⚛️ phenomenology

Optimal-Transport-Based Cell Resampling for Negative and Pathological Event Weights

This paper proposes using Optimal Transport-based metrics, such as the Energy Mover's Distance, to drive cell resampling algorithms that effectively mitigate negative and pathological event weights in LHC Monte Carlo simulations while reducing bias compared to existing techniques.

Original authors: Regan Doherty, Lauren Hay, Rishabh Jain, Matt LeBlanc, Julia Marrinan, Camille Mauceri, Jennifer Roloff

Published 2026-07-10

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Regan Doherty, Lauren Hay, Rishabh Jain, Matt LeBlanc, Julia Marrinan, Camille Mauceri, Jennifer Roloff

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are running a massive simulation of particle collisions, like a digital video game of the universe's most energetic smash-ups. To make these simulations accurate, scientists use complex math that sometimes creates "ghost" events. These are like negative numbers in a bank account: they cancel out real money, but if you have too many of them, your account balance becomes shaky and unreliable. In the world of the Large Hadron Collider (LHC), these "negative weight" events are a huge headache. They force scientists to generate way more data than they need, filling up hard drives and burning through computing power just to get a clear picture.

The problem is that these negative events are scattered throughout the simulation. To fix them, scientists have tried a method called "cell resampling." Think of it like a game of musical chairs, but instead of people, you have events. If a chair (an event) has a negative weight, you look at the people sitting in the chairs right next to it. If the group of neighbors has enough "positive" weight to cover the negative one, you shuffle the numbers around so everyone ends up with a positive balance, without changing the total sum.

But here's the tricky part: How do you decide who is "next to" whom? In the past, scientists used a simple ruler to measure distance, but this ruler was flawed. It would get confused if a particle split into two or if a tiny, invisible particle appeared, even though these changes shouldn't matter to the physics. It was like trying to measure the distance between two cities by counting the number of potholes on the road; a tiny pothole shouldn't change the distance between New York and Boston.

This paper introduces a new, smarter way to measure distance using a concept called "Optimal Transport," specifically a tool called the Energy Mover's Distance (EMD). Imagine you have a pile of sand (energy) in one spot and you want to move it to match a pile in another spot. The EMD calculates the minimum amount of "work" needed to move that sand. Crucially, this method is "infrared and collinear safe." In plain English, this means it doesn't care about tiny, soft particles or particles splitting in half. It looks at the big picture of the energy flow, making it perfect for measuring distance between complex particle events without getting tripped up by tiny details.

The researchers tested this new method on two types of simulated collisions: one where a Z boson is made with some jets (sprays of particles), and another where top quark pairs are created. They tried applying this "sand-moving" distance measure at three different stages of the simulation: right after the initial collision, after the particles start showering, and after they have fully formed into the particles we would actually see (hadronization).

The results showed that waiting until the very end—after the particles have fully formed—gave the best results. It was like waiting for the dust to settle before measuring the room; the final picture was the most stable and introduced the least amount of bias. They also tested different settings for how the distance is calculated. They found that a specific setting (called $\beta = 1$ ) worked the best, balancing accuracy with speed. If they tried to be too simple (ignoring the shape of the energy) or too extreme (only looking at total energy), the results got messy and introduced errors.

To prove their method worked, they didn't just look at one graph; they invented a new "scorecard" called the Cross-Section Mover's Distance ( $\Sigma$ MD). This scorecard measures the total difference between the original messy simulation and the new, cleaned-up version across the entire board, not just in one specific area. They found that their new method, using the Energy Mover's Distance on fully formed particles, kept this score very low, meaning the physics stayed true to the original simulation.

In these simulations, the new method successfully reduced the "negative weight" problem without distorting the results. For example, in the Z+jets sample, which started with about 37.6% negative weights, the method improved the statistical power of the data significantly. The researchers showed that by using this approach, they could achieve the same statistical certainty with far fewer events, saving massive amounts of computing time and storage space.

However, it is important to note that these findings are based on simulations. The paper demonstrates that this approach works well in the digital world of computer models for Z+jets and top quark pairs, but it hasn't been tested on real-world data from the LHC yet. The authors suggest that this could be a powerful tool for the future of high-energy physics, potentially helping experiments handle the massive data loads expected in the High-Luminosity LHC era, but for now, it remains a highly promising simulation-based solution.

Technical Summary: Optimal-Transport-Based Cell Resampling for Negative and Pathological Event Weights

Problem Statement
The pursuit of high-precision physics at the High-Luminosity Large Hadron Collider (HL-LHC) necessitates Monte Carlo (MC) event simulations at next-to-leading order (NLO) and beyond. However, these sophisticated calculations often produce events with negative or pathologically large weights due to infrared subtraction procedures (e.g., Catani-Seymour, FKS) and matching/merging schemes (e.g., MC@NLO, FxFx). Negative weights degrade the statistical significance of a sample, effectively reducing the sample size by a factor of $(1-2f_-)^2$ , where $f_-$ is the fraction of negative weights. This creates a severe computational bottleneck, as larger sample sizes are required to achieve target statistical uncertainties, exacerbating storage and processing demands.

Existing solutions involve "cell resampling," an algorithm that locally redistributes weights among nearby events in phase space to eliminate negative weights while preserving the total cross-section. However, previous implementations relied on modified Euclidean distance metrics that required clustering final-state particles into jets to ensure infrared and collinear (IRC) safety. This introduced a dependence on jet definition parameters (radius, algorithm) and prevented the application of resampling at intermediate stages of event generation (e.g., before hadronization).

Methodology
This paper proposes replacing the Euclidean distance metric with Optimal Transport (OT)-based metrics, specifically the Energy Mover's Distance (EMD) and the Spectral Energy Mover's Distance (sEMD). These metrics are IRC-safe by construction, allowing cell resampling to be applied directly to particle-level information at any stage of event generation (hard-scattering, parton shower, or hadronization) without intermediate jet clustering.

EMD: Defined as the minimum "work" required to rearrange the energy flow of one event into another in the pseudorapidity-azimuth plane. It utilizes a ground metric (Euclidean distance in $\eta$ - $\phi$ ) and an angular exponent $\beta$ to control sensitivity to angular scales.
sEMD: Operates on one-dimensional spectral functions encoding the energy-weighted pairwise angular structure of events. It retains IRC safety and offers computational advantages ( $O(N^2 \log N)$ ) while being invariant under detector isometries.
Cross-Section Mover's Distance ( $\Sigma$ MD): To quantify the global bias introduced by resampling, the authors utilize the $\Sigma$ MD. This metric treats the entire event sample as a distribution of cross-section weights and calculates the minimum transport cost to match a reweighted sample to the original. To handle negative weights in this framework, a constant offset is added to all event weights in both samples, leveraging the cancellation property of the generalized Wasserstein distance.

The study utilizes NLO QCD simulations of $Z$ +jets and $t\bar{t}$ production generated with MadGraph5_aMC@NLO and Pythia 8. Cell resampling is performed at three stages: Hard Scattering (HS), Parton Shower (PS), and Hadronization (HAD).

Key Results

Stage of Application: Resampling performed on fully hadronized events (HAD) consistently yields the lowest bias across kinematic distributions ( $H_T$ , jet multiplicity $N_{jets}$ , angular separations) and minimizes the $\Sigma$ MD compared to resampling at the HS or PS stages. While HS resampling allows for smaller cell radii, it introduces significant biases in jet multiplicity and $H_T$ distributions.
Metric Parameters: The angular exponent $\beta$ $β$ in the EMD significantly impacts performance.
- $\beta = 1$ (equivalent to the first Wasserstein distance) provides the optimal balance of accuracy and computational efficiency.
- $\beta = 0$ and $\beta \to \infty$ introduce substantial biases by failing to account for the spatial distribution of energy within events.
- Intermediate values ( $\beta = 0.5, 2$ ) perform comparably to $\beta=1$ , though $\beta=0.5$ is less computationally efficient due to square root operations.
Metric Comparison: The OT-based metrics (EMD and sEMD) outperform the traditional Euclidean object-based metric (Andersen et al.) used in prior literature. The Euclidean approach shows larger deviations in kinematic distributions, particularly at high resampling fractions ( $f_{rw} = 0.75$ ), and exhibits a significantly higher $\Sigma$ MD. The sEMD performs similarly to the EMD but shows slightly larger deviations in specific kinematic regions.
Generalizability: The trends observed in $Z$ +jets events hold for $t\bar{t}$ events. While $t\bar{t}$ events require larger cell radii due to higher particle multiplicity, the HAD-based resampling with $\beta=1$ EMD remains the most robust configuration.

Significance and Claims
The paper establishes OT-based cell resampling as a practical "afterburner" for mitigating negative weights in collider MC simulations. The primary significance lies in the IRC safety of the proposed metrics, which removes the dependency on jet clustering algorithms and allows resampling to be applied directly to particles at any simulation stage. This flexibility enables the use of more physical event representations (e.g., hadron-level) for the resampling procedure, resulting in lower bias compared to earlier methods.

The authors claim that this approach reduces the observed bias relative to other cell resampling techniques presented in the literature, particularly for NLO samples. Furthermore, the introduction of the $\Sigma$ MD provides a novel, holistic, and unbinned figure of merit for quantifying the bias of any full-phase-space reweighting, applicable beyond just cell resampling. The study concludes that general-purpose algorithms like this, which require no sample-specific training, can complement machine learning approaches that learn corrections tailored to specific MC samples.

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