MC@NLO's negative-weight events get a learned fix: decision trees and normalizing flows cut the negative fraction in most processes

The fraction of negative-weight events can be lowered without altering physical predictions by adding to the MC@NLO radiation-event short-distance cross section a local correction learned by machine learning whose integral over radiation variables is zero. Previously, negative weights were mainly reduced by folding, which integrates over radiation variables before unweighting and maximally reduces negative-weight events but is computationally very expensive; MC@NLO matching itself keeps producing negative weights because unweighting happens in a space that retains redundant radiation variables. Across several electron- and hadron-collision processes, the corrected total cross sections and Born-level distributions agree with the original MC@NLO predictions within statistical uncertainties, while the negative-weight fraction drops in most processes, for example from 20.4% to 3.2% in one electron-collision process.

The correction comes from two model classes: decision trees partition the input space into regions and assign a constant value to each, while normalizing flows reshape a simple distribution into a conditional probability density over radiation variables through invertible transformations. Both model classes are constrained to integrate to zero over radiation variables, so adding them leaves the cross section at fixed Born kinematics unchanged; decision trees enforce this by explicitly computing the radiation integral of their leaves, while normalizing flows rely on the normalization built into their construction. Both model classes were trained on several electron- and hadron-collision processes, and the hadron-collision processes needed larger training samples to reach a comparable reduction in negative weights, consistent with the two extra Bjorken integration variables in the hadronic case.

Whether the method pays off depends on balancing savings against overhead, so a figure of merit is introduced that puts the statistical gain from fewer negative weights and the costs of training, inference and integration on one scale. A reduction in negative weights does not by itself imply a net gain, because training and per-event inference add cost, while downstream showering and detector simulation costs vary widely between applications. Under assumed theoretical and experimental scenario parameters, processes with high negative-weight fractions obtain large figures of merit, and normalizing flows generally give lower negative-weight fractions and larger figures of merit than decision trees.

The demonstrated scope covers several specific electron- and hadron-collision processes, with emphasis on final states containing quarks, which have complicated matching conditions and high negative-weight fractions, all in a fully realistic simulation environment without simplifying assumptions. The authors explicitly frame the work as a proof of concept, and the method is not implemented in a public code.

The authors note that improving the accuracy or complexity of the machine-learning model is not by itself enough to eliminate all residual negative weights: some come from Born-level differential cross sections that remain negative after integrating over radiation variables, and others from cancellations between separately treated FKS sectors or channels that the procedure cannot exploit. The numerical treatment of virtual corrections, the choice of learning variables and the handling of unphysical phase-space points add further limitations, and whether the candidate radiation domain is truly contained in the physical domain is not proven, so any residual bias must be assessed numerically.