%preprint format:
\documentstyle[preprint,aps]{revtex}
%galley format
%\documentstyle[aps]{revtex}
%
\begin{document}

\draft

\preprint{$
\begin{array}{l}
\mbox{AMES-HET-97-07}\\[-3mm]
\mbox{July 1997}\\
\end{array}
$}
%
\title{Searching for an Anomalous $\bar t q \gamma$ Coupling\\
Via Single Top Quark Production at a $\gamma\gamma$ Collider}

\author{K.J. Abraham$^a$, K. Whisnant$^b$, and B.-L. Young$^{b}$}

\address{
{ $^a$Department of Physics, University of Natal,
Pietermaritzburg, SOUTH AFRICA}\\
{$^b$Department of Physics and Astronomy, Iowa State University,
Ames, IA 50011, USA}\\
}
\maketitle

\begin{abstract}

We investigate the potential of a high-energy $\gamma\gamma$ collider
to detect an anomalous $\bar t q \gamma$ coupling from observation of
the reaction $\gamma \gamma \rightarrow t\bar q$, $\bar t q$, where
$q=c$ or $u$. We find that with $b$-tagging and suitable kinematic cuts
this process should be observable if the anomalous coupling
$\kappa/\Lambda$ is no less than about 0.05/TeV, where $\Lambda$ is the
scale of new physics associated with the anomalous interaction. This
improves upon the bound possible from observation of top decays at the
Tevatron.

\end{abstract}
              
%\pacs{??}

\narrowtext

\section{Introduction}

Since the discovery of the top quark at the Fermilab Tevatron by the CDF
and D0 collaborations \cite{CDFD0} there has been much speculation as to
whether or not its interactions are in accordance with Standard Model (SM)
predictions. Because its mass, around 175~GeV, is of the order of the
Fermi scale, the top quark couples quite strongly to the electroweak
symmetry-breaking sector. In the minimal SM the electroweak symmetry-breaking
sector consists of a single complex fundamental Higgs scalar, but
``triviality'' \cite{tri} and ``naturalness'' \cite{natural} of the
scalar sector suggest that in fact the Higgs sector, and therefore the
top quark mass generation mechanism, may be more complicated. It is
therefore plausible to assume that the Higgs sector of the SM
is an effective theory, and that new physics phenomena may be
manifested through effective interactions of the top quark \cite{PZ}.

One interesting sub-set of effective top quark interactions mediates
flavour changing neutral $t$ decays, {\em i.e.}, $t \rightarrow c Z$, $cg$,
and $c \gamma$. The SM predictions for the corresponding branching 
fractions are unobservably small \cite{tcgammaSM}; thus any experimental
evidence for such decays will be an unambiguous signal for 
physics beyond the Standard Model. Furthermore, it has
been argued that these decay rates may be enhanced significantly in many
extensions of the standard model, such as SUSY or other models with
multiple Higgs doublets \cite{tcgammaSM,multi}, models with new
dynamical interactions of the top quark \cite{dyn}, and models where the
top quark has a composite \cite{comp} or soliton structure \cite{sol}.

Many aspects of such anomalous top-quark couplings have already been
investigated in hadron and lepton colliders. These couplings give
contributions to low-energy observables such as the partial width ratio
$R_b = \Gamma(Z \rightarrow b \bar b) / \Gamma(Z\rightarrow {\rm
hadrons})$ measured at LEP-I \cite{zbbb}, or the branching fraction for
$b \rightarrow s \gamma$ \cite{hewett}. Other constraints from
low-energy processes on anomalous couplings of top quark have also been
considered in the literature \cite{HRFY}. Furthermore, such couplings
would also affect top-quark production and decay processes at hadron and
$e^+e^-$ colliders \cite{cmy,akr}. In Ref.~\cite{HPZ}, the experimental
constraints on an anomalous top-quark coupling $\bar t c Z$ and the
experimental observability of the induced rare decay mode $t \rightarrow
Z c$, at the Fermilab Tevatron and the CERN LHC, have been investigated
in detail. The observability of an anomalous coupling $\bar t c g$ has
been studied for $t \rightarrow c g$ decays at the Tevatron
\cite{HWYZgl}, for single top production in association with a charm
quark \cite{tcg,HHWYZ}, and for direct top production, $g q \rightarrow
t$ ($q = c$ or $u$) at the Tevatron and the LHC \cite{HWY}.

In this paper, we examine the possibility of searching for the anomalous
top-quark couplings $\bar t c \gamma$ and $\bar t u \gamma$ at a
high-energy $\gamma\gamma$ collider. Such a collider may be constructed
by the compton scattering of laser light off the $e^+$ and $e^-$ beams
in an $e^+e^-$ linear collider \cite{gg}.  We will consider the
anomalous effective Lagrangian which includes only the lowest dimension,
$CP$-conserving operators which give rise to anomalous $\bar t q \gamma$
vertices, namely
%
\begin{equation}
\Delta {\cal L}^{eff} = {e\over\Lambda}
[\kappa_{c} \bar t \sigma_{\mu\nu} c
+ \kappa_{u} \bar t \sigma_{\mu\nu} u ]
F^{\mu\nu} + {\it h.c.},
\label{Leff}
\end{equation}
%
where $F^{\mu\nu}$ is the electromagnetic field strength tensor, $e$ is
the electromagnetic coupling constant, $\Lambda$ is the cutoff of
the effective theory, which is generally taken to be the order of 1 TeV,
and the parameters $\kappa_c$ and $\kappa_u$ can be interpreted as
the strengths of the anomalous interactions. Here we do not choose a
particular scale, and consider only the ratio $\kappa/\Lambda$.
In principle, one could consider a 
more complex form factor with the tensor structure 
$\sigma_{\mu \nu}(A + B \gamma^{5})$, however, if we consider only polarization
averaged cross-sections, we may set $B = 0$ without any loss of
generality in probing the strength of such couplings.
%

The constraint from the inclusive branching ratio for the process $b
\rightarrow s \gamma$ \cite{cleo} on the anomalous top-quark coupling
$\bar t c \gamma$ gives $\kappa_c/\Lambda < 0.16/$TeV in the absence of an
anomalous $\bar t c g$ coupling \cite{HWYZga} (the limit with a non-zero
$\bar t c g$ coupling is 0.28/TeV). The analysis can be extended to the
$\bar t u \gamma$ coupling by realizing that the anomalous contribution
to $b \rightarrow s \gamma$ will be suppressed by the CKM factor $V_{us}
\approx 0.22$ compared to the $\bar t c \gamma$ case, which translates
into the limit $\kappa_u/\Lambda < 0.72/$TeV.  The current direct
experimental bound comes from CDF data on the top branching fractions to
a jet plus photon \cite{cdftcg}
%
\begin{equation}
BF(t \rightarrow c \gamma) + BF(t \rightarrow u \gamma) < 2.9\%,
\end{equation}
%
at 95\% Confidence Level (CL), which translates into the limit
%
\begin{equation}
\kappa/\Lambda \equiv \sqrt{\kappa_c^2 + \kappa_u^2}/\Lambda <
(0.73/{\rm TeV})/\sqrt{BF(t \rightarrow b W)}.
\end{equation}
%
This experimental limit therefore does not improve on the $b \rightarrow
s \gamma$ constraint in either case, although the bounds on $\kappa_u$
are numerically about the same.

By searching for the decay $t \rightarrow q \gamma$ in $t \bar t$
production \cite{HWYZga} one can potentially observe the anomalous
couplings down to $\kappa/\Lambda = 0.12/$TeV at the Tevatron Upgrade
with 10~fb$^{-1}$ of integrated luminosity and to $\kappa/\Lambda =
0.01/$TeV at the LHC with 100~fb$^{-1}$ of integrated luminosity,
assuming that the light quark jet can not be identified. We will find
that the $\gamma\gamma$ collider may be able to improve upon the
potential Tevatron limit on these anomalous couplings by roughly a
factor of 2.5. The relatively clean environment of a $\gamma\gamma$ collider 
presents an opportunity for differentiating between the $\bar t c \gamma$ and
$\bar t u \gamma$ couplings, assuming $c$-tagging by the identification of 
D mesons produced by hadronization, such as is done in various
$e^+e^-$ experiments \cite{LEP}, is possible. This particular
option does not seem to be feasible at hadron colliders.

\section{Anomalous single top production at a $\gamma \gamma$ collider}

The anomalous vertices under consideration here lead to the interaction
$\gamma \gamma \rightarrow t \bar q$, where $q = c$ or $u$. There are
four diagrams which contribute, each with one anomalous vertex and one
SM vertex: either the top quark or light quark can be exchanged in the
$t$ or $u$ channel. For very high energies, in the limit that both
fermion masses can be ignored, the total cross section approaches
$\sigma(\gamma \gamma \rightarrow t \bar q) = 64 \pi \alpha^2
(\kappa/\Lambda)^2$. This result is finite despite the $t$ and $u$-channel
poles because the momentum coupling in Eq.~\ref{Leff} regulates the
divergence when the fermion masses vanish. For $\kappa/\Lambda = 0.16/$TeV,
the maximal value allowed for the $\bar t c \gamma$ coupling in the
absence of a $\bar t q g$ coupling, this gives about 120~fb (for the
corresponding $\bar t u \gamma$ coupling this is
increased by roughly a factor of 20). In this paper we will
use $\kappa/\Lambda=0.16$/TeV in our calculations of the
signal unless noted otherwise. The cross section for single
antitop production $\gamma\gamma\rightarrow\bar t q$ is the same;
henceforth in this paper all signal rates and discussions will include
the sum of the single top and antitop signals. The cleanest signal
occurs when the top quark decays semileptonically, which gives the
%signature $b j \ell \rlap/p_T$, where $j$ is a light quark jet and $\ell
signature $b j \ell \rlap/{p_{T}}$, where $j$ is a light quark jet and 
$\ell
= e$ or $\mu$.

Since a likely $\gamma\gamma$ collider will have a maximum CM energy of
500~GeV to 1~TeV \cite{gaga}, the top mass cannot be completely ignored,
and we have calculated the full matrix element for the $2 \rightarrow 2$
process, assuming the top quark is on shell. Because various cuts must
be introduced to simulate a detector and to eliminate backgrounds, we
have then calculated the top quark decay, using the exact matrix element
for the top 3-body semileptonic decay, assuming an on-shell $W$. Because
these couplings do not favor one helicity, the top quark is produced
unpolarized and we therefore do not have to worry about spin
correlations between the top production and decay. The cross sections
were calculated via a Monte Carlo program, using both helicity
amplitudes and Dirac matrices, and agreement was found to within
1\% between the two methods. We have also assumed that $BF(t
\rightarrow b W) \approx 1$, which would be the case if there were
no non-standard top decays other than $t \rightarrow q \gamma$. If there
are non-standard top decays with a significant branching fraction, the
signal results quoted in this paper would be reduced at most by a factor
of two \cite{HWYZga}. For $\sqrt{s} = 500$~GeV and $m_t = 175$~GeV we
find a total signal cross section of
%
\begin{equation}
\sigma(\gamma\gamma \rightarrow t \bar q + \bar t q
\rightarrow b c \ell \nu) = 76.4 {\rm~fb}
\left({\kappa/\Lambda \over 0.16/{\rm TeV}}\right)^2,
\end{equation}
%
which for a $\bar t c \gamma$ ($\bar t u \gamma$) coupling of
$\kappa/\Lambda = 0.16/$TeV (0.72/TeV) translates into
764 (15280) events with an integrated luminosity of 10~fb$^{-1}$.
Therefore we see that this is a viable signal.

\section{Acceptance cuts and backgrounds}

There are several potentially severe SM backgrounds to the signal. The
largest is $\gamma\gamma \rightarrow W^+W^-$, which has a cross section
of about 88~pb at $\sqrt{s} = 500$~GeV \cite{WWprod}. This will mimic
the signal when one $W$ decays leptonically and the other decays into
two light quark jets, giving a cross section of about 26~pb. To make a
quantitative analysis of the experimental sensitivity to the anomalous
couplings, we have done a monte carlo study of $\gamma\gamma
\rightarrow W^+W^- \rightarrow \ell \nu
q^\prime \bar q^{\prime\prime}$ using the full helicity amplitudes
\cite{WWhel}, imposing the following basic acceptance cuts
%
\begin{eqnarray}
& p_T^\ell> 15~{\rm GeV}, \qquad p_T^j> 15~{\rm GeV}, 
\qquad E_T^{miss} > 15 {\rm~GeV},  \nonumber\\
& |\eta^\ell|, |\eta^j|<1.5, \qquad 
\Delta R_{\ell j}, \Delta R_{jj} > 0.4,
\label{EQ:BASIC}
\end{eqnarray}
%
where $p_T$ denotes transverse momentum, $\eta$ denotes pseudo-rapidity,
and $\Delta R$ denotes the separation in the azimuthal angle-pseudo rapidity
plane. With these basic cuts, the signal rate is reduced to about 33~fb,
while the $W^+W^-$ background is now about 4.8~pb. 

A significant reduction in the background is possible if $b$-tagging is
employed. We assume a 50\% efficiency for detecting a $b$-quark jet, and
a 0.4\% mis-tagging rate where a light quark jet is misidentified as a
$b$-quark \cite{btag,cmsatlas}. The signal is then halved, while
the $W^+W^-$ background is reduced a level comparable with the signal.

Next we can then use the fact that for this background the two jets
should reconstruct to the $W$ mass, while for the signal the $m(b \bar
c)$ invariant mass distribution is peaked towards its maximal value of
$\sqrt{s}-m_W$=420~GeV, which can be understood by realizing that the
charm jet and the $b$ jet from top decay tend to be back-to-back. If we
exclude values of $m(jj)$ which include the $W$ resonance, we can
substantially reduce this background. To quantify this in our
calculation, we assume a Gaussian energy smearing for the
electromagnetic and hadronic calorimetry as follows
%
\begin{eqnarray}
\Delta E/E &=& 30\%/\sqrt E \oplus 1\%,
\quad {\rm for \ \ lepton \ and \ photon}
\nonumber
\\ &=& 80\%/\sqrt E \oplus 5\%,
\quad {\rm for \ \ jets,}
\label{smear}
\end{eqnarray}
%
where the $\oplus$ indicates that the $E$-dependent and $E$-independent
errors are to be added in quadrature, and $E$ is measured in GeV.
If we impose the cut
%
\begin{equation}
m(jj) > 106{\rm~GeV},
\label{eq:mjj}
\end{equation}
%
then we find that the resonant contributions are reduced to 
${\cal O}(10^{-2})$~fb, effectively eliminating this background. The signal
is affected very little by the $m(jj)$ cut, as can be seen from Fig. 1.

In addition to resonant $W^+W^-$ production, there is a
set of Feynman graphs contributing to
nonresonant $\gamma\gamma \rightarrow \ell\nu q^\prime \bar q$
production, where the invariant mass of the lepton and quark pairs are
not both at the $W$ mass peak. 
% My insertion
Due to the large number of different graphs involved, a precise
analytical treatment is beyond the scope of this letter, and indeed has
not yet been carried out. However, we will make use of the results for
resonant and nonresonant contributions analyzed numerically in
Ref.~\cite{ggffff} to show how this background can be suppressed.
% end of insertion
Most of the nonresonant cross section occurs when
one pair of fermions is on a $W$ peak and one of the two other fermions
is roughly collinear with an incoming photon. The pseudo-rapidity cut in
Eq.~\ref{smear}, larger than likely required from detector geometry, has
been chosen to reduce these forward contributions. From
Ref.~\cite{ggffff} the cross section for $\gamma\gamma \rightarrow
\ell\nu q^\prime \bar q^{\prime\prime}$
% My insertion 
at $ \sqrt{s} = 500$~GeV, excluding the $W^+W^-$ resonant
contributions, is about 67~fb after the $\eta$ cuts (which corresponds to
cuts on the charged particles of about $|\cos\theta|<0.92$ in the 
COM frame of the incoming photons). We then
multiplied this result by 8 to account for two generations each of
leptons and quarks and for positive and negative charged leptons in the
final state, and divided by 2 since we expect that in roughly half of
these events the $W$ has on-shell decays into quarks, which will be
eliminated by the $m(jj)$ cut. The net result is a $Wjj\rightarrow
\ell\nu jj$ background of about 270~fb, which will be reduced by
more than two orders of magnitude by $b$-tagging alone.

A further reduction of the nonresonant background can be made by
determining the invariant mass $m(bW)$, which for the signal should be
strongly peaked near $m_t$ and for the background should be much
flatter. In order to do this reconstruction one must determine the $W$
momentum. The neutrino transverse momentum can be inferred from the
missing $p_T$, but its longitudinal momentum is undetermined since in
general the exact initial photon energies may not be known. However, if
we assume that the $W$ which decays leptonically is on mass shell, one
can determine the neutrino longitudinal momentum up to a two-fold ambiguity
\cite{pnu}. We can then take the solution which gives $m(bW)$ closest to
$m_t$, where the $b$ quark is identified via a $b$-tag. Applying the cut
%
\begin{equation}
|m(bW) - m_t| < 30 \rm{~GeV},
\end{equation}
%
will then provide a strong constraint on the background, while the
signal is generally reduced by only about 10-20\% \cite{HWYZgl,HWYZga}.

To simulate the effect of the remaining cuts (in $p_T$, $\Delta R$,
$m(jj)$, and $m(bW)$) on the nonresonant background, we have assumed
that the distributions that survive the $\eta$ cuts are relatively flat. 
% My insertion
This assumption is reasonable as the region close to the beam pipe, where
the bulk of the non-resonant cross-section arises \cite{ggffff}, has been 
excluded. The fraction of the non-resonant cross-section which survives is
then proportional to the volume of phase space allowed after our additional 
cuts, which is reduced by a factor of 7. 
If we assume that the cross section is correspondingly reduced, 
we get a nonresonant background of about .16~fb.  
The successive effect of all the cuts on the signal
and primary backgrounds are summarized in Table~1.

There are also SM final states $b c \ell^\pm \nu$ from both resonant and
nonresonant backgrounds. However, these processes are suppressed by a
factor $|V_{bc}|^2\approx 1/400$ compared to the final states $u d
\ell^\pm \nu$ and $c s \ell^\pm \nu$ . Although they are reduced only by
a factor of 2 when $b$-tagging is employed, they are still more than a
factor of 3 smaller than the corresponding processes with light quarks
after $b$-tagging.  We can account for these backgrounds by multiplying
the light quark background cross section by 21/16, which gives an
overall estimated background at the 0.2~pb level, which corresponds to 2
events for 10~fb$^{-1}$ integrated luminosity.

The signal cross section after all cuts is about 23.4~fb for
$\kappa/\Lambda=0.16$/TeV. However, about half the signal is lost due to
the requirement of $b$-tagging, still leaving a cross-section large enough
to be phenomenologically viable, assuming $ {\cal L} = 10^{-1}$ fb. At
realistic $\gamma \gamma$ colliders, of course, $\sqrt{s}$ is not 500
GeV as we assumed above, but lower; taking $\sqrt{s} = 400$~GeV, 
as for example in \cite{WWprod}, leads to a reduction of
$ \sim 10\%$ in the signal cross-section and should have no
significant influence on possible discovery bounds.

\section{Anomalous coupling limits and discussion}

To estimate the sensitivity to the anomalous couplings for a given
integrated luminosity, we require that the signal be observed at the
3-$\sigma$ level,
%
\begin{equation}
S \ge 3 \sqrt{S+B},
\end{equation}
%
where $S$ and $B$ are the number of signal and background events,
respectively, after all cuts are made. In our case ($B\approx2$),
this corresponds to at least 11 signal events. For
$\sqrt{s} = 500$~GeV and an integrated luminosity of 10~fb$^{-1}$,
the discovery limit for $\kappa/\Lambda$ is then about 0.048/TeV. For
a more realistic CM energy of $\sqrt{s}=400$~GeV, the
discovery limit can be written approximately as
%
\begin{equation}
\kappa/\Lambda \le
{0.051/{\rm TeV} \over \sqrt{{\cal L}/10{\rm~fb}^{-1}}},
\end{equation}
%
when the integrated luminosity ${\cal L}$ lies in the range
5-20~fb$^{-1}$. A $\gamma\gamma$ collider with maximum CM energy of
1~TeV and an effective $\sqrt{s}=800$~GeV may be able to reduce this by
about 10\%. We note that if the background has been underestimated by a
factor of 3, these discovery limits are raised by only about 10\%.

We have also examined the possibility of looking for $e^+e^- \rightarrow
t \bar q + \bar t q$ in an electron-positron collider. However, the
production cross section is much lower than for a $\gamma\gamma$
collider. We find that for $\kappa/\Lambda=0.16$/TeV and $m_t=175$~GeV
the total cross section for $e^+e^- \rightarrow t \bar q + \bar t q$
before cuts is about 1.8~fb at $\sqrt{s}=200$~GeV and 4.4~fb at
$\sqrt{s}=500$~GeV. These are more than an order of magnitude smaller
than the corresponding $\gamma\gamma$ cross sections, and means that an
$e^+e^-$ collider cannot effectively probe the $\bar t q \gamma$ coupling.

In summary, we have shown that a $\gamma\gamma$ collider with $\sqrt{s} =
500$~GeV can probe an anomalous $\bar t c \gamma$ or $\bar t u \gamma$
coupling down to the level of $\kappa/\Lambda \sim 0.05/$TeV for the
integrated luminosities expected at such machines. As noted previously,
a reduction in $\sqrt{s}$ to 400~GeV does not significantly affect this
limit. This will be much more sensitive than looking for
$e^+e^-\rightarrow t \bar c$ at a similar energy, and is a factor of
about 2.5 better than the limit expected to be obtained from studying
top decays at the upgraded Tevatron, where backgrounds are not so easy
to suppress. If charm tagging is available it also offers the
possibility of distinguishing between the anomalous $\bar t c \gamma$ and
$\bar t u \gamma$ couplings, which plausibly could be quite different.

\section{Acknowledgments}

This work was supported in part by the U.S.~Department of Energy under
Contract DE-FG02-94ER40817 (KW and BLY). KJA wishes to acknowledge the
generous support of IITAP during the course of this investigation.
  
\begin{references}

\bibitem{CDFD0}
F. Abe {\it et al.} (CDF Collaboration), 
%Preprint FNAL-PUB-94/022-E (1995);
Phys. Rev. Lett. {\bf 74},  2626 (1995);
S. Abachi {\it et al.} (D0 Collaboration),
%Preprint FNAL-PUB-94/028-E (1995).
Phys. Rev. Lett. {\bf 74},  2632 (1995).

\bibitem{tri}
For a recent review, see, {\it e. g.}, H. Neuberger, in {\it{ Proceedings of
the XXVI International Conference on High Energy Physics}}, Dallas, Texas,
1992, edited by J. Sanford, AIP Conf. Proc. No. 272 (AIP, New York, 1992), Vol.
II, p. 1360.

\bibitem{natural}
G. 't Hooft, in {\it Recent Developments in Gauge Theories}, Proceedings
of the Cargese Summer Institute, Cargese, France, 1979, edited by
G. 't Hooft {\it et al.}, NATO Advanced Study Institute Series B;
Physics Vol. 59 (Plenum, New York, 1980).

\bibitem{PZ}
R. D. Peccei and X. Zhang, Nucl. Phys. {\bf B337}, 269 (1990).  

\bibitem{tcgammaSM}
B. Grzadkowski, J.F. Gunion and P. Krawczyk, Phys. Lett. {\bf B268},
106 (1991);
G. Eilam, J.L. Hewett and A. Soni, Phys. Rev. {\bf D44}, 1473 (1991);
M. Luke and M.J. Savage, Phys. Lett. {\bf B307}, 387 (1993);
G. Couture, C. Hamzaoui and H. K\"onig, Phys. Rev. {\bf D52}, 1713 (1995).

\bibitem{multi}
T.P. Cheng and M. Sher, Phys. Rev. {\bf D35}, 3484 (1987);
W.S. Hou, Phys. Lett. {\bf B296}, 179 (1992);
L.J. Hall and S. Weinberg, Phys. Rev. {\bf D48}, R979 (1993);
D. Atwood, L. Reina, and A. Soni, Phys. Rev. {\bf D 53}, 1199 (1996).

\bibitem {dyn}
C.T. Hill, Phys. Lett. {\bf B266}, 419 (1991);
Phys. Lett. {\bf B345}, 483 (1995);
B. Holdom, Phys. Lett. {\bf B339}, 114 (1994);
Phys. Lett. {\bf B351}, 279 (1995).

\bibitem{comp}
H. Georgi, L. Kaplan, D. Morin, and A. Schenk, Phys. Rev. {\bf D51},
3888 (1995). For a review of composite models, see, {\it e.g.}, R.D. Peccei,
in the proceedings of {\it The 1987 Lake Louise Winter Institute:
Selected Topics in the Electroweak Interactions}, ed. by J.M. Cameron
et al. (World Scientific, Singapore, 1987).

\bibitem{sol}
X. Zhang, Phys. Rev. {\bf D51} 5039 (1995);
J. Berger, A. Blotz, H.C. Kim, and K. Goeke, Phys. Rev. {\bf D 54},
3598 (1996).

\bibitem{zbbb}
R.S. Chivukula, E.H. Simmons and J. Terning, Phys. Lett. {\bf B331},
383 (1994);
C. T. Hill and X. Zhang, Phys. Rev. {\bf D51}, 3563 (1995).

\bibitem{hewett}
J.L. Hewett, SLAC-PUB-6521 (May 1994), in {\it SLAC Summer Institute 1993}.

\bibitem{HRFY}
J.L. Hewett and T. Rizzo, Phys. Rev. {\bf D49}, 319 (1994);
K. Fujikawa and A. Yamada, Phys. Rev. {\bf D49}, 5890 (1994).

\bibitem{cmy}
See, {\it e. g.}, D. O. Carlson, E. Malkawi and C.-P. Yuan,
Phys. Lett. {\bf B337}, 145 (1994); and references therein.

\bibitem{akr}
See, {\it e. g.},  D. Atwood, A. Kagan and T. Rizzo,
Phys. Rev. {\bf D52}, 6264 (1995); and references therein.

\bibitem{HPZ}
T. Han, R.D. Peccei and X. Zhang, Nucl. Phys. {\bf B454}, 527 (1995). 

\bibitem{HWYZgl}
T. Han, K. Whisnant, B.-L. Young and X. Zhang, Phys. Lett. {\bf B385},
311 (1996).

\bibitem{tcg}
E. Malkawi and T. Tait, Phys. Rev. {\bf D54}, 5758 (1996). 

\bibitem{HHWYZ}
T. Han, M. Hosch, K. Whisnant, B.-L. Young, and X. Zhang,
manuscript in preparation.

\bibitem{HWY}
M. Hosch, K. Whisnant, and B.-L. Young, Iowa State U. preprint
AMES-HET-97-02, March 1997,  to be published in
Phys. Rev. {\bf D}.

\bibitem{gg}
C. Akerloff, University of Michigan Report No. UMHE 81-59, 1981;
I.F. Ginzburg {\it et al.}, Nucl. Instrum. Meth. {\bf 205}, 47 (1983);
I.F. Ginzburg {\it et al.}, Nucl. Instrum. Meth. {\bf 219}, 5 (1984);
V. I. Telnov  Nucl. Instrm. Met. {\bf A294}72, (1990).

\bibitem{cleo}
M. Alam {\it et al}. (CLEO collaboration), Phys. Rev. Lett. {\bf 74},
2885 (1995).

\bibitem{cdftcg}
T.J. LeCompte (CDF), reported at the {\it 2nd Rencontres du Vietnam,
Physics at the Frontiers of the Standard Model}, Ho Chi Minh City,
Vietnam, October 1995, FERMILAB-CONF-96/021-E.

\bibitem{HWYZga}
T. Han, K. Whisnant, B.-L. Young and X. Zhang, Phys. Rev. {\bf D 55},
7241 (1997).

\bibitem{LEP}
E. Nakano {\it et al.} (TOPAZ Collaboration), Phys. Lett. {\bf B314},
471 (1993);
R. Akers {\it et al.} (OPAL collaboration), Z.Phys. {\bf C60}, 601 (1993);
P. Abreu {\it et al.} (DELPHI Collaboration), Z. Phys. {\bf C66}, 341 (1994).

\bibitem{gaga}
T. Barklow, SLAC-PUB 5364, Nov. 1990, published in 
``Research Directions for the decade: Snowmass '90",
ed. Edmond L. Berger, World Scientific 1992. 

\bibitem{WWprod}
For a recent review see F.Boudjema,  published in
``Workshop on Physics \& Experiments with Linear Colliders"
Proceedings, ed. A. Miyamoto {\it et al.}, World Scientific 1996.

\bibitem{ggffff}
M. Moretti, Nucl. Phys. {\bf B484}, 3 (1997).

\bibitem{btag}
F. Abe {\it et al.} (CDF Collaboration), Phys. Rev. {\bf D52}, 2605 (1995).

\bibitem{cmsatlas}
CMS Collaboration, Technical Proposal, CERN/LHCC/94-38;
ATLAS Collaboration, Technical Proposal, CERN/LHCC/94-43.

\bibitem{WWhel}
G. Belanger and G. Couture, Phys. ReV. {\bf D49}, 5720 (1994). 

\bibitem{pnu}
J.~Stroughair and C.~Bilchak, Z. Phys. {\bf C26}, 415 (1984); 
J.~Gunion, Z.~Kunszt, and M.~Soldate, Phys. Lett. {\bf B163}, 389 (1985); 
J.~Gunion and M.~Soldate, Phys. Rev. {\bf D34}, 826 (1986); 
W.~J.~Stirling {\it et al.}, Phys. Lett. {\bf B163}, 261 (1985);
J.~Cortes, K.~Hagiwara, and F.~Herzog, 
Nucl. Phys. {\bf B278}, 26 (1986).   % p p -> W gamma anomalous couplings

\end{references}
%
\vfill
\eject

\begin{table}[htb]
%\begin{table}[h]
\centering
\caption[]{Cross sections in units of fb for the 
$t \bar q, \bar t q \rightarrow bq \ell^\pm \rlap/{p_T}$ signal 
with $\kappa/\Lambda=0.16/TeV$ and the SM backgrounds. The successive
effect of the cuts on the signal and $W^+W^-$ background have been
explicitly calculated, while the effects on the nonresonant background
values have been estimated as discussed in the text. The dashes indicate
cross sections too small to be of interest.}
%
\begin{tabular}{|c|c|c|c|} 
(a). Cuts & signal $t \bar c, \bar t c \rightarrow b c \ell^\pm \nu$~~~ & 
$WW \rightarrow jj\ell^\pm \nu$~~~  & 
$Wjj \to jj \ell^\pm \nu$~~~ \\ \hline \hline
none        & 76.4   & 26000 &  2140 \\ \hline
$\eta$ only & 38.2   &  6928 &  270 \\ \hline
basic+smear & 33.5   &  4786 &  175 \\ \hline
b-tag       & 16.75  &  19.0  & 0.70 \\ \hline
m(jj)       & 15.7   &   -   & 0.65 \\ \hline
m(bW)       & 11.7   &   -   & 0.16 \\
\end{tabular}
\label{T:ONE}
\end{table}

\vfill

\centerline{FIGURE CAPTIONS}
%
FIG.~1 The $\gamma\gamma \rightarrow t \bar q + \bar t q \rightarrow
bq\ell^\pm\rlap/{p_T}$ cross section versus the jet-jet invariant mass
$m(jj)$ at $\sqrt{s}=500$~GeV after the basic cuts have been implemented.
The solid (dotted) histogram corresponds to the result before (after)
the effects of detector smearing are included. 

\vfill

\end{document}

