xicut.tex 2.27 KB
 ulrich_y committed Jul 03, 2020 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 %!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.