% Selecting Gauge Theories on an Interval \\ by 5D %Five-dimensional Gauge Transformations
\abst{% Gauge symmetry breaking by boundary conditions is studied in a general warped geometry in five dimensions. It has been %advocated suggested that a wider class of boundary conditions %are is allowed by requiring only vanishing surface terms %to when deriving %e the field equations for gauge theories %in on an interval (i.e., employing a variational principle), %compared in comparison to the twist in orbifolding with automorphisms of the Lie algebra. We find that there are classes of boundary conditions allowed by the variational principle which violate the Ward-Takahashi identity and give %the four-point tree amplitudes %growing that increase with energy in channels %so far that have not yet been %un explored, leading to cross sections %increasing that increase as powers of the energy (%the violation of which violates the tree level unitarity). We also find that such boundary conditions are forbidden by %requiring the consistency of defining the requirement that the definitions of the restricted class of five-dimensional (5D) gauge transformations be consistent. } %
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%\pubinfo{Vol.~11X, No.~X, Mmmmm YYYY}%Editorial Office will fill in this.
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%%%%% PREPRINT NUMBERS %%%%%%
%\begin{flushright}
%TIT/HEP-552 \\ 
%hep-th/0604121\\
%April, 2006 \\
%\end{flushright}
\preprintnumber{%
IT/HEP-552 \\
hep-th/0604121 \\
April, 2006 \\
}
%\vspace*{2cm}
\markboth{Norisuke Sakai and Nobuhiro Uekusa}{Selecting Gauge Theories on an Interval by 5D %Five-dimensional 
Gauge Transformations}
%%%%%%%%%%%%%%%%%%% TITLE %%%%%%%%%%%%%%%%%%
%\begin{center}{\large\bf 
%Selecting Gauge Theories on an Interval \\
%by 5D Gauge Transformations 
%}\end{center}
\title{%
Selecting Gauge Theories on an Interval \\
by 5D %Five-dimensional 
Gauge Transformations
}
%%%%%%%%%%%%%%%% AUTHORS %%%%%%%%%%%%%%%%%%%%%%%
%\vspace{1cm}
%\begin{center}
%{\bf Norisuke Sakai}
%\footnote{E-mail: nsakai@th.phys.titech.ac.jp}
%and
%{\bf Nobuhiro Uekusa}
%\footnote{E-mail: uekusa@th.phys.titech.ac.jp}
%\end{center}
\author{%
Norisuke \textsc{Sakai}
\footnote{E-mail: nsakai@th.phys.titech.ac.jp}
and
Nobuhiro \textsc{Uekusa}
\footnote{E-mail: uekusa@th.phys.titech.ac.jp}
}
%%%%%%%%%%%%%%%%%%%%%%% AFFILIATION %%%%%%%%%%%%
%\vspace{0.2cm}
%\begin{center}
%%\small
%{\it Department of Physics, Tokyo Institute of% Technology, 
%Tokyo 152-8551, Japan}
%%%%%%
%%%%%%%%
%\end{center}
\inst{%
Department of Physics, Tokyo Institute of Technology, 
Tokyo 152-8551, Japan
}
%\publishedin{%         %Write this ONLY in cases of addenda and errata
%Prog.~Theor.~Phys.\ \textbf{XX} (19YY), page.}

%\recdate{Mmmmm DD, YYYY}%            %Editorial Office will fill in
%this.
\recdate{March 9, 2007}%            %Editorial Office will fill in
%this.
%%%%%%%%%%%%%%%%%% ABSTRACT %%%%%%%%%%%%%%%
%\vspace{1cm}
%\begin{abstract}
\abst{%
Gauge symmetry breaking by boundary conditions is studied 
in a general warped geometry in five dimensions. 
It has been %advocated 
suggested
that a wider class 
of boundary 
conditions %are 
is
allowed by requiring only vanishing surface 
terms %to 
when deriving %e 
the field equations for gauge theories %in
on 
an interval (i.e., employing a 
variational principle), %compared
in comparison
to the twist 
in orbifolding with automorphisms of the Lie algebra. 
We find that there are classes of boundary conditions 
allowed by the variational principle which violate the 
Ward-Takahashi identity and give %the 
four-point tree 
amplitudes %growing 
that increase 
with energy in channels %so far 
that have not yet been 
%un
explored, leading to cross sections %increasing 
that increase
as 
powers of the
energy (%the violation of 
which violates
the tree level 
unitarity). 
We also find that such boundary conditions are forbidden 
by %requiring the consistency of defining 
the requirement that the definitions of 
the restricted
class of five-dimensional (5D) 
gauge transformations
be consistent. 
}

%\end{abstract}
%\end{titlepage}
%%%%%%%%%%
%\tableofcontents
%%%%%%%%%%%%
%\newpage

\begin{document}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%

In models with extra dimensions, there are various 
possibilities for gauge fields. 
The %I
initial proposal for the large extra dimensions assumes 
that all particles in the standard model are localized 
on a brane with 
a
four-dimensional (4D) 
world volume. 
\cite{Arkani-Hamed:1998rs}${}^{-}$\cite{Antoniadis:1990ew}
Formulating the 
%L
localization of gauge fields on a wall is a challenging 
problem, which can be %achieved 
realized in certain models. 
\cite{Dvali:1996xe}${}^{,}$ 
\cite{Isozumi:2003uh} 
However, other interesting possibilities arise if the 
gauge fields are propagating in the higher-dimensional 
bulk spacetime. 
The extra-dimensional component of gauge fields can act 
as a Higgs scalar field to break the gauge symmetry 
\cite{Hatanaka:1998yp}. 
The Wilson line dynamics can provide another source of 
gauge symmetry breaking, namely, the Hosotani mechanism 
\cite{Hosotani:1983xw}. 
If the extra dimensions %is
are compactified on a topologically 
nontrivial manifold, such as $S^1$, %the 
twisting can be 
%accommodated  
realized, 
and
%to give 
the Scherk-Schwarz symmetry breaking 
mechanism thereby appears \cite{Scherk:1979zr}. 
%These
The key %of 
to these mechanisms 
can be summarized as a nontrivial holonomy along 
a nontrivial cycle, %circle, 
which can also be understood as 
vacuum expectation values of adjoint scalar fields coming 
from gauge field components along the extra dimensions.
%\cite{Hosotani:1983xw}. 
If orbifolds are introduced, one can also impose 
boundary conditions at the fixed points of the orbifold 
to break %gauge symmetry,
all or part of a gauge group,
usually using the automorphisms 
of the Lie algebra. 
\cite{Kawamura:1999nj}${}^{-}$\cite{Quiros:2003gg}
Combined with the Wilson lines, the orbifold models have 
recently gained much attention. 
\cite{Hall:2001tn}${}^{-}$\cite{Hosotani:2006qp}
A %wider 
class of boundary conditions 
wider
than 
the orbifolding with automorphisms 
has been pursued to obtain more realistic 
models, in particular, to reduce the rank of the gauge 
group \cite{Hebecker:2001jb}. 
One notable proposal %was
is 
to consider gauge theories on 
an interval and to require that the surface terms must 
vanish in order for the variational principle 
 \cite{Csaki:2003dt} to give 
field equations.\footnote{
Alternatively, the hermiticity of the supercharges of %the 
supersymmetric quantum mechanics has also been  proposed 
to determine the boundary conditions \cite{Nagasawa:2004xk}. 
If applied merely at the quadratic level, as in 
Ref.%\cite{Nagasawa:2004xk}, 
22),
it also allows the undesirable boundary 
conditions that we found to violate the Ward-Takahashi 
identity and %the 
tree level unitarity. 
However, imposing %the conditions 
this condition of hermiticity
at the nonlinear level, 
one finds that %their condition 
it
eventually leads %to the 
%same 
a
condition %as 
that is identical to
our condition of the restricted class of 5D 
gauge transformations.
} 
By imposing boundary conditions, part of 
the
five-dimensional 
(5D) gauge 
invariance is explicitly broken, although the 5D 
%five-dimensional 
gauge invariance is %valid 
preserved
in the bulk. 
It is %a common wisdom in %four-dimensions 
commonly believed that
in 4D,
%that %the 
gauge 
invariance is %vital to assure 
%necessary 
vital
to guarantee
the Ward-Takahashi 
identity and %the 
unitarity. 
Therefore it is %of great interest to know 
important to determine
whether or not 
the wider %varieties 
range
of boundary conditions allowed by 
the variational principle %have troubles in these aspects. 
introduces difficulties in regard to these matters.
It should also be useful to examine if 
a
certain restricted 
class of 5D %five-dimensional 
gauge transformations (%defined by 
%certain 
%boundary conditions on gauge transformation functions) 
with gauge transformation functions restricted by 
certain boundary conditions)
can 
be consistently defined as %invariances 
the 
invariance of the theories with 
%such boundary conditions. 
boundaries.
%The 5D gauge transofrmations broken only at 
%boundaries are sometimes called the restricted 5D gauge 
%transformations. 
%Namely the gauge transformation functions are 
%well-defined if we ignore boundaries. 
%We will adopt the terminology to distinguish it from 
%the full 5D or 4D gauge transformations. 

Higgs bosons are needed in %four dimensions 
4D
to cancel 
terms %growing 
that increase
with energy in 
the
scattering amplitudes of 
longitudinal massive gauge bosons \cite{LlewellynSmith:1973ey}. 
The unitarity relation %evaluated 
between %the same states 
$\langle n \vert$ and $\vert n \rangle$ 
corresponding to a single state,
\begin{equation}
-i\langle n \vert (T-T^\dagger)\vert n \rangle =
\sum_m \langle n \vert T^\dagger \vert m \rangle 
\langle m \vert T \vert n \rangle , 
\label{eq:unitarity}
\end{equation}
gives the imaginary part of the elastic 
scattering amplitude as a sum of cross sections of 
various channels, including elastic ($m=n$) %as well as 
and
inelastic ones ($m\not=n$). 
Any two-to-two elastic or inelastic scattering amplitudes 
%growing 
that increase
with energy lead %s 
to %a growing 
an increasing
elastic 
scattering amplitude, 
%since 
because the
contributions of each channel in 
Eq.(\ref{eq:unitarity}) %is 
are
nonnegative. 
In %that 
this
case, the unitarity bound for the 
elastic scattering amplitude is violated. 
This is called %the 
tree level unitarity, which can be 
tested even in higher-dimensional gauge 
theories. %\cite{SekharChivukula:2001hz}. 
It has been shown that scattering amplitudes 
%growing 
that increase
with energy are canceled by the 
exchange of 
Kaluza-Klein (KK) gauge bosons in the higher-dimensional 
gauge theories compactified on the torus $S^1$, 
and thus the tree level unitarity 
is maintained, in spite of %the 
absence of the explicit 
Higgs scalars \cite{SekharChivukula:2001hz}. 
Even with the wider class 
of boundary conditions allowed 
by the variational principle, 
it has been shown that 
the contributions of the 
elastic scattering of longitudinal massive gauge bosons %at 
%the identical 
in the same
KK excitation level do not 
%grow 
increase
with energy. 
Therefore, these boundary conditions %have passed 
pass
the 
consistency test of the tree level unitarity, at least 
for the contributions from %the 
elastic scattering 
\cite{Csaki:2003dt}. 


The purpose of %our 
this
paper is to point out that 
the variational principle allows certain 
simple classes of boundary conditions which violate 
the Ward-Takahashi identity %as well as 
and
the tree level unitarity.\footnote{
The %A
authors 
have
learned that Masaharu Tanabashi also knew of 
examples of boundary conditions which are allowed 
by the variational principle, and 
violate the equivalence theorem, and consequently 
tree level unitarity. 
} 
%Contrary to previous works, 
In contrast to %the 
previous works
\cite{Abe:2003vg,Csaki:2003dt,SekharChivukula:2001hz,Ohl:2003dp,Muck:2004br},
we compute scattering amplitudes 
in various channels, including the massless modes %as well as 
and
the KK modes at various levels, %especially 
in particular
inelastic scattering amplitudes involving different 
excitation levels. 
We %will 
also show that the %consistency of defining
condition that the definition of 
%boundary conditions for 5D %five-dimensional 
%gauge transformation functions 
%for the restricted class of 5D %five-dimensional 
%gauge transformations forbids 
%such boundary conditions. 
the restricted class of 5D gauge transformations be consistent
forbids such classes of boundary conditions.
%show that the consistency 
%of defining 5D gauge transformations ensures that 
%the boundary conditions to break gauge symmetry is 
%consistent, namely does not violate the Ward-Takahashi 
%identity and the tree level unitarity. 
%We find that 


The  5D %five-dimensional 
gauge transformations have been examined by 
consistently gauging the global symmetry 
\cite{Hall:2003yc}. 
In %four dimensions, 
4D, 
BRST invariance is useful 
as a sophisticated formulation of gauge theories 
after the gauge fixing 
\cite{Becchi:1974md, Kugo:1977zq, Kugo:1981hm}. 
The BRST approach for higher-dimensional gauge theories 
has been used to %discuss 
study
the deconstruction 
approach \cite{He:2004zr} %, 
and orbifold models. 
\cite{Hall:2003yc}%{Muck:2001yv}
${}^{-}$\cite{Muck:2004br} 
We find that %the consistency of defining gauge 
%transformations to select boundary conditions 
%can be rewritten equally well in terms 
%of the BRST formulation.
the gauge transformations to select boundary conditions can
be consistently defined in terms of
the BRST formulation as well.\footnote{
%Since 
Because
the problem of defining the 
gauge transformations %occurs 
arises
at the tree level rather 
than %at 
the loop level, 
using the BRST formulation in selecting the 
boundary conditions is %just 
equivalent to using 
the classical gauge transformations. 
}


In %Sec.
\S\ref{sc:bound_cond_var_pr}, 
the variational principle is briefly reviewed. 
In %Sec.
\S\ref{sc:scatt_amp}, we compute the scattering 
amplitudes and find 
that
%possible 
terms 
%boundary contributions 
%violating 
that violate
the Ward-Takahashi identity 
and the tree level unitarity 
can be reduced to %a 
functions of the values of the mode 
functions at the boundaries. 
In %Sec.
\S\ref{sc:bound_cond_WT}, we demonstrate that the 
variational principle allows %the 
boundary conditions 
%violating 
that violate
both the 
Ward-Takahashi identity and %the 
unitarity. 
We %call 
refer to
%the 
such
boundary conditions as the 
coset-N/subgroup-D boundary conditions. 
In %Sec.
\S\ref{sc:brst}, we show that the 
coset-N/subgroup-D boundary conditions do %es 
not allow 
a consistent definition of 
a restricted class of 5D gauge transformations. 
We also %mention 
discuss
the BRST formulation. 
Useful formulas for mode functions are summarized in 
Appendix \ref{Ap:KK}. 
Some details of the 
computation of scattering amplitudes are given in 
Appendix \ref{Ap:amp}. 


%%%%%%%%%%%%%%%%%%%%
\section{Variational principle and scattering amplitudes}
\label{sc:var_pr_sct}
%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%
\subsection{Boundary conditions obtained from the variational principle}
\label{sc:bound_cond_var_pr}
%%%%%%%%%%%%%%%%%%%%

To a good approximation, our 4D spacetime is flat 
(except for a %tiny 
very small positive cosmological constant). 
Assuming %the 
flat 4D %four-dimensional 
slices in 5D %five-dimensional 
spacetime, we parameterize the generic metric in 
terms of two\footnote{
%One can reparametrize t
The extra-dimensional coordinate $y$ can be reparameterized 
%in such a way to make 
such that
either one of the two positive 
functions $e^{-4W(y)}$ or $g_{55}(y)$ %to a 
is
constant; %, 
%say 
for example, we could choose
$g_{55}(y)=1$. 
%We 
However, we leave these two functions %as a flexible parameterization
unfixed, in order to 
to accommodate various coordinate systems. 
} arbitrary functions, $W(y)$ and $g_{55}(y)$, 
of the extra-dimensional coordinate $x^5 \equiv y$: 
\begin{eqnarray}
 ds^2=g_{MN}dx^Mdx^N=
e^{-4W(y)}\eta_{\mu\nu}dx^\mu dx^\nu+g_{55}(y)dy^2 .  %,
\end{eqnarray}
%where 
Here,
the %capital 
upper-case
Latin indices $M,N,\cdots=0, 1, 2, 3, 5$ 
run over the five spacetime dimensions, $g_{MN}$ is the 
5D %five-dimensional 
metric, and the 4D %four-dimensional 
spacetime is flat: $\eta_{\mu\nu}=\textrm{diag}(-,+,+,+)$ 
with $\mu, \nu =0, 1,2,3$. 
We assume the extra dimension to %be 
consistent of
an interval, 
$0 \le y \le \pi R$, and consider appropriate boundary 
conditions at $y=0$ and $y=\pi R$.  

As a simple illustrative example, we %take the 
consider
pure gauge 
theory, and choose the $SU(N)$ gauge group %if 
in the case that
we need to 
specify the gauge group. 
Introducing the gauge fixing term with the parameter $\xi$, 
we obtain the action in the $R_\xi$ gauge as
%
\begin{eqnarray}
        \label{eq:lagr}
\mathcal{S}+\mathcal{S}_{\rm GF}=
\int d^4x\,\int_0^{\pi R} dy \sqrt{-g(y)}\left(
-\frac{1}{4} F_{MN}^a F_{PQ}^a g^{MP}g^{NQ}
-\frac{1}{2\xi} (G^a{})^2\right). 
\end{eqnarray}
%
The field strength is given by 
$F^a_{MN}=\partial_M A_N^a -  \partial_N A_M^a
+g_5 f^{abc}\, A^b_M A^c_N$, where $f^{abc}$ 
is the structure constant of the gauge group, and $g_5$ 
is the 5D gauge coupling. 
We %will 
choose the gauge fixing function $G^a$ 
%in order 
so as
to cancel the cross terms 
between the 4D components $A_\mu^a$ and the 
extra-dimensional component $A_5^a$ 
(4D scalar): \cite{Muck:2001yv} 
\begin{eqnarray}
 G^a=e^{4W}\left(\eta^{\mu\nu}\partial_\mu A_\nu^a
   +\xi {1\over \sqrt{g_{55}}}
\partial_5 {e^{-4W}\over \sqrt{g_{55}}}
    A_5^a\right) . 
\end{eqnarray}
We omit %to write 
the corresponding ghost fields,
%since 
as
they do not 
appear in tree level scattering amplitudes. 


Let us briefly review the variational principle used to 
determine the boundary conditions \cite{Csaki:2003dt}. 
The action~(\ref{eq:lagr}) should %reach a 
realize its
minimum with 
respect to variation %s around 
about
the configuration satisfying 
the field equations in the bulk. 
To %arrive at 
obtain
the field equations, %one needs to 
we must 
perform 
integrations by parts, %resulting 
which results
in the boundary terms 
\begin{eqnarray}
(\delta \mathcal{S}
+\delta \mathcal{S}_{\rm GF})_{\textrm{\scriptsize boundary}}  = 
-\int d^4x \left[ {e^{-4W}\over \sqrt{g_{55}}}
  \eta^{\mu\nu} F^{a}_{5 \mu} \, \delta A_{\nu}^a  
 +{e^{-8W}\over \sqrt{g_{55}}}G^a\delta A_5^a 
 \right]_{y=0}^{y=\pi R} . 
\end{eqnarray}
These boundary terms %are required to 
must
vanish 
for the variational principle to be well-defined.
The %S
simplest way to satisfy this condition is to use 
the Neumann, $\partial_5 \left. A_{\mu}^a\right|=0$ 
($\partial_5 {e^{-4W}\over \sqrt{g_{55}}} \left. A_5^a\right|=0$), 
or 
Dirichlet, $\left. A_{\mu, 5}^a\right|=0$, boundary 
conditions. 
It has been found that the 
following three %patterns 
forms
of the Neumann and/or 
Dirichlet boundary conditions %are available as 
provide
the solution: 
 \cite{Csaki:2003dt} 
\begin{eqnarray}
&& \left. A_\mu^a\right|=0, \qquad \left. A_5^a\right|=0, 
\label{eq:var_bnd_cod1}
%\nonumber 
\\
&& \left. A_\mu^a\right|=0, \qquad 
\partial_5 {e^{-4W}\over \sqrt{g_{55}}} \left. A_5^a\right|=0, 
\label{eq:var_bnd_cod2}
%\nonumber 
\\
&& \partial_5 \left. A_\mu^a\right|=0, \qquad 
\left. A_5^a\right|=0. 
\label{eq:var_bnd_cod}
\end{eqnarray}
The Neumann conditions $\partial_5 \left. A_\mu^a\right|=0$ 
for the 4D %four-dimensional 
vector $A^a_{\mu}$ at both 
boundaries, 
%$y=0,\pi R$
$y=0$ and $\pi R$,
are necessary for the existence of a zero mode 
%(the unbroken gauge symmetry) 
with %that gauge generator 
the color
$a$ 
in the low-energy effective theory. 
It %has also been pointed out 
is also known
that the choice of these three 
%patterns 
types
of boundary conditions can be made independently 
for each %gauge generator 
color
$a$ and for the 4D %four-dimensional 
vector $A^a_{\mu}$ and the 4D %four-dimensional 
scalar $A^a_5$. 
The 4D Lorentz invariance also %dictates 
implies that the 
gauge generators of the 4D 
massless gauge fields 
$A^a_{\mu}$ 
%should 
form a group, as argued in Ref.%\cite{Csaki:2003dt}
21):
\begin{eqnarray}
\partial_5 \left. A_\mu^a\right|=0, \qquad 
a \in H \subseteq G , 
\qquad 
y=0, \pi R ,
\label{eq:gr_bnd_cod}
\end{eqnarray}
where $H$ is a (sub)group %in 
of the gauge group $G$. 
%Let us note that the 
%Neumann boundary condition should be imposed at both 
%boundaries $y=0, \pi R$ for massless 
%%unbroken 
%gauge fields.
%%generators $a$. 


In the case of the second 
choice of the boundary conditions,
(\ref{eq:var_bnd_cod2}), and the third choice, 
(\ref{eq:var_bnd_cod}), %choices, 
mode functions 
for $A_\mu$ and $A_5$ satisfy simple relations, such as 
(\ref{mtb}) in Appendix \ref{Ap:KK}. %, 
%which 
This simplicity
plays an important role 
in %discussing 
the investigation of
possible violations of the Ward-Takahashi 
identity and unitarity. 
In the case of the first choice, (\ref{eq:var_bnd_cod1}), 
the mode functions for $A_\mu$ and $A_5$ satisfy a different 
relation, and %requires 
this results in the necessity of
a separate treatment. 
Therefore, we %will 
consider the second 
%(\ref{eq:var_bnd_cod2}) 
and %the 
third 
%(\ref{eq:var_bnd_cod}) 
choices, (\ref{eq:var_bnd_cod2}) and (\ref{eq:var_bnd_cod}),
in this section and 
discuss the first choice, (\ref{eq:var_bnd_cod1}), only 
briefly in %sect.
%Sec.
\S\ref{sc:brst}. 
We %will %only 
denote the Neumann or 
Dirichlet boundary conditions for the 4D %four-dimensional 
vector $A_\mu^a$ at each %boundaries 
boundary with the %gauge 
%group generator 
color
$a$ as $D(a)$. 
(%t
The opposite 
boundary conditions for the scalar $A_5^a$ are implied.) %. 
%By u
Using these boundary conditions, %one
we can obtain the 
$n$-th mode functions $f_n^{D(a)}(y)$ %(
and $g_n^{D(a)}(y)$ %) 
for the vector $A_{\mu}^a(x,y)$ %(
and scalar  $A_{5}^a(x,y)$, respectively, %) 
and can decompose %it 
them into the KK effective 
4D %four-dimensional 
fields $A_{\mu n}^a(x)$ %(
and $A_{5 n}^a(x)$ as %) 
\begin{eqnarray}
 A_{\mu}^a(x,y)&=&\sum_{n=0}^{\infty} A_{\mu n}^a(x) f_n^{D(a)}(y), 
\label{eq:kk_vector}
\\ 
A_5^a(x,y)&=&\sum_{n=0}^{\infty} A_{5 n}^a(x) g_n^{D(a)}(y) .
\label{eq:kk_scalar}
\end{eqnarray}
Important properties of the  mode functions are given in 
Appendix~\ref{Ap:KK}. 

%If we consider each gauge generators $a$ 
%separately, we find that the second (\ref{eq:var_bnd_cod2}) 
%and the third (\ref{eq:var_bnd_cod}) choices are 
%consistent with the orbifold projection under the 
%reflection $Z_2$: 
%$y\to -y$. 
%On the other hand, the first choice (\ref{eq:var_bnd_cod1}) 
%contradicts with the orbifoldings. 


%%%%%%%%%%%%%%%%%%%%
\subsection{Scattering amplitudes}
\label{sc:scatt_amp}
%%%%%%%%%%%%%%%%%%%%

Now, we %would like to 
examine scattering amplitudes 
%when 
for various choices of
%the 
Dirichlet %or 
and/or
Neumann boundary conditions 
%are variously chosen 
at each boundary and for each color.
We consider the gauge boson scattering 
$A_n^a A_m^b\to A_l^c A_m^d$, 
where the subscripts $l, n$ and $m$ %are 
indicate the
KK levels, 
and the superscripts $a,b,c$ and $d$ indicate the colors, 
as illustrated in Fig.~\ref{ampfig}. 
For %technical 
simplicity, we choose the KK level and boundary 
conditions for the gauge bosons with the %generators 
%color
colors
$b$ and $d$ 
%to be 
identical. 
In the center-of-mass frame,
the scattering angle and the total energy are denoted as $\theta$ and 
$E\equiv E_n+E_m$, respectively.
To examine the possible growth of scattering amplitudes at a
fixed scattering angle as the total energy $E$ %grows,
increase, 
we choose polarization vectors $\epsilon$ 
for gauge bosons to be longitudinal, except for the 
gauge boson $A_l^c$, as tabulated in Table~\ref{Tab:kin}. %\ref{ampfig}. 
%
\begin{figure}[b]
\begin{center}
\includegraphics{nmlm.eps}
\caption{$A_n^a A_m^b\to A_l^c A_m^d$ \label{ampfig}}
\end{center}
\end{figure}
%
We %will 
examine both longitudinal and transverse 
polarizations and massive as well as massless modes 
for $A_l^c$. 
%
%\begin{figure}[h]
%\begin{center}
%\includegraphics{nmlm.eps}
%\caption{$A_n^a A_m^b\to A_l^c A_m^d$ \label{ampfig}}
%\end{center}
%\end{figure}
%
%
\begin{table}[h]
\caption{Kinematics: The energy of each mode  
is given %as %$E_n=\sqrt{p^2+m_n^2}$.
by $E_n=\sqrt{p^2+M_n^2}$. 
Both the longitudinal and transverse polarizations 
$\epsilon(p_3)$ are explored for the gauge 
field $A_l^c$.\label{Tab:kin}}
\begin{tabular}{ll}\hline\hline
 $p_1=(E_n,0,0,p)$  &
           %$\epsilon(p_1)=(p/m_n,0,0,E_n/m_n)$
        $\epsilon(p_1)=(p/M_n,0,0,E_n/M_n)$ \\
 $p_2=(E_m,0,0,-p)$  &
           %$\epsilon(p_2)=(p/m_m,0,0,-E_m/m_m)$
  $\epsilon(p_2)=(p/M_m,0,0,-E_m/M_m)$ \\ 
 $p_3=(E_l,p'\sin\theta,0,p'\cos\theta)$  &
           %$\epsilon(p_3)$ 
  $\epsilon(p_3)=$ 
  Eq.(\ref{polart}) (for transverse) or
  Eq.(\ref{polarl}) (for longitudinal)\\
 $p_4=(E_m',-p'\sin\theta,0,-p'\cos\theta)$  &
           %$\epsilon(p_4)=(p'/m_m,-E_m'\sin\theta/m_m,0,
   %-E_m'\cos\theta/m_m)$
   $\epsilon(p_4)=(p'/M_m,-E_m'\sin\theta/M_m,0,
   -E_m'\cos\theta/M_m)$ \\
\hline 
 \end{tabular}
\end{table}
%
Some details of the scattering amplitude calculation 
with arbitrary gauge parameters $\xi$ 
are given 
in Appendix~\ref{Ap:amp}. 
To test the tree level unitarity, we study 
the scattering amplitude at %high 
%enough 
energies %where 
high enough that
the total energy $E$ is much larger 
than any mass %$m_k$ 
$M_k$
for the 
intermediate states as well as the external states. 
%With l
Leaving the polarization $\epsilon(p_3)$ for $A_l^c$ 
unspecified,  %yet, 
the terms %growing %in
that increase 
with
$E$ in the invariant matrix 
element for the gauge parameter $\xi\ll E^2/M_k^2$ 
%is 
are
given by 
\begin{eqnarray}
 && ig_5^2 f^{abe}f^{cde} {E^3\over 8M_n M_m^2}
\epsilon_{\mu}^*(p_3)
 \sum_k \Bigg\{ F_{lmk}^{D(cde)}F_{nmk}^{D(abe)}
 {\cal A}^{\mu}
\nonumber\\
&&
 +{2\over E^2} \left(M_l T_{mkl}^{D(dec)}
-M_m T_{lkm}^{D(ced)}\right)
    \left(M_n T_{mkn}^{D(bea)}-M_m T_{nkm}^{D(aeb)}\right)
      \left(0,-\sin\theta,0,1-\cos\theta\right)^{\mu}
   \Bigg\} ,
\nonumber\\   \label{invm}
\end{eqnarray}
with the four-vector ${\cal A}^{\mu}$ whose components are 
${\cal A}^{\mu=2}= 0$ and 
\begin{eqnarray}
 {\cal A}^{0}
 &\!=\!& 
1+{3M_n^2-3M_m^2+2M_l^2-4M_k^2\over E^2}
    -\left(1+{M_n^2+M_m^2\over E^2}\right)\cos\theta,
\\
 {\cal A}^{1}
 &\!=\!& 
    \left(1+{M_n^2-7M_m^2-2M_l^2+2M_k^2\over E^2}\right)
\sin\theta
\nonumber\\
&&\quad
    -\left(1+{M_n^2+M_m^2-2M_l^2\over E^2}\right)
\sin\theta\cos\theta,
%\\ {\cal A}^{2} &\!=\!&  0,
\\
 {\cal A}^{3}
 &\!=\!& 
     {2M_n^2+4M_m^2+2M_l^2-6M_k^2\over E^2}
     +\left(1+{M_n^2-7M_m^2-2M_l^2+2M_k^2\over E^2}\right)
\cos\theta
\nonumber\\
&&\quad
     -\left(1+{M_n^2+M_m^2-2M_l^2\over E^2}\right)
\cos^2\theta . 
\end{eqnarray}
Here,
%W
we have ignored terms which do not %grow 
increase
with energies 
%(
[to ${\cal O}((E/M_k)^0)$]. %). 
The overlap functions $F_{lmk}^{D(cde)}$ and 
$T_{mkl}^{D(dec)}$ are 
given in Eq.(\ref{overlap}) in Appendix \ref{Ap:KK}. 

Let us consider the case with %the
a massive mode for the 
external $A_l^c$ boson. 
%T
Then, the transverse polarization is given by 
\begin{equation}
  \epsilon^*(p_3)=(0,\cos\theta,0,-\sin\theta) 
~~\textrm{or}~~ (0,0,1,0) .
  \label{polart}
\end{equation}
The latter polarization gives a vanishing result upon 
multiplication  %with 
by ${\cal A}^\mu$. 
For the former polarization, the invariant matrix element 
(\ref{invm}) becomes 
\begin{eqnarray}
 && -ig_5^2 f^{abe}f^{cde} {E\sin\theta \over 4M_n M_m^2}{\cal K} ,
   \label{tk}
\end{eqnarray}
where
\begin{eqnarray}
 {\cal K}&=&\sum_k \Bigg\{ \left(M_n^2+2M_m^2-3M_k^2+M_l^2\right)
  F_{lmk}^{D(cde)}F_{nmk}^{D(abe)}
\nonumber\\
&&\qquad
 + \left(M_l T_{mkl}^{D(dec)}-M_m T_{lkm}^{D(ced)}\right)
    \left(M_n T_{mkn}^{D(bea)}-M_m T_{nkm}^{D(aeb)}\right)
   \Bigg\}
\nonumber\\ 
    &=&-3\sum_k \left[{e^{-4W}\over \sqrt{g_{55}}}
       g_k^{D(e)} f_m^{D(b)} f_l^{D(c)}\right]_0^{\pi R}
    \left[{e^{-4W}\over \sqrt{g_{55}}}
       g_k^{D(e)} f_m^{D(d)} f_n^{D(a)}\right]_0^{\pi R}
\nonumber\\
  &&\qquad
    +2\left[{e^{-4W}\over \sqrt{g_{55}}}
       \left(f_n^{D(a)} f_m^{D(b)} f_m^{D(d)} f_l^{D(c)}\right)'
      \right]_0^{\pi R} . 
   \label{kb}
\end{eqnarray} 
For the longitudinal polarization, which is given by
\begin{equation}
 \epsilon^*(p_3)=\left(p'/M_l,E_l \sin\theta/M_l,0,
       E_l \cos\theta/M_l \right) ,   \label{polarl}
\end{equation}
the invariant matrix element is 
\begin{eqnarray}  
 && -ig_5^2 f^{abe}f^{cde} 
{E^2(1-\cos\theta)\over 8M_n M_m^2 M_l}{\cal K} .
    \label{lk} 
\end{eqnarray}
The amplitude %growing 
that increases with energy has the 
coefficient ${\cal K}$ %in Eq.(\ref{kb})
for both transverse and longitudinal polarizations. 
We emphasize that the coefficient ${\cal K}$ in 
Eq.~(\ref{kb}) 
%is reduced 
reduces to a function of the values of the mode 
functions at the boundaries. 
Therefore, it is directly determined by the choices of 
the boundary conditions. 
If the coefficient ${\cal K}$ of the %(in-)elastic 
elastic (inelastic)
scattering amplitudes does not vanish, 
the unitarity relation (\ref{eq:unitarity}) gives the 
imaginary part of the elastic amplitude that %grows
increases 
with energy and violates the unitarity bound. 




%Even if we choose the zero mode for the gauge boson $A_l^c$ 
%($l\to 0$), we find that Eq.(\ref{tk}) is nonvanishing. 
%Eq.
Equation (\ref{tk}) is applicable 
%available 
also for $l\to 0$.
If we %take 
choose the zero mode ($l=0$) for the gauge boson $A_l^c$, 
the scattering amplitude should satisfy the (on-shell) 
Ward-Takahashi identity: 
The amplitude must vanish if we substitute 
the momentum of the zero mode for 
the polarization vector $\epsilon$. 
%By substituting
Making the substitution 
\begin{equation}
   \epsilon^*(p_3) \to p_3=p'(1,\sin\theta,0,\cos\theta) ,
\end{equation}
we obtain the amplitude (\ref{invm}) as 
\begin{eqnarray}
ig_5^2 f^{abe}f^{cde} (1-\cos\theta)   {E^2\over 8M_n M_m^2}
 f_{l=0}^{D(c)} \left[{e^{-4W}\over \sqrt{g_{55}}}
 (f_m^{D(b)} f_m^{D(d)} f_n^{D(a)}{}'-f_n^{D(a)} f_m^{D(b)}
 f_m^{D(d)}{}')
 \right]_0^{\pi R} ,
\nonumber\\
\label{eq:ward_takahashi_id}
\end{eqnarray}
which %is 
also reduces %d 
to a function of the values of the 
mode functions at the boundaries. 


We thus find that the violation of both the Ward-Takahashi 
identity and the tree level unitarity are proportional to 
the functions given in (\ref{eq:ward_takahashi_id}) and (\ref{kb}),
which are functions 
of the values of the mode functions at the boundaries.  
In the next subsection, we will %provide concrete 
present explicit
examples 
of boundary conditions that are allowed by the variational 
principle and give nonvanishing functions 
(\ref{eq:ward_takahashi_id}) %leading 
that lead
to the violation 
of the Ward-Takahashi identity and (\ref{kb}) %leading 
that lead to the 
violation of %the 
tree level unitarity. 





%%%%%%%%%%%%%%%%%%%%
\subsection{The coset-N/subgroup-D boundary conditions 
%Violating Ward-Takahashi identity and Tree-Level Unitarity
}
\label{sc:bound_cond_WT}
%%%%%%%%%%%%%%%%%%%%



In order to illustrate %the
boundary conditions %which 
that
violate 
the Ward-Takahashi identity and %the 
tree level unitarity, 
we %take 
consider the $SU(N)$ gauge group as an example. 
To obtain a semi-realistic symmetry breaking pattern, let us 
consider the symmetry breaking in two steps: 
two different boundary conditions at $y=0$ and $y=\pi R$, 
such as $SU(5) \to SU(3)\times SU(2)\times U(1) \to 
SU(3)\times U(1)$. 
The warp factor $e^{-4W(y)}$ and $g_{55}(y)$ can provide 
two vastly different mass scales for 
%two 
the boundaries $y=0$ and $y=\pi R$. 
Without loss of generality, let us take one of the 
boundaries, say $y=0$, to be associated with the high 
energy scale, $M_{\rm GUT}$. 
At this high energy scale, we break the gauge %symmetry 
group as
\begin{eqnarray}
SU(N) \to SU(N_1)\times SU(N-N_1)\times U(1) 
\label{eq:1st_syM_br}
\end{eqnarray}
by imposing the Neumann boundary conditions 
(\ref{eq:var_bnd_cod}) for the 
4D vectors $A_\mu^a$ 
in the subgroups $SU(N_1)$, $SU(N-N_1)$ and $U(1)$, 
and by imposing the Dirichlet boundary conditions 
(\ref{eq:var_bnd_cod2}) for the coset 
$SU(N)/[SU(N_1)\times SU(N-N_1)\times U(1)]$. 
The boundary conditions %is 
are written in %a 
matrix form
as
\begin{eqnarray}
 \left(\begin{array}{l|l}
  \textrm{N} & \textrm{D}\\ \hline
  \textrm{D} & \textrm{N}\\
       \end{array}\right) 
 \!\!\!\!
 \begin{array}{l}
   \} ~\textrm{\scriptsize $N_1$} \\  
   \} ~\textrm{\scriptsize $N-N_1$} \\
 \end{array}
  ~~ \textrm{at}~~ y=0 , \nonumber
\end{eqnarray}
where N and D denote the Neumann and Dirichlet boundary 
conditions, respectively. 
This set of boundary conditions %does not 
violates 
%gives rise to neither the 
%boundary contributions (\ref{eq:ward_takahashi_id}) 
%violating 
neither
the Ward-Takahashi identity nor 
%(\ref{kb}) violating 
%the 
tree level unitarity. 
It is realizable 
%by 
with
automorphisms of the 
Lie algebra in orbifoldings and %are 
is
often 
used. 
We denote the generators of the subgroups and the coset %as 
by
\begin{eqnarray}
&&G \in SU(N_1), \qquad 
W \in SU(N-N_1), 
\qquad 
B \in U(1), \nonumber \\
&&X \in SU(N)/[SU(N_1)\times SU(N-N_1)\times U(1)] .
\label{eq:decomp_generators}
\end{eqnarray}


Now we consider the boundary condition at $y=\pi R$, which 
is associated with the lower energy scale, $M_{\rm W}$. 
In order to preserve 
gauge invariance with respect to 
$SU(N_1)$ and $U(1)$, %gauge symmetries, 
we impose the Neumann boundary condition for 
the generators
$G \in SU(N_1)$ and $B \in U(1)$ %generators 
at both 
boundaries, $y=0$ and $\pi R$. 
To break the gauge %symmetry 
group 
$SU(N-N_1)$ by means of 
boundary conditions, we impose the Dirichlet boundary 
condition (\ref{eq:var_bnd_cod2}) for $W \in SU(N-N_1)$ 
generators. 
%The variational principle allows the Dirichlet boundary 
%condition for the generators of the subgroups. 
%This assignment reduces the rank of the unbroken gauge group. 

The remaining generators belong to the coset 
$X \in SU(N)/[SU(N_1)\times SU(N-N_1)\times U(1)]$. 
Usually, the Neumann boundary conditions %is 
are not imposed 
%to %four-dimensional 
on 4D vectors in the coset at both boundaries,
%since 
because massless 
%unbroken 
gauge fields have to form a group,
as indicated in 
Eq.(\ref{eq:gr_bnd_cod}). 
We have %a 
the freedom %to 
%choose 
of choosing the Neumann boundary conditions here, 
%since 
as it is clear that 
the 4D vectors with colors in the coset 
are massive at the high 
energy scale $M_{\textrm{\scriptsize GUT}}$ by the boundary condition 
at $y=0$. 
Thus, we can assign the  boundary condition in %the 
matrix 
form as 
\begin{eqnarray}
 \left(\begin{array}{l|l}
  \textrm{N} & \textrm{N}\\ \hline
  \textrm{N} & \textrm{D}\\
       \end{array}\right)
 \!\!\!\!
 \begin{array}{l}
   \} ~\textrm{\scriptsize $N_1$} \\  
   \} ~\textrm{\scriptsize $N-N_1$} \\
 \end{array} 
  ~~ \textrm{at}~~ y=\pi R , \nonumber
\end{eqnarray}
with unbroken $U(1)$ %obeying 
satisfying the Neumann boundary conditions.
Let us %call 
refer to
this assignment of the Neumann 
boundary conditions for coset and the Dirichlet boundary 
conditions for a subgroup as the coset-N/subgroup-D 
boundary conditions. 
We %will 
show that %this 
these
coset-N/subgroup-D boundary conditions 
violate both the Ward-Takahashi 
identity and %the 
tree level 
unitarity. 
We %will 
also show in %Sec.
\S\ref{sc:brst}, that this choice 
of boundary conditions %will be
is forbidden if 
we require the compatibility with 
the restricted class of 5D 
gauge 
transformations. 


Nonvanishing values of Eqs.~(\ref{tk}) and (\ref{lk}) 
violate %the 
tree level unitarity, whereas %the 
nonvanishing values of 
Eq.~(\ref{eq:ward_takahashi_id}) violate %s 
the Ward-Takahashi identity. 
%Let us observe 
Note that 
all of them 
%Eqs.~(\ref{tk}), (\ref{lk}) and (\ref{eq:ward_takahashi_id}) 
have a common factor of $f^{abe}f^{cde}$. 
%One should 
Also note that our decomposition in 
Eq.~(\ref{eq:decomp_generators}) 
has the following commutation relations,
as follows from the group 
structure 
\begin{eqnarray}
[G, G] = G, \qquad 
[W, W] = W, 
\qquad 
[X, X] = G + W + B . %, 
\label{eq:com_rel_generators}
\end{eqnarray}
This implies %ying 
the following nonvanishing structure constants: 
\begin{eqnarray}
f^{GGG}, \quad f^{WWW}, \quad f^{XXG}, \quad 
f^{XXW}, \quad f^{XXB}. 
\label{eq:nonzero_structure_const}
\end{eqnarray}
Therefore, the nonvanishing group theory factor 
$f^{abe}f^{cde}$ can be classified into 
the following three types: among the labels $a,b,c,d$ and $e$ 
the coset generators $X$ appear %either 
nowhere, 
only in three labels, $(a,c, e), (a,d,e), (b,c,e)$ or $(b,d,e)$, 
or only in four labels, $a,b,c$ and $d$. 
%We will use these group structure in the following.


We first examine the violation of the Ward-Takahashi 
identity, which is given by Eq.~(\ref{eq:ward_takahashi_id}) 
and is proportional to 
\begin{eqnarray}
f^{abe}f^{cde} f_{l=0}^{D(c)} \left[{e^{-4W}\over \sqrt{g_{55}}}
 (f_m^{D(b)} f_m^{D(d)} f_n^{D(a)}{}'-f_n^{D(a)} f_m^{D(b)}
 f_m^{D(d)}{}')
 \right]_0^{\pi R} . 
\label{eq:viol_wt_id}
\end{eqnarray}
Let us consider the processes $WX \to BX$ and $WX \to GX$, 
with the zero mode $l=0$ for the massless gauge bosons $B$ 
and $G$, by choosing the colors 
$(a, b, c, d)=(W, X, B, X)$ and $(W, X, G, X)$. 
We find that the 
coset generator $X$ can contribute to the intermediate 
state $e=X$ and that 
the coset-N/subgroup-D boundary conditions 
at $y=\pi R$ give %s 
\begin{eqnarray}
f^{WXX} f^{BXX}\not=0, \; 
 f_n^{D(a=W)'}\not=0, \; f_m^{D(b=X)}\not=0, \; 
f_{l=0}^{D(c=B, G)}\not=0, \; f_m^{D(d=X)}\not=0. 
\nonumber\\
\label{eq:bc_viol_wt_id}
\end{eqnarray}
The first term %of
in Eq.~(\ref{eq:viol_wt_id}) does not vanish, 
whereas the second term vanishes for %this 
these boundary conditions. 
Therefore, the Ward-Takahashi identity is violated by the 
coset-N/subgroup-D boundary conditions 
in %this 
these processes, $WX\to BX$ and $WX\to GX$. 



We next consider the coefficient 
${\cal K}$ in Eq.~(\ref{kb}), %that 
which appears in 
Eqs.~(\ref{tk}) and 
(\ref{lk}), for the violation of %the 
tree level unitarity. 
The coefficient $f^{abe}f^{cde}{\cal K}$ consists of two pieces, 
\begin{eqnarray}
f^{abe}f^{cde} \sum_k \left[{e^{-4W}\over \sqrt{g_{55}}}
       g_k^{D(e)} f_m^{D(b)} f_l^{D(c)}\right]_0^{\pi R}
    \left[{e^{-4W}\over \sqrt{g_{55}}}
       g_k^{D(e)} f_m^{D(d)} f_n^{D(a)}\right]_0^{\pi R} , 
\label{eq:bundary_contr1}
\end{eqnarray}
and 
\begin{eqnarray}
f^{abe}f^{cde} \left[{e^{-4W}\over \sqrt{g_{55}}}
       \left(f_n^{D(a)} f_m^{D(b)} f_m^{D(d)} f_l^{D(c)}\right)'
      \right]_0^{\pi R} .
\label{eq:boundary_contr2}
\end{eqnarray}
Let us choose the same processes, $WX\to BX$ and $WX\to GX$, 
as in the case of the 
violation of the Ward-Takahashi identity, 
without %restricting 
restriction to the zero 
mode for $B$ and $G$ in this case. 
For %this process,
these processes, the first term, (\ref{eq:bundary_contr1}), 
vanishes, but the second term, (\ref{eq:boundary_contr2}), 
%is 
does not %nonvanishing
vanish,\footnote{
The first term, (\ref{eq:bundary_contr1}), %will be
is 
nonvanishing with the coset-N/subgroup-D 
boundary conditions if we consider %another
the process $XX\to XX$. 
In order to ascertain %that 
whether
%the
tree level unitarity is 
indeed violated in this process, we need to take account of 
another $s$-channel Feynman diagram, in addition to those 
computed in %the 
Appendix.\ref{Ap:amp}. 
We %are computing
compute 
inelastic scattering $XX\to XX$ with different KK levels 
for all $X$, %contrary to Ref.\cite{Csaki:2003dt} 
in contrast to Ref.21),
where only 
the elastic scattering %have been 
is computed with identical 
KK levels for all gauge bosons $X$. 
} because of Eq.~(\ref{eq:bc_viol_wt_id}). 
This establishes that the coset-N/subgroup-D boundary 
conditions violate %s 
%the 
tree level unitarity 
in the processes $WX\to BX$ and $WX\to GX$. 







We %explicitly 
illustrate typical Feynman diagrams 
in Fig.\ref{typfd} 
%for the process in Fig.\ref{ampfig}
that 
contribute %ing 
to the 
violation (\ref{eq:ward_takahashi_id}) of the 
Ward-Takahashi identity and the violation (\ref{kb}) of 
%the 
tree level unitarity. 
\begin{figure}[t]
\begin{center}
\includegraphics[height=2.5cm]{typfd.eps}
\end{center}
\caption{Typical Feynman diagrams contributing to the 
violation (\ref{eq:ward_takahashi_id}) of the 
Ward-Takahashi identity and the violation (\ref{kb}) of 
%the 
tree level unitarity.
\label{typfd}} 
\end{figure}
The colors of the external gauge bosons with momentum 
assignments in Fig.~\ref{ampfig} are chosen as follows. %: 
%t
The gauge boson with %the
momentum $p_1$ is 
assigned to %be
$W \in SU(N-N_1)$, those with $p_2, p_4$ 
to %be 
$X \in SU(N)/[SU(N_1)\times SU(N-N_1)\times U(1)]$, 
and that with $p_3$ to %be 
$B \in U(1)$ or $G\in SU(N_1)$. 
We can choose the color of the internal gauge bosons 
to be $X \in SU(N)/[SU(N_1)\times SU(N-N_1)\times U(1)]$. 


Thus, we conclude that 
the coset-N/subgroup-D boundary conditions give %s 
%the 
a four gauge boson scattering 
amplitude %which grows 
that increases
as powers of the center-of-mass 
energies. 
More specifically, this breakdown of %the 
tree level unitarity 
occurs if we take one of the external gauge bosons 
in a subgroup with the Dirichlet boundary conditions, 
the other two gauge bosons in the coset 
with the Neumann boundary conditions, and another gauge boson 
in %the 
an
unbroken subgroup %$U(1)$ or $SU(N_1)$ 
with 
the Neumann boundary conditions. 
%At the same time, t
The Ward-Takahashi identity is also 
broken with this choice of the boundary conditions. 



%Let us n
Note that we should not use the unitary gauge 
in the calculation of the tree diagram. 
As we have used the condition $\xi \ll E^2/M_k^2$ in 
computing the scattering amplitude, our calculation is not 
valid for the unitary gauge, %where 
for which $\xi \rightarrow \infty$. 
The part of the scattering amplitude %growing 
that increase with the energy 
does not depend on the gauge parameter $\xi$ for 
finite values of $\xi$. 
For the unitary gauge, i.e. $\xi= \infty$, the 
extra-dimensional component of the gauge boson 
drops out, as seen from the propagator (\ref{a5prop}). 
Then the breakdown %s 
of %the 
tree level unitarity and the 
Ward-Takahashi identity seems to be invisible, even for the 
choice of the coset-N/subgroup-D boundary 
conditions.\footnote{
In %Sec.
\S\ref{sc:brst}, we show that the 5D %five-dimensional 
gauge 
transformations are ill-defined if the coset-N/subgroup-D 
boundary conditions %is
are used. 
Therefore, the different ``gauge choice'' may not 
give the same physical results. 
}


The breakdown %s 
of %the 
tree level unitarity and the 
Ward-Takahashi identity %are 
is a serious problem for %the 
gauge theories in higher dimensions. 
%Therefore 
For this reason,
we look for an additional requirement 
which forbids such a choice of boundary conditions that 
violate %the 
tree level unitarity and the Ward-Takahashi 
identity in the scattering amplitudes. 
It is most likely that this problem is related to the 
gauge transformation property of gauge fields 
%around 
near the boundary. 
We %will 
examine the 5D %five-dimensional 
gauge transformations in the 
next section.



%%%%%%%%%%%%%%%%%%%%
\section{%Consistency of 
5D %five-dimensional 
gauge transformations and boundary conditions 
%and Discussion
}
\label{sc:brst}
%%%%%%%%%%%%%%%%%%%%


\subsubsection*{\underline{\it 5D %five-dimensional 
gauge transformations}}

Let us consider 5D %five-dimensional 
gauge transformations %for 
of the 
%four-dimensional 
4D vector component $A_\mu^a(x,y)$,
which are expressed as 
\begin{eqnarray}
\delta A_\mu^a(x,y)=
  \partial_\mu  \epsilon^a(x,y) 
+ g_5 f^{abc} A_\mu^b(x,y) \epsilon^c(x,y) .
\label{eq:gauge_tr_vector}
\end{eqnarray}
The first term %in
on the right-hand side is inhomogeneous 
in the gauge field and is independent of the gauge 
coupling constant $g_5$. 
The second nonlinear term is %of the 
first order in $g_5$.
%
In the bulk, the ordinary 5D gauge invariance is preserved, 
and the transformation function $\epsilon^a$ is an 
arbitrary function of $x$ and $y$.
%On the other hand, 
By contrast,
on the boundaries, $\epsilon^a$, as well as $A_\mu^a$ and $A_5^a$,
%are 
is subject to %some 
certain boundary conditions.
Only if appropriate boundary conditions are chosen %,
is the
theory %is 
invariant under the transformation (\ref{eq:gauge_tr_vector}),
with the transformation function restricted by the boundary conditions.
We call 
such transformations 
the restricted class of 5D gauge transformations.

We first examine the relation between the boundary 
conditions for $A_\mu^a$ and $A_5^a$ to %the 
leading 
order (i.e., in the limit of a
vanishing gauge coupling, %limit 
$g_5\to 0$). 
The first (inhomogeneous) term %in 
on the right-hand side of 
Eq.~(\ref{eq:gauge_tr_vector}) 
implies that the 
gauge transformation function $\epsilon^a(x, y)$ %should 
%have
has the same boundary condition as the corresponding 
%four-dimensional 
4D vector component, $A_\mu^a(x, y)$. 
%
The gauge transformation of 
%four-dimensional 
the 4D scalar component 
$A_5^a(x,y)$ is given by 
\begin{eqnarray}
\delta A_5^a(x,y)=
  \partial_5  \epsilon^a(x,y) 
+ g_5 f^{abc} A_5^b(x,y) \epsilon^c(x,y) . 
\label{eq:gauge_tr_scalar}
\end{eqnarray}
The first (inhomogeneous) term implies that 
the boundary condition for 
$A_5^a(x,y)$ %should be 
is the same as $\partial_5\epsilon^a$.
%Namely, 
In other words, the Neumann (Dirichlet) boundary conditions for 
$A_5^a(x,y)$ 
imply %ies 
the Dirichlet (Neumann) boundary 
conditions for $\epsilon^a(x,y)$. 

\vspace{0.2cm}

\subsubsection*{\underline{\it First choice of the boundary conditions, (\ref{eq:var_bnd_cod1})}}

We can now examine the first choice of the boundary 
conditions, (\ref{eq:var_bnd_cod1}). 
The restricted class of 5D 
gauge transformations 
for $A_\mu^a$ in 
Eq.~(\ref{eq:gauge_tr_vector}) implies that the boundary 
conditions for $A_\mu^a$ and for the gauge transformation 
function $\epsilon^a$ %should be 
are the same, 
whereas the restricted class of 5D 
gauge transformations for $A_5^a$ in 
Eq.(\ref{eq:gauge_tr_scalar}) implies that 
the boundary conditions for $A_5^a$ 
%should be 
are opposite to %that 
those for $\epsilon$. 
Therefore, the restricted class of 5D 
gauge transformations 
(at the order of the inhomogeneous term) implies that the 
boundary conditions for $A_\mu^a$ and %for 
$A_5^a$ %should 
%be 
are opposite.  
The first choice of boundary conditions, 
(\ref{eq:var_bnd_cod1}), allowed by the variational principle 
contradicts\footnote{
The same conclusion %has been 
was reached previously %by 
with a 
different argument using the unitary gauge \cite{Csaki:2005vy}. 
} %with 
the gauge 
transformation properties %y 
of $A_\mu^a(x,y)$ and %that of 
$A_5^a(x,y)$. 
%Namely
Specifically, the first choice of boundary conditions does not 
allow a consistent definition of the 5D %five-dimensional 
gauge transformation 
functions $\epsilon^a(x,y)$. 


\vspace{0.2cm}


\subsubsection*{\underline{\it Coset-N/subgroup-D boundary conditions}}

Next, let us study %if
whether the coset-N/subgroup-D boundary conditions 
considered in %Sec.
\S\ref{sc:bound_cond_WT} 
%is
are consistent with the restricted class of 5D 
gauge transformations 
(\ref{eq:gauge_tr_vector}) and 
(\ref{eq:gauge_tr_scalar}). 
We need to examine the nonlinear terms with the structure 
constant $f^{abc}$ in these gauge transformations. 
The group structure given  in 
Eq.(\ref{eq:nonzero_structure_const}) implies the 
following. 
If $a$ is in a subgroup, 
either both $b$ and $c$ belong to the subgroup %, 
or both belong to the 
coset. 
If $a$ is in a coset, 
%one of $b, c$ should 
then either $b$ or $c$
belongs to the subgroup and the other 
to the coset. 

Let us consider the restricted class of 5D 
gauge transformations for 
the generators in the coset 
$X \in SU(N)/[SU(N_1)\times SU(N-N_1)\times U(1)]$ with 
the Neumann boundary conditions for $A_\mu^a(x,y)$ (and 
the Dirichlet boundary conditions for $A_5^a(x,y)$). 
The nonlinear term $g_5 f^{abc} A_\mu^b \epsilon^c$ in 
the restricted class of 5D 
gauge transformations  $\delta A_\mu^a$ in 
Eq.~(\ref{eq:gauge_tr_vector}) for %four-dimensional 
the 4D vector contains contributions with $A_\mu^b$ 
in the subgroup (coset) and $\epsilon^c$ 
in the coset (subgroup). 
The boundary conditions for $A_\mu^b$ and $\epsilon^b$ with 
the same %gauge generator
color
$b$ are the same, whereas the %those 
generators in 
the subgroup and in the coset have %the 
opposite boundary 
conditions. 
Hence, %either 
one
of the following two cases %are 
is realized: 
\begin{eqnarray}
A_\mu^b(x,y=\pi R)=0, 
\quad  
\partial_5 \epsilon^c(x,y=\pi R)= 
0 ,  
\label{eq:cst_N_sub_D1}
\end{eqnarray}
\begin{eqnarray}
\partial_5 A_\mu^b(x,y=\pi R)&=&0, 
\quad  
 \epsilon^c(x,y=\pi R)= 
0 .  %,  
\label{eq:cst_N_sub_D}
\end{eqnarray}
Then the nonlinear term $g_5 f^{abc} A_\mu^b \epsilon^c$ 
consists of a product of functions with the Dirichlet and 
the Neumann boundary conditions: %. 
\begin{eqnarray}
&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\partial_5 \left. \left( A_\mu^b(x,y) 
\epsilon^c(x,y)\right)\right|_{y=\pi R} \nonumber \\
&=&
%\left(
\partial_5 A_\mu^b(x,y=\pi R)
%\right) 
\epsilon^c(x,y=\pi R)
+A_\mu^b(x,y=\pi R)
\partial_5 \epsilon^c(x,y=\pi R) 
\not=0 . 
\label{eq:prod_D_N}
\end{eqnarray}
Since this nonlinear term does not satisfy the Neumann 
boundary conditions, the boundary conditions for 
$A_\mu^a$ %is
are not %respected. 
satisfied.


Similarly, we can also examine 
the restricted class of 5D 
gauge transformations for 
the generators in the 
subgroup $W \in SU(N-N_1)$ with 
the Neumann boundary conditions for $A_5^a(x,y)$ 
(and the Dirichlet boundary conditions for 
$A_\mu^a(x,y)$). 
The nonlinear term $g_5 f^{abc} A_5^b \epsilon^c$ in 
the restricted class of 5D 
gauge transformations  $\delta A_5^a$ in 
Eq.~(\ref{eq:gauge_tr_scalar}) for %four-dimensional 
4D scalar contains contributions from $b$ and $c$ in the 
coset. 
The boundary conditions in the coset 
are Neumann for the gauge function $\epsilon$ %, 
and Dirichlet for the 4D scalar $A_5$: %, 
%respectively : 
$A_5^b(x,y=\pi R)=0$, 
$\partial_5 \epsilon^c(x,y=\pi R)=0$. 
%\label{eq:cst_N_sub_D}\end{eqnarray}
Therefore, the product does not %obey 
satisfy the Neumann boundary 
condition. 
%\begin{eqnarray}\partial_5 \left. \left( A_5^b(x,y) 
%\epsilon^c(x,y)\right)\right|_{y=\pi}=\left(\partial_5 A_5^b(x,y=\pi)\right) 
%\epsilon^c(x,y=\pi)\not=0 . \label{eq:prod_D_N}\end{eqnarray}

In both cases, there is no way to define the gauge 
transformation function $\epsilon^a(x,y)$ consistently. 
Therefore, the coset-N/subgroup-D boundary conditions do not 
allow a consistent definition of 
the restricted class of 5D 
gauge 
transformations.\footnote{
More generally, Dirichlet boundary conditions for $A_\mu^a$ 
in the (non-Abelian) subgroup give %s 
a similar 
inconsistency for $\delta A_5^a$, although it does not show 
up in Eqs.~(\ref{kb}) and (\ref{eq:ward_takahashi_id}) 
for the 2-to-2 %(in-)
elastic (inelastic)
scattering amplitudes. 
} 

\vspace{0.2cm}

\subsubsection*{\underline{\it Mode expansions and 4D %Four-dimensional 
gauge symmetry 
%Breaking
}}

To identify the 4D %four-dimensional 
gauge symmetry, 
it is useful to perform mode 
expansions for the 5D %five-dimensional 
gauge transformation functions 
$\epsilon^a(x, y)$ as well as the 5D %five-dimensional 
gauge fields 
$A^a_M(x, y)$. 
Once the boundary conditions for the gauge transformation 
functions and the gauge fields 
are defined consistently, 
we can perform mode expansions of 
the restricted class of 5D 
gauge transformations of the 
%four-dimensional 
4D vector component $A^a_\mu(x, y)$ in Eq.~(\ref{eq:gauge_tr_vector}) 
as 
\begin{eqnarray}
\sum_{n=0}^{\infty} \delta A_{\mu n}^a(x) f_n^{D(a)}(y)
&=&
\sum_{n=0}^{\infty} \partial_\mu \epsilon_n^a(x) f_n^{D(a)}(y) 
\nonumber \\ 
&+& g_5 f^{abc} \sum_{n=0}^{\infty} A_{\mu n}^b(x) f_n^{D(b)}(y) 
 \sum_{m=0}^{\infty} \epsilon_m^c(x) f_m^{D(c)}(y) .
\label{eq:mode_ex_gauge_tr_v}
\end{eqnarray}
If the zero mode is present for the gauge transformation 
function $\epsilon^a(x, y)$, 
the %four-dimensional 
4D gauge invariance is maintained, and the corresponding 
%four-dimensional 
4D gauge field $A^a_{\mu n}(x)$ contains a massless mode. 
If the zero mode is absent for 
 the gauge transformation function $\epsilon^a(x, y)$, 
the %four-dimensional 
4D gauge invariance is broken, and
the modes of the corresponding %four-dimensional 
4D gauge field 
$A^a_{\mu n}(x)$ are all massive %without 
with no
zero mode.  %at all. 



\vspace{0.2cm}

\subsubsection*{\underline{\it Orbifoldings}}


We %will 
now confirm that the boundary conditions 
realized as the (inner) automorphism of the Lie 
algebra in orbifoldings %is
are consistent with 
the restricted class of 5D 
gauge transformations. 
We %call 
refer to the automorphism at the boundary $y=0$ 
as $P_0$ 
\begin{eqnarray}
 A_\mu (x,-y)=P_0 A_\mu(x,y)P_0^{-1},  
\label{eq:automorphism}
\end{eqnarray}
with $P_0=P_0^\dagger=P_0^{-1}$. 
Similarly, the automorphism $P_1$ is defined at $y=\pi R$. 
To obtain the Dirichlet or Neumann boundary 
conditions, we restrict ourselves to diagonal matrices 
for the automorphisms. 
Then we never obtain %the 
boundary conditions 
%which
that correspond to the reduction of the 
rank of the group.
\footnote{
More precisely, %speaking, 
here we are considering %here 
inner 
automorphisms of the Lie algebra excluding the possibility 
of outer automorphisms \cite{Hebecker:2001jb}. 
}
Combining the automorphism (\ref{eq:automorphism}) 
with the gauge transformation %law  
\begin{eqnarray}
 \delta A_\mu(x,-y)=\partial_\mu \epsilon (x,-y)
         -ig_5 [A_\mu(x,-y),\epsilon(x,-y)] ,
\end{eqnarray}
we find that the gauge transformation function %to 
transforms 
under the automorphism 
\begin{eqnarray}
   \epsilon(x,-y)=P_0 \epsilon(x,y) P_0^{-1}. 
\end{eqnarray}
On the other hand, the gauge transformation 
dictates the transformation of $A_5$ under the 
 automorphism 
\begin{eqnarray}
   \delta A_5(x,-y)&=&-\partial_5 \epsilon(x,-y)
                   -ig_5[A_5(x,-y),\epsilon(x,-y)] ,
%\end{eqnarray}
\\ %\begin{eqnarray}
  A_5(x,-y)&=&-P_0 A_5(x,y)P_0^{-1} .
\end{eqnarray}
We can repeat the same analysis at $y=\pi R$. 
This shows that 
a consistent transformation property under the automorphism 
can be assigned for the gauge 
transformation functions. 
This consideration is in accord with the previous result: 
%t
The automorphism of the 
Lie algebra in orbifolding satisfies %the 
tree level unitarity 
\cite{Ohl:2003dp}. 



\vspace{0.2cm}


\subsubsection*{\underline{\it BRST formulation}}

BRST transformations replace the classical gauge 
transformations after the gauge fixing. 
The BRST invariance is %just 
equivalent to 
the classical gauge invariance and gives no extra 
advantage at the tree level, %where
which we are considering. 
We %will
briefly consider the BRST 
formulation in order to see that 
we can obtain
the same result %can be 
%obtained 
as in the case of the gauge transformations. 
As long as the boundary conditions for the 
5D
gauge 
transformation function $\epsilon^a$ in %five dimensions 
\begin{eqnarray}
\delta A_M^a(x,y)=
  \partial_M  \epsilon^a(x,y) 
+ g_5 f^{abc} A_M^b(x,y) \epsilon^c(x,y) 
\label{eq:gauge_tr_5D}
\end{eqnarray}
are defined consistently with the boundary conditions for 
the gauge fields $A_M^a$, 
(the restricted class of 5D) 
%(5D restricted)
BRST transformations can be defined 
straightforwardly by promoting the gauge 
transformation function $\epsilon^a(x,y)$ to the ghost 
field $c^a(x,y)$ multiplied by an anti-commuting parameter 
$\varepsilon$. 
%One should 
We must also add the anti-ghost field $\bar c^a(x,y)$ 
and the Nakanishi-Lautrup field $B^a(x,y)$. 
The restricted class of 5D 
BRST transformations $\delta$ %in five dimensions 
are 
given by 
\begin{eqnarray}
 \varepsilon \delta A_M^a(x,y)&=&
  \varepsilon \left[ \partial_M c^a(x,y) 
+g_5 f^{abc}A_M^b(x,y) c^c(x,y)\right] , 
\\
 \varepsilon \delta c^a(x,y)
&=&-{1\over 2} g_5 \varepsilon f^{abc}c^b(x,y) c^c(x,y)  ,
\\
 \varepsilon \delta\bar{c}^a(x,y)&=& i \varepsilon B^a(x,y) ,
\\
 \varepsilon \delta B^a(x,y)&=&0 , %. ver.34
\end{eqnarray}
%If we perform the 
%mode expansion,\cite{Muck:2001yv}--\cite{Muck:2004br} 
%we obtain the 
%mode decomposition of the restricted class of 5D 
%BRST transformations. 
where the functions
$c^a(x, y)$ %obey 
satisfy the same boundary conditions as the
corresponding $\epsilon^a(x, y)$. %
\cite{Muck:2001yv}${}^{-}$\cite{Muck:2004br} %30)-32).


\vspace{0.2cm}

\subsubsection*{\underline{\it Summary and prospects}}


Summarizing our results, we have examined the choice of 
the Neumann %or
and Dirichlet boundary conditions to break the 
gauge symmetry in 5D %five-dimensional 
gauge theory. 
The variational principle allows %the 
(coset-N/subgroup-D) 
boundary conditions that violate the Ward-Takahashi 
identity and %the 
tree level unitarity. 
We have observed that the 
5D %five-dimensional 
gauge transformation functions 
$\epsilon(x, y)$, %besides 
in addition to
the gauge fields 
$A_M(x, y)$, 
%should 
must
be given %a 
boundary conditions %which is
that are 
consistent with the restricted class of 5D 
gauge transformations. 
This condition provides a stringent constraint 
and forbids the coset-N/subgroup-D boundary conditions. 
%that violate 
%the tree level unitarity and the Ward-Takahashi 
%identity. 


In considering %the 
orbifoldings, it is possible to use nondiagonal 
automorphisms of the Lie algebra which give 
%more involved 
boundary conditions that are more involved than Neumann or Dirichlet 
conditions. 
More generally, there are possibilities of nontrivial 
Wilson lines which break the gauge %symmetry 
group
\cite{Hosotani:1983xw}. 
It has been found that different boundary conditions are 
sometimes related by gauge transformations. 
%In 
Under such circumstances, the effective potential at the 1-loop level 
determines  the value of the Wilson lines and 
the symmetry breaking pattern \cite{Haba:2003ux}. 
The elucidation of 
%S
such a possibility is an interesting %future 
problem. 

We have considered pure gauge theory without matter 
fields in 5D. 
We need to re-analyze the boundary conditions 
%if 
in the case that there %are
exist 
matter fields, especially if they are localized on the 
boundary. 

%D
The deconstruction approach %uses %four-dimensional 
employs
4D gauge theories 
to build higher-dimen-sional gauge models in a discretized 
version \cite{Arkani-Hamed:2001ca}. 
Various boundary conditions, such as the automorphisms 
of the Lie algebra in the orbifolding, 
can be obtained as appropriate limits from the 
deconstruction \cite{He:2004zr}.
It has been shown that 
%T
these boundary conditions %are shown to 
satisfy %the
tree level 
unitarity automatically. 
The deconstruction can provide realistic models 
if %one allows 
some amount of fine tuning is allowed.
\cite{Cacciapaglia:2004rb}
It is an interesting %future 
problem %if 
to determine whether %the 
boundary conditions that
reduce %ing 
the rank of the group
can be 
forbidden as a limit of the discretized gauge theories 
%like 
as in the deconstruction approach \cite{Arkani-Hamed:2001ca}. 

It is also an interesting %future 
problem to build a realistic 
model using the 5D %five-dimensional 
gauge theories with 
boundaries. 
%The l
Left-right symmetric models have been %much 
extensively studied in 
model building \cite{Csaki:2005vy}. 
It %is
would be interesting to obtain %the 
a reduction of the rank of 
the gauge group with such a left-right symmetric model. 
Although they %use 
employ linear combinations of generators of 
the factor group, %one 
it should  be examined carefully %if
whether the 
rank reduction is %really 
indeed
compatible with 
the restricted class of 5D 
gauge transformations.


%There have been arguments to have additional boundary kinetic 
%terms for gauge fields \cite{Georgi:2000ks}. 
%In the case of automorphisms of Lie 
%algebras, it has been shown 
%that the BRST transformation is still 
%nilpotent, even in the presence of 
%the boundary kinetic terms \cite{Muck:2004br}. 
%Although we still have to show that the nilpotency of BRST 
%transformation is still valid for our new boundary conditions, 
%we expect that such a behavior may not be strongly influenced 
%by the presence of localized boundary kinetic terms as 
%pointed out in Ref.\cite{Muck:2004br}. 
%
%
%It has been observed that the supersymmetric quantum 
%mechanics (SQM) is useful to formulate the 
%higher-dimensional gauge theories  \cite{Nagasawa:2004xk}. 
%They have even advocated the hermiticity of the 
%supercharge of SQM as a fundamental 
%principle to formulate the higher-dimensional gauge 
%theories, although its precise relation to gauge 
%invariance does not appear obvious. 
%On the other hand, the gauge invariance of the 
%higher dimensional gauge theory is 
%precisely and directly expressed as a global symmetry in the 
%form of the BRST invariance, rather than SQM. 
%Our view is that the BRST transformations are the 
%fundamental principle, whereas SQM is a technical 
%device to formulate the BRST transformation, even though 
%we also find it useful. 



\vspace*{10mm}
%%%%%%%%%%%%%%%%%%%%
%\begin{center}
%{\bf Acknowledgements}
%\end{center}  
\section*{Acknowledgements}
We wish to thank Nobuhito Maru and Makoto Sakamoto for %a 
useful discussions, and Masaharu Tanabashi for %a 
discussions as 
well as for informing us of his unpublished results and 
literature %s 
on the deconstruction approach. 
This work is supported in part by Grants-in-Aid for Scientific 
Research from the Ministry of Education, Culture, Sports, 
Science and Technology %, 
of Japan [Nos.17540237 (N.S.) 
and 16028203 for the %p
Priority %a
Area ``%o
Origin of %m
Mass'' 
(N.S.~and N.U.)]. 

%%%%%%%%%%%

%\newpage
\begin{appendix}
\section{Formulas for Mode Functions and Overlap Integrals\label{Ap:KK}}
In this appendix, we summarize properties of the mode 
functions $f_n^{D(a)}(y)$ and $g_m^{D(b)}(y)$ and overlap 
functions %made of 
constructed from them.

The mode functions $f_n^{D(a)}$ and $g_n^{D(a)}$ are 
defined by the eigenvalue equations with the mass 
eigenvalue $M_n$ as 
\begin{eqnarray}
 -{1\over \sqrt{g_{55}}}\partial_5 {e^{-4W}\over \sqrt{g_{55}}}
   \partial_5 f_n^{D(a)}(y)
&=&M_n^2 f_n^{D(a)}(y) ,
\\ 
 -\partial_5 {1\over \sqrt{g_{55}}}\partial_5
    {e^{-4W}\over \sqrt{g_{55}}} g_n^{D(a)}(y)&=&M_n^2 g_n^{D(a)}(y) ,
\end{eqnarray}
which can be rewritten %into
as the following 
coupled first-order differential equations 
\begin{equation}
 \partial_5 f_n^{D(a)}=M_ng_n^{D(a)},
 ~~~~~
-{1\over \sqrt{g_{55}}}\partial_5 {e^{-4W}\over
  \sqrt{g_{55}}}g_n^{D(a)}=M_n f_n^{D(a)} .
\label{eq:coupled_eq}
\end{equation}
We choose them to be orthonormal, 
\begin{eqnarray}
 \int_0^{\pi R} dy \sqrt{g_{55}} f_n^{D(a)}f_m^{D(a)}=\delta_{nm} ,
 &&
 \int_0^{\pi R} dy \sqrt{g_{55}}^{-1}e^{-4W}
    g_n^{D(a)}g_m^{D(a)} 
  =\delta_{nm} ,
\end{eqnarray}
and assume them to be complete 
\begin{eqnarray}
\sum_n f_n^{D(a)}(y) f_n^{D(a)}(y')&=&{1\over \sqrt{g_{55}}}\delta(y-y') ,
\\
\sum_n g_n^{D(a)}(y) g_n^{D(a)}(y')&=&e^{4W}\sqrt{g_{55}}\delta(y-y')   .
\end{eqnarray}


Using the KK decompositions in 
Eqs.(\ref{eq:kk_vector}) and (\ref{eq:kk_scalar}) and 
integrating over $y$, 
we obtain the action for the field strengths without the 
gauge fixing term in Eq.(\ref{eq:lagr}) as 
\begin{eqnarray}
 {\cal S}&\!\!\!=&\!\!\!\int d^4x \bigg[
  -{1\over 4}(\partial_{\mu}A_{\nu n}^a-\partial_{\nu}A_{\mu
  n}^a)
     (\partial^{\mu}A_n^{\nu a}-\partial^{\nu}A_n^{\mu a})
    -{1\over 2} M_n^2 A_{\mu n}^a A_n^{\mu a}
%
 +M_n(\partial_{\mu} A_{5n}^a)A_n^{\mu a}
\nonumber
\\ 
   &&\qquad
    -g_5 f^{abc}A_{\mu n}^b A_{\nu m}^c \partial^{\mu}A_l^{\nu a}
    F_{nml}^{D(bca)}
% 
   -{1\over 4}g_5^2 f^{abc}f^{ade}A_{\mu m}^b A_{\nu n}^c
    A_l^{\mu d}A_k^{\nu e}F_{mnlk}^{D(bcde)} 
\nonumber
\\
  &&\qquad
  -{1\over 2}\partial_{\mu}A_{5n}^a \partial^{\mu} A_{5n}^a
%
   -g_5 f^{abc}A_{\mu n}^b A_{5m}^c \partial^{\mu}A_{5l}^a
     T_{nml}^{D(bca)}
\nonumber
\\ 
 &&\qquad
  +M_l g_5 f^{abc}A_{\mu n}^b A_{5m}^c A_l^{\mu a}
   T_{nml}^{D(bca)}
%
  -{1\over 2}g_5^2 f^{abc}f^{ade}A_{\mu m}^b A_{5 n}^c
       A_l^{\mu d}A_{5 k}^e T_{mnlk}^{D(bcde)}
 \bigg] \!,
  \label{KKaction}
\end{eqnarray}
where the overlap integrals are given by 
\begin{eqnarray}
 F_{nml}^{D(bca)}&=&\int_0^{\pi R} dy
  \sqrt{g_{55}}f_n^{D(b)}f_m^{D(c)}f_l^{D(a)} ,
\nonumber
  \\
T_{nml}^{D(bca)}&=&\int_0^{\pi R} dy {e^{-4W}\over \sqrt{g_{55}}}
     f_n^{D(b)} g_m^{D(c)} g_l^{D(a)} ,
\nonumber
\\
   F_{mnlk}^{D(bcde)}
  &=&\int_0^{\pi R} dy \sqrt{g_{55}}f_m^{D(b)} f_n^{D(c)} f_l^{D(d)}
     f_k^{D(e)} ,
\nonumber
\\
 T_{mnlk}^{D(bcde)}
     &=&\int_0^{\pi R} dy {e^{-4W}\over\sqrt{g_{55}}}
  f_m^{D(b)}g_n^{D(c)} f_l^{D(d)} g_k^{D(e)} .
\label{overlap}
\end{eqnarray}

By repeatedly using the defining equations 
(\ref{eq:coupled_eq}), we obtain %a
the 
useful identity 
\begin{eqnarray}
&& M_m(M_n T_{mmn}^{D(bda)}-M_m T_{nmm}^{D(abd)})
  =(M_n^2-M_m^2)F_{nmm}^{D(abd)}
\nonumber
\\    
& &+\left[{e^{-4W}\over \sqrt{g_{55}}}(f_m^{D(b)} f_m^{D(d)}
       f_n^{D(a)}{}'
         -f_n^{D(a)} f_m^{D(b)} f_m^{D(d)}{}')\right]_0^{\pi R} . 
 \label{mtb}
\end{eqnarray}


%%%%%%%%%%%%%%%%%%%%%%
\section{Calculation of Scattering Amplitude \label{Ap:amp}}
%%%%%%%%%%%%%%%%%%%%%%

In this appendix, we give some details %of calculating
concerning the calculation of  
the scattering amplitude $A_n^a A_m^b\to A_l^c A_m^d$ at %the
tree 
level with the high energy approximation, %where 
in which the
total energy $E$ is sufficiently large %enough 
compared to the mass 
of any external %and 
or intermediate state. %s.
The tree level diagrams %is 
are %shown
presented\footnote{ 
%Since
Because we are interested in the case %where
in which the gauge 
boson $A_l^c$ can have a zero mode  ($l=0$),
we %are 
list %ing 
only the diagrams appropriate for that case. 
} 
in Fig.~\ref{tree}.
\begin{figure}[h]
\begin{center}
\includegraphics{tree.eps}
\end{center}
\caption{The tree level diagrams: The  $A_\mu$ and $A_5$ components 
of the gauge bosons are %drawn as 
represented by wavy lines and straight lines, 
respectively.\label{tree}} 
\end{figure}


In the $R_{\xi}$ gauge, the propagators for $A_{\mu}$ and $A_5$ 
have a $\xi$ dependent mass: 
\begin{eqnarray}
&&
\left\langle A_{\mu m}^a A_{\nu k}^b\right\rangle
    ={-i\delta^{ab}\delta_{mk}\over p^2+M_k^2}
  \left(\eta_{\mu\nu}-(1-\xi){p_{\mu}p_{\nu}\over p^2 + \xi
     M_k^2}\right) , 
\\
  && \left\langle A_{5 m}^a A_{5 k}^b\right\rangle
    ={-i\delta^{ab}\delta_{mk}\over p^2+\xi M_k^2} .
  \label{a5prop}
\end{eqnarray}
The %V
vertices can be read off %from
of the KK decomposed 
action (\ref{KKaction}), 
as given in Fig.~\ref{vertex}.
\begin{figure}[h]
\begin{center}
\includegraphics[height=3.5cm]{v3.eps} \\
\includegraphics[height=3cm]{v4.eps} \\
\includegraphics[height=3cm]{v5.eps}
\end{center}
\caption{Vertices. \label{vertex}} 
\end{figure}
We use %the 
an expansion in powers of $M_k/E$. 
As seen from the propagators, this expansion requires 
the condition
$\xi\ll E^2/M_k^2$, which is possible except in the 
unitary gauge, where we have $\xi \to \infty$. 



Assuming $\xi\ll E^2/M_k^2$, 
the contribution %s 
of each diagram in Fig.\ref{tree}
to the invariant matrix element %are shown
is listed in Table~\ref{Tab:xi1}. 
%
\begin{table}[b]
\caption{Contribution from each diagram to 
the invariant matrix element %for
in the case $\xi\ll E^2/M_k^2$.
Each contribution should be multiplied by the factor 
$-ig_5^2 f^{abe}f^{cde} E^3/(8M_n M_m^2)\epsilon_{\mu}^*(p_3)$.
Here, %T
the Jacobi identity 
$f^{ade}f^{cbe}=f^{abe}f^{cde}$ %is
has been used.\label{Tab:xi1}}
\begin{eqnarray}
c &\Bigg|&
 F_{lmnm}^{D(cdab)}\left(1-{M_l^2\over E^2}\right)
 \Bigg(1-{M_m^2\over E^2}+\left(3-{4M_n^2-M_m^2\over
			     E^2}\right)\cos\theta,
   -2\left(1-{M_n^2\over E^2}\right)\sin\theta,
\nonumber\\
  &&\qquad\qquad\qquad\qquad
  0,-3+{2M_n^2+7M_m^2\over E^2}
    -\left(1-{2M_n^2+M_m^2\over E^2}\right)\cos\theta\Bigg)^{\mu} .
\nonumber\\
s &\Bigg|&
  -2\sum_k F_{lmk}^{D(cde)}F_{nmk}^{D(abe)} \left(1-{M_l^2\over E^2}\right)
  \Bigg({M_n^2-M_m^2\over E^2}+4{M_m^2\over E^2}\cos\theta,
     {M_n^2-M_m^2\over E^2}\sin\theta ,
\nonumber\\
  &&\qquad\qquad\qquad\qquad
   0,-2\left(1-{M_m^2-M_k^2\over E^2}\right)
     +{M_n^2-M_m^2\over E^2}\cos\theta \Bigg)^{\mu} .
\nonumber\\
u &\Bigg|&
  -\sum_k F_{mnk}^{D(dae)}F_{mlk}^{D(bce)}\left(1-{M_l^2\over
					   E^2}\right)
  %\times
\nonumber
\\
  && \times
   \Bigg(2+{M_n^2-2M_m^2+3M_l^2-4M_k^2\over E^2}
     +\left(2-{5M_n^2+8M_m^2+M_l^2\over E^2}\right)\cos\theta,
\nonumber\\
  &&
     -\left(1-{M_n^2-5M_m^2-M_l^2+2M_k^2\over E^2}\right)\sin\theta
      -\left(1+{M_n^2+M_m^2-M_l^2\over
	E^2}\right)\sin\theta\cos\theta,0,
\nonumber\\
  &&
    1+{4M_n^2+5M_m^2+2M_l^2-2M_k^2\over E^2}
   +\left({M_n^2-4M_m^2-M_l^2+2M_k^2\over E^2}\right)\cos\theta
\nonumber\\
  &&
    -\left(1+{M_n^2+M_m^2-M_l^2\over E^2}\right)\cos^2\theta\Bigg)^{\mu} .
\nonumber\\
s5 &\Bigg|&
 -{2\over E^2}\sum_k \left(M_l T_{mkl}^{D(dec)}-M_m T_{lkm}^{D(ced)}\right)
    \left(M_n T_{mkn}^{D(bea)}-M_m T_{nkm}^{D(aeb)}\right)
      \left(1,-\sin\theta, \right.
\nonumber\\
  &&\qquad\qquad\qquad\qquad \left.
 0,-\cos\theta\right)^{\mu} .
\nonumber\\
u5 &\Bigg|&
 -{2\over E^2}\sum_k \left(M_n T_{mkn}^{D(dea)}-M_m T_{nkm}^{D(aed)}\right)
    \left(M_l T_{mkl}^{D(bec)}-M_m T_{lkm}^{D(ceb)}\right)
      \left(-1,0,0,1\right)^{\mu} .
\nonumber
\end{eqnarray}
\end{table}
%
Summing all the contributions in Table~\ref{Tab:xi1},
we obtain the total invariant matrix element as
\begin{eqnarray}
 && ig_5^2 f^{abe}f^{cde} {E^3\over 8M_n M_m^2}\epsilon_{\mu}^*(p_3)
 \sum_k \Bigg\{ F_{lmk}^{D(cde)}F_{nmk}^{D(abe)}
 {\cal A}^{\mu}
\nonumber\\
&&
 +{2\over E^2} \left(M_l T_{mkl}^{D(dec)}-M_m T_{lkm}^{D(ced)}\right)
    \left(M_n T_{mkn}^{D(bea)}-M_m T_{nkm}^{D(aeb)}\right)
      \left(0,-\sin\theta,0,1-\cos\theta\right)^{\mu}
   \Bigg\}
\nonumber\\   \label{ime}
\end{eqnarray}
with 
\begin{eqnarray}
 {\cal A}^{\mu}
 &\!=\!& \Bigg(1+{3M_n^2-3M_m^2+2M_l^2-4M_k^2\over E^2}
    -\left(1+{M_n^2+M_m^2\over E^2}\right)\cos\theta,
\nonumber\\
&&\quad
    \left(1+{M_n^2-7M_m^2-2M_l^2+2M_k^2\over E^2}\right)\sin\theta
    -\left(1+{M_n^2+M_m^2-2M_l^2\over E^2}\right)\sin\theta\cos\theta,
    0,
\nonumber\\
&&\quad
     {2M_n^2+4M_m^2+2M_l^2-6M_k^2\over E^2}
     +\left(1+{M_n^2-7M_m^2-2M_l^2+2M_k^2\over E^2}\right)\cos\theta
\nonumber\\
&&\quad
     -\left(1+{M_n^2+M_m^2-2M_l^2\over
       E^2}\right)\cos^2\theta\Bigg)^{\mu} ,
 \label{cala}
\end{eqnarray}
where $F_{lmnm}^{D(cdab)}=\sum_k F_{lmk}^{D(cde)}F_{nmk}^{D(abe)}$.
We have not yet specified the polarization $\epsilon_\mu(p_3)$ 
of the  external $A_l^c$ boson %in order to be able to 
so that we are free to 
choose 
a transverse or longitudinal polarization as well as 
a massless or massive $A_l^c$ 
in the following. 



Let us first consider the case %where
in which the $A_l^c$ boson 
is a massive KK mode.
%T
Then, the on-shell polarization can be transverse, as in 
Eq.~(\ref{polart}), or longitudinal, as in Eq.~(\ref{polarl}). 
If we choose %one of 
the transverse polarization 
$\epsilon^*(p_3)=(0,\cos\theta,0,-\sin\theta)$, 
we find that the 
invariant matrix element (\ref{ime}) reduces to 
Eq.~(\ref{tk}), whereas the other transverse polarization, 
$\epsilon^*(p_3)=(0,0,1,0)$, makes the invariant matrix 
element vanish. 
If we choose the longitudinal 
polarization in Eq.~(\ref{polarl}), 
we find that the 
invariant matrix element (\ref{ime}) reduces to 
%Eq.(\ref{kb}). 
Eq.~(\ref{lk}).


Let us next consider the case %where
in which the external 
$A_l^c$ boson is massless. 
Then, only the state with $M_k=M_m$ is allowed in the 
intermediate state, 
since $F_{lmk}^{D(cde)}$ and
$T_{lkm}^{D(ced)}$ reduce to $f_l^{D(c)}\delta_{mk}$. 
Before taking the high energy limit, 
we obtain the result for arbitrary $\xi$ by 
substituting the polarization of $A_l^c$ with its momentum, 
\begin{eqnarray}
 &&ig_5^2 f^{abe}f^{cde} \times
\\
 &&\bigg[
   f_l^{D(c)}\{(M_n^2-M_m^2)F_{nmm}^{D(abd)}
     -M_m(M_n T_{mmn}^{D(bda)}-M_m T_{nmm}^{(abd)})\}
     {p_3^\sigma\eta^{\mu\nu}\over (p_3+p_4)^2+\xi M_m^2}
\nonumber\\
 && + f_l^{D(c)}\{(M_n^2-M_m^2)F_{nmm}^{D(abd)}
     -M_m(M_n T_{mmn}^{D(dba)}-M_m T_{nmm}^{(abd)})\}
     {p_3^\nu\eta^{\mu\sigma}\over (p_4-p_1)^2+\xi M_m^2}
 \bigg]. 
\nonumber
\end{eqnarray}
%Eq.
Equation (\ref{mtb}) implies that this amplitude is proportional 
to a function of the values of the mode functions at the 
boundaries.  
Moreover, it has $\xi$ in %the
its denominator. 
Therefore, the unitary gauge, for which $\xi \to \infty$, clearly 
misses this possible violation of the Ward-Takahashi identity. 
Returning to the high energy approximation, we find 
by using Eq.~(\ref{mtb}) that 
the invariant matrix element in Eq.~(\ref{ime}) 
reduces to Eq.~(\ref{eq:ward_takahashi_id}). 



\end{appendix}

\vspace*{10mm}
%%%%%%%%%%%%% BIBLIOGRAPHY %%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{99}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\cite{Arkani-Hamed:1998rs}
\bibitem{Arkani-Hamed:1998rs}
  N.~Arkani-Hamed, S.~Dimopoulos and G.~R.~Dvali,
  %``The hierarchy problem and new dimensions at a millimeter,''
  Phys.\ Lett.\ B {\bf 429}, 263 (1998)
  [arXiv:hep-ph/9803315];
  %%CITATION = HEP-PH 9803315;%%
%
%\cite{Antoniadis:1998ig}
%\bibitem{Antoniadis:1998ig}
  I.~Antoniadis, N.~Arkani-Hamed, S.~Dimopoulos and G.~R.~Dvali,
  %``New dimensions at a millimeter to a Fermi and superstrings at a TeV,''
  Phys.\ Lett.\ B {\bf 436}, 257 (1998)
  [arXiv:hep-ph/9804398].
  %%CITATION = HEP-PH 9804398;%%

%\cite{Randall:1999ee}
\bibitem{Randall:1999ee}
  L.~Randall and R.~Sundrum,
  %``A large mass hierarchy from a small extra dimension,''
  Phys.\ Rev.\ Lett.\  {\bf 83}, 3370 (1999)
  [arXiv:hep-ph/9905221];
  %%CITATION = HEP-PH 9905221;%%
%
%\cite{Randall:1999vf}
%\bibitem{Randall:1999vf}
%  L.~Randall and R.~Sundrum,
  %``An alternative to compactification,''
  Phys.\ Rev.\ Lett.\  {\bf 83}, 4690 (1999)
  [arXiv:hep-th/9906064].
  %%CITATION = HEP-TH 9906064;%%

%\cite{Horava:1996ma}
\bibitem{Horava:1996ma}
  P.~Horava and E.~Witten,
%  ``Eleven-Dimensional Supergravity on a Manifold with Boundary,''
  Nucl.\ Phys.\ B {\bf 475}, 94 (1996)
  [arXiv:hep-th/9603142].
  %%CITATION = HEP-TH 9603142;%%
  %%Cited 557 times in SPIRES-HEP


%\cite{Antoniadis:1990ew}
\bibitem{Antoniadis:1990ew}
  I.~Antoniadis,
  %``A Possible new dimension at a few TeV,''
  Phys.\ Lett.\ B {\bf 246}, 377 (1990).
  %%CITATION = PHLTA,B246,377;%%


%\cite{Dvali:1996xe}
\bibitem{Dvali:1996xe}
  G.~R.~Dvali and M.~A.~Shifman,
  %``Domain walls in strongly coupled theories,''
  Phys.\ Lett.\ B {\bf 396}, 64 (1997)
  [Erratum-ibid.\ B {\bf 407}, 452 (1997)]
  [arXiv:hep-th/9612128].
  %%CITATION = HEP-TH 9612128;%%

%\cite{Isozumi:2003uh}
\bibitem{Isozumi:2003uh}
  Y.~Isozumi, K.~Ohashi and N.~Sakai,
  %``Massless localized vector field on a wall in D = 5 SQED with tensor
  %multiplets,''
  JHEP {\bf 0311}, 061 (2003)
  [arXiv:hep-th/0310130];
  %%CITATION = HEP-TH 0310130;%%
%\cite{Isozumi:2003rp}\bibitem{Isozumi:2003rp}
  Y.~Isozumi, K.~Ohashi and N.~Sakai,
  %``Exact wall solutions in 5-dimensional SUSY QED at finite coupling,''
  JHEP {\bf 0311}, 060 (2003)
  [arXiv:hep-th/0310189].
  %%CITATION = HEP-TH 0310189;%%

%\cite{Hatanaka:1998yp}
\bibitem{Hatanaka:1998yp}
  H.~Hatanaka, T.~Inami and C.~S.~Lim,
  %``The gauge hierarchy problem and higher dimensional gauge theories,''
  Mod.\ Phys.\ Lett.\ A {\bf 13}, 2601 (1998)
  [arXiv:hep-th/9805067].
  %%CITATION = HEP-TH 9805067;%%

%\cite{Hosotani:1983xw}
\bibitem{Hosotani:1983xw}
  Y.~Hosotani,
  %``Dynamical Mass Generation By Compact Extra Dimensions,''
  Phys.\ Lett.\ B {\bf 126}, 309 (1983);
  %%CITATION = PHLTA,B126,309;%%
%\cite{Hosotani:1988bm}\bibitem{Hosotani:1988bm}
  Y.~Hosotani,
  %``Dynamics Of Nonintegrable Phases And Gauge Symmetry Breaking,''
  Annals Phys.\  {\bf 190}, 233 (1989).
  %%CITATION = APNYA,190,233;%%

%\cite{Scherk:1979zr}
\bibitem{Scherk:1979zr}
  J.~Scherk and J.~H.~Schwarz,
  %``How To Get Masses From Extra Dimensions,''
  Nucl.\ Phys.\ B {\bf 153}, 61 (1979);
  %%CITATION = NUPHA,B153,61;%%
%\cite{Scherk:1978ta}\bibitem{Scherk:1978ta}
  J.~Scherk and J.~H.~Schwarz,
  %``Spontaneous Breaking Of Supersymmetry Through Dimensional Reduction,''
  Phys.\ Lett.\ B {\bf 82}, 60 (1979).
  %%CITATION = PHLTA,B82,60;%%

%\cite{Kawamura:1999nj}
\bibitem{Kawamura:1999nj}
  Y.~Kawamura,
  %``Gauge symmetry reduction from the extra space S(1)/Z(2),''
  Prog.\ Theor.\ Phys.\  {\bf 103}, 613 (2000)
  [arXiv:hep-ph/9902423];
  %%CITATION = HEP-PH 9902423;%%
%\cite{Kawamura:2000ev}\bibitem{Kawamura:2000ev}
  Y.~Kawamura,
  %``Triplet-doublet splitting, proton stability and extra dimension,''
  Prog.\ Theor.\ Phys.\  {\bf 105}, 999 (2001)
  [arXiv:hep-ph/0012125].
  %%CITATION = HEP-PH 0012125;%%

%\cite{Hall:2001pg}
\bibitem{Hall:2001pg}
  L.~J.~Hall and Y.~Nomura,
  %``Gauge unification in higher dimensions,''
  Phys.\ Rev.\ D {\bf 64}, 055003 (2001)
  [arXiv:hep-ph/0103125].
  %%CITATION = HEP-PH 0103125;%%

%\cite{Hebecker:2001jb}
\bibitem{Hebecker:2001jb}
  A.~Hebecker and J.~March-Russell,
  %``The structure of GUT breaking by orbifolding,''
  Nucl.\ Phys.\ B {\bf 625}, 128 (2002)
  [arXiv:hep-ph/0107039].
  %%CITATION = HEP-PH 0107039;%%

%\cite{Abe:2003vg}
\bibitem{Abe:2003vg}
  Y.~Abe, N.~Haba, Y.~Higashide, K.~Kobayashi and M.~Matsunaga,
  %``Unitarity in gauge symmetry breaking on orbifold,''
  Prog.\ Theor.\ Phys.\  {\bf 109}, 831 (2003)
  [arXiv:hep-th/0302115].
  %%CITATION = HEP-TH 0302115;%%

%\cite{Abe:2004wv}
\bibitem{Abe:2004wv}
  Y.~Abe, N.~Haba, K.~Hayakawa, Y.~Matsumoto, M.~Matsunaga and K.~Miyachi,
  %``4D equivalence theorem and gauge symmetry on orbifold,''
  Prog.\ Theor.\ Phys.\  {\bf 113}, 199 (2005)
  [arXiv:hep-th/0402146].
  %%CITATION = HEP-TH 0402146;%%

%\cite{Quiros:2003gg}
\bibitem{Quiros:2003gg}
  M.~Quiros,
  %``New ideas in symmetry breaking,''
  arXiv:hep-ph/0302189.
  %%CITATION = HEP-PH 0302189;%%



%\cite{Hall:2001tn}
\bibitem{Hall:2001tn}
  L.~J.~Hall, H.~Murayama and Y.~Nomura,
  %``Wilson lines and symmetry breaking on orbifolds,''
  Nucl.\ Phys.\ B {\bf 645}, 85 (2002)
  [arXiv:hep-th/0107245].
  %%CITATION = HEP-TH 0107245;%%


%\cite{Haba:2003ux}
\bibitem{Haba:2003ux}
  N.~Haba, Y.~Hosotani and Y.~Kawamura,
  %``Classification and dynamics of equivalence classes in SU(N) gauge  theory
  %on the orbifold S(1)/Z(2),''
  Prog.\ Theor.\ Phys.\  {\bf 111}, 265 (2004)
  [arXiv:hep-ph/0309088].
  %%CITATION = HEP-PH 0309088;%%
  %%Cited 17 times in SPIRES-HEP

%\cite{Oda:2004rm}
\bibitem{Oda:2004rm}
  K.~y.~Oda and A.~Weiler,
  %``Wilson lines in warped space: Dynamical symmetry breaking and
  %restoration,''
  Phys.\ Lett.\ B {\bf 606}, 408 (2005)
  [arXiv:hep-ph/0410061].
  %%CITATION = HEP-PH 0410061;%%
  %%Cited 10 times in SPIRES-HEP

%\cite{Agashe:2004rs}
\bibitem{Agashe:2004rs}
  K.~Agashe, R.~Contino and A.~Pomarol,
  %``The minimal composite Higgs model,''
  Nucl.\ Phys.\ B {\bf 719}, 165 (2005)
  [arXiv:hep-ph/0412089].
  %%CITATION = HEP-PH 0412089;%%
  %%Cited 35 times in SPIRES-HEP


%\cite{Hosotani:2006qp}
\bibitem{Hosotani:2006qp}
  Y.~Hosotani, S.~Noda, Y.~Sakamura and S.~Shimasaki,
  %``Gauge-Higgs unification and quark-lepton phenomenology in the warped
  %spacetime,''
  arXiv:hep-ph/0601241.
  %%CITATION = HEP-PH 0601241;%%
  %%Cited 0 times in SPIRES-HEP



%\cite{Csaki:2003dt}
\bibitem{Csaki:2003dt}
  C.~Csaki, C.~Grojean, H.~Murayama, L.~Pilo and J.~Terning,
  %``Gauge theories on an interval: Unitarity without a Higgs,''
  Phys.\ Rev.\ D {\bf 69}, 055006 (2004)
  [arXiv:hep-ph/0305237].
  %%CITATION = HEP-PH 0305237;%%


%\cite{Nagasawa:2004xk}
\bibitem{Nagasawa:2004xk}
  T.~Nagasawa and M.~Sakamoto,
  %``Higgsless gauge symmetry breaking with a large mass hierarchy,''
  Prog.\ Theor.\ Phys.\  {\bf 112}, 629 (2004)
  [arXiv:hep-ph/0406024];
  %%CITATION = HEP-PH 0406024;%%
%\cite{Lim:2005rc}\bibitem{Lim:2005rc}
  C.~S.~Lim, T.~Nagasawa, M.~Sakamoto and H.~Sonoda,
  %``Supersymmetry in gauge theories with extra dimensions,''
  Phys.\ Rev.\ D {\bf 72}, 064006 (2005)
  [arXiv:hep-th/0502022].
  %%CITATION = HEP-TH 0502022;%%


%\cite{LlewellynSmith:1973ey}
\bibitem{LlewellynSmith:1973ey}
  C.~H.~Llewellyn Smith,
  %``High-Energy Behavior And Gauge Symmetry,''
  Phys.\ Lett.\ B {\bf 46}, 233 (1973);
  %%CITATION = PHLTA,B46,233;%%
%\cite{Dicus:1992vj}\bibitem{Dicus:1992vj}
  D.~A.~Dicus and V.~S.~Mathur,
  %``Upper bounds on the values of masses in unified gauge theories,''
  Phys.\ Rev.\ D {\bf 7}, 3111 (1973);
  %%CITATION = PHRVA,D7,3111;%%
%\cite{Cornwall:1973tb}\bibitem{Cornwall:1973tb}
  J.~M.~Cornwall, D.~N.~Levin and G.~Tiktopoulos,
  %``Uniqueness Of Spontaneously Broken Gauge Theories,''
  Phys.\ Rev.\ Lett.\  {\bf 30}, 1268 (1973)
  [Erratum-ibid.\  {\bf 31}, 572 (1973)];
  %%CITATION = PRLTA,30,1268;%%
%\cite{Cornwall:1974km}\bibitem{Cornwall:1974km}
%  J.~M.~Cornwall, D.~N.~Levin and G.~Tiktopoulos,
%``Derivation Of Gauge Invariance From High-Energy Unitarity Bounds On The S -
  %Matrix,''
  Phys.\ Rev.\ D {\bf 10}, 1145 (1974)
  [Erratum-ibid.\ D {\bf 11}, 972 (1975)];
  %%CITATION = PHRVA,D10,1145;%%
%\cite{Lee:1977yc}\bibitem{Lee:1977yc}
  B.~W.~Lee, C.~Quigg and H.~B.~Thacker,
  %``The Strength Of Weak Interactions At Very High-Energies And The Higgs Boson
  %Mass,''
  Phys.\ Rev.\ Lett.\  {\bf 38}, 883 (1977);
  %%CITATION = PRLTA,38,883;%%
%\cite{Chanowitz:1985hj}\bibitem{Chanowitz:1985hj}
  M.~S.~Chanowitz and M.~K.~Gaillard,
  %``The Tev Physics Of Strongly Interacting W's And Z's,''
  Nucl.\ Phys.\ B {\bf 261}, 379 (1985).
  %%CITATION = NUPHA,B261,379;%%


%\cite{SekharChivukula:2001hz}
\bibitem{SekharChivukula:2001hz}
  R.~Sekhar Chivukula, D.~A.~Dicus and H.~J.~He,
  %``Unitarity of compactified five dimensional Yang-Mills theory,''
  Phys.\ Lett.\ B {\bf 525}, 175 (2002)
  [arXiv:hep-ph/0111016];
  %%CITATION = HEP-PH 0111016;%%
%\cite{Chivukula:2003kq}\bibitem{Chivukula:2003kq}
  R.~S.~Chivukula, D.~A.~Dicus, H.~J.~He and S.~Nandi,
  %``Unitarity of the higher dimensional standard model,''
  Phys.\ Lett.\ B {\bf 562}, 109 (2003)
  [arXiv:hep-ph/0302263].
  %%CITATION = HEP-PH 0302263;%%


%\cite{Hall:2003yc}
\bibitem{Hall:2003yc}
  L.~J.~Hall, Y.~Nomura, T.~Okui and S.~J.~Oliver,
  %``Explicit supersymmetry breaking on boundaries of warped extra
  %dimensions,''
  arXiv:hep-th/0302192.
  %%CITATION = HEP-TH 0302192;%%


%\cite{Becchi:1974md}
\bibitem{Becchi:1974md}
  C.~Becchi, A.~Rouet and R.~Stora,
  %``Renormalization Of The Abelian Higgs-Kibble Model,''
  Commun.\ Math.\ Phys.\  {\bf 42}, 127 (1975);
  %%CITATION = CMPHA,42,127;%%
%\cite{Becchi:1975nq}\bibitem{Becchi:1975nq}
  C.~Becchi, A.~Rouet and R.~Stora,
  %``Renormalization Of Gauge Theories,''
  Annals Phys.\  {\bf 98}, 287 (1976); 
  %%CITATION = APNYA,98,287;%%
I.~V.~Tyutin, Lebedev Institute preprint N39 (1975). 

%\cite{Kugo:1977zq}
\bibitem{Kugo:1977zq}
  T.~Kugo and I.~Ojima,
  %``Manifestly Covariant Canonical Formulation Of Yang-Mills Field Theories:
  %Physical State Subsidiary Conditions And Physical S Matrix Unitarity,''
  Phys.\ Lett.\ B {\bf 73}, 459 (1978);
  %%CITATION = PHLTA,B73,459;%%
Prog.\ Theor.\ Phys.\ Suppl.\ {\bf 66}, 1 (1979). 

%\cite{Kugo:1981hm}
\bibitem{Kugo:1981hm}
  T.~Kugo and S.~Uehara,
  %``General Procedure Of Gauge Fixing Based On Brs Invariance Principle,''
  Nucl.\ Phys.\ B {\bf 197}, 378 (1982).
  %%CITATION = NUPHA,B197,378;%%

%\cite{He:2004zr}
\bibitem{He:2004zr}
  H.~J.~He,
  %``Higgsless deconstruction without boundary condition,''
  Int.\ J.\ Mod.\ Phys.\ A {\bf 20}, 3362 (2005)
  [arXiv:hep-ph/0412113].
  %%CITATION = HEP-PH 0412113;%%

%\cite{Muck:2001yv}
\bibitem{Muck:2001yv}
  A.~Muck, A.~Pilaftsis and R.~Ruckl,
  %``Minimal higher-dimensional extensions of the standard model and
  %electroweak observables,''
  Phys.\ Rev.\ D {\bf 65}, 085037 (2002)
  [arXiv:hep-ph/0110391].
  %%CITATION = HEP-PH 0110391;%%

%\cite{Ohl:2003dp}
\bibitem{Ohl:2003dp}
  T.~Ohl and C.~Schwinn,
%``Unitarity, BRST symmetry and Ward identities in orbifold gauge  theories,''
  Phys.\ Rev.\ D {\bf 70}, 045019 (2004)
  [arXiv:hep-ph/0312263];
  %%CITATION = HEP-PH 0312263;%%
%\cite{Schwinn:2004xa}\bibitem{Schwinn:2004xa}
  C.~Schwinn,
  %``Higgsless fermion masses and unitarity,''
  Phys.\ Rev.\ D {\bf 69}, 116005 (2004)
  [arXiv:hep-ph/0402118].
  %%CITATION = HEP-PH 0402118;%%


%\cite{Muck:2004br}
\bibitem{Muck:2004br}
  A.~Muck, L.~Nilse, A.~Pilaftsis and R.~Ruckl,
%``Quantization and high energy unitarity of 5D orbifold theories with  brane
  %kinetic terms,''
  Phys.\ Rev.\ D {\bf 71}, 066004 (2005)
  [arXiv:hep-ph/0411258].
  %%CITATION = HEP-PH 0411258;%%


%\cite{Arkani-Hamed:2001ca}
\bibitem{Arkani-Hamed:2001ca}
  N.~Arkani-Hamed, A.~G.~Cohen and H.~Georgi,
  %``(De)constructing dimensions,''
  Phys.\ Rev.\ Lett.\  {\bf 86}, 4757 (2001)
  [arXiv:hep-th/0104005];
  %%CITATION = HEP-TH 0104005;%%
%\cite{Hill:2000mu}\bibitem{Hill:2000mu}
  C.~T.~Hill, S.~Pokorski and J.~Wang,
  %``Gauge invariant effective Lagrangian for Kaluza-Klein modes,''
  Phys.\ Rev.\ D {\bf 64}, 105005 (2001)
  [arXiv:hep-th/0104035].
  %%CITATION = HEP-TH 0104035;%%


%\cite{Cacciapaglia:2004rb}
\bibitem{Cacciapaglia:2004rb}
  G.~Cacciapaglia, C.~Csaki, C.~Grojean and J.~Terning,
  %``Curing the ills of Higgsless models: The S parameter and
%  unitarity,''
  Phys.\ Rev.\ D {\bf 71}, 035015 (2005)
  [arXiv:hep-ph/0409126];
  %%CITATION = HEP-PH 0409126;%%
%\cite{Foadi:2004ps}\bibitem{Foadi:2004ps}
  R.~Foadi, S.~Gopalakrishna and C.~Schmidt,
  %``Effects of fermion localization in Higgsless theories and
%  electroweak
  %constraints,''
  Phys.\ Lett.\ B {\bf 606}, 157 (2005)
  [arXiv:hep-ph/0409266];
  %%CITATION = HEP-PH 0409266;%%
%\cite{Chivukula:2005bn}\bibitem{Chivukula:2005bn}
  R.~S.~Chivukula, E.~H.~Simmons, H.~J.~He, M.~Kurachi and
  M.~Tanabashi,
  %``Deconstructed Higgsless models with one-site delocalization,''
  Phys.\ Rev.\ D {\bf 71}, 115001 (2005)
  [arXiv:hep-ph/0502162];
  %%CITATION = HEP-PH 0502162;%%
%\cite{Casalbuoni:2005rs}\bibitem{Casalbuoni:2005rs}
  R.~Casalbuoni, S.~De Curtis, D.~Dolce and D.~Dominici,
  %``Playing with fermion couplings in Higgsless models,''
  Phys.\ Rev.\ D {\bf 71}, 075015 (2005)
  [arXiv:hep-ph/0502209];
  %%CITATION = HEP-PH 0502209;%%
%\cite{SekharChivukula:2005xm}\bibitem{SekharChivukula:2005xm}
  R.~Sekhar Chivukula, E.~H.~Simmons, H.~J.~He, M.~Kurachi and
  M.~Tanabashi,
  %``Ideal fermion delocalization in Higgsless models,''
  Phys.\ Rev.\ D {\bf 72}, 015008 (2005)
  [arXiv:hep-ph/0504114].
  %%CITATION = HEP-PH 0504114;%%


%\cite{Csaki:2005vy}
\bibitem{Csaki:2005vy}
  C.~Csaki, J.~Hubisz and P.~Meade,
  %``TASI lectures on electroweak symmetry breaking from extra dimensions,''
  arXiv:hep-ph/0510275.
  %%CITATION = HEP-PH 0510275;%%


%%%%%%%%%%%%%%%%%%%%%
\end{thebibliography}
%%%%%%%%%%%%%%%%%%%%%

\end{document}

hep-th0604121.extracted_bib
\begin{thebibliography}{99}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\cite{Arkani-Hamed:1998rs}
\bibitem{Arkani-Hamed:1998rs}
  N.~Arkani-Hamed, S.~Dimopoulos and G.~R.~Dvali,
  %``The hierarchy problem and new dimensions at a millimeter,''
  Phys.\ Lett.\ B {\bf 429}, 263 (1998)
  [arXiv:hep-ph/9803315];
  %%CITATION = HEP-PH 9803315;%%
%
%\cite{Antoniadis:1998ig}
%\bibitem{Antoniadis:1998ig}
  I.~Antoniadis, N.~Arkani-Hamed, S.~Dimopoulos and G.~R.~Dvali,
  %``New dimensions at a millimeter to a Fermi and superstrings at a TeV,''
  Phys.\ Lett.\ B {\bf 436}, 257 (1998)
  [arXiv:hep-ph/9804398].
  %%CITATION = HEP-PH 9804398;%%

%\cite{Randall:1999ee}
\bibitem{Randall:1999ee}
  L.~Randall and R.~Sundrum,
  %``A large mass hierarchy from a small extra dimension,''
  Phys.\ Rev.\ Lett.\  {\bf 83}, 3370 (1999)
  [arXiv:hep-ph/9905221];
  %%CITATION = HEP-PH 9905221;%%
%
%\cite{Randall:1999vf}
%\bibitem{Randall:1999vf}
%  L.~Randall and R.~Sundrum,
  %``An alternative to compactification,''
  Phys.\ Rev.\ Lett.\  {\bf 83}, 4690 (1999)
  [arXiv:hep-th/9906064].
  %%CITATION = HEP-TH 9906064;%%

%\cite{Horava:1996ma}
\bibitem{Horava:1996ma}
  P.~Horava and E.~Witten,
%  ``Eleven-Dimensional Supergravity on a Manifold with Boundary,''
  Nucl.\ Phys.\ B {\bf 475}, 94 (1996)
  [arXiv:hep-th/9603142].
  %%CITATION = HEP-TH 9603142;%%
  %%Cited 557 times in SPIRES-HEP


%\cite{Antoniadis:1990ew}
\bibitem{Antoniadis:1990ew}
  I.~Antoniadis,
  %``A Possible new dimension at a few TeV,''
  Phys.\ Lett.\ B {\bf 246}, 377 (1990).
  %%CITATION = PHLTA,B246,377;%%


%\cite{Dvali:1996xe}
\bibitem{Dvali:1996xe}
  G.~R.~Dvali and M.~A.~Shifman,
  %``Domain walls in strongly coupled theories,''
  Phys.\ Lett.\ B {\bf 396}, 64 (1997)
  [Erratum-ibid.\ B {\bf 407}, 452 (1997)]
  [arXiv:hep-th/9612128].
  %%CITATION = HEP-TH 9612128;%%

%\cite{Isozumi:2003uh}
\bibitem{Isozumi:2003uh}
  Y.~Isozumi, K.~Ohashi and N.~Sakai,
  %``Massless localized vector field on a wall in D = 5 SQED with tensor
  %multiplets,''
  JHEP {\bf 0311}, 061 (2003)
  [arXiv:hep-th/0310130];
  %%CITATION = HEP-TH 0310130;%%
%\cite{Isozumi:2003rp}\bibitem{Isozumi:2003rp}
  Y.~Isozumi, K.~Ohashi and N.~Sakai,
  %``Exact wall solutions in 5-dimensional SUSY QED at finite coupling,''
  JHEP {\bf 0311}, 060 (2003)
  [arXiv:hep-th/0310189].
  %%CITATION = HEP-TH 0310189;%%

%\cite{Hatanaka:1998yp}
\bibitem{Hatanaka:1998yp}
  H.~Hatanaka, T.~Inami and C.~S.~Lim,
  %``The gauge hierarchy problem and higher dimensional gauge theories,''
  Mod.\ Phys.\ Lett.\ A {\bf 13}, 2601 (1998)
  [arXiv:hep-th/9805067].
  %%CITATION = HEP-TH 9805067;%%

%\cite{Hosotani:1983xw}
\bibitem{Hosotani:1983xw}
  Y.~Hosotani,
  %``Dynamical Mass Generation By Compact Extra Dimensions,''
  Phys.\ Lett.\ B {\bf 126}, 309 (1983);
  %%CITATION = PHLTA,B126,309;%%
%\cite{Hosotani:1988bm}\bibitem{Hosotani:1988bm}
  Y.~Hosotani,
  %``Dynamics Of Nonintegrable Phases And Gauge Symmetry Breaking,''
  Annals Phys.\  {\bf 190}, 233 (1989).
  %%CITATION = APNYA,190,233;%%

%\cite{Scherk:1979zr}
\bibitem{Scherk:1979zr}
  J.~Scherk and J.~H.~Schwarz,
  %``How To Get Masses From Extra Dimensions,''
  Nucl.\ Phys.\ B {\bf 153}, 61 (1979);
  %%CITATION = NUPHA,B153,61;%%
%\cite{Scherk:1978ta}\bibitem{Scherk:1978ta}
  J.~Scherk and J.~H.~Schwarz,
  %``Spontaneous Breaking Of Supersymmetry Through Dimensional Reduction,''
  Phys.\ Lett.\ B {\bf 82}, 60 (1979).
  %%CITATION = PHLTA,B82,60;%%

%\cite{Kawamura:1999nj}
\bibitem{Kawamura:1999nj}
  Y.~Kawamura,
  %``Gauge symmetry reduction from the extra space S(1)/Z(2),''
  Prog.\ Theor.\ Phys.\  {\bf 103}, 613 (2000)
  [arXiv:hep-ph/9902423];
  %%CITATION = HEP-PH 9902423;%%
%\cite{Kawamura:2000ev}\bibitem{Kawamura:2000ev}
  Y.~Kawamura,
  %``Triplet-doublet splitting, proton stability and extra dimension,''
  Prog.\ Theor.\ Phys.\  {\bf 105}, 999 (2001)
  [arXiv:hep-ph/0012125].
  %%CITATION = HEP-PH 0012125;%%

%\cite{Hall:2001pg}
\bibitem{Hall:2001pg}
  L.~J.~Hall and Y.~Nomura,
  %``Gauge unification in higher dimensions,''
  Phys.\ Rev.\ D {\bf 64}, 055003 (2001)
  [arXiv:hep-ph/0103125].
  %%CITATION = HEP-PH 0103125;%%

%\cite{Hebecker:2001jb}
\bibitem{Hebecker:2001jb}
  A.~Hebecker and J.~March-Russell,
  %``The structure of GUT breaking by orbifolding,''
  Nucl.\ Phys.\ B {\bf 625}, 128 (2002)
  [arXiv:hep-ph/0107039].
  %%CITATION = HEP-PH 0107039;%%

%\cite{Abe:2003vg}
\bibitem{Abe:2003vg}
  Y.~Abe, N.~Haba, Y.~Higashide, K.~Kobayashi and M.~Matsunaga,
  %``Unitarity in gauge symmetry breaking on orbifold,''
  Prog.\ Theor.\ Phys.\  {\bf 109}, 831 (2003)
  [arXiv:hep-th/0302115].
  %%CITATION = HEP-TH 0302115;%%

%\cite{Abe:2004wv}
\bibitem{Abe:2004wv}
  Y.~Abe, N.~Haba, K.~Hayakawa, Y.~Matsumoto, M.~Matsunaga and K.~Miyachi,
  %``4D equivalence theorem and gauge symmetry on orbifold,''
  Prog.\ Theor.\ Phys.\  {\bf 113}, 199 (2005)
  [arXiv:hep-th/0402146].
  %%CITATION = HEP-TH 0402146;%%

%\cite{Quiros:2003gg}
\bibitem{Quiros:2003gg}
  M.~Quiros,
  %``New ideas in symmetry breaking,''
  arXiv:hep-ph/0302189.
  %%CITATION = HEP-PH 0302189;%%



%\cite{Hall:2001tn}
\bibitem{Hall:2001tn}
  L.~J.~Hall, H.~Murayama and Y.~Nomura,
  %``Wilson lines and symmetry breaking on orbifolds,''
  Nucl.\ Phys.\ B {\bf 645}, 85 (2002)
  [arXiv:hep-th/0107245].
  %%CITATION = HEP-TH 0107245;%%


%\cite{Haba:2003ux}
\bibitem{Haba:2003ux}
  N.~Haba, Y.~Hosotani and Y.~Kawamura,
  %``Classification and dynamics of equivalence classes in SU(N) gauge  theory
  %on the orbifold S(1)/Z(2),''
  Prog.\ Theor.\ Phys.\  {\bf 111}, 265 (2004)
  [arXiv:hep-ph/0309088].
  %%CITATION = HEP-PH 0309088;%%
  %%Cited 17 times in SPIRES-HEP

%\cite{Oda:2004rm}
\bibitem{Oda:2004rm}
  K.~y.~Oda and A.~Weiler,
  %``Wilson lines in warped space: Dynamical symmetry breaking and
  %restoration,''
  Phys.\ Lett.\ B {\bf 606}, 408 (2005)
  [arXiv:hep-ph/0410061].
  %%CITATION = HEP-PH 0410061;%%
  %%Cited 10 times in SPIRES-HEP

%\cite{Agashe:2004rs}
\bibitem{Agashe:2004rs}
  K.~Agashe, R.~Contino and A.~Pomarol,
  %``The minimal composite Higgs model,''
  Nucl.\ Phys.\ B {\bf 719}, 165 (2005)
  [arXiv:hep-ph/0412089].
  %%CITATION = HEP-PH 0412089;%%
  %%Cited 35 times in SPIRES-HEP


%\cite{Hosotani:2006qp}
\bibitem{Hosotani:2006qp}
  Y.~Hosotani, S.~Noda, Y.~Sakamura and S.~Shimasaki,
  %``Gauge-Higgs unification and quark-lepton phenomenology in the warped
  %spacetime,''
  arXiv:hep-ph/0601241.
  %%CITATION = HEP-PH 0601241;%%
  %%Cited 0 times in SPIRES-HEP



%\cite{Csaki:2003dt}
\bibitem{Csaki:2003dt}
  C.~Csaki, C.~Grojean, H.~Murayama, L.~Pilo and J.~Terning,
  %``Gauge theories on an interval: Unitarity without a Higgs,''
  Phys.\ Rev.\ D {\bf 69}, 055006 (2004)
  [arXiv:hep-ph/0305237].
  %%CITATION = HEP-PH 0305237;%%


%\cite{Nagasawa:2004xk}
\bibitem{Nagasawa:2004xk}
  T.~Nagasawa and M.~Sakamoto,
  %``Higgsless gauge symmetry breaking with a large mass hierarchy,''
  Prog.\ Theor.\ Phys.\  {\bf 112}, 629 (2004)
  [arXiv:hep-ph/0406024];
  %%CITATION = HEP-PH 0406024;%%
%\cite{Lim:2005rc}\bibitem{Lim:2005rc}
  C.~S.~Lim, T.~Nagasawa, M.~Sakamoto and H.~Sonoda,
  %``Supersymmetry in gauge theories with extra dimensions,''
  Phys.\ Rev.\ D {\bf 72}, 064006 (2005)
  [arXiv:hep-th/0502022].
  %%CITATION = HEP-TH 0502022;%%


%\cite{LlewellynSmith:1973ey}
\bibitem{LlewellynSmith:1973ey}
  C.~H.~Llewellyn Smith,
  %``High-Energy Behavior And Gauge Symmetry,''
  Phys.\ Lett.\ B {\bf 46}, 233 (1973);
  %%CITATION = PHLTA,B46,233;%%
%\cite{Dicus:1992vj}\bibitem{Dicus:1992vj}
  D.~A.~Dicus and V.~S.~Mathur,
  %``Upper bounds on the values of masses in unified gauge theories,''
  Phys.\ Rev.\ D {\bf 7}, 3111 (1973);
  %%CITATION = PHRVA,D7,3111;%%
%\cite{Cornwall:1973tb}\bibitem{Cornwall:1973tb}
  J.~M.~Cornwall, D.~N.~Levin and G.~Tiktopoulos,
  %``Uniqueness Of Spontaneously Broken Gauge Theories,''
  Phys.\ Rev.\ Lett.\  {\bf 30}, 1268 (1973)
  [Erratum-ibid.\  {\bf 31}, 572 (1973)];
  %%CITATION = PRLTA,30,1268;%%
%\cite{Cornwall:1974km}\bibitem{Cornwall:1974km}
%  J.~M.~Cornwall, D.~N.~Levin and G.~Tiktopoulos,
%``Derivation Of Gauge Invariance From High-Energy Unitarity Bounds On The S -
  %Matrix,''
  Phys.\ Rev.\ D {\bf 10}, 1145 (1974)
  [Erratum-ibid.\ D {\bf 11}, 972 (1975)];
  %%CITATION = PHRVA,D10,1145;%%
%\cite{Lee:1977yc}\bibitem{Lee:1977yc}
  B.~W.~Lee, C.~Quigg and H.~B.~Thacker,
  %``The Strength Of Weak Interactions At Very High-Energies And The Higgs Boson
  %Mass,''
  Phys.\ Rev.\ Lett.\  {\bf 38}, 883 (1977);
  %%CITATION = PRLTA,38,883;%%
%\cite{Chanowitz:1985hj}\bibitem{Chanowitz:1985hj}
  M.~S.~Chanowitz and M.~K.~Gaillard,
  %``The Tev Physics Of Strongly Interacting W's And Z's,''
  Nucl.\ Phys.\ B {\bf 261}, 379 (1985).
  %%CITATION = NUPHA,B261,379;%%


%\cite{SekharChivukula:2001hz}
\bibitem{SekharChivukula:2001hz}
  R.~Sekhar Chivukula, D.~A.~Dicus and H.~J.~He,
  %``Unitarity of compactified five dimensional Yang-Mills theory,''
  Phys.\ Lett.\ B {\bf 525}, 175 (2002)
  [arXiv:hep-ph/0111016];
  %%CITATION = HEP-PH 0111016;%%
%\cite{Chivukula:2003kq}\bibitem{Chivukula:2003kq}
  R.~S.~Chivukula, D.~A.~Dicus, H.~J.~He and S.~Nandi,
  %``Unitarity of the higher dimensional standard model,''
  Phys.\ Lett.\ B {\bf 562}, 109 (2003)
  [arXiv:hep-ph/0302263].
  %%CITATION = HEP-PH 0302263;%%


%\cite{Hall:2003yc}
\bibitem{Hall:2003yc}
  L.~J.~Hall, Y.~Nomura, T.~Okui and S.~J.~Oliver,
  %``Explicit supersymmetry breaking on boundaries of warped extra
  %dimensions,''
  arXiv:hep-th/0302192.
  %%CITATION = HEP-TH 0302192;%%


%\cite{Becchi:1974md}
\bibitem{Becchi:1974md}
  C.~Becchi, A.~Rouet and R.~Stora,
  %``Renormalization Of The Abelian Higgs-Kibble Model,''
  Commun.\ Math.\ Phys.\  {\bf 42}, 127 (1975);
  %%CITATION = CMPHA,42,127;%%
%\cite{Becchi:1975nq}\bibitem{Becchi:1975nq}
  C.~Becchi, A.~Rouet and R.~Stora,
  %``Renormalization Of Gauge Theories,''
  Annals Phys.\  {\bf 98}, 287 (1976); 
  %%CITATION = APNYA,98,287;%%
I.~V.~Tyutin, Lebedev Institute preprint N39 (1975). 

%\cite{Kugo:1977zq}
\bibitem{Kugo:1977zq}
  T.~Kugo and I.~Ojima,
  %``Manifestly Covariant Canonical Formulation Of Yang-Mills Field Theories:
  %Physical State Subsidiary Conditions And Physical S Matrix Unitarity,''
  Phys.\ Lett.\ B {\bf 73}, 459 (1978);
  %%CITATION = PHLTA,B73,459;%%
Prog.\ Theor.\ Phys.\ Suppl.\ {\bf 66}, 1 (1979). 

%\cite{Kugo:1981hm}
\bibitem{Kugo:1981hm}
  T.~Kugo and S.~Uehara,
  %``General Procedure Of Gauge Fixing Based On Brs Invariance Principle,''
  Nucl.\ Phys.\ B {\bf 197}, 378 (1982).
  %%CITATION = NUPHA,B197,378;%%

%\cite{He:2004zr}
\bibitem{He:2004zr}
  H.~J.~He,
  %``Higgsless deconstruction without boundary condition,''
  Int.\ J.\ Mod.\ Phys.\ A {\bf 20}, 3362 (2005)
  [arXiv:hep-ph/0412113].
  %%CITATION = HEP-PH 0412113;%%

%\cite{Muck:2001yv}
\bibitem{Muck:2001yv}
  A.~Muck, A.~Pilaftsis and R.~Ruckl,
  %``Minimal higher-dimensional extensions of the standard model and
  %electroweak observables,''
  Phys.\ Rev.\ D {\bf 65}, 085037 (2002)
  [arXiv:hep-ph/0110391].
  %%CITATION = HEP-PH 0110391;%%

%\cite{Ohl:2003dp}
\bibitem{Ohl:2003dp}
  T.~Ohl and C.~Schwinn,
%``Unitarity, BRST symmetry and Ward identities in orbifold gauge  theories,''
  Phys.\ Rev.\ D {\bf 70}, 045019 (2004)
  [arXiv:hep-ph/0312263];
  %%CITATION = HEP-PH 0312263;%%
%\cite{Schwinn:2004xa}\bibitem{Schwinn:2004xa}
  C.~Schwinn,
  %``Higgsless fermion masses and unitarity,''
  Phys.\ Rev.\ D {\bf 69}, 116005 (2004)
  [arXiv:hep-ph/0402118].
  %%CITATION = HEP-PH 0402118;%%


%\cite{Muck:2004br}
\bibitem{Muck:2004br}
  A.~Muck, L.~Nilse, A.~Pilaftsis and R.~Ruckl,
%``Quantization and high energy unitarity of 5D orbifold theories with  brane
  %kinetic terms,''
  Phys.\ Rev.\ D {\bf 71}, 066004 (2005)
  [arXiv:hep-ph/0411258].
  %%CITATION = HEP-PH 0411258;%%


%\cite{Arkani-Hamed:2001ca}
\bibitem{Arkani-Hamed:2001ca}
  N.~Arkani-Hamed, A.~G.~Cohen and H.~Georgi,
  %``(De)constructing dimensions,''
  Phys.\ Rev.\ Lett.\  {\bf 86}, 4757 (2001)
  [arXiv:hep-th/0104005];
  %%CITATION = HEP-TH 0104005;%%
%\cite{Hill:2000mu}\bibitem{Hill:2000mu}
  C.~T.~Hill, S.~Pokorski and J.~Wang,
  %``Gauge invariant effective Lagrangian for Kaluza-Klein modes,''
  Phys.\ Rev.\ D {\bf 64}, 105005 (2001)
  [arXiv:hep-th/0104035].
  %%CITATION = HEP-TH 0104035;%%


%\cite{Cacciapaglia:2004rb}
\bibitem{Cacciapaglia:2004rb}
  G.~Cacciapaglia, C.~Csaki, C.~Grojean and J.~Terning,
  %``Curing the ills of Higgsless models: The S parameter and
%  unitarity,''
  Phys.\ Rev.\ D {\bf 71}, 035015 (2005)
  [arXiv:hep-ph/0409126];
  %%CITATION = HEP-PH 0409126;%%
%\cite{Foadi:2004ps}\bibitem{Foadi:2004ps}
  R.~Foadi, S.~Gopalakrishna and C.~Schmidt,
  %``Effects of fermion localization in Higgsless theories and
%  electroweak
  %constraints,''
  Phys.\ Lett.\ B {\bf 606}, 157 (2005)
  [arXiv:hep-ph/0409266];
  %%CITATION = HEP-PH 0409266;%%
%\cite{Chivukula:2005bn}\bibitem{Chivukula:2005bn}
  R.~S.~Chivukula, E.~H.~Simmons, H.~J.~He, M.~Kurachi and
  M.~Tanabashi,
  %``Deconstructed Higgsless models with one-site delocalization,''
  Phys.\ Rev.\ D {\bf 71}, 115001 (2005)
  [arXiv:hep-ph/0502162];
  %%CITATION = HEP-PH 0502162;%%
%\cite{Casalbuoni:2005rs}\bibitem{Casalbuoni:2005rs}
  R.~Casalbuoni, S.~De Curtis, D.~Dolce and D.~Dominici,
  %``Playing with fermion couplings in Higgsless models,''
  Phys.\ Rev.\ D {\bf 71}, 075015 (2005)
  [arXiv:hep-ph/0502209];
  %%CITATION = HEP-PH 0502209;%%
%\cite{SekharChivukula:2005xm}\bibitem{SekharChivukula:2005xm}
  R.~Sekhar Chivukula, E.~H.~Simmons, H.~J.~He, M.~Kurachi and
  M.~Tanabashi,
  %``Ideal fermion delocalization in Higgsless models,''
  Phys.\ Rev.\ D {\bf 72}, 015008 (2005)
  [arXiv:hep-ph/0504114].
  %%CITATION = HEP-PH 0504114;%%


%\cite{Csaki:2005vy}
\bibitem{Csaki:2005vy}
  C.~Csaki, J.~Hubisz and P.~Meade,
  %``TASI lectures on electroweak symmetry breaking from extra dimensions,''
  arXiv:hep-ph/0510275.
  %%CITATION = HEP-PH 0510275;%%


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\end{thebibliography}