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%!TEX root=manual
\subsection{Study of \texorpdfstring{$\xc$}{xic} dependence}\label{sec:xicut}

When performing calculations with \mcmule{}, we need to check that the
dependence of the unphysical $\xc$ parameter introduced in the FKS
scheme (cf. Appendix~\ref{sec:fks}) actually drops out at \ac{NLO} and \ac{NNLO}.
In principle it is sufficient to do this once during the development
phase.  However, we consider it good practice to also do this (albeit
with a reduced range of $\xc$) for production runs.

Because the $\xc$ dependence is induced through terms as
$\xc^{-2\epsilon}/\epsilon$, we know the functional dependence
of $\sigma^{(\ell)}_{n+j}$. For example, at \ac{NLO} we have
\begin{subequations}
\begin{align}\begin{split}
\sigma^{(1)}_{n  }(\xc) &= a_{0,0} + a_{0,1}\log(\xc)\,,\\
\sigma^{(1)}_{n+1}(\xc) &= a_{1,0} + a_{1,1}\log(\xc)\,,
\end{split}\end{align}
where $\xc$ independence of $\sigma^{(1)}$ of course requires
\begin{align}
a_{0,1}+ a_{1,1} = 0\,.
\end{align}
\label{eq:xinlo}%
\end{subequations}
\begin{subequations}
At \ac{NNLO} we have
\begin{align}\begin{split}
\sigma^{(2)}_{n  }(\xc) &= a_{0,0} + a_{0,1}\log(\xc) + a_{0,2}\log(\xc)^2\,,\\
\sigma^{(2)}_{n+1}(\xc) &= a_{1,0} + a_{1,1}\log(\xc) + a_{1,2}\log(\xc)^2\,,\\
\sigma^{(2)}_{n+2}(\xc) &= a_{2,0} + a_{2,1}\log(\xc) + a_{2,2}\log(\xc)^2\,.
\end{split}\end{align}
We require
\begin{align}
a_{0,i} + a_{1,i} + a_{2,i} = 0
\end{align}
for $i=1,2$. However, the \ac{IR} structure allows for an even
stronger statement for the $a_{j,2}$ terms
\begin{align}
a_{0,2} = a_{2,2} = -\frac{a_{1,2}}2\,.
\end{align}
\label{eq:xinnlo}%
\end{subequations}
Of course we cannot directly calculate any of the $a_{1,i}$ or
$a_{2,i}$ because we use numerical integration to obtain the
$\sigma^{(\ell)}_{n+j}$. Still, knowing the coefficients can be
extremely helpful when debugging the code or to just quantify how well
the $\xc$ dependence vanishes. Hence, we use a fitting routine to fit
the Monte Carlo results \emph{after} any phase-space partitioning has
been undone. Sometimes non of this is sufficient to pin-point the
source of a problem to any one integrand. However, if the goodness of,
for example, $\sigma^{(2)}_{n+2}(\xc)$ is much worse than the one for
$\sigma^{(2)}_{n+1}(\xc)$, a problem in the double-real corrections
can be expected.