\topheading{CuTe-MCFM} \includegraphics[width=0.4\textwidth]{./sections/cute-mcfmpic.png} \midheading{N$^3$LL and N$^4$LL $q_T^2$ resummation for color-singlet processes in MCFM} Based on \href{https://arxiv.org/abs/2009.11437}{arXiv:2009.11437} (Becher, Neumann '20). The $q_T$ resummation in CuTe-MCFM is available for color-singlet processes and based on a factorization theorem in SCET. It is fully differential in the Born kinematics and matches to large-$q_T^2$ fixed-order predictions at relative order $\alpha_s^2$. It provides an efficient way to estimate uncertainties from fixed-order truncation, resummation, and parton distribution functions. In addition to $W, Z$ and $H$ production, also the diboson processes $\gamma\gamma$, $Z\gamma$, $ZH$, $WH$, $WW$, $WZ$ and $ZZ$ are available, including decays. While CuTe-MCFM can calculate $q_T$-resummed results without using pregenerated beam functions grids, we recommend that LHAPDF grid files are generated for the beam functions beforehand for a choice of a PDF set. This \emph{significantly} accelerates the evaluation of the beam functions and the integration. CuTe-MCFM ships with pregenerated beamfunction grids for the central values of \texttt{CT14nnlo} and \texttt{NNPDF31\_nnlo\_as\_0118}, which are included in the \texttt{Bin/PDFs} directory. This path is automatically used as the preferred path for LHAPDF grid files. With these pregenerated grids the example input files work out of the box. For other PDF sets or when using PDF errors, the first run of CuTe-MCFM should be with the setting \texttt{makegrid=.true.}. Additionally the input and output directories for the PDF grids have to be specified. For example the input directory is typically \texttt{/usr/local/share/LHAPDF/} (or the \texttt{PDFs/} directory relative to the mcfm executable in \texttt{Bin}) and the output directory should be a user-writeable directory like \texttt{/home/user/gridout/} (or \texttt{PDFs/}). Note the trailing slashes. When calling mcfm with \texttt{makegrid=.true.} only the beam function grids are written during that run, and mcfm exits afterwards. We recommend to use \texttt{PDFs/} as the gridout path, since this path is automatically added to the LHAPDF search paths, and you won't have to copy the generated grid directories to your LHAPDF grid directory or set the \texttt{LHAPDF\_DATA\_PATH} environment variable to the gridout path. For example for the set CT14nnlo the grid directories \texttt{CT14nnlo\_B00}, \texttt{CT14nnlo\_B10}, \texttt{CT14nnlo\_B11}, \texttt{CT14nnlo\_B20}, \texttt{CT14nnlo\_B21}, \texttt{CT14nnlo\_B22} and \texttt{CT14nnlo\_G10} are written and have to be copied to the directory where LHAPDF searches for the grid files. When the gridout path is chosen as \texttt{PDFs/} no further action is necessary. The LHAPDF grid file search path can be modified by setting the shell environment variable \texttt{LHAPDF\_DATA\_PATH} to the desired directory, but the \texttt{PDFs} directory is always used as the preferred directory. The next run of mcfm should be done with \texttt{makegrid=.false.} and \texttt{usegrid=.true.}. Other important parameters for the resummation are \texttt{res\_range}, determining the integration range of the purely resummed part, \texttt{resexp\_range}, determining the integration range of the fixed-order expanded resummed part, and \texttt{fo\_cutoff} which sets the lower $q_T$ cutoff for the fixed-order part. Typically this cutoff should agree with the lower range of \texttt{resexp\_range}. For example for $Z$ production one can integrate up to $m_Z$ with a cutoff of 1 GeV: \texttt{res\_range = 0.0 90.0}, \texttt{resexp\_range = 1.0 90.0}, \texttt{qt\_cutoff = 1.0}. For details regarding these parameters see the next section. The transition function is also discussed below. \hypertarget{input-file-parameters}{% \midheading{Input file parameters}\label{input-file-parameters}} The \texttt{[resummation]} section has been added to the input file to control the resummation. The following keys are available: \begin{longtable}[]{@{}ll@{}} % \toprule \begin{minipage}[b]{0.24\columnwidth}\raggedright Key\strut \end{minipage} & \begin{minipage}[b]{0.71\columnwidth}\raggedright Description\strut \end{minipage}\tabularnewline % \midrule \endhead \begin{minipage}[t]{0.24\columnwidth}\raggedright \texttt{usegrid}\strut \end{minipage} & \begin{minipage}[t]{0.71\columnwidth}\raggedright \texttt{.true.} or \texttt{.false.} determines whether pregenerated LHAPDF interpolation grids should be used for the resummation beam functions.\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.24\columnwidth}\raggedright \texttt{makegrid}\strut \end{minipage} & \begin{minipage}[t]{0.71\columnwidth}\raggedright If \texttt{.true.}, then MCFM runs in grid generation mode. This generates LHAPDF grid files in the directory \texttt{gridoutpath} from LHAPDF grids in the directory \texttt{gridinpath}. After the grid generation MCFM stops and should be run subsequently with \texttt{makegrid = .false.} and \texttt{usegrid = .true.}. When \texttt{lhapdf\%dopdferrors=.true.} then also grids for the error sets are generated.\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.24\columnwidth}\raggedright \texttt{gridoutpath}\strut \end{minipage} & \begin{minipage}[t]{0.71\columnwidth}\raggedright Output directory for LHAPDF grid files, for example \texttt{/home/tobias/local/share/LHAPDF/}\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.24\columnwidth}\raggedright \texttt{gridinpath}\strut \end{minipage} & \begin{minipage}[t]{0.71\columnwidth}\raggedright Input directory for LHAPDF grid files, for example \texttt{/home/tobias/local/share/LHAPDF/}\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.24\columnwidth}\raggedright \texttt{res\_range}\strut \end{minipage} & \begin{minipage}[t]{0.71\columnwidth}\raggedright Integration range of purely resummed part, for example \texttt{0.0 80.0} for $q_T$ integration between 0 and 80 GeV.\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.24\columnwidth}\raggedright \texttt{resexp\_range}\strut \end{minipage} & \begin{minipage}[t]{0.71\columnwidth}\raggedright Integration range of fixed-order expanded resummed part, for example \texttt{1.0 80.0} for $q_T$ integration between 1 and 80 GeV.\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.24\columnwidth}\raggedright \texttt{fo\_cutoff}\strut \end{minipage} & \begin{minipage}[t]{0.71\columnwidth}\raggedright Lower $q_T$ cutoff $q_0$ for the fixed-order part, see eq.~\eqref{eq:matchingmod} below. Typically the value should agree with the lower range of \texttt{resexp\_range}.\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.24\columnwidth}\raggedright \texttt{transitionswitch}\strut \end{minipage} & \begin{minipage}[t]{0.71\columnwidth}\raggedright Parameter passed to the plotting routine to modify the transition function, see text.\strut \end{minipage}\tabularnewline % \bottomrule \end{longtable} We strongly recommend to calculate resummed results with pregenerated grids, see the previous section. The integration range for the purely resummed part can be controlled with the key \texttt{res\_range} and should typically be between $0$ and some upper value. For example for $W^\pm, Z$ or $H$ production this can just be the boson mass. For other processes there can be thresholds and this number must be selected more carefully to not run into numerical issues, see arXiv:2009.11437. The setting \texttt{resexp\_range} and \texttt{fo\_cutoff} are relevant for the matching corrections. The values of the \texttt{resexp\_range} determine the integration range for the fixed-order expansion of the resummed part. The minimum should typically be at least one GeV for numerical stability. For smaller values significantly more time goes into the integration, and the minimum number of Vegas calls might need to increased. For single boson processes the maximum value can again be the boson mass, although it can be set to a value where the implemented transition function fully switches to zero. The fixed-order cutoff \texttt{fo\_cutoff} determines the minimum $q_T$ for the fixed-order calculation. This should typically agree with the lower range of the \texttt{resexp\_range}. Lastly, the parameter \texttt{transitionswitch} is passed for convenience to the plotting routines where the transition function is implemented. It can be used for for an easy control of the transition region as described in the following. \input{sections/CMplottingandtransition.tex} \input{sections/CMmodifyingplotting.tex}