Thanks to the effective shielding scheme of the EDELWEISS experiment, the residual background mainly originates from radioactive materials inside the cryostat such as connectors, holding structure and detector copper casings as well as from decays of cosmogenically activated isotopes within the detectors [14, 15]. Each background component is modelled with a data-driven approach: unblinded data from outside the region of interest (sideband data), acquired in the same WIMP run, are fitted and extrapolated to the low-energy RoI considered in the analysis. In order to construct a likelihood model describing the data for each of the eight detectors, a PDF is calculated for each different background component i. This PDF \({\mathcal {P}}_{\mathrm{i}}\) describes a recoil spectrum \(\rho _{\mathrm{i}}(E_{\mathrm{r}})\) in the two observables heat and ionization energy. It takes into account the ionization yield \(Q_{\mathrm{i}}\) for each background, the efficiency of the trigger on the heat channel \(\varepsilon (E_{\mathrm{heat}})\) and the efficiency of the fiducial cut \(\varepsilon ^\mathrm{fid}(E_{\mathrm{r}})\) as well as a Gaussian smearing due to the degraded, energy-dependent resolutions \(\sigma _{\mathrm{heat}}\) and \(\sigma _{\mathrm{ion}}\) of a given detector. In the energy range covered by this analysis, the intrinsic widths of the \(Q_{\mathrm{i}}\)-distributions of the different populations are small compared to the effect of \(\sigma _{\mathrm{heat}}\) and \(\sigma _{\mathrm{ion}}\), and are neglected. Before normalization, the PDF can be written as:
$$\begin{aligned}&{\mathcal {P}}_{\mathrm{i}} (E_{\mathrm{heat}}, E_{\mathrm{ion}}) = \frac{ \varepsilon (E_{\mathrm{heat}}) }{ 2 \pi \sigma _{\mathrm{heat}} \sigma _{\mathrm{ion}} } \int _{0}^{\infty } \mathrm{d}E_{\mathrm{r}} \, \rho _{\mathrm{i}}(E_{\mathrm{r}}) \, \varepsilon _{\mathrm{i}}^\mathrm{fid}(E_{\mathrm{r}}) \nonumber \\&\quad \times \;\exp \left[ - \frac{(E_{\mathrm{heat}}- f_{\mathrm{i}}(E_{\mathrm{r}}))^2 }{2 \sigma _{\mathrm{heat}}^2} - \frac{ (E_{\mathrm{ion}}-Q_{\mathrm{i}} \cdot E_{\mathrm{r}} )^2 }{2 \sigma _{\mathrm{ion}}^2}\right] \end{aligned}$$
(1)
where the function \(f_{\mathrm{i}}(E_{\mathrm{r}})\) allows to calculate the observed heat signal of a given recoil energy. It includes the additional heating via the Neganov–Luke effect [16, 17], produced by the scattering of charges which are collected by electrodes with a differential voltage U (in volts):
$$\begin{aligned} f_{\mathrm{i}}(E_{\mathrm{r}}) = \frac{1 + Q_{\mathrm{i}}(E_{\mathrm{r}}) \frac{U_{\mathrm{i}}}{3} }{1 + \frac{U_{\mathrm{ref}}}{3}} \cdot E_{\mathrm{r}} \end{aligned}$$
(2)
The selected detectors have an electric potential of \(U_{\mathrm{ref}} = 8\,\mathrm{V}\) between the fiducial electrodes and bulk ER-events were used to calibrate the energy scale of all heat and fiducial ionization channels. For charges created in the near-surface volume, the Neganov–Luke contribution to the heat energy is smaller, due to the reduced potential of only \(5.5\,\mathrm{V}\) between fiducial and veto electrodes. It reduces the measured heat energy for surface events, as can be seen in Fig. 1 for the group of 10 keV cosmogenic peaks at the surface which are observed at \(E_{\mathrm{heat}} \approx 7.7\,\mathrm{keV}_{\mathrm{ee}}\). The measured average value of \(Q_{\mathrm{i}}\) for those events is 0.9. For surface backgrounds from \(\beta \)’s and \(^{206}\mathrm{Pb}\)-recoils as well as so-called heat-only events, the spectrum in heat energy was directly extracted from sideband data. For those components the smearing in heat energy is already included and the PDF before normalization can be directly expressed as:
$$\begin{aligned} {\mathcal {P}}_{\mathrm{i}} (E_{\mathrm{heat}}, E_{\mathrm{ion}})= & {} \frac{ \varepsilon (E_{\mathrm{heat}}) }{ \sqrt{2 \pi } \sigma _{\mathrm{ion}} } \, \rho _{\mathrm{i}}(E_{\mathrm{heat}}) \, \varepsilon _{\mathrm{i}}^\mathrm{fid} \left( f_{\mathrm{i}}^{-1}\left( E_{\mathrm{heat}} \right) \right) \nonumber \\&\times \; \exp \left[ - \frac{ (E_{\mathrm{ion}}-Q_{\mathrm{i}} \cdot f_{\mathrm{i}}^{-1}\left( E_{\mathrm{heat}} \right) )^2 }{2 \sigma _{\mathrm{ion}}^2} \right] \nonumber \\ \end{aligned}$$
(3)
where the average measured value of \(Q_{\mathrm{i}}\) for surface \(\beta \)’s and \(^{206}\mathrm{Pb}\)-recoils are 0.4 and 0.1, respectively. The suppression of surface events via the fiducial cut decreases at low energies due to the finite resolution of the ionization channels: the veto energy of a surface event can be smaller than the noise on the veto electrode, thus the event will not be rejected. To a small extent, our data selection is therefore polluted by surface events with heat energies just above the analysis threshold. To build the PDF for these events we take into consideration the efficiency \(\varepsilon ^\mathrm{fid} (E_{\mathrm{r}})\) as a function of recoil energy. For surface events the survival probability after the fiducial cut is highly reduced, as is shown in Fig. 2 for different background components. It is calculated for each of the detector sides considering the baseline resolution \(\sigma _{\mathrm{veto}}\) and the measured energy \(E_{\mathrm{veto}}\) of the corresponding veto electrode:
$$\begin{aligned} \varepsilon ^\mathrm{fid}_{\mathrm{surf}} (E_{\mathrm{r}})= & {} \frac{1}{ \sqrt{2 \pi } \sigma _{\mathrm{veto}} } \int _{-1.64 \sigma _{\mathrm{veto}}}^{+1.64 \sigma _{\mathrm{veto}}} \text {d}E_{\mathrm{veto}} \nonumber \\&\times \; \exp \left[ - \frac{(E_{\mathrm{veto}} - Q_{\mathrm{i}} E_{\mathrm{r}})^2}{2 \sigma _{\mathrm{veto}}^2} \right] \end{aligned}$$
(4)
For events originating in the bulk of the crystal, no signal is measured on the veto electrodes and only noise is reconstructed. The efficiency of the fiducial cut is \(\varepsilon ^\mathrm{fid}_{\mathrm{bulk}} = 81\,\%\) as described in Sect. 2. The fraction of surface nuclear recoils leaking into the acceptance below \(5\,\mathrm {keV}\) (Fig. 2), and increasing further the WIMP efficiency, is neglected. With the definition of the PDF mentioned above, the WIMP signal and the following background components can be fully described.
WIMP signal
A signal PDF is constructed for each WIMP mass \(m_\chi \) independently, using Eq. 1. The parametrization for the ionization yield \(Q_{\mathrm{NR}}\) for nuclear recoils has been validated to a precision of 5 % using neutron calibration data taken during the same run. In the description of the signal PDF, \(Q_{\mathrm{NR}}\) is a nuisance parameter and constrained with its systematic uncertainty. The recoil spectrum for the scattering of WIMPs on natural germanium with an average of \(A = 72.6\) nucleons is calculated following [18]. For all astrophysical parameters we use values corresponding to the Standard Halo Model (SHM), i.e. \(\rho _{\mathrm{DM}}^\mathrm{local} = 0.3\,\mathrm{GeV}/\mathrm{c}^2/\mathrm{cm}^3\), \(v_{\mathrm{0}} = 220\,\mathrm{km/s}\), \(v_{\mathrm{earth}} = 230\,\mathrm{km/s}\) and \(v_{\mathrm{esc}} = 544\,\mathrm{km/s}\). With the cuts described in Sect. 2 a potential WIMP signal is reduced to \({\sim }60\,\%\) for \(m_\chi = 30\,\mathrm{GeV/}c^2\). Detector FID824 has the highest sensitivity for a WIMP signal due to its good baseline of the heat channel and the resulting low heat threshold \(E_{\mathrm{heat}}^\mathrm{min} = 0.9\,\mathrm{keV}_{\mathrm{ee}}\). For this detector, the signal fraction after cuts decreases to \(2 \cdot 10^{-4}\) for a \(m_\chi = 4\,\mathrm{GeV/}c^2\) signal but is above 1 % for masses \(m_\chi > 5\,\mathrm{GeV/}c^2\).
Heat-only events
The dominant background in the EDELWEISS-III low-energy data are heat-only events. They are present in all detectors with different intensity and constitute between 85 and 95 % of the events in the RoI after all cuts. For those events the data acquisition was triggered by a clear signal on one or both NTD heat sensors, while only noise can be seen on each of the four ionization channels, and the signals of the two NTDs are compatible. The heat energy spectrum of those events shows an exponential decrease (e.g. Fig. 3) for all detectors and overlaps with randomly triggered noise fluctuations near the heat threshold of a detector. The variation of the heat-only event rate shows a common behaviour for all detectors: a simultaneous burst of the rate which coincides with a period of unstable operating conditions due to the cryogenic system followed by an exponential decay with a time constant of around 20 days which is not compatible with any of the known radioactive isotopes in the setup. A particle origin, e.g. from \(^{206}\mathrm{Pb}\)-recoils absorbed in one of the electrodes and producing no ionization signal, can be excluded due to the high rate and temporal behaviour. Internal radiation within the NTD heat sensors is rejected by a cut requiring a coincident signal in both NTDs described in [1]. The source of heat-only events is yet unknown, but possible explanations are the creation of phonons from friction of the detector with the holders, or stress near the NTD gluing spot. Several strategies are pursued to identify the origin of those events and to significantly reduce them in future runs. We use the sideband with negative ionization energy to model heat-only events in the RoI. In the absence of a theory to describe the shape of the heat-only energy spectrum, we use a Kernel density estimation (KDE) function of the data in this sideband to model this background. The ionization energy spectrum has a gaussian shape with a width given by the average baseline noise for the ionization channels. Fitting the distribution of sideband data in \(E_{\mathrm{ion}}\) with a gaussian indicates a small possible shift of the mean with respect to \(E_{\mathrm{ion}} = 0\,\mathrm{keV}_{\mathrm{ee}}\). That shift is only statistically significant for some of the detectors and is related to a small fraction of \({<}1\,\%\) uncorrected cross-talk between heat and fiducial ionization channels. The effect of a possible shift on the number of expected events for this background is taken as a systematic uncertainty and ranges between 0.4 and 14.9 % and is considered in the constraint for this background. For the most sensitive detector, FID824, Fig. 3 shows heat and ionization energy spectra of the sideband data with the respective models. In principle, the heat-only sideband can be contaminated by underfluctuations of the ionization energy from low-energy event populations with small ionization yield, such as \(^{206}\mathrm{Pb}\)-recoils and \(\beta \)-particles. Considering the low number of expected events for these components (\(\mathcal {O}(10)\) events above the analysis threshold per detector) compared to the high rate of heat-only events, the effect on the extracted spectrum is negligible. It was also checked that the number of events for a possible WIMP signal of mass \(m_\chi \) in the heat-only sideband is negligible for the cross section excluded in the following. The heat-only sideband data (\(E_{\mathrm{ion}} < 0\)) and modelled PDF in the RoI (\(E_{\mathrm{ion}} > 0\)) are shown in Fig. 4, together with WIMP signals for two different masses.
Electron recoils from Gammas and Betas
The energy spectrum of electron recoils in the fiducial volume up to \(15\,\mathrm{keV}_{\mathrm{ee}}\) consists of a set of peaks on top of a continuous component. This component is due to the Compton scattering of gamma rays from external radioactive sources and to betas from the decay of \(^{3}\mathrm{H}\) inside the detectors [19]. The observed peaks are produced by mono-energetic gammas from electron capture reactions within the crystal and result from the activation of different isotopes due to cosmic rays or neutron calibration. The intensity of these peaks is different for each detector and depends on its age and exposure to cosmic rays before installation underground. In the energy range between \(5\,\mathrm{keV}_{\mathrm{ee}}\) and \(7.7\,\mathrm{keV}_{\mathrm{ee}}\) X-rays from the K-shell EC of the isotopes \(^{49}\mathrm{V}\) (\(E = 4.97\,\mathrm{keV}\)), \(^{51}\mathrm{Cr}\) (\(5.46\,\mathrm{keV}\)), \(^{54}\mathrm{Mn}\) (\(5.99\,\mathrm{keV}\)), \(^{55}\mathrm{Fe}\) (\(6.54\,\mathrm{keV}\)), \(^{56,57,58}\mathrm{Co}\) (\(7.11\,\mathrm{keV}\)) and \(^{56}\mathrm{Ni}\) (\(7.71\,\mathrm{keV}\)) are included in the fit as potential peaks. Around \(10\,\mathrm{keV}_{\mathrm{ee}}\) a triplet of \(^{65}\mathrm{Zn}\) (\(8.98\,\mathrm{keV}\)), \(^{68}\,\mathrm{Ga}\) (\(9.66\,\mathrm{keV}\)) and \(^{68}\mathrm{Ge}\) (\(10.37\,\mathrm{keV}\)) can be resolved, which has corresponding L-shell peaks at 1.10, 1.19 and \(1.30\,\mathrm{keV}\) (Fig. 5, light blue). While the K-shell peaks are well separated from a WIMP signal in the analysis parameter space, the 3 L-shell peaks can have significant overlap with a signal for the lowest WIMP masses probed. Depending on the analysis threshold \(E_\text {heat}^\mathrm{min}\) of each detector, the fraction of those peaks in the RoI can vary significantly, from almost full coverage to only a tail of the gaussian peak. With the known L/K-shell ratio of 11 % [20] and the calculated peak fraction above threshold we extrapolate the rate of L-shell X-ray events. For this we perform a sideband fit of fiducial events in the electron recoil band \(3\,\mathrm{keV}_{\mathrm{ee}}< E_{\mathrm{heat}}, E_{\mathrm{ion}} < 30\,\mathrm{keV}_{\mathrm{ee}}\) with a separate likelihood model including all K-shell peaks, Compton gammas and tritium \(\beta \)-events. We find the extrapolated rate of tritium decay for each detector to be in agreement within uncertainties with the rates found in [19]. Systematic uncertainties for all ER-components in the RoI are propagated from the errors of this sideband fit and are typically \(\mathcal {O}(30\,\%)\).
Unrejected surface events
At higher energies, the fiducial cut allows the rejection of all surface events, as they would induce a clear signal \(E_{\mathrm{veto}}\) on one of the two veto electrodes. For low ionization energies however, the rejection can fail. If the ionization energy of a surface event is low enough, so that \(E_{\mathrm{veto}} < 1.64 \, \sigma _{\mathrm{veto}}\), the event passes the cut. For particle types with low ionization yield \(Q_{\mathrm{i}}\), the produced ionization energy is smaller, and therefore less charge is collected on the veto electrodes to reject surface events. The surface events in this analysis are mostly \(^{206}\mathrm{Pb}\)-recoils and \(\beta \)-particles originating from the \(^{238}\mathrm{U}\) decay chain of surrounding materials such as \(^{222}\mathrm{Rn}\) daughter isotopes [15]. Those particles have a small penetration depth or even scatter on the crystal surface. Another possible component would originate from the electron recoils described in Sect. 3.3, which are also produced in the near-surface volume. However, due to their high ionization yield of \(Q_{\mathrm{ER}} \approx 1\), the rejection of these surface events above the heat threshold \(E_{\mathrm{heat}}^\mathrm{min}\) is very efficient: the expected number of events in the RoI after the applied fiducial cut was calculated to be well below \(10^{-2}\) for all detectors and these events are therefore not considered in the analysis. For both \(\beta \)’s and \(^{206}\mathrm{Pb}\)-recoils, the spectrum in heat energy is extracted from a clear selection of surface events with energies \(E_{\mathrm{veto}} > 5\,\sigma _{\mathrm{veto}}\) and then extrapolated to the lower heat threshold within the RoI. The ionization yield of the events is fitted from the same sideband data. We do not include any uncertainty on the fitted \(Q_i\) as it is negligible with respect to the smearing due to the energy resolutions. Both energy spectra and ionization yield are determined for top and bottom surface of each detector independently.
Nuclear recoils from neutrons
Neutron background can mimic a WIMP signal, as neutrons can produce single scatter nuclear recoils with the same ionization yield \(Q_{\mathrm{NR}}\) as WIMPs, according to an exponential energy spectrum. We distinguish between two different sources of neutrons in our detectors: muon-induced and radiogenic neutrons. Simulations showed that in the energy range of this analysis, the number of single scattering neutrons induced by muons is compatible with zero after vetoing [21]. For radiogenic neutrons coming from radioactivity due to (\(\alpha \), n) reactions and spontaneous fissions within the cryostat, Monte Carlo simulations have been performed with all known sources to derive their energy spectrum down to the lowest energies. The spectral shape of the radiogenic neutron background shows little dependence on the exact location of individual sources and can be fitted and parametrized by a double exponential law in the energy range of 2–20 \(\mathrm{keV}_{\mathrm{nr}}\), calibrated for nuclear recoil interactions. The normalization of the spectrum is derived from data taken with 17 detectors during the same EDELWEISS-III physics run. In the energy range of 10–100 \(\mathrm{keV}_{\mathrm{nr}}\), nine multiple scattering events are found in the \(90\,\%\,\mathrm{C.L.}\) nuclear recoil band for a fiducial exposure of 1309 kg-days. This number cannot be reproduced with the simulation of all known sources and hints at an additional neutron source in the experiment. The Monte Carlo simulation however is able to reproduce the measured single-over-multiple-ratio within uncertainties. We derive the normalized neutron spectrum for each detector by weighting it with corresponding exposure in the present data set, as well as the single-over-multiple-ratio of 0.45 from simulations. After all cuts and efficiency corrections, the expected background from single scatter neutrons in the RoI is similar for all detectors and has an average value of \(\mu ^\mathrm{exp}_{\mathrm{neutron}} = 0.20 \pm 0.07\,\mathrm{events}\) (Table 1). Expected rates for individual detectors have a combined uncertainty of \(45\,\%\) coming from the single-over-multiple ratio uncertainty and the statistical error from the measurement of multiples.
Table 1 Rate of expected events for different types of backgrounds for detector FID824 and all detectors combined. Event rates for components of the same type have been summed up for demonstration purposes only with propagated systematic errors. During fitting all components are considered as separate PDFs with individual constraints. The background model is clearly dominated by heat-only events
With respect to the BDT based analysis [1], most of the background components listed above are identical. Deviations are mainly related to the different fiducial cut and the resulting survival probability of background components. The preselection applied before the BDT analysis accepts more surface beta and gamma events than the present stricter fiducial cuts, leaving the BDT a larger population of these events to optimize its multi-parametric selection. The present fiducial cut effectively removes most of them. For the same reason we do not include so-called triple events with a signal on both fiducial and one veto electrode. Lastly we intentionally differentiate between bulk events from Compton \(\gamma \)’s and tritium \(\beta \)’s as two separate components in the likelihood analysis although their energy spectra are approximately degenerated in the RoI. An overview of the expected event rates summarized for different types of backgrounds is given in Table 1. The total background for both detector FID824 and all detectors combined is within 1–2 % agreement with the observed number of events.