ExSample -- A Library for Sampling Sudakov-Type Distributions

TL;DR

This work introduces ExSample, a C++ header-only library for adaptive sampling of Sudakov-type distributions where the splitting kernel P may be known only numerically and depend on multiple external parameters. It builds a dynamic overestimate R via a binary-tree of cells, with inequality-exceeding maxima handled through a compensation mechanism to maintain correct distributions. The approach enables efficient, parameter-aware sampling in parton showers and POWHEG-style NLO matching, validated against toy kernels and realistic scenarios, and demonstrates substantial gains in sampling efficiency as the cell-tree adapts. Availability and design choices emphasize portability and compatibility with existing Monte Carlo workflows, leveraging Boost and GPLv2 licensing. The method offers a flexible tool for advancing radiation generation in high-energy physics simulations.

Abstract

Sudakov-type distributions are at the heart of generating radiation in parton showers as well as contemporary NLO matching algorithms along the lines of the POWHEG algorithm. In this paper, the C++ library ExSample is introduced, which implements adaptive sampling of Sudakov-type distributions for splitting kernels which are in general only known numerically. Besides the evolution variable, the splitting kernels can depend on an arbitrary number of other degrees of freedom to be sampled, and any number of further parameters which are fixed on an event-by-event basis.

Figures (6)

• Figure 1: A sketch of the algorithm for an evolution variable $q$, one additional variable $z$, and no further parameters $\xi$. The top of the figure shows how the leaf cells (in the third plane from the top, shown here after two cell splits) are organized in a binary tree structure starting from the root cell $U_{((12)3)}$. The bottom of the figure sketches the overestimate $R$. To the left of the overestimate, the Sudakov exponent corresponding to $R$, $F(q)=-\ln \Delta_R(q|1)$ is shown. Here we assume that the absolute upper bound on the evolution variable is $q<1$, thus the first step to draw an event starting from a scale $Q$ is to solve $s(Q)=-\ln {\bf rnd}+F(Q)=F(q)$ for q (indicated by the dashed blue line). A $z$ value is then sampled in the cells containing the $q$ value just chosen: The cell integrals over $z$ are computed to only reflect the subtree consisting of the black arrows, and the tree structure is traversed only along the corresponding paths, selecting either of the children cells with weight given by the respective integral. Within the boundaries of a leaf cell selected by this procedure, a $z$ value is drawn flat. This corresponds to drawing a $z$ value from the distribution sketched by the solid blue line, the overestimate $R$ at fixed $q$.
• Figure 2: A sketch of the algorithm in a setup similar to figure \ref{['figures:trees-sudakov']}, now sketching the situation upon encounter of a new overestimate. The new overestimate gave rise to different numbers of events expected in each cell (solid rectangles in the lower part), as compared to the number of events expected with the old overestimate (dotted rectangles). The difference between these determines the number of missing events per cell (see text for more details). In the sketch given here, the cells $U_2$ and $U_3$ would receive a positive number of missing events (forcing sampling in these cells as indicated by the black arrows), whereas cell $U_1$ would contain a negative count of missing events, triggering vetoes of attempts to sample points in this cell (indicated by the red arrow).
• Figure 3: A Sudakov-type distribution with a QCD splitting function type kernel density as sampled by ExSample using the adaptive Sudakov veto algorithm. The vertical axis corresponds to the evolution variable $q$, the horizontal to a variable similar to a momentum fraction. Shown are few sampled events, projections of the generated distribution versus the result from a numerical integration, and the the cell grid produced.
• Figure 4: The same distribution as shown in the upper left panel of figure \ref{['figures:samplesudakov']}, now sampled with a different number of presampling points proving functionality of the compensation procedure.
• Figure 5: Distributions for a Sudakov-type distribution using a quark splitting function, multiplied by a power law in $x/z$. Shown are the sampled distribution for the evolution variable $q$ and momentum fraction $z$ in various bins of the additional parameter $x$. The distributions are compared to a numerical integration, proving full functionality of the sampling in presence of additional parameters.