Energy Calibration and Performance of HPGe Detectors in the LEGEND-200 Experiment

This paper details the energy calibration procedures and performance of HPGe detectors in the LEGEND-200 experiment, demonstrating a high-resolution energy reconstruction of $(2.47 \pm 0.08)$~keV at the $Q_{\beta\beta}$ value and exceptional long-term stability essential for the search for neutrinoless double beta decay.

The Gist

The Big Picture: Listening for a Whisper in a Storm

Imagine the universe is a giant, noisy concert hall. Scientists are trying to hear a single, specific whisper (a rare particle event called "neutrinoless double beta decay") that could explain why our universe is made of matter instead of antimatter. The problem is that the "concert hall" is incredibly loud with background noise.

To hear that whisper, the LEGEND-200 experiment uses a team of 142 "super-listeners" (High-Purity Germanium detectors). These detectors are like incredibly sensitive microphones buried deep underground to block out the noise of the surface world.

This paper isn't about finding the whisper yet; it's about tuning the microphones. The authors explain how they calibrated these detectors to ensure that when they do hear a sound, they know exactly what note it is and how loud it is, down to the tiniest fraction of a second.

The Detectors: The "Super-Microphones"

The experiment uses four different types of germanium crystals (IC, BEGe, PPC, and Coax). Think of these as different models of microphones. Some are big and bulky (IC), some are small and pointy (PPC), and some are in between.

• The Job: When a particle hits a crystal, it creates a tiny electrical pulse.
• The Challenge: These pulses can get distorted. Imagine shouting into a microphone that has a sticky diaphragm; the sound might get muffled or lose some volume. In the crystals, this is called "charge trapping." Some of the electrical signal gets stuck in the crystal lattice before it reaches the readout.

The Solution: Digital Signal Processing (The "Audio Engineer")

To fix the distorted sounds, the team uses a sophisticated digital audio engineer (software called pygama). They apply three main tricks:

1. The Shaping Filter (The Equalizer):
   The raw signal looks like a messy spike. The team uses a "cusp filter" (shaped like a mountain peak with a flat top) to smooth it out. Imagine taking a jagged rock and sanding it down until it's a perfect, smooth sphere. This makes it much easier to measure the exact size of the signal.

2. Charge Trapping Correction (The Volume Booster):
   Since some signals get "stuck" and lose volume, the software estimates how much signal was lost based on how long it took the signal to arrive. It then adds that missing volume back in. It's like a sound engineer realizing a singer was too far from the mic and digitally boosting their volume to match the others.

3. The Result:
   After this digital surgery, the detectors can distinguish between two sounds that are incredibly close in pitch. The paper reports that the "blur" (energy resolution) at the critical frequency is about 2.5 keV. To put that in perspective, if the energy scale were a ruler measuring a football field, the error would be smaller than the width of a human hair.

The Calibration: Tuning the Piano

Even with perfect digital processing, the detectors need to be "tuned" regularly, just like a piano.

• The Tuning Fork: Once a week, the team inserts a radioactive source (Thorium-228) into the liquid argon bath surrounding the detectors. This source emits gamma rays at very specific, known energies (like specific musical notes: 583 keV, 2614 keV, etc.).
• The Two-Stage Tuning:
  1. Weekly Gain (The Volume Knob): They check if the overall volume has shifted slightly this week. They adjust a linear "gain" factor to make sure the 2614 keV note still lands exactly on 2614.
  2. Long-Term Non-Linearity (The Stretchy String): Sometimes, the relationship between the input and output isn't perfectly straight (like a guitar string that stretches differently at high notes). They use a massive amount of data collected over months to fix this "curvature" in the scale.

The Stability: The paper shows that this tuning is incredibly stable. The "notes" the detectors hear shift by less than 0.05 keV from week to week. That's like a piano staying perfectly in tune for months without a tuner touching it.

The Performance: Are They Ready?

The team tested their work by looking at the "background noise" (natural radiation from potassium in the rocks) to see if their tuning held up in real life.

• Resolution: The average clarity of the signal across all detectors is 2.47 keV. This meets the strict goal set for the experiment.
• Bias: They checked if the "notes" were slightly off-key (biased). They found a tiny shift (about 0.25 keV), but they have a map of exactly where that shift is, so they can correct for it in their final analysis.

The Bottom Line

This paper is the "quality control report" for the LEGEND-200 experiment. It proves that the team has successfully built a system of super-sensitive detectors that are:

1. Sharp: They can separate signals that are very close together.
2. Stable: They don't drift out of tune over time.
3. Accurate: They know exactly where the "target" energy is.

With this foundation, the experiment is now ready to start the actual search for the rare particle decay, confident that if they hear a signal, it's real and not just a glitch in the tuning.

Technical Summary

Technical Summary: Energy Calibration and Performance of HPGe Detectors in the LEGEND-200 Experiment

Problem Statement
The LEGEND-200 experiment is designed to search for neutrinoless double beta ($0\nu\beta\beta$) decay in $^{76}$Ge, a process that, if observed, would confirm the Majorana nature of neutrinos and explain the matter-antimatter asymmetry of the universe. The sensitivity of this search is critically dependent on the energy resolution and the stability of the energy scale of the High-Purity Germanium (HPGe) detectors. To achieve a "background-free" condition where the expected background events in the region of interest (ROI) are less than one, the experiment requires exceptional energy resolution to distinguish the narrow signal peak at the $Q$-value ($Q_{\beta\beta} = 2039$ keV) from the continuous background. Furthermore, a precise and stable energy scale is necessary to correctly model the background and identify the ROI. This paper addresses the methodologies and performance metrics required to meet these stringent experimental demands during the first data-taking campaign (March 2023 – February 2024).

Methodology
The paper details the digital signal processing (DSP) pipeline, energy calibration procedures, and performance validation for the 142 kg array of HPGe detectors, which includes Inverted Coaxial (IC), Broad Energy Germanium (BEGe), P-type Point-Contact (PPC), and semi-coaxial (Coax) types.

• Data Acquisition and Signal Processing: Detector signals are digitized by FlashCam-based DAQ systems at 62.5 MHz. Offline analysis utilizes the pygama framework. Energy reconstruction employs three shaping filter methods: a symmetric trapezoidal filter, a cusp-like filter, and a Zero Area Cusp (ZAC) filter. The cusp filter was selected as the reference method for the main analysis due to its superior average energy resolution.
• Charge Trapping Correction (CTC): To mitigate energy degradation caused by charge carrier trapping in the crystal lattice (particularly relevant for large IC detectors), a CTC is applied. This involves extracting the effective drift time ($d t_{eff}$) from the signal leading edge and applying a linear correction: $E_{CTC} = E(1 + \alpha \times d t_{eff})$. Parameters for shaping filters and the CTC coefficient ($\alpha$) are optimized per detector using a grid search and Bayesian optimization to minimize the Full Width at 20% Maximum (FW20M).
• Two-Stage Energy Calibration: To ensure long-term stability and robustness, a two-stage calibration approach is employed using weekly $^{228}$Th source runs:
  1. Stable Parameters: The constant offset ($a$) and quadratic non-linearity term ($c$) are calculated once per "partition" (a stable time-span) using a high-statistics dataset combining all calibration runs.
  2. Time-Varying Gain: The linear gain term ($b$) is extracted weekly from the 2614.5 keV Full Energy Peak (FEP) to account for temporal gain drifts.
• Peak Shape Modeling: Calibration peaks are fitted using a model comprising a Gaussian function, a step function for continuum background, and an exponential low-energy tail to account for incomplete charge collection. Model selection is governed by statistical criteria (p-values) and parameter constraints to avoid overfitting.

Key Results

• Energy Resolution: The optimized reconstruction achieves a combined average energy resolution of $(2.47 \pm 0.08)$ keV at $Q_{\beta\beta}$. Breakdown by detector type shows:
  • BEGe: $(2.09 \pm 0.08)$ keV
  • PPC: $(2.55 \pm 0.21)$ keV
  • IC (the baseline design): $(2.63 \pm 0.48)$ keV
  • Coax: $(4.42 \pm 0.40)$ keV
    The majority of BEGe, PPC, and IC detectors meet or exceed the LEGEND experimental goal of 2.5 keV.
• Calibration Stability: The weekly variation of calibration peak positions is below 0.05 keV for energies up to 2614.5 keV. The gain parameters exhibit temporal stability with variations smaller than 0.1%.
• Energy Bias and Linearity: A systematic non-linearity persists around the 2 MeV region, with a mean deviation of 0.24 keV at the 2103.5 keV Single Escape Peak. The overall energy bias at $Q_{\beta\beta}$ is measured at $(0.25 \pm 0.01)$ keV. These deviations are corrected in the final analysis to minimize systematic uncertainties.
• Validation: The energy resolution model derived from calibration sources is validated against physics background data (specifically the $^{40}$K and $^{42}$K lines), showing a Pearson correlation coefficient of 0.99 and confirming the robustness of the resolution model across different data-taking periods.

Significance and Claims
The paper claims that the LEGEND-200 detector system has successfully demonstrated state-of-the-art performance in energy reconstruction and calibration. The primary significance lies in the validation that the HPGe array meets the rigorous experimental requirements for the first unblinding of the $0\nu\beta\beta$ decay search. The implementation of the two-stage calibration and advanced DSP techniques ensures a stable, linear, and high-resolution energy scale. The authors state that these performance metrics provide the necessary foundation for LEGEND-200 to significantly advance the search for neutrinoless double beta decay, with the system operating at a level capable of achieving the experiment's sensitivity goals. The work establishes the procedures and confirms the stability required for the subsequent physics analysis described in the companion paper [7].