
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Just a Sample heading for an Appendix}
\label{CC03MatElem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{equation}
a = 1-4\sin^2\Theta_{\rm W}, \quad\quad b = -1
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The kinematical functions $G$ are given by:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
G_1(s,s_1,s_2) &=& -\lambda^{3/2}\left[\frac{\lambda}{6}+
                 2( ss_1 + s_1s_2 + s_2s )\right] \\
*[4mm]
G_2(s,s_1,s_2) &=& -\lambda^{1/2}\left[\frac{\lambda}{6}+
                 2( ss_1 - 4s_1s_2 + s_2s )\right] \\ 
*[4mm]\nonumber&&\quad\quad\quad\quad\quad+\:4s_1s_2(s-s_1-s_2)\:{\rm ln}\:f\\
*[4mm]
G_3(s,s_1,s_2) &=&-\lambda^{1/2}\Bigg[ (s+11s_1+11s_2)\frac{\lambda}{6} + \\
*[4mm]\nonumber&&\quad\quad\quad\quad\quad2\:(s_1^2+3s_1s_2+s_2^2)s
                                                     -2\:(s_1^3+s_2^3)\Bigg] \\
*[4mm]\nonumber&&-\:4s_1s_2( ss_1 + s_1s_2 + s_2s )\:{\rm ln}\: f
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\lambda &\equiv& \lambda(s,s_1,s_2) = s^2+s_1^2+s_2^2-2(ss_1+s_1s_2+s_2s) \\
*[4mm]
f &\equiv& f(s,s_1,s_2)=\frac{s-s_1-s_2-\sqrt{\lambda}}{s-s_1-s_2+
               \sqrt{\lambda}}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

A couple of notes are of interest. First, the $\lambda$ functions vanish
at threshold where $\sqrt{s}=\sqrt{s_1}+\sqrt{s_2}$.  Since all $s$-channel
diagrams are proportional to the $G_1$ kinematic function, the WW 
cross-sections involving the triple gauge bosons couplings are zero at 
threshold and the dominant production mechanism is through the $t$-channel 
neutrino exchange.

Second, the $t$-channel cross-section involves the $G_2$ kinematic function 
which grows without bound and eventually violates unitarity.
However, it is shown that the 
$s$-channel diagrams destructively interfere with the $t$-channel
diagrams through the $G_3$ kinematic function.  In the high energy limit, it 
is this delicate cancellation 
involving the triple gauge couplings which restores unitarity.


