An assay-based background projection for the MAJORANA DEMONSTRATOR using Monte Carlo Uncertainty Propagation

This paper presents a novel Bayesian analysis framework utilizing Monte Carlo uncertainty propagation to project the background index of the MAJORANA DEMONSTRATOR experiment, resulting in a calculated mean background index of $[8.95 \pm 0.36] \times 10^{-4}$ cts/(keV kg yr) from thorium and uranium contaminants.

The Gist

Imagine you are trying to hear a single, tiny whisper in a room that is supposed to be completely silent. That whisper is a rare event in physics called neutrinoless double-beta decay. If scientists can hear it, it proves that neutrinos are their own antiparticles, which would rewrite the rules of the universe.

But there's a problem: the room isn't perfectly silent. There's background noise—creaks, drafts, and distant traffic. In physics, this "noise" is called background radiation. It comes from tiny amounts of natural radioactivity (like Uranium and Thorium) hiding in the copper, plastic, and steel used to build the experiment.

This paper is about a new, smarter way to predict exactly how loud that background noise will be before the experiment even starts, using a method called Monte Carlo Uncertainty Propagation.

Here is the breakdown using simple analogies:

1. The Problem: The "Guessing Game" of Noise

In the past, when scientists built the Majorana Demonstrator (a giant, ultra-sensitive detector buried deep underground in a mine), they tried to predict the background noise.

• The Old Way: They looked at their materials. If a piece of copper had a tiny bit of Uranium, they measured it. If the measurement said "less than 5 units," they just added up all the "less than" numbers and the "exact" numbers to get a total.
• The Flaw: This was like trying to guess the total weight of a bag of marbles by adding up the heaviest possible weight for every single marble. It ignored the fact that some measurements were just "upper limits" (we know it's at most this much, but it could be zero). It also ignored the fact that our measuring tools have errors. The result was a prediction that didn't quite match reality, and it didn't tell them how unsure they were.

2. The New Solution: The "Dice Rolling" Simulator

The authors of this paper created a new framework. Instead of adding up single numbers, they treat every measurement as a range of possibilities.

Think of it like this:

• The Old Way: You have a bag of marbles. You weigh one and say, "It weighs 10 grams." You weigh another and say, "It weighs less than 5 grams." You add them up: 15 grams.
• The New Way (Monte Carlo): You realize that "10 grams" might actually be 9.8 or 10.2. And "less than 5" could be 0, 2, or 4.9.
  • You build a computer simulation.
  • You roll a digital die 1,000,000 times.
  • In every roll, you pick a random weight for the first marble (based on its measurement range) and a random weight for the second (based on its "less than" range).
  • You add them up and record the total.
  • After a million rolls, you don't get one single number. You get a curve (a distribution) showing the most likely total weight and how much it might vary.

3. Handling the "Upper Limits" (The "Silent" Marbles)

A tricky part of this experiment is that many materials are so clean that the detectors can't find any radioactivity. They just say, "It's less than X."

• The Analogy: Imagine you are looking for a needle in a haystack. You look and don't see it. You say, "There are 0 needles." But really, there might be 1 needle hidden deep down that you missed.
• The Fix: The paper uses a clever statistical trick (truncated-at-zero Gaussian). It treats "0 needles" not as a hard zero, but as a probability curve that starts at zero and tapers off. This ensures that even if you didn't see the radioactivity, the simulation knows it could be there, just very faintly. This prevents the scientists from accidentally thinking the room is perfectly silent when it's not.

4. The Result: A Clearer Picture

When they applied this new "dice-rolling" method to the Majorana Demonstrator:

• They calculated the background noise coming from Thorium and Uranium in all the parts of the machine (copper shields, cables, plastic connectors, etc.).
• They found the average background noise to be 8.95 units (with a small margin of error).
• Why this matters: This number is crucial. If the background noise is too high, it drowns out the "whisper" of the neutrinoless decay. By knowing the noise level and the uncertainty of that noise, scientists can accurately predict how long they need to run the experiment to hear the whisper.

5. The Takeaway

This paper isn't just about one experiment; it's about how to do better math for the future.

• Standardization: It gives the whole scientific community a unified rulebook for how to combine messy data (some precise, some just "upper limits").
• Resource Saving: It tells engineers exactly how much computer power they need to simulate the noise. If the simulation shows the noise is negligible, they don't need to waste time simulating it further.
• Future Proofing: As the next generation of experiments (like LEGEND or nEXO) are built, they will use this method to design machines that are cleaner and more sensitive, ensuring they don't get fooled by background noise.

In a nutshell: The scientists stopped guessing the background noise with a ruler and started simulating it with a million dice rolls. This gave them a much more accurate, honest, and useful prediction of how "quiet" their experiment really is, bringing them one step closer to hearing the universe's most elusive whisper.

Technical Summary

1. Problem Statement

In experiments searching for neutrinoless double-beta decay ($0\nu\beta\beta$), the Background Index (BI) is a critical metric used to project the experiment's sensitivity to the decay half-life. The BI represents the background rate in counts per keV, per kg of detector mass, per year (cts/(keV kg yr)).

The Majorana Demonstrator, a prototype experiment using high-purity germanium (HPGe) detectors enriched in $^{76}$Ge, previously reported a measured BI of $6.2^{+0.6}_{-0.5} \times 10^{-3}$ cts/(keV kg yr). This was significantly higher than the originally projected BI of $< 8.75 \times 10^{-4}$ cts/(keV kg yr).

The core problems addressed in this work are:

• Uncertainty Propagation: Previous projections treated specific activities (from radioassays), component masses, and simulation efficiencies as single-valued numbers. This failed to capture the significant uncertainties inherent in assay data, particularly when measurements disagreed or resulted in upper limits.
• Handling Upper Limits: Standard methods often struggled to combine multiple assay results where some were measured values and others were upper limits (non-detections), often leading to unrealistic lower bounds or ignored contributions.
• Null Efficiencies: Components far from the detector array often had "zero" simulated efficiency in deterministic models. However, due to statistical fluctuations in Monte Carlo simulations, these components could still contribute to the background, a factor previously unaccounted for.
• Data Discrepancy: Duplicate samples of the same material often yielded specific activities that did not agree within their stated error bars, requiring a robust statistical method to combine them.

2. Methodology

The authors developed a Bayesian framework utilizing Monte Carlo (MC) uncertainty propagation to generate a probability distribution for the BI rather than a single point estimate.

A. Unified Approach to Combining Assay Results

To handle multiple assay measurements (including upper limits) for specific materials:

• Weighted Averaging: They adopted the Particle Data Group (PDG) method for unconstrained averaging. Specific activities ($a_k$) are weighted by the inverse square of their uncertainties ($w_k = 1/\delta a_k^2$).
• Handling Upper Limits: Upper limits are treated as truncated-at-zero Gaussian distributions. Only the most stringent 90% Confidence Level (C.L.) upper limit is included in the average to avoid artificially lowering the limit through repeated non-detections.
• Scaling Factor ($\Sigma$): If the measurements show significant scatter ($\chi^2/(N_a-1) > 1$), the uncertainty is scaled by $\Sigma = \sqrt{\chi^2/(N_a-1)}$ to account for unmodeled systematic variations or sample-to-sample differences.

B. Promoting Single Values to Distributions

The framework converts single-valued inputs into Probability Density Functions (PDFs):

1. Specific Activity ($a$) and Mass ($m$): Modeled as truncated-at-zero Gaussian distributions. Mass uncertainties range from 1% (direct measurement) to 10% (geometric estimation).
2. Simulation Efficiency ($\epsilon$):
   • Instead of a single efficiency value derived from $N_{counts}/N_{simulated}$, the efficiency is treated as a random variable.
   • Using Bayes' theorem with an uninformative prior, the probability of the true expectation value $\lambda$ given observed counts $n$ follows a Gamma distribution.
   • This allows components with zero observed counts in the simulation (null efficiency) to still have a non-zero probability of contributing to the background, effectively setting a statistical upper limit based on the number of simulated decays.

C. Monte Carlo Uncertainty Propagation

The final BI is calculated by:

1. Randomly drawing samples from the PDFs of specific activity, mass, and efficiency for every component.
2. Calculating the BI contribution for each component using the formula: $BI_i \propto a_i \times m_i \times \epsilon_i$.
3. Repeating this process $10^6$ times to generate a full BI distribution for each component group and the total experiment.
4. Summing the distributions to obtain the total projected background.

3. Key Contributions

• Novel Framework: Introduction of a unified Bayesian framework that treats assay data, mass, and simulation statistics as distributions rather than point estimates.
• Robust Handling of Upper Limits: A rigorous method for incorporating assay upper limits into background projections without biasing the result downward.
• Statistical Treatment of Null Efficiencies: The ability to quantify background contributions from components with "zero" simulated counts by modeling the efficiency as a Gamma distribution, preventing the underestimation of background from distant or low-activity components.
• Standardization: The methodology provides a template for standardizing assay reporting and uncertainty propagation for the broader low-background physics community.

4. Results

The framework was applied to the Majorana Demonstrator's "as-built" geometry, considering only $^{232}$Th and $^{238}$U contamination (excluding $^{40}$K as its gamma rays are below the region of interest).

• Projected Background Index: The mean projected assay-based BI is:
  $$BI_{assay} = [8.95 \pm 0.36] \times 10^{-4} \text{ cts/(keV kg yr)}$$
  • Breakdown:
    • $^{232}$Th contribution: $[7.37 \pm 0.24] \times 10^{-4}$
    • $^{238}$U contribution: $[1.58 \pm 0.11] \times 10^{-4}$
• Uncertainty Sources: The uncertainty is dominated by the specific activity measurements (act.), not the simulation statistics (stat.).
• Comparison to Previous Projections:
  • The new projection ($8.95 \times 10^{-4}$) is approximately 44% higher than the original design geometry projection ($6.4 \times 10^{-4}$, derived from Ref [15] which used a direct sum of upper limits).
  • This increase is attributed to the proper inclusion of uncertainties and the contribution of components with null single-valued efficiencies.
• Component Contributions: The largest contributors to the background were identified as the Front Ends (electronics), Cables, and Pb Shielding.

5. Significance

• Realistic Sensitivity Projections: By properly propagating uncertainties, this method provides a more realistic and conservative estimate of the background index, which is crucial for calculating the true physics reach (half-life sensitivity) of current and future $0\nu\beta\beta$ experiments (e.g., LEGEND, nEXO).
• Optimization of Resources: The framework allows analysts to determine how many simulated decays are necessary to reduce statistical uncertainties to a negligible level, optimizing computational resources.
• Community Standard: The unified approach to averaging assay results (handling upper limits and scatter) offers a standardized method for the low-background community, facilitating better data sharing and comparison between different experiments.
• Explanation of Discrepancies: The study demonstrates that the discrepancy between the Demonstrator's original design projection and its actual performance is largely due to the underestimation of uncertainties and the exclusion of "zero-efficiency" components in previous deterministic models.

In conclusion, this paper establishes a rigorous statistical foundation for background projection in rare-event searches, moving away from deterministic upper limits toward probabilistic distributions that fully account for experimental and simulation uncertainties.