
\documentstyle[aps,floats,preprint,epsfig]{revtex}
\tighten
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\begin{document}
\draft
\preprint{\vbox{\hbox{OCIP/C 98-02}
\hbox{MADPH-98-1066}
\hbox
\hbox{October 1998}}}
\title{Discovery Limits for Techni-Omega Production in $e\gamma$
Collisions}
\author{Stephen Godfrey$^1$, Tao Han$^2$ and Pat Kalyniak$^1$}
\address{
$^1$Ottawa-Carleton Institute for Physics \\
Department of Physics, Carleton University, Ottawa CANADA, K1S 5B6 \\ 
$^2$Department of Physics, University of Wisconsin, Madison,  WI 53706}

\maketitle

% my defs
\def\ot{\omega_T^{}}
\def\mot{M_{\omega_T}}
\def\got{\Gamma_{\omega_T}}
\def\gwwz{\Gamma_{WWZ}}
\def\zl{Z_L^{}}
%

\begin{abstract}
In a strongly-interacting electroweak 
sector with an isosinglet
vector state, such as the techni-omega, $\ot$,
the direct $ \omega_T Z \gamma $ coupling
implies that an $\ot$ can be produced by $Z \gamma$ fusion in $e
\gamma$ collisions. This is a unique feature for high energy $e^+e^-$ or
$e^-e^-$ colliders operating in an $e\gamma$ mode. We consider the
processes $e^- \gamma \to e^- Z\gamma$ and $e^- \gamma \to e^- W^+ W^-
Z$, both of which proceed via an intermediate $\omega_T$. 
We find that at a 1.5~TeV $e^+e^-$ linear 
collider operating in an $e\gamma$ mode with an integrated luminosity 
of 200 fb$^{-1}$, 
an $\omega_T$ can be discovered for a broad range of masses and widths.

\end{abstract}
\pacs{PACS numbers: }

\section{Introduction}

The mechanism for electroweak symmetry breaking remains the most
prominent mystery in elementary particle physics. In the Standard Model 
(SM), a neutral scalar fundamental Higgs boson is expected with a mass
($m_H$) 
less than about 800 GeV. In the weakly coupled supersymmetric
extension of the SM, the lightest Higgs boson is expected to be lighter
than
about 140 GeV. Searching for the Higgs bosons is
a primary goal of current and future collider 
experiments \cite{higgs}. However, if no light fundamental Higgs boson is 
found for $m_H< 800$ GeV, one would anticipate that the interactions 
among the longitudinal vector bosons become strong \cite{sews}.  
This is the case when strongly interacting dynamics 
is responsible for electroweak symmetry breaking, such as
in Technicolor models \cite{TC}.

Without knowing the underlying dynamics of the strongly-interacting 
electroweak sector (SEWS), it is instructive to
parametrize the physics with an effective theory 
for the possible low-lying resonant states.
This typically includes an isosinglet scalar meson $(H)$ and an
isotriplet vector meson $(\rho_T)$  \cite{technirho}.  
However, in many dynamical electroweak symmetry breaking models there 
exist other resonant states such as an isosinglet vector ($\omega_T$),
and isotriplet vector ($a_{1_{T}}$) \cite{chivukula90,rr}.  In fact it has
been 
argued that to preserve good high energy behavior for strong
scattering amplitudes in a SEWS, it is 
necessary for all the above resonant states to coexist \cite{han96}.  
It is therefore wise to keep an open mind and to consider other 
characteristic resonant states when studying the physics of a SEWS at 
high energy colliders.

Among those heavy resonant states, the isosinglet vector state
$\omega_T$ has rather unique features.
Due to its isosinglet nature, it does not
have strong coupling to two gauge bosons.
It couples strongly to three longitudinal gauge bosons $W^+_LW^-_LZ^{}_L$
(or equivalently, electroweak Goldstone bosons $w^+w^-z$) and 
electroweakly to $Z \gamma$, $Z Z$, and $W^+ W^-$.
It may mix with the $U(1)$ gauge boson $B$, depending
on its hypercharge assignment in the model. The signal for $\ot$
production was studied for $pp$ collisions at 40 TeV and 17 TeV
\cite{chivukula90,rr}. It appears to be difficult to observe the
$\ot$ signal at the LHC, largely due to the SM backgrounds. 
On the other hand, the direct $\omega_T Z_L^{} \gamma$ 
coupling implies that an $\ot$ can be effectively produced by 
$\zl\gamma$ fusion in $e\gamma$ collisions.  This is a unique feature 
for high energy $e^+ e^-$ or $e^- e^-$ colliders operating in an 
$e\gamma$ mode. In this paper we concentrate on $\ot$ production
at $e\gamma$ linear colliders. We first describe a 
SEWS scenario in Sec.~II in terms of an effective Lagrangian involving 
$\ot$ interactions. We then present our results in Sec.~III
for the production and decay of an $\omega_T$ in $e\gamma$ colliders.
We show that a high energy $e\gamma$ linear collider 
will have great potential for the discovery of an $\omega_T$ with a mass
of order 1 TeV. We conclude in Sec.~IV.

\section{$\ot$ Interactions}
For an isosinglet vector, a Techniomega-like state $\ot$,
the leading strong interaction can be parameterized by
%
\begin{equation}
{\cal L}_{\rm strong} = \frac{g_\omega}{v\Lambda^2}\ 
\varepsilon_{\mu\nu\rho\sigma}\ \omega_T^\mu \ 
\partial^\nu{w^+} \partial^\rho{w^-} \partial^\sigma z,
\label{otwww}
\end{equation}
%
where $v=246$~GeV is the scale of electroweak symmetry breaking and 
$\Lambda$ is the new physics scale at which the strong dynamics
sets in. The effective coupling $g_\omega$ is of strong coupling
strength, and is model-dependent. It governs the partial decay
width $\Gamma(\ot \to w^+w^-z) \equiv \gwwz$. To study the $\ot$ signal in
a model-independent way, we will use this physical 
partial width as the input parameter to extract the factor
$(g_\omega/v\Lambda^2)$. With the interaction Eq.~(\ref{otwww}),
the spin-averaged amplitude squared for the decay 
$\ot \to W^+W^-Z$ is calcuated to be
%
\begin{eqnarray}
\overline{\Sigma}| {\cal M}(\ot \to WWZ)|^2 & = & \frac{1}{3}
\frac{g^2_\omega}{v^2 \Lambda^4} \{M^4_W M^2_Z + 2q_1 \cdot q_2 \,
q_2 \cdot q_3 \, q_3 \cdot q_1 \nonumber \\
& & -M^2_Z (q_1 \cdot q_2)^2 - M^2_W ((q_2 \cdot q_3)^2 + (q_1 \cdot
q_3)^2) \}. \nonumber
\end{eqnarray}
%
The four-momenta in this expression correspond to the
labelling of Fig.~\ref{fig:vertex}a.
As the full expression for $\gwwz$ is rather
complicated, we evaluated the phase space integral numerically to relate
$(g_\omega/v\Lambda^2)$ to $\gwwz$. 

The effective Lagrangian describing the electroweak 
interactions of the $\omega_T$ 
with the electroweak gauge bosons can be written as \cite{chivukula90}:
%
\begin{equation}
{\cal L}_{\rm e.w.} = \chi \varepsilon_{\mu\nu\rho\sigma} 
[ -g \hbox{Tr} ( \frac{\vec{\sigma}}{2}\cdot \vec{W}^{\rho\sigma} 
\left\{  { \Sigma D^\nu \Sigma^{\dag} , \omega_T^\mu  } 
\right\} ) + g' \hbox{Tr} ( \frac{\sigma^3}{2} {B}^{\rho\sigma} 
\{ \Sigma^{\dag} D^\nu \Sigma , \omega_T^\mu \} ) ],
\end{equation}
%
where  the covariant derivative is defined by
\begin{equation}
D^\mu\Sigma  = \partial ^\mu\Sigma -
ig\ \vec{W}^\mu \cdot \frac{\vec{\sigma}}{2}\Sigma + 
ig'\ \Sigma B^\mu \frac{\sigma_3}{2},
\end{equation}
$g$ and $g'$ are the $SU(2)_L$ and $U(1)_Y$ coupling constants, 
$\Sigma$ is the non-linearly realized representation of the 
Goldstone boson fields and transforms like 
$\Sigma \to L\Sigma R^{\dag}$. 
%The fields $\xi$ are defined by $\xi \xi =\Sigma$ 
%so that $\xi$ transforms as $\xi \to L \xi U^{\dag} 
%= U \xi R^{\dag}$. and $M^\mu$ is given by
%\begin{equation}
%M^\mu = \omega_T^\mu I + i\vec{\sigma}\cdot \vec{\rho}_T^\mu
%\end{equation}
%which transforms as $M^\mu \to U M U^{\dag}$.
In the Unitary gauge $\Sigma \to 1$. 
With these substitutions ${\cal L}_{e.w.}$ leads to
%
\begin{eqnarray}
{\cal L}_{\rm e.w.} & \sim &  
2i e^2 \chi \varepsilon_{\mu\nu\rho\sigma}\  
\omega^\mu_T \left[ 
\frac{1}{\sin^2\theta_w} (\partial^\rho W^{+\sigma} W^{-\nu} +
\partial^\rho W^{-\sigma} W^{+\nu}) \right.
\nonumber \\
& &   \left. { + \left( { \frac{\cos^2\theta_w}{\sin^2\theta_w}
- \frac{\sin^2\theta_w}{\cos^2\theta_w} } \right) 
\partial^\rho Z^\sigma Z^\nu + 
\frac{2}{\sin\theta_w\cos\theta_w} \partial^\rho 
\gamma^\sigma Z^\nu } \right] + \cdots
\label{lew}
\end{eqnarray}
 
%
\noindent where $\theta_w$ is the electroweak mixing angle.
The last term gives the $\omega_T Z\gamma$ vertex, of importance for
production of the $\ot$ in $e\gamma$ colliders.
It involves the unknown coupling parameter $\chi$, where $e\chi^{1/2}$ is
expected to be  of electroweak strength. Similarly, the $\ot W^+
W^-$ and $\ot ZZ$ vertices, corresponding to the first and second terms,
respectively, of Eq.~(\ref{lew}) above, are proportional to $\chi$. 
The Feynman rules for the effective interactions of the $\ot$, represented
in  Fig.~\ref{fig:vertex}, are given  in Table \ref{Frules}.
We take the partial width of the $\ot$ into these two body states,
$$\Gamma_{2-body} = \Gamma(\ot \to Z\gamma) + \Gamma(\ot \to W^+ W^-)
+\Gamma(\ot \to ZZ)$$ 
as input to determine this coupling $\chi$.
The two body partial widths are:
%
\begin{equation}
\Gamma(\ot \to Z\gamma) =
\frac{8\pi\alpha^2\chi^2}{3\cos^2\theta_w\sin^2\theta_w}
\frac{M^3_{\ot}}{M^2_Z}\ \left(1-\frac{M^2_Z}{M^2_{\ot}}\right)^3
\left(1+\frac{M^2_Z}{M^2_{\ot}}\right),
\end{equation}
%
\begin{equation}
\Gamma(\ot \to W^+W^-) =
\frac{4\pi\alpha^2\chi^2 }{3\sin^4\theta_w}
\frac{M^3_{\ot}}{M^2_W}\ 
\sqrt{1-4\frac{M^2_W}{M^2_{\ot}}}
\left(1-4\frac{M^2_W}{M^2_{\ot}}\right)^2,
\end{equation}

\begin{eqnarray}
\Gamma(\ot \to ZZ) & = &
\frac{8\pi\alpha^2\chi^2}{3}(\cot^2\theta_w-\tan^2\theta_w)^2
\frac{\mot^3}{M^2_Z}\ \sqrt{1-4\frac{M^2_Z}{\mot^2}}
\nonumber \\
& &\times\left[6\frac{M^4_Z}{\mot^4}+\frac{1}{2}\left(1+\frac{M^2_Z}
{\mot^2}\right)\left(1-4\frac{M^2_Z}{\mot^2}\right)\right].
\end{eqnarray}


%
%\begin{equation}
%i \frac{4 \chi e^2}{\sin\theta_w\cos\theta_w} 
%(\frac{N_T}{3})^{1/2}
%\varepsilon_{\mu\nu\rho\sigma} 
%\varepsilon_{\omega_T}^\mu \varepsilon^\nu_Z 
%p_\gamma^\rho \varepsilon^\sigma_\gamma
%\end{equation}

\begin{table}
\begin{center}
\caption{Feynman rules for the effective interactions of $\ot$}
\label{Frules}
\vspace{0.2cm}
\begin{tabular}{l|l}
%\hline
\raisebox{0pt}[12pt][6pt]{Vertex} & 
\raisebox{0pt}[12pt][6pt]{Feynman rule} \\
\hline
\raisebox{0pt}[12pt][6pt]{$\ot w^+w^-z$} & 
\raisebox{0pt}[12pt][6pt]{$i g_{\ot}/({v\Lambda^2})\ 
\epsilon_{\mu\nu\rho\sigma}
\epsilon^\mu(\omega)q^{\nu}_1 q^{\rho}_2 q^{\sigma}_3$}
\\
\hline
\raisebox{0pt}[12pt][6pt]{$\ot Z\gamma$} & 
\raisebox{0pt}[12pt][6pt]{$i 4 e^2\chi/({\sin\theta_w
\cos\theta_w})\ \epsilon_{\mu\nu\rho\sigma}\epsilon^\mu(\omega)
\epsilon^\nu(Z)p^\rho_\gamma\epsilon^\sigma(\gamma)$} \\
\hline
\raisebox{0pt}[12pt][6pt]{$\ot ZZ$} & 
\raisebox{0pt}[12pt][6pt]{$i4e^2\chi(\cot^2\theta_w
-\tan^2\theta_w)\ \epsilon_{\mu\nu\rho\sigma}\epsilon^\mu(\omega)
\epsilon^\nu_1p^{\rho}_{2}\epsilon^\sigma_2$} \\
\hline
\raisebox{0pt}[12pt][6pt]{$\ot W^+ W^-$} & 
\raisebox{0pt}[12pt][6pt]{$i 2e^2\chi/
\sin^2\theta_w\ \epsilon_{\mu\nu\rho\sigma}\epsilon^\mu(\omega)
\epsilon^\nu_-(p^\rho_+-p^\rho_-)
\epsilon^\sigma_+$} \\
%\hline
\end{tabular}
\end{center}
%\label{Frules}
\end{table}
\vspace*{3pt}

\begin{figure}[tbh]
\centering
\leavevmode
\epsfxsize=4.8in\epsffile{vertex2.ps}
\caption{ Effective interactions of $\ot$ with (a)
$w^+w^-z$ and (b) two vector bosons.}
\label{fig:vertex}
\end{figure}


\section{Calculation and Results}


We consider the two signal processes which proceed via an intermediate 
$\omega_T$:
%
\begin{eqnarray}
\label{zgm}
e^- \gamma & \to & e^- \ot \to e^- Z \gamma, \\
\label{wwz}
e^- \gamma & \to & e^- \ot \to e^- W^+ W^- Z .
\end{eqnarray}
%
Depending upon the  $\omega_T Z\gamma$ coupling, the signal 
cross section can be fairly large. The cross section expressions
are lengthy and we do not present them here. We choose to look at these
$Z\gamma$ and $WWZ$ final states based on the distinctive signature of the
first process and the potential
enhancement of the second arising from its dependence on the strong
coupling $g_\omega$.


The SM background to the process $e\gamma \to e Z \gamma$ is the 
bremsstrahlung of photons and $Z$'s from the electron. The
background process to the $e^- W^+ W^- Z$ final state
has a complicated structure, with 90 diagrams. The main contribution
comes from the subprocesses
$e^+ e^- \to W^+ W^-, \gamma\gamma \to W^+ W^-$ with 
a radiated $Z$. We use the MADGRAPH package \cite{madg}
to evaluate the full SM amplitudes for the background processes.

In calculating the total cross sections for the $e Z \gamma$ signal 
and the
backgrounds, we impose the following ``basic cuts'' to roughly
simulate the detector coverage:
%
\begin{equation}
170^\circ >\theta_e > 10^\circ, \quad 165^\circ > \theta_\gamma > 15^\circ, 
\quad  \theta_{e\gamma} > 30^\circ, \quad E_\gamma > 50\ {\rm GeV},
\label{Basic}
\end{equation}
%
where $\theta_e$ is the polar angle with respect to the $e^-$ beam
direction in the lab ($e^+e^-$ c. m.) frame and $\theta_{e\gamma}$ is the 
angle between the outgoing $e^-$ and $\gamma$.
Only the cut on $\theta_e$ is relevant to the $e^- W^+ W^- Z$ 
process. The cuts on photons also regularize the infrared and
collinear divergences in the tree-level background calculations.

We have also implemented the back-scattered laser spectrum for
the photon beam \cite{Photon}. For simplicity, we have ignored 
the possible polarization for the electron and photon beams, 
although an appropriate choice of photon beam polarization may 
enhance the signal and suppress the backgrounds.

\begin{figure}[tbh]
\centering
\leavevmode
%\psfig{figure= fig2a.eps,height=1.5in}
%\epsfxsize=3.0in\epsffile{fig2b.eps}
\epsfig{file=fig2a.eps,width=3.0in,clip=}
\epsfig{file=fig2b.eps,width=3.0in,clip=}
\caption[]{Cross sections versus the $e^+e^-$ c. m. energy
for the signal $e^-\gamma\to e^-\ot$ with
$\ot \to W^+W^-Z$ and $Z\gamma$ and the SM backgrounds.
The solid lines are for $M_{\omega_T}=0.8$~TeV 
and the long-dashed lines for $M_{\omega_T}=1.0$~TeV. 
In each case the curve with the larger 
cross section is for the $W^+W^-Z$ final state and the lower
for the $Z\gamma$ final state.  The dash-dot line is for 
the $e^-W^+W^-Z$ SM background and the dotted line is 
for the $e^-Z \gamma$ SM background.
In (a), $\gwwz=4\Gamma_{2-body}$ and in (b), 
$\gwwz=2\Gamma_{2-body}$. The chosen values of the partial widths are
given in
the text.}
\label{fig:xsecE}
\end{figure}


We present the total cross section for the signal and background
versus the $e^+e^-$ c. m. energy $\sqrt {s_{e^+e^-}}$ in 
Fig.~\ref{fig:xsecE} with various choices of $\ot$ mass and partial
widths. We have taken representative values for
$\mot$ of 0.8 (1.0)~TeV.
In Fig.~\ref{fig:xsecE}(a), we use partial widths 
$\Gamma_{2-body}=5$~(20)~GeV and $\gwwz=20$~(80)~GeV,
 setting $\gwwz=4\Gamma_{2-body}$. In Fig.~\ref{fig:xsecE}(b),
we
take $\Gamma_{2-body}=15$~(40)~GeV and $\gwwz=30$~(80)~GeV,
such that $\gwwz=2\Gamma_{2-body}$.
As noted above, the values for the couplings $\chi$ and $g_\omega$ are 
obtained using these partial widths as input. In Fig.~\ref{fig:xsecE}(b),
the 2-body decay modes represent a larger fraction of the total width and,
hence, the cross sections for the signal processes, which go via $Z\gamma$
fusion, are enhanced due to the larger value of $\chi$.
 We see that, for the parameters considered,  the signal cross sections
for the
$e^- W^+ W^- Z$ channel
are about $10-100$ fb once above the $\ot$ mass threshold and overtake the
background rates by as much as an order of magnitude. Such
high production rates imply that the linear collider would
have great potential to discover and study the $\ot$.
The cross sections for the $Z\gamma$ final state are lower,
as expected, and lie below the background when only the cuts of 
Eq.~(\ref{Basic}) are imposed.

The reason that the cross section for $\mot=1.0$ TeV becomes
larger than that for 0.8 TeV in Fig.~\ref{fig:xsecE}(a) is because we have
chosen
relatively larger couplings for the 1.0 TeV $\ot$,
based on the larger input partial widths.


In Fig.~\ref{fig:xsecM}, we show the total signal cross sections
versus  $\mot$ for $\sqrt{s_{e^+e^-}}$=1.5~TeV.
The couplings are the values obtained by taking
$\Gamma_{2-body}=1\% M_{\omega_T}$ and 
$\gwwz=4\Gamma_{2-body}$. As expected, below the
mass threshold $\sqrt{s_{e\gamma}}<\mot$, the signal cross
section drops sharply. However, depending on the broadness
of the resonance, there is still non-zero signal cross section.
%
\begin{figure}[tb]
\centering
\leavevmode
\epsfig{file=fig3n.eps,width=4.8in,clip=}
\caption[]{
Cross sections versus the $\mot$ 
for the signal $e^-\gamma \to e^-\ot$ for
$\sqrt{s_{e^+e^-}}=1.5$~TeV
with $\ot \to W^+W^-Z$ (dashed line) and $Z\gamma$ (solid line).}
\label{fig:xsecM}
\end{figure}

%

We have chosen partial decay widths of $\ot$ as input parameters
to characterize its coupling strength. It is informative to explore
how the cross section changes with the widths. Fig.~\ref{xsecW}
demonstrates this point, for $\sqrt{s_{e^+e^-}}$=1.5~TeV and 
$\mot$=1~TeV.
We vary $\Gamma_{2-body}$ and take
$\gwwz=4\Gamma_{2-body}$. Measurement of the signal cross section
rates and relative branching fractions for the two channels 
would reveal important information on the underlying SEWS
dynamics.

%
\begin{figure}[tb]
\centering
\leavevmode
\epsfig{file=fig4n.eps,width=4.8in,clip=}
\caption[]{
Cross sections versus the partial width $\Gamma_{2-body}$
for the signal $e^-\gamma\to e^-\ot$ 
for $\sqrt{s_{e^+e^-}}=1.5$~TeV and $\mot=1$~TeV
with $\ot \to W^+W^-Z$ (dashed line) and $Z\gamma$ (solid line).}
\label{xsecW}
\end{figure}

%


Although the SM backgrounds seem to be larger or, at best comparable, to
the signal
rate for the $Z\gamma$ channel,
the final state kinematics is very different between them.
Because the final state vector bosons in the signal are from the
decay of a very massive particle, they are generally very energetic
and fairly central. We thus impose further cuts to reduce the
backgrounds at little cost to the signal:
%
\begin{equation}
15^\circ < \theta_{\gamma,Z,W} < 165^\circ, \quad 
E_{\gamma,Z,W} > 150 \ {\rm GeV}.
\label{Cutii}
\end{equation}
%

The most distinctive feature for the signal is the resonance
in the invariant mass spectrum for $W^+W^-Z$ and $Z\gamma$ final
states. We demonstrate this in Fig.~\ref{Mass} for both the $W^+W^-Z$
and $Z\gamma$ modes. Results for $\mot$ of 0.8~TeV (dotted lines) and 1.0
TeV (dashed lines) are shown
for $\sqrt{s_{e^+e^-}}=1.5$~TeV. We take $\gwwz=2\Gamma_{2-body}$. The
cuts in both
Eqs.~(\ref{Basic}) and (\ref{Cutii}) 
are imposed. We see that a resonant structure at $\mot$ is evident 
and the SM backgrounds (solid lines) after cuts (\ref{Cutii}) are
essentially
negligible for the particular choice of parameters shown. 

%
\begin{figure}[tb]
\centering
\leavevmode
\epsfxsize=4.0in\epsffile{fig5n.eps}
%\epsfxsize=4.0in\epsffile{fig5nb.eps}
\caption[]{
Differential cross sections for $\sqrt{s_{e^+e^-}}=1.5$~TeV as a function of
the invariant mass of the $\ot$ decay products
$M(WWZ)$ and $M(Z\gamma)$
for the signal $e^-\gamma \to e^-\ot$ with
$\ot \to W^+W^-Z$ and $Z\gamma$.  The dotted lines are for 
${\mot}=0.8$~TeV and the dashed lines for ${\mot}=1.0$~TeV, with 
$\gwwz=2\Gamma_{2-body}$ in each case.  The SM 
background relevant to each case are given by the solid lines.}

\label{Mass}
\end{figure}
%

To further assess the discovery potential, we explore the 
parameter space for $\mot$ and $\got$ at a 1.5~TeV
linear collider. The signal for the $WWZ$ mode consists of both $W$'s
decaying hadronically and the $Z$ decaying hadronically or into electrons
or muons. The same $Z$ decay modes provide the signal for the $Z \gamma$
channel, along with the detected photon.  We assume an 80\%
detection efficiency for each of the $W$, $Z$, and $\gamma$ and
an integrated luminosity of 200 fb$^{-1}$. In
Fig.~\ref{fig:contour}, we give contours representing 3$\sigma$ and
5$\sigma$ discovery 
limits for the two cases of $\gwwz=2\Gamma_{2-body}$ 
(solid lines) and $\gwwz=4\Gamma_{2-body}$ (dashed lines). 
For these results, both
cuts (\ref{Basic}) and (\ref{Cutii}) are imposed. 
In addition, a cut about the invariant masses $M_{WWZ},M_{Z\gamma} 
\pm \got$ is made for the signals and backgrounds. 
The two channels are combined by convolution of the product of 
their respective Gaussian probability functions. 
We present the results for the techniomega mass and width range 
at the limit of its resonance production at a 1.5~TeV collider.
As an example, from Fig.~\ref{fig:contour}, we see that for $\Gamma_{\ot}$
of 100~GeV, an $\ot$ can be detected at the 3$\sigma$ level up to $\mot$
of about 1340~GeV for $\gwwz=4\Gamma_{2-body}$  
and up to 1355~GeV for $\gwwz=2\Gamma_{2-body}$.


\begin{figure}[tbh]
\centering
\leavevmode
\epsfig{file=prob.ps,width=4.0in,clip=}
\caption[]{
Contours representing 3$\sigma$ and 5$\sigma$ $\ot$ discovery 
limits in the parameter
space for $\mot$ and $\got$ with $\sqrt{s_{e^+e^-}}=1.5$~TeV and an
integrated 
luminosity of 200 fb$^{-1}$.
Two choices of the ratio of
$\gwwz$ to $\Gamma_{2-body}$ are shown.}
\label{fig:contour}
\end{figure}





\section{Conclusions}

A high energy $e\gamma$ collider is unique in producing an
isosinglet vector state such as $\ot$. We calculated the signal
cross sections for processes (\ref{zgm}) and (\ref{wwz})
in an effective Lagrangian framework. We found that signal rates
can be fairly large once above the $\mot$ threshold, although the
determining factor is the effective electroweak coupling of the $\ot$,
parametrized by $\chi$. The signal
characteristics are very different from the SM backgrounds,
making the discovery and further study of $\ot$ physics
very promising at the linear collider. With an integrated 
luminosity of 200 fb$^{-1}$ at $\sqrt{s_{e^+e^-}}=1.5$~TeV, 
one may discover an $\ot$ for a broad range of masses and widths, as
indicated in Fig.~\ref{fig:contour}.


\acknowledgments

This research was supported in part by the Natural Sciences and Engineering 
Research Council of Canada, and in part by the U. S. Department of Energy
under Grant No. DE-FG02-95ER40896. Further support for T.H. was provided
by the University of Wisconsin Research Committee, 
with funds granted by the Wisconsin Alumni Research Foundation.
P.K. thanks Dean Karlen and Bob Carnegie for useful discussions.

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\end{references}


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