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\begin{document}
\hfill$\vcenter{
 %\hbox{\bf BARI-TH 426/01}
\hbox{\bf FIRENZE-DDF-392/07/02}
 %\hbox{\bf UGVA-DPT-2001 11/1097}}
 }$
%
\begin{center}
{\Large\bf\boldmath {The linear BESS model and the
double\vskip.5cm Higgs-strahlung production}}
\\ \rm \vskip1pc {\large
R. Casalbuoni and  L. Marconi}\\ \vspace{5mm} {\it{Dipartimento di
Fisica, Universit\`a di Firenze, I-50125 Firenze, Italia
\\
I.N.F.N., Sezione di Firenze, I-50125 Firenze, Italia}}
\end{center}
%%% ----------------------------------------------------------------------

\begin{abstract}
In this paper we evaluate, in the context of  the linear BESS
model  the cross-section for the double Higgs-strahlung process.
We find that, within the bounds given by the actual experimental
data, significant deviations with  respect to the SM may arise. In
the linear BESS model not only the self-couplings of the Higgs
are modified, but also the $Z$-Higgs couplings. We think that
this is a generic feature of any extension of the SM and, in our
opinion, it should be kept in mind in analyzing the future data on
the process studied here.
\end{abstract}


%INTRODUZIONE
\section{Introduction}

During the last decade the Standard Model (SM) has been verified
with great accuracy and today there is no clear evidence  for any
deviation of the experimental data from the theoretical
expectations. However, as it is well known, there is more than
one reason to look for extensions of the SM. The main scenarios
considered for such extensions are  supersymmetry
\cite{supersimm} and technicolor \cite{technicolor}. The last
type of models are somewhat disfavored, considered the
experimental data, mainly because usually there is no natural
limit in which they reduce to the SM. On the other hand the
supersymmetric models are generally such that making heavy the
supersymmetric partners of the SM particles they decouple giving
the SM in the limit. The linear BESS model, although belonging
naturally to a technicolor type scenario, enjoys the decoupling
property \cite{BESSlineare}. As such it turns out to be
compatible with the actual experimental data. In the model there
are six new heavy vector bosons and two heavy scalars and it is
characterized by a heavy scale of order of a few $TeV$.
Decoupling the heavy particles, as we will do in this paper in
order to get an effective lagrangian for the SM particles, has
quite different results in the low-energy physics and in the
physics associated with the standard Higgs. As far as low-energy
is concerned the effects of new physics are very soft  leading to
corrections ${\cal O}(1/u^2)$ ($u$ being the heavy scale).
However there are potentially significant corrections to the
Higgs potential and to the couplings of the Higgs to the $Z$.
Therefore in this work we study  the process of double
Higgs-strahlung production where all these couplings are present.
The main difference of this paper with previous ones making
similar analysis is that we take into account  corrections to the
SM cross-section arising  from the couplings of the Higgs to the
$Z$ boson. We think that this is an important point, since it is
difficult to imagine a generic model where the only difference
with respect to the SM is in the Higgs self-couplings. We have
organized the paper starting in Section 2 with a presentation of
the cross-section for the double Higgs-strahlung where we leave
all the Higgs couplings arbitrary for the reasons explained
above. In Section 3 we give the expressions for the Higgs
self-coupling and for the relevant $Z$ Higgs couplings within the
linear BESS model. These couplings, as already said, have been
evaluated by eliminating all the heavy fields from the BESS
lagrangian obtaining an effective low-energy description of the
SM fields. Finally, in Section 4, we present our numerical
results showing that, under the constraint of compatibility with
the low-energy and LEP/SLC data, significant deviations from the
SM expectation of the  double Higgs-strahlung cross-section might
arise.

%SEZIONE I
\section{Double Higgs-strahlung production}\label{sez_urto}

We give here the expression for the unpolarized differential
cross-section for the process of double Higgs-strahlung
$e^+e^-\longrightarrow ZHH$ \cite{Zerwas1} assuming  arbitrary
trilinear Higgs self-coupling, $\lambda_{HHH}$, and
couplings $\lambda_{HZZ}$ and $\lambda_{HHZZ}$ of the Higgs to
the  $Z$ boson. After integration over the angular variables one
gets:
\begin{equation}\label{sezione}
\left(\frac{d\sigma}{dx_1dx_2}\right) =
\left(\frac{d\sigma_0}{dx_1dx_2}+
 \frac{d\sigma_1}{dx_1dx_2}+\frac{d\sigma_2}{dx_1dx_2}+\frac{d\sigma_3}{dx_1dx_2}
\right) + (y_1 \leftrightarrow y_2),
\end{equation}
where
\begin{eqnarray}
 \frac{d\sigma_0}{dx_1dx_2} & = &  \frac{\sqrt{2}G_F^3M_Z^6}{384\pi^3}\frac
    {(2c_V)^2 +(2c_A)^2}{s(1-\mu_Z)^2}\frac{2}{G_F^2M_Z^8}a^2f_0,\nonumber \\
\frac{d\sigma_1}{dx_1dx_2} & = &
\frac{\sqrt{2}G_F^3M_Z^6}{384\pi^3}\frac
    {(2c_V)^2 +(2c_A)^2}{s(1-\mu_Z)^2}\frac{\lambda_{HZZ}^4}{2G_F^2M_Z^8}
    \frac{f_1}{4\mu_Z(y_1+\mu_{HZ})^2},\nonumber \\
\frac{d\sigma_2}{dx_1dx_2} & = &
\frac{\sqrt{2}G_F^3M_Z^6}{384\pi^3}\frac
    {(2c_V)^2 +(2c_A)^2}{s(1-\mu_Z)^2}\frac{\lambda_{HZZ}^4}{2G_F^2M_Z^8}
    \frac{f_2}{4\mu_Z(y_1+\mu_{HZ})(y_2+\mu_{HZ})},\nonumber \\
\frac{d\sigma_3}{dx_1dx_2} & = &
\frac{\sqrt{2}G_F^3M_Z^6}{384\pi^3}\frac
    {(2c_V)^2 +(2c_A)^2}{s(1-\mu_Z)^2}\frac{\lambda_{HZZ}^2}{G_F^2M_Z^8}
    \frac{af_3}{2(y_1+\mu_{HZ})}. \label{2}
\end{eqnarray}
We have introduced the scaled energies of the two Higgs particles
$x_{1,2}=2E_{1,2}/\sqrt{s}$,  $x_3=2-x_1-x_2$ the scaled energy
of the  $Z$ boson and $y_i=1-x_i$. We have also introduced scaled
square masses $\mu_i=M^2_i/s$, $i=Z,H$, and
$\mu_{ij}=\mu_i-\mu_j$. For the moment being we have also  left
unspecified the vector and axial coupling of the $Z$ to the
fermions, $c_V$ and $c_A$. The coefficient $a$ turns out to be
\begin{equation}\label{def_a}
a=\frac{3\lambda_{HHH}\lambda_{HZZ}}{y_3-\mu_{HZ}}+\frac{\lambda_{HZZ}^2}
         {y_1+\mu_{HZ}}+\frac{\lambda_{HZZ}^2}{y_2+\mu_{HZ}}+\lambda_{HHZZ}s,\label{3}
\end{equation}
and the coefficients $f_i$ are:
\begin{eqnarray}\label{def_f_i}
f_0 & = & \mu_Z[(y_1+y_2)^2+8\mu_z]/8,\nonumber \\
f_1 & = & (y_1-1)^2(\mu_Z-y_1)^2 -4\mu_Hy_1(y_1+y_1\mu_Z-4\mu_Z)+\nonumber \\
    &   & +\mu_Z(\mu_Z-4\mu_H)(1-4\mu_H)-\mu^2_Z, \nonumber \\
f_2 & = & [\mu_Z(y_3+\mu_Z-8\mu_H)-(1+\mu_Z)y_1y_2](1+y_3+2\mu_Z)+ \nonumber \\
    &   & +y_1y_2[y_1y_2+1+\mu_Z^2+4\mu_H(1+\mu_Z)] +4\mu_H\mu_Z(1+
          \mu_Z+4\mu_H)+\mu_Z^2, \nonumber \\
f_3 & = & y_1(y_1-1)(\mu_Z-y_1)-y_2(y_1+1)(y_1+\mu_Z)+\nonumber \\
    &   &    +2\mu_Z(1+\mu_Z-4\mu_H).
\end{eqnarray}
As we have said we have left unspecified the various couplings
appearing in the differential cross-section. The reason is that
for any specified model,  all the couplings are generally
different with respect to the ones of the SM, although some of the
couplings, as for instance, $c_V$ and $c_A$ are strongly
constrained by the LEP and SLC data. In the next Section we will
consider a model for which the parameters can be chosen in such a
way to respect the experimental bounds, but nevertheless able to
produce strong deviations in the cross-section we are considering
here.
%SEZIONE II
\section{The  linear BESS model}\label{fatt_vertici}

The linear BESS model \cite{BESSlineare} is an extension of the
SM involving six more vector bosons, $V_L^i$ and $V_R^i$,
$i=1,2,3$, and two more scalar particles $\rho_L$ and $\rho_R$.
Furthermore it is characterized by a large mass scale, $u$, such
that for $u\to\infty$ all the new particles decouple from the SM
modes. Therefore, by appropriate choice of the scale, it is
possible to satisfy the experimental bounds  from low energy and
LEP/SLC experiments. The masses of the new particles are of order
$u$ (heavy particles), and the deviations from the SM can be
discussed in terms of the parameter expansion $v^2/u^2$ ($v^2=
1/\sqrt{2} G_F$). The new particles modify the double
Higgs-strahlung cross-section since they are  coupled to the
particles of the SM (see \cite{BESSlineare}). Therefore there are
many more diagram involved in this process than the ones coming
from the SM alone. A way of treat this problem is the  use of an
effective lagrangian describing the linear BESS model at energies
lower than the heavy scale $u$. This lagrangian  can be evaluated
solving the classical equations of motion for the heavy fields in
the low-energy limit. The effective lagrangian obtained in this
way has exactly the same structure of the SM lagrangian, except
that the couplings are modified by the heavy physics. We give
here  the results of this calculation concerning the couplings
which are relevant in the process we are interested in. These are
nothing but the couplings appearing in eqs. (\ref{2}) and
(\ref{3}). First we recall the expressions for the couplings in
the SM:

\begin{eqnarray}\label{vert_SM}
&\lambda_{HHH_{SM}}  =  \dd{\frac{1}{2}\sqrt{\sqrt{2}G_F}M^2_H},& \nonumber \\
&\lambda_{HZZ_{SM}}  = \dd{\frac{1}{4\sqrt{\sqrt{2}G_F}}\frac{e^2}
                        {s^2_{\theta_W}c^2_{\theta_W}}},~~~
\lambda_{HHZZ_{SM}} =    \dd{\frac{1}{8}\frac{e^2}
                        {s^2_{\theta_W}c^2_{\theta_W}}},&\nonumber \\
&c_{V_{SM}}  =  -\dd{\frac{1}{2}+2s^2_{\theta_W}},~~~ c_{A_{SM}}
= -\dd{\frac{1}{2}},&
\end{eqnarray}
where $\: s_{\theta_W} \:$ is defined by
\begin{equation}\label{COS_THETA}
s^2_{\theta_W}=\frac{1}{2}-\sqrt{\frac{1}{4}-\frac{\pi\alpha}
                  {\sqrt{2}G_FM^2_Z}}.
\end{equation}
Then we find:
\begin{eqnarray}\label{vert_BESS}
\bullet &\; HHH:& \lambda_{HHH_{SM}} \left(1-\Delta_{HHH}\epsilon
              \right), \nonumber \\
\bullet &\; HZZ:&  \lambda_{HZZ_{SM}}\left(1-
                    \Delta_{HZZ}\epsilon\right),\nonumber \\
\bullet &\; HHZZ:&
\lambda_{HHZZ_{SM}}\left(1-\Delta_{HHZZ}\epsilon
                      \right),\nonumber \\
\bullet &\; \Psi\overline{\Psi}Z:&
-\frac{e}{s_{\theta_W}c_{\theta_W}}
                         \gamma^\mu\frac{1}{2}[c_{V_{SM}} +\Delta c_V\epsilon -
                       (c_{A_{SM}} +\Delta c_A\epsilon)\gamma_5],
\end{eqnarray}
where $\epsilon = v^2/u^2$  is our expansion parameter and:
\begin{eqnarray}\label{delte}
\Delta_{HHH} & = & q^2+\frac{16q^3f}{\sqrt{2}G_FM^2_H},\nonumber \\
\Delta_{HZZ} & = &q^2
+\frac{e^4}{g^4_2}\frac{1-2c^2_{\theta_W}+2c^4_{
              \theta_W}}{s^4_{\theta_W}c^4_{\theta_W}}, \nonumber \\
\Delta_{HHZZ} & = &  4q^2 +\frac{e^4}{g^4_2}
       \frac{9c^4_{\theta_W}+s^4_{\theta_W}}{s^4_{\theta_W}c^4_{\theta_W}}
      +\frac{e^5}{g^5_2}\frac{8}{s_{\theta_W}c^4_{\theta_W}},\nonumber \\
\Delta c_V & = & \frac{1}{4}\frac{e^4}{g^4_2}
                 \frac{3-16c^2_{\theta_W}+18c^4_{\theta_W}-4c^6_{\theta_W}}
                      {s^4_{\theta_W}c^4_{\theta_W}(c^2_{\theta_W}-
                      s^2_{\theta_W})},\nonumber \\
\Delta c_A & = & \frac{1}{4}\frac{e^4}{g^4_2}\frac{s^4_{\theta_W}
+
           c^4_{\theta_W}}{s^4_{\theta_W}c^4_{\theta_W}},
\end{eqnarray}
where $M_H$ is the Higgs mass in the linear BESS model, $g_2$ is
the gauge coupling of the fields $V_L$ and $V_R$, whereas  $q$
and $f$ are two parameters appearing in the scalar sector of the
BESS model \cite{BESSlineare}. Notice that the parameters $q$ and
$f$ modify the $HHH$ coupling and also the couplings $HZZ$ and
$HHZZ$. This is quite important because it shows that fitting the
data to a theoretical form of the cross-section, where only the
trilinear coupling of the Higgs is left as a free parameter, might
not be the right thing to do. In fact this example shows the
possibility of correlations among  different Higgs couplings.

For a better comparison with  future data is better to use a set
of parameters  related to more practical  quantities. Therefore
we will trade the  parameters $M_H$, $g_2$, $q$, $f$ and
$\epsilon$ with the set: \be
M_H,~~M_{\rho_{L,R}},~~\frac{M_W}{M_V},~~\frac{g_{SM}}{g_2},~~q\ee
In particular: \be
\frac{M_W^2}{M_V^2}=\epsilon\frac{g_{SM}^2}{g_2^2},~~~f=\frac{q}{8v^2}
M^2_{\rho_{L,R}}\epsilon\ee $M_V$ is the mass scale of the new
vector bosons, whereas $M_{\rho_{L,R}}$ is the mass scale of the
new scalars. We will consider the expansion of the cross-section
around its SM value. For this we need to keep under control the
parameter $q$. Since in our numerical analysis we will take
$M_H\approx M_Z$ and $M_{\rho_{L,R}}\approx M_V$, the condition
$q$ must satisfy is
\begin{equation}\label{limite_teorico} q\lesssim
\frac{g_{SM}}{g_2}\sqrt{\frac{M_V}{M_{W^\pm}}}.
\end{equation}

%SEZIONE IV
\section{Numerical results}\label{differenza}

At  first order in the expansion parameter  $\epsilon=v^2/u^2$
the difference between the SM and BESS differential cross-section
is:
\begin{eqnarray}\label{DELTA}
&& \Delta\left(\frac{d\sigma}{dx_1dx_2}\right)  =
     \frac{\sqrt{2}G_F^3M_Z^6}{384\pi^3s(1-\mu_Z)^2}
     \frac{1}{G_F^2M_Z^8}\cdot \nonumber \\
& & \cdot\left\{2f_0\left[8\left(c_{V_{SM}} \Delta c_V+
     c_{A_{SM}}\Delta c_A\right)a^2
  -2aa_1\left((2c_{V_{SM}})^2+(2c_{A_{SM}})^2\right)\right] \right.+\nonumber\\
& & +\frac{1}{4\mu_Z(y_1+\mu_{HZ})}\left(\frac{f_1}{y_1+
          \mu_{HZ}}+\frac{f_2}{y_2+\mu_{HZ}}\right)
         \frac{\lambda_{HZZ_{SM}}^4}{2} \cdot\nonumber\\
& & \cdot\left[8\left(c_{V_{SM}} \Delta c_V+ c_{A_{SM}}\Delta
c_A\right)
      -4 \Delta\lambda_{HZZ}\left((2c_{V_{SM}})^2+(2c_{A_{SM}})^2\right)
     \right]+\nonumber \\
& & + \frac{f_3}{2(y_1+\mu_{HZ})}\lambda^2_{HZZ_{SM}}
\left.\left[8\left(c_{V_{SM}} \Delta c_V+c_{A_{SM}}\Delta
c_A\right)a
      -(a_1 + 2a\Delta\lambda_{HZZ})\left((2c_{V_{SM}})^2+\right.\right.
       \right.\nonumber\\
& &       \left.\left.\left.+(2c_{A_{SM}})^2\right)\right]\right\}
      \frac{M^2_{W^\pm}}{M^2_V}\frac{1}{(g_{SM}/g_2)^2},
\end{eqnarray}
where $a,f_i$ are defined  in Eq.~(\ref{def_a}),~(\ref{def_f_i}),
and $a_1$ is given by
\begin{eqnarray}\label{def_a_1}
a_1&=&
\frac{3\lambda_{HHH_{SM}}\lambda_{HZZ_{SM}}(\Delta\lambda_{HHH}+
    \Delta\lambda_{HZZ})}{y_3-\mu_{HZ}}+\frac{2\lambda_{HZZ_{SM}}^2
      \Delta\lambda_{HZZ} }
         {y_1+\mu_{HZ}}+\nonumber \\
& &+\frac{2\lambda_{HZZ_{SM}}^2 \Delta\lambda_{HZZ}}{y_2+\mu_{HZ}}
   +\lambda_{HHZZ_{SM}}\Delta\lambda_{HHZZ}s.
\end{eqnarray}
%tabella
\begin{table}[here]
\begin{center}
\begin{tabular}{|l|c|c|c|}
\hline
$M_H(GeV)$ & $120$ & $130$ & $140$ \\
\hline \hline
$N_{HHZ}$ & $80$ & $64$ & $44$ \\
Efficiency & $0.43$ & $0.43$ & $0.49$ \\
\hline
$\delta\sigma/\sigma$ & $\pm0.17$ & $\pm0.19$ & $\pm0.23$\\
\hline
\end{tabular}
\end{center}
\caption{\small{\textit{Number of events, efficiency and
cross-section uncertainty of  double Higgs-strahlung
production with an integrated luminosity of $1000fb^{-1}$
}\rm{\cite{TESLA}}.}} \label{tab:err_TESLA}
\end{table}
The    analysis made by TESLA \cite{TESLA} (see Table 1) shows
that the estimated uncertainty in measuring the total
cross-section for the double Higgs-strahlung  is around 20\%.
Therefore we will study the region in the parameter space defined
by the inequality
\begin{equation}\label{varianza}
\left|\frac{\Delta\sigma}{\sigma_{SM}}\right|=
\left|\frac{\sigma_{BESS}-\sigma_{SM}}{\sigma_{SM}}\right|\le 0.2.
\end{equation}
In order to simplify our study we will consider various choices
of the parameters. We start considering $(g_{SM}/g_2,M_V)$ (or
$(g_{SM}/g_2,M_W/M_V)$). These parameters are the ones accessible
at low-energy physics and the corresponding restrictions have
been studied in \cite{restrizione_par}. The results of this
analysis, which has been made using the data of ref.
\cite{exp_epsilon}, are given in Fig. 1, where the 95\% CL allowed
region is shown. For the aim of this paper we have selected seven
points ranging in the allowed region in a range of heavy vector
masses between 500 and 1800 $GeV$. Then we have considered two
different scenarios corresponding to two possible values of the
ratio of the scalar mass to the vector mass (but always of the
same order of magnitude) \be
 \frac{M_{\rho_{L,R}}}{M_V}  =  0.5,~~~~\frac{M_{\rho_{L,R}}}{M_V}  =  2.
\ee In Figs. 2, 3 and 4 we have fixed the mass of the Higgs to 140
$GeV$ and the center of mass energy to $\sqrt{s}=500~GeV$. The
only free parameter is now $q$ and we have plotted the expression
(\ref{varianza}) as a function of this variable for the various
points of Fig. 1. In Figs. 2 and 3 we have also plotted the line
corresponding to a variation of 20\%, which is assumed as the
uncertainty at the TESLA machine. In particular we see that the
sensitivity of the process considered to the parameter $q$
increases with  the ratio $M_{{\rho}_{L,R}}/M_V$. This is made
clear in Fig. 4 where the maximum value of $q$ compatible with the
SM at 20\% level is plotted against $M_{{\rho}_{L,R}}/M_V$.

In our numerical analysis we have  varied the TESLA energy from
500 $GeV$ to 1 $TeV$ but the results are practically insensitive
to this variation of the energy. We have also done some more
analysis fixing the ratio $M_{{\rho}_{L,R}}/M_V$ to one and
varying the Higgs mass from 130 $GeV$ to 180 $GeV$. The
corresponding results are given in Figs. 5 and 6. We
see that the sensitivity to $q$ decreases slightly increasing
$M_H$.

It is natural to ask if our results about the range of
compatibility of $q$ with respect to the SM is consistent with
the limitation given in eq. (\ref{limite_teorico}). In Table 2 we
compare, for each point given in Fig. 1, the limiting value
$q_{lim}$ from eq. (\ref{limite_teorico}) with the value
$q_{MAX}$. We see that we have always $q_{lim}>q_{MAX}$.

\begin{table}[here]
\begin{center}
\begin{tabular}{|c||c|r|c||c|}
\hline
  & $g_{SM}/g_2$ & $M_V(GeV)$ & $q_{lim}$ & $q_{MAX}$ \\
\hline \hline
A & $0.2$ & $500$ &$0.50$ & $0.20\div0.23$ \\
\hline
B & $0.3$ & $800$ & $0.95$ & $0.44\div0.53$\\
\hline
C & $0.4$ & $1100$ & $1.48$ & $0.73\div0.96$\\
\hline
D & $0.3$ & $1400$ & $1.25$ & $0.65\div0.91$\\
\hline
E & $0.6$ & $1500$ & $2.60$ & $1.34\div1.90$\\
\hline
F & $0.4$ & $1800$ & $1.90$ & $1.01\div1.50$\\
\hline
G & $0.8$ & $1800$ & $3.80$ & $1.99\div2.94$\\
\hline
\end{tabular}
\end{center}
\caption{\small{\textit{For each point chosen in the plane $(M_V,
g_{SM}/g_2)$ we compare the value $q_{lim}$ (see text) with the
maximal value of $q$ ($q_{MAX}$) compatible with the SM.}}}
\end{table}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=12cm]{PUNTI.eps}
\end{center}
\caption{\small{\textit{95\% CL allowed region in the plane
$(M_V,g_{SM}/g_2)$ from  low-energy and LEP/SLC data. We have
shown the points chosen for the present analysis.}}}
\label{fig:punti}
%\end{figure}
%\begin{figure}[htb]
\begin{center}
\includegraphics[width=12cm]{DELTA_05_500.eps}
\end{center}
\caption{\small{\textit{For $\:\sqrt{s}=500~GeV$,
$\:M_H=140~GeV\:$ and $\:M_{\rho_{L,R}}/M_V=0.5$, we plot
$\:\left|\Delta\sigma/\sigma_{SM}\right|\:$ as a function of  $q$
for each point chosen in the plane $\:(M_V,g_{SM}/g_2)$.}}}
\label{fig:0.5_500}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=12cm]{DELTA_2_500.eps}
\end{center}
\caption{\small{\textit{For $\sqrt{s}=500~GeV$, $\:M_H=140~GeV\:$
and $M_{\rho_{L,R}}/M_V=2$, we plot
$\left|\Delta\sigma/\sigma_{SM}\right|$ as a function of  $q$ for
each point chosen in the plane $\:(M_V,g_{SM}/g_2)$.}}}
\label{fig:2_500}
%\end{figure}
%\clearpage \hfill\break
%\begin{figure}[here]
\begin{center}
\includegraphics[width=10cm]{q_MAX_500_MR_MV.eps}
\end{center}
\caption{\small{\textit{ We plot the maximal value of the
parameter $q$ ($q_{MAX}$) compatible with the SM against
$M_{\rho_{L,R}}/M_V$. Here $\sqrt{s}=500~GeV$ and $M_H=140~GeV$.
The different curves correspond to the points chosen in the plane
$(M_V,g_{SM}/g_2)$.}}} \label{fig:q_MAX_500_MR_MV}
\end{figure}
\hfill\break
%figura
\begin{figure}[here]
\begin{center}
\includegraphics[width=12cm]{DELTA_130_500.eps}
\end{center}
\caption{\small{\textit{For $\sqrt{s}=500~GeV$, $M_H=130~GeV$ and
$M_{\rho_{L,R}}/M_V=1$, we plot
$\left|\Delta\sigma/\sigma_{SM}\right|$ as a function of  $q$ for
each point chosen in the plane $(M_V,g_{SM}/g_2)$.}}}
\label{fig:130_500}
%\end{figure}
%figura
%\begin{figure}[here]
\begin{center}
\includegraphics[width=12cm]{DELTA_180_500.eps}
\end{center}
\caption{\small{\textit{For $\sqrt{s}=500~GeV$, $M_H=180~GeV$ and
$M_{\rho_{L,R}}/M_V=1$, we plot
$\left|\Delta\sigma/\sigma_{SM}\right|$ as a function of  $q$ for
each point chosen in the plane $(M_V,g_{SM}/g_2)$.}}}
\label{fig:180_500}
\end{figure}
\section{Conclusions}


In this paper we have evaluated the cross-section for the double
Higgs-strahlung in the context of the linear BESS model. The main
property of this model is  decoupling.  This makes possible the
compatibility of the model with the actual experimental data.
Decoupling derives from the presence in the model of a heavy scale
$u$ (of the order or larger than some $TeV$) such that the
deviations (as far as the low-energy and LEP/SLC physics are
concerned) with respect to the SM are of the order $v^2/u^2$. The
model makes clear that new physics may affect the Higgs sector in
a way which is not under experimental control at low energy and,
more important, has its effects in all the Higgs couplings. That
is, not only the self-couplings, but also the couplings of the
Higgs to the $Z$. This, of course, creates correlations in the
amplitudes of processes like the one studied here, and it should
be taken in due care in all future analysis.
\newpage
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%9




\end{thebibliography}




\end{document}

