Measurement of the 2νββ Decay Half-life of ^{130}Te with CUORE
CUORE Collaboration, D. Q. Adams, C. Alduino, K. Alfonso, F. T. Avignone III, O. Azzolini, G. Bari, F. Bellini, G. Benato, M. Biassoni, A. Branca, C. Brofferio, C. Bucci, J. Camilleri, A. Caminata, A. Campani, L. Canonica, X. G. Cao, S. Capelli, L. Cappelli, L. Cardani, P. Carniti, N. Casali, D. Chiesa, M. Clemenza, S. Copello, C. Cosmelli, O. Cremonesi, R. J. Creswick, A. D'Addabbo, I. Dafinei, C. J. Davis, S. Dell'Oro, S. Di Domizio, V. Dompè, D. Q. Fang, G. Fantini, M. Faverzani, E. Ferri, F. Ferroni, E. Fiorini, M. A. Franceschi, S. J. Freedman, S.H. Fu, B. K. Fujikawa, A. Giachero, L. Gironi, A. Giuliani, P. Gorla, C. Gotti, T. D. Gutierrez, K. Han, K. M. Heeger, R. G. Huang, H. Z. Huang, J. Johnston, G. Keppel, Yu. G. Kolomensky, C. Ligi, L. Ma, Y. G. Ma, L. Marini, R. H. Maruyama, D. Mayer, Y. Mei, N. Moggi, S. Morganti, T. Napolitano, M. Nastasi, J. Nikkel, C. Nones, E. B. Norman, A. Nucciotti, I. Nutini, T. O'Donnell, J. L. Ouellet, S. Pagan, C. E. Pagliarone, L. Pagnanini, M. Pallavicini, L. Pattavina, M. Pavan, G. Pessina, V. Pettinacci, C. Pira, S. Pirro, S. Pozzi, E. Previtali, A. Puiu, C. Rosenfeld, C. Rusconi, M. Sakai, S. Sangiorgio, B. Schmidt, N. D. Scielzo, V. Sharma, V. Singh, M. Sisti, D. Speller, P.T. Surukuchi, et al. (13 additional authors not shown)
MMeasurement of the 2 νββ
Decay Half-life of
Te with CUORE
D. Q. Adams, C. Alduino, K. Alfonso, F. T. Avignone III, O. Azzolini, G. Bari, F. Bellini,
5, 6
G. Benato, M. Biassoni, A. Branca,
9, 8
C. Brofferio,
9, 8
C. Bucci, J. Camilleri, A. Caminata, A. Campani,
12, 11
L. Canonica,
13, 7
X. G. Cao, S. Capelli,
9, 8
L. Cappelli,
7, 15, 16
L. Cardani, P. Carniti,
9, 8
N. Casali, D. Chiesa,
9, 8
M. Clemenza,
9, 8
S. Copello,
12, 11
C. Cosmelli,
5, 6
O. Cremonesi, R. J. Creswick, A. D’Addabbo,
17, 7
I. Dafinei, C. J. Davis, S. Dell’Oro,
9, 8
S. Di Domizio,
12, 11
V. Domp`e,
17, 7
D. Q. Fang, G. Fantini,
5, 6
M. Faverzani,
9, 8
E. Ferri,
9, 8
F. Ferroni,
17, 6
E. Fiorini,
8, 9
M. A. Franceschi, S. J. Freedman,
16, 15, ∗ S.H. Fu, B. K. Fujikawa, A. Giachero,
9, 8
L. Gironi,
9, 8
A. Giuliani, P. Gorla, C. Gotti, T. D. Gutierrez, K. Han, K. M. Heeger, R. G. Huang, H. Z. Huang, J. Johnston, G. Keppel, Yu. G. Kolomensky,
15, 16
C. Ligi, L. Ma, Y. G. Ma, L. Marini,
15, 16
R. H. Maruyama, D. Mayer, Y. Mei, N. Moggi,
23, 4
S. Morganti, T. Napolitano, M. Nastasi,
9, 8
J. Nikkel, C. Nones, E. B. Norman,
25, 26
A. Nucciotti,
9, 8
I. Nutini,
9, 8
T. O’Donnell, J. L. Ouellet, S. Pagan, C. E. Pagliarone,
7, 27
L. Pagnanini,
17, 7
M. Pallavicini,
12, 11
L. Pattavina, M. Pavan,
9, 8
G. Pessina, V. Pettinacci, C. Pira, S. Pirro, S. Pozzi,
9, 8
E. Previtali,
9, 8
A. Puiu,
17, 7
C. Rosenfeld, C. Rusconi,
1, 7
M. Sakai, S. Sangiorgio, B. Schmidt, N. D. Scielzo, V. Sharma, V. Singh, M. Sisti, D. Speller, P.T. Surukuchi, L. Taffarello, F. Terranova,
9, 8
C. Tomei, K. J. Vetter,
15, 16
M. Vignati, S. L. Wagaarachchi,
15, 16
B. S. Wang,
25, 26
B. Welliver, J. Wilson, K. Wilson, L. A. Winslow, S. Zimmermann, and S. Zucchelli
23, 4 Department of Physics and Astronomy, University of South Carolina, Columbia, SC 29208, USA Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA INFN – Laboratori Nazionali di Legnaro, Legnaro (Padova) I-35020, Italy INFN – Sezione di Bologna, Bologna I-40127, Italy Dipartimento di Fisica, Sapienza Universit`a di Roma, Roma I-00185, Italy INFN – Sezione di Roma, Roma I-00185, Italy INFN – Laboratori Nazionali del Gran Sasso, Assergi (L’Aquila) I-67100, Italy INFN – Sezione di Milano Bicocca, Milano I-20126, Italy Dipartimento di Fisica, Universit`a di Milano-Bicocca, Milano I-20126, Italy Center for Neutrino Physics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, USA INFN – Sezione di Genova, Genova I-16146, Italy Dipartimento di Fisica, Universit`a di Genova, Genova I-16146, Italy Massachusetts Institute of Technology, Cambridge, MA 02139, USA Key Laboratory of Nuclear Physics and Ion-beam Application (MOE),Institute of Modern Physics, Fudan University, Shanghai 200433, China Department of Physics, University of California, Berkeley, CA 94720, USA Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA Gran Sasso Science Institute, L’Aquila I-67100, Italy Wright Laboratory, Department of Physics, Yale University, New Haven, CT 06520, USA INFN – Laboratori Nazionali di Frascati, Frascati (Roma) I-00044, Italy Universit´e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France Physics Department, California Polytechnic State University, San Luis Obispo, CA 93407, USA INPAC and School of Physics and Astronomy,Shanghai Jiao Tong University; Shanghai Laboratory for Particle Physics and Cosmology, Shanghai 200240, China Dipartimento di Fisica e Astronomia, Alma Mater Studiorum – Universit`a di Bologna, Bologna I-40127, Italy IRFU, CEA, Universit´e Paris-Saclay, F-91191 Gif-sur-Yvette, France Lawrence Livermore National Laboratory, Livermore, CA 94550, USA Department of Nuclear Engineering, University of California, Berkeley, CA 94720, USA Dipartimento di Ingegneria Civile e Meccanica,Universit`a degli Studi di Cassino e del Lazio Meridionale, Cassino I-03043, Italy Department of Physics and Astronomy, The Johns Hopkins University,3400 North Charles Street Baltimore, MD, 21211 INFN – Sezione di Padova, Padova I-35131, Italy Engineering Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA (Dated: December 23, 2020)We measured two-neutrino double beta decay of
Te using an exposure of 300.7 kg · yr accumu-lated with the CUORE detector. Using a Bayesian analysis to fit simulated spectra to experimentaldata, it was possible to disentangle all the major background sources and precisely measure thetwo-neutrino contribution. The half-life is in agreement with past measurements with a strongly a r X i v : . [ nu c l - e x ] D ec reduced uncertainty: T ν / = 7 . +0 . − . (stat . ) +0 . − . (syst . ) × yr. This measurement is the mostprecise determination of the Te 2 νββ decay half-life to date.
INTRODUCTION
Two-neutrino double beta (2 νββ ) decay is a nucleartransition with the longest lifetime experimentally mea-sured. This process occurs when two neutrons in a nu-cleus simultaneously decay emitting two anti-neutrinosand two electrons. This decay can act as backgroundfor a hypothetical process called neutrinoless double beta(0 νββ ) decay [1], which may occur if the neutrino were aMajorana fermion [2], and which violates lepton numberconservation [3]. 0 νββ decay would be new physics andcould explain the origin and nature of the neutrino massstates [4–7].Precision measurements of the 2 νββ decay half-lifeand studies of the 2 νββ decay spectral shape can pro-vide important input for nuclear models [8–11]. Mea-surements are available in literature for the 2 νββ de-cay of various isotopes such as
Cd (Aurora [12]), Ge(GERDA [13]),
Mo (CUPID-Mo [14]),
Nd (NEMO-3 [15]), Se (NEMO-3 [16], CUPID-0 [17]),
Xe (EXO-200 [18], KamLAND-Zen [19, 20]), and Zr (NEMO-3 [21]). This paper will discuss the first measurementof the
Te 2 νββ decay half-life performed with the un-precedented statistics of the Cryogenic Underground Ob-servatory for Rare Events (CUORE) experiment.The CUORE experiment primarily searches for neutri-noless double beta decay (0 νββ ) of Te → Xe + 2 e − [22, 23], however other searches are possible [24–27]. Thisletter will outline first the CUORE detector, the datacollection, and the analysis of the 2 νββ decay of Te.Thereafter, we provide a description of the techniqueused to fit the experimental data (comprised of eventsfrom both the 2 νββ decay and the background sources),and finally we present a discussion of the fit results.
CUORE DETECTOR
CUORE is located underground at the Gran SassoNational Laboratory of INFN, Italy, with ∼ crystals [30] arranged in 19 towers made of 13floors of 4 crystals, read out as individual channels. Eachcrystal is 5 × × in size and ∼
750 g in mass, with anatural abundance of ∼
34% for
Te. The crystals arecooled to ∼
10 mK at which point they have a low heatcapacity and can be operated as cryogenic bolometers.The energy deposited in the crystal by particle inter-action causes a temperature increase which is measuredvia a neutron transmutation doped (NTD) Ge thermis-tor [31]. The signal rise time is ∼
100 ms, and the high energy resolution is second only to that of Ge detectors.Si heaters are used to inject regular reference pulses fora detector-based correction of thermal gain drifts fromlong term temperature variations.The detector is housed in a large cryogen-free cryostat,cooled to ∼
10 mK by a dilution refrigerator [32]. Thecryostat and the detector were constructed with strictradiopurity controls [30] in order to reduce the α and γ backgrounds seen in Cuoricino [33], and further reducethe γ background seen in CUORE-0 [34]. The cryostatis equipped with two lead shields: a 6-cm thick shieldof ancient Roman lead [35] at ∼ ∼
50 mK lo-cated above the detector. An additional external shieldis comprised of a 25-cm thick layer of lead surroundedby a 20-cm thick layer of polyethylene. A layer (2 cmthick) of boric acid that absorbs thermalized neutrons islocated between the two shields.When 2 νββ decay occurs inside a bolometer the neu-trinos escape without interacting, thus we detect the twoelectrons sum kinetic energy forming a continuous, β -like spectrum from 0 keV up to the Q-value of the decay( Q ββ = 2527 keV [36–38]). Background contributions tothis spectrum originate from radioactivity in the detec-tor and cryostat components. These backgrounds can bedisentangled and quantified via careful analysis of the ob-served spectral shape and topological information in thesegmented CUORE detector in comparison to a detailedbackground model [39, 40]. DATA COLLECTION
CUORE began taking data in early 2017. The datacollected through mid 2019 are analyzed in this workand are grouped into 7 datasets (physics data boundedby calibration data). The 2 νββ decay analysis requireshigh quality data over the whole energy range, specifi-cally channels need to be both well-performing and well-calibrated. This led us to exclude 2 datasets from theanalysis given that the large majority of channels did notsatisfy these criteria. With this choice of dataset-channelwe have 300.7 kg · yr of TeO exposure (102.7 kg · yr of Te exposure).The data itself are a collection of events, correspondingto a triggered waveform on a single bolometer. The mod-ularity of the detector allows us to reconstruct the eventtopology via a time based coincidence analysis. Eventsare grouped into multiplets, M i ( i = number of trig-gered bolometers) if they occur within a ±
30 ms windowon bolometers that are ≤
15 cm apart from each other,with a minimum energy of 70 keV. Given the extremely
500 1000 1500 2000 2500Energy (keV)10 C oun t s / k e V y) • (300.7 kg CUORE data, M Fit reconstruction bbn
Te 2
Fit reconstruction M Ac Co Co K Bi Tl Bi Bi Tl FIG. 1. The observed M spectrum ( black ) compared with its reconstruction as obtained by the background model ( blue ). Thereconstructed 2 νββ decay component is shown in yellow for comparison. We observe that from 900 to 2000 keV more than 50%of the M spectrum counts originate from the 2 νββ decay process. The experimental data and the spectra reconstructed by thefit have been converted back to 1 keV binning for illustrative purposes. Selected gamma lines from background contaminantsare labeled. low trigger rate of CUORE bolometers ( ∼ i > M ) spectrum comprised of events whereenergy was deposited into a single bolometer, a mul-tiplicity 2 ( M ) spectrum comprised of the single en-ergies detected by each of the two bolometers simulta-neously triggered, and a Σ spectrum comprised of thesum energy of the M events. The energy of a 2 νββ decay event is deposited into a single bolometer with aprobability obtained from Monte Carlo simulations of ∼ γ ’s that scatter from onecrystal into another or α decays that occur on a surfacebetween two crystals), making the M and Σ spectrauseful for understanding backgrounds. Events with mul-tiplicity higher than 2 are not considered in this analysissince they do not add new information. SPECTRAL FIT
We analyze the events with energies from a thresh-old of 350 keV to 2.8 MeV, where the M spectrum isdominated by 2 νββ decay (between 900 to 2000 keV thecontribution exceeds 50% of total M events) along with γ/β emissions from radioactive contaminants. To disen-tangle the 2 νββ decay signal we construct a backgroundmodel (BM) that describes the data via a comprehen-sive list of possible sources. Guidelines for this work aretaken from the CUORE-0 BM [39] and the CUORE back-ground budget [41]. The background sources are radioac-tive contaminations located both in the bulk of the detec-tor and cryostat components, on the surfaces of crystals,and materials with a line of sight to them. We also in-clude cosmogenic muons.We developed a G eant νββ decay in the crystals, and 60 others refer to dif-ferent contaminants in the 9 elements listed above. Theseinclude bulk and surface U and
Th contaminations(allowing for secular equilibrium breaks), bulk Co, K,and a few other long lived isotopes, as indicated in Fig. 1.All these isotopes are identified from the presence of oneor more characteristic γ lines in the observed spectra.The only exception is Sr, a long-lived pure β emitter,that could be present due to a hypothesized contami-nation by radioactive fallout. The remaining simulation(number 62) is the cosmogenic muon flux. As describedin [39] a variable binning is applied to all the spectra:the minimum bin size is 15 keV, and bins with less than30 counts are merged. All counts belonging to a single γ line are combined into a single bin to avoid systematicsfrom the modeling of the γ peak shapes in MC simula-tions. Finally, the trigger efficiency vs. energy and theefficiency of quality cuts are included in the analysis asglobal parameters.The observed spectrum is reconstructed by simultane-ously fitting a linear combination of the 62 MC simu-lated spectra to the M , M , and Σ data. The fit isdone with a Bayesian approach using a Markov-ChainMonte Carlo (MCMC), implemented in the JAGS soft-ware package, to sample the joint posterior probabilitydensity function (PDF) of the fit parameters [43–46]. Thelikelihood is a product, over bins and spectra, of Poissondistributions that give the probability of drawing the ex-perimental counts as a function of the MC spectra nor-malizations. To prevent bias while tuning data qualitycuts and setting the fitting procedure, the MC normaliza-tion coefficient was blinded to keep the extracted 2 νββ decay half-life in terms of a nonphysical ratio that couldnot be compared to previous results.For each source, except cosmogenic muons, a uniformprior is used. For muons, additional information is gainedfrom the high multiplicity spectra ( M >
5) where muonsbecome dominant, and is used to extract a Gaussian priorfor the BM fit. The fit result is a joint posterior PDF forthe 62 parameters from which we extract the marginal-ized posterior PDF for the 2 νββ decay rate. The fittingprocedure closely follows the description in [39] and apaper detailing the CUORE BM is in preparation.
MODEL SYSTEMATICS
The background model is able to reproduce the ma-jor features of the observed spectra (see Fig. 1) witha global χ /d.o.f of 681/365 ( χ = 1.87). The sub- optimal agreement between the data and the MC likelyarises from an imperfect modelling of source position anddistribution. Increased statistics from more data will al-low for refinement of the background model by betteridentifying source locations or additional sub-dominantcontaminants.In order to check the stability of the 2 νββ decay half-life result we run multiple fits over the whole datasetvarying aspects of the background model. In particularwe test two different models for the 2 νββ decay spectralshape, we alter the list of background sources used andwe remove the Sr source. As additional probes of oursensitivity to various aspects of the BM sources we fitsubsets of data in which we split the detector in half indifferent ways (see Fig. 2) and perform the fit on singledatasets. E v e n O d dL a s t d i g i t: -
4L a s t d i g i t: - F i r s t h a l f S e c o n d h a l f E v e n O d d F i r s t h a l f S e c o n d h a l f E v e n O d d · ( y r) / T n Channel Floor Tower
FIG. 2. Results of the 2 νββ decay half-life from the tests inwhich data are fit by splitting the detector in half in differentways
Channel:
Detector split based on channel numbers as-signed to the individual crystals (even, odd, last digits: 0-4,5-9).
Floor:
Detector split based on tower floor number (firstand second halves, even, odd).
Tower:
Detector split basedon tower number (first and second halves, even, odd). Theresults are compared to the one obtained in the full statisticsfit (red solid line) and its statistical uncertainty range (reddashed lines) – νββ Decay Model
There are two competingmodels for 2 νββ decay, yielding slightly differentspectral shapes [10, 11, 47]. The Single State Dom-inance (SSD) is the default used in this analysis.It results in a better fit quality, and might be thefirst experimental hint for SSD dominance in
Te2 νββ decay. The alternate mechanism, HigherState Dominance (HSD), yields a slightly worse fitand a few percent increase of the 2 νββ half-life. Asthis is the only shape test we perform on the 2 νββ decay spectrum, we conservatively assume this sys-tematic to be double-sided and estimate the uncer-tainty as 68% of the difference between the fits withHSD and SSD: ± Energy Threshold
The energy threshold used inthis analysis is 350 keV. If we vary this thresholdin the range of 300–800 keV we observe an increaseof the 2 νββ decay rate. We assume this systematicto be uniform between the best fit and the valuethat deviates the most. We symmetrize this uncer-tainty around the best fit to account for possibledeviations given by untested threshold values, witha result of ± Geometrical effect
All contaminants in themodel are uniformly distributed in the 9 simulatedelements. To investigate possible biases we com-pare fits done by splitting the detector according tocrystal, floor, or tower number (see Fig. 2). Eachpair of results are statistically compatible with eachother to within two sigma. We take the pair withthe largest splitting, subtract the statistical errorand interpret the result as the 1 σ uncertainty of aflat distribution: ± Sr As mentioned, a background source due topossible crystal contamination with Sr was intro-duced. This is the only long-lived pure β emitterproduced by fission that produces a background(via its daughter Y) extending up to 2.2 MeV,without any associated gamma emission that wouldallow us to constrain its activity [17]. The 2 νββ de-cay result is weakly sensitive to this contaminant,as upon its removal the counts ascribed to 2 νββ decay increase resulting in a slightly shorter half-life. Since Sr has no clear signature we use it as aproxy for the removal or addition of components tothe BM. We take the systematic to be symmetricand 68% of the difference between the best fit andthe fit without, giving ± Datasets
We investigated the 2 νββ decay resultstability in time by fitting separately each of the5 datasets used in this analysis and observed onlystatistical variations in the fit result. We also fit thetwo excluded datasets and use the result to quantifythe bias introduced by their removal. This yieldsan asymmetric uncertainty of +0.3% and -1.1%.–
List of background sources
In the referencefit we use 62 sources, selected to be comprehen-sive, as extracted from the CUORE-0 backgroundmodel [39]. Given the limited statistics used inthis work and the choice of fitting only the γ re-gion some of the sources could be degenerate witheach other while others cannot be identified eas-ily. To check how the fit performs with a differentbackground source list, all components with con-tributions compatible with 0, including Sr, areremoved. This reduces the number of distinct back-ground source components down to 25. This has animpact on the 2 νββ decay fit result compatible tothat resulting from the removal of the Sr alone. As a result of this study we can conclude that all thesystematics we explored are at most in the range of 1%.The dominant contribution comes from the uncertaintyin the decay model (SSD vs HSD) which may be improvedwith increased statistics or theoretical input. Finally,other sources of uncertainties such as the efficiency of ourcoincidence selection, the chance of mixing up M with M events, or the efficiency of pulse shape cuts, have anoverall impact on the final error that is lower than 0.1%.The Monte Carlo statistics, though optimized to yieldnegligible error, is properly accounted for in the fittingprocedure. νββ DECAY RESULTS AND DISCUSSION
To extract a robust estimate of the 2 νββ decay half-life, we combine our systematics in quadrature. Throughthe unblinding of the correct normalization coefficient forthe MC spectrum, we obtain the measurement of the2 νββ decay half-life of
Te. Though the posterior for Sr is compatible with null activity, the insertion of Srin the BM does weakly distort the 2 νββ decay posterior.Removal of the Sr source results in a symmetric poste-rior, however we choose to include this source due to thehigh anti-correlation with 2 νββ decay.We use an isotopic abundance of
Te of (34.167 ± T ν / = 7 . +0 . − . (stat . ) +0 . − . (syst . ) × yr, a valueconsistent with previous measurements (see Table I).This result is the most precise measurement of the 2 νββ decay half-life of Te to date and one of the most pre-cise measurements of a 2 νββ decay half-life. It repre-sents a substantial improvement over previous measure-ments from NEMO-3 [49] and CUORE-0 [39] owing tothe CUORE strict radiopurity controls, the improvedsignal-to-noise ratio, the increased statistics, and the ro-bust background model.
TABLE I. Chronology of T ν / measurement in Te. Therelative uncertainty refers to statistical and systematic errorssummed in quadrature. T ν / (10 yr) Relative Uncert. Ref.MiBeta 6 . ± . +2 . − .
57% 2003 [50]NEMO-3 7 . ± . ± . . ± . ± . . +0 . . − . − . CONCLUSION
In this paper we described the analysis of the
Te2 νββ decay measured with CUORE. We exploit the ge-ometry of the CUORE detector to tag single scatter andmultiple scatter events to obtain separate spectra dom-inated by 2 νββ decay and background events, respec-tively. The
Te 2 νββ decay half-life is measured to be T ν / = 7 . +0 . − . (stat . ) +0 . − . (syst . ) × yr. Comparedto previous results (Table I) this is the most precise de-termination of the 2 νββ decay half-life in Te. Thepresent result is dominated by a ∼
2% systematic uncer-tainty. Further improvement will require a better un-derstanding of the background sources localization, asindicated by the observed BM variations with differentgeometrical detector splittings. This refinement, as wellas an improved study of the SSD vs HSD models, is feasi-ble in the near future given the increased statistics beingcollected by CUORE.
ACKNOWLEDGMENTS
The CUORE Collaboration thanks the directors andstaff of the Laboratori Nazionali del Gran Sasso andthe technical staff of our laboratories. This workwas supported by the Istituto Nazionale di FisicaNucleare (INFN); the National Science Foundationunder Grant Nos. NSF-PHY-0605119, NSF-PHY-0500337, NSF-PHY-0855314, NSF-PHY-0902171, NSF-PHY-0969852, NSF-PHY-1307204, NSF-PHY-1314881,NSF-PHY-1401832, and NSF-PHY-1913374; and YaleUniversity. This material is also based upon worksupported by the US Department of Energy (DOE)Office of Science under Contract Nos. DE-AC02-05CH11231 and DE-AC52-07NA27344; by the DOE Of-fice of Science, Office of Nuclear Physics under ContractNos. DE-FG02-08ER41551, DE-FG03-00ER41138, DE-SC0012654, DE-SC0020423, DE-SC0019316; and by theEU Horizon2020 research and innovation program un-der the Marie Sklodowska-Curie Grant Agreement No.754496. This research used resources of the NationalEnergy Research Scientific Computing Center (NERSC).This work makes use of both the DIANA data anal-ysis and APOLLO data acquisition software packages,which were developed by the CUORICINO, CUORE,LUCIFER and CUPID-0 Collaborations. ∗ Deceased[1] W. H. Furry, Phys. Rev. , 1184 (1939).[2] J. D. Vergados, H. Ejiri, and F. ˇSimkovic,Rep. Prog. Phys. , 106301 (2012).[3] M. Fukugita and T. Yanagida, Phys. Lett. B , 45(1986).[4] T. Kajita, Rev. Mod. Phys. , 030501 (2016).[5] A. B. McDonald, Rev. Mod. Phys. , 030502 (2016).[6] K. Eguchi et al. , Phys. Rev. Lett. , 021802 (2003).[7] F. P. An et al. , Phys. Rev. D , 072006 (2017). [8] J. T. Suhonen, Front. Phys. , 55 (2017).[9] F. ˇSimkovic, R. Dvornick´y, D. c. v. ˇStef´anik, andA. Faessler, Phys. Rev. C , 034315 (2018).[10] O. Moreno, R. ´Alvarez-Rodr´ıguez, P. Sarriguren, E. M.de Guerra, F. ˇSimkovic, and A. Faessler, J. Phys. G ,015106 (2008).[11] P. Domin, S. Kovalenko, F. ˇSimkovic, and S. Semenov,Nucl. Phys. A , 337 (2005).[12] A. S. Barabash et al. , Phys. Rev. D , 092007 (2018).[13] M. Agostini et al. , Eur. Phys. J. C , 416 (2015).[14] Armengaud, E. et al. , Eur. Phys. J. C , 674 (2020).[15] R. Arnold et al. , Phys. Rev. D , 072003 (2016).[16] R. Arnold et al. , Eur. Phys. J. C , 440 (2019).[17] O. Azzolini et al. , Phys. Rev. Lett. , 262501 (2019).[18] J. B. Albert et al. , Phys. Rev. C , 015502 (2014).[19] A. Gando et al. , Phys. Rev. Lett. , 082503 (2016).[20] A. Gando et al. , Phys. Rev. Lett. , 192501 (2019).[21] J. Argyriades et al. , Nucl. Phys. A , 168 (2010).[22] C. Alduino et al. , Phys. Rev. Lett. , 132501 (2018).[23] D. Q. Adams et al. , Phys. Rev. Lett. , 122501 (2020).[24] C. Alduino et al. , Phys. Rev. C , 055502 (2018).[25] C. Alduino et al. , Int. J. Mod. Phys. A , 1843002(2018).[26] C. Alduino et al. , Eur. Phys. J. C , 795 (2019).[27] C. Alduino et al. , Eur. Phys. J. C , 857 (2017).[28] M. Aglietta et al. , Phys. Rev. D , 092005 (1998).[29] F. Bellini, C. Bucci, S. Capelli, O. Cremonesi, L. Gironi,M. Martinez, M. Pavan, C. Tomei, and M. Vignati, As-tropart. Phys. , 169 (2010).[30] C. Arnaboldi et al. , J. Cryst. Growth , 2999 (2010).[31] E. E. Haller, N. P. Palaio, M. Rodder, W. L. Hansen,and E. Kreysa, NTD Germanium: A Novel Material forLow Temperature Bolometers, in Neutron Transmuta-tion Doping of Semiconductor Materials , edited by R. D.Larrabee (Springer US, Boston, MA, 1984) pp. 21–36.[32] C. Alduino et al. , Cryogenics , 9 (2019).[33] E. Andreotti et al. , Astropart. Phys. , 822 (2011).[34] C. Alduino et al. , J. Instrum. , P07009 (2016).[35] A. Alessandrello et al. , Nucl. Instrum. MethodsPhys. Res. B , 163 (1998).[36] M. Redshaw, B. J. Mount, E. G. Myers, and F. T. Avi-gnone, Phys. Rev. Lett. , 212502 (2009).[37] N. D. Scielzo et al. , Phys. Rev. C , 025501 (2009).[38] S. Rahaman, V.-V. Elomaa, T. Eronen, J. Hakala,A. Jokinen, A. Kankainen, J. Rissanen, J. Suhonen,C. Weber, and J. ¨Ayst¨o, Phys. Lett. B , 412 (2011).[39] C. Alduino et al. , Eur. Phys. J. C , 13 (2017).[40] O. Azzolini et al. , Eur. Phys. J. C , 583 (2019).[41] C. Alduino et al. , Eur. Phys. J. C , 543 (2017).[42] S. Agostinelli et al. , Nucl. Instrum. Methods Phys. Res. A , 250 (2003).[43] A. Gelman, J. B. Carlin, H. S. Stern, and D. B. Rubin, Bayesian Data Analysis , 2nd ed., Texts in Statistical Sci-ence (Chapman and Hall/CRC Press, Boca Raton, FL,2014).[44] M. Plummer,
JAGS User’s Manual version 3.3.0 (2012).[45] D. Chiesa, E. Previtali, and M. Sisti, Ann. Nucl. Energy , 157 (2014).[46] M. Plummer, 3rd International Workshop on DistributedStatistical Computing (DSC 2003) (2003).[47] J. Barea, J. Kotila, and F. Iachello, Phys. Rev. C ,014315 (2013). [48] M. A. Fehr, M. Rehk¨amper, and A. N. Halliday,Int. J. Mass Spectrom. , 83 (2004).[49] R. Arnold et al. , Phys. Rev. Lett. , 062504 (2011).[50] C. Arnaboldi et al. , Phys. Lett. B , 167 (2003). Supplemental Materials
Here we present additional plots to illustrate key re-sults from the CUORE background model fits with re-spect to the 2 νββ decay half-life result. In particular weshow the fit results for the 3 spectra used in the fits ( M , M , Σ ) which show good agreement between data andmodel. We also include a comparison of the 2 νββ decayposterior with and without the Sr source to illustratethat, while there is some slight distortion, the overall im-pact is quite negligible. This is further supported by theposterior of the Sr contribution.The M spectrum (Fig. 3) is comprised of single-crystal events which contain a significant contributionfrom 2 νββ decay. The fit residuals show that in the re-gion of 1-2 MeV the reconstructed spectrum matches theobserved data quite well.In Figs. 4 and 5 we show two views of the M data:a spectrum from the individual components of the M multiplets (i.e. the M spectrum), and a spectrum fromthe sum of the two components (i.e., the Σ spectrum).The M spectrum shown in Fig. 4 displays the energyspectra from individual components of M multiplets.Events in the M spectrum provide useful informationon the localization of contaminations, as the strict coinci-dence criterion described earlier limits these to the near-est neighbor crystals. The Σ spectrum in Fig. 5 showsthe summed energy of each member of an M multiplet. The Σ events show more clearly the γ lines that producethe physical interactions in two crystals. We see here,that in the γ region the background model accurately re-constructs the observed spectrum. Minor disagreementsin some of the peaks is attributed to a potential for fur-ther improvement in source localization throughout thedetector.There is a possible contribution to the backgroundfrom a fission product, Sr. This is a pure β emitter witha decay energy of 0.564 MeV to Y , which is anothernearly pure β emitter with a 2.28 MeV endpoint allow-ing the decay chain to contribute to the background upthrough this energy. The net effect is an anti-correlationin the fit between the Sr rate and the 2 νββ decay rate.Systematics checks (described earlier) show that togglingthis potential source off alters the 2 νββ decay rate by ∼ Sr source causes a slightasymmetry in the 2 νββ decay posterior (Fig. 6(a)). Byfitting this distribution with a 2-sided Gaussian we getthe left and right uncertainty range. Without Sr theposterior is far more symmetric requiring only a singleGaussian to fit. The half-life from this analysis with the Sr source is 7 . +0 . − . × yr, compared to the re-sult without the source: 7.67 ± × yr. As thereis an anti-correlation between the 2 νββ and Sr decayrates we keep this source in the final fit to get a conser-vative result. An examination of the Sr posterior itself(Fig. 6(b)) shows that this is a conservative approach asthe posterior indicates a most probable value of zero.
500 1000 1500 2000 2500Energy [keV] C oun t s / k e V CUORE data (300.7 kgy)JAGS reconstruction
500 1000 1500 2000 2500Energy (keV)0.60.811.21.41.6 D a t a / M ode l r a t i o Data/Model ratio s s s FIG. 3. Top: The measured M spectrum ( blue ) and its reconstruction ( red ). The spectra are binned with an adaptivebinning to contain peaks into a single bin (to avoid dependence on the peak shape), while also achieving good resolution of thecontinuum shape. Bottom: The ratio of the data to the reconstructed model with 1 σ , 2 σ and 3 σ error bars. It is clear fromthe data that we are able to faithfully reconstruct the continuum and peaks from sources.
500 1000 1500 2000 2500Energy [keV] C oun t s / k e V CUORE data (300.7 kgy)JAGS reconstruction
500 1000 1500 2000 2500Energy (keV)0.20.40.60.811.21.41.61.822.2 D a t a / M ode l r a t i o Data/Model ratio s s s FIG. 4. The M spectrum provides an indication of the localization of sources as they are populated by coincident eventsfrom neighboring crystals. Top: The measured M spectrum ( blue ) and its reconstruction ( red ). The spectra are binned withan adaptive binning to contain peaks into a single bin (to avoid dependence on the peak shape), while also achieving goodresolution of the continuum shape. Bottom: The ratio of the data to the reconstructed model with 1 σ , 2 σ and 3 σ error bars.
800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800Energy [keV] C oun t s / k e V CUORE data (300.7 kgy)JAGS reconstruction
800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800Total energy (keV)0.20.40.60.811.21.41.61.822.2 D a t a / M ode l r a t i o Data/Model ratio s s s FIG. 5. The Σ spectrum better indicates the contributions to the background that originate from various γ lines as a resultof interactions with two crystals. Top: The measured Σ spectrum ( blue ) and its reconstruction ( red ). The spectra are binnedwith an adaptive binning to contain peaks into a single bin (to avoid dependence on the peak shape), while also achieving goodresolution of the continuum shape. Bottom: The ratio of the data to the reconstructed model with 1 σ , 2 σ and 3 σ error bars. · (yr) T bbn Te 2 p . d .f. bbn - no bbn (a) - · Sr activity (Bq/kg) p . d .f. (b) FIG. 6. (a) Posterior of the 2 νββ decay reference fit (red line) compared with the posterior of the 2 νββ decay fit with the Srsource turned off (blue line). The latter posterior is far more symmetric due to the anti-correlation between the two sourcescausing a distortion in the reference fit. (b) Posterior for the contribution from Sr in the reference fit, with the 90% C.I. inyellow. The posterior peaks at a value consistent with 0 activity, indicating that the contribution from this source on the 2 νββ decay half-life measurement is negligible. Since there is a slight anti-correlation and distortion of the 2 νββ posterior we makea conservative choice to include it in the model for the 2 νββνββ