Analysis techniques for the evaluation of the neutrinoless double-$\beta$ decay lifetime in ${}^{130}\mathrm{Te}$ with the CUORE-0 detector

We describe in detail the methods used to obtain the lower bound on the lifetime of neutrinoless double-beta ($0\nu \beta \beta$) decay in ${}^{130}\mathrm{Te}$ and the associated limit on the effective Majorana mass of the neutrino using the CUORE-0 detector. CUORE-0 is a bolometric detector array located at the Laboratori Nazionali del Gran Sasso that was designed to validate the background reduction techniques developed for CUORE, a next-generation experiment scheduled to come online in 2016. CUORE-0 is also a competitive $0\nu \beta \beta$ decay search in its own right and functions as a platform to further develop the analysis tools and procedures to be used in CUORE. These include data collection, event selection and processing, as well as an evaluation of signal efficiency. In particular, we describe the amplitude evaluation, thermal gain stabilization, energy calibration methods, and the analysis event selection used to create our final $0\nu \beta \beta$ search spectrum. We define our high level analysis procedures, with emphasis on the new insights gained and challenges encountered. We outline in detail our fitting methods near the hypothesized $0\nu \beta \beta$ decay peak and catalog the main sources of systematic uncertainty. Finally, we derive the $0\nu \beta \beta$ decay half-life limits previously reported for CUORE-0, $T_{1/2}^{0\nu }>2.7\times {10}^{24}\mathrm{yr}$, and in combination with the Cuoricino limit, $T_{1/2}^{0\nu }>4.0\times {10}^{24}\mathrm{yr}$.

Figure 1

(a) CUORE-0 tower array rendering. The tower consists of 13 floors of four bolometers, mounted in a copper frame. (b) Schematic of a single CUORE-0 bolometer showing the thermistor (T), the heater (H), and the weak thermal link (L) between ${\mathrm{TeO}}_2$ crystal and copper thermal bath (not to scale). (c) An example of a bolometer signal with an energy of approximately 2615 keV. The rise and fall times of this signal are 0.05 s and 0.2 s, respectively. Figure from [21].

Figure 2

The distribution of amplitudes of the noise pulses collected from a single channel during the physics data of a dataset from Campaign I. The widths of the above distributions are indicative of the amount of noise remaining after filtering. The channel presented is one where the DOF performed well. The raw unfiltered rms is 2.7 mV (black dotted histogram); the rms after OF is 1.1 mV (blue solid histogram); the rms after DOF is 0.8 mV (red dashed histogram).

Figure 3

Example of the calibration-TGS. The points are taken during the calibration runs for one of the CUORE-0 datasets. The cluster of points on the right are from the calibration runs which opened the dataset, while the cluster on the left are from the closing calibration runs. The solid blue points have energies around the 2615 keV ${}^{208}\mathrm{Tl}$ peak and are used for calibration-TGS. By regressing the measured amplitudes of these points against the NTD voltage we can determine a stabilization curve (red dashed line) which is then applied to the physics runs taken between calibrations.

Figure 4

Plot of the two absorbed energies in double-crystal coincidences during physics data collection. The diagonal lines correspond to events where a $\gamma$ scatters in one crystal and is then fully absorbed in another. The vertical and horizontal lines are cascade events where one $\gamma$ is fully absorbed and the other is scattered. This can be seen for the two ${}^{60}\mathrm{Co}\gamma$ rays, ${}^{208}\mathrm{Tl}$ 2615 keV $+583$ keV $\gamma$ rays, and ${}^{208}\mathrm{Tl}$ pair production events where one annihilation photon escapes and is absorbed in another bolometer.

Figure 5

Distribution of measured time between coincident events before correcting for different detectors' rise time (black solid, $\mathrm{\Delta }{T}_{\mathrm{Raw}}$) and after (red dashed, $\mathrm{\Delta }{T}_{\mathrm{Sync}}$).

Figure 6

The final CUORE-0 physics spectrum (blue) and calibration spectrum (red). The calibration spectrum has been normalized to match the rate of the physics spectrum around the 2615 keV ${}^{208}\mathrm{Tl}$ peak. The most prominent peaks in the physics spectrum are from the decay of known radioactive backgrounds: (1) $e^{+}{e}^{-}$ annihilation, (2) ${}^{214}\mathrm{Bi}$, (3) ${}^{40}\mathrm{K}$, (4) ${}^{208}\mathrm{Tl}$, (5) ${}^{60}\mathrm{Co}$, and (6) ${}^{228}\mathrm{Ac}$. Figure adapted from [11].

Figure 7

(a) The CUORE-0 spectrum around the ROI. This is a zoomed view of Fig. 6. The shaded region corresponds to the energy range used in the ROI fit. The background in the ROI is now dominated by the scattered-$\gamma$ background rather than the flat $\alpha$ background. (b) Comparison of the blinded (dashed lines) and unblinded (solid lines) spectra in the ROI. The peak in the dashed spectrum is the artificial peak created by the blinding procedure. $Q_{\beta \beta }$ is indicated by the dotted line.

Figure 8

(a) Best-fit model to the calibration 2615 keV peak for a single bolometer dataset. The solid line shows the full summed model, and the dashed lines show the individual components: (i) the primary peak of the detector response; (ii) the secondary peak of the detector response; (iii) the smeared step-function; (iv) the x-ray escape peak; (v) flat background. (b) The calibration spectrum summed over all BoDs pairs, with the summed best-fit model. Figure adapted from [11].

Figure 9

(a) Residuals of the best-fit reconstructed peak energy and expected peak energy (fit–expected) for the physics data (blue circles) and calibration data (red triangles)—the statistical errors on the calibration points are negligible. The blue curve and shaded band are the fit to the physics peak residuals $\mathrm{\Delta }\mu \left(E\right)$ and the $1\sigma$ uncertainty band. Figure adapted from [11]. (b) Best-fit resolution scaling parameter, ${\alpha }_{\sigma }$, for a few of the peaks in the physics spectrum, as well as the best-fit interpolation.

Figure 10

Bottom panel: The best-fit model (solid blue lines) overlaid on the CUORE-0 energy spectrum (data points). For simplicity, the data are shown with Gaussian error bars. The peak at 2507 keV is due to ${}^{60}\mathrm{Co}$ and the dash-dotted line indicates the position at which we expect a potential $0\nu \beta \beta$ decay signal. The dashed black line indicates the continuum background component in the ROI. Top panel: The normalized residuals of the best fit model and the binned data points. Figure from [11].

Figure 11

The distribution of FWHM values for each BoDs for CUORE-0 (red–solid lines) compared to the similar distribution for Cuoricino (blue–dashed lines).

Figure 12

The $\alpha$-continuum region for CUORE-0 (red lower line) and Cuoricino (blue upper line). The regions over which the $\alpha$ continuum is evaluated are the shaded regions from 2700–3900 excluding the ${}^{190}\mathrm{Pt}$ peak. Note that the ${}^{190}\mathrm{Pt}$ peak reconstructs $\sim 40\mathrm{keV}$ too high due to the quenching of $\alpha$ particles in the bolometers; this is discussed further in [24].

Figure 13

The spectrum from the dedicated ${}^{60}\mathrm{Co}$ calibration measurement. The single-crystal coincidence peak (red shaded) reconstructs 2 keV higher than the summed energy peak from the two-crystal coincidence (blue).

Figure 14

Combined negative log-likelihood (NLL) curves for CUORE-0, Cuoricino, and their combined curves. Figure from [11].

Figure 15

Plot of the allowed regions for $m_{\beta \beta }$ as a function of the lightest neutrino mass, $m_{\mathrm{lightest}}$, assuming the light Majorana neutrino exchange model of $0\nu \beta \beta$ decay. The exclusion bands represent 90% C.L., with the vertical width coming from the uncertainty in the NME. The exclusion from ${}^{76}\mathrm{Ge}$ is from [8]. The exclusion from ${}^{136}\mathrm{Xe}$ is the combined limits from [9, 10]. The exclusion from ${}^{130}\mathrm{Te}$ is the result of this paper. Figure from [11].