\documentstyle[twoside,epsfig]{hsproc}
% Define next two lines for running header:
\def\runauthor{D. Bruncko, M. Gintner, I. Melo}
\def\shorttitle{Strong ESB in $e^+e^- \rightarrow \nu {\bar
\nu} t{\bar t}$}
%
\begin{document}
%
\title{Strong Electroweak Symmetry Breaking in $e^+e^- \rightarrow \nu {\bar
\nu} t{\bar t}$}
%\author{D. Bruncko\email{bruncko@saske.sk}}{IEP SAS, Ko\v sice}
%\author{M. Gintner\email{gintner@fel.utc.sk},I. Melo\email{melo@fel.utc.sk}}
%{Dept. of Physics, University of \v Zilina}
%\author{I. Melo\email{melo@fel.utc.sk}}{Dept. of Physics , University of \v Zilina}

%		etc
%	for e-mail address, use macro \email{} in \author - macro:
%	\author{name of the author\email{e-mail address}}{address of the author}
%
%\abstract{We study new strong resonances associated with the physics
%responsible for the strong electroweak symmetry breaking in chiral Lagrangian
%approach. We calculate signals for vector $\rho$-resonance and scalar $S$-resonance 
%in the $WW t\bar{t}$ scattering in the process
%$e^+e^- \rightarrow \nu {\bar \nu} t {\bar t}$ at a future Linear
%Collider at $\sqrt{s} = 1500$~GeV and $3000$~GeV.
%We also find and discuss the relevant backgrounds.}
%
\section{$\rho$ -resonance Lagrangian}
The gauged $\rho$-resonance Lagrangian is given by 
\begin{eqnarray}
\label{a1}
{\cal L}(\rho) & = &  {\cal L}_{kin} + {\cal L}_{bos} + {\cal L}_{f}^{3}
  \end{eqnarray}
where ${\cal L}_{kin}$ includes kinetic terms for the $\rho$ resonance,
gauge fields and fermion fields, ${\cal L}_{bos}$ describes bosons,
fermionic term ${\cal L}_{f}^{3}$ describes 3rd 
generation quarks. Remaining fermions (1st and
2nd generation fermions plus 3rd generation leptons) are represented by the
relevant terms of the SM lagrangian which we do not show here .
The terms of Eq. \ref{a1} are given by 
\begin{eqnarray}
\label{a2}
{\cal L}_{kin} & = & \frac{1}{2{g''}^2} \mbox{Tr}[{\bf \rho}_{\mu\nu} {\bf \rho}^{\mu\nu}]
 + \frac{1}{2{g}^2} \mbox{Tr}[{\bf \tilde{W}}_{\mu\nu} {\bf \tilde{W}}^{\mu\nu}]
 + \frac{1}{2{g'}^2} \mbox{Tr}[{\bf \tilde{Y}}_{\mu\nu} {\bf \tilde{Y}}^{\mu\nu}]
 + {\cal L}_{kin}(f)
 \nonumber \\
{\cal L}_{bos} & = & - v^2 \mbox{Tr}[{\cal A_{\mu}}{\cal A^{\mu}}]
- a \frac{v^2}{4} \mbox{Tr}[(\omega_\mu + \rho_\mu)(\omega^\mu + \rho^\mu)] \nonumber \\
{\cal L}_{f}^{3} & = & - {\bar \psi}_L u^{\dagger} u^{\dagger} M \psi_R + h.c.
 + I_C^L + b_1 I_b^L +\lambda_1 I_{\lambda}^L 
 + \kappa_1 I_{\kappa}^L  
 \nonumber \\
 & &
 + I_C^R + b_2 I_b^R +\lambda_2 I_{\lambda}^R 
 + \kappa_2 I_{\kappa}^R 
 \end{eqnarray}

 where
 \begin{eqnarray}
\label{a3}
{\bf \rho}_{\mu\nu} & = &  \partial_{\mu} {\bf \rho}_{\nu} - \partial_{\nu} {\bf \rho}_{\mu}
                                 - [{\bf \rho}_{\mu}, {\bf \rho}_{\nu}] 
				 \nonumber \\
\rho_{\mu} & = & i g''\;{\vec \rho_{\mu}}. {\vec \tau}/2 \nonumber \\ 
{\bf \tilde{W}}_{\mu\nu} & = & \partial_{\mu} {\bf \tilde{W}}_{\nu} - \partial_{\nu} {\bf \tilde{W}}_{\mu}
                                 + [{\bf \tilde{W}}_{\mu}, {\bf \tilde{W}}_{\nu}]
				 \nonumber \\
{\bf \tilde{W}}_{\mu} & = & i g \; \vec{W}_{\mu} . \vec{\tau}/2 \nonumber \\				 
{\bf \tilde{Y}}_{\mu\nu} & = & \partial_{\mu} {\bf \tilde{Y}}_{\nu} - \partial_{\nu} {\bf \tilde{Y}}_{\mu}
\nonumber \\
{\bf \tilde{Y}}_{\mu} & = & i g' \; Y_{\mu} \tau^{3}/2 \nonumber \\
{\cal A}_{\mu} & = & [u^{\dagger}(\partial_{\mu}
+ {\bf \tilde{Y}}_{\mu})u
- u (\partial_{\mu} + {\bf \tilde{W}}_{\mu})u^{\dagger}]/2 \nonumber \\ 
\omega_{\mu} & = & [u^{\dagger}(\partial_{\mu}
+ {\bf \tilde{Y}}_{\mu}
)u + u(\partial_{\mu} + {\bf \tilde{W}}_{\mu}) u^{\dagger}]/2 \nonumber \\
u & = & \exp(i \vec{\pi} .\vec{\tau}/{2 v}) \nonumber \\
I_C^L & = &
  {\bar \psi}_L \;i \gamma^{\mu}
 \big[\partial_{\mu}  + {\bf \tilde{W}}_{\mu} 
 + \frac{i}{2}g'(B-L)Y_{\mu}\big]\psi_L  \nonumber \\
 I_C^R & = & {\bar \psi}_R \;i \gamma^{\mu}
 \big[\partial_{\mu}  + {\bf \tilde{Y}}_{\mu} 
 + \frac{i}{2}g'(B-L)Y_{\mu}\big]\psi_R  \nonumber \\
 I_b^L & = &
  {\bar \psi}_L \, i \gamma^{\mu}\big[u \partial_{\mu} - u \rho_{\mu}
 +
 u \frac{i}{2} g' (B - L) Y_{\mu}\big] \,u^{\dagger}\, \psi_L \nonumber \\
 I_b^R & = &
  {\bar \psi}_R \, P_b\, i \gamma^{\mu}\big[u \partial_{\mu} - u \rho_{\mu}
 + u \frac{i}{2} g' (B - L) Y_{\mu}\big] \,u^{\dagger}\,P_b\, \psi_R  \nonumber \\
I_{\lambda}^L & = & 
 {\bar \psi}_L \, i \gamma^{\mu}\; u^{\dagger}\; {\cal A_{\mu}}\; \gamma^{5} u
 \;\psi_L \;\; \nonumber \\
I_{\lambda}^R & = &
{\bar \psi}_R \, P_{\lambda}\, i \gamma^{\mu} u \;{\cal A_{\mu}}\;
\gamma^{5} \; u^{\dagger}\;P_{\lambda}\;\psi_R
\end{eqnarray}

Here ${\vec \rho_{\mu}}$ is the vector $(I=J=1)$ ESB $\rho$-resonance field,
$W_\mu$ is the $SU(2)_L$
 gauge field, 
and $Y_\mu$ is the $U(1)_Y$ gauge field ;
$\vec{\pi}$ is the
isospin triplet of the EW pions,
$\psi$ denotes the 3rd-generation quark field doublet, $(t,b)$;
the $B-L$ is the baryon minus lepton number operator, 
the $M = \mbox{diag}(m_t,m_b)$ is the fermion doublet mass matrix;
the $P_b = \mbox{diag}(1,p)$, $P_{\lambda} = \mbox{diag}(1,r)$ are  
matrices which ensure that (depending on p and r) either the $\rho$ couples to the 
right-handed top and bottom quarks with equal strength (if p=r=1), 
or (if p=r=0)the $\rho$ only 
couples to the right-handed top quark but not to the right-handed bottom quark
(to avoid constraints on the $\rho$ couplings from the
$Zb{\bar b}$ vertex) \footnote{note that p=r=0 breaks the
custodial symmetry in much the same fashion as does the top-bottom
mass splitting}.
Finally, the $a$, $g''$, $b_1$, $b_2$, $\lambda_1$ and $\lambda_2$ are free parameters,
$\vec{\tau}$  are
Pauli matrices and $v$ is the vacuum expectation value of the Higgs field;



To derive the low energy Lagrangian 
we have to integrate out the $\rho$ from the spectrum by taking $M_{\rho} = av^2 {g''}^2/4 \rightarrow \infty$, 
which is
accomplished by $a \rightarrow \infty$ while keeping $g''$ fixed, as in BESS model studies (Casalbuoni 9410377).
In this limit the equation of motion for the $\rho$ field ($\partial {\cal L}_{\rho}/\partial{\rho}_{\mu} =
0$) has the solution $\rho_{\mu} = - ({\bf \tilde{W}}_{\mu} + {\bf \tilde{Y}}_{\mu})/2$. When we substitute
this solution into the right-handed part of the ${\cal L}_f^3$
 (${\cal L}_f^R3 = I_C^R + b_R I_b^R + \lambda_R I_{\lambda}^R$) we obtain in the unitary gauge ($u = 1$)
the following expression: 
\begin{eqnarray}
\label{d1}
{\cal L}_f^R3
& = &
{\bar t}_R i \gamma^{\mu} \partial_{\mu} t_R + {\bar b}_R i \gamma^{\mu} \partial_{\mu} b_R \nonumber \\
& &
- \frac{g'}{2} {\bar t}_R \gamma^{\mu} t_R Y_{\mu} +  \frac{g'}{2} {\bar b}_R \gamma^{\mu} b_R Y_{\mu}
\nonumber \\
& &
- \frac{g'}{2} (B-L) {\bar t}_R \gamma^{\mu} t_R Y_{\mu} 
- \frac{g'}{2} (B-L) {\bar b}_R \gamma^{\mu} b_R Y_{\mu} \nonumber \\
& &
+ b_2 {\Big \{} {\bar t}_R i \gamma^{\mu} \partial_{\mu} t_R  
+ p^2 {\bar b}_R i \gamma^{\mu} \partial_{\mu} b_R  \nonumber \\
& & 
- p \frac{g}{2\sqrt{2}} {\bar t}_R  \gamma^{\mu} b_R W_{\mu}^{+}
- p \frac{g}{2\sqrt{2}} {\bar b}_R  \gamma^{\mu} t_R W_{\mu}^{-} \nonumber \\
& &
- \frac{g}{4} {\bar t}_R  \gamma^{\mu} t_R W_{\mu}^3
+ p^2 \frac{g}{4} {\bar b}_R  \gamma^{\mu} b_R W_{\mu}^3 \nonumber \\
& &
- \frac{g'}{4} {\bar t}_R  \gamma^{\mu} t_R Y_{\mu} 
+ p^2 \frac{g'}{4} {\bar b}_R  \gamma^{\mu} b_R Y_{\mu} \nonumber \\
& &
- \frac{g'}{2} (B-L) {\bar t}_R  \gamma^{\mu} t_R Y_{\mu} 
- p^2 \frac{g'}{2} (B-L) {\bar b}_R  \gamma^{\mu} b_R Y_{\mu} {\Big \}} \nonumber \\
& &
+ \lambda_2 {\Big \{} - \frac{g'}{4} {\bar t}_R  \gamma^{\mu} \gamma^5 t_R Y_{\mu}
+ r^2 \frac{g'}{4} {\bar b}_R  \gamma^{\mu} \gamma^5 b_R Y_{\mu} \nonumber \\
& &
+ r \frac{g}{2\sqrt{2}} {\bar t}_R  \gamma^{\mu} \gamma^5 b_R W_{\mu}^{+}
+ r \frac{g}{2\sqrt{2}} {\bar b}_R  \gamma^{\mu} \gamma^5 t_R W_{\mu}^{-} \nonumber \\
& &
+ \frac{g}{4} {\bar t}_R  \gamma^{\mu} \gamma^5 t_R W_{\mu}^3
- r^2 \frac{g}{4} {\bar b}_R  \gamma^{\mu} \gamma^5 b_R W_{\mu}^3 {\Big \}}
\end{eqnarray}
Kinetic terms of Eq.\ref{d1} can be written in the canonical form:
\begin{eqnarray}
\label{d2}
{\cal L}_{kin}^R & = &  (1 + b_2){\bar t}_R i \gamma^{\mu} \partial_{\mu} t_R 
+ (1+ p^2 b_2){\bar b}_R i \gamma^{\mu} \partial_{\mu} b_R \nonumber \\
& = &  
{\bar t'}_R i \gamma^{\mu} \partial_{\mu} t'_R 
+ {\bar b'}_R i \gamma^{\mu} \partial_{\mu} b'_R
\end{eqnarray}
where we defined new fields $t'_R, b'_R$. Their relation to old fields $t_R, b_R$ is given by
\begin{eqnarray}
\label{d3}
t_R & = & \frac{t'_R}{\sqrt{1+b_2}} \nonumber \\
b_R & = & \frac{b'_R}{\sqrt{1+p^2 b_2}}
\end{eqnarray}
If we now plug Eq. \ref{d3} back into nonkinetic terms of Eq. \ref{d1}, use
\begin{eqnarray}
\label{d4}
Y & = & - c_W A + s_W Z \nonumber \\
W^3 & = & - s_W A - c_W Z \nonumber \\
B - L & = & \frac{1}{3},
\end{eqnarray}
we obtain after some tedious steps the following
low-E Lagrangian describing interactions of A,Z and W bosons with top and bottom quarks
(terms such as $(1+p^2 b_2)^{-1}$ were expanded as $(1+p^2 b_2)^{-1} \approx 1 - p^2 b_2$;
we also drop primes on $t_R, b_R$ fields for simplicity):
\begin{eqnarray}
\label{d5}
{\cal L}_{lowE}^R  & = & 
+ \frac{2}{3} e {\bar t}_R \gamma^{\mu} t_R A_{\mu}
- \frac{e}{3} {\bar b}_R \gamma^{\mu} b_R A_{\mu} \nonumber \\
& &
+ \frac{g}{2 c_W} \Big(-\frac{4}{3} s_W^2 + \frac{b_2}{2}-\frac{\lambda_2}{2}\Big)
{\bar t}_R \gamma^{\mu} t_R Z_{\mu} \nonumber \\
&  & +   \frac{g}{2 c_W} \Big(\frac{2}{3} s_W^2 - p^2 \frac{b_2}{2} + r^2 \frac{\lambda_2}{2}\Big)
{\bar b}_R \gamma^{\mu} b_R Z_{\mu} \nonumber \\
&  & + \frac{g}{\sqrt{2}} \Big(  - p \frac{b_2}{2} + r \frac{\lambda_2}{2}\Big)
{\bar t}_R \gamma^{\mu} b_R W_{\mu}^+ + h.c.;
\end{eqnarray}
By analogy we obtain for the left-handed part of the ${\cal L}_f^3$ the following low energy limit:
\begin{eqnarray}
\label{d6}
{\cal L}_{lowE}^L & = & 
+ \frac{2}{3} e {\bar t}_L \gamma^{\mu} t_L A_{\mu}
- \frac{e}{3} {\bar b}_L \gamma^{\mu} b_L A_{\mu} \nonumber \\
& &
\frac{g}{2 c_W} \Big(1-\frac{4}{3} s_W^2 - \frac{b_1}{2}+\frac{\lambda_1}{2}\Big)
{\bar t}_L \gamma^{\mu} t_L Z_{\mu} \nonumber \\
& & + \frac{g}{2 c_W} \Big(-1+\frac{2}{3} s_W^2 + \frac{b_1}{2}-\frac{\lambda_1}{2}\Big)
{\bar b}_L \gamma^{\mu} b_L Z_{\mu} \nonumber \\
&  & - \frac{g}{\sqrt{2}} \Big(1 - \frac{b_1}{2} + \frac{\lambda_1}{2}\Big)
{\bar t}_L \gamma^{\mu} b_L W_{\mu}^+ + h.c.; 
\end{eqnarray}

We compare this with Larios/Yuan hep-ph/9903394:
\begin{eqnarray}
\label{d7}
{\cal L}_{lowE}^{L,R}  & = & 
\frac{g}{2 c_W} \Big(1-\frac{4}{3} s_W^2 + \kappa_L^{NC}  \Big)
{\bar t}_L \gamma^{\mu} t_L Z_{\mu} \nonumber \\
&  &
+ \frac{g}{2 c_W} \Big(-\frac{4}{3} s_W^2 + \kappa_R^{NC} \Big)
{\bar t}_R \gamma^{\mu} t_R Z_{\mu} \nonumber \\
&  & + \frac{g}{\sqrt{2}} \Big(1 + \kappa_L^{CC} \Big)
{\bar t}_L \gamma^{\mu} b_L W_{\mu}^+ + h.c. \nonumber \\
&  & + \frac{g}{\sqrt{2}} \kappa_R^{CC} 
{\bar t}_R \gamma^{\mu} b_R W_{\mu}^+ + h.c.;
\end{eqnarray}
and obtain:
\begin{eqnarray}
\label{d8}
\kappa_R^{CC} & = & - p \frac{b_2}{2} + r \frac{\lambda_2}{2} \nonumber \\
\kappa_L^{CC} & = & +- -  \frac{b_1}{2} +  \frac{\lambda_1}{2} \nonumber \\
\kappa_R^{NC} & = & \frac{b_2}{2} - \frac{\lambda_2}{2} \nonumber \\
\kappa_L^{NC} & = & - \frac{b_1}{2} + \frac{\lambda_1}{2} 
\end{eqnarray}

The right-handed coupling $\kappa_R^{CC}$ 
is constrained by $b \rightarrow s \gamma$ (this process is almost insensitive to 
$\kappa_L^{CC}, \kappa_R^{NC}$ and $\kappa_L^{NC}$) at the $2\sigma$ level as (see 9903394)
$|\kappa_R^{CC}| < 0.004$ which translates into
\begin{equation}
\label{d9}
|-p b_2 + r \lambda_2| < 0.008
\end{equation}
This is rather strict limit which can be avoided in two different ways: either (i) assuming 
cancellation $p b_2 = r \lambda_2$ or (ii) setting $p = r = 0$. 

We will find further limits from $\epsilon$ parameters assuming "individual" fits, e.g.,
to find limit on $b_1 - \lambda_1$ we put $b_2 = \lambda_2 = 0$.
The $Zbb$ vertex (see Eqs. \ref{d5,d6}) can be rewritten as
\begin{eqnarray}
\label{d10}
{\cal L}_{Zbb} & = & 
\frac{g}{2 c_W} \Big(\frac{2}{3} s_W^2 -\frac{1}{2} + \frac{b_1}{4}-\frac{\lambda_1}{4}\Big)
{\bar b} \gamma^{\mu} b Z_{\mu} \nonumber \\
& &
- \frac{g}{2 c_W} \Big(-\frac{1}{2} + \frac{b_1}{4}-\frac{\lambda_1}{4}\Big)
{\bar b} \gamma^{\mu} \gamma^5 b Z_{\mu}
\end{eqnarray}
From here we extract the vector and axial-vector couplings of the Zbb vertex:
\begin{eqnarray}
\label{d11}
g_b^V & = &  \frac{2}{3} s_W^2 - \frac{1}{2} + \frac{b_1}{4} - \frac{\lambda_1}{4} \nonumber \\
g_b^A & = & - \frac{1}{2} + \frac{b_1}{4} - \frac{\lambda_1}{4}
\end{eqnarray}
$\epsilon_b$ parameter is defined by
\begin{equation}
\label{d12}
g_b^A = - \frac{1}{2} \Big(1 + \frac{\Delta \rho}{2}\Big)\Big(1 + \epsilon_b\Big)
\end{equation}
where $\Delta_{\rho}$ is the nonstandard correction to $g_l^A$, 
the axial-vector coupling of leptons to Z. At the tree-level our model has
$\Delta_{\rho} = 0$ since all nonstandard effects are restricted to the quarks of the
3rd generation. Hence we obtain 
\begin{equation}
\label{d13}
\epsilon_b^{tree} = - \frac{b_1}{2} + \frac{\lambda_1}{2}
\end{equation}
At one loop level we pick additional contributions
\begin{eqnarray}
\label{d14}
\epsilon_b & = & 
\epsilon_b^{tree} + \epsilon_b^{SM} + \delta \epsilon_b 
\nonumber \\
& = & - \frac{b_1}{2} + \frac{\lambda_1}{2}  + \frac{m_t^2 G_F}{2 \sqrt{2} \pi^2} 
\Big(- \frac{b_1}{2} + \frac{\lambda_1}{2} \Big) \ln \frac{\Lambda^2}{m_t^2}
+ \epsilon_b^{SM}
\end{eqnarray}
with $\epsilon_b^{SM} = - 6.5 x 10^{-3}$ (see ???) 
We now use the experimental value of $\epsilon_b$ to constrain the theoretical prediction at $2\sigma$ 
level (see ???)
\begin{equation}
\label{d15}
- 8.32 x 10^{-3} \leq \epsilon_b \leq - 0.88 x 10^{-3}
\end{equation}
finally yielding
\begin{equation}
\label{d16}
|b_1 - \lambda_1| \leq \sim 0.01
\end{equation}
Again, this constraint can be escaped assuming cancellation $b_1 = \lambda_1$.

Finally, we examine constraint on $b_2 - \lambda_2$ in the scenario $p = r = 0$ assuming
$b_1 = \lambda_1 = 0$. There is no tree level contribution to $\epsilon$ parameters in this case.
There is, however, an important one-loop contribution to $\epsilon_1$ from the $Ztt$  coupling.
It comes from the diagrams such as, e.g., $t{\bar t}$ loop in the Z boson self-energy, famous for its $m_t^2$
dependence which led to the succesful prediction of the top quark mass before the top quark was actually
discovered. The analysis of Ref. ??,  yields
\begin{equation}
\label{d17}
\delta \epsilon_1 = \frac{3 m_t^2 G_F}{2 \sqrt{2} \pi^2} 
\Big(\frac{b_2}{2} - \frac{\lambda_2}{2} \Big) \ln \frac{\Lambda^2}{m_t^2}
\end{equation}
where $\Lambda$ is the cut-off scale of the divergent one-loop graphs. Wer now use the experimental value of
$\epsilon_1$ to constrain the theoretical prediction at $2\sigma$ level (see ???):
\begin{equation}
\label{d18}
1.5 x 10^{-3} \leq \epsilon_1^{SM} + \delta \epsilon_1 \leq 5.9 x 10^{-3}
\end{equation}
where $\epsilon_1^{SM} =  3.32 x 10^{-3}$ ($M_H = 1$ TeV, $m_t = 170$ GeV). This implies
($\Lambda = 1$ TeV)
\begin{equation}
\label{d19}
|b_2 - \lambda_2| \leq 0.04
\end{equation}

Limits on free parameters are given by
\begin{eqnarray}
\label{d20}
g'' & > & 10 \nonumber \\
|b_1-\lambda_1| & < & 0.01     \nonumber \\
|b_2-\lambda_2| & < & 0.04   \; (0.01)
\end{eqnarray}   

The most natural Lagrangian we can write down for interactions of $\rho$ resonance with scalar pions
and top quarks is given by
\begin{eqnarray}
\label{b1}
{\cal L}_{\rho \pi \pi} & = & + i g_{\pi} \frac{M_{\rho}}{v} 
\big( \pi^{-} \partial^{\mu} \pi^{+} - \pi^{+} \partial^{\mu} \pi^{-} \big) \rho_{\mu}^{0} \nonumber \\
{\cal L}_{\rho t t} & = & g^V {\bar t} \gamma^{\mu} t \;\rho_{\mu}^{0} 
+ g^A {\bar t} \gamma^{\mu} \gamma^5  t \;\rho_{\mu}^{0}; 
\end{eqnarray}
with important amplitudes and partial widths given as
\begin{eqnarray}
\label{b2}
i {\cal M}_{\rho}^{+-} & = &  g_{\pi} \frac{M_{\rho}}{v}  \frac{g^V s + 4 g^A p_3 E}{s - M_{\rho}^2} 
\sin{\Theta} \nonumber \\
\Gamma_{\rho t t} & = & N_c \frac{\sqrt{\frac{1}{4} M_{\rho}^2 - m_t^2}}{8 \pi M_{\rho}^2} \frac{4}{3}
\Big[{g^V}^2 (M_{\rho}^2 + 2 m_t^2) + {g^A}^2(M_{\rho}^2 - 4 m_t^2)\Big] \nonumber \\
\Gamma_{\rho \pi \pi} & = & \frac{4 g_{\pi}^2}{192 \pi} \frac{M_{\rho}^3}{v^2}
\end{eqnarray}
where $g_{\pi}$, $g^V$ and $g^A$ are free parameters.
When expanded, Lagrangian of Eqs \ref{a1},\ref{a2}, \ref{a3} also assumes the form  given by Eqs. \ref{b1},
\ref{b2}
however, $g_{\pi}$, $g^V$ and $g^A$ are no longer free parameters but can be expressed in terms of the
parameters of Lagrangian of Eqs \ref{a1}, \ref{a2}, \ref{a3}
\begin{eqnarray}
\label{c1}
g^V & = & \frac{1}{2}(c_L +c_R) \nonumber \\
g^A & = & \frac{1}{2}(c_R - c_L) \nonumber \\
c_L & = & \frac{1}{1+b_1}\Big(-\frac{g}{2} O_{32} - \frac{g'}{6} O_{31} +b_1 \frac{g''}{2} O_{33}
- b_1 \frac{g'}{6} O_{31} + \lambda_1 \frac{g'}{4} O_{31} - \lambda_1 \frac{g}{4} O_{32}\Big) \nonumber \\
c_R & = & \frac{1}{1+b_2}\Big(-\frac{g'}{2} O_{31} - \frac{g'}{6} O_{31} +b_2 \frac{g''}{2} O_{33}
- b_2 \frac{g'}{6} O_{31} - \lambda_2 \frac{g'}{4} O_{31} + \lambda_2 \frac{g}{4} O_{32}\Big) \nonumber \\
g^V & \doteq & \frac{g''}{4}\Big(\frac{b_2}{1+b_2} + \frac{b_1}{1+b_1}\Big)
 \nonumber \\
g^A & \doteq & \frac{g''}{4}\Big(\frac{b_2}{1+b_2} - \frac{b_1}{1+b_1}\Big)
 \nonumber \\ 
g_{\pi} & = & \frac{M_{\rho}}{2 v} \frac{1}{g''} \nonumber \\
M_{\rho}^2 & = & \frac{a g''^2 v^2}{4} \nonumber \\
{\cal L}_X^R & = & \frac{-i}{4 v^2} \frac{b_2-2 \lambda_2}{1+b_2} {\bar t}_R \gamma^{\mu} t_R
\Big(\partial_{\mu} \pi^+ \pi^- - \partial_{\mu} \pi^- \pi^+\Big) + \frac{m_t}{v^2} \pi^+ \pi^- {\bar t} t
\nonumber \\
{\cal L}_X^L & = &  ? \nonumber \\
i {\cal M}_{\rho}^{-+} & = & 0 \nonumber \\
b_1 & = & \lambda_1  =  0; \;\;\; \lambda_2 \neq 0
  \end{eqnarray}

\section{Introduction}
\label{sec:intro}

One of the most important problems of today's particle physics is the
mechanism of electroweak symmetry breaking (ESB).
The mechanism of ESB is responsible for giving the W and Z bosons
their masses.
Despite some progress in the experimental limits on the Higgs mass
\cite{higgsmass}, the ESB sector of the Standard Model (SM)
is still rather weakly constrained by the
experimental data and the physics behind the mechanism of ESB remains
unknown.

In this work we study some phenomenological consequences of
the {\it strongly-coupled} ESB sector.
Typical representatives of the strongly-coupled scenario are technicolor models built
in analogy to chiral symmetry breaking in QCD.
The strong ESB gives rise to massless Goldstone bosons
--- electroweak (EW) pions --- which, just like QCD pions, are assumed
to be bound states of some
more fundamental strongly interacting objects (e.g. technifermions).
According to Higgs mechanism, EW pions are absorbed by $W$ and $Z$ bosons and become
their longitudinal components ($W_L,Z_L$).
Thus any collider process which involves the longitudinal weak
gauge bosons can in principle give us an access to the
interactions connected to the mechanism of ESB \cite{snowmass}.

Since $V_L V_L$ ($V=W,Z$) scattering amplitudes start to violate
unitarity at
1-3~TeV range we expect new strongly interacting ESB resonances (equivalent of lowest
QCD resonances such as $\rho$ and $\omega$) to
appear at or below this scale.
We will not be able to see the substructure of these new resonances
and EW pions at the Large Hadron Colider (LHC) and at a future $e^+e^-$ Linear
Collider (LC) operating at 1-3~TeV range, but the question is if we
can distinguish at least the resonances themselves and measure their
masses and couplings.

A lot of attention has been devoted to the testing of the
strongly-interacting scenario in the longitudinal vector boson
scattering $V_L V_L \rightarrow V_L V_L$ \cite{review}.
The studies concentrated on signatures of either a new $I=J=1$
 ESB $\rho$-resonance or a new $I=J=0$ ESB $S$-resonance
(parameters of the $S$-resonance can be tuned to immitate
the SM Higgs boson --- SM-like $S$-resonance) at either
the LC or the LHC.
The results have shown that it will be possible to establish the
presence of strong ESB at the LHC and the LC and that,
to a degree, it will
be possible to distinguish new strong resonances \cite{snowmass}.

Another potentially powerful process for the study of strong ESB is
the scattering of longitudinal vector bosons to top quarks,
$V_L V_L \rightarrow t \bar{t}$.
Its main appeal is in the possibility to test whether the
extraordinarily large top quark
mass is generated by the same new strong interactions which are
responsible for ESB, or by yet additional new strong interactions
introduced just for that sake.
In the former case (represented, e.g., by the extended
technicolor theories \cite{Lane}) 
we expect the top quark to couple
significantly to the resonances which unitarize
$V_L V_L \rightarrow V_L V_L$ scattering \cite{Morales,Lee}.
This could lead to significant event rates in
$V_L V_L \rightarrow t \bar{t}$.
In the latter case, when the mechanism of the top mass generation is
different from the $W$ mass generation (as in topcolor-assisted
technicolor models \cite{Lane}), 
we expect that the top quark does not couple
significantly to the new resonances of the strong ESB sector.
This would imply that the new resonances observed in the
$V_L V_L \rightarrow V_L V_L$ channel are suppressed in the
$V_L V_L \rightarrow t \bar{t}$ channel  \cite{snowmass,Morales}.

For these reasons the $V_L V_L \rightarrow t \bar{t}$ process at
the LC has recently attracted growing interest.
There have been studies within the SM
\cite{Kauffman,Barklow,Godfrey}, within the Higgsless SM below
the scale of new physics (no-resonance model) \cite{Lee,Larios},
and also within models above the scale of a new $S$-resonance and
$\rho$-resonance \cite{snowmass,Morales,Melogintner,Han}.

In this work we study the new 
$\rho$ and $S$ resonances in chiral Lagrangian approach and calculate 
their signals in the
 process $W_L W_L \rightarrow t \bar{t}$ 
which is being
 considered as a subprocess of
$e^+e^- \rightarrow \nu {\bar \nu} t {\bar t}$
at the LC with the CM energy of $1.5$~TeV and 3~TeV.
Three cases, the no-resonance, the $S$-resonance, and
the $\rho$-resonance are considered with various
values of parameters.
We show the importance of low energy constraints for
$\rho$-resonance signals at LC.
We calculate total and differential cross sections using the
Effective-$W$ Approximation and Equivalence theorem (ET).
The number of events obtained is for the assumed integrated
luminosity of 200~fb$^{-1}$.
We also calculate the relevant backgrounds.

This contribution is organized as follows.
In Section \ref{sec:models} we introduce the lowest order
effective chiral Lagrangians describing models with
$\rho$-resonance,  $S$-resonance and no resonance.
Our calculations and results obtained for the models under
consideration are presented in Section \ref{sec:signals} and conclusions
in Section \ref{sec:concl}.

	\section{$\rho$-\ , $S$-\ and no-\ resonance models}
	\label{sec:models}

Due to our ignorance of details of new strong physics behind ESB
the most convenient approach to the analysis of its possible
consequences is the effective field theory framework.
In this approach, if energy available is below the threshold of new
resonances production, one starts with EW pions as the only
particles in the spectrum which are subject to new strong
interactions.
The Lagrangian of EW pions is the familiar nonlinear $\sigma$-model
based on global $SU(2)_L \times SU(2)_R$ spontaneously broken down to
$SU(2)_V$ custodial or isospin symmetry.
This symmetry breaking pattern is supported by the relation
$M_W/M_Z = \cos \theta_W$ which is satisfied to high accuracy.

If we assume to have enough energy to produce a new resonance it
must be added to the set of building elements of the effective
Lagrangian.
In our work beside the no-resonance case we also assume
the production of either the $S$-resonance or
$\rho$-resonance.
They are added to the Lagrangian
respecting chiral $SU(2)_L \times SU(2)_R$ symmetry.
For $S$-resonance it is a straightforward procedure, for
$\rho$-resonance one can follow either Weinberg \cite{Weinberg} or
hidden symmetry approach \cite{Bando}.
The gauge interactions of the SM are introduced by requiring
the $S$ and $\rho$ Lagrangians to be gauge invariant under
$SU(2)_L \times U(1)_Y$.

Let us begin with the gauged no-resonance Lagrangian given by
\begin{eqnarray}
\label{Lagr0}
{\cal L}_0 & = &  
- \overbrace{v^2 Tr \{{\cal A_{\mu}}{\cal A^{\mu}}\}}^{{\rm nonlinear}
\;\sigma\;{\rm model}} 
- \overbrace{{\bar \psi}_L u^{\dagger} u^{\dagger} M \psi_R}^{t\;{\rm couples}\;
 {\rm to}\;\pi ;\;\;m_t,m_b} + h.c. 
 + {\cal L}_{kin}({\bf \tilde{W}},{\bf \tilde{Y}}) \nonumber \\
 & &
 + {\bar \psi}_L \;i \gamma^{\mu}
 \big[\partial_{\mu}  + {\bf \tilde{W}}_{\mu} 
 + \frac{i}{2}g'(B-L)Y_{\mu}\big]\psi_L  \nonumber \\
 & & + {\bar \psi}_R \;i \gamma^{\mu}
 \big[\partial_{\mu}  + {\bf \tilde{Y}}_{\mu} 
 + \frac{i}{2}g'(B-L)Y_{\mu}\big]\psi_R  
\end{eqnarray}
where
${\cal A}_{\mu} = [u^{\dagger}(\partial_{\mu}
+ {\bf \tilde{Y}}_{\mu})u
- u (\partial_{\mu} + {\bf \tilde{W}}_{\mu})u^{\dagger}]/2$ ;
$u = \exp(i \vec{\pi} .\vec{\tau}/{2 v})$ with $v = 246$~GeV,
$\vec{\pi}$ is the
isospin triplet of the EW pions, and $\vec{\tau}$  are
Pauli matrices;
$\psi$ denotes fermion field doublet, $(t,b)$,
the $B-L$ is the baryon minus lepton number operator, and
the $M = \mbox{diag}(m_t,m_b)$ is the fermion doublet mass matrix.
${\cal L}_{kin}({\bf \tilde{W}},{\bf \tilde{Y}})$ is the kinetic energy term 
of the $SU(2)_L$
gauge field, $W_\mu$
and the $U(1)_Y$ gauge field, $Y_\mu$;
${\bf \tilde{W}}_{\mu} = i g \vec{W}_{\mu} . \vec{\tau}/2$;
${\bf \tilde{Y}}_{\mu}  =  i g' Y_{\mu} \tau^{3}/2$. 

The gauged $S$-resonance model can be described by the following
leading order effective Lagrangian
\begin{eqnarray} 
\label{LagrS}
{\cal L}_S & = & {\cal L}_0 + {\cal L}_{kin}(S)  - \frac{1}{2} M_S^2 S^2
- 2 g''' v  Tr \{{\cal A_{\mu}}{\cal A^{\mu}}\}\;S \nonumber \\
 & & 
 - \frac{\alpha}{v} {\bar \psi}_L u^{\dagger} u^{\dagger} M \psi_R
 S
\;\;+ \;h.c.
\end{eqnarray}
where ${\cal L}_0$ is given by (\ref{Lagr0}), the ${\cal L}_{kin}(S)$
is the kinetic energy term of the $S$-resonance, 
$S$ denotes the scalar $(I=J=0)$ ESB resonance field,
$M_S$ is its mass.
The $g'''$, $\alpha$ and $M_S$ are free parameters.

The gauged $\rho$-resonance model is given by
\begin{eqnarray}
\label{LagrR}
{\cal L}_\rho & = & {\cal L}_0 +  {\cal L}_{kin}(\rho)
- a \frac{v^2}{4} \mbox{Tr}[(\omega_\mu + \rho_\mu)(\omega^\mu + \rho^\mu)] \nonumber \\
& &
 + b_1 \,{\bar \psi}_L \, i \gamma^{\mu}\big[u \partial_{\mu} - u \rho_{\mu}
 +
 u \frac{i}{2} g' (B - L) Y_{\mu}\big] \,u^{\dagger}\, \psi_L \nonumber \\
 & &
 + b_2 \,{\bar \psi}_R \, P\, i \gamma^{\mu}\big[u \partial_{\mu} - u \rho_{\mu}
 + u \frac{i}{2} g' (B - L) Y_{\mu}\big] \,u^{\dagger}\,P\, \psi_R  \nonumber \\
& & 
 + \lambda_1 \,{\bar \psi}_L \, i \gamma^{\mu}\; u^{\dagger}\; {\cal A_{\mu}}\; \gamma^{5} u
 \;\psi_L \;\;
+ \lambda_2 \,{\bar \psi}_R \, P\, i \gamma^{\mu} u \;{\cal A_{\mu}}\;
\gamma^{5} \; u^{\dagger}\;P\;\psi_R
 \end{eqnarray}
where
$\omega_{\mu} = [u^{\dagger}(\partial_{\mu}
+ {\bf \tilde{Y}}_{\mu}
)u + u(\partial_{\mu} + {\bf \tilde{W}}_{\mu}) u^{\dagger}]/2$;
$\rho_{\mu} = i g''{\vec \rho_{\mu}}. {\vec \tau}/2$, 
${\vec \rho_{\mu}}$ denotes 
the vector $(I=J=1)$ ESB $\rho$-resonance field,
${\cal L}_0$ is given by (\ref{Lagr0}),
the ${\cal L}_{kin}(\rho)$ is the kinetic energy term of
the $\rho$-resonance, 
the $a$, $g''$, $b_1$, $b_2$, $\lambda_1$ and $\lambda_2$ are free parameters
and $P = \mbox{diag}(1,0)$.
The $P$ matrix ensures that the $\rho$ only couples to the
right-handed top quark but not to the right-handed bottom quark
(to avoid constraints on the $\rho$ couplings from the
$Zb{\bar b}$ vertex), which spoils the
custodial symmetry in much the same fashion as does the top-bottom
mass splitting.

The low energy experiments place the following constraints on the parameters of
the $\rho$-resonance Lagrangian \cite{APS}:
$g''  \stackrel{>}{\sim}  10$,
$|b_2 - \lambda_2|  \stackrel{<}{\sim}  0.04$ and
$|b_1 - \lambda_1|  \stackrel{<}{\sim}  0.01$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%
%   etc
%
%  for pictures use \epsfig command
%  for instance:

%


	\section{Signals and backgrounds}
	\label{sec:signals}

%	\begin{figure}[t]
%	\begin{center}
%        \includegraphics[width=.4\textwidth]{fd1.eps}
%	\end{center}
%	\caption[]{Feynman diagrams for
%	({\bf a}) $\;e^+e^- \rightarrow \nu {\bar \nu} t {\bar t}$,
%	 ({\bf b-f}) $\;W_LW_L \rightarrow t{\bar t}$}
%	\label{fig:feynman}
%	\end{figure}
%


We have chosen to calculate signals for two $\rho$-resonances:
one with mass $M_{\rho} = 1000$~GeV and width
$\Gamma_{\rho} = 9.4$~GeV ($b_2 = 0.08, g'' = 10, b_1=\lambda_1=\lambda_2=0$),
and another one with $M_{\rho} = 750$~GeV and width
$\Gamma_{\rho} = 2.0$~GeV ($b_2 = 0.04, g'' = 10, b_1=\lambda_1=\lambda_2=0$).
For comparison we also calculate signals from an $S$-resonance
--- $800$~GeV SM-like 
Higgs boson with the width $\Gamma_H = 285$~GeV
($g''' = \alpha = 1$) 
--- and from the no-resonance model.

\begin{figure}
\epsfig{figure=fd.eps,width=0.9\textwidth,height=0.2\textheight}
\caption{Feynman diagrams for
	({\bf a}) $\;e^+e^- \rightarrow \nu {\bar \nu} t {\bar t}$,
	 ({\bf b-f}) $\;W_LW_L \rightarrow t{\bar t}$}
\label{fig:feynman}
\end{figure}

$W_L W_L \rightarrow t {\bar t}$ scattering is a subprocess of the process
$e^+e^- \rightarrow \nu {\bar \nu} t {\bar t}$, see Fig.~1.
To calculate cross sections of the process
$e^+e^- \rightarrow \nu {\bar \nu} t {\bar t}$
we used Effective-$W$ Approximation (EWA) \cite{Dawson}
in which one convolutes the cross section for the subprocess
$W_L W_L \rightarrow t {\bar t}$ with the corresponding
distribution functions of the $W_L$ boson.
For the EWA to be valid the invariant energy scales of the process
should be much larger than the $W$ mass (we impose cut $m_{tt} > 500$~GeV).
To calculate the subprocess cross sections we used the ET theorem
according to which we replaced $W_L$ boson with the corresponding
EW pion $\pi$.
The subprocess diagrams are shown in Figs. 1b -- 1f.
We implemented our calculations in MC program Pythia \cite{Pythia}
in a way which closely mimics the production of a heavy Higgs boson in the
process 76. With Pythia we can both calculate cross sections ($\sigma \simeq 1-2 fb$ for
$\rho$ and $S$ at $\sqrt{s} = 1500$~GeV) and generate events. 

Since the sensitivity of $W_T$ channels to the ESB sector is much weaker than 
the sensitivity of $W_L$ channels, we treat $W_T W_T$ contribution to 
$e^+e^- \rightarrow \nu {\bar \nu} t {\bar t}$ as irreducible background
with $\sigma \simeq \sigma_{SM}(M_H =$~100 GeV) \cite{Han}. We computed this cross 
section with CompHEP \cite{CompHEP}, $\sigma = 1.79 fb$ at 1500 GeV .
 The backgrounds include two other processes which we generated with Pythia:  
$e^+e^- \rightarrow t {\bar t} (\gamma)$ with  
$\sigma \simeq 100 fb$ at 1500 GeV and 
$e^+e^- \rightarrow e^+ e^- t {\bar t}$ with $\sigma = 5.28 fb$ at 1500 GeV.

In order to suppress the backgrounds and enhance signals we impose 
the following cuts:
$m_{tt} > 500$~GeV, $p_T(t{\bar t}) > 30$~GeV, $m_{\rm recoil} > 200$~GeV, 
$|\cos \theta_{t,{\bar t}}| < 0.8$, 
$p_T(t),p_T({\bar t}) > 50$~GeV. These cuts reduce 
the backgrounds to a total $\sigma = 0.94 fb$ at 1500 GeV which is of the order of
$\rho$ and $S$ resonance signals.

The results of our calculations are shown in Fig. 2. and Table 1.
Fig.2 shows the number of events/20 GeV we expect from signals 
and backgrounds 
at a  future $e^+e^-$ collider after one year run at the energy of 1500 GeV 
and luminosity of 200 $fb^{-1}$. Event numbers are plotted as a function of 
invariant mass of the $t{\bar t}$ pair. 
Fig. 2({\bf a}) shows the signal from $\rho$ resonance with mass $M_{\rho}$=1000
	GeV , Fig. 2({\bf b}) signal from $\rho$ resonance with mass $M_{\rho}$=
	750 GeV, Fig.2({\bf c}) from an S resonance with mass $M_S$ = 800 GeV
	and Fig.2({\bf d}) from the no-resonance case.
Signals are plotted against the total background drawn as the dark-shaded distribution.

Corresponding numerical results can be found in Table 1. The numbers shown are for
$\sqrt{s}= 1500$~GeV, the exception is the last column which is for $\sqrt{s}= 3000$~GeV. 
The cross sections are after the cuts and event numbers assume luminosity $200 fb^{-1}$. 
These events could be fully reconstructed  (including $b$-tagging) with
the efficiency of about  $\epsilon_{rec} = 30$~\% \cite{Morales}; this loss could be partially or
fully compensated through the use of polarized electron and/or positron beams.
The significance of events is defined as $S/\sqrt{B}$ and the discovery criterion is 
$S/\sqrt{B} > 5$.



%
\begin{figure}
\epsfig{figure=cuts.eps,width=1.0\textwidth,height=0.4\textheight}
\caption{Number of events/20 GeV expected from signals and background at a 
future $e^+e^-$ collider after one year run at the energy of 1500 GeV and 
luminosity of 200 $fb^{-1}$. Event numbers are plotted against the invariant mass 
of the $t{\bar t}$ pair. 
Fig.2({\bf a}) shows the signal from $\rho$ resonance with mass $M_{\rho}$=1000
	GeV , Fig. 2({\bf b}) signal from $\rho$ resonance with mass $M_{\rho}$=
	750 GeV, Fig.2({\bf c}) from an S resonance with mass $M_S$ = 800 GeV
	and Fig.2({\bf d}) from the no-resonance case.
The total background is plotted on each figure as dark-shaded distribution.	 }
\label{fig:results}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}
\caption{Numerical results}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\setlength\tabcolsep{5pt}
\begin{tabular}{lllllll}
\hline\noalign{\smallskip}
& $M$(GeV) & $\Gamma $(GeV) & $\sigma $(fb) & N/year
  & S/$\sqrt{B}$  &  S/$\sqrt{B}$   \\
& & & & & (1.5 TeV) & (3 TeV) \\  
\noalign{\smallskip}
\hline
\noalign{\smallskip}
 $\rho$ & 1000 & 9.4 & 0.83 & 166 & 12.1 & 50.7\\
 $\rho$ & 750 & 2.0 & 0.90 & 180 & 13.1 & 42.0\\
$S$ & 800 & 285 & 1.94 & 388 & 28.3 & 86.8 \\
no resonance & - & - & 0.58 & 117 & 9.5 & 30.7 \\
Background & - & - & 0.94 & 188 & - & - \\
\hline
\end{tabular}
\end{center}
\label{Tab1a}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

	\section{Conclusions}
	\label{sec:concl}

We have studied the new strong resonances associated with
the physics responsible for the strong electroweak symmetry breaking
and signals of these resonances in the process
$e^+ e^- \rightarrow \nu {\bar \nu} t {\bar t}$.
This process might shed light on the role of the top quark in ESB.
We have provided the comprehensive treatment of the $\rho$ Lagrangian
including the low-energy constraints on its parameters.
The analysis presented in this work shows that the
significance of the signal from the $\rho$ and $S$ resonances 
at the energy of $1.5$~TeV and the integrated luminosity
of 200~fb$^{-1}$ could be well above the discovery limit
$S/\sqrt{B} > 5$. On the other hand more detailed study is required 
to see if we can distinguish the masses and widths of the $\rho$ and 
$S$ resonances at the energy of 1.5 TeV or if a higher energy will be 
needed.

This work was supported by grant VEGA 1/8307/01.


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%
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%
%	\refer{X.Y. Author, Z.T. Anotherauthor}{Phys. Rev. Lett.}{1}{1961}{1}
%	for book references, use
%	A.B. Author: {\sl name of book}, Publishing House, year.
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\end{thebibliography}
\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
