Analytical and Machine Learning Methods for Model Discernment at CE$\nu$NS Experiments

This paper demonstrates that leveraging multidimensional correlations in baseline, recoil energy, and timing through both conventional likelihood analyses and machine learning techniques enables robust discrimination between sterile neutrino and non-standard interaction models in CE$\nu$NS experiments, even when total event rates are obscured by systematic uncertainties.

The Gist

Imagine you are a detective trying to solve a mystery at a busy train station. You see a crowd of people (neutrinos) arriving, but you can't see their faces or read their tickets. You only know how many people arrived, how heavy their luggage was (recoil energy), and what time they stepped off the train (timing).

For years, scientists have been looking at these "neutrino trains" to see if something strange is happening. Usually, they just count the total number of people. But here's the problem: two different criminals (new physics theories) could send the exact same number of people, making it impossible to tell who did it just by counting.

This paper, written by a team of physicists and statisticians, asks a new question: If we can't just count the people, can we look at how they are arranged to solve the mystery?

Here is the breakdown of their investigation using simple analogies:

1. The Setting: The "Stopped-Pion" Train Station

The scientists are studying a specific type of experiment where protons hit a target, creating pions (like unstable particles) that stop and decay. This creates a very specific "train schedule":

• The Express Train (Prompt): A burst of neutrinos arrives immediately (like a fast train).
• The Local Train (Delayed): A slower stream of different neutrinos arrives a split second later (like a local train stopping at every station).

Because these two "trains" arrive at different times, scientists can separate them. But the real magic happens when you look at three things together:

1. Where they came from (Baseline): How far the detector is from the source.
2. How hard they hit (Recoil Energy): How much energy they transferred when they bumped into the detector.
3. When they arrived (Timing): The split-second difference between the "Express" and "Local" trains.

2. The Suspects: Two Different "Criminals"

The scientists want to distinguish between two possible explanations for strange signals they might see:

• Suspect A: The "Ghost" (Sterile Neutrinos). Imagine a ghost that steals people from the crowd. If this is the culprit, the total number of people drops, but the pattern of who is missing changes in a wavy, rhythmic way depending on the distance and energy. It's like a ghost that only steals people wearing red hats at specific times.
• Suspect B: The "Strongman" (Non-Standard Interactions). Imagine a strongman who makes the people bump into the walls harder or softer. This changes the strength of the interaction. It's like a strongman who makes everyone's luggage heavier or lighter, changing the overall weight of the crowd without necessarily stealing anyone.

The Problem: If you just count the total number of people, both suspects might look identical. The Ghost might steal 10% of the crowd, and the Strongman might make the remaining 90% look like 10% less. It's a "degeneracy"—they look the same from a distance.

3. The Old Detective Method: The Likelihood Analysis

First, the team used traditional math (Likelihood-based analysis). They looked at the data and asked, "Does the wavy pattern of missing people fit the Ghost theory better than the Strongman theory?"

The Result: They found that if you look at the shape of the data (the wavy patterns across distance and energy), you can tell them apart. But if you just look at the total count (like putting everyone in one big bucket), you can't tell the difference. The "shape" is the key.

4. The New Detective Tool: The Machine Learning "AI Eye"

Then, the team tried something cooler. They built a Convolutional Neural Network (CNN). Think of this as a super-smart AI camera that looks at the data not as numbers, but as a 3D image.

• The X-axis is distance.
• The Y-axis is energy.
• The Z-axis is time.

They trained this AI to look at these "images" and guess which suspect is responsible.

The Big Twist: To make the test harder, they removed the total count from the AI's input. They told the AI, "Don't tell me how many people there are. Just look at the pattern."

The Result: Even without knowing the total number of people, the AI could still distinguish between the Ghost and the Strongman with high accuracy! This proves that the "fingerprint" of the new physics is hidden in the shape and rhythm of the data, not just the volume.

5. The Final Challenge: Finding the Exact Address

Finally, they asked the AI an even harder question: "If it is the Ghost, exactly which version of the Ghost is it?" (i.e., What are the specific settings of the Ghost's powers?)

They trained the AI to act like a GPS. Instead of just saying "It's the Ghost," it tried to pinpoint the location on a map of possible Ghost settings.

• Success: In areas where the "Ghost" effects are strong, the AI could pinpoint the location quite well.
• Limitation: In areas where the effects are weak or the "waves" are too messy, the AI got confused.

Why Does This Matter?

This paper is a roadmap for the future of particle physics.

1. It's not just about finding "something new." It's about figuring out what that new thing is.
2. Shape matters more than size. Even if we are unsure about the total number of particles (due to messy equipment or unknown factors), looking at the pattern can still solve the mystery.
3. AI is a powerful partner. Machine learning isn't just a hype; it can find clues in complex, multi-dimensional data that traditional math might miss or struggle to visualize.

In a nutshell: The scientists showed that by looking at the "dance moves" (patterns in time, distance, and energy) of neutrinos rather than just counting the dancers, we can tell if a "Ghost" or a "Strongman" is messing with the show. And we can use AI to read those dance moves even if we don't know exactly how many dancers are on stage.

Technical Summary

1. Problem Statement

Neutrino experiments, particularly those searching for Beyond-the-Standard-Model (BSM) physics, often face limitations due to low statistics, large systematic uncertainties (especially in flux normalization), and coarse observable binning. These factors hinder the ability to distinguish between competing theoretical models that may produce similar total event rates (inclusive yields).

Specifically, the paper addresses the challenge of model discernment in Coherent Elastic Neutrino-Nucleus Scattering (CEνNS) experiments at stopped-pion sources. The core problem is determining whether an observed anomaly is caused by:

1. Sterile Neutrino Oscillations (specifically a $3+1$ framework), which deplete the active neutrino flux via oscillation.
2. Non-Standard Interactions (NSI), which modify the interaction strength of active neutrinos with matter.

Traditional analyses relying solely on total event rates are vulnerable to flux normalization uncertainties and degeneracies where different models mimic each other. The authors investigate how much discriminatory power can be recovered by utilizing multidimensional shape information (correlations in baseline, recoil energy, and timing) rather than just total rates.

2. Methodology

The study employs a two-pronged approach: a conventional Likelihood-Based Analysis and a Machine Learning (ML) approach using Convolutional Neural Networks (CNNs).

A. Benchmark Physics Setup

• Environment: A hypothetical stopped-pion source (similar to COHERENT or CCM experiments) with multiple CsI detectors placed at baselines ranging from 5m to 25m.
• Signal: The neutrino flux consists of a prompt $\nu_\mu$ component (from $\pi^+$ decay) and delayed $\nu_e, \bar{\nu}_\mu$ components (from $\mu^+$ decay).
• Observables: The analysis utilizes a 3D binned distribution:
  1. Baseline ($L$): Distance from source.
  2. Recoil Energy ($E_{rec}$): Nuclear recoil energy.
  3. Timing ($t$): Distinguishing prompt vs. delayed components.
• BSM Scenarios:
  • Scenario I ($3+1$ Sterile): Characterized by mixing parameters $|U_{e4}|^2, |U_{\mu4}|^2$ and mass splitting $\Delta m^2_{41}$. The signal manifests as an oscillatory depletion of the flux.
  • Scenario II (Neutral-Current NSI): Characterized by effective couplings $\epsilon_{ee}^{uV}, \epsilon_{ee}^{dV}$. The signal manifests as a modification of the effective weak charge and scattering rate.

B. Statistical Framework

• Likelihood Analysis: The authors construct a binned likelihood function incorporating a nuisance parameter for flux normalization uncertainty ($\eta$). They use the Bayesian Information Criterion (BIC) to compare the fit of the sterile hypothesis against the best-fit NSI hypothesis.
• Machine Learning Architecture:
  • Input: The binned event distributions are treated as "images" where the baseline-energy plane forms the spatial dimensions and timing bins act as color channels ($L \times E \times T$).
  • Model: A shallow Convolutional Neural Network (CNN) with ReLU activation, batch normalization, max-pooling, and dropout.
  • Crucial Preprocessing: To test if discrimination relies on shape rather than total rate, all pseudo-datasets are renormalized to a common total event rate before being fed into the network. This explicitly removes flux normalization information.
  • Tasks:
    1. Binary Classification: Distinguish between a sterile-generated dataset and its best-fit NSI counterpart.
    2. Multi-class Classification: Identify the specific location of a sterile benchmark point within a discretized parameter space grid ($20 \times 20$).

3. Key Contributions

1. Quantification of Shape Information: The paper demonstrates that multidimensional shape information (baseline, energy, timing) contains genuine discriminatory power that is independent of total event rates.
2. ML as a Rate-Insensitive Tool: By training CNNs on rate-normalized data, the authors prove that machine learning can effectively distinguish between oscillation-based (sterile) and interaction-based (NSI) new physics without relying on flux normalization, which is often the dominant systematic uncertainty.
3. Parameter Localization: The study extends beyond simple model selection to show that ML can perform approximate parameter localization (identifying the specific $|U_{e4}|^2$ and $\Delta m^2_{41}$ values) within the sterile parameter space, provided the oscillation signal is strong enough.
4. Complementarity of Methods: The work establishes a framework where conventional likelihood methods and ML methods play complementary roles, with ML offering robustness against normalization uncertainties.

4. Key Results

A. Likelihood-Based Discrimination

• Inclusive Binning Failure: When data is collapsed into a single bin ({1,1,1}), the discrimination power is negligible ($L_{NSI} \approx 1$), as both models can mimic each other via rate adjustments.
• Shape Sensitivity: Introducing multidimensional binning significantly improves discrimination.
  • Baseline and Energy: These are the most critical discriminators. The oscillatory pattern of the sterile neutrino (dependent on $L/E_\nu$) cannot be replicated by the smooth rate modifications of NSI.
  • Timing: Provides a secondary but useful improvement by separating prompt and delayed components.
• Degeneracy: In regions with small mixing angles or very large mass splittings (where oscillations wash out), the NSI model can mimic the sterile signal, leading to poor discrimination.

B. Machine Learning Performance

• Binary Classification:
  • The CNN achieves high accuracy (up to ~0.9–1.0) in distinguishing sterile vs. NSI scenarios in regions where the likelihood analysis also shows strong discrimination.
  • Rate Independence: Crucially, this high accuracy is maintained even when the total event rate is removed from the input. This confirms that the neural network learns the oscillatory shape features rather than normalization differences.
• Multi-class Classification (Localization):
  • The network can successfully identify the approximate location of the sterile benchmark point in the parameter space.
  • Directional Sensitivity: Reconstruction is more accurate for the mixing parameter $|U_{e4}|^2$ (horizontal axis) than for the mass splitting $\Delta m^2_{41}$ (vertical axis). This is physically intuitive: mixing controls the amplitude of the distortion (easier to see), while mass splitting controls the frequency/wavelength (easier to smear out with binning and energy resolution).
  • Binning Dependence: Finer binning in baseline and energy significantly improves localization accuracy, while timing adds a marginal benefit.

5. Significance and Implications

• From Anomaly to Interpretation: The paper provides a pathway for moving neutrino physics from merely detecting an anomaly to interpreting its physical origin. It shows that even with limited statistics and large systematics, the structure of the data can reveal the underlying mechanism.
• Robustness to Systematics: The ML approach demonstrates that rate-insensitive strategies are viable. This is critical for future experiments where flux normalization may be difficult to constrain (e.g., reactor or stopped-pion sources).
• Experimental Design: The results highlight the importance of multi-baseline and high-granularity energy measurements. Experiments with a single detector or coarse binning will struggle to distinguish between sterile neutrinos and NSI.
• Broader Applicability: The techniques developed here are applicable to other dark sector searches, such as distinguishing light dark matter signals from NSI or sterile neutrino interpretations.

In conclusion, the paper establishes that correlated multidimensional information in CEνNS experiments is a powerful tool for model discernment. By leveraging both traditional statistical methods and modern machine learning, researchers can effectively disentangle competing BSM scenarios and localize new physics parameters, even in the presence of significant normalization uncertainties.