%*********************MASter File: Margarete ************************* 
\newcommand{\sla}[1]{#1\!\!\!/}
\newcommand{\ewwhh}{\eta_{\ensuremath{\mbox{\tiny WWHH}}}}
\newcommand{\ewwaa}{\eta_{\ensuremath{\mbox{\tiny WWAA}}}}
\newcommand{\ewwH}{\eta_{\ensuremath{\mbox{\tiny WWH}}}}
\newcommand{\ehhh}{\eta_{\ensuremath{\mbox{\tiny HHH}}}}
\newcommand{\ewwh}{\eta_{\ensuremath{\mbox{\tiny WWh}}}}
\newcommand{\eHHh}{\eta_{\ensuremath{\mbox{\tiny HHh}}}}
\newcommand{\ehhH}{\eta_{\ensuremath{\mbox{\tiny hhH}}}}
\newcommand{\eHaa}{\eta_{\ensuremath{\mbox{\tiny HAA}}}}
\newcommand{\ehaa}{\eta_{\ensuremath{\mbox{\tiny hAA}}}}
\newcommand{\ewgh}{\eta_{\ensuremath{\mbox{\tiny $W^+G^-H$}}}}
\newcommand{\ewghh}{\eta_{\ensuremath{\mbox{\tiny $W^+G^-h$}}}}
\newcommand{\ewhmh}{\eta_{\ensuremath{\mbox{\tiny $W^+H^-H$}}}}
\newcommand{\ewhmhh}{\eta_{\ensuremath{\mbox{\tiny $W^+H^-h$}}}}
\newcommand{\ezah}{\eta_{\ensuremath{\mbox{\tiny ZAH}}}}
\newcommand{\ezahh}{\eta_{\ensuremath{\mbox{\tiny ZAh}}}}
\newcommand{\ewhma}{\eta_{\ensuremath{\mbox{\tiny $W^+H^-A$}}}}
\newcommand{\bw}{\beta_{\ensuremath{\mbox{\tiny W}}}}
\newcommand{\bh}{\beta_{\ensuremath{\mbox{\tiny H}}}}
\newcommand{\ba}{\beta_{\ensuremath{\mbox{\tiny A}}}}
\newcommand{\bp}{\beta_+}
\newcommand{\xh}{x_{\ensuremath{\mbox{\tiny H}}}}
\newcommand{\xhpm}{x_{\ensuremath{\mbox{\tiny $H^\pm$}}}}
\newcommand{\aw}{a_{\ensuremath{\mbox{\tiny W}}}}
\newcommand{\aH}{a_{\ensuremath{\mbox{\tiny H}}}}
\newcommand{\ah}{a_h}
\newcommand{\az}{a_{\ensuremath{\mbox{\tiny Z}}}}
\newcommand{\aA}{a_{\ensuremath{\mbox{\tiny A}}}}
\newcommand{\ahpm}{a_{\ensuremath{\mbox{\tiny $H^\pm$}}}}
\newcommand{\xn}{x_0}
\newcommand{\yn}{y_0}
\newcommand{\zn}{z_0}
\newcommand{\bb}{\beta}
\newcommand{\In}[1]{\ensuremath{\mbox{\tiny#1}}}
\newcommand{\lHh}{\lambda_{\ensuremath{\mbox{\tiny Hh}}}}
\newcommand{\mz}{M_{\ensuremath{\mbox{\tiny Z}}}}
\newcommand{\mw}{M_{\ensuremath{\mbox{\tiny W}}}}
\newcommand{\mH}{M_{\ensuremath{\mbox{\tiny H}}}}
\newcommand{\mh}{M_{\ensuremath{\mbox{\tiny h}}}}
\newcommand{\lij}{\lambda_{ij}}
\newcommand{\lia}{\lambda_{\ensuremath{\mbox{\tiny iA}}}}
\newcommand{\lhAA}{\lambda_{\ensuremath{\mbox{\tiny hAA}}}}
\newcommand{\lHAA}{\lambda_{\ensuremath{\mbox{\tiny HAA}}}}
\newcommand{\lhhphm}{\lambda_{\ensuremath{\mbox{\tiny $hH^+H^-$}}}}
\newcommand{\lHhphm}{\lambda_{\ensuremath{\mbox{\tiny $HH^+H^-$}}}}
\newcommand{\xp}{x_+}
\newcommand{\xw}{x_{\ensuremath{\mbox{\tiny W}}}}
\newcommand{\xa}{x_{\ensuremath{\mbox{\tiny A}}}}
\newcommand{\Xh}{x_{\ensuremath{\mbox{\tiny h}}}}
\newcommand{\XH}{x_{\ensuremath{\mbox{\tiny H}}}}
\newcommand{\sw}{\sin\theta_W}
\newcommand{\cw}{\cos\theta_W}
\newcommand{\sh}{\hat{s}}
\newcommand{\lh}{\lambda_{\ensuremath{\mbox{\tiny $hH_i H_j$}}}}
\newcommand{\lH}{\lambda_{\ensuremath{\mbox{\tiny $HH_i H_j$}}}}
\newcommand{\ct}{\cos\theta}
\newcommand{\ctt}{\cos^2 \theta}
\newcommand{\eep}{e^+ e^-}
\newcommand{\ra}{\rightarrow}

% ********************************************************************* 

% ********************************************************************* 

\subsection*{Appendix A: Double Higgs-strahlung Processes} 

\setcounter{equation}{0}
\renewcommand{\theequation}{A.\arabic{equation}}

In this appendix, we present the cross sections for the pair
production of the heavy MSSM Higgs bosons in the Higgs-strahlung
processes, $\eep \to Z H_i H_j $ and $ZAA$ with $H_{i,j}=h,H$. The
mixed process $\eep \ra Z H_iA $ is generated at the tree level by
gauge couplings only.  The notation is the same as in Section 2.1. The
trilinear couplings have been introduced earlier. Modifications of the
SM Higgs-gauge coupling in the MSSM are accounted for by the mixing
parameters:
\beq
\begin{array}{l@{:\;\;}l@{=\;}l l@{:\;\;}l@{=\;}l l@{:\;\;}l@{=\;}l}
VVh & d_1 & \sin(\beta-\alpha)\;\;\;\; & VVH & d_2 & 
\cos(\beta-\alpha) \;\;\;\;& VVA & d_3 & 0 \\
VAh & c_1 & \cos(\beta-\alpha) \;\;\;\;& VAH & c_2 & 
-\sin(\beta-\alpha) \;\;\;\;& WAH & c_3 & 1
\end{array}
\eeq
for $V=Z$ and $W$. The Higgs bosons are neutral except in the last 
entry.

\subsubsection*{A1: $\eep \to Z H_i H_j$} 

The double differential cross section of the process $e^+ e^- \to ZH_i
H_j$ is given for unpolarized beams by the expression
\beq
\frac{d\sigma[e^+ e^- \to ZH_i H_j]}{dx_1 dx_2} &=&  
\frac{ \sqrt{2} \, G_F^3 \, M_Z^6}{ 384\, \pi^3 s\,} 
\frac{ v_e^2+a_e^2}{(1-\mu_Z)^2} \  {\cal Z}_{ij} 
\label{eq:sigma}
\eeq
In terms of the variables $y_1, y_2, y_3$ defined in Section 2.1, and
with $\mu_i=M_{H_i}^2/s,$ $\mu_{ij}=\mu_i-\mu_j,$ etc., the 
coefficient ${\cal Z}_{ij}$ in the cross sections can be written as
\beq
{\cal Z}_{ij} &=& {\mathfrak a}^2_{ij}\,f_0 
+ \frac{{\mathfrak a}_{ij}}{2} \, \left[
  \frac{ d_i d_j \, f_3 }{y_1+\mu_{iZ}} 
+ \frac{ c_i c_j \, f_3}{y_1+\mu_{iA}} 
   \right] +\frac{ (d_i d_j)^2}{4\mu_Z (y_1+\mu_{iZ})}
\left[\frac{f_1}{y_1+\mu_{iZ}}+ \frac{f_2}{y_2+\mu_{jZ}} \right]\non \\
&+&  \!\!\!
\frac{ (c_i c_j )^2 }{4\mu_Z (y_1+\mu_{iA})}
\left[
\frac{f_1}{y_1+\mu_{iA}} 
+ \frac{f_2}{y_2+\mu_{jA}}\right] 
+ \frac{ d_i d_j c_i c_j} {2 \mu_Z (y_1+\mu_{iA})}
\left[
\frac{f_1}{y_1+\mu_{iZ}}
+\frac{f_2}{y_2+\mu_{jZ}} \right] 
\non\\
&+& \!\!\!
\Bigg\{ (y_1,\mu_i) \leftrightarrow (y_2,\mu_j) \Bigg\} 
\eeq
with 
\beq
{\mathfrak a}_{ij} = 
 \left[ \frac{ d_1 \lambda_{hH_iH_j}}{y_3 - \mu_{1Z}} + 
\frac{ d_2 \lambda_{HH_iH_j}}{y_3 - \mu_{2Z}} \right] + 
\frac{2 d_i d_j}{y_1+\mu_{iZ}} + \frac{2 d_i d_j}{y_2+\mu_{jZ}}
+ \frac{\delta_{ij}}{\mu_Z}
\eeq
The coefficients $f_{0}$ to $f_3$ are given by
\beq
f_0 &=& \mu_Z [(y_1+y_2)^2 + 8\mu_Z]/8 \non\\
f_1 &=& (y_1-1)^2(\mu_Z-y_1)^2-4\mu_iy_1(y_1+y_1\mu_Z-4\mu_Z) \non\\
& & + \mu_Z(\mu_Z-4\mu_i)(1-4\mu_i)-\mu_Z^2 + (\mu_i-\mu_j)^2 
[y_1(y_1-2)+1-4\mu_i] 
\non\\
& & + (\mu_i-\mu_j)
[8\mu_i(-y_1-\mu_Z)+2y_1 \mu_Z(y_1-2) + 2\mu_Z + 2y_1(y_1-1)^2]
\non\\
f_2 &=& [\mu_Z(1+\mu_Z - y_1 -y_2 - 8\mu_i)-(1+\mu_Z)y_1 y_2]
(2+2\mu_Z -y_1-y_2)
\non\\
& & + y_1 y_2[y_1 y_2 + \mu_Z^2+1+4\mu_i (1+\mu_Z)]
+ 4\mu_i \mu_Z(1+\mu_Z+4\mu_i)+ 
\mu_Z^2 \non\\
& & -2(\mu_i-\mu_j)^3-(\mu_i-\mu_j)^2[y_2(y_1-1)+10\mu_Z +4 \mu_j 
+3y_1 -1]\non\\
& & +(\mu_i-\mu_j)[\mu_Z(2(-y_1 y_2-y_1-8\mu_j)+6(\mu_Z+1-y_2))\non\\
& & +y_1((y_2-1)^2 -
y_1(1+y_2))+y_2(y_2-1)-4\mu_j(y_1-y_2)] \non\\
f_3 &=& y_1(y_1-1)(\mu_Z-y_1)-y_2(y_1+1)(y_1+\mu_Z)
+2\mu_Z(\mu_Z+1-4\mu_i) \non\\
& & + 2(\mu_i-\mu_j)^2 - (\mu_i-\mu_j)[y_2+y_1^2-3y_1+y_1y_2-4\mu_Z]
\eeq
For resonance contributions, propagator products have to be
substituted by $\pi_1(\mu_i) \pi_2(\mu_j) \to Re~\{\pi_1(\mu_i)
\pi_2(\mu_j^*) \}$ with $\mu_i \to \mu_i - i \gamma_i$ and $\gamma_i =
M_{H_i} \Gamma_{H_i}/s$.  Setting $i=j=1$, one recovers the
expressions eqs.~(\ref{zhh1}--\ref{zhh3}) for the process $\eep \to Zhh$.

\subsubsection*{A2: $\eep \to ZAA$} 

The differential cross section of the process $e^+ e^- \to ZAA$ 
is given by inserting ${\cal Z}_{33}$ in eq.~(\ref{eq:sigma}): 
\beq
{\cal Z}_{33} &=&  {\mathfrak a}^2_{33} \,  
f_0 + \frac{{\mathfrak a}_{33}}{2} \, \left[
\frac{c_1^2}{y_1-\mu_{1A}}+\frac{ c_2^2}{y_1-\mu_{2A}} 
\right]f_3  
\non \\
& & 
+ \frac{1}{4 \mu_Z} \left[ \frac{c_1^2} {y_1-\mu_{1A}}+ 
\frac{c_2^2}{y_1-\mu_{2A}} \right] \left[ 
\frac{c_1^2}{y_2-\mu_{1A}}+ 
\frac{c_2^2}{y_2-\mu_{2A}} \right]f_2  
\non \\
& &  
+ \frac{1}{4\mu_Z} \left[ 
\frac{c_1^2}{y_1-\mu_{1A}}+ 
\frac{c_2^2}{y_1-\mu_{2A}} 
\right]^2 f_1  
+ \Bigg\{ y_1 \leftrightarrow y_2 \Bigg\} 
\eeq
where 
\beq
{\mathfrak a}_{33} = 
\left[ \frac{ d_1 \lambda_{hAA}}{y_3 - \mu_{1Z}} + 
\frac{ d_2 \lambda_{HAA}}{y_3 - \mu_{2Z}} \right]
+  \frac{1}{\mu_Z} 
\eeq 
The coefficients $f_0$ to $f_3$ follow from the previous subsection 
after substituting $\mu_1,$ $\mu_2\to \mu_A$.

% **********************************************************************

% **********************************************************************

\subsection*{Appendix B: Triple Higgs Boson Production} 

\setcounter{equation}{0}
\renewcommand{\theequation}{B.\arabic{equation}}

The cross sections for the triple Higgs boson production of MSSM Higgs
bosons, $\eep \to A H_i H_j$ and $\eep \ra AAA$ with $H_{i,j}=h,H$ are
presented in this second appendix. The remaining process $\eep \to
H_iA A$ does not occur at tree level because of CP--invariance.

\subsubsection*{B1: $\eep \to A H_i H_j$} 

The double differential cross section of the process $e^+ e^- \to A
H_i H_j$ for unpolarized beams reads in the same notation as above:
\beq 
\frac{d\sigma}{dx_1 dx_2} = 
\frac{G_F^3 M_Z^6}{768 \sqrt{2} \pi^3 s} \, 
\frac{v_e^2+a_e^2}{(1-\mu_Z)^2} \, {\mathfrak A}_{ij} 
\eeq
with the function ${\mathfrak A}_{ij}$ 
\beq
{\mathfrak A}_{ij} &=&  \left[ 
\frac{ \lambda_{hH_iH_j} c_1} {y_3-\mu_{1A}}
+ \frac{ \lambda_{H H_iH_j} c_2} {y_3-\mu_{2A}} \right]^2 g_0
+ \frac{\lambda^2_{H_jAA} c_i^2} {(y_1+\mu_{iA})^2} g_1
+ \frac{ \lambda^2_{H_iAA} c_j^2} {(y_2+\mu_{jA})^2} g_1' \non \\
&& + \frac{c_j^2 d_i^2 } {(y_1+\mu_{iZ})^2} g_2 +
\frac{c_i^2 d_j^2 } {(y_2+\mu_{jZ})^2} g_2'+
\left[ \frac{\lambda_{hH_iH_j} c_1 } {y_3-\mu_{1A}}
+ \frac{ \lambda_{H H_iH_j} c_2 } {y_3-\mu_{2A}} \right] \non \\
&& \times 
\left[ \frac{ \lambda_{H_jAA} c_i} {y_1+\mu_{iA}} g_3
+ \frac{ \lambda_{H_iAA} c_j} {y_2+\mu_{jA}} g_3' 
+ \frac{c_j d_i } {y_1+\mu_{iZ}} g_4 +
\frac{c_i d_j } {y_2+\mu_{jZ}} g_4' \right] \non \\
&&+ 
\frac{\lambda_{H_i AA} \lambda_{H_j AA} c_ic_j}{(y_1+\mu_{iA})(y_2+\mu_{jA})}g_5
+ \frac{c_ic_j d_i d_j}{(y_1+\mu_{iZ})(y_2+\mu_{jZ})}g_8 
\non \\
&& + 
\frac{ \lambda_{H_j AA} c_ic_j d_i}{(y_1+\mu_{iA})(y_1+\mu_{iZ})}g_6
+ \frac{\lambda_{H_i AA} c_ic_j d_j}{(y_2+\mu_{jA})(y_2+\mu_{jZ})}
g_6' 
\non \\
&& 
+ \frac{\lambda_{H_j AA} c_i^2 d_j}{(y_1+\mu_{iA})(y_2+\mu_{jZ})} 
g_7 
+ \frac{\lambda_{H_i AA} c_j^2 d_i}{(y_2+\mu_{jA})(y_1+\mu_{iZ})}
g_7'
\eeq
The following expressions must be inserted for the coefficients $g_k$:
\beq
g_0 &=& \mu_Z[(y_1+y_2)^2-4\mu_A] \non \\
g_1 &=& \mu_Z(y_1^2-2y_1-4\mu_i+1) \non \\
g_2 &=& \mu_Z (2y_1 + y_1^2 -4y_2 +4y_2^2 + 4y_1 y_2 +1+ 4\mu_i 
- 8\mu_j - 8\mu_A) + (\mu_j-\mu_A)^2 \non\\
&& [8+(-2y_1+y_1^2-4\mu_i+1)/\mu_Z] + 2(\mu_j-\mu_A) 
(2y_1 y_2+y_1^2+2y_2-1)\non\\
g_3 &=& 2\mu_Z(y_1^2+y_1 y_2- y_1+y_2+2\mu_j-2\mu_i-2\mu_A) \non \\
g_4 &=& 2\mu_Z(y_1-y_2+y_1^2+2y_2^2+3y_1 y_2 -2\mu_j +2 \mu_i-6 
\mu_A) \non\\ 
&&+2(\mu_j-\mu_A)(-y_1+y_2+ y_1^2 + y_1y_2+2\mu_j-2\mu_A-2\mu_i) 
    \non \\
g_5 &=& 2\mu_Z(y_1+ y_2+y_1y_2+2\mu_j+ 2\mu_i-2\mu_A-1) \non \\
g_6 &=& 2\mu_Z( y_1^2+2 y_1y_2+2y_2+4\mu_j-4\mu_A -1) \non \\
 && + 2(\mu_j-\mu_A)(y_1^2-2y_1-4\mu_i+1) \non \\
g_7 &=& 2[ \mu_Z(2y_1^2+y_1y_2+y_2-3y_1+2\mu_j-6\mu_i-2\mu_A+1) 
\non \\
 && + (\mu_i-\mu_A)(y_1+y_2+y_1y_2+2\mu_j+2\mu_i-2\mu_A-1)] \non \\
g_8 &=& 2 \left\{\mu_Z(y_1+y_2+2y_1^2+2y_2^2+5y_1 y_2 -1 + 2\mu_j 
+ 2\mu_i -10\mu_A)  \right. \non \\
&& +2(\mu_i-\mu_A)(\mu_i-3\mu_j-\mu_A-2y_2+1) 
+2(\mu_j-\mu_A)(\mu_j-3\mu_i-\mu_A-2y_1+1)  \non \\
&& + [(\mu_j-\mu_A)((1+y_1)(y_2+2y_1-1)\mu_Z
+2\mu_i^2+4\mu_A^2 +\mu_A-\mu_i)\non \\
&& + (\mu_i-\mu_A) ((1+y_2) (y_1+2y_2-1)\mu_Z 
+ 2\mu_j^2+4\mu_A^2+\mu_A-\mu_j) \non \\
&& + \left. 6\mu_A(\mu_A^2-\mu_i\mu_j) + 
(\mu_i-\mu_A)(\mu_j-\mu_A) (1+y_1)(1+y_2)]/\mu_Z 
\right\}
\eeq
and 
\beq
g_k' (y_1,y_2,\mu_i,\mu_j) = g_k(y_2,y_1,\mu_j,\mu_i) 
\eeq

% **********************************************************************

% **********************************************************************

\subsection*{Appendix C: Heavy Higgs Production in $W_L W_L$ Fusion} 

\setcounter{equation}{0}
\renewcommand{\theequation}{C.\arabic{equation}}

In this third appendix we list the amplitudes and cross sections for the
pair production of the CP--even Higgs bosons in the longitudinal $W$
approximation $W_L W_L \ra H_i H_j$, as well as for $W_L W_L \ra AA$.
The notation is the same as in section 2.2.

\subsubsection*{C1: $W_L W_L \to H_i H_j$ } 

The amplitudes for the process $W_L W_L \to H_i H_j$ are given by: 
\beq
{\cal M}_{LL} &=& \frac{G_F \hat{s}}{\sqrt{2}} \left\{ 
(1+\beta_W^2)\left[\delta_{ij} + 
\frac{\lambda_{hH_iH_j} d_1}{(\hat{s} -M_h^2)/M_Z^2}
+ \frac{\lambda_{H H_iH_j} d_2}{(\hat{s}-M_H^2)/M_Z^2} \right] \right. 
\non \\
&&+ \frac{d_i d_j}{\beta_W \lambda_{ij}} \left[\frac{  r_W
+ (\beta_W -\lambda_{ij} \cos \theta )^2} {\cos \theta-x_W} -
\frac{ r_W + (\beta_W +\lambda_{ij} \cos \theta )^2}{\cos \theta+x_W} 
\right] \non \\
&&+ \left. \frac{c_i c_j}{\beta_W \lambda_{ij}} \left[ \frac{ 
r_+ + (\beta_W -\lambda_{ij} \cos \theta )^2 }{\cos \theta-x_+} - 
\frac{ r_+ + (\beta_W +\lambda_{ij} \cos \theta )^2 }{\cos \theta+x_+} 
\right] \right\}
\eeq
where $\mu_{i,j}= M_{H_{i,j}}^2/\hat{s}$,
$\beta_W=(1-4M_W^2/\hat{s})^{1/2}$ and $\lambda_{ij}$ is the usual
two--body phase space function, $\lambda^2_{ij} = (1-\mu_i -
\mu_j)^2-4\mu_i\mu_j$. Furthermore,
\beq
\begin{array}{l@{\quad\quad}l}
\hspace{-0.7cm}x_W= (1-\mu_i -\mu_j)/(\beta_W \lambda_{ij}) &
\hspace{-0.4cm}r_W= 1-\beta_W^4 -\beta^2_W (\mu_i-\mu_j)^2 \\[0.1cm]
\hspace{-0.7cm}x_+= (1-\mu_i -\mu_j+2 M_{H^\pm}^2/\hat{s} 
-2M_W^2/\hat{s})/(\beta_W\lambda_{ij}) &
\hspace{-0.4cm}r_+ =  -\beta^2_W (\mu_i-\mu_j)^2 
\end{array}
\eeq
The total cross section of the subprocess is obtained by integrating
the squared amplitude over the scattering angle; the result may be
written:
\beq
\sigma_{LL} (H_i H_j) &=& \frac{1}{1+\delta_{ij}} 
\frac{G_F^2 M_W^4}{2\pi \hat{s}}
\frac{\lambda_{ij}}{ \beta_W (1-\beta_W^2)^2} \non \\
&& \Bigg\{ (1+\beta_W^2)^2 \left[ \delta_{ij} 
+ \frac{\lambda_{hH_iH_j} d_1}{(\hat{s} -M_h^2)/M_Z^2} 
+ \frac{\lambda_{H H_iH_j} d_2}{(\hat{s} -M_H^2)/M_Z^2} \right]^2  
\non \\
&& +  \frac{2(1+\beta_W^2)} {\beta_W \lambda_{ij} }  \left[
\delta_{ij}  + \frac{\lambda_{hH_iH_j}d_1}{(\hat{s} -M_h^2)/M_Z^2}
 + \frac{\lambda_{H H_iH_j}d_2}{(\hat{s} -M_H^2)/M_Z^2} \right] \left[ 
d_i d_j a_1^W  + c_i c_j a_1^+   \right] \non \\
&&+ \left( \frac{d_i d_j}{\beta_W \lambda_{ij} } \right)^2 a_2^W  
+ \left( \frac{c_i c_j}{\beta_W \lambda_{ij} } \right)^2 a_2^+  + 4 
\left(\frac{d_i d_jc_i c_j}{\beta_W^2 \lambda^2_{ij} } \right) [a_3^W 
+ a_3^+ ] \Bigg\} 
\eeq
with 
\beq
a_1^W  &=& [ (x_W \lambda_{ij} -\beta_W)^2 + r_W ] \log \frac
{x_W-1}{x_W+1} +2 \lambda_{ij} (x_W \lambda_{ij} -2\beta_W ) \non \\
%
a_2^W &=& \left[ \frac{1}{x_W} \log \frac{x_W-1}{x_W+1} + \frac{2}
{x_W^2-1} \right] \bigg[ x_W^2 \lambda_{ij}^2 (3 \lambda_{ij}^2 x_W^2 
+2 r_W +14 \beta_W^2) \non \\
&&  -(\beta_W^2+ r_W)^2   -4 \lambda_{ij} \beta_W x_W 
 (3 \lambda_{ij}^2 x_W^2 +\beta_W^2+r_W ) \bigg] \non \\
&& - \frac{4}{x_W^2-1} \left[ \lambda_{ij}^2 (\lambda_{ij}^2 x_W^2 
+4 \beta^2_W - 4 \lambda_{ij} x_W \beta_W) - (\beta_W^2 +r_W)^2 
\right] \non \\
a_3^W&=& \frac{1}{x_+^2 - x_W^2} \, \log \frac{x_W-1}{x_W+1} \bigg[
2 \beta_W \lambda_{ij} x_W [(\beta_W^2+x_W^2 \lambda_{ij}^2)(x_W+x_+)
+ x_W r_W +x_+ r_+]  \non \\
&&  -x_+( r_+ + r_W + \lambda_{ij}^2 x_W^2)(\beta_W^2 + 
\lambda_{ij}^2 x_W^2)
-\beta_W^2( x_+ \beta_W^2 +4\lambda_{ij}^2 x_W^3+
x_+ x_W^2 \lambda_{ij}^2 ) \non \\
&&- x_+ r_W r_+  \bigg] + \lambda_{ij}^2 
\left[ \lambda_{ij}^2 x_+ x_W -2 
\beta_W \lambda_{ij} (x_W+x_+) + 4\beta_W^2   \right] \non 
\eeq
\beq
a_i ^+ \equiv a_i^W \ (x_W \leftrightarrow x_+ \ , 
\ r_W \leftrightarrow r_+ )
\eeq

\subsubsection*{C2: $W_L W_L \to AA $}

For pseudoscalar Higgs bosons, amplitude and cross section are
significantly simpler since only a few diagrams contribute to the $AA$
final state and, moreover, the masses of the final-state particles are
equal:
\beq
{\cal M}_{LL} &=& \frac{G_F \hat{s}}{\sqrt{2}}\left\{ 
(1+\beta_W^2) \left[1 + 
\frac{\lambda_{hAA} d_1} {(\hat{s} -M_h^2)/M_Z^2}
+ \frac{\lambda_{H AA} d_2} {(\hat{s} -M_H^2)/M_Z^2} \right] \right. 
\non \\
&&+ \left. \frac{1}{\beta_W \beta_A} \left[ 
\frac{ (\beta_W - \beta_A \cos \theta )^2}{\cos \theta-x_A} -  
\frac{ (\beta_W +\beta_A \cos\theta)^2}{\cos \theta+x_A} 
\right] \right\}
\eeq
with 
\beq
\beta_A=(1-4M_A^2/\hat{s})^{1/2} \quad \mathrm{and} \quad 
x_A=(1-2 M_A^2/\hat{s} +2 M_{H^\pm}^2/\hat{s} -2 M_W^2/\hat{s})
/(\beta_W \beta_A)
\eeq
The total cross section for the subprocess $W_L W_L \to AA$ may be cast 
into the form
\beq 
\!\!\!\!\!\!\!\!\!\!\!\!\sigma_{LL} (AA) 
\!\!&=&\!\! \frac{G_F^2 M_W^4}{4\pi\hat{s}}\frac{\beta_A}
{\beta_W (1-\beta_W^2)^2} \Bigg\{ (1+\beta_W^2)^2 \left[ 1 + 
\frac{\lambda_{hAA} d_1 } {(\hat{s} -M_h^2)/M_Z^2} + 
\frac{\lambda_{HAA} d_2 }{(\hat{s} -M_H^2)/M_Z^2} \right]^2  \non \\
&&+ 2(1+\beta_W^2) \left[ 1 + 
\frac{\lambda_{hAA} d_1 }{(\hat{s} -M_h^2)/M_Z^2}
\; + \; \frac{\lambda_{HAA} d_2 }{(\hat{s} -M_H^2)/M_Z^2} 
\; \right] \ \\
&& \times \frac{1}{\beta_W \beta_A}
\bigg[ (x_A \beta_A - \beta_W)^2 \log \frac{x_A-1}{x_A+1} 
 + 2\beta_A (x_A \beta_A -2\beta_W) \bigg] + \frac{1}{\beta_A^2 \beta_W^2} 
\times \non \\
&& \Bigg(
\log \frac{x_A-1}{x_A+1} \left[ 3 \beta_A^2 x_A (\beta_A x_A-2 \beta_W)^2
+\beta_W^2(2\beta_A^2 x_A -4 \beta_W \beta_A -\beta_W^2/x_A) \right] \non \\
&& 
+ \frac{2}{x_A^2-1} \left[ (3x_A^2 \beta_A^2-2\beta_A^2+\beta_W^2)
(\beta_Ax_A -2\beta_W)^2 + \beta_W^2 (\beta_A^2 x_A^2-3 \beta_W^2) 
\right] \Bigg) \Bigg\}  \non 
\eeq

\subsubsection*{C3: Asymptotia}

For asymptotic energies, the leading part of the $WW$ cross sections
does not involve the trilinear couplings $H_i H_j H_k$ or $H_i H_j A$.
Note however that the convoluted leptonic cross sections \ee
$\longrightarrow \bar{\nu}_e \nu_e H_i H_j$ and $AA$ are dominated by
the threshold regions, also for asymptotic \ee ~energies, so that the
observable leptonic high-energy cross sections are indeed sensitive in
leading order to the trilinear Higgs couplings.

