Calibration of the Gerda experiment

Introduction

Neutrinoless double-$\beta$ ($0\nu \beta \beta$) decay is a hypothetical, second-order weak interaction process in which a nucleus changes its charge number by two units with the emission of two electrons but without accompanying anti-neutrinos. This lepton-number violating process is only permitted if neutrinos are massive Majorana fermions, i.e. if there is a Majorana mass term in the Lagrangian of the underlying theory. Such a term appears in many extensions of the Standard Model of particle physics and could explain why neutrino masses are much smaller than those of all other fermions [1]. The GERmanium Detector Array (Gerda) collaboration searched for the $0\nu \beta \beta$ decay of the isotope ${}^{76}$Ge with a Q-value of $Q_{\beta \beta }=2039.061(7)$ keV [2] by operating high-purity germanium (HPGe) detectors isotopically enriched to >86% in ${}^{76}$Ge, making them also the potential source of $0\nu \beta \beta$ decay.

We used three types of enriched germanium detectors: 30 broad energy germanium (BEGe) detectors, 7 coaxial detectors, and 5 newer inverted coaxial (IC) detectors. The BEGe detectors are smaller (average 0.7 kg) but offer superior energy resolution and pulse shape discrimination (PSD) properties compared to the coaxial detectors [3], while the IC detectors provide energy resolution and PSD properties similar to the BEGe detectors [4] but with a larger mass (average 1.9 kg) comparable to that of the coaxial detectors (average 2.3 kg), allowing for the easier design of larger germanium arrays.

The array of germanium detectors was immersed in a cryostat filled with 64 m${}^3$ of liquid argon (LAr). The top of the cryostat and the surrounding water tank houses a clean room containing a glove box and lock system for deploying the HPGe detectors and calibration sources. The entire setup was located underground at the Laboratori Nazionali del Gran Sasso (LNGS) of INFN, Italy, and is described in detail in [5].

The first phase of the experiment was operated with 18 kg of coaxial detectors (inherited from the Heidelberg-Moscow [6] and Igex [7] collaborations) between November 2011 and September 2013. Phase II started in December 2015, after 20 kg of BEGe detectors produced for the Gerda experiment were added and the liquid argon volume around the detector array was instrumented with photosensors as a veto against radioactivity [5]. During an upgrade in mid-2018, IC detectors with a total mass of 9.6 kg were added, and the LAr instrumentation was upgraded. Phase II data taking ended in November 2019. While the calibration procedure of Phase I data has been discussed in [8, 9], the focus of this paper is the calibration of the Phase II data.

In all recent $0\nu \beta \beta$ decay experiments, the signature of the rare nuclear transition is a monoenergetic peak in the measured energy spectrum of the two electrons at $Q_{\beta \beta }$. Consequently, a crucial parameter to distinguish a signal from the background is the energy estimator. The better the energy resolution of the detectors, the narrower the signal energy region effectively becomes, and an excess over the continuous background can be more clearly identified. One strength of HPGe detectors is their unparalleled energy resolution (typically $\sigma$/E$\sim$0.1% at $Q_{\beta \beta }$). It permits the almost complete rejection of background events from regular two-neutrino-accompanied double-$\beta$ decays [10], an otherwise irreducible background in $0\nu \beta \beta$ decay searches [11, 12].

Given the central role of the energy observable, adequate measures must be taken to accurately determine the energy scale and resolution, monitor their stability over the full data acquisition period, and determine the relevant uncertainties entering the statistical analysis for the $0\nu \beta \beta$ decay search. In Sect. 2 we detail the calibration procedure, while in Sect. 3 we discuss the analysis of the calibration data and the energy scale determination, including the procedures employed to monitor and maintain the stability of the HPGe detectors over time. In Sect. 4 we describe the determination of the energy resolution for the $0\nu \beta \beta$ decay analysis, and in Sect. 5 we provide an evaluation of the associated uncertainties. In Sect. 6, we discuss the determination of the residual energy bias and its uncertainty. In Sect. 7, we compare the results from calibration data with those in the physics data (data used for the $0\nu \beta \beta$ decay search) for the resolutions of the lines from decays of ${}^{40}$K and ${}^{42}$K. We close in Sect. 8 with a summary and a discussion of our main results.

Energy calibration process

To perform the calibrations, we regularly exposed the HPGe detectors to three custom-made low-neutron emission ${}^{228}$Th calibration sources [13], each with an activity of about 10 kBq. These sources were stored within shielding above the lock system, at a vertical distance of at least 8 m to the HPGe detector array, during physics data acquisition. Since ${}^{228}$Th has a half-life of 1.9 yr, the sources were replaced during Phase II to ensure a sufficient level of activity.

During calibration runs of the HPGe detectors, the ${}^{228}$Th sources were lowered into the LAr cryostat to reach the level of the detector array by three source insertion systems [14, 15]. Each of these deploy a single source, placed on tantalum absorbers (h = 60 mm, $⌀$ = 32 mm). During calibration, each source was placed at three different heights to expose the detector array more homogeneously, and data were acquired at each location for up to 30 min. With this exposure, typically around $(1-3)\times {10}^3$ events are observed in the prominent ${}^{208}$Tl $\gamma$ line at 2614.5 keV in a BEGe detector, and $(0.6-1)\times {10}^4$ events in a coaxial or IC detector. Calibration data were acquired every 7–10 days with a total of 142 calibration runs used for the analysis of the Phase II data.

The triggering energy threshold for this acquisition during calibration corresponds to $\sim$400 keV. This threshold was set to include the strong $\gamma$ line of the ${}^{228}$Th spectrum at 583.2 keV while keeping the event rate at a manageable level for the data acquisition system. The detector signals were read out with charge sensitive amplifiers, and digitized by a 100 MHz 14-bit flash analog-to-digital converter (FADC). As for physics data acquisition, for each trigger a 160 $\mathrm{\mu }$s long waveform is recorded at a sampling rate of 25 MHz, centered around the trigger time and covering an energy range up to $\sim$6 MeV. During calibration, every 2 s a test pulse was injected into the amplifier electronics of all germanium detectors to monitor the stability of their gains. Between successive calibration runs, i.e. during physics data acquisition, test pulsers were injected every 20 s for the same purpose.

Data from the FADCs are transformed into the MGDO (ROOT-based) format [16, 17] and processed to analyze properties of the recorded waveforms using the Gelatio software [18], as is the case for physics data. The energy is estimated from the amplitude of the waveform after applying a digital filter which reduces the impact of noise and thus improves the resolution. As a fast first estimate for monitoring and cross-check purposes, a pseudo-Gaussian filter is applied to obtain an energy estimator as part of the online analysis [19]. An improved energy resolution for the final data analysis is achieved with a zero area cusp (ZAC) filter [20], which removes the effect of low-frequency noise. This filter is optimized offline for each calibration run and HPGe detector to minimise the resolution of the highest energy $\gamma$ line in the ${}^{228}$Th spectrum [21].

A set of heuristic event selection criteria is applied to ensure that events recorded during calibration are of a physical origin, and to reduce pile-up events. The underestimation of energy for these events cause low-energy tails in the spectra of $\gamma$ lines and can bias the estimated energy resolution. These selection criteria are based on the properties of the waveform, such as the baseline stability and slope, trigger time, number of triggered events, and rise time of the pulse. The probability of rejecting physical interactions, estimated with events from the regularly injected test pulses, is below 0.1% [22].

Analysis of energy spectra

Nuclei of the ${}^{228}$Th isotope decay in a chain via $\alpha$ and $\beta$ decays to the stable ${}^{208}$Pb with the emission of multiple monoenergetic $\gamma$ rays. These result in sharp peaks in the recorded energy spectra, as shown in Fig. 1 in the combined spectra of each detector type. The pattern of observed peaks is used to identify the $\gamma$ lines and thereby determine their energy and resolution.

The TSpectrum class of ROOT is used to find peak positions in the uncalibrated spectrum, such that all peaks with amplitudes exceeding 1/20 of the amplitude of the most prominent peak are found. This threshold was chosen to avoid the detection of spurious peaks. The peak with the highest energy is identified as the full energy peak (FEP) of the $\gamma$ ray from the decay of ${}^{208}$Tl, a daughter of ${}^{228}$Th, at $E_{\mathrm{FEP}}=2614.5$ keV. A preliminary calibration for the energy estimator T is applied assuming a linear energy scale without offset:

A candidate peak is confirmed if its preliminary estimated energy is consistent within 6 keV with the energy of a known line in the ${}^{228}$Th spectrum. The 6 keV value permits the accurate identification of peaks while allowing for some non-linearity of the energy scale. The known peaks correspond to $\gamma$ rays from isotopes in the ${}^{228}$Th decay chain with energies above 500 keV and branching ratios above 0.3%, including the detector specific single escape peak (SEP) at 2103.5 keV and double escape peak (DEP) at 1592.5 keV resulting from the 2.6 MeV $\gamma$ ray of ${}^{208}$Tl  decays. In the context of this paper, without ambiguity, FEP, SEP, and DEP always refer to those of ${}^{208}$Tl. The double peak due to the 511.0 keV annihilation line and 510.7 keV $\gamma$ line from ${}^{208}$ Tl is excluded from the analysis, in particular since the resolution of the annihilation peak is broadened due to the Doppler effect [23].

Energy resolutions from the combined calibration spectra

Depending on the specific physics analysis, we calculated the energy resolution either by partition, described in the previous section, or by detector type. For the $0\nu \beta \beta$ decay search reported in [26], where the data before the 2018 upgrade were analyzed, an effective resolution for each detector type was employed. For the final $0\nu \beta \beta$ decay search of Gerda reported in [25], where all Gerda data were analyzed, we calculated a resolution for each partition, a much more fine-grained approach. At the expense of increased complexity, the partition approach improves the physics result by capturing the variations among the detectors as well as the variation over time.

Since both methods are applicable for Gerda  and any other experiment with a modular detector setup, here, we discuss both approaches. While the detector type approach was replaced in favor of partitioning for the final $0\nu \beta \beta$ decay search, the former gives an overview of the overall detector performance. For this reason all the illustrations and calibration parameters are provided by detector type. Here, we refer to a dataset containing all the partitions of detectors of the same type as a DTD (detector type dataset).

Energy resolution uncertainty at $Q_{\beta \beta }$

The statistical uncertainty on the energy resolution decreases with rising statistics over time, and is on the order of only a few eV. As such, the uncertainty on the energy resolution is dominated by systematic effects. We consider various sources of systematic uncertainty, given here in decreasing order of their contribution: (i) resolution shifts over time; (ii) energy scale shifts over time; (iii) choice of the resolution fitting function. Due to the nature of these uncertainties, their magnitude will not decrease over time, but could change if the detector setup or analysis methods change.

In the following sections, we explain how individual contributions to the systematic uncertainty were determined (Sects. 5.1–5.3), and how they are combined together to give a total uncertainty per partition (Sect. 5.4).

Energy bias and uncertainty

Due to the different assumptions and approximations in the calibration procedure, slight biases in the energy scale may remain. Such biases may, for example, be caused by the integral non-linearity of the FADCs [30]. Small non-linearities in the energy scale are for example neglected due to the use of a linear calibration function. Therefore a peak from a $\gamma$ ray with well defined energy might be displaced towards higher or lower energies. Correspondingly, for each individual event, while its reconstructed energy will fluctuate according to the resolution, it may also be systematically displaced.

To evaluate the energy bias per partition near $Q_{\beta \beta }$, we look at the residual at the SEP defined in Sect. 3.3 in the combined calibration spectrum, since the SEP is very close to $Q_{\beta \beta }$ with a difference of 64.5 keV. The statistics is sufficient to reach a precision of $\mathcal{O}$(0.01 keV) for the SEP position. The average bias is found to be $-0.07$ keV, with a standard deviation of 0.29 keV among the partitions. Since the $0\nu \beta \beta$ decay search is extremely sensitive to the energy of the events close to $Q_{\beta \beta }$, in the final Gerda analysis [25], we correct for the energy bias of the events that fall into the energy range considered for the $0\nu \beta \beta$ decay search (1930 keV to 2190 keV), by adding the amount of bias to the calibrated event energy. This approach is justified by studying the residuals at the ${}^{42}$K peak (1525 keV) and the DEP (1592.5 keV), which are two closely located peaks with the former appearing in the physics data [10] and the latter in the calibration data. The relation between them is consistent with that between $Q_{\beta \beta }$ and the SEP.

For the uncertainty of the bias, we use the residual fluctuations of the SEP over time near $Q_{\beta \beta }$. We additionally include a systematic uncertainty of 0.02 keV accounting for the potential difference between the bias at SEP and that at $Q_{\beta \beta }$. It was estimated by performing a linear interpolation between the residuals at the DEP and the SEP which are on the two sides of $Q_{\beta \beta }$. In total, the average bias uncertainty is 0.17 keV.

Comparison to physics data

The two strongest $\gamma$ lines in our physics data spectrum are those due to ${}^{40}$K (1460.8 keV) and ${}^{42}$K (1524.7 keV) decays [25]. The measured resolution of these peaks allows for a cross-check to the conclusions drawn solely from calibration data. For every partition, the background energy spectrum around each of these lines is fitted using a Gaussian for the signal and a linear function for the background. The background rate was constrained to be non-negative across the fitting window. Partitions with potassium peaks with low counting statistics, i.e. those whose best-fit is compatible with zero counts, are excluded from further analysis.

Given their proximity in energy, the extracted resolution for each of the two lines is expected to coincide within 0.05 keV. Indeed, no significant difference between the resolutions of the two peaks was found.

We compared the resolution obtained in the potassium lines with the one predicted from the resolution curves extracted from the combined energy spectra (see Sect. 4), as shown in Fig. 7. The systematic uncertainty for the calibration resolution is calculated in the same way as described in Sect. 5. The measured resolutions and predicted values from calibration data show a high degree of correlation, with a Pearson correlation coefficient of 0.92, and with 66% compatibility within one $\sigma$. Similar results are obtained for the ${}^{40}$K line.

Conclusions

A reliable and stable energy scale is crucial to the search for $0\nu \beta \beta$ decay of ${}^{76}$Ge performed with the Gerda experiment. The event energies are reconstructed using the ZAC filter to minimize the effects of low-frequency noise. To preserve the excellent energy resolution of the germanium detectors when combining data over a long period of time, they are calibrated weekly using ${}^{228}$Th sources. By identifying $\gamma$ peaks in the recorded spectrum the energy scale and energy resolution can be determined.

For each calibration, the stability of the energy scale and resolution is monitored via the 2.6 MeV FEP from ${}^{208}$Tl decays. Between successive calibration runs the energy scale is monitored via test pulser events injected into the readout electronics of the HPGe detectors. Data with short-term instabilities are discarded from further analysis.

To more accurately reflect the properties of a detector at a certain time, we have introduced the partitioning of the detectors’ data into stable sub-periods. The stability is based on the long-term changes of the energy resolution at the FEP and the residual at the SEP.

For each partition, a combined calibration analysis is performed to calculate the energy resolution used for the $0\nu \beta \beta$ decay analysis. For this purpose, calibration data in a partition are combined into a single spectrum. The resolution curve is obtained by fitting a resolution model function to the obtained resolutions of individual peaks in the combined spectrum. Among the partitions, the calculated resolutions at ${\text{Q}}_{\beta \beta }$ range from 2.3 to 8.8 keV, with an exposure-weighted mean (standard deviation) of 3.0 (0.8) keV.

Alternatively, effective resolution curves per detector type are calculated by modeling the signal by a single Gaussian with a width according to the standard deviation of a Gaussian mixture of the individual detector partition contributions. Over Phase II we obtained exposure-weighted average resolutions at ${\text{Q}}_{\beta \beta }$ for the BEGe/coaxial/IC detectors of ($2.8\pm 0.3$) keV, ($4.0\pm 1.3$) keV, and ($2.9\pm 0.1$) keV respectively.

Dedicated studies were performed to determine the resolution systematic uncertainties for the $0\nu \beta \beta$ decay analysis. Various sources of systematic uncertainty on the resolution were considered: the fluctuations of the resolution and energy scale over time, and the choice of resolution function. The average total systematic uncertainty across all partitions is 0.13 keV.

The energy bias for the events near $Q_{\beta \beta }$ is estimated and corrected based on the residual of the SEP. Among the partitions, the average bias is -0.07 keV with a standard deviation of 0.29 keV. The average uncertainty of these biases is 0.17 keV.

The energy scale, partitioning, resolutions, and energy biases discussed in this paper are essential to the final search for $0\nu \beta \beta$ decay with Gerda described in [25]. The success of the Gerda program in reaching the world’s most stringent $0\nu \beta \beta$ decay half-life constraint given by $T_{1/2}^{0\nu }>1.8·{10}^{26}$ yr at 90% C.L, was achieved in part due to the excellent energy resolution offered by germanium detectors and the analysis described in this work. This is an important step towards Legend in developing the next generation of $0\nu \beta \beta$ decay ${}^{76}$Ge experiments [31].

Data Availability Statement

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: The energy resolutions at Q${}_{\beta \beta }$ are listed for each detector and partition in the Supplemental Material of Phys. Rev. Lett. 125, 252502 (2020)[25]. For further information contact the GERDA collaboration ([email protected]).]