}}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{
} {\pgfsys@beginscope\pgfsys@invoke{ } {{}}
{{}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{28.45276pt}{1.44pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}
{{{}}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{{}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$ (81)

The external momenta are again in the configuration (74) with Mandelstams
(75). The optical theorem tells us to cut the internal lines of the diagram,
putting the corresponding legs on-shell. This implies the following equation
for the discontinuity of the amplitude

$2\mathop{\mathrm{Im}}A_{\text{1-loop}}(s,q)=\\!\sum\limits_{\begin{subarray}{c}m_{5},\rho_{5},\epsilon_{5}\\\
m_{6},\rho_{6},\epsilon_{6}\end{subarray}}\int\\!\frac{dl_{1}}{(2\pi)^{D}}\,2\pi
i\delta(k_{5}^{2}+m_{5}^{2})\,2\pi
i\delta(k_{6}^{2}+m_{6}^{2})A_{\text{tree}}^{3652}(s,l_{2})^{*}A_{\text{tree}}^{1564}(s,l_{1})\,,$
(82)

where one sums over all possible particles 5 and 6 with masses $m$, in Little
group representations $\rho$ and with polarization tensors $\epsilon$. The
sums over polarizations can be evaluated using completeness relations. In
order to remove the delta functions we express $k_{5}$ and $k_{6}$ in terms of
$l_{1}$ and the external momenta (74). Then we write the loop momentum as
$l_{1}^{\mu}=(l^{u},l^{v},q_{1})$ and use the delta-functions to fix the
forward components of the loop momentum $l_{u}$ and $l_{v}$ to

$\displaystyle l^{u}=\frac{m_{6}^{2}+q_{1}^{2}+q\cdot
q_{1}}{k^{v}}\,,\qquad\qquad l^{v}=-\frac{m_{5}^{2}+q_{1}^{2}-q\cdot
q_{1}}{k^{u}}\,,$ (83)

leaving only the transverse integration over $q_{1}$. We arrive at the
equation

$\mathop{\mathrm{Im}}A_{\text{1-loop}}(s,q)=\sum\limits_{\begin{subarray}{c}m_{5},\rho_{5},\epsilon_{5}\\\
m_{6},\rho_{6},\epsilon_{6}\end{subarray}}\int\frac{dq_{1}dq_{2}}{(2\pi)^{D-2}}\,\frac{\delta(q-q_{1}-q_{2})}{4s}A_{\text{tree}}^{3652}(s,q_{2})^{*}A_{\text{tree}}^{1564}(s,q_{1})\,,$
(84)

where we introduced the transverse momentum $q_{2}=q-q_{1}$ to write the
expression in a more symmetrical way. Using that the tree-level amplitudes are
given in the Regge limit by Pomeron exchange, we can write (84)
diagrammatically as in figure 2.

$\leavevmode\hbox to120.48pt{\vbox
to126.92pt{\pgfpicture\makeatletter\hbox{\hskip
60.23853pt\lower-60.23853pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}{{{}}
{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,1,1}\pgfsys@color@cmyk@fill{1}{0}{0}{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.4}\pgfsys@invoke{
}{}\pgfsys@moveto{-22.76228pt}{2.84544pt}\pgfsys@moveto{-15.6491pt}{2.84544pt}\pgfsys@curveto{-15.6491pt}{15.41685pt}{-18.83374pt}{25.60773pt}{-22.76228pt}{25.60773pt}\pgfsys@curveto{-26.69083pt}{25.60773pt}{-29.87547pt}{15.41685pt}{-29.87547pt}{2.84544pt}\pgfsys@curveto{-29.87547pt}{-9.72597pt}{-26.69083pt}{-19.91684pt}{-22.76228pt}{-19.91684pt}\pgfsys@curveto{-18.83374pt}{-19.91684pt}{-15.6491pt}{-9.72597pt}{-15.6491pt}{2.84544pt}\pgfsys@closepath\pgfsys@moveto{-22.76228pt}{2.84544pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}
{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,1,1}\pgfsys@color@cmyk@fill{1}{0}{0}{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.4}\pgfsys@invoke{
}{}\pgfsys@moveto{22.76228pt}{2.84544pt}\pgfsys@moveto{29.87547pt}{2.84544pt}\pgfsys@curveto{29.87547pt}{15.41685pt}{26.69083pt}{25.60773pt}{22.76228pt}{25.60773pt}\pgfsys@curveto{18.83374pt}{25.60773pt}{15.6491pt}{15.41685pt}{15.6491pt}{2.84544pt}\pgfsys@curveto{15.6491pt}{-9.72597pt}{18.83374pt}{-19.91684pt}{22.76228pt}{-19.91684pt}\pgfsys@curveto{26.69083pt}{-19.91684pt}{29.87547pt}{-9.72597pt}{29.87547pt}{2.84544pt}\pgfsys@closepath\pgfsys@moveto{22.76228pt}{2.84544pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-21.34798pt}{-19.91684pt}\pgfsys@curveto{-21.34798pt}{-19.13574pt}{-21.98119pt}{-18.50253pt}{-22.76228pt}{-18.50253pt}\pgfsys@curveto{-23.54338pt}{-18.50253pt}{-24.17659pt}{-19.13574pt}{-24.17659pt}{-19.91684pt}\pgfsys@curveto{-24.17659pt}{-20.69794pt}{-23.54338pt}{-21.33115pt}{-22.76228pt}{-21.33115pt}\pgfsys@curveto{-21.98119pt}{-21.33115pt}{-21.34798pt}{-20.69794pt}{-21.34798pt}{-19.91684pt}\pgfsys@closepath\pgfsys@moveto{-22.76228pt}{-19.91684pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-22.76228pt}{-19.91684pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-21.34798pt}{25.6073pt}\pgfsys@curveto{-21.34798pt}{26.3884pt}{-21.98119pt}{27.0216pt}{-22.76228pt}{27.0216pt}\pgfsys@curveto{-23.54338pt}{27.0216pt}{-24.17659pt}{26.3884pt}{-24.17659pt}{25.6073pt}\pgfsys@curveto{-24.17659pt}{24.8262pt}{-23.54338pt}{24.193pt}{-22.76228pt}{24.193pt}\pgfsys@curveto{-21.98119pt}{24.193pt}{-21.34798pt}{24.8262pt}{-21.34798pt}{25.6073pt}\pgfsys@closepath\pgfsys@moveto{-22.76228pt}{25.6073pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-22.76228pt}{25.6073pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{24.17659pt}{25.6073pt}\pgfsys@curveto{24.17659pt}{26.3884pt}{23.54338pt}{27.0216pt}{22.76228pt}{27.0216pt}\pgfsys@curveto{21.98119pt}{27.0216pt}{21.34798pt}{26.3884pt}{21.34798pt}{25.6073pt}\pgfsys@curveto{21.34798pt}{24.8262pt}{21.98119pt}{24.193pt}{22.76228pt}{24.193pt}\pgfsys@curveto{23.54338pt}{24.193pt}{24.17659pt}{24.8262pt}{24.17659pt}{25.6073pt}\pgfsys@closepath\pgfsys@moveto{22.76228pt}{25.6073pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{22.76228pt}{25.6073pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{24.17659pt}{-19.91684pt}\pgfsys@curveto{24.17659pt}{-19.13574pt}{23.54338pt}{-18.50253pt}{22.76228pt}{-18.50253pt}\pgfsys@curveto{21.98119pt}{-18.50253pt}{21.34798pt}{-19.13574pt}{21.34798pt}{-19.91684pt}\pgfsys@curveto{21.34798pt}{-20.69794pt}{21.98119pt}{-21.33115pt}{22.76228pt}{-21.33115pt}\pgfsys@curveto{23.54338pt}{-21.33115pt}{24.17659pt}{-20.69794pt}{24.17659pt}{-19.91684pt}\pgfsys@closepath\pgfsys@moveto{22.76228pt}{-19.91684pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{22.76228pt}{-19.91684pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-56.90552pt}{2.84544pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{56.90552pt}{2.84544pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-36.64322pt}{-56.90552pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-36.64322pt}{56.90552pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$2$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{31.64322pt}{56.90552pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$3$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{31.64322pt}{-56.90552pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$4$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-5.30452pt}{-31.2982pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$P_{1}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-5.30452pt}{14.22638pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$P_{2}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-31.86751pt}{-46.92809pt}\pgfsys@lineto{-23.27788pt}{-21.44658pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-23.26776pt}{27.14041pt}\pgfsys@lineto{-31.91667pt}{53.37251pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{31.91698pt}{53.37251pt}\pgfsys@lineto{23.26773pt}{27.14043pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{23.27792pt}{-21.44658pt}\pgfsys@lineto{31.8674pt}{-46.92809pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}}
}}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}}
}{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}}
{{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}}
{{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}}
}}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}}
}{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}}
{{}{}{}{}} }{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}} }}{{}} {{}} }
\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-21.14798pt}{-19.91684pt}\pgfsys@curveto{-20.39798pt}{-19.91684pt}{-20.02298pt}{-18.91684pt}{-19.27298pt}{-18.91684pt}\pgfsys@curveto{-18.72998pt}{-18.91684pt}{-18.262pt}{-19.40485pt}{-17.77298pt}{-19.91684pt}\pgfsys@curveto{-17.28398pt}{-20.42883pt}{-16.81598pt}{-20.91684pt}{-16.27298pt}{-20.91684pt}\pgfsys@curveto{-15.72998pt}{-20.91684pt}{-15.262pt}{-20.42883pt}{-14.77298pt}{-19.91684pt}\pgfsys@curveto{-14.28398pt}{-19.40485pt}{-13.81598pt}{-18.91684pt}{-13.27298pt}{-18.91684pt}\pgfsys@curveto{-12.72998pt}{-18.91684pt}{-12.262pt}{-19.40485pt}{-11.77298pt}{-19.91684pt}\pgfsys@curveto{-11.28398pt}{-20.42883pt}{-10.81598pt}{-20.91684pt}{-10.27298pt}{-20.91684pt}\pgfsys@curveto{-9.72998pt}{-20.91684pt}{-9.262pt}{-20.42883pt}{-8.77298pt}{-19.91684pt}\pgfsys@curveto{-8.28398pt}{-19.40485pt}{-7.81598pt}{-18.91684pt}{-7.27298pt}{-18.91684pt}\pgfsys@curveto{-6.72998pt}{-18.91684pt}{-6.262pt}{-19.40485pt}{-5.77298pt}{-19.91684pt}\pgfsys@curveto{-5.28398pt}{-20.42883pt}{-4.81598pt}{-20.91684pt}{-4.27298pt}{-20.91684pt}\pgfsys@curveto{-3.72998pt}{-20.91684pt}{-3.262pt}{-20.42883pt}{-2.77298pt}{-19.91684pt}\pgfsys@curveto{-2.28398pt}{-19.40485pt}{-1.81598pt}{-18.91684pt}{-1.27298pt}{-18.91684pt}\pgfsys@curveto{-0.72998pt}{-18.91684pt}{-0.262pt}{-19.40485pt}{0.22702pt}{-19.91684pt}\pgfsys@curveto{0.71602pt}{-20.42883pt}{1.18402pt}{-20.91684pt}{1.72702pt}{-20.91684pt}\pgfsys@curveto{2.27002pt}{-20.91684pt}{2.738pt}{-20.42883pt}{3.22702pt}{-19.91684pt}\pgfsys@curveto{3.71602pt}{-19.40485pt}{4.18402pt}{-18.91684pt}{4.72702pt}{-18.91684pt}\pgfsys@curveto{5.27002pt}{-18.91684pt}{5.738pt}{-19.40485pt}{6.22702pt}{-19.91684pt}\pgfsys@curveto{6.71602pt}{-20.42883pt}{7.18402pt}{-20.91684pt}{7.72702pt}{-20.91684pt}\pgfsys@curveto{8.27002pt}{-20.91684pt}{8.738pt}{-20.42883pt}{9.22702pt}{-19.91684pt}\pgfsys@curveto{9.71602pt}{-19.40485pt}{10.18402pt}{-18.91684pt}{10.72702pt}{-18.91684pt}\pgfsys@curveto{11.27002pt}{-18.91684pt}{11.738pt}{-19.40485pt}{12.22702pt}{-19.91684pt}\pgfsys@curveto{12.71602pt}{-20.42883pt}{13.18402pt}{-20.91684pt}{13.72702pt}{-20.91684pt}\pgfsys@curveto{14.47702pt}{-20.91684pt}{14.85202pt}{-19.91684pt}{15.60202pt}{-19.91684pt}\pgfsys@lineto{21.14798pt}{-19.91684pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}}
}}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}}
}{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}}
{{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}}
{{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}}
}}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}}
}{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}}
{{}{}{}{}} }{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}} }}{{}} {{}} }
\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-21.14798pt}{25.6073pt}\pgfsys@curveto{-20.39798pt}{25.6073pt}{-20.02298pt}{26.6073pt}{-19.27298pt}{26.6073pt}\pgfsys@curveto{-18.72998pt}{26.6073pt}{-18.262pt}{26.1193pt}{-17.77298pt}{25.6073pt}\pgfsys@curveto{-17.28398pt}{25.0953pt}{-16.81598pt}{24.6073pt}{-16.27298pt}{24.6073pt}\pgfsys@curveto{-15.72998pt}{24.6073pt}{-15.262pt}{25.0953pt}{-14.77298pt}{25.6073pt}\pgfsys@curveto{-14.28398pt}{26.1193pt}{-13.81598pt}{26.6073pt}{-13.27298pt}{26.6073pt}\pgfsys@curveto{-12.72998pt}{26.6073pt}{-12.262pt}{26.1193pt}{-11.77298pt}{25.6073pt}\pgfsys@curveto{-11.28398pt}{25.0953pt}{-10.81598pt}{24.6073pt}{-10.27298pt}{24.6073pt}\pgfsys@curveto{-9.72998pt}{24.6073pt}{-9.262pt}{25.0953pt}{-8.77298pt}{25.6073pt}\pgfsys@curveto{-8.28398pt}{26.1193pt}{-7.81598pt}{26.6073pt}{-7.27298pt}{26.6073pt}\pgfsys@curveto{-6.72998pt}{26.6073pt}{-6.262pt}{26.1193pt}{-5.77298pt}{25.6073pt}\pgfsys@curveto{-5.28398pt}{25.0953pt}{-4.81598pt}{24.6073pt}{-4.27298pt}{24.6073pt}\pgfsys@curveto{-3.72998pt}{24.6073pt}{-3.262pt}{25.0953pt}{-2.77298pt}{25.6073pt}\pgfsys@curveto{-2.28398pt}{26.1193pt}{-1.81598pt}{26.6073pt}{-1.27298pt}{26.6073pt}\pgfsys@curveto{-0.72998pt}{26.6073pt}{-0.262pt}{26.1193pt}{0.22702pt}{25.6073pt}\pgfsys@curveto{0.71602pt}{25.0953pt}{1.18402pt}{24.6073pt}{1.72702pt}{24.6073pt}\pgfsys@curveto{2.27002pt}{24.6073pt}{2.738pt}{25.0953pt}{3.22702pt}{25.6073pt}\pgfsys@curveto{3.71602pt}{26.1193pt}{4.18402pt}{26.6073pt}{4.72702pt}{26.6073pt}\pgfsys@curveto{5.27002pt}{26.6073pt}{5.738pt}{26.1193pt}{6.22702pt}{25.6073pt}\pgfsys@curveto{6.71602pt}{25.0953pt}{7.18402pt}{24.6073pt}{7.72702pt}{24.6073pt}\pgfsys@curveto{8.27002pt}{24.6073pt}{8.738pt}{25.0953pt}{9.22702pt}{25.6073pt}\pgfsys@curveto{9.71602pt}{26.1193pt}{10.18402pt}{26.6073pt}{10.72702pt}{26.6073pt}\pgfsys@curveto{11.27002pt}{26.6073pt}{11.738pt}{26.1193pt}{12.22702pt}{25.6073pt}\pgfsys@curveto{12.71602pt}{25.0953pt}{13.18402pt}{24.6073pt}{13.72702pt}{24.6073pt}\pgfsys@curveto{14.47702pt}{24.6073pt}{14.85202pt}{25.6073pt}{15.60202pt}{25.6073pt}\pgfsys@lineto{21.14798pt}{25.6073pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-53.37251pt}{2.84544pt}\pgfsys@lineto{53.37251pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\sim\sum\limits_{\begin{subarray}{c}m_{5},\rho_{5},\epsilon_{5}\\\
m_{6},\rho_{6},\epsilon_{6}\end{subarray}}\int\leavevmode\hbox to79.95pt{\vbox
to81.4pt{\pgfpicture\makeatletter\hbox{\hskip 39.97623pt\lower-37.47623pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-21.34798pt}{2.84544pt}\pgfsys@curveto{-21.34798pt}{3.62654pt}{-21.98119pt}{4.25975pt}{-22.76228pt}{4.25975pt}\pgfsys@curveto{-23.54338pt}{4.25975pt}{-24.17659pt}{3.62654pt}{-24.17659pt}{2.84544pt}\pgfsys@curveto{-24.17659pt}{2.06435pt}{-23.54338pt}{1.43114pt}{-22.76228pt}{1.43114pt}\pgfsys@curveto{-21.98119pt}{1.43114pt}{-21.34798pt}{2.06435pt}{-21.34798pt}{2.84544pt}\pgfsys@closepath\pgfsys@moveto{-22.76228pt}{2.84544pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-22.76228pt}{2.84544pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{24.17659pt}{2.84544pt}\pgfsys@curveto{24.17659pt}{3.62654pt}{23.54338pt}{4.25975pt}{22.76228pt}{4.25975pt}\pgfsys@curveto{21.98119pt}{4.25975pt}{21.34798pt}{3.62654pt}{21.34798pt}{2.84544pt}\pgfsys@curveto{21.34798pt}{2.06435pt}{21.98119pt}{1.43114pt}{22.76228pt}{1.43114pt}\pgfsys@curveto{23.54338pt}{1.43114pt}{24.17659pt}{2.06435pt}{24.17659pt}{2.84544pt}\pgfsys@closepath\pgfsys@moveto{22.76228pt}{2.84544pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{22.76228pt}{2.84544pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-36.64322pt}{-34.14322pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-36.64322pt}{34.14322pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$5$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{31.64322pt}{34.14322pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$6$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{31.64322pt}{-34.14322pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$4$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-5.30452pt}{-11.38092pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$P_{1}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-31.86751pt}{-24.16579pt}\pgfsys@lineto{-23.27788pt}{1.3157pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-23.26776pt}{4.37856pt}\pgfsys@lineto{-31.91667pt}{30.61021pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}}
}}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}}
}{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}}
{{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}}
{{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}}
}}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}}
}{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}}
{{}{}{}{}} }{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}} }}{{}} {{}} }
\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-21.14798pt}{2.84544pt}\pgfsys@curveto{-20.39798pt}{2.84544pt}{-20.02298pt}{3.84544pt}{-19.27298pt}{3.84544pt}\pgfsys@curveto{-18.72998pt}{3.84544pt}{-18.262pt}{3.35744pt}{-17.77298pt}{2.84544pt}\pgfsys@curveto{-17.28398pt}{2.33345pt}{-16.81598pt}{1.84544pt}{-16.27298pt}{1.84544pt}\pgfsys@curveto{-15.72998pt}{1.84544pt}{-15.262pt}{2.33345pt}{-14.77298pt}{2.84544pt}\pgfsys@curveto{-14.28398pt}{3.35744pt}{-13.81598pt}{3.84544pt}{-13.27298pt}{3.84544pt}\pgfsys@curveto{-12.72998pt}{3.84544pt}{-12.262pt}{3.35744pt}{-11.77298pt}{2.84544pt}\pgfsys@curveto{-11.28398pt}{2.33345pt}{-10.81598pt}{1.84544pt}{-10.27298pt}{1.84544pt}\pgfsys@curveto{-9.72998pt}{1.84544pt}{-9.262pt}{2.33345pt}{-8.77298pt}{2.84544pt}\pgfsys@curveto{-8.28398pt}{3.35744pt}{-7.81598pt}{3.84544pt}{-7.27298pt}{3.84544pt}\pgfsys@curveto{-6.72998pt}{3.84544pt}{-6.262pt}{3.35744pt}{-5.77298pt}{2.84544pt}\pgfsys@curveto{-5.28398pt}{2.33345pt}{-4.81598pt}{1.84544pt}{-4.27298pt}{1.84544pt}\pgfsys@curveto{-3.72998pt}{1.84544pt}{-3.262pt}{2.33345pt}{-2.77298pt}{2.84544pt}\pgfsys@curveto{-2.28398pt}{3.35744pt}{-1.81598pt}{3.84544pt}{-1.27298pt}{3.84544pt}\pgfsys@curveto{-0.72998pt}{3.84544pt}{-0.262pt}{3.35744pt}{0.22702pt}{2.84544pt}\pgfsys@curveto{0.71602pt}{2.33345pt}{1.18402pt}{1.84544pt}{1.72702pt}{1.84544pt}\pgfsys@curveto{2.27002pt}{1.84544pt}{2.738pt}{2.33345pt}{3.22702pt}{2.84544pt}\pgfsys@curveto{3.71602pt}{3.35744pt}{4.18402pt}{3.84544pt}{4.72702pt}{3.84544pt}\pgfsys@curveto{5.27002pt}{3.84544pt}{5.738pt}{3.35744pt}{6.22702pt}{2.84544pt}\pgfsys@curveto{6.71602pt}{2.33345pt}{7.18402pt}{1.84544pt}{7.72702pt}{1.84544pt}\pgfsys@curveto{8.27002pt}{1.84544pt}{8.738pt}{2.33345pt}{9.22702pt}{2.84544pt}\pgfsys@curveto{9.71602pt}{3.35744pt}{10.18402pt}{3.84544pt}{10.72702pt}{3.84544pt}\pgfsys@curveto{11.27002pt}{3.84544pt}{11.738pt}{3.35744pt}{12.22702pt}{2.84544pt}\pgfsys@curveto{12.71602pt}{2.33345pt}{13.18402pt}{1.84544pt}{13.72702pt}{1.84544pt}\pgfsys@curveto{14.47702pt}{1.84544pt}{14.85202pt}{2.84544pt}{15.60202pt}{2.84544pt}\pgfsys@lineto{21.14798pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{31.91698pt}{30.61021pt}\pgfsys@lineto{23.26773pt}{4.37857pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{23.27792pt}{1.3157pt}\pgfsys@lineto{31.8674pt}{-24.16579pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\times\leavevmode\hbox
to79.95pt{\vbox to81.4pt{\pgfpicture\makeatletter\hbox{\hskip
39.97623pt\lower-37.47623pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-21.34798pt}{2.84544pt}\pgfsys@curveto{-21.34798pt}{3.62654pt}{-21.98119pt}{4.25975pt}{-22.76228pt}{4.25975pt}\pgfsys@curveto{-23.54338pt}{4.25975pt}{-24.17659pt}{3.62654pt}{-24.17659pt}{2.84544pt}\pgfsys@curveto{-24.17659pt}{2.06435pt}{-23.54338pt}{1.43114pt}{-22.76228pt}{1.43114pt}\pgfsys@curveto{-21.98119pt}{1.43114pt}{-21.34798pt}{2.06435pt}{-21.34798pt}{2.84544pt}\pgfsys@closepath\pgfsys@moveto{-22.76228pt}{2.84544pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-22.76228pt}{2.84544pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{24.17659pt}{2.84544pt}\pgfsys@curveto{24.17659pt}{3.62654pt}{23.54338pt}{4.25975pt}{22.76228pt}{4.25975pt}\pgfsys@curveto{21.98119pt}{4.25975pt}{21.34798pt}{3.62654pt}{21.34798pt}{2.84544pt}\pgfsys@curveto{21.34798pt}{2.06435pt}{21.98119pt}{1.43114pt}{22.76228pt}{1.43114pt}\pgfsys@curveto{23.54338pt}{1.43114pt}{24.17659pt}{2.06435pt}{24.17659pt}{2.84544pt}\pgfsys@closepath\pgfsys@moveto{22.76228pt}{2.84544pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{22.76228pt}{2.84544pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-36.64322pt}{-34.14322pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$5$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-36.64322pt}{34.14322pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$2$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{31.64322pt}{34.14322pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$3$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{31.64322pt}{-34.14322pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$6$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-5.30452pt}{-11.38092pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$P_{2}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-31.86751pt}{-24.16579pt}\pgfsys@lineto{-23.27788pt}{1.3157pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-23.26776pt}{4.37856pt}\pgfsys@lineto{-31.91667pt}{30.61021pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}}
}}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}}
}{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}}
{{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}}
{{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}}
}}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}}
}{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}}
{{}{}{}{}} }{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}} }}{{}} {{}} }
\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-21.14798pt}{2.84544pt}\pgfsys@curveto{-20.39798pt}{2.84544pt}{-20.02298pt}{3.84544pt}{-19.27298pt}{3.84544pt}\pgfsys@curveto{-18.72998pt}{3.84544pt}{-18.262pt}{3.35744pt}{-17.77298pt}{2.84544pt}\pgfsys@curveto{-17.28398pt}{2.33345pt}{-16.81598pt}{1.84544pt}{-16.27298pt}{1.84544pt}\pgfsys@curveto{-15.72998pt}{1.84544pt}{-15.262pt}{2.33345pt}{-14.77298pt}{2.84544pt}\pgfsys@curveto{-14.28398pt}{3.35744pt}{-13.81598pt}{3.84544pt}{-13.27298pt}{3.84544pt}\pgfsys@curveto{-12.72998pt}{3.84544pt}{-12.262pt}{3.35744pt}{-11.77298pt}{2.84544pt}\pgfsys@curveto{-11.28398pt}{2.33345pt}{-10.81598pt}{1.84544pt}{-10.27298pt}{1.84544pt}\pgfsys@curveto{-9.72998pt}{1.84544pt}{-9.262pt}{2.33345pt}{-8.77298pt}{2.84544pt}\pgfsys@curveto{-8.28398pt}{3.35744pt}{-7.81598pt}{3.84544pt}{-7.27298pt}{3.84544pt}\pgfsys@curveto{-6.72998pt}{3.84544pt}{-6.262pt}{3.35744pt}{-5.77298pt}{2.84544pt}\pgfsys@curveto{-5.28398pt}{2.33345pt}{-4.81598pt}{1.84544pt}{-4.27298pt}{1.84544pt}\pgfsys@curveto{-3.72998pt}{1.84544pt}{-3.262pt}{2.33345pt}{-2.77298pt}{2.84544pt}\pgfsys@curveto{-2.28398pt}{3.35744pt}{-1.81598pt}{3.84544pt}{-1.27298pt}{3.84544pt}\pgfsys@curveto{-0.72998pt}{3.84544pt}{-0.262pt}{3.35744pt}{0.22702pt}{2.84544pt}\pgfsys@curveto{0.71602pt}{2.33345pt}{1.18402pt}{1.84544pt}{1.72702pt}{1.84544pt}\pgfsys@curveto{2.27002pt}{1.84544pt}{2.738pt}{2.33345pt}{3.22702pt}{2.84544pt}\pgfsys@curveto{3.71602pt}{3.35744pt}{4.18402pt}{3.84544pt}{4.72702pt}{3.84544pt}\pgfsys@curveto{5.27002pt}{3.84544pt}{5.738pt}{3.35744pt}{6.22702pt}{2.84544pt}\pgfsys@curveto{6.71602pt}{2.33345pt}{7.18402pt}{1.84544pt}{7.72702pt}{1.84544pt}\pgfsys@curveto{8.27002pt}{1.84544pt}{8.738pt}{2.33345pt}{9.22702pt}{2.84544pt}\pgfsys@curveto{9.71602pt}{3.35744pt}{10.18402pt}{3.84544pt}{10.72702pt}{3.84544pt}\pgfsys@curveto{11.27002pt}{3.84544pt}{11.738pt}{3.35744pt}{12.22702pt}{2.84544pt}\pgfsys@curveto{12.71602pt}{2.33345pt}{13.18402pt}{1.84544pt}{13.72702pt}{1.84544pt}\pgfsys@curveto{14.47702pt}{1.84544pt}{14.85202pt}{2.84544pt}{15.60202pt}{2.84544pt}\pgfsys@lineto{21.14798pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{31.91698pt}{30.61021pt}\pgfsys@lineto{23.26773pt}{4.37857pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{23.27792pt}{1.3157pt}\pgfsys@lineto{31.8674pt}{-24.16579pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$
Figure 2: Optical theorem in the Regge limit in terms of Feynman diagrams. The
tree-level correlators are dominated by $s$-channel Pomeron exchange. The
ellipses on the l.h.s. indicate that all string excitations are taken into
account.

The optical theorem can be simplified even further by transforming it to
impact parameter space. To this end the amplitude is expressed in terms of the
impact parameter $b$, which is a vector in the transverse impact parameter
space $\mathbb{R}^{D-2}$, using the following transformation

$\delta(s,b)=\frac{1}{2s}\int\frac{dq}{(2\pi)^{D-2}}\,e^{iq\cdot b}A(s,t)\,.$
(85)

We can use this definition together with (84) to compute

$\displaystyle\mathop{\mathrm{Im}}\delta_{\text{1-loop}}(s,b)={}$
$\displaystyle\frac{1}{2}\sum\limits_{\begin{subarray}{c}m_{5},\rho_{5},\epsilon_{5}\\\
m_{6},\rho_{6},\epsilon_{6}\end{subarray}}\delta_{\text{tree}}^{3652}(s,-b)^{*}\,\delta_{\text{tree}}^{1564}(s,b)\,.$
(86)

We conclude that the impact parameter representation absorbs the remaining
phase space integrals in the optical theorem, resulting in a purely
multiplicative formula. In fact, in the case where the particles on the left
and right of the diagram do not change (i.e. 1,5,2 and 3,6,4 are identical
particles), such a statement holds to all-loops, leading to exponentiation of
the tree-level phase shift, which is the basis for the famous eikonal
approximation.

###  3.3 Vertex function

Another notion we will use is that of the vertex function, which arises when
combining the optical theorem (84) with the factorization of the tree-level
amplitudes (78) into three-point amplitudes. By combining the two results one
sees that the sums over particles and their polarizations factorize into
separate sums for particles 5 and 6, which we call the vertex function $V$

$V(q_{1},q_{2})\equiv\sum\limits_{m_{5},\rho_{5},\epsilon_{5}}A^{15P_{1}}(q_{1})A^{25P_{2}}(q_{2})\,.$
(87)

Moreover, such a sum over representations and polarizations for each mass is
given by tree-level unitarity as the residue of the four-point amplitudes with
two external Pomerons

$\underset{k_{5}^{2}=-m_{5}^{2}}{\mathop{\mathrm{Res}}}A^{12P_{1}P_{2}}(k_{5},q_{1},q_{2})=\sum\limits_{\rho_{5},\epsilon_{5}}A^{15P_{1}}(q_{1})A^{25P_{2}}(q_{2})\,.$
(88)

In terms of diagrams this reads

$V(q_{1},q_{2})\equiv\sum\limits_{m_{5},\rho_{5},\epsilon_{5}}\leavevmode\hbox
to82.76pt{\vbox to63.28pt{\pgfpicture\makeatletter\hbox{\hskip
39.97623pt\lower-27.89973pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{25.6073pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$5$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{1.4143pt}{-8.5359pt}\pgfsys@curveto{1.4143pt}{-7.7548pt}{0.7811pt}{-7.1216pt}{0.0pt}{-7.1216pt}\pgfsys@curveto{-0.7811pt}{-7.1216pt}{-1.4143pt}{-7.7548pt}{-1.4143pt}{-8.5359pt}\pgfsys@curveto{-1.4143pt}{-9.317pt}{-0.7811pt}{-9.95021pt}{0.0pt}{-9.95021pt}\pgfsys@curveto{0.7811pt}{-9.95021pt}{1.4143pt}{-9.317pt}{1.4143pt}{-8.5359pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-8.5359pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-8.5359pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-36.64322pt}{-22.76228pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{28.8387pt}{-22.76228pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$P_{1}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-28.11021pt}{-17.59659pt}\pgfsys@lineto{-1.53648pt}{-9.0311pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{22.0743pt}\pgfsys@lineto{0.0pt}{-6.9216pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}}
}}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}}
}{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}}
{{}{}{}{}} }{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}} }}{{}} {{{}}} }
\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{1.52695pt}{-9.05972pt}\pgfsys@curveto{2.23636pt}{-9.30309pt}{2.91556pt}{-8.47888pt}{3.62498pt}{-8.72224pt}\pgfsys@curveto{4.1386pt}{-8.89844pt}{4.42293pt}{-9.51189pt}{4.71933pt}{-10.15488pt}\pgfsys@curveto{5.01573pt}{-10.79784pt}{5.30005pt}{-11.4113pt}{5.81367pt}{-11.58751pt}\pgfsys@curveto{6.32729pt}{-11.7637pt}{6.9283pt}{-11.45396pt}{7.557pt}{-11.12836pt}\pgfsys@curveto{8.18568pt}{-10.80275pt}{8.78671pt}{-10.493pt}{9.30034pt}{-10.6692pt}\pgfsys@curveto{9.81395pt}{-10.8454pt}{10.09828pt}{-11.45885pt}{10.39468pt}{-12.10184pt}\pgfsys@curveto{10.69109pt}{-12.7448pt}{10.9754pt}{-13.35826pt}{11.48903pt}{-13.53447pt}\pgfsys@curveto{12.00264pt}{-13.71066pt}{12.60365pt}{-13.40092pt}{13.23236pt}{-13.07532pt}\pgfsys@curveto{13.86104pt}{-12.74971pt}{14.46207pt}{-12.43996pt}{14.9757pt}{-12.61617pt}\pgfsys@curveto{15.6851pt}{-12.85953pt}{15.71532pt}{-13.92711pt}{16.42474pt}{-14.17047pt}\pgfsys@lineto{25.30576pt}{-17.21715pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\times\leavevmode\hbox
to82.76pt{\vbox to61.87pt{\pgfpicture\makeatletter\hbox{\hskip
39.97623pt\lower-26.09529pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{1.4143pt}{11.38092pt}\pgfsys@curveto{1.4143pt}{12.16202pt}{0.7811pt}{12.79523pt}{0.0pt}{12.79523pt}\pgfsys@curveto{-0.7811pt}{12.79523pt}{-1.4143pt}{12.16202pt}{-1.4143pt}{11.38092pt}\pgfsys@curveto{-1.4143pt}{10.59982pt}{-0.7811pt}{9.96661pt}{0.0pt}{9.96661pt}\pgfsys@curveto{0.7811pt}{9.96661pt}{1.4143pt}{10.59982pt}{1.4143pt}{11.38092pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{11.38092pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{11.38092pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-22.76228pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$5$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-36.64322pt}{25.6073pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$2$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{28.8387pt}{25.6073pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$P_{2}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-1.43745pt}{12.11554pt}\pgfsys@lineto{-28.11021pt}{25.7472pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{9.76662pt}\pgfsys@lineto{0.0pt}{-12.78485pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}}
}}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}}
}{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}} }}{{}} {{{}}} }
\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{25.3057pt}{23.78983pt}\pgfsys@curveto{24.63231pt}{23.45963pt}{24.73589pt}{22.39667pt}{24.0625pt}{22.06645pt}\pgfsys@curveto{23.57497pt}{21.8274pt}{22.93994pt}{22.05951pt}{22.27544pt}{22.3039pt}\pgfsys@curveto{21.61098pt}{22.5483pt}{20.97592pt}{22.78041pt}{20.48837pt}{22.54134pt}\pgfsys@curveto{20.00084pt}{22.30228pt}{19.7955pt}{21.65808pt}{19.58186pt}{20.98306pt}\pgfsys@curveto{19.36824pt}{20.30809pt}{19.1629pt}{19.66386pt}{18.67535pt}{19.42479pt}\pgfsys@curveto{18.00197pt}{19.09459pt}{17.22499pt}{19.82735pt}{16.5516pt}{19.49713pt}\pgfsys@lineto{1.44937pt}{12.09146pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\sum\limits_{m_{5}}\underset{k_{5}^{2}=-m_{5}^{2}}{\mathop{\mathrm{Res}}}\leavevmode\hbox
to82.76pt{\vbox to83.59pt{\pgfpicture\makeatletter\hbox{\hskip
39.97623pt\lower-39.28067pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{1.4143pt}{19.91684pt}\pgfsys@curveto{1.4143pt}{20.69794pt}{0.7811pt}{21.33115pt}{0.0pt}{21.33115pt}\pgfsys@curveto{-0.7811pt}{21.33115pt}{-1.4143pt}{20.69794pt}{-1.4143pt}{19.91684pt}\pgfsys@curveto{-1.4143pt}{19.13574pt}{-0.7811pt}{18.50253pt}{0.0pt}{18.50253pt}\pgfsys@curveto{0.7811pt}{18.50253pt}{1.4143pt}{19.13574pt}{1.4143pt}{19.91684pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{19.91684pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{19.91684pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{1.4143pt}{-19.91684pt}\pgfsys@curveto{1.4143pt}{-19.13574pt}{0.7811pt}{-18.50253pt}{0.0pt}{-18.50253pt}\pgfsys@curveto{-0.7811pt}{-18.50253pt}{-1.4143pt}{-19.13574pt}{-1.4143pt}{-19.91684pt}\pgfsys@curveto{-1.4143pt}{-20.69794pt}{-0.7811pt}{-21.33115pt}{0.0pt}{-21.33115pt}\pgfsys@curveto{0.7811pt}{-21.33115pt}{1.4143pt}{-20.69794pt}{1.4143pt}{-19.91684pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-19.91684pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-19.91684pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-36.64322pt}{-34.14322pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-36.64322pt}{34.14322pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$2$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{28.8387pt}{34.14322pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$P_{2}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{28.8387pt}{-34.14322pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$P_{1}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-11.0359pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$5$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-28.11021pt}{-28.97752pt}\pgfsys@lineto{-1.53648pt}{-20.41203pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-1.43745pt}{20.65146pt}\pgfsys@lineto{-28.11021pt}{34.28311pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{18.30254pt}\pgfsys@lineto{0.0pt}{-18.30254pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}}
}}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}}
}{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}} }}{{}} {{{}}} }
\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{25.3057pt}{32.32574pt}\pgfsys@curveto{24.63231pt}{31.99554pt}{24.73589pt}{30.93259pt}{24.0625pt}{30.60237pt}\pgfsys@curveto{23.57497pt}{30.36331pt}{22.93994pt}{30.59543pt}{22.27544pt}{30.83981pt}\pgfsys@curveto{21.61098pt}{31.08421pt}{20.97592pt}{31.31633pt}{20.48837pt}{31.07726pt}\pgfsys@curveto{20.00084pt}{30.8382pt}{19.7955pt}{30.194pt}{19.58186pt}{29.51898pt}\pgfsys@curveto{19.36824pt}{28.84401pt}{19.1629pt}{28.19978pt}{18.67535pt}{27.96071pt}\pgfsys@curveto{18.00197pt}{27.63051pt}{17.22499pt}{28.36327pt}{16.5516pt}{28.03305pt}\pgfsys@lineto{1.44939pt}{20.6274pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}}
}}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}}
}{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}}
{{}{}{}{}} }{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}} }}{{}} {{{}}} }
\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{1.52695pt}{-20.44066pt}\pgfsys@curveto{2.23636pt}{-20.68402pt}{2.91556pt}{-19.85982pt}{3.62498pt}{-20.10318pt}\pgfsys@curveto{4.1386pt}{-20.27937pt}{4.42293pt}{-20.89282pt}{4.71933pt}{-21.53581pt}\pgfsys@curveto{5.01573pt}{-22.17877pt}{5.30005pt}{-22.79224pt}{5.81367pt}{-22.96844pt}\pgfsys@curveto{6.32729pt}{-23.14464pt}{6.9283pt}{-22.8349pt}{7.557pt}{-22.5093pt}\pgfsys@curveto{8.18568pt}{-22.18369pt}{8.78671pt}{-21.87393pt}{9.30034pt}{-22.05014pt}\pgfsys@curveto{9.81395pt}{-22.22633pt}{10.09828pt}{-22.83978pt}{10.39468pt}{-23.48277pt}\pgfsys@curveto{10.69109pt}{-24.12573pt}{10.9754pt}{-24.7392pt}{11.48903pt}{-24.9154pt}\pgfsys@curveto{12.00264pt}{-25.0916pt}{12.60365pt}{-24.78186pt}{13.23236pt}{-24.45625pt}\pgfsys@curveto{13.86104pt}{-24.13065pt}{14.46207pt}{-23.82089pt}{14.9757pt}{-23.9971pt}\pgfsys@curveto{15.6851pt}{-24.24046pt}{15.71532pt}{-25.30804pt}{16.42474pt}{-25.5514pt}\pgfsys@lineto{25.30576pt}{-28.59808pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$ (89)

The vertex function combines all information about the exchanges of possibly
spinning particles 5 and 6 into a single scalar function. In terms of the
vertex function the optical theorem (84) in the Regge limit becomes

$\displaystyle\mathop{\mathrm{Im}}A_{\text{1-loop}}(s,q)=-\frac{1}{4s}$
$\displaystyle\int\frac{dq_{1}dq_{2}}{(2\pi)^{D-2}}\,\delta(q-q_{1}-q_{2})$
(90)
$\displaystyle\left(\frac{8}{\alpha^{\prime}}\right)^{2}\beta(t_{1})^{*}\beta(t_{2})V(q_{1},q_{2})^{2}\left(\frac{\alpha^{\prime}s}{4}\right)^{j(t_{1})+j(t_{2})}\,,$

where $t_{i}=-q_{i}^{2}$.

###  3.4 Spinning three-point amplitudes

Since it will be important later to compare tensor structures in AdS and flat
space, we will provide here some more details on the tensor structures of the
three-point amplitudes that appear in the unitarity cut of the four-point
amplitude $A^{12P_{1}P_{2}}$ discussed above. The external momentum $k_{1}$
and the exchanged momentum $l_{1}$, with light-cone components given in the
Regge limit by (83), fix the momentum $k_{5}=k_{1}-l_{1}$ as shown in the
figure below. We may, however, change frame such that $k_{5}$ has no
transverse momentum DAppollonio:2013mgj . Such change of frame does not alter
the fact that the light-cone components of $l_{1}$ are subleading. The same
applies to $l_{2}$. Thus in the Regge limit we can safely write

$\leavevmode\hbox to82.1pt{\vbox to83.7pt{\pgfpicture\makeatletter\hbox{\hskip
41.63664pt\lower-39.28067pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{1.4143pt}{19.91684pt}\pgfsys@curveto{1.4143pt}{20.69794pt}{0.7811pt}{21.33115pt}{0.0pt}{21.33115pt}\pgfsys@curveto{-0.7811pt}{21.33115pt}{-1.4143pt}{20.69794pt}{-1.4143pt}{19.91684pt}\pgfsys@curveto{-1.4143pt}{19.13574pt}{-0.7811pt}{18.50253pt}{0.0pt}{18.50253pt}\pgfsys@curveto{0.7811pt}{18.50253pt}{1.4143pt}{19.13574pt}{1.4143pt}{19.91684pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{19.91684pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{19.91684pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{1.4143pt}{-19.91684pt}\pgfsys@curveto{1.4143pt}{-19.13574pt}{0.7811pt}{-18.50253pt}{0.0pt}{-18.50253pt}\pgfsys@curveto{-0.7811pt}{-18.50253pt}{-1.4143pt}{-19.13574pt}{-1.4143pt}{-19.91684pt}\pgfsys@curveto{-1.4143pt}{-20.69794pt}{-0.7811pt}{-21.33115pt}{0.0pt}{-21.33115pt}\pgfsys@curveto{0.7811pt}{-21.33115pt}{1.4143pt}{-20.69794pt}{1.4143pt}{-19.91684pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{-19.91684pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-19.91684pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-38.30363pt}{-34.14322pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$k_{1}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-38.30363pt}{34.14322pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$k_{2}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{31.15294pt}{34.14322pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$l_{2}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{31.15294pt}{-34.14322pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$l_{1}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-12.69632pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$k_{5}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-26.4498pt}{-28.94728pt}\pgfsys@lineto{-1.52771pt}{-20.4384pt}\pgfsys@stroke\pgfsys@invoke{
}{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}}
}}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{
} {\pgfsys@beginscope\pgfsys@invoke{ } {{}}
{{}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.94637}{0.3231}{-0.3231}{0.94637}{-15.35155pt}{-25.15816pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}
{{{}}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{{}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-1.4485pt}{20.62944pt}\pgfsys@lineto{-26.4498pt}{32.92943pt}\pgfsys@stroke\pgfsys@invoke{
}{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}}
}}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{
} {\pgfsys@beginscope\pgfsys@invoke{ } {{}}
{{}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.8973}{0.44144}{-0.44144}{-0.8973}{-12.65681pt}{26.14354pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}
{{{}}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{{}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-18.30254pt}\pgfsys@lineto{0.0pt}{18.30254pt}\pgfsys@stroke\pgfsys@invoke{
}{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}}
}}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{
} {\pgfsys@beginscope\pgfsys@invoke{ } {{}}
{{}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{0.0pt}{-1.44pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}
{{{}}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{{}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{1.4485pt}{20.62944pt}\pgfsys@lineto{27.61993pt}{33.50493pt}\pgfsys@stroke\pgfsys@invoke{
}{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}}
}}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{
} {\pgfsys@beginscope\pgfsys@invoke{ } {{}}
{{}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.8973}{0.44144}{-0.44144}{0.8973}{13.24203pt}{26.43144pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}
{{{}}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{{}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{1.52771pt}{-20.43842pt}\pgfsys@lineto{27.61993pt}{-29.34679pt}\pgfsys@stroke\pgfsys@invoke{
}{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}}
}}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{
} {\pgfsys@beginscope\pgfsys@invoke{ } {{}}
{{}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.94637}{-0.3231}{0.3231}{0.94637}{13.21101pt}{-24.42729pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}
{{{}}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{{}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\qquad\quad\begin{aligned}
k_{5}&\approx\left(k_{5}^{u},\frac{m_{5}^{2}}{k_{5}^{u}},0\right)\,,&\\\
l_{2}&\approx(0,0,q_{2})\,,\qquad&k_{2}=k_{5}-l_{2}\,,\\\
l_{1}&\approx(0,0,q_{1})\,,\qquad&k_{1}=k_{5}+l_{1}\,.\end{aligned}$ (91)

We focus on the three-point amplitude $A^{15P_{1}}(q_{1})$, which is related
to the four-point amplitude via the tree-level unitarity (88). In this
relation we have a sum over a basis of possible polarizations $\epsilon_{5}$,
which can be evaluated using completeness relations, e.g. for massive bosons
Boels:2014dka

$\sum\limits_{\epsilon_{5}}\epsilon_{5}^{\mu_{1}\ldots\mu_{|\rho|}}\epsilon_{5}^{\nu_{1}\ldots\nu_{|\rho|}}=P^{\mu_{1}}_{5\gamma_{1}}\ldots
P^{\mu_{\rho}}_{5\gamma_{\rho}}\pi_{\rho}^{\gamma_{1}\ldots\gamma_{|\rho|};\,\sigma_{1}\ldots\sigma_{|\rho|}}P^{\nu_{1}}_{5\sigma_{1}}\ldots
P^{\nu_{\rho}}_{5\sigma_{\rho}}\,,$ (92)

where $\pi_{\rho}$ is the projector to the irreducible $SO(D-1)$
representation $\rho$ and

$P_{5\nu}^{\mu}=\delta^{\mu}_{\nu}-\frac{k_{5}^{\mu}k_{5\nu}}{k_{5}^{2}}\,,$
(93)

is a projector to the space transverse to $k_{5}$. We will always absorb the
projectors $P_{5\nu}^{\mu}$ into the three-point amplitudes, i.e. consider
amplitudes in a transverse configuration. That means that the indices
corresponding to particle 5 have to be constructed from the projected momenta
of the other particles, which are identical

$P_{5\nu}^{\mu}l_{1\mu}=P_{5\nu}^{\mu}k_{1\mu}\,.$ (94)

Apart from that, massive particles can also have a longitudinal polarization
$v$ which satisfies

$v\cdot k_{5}=0\,,\qquad v^{2}=1\,,$ (95)

and is given in this frame explicitly by

$v_{\mu}=\frac{1}{m_{5}}\left(k_{5}^{u},-\frac{m_{5}^{2}}{k_{5}^{u}},0\right)\,.$
(96)

For the case that particle 1 is a scalar, we can then construct $A^{15P}$ in
terms of the following manifestly transverse tensor structures

$A^{15P}_{m_{5},\rho_{5},\bm{\mu}}=\sum\limits_{k=0}^{|\rho_{5}|}a_{m_{5},\rho_{5}}^{k}(t_{1})\,i^{|\rho_{5}|}\sqrt{\alpha^{\prime}}^{|\rho_{5}|-k}v_{\mu_{1}}\ldots
v_{\mu_{k}}q_{1\mu_{k+1}}\ldots q_{1\mu_{|\rho_{5}|}}\,,$ (97)

where we introduced boldface indices $\bm{\mu}$ as multi-indices that stand
for the $|\rho|$ indices for the irrep $\rho$. By abuse of language we defined
the vector $q_{1}\equiv(0,0,q_{1})$, since $q_{1}$ is transverse. If particle
1 carries spin as well, as will be the case for the gravitons considered later
on, we construct the polarization tensors out of the vector
$\xi_{1}=(\xi_{1}^{u},\xi_{1}^{v},\epsilon_{1})$. In this case, again defining
$\epsilon_{1}\equiv(0,0,\epsilon_{1})$, the amplitudes take the following form
in the Regge limit

$\displaystyle
A^{15P}_{m_{5},\rho_{5},\bm{\mu}}=\sum\limits_{n=0}^{\ell_{1}}\sum\limits_{k=0}^{|\rho_{5}|-n}$
$\displaystyle
a_{m_{5},\rho_{5}}^{k,n}(t_{1})\,i^{|\rho_{5}|}\sqrt{\alpha^{\prime}}^{|\rho_{5}|+\ell_{1}-2n-k}(\epsilon_{1}\cdot
q_{1})^{\ell_{1}-n}$ (98)
$\displaystyle\epsilon_{1\mu_{1}}\ldots\epsilon_{1\mu_{n}}v_{\mu_{n+1}}\ldots
v_{\mu_{n+k}}q_{1\mu_{n+k+1}}\ldots q_{1\mu_{|\rho_{5}|}}\,,$

as can be checked by comparing with the explicit amplitudes computed in
DAppollonio:2013mgj . These choices for the tensor structures are particularly
convenient since $q_{1}\cdot v=\epsilon_{1}\cdot v=0$. Contact with the
momentum frame used in the previous subsections is made by identifying the
Lorentz invariant $A^{12P_{1}P_{2}}$.

##  4 AdS impact parameter space

Our goal in this section is to compute the Regge limit of a scalar four-point
function in a perturbative large $N$ CFT at one-loop and finite
$\Delta_{\text{gap}}$. At finite $\Delta_{\text{gap}}$ we have to consider the
$t$-channel exchange of all possible double-trace operators and also single-
trace operators, which are respectively dual to tidal excitations of the
external scattering states and to long-string creation in the string theory
context. It was shown in Meltzer:2019pyl  that the exchange of single-trace
operators dual to the long-string creation effects is subleading in the Regge
limit. Therefore we only need to consider the exchange of double-trace
operators. This is where the new perturbative CFT optical theorem (5) takes a
central role, as it allows us to compute the contributions of double-trace
operators to the correlator starting from the corresponding tree-level
correlators. The contribution of the leading Regge trajectory to the scalar
tree-level correlators is known to leading order in the Regge limit
Cornalba:2007fs ; Costa:2012cb .

In this section we will therefore study (5) in the Regge limit, and this time
we expand the tree-level correlators in the $s$-channel. In the Regge limit
the four external points are in Lorentzian kinematics as depicted in figure 3.

$y_{1}$$y_{4}$$y_{3}$$y_{2}$$y^{+}$$y^{-}$ Figure 3: Kinematics in the central
Poincaré patch with coordinates $y_{i}$. Time is on the vertical axis,
transverse directions are suppressed.

In this configuration all distances between points are spacelike except for
$y_{14}^{2},y_{23}^{2}<0$. The Regge limit is reached by sending the four-
points to infinity along the light cones

$y_{1}^{+}\rightarrow-\infty,\quad y_{2}^{+}\rightarrow+\infty,\quad
y_{3}^{-}\rightarrow-\infty,\quad y_{4}^{-}\rightarrow+\infty\,.$ (99)

The Regge limit can be directly applied to the left hand side of (5). The
terms on the Regge sheets $A^{\circlearrowleft}(y_{i})$ and
$A^{\circlearrowright}(y_{i})$ are dominant over the Euclidean terms in this
limit. However, we cannot apply the Regge limit directly to the right hand
side of (5) as the shadow integrals range over Euclidean configurations. Hence
we will apply Wick rotations on the points $y_{5},\,y_{6},\,y_{7},\,y_{8}$ to
obtain a gluing of the discontinuities of Lorentzian correlators. We will
assume that in the Regge limit the dominant contribution to the gluing formula
comes from the domain where the individual tree-level correlators are in the
Regge limit themselves. We do not provide a proof of this assumption but we
justify it in section 4.1.

When each four-point function in (61) is in the Regge limit, the points
$y_{5},\,y_{6},\,y_{7},\,y_{8}$ are placed in the same positions as
$y_{1},\,y_{4},\,y_{2},\,y_{3}$ in fig. 3, respectively. Thus $y_{7}$ is in
the future of $y_{8}$ and this pair is spacelike from
$y_{1},\,y_{4},\,y_{5},\,y_{6}$. Similarly, $y_{6}$ is in the future of
$y_{5}$ and is spacelike from $y_{2},\,y_{3},\,y_{7},\,y_{8}$. For the chosen
kinematics we put the pair $y_{5},\,y_{6}$ in anti-time order using the
epsilon prescription of (58) with $\epsilon_{5}>\epsilon_{6}$, and the pair
$y_{7},\,y_{8}$ in time order using $\epsilon_{7}>\epsilon_{8}$. Applying this
on (70) gives the following formula for the Regge limit of the double
discontinuity

$\displaystyle\text{dDisc}_{t}A_{\text{1-loop}}(y_{k})\Big{|}_{\text{d.t.}}=-\frac{1}{2}\sum\limits_{\mathcal{O}_{5},\mathcal{O}_{6}}\frac{1}{N_{\mathcal{O}_{5}}N_{\mathcal{O}_{6}}}\int$
$\displaystyle
dy_{5}dy_{6}dy_{7}dy_{8}\,\langle[\mathcal{O}_{2},\mathcal{O}_{3}]\mathcal{O}_{5}\mathcal{O}_{6}\rangle_{\text{tree}}\langle\widetilde{\mathcal{O}}_{5}\widetilde{\mathcal{O}}_{7}^{{\dagger}}\rangle$
$\displaystyle\langle\widetilde{\mathcal{O}}_{6}\widetilde{\mathcal{O}}_{8}^{{\dagger}}\rangle\langle\mathcal{O}_{7}\mathcal{O}_{8}[\mathcal{O}_{4},\mathcal{O}_{1}]\rangle_{\text{tree}}\Big{|}_{\left[\mathcal{O}_{5}\mathcal{O}_{6}\right]}\,.$
(100)

The relative ordering between $\epsilon_{5},\epsilon_{7}$ and
$\epsilon_{6},\epsilon_{8}$ is irrelevant as the pairs, appearing in the two-
point functions on the right hand side of (100), are spacelike separated in
the Regge configuration.

In this section we define the discontinuities as the commutators inserted into
the fully Lorentzian correlators

$\displaystyle\operatorname{Disc}_{14}A^{1874}(y_{i})$
$\displaystyle\coloneqq\langle\mathcal{O}_{7}\mathcal{O}_{8}[\mathcal{O}_{4},\mathcal{O}_{1}]\rangle=A^{1874}{}^{\circlearrowleft}(y_{i})-A^{1874}_{\text{Euc}}(y_{i})\,,$
(101) $\displaystyle\operatorname{Disc}_{23}A^{3652}(y_{i})$
$\displaystyle\coloneqq\langle[\mathcal{O}_{2},\mathcal{O}_{3}]\mathcal{O}_{5}\mathcal{O}_{6}\rangle=A^{3652}_{\text{Euc}}(y_{i})-A^{3652}{}^{\circlearrowright}(y_{i})\,.$

This definition differs slightly from the one in (22). Stripping out the
appropriate pre-factors from (101), one can check that these single
discontinuities can be equivalently defined as

$\displaystyle\operatorname{Disc}_{14}\mathcal{A}^{1234}(z,\overline{z})$
$\displaystyle\coloneqq
e^{i\pi(a+b)}\mathcal{A}^{1234}(z,\overline{z}^{\circlearrowleft})-e^{-i\pi(a+b)}\mathcal{A}^{1234}(z,\overline{z})\,,$
(102)
$\displaystyle\operatorname{Disc}_{23}\mathcal{A}^{1234}(z,\overline{z})$
$\displaystyle\coloneqq\mathcal{A}^{1234}(z,\overline{z})-\mathcal{A}^{1234}(z,\overline{z}^{\circlearrowright})\,,$
$\displaystyle\operatorname{Disc}_{23}\mathcal{A}^{3412}(z,\overline{z})$
$\displaystyle=e^{-i\pi(a+b)}\mathcal{A}^{3412}(z,\overline{z})-e^{i\pi(a+b)}\mathcal{A}^{3412}(z,\overline{z}^{\circlearrowright})\,.$

Starting from the discontinuity defined in (22), these expressions result from
continuing another half circle in $\overline{z}$, so that the different terms
are either evaluated at the original position or continued a full circle
around 1. The extra phase comes from the additional Wick rotations. The final
result matches the definition of the discontinuity in Caron-Huot:2020nem .
Note that $\overline{z}$ is continued an extra half circle in opposite
directions for the first two lines in (102), so that with these definitions
the relation to the dDisc in (21) remains valid.444For $t$-channel blocks, the
new definitions for the discontinuities in (102) are related to the old
definition in (22) by a phase, for example, for $\operatorname{Disc}_{14}$ the
relative phase is $e^{i\pi\tau_{\mathcal{O}}/2}$. Therefore the optical
theorem in the Regge limit can still be expressed as

$\displaystyle\text{dDisc}_{t}A_{\rm
1-loop}(y_{i})\Big{|}_{\text{d.t.}}\\!=-\frac{1}{2}\sum\limits_{\begin{subarray}{c}\mathcal{O}_{5},\mathcal{O}_{6}\\\
\in\ s.t.\end{subarray}}\\!\int dy_{5}dy_{6}\,\operatorname{Disc}_{23}A_{\rm
tree}^{3652}(y_{k})\,\mathbf{S}_{5}\mathbf{S}_{6}\operatorname{Disc}_{14}A_{\rm
tree}^{1564}(y_{k})\Big{|}_{[\mathcal{O}_{5}\mathcal{O}_{6}]}\,,$ (103)

with the discontinuities as defined in the first and third lines of (102), and
the gluing and shadow integrals now ranging over Minkowski space. This formula
is also depicted in figure 4 in terms of Witten diagrams.

$\text{dDisc}_{t}\,\leavevmode\hbox to97.14pt{\vbox
to97.14pt{\pgfpicture\makeatletter\hbox{\hskip 48.5696pt\lower-48.5696pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}{{{}}
{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,1,1}\pgfsys@color@cmyk@fill{1}{0}{0}{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.4}\pgfsys@invoke{
}{}\pgfsys@moveto{-17.07182pt}{0.0pt}\pgfsys@moveto{-11.38136pt}{0.0pt}\pgfsys@curveto{-11.38136pt}{9.42862pt}{-13.92903pt}{17.07182pt}{-17.07182pt}{17.07182pt}\pgfsys@curveto{-20.21461pt}{17.07182pt}{-22.76228pt}{9.42862pt}{-22.76228pt}{0.0pt}\pgfsys@curveto{-22.76228pt}{-9.42862pt}{-20.21461pt}{-17.07182pt}{-17.07182pt}{-17.07182pt}\pgfsys@curveto{-13.92903pt}{-17.07182pt}{-11.38136pt}{-9.42862pt}{-11.38136pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-17.07182pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}
{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,1,1}\pgfsys@color@cmyk@fill{1}{0}{0}{0}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.4}\pgfsys@invoke{
}{}\pgfsys@moveto{17.07182pt}{0.0pt}\pgfsys@moveto{22.76228pt}{0.0pt}\pgfsys@curveto{22.76228pt}{9.42862pt}{20.21461pt}{17.07182pt}{17.07182pt}{17.07182pt}\pgfsys@curveto{13.92903pt}{17.07182pt}{11.38136pt}{9.42862pt}{11.38136pt}{0.0pt}\pgfsys@curveto{11.38136pt}{-9.42862pt}{13.92903pt}{-17.07182pt}{17.07182pt}{-17.07182pt}\pgfsys@curveto{20.21461pt}{-17.07182pt}{22.76228pt}{-9.42862pt}{22.76228pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{17.07182pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-15.65752pt}{17.07182pt}\pgfsys@curveto{-15.65752pt}{17.85292pt}{-16.29073pt}{18.48613pt}{-17.07182pt}{18.48613pt}\pgfsys@curveto{-17.85292pt}{18.48613pt}{-18.48613pt}{17.85292pt}{-18.48613pt}{17.07182pt}\pgfsys@curveto{-18.48613pt}{16.29073pt}{-17.85292pt}{15.65752pt}{-17.07182pt}{15.65752pt}\pgfsys@curveto{-16.29073pt}{15.65752pt}{-15.65752pt}{16.29073pt}{-15.65752pt}{17.07182pt}\pgfsys@closepath\pgfsys@moveto{-17.07182pt}{17.07182pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-17.07182pt}{17.07182pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{18.48613pt}{17.07182pt}\pgfsys@curveto{18.48613pt}{17.85292pt}{17.85292pt}{18.48613pt}{17.07182pt}{18.48613pt}\pgfsys@curveto{16.29073pt}{18.48613pt}{15.65752pt}{17.85292pt}{15.65752pt}{17.07182pt}\pgfsys@curveto{15.65752pt}{16.29073pt}{16.29073pt}{15.65752pt}{17.07182pt}{15.65752pt}\pgfsys@curveto{17.85292pt}{15.65752pt}{18.48613pt}{16.29073pt}{18.48613pt}{17.07182pt}\pgfsys@closepath\pgfsys@moveto{17.07182pt}{17.07182pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{17.07182pt}{17.07182pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-15.65752pt}{-17.07182pt}\pgfsys@curveto{-15.65752pt}{-16.29073pt}{-16.29073pt}{-15.65752pt}{-17.07182pt}{-15.65752pt}\pgfsys@curveto{-17.85292pt}{-15.65752pt}{-18.48613pt}{-16.29073pt}{-18.48613pt}{-17.07182pt}\pgfsys@curveto{-18.48613pt}{-17.85292pt}{-17.85292pt}{-18.48613pt}{-17.07182pt}{-18.48613pt}\pgfsys@curveto{-16.29073pt}{-18.48613pt}{-15.65752pt}{-17.85292pt}{-15.65752pt}{-17.07182pt}\pgfsys@closepath\pgfsys@moveto{-17.07182pt}{-17.07182pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-17.07182pt}{-17.07182pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{18.48613pt}{-17.07182pt}\pgfsys@curveto{18.48613pt}{-16.29073pt}{17.85292pt}{-15.65752pt}{17.07182pt}{-15.65752pt}\pgfsys@curveto{16.29073pt}{-15.65752pt}{15.65752pt}{-16.29073pt}{15.65752pt}{-17.07182pt}\pgfsys@curveto{15.65752pt}{-17.85292pt}{16.29073pt}{-18.48613pt}{17.07182pt}{-18.48613pt}\pgfsys@curveto{17.85292pt}{-18.48613pt}{18.48613pt}{-17.85292pt}{18.48613pt}{-17.07182pt}\pgfsys@closepath\pgfsys@moveto{17.07182pt}{-17.07182pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{17.07182pt}{-17.07182pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-36.98866pt}{-36.98866pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-36.98866pt}{36.98866pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{36.98866pt}{-36.98866pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{36.98866pt}{36.98866pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-42.33368pt}{-42.67914pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-42.33368pt}{36.98866pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$2$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{37.33368pt}{-42.67914pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$4$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{37.33368pt}{36.98866pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$3$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-5.30452pt}{3.86006pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$P_{2}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-5.30452pt}{-30.28316pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$P_{1}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-18.21333pt}{-18.21327pt}\pgfsys@lineto{-33.4562pt}{-33.45566pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-18.21333pt}{18.21327pt}\pgfsys@lineto{-33.4562pt}{33.45566pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}}
}}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}}
}{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}}
{{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}}
{{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}}
}}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}} }}{{}}
{{}} } \pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-15.45752pt}{17.07182pt}\pgfsys@curveto{-14.70752pt}{17.07182pt}{-14.33252pt}{18.07182pt}{-13.58252pt}{18.07182pt}\pgfsys@curveto{-13.03952pt}{18.07182pt}{-12.57153pt}{17.58382pt}{-12.08252pt}{17.07182pt}\pgfsys@curveto{-11.59352pt}{16.55983pt}{-11.12552pt}{16.07182pt}{-10.58252pt}{16.07182pt}\pgfsys@curveto{-10.03952pt}{16.07182pt}{-9.57153pt}{16.55983pt}{-9.08252pt}{17.07182pt}\pgfsys@curveto{-8.59352pt}{17.58382pt}{-8.12552pt}{18.07182pt}{-7.58252pt}{18.07182pt}\pgfsys@curveto{-7.03952pt}{18.07182pt}{-6.57153pt}{17.58382pt}{-6.08252pt}{17.07182pt}\pgfsys@curveto{-5.59352pt}{16.55983pt}{-5.12552pt}{16.07182pt}{-4.58252pt}{16.07182pt}\pgfsys@curveto{-4.03952pt}{16.07182pt}{-3.57153pt}{16.55983pt}{-3.08252pt}{17.07182pt}\pgfsys@curveto{-2.59352pt}{17.58382pt}{-2.12552pt}{18.07182pt}{-1.58252pt}{18.07182pt}\pgfsys@curveto{-1.03952pt}{18.07182pt}{-0.57153pt}{17.58382pt}{-0.08252pt}{17.07182pt}\pgfsys@curveto{0.40648pt}{16.55983pt}{0.87448pt}{16.07182pt}{1.41748pt}{16.07182pt}\pgfsys@curveto{1.96048pt}{16.07182pt}{2.42847pt}{16.55983pt}{2.91748pt}{17.07182pt}\pgfsys@curveto{3.40648pt}{17.58382pt}{3.87448pt}{18.07182pt}{4.41748pt}{18.07182pt}\pgfsys@curveto{4.96048pt}{18.07182pt}{5.42847pt}{17.58382pt}{5.91748pt}{17.07182pt}\pgfsys@curveto{6.40648pt}{16.55983pt}{6.87448pt}{16.07182pt}{7.41748pt}{16.07182pt}\pgfsys@curveto{7.96048pt}{16.07182pt}{8.42847pt}{16.55983pt}{8.91748pt}{17.07182pt}\pgfsys@curveto{9.40648pt}{17.58382pt}{9.87448pt}{18.07182pt}{10.41748pt}{18.07182pt}\pgfsys@curveto{11.16748pt}{18.07182pt}{11.54248pt}{17.07182pt}{12.29248pt}{17.07182pt}\pgfsys@lineto{15.45752pt}{17.07182pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}}
}}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}}
}{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}}
{{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}}
{{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}}
}}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}} }}{{}}
{{}} } \pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-15.45752pt}{-17.07182pt}\pgfsys@curveto{-14.70752pt}{-17.07182pt}{-14.33252pt}{-16.07182pt}{-13.58252pt}{-16.07182pt}\pgfsys@curveto{-13.03952pt}{-16.07182pt}{-12.57153pt}{-16.55983pt}{-12.08252pt}{-17.07182pt}\pgfsys@curveto{-11.59352pt}{-17.58382pt}{-11.12552pt}{-18.07182pt}{-10.58252pt}{-18.07182pt}\pgfsys@curveto{-10.03952pt}{-18.07182pt}{-9.57153pt}{-17.58382pt}{-9.08252pt}{-17.07182pt}\pgfsys@curveto{-8.59352pt}{-16.55983pt}{-8.12552pt}{-16.07182pt}{-7.58252pt}{-16.07182pt}\pgfsys@curveto{-7.03952pt}{-16.07182pt}{-6.57153pt}{-16.55983pt}{-6.08252pt}{-17.07182pt}\pgfsys@curveto{-5.59352pt}{-17.58382pt}{-5.12552pt}{-18.07182pt}{-4.58252pt}{-18.07182pt}\pgfsys@curveto{-4.03952pt}{-18.07182pt}{-3.57153pt}{-17.58382pt}{-3.08252pt}{-17.07182pt}\pgfsys@curveto{-2.59352pt}{-16.55983pt}{-2.12552pt}{-16.07182pt}{-1.58252pt}{-16.07182pt}\pgfsys@curveto{-1.03952pt}{-16.07182pt}{-0.57153pt}{-16.55983pt}{-0.08252pt}{-17.07182pt}\pgfsys@curveto{0.40648pt}{-17.58382pt}{0.87448pt}{-18.07182pt}{1.41748pt}{-18.07182pt}\pgfsys@curveto{1.96048pt}{-18.07182pt}{2.42847pt}{-17.58382pt}{2.91748pt}{-17.07182pt}\pgfsys@curveto{3.40648pt}{-16.55983pt}{3.87448pt}{-16.07182pt}{4.41748pt}{-16.07182pt}\pgfsys@curveto{4.96048pt}{-16.07182pt}{5.42847pt}{-16.55983pt}{5.91748pt}{-17.07182pt}\pgfsys@curveto{6.40648pt}{-17.58382pt}{6.87448pt}{-18.07182pt}{7.41748pt}{-18.07182pt}\pgfsys@curveto{7.96048pt}{-18.07182pt}{8.42847pt}{-17.58382pt}{8.91748pt}{-17.07182pt}\pgfsys@curveto{9.40648pt}{-16.55983pt}{9.87448pt}{-16.07182pt}{10.41748pt}{-16.07182pt}\pgfsys@curveto{11.16748pt}{-16.07182pt}{11.54248pt}{-17.07182pt}{12.29248pt}{-17.07182pt}\pgfsys@lineto{15.45752pt}{-17.07182pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{18.21333pt}{-18.21327pt}\pgfsys@lineto{33.4562pt}{-33.45566pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{18.21333pt}{18.21327pt}\pgfsys@lineto{33.4562pt}{33.45566pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{48.3696pt}{0.0pt}\pgfsys@curveto{48.3696pt}{26.71413pt}{26.71413pt}{48.3696pt}{0.0pt}{48.3696pt}\pgfsys@curveto{-26.71413pt}{48.3696pt}{-48.3696pt}{26.71413pt}{-48.3696pt}{0.0pt}\pgfsys@curveto{-48.3696pt}{-26.71413pt}{-26.71413pt}{-48.3696pt}{0.0pt}{-48.3696pt}\pgfsys@curveto{26.71413pt}{-48.3696pt}{48.3696pt}{-26.71413pt}{48.3696pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\sim\sum\limits_{\mathcal{O}_{5},\mathcal{O}_{6}}\int\operatorname{Disc}_{23}\leavevmode\hbox
to79.95pt{\vbox to81.4pt{\pgfpicture\makeatletter\hbox{\hskip
39.97623pt\lower-40.32167pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-12.81207pt}{0.0pt}\pgfsys@curveto{-12.81207pt}{0.7811pt}{-13.44528pt}{1.4143pt}{-14.22638pt}{1.4143pt}\pgfsys@curveto{-15.00748pt}{1.4143pt}{-15.64069pt}{0.7811pt}{-15.64069pt}{0.0pt}\pgfsys@curveto{-15.64069pt}{-0.7811pt}{-15.00748pt}{-1.4143pt}{-14.22638pt}{-1.4143pt}\pgfsys@curveto{-13.44528pt}{-1.4143pt}{-12.81207pt}{-0.7811pt}{-12.81207pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-14.22638pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{15.64069pt}{0.0pt}\pgfsys@curveto{15.64069pt}{0.7811pt}{15.00748pt}{1.4143pt}{14.22638pt}{1.4143pt}\pgfsys@curveto{13.44528pt}{1.4143pt}{12.81207pt}{0.7811pt}{12.81207pt}{0.0pt}\pgfsys@curveto{12.81207pt}{-0.7811pt}{13.44528pt}{-1.4143pt}{14.22638pt}{-1.4143pt}\pgfsys@curveto{15.00748pt}{-1.4143pt}{15.64069pt}{-0.7811pt}{15.64069pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.22638pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-28.45276pt}{-28.45276pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-28.45276pt}{28.45276pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{28.45276pt}{-28.45276pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{28.45276pt}{28.45276pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-36.64322pt}{-36.98866pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$5$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-36.64322pt}{31.2982pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$2$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{31.64322pt}{-36.98866pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$6$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{31.64322pt}{31.2982pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$3$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-5.30452pt}{-13.21176pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$P_{2}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.94832pt}{-1.44386pt}\pgfsys@lineto{-26.68654pt}{-24.91975pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.94832pt}{1.44386pt}\pgfsys@lineto{-26.68654pt}{24.91975pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}}
}}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}}
}{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}}
{{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}}
{{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}} }}{{}} {{}} }
\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-12.61208pt}{0.0pt}\pgfsys@curveto{-11.86208pt}{0.0pt}{-11.48708pt}{1.0pt}{-10.73708pt}{1.0pt}\pgfsys@curveto{-10.19408pt}{1.0pt}{-9.72609pt}{0.512pt}{-9.23708pt}{0.0pt}\pgfsys@curveto{-8.74808pt}{-0.512pt}{-8.28008pt}{-1.0pt}{-7.73708pt}{-1.0pt}\pgfsys@curveto{-7.19408pt}{-1.0pt}{-6.72609pt}{-0.512pt}{-6.23708pt}{0.0pt}\pgfsys@curveto{-5.74808pt}{0.512pt}{-5.28008pt}{1.0pt}{-4.73708pt}{1.0pt}\pgfsys@curveto{-4.19408pt}{1.0pt}{-3.72609pt}{0.512pt}{-3.23708pt}{0.0pt}\pgfsys@curveto{-2.74808pt}{-0.512pt}{-2.28008pt}{-1.0pt}{-1.73708pt}{-1.0pt}\pgfsys@curveto{-1.19408pt}{-1.0pt}{-0.72609pt}{-0.512pt}{-0.23708pt}{0.0pt}\pgfsys@curveto{0.25192pt}{0.512pt}{0.71992pt}{1.0pt}{1.26292pt}{1.0pt}\pgfsys@curveto{1.80592pt}{1.0pt}{2.27391pt}{0.512pt}{2.76292pt}{0.0pt}\pgfsys@curveto{3.25192pt}{-0.512pt}{3.71992pt}{-1.0pt}{4.26292pt}{-1.0pt}\pgfsys@curveto{4.80592pt}{-1.0pt}{5.27391pt}{-0.512pt}{5.76292pt}{0.0pt}\pgfsys@curveto{6.25192pt}{0.512pt}{6.71992pt}{1.0pt}{7.26292pt}{1.0pt}\pgfsys@curveto{8.01292pt}{1.0pt}{8.38792pt}{0.0pt}{9.13792pt}{0.0pt}\pgfsys@lineto{12.61208pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{14.94832pt}{-1.44386pt}\pgfsys@lineto{26.68654pt}{-24.91975pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{14.94832pt}{1.44386pt}\pgfsys@lineto{26.68654pt}{24.91975pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{20.42853pt}{20.42853pt}{36.98866pt}{0.0pt}{36.98866pt}\pgfsys@curveto{-20.42853pt}{36.98866pt}{-36.98866pt}{20.42853pt}{-36.98866pt}{0.0pt}\pgfsys@curveto{-36.98866pt}{-20.42853pt}{-20.42853pt}{-36.98866pt}{0.0pt}{-36.98866pt}\pgfsys@curveto{20.42853pt}{-36.98866pt}{36.98866pt}{-20.42853pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\operatorname{Disc}_{14}\leavevmode\hbox
to80.51pt{\vbox to82.18pt{\pgfpicture\makeatletter\hbox{\hskip
40.25401pt\lower-40.32167pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-12.81207pt}{0.0pt}\pgfsys@curveto{-12.81207pt}{0.7811pt}{-13.44528pt}{1.4143pt}{-14.22638pt}{1.4143pt}\pgfsys@curveto{-15.00748pt}{1.4143pt}{-15.64069pt}{0.7811pt}{-15.64069pt}{0.0pt}\pgfsys@curveto{-15.64069pt}{-0.7811pt}{-15.00748pt}{-1.4143pt}{-14.22638pt}{-1.4143pt}\pgfsys@curveto{-13.44528pt}{-1.4143pt}{-12.81207pt}{-0.7811pt}{-12.81207pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-14.22638pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{15.64069pt}{0.0pt}\pgfsys@curveto{15.64069pt}{0.7811pt}{15.00748pt}{1.4143pt}{14.22638pt}{1.4143pt}\pgfsys@curveto{13.44528pt}{1.4143pt}{12.81207pt}{0.7811pt}{12.81207pt}{0.0pt}\pgfsys@curveto{12.81207pt}{-0.7811pt}{13.44528pt}{-1.4143pt}{14.22638pt}{-1.4143pt}\pgfsys@curveto{15.00748pt}{-1.4143pt}{15.64069pt}{-0.7811pt}{15.64069pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.22638pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-28.45276pt}{-28.45276pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-28.45276pt}{28.45276pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{28.45276pt}{-28.45276pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{28.45276pt}{28.45276pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-36.64322pt}{-36.98866pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-36.921pt}{31.2982pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\widetilde{5}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{31.64322pt}{-36.98866pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$4$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{31.36543pt}{31.2982pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\widetilde{6}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-5.30452pt}{-13.21176pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$P_{1}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.94832pt}{-1.44386pt}\pgfsys@lineto{-26.68654pt}{-24.91975pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.94832pt}{1.44386pt}\pgfsys@lineto{-26.68654pt}{24.91975pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}}
}}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}}
}{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}}
{{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}}
{{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}} }}{{}} {{}} }
\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-12.61208pt}{0.0pt}\pgfsys@curveto{-11.86208pt}{0.0pt}{-11.48708pt}{1.0pt}{-10.73708pt}{1.0pt}\pgfsys@curveto{-10.19408pt}{1.0pt}{-9.72609pt}{0.512pt}{-9.23708pt}{0.0pt}\pgfsys@curveto{-8.74808pt}{-0.512pt}{-8.28008pt}{-1.0pt}{-7.73708pt}{-1.0pt}\pgfsys@curveto{-7.19408pt}{-1.0pt}{-6.72609pt}{-0.512pt}{-6.23708pt}{0.0pt}\pgfsys@curveto{-5.74808pt}{0.512pt}{-5.28008pt}{1.0pt}{-4.73708pt}{1.0pt}\pgfsys@curveto{-4.19408pt}{1.0pt}{-3.72609pt}{0.512pt}{-3.23708pt}{0.0pt}\pgfsys@curveto{-2.74808pt}{-0.512pt}{-2.28008pt}{-1.0pt}{-1.73708pt}{-1.0pt}\pgfsys@curveto{-1.19408pt}{-1.0pt}{-0.72609pt}{-0.512pt}{-0.23708pt}{0.0pt}\pgfsys@curveto{0.25192pt}{0.512pt}{0.71992pt}{1.0pt}{1.26292pt}{1.0pt}\pgfsys@curveto{1.80592pt}{1.0pt}{2.27391pt}{0.512pt}{2.76292pt}{0.0pt}\pgfsys@curveto{3.25192pt}{-0.512pt}{3.71992pt}{-1.0pt}{4.26292pt}{-1.0pt}\pgfsys@curveto{4.80592pt}{-1.0pt}{5.27391pt}{-0.512pt}{5.76292pt}{0.0pt}\pgfsys@curveto{6.25192pt}{0.512pt}{6.71992pt}{1.0pt}{7.26292pt}{1.0pt}\pgfsys@curveto{8.01292pt}{1.0pt}{8.38792pt}{0.0pt}{9.13792pt}{0.0pt}\pgfsys@lineto{12.61208pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{14.94832pt}{-1.44386pt}\pgfsys@lineto{26.68654pt}{-24.91975pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{14.94832pt}{1.44386pt}\pgfsys@lineto{26.68654pt}{24.91975pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{20.42853pt}{20.42853pt}{36.98866pt}{0.0pt}{36.98866pt}\pgfsys@curveto{-20.42853pt}{36.98866pt}{-36.98866pt}{20.42853pt}{-36.98866pt}{0.0pt}\pgfsys@curveto{-36.98866pt}{-20.42853pt}{-20.42853pt}{-36.98866pt}{0.0pt}{-36.98866pt}\pgfsys@curveto{20.42853pt}{-36.98866pt}{36.98866pt}{-20.42853pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ Figure 4: Optical
theorem in the Regge limit in terms of Witten diagrams. The tree-level
correlators are dominated by $s$-channel Pomeron exchange. The external
operators are scalars, while $\mathcal{O}_{5}$ and $\mathcal{O}_{6}$ are
summed over all states that couple to the external scalars and the Pomeron
(tidal excitations). The ellipses on the l.h.s. indicate that all string
excitations are taken into account.

We should also note that for real $z,\overline{z}$, $\operatorname{Disc}_{23}$
in the third line of (102) is related to $\operatorname{Disc}_{14}$ in the
first line by

$\operatorname{Disc}_{23}\mathcal{A}^{3412}(z,\overline{z})=-\left.\left(\operatorname{Disc}_{14}\mathcal{A}^{1234}(z,\overline{z})\right)^{*}\right|_{(a,b)\to(-b,-a)}\,,\quad
0<z,\overline{z}<1\,.$ (104)

Applied to the correlators appearing in (103) this reads

$\operatorname{Disc}_{23}\mathcal{A}^{3652}(z,\overline{z})=-\left.\left(\operatorname{Disc}_{14}\mathcal{A}^{1564}(z,\overline{z})\right)^{*}\right|_{1564\to
3652}\,.$ (105)

To benefit from this useful relation, we will always strip out a pre-factor
such that we obtain the correlator $\mathcal{A}^{3652}(z,\overline{z})$ on the
right hand side of (103). Otherwise we would have to use the second line of
(102) for $\operatorname{Disc}_{23}$. Finally, in the Regge limit the
analytically continued correlators are dominant over the Euclidean
contributions so that we have

$\displaystyle\operatorname{Disc}_{14}\mathcal{A}^{1234}(z,\overline{z})$
$\displaystyle\approx
e^{i\pi(a+b)}\mathcal{A}^{1234}(z,\overline{z}^{\circlearrowleft})\,,$ (106)
$\displaystyle\operatorname{Disc}_{23}\mathcal{A}^{3412}(z,\overline{z})$
$\displaystyle\approx-e^{i\pi(a+b)}\mathcal{A}^{3412}(z,\overline{z}^{\circlearrowright})\,,$
$\displaystyle\text{dDisc}_{t}\,\mathcal{A}^{1234}(z,\overline{z})$
$\displaystyle\approx-\frac{1}{2}\left(e^{i\pi(a+b)}\mathcal{A}^{1234}(z,\overline{z}^{\circlearrowleft})+e^{-i\pi(a+b)}\mathcal{A}^{1234}(z,\overline{z}^{\circlearrowright})\right)\,,$

where the $\approx$ sign means we took the Regge limit. In order to account
for the tidal excitations, the operators $\mathcal{O}_{5}$ and
$\mathcal{O}_{6}$ can carry spin, in which case their indices are contracted
with the ones of $\widetilde{\mathcal{O}}_{5}$ and
$\widetilde{\mathcal{O}}_{6}$ and sums over tensor structures are implied. In
subsections 4.1, 4.2 and 4.3 below we will mostly suppress the aspect of
spinning correlators. We will come back to this issue in subsection 4.4.

###  4.1 Regge limit

To obtain the impact parameter representation, we first change the coordinate
system placing each point on a different Poincaré patch as shown in figure 5.
We use the following coordinate transformations

$\displaystyle x_{i}$
$\displaystyle=\left(x_{i}^{+},x_{i}^{-},x_{i\perp}\right)=-\frac{1}{y_{i}^{+}}\left(1,y_{i}^{2},y_{i\perp}\right),$
$\displaystyle i=1,2,5,7\,,$ (107) $\displaystyle x_{i}$
$\displaystyle=\left(x_{i}^{+},x_{i}^{-},x_{i\perp}\right)=-\frac{1}{y_{i}^{-}}\left(1,y_{i}^{2},y_{i\perp}\right),$
$\displaystyle i=3,4,6,8\,.$