We base the performance estimate of a COSINUS detector module on the following assumptions:
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light detector energy resolution from standard performance of silicon beaker light detector operated in CRESST-II phase 2 (\(\sigma =0.11\,{\mathrm{keV}_{ee}}\))
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phonon detector resolution of \(\sigma =0.2~\,\mathrm{keV}\) (corresponding to a threshold of \(5\sigma =1\,\mathrm{keV}\)),
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4 % of deposited energy in the NaI detected in the light detector,
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a gross exposure (before cuts) of 100 kg-days. Based on results from CRESST-II [19, 24], we conservatively estimate an efficiency of 50 % down to an energy of 2 keV and a linear decrease to 20 % efficiency at the threshold of 1 keV,
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a flat background contribution down to energy threshold of 1 count/(keV kg days) and an additional contribution from \(^{\text {40}}\)K at a level of 600 \(\upmu \)Bq.
In the following the above assumptions will be motivated in more detail.
In COSINUS we aim to bring the performance of a NaI-based detector in line with other scintillating bolometers, rendering an energy resolution of
\(\sigma =0.2\,\mathrm{keV}\), corresponding to an energy threshold of 1 keV, feasible. From our understanding the key points to be addressed are the optimization of the sensitivity of the TES itself and of the interface between NaI-crystal and carrier crystal.
In [21] we measured two CsI-crystals; in average around 7 % of the energy deposited in the crystal were determined as scintillation light (for an e\(^{-}\)/\(\gamma \)-event). From measurements in [13] it is known that CsI roughly emits 3.5 times more energy in form of scintillation light as NaI. However, for COSINUS beaker-shaped light detectors will be used which are found to collect more than twice as much light the light detector of conventional CRESST design used for the measurement in [21]. Thus, in total we expect \(7\,\%/3.5\times 2 = 4\,\%\) of the deposited energy in a NaI-crystal to be detected in the light detector.
Using the above assumptions a simulation is carried out – the outcome is depicted in the light yield-energy plane in Fig. 2 with data points arising from simulated contributions of e\(^-\)/\(\gamma \)-events in black and a potential signal in red. To stick to the DAMA/LIBRA background budget, an overall flat background contribution of 1 count/(keV kg days) is assumed.
NaI-crystals typically show a contamination with \(^{\text {40}}\)K which undergoes an electron capture, either directly to the ground state of \(^{\text {40}}\)Ar (0.2 % branching ratio) or to an excited state of \(^{\text {40}}\)Ar de-exciting by the emission of an 1.46 MeV gamma (10.55 % branching ratio). Both processes are accompanied by the energy releases of 3.2 and 3 keV. This contamination makes up for the line at about 3 keV in the e\(^-\)/\(\gamma \)-band in Fig. 2 corresponding to a \(^{\text {40}}\)K-activity of 600 \(\upmu \)Bq as measured by the DAMA/LIBRA collaboration [25].
The bands in Fig. 2 are calculated on the basis of the bands determined in the CsI measurement (a detailed description on the underlying model is e.g. given in [22]), the experience of beaker-shaped light detectors operated in CRESST and the assumptions listed above. Thereby, the lines depict the lower and upper 90 %-boundaries for the bands corresponding to different event classes to be observed in NaI (electron recoils in black, nuclear recoils off Na and I in blue and green, respectively). Thus, in between two corresponding lines 80 % of the events of the respective event class are expected. The widths of the bands is dominated by two effects: firstly by the finite baseline resolution of phonon and light detector and secondly by Poissonian fluctuations in the number of scintillation photons produced. The mean of a band obviously depends on the type of particle and the respective quenching factor. For scatterings off Na and I we use the energy-dependent values reported by Tretyak [23].
In summary, the overall e\(^-\)/\(\gamma \)-background (black dots) is given by the \(^{\text {40}}\)K contamination on top of the constant background level of 1 count/(keV kg days). With the performance and backgrounds assumed, we expect five counts below the mean of the Na-band for a gross exposure of 100 kg-days and an energy threshold of 1 keV. Thereby, one half of the counts originates from the constant background and the other half from the \(^{\text {40}}\)K contamination, which roughly corresponds to 0.9 % of the activity in the full double-peak. The expected leakage quickly drops with increasing energy, due to the enhanced separation between e\(^-\)/\(\gamma \)-band and Na-band. Additionally, the above numbers show that the leakage is to a large extent caused by the \(^{\text {40}}\)K contamination yielding a considerably diminishing leakage for energies above 3 keV, where we expect less than one leakage event (for 100 kg-days before cuts), which is commonly denoted background-free. Considering leakage to the full 80 % Na-band (as depicted in Fig. 2) the values change to 26 events expected above the threshold of 1 keV, out of which 13 events are attributed to \(^{\text {40}}\)K-origin. For the complete Na-band the anticipated leakage drops below one event at 3.9 keV.
Because of the high discrimination power we only anticipate a few ten leakage events in the full nuclear recoil bands for the radiopurity levels reported by the DAMA/LIBRA experiment (1 count/(keV kg days) \(\,+\,^{\text {40}}\)K: 600 \(\upmu \) Bq), while at the same time expecting \(\mathcal {O}(10^3)\) signal events. Hence, we can tolerate a higher level of e\(^-\)/\(\gamma \)-background (as e.g. the two times higher background recently reported for NaI-crystals by the KIMS collaboration [26]), in particular originating from \(^{\text {40}}\)K, while still maintaining high sensitivity for a nuclear recoil signal – a distinct feature of COSINUS compared to other NaI-based dark matter searches [6, 9–11].
The red events in Fig. 2 result from a contribution of a hypothetical WIMP of 10 GeV/\(c^{2}\) and 0.0002 pb which is consistent with the interpretation of the DAMA/LIBRA modulation signal by Savage et al. [27]. The blue-colored islands in Fig. 3 correspond to the DAMA/LIBRA signal regions for scatterings off Na and I. For clarity we indicated the assumed dark matter particle mass and nucleon cross-section for the simulated data presented here in form of a blue benchmark point (see Fig. 3).
What concerns the distribution of the hypothetical WIMP events we find in the simulation a total of 2386 events in the energy window 1–6 keV. About 45 % of these events are present in the energy interval from 1 to 2 keV and about 53 % within 2–6 keV. These numbers clearly show the benefit of a low threshold. Despite the conservative assumption on the cut efficiency being 20 % at threshold energy, the number of expected events roughly doubles when lowering the threshold from 2 to 1 keV, which is attributed to the anticipated exponential rise of the dark matter recoil spectrum. Above an energy of 6 keV (up to 100 keV) we find a total of 46 events.
Since WIMPs are expected to scatter coherently off the nucleus as a whole, the cross-section scales quadratically with the atomic mass number (A\(^2\)), thus preferring the heavy I over the light Na. As a consequence, the WIMP events in Fig. 2 appear almost symmetric to the I recoil band. However, the energy transferred in a scattering for the rather light WIMP considered here is enhanced for light nuclei due to kinematic reasons. Consequently, the majority of the 46 events above 6 keV is found in the Na recoil band. For this reason, cryogenic detectors providing low thresholds in combination with light target nuclei currently lead the field of direct dark matter searches in the low mass regime [24, 28].
At last we want to discuss and underline the prominent features of such COSINUS detector in contrast to a purely scintillating detector considering the DAMA/LIBRA experiment as representative. In Fig. 2 we display magenta colored boxes indicating the different regions that may contribute to the positive modulation signal observed by DAMA/LIBRA in an energy window from 2 to 6 keV\(_{ee}\). Due to the lack of particle discrimination it remains unknown if the positive modulation signal is made from particles scattering off the electrons or purely off the nuclei in NaI(Tl).
For interactions with electrons the box extends from 2 to 6 keV (e\(^-\)/\(\gamma \)-band), the same energy range as for DAMA/LIBRA. Instead, interactions on nuclei are quenched in the light channel, hence the energy interval for Na and I recoil events has to be corrected for by the respective quenching factor (energy-dependent QF are taken from [23]). For Na-recoils (QF \(\approx \) 0.3) the region is restricted to about 6–20 keV as qualitatively indicated by the magenta box in the Na-recoil band. For I-recoils, due to the even higher light quenching effect (QF \(\approx \) 0.1), the box is confined in an energy region of about 20–60 keV.
As already mentioned, a distinctive feature of the COSINUS technology is the unquenched phonon channel directly measuring the deposited energy quasi independent of the type of interacting particle. Taking into account the aimed for energy threshold of 1 keV, this would result in an improvement in detection threshold by a factor of about six for Na-recoil events and a factor of about 20 for I-recoils in comparison to the sensitivity demonstrated by DAMA/LIBRA.
It deserves mentioning that we already achieved, by operating two scintillating calorimeters based on CsI an energy threshold as low as 4.7 and 3.5 keV, respectively [21]. Thus, such detectors already indicate an increased sensitivity for nuclear recoil events in comparison to the DAMA/LIBRA experiment with further refinements anticipated for the future [21].