Optimized filtering for pulse-shape based pile-up rejection applied to $0\nu\beta\beta$ search with $^{100}$Mo

This paper presents an optimized digital filtering algorithm for cryogenic detectors that significantly reduces pile-up induced background in neutrinoless double beta decay searches by 31% compared to previous methods, while maintaining high signal efficiency.

The Gist

The Big Picture: Listening for a Whisper in a Storm

Imagine you are trying to hear a single, specific whisper (a rare nuclear event called neutrinoless double beta decay) in a very noisy room. The problem is that the room is full of people talking constantly (background noise).

Even worse, sometimes two people talk at almost the exact same time. Their voices overlap, creating a sound that sounds exactly like the special whisper you are looking for. In physics, this overlap is called "pile-up."

If you can't tell the difference between the real whisper and the overlapping chatter, you will get false alarms, and your experiment will fail. This paper presents a new, smarter way to filter out that overlapping chatter so scientists can hear the real signal.

The Problem: The "Double-Clap" Confusion

The scientists are using a special type of detector made of crystals (specifically Molybdenum-100). These crystals are like super-sensitive microphones that get slightly warmer when a particle hits them.

• The Real Signal: A single particle hits, creating a clean, smooth "blip" of heat.
• The Pile-Up: Two particles hit the crystal almost instantly (within less than a millisecond). The detector sees one big, messy "blip" that looks like a single event but is actually two events squashed together.

Because the "messy blip" looks so much like the "real signal," the old methods of filtering it out weren't good enough. They were throwing away too many real signals just to catch a few fake ones.

The Solution: A Smart "Sound Engineer"

The authors created a new digital filter (a mathematical tool) that acts like a super-smart sound engineer. Instead of just listening to the volume, this engineer analyzes the shape and texture of the sound wave.

Here is how their new method works, using an analogy:

The "Two-Filter" Trick
Imagine you are trying to distinguish between a single drum hit and two drum hits that happened so fast they sounded like one.

1. Filter A is tuned to hear the "smoothness" of the sound. It loves the clean, single drum hit.
2. Filter B is tuned to hear the "roughness" or the sharp edges that happen when two hits overlap.

The new algorithm doesn't just look at the result of Filter A or Filter B alone. Instead, it takes the ratio (the comparison) of the two.

• If the sound is a single hit, the ratio stays steady.
• If the sound is a messy overlap, the ratio changes dramatically.

By constantly comparing these two perspectives, the algorithm can spot the "fake" overlapping events with much higher precision than before.

How They Built It

The scientists didn't just guess the settings for this filter. They used a computer to "train" it, similar to how you might train a dog.

• They fed the computer thousands of examples of real signals and fake overlapping signals.
• The computer adjusted its internal knobs (mathematical weights) millions of times a second to find the perfect setting that catches the most fakes while letting the most real signals pass through.
• They used a powerful technique called gradient descent (a method of slowly walking down a hill to find the lowest point) to find the absolute best filter settings.

The Results: A Clearer Signal

When they tested this new filter on data from the CROSS experiment (a test run for the future CUPID experiment):

• The Goal: They wanted to keep 90% of the real signals (efficiency).
• The Win: With the new filter, they were able to reject 31% more of the fake overlapping signals compared to the old method.

Think of it like this: If the old method caught 100 fake signals, the new method catches 131, without losing any of the real whispers. This is a huge improvement for an experiment where every single false alarm counts.

Why This Matters

The paper focuses specifically on cryogenic detectors (detectors that must be kept near absolute zero) looking for neutrinoless double beta decay.

• The authors state that this method is broadly applicable to other detector technologies, but their specific results and numbers are strictly about improving the search for this specific nuclear decay using Molybdenum crystals.
• They did not claim this works for medical imaging or other fields in this paper; the focus is entirely on making physics experiments more sensitive.

Summary

The paper introduces a new mathematical "ear" for scientists. By comparing two different ways of listening to a signal, this new tool can spot when two events have accidentally merged into one. This allows them to ignore the noise and focus on the rare, precious signals that could unlock secrets about the universe, specifically regarding the nature of neutrinos.

Technical Summary

Technical Summary: Optimized Filtering for Pulse-Shape Based Pile-up Rejection in $0\nu\beta\beta$ Search with $^{100}$Mo

Problem Statement
In rare-event searches, such as the search for neutrinoless double beta decay ($0\nu\beta\beta$) using cryogenic calorimeters, pile-up events—where distinct signals temporally overlap—pose a critical challenge. For detectors utilizing $^{100}$Mo, such as the CUPID experiment, the dominant background source is expected to arise from the pile-up of two-neutrino double beta decay ($2\nu\beta\beta$) events. These events can mimic the mono-energetic $0\nu\beta\beta$ signal at 3034 keV. While hardware improvements (e.g., faster sensors like TESs and MMCs) aim to reduce pulse durations, residual pile-up remains unavoidable in large-scale experiments. Existing analysis techniques, including standard optimal filtering and $\chi^2$-based rejection, lose discrimination power when the time separation between pulses is shorter than the signal rise time (specifically $\Delta t < 0.8$ ms for the detectors studied). This necessitates advanced software-based discrimination methods to mitigate the background without sacrificing signal efficiency.

Methodology
The authors propose a framework to derive an optimized digital filter for pile-up discrimination based on a frequency-domain description of the signal. The method assumes:

1. A fully known, energy-independent impulse response $s(t)$ of the detector.
2. Stationary, zero-mean Gaussian noise characterized by a power spectral density (PSD) $P_N(\omega)$.
3. Pile-up events modeled as the sum of two pulses with relative amplitude $r$ and time separation $\Delta t$.

The core of the method involves constructing a discriminant parameter $Y$ based on the ratio of two amplitude estimators, $X[m_1]$ and $X[m_2]$, derived from the measured signal $\hat{x}(\omega)$:
$$Y = \frac{X[m_1]}{X[m_2]}$$
where $X[m_i]$ is the maximum of the real part of the inverse Fourier transform of the signal filtered by a transfer function $H(\omega)$ and a weighting function $m_i(\omega)$. The standard optimal filter $H(\omega)$ maximizes the signal-to-noise ratio (SNR), while the weighting functions $m_1(\omega)$ and $m_2(\omega)$ are the variables to be optimized.

The optimization goal is to minimize the global pile-up misidentification rate $\langle M \rangle$, defined as the probability that a pile-up event is incorrectly classified as a single pulse, subject to a fixed single-pulse acceptance efficiency $\epsilon$. The authors utilize PyTorch and automatic differentiation to perform gradient descent on the weighting functions $m_1(\omega)$ and $m_2(\omega)$, treating them as trainable parameters discretized over frequency bins. The cost function is evaluated over a grid of time separations and amplitude ratios, weighted by the expected distributions of $2\nu\beta\beta$ events.

Key Contributions

• Optimization Framework: Development of a general, quantitative framework for modeling, identifying, and rejecting pile-up events in cryogenic calorimeters using optimized frequency-domain filters.
• Ratio-Based Discriminant: Implementation of a ratio-based discriminant parameter, which leverages the correlation between two filtered observables to reduce sensitivity to amplitude fluctuations and improve discrimination power compared to single-amplitude methods.
• Computational Implementation: A modular Python implementation using PyTorch that enables efficient training of filters via GPU acceleration, capable of handling large parameter grids.
• Systematic Validation: A comprehensive study of systematic uncertainties arising from pulse template selection, processing window size, pulse position, and grid discretization.

Results
The method was validated using data from the CROSS experiment, which operated a tower of scintillating $Li_2^{100}MoO_4$ crystals coupled to light detectors (LDs) with Neganov–Trofimov–Luke (NTL) amplification.

• Performance Improvement: Compared to a reference method (previously presented in Eur. Phys. J. C 83(5), 373 (2023)), the new optimized filter reduces the pile-up induced background index (BI) by an average of 31% at 90% single-pulse efficiency.
• Background Index: The extrapolated mean residual background index using the new method is $5.9 \times 10^{-5}$ cts/keV/kg/yr (median: $5.7 \times 10^{-5}$), approaching the CUPID design goal of $5.0 \times 10^{-5}$ cts/keV/kg/yr.
• Filter Characteristics: The optimized filters $m_1(\omega)$ and $m_2(\omega)$ exhibit complementary frequency weighting. $m_1(\omega)$ shows a rising response in the 1–3 kHz range to amplify high-frequency features essential for resolving pile-up, while $m_2(\omega)$ minimizes gain in this region to emphasize the bulk signal shape.
• Systematics: The study identified that the choice of pulse template significantly impacts results (up to 13% variation in BI). The authors recommend using an "average pulse" template as a conservative option to avoid artifacts from high-frequency noise imprints in single high-energy pulses.
• Detector Dependence: A preliminary study on detector working points (bias current) suggests that individual optimization of the operating point can further improve pile-up rejection, with variations in background rejection reaching a factor of two for different detectors.

Significance and Claims
The paper claims that the proposed ratio-based discriminant captures the dominant pile-up features relevant for cryogenic calorimeters with slow thermal responses. It provides a robust, computationally efficient tool for optimizing analysis techniques in the absence of hardware changes. The authors state that while the method relies on assumptions of stationary noise and pulse shapes, it demonstrates significant and robust background reduction across diverse detectors.

The work positions this technique as a complementary "software handle" to hardware developments (such as faster sensors or improved NTL amplification coverage). The authors note that the extrapolation of results to the full CUPID experiment depends on the noise environment, particularly in the O(1) kHz range, and that future upgrades to CUPID's cryogenic infrastructure and light detector performance are expected to yield further improvements. The framework is presented as broadly applicable to various detector technologies and experimental contexts beyond the specific $^{100}$Mo search.