\begin{picture}(0,0)%
\special{psfile=2b1.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(249,324)(589,-73)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=2b2.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(249,324)(589,-73)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=2t1.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(290,324)(548,-73)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=2t2.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(290,324)(364,-73)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=2to1gluon1.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(774,729)(964,-1150)
\put(1566,-1101){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$\kf_j$}}}
\put(1301,-556){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$k$}}}
\put(981,-1101){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$\kf_i$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=2to1gluon2.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(774,745)(964,-1180)
\put(1091,-1131){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$\kf_i+\kf_j$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=3a2-1.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(324,324)(889,-73)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=3an2f12.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(249,324)(1714,-973)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=3an2f12mab.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(249,739)(1714,-1169)
\put(1801,-1111){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}{\scriptsize $\{a\}$}}}}
\put(1801,-586){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{rm}{\scriptsize $\{b\}$}}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=3b12.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(324,399)(514,-148)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=3b23.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(324,399)(589,-148)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=3s1-2.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(399,399)(1039,-148)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=3t12.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(324,399)(514,-148)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=4a23-1.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(324,324)(889,-73)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=4b123.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(324,399)(514,-148)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=4b234.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(324,399)(589,-148)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=5a234-1.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(324,324)(889,-73)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=5an4f12.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(399,324)(1714,-973)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=5an4f13.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(399,324)(1864,-973)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=5an4f14l.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(399,324)(1264,-973)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=5an4f15l.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(421,324)(1264,-973)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=5an4f23.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(399,324)(1639,-973)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=5an4f24l.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(399,324)(1189,-973)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=5an4f25l.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(414,324)(1189,-973)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=5an4f34.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(399,324)(1564,-973)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=5an4f35l.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(420,324)(1489,-973)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=5an4f45.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(399,324)(1489,-973)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=5b1234.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(348,399)(490,-148)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=5b2345.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(348,399)(814,-148)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=6a2345-1.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(324,324)(889,-73)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=6b12345.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(399,399)(439,-148)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=6b23456.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(399,399)(814,-148)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=c12.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(324,324)(589,-673)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=c123.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(324,324)(589,-673)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=c1234.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(324,331)(589,-680)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=c12345.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(324,324)(589,-673)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=c123456.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(331,324)(589,-673)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=color2to2.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(216,324)(868,-223)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=color2to3.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(366,324)(868,-223)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=color2to3ex12.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(366,324)(868,-223)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=color2to3ex23.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(366,324)(518,-223)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=color2to4.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(516,324)(868,-223)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=color2to5.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(666,324)(868,-223)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=color2to6.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(816,324)(868,-223)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=d1234.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(474,245)(964,-673)
\put(1160,-553){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{rm}$d$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=d12cdc34.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(399,205)(1114,-373)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=d12d34.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(394,104)(591,-369)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=d20diag1.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(1150,624)(776,-1348)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=d20diag2.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(1150,624)(776,-1348)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=d20diag3.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(1150,624)(776,-1348)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=d20diag4.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(1150,624)(776,-1348)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=d4fff.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(699,470)(814,-898)
\put(1160,-553){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{rm}$d$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=d_abc.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(675,597)(1126,-298)
\put(1351,164){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$a$}}}
\put(1126,-286){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$b$}}}
\put(1801,-286){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$c$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=d_acb.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(675,597)(1126,-298)
\put(1351,164){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$a$}}}
\put(1126,-286){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$b$}}}
\put(1801,-286){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$c$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=dddd.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(849,205)(1114,-373)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=dddel.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(621,205)(1114,-373)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=ddf.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(624,205)(1114,-373)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=ddff.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(849,205)(1114,-373)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=del_abf_cde.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(547,199)(816,-373)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=del_acf_bde.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(470,199)(893,-373)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=delab.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(687,301)(526,-211)
\put(526,-211){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$a$}}}
\put(1201,-211){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$b$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=deldd.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(622,205)(891,-373)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=deldeldel.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(619,104)(591,-369)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=delff.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(622,199)(891,-373)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=dfd.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(624,205)(1114,-373)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=dfdf.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(849,205)(1114,-373)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=dffd.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(849,205)(1114,-373)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=diagexample.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(3300,1485)(376,-1411)
\put(1726,-1411){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$\kf_1$}}}
\put(2626,-1411){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$\kf_3$}}}
\put(3076,-61){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$\sum_{j=1}^4 \kf_j -\lf$}}}
\put(3076,-1411){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$\kf_4$}}}
\put(1651,-61){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$\lf$}}}
\put(3676,-661){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}propagator}}}
\put(3676,-856){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$(\lf-\kf_1-\kf_2-\kf_3)^{-2}$}}}
\put(376,-856){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$(\lf-\kf_1)^{-2}$}}}
\put(376,-661){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}propagator}}}
\put(2176,-1411){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$\kf_2$}}}
\put(2101,-181){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}vertex A}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=expsum2at85.pstex}%
\end{picture}%
\setlength{\unitlength}{3355sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(976,789)(578,-973)
\put(976,-511){\makebox(0,0)[lb]{\smash{\SetFigFont{11}{13.2}{rm}$D_3$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=expsum3at85.pstex}%
\end{picture}%
\setlength{\unitlength}{3355sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(976,789)(578,-973)
\put(976,-511){\makebox(0,0)[lb]{\smash{\SetFigFont{11}{13.2}{rm}$D_3$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=expsum4at85.pstex}%
\end{picture}%
\setlength{\unitlength}{3355sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(976,789)(578,-973)
\put(976,-511){\makebox(0,0)[lb]{\smash{\SetFigFont{11}{13.2}{rm}$D_3$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=f12345.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(624,245)(889,-673)
\put(1160,-546){\makebox(0,0)[lb]{\smash{\SetFigFont{8}{9.6}{rm}$f$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=f12cfc34.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(399,199)(1114,-373)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=f_abc.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(675,597)(1126,-298)
\put(1801,-286){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$c$}}}
\put(1351,164){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$a$}}}
\put(1126,-286){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$b$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=f_abcdel_de.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(546,199)(1039,-373)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=f_acb.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(675,597)(1126,-298)
\put(1801,-286){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$c$}}}
\put(1351,164){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$a$}}}
\put(1126,-286){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$b$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=f_adedel_bc.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(549,274)(814,-373)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=fdd.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(624,205)(1114,-373)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=fddf.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(849,205)(1114,-373)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=fdfd.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(849,205)(1114,-373)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=ffdd.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(849,205)(1114,-373)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=ffdel.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(621,199)(1114,-373)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=fff.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(624,199)(1114,-373)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=ffff.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(849,199)(1114,-373)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=gld21at85.pstex}%
\end{picture}%
\setlength{\unitlength}{3355sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(905,789)(559,-973)
\put(909,-511){\makebox(0,0)[lb]{\smash{\SetFigFont{11}{13.2}{rm}$D_2$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=gld22at85.pstex}%
\end{picture}%
\setlength{\unitlength}{3355sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(734,665)(645,-898)
\put(811,-511){\makebox(0,0)[lb]{\smash{\SetFigFont{9}{10.8}{rm}$D_{(2;0)}$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=gld23at85.pstex}%
\end{picture}%
\setlength{\unitlength}{3355sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(905,789)(559,-73)
\put(909,389){\makebox(0,0)[lb]{\smash{\SetFigFont{11}{13.2}{rm}$D_2$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=gld31at85.pstex}%
\end{picture}%
\setlength{\unitlength}{3355sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(905,789)(559,-973)
\put(909,-511){\makebox(0,0)[lb]{\smash{\SetFigFont{11}{13.2}{rm}$D_3$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=gld32at85.pstex}%
\end{picture}%
\setlength{\unitlength}{3355sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(734,665)(645,-898)
\put(811,-511){\makebox(0,0)[lb]{\smash{\SetFigFont{9}{10.8}{rm}$D_{(3;0)}$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=gld33at85.pstex}%
\end{picture}%
\setlength{\unitlength}{3355sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(905,789)(559,-73)
\put(909,389){\makebox(0,0)[lb]{\smash{\SetFigFont{11}{13.2}{rm}$D_2$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=gld34at85.pstex}%
\end{picture}%
\setlength{\unitlength}{3355sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(905,789)(559,-73)
\put(909,389){\makebox(0,0)[lb]{\smash{\SetFigFont{11}{13.2}{rm}$D_3$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=gld41at85.pstex}%
\end{picture}%
\setlength{\unitlength}{3355sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(976,789)(578,-973)
\put(976,-511){\makebox(0,0)[lb]{\smash{\SetFigFont{11}{13.2}{rm}$D_4$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=gld42at85.pstex}%
\end{picture}%
\setlength{\unitlength}{3355sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(778,669)(675,-898)
\put(879,-496){\makebox(0,0)[lb]{\smash{\SetFigFont{9}{10.8}{rm}$D_{(4;0)}$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=gld43at85.pstex}%
\end{picture}%
\setlength{\unitlength}{3355sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(976,789)(578,-973)
\put(976,-511){\makebox(0,0)[lb]{\smash{\SetFigFont{11}{13.2}{rm}$D_2$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=gld44at85.pstex}%
\end{picture}%
\setlength{\unitlength}{3355sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(976,789)(578,-973)
\put(976,-511){\makebox(0,0)[lb]{\smash{\SetFigFont{11}{13.2}{rm}$D_3$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=gld45at85.pstex}%
\end{picture}%
\setlength{\unitlength}{3355sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(976,789)(578,-973)
\put(976,-511){\makebox(0,0)[lb]{\smash{\SetFigFont{11}{13.2}{rm}$D_4$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=gld51at85.pstex}%
\end{picture}%
\setlength{\unitlength}{3355sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(1130,789)(559,-973)
\put(1044,-511){\makebox(0,0)[lb]{\smash{\SetFigFont{11}{13.2}{rm}$D_5$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=gld52at85.pstex}%
\end{picture}%
\setlength{\unitlength}{3355sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(931,654)(664,-898)
\put(953,-496){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$D_{(5;0)}$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=gld53at85.pstex}%
\end{picture}%
\setlength{\unitlength}{3355sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(1130,789)(559,-73)
\put(1044,389){\makebox(0,0)[lb]{\smash{\SetFigFont{11}{13.2}{rm}$D_2$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=gld54at85.pstex}%
\end{picture}%
\setlength{\unitlength}{3355sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(1130,789)(559,-73)
\put(1044,389){\makebox(0,0)[lb]{\smash{\SetFigFont{11}{13.2}{rm}$D_3$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=gld55at85.pstex}%
\end{picture}%
\setlength{\unitlength}{3355sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(1130,789)(559,-73)
\put(1044,389){\makebox(0,0)[lb]{\smash{\SetFigFont{11}{13.2}{rm}$D_4$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=gld56at85.pstex}%
\end{picture}%
\setlength{\unitlength}{3355sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
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\title{Unitarity Corrections in High Energy QCD
  \thanks{Work supported in part by the EU Fourth Framework Programme
`Training and Mobility of Researchers', Network `Quantum Chromodynamics
and the Deep Structure of Elementary Particles',
contract FMRX-CT98-0194 (DG 12 - MIHT). 
CE was also supported in part by Deutsche 
Forschungsgemeinschaft and by the German Bundesministerium f\"ur 
Bildung, Wissenschaft, Forschung und Technologie.}
}

\author{Jochen Bartels\\
                II.\ Institut f\"ur Theoretische Physik, 
                Universit\"at Hamburg,\\
                Luruper Chaussee 149, D-22761 Hamburg, Germany\\
                E-mail: \email{bartels@x4u.desy.de}}
\author{Carlo Ewerz\\
             Cavendish Laboratory, Cambridge University,\\
             Madingley Road, Cambridge CB3 0HE, U.\,K.\\
             and\\
             DAMTP, Cambridge University,\\
             Silver Street, Cambridge CB3 9EW, U.\,K.\\ 
             E-mail: \email{carlo@hep.phy.cam.ac.uk}}

\abstract{
The high energy limit of QCD is investigated in the 
generalized leading logarithmic approximation. 
We study unitarity corrections to the BFKL Pomeron 
containing $t$-channel states with up to six gluons. 
Special attention is given to the field theory structure 
of the corresponding multi--gluon amplitudes. 
We discuss the transition from 
two to six gluons in the $t$-channel. 
}

\keywords{QCD, Deep Inelastic Scattering}

\preprint{
Cavendish-HEP-99/08\\
DAMTP-1999-94\\
DESY 99-119}

\begin{document} 

\section{Introduction}
\label{intro}

The scattering of hadrons at high energy and small 
momentum transfer is very interesting. 
On the one hand, hadronic scattering in this Regge limit 
has been very successfully described \cite{softpom} 
by Regge theory \cite{Collins}, 
based on analyticity and unitarity of the $S$-matrix. 
The behaviour of hadronic scattering amplitudes is  
encoded in the positions of so--called Regge poles and 
cuts in the plane of complex angular momentum. 
With Gribov's reggeon 
calculus \cite{Gribov} a consistent field theory of interacting 
reggeons was found and extensively studied \cite{BakerTerMar}. 
To the present day, Regge theory remains one of the deep truths of 
particle physics.
The positions of the Regge singularities, however, cannot be calculated 
in the framework of Regge theory. 

On the other hand, QCD has been firmly established as the 
quantum field theory describing the physics of strong 
interactions. It should therefore -- at least 
in principle -- be possible to derive Regge theory from QCD 
and to understand it in terms of quark and gluon 
degrees of freedom. This important and fundamental problem 
has not yet been solved. 
The main difficulty lies in the fact that the Regge limit 
is characterized by high parton densities. In addition, 
many hadronic scattering processes in the Regge kinematics 
are dominated by small momentum scales. Therefore 
non--perturbative effects are expected to become important. 

Fortunately, there are a few scattering processes that can 
be treated perturbatively even in the Regge limit. These processes 
involve the scattering of small color dipoles. An example 
is the scattering of highly virtual photons, 
in which the virtuality of the photons provides a hard scale. 
This process is of phenomenological interest \cite{gamgam,Brodsky}, 
as it can be measured at LEP \cite{L3} and, even better, 
at a future Linear Collider. For our present investigation we will 
look at it from a more theoretical perspective. 
The basic idea is to approach the difficult dynamics of QCD in 
the Regge limit from a perturbative starting point, namely highly 
virtual $\gamma^* \gamma^*$ scattering. Our hope is that the 
results obtained in the study of this special process are of more 
general relevance to the general dynamics of QCD in the Regge limit. 
In a later step, one should then of course try to understand the effects 
of non--perturbative dynamics on the emerging picture. 
One such non--perturbative effect would be the diffusion of 
momenta into the infra--red \cite{HansZigarre,Muellergeneral}, 
which can be controlled to a certain extent 
in $\gamma^* \gamma^*$ scattering 
by demanding sufficiently large photon virtualities. 

The first step towards a QCD-based description of hadronic 
scattering in the high energy limit was done when the 
BFKL (or perturbative) Pomeron was derived \cite{FKL,BL}. 
It describes a $t$-channel exchange carrying vacuum 
quantum numbers and resums the leading logarithms 
of the squared energy $s$ which compensate the smallness of the 
strong coupling constant. The BFKL Pomeron can be understood 
as the $t$-channel exchange of two (reggeized) gluons. It is 
expected to apply to processes which are governed by a single 
hard momentum scale. 

The BFKL Pomeron results in a power--like growth 
$\sigma \sim s^{0.5}$ 
of the total cross section at high energy. This eventually 
leads to a violation of unitarity, since according to the 
Froissart bound \cite{Froissartcite,Martin} 
the growth of total cross--sections 
can at most be logarithmic with the energy. 
Even though the actual violation of that bound would 
occur only well above the Planck scale it makes the 
leading logarithmic approximation inconsistent as far 
as our main goal is concerned. Namely, we cannot 
expect to derive a consistent picture of the 
Regge limit from an approximation 
that does not respect unitarity. 

Unitarity can be restored by including exchanges with more than 
two gluons in the $t$-channel. 
A set of so--called unitarity 
corrections can be identified that results in a unitary amplitude. 
This leads to the generalized leading logarithmic approximation 
\cite{Bartelsnuclphys,Bartelskernels,Bartelsinteq}. 
In order to satisfy unitarity 
also in all sub--channels, the number of gluons in the $t$-channel 
should not be fixed, and arbitrary numbers of gluons should be 
taken into account. The natural objects to consider are therefore 
amplitudes describing the production of $n$ gluons in the $t$-channel. 
These $n$-gluon amplitudes obey a tower of coupled integral 
equations, the first of which ($n=2$) coincides with the BFKL equation. 
In the Regge limit the dynamics of the interaction is effectively 
reduced to the two transverse dimension. Consequently, the integral 
equations are equations in two--dimensional 
tranverse momentum space. The complex angular momentum 
acquires the meaning of an energy--like variable, whereas its 
conjugate variable, i.\,e.\ rapidity, can be interpreted as a 
time--like variable. This is in agreement with the fact that 
real gluon emissions in the (generalized) leading logarithmic 
approximation are strongly ordered in rapidity. 

So far, the amplitudes have been investigated for up to $n=4$ gluons. 
Although a complete analytic solution for the four--gluon amplitude 
is still missing, a series of remarkable properties 
of the amplitudes has been found. 
The three--gluon amplitude and 
a part of the four--gluon amplitude can be written as superpositions of 
two--gluon amplitudes. This is called the reggeization of that parts, 
generalizing the well--known reggeization of the gluon 
in non--abelian gauge theories \cite{Lipatovreggeierung}. 
A consequence of this phenomenon is that the amplitudes exhibit 
a very interesting field theory structure. They consist of only very 
few building blocks: states of two and four interacting $t$-channel 
gluons and a vertex coupling the two-- to the four--gluon state 
\cite{BPLB,BW}. 
The latter is a number--changing element and turns the 
quantum--mechanical problem of the $n$-gluon states 
\cite{Bartelskernels,BKP} into a field 
theory of unitarity corrections. A further striking feature is 
observed after a Fourier transformation to two--dimensional 
impact parameter space. The two--gluon state \cite{Lipatov86}, 
the four--gluon states as well
as the two--to--four vertex are invariant under conformal 
transformations of the gluon coordinates \cite{BLW}. 
These properties indicate that the whole set of unitarity corrections 
in the generalized leading logarithmic approximation 
can be described by an effective conformal field theory in 
two--dimensional impact parameter space with 
rapidity as an additional parameter. It would be a great step 
forward to identify this conformal field theory. 
That would allow one to apply the extremely powerful 
methods of conformally invariant field theory in two dimensions 
\cite{BPZconf} (for a review see \cite{Ginsparg}). 

These interesting properties of the unitarity corrections have 
until now only been observed in the amplitudes with up to four gluons, 
and only the most basic elements of the potential field theory have 
been calculated explicitly. 
More generally, the existence of such a field theory structure 
has not yet been proved beyond four gluons. 
To find further evidence for the conjectured field theory structure 
and to extract its general properties requires the investigation 
of higher $n$-gluon amplitudes. 
A series of questions arise naturally in this program. 
An important goal is of course to find out about the existence of 
new elements of the effective field theory, like a possible 
two--to--six vertex, and to determine 
their properties. A further natural question is 
whether the two--to--four vertex can be generalized to the 
case in which the two incoming gluons are not in a color singlet
state, and how this generalization looks like. Closely related is 
the question of how a repeated two--to--four transition 
takes place. The study of the 
six--gluon amplitude will provide an answer to the 
question whether a Pomeron can split into two Odderons, 
the Odderon being the $C=-1$ partner of the Pomeron. 
An important issue will also be to test the conformal invariance 
of the higher $n$-gluon amplitudes in impact parameter space. 

In the present paper we study the amplitudes with five 
and six gluons in the $t$-channel. 
We show that the five--gluon amplitude reggeizes completely, i.\,e.\ 
is a superposition of two-- and four--gluon amplitudes. 
The corresponding mechanism appears to apply to each $n$-gluon 
amplitude with odd $n$. We extract a reggeizing part from the 
six--gluon amplitude and derive an integral equation for the 
remaining part. This equation contains a term which gives rise to 
at least one new element in the effective field theory. The other terms 
in the equation can be expressed through the known two--to--four 
vertex, indicating that further reggeization takes place in the 
six--gluon amplitude. We discuss our findings in the light of 
the potential effective field theory. 
The six--gluon amplitude allows us for the 
first time to determine the perturbative Pomeron--Odderon--Odderon 
vertex. 
Its precise form and properties, as well as those of the generalization 
of the two--to--four vertex to the color non--singlet will be 
discussed in a separate publication. 
Also the conformal invariance of the five-- and 
six--gluon amplitudes will be investigated in a separate publication. 

The problem of unitarity in high energy QCD has been addressed 
in many different ways, among them the following. 
The first attempt was made in ref.\ \cite{GLR}, 
but in that approach 
a set of diagrams has been left out which is included in our 
approach. These diagrams turn out to be essential for 
the emergence of higher $n$-gluon states. 
Considerable progress has been made in investigating these 
$n$-gluon states in the $t$-channel. 
Much interest was attracted by the symmetric three--gluon state 
(the Odderon) \cite{Lipatovodderon,Odderonpapers}. 
More generally, the large $N_c$ limit of the $n$-gluon states 
has been shown to 
be equivalent to a completely integrable model \cite{LipatovXXX}, 
namely the XXX Heisenberg model with non--compact 
$\mathrm{SL}(2,\C)$ spin zero \cite{GregoryFaddeev}. This 
remarkable result was subsequently used to obtain further interesting 
results \cite{Gregory}, including the Odderon as a special case. 
These results are relevant for our present approach since the 
$n$-gluon states -- although with finite $N_c$ -- 
are elements of the potential effective field theory. 
The two--to--four transition vertex and its relation to the 
triple--Pomeron vertex has been studied in \cite{BraunVacca,Vacca}. 
The latter can be obtained from the two--to--four vertex 
after projection on BFKL Pomerons. It has been 
shown to have the structure of a conformal 
three--point function in \cite{Hans}. That reference also 
discusses the analytic structure of the four--gluon state 
for finite $N_c$. 
An alternative approach to QCD at high energy is Mueller's 
dipole picture \cite{dipole}, which has been proved to be equivalent 
to the BFKL formalism \cite{WallonNavelet,Peschequiv}. 
The dipole picture has been used to study unitarity effects as well 
\cite{Muellerunitarity,Salam,Peschunitarity}. 
The value of the triple--Pomeron coupling obtained in the dipole 
picture \cite{Peschanski3p} 
agrees with the one calculated from the two--to--four 
vertex arising in our present approach \cite{Gregory3p}. 
Other related approaches are the formulation of an effective 
action for the Regge limit \cite{effaction,Lipatovphysrep}, 
a similar approach also aiming at a simplified effective 
theory for high energy scattering \cite{Verlinde}, and 
the method of operator expansion \cite{Balitsky}.
Renormalization group methods have been used to study 
unitarization in \cite{Weigert}. In \cite{Whitepreprint} 
a supercritical phase of reggeon field theory is studied 
in which the Pomeron consists of a single gluon and a 
'wee parton'. 
Approaches of more non--perturbative nature are 
for example the eikonal 
approximation in a soft gluon background \cite{Nachtmann}, 
the model of the stochastic vacuum \cite{Dosch}, and the 
semiclassical approach \cite{Buchmueller}. 
Also the recently conjectured AdS/CFT correspondence 
\cite{Maldacena} has been applied to the Regge limit 
\cite{RhoAdS,PeschAdS}. 
This short overview is of course not complete. 
All the different approaches should be regarded as complementary. 
As far as this has been tested, different approaches often lead to 
equivalent or similar results. The most difficult problem will 
eventually be the combination of the perturbative and non--perturbative 
approaches. 

The paper is organized as follows. In section \ref{BFKLchap} 
we recall important facts about the BFKL equation and  
briefly describe the violation of unitarity and the reggeization 
of the gluon. In section \ref{ngluon} we motivate the use of 
$n$-gluon amplitudes and outline their formal definition. 
The integral equations for the amplitudes are described in 
detail, including color algebra and the inhomogeneous terms 
consisting of quark loops with $n$ gluons attached. We introduce 
reggeon momentum diagrams and use them to classify the 
occurring momentum space integrals in terms of a small set 
of standard integrals. Section \ref{34V} contains a review 
of the known results about the three-- and four--gluon 
amplitudes as well as a discussion of the field theory structure 
found in these amplitudes. In section \ref{5gluons} we 
solve the equation for the five--gluon amplitude and discuss 
the general mechanism that is expected to cause complete 
reggeization in all amplitudes with an odd number of gluons. 
Section \ref{d6} is devoted to the study of the six--gluon amplitude. 
We extract a reggeizing part and derive the equation for 
the remaining part which is then discussed in detail. From that 
equation we also derive the Pomeron--Odderon--Odderon vertex. 
We conclude with a summary and an outlook. 
The two appendices contain algorithms for 
computing contractions of color tensors (appendix \ref{colorapp}) 
and for bringing the momentum space integrals to their 
standard forms (appendix \ref{momentumapp}). 

\section{The BFKL equation and violation of unitarity}
\label{BFKLchap}

In this section we give a very brief account of the 
basic properties of the BFKL equation. 
For more detailed accounts of the BFKL theory we 
refer the reader to the reviews 
\cite{Lipatovphysrep,Jeff,LipatovinMueller}. 

\subsection{The BFKL equation}
\label{BFKLsection}

The total hadronic cross section can be related to the 
elastic forward scattering amplitude via the 
optical theorem, 
\be
  \sigma_{\mbox{\scriptsize tot}} = 
  \frac{1}{s}\, \mbox{Im}\, A_{\mbox{\scriptsize el}}(s,t=0) 
\,,
\label{opttheorem}
\ee
where $t$ is the momentum transfer. 
It is convenient to use partial wave amplitudes, which  
in the high energy limit amounts to performing 
a Mellin transformation 
\be
  A(s,t) = i s \int_{\delta-i \infty}^{\delta+i \infty} 
           \frac{d\omega}{2 \pi i} \left( \frac{s}{M^2} \right)^\omega
           A(\omega,t) 
\,,
\label{Mellin}
\ee
thereby changing from squared energy $s$ to complex angular 
momentum $\omega$. 
The high energy behavior of the total cross section is then 
determined by the singularities of $A(\omega,t)$ in 
the $\omega$-plane, i.\,e.\ Regge poles and 
Regge cuts. The rightmost singularity gives the leading 
contribution and is identified with the Pomeron. 
As it describes an elastic scattering process 
(see (\ref{opttheorem})) it carries vacuum quantum numbers. 

If there is a hard momentum scale in the process the use 
of perturbation theory is justified. The smallness of the 
strong coupling constant $\alpha_s$ at large momentum 
scales can at high energy be compensated by large 
logarithms of the energy. 
The resummation of Feynman diagrams contributing 
to the leading logarithmic approximation (LLA)  
--- $\alpha_s \ll 1$ and $\alpha_s \log(s) \sim 1$ ---
results in the BFKL equation \cite{FKL,BL}. 
The longitudinal degrees of freedom decouple in 
high energy scattering, and the dynamics takes place 
in transverse space only. 
The full amplitude can be written in factorized form 
\be
A(\omega,t) = \int \frac{d^2\kf}{(2\pi)^3} 
\frac{d^2\kf'}{(2\pi)^3} 
\phi_\omega(\kf,\kf';\qf) \phi_1(\kf,\qf)
\phi_2(\kf',\qf)
\,,
\ee
$\phi_1$, $\phi_2$ being the impact factors 
of the scattered colorless states. The color neutrality 
implies 
\be
\phi_{1,2}(\kf=0,\qf) = \phi_{1,2}(\kf=\qf,\qf) = 0
\,,
\ee
which is important for the infrared finiteness of the amplitude. 
The function $\phi_\omega$ can be interpreted as the 
partial wave amplitude for the scattering of virtual 
gluons with virtualities $-\kf^2$, $-(\qf-\kf)^2$, 
$-\kf'^2$, and $-(\qf-\kf')^2$ respectively. 
It is described by the BFKL equation, which is an integral equation in 
the two--dimensional space of transverse momenta 
and of Bethe--Salpeter type. In detail it has the form 
\be
  \omega \phi_\omega(\kf,\kf';\qf) = 
 \phi^0(\kf,\kf';\qf) 
 + \int \frac{d^2\lf}{(2\pi)^3} \,
 \frac{1}{\lf^2 (\qf-\lf)^2} 
 K_{\mbox{\scriptsize BFKL}}(\lf,\qf-\lf;\kf,\qf-\kf) 
\,\phi_\omega(\lf,\kf';\qf) 
\,.
\label{BFKLeq}
\ee
$\phi^0$ is an inhomogeneous term, 
depending on the process under consideration. 
The integral kernel, the so-called 
BFKL or Lipatov kernel, is given by 
\bea
K_{\mbox{\scriptsize BFKL}}(\lf,\qf-\lf;\kf,\qf-\kf) &=& 
 -  N_c g^2 \left[ \qf^2 - \frac{\kf^2(\qf-\lf)^2}{(\kf-\lf)^2} 
 - \frac{(\qf-\kf)^2 \lf^2}{(\kf-\lf)^2} \right] \nonumber \\
 & & +(2\pi)^3 \kf^2 (\qf-\kf)^2 
 \left[ \,\beta(\kf) + \beta(\qf - \kf) \right] 
 \delta^{(2)}(\kf-\lf).
\label{Lipatovkernel}
\eea
The strong coupling constant is normalized to 
$\alpha_s= \frac{g^2}{4\pi}$ and is kept fixed in LLA. 
The function $\beta$ in 
the kernel is defined as 
\be
  \beta(\kf^2) = \frac{N_c}{2} g^2  \int \frac{d^2\lf}{(2 \pi)^3} 
          \frac{\kf^2}{\lf^2 (\lf -\kf)^2} 
\,.
\label{traject}
\ee
The function $\alpha(\kf^2) = 1 + \beta(\kf^2)$ 
is known as the gluon trajectory function. It passes through 
the physical 
spin $1$ of the gluon at vanishing argument $\kf^2=0$. 
Since it is the function $\beta$ that will frequently occur 
throughout this paper we call it (in obvious abuse of language) 
the trajectory function of the gluon as well. 

The factor $(-N_c)$ in the BFKL kernel is a color factor. If the 
two gluons entering the amplitude $\phi_\omega$ are not in a 
color singlet state the color factor $C_I$ will be 
different. 
(In that case the amplitude is not infrared finite and has to be 
regularized.) 
In general, $C_I$ depends on the 
irreducible representation $I$ of the two gluons. If 
$N_c=3$, the factor $C_I$ equals $-3$, $-\frac{3}{2}$, 
$-\frac{3}{2}$, $0$, $1$ 
for the irreducible representations ${\bf 1}$, ${\bf 8_A}$, ${\bf 8_S}$, 
${\bf 10}+{\bf \overline{10}}$, ${\bf 27}$, respectively.
We will use the symbol $K_{\mbox{\scriptsize BFKL}}$ 
for the BFKL kernel only if the two gluons are in a color singlet
state. 

The complex angular momentum $\omega$ acts as an energy 
variable in the BFKL equation. It can be shown to be conjugate 
to rapidity which thus acquires the meaning of a time variable 
in the BFKL equation. 

The general form of the solution of the BFKL equation 
can be derived from the integral equation by iteration. 
Accordingly, the elastic scattering amplitude at high energies 
has in LLA the structure of a gluon ladder in the $t$-channel, 
\be
 s \rarr \infty \quad \picbox{limit2to2.pstex_t} 
= \sum_{\mbox{\tiny number} \atop \mbox{\tiny of rungs}} 
 \,\,\picbox{ladder.pstex_t}
\,,
\label{ladder}
\ee
and the ladder rungs represent BFKL kernels. 
The BFKL equation can be solved analytically, but 
we will not make use of the explicit form of the solution. 
In the present paper we will also not need  
the conformal invariance \cite{Lipatov86} 
of the BFKL equation in impact parameter space. 

Finally, we mention the $t$-channel 
reggeon unitarity relation for the BFKL amplitude. 
Let $C(\omega;\kf,\kf';\qf)$ be the amputated 
BFKL amplitude, i.\,e.\ the amplitude 
$\phi_\omega$ without the reggeon propagator 
$(\omega-\beta(\kf^2) - \beta((\qf-\kf)^2))^{-1}$, 
\be
 C(\omega;\kf,\kf';\qf)=
\left(
\omega-\beta(\kf^2) - \beta((\qf-\kf)^2)
\right)
\phi_\omega(\kf,\kf';\qf)
\ee
with $t=-\qf^2$. After a continuation to the physical 
region of the $t$-channel it is possible to show \cite{FKL} 
with the help of the BFKL equation that 
\be
\mbox{disc}_\omega C =
\int d^2\lf \,(2\pi)^3
\frac{1}{\lf^2 (\qf-\lf)^2} 
\delta \left(\omega-\beta(\lf^2) - \beta((\qf-\lf)^2)\right)
 C(\omega;\kf,\lf;\qf) C^*(\omega;\lf,\kf';\qf) 
\,.
\label{tunitaryBFKL}
\ee
The right hand side can be understood as a unitarity 
integral for the two--gluon amplitude $C$. 

\subsection{Violation of unitarity}
\label{violation}

The leading singularity in the $\omega$-plane 
can be determined analytically from the BFKL 
equation, and leads a 
power--like growth of the amplitude 
$A \sim s^{(1+\omega_{\mbox{\tiny BFKL}})}$ 
with the exponent 
$\omega_{{\mbox{\scriptsize BFKL}}} 
= \alpha_s N_c \,4 \ln 2/\pi \simeq 0.5 $. 
Consequently, the total cross section in the leading logarithmic 
approximation grows like 
$\sigma_{\mbox{\scriptsize tot}}\sim s^{\omega_{\mbox {\tiny BFKL}}}$. 
This result is in conflict with the 
Froissart--Martin theorem \cite{Froissartcite,Martin} 
which derives a bound on the total hadronic 
cross section from unitarity. 
In detail, the Froissart--Martin bound is 
$\sigma_{\mbox{\scriptsize tot}} \le \mbox{const.} \,\log^2(s)$. 
A power--like growth will eventually violate 
this bound --- and thus unitarity --- 
at asymptotically large energies.
This observation is the starting point for the considerations 
in this paper. 

\subsection{Reggeization of the gluon}
\label{reggeization}

The phenomenon of reggeization 
in non--abelian gauge theories \cite{Lipatovreggeierung} is the following. 
The $t$-channel exchange in the BFKL equation carrying 
the quantum numbers of a gluon, i.\,e.\ a color octet\footnote{We 
speak of 'octet' to mean the adjoint representation 
also for general $N_c$.} exchange, gives rise to a special solution. 
For antisymmetric 
color octet exchange the color factor in the BFKL kernel 
is $N_c/2$ instead of $N_c$. 
(In this color representation the amplitude is not infrared finite 
and a regularization has to be applied.) 
Let us further assume that the inhomogeneous term $\phi_0$ 
is a function of $(\kf_1+\kf_2)$. 
Then the equation exhibits the solution 
\be
\phi^{\bf 8_A} (\kf_1+\kf_2) = 
\frac{\phi^{\bf 8_A}_0 (\kf_1+\kf_2)}{\omega -\beta(\kf_1+\kf_2)} 
\,.
\label{Pole}
\ee
This solution has a pole and can thus be interpreted as describing 
the propagation of a single particle with momentum $(\kf_1+\kf_2)$ 
and the quantum numbers of a gluon. 
In a sense the gluon turns out to be a bound state of two gluons here. 
The fact that the gluon is a composite state of gluons is 
often termed 'bootstrap'. 
It indicates that the correct degrees 
of freedom in high energy QCD are not elementary gluons but 
so--called reggeized gluons. The reggeized gluon can be understood as a 
collective excitation of the gauge field. 

When we interchange the two gluons in the color octet amplitude 
above we find that its sign changes. This fact gives rise to the notion 
of signature. It characterizes the behavior under the exchange 
of two gluons, that is the simultaneous interchange of color and 
momentum labels. The reggeized gluons obviously carries negative 
signature. 

\boldmath
\section{Integral equations for $n$-gluon amplitudes}
\unboldmath
\label{ngluon}

\boldmath
\subsection{The $n$-gluon amplitudes}
\unboldmath
\label{amplitudes}

The method suited to restore unitarity in the perturbative approach 
is known as the generalized leading logarithmic approximation (GLLA).  
It constitutes an approximation scheme in which 
a minimal set of non--leading corrections is identified that 
leads to a unitary amplitude. The minimal set of 
contributions required here comprises 
subleading corrections with a larger number $n$ of 
reggeized gluons in the $t$-channel. These are what 
we call unitarity corrections to the BFKL Pomeron. 
It is necessary to include all possible $n$ to eventually fulfill the 
requirement of unitarity. Quark exchanges in the $t$-channel 
are always suppressed by powers of the energy $s$ 
with respect to the corresponding gluon exchanges and are  
not taken into account in GLLA. 

The most complete approach to a systematic treatment 
of unitarity corrections in a perturbative framework 
was formulated 
in \cite{Bartelsnuclphys,Bartelskernels,Bartelsinteq}. 
Its aim is to arrive at an effective description of QCD in the Regge 
limit in the spirit of a reggeon field theory 
\cite{Gribov,BakerTerMar}, the requirement of unitarity 
being built in from the very beginning. 
Of course, the BFKL amplitude with its two $t$-channel gluons 
should be incorporated into the whole approach as the 
lowest order contribution. 
It appears natural to define partial wave amplitudes 
similar to the BFKL amplitude but now with $n$ reggeized gluons 
in the $t$-channel. In eq.\ (\ref{tunitaryBFKL}) we have seen the 
$t$-channel reggeon unitarity equation for the BFKL amplitude. 
%It is valid when the amplitudes are 
%continued to positive $t$, that is to the physical region of the $t$-channel. 
It can be summarized symbolically as 
\be
 \mbox{disc}_\omega A(\omega,t) \sim C_2 C_2^* 
\,, 
\label{tunitarity2}
\ee
where $C_2$ is an amputated amplitude. We include a 
reggeon propagator to arrive at 
$D_2=C_2 [\omega-\beta(\kf_1)-\beta(\kf_2)]^{-1}$. 
We have chosen a new symbol $D_2$ for the BFKL amplitude here 
since we now want to consider the special physical process 
of  $\gamma^* \gamma^*$-scattering. 
The amplitude $D_2$ obeys the BFKL 
equation (\ref{BFKLeq}) with a special choice of 
the inhomogeneous term, namely the coupling of the two gluons 
to the photons via a quark loop. 
The unitarity relation (\ref{tunitarity2}) can  
be generalized to include $n$-gluon intermediate states. 
Symbolically, the generalization has the form 
\be
 \mbox{disc}_\omega A(\omega,t) \sim \sum_{n=2}^\infty C_n C_n^* 
\,.
\label{tunitarityn}
\ee
We include a reggeon propagator to find 
$D_n=C_n (\omega -\sum_{i=1}^n \beta(\kf_i))^{-1}$. 
Here the $D_n$ describe the production of $n$ on--shell gluons 
in the $t$-channel. They are non--amputated amplitudes, 
i.\,e.\ have propagators on the external gluon lines. 
The correct treatment of $t$-channel unitarity relations 
including multi--particle amplitudes is highly non--trivial. 
To our knowledge the most complete survey of this 
extensive technology is \cite{Abarbanel}, 
and the reader is referred to 
that reference for the details. We will content ourselves here with 
having motivated the physical meaning of the $n$-gluon 
amplitudes $D_n$ we are going to study. 
Further below we will briefly outline the formal definition of 
the amplitudes $D_n$.  

Once we take into account subleading 
corrections with more reggeized gluons in the $t$-channel 
and consider multi--particle amplitudes like the 
$n$-gluon amplitudes $D_n$ 
there exist of course subchannels of the scattering 
amplitude and we have to care about unitary also 
in the subchannels. The approach initiated in 
\cite{Bartelsnuclphys,Bartelskernels,Bartelsinteq} 
is designed to ensure unitarity not 
only in the direct channel but also in all subchannels. 
This implies that the number of gluons in the $t$-channel 
gluons is not conserved. Due to that the set of integral 
equations for the $n$-gluon amplitudes is turned 
into a tower of coupled equations including number--changing 
integral kernels. (A detailed 
description of the integral equations will follow in 
section \ref{inteqsection}.) 

The non--conservation of the number of reggeized 
gluons in the $t$-channel evolution contrasts 
sharply with the situation in the 
Bartels--Kwieci\'nski--Prasza{\l}owicz 
(BKP) equations \cite{Bartelskernels,BKP}. 
The latter describe the $t$-channel evolution of a compound 
state of a fixed number of reggeized gluons in the Regge limit. 
Their large-$N_c$ limit turned out to be 
equivalent to a completely integrable model 
\cite{LipatovXXX}, 
namely the XXX--Heisenberg model with conformal 
$\mbox{SL}(2,\C)$ spin $s=0$ \cite{GregoryFaddeev}. 
Although the BKP equations do not apply directly 
to our $n$-gluon amplitudes $D_n$ they will play an 
important role in the effective field theory of 
unitarity corrections that we are heading for. 
As we will explain in more detail in section \ref{4ftconf} 
there will be different elements in the effective field 
theory. 
First we will have $n$-particle Green functions, 
which are number--conserving elements. 
Their behavior will be governed by the 
BKP equations. In addition, there will be 
number--changing elements which we will call vertices. 
They arise as a unique feature of the approach pursued here, 
and turn the quantum mechanical problem described by 
the BKP equations into a quantum field theory. 

We will now outline the formal definition 
of the $n$-gluon amplitudes $D_n$. 
The way the amplitudes $D_n$ are defined 
is inspired by Regge theory. A condensed but still rather 
extensive description of the methods that have to be used 
here can be found in 
\cite{Whitepreprint}, more complete 
reviews are contained in \cite{BrowerDeTar} and 
\cite{WhiteIntJ}. 
The procedure starts from a physical $2+n$ multi--particle 
scattering process. One identifies certain 
kinematical variables with the use of so--called Toller 
diagrams and hexagraphs, and after defining partial waves 
one can eventually get the desired amplitude by taking 
an appropriate mixed Regge and helicity--pole limit. 
Although it is in principle possible, we will not carry out 
this program for the amplitudes under consideration 
in this paper. 
The procedure becomes technically complicated very 
quickly with the increasing number of gluons in the 
amplitude. Moreover, the physical processes we would 
have to start with for larger $n$ are rather artifical. 
However, the method outlined here appears to be very 
natural for a special phenomenological application 
of the four--gluon amplitude. In \cite{BW} a part of 
the four--gluon amplitude $D_4$ in which the 
four gluons form two pairs of color singlets was 
used for the description of the process of 
high-mass diffractive dissociation in deep 
inelastic electron--proton scattering. 
The rapidity gap between the proton and the 
diffractively produced system is caused by a 
colorless exchange between the proton and the 
photon. The latter is modelled by a two--gluon 
exchange. In the amplitude for this process 
the initial state is therefore indeed a three--particle 
state, and the cross--section takes the form of 
a three--to--three scattering process. 
In \cite{BW} the appropriate limit for this process 
was identified 
as the well--known triple Regge limit in which 
$s \gg M^2 \gg Q^2 \gg \Lambda^2_{\mbox{\scriptsize QCD}}$, 
where $M^2$ is the invariant mass of the diffractively produced 
particles. 

The $n$-gluon amplitudes are defined to have 
as external lines two photons and $n$ gluons. 
The photons are coupled to the gluons via their 
splitting into a quark--antiquark pair to 
which the gluons are attached. 
Being $n$-gluon amplitudes the $D_n$ carry as 
arguments $n$ color labels $a_i$ in addition to the transverse 
momenta $\kf_i$ of the gluons. The color labels correspond 
to generators $t^{a_i}$ of the gauge group $\mbox{SU}(N_c)$ 
in the adjoint representation. 
As partial waves the $D_n$ have also as an argument  
the complex angular momentum $\omega$. 
Since all $D_n$ will carry the same argument $\omega$ we 
will suppress it in our notation. 
In our notation we will suppress the photon momenta as well. 
The $n$-gluon amplitudes are thus characterized as 
\be
 D_n^{a_1\dots a_n}(\kf_1, \dots, \kf_n) 
\,.
\ee
The transverse gluon momenta $\kf_i$ in the amplitude 
are all chosen to be outgoing. 
The $D_n$ are non--amputated amplitudes, i.\,e.\ they have 
propagators for the outgoing reggeized gluons. 
Further they are multiply--cut amplitudes. We take 
discontinuities in the $n-1$ energy variables defined from 
one photon and the $i$ first gluons ($1\le i \le n-1$), 
\be
 s_i = \left( p_{\gamma^*;1} + 
\sum_{j=1}^i p_j \right)^2
\,.
\label{energiesforcuts}
\ee
Here $p_{\gamma^*;1}$ and $p_j$ are the 
four--momenta of the left photon and the 
gluons, respectively. 
The amplitudes $D_n$ can be defined for the non--forward 
direction 
\be
  \sum_{i=1}^n \kf_i \neq 0 
\ee
as well. All results in this paper will hold for the non--forward 
direction, and we will not mention this in each case separately. 

The simplest of the $n$-gluon amplitudes is $D_2$. 
It is identical with the well--known BFKL amplitude 
discussed in section \ref{BFKLsection}. 
There the inhomogeneous term in the BFKL equation was not 
specified. In $D_2$ it is given by the 
lowest order coupling of the two $t$-channel gluons to the virtual 
photons through a quark loop. 
The two outgoing gluons in the BFKL amplitude $D_2^{a_1a_2}$ 
are in a color singlet such that we can factorize the color 
structure and define the momentum part $D_2$ by
\be
  D_2^{a_1a_2} (\kf_1,\kf_2) = \delta_{a_1a_2} D_2 (\kf_1,\kf_2) 
\,.
\label{splitd2}
\ee
We recall two simple but very important properties of 
the momentum part $D_2$ of the BFKL amplitude. 
The first is that it vanishes when one of its momentum 
argument vanishes, 
\be
  D_2 (\kf_1,\kf_2) |_{\kf_1=0} = D_2 (\kf_1,\kf_2) |_{\kf_2=0} = 0
\label{nullstelld2}
\,.
\ee
The second is the symmetry in its two momentum arguments, 
\be
  D_2 (\kf_1,\kf_2) = D_2 (\kf_2,\kf_1) 
\label{symmetryd2}
\,.
\ee

Concluding this section, we introduce a shorthand notation 
for the arguments of $D_2$. Later we will use it for 
the arguments of other functions as well. 
In the case that an argument of a function, say $D_2$, 
is a sum of two or more transverse momenta 
we will only give the indices of these momenta, and 
a string of indices stands for the sum of the corresponding 
momenta.
So we have for example 
\be
  D_2(12,3) = D_2(\kf_1+\kf_2,\kf_3) \,.
\label{123notation}
\ee

\subsection{Integral equations}
\label{inteqsection}
The $n$-gluon amplitudes $D_n$ obey a tower of 
coupled integral equations. These have been derived in 
\cite{Bartelsinteq} by means of $s$-channel 
dispersion relations. 
Like the BFKL equation they are equations in two--dimensional 
transverse momentum space describing the $t$-channel evolution of 
the amplitudes under investigation. In this evolution, the complex 
angular momentum $\omega$ again plays the role of an 
energy variable. Its conjugate, i.\,e.\ rapidity, acquires the meaning 
of the time--like variable in the evolution. 

The integral equation for the two--gluon amplitude $D_2^{a_1a_2}$ 
is of course identical to the BFKL equation, 
\be
   \left( \omega - \sum_{i=1}^2\beta(\kf_i)\right) D_2^{a_1a_2} = 
D_{(2;0)}^{a_1a_2} 
+ K^{ \{b\} \rarr \{a\} }_{2\rarr 2} \otimes D_2^{b_1b_2} 
\,. 
\label{inteq2}
\ee
We have moved the trajectory functions to the left hand side of 
the equation to make the generalization to larger $n$ more transparent. 
The inhomogeneous term $D_{(2;0)}$ denotes the lowest order 
coupling of the two gluons to the photons via the quark loop. 
The quark loop will  be the subject of section \ref{quarkloop}. 
The integral kernel $K^{ \{b\} \rarr \{a\} }_{2\rarr 2}$ is, 
roughly speaking, the BFKL kernel (\ref{Lipatovkernel}) without 
the gluon trajectory functions $\beta$. An exact definition of the 
integral kernels will be given in section \ref{kernels}. 
The superscript $\{b\} \rarr \{a\}$ corresponds to the color 
labels of the in-- and outgoing gluons. 
The convolution symbol $\otimes$ stands for 
an integral $\int \frac{d^2 \lf}{(2 \pi)^3}$ over the loop momentum 
and a propagator $\frac{1}{\lf^2}$ for each of the two 
gluons entering the kernel from above. 

The integral equation for the three--gluon amplitude $D_3^{a_1a_2a_3}$ 
has the form 
\be
  \left( \omega - \sum_{i=1}^3 \beta(\kf_i) \right) D_3^{a_1a_2a_3} = 
D_{(3;0)}^{a_1a_2a_3} 
+ K^{ \{b\} \rarr \{a\} }_{2\rarr 3} \otimes D_2^{b_1b_2} 
+ \sum K^{ \{b\} \rarr \{a\} }_{2\rarr 2} \otimes D_3^{b_1b_2b_3}
\,.
\label{inteq3}
\ee
The inhomogeneous term $D_{(3;0)}$ is now the quark loop with 
three gluons attached to it. 
In (\ref{inteq3}) we find for the first time 
a new kernel in the equation. 
$K^{ \{b\} \rarr \{a\} }_{2\rarr 3}$ is a transition 
kernel from two to three reggeized gluons. 
The second term on the right hand side of the equation therefore 
tells us that at some point in the $t$-channel evolution we can 
have a transition from two to three gluons. 
The last term describes the evolution of a system of three gluons, 
and the sum extends over all pairwise interactions of the three 
reggeized gluons via the kernel $K^{ \{b\} \rarr \{a\} }_{2\rarr 2}$. 
Let us look at the term in which the first and second gluon 
interact via a kernel. In this term the momentum and the color 
label of the third gluon are not affected. The kernel should 
thus be understood to contain a factor $\delta_{a_3b_3}$. 
The symbol $\otimes$ denotes again the integration over 
the loop momentum in the first two gluons and propagator 
factors for each of them. 
The other terms are obtained in analogy. 

The equations for higher $n$ are built in a very similar way. 
They contain as the respective inhomogeneous term the lowest 
order coupling of $n$ gluons to the quark loop. We denote this 
lowest order term as $D_{(n;0)}$. A detailed 
discussion of the quark loop and explicit formulae for 
$n \le 6$ will follow in section \ref{quarkloop}. 
In addition, the higher equations contain also higher transition 
kernels $K^{ \{b\} \rarr \{a\} }_{2\rarr m}$ 
from two to $m$ gluons. A general formula for arbitrary $m$ as well 
as the explicit formulae for $m\le 6$ are contained in 
section \ref{kernels}. 

Since we will  make use of the integral equations for up to $n=6$ 
in this paper, we now state them explicitly. The general rule 
should then be obvious. 
For $n=4$ we have
\bea
  \left( \omega - \sum_{i=1}^4 \beta(\kf_i) \right) D_4^{a_1a_2a_3a_4} &=& 
D_{(4;0)}^{a_1a_2a_3a_4} 
+ K^{ \{b\} \rarr \{a\} }_{2\rarr 4} \otimes D_2^{b_1b_2} 
+ \sum K^{ \{b\} \rarr \{a\} }_{2\rarr 3} \otimes D_3^{b_1b_2b_3} \nn \\
&&+ \sum K^{ \{b\} \rarr \{a\} }_{2\rarr 2} \otimes D_4^{b_1b_2b_3b_4}
\,,
\label{inteq4}
\eea
for $n=5$ the equation is 
\bea
 \left(\omega - \sum_{i=1}^5 \beta(\kf_i) \right) 
 D_5^{a_1a_2a_3a_4a_5} &=& 
D_{(5;0)}^{a_1a_2a_3a_4a_5} 
+ K^{ \{b\} \rarr \{a\} }_{2\rarr 5} \otimes D_2^{b_1b_2} \nn \\
&&+ \sum K^{ \{b\} \rarr \{a\} }_{2\rarr 4} \otimes D_3^{b_1b_2b_3} 
+ \sum K^{ \{b\} \rarr \{a\} }_{2\rarr 3} \otimes D_4^{b_1b_2b_3b_4} \nn \\
&&+ \sum K^{ \{b\} \rarr \{a\} }_{2\rarr 2} \otimes D_5^{b_1b_2b_3b_4b_5}
\,,
\label{inteq5}
\eea
and finally for $n=6$ the integral equation reads 
\bea
 \left(\omega - \sum_{i=1}^6 \beta(\kf_i) \right) 
 D_6^{a_1a_2a_3a_4a_5a_6} &=& 
D_{(6;0)}^{a_1a_2a_3a_4a_5a_6} 
+ K^{ \{b\} \rarr \{a\} }_{2\rarr 6} \otimes D_2^{b_1b_2} \nn \\
&&+ \sum K^{ \{b\} \rarr \{a\} }_{2\rarr 5} \otimes D_3^{b_1b_2b_3} 
+ \sum K^{ \{b\} \rarr \{a\} }_{2\rarr 4} \otimes D_4^{b_1b_2b_3b_4} \nn \\
&&+ \sum K^{ \{b\} \rarr \{a\} }_{2\rarr 3} \otimes D_5^{b_1b_2b_3b_4b_5}
\nn \\
&&+ \sum K^{ \{b\} \rarr \{a\} }_{2\rarr 2} \otimes D_6^{b_1b_2b_3b_4b_5b_6}
\,.
\label{inteq6}
\eea
Here we again have to explain the meaning of the convolutions 
and the summation symbols. In short, the sums contain all 
combinations of the respective amplitudes and kernels in which 
the $t$-channel gluons do not cross. Before we give an example 
of the combinatorics we write the integral equations in 
pictorial language,  which makes them easier to understand. 
\bea
\left( \omega - \sum_{i=1}^2 \beta(\kf_i)\right) 
\pichallo{gld21at85.pstex_t}{3} &=& \pichallo{gld22at85.pstex_t}{3}
 + \pichallo{gld23at85.pstex_t}{3} 
\label{inteq2diag}
\\
 \left( \omega - \sum_{i=1}^3 \beta(\kf_i)\right) 
\pichallo{gld31at85.pstex_t}{3} &=& \pichallo{gld32at85.pstex_t}{3} 
+ \pichallo{gld33at85.pstex_t}{3} + \sum \pichallo{gld34at85.pstex_t}{3} 
\label{inteq3diag}
\\
 \left( \omega - \sum_{i=1}^4 \beta(\kf_i)\right) 
\pichallo{gld41at85.pstex_t}{3} &=& \pichallo{gld42at85.pstex_t}{3} 
 + \pichallo{gld43at85.pstex_t}{3} + \sum \pichallo{gld44at85.pstex_t}{3} 
\nn \\
&&+ \sum \pichallo{gld45at85.pstex_t}{3} 
\label{inteq4diag}
\\
\left( \omega - \sum_{i=1}^5 \beta(\kf_i)\right) 
\pichallo{gld51at85.pstex_t}{3} &=&
\pichallo{gld52at85.pstex_t}{3} + \pichallo{gld53at85.pstex_t}{3} 
 +\sum \pichallo{gld54at85.pstex_t}{3} +\nn \\
&& +\sum \pichallo{gld55at85.pstex_t}{3} 
+\sum \pichallo{gld56at85.pstex_t}{3} 
\label{inteq5diag}
\eea

\bea
\left( \omega - \sum_{i=1}^6 \beta(\kf_i)\right) 
\pichallo{gld61at85.pstex_t}{3} &=&
\pichallo{gld62at85.pstex_t}{3} + \pichallo{gld63at85.pstex_t}{3} 
+ \sum \pichallo{gld64at85.pstex_t}{3}
\nn \\
&& + \sum \pichallo{gld65at85.pstex_t}{3}
  + \sum \pichallo{gld66at85.pstex_t}{3} \nn \\
&&+ \sum \pichallo{gld67at85.pstex_t}{3}
\label{inteq6diag}
\eea
In each diagram only two gluon lines from the amplitudes 
enter a kernel. 
An integration $\int \frac{d^2 \lf}{(2 \pi)^3}$ over the 
loop momentum and a propagator $\frac{1}{\lf^2}$ for 
each of the two gluons entering the kernel from above 
are implied again. 
The momenta and color labels of the other 
gluons are not changed. 
With the help of the pictorial notation it is also very easy 
to understand which combinations of amplitudes and 
kernels have to be convoluted such that $t$-channel 
gluons do not cross. For example, the 
sum in the last but one term in the 
equation (\ref{inteq4diag}) for the four--gluon amplitude 
extends over the four convolutions 
\be
\sum \pichallo{gld44at85.pstex_t}{3}  
= \pichallo{gld44at85.pstex_t}{3}  
+ \pichallo{expsum2at85.pstex_t}{3} 
+ \pichallo{expsum3at85.pstex_t}{3} 
+ \pichallo{expsum4at85.pstex_t}{3} 
\,.
\ee

We will now in turn discuss the elements occurring 
in the integral equations: 
the inhomogeneous terms $D_{(n;0)}$ representing 
the coupling of $n$ gluons to a quark loop and 
the integral kernels $K_{2 \rarr n}$. 
But before doing so, we first have to consider some 
color algebra. 

\subsection{Color structure}
\label{color}

In this section we collect some essential facts about color algebra. 
While doing so we also introduce the so--called birdtrack 
notation\footnote{A more complete account of this notation can 
be found in \protect\cite{Cvitanovic} where it is also applied to 
other Lie groups. Our normalization convention 
slightly deviates from \protect\cite{Cvitanovic}.} for 
structure constants. This diagrammatic 
notation is very useful for the problem of contracting indices of 
arbitrary color tensors. 
Such a powerful tool is needed here 
since tensor contractions constitute an essential part of the 
computations in the study of the integral equations. 
The diagrammatic method that serves this purpose is described in detail 
in appendix \ref{colorapp}. 

We are interested in the structure of the gauge group $G=\mbox{SU}(N_c)$ 
with generators $t^a$ ($a=1,\dots,N_c^2-1$) in the Lie algebra 
$\mbox{su}(N_c)$. 
The algebra is 
\be
  [ t^a, t^b ] = i f_{abc} t^c  \,.  \label{algebra}
\ee
For the case of $\mbox{su}(3)$ the $t^a$ are given by the well--known 
Gell--Mann matrices $\lambda^a$, $t^a= \lambda^a / 2$. 
The antisymmetric structure constants $f_{abc}$ can be expressed 
in terms of generators as 
\be
  f_{abc} = - f_{acb} = 
             - 2 i \, [ \mbox{tr} ( t^a t^b t^c ) 
                          - \mbox{tr} ( t^c t^b t^a ) ] \,,
\label{fabcdef}
\ee
diagrammatically 
\be
  f_{abc} = 
     \picbox{f_abc.pstex_t} =  \,- \hspace{-.2cm} \picbox{f_acb.pstex_t}
     =  - 2 i \left[ \, \picbox{trabc.pstex_t} - 
                  \,\picbox{tracb.pstex_t} \, \right] \,.
\ee
The thicker oriented lines stand for  quark color representations, 
the unorientated lines correspond to gluon color lines. 
The $f_{abc}$ are obviously invariant under cyclic permutations of 
the indices. Normalization of generators is such that for the 
Killing form 
\be
\mbox{tr} ( t^a t^b ) = \frac{1}{2} \delta_{ab} \hspace{.7cm} 
  \mbox{or} \hspace{.7cm} \picbox{trab.pstex_t}
              = \frac{1}{2} \,\, \picbox{delab.pstex_t} \,.
\label{deltanorm}
\ee
Using birdtrack notation the algebra (\ref{algebra}) becomes 
\be
   \picbox{lambdaab.pstex_t} \,-\, \picbox{lambdaba.pstex_t} = 
   i \, \picbox{lambdacf_abc.pstex_t} \,.
\label{algebrabird}
\ee
The anticommutator of two generators is 
\be
  \{ t^a, t^b \} = \fr{1}{N_c} \delta_{ab} + d_{abc} t^c \,,
\label{anticommab}
\ee
and the symmetric structure constants $d_{abc}$ are expressed 
in terms of generators as 
\be
  d_{abc}  = d_{acb} = 2 \, [ \mbox{tr} ( t^a t^b t^c ) 
                          + \mbox{tr} ( t^c t^b t^a ) ] \,,
\label{dabcdef}
\ee
in diagrams
\be
  d_{abc}  = \picbox{d_abc.pstex_t} = \picbox{d_acb.pstex_t}
           = 2 \left[ \, \picbox{trabc.pstex_t} +  
                  \,\picbox{tracb.pstex_t} \, \right] \,.
\ee
With this we have collected all basic elements of the 
birdtrack notation. 
We will slightly extend the birdtrack notation by the 
definition 
\be
  \left[ \picbox{3an2f12.pstex_t} \right] \star \Theta^{\{b\}}
   = \picbox{3an2f12mab.pstex_t} \star \Theta^{\{b\}}
   = f_{a_1a_2b_1} \delta_{a_3b_2} \Theta^{b_1b_2}
\ee
for the contraction of the set of color labels $\{b\}$ 
of an arbitrary color tensor $\Theta^{\{b\}}$. 
The symbol $\star$ stands for the contraction of the set $\{b\}$ 
in color space. The extension to more than two elements in the 
set $\{b\}$ is straightforward. 

Generalizing (\ref{fabcdef}) and (\ref{dabcdef}) we 
define further tensors that are built from traces of 
generators in the following way\footnote{In appendix 
\ref{colorapp} we will for brevity refer to tensors built from 
traces over generators in this way as 'standard 
tensors'. To the best of our knowledge this term 
does not carry a fixed meaning in the literature on 
Lie algebras.}  
\bea
  d^{b_1 b_2 \dots b_n} &=& 
           \mbox{tr} (t^{b_1} t^{b_2} \dots t^{b_n} )  
         + \mbox{tr} (t^{b_n} \dots t^{b_2} t^{b_1} ) \label{dallgdef} \\
  f^{b_1 b_2 \dots b_n} &=& 
    \fr{1}{i} \,  [ \mbox{tr} (t^{b_1} t^{b_2}\dots t^{b_n}) 
         - \mbox{tr} (t^{b_n} \dots t^{b_2} t^{b_1} ) ] 
    \label{fallgdef} 
\,.
\eea
The definitions are valid for any $n \in \N$. 
For $n=2,3$, however, the tensors arising from (\ref{dallgdef}),
(\ref{fallgdef}) are proportional\footnote{
In detail we have according to (\protect\ref{deltanorm}), 
(\protect\ref{dabcdef}), and (\protect\ref{fabcdef}) 
\be
 d^{ab} = \delta_{ab}\,; 
\quad \quad
\protect\label{differentnorm1}
 d^{abc} = \frac{1}{2} d_{abc}\,; 
\protect\label{differentnorm2} 
\quad \quad
 f^{abc} = \frac{1}{2} f_{abc}
\protect\label{differentnorm3}
\,.
\ee
} 
to $\delta_{b_1b_2}$, $f_{b_1b_2b_3}$, 
and $d_{b_1b_2b_3}$ respectively. 
For the cases $n=2,3$ we will stick to the conventional definitions of 
structure constants given earlier in (\ref{fabcdef}), (\ref{dabcdef}). 

The following three tensors are special cases 
of (\ref{dallgdef}), (\ref{fallgdef}). We will make extensive use of 
them throughout this paper.
The tensor 
\be
  d^{a b c d} = \picbox{d1234.pstex_t} =
           \mbox{tr} (t^{a} t^{b} t^{c} t^{d} )  
         + \mbox{tr} (t^{d} t^{c} t^{b} t^{a} )  
    = \picbox{tr1234.pstex_t} +\, \picbox{tr4321.pstex_t} 
\label{d4def} 
\ee
was used already in \cite{BW} in the investigation 
of the four--gluon amplitude. We now add to this the two tensors 
\be
  f^{a b c d e} = \picbox{f12345.pstex_t} =
    \fr{1}{i} \,  [ \mbox{tr} (t^{a} t^{b} t^{c} t^{d} t^{e}) 
         - \mbox{tr} (t^{e} t^{d} t^{c} t^{b} t^{a} ) ] 
    \label{f5def} 
    = \fr{1}{i} \, 
   \left[ \,\picbox{tr12345.pstex_t} 
      - \,\picbox{tr54321.pstex_t} \right]
\ee
and 
\be
d^{abcdef} = \mbox{tr} (t^{a} t^{b} t^{c} t^{d} t^{e}t^{f}) 
 +  \mbox{tr} (t^{f}t^{e} t^{d} t^{c} t^{b} t^{a} ) 
\,.
\ee
All three are invariant under cyclic permutations 
of the color labels (as are obviously all tensors defined according 
to (\ref{dallgdef}), (\ref{fallgdef})). 
It will turn out that these color tensors are very well suited 
for the whole problem of solving the integral equations. 
When interpreting the results in terms of reggeon color 
representations, the decomposition of these tensors into 
the lower order tensors $f_{abc}$, $d_{abc}$ and $\delta_{ab}$
is also useful:
\bea
  d^{a b c d} &=&  \frac{1}{2 N_c} \delta_{ab} \delta_{cd} 
                   + \frac{1}{4} (d_{abk}d_{kcd} - f_{abk}f_{kcd} ) 
                   \label{d4decomp} \\
              &=&  \frac{1}{2 N_c} \,\picbox{d12d34.pstex_t} 
                   + \frac{1}{4} \left( \, \picbox{d12cdc34.pstex_t} 
                     - \picbox{f12cfc34.pstex_t} \right) 
                   \label{d4decompdiag} \,,
\eea
as is easily derived using (\ref{algebra}) and (\ref{anticommab}). 
From this we get by cyclic permutation 
\be            
 d^{a b c d} =  \frac{1}{2 N_c} \delta_{ad} \delta_{bc} 
                   + \frac{1}{4} (d_{adk}d_{kbc} + f_{adk}f_{kbc} ) \,.
\ee
We further mention the property $d^{bacd} = d^{abdc}$, 
which turns out to be useful for calculational purposes. 
For $f^{a b c d e}$ we have 
\bea
  f^{a b c d e} &=&  \frac{1}{4N_c} ( \delta_{ab} f_{cde} 
                   + f_{abc} \delta_{de} ) \nn \\
             & &   + \frac{1}{8} ( f_{abk} d_{kcl} d_{lde}
                   + d_{abk} f_{kcl} d_{lde}
                   + d_{abk} d_{kcl} f_{lde}
                   - f_{abk} f_{kcl} f_{lde}) 
                   \label{f5decomp} \\
             &=& \frac{1}{4N_c} 
                         \left( \, \picbox{del_abf_cde.pstex_t} 
                                    + \picbox{f_abcdel_de.pstex_t} \right)
                         \nn \\
             & & + \frac{1}{8} 
                         \left(\, \picbox{fdd.pstex_t} 
                                    + \picbox{dfd.pstex_t} 
                                    + \picbox{ddf.pstex_t} 
                                    - \picbox{fff.pstex_t} \right)
                   \label{f5decompdiag}
\eea
and further identities can be obtained 
from (\ref{f5decomp}) by making use of the invariance 
under cyclic permutations. 
The tensor $d^{abcdef}$ can be decomposed in a similar way, 
\bea
%
% PLEASE do not change the layout of this equation.
%
 d^{a b c d e f} &=& 
 \frac{1}{4 N_c^2}\, \delta_{ab} \delta_{cd} \delta_{ef} \nn \\
& &\,+ \frac{1}{8 N_c} \,
( \delta_{ab} d_{cdk} d_{kef} 
  - \delta_{ab} f_{cdk} f_{kef} 
  + d_{abk} d_{kcd} \delta_{ef}
  - f_{abk} f_{kcd} \delta_{ef}) \nn \\
& &\,+ \frac{1}{16}\, 
 ( d_{abk} d_{kcl} d_{ldm} d_{mef} 
- d_{abk} d_{kcl} f_{ldm} f_{mef} 
- d_{abk} f_{kcl} d_{ldm} f_{mef} \nn \\
& & \hspace{.9cm}- d_{abk} f_{kcl} f_{ldm} d_{mef} 
- f_{abk} d_{kcl} d_{ldm} f_{mef} 
- f_{abk} d_{kcl} f_{ldm} d_{mef} \nn \\
& &\hspace{.9cm}- f_{abk} f_{kcl} d_{ldm} d_{mef} 
+ f_{abk} f_{kcl} f_{ldm} f_{mef} )
\,.
\label{d6decomp} 
\eea
In birdtracks it becomes
\bea
%
% PLEASE do not change the layout of this equation.
%
 d^{a b c d e f} 
&=& \frac{1}{4 N_c^2}\, \picbox{deldeldel.pstex_t} \nn \\
& &\,+ \frac{1}{8 N_c} \, 
 \left( \, \picbox{deldd.pstex_t} 
- \picbox{delff.pstex_t}
+ \picbox{dddel.pstex_t}
- \picbox{ffdel.pstex_t} \right) \nn \\
& &\,+ \frac{1}{16}\, 
 \left( \picbox{dddd.pstex_t}
- \picbox{ddff.pstex_t}
- \picbox{dfdf.pstex_t} 
- \picbox{dffd.pstex_t} \right. \nn \\
& & \hspace{.9cm} \left.
- \picbox{fddf.pstex_t} 
- \picbox{fdfd.pstex_t} 
- \picbox{ffdd.pstex_t} 
+ \picbox{ffff.pstex_t} \right)
\,.
\label{d6decompdiag}
\eea
and again other possible decompositions of $d^{abcdef}$ are 
obtained from this by cyclic permutations. 

To conclude this section about color tensors, we mention the 
well--known Jacobi identity 
\be
 f_{abk}f_{kcd} + f_{ack}f_{bkd} + f_{adk}f_{kbc} = 0
\,.
\label{Jacobi}
\ee

\subsection{The quark loop}
\label{quarkloop}
Let us now consider the inhomogeneous terms in the integral 
equations (\ref{inteq2})--(\ref{inteq6}). 
The terms $D_{(n;0)}^{a_1\dots a_n}$ describe 
the lowest order coupling of $n$ gluons to the quark loop. 
$D_{(2;0)}^{a_1a_2}$ is the sum of four cut diagrams 
\be
  D_{(2;0)}^{a_1a_2}(\kf_1,\kf_2) = \picbox{d20diag1.pstex_t} + 
      \picbox{d20diag2.pstex_t} + \picbox{d20diag3.pstex_t} +
      \picbox{d20diag4.pstex_t} 
  \,.
\ee
The two gluons are attached to the quark loop in all 
possible ways to preserve gauge--invariance. 
The $s$-channel cut (indicated in each diagram by the 
dotted vertical line) 
implies that the cut quark lines are set on--shell. 
$D_{(2;0)}^{a_1a_2}$ depends on the transverse momenta of the two gluons 
and on their color indices. The latter dependence is of course 
trivial and we can define the momentum part $D_{(2;0)}(\kf_1,\kf_2)$ 
of this amplitude by 
\be
  D_{(2;0)}^{a_1a_2}(\kf_1,\kf_2) = 
              \delta_{a_1a_2} D_{(2;0)}(\kf_1,\kf_2) \,.
\label{splitD20}
\ee
An analytic expression for $D_{(2;0)}(\kf_1,\kf_2)$ 
was first found in \cite{LevinRyskin}. 
The explicit formula for  $D_{(2;0)}$ will not be needed in the following. 
We will only make use of  the fact that $D_{(2;0)}$ is symmetric 
under the exchange of its transverse momentum arguments
\be
 D_{(2;0)}(\kf_1,\kf_2) = D_{(2;0)}(\kf_2,\kf_1) 
\ee
and vanishes if one of its two arguments vanishes, 
\be
\left. D_{(2;0)}(\kf_1,\kf_2) \right|_{\kf_1=0} = 
\left. D_{(2;0)}(\kf_1,\kf_2) \right|_{\kf_2=0} = 0
\,.
\ee
In addition, it will be important for the consistency of the 
integral equations that 
\be
D_{(2;0)}(\kf_1,\kf_2) < \mbox{const.}\, \log \kf_i^2
\ee
in the ultraviolet region, i.\,e.\ the growth with the momenta 
is not stronger than logarithmic. 

For the amplitudes $D_{(n;0)}$ with $n>2$ we again have to attach 
the $n$ gluons to the quark loop in all possible ways to preserve gauge 
invariance. But the fact that we are dealing with multiply--cut 
amplitudes reduces the number of diagrams to consider. 
The cuts forbid the crossing of $t$-channel gluons as indicated 
in Fig.~\ref{fig:nocrossing}. 
\FIGURE{
\input{nocross.pstex_t}
\caption{Cut amplitude contributing to the coupling of $n=4$ 
gluons to a quark loop}
\label{fig:nocrossing}
}
The ordering of the gluons along the loop is therefore fixed up to 
the possibility of coupling the gluons to the quark or the 
antiquark. We thus have $2^n$ cut diagrams 
for the inhomogeneous term $D_{(n;0)}$. 

It turns out that all the amplitudes $D_{(n;0)}^{a_1 \dots a_n}$ 
can be expressed in terms of $D_{(2;0)}$, the momentum part of 
$D_{(2;0)}^{a_1a_2}$ as defined in (\ref{splitD20}). 
To see how this reduction mechanism works let us have a look at 
two neighbouring gluons along the quark  loop. 
(They are not necessarily neighbouring as arguments 
of the amplitude, c.\,f.\ gluons 1 and 3 in Fig.~\ref{fig:nocrossing}.) 
In the high energy limit 
the quark--gluon vertices have to be contracted with a 
longitudinal momentum $p$. 
The Dirac trace over the quark loop then contains 
\be
  \picbox{2to1gluon1.pstex_t} \sim\,
     \mbox{tr}(\dots \slash p \slash k \slash p \dots) 
    \,\delta(k^2)  \,.
\ee
The $\delta(k^2)$ comes in since the quark has to be set on--shell. 
Using a Sudakov decomposition $k=\alpha q^\prime + \beta p + k_t$ 
with $q^\prime= q +xp$, $q^{\prime\,2}=p^2=0$, $2 p \cdot q = s$, and 
$k_t^2=-\kf^2$ one finds for this expression 
\be
  \mbox{tr}(\dots \slash p \dots) \, 
  \delta(\beta - \kf^2/(\alpha s)) 
  \, \simeq \picbox{2to1gluon2.pstex_t}\,.
\ee
This means that due to energy--momentum conservation 
only the sum $\kf_i + \kf_j$ of the two momenta enters. 
We can apply this to all gluons along the quark loop and 
thereby reduce the momentum part of each 
diagram to one corresponding to a 
diagram in which only two gluons are coupled to the quark loop. 
The color structure is not affected by this reduction. 
A remark is in order concerning the contribution 
in which all gluons are coupled to the 
quark line or the antiquark line. This term 
acts as a regularization term. As we will see below 
it can be added and subtracted in such a way that the 
full $D_{(n;0)}^{a_1 \dots a_n}$ can be 
expressed in terms of $D_{(2;0)}$. 

Let us now see how the color structure of the 
quark loop amplitudes builds up. 
Each diagram contributing to $D_{(n;0)}$ 
contains a trace over $n$ $\mbox{su}(N_c)$ generators. 
The $2^n$ diagrams come in pairs in the following sense. 
Consider a diagram with $k$ gluons coupled to the 
quark and $n-k$ gluons coupled to the antiquark. 
Then there is also a diagram with the $k$ gluons 
coupled to the antiquark instead of the quark and the 
other $n-k$ gluons now coupled to the quark. 
The momentum structure of the two diagrams is the 
same up to a factor $(-1)^n$. 
(This is because the coupling of a gluon to a quark 
effectively differs from that to an antiquark by a sign.) 
The color part of the second diagram is again 
a trace over generators $t^{a_i}$, but in the trace they now appear 
in reversed order compared to the first diagram. 
Adding the two diagrams one thus finds a color tensor of the 
kind $d^{a_1\dots d_n}$ for an even number $n$ of gluons 
and a tensor of the kind $f^{a_1\dots d_n}$ for an odd 
number $n$ of gluons 
(cf.\ (\ref{dallgdef}), (\ref{fallgdef}) for the definition of 
the $d$- and $f$-tensors). 
There are $2^{n-1}$ pairs of such diagrams. 
Having in mind that due to the photons at the two ends of the loop 
the color tensor is not altered if the first 
or $n$th gluon is coupled to the quark instead of the antiquark 
and vice versa, 
we conclude that the number of different color tensors 
contributing to the coupling of $n$ gluons to the quark loop is 
in general $2^{n-3}$ if $n \ge 3$. 
The color structures for two and three gluons attached to the 
quark loop are more or less trivial: in both cases there is only one 
color tensor ($\delta_{a_1a_2}$ and $f_{a_1a_2a_3}$, respectively). 

We have seen that the diagrams contributing to the quark loop 
come in $2^{n-1}$ pairs. Among them is a special pair, 
namely the one consisting of the two regularization terms 
mentioned earlier, in which all gluons couple to the quark line 
or the antiquark line. This pair can 
be added and subtracted with different color structures 
in such a way that the remaining $2^{n-1}-1$ pairs are 
regularized to give a $D_{(2;0)}$ each. Therefore the quark loop 
$D_{(n;0)}^{a_1 \dots a_n}$ can be expressed as a sum of 
$2^{n-1}-1$ amplitudes $D_{(2;0)}$ . 

In this paper, we will need the expressions 
for the quark loop with up to six gluons attached. 
For three gluons coupled to the quark loop we find 
\be
  D_{(3;0)}^{a_1a_2a_3}(\kf_1,\kf_2,\kf_3) 
= \fr{1}{2} g f_{a_1a_2a_3} \,
[ D_{(2;0)}(12,3) - D_{(2;0)}(13,2) + D_{(2;0)}(1,23) ]  \,, 
\label{d30}
\ee
where we use the notation introduced in (\ref{123notation}). 
In the case of four gluons the amplitude contains 
two different color structures, 
\bea
%\lefteqn{
D_{(4;0)}^{a_1a_2a_3a_4}(\kf_1,\kf_2,\kf_3,\kf_4)
% } \nn \\
 &=& \! - g^2 d^{a_1a_2a_3a_4} \, [ D_{(2;0)}(123,4) + D_{(2;0)}(1,234) 
                                  - D_{(2;0)}(14,23) ] 
   \nn \\
 & & \! - g^2 d^{a_2a_1a_3a_4}  \, [ D_{(2;0)}(134,2) + D_{(2;0)}(124,3)
                                 - D_{(2;0)}(12,34) \nn \\
 && \hspace{2.4cm} - D_{(2;0)}(13,24) ]
\,.
\label{d40}
\eea
When five gluons are coupled to the quark loop 
there appear four different color structures in the corresponding 
amplitude, 
\bea
\lefteqn{D_{(5;0)}^{a_1a_2a_3a_4a_5}(\kf_1,\kf_2,\kf_3,\kf_4,\kf_5)= } 
%  \nn 
\\
  &=&  - g^3 \{ f^{a_1a_2a_3a_4a_5} \, [ 
                  D_{(2;0)}(1234,5) + D_{(2;0)}(1,2345) - D_{(2;0)}(15,234)] 
   \nn \\
  & &  \,
              + f^{a_2a_1a_3a_4a_5} \, [ 
                  D_{(2;0)}(1345,2) - D_{(2;0)}(12,345)
                  + D_{(2;0)}(125,34) - D_{(2;0)}(134,25) ]
   \nn \\
  & &  \,
              + f^{a_1a_2a_3a_5a_4} \, [ 
                  D_{(2;0)}(1235,4) - D_{(2;0)}(14,235)
                  + D_{(2;0)}(145,23) - D_{(2;0)}(123,45) ]
   \nn \\
  & &  \,
              + f^{a_1a_2a_4a_5a_3} \, [ 
                  D_{(2;0)}(1245,3) - D_{(2;0)}(13,245)
                  + D_{(2;0)}(135,24) 
%    \nn \\
%&& \hspace{2.5cm} 
- D_{(2;0)}(124,35) ] \}. \nn
\label{d50}
\eea
For six gluons attached to the quark loop we 
find the following result. Now eight different 
color structures contribute, 
\bea
%
% PLEASE do not change the layout of this equation.
%
\lefteqn{D_{(6;0)}^{a_1a_2a_3a_4a_5a_6}
   (\kf_1,\kf_2,\kf_3,\kf_4,\kf_5,\kf_6)= } \nn \\
&=&  g^4 \{ d^{a_1a_2a_3a_4a_5a_6} \,[ 
  D_{(2;0)}(12345,6) + D_{(2;0)}(1,23456) - D_{(2;0)}(16,2345) ] \nn \\
& &\,+\,d^{a_2a_1a_3a_4a_5a_6} \,[ 
  D_{(2;0)}(13456,2) - D_{(2;0)}(1345,26) 
+ D_{(2;0)}(126,345) \nn \\
&&\hspace{3cm} -  D_{(2;0)}(12,3456) ] \nn \\
& &\, +\,d^{a_1a_2a_3a_4a_6a_5} \,[ 
 D_{(2;0)}(12346,5) - D_{(2;0)}(1234,56) 
+ D_{(2;0)}(156,234) \nn \\
&&\hspace{3cm} - D_{(2;0)}(15,2346) ] \nn \\
& &\, +\,d^{a_2a_1a_3a_4a_6a_5} \,[
 - D_{(2;0)}(1256,34) - D_{(2;0)}(1346,25) 
 + D_{(2;0)}(125,346) \nn \\
&& \hspace{3cm} + D_{(2;0)}(134,256) ] \nn \\
& &\, +\,d^{a_3a_1a_2a_4a_5a_6} \,[
 D_{(2;0)}(12456,3) - D_{(2;0)}(1245,36) 
 + D_{(2;0)}(136,245) +\nn \\
&&\hspace{3cm} - D_{(2;0)}(13,2456) ] \nn \\
& &\,+\,d^{a_1a_2a_3a_5a_6a_4} \,[
 D_{(2;0)}(12356,4) - D_{(2;0)}(1235,46) 
+ D_{(2;0)}(146,235) \nn \\
&&\hspace{3cm} - D_{(2;0)}(14,2356) ] \nn \\
& &\, +\,d^{a_2a_1a_3a_5a_6a_4} \,[
- D_{(2;0)}(1246,35) - D_{(2;0)}(1356,24)
+ D_{(2;0)}(124,356) \nn \\
&&\hspace{3cm} + D_{(2;0)}(135,246) ] \nn \\
& &\, +\,d^{a_1a_2a_3a_6a_5a_4} \,[
- D_{(2;0)}(1236,45) - D_{(2;0)}(1456,23) 
+ D_{(2;0)}(123,456) \nn \\
& & \hspace{3cm} + D_{(2;0)}(145,236) ] 
\}
\,.
\label{d60}
\eea

\subsection{Reggeon momentum diagrams}
\label{momentumdiags}
We now introduce a further diagrammatic notation for the momentum 
space integrals occurring in our analysis of the integral equations. 
It will be applied in the next section where we will present 
the integral kernels $K^{ \{b\} \rarr \{a\} }_{2\rarr m}$. 
With the help of so--called 
reggeon momentum diagrams we hope to make our results more 
transparent and easier to read. 
A reggeon momentum integral looks like the following 
example:
\be
  \picbox{diagexample.pstex_t} \hspace{2cm}
\label{diagex}
\ee
We now state the rules for translating these 
diagrams back to explicit integrals. At the same 
time we apply the rules step by step to the above 
example. 
\begin{itemize}
\item[(i)] Assign momentum variables to all lines according 
            to momentum conservation (see diagram). 
\item[(ii)] Write down an integral $\int \fr{d^2\lf}{(2 \pi)^3}$ 
            over the loop momentum. (There is always a loop since 
            some amplitude, for instance $D_2$, has to be attached to 
            the upper two lines and thus be written under the integral.) 
\item[(iii)] Find in the diagram the vertex which has two lines 
            attached from above (vertex A in (\ref{diagex})). 
\item[(iv)] Write down the square of the sum of the momenta 
           attached to this vertex from below. 
           (In our example $(\kf_2+\kf_3)^2$.) 
\item[(v)] Write down propagators for the two lines attached 
           to this vertex from above. 
           (In our example 
            $(\lf-\kf_1)^{-2} (\lf-\kf_1-\kf_2-\kf_3)^{-2}$.)

\end{itemize}
For the diagram (\ref{diagex}) this results in 
\be
   \picbox{4a23-1.pstex_t} = 
    \int \fr{d^2\lf}{(2 \pi)^3} 
    \fr{(\kf_2+\kf_3)^2}{(\lf-\kf_1)^2 (\lf-\kf_1-\kf_2-\kf_3)^2} \,.
\ee
These rules can easily be inverted in order to construct the reggeon 
momentum diagram from a given momentum space integral. 
The reggeon momentum diagrams have to be understood 
as integral  operators. The integration has to be carried 
out with a function of the two upper momenta. 
We emphasize that our notation implies 
only two propagators for a given reggeon momentum diagram. 

A few more examples of the diagrammatic notation for 
momentum space integrals are contained in section \ref{standardint}. 

\subsection{Integral kernels}
\label{kernels}
The integral kernels $K^{ \{b\} \rarr \{a\} }_{2\rarr m}$ 
were calculated in \cite{Bartelskernels} by means of 
$s$-channel dispersion relations. 
As explained in section \ref{inteqsection} the kernels are 
convoluted with different amplitudes in the integral equations. 
Only two of the gluons in the respective amplitude 
actually interact with each other. The kernel acts trivially 
on the momenta and color labels of the other gluons. We 
will therefore discuss only the non--trivial action of the 
kernel here. 

The kernel for the transition 
from two gluons with transverse momenta $\qf_1,\qf_2$ 
and color labels $\{b\} = \{b_1,b_2\}$ 
to $m$ gluons with transverse momenta $\kf_1,\dots,\kf_m$ and 
color labels $\{a\} = \{ a_1,\dots ,a_m\} $ is given 
by 
\bea
\lefteqn{K^{ \{b\} \rarr \{a\} }_{2\rarr m} 
              (\qf_1,\qf_2;\kf_1,\dots,\kf_m) = \picbox{kernel2n.pstex_t}
        } \nn \\
     &=&   f_{b_1a_1c_1} f_{c_1a_2c_2} \dots f_{c_{m-1}a_mb_2} \;
    g^m  \left[ (\kf_1 + \dots + \kf_m)^2 
               - \fr{\qf_2^2(\kf_1+\dots+\kf_{m-1})^2}{(\kf_m-\qf_2)^2}
      \right. \nn \\
     & & \hspace{3.5cm} \left. 
               - \fr{\qf_1^2(\kf_2+\dots+\kf_m)^2}{(\kf_1-\qf_1)^2}
               + \fr{\qf_1^2\qf_2^2 
                (\kf_2+\dots+\kf_{m-1})^2}{(\kf_1-\qf_1)^2(\kf_m-\qf_2)^2}
        \right]
\,.
\label{Kn}
\eea
For the kernels that are needed for up to six gluons in the $t$-channel 
this means in our diagrammatic notation for color tensors 
and momentum integral kernels 
\bea
%
% PLEASE do not change the layout of this equation.
%
   K^{ \{b\} \rarr \{a\} }_{2\rarr 2} 
              (\qf_1,\qf_2;\{ \kf_i\} )   &=& 
       \picbox{color2to2.pstex_t}  g^2  \qf_1^2 \qf_2^2 
      \left[ \; \picbox{c12.pstex_t} - \picbox{2b1.pstex_t} 
             - \picbox{2b2.pstex_t}  \;  \right]
   \\
   K^{ \{b\} \rarr \{a\} }_{2\rarr 3} 
              (\qf_1,\qf_2;\{ \kf_i\} )   &=& 
       \picbox{color2to3.pstex_t}  g^3 \qf_1^2 \qf_2^2
      \left[ \; \picbox{c123.pstex_t} - \picbox{3b12.pstex_t} 
             -  \picbox{3b23.pstex_t} +  \picbox{3a2-1.pstex_t} \; \right]
\label{k23}
   \\
   K^{ \{b\} \rarr \{a\} }_{2\rarr 4} 
              (\qf_1,\qf_2;\{ \kf_i\} )   &=& 
       \picbox{color2to4.pstex_t} g^4 \qf_1^2 \qf_2^2
      \left[ \; \picbox{c1234.pstex_t} - \picbox{4b123.pstex_t} 
             - \picbox{4b234.pstex_t} + \picbox{4a23-1.pstex_t} \; \right]
   \\
   K^{ \{b\} \rarr \{a\} }_{2\rarr 5} 
              (\qf_1,\qf_2;\{ \kf_i\} )   &=& 
       \picbox{color2to5.pstex_t} g^5 \qf_1^2 \qf_2^2
      \left[ \;\picbox{c12345.pstex_t} - \picbox{5b1234.pstex_t} 
      -  \picbox{5b2345.pstex_t} + \picbox{5a234-1.pstex_t}\; \right] 
\\
 K^{ \{b\} \rarr \{a\} }_{2\rarr 6} 
              (\qf_1,\qf_2;\{ \kf_i\} )   &=&
       \picbox{color2to6.pstex_t} g^6 \qf_1^2 \qf_2^2
      \left[ \;\picbox{c123456.pstex_t} - \picbox{6b12345.pstex_t} 
      -  \picbox{6b23456.pstex_t} + \picbox{6a2345-1.pstex_t}
\; \right] \hspace*{.4cm}
\eea
Here the inverse propagators $\qf_1^2 \qf_2^2$ are required in order to 
cancel the propagators implied by our graphical 
notation. This is necessary because the kernels 
$K^{ \{b\} \rarr \{a\} }_{2\rarr m}$ in the integral equations are 
defined without propagators; there the propagators come in via 
the convolution denoted by the symbol $\otimes$.

We emphasize that the kernel 
$K^{ \{b\} \rarr \{a\} }_{2\rarr 2}$ is not identical to the full 
BFKL kernel (\ref{Lipatovkernel}) as it does not contain the 
trajectory functions $\beta$. In equation (\ref{inteq2}) 
they have been moved to the left--hand side to make the 
generalization of this equation to $n>2$ more transparent. 
Also the kernels $K^{ \{b\} \rarr \{a\} }_{2\rarr n}$ for $n>2$ 
do not contain any trajectories and are not infrared finite. 
It is only in the full integral equations that the infrared 
divergences cancel. 

\subsection{Standard integrals}
\label{standardint}

When convoluting the amplitudes $D_n$ with the kernels 
$K_{2\rarr m}$ according to the integral equations, we have 
to deal with a large number of momentum space integrals. 
It turns out that all these integrals can be classified by 
a small number of standard integrals. These five 
standard types 
of integrals are even sufficient for addressing the case 
of $n$-gluon amplitudes for any value of $n$. 
(Strictly speaking this applies to the so--called reggeizing 
pieces of the amplitudes as will be explained in the next section.) 
We will therefore give a complete list of the 
five different standard integrals (or diagrams) which 
at the same time serve as a few more examples of 
our notation. 

All five standard integrals occur already when dealing 
with three outgoing gluons. We therefore give the list 
for $n=3$ here, the generalization to larger $n$ being obvious. 
First we have 
\be
  a(2,1) = \picbox{3a2-1.pstex_t}  
    = \int \fr{d^2\lf}{(2 \pi)^3}  
       \fr{\kf_2^2}{(\lf-\kf_1)^2 [\lf-(\kf_1+\kf_2)]^2}
      \,D_2\!\left(\lf, \sum_{j=1}^{3}\kf_j-\lf \right) \,.
\label{adiag}
\ee
The first argument of the function $a$ is the index of the momentum 
attached to the vertex A from below. (As in section \ref{momentumdiags} 
vertex A is the vertex 
in the diagram with two lines attached from above.)
The momentum structure of the diagram is then fixed 
by giving one of the other two outgoing momenta
since the diagram has to be folded with a symmetric 
function of the upper two momenta, 
namely with the BFKL amplitude $D_2$. 
We choose the outgoing momentum with 
the lowest index as the second argument of the function $a$ 
thereby completely fixing the corresponding momentum integral 
since the momentum carried by the third outgoing gluon line 
can easily be inferred from the total number of gluons. 
Therefore our notation for $a$ has to be supplied with the 
total number of $t$-channel gluons it describes. 
(In the case of more than three gluons we here choose the 
group of momenta containing the lowest index as the second 
argument of $a$.) 
Applying this notation, 
our earlier example given in (\ref{diagex}) 
would be assigned the expression $a(23,1)$ when applied 
to $D_2$. In section \ref{momentumdiags} the reggeon 
momentum diagrams stood for integral operators. In this 
and in the following sections we use the same diagrams also 
for the convolution with BFKL amplitudes. The meaning 
should be clear from the context. The function $a$ and the 
functions $b$, $c$, $s$, $t$ to be defined below always mean 
the integrals with $D_2$ included. 
The second type of diagram is 
\be
  b(12) =  \picbox{3b12.pstex_t} 
      = \int \fr{d^2\lf}{(2 \pi)^3}  
         \fr{(\kf_1+\kf_2)^2}{\lf^2 [\lf-(\kf_1+\kf_2)]^2} 
    \,D_2\!\left(\lf, \sum_{j=1}^{3}\kf_j-\lf \right)\,.
\label{bdiag}
\ee
The third one is the contact term 
\be
  c(123) = \picbox{c123.pstex_t} = \int \fr{d^2\lf}{(2 \pi)^3}  
 \fr{(\kf_1+\kf_2+\kf_3)^2}{\lf^2\left(\lf-\sum_{j=1}^3\kf_j\right)^2}
 \,D_2\!\left(\lf, \sum_{j=1}^{3}\kf_j-\lf \right)\,, 
\label{cdiag}
\ee
which is local in impact parameter space. 
The integrals $a$, $b$, and $c$ correspond to real 
corrections, that is to $s$-channel gluon production. 

Further we have two integral types corresponding to 
virtual corrections. These are factorizing and 
are connected with what we already know as 
the trajectory function $\beta$ of the reggeized gluon: 
\bea
  t(12) =  \picbox{3t12.pstex_t}
      &=& \int \fr{d^2\lf}{(2 \pi)^3}  
       \fr{(\kf_1+\kf_2)^2}{\lf^2 [\lf-(\kf_1+\kf_2)]^2}
     D_2(\kf_1+\kf_2,\kf_3) 
\nn \\
    &=&   \fr{2}{N_c g^2} \beta(\kf_1+\kf_2) D_2(\kf_1+\kf_2,\kf_3) 
   \label{tdiag}
\eea
and 
\bea
  s(1,2) =  \picbox{3s1-2.pstex_t}
   &=& \int \fr{d^2\lf}{(2 \pi)^3} 
    \fr{\kf_1^2}{\lf^2(\lf-\kf_1)^2}
     D_2(\kf_1+\kf_2,\kf_3)   
\nn \\
    &=& \fr{2}{N_c g^2} \beta(\kf_1)  D_2(\kf_1+\kf_2,\kf_3) \,.
\label{sdiag}
\eea
This means that the bubble diagram corresponds to a gluon 
trajectory function $\beta$ for the line it is drawn on. 

It suggests itself here to write the BFKL kernel 
using the standard integrals introduced above. 
When the kernel is applied to the BFKL amplitude $D_2$ 
we find
\bea
K_{\mbox{\scriptsize BFKL}} \otimes D_2 &=& 
N_c g^2 \left[ \,c(12) - b(1) - b(2) + \frac{1}{2} t(1) 
+\frac{1}{2} t(2) \right]  \\
&=& N_c g^2 \left[ \,\picbox{c12.pstex_t} 
- \picbox{2b1.pstex_t} - \picbox{2b2.pstex_t} 
+ \frac{1}{2} \,\picbox{2t1.pstex_t} +\frac{1}{2}\,\picbox{2t2.pstex_t} 
\right]
\,.
\eea

The notation introduced here makes implicit use of the 
fact that the BFKL amplitude $D_2$ is symmetric in 
its two momentum arguments. Without this assumption 
it would, for example, be necessary to give explicitly a third 
argument to fully specify the integral $a$ in (\ref{adiag}). 
We have chosen to restrict the short--hand notation to 
the case of a symmetric function since 
we will only apply it to the case of $D_2$. 

As an example of how the convolutions of $n$-gluon 
amplitudes with kernels can be reduced to the standard 
integrals may serve 
\be
  D_2(\lf_1+\lf_2,\lf_3) 
  \otimes K_{2\rarr 3}(\lf_2,\lf_3;\kf_2,\kf_3,\kf_4)= 
  g^3[ b(234) - a(23,1) - b(34) + a(3,12) ] \,. 
\ee
This can be easily checked using the explicit definition (\ref{k23}) 
or (\ref{Kn}) of the kernel. 
When studying the $n$-gluon amplitudes $D_n$ we 
encounter a problem connected with these convolutions 
in momentum space. For each convolution it is relatively easy 
to find a representation in terms of the standard integrals, 
as the above example shows. The actual problem is the 
rapidly increasing number of convolutions we have to deal with 
when coming to larger $n$. We will explain this problem in more 
detail in the following sections. For $n=5$ gluons the problem is 
at the edge of being tractable by hand. For $n=6$ gluons the 
problem has to be attacked with the help of a computer. 
It is exactly for this reason that we introduce the 
classification of momentum space integrals in this section. 
In appendix \ref{momentumapp} we give an algorithm 
suited for implementation on a computer, for instance 
in the PERL script language. 
The example above is intended to illustrate  that the notation 
indeed allows us to handle the rather complicated integrals in 
compact form. 

Closing this section, we remark that even the five standard types 
of integrals mentioned above are not completely independent if we take 
our definitions (\ref{adiag})--(\ref{sdiag}) literally. 
Relations between them occur in the case that one of the outgoing 
legs of the diagrams has zero momentum. 
One finds, for example, that the function $b$ emerges 
from $a$ when the two arguments 
of $a$ exhaust the outgoing gluons, that is for three gluons we would 
get 
\be
  a(1,23) = b(1)\,.
\ee
Similarly, for three outgoing gluons
\be
  b(123) = c(123) \,.
\ee
In addition, we find the function $t$ from $s$ when the second 
argument of $s$ vanishes, for example 
\be
  s(1,-) = t(1) \,.
\ee
In spite of these relations we prefer to treat the five integral 
types as fundamentally different because they correspond 
to very different locality properties in impact parameter space. 
The function $c$, for instance, corresponds to a contact 
interaction whereas the function $a$ contains a non--locality.  

This concludes our discussion of the elements of the integral 
equations and we can now proceed to solving them (at least partially). 

\boldmath
\section{Three and four gluons, 
         the transition vertex $V_{2 \rightarrow 4}$}
\unboldmath
\label{34V}

\subsection{The three-gluon amplitude}
\label{threegluon}

The amplitudes with three and four gluons, $D_3$ and $D_4$, 
were first investigated in \cite{BPLB,BW}. 
It was found that the integral equation (\ref{inteq3}) for the 
three gluon amplitude can be solved, the solution being 
\be
D_3^{a_1a_2a_3}(\kf_1,\kf_2,\kf_3) 
= \fr{1}{2} g f_{a_1a_2a_3} \,
[ D_2(12,3) - D_2(13,2) + D_2(1,23) ]  
\,,
\label{d3}
\ee
which can be shown by direct computation. In addition 
to performing the color algebra contractions and the convolutions 
in momentum space one has to make use of the integral equation 
(\ref{inteq2}) for the two--gluon amplitude $D_2$. 

The result (\ref{d3}) means that an actual 
three--gluon state does not appear.\footnote{This 
does not affect the existence of the Odderon, since in 
our case the three--gluon system has even $C$-parity.}  
In the contrary, the amplitude turns out to be a superposition of 
two--gluon states. We call this phenomenon 
the reggeization of the amplitude. It generalizes the notion 
of reggeization previously attributed to the fact that the BFKL 
equation in the color octet channel can be solved by a pole--ansatz 
and thus describes a one--reggeon state. In the case of the three--gluon 
amplitude reggeization again occurs in a channel corresponding to the 
adjoint representation, i.\,e.\ in the octet channel. 
We should emphasize that the reggeization of $D_3$ is a property of the 
momentum space part of the amplitude. The analytic 
properties correspond to those of a superposition of two--gluon 
compound states. Nevertheless, $D_3$ remains a three--gluon 
amplitude carrying three color labels, i.\,e.\ the color part of the 
amplitude is not affected. 

It is worth noting that the solution (\ref{d3}) 
is obtained from the lowest order term (\ref{d30}) by replacing 
the quark loop amplitudes $D_{(2;0)}$ by the full BFKL amplitudes 
$D_2$ while keeping the color and momentum structure. 

We do not give a proof of (\ref{d3}) here. In section \ref{5gluons} 
we will discover that the reggeization of the three--gluon 
amplitude $D_3$ is actually a special case of a more general 
identity that we will discuss in detail. 

\boldmath
\subsection{The four-gluon amplitude and the two-to-four transition 
vertex $V_{2 \rightarrow 4}$}
\unboldmath
\label{fourgluon}

Our method for analyzing the structure of the $n$-gluon 
amplitudes is the following. 
One starts with an educated guess for the solution or at least a part 
of it, assuming the full solution 
to be a sum of the part we have guessed and a remaining 
term. That ansatz is inserted into the integral equation and 
a new integral equation for the unknown part is derived. 
If the guess was good the new integral equation is 
in a certain sense simple, and allows one to gain further 
information about the unknown part. 
If the guess was not optimal, on the other hand, 
the new integral equation 
will be complicated and not allow us to extract 
further information. 
Of course, this procedure is not uniquely determined. 
As we will see, the quark loop contains very useful 
information that can be used for choosing a promising 
ansatz.  
Let us now see how this method works in practice. 

In \cite{BPLB,BW} the four--gluon amplitude was split into two parts, 
\be
  D_4 = D_4^R + D_4^I \,,
\label{d4ri}
\ee
a reggeizing part $D_4^R$ and a part $D_4^I$ that for 
reasons to be explained below is called the irreducible part of the 
four--gluon amplitude. 
The reggeizing part is --- in analogy with the three--gluon case --- 
chosen as the superposition of two--gluon amplitudes, 
\bea
\lefteqn{D_4^{R\,a_1a_2a_3a_4}(\kf_1,\kf_2,\kf_3,\kf_4)= } \nn \\
  &=&  - g^2 d^{a_1a_2a_3a_4} \, [ D_2(123,4) + D_2(1,234) 
                                  - D_2(14,23) ] 
   \nn \\
 & & - g^2 d^{a_2a_1a_3a_4}  \, [ D_2(134,2) + D_2(124,3)
                                 - D_2(12,34) - D_2(13,24) ] \,.
\label{d4r}
\eea
Again, this is obtained from the lowest order term 
$D_{(4;0)}$, see (\ref{d40}),  
by the replacement $D_{(2;0)} \to D_2$. 
The ansatz for the reggeizing part and thus the 
decomposition (\ref{d4ri}) is to some extend arbitrary. 
Recently, a different ansatz for the reggeizing part was 
investigated in \cite{BraunVacca,Vacca}. That ansatz, as also 
discussed in \cite{BW}, is more convenient for the 
analysis of high mass diffractive processes. 
We will not further discuss other possible choices for the 
splitting (\ref{d4ri}) of the amplitude here. 
All choices will, of course, lead to 
equivalent results for the complete amplitude. 
The choice given above is singled out because it leads to a 
simple picture for the remaining part $D_4^I$, especially 
when interpreted in view of a field theory structure 
of unitarity corrections. 

The next step is to derive a new integral equation for 
the unknown irreducible part $D_4^I$. To this end we 
insert the ansatz (\ref{d4ri}),(\ref{d4r}) for the full 
amplitude into the original integral equation (\ref{inteq4}). 
The known result (\ref{d3}) for the three gluon amplitude 
is inserted as well. 
Due to the choice of $D_4^R$ we can then apply 
the equation (\ref{inteq2}) for the two--gluon amplitude 
to the expression $\omega  D_4^R$ 
on the left hand side. 
That exactly eliminates the lowest order term $D_{(4;0)}$ 
on the right hand side and produces additional terms 
containing only the convolution of $D_2$ amplitudes with 
the kernel $K^{ \{b\} \rarr \{a\} }_{2\rarr 2}$ or 
products of $D_2$ with trajectory functions $\beta$. 
Besides the terms containing $D_4^I$ all other contributions to 
the right hand side consist of convolutions of two--gluon 
amplitudes $D_2$ with the integral kernels. 
Our new equation thus takes the form 
\be
  \left(\omega - \sum_{i=1}^4 \beta(\kf_i) \right)
  D_4^{I\,a_1a_2a_3a_4} (\kf_1,\kf_2,\kf_3,\kf_4) = 
   V_{2 \rightarrow 4}^{a_1a_2a_3a_4} D_2
  +\sum K^{ \{b\} \rarr \{a\} }_{2\rarr 2} \otimes D_4^{I\,b_1b_2b_3b_4}
   \,.
\label{inteq4i}
\ee
The sum on the right--hand side of this new equation extends over 
all pairwise interactions of the four gluons. 
In the inhomogeneous term $V_{2 \rightarrow 4} D_2$ 
we collect all the terms containing $D_2$, 
hence the notation. $V_{2 \rightarrow 4}$ should be 
understood as an integral operator acting on the two--gluon 
amplitude. As we will explain in more detail below, 
it describes the transition from the two--gluon 
state to a fully interacting four--gluon system in the $t$-channel. 
The explicit expression for the two--to--four transition vertex 
was computed in \cite{BPLB,BW}. To arrive at this explicit result the 
following steps have to be done. First we have to contract 
the color tensors of the amplitudes with those of the kernels. 
This is done along the lines described in appendix \ref{colorapp}. 
The second step is to bring the momentum space integrals to 
their standard form, that is to classify them according to 
the scheme explained in section \ref{standardint}. 
Both steps result in lengthy calculations because of 
the large number of contractions involved. The results of 
the $\mbox{su}(N_c)$ tensor contractions are then multiplied with 
the corresponding momentum space integrals. 
Finally, all terms can be collected to give the vertex 
$V_{2 \rightarrow 4}$. Due to cancellations 
the resulting expression is comparatively compact. 
Remarkably, all terms belonging to the color tensors $d^{a_1a_2a_3a_4}$ 
and $d^{a_2a_1a_3a_4}$ drop out. One finds 
the following color and momentum structure for the vertex: 
\bea
 V_{2 \rightarrow 4}^{a_1a_2a_3a_4}(\{ {\bf q}_j\},\kf_1,\kf_2,\kf_3,\kf_4) 
 &=& 
  \delta_{a_1a_2} \delta_{a_3a_4} 
V(\{ {\bf q}_j\},\kf_1,\kf_2;\kf_3,\kf_4)
 \nonumber \\
 & & 
 +\, \delta_{a_1a_3} \delta_{a_2a_4} 
V(\{ {\bf q}_j\},\kf_1,\kf_3;\kf_2,\kf_4)
 \nonumber \\
 & &
 +\,\delta_{a_1a_4} \delta_{a_2a_3} 
V(\{ {\bf q}_j\},\kf_1,\kf_4;\kf_2,\kf_3)
\,.
\label{colV}
\eea
The function $V$ is the same in all three terms on the right hand side. 
The $\qf_j$ are the two momenta entering from above. Since 
throughout this paper 
$V_{2 \rightarrow 4}$ is almost always contracted with a BFKL 
amplitude $D_2$ from above, we will omit these arguments 
in the following and consider the quantity $V_{2 \rightarrow 4}D_2$. 
The vertex function $V(\kf_1,\kf_2;\kf_3,\kf_4)$ 
has the explicit form
\bea
%
% PLEASE do not change the layout of this equation.
%
(V D_2)  (\kf_1,\kf_2;\kf_3,\kf_4) &=& \frac{g^4}{4} \times 
\{\,2  \,[\, c(1234)   \nn \\
&& \hspace{1.5cm}
-\, b(123) - b(124) - b(134) - b(234) + b(12) + b(34) \nn \\
&& \hspace{1.5cm}+\, a(13,2) + a(14,2) + a(23,1) + a(24,1) \nn \\
&& \hspace{1.5cm}-\, a(1,2) - a(2,1) - a(3,12) - a(4,12) ]  \nn \\
&& \hspace{1.2cm}+ [\, t(123) + t(124) + t(134) + t(234) -t(12) -t(34) 
\nn \\
&& \hspace{1.5cm}-\, s(13,2) - s(13,4) - s(14,2) - s(14,3) \nn \\
&& \hspace{1.5cm}-\, s(23,1) -s(23,4) - s(24,1) - s(24,3) \nn \\
&&  \hspace{1.5cm}+\, s(1,2) + s(1,34) + s(2,1) + s(2,34) \nn \\
&& \hspace{1.5cm}+ \,s(3,12) + s(3,4)+ s(4,12) + s(4,3) ]  \} 
\label{v24}
\eea
where we have made use of the notation introduced in 
section \ref{standardint}. 

Let us now describe the known properties of the vertex function $V$ 
and of the full transition vertex $V_{2 \rightarrow 4}$. The first 
observation is that 
$V(\kf_1,\kf_2;\kf_3,\kf_4)$ is symmetric in its first two and in 
its last two arguments
\bea
%
% PLEASE do not change the layout of this equation.
%
V(\kf_1,\kf_2;\kf_3,\kf_4) &=& V(\kf_2,\kf_1;\kf_3,\kf_4) \nn \\
                             &=& V(\kf_1,\kf_2;\kf_4,\kf_3) \,,
\label{vsymm12}
\eea
and symmetric under the exchange of the first pair of arguments 
and the second pair of arguments (that is why our notation separates these 
pairs by a semicolon) 
\be
V(\kf_1,\kf_2;\kf_3,\kf_4) = V(\kf_3,\kf_4;\kf_1,\kf_2) \,.
\label{vsymm12,34}
\ee
Therefore, according to (\ref{colV}) the full vertex 
$V_{2 \rightarrow 4}$ is completely symmetric under the simultaneous 
exchange of color and momentum of the gluons. 

The combination of integrals in $VD_2$ vanishes when one of the 
outgoing momenta vanishes, 
\be
\left.
(V D_2) (\kf_1,\kf_2;\kf_3,\kf_4) 
\right|_{\kf_i = 0} = 0  \;\;\;\;\;\;  (i\in\{1,\dots,4\}) 
\,.
\ee
This result can be proven easily using 
identities of the kind mentioned at the end of section \ref{standardint} 
and the fact that the BFKL amplitude $D_2$ 
vanishes at zero--momentum argument. 
This property of $V D_2$ is carried over to the full vertex, 
\be
\left. 
(V_{2 \rightarrow 4} D_2)^{a_1a_2a_3a_4}(\kf_1,\kf_2,\kf_3,\kf_4) 
\right|_{\kf_i = 0} = 0  \;\;\;\;\;\;  (i\in\{1,\dots,4\}) 
\,.
\label{nullstellvertex}
\ee
Further, the function $VD_2$ is infrared finite, i.\,e.\ the 
infrared divergences in the different integrals contributing to 
(\ref{v24}) cancel in the sum. 
This can be easily seen after noticing that already certain 
combinations of very few standard integrals are infrared 
finite, nicely showing the cancellation of divergences between 
real and virtual corrections. The combination 
\be
  b(\lf) - \frac{1}{2} \,t(\lf)
\label{inffinite1}
\ee
is infrared finite for any sum of momenta that is 
substituted for $\lf$, as is clear from the corresponding 
integrals (see section \ref{standardint}). 
The factor $1/2$ comes about because the integrand of the 
trajectory function $\beta$ (see (\ref{tdiag}) and 
(\ref{traject})) exhibits two divergences of the same form. 
Similarly, one can show that the combination 
\be
 a(\lf_2,\lf_1) 
-\frac{1}{2} \,s(\lf_2,\lf_1) - \frac{1}{2}\,s(\lf_2,\lf_3) 
\label{inffinite2}
\ee
is infrared finite separately for any partition of the momenta 
$\{ \kf_i \}$ into three sums $\lf_1,\lf_2,\lf_3$. 
Finally, the term $c(\lf)$ is infrared finite separately 
since in this term (see (\ref{cdiag})) the poles of the 
propagators are cancelled by the zeros of the BFKL 
amplitude $D_2$ in the integral. The finiteness of these 
three groups is independent of the total 
number of momenta $\kf_i$ that are split into the groups 
denoted by $\lf_j$. 
It should be obvious from equation (\ref{v24}) that 
all integrals in the vertex function come in exactly 
these infrared finite combinations. 

We now come back to 
the main problem of understanding the full 
four--gluon amplitude $D_4$. When the ansatz 
(\ref{d4ri}) was made the goal was to arrive at a 
simple equation for the yet unknown part $D_4^I$. 
The new integral equation (\ref{inteq4i}) 
is in fact simple: It contains only the vertex 
$V_{2 \rightarrow 4}$ and a homogeneous part, 
and can therefore now be iterated.
The structure arising from this is 
\be
  D_4^I = G_4 \cdot V_{2 \rightarrow 4} \cdot D_2 \,,
\label{d4istruct}
\ee
$G_4$ being the Green function of the four--gluon state. 
The Green function obeys the BKP equation with 
four $t$-channel gluons, which is a four--particle 
Schr\"odinger equation. 
Its Hamiltonian is given by the homogeneous part of the 
integral equation (\ref{inteq4i}), 
i.\,e.\ by the sum of all pairwise interactions 
$K_{2 \to 2}$ of the four gluons. 
Unfortunately, the eigenvalues and eigenfunctions 
of the Hamiltonian are not known and (\ref{d4istruct}) 
remains a formal solution only. Though, some 
properties of the four--gluon state have been worked 
out in \cite{Hans}. 
We will return to the interpretation of the structure 
inherent in (\ref{d4istruct}) momentarily. 

Even without knowing an analytic formula for $D_4^I$ 
we can deduce two important properties. Like the 
two--gluon amplitude, the irreducible part of the 
four--gluon amplitude vanishes (modulo logarithms) 
when one of the outgoing 
gluon momenta is set to zero,
\be
\left. 
  D_4^{I\,a_1a_2a_3a_4}(\kf_1,\kf_2,\kf_3,\kf_4) 
\right|_{\kf_i = 0} = 0  \;\;\;  (i\in\{1,\dots,4\}) 
\,.
\label{nullstelld4i}
\ee
To prove this we proceed order by order in the iteration of the 
Hamiltonian $\sum K_{2 \to 2}$. The identity holds 
in lowest order since the vertex itself has this property 
(see (\ref{nullstellvertex})). 
In the next order, (\ref{nullstelld4i}) holds because 
$K_{2 \to 2}$ also vanishes if one of the outgoing 
momenta becomes zero, etc. 
Similarly, we can show that the irreducible part $D_4^I$ is completely 
symmetric in the four gluons, that is under the simultaneous exchange 
of color and momentum, 
\bea
D_4^{I\,a_1a_2a_3a_4}(\kf_1,\kf_2,\kf_3,\kf_4) 
&=& D_4^{I\,a_2a_1a_3a_4}(\kf_2,\kf_1,\kf_3,\kf_4) \nn \\
&=& D_4^{I\,a_3a_2a_1a_4}(\kf_3,\kf_2,\kf_1,\kf_4) \nn \\
&=& D_4^{I\,a_4a_2a_3a_1}(\kf_4,\kf_2,\kf_3,\kf_1) 
\,.
\label{complsymmd4i}
\eea

\subsection{Field theory structure}
\label{4ftconf}

Although we do not have an analytic expression for the 
irreducible part, we have 
gathered by now quite some knowledge about the structure of the 
four--gluon amplitude $D_4$. Neglecting for a moment 
color and normalization factors, this structure is 
illustrated in the following diagram 
\be
  D_4(\kf_1,\kf_2,\kf_3,\kf_4) 
 = \sum \picbox{solutiond41.pstex_t} + \picbox{solutiond42.pstex_t} 
\,.
\label{solutiond4diag}
\ee
The first part is the superposition of two--gluon states 
coupled to the quark loop (the reggeizing part $D_4^R$). 
In the second (irreducible) part $D_4^I$ a two--gluon system couples 
to the quark loop and then undergoes a transition to a 
four--gluon compound state via the vertex $V_{2 \rightarrow 4}$. 
From this we learn that the complete amplitude consists of only a few 
basic building blocks: a quark loop allowing the coupling of the 
$t$-channel gluons to external particles, 
the two--gluon Green function (BFKL 
amplitude), the four--gluon Green function, and the two--to--four 
transition vertex. The Green functions describe the quantum mechanical 
propagation of bound states of $t$-channel gluons. 
With the transition vertex we have in addition a number--changing 
element connecting different $n$-reggeon states. 
This turns the quantum mechanical problem of $n$-gluon 
states into that of a quantum field theory of reggeized gluons. 
All this takes place in the $2$-dimensional space of transverse momenta. 
The complex angular momentum $\omega$ plays the role 
of an energy variable. Its conjugate variable, i.\,e.\ rapidity, plays 
the role of the time variable. 

We would like to emphasize that phenomenon of reggeization 
is crucial for the emergence of a field theory structure. 
Without reggeization in the three--gluon amplitude and 
in the part $D_4^R$ of the four--gluon 
amplitude one would not have been able to arrive at the 
simple structure (\ref{solutiond4diag}). 

So far only the simplest elements of a potential effective 
field theory have been identified: the two--gluon compound state, the 
two--to--four transition vertex $V_{2 \rightarrow 4}$ 
and the four--gluon compound state. An analytic formula 
for the latter is still missing. The concept of an effective field 
theory has not been derived from first principles. It has to be 
tested and further elements should be derived. To achieve this 
is seems natural to proceed to higher $n$-gluon amplitudes. 
This constitutes the main goal of this paper, namely 
to deepen our understanding of the field theory structure 
of unitarity corrections by studying the five-- and six--gluon 
amplitudes. 

\section{Five gluons}
\label{5gluons}

\subsection{A reggeizing part and the integral equation for the 
remaining part}
\label{splitd5}

In the first step, our analysis of the five--gluon amplitude $D_5$ 
follows the same lines as 
the study of the three-- and four--gluon amplitudes. 
To get started we identify a reggeizing part $D_5^R$ of 
the amplitude and split the amplitude accordingly, 
\be
  D_5 = D_5^R + D_5^I
\,.
\label{d5ri}
\ee
With a well--chosen $D_5^R$ we will come to a new integral 
equation for the yet unknown quantity $D_5^I$. Again, this 
decomposition is not unique. Our ansatz will lead to 
an equation for $D_5^I$ that can even be solved. 
This situation is the best we can hope for and 
further justification for the ansatz is certainly not needed. 
The natural choice for $D_5^R$ is once more suggested by the 
inhomogeneous term $D_{(5;0)}$. This means that our ansatz 
has exactly the 
same color and momentum structure as $D_{(5;0)}$ 
in (\ref{d50}), but we replace $D_{(2;0)}$ by $D_2$ 
resulting in 
\bea
\lefteqn{D_5^{R\,a_1a_2a_3a_4a_5}(\kf_1,\kf_2,\kf_3,\kf_4,\kf_5)= } 
%\nn 
\\
  &=&  - g^3 \{ f^{a_1a_2a_3a_4a_5} \, [ 
                  D_2(1234,5) + D_2(1,2345) - D_2(15,234)] 
   \nn \\
  & &  \phantom{ - g^3 } 
              + f^{a_2a_1a_3a_4a_5} \, [ 
                  D_2(1345,2) - D_2(12,345)
                  + D_2(125,34) - D_2(134,25) ]
   \nn \\
  & &  \phantom{ - g^3 } 
              + f^{a_1a_2a_3a_5a_4} \, [ 
                  D_2(1235,4) - D_2(14,235)
                  + D_2(145,23) - D_2(123,45) ]
   \nn \\
  & &  \phantom{ - g^3 } 
              + f^{a_1a_2a_4a_5a_3} \, [ 
                  D_2(1245,3) - D_2(13,245)
                  + D_2(135,24) 
\nn 
%\\
%&& \hspace{3.2cm}
- D_2(124,35) ] \} \,.
\nn
\label{d5r}
\eea
This is inserted into the integral equation (\ref{inteq5}). 
We insert into that equation 
the expressions (\ref{d3}), (\ref{d4ri}), (\ref{d4r}) for $D_3$ and $D_4$ 
as well. In order to find the new integral equation for $D_5^I$ we 
have to simplify and collect all terms not involving $D_5^I$ and 
$D_4^I$. These terms will contribute to the inhomogeneous term 
of the new equation for $D_5^I$ and we will now discuss them. 
From the left--hand side of (\ref{inteq5}) we get 
$\omega D_5^R$, which can be treated using the 
BFKL equation (\ref{inteq2}). Due to this, the inhomogeneous term 
$D_{(5;0)}$ in (\ref{inteq5}) is exactly cancelled and we get 
further terms involving only convolutions of $D_2$ functions 
with kernels $K_{2 \rarr 2}$ or trajectories $\beta$. 
From the right--hand side of 
(\ref{inteq5}) we get contributions of the type 
$K_{2 \rarr 5} \otimes D_2$, 
$\sum K_{2 \rarr 4} \otimes D_3$, $\sum K_{2 \rarr 3} \otimes D_4^R$, 
and $\sum K_{2 \rarr 2} \otimes D_5^R$. All of these can be written 
as sums of convolutions of $D_2$ with the kernels $K_{2 \rarr m}$.  
The corresponding contractions of 
color tensors are performed using the diagrammatic method described 
in appendix \ref{colorapp}. In that appendix we also give the 
explicit formulae for some of the contractions required. 
The total number 
of contractions needed here is close to one hundred, and they can be 
easily obtained from those in the appendix. 
The momentum integrals are brought 
to their standard forms as classified in section \ref{standardint}. 
The respective momentum integrals and color contractions 
are then multiplied and can be collected. 
This last step amounts to collecting several thousand terms 
and sorting them according to the different color tensors, 
and we do this with the help of a computer algebra program. 

In the derivation of the new integral equation 
the terms involving $D_4^I$ remain unchanged. 
They will be treated at a later stage of the analysis. 
The same is true for the homogeneous term containing $D_5^I$. 
The combinations of the $D_4^I$ and $D_5^I$ amplitudes with 
the kernels are therefore the same as in the 
original equation (\ref{inteq5}). 
We thus find the following equation for the unknown part $D_5^I$ 
of the five--gluon amplitude:
\bea
 \left(\omega - \sum_{i=1}^5 \beta(\kf_i)\right) D_5^I &=& 
 \sum   f_{a_1a_2a_3} \delta_{a_4a_5} H(1,2,3;4,5) 
 + \sum K^{ \{b\} \rarr \{a\} }_{2 \rarr 3} \otimes D_4^{I\,b_1b_2b_3b_4} 
\nn \\
&& 
+ \sum K^{ \{b\} \rarr \{a\} }_{2 \rarr 2} \otimes D_5^{I\,b_1b_2b_3b_4b_5}
\,.
\label{inteq5i}
\eea
The first term on the right hand side is the result of the computation 
described above. We will now treat it in more detail. 

The first interesting observation concerns its color structure. 
All terms proportional to 
$f^{a_1a_2a_3a_4a_5}$ (and the other three permutations of this 
occurring in (\ref{d5r})) 
are cancelled between the different contributions to this inhomogeneous 
term and drop out. Something similar happened in the case of $D_4$ 
where there is no term proportional to $d^{a_1a_2a_3a_4}$ 
in the vertex $V_{2 \rarr 4}$ and only lower tensors (i.\,e.\ products 
of $\delta$-tensors) contribute.  

Secondly, we observe the following symmetry of the new inhomogeneous 
term that we have calculated. 
The sum extends over all (ten) 
possibilities to have a pair of gluons in a color singlet. 
For each of these permutations of the gluons color and momentum 
labels are exchanged simultaneously, i.\ e.\ the sum in 
(\ref{inteq5i}) stands for 
\bea
  \sum  f_{a_1a_2a_3} \delta_{a_4a_5} H(1,2,3;4,5) 
 &=& f_{a_1a_2a_3} \delta_{a_4a_5} H(1,2,3;4,5) + 
 f_{a_1a_2a_4} \delta_{a_3a_5} H(1,2,4;3,5) \nn \\
&& + \dots 
 +  \delta_{a_1a_2} f_{a_3a_4a_5} H(3,4,5;1,2)
\,.
\label{explsumoverH}
\eea
The function $H$ is the same in all ten permutations. 
This symmetry is an outcome of our computation, and it has not 
been used to derive (\ref{inteq5i}). On the other hand, 
it is not an unexpected property 
of the inhomogeneous term. Already in the corresponding 
equation (\ref{inteq4i}) in the four--gluon case the 
inhomogeneous term, i.\,e.\ the vertex $V_{2 \rightarrow 4}$, 
had a similar symmetry. 

A closer inspection of the function $H$ reveals that it is actually 
a superposition of vertex functions $V$ which we encountered in the 
discussion of the two--to--four vertex $V_{2 \rightarrow 4}$. 
Namely,
\be
  H(1,2,3;4,5) = \frac{g}{2}
 [(VD_2)(12,3;4,5) - (VD_2)(13,2;4,5) + (VD_2)(1,23;4,5)] 
\,.
\label{h=sumv}
\ee
To obtain this striking result is was necessary to go through 
the full calculation of all convolutions of amplitudes with kernels 
as described. It is only afterwards that we are able to discover the 
simple structure in terms of $V$. 
Unfortunately, we do not know a way leading 
to (\ref{inteq5i}), (\ref{h=sumv}) that avoids this tedious 
calculation. 

\subsection{Solving the equation for the remaining part}
\label{solving5i}
Up to this point, our analysis of the five--gluon amplitude 
followed essentially the same lines as in the case of four 
gluons. Whereas there the new integral equation could 
simply be iterated, this is not possible here. 
To find the solution for $D_5^I$ we now have to go beyond the 
procedure applied for $n=3$ and $4$ gluons. 

Taking a close look at the integral equation (\ref{inteq5i}) 
for $D_5^I$ we discover that its structure bears a strong 
resemblance to the equation (\ref{inteq3}) for the three--gluon 
amplitude $D_3$. 
In the second term on the right hand side 
of (\ref{inteq5i}) a pair of gluons of the amplitude $D_4^I$ is 
convoluted with a 
two--to--three kernel. In the corresponding term in 
(\ref{inteq3}) it was the two--gluon (BFKL) amplitude $D_2$ that 
was convoluted with the same kernel. There the first 
term on the right hand side, i.\,e.\ $D_{(3;0)}$, 
was the superposition of quark loop amplitudes $D_{(2;0)}$ 
that are the lowest order terms in the ladder 
expansion of $D_2$. In (\ref{inteq5i}) the corresponding 
term (\ref{explsumoverH}) is, according to (\ref{h=sumv}), 
the superposition of functions $VD_2$. (In fact it is even 
the superposition of full two--to--four reggeon vertices 
$V_{2 \rightarrow 4}D_2$ as we will see below.) 
These vertex functions, in turn, constitute the lowest 
order terms in the ladder expansion\footnote{This statement 
has to be taken with some care, since the terms $VD_2$ are 
of course not of lowest order in the coupling constant $g$. 
In the contrary, $D_2$ already contains an infinite series 
of ladder diagrams. What is meant here is that each diagram 
in the ladder expansion of $D_4^I$ starts with a full two--gluon ladder 
and a vertex attached to this.} 
of the irreducible 
amplitude $D_4^I$, cf.\ (\ref{d4istruct}). 
More specifically, it is for each of the ten terms in (\ref{explsumoverH}) 
that we find in a three--gluon subsystem exactly the 
momentum structure that also determines $D_{(3;0)}$. 
The similarity is also evident for the color structure, 
namely the three--gluon subsystem comes with 
a tensor $f_{abc}$. 

Clearly, this suggests to construct a solution $D_5^I$ 
in analogy with the three--gluon amplitude. 
While $D_3$ is 
a superposition of BFKL amplitudes $D_2$ 
we now should choose  $D_5^I$ as a similar superposition of irreducible 
four--gluon amplitudes $D_4^I$. 
The following combination of $D_4^I$ amplitudes is of this kind 
and in fact is a solution to equation (\ref{inteq5i}). 
We will outline the proof of this fact momentarily. 
\bea
%
% PLEASE do not change the layout of this equation.
%
\lefteqn{D_5^{I\,a_1a_2a_3a_4a_5}(\kf_1,\kf_2,\kf_3,\kf_4,\kf_5) =
 \frac{g}{2} \times} \nn \\
&&\times \left\{ f_{a_1a_2c} D_4^{I\,ca_3a_4a_5}(12,3,4,5) 
+ f_{a_1a_3c} D_4^{I\,ca_2a_4a_5}(13,2,4,5) \right. \nn \\
&& \hspace{.5cm} 
+ \,f_{a_1a_4c} D_4^{I\,ca_2a_3a_5}(14,2,3,5) 
+ f_{a_1a_5c} D_4^{I\,ca_2a_3a_4}(15,2,3,4) \nn \\
&& \hspace{.5cm} 
+ \,f_{a_2a_3c} D_4^{I\,a_1ca_4a_5}(1,23,4,5) 
+ f_{a_2a_4c} D_4^{I\,a_1ca_3a_5}(1,24,3,5) \nn \\
&& \hspace{.5cm} 
+\, f_{a_2a_5c} D_4^{I\,a_1ca_3a_4}(1,25,3,4) 
+ f_{a_3a_4c} D_4^{I\,a_1a_2ca_5}(1,2,34,5) \nn \\
&& \hspace{.5cm} \left.
+ \,f_{a_3a_5c} D_4^{I\,a_1a_2ca_4}(1,2,35,4) 
+ f_{a_4a_5c} D_4^{I\,a_1a_2a_3c}(1,2,3,45)
\right\}
\label{d5isolution}
\eea
In each of the terms one pair $(i,j)$ of gluons is 
merged\footnote{Depending on the context one would like to use different 
words for the formula (\protect\ref{d5isolution}). 
From the point of view of constructing the  solution 
it is clearly a 'merging' of two gluons, with the concept of a 
$t$-channel evolution in mind one would prefer to speak of a 'splitting' 
of gluons.} 
into one gluon which then enters the irreducible four--gluon 
amplitude from below. 
This gluon in $D_4^I$ has momentum $(\kf_i + \kf_j)$ and color label $c$. 
The merging of the two gluons in color space happens via 
a $f_{a_ia_jc}$ tensor ($i<j$). 
The position in the amplitude $D_4^I$ at which the 'composite' 
gluon with color $c$ 
and momentum $(\kf_i + \kf_j)$ enters does not matter since $D_4^I$ is 
completely symmetric in the four gluons, cf.\ (\ref{complsymmd4i}). 
All possible pairs of gluons are treated in the same way. 
The way pairs of gluons are merged (or arise from splittings) 
becomes more transparent when (\ref{d5isolution}) is written 
using birdtrack notation, 
\bea
%
% PLEASE do not change the layout of this equation.
%
\lefteqn{D_5^{I\,a_1a_2a_3a_4a_5}(\kf_1,\kf_2,\kf_3,\kf_4,\kf_5) =
 \frac{g}{2} \times} \nn \\
&&\times \Bigg\{ \left[ \, \picbox{5an4f12.pstex_t} \right] 
 \star D_4^{I\,b_1b_2b_3b_4}(12,3,4,5) +
\left[ \, \picbox{5an4f13.pstex_t} \right] 
 \star D_4^{I\,b_1b_2b_3b_4}(13,2,4,5) \nn \\
&& \hspace{.5cm} + \left[ \, \picbox{5an4f14l.pstex_t} \right] 
\star D_4^{I\,b_1b_2b_3b_4}(14,2,3,5) 
+ \left[ \, \picbox{5an4f15l.pstex_t} \right] 
\star D_4^{I\,b_1b_2b_3b_4}(15,2,3,4) \nn \\
&& \hspace{.5cm} + \left[ \, \picbox{5an4f23.pstex_t} \right] 
\star D_4^{I\,b_1b_2b_3b_4}(1,23,4,5) 
+ \left[ \, \picbox{5an4f24l.pstex_t} \right] 
\star D_4^{I\,b_1b_2b_3b_4}(1,24,3,5) \nn \\
&& \hspace{.5cm} + \left[ \, \picbox{5an4f25l.pstex_t} \right] 
\star D_4^{I\,b_1b_2b_3b_4}(1,25,3,4) 
+ \left[ \, \picbox{5an4f34.pstex_t} \right] 
\star D_4^{I\,b_1b_2b_3b_4}(1,2,34,5) \nn \\
&& \hspace{.5cm} + \left[ \, \picbox{5an4f35l.pstex_t} \right] 
\star D_4^{I\,b_1b_2b_3b_4}(1,2,35,4) 
+ \left[ \, \picbox{5an4f45.pstex_t} \right] 
\star D_4^{I\,b_1b_2b_3b_4}(1,2,3,45) \Bigg\}
\,.
\label{d5isolutiondiag}
\eea
We will come to the interpretation of this structure 
in section \ref{interpret5i}. 

Now we explain how (\ref{d5isolution}) can be 
shown to solve the integral equation (\ref{inteq5i}). 
The only pieces of information about the irreducible four--gluon 
amplitude $D_4^I$ we need for the purpose of this proof are its complete 
symmetry in the four outgoing gluons, cf.\ (\ref{complsymmd4i}), 
and the integral equation (\ref{inteq4i}) it fulfills. Fortunately, 
an analytic solution of the latter is not required. 

We start from the integral equation (\ref{inteq5i}) derived previously 
and insert the conjectured solution (\ref{d5isolution}). On the left 
hand side we then find ten terms of the kind $\omega D_4^I$. To these 
we apply (\ref{inteq4i}). Thereby we produce different terms and 
we first concentrate on the terms involving the 
vertex function $V$. For example, applying (\ref{inteq4i}) to 
\be
\omega f_{a_1a_2c} D_4^{I\,ca_3a_4a_5}(12,3,4,5) 
\label{onetermind5ilhs}
\ee
produces, due to (\ref{colV}), the expression 
\bea
%
% PLEASE do not change the layout of this equation.
%
f_{a_1a_2c} (V_ {2 \rightarrow 4}D_2)^{ca_2a_3a_4} (12,3,4,5) &=& 
 f_{a_1a_2a_3} \delta_{a_4a_5} (VD_2)(12,3;4,5) \nn \\
&&+ f_{a_1a_2a_4} \delta_{a_3a_5} (VD_2)(12,4;3,5) \nn \\
&&+ f_{a_1a_2a_5} \delta_{a_3a_4} (VD_2)(12,5;3,4) 
\eea
containing three different vertex functions $V$. 
Similar expressions are obtained from the other $\omega D_4^I$ 
terms on the left hand side. In some cases a minus sign arises due to 
the antisymmetry of the structure constant $f_{abc}$. For instance, 
from the second term in (\ref{d5isolution}),
\be
 f_{a_1a_3c} D_4^{I\,ca_2a_4a_5}(13,2,4,5) 
\,,
\ee
in which the pair (1,3) 
of gluons is merged we get
\bea
&&- f_{a_1a_2a_3} \delta_{a_4a_5} (VD_2)(13,2;4,5) 
+ f_{a_1a_3a_4} \delta_{a_2a_5} (VD_2)(13,4;2,5) \nn \\
&& \hspace{4mm}
+ f_{a_1a_3a_5} \delta_{a_2a_4} (VD_2)(13,5;2,4) 
\,. 
\eea
Therefore we find exactly the same 
thirty vertex functions that occur also on the 
right hand side of (\ref{inteq5i}) according 
to (\ref{explsumoverH}),(\ref{h=sumv}). 
We have thus confirmed that the conjectured solution (\ref{d5isolution}) 
indeed reproduces the correct lowest order term in the integral equation, 
namely the combination of vertex functions $VD_2$ given above. 
Moreover, we see that the first term on the right hand side of 
the integral equation (\ref{inteq5i}) 
is not only a superposition of vertex functions $VD_2$ but of full 
transition vertices $V_ {2 \rightarrow 4}$ (applied to $D_2$ as usual). 

Let us now consider further terms in the integral equation (\ref{inteq5i}) 
that we have not treated yet, namely those  
involving the irreducible four--gluon amplitude $D_4^I$. 
Having applied (\ref{inteq4i}) to the $\omega D_4^I$ terms 
on the left hand side the homogeneous term of that equation 
produces convolutions of $D_4^I$ amplitudes with kernels 
$K_{2 \rightarrow 2}$. In these, a kernel acts on $D_4^I$ first 
and then the splitting of one gluon into a pair happens according 
to the combinations in (\ref{d5isolution}). 
On the right hand side of the integral equation (\ref{inteq5i}) the last 
term also gives us convolutions of $D_4^I$ amplitudes with kernels 
$K_{2 \rightarrow 2}$, but here the order of the convolution and 
the splitting of gluons is interchanged: first one gluons splits into 
two and then two of the now five gluons interact via a kernel 
$K_{2 \rightarrow 2}$. 
Among the terms just mentioned a subclass cancels immediately. 
Consider the case that the two--to--two kernel acts between two gluons 
none of which undergoes a splitting (LHS) or has emerged from a 
splitting (RHS). Then the order of interaction and splitting along 
the $t$-channel evolution is irrelevant 
and these terms are in fact the same on both sides. 

The next terms in the integral equation that we look at are the products 
of $D_4^I$'s with trajectories $\beta$. These arise on the 
left hand side either from $\omega D_4^I$ via (\ref{inteq4i}) or from 
the original $\left[ \sum_{i=i}^5 \beta(\kf_i)\right] D_5^I$ 
after (\ref{d5isolution}) 
is inserted. Those in which the argument of the trajectory function 
does not correspond to a gluon undergoing or arising from a splitting 
cancel directly between these contributions. It can be easily checked that 
the others are exactly cancelled by the terms from the right hand side 
in which the two gluons emerging from a splitting interact with each 
other via a kernel $K_{2 \rightarrow 2}$. 

It is a bit more complicated to study the expressions still left in the 
integral equation after the cancellations discussed so far. 
These are $D_4^I$'s undergoing a two--to--three transition 
via the kernel $K_{2 \rightarrow 3}$ on the RHS and 
convolutions of $D_4^I$ functions 
with kernels $K_{2 \rightarrow 2}$ 
in which one of the gluons undergoing or emerging from a 
splitting is involved in the interaction (both sides). The latter 
do no longer include such convolutions in which the 
interaction is between the two gluons emerging from the 
splitting. In the terms under consideration three of the five 
outgoing gluons participate in the splitting or in the interaction. 
The other two gluons do not interact and can be in an arbitrary 
color state. Among the five gluons there can be a total of ten 
different three--gluon subgroups, and we will argue that the cancellation 
takes place in each of these subgroups separately. To this end let us 
concentrate on one of these subgroups, say the one with the first three 
of the outgoing gluons affected. 

The mechanism that makes these contributions cancel between the 
two sides of the integral equation is the same that already caused 
reggeization in the three--gluon amplitude $D_3$. 
This does not come as a surprise since 
it was just the similarity of the corresponding integral equations that 
lead us to the ansatz (\ref{d5isolution}). 
The identity actually bringing about the reggeization of the 
three--gluon subsystem is in pictorial language 
\bea
 \picbox{reggel1.pstex_t} + \picbox{reggel2.pstex_t} 
+ \picbox{reggel3.pstex_t} &=&
 \frac{2}{g}\,\picbox{regger1.pstex_t} + \picbox{regger2.pstex_t}
+ \picbox{regger3.pstex_t}
\nn \\
&& + \picbox{regger4.pstex_t}+ \picbox{regger5.pstex_t}
+ \picbox{regger6.pstex_t}+ \picbox{regger7.pstex_t}
\,.
\label{reggeizebilder}
\eea
Only the three--gluon subsystem is shown, and the horizontal lines 
at the top are meant to suggest the irreducible amplitude $D_4^I$ 
that the gluons enter. The arrows indicate the symmetry of this 
amplitude under the simultaneous exchange of color and momentum 
of the two gluons. The kernels are the ones defined in 
section \ref{kernels}. 

The splitting of a gluon is depicted here by the corresponding 
color diagram and is meant to indicate the behavior in 
momentum space\footnote{Strictly speaking, this 
is done in abuse of our notation that usually separates momentum 
and color space. Confusion should hardly be possible here as 
all terms have been described in detail before.} as well. 
The terms on the left (right) hand side of (\ref{reggeizebilder}) 
are exactly the ones that occur on the left (right) hand side of 
the integral equation (\ref{inteq5i}). 
To prove (\ref{reggeizebilder}), the convolutions 
are evaluated as described in the preceding sections. 
However, here the situation is slightly complicated by the fact that 
the two gluons entering from above can be in an arbitrary color state. 
In the case of $D_3$ these two gluons were in a color singlet state, 
effectively reducing all color tensors to an overall $f_{a_1a_2a_3}$. 
Here we have to be more careful and treat three independent color classes 
separately. (Of course, this could have been done already for $D_3$ 
but there it was not necessary.) The three classes are 
\be
 \picbox{color2to3.pstex_t}\,,
 \qquad\qquad \picbox{color2to3ex12.pstex_t}\,, 
 \qquad \qquad \picbox{color2to3ex23.pstex_t}
\,.
\label{threecolors}
\ee
For some of the terms in (\ref{reggeizebilder}) it is necessary to 
use the symmetry in the upper two gluons which 
is a property of $D_4^I$. 
Each of the three terms on the left hand side of 
(\ref{reggeizebilder}) contributes to two of the three color classes 
in (\ref{threecolors}) via the Jacobi identity (\ref{Jacobi}). 
Having dissected the integral equation this far, it is finally a 
comparatively short calculation to check that (\ref{reggeizebilder}) 
holds. Thereby we have finished 
the proof that $D_5^I$, as given in (\ref{d5isolution}), in fact is 
the solution of the integral equation. 

\subsection{Interpretation of the result}
\label{interpret5i}

In the preceding section we have been able to solve the 
equation for the five--gluon amplitude. 
Now we want to interpret our findings in view of a possible 
field theory of unitarity corrections. 
Let us first summarize the essential results we have obtained. 
We have split the five--gluon amplitude into 
two parts. The first part was the reggeizing part $D_5^R$ 
that is the superposition of two--gluon amplitudes $D_2$. 
We have found an integral equation for the remaining 
part $D_5^I$ and have solved it. It turned 
out that the remaining part is the superposition of 
irreducible four--gluon amplitudes $D_4^I$. 
Neglecting all normalization factors and color tensors, 
this situation can be sketched in the following way:
\be
  D_5(\kf_1,\kf_2,\kf_3,\kf_4,\kf_5) =
\sum 
 \picbox{solutiond51.pstex_t} + \sum \picbox{solutiond52.pstex_t}
\,.
\label{solutiond5diag}
\ee
The first term on the right hand side is the reggeizing 
part $D_5^R$ of the amplitude (see (\ref{d5r})). The 
sum extends over all partitions of the five gluons into 
two groups. 
The second term is the one found in the 
preceding section, see (\ref{d5isolution}).  
Here the sum includes all possible 
pairs of gluons that then merge into one. 

In the five--gluon amplitude, 
we only find elements that are already known: 
the amplitudes $D_2$ and $D_4^I$, and the 
two--to--four transition vertex $V_ {2 \rightarrow 4}$. 
There is no new transition vertex and no new irreducible amplitude 
that would include a five--gluon compound state. 
The absence of new elements is an intriguing result. 
It shows that the five--gluon amplitude reggeizes 
completely. This result (\ref{solutiond5diag}) 
clearly constitutes a generalization of the concept 
of reggeization and proves that reggeization also takes place in 
more complicated amplitudes. 
Especially interesting is the reggeization 
in the second part that turns out to be a superposition 
of irreducible four--gluon amplitudes. 
The mechanism at work here is exactly 
the same as in the three--gluon amplitude. 

Reggeization was an important prerequisite 
for the emergence of the two--to--four 
vertex and thus of the field theory structure 
in the four--gluon amplitude. That the phenomenon 
of reggeization also occurs in the five--gluon amplitude 
gives us confidence that 
the idea of a field theory structure will be a good 
guiding line also for the investigation of the six--gluon 
amplitude. 

Given that the three-- and five--gluon 
amplitudes exhibit complete reggeization caused by the 
same mechanism, one is 
naturally lead to the question whether the same 
is true for each odd number $n$ of gluons. Indeed, as we have seen, 
the mechanism leading to reggeization in a three--gluon 
subsystem is very general.  It is completely independent of the 
structure of the quark loop that we started with 
in the analysis of the integral equations. 
In deriving (\ref{reggeizebilder}) 
we only made use of the fact that the amplitude to which the 
upper two gluons are attached --- in this case $D_4^I$ --- 
is symmetric in the two gluons. Therefore we can conclude 
that for a given odd $n$ one important condition for 
the reggeization of the $n$-gluon amplitude is 
fulfilled as soon as the irreducible part of the $(n-1)$-gluon 
amplitude is symmetric. 
However, we have to keep in mind that this is only one 
of the two conditions leading to complete reggeization. 
The second condition necessary for the reggeization 
of $D_5^I$ was that the inhomogeneous term in 
the integral equation (\ref{inteq5i}) had the 
specific form (\ref{h=sumv}), i.\,e.\ could be written 
as a special superposition of transition vertices 
$V_{2 \rightarrow 4}$. 
For a part of an arbitrary $n$-gluon 
amplitude with odd $n$ to reggeize it is obviously 
necessary that the respective 
inhomogeneous term has a very specific form. 
With our present knowledge, we are not able to derive this 
specific form of the inhomogeneous term for general $n$. 
The complete reggeization of 
amplitudes with an odd number of gluons 
is therefore at present only a (plausible) conjecture. 

We would like to add a remark that concerns our choice of notation, 
but actually goes beyond a pure issue of notation. 
In the splitting of the four--gluon amplitude into two parts 
in (\ref{d4ri}) the superscript $I$ in $D_4^I$ was meant to 
indicate that this part of the amplitude is irreducible --- 
in contrast to the other part. For four gluons this was 
a good choice of notation since that part in fact contains 
a new irreducible compound state of four reggeized gluons. 
In the case of five gluons, we again split the amplitude 
into two parts, cf.\ (\ref{d5ri}). The first part $D_5^R$ 
is a reggeizing one as the superscript $R$ indicates. 
But now we have discovered that the remaining part 
$D_5^I$ reggeizes as well. (For this reason we have 
avoided to call this part 'irreducible'.) 
Nevertheless our notation makes perfect sense when 
extended in an appropriate way. 
The first superscript $I$ or $R$ should be understood 
as specifying the (non--)reggeization of the respective 
part with respect to the two--gluon state. 
We can then introduce a second superscript to indicate 
the (non--)reggeization with respect to the four--gluon 
state. (We will actually be forced to do so when considering 
the six--gluon amplitude in the next section.) 
For a proper notation we should thus identify 
$D_5^I = D_5^{I,\,R}$, 
the second superscript now indicating that this part 
reggeizes with respect to the four--gluon state. 
The notation can easily be extended to accommodate 
reggeization with respect to a potential six--gluon state 
or even higher compound states. 

\section{Six gluons}
\label{d6}

\subsection{A reggeizing part}
\label{regge6}

Encouraged by the success we have had so far with 
that procedure we again 
use the quark loop amplitude $D_{(6;0)}$ to construct 
from it a reggeizing part $D_6^R$ as a superposition 
of two--gluon (BFKL) amplitudes. The full six--gluon 
amplitude is then split into two parts, 
\be
 D_6 = D_6^R + D_6^I
\,,
\label{d6split}
\ee
and it will be our first task to find a new integral 
equation for the remaining part $D_6^I$. In detail, 
the reggeizing part $D_6^R$ is 
\bea
\lefteqn{D_6^{R\,a_1a_2a_3a_4a_5a_6}
   (\kf_1,\kf_2,\kf_3,\kf_4,\kf_5,\kf_6)= } 
%\nn 
\\
&=&  g^4 \{ d^{a_1a_2a_3a_4a_5a_6} \,[ 
  D_2(12345,6) + D_2(1,23456) - D_2(16,2345) ] \nn \\
& &\,+\,d^{a_2a_1a_3a_4a_5a_6} \,[ 
  D_2(13456,2) - D_2(1345,26) 
+ D_2(126,345) -  D_2(12,3456) ] \nn \\
& &\, +\,d^{a_1a_2a_3a_4a_6a_5} \,[ 
 D_2(12346,5) - D_2(1234,56) 
+ D_2(156,234) - D_2(15,2346) ] \nn \\
& &\, +\,d^{a_2a_1a_3a_4a_6a_5} \,[
 - D_2(1256,34) - D_2(1346,25) 
 + D_2(125,346) + D_2(134,256) ] \nn \\
& &\, +\,d^{a_3a_1a_2a_4a_5a_6} \,[
 D_2(12456,3) - D_2(1245,36) 
 + D_2(136,245) - D_2(13,2456) ] \nn \\
& &\,+\,d^{a_1a_2a_3a_5a_6a_4} \,[
 D_2(12356,4) - D_2(1235,46) 
+ D_2(146,235) - D_2(14,2356) ] \nn \\
& &\, +\,d^{a_2a_1a_3a_5a_6a_4} \,[
- D_2(1246,35) - D_2(1356,24)
+ D_2(124,356) + D_2(135,246) ] \nn \\
& &\, +\,d^{a_1a_2a_3a_6a_5a_4} \,[
- D_2(1236,45) - D_2(1456,23) 
+ D_2(123,456)  
%\nn \\
%&&\hspace{3cm}
+ D_2(145,236) ] 
\}
\nn
\label{d6r}
\eea
as obtained from (\ref{d60}) by the replacement 
$D_{(2;0)} \rarr D_2$ while keeping the color 
and momentum structure. 
This expression already indicates one of the major 
difficulties we have to overcome during the treatment of 
the six--gluon amplitude: the large number of 
terms we have to take care of. 

\subsection{The integral equation for the remaining part}
\label{inteq6isection}

The original integral equation (\ref{inteq6}) for the 
six--gluon amplitude is now used to 
derive a new integral equation for the unknown part $D_6^I$. 
The method in this step is exactly the same as for the 
four-- and five--gluon amplitudes. We 
insert into the integral equation our complete knowledge 
about the reggeizing parts $D_n^R$ 
of the amplitudes $D_n$ with up to 
$n=6$ gluons, including the ansatz (\ref{d6split}), (\ref{d6r}) 
for the six--gluon amplitude. 
The corresponding formulae for $n\le5$ can be found in the 
preceding sections. 
Then we apply the BFKL equation 
(\ref{inteq2}) to the expression $\omega D_6^R$ 
on the left hand side. This is possible because 
$D_6^R$ was chosen as a superposition of BFKL amplitudes, 
cf.\ (\ref{d6r}). 
We thereby produce convolutions of $D_2$ 
amplitudes with two--to--two 
kernels and products of $D_2$ amplitudes 
with trajectory functions $\beta$. 
The insertion of the reggeizing parts $D_n^R$ of the 
amplitudes on the right hand side leads to convolutions 
of $D_2$ amplitudes with the integral kernels. 
We have to perform the corresponding contractions 
of color tensors and have to bring the integrals 
to their standard forms as classified in section \ref{standardint}. 
The main problem consists in the huge 
number of combinations of amplitudes with kernels. 
We have to perform close to $250$ contractions 
of color tensors, and we have to find the standard form of 
more than $3500$ integrals. Whereas the color tensors 
can still be calculated by hand this is no longer possible 
for the huge number of momentum space integrals. 
We have therefore developed an algorithm for this purpose 
that is suited for the implementation on a computer. 
The algorithm is explained in detail in appendix 
\ref{momentumapp}. We have written a PERL script 
based on this algorithm that produces an output which 
can directly be used as an input for a computer algebra 
program like MAPLE. The tensor contractions are calculated 
with the help of the method described in appendix \ref{colorapp}. 
Some of the contractions are given explicitly in that 
appendix. Many other contractions are obtained from 
these by permutations of the gluon color labels. 
The computer algebra program is then used to multiply 
the resulting sums of elementary tensors with the 
corresponding integrals, and to finally collect all terms. 
In the final step more than $2 \cdot 10^4$ integrals 
have to be sorted according to their color tensor coefficients. 
(This shows that our method of dealing with the integral 
equations will in its practical applicability be limited 
to relatively small numbers $n$ of gluons.) 
Having collected all terms in the equation which contain 
the amplitude $D_2$  we have found the inhomogeneous term 
of the new integral equation for $D_6^I$. 

In the derivation of the new integral equation the terms containing 
the irreducible four--gluon amplitude $D_4^I$ and the second part 
$D_5^I$ of the five--gluon amplitude remain unchanged. 
Their combinations with the kernels are the same as in the 
original equation (\ref{inteq6}). 
The resulting integral equation for $D_6^I$ 
is then found to have the form
\bea
%
% PLEASE do not change the layout of this equation.
%
\lefteqn{
\left(\omega - \sum_{i=1}^6 \beta(\kf_i)\right) 
D_6^{I\,a_1a_2a_3a_4a_5a_6}(\kf_1,\kf_2,\kf_3,\kf_4,\kf_5\kf_6) 
= } \nn \\
&=&
 ( W^{a_1a_2a_3a_4a_5a_6}D_2)(\kf_1,\kf_2,\kf_3,\kf_4,\kf_5\kf_6) 
\nn \\
&&
 + \sum f_{a_1a_2a_3} f_{a_4a_5a_6} L(1,2,3;4,5,6) \nn \\
&&
 + \sum d^{a_1a_2a_3a_4} \delta_{a_5a_6} I(1,2,3,4;5,6)
\nn\\
&&
 + \sum d^{a_2a_1a_3a_4} \delta_{a_5a_6} J(1,2,3,4;5,6) \nn \\
&&
+ \sum K^{ \{b\} \rarr \{a\} }_{2 \rarr 4} \otimes D_4^{I\,b_1b_2b_3b_4} 
+ \sum K^{ \{b\} \rarr \{a\} }_{2 \rarr 3}\otimes D_5^{I\,b_1b_2b_3b_4b_5}
\nn \\
&& 
+ \sum K^{ \{b\} \rarr \{a\} }_{2 \rarr 2} 
\otimes D_6^{I\,b_1b_2b_3b_4b_5b_6}
\,.
\label{inteq6i}
\eea
The first four terms on the right hand side are the result of the 
computation outlined above. We will now describe them in detail. 

The first observation we make is again that certain color structures 
are completely cancelled in the equation. All terms proportional 
to $d^{a_1a_2a_3a_4a_5a_6}$ (and the other seven permutations 
of this occurring in (\ref{d6r})) are cancelled between the different 
contributions to the inhomogeneous term and drop out. The 
same was observed 
for the tensors $d^{a_1a_2a_3a_4}$ and $f^{a_1a_2a_3a_4a_5}$ 
in the equations for the parts 
$D_4^I$ and $D_5^I$ of the four-- and five--gluon amplitudes, 
respectively. 

As in the integral equations for $D_4^I$ and $D_5^I$ the 
inhomogeneous term has a high degree of symmetry which 
we will explain for each of the terms below. 
This symmetry is not only nice by itself, but it is also a 
possibility to check our calculation. 

The first term on the right hand side of (\ref{inteq6i}) 
differs in its structure from the other terms and will be treated 
separately in section \ref{v26}. Here we mention already that 
it is symmetric in the sense that it is the sum of terms that 
are obtained from each other by permutations of the gluons. 

The same is true for the second term on the right hand side 
of the new integral equation. The sum extends over all 
partitions of the six gluons into two groups each of which contains  
three gluons,  
\bea
\sum f_{a_1a_2a_3} f_{a_4a_5a_6} L(1,2,3;4,5,6) &=& 
f_{a_1a_2a_3} f_{a_4a_5a_6} L(1,2,3;4,5,6) 
\nn \\
&& + \, f_{a_1a_2a_4} f_{a_3a_5a_6} L(1,2,4;3,5,6) 
\nn \\
&& + \dots +  f_{a_1a_5a_6} f_{a_2a_3a_4} L(1,5,6;2,3,4) 
\,.
\label{explsumoverL}
\eea
The function $L$ is the same in all terms in the sum 
and only its arguments are exchanged in the different 
terms. 
A closer inspection reveals that the function $L$ 
permits a decomposition into 
vertex functions $V$ known from the two--to--four 
transition vertex (see section \ref{34V}), 
\bea
\lefteqn{
L(1,2,3;4,5,6) = \frac{g^2}{4} \times } \nn \\
&&\times [(VD_2)(12,3;45,6) - (VD_2)(12,3;46,5) 
+ (VD_2)(12,3;4,56) \nn \\
&&\hspace{.5cm} 
- (VD_2)(13,2;45,6) + (VD_2)(13,2;46,5) - (VD_2)(13,2;4,56) \nn \\
&&\hspace{.5cm} 
+ (VD_2)(1,23;45,6) - (VD_2)(1,23;46,5) 
%\nn \\
%&&\hspace{.5cm} 
+ (VD_2)(1,23;4,56) ]
\,.
\label{l=sumv}
\eea

The sum in the third term on the right hand side of the 
integral equation extends over all partitions of the six gluons 
into one group containing four and one group containing 
two gluons, 
\bea
\sum d^{a_1a_2a_3a_4} \delta_{a_5a_6} I(1,2,3,4;5,6) &=& 
d^{a_1a_2a_3a_4} \delta_{a_5a_6} I(1,2,3,4;5,6) 
\nn \\
&&+ \,d^{a_1a_2a_3a_5}\delta_{a_4a_6} I(1,2,3,5;4,6) 
\nn \\
&&+ \dots + \delta_{a_1a_2} d^{a_3a_4a_5a_6} I(3,4,5,6;1,2) 
\,.
\label{explsumoverI}
\eea
Also in this case we find that the function $I$ is the 
same in all terms in the sum. Remarkably, also this function 
can be written in terms of the vertex function $V$, 
\be
I(1,2,3,4;5,6) = - g^2 [ 
(VD_2)(1,234;5,6) + (VD_2)(123,4;5,6) - (VD_2)(14,23;5,6) ]
\,.
\label{i=sumv}
\ee
The sum in the fourth term on the right hand side of 
the new integral equation (\ref{inteq6i}), 
\be
\sum d^{a_2a_1a_3a_4} \delta_{a_5a_6} J(1,2,3,4;5,6) 
\,,
\label{explsumoverJ}
\ee
extends over the same permutations of gluons as the 
term discussed before (see (\ref{explsumoverI})). 
Again the function $J$ is the same in all terms in the sum, 
and it can be written as a superposition 
of vertex functions $V$ as 
\bea
J(1,2,3,4;5,6) &=& - g^2 [ 
(VD_2)(134,2;5,6) + (VD_2)(124,3;5,6) \nn \\
&&
\hspace{.8cm} - (VD_2)(12,34;5,6) - (VD_2)(13,24;5,6)]
\,.
\label{j=sumv}
\eea

We would like to emphasize that the symmetry of the 
sums contributing to the inhomogeneous term of the 
new integral equation is an outcome of our calculation. 
We have not used it to derive the new equation. That 
we find the symmetry in the resulting equation gives 
us confidence that we did not make any errors in the 
long and tedious calculation leading to (\ref{inteq6i}). 
We also would like to stress that the representation 
of a part of the inhomogeneous term as a superposition 
of well--known 
vertex function $V$ is an outcome of our calculation and 
was not used to derive the new equation. Unfortunately, we 
do not know a way that directly leads to the comparatively 
simple structure arising in the terms discussed above.  

\subsection{A new piece in the field theory}
\label{v26}

The first term on the right hand side of 
the integral equation (\ref{inteq6i}) differs in its 
structure from the other terms.  We will therefore 
discuss it separately in this section. The study of the other terms 
will be resumed in section \ref{d6ir}. 
We start by giving an explicit representation of the term 
under consideration and proceed by listing its properties. 
After that we will 
elaborate on the question how the new term 
has to be interpreted in the context of the effective 
field theory structure of unitarity corrections. 

\subsubsection{Explicit representation and properties}
\label{expl26}

The first term on the right hand side in (\ref{inteq6i}) 
has the following color and momentum structure: 
\be
 (W^{a_1a_2a_3a_4a_5a_6}D_2)(\kf_1,\kf_2,\kf_3,\kf_4,\kf_5,\kf_6) = 
\sum   d_{a_1a_2a_3} d_{a_4a_5a_6} (W D_2)(1,2,3;4,5,6) 
\,.
\label{newpiece}
\ee
The sum extends over all (ten) partitions of the six gluons 
into two groups containing three gluons each, 
\bea
\sum   d_{a_1a_2a_3} d_{a_4a_5a_6} (W D_2)(1,2,3;4,5,6) 
&=& d_{a_1a_2a_3} d_{a_4a_5a_6} (W D_2)(1,2,3;4,5,6) 
\nn \\
&&+ \, d_{a_1a_2a_4} d_{a_3a_5a_6} (W D_2)(1,2,4;3,5,6) 
+ \dots 
\nn \\
&&
+ \,d_{a_1a_5a_6} d_{a_2a_3a_4} (W D_2)(1,5,6;2,3,4) 
\,.
\label{explsumoverw}
\eea
The function $W D_2$ is the same in all permutations. 
(Again this is an outcome of our calculation and was not 
assumed at any stage when the equation (\ref{inteq6i}) 
was derived.) 
The sum thus contains the same permutations of the 
six gluons as the second 
term (\ref{explsumoverL}) on the right hand 
side of (\ref{inteq6i}). 
For the notation to be consistent 
the function $W^{a_1a_2a_3a_4a_5a_6}$ should be 
understood as an integral operator acting on a BFKL 
amplitude $D_2$. It thus carries two more momentum 
arguments $\qf_j$ for the momenta entering from 
above. We will again suppress these two momenta in 
the following. 

In contrast to the other 
terms in (\ref{inteq6i}) 
discussed so far the function $W$ does not permit 
a decomposition into the vertex functions $V$ known from the 
two--to--four transition vertex. 
We therefore give its full momentum space representation 
as we have obtained it as a result of our calculation. We again use the 
standard integrals defined in section \ref{standardint}. 
Then $WD_2$ has the explicit representation 
\bea
%
% PLEASE do not change the layout of this equation.
%
\lefteqn{(WD_2)(\kf_1,\kf_2,\kf_3;\kf_4,\kf_5,\kf_6) = 
  \fr{g^6}{16} \,\times } \nn \\
&& \{  2 \,[ \,  c(123456) \nn \\
&&\quad\, -\,\, b(12345) - b(12346) - b(12356)- b(12456)- b(13456)
- b(23456) \nn \\
&&\quad\, +\, b(1234) + b(1235) + b(1236) + b(1456) + b(2456) + b(3456)  
\nn \\
&&\quad\, +\, a(1245,3) + a(1246,3) + a(1256,3) + a(1345,2) + a(1346,2) 
+ a(1356,2) \nn \\
&&\quad\, +\, a(2345,1) + a(2346,1) + a(2356,1) \nn \\
&&\quad\, -\,\, b(123) - b(456) \nn \\
&&\quad\, -\, a(124,3) - a(125,3) - a(126,3) - a(134,2) - a(135,2) 
- a(136,2) +\nn \\
&&\quad\, -\, a(234,1) - a(235,1) - a(236,1) \nn \\
&&\quad\, -\, a(145,23) - a(146,23) - a(156,23) - a(245,13) - a(246,13) 
- a(256,13) \nn \\
&&\quad\, -\, a(345,12) - a(346,12) - a(356,12)\nn \\
&&\quad\, +\, a(12,3) + a(13,2) + a(23,1) + a(45,123) + a(46,123) 
+ a(56,123) \nn \\
&&\quad\, +\, a(14,23) + a(15,23) + a(16,23) + a(24,13) + a(25,13) 
+ a(26,13) \nn \\
&&\quad\, +\, a(34,12) + a(35,12) + a(36,12) \nn \\
&&\quad\, -\, a(1,23) - a(2,13) - a(3,12) - a(4,123) - a(5,123) - a(6,123) ] 
\nn \\
&& +[\, t(12345) + t(12346) + t(12356) + t(12456) + t(13456) + t(23456) 
\nn \\
&& \quad\, -\, t(1234) - t(1235) - t(1236) - t(1456) - t(2456) - t(3456) \nn \\
&& \quad\, -\, s(1245,3) - s(1245,6) - s(1246,3) 
- s(1246,5) - s(1256,3) - s(1256,4) 
\nn \\
&& \quad\, -\, s(1345,2) - s(1345,6) - s(1346,2) 
- s(1346,5) - s(1356,2) - s(1356,4) 
\nn \\
&& \quad\, -\, s(2345,1) - s(2345,6) - s(2346,1) 
- s(2346,5) - s(2356,1) - s(2356,4) 
\nn \\
&& \quad\, +\, t(123) + t(456) \nn \\
&& \quad\, +\, s(124,3) + s(124,56) + s(125,3) 
+ s(125,46) + s(126,3) + s(126,45) 
\nn \\
&& \quad\, +\, s(134,2) + s(134,56) + s(135,2) 
+ s(135,46) + s(136,2) + s(136,45) 
\nn \\
&& \quad\, +\, s(234,1) + s(234,56) + s(235,1) 
+ s(235,46) + s(236,1) + s(236,45) 
\nn \\
&& \quad\, +\, s(145,23) + s(145,6) + s(146,23) 
+ s(146,5) + s(156,23) + s(156,4) 
\nn \\
&& \quad\, +\, s(245,13) + s(245,6) + s(246,13) 
+ s(246,5) + s(256,13) + s(256,4) 
\nn \\
&& \quad\, +\, s(345,12) + s(345,6) + s(346,12) 
+ s(346,5) + s(356,12) + s(356,4) 
\nn \\
&& \quad\, -\, s(12,3) - s(12,456) - s(13,2) 
- s(13,456) - s(23,1) - s(23,456) \nn \\
&& \quad\, -\, s(45,123) - s(45,6) - s(46,123) 
- s(46,5) - s(56,123) - s(56,4) \nn \\
&& \quad\, -\, s(14,23) - s(14,56) - s(15,23) 
- s(15,46) - s(16,23) - s(16,45) \nn \\
&& \quad\, -\, s(24,13) - s(24,56) - s(25,13) 
- s(25,46) - s(26,13) - s(26,45) \nn \\
&& \quad\, -\, s(34,12) - s(34,56) - s(35,12) 
- s(35,46) - s(36,12) - s(36,45) \nn \\
&& \quad\, +\, s(1,23) + s(1,456) + s(2,13) 
+ s(2,456) + s(3,12) + s(3,456) \nn \\
&& \quad\, +\, s(4,123) + s(4,56) + s(5,123) 
+ s(5,46) + s(6,123) + s(6,45) 
] \}
\,.
\label{w123456}
\eea
On first sight this expression appears to be very complicated. 
Closer inspection reveals that it has a series of very 
interesting properties. Some of them very much resemble 
those of the function $V$ we have described in section 
\ref{fourgluon}. 

Let us first have a look at the symmetry properties of the 
function $W$. We find that 
$W$ is fully symmetric in its first three arguments
\bea
W(\kf_1,\kf_2, \kf_3;\kf_4,\kf_5,\kf_6) 
&=& W(\kf_2,\kf_1, \kf_3;\kf_4,\kf_5,\kf_6) \nn \\
&=& W(\kf_3,\kf_2, \kf_1;\kf_4,\kf_5,\kf_6)
\label{wsymmetry123}
\eea
as well as in its last three arguments,
\bea
W(\kf_1,\kf_2, \kf_3;\kf_4,\kf_5,\kf_6) 
&=& W(\kf_1,\kf_2, \kf_3;\kf_5,\kf_4,\kf_6) \nn \\
&=& W(\kf_1,\kf_2, \kf_3;\kf_6,\kf_5,\kf_4)
\,.
\label{wsymmetry456}
\eea
Notably, the color structure corresponding to 
that permutation of momenta, i.\,e.\ the tensor 
$d_{a_1a_2a_3}d_{a_4a_5a_6}$, has exactly the 
same symmetry properties.  
Further, $W$ is symmetric under the exchange 
of the first three and last three arguments, 
\be
W(\kf_1,\kf_2, \kf_3;\kf_4,\kf_5,\kf_6) = 
W(\kf_4,\kf_5, \kf_6;\kf_1,\kf_2,\kf_3)
\,.
\ee
From these symmetries of the function $W$ and 
from the permutations that enter 
the sum in (\ref{newpiece}) we can conclude that the full 
expression $W^{a_1a_2a_3a_4a_5a_6}(1,2,3,4,5,6)$ 
is completely symmetric in the six outgoing gluons, i.\,e.\ 
under the simultaneous exchange of color labels and 
momentum arguments. 

Next we look at the behavior of $W$ when 
one of its momentum arguments vanishes. 
Not unexpectedly, we find that 
$W$ vanishes whenever one of the six gluons 
carries zero transverse momentum, 
\be
\left.
W(\kf_1,\kf_2, \kf_3;\kf_4,\kf_5,\kf_6) 
\right|_{\kf_i=0} = 0 
\;\;\;\;\;\;  (i\in\{1,\dots,6\}) 
\,.
\label{nullstellw}
\ee
Starting from the explicit representation (\ref{w123456}) 
the proof is straightforward. 
Of course, we again have to use the relations 
between the different standard integrals 
mentioned at the end of section \ref{standardint} and 
the fact that the gluon trajectory function $\beta(\kf)$ 
vanishes for $\kf=0$. 

Further we find that the function $W$ is infrared finite. 
The proof requires just a close inspection of the 
standard integrals occurring in (\ref{w123456}). 
It is easily seen that the integrals come in the 
infrared finite combinations (\ref{inffinite1}) and 
(\ref{inffinite2}) discussed already in section 
\ref{fourgluon}. (Again, the integrals 
have been arranged in (\ref{w123456}) 
in a way that hopefully makes 
this transparent.) 

We expect the function $W$ to be conformally 
invariant in impact parameter space, 
but we will not discuss this issue further here. 

\subsubsection{Interpretation in view of an effective field theory}
\label{interpret26}

It is now natural to ask where the new piece 
$(W D_2)^{a_1a_2a_3a_4a_5a_6}$ finds its place 
in the effective field theory of unitarity corrections. 
At present, we are not yet able to give a conclusive 
answer to this question. However, the following two 
possible answers naturally arise. 

As we have seen in the preceding section the new piece 
has properties 
that very much resemble those of the two--to--four 
transition vertex $V_{2 \rightarrow 4}$. 
It is fully symmetric in the six 
gluons, it is infrared finite, and it vanishes when one of the 
gluon momenta vanishes. 
It is well possible that the new piece 
$W^{a_1a_2a_3a_4a_5a_6}$ is a new two--to--six 
gluon transition vertex $V_{2 \rightarrow 6}$, and 
that it constitutes a new element 
of the effective field theory. 

The second possibility is the following. 
A coupling scheme can be 
chosen in the first three gluons and in the remaining 
three gluons. Then the function $W$ is split into 
several parts according to the symmetry or antisymmetry 
under the exchange of the gluons $1$ and $2$, say, and under the 
exchange of the pair $(12)$ of gluons with gluon $3$, and analogously 
for the other three gluons. Based on the result of this procedure 
it is possible to define new two--to--four vertices 
with symmetry properties different from those of the 
well--known two--to--four vertex $V_{2 \rightarrow 4}$. 
Based on these one can in turn define new four--gluon amplitudes 
that then become basic elements of the effective field theory. 
Due to this a direct transition from two to six reggeized gluons 
is avoided, and the new piece becomes a superposition of different 
two--to--four vertices. However, 
this second possibility has not yet been fully investigated. 

The two possibilities have considerably different implications 
for the emerging field theory of unitarity corrections, especially 
in view of the necessary extension to amplitudes with more 
gluons in the $t$-channel. The first possibility corresponds to a 
picture in which an infinite number of new transition vertices 
$V_{2 \rightarrow 2n}$ occurs 
in the field theory, one for each even number of gluons. 
The second possibility, in contrast, leads to a picture in which 
there are only four two--to--four vertices with different 
symmetry properties, and no other vertices of the type 
$V_{2 \rightarrow 2n}$. Both possibilities should be 
investigated in more detail. 
We expect that especially a better understanding of the 
conformal invariance of the expected effective field theory 
will help to clarify the status of the new piece. 

\subsection{Further reggeization}
\label{d6ir}

Now we come to discuss the other terms that are present in 
the new integral equation (\ref{inteq6i}) 
for the part $D_6^I$ of the six--gluon amplitude. 
In section \ref{inteq6isection} we have already shown 
that these terms can be written as superpositions of 
well--known vertex functions $V$. We will in this 
section disregard the new piece discussed in the 
previous section. 

Already in section \ref{5gluons} we encountered 
a situation similar to the one which we find here in 
the integral equation (\ref{inteq6i}). 
Also there the inhomogeneous term of the integral 
equation for $D_5^I$ could be written as a superposition 
of vertex functions $V$. It was a characteristic 
indication for the occurrence of a further reggeization 
of the amplitude $D_5^I$ with respect to the 
irreducible four--gluon amplitude $D_4^I$. 
This idea even allowed 
us to find the exact solution of the equation. 
We cannot expect that the remaining part $D_6^I$ 
of the six--gluon amplitude reggeizes completely. 
But the occurrence of the vertex functions in the 
inhomogeneous term of its equation strongly 
suggests that a part of $D_6^I$ will reggeize. 
To gain further insight we should therefore 
construct an ansatz for the remaining part in order 
to simplify the integral equation (\ref{inteq6i}). 
The remaining part $D_6^I$ should thus be split 
into a reggeizing part and an irreducible part, 
\be
D_6^I = D_6^{I,\,R} + D_6^{I,\,I} 
\,,
\label{d6iirii}
\ee
where this time the term 'reggeizing' refers to 
the reggeization with respect to the four--gluon 
compound state, cf.\ the discussion at the end of 
section \ref{interpret5i}.
The reggeizing part should be a superposition of 
irreducible four--gluon amplitudes, symbolically 
\be
  D_6^{I,\,R} = \sum D_4^I
\label{d6ird4i}
\,.
\ee
The problem is now to find the correct color and momentum 
structure for the right hand side of this symbolic equation. 

We should have in mind that the inhomogeneous term in the 
integral equations for $D_n$, i.\,e.\ the quark loop, 
always suggests the best choice 
of a reggeizing part $D_n^R$. 
To make a good guess for the reggeizing part $D_6^{I,\,R}$ 
we should therefore have a close look at the inhomogeneous 
term of the new integral equation (\ref{inteq6i}). 

Let us first look at the terms (\ref{i=sumv}) and 
(\ref{j=sumv}) containing the functions $I$ and $J$. 
We will pick one permutation in the sums (\ref{explsumoverI}) 
and (\ref{explsumoverJ}) only, the other permutations 
can then be treated in analogy. We see immediately that 
the color and momentum structure in the first four gluons 
in the terms 
\be
 d^{a_1a_2a_3a_4} \delta_{a_5a_6} I(1,2,3,4;5,6) 
\ee
and 
\be
d^{a_2a_1a_3a_4} \delta_{a_5a_6} J(1,2,3,4;5,6) 
\ee
is exactly the same as in $D_4^R$. 

A second observation is a certain mismatch 
between $f_{abc}$ and $d_{abc}$ tensors. 
While there are terms of the kind 
\be
\sum f_{a_1a_2a_3} f_{a_4a_5a_6} L(1,2,3;4,5,6)
\ee
present in the equation which can be written 
in terms of $V$, the 
corresponding terms with $d_{abc}$ tensors 
(the new piece, see section \ref{v26}) cannot be 
written in a similar way. This already indicates that 
the $f$- and $d$-tensors have to be treated differently. 

In order to come from an ansatz of the form (\ref{d6ird4i}) 
to the cancellation of the inhomogeneous term in 
the integral equation we have to use the integral 
equation (\ref{inteq4i}) for the irreducible part $D_4^I$ 
of the four--gluon amplitude. We want to write 
the arguments of the $D_4^I$'s in the ansatz 
in such a way that they have exactly the same 
momentum structure which 
we find in the vertex functions $V$ in 
(\ref{inteq6i}). This is completely analogous to 
the parts $D_n^R$ in which the momentum 
structure was taken from the quark loop. 
In the case of $D_n^R$ we could also 
keep the color structure. This was possible because 
the color structure of the two--gluon amplitude $D_2$ was 
trivial, i.\,e.\ the two gluons were always in a color singlet 
state. That allowed us to factorize the two--gluon 
amplitude into a color part ($\delta_{a_1a_2}$) and 
a momentum part. 
Now the situation is more complicated since such a factorization 
is not possible for the irreducible four--gluon amplitude. 
We have to use the full amplitude $D_4^{I\,b_1b_2b_3b_4}$. 
Since we certainly need the six color labels 
$a_1,\dots,a_6$ in the ansatz (\ref{d6ird4i}), 
the technical procedure we have to use is the 
contraction with a tensor $\Theta$, 
\be
 \Theta^{a_1a_2a_3a_4a_5a_6;b_1b_2b_3b_4} 
 D_4^{I\,b_1b_2b_3b_4} 
\,.
\ee
The tensor $\Theta$ is an invariant tensor in the ten--fold 
tensor product $\bigotimes_{i=1}^{10} [\mbox{su}(N_c)]$ 
of the Lie algebra. 
It will obviously be very difficult to find the correct 
tensors for the contractions in this huge tensor space 
without having additional information.  
We have to hope that the situation is in a certain sense 
more simple. It will be necessary to find restrictions on the 
tensors from the inhomogeneous term in the new integral 
equation. 
Now a problem arises. In the term 
\be
d^{a_1a_2a_3a_4} \delta_{a_5a_6} V(123,4;5,6) 
\ee
for example the first four gluons are in an overall 
color singlet state, as are the last two gluons. 
Obviously, the color tensor necessary for the contraction 
with $D_4^{I\,b_1b_2b_3b_4}$ is fixed by the inhomogeneous 
term only in the case in which the gluons with labels $b_1$ and 
$b_2$ are in a color singlet state. For the other irreducible 
representations we have no hint from the inhomogeneous 
term which would restrict the tensor $\Theta$. 
This problem seems to be a conceptual one in our approach. 
In the first step, that is for identifying a reggeizing part $D_n^R$ 
in the $n$-gluon amplitude, the quark loop was sufficient 
to fix the reggeizing part. In the present situation 
it is not completely excluded that the correct 
color tensor for the non--singlet states cannot be fixed 
unambiguously. Possibly the solution of this problem 
requires a better knowledge of the irreducible four--gluon 
amplitude $D_4^I$. 
In spite of this conceptual problem we expect that 
one can find a simple ansatz that leads to further insight. 

Unfortunately, we have not yet been able to find a 
satisfying and unique solution. It appears that a better 
understanding of the phenomenon of reggeization, 
especially in the irreducible parts of the amplitudes, 
will help to resolve this problem. First steps in this 
direction have been done in \cite{Carlothesis}. 

In view of the discussion above and in section 
\ref{interpret26} it is certainly to early to 
draw final conclusions concerning the field theory 
structure of the six--gluon amplitude. 
The most important result of this section is contained 
in the integral equation (\ref{inteq6i}) 
for the remaining part $D_6^I$ 
of the six--gluon amplitude. The occurrence of the vertex 
function $V$ in this equation is an extremely strong 
indication for the fact that further reggeization with 
respect to the four--gluon compound state takes place 
in the six--gluon amplitude. Exactly this is 
the necessary condition for the emergence of the 
field theory structure in the unitarity corrections. 
We regard this as strong evidence for the existence of 
an effective field theory of unitarity corrections. 

\subsection{The Pomeron-Odderon-Odderon vertex}
\label{POO}

We now turn to the question which place the Odderon 
finds in the effective field theory. 
The Odderon is the $C=-1$ partner of the Pomeron, i.\,e.\ 
it carries negative charge parity. In perturbative QCD 
it consists of a compound state of three reggeized gluons 
described by the three--particle BKP equation 
\cite{Bartelskernels,BKP}.
The three gluons are in a completely symmetric state, and the 
color part of the wavefunction is a $d_{abc}$ tensor. 
Given the quantum numbers of these states, it is a natural question 
whether a BFKL Pomeron can be coupled to two Odderons. 
The six--gluon amplitude is the obvious place to look 
for such a Pomeron--Odderon--Odderon vertex. 
The triple Pomeron vertex is obtained from the 
two--to--four 
gluon transition vertex $V_{2 \rightarrow 4}$ 
by projecting it onto three 
BFKL eigenfunctions \cite{Hans,Gregory3p}. 
We can therefore in analogy try to project the inhomogeneous 
term of the new integral equation (\ref{inteq6i}) onto 
two Odderon wavefunctions from below. 
The inhomogeneous term in that equation consists of 
several contributions, and we will concentrate on the 
piece discussed in section \ref{v26}. We therefore ask 
whether the integral 
\bea
V_{POO}
&=&\int \left( \prod_{i=1}^6 d^2\kf_i \right) 
  (W^{a_1a_2a_3a_4a_5a_6}D_2)(\kf_1,\kf_2,\kf_3,\kf_4,\kf_5,\kf_6) 
\nn \\
&&\hspace{.9cm}
\times d_{a_1a_2a_3} d_{a_4a_5a_6}
\Psi_1(\kf_1,\kf_2,\kf_3) \Psi_2(\kf_4,\kf_5,\kf_6) 
\label{poointegral}
\eea
is different from zero. Unfortunately, the wavefunction 
$\Psi$ of the Odderon is not known explicitly, but 
conformal invariance places strong constraints on it. 
In \cite{Lipatovodderon} the wavefunction of the Odderon 
in impact parameter space was found to have the general form 
\be
 \Psi(\rho_1, \rho_2, \rho_3) = 
\left( \frac{\rho_{12}\rho_{13}\rho_{23}}{\rho_{10}^2
\rho_{20}^2\rho_{30}^2} \right)^{h/3}
\psi(x)
\label{odderonwavefct}
\,.
\ee
$h$ is the conformal weight of the Odderon state 
and $x$ is  the anharmonic ratio
\be
  x =\frac{\rho_{12}\rho_{30}}{\rho_{13}\rho_{20}}
\ee
with 
$\rho_{ij} = \rho_i - \rho_j$, the $\rho_i$ being 
the two--dimensional coordinates in impact 
parameter space. 
The wavefunction $\Psi$ vanishes when two of the 
coordinates of the three gluons in the Odderon 
coincide. This property of the Odderon drastically reduces 
the number of terms in 
$(W^{a_1a_2a_3a_4a_5a_6}D_2)$ that can give 
a non--vanishing contribution 
to the integral (\ref{poointegral}). 
If a term in $(W D_2)$ depends only on the 
sum of two momenta, say $\kf_1$ and $\kf_2$, then this 
term is after Fourier transformation to impact parameter 
space proportional to a delta--function of the two 
corresponding coordinates, i.\,e.\ proportional to 
$\delta(\rho_1-\rho_2)$. 
This implies a zero in the Odderon wavefunction and the 
corresponding term does not contribute to (\ref{poointegral}). 
Only very few standard integrals in $(W D_2)$ can actually give 
non--vanishing contributions. Some of them give 
identical results in the integral (\ref{poointegral}) due 
to the symmetry of the Odderon wavefunction. 
(They are in this sense equivalent to each other 
as far as their contribution to (\ref{poointegral}) is concerned.) 
The possible contributions to the above integral can 
in this way be reduced to the following infrared finite 
combination of standard integrals:
\be
2 a(14,25) - s(14,25) - s(14,36) 
\,.
\label{nonvanishterms}
\ee
We now have to look for these terms in the new piece 
$(W^{a_1a_2a_3a_4a_5a_6}D_2)$. 
Interestingly, the first of the permutations in 
(\ref{explsumoverw}) --- the color tensor of which 
exactly matches the color structure of the two 
Odderons --- does not contain terms 
equivalent to the above combination. 
But the other 9 permutations do contain such terms. 
The term 
\be
  d_{a_1a_2a_5} d_{a_3a_4a_6} (W D_2)(1,2,5;3,4,6) \,
\ee
for example, 
contains exactly the terms (\ref{nonvanishterms}). 
As can be easily shown, each of those other 
permutations contains exactly four combinations 
equivalent to (\ref{nonvanishterms}). 
The corresponding color contraction gives a factor 
\be
d_{a_1a_2a_5} d_{a_3a_4a_6}
d_{a_1a_2a_3} d_{a_4a_5a_6} = 
\left( \frac{N_c^2-4}{N_c} \right) ^2 (N_c^2 -1) 
\,.
\ee
We can therefore conclude that a perturbative 
Pomeron--Odderon--Odderon vertex exists, and collecting 
all terms it becomes 
\bea
V_{POO}&=&
9 \,\cdot \,4 \cdot \,\frac{g^6}{16} 
\left( \frac{N_c^2-4}{N_c} \right) ^2 (N_c^2 -1) 
\int  \!\left( \prod_{i=1}^6 d^2\kf_i \right) \times \\
&&\times [ 2 a(14,25) - s(14,25) - s(14,36) ] 
\Psi_1(\kf_1,\kf_2,\kf_3) \Psi_2(\kf_4,\kf_5,\kf_6) 
\,. 
\nn
\label{POOvertex}
\eea
This vertex certainly deserves further study.
For example, it should be possible to write 
it in the form of a conformal three--point 
function as it was possible for the triple--Pomeron vertex. 
A more detailed knowledge of the Odderon wavefunction 
will eventually allow one to determine the numerical 
value of the integral (\ref{POOvertex}). 
Another interesting question is whether the 
Pomeron--Odderon--Odderon vertex can also be 
calculated in the dipole picture of high energy QCD, 
as it was possible for the coupling of three Pomerons. 
This immediately raises the more fundamental 
question of how the Odderon arises in the dipole 
picture at all. 

\section{Summary and outlook}

We have studied unitarity corrections in high energy QCD 
in the generalized leading logarithmic approximation. 
The objects of interest in this framework are 
amplitudes describing the production of $n$ gluons in the 
$t$-channel. These $n$-gluon amplitudes obey a tower of 
coupled integral equations. The equation for the two--gluon 
amplitudes coincides with the BFKL equation, and 
each integral equation involves all amplitudes with a lower 
number of gluons. The equation for a given $n$-gluon 
amplitude can therefore only be approached after the 
equations with less gluons have been solved. 

A systematic approach to solving the equations 
successively has been presented. 
The first step is the identification 
of a reggeizing part $D_n^R$ of the amplitude under 
consideration. It is a superposition of two--gluon (BFKL) 
amplitudes and can be obtained from the lowest order term 
in the corresponding integral equation, i.\,e.\ the quark 
loop with $n$ gluons attached to it. We have given the 
corresponding expressions explicitly for up to six gluons. 
In the case of the three--gluon amplitude this part 
already solves the integral equation, whereas for $n \ge 4$ 
a new integral equation for the remaining part has to 
be derived. That derivation is technically involved 
since the number of convolutions of amplitudes with 
integral kernels increases rapidly with the 
number of gluons. We have identified a small set of 
standard integrals that can be used to classify all 
momentum space integrals occurring in the derivation. 
A combinatorial method suitable for implementation 
on a computer has been developed to bring all integrals 
to their standard form. Birdtrack notation is used for performing 
the contractions in color space. The resulting equation can then 
be studied in a second step. 

We have reviewed 
the known results about the three-- and four--gluon amplitudes, 
and discussed the field theory structure discovered in these 
amplitudes. The three--gluon amplitude reggeizes completely 
and is a superposition of BFKL amplitudes. The four--gluon 
amplitude consists of a reggeizing part (a superposition 
of BFKL amplitudes) and an irreducible part. The $t$-channel 
evolution of the latter starts with a two--gluon state that couples 
to a four--gluon state via the two--to--four vertex 
$V_{2 \rightarrow 4}$. The emerging picture is that of an 
effective field theory with $n$-gluon states coupled to each other 
via number--changing vertices. 

Using the methods described above we have then investigated 
the amplitudes with five and six gluons in the $t$-channel. 
After extracting a reggeizing part, i.\,e.\ a superposition 
of BFKL amplitudes, from the five--gluon amplitude we 
have derived the equation for the remaining part. The solution 
of this equation can be found as a superposition of irreducible 
four--gluon amplitudes. Thus the five--gluon amplitude 
again reggeizes completely. The mechanism causing this 
is the same as in the case of the three--gluon amplitude. 
It appears therefore very likely that every amplitude with 
an odd number of gluons exhibits complete reggeization. 
This would imply that only $t$-channel states with even 
numbers of gluons occur in the effective field theory. 

A part of the six--gluon amplitude reggeizes again 
and is a superposition of BFKL amplitudes. As one of our 
main results, the integral equation for the remaining part 
has been derived, see eq.~(\ref{inteq6i}). 
It contains a new piece with very 
interesting symmetry properties. However, we have not 
yet been able to clarify whether it should be interpreted as 
a new two--to--six vertex or as a superposition of new 
two--to--four vertices. 
The other terms in the equation have been shown to be 
superpositions of two--to--four vertices $V_{2 \rightarrow 4}$. 
We have calculated the perturbative Pomeron--Odderon--Odderon 
vertex from the new piece. 

The emergence of the vertex 
$V_{2 \rightarrow 4}$ in some of the terms in the 
equation strongly suggests 
that a further part of the six--gluon amplitude is a superposition 
of irreducible four--gluon states, i.\,e.\ reggeizes with 
respect to the four--gluon state. 
We have not yet been able to find the correct color structure 
in this superposition. 
Here we encounter for the first time a situation in which 
reggeization occurs in a subsystem of two gluons which do not 
form a color singlet. In contrary to the first stage of reggeization 
(with respect to the two--gluon state) our equation does not 
fix that color structure uniquely. Knowing it would 
immediately allow one to generalize also the two--to--four 
vertex to the non--singlet case. 
The solution of this problem will probably require a better 
and more general understanding of the phenomenon 
of reggeization. Also information gained in other 
approaches could be very helpful here, especially the 
the dipole picture seems to be promising in this respect. 

The most important property of the five-- and six--gluon 
amplitudes found in the present paper is that they exhibit 
reggeization. On the other hand, we have seen that exactly 
reggeization is the necessary condition for the emergence 
of the field theory structure in the unitarity corrections. 
Therefore the five-- and six--gluon amplitudes 
fit nicely into the picture of a potential effective field theory. 
We regard this as  strong evidence for the conjecture 
that the whole set of unitarity 
corrections can be formulated as an effective field theory 
in $2+1$ dimensions, with rapidity acting as the time--like 
variable. 

An important step will be to prove the conformal invariance 
of that effective field theory in impact parameter space also 
for the amplitudes with more than four gluons. The 
proof should obviously start with the five-- and six--gluon 
amplitudes, but can hopefully be extended to all possible 
elements of the theory. We hope that the conformal 
symmetry can also help to answer the open questions about 
reggeization, about the two--to--four vertex in the color 
non--singlet, and about 
the meaning of the new piece found in the six--gluon amplitude. 
In summary, we expect that the unitarity corrections can be cast into 
the form of a conformal field theory. This opens the 
fascinating possibility of applying the powerful methods 
of conformal field theory and --- once the effective 
conformal theory is identified --- to derive the general 
properties of high energy QCD, now bypassing the 
laborious explicit calculation of higher $n$-gluon amplitudes. 
These general features should certainly not depend on the 
process under consideration. An effective field theory found 
in the amplitudes describing virtual photon--photon scattering 
is expected to be relevant also to other scattering processes 
in the high energy limit. 

The NLO corrections to the BFKL equation have recently become 
available \cite{FLNLO,Ciafaloni}, and the understanding 
of these corrections has been rapidly improved, see for example 
\cite{GavinNLO}. 
On a long--term basis it would be interesting to compute 
the NLO order corrections to all elements of the 
effective field theory of unitarity corrections, although 
this appears very difficult. 

In the present paper we have concentrated on more 
theoretical aspects of the unitarity corrections. We would 
like to point out that unitarity corrections are also 
interesting from a phenomenological point of view. 
Unitarity is violated only at asymptotically 
high energies in any measurable quantity calculated in 
the BFKL formalism. However, this does by not means 
imply that the unitarity corrections, for example the four--gluon 
state, give a negligible contribution to observable quantities 
at presently accessible energies. The 
presence of the four--gluon state might well  
have a sizable effect on many observables, among them 
the total cross--section in virtual photon--photon scattering. 
A study of these effects would be very valuable. 

\acknowledgments
We would like to thank Gregory Korchemsky, Lev Lipatov, 
Alan White, and especially Hans Lotter and Mark W\"usthoff 
for most helpful discussions. We have also benefitted from 
very enjoyable and encouraging 
discussions with the late Vladimir Gribov. 

\appendix
\section{Colors}
\label{colorapp}

In this appendix we focus on more technical details 
of color algebra. 
In section \ref{colormethod} we explain 
how the notation introduced in section \ref{color} can be used to 
contract $\mbox{su}(N_c)$ tensors of arbitrary rank. 
We have applied the method for tensor contractions with 
up to six external gluon lines. Some of the basic results needed for 
the investigation of the $n$-gluon amplitudes are collected 
in section \ref{formcolorcont}. 

\boldmath
\subsection{A method for contractions in $\mbox{su}(N_c)$ algebra}
\label{colormethod}
\unboldmath
When doing the calculations sketched in sections \ref{34V} 
through \ref{d6}
a standard task would be to calculate contractions of the type 
\be
  \picbox{d4fff.pstex_t} = d^{knde} f_{kal} f_{lbm} f_{mcn}  \,.
\label{contractionexample}
\ee
We will now outline an algorithm to solve problems of this kind. 
We restrict ourselves to contractions in which the outgoing 
lines correspond to gluon color representations. The method is, 
however, readily extended to arbitrary tensors involving 
quark representations as well. 
The following prescription can be 
carried out diagrammatically. 

Let us  call 'standard tensors' such tensors that are 
(up to overall factors) 
the sum or the difference of traces of generators of 
the form 
\be
 \mbox{tr} (t^a\dots t^z) \pm \mbox{tr} (t^z\dots t^a)
\label{standardtensors}
\ee
like the ones defined in (\ref{dallgdef}), (\ref{fallgdef}). 
The typical examples that occur in the analysis of the 
integral equations are  $f_{abc}$, $d_{abc}$, $d^{abcd}$, $f^{abcde}$, and $d^{abcdef}$.  
The first step is to express all 
standard tensors occurring in the diagram 
by their representation in terms of generators, that is -- 
diagrammatically speaking -- by quark loops according to 
their respective definitions. 
(Here the terms 'quark line' and 'gluon line' refer to their 
respective color representation only.) 
For each of the standard tensors we then get two quark loops. The whole 
diagram is thus transferred to a sum of $2^m$ diagrams, $m$ 
being the number of standard tensors involved. 
Each of these diagrams contains only gluon lines and closed 
quark lines. 
It is natural to call all gluon lines starting on some closed quark line 
and ending on some closed quark line 'inner' gluon lines. 

The key ingredient for our method is the decomposition 
of a quark--antiquark state into a singlet and an adjoint 
representation, also known as the Fierz identity, 
\be
   \delta^\alpha_\gamma \delta^\delta_\beta
   = 2 (t^a)^\alpha_\beta (t^a)^\delta_\gamma 
       + \fr {1}{N_c} \delta^\alpha_\beta \delta^\delta_\gamma
\,, 
\label{singoctlett}
\ee
$\alpha,\dots,\delta$ being color labels in the 
fundamental representation. 
After rearranging terms, it is in birdtracks 
\be
   \picbox{singoctoct.pstex_t} = 
     \fr{1}{2}\, \picbox{singoct1.pstex_t} \,
     - \fr{1}{2N_c}\, \picbox{singoctsing.pstex_t}   \,,
\label{singoctrev}
\ee
which is applied to all inner gluon lines. 
To do this properly one has to draw all 
quark loops in the diagrams counterclockwise before, 
which is not a mathematical operation although it 
might be quite some exercise in drawing. 
Applying (\ref{singoctrev}) again considerably increases 
the number of diagrams, but we now can read off the result. 
The reason is the following. The use of (\ref{singoctrev}) 
replaces each inner gluon line by two diagrams. 
In one of them the two quark loops\footnote{They might 
be one and the same quark line, but that does not change our argument.}  
joined by the gluon line are disconnected, in the other one they 
are joined to one closed quark loop. 
Having applied (\ref{singoctrev}) to all inner gluon lines, 
we are left with diagrams that 
only contain closed quark loops on which the outer gluon lines end. 
We can now join the diagrams back into rather compact 
expressions using the identity
\be
  \picbox{tr1.pstex_t} = \mbox{tr} \,{\bf 1} = N_c \,,
\ee
the vanishing of the trace of $\mbox{su}(N_c)$ generators, 
\be
  \picbox{trta.pstex_t} = \mbox{tr} \,t^a = 0 \,,
\ee
and the definitions of the standard tensors given in section 
\ref{color}. The latter are supplied by 
\bea
d_{abc} d_{def} + f_{abc} f_{def} &=& 
8\, [ \mbox{tr}(t^at^bt^c) \mbox{tr}(t^ft^et^d) 
+ \mbox{tr}(t^ct^bt^a) \mbox{tr}(t^dt^et^f)]
\\
d_{abc} d_{def} - f_{abc} f_{def} &=& 
8\, [ \mbox{tr}(t^at^bt^c) \mbox{tr}(t^dt^et^f) 
+ \mbox{tr}(t^ct^bt^a) \mbox{tr}(t^ft^et^d)]
\eea
which is readily proved using the definition 
of the structure constants. 
In general, it is also necessary to use standard 
tensors of the type 
\bea
f^{abcd}&=& -i \, [  \mbox{tr}(t^at^bt^ct^d) - \mbox{tr}(t^dt^ct^bt^a) ] \\
d^{abcde} &=& \mbox{tr}(t^at^bt^ct^dt^e) + \mbox{tr}(t^et^dt^ct^bt^a) \, 
\eea
--- that is $f$-type tensors with an even number of color labels or 
$d$-type tensors with an odd number of color labels --- 
to the ones mentioned below equation (\ref{standardtensors}). 
For the identities needed in this paper (see next section), however, 
this is not necessary. 

In the case of the above example (\ref{contractionexample}) 
the result of this procedure is 
\be
  \picbox{d4fff.pstex_t} 
   = - \fr{N_c}{2} \,\picbox{f12345.pstex_t} 
     - \fr{1}{8} \,\picbox{del_abf_cde.pstex_t} 
     - \fr{1}{8} \,\picbox{del_acf_bde.pstex_t}
     - \fr{1}{8} \,\picbox{f_adedel_bc.pstex_t}
     - \fr{1}{8} \,\picbox{f_abcdel_de.pstex_t} \,.
\ee

In many cases the above prescription can be shortened: 
if a subdiagram can be reduced or vanishes, if the whole 
diagram can be obtained from a known diagram by permutation 
of outgoing gluon lines, or using 
the invariance of subdiagrams under cyclic permutations. 

\subsection{Useful contractions of color tensors}
\label{formcolorcont}

In this section we collect a series of $\mbox{su}(N)$ 
identities\footnote{To avoid possible confusion of the 
subscript $c$ in $N_c$ with a color label $c$ we 
omit the subscript and give all results for $\mbox{su}(N)$ 
in this appendix.} 
obtained with the help of the method 
explained above. 
The list does not exhaust the contractions 
needed for the calculations described in this paper. 
Instead, we provide a list of identities from 
which many others can be easily derived.

For two external gluons we have 
\bea
 f_{lak} f_{kbl} &=& - N \delta_{ab} \\
 d_{lak} d_{kbl} &=& \frac{N^2-4}{N} \,\delta_{ab} \,.
\eea
For three external gluons the following identities hold: 
\bea
f_{kal} f_{lbm} f_{mck} &=& - \frac{N}{2} f_{abc} \\
d_{kal} f_{lbm} f_{mck} &=& - \frac{N}{2} \,d_{abc}\\
d_{kal} d_{lbm} f_{mck} &=& \frac{N^2-4}{2N} f_{abc}\\
d_{kal} d_{lbm} d_{mck} &=& \frac{N^2-12}{2N} \,d_{abc} \,.
\eea
The use of the last two identities can be avoided for the 
problems under consideration in this paper. They have been added 
for the sake of completeness here. 
For considering the case of four external gluons the following 
identities are helpful: 
\bea
f_{kal} f_{lbm} f_{mcn}f_{ndk} &=& N d^{abcd} + \frac{1}{2} 
 ( \delta_{ab}\delta_{cd} + \delta_{ac}\delta_{bd} 
 + \delta_{ad}\delta_{bc} ) \\
d^{klcd} f_{kam}f_{mbl} &=& -\frac{N}{2} d^{abcd} 
 - \frac{1}{4} \delta_{ab}\delta_{cd} \\
d^{kbld} f_{kam}f_{mcl} &=& \frac{1}{4} (\delta_{ab}\delta_{cd}
+ \delta_{ad}\delta_{bc} ) 
\eea
The following identities apply to five external gluons: 
\bea
f_{kal} f_{lbm} f_{mcn}f_{ndo}f_{oek} &=& 
 N f^{abcde} \nn \\
&&+ \frac{1}{4} ( \delta_{ab}f_{cde} 
+ \delta_{ac}f_{bde} + \delta_{ad}f_{bce} 
+ \delta_{ae}f_{bcd} \nn \\
&& \hspace{0.7cm}
+ f_{ade}\delta_{bc} + f_{ace}\delta_{bd} 
+ f_{acd}\delta_{be} + f_{abe}\delta_{cd}\nn \\
&& \hspace{0.7cm}
+ f_{abd}\delta_{ce} + f_{abc}\delta_{de})\\
d^{klde} f_{kam} f_{mbn}f_{ncl} &=& 
- \frac{N}{2} f^{abcde} \nn \\
&&- \frac{1}{8} ( \delta_{ab}f_{cde} 
+ \delta_{ac}f_{bde} + f_{ade}\delta_{bc}
+ f_{abc}\delta_{de}) \\
d^{kcle}f_{kam} f_{mbn}f_{ndl} &=&
\frac{1}{8} ( \delta_{ac}f_{bde}
- \delta_{ae}f_{bcd} + f_{ade}\delta_{bc}
- f_{acd}\delta_{be} \nn \\
&& \hspace{.4cm}
+ f_{abe}\delta_{cd} + f_{abc}\delta_{de}) \\
f^{klcde} f_{kam}f_{mbl} &=& 
-\frac{N}{2} f^{abcde} - \frac{1}{8} \delta_{ab}f_{cde} \\
f^{kblde}f_{kam}f_{mcl}&=& 
\frac{1}{8} ( \delta_{ab}f_{cde} 
+ f_{ade}\delta_{bc} - f_{abc}\delta_{de}) 
\eea
Six external gluons require these identities: 
\bea
f_{kal} f_{lbm} f_{mcn}f_{ndo}f_{oep}f_{pfk} &=& 
- N d^{abcdef} 
%\nn 
\\
&& -\frac{1}{2} ( \delta_{ab} d^{cdef}
+ \delta_{ac} d^{bdef} + \delta_{ad} d^{bcef} 
+ \delta_{ae} d^{bcdf} \nn \\
&& \hspace{.7cm}
+ \delta_{af} d^{bcde} + d^{adef} \delta_{bc} 
+ d^{acef} \delta_{bd} + d^{acdf} \delta_{be} \nn \\
&& \hspace{.7cm}
+ d^{acde} \delta_{bf} + d^{abef} \delta_{cd} 
+ d^{abdf} \delta_{ce} + d^{abde} \delta_{cf} \nn \\
&& \hspace{.7cm}
+ d^{abcf} \delta_{de} + d^{abce} \delta_{df} 
+ d^{abcd} \delta_{ef} )
\nn \\
&& + \frac{1}{8} [ 
(d_{abc} d_{def} + f_{abc} f_{def}) 
+ (d_{abd} d_{cef} + f_{abd} f_{cef}) \nn \\
&& \hspace{.7cm}
+ (d_{abe} d_{cdf} + f_{abe} f_{cdf})
+ (d_{abf} d_{cde} + f_{abf} f_{cde})\nn \\
&& \hspace{.7cm}
+ (d_{acd} d_{bef} + f_{acd} f_{bef})
+ (d_{ace} d_{bdf} + f_{ace} f_{bdf})\nn \\
&& \hspace{.7cm}
+ (d_{acf} d_{bde} + f_{acf} f_{bde})
+ (d_{ade} d_{bcf} + f_{ade} f_{bcf})\nn \\
&& \hspace{.7cm}
+ (d_{adf} d_{bce} + f_{adf} f_{bce})
%\nn \\
%&& \hspace{.7cm}
+ (d_{aef} d_{bcd} + f_{aef} f_{bcd})
]
\nn
\\
d^{klef}f_{kam}f_{mbn}f_{nco}f_{odl} &=&
\frac{N}{2} d^{abcdef}+ 
%\nn 
\\
&& 
+ \frac{1}{4} ( \delta_{ab} d^{cdef} + \delta_{ac} d^{bdef}
+ \delta_{ad} d^{bcef} + d^{adef} \delta_{bc} \nn \\
&& \hspace{0.7cm}
+ d^{acef} \delta_{bd}+ d^{abef} \delta_{cd}
+ d^{abcd} \delta_{ef}) \nn \\
&& - \frac{1}{16} [
(d_{abc} d_{def} + f_{abc} f_{def}) 
+ (d_{aef} d_{bcd} + f_{aef} f_{bcd}) \nn \\
&&\hspace{.8cm} 
+ (d_{acd} d_{bef} + f_{acd} f_{bef}) 
%\nn \\
%&&\hspace{.8cm} 
+ (d_{abd} d_{cef} + f_{abd} f_{cef}) ] 
\nn
\\
d^{kdlf}f_{kam}f_{mbn}f_{nco}f_{oel}&=&
- \frac{1}{4} 
( \delta_{ad} d^{bcef}+ d^{acef} \delta_{bd}
+ d^{abef} \delta_{cd} + d^{abcf} \delta_{de} \nn \\
&&\hspace{.7cm}
+ d^{abcd} \delta_{ef} + \delta_{af} d^{cbde}
+ d^{bade} \delta_{cf} + d^{cade} \delta_{bf} ) \nn \\
&& + \frac{1}{16} [
(d_{abd} d_{cef} + f_{abd} f_{cef})
+ (d_{acd} d_{bef} + f_{acd} f_{bef}) \nn \\
&& \hspace{.8cm} 
+ (d_{aef} d_{bcd} + f_{aef} f_{bcd})
+ (d_{ade} d_{bcf} - f_{ade} f_{bcf}) \nn \\
&& \hspace{.8cm} 
+ (d_{acf} d_{bde} - f_{acf} f_{bde})\nn \\
&& \hspace{.8cm} 
+ (d_{abf} d_{cde} - f_{abf} f_{cde})] \\
f^{kldef} f_{kam} f_{mbn}f_{ncl} &=&
\frac{N}{2} d^{abcdef} 
+ \frac{1}{4} ( \delta_{ab} d^{cdef} + \delta_{ac} d^{bdef}
+ d^{adef} \delta_{bc} ) \nn \\
&&- \frac{1}{16} ( d_{abc} d_{def} + f_{abc} f_{def}) 
\\
f^{kclef}f_{kam} f_{mbn}f_{ndl} &=&
-\frac{1}{4} ( \delta_{ac} d^{bdef}
+ d^{adef} \delta_{bc}+ d^{abef} \delta_{cd}
+ d^{bacd} \delta_{ef}) \nn \\
&&
+ \frac{1}{16} [ (d_{abc} d_{def} + f_{abc} f_{def}) 
+  (d_{acd} d_{bef} - f_{acd} f_{bef}) \nn \\
&&\hspace{.8cm}
+  (d_{aef} d_{bcd} - f_{aef} f_{bcd}) ]
\\
d^{klcdef} f_{kam} f_{mbl} &=&
- \frac{N}{2} d^{abcdef} - \frac{1}{4} \delta_{ab} d^{cdef} 
\\
d^{kbldef} f_{kam} f_{mcl} &=&
\frac{1}{4} (\delta_{ab} d^{cdef} + \delta_{bc} d^{adef}) \nn \\
&&- \frac{1}{16} (d_{abc} d_{def} - f_{abc} f_{def}) 
\\
d^{kbclef} f_{kam} f_{mdl} &=&
- \frac{1}{4} (d^{abcd} \delta_{ef}+ \delta_{bc} d^{adef}) 
%\nn 
\\
&& + \frac{1}{16} [
(d_{abc} d_{def} - f_{abc} f_{def}) 
%\nn \\
%&& \hspace{.8cm}
+ (d_{aef} d_{bcd} - f_{aef} f_{bcd}) 
]
\nn
\eea

\section{A combinatorial method for the momentum space integrals}
\label{momentumapp}

A main step in our investigation of a given 
$n$--gluon amplitude $D_n$ is 
to split it into a reggeizing part $D_n^R$ and 
a remaining part. The reggeizing part is a superposition of 
BFKL amplitudes $D_2$. Starting from this ansatz a new integral equation 
for the remaining part of the amplitude is derived. 
In order to calculate its inhomogeneous term it is necessary to convolute 
the reggeizing parts $D_l^R$ of the $l$-gluon amplitudes 
(with $l\le n$) with the integral kernels 
$K^{ \{b\} \rarr \{a\} }_{2\rarr m}$ ($l+m-2=n$) 
according to the original integral equations. 
In this appendix we present an algorithm for performing 
the momentum space integrals. 
It relies on the classification of possible momentum 
space integrals given in section \ref{standardint}, and allows 
us to bring all occuring integrals to their standard form. 
For obtaining a part of the results in this paper we have implemented 
the algorithm in the PERL script language. 
After that a computer algebra program (like MAPLE for example) 
can be used to multiply the integrals by the corresponding color tensors 
and to finally collect all terms. 

The main purpose of our method is to reduce the problem 
of convoluting amplitudes with kernels to a purely combinatorial 
task. We therefore use notation known from the theory of sets 
in this appendix. Below we will give a rule for the treatment 
of the convolution of one specific term in the amplitude $D_l^R$ 
with the transition kernel $K^{ \{b\} \rarr \{a\} }_{2\rarr m}$ 
($l+m-2=n$), and only the momentum part of the kernel will 
be of interest in this appendix. The method can then successively be 
applied to all possible convolutions of individual terms in the 
reggeizing parts of the amplitudes with the integral kernels. 

Let us now consider one specific term in the reggeizing part $D_l^R$ 
of the $l$-gluon amplitude. It is given by a BFKL amplitude that 
has two momentum arguments. Each of them is the sum 
of a subset of the $l$ momenta $\qf_j$. Let us call these two subsets 
$\ca$ and $\cb$, respectively. Their union exhausts the $l$ momenta, 
\be
 \ca \cup \cb = \{ \qf_1,\dots,\qf_l \}
\,,
\ee
and each of them contains at least one element,
\be
 1 \le \#\ca,\,\#\cb \le l-1
\,.
\ee
We will in the following identify a set of momenta with the sum 
of its elements. With this identification 
the term in the amplitude $D_l^R$ we want to consider is 
\be
 D_2(\ca,\cb) = 
D_2\left(\sum_{r=1}^{\# \ca} \qf_{j_r}, 
\sum_{s=1}^{\# \cb} \qf_{j_s}\right) 
\,.
\ee
Now we want to convolute this term with an integral kernel. 
Only two of the $l$ momenta $\qf_j$ will actually be affected 
by the convolution. Let us call these two momenta 
$\vf$ and $\wf$. 
The kernel $K^{ \{b\} \rarr \{a\} }_{2\rarr m}$ was given 
explicitly in section \ref{kernels}. We neglect the coupling 
constant $g$ and the color tensor for the purpose of this 
appendix. The momentum part of the kernel is according 
to (\ref{Kn}) 
\bea
\lefteqn{
K_{2\rarr m}(\vf,\wf;\kf_{i_1},\dots,\kf_{i_m}) =
(\kf_{i_1} + \dots + \kf_{i_m})^2 
- \fr{\wf^2(\kf_{i_1}+\dots+\kf_{i_{m-1}})^2}{(\kf_{i_m}-\wf)^2}
\rule{3cm}{0cm}
}
\nn \\
&& \hspace{3.5cm}
- \fr{\vf^2(\kf_{i_2}+\dots+\kf_{i_m})^2}{(\kf_{i_1}-\vf)^2}
+ \fr{\vf^2\wf^2 
 (\kf_{i_2}+\dots+\kf_{i_{m-1}})^2}{(\kf_{i_1}-\vf)^2(\kf_{i_m}-\wf)^2}
\,.
\label{4kernel}
\eea
The last term is not present if $m=2$. 
The momenta $\kf_{i_t}$ with ($t\in \{ 1,\dots,m\} $) 
are $m$ of the $n$ momenta that occur in the integral 
equation for $D_n$. 
Due to the condition that $t$-channel gluons do not cross 
in the integral equations (see section \ref{inteqsection})  
they are ordered:
\be
 1 \le i_1 < \dots < i_m \le n
\,.
\ee
Which $m$ of the $n$ gluons in the integral equation 
enter the kernel from below depends of course on 
the term we have chosen in the sums on the right 
hand side of the integral equations (\ref{inteq2})--(\ref{inteq6}). 
The quantity we want to calculate here is the convolution 
\be
  K_{2\rarr m}(\vf,\wf;\kf_{i_1},\dots,\kf_{i_m}) 
\otimes D_2(\ca,\cb) 
\,,
\ee
where the symbol $\otimes$ again includes an integral 
over the loop momentum and the two propagators 
$\frac{1}{\vf^2}\frac{1}{\wf^2}$. 
The kernel acts trivially on the other $l-2$ momenta 
in the term $D_2(\ca,\cb)$. 
Our algorithm will leave them unchanged, that is after 
its application we are still left with some $\qf_j$'s in the 
standard integral. They have to be replaced in the end 
by the respective $\kf_i$'s. 
Mathematically speaking this is done by the one--to--one 
map
\be
 \ca \cup \cb \setminus \{ \vf,\wf \} \longrightarrow 
\{ \kf_1, \dots, \kf_n\} \setminus \{ \kf_{i_1}, \dots, \kf_{i_m} \}
\,
\ee
which has to be applied in ascending order according 
to the occurrence of the momenta on both sides. 
On the left hand side we have the $l-2$ momenta in $D_2(\ca,\cb)$ 
not affected by the kernel, on the right hand side we find the 
$n-m=l-2$ momenta that are not attached to the kernel 
from below. 

In addition, one more step has to be performed to finish the result 
after the rules below have been applied. 
This is connected with the definition of the second argument 
of the standard integral $a$. 
As described in section \ref{standardint} the second argument 
of the function $a$ is a sum of momenta which has to be chosen 
out of two sums that occur in the integral. According to our definition 
the second argument of $a$ is the group of momenta that contains the 
momentum $\kf$ with the lowest index. 
In our general treatment in this appendix 
it is not convenient to implement this condition from the beginning. 
Instead we adjust the resulting standard integrals in the end. 
This is done very easily. For example, if we have $n=4$ and the 
algorithm below leads to the result $a(2,34)$ then this should be 
replaced by $a(2,1)$. 

We will treat the four parts of the kernel in (\ref{4kernel}) 
separately now. The resulting standard integrals 
have to be added in the end. 

\subsection*{First part of the kernel}
The first term in the kernel (\ref{4kernel}) is 
\be
  P(\vf,\wf;\kf_{i_1},\dots,\kf_{i_m}) = 
(\kf_{i_1} + \dots + \kf_{i_m})^2 
\,, 
\ee
and we want to bring the convolution 
\be
P(\vf,\wf;\kf_{i_1},\dots,\kf_{i_m}) \otimes 
D_2(\ca,\cb) 
\ee
to its standard form. 
Let the set $\cx$ be $\cx=\{\kf_{i_1},\dots,\kf_{i_m}\}$. 
Then the different possible cases are 
\begin{enumerate}
\item
$\vf$ and $\wf$ are elements of the same set $\ca$ or $\cb$. 
We then denote this set ($\ca$ or $\cb$) by $\ccal$. 
      \begin{enumerate}
        \item
          $\# \ccal = 2$: 
          The integral is $t(\cx)$. 
        \item
          $\# \ccal > 2$: 
          The integral is $s(\cx,\ccal \setminus \{ \vf,\wf \} )$.
      \end{enumerate}
\item
$\vf$ and $\wf$ are {\sl not} elements of the same set $\ca$ or $\cb$. 
      \begin{enumerate}
        \item
          $(\# \ca = 1) \wedge (\# \cb = 1)$: 
          The integral is $c(\cx)$.
        \item
          $(\# \ca = 1) \wedge (\# \cb > 1)$:
          The integral is $b(\cx)$.
        \item
         $(\# \ca > 1) \wedge (\# \cb = 1)$:
          The integral is $b(\cx)$.
        \item
         $(\# \ca > 1) \wedge (\# \cb > 1)$:
         The integral is $a(\cx,\ca \setminus \{ \vf,\wf \} )$.
      \end{enumerate}
\end{enumerate}

\subsection*{Second part of the kernel}
The second term in the kernel (\ref{4kernel}) is 
\be
  Q(\vf,\wf;\kf_{i_1},\dots,\kf_{i_m}) = 
\fr{\wf^2(\kf_{i_1}+\dots+\kf_{i_{m-1}})^2}{(\kf_{i_m}-\wf)^2}
\,, 
\ee
and we want to bring the convolution 
\be
Q(\vf,\wf;\kf_{i_1},\dots,\kf_{i_m}) \otimes 
D_2(\ca,\cb) 
\ee
to its standard form. 
Let now the set $\cx$ denote 
$\cx=\{\kf_{i_1},\dots,\kf_{i_{m-1}}\}$,
and let the set $\cy$ be 
$\cy = \{\kf_{i_m} \}$. 
Then the different possible cases are 
\begin{enumerate}
\item
$\vf$ and $\wf$ are elements of the same set $\ca$ or $\cb$. 
We then denote this set ($\ca$ or $\cb$) by $\ccal$. 
The integral is $s(\cx, (\ccal \setminus \{ \vf,\wf \} ) \cup \cy)$. 
\item
$\vf$ and $\wf$ are {\sl not} elements of the same set $\ca$ or $\cb$. 
Let the set ($\ca$ or $\cb$) containing $\vf$ be $\ccal$. 
     \begin{enumerate}
     \item
       $\# \ccal = 1$: 
       The integral is $b(\cx)$. 
     \item
       $\# \ccal > 1$:
       The integral is $a(\cx, \ccal \setminus \{ \vf \} )$. 
     \end{enumerate}
\end{enumerate}

\subsection*{Third part of the kernel}
The third term in the kernel (\ref{4kernel}) is 
\be
  R(\vf,\wf;\kf_{i_1},\dots,\kf_{i_m}) = 
\fr{\vf^2(\kf_{i_2}+\dots+\kf_{i_m})^2}{(\kf_{i_1}-\vf)^2}
\,, 
\ee
and we want to bring the convolution 
\be
R(\vf,\wf;\kf_{i_1},\dots,\kf_{i_m}) \otimes 
D_2(\ca,\cb) 
\ee
to its standard form. 
Let now the set $\cx$ denote 
$\cx=\{\kf_{i_2},\dots,\kf_{i_m}\}$, 
and let now the set $\cy$ be 
$\cy = \{\kf_{i_1} \}$. 
Then the different possible cases are 
\begin{enumerate}
\item
$\vf$ and $\wf$ are elements of the same set $\ca$ or $\cb$. 
We then denote this set ($\ca$ or $\cb$) by $\ccal$. 
The integral is $s(\cx, (\ccal \setminus \{ \vf,\wf \} ) \cup \cy)$. 
\item
$\vf$ and $\wf$ are {\sl not} elements of the same set $\ca$ or $\cb$. 
Let the set ($\ca$ or $\cb$) containing $\wf$ be $\ccal$. 
     \begin{enumerate}
     \item
       $\# \ccal = 1$: 
       The integral is $b(\cx)$. 
     \item
       $\# \ccal > 1$: 
       The integral is $a(\cx, \ccal \setminus \{ \wf \} )$. 
     \end{enumerate}
\end{enumerate}

\subsection*{Fourth part of the kernel}
The fourth term in the kernel (\ref{4kernel}) is 
\be
  S(\vf,\wf;\kf_{i_1},\dots,\kf_{i_m}) = 
\fr{\vf^2\wf^2 
 (\kf_{i_2}+\dots+\kf_{i_{m-1}})^2}{(\kf_{i_1}-\vf)^2(\kf_{i_m}-\wf)^2}
\,, 
\ee
and we want to bring the convolution 
\be
S(\vf,\wf;\kf_{i_1},\dots,\kf_{i_m}) \otimes 
D_2(\ca,\cb) 
\ee
to its standard form. 
Let now the set $\cx$ denote 
$\cx=\{\kf_{i_2},\dots,\kf_{i_{m-1}}\}$,
and let now the set $\cy$ be 
$\cy = \{\kf_{i_1} \}$. 
Let in addition the set $\cz$ be
$\cz = \{\kf_{i_m} \}$. 
Then the different possible cases are 
\begin{enumerate}
\item
$\vf$ and $\wf$ are elements of the same set $\ca$ or $\cb$. 
We then denote this set ($\ca$ or $\cb$) by $\ccal$. 
The integral is 
$s(\cx, (\ccal \setminus \{ \vf,\wf \} ) \cup \cy \cup \cz)$. 
\item
$\vf$ and $\wf$ are {\sl not} elements of the same set $\ca$ or $\cb$. 
Let the set ($\ca$ or $\cb$) containing $\vf$ be $\ccal$. 
The integral is $a(\cx,(\ccal \setminus \{ \vf \} ) \cup \cy)$.
\end{enumerate}

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\end{document}
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\begin{picture}(0,0)%
\special{psfile=regger6.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(549,576)(964,-73)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=regger7.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(549,576)(964,-73)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=singoct1.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(494,421)(804,-158)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=singoctoct.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(480,255)(1486,-76)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=singoctsing.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(480,255)(1486,-76)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=solutiond41.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(865,966)(2373,-2016)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=solutiond42.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(865,1333)(3691,-2237)
\put(4427,-1861){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$V_{2 \rightarrow 4}$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=solutiond51.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(865,966)(2373,-2016)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=solutiond52.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(865,1460)(3814,-2364)
\put(4576,-1936){\makebox(0,0)[lb]{\smash{\SetFigFont{11}{13.2}{rm}$V_{2 \rightarrow 4}$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=tr1.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(230,279)(936,-175)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=tr1234.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(480,300)(961,-673)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=tr12345.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(630,304)(886,-673)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=tr4321.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(480,304)(961,-673)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=tr54321.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(630,304)(886,-673)
\end{picture}
\begin{picture}(0,0)%
\special{psfile=trab.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(987,315)(526,-211)
\put(526,-211){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$a$}}}
\put(1501,-211){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$b$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=trabc.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(675,660)(676,-361)
\put(1351,-361){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$c$}}}
\put(901,164){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$a$}}}
\put(676,-361){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$b$}}}
\end{picture}
\begin{picture}(0,0)%
\special{psfile=tracb.pstex}%
\end{picture}%
\setlength{\unitlength}{3947sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
\begin{picture}(675,660)(1126,-361)
\put(1801,-361){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$c$}}}
\put(1351,164){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$a$}}}
\put(1126,-361){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{rm}$b$}}}
\end{picture}
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\special{psfile=trta.pstex}%
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%
\begingroup\makeatletter\ifx\SetFigFont\undefined
% extract first six characters in \fmtname
\def\x#1#2#3#4#5#6#7\relax{\def\x{#1#2#3#4#5#6}}%
\expandafter\x\fmtname xxxxxx\relax \def\y{splain}%
\ifx\x\y   % LaTeX or SliTeX?
\gdef\SetFigFont#1#2#3{%
  \ifnum #1<17\tiny\else \ifnum #1<20\small\else
  \ifnum #1<24\normalsize\else \ifnum #1<29\large\else
  \ifnum #1<34\Large\else \ifnum #1<41\LARGE\else
     \huge\fi\fi\fi\fi\fi\fi
  \csname #3\endcsname}%
\else
\gdef\SetFigFont#1#2#3{\begingroup
  \count@#1\relax \ifnum 25<\count@\count@25\fi
  \def\x{\endgroup\@setsize\SetFigFont{#2pt}}%
  \expandafter\x
    \csname \romannumeral\the\count@ pt\expandafter\endcsname
    \csname @\romannumeral\the\count@ pt\endcsname
  \csname #3\endcsname}%
\fi
\fi\endgroup
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