Optimal Transport for $e/π^0$ Particle Classification in LArTPC Neutrino Experiments

This paper demonstrates that Optimal Transport methods achieve state-of-the-art performance in distinguishing electrons from neutral pions in Liquid Argon Time Projection Chamber neutrino experiments using MicroBooNE simulated data, highlighting the technique's broader potential for LArTPC-based reconstruction.

The Gist

The Big Picture: Sorting the "Messy" from the "Clean"

Imagine you are a detective trying to solve a crime in a very large, foggy room. You have a camera that takes incredibly high-resolution photos of everything that happens.

In the world of particle physics, specifically in experiments using Liquid Argon Time Projection Chambers (LArTPCs), scientists are looking for a very specific type of "crime": a neutrino hitting an atom and creating an electron. This is a golden signal that helps us understand the universe.

However, there is a massive problem: the room is full of "fake" signals. Specifically, neutral pions ($\pi^0$) often decay into two photons, which look almost exactly like an electron in the camera. It's like trying to find a single, perfect red apple in a pile of red balls that look identical.

For years, scientists have used standard "reconstruction" tools to try to separate the apples from the balls. But these tools are often clumsy. They try to trace every single line and curve, and if they miss a tiny detail (like a faint spark), they get confused and throw away the good apples along with the bad balls.

The Solution: This paper introduces a new mathematical tool called Optimal Transport (OT). Instead of trying to trace every single line, OT looks at the "shape" and "weight" of the whole picture to decide what it is.

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The Analogy: Moving Dirt vs. Tracing Lines

To understand Optimal Transport, imagine you have two piles of dirt on a field.

• Pile A is a perfect circle.
• Pile B is a long, thin snake.

The Old Way (Traditional Reconstruction):
The old method tries to draw a line around Pile A and a line around Pile B. It measures the length of the lines and the angles. If the lines are messy or broken, the computer gets confused and can't tell them apart.

The New Way (Optimal Transport):
Optimal Transport asks a different question: "How much work would it take to move the dirt from Pile A to look exactly like Pile B?"

• If you have two circles, it takes very little work to move the dirt from one to the other. The "distance" is small.
• If you have a circle and a snake, you have to move a lot of dirt a long way to reshape the circle into a snake. The "distance" is huge.

In this paper, the "dirt" is the energy deposited by particles in the detector.

• An electron leaves a single, fuzzy trail (one pile of dirt).
• A pion leaves two separate trails (two piles of dirt).

The OT method calculates the "work" required to turn a pion's two trails into an electron's single trail. If the work is high, it's a pion. If the work is low, it's an electron.

Why This is a Game-Changer

The authors tested this method using data from the MicroBooNE experiment (a real neutrino detector). Here is what they found:

1. It's Smarter: The OT method is much better at spotting the difference between an electron and a pion than the current "gold standard" tools (like a software called Pandora).
2. It Doesn't Get Confused by Angles: Traditional tools get confused if a particle hits the detector at a weird angle. OT is like a sculptor who can rotate the clay in their hands to see the shape better, regardless of which way it's facing.
3. It Works with "Simple" Math: The authors combined OT with simple, easy-to-understand machine learning (like Support Vector Machines). They didn't need a "black box" AI that no one understands; they used a method where they can actually see why the decision was made.

The "Magic" of the Method

Think of the detector image as a 3D cloud of glowing dust.

• Electrons look like a single, fluffy cloud.
• Pions look like two clouds that might be touching or slightly separated.

The old tools try to count the grains of dust and trace their paths. If a few grains are missing (because of noise or detector limits), the count is wrong, and the identification fails.

The Optimal Transport method ignores the missing grains. It looks at the overall shape and weight of the cloud. Even if the cloud is a bit fuzzy, the math can tell: "This shape is too heavy and spread out to be a single electron; it must be two things stuck together."

The Results

The paper shows that using this "Moving Dirt" math:

• They can correctly identify electrons 80% to 90% of the time.
• The old methods only get it right about 40% of the time in difficult situations.
• This is huge because it means scientists can stop throwing away good data. They can be more confident in their discoveries about neutrinos and potentially find new, unknown particles (Dark Matter candidates) that were previously hidden by the "noise" of pions.

In a Nutshell

This paper is about teaching computers to look at the big picture rather than getting lost in the tiny details. By using a mathematical concept called Optimal Transport (which measures the "effort" to reshape one image into another), the researchers created a much better filter for sorting neutrino signals. It's like upgrading from a magnifying glass that only sees scratches to a pair of eyes that sees the whole face.

Technical Summary

1. Problem Statement

In Liquid Argon Time Projection Chamber (LArTPC) neutrino experiments (such as MicroBooNE, SBN, and DUNE), a critical challenge is distinguishing between electrons ($e^-$) and neutral pions ($\pi^0$).

• Physics Context: Electrons are the primary signal for $\nu_\mu \to \nu_e$ oscillations and searches for Beyond Standard Model (BSM) physics. However, $\pi^0$ mesons produced in neutrino interactions decay into two photons ($\pi^0 \to \gamma\gamma$), creating electromagnetic (EM) showers that mimic single electrons.
• Current Limitations: Traditional reconstruction algorithms (e.g., Pandora, Wire-Cell) and standard Machine Learning (ML) approaches often struggle with this classification. They rely on reconstructing individual particle tracks and showers. This process is inefficient when:
  • The two photons from a $\pi^0$ are collinear (small opening angle).
  • One photon is very low energy (sub-leading photon).
  • One photon escapes the detector volume.
  • Reconstruction inefficiencies lead to missing shower components, causing $\pi^0$ events to be misidentified as single-electron events.
• Goal: Develop a method to classify $e$ vs. $\pi^0$ events that bypasses complex low-level reconstruction, leveraging the high-resolution calorimetric data of LArTPCs directly to improve background rejection and signal efficiency.

2. Methodology

The authors propose using Optimal Transport (OT), a mathematical framework that calculates the minimal "work" required to transform one probability distribution into another.

A. Data Representation

• Input: Instead of reconstructing tracks, the energy deposits in the detector are treated as discrete probability distributions in 3D space.
• Normalization: Events are normalized to have the same total charge (balanced OT) to focus on topological differences (shape) rather than total energy magnitude.
• Binning: To avoid classification bias based on particle energy, events are separated into 9 energy bins (ranging from 0.05 GeV to 0.9 GeV).

B. Pre-processing

To ensure the OT distance reflects physical differences rather than arbitrary detector orientation:

1. Event Filtering: Events with showers extending outside the detector volume are removed to prevent "missing shower" artifacts from skewing results.
2. 3D Reconstruction: Hits are reconstructed into 3D "spacepoints" using the SpacePointSolver algorithm.
3. Down-sampling: Spacepoints within 1 cm are merged to reduce computational cost ($O(n^3)$ scaling) while preserving shower topology.
4. Spatial Alignment:
   • Principal Axis Alignment: Weighted Principal Component Analysis (WPCA) is applied to the largest cluster of energy deposits to align the primary axis of the event.
   • Rotation: Events are rotated to align their principal axes and centers of mass, removing dependencies on the particle's drift direction.
   • Note: The authors also tested 2D projections (analogous to LHC jet images) but found 3D reconstruction superior when combined with ML.

C. Classification Strategies

The paper evaluates three distinct strategies using OT distances:

1. Minimum OT (min(OT)): An unsupervised approach where a test event is compared to a reference set of known electron events. The score is the minimum OT distance to any reference electron. If the distance is small, it is classified as an electron; if large, a $\pi^0$.
2. OT + k-Nearest Neighbors (kNN): A supervised method where the full set of pairwise OT distances between training events forms a feature space. The algorithm classifies test events based on the majority vote of their $k$ nearest neighbors in this metric space.
3. OT + Support Vector Machine (SVM): A supervised method using OT distances as input features. An SVM finds a hyperplane to separate $e$ and $\pi^0$ distributions in the OT distance space. The authors used a Gaussian RBF kernel.

3. Key Contributions

• First Application to LArTPC: This is the first successful implementation of Optimal Transport for particle classification in Liquid Argon detectors, extending its previous success from High Energy Physics (LHC) collider data.
• Topology-Based Classification: The method bypasses the need to explicitly reconstruct individual photon showers. It treats the entire event as a single topological object, making it robust against cases where one photon is missing or collinear.
• Interpretability: Unlike "black box" deep learning models (e.g., CNNs), OT distances provide a physically interpretable metric (the "work" to transform shapes), and when coupled with SVM/kNN, the classification logic remains transparent.
• Benchmarking: The study provides a rigorous benchmark against the state-of-the-art Pandora reconstruction framework using the publicly available MicroBooNE Open Dataset.

4. Results

The performance was evaluated using Area Under the Curve (AUC), overall accuracy, and electron efficiency at 80% $\pi^0$ rejection.

• Superior Performance: All OT-based methods significantly outperformed the traditional Pandora reconstruction.
  • Pandora: Achieved $\sim$40% efficiency for identifying $\pi^0$s (finding $\ge 2$ showers) and $\sim$40% efficiency for electrons.
  • OT + SVM (3D): Achieved the best performance, with AUC values ranging from 0.82 to 0.94 and electron efficiencies of 74%–87% at 80% $\pi^0$ rejection, depending on the energy bin.
• Robustness in Challenging Topologies:
  • Small Opening Angles: For $\pi^0$ events with very small opening angles (where photons merge), Pandora's efficiency dropped to $\sim$10%. The OT+SVM method maintained high efficiency ($\sim$80%), successfully distinguishing the subtle topological differences.
  • Low Momentum Ratio: For events with a very low-energy sub-leading photon, OT methods improved classification efficiency from $\sim$15% (Pandora) to $\sim$60%.
• 3D vs. 2D: While 2D projections performed well with min(OT), 3D reconstruction combined with SVM yielded the highest overall performance, proving that full 3D calorimetric information is crucial for LArTPC classification.
• Computational Cost: Standard OT calculation scales as $O(n^3)$. While currently slower than simple cuts, the authors note that linearized OT (LOT) and approximate methods (like Sinkhorn) can reduce this to near-linear scaling, making real-time deployment feasible.

5. Significance and Future Outlook

• Physics Impact: Improved $e/\pi^0$ separation directly enhances the sensitivity of neutrino oscillation measurements and BSM searches (e.g., dark sector searches) by reducing background contamination without sacrificing signal efficiency.
• Methodological Shift: The success of OT suggests a paradigm shift in LArTPC analysis: moving away from rigid, step-by-step reconstruction pipelines toward holistic, distribution-based classification that leverages the detector's full granular resolution.
• Future Work:
  • Integration of OT into the full reconstruction pipelines of DUNE and SBN.
  • Application to other tasks, such as $e/\gamma$ separation and anomaly detection.
  • Optimization of computational costs using Linearized OT (LOT) to enable deployment in high-throughput environments.

In conclusion, the paper demonstrates that Optimal Transport provides a mathematically rigorous, interpretable, and highly effective framework for solving one of the most persistent challenges in LArTPC neutrino physics, outperforming current state-of-the-art reconstruction tools.