# Development of Cryogenic Detectors for Neutrinoless Double Beta Decay Searches with CUORE and CUPID
## 1 Introduction
The peak to be searched has to be distinguished from two main background categories. On the one hand, spurious events due to radioactivity and cosmic radiation can hide the peak. On the other, the \(2\
u\beta\beta\) continuous spectrum smeared by the experimental resolution can leak in the ROI. The latter background category is unavoidable, since it comes from the \(0\
u\beta\beta\) source itself. The possibility to detect the \(0\
u\beta\beta\) signal depends on different detector parameters. In particular, the FWHM energy resolution (\(\Delta\)) and the background level (\(B\)) in the ROI play a crucial role in determining the effectiveness of a given experimental strategy. The performances of a given \(0\
u\beta\beta\) detector are parameterized with the experimental sensitivity, \(F_{D}^{0\
u}\), defined as the process half-life corresponding to the maximum signal that can be observed at a given statistical confidence level (CL). At a \(1\sigma\) level this is given by:  
\[F_{D}^{0\
u}\ =\ ln2\frac{\eta eN_{A\
u}}{A}\sqrt{\frac{T_{\rm m}M}{B\ \Delta}}\ (68\%\ {\rm C.L.}) \tag{1}\]  
where \(\eta\) is the isotopic abundance of the candidate isotope, \(\epsilon\) is the detection efficiency, \(N_{A\
u}\) is the Avogadro number, \(A\) is the mass number, \(T_{\rm m}\) is the measurement time and \(M\) is the total detector mass. The expression in Equation (1) holds when the total background level is not compatible with zero. Such a condition is met if \(M\times T_{\rm m}\times B\times\Delta>1\), that is to say when the expected number of events during the experiment measurement time is higher than one [17]. When the experiment parameters do not satisfy this boundary, the sensitivity is better represented assuming \(B\sim 0\), resulting in the following expression:  
\[F_{D}^{0\
u}(ZB)\ =\ ln2\frac{\eta eN_{A\
u}}{A}\frac{T_{\rm m}M}{n_{L}} \tag{2}\]  
where \(n_{L}\) is a constant depending on the chosen confidence level and on the actual number of observed events. In this condition, sensitivity is directly proportional to time and mass, making it possible to scale up the experiment's reach by either increasing the mass or extending the measurement period [14]. Equations (1) and (2) summarize efficiently the most important design criteria for \(0\
u\beta\beta\) detectors:  
* Minimization of the continuous background (i.e., lowering \(B\) factor), achievable by :
* Placing the experiment underground, in order to reduce the cosmic ray contribution to the experimental background;
* Building the detector with radiopure materials, in order to minimize the \(\gamma\) and \(\beta\) contribution to the background;
* Cleaning the surface of materials from radioactive contaminations, in order to reduce the degraded \(\alpha\) particle contribution to the background;
* Shielding of the detector's active volume with lead and copper layers, in order to reduce the external and setup radioactivity;
* Developing particle identification techniques, in order to discriminate degraded \(\alpha\) particles from electrons signals;
* Choosing a \(0\
u\beta\beta\) candidate isotope with high \(Q_{\beta\beta}\), in order to reduce \(\beta\) and \(\gamma\) events from the ROI and enhance the \(0\
u\beta\beta/2\
u\beta\beta\) signal ratio;
* Observation of large isotope mass (i.e., increasing \(M\) factor), achievable by:
* Choosing a \(0\
u\beta\beta\) candidate isotope with high natural isotopic abundance (high \(\eta\));
Choosing a detector technology that allows easy mass scalability, that is, to have high masses without substantial technological issues;
* Achieve good energy resolution (i.e., decreasing the \(\Delta\) factor), to reduce the \(2\
u\beta\beta\) background in the ROI;
* Long time of observation (i.e., increasing the \(T_{m}\) factor), achievable by choosing stable detectors with low maintenance issues. In this complex optimization problem, cryogenic calorimeters can play an important role, given their native characteristics [18].