text:
\section{Advantageous Deconstruction}

\renewcommand{\thefootnote}{\arabic{footnote}}

Deconstruction\cite{DC}
is a powerful means to explore the rich
dynamics of gauge theories in four and higher dimensions.
A compactified higher dimensional theory may be deconstructed
into a proper moose representation\cite{moose} or the
equivalent transverse Wilson lattice\cite{Tlattice}.
An essential advantage of the deconstruction
is that it allows us to formulate the often involved
higher dimensional gauge symmetry breaking (without/with
gauge group rank reduction) in terms
of the conventional 4D gauged nonlinear sigma model \`{a} la
CCWZ\cite{CCWZ},
where no extra boundary condition (BC) is required {\it a priori.}
Furthermore, deconstruction allows us to identify and analyze
the general 4D moose theories with arbitrary inputs of gauge
couplings $g_j^{~}$ and Goldstone decay constants $f_k^{~}$
or with only a few Kaluza-Klein (KK)\cite{KK} modes.
Such theories need not resemble any known higher-D geometry,
but as we will show,
they can still exhibit {\it delayed unitarity violation}
(effective unitarity) as a {\it collective effect} of many
participating gauge groups,  which was originally revealed
for the deconstruction of specified 5D geometries\cite{CH,SUSY03}.
The effective unitarity in the compactified or deconstructed
5D Yang-Mills theories is ensured by {\it the presence of spin-1
vector-bosons (gauge KK modes)}\,\cite{CDH,CH,SUSY03}
rather than the conventional Higgs scalar\cite{Higgs,SM-UB},
which is the key for seeking realistic Higgsless models
of electroweak symmetry breaking (EWSB)
in the recent literature\cite{HLxx,HLmoose}.\,
These new vector bosons also provide discovery signatures
at the CERN LHC.

\section{Eaten Goldstone Bosons
         in General Higgsless Moose Theory}

Consider a most general linear moose theory,
consisting of the replicated gauge group
\,$G_A^{N+1}\otimes G_B^{M+1}\otimes G_C^{L+1}\otimes\cdots$\,
which is {\it spontaneously broken} to the diagonal
subgroup $\,G_D\,$ by the link fields $\,U_j\,$
transforming as
bi-fundamentals under the two adjacent gauge groups.
Without losing generality,  we will set\cite{HLmoose}
\,$G_A=SU(2)$,\, \,$G_B=U(1)$\, and \,$G_C = \cdots =0$\,
for convenience of analyzing the EWSB,
so the residual gauge symmetry is
$\,G_D=U(1)_{\rm em}\,$\, with a massless photon,
as depicted in Fig.\,1\,\cite{HLmoose}.

\begin{figure}[h]
\label{fig:Fig1}
\begin{center}
\vspace*{-5mm}
\centerline{\psfig{file=fig1.eps,width=13cm}}
\setlength{\unitlength}{1mm}
\begin{picture}(0,0)
\end{picture}
\vspace*{-5mm}
\caption{A generic Higgsless deconstruction of linear moose theory
with arbitrary product gauge group
\,$G_A^{N+1}\otimes G_B^{M+1}$\,,\, where the
most general inputs of gauge couplings and Goldstone decay constants
$\{g_j^{~},\,f_k^{~}\}$ $(j=0,1,\cdots,K+1;~k=1,2,\cdots,K+1;
~K\equiv N+M)$ are allowed.
The open circles represent the group $G_A$
and shaded circles the group $G_B$.}
\end{center}
\vspace*{-4mm}
\end{figure}

The Lagrangian for this linear moose theory is
\begin{equation}
\label{eq:L}
\displaystyle
  {\cal L} \,~=~\,
  \sum_{j=0}^{K+1} -\dfrac{1}{\,2\,}
                   \mbox{Tr}\(F_{j\mu\nu} F_j^{\mu\nu}\)
~+\,
  \sum_{j=1}^{K+1} \df{\,f_j^2\,}{4} \mbox{Tr}\[
                   (D_\mu U_j)^\dagger (D^\mu U_j) \],
\end{equation}
with $~
\displaystyle
  D^\mu U_j = \partial^\mu U_j - i g_{j-1}^{~}\mathbf A_{j-1}^\mu U_j
                               + i g_j^{~}U_j \mathbf A_{j}^\mu \,,
~$
and
$~U_j = \exp\[i2\mbox {\boldmath$\pi$}_j^{~}/f_j^{~}\] \,$,\,
where
\,$\mathbf A_j^\mu\equiv A_j^{a\mu}T^a \in SU(2)_j$\,
for $j=0,1,\cdots, N$,
and
\,$\mathbf A^\mu_j\equiv A^\mu_jT^3\in U(1)_j$ $(j=N+1,\cdots,K+1)$.\,
Also,
\,$\mbox {\boldmath$\pi$}_j \equiv \pi^a_jT^a$  ($1\leqslant j \leqslant N+1$)\,
and
\,$\mbox {\boldmath$\pi$}_j \equiv \pi^3_jT^3$  ($N+2 \leqslant j \leqslant K+1$).\,
The $(N+1)\times (N+1)$ mass matrix of charged gauge bosons is
given by\cite{HLmoose}
{\small
\begin{equation}
\label{eq:M_CC}
\mathbb{M}_{\,W}^{\,2} = \df{1}{4}
\left(
\begin{array}{c|c|c|c|c}
&&&&
\\[-2mm]
g_0^2 f_1^2    & -g_0^{~} g_1^{~} f_1^2     &      &
\\[1.5mm]
\hline
&&&&
\\[-2mm]
-g_0^{~} g_1^{~} f_1^2  &  g_1^2(f_1^2\!+\!f_2^2)
& -g_1^{~} g_2^{~} f_2^2         &
\\[1.5mm]
\hline
&&&&
\\[-2mm]
& -g_1^{~} g_2^{~} f_2^2     &  g_2^2 (f_2^2 \!+\! f_3^2) &
                 - g_2^{~} g_3^{~} f_3^2
\\[1.5mm]
\hline
&&&&
\\[-4mm]
 & & \ddots & \ddots & \ddots
\\[0.5mm]
\hline
&&&&
\\[-2mm]
 & & & -g_{N-1}^{~} g_{N}^{~} f_N^2         &
          g_N^2(f_N^2\!+\!f_{N+1}^2)
\\[1.5mm]
\end{array}
\right),
\end{equation}
}
\noindent\hspace*{-2mm}
which may be denoted as $\,\mathbb{M}_{W}^{\,2} =\mathbb{M}_{[0,N+1)}^{\,2}\,$
by using the notation of open/closed intervals\cite{HLmoose}.
Similarly the $(K+2)\times (K+2)$ mass matrix of neutral
gauge bosons is
$\,\mathbb{M}_{Z}^{\,2} =\mathbb{M}_{[0,K+1]}^{\,2}\,$  and can be obtained
from the matrix $\,\mathbb{M}_{[0,N+1)}^{\,2}\,$
with simple replacements,
$\,N\rightarrow K+1\,$ and \,$f_{K+2}\rightarrow 0$\,.\,
The eigenvalues of $\mathbb{M}_{W}^{\,2}$ are denoted by
$M_W^2$ for light $W^\pm_0\equiv W^\pm$ boson and
$M_{Ww}^2$ ($w=1,2,\cdots,N$) for heavy ``KK'' states $W^{\pm}_w$,
while $\mathbb{M}_{Z}^{\,2}$ has eigenvalues $0$ ($M_Z^2$) for photon $A_\gamma^0$
(light $Z^0_0\equiv Z^0$ boson) and $M_{Zz}^2$ ($z=1,2,\cdots,K$)
for heavy ``KK'' states $Z^0_{z}$.
With the diagonal matrices
\,$\mathcal F_{n+1}
   ={\rm diag}\(f_1^{~},f_2^{~},\cdots,f_{n+1}^{~}\)/2$\, and
\,$\mathcal G_{n+1}
   ={\rm diag}\(g_0^{~},g_1^{~},\cdots,g_n^{~}\)$\,,\,
it is convenient to define the notations,
$\,\mathbb{Q}_{\,W} =\mathcal F_{N+1}\mathcal D_W\mathcal G_{N+1}\,$ and
$\,\mathbb{Q}_Z =\mathcal F_{K+1}\mathcal D_Z\mathcal G_{K+2}\,$, where the
$(N+1)\times (N+1)$ and $(K+1)\times (K+2)$
non-diagonal matrices $\mathcal D_W$ and $\mathcal D_Z$ are
\begin{equation}
\label{eq:D}
\mathcal D_W = \displaystyle
\left(
\begin{array}{c|r|c|c|r}
&&&&
\\[-3.5mm]
\,1\,   &  -1     &    & &
\\[0mm]
\hline
&&&&
\\[-4.5mm]
  & 1  & -1  &&
\\[0mm]
\hline
&&&&
\\[-5mm]
 & & \ddots & \ddots   &
\\[0mm]
\hline
&&&&
\\[-4.5mm]
 & &  & 1         &   -1
\\[0mm]
\hline
&&&&
\\[-4mm]
 & &  & &   1
\\[-1mm]
\end{array}
\right),
~~~~~~~~~~
\mathcal D_Z = \displaystyle
\left(
\begin{array}{c|r|c|c|r|c}
&&&&
\\[-3.5mm]
\,1\,   &  -1     &    & & &
\\[0mm]
\hline
&&&&&
\\[-4.5mm]
  & 1  & -1  &&&
\\[0mm]
\hline
&&&&&
\\[-5mm]
 & & \ddots & \ddots   &&
\\[0mm]
\hline
&&&&&
\\[-4.5mm]
 & &  & 1         &   -1 &
\\[0mm]
\hline
&&&&
\\[-4mm]
 & &  & &   1    & -1
\\[-1mm]
\end{array}
\right).
\end{equation}
So the mass matrices ($\mathbb{M}_{W}^{\,2},\,\mathbb{M}_{Z}^{\,2}$) and
their duals ($\widetilde{\mathbb{M}}_{W}^{\,2},\,\widetilde{\mathbb{M}}_{Z}^{\,2}$)
can be written as\cite{HLmoose,dualm}
\begin{equation}
\begin{array}{llll}
\mathbb{M}_{W}^{\,2} = \mathbb{Q}_W^T\mathbb{Q}_W^{~}\,,~~ &
\mathbb{M}_{Z}^{\,2} = \mathbb{Q}_Z^T\mathbb{Q}_Z^{~}\,;~~ &
\widetilde{\mathbb{M}}_{W}^{\,2} = \mathbb{Q}_W^{~}\mathbb{Q}_W^T\,,~~ &
\widetilde{\mathbb{M}}_{Z}^{\,2} = \mathbb{Q}_Z^{~}\mathbb{Q}_Z^T\,;
\end{array}
\end{equation}
where the dual mass matrices \,$\widetilde{\mathbb{M}}_{W}^{\,2}$ and $\widetilde{\mathbb{M}}_{Z}^{\,2}$\, share
the same nonzero eigenvalues as
\,$\mathbb{M}_{W}^{\,2}$\, and\, $\mathbb{M}_{Z}^{\,2}$\,,\, respectively.
This is because $\mathbb{M}_a^2$ and $\widetilde{\mathbb{M}}_a^2$
($a=W,Z$) are real symmetric matrices
and $\mathbb{Q}_a$ can be diagonalized by the
bi-orthogonal rotations,
\begin{equation}
\label{eq:Q-diag}
\widetilde{\mathbb{R}}_a^T\mathbb{Q}_a\mathbb{R}_a = \mathbb{Q}_a^{\rm diag} \,,
\end{equation}
which leads to
\begin{equation}
\label{eq:M2-diag}
\begin{array}{ll}
\mathbb{R}_a^T\mathbb{M}_a^2\mathbb{R}_a
= (\mathbb{Q}_a^{\rm diag})^T(\mathbb{Q}_a^{\rm diag})
\equiv (\mathbb{M}_a^2)^{\rm diag} \,,
\\[2mm]
\widetilde{\mathbb{R}}_a^T\widetilde{\mathbb{M}}_a^2\widetilde{\mathbb{R}}_a
= (\mathbb{Q}_a^{\rm diag})(\mathbb{Q}_a^{\rm diag})^T
\equiv (\widetilde{\mathbb{M}}_a^2)^{\rm diag} \,.
\end{array}
\end{equation}
The $(N+1)\times(N+1)$ matrices
$\mathbb{M}_W^2$ and $\widetilde{\mathbb{M}}_W^2$ must have identical eigenvalues
(which are all nonzero) because of
$\,(\mathbb{M}_{W}^{\,2} )^{\rm diag} =
   (\mathbb{Q}_W^{\rm diag})^T(\mathbb{Q}_W^{\rm diag})=
   (\mathbb{Q}_W^{\rm diag})(\mathbb{Q}_W^{\rm diag})^T
  =(\widetilde{\mathbb{M}}_{W}^{\,2})^{\rm diag}\,$.\,
On the other hand, the $(K+2)\times(K+2)$ matrix
$\,(\mathbb{Q}_Z^{\rm diag})^T(\mathbb{Q}_Z^{\rm diag})=
   (\mathbb{M}_{Z}^{\,2})^{\rm diag}\,$
has one zero-eigenvalue corresponding to the photon mass,
and it is thus clear that the $(K+1)\times(K+1)$ matrix
$\,(\mathbb{Q}_Z^{\rm diag})(\mathbb{Q}_Z^{\rm diag})^T
  =(\widetilde{\mathbb{M}}_{Z}^{\,2})^{\rm diag}\,$
has $K+1$ massive eigenvalues identical to those in
$(\mathbb{M}_{Z}^{\,2})^{\rm diag}\,$.\,

Expanding the Lagrangian (\ref{eq:L}) gives
the gauge-Goldstone boson mixing term,\cite{HLmoose}
\begin{equation}
\label{eq:A-GB}
\begin{array}{lcl}
{\cal L}_{\rm GB}^{\rm mix} &~=~&
\[-{{\mathbb A}^{+}_{\mu}}^T\mathbb{Q}_{\,W}^{\,T}
    \partial^\mu\Pi^- + {\rm h.c.} \]
  \,-\,{{\mathbb A}^{3}_{\mu}}^T\mathbb{Q}_{\,Z}^{\,T}
    \partial^\mu\Pi^0
\\[2mm]
&~=~&
\[-{\mathbb{W}^{+}_{\mu}}^T\mathbb{M}_{W}^{\rm \,diag}
    \partial^\mu\widetilde{\Pi}^- + {\rm h.c.} \]
 \,-\,{\mathbb{Z}^{0}_{\mu}}^T\widetilde{\mathbb{M}}_{Z}^{\rm \,diag}
    \partial^\mu\widetilde{\Pi}^0 \,,
\end{array}
\end{equation}
with
\begin{equation}
\begin{array}{ll}
\mathbb{A}^{\pm\mu}\,=\,
(A_0^{\pm\mu},\,A_1^{\pm\mu},\,\cdots,\,A_N^{\pm\mu}\,)^T,
~~&~~
\mathbb{A}^{3\mu}\,=\,
(A_0^{3\mu},\,A_1^{3\mu},\,\cdots,\,A_{K+1}^{3\mu})^T;
\\[2mm]
\mathbb{W}^{\pm\mu}\,=\,
(W_0^{\pm\mu},\,W^{\pm\mu}_1,\,\cdots,\,W^{\pm\mu}_N\,)^T,
~~&~~
\mathbb{Z}^{0\mu}\,=\,
(Z_0^{0\mu},\,Z^{0\mu}_1,\,\cdots,\,Z^{0\mu}_{K})^T;
\\[2mm]
\Pi^{\pm}\,=\,
(\pi_1^{\pm},\,\pi^{\pm}_2,\,\cdots,\,\pi^{\pm}_{N+1}\,)^T,
~~&~~
\Pi^{0}\,=\,
(\pi_1^{0},\,\pi^{0}_2,\,\cdots,\,\pi^{0}_{K+1})^T;
\\[2mm]
\widetilde{\Pi}^{\pm}\,=\,
(\widetilde{\pi}_0^{\pm},\,\widetilde{\pi}^{\pm}_1,\,\cdots,\,\widetilde{\pi}^{\pm}_N\,)^T,
~~&~~
\widetilde{\Pi}^{0}\,=\,
(\widetilde{\pi}_0^{0},\,\widetilde{\pi}^{0}_1,\,\cdots,\,\widetilde{\pi}^{0}_K)^T;
\end{array}
\end{equation}
where $\,(\mathbb{W}^{\pm\mu},\,\mathbb{Z}^{0\mu})\,$
are mass-eigenbasis fields,  while
$\,(\widetilde{\Pi}^\pm,\,\widetilde{\Pi}^0)\,$ are ``eaten'' Goldstone fields,
which are connected to the site Goldstone states
$\,(\Pi^\pm,\,\Pi^0)\,$ via
\begin{equation}
\label{eq:PI-rotation}
\widetilde{\Pi}^\pm \,=\, \widetilde{\mathbb{R}}_W^T\,\Pi^\pm \,, ~~~~~~
\widetilde{\Pi}^0   \,=\, \widetilde{\mathbb{R}}_Z^T\,\Pi^0 \,.
\end{equation}
Hence, the ``eaten'' Goldstones are exactly aligned with the
``gauge boson'' mass-eigenstates of the dual moose.
The gauge-Goldstone mixing (\ref{eq:A-GB}) can be
removed by the familiar
$R_\xi$ gauge-fixing term,
\begin{equation}
\label{eq:GF}
\begin{array}{l}
{\cal L}_{\rm gf} \,=\, \displaystyle
\sum_{n=0}^N
- \frac{1}{\,\xi_W^{~}}F_n^+ F_n^- \,+\,
\sum_{n=0}^{K+1} -\frac{1}{\,2\xi_Z^{~}\,}(F_n^0)^2
\,,
\\[6mm]
\displaystyle
F_n^a \,=\,
\partial_\mu V_n^{a\mu}  +
\xi_a M_{an}\widetilde{\pi}_n^a  \,,
\end{array}
\end{equation}
where
$\,V_n^{a\mu}\in (W_n^{\pm\mu},\,Z_n^{0\mu}),~
   M_{an}\in (M_{Wn},M_{Zn})$\, and
$\,\xi_a^{~}\in (\xi_W^{~},\,\xi_Z^{~})\,$.\,
Also, $\,Z^{0\mu}_{K+1}\equiv A_{\gamma}^{0\mu}\,$,\,
$\,M_{Z(K+1)}\equiv M_{\gamma}=0\,$ and
$\,\widetilde{\pi}^0_{K+1} \equiv 0\,$.\,
From Eq.\,(\ref{eq:GF}), the ``eaten''  Goldstone boson mass is
given by  $~M_{\widetilde{\pi}^a_n}^2=\xi_a^{~}M_{an}^2\,$.

\vspace*{3mm}
\section{4D Higgs Mechanism and Geometrization
         in 5D Continuum}

Corresponding to the gauge-fixing (\ref{eq:GF}), we
derive Faddeev-Popov ghost term,
\begin{equation}
\label{eq:FP}
{\cal L}_{\rm FP} \,=\, \displaystyle
\sum_{n=0}^N
\[\overline{c}_n^+\widehat{s} F_n^- + \overline{c}_n^-\widehat{s} F_n^+\]
+ \sum_{n=0}^{K+1}
\overline{c}_n^Z\widehat{s} F_n^0 \,,
\end{equation}
which ensures BRST\cite{BRST} invariance
at quantum level.
Extending the notations of [\refcite{ET}], we write the BRST
transformations for the moose theory,
\begin{equation}
\label{eq:BRST}
\begin{array}{ll}
\displaystyle
\widehat{s} c_n^a      = \displaystyle -\frac{1}{2}g_n^{~}C^{abc}c_n^bc_n^c\,,~~~
&\displaystyle
\widehat{s}\,\overline{c}_n^a   = -\xi_a^{-1}F_n^a \,,
\\[2mm]
\displaystyle
\widehat{s} V_n^{a\mu} = D^{a\mu}_{nb}(V)c_n^b \,,~~~
&\displaystyle
\widehat{s} \pi_n^a    = D^{a\pi}_{nb}(\pi)c_n^b \,,
\\[2mm]
\displaystyle
\widehat{s} \mathbb{V}^{a\mu} = \mathbb{R}_a^T\mathbb{D}^{a\mu}_{b}\mathbb{C}_n^b \,,~~~
&\displaystyle
\widehat{s} \widetilde{\Pi}^{a} = \widetilde{\mathbb{R}}_a^T\mathbb{D}^{a\pi}_{b}\mathbb{C}_n^b \,,
\end{array}
\end{equation}
where
$\,\mathbb{V}^{a\mu}=(V_0^{a\mu},\,\cdots,\,V_{P}^{a\mu})^T\,$,\,
$\,\widetilde{\Pi}^{a}=(\widetilde{\pi}_0^{a},\,\cdots,\,\widetilde{\pi}_{P'}^{a})^T\,$,\,
$\,\mathbb{C}^{a}=(c_0^{a},\,\cdots,\,c_P^{a})^T\,$,\,
and $P,P'=N ~(P=K+1,\,P'=K)$ for charged (neutral) fields.
In (\ref{eq:BRST}),
$\,D^{a\mu}_{nb}(V)\,$ and $\,D^{a\pi}_{nb}(\pi)\,$
are given by the gauge transformations of
\,$V_n^{a\mu}$\, and $\,\pi_n^a$,\, respectively; also
we define the matrix
$\,\mathbb{D}^{i}_b = {\rm diag}(D^i_{0b},\cdots,D^i_{Pb})^T\,$.\,
Following [\refcite{CH,SUSY03}],
we directly extend
Appendix-A of the fourth paper in [\refcite{ET}] to
derive a 4D Slavnov-Taylor (ST) identity for the moose
theory (\ref{eq:L}) with replicated gauge group,
\begin{equation}
\label{eq:4D-ETI1}
\langle 0|T
F^{a_1^{~}n_1^{~}}(x_1^{~})
F^{a_2^{~}n_2^{~}}(x_2^{~})
\cdots
F^{a_\ell^{~}n_\ell^{~}}(x_\ell^{~})\Phi_{\alpha} |0\rangle \,=\, 0\,,
\end{equation}
which is due to the
{\it manifest gauge invariance} of the moose
action \,$\int dx^4{\mathcal L}$\, from (\ref{eq:L}) or
the equivalent BRST invariance of the effective action
\,$\int dx^4 \[{\mathcal L}+{\mathcal L}_{\rm gf}+{\mathcal L}_{\rm FP}\]$\,.\,
In (\ref{eq:4D-ETI1})
$\,\Phi_{\alpha}$\, denotes other possible amputated physical fields.
We stress that (\ref{eq:4D-ETI1}) generally holds
for {\it any} possible form of the gauge-fixing function
$\,F^{an}$.\,
Compared to the SM case\cite{ET}, the main complication
below comes from amputating the external fields $F^a_n$
in (\ref{eq:4D-ETI1}), due to the rotations in
(\ref{eq:Q-diag})-(\ref{eq:M2-diag}).
We find it convenient to introduce a matrix notation for
all relevant gauge-fixing functions,
\begin{equation}
\label{eq:F-matrix}
\mathbb{F}^a = {\underline{\mathbb{K}}^a}^T\underline{\mathbb{V}}^a \,,~~~~~
\underline{\mathbb{V}}^a \equiv
\(\begin{array}{c}\mathbb{V}^{a\mu}  \\[1mm] \widetilde{\Pi}^a \end{array}
\),~~~~~
\underline{\mathbb{K}}^a \equiv
\(\begin{array}{c} \underline{I}_a\partial^\mu \\[1mm]
         \xi_a \underline{M}_a \end{array}
\),
\end{equation}
where \,$\underline{I}_a={\rm diag}(1,\cdots,1)$\, and
$\,\underline{M}_a ={\rm diag}(M_{a0},\cdots,M_{aP})\,$
are $(1+P)\times (1+P)$ diagonal matrices with
$\,P=N\,(P=K+1)\,$ for charged (neutral) sector.
In (\ref{eq:F-matrix}) we have extended
the column $\widetilde{\Pi}^a$ to \,$1\!+\!P$\, dimensional with
$\,\widetilde{\pi}^0_{K+1}\equiv 0\,$.\,
Introducing the external source term
$\,\int d^4x \[{\underline{\mathbb{J}}^a}^T\underline{\mathbb{V}}^a
               +\overline{\mathbb{I}}^a\mathbb{C}^a
               +\overline{\mathbb{C}}^a\mathbb{I}^a\]\,$,\,
we derive a generating equation for connected Green
functions,
\begin{equation}
{\underline{\mathbb{J}}^a}^T\langle 0|T\widehat{s}\,\underline{\mathbb{V}}^a(x)|0\rangle
+\overline{\mathbb{I}}^a(x)\langle 0|T\widehat{s}\mathbb{C}^a(x)|0\rangle
-\langle 0|T\widehat{s}\overline{\mathbb{C}}^a(x)|0\rangle\mathbb{I}^a(x) ~=~ 0\,,
\end{equation}
from which we deduce an ST identity for the matrix
propagator of \,$\underline{\mathbb{V}}^a$,
\begin{equation}
\label{eq:ST-D}
{\underline{\mathbb{K}}^a}^T\underline{\mathcal D}^{ab}(p) = -[\underline{\Omega}^{ab}](p)^T   \,,
\end{equation}
with
\begin{equation}
\label{eq:ST-Dx}
\begin{array}{l}
\underline{\mathcal D}^{ab}(p) =\lan0|T{\underline{\mathbb{V}}^a}{\underline{\mathbb{V}}^b}^T|0\rangle(p) \,,~~~
{\mathcal S}(p)\delta^{ab}
=\lan0|T\mathbb{C}^{\,a}\overline{\mathbb{C}}^b|0\rangle(p) \,,
\\[2mm]
\underline{\Omega}^{ab}(p) \equiv \underline{\widehat{\Omega}}^{ab}\!(p)\,
                       {\mathcal S}(p)\,,
\\[3mm]
\underline{\widehat{\Omega}}^{ab}(p) \equiv
\langle 0|T\widehat{s}\underline{\mathbb{V}}^b|\overline{\mathbb{C}}^a\rangle (p)
\equiv \!
\(\begin{array}{c} \langle 0|T\widehat{s}\mathbb{V}^{b\mu}|\overline{\mathbb{C}}^a\rangle
\\[1mm]  \langle 0|T\widehat{s}\,\widetilde{\Pi}^b|\overline{\mathbb{C}}^a\rangle
\end{array}
\)_{\!\!(p)}  \!\!\! \equiv\!
\(\!\begin{array}{c}
-ip^\mu\delta^{ab}\mathbb{R}_a^T [\mathbf{1}\!+\!\Delta_V^a(p^2)]
\\[1mm]
\delta^{ab}\widetilde{\mathbb{R}}_a^T\mathbb{Q}_a [\mathbf{1}\!+\!\Delta_\pi^a(p^2)]
\end{array}
\)\!.
\end{array}
\end{equation}
Using (\ref{eq:ST-D}) and collecting $F^{an}$ in the matrix
form $\mathbb{F}^a$ for each external line in (\ref{eq:4D-ETI1}),
we make an amputation for (\ref{eq:4D-ETI1}),
$\,0=G[\mathbb{F}^a(p),\cdots]
    =-[\underline{\Omega}^{ab}]^T{\cal T} [\underline{\mathbb{V}}^a(p),\cdots],
\,$
which leads to
\begin{equation}
\begin{array}{cl}
0  & =~
{\cal T} [\,p_\mu\!\mathbb{V}^{a\mu}(p)-\widetilde{\mathbb{M}}_a^{\rm diag}
     {\mathbf C}^a\widetilde{\Pi}^a(p),\,
     \cdots\cdots \,]  \,,
\\[3mm]
\mathbf{C}^a &\equiv~ -i\,
(\mathbb{Q}_a^T\widetilde{\mathbb{R}}_a)^{-1}
\(\mathbf{1}\!+\!{\Delta_V^a}(p^2)^T\)^{-1} \!
\(\mathbf{1}\!+\!{\Delta_\pi^a}(p^2)^T\)
(\mathbb{Q}_a^T\widetilde{\mathbb{R}}_a)
\\[3mm]
& =~ -i\[\,\mathbf{1}+O({\rm loop})\,\]
\,.
\end{array}
\end{equation}
Repeating this amputation for all external lines in
(\ref{eq:4D-ETI1}), we arrive at a matrix identity
for $S$-matrix elements,
\begin{equation}
\label{eq:4D-ETI2}
 {\cal T}\!\[\,\overline{\mathbb{F}}^{a_1^{~}}(p_1^{~}),\,
         \overline{\mathbb{F}}^{a_2^{~}}(p_2^{~}),\, \cdots, \,
         \overline{\mathbb{F}}^{a_{\ell}}(p_{\ell}^{~}),\,\Phi_{\alpha}\]
     \,=~ 0\,,
\end{equation}
where
$\,\overline{\mathbb{F}}^{a}(p)
\equiv p^\mu \mathbb{V}_\mu^{a}-\widetilde{\mathbb{M}}_a^{\rm diag}\mathbf{C}^{a}\widetilde{\Pi}^a
=\widetilde{\mathbb{M}}_a^{\rm diag}(\mathbb{V}_S^{a}-\mathbf{C}^{a}\widetilde{\Pi}^a)\,$
with $\,\mathbb{V}_S^{a}=\epsilon^\mu_S\mathbb{V}_\mu^{a}\,$
($\epsilon_S^\mu \equiv p^\mu/\widetilde{\mathbb{M}}_a^{\rm diag}$),\,
$\mathbb{V}^{a\mu} =\mathbb{W}^{\pm\mu} ,\,\mathbb{Z}^{0\mu}$,\,
and
\,$\mathbf{C}^a = -i[1+O({\rm loop})]$.\,
The identity (\ref{eq:4D-ETI2}) states
that in the $S$-matrix element
the unphysical eaten ``KK'' Goldstones
$\widetilde{\Pi}^{a}$
and the unphysical ``KK'' scalar gauge-components
$\mathbb{V}_S^{a}$ are {\it confined,} so they together
have no net contribution to any physical process ---
a quantitative formulation of the 4D Higgs mechanism at
the $S$-matrix level, which holds even without a physical
Higgs boson such as in the present moose theory (gauged
nonlinear sigma model) with replicated gauge groups.
Under high energy expansion,
Eq.\,(\ref{eq:4D-ETI2}) results in
a {\it generalized form} of
the equivalence theorem (ET)\cite{ET,CH}, which connects the
high energy longitudinal gauge boson scattering amplitude
to that of the ``eaten'' Goldstone bosons,
\begin{equation}
\label{eq:ETm}
\displaystyle
{\cal T}\!\[V_L^{a_1^{~}n_1^{~}},
      V_L^{a_2^{~}n_2^{~}},\,
\cdots\,\] =\, C_{\rm mod}^{n_1m_1,n_2m_2\cdots}\,
{\cal T}\!\[\widetilde{\pi}^{a_1^{~}m_1^{~}},
      \widetilde{\pi}^{a_2^{~}m_2^{~}},\,
      \cdots\,\]
      \,+\, O\!\(\!\frac{M_{an}}{E_n}\!\)\!,
\end{equation}
where $\,V_n^a=W^\pm_{n},\,Z^0_{n}\,$
($n=0,1,2,\cdots$),\, and sums over repeated indices
$(m_1,m_2,\cdots)$ are implied.
The exact expression of the $\,O(M_{an}/E_n)\,$ term is
obtained from directly expanding the ET identity
(\ref{eq:4D-ETI2}),
in the {\it same} way as in [\refcite{ET}].
The radiative modification factor is
\begin{equation}
\label{eq:Cmod}
C_{\rm mod}^{n_1m_1,n_2m_2\cdots}
   \equiv\, \mathbf{C}^{a_1}_{n_1m_1}\mathbf{C}^{a_2}_{n_2m_2}\cdots
   \,=\,(-i)^{\ell}
        \[(\delta_{n_1m_1}\delta_{n_2m_2}\cdots)+O({\rm loop})\],
\end{equation}
which extends
Ref.\,[\refcite{ET}] to the case of replicated gauge group.\,

Analyzing both sides of (\ref{eq:ETm}),
we observed\,\cite{CH} that the non-canceled leading
$E^2$-terms in the usual nonlinear gauged sigma model
are now suppressed by
large $N$ due to the {\it collective effect} of
many ``KK'' modes in the deconstruction theory.

To take the 5D continuum limit we redefine the
lattice link field (Wilson line),
\begin{equation}
\label{eq:U-lattice}
U_j(x)=U(x,x^5_j)
 =\exp\!\[i\!\int_{x^5_{j-1}}^{x^5_j}\!\!\!\!\! dx^5
  g_j^{~}\mathbf{A}_5(x,x^5)\]\!
 =\exp\!\[i\,\mathbf{a} g_j^{~}\mathbf{A}_5(x,x^5_j)\]
\end{equation}
where $\,\mathbf{a}=\Delta x^5_j\,$ is lattice spacing and
\,$A_5^a(x,x^5_j)$\, is related to the site-Goldstone boson
$\pi^a_j(x)$ in (\ref{eq:L}),
$\,A_5^a(x,x^5_j)
  =\pi^a_j(x)(2/g_j^{~}f_{j\,}\mathbf{a})\,$.\,
So the site-Goldstone field $\pi^{a}_j$
will geometrize as $\widehat{A}^a_5$ component of the 5D gauge field
$\,\widehat{A}^a_J = (\widehat{A}^a_\mu,\,\widehat{A}^a_5)\,$,\,
where $\,J,I,\ldots\in (\mu,\,5)\,$
denote the 5D Lorentz indices.
Following [\refcite{DeRS}] and ignoring the $U(1)^{M+1}$
part of Fig.\,1 for simplicity,
we define the position-dependent couplings
$\,g_j^{~} = g_5^{~}\kappa_j^{~}/\sqrt{\mathbf a}\,$
and decay constants
$\,f_j^{~}=h_j^{~}f\,$, where $g_5^{~}$ and $f$ are pure
constants of mass-dimension $-\frac{1}{2}$ and $1$, respectively.
Thus we can derive,
$\,D^{\mu}U_j
= -i\,\mathbf{a}g_j^{~}F^{a\mu 5}_j
+O(\mathbf{a}^2)
= -i\,\mathbf{a}g_5^{~}\widehat{F}^{a\mu 5}(x,x^5_j)
+O(\mathbf{a}^2)\,$, where the identification
$\, A^{a\mu,5}_j(x) \rightarrow
    (\sqrt{\mathbf{a}}/\kappa_j^{~})
    \widehat{A}^{a J}(x,x^5_j)
 = (g_5^{~}/g_j^{~})\widehat{A}^{a J}(x,x^5_j)
$\,
is made under $\,\mathbf{a}=L/N\rightarrow 0\,$.\,
As such, we see that the Goldstone Lagrangian in the second
term of (\ref{eq:L}) just gives the continuum $\widehat{A}^a_5$-Lagrangian,
$\,\int_0^L dx^5
   \[-\frac{1}{2}h^2(x^5)\widehat{F}^{a\mu 5}\widehat{F}^a_{\mu 5}\]\,$,\,
after imposing a normalization condition
$\,\mathbf{a}=(2/g_5^{~}f)^2\,$.\,
With these, we reproduce a
bulk $SU(2)_{\rm 5D}$ gauge theory from (\ref{eq:L})
in the continuum limit,
\begin{equation}
\label{eq:SU2-5D-L}
S_5^{~} ~=~ \displaystyle\int\! d^5\widehat{x}\, \sqrt{-\widehat{g}~}~\widehat{g}^{IK}\widehat{g}^{JL}\,
\frac{-1}{\,4\kappa^2(x^5)\,}\,\widehat{F}^a_{IJ}\widehat{F}^a_{KL} \,,
\end{equation}
defined on a general 5D background,
\begin{eqnarray}
ds^2 &~=~& [\kappa(x^5)h(x^5)]^2\eta_{\mu\nu}^{~}
         dx^\mu dx^\nu -\, dx^5 dx^5 \,,
\end{eqnarray}
where the metric
$\,\widehat{g}_{IK}^{~} =
 {\rm diag}\(\eta_{\mu\nu}^{~}(\kappa h)^2,\,-1\)\,$.\,
For the simplest case $\,\kappa = h =1\,$, it reduces
to flat 5D geometry, while for the case
$\,\kappa =1\,$ and $\,h=\exp [-k|x^5|]\,$
it reduces to the familiar warped RS1\cite{RS}.
Various extended continuum models from (\ref{eq:SU2-5D-L})
can be derived from the moose theory (\ref{eq:L})
by including the $U(1)^{M+1}$ groups and/or proper
folding(s) of the moose chain in Fig.\,1.
These together with the {\it induced} 5D BCs
and possible brane kinetic terms
will be analyzed in Sec.\,4.

Analyzing (\ref{eq:SU2-5D-L}) we construct
an $R_\xi$ gauge-fixing term to remove the
\,$\widehat{A}^{a\mu}-\widehat{A}^{a5}$\, mixing,
\begin{equation}
\displaystyle
\widehat{\mathcal{L}}_{\rm gf} \,=\, -\frac{1}{\,2\xi\,}(\widehat{F}^a)^2\,,
~~~~~\widehat{F}^a =  \kappa^{-1}\partial_\mu\widehat{A}^{a\mu}
            +\xi\kappa\,\partial_5(h^2\widehat{A}^{a5}) \,,
\end{equation}
The corresponding ghost term is given by
\begin{equation}
\label{eq:FP-5D}
\widehat{\mathcal{L}}_{\rm FP} ~=~ \displaystyle \widehat{\overline{c}}^a\widehat{s}\widehat{F}^a \,,
\end{equation}
and the 5D BRST transformations are
\begin{equation}
\label{eq:BRST-5D}
\widehat{s}\widehat{A}^{aJ}   = D^{aJ}_b(\widehat{A})\widehat{c}^b ,~~~~
\widehat{s} \widehat{c}^a     = \displaystyle-\frac{1}{2}g_5^{~}C^{abc}\widehat{c}^b\widehat{c}^c ,~~~~
\displaystyle
\widehat{s}\,\widehat{\overline{c}}^a   = -\xi^{-1}\widehat{F}^a .
\end{equation}
Using 5D gauge invariance of the action $\,S_5\,$, we
derive an ST identity\cite{SUSY03} in parallel
to the 4D result (\ref{eq:4D-ETI1}),
\begin{equation}
\label{eq:5D-ETI1}
\langle 0|T
\widehat{F}^{a_1^{~}}(\widehat{x}_1^{~})
\widehat{F}^{a_2^{~}}(\widehat{x}_2^{~})
\cdots
\widehat{F}^{a_\ell}(\widehat{x}_{\ell})\widehat{\Phi}_{\alpha} |0\rangle \,=\, 0\,.
\end{equation}
The 5D equation of motion (EOM) for free field
$\,\widehat{A}^{a}_{\mu 5}\,$ is
\begin{equation}
\label{eq:5D-EOM}
\begin{array}{l}
\(\partial_{\mu}^2 -\kappa^2\partial_5h^2\partial_5\)\widehat{A}^a_{\nu} -
\(1-\xi^{-1}\)\partial_{\nu}\partial_{\mu}\widehat{A}^{a\mu} \,=\, 0 \,,
\\[2mm]
\partial_5(\kappa^2\partial_5 h^2\widehat{A}^a_5)
 -\xi^{-1}\partial_{\mu}^2\widehat{A}^a_5 \,=\, 0 \,,
\end{array}
\end{equation}
where we have made integration by part in
the action $S_5+S_{\rm gf}+S_{\rm FP}$ and verified that all
surface terms vanish under the induced consistent BCs
in Sec.\,4.
Then, we define the KK expansions,
\begin{equation}
\label{eq:KK-expansion}
\widehat{A}^{a\mu}(\widehat{x})
=\displaystyle\frac{1}{\sqrt{L\,}\,}
 \sum_{n=0}^{\infty} V^{a\mu}_n(x){\mathcal X}_n(x^5)\,,
~~~\,
\widehat{A}^{a5}(\widehat{x})
=\displaystyle\frac{1}{\sqrt{L\,}\,}
 \sum_{n=0}^{\infty} V^{a5}_n(x)\widetilde{\mathcal X}_n(x^5)\,.
\end{equation}
As in [\refcite{randall}] we find it convenient to
work in the momentum space for 4D KK fields $V^{a\mu,5}_n$ and
position space for 5D wavefunction (${\mathcal X}_n,\widetilde{\mathcal X}_n$).
Imposing the 4D EOMs for the free KK fields as usual,
$\,(\partial_\mu^2+M_{an}^2)V^{a\nu}_n(x) -
   (1-\xi^{-1})\partial^\mu\partial^\nu V^{a}_{n\mu}(x)=0\,$
and
$\,(\partial_\mu^2+\widetilde{M}_{an}^2)V^{a5}_n(x)=0\,$,\,
we derive from (\ref{eq:5D-EOM}),
\begin{equation}
\label{eq:5D-EOM-X}
-\kappa^2\partial_5\(h^2\partial_5{\mathcal X}_n\) \,=\, M_{an}^2{\mathcal X}_n \,,~~~~~
-\partial_5(\kappa^2\partial_5h^2\widetilde{\mathcal X}_n) \,=\, \xi^{-1}\widetilde{M}_{an}^2\widetilde{\mathcal X}_n\,,
\end{equation}
where
\,$(M_{an},\,\widetilde{M}_{an})$\, are mass-eigenvalues of
4D KK fields $(V^{a\mu}_n,\,V^{a5}_n)$\,.\,
Acting $\partial_5$ on the first equation of (\ref{eq:5D-EOM-X})
we see, \,$\widetilde{\mathcal X}_n\propto \partial_5 {\mathcal X}_n$\, and
$\,\widetilde{M}^2_{an}=\xi M^2_{an}\,$.\,
So, with the normalization conditions of
\,$({\mathcal X}_n,\widetilde{\mathcal X}_n)$\,,\, we deduce from (\ref{eq:5D-EOM-X}),
\begin{equation}
\label{eq:Xn-XTn}
\partial_5 {\mathcal X}_n \,=\,\displaystyle M_{an}\widetilde{\mathcal X}_n \,,~~~~~~
\partial_5 (h^2\widetilde{\mathcal X}_n) \,=\, -\kappa^{-2}M_{an}{\mathcal X}_n \,.
\end{equation}
With these we decompose the 5D gauge-fixing function
$\widehat{F}^a$ as
\begin{equation}
\label{eq:F_an}
\begin{array}{cl}
\widehat{F}^a(\widehat{x} ) & \,=\,\displaystyle
\frac{1}{\sqrt{L\,}\,}
\sum_{n=0}^{\infty} F^{an}(x)\, {\mathcal X}_n(x^5)/\kappa(x^5) \,,
\\[5mm]
F^{an}(x) & \,=~ \partial_\mu V_n^{a\mu}(x) -
                 \xi M_{an} V_n^{a5}(x)
            \,\equiv\, \underline{K}_{an}^T \underline{V_n^a}
\,,
\end{array}
\end{equation}
where $\,\underline{K}_{an}=(\partial_\mu,\,-\xi M_{an})^T\,$ and
$\,\underline{V_n^a} =(V_n^{a\mu},\,V^{a5}_n)^T\,$.\,
Thus we derive a 4D ST identity from
(\ref{eq:5D-ETI1}),
\begin{equation}
\label{eq:5D-ETI1-KK}
\langle 0|T
F^{a_1n_1}(x_1^{~})
F^{a_2n_2}(x_2^{~})
\cdots
F^{a_{\ell}n_{\ell}}(x_{\ell})\Phi_{\alpha} |0\rangle \,=\, 0\,.
\end{equation}
Adding the KK expansions for ghost fields,
\begin{equation}
\displaystyle
\widehat{c}^a(\widehat{x}) = \frac{1}{\sqrt{L}\,}
\sum_{n=0}^{\infty} c^a_n(x){\mathcal X}_n(x^5)\,,~~~~~
\widehat{\overline{c}}^a(\widehat{x})= \frac{1}{\sqrt{L}\,}
\sum_{n=0}^{\infty}\overline{c}^a_n(x){\mathcal X}_n(x^5)/\kappa(x^5)\,,
\end{equation}
we derive the BRST transformations for the KK fields,
\begin{equation}
\label{eq:BRST-4D-KK}
\begin{array}{ll}
\widehat{s} V^{a\mu}_n = D^{ab,\mu}_{nm}c^b_m \,,~~~~
&
\widehat{s} V^{a5}_n   = D^{ab,5}_{nm}  c^b_m \,,
\\[2mm]
\widehat{s} c^a_n      =
           -\frac{1}{2}gC^{abc}
           \mathfrak{D}_n^{m\ell}c^b_mc^c_{\ell}\,,~~~~
&
\widehat{s} \overline{c}^a_n    = -\xi^{-1}F^{an} \,,
\end{array}
\end{equation}
where
\begin{equation}
\label{eq:BRST-Dfunc}
\begin{array}{ll}
\displaystyle
D^{ab,\mu}_{nm} & =\displaystyle
-\delta^{ab}\delta_{nm}\partial^\mu
+\[\frac{1}{L}\int_0^L\!\! dx^5\kappa^{-2}{\mathcal X}_n{\mathcal X}_m{\mathcal X}_{\ell}\]
gC^{abc}V_{\ell}^{c\mu}  \,,
\\[4mm]
\displaystyle
D^{ab,5}_{nm} & =\displaystyle
-\delta^{ab}\delta_{nm}M_{am}
+\[\frac{1}{L}\int_0^L\!\! dx^5 h^2\widetilde{\mathcal X}_n{\mathcal X}_m\widetilde{\mathcal X}_{\ell}\]
gC^{abc}V_{\ell}^{c5}  \,,
\\[4mm]
\displaystyle
\mathfrak{D}_n^{m\ell} & =\displaystyle
\frac{1}{L}\int_0^L\!\! dx^5\, \kappa^{-2}{\mathcal X}_n{\mathcal X}_m{\mathcal X}_{\ell} \,,
\end{array}
\end{equation}
and repeated indices are summed up.
To amputate the external fields $F^{an}$ in
(\ref{eq:5D-ETI1-KK}),
we deduce an ST identity for the KK propagator
of \,$V^a_n$, similar to Eq.\,(\ref{eq:ST-D}),
\begin{equation}
\label{eq:ST-D-5D}
\underline{K}_{an}^T\underline{\mathcal D}_{nm}^{ab}(p)
= -[\underline{\Omega}_{nm}^{ab}](p)^T   \,,
\end{equation}
with
\begin{equation}
\label{eq:ST-Dx-5D}
\begin{array}{l}
\mathcal D_{nm}^{ab} (p)
=\lan0|T{\underline{V^a_n}}\,\underline{V^b_m}^T|0\rangle (p) \,,~~~
\mathcal{S}_{nm} (p)\delta^{ab}
=\lan0|Tc^a_n\overline{c}_m^b|0\rangle (p) \,,
\\[2mm]
\underline{\Omega}_{nm}^{ab}(p)
\equiv \underline{\widehat{\Omega}}_{mj}^{ab}(p)
                        \mathcal{S}_{jn} (p) \,,
\\[3mm]
\underline{\widehat{\Omega}}_{mj}^{ab} (p) \equiv
\langle 0|T\widehat{s}\underline{V^b_m}|\overline{c}^a_j\rangle (p)
\equiv \!
\(\begin{array}{c} \langle 0|T\widehat{s} V_m^{b\mu}|\overline{c}^a_j\rangle
\\[1mm]  \langle 0|T\widehat{s} V_m^{b5}|\overline{c}^a_j\rangle
\end{array}
\)_{\!\!\!(p)}\!\! \equiv\!
\(\!\begin{array}{c}
-ip^\mu\delta^{ab}
    [\delta_{jm}\!+\!\Delta^a_{jm}(p^2)]
\\[1mm]
-M_{am}\delta^{ab}
    [\delta_{jm}\!+\!\widetilde{\Delta}^a_{jm}(p^2)]
\end{array}
\)\!.
\end{array}
\end{equation}
Extension of the above formulation to include possible brane
term is straightforward. Consider, for simplicity, a brane term
similar to (\ref{eq:MooseB-BK})-(\ref{eq:MooseB-BKx}) in Sec.\,4,
at $\,y=0,\,{\rm or}~L$,\,
\begin{equation}
\label{eq:BT0}
\! S_{\rm BT}  = \!\!
\displaystyle\int\!\! d^5\widehat{x} ~\delta(x^5\!\!-\!y)\,
\frac{-\zeta\,}{4} \widehat{F}^{a\mu\nu}\widehat{F}^a_{\mu\nu}
               = \!\!
\displaystyle\int\!\! d^5\widehat{x}~
\delta (x^5\!\!-\!y)\sqrt{-\widehat{g}\,}\,\widehat{g}^{IK}\widehat{g}^{JL}
\frac{-\widehat{\zeta}\,}{\,4\kappa^2} \widehat{F}^a_{IJ}\widehat{F}^a_{KL}\,,\,
\end{equation}
where $\,\widehat{\zeta} =\zeta\, \kappa^2(y)\,$.
In (\ref{eq:BT0}) the second expression is
equivalent to the first one under the covariant
BC $\widehat{F}^a_{\mu 5}=0$ [cf. (\ref{eq:5D-covBC})], and appears
more convenient for direct extension of the above derivation.
The brane term (\ref{eq:BT0}) may vary, depending
on the residual symmetry at the boundary.
We stress that the total action $\,S_5+S_{\rm BT}\,$
is invariant under the 5D gauge symmetry supplemented by the
BCs [which are {\it induced from the continuum limit of the
gauge-invariant moose theory (\ref{eq:L})}, cf. Sec.\,4].
The quantized action may be written as
$\,S_5+S_{\rm BT}+S_{\rm gf}+S_{\rm FP}\,$ with
$\,S_{\rm gf}+S_{\rm FP}
   =\int\! d^5\widehat{x} \,[1+\widehat{\zeta}\delta(x^5\!-y)]
    (\widehat{\mathcal{L}}_{\rm gf}+\widehat{\mathcal{L}}_{\rm FP})\,$,\,
which is expected from the deconstruction viewpoint since
the gauge group at each lattice site of the 5D
(including boundary sites)
can be quantized with its own gauge-fixing and ghost terms as shown
earlier in this section.
With these and a prescription of the vanishing total derivative
$\,\partial_5(\cdots)\,$ under the integration
$\,\int\! dx^5[1+\widehat{\zeta}\delta(x^5-y)]\,$,\,
we readily verify that the above
Eqs.\,(\ref{eq:BRST-5D})-(\ref{eq:ST-Dx-5D}) remain,
except that all integrals in (\ref{eq:BRST-Dfunc}) will contain
an extra brane-term-factor \,$[1+\widehat{\zeta}\delta(x^5-y)]$.\,
Accordingly, the 5D wavefunctions $({\mathcal X}_n,\,\widetilde{\mathcal X}_n)$ satisfy
the normalization conditions,
\begin{equation}
\label{eq:NCkin}
\begin{array}{rcl}
L^{-1}\!\int\!dx^5[1+\widehat{\zeta}\delta(x^5\!-\!y)]
    \kappa^{-2}{\mathcal X}_n{\mathcal X}_m &~=~& \delta_{nm} \,,
\\[2mm]
L^{-1}\!\int\!dx^5[1+\widehat{\zeta}\delta(x^5\!-\!y)]h^2
    \widetilde{\mathcal X}_n\widetilde{\mathcal X}_m     &~=~& \delta_{nm} \,,
\end{array}
\end{equation}
on the general 5D background.
The corresponding normalization conditions for
the KK mass-terms are,
\begin{equation}
\label{eq:NCmass}
\begin{array}{rcl}
  L^{-1}\!\int\!dx^5[1+\widehat{\zeta}\delta(x^5\!-\!y)]
      h^2\partial_5{\mathcal X}_n\partial_5{\mathcal X}_m
&~=~& M_{an}^2\delta_{nm} \,,
\\[2mm]
 L^{-1}\!\int\!dx^5[1+\widehat{\zeta}\delta(x^5\!-\!y)]\kappa^2
    \partial_5(h^2\widetilde{\mathcal X}_n)\partial_5(h^2\widetilde{\mathcal X}_m)
&~=~& M_{an}^2\delta_{nm} \,.
\end{array}
\end{equation}
Under Eq.\,(\ref{eq:Xn-XTn}) we see that (\ref{eq:NCmass})
consistently reduces to (\ref{eq:NCkin}).

With these given, we are ready to amputate the external fields in
(\ref{eq:5D-ETI1-KK}) and derive an identity
for $S$-matrix elements,
\begin{equation}
\label{eq:5D-ETI2}
{\cal T}\[\overline{F}^{a_1n_1}\!(p_1^{~}),\,
    \overline{F}^{a_2n_2}\!(p_2^{~}),\,\cdots,\,
    \overline{F}^{a_{\ell}n_{\ell}}\!(p_{\ell}^{~}),\Phi_{\alpha}\]
~=~0 \,,
\end{equation}
where
$\,\overline{F}^{an}=  p^\mu V^{an}_{\mu}
             -M_{an} \widehat{C}^a_{nm} V_5^{am}
           =  M_{an} (V^{an}_S-\widehat{C}^a_{nm} V_5^{am})
\,$,\,  $V_S^{an}=\epsilon_S^{\mu}V_{\mu}^{an}$\,
[$\epsilon_S^{\mu}\equiv k^{\mu}/M_{an}
  =\epsilon_L^{\mu}-v^{\mu}$,\,
 $v^{\mu}=O(M_{an}/E_n)$\,],\,
and
$\,\widehat{C}^a_{nm} \equiv i(M_{am}/M_{an})
  [(\mathbf{1}+\widetilde{\Delta}^a(p^2))
   (\mathbf{1}+\Delta^a(p^2))^{-1}]_{mn}$\,.\,
Similar to (\ref{eq:4D-ETI2}) for the deconstruction theory,
our ET identity (\ref{eq:5D-ETI2}) shows that
the unphysical scalar-KK-component $V^{an}_S$ and the fifth
gauge-KK-component $V^{an}_5$ (or its linear composition
$\widehat{C}^a_{nm}V_5^{am}$) are {\it confined} at the $S$-matrix level,
so they together have zero contribution to any physical process.
This is just the quantitative $S$-matrix formulation of the 5D
{\it geometric Higgs mechanism} (GHM)\cite{SUSY03,CDH},
where $V^{an}_5$\,'s serve as the
would-be Goldstone bosons and get converted to the
longitudinal gauge KK-modes $V_L^{an}$\,'s.
Expanding the ET identity (\ref{eq:5D-ETI2}),
we thus derive a {\it general Kaluza-Klein ET}
(KK-ET\cite{CDH}) at asymptotic energy,
\begin{equation}
\label{eq:KK-ET}
\displaystyle
{\cal T}\!\[V_L^{a_1^{~}n_1^{~}},\,
      V_L^{a_2^{~}n_2^{~}},
\cdots\,\] ~=~
\widehat{C}_{\rm mod}^{n_1m_1,n_2m_2\cdots}\,
{\cal T}\!\[V_5^{a_1^{~}m_1^{~}},\,
      V_5^{a_2^{~}m_2^{~}},
      \cdots\,\]
      \,+\, O\!\(\!\frac{M_{an}}{E_n}\!\) \!,
\end{equation}
where
\,$\widehat{C}_{\rm mod}^{n_1m_1,n_2m_2\cdots}
   =\widehat{C}^{a_1}_{n_1m_1}\cdots
    \widehat{C}^{a_{\ell}}_{n_{\ell}m_{\ell}}
   = i^{\ell}
    [(\delta_{n_1m_1}\delta_{n_2m_2}\cdots )
     +O({\rm loop})]$\,,\,
and the external momenta are put on-shell
($p_n^2=M_{an}^2$).\,
The exact expression of the suppressed term
$\,O(M_{an}/E_n)\,$ in (\ref{eq:KK-ET})
is tedious and is directly obtained from
expanding the ET identity (\ref{eq:5D-ETI2}) by
\,$\epsilon_S^{\mu}= \epsilon_L^{\mu}-v^{\mu}$,\,
in the {\it same} way as in [\refcite{ET}].\footnote{As
well-known\cite{ET,CDH,SUSY03}, the exact ET identity
(\ref{eq:5D-ETI2}) ensures the sum of the
Goldstone amplitude and the lengthy $O(M_{an}/E_n)$ term to
precisely equal the amplitude of longitudinal gauge bosons.}
We stress that {\it our KK-ET (\ref{eq:KK-ET})
and its KK-ET identity (\ref{eq:5D-ETI2})
are valid for arbitrary geometry with any consistent BC
and possible brane term such as (\ref{eq:BT0})
(cf. Sec.\,4 for detail).}\footnote{After the
completion of this manuscript, a preprint\cite{muck}
appeared which considered a generalization of the 5D ET
in Refs.\,[\refcite{CDH,SUSY03}] to include brane term
in the special case of flat $S^1/Z_2$.}
The KK-ET (\ref{eq:KK-ET}) is the manifestation of the 5D
geometric Higgs mechanism\cite{SUSY03,CDH} at
the $S$-matrix level, where
the conversion $\,V^{an}_5\Longrightarrow V^{an}_L\,$\,
is realized.
An essential advantage of this 5D formulation is that we
start with the well-defined 4D moose theory (gauged nonlinear
sigma model) which is manifestly gauge-invariant without any
BC.   The consistent BCs and possible brane terms
are {\it automatically induced}
by taking proper continuum limits (cf. Sec.\,4),
which also ensure the gauge-invariance of the
5D action at the boundaries.

The gauge interactions of $V_5^{an}$ arise from
$\,\int d^5\widehat{x} \frac{-h^2}{2}\widehat{F}^a_{\mu 5}\widehat{F}^{a\mu 5}\,$
under KK expansion.
Examining this\cite{CDH,SUSY03} and
applying power counting\cite{PC},
we find that scalar amplitude
$\,{\cal T}[V_5^{a_1^{~}n_1^{~}},\,V_5^{a_2^{~}n_2^{~}},
         \cdots\cdots ]=O(E^0)$\,
generally holds\cite{SUSY03},\,
{\it independent of the detail of
any particular compactification/BC and
possible brane term.}
Because of this and Eq.\,(\ref{eq:KK-ET}),
we conclude that the longitudinal amplitude
$\,{\cal T}[V_L^{a_1^{~}n_1^{~}},\,
      V_L^{a_2^{~}n_2^{~}},\cdots\cdots ]=O(E^0)\,$,\,
generally guaranteeing the $E$-power cancellations down to
a constant.

Finally, we note that the nonlinearly realized gauge symmetry
in the moose theories
may be regarded as a (spontaneously) broken phase formulation,
but the crucial difference from
an arbitrary or random {\it explicit symmetry breaking} is that
our formalism (including the continuum 5D action)
is manifestly gauge-invariant or BRST-invariant
in which the would-be Goldstone bosons
and the corresponding ``Higgs'' mechanism are consistently
embedded, so the various Slavnov-Taylor and Ward-Takahashi
identities can be derived even in the broken phase\cite{BWLee}.\,
It is such {\it general} identities and their resulting
ET (\ref{eq:ETm}) or KK-ET (\ref{eq:KK-ET})
that have directly guaranteed the relevant $E$-cancellations
in {\it all} \,\,$2\rightarrow n$ ($n\geqslant 2$)\,\,
longitudinal gauge boson scatterings for {\it any}
4D or compactified higher-D theory,
leading to the effective unitarity.\cite{CH,CDH,SUSY03}

\section{Deconstruction, (Non)Geometric Gauge Symmetry Breaking
         and Induced Boundary Conditions}

Various compactified 5D theories can be derived by taking
proper continuum limit of our general moose theory (Fig.\,1).
The bulk gauge symmetry and its breaking at the boundary
are essentially determined by the structure of 4D moose
and its spontaneous symmetry breaking pattern.
In comparison with the traditional 4D Higgs mechanism\cite{Higgs}
which spontaneously breaks the gauge symmetry by nontrivial
vacuum, the 5D gauge symmetry can be spontaneously broken
by the proper compactification at boundaries which arise
from the ``boundary'' groups (sites) of the moose theory
(Fig.\,1) under possible folding(s).
This provides us a
conceptually clean and elegant formulation of the gauge
symmetry breaking (without/with gauge-group rank reduction)
in compactified 5D theories,
without inputting the BCs {\it a priori} or
relying on the technique of adding
extra boundary Higgs fields\cite{Heb}.\,
Our generic moose analysis below also reveals that the
induced 5D BCs for $\widehat{A}^{a\mu}$ do not
depend on particular choice of bulk geometry,
while the same is not generally true for $\widehat{A}^{a5}$.

\begin{figure}[h]
\label{fig:Fig2}
\begin{center}
\vspace*{-5mm}
\centerline{\psfig{file=fig2.eps,width=11cm}}
\setlength{\unitlength}{1mm}
\begin{picture}(0,0)
\end{picture}
\vspace*{-8mm}
\caption{Higgsless Moose-A: a special case of Fig.\,1
with \,$M=0$\, and \,$f_{N+1}^{~}=0$\,.\,
The gauge symmetry breaking pattern is
$\,SU(2)^{N+1}\rightarrow SU(2)_D\,$,
with massless zero modes.}
\end{center}
\vspace*{-5mm}
\end{figure}

We start by considering the special case of Fig.\,1 with
\,$M=0$\, and \,$f_{N+1}^{~}=0$\,,\, whose gauge symmetry
breaking pattern is $\,SU(2)^{N+1}\rightarrow SU(2)_D\,$,\,
as shown in Fig.\,2.
The $(N+1)\times (N+1)$ gauge boson mass-matrix
$\mathbb{M}_a^2 =\mathbb{M}_{[0,N]}^2$
is given in (\ref{eq:M_CC}) with \,$f_{N+1}^{~}=0$\,,\,
and its orthonormal eigenvectors are denoted as
\,$\mathbb{X}_n = (X_{0n},\,X_{1n},\,\cdots,\,X_{Nn})^T$\,
($n=0,1,2,\cdots,N$).   So the gauge-eigenbasis field
$\mathbb{A}^{a\mu}$ and mass-eigenbasis field
$\mathbb{V}^{a\mu}$ are connected
by   $\,\mathbb{A}^{a\mu} = \mathbb{R}_a^{~}\mathbb{V}^{a\mu}\,$
with rotation
$\,\mathbb{R}_a^{~} = \(\mathbb{X}_0,\,\mathbb{X}_1,\,\cdots,\,\mathbb{X}_N\)\,$,\,
i.e.,
\begin{equation}
\label{eq:X-X5D}
\displaystyle
A^{a\mu}_j ~=
\sum_{n=0}^N V_n^{a\mu} X_{jn} \,,
~~~\Longrightarrow~~~
\widehat{A}^{a\mu}(\widehat{x} ) ~= \displaystyle \frac{1}{\sqrt{L\,}\,}
\sum_{n=0}^{\infty} V_n^{a\mu}(x) {\mathcal X}_{n}(x^5) \,,
\end{equation}
where the second relation is
the corresponding continuum KK-expansion
under \,$N\rightarrow \infty$\, and $\,j\,{\bf a}\rightarrow x^5\,$.\,
Analyzing the eigenvalue equation
$\,\mathbb{M}_a^2 \mathbb{X}_n =\lambda_n \mathbb{X}_n\,$,\,
we find that the eigenvectors satisfy the
following consistency relations,
\begin{equation}
\label{eq:4D-BC0}
\displaystyle
g_0^{~}X_{0,n} - g_{-1}^{~} X_{-1,n} \,=\, 0\,, ~~~~~
g_{N+1}^{~} X_{N+1,n} - g_N^{~} X_{N,n} \,=\, 0\,.
\end{equation}
As shown above Eq.\,(\ref{eq:SU2-5D-L}), taking the
5D continuum limit leads to the identification,
 $\, A^{a\mu}_j(x) \rightarrow
     (g_5^{~}/g_j^{~})\widehat{A}^{a\mu}(x,x^5_j)
 $\,.\,
Combining this with (\ref{eq:X-X5D}) we find that the eigenvector
$\,X_{jn}\,$ is related to the 5D KK-wavefunction ${\mathcal X}_n(x^5)$
via $\,g_j^{~}X_{jn}\rightarrow (g_5^{~}/\sqrt{L}\,){\mathcal X}_n(x_j^5)$\,.\,
Hence, we rewrite (\ref{eq:4D-BC0}) as conditions for the
5D KK-wavefunction $\,{\mathcal X}_n(x_j^5)\,$,
\begin{equation}
\label{eq:5D-BC0j}
\displaystyle
{\mathcal X}_{n}(x_0^5)     - {\mathcal X}_{n}(x_{-1}^5) \,=\, 0\,, ~~~~~
{\mathcal X}_{n}(x_{N+1}^5) - {\mathcal X}_{n}(x_N^5)    \,=\, 0\,.
\end{equation}
From (\ref{eq:5D-BC0j}),
we thus derive the {\it induced boundary conditions (BCs)}
of Neumann type in the 5D continuum limit,
\begin{equation}
\label{eq:5D-BC0}
\left.
\partial_5^{~}\widehat{A}^{a}_\mu \right|_{x^5=0} \,=\,0\,,
~~~~~~
\left.
\partial_5^{~}\widehat{A}^{a}_\mu \right|_{x^5=L} \,=\,0\,,
\end{equation}
where $\,L=N\mathbf a\,$ is the length of 5D.
This compactified 5D theory has an unbroken
$SU(2)$ gauge symmetry survived in 4D (corresponding to the
residual diagonal group $SU(2)_D$ of the Moose-A in Fig.\,2),
which ensures the massless zero KK-modes.
So, the Neumann BCs (\ref{eq:5D-BC0})
break a 5D $SU(2)$ into a 4D $SU(2)$
(with massive gauge KK towers as well as massless zero modes),
and preserve the rank of 5D gauge group, precisely matching
the symmetry breaking structure of its parent Moose-A
in Fig.\,2.

We can further derive the BCs for $\widehat{A}^{a5}$. In the continuum
limit the site-Goldstone fields $\pi^a_j$ will geometrize as $\widehat{A}^{a5}$
[cf. (\ref{eq:U-lattice})],
while the eaten Goldstones $\widetilde{\pi}^a_n$ will geometrize
as the KK modes $V^{a5}_n$
[cf. (\ref{eq:pi-A5-geo}) below].
From the mass-diagonalization in Eqs.\,(\ref{eq:M2-diag}) and
(\ref{eq:PI-rotation}) and
the $R_\xi$ gauge-fixing in Eq.\,(\ref{eq:GF}),
we see that the eaten Goldstone fields $\widetilde{\Pi}^a$ have mass matrix
$\xi_a(\widetilde{\mathbb{M}}_a^2)^{\rm diag}$ and the site-Goldstones $\Pi^a$ have
mass matrix $\xi_a\widetilde{\mathbb{M}}_a^2$.\,  For the Moose-A (Fig.\,2), we
derive the $N\times N$ mass matrix $\widetilde{\mathbb{M}}_a^2$,
{\small
\begin{equation}
\label{eq:MA-dual}
\widetilde{\mathbb{M}}_a^{2} = \df{1}{4}
\left(
\begin{array}{c|c|c|c|c}
&&&&
\\[-2mm]
(g_0^2+g_1^2) f_1^2    & -g_1^2f_1^{~}f_2^{~}     &      &
\\[1.5mm]
\hline
&&&&
\\[-2mm]
   -g_1^2f_1^{~}f_2^{~}
&  (g_1^2\!+\!g_2^2)f_2^2
&  -g_2^2f_2^{~}f_3^{~}         &
\\[1.5mm]
\hline
&&&&
\\[-2mm]
& -g_2^2f_2^{~}f_3^{~}
& (g_2^2 \!+\! g_3^2)f_3^2
& -g_3^2f_3^{~}f_4^{~}
\\[1.5mm]
\hline
&&&&
\\[-4mm]
 & & \ddots & \ddots & \ddots
\\[0.5mm]
\hline
&&&&
\\[-2mm]
 & & &  -g_{N\!-\!1}^2 f_{N\!-\!1}^{~}f_N^{~}    &
        (g_{N\!-\!1}^2\!+\!g_N^2)f_N^2
\\[1.5mm]
\end{array}
\right)\!,~
\end{equation}
}
\noindent\hspace*{-2mm}
whose orthonormal eigenvectors are denoted by
$\,\widetilde{\mathbb{X}}_n=(\widetilde{X}_{1n},\,\widetilde{X}_{2n},\,\cdots,\,\widetilde{X}_{Nn})^T$\,
($n=1,2,\cdots,N$).\,
The orthogonal rotation matrix in (\ref{eq:PI-rotation})
is thus given by
$\,\mathbb{R}_a^{~} = (\widetilde{\mathbb{X}}_1,\,\widetilde{\mathbb{X}}_2,\,\cdots,\,\widetilde{\mathbb{X}}_N)\,$.\,
So, in the continuum limit we have the following geometrization,
\begin{equation}
\label{eq:pi-A5-geo}
\displaystyle
\pi^{a}_j ~=
\sum_{n=0}^N \widetilde{\pi}_n^a \widetilde{X}_{jn} \,,
~~~\Longrightarrow~~~
\widehat{A}^{a5}(\widehat{x} ) ~= \displaystyle
\frac{1}{\sqrt{L\,}\,}
\sum_{n=0}^{\infty} V_n^{a5}(x) \widetilde{\mathcal X}_{n}(x^5) \,.
\end{equation}
As shown below Eq.\,(\ref{eq:U-lattice}), the site-Goldstone field
$\,\pi^a_j = (g_j^{~}f_j^{~}\mathbf a/2)A^a_{5}(x,x^5_j)\,$
is connected to the 5D gauge field $\,\widehat{A}^a_5(x,x_j^5)\,$ via
$\,\pi^a_j \rightarrow (\mathbf a g_5^{~}f_j^{~}/2)\widehat{A}^a_5(x,x_j^5)
$\, in the continuum limit.
So, from (\ref{eq:pi-A5-geo}) we find that $\widetilde{X}_{jn}$ and
$\widetilde{\mathcal X}_n(x^5_j)$ are connected via
$\,\widetilde{X}_{jn}\rightarrow (\mathbf a g_5^{~}f_j^{~}/2\sqrt{L})\widetilde{\mathcal X}_n(x_j^5)
\,$.\,
Analyzing the eigenvalue equation
$\,\widetilde{\mathbb{M}}_a^2\widetilde{X}_n = \lambda_n \widetilde{X}_n\,$,\,
we find that the eigenvectors must satisfy the following
consistency conditions,
\begin{equation}
\label{eq:MooseA-XTn-BC}
\widetilde{X}_{0,n}=\,0\,, ~~~~~
\widetilde{X}_{N+1,n}=\,0\,; ~~~\Longrightarrow~~~
\widetilde{\mathcal X}_{n}(x_0^5)\,=\,0\,, ~~~~~
\widetilde{\mathcal X}_{n}(x_{N+1}^5)\,=\,0\,.
\end{equation}
In the continuum limit, they result in the {\it induced} 5D BCs
for $\widehat{A}^a_5$,
\begin{equation}
\label{eq:5D-BC0-A5}
\left.\widehat{A}^a_5\right|_{x^5=0}\,=~0\,, ~~~~~~
\left.\widehat{A}^a_5\right|_{x^5=L}\,=~0\,,
\end{equation}
which are both of Dirichlet type, unlike the
Neumann BCs for $\widehat{A}^{a\mu}$ in (\ref{eq:5D-BC0}).
This is fully consistent with
the relation $\,\partial_5{\mathcal X}_n \propto \widetilde{\mathcal X}_n\,$
in (\ref{eq:Xn-XTn}) for 5D KK-wavefunctions.
We also note that the Neumann BCs (\ref{eq:5D-BC0})
for $\widehat{A}^{a\mu}$ and the Dirichlet BCs (\ref{eq:5D-BC0-A5})
for $\widehat{A}^{a5}$ can be combined into a gauge-covariant form,
\begin{equation}
\label{eq:5D-covBC}
\left.\widehat{F}^a_{\mu 5}\right|_{x^5=0}\,=~0\,, ~~~~~~
\left.\widehat{F}^a_{\mu 5}\right|_{x^5=L}\,=~0\,,
\end{equation}
which explicitly retains the gauge-invariance of the 5D action
at the corresponding boundary.

\vspace*{3mm}
We then turn to a different gauge symmetry
breaking structure. This is most cleanly described by
the Moose-B in Fig.\,3 (which we called ``enlarged moose''
in 2001\cite{SUSY03}).
This is a special case of our general moose theory in
Fig.\,1 with \,$M=0$\, and \,$g_{N+1}^{~}=0$\,,\,
with gauge group breaking pattern
$\,SU(2)^{N+1}\!\rightarrow\,$nothing\,.\,
As we will show, it {\it induces} the
Dirichlet type BCs in 5D continuum limit and
reduces the rank of the 5D gauge group.

\begin{figure}[h]
\label{fig:Fig3}
\begin{center}
\vspace*{-7mm}
\centerline{\psfig{file=fig3.eps,width=12cm}}
\setlength{\unitlength}{1mm}
\begin{picture}(0,0)
\end{picture}
\vspace*{-8mm}
\caption{Higgsless Moose-B: a special case of Fig.\,1
with \,$M=0$\, and \,$g_{N+1}^{~}=0$\,.\,
The gauge symmetry breaking pattern is
$\,SU(2)^{N+1}\!\rightarrow\,$\,nothing,
without massless ``zero modes''.}
\end{center}
\vspace*{-5mm}
\end{figure}

The gauge boson mass matrix
$\,\mathbb{M}_a^2=\mathbb{M}_{[0,N+1)}^2\,$
is just given by
Eq.\,(\ref{eq:M_CC}),
which has massive eigenvalues only.
Analyzing the eigenvalue equation
$\,\mathbb{M}_a \mathbb{X}_n =\lambda_n \mathbb{X}_n\,$,\,
we observe that the eigenvectors satisfy the
following consistency relations,
\begin{equation}
\label{eq:4D-BC1}
\begin{array}{lll}
&  g_0^{~}X_{0,n} - g_{-1}^{~}X_{-1,n} \,=\, 0\,, ~~~&~~~
   X_{N+1,n} \,=\, 0\,;
\\[2.5mm]
\Longrightarrow~~ &
{\mathcal X}_{n}(x_0^5) - {\mathcal X}_{n}(x_{-1}^5) \,=\, 0\,, ~~~&~~~
{\mathcal X}_{n}(x_{N+1}^5)                \,=\, 0\,.
\end{array}
\end{equation}
Taking the continuum limit as before, we derive the induced
BCs,
\begin{equation}
\label{eq:5D-BC1}
\left.
\partial_5^{~}\widehat{A}^{a}_\mu \right|_{x^5=0} \,=\,0\,,
~~~~~~~~
\left.
\widehat{A}^{a}_\mu \right|_{x^5=L} \,=\,0\,,
\end{equation}
where we see that similar to (\ref{eq:5D-BC0})
the BC at $x^5=0$ remains the Neumann type, but
the new condition at $x^5=L$ is a Dirichlet BC.
The moose representation of Fig.\,3 clearly shows
that it is this induced Dirichlet BC at $x^5=L$
that completely breaks the 5D $SU(2)$ gauge group
and makes all zero KK-modes massive.
The above formulation based on the Moose-A and -B
can be directly generalized to any
5D gauge theory with an arbitrary
simple group \,$G_A$\,.

We have also derived the BCs for $\widehat{A}^{a5}$
in the Moose-B.  Consider the mass matrix
$\xi_a\widetilde{\mathbb{M}}_a^2$ for site-Goldstone fields $\Pi^a$,
where the $(N+1)\times (N+1)$ matrix $\widetilde{\mathbb{M}}_a^2$
can be obtained from (\ref{eq:MA-dual}) by simple
replacements $\,N\rightarrow N+1\,$ and \,$g_{N+1}^{~}\rightarrow 0$\,.\,
Analyzing the eigenvalue equation
$\,\widetilde{\mathbb{M}}_a^2\widetilde{\mathbb{X}}_n=\lambda_n\widetilde{\mathbb{X}}_n\,$, we deduce the
following consistency relations for the eigenvectors,
\begin{equation}
\label{eq:MooseB-XTn-BC}
\widetilde{X}_{0,n} \,=~0\,,~~~~~~~~
f_{N+2}^{~}\widetilde{X}_{N+2,n} - f_{N+1}^{~}\widetilde{X}_{N+1,n} \,=~0\,.
\end{equation}
As before, we have the identification,
$\,\widetilde{X}_{jn}\rightarrow (\mathbf a g_5^{~}f_j^{~}/2\sqrt{L})\widetilde{\mathcal X}_n(x_j^5)
\,$,\, in the continuum limit.
So we can rewrite (\ref{eq:MooseB-XTn-BC}) as conditions
for $\widetilde{\mathcal X}_n(x_j^5)$,
\begin{equation}
\label{eq:5D-BC1-A5j}
\widetilde{\mathcal X}_{n}(x_0^5) \,=~0\,,~~~~~~~~
f_{N+2}^{2}\widetilde{\mathcal X}_n(x_{N+2}^5) -
f_{N+1}^{2}\widetilde{\mathcal X}_n(x_{N+1}^5) \,=~0\,.
\end{equation}
With the definition $\,f_j^{~}=h_jf=h(x^5_j)f\,$,\,
we derive the {\it induced BCs}
from (\ref{eq:MooseB-XTn-BC})
in the 5D continuum limit,
\begin{equation}
\label{eq:5D-BC1-A5}
\left.
\widehat{A}^{a}_5 \right|_{x^5=0} \,=\,0    \,,
~~~~~~~~
\left.
\partial_5 (h^2\widehat{A}^a_5 ) \right|_{x^5=L} \,=\,0 \,.
\end{equation}
They are fully consistent with the BCs for $\widehat{A}^a_{\mu}$
in (\ref{eq:5D-BC1}) according to Eq.\,(\ref{eq:Xn-XTn})
which gives
$\,\partial_5{\mathcal X}_n        \propto\widetilde{\mathcal X}_n\,$ and
$\,\partial_5(h^2\widetilde{\mathcal X}_n) \propto{\mathcal X}_n\,$.
Note that, unlike the Dirichlet BC for $\widehat{A}^a_5$,
the Neumann BC for $\widehat{A}^a_5$ does depend on the bulk
geometry via the function \,$h(x^5)$\,.
Implications of this will be
explored elsewhere\cite{HLmoosenew}.

\begin{figure}[h]
\label{fig:Fig4}
\begin{center}
\vspace*{-3mm}
\centerline{\psfig{file=fig4.eps,width=12cm}}
\setlength{\unitlength}{1mm}
\begin{picture}(0,0)
\end{picture}
\vspace*{-5mm}
\caption{Two dual mooses:
(a). Moose-$\rm\widetilde{A}$ is the dual of Moose-A in Fig.\,2;
(b). Moose-$\rm\widetilde{B}$ is the dual of Moose-B in Fig.\,3.
}
\end{center}
\vspace*{-5mm}
\end{figure}

It is also instructive to view the Goldstone mass matrix
$\widetilde{\mathbb{M}}_a^2$ as the ``gauge boson'' mass matrix generated
in the corresponding dual moose\cite{dualm,HLmoose}
which is defined by the exchange
\,$\{g_n^{~}\}\longleftrightarrow
   \{f_n^{~}\}$\, from the original moose.
We show the dual versions of the Moose-A and -B in Fig.\,4.
With the notations of open/closed intervals, it may be
convenient to denote the dual mass matrix as
\,$\widetilde{\mathbb{M}}_a^2=\widetilde{\mathbb{M}}_{(0,N+1)}^2$\, for Moose-${\rm\widetilde{A}}$
(where the sites \,$j=0,\,N\!+\!1$\, themselves
have no contribution due to $\,f_0^{~}=f_{N+1}^{~}=0$\,),
and  \,$\widetilde{\mathbb{M}}_a^2=\widetilde{\mathbb{M}}_{(0,N+1]}^2$\,
for Moose-${\rm\widetilde{B}}$ (where the site $j=0$ has
no contribution due to $\,f_0^{~}=0$\,).\,
Comparing the dual Moose-${\rm\widetilde{A}}$ [Fig.\,4(a)] with
Moose-A [Fig.\,2], we can now intuitively see why the
BCs for $\widetilde{\mathbb{X}}_n$ and $\widehat{A}^{a5}$ in
(\ref{eq:MooseA-XTn-BC})-(\ref{eq:5D-BC0-A5})
just take Dirichlet type, different from the Neumann BCs
(\ref{eq:4D-BC0})-(\ref{eq:5D-BC0})
for $\mathbb{X}_n$ and $\widehat{A}^{a\mu}$.\,
Similarly, comparison of the dual Moose-$\rm\widetilde{B}$ [Fig.\,4(b)]
with Moose-B [Fig.\,3] explains the difference between the
BCs (\ref{eq:5D-BC1-A5j})-(\ref{eq:5D-BC1-A5}) and
(\ref{eq:4D-BC1})-(\ref{eq:5D-BC1}).

\vspace*{2mm}
Another advantage of our general deconstruction
formalism is that we can also derive the proper
brane kinetic terms in the continuum limit.
Considering our Moose-B for instance, we quantify
how a brane term at \,$x^5=0$\,
is reconstructed from localizing
the boundary group \,$SU(2)_0$\, at site \,$j=0$\,.\,
Setting $\,f_1^2\gg f_{j}^2\,$ ($j\geqslant 2$)
for taking the continuum limit
(which prevents the site-$0$ from
joining the 5D geometry), we examine the equation
of motion (EOM) for $A^{a\mu}_0$ and derive the following
relation,
\begin{equation}
\label{eq:W3_0}
A_{0}^{a\mu} ~=~
\displaystyle\({g_1^{~}}/{g_{0}^{~}}\)A_1^{a\mu} \,.
\end{equation}
In the continuum theory,
this localizes \,$SU(2)_0$\, as a brane kinetic term at
$\,x^5=0\,$,
\begin{equation}
\label{eq:MooseB-BK}
S_{\rm BT}^{~} ~= \displaystyle\int\!\! d^5\widehat{x}~
\delta (x^5\!-0)\,
\frac{\,-(\widehat{g}^{~}\!\!/g_{0}^{~})^2\,}{4}
\widehat{F}^a_{\mu\nu}\widehat{F}^{a\mu\nu} \,,
\end{equation}
where \,$\widehat{g} \equiv g_5^{~}$ denotes the 5D gauge coupling.\,
The above brane term may be viewed as being actually localized at
$\,x^5=0^+$,\, the {\it right-neighborhood} of the point
\,$x^5=0$\, at which our BC
\,$\,\partial_5^{~}\widehat{A}^{a}_\mu |_{x^5=0} \!=\,0\,$\,
is imposed. This is because our
deconstruction procedure of deriving this BC from (\ref{eq:4D-BC1})
{\it includes} the boundary site \,$j=0$\, itself
plus its left-neighborhood.
We also note that the theory obeys a covariant BC
$\,\widehat{F}^a_{5\mu}=0\,$ at $x^5=0$ [similar to
(\ref{eq:5D-covBC})],
so we may rewrite the brane term (\ref{eq:MooseB-BK})
to have explicit invariance under the 5D gauge group
at $x^5=0$,
\begin{equation}
\label{eq:MooseB-BKx}
S_{\rm BT} ~= \displaystyle\int\!\! d^5\widehat{x}~
\delta (x^5\!-0)\,\sqrt{-\widehat{g}\,}\,\widehat{g}^{IK}\widehat{g}^{JL}
\frac{\,-\widehat{\zeta}\,}{\,4\kappa^2\,}
\widehat{F}^a_{IJ}\widehat{F}^a_{KL} \,,
\end{equation}
where $\,\widehat{\zeta} = (\widehat{g}/g_0)^2\kappa^2 (0)\,$.
Other brane terms may be rewritten in the similar way.

\vspace*{3mm}
Next, we analyze the Moose-C in Fig.\,5, which
is a special case of Fig.\,1 with
\,$M=0$\, and \,$g_{N+1}^{~}\neq 0$\,.\,
As a slight variation of Moose-B,
Moose-C has gauged its subgroup $U(1)_{N+1}$
and has the symmetry breaking pattern
$\,SU(2)^{N+1}\otimes U(1)\rightarrow U(1)_{\rm em}\,$,\,
as a straightforward extension of
Ref.\,[\refcite{GBESS}] which studied
the nontrivial minimal models
\,$SU(2)^3\otimes U(1)\rightarrow U(1)_{\rm em}$
($N+1= 3$) and
\,$SU(2)^2\otimes U(1)^2\rightarrow U(1)_{\rm em}$
($N+1=M+1= 2$)\, with general inputs
\,$(g_j^{~},\,f_k^{~})$\,.\,

\begin{figure}[h]
\label{fig:Fig5}
\begin{center}
\vspace*{-6mm}
\centerline{\psfig{file=fig5.eps,width=12cm}}
\setlength{\unitlength}{1mm}
\begin{picture}(0,0)
\end{picture}
\vspace*{-6mm}
\caption{Higgsless Moose-C: a special case of Fig.\,1
with \,$M=0$\, and \,$g_{N+1}^{~}\neq 0$\,.\,
The gauge symmetry breaking pattern is
$\,SU(2)^{N+1}\otimes U(1)\rightarrow U(1)_{\rm em}\,$,\,
extended from Ref.\,[22].}
\end{center}
\vspace*{-5mm}
\end{figure}

The charged gauge boson mass matrix
$\mathbb{M}_{W}^{\,2} =\mathbb{M}^2_{[0,N+1)}$
is given in (\ref{eq:M_CC}), the same as in
Moose-B, while the neutral gauge boson mass matrix
$\mathbb{M}_{Z}^{\,2} =\mathbb{M}_{[0,N+1]}^2$ can be obtained from
$\mathbb{M}_{Z}^{\,2}$[Moose-A] with a replacement $N\rightarrow N+1$.
In consequence we find that the consistency
relations for the eigenvectors
$\{\mathbb{X}^\pm_n\}$ and $\{\mathbb{X}^3_n\}$
are just a mixture
of Moose-A and -B in (\ref{eq:4D-BC0}) and
(\ref{eq:4D-BC1}),
\begin{equation}
\label{eq:4D-BC2}
\begin{array}{ll}
g_0^{~}X_{0,n}^\pm - g_{-1}^{~}X_{-1,n}^\pm \,=\, 0\,, ~~~&~~
X_{N+1,n}^\pm \,=\, 0\,;
\\[2mm]
g_0^{~}X_{0,n}^3 - g_{-1}^{~}X_{-1,n}^3 \,=\, 0\,, ~~~&~~
g_{N+2}^{~}X_{N+2,n}^3 - g_{N+1}^{~}X_{N+1,n}^3 \,=\, 0\,.
\end{array}
\end{equation}
So, with the same reasoning as before,
we obtain the induced continuum BCs
which combine (\ref{eq:5D-BC1}) for charged sector
and (\ref{eq:5D-BC0}) for neutral sector,
\begin{equation}
\label{eq:5D-BC2}
\begin{array}{l}
\left.\partial_5^{~}\widehat{A}^{\pm}_\mu \right|_{x^5=0} \!=\,0\,,
~~~
\left.\partial_5^{~}\widehat{A}^{3}_\mu \right|_{x^5=0} \!=\,0\,;
~~~
\left.\widehat{A}^{\pm}_\mu \right|_{x^5=L} \!=\,0\,,
~~~
\left.\partial_5^{~}\widehat{A}^{3}_\mu \right|_{x^5=L} \!=\,0\,.
\end{array}
\end{equation}
In the continuum limit, the boundary group
$U(1)_{N+1}$ will be localized at \,$x^5=L\,$,\,
similar to the brane term which we reconstructed in
(\ref{eq:MooseB-BK}).
So, we can localize $U(1)_{N+1}$
by setting $\,f_{N+1}^{2}\gg f_{j}^{2}\,$  ($j\leqslant N$)\,
for the continuum limit.  We quantify this by
examining the EOM for $A_{N+1}^{3\mu}$
under large $f_{N+1}^{2}$ limit, thus we derive a relation,
\begin{equation}
\label{eq:W3_N+1}
A_{N+1}^{3\mu} ~=~
\displaystyle\({g_N^{~}}/{g_{N+1}^{~}}\)A_N^{3\mu} \,,
\end{equation}
which generates the localized brane kinetic term at
\,$x^5=L$\,,
\begin{equation}
\label{eq:MooseC-BK}
S_{\rm BT}^{~} ~= \displaystyle\int\! d^5\widehat{x}~
\delta (x^5\!-\! L)
\frac{\,-(\widehat{g}^{~}\!/g_{N+1}^{~})^2\,}{4}
\widehat{F}^3_{\mu\nu}\widehat{F}^{3\mu\nu} \,.
\end{equation}
This brane term is actually localized at $\,x^5=L^-$,\,
the left-neighborhood of \,$x^5=L$\, at which
the BC $\,\partial_5^{~}\widehat{A}^{3}_\mu |_{x^5=L} \!=\,0\,$
is imposed, since our
deconstruction procedure of deriving this BC
from (\ref{eq:4D-BC2}) includes the
boundary site $j=N+1$ plus its right-neighborhood.

\vspace*{3mm}
We then consider the Moose-D in Fig.\,6
which is obtained from Fig.\,1 by folding
the $SU(2)$ lattice chain once at the middle
with renumbering \,$N\rightarrow 2N$\, and setting $\,M=0\,$.\,
The gauge symmetry breaking structure is
$\,SU(2)^{N+1}_L\otimes SU(2)_R^N \otimes U(1)
   \rightarrow U(1)_{\rm em}\,$,\, where
$SU(2)_L^{N+1}$ part has inputs $\{g_j^{~},\,f_k^{~}\}$
and $SU(2)_R^{N}\otimes U(1)$ part has inputs
$\{\widetilde{g}_j^{~},\,\widetilde{f}_k^{~}\}$.

\begin{figure}[t]
\label{fig:Fig6}
\begin{center}
\vspace*{-7mm}
\centerline{\psfig{file=fig6.eps,width=12cm}}
\setlength{\unitlength}{1mm}
\begin{picture}(0,0)
\end{picture}
\vspace*{-8mm}
\caption{Higgsless Moose-D:
Folding the $SU(2)^{N+1}$ moose chain once produces
a variation of Higgsless deconstruction from Fig.\,1
with $\,M=0\,$ and by renumbering \,$N\rightarrow 2N$.\,
The gauge symmetry breaking pattern is
$\,SU(2)^{N+1}_L\otimes SU(2)_R^N \otimes U(1)
   \rightarrow U(1)_{\rm em}\,$,\,
whose continuum limit has a bulk gauge group
$\,[SU(2)_L\otimes SU(2)_R]_{\rm 5D}\,$.}
\end{center}
\vspace*{-5mm}
\end{figure}

The continuum limit of Moose-D has the bulk gauge group
$\,[SU(2)_L\otimes SU(2)_R]_{\rm 5D}\,$
(with 5D gauge couplings $\widehat{g}_L^{~}$ and $\widehat{g}_R^{~}$),
plus a $U(1)_0$ brane term localized at \,$x^5=0$\,
[similar to (\ref{eq:MooseC-BK}) of Moose-C].
The other brane terms at \,$x^5=0$\, for $SU(2)_L$
[similar to (\ref{eq:MooseB-BK}) of Moose-C]
and at \,$x^5=L$\, for $SU(2)_D$ are also possible.
From our moose formulation,
it is clear that at \,$x^5=0$\,
the BCs for $SU(2)_L$ and $SU(2)_R$
are essentially the same as those
of Moose-C at \,$x^5=0$\, and \,$x^5=L$,\, respectively,
i.e.,
\begin{equation}
\label{eq:5D-BC3-x50}
\left.\partial_5^{~}\widehat{A}_L^{a\mu}\right|_{x^5=0} =0\,;
~~~~
\left.\widehat{A}_R^{\pm\mu}\right|_{x^5=0} =0\,,
~~~~
\left.\partial_5^{~}\widehat{A}_R^{3\mu}\right|_{x^5=0} =0\,.
\end{equation}
We then derive the new BCs at \,$x^5=L$\,.\,
Ordering the sites
\,$j=0,1,\cdots,N,\linebreak
     \widetilde{N},\widetilde{N\!\!+\!\!1},
     \cdots,\widetilde{1},\widetilde{0}$,\,
we can write down the charged gauge boson mass matrix
$\,\mathbb{M}_{W}^{\,2} =\mathbb{M}_{[0,\widetilde{0} )}^2\,$ and the
neutral gauge boson mass matrix
$\,\mathbb{M}_{Z}^{\,2} =\mathbb{M}_{[0,\widetilde{0} ]}^2\,$.\,
Inspecting their eigenvalue equations we derive the
consistency relations for the eigenvectors
\,$\mathbb{X}_n\in SU(2)_L$\, and \,$\widetilde{\mathbb{X}}_n\in SU(2)_R$,\,
related to the boundary \,$x^5=L$\,,
\begin{equation}
\label{eq:4D-BC3}
\begin{array}{rr}
\displaystyle
F^2(g_N^{~}X_{N,n}-\widetilde{g}_N^{~}\widetilde{X}_{N,n}) +
f_{N+1}^2(g_{N+1}^{~}X_{N+1,n}-g_N^{~}X_{N,n}) & \,=~ 0 \,,
\\[2mm]
\displaystyle
-F^2(g_N^{~}X_{N,n}-\widetilde{g}_N^{~}\widetilde{X}_{N,n}) +
\ft_{N+1}^2(\widetilde{g}_{N+1}^{~}\widetilde{X}_{N+1,n}-\widetilde{g}_N^{~}\widetilde{X}_{N,n}) & \,=~ 0 \,.
\end{array}
\end{equation}
The eigenvectors $\,(X_{jn},\,\widetilde{X}_{jn})\,$
are related to the latticized 5D KK-wavefunctions
$\,\displaystyle \({\mathcal X}_n(x^5_j),\,\widetilde{\mathcal X}_n(x^5_j)\)\equiv ({\mathcal X}_{jn},\,\widetilde{\mathcal X}_{jn})
=\sqrt{L}\(\frac{g_j}{\widehat{g}_L}X_{jn},\,\frac{\widetilde{g}_j}{\widehat{g}_R}\widetilde{X}_{jn}\)\,$.\,
So we rewrite (\ref{eq:4D-BC3}) as conditions for
\,$({\mathcal X}_{jn},\,\widetilde{\mathcal X}_{jn})$\,,
\begin{equation}
\label{eq:5D-BC3j}
\begin{array}{rr}
\displaystyle
F^2(\widehat{g}_L^{~}{\mathcal X}_{N,n}-\widehat{g}_R^{~}\widetilde{\mathcal X}_{N,n}) +
\widehat{g}_L^{~}f_{N+1}^2({\mathcal X}_{N+1,n}-{\mathcal X}_{N,n}) & \,=~ 0 \,,
\\[2mm]
\displaystyle
-F^2(\widehat{g}_L^{~}{\mathcal X}_{N,n}-\widehat{g}_R^{~}\widetilde{\mathcal X}_{N,n}) +
\widehat{g}_R^{~}\ft_{N+1}^2
(\widetilde{\mathcal X}_{N+1,n}-\widetilde{\mathcal X}_{N,n}) & \,=~ 0 \,,
\end{array}
\end{equation}
which are equivalent to
\begin{equation}
\label{eq:5D-BC3j2}
\begin{array}{l}
\displaystyle
\widehat{g}_L^{~}{\mathcal X}_{N,n}-\widehat{g}_R^{~}\widetilde{\mathcal X}_{N,n}   ~=~
\widehat{g}_L^{~}(f_{N+1}/F)^2({\mathcal X}_{N,n}-{\mathcal X}_{N+1,n})   \,,
\\[2mm]
\displaystyle
\widehat{g}_R^{~} ({\mathcal X}_{N+1,n}-{\mathcal X}_{N,n}) +
\zeta^2\,\widehat{g}_L^{~}(\widetilde{\mathcal X}_{N+1,n}-\widetilde{\mathcal X}_{N,n}) \,=~ 0 \,,
\end{array}
\end{equation}
where
$\,\zeta \equiv(\widehat{g}_R^{~}\ft_{N+1}^{~})/(\widehat{g}_L^{~}f_{N+1}^{~})\,$.\,
With the definitions $\,f_j^{~}= h_jf_L^{~}\,$ and
$\,\ft_j^{~}=h_jf_R^{~}\,$, we have the lattice spacing
$\,\mathbf a \equiv (2/\widehat{g}_Lf_L)^2 = (2/\widehat{g}_Rf_R)^2$ [as shown above
(\ref{eq:SU2-5D-L})], which leads to a geometric equality
$\,\widehat{g}_L^{~}f_L^{~} = \widehat{g}_R^{~}f_R^{~}\,$.\,
Hence the 5D geometry ensures $\,\zeta = 1\,$.\,
Taking continuum limit,
we derive the induced 5D BCs from (\ref{eq:5D-BC3j2}),
\begin{equation}
\label{eq:5D-BC3-x5L}
\left.\(\widehat{g}_L^{~}\widehat{A}_L^{a\mu}
       -\widehat{g}_R^{~}\widehat{A}_R^{a\mu}\)\right|_{x^5=L} =\,0\,,
~~~~~~
\left.\partial_5^{~}\!\(\widehat{g}_R^{~}\widehat{A}_L^{a\mu}
                  + \widehat{g}_L^{~}\widehat{A}_R^{a\mu}\)
                   \right|_{x^5=L} =\,0\,.
\end{equation}

We may also reconstruct a brane term at \,$x^5=L$\,.\,
This can be done by taking $\,f_N^2\gg f_{j<N}^2\,$
and $\,\ft_N^2\gg \ft_{j<N}^2\,$ together with
$\,F^2\rightarrow \infty\,$.\,
This will localize the diagonal group $SU(2)_N^D$
at \,$x^5=L$\,.\,
By definition, $\,g_j^{~}=\widehat{g}_L^{~}\kappa_j/\sqrt{\mathbf a}\,$
and $\,\widetilde{g}_j^{~}=\widehat{g}_R^{~}\kappa_j/\sqrt{\mathbf a}\,$, which
contain the same $\kappa_j$ because of the 5D geometry
in the continuum limit of Moose-D. Thus,
$\,\widehat{g}_L^{~}/g_N^{~}=\widehat{g}_R^{~}/\widetilde{g}_N^{~}
  =\sqrt{\mathbf a}/\kappa_N^{~}\equiv\eta\,$.\,
So we can write the brane coupling of  $SU(2)_N^D$  as
$\,g_D^{~}=1/\sqrt{1/g_N^{2}+1/\widetilde{g}_N^{2}}
=\eta^{-1}\widehat{g}_L^{~}\widehat{g}_R^{~}/\sqrt{\widehat{g}_L^2+\widehat{g}_R^2}\,$.\,
The brane gauge field is
$\,A^{a\mu}_{N,D}
=(\widetilde{g}_N A^{a\mu}_N +g_N\widetilde{A}^{a\mu}_N)/\sqrt{g_N^2+\widetilde{g}_N^2}\,$.\,
In the continuum limit, it becomes
$\,A^{a\mu}_{N,D}
=\eta (\widehat{g}_R^{~}\widehat{A}^{a\mu}_L +
  \widehat{g}_L^{~}\widehat{A}^{a\mu}_R )/\sqrt{\widehat{g}_L^2+\widehat{g}_R^2}\,$
at \,$x^5=L$\,,\,
and leads to the Yang-Mills term
$\,\displaystyle -\frac{1}{4}(F_{N,D}^{a\mu\nu})^2 =
   -\frac{\eta^2}{4(\widehat{g}_L^2+\widehat{g}_R^2)}
[\widehat{g}_R^{~}\widehat{F}_L^{a\mu\nu}+\widehat{g}_L^{~}\widehat{F}_R^{a\mu\nu}]^2
$.~
Hence, we derive the brane localized kinetic term
at \,$x^5=L$\,,
\begin{equation}
\label{eq:MooseD-BK}
S_{\rm BT} ~= \displaystyle\int\! d^5\widehat{x}~
\delta (x^5\!-\! L)
\frac{-\eta^2}{\,4(\widehat{g}_L^2+\widehat{g}_R^2)\,}
\[\widehat{g}_R^{~}\widehat{F}_L^{a\mu\nu}+\widehat{g}_L^{~}\widehat{F}_R^{a\mu\nu}\]^2
\,.
\end{equation}
The possible brane terms at \,$x^5=0$\, can be similarly
reconstructed as in our Moose-B and Moose-C.

\begin{figure}[h]
\label{fig:Fig7}
\begin{center}
\vspace*{-4mm}
\centerline{\psfig{file=fig7.eps,width=13cm}}
\setlength{\unitlength}{1mm}
\begin{picture}(0,0)
\end{picture}
\vspace*{-7mm}
\caption{Higgsless Moose-E with double-folding
from the general moose theory in Fig.\,1.
(a). Moose-E1 in the continuum limit has gauge group
$\,[SU(2)^3]_{\rm 5D}\otimes U(1)_0\,$.\,
(b). Moose-E2 in the continuum limit has gauge group
$\,[SU(2)_L\otimes SU(2)_R\otimes U(1)_X]_{\rm 5D}\,$.
They are both broken to $U(1)_{\rm em}$.}
\end{center}
\vspace*{-4mm}
\end{figure}

A further extension of the Moose-D is to fold the
$SU(2)$-chain twice, called Moose-E1 in Fig.\,7(a).
This leads to a 5D continuum Higgsless theory with
symmetry $\,[SU(2)^3]_{\rm 5D}\otimes U(1)_0$\, whose BCs
can be similarly derived as in Moose-D.
The gauge group breaks as
$\,[SU(2)^3]_{\rm 5D}\otimes U(1)_0 \rightarrow U(1)_{\rm em}$\,
under compactification.
Another variation is to add many $U(1)$'s to Moose-D
and form a separate $U(1)$-chain,
or, from Fig.\,1 we can fold
the $SU(2)$-chain once in its middle as well as
folding the moose at
the intersection of the $SU(2)$ and $U(1)$ chains,
which we call Moose-E2 in Fig.\,7(b).
Its 5D continuum model has the
bulk gauge group
$\,[SU(2)_L\otimes SU(2)_R\otimes U(1)_X]_{\rm 5D}$\,
which, for the case of warped geometry, gives precisely
the 5D Higgsless model proposed by
Cs\'{a}ki et al [\refcite{HLxx}].
Possible brane terms can be generated in
the continuum limit as we discussed before.
With the above method,
we have also derived all the {\it induced} BCs in the
continuum limit of Moose-E2,
and found that the BCs for
$(\widehat{A}^{a\mu}_L,\,\widehat{A}^{a\mu}_R,\,\widehat{B}^{\mu})$ agree
with the BCs of Cs\'{a}ki et al (based on the technique\cite{Heb}
of adding extra boundary Higgs fields with
large VEV).
Systematic elaboration of these will be given
elsewhere.\cite{HLmoosenew}

\section{$\!\!$Effective\,Unitarity\,of\,Higgsless\,Deconstruction\,Without~Geometry}

In this section we will demonstrate a conceptual point that
the delayed unitarity violation (effective unitarity) in
the general moose theory can be realized without
resorting to any known 5D geometry.
We observe that the delay of unitarity violation is
essentially a {\it collective effect}
due to the participation in the EWSB
from many gauge groups whose own symmetry breaking scales
$\{f_j^{~}\}$ are higher than the SM EWSB scale,
\,$v=(\sqrt{2}G_F)^{-1/2}\simeq 246$\,GeV and whose gauge
couplings $\{g_j^{~}\}$ are larger than $g\in SU(2)_{\rm SM}$.
Such a collective effect {\it does not} necessarily require
any exact 5D geometry, and can be realized in general
non-geometric moose settings with relevant input parameters
$\,f_j^{~} > v\,$ and/or $\,g_j^{~} > g\,$.

\vspace*{-2mm}
\subsection{Formulation of the Effective Unitarity}

To analyze the unitarity it is enough to consider the
\,$g_{N+1}^{~}\rightarrow 0$\, limit, so the nontrivial part of Fig.\,1
reduces to our Moose-B (Fig.\,3), corresponding to the
custodial symmetry limit. The other settings such as Moose-C,D,E
also reduce to Moose-B at the zeroth order of \,$g_{N+1}^{~}$\,.\,
From Eq.\,(\ref{eq:L}) we first derive the relevant
unitary-gauge Lagrangian,
\begin{equation}
\label{eq:B-Lu}
\begin{array}{ll}
{\cal L} & ~=~ \displaystyle
\sum_{j=0}^{N}
-\frac{1}{4}{F}^a_{j\mu\nu}{F}_j^{a\mu\nu}
+\frac{1}{2}{\mathbb{A}_\mu^a}^T\mathbb{M}_W^2\mathbb{A}^{a\mu}
\\[4mm]
& ~=~ \displaystyle
\sum_{j=0}^{N}
-\frac{1}{4}\overline{W}^a_{j\mu\nu}\overline{W}_j^{a\mu\nu}
+\frac{1}{2}{\mathbb{W}_\mu^a}^T{\mathbb{M}_W^2}^{\!\!\rm diag}\mathbb{W}^{a\mu}
+{\cal L}_G^{\rm int}
\end{array}
\end{equation}
with the Yang-Mills interactions,
\begin{equation}
\label{eq:YM-int}
\begin{array}{ll}
{\cal L}_G^{\rm int}
& \displaystyle ~=\,
\sum_{j=0}^N \[
-\frac{\,g_j^{~}}{2}C^{abc}
 \overline{F}^{a\mu\nu}_jA^b_{j\mu}A^c_{j\nu}
-\frac{\,g_j^2}{4}C^{abc}C^{ade}
  A^{b\mu}_jA^{c\nu}_jA^d_{j\mu}A^e_{j\nu}
\]
\\[5mm]
& \displaystyle ~=\,
-\frac{\,G_3^{kmn}\,}{2}C^{abc}\overline{W}^{a\mu\nu}_kW^b_{m\mu}W^c_{n\nu}
-\frac{\,G_4^{k\ell mn}\,}{4}C^{abc}C^{ade}
   W^{b\mu}_kW^{c\nu}_{\ell}W^d_{m\mu}W^e_{n\nu} \,,
\end{array}
\end{equation}
where
$\,\overline{F}^{a\mu\nu}_j \equiv
   \partial^\mu A_j^{a\nu} -\partial^\nu A_j^{a\mu}\,$,\,
$\,\overline{W}^{a\mu\nu}_j \equiv
   \partial^\mu W_j^{a\nu} -\partial^\nu W_j^{a\mu}\,$,\,
and
$\,\mathbb{W}^{a\mu}=(W^{a\mu}_0,\,\cdots,\,W^{a\mu}_N)^T\,$
denotes the mass-eigenbasis fields.
Also the sums over repeated indices
\,$k,\ell,m,n=(0,1,\cdots, N)\,$ are implied.
In Eq.\,(\ref{eq:YM-int})
the effective cubic and quartic gauge couplings are given by
\begin{equation}
\label{eq:g-3W-4W}
G_3^{kmn} \,=\,\sum_{j=0}^N g_j^{~}\,
             \mathbb{R}_{jk}^a\mathbb{R}_{jm}^b\mathbb{R}_{jn}^c\,,~~~~~~
G_4^{k\ell mn} \,=\,\sum_{j=0}^N g_j^2\,
             \mathbb{R}_{jk}^b\mathbb{R}_{j\ell}^d\mathbb{R}_{jm}^c\mathbb{R}_{jn}^e\,,
\end{equation}
with  \,$\mathbb{R}^{a} = \mathbb{R}_W$\,
determined from the diagonalization
$~\mathbb{R}_W^T\mathbb{M}_{W}^{\,2}\mathbb{R}_W\,= (\mathbb{M}_W^2\!)^{\rm diag}\,$.\,

The longitudinal gauge boson scattering
$\,W^a_{mL}W^b_{nL}\rightarrow W^c_{kL}W^d_{\ell L}\,$
has the following structure,
\begin{equation}
\label{eq:WW-WW}
\begin{array}{l}
{\cal T}^{ab,cd}_{mn,k\ell} ~ =
{\cal T}^{ab,cd}_{mn,k\ell [c]} +
{\cal T}^{ab,cd}_{mn,k\ell [s]} +
{\cal T}^{ab,cd}_{mn,k\ell [t]} +
{\cal T}^{ab,cd}_{mn,k\ell [u]}
\\[5mm]
~= \displaystyle
G_4^{mnk\ell}{\cal A}^{abcd}_{[c]} +
\sum_{j=0}^N \[
\frac{G_3^{mnj}G_3^{k\ell j}}{\,s-M_{Wj}^2\,}
{\cal A}^{abcd}_{[s]}  +
\frac{G_3^{mkj}G_3^{n\ell j}}{\,t-M_{Wj}^2\,}
{\cal A}^{abcd}_{[t]}  +
\frac{G_3^{m\ell j}G_3^{nkj}}{\,u-M_{Wj}^2\,}
{\cal A}^{abcd}_{[u]}
\],
\end{array}
\end{equation}
which is a sum of possible diagrams from the contact
interactions (denoted as $[c]$)
and the gauge boson exchanges via
$(s,t,u)$-channels (denoted as $[s],[t],[u]$).
Here the quantities \,${\cal A}^{abcd}_{[c],[s],[t],[u]}$\,
contain an overall factor of relevant $SU(2)$
structure constants, and
are functions of the energy $\sqrt{s}$ and
scattering angle $\theta$, whose analytical
structures are identical to that of the (Higgsless) SM
(with $g'\simeq 0$) up to a simple overall factor $g^2$.\,
Taking the asymptotic energy \,$s\gg M_{Wj}^2$\, and
expanding the amplitude (\ref{eq:WW-WW}),
we observe that all $O(E^4)$ terms exactly cancel
as enforced by the ET (\ref{eq:ETm}),
while the nonzero $O(E^2)$ terms are {\it suppressed} by
large \,$N\!+\!1$\,,\, leading to the {\it delayed unitarity
violation}\cite{CH}\,   in deconstruction theories.

To understand the suppressed $O(E^2)$ terms, it is very
useful to examine the pure Goldstone interactions derived
from (\ref{eq:L}), with general inputs for Moose-B,
\begin{eqnarray}
\label{eq:L-pit-int}
{\mathcal L}_{\widetilde{\pi}}^{\rm int}
\,&=&\,\displaystyle
\frac{~{\mathcal C}_{k{\ell} mn}~}{\,6v^2\,} \[
(\widetilde{\pi}_k^a\partial_\mu\widetilde{\pi}^a_{{\ell}})
(\widetilde{\pi}_m^b\partial^\mu\widetilde{\pi}^b_{n})
- (\widetilde{\pi}_k^a \widetilde{\pi}_{{\ell}}^a)
(\partial_\mu\widetilde{\pi}_m^b \partial^\mu\widetilde{\pi}_n^b)   \]
+O\!\(\widetilde{\pi}_j^6\) ,
\end{eqnarray}
where \,$(k,\ell ,m,n)=0,1,2,\cdots,N$\, and
sum over repeated indices is implied.
The quartic coupling of the eaten Goldstone bosons is
\begin{eqnarray}
\label{eq:C}
{\mathcal C}_{k\ell mn}^{~} ~&=&~ \displaystyle
\sum_{j=1}^{N+1} \frac{v^2}{\,f_j^2\,}
\widetilde{\mathbb{R}}_{jk}^a\widetilde{\mathbb{R}}_{j\ell}^a\widetilde{\mathbb{R}}_{jm}^b\widetilde{\mathbb{R}}_{jn}^b \leqslant 1 \,,
\end{eqnarray}
with  \,$\widetilde{\mathbb{R}}^{a} = \widetilde{\mathbb{R}}_W$\,
determined from the diagonalization
$~\widetilde{\mathbb{R}}_W^T\widetilde{\mathbb{M}}_{W}^{\,2}\widetilde{\mathbb{R}}_W\,=\,
  (\widetilde{\mathbb{M}}_W^2\!)^{\rm diag}\,$.\,
For the special case of the Higgsless SM
($N\!+\!1=M\!+\!1=1$),\, the coupling
$\,{\mathcal C}_{0000}^{\rm SM} \equiv 1$\,,\, so there is
no suppression for the eaten Goldstone interactions.
Computing the leading amplitude of the Goldstone scattering
~$\widetilde{\pi}_{n}^a \widetilde{\pi}_{n}^b\rightarrow \widetilde{\pi}_{n}^c\widetilde{\pi}_{n}^d$\,
($n\geqslant 0$),\,  we arrive at
\begin{equation}
{\cal T}[\widetilde{\pi}_n^a\widetilde{\pi}_n^b\rightarrow\widetilde{\pi}_n^c\widetilde{\pi}_n^d] ~=~
\displaystyle\frac{\,{\mathcal C}_{nnnn}\,}{v^2}
\[s\,\delta^{ab}\delta^{cd} + t\,\delta^{ac}\delta^{bd} + u\,\delta^{ad}\delta^{bc}\],
\end{equation}
as compared to the unsuppressed result in the Higgsless SM,
\begin{equation}
{\cal T}[\widetilde{\pi}_0^a\widetilde{\pi}_0^b\rightarrow\widetilde{\pi}_0^c\widetilde{\pi}_0^d]_{\rm SM} ~=~
\displaystyle\frac{1}{\,v^2\,}
\[s\,\delta^{ab}\delta^{cd} + t\,\delta^{ac}\delta^{bd} + u\,\delta^{ad}\delta^{bc}\].
\end{equation}
Thus, we find that our general Moose-B has a {\it delayed}
unitarity violation scale for the Goldstone scattering
~$\widetilde{\pi}_{n}^a \widetilde{\pi}_{n}^b\rightarrow \widetilde{\pi}_{n}^c\widetilde{\pi}_{n}^d$\,
($n\geqslant 0$),\,
relative to the \,$n=N=0$\, case of the Higgsless SM,
\begin{equation}
\label{eq:UB-pipi}
\displaystyle
\mathscr{D}_{\rm U}^{~} \,\equiv\,\frac{E^{\star}}{\,E^{\star}_{\rm SM}\,} \,=\,
\frac{1}{~\sqrt{{\mathcal C}_{nnnn}^{~}\,}~} > 1 \,,
~~~~~~{\rm with}~~~~~
{\mathcal C}_{nnnn}^{~} \,=\,
\sum_{j=1}^{N+1} \frac{v^2}{\,f_j^2\,}\,
\widetilde{\mathbb{R}}_{jn}^{a\,4}  \,.
\end{equation}
The original analysis
[\refcite{CH}] found that for a flat-geometry input
\,$f_0^{~}=\cdots =f_N^{~}\equiv f$\, and
\,$g_0^{~}=\cdots =g_N^{~}\equiv\widetilde{g}$\,,\,
the delay factor is
$\,\mathscr{D}_{\rm U}^{~} \simeq (f/v)\sqrt{N\!+\!1}\,$
for large $\,N\!+\!1$,\,
where $f$ scales like\cite{CH}
$\,\sqrt{N\!+\!1}\,$ when holding $g$ and $1/R$\,.\,
This means\cite{CH} that for large $\,N\!+\!1$\,,\,
the delay factor $\mathscr{D}_{\rm U}^{~}$ scales as \,$N\!+\!1$\,.

To analyze essential features of the delay factor
$\mathscr{D}_{\rm U}^{~}$, we first consider the simplest flat version of
Moose-B (Fig.\,3) with massive zero-modes,
by setting the inputs
\,$f_0^{~}=\cdots =f_N^{~}=f_{N+1}^{~}\equiv f$\, and
\,$g_0^{~}=\cdots =g_N^{~}\equiv\widetilde{g}$\,.\,
The Goldstone decay constants $f_j^{~}$ are connected to
the EWSB vacuum expectation value (VEV) $v$ via
the low energy four-fermion interactions,\cite{HLmoose}
\begin{eqnarray}
\label{eq:v}
\[\sum_{j=p+1}^{N+1}\df{1}{\,f_j^2\,}\]^{-\frac{1}{2}}
&=&~ v ~=~ \(\sqrt{2}G_F\)^{-\frac{1}{2}}
\,\simeq~ 246\,{\rm GeV}\,,
\end{eqnarray}
where the site \,$j=p\in [0,N]$\,
is the location of left-handed SM fermions.
Thus, inputting $v$ makes one of $\{f_j^{~}\}$'s non-independent,
and for the flat (equal) $f_j^{~}$\,'s we deduce
$~f=v\sqrt{N+1-p~}~$.\,
The symmetry breaking pattern also imposes a relation
$\,g_0^{-2}+\cdots +g_N^{-2}=g^{-2}\,$  with
$\,g\in SU(2)_W$,\, so for the flat (equal)
$g_j^{~}$\,'s we have
$\,g_j^{~}\equiv\widetilde{g} =g\sqrt{N\!+\!1\,}\,$.\,
With these we exactly solve all the mass eigenvalues
and eigenvectors. In particular we find that
site-Goldstones \,$\Pi^a=(\pi_1^a,\cdots,\pi_{N+1}^a)^T$\,
are connected to the eaten Goldstones
\,$\widetilde{\Pi}^a=(\widetilde{\pi}_0^a,\cdots,\widetilde{\pi}_{N}^a)^T$\, by
the orthogonal rotation $\,\Pi^a = \widetilde{\mathbb{R}}^a \widetilde{\Pi}^a$\, with
\begin{equation}
\label{eq:MooseB-Rpi}
\widetilde{\mathbb{R}}_{jn}^a  ~=~
\sqrt{\frac{2}{\,N+\frac{3}{2}\,}}\sin\[(j+1)\alpha_n\],~~~~~
\( j,n=0,1,\cdots,N\) ,
\end{equation}
where $\,\alpha_n \equiv \(n+\frac{1}{2}\)\pi/\(N+\frac{3}{2}\)\,$.\,
In the gauge sector we deduce the transformation from
the site-fields \,$\{A_j^{a\mu}\}$\, to the mass-eigenbasis fields
$\,\{W_n^{a\mu}\}\,$ via $\,\mathbb{A}^{a\mu}=\mathbb{R}^a\mathbb{W}^{a\mu}\,$
with
\begin{equation}
\label{eq:MooseB-Rw}
\mathbb{R}^a_{jn} = \displaystyle
\sqrt{\frac{2}{\,N+\frac{3}{2}\,}}\cos\[\(j+\frac{1}{2}\)\alpha_n\],~~~~~
\( j,n=0,1,\cdots,N \) .
\end{equation}
Substituting (\ref{eq:MooseB-Rpi}) and
the relation $~f=v\sqrt{N+1-p\,}~$ into
Eq.\,(\ref{eq:UB-pipi}), we compute the delay factor
for a flat Moose-B,
\begin{equation}
\mathscr{D}_{\rm U}^{~} \,=\, \displaystyle
\frac{f}{\,v\,}
\[\sum_{j=0}^{N}\widetilde{\mathbb{R}}_{jn}^{a\,4}\]^{-\frac{1}{2}}
=~\[\frac{2}{3}\(N+\frac{3}{2}\)\(N+1-p\)\]^{\frac{1}{2}} \,,
\end{equation}
which scales as \,$N/\sqrt{1.5}$\, for large
$\,N\!+\!1$\, and small \,$p =0-O(1)$\,.\,
For large $\,p\sim N\,$,\, the factor $\mathscr{D}_{\rm U}^{~}$ will be
reduced, scaling like $\,\sim\! \sqrt{N\,}\,$.\,

From the above, we see that a sizable delay factor
$\,\mathscr{D}_{\rm U}^{~} > 1\,$
is essentially due to the {\it collective effect}
of the many gauge groups participating in the EWSB via
{\bf (i)}
the composition of the eaten Goldstone state
$\widetilde{\pi}^a_n=\sum_j\widetilde{\mathbb{R}}_{jn}^a\pi_j^a$
in terms of all $N\!+\!1$ site-Goldstones
$\{\pi^a_j\}$; and
{\bf (ii)}
many {\it comparable} symmetry breaking scales $\,f_j^{~}>v\,$
under the condition (\ref{eq:v}).
We observe that the realization of this collective effect
does not necessarily require the moose inputs
\,$(g_j^{~},\,f_k^{~})$\, to obey any exact 5D geometry.
In fact, for the general parameters
\,$(g_j^{~},\,f_k^{~})$\,,\, we can achieve a visible delay
factor $\,\mathscr{D}_{\rm U}^{~} > 1\,$ as long as $\,N\!+\!1\,$ is reasonably
large (implying that many site-Goldstones participate
in the collective symmetry breaking) and all relevant
$\,(g_j^{~},\,f_k^{~})\,$ are significantly larger than
\,$(g,\,v)$\, of the SM.
With these two requirements satisfied, we find that, even if
the precise pattern of the inputs \,$(g_j^{~},\,f_k^{~})$\,
are very non-geometric and random-like,
a nearly maximal delay factor can be reached for each
given \,$p$\,, which roughly scales like
\begin{equation}
\label{eq:UB-max}
\displaystyle
\mathscr{D}_{\rm U}^{~} ~\sim~
\mathscr{D}_{\rm U}^{\max} ~\longrightarrow~
O(1)\[(N+1)(N+1-p)\]^{\frac{1}{2}}  \,.
\end{equation}

Finally, we note that the above instructive estimate of the delay
factor $\mathscr{D}_{\rm U}^{~}$ is based on the leading Goldstone amplitude
at \,$O(E^2)$.\,
The exact longitudinal gauge boson amplitude
(\ref{eq:WW-WW}) may result in a stronger
unitarity limit $E^{\star}$ and thus smaller delay factor
$\mathscr{D}_{\rm U}^{~}$, since the analysis of leading $E^2$-amplitude of
the Goldstone scattering has ignored all the subleading
terms of $O(E^0)$  and  $O(M_{Wn}^2/E^2)$ which
could be visible when $\,N\!+\!1\,$ is not too
large and the limit $E^{\star}$ is only around a few TeV.
So, in the explicit analysis below,
we will directly compute the gauge boson amplitude
(\ref{eq:WW-WW}) for deriving the unitarity bound
\,$E^{\star}$\,.

\subsection{Explicit Analysis of Effective Unitarity Without Geometry}

Next, we explicitly demonstrate how
the effective unitarity can be realized in various minimal
moose theories without geometry.  For the explicit analysis,
we will focus on the zero-mode amplitude
$\,{\cal T}[W^a_{0L}W^b_{0L}\rightarrow W^c_{0L}W^d_{0L}]\,$
which is also to be tested at the LHC.
The unitarity requires its $s$-wave amplitude to be
bounded from above,
$\,\left|a_{0000}^{0}[ab,cd]\right|< 1/2\,$,\,
where the possible effect of final state identical particles
can be included\cite{DH}.\,
We know that the naive Higgsless SM
($N\!+\!1=M\!+\!1=1$) predicts the unitarity violation
scale in the $W_{0L}W_{0L}$ scattering as,
$\,E^\star_{\rm SM} \simeq \sqrt{8\pi}v\simeq
   1.2\,$TeV\cite{SM-UB,CG,DH}.\,
This is significantly lower than the conventional cutoff
scale \,$\Lambda \simeq 4\pi v\simeq 3$TeV as estimated by
the consistency of chiral perturbation theory\cite{NDA}.\,
So we will identify and analyze the minimal moose (MM) models
that can exhibit an effective unitarity at least
up to a scale $E^\ast \gtrsim 3$\,TeV.

We perform the numerical analysis
at the zeroth order of $U(1)$ coupling $g_{N+1}^{~}$.
We ensure the light mass $M_W=M_{W0}\simeq 80$\,GeV
and the lowest new gauge boson mass
$M_{W1}\gtrsim 800$\,GeV
by the phenomenological consideration.
We will consider one minimal moose $SU(2)^3\otimes U(1)$
(called MM3, $N\!+\!1=3$ and $M=0$) with the
left(right)-handed fermions
coupled to the site $j=p=0$ ($j=q=3$) in Fig.\,1,
and another minimal moose
(called MM4, $N\!+\!1=4$ and $M=0$) with the left(right)-handed
fermions coupled to the site $j=p=1$ ($j=q=4$).
We compute the unitarity limit $E^\star$ for
each model by scanning the parameter space in Fig.\,8, where the
$E^\star$ with ``non-geometric'' inputs
(of random-like pattern) are shown by solid curves and
are compared to the ``geometric'' inputs
(with relevant $g_j^{~}$'s and/or
$f_j^{~}$'s being equal) as depicted by dotted curves.
We have chosen the scattering
$\,W^+_{0L}W^-_{0L}\rightarrow W^+_{0L}W^-_{0L}\,$ in Fig.\,8
for illustration since other channels are found to
exhibit similar features.
For this process the $u$-channel term
in (\ref{eq:WW-WW}) is absent.
[The exact tree-level amplitude of this scattering in the
Higgsless SM ($N\!+\!1=M\!+\!1=1$)
may be found in Eq.\,(3.17)
of the first paper in [\refcite{ET}] which does not pose
delayed unitarity violation.]
\begin{figure}[h]
\label{fig:8}
\vspace*{-6mm}
\centerline{\psfig{file=fig8.eps,width=13cm}}
\setlength{\unitlength}{1mm}
\begin{picture}(0,0)
\put(20,110){$(g_1^{~},\,g_2^{~})=(3,4)$}
\put(20,78){Top to bottom (solid): $f_1^{~}=0.4,0.5,0.8$TeV}
\put(20,35.4){$(g_1^{~},\,g_2^{~},\,g_3^{~})=(3,4.2,3.5)$, $f_1^{~}=1$TeV}
\put(20,67.5){Top to bottom (solid): $f_2^{~}=0.4,0.5,0.8$TeV}
\end{picture}
\vspace*{-18.5mm}
\caption{Unitarity limit $E^\star$ is shown
for the moose $SU(2)^3\otimes U(1)$ (MM3)
with $p=0$ in plot (a) as a function of $f_2^{~}$,
and for the moose $SU(2)^4\otimes U(1)$ (MM4) with
$p=1$ in plot (b) as a function of $f_3^{~}$.
The solid curves are shown for three sets of
non-geometric inputs which appear random-like.
As a comparison,
the dotted curves come from two sets of
typical geometric inputs:
$g_1^{~}=g_2^{~}=4$, $f_1^{~}=\sqrt{3}v\simeq 0.43$\,TeV
in plot (a); and
$g_1^{~}=g_2^{~}=g_3^{~}=4$,
$f_1^{~}=1$\,TeV, $f_2^{~}=\sqrt{3}v$ in plot (b).
}
\vspace*{-2mm}
\end{figure}

Fig.\,8 shows that either solid or dotted curves
are generally quite flat
and there exists no sharp ``peak'' when we vary
$f_2^{~}$ in the plot (a) and $f_3^{~}$ in the plot (b).
There is a sizable region on each curve to realize
a nearly maximal scale \,$E^{\star}\sim E^{\star}_{\max}$\,.\,
The inputs for dotted curves mimic closely for
certain flat-geometry settings, but for realizing
\,$E^{\star}\sim E^{\star}_{\max}$\,
they clearly do not exhibit any real
advantage over other non-geometric
inputs (shown as solid curves).
We also find $E^{\star}$ to be less sensitive to the variation
of the pattern of gauge couplings.
To have a direct feeling,
we explicitly list two sample inputs/outputs
for each model. For the MM3 model with $p=0$,
inputting \,$(g_1^{~},\,g_2^{~})=(3.1,\,4)$\, and
\,$(f_1^{~},\,f_2^{~})=(0.5,\,0.4)$\,TeV, we derive
the gauge boson mass-spectrum
$\,(M_{W0},\,M_{W1},\,M_{W2})\simeq (0.08,\,0.8,\,1.3)\,$TeV,
and the unitarity violation scale
$\,E^{\star} \simeq 3.1$\,TeV.
For the MM4 model with $p=1$, inputting
\,$(g_1^{~},\,g_2^{~},\,g_3^{~})=(3,\,4.2,\,3.5)$\, and
\,$(f_1^{~},\,f_2^{~},\,f_3^{~})=(1,\,0.5,\,0.38)$\,TeV,
we derive the gauge boson mass-spectrum
$\,(M_{W0},\,M_{W1},\,M_{W2},\,M_{W3})\simeq
   (0.08,\,0.81,\,1.3,\,1.8)\,$TeV,
and the unitarity violation scale
$\,E^{\star} \simeq 3.0$\,TeV; if we reduce the input values of
$\,(g_1^{~},\,g_2^{~})\,$ to
$\,(g_1^{~},\,g_2^{~})=(2.5,\,4)\,$ and keep other inputs
unchanged, we obtain a very similar mass-spectrum
$\,(M_{W0},\,M_{W1},\,M_{W2},\,M_{W3})\simeq
   (0.08,\,0.8,\,1.2,\,1.6)\,$TeV and a unitarity limit
$\,E^{\star} \simeq 3.1$\,TeV.

We have also analyzed other moose models with different
inputs of \,$N\!+\!1$\, and/or \,$p$\,
and found similar features, although the limit $E^{\star}$
for each scattering process
generally increases for larger \,$N\!+\!1$\, and smaller
\,$p$\,,\, as expected.
More systematic analysis including the effect of
delocalized fermions will be given elsewhere.\cite{HLmoosenew}

\section{Discussion and Conclusion}

In conclusion,
using the powerful deconstruction approach has revealed that
4D gauge theories exhibit far richer dynamics than
naive expectation.
With this conceptually clean tool,
we have formulated and classified
the compactified 5D gauge symmetry
breaking (without/with gauge group rank reduction)
via spontaneous symmetry breaking in
general gauge-invariant 4D moose theories.
We demonstrate that
various consistent 5D boundary conditions (BCs) as well
as possible brane terms are automatically {\it induced}
from taking proper continuum limits of the general
gauge-invariant moose theory.

We observe that the {\it effective unitarity}
(delayed unitarity violation)
is essentially a {\it collective effect}
due to the participation in the EWSB
from many gauge groups whose symmetry breaking scales
($f_j^{~}$)
are higher than the SM EWSB scale
$v=(\sqrt{2}G_F)^{-1/2}$ and whose gauge couplings
$g_j^{~}$ are larger than the SM $SU(2)_W$ coupling $g$\,.\,
We find that the effective unitarity
can be naturally realized in a wide class of 4D moose theories
even with a few extra gauge bosons,
without resorting to any known 5D geometry.
As shown in Sec.\,5, an important advantage of the
advocated non-geometric moose theories is
that they contain much larger parameter space than the
special geometric setting, so they open up much wider windows
for accommodating experimental constraints while retaining
effective unitarity.
More elaboration including the effect of delocalized fermions
will be given elsewhere\cite{HLmoosenew}.

The precision constraints on our general moose theory
(Fig.\,1) with the SM fermions localized
at arbitrary sites $j=p$ ($0\leqslant p\leqslant N$) and
$j=q$ ($N+1\leqslant q\leqslant K+1$)
have been systematically analyzed\cite{HLmoose}.
In particular, by requiring a sizable gap between the
light masses \,$M_{W,Z}$\, and the first heavy masses
\,$M_{W',Z'} \,(\,\lesssim \sqrt{8\pi}\,v\,)$,\,
a sum rule for the precision parameters
(corresponding to $\widehat{S}$\,) can be derived,\cite{HLmoose}
\begin{equation}
\label{eq:Shat}
\!\!\widehat{S} \,=\,\displaystyle
\frac{1}{4s_Z^2}\[\alpha S\!+\! 4c_Z^2\(\Delta\rho \!-\!\alpha T\)
\!+\!\frac{\alpha\delta}{c_Z^2}\]
\geqslant \sum_{r=p+1}^{N}\!\frac{\,M_W^2\,}{M_r^2}
\,\gtrsim\, \frac{M_W^2}{\,8\pi v^2\,}
\,\simeq\, 4\!\times\! 10^{-3},
\end{equation}
where $\,p\in [0,N)\,$ and
$\,\{M_r^2\}$\, are eigenvalues of mass matrix
$\,\mathbb{M}_{(p,N+1)}^2$.\,
Although the precision data seem not to favor (\ref{eq:Shat}),
this phenomenological constraint may be evaded in a number
of ways, {\it e.g.,} by smearing the
fermion location out of a single site in the general moose
theory,\cite{HLmoosenew}\, similar to the use of
delocalized bulk fermions in certain 5D models\cite{HLyy}.

\vspace*{4mm}
\noindent
{\bf Acknowledgments}~\,
I am grateful to R. S. Chivukula, E. H. Simmons, M. Tanabashi and
M. Kurachi for fruitful collaborations and helpful suggestions
to the manuscript.
I also thank D. A. Dicus, A. Hebecker, R. Sundrum, J. Terning and
U. Varadarajan for useful discussions.
This work was supported by U.S. Department of Energy under
grant No.~DE-FG03-93ER40757.
\vspace*{-1mm}

redpajama_set_name: RedPajamaArXiv