CUORE sensitivity to \(0\nu \beta \beta \) decay
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Abstract
We report a study of the CUORE sensitivity to neutrinoless double beta (\(0\nu \beta \beta \)) decay. We used a Bayesian analysis based on a toy Monte Carlo (MC) approach to extract the exclusion sensitivity to the \(0\nu \beta \beta \) decay halflife (\(T_{1/2}^{\,0\nu }\)) at \(90\%\) credibility interval (CI) – i.e. the interval containing the true value of \(T_{1/2}^{\,0\nu }\) with \(90\%\) probability – and the \(3~\sigma \) discovery sensitivity. We consider various background levels and energy resolutions, and describe the influence of the data division in subsets with different background levels. If the background level and the energy resolution meet the expectation, CUORE will reach a \(90\%\) CI exclusion sensitivity of \(2\cdot 10^{25}\) year with 3 months, and \(9\cdot 10^{25}\) year with 5 years of live time. Under the same conditions, the discovery sensitivity after 3 months and 5 years will be \(7\cdot 10^{24}\) year and \(4\cdot 10^{25}\) year, respectively.
1 Introduction
Neutrinoless double beta decay is a non Standard Model process that violates the total lepton number conservation and implies a Majorana neutrino mass component [1, 2]. This decay is currently being investigated with a variety of double beta decaying isotopes. A recent review can be found in Ref. [3]. The cryogenic underground observatory for rare events (CUORE) [4, 5, 6] is an experiment searching for \(0\nu \beta \beta \) decay in \(^{130}\)Te. It is located at the Laboratori Nazionali del Gran Sasso of INFN, Italy. In CUORE, 988 TeO\(_2\) crystals with natural \(^{130}\)Te isotopic abundance and a 750 g average mass are operated simultaneously as source and bolometric detector for the decay. In this way, the \(0\nu \beta \beta \) decay signature is a peak at the Qvalue of the reaction (\(Q_{\beta \beta }\), 2527.518 keV for \(^{130}\)Te [7, 8, 9]). Bolometric crystals are characterized by an excellent energy resolution (\({\sim }0.2\%\) Full Width at Half Maximum, FWHM) and a very low background at \(Q_{\beta \beta }\), which is expected to be at the \(10^{\text{ }2}\) cts\(/(\)keV\(\cdot \)kg\(\cdot \)yr\()\) level in CUORE [10].
The current best limit on \(0\nu \beta \beta \) decay in \(^{130}\)Te comes from a combined analysis of the CUORE0 [11, 12] and Cuoricino data [13, 14]. With a total exposure of 29.6 kg\(\cdot \)year, a limit of \(T_{1/2}^{0\nu }>4.0\cdot 10^{24}\) year (\(90\%\) CI) is obtained [15] for the \(0\nu \beta \beta \) decay half life, \(T_{1/2}^{\,0\nu }\).
After the installation of the detector, successfully completed in the summer 2016, CUORE started the commissioning phase at the beginning of 2017. The knowledge of the discovery and exclusion sensitivity to \(0\nu \beta \beta \) decay as a function of the measurement live time can be exploited to set the criteria for the unblinding of the data and the release of the \(0\nu \beta \beta \) decay analysis results.
In this work, we dedicate our attention to those factors which could strongly affect the sensitivity, such as the background index (\(BI\)) and the energy resolution at \(Q_{\beta \beta }\). In CUORE, the crystals in the outer part of the array are expected to show a higher \(BI\) than those in the middle [10]. Considering this and following the strategy already implemented by the Gerda Collaboration [16, 17], we show how the division of the data into subsets with different \(BI\) could improve the sensitivity.
The reported results are obtained by means of a Bayesian analysis performed with the Bayesian analysis toolkit (BAT) [18]. The analysis is based on a toyMC approach. At a cost of a much longer computation time with respect to the use of the median sensitivity formula [19], this provides the full sensitivity probability distribution and not only its median value.
In Sect. 2, we review the statistical methods for the parameter estimation, as well as for the extraction of the exclusion and discovery sensitivity. Section 3 describes the experimental parameters used for the analysis while its technical implementation is summarized in Sect. 4. Finally, we present the results in Sect. 5.
2 Statistical method

H indicates both a hypothesis and the corresponding model;

\(H_0\) is the backgroundonly hypothesis, according to which the known physics processes are enough to explain the experimental data. In the present case, we expect the CUORE background to be flat in a 100 keV region around \(Q_{\beta \beta }\), except for the presence of a \(^{60}\)Co summation peak at 2505.7 keV. Therefore, \(H_0\) is implemented as a flat background distribution plus a Gaussian describing the \(^{60}\)Co peak. In CUORE0, this peak was found to be centered at an energy \(1.9\pm 0.7\) keV higher than that tabulated in literature [15]. This effect, present also in Cuoricino [14], is a feature of all gamma summation peaks. Hence, we will consider the \(^{60}\)Co peak to be at 2507.6 keV.

\(H_1\) is the backgroundplussignal hypothesis, for which some new physics is required to explain the data. In our case, the physics involved in \(H_1\) contains the background processes as well as \(0\nu \beta \beta \) decay. The latter is modeled as a Gaussian peak at \(Q_{\beta \beta }\).

\(\mathbf {E}\) represents the data. It is a list of N energy bins centered at the energy \(E_i\) and containing \(n_i\) event counts. The energy range is [2470; 2570] keV. This is the same range used for the CUORE0 \(0\nu \beta \beta \) decay analysis [15], and is bounded by the possible presence of peaks from \(^{214}\)Bi at 2447.7 keV and \(^{208}\)Tl Xray escape at \({\sim }2585\) keV [15]. While an unbinned fit allows to fully exploit the information contained in the data, it can result in a long computation time for large data samples. Given an energy resolution of \({\sim }5\) keV FWHM and using a 1 keV bin width, the \(\pm 3\) sigma range of a Gaussian peak is contained in 12.7 bins. With the 1 keV binning choice, the loss of information with respect to the unbinned fit is negligible.
 \(\Gamma ^{0\nu }\) is the parameter describing the \(0\nu \beta \beta \) decay rate for \(H_1\):$$\begin{aligned} \Gamma ^{0\nu } = \frac{\ln {2}}{T_{1/2}^{0\nu }}. \end{aligned}$$(1)

\(\mathbf {\theta }\) is the list of nuisance parameters describing the background processes in both \(H_0\) and \(H_1\);

\(\Omega \) is the parameter space for the parameters \(\mathbf {\theta }\).
2.1 Parameter estimation
2.2 Exclusion sensitivity
In the Bayesian approach, the limit is a statement regarding the true value of the considered physical quantity. In our case, a \(90\%\) CI limit on \(T_{1/2}^{\,0\nu }\) is to be interpreted as the value above which, given the current knowledge, the true value of \(T_{1/2}^{\,0\nu }\) lies with \(90\%\) probability. This differs from a frequentist \(90\%\) C.L. limit, which is a statement regarding the possible results of the repetition of identical measurements and should be interpreted as the value above which the bestfit value of \(T_{1/2}^{\,0\nu }\) would lie in the \(90\%\) of the imaginary identical experiments.
In order to extract the exclusion sensitivity, we generate a set of N toyMC spectra according to the backgroundonly model, \(H_0\). We then fit spectra with the backgroundplussignal model, \(H_1\), and obtain the \(T_{1/2}^{\,0\nu }\left( 90\%\ CI \right) \) distribution (Fig. 1, bottom). Its median \(\hat{T}_{1/2}^{\,0\nu }\left( 90\%\ CI \right) \) is referred as the median sensitivity. For a real experiment, the experimental \(T_{1/2}^{\,0\nu }\) limit is expected to be above/below \(\hat{T}_{1/2}^{\,0\nu }\left( 90\%\ CI \right) \) with 50% probability. Alternatively, one can consider the mode of the distribution, which corresponds to the most probable \(T_{1/2}^{\,0\nu }\) limit.

for each subset, we generate a random number of background events \(N_d^{bkg}\) according to a Poisson distribution with mean \(\lambda ^{bkg}_d\);

for each subset, we generate \(N_d^{bkg}\) events with an energy randomly distributed according to \(f_{bkg}(E)\);

we repeate the procedure for the \(^{60}\)Co contribution;

we fit the toyMC spectrum with the \(H_1\) model (Eq. 2), and marginalize the likelihood with respect to the parameters \(BI_d\) and \(R_d^{Co}\) (Eq. 3);

we extract the \(90\%\) CI limit on \(T_{1/2}^{\,0\nu }\);

we repeat the algorithm for N toyMC experiments, and build the distribution of \(T_{1/2}^{\,0\nu }\left( 90\%\ CI \right) \).
2.3 Discovery sensitivity
The discovery sensitivity provides information on the required strength of the signal amplitude for claiming that the known processes alone are not sufficient to properly describe the experimental data. It is computed on the basis of the comparison between the backgroundonly and the backgroundplussignal models. A method for the calculation of the Bayesian discovery sensitivity was introduced in Ref. [21]. We report it here for completeness.

we produce a toyMC spectrum according to the \(H_1\) model with an arbitrary value of \(T_{1/2}^{\,0\nu }\);

we fit the spectrum with both \(H_0\) and \(H_1\);

we compute \(P(H_0  \mathbf {E})\);

we repeat the procedure for N toyMC spectra using the same \(T_{1/2}^{\,0\nu }\);

we repeat the routine with different values of \(T_{1/2}^{\,0\nu }\) until the condition of Eq. 20 is satisfied. The iteration is implemented using the bisection method until a \(5\cdot 10^{\text{ }5}\) precision is obtained on the median \(P(H_0  \mathbf {E})\).
3 Experimental parameters
The fit parameters of the \(H_1\) model are \(BI\), \(R^{Co}\) and \(\Gamma ^{0\nu }\), while only the first two are present for \(H_0\). If the data are divided in subsets, different \(BI\) and \(R^{Co}\) fit parameter are considered for each subset. On the contrary, the inverse \(0\nu \beta \beta \) halflife is common to all subsets.
Prior to the assembly of the CUORE crystal towers, we performed a screening survey of the employed materials [22, 23, 24, 25, 26, 27, 28, 29]. From these measurements, either a nonzero activity was obtained, or a \(90\%\) confidence level (C.L.) upper limit was set. Additionally, the radioactive contamination of the crystals and holders was also obtained from the CUORE0 background model [30]. We developed a full MC simulation of CUORE [10], and we used the results of the screening measurements and of the CUORE0 background model for the normalization of the simulated spectra. We then computed the \(BI\) at \(Q_{\beta \beta }\) using the output of the simulations. In the present study, we consider only those background contributions for which a nonzero activity is obtained from the available measurements. The largest background consists of \(\alpha \) particles emitted by U and Th surface contaminations of the copper structure holding the crystals. Additionally, we consider a \(^{60}\)Co contribution normalized to the \(90\%\) C.L. limit from the screening measurement. In this sense, the effect of a \(^{60}\)Co background on the CUORE sensitivity is to be held as an upper limit. Given the \(^{60}\)Co importance especially in case of suboptimal energy resolution, we preferred to maintain a conservative approach in this regard. In the generation of the toyMC spectra, we take into account the \(^{60}\)Co half life (5.27 year), and set the start of data taking to January 2017.
Input parameters for the production of toyMC spectra
\(BI\) [cts / (keV\(\cdot \)kg\(\cdot \)year)]  \(R^{Co}\) [cts / (kg\(\cdot \)year)] 

\(\left( 1.02\pm 0.03(\text {stat})^{+0.23}_{0.10}(\text {syst})\right) \cdot 10^{\text{ }2}\)  0.428 
Crystal subsets with different expected \(\alpha \) background in CUORE. The values of BI and \(R^{Co}\) are taken from [10]
Subset name  Free sides  Number of crystals  \(BI\) [cts/(keV\(\cdot \)kg\(\cdot \)year)]  \(R^{Co}\) [cts(kg\(\cdot \)year)] 

Inner  0  528  \(0.82(2)\cdot 10^{\text{ }2}\)  0.40 
Middle1  1  272  \(1.17(4)\cdot 10^{\text{ }2}\)  0.47 
Middle2  2  164  \(1.36(4)\cdot 10^{\text{ }2}\)  0.43 
Outer  3  24  \(1.78(7)\cdot 10^{\text{ }2}\)  0.59 
A major ingredient of a Bayesian analysis is the choice of the priors. In the present case, we use a flat prior for all parameters. In particular, the prior distribution for \(\Gamma ^{0\nu }\) is flat between zero and a value large enough to contain \({>}99.999\%\) of its posterior distribution. This corresponds to the most conservative choice. Any other reasonable prior, e.g. a scale invariant prior on \(\Gamma ^{0\nu }\), would yield a stronger limit. A different prior choice based on the real characteristic of the experimental spectra might be more appropriate for \(BI\) and \(R^{Co}\) in the analysis of the CUORE data. For the time being the lack of data prevents the use of informative priors. As a crosscheck, we performed the analysis using the \(BI\) and \(^{60}\)Co rate uncertainties obtained by the background budget as the \(\sigma \) of a Gaussian prior. No significant difference was found on the sensitivity band because the Poisson fluctuations of the generated number of background and \(^{60}\)Co events are dominant for the extraction of the \(\Gamma ^{0\nu }\) posterior probability distribution.
Table 3 lists the constant quantities present in the formulation of \(H_0\) and \(H_1\). All of them are fixed, with the exception of the live time t and the FWHM of the \(0\nu \beta \beta \) decay and \(^{60}\)Co Gaussian peaks. We perform the analysis with a FWHM of 5 and 10 keV, corresponding to a \(\sigma \) of 2.12 and 4.25 keV, respectively. Regarding the efficiency, while in the toyMC production the BI and \(R^{Co}\) are multiplied by the instrumental efficiency,^{1} in the fit the total efficiency is used. This is the product of the containment and instrumental efficiency. Also in this case, we use the same value as for CUORE0, i.e. \(81.3\%\) [15]. We evaluate the exclusion and discovery sensitivities for different live times, with t ranging from 0.1 to 5 year and using logarithmically increasing values: \(t_{i} = 1.05\cdot t_{i1}\).
4 Fit procedure
We perform the analysis with the software BAT v1.1.0DEV [21], which internally uses CUBA [31] v4.2 for the integration of multidimensional probabilities and the MetropolisHastings algorithm [32] for the fit. The computation time depends on the number of samples drawn from the considered probability distribution.
For the exclusion sensitivity, we draw \(10^5\) likelihood samples for every toyMC experiment, while, due to the higher computational cost, we use only \(10^3\) for the discovery sensitivity.
Constants used in \(H_0\) and \(H_1\)
Constant  Symbol  Value 

Detector mass  \(m_d\)  741.67 kg 
Avogadro number  \(N_A\)  \(6.022\cdot 10^{23}\) mol\(^{1}\) 
Molar mass  \(m_A\)  159.6 g/mol 
Live time  \(t_d\)  0.1–5 year 
Efficiency  \(\varepsilon _{\mathrm{tot}}\)  81.3% 
\(^{130}\)Te abundance  \(f_{130}\)  0.34167 
\(0\nu \beta \beta \) Qvalue  \(Q_{\beta \beta }\)  2527.518 keV 
\(^{60}\)Co peak position  \(\mu _{Co}\)  \((2505.692+1.9)\) keV 
Energy resolution  FWHM  5,10 keV 
5 Results and discussion
5.1 Exclusion sensitivity
Median exclusion sensitivity for different energy resolutions and different subset numbers
FWHM [keV]  \(N_d\)  \(\hat{T}_{1/2}^{\,0\nu }\) at 0.25 year [year]  \(\hat{T}_{1/2}^{\,0\nu }\) at 5 year [year] 

5  1  \(1.7\cdot 10^{25}\)  \(8.9\cdot 10^{25}\) 
10  1  \(1.2\cdot 10^{25}\)  \(6.1\cdot 10^{25}\) 
5  4  \(1.8\cdot 10^{25}\)  \(9.1\cdot 10^{25}\) 
10  4  \(1.3\cdot 10^{25}\)  \(6.2\cdot 10^{25}\) 
Ideally, the final CUORE \(0\nu \beta \beta \) decay analysis should be performed keeping the spectra collected by each crystal separate, additionally to the usual division of the data into data sets comprised by two calibration runs [15]. Assuming an average frequency of one calibration per month, the total number of energy spectra would be \({\sim }6\cdot 10^4\). Assuming a different but stationary \(BI\) for each crystal, and using the same \(^{60}\)Co rate for all crystals, the fit model would have \({\sim }10^3\) parameters. This represents a major obstacle for any existing implementation of the MetropolisHastings or Gibbs sampling algorithm. A possible way to address the problem might be the use of different algorithms, e.g. nested sampling [35, 36], or a partial analytical solution of the likelihood maximization.
We perform two further crosschecks in order to investigate the relative importance of the flat background and the \(^{60}\)Co peak. In the first scenario we set the \(BI\) to zero, and do the same for the \(^{60}\)Co rate in the second one. In both cases, the data are not divided into subsets, and resolutions of 5 and 10 keV are considered. With no flat background and a 5 keV resolution, no \(^{60}\)Co event leaks in the \(\pm 3\sigma \) region around \(Q_{\beta \beta }\) even after 5 year of measurement. As a consequence, the \(90\%\) CI limits are distributed on a very narrow band, and the median sensitivity reaches \(1.2\cdot 10^{27}\) year after 5 year of data collection. On the contrary, if we assume a 10 keV FWHM, some \(^{60}\)Co events fall in the \(0\nu \beta \beta \) decay ROI from the very beginning of the data taking. This results in a strong asymmetry of the sensitivity band. In the second crosscheck, we keep the \(BI\) at \(1.02\cdot 10^{\text{ }2}\) cts\(/(\)keV\(\cdot \)kg\(\cdot \)yr\()\), but set the \(^{60}\)Co rate to zero. In both cases, the difference with respect to the standard scenario is below \(1\%\). We can conclude that the \(^{60}\)Co peak with an initial rate of 0.428 cts/(kg\(\cdot \)yr) is not worrisome for a resolution of up to 10 keV, and that the lower sensitivity obtained with 10 keV FWHM with respect to the 5 keV case is ascribable to the relative amplitude of \(\lambda ^{bkg}_{di}\) and \(\lambda ^{0\nu }_{di}\) only (Eqs. 9 and 13). This is also confirmed by the computation of the sensitivity for the optimistic scenario without the 1.9 keV shift of the \(^{60}\)Co peak used in the standard case.
5.2 Discovery sensitivity
The extraction of the discovery sensitivity involves fits with the backgroundonly and the backgroundplussignal models. Moreover, two multidimensional integrations have to be performed for each toyMC spectrum, and a loop over the \(0\nu \beta \beta \) decay halflife has to be done until the condition of Eq. 20 is met. Due to the high computation cost, we compute the \(3~\sigma \) discovery sensitivity for a FWHM of 5 and 10 keV with no crystal subdivision. As shown in Fig. 3, with a 5 keV energy resolution CUORE has a \(3~\sigma \) discovery sensitivity superior to the limit obtained from the combined analysis of Cuore0 and Cuoricino data [15] after less than one month of operation, and reaches \(3.7\cdot 10^{25}\) year with 5 year of live time.
Also in this case, the pulls are characterized by an RMS smaller than expected, but no bias is present due to the use of \(H_1\) for both the generation and the fit of the toyMC spectra.
6 Conclusion and outlook
We implemented a toyMC method for the computation of the exclusion and discovery sensitivity of CUORE using a Bayesian analysis. We have highlighted the influence of the \(BI\) and energy resolution on the exclusion sensitivity, showing how the achievement of the expected 5 keV FWHM is desirable. Additionally, we have shown how the division of the data into subsets with different \(BI\) could yield an improvement in exclusion sensitivity.
Once the CUORE data collection starts and the experimental parameters are available, the sensitivity study can be repeated in a more detailed way. As an example, nonGaussian spectral shapes for the \(0\nu \beta \beta \) decay and \(^{60}\)Co peaks can be used, and the systematics of the energy reconstruction can be included.
Footnotes
Notes
Acknowledgements
The CUORE Collaboration thanks the directors and staff of the Laboratori Nazionali del Gran Sasso and the technical staff of our laboratories. CUORE is supported by The Istituto Nazionale di Fisica Nucleare (INFN); The National Science Foundation under Grant Nos. NSFPHY0605119, NSFPHY0500337, NSF PHY0855314, NSFPHY0902171, NSFPHY0969852, NSFPHY1307204, NSFPHY1314881, NSFPHY 1401832, and NSFPHY1404205; The Alfred P. Sloan Foundation; The University of Wisconsin Foundation; Yale University; The US Department of Energy (DOE) Office of Science under Contract Nos. DEAC0205CH11231, DEAC5207NA27344, and DESC0012654; The DOE Office of Science, Office of Nuclear Physics under Contract Nos. DEFG0208ER41551 and DEFG0300ER41138; The National Energy Research Scientific Computing Center (NERSC).
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