\section{Discussion and Comparison with \sherpa}%
\label{sec:compsher}

The result obtained in \cref{sec:qqggcalc} shall now be verified by the
monte-carlo event generator \sherpa{}~\cite{Gleisberg:2008ta}. To
facilitate this, an expression for the total cross section for a range
of \(\theta\) or \(\eta\) has to be obtained. Using the golden rule
for \(2\rightarrow 2\) processes and observing that the initial and
final momenta are equal (\(p_i=p_f\)) and \(g=\sqrt{4\pi\alpha}\), the
result \cref{eq:crossec} arises. The differential cross section has
also been calculated in terms of the pseudo-rapidity
in \cref{eq:xs-eta}.

An additional factor of \(\frac{1}{2}\) comes in due to there being
two identical photons in the final state.
\begin{equation}
  \label{eq:crossec}
  \dv{\sigma}{\Omega} =
  \frac{1}{2}\frac{1}{(8\pi)^2}\cdot\frac{\abs{\mathcal{M}}^2}{\ecm^2}\cdot\frac{\abs{p_f}}{\abs{p_i}}
  = \underbrace{\frac{\alpha^2Z^4}{6\ecm^2}}_{\mathfrak{C}}\frac{1+\cos^2(\theta)}{\sin^2(\theta)}
\end{equation}

\begin{equation}
  \label{eq:xs-eta}
  \dv{\sigma}{\eta} = 2\pi\cdot\frac{\alpha^2Z^4}{6\ecm^2}\cdot\qty(\tanh(\eta)^2 + 1)
\end{equation}

\begin{figure}[ht]
  \centering
  \begin{subfigure}[c]{.45\textwidth}
    \centering \plot{xs/diff_xs}
    \caption[Plot of the differential cross section of the \(\qqgg\)
    process.]{\label{fig:diffxs} The differential cross section as a
      function of the polar angle \(\theta\). }
  \end{subfigure}
  \begin{subfigure}[c]{.45\textwidth}
  \centering
  \plot{xs/total_xs}
  \caption[Plot of the total cross section of the \(\qqgg\)
  process.]{\label{fig:totxs} The total cross section
    (\cref{eq:total-crossec}) of the process for a pseudo-rapidity
    range of \([-\eta, \eta]\).}
  \end{subfigure}
\end{figure}

The differential cross section \cref{eq:crossec} (see
also \cref{fig:diffxs}) is divergent for angles near zero or
\(\pi\). Allowing finite mass in the calculation may regularize this
divergence. Because \(m=0\) is the limit for
\(\ecm\rightarrow\infty\), the cross section would still have strong
peaks for angles near \(0,\pi\) at high energies so that the result is
not altogether nonphysical. The divergence of the cross section itself
is also not the problem here, because it can be transformed into a
form where the divergence does not occur (see \cref{eq:xs-eta}).

The differential cross section is clearly symmetric around
\(\theta=\frac{\pi}{2}\) as was to be expected\footnote{Such
  properties are very handy and intuitive checks for calculations
  along the way.}, because the photons are indistinguishable. To
compare the cross section to experiment and to simulation an interval
around \(\theta=\frac{\pi}{2}\) has to be chosen, where the first
order, mass-less approximation may yield a physical result.

The total cross section in such an interval, given by
integrating \cref{eq:crossec} for \(\theta\in [\theta_1, \theta_2]\)
or \(\eta\in [\eta_1, \eta_2]\) is given
in \cref{eq:total-crossec}.

\begin{equation}
  \label{eq:total-crossec}
  \begin{split}
  \sigma &=
  2\pi\mathfrak{C}\cdot\qty{\cos(\theta_2)-\cos(\theta_1)+2\qty[\artanh(\cos(\theta_1))
    - \artanh(\cos(\theta_2))]} \\
  &=2\pi\mathfrak{C}\cdot\qty[\tanh(\eta_2) - \tanh(\eta_1) + 2(\eta_1
  - \eta_2))] \\
  &={\frac{\pi\alpha^2Z^4}{3\ecm^2}}\cdot\qty[\tanh(\eta_2) - \tanh(\eta_1) + 2(\eta_1
  - \eta_2))]
  \end{split}
\end{equation}

As can be seen in \cref{fig:totxs}, the cross section, integrated over
an interval of \([-\eta, \eta]\), is dominated by the linear
contributions in \cref{eq:total-crossec} and would result in an
infinity if no cut on \(\eta\) would be made.  Choosing
\(\eta\in [-2.5,2.5]\) and \(\ecm=\SI{100}{\giga\electronvolt}\) the
process was monte carlo integrated in \sherpa\ using the runcard
in \cref{sec:qqggruncard}. This runcard describes the exact same (first
order) process as the calculated cross section.

\sherpa\ arrives at the cross section \result{xs/sherpa_xs}. Plugging
the same parameters into \cref{eq:total-crossec} gives
\result{xs/python/xs} which is within the uncertainty range of the
\sherpa\ value. This verifies the result for the total cross section.

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