Improved muon energy estimation using a detailed model of multiple Coulomb scattering in the MicroBooNE LArTPC

This paper presents an improved technique for estimating muon energy in the MicroBooNE LArTPC by modeling multiple Coulomb scattering with enhanced detector non-idealizations, achieving significantly better resolution and reduced bias compared to previous methods while demonstrating strong agreement between data and simulations.

The Gist

Imagine you are trying to guess the speed of a car driving through a dense, foggy forest. You can't see the car's speedometer, and the car is moving so fast it leaves the forest before you can time how long it takes to cross. However, you can see the path the car leaves behind.

If the car is driving slowly, it swerves a lot to avoid the trees. If it's driving fast, it barely swerves at all. By measuring how much the car "wobbles" or scatters as it hits the trees, you can estimate its speed.

This is exactly what the MicroBooNE Collaboration did, but instead of a car and trees, they were tracking muons (tiny, ghost-like particles) moving through a giant tank of liquid argon (a super-cold, invisible fluid).

Here is a breakdown of their new method, using simple analogies:

The Problem: The "Exit" Strategy

In their giant detector, many muons zip through so fast that they don't stop inside; they fly right out the other side.

• Old Method: Previous ways to guess the muon's energy were like trying to guess a runner's speed by measuring how far they ran. If the runner leaves the track before finishing, you can't measure the distance, so you can't guess the speed.
• The New Idea: Instead of measuring distance, they measure the wobble. As the muon flies through the liquid argon, it bumps into atoms, causing it to scatter slightly. The faster the muon, the straighter the path. The slower it is, the more it zig-zags.

The Old "Wobble" Calculator Was Flawed

The team had a previous tool to measure this wobble, but it was like using a blurry, low-resolution camera. It made two main mistakes:

1. It ignored the "weird" wobbles: Sometimes a muon hits a stray electron or gets hit by a "delta ray" (a tiny particle knocked loose), causing a sudden, huge jump in its path. The old model assumed all wobbles were smooth and predictable (like a bell curve). When a huge, unexpected jump happened, the old model got confused and guessed the muon was much slower than it actually was.
2. It treated all directions the same: The detector is built with wires in specific directions. The "blur" or error in measuring the muon's position is different depending on which way the muon is traveling relative to the wires. The old model used one single "blur" number for everything, which wasn't accurate.

The New "High-Definition" Model

The team built a new, smarter calculator with four key upgrades:

1. The "Double-Gaussian" Lens
Instead of assuming the muon's path is a perfect, smooth curve, they realized the path is usually smooth but occasionally has "spikes."

• Analogy: Imagine a crowd of people walking down a hallway. Most walk in a straight line (the main group). But every now and then, someone bumps into a doorframe and stumbles wildly (the tail).
• The Fix: Their new model uses a "double-Gaussian" function. It has one curve for the smooth walkers and a second, wider curve for the wild stumblers. This allows them to account for the weird jumps without getting confused and guessing the wrong speed.

2. Separating the "Drift" from the "Wires"
The detector has a "drift" direction (where electrons float) and "wire" directions (where they are caught). The measurement error is different in each direction.

• Analogy: Imagine trying to measure the path of a ball rolling on a grid. If you measure along the grid lines, your ruler is very precise. If you measure diagonally across the grid, your ruler is a bit fuzzier.
• The Fix: They split the measurement into two separate angles: one that is very sensitive to the "drift" fuzziness and one that is sensitive to the "wire" fuzziness. They treat them as two different problems with two different solutions, rather than mixing them into one messy average.

3. The "Track Orientation" Tuning
The quality of the measurement changes depending on the angle of the muon's path.

• Analogy: Think of taking a photo of a moving car. If the car drives straight toward the camera, it's easy to track. If it drives straight across the camera's view, it's harder to track because of motion blur.
• The Fix: They created five different "settings" for their calculator based on how the muon is angled relative to the detector. They tune the math specifically for each angle, ensuring the "blur" is calculated correctly no matter which way the muon is flying.

4. Learning from the "Fastest" Runners
To figure out exactly how "blurry" their camera is (the detector resolution), they looked at the fastest muons (those with the most energy).

• Analogy: If you want to know how shaky your hand is when drawing, look at someone drawing a straight line while holding a heavy weight. If the line is still straight, your hand is steady. If it's wobbly, your hand is shaky.
• The Fix: High-energy muons barely wobble at all due to physics. So, any wobble they do see is purely due to the detector's imperfections. They used these "perfect" tracks to measure the detector's exact error rate, rather than guessing.

The Results: Sharper, Faster, and Fairer

When they tested this new method against their simulations and real data:

• Less Bias: The old method often guessed the muon was 20% slower than it really was. The new method is accurate within 1% to 2%.
• Better Resolution: The "fuzziness" of the guess dropped significantly. For muons that stay inside the tank, the guess is now accurate to within 4.3%. For muons that fly out, it's accurate to within 7% to 17%.
• Real-World Check: When they compared their new calculator's predictions against actual data from the detector, the numbers matched perfectly. The "blur" in their model explained the real-world data exactly as expected.

Why It Matters

This new tool allows scientists to accurately measure the energy of muons that fly out of the detector. Previously, these "exit" muons were a blind spot. Now, scientists can study them with high precision, opening up new ways to understand how neutrinos interact with matter. It's like upgrading from a blurry security camera to a high-definition one, allowing them to see the details of the universe that were previously hidden in the fog.

Technical Summary

Technical Summary: Improved Muon Energy Estimation using Multiple Coulomb Scattering in MicroBooNE

Problem Statement
In Liquid Argon Time Projection Chamber (LArTPC) detectors like MicroBooNE, reconstructing the energy of muons that exit the detector volume (partially contained, or PC events) presents a significant challenge. Traditional energy estimators relying on calorimetric energy or residual range require the full muon track to stop within the detector. Since over half of the muons from $\nu_\mu$ charged current interactions in the Booster Neutrino Beam exit the MicroBooNE detector, these standard methods fail for a large fraction of events, particularly at higher energies. While Multiple Coulomb Scattering (MCS) offers a pathway to estimate energy without requiring a full track stop, previous implementations at MicroBooNE [1] suffered from limited accuracy due to simplified modeling of detector non-idealizations, resulting in large biases (up to 20%) and poor resolution (twice that of the new method).

Methodology
This work presents a refined MCS-based muon energy estimator ($E_\mu^{MCS}$) that addresses the limitations of previous models through four key innovations in the modeling of scattering angle distributions and detector resolution:

1. Double-Gaussian Probability Density Function (PDF): Instead of a single Gaussian distribution derived from the Highland formula, the authors implement a double-Gaussian function. This model includes a "primary" Gaussian (describing the core $\sim$90% of the distribution) and a "secondary" Gaussian with a wider width to accurately model the non-Gaussian tails. This is critical for Maximum Likelihood Estimation (MLE), as a single Gaussian significantly under-predicts the likelihood of large-angle tails (caused by reconstruction flaws like delta rays), leading to biased energy estimates.
2. Decomposition of Scattering Angles: The authors separate the planar scattering angles into two components: $\theta_{xz}$ (sensitive to the drift direction) and $\theta_{yz}$ (sensitive to the wire-plane directions). This separation accounts for the asymmetric reconstruction resolutions inherent to the LArTPC geometry, where ionization charge diffusion differs between the drift and wire directions.
3. Track-Oriented Tuning: The model introduces a dependence on the muon track's velocity component along the drift direction ($|v_x|$). Tracks are binned by $|v_x|$ to account for variations in reconstruction quality (e.g., isochronous vs. prolonged topologies), allowing for independent tuning of resolution parameters for different track orientations.
4. Direct Resolution Extraction: Rather than tuning resolution parameters solely to minimize energy bias, the authors extract detector angular resolutions directly from high-energy muon simulations where the MCS contribution is minimized. This provides a more direct measurement of the detector's intrinsic resolution.

The energy estimation process utilizes an iterative tuning procedure to determine the parameters of the double-Gaussian PDF ($\sigma_1$, $\sigma_2$, and area fraction $A$) as functions of muon energy. The final energy estimate is obtained by maximizing the total likelihood of the observed scattering angles given the tuned PDFs.

Key Results
Using independent simulation samples for validation, the new estimator demonstrates significant performance improvements over the previous MicroBooNE MCS work [1]:

• Fully Contained (FC) Events: For muons stopping within the detector, the estimated bias is within 1% across the energy range of 0.1 GeV to 2 GeV. The energy resolution improves from 4.3% to 10% as energy increases from 0.1 GeV to 2 GeV.
• Partially Contained (PC) Events: For exiting muons with at least 1 meter of reconstructed track and energy below 2 GeV, the estimated bias is less than 2%, and the resolution varies between 7% and 17%.
• Comparison to Previous Work: These results represent a substantial improvement over the prior estimator, which exhibited a resolution twice as large and a bias of 20% in the same energy region.
• Data-Model Agreement: Validation against real MicroBooNE data (6.4 $\times$ 10$^{20}$ POT) shows excellent agreement. Goodness-of-fit tests ($\chi^2$/ndf) for FC and PC distributions, as well as comparisons of fractional differences between $E_\mu^{MCS}$ and residual range-based estimates ($E_\mu^{RR}$), yield $p$-values well above 0.9. This indicates that the simulation uncertainties fully cover the observed data-model differences.

Significance and Claims
The paper claims that this work successfully demonstrates a refined model of MCS in LArTPC detectors that yields reduced bias, improved resolution, and robust data-model agreement. By enabling reliable energy reconstruction for exiting muons, this method extends the accessible phase space for neutrino interaction analyses to higher energies and wider angles. The authors posit that this approach is competitive with other reconstruction methods (such as those using residual neural networks) and serves as a valuable complementary estimator for validating high-energy muon tracks. The techniques developed are presented as potentially applicable to other LArTPC experiments, including those in the short-baseline and deep-underground neutrino programs.