gamma}}{ \left( e^{\sigma} + e^{\gamma} - 1 \right)^{5}} 
\partial_{\phi} + \partial_{\phi} \frac{e^{\gamma}}{ \left( e^{\sigma} +
 e^{\gamma} - 1 \right)^{5}} \right) \psi_{R+} \\
 & & +\frac{i}{2} \overline{\psi_{R-}} \left( 
\frac{e^{\gamma}}{ \left( e^{\sigma} + e^{\gamma} - 1 \right)^{5}} 
\partial_{\rho} + \partial_{\rho} \frac{e^{\gamma}}{ \left( e^{\sigma} +
 e^{\gamma} - 1 \right)^{5}} \right) \psi_{L-} \\
 & &  +\frac{i}{2} \overline{\psi_{L-}} \left( 
\frac{e^{\gamma}}{ \left( e^{\sigma} + e^{\gamma} - 1 \right)^{5}} 
\partial_{\rho} + \partial_{\rho} \frac{e^{\gamma}}{ \left( e^{\sigma} +
 e^{\gamma} - 1 \right)^{5}} \right) \psi_{R-} \\
 & & -\frac{i}{2} \overline{\psi_{R+}} \left( 
\frac{e^{\gamma}}{ \left( e^{\sigma} + e^{\gamma} - 1 \right)^{5}} 
\partial_{\rho} + \partial_{\rho} \frac{e^{\gamma}}{ \left( e^{\sigma} +
 e^{\gamma} - 1 \right)^{5}} \right) \psi_{L+} \\
 & & -\frac{i}{2} \overline{\psi_{L+}} \left( 
\frac{e^{\gamma}}{ \left( e^{\sigma} + e^{\gamma} - 1 \right)^{5}} 
\partial_{\rho} + \partial_{\rho} \frac{e^{\gamma}}{ \left( e^{\sigma} +
 e^{\gamma} - 1 \right)^{5}} \right) \psi_{R+} \\
 & & \left. -m i \left( \frac{\sigma^{\prime}}{k} \right) 
 \left( \frac{\dot{\gamma}}{k} \right) 
\frac{i e^{\sigma} e^{\gamma} }{\left( e^{\sigma} + e^{\gamma} - 1 \right)^{5}}
\left( \overline{\psi_{L+}} \psi_{R-} + \overline{\psi_{R+}} \psi_{L-} -
\overline{\psi_{L-}} \psi_{R+} - \overline{\psi_{R-}} \psi_{L+} \right) \right)
\end{eqnarray*}

In order to extract the four dimensional physics, we 
write the action in terms of a sum over KK modes.
Ultimately, we will only be interested in the zero modes.

\begin{eqnarray*}
S & = & \sum_{m}  \sum_{n}
 \int \mbox{d}^{4} x  \left\{ i \overline{\psi_{n,m+}} \left( x 
\right)
\gamma^{\mu} \partial_{\mu} \psi_{n,m+} \left( x \right) + 
i \overline{\psi_{n,m-}} \left( x 
\right)
\gamma^{\mu} \partial_{\mu} \psi_{n,m-} \left( x \right) \right. \\
 & & \left.
 -m_{n,m+} \overline{\psi_{n,m+}} \left( x \right) \psi_{n,m+} \left( x \right)
 -m_{n,m-} \overline{\psi_{n,m-}} \left( x \right) \psi_{n,m-} \left( x \right)
 \right\}
\end{eqnarray*}

The decomposition of the KK modes is simplified when we
express
 $\psi_{R+}$ and $\psi_{L+}$ in the form:

\begin{equation}
\Psi_{(R,L)+} \left( x, \phi, \rho \right) = \sum_{m}  \sum_{n} 
\psi_{n,m+}^{R,L} \left( x \right) \left(
\frac{ e^{\sigma} e^{\gamma} }{ \left( e^{\sigma} + e^{\gamma} -1 \right)^{6}}
\right)^{-\frac{1}{2}} f_{n,m+}^{R,L} \left( \phi, \rho \right)
\end{equation}

and $\psi_{R-}$ and $\psi_{L-}$ in the form:

\begin{equation}
\Psi_{(R,L)-} \left( x, \phi, \rho \right) =  \sum_{m} \sum_{n} 
\psi_{n,m-}^{R,L} \left( x \right) \left(
\frac{ e^{\sigma} e^{\gamma} }{ \left( e^{\sigma} + e^{\gamma} -1 \right)^{6}}
\right)^{-\frac{1}{2}} f_{n,m-}^{R,L} \left( \phi, \rho \right)
\end{equation}

To reproduce the standard 4-d kinetic terms, we require the normalization
conditions

\begin{equation}
\int \sum_{k,j}  \sum_{n,m} \left( e^{\sigma} + e^{\gamma} - 1 \right) 
f_{n,k+}^{R^{\ast}} \left( \phi, \rho \right) f_{m,j+}^{R} \left(
\phi, \rho \right) \mbox{d} \phi \mbox{d} \rho = \delta_{mn}, \delta_{kj}
\end{equation}

\begin{equation}
\int \sum_{k,j}  \sum_{n,m} \left( e^{\sigma} + e^{\gamma} - 1 \right) 
f_{n,k-}^{R^{\ast}} \left( \phi, \rho \right) f_{m,j-}^{R} \left(
\phi, \rho \right) \mbox{d} \phi \mbox{d} \rho = \delta_{mn}, \delta_{kj}
\end{equation}

\begin{equation}
\int \sum_{k,j} \sum_{n,m} \left( e^{\sigma} + e^{\gamma} - 1 \right) 
f_{n,k+}^{L^{\ast}} \left( \phi, \rho \right) f_{m,j+}^{L} \left(
\phi, \rho \right) \mbox{d} \phi \mbox{d} \rho = \delta_{mn}, \delta_{kj}
\end{equation}

\begin{equation}
\int \sum_{k,j} \sum_{n,m} \left( e^{\sigma} + e^{\gamma} - 1 \right) 
f_{n,k-}^{L^{\ast}} \left( \phi, \rho \right) f_{m,j-}^{L} \left(
\phi, \rho \right) \mbox{d} \phi \mbox{d} \rho = \delta_{mn}, \delta_{kj}
\end{equation}

In order to read off the equations of motion that the $f$'s must solve,
we need to simplify some terms in the action.
For example,

\begin{eqnarray*}
\frac{1}{2} \overline{\psi_{L+}} 
\left( 
\frac{e^{\gamma}}{ \left( e^{\sigma} + e^{\gamma} - 1 \right)^{5}} 
\partial_{\phi} + \partial_{\phi} \frac{e^{\gamma}}{ \left( e^{\sigma} +
 e^{\gamma} - 1 \right)^{5}} \right) \psi_{R+} & = & \\
 -\frac{1}{2} \sum_{l,m=0}^{\infty} \overline{\psi_{L+l,m}} \left( x \right)
f_{L+l,m}^{\ast} \left( \phi, \rho \right) \frac{ \left( e^{\sigma} + 
e^{\rho} - 1 \right) e^{\gamma} \sigma^{\prime} }{ e^{\sigma} 
e^{\gamma} } \sum_{n,p=0}^{\infty} \psi_{R+n,p} \left( x \right) 
f_{R+n,p} \left( \phi, \rho \right) & & \\
+ 3 \sum_{l,m=0}^{\infty} \overline{\psi_{L+l,m}} \left( x \right) 
f_{L+l,m}^{\ast} \left( \phi, \rho \right) e^{\sigma} e^{\gamma} 
\sigma^{\prime} \sum_{n,p=0}^{\infty} \psi_{R+n,p} \left( x \right) f_{R+n,p}
\left( \phi, \rho \right) & & \\
+ \sum_{l,m=0}^{\infty} \overline{\psi_{L+l,m}} \left( x \right)
f_{L+l,m}^{\ast} \left( \phi, \rho \right) \frac{
\left( e^{\sigma} + e^{\gamma} - 1 \right)}{ e^{\sigma}} \sum_{n,p=0}^{\infty}
 \psi_{R+n,p} \left( x \right) f_{R+n,p}^{\prime} \left( \phi, \rho \right)
 & & \\
-\frac{5}{2} \sum_{l,m=0}^{\infty} \overline{\psi_{L+l,m}} \left( x \right)
f_{L+l,m}^{\ast} \left( \phi, \rho \right) \sigma^{\prime} 
\sum_{n,p=0}^{\infty} \psi_{R+n,p} \left( x \right) f_{R+n,p} 
\left( \phi, \rho \right) & &
\end{eqnarray*}
with similar expressions holding for the other terms.

We are interested in the equations for the zero modes
$( m_{n} = 0 )$,
found by means of varying $S$ with respect to $f_{R+0}^{\ast},
f_{L+0}^{\ast}, f_{L-0}^{\ast}, f_{R-0}^{\ast}$.

The normalization conditions reproduce the desired
 four dimensional kinetic energy terms, hence it is only necessary
to vary the remaining parts of $S$.
For example, the remaining
terms involving $f_{L+0}^{\ast}$ are:

\begin{eqnarray*}
-\frac{1}{2} \overline{\psi_{L+0}} \left( x \right) f_{L+0}^{\ast} 
\left( \phi, \rho \right) \frac{ \left( e^{\sigma} + e^{\gamma} 
-1 \right) e^{\gamma} }{ e^{\sigma} e^{\gamma} } \sigma^{\prime}
\psi_{R+0} \left( x \right) f_{R+0} \left( \phi, \rho \right) & & \\
+ 3 \overline{\psi_{L+0}} \left( x \right) f_{L+0}^{\ast} \left( \phi, 
\rho \right) e^{\sigma} e^{\gamma} \sigma^{\prime} \psi_{R+0} \left( x
\right) f_{R+0} \left( \phi, \rho \right) & & \\
+ \overline{\psi_{L+0}} \left( x \right) f_{L+0}^{\ast} \left( \phi, 
\rho \right) \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{
e^{\sigma} } \psi_{R+0} \left( x \right) f_{R+0}^{\prime} \left(
\phi, \rho \right) & & \\
-\frac{5}{2} \overline{\psi_{L+0}} \left( x \right) f_{L+0}^{\ast}
\left( \phi, \rho \right) \sigma^{\prime} \psi_{R+0} \left( x \right)
f_{R+0} \left( \phi, \rho \right) & & \\
+\frac{i}{2} \overline{\psi_{L+0}} \left( x \right) f_{L+0}^{\ast} \left(
\phi, \rho \right) \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) 2
 e^{\sigma} \dot{\gamma} }{ e^{\sigma} e^{\gamma} } \psi_{R+0} \left(
 x \right) f_{R+0} \left( \phi, \rho \right)  & & \\
 - 3 i \overline{\psi_{L+0}} \left( x \right) f_{L+0}^{\ast} \left(
\phi, \rho \right) e^{\gamma} e^{\sigma} \dot{\gamma} \psi_{R+0} \left(
x \right) f_{R+0} \left( \phi, \rho \right) & & \\
-i \overline{\psi_{L+0}} \left( x \right) f_{L+0}^{\ast} \left( \phi, 
\rho \right) \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{
e^{\gamma} } \psi_{R+0} \left( x \right) \dot f_{R+0} \left( \phi,
 \rho \right) & & \\
+ \frac{i5}{2} \overline{\psi_{L+0}} \left( x \right) f_{L+0}^{\ast} \left(
\phi, \rho \right) \dot{\gamma} \psi_{R+0} \left( x \right) 
 f_{R+0} \left( \phi, \rho \right) & & \\
- i m \left( \frac{ \sigma^{\prime} }{ k } \right)
\left( \frac{ \dot{\gamma} }{ k } \right) \overline{\psi_{L+0}} \left(
x \right) f_{L+0}^{\ast} \left( \phi, \rho \right) \psi_{R-0} \left( x
 \right) f_{R-0} \left( \phi, \rho \right) & &
\end{eqnarray*}

Noting that the two four dimensional right-handed and 
left-handed spinors $\psi_{+}$ and $\psi_{-}$ are 
complex conjugates of each other 
\cite{Hung:2001hw}, we arrive at the 
following equation for the right handed zero mode
(recall that dots denote derivatives with respect to $\rho$ and
primes denote derivatives with respect to $\phi$):


\begin{eqnarray*}
-\frac{1}{2} \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }
{ e^{\sigma} e^{\gamma} } e^{\gamma} \sigma^{\prime} f_{R+0}
+ 3 e^{\sigma} e^{\gamma} \sigma^{\prime} f_{R+0} & & \\
 + \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\sigma} }
f_{R+0}^{\prime} - \frac{5}{2} \sigma^{\prime} f_{R+0} + \frac{i}{2}
 \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) e^{\sigma} 
\dot{\gamma} }{ e^{\sigma} e^{\gamma} } f_{R+0} & & \\
- 3 i e^{\gamma} e^{\sigma} \dot{\gamma} f_{R+0} -
 i \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\gamma} }
 \dot{f}_{R+0} & & \\
+ i \frac{5}{2} \dot{\gamma} f_{R+0} - i m \left( \frac{
\sigma^{\prime} }{ k } \right) \left( \frac{ \dot{\gamma} }{ k } \right)
f_{R-0} & = & 0
\end{eqnarray*}

Similarly, we find the following remaining equations for the 
other zero modes:

\begin{eqnarray*}
-\frac{1}{2} \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }
{ e^{\sigma} e^{\gamma} } e^{\gamma} \sigma^{\prime} f_{R-0}
+ 3 e^{\sigma} e^{\gamma} \sigma{\prime} f_{R-0} & & \\
 + \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\sigma} }
f_{R-0}^{\prime} - \frac{5}{2} \sigma^{\prime} f_{R-0} - \frac{i}{2}
 \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) e^{\sigma} 
\dot{\gamma} }{ e^{\sigma} e^{\gamma} } f_{R-0} & & \\
+ 3 i e^{\gamma} e^{\sigma} \dot{\gamma} f_{R-0} +
 i \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\gamma} }
 \dot f_{R-0} & & \\
- i \frac{5}{2} \dot{\gamma} f_{R-0} + i m \left( \frac{
\sigma^{\prime} }{ k } \right) \left( \frac{ \dot{\gamma} }{ k } \right)
f_{R+0} & = & 0
\end{eqnarray*}


\begin{eqnarray*}
+\frac{1}{2} \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }
{ e^{\sigma} e^{\gamma} } e^{\gamma} \sigma^{\prime} f_{L+0}
- 3 e^{\sigma} e^{\gamma} \sigma{\prime} f_{L+0} & & \\
 - \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\sigma} }
f_{L+0}^{\prime} + \frac{5}{2} \sigma^{\prime} f_{L+0} + \frac{i}{2}
 \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) e^{\sigma} 
\dot{\gamma} }{ e^{\sigma} e^{\gamma} } f_{L+0} & & \\
- 3 i e^{\gamma} e^{\sigma} \dot{\gamma} f_{L+0} -
 i \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\gamma} }
 \dot f_{L+0} & & \\
+ i \frac{5}{2} \dot{\gamma} f_{L+0} - i m \left( \frac{
\sigma^{\prime} }{ k } \right) \left( \frac{ \dot{\gamma} }{ k } \right)
f_{L-0} & = & 0
\end{eqnarray*}


\begin{eqnarray*}
+\frac{1}{2} \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }
{ e^{\sigma} e^{\gamma} } e^{\gamma} \sigma^{\prime} f_{L-0}
- 3 e^{\sigma} e^{\gamma} \sigma^{\prime} f_{L-0} & & \\
 - \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\sigma} }
f_{L-0}^{\prime} + \frac{5}{2} \sigma^{\prime} f_{L-0} - \frac{i}{2}
 \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) e^{\sigma} 
\dot{\gamma} }{ e^{\sigma} e^{\gamma} } f_{L-0} & & \\
+ 3 i e^{\gamma} e^{\sigma} \dot{\gamma} f_{L-0} +
 i \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\gamma} }
 \dot f_{L-0} & & \\
- i \frac{5}{2} \dot{\gamma} f_{L-0} + i m \left( \frac{
\sigma^{\prime} }{ k } \right) \left( \frac{ \dot{\gamma} }{ k } \right)
f_{L+0} & = & 0
\end{eqnarray*}


Because $f_{L+0}^{\ast} = f_{L-0}$, this is the complex conjugate
of the previous equation.


The equations for the right-handed zero modes are

\begin{eqnarray*}
-\frac{1}{2} \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }
{ e^{\sigma} e^{\gamma} } e^{\gamma} \sigma^{\prime} f_{R-} + 
3 e^{\sigma} e^{\gamma} \sigma^{\prime} f_{R-} + \frac{ \left(
e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\sigma} } f_{R-}^{\prime}
 & & \\
-\frac{5}{2} \sigma^{\prime} f_{R-} - i\frac{1}{2} \frac{ \left(
 e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\sigma} e^{\gamma} }
e^{\sigma} \dot{\gamma} f_{R-} + 3i e^{\gamma} e^{\sigma} 
\dot{\gamma} f_{R-} & & \\
+ i \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{
e^{\gamma} } \dot{f_{R-}} - i \frac{5}{2} \dot{\gamma} f_{R-} + 
im \left( \frac{ \sigma^{\prime} }{k} \right)
 \left( \frac{ \dot{\gamma} }{ k } \right) f_{R+} & = & 0
\end{eqnarray*}


\begin{eqnarray*}
-\frac{1}{2} \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }
{ e^{\sigma} e^{\gamma} } e^{\gamma} \sigma^{\prime} f_{R+} + 
3 e^{\sigma} e^{\gamma} \sigma^{\prime} f_{R+} + \frac{ \left(
e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\sigma} } f_{R+}^{\prime}
 & & \\
-\frac{5}{2} \sigma^{\prime} f_{R+} + i\frac{1}{2} \frac{ \left(
 e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\sigma} e^{\gamma} }
e^{\sigma} \dot{\gamma} f_{R+} - 3i e^{\gamma} e^{\sigma} 
\dot{\gamma} f_{R+} & & \\
- i \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{
e^{\gamma} } \dot{f_{R+}} + i \frac{5}{2} \dot{\gamma} f_{R+} - 
im \left( \frac{ \sigma^{\prime} }{k} \right)
 \left( \frac{ \dot{\gamma} }{ k } \right) f_{R-} & = & 0
\end{eqnarray*}


Because $f_{R+} = f_{R-}^{\ast}$, 
we are free
to write
\begin{eqnarray}
f_{R+}  & = & U + i V \\
f_{R-} & = & U - i V
\end{eqnarray}

where
$U$ and $V$ are real.
Setting the real and imaginary parts of this zero mode equation
to zero, we find 
equations:

\begin{eqnarray*}
-\frac{1}{2} \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{
 e^{\sigma} e^{\gamma} } e^{\gamma} \sigma^{\prime} U_{R} + 
3 e^{\sigma} e^{\gamma} \sigma^{\prime} U_{R} + \frac{ \left(
 e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\sigma} } U^{\prime}_{R} 
 & & \\
 -\frac{5}{2} \sigma^{\prime} U_{R} - \frac{1}{2} \frac{ \left(
e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\sigma} e^{\gamma} }
e^{\sigma} \dot{\gamma} V_{R} + 3 e^{\gamma} e^{\sigma} \dot{\gamma} V_{R}
 + \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{
e^{\gamma} } \dot{V}_{R} & & \\
-\frac{5}{2} \dot{\gamma} V_{R} - m \left( \frac{ \sigma^{\prime} }{
k} \right) \left( \frac{ \dot{\gamma} }{ k } \right) V_{R} & = & 0
\end{eqnarray*}

and
\begin{eqnarray*}
\frac{1}{2} \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{
 e^{\sigma} e^{\gamma} } e^{\gamma} \sigma^{\prime} V_{R} - 
3 e^{\sigma} e^{\gamma} \sigma^{\prime} V_{R} - \frac{ \left(
 e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\sigma} } V^{\prime}_{R} 
 & & \\
 +\frac{5}{2} \sigma^{\prime} V_{R} - \frac{1}{2} \frac{ \left(
e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\sigma} e^{\gamma} }
e^{\sigma} \dot{\gamma} U_{R} + 3 e^{\gamma} e^{\sigma} \dot{\gamma} U_{R}
 + \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{
e^{\gamma} } \dot{U}_{R} & & \\
-\frac{5}{2} \dot{\gamma} U_{R} + m \left( \frac{ \sigma^{\prime} }{
k} \right) \left( \frac{ \dot{\gamma} }{ k } \right) U_{R} & = & 0
\end{eqnarray*}


Similarly
setting the real and imaginary parts of the
left-handed 
 zero mode
equation to zero, we find:
\begin{eqnarray*}
\frac{1}{2} \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{
 e^{\sigma} e^{\gamma} } e^{\gamma} \sigma^{\prime} U_{L} - 
3 e^{\sigma} e^{\gamma} \sigma^{\prime} U_{L} - \frac{ \left(
 e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\sigma} } U^{\prime}_{L} 
 & & \\
 +\frac{5}{2} \sigma^{\prime} U_{L} - \frac{1}{2} \frac{ \left(
e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\sigma} e^{\gamma} }
e^{\sigma} \dot{\gamma} V_{L} + 3 e^{\gamma} e^{\sigma} \dot{\gamma} V_{L}
 + \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{
e^{\gamma} } \dot{V}_{L} & & \\
-\frac{5}{2} \dot{\gamma} V_{L} - m \left( \frac{ \sigma^{\prime} }{
k} \right) \left( \frac{ \dot{\gamma} }{ k } \right) V_{L} & = & 0
\end{eqnarray*}

and

\begin{eqnarray*}
-\frac{1}{2} \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{
 e^{\sigma} e^{\gamma} } e^{\gamma} \sigma^{\prime} V_{L} + 
3 e^{\sigma} e^{\gamma} \sigma^{\prime} V_{L} + \frac{ \left(
 e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\sigma} } V^{\prime}_{L} 
 & & \\
 -\frac{5}{2} \sigma^{\prime} V_{L} - \frac{1}{2} \frac{ \left(
e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\sigma} e^{\gamma} }
e^{\sigma} \dot{\gamma} U_{L} + 3 e^{\gamma} e^{\sigma} \dot{\gamma} U_{L}
 + \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{
e^{\gamma} } \dot{U}_{L} & & \\
-\frac{5}{2} \dot{\gamma} U_{L} + m \left( \frac{ \sigma^{\prime} }{
k} \right) \left( \frac{ \dot{\gamma} }{ k } \right) U_{L} & = & 0
\end{eqnarray*}

In order to find solutions that respect the symmetries
$\phi \rightarrow -\phi$ and $\rho \rightarrow -\rho$
of this brane background, we need to know how these two $Z_{2}$
symmetries are realized on fermions in this background:
%The above symmetries are realized in the following way on
%the fermions:
\begin{equation}
\psi \left( x^{\mu}, \phi, \rho \right) 
\rightarrow \Psi \left( x^{\mu}, \phi, \rho \right) = \pm
\Gamma^{0} \Gamma^{1} \Gamma^{2} \Gamma^{3} \Gamma^{\rho}
\psi \left( x^{\mu}, \phi_{c} - \phi, \rho \right)
\end{equation}

and

\begin{equation}
\psi \left( x^{\mu}, \phi, \rho \right) 
\rightarrow \Psi \left( x^{\mu}, \phi, \rho \right) = \pm
\Gamma^{\rho} \Gamma^{7} 
\psi \left( x^{\mu}, \phi, \rho_{c} - \rho \right)
\end{equation}

Because only fermion bilinears appear in the action,
the choice of sign is arbritrary.
In order to derive definite boundary conditions,
we choose the signs given below.
One can easily show that
\begin{equation}
S_{\phi} = -\Gamma^{0} \Gamma^{1} \Gamma^{2} \Gamma^{3} \Gamma^{\rho}
\end{equation}
satisfies
\begin{equation}
S_{\phi}^{-1} \Gamma^{M} S_{\phi} 
= 
\Lambda_{N}^{M} \Gamma^{N}
\end{equation}
where $\Lambda$ corresponds to the Lorentz transformation
\begin{equation}
\Lambda = \left( \begin{array}{cccccc}
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 
\end{array} \right)
\end{equation}
and 
\begin{equation}
L = i \overline{\psi} \Gamma^{N} \partial_{N} \psi
\end{equation}
is invariant.


Similarly,
\begin{equation}
S_{\rho} = -\Gamma_{\rho} \Gamma_{7}
\end{equation}
satisfies
\begin{equation}
S_{\rho}^{-1} \Gamma^{M} S_{\rho} =
\Lambda_{N}^{M} \Gamma^{N}
\end{equation}
where $\Lambda$ corresponds to the L.T.
\begin{equation}
\Lambda = \left( \begin{array}{cccccc}
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & -1 
\end{array} \right)
\end{equation}
and 
\begin{equation}
L = i \overline{\psi} \Gamma^{N} \partial_{N} \psi
\end{equation}
is invariant.


With $S_{\rho} = -\Gamma_{\rho} \Gamma_{7}$, this 
symmetry translates into
\begin{equation}
\left( \begin{array}{c}
\psi_{+}^{\prime} \\
\psi_{-}^{\prime} \end{array} \right)
= -\Gamma_{\rho} \Gamma_{7} \left( \begin{array}{c}
\psi_{+} \\
\psi_{-} \end{array}
\right)
\end{equation}

\begin{equation}
\left( \begin{array}{c}
\psi_{+}^{\prime} \\
\psi_{-}^{\prime} \end{array} \right)
= 
\left( \begin{array}{c}
+i \psi_{-}^{\prime} \\
-i \psi_{+}^{\prime} \end{array} \right)
\end{equation}

\begin{equation}
\psi_{+}^{\prime} \left( x^{\mu}, \phi, \rho \right) \rightarrow
\Psi_{+} \left( x^{\mu}, \phi, \rho \right) = +i \psi_{-}
\left( x^{\mu}, \phi, \rho_{c} - \rho \right) 
\end{equation}
\begin{equation}
\psi_{-}^{\prime} \left( x^{\mu}, \phi, \rho \right) \rightarrow
\Psi_{-} \left( x^{\mu}, \phi, \rho \right) = 
- i \psi_{+} \left( x^{\mu}, \phi, \rho_{c} - \rho \right)
\end{equation}
Our periodic boundary condition is
\begin{equation}
\psi_{\pm} \left( x^{\mu}, \phi, \rho \right) = 
\Psi_{\pm} \left( x^{\mu}, \rho_{c} + \rho \right)
\end{equation}
\begin{equation}
\psi_{\pm} \left( x^{\mu}, \phi, \rho \right) =
\psi_{\pm} \left( x^{\mu}, \phi, 2 \rho_{c} + \rho \right)
\end{equation}

\begin{eqnarray*}
\psi_{\pm} \left( x^{\mu}, \phi, -\rho \right) & = &
\Psi_{\pm} \left( x^{\mu}, \phi, \rho_{c} - \rho \right) \\
 & = & \pm i \psi_{\mp} \left( x^{\mu}, \phi, \rho \right)
\end{eqnarray*}

  
\begin{eqnarray*}
\psi_{\pm} \left( x^{\mu}, \phi, \rho_{c} + \rho \right) & = &
\Psi_{\pm} \left( x^{\mu}, \phi, \rho \right) \\
 & = & \pm i \psi_{\mp} \left( x^{\mu}, \phi,\rho_{c} - \rho \right)
\end{eqnarray*}
We immediately recognize $\rho = 0, \rho_{c}$ to be the
fixed points of the orbifold.

It is convenient to rewrite $\psi_{\pm}$ as
\begin{equation}
\psi_{\pm} = U \pm i V
\end{equation}
It then follows that

\begin{eqnarray*}
U \left( x^{\mu}, \phi, -\rho \right) & = & 
 V \left( x^{\mu}, \phi, \rho \right) \\
V \left( x^{\mu}, \phi, -\rho \right) & = & 
 U \left( x^{\mu}, \phi, \rho \right)
\end{eqnarray*}
and 

\begin{eqnarray*}
U \left( x^{\mu}, \phi, \rho + \rho_{c} \right) & = & 
 V \left( x^{\mu}, \phi, \rho_{c} - \rho \right) \\
V \left( x^{\mu}, \phi, \rho_{c} + \rho \right) & = & 
 U \left( x^{\mu}, \phi, \rho_{c} - \rho \right)
\end{eqnarray*}

With
\begin{equation}
S_{\phi} = - \Gamma^{0} \Gamma^{1} \Gamma^{2} 
\Gamma^{3} \Gamma^{\rho}
\end{equation}
the $\phi \rightarrow -\phi$ symmetry translates to
\begin{equation}
\psi_{\pm}^{\prime} \left( x^{\mu}, \phi, \rho \right) 
\rightarrow \Psi_{\pm} \left( x^{\mu}, \phi, \rho \right)
= -\gamma^{5} \psi_{\pm} \left( x^{\mu}, \phi_{c} - \phi,
\rho \right)
\end{equation}
Combine this symmetry with the periodic b.c.
\begin{eqnarray*}
\psi_{\pm} \left( x^{\mu}, \phi, \rho \right) & = & 
\Psi_{\pm} \left( x^{\mu}, \phi + \phi_{c}, \rho \right) \\
 & = & \psi_{\pm} \left( x^{\mu}, 2 \phi_{c} + \phi, \rho \right)
\end{eqnarray*}
and
\begin{eqnarray*}
\psi_{\pm} \left( x^{\mu}, -\phi, \rho \right) & = & 
\Psi_{\pm} \left( x^{\mu}, \phi_{c} - \phi, \rho \right) \\
 & = & -\gamma_{5} \psi_{\pm} \left( x^{\mu},
  \phi_{c} - \left( \phi_{c} - \phi \right), \rho \right)
\end{eqnarray*}
and
\begin{eqnarray*}
\psi_{\pm} \left( x^{\mu}, \phi + \phi_{c}, \rho \right) & = & 
\Psi_{\pm} \left( x^{\mu}, \phi, \rho \right) \\
 & = & -\gamma_{5} \psi_{\pm} \left( x^{\mu},
  \phi_{c} - \phi, \rho \right)
\end{eqnarray*}
This shows that $\phi = 0, \phi_{c}$ are fixed points.

One can subsequently define the chiral components
of $\psi_{+}, \psi_{-}$ by using the usual operators
\begin{equation}
P_{R,L} = \frac{ 1 \pm \gamma^{5} }{2}
\end{equation}
with
\begin{eqnarray*}
\psi_{+,R} & = & P_{R} \psi_{+} \\
\psi_{+,L} & = & P_{L} \psi_{+} \\
\psi_{-,R} & = & P_{R} \psi_{-} \\
\psi_{-,L} & = & P_{L} \psi_{-} 
\end{eqnarray*}
It follows that
\begin{eqnarray*}
U \left( x^{\mu}, -\phi, \rho \right) & = & 
-\gamma_{5} U \left( x^{\mu}, \phi, \rho \right) \\
V \left( x^{\mu}, -\phi, \rho \right) & = & 
-\gamma_{5} V \left( x^{\mu}, \phi, \rho \right) \\
U \left( x^{\mu}, \phi + \phi_{c}, \rho \right) & = & 
-\gamma_{5} U \left( x^{\mu}, \phi_{c} - \phi, \rho \right) \\
V \left( x^{\mu}, \phi + \phi_{c}, \rho \right) & = & 
-\gamma_{5} V \left( x^{\mu}, \phi - \phi_{c}, \rho \right)
\end{eqnarray*}

Summarizing all the b.c.:

\begin{eqnarray*}
U_{L} \left( x^{\mu}, -\phi, \rho \right) & = & 
U_{L} \left( x^{\mu}, \phi, \rho \right) \\
U_{R} \left( x^{\mu}, -\phi, \rho \right) & = & 
-U_{R} \left( x^{\mu}, \phi, \rho \right) \\
V_{L} \left( x^{\mu}, -\phi, \rho \right) & = & 
V_{L} \left( x^{\mu}, \phi, \rho \right) \\
V_{R} \left( x^{\mu}, -\phi, \rho \right) & = & 
-V_{R} \left( x^{\mu}, \phi, \rho \right) \\
U_{L} \left( x^{\mu}, \phi + \phi_{c}, \rho \right) & = & 
U_{L} \left( x^{\mu}, \phi_{c} -\phi, \rho \right) \\
U_{R} \left( x^{\mu}, \phi + \phi_{c}, \rho \right) & = & 
-U_{R} \left( x^{\mu}, \phi_{c} - \phi, \rho \right) \\
V_{L} \left( x^{\mu}, \phi + \phi_{c}, \rho \right) & = & 
V_{L} \left( x^{\mu}, \phi_{c} - \phi, \rho \right) \\
V_{R} \left( x^{\mu}, \phi + \phi_{c}, \rho \right) & = &
-V_{R} \left( x^{\mu}, \phi_{c} - \phi, \rho \right) \\
U_{L} \left( x^{\mu}, \phi, -\rho \right) & = &
V_{L} \left( x^{\mu}, \phi, \rho \right) \\
U_{R} \left( x^{\mu}, \phi, -\rho \right) & = &
V_{R} \left( x^{\mu}, \phi, \rho \right) \\
V_{L} \left( x^{\mu}, \phi, -\rho \right) & = & 
U_{L} \left( x^{\mu}, \phi, \rho \right) \\
V_{R} \left( x^{\mu}, \phi, -\rho \right) & = & 
U_{R} \left( x^{\mu}, \phi, \rho \right) \\
U_{L} \left( x^{\mu}, \phi, \rho + \rho_{c} \right) & = &
V_{L} \left( x^{\mu}, \phi, \rho_{c} - \rho \right) \\
U_{R} \left( x^{\mu}, \phi, \rho + \rho_{c} \right) & = &
V_{R} \left( x^{\mu}, \phi, \rho_{c} - \rho \right) \\
V_{L} \left( x^{\mu}, \phi, \rho_{c} + \rho \right) & = &
U_{L} \left( x^{\mu}, \phi, \rho_{c} - \rho \right) \\
V_{R} \left( x^{\mu}, \phi, \rho_{c} + \rho \right) & = &
U_{R} \left( x^{\mu}, \phi, \rho_{c} - \rho \right) \\
\end{eqnarray*}

Note that the representations
for the two $Z_{2}$ symmetries $\phi \rightarrow
-\phi$ and $\rho \rightarrow -\rho$ acting on
the fermions do not commute.
Because $\left[ S_{\phi}, S_{\rho} \right] \neq 0$,
we cannot find a decomposition of the fermion field
$\psi$ such that its components have definite
$Z_{2}$ transformation properties (even or odd)
under both $Z_{2}$'s.
This fact may seem odd, given our intuition
that these two $Z_{2}$ symmetries are independent.
In \cite{Hung:2001hw}, the fermion representation
of these $Z_{2}$'s do commute, and boundary
conditions are derived for component fields
with definite $Z_{2}$ transformation properties
under both symmetries.
As we demonstrate below, this treatment is
fundamentally not possible.
It follows that the action is not invariant under 
both symmetries in the inconsistent representation
given in \cite{Hung:2001hw}, as the reader
may verify.
With incorrect boundary conditions, the
resultant solutions are not trustworthy.
We now demonstrate why $S_{\phi}$ and $S_{\rho}$
must anticommute.

As is well-known, fermions have the odd
property of going into minus themselves when rotated
by $ 2 \pi$ (pun intended).
Applying successively the discrete transformations
$\phi \rightarrow -\phi$ followed by
$\rho \rightarrow -\rho$ is equivalent to
a rotation of $\pi$ radians in the $\phi - \rho$ 
plane.
Applying these discrete transformations
in the opposite order is equivalent to a rotation
of $-\pi$ radians in the same plane.
The difference in angles of these two rotations
is $2 \pi$.
Applying both transformations, first in the order
$S_{\rho} S_{\phi}$ and then in the order
$S_{\phi} S_{\rho}$, is equivalent to a rotation
of $2 \pi$ in the $\phi - \rho$ plane, and must
result in an overall minus sign for the fermion
field.
Therefore we must have $\left\{ S_{\phi}, S_{\rho} \right\} = 
0$ in order to be consistent with the spinor 
nature of fermions.
 

From these boundary conditions we see that
$U_{R} = V_{R} = 0$ on the entire boundary of rectangular region of 
dimensions $\phi_{c} \times \rho_{c}$ of our
fundamental cell.
It also follows that both $U_{L}$ and $V_{L}$ are even 
functions of the $\phi$ coordinate and that $U_{L} = V_{L}$ on
the $\rho = 0$ and $\rho = \rho_{c}$ boundaries.


\section{Scalar Field Zero Mode and Quark Mass Matrices}
In this section, we solve for the zero mode of a 
six dimensional Higgs scalar field.
As in the fermionic case, we also include a mass term for
the scalar in the six dimensional action.

We consider a real scalar field $\Phi$ propagating in a six
dimensional curved background described by the metric~(\ref{metric}).
%\begin{equation}
%ds^{2} = A^{2} \left( \phi, \rho \right) \eta_{\mu \nu} dx^{\mu} dx^{\nu} -
%B^{2} \left( \phi, \rho \right) d \phi^{2} - C^{2} \left( \phi, \rho \right)^{2}
%\end{equation}
%\begin{equation}
%A = \frac{1}{e^{\sigma} + e^{ \gamma} -1}
%\end{equation}
%\begin{equation}
%B = e^{\sigma} A
%\end{equation}
%\begin{equation}
%C = e^{\gamma} A
%\end{equation}
%\begin{equation}
%\sqrt{g} = \frac{ e^{\sigma} e^{\gamma} }{\left( e^{\sigma} + e^{\gamma} - 1\right)^{6}}
%\end{equation}
Including a six dimensional mass term, the action is:

\begin{equation}
S = \frac{1}{2} \int \mbox{d}^{4}x \int \mbox{d} \phi \mbox{d} \rho 
\sqrt{G} \left( G^{AB} \partial_{A} \Phi \partial_{B} \Phi -
m_{\Phi}^{2} \Phi^{2} \right)
\end{equation}
Integrating by parts, one obtains:
\begin{eqnarray*}
S & = & \frac{1}{2} \int \mbox{d}^{4}x \int \mbox{d} \phi \mbox{d} \rho
\left( \frac{ e^{\sigma} e^{\gamma} }{ \left( e^{\sigma} + e^{\gamma}
 - 1 \right)^{4} } \eta^{\mu \nu} \partial_{\mu} \Phi \partial_{\nu}
\Phi + \Phi \partial_{\phi} \left( \frac{ e^{\gamma} }{
e^{\sigma} \left( e^{\sigma} + e^{\gamma} - 1 \right)^{4} } \partial_{\phi}
 \Phi \right) \right. \\
& & + \left. \Phi \partial_{\rho} \left( \frac{ e^{\sigma} }{ e^{\gamma}
\left( e^{\sigma} + e^{\gamma} - 1 \right)^{4} } \partial_{\rho}
 \Phi \right) - m_{\Phi}^{2} \Phi^{2} \sqrt{G} \right) 
\end{eqnarray*}

In order to give a four dimensional interpretation to this action, we
go through the dimensional reduction procedure. 
Thus we decompose the six dimensional field into KK modes
\begin{equation}
\Phi \left( x, \phi, \rho \right) = \sum_{n,m} \phi_{n,m} \left( x
 \right) f_{n,m} \left( \phi, \rho \right)
\end{equation}
Using this decomposition, the above action becomes a sum over
KK modes.

\begin{equation}
S = \frac{1}{2} \sum_{n,m} \int \mbox{d}^{4}x \left\{ \eta^{\mu \nu}
 \partial_{\mu} \phi_{n,m} \left( x \right) \partial_{\nu}
\phi_{n,m} \left( x \right) - m_{n,m}^{2} \phi_{n,m}^{2} \left( x
 \right) \right\}
\end{equation}

In order to reproduce the canonical four dimensional
kinetic terms,
we need to impose the orthogonality relations:
\begin{equation}
\int \int \mbox{d} \phi \mbox{d} \rho \frac{ e^{\sigma} e^{\gamma} }{
 \left( e^{\sigma} + e^{\gamma} - 1 \right)^{4} } f_{m,i}^{\ast} \left(
\phi, \rho \right) f_{n,k} \left( \phi, \rho \right) = \delta_{mn},
\delta_{ik}
\end{equation}
Varying the action with respect to $f_{n}$ leads to
the following equation for the zero mode:
\begin{equation}
\partial_{\phi} \left( \frac{ e^{\gamma} }{ e^{\sigma} \left(
e^{\sigma} + e^{\gamma} - 1 \right)^{4} } \partial_{\phi} f_{0} \right)
  +  \partial_{\rho} \left( \frac{ e^{\sigma} }{ e^{\gamma} \left(
e^{\sigma} + e^{\gamma} - 1 \right)^{4} } \partial_{\rho} f_{0} \right)
 -\frac{ m^{2} e^{\sigma} e^{\gamma} }{ \left( e^{\sigma}
+ e^{\gamma} - 1 \right)^{6} } f_{0} = 0
\end{equation}
From the form of this equation, we expect the 
$\phi$ and $\rho$ dependence of $f_{0}$ to be the same.
Hence, we write

\begin{eqnarray*}
\partial_{\phi} \left( \frac{ e^{\gamma} }{ e^{\sigma} \left(
e^{\sigma} + e^{\gamma} - 1 \right)^{4} } \partial_{\phi} f_{0} \right)
& = & 
\frac{ m^{2} e^{\sigma} e^{\gamma} }{ 2 \left( e^{\sigma}
+ e^{\gamma} - 1 \right)^{6} } f_{0} \\
   \partial_{\rho} \left( \frac{ e^{\sigma} }{ e^{\gamma} \left(
e^{\sigma} + e^{\gamma} - 1 \right)^{4} } \partial_{\rho} f_{0} \right)
& = & 
\frac{ m^{2} e^{\sigma} e^{\gamma} }{ 2 \left( e^{\sigma}
+ e^{\gamma} - 1 \right)^{6} } f_{0}
\end{eqnarray*}


Any solution to this pair of equations automatically
satisfies the original equation.

The first of the above equations can be written as

\begin{equation}
\left( \frac{ 4 \sigma^{\prime} e^{\gamma} }{ \left( e^{\sigma}
+ e^{\gamma} - 1 \right) } + \frac{ \sigma^{\prime} e^{\gamma} }{
 e^{\sigma} } \right) \partial_{\phi} f_{0}  - 
\frac{ e^{\gamma} }{ e^{\sigma} } \partial_{\phi}^{2} f_{0} -
\frac{ m^{2} e^{\sigma} e^{\gamma} f_{0} }{ 2 \left(
e^{\sigma} + e^{\gamma} - 1 \right)^{2} } = 0
\end{equation}

Substituting in the forms for $\sigma$ and $\gamma$, we arrive at
the two equations:
\begin{equation}
\left( \frac{ 4 k e^{ k \rho} }{ \left( e^{k \phi}
+ e^{ k \rho} - 1 \right) } + \frac{ k e^{k \rho} }{
 e^{ k \phi} } \right) \partial_{\phi} f_{0}  - 
\frac{ e^{k \rho} }{ e^{k \phi} } \partial_{\phi}^{2} f_{0} -
\frac{ m^{2} e^{k \phi} e^{k \rho} f_{0} }{ 2 \left(
e^{k \phi} + e^{k \rho} - 1 \right)^{2} } = 0
\end{equation}

\begin{equation}
\left( \frac{ 4 k e^{ k \sigma} }{ \left( e^{k \phi}
+ e^{ k \rho} - 1 \right) } + \frac{ k e^{k \sigma} }{
 e^{ k \rho} } \right) \partial_{\rho} f_{0}  - 
\frac{ e^{k \phi} }{ e^{k \rho} } \partial_{\rho}^{2} f_{0} -
\frac{ m^{2} e^{k \phi} e^{k \rho} f_{0} }{ 2 \left(
e^{k \phi} + e^{k \rho} - 1 \right)^{2} } = 0
\end{equation}
 

 For the special case when 

\begin{equation}
2 k^{2} m^{2} = 25 k^{4}
\end{equation}

we can find an analytic solution:

\begin{equation}
f \left( \phi, \rho \right) = e^{ \left( \frac{ \ln \left( e^{k \phi} + 
e^{k \rho} - 1 \right) }{ 2 k^{2} } \right) }
\end{equation}
 
%Here is a plot of this function for the case $k = 1$:

\vglue 5cm
\hglue 5.5cm
\psfig{figure=plot.ps,height=8cm,angle=0}
\begin{quote}
\scriptsize Fig. (4): A plot of the
 scalar profile for the case $k=1$.
\end{quote}

In Fig. (4) we plot the scalar profile for the case $k=1$.


Having presented the equations for the fermion
field profiles in the extra dimensions
and having demonstrated the complex nature of the solutions,
we may now see how CP violation emerges naturally.
From the fermions and the 
Higgs scalar, we construct
a six dimensional lorentz invariant Yukawa interaction from a term
such as
\begin{equation}
\sqrt{-G} \lambda_{ij}^{(6)} H \overline{\psi_{i}} \psi_{j}
\end{equation}
where the fermion fields are eight component objects and their
indices $i$ and $j$ are generation labels with the 
Yukawa coupling matrices $\lambda_{ij}^{(6)}$ 
connecting only fermions in combinations consistent with
gauge group invariance.
The fermion bilinear may be expressed as
\begin{eqnarray*}
\overline{\psi} \psi & =  &
-i \overline{\psi_{-}} \psi_{+} + i 
\overline{\psi_{+}} \psi_{-} \\
 & = & i \overline{\psi_{L+}} \psi_{R-} +
 i \overline{\psi_{R+}} \psi_{L-} \\
 & & -i \overline{\psi_{L-}} \psi_{R+} - i
 \overline{\psi_{R-}} \psi_{L+}
\end{eqnarray*}
where the conjugate fields are formed with the usual $\gamma_{0}$
of the four dimensional Dirac algebra.

One may write the exact same form of the Yukawa
coupling in the six dimensional action as in the
SM action:
\begin{equation}
\int \mbox{d}^{4} x \int \mbox{d} \phi \mbox{d} \rho
\sqrt{-G} \lambda_{ij}^{(6)} H \overline{\psi_{i}} \psi_{j}
\end{equation}
Recall, however, that the motivation for considering the 
implications of extra dimensions is to see if their existence
can help to reduce or simplify 
the redundancy inherent in the four dimensional Yukawa coupling
matrices.
Allowing $\lambda_{ij}^{(6)}$ and the six dimensional fermion mass
terms $m_{i}$ to be arbitrary parameters only 
increases the redundancy of physical information contained 
in the parameters of the action, and, from the
perspective of attempting to gain deeper understanding
of the fermion mass hierarchy, considerably weakens the
motivation of considering extra dimensions.
However, one
starting point often adopted in attempts to study
the fermion mass hierarchy is to start with a discrete
flavor symmetry which leads to the so-called democratic mass
matrix, with all entries being the same.
This simplest ans\"{a}tz leads to one massive eigenstate
and two degenerate massless eigenstates and is thus
considered a reasonably successful approximation
given the simplicity of the ans\"{a}tz.
In \cite{Hung:2001hw}, the introduction of a single flat extra 
dimension provides a theoretical explanation of the
democratic ans\"{a}tz itself in terms of higher
dimensional geography rather than some additional
flavor physics.
The introduction of an additional flat extra dimension and
its associated Dirac algebra then allow for the 
possibility to realize a more realistic spectrum than that 
provided by the democratic mass matrices.
Because the extra dimensions are taken to be flat, the
fermion profiles may be separable
functions of each of the extra dimensions.
This property transforms the effective
democratic quark mass matrices one 
obtains from dimensionally reducing from five to four
dimensions into pure phase mass matrices.

The success of this approach in the flat space scenario 
cannot be carried over without modification to the case
of two warped extra dimensions.
Because the equations for the fermion profiles in
the warped scenario we are considering
are not separable,
it is not possible to automatically achieve pure
phase effective four dimensional quark mass matrices within
our context.

If we set $\lambda_{ij}^{(6)}$ equal to the democratic
mass matrix in both the up and down quark sectors, we
arrive at essentially the same level of understanding as
when the democratic form is adopted in flavor
symmetry approaches to the problem in four dimensions.
In that case, a longstanding difficulty has been to find
a successful implementation of a breaking of this symmetry,
(sometimes involving the additional physics of a ``flavon''
field) with the additional complication of 
ultimately arriving at complex mass matrices.

The principle observation of this work is that the presence 
of the extra dimensions serves to break the 
democratic form of the $\lambda_{ij}^{(6)}$, which is now
 controlled by the
six dimensional fermion mass terms down to the 
effective $\lambda_{ij}^{(4)}$.
Geography in the extra warped dimensions is an
alternative to the flavon field method of breaking the 
flavor symmetry of the Yukawa terms.
In addition, this geometric method of breaking the 
flavor symmetry naturally leads to complex 
effective four dimensional mass matrices.


Adopting the democratic form for $\lambda_{ij}^{(6)}$ in
each quark sector does not result in a calculable
model of flavor mixing and masses.
It does, however, correspond to a so-called
minimal parameter basis.
Six quark masses and four flavor mixing parameters are
derived from ten six-dimensional parameters.
This minimal parameter set arises from the 
following considerations.
The SM gauge symmetries allow for the existence of nine
different six-dimensional fermion masses $m_{i}$.
The left-handed up-type and down-type quarks of the same
generation must have the same six-dimensional mass
parameter $m$ because together they form an $SU \left( 2 \right)$
doublet.
The right-handed components of the up and down type
quarks for each generation have different mass parameters.
From the four dimensional perspective, a massive fermion 
field is formed only after electroweak symmetry breaking.
In addition to these nine independent mass parameters,
we also have the freedom associated with the 
curvature $k$ of the bulk AdS space.
We are assuming equal magnitudes for all the 4-brane
tensions and so have only one dimension-full parameter $k$, instead
of two as would be the case if we allowed arbitrary 
4-brane tensions.
Requiring that this setup reproduce the RS resolution 
of the gauge hierarchy problem then fixes the coordinate
length of the fundamental cell we are considering.


\section{Conclusion}

We first remind the reader of some of the 
promising results already attained in addressing the fermion
mass and mixing hierarchy problems within the context
of extra dimensions \cite{Hung:2001hw,Huber:2000ie,Arkani-Hamed:1999dc,Mouslopoulos:2001uc}.
Considering the $2 \times 2$ matrix with columns labelled
``one extra dimension'' and ``two extra dimensions'' and
rows labelled ``flat extra space'' and ``warped extra space'', 
we have shown that the $( 2, 2 )$ element is also a possible
arena in which to study the problem.
But the principle motivations for considering the
two warped extra dimensions scenario are not just 
for the sake of completeness.
The possibility of attaining the CP violating phase in the
quark flavor-mixing matrix via this higher dimensional
mechanism is a property of the Dirac algebra in six
dimensions and remains a possibility whether or not the two
extra dimensions are flat or warped.
Independent of considerations of the gauge hierarchy
problem, one advantage of the warped case over the flat 
case is that no
additional physics is required in order to
localize the fermions.
In the flat space case, the localization mechanism
of the fermions involves the introduction of a scalar field and
 specific forms for the scalar field must be assumed. 
 Even 
then the fermion profiles are solved analytically only after making 
approximations to the given scalar field \cite{Arkani-Hamed:1999dc}.


We have shown in this work that CP violation may be understood
as a natural consequence of the Dirac algebra in 
six dimensions.
Another motivation for going to six dimensions
concerns the mystery of the generation index.
As shown in \cite{Kogan:2001wp}, multibrane world scenarios imply the 
existence of light KK modes that are suggestive of
family replication.
Going to six dimensions may alleviate some of the 
difficulties encountered in trying to implement this
program.
This possibility is currently under investigation.

\section*{Acknowledgements}
We would like to thank L.~Brown, C.~Burgess, J.~Cline, P.~Hung and M.~Seco
for useful discussions.  D.D. and D.E. are supported in part
by NSERC (Canada) and FCAR (Quebec). K.K. is supported by
the US Department of Energy under Contract DE-FG0291ER40688, TASK A.

%\bibliography{one,two,three,four,five,six,seven,cp}
\bibliography{reff}
\end{document}































\documentclass{article}
\usepackage[dvips]{color}
\usepackage{psfig}
\begin{document}

\title{ Geometric Origin of CP Violation}

\author{Jim Cline\thanks{email:
 jcline@hep.physics.mcgill.ca} \\
McGill University \\
Montreal, Quebec, Canada \\
H3A 2T8
\and
 David Dooling
\thanks{email:
 dooling@hep.physics.mcgill.ca} \\
McGill University \\
Montreal, Quebec, Canada \\
H3A 2T8
\and
 Damien Easson
\thanks{email:
 easson@hep.physics.mcgill.ca} \\
McGill University \\
Montreal, Quebec, Canada \\
H3A 2T8
\and
 Kyungsik Kang
\thanks{email:
kang@het.brown.edu} \\
Brown University \\
Providence, RI, USA \\
02916}

\maketitle

\begin{abstract}
The fermion mass hierarchy and finding a predictive mechanism
of the flavor mixing parameters remains one of the least
understood puzzles facing particle physics today.
In this work, we demonstrate how the realization of the Dirac
algebra in the presence of two extra spatial
dimensions leads to complex fermion field profiles in
the extra dimensions.
Dimensionally reducing to four dimensions then leads to 
complex quark mass matrices in such a fashion that CP
violation necessarily follows.
We also present the generalization of the Randall-Sundrum
scenario to the case of a multi-brane, six-dimensional
brane-world and discuss how multi-brane worlds may shed light on 
the generation index of the SM matter content.
\end{abstract}

\bibliographystyle{plain}
\section{Introduction}

A fundamental explanation of the quark flavor-mixing matrix, the 
fermion masses and their hierarchical structure persists to be one of the
most challenging and outstanding problems of particle physics today.
A predictive mechanism for fermion mass generation is currently
lacking.
After spontaneous symmetry breaking, the quark mass term in
the langrangian reads:
\begin{equation}
\mathcal{L}_{\mathnormal \mbox{mass} } \mathnormal =
\frac{v}{\sqrt{2}} \left( \overline{u_{L_{i}}} h_{ij}^{(u)} u_{R_{j}} +
 \overline{d_{L_{i}}} h_{ij}^{(d)} d_{R_{j}} \right) + \mbox{h.c.}
\end{equation}
where the $h_{ij}$ are arbitrary $3 \times 3$ complex Yukawa
coupling matrices.
Within the standard model (SM) the fermion masses, the quark 
flavor-mixing angles and the CP violating phase are free parameters and
no relation exists among them.
The SM can accommodate the observed mass spectrum of the fermions but
unfortunately does not predict it.
Thus the calculability of the fermion masses remains an outstanding
theoretical challenge.
It is hoped there can be found some predictive mechanism of fermion
mass generation so as to place the understanding of fermion mass on a 
par with that of gauge boson mass; i.e., to find an overriding
principle that predicts the values of the $h_{ij}$ to be what they 
are.
The number of free parameters in the Yukawa sector eliminates any real
predictive power of this sector of the SM.
A first modest step one may take is to modify the Yukawa sector
in such a way that the four quark flavor-mixing parameters depend
solely on the quark masses themselves.
As an attempt to derive a relationship between the quark masses and
flavor-mixing parameters, mass matrix ans\"{a}tze based on 
flavor democracy with a suitable breaking so as to allow mixing 
between the quarks of nearest kinship
via nearest neighbor interactions was suggested about two decades
ago \cite{Weinberg:1977hb,Wilczek:1977uh,Rothman:1979ft,Kang:1981yg,Fritzsch:1978vd,Fritzsch:1979zq,DeRujula:1977ry,Georgi:1979dq}.
These early attempts are the first examples of ``strict calculability'';
i.e., mass matrices such that all flavor-mixing parameters 
depend soley on, and are determined by, the quark masses.
But the simple symmetric NNI texture leads to the experimentally
violated inequality $M_{top} < 110$ GeV, prompting
consideration of a less restricted form for the mass matrices
so as to still achieve calculability, yet be consistent
with experiment \cite{Kang:1997uv}.


After implementing successfully this modest first step towards
gaining a deeper understanding of the Yukawa sector of the SM in the
guise of calculability, one may then attempt with a little more
confidence to confront the fundamental problem of explaining
the fermion mass hierarchy itself.
In this paper, we will address neither of the above issues, but
instead shall retreat even further into a simpler domain of the 
overall problem.
The quark mixing matrix contains four physically meaningful 
parameters, the three mixing angles and the single CP violating
phase of the Cabibo-Kobayashi-Maskawa (CKM) matrix.
In this paper, we put forth the possibility that the CP phase
in the quark flavor-mixing matrix is a result of the existence of 
extra dimensions and the Dirac algebra realized in the presence
of these extra dimensions.
The notion of CP violation arising from the presence of extra
dimensions is not new, but in fact was studied long ago
 \cite{Thirring:1972de}, and more recently in
\cite{Casadio:2001fe}.
Our work is largely inspired by the papers \cite{Hung:2001hw,Huber:2000ie,Arkani-Hamed:1999dc,Mouslopoulos:2001uc} in which the fermion
mass and mixing hierarchies have been addressed with the context
of Large Extra Dimensions (LED), both in the five- and six-dimensional
cases, as well as within the context of a single extra dimension
with warped geometry.
This paper addresses the same problem within the context of two
extra dimensions with warped geometry, so as to explain the
existence of the CP phase in the quark-flavor mixing matrix.
Six dimensional extensions of the RS scenario have been
studied in \cite{Chodos:1999zt,Gherghettaet:2000,Collins:2001ni,Kogan:2001yr,Kim:2001rm}.



Another major theoretical problem of physics that has been addressed
in novel ways recently is that
of the hierarchy between the Planck scale 
(the rest mass of a flea) and particle physics scales
, such as the masses of the $W$ and $Z$ bosons.
String/M-theory seems to be the most promising candidate to form
a tight conceptual connection between gravity and particle
physics, with its attendant extra dimensions.
The novel ideas in the recent history of solutions to the gauge
hiararchy problem are rooted in the possibility that
the hierarchy is controlled by exotic features of the extra 
dimensions; namely, either they are very large and so the
hierarchy is generated by the large volume of the extra
dimensions \cite{Arkani-Hamed:1998rs,Arkani-Hamed:1998nn,
Antoniadis:1998ig}.
, or that the extra dimensions are warped, with the 
hierarchy generated by an exponential damping
\cite{Gogberashvili:1998vx,Randall:1999ee,Randall:1999vf}.

Extra dimensions are well-motivated by string theory, but are
not so well-motivated as a context within which to study the 
fermion mass hierarchy.
Without the recent ADD and RS developments, it is very unlikely
this approach would have developed solely from particle
physics considerations.
As is well-known, string theory has little data to constrain
the ideas of its metatopology.
Less constraints promote creative ideas.
In particle physics, and in particular in the quark sector, we
are blessed with increasingly precise data, which
however serve as increasingly tighter constraints on any
further theoretical understanding.
As ideas flow from free areas to prison cells, we explore
the implications of extra dimensions for the fermion
mass and mixing problem.
We acknowledge that revolutions flow the other way, but
the light this investigation sheds on the source of CP
violation makes this exchange of ideas between these two
branches of theoretical physics more promising and worthy
of continued investigation.


The organization of this paper is as follows.
We will briefly review the Randall-Sundrum scenario and introduce
its extension to two extra dimensions.
We will then introduce fundamental six-dimensional
(eight-component object), as well as a fundamental
Higgs scalar.
We will then discuss how to localize these various fields in 
the extra dimensions as in 
\cite{Kogan:2001wp}, where six-dimensional masses
can allow one to override the tendency of the AdS geometry to
localize the fields in other regions of the extra spacetime
that are less appealing given our notion that the SM fields
should be localized within a small region of the extra space.
The possibility of bulk SM fields within the RS scenario
has been extensively studied in \cite{Chang:1999nh}.
Finally, we show how the presence of two extra dimensions leads to
complex fermion field profiles in the extra dimension and 
illustrate how this leads to CP violation.
We then conclude and discuss briefly future avenues to be
investigated.


\section{Many-Brane , Six-Dimensional Extension of the 
Randall-Sundrum Solution}

We present a simple generalization of teh RS scenario to the six
dimension, multi-brane case.
In what follows, we will be using one fundamental cell of the 
brane lattice to be described below, but for completeness we
write down the brane-lattice solution.
Ultimately, one wants to understand not only the fermion mass
hierarchy and the flavor mixing parameters, but also the very
existence of the three fermion generations of the SM.
An interesting observation of Kogan et al \cite{Kogan:2001wp} is that
in multi-brane worlds, there exist ultra-light localized
and strongly coupled bulk fermion KK modes.
This observation leads one to entertain the notion that for a 
given fundamental bulk 
fermion field with given SM gauge group transformation
properties, the generation index may be associated with KK
mode number, so that ultimately there is only one six-dimensional
species of up-type quarks, for example, and that the generation
structure is just a reflection of the existence of ultra-light
KK modes arising from the brane-setup and geometry of the
extra dimensions.
One stumbling blobk confronting this mode of 
understanding fermion generation structure is that the very
multi-brane setup that gives arise to the family
structure also gives rise to the same number of ultralight
KK modes for al other fields, including the graviton.
This approach to the family index is currently under investigation,
though in the applications to follow, we will consider only one 
fundamental cell of this brane crystal and hence will not
have any ultralight KK modes.

We now present a solution for the metric corresponding to a 
six dimensional, multi-brane extension of the RS scenario.
A five dimensional multi-brane extension has been 
presented in \cite{Hatanaka:1999ac}.


To generalize to six dimensions, we consider a 
N X M lattice of N parallel 4-branes localized in the $\phi$ dimension 
orthogonal to M parallel 4-branes localized in the $\rho$ dimension.
3-Branes reside at their loci of intersection.
The action describing this set-up is given by the following three
terms:
\normalcolor
\begin{displaymath}
S = S_{gravity} + \sum_{i=1}^{N} S_{i} + \sum_{j=1}^{M} S_{j}
\end{displaymath}
%\Large
where $S_{grav}$ is given by
\begin{displaymath}
S_{grav} = \int d^{4} x \int_{0}^{2 \pi} d \phi \int_{o}^{2 \pi} d
\rho \sqrt{-g} \left( \frac{1}{ \kappa_{6}^{2} } R - \sum_{i,j} 
\Lambda_{ij} \left[ \Theta \left( \phi - \phi_{i} \right) - \Theta
 \left( \phi - \phi_{i+1} \right) \right] \times \right.
\end{displaymath}
\begin{displaymath}
\left.
 \left[ \Theta \left( \rho
- \rho_{j} \right) - \Theta \left( \rho - \rho_{j+1} \right) \right] \right)
\end{displaymath}
%\large
and the the terms in the action representing the 4-branes
are given by
\begin{displaymath}
S_{i} = - \int d^{4} x \int_{0}^{2 \pi} d \rho \sqrt{g^{\left( \phi = \phi_{i} \right)}} T_{\phi_{i}}
\end{displaymath}
and
\begin{displaymath}
S_{j} = -\int d^{4} x \int_{0}^{2 \pi} d \phi \sqrt{g^{\left( \rho = \rho_{j} \right)}} T_{\rho_{j}}
\end{displaymath}
%\Large
We are considering the case of both extra dimensions being compact
and impose the following periodicity conditions on the 
extra coordinates:
\begin{displaymath}
\rho_{M+1} = 2 \pi
\end{displaymath}
\begin{displaymath}
\phi_{N+1} = 2 \pi
\end{displaymath}
As in \cite{Hatanaka:1999ac}, we may use the $S^{1}$ symmetry(ies) to 
define the position of the first set of 
 brane sources to be at the origin of the extra dimensions.
\begin{displaymath}
\phi_{1} = 0 = \rho_{1}
\end{displaymath}
In the above expressios for the 4-brane actions, 
the induced metric is given by
\begin{displaymath}
g_{ab}^{\rho = \rho_{j}} = g_{ab} \left( x^{\mu}, \phi, \rho = \rho_{j} \right)
\end{displaymath}
for the 4-branes localized in the $\phi$ direction and by
\begin{displaymath}
g_{\alpha \beta}^{\phi = \phi_{i}} = g_{\alpha \beta} \left( x^{\mu}, \phi = 
\phi_{i}, \rho \right)
\end{displaymath}
for the 4-branes localized in the $\rho$ direction.
%\newpage
The six dimensional Einstein equations are given by
\begin{displaymath}
R_{N}^{M} - \frac{1}{2} \delta_{N}^{M} R = \frac{\kappa_{6}^{2}}{2} T_{N}^{M}
\end{displaymath}
where
\begin{eqnarray*}
T_{N}^{M} & = & -\sum_{i,j}^{N,M} \Lambda_{ij} \left[ \Theta \left(
\phi - \phi_{i} \right) - \Theta \left( \phi - \phi_{i+1} \right) 
\right] \left[ \Theta \left( \rho - \rho_{j} \right) - \Theta
 \left( \rho - \rho_{j+1} \right) \right] \delta_{N}^{M} \\
& = & -\sum_{i}^{N} \sqrt{\frac{-\mbox{det} g^{\phi = \phi_{i}}}{\mbox{det} g}} T_{\phi_{i}} \delta \left( \phi - \phi_{i} \right) \delta_{a}^{M} \delta^{a}_{N} \\
& = & -\sum_{j=1}^{M} \sqrt{\frac{-\mbox{det} g^{\rho = \rho_{j}}}{\mbox{det} g}} T_{\rho_{j}} \delta \left( \rho - \rho_{j} \right) \delta_{\alpha}^{M} \delta_{N}^{\alpha}
\end{eqnarray*}
%\Huge
We are not addressing any cosmological issues in this work, and for 
simplicity consider adopt the following
static ans\"{a}tze for the metric:
\normalcolor
%\Large
\begin{displaymath}
ds^{2} = A^{2} \left( \phi, \rho \right) \eta_{\mu \nu} dx^{\mu}
dx^{\nu} - B^{2} \left( \phi, \rho \right) d \phi^{2} - C^{2} \left(
\phi, \rho \right) d \rho^{2}
\end{displaymath}


With this ans\"{a}tz, the left-hand side of the Einstein 
equations are given by the following expressions, where dots
denote derivatives with respect to the $\rho \left( \tilde{\rho}
\right)$ coordinate and
primes denote derivatives with respect to the $\phi \left(
\tilde{\phi} \right)$ coordinate.
%\begin{displaymath}
\begin{eqnarray*}
G_{\nu}^{\mu} & = & \frac{2}{A^{2}} \left[ \left(
\frac{\dot{A}}{A} \right)^{2} + \left( \frac{A^{\prime}}{A} \right)^{2} 
+ 2 \left( \frac{\dot{\dot{A}}}{A} + \frac{A^{\prime \prime}}{A} \right)
\right] \delta_{\nu}^{\mu} \\
G_{\tilde{\phi}}^{\tilde{\phi}} & = & \frac{2}{A^{2}} \left[ 5 \left( \frac{A^{\prime}}{A} \right)^{2} + 2 \frac{\dot{\dot{A}}}{A} + \left(
\frac{\dot{A}}{A} \right)^{2} \right] \\
%G_{\tilde{\rho}}^{\tilde{\rho}} & =  & \frac{2}{A^{2}} \left[ 5 \left(
%\frac{\dot{A}}{A} \right)^{2} + 2 \frac{ A^{\prime \prime}}{A} + 
%\left( \frac{A^{\prime}}{A} \right)^{2} \right] \\
%G_{\tilde{\rho}}^{\tilde{\phi}} & = & \frac{4}{A^{4}} \left[
%-A\dot{A^{\prime}} + 2 \dot{A} A^{\prime} \right] \\
%G_{\tilde{\phi}}^{\tilde{\rho}} & = & G_{tilde{\rho}}^{\tilde{\phi}}
\end{eqnarray*}
%\newpage
\begin{eqnarray*}
G_{\tilde{\rho}}^{\tilde{\rho}} & =  & \frac{2}{A^{2}} \left[ 5 \left(
\frac{\dot{A}}{A} \right)^{2} + 2 \frac{ A^{\prime \prime}}{A} + 
\left( \frac{A^{\prime}}{A} \right)^{2} \right] \\
G_{\tilde{\rho}}^{\tilde{\phi}} & = & \frac{4}{A^{4}} \left[
-A\dot{A^{\prime}} + 2 \dot{A} A^{\prime} \right] \\
G_{\tilde{\phi}}^{\tilde{\rho}} & = & G_{\tilde{\rho}}^{\tilde{\phi}}
\end{eqnarray*}
%\end{displaymath}
Taking the warp factor to be
\normalcolor

\begin{displaymath}
A = \frac{1}{k_{\phi} | \tilde{\phi} | + k_{\rho} | \tilde{\rho} |
 + 1}
\end{displaymath}
and
\begin{displaymath}
B \left( \phi, \rho \right) = A \left( \phi, \rho \right) e^{ \sigma \left( \phi \right)}
\end{displaymath}
\begin{displaymath}
C \left( \phi, \rho \right) = A \left( \phi, \rho \right) e^{ \gamma \left( \rho \right)}
\end{displaymath}
one can easily check that the nondiagonal elements of the Einstein tensor
vanish,
$G_{\tilde{\rho}}^{\tilde{\phi}} = 0$, and thus the $\tilde{\phi} - 
\tilde{\rho}$ component of the Einstein equations are trivially 
satisfied.


The remaining equations will be satisifed if the following relations
are fulfilled:
\normalcolor
\begin{displaymath}
10 \left( k_{\phi_{i}}^{2} + k_{\rho_{J}}^{2} \right) =
-\frac{\kappa_{6}^{2}}{2} \sum_{i,j} \Lambda_{ij} \left[ \Theta \left(
\phi - \phi_{i} \right) - \Theta \left( \phi - \phi_{i+1} \right)
\right] \times
\end{displaymath}
\begin{displaymath}
 \left[ \Theta \left( \rho - \rho_{j} \right) - \Theta \left(
\rho - \rho_{j} \right) \right]
\end{displaymath}
\begin{displaymath}
8 \left( k_{\rho_{j}} - k_{\rho_{j-1}} \right) = \frac{\kappa_{6}^{2}}{2}
T_{\rho_{j}}
\end{displaymath}
\begin{displaymath}
8 \left( k_{\phi_{i}} - k_{\phi_{i-1}} \right) = \frac{\kappa_{6}^{2}}{2}
T_{\phi_{i}}
\end{displaymath}
%\newpage
So we have finally
\normalcolor
\begin{displaymath}
A \left( \phi, \rho \right) = \frac{1}{\left( e^{ \sigma \left( \phi \right) } + e^{\gamma \left( \rho \right)} -1 \right)}
\end{displaymath}
\begin{displaymath}
B \left( \phi, \rho \right) = A \left( \phi, \rho \right) e^{ \sigma \left( \phi \right)}
\end{displaymath}
\begin{displaymath}
C \left( \phi, \rho \right) = A \left( \phi, \rho \right) e^{ \gamma \left( \rho \right)}
\end{displaymath}
where
\normalcolor
\begin{displaymath}
\sigma \left( \phi \right) = 
k_{\phi_{1}} | \phi - \phi_{1} | \Theta \left( \phi - \phi_{1} \right) + 
\left( k_{\phi_{2}} - k_{\phi_{1}} \right) | \phi - \phi_{2} | \Theta \left( \phi - \phi_{2} \right) +
\end{displaymath}
\begin{displaymath}
 \left( k_{\phi_{3}} - k_{\phi_{2}} \right) | \phi
- \phi_{3} | \Theta \left( \phi - \phi_{3} \right) + \cdots
 +
\left( k_{\phi_{N}} - k_{\phi_{N-1}} \right) | \phi - \phi_{N}| 
\Theta \left( \phi - \phi_{N} \right)
\end{displaymath}
\begin{displaymath}
\gamma \left( \rho \right) =
k_{\rho_{1}} | \rho - \rho_{1} | \Theta \left( \rho - \rho_{1} \right) +
\left( k_{\rho_{2}} - k_{\rho_{1}} \right) | \rho - \rho_{2} | \Theta
\left( \rho - \rho_{2} \right) +
\end{displaymath}
\begin{displaymath}
 \left( k_{\rho_{3}} - k_{\rho_{2}} \right)
| \rho - \rho_{3} | \Theta \left( \rho - \rho_{3} \right) + \cdots
+ \left( k_{\rho_{M}} - k_{\rho_{M-1}} \right) |\rho - \rho_{M} |
\Theta \left( \rho - \rho_{M} \right)
\end{displaymath}
and
\normalcolor
\begin{displaymath}
k_{i} - k_{i-1} = \frac{\kappa_{6}^{2}}{16} T_{i}
\end{displaymath}
\begin{displaymath}
\kappa_{6}^{2} = \frac{ 16 \pi}{M_{6}^{4}}
\end{displaymath}
We will 
work in units where $M_{6}^{-4} = 1$
\normalcolor
\begin{displaymath}
k_{i} - k_{i-1} = \pi T_{i}
\end{displaymath}
and
for simplicity, we will take all the bulk cosmological constants to be the
same and to have the magnitudes of the brane tensions to be the same:
$| T_{\phi_{i}} | = |T_{\rho_{j}}| = T$
Thus we have the relations

\begin{displaymath}
10 \left( 2 \right) k_{1}^{2} = -\frac{ 16 \pi}{\left( 2 \right) M_{6}^{4}} \Lambda
\end{displaymath}
\begin{displaymath}
k_{1} = \sqrt{ -\frac{2 \pi \Lambda}{5}} = k_{\phi_{1}} = k_{\rho_{1}}
\end{displaymath}
%\newpage
$\rightarrow$
%\LARGE
 Our expressions for the functions
$\sigma \left( \phi \right)$ and $\gamma \left( \rho \right)$
then become 
\begin{displaymath}
\sigma \left( \phi \right) = \sqrt{-\frac{2 \pi \Lambda}{5}} \phi \Theta 
\left( \phi \right) + \pi T \left[ -\left( \phi - \phi_{2} \right) 
\Theta \left( \phi - \phi_{2}\right) \right.
\end{displaymath}
\begin{displaymath}
+ \left( \phi - \phi_{3} \right) \Theta \left( \phi - \phi_{3} \right) - 
\left( \phi - \phi_{4} \right) \Theta \left( \phi - \phi_{4} \right)
\end{displaymath}
\begin{displaymath}
+ \left( \phi - \phi_{5} \right) \Theta \left( \phi - \phi_{5} \right) -
\left( \phi - \phi_{6} \right) \Theta \left( \phi - \phi_{6} \right)
\end{displaymath}
\begin{displaymath}
\left. + \left( \phi - \phi_{7} \right) \Theta \left( \phi - \phi_{7}
\right) - \left( \phi - \phi_{8} \right) \Theta \left( \phi - \phi_{8}
\right) \right]
\end{displaymath}
\begin{displaymath}
\sigma \left( \phi_{c} \right) = \sigma \left( 0 \right) = 0 \rightarrow
\end{displaymath}
\begin{displaymath}
\phi_{c} = \frac{ \pi T \left( \phi_{2} - \phi_{3} + \phi_{4} - \phi_{5} +
\phi_{6} - \phi_{7} + \phi_{8} \right)}{\left( \pi T - \sqrt{
-\frac{2 \pi \Lambda}{5}} \right)}
\end{displaymath}
and similarly
\normalcolor
\begin{displaymath}
\gamma \left( \rho \right) = \sqrt{ -\frac{ 2 \pi \Lambda}{5}} \rho
\Theta \left( \rho \right) + \pi T \left[ -\left( \rho - \rho_{2} \right)
\Theta \left( \rho - \rho_{2} \right) \right.
\end{displaymath}
\begin{displaymath}
\left. + \left( \rho - \rho_{3} \right) \Theta \left( \rho - 
\rho_{3} \right) - \left( \rho - \rho_{4} \right) \Theta
\left( \rho - \rho_{4} \right) \right]
\end{displaymath}
\begin{displaymath}
\gamma \left( \rho_{c} \right) = \gamma \left( 0 \right) = 0 \rightarrow
\end{displaymath}
\begin{displaymath}
\rho_{c} = \frac{ \pi T \left( \rho_{2} - \rho_{3} + \rho_{4} \right)}{
\left( \pi T - \sqrt{-\frac{ 2 \pi \Lambda}{5}} \right)}.
\end{displaymath}

\section{Six Dimensional Dirac Fermions}

In the seminal work of \cite{Arkani-Hamed:1999dc}, a framework is introduced for 
understanding both the fermion mass hierarchy and proton
stability without recourse to flavor symmetries, but rather
in terms of higher dimensional geography.
Additional physics must be assumed in order to localize the fermions in
the flat extra dimension, but once this additional
scalar field is introduced, any coupling between chiral fermions
is exponentially suppressed because the two fields are
separated in space.
A key observation made by Huber and Shafi in \cite{Huber:2000ie} is that one
can get this exponential damping automatically from a 
non-factorizable geometry and that there is no need to assume 
additional physics.

 In both \cite{Arkani-Hamed:1999dc} and 
\cite{Huber:2000ie}, the effective four dimensional masses arise
from integrating over the extra dimension Yukawa interactions
between the five dimensional Higgs and chiral fermion
fields.
The resulting four dimensional Yukawa coupling matrices
exhibit phenemenollogically acceptable spectrums and mixing
(excluding CP violation) because of how these overlap
integrals can be tuned depending on the values of the
five dimensional mass term for the fermions in the action.
In the flat space scenario of \cite{Arkani-Hamed:1999dc}, the five dimensional mass
term serves to translate the gaussian profile of the fermions
along the extra dimension, so that the overlap integral of two
chiral fermions and the flat zero mode of the Higgs is itself
a gaussian, thus generating exponentially small effective
four dimensional Yukawa couplings.
In the warped space scenario of \cite{Huber:2000ie},
 with the $\frac{S^{1}}{Z_{2}}$
geometry of Randall-Sundrum, the right-handed zero modes
are peaked at the origin on the positive tension 3-brane, while
the left-handed zero modes are peaked at the other orbifold
fixed point around the negative tension 3-brane.
In this case, varying the value of the five dimensional mass
does not translate the fermion field profiles along the 
extra dimension; the right-handed and left-handed fermions remain
localized around the positive and negative tension 3-branes,
respectively.
What does change, however, is the width of the profile.
Hence, just as in the flat space scenario, small changes in the 
values of the five dimensional mass parameters lead to 
greatly smplified changes in the resulting four dimensional
Yukawa coupling matrices.

In both the flat and warped space scenarios in five dimensions,
the profiles for all fields in the extra dimension are real
and CP violation cannot be realized in a natural way.
While CP violation is not addressed in
\cite{Huber:2000ie}, a numerical
example is given where nine input parameters, essentially the ratios
of the five dimensional masses for the fermions appearing
in the five dimensional action to the AdS curvature scale, 
result in very good agreement with the quark mass spectrum and
absolute values of the elements of the $V_{CKM}$ matrix.
The agreement is in fact striking, with just enough
disagreement in the mixing matrix to lead one to 
suspect that inclusion of CP violation may somehow bring
even closer agreement between this type of model's predictions
and experiment.



\begin{equation} 
S = \int d^{4}x \int d \phi d \rho \sqrt{G} \left\{ E^{A}_{\alpha} \left[ \frac{i}{2} \overline{\psi} \gamma^{\alpha} \left( \stackrel{\rightarrow}{\partial_{A}} - \stackrel{\leftarrow}{\partial_{A}} \right) \psi \right] -m \left( \phi, \rho \right) \overline{\psi} \psi \right\}
\end{equation}
\begin{equation}
ds^{2} = A^{2} \left( \phi, \rho \right) \eta_{\mu \nu} dx^{\mu} dx^{\nu} -
B^{2} \left( \phi, \rho \right) d \phi^{2} - C^{2} \left( \phi, \rho \right)^{2}
\end{equation}
\begin{equation}
A = \frac{1}{e^{\sigma} + e^{ \gamma} -1}
\end{equation}
\begin{equation}
B = e^{\sigma} A
\end{equation}
\begin{equation}
C = e^{\gamma} A
\end{equation}
\begin{equation}
\sqrt{g} = \frac{ e^{\sigma} e^{\gamma} }{\left( e^{\sigma} + e^{\gamma} - 1\right)^{6}}
\end{equation}

We now proceed to show how the introduction
of another extra dimension can lead to a natural explanation of 
CP violation.
In six spacetime dimensions the Dirac algebra is minimally
realized by $8 \times 8$ matrices.
A particularly convenient representation has been given in [],
in which the ideas of [] have been extended to the six dimensional case,
as well as providing a theoretical motivation for
the so-called democratic mass matrices that have served as the 
starting point for many flavor symmetry approaches to the
quark mass hierarchy problem.
This convenient representation is given below, where the
$\gamma_{5}$ in $\Gamma_{\phi}$ is the same $\gamma_{5}$
one would find from the four dimensional $\gamma$ ' s.


\begin{equation}
\Gamma_{0} = \left( \begin{array}{cc}
0 & +i \gamma_{0} \\
-i \gamma^{0} & 0  
\end{array} \right)
\end{equation}
\begin{equation}
\Gamma_{1} = \left( \begin{array}{cc}
0 & +i \gamma_{1} \\
-i \gamma^{1} & 0  
\end{array} \right)
\end{equation}
\begin{equation}
\Gamma_{2} = \left( \begin{array}{cc}
0 & +i \gamma_{2} \\
-i \gamma^{2} & 0  
\end{array} \right)
\end{equation}
\begin{equation}
\Gamma_{3} = \left( \begin{array}{cc}
0 & +i \gamma_{3} \\
-i \gamma^{3} & 0  
\end{array} \right)
\end{equation}
\begin{equation}
\Gamma_{\phi} = \left( \begin{array}{cc}
0 &  \gamma_{5} \\
 -\gamma^{5} & 0  
\end{array} \right)
\end{equation}
\begin{equation}
\Gamma_{\rho} = \left( \begin{array}{cc}
0 & +i_{4 \times 4} \\
+i_{4 \times 4} & 0  
\end{array} \right)
\end{equation}
\begin{equation}
\Gamma_{7} = \left( \begin{array}{cc}
1 & 0 \\
0 & -1  
\end{array} \right)
\end{equation}
\begin{equation}
\gamma_{0} = \left( \begin{array}{cccc}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \end{array} \right)
\end{equation}
\begin{equation}
\gamma_{1} = \left( \begin{array}{cccc}
0 & 0 & 0 & -1 \\
0 & 0 & -1 & 0 \\
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \end{array} \right)
\end{equation}
\begin{equation}
\gamma_{2} = \left( \begin{array}{cccc}
0 & 0 & 0 & i \\
0 & 0 & -i & 0 \\
0 & -i & 0 & 0 \\
i & 0 & 0 & 0 \end{array} \right)
\end{equation}
\begin{equation}
\gamma_{3} = \left( \begin{array}{cccc}
0 & 0 & -1 & 0 \\
0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \end{array} \right)
\end{equation}
\begin{equation}
\gamma_{5} = \left( \begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1 \end{array} \right)
\end{equation}


Both the $\Gamma$ 's and the $\gamma$'s have the mostly minus signature.


Using $\Gamma_{7}$, we can construct the usual projection operators to
get two four-component objects we will call $\psi_{+}$ and $\psi_{-}$,
in perfect analogy with what one can write in four dimensions, 
expressing a four dimensional fermion field in terms of its
left and right-handed components.

\begin{equation}
\psi_{+} = \frac{1}{2} \left( 1 - \Gamma_{7} \right) \psi
\end{equation}
\begin{equation}
\psi_{-} = \frac{1}{2} \left( 1 + \Gamma_{7} \right) \psi
\end{equation}
\begin{equation}
\psi = \psi_{+} + \psi_{-}
\end{equation}

An interesting point to notice in this representation is that
the six dimensional lorentz invariant fermion bilinear
$\overline{\psi} \psi$ has a complex coefficient when
expressed in terms of the four component fields $\psi_{+}$ and
$\psi_{-}$ and their four component conjugate fields
$\overline{\psi_{+}}$ and $\overline{\psi_{-}}$.
This fact leads to the result that a real six dimensional mass
term will appear to be complex when expressed in terms of the 
four component projections $\psi_{+}$ and $\psi_{-}$, the 
six dimensional analogs of $\psi_{L}$ and $\psi_{R}$.


\begin{equation}
\overline{\psi} = \psi^{\dagger} \Gamma_{0} = \left( \psi_{+}^{\dagger}, \psi_{-}^{\dagger} \right) \left( \begin{array}{cc}
0 & i \gamma_{0} \\
-i \gamma_{0} & 0 \end{array} \right) = \left( -i \overline{\psi_{-}} , i \overline{\psi_{+}} \right)
\end{equation}

As in [],
we assume that the higher dimensional fermion mass term is the 
result of the coupling of the fermions with a scalar field
which has a nontrivial stable vacuum.
We assume this VEV to have a multi-kink solution, which we can 
express in terms of the functions $\sigma \left( \phi \right)$
and $\gamma \left( \rho \right)$.

After integrating by parts, we can reexpress the action as:

\begin{eqnarray*}
S & =  & \int \mbox{d}x^{4} \int \mbox{d} \phi \mbox{d} \rho \left(
\frac{i e^{\sigma} e^{\gamma} }{\left( e^{\sigma} + e^{\gamma} - 1 \right)^{5}}
\left[
\overline{\psi_{R+}} \gamma^{\mu} \partial_{\mu} \psi_{R+} + 
\overline{\psi_{L+}} \gamma^{\mu} \partial_{\mu} \psi_{L+} +
\overline{\psi_{R-}} \gamma^{\mu} \partial_{\mu} \psi_{R-} +
\overline{\psi_{L-}} \gamma^{\mu} \partial_{\mu} \psi_{L-} \right] \right. \\
 & & -\frac{1}{2} \overline{\psi_{R-}} \left( 
\frac{e^{\gamma}}{ \left( e^{\sigma} + e^{\gamma} - 1 \right)^{5}} 
\partial_{\phi} + \partial_{\phi} \frac{e^{\gamma}}{ \left( e^{\sigma} +
 e^{\gamma} - 1 \right)^{5}} \right) \psi_{L-} \\
 & &  +\frac{1}{2} \overline{\psi_{L-}} \left( 
\frac{e^{\gamma}}{ \left( e^{\sigma} + e^{\gamma} - 1 \right)^{5}} 
\partial_{\phi} + \partial_{\phi} \frac{e^{\gamma}}{ \left( e^{\sigma} +
 e^{\gamma} - 1 \right)^{5}} \right) \psi_{R-} \\
 & &  -\frac{1}{2} \overline{\psi_{R+}} \left( 
\frac{e^{\gamma}}{ \left( e^{\sigma} + e^{\gamma} - 1 \right)^{5}} 
\partial_{\phi} + \partial_{\phi} \frac{e^{\gamma}}{ \left( e^{\sigma} +
 e^{\gamma} - 1 \right)^{5}} \right) \psi_{L+} \\
 & & +\frac{1}{2} \overline{\psi_{L+}} \left( 
\frac{e^{\gamma}}{ \left( e^{\sigma} + e^{\gamma} - 1 \right)^{5}} 
\partial_{\phi} + \partial_{\phi} \frac{e^{\gamma}}{ \left( e^{\sigma} +
 e^{\gamma} - 1 \right)^{5}} \right) \psi_{R+} \\
 & & +\frac{i}{2} \overline{\psi_{R-}} \left( 
\frac{e^{\gamma}}{ \left( e^{\sigma} + e^{\gamma} - 1 \right)^{5}} 
\partial_{\rho} + \partial_{\rho} \frac{e^{\gamma}}{ \left( e^{\sigma} +
 e^{\gamma} - 1 \right)^{5}} \right) \psi_{L-} \\
 & &  +\frac{i}{2} \overline{\psi_{L-}} \left( 
\frac{e^{\gamma}}{ \left( e^{\sigma} + e^{\gamma} - 1 \right)^{5}} 
\partial_{\rho} + \partial_{\rho} \frac{e^{\gamma}}{ \left( e^{\sigma} +
 e^{\gamma} - 1 \right)^{5}} \right) \psi_{R-} \\
 & & -\frac{i}{2} \overline{\psi_{R+}} \left( 
\frac{e^{\gamma}}{ \left( e^{\sigma} + e^{\gamma} - 1 \right)^{5}} 
\partial_{\rho} + \partial_{\rho} \frac{e^{\gamma}}{ \left( e^{\sigma} +
 e^{\gamma} - 1 \right)^{5}} \right) \psi_{L+} \\
 & & -\frac{i}{2} \overline{\psi_{L+}} \left( 
\frac{e^{\gamma}}{ \left( e^{\sigma} + e^{\gamma} - 1 \right)^{5}} 
\partial_{\rho} + \partial_{\rho} \frac{e^{\gamma}}{ \left( e^{\sigma} +
 e^{\gamma} - 1 \right)^{5}} \right) \psi_{R+} \\
 & & \left. -m i \left( \frac{\sigma^{\prime}}{k} \right) 
 \left( \frac{\dot{\gamma}}{k} \right) 
\frac{i e^{\sigma} e^{\gamma} }{\left( e^{\sigma} + e^{\gamma} - 1 \right)^{5}}
\left( \overline{\psi_{L+}} \psi_{R-} + \overline{\psi_{R+}} \psi_{L-} -
\overline{\psi_{L-}} \psi_{R+} - \overline{\psi_{R-}} \psi_{L+} \right) \right)
\end{eqnarray*}

In order to easily identify the four dimensional physics, we need to
write the action in terms of a sum over KK modes.
Ultimately, we will only be interested in the zero modes.

\begin{eqnarray*}
S & = & \sum_{n} \int \mbox{dx}^{4} \left\{ i \overline{\psi_{n+}} \left( x 
\right)
\gamma^{\mu} \partial_{\mu} \psi_{n+} \left( x \right) + 
i \overline{\psi_{n-}} \left( x 
\right)
\gamma^{\mu} \partial_{\mu} \psi_{n-} \left( x \right) \right. \\
 & & \left.
 -m_{n+} \overline{\psi_{n+}} \left( x \right) \psi_{n+} \left( x \right)
 -m_{n-} \overline{\psi_{n-}} \left( x \right) \psi_{n-} \left( x \right)
 \right\}
\end{eqnarray*}

The decomposition of the KK modes is simplified if we choose to
express
 $\psi_{R+}$ and $\psi_{L+}$ in the form:

\begin{equation}
\Psi_{(R,L)+} \left( x, \phi, \rho \right) = \sum_{n} 
\psi_{n+}^{R,L} \left( x \right) \left(
\frac{ e^{\sigma} e^{\gamma} }{ \left( e^{\sigma} + e^{\gamma} -1 \right)^{6}}
\right)^{-\frac{1}{2}} f_{n+}^{R,L} \left( \phi, \rho \right)
\end{equation}

and $\psi_{R-}$ and $\psi_{L-}$ in the form:

\begin{equation}
\Psi_{(R,L)-} \left( x, \phi, \rho \right) = \sum_{n} 
\psi_{n-}^{R,L} \left( x \right) \left(
\frac{ e^{\sigma} e^{\gamma} }{ \left( e^{\sigma} + e^{\gamma} -1 \right)^{6}}
\right)^{-\frac{1}{2}} f_{n-}^{R,L} \left( \phi, \rho \right)
\end{equation}

To give the standard 4-d kinetic terms, we need the normalization
conditions

\begin{equation}
\int \sum_{n,m} \left( e^{\sigma} + e^{\gamma} - 1 \right) 
f_{n+}^{R^{\ast}} \left( \phi, \rho \right) f_{m+}^{R} \left(
\phi, \rho \right) \mbox{d} \phi \mbox{d} \rho = \delta_{mn}
\end{equation}

\begin{equation}
\int \sum_{n,m} \left( e^{\sigma} + e^{\gamma} - 1 \right) 
f_{n-}^{R^{\ast}} \left( \phi, \rho \right) f_{m-}^{R} \left(
\phi, \rho \right) \mbox{d} \phi \mbox{d} \rho = \delta_{mn}
\end{equation}

\begin{equation}
\int \sum_{n,m} \left( e^{\sigma} + e^{\gamma} - 1 \right) 
f_{n+}^{L^{\ast}} \left( \phi, \rho \right) f_{m+}^{L} \left(
\phi, \rho \right) \mbox{d} \phi \mbox{d} \rho = \delta_{mn}
\end{equation}

\begin{equation}
\int \sum_{n,m} \left( e^{\sigma} + e^{\gamma} - 1 \right) 
f_{n-}^{L^{\ast}} \left( \phi, \rho \right) f_{m-}^{L} \left(
\phi, \rho \right) \mbox{d} \phi \mbox{d} \rho = \delta_{mn}
\end{equation}

In order to read off the equations of motion that the f's must solve,
we need to simplifym some terms in the action.
For example,

\begin{eqnarray*}
\frac{1}{2} \overline{\psi_{L+}} 
\left( 
\frac{e^{\gamma}}{ \left( e^{\sigma} + e^{\gamma} - 1 \right)^{5}} 
\partial_{\phi} + \partial_{\phi} \frac{e^{\gamma}}{ \left( e^{\sigma} +
 e^{\gamma} - 1 \right)^{5}} \right) \psi_{R+} & = & \\
 -\frac{1}{2} \sum_{l=0}^{\infty} \overline{\psi_{L+l}} \left( x \right)
f_{L+l}^{\ast} \left( \phi, \rho \right) \frac{ \left( e^{\sigma} + 
e^{\rho} - 1 \right) e^{\gamma} \sigma^{\prime} }{ e^{\sigma} 
e^{\gamma} } \sum_{n=0}^{\infty} \psi_{R+n} \left( x \right) 
f_{R+n} \left( \phi, \rho \right) & & \\
+ 3 \sum_{l=0}^{\infty} \overline{\psi_{L+l}} \left( x \right) 
f_{L+l}^{\ast} \left( \phi, \rho \right) e^{\sigma} e^{\gamma} 
\sigma^{\prime} \sum_{n=0}^{\infty} \psi_{R+n} \left( x \right) f_{R+n}
\left( \phi, \rho \right) & & \\
+ \sum_{l=0}^{\infty} \overline{\psi_{L+l}} \left( x \right)
f_{L+l}^{\ast} \left( \phi, \rho \right) \frac{
\left( e^{\sigma} + e^{\gamma} - 1 \right)}{ e^{\sigma}} \sum_{n=0}^{\infty}
 \psi_{R+n} \left( x \right) f_{R+n}^{\prime} \left( \phi, \rho \right)
 & & \\
-\frac{5}{2} \sum_{l=0}^{\infty} \overline{\psi_{L+l}} \left( x \right)
f_{L+l}^{\ast} \left( \phi, \rho \right) \sigma^{\prime} 
\sum_{n=0}^{\infty} \psi_{R+n} \left( x \right) f_{R+n} 
\left( \phi, \rho \right) & &
\end{eqnarray*}
with similar expressions holding for the other terms.

We are interested in the equations (4 of them) for the zero modes
$( m_{n} = 0 )$,
found by means of varying $S$ with respect to $f_{R+0}^{\ast},
f_{L+0}^{\ast}, f_{L-0}^{\ast}, f_{R-0}^{\ast}$.

The normalization conditions reproduce the desired
 four dimensional K.E. terms, so we
only need to vary the remaining parts of $S$.
For example, the remaining
terms involving $f_{L+0}^{\ast}$ are:

\begin{eqnarray*}
-\frac{1}{2} \overline{\psi_{L+0}} \left( x \right) f_{L+0}^{\ast} 
\left( \phi, \rho \right) \frac{ \left( e^{\sigma} + e^{\gamma} 
-1 \right) e^{\gamma} }{ e^{\sigma} e^{\gamma} } \sigma^{\prime}
\psi_{R+0} \left( x \right) f_{R+0} \left( \phi, \rho \right) & & \\
+ 3 \overline{\psi_{L+0}} \left( x \right) f_{L+0}^{\ast} \left( \phi, 
\rho \right) e^{\sigma} e^{\gamma} \sigma^{\prime} \psi_{R+0} \left( x
\right) f_{R+0} \left( \phi, \rho \right) & & \\
+ \overline{\psi_{L+0}} \left( x \right) f_{L+0}^{\ast} \left( \phi, 
\rho \right) \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{
e^{\sigma} } \psi_{R+0} \left( x \right) f_{R+0}^{\prime} \left(
\phi, \rho \right) & & \\
-\frac{5}{2} \overline{\psi_{L+0}} \left( x \right) f_{L+0}^{\ast}
\left( \phi, \rho \right) \sigma^{\prime} \psi_{R+0} \left( x \right)
f_{R+0} \left( \phi, \rho \right) & & \\
+\frac{i}{2} \overline{\psi_{L+0}} \left( x \right) f_{L+0}^{\ast} \left(
\phi, \rho \right) \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) 2
 e^{\sigma} \dot{\gamma} }{ e^{\sigma} e^{\gamma} } \psi_{R+0} \left(
 x \right) f_{R+0} \left( \phi, \rho \right)  & & \\
 - 3 i \overline{\psi_{L+0}} \left( x \right) f_{L+0}^{\ast} \left(
\phi, \rho \right) e^{\gamma} e^{\sigma} \dot{\gamma} \psi_{R+0} \left(
x \right) f_{R+0} \left( \phi, \rho \right) & & \\
-i \overline{\psi_{L+0}} \left( x \right) f_{L+0}^{\ast} \left( \phi, 
\rho \right) \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{
e^{\gamma} } \psi_{R+0} \left( x \right) \dot{ f_{R+0} } \left( \phi,
 \rho \right) & & \\
+ \frac{i5}{2} \overline{\psi_{L+0}} \left( x \right) f_{L+0}^{\ast} \left(
\phi, \rho \right) \dot{\gamma} \psi_{R+0} \left( x \right) 
 f_{R+0} \left( \phi, \rho \right) & & \\
- i m \left( \frac{ \sigma^{\prime} }{ k } \right)
\left( \frac{ \dot{\gamma} }{ k } \right) \overline{\psi_{L+0}} \left(
x \right) f_{L+0}^{\ast} \left( \phi, \rho \right) \psi_{R-0} \left( x
 \right) f_{R-0} \left( \phi, \rho \right) & &
\end{eqnarray*}

Noting that the two four dimensional right-handed and 
left-handed spinors $\psi_{+}$ and $\psi_{-}$ are 
complex conjugates of each other [], we arrive at the 
following equation for the zero mode $f_{R_{0}}$
(where dots denote derivatives with respect to $\rho$ and
primes denote derivatives with respect to $\phi$):



\begin{eqnarray*}
-\frac{1}{2} \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }
{ e^{\sigma} e^{\gamma} } e^{\gamma} \sigma^{\prime} f_{R+0}
+ 3 e^{\sigma} e^{\gamma} \sigma{\prime} f_{R+0} & & \\
 + \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\sigma} }
f_{R+0}^{\prime} - \frac{5}{2} \sigma^{\prime} f_{R+0} + \frac{i}{2}
 \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) e^{\sigma} 
\dot{\gamma} }{ e^{\sigma} e^{\gamma} } f_{R+0} & & \\
- 3 i e^{\gamma} e^{\sigma} \dot{\gamma} f_{R+0} -
 i \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\gamma} }
 \dot{f_{R+0}} & & \\
+ i \frac{5}{2} \dot{\gamma} f_{R+0} - i m \left( \frac{
\sigma^{\prime} }{ k } \right) \left( \frac{ \dot{\gamma} }{ k } \right)
f_{R_0} & = & 0
\end{eqnarray*}

Similary, we find the following remaining equations for the 
other zero modes:

\begin{eqnarray*}
-\frac{1}{2} \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }
{ e^{\sigma} e^{\gamma} } e^{\gamma} \sigma^{\prime} f_{R-0}
+ 3 e^{\sigma} e^{\gamma} \sigma{\prime} f_{R-0} & & \\
 + \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\sigma} }
f_{R-0}^{\prime} - \frac{5}{2} \sigma^{\prime} f_{R-0} - \frac{i}{2}
 \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) e^{\sigma} 
\dot{\gamma} }{ e^{\sigma} e^{\gamma} } f_{R-0} & & \\
+ 3 i e^{\gamma} e^{\sigma} \dot{\gamma} f_{R-0} +
 i \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\gamma} }
 \dot{f_{R-0}} & & \\
- i \frac{5}{2} \dot{\gamma} f_{R-0} + i m \left( \frac{
\sigma^{\prime} }{ k } \right) \left( \frac{ \dot{\gamma} }{ k } \right)
f_{R+0} & = & 0
\end{eqnarray*}


\begin{eqnarray*}
+\frac{1}{2} \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }
{ e^{\sigma} e^{\gamma} } e^{\gamma} \sigma^{\prime} f_{L+0}
- 3 e^{\sigma} e^{\gamma} \sigma{\prime} f_{L+0} & & \\
 - \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\sigma} }
f_{L+0}^{\prime} + \frac{5}{2} \sigma^{\prime} f_{L+0} + \frac{i}{2}
 \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) e^{\sigma} 
\dot{\gamma} }{ e^{\sigma} e^{\gamma} } f_{L+0} & & \\
- 3 i e^{\gamma} e^{\sigma} \dot{\gamma} f_{L+0} -
 i \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\gamma} }
 \dot{f_{L+0}} & & \\
+ i \frac{5}{2} \dot{\gamma} f_{L+0} - i m \left( \frac{
\sigma^{\prime} }{ k } \right) \left( \frac{ \dot{\gamma} }{ k } \right)
f_{L-0} & = & 0
\end{eqnarray*}


\begin{eqnarray*}
+\frac{1}{2} \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }
{ e^{\sigma} e^{\gamma} } e^{\gamma} \sigma^{\prime} f_{L-0}
- 3 e^{\sigma} e^{\gamma} \sigma{\prime} f_{L-0} & & \\
 - \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\sigma} }
f_{L-0}^{\prime} + \frac{5}{2} \sigma^{\prime} f_{L-0} - \frac{i}{2}
 \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) e^{\sigma} 
\dot{\gamma} }{ e^{\sigma} e^{\gamma} } f_{L-0} & & \\
+ 3 i e^{\gamma} e^{\sigma} \dot{\gamma} f_{L-0} +
 i \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\gamma} }
 \dot{f_{L-0}} & & \\
- i \frac{5}{2} \dot{\gamma} f_{L-0} + i m \left( \frac{
\sigma^{\prime} }{ k } \right) \left( \frac{ \dot{\gamma} }{ k } \right)
f_{L+0} & = & 0
\end{eqnarray*}


Because $f_{L+0}^{\ast} = f_{L-0}$, this is the complex conjugate
of the previous equation.


The equation for the right-handed zero modes is

\begin{eqnarray*}
-\frac{1}{2} \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }
{ e^{\sigma} e^{\gamma} } e^{\gamma} \sigma^{\prime} f_{R-} + 
3 e^{\sigma} e^{\gamma} \sigma^{\prime} f_{R-} + \frac{ \left(
e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\sigma} } f_{R-}^{\prime}
 & & \\
-\frac{5}{2} \sigma^{\prime} f_{R-} - i\frac{1}{2} \frac{ \left(
 e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\sigma} e^{\gamma} }
e^{\sigma} \dot{\gamma} f_{R-} + 3i e^{\gamma} e^{\sigma} 
\dot{\gamma} f_{R-} & & \\
+ i \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{
e^{\gamma} } \dot{f_{R-}} - i \frac{5}{2} \dot{\gamma} f_{R-} + 
im \left( \frac{ \sigma^{\prime} }{k} \right)
 \left( \frac{ \dot{\gamma} }{ k } \right) f_{R+} & = & 0
\end{eqnarray*}


\begin{eqnarray*}
-\frac{1}{2} \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }
{ e^{\sigma} e^{\gamma} } e^{\gamma} \sigma^{\prime} f_{R+} + 
3 e^{\sigma} e^{\gamma} \sigma^{\prime} f_{R+} + \frac{ \left(
e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\sigma} } f_{R+}^{\prime}
 & & \\
-\frac{5}{2} \sigma^{\prime} f_{R+} + i\frac{1}{2} \frac{ \left(
 e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\sigma} e^{\gamma} }
e^{\sigma} \dot{\gamma} f_{R+} - 3i e^{\gamma} e^{\sigma} 
\dot{\gamma} f_{R+} & & \\
- i \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{
e^{\gamma} } \dot{f_{R+}} + i \frac{5}{2} \dot{\gamma} f_{R+} - 
im \left( \frac{ \sigma^{\prime} }{k} \right)
 \left( \frac{ \dot{\gamma} }{ k } \right) f_{R-} & = & 0
\end{eqnarray*}


Because $f_{R+} = f_{R-}^{\ast}$, 
we are free
to write
\begin{eqnarray}
f_{R+}  & = & U + i V \\
f_{R-} & = & U - i V
\end{eqnarray}

where
$U$ and $V$ are real.
Setting the real and imaginary parts of this zero mode equation
separately to zero, we find the following pair of
equations:

\begin{eqnarray*}
-\frac{1}{2} \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{
 e^{\sigma} e^{\gamma} } e^{\gamma} \sigma^{\prime} U + 
3 e^{\sigma} e^{\gamma} \sigma^{\prime} U + \frac{ \left(
 e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\sigma} } U^{\prime} 
 & & \\
 -\frac{5}{2} \sigma^{\prime} U - \frac{1}{2} \frac{ \left(
e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\sigma} e^{\gamma} }
e^{\sigma} \dot{\gamma} V + 3 e^{\gamma} e^{\sigma} \dot{\gamma} V
 + \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{
e^{\gamma} } \dot{V} & & \\
-\frac{5}{2} \dot{\gamma} V - m \left( \frac{ \sigma^{\prime} }{
k} \right) \left( \frac{ \dot{\gamma} }{ k } \right) V & = & 0
\end{eqnarray*}

and
\begin{eqnarray*}
\frac{1}{2} \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{
 e^{\sigma} e^{\gamma} } e^{\gamma} \sigma^{\prime} V - 
3 e^{\sigma} e^{\gamma} \sigma^{\prime} V - \frac{ \left(
 e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\sigma} } V^{\prime} 
 & & \\
 +\frac{5}{2} \sigma^{\prime} V - \frac{1}{2} \frac{ \left(
e^{\sigma} + e^{\gamma} - 1 \right) }{ e^{\sigma} e^{\gamma} }
e^{\sigma} \dot{\gamma} U + 3 e^{\gamma} e^{\sigma} \dot{\gamma} U
 + \frac{ \left( e^{\sigma} + e^{\gamma} - 1 \right) }{
e^{\gamma} } \dot{U} & & \\
-\frac{5}{2} \dot{\gamma} U + m \left( \frac{ \sigma^{\prime} }{
k} \right) \left( \frac{ \dot{\gamma} }{ k } \right) U & = & 0
\end{eqnarray*}


So far, we have been comlpetely general.
In order to find an analytic sollution, we
 look at the restricted class of solutions with
$U = V$.
One then finds the following equation for U:

\begin{eqnarray*}
{\frac {\partial }{\partial \phi}}u(\phi,\rho)+{\frac {\partial }{
\partial \rho}}u(\phi,\rho)-\left ({\frac {m{e^{k\phi}}}{{e^{k\phi}}+{
e^{k\rho}}-1}}+5\,{\frac {k{e^{k\phi}}}{2\,{e^{k\phi}}+2\,{e^{k\rho}}-
2}}+k \right. & &  \\
\left.
-3\,{\frac {{e^{2\,k\phi}}{e^{k\rho}}k}{{e^{k\phi}}+{e^{k\rho}}-1
}}+5/2\,{\frac {k{e^{k\rho}}}{{e^{k\phi}}+{e^{k\rho}}-1}}-3\,{\frac {k
{e^{2\,k\rho}}{e^{k\phi}}}{{e^{k\phi}}+{e^{k\rho}}-1}}\right )u(\phi,
\rho) & =& 0
\end{eqnarray*}

The solution to this equation is:
\begin{eqnarray*}
u(\phi,\rho) & = & {\it \_F1}(\rho-\phi) \\
e^{-\frac{1}{2 \left( 1 + e^{k \left( \rho - \phi \right)} \right)^{2}
k} \times \left( \right. }  & & \\
-7 k \ln \left( 2 \right) e^{\left( 2 k \left( \rho - 
\phi \right) \right)} - 2 \ln \left( e^{k \phi} \right) k
 e^{ 2 k \left( \rho - \phi \right)} & & \\
-5 e^{2k \left( \rho - \phi \right) } \ln \left(
e^{k \phi} + e^{k \rho} - 1 \right) k +
3 e^{k \left( 3 \rho - \phi \right)} k +
6 e^{2 k \rho} k & & \\
-14 k \ln \left( 2 \right) e^{k \left( \rho - \phi \right)} - 2m
 \ln \left( e^{k \phi} + e^{k \rho} - 1 \right) e^{k \left(
 \rho - \phi \right)} - 4 \ln \left( e^{k \phi} \right) k
 e^{k \left( \rho - \phi \right)} & & \\
-2m \ln \left( 2 \right) e^{k \left( \rho - \phi \right)}
-4k \ln \left( e^{k \phi} + e^{k \rho} - 1 \right) e^{k \left(
\rho - \phi \right)} + 
3k e^{k \left( \rho + \phi \right) } & & \\
-2 \ln \left( e^{k \phi} \right) k
 - 7 k \ln \left( 2 \right) + 6k e^{ k \left( 2 \rho - \phi \right)} +
6 k e^{k \rho} & & \\
\left.
- 2m \ln \left( e^{k \phi} + e^{k \rho} - 1 \right)
- 5 k \ln \left( e^{k \phi} + e^{k \rho} - 1 \right)
 - 2m \ln \left( 2 \right) \right)
\end{eqnarray*}
where $F1$ is an arbitrary function of $\left( \rho - \phi \right)$.


Similary, for the left-handed sector we find the following
equation:
\begin{eqnarray*}
{\frac {\partial }{\partial \phi}}u(\phi,\rho)+{\frac {\partial }{
\partial \rho}}u(\phi,\rho)-\left (-{\frac {m{e^{k\phi}}}{{e^{k\phi}}+
{e^{k\rho}}-1}}+5\,{\frac {k{e^{k\phi}}}{2\,{e^{k\phi}}+2\,{e^{k\rho}}
-2}}+k \right. & & \\
\left.
-3\,{\frac {{e^{2\,k\phi}}{e^{k\rho}}k}{{e^{k\phi}}+{e^{k\rho}}-
1}}+5/2\,{\frac {k{e^{k\rho}}}{{e^{k\phi}}+{e^{k\rho}}-1}}-3\,{\frac {
k{e^{2\,k\rho}}{e^{k\phi}}}{{e^{k\phi}}+{e^{k\rho}}-1}}\right )u(\phi,
\rho) & = & 0
\end{eqnarray*}


The solution to this equation is:
\begin{eqnarray*}
u(\phi,\rho) & = & {\it \_F1}(\rho-\phi) \\
e^{-\frac{1}{2 \left( 1 + e^{k \left( \rho - \phi \right)} \right)^{2}
k} \times \left( \right. }  & & \\
-2 \ln \left( e^{k \phi} \right) k e^{2k \left( \rho - \phi 
\right)} - 7 k \ln \left( 2 \right) e^{2k \left( \rho - \phi \right)} -
5 k e^{2k \left( \rho - \phi \right)} \ln \left( e^{k \phi} +
e^{k \rho} - 1 \right) & & \\
+ 2m \ln \left( 2 \right) e^{k \left( \rho - \phi \right) } +
6k e^{2k \rho} + 2m \ln \left( e^{k \phi} + e^{k \rho} - 1 \right) 
e^{k \left( \rho - \phi \right)} - 14 k \ln \left( 2 \right)
e^{k \left( \rho - \phi \right)} & & \\
-4k \ln \left( e^{k \phi} + e^{k \rho} - 1 \right) 
e^{k \left( \rho - \phi \right) } + 3k e^{k \left( 3 \rho - \phi
\right)} - 4 k \ln \left( e^{k \phi} \right) e^{k \left(
\rho - \phi \right)} & & \\
+ 3 k e^{k \left( \rho + \phi \right)} + 6k e^{ k \left( 
2 \rho - \phi \right)} - 2 k \ln \left( e^{k \phi} \right) +
2m \ln \left( e^{k \phi} + e^{k \rho} - 1 \right) & & \\
+ 6k e^{k \rho} + 2m \ln \left( 2 \right) - 7k \ln \left( 2 \right) & & \\
\left.
-5 k \ln \left( e^{k \phi} + e^{k \rho} - 1 \right) \right)
\end{eqnarray*}

Here is a plot of this function for the special cases of 
$k=1, m=1$ and $F\left( \phi - \phi \right) = 1$:

\psfig{figure=lefty.ps,height=8cm,angle=0}

The positive tension 3-brane diagonally opposite the negative
tension 3-brane in the $\phi - \rho$ plane is where the SM
fields are localized, or possibly on one of the other corners
of this plane, where two 4-branes of opposite tension intersect
to form a 3-brane of zero tension.
As one may readily verify, this function $u_{L}$ only
becomes more strongly peaked around the
positive tension 3-brane as $m$ is increased.
Thus, if we wish to localize the SM fermion field profiles away
from the positive tension 3-brane, we must consider
solutions other than those considered above.

We want CP violation to be coming solely from the Dirac algebra;
that is the possibility we are investigating.
Thus, in order to realize this scenario, we have several choices
on how to arrive at analytical solutions for the fermions.
We need to have either both $f_{L}$ and $f_{R}$ be real, or
both complex.
This condition will become clear in the last section, where we 
construct the effective four dimensional quark mass matrices that
are complex.
Hence, we must have either $V_{L} \neq 0$ and $U_{L} = 0$ with
$V_{R} \neq 0$ and $U_{R} = 0$ or $U_{L} \neq 0$ and 
$V_{L} = 0$ with $U_{R} \neq 0$ and $V_{R} = 0$.
The positive tension 3-brane diagonally opposite the negative
tension 3-brane in the $\phi - \rho$ plane is where the SM
fields are localized, or possibly on one of the other corners
of this plane, where two 4-branes of opposite tension intersect
to form a 3-brane of zero tension.
As one may readily verify, this function $U_{L}$ only
becomes more strongly peaked around the
positive tension 3-brane as $m$ is increased.
Thus, if we wish to localize the SM fermion field profiles away
from the positive tension 3-brane and have CP violation coming
solely from the Dirac algebra $ \left( \Gamma_{0} \right)$,
we must make the choices
\begin{eqnarray*}
f_{L0} &  = & i V_{L0} \\
f_{R0} &  = & i V_{R0}
\end{eqnarray*}
 
This leads to the following equation for both $V_{L} = V_{R} \equiv V$:
\begin{eqnarray*}
{\frac {\partial }{\partial \phi}}V(\phi,\rho)+{\frac {\partial }{
\partial \rho}}V(\phi,\rho)-\left ({\frac {m{e^{k\phi}}}{{e^{k\phi}}+{
e^{k\rho}}-1}}+5\,{\frac {k{e^{k\phi}}}{2\,{e^{k\phi}}+2\,{e^{k\rho}}-
2}}+k \right. & &  \\
\left.
-3\,{\frac {{e^{2\,k\phi}}{e^{k\rho}}k}{{e^{k\phi}}+{e^{k\rho}}-1
}}+5/2\,{\frac {k{e^{k\rho}}}{{e^{k\phi}}+{e^{k\rho}}-1}}-3\,{\frac {k
{e^{2\,k\rho}}{e^{k\phi}}}{{e^{k\phi}}+{e^{k\rho}}-1}}\right )V(\phi,
\rho) & =& 0
\end{eqnarray*}

This equation is the same as the one found for $U_{R}$ in the
previously considered case.
We plot in Fig [] the (unnormalized) solution for the case
$k=1$ and $m=0$.

\psfig{figure=POS.ps,height=8cm,angle=0}

As expected, it is peaked at the origin of the extra
dimensions, just as what occurs in the case of five
dimensions, where right-handed fermion fields are peaked
around the Planck brane when no five dimensional mass term
is present.

In Fig. [], we plot the same function for the parameter choice
$k= \frac{1}{2}$ and $m=1$.

\psfig{figure=SM.ps,height=8cm,angle=0}

One deduces that increasing the value of $m$ translates
the peak of the profile along the $\phi$ direction towards
the 3-brane found at the intersection of two 4-branes with 
opposite sign tensions of equal magnitudes.
Increasing the value of $k$ serves to decrease the
width of the profile.
We note that in this subclass of solutions with
$U_{L} = U_{R} = 0$, both $V_{L}$ and $V_{R}$ can
be localized around the same region of the 
orbifold; there is no need to work with fields of only
one chirality and introduce the higher-dimensional
charge conjugation operator in order to achieve the 
clustering of SM fields in a safely localized
region in the extra dimensions.
We note however that we have only been able to construct 
analytic solutions within this subclass of solutions.
It seems intuitively obvious that the full four 
independent general solutions for 
$f_{L0+}, f_{L0-}, f_{R0+}$ and $f_{R0-}$ will be peaked
at each of the four corners of the
$\phi - \rho$ plane.

\section{Scalar Field Zero Mode and Quark Mass Matrices}
In this section, we solve for the zero mode of a 
six dimensional Higgs scalar field.
As in the fermion case, we also include a mass term for
the scalar in the six dimensional action.

We consider a real scalar field propagating in a six
dimensional curved background described by the metric

\begin{equation}
ds^{2} = A^{2} \left( \phi, \rho \right) \eta_{\mu \nu} dx^{\mu} dx^{\nu} -
B^{2} \left( \phi, \rho \right) d \phi^{2} - C^{2} \left( \phi, \rho \right)^{2}
\end{equation}
\begin{equation}
A = \frac{1}{e^{\sigma} + e^{ \gamma} -1}
\end{equation}
\begin{equation}
B = e^{\sigma} A
\end{equation}
\begin{equation}
C = e^{\gamma} A
\end{equation}
\begin{equation}
\sqrt{g} = \frac{ e^{\sigma} e^{\gamma} }{\left( e^{\sigma} + e^{\gamma} - 1\right)^{6}}
\end{equation}

Including a six dimensional mass term, the action becomes:

\begin{equation}
S = \frac{1}{2} \int \mbox{dx}^{4} \int \mbox{d} \phi \mbox{d} \rho 
\sqrt{G} \left( G^{AB} \partial_{A} \Phi \partial_{B} \Phi -
m_{\Phi}^{2} \Phi^{2} \right)
\end{equation}

Integrating by parts, we can write $S$ as:
\begin{eqnarray*}
S & = & \frac{1}{2} \int \mbox{dx}^{4} \int \mbox{d} \phi \mbox{d} \rho
\left( \frac{ e^{\sigma} e^{\gamma} }{ \left( e^{\sigma} + e^{\gamma}
 - 1 \right)^{4} } \eta^{\mu \nu} \partial_{\mu} \Phi \partial_{\nu}
\Phi + \Phi \partial_{\phi} \left( \frac{ e^{\gamma} }{
e^{\sigma} \left( e^{\sigma} + e^{\gamma} - 1 \right)^{4} } \partial_{\phi}
 \Phi \right) \right. \\
& & + \left. \Phi \partial_{\rho} \left( \frac{ e^{\sigma} }{ e^{\gamma}
\left( e^{\sigma} + e^{\gamma} - 1 \right)^{4} } \partial_{\rho}
 \Phi \right) - m_{\Phi}^{2} \Phi^{2} \sqrt{G} \right) 
\end{eqnarray*}

In order to give a four dimensional interpretation to this action, we
go through the dimensional reduction procedure. 
Thus we decompose the six dimensional field into KK modes
\begin{equation}
\Phi \left( x, \phi, \rho \right) = \sum_{n} \phi_{n} \left( x
 \right) f_{n} \left( \phi, \rho \right)
\end{equation}
Using this decomposition, the above action can be brought to the 
form

\begin{equation}
S = \frac{1}{2} \sum_{n} \int \mbox{dx}^{4} \left\{ \eta^{\mu \nu}
 \partial_{\mu} \phi_{n} \left( x \right) \partial_{\nu}
\phi_{n} \left( x \right) - m_{n}^{2} \phi_{n}^{2} \left( x
 \right) \right\}
\end{equation}

In order to reproduce the canonical four dimensional
kinetic terms,
we need to impose the orthogonality relations:
\begin{equation}
\int \int \mbox{d} \phi \mbox{d} \rho \frac{ e^{\sigma} e^{\gamma} }{
 \left( e^{\sigma} + e^{\gamma} - 1 \right)^{4} } f_{m}^{\ast} \left(
\phi, \rho \right) f_{n} \left( \phi, \rho \right) = \delta_{mn}
\end{equation}

Varying the action with respect to $f_{n}$ leads to
 the following equation for the zero mode:

\begin{equation}
\partial_{\phi} \left( \frac{ e^{\gamma} }{ e^{\sigma} \left(
e^{\sigma} + e^{\gamma} - 1 \right)^{4} } \partial_{\phi} f_{0} \right)
  +  \partial_{\rho} \left( \frac{ e^{\sigma} }{ e^{\gamma} \left(
e^{\sigma} + e^{\gamma} - 1 \right)^{4} } \partial_{\rho} f_{0} \right)
 -\frac{ m^{2} e^{\sigma} e^{\gamma} }{ \left( e^{\sigma}
+ e^{\gamma} - 1 \right)^{6} } f_{0} = 0
\end{equation}

From the form of this equation, we expect the 
$\phi$ and $\rho$ dependence of $f_{0}$ to be the same.
Hence, we write

\begin{eqnarray*}
\partial_{\phi} \left( \frac{ e^{\gamma} }{ e^{\sigma} \left(
e^{\sigma} + e^{\gamma} - 1 \right)^{4} } \partial_{\phi} f_{0} \right)
& = & 
\frac{ m^{2} e^{\sigma} e^{\gamma} }{ 2 \left( e^{\sigma}
+ e^{\gamma} - 1 \right)^{6} } f_{0} \\
   \partial_{\rho} \left( \frac{ e^{\sigma} }{ e^{\gamma} \left(
e^{\sigma} + e^{\gamma} - 1 \right)^{4} } \partial_{\rho} f_{0} \right)
& = & 
\frac{ m^{2} e^{\sigma} e^{\gamma} }{ 2 \left( e^{\sigma}
+ e^{\gamma} - 1 \right)^{6} } f_{0}
\end{eqnarray*}


Any solution to this pair of equations automatically
satisifies the orginal equation.

The first of the above equations can be written as

\begin{equation}
\left( \frac{ 4 \sigma^{\prime} e^{\gamma} }{ \left( e^{\sigma}
+ e^{\gamma} - 1 \right) } + \frac{ \sigma^{\prime} e^{\gamma} }{
 e^{\sigma} } \right) \partial_{\phi} f_{0}  - 
\frac{ e^{\gamma} }{ e^{\sigma} } \partial_{\phi}^{2} f_{0} -
\frac{ m^{2} e^{\sigma} e^{\gamma} f_{0} }{ 2 \left(
e^{\sigma} + e^{\gamma} - 1 \right)^{2} } = 0
\end{equation}

Substituting in the forms for $\sigma$ and $\gamma$, we arrive at
the two equations:
\begin{equation}
\left( \frac{ 4 k e^{ k \rho} }{ \left( e^{k \phi}
+ e^{ k \rho} - 1 \right) } + \frac{ k e^{k \rho} }{
 e^{ k \phi} } \right) \partial_{\phi} f_{0}  - 
\frac{ e^{k \rho} }{ e^{k \phi} } \partial_{\phi}^{2} f_{0} -
\frac{ m^{2} e^{k \phi} e^{k \rho} f_{0} }{ 2 \left(
e^{k \phi} + e^{k \rho} - 1 \right)^{2} } = 0
\end{equation}

\begin{equation}
\left( \frac{ 4 k e^{ k \sigma} }{ \left( e^{k \phi}
+ e^{ k \rho} - 1 \right) } + \frac{ k e^{k \sigma} }{
 e^{ k \rho} } \right) \partial_{\rho} f_{0}  - 
\frac{ e^{k \phi} }{ e^{k \rho} } \partial_{\rho}^{2} f_{0} -
\frac{ m^{2} e^{k \phi} e^{k \rho} f_{0} }{ 2 \left(
e^{k \phi} + e^{k \rho} - 1 \right)^{2} } = 0
\end{equation}
 

 For the special case when 

\begin{equation}
2 k^{2} m^{2} = 25 k^{4}
\end{equation}

we can find an analytic solution.
For this case , the solution is

\begin{equation}
f \left( \phi, \rho \right) = e^{ \left( \frac{ \ln \left( e^{k \phi} + 
e^{k \rho} - 1 \right) }{ 2 k^{2} } \right) }
\end{equation}
 
%Here is a plot of this function for the case $k = 1$:



\psfig{figure=plot.ps,height=8cm,angle=0}

Here is a plot of this function for the case $k=1$.



Having now solved for the fermion field profiles in the extra
dimensions as wellas that for the Higgs scalar, we may construct
a six dimensional lorentz invariant Yukawa interaction from a term
such as
\begin{equation}
\sqrt{-G} \lambda_{ij}^{(6)} H \overline{\psi_{i}} \psi_{j}
\end{equation}
where the fermion fields are eight component objects and their
indices $i$ and $j$ are generation labels with the 
Yukawa coupling matrices $\lambda_{ij}^{(6)}$ 
connecting only fermions in combinations consistent with
gauge group invariance.
One sees that the fermion bilinear may be expressed as
\begin{eqnarray*}
\overline{\psi} \psi & =  &
-i \overline{\psi_{-}} \psi_{+} + i 
\overline{\psi_{+}} \psi_{-} \\
 & = & i \overline{\psi_{L+}} \psi_{R-} +
 i \overline{\psi_{R+}} \psi_{L-} \\
 & & -i \overline{\psi_{L-}} \psi_{R+} - i
 \overline{\psi_{R-}} \psi_{L+}
\end{eqnarray*}
where here the conjugate fields are formed with the usual $\gamma_{0}$
of the four dimensional Dirac algebra.
From a given six dimensional fermion field, one may use the 
total orbifold symmetry of the extra dimensions to project out
an appropriate two component, four-dimensional fermion field with the
desired SM gauge group transformation properties.

If one wishes to write down the exact same form of the Yukawa
coupling in the six dimensional action as one finds in the
SM action, we then write down a general term of the form

\begin{equation}
\int \mbox{d}^{4} x \int \mbox{d} \phi \mbox{d} \rho
\sqrt{-G} \lambda_{ij}^{(6)} H \overline{\psi_{i}} \psi_{j}
\end{equation}

Recall, however, that the motivation for considering the 
implications of extra dimensions is to see if their existence
can help in any way to reduce or simplify in an organized way
the redundancy inherent in the four dimensional Yukawa coupling
matrices.
Allowing $\lambda_{ij}^{(6)}$ and the six dimensional fermion mass
terms $m_{i}$ to be arbitary parameters only 
increases the redundancy of physical information contained 
in the parameters of the action, and, from the
perspective of attempting to gain deeper understanding
of the fermion mass hierarchy, considerably weakens the
motivation of considerating extra dimensions.
However, as mentioned in the introduction, one
starting point often adopted in various attempts to study
the fermion mass hierarchy is to start from a discrete
flavor symmetry which leads to teh so-called democratic mass
matrix, with all entries being the same.
This simplest of ans\"{a}tz leads to one massive eigenstate
and two degenerate massless eigenstates and is thus
considered a reasonably successful approximation
given the simplicity of the ans\"{a}tz.
In [], the introduction of a single flat extra 
dimension provides a theoretical explanation of the
democratic ans\"{a}tz itself in terms of higher
dimensional geography rather than some additional
flavor physics.
The introduction of an additional flat extra dimension and
its associated Dirac algebra then allow for the 
possibility to realize a more realistic spectrum than that 
provided by the democratic mass matrices.
Because the extra dimensions are taken to be flat, the
fermion profiles can be taken to be separable
functions of each of the extra dimensions.
This property than transforms the effective
democratic quark mass matrices one 
obtains from dimensionally reducing from five to four
dimensions into pure phase mass matrices.

The success of this approach in the flat space scenario 
cannot be carried over without modification to the case
of two warped extra dimensions.
Because the equations for the fermion profiles in
the warped scenario we are considering
are not separable,
it is not possible to automatically achieve pure
phase effective four dimensional quark mass matrices within
the context of two warped extra dimensions.

If we set $\lambda_{ij}^{(6)}$ equal to the democratic
mass matrix in both the up and down quark sectors, we then
arrive at essentially the same level of understanding as
when the democratic form is adopted in flavor
symmetry approaches to the problem in four dimensions.
In that case, a longstanding problem has been to find
a successful implementation of a breaking of this symmetry,
(sometimes involving the additional physics of a ``flavon''
field) with the additional complication of 
ultimately arriving at complex mass matrices.

The principle observation of this work is that the presence 
of the extra dimensions serves to break the 
democratic form of the $\lambda_{ij}^{(6)}$, which is now
communicated in some adulterated form controlled by the
six dimensional fermion mass terms down to the 
effective $\lambda_{ij}^{(4)}$.
Geography in the extra warped dimensions is an
alternative to the flavon field method of breaking the 
flavor symmetry of the Yukawa terms.
In addition, this geometric method of breaking the 
flavor symmetry naturally leads to complex 
effective four dimensional mass matrices.


Adopting the democratic form for $\lambda_{ij}^{(6)}$ in
each quark sector does not result in a calculable
model of flavor mixing and masses.
It does, however, correspond to a so-called
minimal parameter basis.
Six quark masses and four flavor mixing parameters are
derived from ten six-dimensional parameters.
This minimal parameter set arises from the 
following considerations.
The SM gauge symmetries allow for the existence of nine
different six-dimensional fermion masses $m_{i}$.
The left-handed up-type and down-type quarks of the same
generation must have the same six-dimensional mass
parameter $m$ because together they form an $SU \left( 2 \right)$
doublet.
The right-handed components of the up and down type
quarks for each generation have different mass parameters.
From the four dimensional perspective, a massive fermion 
field is formed only after electroweak symmetry breaking.
In addition to these nine independent mass paraemters,
we also have the freedom associated with the 
curvature $k$ of the bulk AdS space.
We are assuming eeequal magnitudes for all the 4-brane
tensions and so have only one dimenionful parameter k, instead
of two as would be the case if we allowed arbitrary 
4-brane tensions.
Requiring that this setup reproduce the RS resolution 
of the gauge hierarchy problem then fixes the coordinate
length of the fundamental cell we are considering.


\section{Conclusion}

To conclude, we first remind the reader of some of the 
promising results already attained in addressing the fermion
mass and mixing hierarchy problems within the context
of extra dimensions [].
Considering the $2 \times 2$ matrix with columns labeled
``one extra dimension'' and ``two extra dimensions'' and
rows labeled ``flat extra space'' and ``warped extra space'', 
we have shown that the $( 2, 2 )$ element is also a possible
arena in which to study the problem.
But the principle motivations for considering the
two wapred extra dimensions scenario are not just 
for the sake of completeness.
The possibility of attaining the CP violating phase in the
quark flavor-mixing matrix via this higher dimensional
mechanism is a property of the Dirac algebra in six
dimensions and remains a possibility whether or not the two
extra dimensions are flat or warped.
Independent of considerations of the gauge hierarchy
problem, one advantage of the warped case over the flat 
case is that in the warped case, analytic solutions
for the relevant fields can be found and hence trustworthy
numerical tests of the model can in principle be made.
In the flat space case, the localization mechanism
of the fermions involves the introduction of this scalar field and
 specific forms for the scalar field must be assumed and even 
then the fermion profiles are solved analytically only after making 
approximations to the given scalar field [].
We have shown in this work that analytic solutions (for the zero
modes)
are also possible in the six dimensional warped case.
Another motivation for going to six dimensions
concerns the mystery of the generation index.
As shown in [], multibrane world scenarios imply the 
existence of light KK modes that are suggestive of
family replication.
Going to six dimensions may alleviate some of the 
difficulties encountered in trying to implement this
program.
This possibility is currently under investigation.
\bibliography{pap}
\end{document}

\begin{thebibliography}{10}

\bibitem{Weinberg:1977hb}
Steven Weinberg.
\newblock The problem of mass.
\newblock {\em Trans. New York Acad. Sci.}, 38:185--201, 1977.

\bibitem{Wilczek:1977uh}
Frank Wilczek and A.~Zee.
\newblock Discrete flavor symmetries and formula for the cabibbo angle.
\newblock {\em Phys. Lett.}, B70:418, 1977.

\bibitem{Rothman:1979ft}
Arthur~C. Rothman and Kyungsik Kang.
\newblock Calculability: The mass matrix and cp violation.
\newblock {\em Phys. Rev. Lett.}, 43:1548, 1979.

\bibitem{Kang:1981yg}
Kyungsik Kang and Arthur~C. Rothman.
\newblock Generalized mixing angles in gauge theories with natural flavor
  conservation.
\newblock {\em Phys. Rev.}, D24:167, 1981.

\bibitem{Fritzsch:1978vd}
H.~Fritzsch.
\newblock Weak interaction mixing in the six - quark theory.
\newblock {\em Phys. Lett.}, B73:317--322, 1978.

\bibitem{Fritzsch:1979zq}
Harald Fritzsch.
\newblock Quark masses and flavor mixing.
\newblock {\em Nucl. Phys.}, B155:189, 1979.

\bibitem{DeRujula:1977ry}
A.~De Rujula, Howard Georgi, and S.~L. Glashow.
\newblock A theory of flavor mixing.
\newblock {\em Ann. Phys.}, 109:258, 1977.

\bibitem{Georgi:1979dq}
Howard Georgi and D.~V. Nanopoulos.
\newblock Ordinary predictions from grand principles: t quark mass in o(10).
\newblock {\em Nucl. Phys.}, B155:52, 1979.

\bibitem{Kang:1997uv}
Kyungsik Kang and Sin~Kyu Kang.
\newblock New class of quark mass matrix and calculability of flavor mixing
  matrix.
\newblock {\em Phys. Rev.}, D56:1511--1514, 1997.

\bibitem{Thirring:1972de}
W.~E. Thirring.
\newblock Fivedimensional theories and cp-violation.
\newblock {\em Acta Phys. Austriaca Suppl.}, 9:256--271, 1972.

\bibitem{Casadio:2001fe}
R.~Casadio and A.~Gruppuso.
\newblock Discrete symmetries and localization in a brane-world.
\newblock {\em Phys. Rev.}, D64:025020, 2001.

\bibitem{Chang:2001uk}
Darwin Chang, Wai-Yee Keung, and Rabindra~N. Mohapatra.
\newblock Models for geometric cp violation with extra dimensions.
\newblock {\em Phys. Lett.}, B515:431--441, 2001.

\bibitem{Chang:2001yn}
Darwin Chang and R.~N. Mohapatra.
\newblock Geometric cp violation with extra dimensions.
\newblock {\em Phys. Rev. Lett.}, 87:211601, 2001.

\bibitem{Huang:2001np}
Chao-Shang Huang, Tian-jun Li, Wei Liao, and Qi-Shu Yan.
\newblock Cp violation and extra dimensions.
\newblock 2001.

\bibitem{Branco:2000rb}
G.~C. Branco, Andre de~Gouvea, and M.~N. Rebelo.
\newblock Split fermions in extra dimensions and cp violation.
\newblock {\em Phys. Lett.}, B506:115--122, 2001.

\bibitem{Sakamura:2000ik}
Yutaka Sakamura.
\newblock Cp violation due to the existence of other wall.
\newblock 2000.

\bibitem{Sakamura:1999fa}
Yutaka Sakamura.
\newblock Spontaneous cp violation in large extra dimensions.
\newblock 1999.

\bibitem{Hung:2001hw}
P.~Q. Hung and M.~Seco.
\newblock Pure phase mass matrices from six dimensions.
\newblock {\em hep-ph/0111013}, 2001.

\bibitem{Huber:2000ie}
Stephan~J. Huber and Qaisar Shafi.
\newblock Fermion masses, mixings and proton decay in a randall- sundrum model.
\newblock {\em Phys. Lett.}, B498:256--262, 2001.

\bibitem{Arkani-Hamed:1999dc}
Nima Arkani-Hamed and Martin Schmlatz.
\newblock Hierarchies without symmetries from extra dimensions.
\newblock {\em Phys. Rev.}, D61:033005, 2000.

\bibitem{Mouslopoulos:2001uc}
Stavros Mouslopoulos.
\newblock Bulk fermions in multi-brane worlds.
\newblock {\em JHEP}, 05:038, 2001.

\bibitem{Collins:2001ni}
Hael Collins and Bob Holdom.
\newblock The randall-sundrum scenario with an extra warped dimension.
\newblock {\em Phys. Rev.}, D64:064003, 2001.

\bibitem{Kogan:2001yr}
Ian~I. Kogan, Stavros Mouslopoulos, Antonios Papazoglou, and Graham~G. Ross.
\newblock Multigravity in six dimensions: Generating bounces with flat positive
  tension branes.
\newblock {\em Phys. Rev.}, D64:124014, 2001.

\bibitem{Burgess:2001bn}
C.~P. Burgess, James~M. Cline, Neil~R. Constable, and Hassan Firouzjahi.
\newblock Dynamical stability of six-dimensional warped brane- worlds.
\newblock 2001.

\bibitem{Kim:2001rm}
Jihn~E. Kim, Bumseok Kyae, and Hyun~Min Lee.
\newblock Localized gravity and mass hierarchy in d = 6 with gauss- bonnet
  term.
\newblock {\em Phys. Rev.}, D64:065011, 2001.

\bibitem{Chodos:1999zt}
Alan Chodos and Erich Poppitz.
\newblock Warp factors and extended sources in two transverse dimensions.
\newblock {\em Phys. Lett.}, B471:119--127, 1999.

\bibitem{Arkani-Hamed:1998rs}
Nima Arkani-Hamed, Savas Dimopoulos, and Gia Dvali.
\newblock The hierarchy problem and new dimensions at a millimeter.
\newblock {\em Phys. Lett.}, B429:263--272, 1998.

\bibitem{Arkani-Hamed:1998nn}
Nima Arkani-Hamed, Savas Dimopoulos, and G.~R. Dvali.
\newblock Phenomenology, astrophysics and cosmology of theories with
  sub-millimeter dimensions and tev scale quantum gravity.
\newblock {\em Phys. Rev.}, D59:086004, 1999.

\bibitem{Antoniadis:1998ig}
Ignatios Antoniadis, Nima Arkani-Hamed, Savas Dimopoulos, and Gia Dvali.
\newblock New dimensions at a millimeter to a fermi and superstrings at a tev.
\newblock {\em Phys. Lett.}, B436:257--263, 1998.

\bibitem{Gogberashvili:1998vx}
Merab Gogberashvili.
\newblock Hierarchy problem in the shell-universe model.
\newblock 1998.

\bibitem{Randall:1999ee}
Lisa Randall and Raman Sundrum.
\newblock A large mass hierarchy from a small extra dimension.
\newblock {\em Phys. Rev. Lett.}, 83:3370--3373, 1999.

\bibitem{Randall:1999vf}
Lisa Randall and Raman Sundrum.
\newblock An alternative to compactification.
\newblock {\em Phys. Rev. Lett.}, 83:4690--4693, 1999.

\bibitem{Kogan:2001wp}
Ian~I. Kogan, Stavros Mouslopoulos, Antonios Papazoglou, and Graham~G. Ross.
\newblock Multi-localization in multi-brane worlds.
\newblock {\em Nucl. Phys.}, B615:191--218, 2001.

\bibitem{Chang:1999nh}
Sanghyeon Chang, Junji Hisano, Hiroaki Nakano, Nobuchika Okada, and Masahiro
  Yamaguchi.
\newblock Bulk standard model in the randall-sundrum background.
\newblock {\em Phys. Rev.}, D62:084025, 2000.

\bibitem{Hatanaka:1999ac}
Hisaki Hatanaka, Makoto Sakamoto, Motoi Tachibana, and Kazunori Takenaga.
\newblock Many-brane extension of the randall-sundrum solution.
\newblock {\em Prog. Theor. Phys.}, 102:1213, 1999.

\bibitem{Grossman:1999ra}
Yuval Grossman and Matthias Neubert.
\newblock Neutrino masses and mixings in non-factorizable geometry.
\newblock {\em Phys. Lett.}, B474:361--371, 2000.

\end{thebibliography}



\end{document}

















toolj.bbl
\begin{thebibliography}{10}

\bibitem{Weinberg:1977hb}
Steven Weinberg.
\newblock The problem of mass.
\newblock {\em Trans. New York Acad. Sci.}, 38:185--201, 1977.

\bibitem{Wilczek:1977uh}
Frank Wilczek and A.~Zee.
\newblock Discrete flavor symmetries and formula for the cabibbo angle.
\newblock {\em Phys. Lett.}, B70:418, 1977.

\bibitem{Rothman:1979ft}
Arthur~C. Rothman and Kyungsik Kang.
\newblock Calculability: The mass matrix and cp violation.
\newblock {\em Phys. Rev. Lett.}, 43:1548, 1979.

\bibitem{Kang:1981yg}
Kyungsik Kang and Arthur~C. Rothman.
\newblock Generalized mixing angles in gauge theories with natural flavor
  conservation.
\newblock {\em Phys. Rev.}, D24:167, 1981.

\bibitem{Fritzsch:1978vd}
H.~Fritzsch.
\newblock Weak interaction mixing in the six - quark theory.
\newblock {\em Phys. Lett.}, B73:317--322, 1978.

\bibitem{Fritzsch:1979zq}
Harald Fritzsch.
\newblock Quark masses and flavor mixing.
\newblock {\em Nucl. Phys.}, B155:189, 1979.

\bibitem{DeRujula:1977ry}
A.~De Rujula, Howard Georgi, and S.~L. Glashow.
\newblock A theory of flavor mixing.
\newblock {\em Ann. Phys.}, 109:258, 1977.

\bibitem{Georgi:1979dq}
Howard Georgi and D.~V. Nanopoulos.
\newblock Ordinary predictions from grand principles: t quark mass in o(10).
\newblock {\em Nucl. Phys.}, B155:52, 1979.

\bibitem{Kang:1997uv}
Kyungsik Kang and Sin~Kyu Kang.
\newblock New class of quark mass matrix and calculability of flavor mixing
  matrix.
\newblock {\em Phys. Rev.}, D56:1511--1514, 1997.

\bibitem{Thirring:1972de}
W.~E. Thirring.
\newblock Fivedimensional theories and cp-violation.
\newblock {\em Acta Phys. Austriaca Suppl.}, 9:256--271, 1972.

\bibitem{Casadio:2001fe}
R.~Casadio and A.~Gruppuso.
\newblock Discrete symmetries and localization in a brane-world.
\newblock {\em Phys. Rev.}, D64:025020, 2001.

\bibitem{Chang:2001uk}
Darwin Chang, Wai-Yee Keung, and Rabindra~N. Mohapatra.
\newblock Models for geometric cp violation with extra dimensions.
\newblock {\em Phys. Lett.}, B515:431--441, 2001.

\bibitem{Chang:2001yn}
Darwin Chang and R.~N. Mohapatra.
\newblock Geometric cp violation with extra dimensions.
\newblock {\em Phys. Rev. Lett.}, 87:211601, 2001.

\bibitem{Huang:2001np}
Chao-Shang Huang, Tian-jun Li, Wei Liao, and Qi-Shu Yan.
\newblock Cp violation and extra dimensions. \it hep-ph/0101002, \rm
\newblock 2001.

\bibitem{Branco:2000rb}
G.~C. Branco, Andre de~Gouvea, and M.~N. Rebelo.
\newblock Split fermions in extra dimensions and cp violation.
\newblock {\em Phys. Lett.}, B506:115--122, 2001.

\bibitem{Sakamura:2000ik}
Yutaka Sakamura.
\newblock Our wall as the origin of cp violation.
\newblock{\em hep-th/0011098}, 2000.

\bibitem{Sakamura:1999fa}
Yutaka Sakamura.
\newblock Spontaneous cp violation in large extra dimensions.
\newblock {\em hep-ph/9909454},  1999.

\bibitem{Hung:2001hw}
P.~Q. Hung and M.~Seco.
\newblock Pure phase mass matrices from six dimensions.
\newblock {\em hep-ph/0111013}, 2001.

\bibitem{Huber:2000ie}
Stephan~J. Huber and Qaisar Shafi.
\newblock Fermion masses, mixings and proton decay in a randall- sundrum model.
\newblock {\em Phys. Lett.}, B498:256--262, 2001.

\bibitem{Arkani-Hamed:1999dc}
Nima Arkani-Hamed and Martin Schmaltz.
\newblock Hierarchies without symmetries from extra dimensions.
\newblock {\em Phys. Rev.}, D61:033005, 2000.

\bibitem{Mouslopoulos:2001uc}
Stavros Mouslopoulos.
\newblock Bulk fermions in multi-brane worlds.
\newblock {\em JHEP}, 05:038, 2001.

\bibitem{Kaplanet:2001}
David E. Kaplan and Tim M.P. Tait
\newblock New tools for fermion masses from extra dimensions.
\newblock {\em JHEP}, 051:0111, 2001.

\bibitem{Chodos:1999zt}
Alan Chodos and Erich Poppitz.
\newblock Warp factors and extended sources in two transverse dimensions.
\newblock {\em Phys. Lett.}, B471:119--127, 1999.

\bibitem{Gherghettaet:2000}
Tony Gherghetta and Mikhail Shaposhnikov.
\newblock Localizing gravity on a string-like defect in six dimensions.
\newblock {\em Phys. Rev. Lett.}, 85,  240-243, 2000.

\bibitem{Collins:2001ni}
Hael Collins and Bob Holdom.
\newblock The randall-sundrum scenario with an extra warped dimension.
\newblock {\em Phys. Rev.}, D64:064003, 2001.

\bibitem{Kogan:2001yr}
Ian~I. Kogan, Stavros Mouslopoulos, Antonios Papazoglou, and Graham~G. Ross.
\newblock Multigravity in six dimensions: Generating bounces with flat positive
  tension branes.
\newblock {\em Phys. Rev.}, D64:124014, 2001.

\bibitem{Burgess:2001bn}
C.~P. Burgess, James~M. Cline, Neil~R. Constable, and Hassan Firouzjahi.
\newblock Dynamical stability of six-dimensional warped brane- worlds.
\newblock {\em JHEP} 0201:014, 2002.

\bibitem{Kim:2001rm}
Jihn~E. Kim, Bumseok Kyae, and Hyun~Min Lee.
\newblock Localized gravity and mass hierarchy in d = 6 with gauss- bonnet
  term.
\newblock {\em Phys. Rev.}, D64:065011, 2001.

\bibitem{Arkani-Hamed:1998rs}
Nima Arkani-Hamed, Savas Dimopoulos, and G.~R. Dvali.
\newblock The hierarchy problem and new dimensions at a millimeter.
\newblock {\em Phys. Lett.}, B429:263--272, 1998.

\bibitem{Arkani-Hamed:1998nn}
Nima Arkani-Hamed, Savas Dimopoulos, and G.~R. Dvali.
\newblock Phenomenology, astrophysics and cosmology of theories with
  sub-millimeter dimensions and tev scale quantum gravity.
\newblock {\em Phys. Rev.}, D59:086004, 1999.

\bibitem{Antoniadis:1998ig}
Ignatios Antoniadis, Nima Arkani-Hamed, Savas Dimopoulos, and G.~R. Dvali.
\newblock New dimensions at a millimeter to a fermi and superstrings at a tev.
\newblock {\em Phys. Lett.}, B436:257--263, 1998.

\bibitem{Gogberashvili:1998vx}
Merab Gogberashvili.
\newblock Hierarchy problem in the shell-universe model.
\newblock {\em hep-ph/9812296}, 1998.

\bibitem{Randall:1999ee}
Lisa Randall and Raman Sundrum.
\newblock A large mass hierarchy from a small extra dimension.
\newblock {\em Phys. Rev. Lett.}, 83:3370--3373, 1999.

\bibitem{Randall:1999vf}
Lisa Randall and Raman Sundrum.
\newblock An alternative to compactification.
\newblock {\em Phys. Rev. Lett.}, 83:4690--4693, 1999.

\bibitem{Chang:1999nh}
Sanghyeon Chang, Junji Hisano, Hiroaki Nakano, Nobuchika Okada, and Masahiro
  Yamaguchi.
\newblock Bulk standard model in the randall-sundrum background.
\newblock {\em Phys. Rev.}, D62:084025, 2000.

\bibitem{Kogan:2001wp}
Ian~I. Kogan, Stavros Mouslopoulos, Antonios Papazoglou, and Graham~G. Ross.
\newblock Multi-localization in multi-brane worlds.
\newblock {\em Nucl. Phys.}, B615:191--218, 2001.

\bibitem{Hatanaka:1999ac}
Hisaki Hatanaka, Makoto Sakamoto, Motoi Tachibana, and Kazunori Takenaga.
\newblock Many-brane extension of the randall-sundrum solution.
\newblock {\em Prog. Theor. Phys.}, 102:1213--1218, 1999.

\bibitem{Grossman:1999ra}
Yuval Grossman and Matthias Neubert.
\newblock Neutrino masses and mixings in non-factorizable geometry.
\newblock {\em Phys. Lett.}, B474:361--371, 2000.

\end{thebibliography}
