The High W Challenge: Robust Neutrino Energy Estimators for LArTPCs

This paper introduces and evaluates a new W$^2$-based neutrino energy estimator for liquid-argon time-projection chambers, demonstrating that while it offers superior robustness against modeling uncertainties and minimal bias in the transition region between scattering regimes, its slightly worse energy resolution under ideal conditions suggests a complementary role alongside more exclusive methods for future oscillation analyses.

The Gist

Imagine you are a detective trying to solve a mystery: What was the speed of a car that crashed into a wall?

You can't see the car anymore. All you have left are the scattered pieces of the bumper, the dented wall, and a single skid mark. Your job is to look at these "final state" clues and guess how fast the car was going before it hit.

This is exactly the challenge physicists face with neutrinos. Neutrinos are ghostly, tiny particles that zip through the universe. When they hit an atom inside a giant tank of liquid argon (a detector), they create a messy explosion of other particles. Scientists need to know the original energy of the neutrino to understand how it changes flavor (oscillates) as it travels, but they can only see the debris left behind.

This paper, titled "The High W Challenge," is about testing different "detective tools" (mathematical formulas) to figure out that original energy as accurately as possible.

The Setting: The Liquid Argon Time Projection Chamber (LArTPC)

Think of the detector as a giant, 3D camera made of liquid argon. When a neutrino hits an argon atom, it leaves a trail of ionization (like a glowing trail of smoke). The scientists can see every charged particle that comes out of the crash.

The problem is that neutrinos hit atoms in many different ways:

• The "Clean" Hit: The neutrino hits a single proton and bounces off (like a billiard ball).
• The "Messy" Hit: The neutrino hits the nucleus, shatters it, and creates a shower of pions and other particles (like a car crash that sends parts flying everywhere).

Most previous methods tried to guess the energy by assuming the hit was always "clean." But in the real world, especially with the high-energy beams used in experiments like DUNE, the hits are often messy.

The Five "Detective Tools" (Estimators)

The authors tested five different ways to calculate the energy:

1. The "Old School" Guess (CCQE-like): This tool assumes every crash was a clean, simple hit. It looks at the main particle (the muon) and guesses the energy.
   • The Flaw: If the crash was actually messy (creating pions), this tool gets the answer wrong because it ignores the extra debris.
2. The "Proton Counter" (Proton-based): This tool adds up the energy of the muon and any protons it can see.
   • The Flaw: It only works if no pions (other particles) are created. If pions are there, it misses a huge chunk of the energy.
3. The "Total Bill" (Calorimetric): This tool tries to weigh everything that came out of the crash, regardless of what it is. It's like weighing all the debris in the room to guess the car's speed.
   • The Flaw: It's great at catching everything, but if your scale isn't perfect (detector resolution), the total weight is a bit fuzzy.
4. The "Specialist" (Sobczyk-Furmanski): This is a very strict tool that only looks at crashes with exactly one proton and one muon.
   • The Flaw: It's very accurate when it works, but it throws away 90% of the data because most crashes aren't that simple.
5. The "New Hero" (The $W^2$ Estimator): This is the star of the paper. It uses a clever math trick involving the Invariant Mass (a fancy way of saying "the total mass of the debris system").
   • How it works: Instead of assuming the crash was simple, it measures the "heaviness" of the debris cloud ($W$). If the cloud is heavy, it knows the neutrino hit hard. It adjusts its calculation based on how many protons and pions it sees. It's like a detective who looks at the pattern of the debris to realize, "Ah, this wasn't a simple hit; the car was carrying a heavy load, so it must have been going faster."

The Results: Who Wins?

The authors ran thousands of simulations to see which tool made the fewest mistakes.

• The "Old School" and "Proton Counter" tools were often biased. They consistently guessed the energy was lower than it actually was because they ignored the messy parts of the crash.
• The "Specialist" tool was very precise but threw away too much data to be useful for big experiments.
• The "Total Bill" (Calorimetric) was good, but it was very sensitive to measurement errors. If the detector was slightly off, the whole calculation wobbled.
• The "New Hero" ($W^2$ Estimator) was the winner.
  • It was the most honest: It had the smallest "bias," meaning its average guess was closest to the true energy.
  • It was the most resilient: Even when the physics models were slightly wrong (simulating different types of nuclear interactions), the $W^2$ tool didn't get confused. It handled the "messy" crashes better than anyone else.
  • The Trade-off: The only downside was that if the detector was perfect, the "Total Bill" method was slightly sharper. But in the real world, where detectors aren't perfect and physics models are imperfect, the $W^2$ tool is much more robust.

Why Does This Matter?

Neutrino experiments are trying to solve the biggest mysteries in physics: Why is there more matter than antimatter in the universe? To do this, they need to measure tiny differences in how neutrinos change.

If your "detective tool" for measuring energy is biased (always guessing too low or too high), you will draw the wrong conclusions about the universe.

The paper concludes that for future giant detectors (like DUNE), we shouldn't just rely on one method. Instead, we should use the $W^2$ estimator as our primary tool because it is the most reliable "Swiss Army Knife" for the messy, complex reality of neutrino collisions. It allows scientists to use more data (including the messy crashes) without sacrificing accuracy, giving them a clearer window into the secrets of the universe.

In short: The authors found a new way to calculate neutrino energy that is less likely to get fooled by messy particle crashes, making our future measurements of the universe much more reliable.

Technical Summary

1. Problem Statement

Accurate reconstruction of neutrino energy ($E_{\nu}$) is the cornerstone of precision neutrino oscillation measurements (e.g., determining $\delta_{CP}$ and mass ordering). Current and future experiments like DUNE utilize broadband neutrino beams that span the transition region between Shallow Inelastic Scattering (SIS) and Deep Inelastic Scattering (DIS).

In this energy regime (0.5–6 GeV), neutrino-nucleus interactions produce complex final states involving multiple protons, pions, and nuclear effects (Final State Interactions, or FSI). Traditional energy estimators often fail because:

• Exclusive methods (e.g., assuming Charged-Current Quasi-Elastic, CCQE, kinematics) suffer from large biases when applied to non-CCQE events (resonances, multi-pion production).
• Calorimetric methods are inclusive but suffer from poor energy resolution due to detector limitations and missing energy (neutrons).
• Model dependence: Different event generators (GENIE, NuWro, NEUT, GiBUU) predict different cross-sections and final state topologies, leading to systematic uncertainties in energy reconstruction.

The paper addresses the need for an estimator that balances inclusivity (retaining high statistics) with robustness (minimizing bias from model mismatches and FSI).

2. Methodology

The authors compare five distinct neutrino energy estimators using a controlled simulation environment based on the DUNE far detector flux and idealized Liquid Argon Time-Projection Chamber (LArTPC) reconstruction.

Simulation Setup:

• Generators: Four event generators were used: GENIE (v3.6.2), NuWro (v25.03.1), NEUT (v5.8.0), and GiBUU (2025).
• Reconstruction Assumptions: Idealized particle identification (100% efficiency above thresholds) and perfect kinematic reconstruction. Neutrons are assumed undetected.
• Energy Range: 0.5–6 GeV.

The Five Estimators Compared:

1. CCQE-like Estimator: Assumes all events are CCQE. Uses lepton kinematics only. Tested in three forms: inclusive, restricted to 1-proton/0-pion events, and with a $\Delta$-mass correction for pion events.
2. $W^2$-based Estimator (New): Calculates the visible hadronic invariant mass ($W_{vis}^2$) from all visible final-state particles (protons and pions). It modifies the CCQE formula to account for multi-nucleon knockout and meson production. Requires at least one visible proton.
3. Proton-Based Estimator: Sum of lepton energy and kinetic energy of all visible protons. Restricted to exclusive 1-proton/0-pion topologies.
4. Calorimetric Method: Sum of total lepton energy, proton kinetic energy, and total pion energy. Mimics a pure calorimeter approach without particle identification.
5. Sobczyk–Furmanski (SF) Estimator: Uses transverse and longitudinal momentum balance to infer the struck nucleon's motion. Restricted to exclusive 1-proton/0-pion topologies.

Analysis Metrics:

• Bias and Variance: Calculated as a function of true neutrino energy ($E_{true}$).
• Secondary Variable Dependence: Sensitivity to mismodeling of visible hadronic invariant mass ($W_{vis}$), missing hadronic energy, and lepton scattering angle ($\theta$).
• Oscillation Impact: "Closure tests" were performed to see how mismodeling (e.g., turning FSI on/off) affects the extraction of $\delta_{CP}$ and $\Delta m^2_{23}$.

3. Key Contributions

• Introduction of the $W^2$ Estimator: A novel approach that leverages the hadronic invariant mass to extend the validity of kinematic reconstruction to inclusive channels (events with pions and multiple protons), bridging the gap between exclusive kinematics and calorimetry.
• Comprehensive Benchmarking: A systematic comparison of five estimators across four different event generators, isolating the effects of nuclear modeling, FSI, and detector resolution.
• Quantification of Inclusivity vs. Resolution: The paper demonstrates the trade-off between statistical power (inclusivity) and systematic bias, showing that the $W^2$ estimator offers the best compromise for broadband beams.
• Secondary Variable Analysis: Detailed mapping of how specific kinematic mismodeling (e.g., resonance peaks in $W_{vis}$) propagates to oscillation parameter biases.

4. Key Results

A. Bias and Variance Performance

• Lowest Bias: The $W^2$-based estimator exhibits the smallest overall bias as a function of true energy. It is particularly stable against mismodeling of lepton angles, missing energy, and FSI.
• Variance: The SF estimator offers the lowest variance (best resolution) but only for a small subset of events (exclusive 1p0$\pi$), leading to a loss of statistical power.
• Resolution Effects: When detector resolution smearing is applied, the SF method remains the most precise, followed closely by the proton-based method. The $W^2$ method shows slightly worse resolution than SF but significantly better bias stability.

B. Robustness to Final State Interactions (FSI)

• The CCQE-like and $W^2$ estimators are the most robust against FSI mismodeling.
• The Proton-based and Calorimetric methods show significant bias shifts when FSI is toggled, as they rely heavily on the visible hadronic energy which is easily altered by nucleon rescattering and absorption.
• The SF method is designed to mitigate FSI via transverse kinematics but suffers from low statistics in inclusive samples.

C. Sensitivity to Secondary Variables

• Hadronic Invariant Mass ($W_{vis}$): The CCQE-like estimator shows periodic bias spikes corresponding to resonance peaks (integer multiples of nucleon mass). The $W^2$ estimator remains flat and stable across these regions.
• Lepton Scattering Angle: The CCQE-like estimator degrades rapidly at large angles (high inelasticity). The $W^2$ estimator maintains a flat bias, making it superior for broadband beams where large-angle scattering is common.

D. Impact on Oscillation Measurements

• Exclusion Power: In $\nu_\mu$ disappearance and $\nu_e$ appearance tests, the Calorimetric and $W^2$ estimators provided the best sensitivity due to their inclusivity. The exclusive SF method suffered from reduced sensitivity due to low event rates.
• Closure Tests: When fitting spectra generated with FSI turned off against data with FSI on, the $W^2$ estimator produced the smallest bias in the extracted $\Delta m^2_{23}$ and $\delta_{CP}$ values, demonstrating superior resilience to nuclear modeling uncertainties.

5. Significance and Conclusion

This work establishes the $W^2$-based estimator as a critical tool for future LArTPC experiments like DUNE.

• Optimal Trade-off: It successfully combines the statistical advantage of inclusive analysis (retaining events with pions) with the kinematic rigor of exclusive methods.
• Model Independence: It is less sensitive to the specific nuclear models and FSI implementations used in different event generators compared to traditional CCQE or pure calorimetric approaches.
• Future Strategy: The authors conclude that future oscillation analyses should not rely on a single estimator. Instead, a combined approach is recommended: using the $W^2$ estimator for the bulk of the inclusive sample to minimize bias, while potentially using exclusive methods (like SF) for high-resolution subsets, provided the statistical loss is accounted for.

The paper provides a roadmap for maximizing the physics reach of broadband neutrino experiments by moving beyond the "CCQE-only" paradigm toward robust, invariant-mass-based reconstruction.