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\begin{document}
%\begin{titlepage}
\hfill PRA-HEP 97/15

\hfill October 2, 1997\\[8ex]
\begin{center}
{\Large\bf Vector Boson Scattering in the Standard Model -- \\[0.2cm]
                 an Overview of Formulae\\[3ex]}
Tom\'{a}\v{s} Bahn\'{\i}k\footnote{e-mail: tomas.bahnik@vslib.cz}\\[2ex]
{\em Department of Physics, Technical University Liberec,\\
H\'{a}lkova 6, 461 17 Liberec, Czech republic\\[2ex]}
and\\[2ex]
{\em Nuclear Centre, Faculty of Mathematics and Physics,\\
Charles University, V Hole\v{s}ovi\v{c}k\'{a}ch
2, 180 00 Prague 8, Czech republic\\[5ex]}
\end{center}
\begin{abstract}
Tree-level scattering amplitudes of longitudinally polarized electroweak
vector bosons in the Standard Model are calculated using {\em Mathematica}
package {\em Feyncalc}. The modifications of low-energy theorems for
longitudinally polarized $W$ and $Z$ in the Standard Model are
discussed.
\end{abstract}
%\end{titlepage}
%%%%%%%% Intro %%%%%%%%%%%%
\section{Introduction}
%
One of the open questions in high energy physics is the mechanism of
electroweak symmetry breaking (EWSB). The physics that breaks
electroweak symmetry is responsible for giving the $W$ and $Z$ their
masses. Since a massive spin-one particle has three polarizations,
rather than the two of a massless mode, the new physics must supply
degrees of freedom to be swallowed by the $W$ and $Z$. These new degrees
of freedom are the longitudinal polarizations $W_L$ and $Z_L$ of the
vector bosons. Therefore the interactions of the longitudinal components
of the vector bosons could provide a good way to probe the interactions
of the symmetry breaking sector \cite{gaillard}.

The interactions of $W_L$ and $Z_L$ in high energy processes are usually
studied by means of different production mechanisms of vector bosons
followed by their purely leptonic decays  {\em e.g.} $W^{\pm} \to
l^{\pm}\nu$ and $Z\to l^+l^- (l=e,\mu)$, referred to as gold-plated
channels. One of the production  mechanism is through light fermion
anti-fermion {\em i.e.} $q\bar{q}$ or $e^+ e^-$ annihilation. This
yields vector boson pairs that are mostly transversely polarized and is
usually a background to the other processes. The important exception is
the production of longitudinally polarized vector bosons through new
vector resonance.

A second mechanism for producing longitudinal vector boson pairs in
hadron colliders is gluon fusion. The initial gluons turn into two
vector bosons via an intermediate state that couples to both gluons and
electroweak vector bosons like the top quark or new colored particles of
a technicolor model \cite{jbagger}.

Finally, there is the vector-boson fusion process when vector bosons are
radiated by colliding fermions and then rescattered. When the fermions
are quarks then the process of vector boson scattering is considered as
a subprocess of subprocess in hadron collision. Sensitivity of different
types of colliders to the above mentioned processes has been discussed
in a series of articles \cite{SISBS}.

In this paper I calculate exact tree-level scattering amplitudes of
longitudinally polarized electroweak vector bosons in the Standard Model.
The aim of the present paper is to check independently the existing
results \cite{dutta,bento,barger,duncan}, in particular because I think
there has been a minor error in an earlier paper \cite{bento}. As a
consistency test I use the high-energy ($E\gg m_W$) behaviour of a
pure gauge amplitudes. Their quadratic growth ($\sim E^2$)
should be canceled by introducing Standard Model Higgs boson.

It has been shown that some universal low-energy theorems (LET) for
the scattering of longitudinally polarized $W$ and $Z$ hold \cite{LET}.
These theorems are valid below the scale $\Lambda_{SB} \sim\,$1\,TeV,
provided that the symmetry breaking sector contains no particles much
lighter than $\Lambda_{SB}$. The derivation in \cite{LET} shows that in
this case the LET are given by $SU(2)_L \times U(1)_Y$ gauge interactions
of vector bosons alone. The particle which modifies pure gauge
amplitudes (and LET) in the SM is the Higgs boson. To see this
modifications, I plot the complete amplitudes for different values of Higgs
boson mass and compare them with the pure gauge contributions.

%%%%%%%%%%%%%% Scat ampl %%%%%%%%%%%%%%%%%
\section{Scattering Amplitudes}
%
Calculation of the scattering amplitudes of the gauge bosons by hand is
a tedious task. The use of function {\tt SpecificPolarization} of the
{\em Mathematica} package {\em FeynCalc} \cite{mertig} has substantially
reduced algebraical manipulations.

Longitudinal polarization picks up a specific direction in space.
Thus the amplitudes written in terms of Mandelstam variables $s,t,u$
do not describe scattering of {\em longitudinally}  polarized
particles in {\em all} Lorentz frames. For the process
$V(p_1) + V(p_2)\to V(k_1) + V(k_2)$ the function
{\tt SpecificPolarization} uses the following representation of the
longitudinal polarization vectors $\varepsilon (p,L)$
\ber
   \varepsilon^{\mu}(p_1,L) &=& F^{\mu }_{L}(p_1,\,k_1,k_2)\nn\\
   \varepsilon^{\mu}(p_2,L) &=& F^{\mu }_{L}(p_2,\,k_1,k_2)\nn\\
   \varepsilon^{\mu}(k_1,L) &=& F^{\mu }_{L}(k_1,\,p_1,p_2)\nn\\
   \varepsilon^{\mu}(k_2,L) &=& F^{\mu }_{L}(k_2,\,p_1,p_2)
\label{longpolvec}
\eer
where
\be
\nn F_{L}^{\mu }(r,\,a,b) = \frac{ r^{\mu}(b\cdot r+a\cdot r) -
   (a+b)^{\mu }r^{2}}{\sqrt{r^{2}[(b\cdot r+a\cdot r)^{2}-
   r^{2}(b+a)^{2}]}}
\ee
Using this function, we get the longitudinal polarization only in
the CM system, where $\bp_1 + \bp_2 = \bk_1 + \bk_2 = 0$.

Table\,\ref{tab:indproc} summarizes tree-level contributions (contact
graphs are not listed) to the scattering amplitudes of the gauge bosons
in the SM in the $U$-gauge.
%%%%%%%%%%%%% TABULKA %%%%%%%%%%%%%%%%
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}\hline
process \# & process & $s$ & $t$ & $u$ & references \\ \hline\hline
\rule[-3mm]{0cm}{8mm} 1 & $W^+_1 W^-_2 \to Z_3 Z_4$ & $H$ & $W$ & $W$ &
\cite{dutta,bento}\\ \hline
\rule[-3mm]{0cm}{8mm} 2 & $W^+_1 Z_2 \to Z_3 W^+_4$ & $W$ & $W$ & $H$ &
\cite{bento}
  \\ \hline
\rule[-3mm]{0cm}{8mm} 3 & $W^+_1 W^+_2 \to W^+_3 W^+_4$ &  &
$Z, \gamma, H$ & $Z, \gamma, H$ & \cite{barger} \\ \hline
\rule[-3mm]{0cm}{8mm} 4 & $W^+_1 W^-_2 \to W^+_3 W^-_4$ & $Z, \gamma, H$
& $Z, \gamma, H$ & & \cite{duncan} \\ \hline
\end{tabular}
\caption{\small Individual processes together with exchanged particles in
different channels ($s, t, u$). $W, Z, \gamma$ are electroweak gauge bosons and
photon and $H$ is the SM Higgs boson. Contact graphs contribute to
each of the process and are not listed.
\label{tab:indproc}}
\end{center}
\end{table}
Contributions to the process \#$n$ from graph with  $V$-boson exchange in the
$k$-channel  are denoted as
\be \M^{(n)}_{kV} = -\frac{g_1 g_2}{k - m_V^2} A^{(n)}_{k} \label{mnkv} \ee
and from the contact graph
\be \M^{(n)}_c = g\, A^{(n)}_c  \label{mnc} \ee
with $g_1, g_2$ and $g$ denoting relevant coupling constants. Amplitudes
given by the low-energy theorems are denoted as $\M^{(n)}_{LET}$. Note
that in the paper of Bento and Llewellyn Smith \cite{bento} the $WZ \to
WZ$ amplitude is claimed to be (in the current notation) \[\M^{(2)} =
\M^{(2)}_c+\M^{(2)}_{uH}+\M^{(2)}_{uW}+\M^{(2)}_{tW}\qquad\mbox{(wrong)}\]
and is also calculated in this way. This is wrong, because $W$ is
exchanged in $s$ and $t$ (or $u$) channel\footnote{There is the
interchange $t \leftrightarrow u$ in \cite{bento} in comparison with
this paper.}.

Explicit formulae for the scattering amplitudes are somewhat difficult to
read and are postponed to the appendix\,\ref{exactformulae}.
Table\,\ref{tab:relA} summarizes
relations among different $A$ parts of gauge amplitudes as defined in
(\ref{mnkv}) and (\ref{mnc}). These relation are almost obvious, nevertheless
they can be used as a check.
%%%%%%%%%%%%% TABULKA2 %%%%%%%%%%%%%%%%
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|c|}\hline
process \# & Gauge boson exchange & Contact graph  \\ \hline\hline
\rule[-3mm]{0cm}{8mm} 1 & $A^{(1)}_{tW} (\cos{\theta_{cm}}) =
A^{(1)}_{uW} (- \cos{\theta_{cm}})$
 & $A^{(1)}_c $  \\ \hline
\rule[-3mm]{0cm}{8mm} 2 & $A^{(2)}_{sW},\ A^{(2)}_{tW}$
  &$A^{(2)}_c$  \\ \hline
\rule[-3mm]{0cm}{8mm} 3 & $A^{(3)}_{(t,u)(\gamma,Z)} = -
A^{(1)}_{(t,u)W}\big|_{m_Z=m_W}$ & $A^{(3)}_c =
A^{(1)}_c\big|_{m_Z=m_W}$  \\ \hline
\rule[-3mm]{0cm}{8mm} 4 & $A^{(4)}_{(s,t)(\gamma,Z)} = -
A^{(2)}_{(s,t)W}\big|_{m_Z=m_W}$ & $A^{(4)}_c = A^{(2)}_c\big|_{m_Z=m_W}$
 \\ \hline
\end{tabular}
\caption{\small Relations among the $A$ parts of the scattering
amplitudes.
\label{tab:relA}}
\end{center}
\end{table}
Besides these relations we have  $A^{(1)}_{tW}\big|_{m_Z=m_W} = -
A^{(2)}_{tW}\big|_{m_Z=m_W}$ but $A^{(2)}_{tW} \neq - A^{(1)}_{tW}$.
Note that the $A^{(n)}_{\gamma} = A^{(n)}_{Z}$ because the longitudinal
term in propagator ($\sim q^{\alpha} q^{\beta}$) does not contribute to
the amplitudes with exchange of neutral gauge boson. 

%*************** WWZZ *******************
\subsection{$W^+(k_1) +  W^-(k_2) \to Z(k_3)+ Z(k_4)$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
On the tree-level the pure gauge contributions are from $W$-exchange in $t$ and $u$
channels and the contact graph
\be  \M^{(1)}_{kW}= \frac{-  g^2\cos^2{\theta_W}}{k - m_W^2}
  A^{(1)}_{kW}\qquad k=t,u  \ee
\be  \M^{(1)}_c= -  g^2\cos^2{\theta_W} A^{(1)}_c \ee
Complete gauge amplitude grows linearly with $s$
\ber \M^{(1)}_{gauge} &=&
\M^{(1)}_{tW} + \M^{(1)}_{uW} + \M^{(1)}_c \nn\\
%&=& g^2 \cos^2{\theta_W} \left
%[\frac{m_Z^2}{4 m_W^4} s + \frac{m_Z^2 (2 m_W^2 (1+x^2)
%-m_Z^2(1-x^2))}{2 m_W^4 (1-x^2)} +  O\left(\frac{m_W^2}{s}\right) \right
%]\nn\\
&=& \M^{(1)}_{LET} - g^2\frac{(1-x^2) - 2 \cos^2{\theta_W} \rho (1+x^2)}
{2 \rho^2 \cos^2{\theta_W}(1-x^2)} +  O\left(\frac{m_W^2}{s}\right)
\label{m1gauge}
\eer
where
\[\M^{(1)}_{LET} = \frac{g^2\,s}{4\,\rho m_W^2}, \quad \rho =
\frac{m_W^2}{m_Z^2 \cos^2{\theta_W}} \quad\mbox{and}\quad\ x \equiv
\cos{\theta_{cm}}.\]
with $\theta_{cm}$ an angle between $\bk_1$ and $\bk_3$ in  CMS.

In the SM this growth is canceled by exchange of the Higgs boson in
the $s$-channel
\be \M^{(1)}_{sH} = - \frac{g^2 m_W m_Z}{\cos{\theta_W}}
    \frac{(\ep_1\cdot\ep_2) (\ep^*_3\cdot\ep^*_4)}{s - m_H^2 + i m_H\Gamma_H}
\label{m1sh}
\ee
The amplitude $\M^{(1)}_{sH}$ has, for longitudinal polarizations, the
high-energy  ($E\gg m_H, m_W$) expansion (for exact formulae see appendix)
\[\M^{(1)}_{sH} = - \frac{g^2 m_W m_Z}{\cos{\theta_W}} \left [ \frac{s}{4 m_Z^2 m_W^2}
+ O(s^0) \right ] = -\frac{g^2\sqrt{\rho}}{4 m_W^2}\,s + O(s^0)\]
so the cancellation occurs for $\rho = 1$.

Figure\,\ref{figwwzz} shows dependence of the complete amplitude $\M^{(1)} =
\M^{(1)}_{gauge} + \M^{(1)}_{sH}$ on the Higgs boson mass, $m_H$, in the
region $m_W < \sqrt{s} < 1\,{\rm TeV}$ and compares it with
$\M^{(1)}_{gauge}$ and $\M^{(1)}_{LET}$.
Note that $\M^{(1)}_{gauge}$ differs from $\M^{(1)}_{LET}$ in the limit $s
\to \infty$ by a constant term. Numerical values of all parameters are
the same throughout the paper.

%%%%%%%%%% fig 1 %%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
\includegraphics[height=8cm,width=10cm]{figure1.eps}
\end{center}
\caption{\small Tree-level amplitudes of the process $WW\to ZZ$
as a function of $\sqrt{s}$ for different values of $m_H$. The Higgs
boson width, $\Gamma_H$, is set to zero, $\cos{\theta_{cm}} = 0.5$,
$\rho=1$, $m_Z=91.2$\,GeV, $\sin^2{\theta_W} = 0.231$,
$g^2=\frac{4\pi\alpha_{em}}{\sin^2{\theta_W}}\doteq 0.42$.
\label{figwwzz}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%************** WZ->WZ *************
\subsection{$W^+(k_1)+ Z(k_2) \to Z(k_3) + W^+(k_4)$}
%*****************************************
%%%%%%%%%% fig 2 %%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
\includegraphics[height=8cm,width=10cm]{figure2.eps}
\end{center}
\caption{\small Tree-level amplitudes of
the process $WZ\to WZ$.
\label{figwzwz}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
On the tree-level the pure gauge contributions are from $W$-exchange in
$s$ and $t$
channels and the contact graph
\be  \M^{(2)}_{kW}= \frac{-  g^2\cos^2{\theta_W}}{k - m_W^2}
A^{(2)}_{kW} \quad k = s, t \ee
\be  \M^{(2)}_c= -  g^2\cos^2{\theta_W} A^{(2)}_c \ee
The asymptotic behaviour of the complete tree-level gauge amplitude is
\ber
\M^{(2)}_{gauge} &=& \M^{(2)}_{sW} + \M^{(2)}_{tW} + \M^{(2)}_c \nn\\
&=& \M^{(2)}_{LET} -\frac{g^2 (2x(1-x) +
\rho\cos^2{\theta_W}(3+2x-x^2))}{4 \rho^2\cos^2{\theta_W}
(1-x)} + O\left(\frac{m_W^2}{s}\right)
%-\frac{g^2 s}{8\rho m_W^2}\,(1+\cos{\theta_{cm}})
%- g^2\frac{2 m_W^2 (1+x)-m_Z^2 (1-x)^2}{4 \rho m_W^2 (1-x)}+
%O\left(\frac{m_W^2}{s}\right)
\label{m2gauge}
\eer
where a low-energy amplitude is usually defined as
\[\M^{(2)}_{LET} =  \frac{g^2 u}{4\rho m_W^2}=
-\frac{g^2 s}{8\rho m_W^2}\,(1+\cos{\theta_{cm}})+O(s^0)\ .
\]
The exchange of the Higgs boson in the $u$-channel
\be
\M^{(2)}_{uH} = - g^2 \frac{m_W m_Z}{\cos{\theta_W}}
\frac{(\ep_1\cdot\ep^*_4)\,(\ep_2\cdot\ep^*_3)}{u - M^2_H}
\label{m2uh}
\ee
has for longitudinal polarizations the high-energy expansion
\[ \M^{(2)}_{uH} =
%- g^2 \frac{m_W m_Z}{\cos{\theta_W}}
%\left[-\frac{(1+\cos{\theta_{cm}})}{8 m_Z^2 m_W^2}\,s + O(s^0)\right] =
\frac{g^2 \sqrt{\rho}}{8 m_W^2} \,(1+\cos{\theta_{cm}})\,s +O(s^0)=
- \frac{g^2 \sqrt{\rho}}{4 m_W^2}\,u + O(s^0)\ .
\]
Figure\,\ref{figwzwz} shows complete tree-level amplitude $\M^{(2)}$
for different $m_H$ and compares it with $\M^{(2)}_{gauge}$ and
$\M^{(2)}_{LET}$.

%***************** WWWW ***************
\subsection{$W^+(k_1) + W^+(k_2) \to W^+(k_3) + W^+(k_4)$}
%************************************************************
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
\includegraphics[height=8cm,width=10cm]{figure3.eps}
\end{center}
\caption{\small Tree-level amplitudes of the process $W^+W^+\to
W^+W^+$.
\label{figwpwp}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
On the tree-level graphs with $\gamma$ and $Z$ exchange in $t$ and $u$
channels and contact graph contribute. In the Standard Model,
Higgs boson exchange in $t$ and $u$
channels gives the desired high energy behaviour.
\[\M^{(3)}_{kZ}= \frac{-g^2\cos^2{\theta_W}}{k - m_Z^2}
A^{(3)}_{kZ}\qquad
\M^{(3)}_{k\gamma}= \frac{-e^2}{k} A^{(3)}_{k\gamma} \qquad k = t,u\ .\]
Because longitudinal term in the $Z$ boson propagator does not
contribute we have
\[A^{(3)}_{k\gamma} = A^{(3)}_{kZ} \ .\]
Contact graph amplitude for longitudinally polarized gauge bosons has
the form
\be \M^{(3)}_c = g^2 A^{(3)}_c = \frac{g^2 s}{8\,m_W^4}\,
(-8\,m_W^2 + 3\,s - s\,\cos^2{\theta_{cm}})
\ee
The gauge amplitude can be written as
\be \M^{(3)}_{gauge} = - g^2\cos^2{\theta_W}\left[\frac{A^{(3)}_{tZ}}{t -
m_Z^2} + \frac{A^{(3)}_{uZ}}{u - m_Z^2}\right] - g^2\sin^2{\theta_W}
\left[\frac{A^{(3)}_{t\gamma}}{t} + \frac{A^{(3)}_{u\gamma}}{u}\right] +
\M^{(3)}_c
\label{m3gauge}
\ee
Expanding this expression in powers of $s$ gives
\ber \M^{(3)}_{gauge} = &-& g^2\cos^2{\theta_W}\left[\frac{3-\cos^2{\theta_{cm}}}{8 m_W^4}\,s^2
- \frac{3 m_Z^2}{4 m_W^4}\,s + O(s^0)\right]\nn \\
&-&
g^2\sin^2{\theta_W}\,\left[\frac{3-\cos^2{\theta_{cm}}}{8 m_W^4}\,s^2 +O(s^0)\right] \nn \\
&+& g^2 \left[\frac{3-\cos^2{\theta_{cm}}}{8 m_W^4}\,s^2 - \frac{s}{m_W^2}\right]
\eer
Quadratic (in $s$) divergencies are canceled and including constant
terms of the order $O(s^0)$ we get
%\be
%\M^{(3)}_{gauge} = \M^{(3)}_{LET}-g^2 \frac{[4m_W^2 (1+3x^2) - \frac{3}{\rho}(2m_W^2 -m_Z^2) -
%\frac{x^2}{\rho}(10 m_W^2 - m_Z^2)]}{2m_W^2 (1 -x^2)} +
%O\left(\frac{m_W^2}{s}\right)
%\ee
\be
\M^{(3)}_{gauge} = \M^{(3)}_{LET}- g^2\,\frac{
\left( 3 + {x^2} - \rho\,\cos^2{\theta_W} (6 -
       4\,{\rho} + 10{x^2} - 12\,{\rho}\,{x^2}\right)}
   {2\,{\rho^2}\,\left( 1 - {x^2} \right) \,{\cos^2{\theta_W}^2}} +
   O\left(\frac{m_W^2}{s}\right)
\label{asym3G}
\ee
where
\be
\M^{(3)}_{LET} = - \frac{g^2\,s}{4 m_W^2}\left(4 -
\frac{3}{\rho}\right) \ .
\label{let3}
\ee
This linear divergence should be canceled by Higgs exchange in $t$ and
$u$ channels
\be
\M^{(3)}_{H} = - g^2 m_W^2 \left[\frac{(\ep_1\cdot\ep^*_3)
(\ep_2\cdot\ep^*_4)}{t-m_H^2} +\frac{(\ep_1\cdot\ep^*_4)
(\ep_2\cdot\ep^*_3)}{u - m_H^2}\right]
\label{m3h}
\ee
%For longitudinal polarization
%\[\M^{(3)}_{H} = - \frac{g^2}{8 m_W^2}\left[\frac{(4 m_W^2 - s + s \cos{\theta_{cm}})^2}
%{(4 m_W^2 - 2 m_H^2 - s - 4 m_W^2 \cos{\theta_{cm}} + s \cos{\theta_{cm}})} + (\cos{\theta_{cm}}\to -\cos{\theta_{cm}}) \right] \ .\]
with high-energy expansion
\be \M^{(3)}_{H} = \frac{g^2\,s}{4 m_W^2} + O(s^0) \label{asym3H} \ .\ee
Comparing (\ref{let3}) and (\ref{asym3H}) we see that $\rho = 1$
ensures desired cancellation.


%%%%% WPWM %%%%%%%%%%%%%%%%%%%%%
\subsection{$W^+(k_1)+ W^-(k_2) \to W^+(k_3)+ W^-(k_4)$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
\includegraphics[height=8cm,width=10cm]{figure4.eps}
\end{center}
\caption{\small Tree-level amplitudes of the process $W^+W^-\to
W^+W^-$.
\label{figwpwm}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this case we have to consider  $Z$, $\gamma$ and Higgs boson exchange
in  $s$ and $t$ channels and contact graph.
As in the previous sections we write
\[\M^{(4)}_{kZ}= \frac{-g^2\cos^2{\theta_W}}{k - m_Z^2}
A^{(4)}_{kZ} \qquad \M^{(4)}_{k\gamma}= \frac{-e^2}{k}
A^{(4)}_{k\gamma}\qquad k= s,t\]
The results for the $A$
parts of the amplitude can be obtained directly from corresponding
formulae for the process $W^+ Z \to W ^+ Z$ by setting $m_Z = m_W$
\[A^{(4)}_{(s,t)(Z,\gamma)} = - A^{(2)}_{(s,t)W}\big|_{m_Z = m_W}\]
or we can also notice that $A^{(4)}_{tZ} = - A^{(3)}_{tZ}$. Again we
have $A^{(4)}_{kZ} =  A^{(4)}_{k\gamma}$.
%\[ i M_{sZ} = -i g^2 \cos^2{\theta_w} \left (
%V_{\mu\nu\alpha}(-k_1,-k_2,q) V_{\sigma\rho\beta}(k_4,k_3,-q)
%D^{\alpha\beta}(q)\varepsilon^{\mu}_1\,
% \varepsilon^{\nu}_2\, \varepsilon^{\ast\rho}_3\, \varepsilon^{\ast\sigma}_4
% \right ) \]
%\[ q=k_1 + k_2 = k_3 + k_4 \]
%For longitudinal polarizations we have
%\[A^{(4)}_{sZ} = - 4m_W x - 3 s x + \frac{s^3 x}{4 m_W^4}\]
%The general amplitude for contact graph
%\[\M^{(4)}_c = g^2 V_{\mu\sigma\nu\rho} \ep^{\mu}(1) \ep^{\nu}(2) \ep^{\rho}(3)
%\ep^{\sigma}(4)\]
In the case of longitudinally polarized gauge bosons
\be \M^{(4)}_c = g^2 A^{(4)}_c = \frac{g^2 s}{16\,m_W^4}\,(8\,m_W^2 - 3\,s - 24\,m_W^2\,x
+ 6\,s\,x + s\,x^2)
\ee
Let us examine high-energy expansion of gauge amplitude
\be \M^{(4)}_{gauge} = -g^2\cos^2{\theta_W}\left[\frac{A^{(4)}_{sZ}}{s-m_Z^2}
+ \frac{A^{(4)}_{tZ}}{t-m_Z^2}\right] -
g^2\sin^2{\theta_W}\left[\frac{A^{(4)}_{s\gamma}}{s} +
\frac{A^{(4)}_{t\gamma}}{t}\right] + \M^{(4)}_c \ .
\label{m4gauge}
\ee

\ber \M^{(4)}_{gauge} = - &g^2 \cos^2{\theta_W}&\left(\frac{x}{4
m_W^4}s^2 + \frac{m_Z^2 x}{4 m_W^4} s + \frac{x^2 +2x -3}{16 m_W^4} s^2 +
\frac{3 m_Z^2 -16 m_W^2 x + m_Z^2 x}{8 m_W^4} s\right)\nn\\
- &g^2 \sin^2{\theta_W}&\left(\frac{x}{4m_W^4}s^2 + \frac{x^2 +2x -3}{16
m_W^4} s^2 - \frac{2 x}{m_W^2} s\right) + O(s^0) \nn\\
+ &g^2& \left(\frac{x^2 + 6x -3}{16 m_W^4} s^2 +\frac{1 - 3x}{2 m_W^2}s
\right)
\eer
After simplification and including terms of order $O(s^0)$ we get
%\ber \M^{(4)}_{gauge} &=& \frac{g^2 s (1+\cos{\theta_{cm}})}{8
%m_W^2}\left(4-\frac{3}{\rho}\right)\nn\\
%&+& g^2\frac{4m_W^2 (1+4x-x^2) -
%\frac{3}{\rho}(2m_W^2 -m_Z^2 +4m_W^2 x) +\frac{x^2}{\rho}(2m_W^2
%+m_Z^2)}{4m_W^2(1-x)}\nn\\
%&+&O\left(\frac{m_W^2}{s}\right)
%\eer
\ber \M^{(4)}_{gauge} &=& \M^{(4)}_{LET}\nn\\ &+&
g^2 \frac{(3 + x^2 - \rho\,\cos^2{\theta_W}(12 -
        12\,\rho + 12\,x - 16\,\rho\,x - 8\,x^2 +
        12\,\rho\,x^2))}
        {4\,\rho^2\,( 1 - x)\,\cos^2{\theta_W}}\nn\\&+&
        O\left(\frac{m_W^2}{s}\right)\nn
\eer
where as in the case of the process \#1 I denote
\[\M^{(4)}_{LET} = -\frac{g^2 u}{4m_W^2}\left(4-\frac{3}{\rho}\right)\]
in accordance with \cite{LET}.
High-energy expansion of the Higgs boson contribution
\be
\M^{(4)}_{H} = -g^2 M^2_W\left[\frac{(\ep_1\cdot\ep_2) (\ep_3^*\cdot\ep_4^*)}
{s-m_H^2} + \frac{(\ep_1\cdot\ep_3^*) (\ep_2^*\cdot\ep_4^*)}{t-m_H^2}\right]
\label{m4h}
\ee
is
\[\M^{(4)}_{H} =
%-g^2 M^2_W\left[\frac{s}{4 m_W^4} -
%\frac{(1 - \cos{\theta_{cm}}) s}{8 m_W^4} + O(s^0) \right]
 - \frac{g^2 s}{8 m_W^2} (1 + \cos{\theta_{cm}}) + O(s^0) =
\frac{g^2 u}{4 m_W^2} + O(s^0)
\]
%%%%%%%%% Exact formulae %%%%%%%%%%%%%%%%%
\begin{appendix}
\section{Exact formulae \label{exactformulae}}
The following Feynman rules and notation is used (all momenta are
outgoing) \cite{horejsi}
\\[1cm]
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$ig m_W g_{\mu\nu}$\\[10mm]
%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$ig\frac{m_Z}{\cos{\theta_W}} g_{\mu\nu}$\\[1cm]
where
\be V_{\lambda\mu\nu}(k,p,q) = (k-p)_{\nu} \,g_{\lambda\mu}
+ (p-q)_{\lambda}\, g_{\mu\nu} + (q-k)_{\mu}\,
g_{\lambda\nu},\quad k+p+q=0
\label{3v}
\ee
\[V_{\mu\nu\rho\sigma} = 2 g_{\mu\nu} g_{\rho\sigma} -
g_{\mu\rho} g_{\nu\sigma} - g_{\nu\rho} g_{\mu\sigma} \ .\]
%The corresponding coupling constants are
%\[g_{WW\gamma} = e,\ g_{WWZ} = g\cos{\theta_W},\ g_{WWWW} = g^2,
%\ g_{WW\gamma\gamma}= - e^2,\ g_{WWZZ}= - g^2\cos^2{\theta_W} \]
Propagators of the gauge  bosons are
%\cite{horejsi}
\[D^{\alpha\beta}_V(q) = \frac{-g^{\alpha\beta} + \frac{q^{\alpha}
q^{\beta}}{m_V^2}}{q^2 -m_V^2} \equiv \frac{P^{\alpha\beta}_V(q)}{q^2
-m_V^2}, \quad D^{\alpha\beta}_{\gamma} = \frac{-g^{\alpha\beta}}{q^2}
\equiv \frac{P^{\alpha\beta}_{\gamma}}{q^2} \ .\]
Mandelstam variables are defined as
\ber
s &=& (k_1 + k_2)^2 = (k_3 + k_4)^2 \nn\\
t &=& (k_1 -k_3)^2 = (k_4 -k_2)^2 \nn\\
u &=& (k_1 -k_4)^2 = (k_3 -k_2)^2 \nn
\eer
The expression containing polarization vectors has in all
amplitudes the form
\[
{\cal E}^{\mu\nu\rho\sigma} =
\varepsilon^{\mu}_1\,\varepsilon^{\nu}_2\,
 \varepsilon^{*\rho}_3\,\varepsilon^{*\sigma}_4
%\label{polvek}
\]
where $\ep_i = \ep(k_i)$. I use the abbreviation $x =
\cos{\theta_{cm}}$, where $\theta_{cm}$ is the angle between $\bk_1$ a
$\bk_3$ in  CMS.

%%%%%%%%%%%%%%%% WWZZ %%%%%%%%%%%%%%%%
\subsection{$W^+(k_1) + W^-(k_2) \to Z(k_3) + Z(k_4)$}
%
\[\M^{(1)} = - g^2\cos^2{\theta_W}\left[\frac{A^{(1)}_{tW}}{t -
m_W^2} + \frac{A^{(1)}_{uW}}{u - m_W^2}  + A^{(1)}_c\right] +
\M^{(1)}_{sH}
\]
\[A^{(1)}_{tW}=
 V_{\mu\alpha\rho}(-k_1,q,k_3)\,V_{\beta\nu\sigma}(-q,-k_2,k_4)
 P^{\alpha\beta}_W(q){\cal E}^{\mu\nu\rho\sigma}
%\label{wwzzt}
\]
\[A^{(1)}_{uW}=
 V_{\mu\alpha\sigma}(-k_1,q,k_4)\,V_{\beta\nu\rho}(-q,-k_2,k_3)
P^{\alpha\beta}_W(q){\cal E}^{\mu\nu\rho\sigma}
%\label{wwzzu}
\]
\[A^{(1)}_c = V_{\rho\sigma\mu\nu}{\cal E}^{\mu\nu\rho\sigma}\]
After contraction
\ber
A^{(1)}_{tW} &=&
      4\,(k_1\cdot\varepsilon_3^*)\,(k_2\cdot\varepsilon_4^*)\,
      (\varepsilon_1\cdot\varepsilon_2) +
      4\,(k_1\cdot\varepsilon_3^*)\,(k_4\cdot\varepsilon_2)\,
      (\varepsilon_1\cdot\varepsilon_4^*)\nn\\
&+&   4\,(k_2\cdot\varepsilon_4^*)\,(k_3\cdot\varepsilon_1)\,
      (\varepsilon_2\cdot\varepsilon_3^*)+
      4\,(k_3\cdot\varepsilon_1)\,(k_4\cdot\varepsilon_2)\,
      (\varepsilon_3^*\cdot\varepsilon_4^*) \nn\\
&-&  2\,(k_4\cdot\varepsilon_2)\,
     ((k_1 + k_3)\cdot \varepsilon_4^*)\,(\varepsilon_1\cdot\varepsilon_3^*)
     - 2\,(k_2\cdot\varepsilon_4^*)\,
     \left((k_1 + k_3)\cdot\varepsilon_2\right) \,
     (\varepsilon_1\cdot\varepsilon_3^*) \nn\\
&-&   2\,(k_1\cdot\varepsilon_3^*)\,((k_2 +k_4)\cdot\varepsilon_1)\,
     (\varepsilon_2\cdot\varepsilon_4^*)- 2\,(k_3\cdot\varepsilon_1)\,
     \left((k_2 +k_4)\cdot\varepsilon_3^*\right) \,
     (\varepsilon_2\cdot\varepsilon_4^*)\nn\\
&+&  \frac{(m_W^2 - m_Z^2)^2}{m_W^2}\,(\varepsilon_1\cdot\varepsilon_3^*)\,
     (\varepsilon_2\cdot\varepsilon_4^*)+
     (s - u) \,(\varepsilon_1\cdot\varepsilon^*_3)\,
     (\varepsilon_2\cdot\varepsilon_4^*).
%\label{a1two}
\eer
\[ A^{(1)}_{uW} = A^{(1)}_{tW} (3\leftrightarrow 4, u\to t) \]
\[ A^{(1)}_c = 2\,(\varepsilon_1\cdot
\varepsilon_2)\,(\varepsilon^*_3\cdot \varepsilon^*_4) -
(\varepsilon_1\cdot
\varepsilon^*_4)\,(\varepsilon_2\cdot\varepsilon^*_3) -
(\varepsilon_1\cdot \varepsilon^*_3)\,(\varepsilon_2\cdot
\varepsilon^*_4)
\]
Replacing all polarization vectors $\varepsilon_i$ by $\varepsilon_i(L)$
given in (\ref{longpolvec}) we get amplitudes for longitudinally
polarized gauge bosons in CMS 
\ber A^{(1)}_{tW}(s,x)&=&
\frac{1}{32\,m_W^4\,m_Z^2}\,[-96\,m_W^4\,m_Z^4 +
    32\,m_W^2\,m_Z^6 + 8\,m_W^4\,m_Z^2\,s + 16\,m_W^2\,m_Z^4\,s\nn\\
&-& 8\,m_Z^6\,s - 4\,m_W^4\,s^2 - 10\,m_W^2\,m_Z^2\,s^2 + 2\,m_Z^4\,s^2 +
    3\,m_W^2\,s^3\nn\\
&+& 16\,m_W^4\,m_Z^2\,\beta_W\,\beta_Z\,s\,x
    + 12\,m_W^4\,\beta_W\,\beta_Z\,s^2\,x
    +24\,m_W^2\,m_Z^2\,\beta_W\,\beta_Z\,s^2\,x\nn\\
&-& 4\,m_Z^4\,\beta_W\,\beta_Z\,s^2\,x - 5\,m_W^2\,\beta_W\,\beta_Z\,s^3\,x
    + 32\,m_W^6\,s\,x^2\nn\\
&+& 96\,m_W^4\,m_Z^2\,s\,x^2 + 32\,m_W^2\,m_Z^4\,s\,x^2 -
    16\,m_W^4\,s^2\,x^2\nn\\
&-& 22\,m_W^2\,m_Z^2\,s^2\,x^2 + 2\,m_Z^4\,s^2\,x^2 +
    m_W^2\,s^3\,x^2 + m_W^2\,\beta_W\,
    \beta_Z\,s^3\,x^3\,]
%\label{a1tw}
\eer
\[A^{(1)}_{uW}(s,x) = A^{(1)}_{tW}(s,-x)\]
and
\[
A^{(1)}_c =\frac{s\,(-4\,m_W^2 - 4\,m_Z^2 + 3\,s - s\,x^2)}{8\,m_W^2\,m_Z^2}
\]
Kinematical variables are related by
\ber t &=& m_W^2 + m_Z^2 - \frac{s}{2} + \frac{s}{2}\,\beta_W\beta_Z \cos{\theta_{cm}}\nn\\
     u &=& m_W^2 + m_Z^2 - \frac{s}{2} - \frac{s}{2}\,\beta_W\beta_Z
     \cos{\theta_{cm}}\nn
\eer
where    
\[\beta_W = \sqrt{1 - \frac{4 m_W^2}{s}}\quad \beta_Z = \sqrt{1 - \frac{4 m_Z^2}{s}}
\]
%%%%%%%%%%% m1g %%%%%%%%%%%%%%%%
%{\frac{ citatel
\[ \M^{(1)}_{gauge} =
\frac{g^2\,m_Z^2\,\cos{\theta_W}^2}{4\,m_W^4}\,\frac{C^{(1)}}{J^{(1)}} \]
\ber  C^{(1)} &=&
       96\,{m_W^4}\,{m_Z^2} -
       32\,{m_W^2}\,{m_Z^4} -
       48\,{m_W^4}\,s +
       8\,{m_W^2}\,{m_Z^2}\,s\nn\\ &+&
       8\,{m_Z^4}\,s + 4\,{m_W^2}\,{s^2} -
       6\,{m_Z^2}\,{s^2} + {s^3} +
       128\,{m_W^6}\,{x^2} +
       32\,{m_W^4}\,s\,{x^2}\nn\\ &-&
       64\,{m_W^2}\,{m_Z^2}\,s\,{x^2} +
       8\,{m_W^2}\,{s^2}\,{x^2} +
       6\,{m_Z^2}\,{s^2}\,{x^2} - {s^3}\,{x^2}\nn
\eer
%%%%%%%%% jmenovatel %%%%%%%%%%%%%%%
\ber    J^{(1)} &=&
       4\,{m_Z^4} - 4\,{m_Z^2}\,s + {s^2} -
       16\,{m_W^2}\,{m_Z^2}\,{x^2}\nn\\ &+&
       4\,{m_W^2}\,s\,{x^2} +
       4\,{m_Z^2}\,s\,{x^2} - {s^2}\,{x^2}\nn
       %}}
\eer
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For the Higgs boson amplitude (\ref{m1sh}) we have
\[\M^{(1)}_{sH} = - \frac{g^2 m_W m_Z}{\cos{\theta_W}}
\frac{(2 m_W^2 - s)\,(2 m_Z^2 - s)}{4 m_W^2 m_Z^2 (s - m_H^2 + i
m_H\Gamma_H)} \]

%%%%%%%%%%%%%% WZWZ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{$W^+(k_1) + Z(k_2) \to Z(k_3) + W^+(k_4)$}
\[\M^{(2)} = - g^2\cos^2{\theta_W}\left[\frac{A^{(2)}_{sW}}{s -
m_W^2} + \frac{A^{(2)}_{tW}}{t - m_W^2}  + A^{(2)}_c\right] +
\M^{(2)}_{uH}
\]
\[A^{(2)}_{sW}=
 V_{\mu\alpha\nu}(-k_1,q,-k_2)\,V_{\beta\sigma\rho}(-q,k_4,k_3)
 P^{\alpha\beta}_W(q){\cal E}^{\mu\nu\rho\sigma}
%\label{wzwzs}\ee
\]
\[A^{(2)}_{tW}=
 V_{\mu\alpha\rho}(-k_1,q,k_3)\,V_{\beta\sigma\nu}(-q,k_4,-k_2)
 P^{\alpha\beta}_W(q){\cal E}^{\mu\nu\rho\sigma}
%\label{wzwzt}\ee
\]
\[A^{(2)}_c = V_{\mu\sigma\nu\rho}{\cal E}^{\mu\nu\rho\sigma}\]
After contraction
%%% Kontrahovany vyraz pro A2sw %%%%%%%%%%%%%
\ber
A^{(2)}_{sW} &=& 4\,(k_1\cdot \varepsilon_2)\,(k_3 \cdot
\varepsilon^*_4)\,(\varepsilon_1 \cdot \varepsilon^*_3) -
4\,(k1\cdot\varepsilon_2)\,(k_4\cdot\varepsilon^*_3)\,(\varepsilon_1
\cdot\varepsilon^*_4)\nn\\
&-& 4\,(k_2\cdot\varepsilon_1)\,(k_3\cdot\varepsilon^*_4)\,
(\varepsilon_2\cdot\varepsilon^*_3) + 4\,(k_2\cdot\varepsilon_1)\,
(k_4\cdot\varepsilon^*_3)\,(\varepsilon_2 \cdot\varepsilon^*_4)\nn\\
&-& 2 \, (k_3 \cdot \varepsilon^*_4)\,(\varepsilon_1 \cdot
\varepsilon_2)\,((k_1 - k_2)\cdot \varepsilon^*_3) + 2\,(k_4 \cdot
\varepsilon^*_3)\,(\varepsilon_1 \cdot \varepsilon_2)\,((k_1 - k_2)
\cdot \varepsilon^*_4)\nn\\
&-& 2 \,(k_1 \cdot \varepsilon_2)\,(\varepsilon^*_3
\cdot \varepsilon^*_4)\,((k_3 - k_4) \cdot \varepsilon_1) + 2 \,(k_2
\cdot \varepsilon_1)\,(\varepsilon^*_3 \cdot \varepsilon^*_4)\,((k_3
- k_4) \cdot \varepsilon_2)\nn\\
&+& \frac{(m_W^2 - m_Z^2)^2}{m_W^2}\,(\varepsilon_1\cdot
\varepsilon_2)\,(\varepsilon^*_3\cdot\varepsilon^*_4) + (u - t)
\,(\varepsilon_1 \cdot \varepsilon_2)\,(\varepsilon^*_3 \cdot
\varepsilon^*_4)
\eer
\[A^{(2)}_{tW} = - A^{(2)}_{sW}(k_2 \leftrightarrow -k_3,
\ep_2 \leftrightarrow \ep_3^*) \]

%%%%%%%%%%%%%%%% konec A2sw %%%%%%%%%%%%%%
\[
 A^{(2)}_c = 2\,(\varepsilon_1\cdot
 \varepsilon^*_4)\,(\varepsilon_2\cdot \varepsilon^*_3) -
 (\varepsilon_1\cdot \varepsilon^*_3)\,(\varepsilon_2\cdot
 \varepsilon^*_4) - (\varepsilon_1\cdot
 \varepsilon_2)\,(\varepsilon^*_3\cdot \varepsilon^*_4)
\]
\[t = m_W^2 + m_Z^2 -\frac{s}{2} + \frac{(m_Z^2 - m_W^2)^2}{2 s} + 2\,k^2
\cos{\theta_{cm}} \]
\[u = - 2 k^2 (1 + \cos{\theta_{cm}}) \]
where in CMS
\[ k^2 = \frac{1}{4 s}\left [s^2 + (m_W^2 - m_Z^2)^2 - 2s
\,(m_W^2 +m_Z^2)\right ] = |\bk_i|^2 \quad i = 1,2,3,4 \]
%%%%%%%%%%%%%%%%%%%% m2g %%%%%%%%%%%%%%%%
\[ \M^{(2)}_{gauge} =
\frac{g^2\,m_Z^2\,\cos{\theta_W}^2}{8\,m_W^4\,s\,
     \left(s - {m_W^2}\right)}\,\frac{C^{(2)}}{J^{(2)}}
\]

\ber C^{(2)} &=& 3\,{m_W^{10}} -
       12\,{m_W^8}\,{m_Z^2} +
       18\,{m_W^6}\,{m_Z^4} -
       12\,{m_W^4}\,{m_Z^6} +
       3\,{m_W^2}\,{m_Z^8} \nn\\ &+&
       17\,{m_W^8}\,s -
       32\,{m_W^6}\,{m_Z^2}\,s +
       10\,{m_W^4}\,{m_Z^4}\,s +
       8\,{m_W^2}\,{m_Z^6}\,s \nn\\ &-&
       3\,{m_Z^8}\,s + 26\,{m_W^6}\,{s^2} +
       32\,{m_W^4}\,{m_Z^2}\,{s^2} -
       30\,{m_W^2}\,{m_Z^4}\,{s^2} \nn\\ &+&
       4\,{m_Z^6}\,{s^2} - 50\,{m_W^4}\,{s^3} +
       16\,{m_W^2}\,{m_Z^2}\,{s^3} +
       2\,{m_Z^4}\,{s^3} + 3\,{m_W^2}\,{s^4} \nn\\ &-&
       4\,{m_Z^2}\,{s^4} + {s^5} +
       6\,{m_W^{10}}\,x -
       24\,{m_W^8}\,{m_Z^2}\,x +
       36\,{m_W^6}\,{m_Z^4}\,x \nn\\ &-&
       24\,{m_W^4}\,{m_Z^6}\,x +
       6\,{m_W^2}\,{m_Z^8}\,x -
       16\,{m_W^8}\,s\,x +
       36\,{m_W^6}\,{m_Z^2}\,s\,x\nn\\ &-&
       28\,{m_W^4}\,{m_Z^4}\,s\,x +
       12\,{m_W^2}\,{m_Z^6}\,s\,x -
       4\,{m_Z^8}\,s\,x + 20\,{m_W^6}\,{s^2}\,x\nn\\ &-&
       44\,{m_W^4}\,{m_Z^2}\,{s^2}\,x -
       20\,{m_W^2}\,{m_Z^4}\,{s^2}\,x +
       12\,{m_Z^6}\,{s^2}\,x -
       16\,{m_W^4}\,{s^3}\,x \nn\\ &-&
       4\,{m_W^2}\,{m_Z^2}\,{s^3}\,x -
       12\,{m_Z^4}\,{s^3}\,x +
       6\,{m_W^2}\,{s^4}\,x +
       4\,{m_Z^2}\,{s^4}\,x +
       3\,{m_W^{10}}\,{x^2}\nn\\ &-&
       12\,{m_W^8}\,{m_Z^2}\,{x^2} +
       18\,{m_W^6}\,{m_Z^4}\,{x^2} -
       12\,{m_W^4}\,{m_Z^6}\,{x^2} +
       3\,{m_W^2}\,{m_Z^8}\,{x^2}\nn\\ &-&
       33\,{m_W^8}\,s\,{x^2} +
       68\,{m_W^6}\,{m_Z^2}\,s\,{x^2} -
       38\,{m_W^4}\,{m_Z^4}\,s\,{x^2} +
       4\,{m_W^2}\,{m_Z^6}\,s\,{x^2}\nn\\ &-&
       {m_Z^8}\,s\,{x^2} +
       122\,{m_W^6}\,{s^2}\,{x^2} +
       12\,{m_W^4}\,{m_Z^2}\,{s^2}\,{x^2} -
       6\,{m_W^2}\,{m_Z^4}\,{s^2}\,{x^2}\nn\\ &+&
       34\,{m_W^4}\,{s^3}\,{x^2} -
       4\,{m_W^2}\,{m_Z^2}\,{s^3}\,{x^2} +
       2\,{m_Z^4}\,{s^3}\,{x^2} +
       3\,{m_W^2}\,{s^4}\,{x^2} - {s^5}\,{x^2}\nn
\eer
%%%%%%%%%% jmenovatel %%%%%%%%%%%%%%
\ber  J^{(2)} &=& {m_W^4} -
       2\,{m_W^2}\,{m_Z^2} + {m_Z^4} +
       2\,{m_Z^2}\,s - {s^2} + {m_W^4}\,x -
       2\,{m_W^2}\,{m_Z^2}\,x\nn\\ &+&
       {m_Z^4}\,x - 2\,{m_W^2}\,s\,x -
       2\,{m_Z^2}\,s\,x + {s^2}\,x\nn
\eer
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Higgs boson amplitude (\ref{m2uh})

%%%%%%%%%%%%%%%%%% m2uh %%%%%%%%%%%%%%%%%%%%%
\[\M^{(2)}_{uH} = \frac{g^2}{8\,m_W\,m_Z\,{\cos{\theta_W}}s}\,\frac{C^{(2)}_{uH}}{J^{(2)}_{uH}}\]
\ber
C^{(2)}_{uH} &=&{m_W^8} - 4\,{m_W^6}\,{m_Z^2} +
       6\,{m_W^4}\,{m_Z^4} -
       4\,{m_W^2}\,{m_Z^6} + {m_Z^8} -
       4\,{m_W^6}\,s + 4\,{m_W^4}\,{m_Z^2}\,s\nn\\ &+&
       4\,{m_W^2}\,{m_Z^4}\,s - 4\,{m_Z^6}\,s +
       6\,{m_W^4}\,{s^2} + 4\,{m_W^2}\,{m_Z^2}\,{s^2} +
       6\,{m_Z^4}\,{s^2} - 4\,{m_W^2}\,{s^3}\nn\\ &-&
       4\,{m_Z^2}\,{s^3} + {s^4} + 2\,{m_W^8}\,x -
       8\,{m_W^6}\,{m_Z^2}\,x+
       12\,{m_W^4}\,{m_Z^4}\,x\nn\\&-&
       8\,{m_W^2}\,{m_Z^6}\,x+
       2\,{m_Z^8}\,x -
       4\,{m_W^6}\,s\,x + 4\,{m_W^4}\,{m_Z^2}\,s\,x+
       4\,{m_W^2}\,{m_Z^4}\,s\,x - 4\,{m_Z^6}\,s\,x\nn\\ &+&
       4\,{m_W^4}\,{s^2}\,x-
       8\,{m_W^2}\,{m_Z^2}\,{s^2}\,x+
       4\,{m_Z^4}\,{s^2}\,x\nn\\ &-&
       4\,{m_W^2}\,{s^3}\,x -
       4\,{m_Z^2}\,{s^3}\,x + 2\,{s^4}\,x + {m_W^8}\,{x^2} -
       4\,{m_W^6}\,{m_Z^2}\,{x^2}+
       6\,{m_W^4}\,{m_Z^4}\,{x^2}\nn\\&-&
       4\,{m_W^2}\,{m_Z^6}\,{x^2}+
       {m_Z^8}\,{x^2} -
       2\,{m_W^4}\,{s^2}\,{x^2} +
       4\,{m_W^2}\,{m_Z^2}\,{s^2}\,{x^2} -
       2\,{m_Z^4}\,{s^2}\,{x^2} + {s^4}\,{x^2}\nn
 %\over
\eer

\ber
J^{(2)}_{uH}&=& {m_W^4} -
       2\,{m_W^2}\,{m_Z^2} + {m_Z^4} +
       2\,{m_H^2}\,s - 2\,{m_W^2}\,s - 2\,{m_Z^2}\,s +
       {s^2} + {m_W^4}\,x\nn\\&-&
       2\,{m_W^2}\,{m_Z^2}\,x+
       {m_Z^4}\,x - 2\,{m_W^2}\,s\,x -
       2\,{m_Z^2}\,s\,x + {s^2}\,x\nn
\eer
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%% WPWP %%%%%%%%%%%%%%
\subsection{$W^+(k_1) + W^+(k_2) \to W^+(k_3) + W^+(k_4)$}
%
\[\M^{(3)} = - g^2\cos^2{\theta_W}\left[\frac{A^{(3)}_{tZ}}{t -
m_Z^2} + \frac{A^{(3)}_{uZ}}{u - m_Z^2}\right] - g^2\sin^2{\theta_W}
\left[\frac{A^{(3)}_{t\gamma}}{t} + \frac{A^{(3)}_{u\gamma}}{u}\right] +
g^2 A^{(3)}_c + \M^{(3)}_{tH} + \M^{(3)}_{uH}\ . \]

\[A^{(3)}_{tZ,\gamma}=
 V_{\mu\rho\alpha}(-k_1,k_3,q)\,V_{\nu\sigma\beta}(-k_2,k_4,-q,)
      P^{\alpha\beta}_{Z,\gamma}(q){\cal E}^{\mu\nu\rho\sigma}
% \label{wpwpt}\ee
\]
\[A^{(3)}_{uZ,\gamma}=
V_{\mu\sigma\alpha}(-k_1,k_4,q)\,V_{\nu\rho\beta}(-k_2,k_3,-q)
  P^{\alpha\beta}_{Z,\gamma}(q){\cal E}^{\mu\nu\rho\sigma}
%\label{wpwpu}\ee
\]

\[A^{(3)}_c = V_{\mu\nu\rho\sigma}{\cal E}^{\mu\nu\rho\sigma}\ .\]
Kinematical variables are given by
\[t = 2 \left( m_W^2 - \frac{s}{4}\right) (1 - x) \qquad
  u = 2 \left( m_W^2 - \frac{s}{4}\right) (1 + x) \]
\ber
A^{(3)}_{tZ} &=&2\,(k_2\cdot\varepsilon^*_4)\,
        \left((k_1\cdot\varepsilon_2) + (k_3\cdot\varepsilon_2) \right) \,
        (\varepsilon_1\cdot\varepsilon^*_3) - 4\,(k_1\cdot\varepsilon^*_3)\,(k_2\cdot \varepsilon^*_4)\,
        (\varepsilon_1\cdot\varepsilon_2) \nn\\
 &+&   2\,\left(  (k_1\cdot\varepsilon^*_4) + (k_3\cdot\varepsilon^*_4) \right) \,
       (k_4\cdot\varepsilon_2)\,(\varepsilon_1\cdot\varepsilon^*_3) -
       4\, (k_1\cdot\varepsilon^*_3)\,(k_4\cdot\varepsilon_2)\,
       (\varepsilon_1\cdot\varepsilon^*_4)\nn\\
 &+&  2\, (k_1\cdot\varepsilon^*_3)\,\left((k_2\cdot \varepsilon_1)
    + (k_4\cdot\varepsilon_1) \right)\,(\varepsilon_2\cdot\varepsilon^*_4)
    - 4\, (k_2\cdot\varepsilon^*_4)\,(k_3\cdot\varepsilon_1)\,
        (\varepsilon_2\cdot\varepsilon^*_3)\nn\\
&+&  2\, (k_3\cdot\varepsilon_1)\,\left((k_2\cdot\varepsilon^*_3) +
      (k_4\cdot\varepsilon^*_3)
     \right)\,(\varepsilon_2\cdot\varepsilon^*_4)\nn\\
&-&  \left( s - u \right)\,(\varepsilon_1\cdot\varepsilon^*_3)\,
    (\varepsilon_2\cdot\varepsilon^*_4) - 4\,(k_3\cdot\varepsilon_1)\,
    (k_4\cdot\varepsilon_2)\,(\varepsilon^*_3\cdot\varepsilon^*_4)\ .
\eer
\[A^{(3)}_{uZ} = A^{(3)}_{tZ} (3\leftrightarrow 4)\]
\[A^{(3)}_{(t,u)\gamma} = A^{(3)}_{(t,u)Z}\]
For relations among different $A$s see table\,2 and below.
\ber A^{(3)}_{tZ}(s,x)=\frac{1}{(32\,m_W^4)}\,&(&64\,m_W^6 - 16\,m_W^4\,s + 12\,m_W^2\,s^2 - 3\,s^3 + 64\,m_W^6\,x\nn\\
&+& 112\,m_W^4\,s\,x - 52\,m_W^2\,s^2\,x + 5\,s^3\,x -
160\,m_W^4\,s\,x^2\nn\\
&+& 36\,m_W^2\,s^2\,x^2 - s^3\,x^2 + 4\,m_W^2\,s^2\,x^3 - s^3\,x^3)
\label{a3tz}
\eer
\[ A^{(3)}_{uZ}(s,x) = A^{(3)}_{tZ}(s,-x)\]
%%%%%%%%%%%%%%%%%% m3g %%%%%%%%%%%%%%%%
\[
\M^{(3)}_{gauge} = {\frac{{g^2}\,s\,\left( -8\,{m_W^2} + 3\,s - s\,{x^2} \right) }
    {8\,{m_W^4}}} +
    \frac{g^2\,\cos^2{\theta_W}}{8\,{m_W^4}}\,\frac{C^{(3)}_Z}{J^{(3)}_Z}
    -
    \frac{g^2\,\sin^2{\theta_W}}{8\,{m_W^4}}\,
    \frac{C^{(3)}_{\gamma}}{J^{(3)}_{\gamma}}
\]
\ber C^{(3)}_Z &=& 256\,{m_W^8} -
        128\,{m_W^6}\,{m_Z^2} -
        128\,{m_W^6}\,s +
        32\,{m_W^4}\,{m_Z^2}\,s +
        64\,{m_W^4}\,{s^2}\nn\\ &-&
        24\,{m_W^2}\,{m_Z^2}\,{s^2} -
        24\,{m_W^2}\,{s^3} + 6\,{m_Z^2}\,{s^3} +
        3\,{s^4} + 256\,{m_W^8}\,{x^2}\nn\\ &-&
        256\,{m_W^6}\,s\,{x^2} +
        320\,{m_W^4}\,{m_Z^2}\,s\,{x^2} -
        16\,{m_W^4}\,{s^2}\,{x^2}\nn\\ &-&
        72\,{m_W^2}\,{m_Z^2}\,{s^2}\,{x^2} +
        32\,{m_W^2}\,{s^3}\,{x^2} +
        2\,{m_Z^2}\,{s^3}\,{x^2}\nn\\ &-&
        4\,{s^4}\,{x^2} +
        16\,{m_W^4}\,{s^2}\,{x^4} -
        8\,{m_W^2}\,{s^3}\,{x^4} + {s^4}\,{x^4}\nn
\eer
\[
J^{(3)}_Z = \left( 4\,{m_W^2} - 2\,{m_Z^2} - s -
        4\,{m_W^2}\,x + s\,x \right) \,
      \left( -4\,{m_W^2} + 2\,{m_Z^2} + s -
        4\,{m_W^2}\,x + s\,x \right)
\]
\ber
C^{(3)}_{\gamma} &=& 64\,{m_W^6} -
        16\,{m_W^4}\,s + 12\,{m_W^2}\,{s^2} -
        3\,{s^3} + 64\,{m_W^6}\,{x^2}\nn\\ &-&
        48\,{m_W^4}\,s\,{x^2} -
        16\,{m_W^2}\,{s^2}\,{x^2} + 4\,{s^3}\,{x^2} +
        4\,{m_W^2}\,{s^2}\,{x^4} - {s^3}\,{x^4}\nn
\eer
\[
J^{(3)}_{\gamma} =\left( s - 4\,{m_W^2} \right) \,\left( {x^2} - 1 \right)
\]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Higgs boson amplitude (\ref{m3h})
%%%%%%%%%%%%%%%%%%%%% m3h %%%%%%%%%%
\[\M^{(3)}_H = g^2\,\frac{C^{(3)}_H}{J^{(3)}_H}\]
\ber C^{(3)}_H &=& 32\,{m_H^2}\,{m_W^4} - 64\,{m_W^6} -
       16\,{m_H^2}\,{m_W^2}\,s + 48\,{m_W^4}\,s\nn\\ &+&
       2\,{m_H^2}\,{s^2} - 12\,{m_W^2}\,{s^2} + {s^3} -
       32\,{m_W^4}\,s\,{x^2} + 2\,{m_H^2}\,{s^2}\,{x^2}\nn\\ &+&
       12\,{m_W^2}\,{s^2}\,{x^2} - {s^3}\,{x^2}
\eer
%\over
\[J^{(3)}_H =  4\,{m_W^2}\,\left( 2\,{m_H^2} - 4\,{m_W^2} + s +
       4\,{m_W^2}\,x - s\,x \right) \,
     \left( 2\,{m_H^2} - 4\,{m_W^2} + s -
       4\,{m_W^2}\,x + s\,x \right)
\]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%% WPWM %%%%%%%%%%%%%%%%%%
\subsection{$W^+(k_1) + W^-(k_2) \to W^+(k_3) + W^-(k_4)$}
%
\[\M^{(4)} = - g^2\cos^2{\theta_W}\left[\frac{A^{(4)}_{sZ}}{s -
m_Z^2} + \frac{A^{(4)}_{tZ}}{t - m_Z^2}\right] - g^2\sin^2{\theta_W}
\left[\frac{A^{(4)}_{s\gamma}}{s} + \frac{A^{(4)}_{t\gamma}}{t}\right] +
g^2 A^{(4)}_c + \M^{(4)}_{sH} + \M^{(4)}_{tH}\ . \]
\[A^{(4)}_{sZ,\gamma}=
  V_{\mu\nu\alpha}(-k_1,-k_2,q)\,V_{\sigma\rho\beta}(k_4,k_3,-q,)
  P^{\alpha\beta}_{Z,\gamma}(q){\cal E}^{\mu\nu\rho\sigma}
%\label{wpwms}\ee
\]
\[A^{(4)}_{tZ,\gamma}=
 V_{\mu\rho\alpha}(-k_1,k_3,q)\,V_{\sigma\nu\beta}(k_4,-k_2,-q)
      P^{\alpha\beta}_{Z,\gamma}(q){\cal E}^{\mu\nu\rho\sigma}
%\label{wpwmt}\ee
\]
\[A^{(4)}_c = g^2
V_{\mu\sigma\rho\nu}{\cal E}^{\mu\nu\rho\sigma}\ .\]
%%%%%%%% m4g %%%%%%%%%%%%%%%%%%%%%
\ber \M^{(4)}_{gauge} &=& {\frac{{g^2}\,s\,\left( 8\,{m_W^2} - 3\,s -
        24\,{m_W^2}\,x + 6\,s\,x + s\,{x^2} \right) }{16\,
      {m_W^4}}}\nn\\ &+&
      \M^{(4)}_{sZ} + \M^{(4)}_{s\gamma}- {g^2}\,{\cos^2{\theta_W}}\,\frac{C^{(4)}_{tZ}}
      {J^{(4)}_{tZ}}  - {g^2}\,{\sin^2{\theta_W}}\,
      \frac{C^{(4)}_{t\gamma}}{J^{(4)}_{t\gamma}}\nn
\eer
\[
\M^{(4)}_{sZ} =  - g^2\,\cos^2{\theta_W}\,\frac{\left( 4\,{m_W^2} - s \right) \,
         \left( 2\,{m_W^2} + s \right)^2\,x}{4\,
         {m_W^4}\,\left( {m_Z^2} - s \right)}
\]
\[
\M^{(4)}_{s\gamma} =  - {g^2}\,\sin^2{\theta_W}\left( -3\,x -
\frac{4\,{m_W^2}\,x}{s} + \frac{s^2\,x}{4\,m_W^4}\right)
\]
\ber
C^{(4)}_{tZ} &=& -64\,{m_W^6} + 16\,{m_W^4}\,s -
         12\,{m_W^2}\,{s^2} + 3\,{s^3} -
         64\,{m_W^6}\,x - 112\,{m_W^4}\,s\,x +
         52\,{m_W^2}\,{s^2}\,x\nn\\ &-&
         5\,{s^3}\,x +
         160\,{m_W^4}\,s\,{x^2} -
         36\,{m_W^2}\,{s^2}\,{x^2} + {s^3}\,{x^2} -
         4\,{m_W^2}\,{s^2}\,{x^3} + {s^3}\,{x^3}\nn
\eer
\[ J^{(4)}_{tZ} = 16\,{m_W^4}\,\left( 4\,{m_W^2} -
           2\,{m_Z^2} - s - 4\,{m_W^2}\,x + s\,x
            \right)
\]
\[
C^{(4)}_{t\gamma} = C^{(4)}_{tZ} \quad
J^{(4)}_{t\gamma} = J^{(4)}_{tZ}|_{m_Z=0}
\]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Higgs boson amplitude (\ref{m4h})
%%%%%%%%%%%%%% m4h %%%%%%%%%%%%%%%%
\[\M^{(4)}_H = g^2\,\frac{C^{(4)}_H}{J^{(4)}_H}\]
\ber
C^{(4)}_H &=& 32\,{m_H^2}\,{m_W^4} - 32\,{m_W^6} -
       24\,{m_H^2}\,{m_W^2}\,s + 24\,{m_W^4}\,s\nn\\ &+&
       5\,{m_H^2}\,{s^2} - 8\,{m_W^2}\,{s^2} + {s^3} +
       32\,{m_W^6}\,x + 8\,{m_H^2}\,{m_W^2}\,s\,x\nn\\ &-&
       40\,{m_W^4}\,s\,x - 2\,{m_H^2}\,{s^2}\,x +
       8\,{m_W^2}\,{s^2}\,x + {m_H^2}\,{s^2}\,{x^2} -
       {s^3}\,{x^2}\nn
%\over
\eer
\[J^{(4)}_H =
   8\,{m_W^2}\,\left( {m_H^2} - s \right) \,
     \left( 2\,{m_H^2} - 4\,{m_W^2} + s +
       4\,{m_W^2}\,x - s\,x \right)
\]
\end{appendix}

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\end{document}

